LMFDB - Embedded newform 7120.2.a.bk.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7120,2,Mod(1,7120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7120 = 2^{4} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7120.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:	$56.8534862392$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 19x^3 - 27x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.50065$ of defining polynomial
Character	$\chi$	$=$	7120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23408 q^{3} -1.00000 q^{5} +4.89614 q^{7} -1.47705 q^{9} +O(q^{10})$	$q-1.23408 q^{3} -1.00000 q^{5} +4.89614 q^{7} -1.47705 q^{9} -5.13396 q^{11} +0.293733 q^{13} +1.23408 q^{15} +4.78896 q^{17} -4.79227 q^{19} -6.04222 q^{21} -5.18988 q^{23} +1.00000 q^{25} +5.52503 q^{27} +8.87657 q^{29} +6.58148 q^{31} +6.33571 q^{33} -4.89614 q^{35} +0.840228 q^{37} -0.362490 q^{39} -2.51195 q^{41} -9.39183 q^{43} +1.47705 q^{45} -8.06738 q^{47} +16.9722 q^{49} -5.90996 q^{51} -11.4348 q^{53} +5.13396 q^{55} +5.91403 q^{57} -1.04378 q^{59} +6.77868 q^{61} -7.23185 q^{63} -0.293733 q^{65} +14.0551 q^{67} +6.40471 q^{69} -2.08800 q^{71} +6.37951 q^{73} -1.23408 q^{75} -25.1366 q^{77} -0.566895 q^{79} -2.38717 q^{81} -6.12572 q^{83} -4.78896 q^{85} -10.9544 q^{87} +1.00000 q^{89} +1.43816 q^{91} -8.12206 q^{93} +4.79227 q^{95} -3.68712 q^{97} +7.58312 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{3} - 8 q^{5} + 6 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 6 * q^3 - 8 * q^5 + 6 * q^7 + 12 * q^9	$ 8 q - 6 q^{3} - 8 q^{5} + 6 q^{7} + 12 q^{9} - 14 q^{11} - 7 q^{13} + 6 q^{15} + 17 q^{17} - 17 q^{19} + q^{23} + 8 q^{25} - 21 q^{27} + 10 q^{29} - q^{31} + 10 q^{33} - 6 q^{35} - 11 q^{37} + 5 q^{39} + 15 q^{41} + 5 q^{43} - 12 q^{45} - 12 q^{47} + 4 q^{49} - 35 q^{51} - q^{53} + 14 q^{55} + 15 q^{57} - 26 q^{59} + 13 q^{61} + 16 q^{63} + 7 q^{65} + 25 q^{67} - 5 q^{69} - 28 q^{71} - 17 q^{73} - 6 q^{75} + 7 q^{79} + 24 q^{81} - 44 q^{83} - 17 q^{85} + 12 q^{87} + 8 q^{89} - 27 q^{91} - 38 q^{93} + 17 q^{95} + q^{97} - q^{99}+O(q^{100}) $ 8 * q - 6 * q^3 - 8 * q^5 + 6 * q^7 + 12 * q^9 - 14 * q^11 - 7 * q^13 + 6 * q^15 + 17 * q^17 - 17 * q^19 + q^23 + 8 * q^25 - 21 * q^27 + 10 * q^29 - q^31 + 10 * q^33 - 6 * q^35 - 11 * q^37 + 5 * q^39 + 15 * q^41 + 5 * q^43 - 12 * q^45 - 12 * q^47 + 4 * q^49 - 35 * q^51 - q^53 + 14 * q^55 + 15 * q^57 - 26 * q^59 + 13 * q^61 + 16 * q^63 + 7 * q^65 + 25 * q^67 - 5 * q^69 - 28 * q^71 - 17 * q^73 - 6 * q^75 + 7 * q^79 + 24 * q^81 - 44 * q^83 - 17 * q^85 + 12 * q^87 + 8 * q^89 - 27 * q^91 - 38 * q^93 + 17 * q^95 + q^97 - q^99

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
$3$	−1.23408	−0.712495	−0.356248	−	0.934392i	$-0.615944\pi$
$3$	−1.23408	−0.712495	−0.356248	+	0.934392i	$0.615944\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.89614	1.85057	0.925284	−	0.379275i	$-0.123827\pi$
$7$	4.89614	1.85057	0.925284	+	0.379275i	$0.123827\pi$
$8$	0	0
$9$	−1.47705	−0.492350
$10$	0	0
$11$	−5.13396	−1.54795	−0.773974	−	0.633218i	$-0.781734\pi$
$11$	−5.13396	−1.54795	−0.773974	+	0.633218i	$0.781734\pi$
$12$	0	0
$13$	0.293733	0.0814670	0.0407335	−	0.999170i	$-0.487031\pi$
$13$	0.293733	0.0814670	0.0407335	+	0.999170i	$0.487031\pi$
$14$	0	0
$15$	1.23408	0.318638
$16$	0	0
$17$	4.78896	1.16149	0.580747	−	0.814084i	$-0.302760\pi$
$17$	4.78896	1.16149	0.580747	+	0.814084i	$0.302760\pi$
$18$	0	0
$19$	−4.79227	−1.09942	−0.549711	−	0.835355i	$-0.685262\pi$
$19$	−4.79227	−1.09942	−0.549711	+	0.835355i	$0.685262\pi$
$20$	0	0
$21$	−6.04222	−1.31852
$22$	0	0
$23$	−5.18988	−1.08216	−0.541082	−	0.840970i	$-0.681985\pi$
$23$	−5.18988	−1.08216	−0.541082	+	0.840970i	$0.681985\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.52503	1.06329
$28$	0	0
$29$	8.87657	1.64834	0.824169	−	0.566344i	$-0.191643\pi$
$29$	8.87657	1.64834	0.824169	+	0.566344i	$0.191643\pi$
$30$	0	0
$31$	6.58148	1.18207	0.591034	−	0.806646i	$-0.298720\pi$
$31$	6.58148	1.18207	0.591034	+	0.806646i	$0.298720\pi$
$32$	0	0
$33$	6.33571	1.10291
$34$	0	0
$35$	−4.89614	−0.827599
$36$	0	0
$37$	0.840228	0.138133	0.0690663	−	0.997612i	$-0.477998\pi$
$37$	0.840228	0.138133	0.0690663	+	0.997612i	$0.477998\pi$
$38$	0	0
$39$	−0.362490	−0.0580449
$40$	0	0
$41$	−2.51195	−0.392301	−0.196150	−	0.980574i	$-0.562844\pi$
$41$	−2.51195	−0.392301	−0.196150	+	0.980574i	$0.562844\pi$
$42$	0	0
$43$	−9.39183	−1.43224	−0.716120	−	0.697977i	$-0.754084\pi$
$43$	−9.39183	−1.43224	−0.716120	+	0.697977i	$0.754084\pi$
$44$	0	0
$45$	1.47705	0.220186
$46$	0	0
$47$	−8.06738	−1.17675	−0.588374	−	0.808589i	$-0.700232\pi$
$47$	−8.06738	−1.17675	−0.588374	+	0.808589i	$0.700232\pi$
$48$	0	0
$49$	16.9722	2.42460
$50$	0	0
$51$	−5.90996	−0.827559
$52$	0	0
$53$	−11.4348	−1.57070	−0.785348	−	0.619055i	$-0.787516\pi$
