LMFDB - Embedded newform 4012.2.b.b.237.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.12
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77972i q^{3} +1.91545i q^{5} +2.45686i q^{7} -0.167415 q^{9} +O(q^{10})$	$q-1.77972i q^{3} +1.91545i q^{5} +2.45686i q^{7} -0.167415 q^{9} -1.63017i q^{11} +2.53859 q^{13} +3.40896 q^{15} +(-3.23137 + 2.56091i) q^{17} +2.17325 q^{19} +4.37253 q^{21} -2.41510i q^{23} +1.33107 q^{25} -5.04122i q^{27} +4.33492i q^{29} +5.11391i q^{31} -2.90125 q^{33} -4.70599 q^{35} +4.50693i q^{37} -4.51800i q^{39} -4.47471i q^{41} -4.13699 q^{43} -0.320674i q^{45} +11.7078 q^{47} +0.963830 q^{49} +(4.55771 + 5.75094i) q^{51} -14.4131 q^{53} +3.12250 q^{55} -3.86779i q^{57} +1.00000 q^{59} -3.02775i q^{61} -0.411315i q^{63} +4.86254i q^{65} +6.34987 q^{67} -4.29820 q^{69} +6.97310i q^{71} +9.10446i q^{73} -2.36893i q^{75} +4.00511 q^{77} -4.05009i q^{79} -9.47422 q^{81} -14.8812 q^{83} +(-4.90529 - 6.18951i) q^{85} +7.71496 q^{87} +5.92912 q^{89} +6.23698i q^{91} +9.10134 q^{93} +4.16275i q^{95} +11.7962i q^{97} +0.272914i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.77972i		−	1.02752i	−0.857933	−	0.513762i	$-0.828252\pi$
$3$		−	1.77972i		−	1.02752i	0.857933	−	0.513762i	$-0.171748\pi$
$4$		0			0
$5$			1.91545i			0.856614i	0.903633	+	0.428307i	$0.140890\pi$
$5$			1.91545i			0.856614i	−0.903633	+	0.428307i	$0.859110\pi$
$6$		0			0
$7$			2.45686i			0.928607i	0.885676	+	0.464303i	$0.153695\pi$
$7$			2.45686i			0.928607i	−0.885676	+	0.464303i	$0.846305\pi$
$8$		0			0
$9$	−0.167415			−0.0558049
$10$		0			0
$11$		−	1.63017i		−	0.491515i	−0.969331	−	0.245758i	$-0.920963\pi$
$11$		−	1.63017i		−	0.491515i	0.969331	−	0.245758i	$-0.0790367\pi$
$12$		0			0
$13$	2.53859			0.704079			0.352040	−	0.935985i	$-0.385488\pi$
$13$	2.53859			0.704079			0.352040	+	0.935985i	$0.385488\pi$
$14$		0			0
$15$	3.40896			0.880191
$16$		0			0
$17$	−3.23137	+	2.56091i	−0.783722	+	0.621112i
$17$	−3.23137	+	2.56091i	−0.783722	+	0.621112i
$18$		0			0
$19$	2.17325			0.498579			0.249289	−	0.968429i	$-0.419803\pi$
$19$	2.17325			0.498579			0.249289	+	0.968429i	$0.419803\pi$
$20$		0			0
$21$	4.37253			0.954165
$22$		0			0
$23$		−	2.41510i		−	0.503582i	−0.967782	−	0.251791i	$-0.918980\pi$
$23$		−	2.41510i		−	0.503582i	0.967782	−	0.251791i	$-0.0810196\pi$
$24$		0			0
$25$	1.33107			0.266213
$26$		0			0
$27$		−	5.04122i		−	0.970183i
$28$		0			0
$29$			4.33492i			0.804974i	0.915426	+	0.402487i	$0.131854\pi$
$29$			4.33492i			0.804974i	−0.915426	+	0.402487i	$0.868146\pi$
$30$		0			0
$31$			5.11391i			0.918485i	0.888311	+	0.459242i	$0.151879\pi$
$31$			5.11391i			0.918485i	−0.888311	+	0.459242i	$0.848121\pi$
$32$		0			0
$33$	−2.90125			−0.505043
$34$		0			0
$35$	−4.70599			−0.795457
$36$		0			0
$37$			4.50693i			0.740935i	0.928845	+	0.370467i	$0.120802\pi$
$37$			4.50693i			0.740935i	−0.928845	+	0.370467i	$0.879198\pi$
$38$		0			0
$39$		−	4.51800i		−	0.723458i
$40$		0			0
$41$		−	4.47471i		−	0.698833i	−0.936968	−	0.349416i	$-0.886380\pi$
$41$		−	4.47471i		−	0.698833i	0.936968	−	0.349416i	$-0.113620\pi$
$42$		0			0
$43$	−4.13699			−0.630884			−0.315442	−	0.948945i	$-0.602153\pi$
$43$	−4.13699			−0.630884			−0.315442	+	0.948945i	$0.602153\pi$
$44$		0			0
$45$		−	0.320674i		−	0.0478032i
$46$		0			0
$47$	11.7078			1.70775			0.853876	−	0.520476i	$-0.174246\pi$
$47$	11.7078			1.70775			0.853876	+	0.520476i	$0.174246\pi$
$48$		0			0
$49$	0.963830			0.137690
$50$		0			0
$51$	4.55771	+	5.75094i	0.638208	+	0.805293i
$52$		0			0
$53$	−14.4131			−1.97979			−0.989897	−	0.141791i	$-0.954714\pi$
$53$	−14.4131			−1.97979			−0.989897	+	0.141791i	$0.954714\pi$
$54$		0			0
$55$	3.12250			0.421038
$56$		0			0
$57$		−	3.86779i		−	0.512301i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$		−	3.02775i		−	0.387664i	−0.981035	−	0.193832i	$-0.937908\pi$
$61$		−	3.02775i		−	0.387664i	0.981035	−	0.193832i	$-0.0620917\pi$
$62$		0			0
$63$		−	0.411315i		−	0.0518208i
$64$		0			0
$65$			4.86254i			0.603124i
$66$		0			0
$67$	6.34987			0.775760			0.387880	−	0.921710i	$-0.373208\pi$
$67$	6.34987			0.775760			0.387880	+	0.921710i	$0.373208\pi$
$68$		0			0
$69$	−4.29820			−0.517443
$70$		0			0
$71$			6.97310i			0.827555i	0.910378	+	0.413778i	$0.135791\pi$
$71$			6.97310i			0.827555i	−0.910378	+	0.413778i	$0.864209\pi$
$72$		0			0
$73$			9.10446i			1.06560i	0.846242	+	0.532798i	$0.178860\pi$
$73$			9.10446i			1.06560i	−0.846242	+	0.532798i	$0.821140\pi$
$74$		0			0
$75$		−	2.36893i		−	0.273540i
$76$		0			0
$77$	4.00511			0.456424
$78$		0			0
$79$		−	4.05009i		−	0.455670i	−0.973700	−	0.227835i	$-0.926835\pi$
$79$		−	4.05009i		−	0.455670i	0.973700	−	0.227835i	$-0.0731647\pi$
$80$		0			0
$81$	−9.47422			−1.05269
$82$		0			0
$83$	−14.8812			−1.63342			−0.816712	−	0.577045i	$-0.804206\pi$
$83$	−14.8812			−1.63342			−0.816712	+	0.577045i	$0.804206\pi$
$84$		0			0
$85$	−4.90529	−	6.18951i	−0.532053	−	0.671347i
$86$		0			0
$87$	7.71496			0.827130
$88$		0			0
$89$	5.92912			0.628485			0.314243	−	0.949343i	$-0.398249\pi$
$89$	5.92912			0.628485			0.314243	+	0.949343i	$0.398249\pi$
