LMFDB - Embedded newform 350.3.b.c.251.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,3,Mod(251,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.251"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	350.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,16,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.53680925261$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.845277938384896.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 50x^{6} - 52x^{5} + 315x^{4} + 48x^{3} - 334x^{2} - 28x + 98 $ x^8 + 50x^6 - 52x^5 + 315x^4 + 48x^3 - 334x^2 - 28x + 98
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.5
Root			$-0.707107 - 0.181255i$ of defining polynomial
Character	$\chi$	$=$	350.251
Dual form			350.3.b.c.251.8

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421 q^{2} -4.76222i q^{3} +2.00000 q^{4} -6.73480i q^{6} +(-6.84393 + 1.46990i) q^{7} +2.82843 q^{8} -13.6788 q^{9} -16.6788 q^{11} -9.52445i q^{12} -14.9917i q^{13} +(-9.67878 + 2.07875i) q^{14} +4.00000 q^{16} -11.3469i q^{17} -19.3447 q^{18} -3.16043i q^{19} +(7.00000 + 32.5923i) q^{21} -23.5874 q^{22} +22.1731 q^{23} -13.4696i q^{24} -21.2015i q^{26} +22.2814i q^{27} +(-13.6879 + 2.93980i) q^{28} -16.0363 q^{29} +21.7846i q^{31} +5.65685 q^{32} +79.4281i q^{33} -16.0469i q^{34} -27.3576 q^{36} +35.8610 q^{37} -4.46953i q^{38} -71.3939 q^{39} -74.6660i q^{41} +(9.89949 + 46.0925i) q^{42} -11.7680 q^{43} -33.3576 q^{44} +31.3576 q^{46} -11.2272i q^{47} -19.0489i q^{48} +(44.6788 - 20.1198i) q^{49} -54.0363 q^{51} -29.9834i q^{52} +20.8103 q^{53} +31.5107i q^{54} +(-19.3576 + 4.15751i) q^{56} -15.0507 q^{57} -22.6788 q^{58} +93.1210i q^{59} -43.4000i q^{61} +30.8081i q^{62} +(93.6166 - 20.1065i) q^{63} +8.00000 q^{64} +112.328i q^{66} +13.6879 q^{67} -22.6937i q^{68} -105.593i q^{69} +81.3576 q^{71} -38.6894 q^{72} -40.0933i q^{73} +50.7151 q^{74} -6.32086i q^{76} +(114.148 - 24.5162i) q^{77} -100.966 q^{78} -40.6061 q^{79} -17.0000 q^{81} -105.594i q^{82} -42.4477i q^{83} +(14.0000 + 65.1847i) q^{84} -16.6424 q^{86} +76.3686i q^{87} -47.1747 q^{88} -52.7121i q^{89} +(22.0363 + 102.602i) q^{91} +44.3463 q^{92} +103.743 q^{93} -15.8777i q^{94} -26.9392i q^{96} -116.701i q^{97} +(63.1853 - 28.4537i) q^{98} +228.145 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} - 36 q^{9} - 60 q^{11} - 4 q^{14} + 32 q^{16} + 56 q^{21} + 92 q^{29} - 72 q^{36} - 204 q^{39} - 120 q^{44} + 104 q^{46} + 284 q^{49} - 212 q^{51} - 8 q^{56} + 64 q^{64} + 504 q^{71} + 112 q^{74}+ \cdots + 944 q^{99}+O(q^{100}) $ 8 * q + 16 * q^4 - 36 * q^9 - 60 * q^11 - 4 * q^14 + 32 * q^16 + 56 * q^21 + 92 * q^29 - 72 * q^36 - 204 * q^39 - 120 * q^44 + 104 * q^46 + 284 * q^49 - 212 * q^51 - 8 * q^56 + 64 * q^64 + 504 * q^71 + 112 * q^74 - 692 * q^79 - 136 * q^81 + 112 * q^84 - 280 * q^86 - 44 * q^91 + 944 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		−	4.76222i		−	1.58741i	−0.608304	−	0.793704i	$-0.708150\pi$
$3$		−	4.76222i		−	1.58741i	0.608304	−	0.793704i	$-0.291850\pi$
$4$	2.00000			0.500000
$5$		0			0
$5$		0			0
$6$		−	6.73480i		−	1.12247i
$7$	−6.84393	+	1.46990i	−0.977704	+	0.209986i
$7$	−6.84393	+	1.46990i	−0.977704	+	0.209986i
$8$	2.82843			0.353553
$9$	−13.6788			−1.51986
$10$		0			0
$11$	−16.6788			−1.51625			−0.758126	−	0.652108i	$-0.773885\pi$
$11$	−16.6788			−1.51625			−0.758126	+	0.652108i	$0.773885\pi$
$12$		−	9.52445i		−	0.793704i
$13$		−	14.9917i		−	1.15321i	−0.817024	−	0.576604i	$-0.804377\pi$
$13$		−	14.9917i		−	1.15321i	0.817024	−	0.576604i	$-0.195623\pi$
$14$	−9.67878	+	2.07875i	−0.691341	+	0.148482i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	11.3469i		−	0.667463i	−0.942668	−	0.333731i	$-0.891692\pi$
$17$		−	11.3469i		−	0.667463i	0.942668	−	0.333731i	$-0.108308\pi$
$18$	−19.3447			−1.07471
$19$		−	3.16043i		−	0.166338i	−0.996535	−	0.0831692i	$-0.973496\pi$
$19$		−	3.16043i		−	0.166338i	0.996535	−	0.0831692i	$-0.0265042\pi$
$20$		0			0
$21$	7.00000	+	32.5923i	0.333333	+	1.55202i
$22$	−23.5874			−1.07215
$23$	22.1731			0.964050			0.482025	−	0.876158i	$-0.339901\pi$
$23$	22.1731			0.964050			0.482025	+	0.876158i	$0.339901\pi$
$24$		−	13.4696i		−	0.561234i
$25$		0			0
$26$		−	21.2015i		−	0.815442i
$27$			22.2814i			0.825237i
$28$	−13.6879	+	2.93980i	−0.488852	+	0.104993i
$29$	−16.0363			−0.552977			−0.276489	−	0.961017i	$-0.589171\pi$
$29$	−16.0363			−0.552977			−0.276489	+	0.961017i	$0.589171\pi$
$30$		0			0
$31$			21.7846i			0.702730i	0.936239	+	0.351365i	$0.114282\pi$
$31$			21.7846i			0.702730i	−0.936239	+	0.351365i	$0.885718\pi$
$32$	5.65685			0.176777
$33$			79.4281i			2.40691i
$34$		−	16.0469i		−	0.471968i
$35$		0			0
$36$	−27.3576			−0.759932
$37$	35.8610			0.969216			0.484608	−	0.874731i	$-0.338962\pi$
$37$	35.8610			0.969216			0.484608	+	0.874731i	$0.338962\pi$
$38$		−	4.46953i		−	0.117619i
$39$	−71.3939			−1.83061
$40$		0			0
$41$		−	74.6660i		−	1.82112i	−0.413376	−	0.910561i	$-0.635650\pi$
$41$		−	74.6660i		−	1.82112i	0.413376	−	0.910561i	$-0.364350\pi$
$42$	9.89949	+	46.0925i	0.235702	+	1.09744i
$43$	−11.7680			−0.273674			−0.136837	−	0.990594i	$-0.543694\pi$
$43$	−11.7680			−0.273674			−0.136837	+	0.990594i	$0.543694\pi$
$44$	−33.3576			−0.758126
$45$		0			0
$46$	31.3576			0.681686
$47$		−	11.2272i		−	0.238877i	−0.992842	−	0.119439i	$-0.961891\pi$
$47$		−	11.2272i		−	0.238877i	0.992842	−	0.119439i	$-0.0381095\pi$
$48$		−	19.0489i		−	0.396852i
$49$	44.6788	−	20.1198i	0.911812	−	0.410608i
$50$		0			0
$51$	−54.0363			−1.05954
$52$		−	29.9834i		−	0.576604i
$53$	20.8103			0.392648			0.196324	−	0.980539i	$-0.437100\pi$
$53$	20.8103			0.392648			0.196324	+	0.980539i	$0.437100\pi$
$54$			31.5107i			0.583531i
$55$		0			0
$56$	−19.3576	+	4.15751i	−0.345671	+	0.0742412i
$57$	−15.0507			−0.264047
$58$	−22.6788			−0.391014
$59$			93.1210i			1.57832i	0.614187	+	0.789161i	$0.289484\pi$
$59$			93.1210i			1.57832i	−0.614187	+	0.789161i	$0.710516\pi$
$60$		0			0
$61$		−	43.4000i		−	0.711476i	−0.934586	−	0.355738i	$-0.884230\pi$
$61$		−	43.4000i		−	0.711476i	0.934586	−	0.355738i	$-0.115770\pi$
$62$			30.8081i			0.496905i
$63$	93.6166	−	20.1065i	1.48598	−	0.319150i
$64$	8.00000			0.125000
$65$		0			0
$66$			112.328i			1.70194i
$67$	13.6879			0.204296			0.102148	−	0.994769i	$-0.467428\pi$
$67$	13.6879			0.204296			0.102148	+	0.994769i	$0.467428\pi$
$68$		−	22.6937i		−	0.333731i
$69$		−	105.593i		−	1.53034i
$70$		0			0
