LMFDB - Embedded newform 296.2.g.a.73.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [296,2,Mod(73,296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, 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("296.73"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 296 = 2^{3} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	296.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$2.36357189983$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.9
Root			$0.279838 + 0.279838i$ of defining polynomial
Character	$\chi$	$=$	296.73
Dual form			296.2.g.a.73.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.94825 q^{3} -3.19874i q^{5} -3.28371 q^{7} +5.69219 q^{9} +1.70530 q^{11} -1.45481i q^{13} -9.43070i q^{15} +4.77715i q^{17} +0.209741i q^{19} -9.68119 q^{21} +8.73294i q^{23} -5.23196 q^{25} +7.93725 q^{27} -2.65144i q^{29} +9.47491i q^{31} +5.02764 q^{33} +10.5037i q^{35} +(3.65355 - 4.86329i) q^{37} -4.28913i q^{39} -11.3457 q^{41} -8.67366i q^{43} -18.2079i q^{45} +12.6681 q^{47} +3.78272 q^{49} +14.0842i q^{51} -5.11378 q^{53} -5.45481i q^{55} +0.618369i q^{57} -1.65780i q^{59} +1.90947i q^{61} -18.6915 q^{63} -4.65355 q^{65} +1.90947 q^{67} +25.7469i q^{69} -9.51127 q^{71} +6.10278 q^{73} -15.4251 q^{75} -5.59969 q^{77} +3.62473i q^{79} +6.32446 q^{81} -3.20221 q^{83} +15.2809 q^{85} -7.81711i q^{87} -10.1727i q^{89} +4.77715i q^{91} +27.9344i q^{93} +0.670907 q^{95} -10.1905i q^{97} +9.70687 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/296\mathbb{Z}\right)^\times$.

$n$	$113$	$149$	$223$
$\chi(n)$	$-1$	$1$	$1$

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	$-0.175944\pi$
$3$	2.94825			1.70217			0.851087	+	0.525025i	$0.175944\pi$
$4$		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	$-0.417239\pi$
$11$	1.70530			0.514166			0.257083	+	0.966389i	$0.417239\pi$
$12$		0			0
$13$		−	1.45481i		−	0.403490i	−0.979438	−	0.201745i	$-0.935339\pi$
$13$		−	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	$0.196680\pi$
$17$			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
$21$	−9.68119			−2.11261
$22$		0			0
$23$			8.73294i			1.82094i	0.413571	+	0.910472i	$0.364281\pi$
$23$			8.73294i			1.82094i	−0.413571	+	0.910472i	$0.635719\pi$
$24$		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	$0.323926\pi$
$31$			9.47491i			1.70174i	−0.525373	+	0.850872i	$0.676074\pi$
$32$		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	$-0.614232\pi$
$53$	−5.11378			−0.702432			−0.351216	+	0.936295i	$0.614232\pi$
$54$		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	$-0.462788\pi$
$67$	1.90947			0.233278			0.116639	+	0.993174i	$0.462788\pi$
$68$		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	$0.0653640\pi$
$79$			3.62473i			0.407814i	−0.978990	+	0.203907i	$0.934636\pi$
$80$		0			0
$81$	6.32446			0.702717
$82$		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
$87$		−	7.81711i		−	0.838082i
$88$		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
$93$			27.9344i			2.89667i
$94$		0			0
$95$	0.670907			0.0688337
$96$		0			0
$97$		−	10.1905i		−	1.03469i	−0.855778	−	0.517344i	$-0.826921\pi$
$97$		−	10.1905i		−	1.03469i	0.855778	−	0.517344i	$-0.173079\pi$
$98$		0			0
$99$	9.70687			0.975577
$100$		0			0
$101$	−14.7607			−1.46875			−0.734374	−	0.678745i	$-0.762524\pi$
$101$	−14.7607			−1.46875			−0.734374	+	0.678745i	$0.762524\pi$
$102$		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
$105$			30.9676i			3.02213i
$106$		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	$-0.891795\pi$
$109$		−	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	$0.159117\pi$
$113$			10.1905i			0.958641i	−0.877640	+	0.479321i	$0.840883\pi$
$114$		0			0
$115$	27.9344			2.60490
$116$		0			0
$117$		−	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	$-0.462873\pi$
