LMFDB - Embedded newform 624.4.d.b.287.6

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} + 120x^{10} + 5196x^{8} + 96803x^{6} + 702900x^{4} + 976752x^{2} + 254016 $ x^12 + 120x^10 + 5196x^8 + 96803x^6 + 702900x^4 + 976752*x^2 + 254016
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.6
Root			$-1.16673i$ of defining polynomial
Character	$\chi$	$=$	624.287
Dual form			624.4.d.b.287.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.28065 + 4.66890i) q^{3} -4.73978i q^{5} -12.5922i q^{7} +(-16.5973 - 21.2962i) q^{9} +O(q^{10})$	\(q+(-2.28065 + 4.66890i) q^{3} -4.73978i q^{5} -12.5922i q^{7} +(-16.5973 - 21.2962i) q^{9} -56.4872 q^{11} -13.0000 q^{13} +(22.1296 + 10.8098i) q^{15} -33.1129i q^{17} +65.5448i q^{19} +(58.7919 + 28.7184i) q^{21} +21.2486 q^{23} +102.534 q^{25} +(137.283 - 28.9218i) q^{27} +81.2014i q^{29} +235.197i q^{31} +(128.828 - 263.733i) q^{33} -59.6844 q^{35} +139.781 q^{37} +(29.6484 - 60.6957i) q^{39} -332.576i q^{41} +133.819i q^{43} +(-100.940 + 78.6676i) q^{45} +286.498 q^{47} +184.436 q^{49} +(154.601 + 75.5190i) q^{51} -52.6315i q^{53} +267.737i q^{55} +(-306.022 - 149.485i) q^{57} +119.958 q^{59} -376.581 q^{61} +(-268.167 + 208.997i) q^{63} +61.6172i q^{65} +487.985i q^{67} +(-48.4606 + 99.2076i) q^{69} +466.446 q^{71} +461.463 q^{73} +(-233.845 + 478.723i) q^{75} +711.300i q^{77} +533.857i q^{79} +(-178.060 + 706.920i) q^{81} +897.316 q^{83} -156.948 q^{85} +(-379.121 - 185.192i) q^{87} +1511.73i q^{89} +163.699i q^{91} +(-1098.11 - 536.401i) q^{93} +310.668 q^{95} +47.3252 q^{97} +(937.535 + 1202.97i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 54 q^{9}+O(q^{10}) $ 12 * q + 54 * q^9	$ 12 q + 54 q^{9} - 156 q^{13} - 54 q^{21} - 408 q^{25} - 360 q^{33} + 636 q^{37} - 810 q^{45} + 336 q^{49} - 1260 q^{57} + 960 q^{61} - 252 q^{69} + 3216 q^{73} - 2538 q^{81} + 2196 q^{85} - 1116 q^{93} + 4800 q^{97}+O(q^{100}) $ 12 * q + 54 * q^9 - 156 * q^13 - 54 * q^21 - 408 * q^25 - 360 * q^33 + 636 * q^37 - 810 * q^45 + 336 * q^49 - 1260 * q^57 + 960 * q^61 - 252 * q^69 + 3216 * q^73 - 2538 * q^81 + 2196 * q^85 - 1116 * q^93 + 4800 * 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
$3$	−2.28065	+	4.66890i	−0.438911	+	0.898531i
$3$	−2.28065	+	4.66890i	−0.438911	+	0.898531i
$4$		0			0
$5$		−	4.73978i		−	0.423939i	−0.977276	−	0.211970i	$-0.932012\pi$
$5$		−	4.73978i		−	0.423939i	0.977276	−	0.211970i	$-0.0679878\pi$
$6$		0			0
$7$		−	12.5922i		−	0.679916i	−0.940441	−	0.339958i	$-0.889587\pi$
$7$		−	12.5922i		−	0.679916i	0.940441	−	0.339958i	$-0.110413\pi$
$8$		0			0
$9$	−16.5973	−	21.2962i	−0.614714	−	0.788750i
$10$		0			0
$11$	−56.4872			−1.54832			−0.774161	−	0.632989i	$-0.781828\pi$
$11$	−56.4872			−1.54832			−0.774161	+	0.632989i	$0.781828\pi$
$12$		0			0
$13$	−13.0000			−0.277350
$13$	−13.0000			−0.277350
$14$		0			0
$15$	22.1296	+	10.8098i	0.380922	+	0.186072i
$16$		0			0
$17$		−	33.1129i		−	0.472416i	−0.971703	−	0.236208i	$-0.924095\pi$
$17$		−	33.1129i		−	0.472416i	0.971703	−	0.236208i	$-0.0759046\pi$
$18$		0			0
$19$			65.5448i			0.791421i	0.918375	+	0.395711i	$0.129502\pi$
$19$			65.5448i			0.791421i	−0.918375	+	0.395711i	$0.870498\pi$
$20$		0			0
$21$	58.7919	+	28.7184i	0.610925	+	0.298423i
$22$		0			0
$23$	21.2486			0.192637			0.0963183	−	0.995351i	$-0.469293\pi$
$23$	21.2486			0.192637			0.0963183	+	0.995351i	$0.469293\pi$
$24$		0			0
$25$	102.534			0.820276
$26$		0			0
$27$	137.283	−	28.9218i	0.978521	−	0.206148i
$28$		0			0
$29$			81.2014i			0.519956i	0.965615	+	0.259978i	$0.0837153\pi$
$29$			81.2014i			0.519956i	−0.965615	+	0.259978i	$0.916285\pi$
$30$		0			0
$31$			235.197i			1.36266i	0.731975	+	0.681332i	$0.238599\pi$
$31$			235.197i			1.36266i	−0.731975	+	0.681332i	$0.761401\pi$
$32$		0			0
$33$	128.828	−	263.733i	0.679576	−	1.39121i
$34$		0			0
$35$	−59.6844			−0.288243
$36$		0			0
$37$	139.781			0.621076			0.310538	−	0.950561i	$-0.399491\pi$
$37$	139.781			0.621076			0.310538	+	0.950561i	$0.399491\pi$
$38$		0			0
$39$	29.6484	−	60.6957i	0.121732	−	0.249208i
$40$		0			0
$41$		−	332.576i		−	1.26682i	−0.773816	−	0.633410i	$-0.781655\pi$
$41$		−	332.576i		−	1.26682i	0.773816	−	0.633410i	$-0.218345\pi$
$42$		0			0
$43$			133.819i			0.474586i	0.971438	+	0.237293i	$0.0762601\pi$
$43$			133.819i			0.474586i	−0.971438	+	0.237293i	$0.923740\pi$
$44$		0			0
$45$	−100.940	+	78.6676i	−0.334382	+	0.260601i
$46$		0			0
$47$	286.498			0.889148			0.444574	−	0.895742i	$-0.353355\pi$
$47$	286.498			0.889148			0.444574	+	0.895742i	$0.353355\pi$
$48$		0			0
$49$	184.436			0.537714
$50$		0			0
$51$	154.601	+	75.5190i	0.424480	+	0.207348i
$52$		0			0
$53$		−	52.6315i		−	0.136406i	−0.997671	−	0.0682028i	$-0.978274\pi$
$53$		−	52.6315i		−	0.136406i	0.997671	−	0.0682028i	$-0.0217265\pi$
$54$		0			0
$55$			267.737i			0.656394i
$56$		0			0
$57$	−306.022	−	149.485i	−0.711116	−	0.347363i
$58$		0			0
$59$	119.958			0.264698			0.132349	−	0.991203i	$-0.457748\pi$
$59$	119.958			0.264698			0.132349	+	0.991203i	$0.457748\pi$
$60$		0			0
$61$	−376.581			−0.790431			−0.395215	−	0.918589i	$-0.629330\pi$
$61$	−376.581			−0.790431			−0.395215	+	0.918589i	$0.629330\pi$
$62$		0			0
$63$	−268.167	+	208.997i	−0.536284	+	0.417954i
$64$		0			0
$65$			61.6172i			0.117580i
$66$		0			0
$67$			487.985i			0.889803i	0.895579	+	0.444902i	$0.146761\pi$
$67$			487.985i			0.889803i	−0.895579	+	0.444902i	$0.853239\pi$
$68$		0			0
$69$	−48.4606	+	99.2076i	−0.0845503	+	0.173090i
$70$		0			0
$71$	466.446			0.779675			0.389838	−	0.920884i	$-0.372531\pi$
$71$	466.446			0.779675			0.389838	+	0.920884i	$0.372531\pi$
$72$		0			0