$71$	81.3576			1.14588			0.572941	−	0.819597i	$-0.305803\pi$
$71$	81.3576			1.14588			0.572941	+	0.819597i	$0.305803\pi$
$72$	−38.6894			−0.537353
$73$		−	40.0933i		−	0.549223i	−0.961555	−	0.274611i	$-0.911451\pi$
$73$		−	40.0933i		−	0.549223i	0.961555	−	0.274611i	$-0.0885492\pi$
$74$	50.7151			0.685339
$75$		0			0
$76$		−	6.32086i		−	0.0831692i
$77$	114.148	−	24.5162i	1.48245	−	0.318392i
$78$	−100.966			−1.29444
$79$	−40.6061			−0.514001			−0.257001	−	0.966411i	$-0.582734\pi$
$79$	−40.6061			−0.514001			−0.257001	+	0.966411i	$0.582734\pi$
$80$		0			0
$81$	−17.0000			−0.209877
$82$		−	105.594i		−	1.28773i
$83$		−	42.4477i		−	0.511418i	−0.966754	−	0.255709i	$-0.917691\pi$
$83$		−	42.4477i		−	0.511418i	0.966754	−	0.255709i	$-0.0823089\pi$
$84$	14.0000	+	65.1847i	0.166667	+	0.776008i
$85$		0			0
$86$	−16.6424			−0.193517
$87$			76.3686i			0.877801i
$88$	−47.1747			−0.536076
$89$		−	52.7121i		−	0.592271i	−0.955146	−	0.296136i	$-0.904302\pi$
$89$		−	52.7121i		−	0.592271i	0.955146	−	0.296136i	$-0.0956980\pi$
$90$		0			0
$91$	22.0363	+	102.602i	0.242158	+	1.12750i
$92$	44.3463			0.482025
$93$	103.743			1.11552
$94$		−	15.8777i		−	0.168912i
$95$		0			0
$96$		−	26.9392i		−	0.280617i
$97$		−	116.701i		−	1.20311i	−0.798833	−	0.601553i	$-0.794549\pi$
$97$		−	116.701i		−	1.20311i	0.798833	−	0.601553i	$-0.205451\pi$
$98$	63.1853	−	28.4537i	0.644748	−	0.290344i
$99$	228.145			2.30450
$100$		0			0
$101$		−	9.14290i		−	0.0905237i	−0.998975	−	0.0452619i	$-0.985588\pi$
$101$		−	9.14290i		−	0.0905237i	0.998975	−	0.0452619i	$-0.0144122\pi$
$102$	−76.4189			−0.749205
$103$		−	41.4499i		−	0.402427i	−0.979547	−	0.201213i	$-0.935512\pi$
$103$		−	41.4499i		−	0.402427i	0.979547	−	0.201213i	$-0.0644884\pi$
$104$		−	42.4030i		−	0.407721i
$105$		0			0
$106$	29.4302			0.277644
$107$	−123.191			−1.15132			−0.575658	−	0.817691i	$-0.695254\pi$
$107$	−123.191			−1.15132			−0.575658	+	0.817691i	$0.695254\pi$
$108$			44.5628i			0.412619i
$109$	134.182			1.23102			0.615512	−	0.788127i	$-0.288949\pi$
$109$	134.182			1.23102			0.615512	+	0.788127i	$0.288949\pi$
$110$		0			0
$111$		−	170.778i		−	1.53854i
$112$	−27.3757	+	5.87961i	−0.244426	+	0.0524965i
$113$	73.6419			0.651698			0.325849	−	0.945422i	$-0.394350\pi$
$113$	73.6419			0.651698			0.325849	+	0.945422i	$0.394350\pi$
$114$	−21.2849			−0.186709
$115$		0			0
$116$	−32.0727			−0.276489
$117$			205.068i			1.75272i
$118$			131.693i			1.11604i
$119$	16.6788	+	77.6572i	0.140158	+	0.652581i
$120$		0			0
$121$	157.182			1.29902
$122$		−	61.3769i		−	0.503090i
$123$	−355.576			−2.89086
$124$			43.5692i			0.351365i
$125$		0			0
$126$	132.394	−	28.4348i	1.05075	−	0.225673i
$127$	−51.4687			−0.405266			−0.202633	−	0.979255i	$-0.564950\pi$
$127$	−51.4687			−0.405266			−0.202633	+	0.979255i	$0.564950\pi$
$128$	11.3137			0.0883883
$129$			56.0418i			0.434432i
$130$		0			0
$131$			170.440i			1.30107i	0.759478	+	0.650533i	$0.225455\pi$
$131$			170.440i			1.30107i	−0.759478	+	0.650533i	$0.774545\pi$
$132$			158.856i			1.20346i
$133$	4.64552	+	21.6298i	0.0349287	+	0.162630i
$134$	19.3576			0.144459
$135$		0			0
$136$		−	32.0938i		−	0.235984i
$137$	−99.9035			−0.729223			−0.364611	−	0.931160i	$-0.618798\pi$
$137$	−99.9035			−0.729223			−0.364611	+	0.931160i	$0.618798\pi$
$138$		−	149.332i		−	1.08211i
$139$			204.866i			1.47386i	0.675971	+	0.736928i	$0.263725\pi$
$139$			204.866i			1.47386i	−0.675971	+	0.736928i	$0.736275\pi$
$140$		0			0
$141$	−53.4666			−0.379196
$142$	115.057			0.810260
$143$			250.043i			1.74856i
$144$	−54.7151			−0.379966
$145$		0			0
$146$		−	56.7005i		−	0.388359i
$147$	−95.8150	−	212.770i	−0.651803	−	1.44742i
$148$	71.7220			0.484608
$149$	−54.6424			−0.366728			−0.183364	−	0.983045i	$-0.558699\pi$
$149$	−54.6424			−0.366728			−0.183364	+	0.983045i	$0.558699\pi$
$150$		0			0
$151$	27.4666			0.181898			0.0909489	−	0.995856i	$-0.471010\pi$
$151$	27.4666			0.181898			0.0909489	+	0.995856i	$0.471010\pi$
$152$		−	8.93905i		−	0.0588095i
$153$			155.211i			1.01445i
$154$	161.430	−	34.6711i	1.04825	−	0.225137i
$155$		0			0
$156$	−142.788			−0.915306
$157$			163.326i			1.04029i	0.854078	+	0.520146i	$0.174122\pi$
$157$			163.326i			1.04029i	−0.854078	+	0.520146i	$0.825878\pi$
$158$	−57.4257			−0.363454
$159$		−	99.1034i		−	0.623292i
$160$		0			0
$161$	−151.751	+	32.5923i	−0.942556	+	0.202437i
$162$	−24.0416			−0.148405
$163$	213.495			1.30978			0.654892	−	0.755722i	$-0.272714\pi$
$163$	213.495			1.30978			0.654892	+	0.755722i	$0.272714\pi$
$164$		−	149.332i		−	0.910561i
$165$		0			0
$166$		−	60.0301i		−	0.361627i
$167$		−	270.861i		−	1.62192i	−0.585099	−	0.810962i	$-0.698944\pi$
$167$		−	270.861i		−	1.62192i	0.585099	−	0.810962i	$-0.301056\pi$
$168$	19.7990	+	92.1850i	0.117851	+	0.548720i
$169$	−55.7515			−0.329890
$170$		0			0
$171$			43.2308i			0.252812i
$172$	−23.5360			−0.136837
$173$		−	190.423i		−	1.10071i	−0.834931	−	0.550355i	$-0.814492\pi$
$173$		−	190.423i		−	1.10071i	0.834931	−	0.550355i	$-0.185508\pi$
$174$			108.002i			0.620699i
$175$		0			0
$176$	−66.7151			−0.379063
$177$	443.463			2.50544
$178$		−	74.5462i		−	0.418799i
$179$	−96.0727			−0.536719			−0.268359	−	0.963319i	$-0.586481\pi$
$179$	−96.0727			−0.536719			−0.268359	+	0.963319i	$0.586481\pi$
$180$		0			0
$181$			248.774i			1.37444i	0.726449	+	0.687220i	$0.241169\pi$
$181$			248.774i			1.37444i	−0.726449	+	0.687220i	$0.758831\pi$
$182$	31.1641	+	145.101i	0.171231	+	0.797261i
$183$	−206.681			−1.12940
$184$	62.7151			0.340843
$185$		0			0
$186$	146.715			0.788791
$187$			189.252i			1.01204i
$188$		−	22.4545i		−	0.119439i
$189$	−32.7515	−	152.492i	−0.173288	−	0.806838i
$190$		0			0
$191$	−234.254			−1.22646			−0.613231	−	0.789903i	$-0.710131\pi$
$191$	−234.254			−1.22646			−0.613231	+	0.789903i	$0.710131\pi$
$192$		−	38.0978i		−	0.198426i
$193$	−180.917			−0.937391			−0.468696	−	0.883360i	$-0.655276\pi$
$193$	−180.917			−0.937391			−0.468696	+	0.883360i	$0.655276\pi$
$194$		−	165.040i		−	0.850724i
$195$		0			0
$196$	89.3576	−	40.2396i	0.455906	−	0.205304i
$197$	158.495			0.804542			0.402271	−	0.915521i	$-0.368221\pi$
$197$	158.495			0.804542			0.402271	+	0.915521i	$0.368221\pi$
$198$	322.646			1.62953
$199$		−	223.490i		−	1.12307i	−0.827454	−	0.561533i	$-0.810212\pi$
$199$		−	223.490i		−	1.12307i	0.827454	−	0.561533i	$-0.189788\pi$
$200$		0			0
$201$		−	65.1847i		−	0.324302i
$202$		−	12.9300i		−	0.0640099i
