LMFDB - Embedded newform 3432.2.g.d.1585.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3432,2,Mod(1585,3432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3432.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3432.g (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:	$27.4046579737$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 25x^{12} + 236x^{10} + 1040x^{8} + 2124x^{6} + 1676x^{4} + 340x^{2} + 16 $ x^14 + 25x^12 + 236x^10 + 1040x^8 + 2124x^6 + 1676x^4 + 340x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.8
Root			$0.260548i$ of defining polynomial
Character	$\chi$	$=$	3432.1585
Dual form			3432.2.g.d.1585.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.260548i q^{5} +0.131822i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.260548i q^{5} +0.131822i q^{7} +1.00000 q^{9} -1.00000i q^{11} +(0.505941 + 3.56988i) q^{13} +0.260548i q^{15} -5.62460 q^{17} +3.47771i q^{19} +0.131822i q^{21} +3.41102 q^{23} +4.93211 q^{25} +1.00000 q^{27} -8.60214 q^{29} +7.79281i q^{31} -1.00000i q^{33} -0.0343458 q^{35} +3.38024i q^{37} +(0.505941 + 3.56988i) q^{39} -4.91241i q^{41} -11.1569 q^{43} +0.260548i q^{45} +3.12279i q^{47} +6.98262 q^{49} -5.62460 q^{51} +7.03871 q^{53} +0.260548 q^{55} +3.47771i q^{57} +5.95027i q^{59} -4.51051 q^{61} +0.131822i q^{63} +(-0.930123 + 0.131822i) q^{65} -5.57889i q^{67} +3.41102 q^{69} +8.99854i q^{71} +15.8572i q^{73} +4.93211 q^{75} +0.131822 q^{77} -2.51480 q^{79} +1.00000 q^{81} -4.70127i q^{83} -1.46548i q^{85} -8.60214 q^{87} +9.77982i q^{89} +(-0.470587 + 0.0666939i) q^{91} +7.79281i q^{93} -0.906110 q^{95} -3.77857i q^{97} -1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) $ 14 * q + 14 * q^3 + 14 * q^9	$ 14 q + 14 q^{3} + 14 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{23} + 20 q^{25} + 14 q^{27} - 10 q^{29} - 14 q^{35} - 4 q^{39} + 22 q^{43} + 8 q^{49} - 4 q^{51} - 20 q^{53} + 2 q^{55} - 2 q^{61} - 20 q^{65} + 2 q^{69} + 20 q^{75} + 2 q^{77} - 40 q^{79} + 14 q^{81} - 10 q^{87} - 44 q^{91} + 4 q^{95}+O(q^{100}) $ 14 * q + 14 * q^3 + 14 * q^9 - 4 * q^13 - 4 * q^17 + 2 * q^23 + 20 * q^25 + 14 * q^27 - 10 * q^29 - 14 * q^35 - 4 * q^39 + 22 * q^43 + 8 * q^49 - 4 * q^51 - 20 * q^53 + 2 * q^55 - 2 * q^61 - 20 * q^65 + 2 * q^69 + 20 * q^75 + 2 * q^77 - 40 * q^79 + 14 * q^81 - 10 * q^87 - 44 * q^91 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3432\mathbb{Z}\right)^\times$.

$n$	$937$	$1145$	$1717$	$2575$	$2641$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			0.260548i			0.116520i	0.998301	+	0.0582602i	$0.0185553\pi$
$5$			0.260548i			0.116520i	−0.998301	+	0.0582602i	$0.981445\pi$
$6$		0			0
$7$			0.131822i			0.0498239i	0.999690	+	0.0249120i	$0.00793054\pi$
$7$			0.131822i			0.0498239i	−0.999690	+	0.0249120i	$0.992069\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	0.505941	+	3.56988i	0.140323	+	0.990106i
$13$	0.505941	+	3.56988i	0.140323	+	0.990106i
$14$		0			0
$15$			0.260548i			0.0672731i
$16$		0			0
$17$	−5.62460			−1.36417			−0.682083	−	0.731275i	$-0.738926\pi$
$17$	−5.62460			−1.36417			−0.682083	+	0.731275i	$0.738926\pi$
$18$		0			0
$19$			3.47771i			0.797842i	0.916985	+	0.398921i	$0.130615\pi$
$19$			3.47771i			0.797842i	−0.916985	+	0.398921i	$0.869385\pi$
$20$		0			0
$21$			0.131822i			0.0287658i
$22$		0			0
$23$	3.41102			0.711247			0.355623	−	0.934629i	$-0.384269\pi$
$23$	3.41102			0.711247			0.355623	+	0.934629i	$0.384269\pi$
$24$		0			0
$25$	4.93211			0.986423
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−8.60214			−1.59738			−0.798688	−	0.601745i	$-0.794472\pi$
$29$	−8.60214			−1.59738			−0.798688	+	0.601745i	$0.794472\pi$
$30$		0			0
$31$			7.79281i			1.39963i	0.714324	+	0.699815i	$0.246734\pi$
$31$			7.79281i			1.39963i	−0.714324	+	0.699815i	$0.753266\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−0.0343458			−0.00580550
$36$		0			0
$37$			3.38024i			0.555708i	0.960623	+	0.277854i	$0.0896231\pi$
$37$			3.38024i			0.555708i	−0.960623	+	0.277854i	$0.910377\pi$
$38$		0			0
$39$	0.505941	+	3.56988i	0.0810154	+	0.571638i
$40$		0			0
$41$		−	4.91241i		−	0.767189i	−0.923502	−	0.383595i	$-0.874686\pi$
$41$		−	4.91241i		−	0.767189i	0.923502	−	0.383595i	$-0.125314\pi$
$42$		0			0
$43$	−11.1569			−1.70140			−0.850702	−	0.525648i	$-0.823823\pi$
$43$	−11.1569			−1.70140			−0.850702	+	0.525648i	$0.823823\pi$
$44$		0			0
$45$			0.260548i			0.0388402i
$46$		0			0
$47$			3.12279i			0.455505i	0.973719	+	0.227753i	$0.0731377\pi$
$47$			3.12279i			0.455505i	−0.973719	+	0.227753i	$0.926862\pi$
$48$		0			0
$49$	6.98262			0.997518
$50$		0			0
$51$	−5.62460			−0.787601
$52$		0			0
$53$	7.03871			0.966842			0.483421	−	0.875388i	$-0.339394\pi$
$53$	7.03871			0.966842			0.483421	+	0.875388i	$0.339394\pi$
$54$		0			0
$55$	0.260548			0.0351322
$56$		0			0
$57$			3.47771i			0.460634i
$58$		0			0
$59$			5.95027i			0.774660i	0.921941	+	0.387330i	$0.126603\pi$
$59$			5.95027i			0.774660i	−0.921941	+	0.387330i	$0.873397\pi$
$60$		0			0
$61$	−4.51051			−0.577512			−0.288756	−	0.957403i	$-0.593242\pi$
$61$	−4.51051			−0.577512			−0.288756	+	0.957403i	$0.593242\pi$
$62$		0			0
$63$			0.131822i			0.0166080i
$64$		0			0
$65$	−0.930123	+	0.131822i	−0.115368	+	0.0163505i
$66$		0			0
$67$		−	5.57889i		−	0.681570i	−0.940141	−	0.340785i	$-0.889307\pi$
$67$		−	5.57889i		−	0.681570i	0.940141	−	0.340785i	$-0.110693\pi$
