LMFDB - Embedded newform 4012.2.b.a.237.2

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:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.2
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.39

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.07288i q^{3} -1.34148i q^{5} +0.0167420i q^{7} -6.44259 q^{9} +O(q^{10})$	$q-3.07288i q^{3} -1.34148i q^{5} +0.0167420i q^{7} -6.44259 q^{9} +2.90041i q^{11} +0.0774426 q^{13} -4.12221 q^{15} +(2.86279 + 2.96723i) q^{17} +7.12343 q^{19} +0.0514462 q^{21} +2.37172i q^{23} +3.20042 q^{25} +10.5787i q^{27} +3.11810i q^{29} +7.22174i q^{31} +8.91262 q^{33} +0.0224591 q^{35} +3.46789i q^{37} -0.237972i q^{39} +9.14065i q^{41} +12.1177 q^{43} +8.64262i q^{45} -8.20247 q^{47} +6.99972 q^{49} +(9.11795 - 8.79700i) q^{51} -5.09882 q^{53} +3.89085 q^{55} -21.8894i q^{57} -1.00000 q^{59} -11.6482i q^{61} -0.107862i q^{63} -0.103888i q^{65} -4.39795 q^{67} +7.28801 q^{69} +6.23573i q^{71} +0.615590i q^{73} -9.83452i q^{75} -0.0485587 q^{77} +3.90064i q^{79} +13.1792 q^{81} +2.13300 q^{83} +(3.98049 - 3.84038i) q^{85} +9.58153 q^{87} +13.1280 q^{89} +0.00129655i q^{91} +22.1915 q^{93} -9.55595i q^{95} +10.5006i q^{97} -18.6862i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * 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
$3$		−	3.07288i		−	1.77413i	−0.461646	−	0.887064i	$-0.652741\pi$
$3$		−	3.07288i		−	1.77413i	0.461646	−	0.887064i	$-0.347259\pi$
$4$		0			0
$5$		−	1.34148i		−	0.599929i	−0.953950	−	0.299965i	$-0.903025\pi$
$5$		−	1.34148i		−	0.599929i	0.953950	−	0.299965i	$-0.0969749\pi$
$6$		0			0
$7$			0.0167420i			0.00632789i	0.999995	+	0.00316394i	$0.00100712\pi$
$7$			0.0167420i			0.00632789i	−0.999995	+	0.00316394i	$0.998993\pi$
$8$		0			0
$9$	−6.44259			−2.14753
$10$		0			0
$11$			2.90041i			0.874507i	0.899338	+	0.437253i	$0.144049\pi$
$11$			2.90041i			0.874507i	−0.899338	+	0.437253i	$0.855951\pi$
$12$		0			0
$13$	0.0774426			0.0214787			0.0107394	−	0.999942i	$-0.496581\pi$
$13$	0.0774426			0.0214787			0.0107394	+	0.999942i	$0.496581\pi$
$14$		0			0
$15$	−4.12221			−1.06435
$16$		0			0
$17$	2.86279	+	2.96723i	0.694327	+	0.719659i
$17$	2.86279	+	2.96723i	0.694327	+	0.719659i
$18$		0			0
$19$	7.12343			1.63423			0.817113	−	0.576477i	$-0.195573\pi$
$19$	7.12343			1.63423			0.817113	+	0.576477i	$0.195573\pi$
$20$		0			0
$21$	0.0514462			0.0112265
$22$		0			0
$23$			2.37172i			0.494538i	0.968947	+	0.247269i	$0.0795331\pi$
$23$			2.37172i			0.494538i	−0.968947	+	0.247269i	$0.920467\pi$
$24$		0			0
$25$	3.20042			0.640085
$26$		0			0
$27$			10.5787i			2.03587i
$28$		0			0
$29$			3.11810i			0.579016i	0.957176	+	0.289508i	$0.0934916\pi$
$29$			3.11810i			0.579016i	−0.957176	+	0.289508i	$0.906508\pi$
$30$		0			0
$31$			7.22174i			1.29706i	0.761188	+	0.648531i	$0.224616\pi$
$31$			7.22174i			1.29706i	−0.761188	+	0.648531i	$0.775384\pi$
$32$		0			0
$33$	8.91262			1.55149
$34$		0			0
$35$	0.0224591			0.00379628
$36$		0			0
$37$			3.46789i			0.570118i	0.958510	+	0.285059i	$0.0920133\pi$
$37$			3.46789i			0.570118i	−0.958510	+	0.285059i	$0.907987\pi$
$38$		0			0
$39$		−	0.237972i		−	0.0381060i
$40$		0			0
$41$			9.14065i			1.42753i	0.700386	+	0.713765i	$0.253011\pi$
$41$			9.14065i			1.42753i	−0.700386	+	0.713765i	$0.746989\pi$
$42$		0			0
$43$	12.1177			1.84793			0.923964	−	0.382479i	$-0.124929\pi$
$43$	12.1177			1.84793			0.923964	+	0.382479i	$0.124929\pi$
$44$		0			0
$45$			8.64262i			1.28837i
$46$		0			0
$47$	−8.20247			−1.19645			−0.598227	−	0.801327i	$-0.704128\pi$
$47$	−8.20247			−1.19645			−0.598227	+	0.801327i	$0.704128\pi$
$48$		0			0
$49$	6.99972			0.999960
$50$		0			0
$51$	9.11795	−	8.79700i	1.27677	−	1.23183i
$52$		0			0
$53$	−5.09882			−0.700376			−0.350188	−	0.936679i	$-0.613882\pi$
$53$	−5.09882			−0.700376			−0.350188	+	0.936679i	$0.613882\pi$
$54$		0			0
$55$	3.89085			0.524642
$56$		0			0
$57$		−	21.8894i		−	2.89933i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$		−	11.6482i		−	1.49140i	−0.666281	−	0.745701i	$-0.732115\pi$
$61$		−	11.6482i		−	1.49140i	0.666281	−	0.745701i	$-0.267885\pi$
$62$		0			0
$63$		−	0.107862i		−	0.0135893i
$64$		0			0
$65$		−	0.103888i		−	0.0128857i
$66$		0			0
$67$	−4.39795			−0.537295			−0.268647	−	0.963239i	$-0.586577\pi$
$67$	−4.39795			−0.537295			−0.268647	+	0.963239i	$0.586577\pi$
$68$		0			0
$69$	7.28801			0.877373
$70$		0			0
$71$			6.23573i			0.740045i	0.929023	+	0.370022i	$0.120650\pi$
$71$			6.23573i			0.740045i	−0.929023	+	0.370022i	$0.879350\pi$
$72$		0			0
$73$			0.615590i			0.0720494i	0.999351	+	0.0360247i	$0.0114695\pi$
$73$			0.615590i			0.0720494i	−0.999351	+	0.0360247i	$0.988530\pi$
$74$		0			0
$75$		−	9.83452i		−	1.13559i
$76$		0			0
$77$	−0.0485587			−0.00553378
$78$		0			0
$79$			3.90064i			0.438857i	0.975629	+	0.219428i	$0.0704192\pi$
$79$			3.90064i			0.438857i	−0.975629	+	0.219428i	$0.929581\pi$
$80$		0			0
$81$	13.1792			1.46436
$82$		0			0
$83$	2.13300			0.234127			0.117064	−	0.993124i	$-0.462652\pi$
$83$	2.13300			0.234127			0.117064	+	0.993124i	$0.462652\pi$
$84$		0			0
$85$	3.98049	−	3.84038i	0.431745	−	0.416547i
$86$		0			0
