LMFDB - Embedded newform 1200.3.l.y.401.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,3,Mod(401,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.401"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0] 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:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 30x^{8} - 216x^{6} + 1080x^{4} - 5184x^{2} + 46656 $ x^12 - 4x^10 + 30x^8 - 216x^6 + 1080x^4 - 5184*x^2 + 46656
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.10
Root			$-1.79523 + 1.66648i$ of defining polynomial
Character	$\chi$	$=$	1200.401
Dual form			1200.3.l.y.401.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.67109 + 2.49147i) q^{3} +12.7692 q^{7} +(-3.41489 + 8.32698i) q^{9} +12.6296i q^{11} -7.44085 q^{13} +14.0550i q^{17} -31.0176 q^{19} +(21.3386 + 31.8142i) q^{21} -7.50423i q^{23} +(-26.4530 + 5.40707i) q^{27} +15.7298i q^{29} +20.4893 q^{31} +(-31.4663 + 21.1053i) q^{33} +12.9261 q^{37} +(-12.4344 - 18.5387i) q^{39} +13.8451i q^{41} -30.0797 q^{43} +20.2570i q^{47} +114.053 q^{49} +(-35.0176 + 23.4872i) q^{51} +29.1185i q^{53} +(-51.8333 - 77.2795i) q^{57} -47.6333i q^{59} +43.0176 q^{61} +(-43.6054 + 106.329i) q^{63} +0.630153 q^{67} +(18.6966 - 12.5403i) q^{69} -90.4047i q^{71} +46.2193 q^{73} +161.270i q^{77} +37.9610 q^{79} +(-57.6771 - 56.8713i) q^{81} +80.2267i q^{83} +(-39.1904 + 26.2860i) q^{87} +140.923i q^{89} -95.0138 q^{91} +(34.2396 + 51.0486i) q^{93} +10.3429 q^{97} +(-105.166 - 43.1286i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9} + 4 q^{21} + 48 q^{31} - 128 q^{39} + 252 q^{49} - 48 q^{51} + 144 q^{61} - 268 q^{69} + 432 q^{79} - 188 q^{81} - 560 q^{99}+O(q^{100}) $ 12 * q - 8 * q^9 + 4 * q^21 + 48 * q^31 - 128 * q^39 + 252 * q^49 - 48 * q^51 + 144 * q^61 - 268 * q^69 + 432 * q^79 - 188 * q^81 - 560 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.67109	+	2.49147i	0.557032	+	0.830491i
$3$	1.67109	+	2.49147i	0.557032	+	0.830491i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	12.7692			1.82417			0.912086	−	0.409998i	$-0.134471\pi$
$7$	12.7692			1.82417			0.912086	+	0.409998i	$0.134471\pi$
$8$		0			0
$9$	−3.41489	+	8.32698i	−0.379432	+	0.925220i
$10$		0			0
$11$			12.6296i			1.14815i	0.818804	+	0.574073i	$0.194637\pi$
$11$			12.6296i			1.14815i	−0.818804	+	0.574073i	$0.805363\pi$
$12$		0			0
$13$	−7.44085			−0.572373			−0.286187	−	0.958174i	$-0.592388\pi$
$13$	−7.44085			−0.572373			−0.286187	+	0.958174i	$0.592388\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			14.0550i			0.826763i	0.910558	+	0.413381i	$0.135652\pi$
$17$			14.0550i			0.826763i	−0.910558	+	0.413381i	$0.864348\pi$
$18$		0			0
$19$	−31.0176			−1.63250			−0.816252	−	0.577696i	$-0.803952\pi$
$19$	−31.0176			−1.63250			−0.816252	+	0.577696i	$0.803952\pi$
$20$		0			0
$21$	21.3386	+	31.8142i	1.01612	+	1.51496i
$22$		0			0
$23$		−	7.50423i		−	0.326271i	−0.986604	−	0.163135i	$-0.947839\pi$
$23$		−	7.50423i		−	0.326271i	0.986604	−	0.163135i	$-0.0521608\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−26.4530	+	5.40707i	−0.979742	+	0.200262i
$28$		0			0
$29$			15.7298i			0.542407i	0.962522	+	0.271204i	$0.0874217\pi$
$29$			15.7298i			0.542407i	−0.962522	+	0.271204i	$0.912578\pi$
$30$		0			0
$31$	20.4893			0.660945			0.330473	−	0.943816i	$-0.392792\pi$
$31$	20.4893			0.660945			0.330473	+	0.943816i	$0.392792\pi$
$32$		0			0
$33$	−31.4663	+	21.1053i	−0.953525	+	0.639553i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	12.9261			0.349355			0.174677	−	0.984626i	$-0.444112\pi$
$37$	12.9261			0.349355			0.174677	+	0.984626i	$0.444112\pi$
$38$		0			0
$39$	−12.4344	−	18.5387i	−0.318830	−	0.475351i
$40$		0			0
$41$			13.8451i			0.337685i	0.985643	+	0.168843i	$0.0540029\pi$
$41$			13.8451i			0.337685i	−0.985643	+	0.168843i	$0.945997\pi$
$42$		0			0
$43$	−30.0797			−0.699528			−0.349764	−	0.936838i	$-0.613738\pi$
$43$	−30.0797			−0.699528			−0.349764	+	0.936838i	$0.613738\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			20.2570i			0.431000i	0.976504	+	0.215500i	$0.0691382\pi$
$47$			20.2570i			0.431000i	−0.976504	+	0.215500i	$0.930862\pi$
$48$		0			0
$49$	114.053			2.32761
$50$		0			0
$51$	−35.0176	+	23.4872i	−0.686619	+	0.460533i
$52$		0			0
$53$			29.1185i			0.549406i	0.961529	+	0.274703i	$0.0885796\pi$
$53$			29.1185i			0.549406i	−0.961529	+	0.274703i	$0.911420\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−51.8333	−	77.2795i	−0.909356	−	1.35578i
$58$		0			0
$59$		−	47.6333i		−	0.807344i	−0.914904	−	0.403672i	$-0.867734\pi$
$59$		−	47.6333i		−	0.807344i	0.914904	−	0.403672i	$-0.132266\pi$
$60$		0			0
$61$	43.0176			0.705206			0.352603	−	0.935773i	$-0.385297\pi$
$61$	43.0176			0.705206			0.352603	+	0.935773i	$0.385297\pi$
$62$		0			0
$63$	−43.6054	+	106.329i	−0.692149	+	1.68776i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.630153			0.00940527			0.00470264	−	0.999989i	$-0.498503\pi$
$67$	0.630153			0.00940527			0.00470264	+	0.999989i	$0.498503\pi$
$68$		0			0
$69$	18.6966	−	12.5403i	0.270965	−	0.181743i
$70$		0			0
$71$		−	90.4047i		−	1.27330i	−0.771151	−	0.636652i	$-0.780319\pi$
$71$		−	90.4047i		−	1.27330i	0.771151	−	0.636652i	$-0.219681\pi$
$72$		0			0
$73$	46.2193			0.633140			0.316570	−	0.948569i	$-0.397469\pi$
$73$	46.2193			0.633140			0.316570	+	0.948569i	$0.397469\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			161.270i			2.09442i
$78$		0			0
$79$	37.9610			0.480519			0.240260	−	0.970709i	$-0.422767\pi$
$79$	37.9610			0.480519			0.240260	+	0.970709i	$0.422767\pi$
