LMFDB - Embedded newform 3696.2.q.b.769.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3696,2,Mod(769,3696)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3696, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3696.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3696.q (of order $2$, degree $1$, not 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:	$29.5127085871$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6679465984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 88x^{2} + 36 $ x^8 + 14x^6 + 61x^4 + 88*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 462)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.8
Root			$0.829319i$ of defining polynomial
Character	$\chi$	$=$	3696.769
Dual form			3696.2.q.b.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +4.14155i q^{5} +(-0.717333 + 2.54665i) q^{7} -1.00000 q^{9} +(0.170681 - 3.31223i) q^{11} +3.09330 q^{13} -4.14155 q^{15} +3.57621 q^{17} -3.91758 q^{19} +(-2.54665 - 0.717333i) q^{21} -7.71776 q^{23} -12.1524 q^{25} -1.00000i q^{27} -1.65864i q^{29} -9.23485i q^{31} +(3.31223 + 0.170681i) q^{33} +(-10.5471 - 2.97087i) q^{35} -0.869330 q^{37} +3.09330i q^{39} -5.91758 q^{41} +10.2831i q^{43} -4.14155i q^{45} -6.76601i q^{47} +(-5.97087 - 3.65359i) q^{49} +3.57621i q^{51} -1.88261 q^{53} +(13.7178 + 0.706884i) q^{55} -3.91758i q^{57} +10.1866i q^{59} -3.37640 q^{61} +(0.717333 - 2.54665i) q^{63} +12.8111i q^{65} -9.84524 q^{67} -7.71776i q^{69} -1.33817 q^{71} -5.57621 q^{73} -12.1524i q^{75} +(8.31266 + 2.81064i) q^{77} -10.8484i q^{79} +1.00000 q^{81} +5.67271 q^{83} +14.8111i q^{85} +1.65864 q^{87} +10.6245i q^{89} +(-2.21893 + 7.87756i) q^{91} +9.23485 q^{93} -16.2248i q^{95} +4.52797i q^{97} +(-0.170681 + 3.31223i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} + 8 q^{11} - 4 q^{15} - 12 q^{17} - 4 q^{19} - 8 q^{21} + 8 q^{23} - 16 q^{25} + 4 q^{33} - 8 q^{35} + 16 q^{37} - 20 q^{41} - 12 q^{49} + 40 q^{55} + 56 q^{61} - 16 q^{67} - 8 q^{71} - 4 q^{73}+ \cdots - 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 8 * q^11 - 4 * q^15 - 12 * q^17 - 4 * q^19 - 8 * q^21 + 8 * q^23 - 16 * q^25 + 4 * q^33 - 8 * q^35 + 16 * q^37 - 20 * q^41 - 12 * q^49 + 40 * q^55 + 56 * q^61 - 16 * q^67 - 8 * q^71 - 4 * q^73 - 20 * q^77 + 8 * q^81 - 4 * q^83 - 20 * q^91 + 20 * q^93 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3696\mathbb{Z}\right)^\times$.

$n$	$463$	$673$	$1585$	$2465$	$2773$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$			4.14155i			1.85216i	0.377331	+	0.926079i	$0.376842\pi$
$5$			4.14155i			1.85216i	−0.377331	+	0.926079i	$0.623158\pi$
$6$		0			0
$7$	−0.717333	+	2.54665i	−0.271126	+	0.962544i
$7$	−0.717333	+	2.54665i	−0.271126	+	0.962544i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.170681	−	3.31223i	0.0514623	−	0.998675i
$11$	0.170681	−	3.31223i	0.0514623	−	0.998675i
$12$		0			0
$13$	3.09330			0.857928			0.428964	−	0.903322i	$-0.358879\pi$
$13$	3.09330			0.857928			0.428964	+	0.903322i	$0.358879\pi$
$14$		0			0
$15$	−4.14155			−1.06934
$16$		0			0
$17$	3.57621			0.867359			0.433680	−	0.901067i	$-0.357215\pi$
$17$	3.57621			0.867359			0.433680	+	0.901067i	$0.357215\pi$
$18$		0			0
$19$	−3.91758			−0.898754			−0.449377	−	0.893342i	$-0.648354\pi$
$19$	−3.91758			−0.898754			−0.449377	+	0.893342i	$0.648354\pi$
$20$		0			0
$21$	−2.54665	−	0.717333i	−0.555725	−	0.156535i
$22$		0			0
$23$	−7.71776			−1.60926			−0.804632	−	0.593773i	$-0.797638\pi$
$23$	−7.71776			−1.60926			−0.804632	+	0.593773i	$0.797638\pi$
$24$		0			0
$25$	−12.1524			−2.43049
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	1.65864i		−	0.308001i	−0.988071	−	0.154001i	$-0.950784\pi$
$29$		−	1.65864i		−	0.308001i	0.988071	−	0.154001i	$-0.0492158\pi$
$30$		0			0
$31$		−	9.23485i		−	1.65863i	−0.558783	−	0.829314i	$-0.688731\pi$
$31$		−	9.23485i		−	1.65863i	0.558783	−	0.829314i	$-0.311269\pi$
$32$		0			0
$33$	3.31223	+	0.170681i	0.576585	+	0.0297118i
$34$		0			0
$35$	−10.5471	−	2.97087i	−1.78278	−	0.502168i
$36$		0			0
$37$	−0.869330			−0.142917			−0.0714585	−	0.997444i	$-0.522765\pi$
$37$	−0.869330			−0.142917			−0.0714585	+	0.997444i	$0.522765\pi$
$38$		0			0
$39$			3.09330i			0.495325i
$40$		0			0
$41$	−5.91758			−0.924170			−0.462085	−	0.886836i	$-0.652899\pi$
$41$	−5.91758			−0.924170			−0.462085	+	0.886836i	$0.652899\pi$
$42$		0			0
$43$			10.2831i			1.56816i	0.620661	+	0.784079i	$0.286864\pi$
$43$			10.2831i			1.56816i	−0.620661	+	0.784079i	$0.713136\pi$
$44$		0			0
$45$		−	4.14155i		−	0.617386i
$46$		0			0
$47$		−	6.76601i		−	0.986924i	−0.869767	−	0.493462i	$-0.835731\pi$
$47$		−	6.76601i		−	0.986924i	0.869767	−	0.493462i	$-0.164269\pi$
$48$		0			0
$49$	−5.97087	−	3.65359i	−0.852981	−	0.521942i
$50$		0			0
$51$			3.57621i			0.500770i
$52$		0			0
$53$	−1.88261			−0.258596			−0.129298	−	0.991606i	$-0.541272\pi$
$53$	−1.88261			−0.258596			−0.129298	+	0.991606i	$0.541272\pi$
$54$		0			0
$55$	13.7178	+	0.706884i	1.84970	+	0.0953162i
$56$		0			0
$57$		−	3.91758i		−	0.518896i
$58$		0			0
$59$			10.1866i			1.32618i	0.748538	+	0.663092i	$0.230756\pi$
$59$			10.1866i			1.32618i	−0.748538	+	0.663092i	$0.769244\pi$
$60$		0			0
$61$	−3.37640			−0.432304			−0.216152	−	0.976360i	$-0.569351\pi$
$61$	−3.37640			−0.432304			−0.216152	+	0.976360i	$0.569351\pi$
$62$		0			0
$63$	0.717333	−	2.54665i	0.0903754	−	0.320848i
$64$		0			0
$65$			12.8111i			1.58902i
$66$		0			0
$67$	−9.84524			−1.20279			−0.601394	−	0.798953i	$-0.705388\pi$
$67$	−9.84524			−1.20279			−0.601394	+	0.798953i	$0.705388\pi$
$68$		0			0
$69$		−	7.71776i		−	0.929110i
$70$		0			0