$73$	461.463			0.739865			0.369933	−	0.929059i	$-0.379381\pi$
$73$	461.463			0.739865			0.369933	+	0.929059i	$0.379381\pi$
$74$		0			0
$75$	−233.845	+	478.723i	−0.360028	+	0.737043i
$76$		0			0
$77$			711.300i			1.05273i
$78$		0			0
$79$			533.857i			0.760299i	0.924925	+	0.380150i	$0.124127\pi$
$79$			533.857i			0.760299i	−0.924925	+	0.380150i	$0.875873\pi$
$80$		0			0
$81$	−178.060	+	706.920i	−0.244253	+	0.969712i
$82$		0			0
$83$	897.316			1.18667			0.593333	−	0.804957i	$-0.297812\pi$
$83$	897.316			1.18667			0.593333	+	0.804957i	$0.297812\pi$
$84$		0			0
$85$	−156.948			−0.200275
$86$		0			0
$87$	−379.121	−	185.192i	−0.467196	−	0.228214i
$88$		0			0
$89$			1511.73i			1.80049i	0.435388	+	0.900243i	$0.356611\pi$
$89$			1511.73i			1.80049i	−0.435388	+	0.900243i	$0.643389\pi$
$90$		0			0
$91$			163.699i			0.188575i
$92$		0			0
$93$	−1098.11	−	536.401i	−1.22440	−	0.598088i
$94$		0			0
$95$	310.668			0.335515
$96$		0			0
$97$	47.3252			0.0495376			0.0247688	−	0.999693i	$-0.492115\pi$
$97$	47.3252			0.0495376			0.0247688	+	0.999693i	$0.492115\pi$
$98$		0			0
$99$	937.535	+	1202.97i	0.951775	+	1.22124i
$100$		0			0
$101$			971.101i			0.956714i	0.878165	+	0.478357i	$0.158768\pi$
$101$			971.101i			0.956714i	−0.878165	+	0.478357i	$0.841232\pi$
$102$		0			0
$103$			1307.84i			1.25112i	0.780175	+	0.625562i	$0.215130\pi$
$103$			1307.84i			1.25112i	−0.780175	+	0.625562i	$0.784870\pi$
$104$		0			0
$105$	136.119	−	278.661i	0.126513	−	0.258995i
$106$		0			0
$107$	743.046			0.671336			0.335668	−	0.941980i	$-0.391038\pi$
$107$	743.046			0.671336			0.335668	+	0.941980i	$0.391038\pi$
$108$		0			0
$109$	−1489.02			−1.30846			−0.654231	−	0.756295i	$-0.727008\pi$
$109$	−1489.02			−1.30846			−0.654231	+	0.756295i	$0.727008\pi$
$110$		0			0
$111$	−318.791	+	652.623i	−0.272597	+	0.558056i
$112$		0			0
$113$		−	2207.42i		−	1.83767i	−0.394638	−	0.918836i	$-0.629130\pi$
$113$		−	2207.42i		−	1.83767i	0.394638	−	0.918836i	$-0.370870\pi$
$114$		0			0
$115$		−	100.714i		−	0.0816662i
$116$		0			0
$117$	215.765	+	276.851i	0.170491	+	0.218760i
$118$		0			0
$119$	−416.965			−0.321203
$120$		0			0
$121$	1859.81			1.39730
$122$		0			0
$123$	1552.76	+	758.489i	1.13828	+	0.556022i
$124$		0			0
$125$		−	1078.46i		−	0.771686i
$126$		0			0
$127$		−	448.045i		−	0.313052i	−0.987674	−	0.156526i	$-0.949971\pi$
$127$		−	448.045i		−	0.313052i	0.987674	−	0.156526i	$-0.0500294\pi$
$128$		0			0
$129$	−624.787	−	305.194i	−0.426430	−	0.208301i
$130$		0			0
$131$	−2490.66			−1.66115			−0.830573	−	0.556909i	$-0.811987\pi$
$131$	−2490.66			−1.66115			−0.830573	+	0.556909i	$0.811987\pi$
$132$		0			0
$133$	825.354			0.538100
$134$		0			0
$135$	−137.083	−	650.690i	−0.0873944	−	0.414833i
$136$		0			0
$137$		−	843.526i		−	0.526039i	−0.964791	−	0.263019i	$-0.915282\pi$
$137$		−	843.526i		−	0.526039i	0.964791	−	0.263019i	$-0.0847183\pi$
$138$		0			0
$139$			1573.93i			0.960426i	0.877152	+	0.480213i	$0.159441\pi$
$139$			1573.93i			0.960426i	−0.877152	+	0.480213i	$0.840559\pi$
$140$		0			0
$141$	−653.400	+	1337.63i	−0.390257	+	0.798927i
$142$		0			0
$143$	734.334			0.429427
$144$		0			0
$145$	384.877			0.220430
$146$		0			0
$147$	−420.634	+	861.113i	−0.236009	+	0.483152i
$148$		0			0
$149$			1266.77i			0.696498i	0.937402	+	0.348249i	$0.113224\pi$
$149$			1266.77i			0.696498i	−0.937402	+	0.348249i	$0.886776\pi$
$150$		0			0
$151$		−	333.612i		−	0.179794i	−0.995951	−	0.0898972i	$-0.971346\pi$
$151$		−	333.612i		−	0.179794i	0.995951	−	0.0898972i	$-0.0286539\pi$
$152$		0			0
$153$	−705.181	+	549.585i	−0.372618	+	0.290401i
$154$		0			0
$155$	1114.78			0.577687
$156$		0			0
$157$	3007.54			1.52884			0.764420	−	0.644718i	$-0.223025\pi$
$157$	3007.54			1.52884			0.764420	+	0.644718i	$0.223025\pi$
$158$		0			0
$159$	245.731	+	120.034i	0.122565	+	0.0598699i
$160$		0			0
$161$		−	267.567i		−	0.130977i
$162$		0			0
$163$			471.180i			0.226415i	0.993571	+	0.113208i	$0.0361125\pi$
$163$			471.180i			0.226415i	−0.993571	+	0.113208i	$0.963887\pi$
$164$		0			0
$165$	−1250.04	−	610.615i	−0.589790	−	0.288099i
$166$		0			0
$167$	−3229.01			−1.49622			−0.748109	−	0.663576i	$-0.769038\pi$
$167$	−3229.01			−1.49622			−0.748109	+	0.663576i	$0.769038\pi$
$168$		0			0
$169$	169.000			0.0769231
$170$		0			0
$171$	1395.86	−	1087.87i	0.624233	−	0.486498i
$172$		0			0
$173$		−	867.551i		−	0.381264i	−0.981662	−	0.190632i	$-0.938946\pi$
$173$		−	867.551i		−	0.381264i	0.981662	−	0.190632i	$-0.0610537\pi$
$174$		0			0
$175$		−	1291.14i		−	0.557719i
$176$		0			0
$177$	−273.582	+	560.071i	−0.116179	+	0.237839i
$178$		0			0
$179$	1444.75			0.603272			0.301636	−	0.953423i	$-0.402467\pi$
$179$	1444.75			0.603272			0.301636	+	0.953423i	$0.402467\pi$
$180$		0			0
$181$	4377.66			1.79773			0.898864	−	0.438227i	$-0.144394\pi$
$181$	4377.66			1.79773			0.898864	+	0.438227i	$0.144394\pi$
$182$		0			0
$183$	858.849	−	1758.22i	0.346929	−	0.710226i
$184$		0			0
$185$		−	662.531i		−	0.263299i
$186$		0			0
$187$			1870.46i			0.731451i
$188$		0			0
$189$	−364.190	−	1728.69i	−0.140164	−	0.665312i
$190$		0			0
$191$	2307.05			0.873992			0.436996	−	0.899464i	$-0.356042\pi$
$191$	2307.05			0.873992			0.436996	+	0.899464i	$0.356042\pi$
$192$		0			0
$193$	−809.433			−0.301887			−0.150944	−	0.988542i	$-0.548231\pi$
$193$	−809.433			−0.301887			−0.150944	+	0.988542i	$0.548231\pi$
$194$		0			0
$195$	−287.685	−	140.527i	−0.105649	−	0.0516070i
$196$		0			0
$197$			254.930i			0.0921980i	0.998937	+	0.0460990i	$0.0146790\pi$
$197$			254.930i			0.0921980i	−0.998937	+	0.0460990i	$0.985321\pi$
$198$		0			0
$199$		−	4222.68i		−	1.50421i	−0.659043	−	0.752105i	$-0.729039\pi$
