LMFDB - Embedded newform 9360.2.a.db.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9360.a (trivial)

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:	yes
Analytic conductor:	$74.7399762919$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 8 $ x^3 - x^2 - 14*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.50331$ of defining polynomial
Character	$\chi$	$=$	9360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +2.50331 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +2.50331 q^{7} +3.27316 q^{11} +1.00000 q^{13} -4.50331 q^{17} -3.23014 q^{19} -2.50331 q^{23} +1.00000 q^{25} +7.77647 q^{29} +5.00662 q^{31} +2.50331 q^{35} +7.73345 q^{37} -7.73345 q^{41} -4.00000 q^{43} +9.00662 q^{47} -0.733451 q^{49} -6.27978 q^{53} +3.27316 q^{55} +9.27316 q^{61} +1.00000 q^{65} -15.5529 q^{67} +5.73345 q^{71} +15.7765 q^{73} +8.19374 q^{77} +4.72684 q^{79} -11.5529 q^{83} -4.50331 q^{85} +15.2864 q^{89} +2.50331 q^{91} -3.23014 q^{95} +12.5033 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 5 * q^7	$ 3 q + 3 q^{5} - 5 q^{7} + 3 q^{11} + 3 q^{13} - q^{17} - 4 q^{19} + 5 q^{23} + 3 q^{25} + 4 q^{29} - 10 q^{31} - 5 q^{35} + 5 q^{37} - 5 q^{41} - 12 q^{43} + 2 q^{47} + 16 q^{49} + 13 q^{53} + 3 q^{55} + 21 q^{61} + 3 q^{65} - 8 q^{67} - q^{71} + 28 q^{73} - 5 q^{77} + 21 q^{79} + 4 q^{83} - q^{85} - 11 q^{89} - 5 q^{91} - 4 q^{95} + 25 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 5 * q^7 + 3 * q^11 + 3 * q^13 - q^17 - 4 * q^19 + 5 * q^23 + 3 * q^25 + 4 * q^29 - 10 * q^31 - 5 * q^35 + 5 * q^37 - 5 * q^41 - 12 * q^43 + 2 * q^47 + 16 * q^49 + 13 * q^53 + 3 * q^55 + 21 * q^61 + 3 * q^65 - 8 * q^67 - q^71 + 28 * q^73 - 5 * q^77 + 21 * q^79 + 4 * q^83 - q^85 - 11 * q^89 - 5 * q^91 - 4 * q^95 + 25 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.50331	0.946161	0.473081	−	0.881019i	$-0.343142\pi$
$7$	2.50331	0.946161	0.473081	+	0.881019i	$0.343142\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.27316	0.986896	0.493448	−	0.869775i	$-0.335736\pi$
$11$	3.27316	0.986896	0.493448	+	0.869775i	$0.335736\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.50331	−1.09221	−0.546106	−	0.837716i	$-0.683891\pi$
$17$	−4.50331	−1.09221	−0.546106	+	0.837716i	$0.683891\pi$
$18$	0	0
$19$	−3.23014	−0.741046	−0.370523	−	0.928823i	$-0.620821\pi$
$19$	−3.23014	−0.741046	−0.370523	+	0.928823i	$0.620821\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.50331	−0.521976	−0.260988	−	0.965342i	$-0.584048\pi$
$23$	−2.50331	−0.521976	−0.260988	+	0.965342i	$0.584048\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.77647	1.44405	0.722027	−	0.691865i	$-0.243210\pi$
$29$	7.77647	1.44405	0.722027	+	0.691865i	$0.243210\pi$
$30$	0	0
$31$	5.00662	0.899215	0.449607	−	0.893226i	$-0.351564\pi$
$31$	5.00662	0.899215	0.449607	+	0.893226i	$0.351564\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.50331	0.423136
$36$	0	0
$37$	7.73345	1.27137	0.635686	−	0.771948i	$-0.280717\pi$
$37$	7.73345	1.27137	0.635686	+	0.771948i	$0.280717\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.73345	−1.20776	−0.603881	−	0.797074i	$-0.706380\pi$
$41$	−7.73345	−1.20776	−0.603881	+	0.797074i	$0.706380\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.00662	1.31375	0.656875	−	0.754000i	$-0.271878\pi$
$47$	9.00662	1.31375	0.656875	+	0.754000i	$0.271878\pi$
$48$	0	0
$49$	−0.733451	−0.104779
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.27978	−0.862594	−0.431297	−	0.902210i	$-0.641944\pi$
$53$	−6.27978	−0.862594	−0.431297	+	0.902210i	$0.641944\pi$
$54$	0	0
$55$	3.27316	0.441353
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	9.27316	1.18731	0.593654	−	0.804721i	$-0.297685\pi$
$61$	9.27316	1.18731	0.593654	+	0.804721i	$0.297685\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−15.5529	−1.90009	−0.950047	−	0.312106i	$-0.898966\pi$
$67$	−15.5529	−1.90009	−0.950047	+	0.312106i	$0.898966\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.73345	0.680435	0.340218	−	0.940347i	$-0.389499\pi$
$71$	5.73345	0.680435	0.340218	+	0.940347i	$0.389499\pi$
$72$	0	0
$73$	15.7765	1.84650	0.923248	−	0.384204i	$-0.125524\pi$
$73$	15.7765	1.84650	0.923248	+	0.384204i	$0.125524\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.19374	0.933763
$78$	0	0
