LMFDB - Embedded newform 2368.2.g.o.961.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2368,2,Mod(961,2368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2368.961"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,-2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.9085751986$
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			961.1
Root			$0.279838 + 0.279838i$ of defining polynomial
Character	$\chi$	$=$	2368.961
Dual form			2368.2.g.o.961.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.94825 q^{3} -3.19874i q^{5} -3.28371 q^{7} +5.69219 q^{9} -1.70530 q^{11} -1.45481i q^{13} +9.43070i q^{15} -4.77715i q^{17} +0.209741i q^{19} +9.68119 q^{21} -8.73294i q^{23} -5.23196 q^{25} -7.93725 q^{27} -2.65144i q^{29} -9.47491i q^{31} +5.02764 q^{33} +10.5037i q^{35} +(-3.65355 - 4.86329i) q^{37} +4.28913i q^{39} -11.3457 q^{41} -8.67366i q^{43} -18.2079i q^{45} +12.6681 q^{47} +3.78272 q^{49} +14.0842i q^{51} +5.11378 q^{53} +5.45481i q^{55} -0.618369i q^{57} -1.65780i q^{59} +1.90947i q^{61} -18.6915 q^{63} -4.65355 q^{65} -1.90947 q^{67} +25.7469i q^{69} -9.51127 q^{71} +6.10278 q^{73} +15.4251 q^{75} +5.59969 q^{77} -3.62473i q^{79} +6.32446 q^{81} +3.20221 q^{83} -15.2809 q^{85} +7.81711i q^{87} +10.1727i q^{89} +4.77715i q^{91} +27.9344i q^{93} +0.670907 q^{95} +10.1905i q^{97} -9.70687 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 4 q^{7} + 12 q^{9} - 2 q^{11} + 8 q^{21} + 4 q^{25} - 8 q^{27} + 4 q^{33} + 6 q^{37} - 26 q^{41} + 8 q^{47} + 14 q^{49} + 20 q^{53} - 12 q^{63} - 4 q^{65} + 2 q^{67} - 4 q^{71} - 14 q^{73}+ \cdots + 28 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 4 * q^7 + 12 * q^9 - 2 * q^11 + 8 * q^21 + 4 * q^25 - 8 * q^27 + 4 * q^33 + 6 * q^37 - 26 * q^41 + 8 * q^47 + 14 * q^49 + 20 * q^53 - 12 * q^63 - 4 * q^65 + 2 * q^67 - 4 * q^71 - 14 * q^73 + 48 * q^75 + 18 * q^81 + 36 * q^83 - 8 * q^85 + 4 * q^95 + 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times$.

$n$	$705$	$1407$	$1925$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.94825			−1.70217			−0.851087	−	0.525025i	$-0.824056\pi$
$3$	−2.94825			−1.70217			−0.851087	+	0.525025i	$0.824056\pi$
$4$		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	$-0.713096\pi$
$7$	−3.28371			−1.24112			−0.620562	+	0.784157i	$0.713096\pi$
$8$		0			0
$9$	5.69219			1.89740
$10$		0			0
$11$	−1.70530			−0.514166			−0.257083	−	0.966389i	$-0.582761\pi$
$11$	−1.70530			−0.514166			−0.257083	+	0.966389i	$0.582761\pi$
$12$		0			0
$13$		−	1.45481i		−	0.403490i	−0.979438	−	0.201745i	$-0.935339\pi$
$13$		−	1.45481i		−	0.403490i	0.979438	−	0.201745i	$-0.0646613\pi$
$14$		0			0
$15$			9.43070i			2.43500i
$16$		0			0
$17$		−	4.77715i		−	1.15863i	−0.815104	−	0.579315i	$-0.803320\pi$
$17$		−	4.77715i		−	1.15863i	0.815104	−	0.579315i	$-0.196680\pi$
$18$		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
$21$	9.68119			2.11261
$22$		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
$27$	−7.93725			−1.52753
$28$		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	$-0.676074\pi$
$31$		−	9.47491i		−	1.70174i	0.525373	−	0.850872i	$-0.323926\pi$
$32$		0			0
$33$	5.02764			0.875200
$34$		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
$39$			4.28913i			0.686811i
$40$		0			0
$41$	−11.3457			−1.77191			−0.885953	−	0.463774i	$-0.846495\pi$
$41$	−11.3457			−1.77191			−0.885953	+	0.463774i	$0.846495\pi$
$42$		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
$45$		−	18.2079i		−	2.71427i
$46$		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
$51$			14.0842i			1.97219i
$52$		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
$57$		−	0.618369i		−	0.0819050i
$58$		0			0
$59$		−	1.65780i		−	0.215827i	−0.994160	−	0.107914i	$-0.965583\pi$
$59$		−	1.65780i		−	0.215827i	0.994160	−	0.107914i	$-0.0344170\pi$
$60$		0			0
$61$			1.90947i			0.244482i	0.992500	+	0.122241i	$0.0390081\pi$
$61$			1.90947i			0.244482i	−0.992500	+	0.122241i	$0.960992\pi$