$87$	9.58153			1.02725
$88$		0			0
$89$	13.1280			1.39156			0.695782	−	0.718254i	$-0.255058\pi$
$89$	13.1280			1.39156			0.695782	+	0.718254i	$0.255058\pi$
$90$		0			0
$91$			0.00129655i			0.000135915i
$92$		0			0
$93$	22.1915			2.30115
$94$		0			0
$95$		−	9.55595i		−	0.980420i
$96$		0			0
$97$			10.5006i			1.06618i	0.846059	+	0.533089i	$0.178969\pi$
$97$			10.5006i			1.06618i	−0.846059	+	0.533089i	$0.821031\pi$
$98$		0			0
$99$		−	18.6862i		−	1.87803i
$100$		0			0
$101$	−12.2533			−1.21924			−0.609622	−	0.792692i	$-0.708679\pi$
$101$	−12.2533			−1.21924			−0.609622	+	0.792692i	$0.708679\pi$
$102$		0			0
$103$	−5.98419			−0.589639			−0.294820	−	0.955553i	$-0.595260\pi$
$103$	−5.98419			−0.589639			−0.294820	+	0.955553i	$0.595260\pi$
$104$		0			0
$105$		−	0.0690142i		−	0.00673509i
$106$		0			0
$107$			13.6662i			1.32116i	0.750755	+	0.660581i	$0.229690\pi$
$107$			13.6662i			1.32116i	−0.750755	+	0.660581i	$0.770310\pi$
$108$		0			0
$109$			13.6658i			1.30895i	0.756085	+	0.654473i	$0.227110\pi$
$109$			13.6658i			1.30895i	−0.756085	+	0.654473i	$0.772890\pi$
$110$		0			0
$111$	10.6564			1.01146
$112$		0			0
$113$			8.39565i			0.789797i	0.918725	+	0.394898i	$0.129220\pi$
$113$			8.39565i			0.789797i	−0.918725	+	0.394898i	$0.870780\pi$
$114$		0			0
$115$	3.18162			0.296688
$116$		0			0
$117$	−0.498931			−0.0461262
$118$		0			0
$119$	−0.0496774	+	0.0479288i	−0.00455392	+	0.00439363i
$120$		0			0
$121$	2.58762			0.235238
$122$		0			0
$123$	28.0881			2.53262
$124$		0			0
$125$		−	11.0007i		−	0.983935i
$126$		0			0
$127$	−14.6489			−1.29988			−0.649939	−	0.759987i	$-0.725205\pi$
$127$	−14.6489			−1.29988			−0.649939	+	0.759987i	$0.725205\pi$
$128$		0			0
$129$		−	37.2362i		−	3.27846i
$130$		0			0
$131$		−	13.4018i		−	1.17092i	−0.810700	−	0.585462i	$-0.800913\pi$
$131$		−	13.4018i		−	1.17092i	0.810700	−	0.585462i	$-0.199087\pi$
$132$		0			0
$133$			0.119261i			0.0103412i
$134$		0			0
$135$	14.1911			1.22138
$136$		0			0
$137$	−6.05059			−0.516937			−0.258468	−	0.966020i	$-0.583218\pi$
$137$	−6.05059			−0.516937			−0.258468	+	0.966020i	$0.583218\pi$
$138$		0			0
$139$		−	6.45138i		−	0.547199i	−0.961844	−	0.273599i	$-0.911786\pi$
$139$		−	6.45138i		−	0.547199i	0.961844	−	0.273599i	$-0.0882142\pi$
$140$		0			0
$141$			25.2052i			2.12266i
$142$		0			0
$143$			0.224615i			0.0187833i
$144$		0			0
$145$	4.18287			0.347368
$146$		0			0
$147$		−	21.5093i		−	1.77406i
$148$		0			0
$149$	14.0814			1.15360			0.576798	−	0.816887i	$-0.304302\pi$
$149$	14.0814			1.15360			0.576798	+	0.816887i	$0.304302\pi$
$150$		0			0
$151$	−15.0899			−1.22800			−0.613998	−	0.789307i	$-0.710440\pi$
$151$	−15.0899			−1.22800			−0.613998	+	0.789307i	$0.710440\pi$
$152$		0			0
$153$	−18.4438	−	19.1167i	−1.49109	−	1.54549i
$154$		0			0
$155$	9.68783			0.778145
$156$		0			0
$157$	11.3310			0.904314			0.452157	−	0.891938i	$-0.350655\pi$
$157$	11.3310			0.904314			0.452157	+	0.891938i	$0.350655\pi$
$158$		0			0
$159$			15.6681i			1.24256i
$160$		0			0
$161$	−0.0397074			−0.00312938
$162$		0			0
$163$			3.67350i			0.287731i	0.989597	+	0.143866i	$0.0459533\pi$
$163$			3.67350i			0.287731i	−0.989597	+	0.143866i	$0.954047\pi$
$164$		0			0
$165$		−	11.9561i		−	0.930782i
$166$		0			0
$167$		−	4.72732i		−	0.365811i	−0.983130	−	0.182906i	$-0.941450\pi$
$167$		−	4.72732i		−	0.365811i	0.983130	−	0.182906i	$-0.0585503\pi$
$168$		0			0
$169$	−12.9940			−0.999539
$170$		0			0
$171$	−45.8933			−3.50955
$172$		0			0
$173$			10.5672i			0.803413i	0.915768	+	0.401706i	$0.131583\pi$
$173$			10.5672i			0.803413i	−0.915768	+	0.401706i	$0.868417\pi$
$174$		0			0
$175$			0.0535816i			0.00405039i
$176$		0			0
$177$			3.07288i			0.230972i
$178$		0			0
$179$	−8.59770			−0.642622			−0.321311	−	0.946974i	$-0.604124\pi$
$179$	−8.59770			−0.642622			−0.321311	+	0.946974i	$0.604124\pi$
$180$		0			0
$181$		−	25.9804i		−	1.93111i	−0.260201	−	0.965554i	$-0.583789\pi$
$181$		−	25.9804i		−	1.93111i	0.260201	−	0.965554i	$-0.416211\pi$
$182$		0			0
$183$	−35.7936			−2.64594
$184$		0			0
$185$	4.65212			0.342031
$186$		0			0
$187$	−8.60619	+	8.30325i	−0.629347	+	0.607194i
$188$		0			0
$189$	−0.177108			−0.0128827
$190$		0			0
$191$	8.03051			0.581068			0.290534	−	0.956865i	$-0.406167\pi$
$191$	8.03051			0.581068			0.290534	+	0.956865i	$0.406167\pi$
$192$		0			0
$193$			12.9111i			0.929358i	0.885479	+	0.464679i	$0.153830\pi$
$193$			12.9111i			0.929358i	−0.885479	+	0.464679i	$0.846170\pi$
$194$		0			0
$195$	−0.319235			−0.0228609
$196$		0			0
$197$		−	0.609732i		−	0.0434416i	−0.999764	−	0.0217208i	$-0.993086\pi$
$197$		−	0.609732i		−	0.0434416i	0.999764	−	0.0217208i	$-0.00691449\pi$
$198$		0			0
$199$		−	7.17711i		−	0.508772i	−0.967103	−	0.254386i	$-0.918127\pi$
$199$		−	7.17711i		−	0.508772i	0.967103	−	0.254386i	$-0.0818733\pi$
$200$		0			0
$201$			13.5144i			0.953230i
$202$		0			0
$203$	−0.0522032			−0.00366395
$204$		0			0
$205$	12.2620			0.856417
$206$		0			0
$207$		−	15.2800i		−	1.06203i
$208$		0			0
$209$			20.6609i			1.42914i
$210$		0			0
$211$		−	0.185174i		−	0.0127479i	−0.999980	−	0.00637397i	$-0.997971\pi$
$211$		−	0.185174i		−	0.0127479i	0.999980	−	0.00637397i	$-0.00202891\pi$