$71$	−1.33817			−0.158812			−0.0794060	−	0.996842i	$-0.525302\pi$
$71$	−1.33817			−0.158812			−0.0794060	+	0.996842i	$0.525302\pi$
$72$		0			0
$73$	−5.57621			−0.652647			−0.326323	−	0.945258i	$-0.605810\pi$
$73$	−5.57621			−0.652647			−0.326323	+	0.945258i	$0.605810\pi$
$74$		0			0
$75$		−	12.1524i		−	1.40324i
$76$		0			0
$77$	8.31266	+	2.81064i	0.947316	+	0.320302i
$78$		0			0
$79$		−	10.8484i		−	1.22054i	−0.792192	−	0.610272i	$-0.791060\pi$
$79$		−	10.8484i		−	1.22054i	0.792192	−	0.610272i	$-0.208940\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	5.67271			0.622660			0.311330	−	0.950302i	$-0.399225\pi$
$83$	5.67271			0.622660			0.311330	+	0.950302i	$0.399225\pi$
$84$		0			0
$85$			14.8111i			1.60649i
$86$		0			0
$87$	1.65864			0.177825
$88$		0			0
$89$			10.6245i			1.12619i	0.826392	+	0.563095i	$0.190390\pi$
$89$			10.6245i			1.12619i	−0.826392	+	0.563095i	$0.809610\pi$
$90$		0			0
$91$	−2.21893	+	7.87756i	−0.232607	+	0.825793i
$92$		0			0
$93$	9.23485			0.957610
$94$		0			0
$95$		−	16.2248i		−	1.66463i
$96$		0			0
$97$			4.52797i			0.459746i	0.973221	+	0.229873i	$0.0738310\pi$
$97$			4.52797i			0.459746i	−0.973221	+	0.229873i	$0.926169\pi$
$98$		0			0
$99$	−0.170681	+	3.31223i	−0.0171541	+	0.332892i
$100$		0			0
$101$	1.37554			0.136871			0.0684357	−	0.997656i	$-0.478199\pi$
$101$	1.37554			0.136871			0.0684357	+	0.997656i	$0.478199\pi$
$102$		0			0
$103$		−	0.0140682i		−	0.00138618i	−1.00000		0.000693089i	$-0.999779\pi$
$103$		−	0.0140682i		−	0.00138618i	1.00000		0.000693089i	$-0.000220617\pi$
$104$		0			0
$105$	2.97087	−	10.5471i	0.289927	−	1.02929i
$106$		0			0
$107$			1.37554i			0.132978i	0.997787	+	0.0664892i	$0.0211798\pi$
$107$			1.37554i			0.132978i	−0.997787	+	0.0664892i	$0.978820\pi$
$108$		0			0
$109$		−	13.1315i		−	1.25777i	−0.777497	−	0.628886i	$-0.783511\pi$
$109$		−	13.1315i		−	1.25777i	0.777497	−	0.628886i	$-0.216489\pi$
$110$		0			0
$111$		−	0.869330i		−	0.0825132i
$112$		0			0
$113$	−7.30718			−0.687402			−0.343701	−	0.939079i	$-0.611681\pi$
$113$	−7.30718			−0.687402			−0.343701	+	0.939079i	$0.611681\pi$
$114$		0			0
$115$		−	31.9635i		−	2.98061i
$116$		0			0
$117$	−3.09330			−0.285976
$118$		0			0
$119$	−2.56533	+	9.10737i	−0.235164	+	0.834871i
$120$		0			0
$121$	−10.9417	−	1.13067i	−0.994703	−	0.102788i
$122$		0			0
$123$		−	5.91758i		−	0.533570i
$124$		0			0
$125$		−	29.6221i		−	2.64948i
$126$		0			0
$127$			19.4564i			1.72648i	0.504795	+	0.863239i	$0.331568\pi$
$127$			19.4564i			1.72648i	−0.504795	+	0.863239i	$0.668432\pi$
$128$		0			0
$129$	−10.2831			−0.905376
$130$		0			0
$131$	14.2108			1.24160			0.620800	−	0.783969i	$-0.286808\pi$
$131$	14.2108			1.24160			0.620800	+	0.783969i	$0.286808\pi$
$132$		0			0
$133$	2.81021	−	9.97670i	0.243676	−	0.865090i
$134$		0			0
$135$	4.14155			0.356448
$136$		0			0
$137$	13.4938			1.15285			0.576426	−	0.817149i	$-0.304447\pi$
$137$	13.4938			1.15285			0.576426	+	0.817149i	$0.304447\pi$
$138$		0			0
$139$	−14.5521			−1.23430			−0.617148	−	0.786847i	$-0.711712\pi$
$139$	−14.5521			−1.23430			−0.617148	+	0.786847i	$0.711712\pi$
$140$		0			0
$141$	6.76601			0.569801
$142$		0			0
$143$	0.527968	−	10.2457i	0.0441509	−	0.856791i
$144$		0			0
$145$	6.86933			0.570467
$146$		0			0
$147$	3.65359	−	5.97087i	0.301343	−	0.492469i
$148$		0			0
$149$		−	7.24892i		−	0.593855i	−0.954900	−	0.296927i	$-0.904038\pi$
$149$		−	7.24892i		−	0.593855i	0.954900	−	0.296927i	$-0.0959619\pi$
$150$		0			0
$151$			10.3422i			0.841638i	0.907145	+	0.420819i	$0.138257\pi$
$151$			10.3422i			0.841638i	−0.907145	+	0.420819i	$0.861743\pi$
$152$		0			0
$153$	−3.57621			−0.289120
$154$		0			0
$155$	38.2466			3.07204
$156$		0			0
$157$		−	13.0801i		−	1.04391i	−0.852974	−	0.521953i	$-0.825204\pi$
$157$		−	13.0801i		−	1.04391i	0.852974	−	0.521953i	$-0.174796\pi$
$158$		0			0
$159$		−	1.88261i		−	0.149301i
$160$		0			0
$161$	5.53620	−	19.6545i	0.436314	−	1.54899i
$162$		0			0
$163$	3.75513			0.294124			0.147062	−	0.989127i	$-0.453018\pi$
$163$	3.75513			0.294124			0.147062	+	0.989127i	$0.453018\pi$
$164$		0			0
$165$	−0.706884	+	13.7178i	−0.0550308	+	1.06793i
$166$		0			0
$167$	16.9658			1.31285			0.656427	−	0.754389i	$-0.272067\pi$
$167$	16.9658			1.31285			0.656427	+	0.754389i	$0.272067\pi$
$168$		0			0
$169$	−3.43148			−0.263960
$170$		0			0
$171$	3.91758			0.299585
$172$		0			0
$173$	3.75513			0.285497			0.142749	−	0.989759i	$-0.454406\pi$
$173$	3.75513			0.285497			0.142749	+	0.989759i	$0.454406\pi$
$174$		0			0
$175$	8.71733	−	30.9480i	0.658968	−	2.33945i
$176$		0			0
$177$	−10.1866			−0.765672
$178$		0			0
$179$	−18.6627			−1.39491			−0.697457	−	0.716626i	$-0.745685\pi$
$179$	−18.6627			−1.39491			−0.697457	+	0.716626i	$0.745685\pi$
$180$		0			0
$181$		−	14.9216i		−	1.10912i	−0.832145	−	0.554558i	$-0.812887\pi$
$181$		−	14.9216i		−	1.10912i	0.832145	−	0.554558i	$-0.187113\pi$
$182$		0			0
$183$		−	3.37640i		−	0.249591i
$184$		0			0
$185$		−	3.60037i		−	0.264705i
$186$		0			0
$187$	0.610392	−	11.8452i	0.0446363	−	0.866210i
$188$		0			0
$189$	2.54665	+	0.717333i	0.185242	+	0.0521783i
$190$		0			0
$191$	10.5871			0.766055			0.383028	−	0.923737i	$-0.374881\pi$
$191$	10.5871			0.766055			0.383028	+	0.923737i	$0.374881\pi$
$192$		0			0
$193$			11.4355i			0.823147i	0.911377	+	0.411574i	$0.135021\pi$
$193$			11.4355i			0.823147i	−0.911377	+	0.411574i	$0.864979\pi$
$194$		0			0
$195$	−12.8111			−0.917420
$196$		0			0
$197$			13.2107i			0.941223i	0.882341	+	0.470611i	$0.155967\pi$
$197$			13.2107i			0.941223i	−0.882341	+	0.470611i	$0.844033\pi$
$198$		0			0
$199$			14.0077i			0.992979i	0.868043	+	0.496489i	$0.165378\pi$