$62$		0			0
$63$	−18.6915			−2.35490
$64$		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
$69$			25.7469i			3.09956i
$70$		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
$75$	15.4251			1.78114
$76$		0			0
$77$	5.59969			0.638144
$78$		0			0
$79$		−	3.62473i		−	0.407814i	−0.978990	−	0.203907i	$-0.934636\pi$
$79$		−	3.62473i		−	0.407814i	0.978990	−	0.203907i	$-0.0653640\pi$
$80$		0			0
$81$	6.32446			0.702717
$82$		0			0
$83$	3.20221			0.351488			0.175744	−	0.984436i	$-0.443767\pi$
$83$	3.20221			0.351488			0.175744	+	0.984436i	$0.443767\pi$
$84$		0			0
$85$	−15.2809			−1.65744
$86$		0			0
$87$			7.81711i			0.838082i
$88$		0			0
$89$			10.1727i			1.07830i	0.842209	+	0.539151i	$0.181255\pi$
$89$			10.1727i			1.07830i	−0.842209	+	0.539151i	$0.818745\pi$
$90$		0			0
$91$			4.77715i			0.500782i
$92$		0			0
$93$			27.9344i			2.89667i
$94$		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
$99$	−9.70687			−0.975577
$100$		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	$0.102509\pi$
$103$			6.42434i			0.633009i	−0.948591	+	0.316504i	$0.897491\pi$
$104$		0			0
$105$		−	30.9676i		−	3.02213i
$106$		0			0
$107$	10.3907			1.00451			0.502255	−	0.864719i	$-0.332504\pi$
$107$	10.3907			1.00451			0.502255	+	0.864719i	$0.332504\pi$
$108$		0			0
$109$		−	6.96215i		−	0.666853i	−0.942776	−	0.333426i	$-0.891795\pi$
$109$		−	6.96215i		−	0.666853i	0.942776	−	0.333426i	$-0.108205\pi$
$110$		0			0
$111$	10.7716	+	14.3382i	1.02239	+	1.36092i
$112$		0			0
$113$		−	10.1905i		−	0.958641i	−0.877640	−	0.479321i	$-0.840883\pi$
$113$		−	10.1905i		−	0.958641i	0.877640	−	0.479321i	$-0.159117\pi$
$114$		0			0
$115$	−27.9344			−2.60490
$116$		0			0
$117$		−	8.28103i		−	0.765581i
$118$		0			0
$119$			15.6868i			1.43800i
$120$		0			0
$121$	−8.09196			−0.735633
$122$		0			0
$123$	33.4501			3.01609
$124$		0			0
$125$			0.741973i			0.0663641i
$126$		0			0
$127$	−11.1582			−0.990131			−0.495066	−	0.868856i	$-0.664856\pi$
$127$	−11.1582			−0.990131			−0.495066	+	0.868856i	$0.664856\pi$
$128$		0			0
$129$			25.5721i			2.25150i
$130$		0			0
$131$			12.0027i			1.04869i	0.851507	+	0.524343i	$0.175689\pi$
$131$			12.0027i			1.04869i	−0.851507	+	0.524343i	$0.824311\pi$
$132$		0			0
$133$		−	0.688728i		−	0.0597203i
$134$		0			0
$135$			25.3892i			2.18516i
$136$		0			0
$137$	10.7005			0.914206			0.457103	−	0.889414i	$-0.348887\pi$
$137$	10.7005			0.914206			0.457103	+	0.889414i	$0.348887\pi$
$138$		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
$141$	−37.3487			−3.14533
$142$		0			0
$143$			2.48087i			0.207461i
$144$		0			0
$145$	−8.48127			−0.704332
$146$		0			0
$147$	−11.1524			−0.919836
$148$		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	$-0.430773\pi$
$151$	5.30288			0.431542			0.215771	+	0.976444i	$0.430773\pi$
$152$		0			0
$153$		−	27.1925i		−	2.19838i
$154$		0			0
$155$	−30.3078			−2.43438
$156$		0			0
$157$	−20.2359			−1.61500			−0.807500	−	0.589868i	$-0.799180\pi$
$157$	−20.2359			−1.61500			−0.807500	+	0.589868i	$0.799180\pi$
$158$		0			0
$159$	−15.0767			−1.19566
$160$		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
$165$		−	16.0821i		−	1.25199i
$166$		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
$171$			1.19389i			0.0912987i
$172$		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
$177$			4.88761i			0.367376i
$178$		0			0
$179$			21.7157i			1.62311i	0.584276	+	0.811555i	$0.301378\pi$
$179$			21.7157i			1.62311i	−0.584276	+	0.811555i	$0.698622\pi$
$180$		0			0
$181$	4.11182			0.305629			0.152814	−	0.988255i	$-0.451166\pi$
$181$	4.11182			0.305629			0.152814	+	0.988255i	$0.451166\pi$
$182$		0			0
$183$		−	5.62959i		−	0.416151i
$184$		0			0
$185$	−15.5564	+	11.6868i	−1.14373	+	0.859228i
$186$		0			0
$187$			8.14646i			0.595728i
$188$		0			0
$189$	26.0636			1.89585
$190$		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
$195$	13.7198			0.982498