$139$	2.74408			0.232750			0.116375	+	0.993205i	$0.462873\pi$
$140$		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	$-0.322153\pi$
$149$	12.9415			1.06021			0.530105	+	0.847932i	$0.322153\pi$
$150$		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	$-0.200820\pi$
$157$	20.2359			1.61500			0.807500	+	0.589868i	$0.200820\pi$
$158$		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	$-0.968134\pi$
$167$		−	2.58312i		−	0.199888i	0.994993	−	0.0999440i	$-0.0318664\pi$
$168$		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	$-0.410508\pi$
$173$	7.29878			0.554916			0.277458	+	0.960738i	$0.410508\pi$
$174$		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	$-0.548834\pi$
$181$	−4.11182			−0.305629			−0.152814	+	0.988255i	$0.548834\pi$
$182$		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	$-0.991758\pi$
$191$		−	0.715581i		−	0.0517776i	0.999665	−	0.0258888i	$-0.00824159\pi$
$192$		0			0
$193$			10.2430i			0.737310i	0.929566	+	0.368655i	$0.120182\pi$
$193$			10.2430i			0.737310i	−0.929566	+	0.368655i	$0.879818\pi$
$194$		0			0
$195$	−13.7198			−0.982498
$196$		0			0
$197$	−3.68119			−0.262274			−0.131137	−	0.991364i	$-0.541863\pi$
$197$	−3.68119			−0.262274			−0.131137	+	0.991364i	$0.541863\pi$
$198$		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
$201$	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	$-0.450721\pi$
$211$	4.47970			0.308395			0.154198	+	0.988040i	$0.450721\pi$
$212$		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	$-0.641185\pi$
$229$	−12.9883			−0.858290			−0.429145	+	0.903236i	$0.641185\pi$
$230$		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	$0.0531550\pi$
$239$			5.13928i			0.332432i	−0.986089	+	0.166216i	$0.946845\pi$
$240$		0			0
$241$		−	23.2327i		−	1.49655i	−0.663390	−	0.748274i	$-0.730883\pi$
$241$		−	23.2327i		−	1.49655i	0.663390	−	0.748274i	$-0.269117\pi$
$242$		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	$0.145059\pi$
$257$			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
$261$		−	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	$-0.384503\pi$
$269$	11.6427			0.709870			0.354935	+	0.934891i	$0.384503\pi$
$270$		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	$0.245559\pi$
$281$			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
$285$	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	$-0.570108\pi$
$293$	−7.47937			−0.436950			−0.218475	+	0.975843i	$0.570108\pi$
$294$		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	$-0.570921\pi$
$307$	−7.74322			−0.441929			−0.220965	+	0.975282i	$0.570921\pi$
$308$		0			0
$309$		−	18.9406i		−	1.07749i
$310$		0			0
$311$		−	5.25867i		−	0.298192i	−0.988823	−	0.149096i	$-0.952364\pi$
$311$		−	5.25867i		−	0.298192i	0.988823	−	0.149096i	$-0.0476363\pi$
$312$		0			0
$313$		−	12.5165i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$313$		−	12.5165i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$314$		0			0
$315$			59.7892i			3.36874i
$316$		0			0
$317$	−6.80377			−0.382138			−0.191069	−	0.981577i	$-0.561195\pi$
$317$	−6.80377			−0.382138			−0.191069	+	0.981577i	$0.561195\pi$
$318$		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	$-0.525793\pi$
$349$	−3.02425			−0.161884			−0.0809422	+	0.996719i	$0.525793\pi$
$350$		0			0
$351$		−	11.5472i		−	0.616342i
$352$		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
$357$		−	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	$-0.294397\pi$
$373$	23.2506			1.20387			0.601934	+	0.798546i	$0.294397\pi$
$374$		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	$-0.707955\pi$
$379$	−23.6659			−1.21564			−0.607819	+	0.794076i	$0.707955\pi$
$380$		0			0
$381$	−32.8972			−1.68538
$382$		0			0
$383$			11.8483i			0.605420i	0.953083	+	0.302710i	$0.0978913\pi$
