LMFDB - Embedded newform 2340.2.c.d.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2340,2,Mod(181,2340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2340, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2340.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.c (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:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.9144576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 4 $ x^6 + 12x^4 + 36x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.2
Root			$-2.26180i$ of defining polynomial
Character	$\chi$	$=$	2340.181
Dual form			2340.2.c.d.181.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -1.11575i q^{7} +O(q^{10})$	$q-1.00000i q^{5} -1.11575i q^{7} +5.37755i q^{11} +(-3.37755 - 1.26180i) q^{13} -7.90116i q^{19} +6.49330 q^{23} -1.00000 q^{25} -3.63935 q^{29} +3.14605i q^{31} -1.11575 q^{35} -7.40786i q^{37} -4.75510i q^{41} -4.78541 q^{43} -6.16296i q^{47} +5.75510 q^{49} -0.292106 q^{53} +5.37755 q^{55} -11.3776i q^{59} -5.34725 q^{61} +(-1.26180 + 3.37755i) q^{65} -11.8709i q^{67} +3.43816i q^{71} -4.59214i q^{73} +6.00000 q^{77} -10.8157 q^{79} -7.40786i q^{83} +14.8157i q^{89} +(-1.40786 + 3.76850i) q^{91} -7.90116 q^{95} +3.76850i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 12 q^{23} - 6 q^{25} + 12 q^{29} + 12 q^{43} - 6 q^{49} + 12 q^{53} + 12 q^{55} - 12 q^{61} + 6 q^{65} + 36 q^{77} - 24 q^{79} + 12 q^{91}+O(q^{100}) $ 6 * q + 12 * q^23 - 6 * q^25 + 12 * q^29 + 12 * q^43 - 6 * q^49 + 12 * q^53 + 12 * q^55 - 12 * q^61 + 6 * q^65 + 36 * q^77 - 24 * q^79 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	1.11575i		−	0.421714i	−0.977517	−	0.210857i	$-0.932375\pi$
$7$		−	1.11575i		−	0.421714i	0.977517	−	0.210857i	$-0.0676254\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.37755i			1.62139i	0.585467	+	0.810696i	$0.300911\pi$
$11$			5.37755i			1.62139i	−0.585467	+	0.810696i	$0.699089\pi$
$12$		0			0
$13$	−3.37755	−	1.26180i	−0.936764	−	0.349961i
$13$	−3.37755	−	1.26180i	−0.936764	−	0.349961i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		−	7.90116i		−	1.81265i	−0.422582	−	0.906325i	$-0.638876\pi$
$19$		−	7.90116i		−	1.81265i	0.422582	−	0.906325i	$-0.361124\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.49330			1.35395			0.676973	−	0.736007i	$-0.263291\pi$
$23$	6.49330			1.35395			0.676973	+	0.736007i	$0.263291\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.63935			−0.675811			−0.337906	−	0.941180i	$-0.609718\pi$
$29$	−3.63935			−0.675811			−0.337906	+	0.941180i	$0.609718\pi$
$30$		0			0
$31$			3.14605i			0.565048i	0.959260	+	0.282524i	$0.0911716\pi$
$31$			3.14605i			0.565048i	−0.959260	+	0.282524i	$0.908828\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.11575			−0.188596
$36$		0			0
$37$		−	7.40786i		−	1.21784i	−0.793230	−	0.608922i	$-0.791602\pi$
$37$		−	7.40786i		−	1.21784i	0.793230	−	0.608922i	$-0.208398\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.75510i		−	0.742622i	−0.928508	−	0.371311i	$-0.878908\pi$
$41$		−	4.75510i		−	0.742622i	0.928508	−	0.371311i	$-0.121092\pi$
$42$		0			0
$43$	−4.78541			−0.729768			−0.364884	−	0.931053i	$-0.618891\pi$
$43$	−4.78541			−0.729768			−0.364884	+	0.931053i	$0.618891\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.16296i		−	0.898960i	−0.893290	−	0.449480i	$-0.851609\pi$
$47$		−	6.16296i		−	0.898960i	0.893290	−	0.449480i	$-0.148391\pi$
$48$		0			0
$49$	5.75510			0.822158
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.292106			−0.0401238			−0.0200619	−	0.999799i	$-0.506386\pi$
$53$	−0.292106			−0.0401238			−0.0200619	+	0.999799i	$0.506386\pi$
$54$		0			0
$55$	5.37755			0.725109
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	11.3776i		−	1.48123i	−0.671929	−	0.740616i	$-0.734534\pi$
$59$		−	11.3776i		−	1.48123i	0.671929	−	0.740616i	$-0.265466\pi$
$60$		0			0
$61$	−5.34725			−0.684645			−0.342322	−	0.939583i	$-0.611214\pi$
$61$	−5.34725			−0.684645			−0.342322	+	0.939583i	$0.611214\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.26180	+	3.37755i	−0.156507	+	0.418934i
$66$		0			0
$67$		−	11.8709i		−	1.45026i	−0.688614	−	0.725128i	$-0.741781\pi$
$67$		−	11.8709i		−	1.45026i	0.688614	−	0.725128i	$-0.258219\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.43816i			0.408034i	0.978967	+	0.204017i	$0.0653998\pi$
$71$			3.43816i			0.408034i	−0.978967	+	0.204017i	$0.934600\pi$
$72$		0			0
$73$		−	4.59214i		−	0.537470i	−0.963214	−	0.268735i	$-0.913394\pi$
$73$		−	4.59214i		−	0.537470i	0.963214	−	0.268735i	$-0.0866056\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000			0.683763
$78$		0			0
$79$	−10.8157			−1.21686			−0.608431	−	0.793607i	$-0.708201\pi$
$79$	−10.8157			−1.21686			−0.608431	+	0.793607i	$0.708201\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	7.40786i		−	0.813118i	−0.913625	−	0.406559i	$-0.866729\pi$