$90$		0			0
$91$			6.23698i			0.653813i
$92$		0			0
$93$	9.10134			0.943765
$94$		0			0
$95$			4.16275i			0.427089i
$96$		0			0
$97$			11.7962i			1.19773i	0.800851	+	0.598863i	$0.204381\pi$
$97$			11.7962i			1.19773i	−0.800851	+	0.598863i	$0.795619\pi$
$98$		0			0
$99$			0.272914i			0.0274289i
$100$		0			0
$101$	8.17839			0.813780			0.406890	−	0.913477i	$-0.366613\pi$
$101$	8.17839			0.813780			0.406890	+	0.913477i	$0.366613\pi$
$102$		0			0
$103$	9.16265			0.902822			0.451411	−	0.892316i	$-0.350921\pi$
$103$	9.16265			0.902822			0.451411	+	0.892316i	$0.350921\pi$
$104$		0			0
$105$			8.37535i			0.817351i
$106$		0			0
$107$			10.0224i			0.968904i	0.874818	+	0.484452i	$0.160981\pi$
$107$			10.0224i			0.968904i	−0.874818	+	0.484452i	$0.839019\pi$
$108$		0			0
$109$			10.4574i			1.00164i	0.865553	+	0.500818i	$0.166967\pi$
$109$			10.4574i			1.00164i	−0.865553	+	0.500818i	$0.833033\pi$
$110$		0			0
$111$	8.02109			0.761328
$112$		0			0
$113$		−	0.100302i		−	0.00943565i	−0.999989	−	0.00471783i	$-0.998498\pi$
$113$		−	0.100302i		−	0.00943565i	0.999989	−	0.00471783i	$-0.00150174\pi$
$114$		0			0
$115$	4.62599			0.431375
$116$		0			0
$117$	−0.424998			−0.0392911
$118$		0			0
$119$	−6.29181	−	7.93902i	−0.576769	−	0.727769i
$120$		0			0
$121$	8.34254			0.758413
$122$		0			0
$123$	−7.96375			−0.718067
$124$		0			0
$125$			12.1268i			1.08466i
$126$		0			0
$127$	7.30255			0.647997			0.323998	−	0.946058i	$-0.394973\pi$
$127$	7.30255			0.647997			0.323998	+	0.946058i	$0.394973\pi$
$128$		0			0
$129$			7.36269i			0.648249i
$130$		0			0
$131$			18.0951i			1.58098i	0.612475	+	0.790490i	$0.290174\pi$
$131$			18.0951i			1.58098i	−0.612475	+	0.790490i	$0.709826\pi$
$132$		0			0
$133$			5.33938i			0.462983i
$134$		0			0
$135$	9.65618			0.831072
$136$		0			0
$137$	−16.0065			−1.36752			−0.683762	−	0.729705i	$-0.739657\pi$
$137$	−16.0065			−1.36752			−0.683762	+	0.729705i	$0.739657\pi$
$138$		0			0
$139$			15.4708i			1.31222i	0.754667	+	0.656108i	$0.227799\pi$
$139$			15.4708i			1.31222i	−0.754667	+	0.656108i	$0.772201\pi$
$140$		0			0
$141$		−	20.8366i		−	1.75476i
$142$		0			0
$143$		−	4.13834i		−	0.346066i
$144$		0			0
$145$	−8.30330			−0.689552
$146$		0			0
$147$		−	1.71535i		−	0.141480i
$148$		0			0
$149$	9.70814			0.795322			0.397661	−	0.917532i	$-0.369822\pi$
$149$	9.70814			0.795322			0.397661	+	0.917532i	$0.369822\pi$
$150$		0			0
$151$	−14.5573			−1.18466			−0.592330	−	0.805696i	$-0.701792\pi$
$151$	−14.5573			−1.18466			−0.592330	+	0.805696i	$0.701792\pi$
$152$		0			0
$153$	0.540978	−	0.428734i	0.0437355	−	0.0346611i
$154$		0			0
$155$	−9.79541			−0.786787
$156$		0			0
$157$	18.0953			1.44416			0.722080	−	0.691810i	$-0.243186\pi$
$157$	18.0953			1.44416			0.722080	+	0.691810i	$0.243186\pi$
$158$		0			0
$159$			25.6514i			2.03428i
$160$		0			0
$161$	5.93356			0.467630
$162$		0			0
$163$		−	9.15635i		−	0.717180i	−0.933495	−	0.358590i	$-0.883258\pi$
$163$		−	9.15635i		−	0.717180i	0.933495	−	0.358590i	$-0.116742\pi$
$164$		0			0
$165$		−	5.55719i		−	0.432627i
$166$		0			0
$167$		−	22.3032i		−	1.72587i	−0.505314	−	0.862935i	$-0.668623\pi$
$167$		−	22.3032i		−	1.72587i	0.505314	−	0.862935i	$-0.331377\pi$
$168$		0			0
$169$	−6.55554			−0.504272
$170$		0			0
$171$	−0.363834			−0.0278231
$172$		0			0
$173$			6.25558i			0.475603i	0.971314	+	0.237802i	$0.0764268\pi$
$173$			6.25558i			0.475603i	−0.971314	+	0.237802i	$0.923573\pi$
$174$		0			0
$175$			3.27025i			0.247207i
$176$		0			0
$177$		−	1.77972i		−	0.133772i
$178$		0			0
$179$	24.4193			1.82518			0.912590	−	0.408875i	$-0.134079\pi$
$179$	24.4193			1.82518			0.912590	+	0.408875i	$0.134079\pi$
$180$		0			0
$181$			13.2410i			0.984194i	0.870540	+	0.492097i	$0.163770\pi$
$181$			13.2410i			0.984194i	−0.870540	+	0.492097i	$0.836230\pi$
$182$		0			0
$183$	−5.38856			−0.398334
$184$		0			0
$185$	−8.63278			−0.634695
$186$		0			0
$187$	4.17472	+	5.26768i	0.305286	+	0.385211i
$188$		0			0
$189$	12.3856			0.900918
$190$		0			0
$191$	−8.22406			−0.595072			−0.297536	−	0.954711i	$-0.596165\pi$
$191$	−8.22406			−0.595072			−0.297536	+	0.954711i	$0.596165\pi$
$192$		0			0
$193$			4.01119i			0.288732i	0.989524	+	0.144366i	$0.0461142\pi$
$193$			4.01119i			0.288732i	−0.989524	+	0.144366i	$0.953886\pi$
$194$		0			0
$195$	8.65398			0.619724
$196$		0			0
$197$			3.07376i			0.218996i	0.993987	+	0.109498i	$0.0349244\pi$
$197$			3.07376i			0.218996i	−0.993987	+	0.109498i	$0.965076\pi$
$198$		0			0
$199$		−	9.20338i		−	0.652411i	−0.945299	−	0.326205i	$-0.894230\pi$
$199$		−	9.20338i		−	0.652411i	0.945299	−	0.326205i	$-0.105770\pi$
$200$		0			0
$201$		−	11.3010i		−	0.797112i
$202$		0			0
$203$	−10.6503			−0.747504
$204$		0			0
$205$	8.57107			0.598629
$206$		0			0
$207$			0.404322i			0.0281023i
$208$		0			0
$209$		−	3.54278i		−	0.245059i
$210$		0			0
$211$			18.7010i			1.28743i	0.765265	+	0.643715i	$0.222608\pi$
$211$			18.7010i			1.28743i	−0.765265	+	0.643715i	$0.777392\pi$
$212$		0			0
$213$	12.4102			0.850333
$214$		0			0
$215$		−	7.92417i		−	0.540424i
$216$		0			0
$217$	−12.5642			−0.852911
$218$		0			0