$53$	−11.4348	−1.57070	−0.785348	+	0.619055i	$0.787516\pi$
$54$	0	0
$55$	5.13396	0.692263
$56$	0	0
$57$	5.91403	0.783333
$58$	0	0
$59$	−1.04378	−0.135888	−0.0679441	−	0.997689i	$-0.521644\pi$
$59$	−1.04378	−0.135888	−0.0679441	+	0.997689i	$0.521644\pi$
$60$	0	0
$61$	6.77868	0.867921	0.433961	−	0.900932i	$-0.357116\pi$
$61$	6.77868	0.867921	0.433961	+	0.900932i	$0.357116\pi$
$62$	0	0
$63$	−7.23185	−0.911128
$64$	0	0
$65$	−0.293733	−0.0364331
$66$	0	0
$67$	14.0551	1.71710	0.858551	−	0.512727i	$-0.171365\pi$
$67$	14.0551	1.71710	0.858551	+	0.512727i	$0.171365\pi$
$68$	0	0
$69$	6.40471	0.771037
$70$	0	0
$71$	−2.08800	−0.247800	−0.123900	−	0.992295i	$-0.539540\pi$
$71$	−2.08800	−0.247800	−0.123900	+	0.992295i	$0.539540\pi$
$72$	0	0
$73$	6.37951	0.746665	0.373332	−	0.927698i	$-0.378215\pi$
$73$	6.37951	0.746665	0.373332	+	0.927698i	$0.378215\pi$
$74$	0	0
$75$	−1.23408	−0.142499
$76$	0	0
$77$	−25.1366	−2.86458
$78$	0	0
$79$	−0.566895	−0.0637807	−0.0318903	−	0.999491i	$-0.510153\pi$
$79$	−0.566895	−0.0637807	−0.0318903	+	0.999491i	$0.510153\pi$
$80$	0	0
$81$	−2.38717	−0.265241
$82$	0	0
$83$	−6.12572	−0.672385	−0.336192	−	0.941793i	$-0.609139\pi$
$83$	−6.12572	−0.672385	−0.336192	+	0.941793i	$0.609139\pi$
$84$	0	0
$85$	−4.78896	−0.519436
$86$	0	0
$87$	−10.9544	−1.17443
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	1.43816	0.150760
$92$	0	0
$93$	−8.12206	−0.842219
$94$	0	0
$95$	4.79227	0.491676
$96$	0	0
$97$	−3.68712	−0.374370	−0.187185	−	0.982325i	$-0.559936\pi$
$97$	−3.68712	−0.374370	−0.187185	+	0.982325i	$0.559936\pi$
$98$	0	0
$99$	7.58312	0.762132
$100$	0	0
$101$	−6.79147	−0.675777	−0.337888	−	0.941186i	$-0.609713\pi$
$101$	−6.79147	−0.675777	−0.337888	+	0.941186i	$0.609713\pi$
$102$	0	0
$103$	−10.1620	−1.00129	−0.500646	−	0.865652i	$-0.666904\pi$
$103$	−10.1620	−1.00129	−0.500646	+	0.865652i	$0.666904\pi$
$104$	0	0
$105$	6.04222	0.589661
$106$	0	0
$107$	−7.95873	−0.769399	−0.384700	−	0.923042i	$-0.625695\pi$
$107$	−7.95873	−0.769399	−0.384700	+	0.923042i	$0.625695\pi$
$108$	0	0
$109$	−3.91220	−0.374721	−0.187360	−	0.982291i	$-0.559993\pi$
$109$	−3.91220	−0.374721	−0.187360	+	0.982291i	$0.559993\pi$
$110$	0	0
$111$	−1.03691	−0.0984188
$112$	0	0
$113$	−8.18970	−0.770423	−0.385211	−	0.922828i	$-0.625871\pi$
$113$	−8.18970	−0.770423	−0.385211	+	0.922828i	$0.625871\pi$
$114$	0	0
$115$	5.18988	0.483958
$116$	0	0
$117$	−0.433859	−0.0401103
$118$	0	0
$119$	23.4475	2.14942
$120$	0	0
$121$	15.3576	1.39614
$122$	0	0
$123$	3.09995	0.279513
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.03100	0.180222	0.0901110	−	0.995932i	$-0.471278\pi$
$127$	2.03100	0.180222	0.0901110	+	0.995932i	$0.471278\pi$
$128$	0	0
$129$	11.5903	1.02046
$130$	0	0
$131$	−9.71902	−0.849155	−0.424578	−	0.905392i	$-0.639577\pi$
$131$	−9.71902	−0.849155	−0.424578	+	0.905392i	$0.639577\pi$
$132$	0	0
$133$	−23.4636	−2.03455
$134$	0	0
$135$	−5.52503	−0.475519
$136$	0	0
$137$	21.4520	1.83277	0.916386	−	0.400296i	$-0.131093\pi$
$137$	21.4520	1.83277	0.916386	+	0.400296i	$0.131093\pi$
$138$	0	0
$139$	0.595475	0.0505075	0.0252538	−	0.999681i	$-0.491961\pi$
$139$	0.595475	0.0505075	0.0252538	+	0.999681i	$0.491961\pi$
$140$	0	0
$141$	9.95578	0.838428
$142$	0	0
$143$	−1.50802	−0.126107
$144$	0	0
$145$	−8.87657	−0.737159
$146$	0	0
$147$	−20.9450	−1.72752
$148$	0	0
$149$	−16.4598	−1.34844	−0.674220	−	0.738531i	$-0.735520\pi$
$149$	−16.4598	−1.34844	−0.674220	+	0.738531i	$0.735520\pi$
$150$	0	0
$151$	−1.01236	−0.0823849	−0.0411924	−	0.999151i	$-0.513116\pi$
$151$	−1.01236	−0.0823849	−0.0411924	+	0.999151i	$0.513116\pi$
$152$	0	0
$153$	−7.07354	−0.571862
$154$	0	0
$155$	−6.58148	−0.528637
$156$	0	0
$157$	−3.66401	−0.292420	−0.146210	−	0.989254i	$-0.546707\pi$
$157$	−3.66401	−0.292420	−0.146210	+	0.989254i	$0.546707\pi$
$158$	0	0
$159$	14.1115	1.11911
$160$	0	0
$161$	−25.4104	−2.00262
$162$	0	0
$163$	0.396340	0.0310437	0.0155219	−	0.999880i	$-0.495059\pi$
$163$	0.396340	0.0310437	0.0155219	+	0.999880i	$0.495059\pi$
$164$	0	0
$165$	−6.33571	−0.493234
$166$	0	0
$167$	8.91272	0.689687	0.344843	−	0.938660i	$-0.387932\pi$
$167$	8.91272	0.689687	0.344843	+	0.938660i	$0.387932\pi$
$168$	0	0
$169$	−12.9137	−0.993363
$170$	0	0
$171$	7.07842	0.541300
$172$	0	0
$173$	20.4096	1.55171	0.775855	−	0.630911i	$-0.217319\pi$
$173$	20.4096	1.55171	0.775855	+	0.630911i	$0.217319\pi$
$174$	0	0
$175$	4.89614	0.370114
$176$	0	0
$177$	1.28810	0.0968197
$178$	0	0
$179$	3.85954	0.288476	0.144238	−	0.989543i	$-0.453927\pi$
$179$	3.85954	0.288476	0.144238	+	0.989543i	$0.453927\pi$
$180$	0	0
$181$	−4.85674	−0.360998	−0.180499	−	0.983575i	$-0.557771\pi$
$181$	−4.85674	−0.360998	−0.180499	+	0.983575i	$0.557771\pi$
$182$	0	0
$183$	−8.36542	−0.618390
$184$	0	0
$185$	−0.840228	−0.0617748
$186$	0	0
$187$	−24.5864	−1.79793
$188$	0	0
$189$	27.0513	1.96770