$83$		−	7.40786i		−	0.813118i	0.913625	−	0.406559i	$-0.133271\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			14.8157i			1.57046i	0.619203	+	0.785231i	$0.287456\pi$
$89$			14.8157i			1.57046i	−0.619203	+	0.785231i	$0.712544\pi$
$90$		0			0
$91$	−1.40786	+	3.76850i	−0.147583	+	0.395046i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.90116			−0.810642
$96$		0			0
$97$			3.76850i			0.382633i	0.981528	+	0.191317i	$0.0612757\pi$
$97$			3.76850i			0.382633i	−0.981528	+	0.191317i	$0.938724\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.292106			0.0290656			0.0145328	−	0.999894i	$-0.495374\pi$
$101$	0.292106			0.0290656			0.0145328	+	0.999894i	$0.495374\pi$
$102$		0			0
$103$	4.03030			0.397118			0.198559	−	0.980089i	$-0.436374\pi$
$103$	4.03030			0.397118			0.198559	+	0.980089i	$0.436374\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.50670			−0.532353			−0.266176	−	0.963924i	$-0.585760\pi$
$107$	−5.50670			−0.532353			−0.266176	+	0.963924i	$0.585760\pi$
$108$		0			0
$109$		−	1.27871i		−	0.122478i	−0.998123	−	0.0612390i	$-0.980495\pi$
$109$		−	1.27871i		−	0.122478i	0.998123	−	0.0612390i	$-0.0195052\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−19.2787			−1.81359			−0.906794	−	0.421574i	$-0.861478\pi$
$113$	−19.2787			−1.81359			−0.906794	+	0.421574i	$0.861478\pi$
$114$		0			0
$115$		−	6.49330i		−	0.605503i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−17.9181			−1.62891
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	7.96970			0.707196			0.353598	−	0.935397i	$-0.384958\pi$
$127$	7.96970			0.707196			0.353598	+	0.935397i	$0.384958\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.27871			−0.635944			−0.317972	−	0.948100i	$-0.603002\pi$
$131$	−7.27871			−0.635944			−0.317972	+	0.948100i	$0.603002\pi$
$132$		0			0
$133$	−8.81571			−0.764419
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.75510i			0.406256i	0.979152	+	0.203128i	$0.0651107\pi$
$137$			4.75510i			0.406256i	−0.979152	+	0.203128i	$0.934889\pi$
$138$		0			0
$139$	−16.3259			−1.38475			−0.692373	−	0.721540i	$-0.743435\pi$
$139$	−16.3259			−1.38475			−0.692373	+	0.721540i	$0.743435\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.78541	−	18.1630i	0.567424	−	1.51886i
$144$		0			0
$145$			3.63935i			0.302232i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	8.81571i		−	0.722211i	−0.932525	−	0.361106i	$-0.882399\pi$
$149$		−	8.81571i		−	0.722211i	0.932525	−	0.361106i	$-0.117601\pi$
$150$		0			0
$151$		−	19.9012i		−	1.61953i	−0.586752	−	0.809767i	$-0.699594\pi$
$151$		−	19.9012i		−	1.61953i	0.586752	−	0.809767i	$-0.300406\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.14605			0.252697
$156$		0			0
$157$	18.7551			1.49682			0.748410	−	0.663236i	$-0.230818\pi$
$157$	18.7551			1.49682			0.748410	+	0.663236i	$0.230818\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	7.24490i		−	0.570978i
$162$		0			0
$163$		−	13.4417i		−	1.05283i	−0.850227	−	0.526416i	$-0.823535\pi$
$163$		−	13.4417i		−	1.05283i	0.850227	−	0.526416i	$-0.176465\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.91806i		−	0.380571i	−0.981729	−	0.190286i	$-0.939059\pi$
$167$		−	4.91806i		−	0.380571i	0.981729	−	0.190286i	$-0.0609414\pi$
$168$		0			0
$169$	9.81571	+	8.52360i	0.755055	+	0.655662i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	19.5708			1.48794			0.743971	−	0.668212i	$-0.232940\pi$
$173$	19.5708			1.48794			0.743971	+	0.668212i	$0.232940\pi$
$174$		0			0
$175$			1.11575i			0.0843427i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.9866			0.970664			0.485332	−	0.874330i	$-0.338699\pi$
$179$	12.9866			0.970664			0.485332	+	0.874330i	$0.338699\pi$
$180$		0			0
$181$	9.40786			0.699280			0.349640	−	0.936884i	$-0.386304\pi$
$181$	9.40786			0.699280			0.349640	+	0.936884i	$0.386304\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.40786			−0.544636
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−26.5574			−1.92163			−0.960814	−	0.277195i	$-0.910595\pi$
$191$	−26.5574			−1.92163			−0.960814	+	0.277195i	$0.910595\pi$
$192$		0			0
$193$		−	21.8023i		−	1.56936i	−0.619898	−	0.784682i	$-0.712826\pi$
$193$		−	21.8023i		−	1.56936i	0.619898	−	0.784682i	$-0.287174\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	7.57081i		−	0.539398i	−0.962945	−	0.269699i	$-0.913076\pi$
$197$		−	7.57081i		−	0.539398i	0.962945	−	0.269699i	$-0.0869242\pi$
$198$		0			0
$199$	17.5708			1.24556			0.622781	−	0.782396i	$-0.286003\pi$
$199$	17.5708			1.24556			0.622781	+	0.782396i	$0.286003\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			4.06061i			0.284999i
$204$		0			0
$205$	−4.75510			−0.332111
$206$		0			0
$207$		0			0
$208$		0			0
$209$	42.4889			2.93902
$210$		0			0
$211$	−0.0606069			−0.00417235			−0.00208618	−	0.999998i	$-0.500664\pi$
