LMFDB - Embedded newform 5175.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.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:	$41.3225830460$
Analytic rank:	$0$
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:	$\Z[a_1, a_2]$
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	$\chi$	$=$	5175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949 q^{2} +4.00000 q^{4} +1.00000 q^{7} -4.89898 q^{8} +O(q^{10})$	$q-2.44949 q^{2} +4.00000 q^{4} +1.00000 q^{7} -4.89898 q^{8} -2.44949 q^{11} -4.44949 q^{13} -2.44949 q^{14} +4.00000 q^{16} -5.44949 q^{17} +4.44949 q^{19} +6.00000 q^{22} -1.00000 q^{23} +10.8990 q^{26} +4.00000 q^{28} -10.3485 q^{29} +0.101021 q^{31} +13.3485 q^{34} -3.89898 q^{37} -10.8990 q^{38} -5.44949 q^{41} -2.00000 q^{43} -9.79796 q^{44} +2.44949 q^{46} +8.44949 q^{47} -6.00000 q^{49} -17.7980 q^{52} +0.550510 q^{53} -4.89898 q^{56} +25.3485 q^{58} -10.3485 q^{59} +0.651531 q^{61} -0.247449 q^{62} -8.00000 q^{64} +7.00000 q^{67} -21.7980 q^{68} +4.34847 q^{71} +5.34847 q^{73} +9.55051 q^{74} +17.7980 q^{76} -2.44949 q^{77} -4.00000 q^{79} +13.3485 q^{82} +15.2474 q^{83} +4.89898 q^{86} +12.0000 q^{88} +16.8990 q^{89} -4.44949 q^{91} -4.00000 q^{92} -20.6969 q^{94} -3.10102 q^{97} +14.6969 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{4} + 2 q^{7}+O(q^{10}) $ 2 * q + 8 * q^4 + 2 * q^7	$ 2 q + 8 q^{4} + 2 q^{7} - 4 q^{13} + 8 q^{16} - 6 q^{17} + 4 q^{19} + 12 q^{22} - 2 q^{23} + 12 q^{26} + 8 q^{28} - 6 q^{29} + 10 q^{31} + 12 q^{34} + 2 q^{37} - 12 q^{38} - 6 q^{41} - 4 q^{43} + 12 q^{47} - 12 q^{49} - 16 q^{52} + 6 q^{53} + 36 q^{58} - 6 q^{59} + 16 q^{61} + 24 q^{62} - 16 q^{64} + 14 q^{67} - 24 q^{68} - 6 q^{71} - 4 q^{73} + 24 q^{74} + 16 q^{76} - 8 q^{79} + 12 q^{82} + 6 q^{83} + 24 q^{88} + 24 q^{89} - 4 q^{91} - 8 q^{92} - 12 q^{94} - 16 q^{97}+O(q^{100}) $ 2 * q + 8 * q^4 + 2 * q^7 - 4 * q^13 + 8 * q^16 - 6 * q^17 + 4 * q^19 + 12 * q^22 - 2 * q^23 + 12 * q^26 + 8 * q^28 - 6 * q^29 + 10 * q^31 + 12 * q^34 + 2 * q^37 - 12 * q^38 - 6 * q^41 - 4 * q^43 + 12 * q^47 - 12 * q^49 - 16 * q^52 + 6 * q^53 + 36 * q^58 - 6 * q^59 + 16 * q^61 + 24 * q^62 - 16 * q^64 + 14 * q^67 - 24 * q^68 - 6 * q^71 - 4 * q^73 + 24 * q^74 + 16 * q^76 - 8 * q^79 + 12 * q^82 + 6 * q^83 + 24 * q^88 + 24 * q^89 - 4 * q^91 - 8 * q^92 - 12 * q^94 - 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 $

$2$	−2.44949	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−2.44949	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	2.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−4.89898	−1.73205
$9$	0	0
$10$	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	$-0.711666\pi$
$13$	−4.44949	−1.23407	−0.617033	+	0.786937i	$0.711666\pi$
$14$	−2.44949	−0.654654
$15$	0	0
$16$	4.00000	1.00000
$17$	−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	$-0.329501\pi$
$19$	4.44949	1.02078	0.510391	+	0.859942i	$0.329501\pi$
$20$	0	0
$21$	0	0
$22$	6.00000	1.27920
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	10.8990	2.13747
$27$	0	0
$28$	4.00000	0.755929
$29$	−10.3485	−1.92166	−0.960831	−	0.277134i	$-0.910615\pi$
$29$	−10.3485	−1.92166	−0.960831	+	0.277134i	$0.910615\pi$
$30$	0	0
$31$	0.101021	0.0181438	0.00907191	−	0.999959i	$-0.497112\pi$
$31$	0.101021	0.0181438	0.00907191	+	0.999959i	$0.497112\pi$
$32$	0	0
$33$	0	0
$34$	13.3485	2.28924
$35$	0	0
$36$	0	0
$37$	−3.89898	−0.640988	−0.320494	−	0.947250i	$-0.603849\pi$
$37$	−3.89898	−0.640988	−0.320494	+	0.947250i	$0.603849\pi$
$38$	−10.8990	−1.76805
$39$	0	0
$40$	0	0
$41$	−5.44949	−0.851067	−0.425534	−	0.904943i	$-0.639914\pi$
$41$	−5.44949	−0.851067	−0.425534	+	0.904943i	$0.639914\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−9.79796	−1.47710
$45$	0	0
$46$	2.44949	0.361158
$47$	8.44949	1.23248	0.616242	−	0.787557i	$-0.288654\pi$
$47$	8.44949	1.23248	0.616242	+	0.787557i	$0.288654\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−17.7980	−2.46813
