LMFDB - Embedded newform 1575.2.d.e.1324.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.e.1324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{2} -0.438447 q^{4} +1.00000i q^{7} +2.43845i q^{8} -2.56155 q^{11} +4.56155i q^{13} -1.56155 q^{14} -4.68466 q^{16} -4.56155i q^{17} -1.12311 q^{19} -4.00000i q^{22} +5.12311i q^{23} -7.12311 q^{26} -0.438447i q^{28} -5.68466 q^{29} -2.43845i q^{32} +7.12311 q^{34} -6.00000i q^{37} -1.75379i q^{38} +3.12311 q^{41} +9.12311i q^{43} +1.12311 q^{44} -8.00000 q^{46} +3.68466i q^{47} -1.00000 q^{49} -2.00000i q^{52} -3.12311i q^{53} -2.43845 q^{56} -8.87689i q^{58} -4.00000 q^{59} -9.36932 q^{61} -5.56155 q^{64} +6.24621i q^{67} +2.00000i q^{68} -8.00000 q^{71} +4.24621i q^{73} +9.36932 q^{74} +0.492423 q^{76} -2.56155i q^{77} +6.56155 q^{79} +4.87689i q^{82} -4.00000i q^{83} -14.2462 q^{86} -6.24621i q^{88} +7.12311 q^{89} -4.56155 q^{91} -2.24621i q^{92} -5.75379 q^{94} +14.8078i q^{97} -1.56155i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4} - 2 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{19} - 12 q^{26} + 2 q^{29} + 12 q^{34} - 4 q^{41} - 12 q^{44} - 32 q^{46} - 4 q^{49} - 18 q^{56} - 16 q^{59} + 12 q^{61} - 14 q^{64} - 32 q^{71} - 12 q^{74}+ \cdots - 56 q^{94}+O(q^{100}) $ 4 * q - 10 * q^4 - 2 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^19 - 12 * q^26 + 2 * q^29 + 12 * q^34 - 4 * q^41 - 12 * q^44 - 32 * q^46 - 4 * q^49 - 18 * q^56 - 16 * q^59 + 12 * q^61 - 14 * q^64 - 32 * q^71 - 12 * q^74 - 64 * q^76 + 18 * q^79 - 24 * q^86 + 12 * q^89 - 10 * q^91 - 56 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.56155i	1.10418i	0.833783	+	0.552092i	$0.186170\pi$
$2$	1.56155i	1.10418i	−0.833783	+	0.552092i	$0.813830\pi$
$3$	0	0
$3$	0	0
$4$	−0.438447	−0.219224
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	2.43845i	0.862121i
$9$	0	0
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	4.56155i	1.26515i	0.774500	+	0.632574i	$0.218001\pi$
$13$	4.56155i	1.26515i	−0.774500	+	0.632574i	$0.781999\pi$
$14$	−1.56155	−0.417343
$15$	0	0
$16$	−4.68466	−1.17116
$17$	− 4.56155i	− 1.10634i	−0.833069	−	0.553170i	$-0.813418\pi$
$17$	− 4.56155i	− 1.10634i	0.833069	−	0.553170i	$-0.186582\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	5.12311i	1.06824i	0.845408	+	0.534121i	$0.179357\pi$
$23$	5.12311i	1.06824i	−0.845408	+	0.534121i	$0.820643\pi$
$24$	0	0
$25$	0	0
$26$	−7.12311	−1.39696
$27$	0	0
$28$	− 0.438447i	− 0.0828587i
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 2.43845i	− 0.431061i
$33$	0	0
$34$	7.12311	1.22160
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	− 1.75379i	− 0.284502i
$39$	0	0
$40$	0	0
$41$	3.12311	0.487747	0.243874	−	0.969807i	$-0.421582\pi$
$41$	3.12311	0.487747	0.243874	+	0.969807i	$0.421582\pi$
$42$	0	0
$43$	9.12311i	1.39126i	0.718400	+	0.695630i	$0.244875\pi$
$43$	9.12311i	1.39126i	−0.718400	+	0.695630i	$0.755125\pi$
$44$	1.12311	0.169315
$45$	0	0
$46$	−8.00000	−1.17954
$47$	3.68466i	0.537463i	0.963215	+	0.268731i	$0.0866044\pi$
$47$	3.68466i	0.537463i	−0.963215	+	0.268731i	$0.913396\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	− 3.12311i	− 0.428992i	−0.976725	−	0.214496i	$-0.931189\pi$
$53$	− 3.12311i	− 0.428992i	0.976725	−	0.214496i	$-0.0688108\pi$
$54$	0	0
$55$	0	0
$56$	−2.43845	−0.325851
$57$	0	0
$58$	− 8.87689i	− 1.16559i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−9.36932	−1.19962	−0.599809	−	0.800143i	$-0.704757\pi$
$61$	−9.36932	−1.19962	−0.599809	+	0.800143i	$0.704757\pi$
$62$	0	0
$63$	0	0
$64$	−5.56155	−0.695194
$65$	0	0
$66$	0	0
$67$	6.24621i	0.763096i	0.924349	+	0.381548i	$0.124609\pi$
$67$	6.24621i	0.763096i	−0.924349	+	0.381548i	$0.875391\pi$
$68$	2.00000i	0.242536i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	4.24621i	0.496981i	0.968634	+	0.248491i	$0.0799345\pi$
