LMFDB - Embedded newform 9576.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-2.23607 q^{5} +1.00000 q^{7} +3.85410 q^{11} -6.23607 q^{13} +5.85410 q^{17} +1.00000 q^{19} -1.76393 q^{23} +6.61803 q^{29} -4.61803 q^{31} -2.23607 q^{35} -7.47214 q^{37} -9.56231 q^{41} +10.9443 q^{43} -7.00000 q^{47} +1.00000 q^{49} -4.85410 q^{53} -8.61803 q^{55} -2.70820 q^{59} +0.708204 q^{61} +13.9443 q^{65} -3.38197 q^{67} +13.9443 q^{71} -6.85410 q^{73} +3.85410 q^{77} +4.47214 q^{79} +13.6180 q^{83} -13.0902 q^{85} -5.23607 q^{89} -6.23607 q^{91} -2.23607 q^{95} -8.23607 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + q^{11} - 8 q^{13} + 5 q^{17} + 2 q^{19} - 8 q^{23} + 11 q^{29} - 7 q^{31} - 6 q^{37} + q^{41} + 4 q^{43} - 14 q^{47} + 2 q^{49} - 3 q^{53} - 15 q^{55} + 8 q^{59} - 12 q^{61} + 10 q^{65} - 9 q^{67} + 10 q^{71} - 7 q^{73} + q^{77} + 25 q^{83} - 15 q^{85} - 6 q^{89} - 8 q^{91} - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + q^11 - 8 * q^13 + 5 * q^17 + 2 * q^19 - 8 * q^23 + 11 * q^29 - 7 * q^31 - 6 * q^37 + q^41 + 4 * q^43 - 14 * q^47 + 2 * q^49 - 3 * q^53 - 15 * q^55 + 8 * q^59 - 12 * q^61 + 10 * q^65 - 9 * q^67 + 10 * q^71 - 7 * q^73 + q^77 + 25 * q^83 - 15 * q^85 - 6 * q^89 - 8 * q^91 - 12 * 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
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.85410	1.16206	0.581028	−	0.813884i	$-0.302651\pi$
$11$	3.85410	1.16206	0.581028	+	0.813884i	$0.302651\pi$
$12$	0	0
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.85410	1.41983	0.709914	−	0.704288i	$-0.248734\pi$
$17$	5.85410	1.41983	0.709914	+	0.704288i	$0.248734\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.76393	−0.367805	−0.183903	−	0.982944i	$-0.558873\pi$
$23$	−1.76393	−0.367805	−0.183903	+	0.982944i	$0.558873\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.61803	1.22894	0.614469	−	0.788941i	$-0.289370\pi$
$29$	6.61803	1.22894	0.614469	+	0.788941i	$0.289370\pi$
$30$	0	0
$31$	−4.61803	−0.829423	−0.414712	−	0.909953i	$-0.636118\pi$
$31$	−4.61803	−0.829423	−0.414712	+	0.909953i	$0.636118\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.23607	−0.377964
$36$	0	0
$37$	−7.47214	−1.22841	−0.614206	−	0.789146i	$-0.710524\pi$
$37$	−7.47214	−1.22841	−0.614206	+	0.789146i	$0.710524\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.56231	−1.49338	−0.746691	−	0.665171i	$-0.768358\pi$
$41$	−9.56231	−1.49338	−0.746691	+	0.665171i	$0.768358\pi$
$42$	0	0
$43$	10.9443	1.66899	0.834493	−	0.551019i	$-0.185761\pi$
$43$	10.9443	1.66899	0.834493	+	0.551019i	$0.185761\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.85410	−0.666762	−0.333381	−	0.942792i	$-0.608190\pi$
$53$	−4.85410	−0.666762	−0.333381	+	0.942792i	$0.608190\pi$
$54$	0	0
$55$	−8.61803	−1.16206
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.70820	−0.352578	−0.176289	−	0.984338i	$-0.556409\pi$
$59$	−2.70820	−0.352578	−0.176289	+	0.984338i	$0.556409\pi$
$60$	0	0
$61$	0.708204	0.0906762	0.0453381	−	0.998972i	$-0.485563\pi$
$61$	0.708204	0.0906762	0.0453381	+	0.998972i	$0.485563\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.9443	1.72957
$66$	0	0
$67$	−3.38197	−0.413173	−0.206586	−	0.978428i	$-0.566235\pi$
$67$	−3.38197	−0.413173	−0.206586	+	0.978428i	$0.566235\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.9443	1.65488	0.827440	−	0.561554i	$-0.189796\pi$
$71$	13.9443	1.65488	0.827440	+	0.561554i	$0.189796\pi$
$72$	0	0
$73$	−6.85410	−0.802212	−0.401106	−	0.916032i	$-0.631374\pi$
$73$	−6.85410	−0.802212	−0.401106	+	0.916032i	$0.631374\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.85410	0.439216
$78$	0	0
$79$	4.47214	0.503155	0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214	0.503155	0.251577	+	0.967837i	$0.419051\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.6180	1.49477	0.747387	−	0.664389i	$-0.231308\pi$
$83$	13.6180	1.49477	0.747387	+	0.664389i	$0.231308\pi$
$84$	0	0
$85$	−13.0902	−1.41983
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.23607	−0.555022	−0.277511	−	0.960722i	$-0.589510\pi$
$89$	−5.23607	−0.555022	−0.277511	+	0.960722i	$0.589510\pi$
$90$	0	0
$91$	−6.23607	−0.653718
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.23607	−0.229416
$96$	0	0
$97$	−8.23607	−0.836246	−0.418123	−	0.908390i	$-0.637312\pi$