$199$			14.0077i			0.992979i	−0.868043	+	0.496489i	$0.834622\pi$
$200$		0			0
$201$		−	9.84524i		−	0.694430i
$202$		0			0
$203$	4.22397	+	1.18979i	0.296465	+	0.0835072i
$204$		0			0
$205$		−	24.5079i		−	1.71171i
$206$		0			0
$207$	7.71776			0.536422
$208$		0			0
$209$	−0.668656	+	12.9759i	−0.0462519	+	0.897563i
$210$		0			0
$211$		−	7.33903i		−	0.505240i	−0.967566	−	0.252620i	$-0.918708\pi$
$211$		−	7.33903i		−	0.505240i	0.967566	−	0.252620i	$-0.0812922\pi$
$212$		0			0
$213$		−	1.33817i		−	0.0916901i
$214$		0			0
$215$	−42.5880			−2.90447
$216$		0			0
$217$	23.5180	+	6.62446i	1.59650	+	0.449698i
$218$		0			0
$219$		−	5.57621i		−	0.376806i
$220$		0			0
$221$	11.0623			0.744132
$222$		0			0
$223$			7.42146i			0.496978i	0.968635	+	0.248489i	$0.0799339\pi$
$223$			7.42146i			0.496978i	−0.968635	+	0.248489i	$0.920066\pi$
$224$		0			0
$225$	12.1524			0.810162
$226$		0			0
$227$	8.46201			0.561644			0.280822	−	0.959760i	$-0.409393\pi$
$227$	8.46201			0.561644			0.280822	+	0.959760i	$0.409393\pi$
$228$		0			0
$229$		−	1.03178i		−	0.0681817i	−0.999419	−	0.0340909i	$-0.989146\pi$
$229$		−	1.03178i		−	0.0681817i	0.999419	−	0.0340909i	$-0.0108536\pi$
$230$		0			0
$231$	−2.81064	+	8.31266i	−0.184926	+	0.546933i
$232$		0			0
$233$			0.944064i			0.0618477i	0.999522	+	0.0309238i	$0.00984493\pi$
$233$			0.944064i			0.0618477i	−0.999522	+	0.0309238i	$0.990155\pi$
$234$		0			0
$235$	28.0218			1.82794
$236$		0			0
$237$	10.8484			0.704681
$238$		0			0
$239$			20.9394i			1.35446i	0.735772	+	0.677229i	$0.236819\pi$
$239$			20.9394i			1.35446i	−0.735772	+	0.677229i	$0.763181\pi$
$240$		0			0
$241$	−17.8593			−1.15042			−0.575210	−	0.818006i	$-0.695080\pi$
$241$	−17.8593			−1.15042			−0.575210	+	0.818006i	$0.695080\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	15.1315	−	24.7286i	0.966718	−	1.57986i
$246$		0			0
$247$	−12.1183			−0.771066
$248$		0			0
$249$			5.67271i			0.359493i
$250$		0			0
$251$			8.96582i			0.565918i	0.959132	+	0.282959i	$0.0913160\pi$
$251$			8.96582i			0.565918i	−0.959132	+	0.282959i	$0.908684\pi$
$252$		0			0
$253$	−1.31728	+	25.5630i	−0.0828164	+	1.60713i
$254$		0			0
$255$	−14.8111			−0.927505
$256$		0			0
$257$		−	8.52797i		−	0.531960i	−0.963979	−	0.265980i	$-0.914304\pi$
$257$		−	8.52797i		−	0.531960i	0.963979	−	0.265980i	$-0.0856955\pi$
$258$		0			0
$259$	0.623599	−	2.21388i	0.0387486	−	0.137564i
$260$		0			0
$261$			1.65864i			0.102667i
$262$		0			0
$263$			31.2871i			1.92925i	0.263629	+	0.964624i	$0.415080\pi$
$263$			31.2871i			1.92925i	−0.263629	+	0.964624i	$0.584920\pi$
$264$		0			0
$265$		−	7.79692i		−	0.478961i
$266$		0			0
$267$	−10.6245			−0.650206
$268$		0			0
$269$			7.57535i			0.461877i	0.972968	+	0.230939i	$0.0741797\pi$
$269$			7.57535i			0.461877i	−0.972968	+	0.230939i	$0.925820\pi$
$270$		0			0
$271$	−14.9667			−0.909161			−0.454581	−	0.890706i	$-0.650211\pi$
$271$	−14.9667			−0.909161			−0.454581	+	0.890706i	$0.650211\pi$
$272$		0			0
$273$	−7.87756	−	2.21893i	−0.476772	−	0.134296i
$274$		0			0
$275$	−2.07419	+	40.2516i	−0.125078	+	2.42727i
$276$		0			0
$277$		−	29.2498i		−	1.75745i	−0.477329	−	0.878725i	$-0.658395\pi$
$277$		−	29.2498i		−	1.75745i	0.477329	−	0.878725i	$-0.341605\pi$
$278$		0			0
$279$			9.23485i			0.552876i
$280$		0			0
$281$			15.4938i			0.924282i	0.886806	+	0.462141i	$0.152919\pi$
$281$			15.4938i			0.924282i	−0.886806	+	0.462141i	$0.847081\pi$
$282$		0			0
$283$	−0.855262			−0.0508401			−0.0254200	−	0.999677i	$-0.508092\pi$
$283$	−0.855262			−0.0508401			−0.0254200	+	0.999677i	$0.508092\pi$
$284$		0			0
$285$	16.2248			0.961076
$286$		0			0
$287$	4.24487	−	15.0700i	0.250567	−	0.889554i
$288$		0			0
$289$	−4.21069			−0.247688
$290$		0			0
$291$	−4.52797			−0.265434
$292$		0			0
$293$	−10.0319			−0.586067			−0.293033	−	0.956102i	$-0.594665\pi$
$293$	−10.0319			−0.586067			−0.293033	+	0.956102i	$0.594665\pi$
$294$		0			0
$295$	−42.1883			−2.45630
$296$		0			0
$297$	−3.31223	−	0.170681i	−0.192195	−	0.00990392i
$298$		0			0
$299$	−23.8734			−1.38063
$300$		0			0
$301$	−26.1875	−	7.37640i	−1.50942	−	0.425169i
$302$		0			0
$303$			1.37554i			0.0790227i
$304$		0			0
$305$		−	13.9835i		−	0.800695i
$306$		0			0
$307$	2.55213			0.145658			0.0728288	−	0.997344i	$-0.476797\pi$
$307$	2.55213			0.145658			0.0728288	+	0.997344i	$0.476797\pi$
$308$		0			0
$309$	0.0140682			0.000800310
$310$		0			0
$311$		−	15.6136i		−	0.885365i	−0.896678	−	0.442682i	$-0.854027\pi$
$311$		−	15.6136i		−	0.885365i	0.896678	−	0.442682i	$-0.145973\pi$
$312$		0			0
$313$			20.0435i			1.13293i	0.824087	+	0.566463i	$0.191689\pi$
$313$			20.0435i			1.13293i	−0.824087	+	0.566463i	$0.808311\pi$
$314$		0			0
$315$	10.5471	+	2.97087i	0.594261	+	0.167389i
$316$		0			0
$317$	−2.56533			−0.144084			−0.0720418	−	0.997402i	$-0.522951\pi$
$317$	−2.56533			−0.144084			−0.0720418	+	0.997402i	$0.522951\pi$
$318$		0			0
$319$	−5.49379	−	0.283098i	−0.307593	−	0.0158504i
$320$		0			0
$321$	−1.37554			−0.0767751
$322$		0			0
$323$	−14.0101			−0.779542
$324$		0			0
$325$	−37.5911			−2.08518
$326$		0			0
$327$	13.1315			0.726175
$328$		0			0
$329$	17.2307	+	4.85348i	0.949957	+	0.267581i
$330$		0			0
$331$	−8.35145			−0.459037			−0.229519	−	0.973304i	$-0.573715\pi$
$331$	−8.35145			−0.459037			−0.229519	+	0.973304i	$0.573715\pi$
$332$		0			0
$333$	0.869330			0.0476390
$334$		0			0
$335$		−	40.7746i		−	2.22775i
$336$		0			0
$337$			9.93164i			0.541011i	0.962718	+	0.270506i	$0.0871909\pi$
$337$			9.93164i			0.541011i	−0.962718	+	0.270506i	$0.912809\pi$
$338$		0			0
$339$		−	7.30718i		−	0.396872i
$340$		0			0
$341$	−30.5880	−	1.57621i	−1.65643	−	0.0853568i
$342$		0			0