$199$		−	4222.68i		−	1.50421i	0.659043	−	0.752105i	$-0.270961\pi$
$200$		0			0
$201$	−2278.35	−	1112.92i	−0.799515	−	0.390544i
$202$		0			0
$203$	1022.51			0.353526
$204$		0			0
$205$	−1576.34			−0.537055
$206$		0			0
$207$	−352.669	−	452.516i	−0.118416	−	0.151942i
$208$		0			0
$209$		−	3702.44i		−	1.22537i
$210$		0			0
$211$		−	91.0781i		−	0.0297160i	−0.999890	−	0.0148580i	$-0.995270\pi$
$211$		−	91.0781i		−	0.0297160i	0.999890	−	0.0148580i	$-0.00472962\pi$
$212$		0			0
$213$	−1063.80	+	2177.79i	−0.342208	+	0.700562i
$214$		0			0
$215$	634.272			0.201195
$216$		0			0
$217$	2961.65			0.926497
$218$		0			0
$219$	−1052.43	+	2154.53i	−0.324735	+	0.664791i
$220$		0			0
$221$			430.468i			0.131024i
$222$		0			0
$223$		−	598.966i		−	0.179864i	−0.995948	−	0.0899322i	$-0.971335\pi$
$223$		−	598.966i		−	0.179864i	0.995948	−	0.0899322i	$-0.0286650\pi$
$224$		0			0
$225$	−1701.79	−	2183.60i	−0.504235	−	0.646992i
$226$		0			0
$227$	−3843.83			−1.12389			−0.561947	−	0.827173i	$-0.689947\pi$
$227$	−3843.83			−1.12389			−0.561947	+	0.827173i	$0.689947\pi$
$228$		0			0
$229$	664.801			0.191840			0.0959198	−	0.995389i	$-0.469421\pi$
$229$	664.801			0.191840			0.0959198	+	0.995389i	$0.469421\pi$
$230$		0			0
$231$	−3320.99	−	1622.22i	−0.945909	−	0.462054i
$232$		0			0
$233$		−	3381.44i		−	0.950754i	−0.879782	−	0.475377i	$-0.842312\pi$
$233$		−	3381.44i		−	0.950754i	0.879782	−	0.475377i	$-0.157688\pi$
$234$		0			0
$235$		−	1357.94i		−	0.376945i
$236$		0			0
$237$	−2492.53	−	1217.54i	−0.683152	−	0.333704i
$238$		0			0
$239$	−3153.88			−0.853587			−0.426793	−	0.904349i	$-0.640357\pi$
$239$	−3153.88			−0.853587			−0.426793	+	0.904349i	$0.640357\pi$
$240$		0			0
$241$	6192.87			1.65526			0.827630	−	0.561275i	$-0.189689\pi$
$241$	6192.87			1.65526			0.827630	+	0.561275i	$0.189689\pi$
$242$		0			0
$243$	−2894.45	−	2443.58i	−0.764110	−	0.645086i
$244$		0			0
$245$		−	874.187i		−	0.227958i
$246$		0			0
$247$		−	852.082i		−	0.219501i
$248$		0			0
$249$	−2046.46	+	4189.48i	−0.520841	+	1.06626i
$250$		0			0
$251$	5318.24			1.33739			0.668694	−	0.743537i	$-0.266853\pi$
$251$	5318.24			1.33739			0.668694	+	0.743537i	$0.266853\pi$
$252$		0			0
$253$	−1200.27			−0.298263
$254$		0			0
$255$	357.944	−	732.775i	0.0879031	−	0.179954i
$256$		0			0
$257$			6217.84i			1.50918i	0.656199	+	0.754588i	$0.272163\pi$
$257$			6217.84i			1.50918i	−0.656199	+	0.754588i	$0.727837\pi$
$258$		0			0
$259$		−	1760.15i		−	0.422280i
$260$		0			0
$261$	1729.28	−	1347.72i	0.410115	−	0.319624i
$262$		0			0
$263$	810.725			0.190082			0.0950408	−	0.995473i	$-0.469702\pi$
$263$	810.725			0.190082			0.0950408	+	0.995473i	$0.469702\pi$
$264$		0			0
$265$	−249.462			−0.0578277
$266$		0			0
$267$	−7058.13	−	3447.73i	−1.61779	−	0.790253i
$268$		0			0
$269$			8369.06i			1.89692i	0.316900	+	0.948459i	$0.397358\pi$
$269$			8369.06i			1.89692i	−0.316900	+	0.948459i	$0.602642\pi$
$270$		0			0
$271$		−	6015.62i		−	1.34843i	−0.738537	−	0.674213i	$-0.764483\pi$
$271$		−	6015.62i		−	1.34843i	0.738537	−	0.674213i	$-0.235517\pi$
$272$		0			0
$273$	−764.294	−	373.340i	−0.169440	−	0.0827676i
$274$		0			0
$275$	−5791.89			−1.27005
$276$		0			0
$277$	1754.37			0.380542			0.190271	−	0.981732i	$-0.439063\pi$
$277$	1754.37			0.380542			0.190271	+	0.981732i	$0.439063\pi$
$278$		0			0
$279$	5008.81	−	3903.63i	1.07480	−	0.837649i
$280$		0			0
$281$			3709.04i			0.787413i	0.919236	+	0.393707i	$0.128807\pi$
$281$			3709.04i			0.787413i	−0.919236	+	0.393707i	$0.871193\pi$
$282$		0			0
$283$			2133.83i			0.448209i	0.974565	+	0.224104i	$0.0719457\pi$
$283$			2133.83i			0.448209i	−0.974565	+	0.224104i	$0.928054\pi$
$284$		0			0
$285$	−708.525	+	1450.48i	−0.147261	+	0.301470i
$286$		0			0
$287$	−4187.87			−0.861332
$288$		0			0
$289$	3816.53			0.776824
$290$		0			0
$291$	−107.932	+	220.957i	−0.0217426	+	0.0445110i
$292$		0			0
$293$		−	6874.80i		−	1.37075i	−0.728190	−	0.685375i	$-0.759638\pi$
$293$		−	6874.80i		−	1.37075i	0.728190	−	0.685375i	$-0.240362\pi$
$294$		0			0
$295$		−	568.574i		−	0.112216i
$296$		0			0
$297$	−7754.72	+	1633.71i	−1.51507	+	0.319184i
$298$		0			0
$299$	−276.232			−0.0534278
$300$		0			0
$301$	1685.08			0.322678
$302$		0			0
$303$	−4533.97	−	2214.74i	−0.859637	−	0.419912i
$304$		0			0
$305$			1784.91i			0.335095i
$306$		0			0
$307$			3241.62i			0.602635i	0.953524	+	0.301317i	$0.0974263\pi$
$307$			3241.62i			0.602635i	−0.953524	+	0.301317i	$0.902574\pi$
$308$		0			0
$309$	−6106.20	−	2982.73i	−1.12417	−	0.549132i
$310$		0			0
$311$	−6309.24			−1.15037			−0.575183	−	0.818025i	$-0.695069\pi$
$311$	−6309.24			−1.15037			−0.575183	+	0.818025i	$0.695069\pi$
$312$		0			0
$313$	3290.01			0.594129			0.297065	−	0.954857i	$-0.403992\pi$
$313$	3290.01			0.594129			0.297065	+	0.954857i	$0.403992\pi$
$314$		0			0
$315$	990.599	+	1271.05i	0.177187	+	0.227352i
$316$		0			0
$317$			9565.15i			1.69474i	0.531003	+	0.847370i	$0.321815\pi$
$317$			9565.15i			1.69474i	−0.531003	+	0.847370i	$0.678185\pi$
$318$		0			0
$319$		−	4586.84i		−	0.805059i
$320$		0			0
$321$	−1694.63	+	3469.21i	−0.294657	+	0.603216i
$322$		0			0
$323$	2170.38			0.373880
$324$		0			0
$325$	−1332.95			−0.227503
$326$		0			0
$327$	3395.93	−	6952.09i	0.574299	−	1.17569i
$328$		0			0
$329$		−	3607.64i		−	0.604546i
$330$		0			0
$331$		−	4651.42i		−	0.772403i	−0.922414	−	0.386201i	$-0.873787\pi$
$331$		−	4651.42i		−	0.772403i	0.922414	−	0.386201i	$-0.126213\pi$
$332$		0			0
$333$	−2319.98	−	2976.81i	−0.381784	−	0.489874i
$334$		0			0
$335$	2312.94			0.377222
$336$		0			0
$337$	−7306.29			−1.18101			−0.590503	−	0.807036i	$-0.701071\pi$
$337$	−7306.29			−1.18101			−0.590503	+	0.807036i	$0.701071\pi$