$203$	109.752	−	23.5718i	0.540648	−	0.116117i
$204$	−108.073			−0.529768
$205$		0			0
$206$		−	58.6191i		−	0.284559i
$207$	−303.302			−1.46522
$208$		−	59.9669i		−	0.288302i
$209$			52.7121i			0.252211i
$210$		0			0
$211$	407.612			1.93181			0.965905	−	0.258897i	$-0.0833589\pi$
$211$	407.612			1.93181			0.965905	+	0.258897i	$0.0833589\pi$
$212$	41.6206			0.196324
$213$		−	387.443i		−	1.81898i
$214$	−174.218			−0.814103
$215$		0			0
$216$			63.0213i			0.291765i
$217$	−32.0212	−	149.092i	−0.147563	−	0.687062i
$218$	189.762			0.870466
$219$	−190.933			−0.871841
$220$		0			0
$221$	−170.109			−0.769724
$222$		−	241.517i		−	1.08791i
$223$		−	322.714i		−	1.44715i	−0.690247	−	0.723574i	$-0.742498\pi$
$223$		−	322.714i		−	1.44715i	0.690247	−	0.723574i	$-0.257502\pi$
$224$	−38.7151	+	8.31502i	−0.172835	+	0.0371206i
$225$		0			0
$226$	104.145			0.460820
$227$		−	169.259i		−	0.745633i	−0.927905	−	0.372817i	$-0.878392\pi$
$227$		−	169.259i		−	0.745633i	0.927905	−	0.372817i	$-0.121608\pi$
$228$	−30.1014			−0.132024
$229$			307.130i			1.34118i	0.741829	+	0.670589i	$0.233959\pi$
$229$			307.130i			1.34118i	−0.741829	+	0.670589i	$0.766041\pi$
$230$		0			0
$231$	−116.751	−	543.600i	−0.505418	−	2.35325i
$232$	−45.3576			−0.195507
$233$	9.29105			0.0398757			0.0199379	−	0.999801i	$-0.493653\pi$
$233$	9.29105			0.0398757			0.0199379	+	0.999801i	$0.493653\pi$
$234$			290.010i			1.23936i
$235$		0			0
$236$			186.242i			0.789161i
$237$			193.375i			0.815930i
$238$	23.5874	+	109.824i	0.0991065	+	0.461445i
$239$	−32.8968			−0.137644			−0.0688218	−	0.997629i	$-0.521924\pi$
$239$	−32.8968			−0.137644			−0.0688218	+	0.997629i	$0.521924\pi$
$240$		0			0
$241$		−	186.750i		−	0.774894i	−0.921892	−	0.387447i	$-0.873357\pi$
$241$		−	186.750i		−	0.774894i	0.921892	−	0.387447i	$-0.126643\pi$
$242$	222.288			0.918547
$243$			281.490i			1.15840i
$244$		−	86.8001i		−	0.355738i
$245$		0			0
$246$	−502.860			−2.04415
$247$	−47.3803			−0.191823
$248$			61.6162i			0.248452i
$249$	−202.145			−0.811829
$250$		0			0
$251$			298.325i			1.18855i	0.804263	+	0.594274i	$0.202560\pi$
$251$			298.325i			1.18855i	−0.804263	+	0.594274i	$0.797440\pi$
$252$	187.233	−	40.2129i	0.742989	−	0.159575i
$253$	−369.821			−1.46174
$254$	−72.7878			−0.286566
$255$		0			0
$256$	16.0000			0.0625000
$257$			224.463i			0.873398i	0.899608	+	0.436699i	$0.143853\pi$
$257$			224.463i			0.873398i	−0.899608	+	0.436699i	$0.856147\pi$
$258$			79.2550i			0.307190i
$259$	−245.430	+	52.7121i	−0.947607	+	0.203522i
$260$		0			0
$261$	219.358			0.840450
$262$			241.038i			0.919993i
$263$	−333.403			−1.26769			−0.633846	−	0.773459i	$-0.718525\pi$
$263$	−333.403			−1.26769			−0.633846	+	0.773459i	$0.718525\pi$
$264$			224.657i			0.850972i
$265$		0			0
$266$	6.56976	+	30.5891i	0.0246983	+	0.114997i
$267$	−251.027			−0.940176
$268$	27.3757			0.102148
$269$		−	72.1823i		−	0.268336i	−0.990959	−	0.134168i	$-0.957164\pi$
$269$		−	72.1823i		−	0.268336i	0.990959	−	0.134168i	$-0.0428361\pi$
$270$		0			0
$271$		−	21.1078i		−	0.0778887i	−0.999241	−	0.0389443i	$-0.987600\pi$
$271$		−	21.1078i		−	0.0778887i	0.999241	−	0.0389443i	$-0.0123995\pi$
$272$		−	45.3875i		−	0.166866i
$273$	488.615	−	104.942i	1.78980	−	0.384403i
$274$	−141.285			−0.515638
$275$		0			0
$276$		−	211.187i		−	0.765170i
$277$	−22.4218			−0.0809453			−0.0404727	−	0.999181i	$-0.512886\pi$
$277$	−22.4218			−0.0809453			−0.0404727	+	0.999181i	$0.512886\pi$
$278$			289.724i			1.04217i
$279$		−	297.987i		−	1.06805i
$280$		0			0
$281$	−43.6119			−0.155203			−0.0776013	−	0.996984i	$-0.524726\pi$
$281$	−43.6119			−0.155203			−0.0776013	+	0.996984i	$0.524726\pi$
$282$	−75.6132			−0.268132
$283$			59.1956i			0.209172i	0.994516	+	0.104586i	$0.0333517\pi$
$283$			59.1956i			0.209172i	−0.994516	+	0.104586i	$0.966648\pi$
$284$	162.715			0.572941
$285$		0			0
$286$			353.615i			1.23642i
$287$	109.752	+	511.009i	0.382410	+	1.78052i
$288$	−77.3789			−0.268677
$289$	160.249			0.554493
$290$		0			0
$291$	−555.757			−1.90982
$292$		−	80.1865i		−	0.274611i
$293$			371.148i			1.26672i	0.773859	+	0.633358i	$0.218324\pi$
$293$			371.148i			1.26672i	−0.773859	+	0.633358i	$0.781676\pi$
$294$	−135.503	−	300.903i	−0.460894	−	1.02348i
$295$		0			0
$296$	101.430			0.342670
$297$		−	371.627i		−	1.25127i
$298$	−77.2761			−0.259316
$299$		−	332.413i		−	1.11175i
$300$		0			0
$301$	80.5393	−	17.2978i	0.267572	−	0.0574677i
$302$	38.8436			0.128621
$303$	−43.5405			−0.143698
$304$		−	12.6417i		−	0.0415846i
$305$		0			0
$306$			219.502i			0.717327i
$307$			49.4319i			0.161016i	0.996754	+	0.0805079i	$0.0256542\pi$
$307$			49.4319i			0.161016i	−0.996754	+	0.0805079i	$0.974346\pi$
$308$	228.297	−	49.0323i	0.741223	−	0.159196i
$309$	−197.394			−0.638815
$310$		0			0
$311$		−	425.196i		−	1.36719i	−0.729862	−	0.683595i	$-0.760416\pi$
$311$		−	425.196i		−	1.36719i	0.729862	−	0.683595i	$-0.239584\pi$
$312$	−201.932			−0.647219
$313$			407.383i			1.30154i	0.759274	+	0.650771i	$0.225554\pi$
$313$			407.383i			1.30154i	−0.759274	+	0.650771i	$0.774446\pi$
$314$			230.977i			0.735597i
$315$		0			0
$316$	−81.2122			−0.257001
$317$	−515.185			−1.62519			−0.812594	−	0.582830i	$-0.801946\pi$
$317$	−515.185			−1.62519			−0.812594	+	0.582830i	$0.801946\pi$
$318$		−	140.153i		−	0.440734i
$319$	267.467			0.838453
$320$		0			0
$321$			586.662i			1.82761i
$322$	−214.609	+	46.0925i	−0.666487	+	0.143144i
$323$	−35.8610			−0.111025
$324$	−34.0000			−0.104938
$325$		0			0
$326$	301.927			0.926157
$327$		−	639.003i		−	1.95414i
$328$		−	211.187i		−	0.643864i
$329$	16.5029	+	76.8384i	0.0501608	+	0.233551i
$330$		0			0
$331$	72.0727			0.217742			0.108871	−	0.994056i	$-0.465276\pi$
$331$	72.0727			0.217742			0.108871	+	0.994056i	$0.465276\pi$
$332$		−	84.8954i		−	0.255709i
$333$	−490.535			−1.47308
$334$		−	383.056i		−	1.14687i
$335$		0			0
$336$	28.0000	+	130.369i	0.0833333	+	0.388004i
$337$	362.450			1.07552			0.537759	−	0.843098i	$-0.319271\pi$
$337$	362.450			1.07552			0.537759	+	0.843098i	$0.319271\pi$
$338$	−78.8445			−0.233268
$339$		−	350.699i		−	1.03451i
$340$		0			0
$341$		−	363.341i		−	1.06552i
$342$			61.1376i			0.178765i
$343$	−276.204	+	203.372i	−0.805260	+	0.592921i
$344$	−33.2849			−0.0967584
$345$		0			0
$346$		−	269.298i		−	0.778319i
$347$	−310.175			−0.893877			−0.446938	−	0.894565i	$-0.647486\pi$
$347$	−310.175			−0.893877			−0.446938	+	0.894565i	$0.647486\pi$
$348$			152.737i			0.438900i