$79$	4.72684	0.531811	0.265905	−	0.963999i	$-0.414329\pi$
$79$	4.72684	0.531811	0.265905	+	0.963999i	$0.414329\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.5529	−1.26810	−0.634050	−	0.773292i	$-0.718609\pi$
$83$	−11.5529	−1.26810	−0.634050	+	0.773292i	$0.718609\pi$
$84$	0	0
$85$	−4.50331	−0.488452
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.2864	1.62035	0.810177	−	0.586185i	$-0.199371\pi$
$89$	15.2864	1.62035	0.810177	+	0.586185i	$0.199371\pi$
$90$	0	0
$91$	2.50331	0.262418
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.23014	−0.331406
$96$	0	0
$97$	12.5033	1.26952	0.634759	−	0.772710i	$-0.281099\pi$
$97$	12.5033	1.26952	0.634759	+	0.772710i	$0.281099\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.23014	−0.520419	−0.260209	−	0.965552i	$-0.583792\pi$
$101$	−5.23014	−0.520419	−0.260209	+	0.965552i	$0.583792\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.726836	−0.0702659	−0.0351329	−	0.999383i	$-0.511185\pi$
$107$	−0.726836	−0.0702659	−0.0351329	+	0.999383i	$0.511185\pi$
$108$	0	0
$109$	16.7831	1.60753	0.803764	−	0.594948i	$-0.202827\pi$
$109$	16.7831	1.60753	0.803764	+	0.594948i	$0.202827\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.77647	0.355261	0.177630	−	0.984097i	$-0.443157\pi$
$113$	3.77647	0.355261	0.177630	+	0.984097i	$0.443157\pi$
$114$	0	0
$115$	−2.50331	−0.233435
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.2732	−1.03341
$120$	0	0
$121$	−0.286395	−0.0260359
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.54633	0.580893	0.290446	−	0.956891i	$-0.406196\pi$
$127$	6.54633	0.580893	0.290446	+	0.956891i	$0.406196\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.2301	−0.981182	−0.490591	−	0.871390i	$-0.663219\pi$
$131$	−11.2301	−0.981182	−0.490591	+	0.871390i	$0.663219\pi$
$132$	0	0
$133$	−8.08604	−0.701149
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.5529	1.15791	0.578953	−	0.815361i	$-0.303461\pi$
$137$	13.5529	1.15791	0.578953	+	0.815361i	$0.303461\pi$
$138$	0	0
$139$	21.7335	1.84341	0.921704	−	0.387895i	$-0.126798\pi$
$139$	21.7335	1.84341	0.921704	+	0.387895i	$0.126798\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.27316	0.273716
$144$	0	0
$145$	7.77647	0.645801
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.27978	−0.186767	−0.0933834	−	0.995630i	$-0.529768\pi$
$149$	−2.27978	−0.186767	−0.0933834	+	0.995630i	$0.529768\pi$
$150$	0	0
$151$	6.46029	0.525731	0.262865	−	0.964833i	$-0.415333\pi$
$151$	6.46029	0.525731	0.262865	+	0.964833i	$0.415333\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.00662	0.402141
$156$	0	0
$157$	−3.53971	−0.282500	−0.141250	−	0.989974i	$-0.545112\pi$
$157$	−3.53971	−0.282500	−0.141250	+	0.989974i	$0.545112\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.26655	−0.493873
$162$	0	0
$163$	18.7401	1.46784	0.733918	−	0.679238i	$-0.237690\pi$
$163$	18.7401	1.46784	0.733918	+	0.679238i	$0.237690\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.46029	0.643224	0.321612	−	0.946872i	$-0.395775\pi$
$173$	8.46029	0.643224	0.321612	+	0.946872i	$0.395775\pi$
$174$	0	0
$175$	2.50331	0.189232
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.76986	−0.356516	−0.178258	−	0.983984i	$-0.557046\pi$
$179$	−4.76986	−0.356516	−0.178258	+	0.983984i	$0.557046\pi$
$180$	0	0
$181$	15.7335	1.16946	0.584729	−	0.811229i	$-0.301201\pi$
$181$	15.7335	1.16946	0.584729	+	0.811229i	$0.301201\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.73345	0.568575
$186$	0	0
$187$	−14.7401	−1.07790
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.0132	−1.88225	−0.941126	−	0.338057i	$-0.890230\pi$
$191$	−26.0132	−1.88225	−0.941126	+	0.338057i	$0.890230\pi$
$192$	0	0
$193$	−0.503308	−0.0362289	−0.0181144	−	0.999836i	$-0.505766\pi$
$193$	−0.503308	−0.0362289	−0.0181144	+	0.999836i	$0.505766\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.0066	0.784189	0.392094	−	0.919925i	$-0.371751\pi$
$197$	11.0066	0.784189	0.392094	+	0.919925i	$0.371751\pi$
$198$	0	0
$199$	−2.01323	−0.142714	−0.0713571	−	0.997451i	$-0.522733\pi$
$199$	−2.01323	−0.142714	−0.0713571	+	0.997451i	$0.522733\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	19.4669	1.36631
$204$	0	0
$205$	−7.73345	−0.540128
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.5728	−0.731335