$196$		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	$0.0691554\pi$
$199$			6.08150i			0.431106i	−0.976492	+	0.215553i	$0.930845\pi$
$200$		0			0
$201$	5.62959			0.397081
$202$		0			0
$203$			8.70655i			0.611080i
$204$		0			0
$205$			36.2921i			2.53475i
$206$		0			0
$207$		−	49.7096i		−	3.45505i
$208$		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
$213$	28.0416			1.92138
$214$		0			0
$215$	−27.7448			−1.89218
$216$		0			0
$217$			31.1128i			2.11208i
$218$		0			0
$219$	−17.9925			−1.21582
$220$		0			0
$221$	−6.94983			−0.467496
$222$		0			0
$223$	−11.2837			−0.755613			−0.377806	−	0.925885i	$-0.623322\pi$
$223$	−11.2837			−0.755613			−0.377806	+	0.925885i	$0.623322\pi$
$224$		0			0
$225$	−29.7813			−1.98542
$226$		0			0
$227$		−	6.54542i		−	0.434435i	−0.976123	−	0.217217i	$-0.930302\pi$
$227$		−	6.54542i		−	0.434435i	0.976123	−	0.217217i	$-0.0696980\pi$
$228$		0			0
$229$	12.9883			0.858290			0.429145	−	0.903236i	$-0.358815\pi$
$229$	12.9883			0.858290			0.429145	+	0.903236i	$0.358815\pi$
$230$		0			0
$231$	−16.5093			−1.08623
$232$		0			0
$233$	17.6978			1.15942			0.579712	−	0.814821i	$-0.303165\pi$
$233$	17.6978			1.15942			0.579712	+	0.814821i	$0.303165\pi$
$234$		0			0
$235$		−	40.5220i		−	2.64336i
$236$		0			0
$237$			10.6866i			0.694170i
$238$		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
$243$	5.16567			0.331378
$244$		0			0
$245$		−	12.1000i		−	0.773038i
$246$		0			0
$247$	0.305132			0.0194151
$248$		0			0
$249$	−9.44091			−0.598293
$250$		0			0
$251$			14.9383i			0.942896i	0.881894	+	0.471448i	$0.156269\pi$
$251$			14.9383i			0.942896i	−0.881894	+	0.471448i	$0.843731\pi$
$252$		0			0
$253$			14.8923i			0.936268i
$254$		0			0
$255$	45.0519			2.82126
$256$		0			0
$257$		−	14.1109i		−	0.880211i	−0.897946	−	0.440106i	$-0.854941\pi$
$257$		−	14.1109i		−	0.880211i	0.897946	−	0.440106i	$-0.145059\pi$
$258$		0			0
$259$	11.9972	+	15.9696i	0.745468	+	0.992303i
$260$		0			0
$261$		−	15.0925i		−	0.934202i
$262$		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
$267$		−	29.9916i		−	1.83546i
$268$		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	$-0.478138\pi$
$271$	2.25946			0.137252			0.0686261	+	0.997642i	$0.478138\pi$
$272$		0			0
$273$		−	14.0842i		−	0.852418i
$274$		0			0
$275$	8.92204			0.538019
$276$		0			0
$277$		−	7.35320i		−	0.441811i	−0.975295	−	0.220905i	$-0.929099\pi$
$277$		−	7.35320i		−	0.441811i	0.975295	−	0.220905i	$-0.0709012\pi$
$278$		0			0
$279$		−	53.9330i		−	3.22888i
$280$		0			0
$281$		−	23.3735i		−	1.39435i	−0.716902	−	0.697174i	$-0.754441\pi$
$281$		−	23.3735i		−	1.39435i	0.716902	−	0.697174i	$-0.245559\pi$
$282$		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
$285$	−1.97800			−0.117167
$286$		0			0
$287$	37.2561			2.19916
$288$		0			0
$289$	−5.82119			−0.342423
$290$		0			0
$291$		−	30.0441i		−	1.76122i
$292$		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
$297$	13.5354			0.785402
$298$		0			0
$299$	−12.7047			−0.734733
$300$		0			0
$301$			28.4817i			1.64166i
$302$		0			0
$303$	−43.5183			−2.50006
$304$		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
$309$		−	18.9406i		−	1.07749i
$310$		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
$315$			59.7892i			3.36874i
$316$		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
$321$	−30.6345			−1.70985
$322$		0			0
$323$	1.00196			0.0557508
$324$		0			0
$325$			7.61148i			0.422209i
$326$		0			0
$327$			20.5262i			1.13510i
$328$		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
$333$	−20.7967	−	27.6828i	−1.13965	−	1.51701i
$334$		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
$339$			30.0441i			1.63177i
$340$		0			0
$341$			16.1575i			0.874980i
$342$		0			0
$343$	10.5646			0.570434
$344$		0			0
$345$	82.3577			4.43399
$346$		0			0
$347$			11.3844i			0.611146i	0.952169	+	0.305573i	$0.0988479\pi$
$347$			11.3844i			0.611146i	−0.952169	+	0.305573i	$0.901152\pi$
$348$		0			0
