LMFDB - Embedded newform 2368.2.g.p.961.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(961,2368)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
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			961.4
Root			$1.23896 + 1.23896i$ of defining polynomial
Character	$\chi$	$=$	2368.961
Dual form			2368.2.g.p.961.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.706809 q^{3} +1.32106i q^{5} -1.54799 q^{7} -2.50042 q^{9} +O(q^{10})$	$q-0.706809 q^{3} +1.32106i q^{5} -1.54799 q^{7} -2.50042 q^{9} +6.14254 q^{11} +4.11467i q^{13} -0.933737i q^{15} -6.36947i q^{17} -1.27349i q^{19} +1.09413 q^{21} -3.80094i q^{23} +3.25480 q^{25} +3.88775 q^{27} -8.80178i q^{29} +7.10415i q^{31} -4.34160 q^{33} -2.04499i q^{35} +(-4.43573 + 4.16224i) q^{37} -2.90829i q^{39} -3.93531 q^{41} +9.78309i q^{43} -3.30321i q^{45} +8.54883 q^{47} -4.60373 q^{49} +4.50200i q^{51} +6.19011 q^{53} +8.11467i q^{55} +0.900116i q^{57} +13.3253i q^{59} -2.10331i q^{61} +3.87063 q^{63} -5.43573 q^{65} +2.10331 q^{67} +2.68654i q^{69} +6.83223 q^{71} +6.78467 q^{73} -2.30052 q^{75} -9.50859 q^{77} -0.376573i q^{79} +4.75337 q^{81} +8.15071 q^{83} +8.41446 q^{85} +6.22118i q^{87} +11.8388i q^{89} -6.36947i q^{91} -5.02127i q^{93} +1.68236 q^{95} +8.18511i q^{97} -15.3589 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 4 * q^7 + 12 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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
$3$	−0.706809			−0.408076			−0.204038	−	0.978963i	$-0.565407\pi$
$3$	−0.706809			−0.408076			−0.204038	+	0.978963i	$0.565407\pi$
$4$		0			0
$5$			1.32106i			0.590796i	0.955374	+	0.295398i	$0.0954523\pi$
$5$			1.32106i			0.590796i	−0.955374	+	0.295398i	$0.904548\pi$
$6$		0			0
$7$	−1.54799			−0.585085			−0.292543	−	0.956253i	$-0.594501\pi$
$7$	−1.54799			−0.585085			−0.292543	+	0.956253i	$0.594501\pi$
$8$		0			0
$9$	−2.50042			−0.833474
$10$		0			0
$11$	6.14254			1.85205			0.926023	−	0.377466i	$-0.123205\pi$
$11$	6.14254			1.85205			0.926023	+	0.377466i	$0.123205\pi$
$12$		0			0
$13$			4.11467i			1.14121i	0.821226	+	0.570603i	$0.193290\pi$
$13$			4.11467i			1.14121i	−0.821226	+	0.570603i	$0.806710\pi$
$14$		0			0
$15$		−	0.933737i		−	0.241090i
$16$		0			0
$17$		−	6.36947i		−	1.54482i	−0.635122	−	0.772412i	$-0.719050\pi$
$17$		−	6.36947i		−	1.54482i	0.635122	−	0.772412i	$-0.280950\pi$
$18$		0			0
$19$		−	1.27349i		−	0.292159i	−0.989273	−	0.146080i	$-0.953334\pi$
$19$		−	1.27349i		−	0.292159i	0.989273	−	0.146080i	$-0.0466656\pi$
$20$		0			0
$21$	1.09413			0.238759
$22$		0			0
$23$		−	3.80094i		−	0.792551i	−0.918132	−	0.396275i	$-0.870303\pi$
$23$		−	3.80094i		−	0.792551i	0.918132	−	0.396275i	$-0.129697\pi$
$24$		0			0
$25$	3.25480			0.650960
$26$		0			0
$27$	3.88775			0.748197
$28$		0			0
$29$		−	8.80178i		−	1.63445i	−0.576319	−	0.817225i	$-0.695511\pi$
$29$		−	8.80178i		−	1.63445i	0.576319	−	0.817225i	$-0.304489\pi$
$30$		0			0
$31$			7.10415i			1.27594i	0.770060	+	0.637972i	$0.220226\pi$
$31$			7.10415i			1.27594i	−0.770060	+	0.637972i	$0.779774\pi$
$32$		0			0
$33$	−4.34160			−0.755776
$34$		0			0
$35$		−	2.04499i		−	0.345666i
$36$		0			0
$37$	−4.43573	+	4.16224i	−0.729230	+	0.684268i
$37$	−4.43573	+	4.16224i	−0.729230	+	0.684268i
$38$		0			0
$39$		−	2.90829i		−	0.465699i
$40$		0			0
$41$	−3.93531			−0.614593			−0.307296	−	0.951614i	$-0.599424\pi$
$41$	−3.93531			−0.614593			−0.307296	+	0.951614i	$0.599424\pi$
$42$		0			0
$43$			9.78309i			1.49191i	0.665998	+	0.745954i	$0.268006\pi$
$43$			9.78309i			1.49191i	−0.665998	+	0.745954i	$0.731994\pi$
$44$		0			0
$45$		−	3.30321i		−	0.492413i
$46$		0			0
$47$	8.54883			1.24698			0.623488	−	0.781833i	$-0.285715\pi$
$47$	8.54883			1.24698			0.623488	+	0.781833i	$0.285715\pi$
$48$		0			0
$49$	−4.60373			−0.657676
$50$		0			0
$51$			4.50200i			0.630406i
$52$		0			0
$53$	6.19011			0.850277			0.425139	−	0.905128i	$-0.360225\pi$
$53$	6.19011			0.850277			0.425139	+	0.905128i	$0.360225\pi$
$54$		0			0
$55$			8.11467i			1.09418i
$56$		0			0
$57$			0.900116i			0.119223i
$58$		0			0
$59$			13.3253i			1.73481i	0.497603	+	0.867405i	$0.334214\pi$
$59$			13.3253i			1.73481i	−0.497603	+	0.867405i	$0.665786\pi$
$60$		0			0
$61$		−	2.10331i		−	0.269301i	−0.990893	−	0.134650i	$-0.957009\pi$
$61$		−	2.10331i		−	0.269301i	0.990893	−	0.134650i	$-0.0429911\pi$
$62$		0			0
$63$	3.87063			0.487653
$64$		0			0
$65$	−5.43573			−0.674220
$66$		0			0
$67$	2.10331			0.256960			0.128480	−	0.991712i	$-0.458990\pi$
$67$	2.10331			0.256960			0.128480	+	0.991712i	$0.458990\pi$
$68$		0			0
$69$			2.68654i			0.323421i
$70$		0			0
$71$	6.83223			0.810837			0.405418	−	0.914131i	$-0.367126\pi$
$71$	6.83223			0.810837			0.405418	+	0.914131i	$0.367126\pi$
$72$		0			0
$73$	6.78467			0.794085			0.397042	−	0.917800i	$-0.370037\pi$
$73$	6.78467			0.794085			0.397042	+	0.917800i	$0.370037\pi$
$74$		0			0
$75$	−2.30052			−0.265641
$76$		0			0
$77$	−9.50859			−1.08360
$78$		0			0
$79$		−	0.376573i		−	0.0423678i	−0.999776	−	0.0211839i	$-0.993256\pi$
$79$		−	0.376573i		−	0.0423678i	0.999776	−	0.0211839i	$-0.00674355\pi$
$80$		0			0
$81$	4.75337			0.528153
$82$		0			0
$83$	8.15071			0.894657			0.447329	−	0.894370i	$-0.352375\pi$
$83$	8.15071			0.894657			0.447329	+	0.894370i	$0.352375\pi$