$349$			190.418i			0.545609i	0.962069	+	0.272804i	$0.0879512\pi$
$349$			190.418i			0.545609i	−0.962069	+	0.272804i	$0.912049\pi$
$350$		0			0
$351$	334.036			0.951670
$352$	−94.3494			−0.268038
$353$			177.839i			0.503793i	0.967754	+	0.251896i	$0.0810542\pi$
$353$			177.839i			0.503793i	−0.967754	+	0.251896i	$0.918946\pi$
$354$	627.151			1.77161
$355$		0			0
$356$		−	105.424i		−	0.296136i
$357$	369.821	−	79.4281i	1.03591	−	0.222488i
$358$	−135.867			−0.379518
$359$	366.933			1.02210			0.511049	−	0.859552i	$-0.329257\pi$
$359$	366.933			1.02210			0.511049	+	0.859552i	$0.329257\pi$
$360$		0			0
$361$	351.012			0.972332
$362$			351.819i			0.971876i
$363$		−	748.534i		−	2.06208i
$364$	44.0727	+	205.204i	0.121079	+	0.563749i
$365$		0			0
$366$	−292.291			−0.798609
$367$			213.129i			0.580734i	0.956915	+	0.290367i	$0.0937773\pi$
$367$			213.129i			0.580734i	−0.956915	+	0.290367i	$0.906223\pi$
$368$	88.6926			0.241012
$369$			1021.34i			2.76786i
$370$		0			0
$371$	−142.424	+	30.5891i	−0.383893	+	0.0824505i
$372$	207.487			0.557759
$373$	594.278			1.59324			0.796619	−	0.604481i	$-0.206620\pi$
$373$	594.278			1.59324			0.796619	+	0.604481i	$0.206620\pi$
$374$			267.643i			0.715622i
$375$		0			0
$376$		−	31.7554i		−	0.0844558i
$377$			240.412i			0.637698i
$378$	−46.3176	−	215.657i	−0.122533	−	0.570521i
$379$	397.939			1.04997			0.524985	−	0.851111i	$-0.324071\pi$
$379$	397.939			1.04997			0.524985	+	0.851111i	$0.324071\pi$
$380$		0			0
$381$			245.106i			0.643322i
$382$	−331.286			−0.867240
$383$		−	464.822i		−	1.21363i	−0.794841	−	0.606817i	$-0.792446\pi$
$383$		−	464.822i		−	1.21363i	0.794841	−	0.606817i	$-0.207554\pi$
$384$		−	53.8784i		−	0.140308i
$385$		0			0
$386$	−255.855			−0.662836
$387$	160.972			0.415947
$388$		−	233.402i		−	0.601553i
$389$	−564.969			−1.45236			−0.726182	−	0.687503i	$-0.758707\pi$
$389$	−564.969			−1.45236			−0.726182	+	0.687503i	$0.758707\pi$
$390$		0			0
$391$		−	251.596i		−	0.643467i
$392$	126.371	−	56.9074i	0.322374	−	0.145172i
$393$	811.672			2.06532
$394$	224.145			0.568897
$395$		0			0
$396$	456.291			1.15225
$397$		−	460.060i		−	1.15884i	−0.815029	−	0.579421i	$-0.803279\pi$
$397$		−	460.060i		−	1.15884i	0.815029	−	0.579421i	$-0.196721\pi$
$398$		−	316.063i		−	0.794128i
$399$	103.006	−	22.1230i	0.258160	−	0.0554462i
$400$		0			0
$401$	−124.109			−0.309499			−0.154749	−	0.987954i	$-0.549457\pi$
$401$	−124.109			−0.309499			−0.154749	+	0.987954i	$0.549457\pi$
$402$		−	92.1850i		−	0.229316i
$403$	326.589			0.810394
$404$		−	18.2858i		−	0.0452619i
$405$		0			0
$406$	155.212	−	33.3356i	0.382296	−	0.0821074i
$407$	−598.118			−1.46958
$408$	−152.838			−0.374603
$409$			59.8790i			0.146403i	0.997317	+	0.0732017i	$0.0233217\pi$
$409$			59.8790i			0.146403i	−0.997317	+	0.0732017i	$0.976678\pi$
$410$		0			0
$411$			475.763i			1.15757i
$412$		−	82.8999i		−	0.201213i
$413$	−136.879	−	637.313i	−0.331425	−	1.54313i
$414$	−428.933			−1.03607
$415$		0			0
$416$		−	84.8059i		−	0.203860i
$417$	975.618			2.33961
$418$			74.5462i			0.178340i
$419$			552.743i			1.31920i	0.751618	+	0.659598i	$0.229274\pi$
$419$			552.743i			1.31920i	−0.751618	+	0.659598i	$0.770726\pi$
$420$		0			0
$421$	378.400			0.898812			0.449406	−	0.893328i	$-0.351636\pi$
$421$	378.400			0.898812			0.449406	+	0.893328i	$0.351636\pi$
$422$	576.450			1.36600
$423$			153.575i			0.363061i
$424$	58.8605			0.138822
$425$		0			0
$426$		−	547.927i		−	1.28621i
$427$	63.7938	+	297.027i	0.149400	+	0.695613i
$428$	−246.382			−0.575658
$429$	1190.76			2.77567
$430$		0			0
$431$	310.036			0.719342			0.359671	−	0.933079i	$-0.382889\pi$
$431$	310.036			0.719342			0.359671	+	0.933079i	$0.382889\pi$
$432$			89.1256i			0.206309i
$433$			672.484i			1.55308i	0.630067	+	0.776541i	$0.283027\pi$
$433$			672.484i			1.55308i	−0.630067	+	0.776541i	$0.716973\pi$
$434$	−45.2849	−	210.849i	−0.104343	−	0.485826i
$435$		0			0
$436$	268.363			0.615512
$437$		−	70.0767i		−	0.160359i
$438$	−270.020			−0.616485
$439$		−	503.192i		−	1.14622i	−0.819478	−	0.573111i	$-0.805736\pi$
$439$		−	503.192i		−	1.14622i	0.819478	−	0.573111i	$-0.194264\pi$
$440$		0			0
$441$	−611.151	+	275.214i	−1.38583	+	0.624069i
$442$	−240.570			−0.544277
$443$	−143.693			−0.324363			−0.162181	−	0.986761i	$-0.551853\pi$
$443$	−143.693			−0.324363			−0.162181	+	0.986761i	$0.551853\pi$
$444$		−	341.556i		−	0.769271i
$445$		0			0
$446$		−	456.386i		−	1.02329i
$447$			260.220i			0.582147i
$448$	−54.7514	+	11.7592i	−0.122213	+	0.0262482i
$449$	337.042			0.750651			0.375325	−	0.926893i	$-0.377531\pi$
$449$	337.042			0.750651			0.375325	+	0.926893i	$0.377531\pi$
$450$		0			0
$451$			1245.34i			2.76128i
$452$	147.284			0.325849
$453$		−	130.802i		−	0.288746i
$454$		−	239.368i		−	0.527242i
$455$		0			0
$456$	−42.5698			−0.0933547
$457$	719.140			1.57361			0.786805	−	0.617201i	$-0.211734\pi$
$457$	719.140			1.57361			0.786805	+	0.617201i	$0.211734\pi$
$458$			434.347i			0.948357i
$459$	252.824			0.550815
$460$		0			0
$461$		−	310.629i		−	0.673815i	−0.941538	−	0.336908i	$-0.890619\pi$
$461$		−	310.629i		−	0.673815i	0.941538	−	0.336908i	$-0.109381\pi$
$462$	−165.111	−	768.767i	−0.357384	−	1.66400i
$463$	278.960			0.602505			0.301253	−	0.953544i	$-0.402595\pi$
$463$	278.960			0.602505			0.301253	+	0.953544i	$0.402595\pi$
$464$	−64.1454			−0.138244
$465$		0			0
$466$	13.1395			0.0281964
$467$		−	60.1527i		−	0.128807i	−0.997924	−	0.0644033i	$-0.979486\pi$
$467$		−	60.1527i		−	0.128807i	0.997924	−	0.0644033i	$-0.0205144\pi$
$468$			410.137i			0.876360i
$469$	−93.6788	+	20.1198i	−0.199742	+	0.0428994i
$470$		0			0
$471$	777.794			1.65137
$472$			263.386i			0.558021i
$473$	196.276			0.414959
$474$			273.474i			0.576949i
$475$		0			0
$476$	33.3576	+	155.314i	0.0700789	+	0.326291i
$477$	−284.660			−0.596771
$478$	−46.5231			−0.0973287
$479$			344.378i			0.718953i	0.933154	+	0.359476i	$0.117045\pi$
$479$			344.378i			0.718953i	−0.933154	+	0.359476i	$0.882955\pi$
$480$		0			0
$481$		−	537.618i		−	1.11771i
$482$		−	264.104i		−	0.547933i
$483$	155.212	+	722.674i	0.321350	+	1.49622i
$484$	314.363			0.649511
$485$		0			0
$486$			398.088i			0.819110i
$487$	−213.495			−0.438388			−0.219194	−	0.975681i	$-0.570343\pi$
$487$	−213.495			−0.438388			−0.219194	+	0.975681i	$0.570343\pi$
$488$		−	122.754i		−	0.251545i
$489$		−	1016.71i		−	2.07916i
$490$		0			0
$491$	−289.260			−0.589125			−0.294562	−	0.955632i	$-0.595174\pi$