$349$	3.02425			0.161884			0.0809422	−	0.996719i	$-0.474207\pi$
$349$	3.02425			0.161884			0.0809422	+	0.996719i	$0.474207\pi$
$350$		0			0
$351$			11.5472i			0.616342i
$352$		0			0
$353$			32.6984i			1.74036i	0.492732	+	0.870181i	$0.335998\pi$
$353$			32.6984i			1.74036i	−0.492732	+	0.870181i	$0.664002\pi$
$354$		0			0
$355$			30.4241i			1.61474i
$356$		0			0
$357$		−	46.2485i		−	2.44773i
$358$		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
$363$	23.8571			1.25218
$364$		0			0
$365$		−	19.5212i		−	1.02179i
$366$		0			0
$367$	14.1503			0.738639			0.369319	−	0.929303i	$-0.379591\pi$
$367$	14.1503			0.738639			0.369319	+	0.929303i	$0.379591\pi$
$368$		0			0
$369$	−64.5821			−3.36201
$370$		0			0
$371$	−16.7922			−0.871805
$372$		0			0
$373$	−23.2506			−1.20387			−0.601934	−	0.798546i	$-0.705603\pi$
$373$	−23.2506			−1.20387			−0.601934	+	0.798546i	$0.705603\pi$
$374$		0			0
$375$		−	2.18752i		−	0.112963i
$376$		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
$381$	32.8972			1.68538
$382$		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
$387$		−	49.3721i		−	2.50972i
$388$		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
$393$		−	35.3871i		−	1.78504i
$394$		0			0
$395$	−11.5946			−0.583387
$396$		0			0
$397$	22.9153			1.15009			0.575043	−	0.818123i	$-0.304985\pi$
$397$	22.9153			1.15009			0.575043	+	0.818123i	$0.304985\pi$
$398$		0			0
$399$			2.03054i			0.101654i
$400$		0			0
$401$		−	14.7107i		−	0.734618i	−0.930099	−	0.367309i	$-0.880279\pi$
$401$		−	14.7107i		−	0.734618i	0.930099	−	0.367309i	$-0.119721\pi$
$402$		0			0
$403$	−13.7842			−0.686638
$404$		0			0
$405$		−	20.2303i		−	1.00525i
$406$		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
$411$	−31.5478			−1.55614
$412$		0			0
$413$			5.44373i			0.267868i
$414$		0			0
$415$		−	10.2430i		−	0.502811i
$416$		0			0
$417$	8.09025			0.396181
$418$		0			0
$419$	10.3546			0.505857			0.252928	−	0.967485i	$-0.418606\pi$
$419$	10.3546			0.505857			0.252928	+	0.967485i	$0.418606\pi$
$420$		0			0
$421$		−	24.9829i		−	1.21759i	−0.793327	−	0.608796i	$-0.791653\pi$
$421$		−	24.9829i		−	1.21759i	0.793327	−	0.608796i	$-0.208347\pi$
$422$		0			0
$423$	72.1091			3.50607
$424$		0			0
$425$			24.9939i			1.21238i
$426$		0			0
$427$		−	6.27013i		−	0.303433i
$428$		0			0
$429$		−	7.31424i		−	0.353135i
$430$		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
$435$	25.0049			1.19889
$436$		0			0
$437$	1.83166			0.0876200
$438$		0			0
$439$		−	17.4395i		−	0.832341i	−0.909287	−	0.416171i	$-0.863372\pi$
$439$		−	17.4395i		−	0.832341i	0.909287	−	0.416171i	$-0.136628\pi$
$440$		0			0
$441$	21.5320			1.02533
$442$		0			0
$443$	−9.49973			−0.451346			−0.225673	−	0.974203i	$-0.572458\pi$
$443$	−9.49973			−0.451346			−0.225673	+	0.974203i	$0.572458\pi$
$444$		0			0
$445$	32.5398			1.54253
$446$		0			0
$447$	38.1548			1.80466
$448$		0			0
$449$			9.99582i			0.471732i	0.971786	+	0.235866i	$0.0757927\pi$
$449$			9.99582i			0.471732i	−0.971786	+	0.235866i	$0.924207\pi$
$450$		0			0
$451$	19.3478			0.911055
$452$		0			0
$453$	−15.6342			−0.734560
$454$		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
$459$			37.9175i			1.76984i
$460$		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	$0.0685538\pi$
$463$			9.19686i			0.427414i	−0.976898	+	0.213707i	$0.931446\pi$
$464$		0			0
$465$	89.3551			4.14374
$466$		0			0
$467$		−	2.62505i		−	0.121473i	−0.998154	−	0.0607364i	$-0.980655\pi$
$467$		−	2.62505i		−	0.121473i	0.998154	−	0.0607364i	$-0.0193449\pi$
$468$		0			0
$469$	6.27013			0.289528
$470$		0			0
$471$	59.6605			2.74901
$472$		0			0
$473$			14.7912i			0.680098i
$474$		0			0
$475$		−	1.09736i		−	0.0503501i
$476$		0			0
$477$	29.1086			1.33279
$478$		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
$483$		−	84.5453i		−	3.84694i
$484$		0			0
$485$	32.5968			1.48014
$486$		0			0
$487$		−	19.0050i		−	0.861199i	−0.902543	−	0.430599i	$-0.858302\pi$