$383$			11.8483i			0.605420i	−0.953083	+	0.302710i	$0.902109\pi$
$384$		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	$-0.695015\pi$
$397$	−22.9153			−1.15009			−0.575043	+	0.818123i	$0.695015\pi$
$398$		0			0
$399$		−	2.03054i		−	0.101654i
$400$		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
$405$		−	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	$0.146187\pi$
$409$			17.9298i			0.886571i	−0.896380	+	0.443286i	$0.853813\pi$
$410$		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	$-0.581394\pi$
$419$	−10.3546			−0.505857			−0.252928	+	0.967485i	$0.581394\pi$
$420$		0			0
$421$		−	24.9829i		−	1.21759i	−0.793327	−	0.608796i	$-0.791653\pi$
$421$		−	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	$0.312325\pi$
$431$			34.5108i			1.66233i	−0.556028	+	0.831164i	$0.687675\pi$
$432$		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	$0.136628\pi$
$439$			17.4395i			0.832341i	−0.909287	+	0.416171i	$0.863372\pi$
$440$		0			0
$441$	21.5320			1.02533
$442$		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
$447$	38.1548			1.80466
$448$		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
$453$	15.6342			0.734560
$454$		0			0
$455$	15.2809			0.716379
$456$		0			0
$457$			6.49951i			0.304034i	0.988378	+	0.152017i	$0.0485769\pi$
$457$			6.49951i			0.304034i	−0.988378	+	0.152017i	$0.951423\pi$
$458$		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	$-0.931446\pi$
$463$		−	9.19686i		−	0.427414i	0.976898	−	0.213707i	$-0.0685538\pi$
$464$		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	$-0.820922\pi$
$479$		−	23.3471i		−	1.06676i	0.845876	−	0.533379i	$-0.179078\pi$
$480$		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	$0.141698\pi$
$487$			19.0050i			0.861199i	−0.902543	+	0.430599i	$0.858302\pi$
$488$		0			0
$489$		−	51.0338i		−	2.30783i
$490$		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
$495$		−	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	$-0.775165\pi$
$503$		−	29.1135i		−	1.29811i	0.760743	−	0.649053i	$-0.224835\pi$
$504$		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	$-0.460367\pi$
$509$	5.60362			0.248376			0.124188	+	0.992259i	$0.460367\pi$
$510$		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	$-0.967332\pi$
$569$		−	4.88761i		−	0.204899i	0.994738	−	0.102450i	$-0.0326681\pi$
$570$		0			0
$571$	−33.7638			−1.41297			−0.706485	−	0.707728i	$-0.749720\pi$
$571$	−33.7638			−1.41297			−0.706485	+	0.707728i	$0.749720\pi$
$572$		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	$0.218294\pi$
$577$			30.4241i			1.26657i	−0.773918	+	0.633286i	$0.781706\pi$
$578$		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	$0.0125553\pi$
$607$			1.94307i			0.0788666i	−0.999222	+	0.0394333i	$0.987445\pi$
$608$		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	$-0.368779\pi$
$613$	19.8400			0.801329			0.400665	+	0.916225i	$0.368779\pi$
$614$		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	$-0.757621\pi$
$619$	−36.0175			−1.44767			−0.723833	+	0.689975i	$0.757621\pi$
$620$		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	$0.106278\pi$
$631$			16.4641i			0.655427i	−0.944777	+	0.327713i	$0.893722\pi$
$632$		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	$-0.833719\pi$
$647$		−	25.3827i		−	0.997898i	0.866631	−	0.498949i	$-0.166281\pi$
$648$		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	$-0.368941\pi$
$659$	20.5470			0.800397			0.400199	+	0.916428i	$0.368941\pi$
$660$		0			0
$661$		−	42.2553i		−	1.64354i	−0.569820	−	0.821770i	$-0.692987\pi$
$661$		−	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	$-0.177681\pi$
$677$	44.1395			1.69642			0.848209	+	0.529662i	$0.177681\pi$
$678$		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	$-0.465014\pi$