$53$	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
$55$	0	0
$56$	−4.89898	−0.654654
$57$	0	0
$58$	25.3485	3.32842
$59$	−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	$0.486719\pi$
$62$	−0.247449	−0.0314260
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−21.7980	−2.64339
$69$	0	0
$70$	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	$-0.398668\pi$
$73$	5.34847	0.625991	0.312995	+	0.949755i	$0.398668\pi$
$74$	9.55051	1.11022
$75$	0	0
$76$	17.7980	2.04157
$77$	−2.44949	−0.279145
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	13.3485	1.47409
$83$	15.2474	1.67362	0.836812	−	0.547490i	$-0.184416\pi$
$83$	15.2474	1.67362	0.836812	+	0.547490i	$0.184416\pi$
$84$	0	0
$85$	0	0
$86$	4.89898	0.528271
$87$	0	0
$88$	12.0000	1.27920
$89$	16.8990	1.79129	0.895644	−	0.444771i	$-0.146715\pi$
$89$	16.8990	1.79129	0.895644	+	0.444771i	$0.146715\pi$
$90$	0	0
$91$	−4.44949	−0.466433
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−20.6969	−2.13473
$95$	0	0
$96$	0	0
$97$	−3.10102	−0.314861	−0.157430	−	0.987530i	$-0.550321\pi$
$97$	−3.10102	−0.314861	−0.157430	+	0.987530i	$0.550321\pi$
$98$	14.6969	1.48461
$99$	0	0
$100$	0	0
$101$	12.5505	1.24882	0.624411	−	0.781096i	$-0.285339\pi$
$101$	12.5505	1.24882	0.624411	+	0.781096i	$0.285339\pi$
$102$	0	0
$103$	12.6969	1.25107	0.625533	−	0.780198i	$-0.284881\pi$
$103$	12.6969	1.25107	0.625533	+	0.780198i	$0.284881\pi$
$104$	21.7980	2.13747
$105$	0	0
$106$	−1.34847	−0.130975
$107$	−5.44949	−0.526822	−0.263411	−	0.964684i	$-0.584848\pi$
$107$	−5.44949	−0.526822	−0.263411	+	0.964684i	$0.584848\pi$
$108$	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
$111$	0	0
$112$	4.00000	0.377964
$113$	−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
$115$	0	0
$116$	−41.3939	−3.84332
$117$	0	0
$118$	25.3485	2.33352
$119$	−5.44949	−0.499554
$120$	0	0
$121$	−5.00000	−0.454545
$122$	−1.59592	−0.144488
$123$	0	0
$124$	0.404082	0.0362876
$125$	0	0
$126$	0	0
$127$	1.55051	0.137586	0.0687928	−	0.997631i	$-0.478085\pi$
$127$	1.55051	0.137586	0.0687928	+	0.997631i	$0.478085\pi$
$128$	19.5959	1.73205
$129$	0	0
$130$	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
$134$	−17.1464	−1.48123
$135$	0	0
$136$	26.6969	2.28924
$137$	−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	$-0.565225\pi$
$139$	−4.79796	−0.406958	−0.203479	+	0.979079i	$0.565225\pi$
$140$	0	0
$141$	0	0
$142$	−10.6515	−0.893857
$143$	10.8990	0.911418
$144$	0	0
$145$	0	0
$146$	−13.1010	−1.08425
$147$	0	0
$148$	−15.5959	−1.28198
$149$	1.34847	0.110471	0.0552355	−	0.998473i	$-0.482409\pi$
$149$	1.34847	0.110471	0.0552355	+	0.998473i	$0.482409\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−21.7980	−1.76805
$153$	0	0
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	−18.5959	−1.48412	−0.742058	−	0.670336i	$-0.766150\pi$
$157$	−18.5959	−1.48412	−0.742058	+	0.670336i	$0.766150\pi$
$158$	9.79796	0.779484
$159$	0	0
$160$	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	$0.371911\pi$
$164$	−21.7980	−1.70213
$165$	0	0
$166$	−37.3485	−2.89880
$167$	13.3485	1.03294	0.516468	−	0.856307i	$-0.327247\pi$
$167$	13.3485	1.03294	0.516468	+	0.856307i	$0.327247\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	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
$175$	0	0
$176$	−9.79796	−0.738549
$177$	0	0
$178$	−41.3939	−3.10260
$179$	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	$0.607294\pi$
$182$	10.8990	0.807886
$183$	0	0
$184$	4.89898	0.361158
$185$	0	0
$186$	0	0
$187$	13.3485	0.976137
$188$	33.7980	2.46497
$189$	0	0
$190$	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	$-0.422515\pi$
$193$	6.69694	0.482056	0.241028	+	0.970518i	$0.422515\pi$
$194$	7.59592	0.545355
$195$	0	0
$196$	−24.0000	−1.71429
$197$	−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	$-0.532765\pi$
$199$	−2.89898	−0.205503	−0.102752	+	0.994707i	$0.532765\pi$
$200$	0	0
$201$	0	0
$202$	−30.7423	−2.16302
$203$	−10.3485	−0.726320
$204$	0	0
