LMFDB - Embedded newform 5328.2.h.q.2737.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5328,2,Mod(2737,5328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5328.2737"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5328 = 2^{4} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5328.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,4,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0, 0,0,0,0,0,0,-6,0,0,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$42.5442941969$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.49179812660224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 6x^{7} + 53x^{6} - 46x^{5} + 18x^{4} + 12x^{3} + 196x^{2} - 112x + 32 $ x^10 - 6x^7 + 53x^6 - 46x^5 + 18x^4 + 12x^3 + 196x^2 - 112*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 296)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2737.1
Root			$0.279838 + 0.279838i$ of defining polynomial
Character	$\chi$	$=$	5328.2737
Dual form			5328.2.h.q.2737.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.19874i q^{5} +3.28371 q^{7} +1.70530 q^{11} +1.45481i q^{13} +4.77715i q^{17} +0.209741i q^{19} -8.73294i q^{23} -5.23196 q^{25} -2.65144i q^{29} +9.47491i q^{31} -10.5037i q^{35} +(3.65355 + 4.86329i) q^{37} +11.3457 q^{41} -8.67366i q^{43} +12.6681 q^{47} +3.78272 q^{49} +5.11378 q^{53} -5.45481i q^{55} +1.65780i q^{59} -1.90947i q^{61} +4.65355 q^{65} -1.90947 q^{67} -9.51127 q^{71} +6.10278 q^{73} +5.59969 q^{77} +3.62473i q^{79} -3.20221 q^{83} +15.2809 q^{85} -10.1727i q^{89} +4.77715i q^{91} +0.670907 q^{95} +10.1905i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{7} + 2 q^{11} + 4 q^{25} - 6 q^{37} + 26 q^{41} + 8 q^{47} + 14 q^{49} + 20 q^{53} + 4 q^{65} + 2 q^{67} - 4 q^{71} - 14 q^{73} - 36 q^{83} + 8 q^{85} + 4 q^{95}+O(q^{100}) $ 10 * q + 4 * q^7 + 2 * q^11 + 4 * q^25 - 6 * q^37 + 26 * q^41 + 8 * q^47 + 14 * q^49 + 20 * q^53 + 4 * q^65 + 2 * q^67 - 4 * q^71 - 14 * q^73 - 36 * q^83 + 8 * q^85 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$		−	3.19874i		−	1.43052i	−0.698858	−	0.715261i	$-0.746308\pi$
$5$		−	3.19874i		−	1.43052i	0.698858	−	0.715261i	$-0.253692\pi$
$6$		0			0
$7$	3.28371			1.24112			0.620562	−	0.784157i	$-0.286904\pi$
$7$	3.28371			1.24112			0.620562	+	0.784157i	$0.286904\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.70530			0.514166			0.257083	−	0.966389i	$-0.417239\pi$
$11$	1.70530			0.514166			0.257083	+	0.966389i	$0.417239\pi$
$12$		0			0
$13$			1.45481i			0.403490i	0.979438	+	0.201745i	$0.0646613\pi$
$13$			1.45481i			0.403490i	−0.979438	+	0.201745i	$0.935339\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.77715i			1.15863i	0.815104	+	0.579315i	$0.196680\pi$
$17$			4.77715i			1.15863i	−0.815104	+	0.579315i	$0.803320\pi$
$18$		0			0
$19$			0.209741i			0.0481179i	0.999711	+	0.0240589i	$0.00765894\pi$
$19$			0.209741i			0.0481179i	−0.999711	+	0.0240589i	$0.992341\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.73294i		−	1.82094i	−0.413571	−	0.910472i	$-0.635719\pi$
$23$		−	8.73294i		−	1.82094i	0.413571	−	0.910472i	$-0.364281\pi$
$24$		0			0
$25$	−5.23196			−1.04639
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.65144i		−	0.492360i	−0.969224	−	0.246180i	$-0.920825\pi$
$29$		−	2.65144i		−	0.492360i	0.969224	−	0.246180i	$-0.0791754\pi$
$30$		0			0
$31$			9.47491i			1.70174i	0.525373	+	0.850872i	$0.323926\pi$
$31$			9.47491i			1.70174i	−0.525373	+	0.850872i	$0.676074\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	10.5037i		−	1.77545i
$36$		0			0
$37$	3.65355	+	4.86329i	0.600640	+	0.799520i
$37$	3.65355	+	4.86329i	0.600640	+	0.799520i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.3457			1.77191			0.885953	−	0.463774i	$-0.153505\pi$
$41$	11.3457			1.77191			0.885953	+	0.463774i	$0.153505\pi$
$42$		0			0
$43$		−	8.67366i		−	1.32272i	−0.750069	−	0.661360i	$-0.769980\pi$
$43$		−	8.67366i		−	1.32272i	0.750069	−	0.661360i	$-0.230020\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.6681			1.84783			0.923915	−	0.382598i	$-0.124970\pi$
$47$	12.6681			1.84783			0.923915	+	0.382598i	$0.124970\pi$
$48$		0			0
$49$	3.78272			0.540389
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.11378			0.702432			0.351216	−	0.936295i	$-0.385768\pi$
$53$	5.11378			0.702432			0.351216	+	0.936295i	$0.385768\pi$
$54$		0			0
$55$		−	5.45481i		−	0.735526i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1.65780i			0.215827i	0.994160	+	0.107914i	$0.0344170\pi$
$59$			1.65780i			0.215827i	−0.994160	+	0.107914i	$0.965583\pi$
$60$		0			0
$61$		−	1.90947i		−	0.244482i	−0.992500	−	0.122241i	$-0.960992\pi$
$61$		−	1.90947i		−	0.244482i	0.992500	−	0.122241i	$-0.0390081\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.65355			0.577202
$66$		0			0
$67$	−1.90947			−0.233278			−0.116639	−	0.993174i	$-0.537212\pi$
$67$	−1.90947			−0.233278			−0.116639	+	0.993174i	$0.537212\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.51127			−1.12878			−0.564390	−	0.825508i	$-0.690889\pi$