$487$		−	19.0050i		−	0.861199i	0.902543	−	0.430599i	$-0.141698\pi$
$488$		0			0
$489$			51.0338i			2.30783i
$490$		0			0
$491$	35.1468			1.58615			0.793076	−	0.609123i	$-0.208478\pi$
$491$	35.1468			1.58615			0.793076	+	0.609123i	$0.208478\pi$
$492$		0			0
$493$	−12.6663			−0.570463
$494$		0			0
$495$			31.0498i			1.39558i
$496$		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
$501$		−	7.61570i		−	0.340244i
$502$		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
$507$	−32.0874			−1.42505
$508$		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
$513$		−	1.66477i		−	0.0735013i
$514$		0			0
$515$	20.5498			0.905533
$516$		0			0
$517$	−21.6028			−0.950092
$518$		0			0
$519$	21.5186			0.944563
$520$		0			0
$521$	11.8492			0.519125			0.259562	−	0.965726i	$-0.416422\pi$
$521$	11.8492			0.519125			0.259562	+	0.965726i	$0.416422\pi$
$522$		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
$525$	−50.6516			−2.21062
$526$		0			0
$527$	−45.2631			−1.97169
$528$		0			0
$529$	−53.2642			−2.31584
$530$		0			0
$531$		−	9.43652i		−	0.409510i
$532$		0			0
$533$			16.5058i			0.714947i
$534$		0			0
$535$		−	33.2373i		−	1.43697i
$536$		0			0
$537$		−	64.0234i		−	2.76281i
$538$		0			0
$539$	−6.45066			−0.277850
$540$		0			0
$541$			34.5457i			1.48523i	0.669716	+	0.742617i	$0.266416\pi$
$541$			34.5457i			1.48523i	−0.669716	+	0.742617i	$0.733584\pi$
$542$		0			0
$543$	−12.1227			−0.520234
$544$		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
$549$			10.8690i			0.463880i
$550$		0			0
$551$	0.556116			0.0236913
$552$		0			0
$553$			11.9025i			0.506148i
$554$		0			0
$555$	45.8642	−	34.4555i	1.94683	−	1.46256i
$556$		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
$561$		−	24.0178i		−	1.01403i
$562$		0			0
$563$		−	13.3043i		−	0.560711i	−0.959896	−	0.280356i	$-0.909548\pi$
$563$		−	13.3043i		−	0.560711i	0.959896	−	0.280356i	$-0.0904524\pi$
$564$		0			0
$565$	−32.5968			−1.37136
$566$		0			0
$567$	−20.7677			−0.872159
$568$		0			0
$569$			4.88761i			0.204899i	0.994738	+	0.102450i	$0.0326681\pi$
$569$			4.88761i			0.204899i	−0.994738	+	0.102450i	$0.967332\pi$
$570$		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
$573$		−	2.10971i		−	0.0881345i
$574$		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
$579$			30.1990i			1.25503i
$580$		0			0
$581$	−10.5151			−0.436240
$582$		0			0
$583$	−8.72051			−0.361167
$584$		0			0
$585$	−26.4889			−1.09518
$586$		0			0
$587$		−	24.4005i		−	1.00712i	−0.863961	−	0.503559i	$-0.832024\pi$
$587$		−	24.4005i		−	1.00712i	0.863961	−	0.503559i	$-0.167976\pi$
$588$		0			0
$589$	1.98728			0.0818843
$590$		0			0
$591$	−10.8531			−0.446436
$592$		0			0
$593$	22.3306			0.917009			0.458505	−	0.888692i	$-0.348385\pi$
$593$	22.3306			0.917009			0.458505	+	0.888692i	$0.348385\pi$
$594$		0			0
$595$	50.1779			2.05709
$596$		0			0
$597$		−	17.9298i		−	0.733818i
$598$		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
$603$	−10.8690			−0.442622
$604$		0			0
$605$			25.8841i			1.05234i
$606$		0			0
$607$		−	1.94307i		−	0.0788666i	−0.999222	−	0.0394333i	$-0.987445\pi$
$607$		−	1.94307i		−	0.0788666i	0.999222	−	0.0394333i	$-0.0125553\pi$
$608$		0			0
$609$		−	25.6691i		−	1.04016i
$610$		0			0
$611$		−	18.4296i		−	0.745582i
$612$		0			0
$613$	−19.8400			−0.801329			−0.400665	−	0.916225i	$-0.631221\pi$
$613$	−19.8400			−0.801329			−0.400665	+	0.916225i	$0.631221\pi$
$614$		0			0
$615$		−	106.998i		−	4.31459i
$616$		0			0
$617$	16.0166			0.644806			0.322403	−	0.946603i	$-0.395509\pi$
$617$	16.0166			0.644806			0.322403	+	0.946603i	$0.395509\pi$
$618$		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
$621$			69.3156i			2.78154i
$622$		0			0
$623$		−	33.4041i		−	1.33831i
$624$		0			0
$625$	−23.7864			−0.951456
$626$		0			0
$627$			1.05450i			0.0421128i
$628$		0			0
$629$	−23.2327	+	17.4536i	−0.926347	+	0.695919i
$630$		0			0
$631$		−	16.4641i		−	0.655427i	−0.944777	−	0.327713i	$-0.893722\pi$