$491$	−289.260			−0.589125			−0.294562	+	0.955632i	$0.595174\pi$
$492$	−711.152			−1.44543
$493$			181.962i			0.369092i
$494$	−67.0058			−0.135639
$495$		0			0
$496$			87.1385i			0.175682i
$497$	−556.806	+	119.588i	−1.12033	+	0.240619i
$498$	−285.877			−0.574050
$499$	−31.3212			−0.0627680			−0.0313840	−	0.999507i	$-0.509991\pi$
$499$	−31.3212			−0.0627680			−0.0313840	+	0.999507i	$0.509991\pi$
$500$		0			0
$501$	−1289.90			−2.57466
$502$			421.896i			0.840430i
$503$		−	11.3597i		−	0.0225839i	−0.999936	−	0.0112919i	$-0.996406\pi$
$503$		−	11.3597i		−	0.0225839i	0.999936	−	0.0112919i	$-0.00359441\pi$
$504$	264.788	−	56.8696i	0.525373	−	0.112837i
$505$		0			0
$506$	−523.006			−1.03361
$507$			265.501i			0.523671i
$508$	−102.937			−0.202633
$509$		−	732.326i		−	1.43875i	−0.694620	−	0.719377i	$-0.744427\pi$
$509$		−	732.326i		−	1.43875i	0.694620	−	0.719377i	$-0.255573\pi$
$510$		0			0
$511$	58.9332	+	274.396i	0.115329	+	0.536978i
$512$	22.6274			0.0441942
$513$	70.4188			0.137269
$514$			317.439i			0.617586i
$515$		0			0
$516$			112.084i			0.217216i
$517$			187.256i			0.362198i
$518$	−347.091	+	74.5462i	−0.670059	+	0.143912i
$519$	−906.836			−1.74728
$520$		0			0
$521$			10.4965i			0.0201468i	0.999949	+	0.0100734i	$0.00320652\pi$
$521$			10.4965i			0.0201468i	−0.999949	+	0.0100734i	$0.996793\pi$
$522$	310.218			0.594288
$523$		−	359.468i		−	0.687319i	−0.939094	−	0.343659i	$-0.888333\pi$
$523$		−	359.468i		−	0.687319i	0.939094	−	0.343659i	$-0.111667\pi$
$524$			340.879i			0.650533i
$525$		0			0
$526$	−471.503			−0.896393
$527$	247.187			0.469046
$528$			317.712i			0.601728i
$529$	−37.3517			−0.0706082
$530$		0			0
$531$		−	1273.78i		−	2.39883i
$532$	9.29105	+	43.2595i	0.0174644	+	0.0813149i
$533$	−1119.37			−2.10013
$534$	−355.006			−0.664805
$535$		0			0
$536$	38.7151			0.0722297
$537$			457.520i			0.851992i
$538$		−	102.081i		−	0.189742i
$539$	−745.188	+	335.574i	−1.38254	+	0.622586i
$540$		0			0
$541$	383.746			0.709326			0.354663	−	0.934994i	$-0.384596\pi$
$541$	383.746			0.709326			0.354663	+	0.934994i	$0.384596\pi$
$542$		−	29.8510i		−	0.0550756i
$543$	1184.72			2.18180
$544$		−	64.1876i		−	0.117992i
$545$		0			0
$546$	691.006	−	148.410i	1.26558	−	0.271814i
$547$	788.077			1.44073			0.720363	−	0.693598i	$-0.243975\pi$
$547$	788.077			1.44073			0.720363	+	0.693598i	$0.243975\pi$
$548$	−199.807			−0.364611
$549$			593.660i			1.08135i
$550$		0			0
$551$			50.6818i			0.0919814i
$552$		−	298.663i		−	0.541057i
$553$	277.905	−	59.6870i	0.502541	−	0.107933i
$554$	−31.7093			−0.0572370
$555$		0			0
$556$			409.732i			0.736928i
$557$	−64.6592			−0.116085			−0.0580424	−	0.998314i	$-0.518486\pi$
$557$	−64.6592			−0.116085			−0.0580424	+	0.998314i	$0.518486\pi$
$558$		−	421.417i		−	0.755228i
$559$			176.422i			0.315603i
$560$		0			0
$561$	901.260			1.60652
$562$	−61.6766			−0.109745
$563$			656.960i			1.16689i	0.812152	+	0.583446i	$0.198296\pi$
$563$			656.960i			1.16689i	−0.812152	+	0.583446i	$0.801704\pi$
$564$	−106.933			−0.189598
$565$		0			0
$566$			83.7152i			0.147907i
$567$	116.347	−	24.9883i	0.205197	−	0.0440711i
$568$	230.114			0.405130
$569$	−834.375			−1.46639			−0.733194	−	0.680019i	$-0.761971\pi$
$569$	−834.375			−1.46639			−0.733194	+	0.680019i	$0.761971\pi$
$570$		0			0
$571$	−66.4971			−0.116457			−0.0582286	−	0.998303i	$-0.518545\pi$
$571$	−66.4971			−0.116457			−0.0582286	+	0.998303i	$0.518545\pi$
$572$			500.087i			0.874278i
$573$			1115.57i			1.94690i
$574$	155.212	+	722.675i	0.270405	+	1.25902i
$575$		0			0
$576$	−109.430			−0.189983
$577$			31.9127i			0.0553079i	0.999618	+	0.0276540i	$0.00880365\pi$
$577$			31.9127i			0.0553079i	−0.999618	+	0.0276540i	$0.991196\pi$
$578$	226.626			0.392086
$579$			861.565i			1.48802i
$580$		0			0
$581$	62.3939	+	290.509i	0.107391	+	0.500015i
$582$	−785.960			−1.35045
$583$	−347.091			−0.595353
$584$		−	113.401i		−	0.194180i
$585$		0			0
$586$			524.883i			0.895704i
$587$		−	783.466i		−	1.33470i	−0.744746	−	0.667348i	$-0.767430\pi$
$587$		−	783.466i		−	1.33470i	0.744746	−	0.667348i	$-0.232570\pi$
$588$	−191.630	−	425.541i	−0.325901	−	0.723709i
$589$	68.8488			0.116891
$590$		0			0
$591$		−	754.787i		−	1.27714i
$592$	143.444			0.242304
$593$		−	1024.80i		−	1.72816i	−0.503357	−	0.864078i	$-0.667902\pi$
$593$		−	1024.80i		−	1.72816i	0.503357	−	0.864078i	$-0.332098\pi$
$594$		−	525.559i		−	0.884780i
$595$		0			0
$596$	−109.285			−0.183364
$597$	−1064.31			−1.78277
$598$		−	470.104i		−	0.786126i
$599$	1109.11			1.85161			0.925805	−	0.378001i	$-0.123388\pi$
$599$	1109.11			1.85161			0.925805	+	0.378001i	$0.123388\pi$
$600$		0			0
$601$		−	1028.00i		−	1.71048i	−0.518232	−	0.855240i	$-0.673410\pi$
$601$		−	1028.00i		−	1.71048i	0.518232	−	0.855240i	$-0.326590\pi$
$602$	113.900	−	24.4627i	0.189202	−	0.0406358i
$603$	−187.233			−0.310503
$604$	54.9332			0.0909489
$605$		0			0
$606$	−61.5756			−0.101610
$607$		−	581.737i		−	0.958381i	−0.877711	−	0.479190i	$-0.840930\pi$
$607$		−	581.737i		−	0.958381i	0.877711	−	0.479190i	$-0.159070\pi$
$608$		−	17.8781i		−	0.0294048i
$609$	−112.254	−	522.662i	−0.184326	−	0.858229i
$610$		0			0
$611$	−168.315			−0.275475
$612$			310.423i			0.507227i
$613$	303.242			0.494685			0.247342	−	0.968928i	$-0.420443\pi$
$613$	303.242			0.494685			0.247342	+	0.968928i	$0.420443\pi$
$614$			69.9072i			0.113855i
$615$		0			0
$616$	322.860	−	69.3422i	0.524124	−	0.112568i
$617$	−530.544			−0.859877			−0.429938	−	0.902858i	$-0.641465\pi$
$617$	−530.544			−0.859877			−0.429938	+	0.902858i	$0.641465\pi$
$618$	−279.157			−0.451711
$619$		−	171.117i		−	0.276440i	−0.990402	−	0.138220i	$-0.955862\pi$
$619$		−	171.117i		−	0.276440i	0.990402	−	0.138220i	$-0.0441382\pi$
$620$		0			0
$621$			494.049i			0.795569i
$622$		−	601.318i		−	0.966749i
$623$	77.4816	+	360.758i	0.124369	+	0.579066i
$624$	−285.576			−0.457653
$625$		0			0
$626$			576.126i			0.920329i
$627$	251.027			0.400362
$628$			326.651i			0.520146i
$629$		−	406.910i		−	0.646916i
$630$		0			0
$631$	−1021.70			−1.61917			−0.809585	−	0.587003i	$-0.800308\pi$
$631$	−1021.70			−1.61917			−0.809585	+	0.587003i	$0.800308\pi$
$632$	−114.851			−0.181727
$633$		−	1941.14i		−	3.06657i
$634$	−728.581			−1.14918
$635$		0			0
$636$		−	198.207i		−	0.311646i
$637$	−301.630	−	669.811i	−0.473517	−	1.05151i
$638$	378.255			0.592876
$639$	−1112.87			−1.74158
$640$		0			0
$641$	337.212			0.526072			0.263036	−	0.964786i	$-0.415276\pi$