$219$	16.2034			1.09493
$220$		0			0
$221$	−8.20313	+	6.50112i	−0.551802	+	0.437312i
$222$		0			0
$223$	17.6784			1.18383			0.591917	−	0.805999i	$-0.298371\pi$
$223$	17.6784			1.18383			0.591917	+	0.805999i	$0.298371\pi$
$224$		0			0
$225$	−0.222840			−0.0148560
$226$		0			0
$227$		−	19.7808i		−	1.31290i	−0.754369	−	0.656450i	$-0.772057\pi$
$227$		−	19.7808i		−	1.31290i	0.754369	−	0.656450i	$-0.227943\pi$
$228$		0			0
$229$	−18.4331			−1.21809			−0.609046	−	0.793135i	$-0.708448\pi$
$229$	−18.4331			−1.21809			−0.609046	+	0.793135i	$0.708448\pi$
$230$		0			0
$231$		−	7.12798i		−	0.468987i
$232$		0			0
$233$			13.8125i			0.904884i	0.891794	+	0.452442i	$0.149447\pi$
$233$			13.8125i			0.904884i	−0.891794	+	0.452442i	$0.850553\pi$
$234$		0			0
$235$			22.4256i			1.46288i
$236$		0			0
$237$	−7.20803			−0.468212
$238$		0			0
$239$	18.4220			1.19162			0.595810	−	0.803125i	$-0.296831\pi$
$239$	18.4220			1.19162			0.595810	+	0.803125i	$0.296831\pi$
$240$		0			0
$241$		−	24.5213i		−	1.57955i	−0.613394	−	0.789777i	$-0.710196\pi$
$241$		−	24.5213i		−	1.57955i	0.613394	−	0.789777i	$-0.289804\pi$
$242$		0			0
$243$			1.73783i			0.111482i
$244$		0			0
$245$			1.84616i			0.117947i
$246$		0			0
$247$	5.51701			0.351039
$248$		0			0
$249$			26.4844i			1.67838i
$250$		0			0
$251$	17.5505			1.10777			0.553887	−	0.832592i	$-0.313144\pi$
$251$	17.5505			1.10777			0.553887	+	0.832592i	$0.313144\pi$
$252$		0			0
$253$	−3.93702			−0.247518
$254$		0			0
$255$	−11.0156	+	8.73005i	−0.689824	+	0.546697i
$256$		0			0
$257$	−28.3471			−1.76825			−0.884123	−	0.467255i	$-0.845243\pi$
$257$	−28.3471			−1.76825			−0.884123	+	0.467255i	$0.845243\pi$
$258$		0			0
$259$	−11.0729			−0.688037
$260$		0			0
$261$		−	0.725729i		−	0.0449215i
$262$		0			0
$263$	27.9897			1.72592			0.862961	−	0.505271i	$-0.168607\pi$
$263$	27.9897			1.72592			0.862961	+	0.505271i	$0.168607\pi$
$264$		0			0
$265$		−	27.6075i		−	1.69592i
$266$		0			0
$267$		−	10.5522i		−	0.645784i
$268$		0			0
$269$			6.44275i			0.392821i	0.980522	+	0.196411i	$0.0629285\pi$
$269$			6.44275i			0.392821i	−0.980522	+	0.196411i	$0.937071\pi$
$270$		0			0
$271$	27.9996			1.70085			0.850426	−	0.526094i	$-0.176344\pi$
$271$	27.9996			1.70085			0.850426	+	0.526094i	$0.176344\pi$
$272$		0			0
$273$	11.1001			0.671808
$274$		0			0
$275$		−	2.16987i		−	0.130848i
$276$		0			0
$277$			7.48484i			0.449720i	0.974391	+	0.224860i	$0.0721925\pi$
$277$			7.48484i			0.449720i	−0.974391	+	0.224860i	$0.927807\pi$
$278$		0			0
$279$		−	0.856143i		−	0.0512559i
$280$		0			0
$281$	0.510434			0.0304499			0.0152250	−	0.999884i	$-0.495154\pi$
$281$	0.510434			0.0304499			0.0152250	+	0.999884i	$0.495154\pi$
$282$		0			0
$283$			31.2509i			1.85767i	0.370489	+	0.928837i	$0.379190\pi$
$283$			31.2509i			1.85767i	−0.370489	+	0.928837i	$0.620810\pi$
$284$		0			0
$285$	7.40854			0.438844
$286$		0			0
$287$	10.9937			0.648940
$288$		0			0
$289$	3.88347	−	16.5505i	0.228439	−	0.973558i
$290$		0			0
$291$	20.9940			1.23069
$292$		0			0
$293$	−11.2697			−0.658383			−0.329192	−	0.944263i	$-0.606776\pi$
$293$	−11.2697			−0.658383			−0.329192	+	0.944263i	$0.606776\pi$
$294$		0			0
$295$			1.91545i			0.111522i
$296$		0			0
$297$	−8.21805			−0.476859
$298$		0			0
$299$		−	6.13095i		−	0.354562i
$300$		0			0
$301$		−	10.1640i		−	0.585843i
$302$		0			0
$303$		−	14.5553i		−	0.836178i
$304$		0			0
$305$	5.79950			0.332078
$306$		0			0
$307$	−2.78761			−0.159097			−0.0795486	−	0.996831i	$-0.525348\pi$
$307$	−2.78761			−0.159097			−0.0795486	+	0.996831i	$0.525348\pi$
$308$		0			0
$309$		−	16.3070i		−	0.927671i
$310$		0			0
$311$		−	27.7573i		−	1.57397i	−0.616969	−	0.786987i	$-0.711640\pi$
$311$		−	27.7573i		−	1.57397i	0.616969	−	0.786987i	$-0.288360\pi$
$312$		0			0
$313$		−	5.64200i		−	0.318905i	−0.987206	−	0.159452i	$-0.949027\pi$
$313$		−	5.64200i		−	0.318905i	0.987206	−	0.159452i	$-0.0509728\pi$
$314$		0			0
$315$	0.787851			0.0443904
$316$		0			0
$317$		−	5.43933i		−	0.305503i	−0.988265	−	0.152752i	$-0.951187\pi$
$317$		−	5.43933i		−	0.305503i	0.988265	−	0.152752i	$-0.0488134\pi$
$318$		0			0
$319$	7.06666			0.395657
$320$		0			0
$321$	17.8371			0.995572
$322$		0			0
$323$	−7.02258	+	5.56551i	−0.390747	+	0.309673i
$324$		0			0
$325$	3.37904			0.187435
$326$		0			0
$327$	18.6113			1.02920
$328$		0			0
$329$			28.7644i			1.58583i
$330$		0			0
$331$	7.87894			0.433066			0.216533	−	0.976275i	$-0.430525\pi$
$331$	7.87894			0.433066			0.216533	+	0.976275i	$0.430525\pi$
$332$		0			0
$333$		−	0.754526i		−	0.0413478i
$334$		0			0
$335$			12.1628i			0.664526i
$336$		0			0
$337$		−	33.0543i		−	1.80058i	−0.435288	−	0.900291i	$-0.643353\pi$
$337$		−	33.0543i		−	1.80058i	0.435288	−	0.900291i	$-0.356647\pi$
$338$		0			0
$339$	−0.178510			−0.00969536
$340$		0			0
$341$	8.33654			0.451449
$342$		0			0
$343$			19.5660i			1.05647i
$344$		0			0
$345$		−	8.23297i		−	0.443248i
$346$		0			0
$347$		−	36.4162i		−	1.95492i	−0.211110	−	0.977462i	$-0.567708\pi$
$347$		−	36.4162i		−	1.95492i	0.211110	−	0.977462i	$-0.432292\pi$
$348$		0			0
$349$	−13.1050			−0.701496			−0.350748	−	0.936470i	$-0.614073\pi$
