LMFDB - Embedded newform 624.4.d.a.287.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(287,624)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("624.287");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$36.8171918436$
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} + 5x^{10} - 129x^{8} + 8982x^{6} - 94041x^{4} + 2657205x^{2} + 387420489 $ x^12 + 5x^10 - 129x^8 + 8982x^6 - 94041x^4 + 2657205*x^2 + 387420489
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.7
Root			$0.861319 + 5.12427i$ of defining polynomial
Character	$\chi$	$=$	624.287
Dual form			624.4.d.a.287.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.861319 - 5.12427i) q^{3} +19.6032i q^{5} +0.649873i q^{7} +(-25.5163 - 8.82726i) q^{9} +O(q^{10})$	\(q+(0.861319 - 5.12427i) q^{3} +19.6032i q^{5} +0.649873i q^{7} +(-25.5163 - 8.82726i) q^{9} +9.64478 q^{11} +13.0000 q^{13} +(100.452 + 16.8846i) q^{15} -48.1203i q^{17} -103.862i q^{19} +(3.33012 + 0.559748i) q^{21} +170.938 q^{23} -259.284 q^{25} +(-67.2109 + 123.149i) q^{27} +218.703i q^{29} +291.539i q^{31} +(8.30723 - 49.4225i) q^{33} -12.7396 q^{35} -217.098 q^{37} +(11.1971 - 66.6155i) q^{39} +353.147i q^{41} +426.237i q^{43} +(173.042 - 500.199i) q^{45} -105.828 q^{47} +342.578 q^{49} +(-246.582 - 41.4470i) q^{51} +286.136i q^{53} +189.068i q^{55} +(-532.214 - 89.4579i) q^{57} -742.972 q^{59} -238.480 q^{61} +(5.73660 - 16.5823i) q^{63} +254.841i q^{65} +139.685i q^{67} +(147.232 - 875.931i) q^{69} +88.4498 q^{71} +131.451 q^{73} +(-223.326 + 1328.64i) q^{75} +6.26788i q^{77} +907.826i q^{79} +(573.159 + 450.477i) q^{81} -245.398 q^{83} +943.311 q^{85} +(1120.69 + 188.373i) q^{87} -771.180i q^{89} +8.44835i q^{91} +(1493.93 + 251.108i) q^{93} +2036.01 q^{95} -777.823 q^{97} +(-246.099 - 85.1370i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{9}+O(q^{10}) $ 12 * q - 10 * q^9	$ 12 q - 10 q^{9} + 156 q^{13} + 478 q^{21} - 600 q^{25} - 184 q^{33} - 828 q^{37} + 170 q^{45} - 720 q^{49} - 204 q^{57} + 192 q^{61} + 2476 q^{69} + 960 q^{73} + 566 q^{81} + 348 q^{85} + 6388 q^{93} - 7248 q^{97}+O(q^{100}) $ 12 * q - 10 * q^9 + 156 * q^13 + 478 * q^21 - 600 * q^25 - 184 * q^33 - 828 * q^37 + 170 * q^45 - 720 * q^49 - 204 * q^57 + 192 * q^61 + 2476 * q^69 + 960 * q^73 + 566 * q^81 + 348 * q^85 + 6388 * q^93 - 7248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0.861319	−	5.12427i	0.165761	−	0.986166i
$3$	0.861319	−	5.12427i	0.165761	−	0.986166i
$4$		0			0
$5$			19.6032i			1.75336i	0.481074	+	0.876680i	$0.340247\pi$
$5$			19.6032i			1.75336i	−0.481074	+	0.876680i	$0.659753\pi$
$6$		0			0
$7$			0.649873i			0.0350898i	0.999846	+	0.0175449i	$0.00558501\pi$
$7$			0.649873i			0.0350898i	−0.999846	+	0.0175449i	$0.994415\pi$
$8$		0			0
$9$	−25.5163	−	8.82726i	−0.945047	−	0.326936i
$10$		0			0
$11$	9.64478			0.264365			0.132182	−	0.991225i	$-0.457802\pi$
$11$	9.64478			0.264365			0.132182	+	0.991225i	$0.457802\pi$
$12$		0			0
$13$	13.0000			0.277350
$13$	13.0000			0.277350
$14$		0			0
$15$	100.452	+	16.8846i	1.72910	+	0.290638i
$16$		0			0
$17$		−	48.1203i		−	0.686523i	−0.939240	−	0.343262i	$-0.888468\pi$
$17$		−	48.1203i		−	0.686523i	0.939240	−	0.343262i	$-0.111532\pi$
$18$		0			0
$19$		−	103.862i		−	1.25408i	−0.778988	−	0.627039i	$-0.784267\pi$
$19$		−	103.862i		−	1.25408i	0.778988	−	0.627039i	$-0.215733\pi$
$20$		0			0
$21$	3.33012	+	0.559748i	0.0346044	+	0.00581652i
$22$		0			0
$23$	170.938			1.54969			0.774847	−	0.632148i	$-0.217827\pi$
$23$	170.938			1.54969			0.774847	+	0.632148i	$0.217827\pi$
$24$		0			0
$25$	−259.284			−2.07427
$26$		0			0
$27$	−67.2109	+	123.149i	−0.479065	+	0.877780i
$28$		0			0
$29$			218.703i			1.40042i	0.713938	+	0.700209i	$0.246910\pi$
$29$			218.703i			1.40042i	−0.713938	+	0.700209i	$0.753090\pi$
$30$		0			0
$31$			291.539i			1.68910i	0.535480	+	0.844548i	$0.320131\pi$
$31$			291.539i			1.68910i	−0.535480	+	0.844548i	$0.679869\pi$
$32$		0			0
$33$	8.30723	−	49.4225i	0.0438213	−	0.260707i
$34$		0			0
$35$	−12.7396			−0.0615251
$36$		0			0
$37$	−217.098			−0.964611			−0.482306	−	0.876003i	$-0.660201\pi$
$37$	−217.098			−0.964611			−0.482306	+	0.876003i	$0.660201\pi$
$38$		0			0
$39$	11.1971	−	66.6155i	0.0459738	−	0.273513i
$40$		0			0
$41$			353.147i			1.34518i	0.740016	+	0.672589i	$0.234818\pi$
$41$			353.147i			1.34518i	−0.740016	+	0.672589i	$0.765182\pi$
$42$		0			0
$43$			426.237i			1.51164i	0.654779	+	0.755820i	$0.272762\pi$
$43$			426.237i			1.51164i	−0.654779	+	0.755820i	$0.727238\pi$
$44$		0			0
$45$	173.042	−	500.199i	0.573236	−	1.65701i
$46$		0			0
$47$	−105.828			−0.328438			−0.164219	−	0.986424i	$-0.552510\pi$
$47$	−105.828			−0.328438			−0.164219	+	0.986424i	$0.552510\pi$
$48$		0			0
$49$	342.578			0.998769
$50$		0			0
$51$	−246.582	−	41.4470i	−0.677026	−	0.113799i
$52$		0			0
$53$			286.136i			0.741581i	0.928717	+	0.370791i	$0.120913\pi$
$53$			286.136i			0.741581i	−0.928717	+	0.370791i	$0.879087\pi$
$54$		0			0
$55$			189.068i			0.463526i
$56$		0			0
$57$	−532.214	−	89.4579i	−1.23673	−	0.207877i
$58$		0			0
$59$	−742.972			−1.63944			−0.819718	−	0.572768i	$-0.805870\pi$
$59$	−742.972			−1.63944			−0.819718	+	0.572768i	$0.805870\pi$
$60$		0			0
$61$	−238.480			−0.500561			−0.250281	−	0.968173i	$-0.580523\pi$
$61$	−238.480			−0.500561			−0.250281	+	0.968173i	$0.580523\pi$
$62$		0			0
$63$	5.73660	−	16.5823i	0.0114721	−	0.0331615i
$64$		0			0
$65$			254.841i			0.486294i
$66$		0			0
$67$			139.685i			0.254706i	0.991857	+	0.127353i	$0.0406481\pi$
$67$			139.685i			0.254706i	−0.991857	+	0.127353i	$0.959352\pi$
$68$		0			0
$69$	147.232	−	875.931i	0.256879	−	1.52826i
$70$		0			0
$71$	88.4498			0.147846			0.0739229	−	0.997264i	$-0.476448\pi$