$71$	−9.51127			−1.12878			−0.564390	+	0.825508i	$0.690889\pi$
$72$		0			0
$73$	6.10278			0.714277			0.357138	−	0.934051i	$-0.383752\pi$
$73$	6.10278			0.714277			0.357138	+	0.934051i	$0.383752\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.59969			0.638144
$78$		0			0
$79$			3.62473i			0.407814i	0.978990	+	0.203907i	$0.0653640\pi$
$79$			3.62473i			0.407814i	−0.978990	+	0.203907i	$0.934636\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.20221			−0.351488			−0.175744	−	0.984436i	$-0.556233\pi$
$83$	−3.20221			−0.351488			−0.175744	+	0.984436i	$0.556233\pi$
$84$		0			0
$85$	15.2809			1.65744
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.1727i		−	1.07830i	−0.842209	−	0.539151i	$-0.818745\pi$
$89$		−	10.1727i		−	1.07830i	0.842209	−	0.539151i	$-0.181255\pi$
$90$		0			0
$91$			4.77715i			0.500782i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.670907			0.0688337
$96$		0			0
$97$			10.1905i			1.03469i	0.855778	+	0.517344i	$0.173079\pi$
$97$			10.1905i			1.03469i	−0.855778	+	0.517344i	$0.826921\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.7607			1.46875			0.734374	−	0.678745i	$-0.237476\pi$
$101$	14.7607			1.46875			0.734374	+	0.678745i	$0.237476\pi$
$102$		0			0
$103$		−	6.42434i		−	0.633009i	−0.948591	−	0.316504i	$-0.897491\pi$
$103$		−	6.42434i		−	0.633009i	0.948591	−	0.316504i	$-0.102509\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.3907			−1.00451			−0.502255	−	0.864719i	$-0.667496\pi$
$107$	−10.3907			−1.00451			−0.502255	+	0.864719i	$0.667496\pi$
$108$		0			0
$109$			6.96215i			0.666853i	0.942776	+	0.333426i	$0.108205\pi$
$109$			6.96215i			0.666853i	−0.942776	+	0.333426i	$0.891795\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.1905i			0.958641i	0.877640	+	0.479321i	$0.159117\pi$
$113$			10.1905i			0.958641i	−0.877640	+	0.479321i	$0.840883\pi$
$114$		0			0
$115$	−27.9344			−2.60490
$116$		0			0
$117$		0			0
$118$		0			0
$119$			15.6868i			1.43800i
$120$		0			0
$121$	−8.09196			−0.735633
$122$		0			0
$123$		0			0
$124$		0			0
$125$			0.741973i			0.0663641i
$126$		0			0
$127$	11.1582			0.990131			0.495066	−	0.868856i	$-0.335144\pi$
$127$	11.1582			0.990131			0.495066	+	0.868856i	$0.335144\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.0027i		−	1.04869i	−0.851507	−	0.524343i	$-0.824311\pi$
$131$		−	12.0027i		−	1.04869i	0.851507	−	0.524343i	$-0.175689\pi$
$132$		0			0
$133$			0.688728i			0.0597203i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.7005			−0.914206			−0.457103	−	0.889414i	$-0.651113\pi$
$137$	−10.7005			−0.914206			−0.457103	+	0.889414i	$0.651113\pi$
$138$		0			0
$139$	−2.74408			−0.232750			−0.116375	−	0.993205i	$-0.537127\pi$
$139$	−2.74408			−0.232750			−0.116375	+	0.993205i	$0.537127\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.48087i			0.207461i
$144$		0			0
$145$	−8.48127			−0.704332
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.9415			−1.06021			−0.530105	−	0.847932i	$-0.677847\pi$
$149$	−12.9415			−1.06021			−0.530105	+	0.847932i	$0.677847\pi$
$150$		0			0
$151$	−5.30288			−0.431542			−0.215771	−	0.976444i	$-0.569227\pi$
$151$	−5.30288			−0.431542			−0.215771	+	0.976444i	$0.569227\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	30.3078			2.43438
$156$		0			0
$157$	20.2359			1.61500			0.807500	−	0.589868i	$-0.200820\pi$
$157$	20.2359			1.61500			0.807500	+	0.589868i	$0.200820\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	28.6764i		−	2.26002i
$162$		0			0
$163$		−	17.3098i		−	1.35581i	−0.735149	−	0.677906i	$-0.762888\pi$
$163$		−	17.3098i		−	1.35581i	0.735149	−	0.677906i	$-0.237112\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.58312i			0.199888i	0.994993	+	0.0999440i	$0.0318664\pi$
$167$			2.58312i			0.199888i	−0.994993	+	0.0999440i	$0.968134\pi$
$168$		0			0
$169$	10.8835			0.837196
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.29878			−0.554916			−0.277458	−	0.960738i	$-0.589492\pi$
$173$	−7.29878			−0.554916			−0.277458	+	0.960738i	$0.589492\pi$
$174$		0			0
$175$	−17.1802			−1.29870
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	21.7157i		−	1.62311i	−0.584276	−	0.811555i	$-0.698622\pi$
$179$		−	21.7157i		−	1.62311i	0.584276	−	0.811555i	$-0.301378\pi$
$180$		0			0
$181$	−4.11182			−0.305629			−0.152814	−	0.988255i	$-0.548834\pi$
$181$	−4.11182			−0.305629			−0.152814	+	0.988255i	$0.548834\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.5564	−	11.6868i	1.14373	−	0.859228i
$186$		0			0
$187$			8.14646i			0.595728i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			0.715581i			0.0517776i	0.999665	+	0.0258888i	$0.00824159\pi$
$191$			0.715581i			0.0517776i	−0.999665	+	0.0258888i	$0.991758\pi$
$192$		0			0
$193$		−	10.2430i		−	0.737310i	−0.929566	−	0.368655i	$-0.879818\pi$
$193$		−	10.2430i		−	0.737310i	0.929566	−	0.368655i	$-0.120182\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.68119			0.262274			0.131137	−	0.991364i	$-0.458137\pi$
$197$	3.68119			0.262274			0.131137	+	0.991364i	$0.458137\pi$
