LMFDB - Embedded newform 592.2.g.d.369.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,2,Mod(369,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, 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("592.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	592.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:	$4.72714379966$
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			369.7
Root			$1.23896 + 1.23896i$ of defining polynomial
Character	$\chi$	$=$	592.369
Dual form			592.2.g.d.369.8

$q$-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 $\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$.

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

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
$3$	0.706809			0.408076			0.204038	−	0.978963i	$-0.434593\pi$
$3$	0.706809			0.408076			0.204038	+	0.978963i	$0.434593\pi$
$4$		0			0
$5$		−	1.32106i		−	0.590796i	−0.955374	−	0.295398i	$-0.904548\pi$
$5$		−	1.32106i		−	0.590796i	0.955374	−	0.295398i	$-0.0954523\pi$
$6$		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	$-0.876795\pi$
$11$	−6.14254			−1.85205			−0.926023	+	0.377466i	$0.876795\pi$
$12$		0			0
$13$		−	4.11467i		−	1.14121i	−0.821226	−	0.570603i	$-0.806710\pi$
$13$		−	4.11467i		−	1.14121i	0.821226	−	0.570603i	$-0.193290\pi$
$14$		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	$0.0466656\pi$
$19$			1.27349i			0.292159i	−0.989273	+	0.146080i	$0.953334\pi$
$20$		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	$0.304489\pi$
$29$			8.80178i			1.63445i	−0.576319	+	0.817225i	$0.695511\pi$
$30$		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	$-0.731994\pi$
$43$		−	9.78309i		−	1.49191i	0.665998	−	0.745954i	$-0.268006\pi$
$44$		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	$-0.639775\pi$
$53$	−6.19011			−0.850277			−0.425139	+	0.905128i	$0.639775\pi$
$54$		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	$-0.665786\pi$
$59$		−	13.3253i		−	1.73481i	0.497603	−	0.867405i	$-0.334214\pi$
$60$		0			0
$61$			2.10331i			0.269301i	0.990893	+	0.134650i	$0.0429911\pi$
$61$			2.10331i			0.269301i	−0.990893	+	0.134650i	$0.957009\pi$
$62$		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	$-0.541010\pi$
$67$	−2.10331			−0.256960			−0.128480	+	0.991712i	$0.541010\pi$
$68$		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	$-0.647625\pi$
$83$	−8.15071			−0.894657			−0.447329	+	0.894370i	$0.647625\pi$
$84$		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	$-0.589633\pi$
$101$	−5.58538			−0.555766			−0.277883	+	0.960615i	$0.589633\pi$
$102$		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	$-0.652286\pi$
$107$	−9.52439			−0.920757			−0.460379	+	0.887723i	$0.652286\pi$
$108$		0			0
$109$		−	9.64687i		−	0.924003i	−0.886879	−	0.462001i	$-0.847132\pi$
$109$		−	9.64687i		−	0.924003i	0.886879	−	0.462001i	$-0.152868\pi$
$110$		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	$0.0573319\pi$
$131$			4.10073i			0.358282i	−0.983823	+	0.179141i	$0.942668\pi$
$132$		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	$-0.545137\pi$
$139$	−3.33243			−0.282653			−0.141326	+	0.989963i	$0.545137\pi$
$140$		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	$-0.590846\pi$
$149$	−6.87332			−0.563084			−0.281542	+	0.959549i	$0.590846\pi$
$150$		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	$-0.382454\pi$
$157$	9.04524			0.721889			0.360944	+	0.932587i	$0.382454\pi$
$158$		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	$0.0346082\pi$
$163$			2.77074i			0.217021i	−0.994095	+	0.108511i	$0.965392\pi$
$164$		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	$-0.180098\pi$
$173$	22.2065			1.68833			0.844163	+	0.536087i	$0.180098\pi$
$174$		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	$-0.614518\pi$
$179$		−	25.0451i		−	1.87196i	0.352057	−	0.935979i	$-0.385482\pi$
$180$		0			0
$181$	1.92137			0.142814			0.0714070	−	0.997447i	$-0.477251\pi$
$181$	1.92137			0.142814			0.0714070	+	0.997447i	$0.477251\pi$
$182$		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	$-0.444084\pi$
$197$	4.90587			0.349529			0.174764	+	0.984610i	$0.444084\pi$
$198$		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	$-0.658477\pi$
$211$	−13.8738			−0.955112			−0.477556	+	0.878601i	$0.658477\pi$
$212$		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	$-0.958613\pi$
$227$		−	3.90687i		−	0.259308i	0.991559	−	0.129654i	$-0.0413867\pi$
$228$		0			0
$229$	−5.96560			−0.394218			−0.197109	−	0.980382i	$-0.563155\pi$
$229$	−5.96560			−0.394218			−0.197109	+	0.980382i	$0.563155\pi$
$230$		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	$-0.793268\pi$
$251$		−	19.1625i		−	1.20952i	0.796406	−	0.604762i	$-0.206732\pi$