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	12.5331	0.850802
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.50331	−0.302925
$222$	0	0
$223$	−23.7897	−1.59308	−0.796538	−	0.604588i	$-0.793338\pi$
$223$	−23.7897	−1.59308	−0.796538	+	0.604588i	$0.793338\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.0198	−1.52788	−0.763940	−	0.645287i	$-0.776738\pi$
$227$	−23.0198	−1.52788	−0.763940	+	0.645287i	$0.776738\pi$
$228$	0	0
$229$	−7.77647	−0.513884	−0.256942	−	0.966427i	$-0.582715\pi$
$229$	−7.77647	−0.513884	−0.256942	+	0.966427i	$0.582715\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.9702	−1.57034	−0.785170	−	0.619280i	$-0.787425\pi$
$233$	−23.9702	−1.57034	−0.785170	+	0.619280i	$0.787425\pi$
$234$	0	0
$235$	9.00662	0.587527
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.812878	0.0525807	0.0262903	−	0.999654i	$-0.491631\pi$
$239$	0.812878	0.0525807	0.0262903	+	0.999654i	$0.491631\pi$
$240$	0	0
$241$	20.0132	1.28917	0.644583	−	0.764535i	$-0.277031\pi$
$241$	20.0132	1.28917	0.644583	+	0.764535i	$0.277031\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.733451	−0.0468585
$246$	0	0
$247$	−3.23014	−0.205529
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.3162	−0.714271	−0.357136	−	0.934053i	$-0.616247\pi$
$251$	−11.3162	−0.714271	−0.357136	+	0.934053i	$0.616247\pi$
$252$	0	0
$253$	−8.19374	−0.515136
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.3294	−1.20574	−0.602868	−	0.797841i	$-0.705975\pi$
$257$	−19.3294	−1.20574	−0.602868	+	0.797841i	$0.705975\pi$
$258$	0	0
$259$	19.3592	1.20292
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.2368	−0.754551	−0.377275	−	0.926101i	$-0.623139\pi$
$263$	−12.2368	−0.754551	−0.377275	+	0.926101i	$0.623139\pi$
$264$	0	0
$265$	−6.27978	−0.385764
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.76986	0.168881	0.0844406	−	0.996429i	$-0.473090\pi$
$269$	2.76986	0.168881	0.0844406	+	0.996429i	$0.473090\pi$
$270$	0	0
$271$	0.0860423	0.00522670	0.00261335	−	0.999997i	$-0.499168\pi$
$271$	0.0860423	0.00522670	0.00261335	+	0.999997i	$0.499168\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.27316	0.197379
$276$	0	0
$277$	17.5529	1.05465	0.527327	−	0.849662i	$-0.323194\pi$
$277$	17.5529	1.05465	0.527327	+	0.849662i	$0.323194\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.0132	−1.43251	−0.716255	−	0.697839i	$-0.754145\pi$
$281$	−24.0132	−1.43251	−0.716255	+	0.697839i	$0.754145\pi$
$282$	0	0
$283$	−22.0132	−1.30855	−0.654275	−	0.756256i	$-0.727026\pi$
$283$	−22.0132	−1.30855	−0.654275	+	0.756256i	$0.727026\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.3592	−1.14274
$288$	0	0
$289$	3.27978	0.192928
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.99338	0.525399	0.262700	−	0.964878i	$-0.415387\pi$
$293$	8.99338	0.525399	0.262700	+	0.964878i	$0.415387\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.50331	−0.144770
$300$	0	0
$301$	−10.0132	−0.577153
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.27316	0.530980
$306$	0	0
$307$	22.2930	1.27233	0.636165	−	0.771553i	$-0.280520\pi$
$307$	22.2930	1.27233	0.636165	+	0.771553i	$0.280520\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−17.5529	−0.992151	−0.496076	−	0.868279i	$-0.665226\pi$
$313$	−17.5529	−0.992151	−0.496076	+	0.868279i	$0.665226\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.54633	0.255347	0.127674	−	0.991816i	$-0.459249\pi$
$317$	4.54633	0.255347	0.127674	+	0.991816i	$0.459249\pi$
$318$	0	0
$319$	25.4537	1.42513
$320$	0	0
$321$	0	0
$322$	0	0
$323$	14.5463	0.809379
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	22.5463	1.24302
$330$	0	0
$331$	−19.7897	−1.08774	−0.543870	−	0.839169i	$-0.683042\pi$
$331$	−19.7897	−1.08774	−0.543870	+	0.839169i	$0.683042\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.5529	−0.849748
$336$	0	0
$337$	−13.0198	−0.709236	−0.354618	−	0.935011i	$-0.615389\pi$
$337$	−13.0198	−0.709236	−0.354618	+	0.935011i	$0.615389\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.3875	0.887432
$342$	0	0
$343$	−19.3592	−1.04530
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.18712	−0.385825	−0.192912	−	0.981216i	$-0.561793\pi$
$347$	−7.18712	−0.385825	−0.192912	+	0.981216i	$0.561793\pi$
$348$	0	0