$80$		0			0
$81$	−57.6771	−	56.8713i	−0.712063	−	0.702115i
$82$		0			0
$83$			80.2267i			0.966587i	0.875458	+	0.483294i	$0.160560\pi$
$83$			80.2267i			0.966587i	−0.875458	+	0.483294i	$0.839440\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−39.1904	+	26.2860i	−0.450465	+	0.302138i
$88$		0			0
$89$			140.923i			1.58341i	0.610907	+	0.791703i	$0.290805\pi$
$89$			140.923i			1.58341i	−0.610907	+	0.791703i	$0.709195\pi$
$90$		0			0
$91$	−95.0138			−1.04411
$92$		0			0
$93$	34.2396	+	51.0486i	0.368168	+	0.548909i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.3429			0.106628			0.0533138	−	0.998578i	$-0.483022\pi$
$97$	10.3429			0.106628			0.0533138	+	0.998578i	$0.483022\pi$
$98$		0			0
$99$	−105.166	−	43.1286i	−1.06229	−	0.435643i
$100$		0			0
$101$		−	19.2739i		−	0.190830i	−0.995438	−	0.0954152i	$-0.969582\pi$
$101$		−	19.2739i		−	0.190830i	0.995438	−	0.0954152i	$-0.0304179\pi$
$102$		0			0
$103$	−6.97008			−0.0676707			−0.0338353	−	0.999427i	$-0.510772\pi$
$103$	−6.97008			−0.0676707			−0.0338353	+	0.999427i	$0.510772\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	73.7731i		−	0.689468i	−0.938700	−	0.344734i	$-0.887969\pi$
$107$		−	73.7731i		−	0.689468i	0.938700	−	0.344734i	$-0.112031\pi$
$108$		0			0
$109$	−74.0314			−0.679187			−0.339593	−	0.940572i	$-0.610289\pi$
$109$	−74.0314			−0.679187			−0.339593	+	0.940572i	$0.610289\pi$
$110$		0			0
$111$	21.6008	+	32.2051i	0.194602	+	0.290136i
$112$		0			0
$113$			147.215i			1.30279i	0.758739	+	0.651394i	$0.225816\pi$
$113$			147.215i			1.30279i	−0.758739	+	0.651394i	$0.774184\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	25.4096	−	61.9598i	0.217176	−	0.529571i
$118$		0			0
$119$			179.471i			1.50816i
$120$		0			0
$121$	−38.5069			−0.318239
$122$		0			0
$123$	−34.4947	+	23.1364i	−0.280444	+	0.188101i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−90.0171			−0.708796			−0.354398	−	0.935095i	$-0.615314\pi$
$127$	−90.0171			−0.708796			−0.354398	+	0.935095i	$0.615314\pi$
$128$		0			0
$129$	−50.2660	−	74.9428i	−0.389659	−	0.580952i
$130$		0			0
$131$		−	11.2911i		−	0.0861917i	−0.999071	−	0.0430958i	$-0.986278\pi$
$131$		−	11.2911i		−	0.0861917i	0.999071	−	0.0430958i	$-0.0137221\pi$
$132$		0			0
$133$	−396.070			−2.97797
$134$		0			0
$135$		0			0
$136$		0			0
$137$			22.4905i			0.164164i	0.996626	+	0.0820822i	$0.0261570\pi$
$137$			22.4905i			0.164164i	−0.996626	+	0.0820822i	$0.973843\pi$
$138$		0			0
$139$	91.0955			0.655363			0.327682	−	0.944788i	$-0.393733\pi$
$139$	91.0955			0.655363			0.327682	+	0.944788i	$0.393733\pi$
$140$		0			0
$141$	−50.4698	+	33.8514i	−0.357942	+	0.240081i
$142$		0			0
$143$		−	93.9750i		−	0.657168i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	190.593	+	284.159i	1.29655	+	1.93306i
$148$		0			0
$149$		−	228.330i		−	1.53242i	−0.642593	−	0.766208i	$-0.722141\pi$
$149$		−	228.330i		−	1.53242i	0.642593	−	0.766208i	$-0.277859\pi$
$150$		0			0
$151$	74.0390			0.490324			0.245162	−	0.969482i	$-0.421159\pi$
$151$	74.0390			0.490324			0.245162	+	0.969482i	$0.421159\pi$
$152$		0			0
$153$	−117.035	−	47.9961i	−0.764937	−	0.313700i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−245.742			−1.56523			−0.782616	−	0.622504i	$-0.786115\pi$
$157$	−245.742			−1.56523			−0.782616	+	0.622504i	$0.786115\pi$
$158$		0			0
$159$	−72.5481	+	48.6598i	−0.456277	+	0.306037i
$160$		0			0
$161$		−	95.8231i		−	0.595175i
$162$		0			0
$163$	−60.1570			−0.369061			−0.184531	−	0.982827i	$-0.559076\pi$
$163$	−60.1570			−0.369061			−0.184531	+	0.982827i	$0.559076\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	81.5664i		−	0.488421i	−0.969722	−	0.244211i	$-0.921471\pi$
$167$		−	81.5664i		−	0.488421i	0.969722	−	0.244211i	$-0.0785288\pi$
$168$		0			0
$169$	−113.634			−0.672389
$170$		0			0
$171$	105.921	−	258.283i	0.619424	−	1.51043i
$172$		0			0
$173$			167.064i			0.965687i	0.875707	+	0.482843i	$0.160396\pi$
$173$			167.064i			0.965687i	−0.875707	+	0.482843i	$0.839604\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	118.677	−	79.5997i	0.670492	−	0.449716i
$178$		0			0
$179$		−	270.104i		−	1.50896i	−0.656322	−	0.754481i	$-0.727889\pi$
$179$		−	270.104i		−	1.50896i	0.656322	−	0.754481i	$-0.272111\pi$
$180$		0			0
$181$	86.9786			0.480545			0.240272	−	0.970705i	$-0.422763\pi$
$181$	86.9786			0.480545			0.240272	+	0.970705i	$0.422763\pi$
$182$		0			0
$183$	71.8865	+	107.177i	0.392822	+	0.585668i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−177.509			−0.949244
$188$		0			0
$189$	−337.785	+	69.0440i	−1.78722	+	0.365312i
$190$		0			0
$191$			302.223i			1.58232i	0.611610	+	0.791159i	$0.290522\pi$
$191$			302.223i			1.58232i	−0.611610	+	0.791159i	$0.709478\pi$
$192$		0			0
$193$	306.780			1.58953			0.794766	−	0.606916i	$-0.207593\pi$
$193$	306.780			1.58953			0.794766	+	0.606916i	$0.207593\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			289.956i			1.47186i	0.677058	+	0.735930i	$0.263255\pi$
$197$			289.956i			1.47186i	−0.677058	+	0.735930i	$0.736745\pi$
$198$		0			0
$199$	382.595			1.92259			0.961293	−	0.275527i	$-0.0888522\pi$
$199$	382.595			1.92259			0.961293	+	0.275527i	$0.0888522\pi$
$200$		0			0
$201$	1.05305	+	1.57001i	0.00523903	+	0.00781100i
$202$		0			0
$203$			200.857i			0.989445i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	62.4876	+	25.6261i	0.301872	+	0.123798i
$208$		0			0
$209$		−	391.740i		−	1.87435i
$210$		0			0
$211$	321.115			1.52187			0.760937	−	0.648826i	$-0.224740\pi$
$211$	321.115			1.52187			0.760937	+	0.648826i	$0.224740\pi$