$71$	88.4498			0.147846			0.0739229	+	0.997264i	$0.476448\pi$
$72$		0			0
$73$	131.451			0.210755			0.105378	−	0.994432i	$-0.466395\pi$
$73$	131.451			0.210755			0.105378	+	0.994432i	$0.466395\pi$
$74$		0			0
$75$	−223.326	+	1328.64i	−0.343833	+	2.04557i
$76$		0			0
$77$			6.26788i			0.00927651i
$78$		0			0
$79$			907.826i			1.29289i	0.762960	+	0.646446i	$0.223745\pi$
$79$			907.826i			1.29289i	−0.762960	+	0.646446i	$0.776255\pi$
$80$		0			0
$81$	573.159	+	450.477i	0.786226	+	0.617939i
$82$		0			0
$83$	−245.398			−0.324529			−0.162264	−	0.986747i	$-0.551880\pi$
$83$	−245.398			−0.324529			−0.162264	+	0.986747i	$0.551880\pi$
$84$		0			0
$85$	943.311			1.20372
$86$		0			0
$87$	1120.69	+	188.373i	1.38104	+	0.232134i
$88$		0			0
$89$		−	771.180i		−	0.918482i	−0.888312	−	0.459241i	$-0.848121\pi$
$89$		−	771.180i		−	0.918482i	0.888312	−	0.459241i	$-0.151879\pi$
$90$		0			0
$91$			8.44835i			0.00973217i
$92$		0			0
$93$	1493.93	+	251.108i	1.66573	+	0.279986i
$94$		0			0
$95$	2036.01			2.19885
$96$		0			0
$97$	−777.823			−0.814185			−0.407092	−	0.913387i	$-0.633457\pi$
$97$	−777.823			−0.814185			−0.407092	+	0.913387i	$0.633457\pi$
$98$		0			0
$99$	−246.099	−	85.1370i	−0.249837	−	0.0864302i
$100$		0			0
$101$			967.502i			0.953169i	0.879129	+	0.476584i	$0.158125\pi$
$101$			967.502i			0.953169i	−0.879129	+	0.476584i	$0.841875\pi$
$102$		0			0
$103$			171.867i			0.164413i	0.996615	+	0.0822066i	$0.0261967\pi$
$103$			171.867i			0.164413i	−0.996615	+	0.0822066i	$0.973803\pi$
$104$		0			0
$105$	−10.9728	+	65.2809i	−0.0101985	+	0.0606740i
$106$		0			0
$107$	1281.92			1.15820			0.579102	−	0.815255i	$-0.303403\pi$
$107$	1281.92			1.15820			0.579102	+	0.815255i	$0.303403\pi$
$108$		0			0
$109$	315.331			0.277094			0.138547	−	0.990356i	$-0.455757\pi$
$109$	315.331			0.277094			0.138547	+	0.990356i	$0.455757\pi$
$110$		0			0
$111$	−186.990	+	1112.47i	−0.159895	+	0.951267i
$112$		0			0
$113$		−	441.668i		−	0.367687i	−0.982956	−	0.183844i	$-0.941146\pi$
$113$		−	441.668i		−	0.367687i	0.982956	−	0.183844i	$-0.0588540\pi$
$114$		0			0
$115$			3350.92i			2.71717i
$116$		0			0
$117$	−331.711	−	114.754i	−0.262109	−	0.0906756i
$118$		0			0
$119$	31.2721			0.0240900
$120$		0			0
$121$	−1237.98			−0.930111
$122$		0			0
$123$	1809.62	+	304.172i	1.32657	+	0.222978i
$124$		0			0
$125$		−	2632.38i		−	1.88358i
$126$		0			0
$127$		−	1725.99i		−	1.20596i	−0.797756	−	0.602981i	$-0.793979\pi$
$127$		−	1725.99i		−	1.20596i	0.797756	−	0.602981i	$-0.206021\pi$
$128$		0			0
$129$	2184.15	+	367.126i	1.49073	+	0.250571i
$130$		0			0
$131$	1626.29			1.08466			0.542328	−	0.840167i	$-0.317543\pi$
$131$	1626.29			1.08466			0.542328	+	0.840167i	$0.317543\pi$
$132$		0			0
$133$	67.4968			0.0440054
$134$		0			0
$135$	−2414.11	−	1317.55i	−1.53906	−	0.839972i
$136$		0			0
$137$		−	500.538i		−	0.312145i	−0.987746	−	0.156072i	$-0.950117\pi$
$137$		−	500.538i		−	0.312145i	0.987746	−	0.156072i	$-0.0498833\pi$
$138$		0			0
$139$			2063.68i			1.25927i	0.776890	+	0.629636i	$0.216796\pi$
$139$			2063.68i			1.25927i	−0.776890	+	0.629636i	$0.783204\pi$
$140$		0			0
$141$	−91.1517	+	542.291i	−0.0544422	+	0.323895i
$142$		0			0
$143$	125.382			0.0733216
$144$		0			0
$145$	−4287.27			−2.45543
$146$		0			0
$147$	295.069	−	1755.46i	0.165557	−	0.984952i
$148$		0			0
$149$			2062.22i			1.13385i	0.823770	+	0.566924i	$0.191867\pi$
$149$			2062.22i			1.13385i	−0.823770	+	0.566924i	$0.808133\pi$
$150$		0			0
$151$		−	1514.65i		−	0.816295i	−0.912916	−	0.408147i	$-0.866175\pi$
$151$		−	1514.65i		−	0.816295i	0.912916	−	0.408147i	$-0.133825\pi$
$152$		0			0
$153$	−424.771	+	1227.85i	−0.224449	+	0.648797i
$154$		0			0
$155$	−5715.09			−2.96159
$156$		0			0
$157$	2327.26			1.18303			0.591514	−	0.806295i	$-0.298530\pi$
$157$	2327.26			1.18303			0.591514	+	0.806295i	$0.298530\pi$
$158$		0			0
$159$	1466.24	+	246.454i	0.731322	+	0.122925i
$160$		0			0
$161$			111.088i			0.0543785i
$162$		0			0
$163$			638.276i			0.306709i	0.988171	+	0.153355i	$0.0490077\pi$
$163$			638.276i			0.306709i	−0.988171	+	0.153355i	$0.950992\pi$
$164$		0			0
$165$	968.836	+	162.848i	0.457114	+	0.0768345i
$166$		0			0
$167$	−3931.12			−1.82155			−0.910776	−	0.412901i	$-0.864515\pi$
$167$	−3931.12			−1.82155			−0.910776	+	0.412901i	$0.864515\pi$
$168$		0			0
$169$	169.000			0.0769231
$170$		0			0
$171$	−916.813	+	2650.16i	−0.410002	+	1.18516i
$172$		0			0
$173$			954.068i			0.419286i	0.977778	+	0.209643i	$0.0672301\pi$
$173$			954.068i			0.419286i	−0.977778	+	0.209643i	$0.932770\pi$
$174$		0			0
$175$		−	168.501i		−	0.0727858i
$176$		0			0
$177$	−639.936	+	3807.19i	−0.271754	+	1.61676i
$178$		0			0
$179$	2223.21			0.928328			0.464164	−	0.885749i	$-0.346355\pi$
$179$	2223.21			0.928328			0.464164	+	0.885749i	$0.346355\pi$
$180$		0			0
$181$	−2149.81			−0.882840			−0.441420	−	0.897301i	$-0.645525\pi$
$181$	−2149.81			−0.882840			−0.441420	+	0.897301i	$0.645525\pi$
$182$		0			0
$183$	−205.407	+	1222.04i	−0.0829735	+	0.493637i
$184$		0			0
$185$		−	4255.80i		−	1.69131i
$186$		0			0
$187$		−	464.110i		−	0.181493i
$188$		0			0
$189$	−80.0313	−	43.6785i	−0.0308011	−	0.0168103i
$190$		0			0
$191$	2912.01			1.10317			0.551586	−	0.834118i	$-0.314023\pi$
$191$	2912.01			1.10317			0.551586	+	0.834118i	$0.314023\pi$
$192$		0			0
$193$	−3265.82			−1.21802			−0.609012	−	0.793161i	$-0.708434\pi$
$193$	−3265.82			−1.21802			−0.609012	+	0.793161i	$0.708434\pi$
$194$		0			0
$195$	1305.87	+	219.499i	0.479567	+	0.0806086i
$196$		0			0
$197$		−	3173.76i		−	1.14782i	−0.818917	−	0.573912i	$-0.805425\pi$
$197$		−	3173.76i		−	1.14782i	0.818917	−	0.573912i	$-0.194575\pi$
$198$		0			0
$199$			1652.47i			0.588644i	0.955706	+	0.294322i	$0.0950938\pi$