$97$	−8.23607	−0.836246	−0.418123	+	0.908390i	$0.637312\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.00000	−0.0995037	−0.0497519	−	0.998762i	$-0.515843\pi$
$101$	−1.00000	−0.0995037	−0.0497519	+	0.998762i	$0.515843\pi$
$102$	0	0
$103$	3.76393	0.370871	0.185436	−	0.982656i	$-0.440630\pi$
$103$	3.76393	0.370871	0.185436	+	0.982656i	$0.440630\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.94427	0.381307	0.190654	−	0.981657i	$-0.438939\pi$
$107$	3.94427	0.381307	0.190654	+	0.981657i	$0.438939\pi$
$108$	0	0
$109$	−5.76393	−0.552085	−0.276042	−	0.961145i	$-0.589023\pi$
$109$	−5.76393	−0.552085	−0.276042	+	0.961145i	$0.589023\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.85410	0.832924	0.416462	−	0.909153i	$-0.363270\pi$
$113$	8.85410	0.832924	0.416462	+	0.909153i	$0.363270\pi$
$114$	0	0
$115$	3.94427	0.367805
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.85410	0.536645
$120$	0	0
$121$	3.85410	0.350373
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	15.7082	1.39388	0.696939	−	0.717131i	$-0.254545\pi$
$127$	15.7082	1.39388	0.696939	+	0.717131i	$0.254545\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.7984	1.29294	0.646470	−	0.762939i	$-0.276245\pi$
$131$	14.7984	1.29294	0.646470	+	0.762939i	$0.276245\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−2.52786	−0.214411	−0.107205	−	0.994237i	$-0.534190\pi$
$139$	−2.52786	−0.214411	−0.107205	+	0.994237i	$0.534190\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−24.0344	−2.00986
$144$	0	0
$145$	−14.7984	−1.22894
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.4721	−1.75907	−0.879533	−	0.475838i	$-0.842145\pi$
$149$	−21.4721	−1.75907	−0.879533	+	0.475838i	$0.842145\pi$
$150$	0	0
$151$	−2.14590	−0.174631	−0.0873154	−	0.996181i	$-0.527829\pi$
$151$	−2.14590	−0.174631	−0.0873154	+	0.996181i	$0.527829\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.3262	0.829423
$156$	0	0
$157$	3.85410	0.307591	0.153795	−	0.988103i	$-0.450850\pi$
$157$	3.85410	0.307591	0.153795	+	0.988103i	$0.450850\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.76393	−0.139017
$162$	0	0
$163$	4.09017	0.320367	0.160183	−	0.987087i	$-0.448791\pi$
$163$	4.09017	0.320367	0.160183	+	0.987087i	$0.448791\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.4721	−1.04251	−0.521253	−	0.853402i	$-0.674535\pi$
$167$	−13.4721	−1.04251	−0.521253	+	0.853402i	$0.674535\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.291796	−0.0221848	−0.0110924	−	0.999938i	$-0.503531\pi$
$173$	−0.291796	−0.0221848	−0.0110924	+	0.999938i	$0.503531\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.7984	−1.85352	−0.926759	−	0.375657i	$-0.877417\pi$
$179$	−24.7984	−1.85352	−0.926759	+	0.375657i	$0.877417\pi$
$180$	0	0
$181$	−22.6180	−1.68119	−0.840593	−	0.541668i	$-0.817793\pi$
$181$	−22.6180	−1.68119	−0.840593	+	0.541668i	$0.817793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.7082	1.22841
$186$	0	0
$187$	22.5623	1.64992
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.03444	−0.364279	−0.182140	−	0.983273i	$-0.558302\pi$
$191$	−5.03444	−0.364279	−0.182140	+	0.983273i	$0.558302\pi$
$192$	0	0
$193$	0.145898	0.0105020	0.00525099	−	0.999986i	$-0.498329\pi$
$193$	0.145898	0.0105020	0.00525099	+	0.999986i	$0.498329\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.56231	−0.325051	−0.162525	−	0.986704i	$-0.551964\pi$
$197$	−4.56231	−0.325051	−0.162525	+	0.986704i	$0.551964\pi$
$198$	0	0
$199$	−17.9443	−1.27204	−0.636018	−	0.771674i	$-0.719420\pi$
$199$	−17.9443	−1.27204	−0.636018	+	0.771674i	$0.719420\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.61803	0.464495
$204$	0	0
$205$	21.3820	1.49338
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.85410	0.266594
$210$	0	0
$211$	−1.43769	−0.0989749	−0.0494875	−	0.998775i	$-0.515759\pi$
$211$	−1.43769	−0.0989749	−0.0494875	+	0.998775i	$0.515759\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−24.4721	−1.66899
$216$	0	0
$217$	−4.61803	−0.313493
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−36.5066	−2.45570
$222$	0	0
$223$	−10.0557	−0.673381	−0.336691	−	0.941615i	$-0.609308\pi$
$223$	−10.0557	−0.673381	−0.336691	+	0.941615i	$0.609308\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.20163	0.0797547	0.0398774	−	0.999205i	$-0.487303\pi$