$631$		−	16.4641i		−	0.655427i	0.944777	−	0.327713i	$-0.106278\pi$
$632$		0			0
$633$	13.2073			0.524942
$634$		0			0
$635$			35.6923i			1.41640i
$636$		0			0
$637$		−	5.50313i		−	0.218042i
$638$		0			0
$639$	−54.1399			−2.14174
$640$		0			0
$641$	−24.3810			−0.962991			−0.481496	−	0.876448i	$-0.659906\pi$
$641$	−24.3810			−0.962991			−0.481496	+	0.876448i	$0.659906\pi$
$642$		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
$645$	81.7987			3.22082
$646$		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
$651$		−	91.7285i		−	3.59512i
$652$		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
$657$	34.7382			1.35527
$658$		0			0
$659$	−20.5470			−0.800397			−0.400199	−	0.916428i	$-0.631059\pi$
$659$	−20.5470			−0.800397			−0.400199	+	0.916428i	$0.631059\pi$
$660$		0			0
$661$		−	42.2553i		−	1.64354i	−0.569820	−	0.821770i	$-0.692987\pi$
$661$		−	42.2553i		−	1.64354i	0.569820	−	0.821770i	$-0.307013\pi$
$662$		0			0
$663$	20.4898			0.795759
$664$		0			0
$665$	−2.20306			−0.0854311
$666$		0			0
$667$	−23.1549			−0.896560
$668$		0			0
$669$	33.2672			1.28618
$670$		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
$675$	41.5274			1.59839
$676$		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
$681$			19.2975i			0.739483i
$682$		0			0
$683$			22.2020i			0.849537i	0.905302	+	0.424769i	$0.139645\pi$
$683$			22.2020i			0.849537i	−0.905302	+	0.424769i	$0.860355\pi$
$684$		0			0
$685$		−	34.2282i		−	1.30779i
$686$		0			0
$687$	−38.2927			−1.46096
$688$		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
$693$	31.8745			1.21081
$694$		0			0
$695$			8.77761i			0.332954i
$696$		0			0
$697$			54.2003i			2.05298i
$698$		0			0
$699$	−52.1777			−1.97354
$700$		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
$705$			119.469i			4.49946i
$706$		0			0
$707$	−48.4699			−1.82290
$708$		0			0
$709$			40.1321i			1.50719i	0.657338	+	0.753596i	$0.271682\pi$
$709$			40.1321i			1.50719i	−0.657338	+	0.753596i	$0.728318\pi$
$710$		0			0
$711$		−	20.6327i		−	0.773785i
$712$		0			0
$713$	−82.7439			−3.09878
$714$		0			0
$715$	7.93568			0.296778
$716$		0			0
$717$			15.1519i			0.565858i
$718$		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
$723$		−	68.4958i		−	2.54739i
$724$		0			0
$725$			13.8722i			0.515201i
$726$		0			0
$727$			24.8473i			0.921534i	0.887521	+	0.460767i	$0.152426\pi$
$727$			24.8473i			0.921534i	−0.887521	+	0.460767i	$0.847574\pi$
$728$		0			0
$729$	−34.2031			−1.26678
$730$		0			0
$731$	−41.4354			−1.53254
$732$		0			0
$733$	−36.5701			−1.35075			−0.675374	−	0.737476i	$-0.736018\pi$
$733$	−36.5701			−1.35075			−0.675374	+	0.737476i	$0.736018\pi$
$734$		0			0
$735$			35.6737i			1.31585i
$736$		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
$741$	−0.899607			−0.0330479
$742$		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
$747$	18.2276			0.666912
$748$		0			0
$749$	−34.1201			−1.24672
$750$		0			0
$751$	4.33703			0.158260			0.0791302	−	0.996864i	$-0.474786\pi$
$751$	4.33703			0.158260			0.0791302	+	0.996864i	$0.474786\pi$
$752$		0			0
$753$		−	44.0418i		−	1.60497i
$754$		0			0
$755$		−	16.9625i		−	0.617330i
$756$		0			0
$757$			25.2830i			0.918926i	0.888197	+	0.459463i	$0.151958\pi$
$757$			25.2830i			0.918926i	−0.888197	+	0.459463i	$0.848042\pi$
$758$		0			0
$759$		−	43.9061i		−	1.59369i
$760$		0			0
$761$	43.0646			1.56109			0.780545	−	0.625099i	$-0.214941\pi$
$761$	43.0646			1.56109			0.780545	+	0.625099i	$0.214941\pi$
$762$		0			0
$763$			22.8616i			0.827647i
$764$		0			0
$765$	−86.9817			−3.14483
$766$		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
$771$			41.6024i			1.49827i
$772$		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
$777$	−35.3707	−	47.0824i	−1.26892	−	1.68907i
$778$		0			0
$779$		−	2.37967i		−	0.0852604i
$780$		0			0
$781$	16.2195			0.580380
$782$		0			0
$783$			21.0452i			0.752092i
$784$		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