$349$	−17.2301	−0.922308	−0.461154	−	0.887320i	$-0.652565\pi$
$349$	−17.2301	−0.922308	−0.461154	+	0.887320i	$0.652565\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.5662	1.68010	0.840049	−	0.542511i	$-0.182526\pi$
$353$	31.5662	1.68010	0.840049	+	0.542511i	$0.182526\pi$
$354$	0	0
$355$	5.73345	0.304300
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.46029	−0.129849	−0.0649245	−	0.997890i	$-0.520681\pi$
$359$	−2.46029	−0.129849	−0.0649245	+	0.997890i	$0.520681\pi$
$360$	0	0
$361$	−8.56617	−0.450851
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.7765	0.825778
$366$	0	0
$367$	−5.09266	−0.265835	−0.132917	−	0.991127i	$-0.542434\pi$
$367$	−5.09266	−0.265835	−0.132917	+	0.991127i	$0.542434\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.7202	−0.816153
$372$	0	0
$373$	11.0066	0.569901	0.284950	−	0.958542i	$-0.408023\pi$
$373$	11.0066	0.569901	0.284950	+	0.958542i	$0.408023\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.77647	0.400509
$378$	0	0
$379$	26.3360	1.35279	0.676396	−	0.736539i	$-0.263541\pi$
$379$	26.3360	1.35279	0.676396	+	0.736539i	$0.263541\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.5463	0.947673	0.473837	−	0.880613i	$-0.342869\pi$
$383$	18.5463	0.947673	0.473837	+	0.880613i	$0.342869\pi$
$384$	0	0
$385$	8.19374	0.417592
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.7699	0.951670	0.475835	−	0.879535i	$-0.342146\pi$
$389$	18.7699	0.951670	0.475835	+	0.879535i	$0.342146\pi$
$390$	0	0
$391$	11.2732	0.570108
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.72684	0.237833
$396$	0	0
$397$	−0.266549	−0.0133777	−0.00668886	−	0.999978i	$-0.502129\pi$
$397$	−0.266549	−0.0133777	−0.00668886	+	0.999978i	$0.502129\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−35.5662	−1.77609	−0.888045	−	0.459757i	$-0.847937\pi$
$401$	−35.5662	−1.77609	−0.888045	+	0.459757i	$0.847937\pi$
$402$	0	0
$403$	5.00662	0.249397
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.3129	1.25471
$408$	0	0
$409$	0.460287	0.0227597	0.0113799	−	0.999935i	$-0.496378\pi$
$409$	0.460287	0.0227597	0.0113799	+	0.999935i	$0.496378\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−11.5529	−0.567112
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8.32280	−0.406595	−0.203298	−	0.979117i	$-0.565166\pi$
$419$	−8.32280	−0.406595	−0.203298	+	0.979117i	$0.565166\pi$
$420$	0	0
$421$	10.3228	0.503103	0.251551	−	0.967844i	$-0.419059\pi$
$421$	10.3228	0.503103	0.251551	+	0.967844i	$0.419059\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.50331	−0.218443
$426$	0	0
$427$	23.2136	1.12338
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.5662	0.846133	0.423066	−	0.906099i	$-0.360954\pi$
$431$	17.5662	0.846133	0.423066	+	0.906099i	$0.360954\pi$
$432$	0	0
$433$	7.00662	0.336716	0.168358	−	0.985726i	$-0.446153\pi$
$433$	7.00662	0.336716	0.168358	+	0.985726i	$0.446153\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.08604	0.386808
$438$	0	0
$439$	1.25993	0.0601334	0.0300667	−	0.999548i	$-0.490428\pi$
$439$	1.25993	0.0601334	0.0300667	+	0.999548i	$0.490428\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.8261	1.27455	0.637273	−	0.770638i	$-0.280062\pi$
$443$	26.8261	1.27455	0.637273	+	0.770638i	$0.280062\pi$
$444$	0	0
$445$	15.2864	0.724645
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.7401	1.54510	0.772550	−	0.634954i	$-0.218981\pi$
$449$	32.7401	1.54510	0.772550	+	0.634954i	$0.218981\pi$
$450$	0	0
$451$	−25.3129	−1.19194
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.50331	0.117357
$456$	0	0
$457$	33.5960	1.57155	0.785776	−	0.618511i	$-0.212264\pi$
$457$	33.5960	1.57155	0.785776	+	0.618511i	$0.212264\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	41.8327	1.94834	0.974172	−	0.225807i	$-0.0725018\pi$
$461$	41.8327	1.94834	0.974172	+	0.225807i	$0.0725018\pi$
$462$	0	0
$463$	6.05625	0.281458	0.140729	−	0.990048i	$-0.455055\pi$
$463$	6.05625	0.281458	0.140729	+	0.990048i	$0.455055\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.2996	1.26328	0.631638	−	0.775263i	$-0.282383\pi$
$467$	27.2996	1.26328	0.631638	+	0.775263i	$0.282383\pi$
$468$	0	0
$469$	−38.9338	−1.79780
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−13.0927	−0.602001
$474$	0	0
$475$	−3.23014	−0.148209
$476$	0	0
$477$	0	0
$478$	0	0