$198$		0			0
$199$		−	6.08150i		−	0.431106i	−0.976492	−	0.215553i	$-0.930845\pi$
$199$		−	6.08150i		−	0.431106i	0.976492	−	0.215553i	$-0.0691554\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	8.70655i		−	0.611080i
$204$		0			0
$205$		−	36.2921i		−	2.53475i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.357671i			0.0247406i
$210$		0			0
$211$	−4.47970			−0.308395			−0.154198	−	0.988040i	$-0.549279\pi$
$211$	−4.47970			−0.308395			−0.154198	+	0.988040i	$0.549279\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−27.7448			−1.89218
$216$		0			0
$217$			31.1128i			2.11208i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.94983			−0.467496
$222$		0			0
$223$	11.2837			0.755613			0.377806	−	0.925885i	$-0.376678\pi$
$223$	11.2837			0.755613			0.377806	+	0.925885i	$0.376678\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6.54542i			0.434435i	0.976123	+	0.217217i	$0.0696980\pi$
$227$			6.54542i			0.434435i	−0.976123	+	0.217217i	$0.930302\pi$
$228$		0			0
$229$	−12.9883			−0.858290			−0.429145	−	0.903236i	$-0.641185\pi$
$229$	−12.9883			−0.858290			−0.429145	+	0.903236i	$0.641185\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.6978			−1.15942			−0.579712	−	0.814821i	$-0.696835\pi$
$233$	−17.6978			−1.15942			−0.579712	+	0.814821i	$0.696835\pi$
$234$		0			0
$235$		−	40.5220i		−	2.64336i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5.13928i		−	0.332432i	−0.986089	−	0.166216i	$-0.946845\pi$
$239$		−	5.13928i		−	0.332432i	0.986089	−	0.166216i	$-0.0531550\pi$
$240$		0			0
$241$			23.2327i			1.49655i	0.663390	+	0.748274i	$0.269117\pi$
$241$			23.2327i			1.49655i	−0.663390	+	0.748274i	$0.730883\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	12.1000i		−	0.773038i
$246$		0			0
$247$	−0.305132			−0.0194151
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	14.9383i		−	0.942896i	−0.881894	−	0.471448i	$-0.843731\pi$
$251$		−	14.9383i		−	0.942896i	0.881894	−	0.471448i	$-0.156269\pi$
$252$		0			0
$253$		−	14.8923i		−	0.936268i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			14.1109i			0.880211i	0.897946	+	0.440106i	$0.145059\pi$
$257$			14.1109i			0.880211i	−0.897946	+	0.440106i	$0.854941\pi$
$258$		0			0
$259$	11.9972	+	15.9696i	0.745468	+	0.992303i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.20613			−0.0743734			−0.0371867	−	0.999308i	$-0.511840\pi$
$263$	−1.20613			−0.0743734			−0.0371867	+	0.999308i	$0.511840\pi$
$264$		0			0
$265$		−	16.3577i		−	1.00484i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.6427			−0.709870			−0.354935	−	0.934891i	$-0.615497\pi$
$269$	−11.6427			−0.709870			−0.354935	+	0.934891i	$0.615497\pi$
$270$		0			0
$271$	−2.25946			−0.137252			−0.0686261	−	0.997642i	$-0.521862\pi$
$271$	−2.25946			−0.137252			−0.0686261	+	0.997642i	$0.521862\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.92204			−0.538019
$276$		0			0
$277$			7.35320i			0.441811i	0.975295	+	0.220905i	$0.0709012\pi$
$277$			7.35320i			0.441811i	−0.975295	+	0.220905i	$0.929099\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			23.3735i			1.39435i	0.716902	+	0.697174i	$0.245559\pi$
$281$			23.3735i			1.39435i	−0.716902	+	0.697174i	$0.754441\pi$
$282$		0			0
$283$		−	28.8503i		−	1.71497i	−0.514509	−	0.857485i	$-0.672026\pi$
$283$		−	28.8503i		−	1.71497i	0.514509	−	0.857485i	$-0.327974\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	37.2561			2.19916
$288$		0			0
$289$	−5.82119			−0.342423
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.47937			0.436950			0.218475	−	0.975843i	$-0.429892\pi$
$293$	7.47937			0.436950			0.218475	+	0.975843i	$0.429892\pi$
$294$		0			0
$295$	5.30288			0.308746
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.7047			0.734733
$300$		0			0
$301$		−	28.4817i		−	1.64166i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.10789			−0.349737
$306$		0			0
$307$	7.74322			0.441929			0.220965	−	0.975282i	$-0.429079\pi$
$307$	7.74322			0.441929			0.220965	+	0.975282i	$0.429079\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			5.25867i			0.298192i	0.988823	+	0.149096i	$0.0476363\pi$
$311$			5.25867i			0.298192i	−0.988823	+	0.149096i	$0.952364\pi$
$312$		0			0
$313$			12.5165i			0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$313$			12.5165i			0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.80377			0.382138			0.191069	−	0.981577i	$-0.438805\pi$
$317$	6.80377			0.382138			0.191069	+	0.981577i	$0.438805\pi$
$318$		0			0
$319$		−	4.52149i		−	0.253155i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.00196			−0.0557508
$324$		0			0
$325$		−	7.61148i		−	0.422209i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	41.5983			2.29339
$330$		0			0
$331$		−	7.10153i		−	0.390335i	−0.980770	−	0.195168i	$-0.937475\pi$
$331$		−	7.10153i		−	0.390335i	0.980770	−	0.195168i	$-0.0625251\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			6.10789i			0.333710i
$336$		0			0
$337$	5.41838			0.295158			0.147579	−	0.989050i	$-0.452852\pi$