$338$		0			0
$339$	10306.2	+	5034.36i	1.65121	+	0.806575i
$340$		0			0
$341$		−	13285.6i		−	2.10984i
$342$		0			0
$343$		−	6641.59i		−	1.04552i
$344$		0			0
$345$	470.223	+	229.693i	0.0733796	+	0.0358442i
$346$		0			0
$347$	8403.19			1.30002			0.650010	−	0.759926i	$-0.274765\pi$
$347$	8403.19			1.30002			0.650010	+	0.759926i	$0.274765\pi$
$348$		0			0
$349$	−5089.19			−0.780568			−0.390284	−	0.920695i	$-0.627623\pi$
$349$	−5089.19			−0.780568			−0.390284	+	0.920695i	$0.627623\pi$
$350$		0			0
$351$	−1784.67	+	375.984i	−0.271393	+	0.0571753i
$352$		0			0
$353$			12131.1i			1.82910i	0.404472	+	0.914551i	$0.367455\pi$
$353$			12131.1i			1.82910i	−0.404472	+	0.914551i	$0.632545\pi$
$354$		0			0
$355$		−	2210.85i		−	0.330535i
$356$		0			0
$357$	950.951	−	1946.77i	0.140980	−	0.288611i
$358$		0			0
$359$	−6926.28			−1.01826			−0.509130	−	0.860690i	$-0.670033\pi$
$359$	−6926.28			−1.01826			−0.509130	+	0.860690i	$0.670033\pi$
$360$		0			0
$361$	2562.88			0.373653
$362$		0			0
$363$	−4241.57	+	8683.26i	−0.613291	+	1.25552i
$364$		0			0
$365$		−	2187.24i		−	0.313658i
$366$		0			0
$367$		−	4512.36i		−	0.641807i	−0.947112	−	0.320903i	$-0.896014\pi$
$367$		−	4512.36i		−	0.641807i	0.947112	−	0.320903i	$-0.103986\pi$
$368$		0			0
$369$	−7082.62	+	5519.86i	−0.999205	+	0.778733i
$370$		0			0
$371$	−662.748			−0.0927444
$372$		0			0
$373$	5078.10			0.704917			0.352459	−	0.935827i	$-0.385346\pi$
$373$	5078.10			0.704917			0.352459	+	0.935827i	$0.385346\pi$
$374$		0			0
$375$	5035.24	+	2459.60i	0.693384	+	0.338702i
$376$		0			0
$377$		−	1055.62i		−	0.144210i
$378$		0			0
$379$			2969.26i			0.402430i	0.979547	+	0.201215i	$0.0644889\pi$
$379$			2969.26i			0.402430i	−0.979547	+	0.201215i	$0.935511\pi$
$380$		0			0
$381$	2091.88	+	1021.83i	0.281286	+	0.137402i
$382$		0			0
$383$	−6104.12			−0.814376			−0.407188	−	0.913344i	$-0.633491\pi$
$383$	−6104.12			−0.814376			−0.407188	+	0.913344i	$0.633491\pi$
$384$		0			0
$385$	3371.41			0.446293
$386$		0			0
$387$	2849.84	−	2221.03i	0.374329	−	0.291734i
$388$		0			0
$389$		−	2923.45i		−	0.381041i	−0.981683	−	0.190520i	$-0.938983\pi$
$389$		−	2923.45i		−	0.381041i	0.981683	−	0.190520i	$-0.0610175\pi$
$390$		0			0
$391$		−	703.603i		−	0.0910045i
$392$		0			0
$393$	5680.33	−	11628.7i	0.729095	−	1.49259i
$394$		0			0
$395$	2530.37			0.322321
$396$		0			0
$397$	9433.57			1.19259			0.596294	−	0.802766i	$-0.296639\pi$
$397$	9433.57			1.19259			0.596294	+	0.802766i	$0.296639\pi$
$398$		0			0
$399$	−1882.34	+	3853.50i	−0.236178	+	0.483499i
$400$		0			0
$401$			4034.68i			0.502449i	0.967929	+	0.251225i	$0.0808333\pi$
$401$			4034.68i			0.502449i	−0.967929	+	0.251225i	$0.919167\pi$
$402$		0			0
$403$		−	3057.56i		−	0.377935i
$404$		0			0
$405$	3350.65	+	843.968i	0.411099	+	0.103548i
$406$		0			0
$407$	−7895.83			−0.961626
$408$		0			0
$409$	8228.17			0.994759			0.497380	−	0.867533i	$-0.334296\pi$
$409$	8228.17			0.994759			0.497380	+	0.867533i	$0.334296\pi$
$410$		0			0
$411$	3938.34	+	1923.79i	0.472662	+	0.230884i
$412$		0			0
$413$		−	1510.54i		−	0.179972i
$414$		0			0
$415$		−	4253.09i		−	0.503074i
$416$		0			0
$417$	−7348.54	−	3589.59i	−0.862972	−	0.421542i
$418$		0			0
$419$	6070.39			0.707776			0.353888	−	0.935288i	$-0.384859\pi$
$419$	6070.39			0.707776			0.353888	+	0.935288i	$0.384859\pi$
$420$		0			0
$421$	5196.88			0.601616			0.300808	−	0.953685i	$-0.402744\pi$
$421$	5196.88			0.601616			0.300808	+	0.953685i	$0.402744\pi$
$422$		0			0
$423$	−4755.08	−	6101.32i	−0.546572	−	0.701316i
$424$		0			0
$425$		−	3395.22i		−	0.387511i
$426$		0			0
$427$			4741.99i			0.537427i
$428$		0			0
$429$	−1674.76	+	3428.53i	−0.188480	+	0.385853i
$430$		0			0
$431$	−4150.56			−0.463864			−0.231932	−	0.972732i	$-0.574505\pi$
$431$	−4150.56			−0.463864			−0.231932	+	0.972732i	$0.574505\pi$
$432$		0			0
$433$	5614.13			0.623090			0.311545	−	0.950231i	$-0.399154\pi$
$433$	5614.13			0.623090			0.311545	+	0.950231i	$0.399154\pi$
$434$		0			0
$435$	−877.769	+	1796.95i	−0.0967490	+	0.198063i
$436$		0			0
$437$			1392.73i			0.152457i
$438$		0			0
$439$		−	4396.64i		−	0.477996i	−0.971020	−	0.238998i	$-0.923181\pi$
$439$		−	4396.64i		−	0.477996i	0.971020	−	0.238998i	$-0.0768190\pi$
$440$		0			0
$441$	−3061.14	−	3927.79i	−0.330540	−	0.424122i
$442$		0			0
$443$	3049.29			0.327034			0.163517	−	0.986541i	$-0.447716\pi$
$443$	3049.29			0.327034			0.163517	+	0.986541i	$0.447716\pi$
$444$		0			0
$445$	7165.28			0.763297
$446$		0			0
$447$	−5914.45	−	2889.07i	−0.625825	−	0.305701i
$448$		0			0
$449$		−	1995.33i		−	0.209723i	−0.994487	−	0.104861i	$-0.966560\pi$
$449$		−	1995.33i		−	0.209723i	0.994487	−	0.104861i	$-0.0334399\pi$
$450$		0			0
$451$			18786.3i			1.96145i
$452$		0			0
$453$	1557.60	+	760.852i	0.161551	+	0.0789138i
$454$		0			0
$455$	775.898			0.0799443
$456$		0			0
$457$	−7565.40			−0.774386			−0.387193	−	0.921999i	$-0.626555\pi$
$457$	−7565.40			−0.774386			−0.387193	+	0.921999i	$0.626555\pi$
$458$		0			0
$459$	−957.686	−	4545.83i	−0.0973877	−	0.462268i
$460$		0			0
$461$		−	5402.40i		−	0.545802i	−0.962042	−	0.272901i	$-0.912017\pi$
$461$		−	5402.40i		−	0.545802i	0.962042	−	0.272901i	$-0.0879832\pi$
$462$		0			0
$463$			14297.3i			1.43510i	0.696506	+	0.717551i	$0.254737\pi$
$463$			14297.3i			1.43510i	−0.696506	+	0.717551i	$0.745263\pi$
$464$		0			0
$465$	−2542.43	+	5204.81i	−0.253553	+	0.519069i
$466$		0			0
$467$	−93.0283			−0.00921806			−0.00460903	−	0.999989i	$-0.501467\pi$
$467$	−93.0283			−0.00921806			−0.00460903	+	0.999989i	$0.501467\pi$
$468$		0			0
$469$	6144.81			0.604991
$470$		0			0
$471$	−6859.14	+	14041.9i	−0.671025	+	1.37371i
