LMFDB - Embedded newform 5520.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.44949 q^{11} +4.44949 q^{13} +1.00000 q^{15} -5.44949 q^{17} -4.44949 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.3485 q^{29} -0.101021 q^{31} +2.44949 q^{33} -1.00000 q^{35} +3.89898 q^{37} -4.44949 q^{39} +5.44949 q^{41} -2.00000 q^{43} -1.00000 q^{45} -8.44949 q^{47} -6.00000 q^{49} +5.44949 q^{51} +0.550510 q^{53} +2.44949 q^{55} +4.44949 q^{57} -10.3485 q^{59} +0.651531 q^{61} +1.00000 q^{63} -4.44949 q^{65} +7.00000 q^{67} -1.00000 q^{69} +4.34847 q^{71} -5.34847 q^{73} -1.00000 q^{75} -2.44949 q^{77} +4.00000 q^{79} +1.00000 q^{81} -15.2474 q^{83} +5.44949 q^{85} -10.3485 q^{87} -16.8990 q^{89} +4.44949 q^{91} +0.101021 q^{93} +4.44949 q^{95} +3.10102 q^{97} -2.44949 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 10 q^{31} - 2 q^{35} - 2 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 12 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 2 q^{63} - 4 q^{65} + 14 q^{67} - 2 q^{69} - 6 q^{71} + 4 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} - 24 q^{89} + 4 q^{91} + 10 q^{93} + 4 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 10 * q^31 - 2 * q^35 - 2 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 - 12 * q^47 - 12 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^57 - 6 * q^59 + 16 * q^61 + 2 * q^63 - 4 * q^65 + 14 * q^67 - 2 * q^69 - 6 * q^71 + 4 * q^73 - 2 * q^75 + 8 * q^79 + 2 * q^81 - 6 * q^83 + 6 * q^85 - 6 * q^87 - 24 * q^89 + 4 * q^91 + 10 * q^93 + 4 * q^95 + 16 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	4.44949	1.23407	0.617033	−	0.786937i	$-0.288334\pi$
$13$	4.44949	1.23407	0.617033	+	0.786937i	$0.288334\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−5.44949	−1.32170	−0.660848	−	0.750520i	$-0.729803\pi$
$17$	−5.44949	−1.32170	−0.660848	+	0.750520i	$0.729803\pi$
$18$	0	0
$19$	−4.44949	−1.02078	−0.510391	−	0.859942i	$-0.670499\pi$
$19$	−4.44949	−1.02078	−0.510391	+	0.859942i	$0.670499\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.3485	1.92166	0.960831	−	0.277134i	$-0.0893846\pi$
$29$	10.3485	1.92166	0.960831	+	0.277134i	$0.0893846\pi$
$30$	0	0
$31$	−0.101021	−0.0181438	−0.00907191	−	0.999959i	$-0.502888\pi$
$31$	−0.101021	−0.0181438	−0.00907191	+	0.999959i	$0.502888\pi$
$32$	0	0
$33$	2.44949	0.426401
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.89898	0.640988	0.320494	−	0.947250i	$-0.396151\pi$
$37$	3.89898	0.640988	0.320494	+	0.947250i	$0.396151\pi$
$38$	0	0
$39$	−4.44949	−0.712489
$40$	0	0
$41$	5.44949	0.851067	0.425534	−	0.904943i	$-0.360086\pi$
$41$	5.44949	0.851067	0.425534	+	0.904943i	$0.360086\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.44949	−1.23248	−0.616242	−	0.787557i	$-0.711346\pi$
$47$	−8.44949	−1.23248	−0.616242	+	0.787557i	$0.711346\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	5.44949	0.763081
$52$	0	0
$53$	0.550510	0.0756184	0.0378092	−	0.999285i	$-0.487962\pi$
$53$	0.550510	0.0756184	0.0378092	+	0.999285i	$0.487962\pi$
$54$	0	0
$55$	2.44949	0.330289
$56$	0	0
$57$	4.44949	0.589349
$58$	0	0
$59$	−10.3485	−1.34726	−0.673628	−	0.739071i	$-0.735265\pi$
$59$	−10.3485	−1.34726	−0.673628	+	0.739071i	$0.735265\pi$
$60$	0	0
$61$	0.651531	0.0834200	0.0417100	−	0.999130i	$-0.486719\pi$
$61$	0.651531	0.0834200	0.0417100	+	0.999130i	$0.486719\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−4.44949	−0.551891
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	4.34847	0.516068	0.258034	−	0.966136i	$-0.416925\pi$
$71$	4.34847	0.516068	0.258034	+	0.966136i	$0.416925\pi$
$72$	0	0
$73$	−5.34847	−0.625991	−0.312995	−	0.949755i	$-0.601332\pi$
$73$	−5.34847	−0.625991	−0.312995	+	0.949755i	$0.601332\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−2.44949	−0.279145
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.2474	−1.67362	−0.836812	−	0.547490i	$-0.815584\pi$
$83$	−15.2474	−1.67362	−0.836812	+	0.547490i	$0.815584\pi$
$84$	0	0
$85$	5.44949	0.591080
$86$	0	0
$87$	−10.3485	−1.10947
$88$	0	0
$89$	−16.8990	−1.79129	−0.895644	−	0.444771i	$-0.853285\pi$
$89$	−16.8990	−1.79129	−0.895644	+	0.444771i	$0.853285\pi$
$90$	0	0
$91$	4.44949	0.466433
$92$	0	0
$93$	0.101021	0.0104753
$94$	0	0
$95$	4.44949	0.456508
$96$	0	0
$97$	3.10102	0.314861	0.157430	−	0.987530i	$-0.449679\pi$