$211$	−0.0606069			−0.00417235			−0.00208618	+	0.999998i	$0.500664\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.78541i			0.326362i
$216$		0			0
$217$	3.51021			0.238288
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			22.6260i			1.51515i	0.652750	+	0.757573i	$0.273615\pi$
$223$			22.6260i			1.51515i	−0.652750	+	0.757573i	$0.726385\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			4.59214i			0.304791i	0.988320	+	0.152396i	$0.0486988\pi$
$227$			4.59214i			0.304791i	−0.988320	+	0.152396i	$0.951301\pi$
$228$		0			0
$229$			0.986602i			0.0651965i	0.999469	+	0.0325983i	$0.0103782\pi$
$229$			0.986602i			0.0651965i	−0.999469	+	0.0325983i	$0.989622\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.2787			−0.869917			−0.434959	−	0.900450i	$-0.643237\pi$
$233$	−13.2787			−0.869917			−0.434959	+	0.900450i	$0.643237\pi$
$234$		0			0
$235$	−6.16296			−0.402027
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	25.6429i		−	1.65870i	−0.558730	−	0.829349i	$-0.688711\pi$
$239$		−	25.6429i		−	1.65870i	0.558730	−	0.829349i	$-0.311289\pi$
$240$		0			0
$241$		−	14.5236i		−	0.935548i	−0.883848	−	0.467774i	$-0.845056\pi$
$241$		−	14.5236i		−	0.935548i	0.883848	−	0.467774i	$-0.154944\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	5.75510i		−	0.367680i
$246$		0			0
$247$	−9.96970	+	26.6866i	−0.634357	+	1.69803i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.2653			−1.27914			−0.639568	−	0.768735i	$-0.720887\pi$
$251$	−20.2653			−1.27914			−0.639568	+	0.768735i	$0.720887\pi$
$252$		0			0
$253$			34.9181i			2.19528i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.58421			−0.410712			−0.205356	−	0.978687i	$-0.565835\pi$
$257$	−6.58421			−0.410712			−0.205356	+	0.978687i	$0.565835\pi$
$258$		0			0
$259$	−8.26531			−0.513581
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.7854			1.15836			0.579179	−	0.815200i	$-0.303373\pi$
$263$	18.7854			1.15836			0.579179	+	0.815200i	$0.303373\pi$
$264$		0			0
$265$			0.292106i			0.0179439i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.8495			0.905391			0.452696	−	0.891665i	$-0.350462\pi$
$269$	14.8495			0.905391			0.452696	+	0.891665i	$0.350462\pi$
$270$		0			0
$271$			20.8878i			1.26884i	0.772988	+	0.634420i	$0.218761\pi$
$271$			20.8878i			1.26884i	−0.772988	+	0.634420i	$0.781239\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	5.37755i		−	0.324279i
$276$		0			0
$277$	22.2653			1.33779			0.668896	−	0.743356i	$-0.266767\pi$
$277$	22.2653			1.33779			0.668896	+	0.743356i	$0.266767\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			7.57081i			0.451637i	0.974169	+	0.225818i	$0.0725056\pi$
$281$			7.57081i			0.451637i	−0.974169	+	0.225818i	$0.927494\pi$
$282$		0			0
$283$	0.724800			0.0430849			0.0215424	−	0.999768i	$-0.493142\pi$
$283$	0.724800			0.0430849			0.0215424	+	0.999768i	$0.493142\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.30550			−0.313174
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			10.2236i			0.597267i	0.954368	+	0.298634i	$0.0965308\pi$
$293$			10.2236i			0.597267i	−0.954368	+	0.298634i	$0.903469\pi$
$294$		0			0
$295$	−11.3776			−0.662427
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−21.9315	−	8.19326i	−1.26833	−	0.473829i
$300$		0			0
$301$			5.33931i			0.307753i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			5.34725i			0.306183i
$306$		0			0
$307$		−	21.6394i		−	1.23502i	−0.786562	−	0.617512i	$-0.788141\pi$
$307$		−	21.6394i		−	1.23502i	0.786562	−	0.617512i	$-0.211859\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.986602			−0.0559451			−0.0279725	−	0.999609i	$-0.508905\pi$
$311$	−0.986602			−0.0559451			−0.0279725	+	0.999609i	$0.508905\pi$
$312$		0			0
$313$	25.8361			1.46034			0.730172	−	0.683263i	$-0.239440\pi$
$313$	25.8361			1.46034			0.730172	+	0.683263i	$0.239440\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			26.4283i			1.48436i	0.670201	+	0.742180i	$0.266208\pi$
$317$			26.4283i			1.48436i	−0.670201	+	0.742180i	$0.733792\pi$
$318$		0			0
$319$		−	19.5708i		−	1.09576i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	3.37755	+	1.26180i	0.187353	+	0.0699922i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.87632			−0.379104
$330$		0			0
$331$		−	14.8878i		−	0.818305i	−0.912466	−	0.409153i	$-0.865824\pi$
$331$		−	14.8878i		−	0.818305i	0.912466	−	0.409153i	$-0.134176\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.8709			−0.648574
$336$		0			0
$337$	−20.0000			−1.08947			−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000			−1.08947			−0.544735	+	0.838608i	$0.683370\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.9181			−0.916164
$342$		0			0
$343$		−	14.2315i		−	0.768429i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	26.0641			1.39919			0.699597	−	0.714537i	$-0.253363\pi$
$347$	26.0641			1.39919			0.699597	+	0.714537i	$0.253363\pi$
$348$		0			0