$212$		0			0
$213$	225.241	−	151.075i	1.05747	−	0.709271i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	261.632			1.20568
$218$		0			0
$219$	77.2368	+	115.154i	0.352679	+	0.525818i
$220$		0			0
$221$		−	104.581i		−	0.473217i
$222$		0			0
$223$	−292.432			−1.31135			−0.655676	−	0.755042i	$-0.727616\pi$
$223$	−292.432			−1.31135			−0.655676	+	0.755042i	$0.727616\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	370.155i		−	1.63064i	−0.579013	−	0.815319i	$-0.696562\pi$
$227$		−	370.155i		−	1.63064i	0.579013	−	0.815319i	$-0.303438\pi$
$228$		0			0
$229$	381.985			1.66806			0.834028	−	0.551722i	$-0.186029\pi$
$229$	381.985			1.66806			0.834028	+	0.551722i	$0.186029\pi$
$230$		0			0
$231$	−401.800	+	269.498i	−1.73939	+	1.16666i
$232$		0			0
$233$		−	144.262i		−	0.619150i	−0.950875	−	0.309575i	$-0.899813\pi$
$233$		−	144.262i		−	0.619150i	0.950875	−	0.309575i	$-0.100187\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	63.4365	+	94.5789i	0.267665	+	0.399067i
$238$		0			0
$239$			249.478i			1.04384i	0.852994	+	0.521921i	$0.174784\pi$
$239$			249.478i			1.04384i	−0.852994	+	0.521921i	$0.825216\pi$
$240$		0			0
$241$	30.4541			0.126366			0.0631829	−	0.998002i	$-0.479875\pi$
$241$	30.4541			0.126366			0.0631829	+	0.998002i	$0.479875\pi$
$242$		0			0
$243$	45.3095	−	238.738i	0.186459	−	0.982463i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	230.797			0.934401
$248$		0			0
$249$	−199.883	+	134.066i	−0.802742	+	0.538420i
$250$		0			0
$251$		−	68.9183i		−	0.274575i	−0.990531	−	0.137287i	$-0.956162\pi$
$251$		−	68.9183i		−	0.274575i	0.990531	−	0.137287i	$-0.0438384\pi$
$252$		0			0
$253$	94.7755			0.374607
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	362.374i		−	1.41002i	−0.709199	−	0.705009i	$-0.750943\pi$
$257$		−	362.374i		−	1.41002i	0.709199	−	0.705009i	$-0.249057\pi$
$258$		0			0
$259$	165.057			0.637284
$260$		0			0
$261$	−130.982	−	53.7155i	−0.501846	−	0.205807i
$262$		0			0
$263$			23.4485i			0.0891580i	0.999006	+	0.0445790i	$0.0141946\pi$
$263$			23.4485i			0.0891580i	−0.999006	+	0.0445790i	$0.985805\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−351.106	+	235.496i	−1.31500	+	0.882007i
$268$		0			0
$269$			396.738i			1.47486i	0.675421	+	0.737432i	$0.263962\pi$
$269$			396.738i			1.47486i	−0.675421	+	0.737432i	$0.736038\pi$
$270$		0			0
$271$	143.926			0.531092			0.265546	−	0.964098i	$-0.414448\pi$
$271$	143.926			0.531092			0.265546	+	0.964098i	$0.414448\pi$
$272$		0			0
$273$	−158.777	−	236.724i	−0.581601	−	0.867122i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−360.884			−1.30283			−0.651415	−	0.758722i	$-0.725824\pi$
$277$	−360.884			−1.30283			−0.651415	+	0.758722i	$0.725824\pi$
$278$		0			0
$279$	−69.9686	+	170.614i	−0.250784	+	0.611520i
$280$		0			0
$281$			531.655i			1.89201i	0.324149	+	0.946006i	$0.394922\pi$
$281$			531.655i			1.89201i	−0.324149	+	0.946006i	$0.605078\pi$
$282$		0			0
$283$	257.400			0.909541			0.454770	−	0.890609i	$-0.349721\pi$
$283$	257.400			0.909541			0.454770	+	0.890609i	$0.349721\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			176.791i			0.615996i
$288$		0			0
$289$	91.4579			0.316463
$290$		0			0
$291$	17.2839	+	25.7690i	0.0593949	+	0.0885533i
$292$		0			0
$293$			310.456i			1.05958i	0.848130	+	0.529789i	$0.177729\pi$
$293$			310.456i			1.05958i	−0.848130	+	0.529789i	$0.822271\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−68.2892	−	334.091i	−0.229930	−	1.12489i
$298$		0			0
$299$			55.8379i			0.186749i
$300$		0			0
$301$	−384.094			−1.27606
$302$		0			0
$303$	48.0204	−	32.2085i	0.158483	−	0.106299i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	266.169			0.866999			0.433499	−	0.901154i	$-0.357279\pi$
$307$	266.169			0.866999			0.433499	+	0.901154i	$0.357279\pi$
$308$		0			0
$309$	−11.6477	−	17.3658i	−0.0376947	−	0.0561999i
$310$		0			0
$311$		−	186.580i		−	0.599934i	−0.953950	−	0.299967i	$-0.903024\pi$
$311$		−	186.580i		−	0.599934i	0.953950	−	0.299967i	$-0.0969757\pi$
$312$		0			0
$313$	20.5021			0.0655019			0.0327510	−	0.999464i	$-0.489573\pi$
$313$	20.5021			0.0655019			0.0327510	+	0.999464i	$0.489573\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	18.9792i		−	0.0598711i	−0.999552	−	0.0299356i	$-0.990470\pi$
$317$		−	18.9792i		−	0.0598711i	0.999552	−	0.0299356i	$-0.00953021\pi$
$318$		0			0
$319$	−198.661			−0.622763
$320$		0			0
$321$	183.804	−	123.282i	0.572597	−	0.384055i
$322$		0			0
$323$		−	435.951i		−	1.34969i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−123.713	−	184.447i	−0.378328	−	0.564059i
$328$		0			0
$329$			258.666i			0.786219i
$330$		0			0
$331$	−413.193			−1.24832			−0.624159	−	0.781297i	$-0.714558\pi$
$331$	−413.193			−1.24832			−0.624159	+	0.781297i	$0.714558\pi$
$332$		0			0
$333$	−44.1413	+	107.636i	−0.132556	+	0.323230i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	484.733			1.43838			0.719188	−	0.694816i	$-0.244514\pi$
$337$	484.733			1.43838			0.719188	+	0.694816i	$0.244514\pi$
$338$		0			0
$339$	−366.783	+	246.010i	−1.08195	+	0.725694i
$340$		0			0
$341$			258.772i			0.758862i
$342$		0			0
$343$	830.672			2.42178
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	336.214i		−	0.968915i	−0.874815	−	0.484457i	$-0.839017\pi$
$347$		−	336.214i		−	0.968915i	0.874815	−	0.484457i	$-0.160983\pi$
$348$		0			0
$349$	428.261			1.22711			0.613555	−	0.789652i	$-0.289739\pi$
$349$	428.261			1.22711			0.613555	+	0.789652i	$0.289739\pi$
$350$		0			0
$351$	196.833	−	40.2332i	0.560778	−	0.114625i
$352$		0			0