$97$	3.10102	0.314861	0.157430	+	0.987530i	$0.449679\pi$
$98$	0	0
$99$	−2.44949	−0.246183
$100$	0	0
$101$	−12.5505	−1.24882	−0.624411	−	0.781096i	$-0.714661\pi$
$101$	−12.5505	−1.24882	−0.624411	+	0.781096i	$0.714661\pi$
$102$	0	0
$103$	12.6969	1.25107	0.625533	−	0.780198i	$-0.284881\pi$
$103$	12.6969	1.25107	0.625533	+	0.780198i	$0.284881\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	5.44949	0.526822	0.263411	−	0.964684i	$-0.415152\pi$
$107$	5.44949	0.526822	0.263411	+	0.964684i	$0.415152\pi$
$108$	0	0
$109$	11.5505	1.10634	0.553169	−	0.833069i	$-0.313418\pi$
$109$	11.5505	1.10634	0.553169	+	0.833069i	$0.313418\pi$
$110$	0	0
$111$	−3.89898	−0.370075
$112$	0	0
$113$	−1.65153	−0.155363	−0.0776815	−	0.996978i	$-0.524752\pi$
$113$	−1.65153	−0.155363	−0.0776815	+	0.996978i	$0.524752\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	4.44949	0.411355
$118$	0	0
$119$	−5.44949	−0.499554
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	−5.44949	−0.491364
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.55051	0.137586	0.0687928	−	0.997631i	$-0.478085\pi$
$127$	1.55051	0.137586	0.0687928	+	0.997631i	$0.478085\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	19.5959	1.71210	0.856052	−	0.516890i	$-0.172910\pi$
$131$	19.5959	1.71210	0.856052	+	0.516890i	$0.172910\pi$
$132$	0	0
$133$	−4.44949	−0.385820
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−16.8990	−1.44378	−0.721889	−	0.692009i	$-0.756726\pi$
$137$	−16.8990	−1.44378	−0.721889	+	0.692009i	$0.756726\pi$
$138$	0	0
$139$	4.79796	0.406958	0.203479	−	0.979079i	$-0.434775\pi$
$139$	4.79796	0.406958	0.203479	+	0.979079i	$0.434775\pi$
$140$	0	0
$141$	8.44949	0.711575
$142$	0	0
$143$	−10.8990	−0.911418
$144$	0	0
$145$	−10.3485	−0.859394
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−1.34847	−0.110471	−0.0552355	−	0.998473i	$-0.517591\pi$
$149$	−1.34847	−0.110471	−0.0552355	+	0.998473i	$0.517591\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	0	0
$153$	−5.44949	−0.440565
$154$	0	0
$155$	0.101021	0.00811416
$156$	0	0
$157$	18.5959	1.48412	0.742058	−	0.670336i	$-0.233850\pi$
$157$	18.5959	1.48412	0.742058	+	0.670336i	$0.233850\pi$
$158$	0	0
$159$	−0.550510	−0.0436583
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	−2.44949	−0.190693
$166$	0	0
$167$	−13.3485	−1.03294	−0.516468	−	0.856307i	$-0.672753\pi$
$167$	−13.3485	−1.03294	−0.516468	+	0.856307i	$0.672753\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	−4.44949	−0.340261
$172$	0	0
$173$	19.5959	1.48985	0.744925	−	0.667148i	$-0.232485\pi$
$173$	19.5959	1.48985	0.744925	+	0.667148i	$0.232485\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	10.3485	0.777839
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−8.89898	−0.661456	−0.330728	−	0.943726i	$-0.607294\pi$
$181$	−8.89898	−0.661456	−0.330728	+	0.943726i	$0.607294\pi$
$182$	0	0
$183$	−0.651531	−0.0481625
$184$	0	0
$185$	−3.89898	−0.286659
$186$	0	0
$187$	13.3485	0.976137
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	18.2474	1.32034	0.660170	−	0.751117i	$-0.270484\pi$
$191$	18.2474	1.32034	0.660170	+	0.751117i	$0.270484\pi$
$192$	0	0
$193$	−6.69694	−0.482056	−0.241028	−	0.970518i	$-0.577485\pi$
$193$	−6.69694	−0.482056	−0.241028	+	0.970518i	$0.577485\pi$
$194$	0	0
$195$	4.44949	0.318635
$196$	0	0
$197$	−22.8990	−1.63148	−0.815742	−	0.578415i	$-0.803671\pi$
$197$	−22.8990	−1.63148	−0.815742	+	0.578415i	$0.803671\pi$
$198$	0	0
$199$	2.89898	0.205503	0.102752	−	0.994707i	$-0.467235\pi$
$199$	2.89898	0.205503	0.102752	+	0.994707i	$0.467235\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	10.3485	0.726320
$204$	0	0
$205$	−5.44949	−0.380609
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	10.8990	0.753898
$210$	0	0
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	0	0
$213$	−4.34847	−0.297952
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−0.101021	−0.00685772
$218$	0	0
$219$	5.34847	0.361416
$220$	0	0
$221$	−24.2474	−1.63106
$222$	0	0
$223$	−21.5959	−1.44617	−0.723085	−	0.690759i	$-0.757276\pi$
$223$	−21.5959	−1.44617	−0.723085	+	0.690759i	$0.757276\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−21.7980	−1.44678	−0.723391	−	0.690439i	$-0.757417\pi$
$227$	−21.7980	−1.44678	−0.723391	+	0.690439i	$0.757417\pi$
$228$	0	0
$229$	−28.4949	−1.88300	−0.941498	−	0.337019i	$-0.890581\pi$