$84$		0			0
$85$	8.41446			0.912676
$86$		0			0
$87$			6.22118i			0.666980i
$88$		0			0
$89$			11.8388i			1.25491i	0.778652	+	0.627457i	$0.215904\pi$
$89$			11.8388i			1.25491i	−0.778652	+	0.627457i	$0.784096\pi$
$90$		0			0
$91$		−	6.36947i		−	0.667702i
$92$		0			0
$93$		−	5.02127i		−	0.520682i
$94$		0			0
$95$	1.68236			0.172607
$96$		0			0
$97$			8.18511i			0.831072i	0.909577	+	0.415536i	$0.136406\pi$
$97$			8.18511i			0.831072i	−0.909577	+	0.415536i	$0.863594\pi$
$98$		0			0
$99$	−15.3589			−1.54363
$100$		0			0
$101$	5.58538			0.555766			0.277883	−	0.960615i	$-0.410367\pi$
$101$	5.58538			0.555766			0.277883	+	0.960615i	$0.410367\pi$
$102$		0			0
$103$		−	0.410194i		−	0.0404177i	−0.999796	−	0.0202088i	$-0.993567\pi$
$103$		−	0.410194i		−	0.0404177i	0.999796	−	0.0202088i	$-0.00643311\pi$
$104$		0			0
$105$			1.44542i			0.141058i
$106$		0			0
$107$	9.52439			0.920757			0.460379	−	0.887723i	$-0.347714\pi$
$107$	9.52439			0.920757			0.460379	+	0.887723i	$0.347714\pi$
$108$		0			0
$109$			9.64687i			0.924003i	0.886879	+	0.462001i	$0.152868\pi$
$109$			9.64687i			0.924003i	−0.886879	+	0.462001i	$0.847132\pi$
$110$		0			0
$111$	3.13522	−	2.94191i	0.297581	−	0.279234i
$112$		0			0
$113$		−	8.18511i		−	0.769991i	−0.922919	−	0.384995i	$-0.874203\pi$
$113$		−	8.18511i		−	0.769991i	0.922919	−	0.384995i	$-0.125797\pi$
$114$		0			0
$115$	5.02127			0.468236
$116$		0			0
$117$		−	10.2884i		−	0.951165i
$118$		0			0
$119$			9.85988i			0.903853i
$120$		0			0
$121$	26.7308			2.43008
$122$		0			0
$123$	2.78151			0.250801
$124$		0			0
$125$			10.9051i			0.975381i
$126$		0			0
$127$	−1.77250			−0.157284			−0.0786419	−	0.996903i	$-0.525058\pi$
$127$	−1.77250			−0.157284			−0.0786419	+	0.996903i	$0.525058\pi$
$128$		0			0
$129$		−	6.91477i		−	0.608812i
$130$		0			0
$131$		−	4.10073i		−	0.358282i	−0.983823	−	0.179141i	$-0.942668\pi$
$131$		−	4.10073i		−	0.358282i	0.983823	−	0.179141i	$-0.0573319\pi$
$132$		0			0
$133$			1.97135i			0.170938i
$134$		0			0
$135$			5.13595i			0.442032i
$136$		0			0
$137$	−10.8354			−0.925731			−0.462866	−	0.886428i	$-0.653179\pi$
$137$	−10.8354			−0.925731			−0.462866	+	0.886428i	$0.653179\pi$
$138$		0			0
$139$	3.33243			0.282653			0.141326	−	0.989963i	$-0.454863\pi$
$139$	3.33243			0.282653			0.141326	+	0.989963i	$0.454863\pi$
$140$		0			0
$141$	−6.04239			−0.508861
$142$		0			0
$143$			25.2746i			2.11357i
$144$		0			0
$145$	11.6277			0.965627
$146$		0			0
$147$	3.25395			0.268382
$148$		0			0
$149$	6.87332			0.563084			0.281542	−	0.959549i	$-0.409154\pi$
$149$	6.87332			0.563084			0.281542	+	0.959549i	$0.409154\pi$
$150$		0			0
$151$	17.6036			1.43256			0.716279	−	0.697814i	$-0.245844\pi$
$151$	17.6036			1.43256			0.716279	+	0.697814i	$0.245844\pi$
$152$		0			0
$153$			15.9264i			1.28757i
$154$		0			0
$155$	−9.38501			−0.753823
$156$		0			0
$157$	−9.04524			−0.721889			−0.360944	−	0.932587i	$-0.617546\pi$
$157$	−9.04524			−0.721889			−0.360944	+	0.932587i	$0.617546\pi$
$158$		0			0
$159$	−4.37522			−0.346978
$160$		0			0
$161$			5.88382i			0.463710i
$162$		0			0
$163$		−	2.77074i		−	0.217021i	−0.994095	−	0.108511i	$-0.965392\pi$
$163$		−	2.77074i		−	0.217021i	0.994095	−	0.108511i	$-0.0346082\pi$
$164$		0			0
$165$		−	5.73552i		−	0.446510i
$166$		0			0
$167$		−	15.6798i		−	1.21334i	−0.794955	−	0.606669i	$-0.792505\pi$
$167$		−	15.6798i		−	1.21334i	0.794955	−	0.606669i	$-0.207495\pi$
$168$		0			0
$169$	−3.93054			−0.302349
$170$		0			0
$171$			3.18427i			0.243507i
$172$		0			0
$173$	−22.2065			−1.68833			−0.844163	−	0.536087i	$-0.819902\pi$
$173$	−22.2065			−1.68833			−0.844163	+	0.536087i	$0.819902\pi$
$174$		0			0
$175$	−5.03839			−0.380867
$176$		0			0
$177$		−	9.41846i		−	0.707934i
$178$		0			0
$179$			25.0451i			1.87196i	0.352057	+	0.935979i	$0.385482\pi$
$179$			25.0451i			1.87196i	−0.352057	+	0.935979i	$0.614518\pi$
$180$		0			0
$181$	−1.92137			−0.142814			−0.0714070	−	0.997447i	$-0.522749\pi$
$181$	−1.92137			−0.142814			−0.0714070	+	0.997447i	$0.522749\pi$
$182$		0			0
$183$			1.48664i			0.109895i
$184$		0			0
$185$	−5.49857	−	5.85988i	−0.404263	−	0.430827i
$186$		0			0
$187$		−	39.1248i		−	2.86109i
$188$		0			0
$189$	−6.01819			−0.437759
$190$		0			0
$191$		−	1.08096i		−	0.0782157i	−0.999235	−	0.0391079i	$-0.987548\pi$
$191$		−	1.08096i		−	0.0782157i	0.999235	−	0.0391079i	$-0.0124516\pi$
$192$		0			0
$193$		−	10.7676i		−	0.775068i	−0.921855	−	0.387534i	$-0.873327\pi$
$193$		−	10.7676i		−	0.775068i	0.921855	−	0.387534i	$-0.126673\pi$
$194$		0			0
$195$	3.84202			0.275133
$196$		0			0
$197$	−4.90587			−0.349529			−0.174764	−	0.984610i	$-0.555916\pi$
$197$	−4.90587			−0.349529			−0.174764	+	0.984610i	$0.555916\pi$
$198$		0			0
$199$			12.6027i			0.893383i	0.894688	+	0.446692i	$0.147398\pi$
$199$			12.6027i			0.893383i	−0.894688	+	0.446692i	$0.852602\pi$
$200$		0			0
$201$	−1.48664			−0.104859
$202$		0			0
$203$			13.6251i			0.956292i
$204$		0			0
$205$		−	5.19879i		−	0.363099i
$206$		0			0
$207$			9.50395i			0.660571i
$208$		0			0
$209$		−	7.82249i		−	0.541093i
$210$		0			0
$211$	13.8738			0.955112			0.477556	−	0.878601i	$-0.341523\pi$
$211$	13.8738			0.955112			0.477556	+	0.878601i	$0.341523\pi$
$212$		0			0
$213$	−4.82908			−0.330883