$227$	1.20163	0.0797547	0.0398774	+	0.999205i	$0.487303\pi$
$228$	0	0
$229$	−24.8328	−1.64100	−0.820499	−	0.571647i	$-0.806304\pi$
$229$	−24.8328	−1.64100	−0.820499	+	0.571647i	$0.806304\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.7984	−0.707425	−0.353712	−	0.935354i	$-0.615081\pi$
$233$	−10.7984	−0.707425	−0.353712	+	0.935354i	$0.615081\pi$
$234$	0	0
$235$	15.6525	1.02105
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.29180	−0.601036	−0.300518	−	0.953776i	$-0.597160\pi$
$239$	−9.29180	−0.601036	−0.300518	+	0.953776i	$0.597160\pi$
$240$	0	0
$241$	−25.1246	−1.61842	−0.809209	−	0.587521i	$-0.800104\pi$
$241$	−25.1246	−1.61842	−0.809209	+	0.587521i	$0.800104\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.23607	−0.142857
$246$	0	0
$247$	−6.23607	−0.396792
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.9787	1.19793	0.598963	−	0.800777i	$-0.295580\pi$
$251$	18.9787	1.19793	0.598963	+	0.800777i	$0.295580\pi$
$252$	0	0
$253$	−6.79837	−0.427410
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.90983	−0.493402	−0.246701	−	0.969092i	$-0.579346\pi$
$257$	−7.90983	−0.493402	−0.246701	+	0.969092i	$0.579346\pi$
$258$	0	0
$259$	−7.47214	−0.464296
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.3262	−1.68501	−0.842504	−	0.538690i	$-0.818919\pi$
$263$	−27.3262	−1.68501	−0.842504	+	0.538690i	$0.818919\pi$
$264$	0	0
$265$	10.8541	0.666762
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.14590	0.557635	0.278818	−	0.960344i	$-0.410058\pi$
$269$	9.14590	0.557635	0.278818	+	0.960344i	$0.410058\pi$
$270$	0	0
$271$	12.0344	0.731040	0.365520	−	0.930803i	$-0.380891\pi$
$271$	12.0344	0.731040	0.365520	+	0.930803i	$0.380891\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.6525	−0.640045	−0.320023	−	0.947410i	$-0.603691\pi$
$277$	−10.6525	−0.640045	−0.320023	+	0.947410i	$0.603691\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.2361	0.729943	0.364971	−	0.931019i	$-0.381079\pi$
$281$	12.2361	0.729943	0.364971	+	0.931019i	$0.381079\pi$
$282$	0	0
$283$	−5.79837	−0.344678	−0.172339	−	0.985038i	$-0.555132\pi$
$283$	−5.79837	−0.344678	−0.172339	+	0.985038i	$0.555132\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.56231	−0.564445
$288$	0	0
$289$	17.2705	1.01591
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.0000	1.34367	0.671837	−	0.740699i	$-0.265505\pi$
$293$	23.0000	1.34367	0.671837	+	0.740699i	$0.265505\pi$
$294$	0	0
$295$	6.05573	0.352578
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.0000	0.636146
$300$	0	0
$301$	10.9443	0.630817
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.58359	−0.0906762
$306$	0	0
$307$	−21.2705	−1.21397	−0.606986	−	0.794712i	$-0.707622\pi$
$307$	−21.2705	−1.21397	−0.606986	+	0.794712i	$0.707622\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.5066	−0.709183	−0.354591	−	0.935021i	$-0.615380\pi$
$311$	−12.5066	−0.709183	−0.354591	+	0.935021i	$0.615380\pi$
$312$	0	0
$313$	27.5967	1.55986	0.779930	−	0.625867i	$-0.215255\pi$
$313$	27.5967	1.55986	0.779930	+	0.625867i	$0.215255\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.3607	1.92989	0.964944	−	0.262456i	$-0.0845324\pi$
$317$	34.3607	1.92989	0.964944	+	0.262456i	$0.0845324\pi$
$318$	0	0
$319$	25.5066	1.42809
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.85410	0.325731
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.00000	−0.385922
$330$	0	0
$331$	−33.6180	−1.84781	−0.923907	−	0.382617i	$-0.875023\pi$
$331$	−33.6180	−1.84781	−0.923907	+	0.382617i	$0.875023\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.56231	0.413173
$336$	0	0
$337$	−33.8541	−1.84415	−0.922075	−	0.387011i	$-0.873508\pi$
$337$	−33.8541	−1.84415	−0.922075	+	0.387011i	$0.873508\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.7984	−0.963836
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.0344	0.807091	0.403546	−	0.914960i	$-0.367778\pi$
$347$	15.0344	0.807091	0.403546	+	0.914960i	$0.367778\pi$
$348$	0	0
$349$	21.0344	1.12595	0.562974	−	0.826475i	$-0.309657\pi$
$349$	21.0344	1.12595	0.562974	+	0.826475i	$0.309657\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.9787	−1.01014	−0.505068	−	0.863080i	$-0.668532\pi$
$353$	−18.9787	−1.01014	−0.505068	+	0.863080i	$0.668532\pi$
$354$	0	0