$229$	−28.4949	−1.88300	−0.941498	+	0.337019i	$0.890581\pi$
$230$	0	0
$231$	2.44949	0.161165
$232$	0	0
$233$	21.7980	1.42803	0.714016	−	0.700129i	$-0.246874\pi$
$233$	21.7980	1.42803	0.714016	+	0.700129i	$0.246874\pi$
$234$	0	0
$235$	8.44949	0.551184
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−6.55051	−0.423717	−0.211859	−	0.977300i	$-0.567952\pi$
$239$	−6.55051	−0.423717	−0.211859	+	0.977300i	$0.567952\pi$
$240$	0	0
$241$	−29.3485	−1.89050	−0.945251	−	0.326346i	$-0.894183\pi$
$241$	−29.3485	−1.89050	−0.945251	+	0.326346i	$0.894183\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−19.7980	−1.25971
$248$	0	0
$249$	15.2474	0.966268
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	−2.44949	−0.153998
$254$	0	0
$255$	−5.44949	−0.341260
$256$	0	0
$257$	−25.3485	−1.58119	−0.790597	−	0.612337i	$-0.790230\pi$
$257$	−25.3485	−1.58119	−0.790597	+	0.612337i	$0.790230\pi$
$258$	0	0
$259$	3.89898	0.242271
$260$	0	0
$261$	10.3485	0.640554
$262$	0	0
$263$	−27.2474	−1.68015	−0.840075	−	0.542471i	$-0.817489\pi$
$263$	−27.2474	−1.68015	−0.840075	+	0.542471i	$0.817489\pi$
$264$	0	0
$265$	−0.550510	−0.0338176
$266$	0	0
$267$	16.8990	1.03420
$268$	0	0
$269$	−5.44949	−0.332261	−0.166131	−	0.986104i	$-0.553127\pi$
$269$	−5.44949	−0.332261	−0.166131	+	0.986104i	$0.553127\pi$
$270$	0	0
$271$	−31.6969	−1.92545	−0.962726	−	0.270479i	$-0.912818\pi$
$271$	−31.6969	−1.92545	−0.962726	+	0.270479i	$0.912818\pi$
$272$	0	0
$273$	−4.44949	−0.269295
$274$	0	0
$275$	−2.44949	−0.147710
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−0.101021	−0.00604794
$280$	0	0
$281$	−19.3485	−1.15423	−0.577116	−	0.816662i	$-0.695822\pi$
$281$	−19.3485	−1.15423	−0.577116	+	0.816662i	$0.695822\pi$
$282$	0	0
$283$	2.10102	0.124893	0.0624464	−	0.998048i	$-0.480110\pi$
$283$	2.10102	0.124893	0.0624464	+	0.998048i	$0.480110\pi$
$284$	0	0
$285$	−4.44949	−0.263565
$286$	0	0
$287$	5.44949	0.321673
$288$	0	0
$289$	12.6969	0.746879
$290$	0	0
$291$	−3.10102	−0.181785
$292$	0	0
$293$	−21.2474	−1.24129	−0.620645	−	0.784092i	$-0.713129\pi$
$293$	−21.2474	−1.24129	−0.620645	+	0.784092i	$0.713129\pi$
$294$	0	0
$295$	10.3485	0.602511
$296$	0	0
$297$	2.44949	0.142134
$298$	0	0
$299$	4.44949	0.257321
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	12.5505	0.721008
$304$	0	0
$305$	−0.651531	−0.0373065
$306$	0	0
$307$	2.65153	0.151331	0.0756654	−	0.997133i	$-0.475892\pi$
$307$	2.65153	0.151331	0.0756654	+	0.997133i	$0.475892\pi$
$308$	0	0
$309$	−12.6969	−0.722304
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	19.6969	1.11334	0.556668	−	0.830735i	$-0.312079\pi$
$313$	19.6969	1.11334	0.556668	+	0.830735i	$0.312079\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	6.24745	0.350892	0.175446	−	0.984489i	$-0.443863\pi$
$317$	6.24745	0.350892	0.175446	+	0.984489i	$0.443863\pi$
$318$	0	0
$319$	−25.3485	−1.41924
$320$	0	0
$321$	−5.44949	−0.304161
$322$	0	0
$323$	24.2474	1.34916
$324$	0	0
$325$	4.44949	0.246813
$326$	0	0
$327$	−11.5505	−0.638745
$328$	0	0
$329$	−8.44949	−0.465835
$330$	0	0
$331$	−24.5959	−1.35191	−0.675957	−	0.736941i	$-0.736269\pi$
$331$	−24.5959	−1.35191	−0.675957	+	0.736941i	$0.736269\pi$
$332$	0	0
$333$	3.89898	0.213663
$334$	0	0
$335$	−7.00000	−0.382451
$336$	0	0
$337$	−19.7980	−1.07846	−0.539232	−	0.842157i	$-0.681285\pi$
$337$	−19.7980	−1.07846	−0.539232	+	0.842157i	$0.681285\pi$
$338$	0	0
$339$	1.65153	0.0896988
$340$	0	0
$341$	0.247449	0.0134001
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	28.8990	1.55138	0.775689	−	0.631115i	$-0.217402\pi$
$347$	28.8990	1.55138	0.775689	+	0.631115i	$0.217402\pi$
$348$	0	0
$349$	23.4949	1.25765	0.628827	−	0.777546i	$-0.283536\pi$
$349$	23.4949	1.25765	0.628827	+	0.777546i	$0.283536\pi$
$350$	0	0
$351$	−4.44949	−0.237496
$352$	0	0
$353$	−15.5505	−0.827670	−0.413835	−	0.910352i	$-0.635811\pi$
$353$	−15.5505	−0.827670	−0.413835	+	0.910352i	$0.635811\pi$
$354$	0	0
$355$	−4.34847	−0.230793
$356$	0	0
$357$	5.44949	0.288418
$358$	0	0
$359$	3.55051	0.187389	0.0936944	−	0.995601i	$-0.470132\pi$
$359$	3.55051	0.187389	0.0936944	+	0.995601i	$0.470132\pi$
$360$	0	0
$361$	0.797959	0.0419978
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	5.34847	0.279952
$366$	0	0
$367$	−13.6969	−0.714974	−0.357487	−	0.933918i	$-0.616366\pi$
$367$	−13.6969	−0.714974	−0.357487	+	0.933918i	$0.616366\pi$