$349$			10.7551i			0.575707i	0.957674	+	0.287854i	$0.0929417\pi$
$349$			10.7551i			0.575707i	−0.957674	+	0.287854i	$0.907058\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			35.6126i			1.89547i	0.319066	+	0.947733i	$0.396631\pi$
$353$			35.6126i			1.89547i	−0.319066	+	0.947733i	$0.603369\pi$
$354$		0			0
$355$	3.43816			0.182479
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.07205i			0.320470i	0.987079	+	0.160235i	$0.0512253\pi$
$359$			6.07205i			0.320470i	−0.987079	+	0.160235i	$0.948775\pi$
$360$		0			0
$361$	−43.4283			−2.28570
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.59214			−0.240364
$366$		0			0
$367$	26.7854			1.39819			0.699093	−	0.715030i	$-0.253587\pi$
$367$	26.7854			1.39819			0.699093	+	0.715030i	$0.253587\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			0.325917i			0.0169208i
$372$		0			0
$373$	10.8763			0.563154			0.281577	−	0.959539i	$-0.409142\pi$
$373$	10.8763			0.563154			0.281577	+	0.959539i	$0.409142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.2921	+	4.59214i	0.633076	+	0.236507i
$378$		0			0
$379$			5.96176i			0.306235i	0.988208	+	0.153118i	$0.0489313\pi$
$379$			5.96176i			0.306235i	−0.988208	+	0.153118i	$0.951069\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			19.4079i			0.991695i	0.868410	+	0.495848i	$0.165143\pi$
$383$			19.4079i			0.991695i	−0.868410	+	0.495848i	$0.834857\pi$
$384$		0			0
$385$		−	6.00000i		−	0.305788i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.55742			0.433878			0.216939	−	0.976185i	$-0.430393\pi$
$389$	8.55742			0.433878			0.216939	+	0.976185i	$0.430393\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			10.8157i			0.544197i
$396$		0			0
$397$		−	14.9449i		−	0.750061i	−0.927012	−	0.375030i	$-0.877632\pi$
$397$		−	14.9449i		−	0.750061i	0.927012	−	0.375030i	$-0.122368\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			27.1416i			1.35539i	0.735344	+	0.677694i	$0.237021\pi$
$401$			27.1416i			1.35539i	−0.735344	+	0.677694i	$0.762979\pi$
$402$		0			0
$403$	3.96970	−	10.6260i	0.197745	−	0.529317i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	39.8361			1.97460
$408$		0			0
$409$			33.2181i			1.64253i	0.570547	+	0.821265i	$0.306731\pi$
$409$			33.2181i			1.64253i	−0.570547	+	0.821265i	$0.693269\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.6945			−0.624655
$414$		0			0
$415$	−7.40786			−0.363637
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.7079			0.865087			0.432544	−	0.901613i	$-0.357616\pi$
$419$	17.7079			0.865087			0.432544	+	0.901613i	$0.357616\pi$
$420$		0			0
$421$			22.2047i			1.08219i	0.840961	+	0.541096i	$0.181990\pi$
$421$			22.2047i			1.08219i	−0.840961	+	0.541096i	$0.818010\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$			5.96619i			0.288724i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	10.6831i		−	0.514585i	−0.966334	−	0.257292i	$-0.917170\pi$
$431$		−	10.6831i		−	0.514585i	0.966334	−	0.257292i	$-0.0828303\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	51.3046i		−	2.45423i
$438$		0			0
$439$	−5.24490			−0.250325			−0.125163	−	0.992136i	$-0.539945\pi$
$439$	−5.24490			−0.250325			−0.125163	+	0.992136i	$0.539945\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−25.4799			−1.21059			−0.605293	−	0.796002i	$-0.706944\pi$
$443$	−25.4799			−1.21059			−0.605293	+	0.796002i	$0.706944\pi$
$444$		0			0
$445$	14.8157			0.702332
$446$		0			0
$447$		0			0
$448$		0			0
$449$			13.9394i			0.657841i	0.944358	+	0.328920i	$0.106685\pi$
$449$			13.9394i			0.657841i	−0.944358	+	0.328920i	$0.893315\pi$
$450$		0			0
$451$	25.5708			1.20408
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.76850	+	1.40786i	0.176670	+	0.0660013i
$456$		0			0
$457$			12.6945i			0.593823i	0.954905	+	0.296912i	$0.0959567\pi$
$457$			12.6945i			0.593823i	−0.954905	+	0.296912i	$0.904043\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			23.0810i			1.07499i	0.843267	+	0.537495i	$0.180629\pi$
$461$			23.0810i			1.07499i	−0.843267	+	0.537495i	$0.819371\pi$
$462$		0			0
$463$			6.16296i			0.286417i	0.989693	+	0.143208i	$0.0457419\pi$
$463$			6.16296i			0.286417i	−0.989693	+	0.143208i	$0.954258\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	23.5067			1.08776			0.543880	−	0.839163i	$-0.316955\pi$
$467$	23.5067			1.08776			0.543880	+	0.839163i	$0.316955\pi$
$468$		0			0
$469$	−13.2449			−0.611593
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	25.7338i		−	1.18324i
$474$		0			0
$475$			7.90116i			0.362530i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			33.2137i			1.51757i	0.651340	+	0.758786i	$0.274207\pi$
$479$			33.2137i			1.51757i	−0.651340	+	0.758786i	$0.725793\pi$
$480$		0			0
$481$	−9.34725	+	25.0204i	−0.426198	+	1.14083i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.76850			0.171119
$486$		0			0
$487$		−	18.7472i		−	0.849515i	−0.905307	−	0.424758i	$-0.860359\pi$