$252$		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	$-0.748425\pi$
$269$	−23.0798			−1.40720			−0.703599	+	0.710597i	$0.748425\pi$
$270$		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	$-0.760727\pi$
$277$		−	22.7308i		−	1.36576i	0.730531	−	0.682880i	$-0.239273\pi$
$278$		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	$0.0246324\pi$
$283$			2.60104i			0.154616i	−0.997007	+	0.0773078i	$0.975368\pi$
$284$		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	$-0.838750\pi$
$293$	−29.9349			−1.74882			−0.874408	+	0.485192i	$0.838750\pi$
$294$		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	$-0.531138\pi$
$307$	−3.42257			−0.195336			−0.0976681	+	0.995219i	$0.531138\pi$
$308$		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	$-0.288177\pi$
$317$	21.9858			1.23485			0.617423	+	0.786632i	$0.288177\pi$
$318$		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	$0.0643155\pi$
$331$			7.30214i			0.401362i	−0.979657	+	0.200681i	$0.935685\pi$
$332$		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	$0.0428560\pi$
$347$			5.00084i			0.268459i	−0.990950	+	0.134230i	$0.957144\pi$
$348$		0			0
$349$	−21.1745			−1.13344			−0.566721	−	0.823910i	$-0.691788\pi$
$349$	−21.1745			−1.13344			−0.566721	+	0.823910i	$0.691788\pi$
$350$		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	$-0.398496\pi$
$373$	12.1096			0.627013			0.313506	+	0.949586i	$0.398496\pi$
$374$		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	$-0.214175\pi$
$379$	30.4497			1.56410			0.782049	+	0.623217i	$0.214175\pi$
$380$		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	$0.0847633\pi$
$389$			10.3805i			0.526311i	−0.964753	+	0.263156i	$0.915237\pi$
$390$		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	$-0.312309\pi$
$397$	22.1592			1.11214			0.556070	+	0.831135i	$0.312309\pi$
$398$		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	$-0.569731\pi$
$419$	−8.89676			−0.434635			−0.217318	+	0.976101i	$0.569731\pi$
$420$		0			0
$421$			19.9102i			0.970364i	0.874413	+	0.485182i	$0.161247\pi$
$421$			19.9102i			0.970364i	−0.874413	+	0.485182i	$0.838753\pi$
$422$		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	$-0.596548\pi$
$443$	−12.5732			−0.597371			−0.298686	+	0.954352i	$0.596548\pi$
$444$		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	$-0.849472\pi$
$461$		−	19.5586i		−	0.910934i	0.890253	−	0.455467i	$-0.150528\pi$
$462$		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	$0.00753008\pi$
$467$			1.02234i			0.0473085i	−0.999720	+	0.0236542i	$0.992470\pi$
$468$		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	$-0.575986\pi$
$491$	−10.4790			−0.472910			−0.236455	+	0.971642i	$0.575986\pi$
$492$		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	$0.366982\pi$
$499$			40.8321i			1.82789i	−0.405833	+	0.913947i	$0.633018\pi$
$500$		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	$-0.466682\pi$
$509$	4.71436			0.208960			0.104480	+	0.994527i	$0.466682\pi$
$510$		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	$-0.921987\pi$
$523$		−	11.0979i		−	0.485276i	0.970117	−	0.242638i	$-0.0780128\pi$
$524$		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	$0.312231\pi$
$541$			38.6571i			1.66200i	−0.556272	+	0.831000i	$0.687769\pi$
$542$		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	$-0.875632\pi$
$547$		−	17.8146i		−	0.761696i	0.924638	−	0.380848i	$-0.124368\pi$
$548$		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	$0.125474\pi$
$557$			18.1282i			0.768116i	−0.923309	+	0.384058i	$0.874526\pi$
$558$		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	$0.226347\pi$
$563$			30.9722i			1.30532i	−0.757650	+	0.652661i	$0.773653\pi$
$564$		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	$-0.432011\pi$
$571$	10.1304			0.423945			0.211973	+	0.977276i	$0.432011\pi$
$572$		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	$0.0987554\pi$
$587$			14.7935i			0.610592i	−0.952257	+	0.305296i	$0.901245\pi$
$588$		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	$-0.501128\pi$
$613$	−0.175456			−0.00708658			−0.00354329	+	0.999994i	$0.501128\pi$
$614$		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	$-0.563609\pi$
$619$	−9.87749			−0.397010			−0.198505	+	0.980100i	$0.563609\pi$
$620$		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	$-0.884220\pi$
$643$		−	18.0427i		−	0.711534i	0.934575	−	0.355767i	$-0.115780\pi$
$644$		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	$-0.822264\pi$
$653$		−	27.0773i		−	1.05962i	0.848117	−	0.529809i	$-0.177736\pi$
$654$		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	$-0.376605\pi$
$659$	19.4084			0.756042			0.378021	+	0.925797i	$0.376605\pi$