$212$		0			0
$213$	19.1616			1.31293
$214$		0			0
$215$		−	16.2557i		−	1.10863i
$216$		0			0
$217$	−0.120906			−0.00820766
$218$		0			0
$219$	1.89164			0.127825
$220$		0			0
$221$	0.221702	+	0.229790i	0.0149133	+	0.0154574i
$222$		0			0
$223$	19.4180			1.30032			0.650161	−	0.759796i	$-0.274701\pi$
$223$	19.4180			1.30032			0.650161	+	0.759796i	$0.274701\pi$
$224$		0			0
$225$	−20.6190			−1.37460
$226$		0			0
$227$		−	26.8038i		−	1.77903i	−0.456903	−	0.889517i	$-0.651041\pi$
$227$		−	26.8038i		−	1.77903i	0.456903	−	0.889517i	$-0.348959\pi$
$228$		0			0
$229$	−26.7046			−1.76469			−0.882344	−	0.470605i	$-0.844036\pi$
$229$	−26.7046			−1.76469			−0.882344	+	0.470605i	$0.844036\pi$
$230$		0			0
$231$			0.149215i			0.00981764i
$232$		0			0
$233$			11.4330i			0.748998i	0.927227	+	0.374499i	$0.122185\pi$
$233$			11.4330i			0.748998i	−0.927227	+	0.374499i	$0.877815\pi$
$234$		0			0
$235$			11.0035i			0.717787i
$236$		0			0
$237$	11.9862			0.778588
$238$		0			0
$239$	−12.5902			−0.814392			−0.407196	−	0.913341i	$-0.633493\pi$
$239$	−12.5902			−0.814392			−0.407196	+	0.913341i	$0.633493\pi$
$240$		0			0
$241$			8.65137i			0.557284i	0.960395	+	0.278642i	$0.0898843\pi$
$241$			8.65137i			0.557284i	−0.960395	+	0.278642i	$0.910116\pi$
$242$		0			0
$243$		−	8.76214i		−	0.562091i
$244$		0			0
$245$		−	9.39000i		−	0.599905i
$246$		0			0
$247$	0.551657			0.0351011
$248$		0			0
$249$		−	6.55446i		−	0.415372i
$250$		0			0
$251$	−20.8650			−1.31699			−0.658494	−	0.752586i	$-0.728806\pi$
$251$	−20.8650			−1.31699			−0.658494	+	0.752586i	$0.728806\pi$
$252$		0			0
$253$	−6.87896			−0.432477
$254$		0			0
$255$	−11.8010	−	12.2316i	−0.739008	−	0.765970i
$256$		0			0
$257$	14.2455			0.888608			0.444304	−	0.895876i	$-0.353451\pi$
$257$	14.2455			0.888608			0.444304	+	0.895876i	$0.353451\pi$
$258$		0			0
$259$	−0.0580595			−0.00360764
$260$		0			0
$261$		−	20.0886i		−	1.24345i
$262$		0			0
$263$	20.2542			1.24892			0.624462	−	0.781055i	$-0.285318\pi$
$263$	20.2542			1.24892			0.624462	+	0.781055i	$0.285318\pi$
$264$		0			0
$265$			6.83997i			0.420176i
$266$		0			0
$267$		−	40.3407i		−	2.46881i
$268$		0			0
$269$			15.4382i			0.941286i	0.882324	+	0.470643i	$0.155978\pi$
$269$			15.4382i			0.941286i	−0.882324	+	0.470643i	$0.844022\pi$
$270$		0			0
$271$	−22.9093			−1.39164			−0.695820	−	0.718216i	$-0.744959\pi$
$271$	−22.9093			−1.39164			−0.695820	+	0.718216i	$0.744959\pi$
$272$		0			0
$273$	0.00398413			0.000241130
$274$		0			0
$275$			9.28255i			0.559759i
$276$		0			0
$277$		−	5.99119i		−	0.359975i	−0.983669	−	0.179988i	$-0.942394\pi$
$277$		−	5.99119i		−	0.359975i	0.983669	−	0.179988i	$-0.0576058\pi$
$278$		0			0
$279$		−	46.5267i		−	2.78548i
$280$		0			0
$281$	−29.9562			−1.78704			−0.893519	−	0.449025i	$-0.851772\pi$
$281$	−29.9562			−1.78704			−0.893519	+	0.449025i	$0.851772\pi$
$282$		0			0
$283$		−	24.6902i		−	1.46768i	−0.679323	−	0.733839i	$-0.737727\pi$
$283$		−	24.6902i		−	1.46768i	0.679323	−	0.733839i	$-0.262273\pi$
$284$		0			0
$285$	−29.3643			−1.73939
$286$		0			0
$287$	−0.153033			−0.00903325
$288$		0			0
$289$	−0.608921	+	16.9891i	−0.0358189	+	0.999358i
$290$		0			0
$291$	32.2672			1.89154
$292$		0			0
$293$	3.27166			0.191132			0.0955662	−	0.995423i	$-0.469534\pi$
$293$	3.27166			0.191132			0.0955662	+	0.995423i	$0.469534\pi$
$294$		0			0
$295$			1.34148i			0.0781041i
$296$		0			0
$297$	−30.6825			−1.78038
$298$		0			0
$299$			0.183672i			0.0106220i
$300$		0			0
$301$			0.202874i			0.0116935i
$302$		0			0
$303$			37.6528i			2.16310i
$304$		0			0
$305$	−15.6259			−0.894735
$306$		0			0
$307$	30.5331			1.74262			0.871309	−	0.490735i	$-0.163272\pi$
$307$	30.5331			1.74262			0.871309	+	0.490735i	$0.163272\pi$
$308$		0			0
$309$			18.3887i			1.04610i
$310$		0			0
$311$		−	29.7539i		−	1.68719i	−0.536982	−	0.843594i	$-0.680436\pi$
$311$		−	29.7539i		−	1.68719i	0.536982	−	0.843594i	$-0.319564\pi$
$312$		0			0
$313$			5.71630i			0.323104i	0.986864	+	0.161552i	$0.0516500\pi$
$313$			5.71630i			0.323104i	−0.986864	+	0.161552i	$0.948350\pi$
$314$		0			0
$315$	−0.144695			−0.00815264
$316$		0			0
$317$			20.8927i			1.17345i	0.809785	+	0.586727i	$0.199584\pi$
$317$			20.8927i			1.17345i	−0.809785	+	0.586727i	$0.800416\pi$
$318$		0			0
$319$	−9.04376			−0.506353
$320$		0			0
$321$	41.9946			2.34391
$322$		0			0
$323$	20.3928	+	21.1369i	1.13469	+	1.17609i
$324$		0			0
$325$	0.247849			0.0137482
$326$		0			0
$327$	41.9934			2.32224
$328$		0			0
$329$		−	0.137326i		−	0.00757102i
$330$		0			0
$331$	17.8161			0.979261			0.489631	−	0.871930i	$-0.337132\pi$
$331$	17.8161			0.979261			0.489631	+	0.871930i	$0.337132\pi$
$332$		0			0
$333$		−	22.3422i		−	1.22435i
$334$		0			0
$335$			5.89977i			0.322339i
$336$		0			0
$337$		−	13.2533i		−	0.721955i	−0.932575	−	0.360977i	$-0.882443\pi$
$337$		−	13.2533i		−	0.721955i	0.932575	−	0.360977i	$-0.117557\pi$
$338$		0			0
$339$	25.7988			1.40120
$340$		0			0
$341$	−20.9460			−1.13429
$342$		0			0
$343$			0.234384i			0.0126555i
$344$		0			0
$345$		−	9.77674i		−	0.526362i
$346$		0			0
$347$		−	13.5496i		−	0.727381i	−0.931520	−	0.363691i	$-0.881517\pi$