$205$	0	0
$206$	−31.1010	−2.16691
$207$	0	0
$208$	−17.7980	−1.23407
$209$	−10.8990	−0.753898
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	2.20204	0.151237
$213$	0	0
$214$	13.3485	0.912483
$215$	0	0
$216$	0	0
$217$	0.101021	0.00685772
$218$	−28.2929	−1.91623
$219$	0	0
$220$	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
$225$	0	0
$226$	4.04541	0.269097
$227$	21.7980	1.44678	0.723391	−	0.690439i	$-0.242583\pi$
$227$	21.7980	1.44678	0.723391	+	0.690439i	$0.242583\pi$
$228$	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
$231$	0	0
$232$	50.6969	3.32842
$233$	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
$235$	0	0
$236$	−41.3939	−2.69451
$237$	0	0
$238$	13.3485	0.865253
$239$	−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	$0.894183\pi$
$242$	12.2474	0.787296
$243$	0	0
$244$	2.60612	0.166840
$245$	0	0
$246$	0	0
$247$	−19.7980	−1.25971
$248$	−0.494897	−0.0314260
$249$	0	0
$250$	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
$254$	−3.79796	−0.238305
$255$	0	0
$256$	−32.0000	−2.00000
$257$	−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
$261$	0	0
$262$	−48.0000	−2.96545
$263$	27.2474	1.68015	0.840075	−	0.542471i	$-0.182511\pi$
$263$	27.2474	1.68015	0.840075	+	0.542471i	$0.182511\pi$
$264$	0	0
$265$	0	0
$266$	−10.8990	−0.668259
$267$	0	0
$268$	28.0000	1.71037
$269$	5.44949	0.332261	0.166131	−	0.986104i	$-0.446873\pi$
$269$	5.44949	0.332261	0.166131	+	0.986104i	$0.446873\pi$
$270$	0	0
$271$	31.6969	1.92545	0.962726	−	0.270479i	$-0.0871820\pi$
$271$	31.6969	1.92545	0.962726	+	0.270479i	$0.0871820\pi$
$272$	−21.7980	−1.32170
$273$	0	0
$274$	41.3939	2.50070
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	11.7526	0.704871
$279$	0	0
$280$	0	0
$281$	19.3485	1.15423	0.577116	−	0.816662i	$-0.304178\pi$
$281$	19.3485	1.15423	0.577116	+	0.816662i	$0.304178\pi$
$282$	0	0
$283$	2.10102	0.124893	0.0624464	−	0.998048i	$-0.480110\pi$
$283$	2.10102	0.124893	0.0624464	+	0.998048i	$0.480110\pi$
$284$	17.3939	1.03214
$285$	0	0
$286$	−26.6969	−1.57862
$287$	−5.44949	−0.321673
$288$	0	0
$289$	12.6969	0.746879
$290$	0	0
$291$	0	0
$292$	21.3939	1.25198
$293$	−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
$295$	0	0
$296$	19.1010	1.11022
$297$	0	0
$298$	−3.30306	−0.191341
$299$	4.44949	0.257321
$300$	0	0
$301$	−2.00000	−0.115278
$302$	−34.2929	−1.97333
$303$	0	0
$304$	17.7980	1.02078
$305$	0	0
$306$	0	0
$307$	2.65153	0.151331	0.0756654	−	0.997133i	$-0.475892\pi$
$307$	2.65153	0.151331	0.0756654	+	0.997133i	$0.475892\pi$
$308$	−9.79796	−0.558291
$309$	0	0
$310$	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	$-0.687921\pi$
$313$	−19.6969	−1.11334	−0.556668	+	0.830735i	$0.687921\pi$
$314$	45.5505	2.57056
$315$	0	0
$316$	−16.0000	−0.900070
$317$	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
$321$	0	0
$322$	2.44949	0.136505
$323$	−24.2474	−1.34916
$324$	0	0
$325$	0	0
$326$	−24.4949	−1.35665
$327$	0	0
$328$	26.6969	1.47409
$329$	8.44949	0.465835
$330$	0	0
$331$	24.5959	1.35191	0.675957	−	0.736941i	$-0.263731\pi$
$331$	24.5959	1.35191	0.675957	+	0.736941i	$0.263731\pi$
$332$	60.9898	3.34725
$333$	0	0
$334$	−32.6969	−1.78910
$335$	0	0
$336$	0	0
$337$	19.7980	1.07846	0.539232	−	0.842157i	$-0.318715\pi$
$337$	19.7980	1.07846	0.539232	+	0.842157i	$0.318715\pi$
$338$	−16.6515	−0.905724
$339$	0	0
$340$	0	0
$341$	−0.247449	−0.0134001
$342$	0	0
$343$	−13.0000	−0.701934
$344$	9.79796	0.528271
$345$	0	0
$346$	−48.0000	−2.58050
$347$	−28.8990	−1.55138	−0.775689	−	0.631115i	$-0.782598\pi$
$347$	−28.8990	−1.55138	−0.775689	+	0.631115i	$0.782598\pi$
$348$	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
$351$	0	0
$352$	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
$355$	0	0
$356$	67.5959	3.58258
$357$	0	0
$358$	−14.6969	−0.776757
$359$	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
$362$	21.7980	1.14568
$363$	0	0
$364$	−17.7980	−0.932867
$365$	0	0
$366$	0	0
$367$	−13.6969	−0.714974	−0.357487	−	0.933918i	$-0.616366\pi$