$190$	0	0
$191$	−7.44680	−0.538831	−0.269416	−	0.963024i	$-0.586831\pi$
$191$	−7.44680	−0.538831	−0.269416	+	0.963024i	$0.586831\pi$
$192$	0	0
$193$	−7.68593	−0.553245	−0.276623	−	0.960979i	$-0.589215\pi$
$193$	−7.68593	−0.553245	−0.276623	+	0.960979i	$0.589215\pi$
$194$	0	0
$195$	0.362490	0.0259585
$196$	0	0
$197$	−26.9802	−1.92226	−0.961131	−	0.276092i	$-0.910961\pi$
$197$	−26.9802	−1.92226	−0.961131	+	0.276092i	$0.910961\pi$
$198$	0	0
$199$	−5.24242	−0.371625	−0.185813	−	0.982585i	$-0.559492\pi$
$199$	−5.24242	−0.371625	−0.185813	+	0.982585i	$0.559492\pi$
$200$	0	0
$201$	−17.3451	−1.22343
$202$	0	0
$203$	43.4609	3.05036
$204$	0	0
$205$	2.51195	0.175442
$206$	0	0
$207$	7.66571	0.532804
$208$	0	0
$209$	24.6033	1.70185
$210$	0	0
$211$	0.509470	0.0350734	0.0175367	−	0.999846i	$-0.494418\pi$
$211$	0.509470	0.0350734	0.0175367	+	0.999846i	$0.494418\pi$
$212$	0	0
$213$	2.57675	0.176556
$214$	0	0
$215$	9.39183	0.640518
$216$	0	0
$217$	32.2239	2.18750
$218$	0	0
$219$	−7.87281	−0.531995
$220$	0	0
$221$	1.40668	0.0946235
$222$	0	0
$223$	−11.1310	−0.745384	−0.372692	−	0.927955i	$-0.621565\pi$
$223$	−11.1310	−0.745384	−0.372692	+	0.927955i	$0.621565\pi$
$224$	0	0
$225$	−1.47705	−0.0984701
$226$	0	0
$227$	1.50278	0.0997427	0.0498713	−	0.998756i	$-0.484119\pi$
$227$	1.50278	0.0997427	0.0498713	+	0.998756i	$0.484119\pi$
$228$	0	0
$229$	−8.07817	−0.533820	−0.266910	−	0.963721i	$-0.586003\pi$
$229$	−8.07817	−0.533820	−0.266910	+	0.963721i	$0.586003\pi$
$230$	0	0
$231$	31.0205	2.04100
$232$	0	0
$233$	9.45410	0.619359	0.309679	−	0.950841i	$-0.399778\pi$
$233$	9.45410	0.619359	0.309679	+	0.950841i	$0.399778\pi$
$234$	0	0
$235$	8.06738	0.526258
$236$	0	0
$237$	0.699593	0.0454434
$238$	0	0
$239$	9.38903	0.607326	0.303663	−	0.952780i	$-0.401790\pi$
$239$	9.38903	0.607326	0.303663	+	0.952780i	$0.401790\pi$
$240$	0	0
$241$	−1.14337	−0.0736510	−0.0368255	−	0.999322i	$-0.511725\pi$
$241$	−1.14337	−0.0736510	−0.0368255	+	0.999322i	$0.511725\pi$
$242$	0	0
$243$	−13.6291	−0.874310
$244$	0	0
$245$	−16.9722	−1.08431
$246$	0	0
$247$	−1.40765	−0.0895665
$248$	0	0
$249$	7.55962	0.479071
$250$	0	0
$251$	16.7613	1.05796	0.528981	−	0.848634i	$-0.322574\pi$
$251$	16.7613	1.05796	0.528981	+	0.848634i	$0.322574\pi$
$252$	0	0
$253$	26.6446	1.67513
$254$	0	0
$255$	5.90996	0.370096
$256$	0	0
$257$	17.2697	1.07725	0.538626	−	0.842545i	$-0.318944\pi$
$257$	17.2697	1.07725	0.538626	+	0.842545i	$0.318944\pi$
$258$	0	0
$259$	4.11387	0.255624
$260$	0	0
$261$	−13.1111	−0.811560
$262$	0	0
$263$	−27.3161	−1.68438	−0.842192	−	0.539177i	$-0.818735\pi$
$263$	−27.3161	−1.68438	−0.842192	+	0.539177i	$0.818735\pi$
$264$	0	0
$265$	11.4348	0.702437
$266$	0	0
$267$	−1.23408	−0.0755244
$268$	0	0
$269$	−8.56066	−0.521953	−0.260976	−	0.965345i	$-0.584044\pi$
$269$	−8.56066	−0.521953	−0.260976	+	0.965345i	$0.584044\pi$
$270$	0	0
$271$	6.04443	0.367173	0.183587	−	0.983004i	$-0.441229\pi$
$271$	6.04443	0.367173	0.183587	+	0.983004i	$0.441229\pi$
$272$	0	0
$273$	−1.77480	−0.107416
$274$	0	0
$275$	−5.13396	−0.309590
$276$	0	0
$277$	1.06300	0.0638694	0.0319347	−	0.999490i	$-0.489833\pi$
$277$	1.06300	0.0638694	0.0319347	+	0.999490i	$0.489833\pi$
$278$	0	0
$279$	−9.72118	−0.581992
$280$	0	0
$281$	−0.545199	−0.0325239	−0.0162619	−	0.999868i	$-0.505177\pi$
$281$	−0.545199	−0.0325239	−0.0162619	+	0.999868i	$0.505177\pi$
$282$	0	0
$283$	−14.1051	−0.838461	−0.419230	−	0.907880i	$-0.637700\pi$
$283$	−14.1051	−0.838461	−0.419230	+	0.907880i	$0.637700\pi$
$284$	0	0
$285$	−5.91403	−0.350317
$286$	0	0
$287$	−12.2989	−0.725980
$288$	0	0
$289$	5.93418	0.349069
$290$	0	0
$291$	4.55020	0.266737
$292$	0	0
$293$	−17.0930	−0.998586	−0.499293	−	0.866433i	$-0.666407\pi$
$293$	−17.0930	−0.998586	−0.499293	+	0.866433i	$0.666407\pi$
$294$	0	0
$295$	1.04378	0.0607711
$296$	0	0
$297$	−28.3653	−1.64592
$298$	0	0
$299$	−1.52444	−0.0881606
$300$	0	0
$301$	−45.9837	−2.65046
$302$	0	0
$303$	8.38121	0.481488
$304$	0	0
$305$	−6.77868	−0.388146
$306$	0	0
$307$	0.280920	0.0160329	0.00801646	−	0.999968i	$-0.497448\pi$
$307$	0.280920	0.0160329	0.00801646	+	0.999968i	$0.497448\pi$
$308$	0	0
$309$	12.5407	0.713416
$310$	0	0
$311$	12.9003	0.731507	0.365753	−	0.930712i	$-0.380811\pi$
$311$	12.9003	0.731507	0.365753	+	0.930712i	$0.380811\pi$
$312$	0	0
$313$	−3.04911	−0.172346	−0.0861729	−	0.996280i	$-0.527464\pi$
$313$	−3.04911	−0.172346	−0.0861729	+	0.996280i	$0.527464\pi$
$314$	0	0
$315$	7.23185	0.407469
$316$	0	0
$317$	1.96859	0.110567	0.0552835	−	0.998471i	$-0.482394\pi$
$317$	1.96859	0.110567	0.0552835	+	0.998471i	$0.482394\pi$
$318$	0	0
$319$	−45.5720	−2.55154
$320$	0	0
$321$	9.82169	0.548193
$322$	0	0
$323$	−22.9500	−1.27697
$324$	0	0
$325$	0.293733	0.0162934
$326$	0	0
$327$	4.82796	0.266987
$328$	0	0
$329$	−39.4990	−2.17765
$330$	0	0