$641$	337.212			0.526072			0.263036	+	0.964786i	$0.415276\pi$
$642$			829.665i			1.29231i
$643$		−	844.298i		−	1.31306i	−0.754300	−	0.656530i	$-0.772023\pi$
$643$		−	844.298i		−	1.31306i	0.754300	−	0.656530i	$-0.227977\pi$
$644$	−303.503	+	65.1847i	−0.471278	+	0.101218i
$645$		0			0
$646$	−50.7151			−0.0785064
$647$		−	279.281i		−	0.431656i	−0.976431	−	0.215828i	$-0.930755\pi$
$647$		−	279.281i		−	0.431656i	0.976431	−	0.215828i	$-0.0692450\pi$
$648$	−48.0833			−0.0742026
$649$		−	1553.14i		−	2.39313i
$650$		0			0
$651$	−710.012	+	152.492i	−1.09065	+	0.234243i
$652$	426.990			0.654892
$653$	−807.833			−1.23711			−0.618555	−	0.785742i	$-0.712282\pi$
$653$	−807.833			−1.23711			−0.618555	+	0.785742i	$0.712282\pi$
$654$		−	903.687i		−	1.38178i
$655$		0			0
$656$		−	298.664i		−	0.455280i
$657$			548.427i			0.834744i
$658$	23.3387	+	108.666i	0.0354691	+	0.165146i
$659$	114.958			0.174443			0.0872214	−	0.996189i	$-0.472201\pi$
$659$	114.958			0.174443			0.0872214	+	0.996189i	$0.472201\pi$
$660$		0			0
$661$		−	419.383i		−	0.634467i	−0.948347	−	0.317234i	$-0.897246\pi$
$661$		−	419.383i		−	0.634467i	0.948347	−	0.317234i	$-0.102754\pi$
$662$	101.926			0.153967
$663$			810.097i			1.22187i
$664$		−	120.060i		−	0.180813i
$665$		0			0
$666$	−693.721			−1.04162
$667$	−355.576			−0.533098
$668$		−	541.723i		−	0.810962i
$669$	−1536.84			−2.29721
$670$		0			0
$671$			723.860i			1.07878i
$672$	39.5980	+	184.370i	0.0589256	+	0.274360i
$673$	1071.68			1.59240			0.796198	−	0.605037i	$-0.206842\pi$
$673$	1071.68			1.59240			0.796198	+	0.605037i	$0.206842\pi$
$674$	512.581			0.760507
$675$		0			0
$676$	−111.503			−0.164945
$677$		−	592.231i		−	0.874788i	−0.899270	−	0.437394i	$-0.855902\pi$
$677$		−	592.231i		−	0.874788i	0.899270	−	0.437394i	$-0.144098\pi$
$678$		−	495.964i		−	0.731510i
$679$	171.539	+	798.695i	0.252635	+	1.17628i
$680$		0			0
$681$	−806.048			−1.18362
$682$		−	513.842i		−	0.753434i
$683$	93.2784			0.136572			0.0682858	−	0.997666i	$-0.478247\pi$
$683$	93.2784			0.136572			0.0682858	+	0.997666i	$0.478247\pi$
$684$			86.4617i			0.126406i
$685$		0			0
$686$	−390.612	+	287.611i	−0.569405	+	0.419259i
$687$	1462.62			2.12900
$688$	−47.0719			−0.0684185
$689$		−	311.982i		−	0.452805i
$690$		0			0
$691$			48.8749i			0.0707307i	0.999374	+	0.0353653i	$0.0112595\pi$
$691$			48.8749i			0.0707307i	−0.999374	+	0.0353653i	$0.988741\pi$
$692$		−	380.845i		−	0.550355i
$693$	−1561.41	+	335.351i	−2.25312	+	0.483912i
$694$	−438.654			−0.632066
$695$		0			0
$696$			216.003i			0.310349i
$697$	−847.225			−1.21553
$698$			269.291i			0.385804i
$699$		−	44.2460i		−	0.0632991i
$700$		0			0
$701$	1151.83			1.64312			0.821562	−	0.570119i	$-0.193103\pi$
$701$	1151.83			1.64312			0.821562	+	0.570119i	$0.193103\pi$
$702$	472.399			0.672933
$703$		−	113.336i		−	0.161218i
$704$	−133.430			−0.189532
$705$		0			0
$706$			251.502i			0.356235i
$707$	13.4392	+	62.5734i	0.0190087	+	0.0885054i
$708$	886.926			1.25272
$709$	113.249			0.159730			0.0798650	−	0.996806i	$-0.474551\pi$
$709$	113.249			0.159730			0.0798650	+	0.996806i	$0.474551\pi$
$710$		0			0
$711$	555.442			0.781212
$712$		−	149.092i		−	0.209400i
$713$			483.034i			0.677466i
$714$	523.006	−	112.328i	0.732501	−	0.157323i
$715$		0			0
$716$	−192.145			−0.268359
$717$			156.662i			0.218497i
$718$	518.922			0.722732
$719$		−	575.543i		−	0.800477i	−0.916411	−	0.400239i	$-0.868927\pi$
$719$		−	575.543i		−	0.800477i	0.916411	−	0.400239i	$-0.131073\pi$
$720$		0			0
$721$	60.9273	+	283.681i	0.0845039	+	0.393454i
$722$	496.405			0.687542
$723$	−889.343			−1.23007
$724$			497.547i			0.687220i
$725$		0			0
$726$		−	1058.59i		−	1.45811i
$727$			802.036i			1.10321i	0.834104	+	0.551607i	$0.185985\pi$
$727$			802.036i			1.10321i	−0.834104	+	0.551607i	$0.814015\pi$
$728$	62.3282	+	290.203i	0.0856156	+	0.398630i
$729$	1187.52			1.62897
$730$		0			0
$731$			133.530i			0.182667i
$732$	−413.361			−0.564701
$733$			123.726i			0.168794i	0.996432	+	0.0843970i	$0.0268964\pi$
$733$			123.726i			0.168794i	−0.996432	+	0.0843970i	$0.973104\pi$
$734$			301.410i			0.410641i
$735$		0			0
$736$	125.430			0.170422
$737$	−228.297			−0.309765
$738$			1444.39i			1.95717i
$739$	−632.193			−0.855471			−0.427736	−	0.903904i	$-0.640689\pi$
$739$	−632.193			−0.855471			−0.427736	+	0.903904i	$0.640689\pi$
$740$		0			0
$741$			225.636i			0.304501i
$742$	−201.419	+	43.2595i	−0.271454	+	0.0583013i
$743$	895.660			1.20546			0.602732	−	0.797944i	$-0.294079\pi$
$743$	895.660			1.20546			0.602732	+	0.797944i	$0.294079\pi$
$744$	293.430			0.394395
$745$		0			0
$746$	840.436			1.12659
$747$			580.632i			0.777286i
$748$			378.504i			0.506021i
$749$	843.109	−	181.078i	1.12565	−	0.241760i
$750$		0			0
$751$	−461.115			−0.614001			−0.307001	−	0.951709i	$-0.599325\pi$
$751$	−461.115			−0.614001			−0.307001	+	0.951709i	$0.599325\pi$
$752$		−	44.9089i		−	0.0597193i
$753$	1420.69			1.88671
$754$			339.994i			0.450921i
$755$		0			0
$756$	−65.5029	−	304.985i	−0.0866441	−	0.403419i
$757$	833.169			1.10062			0.550310	−	0.834961i	$-0.314510\pi$
$757$	833.169			1.10062			0.550310	+	0.834961i	$0.314510\pi$
$758$	562.771			0.742442
$759$			1761.17i			2.32038i
$760$		0			0
$761$			228.343i			0.300056i	0.988682	+	0.150028i	$0.0479364\pi$
$761$			228.343i			0.300056i	−0.988682	+	0.150028i	$0.952064\pi$
$762$			346.632i			0.454897i
$763$	−918.330	+	197.234i	−1.20358	+	0.258498i
$764$	−468.509			−0.613231
$765$		0			0
$766$		−	657.358i		−	0.858169i
$767$	1396.04			1.82013
$768$		−	76.1956i		−	0.0992130i
$769$			1211.59i			1.57554i	0.615972	+	0.787768i	$0.288764\pi$
$769$			1211.59i			1.57554i	−0.615972	+	0.787768i	$0.711236\pi$
$770$		0			0
$771$	1068.94			1.38644
$772$	−361.833			−0.468696
$773$		−	139.181i		−	0.180053i	−0.995939	−	0.0900267i	$-0.971305\pi$
$773$		−	139.181i		−	0.180053i	0.995939	−	0.0900267i	$-0.0286952\pi$
$774$	227.648			0.294119
$775$		0			0
$776$		−	330.081i		−	0.425362i
$777$	251.027	+	1168.79i	0.323072	+	1.50424i
$778$	−798.988			−1.02698
$779$	−235.977			−0.302923
$780$		0			0
$781$	−1356.94			−1.73745
$782$		−	355.810i		−	0.455000i
$783$		−	357.312i		−	0.456337i
$784$	178.715	−	80.4792i	0.227953	−	0.102652i
$785$		0			0
$786$	1147.88			1.46040
$787$		−	149.466i		−	0.189919i	−0.995481	−	0.0949596i	$-0.969728\pi$
$787$		−	149.466i		−	0.189919i	0.995481	−	0.0949596i	$-0.0302722\pi$
$788$	316.989			0.402271
$789$			1587.74i			2.01234i