$367$	−13.6969	−0.714974	−0.357487	+	0.933918i	$0.616366\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	0.550510	0.0285811
$372$	0	0
$373$	−27.5959	−1.42886	−0.714431	−	0.699706i	$-0.753314\pi$
$373$	−27.5959	−1.42886	−0.714431	+	0.699706i	$0.753314\pi$
$374$	−32.6969	−1.69072
$375$	0	0
$376$	−41.3939	−2.13473
$377$	46.0454	2.37146
$378$	0	0
$379$	25.3939	1.30440	0.652198	−	0.758049i	$-0.273847\pi$
$379$	25.3939	1.30440	0.652198	+	0.758049i	$0.273847\pi$
$380$	0	0
$381$	0	0
$382$	−44.6969	−2.28689
$383$	1.65153	0.0843893	0.0421946	−	0.999109i	$-0.486565\pi$
$383$	1.65153	0.0843893	0.0421946	+	0.999109i	$0.486565\pi$
$384$	0	0
$385$	0	0
$386$	−16.4041	−0.834946
$387$	0	0
$388$	−12.4041	−0.629722
$389$	3.30306	0.167472	0.0837359	−	0.996488i	$-0.473315\pi$
$389$	3.30306	0.167472	0.0837359	+	0.996488i	$0.473315\pi$
$390$	0	0
$391$	5.44949	0.275593
$392$	29.3939	1.48461
$393$	0	0
$394$	56.0908	2.82581
$395$	0	0
$396$	0	0
$397$	13.7980	0.692500	0.346250	−	0.938142i	$-0.387455\pi$
$397$	13.7980	0.692500	0.346250	+	0.938142i	$0.387455\pi$
$398$	7.10102	0.355942
$399$	0	0
$400$	0	0
$401$	13.1010	0.654234	0.327117	−	0.944984i	$-0.393923\pi$
$401$	13.1010	0.654234	0.327117	+	0.944984i	$0.393923\pi$
$402$	0	0
$403$	−0.449490	−0.0223907
$404$	50.2020	2.49764
$405$	0	0
$406$	25.3485	1.25802
$407$	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
$411$	0	0
$412$	50.7878	2.50213
$413$	−10.3485	−0.509215
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	26.6969	1.30579
$419$	−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	$0.682385\pi$
$422$	−26.9444	−1.31163
$423$	0	0
$424$	−2.69694	−0.130975
$425$	0	0
$426$	0	0
$427$	0.651531	0.0315298
$428$	−21.7980	−1.05364
$429$	0	0
$430$	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	$-0.521416\pi$
$433$	−2.79796	−0.134461	−0.0672307	+	0.997737i	$0.521416\pi$
$434$	−0.247449	−0.0118779
$435$	0	0
$436$	46.2020	2.21268
$437$	−4.44949	−0.212848
$438$	0	0
$439$	−36.6969	−1.75145	−0.875725	−	0.482811i	$-0.839616\pi$
$439$	−36.6969	−1.75145	−0.875725	+	0.482811i	$0.839616\pi$
$440$	0	0
$441$	0	0
$442$	−59.3939	−2.82508
$443$	−20.4495	−0.971585	−0.485792	−	0.874074i	$-0.661469\pi$
$443$	−20.4495	−0.971585	−0.485792	+	0.874074i	$0.661469\pi$
$444$	0	0
$445$	0	0
$446$	52.8990	2.50484
$447$	0	0
$448$	−8.00000	−0.377964
$449$	21.2474	1.00273	0.501365	−	0.865236i	$-0.332832\pi$
$449$	21.2474	1.00273	0.501365	+	0.865236i	$0.332832\pi$
$450$	0	0
$451$	13.3485	0.628555
$452$	−6.60612	−0.310726
$453$	0	0
$454$	−53.3939	−2.50590
$455$	0	0
$456$	0	0
$457$	−0.101021	−0.00472554	−0.00236277	−	0.999997i	$-0.500752\pi$
$457$	−0.101021	−0.00472554	−0.00236277	+	0.999997i	$0.500752\pi$
$458$	69.7980	3.26144
$459$	0	0
$460$	0	0
$461$	18.4949	0.861393	0.430697	−	0.902497i	$-0.358268\pi$
$461$	18.4949	0.861393	0.430697	+	0.902497i	$0.358268\pi$
$462$	0	0
$463$	24.9444	1.15926	0.579632	−	0.814878i	$-0.303196\pi$
$463$	24.9444	1.15926	0.579632	+	0.814878i	$0.303196\pi$
$464$	−41.3939	−1.92166
$465$	0	0
$466$	−53.3939	−2.47342
$467$	−34.8434	−1.61236	−0.806179	−	0.591671i	$-0.798468\pi$
$467$	−34.8434	−1.61236	−0.806179	+	0.591671i	$0.798468\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	0	0
$472$	50.6969	2.33352
$473$	4.89898	0.225255
$474$	0	0
$475$	0	0
$476$	−21.7980	−0.999108
$477$	0	0
$478$	16.0454	0.733900
$479$	−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
$482$	71.8888	3.27444
$483$	0	0
$484$	−20.0000	−0.909091
$485$	0	0
$486$	0	0
$487$	18.9444	0.858452	0.429226	−	0.903197i	$-0.358786\pi$
$487$	18.9444	0.858452	0.429226	+	0.903197i	$0.358786\pi$
$488$	−3.19184	−0.144488
$489$	0	0
$490$	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
$494$	48.4949	2.18189
$495$	0	0
$496$	0.404082	0.0181438
$497$	4.34847	0.195056
$498$	0	0
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	0	0
$501$	0	0
$502$	44.0908	1.96787
$503$	7.04541	0.314139	0.157070	−	0.987588i	$-0.449795\pi$