$337$	5.41838			0.295158			0.147579	+	0.989050i	$0.452852\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			16.1575i			0.874980i
$342$		0			0
$343$	−10.5646			−0.570434
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	11.3844i		−	0.611146i	−0.952169	−	0.305573i	$-0.901152\pi$
$347$		−	11.3844i		−	0.611146i	0.952169	−	0.305573i	$-0.0988479\pi$
$348$		0			0
$349$	−3.02425			−0.161884			−0.0809422	−	0.996719i	$-0.525793\pi$
$349$	−3.02425			−0.161884			−0.0809422	+	0.996719i	$0.525793\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	32.6984i		−	1.74036i	−0.492732	−	0.870181i	$-0.664002\pi$
$353$		−	32.6984i		−	1.74036i	0.492732	−	0.870181i	$-0.335998\pi$
$354$		0			0
$355$			30.4241i			1.61474i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11.3351			−0.598242			−0.299121	−	0.954215i	$-0.596693\pi$
$359$	−11.3351			−0.598242			−0.299121	+	0.954215i	$0.596693\pi$
$360$		0			0
$361$	18.9560			0.997685
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	19.5212i		−	1.02179i
$366$		0			0
$367$	−14.1503			−0.738639			−0.369319	−	0.929303i	$-0.620409\pi$
$367$	−14.1503			−0.738639			−0.369319	+	0.929303i	$0.620409\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.7922			0.871805
$372$		0			0
$373$	23.2506			1.20387			0.601934	−	0.798546i	$-0.294397\pi$
$373$	23.2506			1.20387			0.601934	+	0.798546i	$0.294397\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.85733			0.198663
$378$		0			0
$379$	23.6659			1.21564			0.607819	−	0.794076i	$-0.292045\pi$
$379$	23.6659			1.21564			0.607819	+	0.794076i	$0.292045\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	11.8483i		−	0.605420i	−0.953083	−	0.302710i	$-0.902109\pi$
$383$		−	11.8483i		−	0.605420i	0.953083	−	0.302710i	$-0.0978913\pi$
$384$		0			0
$385$		−	17.9120i		−	0.912879i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			23.5738i			1.19524i	0.801780	+	0.597619i	$0.203886\pi$
$389$			23.5738i			1.19524i	−0.801780	+	0.597619i	$0.796114\pi$
$390$		0			0
$391$	41.7186			2.10980
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.5946			0.583387
$396$		0			0
$397$	−22.9153			−1.15009			−0.575043	−	0.818123i	$-0.695015\pi$
$397$	−22.9153			−1.15009			−0.575043	+	0.818123i	$0.695015\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			14.7107i			0.734618i	0.930099	+	0.367309i	$0.119721\pi$
$401$			14.7107i			0.734618i	−0.930099	+	0.367309i	$0.880279\pi$
$402$		0			0
$403$	−13.7842			−0.686638
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.23038	+	8.29335i	0.308829	+	0.411086i
$408$		0			0
$409$		−	17.9298i		−	0.886571i	−0.896380	−	0.443286i	$-0.853813\pi$
$409$		−	17.9298i		−	0.886571i	0.896380	−	0.443286i	$-0.146187\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			5.44373i			0.267868i
$414$		0			0
$415$			10.2430i			0.502811i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.3546			−0.505857			−0.252928	−	0.967485i	$-0.581394\pi$
$419$	−10.3546			−0.505857			−0.252928	+	0.967485i	$0.581394\pi$
$420$		0			0
$421$			24.9829i			1.21759i	0.793327	+	0.608796i	$0.208347\pi$
$421$			24.9829i			1.21759i	−0.793327	+	0.608796i	$0.791653\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	24.9939i		−	1.21238i
$426$		0			0
$427$		−	6.27013i		−	0.303433i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	34.5108i		−	1.66233i	−0.556028	−	0.831164i	$-0.687675\pi$
$431$		−	34.5108i		−	1.66233i	0.556028	−	0.831164i	$-0.312325\pi$
$432$		0			0
$433$	8.87225			0.426373			0.213187	−	0.977012i	$-0.431616\pi$
$433$	8.87225			0.426373			0.213187	+	0.977012i	$0.431616\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.83166			0.0876200
$438$		0			0
$439$			17.4395i			0.832341i	0.909287	+	0.416171i	$0.136628\pi$
$439$			17.4395i			0.832341i	−0.909287	+	0.416171i	$0.863372\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.49973			0.451346			0.225673	−	0.974203i	$-0.427542\pi$
$443$	9.49973			0.451346			0.225673	+	0.974203i	$0.427542\pi$
$444$		0			0
$445$	−32.5398			−1.54253
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9.99582i		−	0.471732i	−0.971786	−	0.235866i	$-0.924207\pi$
$449$		−	9.99582i		−	0.471732i	0.971786	−	0.235866i	$-0.0757927\pi$
$450$		0			0
$451$	19.3478			0.911055
$452$		0			0
$453$		0			0
$454$		0			0
$455$	15.2809			0.716379
$456$		0			0
$457$		−	6.49951i		−	0.304034i	−0.988378	−	0.152017i	$-0.951423\pi$
$457$		−	6.49951i		−	0.304034i	0.988378	−	0.152017i	$-0.0485769\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	9.20085i		−	0.428526i	−0.976776	−	0.214263i	$-0.931265\pi$
$461$		−	9.20085i		−	0.428526i	0.976776	−	0.214263i	$-0.0687350\pi$
$462$		0			0
$463$		−	9.19686i		−	0.427414i	−0.976898	−	0.213707i	$-0.931446\pi$
$463$		−	9.19686i		−	0.427414i	0.976898	−	0.213707i	$-0.0685538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			2.62505i			0.121473i	0.998154	+	0.0607364i	$0.0193449\pi$
$467$			2.62505i			0.121473i	−0.998154	+	0.0607364i	$0.980655\pi$
$468$		0			0
$469$	−6.27013			−0.289528
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	14.7912i		−	0.680098i