$660$		0			0
$661$		−	5.19746i		−	0.202158i	−0.994878	−	0.101079i	$-0.967771\pi$
$661$		−	5.19746i		−	0.202158i	0.994878	−	0.101079i	$-0.0322295\pi$
$662$		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	$-0.619946\pi$
$677$	−19.1484			−0.735933			−0.367967	+	0.929839i	$0.619946\pi$
$678$		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	$0.221891\pi$
$683$			33.5559i			1.28398i	−0.766713	+	0.641990i	$0.778109\pi$
$684$		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	$-0.275245\pi$
$691$	34.1132			1.29773			0.648863	+	0.760905i	$0.275245\pi$
$692$		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	$0.0336491\pi$
$701$			5.58732i			0.211030i	−0.994418	+	0.105515i	$0.966351\pi$
$702$		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	$-0.824478\pi$
$709$		−	27.8996i		−	1.04779i	0.851783	−	0.523895i	$-0.175522\pi$
$710$		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	$-0.647243\pi$
$733$	−24.1639			−0.892514			−0.446257	+	0.894905i	$0.647243\pi$
$734$		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	$-0.748425\pi$
$739$	−38.2540			−1.40720			−0.703598	+	0.710598i	$0.748425\pi$
$740$		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	$0.0808264\pi$
$757$			13.8231i			0.502408i	−0.967934	+	0.251204i	$0.919174\pi$
$758$		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	$-0.504737\pi$
$773$	−0.827395			−0.0297593			−0.0148797	+	0.999889i	$0.504737\pi$
$774$		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	$-0.426819\pi$
$787$	12.7860			0.455771			0.227886	+	0.973688i	$0.426819\pi$
$788$		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	$0.0837540\pi$
$797$			14.6856i			0.520191i	−0.965583	+	0.260095i	$0.916246\pi$
$798$		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	$-0.349175\pi$
$811$	25.9890			0.912596			0.456298	+	0.889827i	$0.349175\pi$
$812$		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	$-0.183630\pi$
$821$	48.0319			1.67632			0.838162	+	0.545421i	$0.183630\pi$
$822$		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	$0.101992\pi$
$827$			18.1152i			0.629926i	−0.949104	+	0.314963i	$0.898008\pi$
$828$		0			0
$829$		−	28.9348i		−	1.00495i	−0.864593	−	0.502473i	$-0.832424\pi$
$829$		−	28.9348i		−	1.00495i	0.864593	−	0.502473i	$-0.167576\pi$
$830$		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	$-0.706939\pi$
$853$		−	46.4969i		−	1.59202i	0.605281	−	0.796012i	$-0.293061\pi$
$854$		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	$0.0554498\pi$
$859$			10.1596i			0.346642i	−0.984865	+	0.173321i	$0.944550\pi$
$860$		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	$-0.561170\pi$
$877$	−11.3121			−0.381984			−0.190992	+	0.981592i	$0.561170\pi$
$878$		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	$0.234844\pi$
$883$			39.9760i			1.34530i	−0.739961	+	0.672649i	$0.765156\pi$
$884$		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	$-0.723800\pi$
$907$		−	45.9485i		−	1.52570i	0.646578	−	0.762848i	$-0.276200\pi$
$908$		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	$-0.496702\pi$
$941$	0.635730			0.0207242			0.0103621	+	0.999946i	$0.496702\pi$
$942$		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	$0.219705\pi$
$947$			39.1873i			1.27342i	−0.771104	+	0.636709i	$0.780295\pi$
$948$		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	$-0.343215\pi$
$971$	29.4707			0.945759			0.472880	+	0.881127i	$0.343215\pi$
$972$		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	$0.0389106\pi$
$997$			7.70039i			0.243874i	−0.992538	+	0.121937i	$0.961089\pi$
$998$		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	592.2.g.d.369.7		10
3.2	odd	2		5328.2.h.q.2737.7		10
4.3	odd	2		296.2.g.a.73.3	✓	10
8.3	odd	2		2368.2.g.o.961.8		10
8.5	even	2		2368.2.g.p.961.4		10
12.11	even	2		2664.2.h.c.73.7		10
37.36	even	2	inner	592.2.g.d.369.8		10
111.110	odd	2		5328.2.h.q.2737.4		10
148.147	odd	2		296.2.g.a.73.4	yes	10
296.147	odd	2		2368.2.g.o.961.7		10
296.221	even	2		2368.2.g.p.961.3		10
444.443	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	4.3	odd	2
296.2.g.a.73.4	yes	10	148.147	odd	2
592.2.g.d.369.7		10	1.1	even	1	trivial
592.2.g.d.369.8		10	37.36	even	2	inner
2368.2.g.o.961.7		10	296.147	odd	2
2368.2.g.o.961.8		10	8.3	odd	2
2368.2.g.p.961.3		10	296.221	even	2
2368.2.g.p.961.4		10	8.5	even	2
2664.2.h.c.73.4		10	444.443	even	2
2664.2.h.c.73.7		10	12.11	even	2
5328.2.h.q.2737.4		10	111.110	odd	2
5328.2.h.q.2737.7		10	3.2	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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.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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