$368$	0	0
$369$	5.44949	0.283689
$370$	0	0
$371$	0.550510	0.0285811
$372$	0	0
$373$	27.5959	1.42886	0.714431	−	0.699706i	$-0.246686\pi$
$373$	27.5959	1.42886	0.714431	+	0.699706i	$0.246686\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	46.0454	2.37146
$378$	0	0
$379$	−25.3939	−1.30440	−0.652198	−	0.758049i	$-0.726153\pi$
$379$	−25.3939	−1.30440	−0.652198	+	0.758049i	$0.726153\pi$
$380$	0	0
$381$	−1.55051	−0.0794350
$382$	0	0
$383$	−1.65153	−0.0843893	−0.0421946	−	0.999109i	$-0.513435\pi$
$383$	−1.65153	−0.0843893	−0.0421946	+	0.999109i	$0.513435\pi$
$384$	0	0
$385$	2.44949	0.124838
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−3.30306	−0.167472	−0.0837359	−	0.996488i	$-0.526685\pi$
$389$	−3.30306	−0.167472	−0.0837359	+	0.996488i	$0.526685\pi$
$390$	0	0
$391$	−5.44949	−0.275593
$392$	0	0
$393$	−19.5959	−0.988483
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−13.7980	−0.692500	−0.346250	−	0.938142i	$-0.612545\pi$
$397$	−13.7980	−0.692500	−0.346250	+	0.938142i	$0.612545\pi$
$398$	0	0
$399$	4.44949	0.222753
$400$	0	0
$401$	−13.1010	−0.654234	−0.327117	−	0.944984i	$-0.606077\pi$
$401$	−13.1010	−0.654234	−0.327117	+	0.944984i	$0.606077\pi$
$402$	0	0
$403$	−0.449490	−0.0223907
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−9.55051	−0.473401
$408$	0	0
$409$	21.8990	1.08283	0.541417	−	0.840754i	$-0.317888\pi$
$409$	21.8990	1.08283	0.541417	+	0.840754i	$0.317888\pi$
$410$	0	0
$411$	16.8990	0.833565
$412$	0	0
$413$	−10.3485	−0.509215
$414$	0	0
$415$	15.2474	0.748468
$416$	0	0
$417$	−4.79796	−0.234957
$418$	0	0
$419$	−33.5505	−1.63905	−0.819525	−	0.573044i	$-0.805763\pi$
$419$	−33.5505	−1.63905	−0.819525	+	0.573044i	$0.805763\pi$
$420$	0	0
$421$	−22.2474	−1.08427	−0.542137	−	0.840290i	$-0.682385\pi$
$421$	−22.2474	−1.08427	−0.542137	+	0.840290i	$0.682385\pi$
$422$	0	0
$423$	−8.44949	−0.410828
$424$	0	0
$425$	−5.44949	−0.264339
$426$	0	0
$427$	0.651531	0.0315298
$428$	0	0
$429$	10.8990	0.526208
$430$	0	0
$431$	24.4949	1.17988	0.589939	−	0.807448i	$-0.299152\pi$
$431$	24.4949	1.17988	0.589939	+	0.807448i	$0.299152\pi$
$432$	0	0
$433$	2.79796	0.134461	0.0672307	−	0.997737i	$-0.478584\pi$
$433$	2.79796	0.134461	0.0672307	+	0.997737i	$0.478584\pi$
$434$	0	0
$435$	10.3485	0.496171
$436$	0	0
$437$	−4.44949	−0.212848
$438$	0	0
$439$	36.6969	1.75145	0.875725	−	0.482811i	$-0.160384\pi$
$439$	36.6969	1.75145	0.875725	+	0.482811i	$0.160384\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	20.4495	0.971585	0.485792	−	0.874074i	$-0.338531\pi$
$443$	20.4495	0.971585	0.485792	+	0.874074i	$0.338531\pi$
$444$	0	0
$445$	16.8990	0.801088
$446$	0	0
$447$	1.34847	0.0637804
$448$	0	0
$449$	−21.2474	−1.00273	−0.501365	−	0.865236i	$-0.667168\pi$
$449$	−21.2474	−1.00273	−0.501365	+	0.865236i	$0.667168\pi$
$450$	0	0
$451$	−13.3485	−0.628555
$452$	0	0
$453$	14.0000	0.657777
$454$	0	0
$455$	−4.44949	−0.208595
$456$	0	0
$457$	0.101021	0.00472554	0.00236277	−	0.999997i	$-0.499248\pi$
$457$	0.101021	0.00472554	0.00236277	+	0.999997i	$0.499248\pi$
$458$	0	0
$459$	5.44949	0.254360
$460$	0	0
$461$	−18.4949	−0.861393	−0.430697	−	0.902497i	$-0.641732\pi$
$461$	−18.4949	−0.861393	−0.430697	+	0.902497i	$0.641732\pi$
$462$	0	0
$463$	24.9444	1.15926	0.579632	−	0.814878i	$-0.303196\pi$
$463$	24.9444	1.15926	0.579632	+	0.814878i	$0.303196\pi$
$464$	0	0
$465$	−0.101021	−0.00468471
$466$	0	0
$467$	34.8434	1.61236	0.806179	−	0.591671i	$-0.201532\pi$
$467$	34.8434	1.61236	0.806179	+	0.591671i	$0.201532\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	−18.5959	−0.856855
$472$	0	0
$473$	4.89898	0.225255
$474$	0	0
$475$	−4.44949	−0.204157
$476$	0	0
$477$	0.550510	0.0252061
$478$	0	0
$479$	−32.9444	−1.50527	−0.752634	−	0.658439i	$-0.771217\pi$
$479$	−32.9444	−1.50527	−0.752634	+	0.658439i	$0.771217\pi$
$480$	0	0
$481$	17.3485	0.791022
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−3.10102	−0.140810
$486$	0	0
$487$	18.9444	0.858452	0.429226	−	0.903197i	$-0.358786\pi$
$487$	18.9444	0.858452	0.429226	+	0.903197i	$0.358786\pi$
$488$	0	0
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−12.5505	−0.566397	−0.283198	−	0.959061i	$-0.591395\pi$
$491$	−12.5505	−0.566397	−0.283198	+	0.959061i	$0.591395\pi$
$492$	0	0
$493$	−56.3939	−2.53985
$494$	0	0
$495$	2.44949	0.110096
$496$	0	0
$497$	4.34847	0.195056