$331$	−19.6542	−1.08029	−0.540147	−	0.841571i	$-0.681631\pi$
$331$	−19.6542	−1.08029	−0.540147	+	0.841571i	$0.681631\pi$
$332$	0	0
$333$	−1.24106	−0.0680096
$334$	0	0
$335$	−14.0551	−0.767912
$336$	0	0
$337$	−19.5743	−1.06628	−0.533140	−	0.846027i	$-0.678988\pi$
$337$	−19.5743	−1.06628	−0.533140	+	0.846027i	$0.678988\pi$
$338$	0	0
$339$	10.1067	0.548922
$340$	0	0
$341$	−33.7891	−1.82978
$342$	0	0
$343$	48.8254	2.63632
$344$	0	0
$345$	−6.40471	−0.344818
$346$	0	0
$347$	−13.2711	−0.712431	−0.356215	−	0.934404i	$-0.615933\pi$
$347$	−13.2711	−0.712431	−0.356215	+	0.934404i	$0.615933\pi$
$348$	0	0
$349$	25.7299	1.37729	0.688646	−	0.725097i	$-0.258205\pi$
$349$	25.7299	1.37729	0.688646	+	0.725097i	$0.258205\pi$
$350$	0	0
$351$	1.62289	0.0866233
$352$	0	0
$353$	33.4472	1.78022	0.890108	−	0.455750i	$-0.150629\pi$
$353$	33.4472	1.78022	0.890108	+	0.455750i	$0.150629\pi$
$354$	0	0
$355$	2.08800	0.110819
$356$	0	0
$357$	−28.9360	−1.53145
$358$	0	0
$359$	−2.49760	−0.131818	−0.0659090	−	0.997826i	$-0.520995\pi$
$359$	−2.49760	−0.131818	−0.0659090	+	0.997826i	$0.520995\pi$
$360$	0	0
$361$	3.96581	0.208727
$362$	0	0
$363$	−18.9524	−0.994744
$364$	0	0
$365$	−6.37951	−0.333919
$366$	0	0
$367$	31.2573	1.63162	0.815810	−	0.578320i	$-0.196292\pi$
$367$	31.2573	1.63162	0.815810	+	0.578320i	$0.196292\pi$
$368$	0	0
$369$	3.71028	0.193150
$370$	0	0
$371$	−55.9866	−2.90668
$372$	0	0
$373$	−35.2179	−1.82351	−0.911757	−	0.410730i	$-0.865274\pi$
$373$	−35.2179	−1.82351	−0.911757	+	0.410730i	$0.865274\pi$
$374$	0	0
$375$	1.23408	0.0637275
$376$	0	0
$377$	2.60735	0.134285
$378$	0	0
$379$	3.50658	0.180121	0.0900605	−	0.995936i	$-0.471294\pi$
$379$	3.50658	0.180121	0.0900605	+	0.995936i	$0.471294\pi$
$380$	0	0
$381$	−2.50641	−0.128407
$382$	0	0
$383$	−21.9576	−1.12198	−0.560991	−	0.827822i	$-0.689580\pi$
$383$	−21.9576	−1.12198	−0.560991	+	0.827822i	$0.689580\pi$
$384$	0	0
$385$	25.1366	1.28108
$386$	0	0
$387$	13.8722	0.705164
$388$	0	0
$389$	−13.3115	−0.674918	−0.337459	−	0.941340i	$-0.609567\pi$
$389$	−13.3115	−0.674918	−0.337459	+	0.941340i	$0.609567\pi$
$390$	0	0
$391$	−24.8541	−1.25693
$392$	0	0
$393$	11.9940	0.605019
$394$	0	0
$395$	0.566895	0.0285236
$396$	0	0
$397$	17.8358	0.895155	0.447578	−	0.894245i	$-0.352287\pi$
$397$	17.8358	0.895155	0.447578	+	0.894245i	$0.352287\pi$
$398$	0	0
$399$	28.9559	1.44961
$400$	0	0
$401$	−3.19387	−0.159494	−0.0797471	−	0.996815i	$-0.525411\pi$
$401$	−3.19387	−0.159494	−0.0797471	+	0.996815i	$0.525411\pi$
$402$	0	0
$403$	1.93320	0.0962996
$404$	0	0
$405$	2.38717	0.118619
$406$	0	0
$407$	−4.31370	−0.213822
$408$	0	0
$409$	−24.5244	−1.21266	−0.606328	−	0.795215i	$-0.707358\pi$
$409$	−24.5244	−1.21266	−0.606328	+	0.795215i	$0.707358\pi$
$410$	0	0
$411$	−26.4735	−1.30584
$412$	0	0
$413$	−5.11048	−0.251470
$414$	0	0
$415$	6.12572	0.300700
$416$	0	0
$417$	−0.734863	−0.0359864
$418$	0	0
$419$	−4.89589	−0.239180	−0.119590	−	0.992823i	$-0.538158\pi$
$419$	−4.89589	−0.239180	−0.119590	+	0.992823i	$0.538158\pi$
$420$	0	0
$421$	4.48965	0.218812	0.109406	−	0.993997i	$-0.465105\pi$
$421$	4.48965	0.218812	0.109406	+	0.993997i	$0.465105\pi$
$422$	0	0
$423$	11.9159	0.579372
$424$	0	0
$425$	4.78896	0.232299
$426$	0	0
$427$	33.1894	1.60615
$428$	0	0
$429$	1.86101	0.0898504
$430$	0	0
$431$	−35.8073	−1.72477	−0.862387	−	0.506249i	$-0.831032\pi$
$431$	−35.8073	−1.72477	−0.862387	+	0.506249i	$0.831032\pi$
$432$	0	0
$433$	−18.7569	−0.901399	−0.450700	−	0.892676i	$-0.648825\pi$
$433$	−18.7569	−0.901399	−0.450700	+	0.892676i	$0.648825\pi$
$434$	0	0
$435$	10.9544	0.525222
$436$	0	0
$437$	24.8713	1.18975
$438$	0	0
$439$	−31.4002	−1.49865	−0.749325	−	0.662202i	$-0.769622\pi$
$439$	−31.4002	−1.49865	−0.749325	+	0.662202i	$0.769622\pi$
$440$	0	0
$441$	−25.0688	−1.19375
$442$	0	0
$443$	25.6079	1.21667	0.608334	−	0.793681i	$-0.291838\pi$
$443$	25.6079	1.21667	0.608334	+	0.793681i	$0.291838\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	0	0
$447$	20.3127	0.960757
$448$	0	0
$449$	−21.3542	−1.00777	−0.503883	−	0.863772i	$-0.668096\pi$
$449$	−21.3542	−1.00777	−0.503883	+	0.863772i	$0.668096\pi$
$450$	0	0
$451$	12.8963	0.607261
$452$	0	0
$453$	1.24933	0.0586988
$454$	0	0
$455$	−1.43816	−0.0674220
$456$	0	0
$457$	−27.2671	−1.27550	−0.637750	−	0.770243i	$-0.720135\pi$
$457$	−27.2671	−1.27550	−0.637750	+	0.770243i	$0.720135\pi$
$458$	0	0
$459$	26.4592	1.23501
$460$	0	0
$461$	−18.9331	−0.881801	−0.440900	−	0.897556i	$-0.645341\pi$
$461$	−18.9331	−0.881801	−0.440900	+	0.897556i	$0.645341\pi$
$462$	0	0
$463$	−30.1659	−1.40193	−0.700965	−	0.713196i	$-0.747247\pi$
$463$	−30.1659	−1.40193	−0.700965	+	0.713196i	$0.747247\pi$
$464$	0	0
$465$	8.12206	0.376652
$466$	0	0
$467$	−34.1721	−1.58130	−0.790649	−	0.612270i	$-0.790257\pi$
$467$	−34.1721	−1.58130	−0.790649	+	0.612270i	$0.790257\pi$
$468$	0	0
$469$	68.8157	3.17762
$470$	0	0