$199$			1652.47i			0.588644i	−0.955706	+	0.294322i	$0.904906\pi$
$200$		0			0
$201$	715.786	+	120.314i	0.251182	+	0.0422203i
$202$		0			0
$203$	−142.129			−0.0491404
$204$		0			0
$205$	−6922.80			−2.35858
$206$		0			0
$207$	−4361.69	−	1508.91i	−1.46453	−	0.506650i
$208$		0			0
$209$		−	1001.72i		−	0.331534i
$210$		0			0
$211$		−	4281.11i		−	1.39679i	−0.715711	−	0.698397i	$-0.753897\pi$
$211$		−	4281.11i		−	1.39679i	0.715711	−	0.698397i	$-0.246103\pi$
$212$		0			0
$213$	76.1835	−	453.240i	0.0245071	−	0.145801i
$214$		0			0
$215$	−8355.59			−2.65045
$216$		0			0
$217$	−189.463			−0.0592701
$218$		0			0
$219$	113.221	−	673.588i	0.0349350	−	0.207840i
$220$		0			0
$221$		−	625.565i		−	0.190407i
$222$		0			0
$223$		−	3317.86i		−	0.996323i	−0.867084	−	0.498162i	$-0.834009\pi$
$223$		−	3317.86i		−	0.996323i	0.867084	−	0.498162i	$-0.165991\pi$
$224$		0			0
$225$	6615.95	+	2288.76i	1.96028	+	0.678152i
$226$		0			0
$227$	6153.68			1.79927			0.899635	−	0.436643i	$-0.143833\pi$
$227$	6153.68			1.79927			0.899635	+	0.436643i	$0.143833\pi$
$228$		0			0
$229$	−2499.23			−0.721196			−0.360598	−	0.932721i	$-0.617427\pi$
$229$	−2499.23			−0.721196			−0.360598	+	0.932721i	$0.617427\pi$
$230$		0			0
$231$	32.1183	+	5.39865i	0.00914818	+	0.00153768i
$232$		0			0
$233$			3281.39i			0.922623i	0.887238	+	0.461311i	$0.152621\pi$
$233$			3281.39i			0.922623i	−0.887238	+	0.461311i	$0.847379\pi$
$234$		0			0
$235$		−	2074.56i		−	0.575870i
$236$		0			0
$237$	4651.95	+	781.928i	1.27501	+	0.214311i
$238$		0			0
$239$	1544.69			0.418067			0.209033	−	0.977908i	$-0.432968\pi$
$239$	1544.69			0.418067			0.209033	+	0.977908i	$0.432968\pi$
$240$		0			0
$241$	2717.55			0.726361			0.363180	−	0.931719i	$-0.381691\pi$
$241$	2717.55			0.726361			0.363180	+	0.931719i	$0.381691\pi$
$242$		0			0
$243$	2802.04	−	2549.02i	0.739716	−	0.672920i
$244$		0			0
$245$			6715.60i			1.75120i
$246$		0			0
$247$		−	1350.20i		−	0.347818i
$248$		0			0
$249$	−211.366	+	1257.48i	−0.0537942	+	0.320039i
$250$		0			0
$251$	7362.85			1.85155			0.925775	−	0.378076i	$-0.123414\pi$
$251$	7362.85			1.85155			0.925775	+	0.378076i	$0.123414\pi$
$252$		0			0
$253$	1648.66			0.409685
$254$		0			0
$255$	812.491	−	4833.78i	0.199530	−	1.18707i
$256$		0			0
$257$		−	5865.64i		−	1.42369i	−0.702336	−	0.711846i	$-0.747859\pi$
$257$		−	5865.64i		−	1.42369i	0.702336	−	0.711846i	$-0.252141\pi$
$258$		0			0
$259$		−	141.086i		−	0.0338481i
$260$		0			0
$261$	1930.55	−	5580.48i	0.457846	−	1.32346i
$262$		0			0
$263$	2548.25			0.597459			0.298730	−	0.954338i	$-0.403437\pi$
$263$	2548.25			0.597459			0.298730	+	0.954338i	$0.403437\pi$
$264$		0			0
$265$	−5609.17			−1.30026
$266$		0			0
$267$	−3951.73	−	664.232i	−0.905776	−	0.152248i
$268$		0			0
$269$			3826.99i			0.867418i	0.901053	+	0.433709i	$0.142795\pi$
$269$			3826.99i			0.867418i	−0.901053	+	0.433709i	$0.857205\pi$
$270$		0			0
$271$		−	3979.32i		−	0.891979i	−0.895038	−	0.445989i	$-0.852852\pi$
$271$		−	3979.32i		−	0.891979i	0.895038	−	0.445989i	$-0.147148\pi$
$272$		0			0
$273$	43.2916	+	7.27672i	0.00959754	+	0.00161321i
$274$		0			0
$275$	−2500.73			−0.548364
$276$		0			0
$277$	2636.56			0.571897			0.285948	−	0.958245i	$-0.407691\pi$
$277$	2636.56			0.571897			0.285948	+	0.958245i	$0.407691\pi$
$278$		0			0
$279$	2573.49	−	7438.99i	0.552226	−	1.59627i
$280$		0			0
$281$			3985.00i			0.845997i	0.906130	+	0.422999i	$0.139023\pi$
$281$			3985.00i			0.845997i	−0.906130	+	0.422999i	$0.860977\pi$
$282$		0			0
$283$			1925.86i			0.404525i	0.979331	+	0.202263i	$0.0648295\pi$
$283$			1925.86i			0.404525i	−0.979331	+	0.202263i	$0.935171\pi$
$284$		0			0
$285$	1753.66	−	10433.1i	0.364483	−	2.16843i
$286$		0			0
$287$	−229.501			−0.0472021
$288$		0			0
$289$	2597.43			0.528686
$290$		0			0
$291$	−669.954	+	3985.77i	−0.134960	+	0.802922i
$292$		0			0
$293$		−	5783.24i		−	1.15311i	−0.817059	−	0.576553i	$-0.804397\pi$
$293$		−	5783.24i		−	1.15311i	0.817059	−	0.576553i	$-0.195603\pi$
$294$		0			0
$295$		−	14564.6i		−	2.87452i
$296$		0			0
$297$	−648.234	+	1187.75i	−0.126648	+	0.232054i
$298$		0			0
$299$	2222.19			0.429808
$300$		0			0
$301$	−277.000			−0.0530432
$302$		0			0
$303$	4957.74	+	833.328i	0.939982	+	0.157998i
$304$		0			0
$305$		−	4674.96i		−	0.877664i
$306$		0			0
$307$		−	536.131i		−	0.0996697i	−0.998757	−	0.0498348i	$-0.984131\pi$
$307$		−	536.131i		−	0.0996697i	0.998757	−	0.0498348i	$-0.0158695\pi$
$308$		0			0
$309$	880.692	+	148.032i	0.162139	+	0.0272533i
$310$		0			0
$311$	−9170.06			−1.67198			−0.835991	−	0.548744i	$-0.815106\pi$
$311$	−9170.06			−1.67198			−0.835991	+	0.548744i	$0.815106\pi$
$312$		0			0
$313$	1384.77			0.250070			0.125035	−	0.992152i	$-0.460096\pi$
$313$	1384.77			0.250070			0.125035	+	0.992152i	$0.460096\pi$
$314$		0			0
$315$	325.066	+	112.455i	0.0581441	+	0.0201147i
$316$		0			0
$317$			10975.1i			1.94456i	0.233820	+	0.972280i	$0.424877\pi$
$317$			10975.1i			1.94456i	−0.233820	+	0.972280i	$0.575123\pi$
$318$		0			0
$319$			2109.34i			0.370221i
$320$		0			0
$321$	1104.14	−	6568.90i	0.191985	−	1.14218i
$322$		0			0
$323$	−4997.85			−0.860953
$324$		0			0
$325$	−3370.69			−0.575299
$326$		0			0
$327$	271.601	−	1615.84i	0.0459314	−	0.273261i
$328$		0			0
$329$		−	68.7747i		−	0.0115248i
$330$		0			0
$331$			2164.05i			0.359357i	0.983725	+	0.179678i	$0.0575057\pi$
$331$			2164.05i			0.359357i	−0.983725	+	0.179678i	$0.942494\pi$
$332$		0			0
$333$	5539.52	+	1916.38i	0.911603	+	0.315366i
$334$		0			0
$335$	−2738.27			−0.446591
$336$		0			0
$337$	−30.9719			−0.00500637			−0.00250319	−	0.999997i	$-0.500797\pi$
$337$	−30.9719			−0.00500637			−0.00250319	+	0.999997i	$0.500797\pi$
$338$		0			0