$355$	−31.1803	−1.65488
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.6180	−1.61596	−0.807979	−	0.589211i	$-0.799439\pi$
$359$	−30.6180	−1.61596	−0.807979	+	0.589211i	$0.799439\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.3262	0.802212
$366$	0	0
$367$	−23.1803	−1.21000	−0.605002	−	0.796224i	$-0.706828\pi$
$367$	−23.1803	−1.21000	−0.605002	+	0.796224i	$0.706828\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.85410	−0.252012
$372$	0	0
$373$	20.0902	1.04023	0.520115	−	0.854096i	$-0.325889\pi$
$373$	20.0902	1.04023	0.520115	+	0.854096i	$0.325889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−41.2705	−2.12554
$378$	0	0
$379$	−21.6525	−1.11221	−0.556106	−	0.831111i	$-0.687705\pi$
$379$	−21.6525	−1.11221	−0.556106	+	0.831111i	$0.687705\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.76393	0.192328	0.0961640	−	0.995366i	$-0.469343\pi$
$383$	3.76393	0.192328	0.0961640	+	0.995366i	$0.469343\pi$
$384$	0	0
$385$	−8.61803	−0.439216
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.32624	−0.219349	−0.109674	−	0.993968i	$-0.534981\pi$
$389$	−4.32624	−0.219349	−0.109674	+	0.993968i	$0.534981\pi$
$390$	0	0
$391$	−10.3262	−0.522220
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.0000	−0.503155
$396$	0	0
$397$	18.7639	0.941735	0.470867	−	0.882204i	$-0.343941\pi$
$397$	18.7639	0.941735	0.470867	+	0.882204i	$0.343941\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.506578	−0.0252973	−0.0126486	−	0.999920i	$-0.504026\pi$
$401$	−0.506578	−0.0252973	−0.0126486	+	0.999920i	$0.504026\pi$
$402$	0	0
$403$	28.7984	1.43455
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.7984	−1.42748
$408$	0	0
$409$	−8.85410	−0.437807	−0.218904	−	0.975746i	$-0.570248\pi$
$409$	−8.85410	−0.437807	−0.218904	+	0.975746i	$0.570248\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.70820	−0.133262
$414$	0	0
$415$	−30.4508	−1.49477
$416$	0	0
$417$	0	0
$418$	0	0
$419$	28.3050	1.38279	0.691394	−	0.722478i	$-0.256997\pi$
$419$	28.3050	1.38279	0.691394	+	0.722478i	$0.256997\pi$
$420$	0	0
$421$	−13.9443	−0.679602	−0.339801	−	0.940497i	$-0.610360\pi$
$421$	−13.9443	−0.679602	−0.339801	+	0.940497i	$0.610360\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.708204	0.0342724
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.8885	−1.34334	−0.671672	−	0.740849i	$-0.734424\pi$
$431$	−27.8885	−1.34334	−0.671672	+	0.740849i	$0.734424\pi$
$432$	0	0
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.76393	−0.0843803
$438$	0	0
$439$	−25.8885	−1.23559	−0.617796	−	0.786338i	$-0.711974\pi$
$439$	−25.8885	−1.23559	−0.617796	+	0.786338i	$0.711974\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.2148	−1.38804	−0.694018	−	0.719958i	$-0.744161\pi$
$443$	−29.2148	−1.38804	−0.694018	+	0.719958i	$0.744161\pi$
$444$	0	0
$445$	11.7082	0.555022
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.3820	−0.489955	−0.244978	−	0.969529i	$-0.578781\pi$
$449$	−10.3820	−0.489955	−0.244978	+	0.969529i	$0.578781\pi$
$450$	0	0
$451$	−36.8541	−1.73539
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.9443	0.653718
$456$	0	0
$457$	40.9230	1.91430	0.957148	−	0.289598i	$-0.0935217\pi$
$457$	40.9230	1.91430	0.957148	+	0.289598i	$0.0935217\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.9787	0.744203	0.372101	−	0.928192i	$-0.378637\pi$
$461$	15.9787	0.744203	0.372101	+	0.928192i	$0.378637\pi$
$462$	0	0
$463$	−23.3607	−1.08566	−0.542831	−	0.839842i	$-0.682648\pi$
$463$	−23.3607	−1.08566	−0.542831	+	0.839842i	$0.682648\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.9098	1.29151	0.645756	−	0.763543i	$-0.276542\pi$
$467$	27.9098	1.29151	0.645756	+	0.763543i	$0.276542\pi$
$468$	0	0
$469$	−3.38197	−0.156165
$470$	0	0
$471$	0	0
$472$	0	0
$473$	42.1803	1.93945
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−35.7426	−1.63312	−0.816562	−	0.577258i	$-0.804123\pi$
$479$	−35.7426	−1.63312	−0.816562	+	0.577258i	$0.804123\pi$
$480$	0	0
$481$	46.5967	2.12463
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.4164	0.836246
$486$	0	0
$487$	8.88854	0.402778	0.201389	−	0.979511i	$-0.435454\pi$
$487$	8.88854	0.402778	0.201389	+	0.979511i	$0.435454\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.9443	0.764684	0.382342	−	0.924021i	$-0.375118\pi$