$68$		0			0
$69$	3.41102			0.410639
$70$		0			0
$71$			8.99854i			1.06793i	0.845506	+	0.533965i	$0.179299\pi$
$71$			8.99854i			1.06793i	−0.845506	+	0.533965i	$0.820701\pi$
$72$		0			0
$73$			15.8572i			1.85594i	0.372651	+	0.927972i	$0.378449\pi$
$73$			15.8572i			1.85594i	−0.372651	+	0.927972i	$0.621551\pi$
$74$		0			0
$75$	4.93211			0.569512
$76$		0			0
$77$	0.131822			0.0150225
$78$		0			0
$79$	−2.51480			−0.282937			−0.141469	−	0.989943i	$-0.545182\pi$
$79$	−2.51480			−0.282937			−0.141469	+	0.989943i	$0.545182\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.70127i		−	0.516031i	−0.966141	−	0.258016i	$-0.916931\pi$
$83$		−	4.70127i		−	0.516031i	0.966141	−	0.258016i	$-0.0830686\pi$
$84$		0			0
$85$		−	1.46548i		−	0.158953i
$86$		0			0
$87$	−8.60214			−0.922246
$88$		0			0
$89$			9.77982i			1.03666i	0.855181	+	0.518329i	$0.173446\pi$
$89$			9.77982i			1.03666i	−0.855181	+	0.518329i	$0.826554\pi$
$90$		0			0
$91$	−0.470587	+	0.0666939i	−0.0493309	+	0.00699143i
$92$		0			0
$93$			7.79281i			0.808076i
$94$		0			0
$95$	−0.906110			−0.0929649
$96$		0			0
$97$		−	3.77857i		−	0.383656i	−0.981429	−	0.191828i	$-0.938558\pi$
$97$		−	3.77857i		−	0.383656i	0.981429	−	0.191828i	$-0.0614415\pi$
$98$		0			0
$99$		−	1.00000i		−	0.100504i
$100$		0			0
$101$	12.7521			1.26888			0.634442	−	0.772971i	$-0.281230\pi$
$101$	12.7521			1.26888			0.634442	+	0.772971i	$0.281230\pi$
$102$		0			0
$103$	−5.97007			−0.588248			−0.294124	−	0.955767i	$-0.595028\pi$
$103$	−5.97007			−0.588248			−0.294124	+	0.955767i	$0.595028\pi$
$104$		0			0
$105$	−0.0343458			−0.00335181
$106$		0			0
$107$	−2.47137			−0.238916			−0.119458	−	0.992839i	$-0.538116\pi$
$107$	−2.47137			−0.238916			−0.119458	+	0.992839i	$0.538116\pi$
$108$		0			0
$109$		−	3.28825i		−	0.314958i	−0.987522	−	0.157479i	$-0.949663\pi$
$109$		−	3.28825i		−	0.314958i	0.987522	−	0.157479i	$-0.0503366\pi$
$110$		0			0
$111$			3.38024i			0.320838i
$112$		0			0
$113$	−3.76391			−0.354079			−0.177039	−	0.984204i	$-0.556652\pi$
$113$	−3.76391			−0.354079			−0.177039	+	0.984204i	$0.556652\pi$
$114$		0			0
$115$			0.888733i			0.0828748i
$116$		0			0
$117$	0.505941	+	3.56988i	0.0467742	+	0.330035i
$118$		0			0
$119$		−	0.741444i		−	0.0679681i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	4.91241i		−	0.442937i
$124$		0			0
$125$			2.58779i			0.231459i
$126$		0			0
$127$	9.28240			0.823680			0.411840	−	0.911256i	$-0.364886\pi$
$127$	9.28240			0.823680			0.411840	+	0.911256i	$0.364886\pi$
$128$		0			0
$129$	−11.1569			−0.982307
$130$		0			0
$131$	7.52607			0.657556			0.328778	−	0.944407i	$-0.393363\pi$
$131$	7.52607			0.657556			0.328778	+	0.944407i	$0.393363\pi$
$132$		0			0
$133$	−0.458438			−0.0397516
$134$		0			0
$135$			0.260548i			0.0224244i
$136$		0			0
$137$		−	3.19421i		−	0.272900i	−0.990647	−	0.136450i	$-0.956431\pi$
$137$		−	3.19421i		−	0.272900i	0.990647	−	0.136450i	$-0.0435693\pi$
$138$		0			0
$139$	4.84062			0.410576			0.205288	−	0.978702i	$-0.434187\pi$
$139$	4.84062			0.410576			0.205288	+	0.978702i	$0.434187\pi$
$140$		0			0
$141$			3.12279i			0.262986i
$142$		0			0
$143$	3.56988	−	0.505941i	0.298528	−	0.0423089i
$144$		0			0
$145$		−	2.24127i		−	0.186127i
$146$		0			0
$147$	6.98262			0.575917
$148$		0			0
$149$			4.64601i			0.380616i	0.981724	+	0.190308i	$0.0609486\pi$
$149$			4.64601i			0.380616i	−0.981724	+	0.190308i	$0.939051\pi$
$150$		0			0
$151$			4.65016i			0.378425i	0.981936	+	0.189212i	$0.0605934\pi$
$151$			4.65016i			0.378425i	−0.981936	+	0.189212i	$0.939407\pi$
$152$		0			0
$153$	−5.62460			−0.454722
$154$		0			0
$155$	−2.03040			−0.163085
$156$		0			0
$157$	−8.68515			−0.693150			−0.346575	−	0.938022i	$-0.612655\pi$
$157$	−8.68515			−0.693150			−0.346575	+	0.938022i	$0.612655\pi$
$158$		0			0
$159$	7.03871			0.558206
$160$		0			0
$161$			0.449646i			0.0354371i
$162$		0			0
$163$			19.6145i			1.53632i	0.640256	+	0.768162i	$0.278828\pi$
$163$			19.6145i			1.53632i	−0.640256	+	0.768162i	$0.721172\pi$
$164$		0			0
$165$	0.260548			0.0202836
$166$		0			0
$167$		−	12.8057i		−	0.990936i	−0.868626	−	0.495468i	$-0.834997\pi$
$167$		−	12.8057i		−	0.990936i	0.868626	−	0.495468i	$-0.165003\pi$
$168$		0			0
$169$	−12.4880	+	3.61229i	−0.960619	+	0.277869i
$170$		0			0
$171$			3.47771i			0.265947i
$172$		0			0
$173$	13.4936			1.02590			0.512951	−	0.858418i	$-0.328552\pi$
$173$	13.4936			1.02590			0.512951	+	0.858418i	$0.328552\pi$
$174$		0			0
$175$			0.650160i			0.0491474i
$176$		0			0
$177$			5.95027i			0.447250i
$178$		0			0
$179$	−15.3725			−1.14899			−0.574496	−	0.818507i	$-0.694802\pi$
$179$	−15.3725			−1.14899			−0.574496	+	0.818507i	$0.694802\pi$
$180$		0			0
$181$	21.1983			1.57566			0.787830	−	0.615892i	$-0.211204\pi$
$181$	21.1983			1.57566			0.787830	+	0.615892i	$0.211204\pi$
$182$		0			0
$183$	−4.51051			−0.333427
$184$		0			0
$185$	−0.880713			−0.0647513
$186$		0			0
$187$			5.62460i			0.411311i
$188$		0			0
$189$			0.131822i			0.00958862i
$190$		0			0
$191$	−14.8102			−1.07163			−0.535813	−	0.844336i	$-0.679995\pi$
$191$	−14.8102			−1.07163			−0.535813	+	0.844336i	$0.679995\pi$
$192$		0			0
$193$			16.1347i			1.16140i	0.814116	+	0.580702i	$0.197222\pi$
$193$			16.1347i			1.16140i	−0.814116	+	0.580702i	$0.802778\pi$
$194$		0			0
$195$	−0.930123	+	0.131822i	−0.0666075	+	0.00943995i
$196$		0			0
$197$			15.0511i			1.07235i	0.844108	+	0.536173i	$0.180130\pi$