$487$		−	18.7472i		−	0.849515i	0.905307	−	0.424758i	$-0.139641\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−26.5574			−1.19852			−0.599260	−	0.800555i	$-0.704538\pi$
$491$	−26.5574			−1.19852			−0.599260	+	0.800555i	$0.704538\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.83612			0.172074
$498$		0			0
$499$		−	26.8539i		−	1.20215i	−0.799193	−	0.601074i	$-0.794740\pi$
$499$		−	26.8539i		−	1.20215i	0.799193	−	0.601074i	$-0.205260\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20.4665			0.912556			0.456278	−	0.889837i	$-0.349182\pi$
$503$	20.4665			0.912556			0.456278	+	0.889837i	$0.349182\pi$
$504$		0			0
$505$		−	0.292106i		−	0.0129985i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	26.9598i		−	1.19497i	−0.801879	−	0.597486i	$-0.796166\pi$
$509$		−	26.9598i		−	1.19497i	0.801879	−	0.597486i	$-0.203834\pi$
$510$		0			0
$511$	−5.12368			−0.226658
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	4.03030i		−	0.177596i
$516$		0			0
$517$	33.1416			1.45757
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.36065			0.366287			0.183143	−	0.983086i	$-0.441373\pi$
$521$	8.36065			0.366287			0.183143	+	0.983086i	$0.441373\pi$
$522$		0			0
$523$	−18.7248			−0.818778			−0.409389	−	0.912360i	$-0.634258\pi$
$523$	−18.7248			−0.818778			−0.409389	+	0.912360i	$0.634258\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	19.1630			0.833172
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.00000	+	16.0606i	−0.259889	+	0.695662i
$534$		0			0
$535$			5.50670i			0.238075i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			30.9484i			1.33304i
$540$		0			0
$541$			42.9439i			1.84630i	0.384435	+	0.923152i	$0.374396\pi$
$541$			42.9439i			1.84630i	−0.384435	+	0.923152i	$0.625604\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.27871			−0.0547738
$546$		0			0
$547$	14.0909			0.602484			0.301242	−	0.953548i	$-0.402599\pi$
$547$	14.0909			0.602484			0.301242	+	0.953548i	$0.402599\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			28.7551i			1.22501i
$552$		0			0
$553$			12.0676i			0.513167i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	0.713359i		−	0.0302260i	−0.999886	−	0.0151130i	$-0.995189\pi$
$557$		−	0.713359i		−	0.0302260i	0.999886	−	0.0151130i	$-0.00481080\pi$
$558$		0			0
$559$	16.1630	+	6.03824i	0.683620	+	0.255390i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.2012			0.514219			0.257110	−	0.966382i	$-0.417230\pi$
$563$	12.2012			0.514219			0.257110	+	0.966382i	$0.417230\pi$
$564$		0			0
$565$			19.2787i			0.811061i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.6260			−0.696996			−0.348498	−	0.937309i	$-0.613308\pi$
$569$	−16.6260			−0.696996			−0.348498	+	0.937309i	$0.613308\pi$
$570$		0			0
$571$	−20.3259			−0.850613			−0.425307	−	0.905049i	$-0.639834\pi$
$571$	−20.3259			−0.850613			−0.425307	+	0.905049i	$0.639834\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.49330			−0.270789
$576$		0			0
$577$			19.1496i			0.797207i	0.917123	+	0.398603i	$0.130505\pi$
$577$			19.1496i			0.797207i	−0.917123	+	0.398603i	$0.869495\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.26531			−0.342903
$582$		0			0
$583$		−	1.57081i		−	0.0650564i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	30.4889i		−	1.25841i	−0.777239	−	0.629205i	$-0.783380\pi$
$587$		−	30.4889i		−	1.25841i	0.777239	−	0.629205i	$-0.216620\pi$
$588$		0			0
$589$	24.8575			1.02423
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	42.3259i		−	1.73812i	−0.494709	−	0.869059i	$-0.664725\pi$
$593$		−	42.3259i		−	1.73812i	0.494709	−	0.869059i	$-0.335275\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−15.5440			−0.635111			−0.317556	−	0.948240i	$-0.602862\pi$
$599$	−15.5440			−0.635111			−0.317556	+	0.948240i	$0.602862\pi$
$600$		0			0
$601$	25.1416			1.02555			0.512774	−	0.858524i	$-0.328618\pi$
$601$	25.1416			1.02555			0.512774	+	0.858524i	$0.328618\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			17.9181i			0.728473i
$606$		0			0
$607$	−10.0909			−0.409577			−0.204789	−	0.978806i	$-0.565651\pi$
$607$	−10.0909			−0.409577			−0.204789	+	0.978806i	$0.565651\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.77643	+	20.8157i	−0.314601	+	0.842113i
$612$		0			0
$613$			37.0204i			1.49524i	0.664127	+	0.747620i	$0.268804\pi$
$613$			37.0204i			1.49524i	−0.664127	+	0.747620i	$0.731196\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.63389i		−	0.347587i	−0.984782	−	0.173794i	$-0.944397\pi$
$617$		−	8.63389i		−	0.347587i	0.984782	−	0.173794i	$-0.0556026\pi$
$618$		0			0
$619$			9.47197i			0.380711i	0.981715	+	0.190355i	$0.0609640\pi$
$619$			9.47197i			0.380711i	−0.981715	+	0.190355i	$0.939036\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.5306			0.662285
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	18.0382i		−	0.718091i	−0.933320	−	0.359045i	$-0.883102\pi$