$339$	−2263.23	−	380.417i	−0.362601	−	0.0609482i
$340$		0			0
$341$			2811.83i			0.446537i
$342$		0			0
$343$			445.538i			0.0701365i
$344$		0			0
$345$	17171.0	+	2886.21i	2.67958	+	0.450401i
$346$		0			0
$347$	−3684.85			−0.570066			−0.285033	−	0.958518i	$-0.592005\pi$
$347$	−3684.85			−0.570066			−0.285033	+	0.958518i	$0.592005\pi$
$348$		0			0
$349$	−5583.18			−0.856334			−0.428167	−	0.903700i	$-0.640841\pi$
$349$	−5583.18			−0.856334			−0.428167	+	0.903700i	$0.640841\pi$
$350$		0			0
$351$	−873.742	+	1600.94i	−0.132869	+	0.243452i
$352$		0			0
$353$			2916.33i			0.439718i	0.975532	+	0.219859i	$0.0705597\pi$
$353$			2916.33i			0.439718i	−0.975532	+	0.219859i	$0.929440\pi$
$354$		0			0
$355$			1733.89i			0.259227i
$356$		0			0
$357$	26.9353	−	160.247i	0.00399318	−	0.0237567i
$358$		0			0
$359$	2375.54			0.349237			0.174619	−	0.984636i	$-0.444131\pi$
$359$	2375.54			0.349237			0.174619	+	0.984636i	$0.444131\pi$
$360$		0			0
$361$	−3928.22			−0.572710
$362$		0			0
$363$	−1066.29	+	6343.73i	−0.154176	+	0.917244i
$364$		0			0
$365$			2576.85i			0.369530i
$366$		0			0
$367$		−	4724.00i		−	0.671910i	−0.941878	−	0.335955i	$-0.890941\pi$
$367$		−	4724.00i		−	0.671910i	0.941878	−	0.335955i	$-0.109059\pi$
$368$		0			0
$369$	3117.32	−	9010.99i	0.439787	−	1.27126i
$370$		0			0
$371$	−185.952			−0.0260220
$372$		0			0
$373$	−9717.33			−1.34891			−0.674456	−	0.738315i	$-0.735622\pi$
$373$	−9717.33			−1.34891			−0.674456	+	0.738315i	$0.735622\pi$
$374$		0			0
$375$	−13489.0	−	2267.32i	−1.85752	−	0.312224i
$376$		0			0
$377$			2843.14i			0.388406i
$378$		0			0
$379$			7270.42i			0.985373i	0.870207	+	0.492687i	$0.163985\pi$
$379$			7270.42i			0.985373i	−0.870207	+	0.492687i	$0.836015\pi$
$380$		0			0
$381$	−8844.45	−	1486.63i	−1.18928	−	0.199901i
$382$		0			0
$383$	6885.10			0.918570			0.459285	−	0.888289i	$-0.348106\pi$
$383$	6885.10			0.918570			0.459285	+	0.888289i	$0.348106\pi$
$384$		0			0
$385$	−122.870			−0.0162651
$386$		0			0
$387$	3762.51	−	10876.0i	0.494209	−	1.42857i
$388$		0			0
$389$			9460.73i			1.23311i	0.787314	+	0.616553i	$0.211471\pi$
$389$			9460.73i			1.23311i	−0.787314	+	0.616553i	$0.788529\pi$
$390$		0			0
$391$		−	8225.58i		−	1.06390i
$392$		0			0
$393$	1400.76	−	8333.57i	0.179794	−	1.06965i
$394$		0			0
$395$	−17796.3			−2.26690
$396$		0			0
$397$	7419.48			0.937968			0.468984	−	0.883207i	$-0.344620\pi$
$397$	7419.48			0.937968			0.468984	+	0.883207i	$0.344620\pi$
$398$		0			0
$399$	58.1363	−	345.872i	0.00729437	−	0.0433966i
$400$		0			0
$401$		−	6828.25i		−	0.850341i	−0.905113	−	0.425170i	$-0.860214\pi$
$401$		−	6828.25i		−	0.850341i	0.905113	−	0.425170i	$-0.139786\pi$
$402$		0			0
$403$			3790.01i			0.468471i
$404$		0			0
$405$	−8830.78	+	11235.7i	−1.08347	+	1.37854i
$406$		0			0
$407$	−2093.86			−0.255009
$408$		0			0
$409$	−12053.0			−1.45717			−0.728585	−	0.684956i	$-0.759821\pi$
$409$	−12053.0			−1.45717			−0.728585	+	0.684956i	$0.759821\pi$
$410$		0			0
$411$	−2564.89	−	431.123i	−0.307826	−	0.0517414i
$412$		0			0
$413$		−	482.837i		−	0.0575275i
$414$		0			0
$415$		−	4810.57i		−	0.569016i
$416$		0			0
$417$	10574.8	+	1777.49i	1.24185	+	0.208738i
$418$		0			0
$419$	10117.3			1.17963			0.589813	−	0.807540i	$-0.299201\pi$
$419$	10117.3			1.17963			0.589813	+	0.807540i	$0.299201\pi$
$420$		0			0
$421$	−16018.4			−1.85437			−0.927184	−	0.374606i	$-0.877778\pi$
$421$	−16018.4			−1.85437			−0.927184	+	0.374606i	$0.877778\pi$
$422$		0			0
$423$	2700.33	+	934.171i	0.310389	+	0.107378i
$424$		0			0
$425$			12476.8i			1.42403i
$426$		0			0
$427$		−	154.982i		−	0.0175646i
$428$		0			0
$429$	107.994	−	642.492i	0.0121539	−	0.0723072i
$430$		0			0
$431$	9489.34			1.06052			0.530262	−	0.847834i	$-0.322094\pi$
$431$	9489.34			1.06052			0.530262	+	0.847834i	$0.322094\pi$
$432$		0			0
$433$	−665.783			−0.0738926			−0.0369463	−	0.999317i	$-0.511763\pi$
$433$	−665.783			−0.0738926			−0.0369463	+	0.999317i	$0.511763\pi$
$434$		0			0
$435$	−3692.70	+	21969.1i	−0.407015	+	2.42147i
$436$		0			0
$437$		−	17753.9i		−	1.94344i
$438$		0			0
$439$		−	1194.73i		−	0.129890i	−0.997889	−	0.0649448i	$-0.979313\pi$
$439$		−	1194.73i		−	0.129890i	0.997889	−	0.0649448i	$-0.0206871\pi$
$440$		0			0
$441$	−8741.30	−	3024.02i	−0.943883	−	0.326533i
$442$		0			0
$443$	−3427.25			−0.367570			−0.183785	−	0.982966i	$-0.558835\pi$
$443$	−3427.25			−0.367570			−0.183785	+	0.982966i	$0.558835\pi$
$444$		0			0
$445$	15117.6			1.61043
$446$		0			0
$447$	10567.3	+	1776.23i	1.11816	+	0.187948i
$448$		0			0
$449$		−	14640.0i		−	1.53876i	−0.638793	−	0.769379i	$-0.720566\pi$
$449$		−	14640.0i		−	1.53876i	0.638793	−	0.769379i	$-0.279434\pi$
$450$		0			0
$451$			3406.03i			0.355618i
$452$		0			0
$453$	−7761.47	−	1304.60i	−0.805002	−	0.135310i
$454$		0			0
$455$	−165.614			−0.0170640
$456$		0			0
$457$	−184.839			−0.0189200			−0.00945998	−	0.999955i	$-0.503011\pi$
$457$	−184.839			−0.0189200			−0.00945998	+	0.999955i	$0.503011\pi$
$458$		0			0
$459$	5925.98	+	3234.21i	0.602616	+	0.328889i
$460$		0			0
$461$			4527.19i			0.457380i	0.973499	+	0.228690i	$0.0734442\pi$
$461$			4527.19i			0.457380i	−0.973499	+	0.228690i	$0.926556\pi$
$462$		0			0
$463$		−	3908.01i		−	0.392269i	−0.980577	−	0.196135i	$-0.937161\pi$
$463$		−	3908.01i		−	0.392269i	0.980577	−	0.196135i	$-0.0628390\pi$
$464$		0			0
$465$	−4922.51	+	29285.6i	−0.490916	+	2.92062i
$466$		0			0
$467$	19539.0			1.93609			0.968047	−	0.250768i	$-0.0806830\pi$
$467$	19539.0			1.93609			0.968047	+	0.250768i	$0.0806830\pi$
$468$		0			0
$469$	−90.7778			−0.00893759
$470$		0			0
$471$	2004.51	−	11925.5i	0.196100	−	1.16666i
$472$		0			0
$473$			4110.96i			0.399624i
$474$		0			0
$475$			26929.6i			2.60129i
$476$		0			0