$343$	13.5875	−	12.5849i	0.733657	−	0.679520i
$344$		0			0
$345$	31.9635			1.72086
$346$		0			0
$347$		−	31.2007i		−	1.67494i	−0.546480	−	0.837472i	$-0.684033\pi$
$347$		−	31.2007i		−	1.67494i	0.546480	−	0.837472i	$-0.315967\pi$
$348$		0			0
$349$	28.7154			1.53710			0.768551	−	0.639789i	$-0.220978\pi$
$349$	28.7154			1.53710			0.768551	+	0.639789i	$0.220978\pi$
$350$		0			0
$351$		−	3.09330i		−	0.165108i
$352$		0			0
$353$		−	5.05065i		−	0.268819i	−0.990926	−	0.134409i	$-0.957086\pi$
$353$		−	5.05065i		−	0.268819i	0.990926	−	0.134409i	$-0.0429137\pi$
$354$		0			0
$355$		−	5.54211i		−	0.294145i
$356$		0			0
$357$	−9.10737	−	2.56533i	−0.482013	−	0.135772i
$358$		0			0
$359$		−	21.3973i		−	1.12931i	−0.825328	−	0.564653i	$-0.809010\pi$
$359$		−	21.3973i		−	1.12931i	0.825328	−	0.564653i	$-0.190990\pi$
$360$		0			0
$361$	−3.65260			−0.192242
$362$		0			0
$363$	1.13067	−	10.9417i	0.0593448	−	0.574292i
$364$		0			0
$365$		−	23.0942i		−	1.20880i
$366$		0			0
$367$			36.3455i			1.89722i	0.316449	+	0.948609i	$0.397509\pi$
$367$			36.3455i			1.89722i	−0.316449	+	0.948609i	$0.602491\pi$
$368$		0			0
$369$	5.91758			0.308057
$370$		0			0
$371$	1.35046	−	4.79435i	0.0701123	−	0.248910i
$372$		0			0
$373$			18.1292i			0.938695i	0.883014	+	0.469347i	$0.155511\pi$
$373$			18.1292i			0.938695i	−0.883014	+	0.469347i	$0.844489\pi$
$374$		0			0
$375$	29.6221			1.52968
$376$		0			0
$377$		−	5.13067i		−	0.264243i
$378$		0			0
$379$	2.87942			0.147906			0.0739530	−	0.997262i	$-0.476439\pi$
$379$	2.87942			0.147906			0.0739530	+	0.997262i	$0.476439\pi$
$380$		0			0
$381$	−19.4564			−0.996783
$382$		0			0
$383$		−	17.7255i		−	0.905728i	−0.891580	−	0.452864i	$-0.850402\pi$
$383$		−	17.7255i		−	0.905728i	0.891580	−	0.452864i	$-0.149598\pi$
$384$		0			0
$385$	−11.6404	+	34.4273i	−0.593249	+	1.75458i
$386$		0			0
$387$		−	10.2831i		−	0.522719i
$388$		0			0
$389$	−9.43467			−0.478357			−0.239178	−	0.970976i	$-0.576878\pi$
$389$	−9.43467			−0.478357			−0.239178	+	0.970976i	$0.576878\pi$
$390$		0			0
$391$	−27.6004			−1.39581
$392$		0			0
$393$			14.2108i			0.716838i
$394$		0			0
$395$	44.9293			2.26064
$396$		0			0
$397$			16.4238i			0.824286i	0.911119	+	0.412143i	$0.135220\pi$
$397$			16.4238i			0.824286i	−0.911119	+	0.412143i	$0.864780\pi$
$398$		0			0
$399$	9.97670	+	2.81021i	0.499460	+	0.140686i
$400$		0			0
$401$	−15.7551			−0.786774			−0.393387	−	0.919373i	$-0.628697\pi$
$401$	−15.7551			−0.786774			−0.393387	+	0.919373i	$0.628697\pi$
$402$		0			0
$403$		−	28.5662i		−	1.42298i
$404$		0			0
$405$			4.14155i			0.205795i
$406$		0			0
$407$	−0.148378	+	2.87942i	−0.00735483	+	0.142728i
$408$		0			0
$409$	7.19662			0.355850			0.177925	−	0.984044i	$-0.443062\pi$
$409$	7.19662			0.355850			0.177925	+	0.984044i	$0.443062\pi$
$410$		0			0
$411$			13.4938i			0.665600i
$412$		0			0
$413$	−25.9417	−	7.30718i	−1.27651	−	0.359563i
$414$		0			0
$415$			23.4938i			1.15326i
$416$		0			0
$417$		−	14.5521i		−	0.712621i
$418$		0			0
$419$			15.6004i			0.762128i	0.924549	+	0.381064i	$0.124442\pi$
$419$			15.6004i			0.762128i	−0.924549	+	0.381064i	$0.875558\pi$
$420$		0			0
$421$	−16.0701			−0.783208			−0.391604	−	0.920134i	$-0.628080\pi$
$421$	−16.0701			−0.783208			−0.391604	+	0.920134i	$0.628080\pi$
$422$		0			0
$423$			6.76601i			0.328975i
$424$		0			0
$425$	−43.4597			−2.10810
$426$		0			0
$427$	2.42200	−	8.59852i	0.117209	−	0.416111i
$428$		0			0
$429$	10.2457	+	0.527968i	0.494669	+	0.0254905i
$430$		0			0
$431$			17.3555i			0.835985i	0.908450	+	0.417993i	$0.137266\pi$
$431$			17.3555i			0.835985i	−0.908450	+	0.417993i	$0.862734\pi$
$432$		0			0
$433$			30.8711i			1.48357i	0.670639	+	0.741784i	$0.266020\pi$
$433$			30.8711i			1.48357i	−0.670639	+	0.741784i	$0.733980\pi$
$434$		0			0
$435$			6.86933i			0.329359i
$436$		0			0
$437$	30.2349			1.44633
$438$		0			0
$439$	−19.3245			−0.922309			−0.461154	−	0.887320i	$-0.652565\pi$
$439$	−19.3245			−0.922309			−0.461154	+	0.887320i	$0.652565\pi$
$440$		0			0
$441$	5.97087	+	3.65359i	0.284327	+	0.173981i
$442$		0			0
$443$	7.71862			0.366723			0.183361	−	0.983046i	$-0.441302\pi$
$443$	7.71862			0.366723			0.183361	+	0.983046i	$0.441302\pi$
$444$		0			0
$445$	−44.0017			−2.08588
$446$		0			0
$447$	7.24892			0.342862
$448$		0			0
$449$	−10.1765			−0.480259			−0.240130	−	0.970741i	$-0.577190\pi$
$449$	−10.1765			−0.480259			−0.240130	+	0.970741i	$0.577190\pi$
$450$		0			0
$451$	−1.01002	+	19.6004i	−0.0475599	+	0.922946i
$452$		0			0
$453$	−10.3422			−0.485920
$454$		0			0
$455$	−32.6253	−	9.18979i	−1.52950	−	0.430824i
$456$		0			0
$457$		−	4.80097i		−	0.224580i	−0.993675	−	0.112290i	$-0.964181\pi$
$457$		−	4.80097i		−	0.224580i	0.993675	−	0.112290i	$-0.0358186\pi$
$458$		0			0
$459$		−	3.57621i		−	0.166923i
$460$		0			0
$461$	39.6386			1.84615			0.923077	−	0.384616i	$-0.125666\pi$
$461$	39.6386			1.84615			0.923077	+	0.384616i	$0.125666\pi$
$462$		0			0
$463$	−37.5474			−1.74498			−0.872488	−	0.488636i	$-0.837495\pi$
$463$	−37.5474			−1.74498			−0.872488	+	0.488636i	$0.837495\pi$
$464$		0			0
$465$			38.2466i			1.77364i
$466$		0			0
$467$			24.5179i			1.13455i	0.823528	+	0.567276i	$0.192003\pi$
$467$			24.5179i			1.13455i	−0.823528	+	0.567276i	$0.807997\pi$
$468$		0			0
$469$	7.06231	−	25.0724i	0.326107	−	1.15774i
$470$		0			0
$471$	13.0801			0.602699
$472$		0			0
$473$	34.0600	+	1.75513i	1.56608	+	0.0807010i
$474$		0			0
$475$	47.6081			2.18441
$476$		0			0
$477$	1.88261			0.0861988
$478$		0			0
$479$	11.2272			0.512982			0.256491	−	0.966547i	$-0.417434\pi$
$479$	11.2272			0.512982			0.256491	+	0.966547i	$0.417434\pi$
$480$		0			0
$481$	−2.68910			−0.122612