$349$	−13.1050			−0.701496			−0.350748	+	0.936470i	$0.614073\pi$
$350$		0			0
$351$		−	12.7976i		−	0.683086i
$352$		0			0
$353$	−4.53985			−0.241632			−0.120816	−	0.992675i	$-0.538551\pi$
$353$	−4.53985			−0.241632			−0.120816	+	0.992675i	$0.538551\pi$
$354$		0			0
$355$	−13.3566			−0.708895
$356$		0			0
$357$	−14.1293	+	11.1977i	−0.747800	+	0.592644i
$358$		0			0
$359$	−10.2595			−0.541475			−0.270738	−	0.962653i	$-0.587268\pi$
$359$	−10.2595			−0.541475			−0.270738	+	0.962653i	$0.587268\pi$
$360$		0			0
$361$	−14.2770			−0.751419
$362$		0			0
$363$		−	14.8474i		−	0.779287i
$364$		0			0
$365$	−17.4391			−0.912804
$366$		0			0
$367$		−	15.5045i		−	0.809331i	−0.914465	−	0.404665i	$-0.867388\pi$
$367$		−	15.5045i		−	0.809331i	0.914465	−	0.404665i	$-0.132612\pi$
$368$		0			0
$369$			0.749132i			0.0389983i
$370$		0			0
$371$		−	35.4110i		−	1.83845i
$372$		0			0
$373$	11.6689			0.604192			0.302096	−	0.953277i	$-0.402314\pi$
$373$	11.6689			0.604192			0.302096	+	0.953277i	$0.402314\pi$
$374$		0			0
$375$	21.5824			1.11451
$376$		0			0
$377$			11.0046i			0.566766i
$378$		0			0
$379$		−	11.1980i		−	0.575203i	−0.957750	−	0.287601i	$-0.907142\pi$
$379$		−	11.1980i		−	0.575203i	0.957750	−	0.287601i	$-0.0928579\pi$
$380$		0			0
$381$		−	12.9965i		−	0.665832i
$382$		0			0
$383$	−1.55995			−0.0797095			−0.0398548	−	0.999205i	$-0.512690\pi$
$383$	−1.55995			−0.0797095			−0.0398548	+	0.999205i	$0.512690\pi$
$384$		0			0
$385$			7.67156i			0.390979i
$386$		0			0
$387$	0.692592			0.0352064
$388$		0			0
$389$	21.5522			1.09274			0.546369	−	0.837545i	$-0.316010\pi$
$389$	21.5522			1.09274			0.546369	+	0.837545i	$0.316010\pi$
$390$		0			0
$391$	6.18485	+	7.80406i	0.312781	+	0.394668i
$392$		0			0
$393$	32.2043			1.62449
$394$		0			0
$395$	7.75772			0.390333
$396$		0			0
$397$			3.56484i			0.178914i	0.995991	+	0.0894570i	$0.0285132\pi$
$397$			3.56484i			0.178914i	−0.995991	+	0.0894570i	$0.971487\pi$
$398$		0			0
$399$	9.50263			0.475726
$400$		0			0
$401$		−	37.3067i		−	1.86301i	−0.363729	−	0.931505i	$-0.618497\pi$
$401$		−	37.3067i		−	1.86301i	0.363729	−	0.931505i	$-0.381503\pi$
$402$		0			0
$403$			12.9821i			0.646686i
$404$		0			0
$405$		−	18.1474i		−	0.901749i
$406$		0			0
$407$	7.34707			0.364181
$408$		0			0
$409$	19.3836			0.958457			0.479228	−	0.877690i	$-0.340917\pi$
$409$	19.3836			0.958457			0.479228	+	0.877690i	$0.340917\pi$
$410$		0			0
$411$			28.4871i			1.40516i
$412$		0			0
$413$			2.45686i			0.120894i
$414$		0			0
$415$		−	28.5041i		−	1.39921i
$416$		0			0
$417$	27.5338			1.34833
$418$		0			0
$419$			5.51292i			0.269324i	0.990892	+	0.134662i	$0.0429948\pi$
$419$			5.51292i			0.269324i	−0.990892	+	0.134662i	$0.957005\pi$
$420$		0			0
$421$	−13.7120			−0.668282			−0.334141	−	0.942523i	$-0.608446\pi$
$421$	−13.7120			−0.668282			−0.334141	+	0.942523i	$0.608446\pi$
$422$		0			0
$423$	−1.96005			−0.0953009
$424$		0			0
$425$	−4.30116	+	3.40874i	−0.208637	+	0.165348i
$426$		0			0
$427$	7.43877			0.359988
$428$		0			0
$429$	−7.36511			−0.355591
$430$		0			0
$431$			10.4744i			0.504533i	0.967658	+	0.252267i	$0.0811760\pi$
$431$			10.4744i			0.504533i	−0.967658	+	0.252267i	$0.918824\pi$
$432$		0			0
$433$	−17.6141			−0.846481			−0.423240	−	0.906017i	$-0.639107\pi$
$433$	−17.6141			−0.846481			−0.423240	+	0.906017i	$0.639107\pi$
$434$		0			0
$435$			14.7776i			0.708531i
$436$		0			0
$437$		−	5.24862i		−	0.251075i
$438$		0			0
$439$		−	21.4576i		−	1.02412i	−0.858951	−	0.512059i	$-0.828883\pi$
$439$		−	21.4576i		−	1.02412i	0.858951	−	0.512059i	$-0.171117\pi$
$440$		0			0
$441$	−0.161359			−0.00768377
$442$		0			0
$443$	23.5184			1.11739			0.558696	−	0.829372i	$-0.311302\pi$
$443$	23.5184			1.11739			0.558696	+	0.829372i	$0.311302\pi$
$444$		0			0
$445$			11.3569i			0.538369i
$446$		0			0
$447$		−	17.2778i		−	0.817212i
$448$		0			0
$449$			37.1905i			1.75513i	0.479461	+	0.877563i	$0.340832\pi$
$449$			37.1905i			1.75513i	−0.479461	+	0.877563i	$0.659168\pi$
$450$		0			0
$451$	−7.29455			−0.343487
$452$		0			0
$453$			25.9080i			1.21727i
$454$		0			0
$455$	−11.9466			−0.560065
$456$		0			0
$457$	14.0684			0.658094			0.329047	−	0.944314i	$-0.393273\pi$
$457$	14.0684			0.658094			0.329047	+	0.944314i	$0.393273\pi$
$458$		0			0
$459$	12.9101	+	16.2900i	0.602592	+	0.760353i
$460$		0			0
$461$	25.6177			1.19313			0.596567	−	0.802564i	$-0.296531\pi$
$461$	25.6177			1.19313			0.596567	+	0.802564i	$0.296531\pi$
$462$		0			0
$463$	−14.8828			−0.691663			−0.345832	−	0.938297i	$-0.612403\pi$
$463$	−14.8828			−0.691663			−0.345832	+	0.938297i	$0.612403\pi$
$464$		0			0
$465$			17.4331i			0.808442i
$466$		0			0
$467$	−16.1674			−0.748139			−0.374070	−	0.927401i	$-0.622038\pi$
$467$	−16.1674			−0.748139			−0.374070	+	0.927401i	$0.622038\pi$
$468$		0			0
$469$			15.6008i			0.720376i
$470$		0			0
$471$		−	32.2046i		−	1.48391i
$472$		0			0
$473$			6.74399i			0.310089i
$474$		0			0
$475$	2.89274			0.132728
$476$		0			0
$477$	2.41297			0.110482
$478$		0			0
$479$			31.9120i			1.45809i	0.684463	+	0.729047i	$0.260037\pi$