$479$	28.2798	1.29214	0.646068	−	0.763280i	$-0.276412\pi$
$479$	28.2798	1.29214	0.646068	+	0.763280i	$0.276412\pi$
$480$	0	0
$481$	7.73345	0.352615
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.5033	0.567746
$486$	0	0
$487$	−24.9636	−1.13121	−0.565604	−	0.824677i	$-0.691357\pi$
$487$	−24.9636	−1.13121	−0.565604	+	0.824677i	$0.691357\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.22353	0.280864	0.140432	−	0.990090i	$-0.455151\pi$
$491$	6.22353	0.280864	0.140432	+	0.990090i	$0.455151\pi$
$492$	0	0
$493$	−35.0198	−1.57721
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.3526	0.643802
$498$	0	0
$499$	19.2301	0.860859	0.430430	−	0.902624i	$-0.358362\pi$
$499$	19.2301	0.860859	0.430430	+	0.902624i	$0.358362\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10.6970	0.476958	0.238479	−	0.971148i	$-0.423351\pi$
$503$	10.6970	0.476958	0.238479	+	0.971148i	$0.423351\pi$
$504$	0	0
$505$	−5.23014	−0.232738
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.740066	−0.0328029	−0.0164014	−	0.999865i	$-0.505221\pi$
$509$	−0.740066	−0.0328029	−0.0164014	+	0.999865i	$0.505221\pi$
$510$	0	0
$511$	39.4934	1.74708
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	29.4801	1.29653
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.5662	1.55818	0.779091	−	0.626911i	$-0.215681\pi$
$521$	35.5662	1.55818	0.779091	+	0.626911i	$0.215681\pi$
$522$	0	0
$523$	−14.0132	−0.612756	−0.306378	−	0.951910i	$-0.599117\pi$
$523$	−14.0132	−0.612756	−0.306378	+	0.951910i	$0.599117\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22.5463	−0.982134
$528$	0	0
$529$	−16.7335	−0.727541
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.73345	−0.334973
$534$	0	0
$535$	−0.726836	−0.0314238
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.40071	−0.103406
$540$	0	0
$541$	11.7765	0.506310	0.253155	−	0.967426i	$-0.418532\pi$
$541$	11.7765	0.506310	0.253155	+	0.967426i	$0.418532\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.7831	0.718908
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−25.1191	−1.07011
$552$	0	0
$553$	11.8327	0.503179
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.0265	1.78072	0.890359	−	0.455259i	$-0.150453\pi$
$557$	42.0265	1.78072	0.890359	+	0.455259i	$0.150453\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.3724	−1.06932	−0.534660	−	0.845067i	$-0.679560\pi$
$563$	−25.3724	−1.06932	−0.534660	+	0.845067i	$0.679560\pi$
$564$	0	0
$565$	3.77647	0.158877
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.0993	−1.51336	−0.756680	−	0.653785i	$-0.773180\pi$
$569$	−36.0993	−1.51336	−0.756680	+	0.653785i	$0.773180\pi$
$570$	0	0
$571$	23.1871	0.970351	0.485175	−	0.874417i	$-0.338756\pi$
$571$	23.1871	0.970351	0.485175	+	0.874417i	$0.338756\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.50331	−0.104395
$576$	0	0
$577$	34.9636	1.45555	0.727777	−	0.685814i	$-0.240554\pi$
$577$	34.9636	1.45555	0.727777	+	0.685814i	$0.240554\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−28.9206	−1.19983
$582$	0	0
$583$	−20.5548	−0.851291
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.9934	0.783941	0.391970	−	0.919978i	$-0.371794\pi$
$587$	18.9934	0.783941	0.391970	+	0.919978i	$0.371794\pi$
$588$	0	0
$589$	−16.1721	−0.666359
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10.5331	−0.432543	−0.216271	−	0.976333i	$-0.569390\pi$
$593$	−10.5331	−0.432543	−0.216271	+	0.976333i	$0.569390\pi$
$594$	0	0
$595$	−11.2732	−0.462155
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.1059	−0.944081	−0.472040	−	0.881577i	$-0.656482\pi$
$599$	−23.1059	−0.944081	−0.472040	+	0.881577i	$0.656482\pi$
$600$	0	0
$601$	−18.8129	−0.767393	−0.383697	−	0.923459i	$-0.625349\pi$
$601$	−18.8129	−0.767393	−0.383697	+	0.923459i	$0.625349\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.286395	−0.0116436
$606$	0	0
$607$	5.09266	0.206705	0.103352	−	0.994645i	$-0.467043\pi$
$607$	5.09266	0.206705	0.103352	+	0.994645i	$0.467043\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.00662	0.364369
$612$	0	0
$613$	27.2004	1.09861	0.549306	−	0.835621i	$-0.314892\pi$
$613$	27.2004	1.09861	0.549306	+	0.835621i	$0.314892\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.460287	0.0185304	0.00926522	−	0.999957i	$-0.497051\pi$
$617$	0.460287	0.0185304	0.00926522	+	0.999957i	$0.497051\pi$