$482$		0			0
$483$	19.6545	+	5.53620i	0.894309	+	0.251906i
$484$		0			0
$485$	−18.7528			−0.851521
$486$		0			0
$487$	−24.9394			−1.13011			−0.565056	−	0.825052i	$-0.691146\pi$
$487$	−24.9394			−1.13011			−0.565056	+	0.825052i	$0.691146\pi$
$488$		0			0
$489$			3.75513i			0.169813i
$490$		0			0
$491$		−	14.9977i		−	0.676835i	−0.940996	−	0.338418i	$-0.890108\pi$
$491$		−	14.9977i		−	0.676835i	0.940996	−	0.338418i	$-0.109892\pi$
$492$		0			0
$493$		−	5.93164i		−	0.267148i
$494$		0			0
$495$	−13.7178	−	0.706884i	−0.616568	−	0.0317721i
$496$		0			0
$497$	0.959915	−	3.40786i	0.0430581	−	0.152863i
$498$		0			0
$499$	−12.2147			−0.546807			−0.273403	−	0.961899i	$-0.588149\pi$
$499$	−12.2147			−0.546807			−0.273403	+	0.961899i	$0.588149\pi$
$500$		0			0
$501$			16.9658i			0.757977i
$502$		0			0
$503$	40.7328			1.81618			0.908092	−	0.418770i	$-0.137539\pi$
$503$	40.7328			1.81618			0.908092	+	0.418770i	$0.137539\pi$
$504$		0			0
$505$			5.69687i			0.253507i
$506$		0			0
$507$		−	3.43148i		−	0.152397i
$508$		0			0
$509$		−	35.5771i		−	1.57693i	−0.615082	−	0.788463i	$-0.710877\pi$
$509$		−	35.5771i		−	1.57693i	0.615082	−	0.788463i	$-0.289123\pi$
$510$		0			0
$511$	4.00000	−	14.2007i	0.176950	−	0.628201i
$512$		0			0
$513$			3.91758i			0.172965i
$514$		0			0
$515$	0.0582640			0.00256742
$516$		0			0
$517$	−22.4106	−	1.15483i	−0.985616	−	0.0507893i
$518$		0			0
$519$			3.75513i			0.164832i
$520$		0			0
$521$			24.1565i			1.05831i	0.848524	+	0.529157i	$0.177492\pi$
$521$			24.1565i			1.05831i	−0.848524	+	0.529157i	$0.822508\pi$
$522$		0			0
$523$	−8.53037			−0.373007			−0.186504	−	0.982454i	$-0.559716\pi$
$523$	−8.53037			−0.373007			−0.186504	+	0.982454i	$0.559716\pi$
$524$		0			0
$525$	30.9480	+	8.71733i	1.35068	+	0.380456i
$526$		0			0
$527$		−	33.0258i		−	1.43863i
$528$		0			0
$529$	36.5639			1.58973
$530$		0			0
$531$		−	10.1866i		−	0.442061i
$532$		0			0
$533$	−18.3049			−0.792871
$534$		0			0
$535$	−5.69687			−0.246297
$536$		0			0
$537$		−	18.6627i		−	0.805354i
$538$		0			0
$539$	−13.1207	+	19.1533i	−0.565146	+	0.824991i
$540$		0			0
$541$			7.49293i			0.322146i	0.986942	+	0.161073i	$0.0514955\pi$
$541$			7.49293i			0.322146i	−0.986942	+	0.161073i	$0.948505\pi$
$542$		0			0
$543$	14.9216			0.640348
$544$		0			0
$545$	54.3849			2.32959
$546$		0			0
$547$			3.57381i			0.152805i	0.997077	+	0.0764026i	$0.0243434\pi$
$547$			3.57381i			0.152805i	−0.997077	+	0.0764026i	$0.975657\pi$
$548$		0			0
$549$	3.37640			0.144101
$550$		0			0
$551$			6.49784i			0.276817i
$552$		0			0
$553$	27.6272	+	7.78193i	1.17483	+	0.330922i
$554$		0			0
$555$	3.60037			0.152827
$556$		0			0
$557$			29.0194i			1.22959i	0.788686	+	0.614796i	$0.210762\pi$
$557$			29.0194i			1.22959i	−0.788686	+	0.614796i	$0.789238\pi$
$558$		0			0
$559$			31.8087i			1.34537i
$560$		0			0
$561$	11.8452	+	0.610392i	0.500107	+	0.0257708i
$562$		0			0
$563$	15.2566			0.642989			0.321495	−	0.946911i	$-0.395815\pi$
$563$	15.2566			0.642989			0.321495	+	0.946911i	$0.395815\pi$
$564$		0			0
$565$		−	30.2631i		−	1.27318i
$566$		0			0
$567$	−0.717333	+	2.54665i	−0.0301251	+	0.106949i
$568$		0			0
$569$		−	27.6740i		−	1.16016i	−0.814561	−	0.580078i	$-0.803022\pi$
$569$		−	27.6740i		−	1.16016i	0.814561	−	0.580078i	$-0.196978\pi$
$570$		0			0
$571$			3.97996i			0.166556i	0.996526	+	0.0832782i	$0.0265390\pi$
$571$			3.97996i			0.166556i	−0.996526	+	0.0832782i	$0.973461\pi$
$572$		0			0
$573$			10.5871i			0.442282i
$574$		0			0
$575$	93.7896			3.91130
$576$		0			0
$577$			20.0382i			0.834202i	0.908860	+	0.417101i	$0.136954\pi$
$577$			20.0382i			0.834202i	−0.908860	+	0.417101i	$0.863046\pi$
$578$		0			0
$579$	−11.4355			−0.475244
$580$		0			0
$581$	−4.06922	+	14.4464i	−0.168820	+	0.599338i
$582$		0			0
$583$	−0.321326	+	6.23564i	−0.0133080	+	0.258254i
$584$		0			0
$585$		−	12.8111i		−	0.529672i
$586$		0			0
$587$		−	22.3796i		−	0.923705i	−0.886957	−	0.461852i	$-0.847185\pi$
$587$		−	22.3796i		−	0.923705i	0.886957	−	0.461852i	$-0.152815\pi$
$588$		0			0
$589$			36.1782i			1.49070i
$590$		0			0
$591$	−13.2107			−0.543415
$592$		0			0
$593$	−29.7492			−1.22165			−0.610826	−	0.791765i	$-0.709162\pi$
$593$	−29.7492			−1.22165			−0.610826	+	0.791765i	$0.709162\pi$
$594$		0			0
$595$	−37.7186	−	10.6245i	−1.54631	−	0.435560i
$596$		0			0
$597$	−14.0077			−0.573297
$598$		0			0
$599$	−24.8484			−1.01528			−0.507640	−	0.861569i	$-0.669482\pi$
$599$	−24.8484			−1.01528			−0.507640	+	0.861569i	$0.669482\pi$
$600$		0			0
$601$	−22.6805			−0.925156			−0.462578	−	0.886579i	$-0.653075\pi$
$601$	−22.6805			−0.925156			−0.462578	+	0.886579i	$0.653075\pi$
$602$		0			0
$603$	9.84524			0.400929
$604$		0			0
$605$	4.68272	−	45.3157i	0.190380	−	1.84235i
$606$		0			0
$607$	26.2075			1.06373			0.531865	−	0.846829i	$-0.321491\pi$
$607$	26.2075			1.06373			0.531865	+	0.846829i	$0.321491\pi$
$608$		0			0
$609$	−1.18979	+	4.22397i	−0.0482129	+	0.171164i
$610$		0			0
$611$		−	20.9293i		−	0.846710i
$612$		0			0
$613$		−	1.56301i		−	0.0631293i	−0.999502	−	0.0315646i	$-0.989951\pi$
$613$		−	1.56301i		−	0.0631293i	0.999502	−	0.0315646i	$-0.0100490\pi$
$614$		0			0
$615$	24.5079			0.988255
$616$		0			0
$617$	−6.87571			−0.276806			−0.138403	−	0.990376i	$-0.544197\pi$
$617$	−6.87571			−0.276806			−0.138403	+	0.990376i	$0.544197\pi$
$618$		0			0
$619$			5.37726i			0.216130i	0.994144	+	0.108065i	$0.0344655\pi$
$619$			5.37726i			0.216130i	−0.994144	+	0.108065i	$0.965534\pi$
$620$		0			0
$621$			7.71776i			0.309703i
$622$		0			0
$623$	−27.0568	−	7.62127i	−1.08401	−	0.305340i
$624$		0			0
$625$	61.9194			2.47677
$626$		0			0