$503$	7.04541	0.314139	0.157070	+	0.987588i	$0.449795\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	0	0
$508$	6.20204	0.275171
$509$	−28.8990	−1.28092	−0.640462	−	0.767990i	$-0.721257\pi$
$509$	−28.8990	−1.28092	−0.640462	+	0.767990i	$0.721257\pi$
$510$	0	0
$511$	5.34847	0.236602
$512$	39.1918	1.73205
$513$	0	0
$514$	62.0908	2.73871
$515$	0	0
$516$	0	0
$517$	−20.6969	−0.910250
$518$	9.55051	0.419625
$519$	0	0
$520$	0	0
$521$	43.8434	1.92081	0.960406	−	0.278603i	$-0.0898713\pi$
$521$	43.8434	1.92081	0.960406	+	0.278603i	$0.0898713\pi$
$522$	0	0
$523$	33.3939	1.46021	0.730106	−	0.683334i	$-0.239471\pi$
$523$	33.3939	1.46021	0.730106	+	0.683334i	$0.239471\pi$
$524$	78.3837	3.42421
$525$	0	0
$526$	−66.7423	−2.91010
$527$	−0.550510	−0.0239806
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	17.7980	0.771639
$533$	24.2474	1.05027
$534$	0	0
$535$	0	0
$536$	−34.2929	−1.48123
$537$	0	0
$538$	−13.3485	−0.575493
$539$	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	$0.410565\pi$
$542$	−77.6413	−3.33498
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−67.5959	−2.88755
$549$	0	0
$550$	0	0
$551$	−46.0454	−1.96160
$552$	0	0
$553$	−4.00000	−0.170097
$554$	−24.4949	−1.04069
$555$	0	0
$556$	−19.1918	−0.813915
$557$	−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
$561$	0	0
$562$	−47.3939	−1.99919
$563$	5.44949	0.229669	0.114834	−	0.993385i	$-0.463366\pi$
$563$	5.44949	0.229669	0.114834	+	0.993385i	$0.463366\pi$
$564$	0	0
$565$	0	0
$566$	−5.14643	−0.216321
$567$	0	0
$568$	−21.3031	−0.893857
$569$	28.2929	1.18610	0.593049	−	0.805166i	$-0.297924\pi$
$569$	28.2929	1.18610	0.593049	+	0.805166i	$0.297924\pi$
$570$	0	0
$571$	13.1464	0.550161	0.275080	−	0.961421i	$-0.411296\pi$
$571$	13.1464	0.550161	0.275080	+	0.961421i	$0.411296\pi$
$572$	43.5959	1.82284
$573$	0	0
$574$	13.3485	0.557154
$575$	0	0
$576$	0	0
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	−31.1010	−1.29363
$579$	0	0
$580$	0	0
$581$	15.2474	0.632571
$582$	0	0
$583$	−1.34847	−0.0558479
$584$	−26.2020	−1.08425
$585$	0	0
$586$	52.0454	2.14998
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	0.449490	0.0185209
$590$	0	0
$591$	0	0
$592$	−15.5959	−0.640988
$593$	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
$595$	0	0
$596$	5.39388	0.220942
$597$	0	0
$598$	−10.8990	−0.445692
$599$	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	$0.467483\pi$
$602$	4.89898	0.199667
$603$	0	0
$604$	56.0000	2.27861
$605$	0	0
$606$	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
$609$	0	0
$610$	0	0
$611$	−37.5959	−1.52097
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−6.49490	−0.262113
$615$	0	0
$616$	12.0000	0.483494
$617$	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	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	29.3939	1.17859
$623$	16.8990	0.677043
$624$	0	0
$625$	0	0
$626$	48.2474	1.92836
$627$	0	0
$628$	−74.3837	−2.96823
$629$	21.2474	0.847191
$630$	0	0
$631$	29.5505	1.17639	0.588194	−	0.808720i	$-0.299839\pi$
$631$	29.5505	1.17639	0.588194	+	0.808720i	$0.299839\pi$
$632$	19.5959	0.779484
$633$	0	0
$634$	−15.3031	−0.607762
$635$	0	0
$636$	0	0
$637$	26.6969	1.05777
$638$	−62.0908	−2.45820
$639$	0	0
$640$	0	0
$641$	−26.9444	−1.06424	−0.532120	−	0.846669i	$-0.678604\pi$
$641$	−26.9444	−1.06424	−0.532120	+	0.846669i	$0.678604\pi$
$642$	0	0
$643$	9.69694	0.382410	0.191205	−	0.981550i	$-0.438760\pi$
$643$	9.69694	0.382410	0.191205	+	0.981550i	$0.438760\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	59.3939	2.33682
$647$	1.95459	0.0768430	0.0384215	−	0.999262i	$-0.487767\pi$
$647$	1.95459	0.0768430	0.0384215	+	0.999262i	$0.487767\pi$
$648$	0	0
$649$	25.3485	0.995014
$650$	0	0
$651$	0	0
$652$	40.0000	1.56652
$653$	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
$655$	0	0
$656$	−21.7980	−0.851067
$657$	0	0
$658$	−20.6969	−0.806851
$659$	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	$0.146200\pi$
$662$	−60.2474	−2.34158