$214$		0			0
$215$	−12.9241			−0.881413
$216$		0			0
$217$		−	10.9971i		−	0.746535i
$218$		0			0
$219$	−4.79546			−0.324047
$220$		0			0
$221$	26.2083			1.76296
$222$		0			0
$223$	6.45201			0.432059			0.216029	−	0.976387i	$-0.430689\pi$
$223$	6.45201			0.432059			0.216029	+	0.976387i	$0.430689\pi$
$224$		0			0
$225$	−8.13837			−0.542558
$226$		0			0
$227$			3.90687i			0.259308i	0.991559	+	0.129654i	$0.0413867\pi$
$227$			3.90687i			0.259308i	−0.991559	+	0.129654i	$0.958613\pi$
$228$		0			0
$229$	5.96560			0.394218			0.197109	−	0.980382i	$-0.436845\pi$
$229$	5.96560			0.394218			0.197109	+	0.980382i	$0.436845\pi$
$230$		0			0
$231$	6.72076			0.442193
$232$		0			0
$233$	−0.652916			−0.0427740			−0.0213870	−	0.999771i	$-0.506808\pi$
$233$	−0.652916			−0.0427740			−0.0213870	+	0.999771i	$0.506808\pi$
$234$		0			0
$235$			11.2935i			0.736709i
$236$		0			0
$237$			0.266165i			0.0172893i
$238$		0			0
$239$		−	24.8471i		−	1.60722i	−0.595153	−	0.803612i	$-0.702909\pi$
$239$		−	24.8471i		−	1.60722i	0.595153	−	0.803612i	$-0.297091\pi$
$240$		0			0
$241$			26.5113i			1.70774i	0.520485	+	0.853871i	$0.325751\pi$
$241$			26.5113i			1.70774i	−0.520485	+	0.853871i	$0.674249\pi$
$242$		0			0
$243$	−15.0230			−0.963723
$244$		0			0
$245$		−	6.08181i		−	0.388552i
$246$		0			0
$247$	5.24001			0.333414
$248$		0			0
$249$	−5.76100			−0.365088
$250$		0			0
$251$			19.1625i			1.20952i	0.796406	+	0.604762i	$0.206732\pi$
$251$			19.1625i			1.20952i	−0.796406	+	0.604762i	$0.793268\pi$
$252$		0			0
$253$		−	23.3474i		−	1.46784i
$254$		0			0
$255$	−5.94741			−0.372441
$256$		0			0
$257$		−	13.1143i		−	0.818049i	−0.912523	−	0.409025i	$-0.865869\pi$
$257$		−	13.1143i		−	0.818049i	0.912523	−	0.409025i	$-0.134131\pi$
$258$		0			0
$259$	6.86647	−	6.44311i	0.426662	−	0.400355i
$260$		0			0
$261$			22.0082i			1.36227i
$262$		0			0
$263$	4.07224			0.251105			0.125553	−	0.992087i	$-0.459930\pi$
$263$	4.07224			0.251105			0.125553	+	0.992087i	$0.459930\pi$
$264$		0			0
$265$			8.17751i			0.502341i
$266$		0			0
$267$		−	8.36779i		−	0.512100i
$268$		0			0
$269$	23.0798			1.40720			0.703599	−	0.710597i	$-0.251575\pi$
$269$	23.0798			1.40720			0.703599	+	0.710597i	$0.251575\pi$
$270$		0			0
$271$	20.7224			1.25880			0.629399	−	0.777082i	$-0.283301\pi$
$271$	20.7224			1.25880			0.629399	+	0.777082i	$0.283301\pi$
$272$		0			0
$273$			4.50200i			0.272473i
$274$		0			0
$275$	19.9927			1.20561
$276$		0			0
$277$			22.7308i			1.36576i	0.730531	+	0.682880i	$0.239273\pi$
$277$			22.7308i			1.36576i	−0.730531	+	0.682880i	$0.760727\pi$
$278$		0			0
$279$		−	17.7634i		−	1.06347i
$280$		0			0
$281$			11.7198i			0.699142i	0.936910	+	0.349571i	$0.113673\pi$
$281$			11.7198i			0.699142i	−0.936910	+	0.349571i	$0.886327\pi$
$282$		0			0
$283$		−	2.60104i		−	0.154616i	−0.997007	−	0.0773078i	$-0.975368\pi$
$283$		−	2.60104i		−	0.154616i	0.997007	−	0.0773078i	$-0.0246324\pi$
$284$		0			0
$285$	−1.18911			−0.0704367
$286$		0			0
$287$	6.09182			0.359589
$288$		0			0
$289$	−23.5702			−1.38648
$290$		0			0
$291$		−	5.78531i		−	0.339141i
$292$		0			0
$293$	29.9349			1.74882			0.874408	−	0.485192i	$-0.161250\pi$
$293$	29.9349			1.74882			0.874408	+	0.485192i	$0.161250\pi$
$294$		0			0
$295$	−17.6036			−1.02492
$296$		0			0
$297$	23.8806			1.38570
$298$		0			0
$299$	15.6396			0.904463
$300$		0			0
$301$		−	15.1441i		−	0.872893i
$302$		0			0
$303$	−3.94779			−0.226795
$304$		0			0
$305$	2.77860			0.159102
$306$		0			0
$307$	3.42257			0.195336			0.0976681	−	0.995219i	$-0.468862\pi$
$307$	3.42257			0.195336			0.0976681	+	0.995219i	$0.468862\pi$
$308$		0			0
$309$			0.289929i			0.0164935i
$310$		0			0
$311$		−	11.4332i		−	0.648315i	−0.946003	−	0.324157i	$-0.894919\pi$
$311$		−	11.4332i		−	0.648315i	0.946003	−	0.324157i	$-0.105081\pi$
$312$		0			0
$313$			7.09207i			0.400868i	0.979707	+	0.200434i	$0.0642352\pi$
$313$			7.09207i			0.400868i	−0.979707	+	0.200434i	$0.935765\pi$
$314$		0			0
$315$			5.11333i			0.288104i
$316$		0			0
$317$	−21.9858			−1.23485			−0.617423	−	0.786632i	$-0.711823\pi$
$317$	−21.9858			−1.23485			−0.617423	+	0.786632i	$0.711823\pi$
$318$		0			0
$319$		−	54.0653i		−	3.02708i
$320$		0			0
$321$	−6.73192			−0.375739
$322$		0			0
$323$	−8.11148			−0.451335
$324$		0			0
$325$			13.3924i			0.742879i
$326$		0			0
$327$		−	6.81849i		−	0.377063i
$328$		0			0
$329$	−13.2335			−0.729587
$330$		0			0
$331$		−	7.30214i		−	0.401362i	−0.979657	−	0.200681i	$-0.935685\pi$
$331$		−	7.30214i		−	0.401362i	0.979657	−	0.200681i	$-0.0643155\pi$
$332$		0			0
$333$	11.0912	−	10.4074i	0.607794	−	0.570320i
$334$		0			0
$335$			2.77860i			0.151811i
$336$		0			0
$337$	−23.8660			−1.30006			−0.650032	−	0.759907i	$-0.725244\pi$
$337$	−23.8660			−1.30006			−0.650032	+	0.759907i	$0.725244\pi$
$338$		0			0
$339$			5.78531i			0.314215i
$340$		0			0
$341$			43.6376i			2.36311i
$342$		0			0
$343$	17.9624			0.969881
$344$		0			0
$345$	−3.54908			−0.191076
$346$		0			0
$347$		−	5.00084i		−	0.268459i	−0.990950	−	0.134230i	$-0.957144\pi$
$347$		−	5.00084i		−	0.268459i	0.990950	−	0.134230i	$-0.0428560\pi$
$348$		0			0
$349$	21.1745			1.13344			0.566721	−	0.823910i	$-0.308212\pi$
$349$	21.1745			1.13344			0.566721	+	0.823910i	$0.308212\pi$
$350$		0			0
$351$			15.9968i			0.853846i