$491$	16.9443	0.764684	0.382342	+	0.924021i	$0.375118\pi$
$492$	0	0
$493$	38.7426	1.74488
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.9443	0.625486
$498$	0	0
$499$	−15.6180	−0.699159	−0.349580	−	0.936907i	$-0.613676\pi$
$499$	−15.6180	−0.699159	−0.349580	+	0.936907i	$0.613676\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	2.23607	0.0995037
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.4721	−1.43930	−0.719651	−	0.694336i	$-0.755698\pi$
$509$	−32.4721	−1.43930	−0.719651	+	0.694336i	$0.755698\pi$
$510$	0	0
$511$	−6.85410	−0.303208
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.41641	−0.370871
$516$	0	0
$517$	−26.9787	−1.18652
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.8885	−0.915144	−0.457572	−	0.889173i	$-0.651281\pi$
$521$	−20.8885	−0.915144	−0.457572	+	0.889173i	$0.651281\pi$
$522$	0	0
$523$	−3.94427	−0.172471	−0.0862355	−	0.996275i	$-0.527484\pi$
$523$	−3.94427	−0.172471	−0.0862355	+	0.996275i	$0.527484\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−27.0344	−1.17764
$528$	0	0
$529$	−19.8885	−0.864719
$530$	0	0
$531$	0	0
$532$	0	0
$533$	59.6312	2.58291
$534$	0	0
$535$	−8.81966	−0.381307
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.85410	0.166008
$540$	0	0
$541$	−2.94427	−0.126584	−0.0632921	−	0.997995i	$-0.520160\pi$
$541$	−2.94427	−0.126584	−0.0632921	+	0.997995i	$0.520160\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.8885	0.552085
$546$	0	0
$547$	5.03444	0.215257	0.107629	−	0.994191i	$-0.465674\pi$
$547$	5.03444	0.215257	0.107629	+	0.994191i	$0.465674\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.61803	0.281938
$552$	0	0
$553$	4.47214	0.190175
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.0902	−0.724134	−0.362067	−	0.932152i	$-0.617929\pi$
$557$	−17.0902	−0.724134	−0.362067	+	0.932152i	$0.617929\pi$
$558$	0	0
$559$	−68.2492	−2.88663
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.2361	−0.768559	−0.384279	−	0.923217i	$-0.625550\pi$
$563$	−18.2361	−0.768559	−0.384279	+	0.923217i	$0.625550\pi$
$564$	0	0
$565$	−19.7984	−0.832924
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.3607	1.14702	0.573510	−	0.819199i	$-0.305581\pi$
$569$	27.3607	1.14702	0.573510	+	0.819199i	$0.305581\pi$
$570$	0	0
$571$	17.0000	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−37.2148	−1.54927	−0.774636	−	0.632408i	$-0.782067\pi$
$577$	−37.2148	−1.54927	−0.774636	+	0.632408i	$0.782067\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.6180	0.564971
$582$	0	0
$583$	−18.7082	−0.774815
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.472136	−0.0194871	−0.00974357	−	0.999953i	$-0.503102\pi$
$587$	−0.472136	−0.0194871	−0.00974357	+	0.999953i	$0.503102\pi$
$588$	0	0
$589$	−4.61803	−0.190283
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−42.7771	−1.75664	−0.878322	−	0.478069i	$-0.841337\pi$
$593$	−42.7771	−1.75664	−0.878322	+	0.478069i	$0.841337\pi$
$594$	0	0
$595$	−13.0902	−0.536645
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.6869	1.13126	0.565628	−	0.824660i	$-0.308634\pi$
$599$	27.6869	1.13126	0.565628	+	0.824660i	$0.308634\pi$
$600$	0	0
$601$	−32.7426	−1.33560	−0.667800	−	0.744341i	$-0.732764\pi$
$601$	−32.7426	−1.33560	−0.667800	+	0.744341i	$0.732764\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.61803	−0.350373
$606$	0	0
$607$	−19.1803	−0.778506	−0.389253	−	0.921131i	$-0.627267\pi$
$607$	−19.1803	−0.778506	−0.389253	+	0.921131i	$0.627267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	43.6525	1.76599
$612$	0	0
$613$	12.3820	0.500103	0.250051	−	0.968233i	$-0.419552\pi$
$613$	12.3820	0.500103	0.250051	+	0.968233i	$0.419552\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.27051	−0.131666	−0.0658329	−	0.997831i	$-0.520970\pi$
$617$	−3.27051	−0.131666	−0.0658329	+	0.997831i	$0.520970\pi$
$618$	0	0
$619$	29.6869	1.19322	0.596609	−	0.802532i	$-0.296514\pi$
$619$	29.6869	1.19322	0.596609	+	0.802532i	$0.296514\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.23607	−0.209779
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−43.7426	−1.74413
$630$	0	0
$631$	45.9443	1.82901	0.914506	−	0.404572i	$-0.132579\pi$
$631$	45.9443	1.82901	0.914506	+	0.404572i	$0.132579\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−35.1246	−1.39388
$636$	0	0
$637$	−6.23607	−0.247082