$197$			15.0511i			1.07235i	−0.844108	+	0.536173i	$0.819870\pi$
$198$		0			0
$199$	−10.2583			−0.727193			−0.363597	−	0.931556i	$-0.618451\pi$
$199$	−10.2583			−0.727193			−0.363597	+	0.931556i	$0.618451\pi$
$200$		0			0
$201$		−	5.57889i		−	0.393504i
$202$		0			0
$203$		−	1.13395i		−	0.0795875i
$204$		0			0
$205$	1.27992			0.0893932
$206$		0			0
$207$	3.41102			0.237082
$208$		0			0
$209$	3.47771			0.240558
$210$		0			0
$211$	−19.9524			−1.37358			−0.686791	−	0.726855i	$-0.740981\pi$
$211$	−19.9524			−1.37358			−0.686791	+	0.726855i	$0.740981\pi$
$212$		0			0
$213$			8.99854i			0.616570i
$214$		0			0
$215$		−	2.90689i		−	0.198248i
$216$		0			0
$217$	−1.02726			−0.0697350
$218$		0			0
$219$			15.8572i			1.07153i
$220$		0			0
$221$	−2.84571	−	20.0791i	−0.191423	−	1.35067i
$222$		0			0
$223$			15.6424i			1.04749i	0.851874	+	0.523747i	$0.175466\pi$
$223$			15.6424i			1.04749i	−0.851874	+	0.523747i	$0.824534\pi$
$224$		0			0
$225$	4.93211			0.328808
$226$		0			0
$227$		−	8.48720i		−	0.563315i	−0.959515	−	0.281658i	$-0.909116\pi$
$227$		−	8.48720i		−	0.563315i	0.959515	−	0.281658i	$-0.0908842\pi$
$228$		0			0
$229$			4.75404i			0.314156i	0.987586	+	0.157078i	$0.0502074\pi$
$229$			4.75404i			0.314156i	−0.987586	+	0.157078i	$0.949793\pi$
$230$		0			0
$231$	0.131822			0.00867323
$232$		0			0
$233$	8.57272			0.561618			0.280809	−	0.959764i	$-0.409397\pi$
$233$	8.57272			0.561618			0.280809	+	0.959764i	$0.409397\pi$
$234$		0			0
$235$	−0.813635			−0.0530757
$236$		0			0
$237$	−2.51480			−0.163354
$238$		0			0
$239$		−	6.43783i		−	0.416429i	−0.978083	−	0.208214i	$-0.933235\pi$
$239$		−	6.43783i		−	0.416429i	0.978083	−	0.208214i	$-0.0667652\pi$
$240$		0			0
$241$			6.76251i			0.435612i	0.975992	+	0.217806i	$0.0698900\pi$
$241$			6.76251i			0.435612i	−0.975992	+	0.217806i	$0.930110\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			1.81931i			0.116231i
$246$		0			0
$247$	−12.4150	+	1.75952i	−0.789948	+	0.111955i
$248$		0			0
$249$		−	4.70127i		−	0.297931i
$250$		0			0
$251$	−4.71189			−0.297412			−0.148706	−	0.988881i	$-0.547511\pi$
$251$	−4.71189			−0.297412			−0.148706	+	0.988881i	$0.547511\pi$
$252$		0			0
$253$		−	3.41102i		−	0.214449i
$254$		0			0
$255$		−	1.46548i		−	0.0917717i
$256$		0			0
$257$	4.39014			0.273850			0.136925	−	0.990581i	$-0.456278\pi$
$257$	4.39014			0.273850			0.136925	+	0.990581i	$0.456278\pi$
$258$		0			0
$259$	−0.445589			−0.0276875
$260$		0			0
$261$	−8.60214			−0.532459
$262$		0			0
$263$	5.58198			0.344200			0.172100	−	0.985079i	$-0.444945\pi$
$263$	5.58198			0.344200			0.172100	+	0.985079i	$0.444945\pi$
$264$		0			0
$265$			1.83392i			0.112657i
$266$		0			0
$267$			9.77982i			0.598515i
$268$		0			0
$269$	10.4390			0.636480			0.318240	−	0.948010i	$-0.396908\pi$
$269$	10.4390			0.636480			0.318240	+	0.948010i	$0.396908\pi$
$270$		0			0
$271$		−	9.83621i		−	0.597507i	−0.954330	−	0.298753i	$-0.903429\pi$
$271$		−	9.83621i		−	0.597507i	0.954330	−	0.298753i	$-0.0965708\pi$
$272$		0			0
$273$	−0.470587	+	0.0666939i	−0.0284812	+	0.00403650i
$274$		0			0
$275$		−	4.93211i		−	0.297418i
$276$		0			0
$277$	9.22235			0.554117			0.277059	−	0.960853i	$-0.410640\pi$
$277$	9.22235			0.554117			0.277059	+	0.960853i	$0.410640\pi$
$278$		0			0
$279$			7.79281i			0.466543i
$280$		0			0
$281$		−	23.3335i		−	1.39196i	−0.718060	−	0.695981i	$-0.754970\pi$
$281$		−	23.3335i		−	1.39196i	0.718060	−	0.695981i	$-0.245030\pi$
$282$		0			0
$283$	−13.4905			−0.801926			−0.400963	−	0.916094i	$-0.631324\pi$
$283$	−13.4905			−0.801926			−0.400963	+	0.916094i	$0.631324\pi$
$284$		0			0
$285$	−0.906110			−0.0536733
$286$		0			0
$287$	0.647562			0.0382244
$288$		0			0
$289$	14.6361			0.860948
$290$		0			0
$291$		−	3.77857i		−	0.221504i
$292$		0			0
$293$		−	25.7032i		−	1.50159i	−0.660533	−	0.750797i	$-0.729670\pi$
$293$		−	25.7032i		−	1.50159i	0.660533	−	0.750797i	$-0.270330\pi$
$294$		0			0
$295$	−1.55033			−0.0902637
$296$		0			0
$297$		−	1.00000i		−	0.0580259i
$298$		0			0
$299$	1.72577	+	12.1769i	0.0998041	+	0.704210i
$300$		0			0
$301$		−	1.47072i		−	0.0847706i
$302$		0			0
$303$	12.7521			0.732590
$304$		0			0
$305$		−	1.17520i		−	0.0672920i
$306$		0			0
$307$		−	29.0430i		−	1.65757i	−0.559566	−	0.828786i	$-0.689032\pi$
$307$		−	29.0430i		−	1.65757i	0.559566	−	0.828786i	$-0.310968\pi$
$308$		0			0
$309$	−5.97007			−0.339625
$310$		0			0
$311$	23.1006			1.30992			0.654958	−	0.755666i	$-0.272687\pi$
$311$	23.1006			1.30992			0.654958	+	0.755666i	$0.272687\pi$
$312$		0			0
$313$	2.11718			0.119670			0.0598351	−	0.998208i	$-0.480943\pi$
$313$	2.11718			0.119670			0.0598351	+	0.998208i	$0.480943\pi$
$314$		0			0
$315$	−0.0343458			−0.00193517
$316$		0			0
$317$			8.87981i			0.498740i	0.968408	+	0.249370i	$0.0802235\pi$
$317$			8.87981i			0.498740i	−0.968408	+	0.249370i	$0.919777\pi$
$318$		0			0
$319$			8.60214i			0.481627i
$320$		0			0
$321$	−2.47137			−0.137938
$322$		0			0
$323$		−	19.5607i		−	1.08839i
$324$		0			0
$325$	2.49536	+	17.6070i	0.138418	+	0.976663i
$326$		0			0
$327$		−	3.28825i		−	0.181841i
$328$		0			0
$329$	−0.411651			−0.0226950
$330$		0			0
$331$		−	11.8226i		−	0.649831i	−0.945743	−	0.324916i	$-0.894664\pi$
$331$		−	11.8226i		−	0.649831i	0.945743	−	0.324916i	$-0.105336\pi$
$332$		0			0
$333$			3.38024i			0.185236i
$334$		0			0
$335$	1.45357			0.0794168
$336$		0			0