$479$			31.9120i			1.45809i	−0.684463	+	0.729047i	$0.739963\pi$
$480$		0			0
$481$			11.4413i			0.521677i
$482$		0			0
$483$		−	10.5601i		−	0.480501i
$484$		0			0
$485$	−22.5951			−1.02599
$486$		0			0
$487$			8.32478i			0.377232i	0.982051	+	0.188616i	$0.0604001\pi$
$487$			8.32478i			0.377232i	−0.982051	+	0.188616i	$0.939600\pi$
$488$		0			0
$489$	−16.2958			−0.736920
$490$		0			0
$491$	−41.6313			−1.87880			−0.939398	−	0.342830i	$-0.888615\pi$
$491$	−41.6313			−1.87880			−0.939398	+	0.342830i	$0.888615\pi$
$492$		0			0
$493$	−11.1013	−	14.0077i	−0.499979	−	0.630876i
$494$		0			0
$495$	−0.522753			−0.0234960
$496$		0			0
$497$	−17.1320			−0.768473
$498$		0			0
$499$			44.1304i			1.97555i	0.155899	+	0.987773i	$0.450173\pi$
$499$			44.1304i			1.97555i	−0.155899	+	0.987773i	$0.549827\pi$
$500$		0			0
$501$	−39.6935			−1.77337
$502$		0			0
$503$			12.3377i			0.550113i	0.961428	+	0.275056i	$0.0886966\pi$
$503$			12.3377i			0.550113i	−0.961428	+	0.275056i	$0.911303\pi$
$504$		0			0
$505$			15.6653i			0.697095i
$506$		0			0
$507$			11.6670i			0.518151i
$508$		0			0
$509$	−33.3709			−1.47914			−0.739570	−	0.673080i	$-0.764971\pi$
$509$	−33.3709			−1.47914			−0.739570	+	0.673080i	$0.764971\pi$
$510$		0			0
$511$	−22.3684			−0.989520
$512$		0			0
$513$		−	10.9558i		−	0.483712i
$514$		0			0
$515$			17.5506i			0.773370i
$516$		0			0
$517$		−	19.0857i		−	0.839386i
$518$		0			0
$519$	11.1332			0.488694
$520$		0			0
$521$			34.9617i			1.53170i	0.643019	+	0.765850i	$0.277682\pi$
$521$			34.9617i			1.53170i	−0.643019	+	0.765850i	$0.722318\pi$
$522$		0			0
$523$	−8.28760			−0.362391			−0.181196	−	0.983447i	$-0.557997\pi$
$523$	−8.28760			−0.362391			−0.181196	+	0.983447i	$0.557997\pi$
$524$		0			0
$525$	5.82013			0.254011
$526$		0			0
$527$	−13.0963	−	16.5249i	−0.570482	−	0.719836i
$528$		0			0
$529$	17.1673			0.746405
$530$		0			0
$531$	−0.167415			−0.00726518
$532$		0			0
$533$		−	11.3595i		−	0.492034i
$534$		0			0
$535$	−19.1974			−0.829977
$536$		0			0
$537$		−	43.4595i		−	1.87542i
$538$		0			0
$539$		−	1.57121i		−	0.0676767i
$540$		0			0
$541$		−	37.1610i		−	1.59768i	−0.601547	−	0.798838i	$-0.705449\pi$
$541$		−	37.1610i		−	1.59768i	0.601547	−	0.798838i	$-0.294551\pi$
$542$		0			0
$543$	23.5653			1.01128
$544$		0			0
$545$	−20.0306			−0.858015
$546$		0			0
$547$			36.8004i			1.57347i	0.617290	+	0.786736i	$0.288231\pi$
$547$			36.8004i			1.57347i	−0.617290	+	0.786736i	$0.711769\pi$
$548$		0			0
$549$			0.506890i			0.0216336i
$550$		0			0
$551$			9.42088i			0.401343i
$552$		0			0
$553$	9.95050			0.423138
$554$		0			0
$555$			15.3640i			0.652164i
$556$		0			0
$557$	39.5920			1.67757			0.838783	−	0.544466i	$-0.183268\pi$
$557$	39.5920			1.67757			0.838783	+	0.544466i	$0.183268\pi$
$558$		0			0
$559$	−10.5021			−0.444193
$560$		0			0
$561$	9.37501	−	7.42985i	0.395813	−	0.313689i
$562$		0			0
$563$	−22.7253			−0.957757			−0.478879	−	0.877881i	$-0.658957\pi$
$563$	−22.7253			−0.957757			−0.478879	+	0.877881i	$0.658957\pi$
$564$		0			0
$565$	0.192124			0.00808271
$566$		0			0
$567$		−	23.2768i		−	0.977535i
$568$		0			0
$569$	−5.98876			−0.251062			−0.125531	−	0.992090i	$-0.540063\pi$
$569$	−5.98876			−0.251062			−0.125531	+	0.992090i	$0.540063\pi$
$570$		0			0
$571$			4.11101i			0.172041i	0.996293	+	0.0860203i	$0.0274150\pi$
$571$			4.11101i			0.172041i	−0.996293	+	0.0860203i	$0.972585\pi$
$572$		0			0
$573$			14.6365i			0.611450i
$574$		0			0
$575$		−	3.21465i		−	0.134060i
$576$		0			0
$577$	−8.01381			−0.333619			−0.166810	−	0.985989i	$-0.553347\pi$
$577$	−8.01381			−0.333619			−0.166810	+	0.985989i	$0.553347\pi$
$578$		0			0
$579$	7.13881			0.296679
$580$		0			0
$581$		−	36.5611i		−	1.51681i
$582$		0			0
$583$			23.4958i			0.973098i
$584$		0			0
$585$		−	0.814060i		−	0.0336573i
$586$		0			0
$587$	−38.3009			−1.58085			−0.790423	−	0.612562i	$-0.790139\pi$
$587$	−38.3009			−1.58085			−0.790423	+	0.612562i	$0.790139\pi$
$588$		0			0
$589$			11.1138i			0.457937i
$590$		0			0
$591$	5.47044			0.225024
$592$		0			0
$593$	−17.0545			−0.700345			−0.350172	−	0.936685i	$-0.613877\pi$
$593$	−17.0545			−0.700345			−0.350172	+	0.936685i	$0.613877\pi$
$594$		0			0
$595$	15.2068	−	12.0516i	0.623417	−	0.494068i
$596$		0			0
$597$	−16.3795			−0.670367
$598$		0			0
$599$	35.7691			1.46149			0.730743	−	0.682653i	$-0.239174\pi$
$599$	35.7691			1.46149			0.730743	+	0.682653i	$0.239174\pi$
$600$		0			0
$601$		−	4.22983i		−	0.172538i	−0.996272	−	0.0862692i	$-0.972505\pi$
$601$		−	4.22983i		−	0.172538i	0.996272	−	0.0862692i	$-0.0274945\pi$
$602$		0			0
$603$	−1.06306			−0.0432912
$604$		0			0
$605$			15.9797i			0.649667i
$606$		0			0
$607$		−	23.5365i		−	0.955316i	−0.878546	−	0.477658i	$-0.841486\pi$
$607$		−	23.5365i		−	0.955316i	0.878546	−	0.477658i	$-0.158514\pi$
$608$		0			0
$609$			18.9546i			0.768078i
$610$		0			0
$611$	29.7213			1.20239
$612$		0			0
$613$	−37.4083			−1.51091			−0.755453	−	0.655202i	$-0.772583\pi$
$613$	−37.4083			−1.51091			−0.755453	+	0.655202i	$0.772583\pi$
$614$		0			0
$615$		−	15.2541i		−	0.615106i
$616$		0			0
$617$		−	7.09868i		−	0.285782i	−0.989738	−	0.142891i	$-0.954360\pi$