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.6180	0.419387	0.209694	−	0.977767i	$-0.432753\pi$
$641$	10.6180	0.419387	0.209694	+	0.977767i	$0.432753\pi$
$642$	0	0
$643$	1.76393	0.0695627	0.0347813	−	0.999395i	$-0.488927\pi$
$643$	1.76393	0.0695627	0.0347813	+	0.999395i	$0.488927\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.3607	0.839775	0.419887	−	0.907576i	$-0.362070\pi$
$647$	21.3607	0.839775	0.419887	+	0.907576i	$0.362070\pi$
$648$	0	0
$649$	−10.4377	−0.409715
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.5967	−0.962545	−0.481273	−	0.876571i	$-0.659825\pi$
$653$	−24.5967	−0.962545	−0.481273	+	0.876571i	$0.659825\pi$
$654$	0	0
$655$	−33.0902	−1.29294
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−51.1591	−1.99287	−0.996437	−	0.0843417i	$-0.973121\pi$
$659$	−51.1591	−1.99287	−0.996437	+	0.0843417i	$0.973121\pi$
$660$	0	0
$661$	−23.3050	−0.906458	−0.453229	−	0.891394i	$-0.649728\pi$
$661$	−23.3050	−0.906458	−0.453229	+	0.891394i	$0.649728\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.23607	−0.0867110
$666$	0	0
$667$	−11.6738	−0.452010
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.72949	0.105371
$672$	0	0
$673$	10.5623	0.407147	0.203573	−	0.979060i	$-0.434744\pi$
$673$	10.5623	0.407147	0.203573	+	0.979060i	$0.434744\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15.0344	0.577821	0.288910	−	0.957356i	$-0.406707\pi$
$677$	15.0344	0.577821	0.288910	+	0.957356i	$0.406707\pi$
$678$	0	0
$679$	−8.23607	−0.316071
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.7771	1.25418	0.627090	−	0.778947i	$-0.284246\pi$
$683$	32.7771	1.25418	0.627090	+	0.778947i	$0.284246\pi$
$684$	0	0
$685$	4.47214	0.170872
$686$	0	0
$687$	0	0
$688$	0	0
$689$	30.2705	1.15321
$690$	0	0
$691$	−28.1803	−1.07203	−0.536015	−	0.844208i	$-0.680071\pi$
$691$	−28.1803	−1.07203	−0.536015	+	0.844208i	$0.680071\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.65248	0.214411
$696$	0	0
$697$	−55.9787	−2.12034
$698$	0	0
$699$	0	0
$700$	0	0
$701$	42.8328	1.61777	0.808887	−	0.587965i	$-0.200071\pi$
$701$	42.8328	1.61777	0.808887	+	0.587965i	$0.200071\pi$
$702$	0	0
$703$	−7.47214	−0.281817
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.00000	−0.0376089
$708$	0	0
$709$	−37.2918	−1.40052	−0.700261	−	0.713887i	$-0.746933\pi$
$709$	−37.2918	−1.40052	−0.700261	+	0.713887i	$0.746933\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.14590	0.305066
$714$	0	0
$715$	53.7426	2.00986
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29.3050	1.09289	0.546445	−	0.837495i	$-0.315981\pi$
$719$	29.3050	1.09289	0.546445	+	0.837495i	$0.315981\pi$
$720$	0	0
$721$	3.76393	0.140176
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25.5410	0.947264	0.473632	−	0.880723i	$-0.342943\pi$
$727$	25.5410	0.947264	0.473632	+	0.880723i	$0.342943\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	64.0689	2.36967
$732$	0	0
$733$	−32.3607	−1.19527	−0.597634	−	0.801769i	$-0.703893\pi$
$733$	−32.3607	−1.19527	−0.597634	+	0.801769i	$0.703893\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.0344	−0.480130
$738$	0	0
$739$	−8.59675	−0.316236	−0.158118	−	0.987420i	$-0.550543\pi$
$739$	−8.59675	−0.316236	−0.158118	+	0.987420i	$0.550543\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.8885	−1.64680	−0.823400	−	0.567461i	$-0.807926\pi$
$743$	−44.8885	−1.64680	−0.823400	+	0.567461i	$0.807926\pi$
$744$	0	0
$745$	48.0132	1.75907
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.94427	0.144121
$750$	0	0
$751$	−23.9098	−0.872482	−0.436241	−	0.899830i	$-0.643690\pi$
$751$	−23.9098	−0.872482	−0.436241	+	0.899830i	$0.643690\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.79837	0.174631
$756$	0	0
$757$	50.3607	1.83039	0.915195	−	0.403011i	$-0.132036\pi$
$757$	50.3607	1.83039	0.915195	+	0.403011i	$0.132036\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.2361	−1.09606	−0.548028	−	0.836460i	$-0.684621\pi$
$761$	−30.2361	−1.09606	−0.548028	+	0.836460i	$0.684621\pi$
$762$	0	0
$763$	−5.76393	−0.208668
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.8885	0.609810
$768$	0	0
$769$	−28.9443	−1.04376	−0.521879	−	0.853020i	$-0.674769\pi$
$769$	−28.9443	−1.04376	−0.521879	+	0.853020i	$0.674769\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.76393	−0.315217	−0.157608	−	0.987502i	$-0.550378\pi$