$474$		0			0
$475$		−	1.09736i		−	0.0503501i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			23.3471i			1.06676i	0.845876	+	0.533379i	$0.179078\pi$
$479$			23.3471i			1.06676i	−0.845876	+	0.533379i	$0.820922\pi$
$480$		0			0
$481$	−7.07514	+	5.31520i	−0.322599	+	0.242352i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	32.5968			1.48014
$486$		0			0
$487$			19.0050i			0.861199i	0.902543	+	0.430599i	$0.141698\pi$
$487$			19.0050i			0.861199i	−0.902543	+	0.430599i	$0.858302\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−35.1468			−1.58615			−0.793076	−	0.609123i	$-0.791522\pi$
$491$	−35.1468			−1.58615			−0.793076	+	0.609123i	$0.791522\pi$
$492$		0			0
$493$	12.6663			0.570463
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−31.2322			−1.40096
$498$		0			0
$499$		−	28.9911i		−	1.29782i	−0.760865	−	0.648910i	$-0.775225\pi$
$499$		−	28.9911i		−	1.29782i	0.760865	−	0.648910i	$-0.224775\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			29.1135i			1.29811i	0.760743	+	0.649053i	$0.224835\pi$
$503$			29.1135i			1.29811i	−0.760743	+	0.649053i	$0.775165\pi$
$504$		0			0
$505$		−	47.2158i		−	2.10107i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.60362			−0.248376			−0.124188	−	0.992259i	$-0.539633\pi$
$509$	−5.60362			−0.248376			−0.124188	+	0.992259i	$0.539633\pi$
$510$		0			0
$511$	20.0397			0.886506
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−20.5498			−0.905533
$516$		0			0
$517$	21.6028			0.950092
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.8492			−0.519125			−0.259562	−	0.965726i	$-0.583578\pi$
$521$	−11.8492			−0.519125			−0.259562	+	0.965726i	$0.583578\pi$
$522$		0			0
$523$			15.1101i			0.660718i	0.943855	+	0.330359i	$0.107170\pi$
$523$			15.1101i			0.660718i	−0.943855	+	0.330359i	$0.892830\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−45.2631			−1.97169
$528$		0			0
$529$	−53.2642			−2.31584
$530$		0			0
$531$		0			0
$532$		0			0
$533$			16.5058i			0.714947i
$534$		0			0
$535$			33.2373i			1.43697i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.45066			0.277850
$540$		0			0
$541$		−	34.5457i		−	1.48523i	−0.669716	−	0.742617i	$-0.733584\pi$
$541$		−	34.5457i		−	1.48523i	0.669716	−	0.742617i	$-0.266416\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	22.2701			0.953947
$546$		0			0
$547$			7.15018i			0.305720i	0.988248	+	0.152860i	$0.0488483\pi$
$547$			7.15018i			0.305720i	−0.988248	+	0.152860i	$0.951152\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.556116			0.0236913
$552$		0			0
$553$			11.9025i			0.506148i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	29.6186i		−	1.25498i	−0.778624	−	0.627491i	$-0.784082\pi$
$557$		−	29.6186i		−	1.25498i	0.778624	−	0.627491i	$-0.215918\pi$
$558$		0			0
$559$	12.6185			0.533705
$560$		0			0
$561$		0			0
$562$		0			0
$563$			13.3043i			0.560711i	0.959896	+	0.280356i	$0.0904524\pi$
$563$			13.3043i			0.560711i	−0.959896	+	0.280356i	$0.909548\pi$
$564$		0			0
$565$	32.5968			1.37136
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	4.88761i		−	0.204899i	−0.994738	−	0.102450i	$-0.967332\pi$
$569$		−	4.88761i		−	0.204899i	0.994738	−	0.102450i	$-0.0326681\pi$
$570$		0			0
$571$	33.7638			1.41297			0.706485	−	0.707728i	$-0.250280\pi$
$571$	33.7638			1.41297			0.706485	+	0.707728i	$0.250280\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			45.6904i			1.90542i
$576$		0			0
$577$		−	30.4241i		−	1.26657i	−0.773918	−	0.633286i	$-0.781706\pi$
$577$		−	30.4241i		−	1.26657i	0.773918	−	0.633286i	$-0.218294\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.5151			−0.436240
$582$		0			0
$583$	8.72051			0.361167
$584$		0			0
$585$		0			0
$586$		0			0
$587$			24.4005i			1.00712i	0.863961	+	0.503559i	$0.167976\pi$
$587$			24.4005i			1.00712i	−0.863961	+	0.503559i	$0.832024\pi$
$588$		0			0
$589$	−1.98728			−0.0818843
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.3306			−0.917009			−0.458505	−	0.888692i	$-0.651615\pi$
$593$	−22.3306			−0.917009			−0.458505	+	0.888692i	$0.651615\pi$
$594$		0			0
$595$	50.1779			2.05709
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.0655			0.411264			0.205632	−	0.978629i	$-0.434075\pi$
$599$	10.0655			0.411264			0.205632	+	0.978629i	$0.434075\pi$
$600$		0			0
$601$	−2.35644			−0.0961213			−0.0480606	−	0.998844i	$-0.515304\pi$
$601$	−2.35644			−0.0961213			−0.0480606	+	0.998844i	$0.515304\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			25.8841i			1.05234i
$606$		0			0
$607$			1.94307i			0.0788666i	0.999222	+	0.0394333i	$0.0125553\pi$
$607$			1.94307i			0.0788666i	−0.999222	+	0.0394333i	$0.987445\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			18.4296i			0.745582i
$612$		0			0
$613$	19.8400			0.801329			0.400665	−	0.916225i	$-0.368779\pi$
$613$	19.8400			0.801329			0.400665	+	0.916225i	$0.368779\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.0166			−0.644806			−0.322403	−	0.946603i	$-0.604491\pi$
$617$	−16.0166			−0.644806			−0.322403	+	0.946603i	$0.604491\pi$