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	13.3485	0.596366
$502$	0	0
$503$	−7.04541	−0.314139	−0.157070	−	0.987588i	$-0.550205\pi$
$503$	−7.04541	−0.314139	−0.157070	+	0.987588i	$0.550205\pi$
$504$	0	0
$505$	12.5505	0.558490
$506$	0	0
$507$	−6.79796	−0.301908
$508$	0	0
$509$	28.8990	1.28092	0.640462	−	0.767990i	$-0.278743\pi$
$509$	28.8990	1.28092	0.640462	+	0.767990i	$0.278743\pi$
$510$	0	0
$511$	−5.34847	−0.236602
$512$	0	0
$513$	4.44949	0.196450
$514$	0	0
$515$	−12.6969	−0.559494
$516$	0	0
$517$	20.6969	0.910250
$518$	0	0
$519$	−19.5959	−0.860165
$520$	0	0
$521$	−43.8434	−1.92081	−0.960406	−	0.278603i	$-0.910129\pi$
$521$	−43.8434	−1.92081	−0.960406	+	0.278603i	$0.910129\pi$
$522$	0	0
$523$	33.3939	1.46021	0.730106	−	0.683334i	$-0.239471\pi$
$523$	33.3939	1.46021	0.730106	+	0.683334i	$0.239471\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	0.550510	0.0239806
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−10.3485	−0.449085
$532$	0	0
$533$	24.2474	1.05027
$534$	0	0
$535$	−5.44949	−0.235602
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	14.6969	0.633042
$540$	0	0
$541$	12.8990	0.554570	0.277285	−	0.960788i	$-0.410565\pi$
$541$	12.8990	0.554570	0.277285	+	0.960788i	$0.410565\pi$
$542$	0	0
$543$	8.89898	0.381892
$544$	0	0
$545$	−11.5505	−0.494769
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	0.651531	0.0278067
$550$	0	0
$551$	−46.0454	−1.96160
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	3.89898	0.165502
$556$	0	0
$557$	−19.0454	−0.806980	−0.403490	−	0.914984i	$-0.632203\pi$
$557$	−19.0454	−0.806980	−0.403490	+	0.914984i	$0.632203\pi$
$558$	0	0
$559$	−8.89898	−0.376387
$560$	0	0
$561$	−13.3485	−0.563573
$562$	0	0
$563$	−5.44949	−0.229669	−0.114834	−	0.993385i	$-0.536634\pi$
$563$	−5.44949	−0.229669	−0.114834	+	0.993385i	$0.536634\pi$
$564$	0	0
$565$	1.65153	0.0694804
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−28.2929	−1.18610	−0.593049	−	0.805166i	$-0.702076\pi$
$569$	−28.2929	−1.18610	−0.593049	+	0.805166i	$0.702076\pi$
$570$	0	0
$571$	−13.1464	−0.550161	−0.275080	−	0.961421i	$-0.588704\pi$
$571$	−13.1464	−0.550161	−0.275080	+	0.961421i	$0.588704\pi$
$572$	0	0
$573$	−18.2474	−0.762298
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	0	0
$579$	6.69694	0.278315
$580$	0	0
$581$	−15.2474	−0.632571
$582$	0	0
$583$	−1.34847	−0.0558479
$584$	0	0
$585$	−4.44949	−0.183964
$586$	0	0
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	0	0
$589$	0.449490	0.0185209
$590$	0	0
$591$	22.8990	0.941938
$592$	0	0
$593$	27.5505	1.13136	0.565682	−	0.824624i	$-0.308613\pi$
$593$	27.5505	1.13136	0.565682	+	0.824624i	$0.308613\pi$
$594$	0	0
$595$	5.44949	0.223407
$596$	0	0
$597$	−2.89898	−0.118647
$598$	0	0
$599$	2.69694	0.110194	0.0550970	−	0.998481i	$-0.482453\pi$
$599$	2.69694	0.110194	0.0550970	+	0.998481i	$0.482453\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	7.00000	0.285062
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	16.2474	0.659464	0.329732	−	0.944075i	$-0.393042\pi$
$607$	16.2474	0.659464	0.329732	+	0.944075i	$0.393042\pi$
$608$	0	0
$609$	−10.3485	−0.419341
$610$	0	0
$611$	−37.5959	−1.52097
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	5.44949	0.219745
$616$	0	0
$617$	12.5505	0.505265	0.252632	−	0.967562i	$-0.418704\pi$
$617$	12.5505	0.505265	0.252632	+	0.967562i	$0.418704\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−16.8990	−0.677043
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.8990	−0.435263
$628$	0	0
$629$	−21.2474	−0.847191
$630$	0	0
$631$	−29.5505	−1.17639	−0.588194	−	0.808720i	$-0.700161\pi$
$631$	−29.5505	−1.17639	−0.588194	+	0.808720i	$0.700161\pi$
$632$	0	0
$633$	11.0000	0.437211
$634$	0	0
$635$	−1.55051	−0.0615301
$636$	0	0
$637$	−26.6969	−1.05777
$638$	0	0
$639$	4.34847	0.172023
$640$	0	0
$641$	26.9444	1.06424	0.532120	−	0.846669i	$-0.321396\pi$
$641$	26.9444	1.06424	0.532120	+	0.846669i	$0.321396\pi$
$642$	0	0
$643$	9.69694	0.382410	0.191205	−	0.981550i	$-0.438760\pi$
$643$	9.69694	0.382410	0.191205	+	0.981550i	$0.438760\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	0	0
$647$	−1.95459	−0.0768430	−0.0384215	−	0.999262i	$-0.512233\pi$
$647$	−1.95459	−0.0768430	−0.0384215	+	0.999262i	$0.512233\pi$
$648$	0	0
$649$	25.3485	0.995014
$650$	0	0
$651$	0.101021	0.00395931
$652$	0	0