$347$		−	13.5496i		−	0.727381i	0.931520	−	0.363691i	$-0.118483\pi$
$348$		0			0
$349$	18.8203			1.00743			0.503714	−	0.863871i	$-0.331967\pi$
$349$	18.8203			1.00743			0.503714	+	0.863871i	$0.331967\pi$
$350$		0			0
$351$			0.819240i			0.0437278i
$352$		0			0
$353$	−11.5163			−0.612949			−0.306474	−	0.951879i	$-0.599149\pi$
$353$	−11.5163			−0.612949			−0.306474	+	0.951879i	$0.599149\pi$
$354$		0			0
$355$	8.36512			0.443974
$356$		0			0
$357$	0.147279	+	0.152653i	0.00779485	+	0.00807924i
$358$		0			0
$359$	0.604100			0.0318832			0.0159416	−	0.999873i	$-0.494925\pi$
$359$	0.604100			0.0318832			0.0159416	+	0.999873i	$0.494925\pi$
$360$		0			0
$361$	31.7432			1.67070
$362$		0			0
$363$		−	7.95143i		−	0.417342i
$364$		0			0
$365$	0.825804			0.0432245
$366$		0			0
$367$		−	30.2307i		−	1.57803i	−0.614375	−	0.789014i	$-0.710592\pi$
$367$		−	30.2307i		−	1.57803i	0.614375	−	0.789014i	$-0.289408\pi$
$368$		0			0
$369$		−	58.8895i		−	3.06566i
$370$		0			0
$371$		−	0.0853645i		−	0.00443190i
$372$		0			0
$373$	12.9586			0.670971			0.335486	−	0.942045i	$-0.391100\pi$
$373$	12.9586			0.670971			0.335486	+	0.942045i	$0.391100\pi$
$374$		0			0
$375$	−33.8039			−1.74563
$376$		0			0
$377$			0.241474i			0.0124365i
$378$		0			0
$379$			31.8666i			1.63688i	0.574594	+	0.818438i	$0.305160\pi$
$379$			31.8666i			1.63688i	−0.574594	+	0.818438i	$0.694840\pi$
$380$		0			0
$381$			45.0142i			2.30615i
$382$		0			0
$383$	30.8966			1.57874			0.789371	−	0.613917i	$-0.210407\pi$
$383$	30.8966			1.57874			0.789371	+	0.613917i	$0.210407\pi$
$384$		0			0
$385$			0.0651407i			0.00331988i
$386$		0			0
$387$	−78.0693			−3.96848
$388$		0			0
$389$	26.6811			1.35278			0.676392	−	0.736542i	$-0.263542\pi$
$389$	26.6811			1.35278			0.676392	+	0.736542i	$0.263542\pi$
$390$		0			0
$391$	−7.03744	+	6.78972i	−0.355899	+	0.343371i
$392$		0			0
$393$	−41.1823			−2.07737
$394$		0			0
$395$	5.23264			0.263283
$396$		0			0
$397$		−	22.6375i		−	1.13614i	−0.822979	−	0.568071i	$-0.807690\pi$
$397$		−	22.6375i		−	1.13614i	0.822979	−	0.568071i	$-0.192310\pi$
$398$		0			0
$399$	0.366473			0.0183466
$400$		0			0
$401$			14.1882i			0.708526i	0.935146	+	0.354263i	$0.115268\pi$
$401$			14.1882i			0.708526i	−0.935146	+	0.354263i	$0.884732\pi$
$402$		0			0
$403$			0.559270i			0.0278592i
$404$		0			0
$405$		−	17.6797i		−	0.878511i
$406$		0			0
$407$	−10.0583			−0.498572
$408$		0			0
$409$	−36.6846			−1.81393			−0.906967	−	0.421201i	$-0.861609\pi$
$409$	−36.6846			−1.81393			−0.906967	+	0.421201i	$0.861609\pi$
$410$		0			0
$411$			18.5927i			0.917112i
$412$		0			0
$413$		−	0.0167420i		−	0.000823821i
$414$		0			0
$415$		−	2.86138i		−	0.140460i
$416$		0			0
$417$	−19.8243			−0.970801
$418$		0			0
$419$		−	2.31826i		−	0.113254i	−0.998395	−	0.0566272i	$-0.981965\pi$
$419$		−	2.31826i		−	0.113254i	0.998395	−	0.0566272i	$-0.0180346\pi$
$420$		0			0
$421$	32.9065			1.60377			0.801883	−	0.597482i	$-0.203832\pi$
$421$	32.9065			1.60377			0.801883	+	0.597482i	$0.203832\pi$
$422$		0			0
$423$	52.8452			2.56942
$424$		0			0
$425$	9.16213	+	9.49640i	0.444429	+	0.460643i
$426$		0			0
$427$	0.195015			0.00943742
$428$		0			0
$429$	0.690216			0.0333240
$430$		0			0
$431$			1.18591i			0.0571233i	0.999592	+	0.0285616i	$0.00909269\pi$
$431$			1.18591i			0.0571233i	−0.999592	+	0.0285616i	$0.990907\pi$
$432$		0			0
$433$	−9.96132			−0.478711			−0.239355	−	0.970932i	$-0.576936\pi$
$433$	−9.96132			−0.478711			−0.239355	+	0.970932i	$0.576936\pi$
$434$		0			0
$435$		−	12.8535i		−	0.616276i
$436$		0			0
$437$			16.8948i			0.808186i
$438$		0			0
$439$		−	32.3671i		−	1.54480i	−0.635137	−	0.772399i	$-0.719056\pi$
$439$		−	32.3671i		−	1.54480i	0.635137	−	0.772399i	$-0.280944\pi$
$440$		0			0
$441$	−45.0963			−2.14744
$442$		0			0
$443$	24.2887			1.15399			0.576995	−	0.816748i	$-0.304225\pi$
$443$	24.2887			1.15399			0.576995	+	0.816748i	$0.304225\pi$
$444$		0			0
$445$		−	17.6110i		−	0.834839i
$446$		0			0
$447$		−	43.2706i		−	2.04663i
$448$		0			0
$449$			9.77875i			0.461488i	0.973015	+	0.230744i	$0.0741160\pi$
$449$			9.77875i			0.461488i	−0.973015	+	0.230744i	$0.925884\pi$
$450$		0			0
$451$	−26.5116			−1.24838
$452$		0			0
$453$			46.3694i			2.17862i
$454$		0			0
$455$	0.00173929			8.15393e−5
$456$		0			0
$457$	8.23722			0.385321			0.192660	−	0.981265i	$-0.438288\pi$
$457$	8.23722			0.385321			0.192660	+	0.981265i	$0.438288\pi$
$458$		0			0
$459$	−31.3894	+	30.2845i	−1.46513	+	1.41356i
$460$		0			0
$461$	−5.41838			−0.252359			−0.126179	−	0.992007i	$-0.540272\pi$
$461$	−5.41838			−0.252359			−0.126179	+	0.992007i	$0.540272\pi$
$462$		0			0
$463$	19.1029			0.887788			0.443894	−	0.896079i	$-0.353597\pi$
$463$	19.1029			0.887788			0.443894	+	0.896079i	$0.353597\pi$
$464$		0			0
$465$		−	29.7695i		−	1.38053i
$466$		0			0
$467$	−3.22970			−0.149453			−0.0747264	−	0.997204i	$-0.523808\pi$
$467$	−3.22970			−0.149453			−0.0747264	+	0.997204i	$0.523808\pi$
$468$		0			0
$469$		−	0.0736305i		−	0.00339994i
$470$		0			0
$471$		−	34.8189i		−	1.60437i
$472$		0			0
$473$			35.1462i			1.61603i
$474$		0			0
$475$	22.7980			1.04604
$476$		0			0
$477$	32.8496			1.50408