$337$	24.7465			1.34803			0.674015	−	0.738718i	$-0.264568\pi$
$337$	24.7465			1.34803			0.674015	+	0.738718i	$0.264568\pi$
$338$		0			0
$339$	−3.76391			−0.204427
$340$		0			0
$341$	7.79281			0.422004
$342$		0			0
$343$			1.84321i			0.0995241i
$344$		0			0
$345$			0.888733i			0.0478478i
$346$		0			0
$347$	−33.1624			−1.78025			−0.890125	−	0.455716i	$-0.849383\pi$
$347$	−33.1624			−1.78025			−0.890125	+	0.455716i	$0.849383\pi$
$348$		0			0
$349$		−	9.06077i		−	0.485012i	−0.970150	−	0.242506i	$-0.922031\pi$
$349$		−	9.06077i		−	0.485012i	0.970150	−	0.242506i	$-0.0779694\pi$
$350$		0			0
$351$	0.505941	+	3.56988i	0.0270051	+	0.190546i
$352$		0			0
$353$			5.97794i			0.318174i	0.987265	+	0.159087i	$0.0508550\pi$
$353$			5.97794i			0.318174i	−0.987265	+	0.159087i	$0.949145\pi$
$354$		0			0
$355$	−2.34455			−0.124436
$356$		0			0
$357$		−	0.741444i		−	0.0392414i
$358$		0			0
$359$			25.7194i			1.35742i	0.734408	+	0.678708i	$0.237460\pi$
$359$			25.7194i			1.35742i	−0.734408	+	0.678708i	$0.762540\pi$
$360$		0			0
$361$	6.90551			0.363448
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	−4.13155			−0.216255
$366$		0			0
$367$	8.29594			0.433045			0.216522	−	0.976278i	$-0.430529\pi$
$367$	8.29594			0.433045			0.216522	+	0.976278i	$0.430529\pi$
$368$		0			0
$369$		−	4.91241i		−	0.255730i
$370$		0			0
$371$			0.927855i			0.0481718i
$372$		0			0
$373$	18.6281			0.964527			0.482264	−	0.876026i	$-0.339815\pi$
$373$	18.6281			0.964527			0.482264	+	0.876026i	$0.339815\pi$
$374$		0			0
$375$			2.58779i			0.133633i
$376$		0			0
$377$	−4.35217	−	30.7086i	−0.224148	−	1.58157i
$378$		0			0
$379$			7.83570i			0.402493i	0.979541	+	0.201246i	$0.0644992\pi$
$379$			7.83570i			0.402493i	−0.979541	+	0.201246i	$0.935501\pi$
$380$		0			0
$381$	9.28240			0.475552
$382$		0			0
$383$		−	26.8722i		−	1.37310i	−0.727081	−	0.686552i	$-0.759124\pi$
$383$		−	26.8722i		−	1.37310i	0.727081	−	0.686552i	$-0.240876\pi$
$384$		0			0
$385$			0.0343458i			0.00175043i
$386$		0			0
$387$	−11.1569			−0.567135
$388$		0			0
$389$	10.0694			0.510537			0.255269	−	0.966870i	$-0.417836\pi$
$389$	10.0694			0.510537			0.255269	+	0.966870i	$0.417836\pi$
$390$		0			0
$391$	−19.1856			−0.970258
$392$		0			0
$393$	7.52607			0.379640
$394$		0			0
$395$		−	0.655225i		−	0.0329679i
$396$		0			0
$397$		−	5.67585i		−	0.284863i	−0.989805	−	0.142431i	$-0.954508\pi$
$397$		−	5.67585i		−	0.284863i	0.989805	−	0.142431i	$-0.0454921\pi$
$398$		0			0
$399$	−0.458438			−0.0229506
$400$		0			0
$401$		−	15.3809i		−	0.768085i	−0.923315	−	0.384042i	$-0.874532\pi$
$401$		−	15.3809i		−	0.768085i	0.923315	−	0.384042i	$-0.125468\pi$
$402$		0			0
$403$	−27.8194	+	3.94270i	−1.38578	+	0.196400i
$404$		0			0
$405$			0.260548i			0.0129467i
$406$		0			0
$407$	3.38024			0.167552
$408$		0			0
$409$			28.5592i			1.41216i	0.708131	+	0.706081i	$0.249538\pi$
$409$			28.5592i			1.41216i	−0.708131	+	0.706081i	$0.750462\pi$
$410$		0			0
$411$		−	3.19421i		−	0.157559i
$412$		0			0
$413$	−0.784375			−0.0385966
$414$		0			0
$415$	1.22490			0.0601282
$416$		0			0
$417$	4.84062			0.237046
$418$		0			0
$419$	−3.41982			−0.167069			−0.0835346	−	0.996505i	$-0.526621\pi$
$419$	−3.41982			−0.167069			−0.0835346	+	0.996505i	$0.526621\pi$
$420$		0			0
$421$			28.1740i			1.37311i	0.727076	+	0.686557i	$0.240879\pi$
$421$			28.1740i			1.37311i	−0.727076	+	0.686557i	$0.759121\pi$
$422$		0			0
$423$			3.12279i			0.151835i
$424$		0			0
$425$	−27.7412			−1.34564
$426$		0			0
$427$		−	0.594583i		−	0.0287739i
$428$		0			0
$429$	3.56988	−	0.505941i	0.172355	−	0.0244270i
$430$		0			0
$431$		−	12.4769i		−	0.600990i	−0.953783	−	0.300495i	$-0.902848\pi$
$431$		−	12.4769i		−	0.600990i	0.953783	−	0.300495i	$-0.0971519\pi$
$432$		0			0
$433$	32.0621			1.54081			0.770403	−	0.637557i	$-0.220055\pi$
$433$	32.0621			1.54081			0.770403	+	0.637557i	$0.220055\pi$
$434$		0			0
$435$		−	2.24127i		−	0.107460i
$436$		0			0
$437$			11.8625i			0.567463i
$438$		0			0
$439$	2.85395			0.136212			0.0681058	−	0.997678i	$-0.478304\pi$
$439$	2.85395			0.136212			0.0681058	+	0.997678i	$0.478304\pi$
$440$		0			0
$441$	6.98262			0.332506
$442$		0			0
$443$	−40.8174			−1.93929			−0.969647	−	0.244509i	$-0.921373\pi$
$443$	−40.8174			−1.93929			−0.969647	+	0.244509i	$0.921373\pi$
$444$		0			0
$445$	−2.54811			−0.120792
$446$		0			0
$447$			4.64601i			0.219749i
$448$		0			0
$449$		−	8.15679i		−	0.384943i	−0.981303	−	0.192471i	$-0.938350\pi$
$449$		−	8.15679i		−	0.384943i	0.981303	−	0.192471i	$-0.0616502\pi$
$450$		0			0
$451$	−4.91241			−0.231316
$452$		0			0
$453$			4.65016i			0.218484i
$454$		0			0
$455$	−0.0173770	−	0.122610i	−0.000814644	−	0.00574806i
$456$		0			0
$457$			0.860609i			0.0402576i	0.999797	+	0.0201288i	$0.00640763\pi$
$457$			0.860609i			0.0402576i	−0.999797	+	0.0201288i	$0.993592\pi$
$458$		0			0
$459$	−5.62460			−0.262534
$460$		0			0
$461$			21.1373i			0.984462i	0.870465	+	0.492231i	$0.163819\pi$
$461$			21.1373i			0.984462i	−0.870465	+	0.492231i	$0.836181\pi$
$462$		0			0
$463$			11.4210i			0.530778i	0.964141	+	0.265389i	$0.0855004\pi$
$463$			11.4210i			0.530778i	−0.964141	+	0.265389i	$0.914500\pi$
$464$		0			0
$465$	−2.03040			−0.0941574
$466$		0			0
$467$	5.45631			0.252488			0.126244	−	0.991999i	$-0.459708\pi$
$467$	5.45631			0.252488			0.126244	+	0.991999i	$0.459708\pi$
$468$		0			0
$469$	0.735418			0.0339585
$470$		0			0
$471$	−8.68515			−0.400191