$627$	−12.9759	−	0.668656i	−0.518208	−	0.0267035i
$628$		0			0
$629$	−3.10891			−0.123960
$630$		0			0
$631$	−4.31123			−0.171628			−0.0858138	−	0.996311i	$-0.527349\pi$
$631$	−4.31123			−0.171628			−0.0858138	+	0.996311i	$0.527349\pi$
$632$		0			0
$633$	7.33903			0.291700
$634$		0			0
$635$	−80.5797			−3.19771
$636$		0			0
$637$	−18.4697	−	11.3017i	−0.731796	−	0.447788i
$638$		0			0
$639$	1.33817			0.0529373
$640$		0			0
$641$	0.869330			0.0343365			0.0171682	−	0.999853i	$-0.494535\pi$
$641$	0.869330			0.0343365			0.0171682	+	0.999853i	$0.494535\pi$
$642$		0			0
$643$		−	0.955731i		−	0.0376903i	−0.999822	−	0.0188452i	$-0.994001\pi$
$643$		−	0.955731i		−	0.0376903i	0.999822	−	0.0188452i	$-0.00599896\pi$
$644$		0			0
$645$		−	42.5880i		−	1.67690i
$646$		0			0
$647$		−	25.9668i		−	1.02086i	−0.859920	−	0.510429i	$-0.829487\pi$
$647$		−	25.9668i		−	1.02086i	0.859920	−	0.510429i	$-0.170513\pi$
$648$		0			0
$649$	33.7404	+	1.73866i	1.32443	+	0.0682484i
$650$		0			0
$651$	−6.62446	+	23.5180i	−0.259633	+	0.921741i
$652$		0			0
$653$	2.93855			0.114994			0.0574971	−	0.998346i	$-0.481688\pi$
$653$	2.93855			0.114994			0.0574971	+	0.998346i	$0.481688\pi$
$654$		0			0
$655$			58.8546i			2.29964i
$656$		0			0
$657$	5.57621			0.217549
$658$		0			0
$659$			16.5763i			0.645720i	0.946447	+	0.322860i	$0.104644\pi$
$659$			16.5763i			0.645720i	−0.946447	+	0.322860i	$0.895356\pi$
$660$		0			0
$661$		−	1.87935i		−	0.0730982i	−0.999332	−	0.0365491i	$-0.988363\pi$
$661$		−	1.87935i		−	0.0730982i	0.999332	−	0.0365491i	$-0.0116365\pi$
$662$		0			0
$663$			11.0623i			0.429625i
$664$		0			0
$665$	41.3190	+	11.6386i	1.60228	+	0.451326i
$666$		0			0
$667$			12.8010i			0.495656i
$668$		0			0
$669$	−7.42146			−0.286930
$670$		0			0
$671$	−0.576288	+	11.1834i	−0.0222473	+	0.431731i
$672$		0			0
$673$		−	51.6158i		−	1.98964i	−0.101647	−	0.994821i	$-0.532411\pi$
$673$		−	51.6158i		−	1.98964i	0.101647	−	0.994821i	$-0.467589\pi$
$674$		0			0
$675$			12.1524i			0.467747i
$676$		0			0
$677$	38.3633			1.47442			0.737210	−	0.675664i	$-0.236143\pi$
$677$	38.3633			1.47442			0.737210	+	0.675664i	$0.236143\pi$
$678$		0			0
$679$	−11.5312	−	3.24806i	−0.442525	−	0.124649i
$680$		0			0
$681$			8.46201i			0.324265i
$682$		0			0
$683$	14.4697			0.553668			0.276834	−	0.960918i	$-0.410715\pi$
$683$	14.4697			0.553668			0.276834	+	0.960918i	$0.410715\pi$
$684$		0			0
$685$			55.8852i			2.13526i
$686$		0			0
$687$	1.03178			0.0393647
$688$		0			0
$689$	−5.82349			−0.221857
$690$		0			0
$691$			8.64622i			0.328918i	0.986384	+	0.164459i	$0.0525878\pi$
$691$			8.64622i			0.328918i	−0.986384	+	0.164459i	$0.947412\pi$
$692$		0			0
$693$	−8.31266	−	2.81064i	−0.315772	−	0.106767i
$694$		0			0
$695$		−	60.2684i		−	2.28611i
$696$		0			0
$697$	−21.1625			−0.801588
$698$		0			0
$699$	−0.944064			−0.0357078
$700$		0			0
$701$			4.37959i			0.165415i	0.996574	+	0.0827074i	$0.0263567\pi$
$701$			4.37959i			0.165415i	−0.996574	+	0.0827074i	$0.973643\pi$
$702$		0			0
$703$	3.40567			0.128447
$704$		0			0
$705$			28.0218i			1.05536i
$706$		0			0
$707$	−0.986720	+	3.50302i	−0.0371094	+	0.131745i
$708$		0			0
$709$	−12.4214			−0.466495			−0.233247	−	0.972417i	$-0.574935\pi$
$709$	−12.4214			−0.466495			−0.233247	+	0.972417i	$0.574935\pi$
$710$		0			0
$711$			10.8484i			0.406848i
$712$		0			0
$713$			71.2724i			2.66917i
$714$		0			0
$715$	42.4332	+	2.18661i	1.58691	+	0.0817744i
$716$		0			0
$717$	−20.9394			−0.781997
$718$		0			0
$719$		−	10.7862i		−	0.402257i	−0.979565	−	0.201129i	$-0.935539\pi$
$719$		−	10.7862i		−	0.402257i	0.979565	−	0.201129i	$-0.0644609\pi$
$720$		0			0
$721$	0.0358267	+	0.0100916i	0.00133426	+	0.000375829i
$722$		0			0
$723$		−	17.8593i		−	0.664195i
$724$		0			0
$725$			20.1565i			0.748593i
$726$		0			0
$727$			15.4215i			0.571950i	0.958237	+	0.285975i	$0.0923175\pi$
$727$			15.4215i			0.571950i	−0.958237	+	0.285975i	$0.907683\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			36.7746i			1.36016i
$732$		0			0
$733$	−44.1026			−1.62897			−0.814484	−	0.580186i	$-0.802980\pi$
$733$	−44.1026			−1.62897			−0.814484	+	0.580186i	$0.802980\pi$
$734$		0			0
$735$	24.7286	+	15.1315i	0.912130	+	0.558135i
$736$		0			0
$737$	−1.68040	+	32.6097i	−0.0618982	+	1.20119i
$738$		0			0
$739$		−	12.8975i		−	0.474441i	−0.971456	−	0.237220i	$-0.923764\pi$
$739$		−	12.8975i		−	0.474441i	0.971456	−	0.237220i	$-0.0762364\pi$
$740$		0			0
$741$		−	12.1183i		−	0.445175i
$742$		0			0
$743$			8.11825i			0.297830i	0.988850	+	0.148915i	$0.0475780\pi$
$743$			8.11825i			0.297830i	−0.988850	+	0.148915i	$0.952422\pi$
$744$		0			0
$745$	30.0218			1.09991
$746$		0			0
$747$	−5.67271			−0.207553
$748$		0			0
$749$	−3.50302	−	0.986720i	−0.127998	−	0.0360539i
$750$		0			0
$751$	−12.4915			−0.455820			−0.227910	−	0.973682i	$-0.573189\pi$
$751$	−12.4915			−0.455820			−0.227910	+	0.973682i	$0.573189\pi$
$752$		0			0
$753$	−8.96582			−0.326733
$754$		0			0
$755$	−42.8328			−1.55885
$756$		0			0
$757$	19.4837			0.708147			0.354074	−	0.935218i	$-0.384796\pi$
$757$	19.4837			0.708147			0.354074	+	0.935218i	$0.384796\pi$
$758$		0			0
$759$	−25.5630	−	1.31728i	−0.927878	−	0.0478141i
$760$		0			0
$761$	−31.6044			−1.14566			−0.572828	−	0.819675i	$-0.694154\pi$
$761$	−31.6044			−1.14566			−0.572828	+	0.819675i	$0.694154\pi$
$762$		0			0
$763$	33.4414	+	9.41967i	1.21066	+	0.341015i
$764$		0			0
$765$		−	14.8111i		−	0.535495i
$766$		0			0
$767$			31.5103i			1.13777i
$768$		0			0
$769$	30.3491			1.09441			0.547207	−	0.836997i	$-0.315691\pi$
$769$	30.3491			1.09441			0.547207	+	0.836997i	$0.315691\pi$
$770$		0			0
$771$	8.52797			0.307127
$772$		0			0