$478$		0			0
$479$		−	4.97426i		−	0.227280i	−0.993522	−	0.113640i	$-0.963749\pi$
$479$		−	4.97426i		−	0.227280i	0.993522	−	0.113640i	$-0.0362510\pi$
$480$		0			0
$481$			0.268563i			0.0122454i
$482$		0			0
$483$			0.122016i			0.00555192i
$484$		0			0
$485$	14.0864			0.639631
$486$		0			0
$487$		−	21.9680i		−	0.995466i	−0.867330	−	0.497733i	$-0.834166\pi$
$487$		−	21.9680i		−	0.995466i	0.867330	−	0.497733i	$-0.165834\pi$
$488$		0			0
$489$	11.2882			0.510472
$490$		0			0
$491$	−16.9723			−0.765948			−0.382974	−	0.923759i	$-0.625100\pi$
$491$	−16.9723			−0.765948			−0.382974	+	0.923759i	$0.625100\pi$
$492$		0			0
$493$	−9.25211	+	8.92644i	−0.416694	+	0.402027i
$494$		0			0
$495$	−25.0672			−1.12669
$496$		0			0
$497$	−0.104399			−0.00468292
$498$		0			0
$499$		−	35.6501i		−	1.59592i	−0.602712	−	0.797959i	$-0.705913\pi$
$499$		−	35.6501i		−	1.59592i	0.602712	−	0.797959i	$-0.294087\pi$
$500$		0			0
$501$	−14.5265			−0.648996
$502$		0			0
$503$			0.160644i			0.00716275i	0.999994	+	0.00358138i	$0.00113999\pi$
$503$			0.160644i			0.00716275i	−0.999994	+	0.00358138i	$0.998860\pi$
$504$		0			0
$505$			16.4375i			0.731461i
$506$		0			0
$507$			39.9290i			1.77331i
$508$		0			0
$509$	2.54798			0.112937			0.0564687	−	0.998404i	$-0.482016\pi$
$509$	2.54798			0.112937			0.0564687	+	0.998404i	$0.482016\pi$
$510$		0			0
$511$	−0.0103062			−0.000455920
$512$		0			0
$513$			75.3564i			3.32707i
$514$		0			0
$515$			8.02768i			0.353742i
$516$		0			0
$517$		−	23.7905i		−	1.04631i
$518$		0			0
$519$	32.4719			1.42536
$520$		0			0
$521$		−	19.0948i		−	0.836559i	−0.908318	−	0.418280i	$-0.862633\pi$
$521$		−	19.0948i		−	0.836559i	0.908318	−	0.418280i	$-0.137367\pi$
$522$		0			0
$523$	−15.4939			−0.677499			−0.338749	−	0.940877i	$-0.610004\pi$
$523$	−15.4939			−0.677499			−0.338749	+	0.940877i	$0.610004\pi$
$524$		0			0
$525$	0.164650			0.00718590
$526$		0			0
$527$	−21.4286	+	20.6743i	−0.933443	+	0.900586i
$528$		0			0
$529$	17.3749			0.755432
$530$		0			0
$531$	6.44259			0.279585
$532$		0			0
$533$			0.707876i			0.0306615i
$534$		0			0
$535$	18.3330			0.792603
$536$		0			0
$537$			26.4197i			1.14009i
$538$		0			0
$539$			20.3021i			0.874472i
$540$		0			0
$541$			12.0635i			0.518649i	0.965790	+	0.259325i	$0.0834999\pi$
$541$			12.0635i			0.518649i	−0.965790	+	0.259325i	$0.916500\pi$
$542$		0			0
$543$	−79.8347			−3.42603
$544$		0			0
$545$	18.3324			0.785275
$546$		0			0
$547$		−	31.4565i		−	1.34498i	−0.740104	−	0.672492i	$-0.765224\pi$
$547$		−	31.4565i		−	1.34498i	0.740104	−	0.672492i	$-0.234776\pi$
$548$		0			0
$549$			75.0447i			3.20283i
$550$		0			0
$551$			22.2115i			0.946243i
$552$		0			0
$553$	−0.0653046			−0.00277703
$554$		0			0
$555$		−	14.2954i		−	0.606806i
$556$		0			0
$557$	44.0496			1.86644			0.933220	−	0.359304i	$-0.116986\pi$
$557$	44.0496			1.86644			0.933220	+	0.359304i	$0.116986\pi$
$558$		0			0
$559$	0.938425			0.0396911
$560$		0			0
$561$	25.5149	+	26.4458i	1.07724	+	1.11654i
$562$		0			0
$563$	25.8016			1.08741			0.543705	−	0.839276i	$-0.317021\pi$
$563$	25.8016			1.08741			0.543705	+	0.839276i	$0.317021\pi$
$564$		0			0
$565$	11.2626			0.473822
$566$		0			0
$567$			0.220647i			0.00926629i
$568$		0			0
$569$	9.08551			0.380884			0.190442	−	0.981698i	$-0.439008\pi$
$569$	9.08551			0.380884			0.190442	+	0.981698i	$0.439008\pi$
$570$		0			0
$571$			11.1961i			0.468544i	0.972171	+	0.234272i	$0.0752706\pi$
$571$			11.1961i			0.468544i	−0.972171	+	0.234272i	$0.924729\pi$
$572$		0			0
$573$		−	24.6768i		−	1.03089i
$574$		0			0
$575$			7.59051i			0.316546i
$576$		0			0
$577$	26.9524			1.12204			0.561021	−	0.827802i	$-0.310409\pi$
$577$	26.9524			1.12204			0.561021	+	0.827802i	$0.310409\pi$
$578$		0			0
$579$	39.6741			1.64880
$580$		0			0
$581$			0.0357107i			0.00148153i
$582$		0			0
$583$		−	14.7887i		−	0.612484i
$584$		0			0
$585$			0.669308i			0.0276725i
$586$		0			0
$587$	−4.04579			−0.166988			−0.0834938	−	0.996508i	$-0.526608\pi$
$587$	−4.04579			−0.166988			−0.0834938	+	0.996508i	$0.526608\pi$
$588$		0			0
$589$			51.4435i			2.11969i
$590$		0			0
$591$	−1.87363			−0.0770710
$592$		0			0
$593$	0.807000			0.0331395			0.0165698	−	0.999863i	$-0.494725\pi$
$593$	0.807000			0.0331395			0.0165698	+	0.999863i	$0.494725\pi$
$594$		0			0
$595$	0.0642956	+	0.0666414i	0.00263586	+	0.00273203i
$596$		0			0
$597$	−22.0544			−0.902626
$598$		0			0
$599$	−40.7773			−1.66612			−0.833058	−	0.553186i	$-0.813412\pi$
$599$	−40.7773			−1.66612			−0.833058	+	0.553186i	$0.813412\pi$
$600$		0			0
$601$			38.4495i			1.56839i	0.620517	+	0.784193i	$0.286923\pi$
$601$			38.4495i			1.56839i	−0.620517	+	0.784193i	$0.713077\pi$
$602$		0			0
$603$	28.3342			1.15386
$604$		0			0
$605$		−	3.47124i		−	0.141126i
$606$		0			0
$607$		−	34.7821i		−	1.41176i	−0.708331	−	0.705881i	$-0.750551\pi$
$607$		−	34.7821i		−	1.41176i	0.708331	−	0.705881i	$-0.249449\pi$
$608$		0			0
$609$			0.160414i			0.00650031i
$610$		0			0
$611$	−0.635221			−0.0256983
$612$		0			0
$613$	−19.3393			−0.781107			−0.390554	−	0.920580i	$-0.627716\pi$
$613$	−19.3393			−0.781107			−0.390554	+	0.920580i	$0.627716\pi$
$614$		0			0