$789$	3.55599			0.126597
$790$		0			0
$791$			33.4626i			1.18979i
$792$		0			0
$793$	2.77790			0.0986462
$794$		0			0
$795$			48.2265i			1.71042i
$796$		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
$801$			57.9048i			2.04597i
$802$		0			0
$803$	−10.4071			−0.367257
$804$		0			0
$805$	91.7285			3.23300
$806$		0			0
$807$	34.3257			1.20832
$808$		0			0
$809$		−	24.7019i		−	0.868471i	−0.900799	−	0.434235i	$-0.857019\pi$
$809$		−	24.7019i		−	0.868471i	0.900799	−	0.434235i	$-0.142981\pi$
$810$		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
$813$	−6.66145			−0.233627
$814$		0			0
$815$	−55.3698			−1.93952
$816$		0			0
$817$	1.81922			0.0636465
$818$		0			0
$819$			27.1925i			0.950181i
$820$		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	$-0.275413\pi$
$823$	37.2061			1.29692			0.648462	+	0.761247i	$0.275413\pi$
$824$		0			0
$825$	−26.3044			−0.915802
$826$		0			0
$827$			25.4952i			0.886556i	0.896384	+	0.443278i	$0.146185\pi$
$827$			25.4952i			0.886556i	−0.896384	+	0.443278i	$0.853815\pi$
$828$		0			0
$829$		−	28.3600i		−	0.984982i	−0.870318	−	0.492491i	$-0.836086\pi$
$829$		−	28.3600i		−	0.984982i	0.870318	−	0.492491i	$-0.163914\pi$
$830$		0			0
$831$			21.6791i			0.752039i
$832$		0			0
$833$		−	18.0706i		−	0.626111i
$834$		0			0
$835$	8.26275			0.285944
$836$		0			0
$837$			75.2048i			2.59946i
$838$		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
$843$			68.9110i			2.37342i
$844$		0			0
$845$		−	34.8137i		−	1.19763i
$846$		0			0
$847$	26.5716			0.913012
$848$		0			0
$849$			85.0578i			2.91918i
$850$		0			0
$851$	−42.4708	+	31.9062i	−1.45588	+	1.09373i
$852$		0			0
$853$		−	13.2081i		−	0.452235i	−0.974100	−	0.226118i	$-0.927397\pi$
$853$		−	13.2081i		−	0.452235i	0.974100	−	0.226118i	$-0.0726034\pi$
$854$		0			0
$855$	3.81893			0.130605
$856$		0			0
$857$		−	30.8046i		−	1.05227i	−0.850402	−	0.526133i	$-0.823641\pi$
$857$		−	30.8046i		−	1.05227i	0.850402	−	0.526133i	$-0.176359\pi$
$858$		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
$861$	−109.840			−3.74335
$862$		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
$867$	17.1623			0.582863
$868$		0			0
$869$			6.18124i			0.209684i
$870$		0			0
$871$			2.77790i			0.0941256i
$872$		0			0
$873$			58.0062i			1.96321i
$874$		0			0
$875$		−	2.43642i		−	0.0823661i
$876$		0			0
$877$	45.7101			1.54352			0.771760	−	0.635914i	$-0.219377\pi$
$877$	45.7101			1.54352			0.771760	+	0.635914i	$0.219377\pi$
$878$		0			0
$879$	−22.0511			−0.743764
$880$		0			0
$881$	−29.6503			−0.998945			−0.499473	−	0.866330i	$-0.666473\pi$
$881$	−29.6503			−0.998945			−0.499473	+	0.866330i	$0.666473\pi$
$882$		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
$885$	15.6342			0.525539
$886$		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
$891$	−10.7851			−0.361314
$892$		0			0
$893$			2.65702i			0.0889137i
$894$		0			0
$895$	69.4630			2.32189
$896$		0			0
$897$	37.4567			1.25064
$898$		0			0
$899$	−25.1222			−0.837871
$900$		0			0
$901$		−	24.4293i		−	0.813858i
$902$		0			0
$903$		−	83.9713i		−	2.79439i
$904$		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
$909$	84.0209			2.78680
$910$		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
$915$	−18.0076			−0.595313
$916$		0			0
$917$		−	39.4135i		−	1.30155i
$918$		0			0
$919$			29.1690i			0.962196i	0.876667	+	0.481098i	$0.159762\pi$
$919$			29.1690i			0.962196i	−0.876667	+	0.481098i	$0.840238\pi$
$920$		0			0
$921$	−22.8290			−0.752240
$922$		0			0
$923$			13.8370i			0.455452i
$924$		0			0
$925$	19.1152	+	25.4445i	0.628504	+	0.836611i
$926$		0			0
$927$			36.5686i			1.20107i
$928$		0			0
$929$	−8.75265			−0.287165			−0.143583	−	0.989638i	$-0.545862\pi$
$929$	−8.75265			−0.287165			−0.143583	+	0.989638i	$0.545862\pi$
$930$		0			0
$931$			0.793392i			0.0260024i
$932$		0			0
$933$		−	15.5039i		−	0.507574i
$934$		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
$939$		−	36.9017i		−	1.20424i