$773$			31.1010i			1.11862i	0.828957	+	0.559312i	$0.188935\pi$
$773$			31.1010i			1.11862i	−0.828957	+	0.559312i	$0.811065\pi$
$774$		0			0
$775$			112.226i			4.03127i
$776$		0			0
$777$	2.21388	+	0.623599i	0.0794225	+	0.0223715i
$778$		0			0
$779$	23.1826			0.830601
$780$		0			0
$781$	−0.228401	+	4.43234i	−0.00817282	+	0.158602i
$782$		0			0
$783$	−1.65864			−0.0592749
$784$		0			0
$785$	54.1719			1.93348
$786$		0			0
$787$	−32.6503			−1.16386			−0.581930	−	0.813239i	$-0.697702\pi$
$787$	−32.6503			−1.16386			−0.581930	+	0.813239i	$0.697702\pi$
$788$		0			0
$789$	−31.2871			−1.11385
$790$		0			0
$791$	5.24168	−	18.6089i	0.186373	−	0.661655i
$792$		0			0
$793$	−10.4442			−0.370886
$794$		0			0
$795$	7.79692			0.276528
$796$		0			0
$797$		−	24.4400i		−	0.865710i	−0.901464	−	0.432855i	$-0.857506\pi$
$797$		−	24.4400i		−	0.865710i	0.901464	−	0.432855i	$-0.142494\pi$
$798$		0			0
$799$		−	24.1967i		−	0.856018i
$800$		0			0
$801$		−	10.6245i		−	0.375397i
$802$		0			0
$803$	−0.951754	+	18.4697i	−0.0335867	+	0.651782i
$804$		0			0
$805$	81.3999	+	22.9285i	2.86897	+	0.808122i
$806$		0			0
$807$	−7.57535			−0.266665
$808$		0			0
$809$			23.7387i			0.834607i	0.908767	+	0.417303i	$0.137025\pi$
$809$			23.7387i			0.834607i	−0.908767	+	0.417303i	$0.862975\pi$
$810$		0			0
$811$	9.56612			0.335912			0.167956	−	0.985794i	$-0.446283\pi$
$811$	9.56612			0.335912			0.167956	+	0.985794i	$0.446283\pi$
$812$		0			0
$813$		−	14.9667i		−	0.524905i
$814$		0			0
$815$			15.5521i			0.544765i
$816$		0			0
$817$		−	40.2848i		−	1.40939i
$818$		0			0
$819$	2.21893	−	7.87756i	0.0775356	−	0.275264i
$820$		0			0
$821$			27.7387i			0.968086i	0.875044	+	0.484043i	$0.160832\pi$
$821$			27.7387i			0.968086i	−0.875044	+	0.484043i	$0.839168\pi$
$822$		0			0
$823$	−11.1041			−0.387065			−0.193532	−	0.981094i	$-0.561994\pi$
$823$	−11.1041			−0.387065			−0.193532	+	0.981094i	$0.561994\pi$
$824$		0			0
$825$	−40.2516	−	2.07419i	−1.40138	−	0.0722140i
$826$		0			0
$827$		−	49.0995i		−	1.70736i	−0.520802	−	0.853678i	$-0.674367\pi$
$827$		−	49.0995i		−	1.70736i	0.520802	−	0.853678i	$-0.325633\pi$
$828$		0			0
$829$			44.4409i			1.54350i	0.635929	+	0.771748i	$0.280617\pi$
$829$			44.4409i			1.54350i	−0.635929	+	0.771748i	$0.719383\pi$
$830$		0			0
$831$	29.2498			1.01466
$832$		0			0
$833$	−21.3531	−	13.0660i	−0.739841	−	0.452711i
$834$		0			0
$835$			70.2648i			2.43161i
$836$		0			0
$837$	−9.23485			−0.319203
$838$		0			0
$839$		−	0.412833i		−	0.0142526i	−0.999975	−	0.00712629i	$-0.997732\pi$
$839$		−	0.412833i		−	0.0142526i	0.999975	−	0.00712629i	$-0.00226839\pi$
$840$		0			0
$841$	26.2489			0.905135
$842$		0			0
$843$	−15.4938			−0.533634
$844$		0			0
$845$		−	14.2116i		−	0.488895i
$846$		0			0
$847$	10.7283	−	27.0537i	0.368628	−	0.929577i
$848$		0			0
$849$		−	0.855262i		−	0.0293525i
$850$		0			0
$851$	6.70929			0.229991
$852$		0			0
$853$	19.4028			0.664340			0.332170	−	0.943220i	$-0.392219\pi$
$853$	19.4028			0.664340			0.332170	+	0.943220i	$0.392219\pi$
$854$		0			0
$855$			16.2248i			0.554878i
$856$		0			0
$857$	−8.86164			−0.302708			−0.151354	−	0.988480i	$-0.548363\pi$
$857$	−8.86164			−0.302708			−0.151354	+	0.988480i	$0.548363\pi$
$858$		0			0
$859$			9.40897i			0.321030i	0.987033	+	0.160515i	$0.0513155\pi$
$859$			9.40897i			0.321030i	−0.987033	+	0.160515i	$0.948685\pi$
$860$		0			0
$861$	15.0700	+	4.24487i	0.513584	+	0.144665i
$862$		0			0
$863$	39.2715			1.33682			0.668409	−	0.743794i	$-0.266975\pi$
$863$	39.2715			1.33682			0.668409	+	0.743794i	$0.266975\pi$
$864$		0			0
$865$			15.5521i			0.528786i
$866$		0			0
$867$		−	4.21069i		−	0.143003i
$868$		0			0
$869$	−35.9325	−	1.85162i	−1.21893	−	0.0628120i
$870$		0			0
$871$	−30.4543			−1.03191
$872$		0			0
$873$		−	4.52797i		−	0.153249i
$874$		0			0
$875$	75.4372	+	21.2489i	2.55024	+	0.718345i
$876$		0			0
$877$		−	29.1113i		−	0.983020i	−0.870872	−	0.491510i	$-0.836445\pi$
$877$		−	29.1113i		−	0.983020i	0.870872	−	0.491510i	$-0.163555\pi$
$878$		0			0
$879$		−	10.0319i		−	0.338366i
$880$		0			0
$881$		−	36.7210i		−	1.23716i	−0.785722	−	0.618580i	$-0.787708\pi$
$881$		−	36.7210i		−	1.23716i	0.785722	−	0.618580i	$-0.212292\pi$
$882$		0			0
$883$	−40.9394			−1.37772			−0.688860	−	0.724894i	$-0.741889\pi$
$883$	−40.9394			−1.37772			−0.688860	+	0.724894i	$0.741889\pi$
$884$		0			0
$885$		−	42.1883i		−	1.41815i
$886$		0			0
$887$	9.08407			0.305013			0.152507	−	0.988302i	$-0.451265\pi$
$887$	9.08407			0.305013			0.152507	+	0.988302i	$0.451265\pi$
$888$		0			0
$889$	−49.5487	−	13.9567i	−1.66181	−	0.468094i
$890$		0			0
$891$	0.170681	−	3.31223i	0.00571803	−	0.110964i
$892$		0			0
$893$			26.5064i			0.887001i
$894$		0			0
$895$		−	77.2924i		−	2.58360i
$896$		0			0
$897$		−	23.8734i		−	0.797109i
$898$		0			0
$899$	−15.3173			−0.510860
$900$		0			0
$901$	−6.73262			−0.224296
$902$		0			0
$903$	7.37640	−	26.1875i	0.245471	−	0.871464i
$904$		0			0
$905$	61.7986			2.05426
$906$		0			0
$907$	−38.6044			−1.28184			−0.640919	−	0.767608i	$-0.721447\pi$
$907$	−38.6044			−1.28184			−0.640919	+	0.767608i	$0.721447\pi$
$908$		0			0
$909$	−1.37554			−0.0456238
$910$		0			0
$911$	−22.6618			−0.750820			−0.375410	−	0.926859i	$-0.622498\pi$
$911$	−22.6618			−0.750820			−0.375410	+	0.926859i	$0.622498\pi$
$912$		0			0
$913$	0.968223	−	18.7893i	0.0320435	−	0.621835i
$914$		0			0
$915$	13.9835			0.462281
$916$		0			0
$917$	−10.1938	+	36.1899i	−0.336630	+	1.19509i
$918$		0			0
$919$			0.294051i			0.00969984i	0.999988	+	0.00484992i	$0.00154378\pi$
$919$			0.294051i			0.00969984i	−0.999988	+	0.00484992i	$0.998456\pi$
$920$		0			0
$921$			2.55213i			0.0840955i
$922$		0			0