$790$		0			0
$791$	−504.000	+	108.246i	−0.637168	+	0.136847i
$792$	645.293			0.814763
$793$	−650.641			−0.820480
$794$		−	650.623i		−	0.819424i
$795$		0			0
$796$		−	446.981i		−	0.561533i
$797$			435.452i			0.546364i	0.961962	+	0.273182i	$0.0880761\pi$
$797$			435.452i			0.546364i	−0.961962	+	0.273182i	$0.911924\pi$
$798$	145.672	−	31.2867i	0.182547	−	0.0392064i
$799$	−127.394			−0.159442
$800$		0			0
$801$			721.038i			0.900172i
$802$	−175.517			−0.218849
$803$			668.707i			0.832761i
$804$		−	130.369i		−	0.162151i
$805$		0			0
$806$	461.866			0.573035
$807$	−343.748			−0.425958
$808$		−	25.8600i		−	0.0320050i
$809$	455.551			0.563104			0.281552	−	0.959546i	$-0.409151\pi$
$809$	455.551			0.563104			0.281552	+	0.959546i	$0.409151\pi$
$810$		0			0
$811$		−	11.6265i		−	0.0143360i	−0.999974	−	0.00716802i	$-0.997718\pi$
$811$		−	11.6265i		−	0.0143360i	0.999974	−	0.00716802i	$-0.00228167\pi$
$812$	219.503	−	47.1437i	0.270324	−	0.0580587i
$813$	−100.520			−0.123641
$814$	−845.866			−1.03915
$815$		0			0
$816$	−216.145			−0.264884
$817$			37.1919i			0.0455225i
$818$			84.6817i			0.103523i
$819$	−301.430	−	1403.47i	−0.368047	−	1.71364i
$820$		0			0
$821$	−485.600			−0.591474			−0.295737	−	0.955269i	$-0.595565\pi$
$821$	−485.600			−0.591474			−0.295737	+	0.955269i	$0.595565\pi$
$822$			672.830i			0.818528i
$823$	−1249.13			−1.51777			−0.758886	−	0.651223i	$-0.774256\pi$
$823$	−1249.13			−1.51777			−0.758886	+	0.651223i	$0.774256\pi$
$824$		−	117.238i		−	0.142279i
$825$		0			0
$826$	−193.576	−	901.297i	−0.234353	−	1.09116i
$827$	−1368.11			−1.65430			−0.827152	−	0.561978i	$-0.810041\pi$
$827$	−1368.11			−1.65430			−0.827152	+	0.561978i	$0.810041\pi$
$828$	−606.603			−0.732612
$829$		−	1111.64i		−	1.34094i	−0.741937	−	0.670469i	$-0.766093\pi$
$829$		−	1111.64i		−	1.34094i	0.741937	−	0.670469i	$-0.233907\pi$
$830$		0			0
$831$			106.778i			0.128493i
$832$		−	119.934i		−	0.144151i
$833$	−228.297	−	506.964i	−0.274066	−	0.608601i
$834$	1379.73			1.65436
$835$		0			0
$836$			105.424i			0.126106i
$837$	−485.392			−0.579919
$838$			781.697i			0.932813i
$839$		−	423.166i		−	0.504369i	−0.967679	−	0.252184i	$-0.918851\pi$
$839$		−	423.166i		−	0.504369i	0.967679	−	0.252184i	$-0.0811490\pi$
$840$		0			0
$841$	−583.836			−0.694216
$842$	535.138			0.635556
$843$			207.690i			0.246370i
$844$	815.224			0.965905
$845$		0			0
$846$			217.188i			0.256723i
$847$	−1075.74	+	231.042i	−1.27006	+	0.272776i
$848$	83.2413			0.0981619
$849$	281.903			0.332041
$850$		0			0
$851$	795.151			0.934373
$852$		−	774.886i		−	0.909490i
$853$		−	723.461i		−	0.848137i	−0.905630	−	0.424068i	$-0.860602\pi$
$853$		−	723.461i		−	0.848137i	0.905630	−	0.424068i	$-0.139398\pi$
$854$	90.2180	+	420.059i	0.105642	+	0.491873i
$855$		0			0
$856$	−348.436			−0.407051
$857$		−	1014.89i		−	1.18423i	−0.805853	−	0.592115i	$-0.798293\pi$
$857$		−	1014.89i		−	1.18423i	0.805853	−	0.592115i	$-0.201707\pi$
$858$	1683.99			1.96270
$859$			97.4114i			0.113401i	0.998391	+	0.0567005i	$0.0180580\pi$
$859$			97.4114i			0.113401i	−0.998391	+	0.0567005i	$0.981942\pi$
$860$		0			0
$861$	2433.54	−	522.662i	2.82641	−	0.607040i
$862$	438.458			0.508652
$863$	480.498			0.556776			0.278388	−	0.960469i	$-0.410200\pi$
$863$	480.498			0.556776			0.278388	+	0.960469i	$0.410200\pi$
$864$			126.043i			0.145883i
$865$		0			0
$866$			951.036i			1.09819i
$867$		−	763.139i		−	0.880207i
$868$	−64.0425	−	298.185i	−0.0737817	−	0.343531i
$869$	677.260			0.779356
$870$		0			0
$871$		−	205.204i		−	0.235596i
$872$	379.523			0.435233
$873$			1596.33i			1.82856i
$874$		−	99.1034i		−	0.113391i
$875$		0			0
$876$	−381.866			−0.435920
$877$	525.779			0.599520			0.299760	−	0.954015i	$-0.403093\pi$
$877$	525.779			0.599520			0.299760	+	0.954015i	$0.403093\pi$
$878$		−	711.620i		−	0.810501i
$879$	1767.49			2.01080
$880$		0			0
$881$			1289.75i			1.46396i	0.681324	+	0.731982i	$0.261405\pi$
$881$			1289.75i			1.46396i	−0.681324	+	0.731982i	$0.738595\pi$
$882$	−864.298	+	389.212i	−0.979930	+	0.441283i
$883$	51.6578			0.0585026			0.0292513	−	0.999572i	$-0.490688\pi$
$883$	51.6578			0.0585026			0.0292513	+	0.999572i	$0.490688\pi$
$884$	−340.218			−0.384862
$885$		0			0
$886$	−203.212			−0.229359
$887$		−	194.121i		−	0.218851i	−0.993995	−	0.109426i	$-0.965099\pi$
$887$		−	194.121i		−	0.218851i	0.993995	−	0.109426i	$-0.0349011\pi$
$888$		−	483.034i		−	0.543957i
$889$	352.249	−	75.6540i	0.396230	−	0.0851001i
$890$		0			0
$891$	283.539			0.318226
$892$		−	645.428i		−	0.723574i
$893$	−35.4829			−0.0397345
$894$			368.006i			0.411640i
$895$		0			0
$896$	−77.4302	+	16.6300i	−0.0864177	+	0.0185603i
$897$	−1583.03			−1.76480
$898$	476.650			0.530790
$899$		−	349.346i		−	0.388594i
$900$		0			0
$901$		−	236.132i		−	0.262078i
$902$			1761.17i			1.95252i
$903$	−82.3759	−	383.546i	−0.0912247	−	0.424746i
$904$	208.291			0.230410
$905$		0			0
$906$		−	184.982i		−	0.204174i
$907$	1386.01			1.52812			0.764060	−	0.645145i	$-0.223203\pi$
$907$	1386.01			1.52812			0.764060	+	0.645145i	$0.223203\pi$
$908$		−	338.517i		−	0.372817i
$909$			125.064i			0.137584i
$910$		0			0
$911$	449.939			0.493896			0.246948	−	0.969029i	$-0.420572\pi$
$911$	449.939			0.493896			0.246948	+	0.969029i	$0.420572\pi$
$912$	−60.2027			−0.0660118
$913$			707.975i			0.775439i
$914$	1017.02			1.11271
$915$		0			0
$916$			614.260i			0.670589i
$917$	−250.530	−	1166.48i	−0.273206	−	1.27206i
$918$	357.547			0.389485
$919$	−310.036			−0.337363			−0.168681	−	0.985671i	$-0.553951\pi$
$919$	−310.036			−0.337363			−0.168681	+	0.985671i	$0.553951\pi$
$920$		0			0
$921$	235.406			0.255598
$922$		−	439.295i		−	0.476459i
$923$		−	1219.69i		−	1.32144i
$924$	−233.503	−	1087.20i	−0.252709	−	1.17662i
$925$		0			0
$926$	394.509			0.426035
$927$			566.985i			0.611634i
$928$	−90.7152			−0.0977535
$929$			568.376i			0.611815i	0.952061	+	0.305908i	$0.0989598\pi$
$929$			568.376i			0.611815i	−0.952061	+	0.305908i	$0.901040\pi$
$930$		0			0
$931$	−63.5873	−	141.204i	−0.0683000	−	0.151669i
$932$	18.5821			0.0199379
$933$	−2024.88			−2.17029
$934$		−	85.0688i		−	0.0910801i
$935$		0			0
$936$			580.021i			0.619680i
$937$		−	610.588i		−	0.651641i	−0.945432	−	0.325821i	$-0.894359\pi$
$937$		−	610.588i		−	0.651641i	0.945432	−	0.325821i	$-0.105641\pi$
$938$	−132.482	+	28.4537i	−0.141239	+	0.0303344i
$939$	1940.05			2.06608
$940$		0			0
$941$		−	1588.92i		−	1.68855i	−0.535912	−	0.844274i	$-0.680032\pi$