$631$		−	18.0382i		−	0.718091i	0.933320	−	0.359045i	$-0.116898\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	7.96970i		−	0.316268i
$636$		0			0
$637$	−19.4382	−	7.26180i	−0.770168	−	0.287723i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.4158			−0.687882			−0.343941	−	0.938991i	$-0.611762\pi$
$641$	−17.4158			−0.687882			−0.343941	+	0.938991i	$0.611762\pi$
$642$		0			0
$643$			15.3472i			0.605236i	0.953112	+	0.302618i	$0.0978607\pi$
$643$			15.3472i			0.605236i	−0.953112	+	0.302618i	$0.902139\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.05072			0.119936			0.0599680	−	0.998200i	$-0.480900\pi$
$647$	3.05072			0.119936			0.0599680	+	0.998200i	$0.480900\pi$
$648$		0			0
$649$	61.1834			2.40166
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.5708			1.00066			0.500332	−	0.865834i	$-0.333211\pi$
$653$	25.5708			1.00066			0.500332	+	0.865834i	$0.333211\pi$
$654$		0			0
$655$			7.27871i			0.284403i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.29211			−0.245106			−0.122553	−	0.992462i	$-0.539108\pi$
$659$	−6.29211			−0.245106			−0.122553	+	0.992462i	$0.539108\pi$
$660$		0			0
$661$			39.5102i			1.53677i	0.639989	+	0.768384i	$0.278939\pi$
$661$			39.5102i			1.53677i	−0.639989	+	0.768384i	$0.721061\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			8.81571i			0.341859i
$666$		0			0
$667$	−23.6314			−0.915012
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	28.7551i		−	1.11008i
$672$		0			0
$673$	1.30550			0.0503235			0.0251617	−	0.999683i	$-0.491990\pi$
$673$	1.30550			0.0503235			0.0251617	+	0.999683i	$0.491990\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.01340			−0.192681			−0.0963403	−	0.995348i	$-0.530714\pi$
$677$	−5.01340			−0.192681			−0.0963403	+	0.995348i	$0.530714\pi$
$678$		0			0
$679$	4.20470			0.161362
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	19.4079i		−	0.742621i	−0.928509	−	0.371310i	$-0.878909\pi$
$683$		−	19.4079i		−	0.742621i	0.928509	−	0.371310i	$-0.121091\pi$
$684$		0			0
$685$	4.75510			0.181683
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.986602	+	0.368580i	0.0375865	+	0.0140418i
$690$		0			0
$691$		−	2.56184i		−	0.0974570i	−0.998812	−	0.0487285i	$-0.984483\pi$
$691$		−	2.56184i		−	0.0974570i	0.998812	−	0.0487285i	$-0.0155169\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16.3259i			0.619277i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.8495			−1.46733			−0.733663	−	0.679513i	$-0.762191\pi$
$701$	−38.8495			−1.46733			−0.733663	+	0.679513i	$0.762191\pi$
$702$		0			0
$703$	−58.5306			−2.20752
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	0.325917i		−	0.0122574i
$708$		0			0
$709$		−	35.3393i		−	1.32720i	−0.748089	−	0.663598i	$-0.769029\pi$
$709$		−	35.3393i		−	1.32720i	0.748089	−	0.663598i	$-0.230971\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			20.4283i			0.765045i
$714$		0			0
$715$	−18.1630	−	6.78541i	−0.679256	−	0.253760i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.11028			−0.227875			−0.113938	−	0.993488i	$-0.536346\pi$
$719$	−6.11028			−0.227875			−0.113938	+	0.993488i	$0.536346\pi$
$720$		0			0
$721$		−	4.49681i		−	0.167470i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.63935			0.135162
$726$		0			0
$727$	1.05072			0.0389689			0.0194845	−	0.999810i	$-0.493798\pi$
$727$	1.05072			0.0389689			0.0194845	+	0.999810i	$0.493798\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	31.5708i		−	1.16609i	−0.812438	−	0.583047i	$-0.801860\pi$
$733$		−	31.5708i		−	1.16609i	0.812438	−	0.583047i	$-0.198140\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	63.8361			2.35143
$738$		0			0
$739$			31.8673i			1.17226i	0.810217	+	0.586130i	$0.199349\pi$
$739$			31.8673i			1.17226i	−0.810217	+	0.586130i	$0.800651\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	39.8550i		−	1.46214i	−0.682304	−	0.731069i	$-0.739022\pi$
$743$		−	39.8550i		−	1.46214i	0.682304	−	0.731069i	$-0.260978\pi$
$744$		0			0
$745$	−8.81571			−0.322983
$746$		0			0
$747$		0			0
$748$		0			0
$749$			6.14410i			0.224500i
$750$		0			0
$751$	9.24490			0.337351			0.168676	−	0.985672i	$-0.446051\pi$
$751$	9.24490			0.337351			0.168676	+	0.985672i	$0.446051\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−19.9012			−0.724277
$756$		0			0
$757$	−22.5912			−0.821092			−0.410546	−	0.911840i	$-0.634662\pi$
$757$	−22.5912			−0.821092			−0.410546	+	0.911840i	$0.634662\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	16.2047i		−	0.587420i	−0.955895	−	0.293710i	$-0.905110\pi$
$761$		−	16.2047i		−	0.587420i	0.955895	−	0.293710i	$-0.0948900\pi$
$762$		0			0
$763$	−1.42672			−0.0516506
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.3562	+	38.4283i	−0.518373	+	1.38756i
$768$		0			0
$769$		−	16.7889i		−	0.605424i	−0.953082	−	0.302712i	$-0.902108\pi$