$352$		0			0
$353$		−	26.2449i		−	1.39688i	−0.715670	−	0.698439i	$-0.753879\pi$
$353$		−	26.2449i		−	1.39688i	0.715670	−	0.698439i	$-0.246121\pi$
$354$		0			0
$355$			9.02580i			0.479040i
$356$		0			0
$357$		−	6.96905i		−	0.368841i
$358$		0			0
$359$	−26.4542			−1.39620			−0.698098	−	0.716002i	$-0.745970\pi$
$359$	−26.4542			−1.39620			−0.698098	+	0.716002i	$0.745970\pi$
$360$		0			0
$361$	17.3782			0.914643
$362$		0			0
$363$	−18.8936			−0.991656
$364$		0			0
$365$			8.96296i			0.469143i
$366$		0			0
$367$	−34.2525			−1.78797			−0.893984	−	0.448099i	$-0.852101\pi$
$367$	−34.2525			−1.78797			−0.893984	+	0.448099i	$0.852101\pi$
$368$		0			0
$369$	9.83994			0.512247
$370$		0			0
$371$	−9.58223			−0.497484
$372$		0			0
$373$	−12.1096			−0.627013			−0.313506	−	0.949586i	$-0.601504\pi$
$373$	−12.1096			−0.627013			−0.313506	+	0.949586i	$0.601504\pi$
$374$		0			0
$375$		−	7.70781i		−	0.398030i
$376$		0			0
$377$	36.2165			1.86524
$378$		0			0
$379$	−30.4497			−1.56410			−0.782049	−	0.623217i	$-0.785825\pi$
$379$	−30.4497			−1.56410			−0.782049	+	0.623217i	$0.785825\pi$
$380$		0			0
$381$	1.25282			0.0641838
$382$		0			0
$383$			21.5104i			1.09913i	0.835450	+	0.549566i	$0.185207\pi$
$383$			21.5104i			1.09913i	−0.835450	+	0.549566i	$0.814793\pi$
$384$		0			0
$385$		−	12.5614i		−	0.640190i
$386$		0			0
$387$		−	24.4618i		−	1.24347i
$388$		0			0
$389$		−	10.3805i		−	0.526311i	−0.964753	−	0.263156i	$-0.915237\pi$
$389$		−	10.3805i		−	0.526311i	0.964753	−	0.263156i	$-0.0847633\pi$
$390$		0			0
$391$	−24.2100			−1.22435
$392$		0			0
$393$			2.89843i			0.146206i
$394$		0			0
$395$	0.497476			0.0250307
$396$		0			0
$397$	−22.1592			−1.11214			−0.556070	−	0.831135i	$-0.687691\pi$
$397$	−22.1592			−1.11214			−0.556070	+	0.831135i	$0.687691\pi$
$398$		0			0
$399$		−	1.39337i		−	0.0697557i
$400$		0			0
$401$			16.7822i			0.838065i	0.907971	+	0.419033i	$0.137631\pi$
$401$			16.7822i			0.838065i	−0.907971	+	0.419033i	$0.862369\pi$
$402$		0			0
$403$	−29.2313			−1.45611
$404$		0			0
$405$			6.27950i			0.312031i
$406$		0			0
$407$	−27.2467	+	25.5668i	−1.35057	+	1.26730i
$408$		0			0
$409$		−	8.90771i		−	0.440458i	−0.975448	−	0.220229i	$-0.929320\pi$
$409$		−	8.90771i		−	0.440458i	0.975448	−	0.220229i	$-0.0706805\pi$
$410$		0			0
$411$	7.65856			0.377769
$412$		0			0
$413$		−	20.6275i		−	1.01501i
$414$		0			0
$415$			10.7676i			0.528560i
$416$		0			0
$417$	−2.35539			−0.115344
$418$		0			0
$419$	8.89676			0.434635			0.217318	−	0.976101i	$-0.430269\pi$
$419$	8.89676			0.434635			0.217318	+	0.976101i	$0.430269\pi$
$420$		0			0
$421$		−	19.9102i		−	0.970364i	−0.874413	−	0.485182i	$-0.838753\pi$
$421$		−	19.9102i		−	0.970364i	0.874413	−	0.485182i	$-0.161247\pi$
$422$		0			0
$423$	−21.3757			−1.03932
$424$		0			0
$425$		−	20.7313i		−	1.00562i
$426$		0			0
$427$			3.25590i			0.157564i
$428$		0			0
$429$		−	17.8643i		−	0.862495i
$430$		0			0
$431$			22.8250i			1.09944i	0.835348	+	0.549721i	$0.185266\pi$
$431$			22.8250i			1.09944i	−0.835348	+	0.549721i	$0.814734\pi$
$432$		0			0
$433$	−33.5110			−1.61044			−0.805218	−	0.592979i	$-0.797952\pi$
$433$	−33.5110			−1.61044			−0.805218	+	0.592979i	$0.797952\pi$
$434$		0			0
$435$	−8.21855			−0.394049
$436$		0			0
$437$	−4.84047			−0.231551
$438$		0			0
$439$		−	17.4260i		−	0.831698i	−0.909434	−	0.415849i	$-0.863484\pi$
$439$		−	17.4260i		−	0.831698i	0.909434	−	0.415849i	$-0.136516\pi$
$440$		0			0
$441$	11.5113			0.548155
$442$		0			0
$443$	12.5732			0.597371			0.298686	−	0.954352i	$-0.403452\pi$
$443$	12.5732			0.597371			0.298686	+	0.954352i	$0.403452\pi$
$444$		0			0
$445$	−15.6398			−0.741398
$446$		0			0
$447$	−4.85812			−0.229781
$448$		0			0
$449$		−	12.8428i		−	0.606090i	−0.952976	−	0.303045i	$-0.901997\pi$
$449$		−	12.8428i		−	0.606090i	0.952976	−	0.303045i	$-0.0980033\pi$
$450$		0			0
$451$	−24.1728			−1.13825
$452$		0			0
$453$	−12.4424			−0.584593
$454$		0			0
$455$	8.41446			0.394476
$456$		0			0
$457$		−	30.5200i		−	1.42767i	−0.700316	−	0.713833i	$-0.746957\pi$
$457$		−	30.5200i		−	1.42767i	0.700316	−	0.713833i	$-0.253043\pi$
$458$		0			0
$459$		−	24.7629i		−	1.15583i
$460$		0			0
$461$			19.5586i			0.910934i	0.890253	+	0.455467i	$0.150528\pi$
$461$			19.5586i			0.910934i	−0.890253	+	0.455467i	$0.849472\pi$
$462$		0			0
$463$		−	12.7087i		−	0.590621i	−0.955401	−	0.295311i	$-0.904577\pi$
$463$		−	12.7087i		−	0.590621i	0.955401	−	0.295311i	$-0.0954231\pi$
$464$		0			0
$465$	6.63341			0.307617
$466$		0			0
$467$		−	1.02234i		−	0.0473085i	−0.999720	−	0.0236542i	$-0.992470\pi$
$467$		−	1.02234i		−	0.0473085i	0.999720	−	0.0236542i	$-0.00753008\pi$
$468$		0			0
$469$	−3.25590			−0.150343
$470$		0			0
$471$	6.39325			0.294586
$472$		0			0
$473$			60.0931i			2.76308i
$474$		0			0
$475$		−	4.14496i		−	0.190184i
$476$		0			0
$477$	−15.4779			−0.708684
$478$		0			0
$479$			21.5439i			0.984365i	0.870492	+	0.492183i	$0.163801\pi$
$479$			21.5439i			0.984365i	−0.870492	+	0.492183i	$0.836199\pi$
$480$		0			0
$481$	−17.1263	−	18.2516i	−0.780891	−	0.832201i
$482$		0			0
$483$		−	4.15873i		−	0.189229i
$484$		0			0
$485$	−10.8130			−0.490995
$486$		0			0
$487$		−	20.2628i		−	0.918197i	−0.888385	−	0.459098i	$-0.848173\pi$
$487$		−	20.2628i		−	0.918197i	0.888385	−	0.459098i	$-0.151827\pi$