$353$		−	558.817i		−	1.58305i	−0.611137	−	0.791525i	$-0.709287\pi$
$353$		−	558.817i		−	1.58305i	0.611137	−	0.791525i	$-0.290713\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−447.147	+	299.913i	−1.25251	+	0.840092i
$358$		0			0
$359$		−	206.915i		−	0.576365i	−0.957576	−	0.288182i	$-0.906949\pi$
$359$		−	206.915i		−	0.576365i	0.957576	−	0.288182i	$-0.0930509\pi$
$360$		0			0
$361$	601.090			1.66507
$362$		0			0
$363$	−64.3487	−	95.9389i	−0.177269	−	0.264295i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−185.425			−0.505246			−0.252623	−	0.967565i	$-0.581293\pi$
$367$	−185.425			−0.505246			−0.252623	+	0.967565i	$0.581293\pi$
$368$		0			0
$369$	−115.288	−	47.2794i	−0.312433	−	0.128128i
$370$		0			0
$371$			371.821i			1.00221i
$372$		0			0
$373$	427.345			1.14570			0.572848	−	0.819661i	$-0.305838\pi$
$373$	427.345			1.14570			0.572848	+	0.819661i	$0.305838\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	117.043i		−	0.310459i
$378$		0			0
$379$	117.727			0.310626			0.155313	−	0.987865i	$-0.450361\pi$
$379$	117.727			0.310626			0.155313	+	0.987865i	$0.450361\pi$
$380$		0			0
$381$	−150.427	−	224.275i	−0.394822	−	0.588649i
$382$		0			0
$383$		−	470.016i		−	1.22720i	−0.789619	−	0.613598i	$-0.789722\pi$
$383$		−	470.016i		−	1.22720i	0.789619	−	0.613598i	$-0.210278\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	102.719	−	250.473i	0.265423	−	0.647217i
$388$		0			0
$389$			128.160i			0.329461i	0.986339	+	0.164730i	$0.0526754\pi$
$389$			128.160i			0.329461i	−0.986339	+	0.164730i	$0.947325\pi$
$390$		0			0
$391$	105.472			0.269749
$392$		0			0
$393$	28.1315	−	18.8685i	0.0715814	−	0.0480115i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	276.316			0.696011			0.348005	−	0.937493i	$-0.386859\pi$
$397$	276.316			0.696011			0.348005	+	0.937493i	$0.386859\pi$
$398$		0			0
$399$	−661.871	−	986.798i	−1.65882	−	2.47318i
$400$		0			0
$401$		−	549.912i		−	1.37135i	−0.727907	−	0.685675i	$-0.759507\pi$
$401$		−	549.912i		−	1.37135i	0.727907	−	0.685675i	$-0.240493\pi$
$402$		0			0
$403$	−152.458			−0.378307
$404$		0			0
$405$		0			0
$406$		0			0
$407$			163.252i			0.401110i
$408$		0			0
$409$	219.166			0.535858			0.267929	−	0.963439i	$-0.413661\pi$
$409$	219.166			0.535858			0.267929	+	0.963439i	$0.413661\pi$
$410$		0			0
$411$	−56.0346	+	37.5838i	−0.136337	+	0.0914448i
$412$		0			0
$413$		−	608.240i		−	1.47273i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	152.229	+	226.962i	0.365058	+	0.544274i
$418$		0			0
$419$		−	204.522i		−	0.488119i	−0.969760	−	0.244060i	$-0.921521\pi$
$419$		−	204.522i		−	0.488119i	0.969760	−	0.244060i	$-0.0784792\pi$
$420$		0			0
$421$	577.186			1.37099			0.685494	−	0.728078i	$-0.259586\pi$
$421$	577.186			1.37099			0.685494	+	0.728078i	$0.259586\pi$
$422$		0			0
$423$	−168.680	−	69.1754i	−0.398770	−	0.163535i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	549.301			1.28642
$428$		0			0
$429$	234.136	−	157.041i	0.545772	−	0.366063i
$430$		0			0
$431$		−	663.363i		−	1.53913i	−0.638571	−	0.769563i	$-0.720474\pi$
$431$		−	663.363i		−	1.53913i	0.638571	−	0.769563i	$-0.279526\pi$
$432$		0			0
$433$	−226.876			−0.523964			−0.261982	−	0.965073i	$-0.584376\pi$
$433$	−226.876			−0.523964			−0.261982	+	0.965073i	$0.584376\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			232.763i			0.532639i
$438$		0			0
$439$	282.524			0.643564			0.321782	−	0.946814i	$-0.395718\pi$
$439$	282.524			0.643564			0.321782	+	0.946814i	$0.395718\pi$
$440$		0			0
$441$	−389.477	+	949.715i	−0.883168	+	2.15355i
$442$		0			0
$443$			455.605i			1.02845i	0.857654	+	0.514227i	$0.171921\pi$
$443$			455.605i			1.02845i	−0.857654	+	0.514227i	$0.828079\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	568.878	−	381.561i	1.27266	−	0.853604i
$448$		0			0
$449$		−	157.206i		−	0.350124i	−0.984557	−	0.175062i	$-0.943987\pi$
$449$		−	157.206i		−	0.350124i	0.984557	−	0.175062i	$-0.0560127\pi$
$450$		0			0
$451$	−174.858			−0.387712
$452$		0			0
$453$	123.726	+	184.466i	0.273126	+	0.407210i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	397.152			0.869041			0.434520	−	0.900662i	$-0.356918\pi$
$457$	397.152			0.869041			0.434520	+	0.900662i	$0.356918\pi$
$458$		0			0
$459$	−75.9962	−	371.797i	−0.165569	−	0.810014i
$460$		0			0
$461$		−	350.730i		−	0.760803i	−0.924821	−	0.380401i	$-0.875786\pi$
$461$		−	350.730i		−	0.760803i	0.924821	−	0.380401i	$-0.124214\pi$
$462$		0			0
$463$	−308.385			−0.666058			−0.333029	−	0.942917i	$-0.608071\pi$
$463$	−308.385			−0.666058			−0.333029	+	0.942917i	$0.608071\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			800.129i			1.71334i	0.515866	+	0.856669i	$0.327470\pi$
$467$			800.129i			1.71334i	−0.515866	+	0.856669i	$0.672530\pi$
$468$		0			0
$469$	8.04656			0.0171568
$470$		0			0
$471$	−410.657	−	612.259i	−0.871884	−	1.29991i
$472$		0			0
$473$		−	379.895i		−	0.803160i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−242.469	−	99.4364i	−0.508321	−	0.208462i
$478$		0			0
$479$		−	630.135i		−	1.31552i	−0.753227	−	0.657761i	$-0.771504\pi$
$479$		−	630.135i		−	1.31552i	0.753227	−	0.657761i	$-0.228496\pi$
$480$		0			0
$481$	−96.1814			−0.199961
$482$		0			0
$483$	238.741	−	160.130i	0.494287	−	0.331531i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−103.952			−0.213454			−0.106727	−	0.994288i	$-0.534037\pi$
$487$	−103.952			−0.213454			−0.106727	+	0.994288i	$0.534037\pi$
$488$		0			0
$489$	−100.528	−	149.879i	−0.205579	−	0.306502i
$490$		0			0
$491$			286.049i			0.582585i	0.956634	+	0.291292i	$0.0940853\pi$