$615$		−	37.6797i		−	1.51939i
$616$		0			0
$617$		−	4.36150i		−	0.175587i	−0.996139	−	0.0877937i	$-0.972018\pi$
$617$		−	4.36150i		−	0.175587i	0.996139	−	0.0877937i	$-0.0279816\pi$
$618$		0			0
$619$		−	13.8075i		−	0.554969i	−0.960730	−	0.277484i	$-0.910499\pi$
$619$		−	13.8075i		−	0.554969i	0.960730	−	0.277484i	$-0.0895007\pi$
$620$		0			0
$621$	−25.0896			−1.00681
$622$		0			0
$623$			0.219789i			0.00880565i
$624$		0			0
$625$	1.24484			0.0497938
$626$		0			0
$627$	63.4884			2.53548
$628$		0			0
$629$	−10.2900	+	9.92784i	−0.410291	+	0.395849i
$630$		0			0
$631$	−43.6715			−1.73853			−0.869267	−	0.494343i	$-0.835409\pi$
$631$	−43.6715			−1.73853			−0.869267	+	0.494343i	$0.835409\pi$
$632$		0			0
$633$	−0.569019			−0.0226165
$634$		0			0
$635$			19.6512i			0.779834i
$636$		0			0
$637$	0.542077			0.0214779
$638$		0			0
$639$		−	40.1742i		−	1.58927i
$640$		0			0
$641$		−	1.97799i		−	0.0781259i	−0.999237	−	0.0390630i	$-0.987563\pi$
$641$		−	1.97799i		−	0.0781259i	0.999237	−	0.0390630i	$-0.0124373\pi$
$642$		0			0
$643$			18.7601i			0.739824i	0.929067	+	0.369912i	$0.120612\pi$
$643$			18.7601i			0.739824i	−0.929067	+	0.369912i	$0.879388\pi$
$644$		0			0
$645$	−49.9517			−1.96685
$646$		0			0
$647$	25.0059			0.983084			0.491542	−	0.870854i	$-0.336434\pi$
$647$	25.0059			0.983084			0.491542	+	0.870854i	$0.336434\pi$
$648$		0			0
$649$		−	2.90041i		−	0.113851i
$650$		0			0
$651$			0.371531i			0.0145614i
$652$		0			0
$653$		−	42.5422i		−	1.66481i	−0.554170	−	0.832403i	$-0.686964\pi$
$653$		−	42.5422i		−	1.66481i	0.554170	−	0.832403i	$-0.313036\pi$
$654$		0			0
$655$	−17.9783			−0.702472
$656$		0			0
$657$		−	3.96600i		−	0.154728i
$658$		0			0
$659$	−18.7740			−0.731330			−0.365665	−	0.930747i	$-0.619158\pi$
$659$	−18.7740			−0.731330			−0.365665	+	0.930747i	$0.619158\pi$
$660$		0			0
$661$	−15.0086			−0.583767			−0.291884	−	0.956454i	$-0.594282\pi$
$661$	−15.0086			−0.583767			−0.291884	+	0.956454i	$0.594282\pi$
$662$		0			0
$663$	0.706118	−	0.681263i	0.0274233	−	0.0264580i
$664$		0			0
$665$	0.159986			0.00620399
$666$		0			0
$667$	−7.39525			−0.286345
$668$		0			0
$669$		−	59.6690i		−	2.30694i
$670$		0			0
$671$	33.7846			1.30424
$672$		0			0
$673$			29.8396i			1.15023i	0.818072	+	0.575116i	$0.195043\pi$
$673$			29.8396i			1.15023i	−0.818072	+	0.575116i	$0.804957\pi$
$674$		0			0
$675$			33.8563i			1.30313i
$676$		0			0
$677$			18.0504i			0.693733i	0.937915	+	0.346866i	$0.112754\pi$
$677$			18.0504i			0.693733i	−0.937915	+	0.346866i	$0.887246\pi$
$678$		0			0
$679$	−0.175802			−0.00674665
$680$		0			0
$681$	−82.3650			−3.15623
$682$		0			0
$683$		−	13.6620i		−	0.522764i	−0.965235	−	0.261382i	$-0.915822\pi$
$683$		−	13.6620i		−	0.522764i	0.965235	−	0.261382i	$-0.0841782\pi$
$684$		0			0
$685$			8.11676i			0.310125i
$686$		0			0
$687$			82.0600i			3.13078i
$688$		0			0
$689$	−0.394866			−0.0150432
$690$		0			0
$691$			49.1335i			1.86913i	0.355795	+	0.934564i	$0.384210\pi$
$691$			49.1335i			1.86913i	−0.355795	+	0.934564i	$0.615790\pi$
$692$		0			0
$693$	0.312844			0.0118840
$694$		0			0
$695$	−8.65441			−0.328280
$696$		0			0
$697$	−27.1224	+	26.1677i	−1.02733	+	0.991173i
$698$		0			0
$699$	35.1321			1.32882
$700$		0			0
$701$	−15.9904			−0.603949			−0.301974	−	0.953316i	$-0.597646\pi$
$701$	−15.9904			−0.603949			−0.301974	+	0.953316i	$0.597646\pi$
$702$		0			0
$703$			24.7033i			0.931702i
$704$		0			0
$705$	33.8123			1.27345
$706$		0			0
$707$		−	0.205144i		−	0.00771524i
$708$		0			0
$709$		−	20.9061i		−	0.785145i	−0.919721	−	0.392573i	$-0.871585\pi$
$709$		−	20.9061i		−	0.785145i	0.919721	−	0.392573i	$-0.128415\pi$
$710$		0			0
$711$		−	25.1302i		−	0.942458i
$712$		0			0
$713$	−17.1279			−0.641446
$714$		0			0
$715$	0.301318			0.0112686
$716$		0			0
$717$			38.6882i			1.44484i
$718$		0			0
$719$			39.2307i			1.46306i	0.681811	+	0.731528i	$0.261193\pi$
$719$			39.2307i			1.46306i	−0.681811	+	0.731528i	$0.738807\pi$
$720$		0			0
$721$		−	0.100187i		−	0.00373117i
$722$		0			0
$723$	26.5846			0.988693
$724$		0			0
$725$			9.97923i			0.370619i
$726$		0			0
$727$	24.1195			0.894544			0.447272	−	0.894398i	$-0.352396\pi$
$727$	24.1195			0.894544			0.447272	+	0.894398i	$0.352396\pi$
$728$		0			0
$729$	12.6127			0.467135
$730$		0			0
$731$	34.6903	+	35.9560i	1.28307	+	1.32988i
$732$		0			0
$733$	6.18605			0.228487			0.114243	−	0.993453i	$-0.463556\pi$
$733$	6.18605			0.228487			0.114243	+	0.993453i	$0.463556\pi$
$734$		0			0
$735$	−28.8543			−1.06431
$736$		0			0
$737$		−	12.7559i		−	0.469868i
$738$		0			0
$739$	−29.9104			−1.10027			−0.550136	−	0.835075i	$-0.685424\pi$
$739$	−29.9104			−1.10027			−0.550136	+	0.835075i	$0.685424\pi$
$740$		0			0
$741$		−	1.69518i		−	0.0622738i
$742$		0			0
$743$			12.4323i			0.456096i	0.973650	+	0.228048i	$0.0732343\pi$
$743$			12.4323i			0.456096i	−0.973650	+	0.228048i	$0.926766\pi$
$744$		0			0
$745$		−	18.8900i		−	0.692076i
$746$		0			0
$747$	−13.7421			−0.502796
$748$		0			0
$749$	−0.228800			−0.00836016
$750$		0			0
$751$		−	25.8425i		−	0.943008i	−0.881864	−	0.471504i	$-0.843711\pi$
$751$		−	25.8425i		−	0.943008i	0.881864	−	0.471504i	$-0.156289\pi$