$471$	4.52168	0.208348
$472$	0	0
$473$	48.2173	2.21703
$474$	0	0
$475$	−4.79227	−0.219884
$476$	0	0
$477$	16.8898	0.773333
$478$	0	0
$479$	−19.0555	−0.870666	−0.435333	−	0.900269i	$-0.643369\pi$
$479$	−19.0555	−0.870666	−0.435333	+	0.900269i	$0.643369\pi$
$480$	0	0
$481$	0.246803	0.0112532
$482$	0	0
$483$	31.3584	1.42686
$484$	0	0
$485$	3.68712	0.167424
$486$	0	0
$487$	−25.7093	−1.16500	−0.582501	−	0.812830i	$-0.697926\pi$
$487$	−25.7093	−1.16500	−0.582501	+	0.812830i	$0.697926\pi$
$488$	0	0
$489$	−0.489114	−0.0221185
$490$	0	0
$491$	−21.1649	−0.955158	−0.477579	−	0.878589i	$-0.658486\pi$
$491$	−21.1649	−0.955158	−0.477579	+	0.878589i	$0.658486\pi$
$492$	0	0
$493$	42.5096	1.91454
$494$	0	0
$495$	−7.58312	−0.340836
$496$	0	0
$497$	−10.2231	−0.458570
$498$	0	0
$499$	−25.0315	−1.12056	−0.560281	−	0.828302i	$-0.689307\pi$
$499$	−25.0315	−1.12056	−0.560281	+	0.828302i	$0.689307\pi$
$500$	0	0
$501$	−10.9990	−0.491399
$502$	0	0
$503$	−5.82665	−0.259797	−0.129899	−	0.991527i	$-0.541465\pi$
$503$	−5.82665	−0.259797	−0.129899	+	0.991527i	$0.541465\pi$
$504$	0	0
$505$	6.79147	0.302217
$506$	0	0
$507$	15.9365	0.707767
$508$	0	0
$509$	−13.7961	−0.611503	−0.305752	−	0.952111i	$-0.598908\pi$
$509$	−13.7961	−0.611503	−0.305752	+	0.952111i	$0.598908\pi$
$510$	0	0
$511$	31.2350	1.38175
$512$	0	0
$513$	−26.4774	−1.16901
$514$	0	0
$515$	10.1620	0.447792
$516$	0	0
$517$	41.4176	1.82154
$518$	0	0
$519$	−25.1870	−1.10559
$520$	0	0
$521$	34.4027	1.50721	0.753604	−	0.657329i	$-0.228314\pi$
$521$	34.4027	1.50721	0.753604	+	0.657329i	$0.228314\pi$
$522$	0	0
$523$	25.0955	1.09735	0.548674	−	0.836036i	$-0.315133\pi$
$523$	25.0955	1.09735	0.548674	+	0.836036i	$0.315133\pi$
$524$	0	0
$525$	−6.04222	−0.263704
$526$	0	0
$527$	31.5185	1.37297
$528$	0	0
$529$	3.93481	0.171079
$530$	0	0
$531$	1.54171	0.0669046
$532$	0	0
$533$	−0.737844	−0.0319596
$534$	0	0
$535$	7.95873	0.344086
$536$	0	0
$537$	−4.76298	−0.205538
$538$	0	0
$539$	−87.1347	−3.75316
$540$	0	0
$541$	16.2525	0.698750	0.349375	−	0.936983i	$-0.386394\pi$
$541$	16.2525	0.698750	0.349375	+	0.936983i	$0.386394\pi$
$542$	0	0
$543$	5.99359	0.257210
$544$	0	0
$545$	3.91220	0.167580
$546$	0	0
$547$	−17.3751	−0.742905	−0.371452	−	0.928452i	$-0.621140\pi$
$547$	−17.3751	−0.742905	−0.371452	+	0.928452i	$0.621140\pi$
$548$	0	0
$549$	−10.0125	−0.427321
$550$	0	0
$551$	−42.5389	−1.81222
$552$	0	0
$553$	−2.77560	−0.118030
$554$	0	0
$555$	1.03691	0.0440142
$556$	0	0
$557$	−43.3855	−1.83830	−0.919152	−	0.393904i	$-0.871124\pi$
$557$	−43.3855	−1.83830	−0.919152	+	0.393904i	$0.871124\pi$
$558$	0	0
$559$	−2.75869	−0.116680
$560$	0	0
$561$	30.3415	1.28102
$562$	0	0
$563$	−14.5950	−0.615108	−0.307554	−	0.951531i	$-0.599510\pi$
$563$	−14.5950	−0.615108	−0.307554	+	0.951531i	$0.599510\pi$
$564$	0	0
$565$	8.18970	0.344543
$566$	0	0
$567$	−11.6879	−0.490846
$568$	0	0
$569$	−25.4644	−1.06752	−0.533762	−	0.845635i	$-0.679222\pi$
$569$	−25.4644	−1.06752	−0.533762	+	0.845635i	$0.679222\pi$
$570$	0	0
$571$	−3.34046	−0.139794	−0.0698969	−	0.997554i	$-0.522267\pi$
$571$	−3.34046	−0.139794	−0.0698969	+	0.997554i	$0.522267\pi$
$572$	0	0
$573$	9.18993	0.383915
$574$	0	0
$575$	−5.18988	−0.216433
$576$	0	0
$577$	−28.9214	−1.20401	−0.602007	−	0.798491i	$-0.705632\pi$
$577$	−28.9214	−1.20401	−0.602007	+	0.798491i	$0.705632\pi$
$578$	0	0
$579$	9.48503	0.394185
$580$	0	0
$581$	−29.9924	−1.24429
$582$	0	0
$583$	58.7060	2.43135
$584$	0	0
$585$	0.433859	0.0179379
$586$	0	0
$587$	−16.5407	−0.682706	−0.341353	−	0.939935i	$-0.610885\pi$
$587$	−16.5407	−0.682706	−0.341353	+	0.939935i	$0.610885\pi$
$588$	0	0
$589$	−31.5402	−1.29959
$590$	0	0
$591$	33.2957	1.36960
$592$	0	0
$593$	−27.0667	−1.11150	−0.555748	−	0.831351i	$-0.687568\pi$
$593$	−27.0667	−1.11150	−0.555748	+	0.831351i	$0.687568\pi$
$594$	0	0
$595$	−23.4475	−0.961252
$596$	0	0
$597$	6.46955	0.264781
$598$	0	0
$599$	−7.11352	−0.290651	−0.145325	−	0.989384i	$-0.546423\pi$
$599$	−7.11352	−0.290651	−0.145325	+	0.989384i	$0.546423\pi$
$600$	0	0
$601$	−42.8950	−1.74972	−0.874861	−	0.484374i	$-0.839047\pi$
$601$	−42.8950	−1.74972	−0.874861	+	0.484374i	$0.839047\pi$
$602$	0	0
$603$	−20.7601	−0.845416
$604$	0	0
$605$	−15.3576	−0.624373
$606$	0	0
$607$	27.9087	1.13278	0.566390	−	0.824137i	$-0.308340\pi$
$607$	27.9087	1.13278	0.566390	+	0.824137i	$0.308340\pi$
$608$	0	0
$609$	−53.6342	−2.17337
$610$	0	0
$611$	−2.36966	−0.0958662
$612$	0	0
$613$	11.2291	0.453538	0.226769	−	0.973949i	$-0.427184\pi$
$613$	11.2291	0.453538	0.226769	+	0.973949i	$0.427184\pi$
$614$	0	0
$615$	−3.09995	−0.125002
$616$	0	0
$617$	−29.7394	−1.19726	−0.598631	−	0.801025i	$-0.704288\pi$
$617$	−29.7394	−1.19726	−0.598631	+	0.801025i	$0.704288\pi$
$618$	0	0
$619$	47.0184	1.88983	0.944915	−	0.327315i	$-0.106144\pi$
$619$	47.0184	1.88983	0.944915	+	0.327315i	$0.106144\pi$
$620$	0	0