$618$	0	0
$619$	1.21691	0.0489119	0.0244559	−	0.999701i	$-0.492215\pi$
$619$	1.21691	0.0489119	0.0244559	+	0.999701i	$0.492215\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	38.2665	1.53312
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−34.8261	−1.38861
$630$	0	0
$631$	−2.01323	−0.0801454	−0.0400727	−	0.999197i	$-0.512759\pi$
$631$	−2.01323	−0.0801454	−0.0400727	+	0.999197i	$0.512759\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.54633	0.259783
$636$	0	0
$637$	−0.733451	−0.0290604
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.37424	0.172772	0.0863861	−	0.996262i	$-0.472468\pi$
$641$	4.37424	0.172772	0.0863861	+	0.996262i	$0.472468\pi$
$642$	0	0
$643$	−3.72022	−0.146711	−0.0733556	−	0.997306i	$-0.523371\pi$
$643$	−3.72022	−0.146711	−0.0733556	+	0.997306i	$0.523371\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.4305	−0.488693	−0.244347	−	0.969688i	$-0.578573\pi$
$647$	−12.4305	−0.488693	−0.244347	+	0.969688i	$0.578573\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	0	0
$655$	−11.2301	−0.438798
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29.3294	1.14251	0.571256	−	0.820772i	$-0.306456\pi$
$659$	29.3294	1.14251	0.571256	+	0.820772i	$0.306456\pi$
$660$	0	0
$661$	2.32280	0.0903465	0.0451732	−	0.998979i	$-0.485616\pi$
$661$	2.32280	0.0903465	0.0451732	+	0.998979i	$0.485616\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.08604	−0.313563
$666$	0	0
$667$	−19.4669	−0.753761
$668$	0	0
$669$	0	0
$670$	0	0
$671$	30.3526	1.17175
$672$	0	0
$673$	−21.1059	−0.813572	−0.406786	−	0.913523i	$-0.633351\pi$
$673$	−21.1059	−0.813572	−0.406786	+	0.913523i	$0.633351\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.1937	1.31417	0.657086	−	0.753816i	$-0.271789\pi$
$677$	34.1937	1.31417	0.657086	+	0.753816i	$0.271789\pi$
$678$	0	0
$679$	31.2996	1.20117
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32.4735	−1.24256	−0.621282	−	0.783587i	$-0.713388\pi$
$683$	−32.4735	−1.24256	−0.621282	+	0.783587i	$0.713388\pi$
$684$	0	0
$685$	13.5529	0.517831
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.27978	−0.239241
$690$	0	0
$691$	34.8095	1.32422	0.662109	−	0.749408i	$-0.269662\pi$
$691$	34.8095	1.32422	0.662109	+	0.749408i	$0.269662\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.7335	0.824397
$696$	0	0
$697$	34.8261	1.31913
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.1507	−0.685543	−0.342772	−	0.939419i	$-0.611366\pi$
$701$	−18.1507	−0.685543	−0.342772	+	0.939419i	$0.611366\pi$
$702$	0	0
$703$	−24.9802	−0.942144
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.0927	−0.492400
$708$	0	0
$709$	−11.2434	−0.422254	−0.211127	−	0.977459i	$-0.567713\pi$
$709$	−11.2434	−0.422254	−0.211127	+	0.977459i	$0.567713\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.5331	−0.469368
$714$	0	0
$715$	3.27316	0.122409
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.53971	−0.355771	−0.177886	−	0.984051i	$-0.556926\pi$
$719$	−9.53971	−0.355771	−0.177886	+	0.984051i	$0.556926\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.77647	0.288811
$726$	0	0
$727$	−36.1125	−1.33934	−0.669669	−	0.742659i	$-0.733564\pi$
$727$	−36.1125	−1.33934	−0.669669	+	0.742659i	$0.733564\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.0132	0.666243
$732$	0	0
$733$	−34.2798	−1.26615	−0.633076	−	0.774089i	$-0.718208\pi$
$733$	−34.2798	−1.26615	−0.633076	+	0.774089i	$0.718208\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−50.9073	−1.87520
$738$	0	0
$739$	−25.8625	−0.951368	−0.475684	−	0.879616i	$-0.657799\pi$
$739$	−25.8625	−0.951368	−0.475684	+	0.879616i	$0.657799\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.55956	−0.167274	−0.0836370	−	0.996496i	$-0.526654\pi$
$743$	−4.55956	−0.167274	−0.0836370	+	0.996496i	$0.526654\pi$
$744$	0	0
$745$	−2.27978	−0.0835247
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.81949	−0.0664828
$750$	0	0
$751$	−35.8327	−1.30755	−0.653777	−	0.756687i	$-0.726817\pi$
$751$	−35.8327	−1.30755	−0.653777	+	0.756687i	$0.726817\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.46029	0.235114
$756$	0	0
$757$	−29.4669	−1.07099	−0.535496	−	0.844538i	$-0.679875\pi$
$757$	−29.4669	−1.07099	−0.535496	+	0.844538i	$0.679875\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.4735	−0.814664	−0.407332	−	0.913280i	$-0.633541\pi$