$941$		−	1588.92i		−	1.68855i	0.535912	−	0.844274i	$-0.319968\pi$
$942$	1099.97			1.16769
$943$		−	1655.58i		−	1.75565i
$944$			372.484i			0.394580i
$945$		0			0
$946$	277.576			0.293420
$947$	595.949			0.629302			0.314651	−	0.949207i	$-0.398112\pi$
$947$	595.949			0.629302			0.314651	+	0.949207i	$0.398112\pi$
$948$			386.751i			0.407965i
$949$	−601.067			−0.633369
$950$		0			0
$951$			2453.43i			2.57984i
$952$	47.1747	+	219.648i	0.0495533	+	0.230722i
$953$	−707.004			−0.741872			−0.370936	−	0.928658i	$-0.620963\pi$
$953$	−707.004			−0.741872			−0.370936	+	0.928658i	$0.620963\pi$
$954$	−402.570			−0.421981
$955$		0			0
$956$	−65.7936			−0.0688218
$957$		−	1273.74i		−	1.33097i
$958$			487.025i			0.508376i
$959$	683.733	−	146.848i	0.712964	−	0.153126i
$960$		0			0
$961$	486.430			0.506171
$962$		−	760.307i		−	0.790339i
$963$	1685.10			1.74984
$964$		−	373.499i		−	0.387447i
$965$		0			0
$966$	219.503	+	1022.02i	0.227229	+	1.05799i
$967$	91.3586			0.0944763			0.0472381	−	0.998884i	$-0.484958\pi$
$967$	91.3586			0.0944763			0.0472381	+	0.998884i	$0.484958\pi$
$968$	444.577			0.459274
$969$			170.778i			0.176242i
$970$		0			0
$971$		−	1137.54i		−	1.17152i	−0.810485	−	0.585759i	$-0.800796\pi$
$971$		−	1137.54i		−	1.17152i	0.810485	−	0.585759i	$-0.199204\pi$
$972$			562.981i			0.579198i
$973$	−301.133	−	1402.09i	−0.309489	−	1.44100i
$974$	−301.927			−0.309987
$975$		0			0
$976$		−	173.600i		−	0.177869i
$977$	−476.101			−0.487309			−0.243654	−	0.969862i	$-0.578346\pi$
$977$	−476.101			−0.487309			−0.243654	+	0.969862i	$0.578346\pi$
$978$		−	1437.85i		−	1.47019i
$979$			879.174i			0.898033i
$980$		0			0
$981$	−1835.44			−1.87099
$982$	−409.076			−0.416574
$983$			777.439i			0.790884i	0.918491	+	0.395442i	$0.129409\pi$
$983$			777.439i			0.790884i	−0.918491	+	0.395442i	$0.870591\pi$
$984$	−1005.72			−1.02207
$985$		0			0
$986$			257.333i			0.260987i
$987$	365.922	−	78.5906i	0.370741	−	0.0796257i
$988$	−94.7606			−0.0959115
$989$	−260.933			−0.263835
$990$		0			0
$991$	−1152.10			−1.16256			−0.581280	−	0.813704i	$-0.697448\pi$
$991$	−1152.10			−1.16256			−0.581280	+	0.813704i	$0.697448\pi$
$992$			123.232i			0.124226i
$993$		−	343.226i		−	0.345646i
$994$	−787.442	+	169.122i	−0.792195	+	0.170143i
$995$		0			0
$996$	−404.291			−0.405914
$997$			990.955i			0.993937i	0.867769	+	0.496968i	$0.165554\pi$
$997$			990.955i			0.993937i	−0.867769	+	0.496968i	$0.834446\pi$
$998$	−44.2949			−0.0443837
$999$			799.033i			0.799833i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.b.c.251.5		8
5.2	odd	4		70.3.d.a.69.5	yes	8
5.3	odd	4		70.3.d.a.69.4	yes	8
5.4	even	2	inner	350.3.b.c.251.4		8
7.6	odd	2	inner	350.3.b.c.251.8		8
15.2	even	4		630.3.h.d.559.2		8
15.8	even	4		630.3.h.d.559.7		8
20.3	even	4		560.3.p.g.209.1		8
20.7	even	4		560.3.p.g.209.7		8
35.2	odd	12		490.3.h.a.129.8		16
35.3	even	12		490.3.h.a.19.8		16
35.12	even	12		490.3.h.a.129.5		16
35.13	even	4		70.3.d.a.69.1	✓	8
35.17	even	12		490.3.h.a.19.1		16
35.18	odd	12		490.3.h.a.19.5		16
35.23	odd	12		490.3.h.a.129.1		16
35.27	even	4		70.3.d.a.69.8	yes	8
35.32	odd	12		490.3.h.a.19.4		16
35.33	even	12		490.3.h.a.129.4		16
35.34	odd	2	inner	350.3.b.c.251.1		8
105.62	odd	4		630.3.h.d.559.3		8
105.83	odd	4		630.3.h.d.559.6		8
140.27	odd	4		560.3.p.g.209.2		8
140.83	odd	4		560.3.p.g.209.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.3.d.a.69.1	✓	8	35.13	even	4
70.3.d.a.69.4	yes	8	5.3	odd	4
70.3.d.a.69.5	yes	8	5.2	odd	4
70.3.d.a.69.8	yes	8	35.27	even	4
350.3.b.c.251.1		8	35.34	odd	2	inner
350.3.b.c.251.4		8	5.4	even	2	inner
350.3.b.c.251.5		8	1.1	even	1	trivial
350.3.b.c.251.8		8	7.6	odd	2	inner
490.3.h.a.19.1		16	35.17	even	12
490.3.h.a.19.4		16	35.32	odd	12
490.3.h.a.19.5		16	35.18	odd	12
490.3.h.a.19.8		16	35.3	even	12
490.3.h.a.129.1		16	35.23	odd	12
490.3.h.a.129.4		16	35.33	even	12
490.3.h.a.129.5		16	35.12	even	12
490.3.h.a.129.8		16	35.2	odd	12
560.3.p.g.209.1		8	20.3	even	4
560.3.p.g.209.2		8	140.27	odd	4
560.3.p.g.209.7		8	20.7	even	4
560.3.p.g.209.8		8	140.83	odd	4
630.3.h.d.559.2		8	15.2	even	4
630.3.h.d.559.3		8	105.62	odd	4
630.3.h.d.559.6		8	105.83	odd	4
630.3.h.d.559.7		8	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	350.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -4.76222i q^{3} +2.00000 q^{4} -6.73480i q^{6} +(-6.84393 + 1.46990i) q^{7} +2.82843 q^{8} -13.6788 q^{9} -16.6788 q^{11} -9.52445i q^{12} -14.9917i q^{13} +(-9.67878 + 2.07875i) q^{14} +4.00000 q^{16} -11.3469i q^{17} -19.3447 q^{18} -3.16043i q^{19} +(7.00000 + 32.5923i) q^{21} -23.5874 q^{22} +22.1731 q^{23} -13.4696i q^{24} -21.2015i q^{26} +22.2814i q^{27} +(-13.6879 + 2.93980i) q^{28} -16.0363 q^{29} +21.7846i q^{31} +5.65685 q^{32} +79.4281i q^{33} -16.0469i q^{34} -27.3576 q^{36} +35.8610 q^{37} -4.46953i q^{38} -71.3939 q^{39} -74.6660i q^{41} +(9.89949 + 46.0925i) q^{42} -11.7680 q^{43} -33.3576 q^{44} +31.3576 q^{46} -11.2272i q^{47} -19.0489i q^{48} +(44.6788 - 20.1198i) q^{49} -54.0363 q^{51} -29.9834i q^{52} +20.8103 q^{53} +31.5107i q^{54} +(-19.3576 + 4.15751i) q^{56} -15.0507 q^{57} -22.6788 q^{58} +93.1210i q^{59} -43.4000i q^{61} +30.8081i q^{62} +(93.6166 - 20.1065i) q^{63} +8.00000 q^{64} +112.328i q^{66} +13.6879 q^{67} -22.6937i q^{68} -105.593i q^{69} +81.3576 q^{71} -38.6894 q^{72} -40.0933i q^{73} +50.7151 q^{74} -6.32086i q^{76} +(114.148 - 24.5162i) q^{77} -100.966 q^{78} -40.6061 q^{79} -17.0000 q^{81} -105.594i q^{82} -42.4477i q^{83} +(14.0000 + 65.1847i) q^{84} -16.6424 q^{86} +76.3686i q^{87} -47.1747 q^{88} -52.7121i q^{89} +(22.0363 + 102.602i) q^{91} +44.3463 q^{92} +103.743 q^{93} -15.8777i q^{94} -26.9392i q^{96} -116.701i q^{97} +(63.1853 - 28.4537i) q^{98} +228.145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 36 q^{9} - 60 q^{11} - 4 q^{14} + 32 q^{16} + 56 q^{21} + 92 q^{29} - 72 q^{36} - 204 q^{39} - 120 q^{44} + 104 q^{46} + 284 q^{49} - 212 q^{51} - 8 q^{56} + 64 q^{64} + 504 q^{71} + 112 q^{74}+ \cdots + 944 q^{99}+O(q^{100}) \) 8 * q + 16 * q^4 - 36 * q^9 - 60 * q^11 - 4 * q^14 + 32 * q^16 + 56 * q^21 + 92 * q^29 - 72 * q^36 - 204 * q^39 - 120 * q^44 + 104 * q^46 + 284 * q^49 - 212 * q^51 - 8 * q^56 + 64 * q^64 + 504 * q^71 + 112 * q^74 - 692 * q^79 - 136 * q^81 + 112 * q^84 - 280 * q^86 - 44 * q^91 + 944 * q^99

Introduction

L-functions

Modular forms

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