$663$	0	0
$664$	−74.6969	−2.89880
$665$	0	0
$666$	0	0
$667$	10.3485	0.400694
$668$	53.3939	2.06587
$669$	0	0
$670$	0	0
$671$	−1.59592	−0.0616097
$672$	0	0
$673$	−22.4495	−0.865364	−0.432682	−	0.901547i	$-0.642433\pi$
$673$	−22.4495	−0.865364	−0.432682	+	0.901547i	$0.642433\pi$
$674$	−48.4949	−1.86795
$675$	0	0
$676$	27.1918	1.04584
$677$	−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
$681$	0	0
$682$	0.606123	0.0232097
$683$	18.2474	0.698219	0.349110	−	0.937082i	$-0.386484\pi$
$683$	18.2474	0.698219	0.349110	+	0.937082i	$0.386484\pi$
$684$	0	0
$685$	0	0
$686$	31.8434	1.21579
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−2.44949	−0.0933181
$690$	0	0
$691$	16.2020	0.616355	0.308177	−	0.951329i	$-0.400281\pi$
$691$	16.2020	0.616355	0.308177	+	0.951329i	$0.400281\pi$
$692$	78.3837	2.97970
$693$	0	0
$694$	70.7878	2.68707
$695$	0	0
$696$	0	0
$697$	29.6969	1.12485
$698$	−57.5505	−2.17832
$699$	0	0
$700$	0	0
$701$	52.0454	1.96573	0.982864	−	0.184332i	$-0.0590123\pi$
$701$	52.0454	1.96573	0.982864	+	0.184332i	$0.0590123\pi$
$702$	0	0
$703$	−17.3485	−0.654310
$704$	19.5959	0.738549
$705$	0	0
$706$	38.0908	1.43357
$707$	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
$711$	0	0
$712$	−82.7878	−3.10260
$713$	−0.101021	−0.00378325
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	0	0
$718$	−8.69694	−0.324567
$719$	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
$722$	−1.95459	−0.0727424
$723$	0	0
$724$	−35.5959	−1.32291
$725$	0	0
$726$	0	0
$727$	16.3031	0.604647	0.302324	−	0.953205i	$-0.402238\pi$
$727$	16.3031	0.604647	0.302324	+	0.953205i	$0.402238\pi$
$728$	21.7980	0.807886
$729$	0	0
$730$	0	0
$731$	10.8990	0.403113
$732$	0	0
$733$	8.59592	0.317497	0.158749	−	0.987319i	$-0.449254\pi$
$733$	8.59592	0.317497	0.158749	+	0.987319i	$0.449254\pi$
$734$	33.5505	1.23837
$735$	0	0
$736$	0	0
$737$	−17.1464	−0.631597
$738$	0	0
$739$	−9.20204	−0.338503	−0.169251	−	0.985573i	$-0.554135\pi$
$739$	−9.20204	−0.338503	−0.169251	+	0.985573i	$0.554135\pi$
$740$	0	0
$741$	0	0
$742$	−1.34847	−0.0495039
$743$	−21.7980	−0.799690	−0.399845	−	0.916583i	$-0.630936\pi$
$743$	−21.7980	−0.799690	−0.399845	+	0.916583i	$0.630936\pi$
$744$	0	0
$745$	0	0
$746$	67.5959	2.47486
$747$	0	0
$748$	53.3939	1.95227
$749$	−5.44949	−0.199120
$750$	0	0
$751$	30.6515	1.11849	0.559245	−	0.829002i	$-0.311091\pi$
$751$	30.6515	1.11849	0.559245	+	0.829002i	$0.311091\pi$
$752$	33.7980	1.23248
$753$	0	0
$754$	−112.788	−4.10749
$755$	0	0
$756$	0	0
$757$	3.69694	0.134368	0.0671838	−	0.997741i	$-0.478599\pi$
$757$	3.69694	0.134368	0.0671838	+	0.997741i	$0.478599\pi$
$758$	−62.2020	−2.25928
$759$	0	0
$760$	0	0
$761$	−14.1464	−0.512808	−0.256404	−	0.966570i	$-0.582538\pi$
$761$	−14.1464	−0.512808	−0.256404	+	0.966570i	$0.582538\pi$
$762$	0	0
$763$	11.5505	0.418157
$764$	72.9898	2.64068
$765$	0	0
$766$	−4.04541	−0.146167
$767$	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
$771$	0	0
$772$	26.7878	0.964112
$773$	−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	0
$776$	15.1918	0.545355
$777$	0	0
$778$	−8.09082	−0.290070
$779$	−24.2474	−0.868755
$780$	0	0
$781$	−10.6515	−0.381142
$782$	−13.3485	−0.477340
$783$	0	0
$784$	−24.0000	−0.857143
$785$	0	0
$786$	0	0
$787$	21.6969	0.773412	0.386706	−	0.922203i	$-0.373613\pi$
$787$	21.6969	0.773412	0.386706	+	0.922203i	$0.373613\pi$
$788$	−91.5959	−3.26297
$789$	0	0
$790$	0	0
$791$	−1.65153	−0.0587217
$792$	0	0
$793$	−2.89898	−0.102946
$794$	−33.7980	−1.19944
$795$	0	0
$796$	−11.5959	−0.411006
$797$	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
$801$	0	0
$802$	−32.0908	−1.13317
$803$	−13.1010	−0.462325
$804$	0	0
$805$	0	0
$806$	1.10102	0.0387818
$807$	0	0
$808$	−61.4847	−2.16302
$809$	−6.55051	−0.230304	−0.115152	−	0.993348i	$-0.536735\pi$
$809$	−6.55051	−0.230304	−0.115152	+	0.993348i	$0.536735\pi$
$810$	0	0
$811$	−42.3939	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$811$	−42.3939	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$812$	−41.3939	−1.45264