$761$	−22.4735	−0.814664	−0.407332	+	0.913280i	$0.633541\pi$
$762$	0	0
$763$	42.0132	1.52098
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−33.4669	−1.20685	−0.603424	−	0.797421i	$-0.706197\pi$
$769$	−33.4669	−1.20685	−0.603424	+	0.797421i	$0.706197\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.0860	0.794380	0.397190	−	0.917736i	$-0.369985\pi$
$773$	22.0860	0.794380	0.397190	+	0.917736i	$0.369985\pi$
$774$	0	0
$775$	5.00662	0.179843
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.9802	0.895007
$780$	0	0
$781$	18.7665	0.671519
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.53971	−0.126338
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.45367	0.336134
$792$	0	0
$793$	9.27316	0.329300
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.1739	0.395800	0.197900	−	0.980222i	$-0.436588\pi$
$797$	11.1739	0.395800	0.197900	+	0.980222i	$0.436588\pi$
$798$	0	0
$799$	−40.5596	−1.43489
$800$	0	0
$801$	0	0
$802$	0	0
$803$	51.6390	1.82230
$804$	0	0
$805$	−6.26655	−0.220867
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−4.68381	−0.164471	−0.0822355	−	0.996613i	$-0.526206\pi$
$811$	−4.68381	−0.164471	−0.0822355	+	0.996613i	$0.526206\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.7401	0.656436
$816$	0	0
$817$	12.9206	0.452034
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.8129	−0.516973	−0.258487	−	0.966015i	$-0.583224\pi$
$821$	−14.8129	−0.516973	−0.258487	+	0.966015i	$0.583224\pi$
$822$	0	0
$823$	−20.0265	−0.698079	−0.349039	−	0.937108i	$-0.613492\pi$
$823$	−20.0265	−0.698079	−0.349039	+	0.937108i	$0.613492\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.9272	1.45795	0.728976	−	0.684540i	$-0.239997\pi$
$827$	41.9272	1.45795	0.728976	+	0.684540i	$0.239997\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.30295	0.114441
$834$	0	0
$835$	−4.00000	−0.138426
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.83934	0.167073	0.0835363	−	0.996505i	$-0.473379\pi$
$839$	4.83934	0.167073	0.0835363	+	0.996505i	$0.473379\pi$
$840$	0	0
$841$	31.4735	1.08529
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−0.716935	−0.0246342
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−19.3592	−0.663625
$852$	0	0
$853$	−34.6673	−1.18698	−0.593492	−	0.804840i	$-0.702251\pi$
$853$	−34.6673	−1.18698	−0.593492	+	0.804840i	$0.702251\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.6026	−0.772089	−0.386045	−	0.922480i	$-0.626159\pi$
$857$	−22.6026	−0.772089	−0.386045	+	0.922480i	$0.626159\pi$
$858$	0	0
$859$	8.16728	0.278664	0.139332	−	0.990246i	$-0.455504\pi$
$859$	8.16728	0.278664	0.139332	+	0.990246i	$0.455504\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.4934	1.75285	0.876427	−	0.481534i	$-0.159920\pi$
$863$	51.4934	1.75285	0.876427	+	0.481534i	$0.159920\pi$
$864$	0	0
$865$	8.46029	0.287658
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.4717	0.524842
$870$	0	0
$871$	−15.5529	−0.526991
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.50331	0.0846272
$876$	0	0
$877$	−53.1323	−1.79415	−0.897076	−	0.441876i	$-0.854313\pi$
$877$	−53.1323	−1.79415	−0.897076	+	0.441876i	$0.854313\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.4471	−0.486734	−0.243367	−	0.969934i	$-0.578252\pi$
$881$	−14.4471	−0.486734	−0.243367	+	0.969934i	$0.578252\pi$
$882$	0	0
$883$	1.98677	0.0668601	0.0334301	−	0.999441i	$-0.489357\pi$
$883$	1.98677	0.0668601	0.0334301	+	0.999441i	$0.489357\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−53.0761	−1.78212	−0.891060	−	0.453885i	$-0.850038\pi$
$887$	−53.0761	−1.78212	−0.891060	+	0.453885i	$0.850038\pi$
$888$	0	0
$889$	16.3875	0.549618
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−29.0927	−0.973549
$894$	0	0
$895$	−4.76986	−0.159439
$896$	0	0
$897$	0	0
$898$	0	0
$899$	38.9338	1.29852
$900$	0	0
$901$	28.2798	0.942136
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.7335	0.522998
$906$	0	0
$907$	7.46690	0.247934	0.123967	−	0.992286i	$-0.460438\pi$
$907$	7.46690	0.247934	0.123967	+	0.992286i	$0.460438\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	56.6456	1.87675	0.938376	−	0.345615i	$-0.112330\pi$
$911$	56.6456	1.87675	0.938376	+	0.345615i	$0.112330\pi$
$912$	0	0