$752$		0			0
$753$			64.1157i			2.33651i
$754$		0			0
$755$			20.2428i			0.736711i
$756$		0			0
$757$	−22.9825			−0.835314			−0.417657	−	0.908605i	$-0.637149\pi$
$757$	−22.9825			−0.835314			−0.417657	+	0.908605i	$0.637149\pi$
$758$		0			0
$759$			21.1382i			0.767269i
$760$		0			0
$761$	3.32223			0.120431			0.0602154	−	0.998185i	$-0.480821\pi$
$761$	3.32223			0.120431			0.0602154	+	0.998185i	$0.480821\pi$
$762$		0			0
$763$	−0.228793			−0.00828287
$764$		0			0
$765$	−25.6447	+	24.7420i	−0.927185	+	0.894548i
$766$		0			0
$767$	−0.0774426			−0.00279629
$768$		0			0
$769$	30.7764			1.10983			0.554913	−	0.831908i	$-0.312752\pi$
$769$	30.7764			1.10983			0.554913	+	0.831908i	$0.312752\pi$
$770$		0			0
$771$		−	43.7746i		−	1.57650i
$772$		0			0
$773$	−38.8252			−1.39644			−0.698222	−	0.715881i	$-0.746025\pi$
$773$	−38.8252			−1.39644			−0.698222	+	0.715881i	$0.746025\pi$
$774$		0			0
$775$			23.1126i			0.830230i
$776$		0			0
$777$			0.178410i			0.00640042i
$778$		0			0
$779$			65.1127i			2.33291i
$780$		0			0
$781$	−18.0862			−0.647174
$782$		0			0
$783$	−32.9853			−1.17880
$784$		0			0
$785$		−	15.2004i		−	0.542524i
$786$		0			0
$787$		−	22.3091i		−	0.795233i	−0.917552	−	0.397617i	$-0.869837\pi$
$787$		−	22.3091i		−	0.795233i	0.917552	−	0.397617i	$-0.130163\pi$
$788$		0			0
$789$		−	62.2386i		−	2.21575i
$790$		0			0
$791$	−0.140560			−0.00499774
$792$		0			0
$793$		−	0.902069i		−	0.0320334i
$794$		0			0
$795$	21.0184			0.745447
$796$		0			0
$797$	7.96502			0.282136			0.141068	−	0.990000i	$-0.454946\pi$
$797$	7.96502			0.282136			0.141068	+	0.990000i	$0.454946\pi$
$798$		0			0
$799$	−23.4819	−	24.3386i	−0.830730	−	0.861039i
$800$		0			0
$801$	−84.5782			−2.98842
$802$		0			0
$803$	−1.78547			−0.0630077
$804$		0			0
$805$			0.0532667i			0.00187741i
$806$		0			0
$807$	47.4398			1.66996
$808$		0			0
$809$		−	48.9625i		−	1.72143i	−0.509088	−	0.860715i	$-0.670017\pi$
$809$		−	48.9625i		−	1.72143i	0.509088	−	0.860715i	$-0.329983\pi$
$810$		0			0
$811$		−	21.6244i		−	0.759335i	−0.925123	−	0.379667i	$-0.876038\pi$
$811$		−	21.6244i		−	0.759335i	0.925123	−	0.379667i	$-0.123962\pi$
$812$		0			0
$813$			70.3975i			2.46895i
$814$		0			0
$815$	4.92794			0.172618
$816$		0			0
$817$	86.3194			3.01993
$818$		0			0
$819$		−	0.00835312i		−	0.000291881i
$820$		0			0
$821$		−	13.5075i		−	0.471416i	−0.971824	−	0.235708i	$-0.924259\pi$
$821$		−	13.5075i		−	0.471416i	0.971824	−	0.235708i	$-0.0757409\pi$
$822$		0			0
$823$			23.2020i			0.808772i	0.914588	+	0.404386i	$0.132515\pi$
$823$			23.2020i			0.808772i	−0.914588	+	0.404386i	$0.867485\pi$
$824$		0			0
$825$	28.5242			0.993084
$826$		0			0
$827$		−	2.79630i		−	0.0972370i	−0.998817	−	0.0486185i	$-0.984518\pi$
$827$		−	2.79630i		−	0.0972370i	0.998817	−	0.0486185i	$-0.0154819\pi$
$828$		0			0
$829$	33.8118			1.17433			0.587166	−	0.809467i	$-0.300244\pi$
$829$	33.8118			1.17433			0.587166	+	0.809467i	$0.300244\pi$
$830$		0			0
$831$	−18.4102			−0.638643
$832$		0			0
$833$	20.0387	+	20.7698i	0.694300	+	0.719630i
$834$		0			0
$835$	−6.34162			−0.219461
$836$		0			0
$837$	−76.3964			−2.64065
$838$		0			0
$839$		−	31.1933i		−	1.07691i	−0.842653	−	0.538457i	$-0.819008\pi$
$839$		−	31.1933i		−	1.07691i	0.842653	−	0.538457i	$-0.180992\pi$
$840$		0			0
$841$	19.2775			0.664741
$842$		0			0
$843$			92.0519i			3.17044i
$844$		0			0
$845$			17.4312i			0.599652i
$846$		0			0
$847$			0.0433219i			0.00148856i
$848$		0			0
$849$	−75.8699			−2.60385
$850$		0			0
$851$	−8.22487			−0.281945
$852$		0			0
$853$			28.6500i			0.980957i	0.871453	+	0.490479i	$0.163178\pi$
$853$			28.6500i			0.980957i	−0.871453	+	0.490479i	$0.836822\pi$
$854$		0			0
$855$			61.5651i			2.10548i
$856$		0			0
$857$			27.3824i			0.935363i	0.883897	+	0.467682i	$0.154911\pi$
$857$			27.3824i			0.935363i	−0.883897	+	0.467682i	$0.845089\pi$
$858$		0			0
$859$	34.6332			1.18167			0.590835	−	0.806792i	$-0.298798\pi$
$859$	34.6332			1.18167			0.590835	+	0.806792i	$0.298798\pi$
$860$		0			0
$861$			0.470252i			0.0160261i
$862$		0			0
$863$	31.1458			1.06021			0.530107	−	0.847931i	$-0.322152\pi$
$863$	31.1458			1.06021			0.530107	+	0.847931i	$0.322152\pi$
$864$		0			0
$865$	14.1758			0.481991
$866$		0			0
$867$	52.2054	+	1.87114i	1.77299	+	0.0635473i
$868$		0			0
$869$	−11.3135			−0.383783
$870$		0			0
$871$	−0.340589			−0.0115404
$872$		0			0
$873$		−	67.6513i		−	2.28965i
$874$		0			0
$875$	0.184174			0.00622623
$876$		0			0
$877$			15.5106i			0.523755i	0.965101	+	0.261877i	$0.0843416\pi$
$877$			15.5106i			0.523755i	−0.965101	+	0.261877i	$0.915658\pi$
$878$		0			0
$879$		−	10.0534i		−	0.339093i
$880$		0			0
$881$			15.3236i			0.516264i	0.966110	+	0.258132i	$0.0831070\pi$
$881$			15.3236i			0.516264i	−0.966110	+	0.258132i	$0.916893\pi$
$882$		0			0
$883$	15.6592			0.526973			0.263486	−	0.964663i	$-0.415128\pi$
$883$	15.6592			0.526973			0.263486	+	0.964663i	$0.415128\pi$
$884$		0			0
$885$	4.12221			0.138567
$886$		0			0
$887$			38.0075i			1.27617i	0.769968	+	0.638083i	$0.220272\pi$
$887$			38.0075i			1.27617i	−0.769968	+	0.638083i	$0.779728\pi$
$888$		0			0
$889$		−	0.245252i		−	0.00822548i
$890$		0			0