$472$		0			0
$473$			11.1569i			0.512993i
$474$		0			0
$475$			17.1525i			0.787010i
$476$		0			0
$477$	7.03871			0.322281
$478$		0			0
$479$			7.04762i			0.322014i	0.986953	+	0.161007i	$0.0514741\pi$
$479$			7.04762i			0.322014i	−0.986953	+	0.161007i	$0.948526\pi$
$480$		0			0
$481$	−12.0670	+	1.71020i	−0.550209	+	0.0779784i
$482$		0			0
$483$			0.449646i			0.0204596i
$484$		0			0
$485$	0.984497			0.0447037
$486$		0			0
$487$			4.68880i			0.212470i	0.994341	+	0.106235i	$0.0338796\pi$
$487$			4.68880i			0.212470i	−0.994341	+	0.106235i	$0.966120\pi$
$488$		0			0
$489$			19.6145i			0.886997i
$490$		0			0
$491$	10.6659			0.481344			0.240672	−	0.970607i	$-0.422632\pi$
$491$	10.6659			0.481344			0.240672	+	0.970607i	$0.422632\pi$
$492$		0			0
$493$	48.3836			2.17909
$494$		0			0
$495$	0.260548			0.0117107
$496$		0			0
$497$	−1.18620			−0.0532085
$498$		0			0
$499$		−	13.4917i		−	0.603970i	−0.953313	−	0.301985i	$-0.902351\pi$
$499$		−	13.4917i		−	0.603970i	0.953313	−	0.301985i	$-0.0976493\pi$
$500$		0			0
$501$		−	12.8057i		−	0.572117i
$502$		0			0
$503$	31.5319			1.40594			0.702970	−	0.711220i	$-0.251857\pi$
$503$	31.5319			1.40594			0.702970	+	0.711220i	$0.251857\pi$
$504$		0			0
$505$			3.32253i			0.147851i
$506$		0			0
$507$	−12.4880	+	3.61229i	−0.554614	+	0.160428i
$508$		0			0
$509$		−	5.63603i		−	0.249813i	−0.992169	−	0.124906i	$-0.960137\pi$
$509$		−	5.63603i		−	0.249813i	0.992169	−	0.124906i	$-0.0398630\pi$
$510$		0			0
$511$	−2.09032			−0.0924704
$512$		0			0
$513$			3.47771i			0.153545i
$514$		0			0
$515$		−	1.55549i		−	0.0685429i
$516$		0			0
$517$	3.12279			0.137340
$518$		0			0
$519$	13.4936			0.592304
$520$		0			0
$521$	−9.14782			−0.400773			−0.200387	−	0.979717i	$-0.564220\pi$
$521$	−9.14782			−0.400773			−0.200387	+	0.979717i	$0.564220\pi$
$522$		0			0
$523$	−39.8509			−1.74256			−0.871280	−	0.490787i	$-0.836709\pi$
$523$	−39.8509			−1.74256			−0.871280	+	0.490787i	$0.836709\pi$
$524$		0			0
$525$			0.650160i			0.0283753i
$526$		0			0
$527$		−	43.8314i		−	1.90933i
$528$		0			0
$529$	−11.3649			−0.494128
$530$		0			0
$531$			5.95027i			0.258220i
$532$		0			0
$533$	17.5367	−	2.48539i	0.759599	−	0.107654i
$534$		0			0
$535$		−	0.643910i		−	0.0278386i
$536$		0			0
$537$	−15.3725			−0.663371
$538$		0			0
$539$		−	6.98262i		−	0.300763i
$540$		0			0
$541$		−	0.417115i		−	0.0179332i	−0.999960	−	0.00896659i	$-0.997146\pi$
$541$		−	0.417115i		−	0.0179332i	0.999960	−	0.00896659i	$-0.00285419\pi$
$542$		0			0
$543$	21.1983			0.909708
$544$		0			0
$545$	0.856747			0.0366990
$546$		0			0
$547$	−9.29618			−0.397476			−0.198738	−	0.980053i	$-0.563684\pi$
$547$	−9.29618			−0.397476			−0.198738	+	0.980053i	$0.563684\pi$
$548$		0			0
$549$	−4.51051			−0.192504
$550$		0			0
$551$		−	29.9158i		−	1.27445i
$552$		0			0
$553$		−	0.331505i		−	0.0140970i
$554$		0			0
$555$	−0.880713			−0.0373842
$556$		0			0
$557$		−	38.6414i		−	1.63729i	−0.574301	−	0.818644i	$-0.694726\pi$
$557$		−	38.6414i		−	1.63729i	0.574301	−	0.818644i	$-0.305274\pi$
$558$		0			0
$559$	−5.64471	−	39.8286i	−0.238746	−	1.68457i
$560$		0			0
$561$			5.62460i			0.237471i
$562$		0			0
$563$	9.04273			0.381105			0.190553	−	0.981677i	$-0.438972\pi$
$563$	9.04273			0.381105			0.190553	+	0.981677i	$0.438972\pi$
$564$		0			0
$565$		−	0.980678i		−	0.0412574i
$566$		0			0
$567$			0.131822i			0.00553599i
$568$		0			0
$569$	1.46539			0.0614322			0.0307161	−	0.999528i	$-0.490221\pi$
$569$	1.46539			0.0614322			0.0307161	+	0.999528i	$0.490221\pi$
$570$		0			0
$571$	5.44435			0.227839			0.113920	−	0.993490i	$-0.463659\pi$
$571$	5.44435			0.227839			0.113920	+	0.993490i	$0.463659\pi$
$572$		0			0
$573$	−14.8102			−0.618704
$574$		0			0
$575$	16.8235			0.701590
$576$		0			0
$577$		−	43.1525i		−	1.79646i	−0.439525	−	0.898231i	$-0.644853\pi$
$577$		−	43.1525i		−	1.79646i	0.439525	−	0.898231i	$-0.355147\pi$
$578$		0			0
$579$			16.1347i			0.670537i
$580$		0			0
$581$	0.619729			0.0257107
$582$		0			0
$583$		−	7.03871i		−	0.291514i
$584$		0			0
$585$	−0.930123	+	0.131822i	−0.0384559	+	0.00545016i
$586$		0			0
$587$		−	27.6308i		−	1.14045i	−0.821490	−	0.570223i	$-0.806857\pi$
$587$		−	27.6308i		−	1.14045i	0.821490	−	0.570223i	$-0.193143\pi$
$588$		0			0
$589$	−27.1011			−1.11668
$590$		0			0
$591$			15.0511i			0.619119i
$592$		0			0
$593$		−	37.6650i		−	1.54672i	−0.633970	−	0.773358i	$-0.718576\pi$
$593$		−	37.6650i		−	1.54672i	0.633970	−	0.773358i	$-0.281424\pi$
$594$		0			0
$595$	0.193182			0.00791967
$596$		0			0
$597$	−10.2583			−0.419845
$598$		0			0
$599$	−4.13130			−0.168801			−0.0844003	−	0.996432i	$-0.526897\pi$
$599$	−4.13130			−0.168801			−0.0844003	+	0.996432i	$0.526897\pi$
$600$		0			0
$601$	−14.4665			−0.590100			−0.295050	−	0.955482i	$-0.595336\pi$
$601$	−14.4665			−0.590100			−0.295050	+	0.955482i	$0.595336\pi$
$602$		0			0
$603$		−	5.57889i		−	0.227190i
$604$		0			0
$605$		−	0.260548i		−	0.0105928i
$606$		0			0
$607$	30.6753			1.24507			0.622537	−	0.782591i	$-0.286102\pi$
$607$	30.6753			1.24507			0.622537	+	0.782591i	$0.286102\pi$
$608$		0			0
$609$		−	1.13395i		−	0.0459499i
$610$		0			0
$611$	−11.1480	+	1.57994i	−0.450998	+	0.0639177i
$612$		0			0
$613$			20.9417i			0.845825i	0.906170	+	0.422913i	$0.138992\pi$
$613$			20.9417i			0.845825i	−0.906170	+	0.422913i	$0.861008\pi$
$614$		0			0
$615$	1.27992			0.0516112
$616$		0			0