$813$	0	0
$814$	−23.3939	−0.819955
$815$	0	0
$816$	0	0
$817$	−8.89898	−0.311336
$818$	−53.6413	−1.87552
$819$	0	0
$820$	0	0
$821$	−21.3031	−0.743482	−0.371741	−	0.928336i	$-0.621239\pi$
$821$	−21.3031	−0.743482	−0.371741	+	0.928336i	$0.621239\pi$
$822$	0	0
$823$	−3.59592	−0.125346	−0.0626729	−	0.998034i	$-0.519962\pi$
$823$	−3.59592	−0.125346	−0.0626729	+	0.998034i	$0.519962\pi$
$824$	−62.2020	−2.16691
$825$	0	0
$826$	25.3485	0.881986
$827$	32.1464	1.11784	0.558920	−	0.829221i	$-0.311216\pi$
$827$	32.1464	1.11784	0.558920	+	0.829221i	$0.311216\pi$
$828$	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
$831$	0	0
$832$	35.5959	1.23407
$833$	32.6969	1.13288
$834$	0	0
$835$	0	0
$836$	−43.5959	−1.50780
$837$	0	0
$838$	82.1816	2.83892
$839$	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
$842$	54.4949	1.87802
$843$	0	0
$844$	44.0000	1.51454
$845$	0	0
$846$	0	0
$847$	−5.00000	−0.171802
$848$	2.20204	0.0756184
$849$	0	0
$850$	0	0
$851$	3.89898	0.133655
$852$	0	0
$853$	−46.0908	−1.57812	−0.789060	−	0.614316i	$-0.789432\pi$
$853$	−46.0908	−1.57812	−0.789060	+	0.614316i	$0.789432\pi$
$854$	−1.59592	−0.0546112
$855$	0	0
$856$	26.6969	0.912483
$857$	−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	$-0.274121\pi$
$859$	38.1918	1.30309	0.651544	+	0.758611i	$0.274121\pi$
$860$	0	0
$861$	0	0
$862$	−60.0000	−2.04361
$863$	6.49490	0.221089	0.110544	−	0.993871i	$-0.464741\pi$
$863$	6.49490	0.221089	0.110544	+	0.993871i	$0.464741\pi$
$864$	0	0
$865$	0	0
$866$	6.85357	0.232894
$867$	0	0
$868$	0.404082	0.0137154
$869$	9.79796	0.332373
$870$	0	0
$871$	−31.1464	−1.05536
$872$	−56.5857	−1.91623
$873$	0	0
$874$	10.8990	0.368663
$875$	0	0
$876$	0	0
$877$	10.4949	0.354388	0.177194	−	0.984176i	$-0.443298\pi$
$877$	10.4949	0.354388	0.177194	+	0.984176i	$0.443298\pi$
$878$	89.8888	3.03360
$879$	0	0
$880$	0	0
$881$	−8.44949	−0.284671	−0.142335	−	0.989819i	$-0.545461\pi$
$881$	−8.44949	−0.284671	−0.142335	+	0.989819i	$0.545461\pi$
$882$	0	0
$883$	37.5505	1.26368	0.631838	−	0.775101i	$-0.282301\pi$
$883$	37.5505	1.26368	0.631838	+	0.775101i	$0.282301\pi$
$884$	96.9898	3.26212
$885$	0	0
$886$	50.0908	1.68283
$887$	−6.49490	−0.218077	−0.109039	−	0.994038i	$-0.534777\pi$
$887$	−6.49490	−0.218077	−0.109039	+	0.994038i	$0.534777\pi$
$888$	0	0
$889$	1.55051	0.0520024
$890$	0	0
$891$	0	0
$892$	−86.3837	−2.89234
$893$	37.5959	1.25810
$894$	0	0
$895$	0	0
$896$	19.5959	0.654654
$897$	0	0
$898$	−52.0454	−1.73678
$899$	−1.04541	−0.0348663
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	−32.6969	−1.08869
$903$	0	0
$904$	8.09082	0.269097
$905$	0	0
$906$	0	0
$907$	−38.7980	−1.28827	−0.644133	−	0.764914i	$-0.722781\pi$
$907$	−38.7980	−1.28827	−0.644133	+	0.764914i	$0.722781\pi$
$908$	87.1918	2.89356
$909$	0	0
$910$	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
$914$	0.247449	0.00818488
$915$	0	0
$916$	−113.980	−3.76599
$917$	19.5959	0.647114
$918$	0	0
$919$	−45.3939	−1.49741	−0.748703	−	0.662906i	$-0.769323\pi$
$919$	−45.3939	−1.49741	−0.748703	+	0.662906i	$0.769323\pi$
$920$	0	0
$921$	0	0
$922$	−45.3031	−1.49198
$923$	−19.3485	−0.636863
$924$	0	0
$925$	0	0
$926$	−61.1010	−2.00790
$927$	0	0
$928$	0	0
$929$	−22.8434	−0.749467	−0.374733	−	0.927133i	$-0.622266\pi$
$929$	−22.8434	−0.749467	−0.374733	+	0.927133i	$0.622266\pi$
$930$	0	0
$931$	−26.6969	−0.874957
$932$	87.1918	2.85606
$933$	0	0
$934$	85.3485	2.79269
$935$	0	0
$936$	0	0
$937$	8.89898	0.290717	0.145358	−	0.989379i	$-0.453566\pi$
$937$	8.89898	0.290717	0.145358	+	0.989379i	$0.453566\pi$
$938$	−17.1464	−0.559851
$939$	0	0
$940$	0	0
$941$	24.8536	0.810203	0.405102	−	0.914272i	$-0.367236\pi$
$941$	24.8536	0.810203	0.405102	+	0.914272i	$0.367236\pi$
$942$	0	0
$943$	5.44949	0.177460
$944$	−41.3939	−1.34726
$945$	0	0
$946$	−12.0000	−0.390154
$947$	−0.494897	−0.0160820	−0.00804100	−	0.999968i	$-0.502560\pi$
$947$	−0.494897	−0.0160820	−0.00804100	+	0.999968i	$0.502560\pi$