$617$		−	7.09868i		−	0.285782i	0.989738	−	0.142891i	$-0.0456398\pi$
$618$		0			0
$619$		−	9.14215i		−	0.367454i	−0.982977	−	0.183727i	$-0.941184\pi$
$619$		−	9.14215i		−	0.367454i	0.982977	−	0.183727i	$-0.0588162\pi$
$620$		0			0
$621$	−12.1750			−0.488567
$622$		0			0
$623$			14.5670i			0.583616i
$624$		0			0
$625$	−16.5729			−0.662917
$626$		0			0
$627$	−6.30516			−0.251804
$628$		0			0
$629$	−11.5419	−	14.5635i	−0.460204	−	0.580687i
$630$		0			0
$631$	−19.0906			−0.759985			−0.379992	−	0.924990i	$-0.624073\pi$
$631$	−19.0906			−0.759985			−0.379992	+	0.924990i	$0.624073\pi$
$632$		0			0
$633$	33.2826			1.32287
$634$		0			0
$635$			13.9876i			0.555083i
$636$		0			0
$637$	2.44677			0.0969447
$638$		0			0
$639$		−	1.16740i		−	0.0461816i
$640$		0			0
$641$			14.0504i			0.554956i	0.960732	+	0.277478i	$0.0894985\pi$
$641$			14.0504i			0.554956i	−0.960732	+	0.277478i	$0.910501\pi$
$642$		0			0
$643$		−	26.8748i		−	1.05984i	−0.848048	−	0.529919i	$-0.822222\pi$
$643$		−	26.8748i		−	1.05984i	0.848048	−	0.529919i	$-0.177778\pi$
$644$		0			0
$645$	−14.1028			−0.555299
$646$		0			0
$647$	28.7167			1.12897			0.564485	−	0.825443i	$-0.309075\pi$
$647$	28.7167			1.12897			0.564485	+	0.825443i	$0.309075\pi$
$648$		0			0
$649$		−	1.63017i		−	0.0639898i
$650$		0			0
$651$			22.3607i			0.876386i
$652$		0			0
$653$			17.4582i			0.683192i	0.939847	+	0.341596i	$0.110968\pi$
$653$			17.4582i			0.683192i	−0.939847	+	0.341596i	$0.889032\pi$
$654$		0			0
$655$	−34.6603			−1.35429
$656$		0			0
$657$		−	1.52422i		−	0.0594655i
$658$		0			0
$659$	−19.5925			−0.763214			−0.381607	−	0.924325i	$-0.624629\pi$
$659$	−19.5925			−0.763214			−0.381607	+	0.924325i	$0.624629\pi$
$660$		0			0
$661$	−9.43303			−0.366902			−0.183451	−	0.983029i	$-0.558727\pi$
$661$	−9.43303			−0.366902			−0.183451	+	0.983029i	$0.558727\pi$
$662$		0			0
$663$	11.5702	+	14.5993i	0.449349	+	0.566990i
$664$		0			0
$665$	−10.2273			−0.396598
$666$		0			0
$667$	10.4692			0.405371
$668$		0			0
$669$		−	31.4627i		−	1.21642i
$670$		0			0
$671$	−4.93576			−0.190543
$672$		0			0
$673$			44.4240i			1.71242i	0.516627	+	0.856211i	$0.327187\pi$
$673$			44.4240i			1.71242i	−0.516627	+	0.856211i	$0.672813\pi$
$674$		0			0
$675$		−	6.71019i		−	0.258275i
$676$		0			0
$677$		−	32.6181i		−	1.25361i	−0.779175	−	0.626807i	$-0.784361\pi$
$677$		−	32.6181i		−	1.25361i	0.779175	−	0.626807i	$-0.215639\pi$
$678$		0			0
$679$	−28.9817			−1.11222
$680$		0			0
$681$	−35.2044			−1.34904
$682$		0			0
$683$		−	17.6943i		−	0.677055i	−0.940956	−	0.338528i	$-0.890071\pi$
$683$		−	17.6943i		−	0.677055i	0.940956	−	0.338528i	$-0.109929\pi$
$684$		0			0
$685$		−	30.6595i		−	1.17144i
$686$		0			0
$687$			32.8058i			1.25162i
$688$		0			0
$689$	−36.5891			−1.39393
$690$		0			0
$691$		−	20.6096i		−	0.784027i	−0.919959	−	0.392013i	$-0.871779\pi$
$691$		−	20.6096i		−	0.784027i	0.919959	−	0.392013i	$-0.128221\pi$
$692$		0			0
$693$	−0.670513			−0.0254707
$694$		0			0
$695$	−29.6335			−1.12406
$696$		0			0
$697$	11.4593	+	14.4594i	0.434053	+	0.547690i
$698$		0			0
$699$	24.5824			0.929790
$700$		0			0
$701$	−48.1164			−1.81733			−0.908665	−	0.417525i	$-0.862898\pi$
$701$	−48.1164			−1.81733			−0.908665	+	0.417525i	$0.862898\pi$
$702$		0			0
$703$			9.79471i			0.369414i
$704$		0			0
$705$	39.9113			1.50315
$706$		0			0
$707$			20.0932i			0.755681i
$708$		0			0
$709$			2.20794i			0.0829209i	0.999140	+	0.0414605i	$0.0132011\pi$
$709$			2.20794i			0.0829209i	−0.999140	+	0.0414605i	$0.986799\pi$
$710$		0			0
$711$			0.678043i			0.0254286i
$712$		0			0
$713$	12.3506			0.462533
$714$		0			0
$715$	7.92677			0.296445
$716$		0			0
$717$		−	32.7861i		−	1.22442i
$718$		0			0
$719$		−	38.7086i		−	1.44359i	−0.692109	−	0.721793i	$-0.743318\pi$
$719$		−	38.7086i		−	1.44359i	0.692109	−	0.721793i	$-0.256682\pi$
$720$		0			0
$721$			22.5114i			0.838367i
$722$		0			0
$723$	−43.6411			−1.62303
$724$		0			0
$725$			5.77006i			0.214295i
$726$		0			0
$727$	36.8702			1.36744			0.683719	−	0.729745i	$-0.260361\pi$
$727$	36.8702			1.36744			0.683719	+	0.729745i	$0.260361\pi$
$728$		0			0
$729$	−25.3298			−0.938141
$730$		0			0
$731$	13.3681	−	10.5945i	0.494438	−	0.391850i
$732$		0			0
$733$	−8.96503			−0.331131			−0.165565	−	0.986199i	$-0.552945\pi$
$733$	−8.96503			−0.331131			−0.165565	+	0.986199i	$0.552945\pi$
$734$		0			0
$735$	3.28566			0.121193
$736$		0			0
$737$		−	10.3514i		−	0.381298i
$738$		0			0
$739$	−19.4158			−0.714221			−0.357111	−	0.934062i	$-0.616238\pi$
$739$	−19.4158			−0.714221			−0.357111	+	0.934062i	$0.616238\pi$
$740$		0			0
$741$		−	9.81875i		−	0.360701i
$742$		0			0
$743$			4.08704i			0.149939i	0.997186	+	0.0749695i	$0.0238859\pi$
$743$			4.08704i			0.149939i	−0.997186	+	0.0749695i	$0.976114\pi$
$744$		0			0
$745$			18.5954i			0.681284i
$746$		0			0
$747$	2.49133			0.0911530
$748$		0			0
$749$	−24.6237			−0.899731
$750$		0			0
$751$			38.1845i			1.39337i	0.717377	+	0.696685i	$0.245343\pi$
$751$			38.1845i			1.39337i	−0.717377	+	0.696685i	$0.754657\pi$
$752$		0			0
$753$		−	31.2350i		−	1.13826i
$754$		0			0