$691$	5.76679			0.219379			0.109690	+	0.993966i	$0.465014\pi$
$692$		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	$-0.847574\pi$
$727$		−	24.8473i		−	0.921534i	0.887521	−	0.460767i	$-0.152426\pi$
$728$		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	$-0.263982\pi$
$733$	36.5701			1.35075			0.675374	+	0.737476i	$0.263982\pi$
$734$		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	$-0.336551\pi$
$739$	26.7073			0.982442			0.491221	+	0.871035i	$0.336551\pi$
$740$		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	$-0.999774\pi$
$769$		−	0.0394155i		−	0.00142136i	1.00000		0.000710679i	$-0.000226216\pi$
$770$		0			0
$771$			41.6024i			1.49827i
$772$		0			0
$773$	−0.727148			−0.0261537			−0.0130768	−	0.999914i	$-0.504163\pi$
$773$	−0.727148			−0.0261537			−0.0130768	+	0.999914i	$0.504163\pi$
$774$		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	$-0.534901\pi$
$787$	−6.13961			−0.218853			−0.109427	+	0.993995i	$0.534901\pi$
$788$		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	$0.142981\pi$
$809$			24.7019i			0.868471i	−0.900799	+	0.434235i	$0.857019\pi$
$810$		0			0
$811$	27.0155			0.948644			0.474322	−	0.880351i	$-0.342693\pi$
$811$	27.0155			0.948644			0.474322	+	0.880351i	$0.342693\pi$
$812$		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	$-0.421168\pi$
$821$	14.0477			0.490269			0.245135	+	0.969489i	$0.421168\pi$
$822$		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	$0.176359\pi$
$857$			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
$861$	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	$-0.780623\pi$
$877$	−45.7101			−1.54352			−0.771760	+	0.635914i	$0.780623\pi$
$878$		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	$0.0163623\pi$
$911$			3.10164i			0.102762i	−0.998679	+	0.0513810i	$0.983638\pi$
$912$		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	$-0.840238\pi$
$919$		−	29.1690i		−	0.962196i	0.876667	−	0.481098i	$-0.159762\pi$
$920$		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	$-0.523411\pi$
$941$	−4.50820			−0.146963			−0.0734815	+	0.997297i	$0.523411\pi$
$942$		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	$0.303169\pi$
$967$			50.6768i			1.62966i	−0.579703	+	0.814828i	$0.696831\pi$
$968$		0			0
$969$	−2.95404			−0.0948976
$970$		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
$975$			22.4406i			0.718673i
$976$		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
$981$		−	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	$0.0977616\pi$
$991$			19.0343i			0.604643i	−0.953206	+	0.302322i	$0.902238\pi$
$992$		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	296.2.g.a.73.9	✓	10
3.2	odd	2		2664.2.h.c.73.10		10
4.3	odd	2		592.2.g.d.369.1		10
8.3	odd	2		2368.2.g.p.961.10		10
8.5	even	2		2368.2.g.o.961.2		10
12.11	even	2		5328.2.h.q.2737.10		10
37.36	even	2	inner	296.2.g.a.73.10	yes	10
111.110	odd	2		2664.2.h.c.73.1		10
148.147	odd	2		592.2.g.d.369.2		10
296.147	odd	2		2368.2.g.p.961.9		10
296.221	even	2		2368.2.g.o.961.1		10
444.443	even	2		5328.2.h.q.2737.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.g.a.73.9	✓	10	1.1	even	1	trivial
296.2.g.a.73.10	yes	10	37.36	even	2	inner
592.2.g.d.369.1		10	4.3	odd	2
592.2.g.d.369.2		10	148.147	odd	2
2368.2.g.o.961.1		10	296.221	even	2
2368.2.g.o.961.2		10	8.5	even	2
2368.2.g.p.961.9		10	296.147	odd	2
2368.2.g.p.961.10		10	8.3	odd	2
2664.2.h.c.73.1		10	111.110	odd	2
2664.2.h.c.73.10		10	3.2	odd	2
5328.2.h.q.2737.1		10	444.443	even	2
5328.2.h.q.2737.10		10	12.11	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:	\( N \)	\(=\)	\( 296 = 2^{3} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	296.g (of order \(2\), degree \(1\), 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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