$472$		0			0
$473$		−	7559.05i		−	0.734811i
$474$		0			0
$475$			6720.60i			0.649183i
$476$		0			0
$477$	−1120.85	+	873.540i	−0.107590	+	0.0838504i
$478$		0			0
$479$	8205.03			0.782667			0.391333	−	0.920249i	$-0.372014\pi$
$479$	8205.03			0.782667			0.391333	+	0.920249i	$0.372014\pi$
$480$		0			0
$481$	−1817.15			−0.172255
$482$		0			0
$483$	1249.24	+	610.227i	0.117687	+	0.0574871i
$484$		0			0
$485$		−	224.311i		−	0.0210009i
$486$		0			0
$487$			1000.78i			0.0931204i	0.998915	+	0.0465602i	$0.0148259\pi$
$487$			1000.78i			0.0931204i	−0.998915	+	0.0465602i	$0.985174\pi$
$488$		0			0
$489$	−2199.89	−	1074.60i	−0.203441	−	0.0993761i
$490$		0			0
$491$	9137.19			0.839828			0.419914	−	0.907564i	$-0.362060\pi$
$491$	9137.19			0.839828			0.419914	+	0.907564i	$0.362060\pi$
$492$		0			0
$493$	2688.81			0.245635
$494$		0			0
$495$	5701.80	−	4443.71i	0.517731	−	0.403495i
$496$		0			0
$497$		−	5873.59i		−	0.530114i
$498$		0			0
$499$			13386.3i			1.20091i	0.799659	+	0.600454i	$0.205013\pi$
$499$			13386.3i			1.20091i	−0.799659	+	0.600454i	$0.794987\pi$
$500$		0			0
$501$	7364.24	−	15075.9i	0.656707	−	1.34440i
$502$		0			0
$503$	21815.8			1.93383			0.966917	−	0.255093i	$-0.0821060\pi$
$503$	21815.8			1.93383			0.966917	+	0.255093i	$0.0821060\pi$
$504$		0			0
$505$	4602.81			0.405589
$506$		0			0
$507$	−385.430	+	789.044i	−0.0337624	+	0.0691177i
$508$		0			0
$509$		−	3193.23i		−	0.278070i	−0.990287	−	0.139035i	$-0.955600\pi$
$509$		−	3193.23i		−	0.278070i	0.990287	−	0.139035i	$-0.0444001\pi$
$510$		0			0
$511$		−	5810.84i		−	0.503046i
$512$		0			0
$513$	1895.67	+	8998.16i	0.163150	+	0.774422i
$514$		0			0
$515$	6198.90			0.530400
$516$		0			0
$517$	−16183.5			−1.37669
$518$		0			0
$519$	4050.51	+	1978.58i	0.342577	+	0.167341i
$520$		0			0
$521$		−	15395.8i		−	1.29463i	−0.762222	−	0.647316i	$-0.775891\pi$
$521$		−	15395.8i		−	1.29463i	0.762222	−	0.647316i	$-0.224109\pi$
$522$		0			0
$523$			17611.8i			1.47249i	0.676715	+	0.736245i	$0.263403\pi$
$523$			17611.8i			1.47249i	−0.676715	+	0.736245i	$0.736597\pi$
$524$		0			0
$525$	6028.19	+	2944.63i	0.501127	+	0.244789i
$526$		0			0
$527$	7788.05			0.643744
$528$		0			0
$529$	−11715.5			−0.962891
$530$		0			0
$531$	−1990.97	−	2554.65i	−0.162714	−	0.208780i
$532$		0			0
$533$			4323.49i			0.351353i
$534$		0			0
$535$		−	3521.88i		−	0.284606i
$536$		0			0
$537$	−3294.97	+	6745.39i	−0.264783	+	0.542058i
$538$		0			0
$539$	−10418.3			−0.832554
$540$		0			0
$541$	−12992.0			−1.03247			−0.516236	−	0.856446i	$-0.672667\pi$
$541$	−12992.0			−1.03247			−0.516236	+	0.856446i	$0.672667\pi$
$542$		0			0
$543$	−9983.90	+	20438.9i	−0.789043	+	1.61531i
$544$		0			0
$545$			7057.64i			0.554709i
$546$		0			0
$547$		−	4116.33i		−	0.321758i	−0.986974	−	0.160879i	$-0.948567\pi$
$547$		−	4116.33i		−	0.321758i	0.986974	−	0.160879i	$-0.0514329\pi$
$548$		0			0
$549$	6250.22	+	8019.76i	0.485889	+	0.623452i
$550$		0			0
$551$	−5322.33			−0.411504
$552$		0			0
$553$	6722.45			0.516940
$554$		0			0
$555$	3093.29	+	1511.00i	0.236582	+	0.115565i
$556$		0			0
$557$			19666.7i			1.49606i	0.663664	+	0.748031i	$0.269000\pi$
$557$			19666.7i			1.49606i	−0.663664	+	0.748031i	$0.731000\pi$
$558$		0			0
$559$		−	1739.64i		−	0.131626i
$560$		0			0
$561$	−8732.98	−	4265.86i	−0.657231	−	0.321042i
$562$		0			0
$563$	7219.27			0.540419			0.270209	−	0.962802i	$-0.412907\pi$
$563$	7219.27			0.540419			0.270209	+	0.962802i	$0.412907\pi$
$564$		0			0
$565$	−10462.7			−0.779062
$566$		0			0
$567$	8901.69	+	2242.18i	0.659323	+	0.166071i
$568$		0			0
$569$			23857.7i			1.75777i	0.477038	+	0.878883i	$0.341711\pi$
$569$			23857.7i			1.75777i	−0.477038	+	0.878883i	$0.658289\pi$
$570$		0			0
$571$			11268.9i			0.825903i	0.910753	+	0.412951i	$0.135502\pi$
$571$			11268.9i			0.825903i	−0.910753	+	0.412951i	$0.864498\pi$
$572$		0			0
$573$	−5261.57	+	10771.4i	−0.383605	+	0.785308i
$574$		0			0
$575$	2178.71			0.158015
$576$		0			0
$577$	−3079.46			−0.222183			−0.111092	−	0.993810i	$-0.535435\pi$
$577$	−3079.46			−0.222183			−0.111092	+	0.993810i	$0.535435\pi$
$578$		0			0
$579$	1846.03	−	3779.16i	0.132502	−	0.271255i
$580$		0			0
$581$		−	11299.2i		−	0.806833i
$582$		0			0
$583$			2973.01i			0.211200i
$584$		0			0
$585$	1312.22	−	1022.68i	0.0927409	−	0.0722778i
$586$		0			0
$587$	−7171.36			−0.504248			−0.252124	−	0.967695i	$-0.581129\pi$
$587$	−7171.36			−0.504248			−0.252124	+	0.967695i	$0.581129\pi$
$588$		0			0
$589$	−15415.9			−1.07844
$590$		0			0
$591$	−1190.24	−	581.406i	−0.0828427	−	0.0404667i
$592$		0			0
$593$			18908.0i			1.30938i	0.755899	+	0.654688i	$0.227200\pi$
$593$			18908.0i			1.30938i	−0.755899	+	0.654688i	$0.772800\pi$
$594$		0			0
$595$			1976.33i			0.136171i
$596$		0			0
$597$	19715.3	+	9630.45i	1.35158	+	0.660215i
$598$		0			0
$599$	−9850.09			−0.671893			−0.335947	−	0.941881i	$-0.609056\pi$
$599$	−9850.09			−0.671893			−0.335947	+	0.941881i	$0.609056\pi$
$600$		0			0
$601$	16503.2			1.12010			0.560048	−	0.828460i	$-0.310783\pi$
$601$	16503.2			1.12010			0.560048	+	0.828460i	$0.310783\pi$
$602$		0			0
$603$	10392.2	−	8099.22i	0.701832	−	0.546975i
$604$		0			0
$605$		−	8815.09i		−	0.592371i
$606$		0			0
$607$			28206.7i			1.88612i	0.332625	+	0.943059i	$0.392066\pi$
$607$			28206.7i			1.88612i	−0.332625	+	0.943059i	$0.607934\pi$
$608$		0			0
$609$	−2331.98	+	4773.98i	−0.155167	+	0.317654i
$610$		0			0
$611$	−3724.47			−0.246605
$612$		0			0
$613$	−14944.3			−0.984656			−0.492328	−	0.870410i	$-0.663854\pi$
$613$	−14944.3			−0.984656			−0.492328	+	0.870410i	$0.663854\pi$
$614$		0			0
$615$	3595.07	−	7359.77i	0.235719	−	0.482560i