$477$	2525.80	−	7301.12i	0.242449	−	0.700829i
$478$		0			0
$479$	2837.41			0.270657			0.135328	−	0.990801i	$-0.456791\pi$
$479$	2837.41			0.270657			0.135328	+	0.990801i	$0.456791\pi$
$480$		0			0
$481$	−2822.27			−0.267535
$482$		0			0
$483$	569.244	+	95.6820i	0.0536263	+	0.00901384i
$484$		0			0
$485$		−	15247.8i		−	1.42756i
$486$		0			0
$487$			11219.2i			1.04393i	0.852968	+	0.521963i	$0.174800\pi$
$487$			11219.2i			1.04393i	−0.852968	+	0.521963i	$0.825200\pi$
$488$		0			0
$489$	3270.70	+	549.759i	0.302466	+	0.0508404i
$490$		0			0
$491$	−8015.30			−0.736711			−0.368356	−	0.929685i	$-0.620079\pi$
$491$	−8015.30			−0.736711			−0.368356	+	0.929685i	$0.620079\pi$
$492$		0			0
$493$	10524.1			0.961419
$494$		0			0
$495$	1668.95	−	4824.31i	0.151543	−	0.438054i
$496$		0			0
$497$			57.4811i			0.00518789i
$498$		0			0
$499$		−	5352.87i		−	0.480215i	−0.970746	−	0.240108i	$-0.922817\pi$
$499$		−	5352.87i		−	0.480215i	0.970746	−	0.240108i	$-0.0771827\pi$
$500$		0			0
$501$	−3385.95	+	20144.1i	−0.301942	+	1.79635i
$502$		0			0
$503$	10387.3			0.920765			0.460383	−	0.887721i	$-0.347712\pi$
$503$	10387.3			0.920765			0.460383	+	0.887721i	$0.347712\pi$
$504$		0			0
$505$	−18966.1			−1.67125
$506$		0			0
$507$	145.563	−	866.001i	0.0127508	−	0.0758589i
$508$		0			0
$509$		−	6102.46i		−	0.531408i	−0.964055	−	0.265704i	$-0.914396\pi$
$509$		−	6102.46i		−	0.531408i	0.964055	−	0.265704i	$-0.0856044\pi$
$510$		0			0
$511$			85.4262i			0.00739536i
$512$		0			0
$513$	12790.5	+	6980.63i	1.10080	+	0.600784i
$514$		0			0
$515$	−3369.13			−0.288275
$516$		0			0
$517$	−1020.69			−0.0868275
$518$		0			0
$519$	4888.90	+	821.757i	0.413485	+	0.0695012i
$520$		0			0
$521$			8742.06i			0.735118i	0.930000	+	0.367559i	$0.119806\pi$
$521$			8742.06i			0.735118i	−0.930000	+	0.367559i	$0.880194\pi$
$522$		0			0
$523$			4745.78i			0.396785i	0.980123	+	0.198393i	$0.0635721\pi$
$523$			4745.78i			0.396785i	−0.980123	+	0.198393i	$0.936428\pi$
$524$		0			0
$525$	−863.447	−	145.134i	−0.0717789	−	0.0120650i
$526$		0			0
$527$	14029.0			1.15960
$528$		0			0
$529$	17052.7			1.40155
$530$		0			0
$531$	18957.9	+	6558.41i	1.54934	+	0.535990i
$532$		0			0
$533$			4590.91i			0.373085i
$534$		0			0
$535$			25129.7i			2.03075i
$536$		0			0
$537$	1914.89	−	11392.3i	0.153880	−	0.915485i
$538$		0			0
$539$	3304.09			0.264039
$540$		0			0
$541$	−23407.2			−1.86017			−0.930087	−	0.367339i	$-0.880269\pi$
$541$	−23407.2			−1.86017			−0.930087	+	0.367339i	$0.880269\pi$
$542$		0			0
$543$	−1851.67	+	11016.2i	−0.146340	+	0.870627i
$544$		0			0
$545$			6181.49i			0.485845i
$546$		0			0
$547$			16442.8i			1.28527i	0.766172	+	0.642636i	$0.222159\pi$
$547$			16442.8i			1.28527i	−0.766172	+	0.642636i	$0.777841\pi$
$548$		0			0
$549$	6085.12	+	2105.13i	0.473054	+	0.163651i
$550$		0			0
$551$	22714.8			1.75623
$552$		0			0
$553$	−589.972			−0.0453674
$554$		0			0
$555$	−21807.8	−	3665.60i	−1.66791	−	0.280353i
$556$		0			0
$557$		−	7550.27i		−	0.574354i	−0.957878	−	0.287177i	$-0.907283\pi$
$557$		−	7550.27i		−	0.574354i	0.957878	−	0.287177i	$-0.0927168\pi$
$558$		0			0
$559$			5541.08i			0.419254i
$560$		0			0
$561$	−2378.23	−	399.747i	−0.178982	−	0.0300844i
$562$		0			0
$563$	−19794.9			−1.48180			−0.740902	−	0.671614i	$-0.765601\pi$
$563$	−19794.9			−1.48180			−0.740902	+	0.671614i	$0.765601\pi$
$564$		0			0
$565$	8658.09			0.644688
$566$		0			0
$567$	−292.753	+	372.481i	−0.0216834	+	0.0275886i
$568$		0			0
$569$		−	11730.7i		−	0.864286i	−0.901805	−	0.432143i	$-0.857758\pi$
$569$		−	11730.7i		−	0.864286i	0.901805	−	0.432143i	$-0.142242\pi$
$570$		0			0
$571$			6554.92i			0.480411i	0.970722	+	0.240206i	$0.0772149\pi$
$571$			6554.92i			0.480411i	−0.970722	+	0.240206i	$0.922785\pi$
$572$		0			0
$573$	2508.17	−	14921.9i	0.182863	−	1.08791i
$574$		0			0
$575$	−44321.4			−3.21448
$576$		0			0
$577$	1566.28			0.113007			0.0565035	−	0.998402i	$-0.482005\pi$
$577$	1566.28			0.113007			0.0565035	+	0.998402i	$0.482005\pi$
$578$		0			0
$579$	−2812.91	+	16734.9i	−0.201901	+	1.20117i
$580$		0			0
$581$		−	159.477i		−	0.0113877i
$582$		0			0
$583$			2759.72i			0.196048i
$584$		0			0
$585$	2249.55	−	6502.59i	0.158987	−	0.459571i
$586$		0			0
$587$	−5943.08			−0.417883			−0.208941	−	0.977928i	$-0.567002\pi$
$587$	−5943.08			−0.417883			−0.208941	+	0.977928i	$0.567002\pi$
$588$		0			0
$589$	30279.7			2.11826
$590$		0			0
$591$	−16263.2	−	2733.62i	−1.13194	−	0.190264i
$592$		0			0
$593$		−	17144.9i		−	1.18728i	−0.804732	−	0.593638i	$-0.797691\pi$
$593$		−	17144.9i		−	1.18728i	0.804732	−	0.593638i	$-0.202309\pi$
$594$		0			0
$595$			613.032i			0.0422384i
$596$		0			0
$597$	8467.68	+	1423.30i	0.580501	+	0.0975742i
$598$		0			0
$599$	−6409.46			−0.437201			−0.218600	−	0.975814i	$-0.570149\pi$
$599$	−6409.46			−0.437201			−0.218600	+	0.975814i	$0.570149\pi$
$600$		0			0
$601$	11290.9			0.766332			0.383166	−	0.923679i	$-0.374834\pi$
$601$	11290.9			0.766332			0.383166	+	0.923679i	$0.374834\pi$
$602$		0			0
$603$	1233.04	−	3564.25i	0.0832724	−	0.240709i
$604$		0			0
$605$		−	24268.3i		−	1.63082i
$606$		0			0
$607$		−	24831.0i		−	1.66039i	−0.557472	−	0.830196i	$-0.688229\pi$
$607$		−	24831.0i		−	1.66039i	0.557472	−	0.830196i	$-0.311771\pi$
$608$		0			0
$609$	−122.418	+	728.307i	−0.00814556	+	0.0484606i
$610$		0			0
$611$	−1375.76			−0.0910924
$612$		0			0
$613$	3472.78			0.228816			0.114408	−	0.993434i	$-0.463503\pi$
$613$	3472.78			0.228816			0.114408	+	0.993434i	$0.463503\pi$
$614$		0			0
$615$	−5962.74	+	35474.3i	−0.390961	+	2.32595i
$616$		0			0
$617$		−	13199.5i		−	0.861251i	−0.902531	−	0.430625i	$-0.858293\pi$
$617$		−	13199.5i		−	0.861251i	0.902531	−	0.430625i	$-0.141707\pi$
$618$		0			0