$621$	−28.6742	−1.15066
$622$	0	0
$623$	4.89614	0.196160
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−30.3624	−1.21256
$628$	0	0
$629$	4.02382	0.160440
$630$	0	0
$631$	4.61993	0.183917	0.0919583	−	0.995763i	$-0.470687\pi$
$631$	4.61993	0.183917	0.0919583	+	0.995763i	$0.470687\pi$
$632$	0	0
$633$	−0.628726	−0.0249896
$634$	0	0
$635$	−2.03100	−0.0805978
$636$	0	0
$637$	4.98531	0.197525
$638$	0	0
$639$	3.08408	0.122004
$640$	0	0
$641$	−0.763355	−0.0301507	−0.0150754	−	0.999886i	$-0.504799\pi$
$641$	−0.763355	−0.0301507	−0.0150754	+	0.999886i	$0.504799\pi$
$642$	0	0
$643$	24.5483	0.968091	0.484046	−	0.875043i	$-0.339167\pi$
$643$	24.5483	0.968091	0.484046	+	0.875043i	$0.339167\pi$
$644$	0	0
$645$	−11.5903	−0.456366
$646$	0	0
$647$	20.9739	0.824567	0.412284	−	0.911056i	$-0.364731\pi$
$647$	20.9739	0.824567	0.412284	+	0.911056i	$0.364731\pi$
$648$	0	0
$649$	5.35871	0.210348
$650$	0	0
$651$	−39.7668	−1.55858
$652$	0	0
$653$	−0.521815	−0.0204202	−0.0102101	−	0.999948i	$-0.503250\pi$
$653$	−0.521815	−0.0204202	−0.0102101	+	0.999948i	$0.503250\pi$
$654$	0	0
$655$	9.71902	0.379754
$656$	0	0
$657$	−9.42285	−0.367621
$658$	0	0
$659$	−1.63679	−0.0637603	−0.0318802	−	0.999492i	$-0.510149\pi$
$659$	−1.63679	−0.0637603	−0.0318802	+	0.999492i	$0.510149\pi$
$660$	0	0
$661$	25.0711	0.975151	0.487576	−	0.873081i	$-0.337881\pi$
$661$	25.0711	0.975151	0.487576	+	0.873081i	$0.337881\pi$
$662$	0	0
$663$	−1.73595	−0.0674188
$664$	0	0
$665$	23.4636	0.909880
$666$	0	0
$667$	−46.0683	−1.78377
$668$	0	0
$669$	13.7365	0.531083
$670$	0	0
$671$	−34.8015	−1.34350
$672$	0	0
$673$	−33.2875	−1.28314	−0.641569	−	0.767065i	$-0.721716\pi$
$673$	−33.2875	−1.28314	−0.641569	+	0.767065i	$0.721716\pi$
$674$	0	0
$675$	5.52503	0.212659
$676$	0	0
$677$	17.5066	0.672834	0.336417	−	0.941713i	$-0.390785\pi$
$677$	17.5066	0.672834	0.336417	+	0.941713i	$0.390785\pi$
$678$	0	0
$679$	−18.0527	−0.692798
$680$	0	0
$681$	−1.85454	−0.0710662
$682$	0	0
$683$	46.2972	1.77151	0.885757	−	0.464150i	$-0.153640\pi$
$683$	46.2972	1.77151	0.885757	+	0.464150i	$0.153640\pi$
$684$	0	0
$685$	−21.4520	−0.819640
$686$	0	0
$687$	9.96909	0.380345
$688$	0	0
$689$	−3.35879	−0.127960
$690$	0	0
$691$	−4.46210	−0.169746	−0.0848732	−	0.996392i	$-0.527049\pi$
$691$	−4.46210	−0.169746	−0.0848732	+	0.996392i	$0.527049\pi$
$692$	0	0
$693$	37.1280	1.41038
$694$	0	0
$695$	−0.595475	−0.0225877
$696$	0	0
$697$	−12.0296	−0.455655
$698$	0	0
$699$	−11.6671	−0.441290
$700$	0	0
$701$	−5.13544	−0.193963	−0.0969815	−	0.995286i	$-0.530919\pi$
$701$	−5.13544	−0.193963	−0.0969815	+	0.995286i	$0.530919\pi$
$702$	0	0
$703$	−4.02659	−0.151866
$704$	0	0
$705$	−9.95578	−0.374956
$706$	0	0
$707$	−33.2520	−1.25057
$708$	0	0
$709$	4.95502	0.186090	0.0930449	−	0.995662i	$-0.470340\pi$
$709$	4.95502	0.186090	0.0930449	+	0.995662i	$0.470340\pi$
$710$	0	0
$711$	0.837333	0.0314024
$712$	0	0
$713$	−34.1571	−1.27919
$714$	0	0
$715$	1.50802	0.0563966
$716$	0	0
$717$	−11.5868	−0.432717
$718$	0	0
$719$	32.2241	1.20176	0.600878	−	0.799341i	$-0.294818\pi$
$719$	32.2241	1.20176	0.600878	+	0.799341i	$0.294818\pi$
$720$	0	0
$721$	−49.7546	−1.85296
$722$	0	0
$723$	1.41101	0.0524760
$724$	0	0
$725$	8.87657	0.329668
$726$	0	0
$727$	31.3167	1.16147	0.580735	−	0.814093i	$-0.302765\pi$
$727$	31.3167	1.16147	0.580735	+	0.814093i	$0.302765\pi$
$728$	0	0
$729$	23.9809	0.888183
$730$	0	0
$731$	−44.9771	−1.66354
$732$	0	0
$733$	−1.84406	−0.0681118	−0.0340559	−	0.999420i	$-0.510842\pi$
$733$	−1.84406	−0.0681118	−0.0340559	+	0.999420i	$0.510842\pi$
$734$	0	0
$735$	20.9450	0.772569
$736$	0	0
$737$	−72.1583	−2.65799
$738$	0	0
$739$	−23.1432	−0.851338	−0.425669	−	0.904879i	$-0.639961\pi$
$739$	−23.1432	−0.851338	−0.425669	+	0.904879i	$0.639961\pi$
$740$	0	0
$741$	1.73715	0.0638158
$742$	0	0
$743$	2.13174	0.0782059	0.0391029	−	0.999235i	$-0.487550\pi$
$743$	2.13174	0.0782059	0.0391029	+	0.999235i	$0.487550\pi$
$744$	0	0
$745$	16.4598	0.603041
$746$	0	0
$747$	9.04800	0.331049
$748$	0	0
$749$	−38.9671	−1.42383
$750$	0	0
$751$	−9.83764	−0.358981	−0.179490	−	0.983760i	$-0.557445\pi$
$751$	−9.83764	−0.358981	−0.179490	+	0.983760i	$0.557445\pi$
$752$	0	0
$753$	−20.6847	−0.753793
$754$	0	0
$755$	1.01236	0.0368436
$756$	0	0
$757$	−38.2109	−1.38880	−0.694400	−	0.719590i	$-0.744330\pi$
$757$	−38.2109	−1.38880	−0.694400	+	0.719590i	$0.744330\pi$
$758$	0	0
$759$	−32.8815	−1.19352
$760$	0	0
$761$	−5.67543	−0.205734	−0.102867	−	0.994695i	$-0.532802\pi$
$761$	−5.67543	−0.205734	−0.102867	+	0.994695i	$0.532802\pi$
$762$	0	0
$763$	−19.1547	−0.693446
$764$	0	0
$765$	7.07354	0.255745
$766$	0	0
$767$	−0.306592	−0.0110704
$768$	0	0
$769$	19.1784	0.691590	0.345795	−	0.938310i	$-0.387609\pi$
$769$	19.1784	0.691590	0.345795	+	0.938310i	$0.387609\pi$
$770$	0	0
$771$	−21.3121	−0.767537
$772$	0	0