$891$			38.2251i			1.28059i
$892$		0			0
$893$	−58.4297			−1.95528
$894$		0			0
$895$			11.5337i			0.385528i
$896$		0			0
$897$	0.564403			0.0188449
$898$		0			0
$899$	−22.5181			−0.751019
$900$		0			0
$901$	−14.5968	−	15.1294i	−0.486291	−	0.504032i
$902$		0			0
$903$	0.623409			0.0207457
$904$		0			0
$905$	−34.8523			−1.15853
$906$		0			0
$907$		−	9.11767i		−	0.302747i	−0.988477	−	0.151374i	$-0.951630\pi$
$907$		−	9.11767i		−	0.302747i	0.988477	−	0.151374i	$-0.0483697\pi$
$908$		0			0
$909$	78.9428			2.61837
$910$		0			0
$911$		−	35.6473i		−	1.18105i	−0.807020	−	0.590525i	$-0.798921\pi$
$911$		−	35.6473i		−	1.18105i	0.807020	−	0.590525i	$-0.201079\pi$
$912$		0			0
$913$			6.18658i			0.204746i
$914$		0			0
$915$			48.0165i			1.58737i
$916$		0			0
$917$	0.224374			0.00740948
$918$		0			0
$919$	−14.3879			−0.474615			−0.237307	−	0.971435i	$-0.576265\pi$
$919$	−14.3879			−0.474615			−0.237307	+	0.971435i	$0.576265\pi$
$920$		0			0
$921$		−	93.8247i		−	3.09163i
$922$		0			0
$923$			0.482911i			0.0158952i
$924$		0			0
$925$			11.0987i			0.364924i
$926$		0			0
$927$	38.5537			1.26627
$928$		0			0
$929$		−	16.4935i		−	0.541134i	−0.962701	−	0.270567i	$-0.912789\pi$
$929$		−	16.4935i		−	0.541134i	0.962701	−	0.270567i	$-0.0872112\pi$
$930$		0			0
$931$	49.8620			1.63416
$932$		0			0
$933$	−91.4301			−2.99329
$934$		0			0
$935$	11.1387	+	11.5451i	0.364273	+	0.377564i
$936$		0			0
$937$	−19.9691			−0.652362			−0.326181	−	0.945307i	$-0.605762\pi$
$937$	−19.9691			−0.652362			−0.326181	+	0.945307i	$0.605762\pi$
$938$		0			0
$939$	17.5655			0.573228
$940$		0			0
$941$			21.2323i			0.692153i	0.938206	+	0.346077i	$0.112486\pi$
$941$			21.2323i			0.692153i	−0.938206	+	0.346077i	$0.887514\pi$
$942$		0			0
$943$	−21.6791			−0.705967
$944$		0			0
$945$			0.237588i			0.00772873i
$946$		0			0
$947$			28.1076i			0.913375i	0.889627	+	0.456687i	$0.150964\pi$
$947$			28.1076i			0.913375i	−0.889627	+	0.456687i	$0.849036\pi$
$948$		0			0
$949$			0.0476729i			0.00154753i
$950$		0			0
$951$	64.2009			2.08186
$952$		0			0
$953$	−15.3842			−0.498343			−0.249171	−	0.968459i	$-0.580158\pi$
$953$	−15.3842			−0.498343			−0.249171	+	0.968459i	$0.580158\pi$
$954$		0			0
$955$		−	10.7728i		−	0.348599i
$956$		0			0
$957$			27.7904i			0.898336i
$958$		0			0
$959$		−	0.101299i		−	0.00327112i
$960$		0			0
$961$	−21.1535			−0.682370
$962$		0			0
$963$		−	88.0458i		−	2.83724i
$964$		0			0
$965$	17.3200			0.557549
$966$		0			0
$967$	31.7200			1.02005			0.510024	−	0.860160i	$-0.329637\pi$
$967$	31.7200			1.02005			0.510024	+	0.860160i	$0.329637\pi$
$968$		0			0
$969$	64.9510	−	62.6648i	2.08653	−	2.01308i
$970$		0			0
$971$	−45.1752			−1.44974			−0.724870	−	0.688886i	$-0.758100\pi$
$971$	−45.1752			−1.44974			−0.724870	+	0.688886i	$0.758100\pi$
$972$		0			0
$973$	0.108009			0.00346261
$974$		0			0
$975$		−	0.761611i		−	0.0243911i
$976$		0			0
$977$	25.5029			0.815909			0.407954	−	0.913002i	$-0.366242\pi$
$977$	25.5029			0.815909			0.407954	+	0.913002i	$0.366242\pi$
$978$		0			0
$979$			38.0765i			1.21693i
$980$		0			0
$981$		−	88.0432i		−	2.81100i
$982$		0			0
$983$		−	23.4580i		−	0.748193i	−0.927390	−	0.374096i	$-0.877953\pi$
$983$		−	23.4580i		−	0.748193i	0.927390	−	0.374096i	$-0.122047\pi$
$984$		0			0
$985$	−0.817945			−0.0260619
$986$		0			0
$987$	−0.421986			−0.0134320
$988$		0			0
$989$			28.7397i			0.913870i
$990$		0			0
$991$			8.76995i			0.278587i	0.990251	+	0.139293i	$0.0444831\pi$
$991$			8.76995i			0.278587i	−0.990251	+	0.139293i	$0.955517\pi$
$992$		0			0
$993$		−	54.7467i		−	1.73734i
$994$		0			0
$995$	−9.62796			−0.305227
$996$		0			0
$997$			0.519864i			0.0164643i	0.999966	+	0.00823213i	$0.00262040\pi$
$997$			0.519864i			0.0164643i	−0.999966	+	0.00823213i	$0.997380\pi$
$998$		0			0
$999$	−36.6857			−1.16068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.2	✓	40
17.16	even	2	inner	4012.2.b.a.237.39	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.2	✓	40	1.1	even	1	trivial
4012.2.b.a.237.39	yes	40	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)

\(f(q)\)	\(=\)	\(q-3.07288i q^{3} -1.34148i q^{5} +0.0167420i q^{7} -6.44259 q^{9} +O(q^{10})\)	\(q-3.07288i q^{3} -1.34148i q^{5} +0.0167420i q^{7} -6.44259 q^{9} +2.90041i q^{11} +0.0774426 q^{13} -4.12221 q^{15} +(2.86279 + 2.96723i) q^{17} +7.12343 q^{19} +0.0514462 q^{21} +2.37172i q^{23} +3.20042 q^{25} +10.5787i q^{27} +3.11810i q^{29} +7.22174i q^{31} +8.91262 q^{33} +0.0224591 q^{35} +3.46789i q^{37} -0.237972i q^{39} +9.14065i q^{41} +12.1177 q^{43} +8.64262i q^{45} -8.20247 q^{47} +6.99972 q^{49} +(9.11795 - 8.79700i) q^{51} -5.09882 q^{53} +3.89085 q^{55} -21.8894i q^{57} -1.00000 q^{59} -11.6482i q^{61} -0.107862i q^{63} -0.103888i q^{65} -4.39795 q^{67} +7.28801 q^{69} +6.23573i q^{71} +0.615590i q^{73} -9.83452i q^{75} -0.0485587 q^{77} +3.90064i q^{79} +13.1792 q^{81} +2.13300 q^{83} +(3.98049 - 3.84038i) q^{85} +9.58153 q^{87} +13.1280 q^{89} +0.00129655i q^{91} +22.1915 q^{93} -9.55595i q^{95} +10.5006i q^{97} -18.6862i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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