$617$		−	2.08744i		−	0.0840370i	−0.999117	−	0.0420185i	$-0.986621\pi$
$617$		−	2.08744i		−	0.0840370i	0.999117	−	0.0420185i	$-0.0133788\pi$
$618$		0			0
$619$		−	49.4099i		−	1.98595i	−0.118325	−	0.992975i	$-0.537752\pi$
$619$		−	49.4099i		−	1.98595i	0.118325	−	0.992975i	$-0.462248\pi$
$620$		0			0
$621$	3.41102			0.136880
$622$		0			0
$623$	−1.28919			−0.0516504
$624$		0			0
$625$	23.9863			0.959453
$626$		0			0
$627$	3.47771			0.138887
$628$		0			0
$629$		−	19.0125i		−	0.758077i
$630$		0			0
$631$		−	0.170357i		−	0.00678179i	−0.999994	−	0.00339089i	$-0.998921\pi$
$631$		−	0.170357i		−	0.00678179i	0.999994	−	0.00339089i	$-0.00107936\pi$
$632$		0			0
$633$	−19.9524			−0.793038
$634$		0			0
$635$			2.41851i			0.0959755i
$636$		0			0
$637$	3.53279	+	24.9271i	0.139974	+	0.987648i
$638$		0			0
$639$			8.99854i			0.355977i
$640$		0			0
$641$	−30.6272			−1.20970			−0.604850	−	0.796339i	$-0.706767\pi$
$641$	−30.6272			−1.20970			−0.604850	+	0.796339i	$0.706767\pi$
$642$		0			0
$643$			34.7767i			1.37146i	0.727856	+	0.685730i	$0.240517\pi$
$643$			34.7767i			1.37146i	−0.727856	+	0.685730i	$0.759483\pi$
$644$		0			0
$645$		−	2.90689i		−	0.114459i
$646$		0			0
$647$	24.5825			0.966439			0.483220	−	0.875499i	$-0.339467\pi$
$647$	24.5825			0.966439			0.483220	+	0.875499i	$0.339467\pi$
$648$		0			0
$649$	5.95027			0.233569
$650$		0			0
$651$	−1.02726			−0.0402615
$652$		0			0
$653$	42.9733			1.68167			0.840837	−	0.541288i	$-0.182063\pi$
$653$	42.9733			1.68167			0.840837	+	0.541288i	$0.182063\pi$
$654$		0			0
$655$			1.96090i			0.0766187i
$656$		0			0
$657$			15.8572i			0.618648i
$658$		0			0
$659$	−21.2804			−0.828967			−0.414483	−	0.910057i	$-0.636038\pi$
$659$	−21.2804			−0.828967			−0.414483	+	0.910057i	$0.636038\pi$
$660$		0			0
$661$		−	14.0203i		−	0.545328i	−0.962109	−	0.272664i	$-0.912095\pi$
$661$		−	14.0203i		−	0.545328i	0.962109	−	0.272664i	$-0.0879048\pi$
$662$		0			0
$663$	−2.84571	−	20.0791i	−0.110518	−	0.779809i
$664$		0			0
$665$		−	0.119445i		−	0.00463188i
$666$		0			0
$667$	−29.3421			−1.13613
$668$		0			0
$669$			15.6424i			0.604771i
$670$		0			0
$671$			4.51051i			0.174126i
$672$		0			0
$673$	31.2070			1.20294			0.601471	−	0.798895i	$-0.294582\pi$
$673$	31.2070			1.20294			0.601471	+	0.798895i	$0.294582\pi$
$674$		0			0
$675$	4.93211			0.189837
$676$		0			0
$677$	−2.43292			−0.0935047			−0.0467524	−	0.998907i	$-0.514887\pi$
$677$	−2.43292			−0.0935047			−0.0467524	+	0.998907i	$0.514887\pi$
$678$		0			0
$679$	0.498097			0.0191152
$680$		0			0
$681$		−	8.48720i		−	0.325230i
$682$		0			0
$683$			28.8498i			1.10391i	0.833875	+	0.551954i	$0.186118\pi$
$683$			28.8498i			1.10391i	−0.833875	+	0.551954i	$0.813882\pi$
$684$		0			0
$685$	0.832244			0.0317984
$686$		0			0
$687$			4.75404i			0.181378i
$688$		0			0
$689$	3.56117	+	25.1273i	0.135670	+	0.957276i
$690$		0			0
$691$		−	19.2288i		−	0.731499i	−0.930713	−	0.365749i	$-0.880813\pi$
$691$		−	19.2288i		−	0.731499i	0.930713	−	0.365749i	$-0.119187\pi$
$692$		0			0
$693$	0.131822			0.00500749
$694$		0			0
$695$			1.26121i			0.0478405i
$696$		0			0
$697$			27.6303i			1.04657i
$698$		0			0
$699$	8.57272			0.324250
$700$		0			0
$701$	−11.3618			−0.429131			−0.214565	−	0.976710i	$-0.568833\pi$
$701$	−11.3618			−0.429131			−0.214565	+	0.976710i	$0.568833\pi$
$702$		0			0
$703$	−11.7555			−0.443367
$704$		0			0
$705$	−0.813635			−0.0306432
$706$		0			0
$707$			1.68101i			0.0632207i
$708$		0			0
$709$		−	47.6358i		−	1.78900i	−0.447068	−	0.894500i	$-0.647532\pi$
$709$		−	47.6358i		−	1.78900i	0.447068	−	0.894500i	$-0.352468\pi$
$710$		0			0
$711$	−2.51480			−0.0943123
$712$		0			0
$713$			26.5814i			0.995482i
$714$		0			0
$715$	0.131822	+	0.930123i	0.00492985	+	0.0347846i
$716$		0			0
$717$		−	6.43783i		−	0.240425i
$718$		0			0
$719$	3.15616			0.117705			0.0588525	−	0.998267i	$-0.481256\pi$
$719$	3.15616			0.117705			0.0588525	+	0.998267i	$0.481256\pi$
$720$		0			0
$721$		−	0.786984i		−	0.0293088i
$722$		0			0
$723$			6.76251i			0.251501i
$724$		0			0
$725$	−42.4267			−1.57569
$726$		0			0
$727$	−31.0198			−1.15046			−0.575229	−	0.817992i	$-0.695087\pi$
$727$	−31.0198			−1.15046			−0.575229	+	0.817992i	$0.695087\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	62.7529			2.32100
$732$		0			0
$733$		−	29.2986i		−	1.08217i	−0.840968	−	0.541084i	$-0.818014\pi$
$733$		−	29.2986i		−	1.08217i	0.840968	−	0.541084i	$-0.181986\pi$
$734$		0			0
$735$			1.81931i			0.0671061i
$736$		0			0
$737$	−5.57889			−0.205501
$738$		0			0
$739$			39.4743i			1.45208i	0.687650	+	0.726042i	$0.258642\pi$
$739$			39.4743i			1.45208i	−0.687650	+	0.726042i	$0.741358\pi$
$740$		0			0
$741$	−12.4150	+	1.75952i	−0.456077	+	0.0646375i
$742$		0			0
$743$		−	8.22618i		−	0.301789i	−0.988550	−	0.150895i	$-0.951785\pi$
$743$		−	8.22618i		−	0.301789i	0.988550	−	0.150895i	$-0.0482154\pi$
$744$		0			0
$745$	−1.21051			−0.0443495
$746$		0			0
$747$		−	4.70127i		−	0.172010i
$748$		0			0
$749$		−	0.325780i		−	0.0119037i
$750$		0			0
$751$	−25.3885			−0.926439			−0.463219	−	0.886244i	$-0.653306\pi$
$751$	−25.3885			−0.926439			−0.463219	+	0.886244i	$0.653306\pi$
$752$		0			0
$753$	−4.71189			−0.171711
$754$		0			0
$755$	−1.21159			−0.0440942
$756$		0			0
$757$	5.01152			0.182147			0.0910735	−	0.995844i	$-0.470970\pi$
$757$	5.01152			0.182147			0.0910735	+	0.995844i	$0.470970\pi$
$758$		0			0
$759$		−	3.41102i		−	0.123812i