$618$		0			0
$619$	36.0175			1.44767			0.723833	−	0.689975i	$-0.242379\pi$
$619$	36.0175			1.44767			0.723833	+	0.689975i	$0.242379\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	33.4041i		−	1.33831i
$624$		0			0
$625$	−23.7864			−0.951456
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−23.2327	+	17.4536i	−0.926347	+	0.695919i
$630$		0			0
$631$			16.4641i			0.655427i	0.944777	+	0.327713i	$0.106278\pi$
$631$			16.4641i			0.655427i	−0.944777	+	0.327713i	$0.893722\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	35.6923i		−	1.41640i
$636$		0			0
$637$			5.50313i			0.218042i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	24.3810			0.962991			0.481496	−	0.876448i	$-0.340094\pi$
$641$	24.3810			0.962991			0.481496	+	0.876448i	$0.340094\pi$
$642$		0			0
$643$			24.9785i			0.985058i	0.870296	+	0.492529i	$0.163927\pi$
$643$			24.9785i			0.985058i	−0.870296	+	0.492529i	$0.836073\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			25.3827i			0.997898i	0.866631	+	0.498949i	$0.166281\pi$
$647$			25.3827i			0.997898i	−0.866631	+	0.498949i	$0.833719\pi$
$648$		0			0
$649$			2.82704i			0.110971i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			37.6994i			1.47529i	0.675188	+	0.737646i	$0.264063\pi$
$653$			37.6994i			1.47529i	−0.675188	+	0.737646i	$0.735937\pi$
$654$		0			0
$655$	−38.3937			−1.50017
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.5470			0.800397			0.400199	−	0.916428i	$-0.368941\pi$
$659$	20.5470			0.800397			0.400199	+	0.916428i	$0.368941\pi$
$660$		0			0
$661$			42.2553i			1.64354i	0.569820	+	0.821770i	$0.307013\pi$
$661$			42.2553i			1.64354i	−0.569820	+	0.821770i	$0.692987\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.20306			0.0854311
$666$		0			0
$667$	−23.1549			−0.896560
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	3.25621i		−	0.125704i
$672$		0			0
$673$	−27.8564			−1.07379			−0.536893	−	0.843650i	$-0.680402\pi$
$673$	−27.8564			−1.07379			−0.536893	+	0.843650i	$0.680402\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−44.1395			−1.69642			−0.848209	−	0.529662i	$-0.822319\pi$
$677$	−44.1395			−1.69642			−0.848209	+	0.529662i	$0.822319\pi$
$678$		0			0
$679$			33.4626i			1.28418i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	22.2020i		−	0.849537i	−0.905302	−	0.424769i	$-0.860355\pi$
$683$		−	22.2020i		−	0.849537i	0.905302	−	0.424769i	$-0.139645\pi$
$684$		0			0
$685$			34.2282i			1.30779i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			7.43955i			0.283424i
$690$		0			0
$691$	−5.76679			−0.219379			−0.109690	−	0.993966i	$-0.534986\pi$
$691$	−5.76679			−0.219379			−0.109690	+	0.993966i	$0.534986\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			8.77761i			0.332954i
$696$		0			0
$697$			54.2003i			2.05298i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			13.4677i			0.508668i	0.967116	+	0.254334i	$0.0818563\pi$
$701$			13.4677i			0.508668i	−0.967116	+	0.254334i	$0.918144\pi$
$702$		0			0
$703$	−1.02003	+	0.766299i	−0.0384712	+	0.0289015i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	48.4699			1.82290
$708$		0			0
$709$		−	40.1321i		−	1.50719i	−0.657338	−	0.753596i	$-0.728318\pi$
$709$		−	40.1321i		−	1.50719i	0.657338	−	0.753596i	$-0.271682\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	82.7439			3.09878
$714$		0			0
$715$	7.93568			0.296778
$716$		0			0
$717$		0			0
$718$		0			0
$719$	46.6461			1.73961			0.869803	−	0.493400i	$-0.164246\pi$
$719$	46.6461			1.73961			0.869803	+	0.493400i	$0.164246\pi$
$720$		0			0
$721$		−	21.0956i		−	0.785643i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			13.8722i			0.515201i
$726$		0			0
$727$		−	24.8473i		−	0.921534i	−0.887521	−	0.460767i	$-0.847574\pi$
$727$		−	24.8473i		−	0.921534i	0.887521	−	0.460767i	$-0.152426\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	41.4354			1.53254
$732$		0			0
$733$	36.5701			1.35075			0.675374	−	0.737476i	$-0.263982\pi$
$733$	36.5701			1.35075			0.675374	+	0.737476i	$0.263982\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.25621			−0.119944
$738$		0			0
$739$	−26.7073			−0.982442			−0.491221	−	0.871035i	$-0.663449\pi$
$739$	−26.7073			−0.982442			−0.491221	+	0.871035i	$0.663449\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−33.4081			−1.22562			−0.612811	−	0.790229i	$-0.709961\pi$
$743$	−33.4081			−1.22562			−0.612811	+	0.790229i	$0.709961\pi$
$744$		0			0
$745$			41.3966i			1.51665i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−34.1201			−1.24672
$750$		0			0
$751$	−4.33703			−0.158260			−0.0791302	−	0.996864i	$-0.525214\pi$
$751$	−4.33703			−0.158260			−0.0791302	+	0.996864i	$0.525214\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			16.9625i			0.617330i
$756$		0			0
$757$		−	25.2830i		−	0.918926i	−0.888197	−	0.459463i	$-0.848042\pi$
$757$		−	25.2830i		−	0.918926i	0.888197	−	0.459463i	$-0.151958\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−43.0646			−1.56109			−0.780545	−	0.625099i	$-0.785059\pi$