$616$		0			0
$617$		−	9362.22i		−	0.610873i	−0.952212	−	0.305436i	$-0.901198\pi$
$617$		−	9362.22i		−	0.610873i	0.952212	−	0.305436i	$-0.0988024\pi$
$618$		0			0
$619$			24425.4i			1.58601i	0.609217	+	0.793003i	$0.291484\pi$
$619$			24425.4i			1.58601i	−0.609217	+	0.793003i	$0.708516\pi$
$620$		0			0
$621$	2917.06	−	614.548i	0.188499	−	0.0397117i
$622$		0			0
$623$	19036.1			1.22418
$624$		0			0
$625$	7705.12			0.493127
$626$		0			0
$627$	17286.3	+	8443.97i	1.10104	+	0.537831i
$628$		0			0
$629$		−	4628.55i		−	0.293406i
$630$		0			0
$631$		−	29822.6i		−	1.88149i	−0.339116	−	0.940744i	$-0.610128\pi$
$631$		−	29822.6i		−	1.88149i	0.339116	−	0.940744i	$-0.389872\pi$
$632$		0			0
$633$	425.235	+	207.717i	0.0267007	+	0.0130427i
$634$		0			0
$635$	−2123.64			−0.132715
$636$		0			0
$637$	−2397.67			−0.149135
$638$		0			0
$639$	−7741.74	−	9933.55i	−0.479277	−	0.614969i
$640$		0			0
$641$		−	6584.30i		−	0.405716i	−0.979208	−	0.202858i	$-0.934977\pi$
$641$		−	6584.30i		−	0.405716i	0.979208	−	0.202858i	$-0.0650230\pi$
$642$		0			0
$643$		−	11578.3i		−	0.710117i	−0.934844	−	0.355059i	$-0.884461\pi$
$643$		−	11578.3i		−	0.710117i	0.934844	−	0.355059i	$-0.115539\pi$
$644$		0			0
$645$	−1446.55	+	2961.36i	−0.0883069	+	0.180780i
$646$		0			0
$647$	−3122.17			−0.189715			−0.0948573	−	0.995491i	$-0.530239\pi$
$647$	−3122.17			−0.189715			−0.0948573	+	0.995491i	$0.530239\pi$
$648$		0			0
$649$	−6776.09			−0.409838
$650$		0			0
$651$	−6754.48	+	13827.7i	−0.406650	+	0.832486i
$652$		0			0
$653$			11624.5i			0.696631i	0.937377	+	0.348315i	$0.113246\pi$
$653$			11624.5i			0.696631i	−0.937377	+	0.348315i	$0.886754\pi$
$654$		0			0
$655$			11805.2i			0.704225i
$656$		0			0
$657$	−7659.03	−	9827.43i	−0.454806	−	0.583569i
$658$		0			0
$659$	−26126.2			−1.54436			−0.772178	−	0.635406i	$-0.780833\pi$
$659$	−26126.2			−1.54436			−0.772178	+	0.635406i	$0.780833\pi$
$660$		0			0
$661$	4266.16			0.251036			0.125518	−	0.992091i	$-0.459941\pi$
$661$	4266.16			0.251036			0.125518	+	0.992091i	$0.459941\pi$
$662$		0			0
$663$	−2009.81	−	981.746i	−0.117730	−	0.0575081i
$664$		0			0
$665$		−	3912.00i		−	0.228122i
$666$		0			0
$667$			1725.42i			0.100162i
$668$		0			0
$669$	2796.52	+	1366.03i	0.161614	+	0.0789445i
$670$		0			0
$671$	21272.0			1.22384
$672$		0			0
$673$	2064.82			0.118266			0.0591330	−	0.998250i	$-0.481166\pi$
$673$	2064.82			0.118266			0.0591330	+	0.998250i	$0.481166\pi$
$674$		0			0
$675$	14076.2	−	2965.48i	0.802657	−	0.169099i
$676$		0			0
$677$			14865.8i			0.843928i	0.906613	+	0.421964i	$0.138659\pi$
$677$			14865.8i			0.843928i	−0.906613	+	0.421964i	$0.861341\pi$
$678$		0			0
$679$		−	595.930i		−	0.0336814i
$680$		0			0
$681$	8766.42	−	17946.5i	0.493289	−	1.00985i
$682$		0			0
$683$	−14535.6			−0.814333			−0.407166	−	0.913354i	$-0.633483\pi$
$683$	−14535.6			−0.814333			−0.407166	+	0.913354i	$0.633483\pi$
$684$		0			0
$685$	−3998.13			−0.223008
$686$		0			0
$687$	−1516.18	+	3103.89i	−0.0842005	+	0.172374i
$688$		0			0
$689$			684.210i			0.0378321i
$690$		0			0
$691$		−	19381.0i		−	1.06699i	−0.845803	−	0.533495i	$-0.820878\pi$
$691$		−	19381.0i		−	1.06699i	0.845803	−	0.533495i	$-0.179122\pi$
$692$		0			0
$693$	15148.0	−	11805.6i	0.830340	−	0.647128i
$694$		0			0
$695$	7460.11			0.407162
$696$		0			0
$697$	−11012.6			−0.598466
$698$		0			0
$699$	15787.6	+	7711.88i	0.854281	+	0.417296i
$700$		0			0
$701$		−	11084.3i		−	0.597215i	−0.954376	−	0.298608i	$-0.903478\pi$
$701$		−	11084.3i		−	0.597215i	0.954376	−	0.298608i	$-0.0965222\pi$
$702$		0			0
$703$			9161.90i			0.491533i
$704$		0			0
$705$	6340.07	+	3096.98i	0.338696	+	0.165445i
$706$		0			0
$707$	12228.3			0.650485
$708$		0			0
$709$	−26108.5			−1.38297			−0.691485	−	0.722391i	$-0.743043\pi$
$709$	−26108.5			−1.38297			−0.691485	+	0.722391i	$0.743043\pi$
$710$		0			0
$711$	11369.2	−	8860.58i	0.599686	−	0.467367i
$712$		0			0
$713$			4997.60i			0.262499i
$714$		0			0
$715$		−	3480.59i		−	0.182051i
$716$		0			0
$717$	7192.88	−	14725.1i	0.374649	−	0.766974i
$718$		0			0
$719$	−34006.7			−1.76389			−0.881945	−	0.471353i	$-0.843766\pi$
$719$	−34006.7			−1.76389			−0.881945	+	0.471353i	$0.843766\pi$
$720$		0			0
$721$	16468.7			0.850659
$722$		0			0
$723$	−14123.7	+	28913.9i	−0.726512	+	1.48730i
$724$		0			0
$725$			8325.94i			0.426507i
$726$		0			0
$727$			17074.9i			0.871075i	0.900171	+	0.435537i	$0.143442\pi$
$727$			17074.9i			0.871075i	−0.900171	+	0.435537i	$0.856558\pi$
$728$		0			0
$729$	18010.1	−	7940.93i	0.915006	−	0.403441i
$730$		0			0
$731$	4431.13			0.224202
$732$		0			0
$733$	−25846.2			−1.30239			−0.651196	−	0.758910i	$-0.725732\pi$
$733$	−25846.2			−1.30239			−0.651196	+	0.758910i	$0.725732\pi$
$734$		0			0
$735$	4081.49	+	1993.71i	0.204827	+	0.100053i
$736$		0			0
$737$		−	27564.9i		−	1.37770i
$738$		0			0
$739$		−	32313.7i		−	1.60849i	−0.594295	−	0.804247i	$-0.702569\pi$
$739$		−	32313.7i		−	1.60849i	0.594295	−	0.804247i	$-0.297431\pi$
$740$		0			0
$741$	3978.29	+	1943.30i	0.197228	+	0.0963413i
$742$		0			0
$743$	19614.0			0.968460			0.484230	−	0.874941i	$-0.339100\pi$
$743$	19614.0			0.968460			0.484230	+	0.874941i	$0.339100\pi$
$744$		0			0
$745$	6004.24			0.295273
$746$		0			0
$747$	−14893.0	−	19109.5i	−0.729460	−	0.935982i
$748$		0			0
$749$		−	9356.60i		−	0.456452i
$750$		0			0
$751$		−	21469.4i		−	1.04318i	−0.853195	−	0.521592i	$-0.825338\pi$
$751$		−	21469.4i		−	1.04318i	0.853195	−	0.521592i	$-0.174662\pi$
$752$		0			0
$753$	−12129.0	+	24830.4i	−0.586995	+	1.20168i
$754$		0			0
$755$	−1581.25			−0.0762219
$756$		0			0
$757$	−4896.90			−0.235113			−0.117557	−	0.993066i	$-0.537506\pi$