$488$		0			0
$489$			1.95838i			0.0885612i
$490$		0			0
$491$	10.4790			0.472910			0.236455	−	0.971642i	$-0.424014\pi$
$491$	10.4790			0.472910			0.236455	+	0.971642i	$0.424014\pi$
$492$		0			0
$493$	−56.0627			−2.52494
$494$		0			0
$495$		−	20.2901i		−	0.911972i
$496$		0			0
$497$	−10.5762			−0.474409
$498$		0			0
$499$		−	40.8321i		−	1.82789i	−0.405833	−	0.913947i	$-0.633018\pi$
$499$		−	40.8321i		−	1.82789i	0.405833	−	0.913947i	$-0.366982\pi$
$500$		0			0
$501$			11.0826i			0.495134i
$502$		0			0
$503$		−	19.3411i		−	0.862377i	−0.902262	−	0.431188i	$-0.858094\pi$
$503$		−	19.3411i		−	0.862377i	0.902262	−	0.431188i	$-0.141906\pi$
$504$		0			0
$505$			7.37863i			0.328345i
$506$		0			0
$507$	2.77814			0.123382
$508$		0			0
$509$	−4.71436			−0.208960			−0.104480	−	0.994527i	$-0.533318\pi$
$509$	−4.71436			−0.208960			−0.104480	+	0.994527i	$0.533318\pi$
$510$		0			0
$511$	−10.5026			−0.464607
$512$		0			0
$513$		−	4.95102i		−	0.218593i
$514$		0			0
$515$	0.541892			0.0238786
$516$		0			0
$517$	52.5116			2.30946
$518$		0			0
$519$	15.6957			0.688965
$520$		0			0
$521$	−0.317478			−0.0139090			−0.00695448	−	0.999976i	$-0.502214\pi$
$521$	−0.317478			−0.0139090			−0.00695448	+	0.999976i	$0.502214\pi$
$522$		0			0
$523$			11.0979i			0.485276i	0.970117	+	0.242638i	$0.0780128\pi$
$523$			11.0979i			0.485276i	−0.970117	+	0.242638i	$0.921987\pi$
$524$		0			0
$525$	3.56118			0.155423
$526$		0			0
$527$	45.2497			1.97111
$528$		0			0
$529$	8.55285			0.371863
$530$		0			0
$531$		−	33.3189i		−	1.44592i
$532$		0			0
$533$		−	16.1925i		−	0.701376i
$534$		0			0
$535$			12.5823i			0.543980i
$536$		0			0
$537$		−	17.7021i		−	0.763901i
$538$		0			0
$539$	−28.2786			−1.21805
$540$		0			0
$541$		−	38.6571i		−	1.66200i	−0.556272	−	0.831000i	$-0.687769\pi$
$541$		−	38.6571i		−	1.66200i	0.556272	−	0.831000i	$-0.312231\pi$
$542$		0			0
$543$	1.35804			0.0582790
$544$		0			0
$545$	−12.7441			−0.545897
$546$		0			0
$547$			17.8146i			0.761696i	0.924638	+	0.380848i	$0.124368\pi$
$547$			17.8146i			0.761696i	−0.924638	+	0.380848i	$0.875632\pi$
$548$		0			0
$549$			5.25915i			0.224455i
$550$		0			0
$551$	−11.2090			−0.477520
$552$		0			0
$553$			0.582931i			0.0247888i
$554$		0			0
$555$	3.88644	+	4.14181i	0.164970	+	0.175810i
$556$		0			0
$557$		−	18.1282i		−	0.768116i	−0.923309	−	0.384058i	$-0.874526\pi$
$557$		−	18.1282i		−	0.768116i	0.923309	−	0.384058i	$-0.125474\pi$
$558$		0			0
$559$	−40.2542			−1.70257
$560$		0			0
$561$			27.6537i			1.16754i
$562$		0			0
$563$		−	30.9722i		−	1.30532i	−0.757650	−	0.652661i	$-0.773653\pi$
$563$		−	30.9722i		−	1.30532i	0.757650	−	0.652661i	$-0.226347\pi$
$564$		0			0
$565$	10.8130			0.454908
$566$		0			0
$567$	−7.35817			−0.309014
$568$		0			0
$569$		−	9.41846i		−	0.394842i	−0.980319	−	0.197421i	$-0.936743\pi$
$569$		−	9.41846i		−	0.394842i	0.980319	−	0.197421i	$-0.0632566\pi$
$570$		0			0
$571$	−10.1304			−0.423945			−0.211973	−	0.977276i	$-0.567989\pi$
$571$	−10.1304			−0.423945			−0.211973	+	0.977276i	$0.567989\pi$
$572$		0			0
$573$			0.764034i			0.0319180i
$574$		0			0
$575$		−	12.3713i		−	0.515919i
$576$		0			0
$577$			9.02580i			0.375749i	0.982193	+	0.187874i	$0.0601598\pi$
$577$			9.02580i			0.375749i	−0.982193	+	0.187874i	$0.939840\pi$
$578$		0			0
$579$			7.61063i			0.316287i
$580$		0			0
$581$	−12.6172			−0.523451
$582$		0			0
$583$	38.0230			1.57475
$584$		0			0
$585$	13.5916			0.561945
$586$		0			0
$587$		−	14.7935i		−	0.610592i	−0.952257	−	0.305296i	$-0.901245\pi$
$587$		−	14.7935i		−	0.610592i	0.952257	−	0.305296i	$-0.0987554\pi$
$588$		0			0
$589$	9.04709			0.372779
$590$		0			0
$591$	3.46751			0.142634
$592$		0			0
$593$	35.4170			1.45440			0.727200	−	0.686425i	$-0.240821\pi$
$593$	35.4170			1.45440			0.727200	+	0.686425i	$0.240821\pi$
$594$		0			0
$595$	−13.0255			−0.533993
$596$		0			0
$597$		−	8.90771i		−	0.364568i
$598$		0			0
$599$	24.3447			0.994697			0.497348	−	0.867551i	$-0.334307\pi$
$599$	24.3447			0.994697			0.497348	+	0.867551i	$0.334307\pi$
$600$		0			0
$601$	31.8342			1.29855			0.649273	−	0.760556i	$-0.275073\pi$
$601$	31.8342			1.29855			0.649273	+	0.760556i	$0.275073\pi$
$602$		0			0
$603$	−5.25915			−0.214169
$604$		0			0
$605$			35.3131i			1.43568i
$606$		0			0
$607$		−	15.2175i		−	0.617659i	−0.951117	−	0.308830i	$-0.900063\pi$
$607$		−	15.2175i		−	0.617659i	0.951117	−	0.308830i	$-0.0999373\pi$
$608$		0			0
$609$		−	9.63032i		−	0.390240i
$610$		0			0
$611$			35.1757i			1.42305i
$612$		0			0
$613$	0.175456			0.00708658			0.00354329	−	0.999994i	$-0.498872\pi$
$613$	0.175456			0.00708658			0.00354329	+	0.999994i	$0.498872\pi$
$614$		0			0
$615$			3.67455i			0.148172i
$616$		0			0
$617$	6.25295			0.251734			0.125867	−	0.992047i	$-0.459829\pi$
$617$	6.25295			0.251734			0.125867	+	0.992047i	$0.459829\pi$
$618$		0			0
$619$	9.87749			0.397010			0.198505	−	0.980100i	$-0.436391\pi$
$619$	9.87749			0.397010			0.198505	+	0.980100i	$0.436391\pi$
$620$		0			0
$621$		−	14.7771i		−	0.592984i
$622$		0			0
$623$		−	18.3264i		−	0.734231i
$624$		0			0
$625$	1.86770			0.0747080
$626$		0			0
$627$			5.52900i			0.220807i
$628$		0			0
$629$	26.5113	+	28.2533i	1.05707	+	1.12653i
$630$		0			0
$631$			8.30179i			0.330489i	0.986253	+	0.165245i	$0.0528414\pi$