$619$		−	27851.7i		−	1.80849i	−0.427016	−	0.904244i	$-0.640435\pi$
$619$		−	27851.7i		−	1.80849i	0.427016	−	0.904244i	$-0.359565\pi$
$620$		0			0
$621$	−11488.9	+	21050.8i	−0.742404	+	1.36029i
$622$		0			0
$623$	501.169			0.0322294
$624$		0			0
$625$	19192.6			1.22832
$626$		0			0
$627$	−5133.09	−	862.802i	−0.326947	−	0.0549553i
$628$		0			0
$629$			10446.8i			0.662228i
$630$		0			0
$631$		−	17001.7i		−	1.07262i	−0.844020	−	0.536312i	$-0.819817\pi$
$631$		−	17001.7i		−	1.07262i	0.844020	−	0.536312i	$-0.180183\pi$
$632$		0			0
$633$	−21937.5	−	3687.40i	−1.37747	−	0.231534i
$634$		0			0
$635$	33834.9			2.11448
$636$		0			0
$637$	4453.51			0.277009
$638$		0			0
$639$	−2256.91	−	780.769i	−0.139721	−	0.0483361i
$640$		0			0
$641$			22175.2i			1.36641i	0.730228	+	0.683204i	$0.239414\pi$
$641$			22175.2i			1.36641i	−0.730228	+	0.683204i	$0.760586\pi$
$642$		0			0
$643$			3121.88i			0.191469i	0.995407	+	0.0957347i	$0.0305200\pi$
$643$			3121.88i			0.191469i	−0.995407	+	0.0957347i	$0.969480\pi$
$644$		0			0
$645$	−7196.83	+	42816.3i	−0.439341	+	2.61378i
$646$		0			0
$647$	3001.73			0.182396			0.0911981	−	0.995833i	$-0.470930\pi$
$647$	3001.73			0.182396			0.0911981	+	0.995833i	$0.470930\pi$
$648$		0			0
$649$	−7165.80			−0.433409
$650$		0			0
$651$	−163.188	+	970.861i	−0.00982467	+	0.0584502i
$652$		0			0
$653$		−	3437.04i		−	0.205975i	−0.994683	−	0.102987i	$-0.967160\pi$
$653$		−	3437.04i		−	0.205975i	0.994683	−	0.102987i	$-0.0328402\pi$
$654$		0			0
$655$			31880.5i			1.90179i
$656$		0			0
$657$	−3354.13	−	1160.35i	−0.199173	−	0.0689033i
$658$		0			0
$659$	−28774.2			−1.70088			−0.850441	−	0.526070i	$-0.823665\pi$
$659$	−28774.2			−1.70088			−0.850441	+	0.526070i	$0.823665\pi$
$660$		0			0
$661$	21113.3			1.24238			0.621189	−	0.783661i	$-0.286650\pi$
$661$	21113.3			1.24238			0.621189	+	0.783661i	$0.286650\pi$
$662$		0			0
$663$	−3205.56	−	538.811i	−0.187773	−	0.0315621i
$664$		0			0
$665$			1323.15i			0.0771572i
$666$		0			0
$667$			37384.6i			2.17022i
$668$		0			0
$669$	−17001.6	−	2857.73i	−0.982540	−	0.165151i
$670$		0			0
$671$	−2300.09			−0.132331
$672$		0			0
$673$	−20196.2			−1.15677			−0.578385	−	0.815764i	$-0.696317\pi$
$673$	−20196.2			−1.15677			−0.578385	+	0.815764i	$0.696317\pi$
$674$		0			0
$675$	17426.7	−	31930.5i	0.993709	−	1.82075i
$676$		0			0
$677$			11518.4i			0.653898i	0.945042	+	0.326949i	$0.106020\pi$
$677$			11518.4i			0.653898i	−0.945042	+	0.326949i	$0.893980\pi$
$678$		0			0
$679$		−	505.486i		−	0.0285696i
$680$		0			0
$681$	5300.28	−	31533.1i	0.298249	−	1.77438i
$682$		0			0
$683$	32987.0			1.84804			0.924022	−	0.382339i	$-0.124881\pi$
$683$	32987.0			1.84804			0.924022	+	0.382339i	$0.124881\pi$
$684$		0			0
$685$	9812.12			0.547302
$686$		0			0
$687$	−2152.64	+	12806.7i	−0.119546	+	0.711219i
$688$		0			0
$689$			3719.77i			0.205678i
$690$		0			0
$691$		−	1876.80i		−	0.103324i	−0.998665	−	0.0516619i	$-0.983548\pi$
$691$		−	1876.80i		−	0.103324i	0.998665	−	0.0516619i	$-0.0164518\pi$
$692$		0			0
$693$	55.3282	−	159.933i	0.00303282	−	0.00876674i
$694$		0			0
$695$	−40454.6			−2.20796
$696$		0			0
$697$	16993.6			0.923497
$698$		0			0
$699$	16814.7	+	2826.33i	0.909859	+	0.152935i
$700$		0			0
$701$			3174.84i			0.171059i	0.996336	+	0.0855293i	$0.0272581\pi$
$701$			3174.84i			0.171059i	−0.996336	+	0.0855293i	$0.972742\pi$
$702$		0			0
$703$			22548.1i			1.20970i
$704$		0			0
$705$	−10630.6	−	1786.86i	−0.567904	−	0.0954568i
$706$		0			0
$707$	−628.753			−0.0334465
$708$		0			0
$709$	32947.2			1.74521			0.872607	−	0.488422i	$-0.162427\pi$
$709$	32947.2			1.74521			0.872607	+	0.488422i	$0.162427\pi$
$710$		0			0
$711$	8013.62	−	23164.3i	0.422692	−	1.22184i
$712$		0			0
$713$			49835.0i			2.61758i
$714$		0			0
$715$			2457.89i			0.128559i
$716$		0			0
$717$	1330.47	−	7915.43i	0.0692991	−	0.412283i
$718$		0			0
$719$	−14586.6			−0.756589			−0.378295	−	0.925685i	$-0.623489\pi$
$719$	−14586.6			−0.756589			−0.378295	+	0.925685i	$0.623489\pi$
$720$		0			0
$721$	−111.692			−0.00576923
$722$		0			0
$723$	2340.68	−	13925.5i	0.120402	−	0.716312i
$724$		0			0
$725$		−	56706.1i		−	2.90484i
$726$		0			0
$727$		−	18354.7i		−	0.936367i	−0.883631	−	0.468184i	$-0.844909\pi$
$727$		−	18354.7i		−	0.936367i	0.883631	−	0.468184i	$-0.155091\pi$
$728$		0			0
$729$	−10648.4	−	16553.9i	−0.540994	−	0.841026i
$730$		0			0
$731$	20510.7			1.03778
$732$		0			0
$733$	19435.7			0.979365			0.489682	−	0.871901i	$-0.337113\pi$
$733$	19435.7			0.979365			0.489682	+	0.871901i	$0.337113\pi$
$734$		0			0
$735$	34412.6	+	5784.28i	1.72697	+	0.290281i
$736$		0			0
$737$			1347.24i			0.0673352i
$738$		0			0
$739$		−	15011.3i		−	0.747227i	−0.927584	−	0.373614i	$-0.878119\pi$
$739$		−	15011.3i		−	0.747227i	0.927584	−	0.373614i	$-0.121881\pi$
$740$		0			0
$741$	−6918.79	−	1162.95i	−0.343007	−	0.0576547i
$742$		0			0
$743$	24884.8			1.22872			0.614358	−	0.789027i	$-0.289415\pi$
$743$	24884.8			1.22872			0.614358	+	0.789027i	$0.289415\pi$
$744$		0			0
$745$	−40425.9			−1.98804
$746$		0			0
$747$	6261.63	+	2166.19i	0.306695	+	0.106100i
$748$		0			0
$749$			833.085i			0.0406412i
$750$		0			0
$751$			14133.3i			0.686724i	0.939203	+	0.343362i	$0.111566\pi$
$751$			14133.3i			0.686724i	−0.939203	+	0.343362i	$0.888434\pi$
$752$		0			0
$753$	6341.76	−	37729.2i	0.306914	−	1.82593i
$754$		0			0
$755$	29691.9			1.43126
$756$		0			0
$757$	−21364.1			−1.02575			−0.512873	−	0.858464i	$-0.671419\pi$
$757$	−21364.1			−1.02575			−0.512873	+	0.858464i	$0.671419\pi$
$758$		0			0
$759$	1420.02	−	8448.16i	0.0679097	−	0.404017i
$760$		0			0
$761$			17506.3i			0.833907i	0.908928	+	0.416954i	$0.136902\pi$
$761$			17506.3i			0.833907i	−0.908928	+	0.416954i	$0.863098\pi$