$913$	−37.8147	−1.25148
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−28.1125	−0.928357
$918$	0	0
$919$	48.7533	1.60822	0.804111	−	0.594479i	$-0.202642\pi$
$919$	48.7533	1.60822	0.804111	+	0.594479i	$0.202642\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.73345	0.188719
$924$	0	0
$925$	7.73345	0.254274
$926$	0	0
$927$	0	0
$928$	0	0
$929$	54.3923	1.78455	0.892276	−	0.451489i	$-0.149107\pi$
$929$	54.3923	1.78455	0.892276	+	0.451489i	$0.149107\pi$
$930$	0	0
$931$	2.36915	0.0776458
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.7401	−0.482052
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.8526	1.07096	0.535482	−	0.844547i	$-0.320130\pi$
$941$	32.8526	1.07096	0.535482	+	0.844547i	$0.320130\pi$
$942$	0	0
$943$	19.3592	0.630423
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.63237	−0.215523	−0.107762	−	0.994177i	$-0.534368\pi$
$947$	−6.63237	−0.215523	−0.107762	+	0.994177i	$0.534368\pi$
$948$	0	0
$949$	15.7765	0.512126
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−40.5298	−1.31289	−0.656444	−	0.754375i	$-0.727940\pi$
$953$	−40.5298	−1.31289	−0.656444	+	0.754375i	$0.727940\pi$
$954$	0	0
$955$	−26.0132	−0.841768
$956$	0	0
$957$	0	0
$958$	0	0
$959$	33.9272	1.09557
$960$	0	0
$961$	−5.93380	−0.191413
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.503308	−0.0162020
$966$	0	0
$967$	−43.3426	−1.39381	−0.696903	−	0.717166i	$-0.745439\pi$
$967$	−43.3426	−1.39381	−0.696903	+	0.717166i	$0.745439\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.78970	0.121617	0.0608087	−	0.998149i	$-0.480632\pi$
$971$	3.78970	0.121617	0.0608087	+	0.998149i	$0.480632\pi$
$972$	0	0
$973$	54.4055	1.74416
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.4669	−0.558816	−0.279408	−	0.960173i	$-0.590138\pi$
$977$	−17.4669	−0.558816	−0.279408	+	0.960173i	$0.590138\pi$
$978$	0	0
$979$	50.0349	1.59912
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.0265	−0.511165	−0.255582	−	0.966787i	$-0.582267\pi$
$983$	−16.0265	−0.511165	−0.255582	+	0.966787i	$0.582267\pi$
$984$	0	0
$985$	11.0066	0.350700
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.0132	0.318402
$990$	0	0
$991$	36.3923	1.15604	0.578019	−	0.816023i	$-0.303826\pi$
$991$	36.3923	1.15604	0.578019	+	0.816023i	$0.303826\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.01323	−0.0638237
$996$	0	0
$997$	−19.0331	−0.602784	−0.301392	−	0.953500i	$-0.597451\pi$
$997$	−19.0331	−0.602784	−0.301392	+	0.953500i	$0.597451\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.db.1.3		3
3.2	odd	2		3120.2.a.bh.1.3		3
4.3	odd	2		4680.2.a.bl.1.1		3
12.11	even	2		1560.2.a.p.1.1	✓	3
60.59	even	2		7800.2.a.bf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.p.1.1	✓	3	12.11	even	2
3120.2.a.bh.1.3		3	3.2	odd	2
4680.2.a.bl.1.1		3	4.3	odd	2
7800.2.a.bf.1.3		3	60.59	even	2
9360.2.a.db.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +2.50331 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +2.50331 q^{7} +3.27316 q^{11} +1.00000 q^{13} -4.50331 q^{17} -3.23014 q^{19} -2.50331 q^{23} +1.00000 q^{25} +7.77647 q^{29} +5.00662 q^{31} +2.50331 q^{35} +7.73345 q^{37} -7.73345 q^{41} -4.00000 q^{43} +9.00662 q^{47} -0.733451 q^{49} -6.27978 q^{53} +3.27316 q^{55} +9.27316 q^{61} +1.00000 q^{65} -15.5529 q^{67} +5.73345 q^{71} +15.7765 q^{73} +8.19374 q^{77} +4.72684 q^{79} -11.5529 q^{83} -4.50331 q^{85} +15.2864 q^{89} +2.50331 q^{91} -3.23014 q^{95} +12.5033 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 5 * q^7	\( 3 q + 3 q^{5} - 5 q^{7} + 3 q^{11} + 3 q^{13} - q^{17} - 4 q^{19} + 5 q^{23} + 3 q^{25} + 4 q^{29} - 10 q^{31} - 5 q^{35} + 5 q^{37} - 5 q^{41} - 12 q^{43} + 2 q^{47} + 16 q^{49} + 13 q^{53} + 3 q^{55} + 21 q^{61} + 3 q^{65} - 8 q^{67} - q^{71} + 28 q^{73} - 5 q^{77} + 21 q^{79} + 4 q^{83} - q^{85} - 11 q^{89} - 5 q^{91} - 4 q^{95} + 25 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 5 * q^7 + 3 * q^11 + 3 * q^13 - q^17 - 4 * q^19 + 5 * q^23 + 3 * q^25 + 4 * q^29 - 10 * q^31 - 5 * q^35 + 5 * q^37 - 5 * q^41 - 12 * q^43 + 2 * q^47 + 16 * q^49 + 13 * q^53 + 3 * q^55 + 21 * q^61 + 3 * q^65 - 8 * q^67 - q^71 + 28 * q^73 - 5 * q^77 + 21 * q^79 + 4 * q^83 - q^85 - 11 * q^89 - 5 * q^91 - 4 * q^95 + 25 * q^97

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