$653$	43.8434	1.71572	0.857862	−	0.513881i	$-0.171793\pi$
$653$	43.8434	1.71572	0.857862	+	0.513881i	$0.171793\pi$
$654$	0	0
$655$	−19.5959	−0.765676
$656$	0	0
$657$	−5.34847	−0.208664
$658$	0	0
$659$	5.14643	0.200476	0.100238	−	0.994963i	$-0.468040\pi$
$659$	5.14643	0.200476	0.100238	+	0.994963i	$0.468040\pi$
$660$	0	0
$661$	46.0908	1.79272	0.896362	−	0.443322i	$-0.146200\pi$
$661$	46.0908	1.79272	0.896362	+	0.443322i	$0.146200\pi$
$662$	0	0
$663$	24.2474	0.941693
$664$	0	0
$665$	4.44949	0.172544
$666$	0	0
$667$	10.3485	0.400694
$668$	0	0
$669$	21.5959	0.834946
$670$	0	0
$671$	−1.59592	−0.0616097
$672$	0	0
$673$	22.4495	0.865364	0.432682	−	0.901547i	$-0.357567\pi$
$673$	22.4495	0.865364	0.432682	+	0.901547i	$0.357567\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−12.5505	−0.482355	−0.241178	−	0.970481i	$-0.577534\pi$
$677$	−12.5505	−0.482355	−0.241178	+	0.970481i	$0.577534\pi$
$678$	0	0
$679$	3.10102	0.119006
$680$	0	0
$681$	21.7980	0.835300
$682$	0	0
$683$	−18.2474	−0.698219	−0.349110	−	0.937082i	$-0.613516\pi$
$683$	−18.2474	−0.698219	−0.349110	+	0.937082i	$0.613516\pi$
$684$	0	0
$685$	16.8990	0.645677
$686$	0	0
$687$	28.4949	1.08715
$688$	0	0
$689$	2.44949	0.0933181
$690$	0	0
$691$	−16.2020	−0.616355	−0.308177	−	0.951329i	$-0.599719\pi$
$691$	−16.2020	−0.616355	−0.308177	+	0.951329i	$0.599719\pi$
$692$	0	0
$693$	−2.44949	−0.0930484
$694$	0	0
$695$	−4.79796	−0.181997
$696$	0	0
$697$	−29.6969	−1.12485
$698$	0	0
$699$	−21.7980	−0.824475
$700$	0	0
$701$	−52.0454	−1.96573	−0.982864	−	0.184332i	$-0.940988\pi$
$701$	−52.0454	−1.96573	−0.982864	+	0.184332i	$0.940988\pi$
$702$	0	0
$703$	−17.3485	−0.654310
$704$	0	0
$705$	−8.44949	−0.318226
$706$	0	0
$707$	−12.5505	−0.472011
$708$	0	0
$709$	−28.7423	−1.07944	−0.539721	−	0.841844i	$-0.681470\pi$
$709$	−28.7423	−1.07944	−0.539721	+	0.841844i	$0.681470\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−0.101021	−0.00378325
$714$	0	0
$715$	10.8990	0.407599
$716$	0	0
$717$	6.55051	0.244633
$718$	0	0
$719$	11.9444	0.445450	0.222725	−	0.974881i	$-0.428505\pi$
$719$	11.9444	0.445450	0.222725	+	0.974881i	$0.428505\pi$
$720$	0	0
$721$	12.6969	0.472859
$722$	0	0
$723$	29.3485	1.09148
$724$	0	0
$725$	10.3485	0.384332
$726$	0	0
$727$	16.3031	0.604647	0.302324	−	0.953205i	$-0.402238\pi$
$727$	16.3031	0.604647	0.302324	+	0.953205i	$0.402238\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.8990	0.403113
$732$	0	0
$733$	−8.59592	−0.317497	−0.158749	−	0.987319i	$-0.550746\pi$
$733$	−8.59592	−0.317497	−0.158749	+	0.987319i	$0.550746\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	−17.1464	−0.631597
$738$	0	0
$739$	9.20204	0.338503	0.169251	−	0.985573i	$-0.445865\pi$
$739$	9.20204	0.338503	0.169251	+	0.985573i	$0.445865\pi$
$740$	0	0
$741$	19.7980	0.727296
$742$	0	0
$743$	21.7980	0.799690	0.399845	−	0.916583i	$-0.369064\pi$
$743$	21.7980	0.799690	0.399845	+	0.916583i	$0.369064\pi$
$744$	0	0
$745$	1.34847	0.0494041
$746$	0	0
$747$	−15.2474	−0.557875
$748$	0	0
$749$	5.44949	0.199120
$750$	0	0
$751$	−30.6515	−1.11849	−0.559245	−	0.829002i	$-0.688909\pi$
$751$	−30.6515	−1.11849	−0.559245	+	0.829002i	$0.688909\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	14.0000	0.509512
$756$	0	0
$757$	−3.69694	−0.134368	−0.0671838	−	0.997741i	$-0.521401\pi$
$757$	−3.69694	−0.134368	−0.0671838	+	0.997741i	$0.521401\pi$
$758$	0	0
$759$	2.44949	0.0889108
$760$	0	0
$761$	14.1464	0.512808	0.256404	−	0.966570i	$-0.417462\pi$
$761$	14.1464	0.512808	0.256404	+	0.966570i	$0.417462\pi$
$762$	0	0
$763$	11.5505	0.418157
$764$	0	0
$765$	5.44949	0.197027
$766$	0	0
$767$	−46.0454	−1.66260
$768$	0	0
$769$	10.4495	0.376818	0.188409	−	0.982091i	$-0.439667\pi$
$769$	10.4495	0.376818	0.188409	+	0.982091i	$0.439667\pi$
$770$	0	0
$771$	25.3485	0.912903
$772$	0	0
$773$	−16.2929	−0.586013	−0.293007	−	0.956110i	$-0.594656\pi$
$773$	−16.2929	−0.586013	−0.293007	+	0.956110i	$0.594656\pi$
$774$	0	0
$775$	−0.101021	−0.00362876
$776$	0	0
$777$	−3.89898	−0.139875
$778$	0	0
$779$	−24.2474	−0.868755
$780$	0	0
$781$	−10.6515	−0.381142
$782$	0	0
$783$	−10.3485	−0.369824
$784$	0	0
$785$	−18.5959	−0.663717
$786$	0	0
$787$	21.6969	0.773412	0.386706	−	0.922203i	$-0.373613\pi$
$787$	21.6969	0.773412	0.386706	+	0.922203i	$0.373613\pi$
$788$	0	0
$789$	27.2474	0.970035
$790$	0	0
$791$	−1.65153	−0.0587217