$761$	−43.0646			−1.56109			−0.780545	+	0.625099i	$0.785059\pi$
$762$		0			0
$763$			22.8616i			0.827647i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.41178			−0.0870842
$768$		0			0
$769$			0.0394155i			0.00142136i	1.00000		0.000710679i	$0.000226216\pi$
$769$			0.0394155i			0.00142136i	−1.00000		0.000710679i	$0.999774\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.727148			0.0261537			0.0130768	−	0.999914i	$-0.495837\pi$
$773$	0.727148			0.0261537			0.0130768	+	0.999914i	$0.495837\pi$
$774$		0			0
$775$		−	49.5723i		−	1.78069i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			2.37967i			0.0852604i
$780$		0			0
$781$	−16.2195			−0.580380
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	64.7294i		−	2.31029i
$786$		0			0
$787$	6.13961			0.218853			0.109427	−	0.993995i	$-0.465099\pi$
$787$	6.13961			0.218853			0.109427	+	0.993995i	$0.465099\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			33.4626i			1.18979i
$792$		0			0
$793$	2.77790			0.0986462
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	31.3278i		−	1.10969i	−0.831954	−	0.554845i	$-0.812778\pi$
$797$		−	31.3278i		−	1.10969i	0.831954	−	0.554845i	$-0.187222\pi$
$798$		0			0
$799$			60.5174i			2.14095i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.4071			0.367257
$804$		0			0
$805$	−91.7285			−3.23300
$806$		0			0
$807$		0			0
$808$		0			0
$809$			24.7019i			0.868471i	0.900799	+	0.434235i	$0.142981\pi$
$809$			24.7019i			0.868471i	−0.900799	+	0.434235i	$0.857019\pi$
$810$		0			0
$811$	−27.0155			−0.948644			−0.474322	−	0.880351i	$-0.657307\pi$
$811$	−27.0155			−0.948644			−0.474322	+	0.880351i	$0.657307\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−55.3698			−1.93952
$816$		0			0
$817$	1.81922			0.0636465
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.0477			−0.490269			−0.245135	−	0.969489i	$-0.578832\pi$
$821$	−14.0477			−0.490269			−0.245135	+	0.969489i	$0.578832\pi$
$822$		0			0
$823$	−37.2061			−1.29692			−0.648462	−	0.761247i	$-0.724587\pi$
$823$	−37.2061			−1.29692			−0.648462	+	0.761247i	$0.724587\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	25.4952i		−	0.886556i	−0.896384	−	0.443278i	$-0.853815\pi$
$827$		−	25.4952i		−	0.886556i	0.896384	−	0.443278i	$-0.146185\pi$
$828$		0			0
$829$			28.3600i			0.984982i	0.870318	+	0.492491i	$0.163914\pi$
$829$			28.3600i			0.984982i	−0.870318	+	0.492491i	$0.836086\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			18.0706i			0.626111i
$834$		0			0
$835$	8.26275			0.285944
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25.8099			−0.891058			−0.445529	−	0.895267i	$-0.646984\pi$
$839$	−25.8099			−0.891058			−0.445529	+	0.895267i	$0.646984\pi$
$840$		0			0
$841$	21.9699			0.757582
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	34.8137i		−	1.19763i
$846$		0			0
$847$	−26.5716			−0.913012
$848$		0			0
$849$		0			0
$850$		0			0
$851$	42.4708	−	31.9062i	1.45588	−	1.09373i
$852$		0			0
$853$			13.2081i			0.452235i	0.974100	+	0.226118i	$0.0726034\pi$
$853$			13.2081i			0.452235i	−0.974100	+	0.226118i	$0.927397\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			30.8046i			1.05227i	0.850402	+	0.526133i	$0.176359\pi$
$857$			30.8046i			1.05227i	−0.850402	+	0.526133i	$0.823641\pi$
$858$		0			0
$859$		−	26.0511i		−	0.888852i	−0.895815	−	0.444426i	$-0.853408\pi$
$859$		−	26.0511i		−	0.888852i	0.895815	−	0.444426i	$-0.146592\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.6663			−0.431167			−0.215583	−	0.976485i	$-0.569165\pi$
$863$	−12.6663			−0.431167			−0.215583	+	0.976485i	$0.569165\pi$
$864$		0			0
$865$			23.3469i			0.793819i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			6.18124i			0.209684i
$870$		0			0
$871$		−	2.77790i		−	0.0941256i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			2.43642i			0.0823661i
$876$		0			0
$877$	−45.7101			−1.54352			−0.771760	−	0.635914i	$-0.780623\pi$
$877$	−45.7101			−1.54352			−0.771760	+	0.635914i	$0.780623\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	29.6503			0.998945			0.499473	−	0.866330i	$-0.333527\pi$
$881$	29.6503			0.998945			0.499473	+	0.866330i	$0.333527\pi$
$882$		0			0
$883$			4.97340i			0.167368i	0.996492	+	0.0836842i	$0.0266687\pi$
$883$			4.97340i			0.167368i	−0.996492	+	0.0836842i	$0.973331\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−26.5299			−0.890786			−0.445393	−	0.895335i	$-0.646936\pi$
$887$	−26.5299			−0.890786			−0.445393	+	0.895335i	$0.646936\pi$
$888$		0			0
$889$	36.6403			1.22888
$890$		0			0
$891$		0			0
$892$		0			0
$893$			2.65702i			0.0889137i
$894$		0			0
$895$	−69.4630			−2.32189
$896$		0			0
$897$		0			0
$898$		0			0
$899$	25.1222			0.837871
$900$		0			0
$901$			24.4293i			0.813858i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			13.1526i			0.437209i
$906$		0			0
$907$		−	35.7385i		−	1.18668i	−0.804953	−	0.593339i	$-0.797809\pi$
$907$		−	35.7385i		−	1.18668i	0.804953	−	0.593339i	$-0.202191\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	3.10164i		−	0.102762i	−0.998679	−	0.0513810i	$-0.983638\pi$