$757$	−4896.90			−0.235113			−0.117557	+	0.993066i	$0.537506\pi$
$758$		0			0
$759$	2737.41	−	5603.97i	0.130911	−	0.267999i
$760$		0			0
$761$			6422.14i			0.305916i	0.988233	+	0.152958i	$0.0488800\pi$
$761$			6422.14i			0.305916i	−0.988233	+	0.152958i	$0.951120\pi$
$762$		0			0
$763$			18750.1i			0.889645i
$764$		0			0
$765$	2604.91	+	3342.41i	0.123112	+	0.157967i
$766$		0			0
$767$	−1559.45			−0.0734140
$768$		0			0
$769$	10539.8			0.494246			0.247123	−	0.968984i	$-0.420515\pi$
$769$	10539.8			0.494246			0.247123	+	0.968984i	$0.420515\pi$
$770$		0			0
$771$	−29030.5	−	14180.7i	−1.35604	−	0.662394i
$772$		0			0
$773$			17792.4i			0.827876i	0.910305	+	0.413938i	$0.135847\pi$
$773$			17792.4i			0.827876i	−0.910305	+	0.413938i	$0.864153\pi$
$774$		0			0
$775$			24115.8i			1.11776i
$776$		0			0
$777$	8217.97	+	4014.28i	0.379431	+	0.185343i
$778$		0			0
$779$	21798.6			1.00259
$780$		0			0
$781$	−26348.2			−1.20719
$782$		0			0
$783$	2348.49	+	11147.5i	0.107188	+	0.508787i
$784$		0			0
$785$		−	14255.1i		−	0.648135i
$786$		0			0
$787$		−	39289.0i		−	1.77955i	−0.456404	−	0.889773i	$-0.650863\pi$
$787$		−	39289.0i		−	1.77955i	0.456404	−	0.889773i	$-0.349137\pi$
$788$		0			0
$789$	−1848.98	+	3785.20i	−0.0834289	+	0.170794i
$790$		0			0
$791$	−27796.4			−1.24946
$792$		0			0
$793$	4895.55			0.219226
$794$		0			0
$795$	568.935	−	1164.71i	0.0253812	−	0.0519599i
$796$		0			0
$797$			4765.27i			0.211787i	0.994377	+	0.105894i	$0.0337703\pi$
$797$			4765.27i			0.211787i	−0.994377	+	0.105894i	$0.966230\pi$
$798$		0			0
$799$		−	9486.77i		−	0.420047i
$800$		0			0
$801$	32194.2	−	25090.6i	1.42013	−	1.10678i
$802$		0			0
$803$	−26066.8			−1.14555
$804$		0			0
$805$	−1268.21			−0.0555262
$806$		0			0
$807$	−39074.3	−	19086.9i	−1.70444	−	0.832578i
$808$		0			0
$809$			11056.5i			0.480500i	0.970711	+	0.240250i	$0.0772294\pi$
$809$			11056.5i			0.480500i	−0.970711	+	0.240250i	$0.922771\pi$
$810$		0			0
$811$		−	12248.1i		−	0.530320i	−0.964204	−	0.265160i	$-0.914575\pi$
$811$		−	12248.1i		−	0.530320i	0.964204	−	0.265160i	$-0.0854248\pi$
$812$		0			0
$813$	28086.4	+	13719.5i	1.21160	+	0.591839i
$814$		0			0
$815$	2233.29			0.0959863
$816$		0			0
$817$	−8771.12			−0.375597
$818$		0			0
$819$	3486.17	−	2716.96i	0.148738	−	0.115920i
$820$		0			0
$821$		−	2241.79i		−	0.0952972i	−0.998864	−	0.0476486i	$-0.984827\pi$
$821$		−	2241.79i		−	0.0952972i	0.998864	−	0.0476486i	$-0.0151728\pi$
$822$		0			0
$823$		−	30297.5i		−	1.28324i	−0.767025	−	0.641618i	$-0.778264\pi$
$823$		−	30297.5i		−	1.28324i	0.767025	−	0.641618i	$-0.221736\pi$
$824$		0			0
$825$	13209.3	−	27041.7i	0.557439	−	1.14118i
$826$		0			0
$827$	36267.0			1.52494			0.762472	−	0.647021i	$-0.223985\pi$
$827$	36267.0			1.52494			0.762472	+	0.647021i	$0.223985\pi$
$828$		0			0
$829$	17040.4			0.713916			0.356958	−	0.934120i	$-0.383814\pi$
$829$	17040.4			0.713916			0.356958	+	0.934120i	$0.383814\pi$
$830$		0			0
$831$	−4001.11	+	8191.00i	−0.167024	+	0.341929i
$832$		0			0
$833$		−	6107.21i		−	0.254024i
$834$		0			0
$835$			15304.8i			0.634305i
$836$		0			0
$837$	6802.32	+	32288.4i	0.280911	+	1.33339i
$838$		0			0
$839$	−33418.3			−1.37512			−0.687561	−	0.726126i	$-0.741319\pi$
$839$	−33418.3			−1.37512			−0.687561	+	0.726126i	$0.741319\pi$
$840$		0			0
$841$	17795.3			0.729646
$842$		0			0
$843$	−17317.2	−	8459.03i	−0.707515	−	0.345604i
$844$		0			0
$845$		−	801.024i		−	0.0326107i
$846$		0			0
$847$		−	23419.1i		−	0.950047i
$848$		0			0
$849$	−9962.65	−	4866.52i	−0.402729	−	0.196724i
$850$		0			0
$851$	2970.15			0.119642
$852$		0			0
$853$	−15972.1			−0.641120			−0.320560	−	0.947228i	$-0.603871\pi$
$853$	−15972.1			−0.641120			−0.320560	+	0.947228i	$0.603871\pi$
$854$		0			0
$855$	−5156.25	−	6616.07i	−0.206246	−	0.264637i
$856$		0			0
$857$		−	2020.39i		−	0.0805312i	−0.999189	−	0.0402656i	$-0.987180\pi$
$857$		−	2020.39i		−	0.0805312i	0.999189	−	0.0402656i	$-0.0128204\pi$
$858$		0			0
$859$		−	19850.2i		−	0.788450i	−0.919014	−	0.394225i	$-0.871013\pi$
$859$		−	19850.2i		−	0.788450i	0.919014	−	0.394225i	$-0.128987\pi$
$860$		0			0
$861$	9551.06	−	19552.8i	0.378048	−	0.773933i
$862$		0			0
$863$	6378.24			0.251585			0.125793	−	0.992057i	$-0.459853\pi$
$863$	6378.24			0.251585			0.125793	+	0.992057i	$0.459853\pi$
$864$		0			0
$865$	−4112.00			−0.161633
$866$		0			0
$867$	−8704.17	+	17819.0i	−0.340956	+	0.698000i
$868$		0			0
$869$		−	30156.1i		−	1.17719i
$870$		0			0
$871$		−	6343.80i		−	0.246787i
$872$		0			0
$873$	−785.470	−	1007.85i	−0.0304515	−	0.0390728i
$874$		0			0
$875$	−13580.3			−0.524682
$876$		0			0
$877$	−45258.3			−1.74260			−0.871302	−	0.490746i	$-0.836724\pi$
$877$	−45258.3			−1.74260			−0.871302	+	0.490746i	$0.836724\pi$
$878$		0			0
$879$	32097.8	+	15679.0i	1.23166	+	0.601638i
$880$		0			0
$881$		−	35315.1i		−	1.35051i	−0.737586	−	0.675253i	$-0.764035\pi$
$881$		−	35315.1i		−	1.35051i	0.737586	−	0.675253i	$-0.235965\pi$
$882$		0			0
$883$		−	36739.2i		−	1.40019i	−0.714047	−	0.700097i	$-0.753140\pi$
$883$		−	36739.2i		−	1.40019i	0.714047	−	0.700097i	$-0.246860\pi$
$884$		0			0
$885$	2654.62	+	1296.72i	0.100829	+	0.0492528i
$886$		0			0
$887$	−9433.23			−0.357088			−0.178544	−	0.983932i	$-0.557139\pi$
$887$	−9433.23			−0.357088			−0.178544	+	0.983932i	$0.557139\pi$
$888$		0			0
$889$	−5641.88			−0.212849
$890$		0			0
$891$	10058.1	−	39931.9i	0.378182	−	1.50143i
$892$		0			0
$893$			18778.4i			0.703691i
$894$		0			0
$895$		−	6847.80i		−	0.255751i
$896$		0			0
$897$	629.988	−	1289.70i	0.0234500	−	0.0480065i
$898$		0			0
$899$	−19098.3			−0.708525
$900$		0			0
$901$	−1742.78			−0.0644401
$902$		0			0