$792$	0	0
$793$	2.89898	0.102946
$794$	0	0
$795$	0.550510	0.0195246
$796$	0	0
$797$	9.24745	0.327561	0.163781	−	0.986497i	$-0.447631\pi$
$797$	9.24745	0.327561	0.163781	+	0.986497i	$0.447631\pi$
$798$	0	0
$799$	46.0454	1.62897
$800$	0	0
$801$	−16.8990	−0.597096
$802$	0	0
$803$	13.1010	0.462325
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	5.44949	0.191831
$808$	0	0
$809$	6.55051	0.230304	0.115152	−	0.993348i	$-0.463265\pi$
$809$	6.55051	0.230304	0.115152	+	0.993348i	$0.463265\pi$
$810$	0	0
$811$	42.3939	1.48865	0.744325	−	0.667817i	$-0.232771\pi$
$811$	42.3939	1.48865	0.744325	+	0.667817i	$0.232771\pi$
$812$	0	0
$813$	31.6969	1.11166
$814$	0	0
$815$	−10.0000	−0.350285
$816$	0	0
$817$	8.89898	0.311336
$818$	0	0
$819$	4.44949	0.155478
$820$	0	0
$821$	21.3031	0.743482	0.371741	−	0.928336i	$-0.378761\pi$
$821$	21.3031	0.743482	0.371741	+	0.928336i	$0.378761\pi$
$822$	0	0
$823$	−3.59592	−0.125346	−0.0626729	−	0.998034i	$-0.519962\pi$
$823$	−3.59592	−0.125346	−0.0626729	+	0.998034i	$0.519962\pi$
$824$	0	0
$825$	2.44949	0.0852803
$826$	0	0
$827$	−32.1464	−1.11784	−0.558920	−	0.829221i	$-0.688784\pi$
$827$	−32.1464	−1.11784	−0.558920	+	0.829221i	$0.688784\pi$
$828$	0	0
$829$	−27.6969	−0.961954	−0.480977	−	0.876733i	$-0.659718\pi$
$829$	−27.6969	−0.961954	−0.480977	+	0.876733i	$0.659718\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	32.6969	1.13288
$834$	0	0
$835$	13.3485	0.461943
$836$	0	0
$837$	0.101021	0.00349178
$838$	0	0
$839$	45.1918	1.56020	0.780098	−	0.625658i	$-0.215169\pi$
$839$	45.1918	1.56020	0.780098	+	0.625658i	$0.215169\pi$
$840$	0	0
$841$	78.0908	2.69279
$842$	0	0
$843$	19.3485	0.666397
$844$	0	0
$845$	−6.79796	−0.233857
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−2.10102	−0.0721068
$850$	0	0
$851$	3.89898	0.133655
$852$	0	0
$853$	46.0908	1.57812	0.789060	−	0.614316i	$-0.210568\pi$
$853$	46.0908	1.57812	0.789060	+	0.614316i	$0.210568\pi$
$854$	0	0
$855$	4.44949	0.152169
$856$	0	0
$857$	−13.5959	−0.464428	−0.232214	−	0.972665i	$-0.574597\pi$
$857$	−13.5959	−0.464428	−0.232214	+	0.972665i	$0.574597\pi$
$858$	0	0
$859$	−38.1918	−1.30309	−0.651544	−	0.758611i	$-0.725879\pi$
$859$	−38.1918	−1.30309	−0.651544	+	0.758611i	$0.725879\pi$
$860$	0	0
$861$	−5.44949	−0.185718
$862$	0	0
$863$	−6.49490	−0.221089	−0.110544	−	0.993871i	$-0.535259\pi$
$863$	−6.49490	−0.221089	−0.110544	+	0.993871i	$0.535259\pi$
$864$	0	0
$865$	−19.5959	−0.666281
$866$	0	0
$867$	−12.6969	−0.431211
$868$	0	0
$869$	−9.79796	−0.332373
$870$	0	0
$871$	31.1464	1.05536
$872$	0	0
$873$	3.10102	0.104954
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−10.4949	−0.354388	−0.177194	−	0.984176i	$-0.556702\pi$
$877$	−10.4949	−0.354388	−0.177194	+	0.984176i	$0.556702\pi$
$878$	0	0
$879$	21.2474	0.716659
$880$	0	0
$881$	8.44949	0.284671	0.142335	−	0.989819i	$-0.454539\pi$
$881$	8.44949	0.284671	0.142335	+	0.989819i	$0.454539\pi$
$882$	0	0
$883$	37.5505	1.26368	0.631838	−	0.775101i	$-0.282301\pi$
$883$	37.5505	1.26368	0.631838	+	0.775101i	$0.282301\pi$
$884$	0	0
$885$	−10.3485	−0.347860
$886$	0	0
$887$	6.49490	0.218077	0.109039	−	0.994038i	$-0.465223\pi$
$887$	6.49490	0.218077	0.109039	+	0.994038i	$0.465223\pi$
$888$	0	0
$889$	1.55051	0.0520024
$890$	0	0
$891$	−2.44949	−0.0820610
$892$	0	0
$893$	37.5959	1.25810
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	−4.44949	−0.148564
$898$	0	0
$899$	−1.04541	−0.0348663
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	2.00000	0.0665558
$904$	0	0
$905$	8.89898	0.295812
$906$	0	0
$907$	−38.7980	−1.28827	−0.644133	−	0.764914i	$-0.722781\pi$
$907$	−38.7980	−1.28827	−0.644133	+	0.764914i	$0.722781\pi$
$908$	0	0
$909$	−12.5505	−0.416274
$910$	0	0
$911$	14.2020	0.470535	0.235267	−	0.971931i	$-0.424403\pi$
$911$	14.2020	0.470535	0.235267	+	0.971931i	$0.424403\pi$
$912$	0	0
$913$	37.3485	1.23605
$914$	0	0
$915$	0.651531	0.0215389
$916$	0	0
$917$	19.5959	0.647114
$918$	0	0
$919$	45.3939	1.49741	0.748703	−	0.662906i	$-0.230677\pi$
$919$	45.3939	1.49741	0.748703	+	0.662906i	$0.230677\pi$
$920$	0	0
$921$	−2.65153	−0.0873709
$922$	0	0
$923$	19.3485	0.636863
$924$	0	0
$925$	3.89898	0.128198
$926$	0	0
$927$	12.6969	0.417022
$928$	0	0
$929$	22.8434	0.749467	0.374733	−	0.927133i	$-0.377734\pi$
$929$	22.8434	0.749467	0.374733	+	0.927133i	$0.377734\pi$
$930$	0	0
$931$	26.6969	0.874957