$769$		−	16.7889i		−	0.605424i	0.953082	−	0.302712i	$-0.0978920\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			16.5921i			0.596778i	0.954444	+	0.298389i	$0.0964493\pi$
$773$			16.5921i			0.596778i	−0.954444	+	0.298389i	$0.903551\pi$
$774$		0			0
$775$		−	3.14605i		−	0.113010i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−37.5708			−1.34611
$780$		0			0
$781$	−18.4889			−0.661584
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	18.7551i		−	0.669398i
$786$		0			0
$787$		−	31.1496i		−	1.11036i	−0.831730	−	0.555181i	$-0.812649\pi$
$787$		−	31.1496i		−	1.11036i	0.831730	−	0.555181i	$-0.187351\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			21.5102i			0.764815i
$792$		0			0
$793$	18.0606	+	6.74717i	0.641351	+	0.239599i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.402391			0.0142534			0.00712670	−	0.999975i	$-0.497731\pi$
$797$	0.402391			0.0142534			0.00712670	+	0.999975i	$0.497731\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	24.6945			0.871450
$804$		0			0
$805$	−7.24490			−0.255349
$806$		0			0
$807$		0			0
$808$		0			0
$809$	55.3652			1.94654			0.973268	−	0.229671i	$-0.0737651\pi$
$809$	55.3652			1.94654			0.973268	+	0.229671i	$0.0737651\pi$
$810$		0			0
$811$		−	27.1461i		−	0.953227i	−0.879113	−	0.476613i	$-0.841864\pi$
$811$		−	27.1461i		−	0.953227i	0.879113	−	0.476613i	$-0.158136\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.4417			−0.470841
$816$		0			0
$817$			37.8102i			1.32281i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			27.8361i			0.971487i	0.874101	+	0.485744i	$0.161451\pi$
$821$			27.8361i			0.971487i	−0.874101	+	0.485744i	$0.838549\pi$
$822$		0			0
$823$	25.2146			0.878925			0.439463	−	0.898261i	$-0.355169\pi$
$823$	25.2146			0.878925			0.439463	+	0.898261i	$0.355169\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.03928i			0.0361391i	0.999837	+	0.0180696i	$0.00575204\pi$
$827$			1.03928i			0.0361391i	−0.999837	+	0.0180696i	$0.994248\pi$
$828$		0			0
$829$	6.97867			0.242379			0.121190	−	0.992629i	$-0.461329\pi$
$829$	6.97867			0.242379			0.121190	+	0.992629i	$0.461329\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−4.91806			−0.170197
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	13.3169i		−	0.459752i	−0.973220	−	0.229876i	$-0.926168\pi$
$839$		−	13.3169i		−	0.459752i	0.973220	−	0.229876i	$-0.0738321\pi$
$840$		0			0
$841$	−15.7551			−0.543279
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.52360	−	9.81571i	0.293221	−	0.337671i
$846$		0			0
$847$			19.9921i			0.686936i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	48.1014i		−	1.64890i
$852$		0			0
$853$		−	30.5653i		−	1.04654i	−0.852168	−	0.523269i	$-0.824712\pi$
$853$		−	30.5653i		−	1.04654i	0.852168	−	0.523269i	$-0.175288\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	52.4203			1.79064			0.895322	−	0.445419i	$-0.146945\pi$
$857$	52.4203			1.79064			0.895322	+	0.445419i	$0.146945\pi$
$858$		0			0
$859$	19.1416			0.653104			0.326552	−	0.945179i	$-0.394113\pi$
$859$	19.1416			0.653104			0.326552	+	0.945179i	$0.394113\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.8575i			1.25464i	0.778761	+	0.627321i	$0.215849\pi$
$863$			36.8575i			1.25464i	−0.778761	+	0.627321i	$0.784151\pi$
$864$		0			0
$865$		−	19.5708i		−	0.665428i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	58.1620i		−	1.97301i
$870$		0			0
$871$	−14.9787	+	40.0944i	−0.507533	+	1.35855i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.11575			0.0377192
$876$		0			0
$877$			37.5370i			1.26753i	0.773524	+	0.633767i	$0.218492\pi$
$877$			37.5370i			1.26753i	−0.773524	+	0.633767i	$0.781508\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15.8212			0.533029			0.266514	−	0.963831i	$-0.414128\pi$
$881$	15.8212			0.533029			0.266514	+	0.963831i	$0.414128\pi$
$882$		0			0
$883$	49.0507			1.65069			0.825344	−	0.564630i	$-0.190981\pi$
$883$	49.0507			1.65069			0.825344	+	0.564630i	$0.190981\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.0909			−0.405973			−0.202987	−	0.979182i	$-0.565065\pi$
$887$	−12.0909			−0.405973			−0.202987	+	0.979182i	$0.565065\pi$
$888$		0			0
$889$		−	8.89218i		−	0.298234i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−48.6945			−1.62950
$894$		0			0
$895$		−	12.9866i		−	0.434094i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	11.4496i		−	0.381866i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	9.40786i		−	0.312728i
$906$		0			0
$907$	9.54051			0.316787			0.158394	−	0.987376i	$-0.449368\pi$
$907$	9.54051			0.316787			0.158394	+	0.987376i	$0.449368\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26.7392			0.885910			0.442955	−	0.896544i	$-0.353930\pi$
$911$	26.7392			0.885910			0.442955	+	0.896544i	$0.353930\pi$
$912$		0			0
$913$	39.8361			1.31838
$914$		0			0
$915$		0			0
$916$		0			0
$917$			8.12121i			0.268186i
$918$		0			0
$919$	51.0810			1.68501			0.842504	−	0.538691i	$-0.181081\pi$