$762$		0			0
$763$			204.925i			0.00972319i
$764$		0			0
$765$	−24069.8	−	8326.85i	−1.13757	−	0.393540i
$766$		0			0
$767$	−9658.63			−0.454698
$768$		0			0
$769$	−37151.9			−1.74217			−0.871087	−	0.491128i	$-0.836585\pi$
$769$	−37151.9			−1.74217			−0.871087	+	0.491128i	$0.836585\pi$
$770$		0			0
$771$	−30057.1	−	5052.19i	−1.40400	−	0.235993i
$772$		0			0
$773$		−	476.121i		−	0.0221538i	−0.999939	−	0.0110769i	$-0.996474\pi$
$773$		−	476.121i		−	0.0221538i	0.999939	−	0.0110769i	$-0.00352596\pi$
$774$		0			0
$775$		−	75591.4i		−	3.50364i
$776$		0			0
$777$	−722.962	−	121.520i	−0.0333798	−	0.00561069i
$778$		0			0
$779$	36678.4			1.68696
$780$		0			0
$781$	853.079			0.0390852
$782$		0			0
$783$	−26933.1	−	14699.2i	−1.22926	−	0.670890i
$784$		0			0
$785$			45621.6i			2.07427i
$786$		0			0
$787$			18331.9i			0.830318i	0.909749	+	0.415159i	$0.136274\pi$
$787$			18331.9i			0.830318i	−0.909749	+	0.415159i	$0.863726\pi$
$788$		0			0
$789$	2194.86	−	13057.9i	0.0990354	−	0.589194i
$790$		0			0
$791$	287.028			0.0129021
$792$		0			0
$793$	−3100.24			−0.138831
$794$		0			0
$795$	−4831.28	+	28742.9i	−0.215532	+	1.28227i
$796$		0			0
$797$			38613.1i			1.71612i	0.513552	+	0.858059i	$0.328329\pi$
$797$			38613.1i			1.71612i	−0.513552	+	0.858059i	$0.671671\pi$
$798$		0			0
$799$			5092.48i			0.225481i
$800$		0			0
$801$	−6807.41	+	19677.6i	−0.300284	+	0.868008i
$802$		0			0
$803$	1267.81			0.0557162
$804$		0			0
$805$	−2177.67			−0.0953451
$806$		0			0
$807$	19610.5	+	3296.26i	0.855419	+	0.143784i
$808$		0			0
$809$			20366.2i			0.885089i	0.896747	+	0.442544i	$0.145924\pi$
$809$			20366.2i			0.885089i	−0.896747	+	0.442544i	$0.854076\pi$
$810$		0			0
$811$		−	34043.1i		−	1.47400i	−0.675892	−	0.737001i	$-0.736241\pi$
$811$		−	34043.1i		−	1.47400i	0.675892	−	0.737001i	$-0.263759\pi$
$812$		0			0
$813$	−20391.1	−	3427.46i	−0.879639	−	0.147855i
$814$		0			0
$815$	−12512.2			−0.537772
$816$		0			0
$817$	44269.6			1.89571
$818$		0			0
$819$	74.5758	−	215.570i	0.00318179	−	0.00919736i
$820$		0			0
$821$		−	9260.05i		−	0.393639i	−0.980440	−	0.196820i	$-0.936939\pi$
$821$		−	9260.05i		−	0.393639i	0.980440	−	0.196820i	$-0.0630614\pi$
$822$		0			0
$823$			21703.2i			0.919229i	0.888119	+	0.459614i	$0.152012\pi$
$823$			21703.2i			0.919229i	−0.888119	+	0.459614i	$0.847988\pi$
$824$		0			0
$825$	−2153.93	+	12814.4i	−0.0908973	+	0.540777i
$826$		0			0
$827$	−27981.5			−1.17656			−0.588278	−	0.808659i	$-0.700194\pi$
$827$	−27981.5			−1.17656			−0.588278	+	0.808659i	$0.700194\pi$
$828$		0			0
$829$	36854.6			1.54404			0.772022	−	0.635596i	$-0.219246\pi$
$829$	36854.6			1.54404			0.772022	+	0.635596i	$0.219246\pi$
$830$		0			0
$831$	2270.92	−	13510.4i	0.0947981	−	0.563985i
$832$		0			0
$833$		−	16485.0i		−	0.685678i
$834$		0			0
$835$		−	77062.3i		−	3.19383i
$836$		0			0
$837$	−35902.8	−	19594.6i	−1.48265	−	0.809186i
$838$		0			0
$839$	27447.1			1.12942			0.564708	−	0.825291i	$-0.308989\pi$
$839$	27447.1			1.12942			0.564708	+	0.825291i	$0.308989\pi$
$840$		0			0
$841$	−23441.9			−0.961168
$842$		0			0
$843$	20420.2	+	3432.36i	0.834294	+	0.140233i
$844$		0			0
$845$			3312.93i			0.134874i
$846$		0			0
$847$		−	804.529i		−	0.0326375i
$848$		0			0
$849$	9868.64	+	1658.78i	0.398929	+	0.0670545i
$850$		0			0
$851$	−37110.2			−1.49485
$852$		0			0
$853$	34680.4			1.39207			0.696035	−	0.718008i	$-0.254946\pi$
$853$	34680.4			1.39207			0.696035	+	0.718008i	$0.254946\pi$
$854$		0			0
$855$	−51951.5	−	17972.4i	−2.07801	−	0.718882i
$856$		0			0
$857$			35211.8i			1.40352i	0.712416	+	0.701758i	$0.247601\pi$
$857$			35211.8i			1.40352i	−0.712416	+	0.701758i	$0.752399\pi$
$858$		0			0
$859$		−	17091.0i		−	0.678858i	−0.940632	−	0.339429i	$-0.889766\pi$
$859$		−	17091.0i		−	0.678858i	0.940632	−	0.339429i	$-0.110234\pi$
$860$		0			0
$861$	−197.673	+	1176.02i	−0.00782426	+	0.0465491i
$862$		0			0
$863$	27598.6			1.08861			0.544303	−	0.838889i	$-0.316794\pi$
$863$	27598.6			1.08861			0.544303	+	0.838889i	$0.316794\pi$
$864$		0			0
$865$	−18702.7			−0.735159
$866$		0			0
$867$	2237.22	−	13309.9i	0.0876354	−	0.521372i
$868$		0			0
$869$			8755.79i			0.341795i
$870$		0			0
$871$			1815.91i			0.0706427i
$872$		0			0
$873$	19847.1	+	6866.04i	0.769443	+	0.266186i
$874$		0			0
$875$	1710.72			0.0660945
$876$		0			0
$877$	−613.560			−0.0236242			−0.0118121	−	0.999930i	$-0.503760\pi$
$877$	−613.560			−0.0236242			−0.0118121	+	0.999930i	$0.503760\pi$
$878$		0			0
$879$	−29634.9	−	4981.21i	−1.13715	−	0.191140i
$880$		0			0
$881$		−	29467.3i		−	1.12688i	−0.826158	−	0.563439i	$-0.809478\pi$
$881$		−	29467.3i		−	1.12688i	0.826158	−	0.563439i	$-0.190522\pi$
$882$		0			0
$883$		−	40698.7i		−	1.55110i	−0.631287	−	0.775549i	$-0.717473\pi$
$883$		−	40698.7i		−	1.55110i	0.631287	−	0.775549i	$-0.282527\pi$
$884$		0			0
$885$	−74632.9	−	12544.8i	−2.83475	−	0.476483i
$886$		0			0
$887$	−30611.1			−1.15876			−0.579379	−	0.815058i	$-0.696705\pi$
$887$	−30611.1			−1.15876			−0.579379	+	0.815058i	$0.696705\pi$
$888$		0			0
$889$	1121.68			0.0423170
$890$		0			0
$891$	5527.99	+	4344.76i	0.207850	+	0.163361i
$892$		0			0
$893$			10991.5i			0.411887i
$894$		0			0
$895$			43582.0i			1.62769i
$896$		0			0
$897$	1914.01	−	11387.1i	0.0712454	−	0.423862i
$898$		0			0
$899$	−63760.4			−2.36544
$900$		0			0
$901$	13769.0			0.509113
$902$		0			0
$903$	−238.585	+	1419.42i	−0.00879249	+	0.0523094i
$904$		0			0
$905$		−	42143.1i		−	1.54794i
$906$		0			0
$907$		−	29969.8i		−	1.09717i	−0.836096	−	0.548583i	$-0.815167\pi$
$907$		−	29969.8i		−	1.09717i	0.836096	−	0.548583i	$-0.184833\pi$
$908$		0			0
$909$	8540.39	−	24687.0i	0.311625	−	0.900789i
$910$		0			0