$760$		0			0
$761$			9.62083i			0.348755i	0.984679	+	0.174377i	$0.0557913\pi$
$761$			9.62083i			0.348755i	−0.984679	+	0.174377i	$0.944209\pi$
$762$		0			0
$763$	0.433463			0.0156924
$764$		0			0
$765$		−	1.46548i		−	0.0529844i
$766$		0			0
$767$	−21.2418	+	3.01049i	−0.766995	+	0.108702i
$768$		0			0
$769$			5.90749i			0.213030i	0.994311	+	0.106515i	$0.0339692\pi$
$769$			5.90749i			0.213030i	−0.994311	+	0.106515i	$0.966031\pi$
$770$		0			0
$771$	4.39014			0.158107
$772$		0			0
$773$		−	29.4490i		−	1.05921i	−0.848245	−	0.529604i	$-0.822341\pi$
$773$		−	29.4490i		−	1.05921i	0.848245	−	0.529604i	$-0.177659\pi$
$774$		0			0
$775$			38.4350i			1.38063i
$776$		0			0
$777$	−0.445589			−0.0159854
$778$		0			0
$779$	17.0839			0.612096
$780$		0			0
$781$	8.99854			0.321993
$782$		0			0
$783$	−8.60214			−0.307415
$784$		0			0
$785$		−	2.26290i		−	0.0807662i
$786$		0			0
$787$		−	13.8644i		−	0.494212i	−0.968988	−	0.247106i	$-0.920520\pi$
$787$		−	13.8644i		−	0.494212i	0.968988	−	0.247106i	$-0.0794796\pi$
$788$		0			0
$789$	5.58198			0.198724
$790$		0			0
$791$		−	0.496165i		−	0.0176416i
$792$		0			0
$793$	−2.28205	−	16.1020i	−0.0810380	−	0.571798i
$794$		0			0
$795$			1.83392i			0.0650425i
$796$		0			0
$797$	35.3904			1.25359			0.626796	−	0.779183i	$-0.284366\pi$
$797$	35.3904			1.25359			0.626796	+	0.779183i	$0.284366\pi$
$798$		0			0
$799$		−	17.5644i		−	0.621384i
$800$		0			0
$801$			9.77982i			0.345553i
$802$		0			0
$803$	15.8572			0.559588
$804$		0			0
$805$	−0.117154			−0.00412915
$806$		0			0
$807$	10.4390			0.367472
$808$		0			0
$809$	47.1005			1.65597			0.827983	−	0.560753i	$-0.189488\pi$
$809$	47.1005			1.65597			0.827983	+	0.560753i	$0.189488\pi$
$810$		0			0
$811$			9.71378i			0.341097i	0.985349	+	0.170549i	$0.0545540\pi$
$811$			9.71378i			0.341097i	−0.985349	+	0.170549i	$0.945446\pi$
$812$		0			0
$813$		−	9.83621i		−	0.344971i
$814$		0			0
$815$	−5.11050			−0.179013
$816$		0			0
$817$		−	38.8004i		−	1.35745i
$818$		0			0
$819$	−0.470587	+	0.0666939i	−0.0164436	+	0.00233048i
$820$		0			0
$821$		−	33.9868i		−	1.18615i	−0.805149	−	0.593073i	$-0.797914\pi$
$821$		−	33.9868i		−	1.18615i	0.805149	−	0.593073i	$-0.202086\pi$
$822$		0			0
$823$	53.4513			1.86319			0.931597	−	0.363493i	$-0.118416\pi$
$823$	53.4513			1.86319			0.931597	+	0.363493i	$0.118416\pi$
$824$		0			0
$825$		−	4.93211i		−	0.171714i
$826$		0			0
$827$		−	47.3592i		−	1.64684i	−0.567431	−	0.823421i	$-0.692063\pi$
$827$		−	47.3592i		−	1.64684i	0.567431	−	0.823421i	$-0.307937\pi$
$828$		0			0
$829$	13.0005			0.451527			0.225764	−	0.974182i	$-0.427512\pi$
$829$	13.0005			0.451527			0.225764	+	0.974182i	$0.427512\pi$
$830$		0			0
$831$	9.22235			0.319920
$832$		0			0
$833$	−39.2745			−1.36078
$834$		0			0
$835$	3.33650			0.115464
$836$		0			0
$837$			7.79281i			0.269359i
$838$		0			0
$839$		−	8.89498i		−	0.307089i	−0.988142	−	0.153544i	$-0.950931\pi$
$839$		−	8.89498i		−	0.307089i	0.988142	−	0.153544i	$-0.0490688\pi$
$840$		0			0
$841$	44.9967			1.55161
$842$		0			0
$843$		−	23.3335i		−	0.803649i
$844$		0			0
$845$	−0.941174	−	3.25373i	−0.0323774	−	0.111932i
$846$		0			0
$847$		−	0.131822i		−	0.00452945i
$848$		0			0
$849$	−13.4905			−0.462992
$850$		0			0
$851$			11.5301i			0.395245i
$852$		0			0
$853$		−	46.7816i		−	1.60177i	−0.598818	−	0.800886i	$-0.704363\pi$
$853$		−	46.7816i		−	1.60177i	0.598818	−	0.800886i	$-0.295637\pi$
$854$		0			0
$855$	−0.906110			−0.0309883
$856$		0			0
$857$	24.0327			0.820941			0.410471	−	0.911874i	$-0.365364\pi$
$857$	24.0327			0.820941			0.410471	+	0.911874i	$0.365364\pi$
$858$		0			0
$859$	−16.7349			−0.570988			−0.285494	−	0.958380i	$-0.592158\pi$
$859$	−16.7349			−0.570988			−0.285494	+	0.958380i	$0.592158\pi$
$860$		0			0
$861$	0.647562			0.0220688
$862$		0			0
$863$			28.7864i			0.979900i	0.871751	+	0.489950i	$0.162985\pi$
$863$			28.7864i			0.979900i	−0.871751	+	0.489950i	$0.837015\pi$
$864$		0			0
$865$			3.51573i			0.119538i
$866$		0			0
$867$	14.6361			0.497069
$868$		0			0
$869$			2.51480i			0.0853087i
$870$		0			0
$871$	19.9159	−	2.82259i	0.674826	−	0.0956397i
$872$		0			0
$873$		−	3.77857i		−	0.127885i
$874$		0			0
$875$	−0.341127			−0.0115322
$876$		0			0
$877$		−	14.3932i		−	0.486025i	−0.970023	−	0.243012i	$-0.921864\pi$
$877$		−	14.3932i		−	0.486025i	0.970023	−	0.243012i	$-0.0781356\pi$
$878$		0			0
$879$		−	25.7032i		−	0.866946i
$880$		0			0
$881$	−2.78115			−0.0936993			−0.0468497	−	0.998902i	$-0.514918\pi$
$881$	−2.78115			−0.0936993			−0.0468497	+	0.998902i	$0.514918\pi$
$882$		0			0
$883$	−21.1390			−0.711383			−0.355691	−	0.934603i	$-0.615755\pi$
$883$	−21.1390			−0.711383			−0.355691	+	0.934603i	$0.615755\pi$
$884$		0			0
$885$	−1.55033			−0.0521138
$886$		0			0
$887$	8.15376			0.273776			0.136888	−	0.990587i	$-0.456290\pi$
$887$	8.15376			0.273776			0.136888	+	0.990587i	$0.456290\pi$
$888$		0			0
$889$			1.22362i			0.0410389i
$890$		0			0
$891$		−	1.00000i		−	0.0335013i
$892$		0			0
$893$	−10.8602			−0.363421
$894$		0			0
$895$		−	4.00526i		−	0.133881i
$896$		0			0
$897$	1.72577	+	12.1769i	0.0576219	+	0.406576i
$898$		0			0
$899$		−	67.0348i		−	2.23573i
$900$		0			0
$901$	−39.5900			−1.31893
$902$		0			0
$903$		−	1.47072i		−	0.0489424i
$904$		0			0
$905$			5.52318i			0.183597i
$906$		0			0
$907$	8.47226			0.281317			0.140658	−	0.990058i	$-0.455078\pi$
$907$	8.47226			0.281317			0.140658	+	0.990058i	$0.455078\pi$