$923$	−4.13938			−0.136249
$924$		0			0
$925$	10.5645			0.347358
$926$		0			0
$927$			0.0140682i			0.000462059i
$928$		0			0
$929$			7.42543i			0.243621i	0.992553	+	0.121810i	$0.0388700\pi$
$929$			7.42543i			0.243621i	−0.992553	+	0.121810i	$0.961130\pi$
$930$		0			0
$931$	23.3913	+	14.3132i	0.766620	+	0.469097i
$932$		0			0
$933$	15.6136			0.511166
$934$		0			0
$935$	49.0577	+	2.52797i	1.60436	+	0.0826734i
$936$		0			0
$937$	−31.9976			−1.04532			−0.522658	−	0.852542i	$-0.675060\pi$
$937$	−31.9976			−1.04532			−0.522658	+	0.852542i	$0.675060\pi$
$938$		0			0
$939$	−20.0435			−0.654095
$940$		0			0
$941$	−10.4033			−0.339139			−0.169570	−	0.985518i	$-0.554238\pi$
$941$	−10.4033			−0.339139			−0.169570	+	0.985518i	$0.554238\pi$
$942$		0			0
$943$	45.6705			1.48723
$944$		0			0
$945$	−2.97087	+	10.5471i	−0.0966423	+	0.343097i
$946$		0			0
$947$	−20.6944			−0.672477			−0.336239	−	0.941777i	$-0.609155\pi$
$947$	−20.6944			−0.672477			−0.336239	+	0.941777i	$0.609155\pi$
$948$		0			0
$949$	−17.2489			−0.559924
$950$		0			0
$951$		−	2.56533i		−	0.0831867i
$952$		0			0
$953$		−	43.1324i		−	1.39720i	−0.715515	−	0.698598i	$-0.753808\pi$
$953$		−	43.1324i		−	1.39720i	0.715515	−	0.698598i	$-0.246192\pi$
$954$		0			0
$955$			43.8470i			1.41885i
$956$		0			0
$957$	0.283098	−	5.49379i	0.00915126	−	0.177589i
$958$		0			0
$959$	−9.67954	+	34.3640i	−0.312568	+	1.10967i
$960$		0			0
$961$	−54.2825			−1.75105
$962$		0			0
$963$		−	1.37554i		−	0.0443262i
$964$		0			0
$965$	−47.3608			−1.52460
$966$		0			0
$967$		−	32.0062i		−	1.02925i	−0.857416	−	0.514624i	$-0.827931\pi$
$967$		−	32.0062i		−	1.02925i	0.857416	−	0.514624i	$-0.172069\pi$
$968$		0			0
$969$		−	14.0101i		−	0.450069i
$970$		0			0
$971$			38.4560i			1.23411i	0.786919	+	0.617057i	$0.211675\pi$
$971$			38.4560i			1.23411i	−0.786919	+	0.617057i	$0.788325\pi$
$972$		0			0
$973$	10.4387	−	37.0592i	0.334650	−	1.18806i
$974$		0			0
$975$		−	37.5911i		−	1.20388i
$976$		0			0
$977$	−39.8771			−1.27578			−0.637891	−	0.770127i	$-0.720193\pi$
$977$	−39.8771			−1.27578			−0.637891	+	0.770127i	$0.720193\pi$
$978$		0			0
$979$	35.1907	+	1.81339i	1.12470	+	0.0579563i
$980$		0			0
$981$			13.1315i			0.419258i
$982$		0			0
$983$			46.8641i			1.49473i	0.664413	+	0.747366i	$0.268682\pi$
$983$			46.8641i			1.49473i	−0.664413	+	0.747366i	$0.731318\pi$
$984$		0			0
$985$	−54.7127			−1.74329
$986$		0			0
$987$	−4.85348	+	17.2307i	−0.154488	+	0.548458i
$988$		0			0
$989$		−	79.3625i		−	2.52358i
$990$		0			0
$991$	26.2631			0.834274			0.417137	−	0.908844i	$-0.363034\pi$
$991$	26.2631			0.834274			0.417137	+	0.908844i	$0.363034\pi$
$992$		0			0
$993$		−	8.35145i		−	0.265025i
$994$		0			0
$995$	−58.0135			−1.83915
$996$		0			0
$997$	−33.5630			−1.06295			−0.531476	−	0.847074i	$-0.678362\pi$
$997$	−33.5630			−1.06295			−0.531476	+	0.847074i	$0.678362\pi$
$998$		0			0
$999$			0.869330i			0.0275044i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.q.b.769.8		8
4.3	odd	2		462.2.e.a.307.4	✓	8
7.6	odd	2		3696.2.q.c.769.1		8
11.10	odd	2		3696.2.q.c.769.8		8
12.11	even	2		1386.2.e.e.307.5		8
28.27	even	2		462.2.e.b.307.1	yes	8
44.43	even	2		462.2.e.b.307.8	yes	8
77.76	even	2	inner	3696.2.q.b.769.1		8
84.83	odd	2		1386.2.e.a.307.8		8
132.131	odd	2		1386.2.e.a.307.1		8
308.307	odd	2		462.2.e.a.307.5	yes	8
924.923	even	2		1386.2.e.e.307.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.e.a.307.4	✓	8	4.3	odd	2
462.2.e.a.307.5	yes	8	308.307	odd	2
462.2.e.b.307.1	yes	8	28.27	even	2
462.2.e.b.307.8	yes	8	44.43	even	2
1386.2.e.a.307.1		8	132.131	odd	2
1386.2.e.a.307.8		8	84.83	odd	2
1386.2.e.e.307.4		8	924.923	even	2
1386.2.e.e.307.5		8	12.11	even	2
3696.2.q.b.769.1		8	77.76	even	2	inner
3696.2.q.b.769.8		8	1.1	even	1	trivial
3696.2.q.c.769.1		8	7.6	odd	2
3696.2.q.c.769.8		8	11.10	odd	2

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:	\( N \)	\(=\)	\( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3696.q (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +4.14155i q^{5} +(-0.717333 + 2.54665i) q^{7} -1.00000 q^{9} +(0.170681 - 3.31223i) q^{11} +3.09330 q^{13} -4.14155 q^{15} +3.57621 q^{17} -3.91758 q^{19} +(-2.54665 - 0.717333i) q^{21} -7.71776 q^{23} -12.1524 q^{25} -1.00000i q^{27} -1.65864i q^{29} -9.23485i q^{31} +(3.31223 + 0.170681i) q^{33} +(-10.5471 - 2.97087i) q^{35} -0.869330 q^{37} +3.09330i q^{39} -5.91758 q^{41} +10.2831i q^{43} -4.14155i q^{45} -6.76601i q^{47} +(-5.97087 - 3.65359i) q^{49} +3.57621i q^{51} -1.88261 q^{53} +(13.7178 + 0.706884i) q^{55} -3.91758i q^{57} +10.1866i q^{59} -3.37640 q^{61} +(0.717333 - 2.54665i) q^{63} +12.8111i q^{65} -9.84524 q^{67} -7.71776i q^{69} -1.33817 q^{71} -5.57621 q^{73} -12.1524i q^{75} +(8.31266 + 2.81064i) q^{77} -10.8484i q^{79} +1.00000 q^{81} +5.67271 q^{83} +14.8111i q^{85} +1.65864 q^{87} +10.6245i q^{89} +(-2.21893 + 7.87756i) q^{91} +9.23485 q^{93} -16.2248i q^{95} +4.52797i q^{97} +(-0.170681 + 3.31223i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} + 8 q^{11} - 4 q^{15} - 12 q^{17} - 4 q^{19} - 8 q^{21} + 8 q^{23} - 16 q^{25} + 4 q^{33} - 8 q^{35} + 16 q^{37} - 20 q^{41} - 12 q^{49} + 40 q^{55} + 56 q^{61} - 16 q^{67} - 8 q^{71} - 4 q^{73}+ \cdots - 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 8 * q^11 - 4 * q^15 - 12 * q^17 - 4 * q^19 - 8 * q^21 + 8 * q^23 - 16 * q^25 + 4 * q^33 - 8 * q^35 + 16 * q^37 - 20 * q^41 - 12 * q^49 + 40 * q^55 + 56 * q^61 - 16 * q^67 - 8 * q^71 - 4 * q^73 - 20 * q^77 + 8 * q^81 - 4 * q^83 - 20 * q^91 + 20 * q^93 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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