$491$			286.049i			0.582585i	−0.956634	+	0.291292i	$0.905915\pi$
$492$		0			0
$493$	−221.082			−0.448442
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1154.40i		−	2.32273i
$498$		0			0
$499$	−528.285			−1.05869			−0.529344	−	0.848407i	$-0.677562\pi$
$499$	−528.285			−1.05869			−0.529344	+	0.848407i	$0.677562\pi$
$500$		0			0
$501$	203.220	−	136.305i	0.405630	−	0.272066i
$502$		0			0
$503$		−	733.418i		−	1.45809i	−0.684467	−	0.729044i	$-0.739965\pi$
$503$		−	733.418i		−	1.45809i	0.684467	−	0.729044i	$-0.260035\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−189.893	−	283.116i	−0.374542	−	0.558413i
$508$		0			0
$509$		−	312.619i		−	0.614183i	−0.951680	−	0.307092i	$-0.900644\pi$
$509$		−	312.619i		−	0.614183i	0.951680	−	0.307092i	$-0.0993558\pi$
$510$		0			0
$511$	590.183			1.15496
$512$		0			0
$513$	820.509	−	167.714i	1.59943	−	0.326928i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−255.838			−0.494851
$518$		0			0
$519$	−416.235	+	279.179i	−0.801995	+	0.537918i
$520$		0			0
$521$		−	715.719i		−	1.37374i	−0.726780	−	0.686870i	$-0.758984\pi$
$521$		−	715.719i		−	1.37374i	0.726780	−	0.686870i	$-0.241016\pi$
$522$		0			0
$523$	109.080			0.208567			0.104283	−	0.994548i	$-0.466745\pi$
$523$	109.080			0.208567			0.104283	+	0.994548i	$0.466745\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			287.977i			0.546445i
$528$		0			0
$529$	472.686			0.893547
$530$		0			0
$531$	396.641	+	162.662i	0.746971	+	0.306332i
$532$		0			0
$533$		−	103.019i		−	0.193282i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	672.958	−	451.370i	1.25318	−	0.840540i
$538$		0			0
$539$			1440.44i			2.67243i
$540$		0			0
$541$	−14.9710			−0.0276729			−0.0138364	−	0.999904i	$-0.504404\pi$
$541$	−14.9710			−0.0276729			−0.0138364	+	0.999904i	$0.504404\pi$
$542$		0			0
$543$	145.350	+	216.705i	0.267679	+	0.399088i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−842.765			−1.54070			−0.770352	−	0.637618i	$-0.779920\pi$
$547$	−842.765			−1.54070			−0.770352	+	0.637618i	$0.779920\pi$
$548$		0			0
$549$	−146.900	+	358.206i	−0.267578	+	0.652471i
$550$		0			0
$551$		−	487.901i		−	0.885482i
$552$		0			0
$553$	484.733			0.876551
$554$		0			0
$555$		0			0
$556$		0			0
$557$			845.989i			1.51883i	0.650606	+	0.759415i	$0.274515\pi$
$557$			845.989i			1.51883i	−0.650606	+	0.759415i	$0.725485\pi$
$558$		0			0
$559$	223.819			0.400391
$560$		0			0
$561$	−296.634	−	442.258i	−0.528759	−	0.788339i
$562$		0			0
$563$		−	336.509i		−	0.597707i	−0.954299	−	0.298853i	$-0.903396\pi$
$563$		−	336.509i		−	0.597707i	0.954299	−	0.298853i	$-0.0966041\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−736.491	−	726.202i	−1.29893	−	1.28078i
$568$		0			0
$569$			552.736i			0.971416i	0.874121	+	0.485708i	$0.161438\pi$
$569$			552.736i			0.971416i	−0.874121	+	0.485708i	$0.838562\pi$
$570$		0			0
$571$	−772.222			−1.35240			−0.676202	−	0.736717i	$-0.736375\pi$
$571$	−772.222			−1.35240			−0.676202	+	0.736717i	$0.736375\pi$
$572$		0			0
$573$	−752.980	+	505.043i	−1.31410	+	0.881401i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−848.557			−1.47064			−0.735318	−	0.677722i	$-0.762967\pi$
$577$	−848.557			−1.47064			−0.735318	+	0.677722i	$0.762967\pi$
$578$		0			0
$579$	512.658	+	764.334i	0.885420	+	1.32009i
$580$		0			0
$581$			1024.43i			1.76322i
$582$		0			0
$583$	−367.755			−0.630798
$584$		0			0
$585$		0			0
$586$		0			0
$587$			575.536i			0.980470i	0.871590	+	0.490235i	$0.163089\pi$
$587$			575.536i			0.980470i	−0.871590	+	0.490235i	$0.836911\pi$
$588$		0			0
$589$	−635.529			−1.07900
$590$		0			0
$591$	−722.419	+	484.544i	−1.22237	+	0.819872i
$592$		0			0
$593$			156.935i			0.264646i	0.991207	+	0.132323i	$0.0422436\pi$
$593$			156.935i			0.264646i	−0.991207	+	0.132323i	$0.957756\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	639.352	+	953.225i	1.07094	+	1.59669i
$598$		0			0
$599$		−	517.564i		−	0.864047i	−0.901862	−	0.432024i	$-0.857800\pi$
$599$		−	517.564i		−	0.864047i	0.901862	−	0.432024i	$-0.142200\pi$
$600$		0			0
$601$	−883.588			−1.47020			−0.735098	−	0.677961i	$-0.762864\pi$
$601$	−883.588			−1.47020			−0.735098	+	0.677961i	$0.762864\pi$
$602$		0			0
$603$	−2.15190	+	5.24727i	−0.00356866	+	0.00870194i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−242.929			−0.400212			−0.200106	−	0.979774i	$-0.564129\pi$
$607$	−242.929			−0.400212			−0.200106	+	0.979774i	$0.564129\pi$
$608$		0			0
$609$	−500.431	+	335.652i	−0.821725	+	0.551152i
$610$		0			0
$611$		−	150.729i		−	0.246693i
$612$		0			0
$613$	157.263			0.256546			0.128273	−	0.991739i	$-0.459057\pi$
$613$	157.263			0.256546			0.128273	+	0.991739i	$0.459057\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	471.040i		−	0.763436i	−0.924279	−	0.381718i	$-0.875333\pi$
$617$		−	471.040i		−	0.763436i	0.924279	−	0.381718i	$-0.124667\pi$
$618$		0			0
$619$	550.320			0.889047			0.444524	−	0.895767i	$-0.353373\pi$
$619$	550.320			0.889047			0.444524	+	0.895767i	$0.353373\pi$
$620$		0			0
$621$	40.5759	+	198.510i	0.0653396	+	0.319662i
$622$		0			0
$623$			1799.48i			2.88841i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	976.009	−	654.634i	1.55663	−	1.04407i
$628$		0			0
$629$			181.676i			0.288834i
$630$		0			0
$631$	347.362			0.550495			0.275248	−	0.961373i	$-0.411240\pi$
$631$	347.362			0.550495			0.275248	+	0.961373i	$0.411240\pi$
$632$		0			0
$633$	536.614	+	800.051i	0.847732	+	1.26390i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−848.649			−1.33226
$638$		0			0