$755$		−	27.8838i		−	1.01480i
$756$		0			0
$757$	−20.0495			−0.728710			−0.364355	−	0.931260i	$-0.618710\pi$
$757$	−20.0495			−0.728710			−0.364355	+	0.931260i	$0.618710\pi$
$758$		0			0
$759$			7.00680i			0.254331i
$760$		0			0
$761$	−0.504268			−0.0182797			−0.00913985	−	0.999958i	$-0.502909\pi$
$761$	−0.504268			−0.0182797			−0.00913985	+	0.999958i	$0.502909\pi$
$762$		0			0
$763$	−25.6924			−0.930126
$764$		0			0
$765$	0.821217	+	1.03621i	0.0296912	+	0.0374644i
$766$		0			0
$767$	2.53859			0.0916633
$768$		0			0
$769$	−36.3548			−1.31099			−0.655494	−	0.755200i	$-0.727540\pi$
$769$	−36.3548			−1.31099			−0.655494	+	0.755200i	$0.727540\pi$
$770$		0			0
$771$			50.4500i			1.81691i
$772$		0			0
$773$	−25.4730			−0.916202			−0.458101	−	0.888900i	$-0.651470\pi$
$773$	−25.4730			−0.916202			−0.458101	+	0.888900i	$0.651470\pi$
$774$		0			0
$775$			6.80695i			0.244513i
$776$		0			0
$777$			19.7067i			0.706974i
$778$		0			0
$779$		−	9.72468i		−	0.348423i
$780$		0			0
$781$	11.3674			0.406756
$782$		0			0
$783$	21.8533			0.780972
$784$		0			0
$785$			34.6605i			1.23709i
$786$		0			0
$787$			11.0753i			0.394791i	0.980324	+	0.197396i	$0.0632484\pi$
$787$			11.0753i			0.394791i	−0.980324	+	0.197396i	$0.936752\pi$
$788$		0			0
$789$		−	49.8140i		−	1.77343i
$790$		0			0
$791$	0.246429			0.00876201
$792$		0			0
$793$		−	7.68624i		−	0.272946i
$794$		0			0
$795$	−49.1338			−1.74260
$796$		0			0
$797$	−9.83717			−0.348450			−0.174225	−	0.984706i	$-0.555742\pi$
$797$	−9.83717			−0.348450			−0.174225	+	0.984706i	$0.555742\pi$
$798$		0			0
$799$	−37.8321	+	29.9825i	−1.33840	+	1.06071i
$800$		0			0
$801$	−0.992621			−0.0350725
$802$		0			0
$803$	14.8418			0.523757
$804$		0			0
$805$			11.3654i			0.400578i
$806$		0			0
$807$	11.4663			0.403633
$808$		0			0
$809$		−	32.0675i		−	1.12743i	−0.825968	−	0.563717i	$-0.809371\pi$
$809$		−	32.0675i		−	1.12743i	0.825968	−	0.563717i	$-0.190629\pi$
$810$		0			0
$811$			26.9284i			0.945582i	0.881175	+	0.472791i	$0.156754\pi$
$811$			26.9284i			0.945582i	−0.881175	+	0.472791i	$0.843246\pi$
$812$		0			0
$813$		−	49.8315i		−	1.74767i
$814$		0			0
$815$	17.5385			0.614347
$816$		0			0
$817$	−8.99072			−0.314545
$818$		0			0
$819$		−	1.04416i		−	0.0364859i
$820$		0			0
$821$		−	23.1629i		−	0.808392i	−0.914672	−	0.404196i	$-0.867551\pi$
$821$		−	23.1629i		−	0.808392i	0.914672	−	0.404196i	$-0.132449\pi$
$822$		0			0
$823$		−	9.16307i		−	0.319405i	−0.987165	−	0.159702i	$-0.948947\pi$
$823$		−	9.16307i		−	0.319405i	0.987165	−	0.159702i	$-0.0510534\pi$
$824$		0			0
$825$	−3.86176			−0.134449
$826$		0			0
$827$		−	41.7676i		−	1.45240i	−0.687482	−	0.726201i	$-0.741284\pi$
$827$		−	41.7676i		−	1.45240i	0.687482	−	0.726201i	$-0.258716\pi$
$828$		0			0
$829$	53.4668			1.85698			0.928490	−	0.371357i	$-0.121108\pi$
$829$	53.4668			1.85698			0.928490	+	0.371357i	$0.121108\pi$
$830$		0			0
$831$	13.3209			0.462098
$832$		0			0
$833$	−3.11449	+	2.46828i	−0.107911	+	0.0855209i
$834$		0			0
$835$	42.7205			1.47840
$836$		0			0
$837$	25.7803			0.891098
$838$		0			0
$839$			4.11365i			0.142019i	0.997476	+	0.0710094i	$0.0226220\pi$
$839$			4.11365i			0.142019i	−0.997476	+	0.0710094i	$0.977378\pi$
$840$		0			0
$841$	10.2085			0.352016
$842$		0			0
$843$		−	0.908431i		−	0.0312880i
$844$		0			0
$845$		−	12.5568i		−	0.431966i
$846$		0			0
$847$			20.4965i			0.704267i
$848$		0			0
$849$	55.6180			1.90880
$850$		0			0
$851$	10.8847			0.373122
$852$		0			0
$853$		−	34.8762i		−	1.19414i	−0.802190	−	0.597069i	$-0.796332\pi$
$853$		−	34.8762i		−	1.19414i	0.802190	−	0.597069i	$-0.203668\pi$
$854$		0			0
$855$		−	0.696905i		−	0.0238337i
$856$		0			0
$857$		−	15.9312i		−	0.544199i	−0.962269	−	0.272099i	$-0.912282\pi$
$857$		−	15.9312i		−	0.544199i	0.962269	−	0.272099i	$-0.0877179\pi$
$858$		0			0
$859$	−26.2313			−0.895000			−0.447500	−	0.894284i	$-0.647686\pi$
$859$	−26.2313			−0.895000			−0.447500	+	0.894284i	$0.647686\pi$
$860$		0			0
$861$		−	19.5658i		−	0.666802i
$862$		0			0
$863$	−26.5634			−0.904228			−0.452114	−	0.891960i	$-0.649330\pi$
$863$	−26.5634			−0.904228			−0.452114	+	0.891960i	$0.649330\pi$
$864$		0			0
$865$	−11.9822			−0.407408
$866$		0			0
$867$	−29.4553	−	6.91149i	−1.00035	−	0.234727i
$868$		0			0
$869$	−6.60233			−0.223969
$870$		0			0
$871$	16.1197			0.546197
$872$		0			0
$873$		−	1.97486i		−	0.0668390i
$874$		0			0
$875$	−29.7939			−1.00722
$876$		0			0
$877$			17.5313i			0.591990i	0.955189	+	0.295995i	$0.0956512\pi$
$877$			17.5313i			0.591990i	−0.955189	+	0.295995i	$0.904349\pi$
$878$		0			0
$879$			20.0570i			0.676505i
$880$		0			0
$881$			29.8560i			1.00587i	0.864323	+	0.502937i	$0.167747\pi$
$881$			29.8560i			1.00587i	−0.864323	+	0.502937i	$0.832253\pi$
$882$		0			0
$883$	−16.5774			−0.557873			−0.278937	−	0.960309i	$-0.589982\pi$
$883$	−16.5774			−0.557873			−0.278937	+	0.960309i	$0.589982\pi$
$884$		0			0
$885$	3.40896			0.114591
$886$		0			0
$887$			5.23541i			0.175788i	0.996130	+	0.0878939i	$0.0280136\pi$
$887$			5.23541i			0.175788i	−0.996130	+	0.0878939i	$0.971986\pi$
$888$		0			0
$889$			17.9414i			0.601734i