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	−13.3485	−0.436542
$936$	0	0
$937$	−8.89898	−0.290717	−0.145358	−	0.989379i	$-0.546434\pi$
$937$	−8.89898	−0.290717	−0.145358	+	0.989379i	$0.546434\pi$
$938$	0	0
$939$	−19.6969	−0.642785
$940$	0	0
$941$	−24.8536	−0.810203	−0.405102	−	0.914272i	$-0.632764\pi$
$941$	−24.8536	−0.810203	−0.405102	+	0.914272i	$0.632764\pi$
$942$	0	0
$943$	5.44949	0.177460
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	0.494897	0.0160820	0.00804100	−	0.999968i	$-0.497440\pi$
$947$	0.494897	0.0160820	0.00804100	+	0.999968i	$0.497440\pi$
$948$	0	0
$949$	−23.7980	−0.772514
$950$	0	0
$951$	−6.24745	−0.202587
$952$	0	0
$953$	10.4041	0.337021	0.168511	−	0.985700i	$-0.446104\pi$
$953$	10.4041	0.337021	0.168511	+	0.985700i	$0.446104\pi$
$954$	0	0
$955$	−18.2474	−0.590474
$956$	0	0
$957$	25.3485	0.819400
$958$	0	0
$959$	−16.8990	−0.545697
$960$	0	0
$961$	−30.9898	−0.999671
$962$	0	0
$963$	5.44949	0.175607
$964$	0	0
$965$	6.69694	0.215582
$966$	0	0
$967$	−54.0454	−1.73798	−0.868992	−	0.494827i	$-0.835231\pi$
$967$	−54.0454	−1.73798	−0.868992	+	0.494827i	$0.835231\pi$
$968$	0	0
$969$	−24.2474	−0.778940
$970$	0	0
$971$	−22.2929	−0.715412	−0.357706	−	0.933834i	$-0.616441\pi$
$971$	−22.2929	−0.715412	−0.357706	+	0.933834i	$0.616441\pi$
$972$	0	0
$973$	4.79796	0.153816
$974$	0	0
$975$	−4.44949	−0.142498
$976$	0	0
$977$	19.6515	0.628708	0.314354	−	0.949306i	$-0.398212\pi$
$977$	19.6515	0.628708	0.314354	+	0.949306i	$0.398212\pi$
$978$	0	0
$979$	41.3939	1.32295
$980$	0	0
$981$	11.5505	0.368779
$982$	0	0
$983$	33.7423	1.07621	0.538107	−	0.842877i	$-0.319140\pi$
$983$	33.7423	1.07621	0.538107	+	0.842877i	$0.319140\pi$
$984$	0	0
$985$	22.8990	0.729622
$986$	0	0
$987$	8.44949	0.268950
$988$	0	0
$989$	−2.00000	−0.0635963
$990$	0	0
$991$	−57.7878	−1.83569	−0.917844	−	0.396941i	$-0.870072\pi$
$991$	−57.7878	−1.83569	−0.917844	+	0.396941i	$0.870072\pi$
$992$	0	0
$993$	24.5959	0.780528
$994$	0	0
$995$	−2.89898	−0.0919038
$996$	0	0
$997$	40.0908	1.26969	0.634844	−	0.772640i	$-0.281064\pi$
$997$	40.0908	1.26969	0.634844	+	0.772640i	$0.281064\pi$
$998$	0	0
$999$	−3.89898	−0.123358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bi.1.1		2
4.3	odd	2		345.2.a.i.1.1	✓	2
12.11	even	2		1035.2.a.k.1.2		2
20.3	even	4		1725.2.b.m.1174.4		4
20.7	even	4		1725.2.b.m.1174.1		4
20.19	odd	2		1725.2.a.y.1.2		2
60.59	even	2		5175.2.a.bl.1.1		2
92.91	even	2		7935.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.i.1.1	✓	2	4.3	odd	2
1035.2.a.k.1.2		2	12.11	even	2
1725.2.a.y.1.2		2	20.19	odd	2
1725.2.b.m.1174.1		4	20.7	even	4
1725.2.b.m.1174.4		4	20.3	even	4
5175.2.a.bl.1.1		2	60.59	even	2
5520.2.a.bi.1.1		2	1.1	even	1	trivial
7935.2.a.t.1.1		2	92.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.44949 q^{11} +4.44949 q^{13} +1.00000 q^{15} -5.44949 q^{17} -4.44949 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.3485 q^{29} -0.101021 q^{31} +2.44949 q^{33} -1.00000 q^{35} +3.89898 q^{37} -4.44949 q^{39} +5.44949 q^{41} -2.00000 q^{43} -1.00000 q^{45} -8.44949 q^{47} -6.00000 q^{49} +5.44949 q^{51} +0.550510 q^{53} +2.44949 q^{55} +4.44949 q^{57} -10.3485 q^{59} +0.651531 q^{61} +1.00000 q^{63} -4.44949 q^{65} +7.00000 q^{67} -1.00000 q^{69} +4.34847 q^{71} -5.34847 q^{73} -1.00000 q^{75} -2.44949 q^{77} +4.00000 q^{79} +1.00000 q^{81} -15.2474 q^{83} +5.44949 q^{85} -10.3485 q^{87} -16.8990 q^{89} +4.44949 q^{91} +0.101021 q^{93} +4.44949 q^{95} +3.10102 q^{97} -2.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 10 q^{31} - 2 q^{35} - 2 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 12 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 2 q^{63} - 4 q^{65} + 14 q^{67} - 2 q^{69} - 6 q^{71} + 4 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} - 24 q^{89} + 4 q^{91} + 10 q^{93} + 4 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 10 * q^31 - 2 * q^35 - 2 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 - 12 * q^47 - 12 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^57 - 6 * q^59 + 16 * q^61 + 2 * q^63 - 4 * q^65 + 14 * q^67 - 2 * q^69 - 6 * q^71 + 4 * q^73 - 2 * q^75 + 8 * q^79 + 2 * q^81 - 6 * q^83 + 6 * q^85 - 6 * q^87 - 24 * q^89 + 4 * q^91 + 10 * q^93 + 4 * q^95 + 16 * q^97

Introduction

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