$639$	752.798	+	308.721i	1.17809	+	0.483132i
$640$		0			0
$641$			333.802i			0.520752i	0.965507	+	0.260376i	$0.0838465\pi$
$641$			333.802i			0.520752i	−0.965507	+	0.260376i	$0.916153\pi$
$642$		0			0
$643$	495.512			0.770625			0.385313	−	0.922786i	$-0.374094\pi$
$643$	495.512			0.770625			0.385313	+	0.922786i	$0.374094\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	149.493i		−	0.231056i	−0.993304	−	0.115528i	$-0.963144\pi$
$647$		−	149.493i		−	0.231056i	0.993304	−	0.115528i	$-0.0368559\pi$
$648$		0			0
$649$	601.590			0.926948
$650$		0			0
$651$	437.212	+	651.850i	0.671601	+	1.00131i
$652$		0			0
$653$			261.846i			0.400990i	0.979695	+	0.200495i	$0.0642550\pi$
$653$			261.846i			0.400990i	−0.979695	+	0.200495i	$0.935745\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−157.833	+	384.867i	−0.240234	+	0.585794i
$658$		0			0
$659$		−	148.718i		−	0.225672i	−0.993614	−	0.112836i	$-0.964006\pi$
$659$		−	148.718i		−	0.225672i	0.993614	−	0.112836i	$-0.0359935\pi$
$660$		0			0
$661$	−535.548			−0.810209			−0.405104	−	0.914270i	$-0.632765\pi$
$661$	−535.548			−0.810209			−0.405104	+	0.914270i	$0.632765\pi$
$662$		0			0
$663$	260.561	−	174.765i	0.393002	−	0.263597i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	118.040			0.176972
$668$		0			0
$669$	−488.681	−	728.586i	−0.730465	−	1.08907i
$670$		0			0
$671$			543.295i			0.809680i
$672$		0			0
$673$	−678.388			−1.00801			−0.504003	−	0.863702i	$-0.668140\pi$
$673$	−678.388			−1.00801			−0.504003	+	0.863702i	$0.668140\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	809.743i		−	1.19608i	−0.801468	−	0.598038i	$-0.795947\pi$
$677$		−	809.743i		−	1.19608i	0.801468	−	0.598038i	$-0.204053\pi$
$678$		0			0
$679$	132.070			0.194507
$680$		0			0
$681$	922.231	−	618.563i	1.35423	−	0.908316i
$682$		0			0
$683$			150.099i			0.219764i	0.993945	+	0.109882i	$0.0350473\pi$
$683$			150.099i			0.219764i	−0.993945	+	0.109882i	$0.964953\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	638.333	+	951.705i	0.929160	+	1.38531i
$688$		0			0
$689$		−	216.667i		−	0.314465i
$690$		0			0
$691$	334.001			0.483359			0.241679	−	0.970356i	$-0.422302\pi$
$691$	334.001			0.483359			0.241679	+	0.970356i	$0.422302\pi$
$692$		0			0
$693$	−1342.89	−	550.719i	−1.93780	−	0.794688i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−194.592			−0.279185
$698$		0			0
$699$	359.425	−	241.076i	0.514199	−	0.344886i
$700$		0			0
$701$		−	924.471i		−	1.31879i	−0.751797	−	0.659394i	$-0.770813\pi$
$701$		−	924.471i		−	1.31879i	0.751797	−	0.659394i	$-0.229187\pi$
$702$		0			0
$703$	−400.937			−0.570324
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	246.112i		−	0.348108i
$708$		0			0
$709$	−797.459			−1.12477			−0.562383	−	0.826877i	$-0.690115\pi$
$709$	−797.459			−1.12477			−0.562383	+	0.826877i	$0.690115\pi$
$710$		0			0
$711$	−129.633	+	316.101i	−0.182324	+	0.444586i
$712$		0			0
$713$		−	153.757i		−	0.215647i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−621.568	+	416.902i	−0.866902	+	0.581453i
$718$		0			0
$719$			907.966i			1.26282i	0.775450	+	0.631409i	$0.217523\pi$
$719$			907.966i			1.26282i	−0.775450	+	0.631409i	$0.782477\pi$
$720$		0			0
$721$	−89.0024			−0.123443
$722$		0			0
$723$	50.8918	+	75.8757i	0.0703897	+	0.104946i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1038.16			1.42801			0.714005	−	0.700140i	$-0.246879\pi$
$727$	1038.16			1.42801			0.714005	+	0.700140i	$0.246879\pi$
$728$		0			0
$729$	670.527	−	286.067i	0.919790	−	0.392410i
$730$		0			0
$731$		−	422.769i		−	0.578344i
$732$		0			0
$733$	−399.107			−0.544485			−0.272242	−	0.962229i	$-0.587765\pi$
$733$	−399.107			−0.544485			−0.272242	+	0.962229i	$0.587765\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			7.95858i			0.0107986i
$738$		0			0
$739$	452.504			0.612319			0.306159	−	0.951980i	$-0.400956\pi$
$739$	452.504			0.612319			0.306159	+	0.951980i	$0.400956\pi$
$740$		0			0
$741$	385.684	+	575.025i	0.520491	+	0.776012i
$742$		0			0
$743$		−	486.909i		−	0.655328i	−0.944794	−	0.327664i	$-0.893739\pi$
$743$		−	486.909i		−	0.655328i	0.944794	−	0.327664i	$-0.106261\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−668.046	−	273.965i	−0.894306	−	0.366754i
$748$		0			0
$749$		−	942.024i		−	1.25771i
$750$		0			0
$751$	470.899			0.627029			0.313515	−	0.949583i	$-0.398494\pi$
$751$	470.899			0.627029			0.313515	+	0.949583i	$0.398494\pi$
$752$		0			0
$753$	171.708	−	115.169i	0.228032	−	0.152947i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	886.264			1.17076			0.585379	−	0.810760i	$-0.300946\pi$
$757$	886.264			1.17076			0.585379	+	0.810760i	$0.300946\pi$
$758$		0			0
$759$	158.379	+	236.131i	0.208668	+	0.311108i
$760$		0			0
$761$			1022.30i			1.34336i	0.740841	+	0.671680i	$0.234427\pi$
$761$			1022.30i			1.34336i	−0.740841	+	0.671680i	$0.765573\pi$
$762$		0			0
$763$	−945.322			−1.23895
$764$		0			0
$765$		0			0
$766$		0			0
$767$			354.432i			0.462102i
$768$		0			0
$769$	−1051.96			−1.36796			−0.683982	−	0.729499i	$-0.739753\pi$
$769$	−1051.96			−1.36796			−0.683982	+	0.729499i	$0.739753\pi$
$770$		0			0
$771$	902.847	−	605.562i	1.17101	−	0.785424i
$772$		0			0
$773$			983.554i			1.27238i	0.771530	+	0.636192i	$0.219492\pi$
$773$			983.554i			1.27238i	−0.771530	+	0.636192i	$0.780508\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	275.825	+	411.234i	0.354987	+	0.529259i
$778$		0			0
$779$		−	429.441i		−	0.551272i
$780$		0			0
$781$	1141.77			1.46194
$782$		0			0
$783$	−85.0522	−	416.101i	−0.108624	−	0.531419i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−984.726			−1.25124			−0.625620	−	0.780128i	$-0.715154\pi$