$908$		0			0
$909$	12.7521			0.422961
$910$		0			0
$911$	5.62473			0.186356			0.0931779	−	0.995649i	$-0.470297\pi$
$911$	5.62473			0.186356			0.0931779	+	0.995649i	$0.470297\pi$
$912$		0			0
$913$	−4.70127			−0.155589
$914$		0			0
$915$		−	1.17520i		−	0.0388510i
$916$		0			0
$917$			0.992099i			0.0327620i
$918$		0			0
$919$	−48.1269			−1.58756			−0.793779	−	0.608206i	$-0.791889\pi$
$919$	−48.1269			−1.58756			−0.793779	+	0.608206i	$0.791889\pi$
$920$		0			0
$921$		−	29.0430i		−	0.957000i
$922$		0			0
$923$	−32.1237	+	4.55273i	−1.05736	+	0.149855i
$924$		0			0
$925$			16.6717i			0.548163i
$926$		0			0
$927$	−5.97007			−0.196083
$928$		0			0
$929$		−	33.9457i		−	1.11372i	−0.830605	−	0.556861i	$-0.812005\pi$
$929$		−	33.9457i		−	1.11372i	0.830605	−	0.556861i	$-0.187995\pi$
$930$		0			0
$931$			24.2836i			0.795862i
$932$		0			0
$933$	23.1006			0.756280
$934$		0			0
$935$	−1.46548			−0.0479262
$936$		0			0
$937$	42.8756			1.40069			0.700343	−	0.713807i	$-0.253031\pi$
$937$	42.8756			1.40069			0.700343	+	0.713807i	$0.253031\pi$
$938$		0			0
$939$	2.11718			0.0690916
$940$		0			0
$941$		−	42.7203i		−	1.39264i	−0.717731	−	0.696321i	$-0.754819\pi$
$941$		−	42.7203i		−	1.39264i	0.717731	−	0.696321i	$-0.245181\pi$
$942$		0			0
$943$		−	16.7563i		−	0.545661i
$944$		0			0
$945$	−0.0343458			−0.00111727
$946$		0			0
$947$			5.29689i			0.172126i	0.996290	+	0.0860629i	$0.0274286\pi$
$947$			5.29689i			0.172126i	−0.996290	+	0.0860629i	$0.972571\pi$
$948$		0			0
$949$	−56.6082	+	8.02280i	−1.83758	+	0.260431i
$950$		0			0
$951$			8.87981i			0.287948i
$952$		0			0
$953$	−39.5150			−1.28001			−0.640007	−	0.768369i	$-0.721069\pi$
$953$	−39.5150			−1.28001			−0.640007	+	0.768369i	$0.721069\pi$
$954$		0			0
$955$		−	3.85876i		−	0.124866i
$956$		0			0
$957$			8.60214i			0.278068i
$958$		0			0
$959$	0.421066			0.0135969
$960$		0			0
$961$	−29.7278			−0.958962
$962$		0			0
$963$	−2.47137			−0.0796388
$964$		0			0
$965$	−4.20387			−0.135327
$966$		0			0
$967$			56.1065i			1.80426i	0.431461	+	0.902131i	$0.357998\pi$
$967$			56.1065i			1.80426i	−0.431461	+	0.902131i	$0.642002\pi$
$968$		0			0
$969$		−	19.5607i		−	0.628382i
$970$		0			0
$971$	22.7818			0.731102			0.365551	−	0.930791i	$-0.380881\pi$
$971$	22.7818			0.731102			0.365551	+	0.930791i	$0.380881\pi$
$972$		0			0
$973$			0.638099i			0.0204565i
$974$		0			0
$975$	2.49536	+	17.6070i	0.0799154	+	0.563877i
$976$		0			0
$977$			6.17006i			0.197398i	0.995117	+	0.0986988i	$0.0314680\pi$
$977$			6.17006i			0.197398i	−0.995117	+	0.0986988i	$0.968532\pi$
$978$		0			0
$979$	9.77982			0.312564
$980$		0			0
$981$		−	3.28825i		−	0.104986i
$982$		0			0
$983$		−	49.6716i		−	1.58428i	−0.610340	−	0.792140i	$-0.708967\pi$
$983$		−	49.6716i		−	1.58428i	0.610340	−	0.792140i	$-0.291033\pi$
$984$		0			0
$985$	−3.92153			−0.124950
$986$		0			0
$987$	−0.411651			−0.0131030
$988$		0			0
$989$	−38.0563			−1.21012
$990$		0			0
$991$	20.6907			0.657263			0.328632	−	0.944458i	$-0.393413\pi$
$991$	20.6907			0.657263			0.328632	+	0.944458i	$0.393413\pi$
$992$		0			0
$993$		−	11.8226i		−	0.375180i
$994$		0			0
$995$		−	2.67278i		−	0.0847329i
$996$		0			0
$997$	20.8544			0.660464			0.330232	−	0.943900i	$-0.392873\pi$
$997$	20.8544			0.660464			0.330232	+	0.943900i	$0.392873\pi$
$998$		0			0
$999$			3.38024i			0.106946i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.g.d.1585.8	yes	14
13.12	even	2	inner	3432.2.g.d.1585.7	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.g.d.1585.7	✓	14	13.12	even	2	inner
3432.2.g.d.1585.8	yes	14	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3432.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.260548i q^{5} +0.131822i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.260548i q^{5} +0.131822i q^{7} +1.00000 q^{9} -1.00000i q^{11} +(0.505941 + 3.56988i) q^{13} +0.260548i q^{15} -5.62460 q^{17} +3.47771i q^{19} +0.131822i q^{21} +3.41102 q^{23} +4.93211 q^{25} +1.00000 q^{27} -8.60214 q^{29} +7.79281i q^{31} -1.00000i q^{33} -0.0343458 q^{35} +3.38024i q^{37} +(0.505941 + 3.56988i) q^{39} -4.91241i q^{41} -11.1569 q^{43} +0.260548i q^{45} +3.12279i q^{47} +6.98262 q^{49} -5.62460 q^{51} +7.03871 q^{53} +0.260548 q^{55} +3.47771i q^{57} +5.95027i q^{59} -4.51051 q^{61} +0.131822i q^{63} +(-0.930123 + 0.131822i) q^{65} -5.57889i q^{67} +3.41102 q^{69} +8.99854i q^{71} +15.8572i q^{73} +4.93211 q^{75} +0.131822 q^{77} -2.51480 q^{79} +1.00000 q^{81} -4.70127i q^{83} -1.46548i q^{85} -8.60214 q^{87} +9.77982i q^{89} +(-0.470587 + 0.0666939i) q^{91} +7.79281i q^{93} -0.906110 q^{95} -3.77857i q^{97} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) \) 14 * q + 14 * q^3 + 14 * q^9	\( 14 q + 14 q^{3} + 14 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{23} + 20 q^{25} + 14 q^{27} - 10 q^{29} - 14 q^{35} - 4 q^{39} + 22 q^{43} + 8 q^{49} - 4 q^{51} - 20 q^{53} + 2 q^{55} - 2 q^{61} - 20 q^{65} + 2 q^{69} + 20 q^{75} + 2 q^{77} - 40 q^{79} + 14 q^{81} - 10 q^{87} - 44 q^{91} + 4 q^{95}+O(q^{100}) \) 14 * q + 14 * q^3 + 14 * q^9 - 4 * q^13 - 4 * q^17 + 2 * q^23 + 20 * q^25 + 14 * q^27 - 10 * q^29 - 14 * q^35 - 4 * q^39 + 22 * q^43 + 8 * q^49 - 4 * q^51 - 20 * q^53 + 2 * q^55 - 2 * q^61 - 20 * q^65 + 2 * q^69 + 20 * q^75 + 2 * q^77 - 40 * q^79 + 14 * q^81 - 10 * q^87 - 44 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more