$890$		0			0
$891$			15.4446i			0.517413i
$892$		0			0
$893$	25.4439			0.851449
$894$		0			0
$895$			46.7738i			1.56347i
$896$		0			0
$897$	−10.9114			−0.364321
$898$		0			0
$899$	−22.1684			−0.739357
$900$		0			0
$901$	46.5741	−	36.9107i	1.55161	−	1.22967i
$902$		0			0
$903$	−18.0891			−0.601968
$904$		0			0
$905$	−25.3624			−0.843074
$906$		0			0
$907$			37.9018i			1.25851i	0.777199	+	0.629255i	$0.216640\pi$
$907$			37.9018i			1.25851i	−0.777199	+	0.629255i	$0.783360\pi$
$908$		0			0
$909$	−1.36918			−0.0454129
$910$		0			0
$911$		−	45.0157i		−	1.49144i	−0.666262	−	0.745718i	$-0.732107\pi$
$911$		−	45.0157i		−	1.49144i	0.666262	−	0.745718i	$-0.267893\pi$
$912$		0			0
$913$			24.2589i			0.802853i
$914$		0			0
$915$		−	10.3215i		−	0.341218i
$916$		0			0
$917$	−44.4573			−1.46811
$918$		0			0
$919$	31.8548			1.05079			0.525397	−	0.850857i	$-0.323917\pi$
$919$	31.8548			1.05079			0.525397	+	0.850857i	$0.323917\pi$
$920$		0			0
$921$			4.96117i			0.163476i
$922$		0			0
$923$			17.7019i			0.582665i
$924$		0			0
$925$			5.99902i			0.197247i
$926$		0			0
$927$	−1.53396			−0.0503819
$928$		0			0
$929$		−	40.9651i		−	1.34402i	−0.740542	−	0.672010i	$-0.765431\pi$
$929$		−	40.9651i		−	1.34402i	0.740542	−	0.672010i	$-0.234569\pi$
$930$		0			0
$931$	2.09465			0.0686493
$932$		0			0
$933$	−49.4004			−1.61730
$934$		0			0
$935$	−10.0900	+	7.99646i	−0.329977	+	0.261512i
$936$		0			0
$937$	41.1250			1.34350			0.671748	−	0.740780i	$-0.265544\pi$
$937$	41.1250			1.34350			0.671748	+	0.740780i	$0.265544\pi$
$938$		0			0
$939$	−10.0412			−0.327682
$940$		0			0
$941$		−	27.9819i		−	0.912184i	−0.889933	−	0.456092i	$-0.849249\pi$
$941$		−	27.9819i		−	0.912184i	0.889933	−	0.456092i	$-0.150751\pi$
$942$		0			0
$943$	−10.8069			−0.351920
$944$		0			0
$945$			23.7239i			0.771739i
$946$		0			0
$947$			29.1771i			0.948127i	0.880491	+	0.474064i	$0.157213\pi$
$947$			29.1771i			0.948127i	−0.880491	+	0.474064i	$0.842787\pi$
$948$		0			0
$949$			23.1125i			0.750265i
$950$		0			0
$951$	−9.68050			−0.313912
$952$		0			0
$953$	−6.70062			−0.217054			−0.108527	−	0.994093i	$-0.534613\pi$
$953$	−6.70062			−0.217054			−0.108527	+	0.994093i	$0.534613\pi$
$954$		0			0
$955$		−	15.7527i		−	0.509747i
$956$		0			0
$957$		−	12.5767i		−	0.406547i
$958$		0			0
$959$		−	39.3257i		−	1.26989i
$960$		0			0
$961$	4.84796			0.156386
$962$		0			0
$963$		−	1.67790i		−	0.0540696i
$964$		0			0
$965$	−7.68322			−0.247332
$966$		0			0
$967$	49.3201			1.58603			0.793013	−	0.609204i	$-0.208511\pi$
$967$	49.3201			1.58603			0.793013	+	0.609204i	$0.208511\pi$
$968$		0			0
$969$	9.90507	+	12.4983i	0.318197	+	0.401502i
$970$		0			0
$971$	55.6067			1.78451			0.892253	−	0.451536i	$-0.149124\pi$
$971$	55.6067			1.78451			0.892253	+	0.451536i	$0.149124\pi$
$972$		0			0
$973$	−38.0096			−1.21853
$974$		0			0
$975$		−	6.01375i		−	0.192594i
$976$		0			0
$977$	48.4637			1.55049			0.775246	−	0.631659i	$-0.217626\pi$
$977$	48.4637			1.55049			0.775246	+	0.631659i	$0.217626\pi$
$978$		0			0
$979$		−	9.66548i		−	0.308910i
$980$		0			0
$981$		−	1.75072i		−	0.0558962i
$982$		0			0
$983$		−	25.0690i		−	0.799575i	−0.916608	−	0.399788i	$-0.869084\pi$
$983$		−	25.0690i		−	0.799575i	0.916608	−	0.399788i	$-0.130916\pi$
$984$		0			0
$985$	−5.88762			−0.187595
$986$		0			0
$987$	51.1926			1.62948
$988$		0			0
$989$			9.99121i			0.317702i
$990$		0			0
$991$		−	33.0435i		−	1.04966i	−0.851206	−	0.524831i	$-0.824128\pi$
$991$		−	33.0435i		−	1.04966i	0.851206	−	0.524831i	$-0.175872\pi$
$992$		0			0
$993$		−	14.0223i		−	0.444986i
$994$		0			0
$995$	17.6286			0.558864
$996$		0			0
$997$			17.7066i			0.560772i	0.959887	+	0.280386i	$0.0904625\pi$
$997$			17.7066i			0.560772i	−0.959887	+	0.280386i	$0.909537\pi$
$998$		0			0
$999$	22.7204			0.718842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.12	✓	46
17.16	even	2	inner	4012.2.b.b.237.35	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.12	✓	46	1.1	even	1	trivial
4012.2.b.b.237.35	yes	46	17.16	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.77972i q^{3} +1.91545i q^{5} +2.45686i q^{7} -0.167415 q^{9} +O(q^{10})\)	\(q-1.77972i q^{3} +1.91545i q^{5} +2.45686i q^{7} -0.167415 q^{9} -1.63017i q^{11} +2.53859 q^{13} +3.40896 q^{15} +(-3.23137 + 2.56091i) q^{17} +2.17325 q^{19} +4.37253 q^{21} -2.41510i q^{23} +1.33107 q^{25} -5.04122i q^{27} +4.33492i q^{29} +5.11391i q^{31} -2.90125 q^{33} -4.70599 q^{35} +4.50693i q^{37} -4.51800i q^{39} -4.47471i q^{41} -4.13699 q^{43} -0.320674i q^{45} +11.7078 q^{47} +0.963830 q^{49} +(4.55771 + 5.75094i) q^{51} -14.4131 q^{53} +3.12250 q^{55} -3.86779i q^{57} +1.00000 q^{59} -3.02775i q^{61} -0.411315i q^{63} +4.86254i q^{65} +6.34987 q^{67} -4.29820 q^{69} +6.97310i q^{71} +9.10446i q^{73} -2.36893i q^{75} +4.00511 q^{77} -4.05009i q^{79} -9.47422 q^{81} -14.8812 q^{83} +(-4.90529 - 6.18951i) q^{85} +7.71496 q^{87} +5.92912 q^{89} +6.23698i q^{91} +9.10134 q^{93} +4.16275i q^{95} +11.7962i q^{97} +0.272914i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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