$787$	−984.726			−1.25124			−0.625620	+	0.780128i	$0.715154\pi$
$788$		0			0
$789$	−58.4214	+	39.1847i	−0.0740449	+	0.0496638i
$790$		0			0
$791$			1879.82i			2.37651i
$792$		0			0
$793$	−320.087			−0.403641
$794$		0			0
$795$		0			0
$796$		0			0
$797$			359.710i			0.451330i	0.974205	+	0.225665i	$0.0724555\pi$
$797$			359.710i			0.451330i	−0.974205	+	0.225665i	$0.927544\pi$
$798$		0			0
$799$	−284.712			−0.356335
$800$		0			0
$801$	−1173.46	−	481.236i	−1.46500	−	0.600794i
$802$		0			0
$803$			583.731i			0.726938i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−988.463	+	662.987i	−1.22486	+	0.821546i
$808$		0			0
$809$		−	1174.75i		−	1.45211i	−0.687638	−	0.726053i	$-0.741352\pi$
$809$		−	1174.75i		−	1.45211i	0.687638	−	0.726053i	$-0.258648\pi$
$810$		0			0
$811$	1000.55			1.23373			0.616863	−	0.787071i	$-0.288403\pi$
$811$	1000.55			1.23373			0.616863	+	0.787071i	$0.288403\pi$
$812$		0			0
$813$	240.514	+	358.588i	0.295835	+	0.441067i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	933.000			1.14198
$818$		0			0
$819$	324.461	−	791.178i	0.396167	−	0.966029i
$820$		0			0
$821$			536.992i			0.654071i	0.945012	+	0.327036i	$0.106050\pi$
$821$			536.992i			0.654071i	−0.945012	+	0.327036i	$0.893950\pi$
$822$		0			0
$823$	563.712			0.684948			0.342474	−	0.939527i	$-0.388735\pi$
$823$	563.712			0.684948			0.342474	+	0.939527i	$0.388735\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			976.487i			1.18076i	0.807126	+	0.590379i	$0.201022\pi$
$827$			976.487i			1.18076i	−0.807126	+	0.590379i	$0.798978\pi$
$828$		0			0
$829$	358.250			0.432147			0.216074	−	0.976377i	$-0.430675\pi$
$829$	358.250			0.432147			0.216074	+	0.976377i	$0.430675\pi$
$830$		0			0
$831$	−603.071	−	899.133i	−0.725718	−	1.08199i
$832$		0			0
$833$			1603.01i			1.92438i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−542.005	+	110.787i	−0.647556	+	0.132362i
$838$		0			0
$839$		−	98.9138i		−	0.117895i	−0.998261	−	0.0589474i	$-0.981226\pi$
$839$		−	98.9138i		−	0.117895i	0.998261	−	0.0589474i	$-0.0187744\pi$
$840$		0			0
$841$	593.573			0.705794
$842$		0			0
$843$	−1324.61	+	888.447i	−1.57130	+	1.05391i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−491.703			−0.580523
$848$		0			0
$849$	430.140	+	641.305i	0.506643	+	0.755366i
$850$		0			0
$851$		−	97.0007i		−	0.113984i
$852$		0			0
$853$	262.197			0.307383			0.153691	−	0.988119i	$-0.450884\pi$
$853$	262.197			0.307383			0.153691	+	0.988119i	$0.450884\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			41.1542i			0.0480212i	0.999712	+	0.0240106i	$0.00764354\pi$
$857$			41.1542i			0.0480212i	−0.999712	+	0.0240106i	$0.992356\pi$
$858$		0			0
$859$	−414.736			−0.482813			−0.241406	−	0.970424i	$-0.577609\pi$
$859$	−414.736			−0.482813			−0.241406	+	0.970424i	$0.577609\pi$
$860$		0			0
$861$	−440.470	+	295.434i	−0.511579	+	0.343129i
$862$		0			0
$863$			1253.21i			1.45216i	0.687613	+	0.726078i	$0.258659\pi$
$863$			1253.21i			1.45216i	−0.687613	+	0.726078i	$0.741341\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	152.835	+	227.865i	0.176280	+	0.262820i
$868$		0			0
$869$			479.433i			0.551706i
$870$		0			0
$871$	−4.68888			−0.00538332
$872$		0			0
$873$	−35.3197	+	86.1249i	−0.0404579	+	0.0986539i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	861.957			0.982847			0.491424	−	0.870921i	$-0.336477\pi$
$877$	861.957			0.982847			0.491424	+	0.870921i	$0.336477\pi$
$878$		0			0
$879$	−773.494	+	518.802i	−0.879970	+	0.590218i
$880$		0			0
$881$			87.0009i			0.0987524i	0.998780	+	0.0493762i	$0.0157233\pi$
$881$			87.0009i			0.0987524i	−0.998780	+	0.0493762i	$0.984277\pi$
$882$		0			0
$883$	−1022.42			−1.15789			−0.578945	−	0.815366i	$-0.696536\pi$
$883$	−1022.42			−1.15789			−0.578945	+	0.815366i	$0.696536\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	669.924i		−	0.755269i	−0.925955	−	0.377634i	$-0.876738\pi$
$887$		−	669.924i		−	0.755269i	0.925955	−	0.377634i	$-0.123262\pi$
$888$		0			0
$889$	−1149.45			−1.29297
$890$		0			0
$891$	718.263	−	728.439i	0.806131	−	0.817552i
$892$		0			0
$893$		−	628.324i		−	0.703610i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−139.119	+	93.3104i	−0.155093	+	0.104025i
$898$		0			0
$899$			322.293i			0.358502i
$900$		0			0
$901$	−409.260			−0.454229
$902$		0			0
$903$	−641.858	−	956.960i	−0.710806	−	1.05976i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1296.52			1.42946			0.714729	−	0.699402i	$-0.246550\pi$
$907$	1296.52			1.42946			0.714729	+	0.699402i	$0.246550\pi$
$908$		0			0
$909$	160.493	+	65.8181i	0.176560	+	0.0724071i
$910$		0			0
$911$			926.622i			1.01715i	0.861018	+	0.508574i	$0.169827\pi$
$911$			926.622i			1.01715i	−0.861018	+	0.508574i	$0.830173\pi$
$912$		0			0
$913$	−1013.23			−1.10978
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	144.179i		−	0.157229i
$918$		0			0
$919$	−478.391			−0.520556			−0.260278	−	0.965534i	$-0.583814\pi$
$919$	−478.391			−0.520556			−0.260278	+	0.965534i	$0.583814\pi$
$920$		0			0
$921$	444.793	+	663.152i	0.482946	+	0.720035i
$922$		0			0
$923$			672.687i			0.728805i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	23.8020	−	58.0397i	0.0256764	−	0.0626102i
$928$		0			0
$929$			1452.00i			1.56297i	0.623923	+	0.781486i	$0.285538\pi$
$929$			1452.00i			1.56297i	−0.623923	+	0.781486i	$0.714462\pi$
$930$		0			0
$931$	−3537.64			−3.79983
$932$		0			0
$933$	464.858	−	311.792i	0.498240	−	0.334182i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1535.86			−1.63912			−0.819562	−	0.572991i	$-0.805783\pi$