$631$			8.30179i			0.330489i	−0.986253	+	0.165245i	$0.947159\pi$
$632$		0			0
$633$	−9.80612			−0.389758
$634$		0			0
$635$		−	2.34158i		−	0.0929227i
$636$		0			0
$637$		−	18.9428i		−	0.750543i
$638$		0			0
$639$	−17.0835			−0.675811
$640$		0			0
$641$	−4.55269			−0.179821			−0.0899103	−	0.995950i	$-0.528658\pi$
$641$	−4.55269			−0.179821			−0.0899103	+	0.995950i	$0.528658\pi$
$642$		0			0
$643$			18.0427i			0.711534i	0.934575	+	0.355767i	$0.115780\pi$
$643$			18.0427i			0.711534i	−0.934575	+	0.355767i	$0.884220\pi$
$644$		0			0
$645$	9.13483			0.359684
$646$		0			0
$647$			3.53345i			0.138914i	0.997585	+	0.0694571i	$0.0221267\pi$
$647$			3.53345i			0.138914i	−0.997585	+	0.0694571i	$0.977873\pi$
$648$		0			0
$649$			81.8514i			3.21295i
$650$		0			0
$651$			7.77288i			0.304643i
$652$		0			0
$653$			27.0773i			1.05962i	0.848117	+	0.529809i	$0.177736\pi$
$653$			27.0773i			1.05962i	−0.848117	+	0.529809i	$0.822264\pi$
$654$		0			0
$655$	5.41731			0.211672
$656$		0			0
$657$	−16.9645			−0.661849
$658$		0			0
$659$	−19.4084			−0.756042			−0.378021	−	0.925797i	$-0.623395\pi$
$659$	−19.4084			−0.756042			−0.378021	+	0.925797i	$0.623395\pi$
$660$		0			0
$661$			5.19746i			0.202158i	0.994878	+	0.101079i	$0.0322295\pi$
$661$			5.19746i			0.202158i	−0.994878	+	0.101079i	$0.967771\pi$
$662$		0			0
$663$	−18.5243			−0.719422
$664$		0			0
$665$	−2.60428			−0.100990
$666$		0			0
$667$	−33.4551			−1.29539
$668$		0			0
$669$	−4.56034			−0.176313
$670$		0			0
$671$		−	12.9197i		−	0.498758i
$672$		0			0
$673$	−16.2646			−0.626956			−0.313478	−	0.949596i	$-0.601494\pi$
$673$	−16.2646			−0.626956			−0.313478	+	0.949596i	$0.601494\pi$
$674$		0			0
$675$	12.6538			0.487046
$676$		0			0
$677$	19.1484			0.735933			0.367967	−	0.929839i	$-0.380054\pi$
$677$	19.1484			0.735933			0.367967	+	0.929839i	$0.380054\pi$
$678$		0			0
$679$		−	12.6705i		−	0.486248i
$680$		0			0
$681$		−	2.76141i		−	0.105817i
$682$		0			0
$683$		−	33.5559i		−	1.28398i	−0.766713	−	0.641990i	$-0.778109\pi$
$683$		−	33.5559i		−	1.28398i	0.766713	−	0.641990i	$-0.221891\pi$
$684$		0			0
$685$		−	14.3142i		−	0.546919i
$686$		0			0
$687$	−4.21654			−0.160871
$688$		0			0
$689$			25.4703i			0.970341i
$690$		0			0
$691$	−34.1132			−1.29773			−0.648863	−	0.760905i	$-0.724755\pi$
$691$	−34.1132			−1.29773			−0.648863	+	0.760905i	$0.724755\pi$
$692$		0			0
$693$	23.7755			0.903156
$694$		0			0
$695$			4.40234i			0.166990i
$696$		0			0
$697$			25.0659i			0.949437i
$698$		0			0
$699$	0.461487			0.0174550
$700$		0			0
$701$		−	5.58732i		−	0.211030i	−0.994418	−	0.105515i	$-0.966351\pi$
$701$		−	5.58732i		−	0.211030i	0.994418	−	0.105515i	$-0.0336491\pi$
$702$		0			0
$703$	5.30059	+	5.64888i	0.199915	+	0.213051i
$704$		0			0
$705$		−	7.98236i		−	0.300633i
$706$		0			0
$707$	−8.64611			−0.325170
$708$		0			0
$709$			27.8996i			1.04779i	0.851783	+	0.523895i	$0.175522\pi$
$709$			27.8996i			1.04779i	−0.851783	+	0.523895i	$0.824478\pi$
$710$		0			0
$711$			0.941592i			0.0353124i
$712$		0			0
$713$	27.0025			1.01125
$714$		0			0
$715$	−33.3892			−1.24869
$716$		0			0
$717$			17.5621i			0.655870i
$718$		0			0
$719$	−24.6403			−0.918927			−0.459464	−	0.888197i	$-0.651958\pi$
$719$	−24.6403			−0.918927			−0.459464	+	0.888197i	$0.651958\pi$
$720$		0			0
$721$			0.634977i			0.0236478i
$722$		0			0
$723$		−	18.7384i		−	0.696889i
$724$		0			0
$725$		−	28.6480i		−	1.06396i
$726$		0			0
$727$			0.137756i			0.00510910i	0.999997	+	0.00255455i	$0.000813140\pi$
$727$			0.137756i			0.00510910i	−0.999997	+	0.00255455i	$0.999187\pi$
$728$		0			0
$729$	−3.64176			−0.134880
$730$		0			0
$731$	62.3131			2.30473
$732$		0			0
$733$	24.1639			0.892514			0.446257	−	0.894905i	$-0.352757\pi$
$733$	24.1639			0.892514			0.446257	+	0.894905i	$0.352757\pi$
$734$		0			0
$735$			4.29867i			0.158559i
$736$		0			0
$737$	12.9197			0.475902
$738$		0			0
$739$	38.2540			1.40720			0.703598	−	0.710598i	$-0.251575\pi$
$739$	38.2540			1.40720			0.703598	+	0.710598i	$0.251575\pi$
$740$		0			0
$741$	−3.70368			−0.136058
$742$		0			0
$743$	33.6707			1.23526			0.617629	−	0.786470i	$-0.288093\pi$
$743$	33.6707			1.23526			0.617629	+	0.786470i	$0.288093\pi$
$744$		0			0
$745$			9.08007i			0.332668i
$746$		0			0
$747$	−20.3802			−0.745673
$748$		0			0
$749$	−14.7436			−0.538721
$750$		0			0
$751$	26.3427			0.961258			0.480629	−	0.876924i	$-0.340408\pi$
$751$	26.3427			0.961258			0.480629	+	0.876924i	$0.340408\pi$
$752$		0			0
$753$		−	13.5442i		−	0.493578i
$754$		0			0
$755$			23.2554i			0.846350i
$756$		0			0
$757$		−	13.8231i		−	0.502408i	−0.967934	−	0.251204i	$-0.919174\pi$
$757$		−	13.8231i		−	0.502408i	0.967934	−	0.251204i	$-0.0808264\pi$
$758$		0			0
$759$			16.5022i			0.598991i
$760$		0			0
$761$	−20.0226			−0.725818			−0.362909	−	0.931825i	$-0.618216\pi$
$761$	−20.0226			−0.725818			−0.362909	+	0.931825i	$0.618216\pi$
$762$		0			0
$763$		−	14.9333i		−	0.540620i
$764$		0			0
$765$	−21.0397			−0.760692
$766$		0			0
$767$	−54.8294			−1.97977
$768$		0			0
$769$		−	47.3669i		−	1.70809i	−0.520197	−	0.854046i	$-0.674141\pi$
$769$		−	47.3669i		−	1.70809i	0.520197	−	0.854046i	$-0.325859\pi$
$770$		0			0
$771$			9.26932i			0.333826i
$772$		0			0