$948$	0	0
$949$	−23.7980	−0.772514
$950$	0	0
$951$	0	0
$952$	26.6969	0.865253
$953$	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
$955$	0	0
$956$	−26.2020	−0.847435
$957$	0	0
$958$	80.6969	2.60720
$959$	−16.8990	−0.545697
$960$	0	0
$961$	−30.9898	−0.999671
$962$	−42.4949	−1.37009
$963$	0	0
$964$	−117.394	−3.78100
$965$	0	0
$966$	0	0
$967$	−54.0454	−1.73798	−0.868992	−	0.494827i	$-0.835231\pi$
$967$	−54.0454	−1.73798	−0.868992	+	0.494827i	$0.835231\pi$
$968$	24.4949	0.787296
$969$	0	0
$970$	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
$974$	−46.4041	−1.48688
$975$	0	0
$976$	2.60612	0.0834200
$977$	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
$981$	0	0
$982$	30.7423	0.981028
$983$	−33.7423	−1.07621	−0.538107	−	0.842877i	$-0.680860\pi$
$983$	−33.7423	−1.07621	−0.538107	+	0.842877i	$0.680860\pi$
$984$	0	0
$985$	0	0
$986$	−138.136	−4.39915
$987$	0	0
$988$	−79.1918	−2.51943
$989$	2.00000	0.0635963
$990$	0	0
$991$	57.7878	1.83569	0.917844	−	0.396941i	$-0.129928\pi$
$991$	57.7878	1.83569	0.917844	+	0.396941i	$0.129928\pi$
$992$	0	0
$993$	0	0
$994$	−10.6515	−0.337846
$995$	0	0
$996$	0	0
$997$	−40.0908	−1.26969	−0.634844	−	0.772640i	$-0.718936\pi$
$997$	−40.0908	−1.26969	−0.634844	+	0.772640i	$0.718936\pi$
$998$	2.44949	0.0775372
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.bl.1.1		2
3.2	odd	2		1725.2.a.y.1.2		2
5.4	even	2		1035.2.a.k.1.2		2
15.2	even	4		1725.2.b.m.1174.4		4
15.8	even	4		1725.2.b.m.1174.1		4
15.14	odd	2		345.2.a.i.1.1	✓	2
60.59	even	2		5520.2.a.bi.1.1		2
345.344	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	15.14	odd	2
1035.2.a.k.1.2		2	5.4	even	2
1725.2.a.y.1.2		2	3.2	odd	2
1725.2.b.m.1174.1		4	15.8	even	4
1725.2.b.m.1174.4		4	15.2	even	4
5175.2.a.bl.1.1		2	1.1	even	1	trivial
5520.2.a.bi.1.1		2	60.59	even	2
7935.2.a.t.1.1		2	345.344	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44949 q^{2} +4.00000 q^{4} +1.00000 q^{7} -4.89898 q^{8} +O(q^{10})\)	\(q-2.44949 q^{2} +4.00000 q^{4} +1.00000 q^{7} -4.89898 q^{8} -2.44949 q^{11} -4.44949 q^{13} -2.44949 q^{14} +4.00000 q^{16} -5.44949 q^{17} +4.44949 q^{19} +6.00000 q^{22} -1.00000 q^{23} +10.8990 q^{26} +4.00000 q^{28} -10.3485 q^{29} +0.101021 q^{31} +13.3485 q^{34} -3.89898 q^{37} -10.8990 q^{38} -5.44949 q^{41} -2.00000 q^{43} -9.79796 q^{44} +2.44949 q^{46} +8.44949 q^{47} -6.00000 q^{49} -17.7980 q^{52} +0.550510 q^{53} -4.89898 q^{56} +25.3485 q^{58} -10.3485 q^{59} +0.651531 q^{61} -0.247449 q^{62} -8.00000 q^{64} +7.00000 q^{67} -21.7980 q^{68} +4.34847 q^{71} +5.34847 q^{73} +9.55051 q^{74} +17.7980 q^{76} -2.44949 q^{77} -4.00000 q^{79} +13.3485 q^{82} +15.2474 q^{83} +4.89898 q^{86} +12.0000 q^{88} +16.8990 q^{89} -4.44949 q^{91} -4.00000 q^{92} -20.6969 q^{94} -3.10102 q^{97} +14.6969 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4} + 2 q^{7}+O(q^{10}) \) 2 * q + 8 * q^4 + 2 * q^7	\( 2 q + 8 q^{4} + 2 q^{7} - 4 q^{13} + 8 q^{16} - 6 q^{17} + 4 q^{19} + 12 q^{22} - 2 q^{23} + 12 q^{26} + 8 q^{28} - 6 q^{29} + 10 q^{31} + 12 q^{34} + 2 q^{37} - 12 q^{38} - 6 q^{41} - 4 q^{43} + 12 q^{47} - 12 q^{49} - 16 q^{52} + 6 q^{53} + 36 q^{58} - 6 q^{59} + 16 q^{61} + 24 q^{62} - 16 q^{64} + 14 q^{67} - 24 q^{68} - 6 q^{71} - 4 q^{73} + 24 q^{74} + 16 q^{76} - 8 q^{79} + 12 q^{82} + 6 q^{83} + 24 q^{88} + 24 q^{89} - 4 q^{91} - 8 q^{92} - 12 q^{94} - 16 q^{97}+O(q^{100}) \) 2 * q + 8 * q^4 + 2 * q^7 - 4 * q^13 + 8 * q^16 - 6 * q^17 + 4 * q^19 + 12 * q^22 - 2 * q^23 + 12 * q^26 + 8 * q^28 - 6 * q^29 + 10 * q^31 + 12 * q^34 + 2 * q^37 - 12 * q^38 - 6 * q^41 - 4 * q^43 + 12 * q^47 - 12 * q^49 - 16 * q^52 + 6 * q^53 + 36 * q^58 - 6 * q^59 + 16 * q^61 + 24 * q^62 - 16 * q^64 + 14 * q^67 - 24 * q^68 - 6 * q^71 - 4 * q^73 + 24 * q^74 + 16 * q^76 - 8 * q^79 + 12 * q^82 + 6 * q^83 + 24 * q^88 + 24 * q^89 - 4 * q^91 - 8 * q^92 - 12 * q^94 - 16 * q^97

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