$940$		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
$945$		−	83.3708i		−	2.71205i
$946$		0			0
$947$			12.5898i			0.409114i	0.978855	+	0.204557i	$0.0655753\pi$
$947$			12.5898i			0.409114i	−0.978855	+	0.204557i	$0.934425\pi$
$948$		0			0
$949$		−	8.87836i		−	0.288204i
$950$		0			0
$951$	−20.0592			−0.650465
$952$		0			0
$953$	15.4612			0.500839			0.250419	−	0.968137i	$-0.419432\pi$
$953$	15.4612			0.500839			0.250419	+	0.968137i	$0.419432\pi$
$954$		0			0
$955$	2.28896			0.0740690
$956$		0			0
$957$		−	13.3305i		−	0.430914i
$958$		0			0
$959$	−35.1373			−1.13464
$960$		0			0
$961$	−58.7740			−1.89593
$962$		0			0
$963$	59.1461			1.90596
$964$		0			0
$965$	−32.7648			−1.05474
$966$		0			0
$967$		−	50.6768i		−	1.62966i	−0.579703	−	0.814828i	$-0.696831\pi$
$967$		−	50.6768i		−	1.62966i	0.579703	−	0.814828i	$-0.303169\pi$
$968$		0			0
$969$	−2.95404			−0.0948976
$970$		0			0
$971$	12.6413			0.405679			0.202839	−	0.979212i	$-0.434983\pi$
$971$	12.6413			0.405679			0.202839	+	0.979212i	$0.434983\pi$
$972$		0			0
$973$	9.01076			0.288872
$974$		0			0
$975$		−	22.4406i		−	0.718673i
$976$		0			0
$977$		−	24.7560i		−	0.792016i	−0.918247	−	0.396008i	$-0.870395\pi$
$977$		−	24.7560i		−	0.792016i	0.918247	−	0.396008i	$-0.129605\pi$
$978$		0			0
$979$		−	17.3474i		−	0.554426i
$980$		0			0
$981$		−	39.6299i		−	1.26528i
$982$		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
$987$	122.642			3.90374
$988$		0			0
$989$	−75.7465			−2.40860
$990$		0			0
$991$		−	19.0343i		−	0.604643i	−0.953206	−	0.302322i	$-0.902238\pi$
$991$		−	19.0343i		−	0.604643i	0.953206	−	0.302322i	$-0.0977616\pi$
$992$		0			0
$993$			20.9371i			0.664419i
$994$		0			0
$995$	19.4532			0.616707
$996$		0			0
$997$		−	41.2547i		−	1.30655i	−0.757121	−	0.653274i	$-0.773395\pi$
$997$		−	41.2547i		−	1.30655i	0.757121	−	0.653274i	$-0.226605\pi$
$998$		0			0
$999$	28.9991	+	38.6012i	0.917492	+	1.22129i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.g.a.73.9	✓	10	296.221	even	2
296.2.g.a.73.10	yes	10	8.5	even	2
592.2.g.d.369.1		10	296.147	odd	2
592.2.g.d.369.2		10	8.3	odd	2
2368.2.g.o.961.1		10	1.1	even	1	trivial
2368.2.g.o.961.2		10	37.36	even	2	inner
2368.2.g.p.961.9		10	4.3	odd	2
2368.2.g.p.961.10		10	148.147	odd	2
2664.2.h.c.73.1		10	24.5	odd	2
2664.2.h.c.73.10		10	888.221	odd	2
5328.2.h.q.2737.1		10	24.11	even	2
5328.2.h.q.2737.10		10	888.443	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.94825 q^{3} -3.19874i q^{5} -3.28371 q^{7} +5.69219 q^{9} -1.70530 q^{11} -1.45481i q^{13} +9.43070i q^{15} -4.77715i q^{17} +0.209741i q^{19} +9.68119 q^{21} -8.73294i q^{23} -5.23196 q^{25} -7.93725 q^{27} -2.65144i q^{29} -9.47491i q^{31} +5.02764 q^{33} +10.5037i q^{35} +(-3.65355 - 4.86329i) q^{37} +4.28913i q^{39} -11.3457 q^{41} -8.67366i q^{43} -18.2079i q^{45} +12.6681 q^{47} +3.78272 q^{49} +14.0842i q^{51} +5.11378 q^{53} +5.45481i q^{55} -0.618369i q^{57} -1.65780i q^{59} +1.90947i q^{61} -18.6915 q^{63} -4.65355 q^{65} -1.90947 q^{67} +25.7469i q^{69} -9.51127 q^{71} +6.10278 q^{73} +15.4251 q^{75} +5.59969 q^{77} -3.62473i q^{79} +6.32446 q^{81} +3.20221 q^{83} -15.2809 q^{85} +7.81711i q^{87} +10.1727i q^{89} +4.77715i q^{91} +27.9344i q^{93} +0.670907 q^{95} +10.1905i q^{97} -9.70687 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 4 q^{7} + 12 q^{9} - 2 q^{11} + 8 q^{21} + 4 q^{25} - 8 q^{27} + 4 q^{33} + 6 q^{37} - 26 q^{41} + 8 q^{47} + 14 q^{49} + 20 q^{53} - 12 q^{63} - 4 q^{65} + 2 q^{67} - 4 q^{71} - 14 q^{73}+ \cdots + 28 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 4 * q^7 + 12 * q^9 - 2 * q^11 + 8 * q^21 + 4 * q^25 - 8 * q^27 + 4 * q^33 + 6 * q^37 - 26 * q^41 + 8 * q^47 + 14 * q^49 + 20 * q^53 - 12 * q^63 - 4 * q^65 + 2 * q^67 - 4 * q^71 - 14 * q^73 + 48 * q^75 + 18 * q^81 + 36 * q^83 - 8 * q^85 + 4 * q^95 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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