$773$	0.827395			0.0297593			0.0148797	−	0.999889i	$-0.495263\pi$
$773$	0.827395			0.0297593			0.0148797	+	0.999889i	$0.495263\pi$
$774$		0			0
$775$			23.1226i			0.830587i
$776$		0			0
$777$	−4.85328	+	4.55404i	−0.174110	+	0.163375i
$778$		0			0
$779$			5.01159i			0.179559i
$780$		0			0
$781$	41.9673			1.50171
$782$		0			0
$783$		−	34.2191i		−	1.22289i
$784$		0			0
$785$		−	11.9493i		−	0.426489i
$786$		0			0
$787$	−12.7860			−0.455771			−0.227886	−	0.973688i	$-0.573181\pi$
$787$	−12.7860			−0.455771			−0.227886	+	0.973688i	$0.573181\pi$
$788$		0			0
$789$	−2.87830			−0.102470
$790$		0			0
$791$			12.6705i			0.450510i
$792$		0			0
$793$	8.65442			0.307328
$794$		0			0
$795$		−	5.77994i		−	0.204993i
$796$		0			0
$797$		−	14.6856i		−	0.520191i	−0.965583	−	0.260095i	$-0.916246\pi$
$797$		−	14.6856i		−	0.520191i	0.965583	−	0.260095i	$-0.0837540\pi$
$798$		0			0
$799$		−	54.4516i		−	1.92636i
$800$		0			0
$801$		−	29.6021i		−	1.04594i
$802$		0			0
$803$	41.6751			1.47068
$804$		0			0
$805$	−7.77288			−0.273958
$806$		0			0
$807$	−16.3130			−0.574244
$808$		0			0
$809$		−	23.9780i		−	0.843021i	−0.906823	−	0.421511i	$-0.861500\pi$
$809$		−	23.9780i		−	0.843021i	0.906823	−	0.421511i	$-0.138500\pi$
$810$		0			0
$811$	−25.9890			−0.912596			−0.456298	−	0.889827i	$-0.650825\pi$
$811$	−25.9890			−0.912596			−0.456298	+	0.889827i	$0.650825\pi$
$812$		0			0
$813$	−14.6468			−0.513686
$814$		0			0
$815$	3.66032			0.128215
$816$		0			0
$817$	12.4587			0.435875
$818$		0			0
$819$			15.9264i			0.556512i
$820$		0			0
$821$	−48.0319			−1.67632			−0.838162	−	0.545421i	$-0.816370\pi$
$821$	−48.0319			−1.67632			−0.838162	+	0.545421i	$0.816370\pi$
$822$		0			0
$823$	−32.4229			−1.13019			−0.565096	−	0.825025i	$-0.691161\pi$
$823$	−32.4229			−1.13019			−0.565096	+	0.825025i	$0.691161\pi$
$824$		0			0
$825$	−14.1310			−0.491980
$826$		0			0
$827$		−	18.1152i		−	0.629926i	−0.949104	−	0.314963i	$-0.898008\pi$
$827$		−	18.1152i		−	0.629926i	0.949104	−	0.314963i	$-0.101992\pi$
$828$		0			0
$829$			28.9348i			1.00495i	0.864593	+	0.502473i	$0.167576\pi$
$829$			28.9348i			1.00495i	−0.864593	+	0.502473i	$0.832424\pi$
$830$		0			0
$831$		−	16.0663i		−	0.557334i
$832$		0			0
$833$			29.3233i			1.01599i
$834$		0			0
$835$	20.7139			0.716836
$836$		0			0
$837$			27.6191i			0.954657i
$838$		0			0
$839$	28.6007			0.987406			0.493703	−	0.869630i	$-0.335643\pi$
$839$	28.6007			0.987406			0.493703	+	0.869630i	$0.335643\pi$
$840$		0			0
$841$	−48.4714			−1.67143
$842$		0			0
$843$		−	8.28362i		−	0.285303i
$844$		0			0
$845$		−	5.19248i		−	0.178627i
$846$		0			0
$847$	−41.3791			−1.42180
$848$		0			0
$849$			1.83844i			0.0630950i
$850$		0			0
$851$	15.8204	+	16.8600i	0.542318	+	0.577952i
$852$		0			0
$853$			46.4969i			1.59202i	0.605281	+	0.796012i	$0.293061\pi$
$853$			46.4969i			1.59202i	−0.605281	+	0.796012i	$0.706939\pi$
$854$		0			0
$855$	−4.20661			−0.143863
$856$		0			0
$857$		−	11.7600i		−	0.401715i	−0.979620	−	0.200858i	$-0.935627\pi$
$857$		−	11.7600i		−	0.401715i	0.979620	−	0.200858i	$-0.0643728\pi$
$858$		0			0
$859$		−	10.1596i		−	0.346642i	−0.984865	−	0.173321i	$-0.944550\pi$
$859$		−	10.1596i		−	0.346642i	0.984865	−	0.173321i	$-0.0554498\pi$
$860$		0			0
$861$	−4.30575			−0.146740
$862$		0			0
$863$	56.0627			1.90840			0.954199	−	0.299174i	$-0.0967111\pi$
$863$	56.0627			1.90840			0.954199	+	0.299174i	$0.0967111\pi$
$864$		0			0
$865$		−	29.3361i		−	0.997457i
$866$		0			0
$867$	16.6596			0.565790
$868$		0			0
$869$		−	2.31312i		−	0.0784671i
$870$		0			0
$871$			8.65442i			0.293244i
$872$		0			0
$873$		−	20.4662i		−	0.692677i
$874$		0			0
$875$		−	16.8810i		−	0.570681i
$876$		0			0
$877$	11.3121			0.381984			0.190992	−	0.981592i	$-0.438830\pi$
$877$	11.3121			0.381984			0.190992	+	0.981592i	$0.438830\pi$
$878$		0			0
$879$	−21.1582			−0.713650
$880$		0			0
$881$	25.0437			0.843744			0.421872	−	0.906655i	$-0.361373\pi$
$881$	25.0437			0.843744			0.421872	+	0.906655i	$0.361373\pi$
$882$		0			0
$883$		−	39.9760i		−	1.34530i	−0.739961	−	0.672649i	$-0.765156\pi$
$883$		−	39.9760i		−	1.34530i	0.739961	−	0.672649i	$-0.234844\pi$
$884$		0			0
$885$	12.4424			0.418245
$886$		0			0
$887$	4.27374			0.143498			0.0717491	−	0.997423i	$-0.477142\pi$
$887$	4.27374			0.143498			0.0717491	+	0.997423i	$0.477142\pi$
$888$		0			0
$889$	2.74381			0.0920244
$890$		0			0
$891$	29.1978			0.978163
$892$		0			0
$893$		−	10.8869i		−	0.364316i
$894$		0			0
$895$	−33.0861			−1.10595
$896$		0			0
$897$	−11.0542			−0.369090
$898$		0			0
$899$	62.5292			2.08547
$900$		0			0
$901$		−	39.4277i		−	1.31353i
$902$		0			0
$903$			10.7040i			0.356207i
$904$		0			0
$905$		−	2.53824i		−	0.0843740i
$906$		0			0
$907$			45.9485i			1.52570i	0.646578	+	0.762848i	$0.276200\pi$
$907$			45.9485i			1.52570i	−0.646578	+	0.762848i	$0.723800\pi$
$908$		0			0
$909$	−13.9658			−0.463216
$910$		0			0
$911$			49.8522i			1.65168i	0.563907	+	0.825839i	$0.309298\pi$
$911$			49.8522i			1.65168i	−0.563907	+	0.825839i	$0.690702\pi$
$912$		0			0
$913$	50.0661			1.65695
$914$		0			0
$915$	−1.96394			−0.0649257
$916$		0			0
$917$			6.34788i			0.209626i
$918$		0			0
$919$		−	23.4603i		−	0.773884i	−0.922104	−	0.386942i	$-0.873531\pi$
$919$		−	23.4603i		−	0.773884i	0.922104	−	0.386942i	$-0.126469\pi$