$903$	−3843.07	+	7867.46i	−0.141627	+	0.289936i
$904$		0			0
$905$		−	20749.2i		−	0.762128i
$906$		0			0
$907$		−	642.884i		−	0.0235354i	−0.999931	−	0.0117677i	$-0.996254\pi$
$907$		−	642.884i		−	0.0235354i	0.999931	−	0.0117677i	$-0.00374586\pi$
$908$		0			0
$909$	20680.8	−	16117.6i	0.754608	−	0.588106i
$910$		0			0
$911$	35614.3			1.29523			0.647614	−	0.761968i	$-0.275767\pi$
$911$	35614.3			1.29523			0.647614	+	0.761968i	$0.275767\pi$
$912$		0			0
$913$	−50686.9			−1.83734
$914$		0			0
$915$	−8333.59	−	4070.76i	−0.301093	−	0.147077i
$916$		0			0
$917$			31363.0i			1.12944i
$918$		0			0
$919$			29451.9i			1.05716i	0.848883	+	0.528580i	$0.177275\pi$
$919$			29451.9i			1.05716i	−0.848883	+	0.528580i	$0.822725\pi$
$920$		0			0
$921$	−15134.8	−	7392.99i	−0.541486	−	0.264503i
$922$		0			0
$923$	−6063.80			−0.216243
$924$		0			0
$925$	14332.3			0.509453
$926$		0			0
$927$	27852.2	−	21706.7i	0.986823	−	0.769083i
$928$		0			0
$929$			8447.63i			0.298340i	0.988812	+	0.149170i	$0.0476601\pi$
$929$			8447.63i			0.298340i	−0.988812	+	0.149170i	$0.952340\pi$
$930$		0			0
$931$			12088.8i			0.425558i
$932$		0			0
$933$	14389.1	−	29457.2i	0.504908	−	1.03364i
$934$		0			0
$935$	8865.57			0.310091
$936$		0			0
$937$	20096.9			0.700678			0.350339	−	0.936623i	$-0.386066\pi$
$937$	20096.9			0.700678			0.350339	+	0.936623i	$0.386066\pi$
$938$		0			0
$939$	−7503.36	+	15360.7i	−0.260770	+	0.533843i
$940$		0			0
$941$			1850.92i			0.0641214i	0.999486	+	0.0320607i	$0.0102070\pi$
$941$			1850.92i			0.0641214i	−0.999486	+	0.0320607i	$0.989793\pi$
$942$		0			0
$943$		−	7066.78i		−	0.244036i
$944$		0			0
$945$	−8193.64	+	1726.18i	−0.282052	+	0.0594209i
$946$		0			0
$947$	14572.9			0.500060			0.250030	−	0.968238i	$-0.419560\pi$
$947$	14572.9			0.500060			0.250030	+	0.968238i	$0.419560\pi$
$948$		0			0
$949$	−5999.02			−0.205202
$950$		0			0
$951$	−44658.8	−	21814.8i	−1.52278	−	0.743840i
$952$		0			0
$953$			36301.7i			1.23392i	0.786994	+	0.616961i	$0.211636\pi$
$953$			36301.7i			1.23392i	−0.786994	+	0.616961i	$0.788364\pi$
$954$		0			0
$955$		−	10934.9i		−	0.370519i
$956$		0			0
$957$	21415.5	+	10461.0i	0.723370	+	0.353349i
$958$		0			0
$959$	−10621.9			−0.357662
$960$		0			0
$961$	−25526.5			−0.856853
$962$		0			0
$963$	−12332.5	−	15824.1i	−0.412680	−	0.529516i
$964$		0			0
$965$			3836.54i			0.127982i
$966$		0			0
$967$		−	17293.4i		−	0.575095i	−0.957766	−	0.287548i	$-0.907160\pi$
$967$		−	17293.4i		−	0.575095i	0.957766	−	0.287548i	$-0.0928399\pi$
$968$		0			0
$969$	−4949.87	+	10133.3i	−0.164100	+	0.335942i
$970$		0			0
$971$	−9064.99			−0.299598			−0.149799	−	0.988716i	$-0.547863\pi$
$971$	−9064.99			−0.299598			−0.149799	+	0.988716i	$0.547863\pi$
$972$		0			0
$973$	19819.3			0.653009
$974$		0			0
$975$	3039.99	−	6223.40i	0.0998538	−	0.204419i
$976$		0			0
$977$			47549.5i			1.55705i	0.627611	+	0.778527i	$0.284033\pi$
$977$			47549.5i			1.55705i	−0.627611	+	0.778527i	$0.715967\pi$
$978$		0			0
$979$		−	85393.5i		−	2.78773i
$980$		0			0
$981$	24713.7	+	31710.6i	0.804330	+	1.03205i
$982$		0			0
$983$	16093.2			0.522171			0.261086	−	0.965316i	$-0.415920\pi$
$983$	16093.2			0.522171			0.261086	+	0.965316i	$0.415920\pi$
$984$		0			0
$985$	1208.31			0.0390864
$986$		0			0
$987$	16843.7	+	8227.76i	0.543203	+	0.265342i
$988$		0			0
$989$			2843.46i			0.0914225i
$990$		0			0
$991$			39905.4i			1.27915i	0.768729	+	0.639575i	$0.220890\pi$
$991$			39905.4i			1.27915i	−0.768729	+	0.639575i	$0.779110\pi$
$992$		0			0
$993$	21717.0	+	10608.3i	0.694028	+	0.339016i
$994$		0			0
$995$	−20014.6			−0.637694
$996$		0			0
$997$	−17774.9			−0.564632			−0.282316	−	0.959321i	$-0.591103\pi$
$997$	−17774.9			−0.564632			−0.282316	+	0.959321i	$0.591103\pi$
$998$		0			0
$999$	19189.5	−	4042.71i	0.607736	−	0.128034i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-2.28065 + 4.66890i) q^{3} -4.73978i q^{5} -12.5922i q^{7} +(-16.5973 - 21.2962i) q^{9} +O(q^{10})\)	\(q+(-2.28065 + 4.66890i) q^{3} -4.73978i q^{5} -12.5922i q^{7} +(-16.5973 - 21.2962i) q^{9} -56.4872 q^{11} -13.0000 q^{13} +(22.1296 + 10.8098i) q^{15} -33.1129i q^{17} +65.5448i q^{19} +(58.7919 + 28.7184i) q^{21} +21.2486 q^{23} +102.534 q^{25} +(137.283 - 28.9218i) q^{27} +81.2014i q^{29} +235.197i q^{31} +(128.828 - 263.733i) q^{33} -59.6844 q^{35} +139.781 q^{37} +(29.6484 - 60.6957i) q^{39} -332.576i q^{41} +133.819i q^{43} +(-100.940 + 78.6676i) q^{45} +286.498 q^{47} +184.436 q^{49} +(154.601 + 75.5190i) q^{51} -52.6315i q^{53} +267.737i q^{55} +(-306.022 - 149.485i) q^{57} +119.958 q^{59} -376.581 q^{61} +(-268.167 + 208.997i) q^{63} +61.6172i q^{65} +487.985i q^{67} +(-48.4606 + 99.2076i) q^{69} +466.446 q^{71} +461.463 q^{73} +(-233.845 + 478.723i) q^{75} +711.300i q^{77} +533.857i q^{79} +(-178.060 + 706.920i) q^{81} +897.316 q^{83} -156.948 q^{85} +(-379.121 - 185.192i) q^{87} +1511.73i q^{89} +163.699i q^{91} +(-1098.11 - 536.401i) q^{93} +310.668 q^{95} +47.3252 q^{97} +(937.535 + 1202.97i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 54 q^{9}+O(q^{10}) \) 12 * q + 54 * q^9	\( 12 q + 54 q^{9} - 156 q^{13} - 54 q^{21} - 408 q^{25} - 360 q^{33} + 636 q^{37} - 810 q^{45} + 336 q^{49} - 1260 q^{57} + 960 q^{61} - 252 q^{69} + 3216 q^{73} - 2538 q^{81} + 2196 q^{85} - 1116 q^{93} + 4800 q^{97}+O(q^{100}) \) 12 * q + 54 * q^9 - 156 * q^13 - 54 * q^21 - 408 * q^25 - 360 * q^33 + 636 * q^37 - 810 * q^45 + 336 * q^49 - 1260 * q^57 + 960 * q^61 - 252 * q^69 + 3216 * q^73 - 2538 * q^81 + 2196 * q^85 - 1116 * q^93 + 4800 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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