$773$	−8.76393	−0.315217	−0.157608	+	0.987502i	$0.550378\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.56231	−0.342605
$780$	0	0
$781$	53.7426	1.92306
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.61803	−0.307591
$786$	0	0
$787$	−36.7214	−1.30898	−0.654488	−	0.756073i	$-0.727116\pi$
$787$	−36.7214	−1.30898	−0.654488	+	0.756073i	$0.727116\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.85410	0.314816
$792$	0	0
$793$	−4.41641	−0.156831
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.7984	−0.949247	−0.474624	−	0.880189i	$-0.657416\pi$
$797$	−26.7984	−0.949247	−0.474624	+	0.880189i	$0.657416\pi$
$798$	0	0
$799$	−40.9787	−1.44972
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26.4164	−0.932215
$804$	0	0
$805$	3.94427	0.139017
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22.2361	0.781779	0.390889	−	0.920438i	$-0.372168\pi$
$809$	22.2361	0.781779	0.390889	+	0.920438i	$0.372168\pi$
$810$	0	0
$811$	32.2492	1.13242	0.566212	−	0.824260i	$-0.308408\pi$
$811$	32.2492	1.13242	0.566212	+	0.824260i	$0.308408\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.14590	−0.320367
$816$	0	0
$817$	10.9443	0.382892
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.8328	−0.517669	−0.258834	−	0.965922i	$-0.583338\pi$
$821$	−14.8328	−0.517669	−0.258834	+	0.965922i	$0.583338\pi$
$822$	0	0
$823$	22.2918	0.777043	0.388522	−	0.921440i	$-0.372986\pi$
$823$	22.2918	0.777043	0.388522	+	0.921440i	$0.372986\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.3050	0.636526	0.318263	−	0.948003i	$-0.396901\pi$
$827$	18.3050	0.636526	0.318263	+	0.948003i	$0.396901\pi$
$828$	0	0
$829$	−39.4164	−1.36899	−0.684494	−	0.729018i	$-0.739977\pi$
$829$	−39.4164	−1.36899	−0.684494	+	0.729018i	$0.739977\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.85410	0.202833
$834$	0	0
$835$	30.1246	1.04251
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.52786	−0.0527477	−0.0263739	−	0.999652i	$-0.508396\pi$
$839$	−1.52786	−0.0527477	−0.0263739	+	0.999652i	$0.508396\pi$
$840$	0	0
$841$	14.7984	0.510289
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−57.8885	−1.99143
$846$	0	0
$847$	3.85410	0.132429
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13.1803	0.451816
$852$	0	0
$853$	54.0476	1.85055	0.925277	−	0.379291i	$-0.123832\pi$
$853$	54.0476	1.85055	0.925277	+	0.379291i	$0.123832\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.8541	−1.08812	−0.544058	−	0.839048i	$-0.683113\pi$
$857$	−31.8541	−1.08812	−0.544058	+	0.839048i	$0.683113\pi$
$858$	0	0
$859$	−14.2148	−0.485002	−0.242501	−	0.970151i	$-0.577968\pi$
$859$	−14.2148	−0.485002	−0.242501	+	0.970151i	$0.577968\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.0344	−0.716021	−0.358010	−	0.933718i	$-0.616545\pi$
$863$	−21.0344	−0.716021	−0.358010	+	0.933718i	$0.616545\pi$
$864$	0	0
$865$	0.652476	0.0221848
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17.2361	0.584694
$870$	0	0
$871$	21.0902	0.714613
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.1803	0.377964
$876$	0	0
$877$	16.2918	0.550135	0.275067	−	0.961425i	$-0.411300\pi$
$877$	16.2918	0.550135	0.275067	+	0.961425i	$0.411300\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.2705	0.682931	0.341465	−	0.939894i	$-0.389077\pi$
$881$	20.2705	0.682931	0.341465	+	0.939894i	$0.389077\pi$
$882$	0	0
$883$	36.1246	1.21569	0.607845	−	0.794056i	$-0.292034\pi$
$883$	36.1246	1.21569	0.607845	+	0.794056i	$0.292034\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.5279	−0.420645	−0.210322	−	0.977632i	$-0.567451\pi$
$887$	−12.5279	−0.420645	−0.210322	+	0.977632i	$0.567451\pi$
$888$	0	0
$889$	15.7082	0.526836
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7.00000	−0.234246
$894$	0	0
$895$	55.4508	1.85352
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30.5623	−1.01931
$900$	0	0
$901$	−28.4164	−0.946688
$902$	0	0
$903$	0	0
$904$	0	0
$905$	50.5755	1.68119
$906$	0	0
$907$	−18.8197	−0.624897	−0.312448	−	0.949935i	$-0.601149\pi$
$907$	−18.8197	−0.624897	−0.312448	+	0.949935i	$0.601149\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.5279	−0.779513	−0.389756	−	0.920918i	$-0.627441\pi$
$911$	−23.5279	−0.779513	−0.389756	+	0.920918i	$0.627441\pi$