$773$	26.8742	0.966598	0.483299	−	0.875455i	$-0.339438\pi$
$773$	26.8742	0.966598	0.483299	+	0.875455i	$0.339438\pi$
$774$	0	0
$775$	6.58148	0.236414
$776$	0	0
$777$	−5.07684	−0.182131
$778$	0	0
$779$	12.0379	0.431304
$780$	0	0
$781$	10.7197	0.383581
$782$	0	0
$783$	49.0433	1.75267
$784$	0	0
$785$	3.66401	0.130774
$786$	0	0
$787$	−11.4331	−0.407546	−0.203773	−	0.979018i	$-0.565320\pi$
$787$	−11.4331	−0.407546	−0.203773	+	0.979018i	$0.565320\pi$
$788$	0	0
$789$	33.7102	1.20012
$790$	0	0
$791$	−40.0980	−1.42572
$792$	0	0
$793$	1.99113	0.0707069
$794$	0	0
$795$	−14.1115	−0.500483
$796$	0	0
$797$	−38.2874	−1.35621	−0.678105	−	0.734965i	$-0.737199\pi$
$797$	−38.2874	−1.35621	−0.678105	+	0.734965i	$0.737199\pi$
$798$	0	0
$799$	−38.6344	−1.36679
$800$	0	0
$801$	−1.47705	−0.0521890
$802$	0	0
$803$	−32.7521	−1.15580
$804$	0	0
$805$	25.4104	0.895598
$806$	0	0
$807$	10.5645	0.371889
$808$	0	0
$809$	−4.98203	−0.175159	−0.0875794	−	0.996158i	$-0.527913\pi$
$809$	−4.98203	−0.175159	−0.0875794	+	0.996158i	$0.527913\pi$
$810$	0	0
$811$	−2.56566	−0.0900924	−0.0450462	−	0.998985i	$-0.514344\pi$
$811$	−2.56566	−0.0900924	−0.0450462	+	0.998985i	$0.514344\pi$
$812$	0	0
$813$	−7.45930	−0.261609
$814$	0	0
$815$	−0.396340	−0.0138832
$816$	0	0
$817$	45.0082	1.57464
$818$	0	0
$819$	−2.12424	−0.0742268
$820$	0	0
$821$	7.39548	0.258104	0.129052	−	0.991638i	$-0.458807\pi$
$821$	7.39548	0.258104	0.129052	+	0.991638i	$0.458807\pi$
$822$	0	0
$823$	−14.2404	−0.496389	−0.248195	−	0.968710i	$-0.579837\pi$
$823$	−14.2404	−0.496389	−0.248195	+	0.968710i	$0.579837\pi$
$824$	0	0
$825$	6.33571	0.220581
$826$	0	0
$827$	22.1225	0.769273	0.384637	−	0.923068i	$-0.374327\pi$
$827$	22.1225	0.769273	0.384637	+	0.923068i	$0.374327\pi$
$828$	0	0
$829$	43.2604	1.50250	0.751248	−	0.660021i	$-0.229452\pi$
$829$	43.2604	1.50250	0.751248	+	0.660021i	$0.229452\pi$
$830$	0	0
$831$	−1.31182	−0.0455067
$832$	0	0
$833$	81.2793	2.81616
$834$	0	0
$835$	−8.91272	−0.308437
$836$	0	0
$837$	36.3629	1.25689
$838$	0	0
$839$	6.71522	0.231835	0.115917	−	0.993259i	$-0.463019\pi$
$839$	6.71522	0.231835	0.115917	+	0.993259i	$0.463019\pi$
$840$	0	0
$841$	49.7935	1.71702
$842$	0	0
$843$	0.672818	0.0231731
$844$	0	0
$845$	12.9137	0.444245
$846$	0	0
$847$	75.1928	2.58365
$848$	0	0
$849$	17.4068	0.597399
$850$	0	0
$851$	−4.36068	−0.149482
$852$	0	0
$853$	−3.93209	−0.134632	−0.0673160	−	0.997732i	$-0.521444\pi$
$853$	−3.93209	−0.134632	−0.0673160	+	0.997732i	$0.521444\pi$
$854$	0	0
$855$	−7.07842	−0.242077
$856$	0	0
$857$	−4.62748	−0.158072	−0.0790358	−	0.996872i	$-0.525184\pi$
$857$	−4.62748	−0.158072	−0.0790358	+	0.996872i	$0.525184\pi$
$858$	0	0
$859$	19.7777	0.674805	0.337403	−	0.941360i	$-0.390452\pi$
$859$	19.7777	0.674805	0.337403	+	0.941360i	$0.390452\pi$
$860$	0	0
$861$	15.1778	0.517257
$862$	0	0
$863$	−8.03916	−0.273656	−0.136828	−	0.990595i	$-0.543691\pi$
$863$	−8.03916	−0.273656	−0.136828	+	0.990595i	$0.543691\pi$
$864$	0	0
$865$	−20.4096	−0.693946
$866$	0	0
$867$	−7.32324	−0.248710
$868$	0	0
$869$	2.91042	0.0987292
$870$	0	0
$871$	4.12845	0.139887
$872$	0	0
$873$	5.44607	0.184321
$874$	0	0
$875$	−4.89614	−0.165520
$876$	0	0
$877$	55.5811	1.87684	0.938420	−	0.345496i	$-0.112289\pi$
$877$	55.5811	1.87684	0.938420	+	0.345496i	$0.112289\pi$
$878$	0	0
$879$	21.0941	0.711488
$880$	0	0
$881$	44.8890	1.51235	0.756174	−	0.654370i	$-0.227066\pi$
$881$	44.8890	1.51235	0.756174	+	0.654370i	$0.227066\pi$
$882$	0	0
$883$	13.5457	0.455848	0.227924	−	0.973679i	$-0.426806\pi$
$883$	13.5457	0.455848	0.227924	+	0.973679i	$0.426806\pi$
$884$	0	0
$885$	−1.28810	−0.0432991
$886$	0	0
$887$	−36.1333	−1.21324	−0.606619	−	0.794992i	$-0.707475\pi$
$887$	−36.1333	−1.21324	−0.606619	+	0.794992i	$0.707475\pi$
$888$	0	0
$889$	9.94406	0.333513
$890$	0	0
$891$	12.2556	0.410579
$892$	0	0
$893$	38.6610	1.29374
$894$	0	0
$895$	−3.85954	−0.129010
$896$	0	0
$897$	1.88128	0.0628141
$898$	0	0
$899$	58.4210	1.94845
$900$	0	0
$901$	−54.7610	−1.82435
$902$	0	0
$903$	56.7475	1.88844
$904$	0	0
$905$	4.85674	0.161443
$906$	0	0
$907$	41.6649	1.38346	0.691730	−	0.722156i	$-0.256849\pi$
$907$	41.6649	1.38346	0.691730	+	0.722156i	$0.256849\pi$
$908$	0	0
$909$	10.0314	0.332719
$910$	0	0
$911$	55.3548	1.83399	0.916993	−	0.398903i	$-0.130609\pi$
$911$	55.3548	1.83399	0.916993	+	0.398903i	$0.130609\pi$
$912$	0	0
$913$	31.4492	1.04082
$914$	0	0
$915$	8.36542	0.276552
$916$	0	0
$917$	−47.5857	−1.57142
$918$	0	0
$919$	−9.44714	−0.311633	−0.155816	−	0.987786i	$-0.549801\pi$
$919$	−9.44714	−0.311633	−0.155816	+	0.987786i	$0.549801\pi$
$920$	0	0
$921$	−0.346677	−0.0114234
$922$	0	0
$923$	−0.613315	−0.0201875
$924$	0	0
$925$	0.840228	0.0276265
$926$	0	0
$927$	15.0098	0.492987
$928$	0	0
$929$	−23.0801	−0.757234	−0.378617	−	0.925553i	$-0.623600\pi$
$929$	−23.0801	−0.757234	−0.378617	+	0.925553i	$0.623600\pi$