$919$	51.0810			1.68501			0.842504	+	0.538691i	$0.181081\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.33828	−	11.6126i	0.142796	−	0.382232i
$924$		0			0
$925$			7.40786i			0.243569i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			11.1237i			0.364956i	0.983210	+	0.182478i	$0.0584119\pi$
$929$			11.1237i			0.364956i	−0.983210	+	0.182478i	$0.941588\pi$
$930$		0			0
$931$		−	45.4720i		−	1.49028i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	15.3055			0.500009			0.250005	−	0.968245i	$-0.419568\pi$
$937$	15.3055			0.500009			0.250005	+	0.968245i	$0.419568\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			28.7124i			0.935999i	0.883729	+	0.467999i	$0.155025\pi$
$941$			28.7124i			0.935999i	−0.883729	+	0.467999i	$0.844975\pi$
$942$		0			0
$943$		−	30.8763i		−	1.00547i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	14.1024i		−	0.458265i	−0.973395	−	0.229132i	$-0.926411\pi$
$947$		−	14.1024i		−	0.458265i	0.973395	−	0.229132i	$-0.0735889\pi$
$948$		0			0
$949$	−5.79438	+	15.5102i	−0.188093	+	0.503483i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.8361			−0.512982			−0.256491	−	0.966547i	$-0.582566\pi$
$953$	−15.8361			−0.512982			−0.256491	+	0.966547i	$0.582566\pi$
$954$		0			0
$955$			26.5574i			0.859378i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.30550			0.171324
$960$		0			0
$961$	21.1024			0.680721
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−21.8023			−0.701841
$966$		0			0
$967$		−	0.713359i		−	0.0229401i	−0.999934	−	0.0114700i	$-0.996349\pi$
$967$		−	0.713359i		−	0.0229401i	0.999934	−	0.0114700i	$-0.00365111\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.8182			0.379263			0.189632	−	0.981855i	$-0.439271\pi$
$971$	11.8182			0.379263			0.189632	+	0.981855i	$0.439271\pi$
$972$		0			0
$973$			18.2156i			0.583966i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			16.9181i			0.541257i	0.962684	+	0.270628i	$0.0872315\pi$
$977$			16.9181i			0.541257i	−0.962684	+	0.270628i	$0.912769\pi$
$978$		0			0
$979$	−79.6722			−2.54634
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	12.5315i		−	0.399694i	−0.979827	−	0.199847i	$-0.935955\pi$
$983$		−	12.5315i		−	0.399694i	0.979827	−	0.199847i	$-0.0640445\pi$
$984$		0			0
$985$	−7.57081			−0.241226
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−31.0731			−0.988067
$990$		0			0
$991$	−49.6543			−1.57732			−0.788660	−	0.614829i	$-0.789225\pi$
$991$	−49.6543			−1.57732			−0.788660	+	0.614829i	$0.789225\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	17.5708i		−	0.557032i
$996$		0			0
$997$	−54.5912			−1.72892			−0.864461	−	0.502700i	$-0.832340\pi$
$997$	−54.5912			−1.72892			−0.864461	+	0.502700i	$0.832340\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.c.d.181.2		6
3.2	odd	2		260.2.f.a.181.2	yes	6
12.11	even	2		1040.2.k.c.961.6		6
13.12	even	2	inner	2340.2.c.d.181.5		6
15.2	even	4		1300.2.d.d.649.5		6
15.8	even	4		1300.2.d.c.649.2		6
15.14	odd	2		1300.2.f.e.701.6		6
39.5	even	4		3380.2.a.n.1.1		3
39.8	even	4		3380.2.a.m.1.1		3
39.38	odd	2		260.2.f.a.181.1	✓	6
156.155	even	2		1040.2.k.c.961.5		6
195.38	even	4		1300.2.d.d.649.2		6
195.77	even	4		1300.2.d.c.649.5		6
195.194	odd	2		1300.2.f.e.701.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.1	✓	6	39.38	odd	2
260.2.f.a.181.2	yes	6	3.2	odd	2
1040.2.k.c.961.5		6	156.155	even	2
1040.2.k.c.961.6		6	12.11	even	2
1300.2.d.c.649.2		6	15.8	even	4
1300.2.d.c.649.5		6	195.77	even	4
1300.2.d.d.649.2		6	195.38	even	4
1300.2.d.d.649.5		6	15.2	even	4
1300.2.f.e.701.5		6	195.194	odd	2
1300.2.f.e.701.6		6	15.14	odd	2
2340.2.c.d.181.2		6	1.1	even	1	trivial
2340.2.c.d.181.5		6	13.12	even	2	inner
3380.2.a.m.1.1		3	39.8	even	4
3380.2.a.n.1.1		3	39.5	even	4

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:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -1.11575i q^{7} +O(q^{10})\)	\(q-1.00000i q^{5} -1.11575i q^{7} +5.37755i q^{11} +(-3.37755 - 1.26180i) q^{13} -7.90116i q^{19} +6.49330 q^{23} -1.00000 q^{25} -3.63935 q^{29} +3.14605i q^{31} -1.11575 q^{35} -7.40786i q^{37} -4.75510i q^{41} -4.78541 q^{43} -6.16296i q^{47} +5.75510 q^{49} -0.292106 q^{53} +5.37755 q^{55} -11.3776i q^{59} -5.34725 q^{61} +(-1.26180 + 3.37755i) q^{65} -11.8709i q^{67} +3.43816i q^{71} -4.59214i q^{73} +6.00000 q^{77} -10.8157 q^{79} -7.40786i q^{83} +14.8157i q^{89} +(-1.40786 + 3.76850i) q^{91} -7.90116 q^{95} +3.76850i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 12 q^{23} - 6 q^{25} + 12 q^{29} + 12 q^{43} - 6 q^{49} + 12 q^{53} + 12 q^{55} - 12 q^{61} + 6 q^{65} + 36 q^{77} - 24 q^{79} + 12 q^{91}+O(q^{100}) \) 6 * q + 12 * q^23 - 6 * q^25 + 12 * q^29 + 12 * q^43 - 6 * q^49 + 12 * q^53 + 12 * q^55 - 12 * q^61 + 6 * q^65 + 36 * q^77 - 24 * q^79 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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