$937$	−1535.86			−1.63912			−0.819562	+	0.572991i	$0.805783\pi$
$938$		0			0
$939$	34.2610	+	51.0805i	0.0364866	+	0.0543988i
$940$		0			0
$941$		−	1237.50i		−	1.31509i	−0.753415	−	0.657545i	$-0.771595\pi$
$941$		−	1237.50i		−	1.31509i	0.753415	−	0.657545i	$-0.228405\pi$
$942$		0			0
$943$	103.897			0.110177
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	603.800i		−	0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$947$		−	603.800i		−	0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$948$		0			0
$949$	−343.911			−0.362393
$950$		0			0
$951$	47.2861	−	31.7160i	0.0497225	−	0.0333501i
$952$		0			0
$953$		−	534.553i		−	0.560916i	−0.959866	−	0.280458i	$-0.909514\pi$
$953$		−	534.553i		−	0.560916i	0.959866	−	0.280458i	$-0.0904863\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−331.982	−	494.959i	−0.346899	−	0.517199i
$958$		0			0
$959$			287.186i			0.299464i
$960$		0			0
$961$	−541.188			−0.563151
$962$		0			0
$963$	614.307	+	251.927i	0.637909	+	0.261606i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	271.601			0.280870			0.140435	−	0.990090i	$-0.455150\pi$
$967$	271.601			0.280870			0.140435	+	0.990090i	$0.455150\pi$
$968$		0			0
$969$	1086.16	−	728.515i	1.12091	−	0.751822i
$970$		0			0
$971$		−	912.176i		−	0.939419i	−0.882821	−	0.469709i	$-0.844359\pi$
$971$		−	912.176i		−	0.939419i	0.882821	−	0.469709i	$-0.155641\pi$
$972$		0			0
$973$	1163.22			1.19550
$974$		0			0
$975$		0			0
$976$		0			0
$977$			333.628i			0.341482i	0.985316	+	0.170741i	$0.0546162\pi$
$977$			333.628i			0.341482i	−0.985316	+	0.170741i	$0.945384\pi$
$978$		0			0
$979$	−1779.80			−1.81798
$980$		0			0
$981$	252.809	−	616.458i	0.257705	−	0.628397i
$982$		0			0
$983$			1600.35i			1.62803i	0.580847	+	0.814013i	$0.302722\pi$
$983$			1600.35i			1.62803i	−0.580847	+	0.814013i	$0.697278\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−644.460	+	432.256i	−0.652948	+	0.437949i
$988$		0			0
$989$			225.725i			0.228236i
$990$		0			0
$991$	69.3757			0.0700057			0.0350029	−	0.999387i	$-0.488856\pi$
$991$	69.3757			0.0700057			0.0350029	+	0.999387i	$0.488856\pi$
$992$		0			0
$993$	−690.485	−	1029.46i	−0.695353	−	1.03672i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−482.497			−0.483949			−0.241974	−	0.970283i	$-0.577795\pi$
$997$	−482.497			−0.483949			−0.241974	+	0.970283i	$0.577795\pi$
$998$		0			0
$999$	−341.936	+	69.8925i	−0.342278	+	0.0699625i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.l.y.401.10		12
3.2	odd	2	inner	1200.3.l.y.401.9		12
4.3	odd	2		600.3.l.g.401.3		12
5.2	odd	4		240.3.c.e.209.9		12
5.3	odd	4		240.3.c.e.209.4		12
5.4	even	2	inner	1200.3.l.y.401.3		12
12.11	even	2		600.3.l.g.401.4		12
15.2	even	4		240.3.c.e.209.3		12
15.8	even	4		240.3.c.e.209.10		12
15.14	odd	2	inner	1200.3.l.y.401.4		12
20.3	even	4		120.3.c.a.89.9	yes	12
20.7	even	4		120.3.c.a.89.4	yes	12
20.19	odd	2		600.3.l.g.401.10		12
40.3	even	4		960.3.c.k.449.4		12
40.13	odd	4		960.3.c.j.449.9		12
40.27	even	4		960.3.c.k.449.9		12
40.37	odd	4		960.3.c.j.449.4		12
60.23	odd	4		120.3.c.a.89.3	✓	12
60.47	odd	4		120.3.c.a.89.10	yes	12
60.59	even	2		600.3.l.g.401.9		12
120.53	even	4		960.3.c.j.449.3		12
120.77	even	4		960.3.c.j.449.10		12
120.83	odd	4		960.3.c.k.449.10		12
120.107	odd	4		960.3.c.k.449.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.c.a.89.3	✓	12	60.23	odd	4
120.3.c.a.89.4	yes	12	20.7	even	4
120.3.c.a.89.9	yes	12	20.3	even	4
120.3.c.a.89.10	yes	12	60.47	odd	4
240.3.c.e.209.3		12	15.2	even	4
240.3.c.e.209.4		12	5.3	odd	4
240.3.c.e.209.9		12	5.2	odd	4
240.3.c.e.209.10		12	15.8	even	4
600.3.l.g.401.3		12	4.3	odd	2
600.3.l.g.401.4		12	12.11	even	2
600.3.l.g.401.9		12	60.59	even	2
600.3.l.g.401.10		12	20.19	odd	2
960.3.c.j.449.3		12	120.53	even	4
960.3.c.j.449.4		12	40.37	odd	4
960.3.c.j.449.9		12	40.13	odd	4
960.3.c.j.449.10		12	120.77	even	4
960.3.c.k.449.3		12	120.107	odd	4
960.3.c.k.449.4		12	40.3	even	4
960.3.c.k.449.9		12	40.27	even	4
960.3.c.k.449.10		12	120.83	odd	4
1200.3.l.y.401.3		12	5.4	even	2	inner
1200.3.l.y.401.4		12	15.14	odd	2	inner
1200.3.l.y.401.9		12	3.2	odd	2	inner
1200.3.l.y.401.10		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.67109 + 2.49147i) q^{3} +12.7692 q^{7} +(-3.41489 + 8.32698i) q^{9} +12.6296i q^{11} -7.44085 q^{13} +14.0550i q^{17} -31.0176 q^{19} +(21.3386 + 31.8142i) q^{21} -7.50423i q^{23} +(-26.4530 + 5.40707i) q^{27} +15.7298i q^{29} +20.4893 q^{31} +(-31.4663 + 21.1053i) q^{33} +12.9261 q^{37} +(-12.4344 - 18.5387i) q^{39} +13.8451i q^{41} -30.0797 q^{43} +20.2570i q^{47} +114.053 q^{49} +(-35.0176 + 23.4872i) q^{51} +29.1185i q^{53} +(-51.8333 - 77.2795i) q^{57} -47.6333i q^{59} +43.0176 q^{61} +(-43.6054 + 106.329i) q^{63} +0.630153 q^{67} +(18.6966 - 12.5403i) q^{69} -90.4047i q^{71} +46.2193 q^{73} +161.270i q^{77} +37.9610 q^{79} +(-57.6771 - 56.8713i) q^{81} +80.2267i q^{83} +(-39.1904 + 26.2860i) q^{87} +140.923i q^{89} -95.0138 q^{91} +(34.2396 + 51.0486i) q^{93} +10.3429 q^{97} +(-105.166 - 43.1286i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9} + 4 q^{21} + 48 q^{31} - 128 q^{39} + 252 q^{49} - 48 q^{51} + 144 q^{61} - 268 q^{69} + 432 q^{79} - 188 q^{81} - 560 q^{99}+O(q^{100}) \) 12 * q - 8 * q^9 + 4 * q^21 + 48 * q^31 - 128 * q^39 + 252 * q^49 - 48 * q^51 + 144 * q^61 - 268 * q^69 + 432 * q^79 - 188 * q^81 - 560 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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