$911$		−	3.10164i		−	0.102762i	0.998679	−	0.0513810i	$-0.0163623\pi$
$912$		0			0
$913$	−5.46071			−0.180723
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	39.4135i		−	1.30155i
$918$		0			0
$919$		−	29.1690i		−	0.962196i	−0.876667	−	0.481098i	$-0.840238\pi$
$919$		−	29.1690i		−	0.962196i	0.876667	−	0.481098i	$-0.159762\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	13.8370i		−	0.455452i
$924$		0			0
$925$	−19.1152	−	25.4445i	−0.628504	−	0.836611i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.75265			0.287165			0.143583	−	0.989638i	$-0.454138\pi$
$929$	8.75265			0.287165			0.143583	+	0.989638i	$0.454138\pi$
$930$		0			0
$931$			0.793392i			0.0260024i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	26.0584			0.852202
$936$		0			0
$937$	43.8869			1.43372			0.716862	−	0.697215i	$-0.245578\pi$
$937$	43.8869			1.43372			0.716862	+	0.697215i	$0.245578\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.50820			0.146963			0.0734815	−	0.997297i	$-0.476589\pi$
$941$	4.50820			0.146963			0.0734815	+	0.997297i	$0.476589\pi$
$942$		0			0
$943$		−	99.0817i		−	3.22654i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	12.5898i		−	0.409114i	−0.978855	−	0.204557i	$-0.934425\pi$
$947$		−	12.5898i		−	0.409114i	0.978855	−	0.204557i	$-0.0655753\pi$
$948$		0			0
$949$			8.87836i			0.288204i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.4612			−0.500839			−0.250419	−	0.968137i	$-0.580568\pi$
$953$	−15.4612			−0.500839			−0.250419	+	0.968137i	$0.580568\pi$
$954$		0			0
$955$	2.28896			0.0740690
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−35.1373			−1.13464
$960$		0			0
$961$	−58.7740			−1.89593
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−32.7648			−1.05474
$966$		0			0
$967$			50.6768i			1.62966i	0.579703	+	0.814828i	$0.303169\pi$
$967$			50.6768i			1.62966i	−0.579703	+	0.814828i	$0.696831\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.6413			−0.405679			−0.202839	−	0.979212i	$-0.565017\pi$
$971$	−12.6413			−0.405679			−0.202839	+	0.979212i	$0.565017\pi$
$972$		0			0
$973$	−9.01076			−0.288872
$974$		0			0
$975$		0			0
$976$		0			0
$977$			24.7560i			0.792016i	0.918247	+	0.396008i	$0.129605\pi$
$977$			24.7560i			0.792016i	−0.918247	+	0.396008i	$0.870395\pi$
$978$		0			0
$979$		−	17.3474i		−	0.554426i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.5440			0.527672			0.263836	−	0.964568i	$-0.415012\pi$
$983$	16.5440			0.527672			0.263836	+	0.964568i	$0.415012\pi$
$984$		0			0
$985$		−	11.7752i		−	0.375189i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−75.7465			−2.40860
$990$		0			0
$991$			19.0343i			0.604643i	0.953206	+	0.302322i	$0.0977616\pi$
$991$			19.0343i			0.604643i	−0.953206	+	0.302322i	$0.902238\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−19.4532			−0.616707
$996$		0			0
$997$			41.2547i			1.30655i	0.757121	+	0.653274i	$0.226605\pi$
$997$			41.2547i			1.30655i	−0.757121	+	0.653274i	$0.773395\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.h.q.2737.1		10
3.2	odd	2		592.2.g.d.369.2		10
4.3	odd	2		2664.2.h.c.73.1		10
12.11	even	2		296.2.g.a.73.10	yes	10
24.5	odd	2		2368.2.g.p.961.9		10
24.11	even	2		2368.2.g.o.961.1		10
37.36	even	2	inner	5328.2.h.q.2737.10		10
111.110	odd	2		592.2.g.d.369.1		10
148.147	odd	2		2664.2.h.c.73.10		10
444.443	even	2		296.2.g.a.73.9	✓	10
888.221	odd	2		2368.2.g.p.961.10		10
888.443	even	2		2368.2.g.o.961.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.g.a.73.9	✓	10	444.443	even	2
296.2.g.a.73.10	yes	10	12.11	even	2
592.2.g.d.369.1		10	111.110	odd	2
592.2.g.d.369.2		10	3.2	odd	2
2368.2.g.o.961.1		10	24.11	even	2
2368.2.g.o.961.2		10	888.443	even	2
2368.2.g.p.961.9		10	24.5	odd	2
2368.2.g.p.961.10		10	888.221	odd	2
2664.2.h.c.73.1		10	4.3	odd	2
2664.2.h.c.73.10		10	148.147	odd	2
5328.2.h.q.2737.1		10	1.1	even	1	trivial
5328.2.h.q.2737.10		10	37.36	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.19874i q^{5} +3.28371 q^{7} +1.70530 q^{11} +1.45481i q^{13} +4.77715i q^{17} +0.209741i q^{19} -8.73294i q^{23} -5.23196 q^{25} -2.65144i q^{29} +9.47491i q^{31} -10.5037i q^{35} +(3.65355 + 4.86329i) q^{37} +11.3457 q^{41} -8.67366i q^{43} +12.6681 q^{47} +3.78272 q^{49} +5.11378 q^{53} -5.45481i q^{55} +1.65780i q^{59} -1.90947i q^{61} +4.65355 q^{65} -1.90947 q^{67} -9.51127 q^{71} +6.10278 q^{73} +5.59969 q^{77} +3.62473i q^{79} -3.20221 q^{83} +15.2809 q^{85} -10.1727i q^{89} +4.77715i q^{91} +0.670907 q^{95} +10.1905i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{7} + 2 q^{11} + 4 q^{25} - 6 q^{37} + 26 q^{41} + 8 q^{47} + 14 q^{49} + 20 q^{53} + 4 q^{65} + 2 q^{67} - 4 q^{71} - 14 q^{73} - 36 q^{83} + 8 q^{85} + 4 q^{95}+O(q^{100}) \) 10 * q + 4 * q^7 + 2 * q^11 + 4 * q^25 - 6 * q^37 + 26 * q^41 + 8 * q^47 + 14 * q^49 + 20 * q^53 + 4 * q^65 + 2 * q^67 - 4 * q^71 - 14 * q^73 - 36 * q^83 + 8 * q^85 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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