$930$	0	0
$931$	−81.3353	−2.66566
$932$	0	0
$933$	−15.9199	−0.521195
$934$	0	0
$935$	24.5864	0.804060
$936$	0	0
$937$	31.4855	1.02859	0.514293	−	0.857615i	$-0.328054\pi$
$937$	31.4855	1.02859	0.514293	+	0.857615i	$0.328054\pi$
$938$	0	0
$939$	3.76284	0.122796
$940$	0	0
$941$	16.7640	0.546492	0.273246	−	0.961944i	$-0.411903\pi$
$941$	16.7640	0.546492	0.273246	+	0.961944i	$0.411903\pi$
$942$	0	0
$943$	13.0367	0.424534
$944$	0	0
$945$	−27.0513	−0.879980
$946$	0	0
$947$	59.7924	1.94299	0.971497	−	0.237053i	$-0.0761816\pi$
$947$	59.7924	1.94299	0.971497	+	0.237053i	$0.0761816\pi$
$948$	0	0
$949$	1.87387	0.0608285
$950$	0	0
$951$	−2.42939	−0.0787785
$952$	0	0
$953$	17.8565	0.578428	0.289214	−	0.957264i	$-0.406606\pi$
$953$	17.8565	0.578428	0.289214	+	0.957264i	$0.406606\pi$
$954$	0	0
$955$	7.44680	0.240973
$956$	0	0
$957$	56.2394	1.81796
$958$	0	0
$959$	105.032	3.39167
$960$	0	0
$961$	12.3159	0.397287
$962$	0	0
$963$	11.7554	0.378814
$964$	0	0
$965$	7.68593	0.247419
$966$	0	0
$967$	31.9918	1.02879	0.514393	−	0.857554i	$-0.328017\pi$
$967$	31.9918	1.02879	0.514393	+	0.857554i	$0.328017\pi$
$968$	0	0
$969$	28.3221	0.909836
$970$	0	0
$971$	−13.8839	−0.445555	−0.222778	−	0.974869i	$-0.571512\pi$
$971$	−13.8839	−0.445555	−0.222778	+	0.974869i	$0.571512\pi$
$972$	0	0
$973$	2.91553	0.0934676
$974$	0	0
$975$	−0.362490	−0.0116090
$976$	0	0
$977$	5.04704	0.161469	0.0807346	−	0.996736i	$-0.474273\pi$
$977$	5.04704	0.161469	0.0807346	+	0.996736i	$0.474273\pi$
$978$	0	0
$979$	−5.13396	−0.164082
$980$	0	0
$981$	5.77852	0.184494
$982$	0	0
$983$	26.6803	0.850971	0.425485	−	0.904965i	$-0.360103\pi$
$983$	26.6803	0.850971	0.425485	+	0.904965i	$0.360103\pi$
$984$	0	0
$985$	26.9802	0.859662
$986$	0	0
$987$	48.7449	1.55157
$988$	0	0
$989$	48.7424	1.54992
$990$	0	0
$991$	37.2339	1.18277	0.591386	−	0.806388i	$-0.298581\pi$
$991$	37.2339	1.18277	0.591386	+	0.806388i	$0.298581\pi$
$992$	0	0
$993$	24.2549	0.769705
$994$	0	0
$995$	5.24242	0.166196
$996$	0	0
$997$	−15.6534	−0.495748	−0.247874	−	0.968792i	$-0.579732\pi$
$997$	−15.6534	−0.495748	−0.247874	+	0.968792i	$0.579732\pi$
$998$	0	0
$999$	4.64228	0.146875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7120.2.a.bk.1.4		8
4.3	odd	2		445.2.a.g.1.1	✓	8
12.11	even	2		4005.2.a.p.1.8		8
20.19	odd	2		2225.2.a.l.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.g.1.1	✓	8	4.3	odd	2
2225.2.a.l.1.8		8	20.19	odd	2
4005.2.a.p.1.8		8	12.11	even	2
7120.2.a.bk.1.4		8	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:	\( N \)	\(=\)	\( 7120 = 2^{4} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7120.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23408 q^{3} -1.00000 q^{5} +4.89614 q^{7} -1.47705 q^{9} +O(q^{10})\)	\(q-1.23408 q^{3} -1.00000 q^{5} +4.89614 q^{7} -1.47705 q^{9} -5.13396 q^{11} +0.293733 q^{13} +1.23408 q^{15} +4.78896 q^{17} -4.79227 q^{19} -6.04222 q^{21} -5.18988 q^{23} +1.00000 q^{25} +5.52503 q^{27} +8.87657 q^{29} +6.58148 q^{31} +6.33571 q^{33} -4.89614 q^{35} +0.840228 q^{37} -0.362490 q^{39} -2.51195 q^{41} -9.39183 q^{43} +1.47705 q^{45} -8.06738 q^{47} +16.9722 q^{49} -5.90996 q^{51} -11.4348 q^{53} +5.13396 q^{55} +5.91403 q^{57} -1.04378 q^{59} +6.77868 q^{61} -7.23185 q^{63} -0.293733 q^{65} +14.0551 q^{67} +6.40471 q^{69} -2.08800 q^{71} +6.37951 q^{73} -1.23408 q^{75} -25.1366 q^{77} -0.566895 q^{79} -2.38717 q^{81} -6.12572 q^{83} -4.78896 q^{85} -10.9544 q^{87} +1.00000 q^{89} +1.43816 q^{91} -8.12206 q^{93} +4.79227 q^{95} -3.68712 q^{97} +7.58312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 8 q^{5} + 6 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 6 * q^3 - 8 * q^5 + 6 * q^7 + 12 * q^9	\( 8 q - 6 q^{3} - 8 q^{5} + 6 q^{7} + 12 q^{9} - 14 q^{11} - 7 q^{13} + 6 q^{15} + 17 q^{17} - 17 q^{19} + q^{23} + 8 q^{25} - 21 q^{27} + 10 q^{29} - q^{31} + 10 q^{33} - 6 q^{35} - 11 q^{37} + 5 q^{39} + 15 q^{41} + 5 q^{43} - 12 q^{45} - 12 q^{47} + 4 q^{49} - 35 q^{51} - q^{53} + 14 q^{55} + 15 q^{57} - 26 q^{59} + 13 q^{61} + 16 q^{63} + 7 q^{65} + 25 q^{67} - 5 q^{69} - 28 q^{71} - 17 q^{73} - 6 q^{75} + 7 q^{79} + 24 q^{81} - 44 q^{83} - 17 q^{85} + 12 q^{87} + 8 q^{89} - 27 q^{91} - 38 q^{93} + 17 q^{95} + q^{97} - q^{99}+O(q^{100}) \) 8 * q - 6 * q^3 - 8 * q^5 + 6 * q^7 + 12 * q^9 - 14 * q^11 - 7 * q^13 + 6 * q^15 + 17 * q^17 - 17 * q^19 + q^23 + 8 * q^25 - 21 * q^27 + 10 * q^29 - q^31 + 10 * q^33 - 6 * q^35 - 11 * q^37 + 5 * q^39 + 15 * q^41 + 5 * q^43 - 12 * q^45 - 12 * q^47 + 4 * q^49 - 35 * q^51 - q^53 + 14 * q^55 + 15 * q^57 - 26 * q^59 + 13 * q^61 + 16 * q^63 + 7 * q^65 + 25 * q^67 - 5 * q^69 - 28 * q^71 - 17 * q^73 - 6 * q^75 + 7 * q^79 + 24 * q^81 - 44 * q^83 - 17 * q^85 + 12 * q^87 + 8 * q^89 - 27 * q^91 - 38 * q^93 + 17 * q^95 + q^97 - q^99

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