$912$	0	0
$913$	52.4853	1.73701
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.7984	0.488685
$918$	0	0
$919$	−42.1935	−1.39183	−0.695917	−	0.718122i	$-0.745002\pi$
$919$	−42.1935	−1.39183	−0.695917	+	0.718122i	$0.745002\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−86.9574	−2.86224
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	22.7426	0.746162	0.373081	−	0.927799i	$-0.378301\pi$
$929$	22.7426	0.746162	0.373081	+	0.927799i	$0.378301\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−50.4508	−1.64992
$936$	0	0
$937$	23.6738	0.773388	0.386694	−	0.922208i	$-0.373617\pi$
$937$	23.6738	0.773388	0.386694	+	0.922208i	$0.373617\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	58.9443	1.92153	0.960764	−	0.277367i	$-0.0894616\pi$
$941$	58.9443	1.92153	0.960764	+	0.277367i	$0.0894616\pi$
$942$	0	0
$943$	16.8673	0.549273
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.5623	−1.31810	−0.659049	−	0.752100i	$-0.729041\pi$
$947$	−40.5623	−1.31810	−0.659049	+	0.752100i	$0.729041\pi$
$948$	0	0
$949$	42.7426	1.38748
$950$	0	0
$951$	0	0
$952$	0	0
$953$	60.2837	1.95278	0.976390	−	0.216016i	$-0.0693065\pi$
$953$	60.2837	1.95278	0.976390	+	0.216016i	$0.0693065\pi$
$954$	0	0
$955$	11.2574	0.364279
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−9.67376	−0.312057
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.326238	−0.0105020
$966$	0	0
$967$	45.8541	1.47457	0.737284	−	0.675583i	$-0.236108\pi$
$967$	45.8541	1.47457	0.737284	+	0.675583i	$0.236108\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.2361	1.03450	0.517252	−	0.855833i	$-0.326955\pi$
$971$	32.2361	1.03450	0.517252	+	0.855833i	$0.326955\pi$
$972$	0	0
$973$	−2.52786	−0.0810396
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.2918	1.48101	0.740503	−	0.672053i	$-0.234587\pi$
$977$	46.2918	1.48101	0.740503	+	0.672053i	$0.234587\pi$
$978$	0	0
$979$	−20.1803	−0.644966
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.1803	0.675548	0.337774	−	0.941227i	$-0.390326\pi$
$983$	21.1803	0.675548	0.337774	+	0.941227i	$0.390326\pi$
$984$	0	0
$985$	10.2016	0.325051
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.3050	−0.613862
$990$	0	0
$991$	31.5967	1.00370	0.501852	−	0.864954i	$-0.332652\pi$
$991$	31.5967	1.00370	0.501852	+	0.864954i	$0.332652\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	40.1246	1.27204
$996$	0	0
$997$	−17.5066	−0.554439	−0.277219	−	0.960807i	$-0.589413\pi$
$997$	−17.5066	−0.554439	−0.277219	+	0.960807i	$0.589413\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bm.1.1		2
3.2	odd	2		1064.2.a.a.1.2	✓	2
12.11	even	2		2128.2.a.n.1.1		2
21.20	even	2		7448.2.a.bc.1.1		2
24.5	odd	2		8512.2.a.bf.1.1		2
24.11	even	2		8512.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.a.1.2	✓	2	3.2	odd	2
2128.2.a.n.1.1		2	12.11	even	2
7448.2.a.bc.1.1		2	21.20	even	2
8512.2.a.i.1.2		2	24.11	even	2
8512.2.a.bf.1.1		2	24.5	odd	2
9576.2.a.bm.1.1		2	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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-2.23607 q^{5} +1.00000 q^{7} +3.85410 q^{11} -6.23607 q^{13} +5.85410 q^{17} +1.00000 q^{19} -1.76393 q^{23} +6.61803 q^{29} -4.61803 q^{31} -2.23607 q^{35} -7.47214 q^{37} -9.56231 q^{41} +10.9443 q^{43} -7.00000 q^{47} +1.00000 q^{49} -4.85410 q^{53} -8.61803 q^{55} -2.70820 q^{59} +0.708204 q^{61} +13.9443 q^{65} -3.38197 q^{67} +13.9443 q^{71} -6.85410 q^{73} +3.85410 q^{77} +4.47214 q^{79} +13.6180 q^{83} -13.0902 q^{85} -5.23607 q^{89} -6.23607 q^{91} -2.23607 q^{95} -8.23607 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + q^{11} - 8 q^{13} + 5 q^{17} + 2 q^{19} - 8 q^{23} + 11 q^{29} - 7 q^{31} - 6 q^{37} + q^{41} + 4 q^{43} - 14 q^{47} + 2 q^{49} - 3 q^{53} - 15 q^{55} + 8 q^{59} - 12 q^{61} + 10 q^{65} - 9 q^{67} + 10 q^{71} - 7 q^{73} + q^{77} + 25 q^{83} - 15 q^{85} - 6 q^{89} - 8 q^{91} - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + q^11 - 8 * q^13 + 5 * q^17 + 2 * q^19 - 8 * q^23 + 11 * q^29 - 7 * q^31 - 6 * q^37 + q^41 + 4 * q^43 - 14 * q^47 + 2 * q^49 - 3 * q^53 - 15 * q^55 + 8 * q^59 - 12 * q^61 + 10 * q^65 - 9 * q^67 + 10 * q^71 - 7 * q^73 + q^77 + 25 * q^83 - 15 * q^85 - 6 * q^89 - 8 * q^91 - 12 * q^97

Introduction

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