$920$		0			0
$921$	−2.41910			−0.0797120
$922$		0			0
$923$			28.1124i			0.925331i
$924$		0			0
$925$	−14.4374	+	13.5473i	−0.474699	+	0.445431i
$926$		0			0
$927$			1.02566i			0.0336871i
$928$		0			0
$929$	−27.4844			−0.901733			−0.450867	−	0.892591i	$-0.648885\pi$
$929$	−27.4844			−0.901733			−0.450867	+	0.892591i	$0.648885\pi$
$930$		0			0
$931$			5.86282i			0.192146i
$932$		0			0
$933$			8.08105i			0.264562i
$934$		0			0
$935$	51.6862			1.69032
$936$		0			0
$937$	1.55340			0.0507475			0.0253738	−	0.999678i	$-0.491922\pi$
$937$	1.55340			0.0507475			0.0253738	+	0.999678i	$0.491922\pi$
$938$		0			0
$939$		−	5.01274i		−	0.163585i
$940$		0			0
$941$	−0.635730			−0.0207242			−0.0103621	−	0.999946i	$-0.503298\pi$
$941$	−0.635730			−0.0207242			−0.0103621	+	0.999946i	$0.503298\pi$
$942$		0			0
$943$			14.9579i			0.487096i
$944$		0			0
$945$		−	7.95039i		−	0.258626i
$946$		0			0
$947$		−	39.1873i		−	1.27342i	−0.771104	−	0.636709i	$-0.780295\pi$
$947$		−	39.1873i		−	1.27342i	0.771104	−	0.636709i	$-0.219705\pi$
$948$		0			0
$949$			27.9167i			0.906214i
$950$		0			0
$951$	15.5398			0.503911
$952$		0			0
$953$	−15.2500			−0.493997			−0.246998	−	0.969016i	$-0.579444\pi$
$953$	−15.2500			−0.493997			−0.246998	+	0.969016i	$0.579444\pi$
$954$		0			0
$955$	1.42802			0.0462096
$956$		0			0
$957$			38.2138i			1.23528i
$958$		0			0
$959$	16.7731			0.541631
$960$		0			0
$961$	−19.4689			−0.628031
$962$		0			0
$963$	−23.8150			−0.767427
$964$		0			0
$965$	14.2246			0.457907
$966$		0			0
$967$		−	4.84372i		−	0.155763i	−0.996963	−	0.0778817i	$-0.975184\pi$
$967$		−	4.84372i		−	0.155763i	0.996963	−	0.0778817i	$-0.0248156\pi$
$968$		0			0
$969$	5.73326			0.184179
$970$		0			0
$971$	−29.4707			−0.945759			−0.472880	−	0.881127i	$-0.656785\pi$
$971$	−29.4707			−0.945759			−0.472880	+	0.881127i	$0.656785\pi$
$972$		0			0
$973$	−5.15856			−0.165376
$974$		0			0
$975$		−	9.46589i		−	0.303151i
$976$		0			0
$977$		−	0.954601i		−	0.0305404i	−0.999883	−	0.0152702i	$-0.995139\pi$
$977$		−	0.954601i		−	0.0305404i	0.999883	−	0.0152702i	$-0.00486084\pi$
$978$		0			0
$979$			72.7205i			2.32416i
$980$		0			0
$981$		−	24.1212i		−	0.770132i
$982$		0			0
$983$	−0.484562			−0.0154551			−0.00772756	−	0.999970i	$-0.502460\pi$
$983$	−0.484562			−0.0154551			−0.00772756	+	0.999970i	$0.502460\pi$
$984$		0			0
$985$		−	6.48095i		−	0.206500i
$986$		0			0
$987$	9.35355			0.297727
$988$		0			0
$989$	37.1849			1.18241
$990$		0			0
$991$			58.8164i			1.86836i	0.356798	+	0.934182i	$0.383869\pi$
$991$			58.8164i			1.86836i	−0.356798	+	0.934182i	$0.616131\pi$
$992$		0			0
$993$			5.16121i			0.163786i
$994$		0			0
$995$	−16.6490			−0.527808
$996$		0			0
$997$		−	7.70039i		−	0.243874i	−0.992538	−	0.121937i	$-0.961089\pi$
$997$		−	7.70039i		−	0.243874i	0.992538	−	0.121937i	$-0.0389106\pi$
$998$		0			0
$999$	−17.2450	+	16.1817i	−0.545608	+	0.511967i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.g.p.961.4		10
4.3	odd	2		2368.2.g.o.961.8		10
8.3	odd	2		296.2.g.a.73.3	✓	10
8.5	even	2		592.2.g.d.369.7		10
24.5	odd	2		5328.2.h.q.2737.7		10
24.11	even	2		2664.2.h.c.73.7		10
37.36	even	2	inner	2368.2.g.p.961.3		10
148.147	odd	2		2368.2.g.o.961.7		10
296.147	odd	2		296.2.g.a.73.4	yes	10
296.221	even	2		592.2.g.d.369.8		10
888.221	odd	2		5328.2.h.q.2737.4		10
888.443	even	2		2664.2.h.c.73.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.g.a.73.3	✓	10	8.3	odd	2
296.2.g.a.73.4	yes	10	296.147	odd	2
592.2.g.d.369.7		10	8.5	even	2
592.2.g.d.369.8		10	296.221	even	2
2368.2.g.o.961.7		10	148.147	odd	2
2368.2.g.o.961.8		10	4.3	odd	2
2368.2.g.p.961.3		10	37.36	even	2	inner
2368.2.g.p.961.4		10	1.1	even	1	trivial
2664.2.h.c.73.4		10	888.443	even	2
2664.2.h.c.73.7		10	24.11	even	2
5328.2.h.q.2737.4		10	888.221	odd	2
5328.2.h.q.2737.7		10	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.706809 q^{3} +1.32106i q^{5} -1.54799 q^{7} -2.50042 q^{9} +O(q^{10})\)	\(q-0.706809 q^{3} +1.32106i q^{5} -1.54799 q^{7} -2.50042 q^{9} +6.14254 q^{11} +4.11467i q^{13} -0.933737i q^{15} -6.36947i q^{17} -1.27349i q^{19} +1.09413 q^{21} -3.80094i q^{23} +3.25480 q^{25} +3.88775 q^{27} -8.80178i q^{29} +7.10415i q^{31} -4.34160 q^{33} -2.04499i q^{35} +(-4.43573 + 4.16224i) q^{37} -2.90829i q^{39} -3.93531 q^{41} +9.78309i q^{43} -3.30321i q^{45} +8.54883 q^{47} -4.60373 q^{49} +4.50200i q^{51} +6.19011 q^{53} +8.11467i q^{55} +0.900116i q^{57} +13.3253i q^{59} -2.10331i q^{61} +3.87063 q^{63} -5.43573 q^{65} +2.10331 q^{67} +2.68654i q^{69} +6.83223 q^{71} +6.78467 q^{73} -2.30052 q^{75} -9.50859 q^{77} -0.376573i q^{79} +4.75337 q^{81} +8.15071 q^{83} +8.41446 q^{85} +6.22118i q^{87} +11.8388i q^{89} -6.36947i q^{91} -5.02127i q^{93} +1.68236 q^{95} +8.18511i q^{97} -15.3589 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 4 * q^7 + 12 * q^9	\( 10 q + 2 q^{3} + 4 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{21} + 4 q^{25} + 8 q^{27} + 4 q^{33} + 6 q^{37} - 26 q^{41} - 8 q^{47} + 14 q^{49} + 20 q^{53} + 12 q^{63} - 4 q^{65} - 2 q^{67} + 4 q^{71} - 14 q^{73} - 48 q^{75} + 18 q^{81} - 36 q^{83} - 8 q^{85} - 4 q^{95} - 28 q^{99}+O(q^{100}) \) 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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