$911$	−23358.6			−0.849510			−0.424755	−	0.905308i	$-0.639640\pi$
$911$	−23358.6			−0.849510			−0.424755	+	0.905308i	$0.639640\pi$
$912$		0			0
$913$	−2366.81			−0.0857940
$914$		0			0
$915$	−23955.8	−	4026.63i	−0.865523	−	0.145482i
$916$		0			0
$917$			1056.88i			0.0380604i
$918$		0			0
$919$		−	18878.3i		−	0.677625i	−0.940854	−	0.338812i	$-0.889975\pi$
$919$		−	18878.3i		−	0.677625i	0.940854	−	0.338812i	$-0.110025\pi$
$920$		0			0
$921$	−2747.28	−	461.780i	−0.0982909	−	0.0165213i
$922$		0			0
$923$	1149.85			0.0410051
$924$		0			0
$925$	56289.9			2.00086
$926$		0			0
$927$	1517.11	−	4385.40i	0.0537525	−	0.155378i
$928$		0			0
$929$			9059.53i			0.319950i	0.987121	+	0.159975i	$0.0511413\pi$
$929$			9059.53i			0.319950i	−0.987121	+	0.159975i	$0.948859\pi$
$930$		0			0
$931$		−	35580.6i		−	1.25253i
$932$		0			0
$933$	−7898.34	+	46989.8i	−0.277149	+	1.64885i
$934$		0			0
$935$	9098.03			0.318222
$936$		0			0
$937$	26517.1			0.924520			0.462260	−	0.886744i	$-0.347039\pi$
$937$	26517.1			0.924520			0.462260	+	0.886744i	$0.347039\pi$
$938$		0			0
$939$	1192.73	−	7095.94i	0.0414518	−	0.246610i
$940$		0			0
$941$		−	9511.14i		−	0.329495i	−0.986336	−	0.164747i	$-0.947319\pi$
$941$		−	9511.14i		−	0.329495i	0.986336	−	0.164747i	$-0.0526809\pi$
$942$		0			0
$943$			60366.2i			2.08462i
$944$		0			0
$945$	856.237	−	1568.87i	0.0294745	−	0.0540055i
$946$		0			0
$947$	28978.4			0.994373			0.497186	−	0.867644i	$-0.334366\pi$
$947$	28978.4			0.994373			0.497186	+	0.867644i	$0.334366\pi$
$948$		0			0
$949$	1708.86			0.0584530
$950$		0			0
$951$	56239.6	+	9453.10i	1.91766	+	0.322332i
$952$		0			0
$953$		−	23462.0i		−	0.797490i	−0.917062	−	0.398745i	$-0.869446\pi$
$953$		−	23462.0i		−	0.797490i	0.917062	−	0.398745i	$-0.130554\pi$
$954$		0			0
$955$			57084.6i			1.93426i
$956$		0			0
$957$	10808.8	+	1816.82i	0.365099	+	0.0613681i
$958$		0			0
$959$	325.286			0.0109531
$960$		0			0
$961$	−55204.1			−1.85305
$962$		0			0
$963$	−32709.8	−	11315.8i	−1.09456	−	0.378658i
$964$		0			0
$965$		−	64020.3i		−	2.13563i
$966$		0			0
$967$		−	46332.4i		−	1.54080i	−0.637564	−	0.770398i	$-0.720058\pi$
$967$		−	46332.4i		−	1.54080i	0.637564	−	0.770398i	$-0.279942\pi$
$968$		0			0
$969$	−4304.75	+	25610.3i	−0.142712	+	0.849043i
$970$		0			0
$971$	−42500.4			−1.40464			−0.702318	−	0.711863i	$-0.747852\pi$
$971$	−42500.4			−1.40464			−0.702318	+	0.711863i	$0.747852\pi$
$972$		0			0
$973$	−1341.13			−0.0441877
$974$		0			0
$975$	−2903.24	+	17272.3i	−0.0953621	+	0.567340i
$976$		0			0
$977$		−	25208.6i		−	0.825480i	−0.910849	−	0.412740i	$-0.864572\pi$
$977$		−	25208.6i		−	0.825480i	0.910849	−	0.412740i	$-0.135428\pi$
$978$		0			0
$979$		−	7437.86i		−	0.242814i
$980$		0			0
$981$	−8046.07	−	2783.51i	−0.261867	−	0.0905919i
$982$		0			0
$983$	−8265.01			−0.268172			−0.134086	−	0.990970i	$-0.542810\pi$
$983$	−8265.01			−0.268172			−0.134086	+	0.990970i	$0.542810\pi$
$984$		0			0
$985$	62215.7			2.01255
$986$		0			0
$987$	−352.420	−	59.2370i	−0.0113654	−	0.00191037i
$988$		0			0
$989$			72860.0i			2.34258i
$990$		0			0
$991$			3171.72i			0.101668i	0.998707	+	0.0508340i	$0.0161880\pi$
$991$			3171.72i			0.101668i	−0.998707	+	0.0508340i	$0.983812\pi$
$992$		0			0
$993$	11089.2	+	1863.94i	0.354386	+	0.0595673i
$994$		0			0
$995$	−32393.5			−1.03210
$996$		0			0
$997$	53321.4			1.69379			0.846894	−	0.531761i	$-0.178470\pi$
$997$	53321.4			1.69379			0.846894	+	0.531761i	$0.178470\pi$
$998$		0			0
$999$	14591.3	−	26735.4i	0.462111	−	0.846716i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.d.a.287.7	yes	12
3.2	odd	2	inner	624.4.d.a.287.5	✓	12
4.3	odd	2	inner	624.4.d.a.287.6	yes	12
12.11	even	2	inner	624.4.d.a.287.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.d.a.287.5	✓	12	3.2	odd	2	inner
624.4.d.a.287.6	yes	12	4.3	odd	2	inner
624.4.d.a.287.7	yes	12	1.1	even	1	trivial
624.4.d.a.287.8	yes	12	12.11	even	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.861319 - 5.12427i) q^{3} +19.6032i q^{5} +0.649873i q^{7} +(-25.5163 - 8.82726i) q^{9} +O(q^{10})\)	\(q+(0.861319 - 5.12427i) q^{3} +19.6032i q^{5} +0.649873i q^{7} +(-25.5163 - 8.82726i) q^{9} +9.64478 q^{11} +13.0000 q^{13} +(100.452 + 16.8846i) q^{15} -48.1203i q^{17} -103.862i q^{19} +(3.33012 + 0.559748i) q^{21} +170.938 q^{23} -259.284 q^{25} +(-67.2109 + 123.149i) q^{27} +218.703i q^{29} +291.539i q^{31} +(8.30723 - 49.4225i) q^{33} -12.7396 q^{35} -217.098 q^{37} +(11.1971 - 66.6155i) q^{39} +353.147i q^{41} +426.237i q^{43} +(173.042 - 500.199i) q^{45} -105.828 q^{47} +342.578 q^{49} +(-246.582 - 41.4470i) q^{51} +286.136i q^{53} +189.068i q^{55} +(-532.214 - 89.4579i) q^{57} -742.972 q^{59} -238.480 q^{61} +(5.73660 - 16.5823i) q^{63} +254.841i q^{65} +139.685i q^{67} +(147.232 - 875.931i) q^{69} +88.4498 q^{71} +131.451 q^{73} +(-223.326 + 1328.64i) q^{75} +6.26788i q^{77} +907.826i q^{79} +(573.159 + 450.477i) q^{81} -245.398 q^{83} +943.311 q^{85} +(1120.69 + 188.373i) q^{87} -771.180i q^{89} +8.44835i q^{91} +(1493.93 + 251.108i) q^{93} +2036.01 q^{95} -777.823 q^{97} +(-246.099 - 85.1370i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{9}+O(q^{10}) \) 12 * q - 10 * q^9	\( 12 q - 10 q^{9} + 156 q^{13} + 478 q^{21} - 600 q^{25} - 184 q^{33} - 828 q^{37} + 170 q^{45} - 720 q^{49} - 204 q^{57} + 192 q^{61} + 2476 q^{69} + 960 q^{73} + 566 q^{81} + 348 q^{85} + 6388 q^{93} - 7248 q^{97}+O(q^{100}) \) 12 * q - 10 * q^9 + 156 * q^13 + 478 * q^21 - 600 * q^25 - 184 * q^33 - 828 * q^37 + 170 * q^45 - 720 * q^49 - 204 * q^57 + 192 * q^61 + 2476 * q^69 + 960 * q^73 + 566 * q^81 + 348 * q^85 + 6388 * q^93 - 7248 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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