$73$	4.24621i	0.496981i	−0.968634	+	0.248491i	$0.920065\pi$
$74$	9.36932	1.08916
$75$	0	0
$76$	0.492423	0.0564847
$77$	− 2.56155i	− 0.291916i
$78$	0	0
$79$	6.56155	0.738232	0.369116	−	0.929383i	$-0.379660\pi$
$79$	6.56155	0.738232	0.369116	+	0.929383i	$0.379660\pi$
$80$	0	0
$81$	0	0
$82$	4.87689i	0.538563i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	−14.2462	−1.53621
$87$	0	0
$88$	− 6.24621i	− 0.665848i
$89$	7.12311	0.755048	0.377524	−	0.926000i	$-0.376776\pi$
$89$	7.12311	0.755048	0.377524	+	0.926000i	$0.376776\pi$
$90$	0	0
$91$	−4.56155	−0.478181
$92$	− 2.24621i	− 0.234184i
$93$	0	0
$94$	−5.75379	−0.593458
$95$	0	0
$96$	0	0
$97$	14.8078i	1.50350i	0.659448	+	0.751750i	$0.270790\pi$
$97$	14.8078i	1.50350i	−0.659448	+	0.751750i	$0.729210\pi$
$98$	− 1.56155i	− 0.157741i
$99$	0	0
$100$	0	0
$101$	−0.246211	−0.0244989	−0.0122495	−	0.999925i	$-0.503899\pi$
$101$	−0.246211	−0.0244989	−0.0122495	+	0.999925i	$0.503899\pi$
$102$	0	0
$103$	1.43845i	0.141734i	0.997486	+	0.0708672i	$0.0225767\pi$
$103$	1.43845i	0.141734i	−0.997486	+	0.0708672i	$0.977423\pi$
$104$	−11.1231	−1.09071
$105$	0	0
$106$	4.87689	0.473686
$107$	− 11.3693i	− 1.09911i	−0.835456	−	0.549557i	$-0.814797\pi$
$107$	− 11.3693i	− 1.09911i	0.835456	−	0.549557i	$-0.185203\pi$
$108$	0	0
$109$	−17.6847	−1.69388	−0.846942	−	0.531686i	$-0.821559\pi$
$109$	−17.6847	−1.69388	−0.846942	+	0.531686i	$0.821559\pi$
$110$	0	0
$111$	0	0
$112$	− 4.68466i	− 0.442659i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	2.49242	0.231416
$117$	0	0
$118$	− 6.24621i	− 0.575010i
$119$	4.56155	0.418157
$120$	0	0
$121$	−4.43845	−0.403495
$122$	− 14.6307i	− 1.32460i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.2462i	− 0.909204i	−0.890695	−	0.454602i	$-0.849781\pi$
$127$	− 10.2462i	− 0.909204i	0.890695	−	0.454602i	$-0.150219\pi$
$128$	− 13.5616i	− 1.19868i
$129$	0	0
$130$	0	0
$131$	9.12311	0.797089	0.398545	−	0.917149i	$-0.369515\pi$
$131$	9.12311	0.797089	0.398545	+	0.917149i	$0.369515\pi$
$132$	0	0
$133$	− 1.12311i	− 0.0973856i
$134$	−9.75379	−0.842599
$135$	0	0
$136$	11.1231	0.953798
$137$	− 8.87689i	− 0.758404i	−0.925314	−	0.379202i	$-0.876199\pi$
$137$	− 8.87689i	− 0.758404i	0.925314	−	0.379202i	$-0.123801\pi$
$138$	0	0
$139$	6.87689	0.583291	0.291645	−	0.956527i	$-0.405797\pi$
$139$	6.87689	0.583291	0.291645	+	0.956527i	$0.405797\pi$
$140$	0	0
$141$	0	0
$142$	− 12.4924i	− 1.04834i
$143$	− 11.6847i	− 0.977120i
$144$	0	0
$145$	0	0
$146$	−6.63068	−0.548759
$147$	0	0
$148$	2.63068i	0.216241i
$149$	−4.24621	−0.347863	−0.173932	−	0.984758i	$-0.555647\pi$
$149$	−4.24621	−0.347863	−0.173932	+	0.984758i	$0.555647\pi$
$150$	0	0
$151$	21.9309	1.78471	0.892354	−	0.451335i	$-0.149052\pi$
$151$	21.9309	1.78471	0.892354	+	0.451335i	$0.149052\pi$
$152$	− 2.73863i	− 0.222133i
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	− 3.75379i	− 0.299585i	−0.988717	−	0.149792i	$-0.952139\pi$
$157$	− 3.75379i	− 0.299585i	0.988717	−	0.149792i	$-0.0478606\pi$
$158$	10.2462i	0.815145i
$159$	0	0
$160$	0	0
$161$	−5.12311	−0.403757
$162$	0	0
$163$	1.12311i	0.0879684i	0.999032	+	0.0439842i	$0.0140051\pi$
$163$	1.12311i	0.0879684i	−0.999032	+	0.0439842i	$0.985995\pi$
$164$	−1.36932	−0.106926
$165$	0	0
$166$	6.24621	0.484800
$167$	21.9309i	1.69706i	0.529146	+	0.848531i	$0.322512\pi$
$167$	21.9309i	1.69706i	−0.529146	+	0.848531i	$0.677488\pi$
$168$	0	0
$169$	−7.80776	−0.600597
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	8.56155i	0.650923i	0.945555	+	0.325461i	$0.105520\pi$
$173$	8.56155i	0.650923i	−0.945555	+	0.325461i	$0.894480\pi$
$174$	0	0
$175$	0	0
$176$	12.0000	0.904534
$177$	0	0
$178$	11.1231i	0.833712i
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	23.6155	1.75533	0.877664	−	0.479276i	$-0.159101\pi$
$181$	23.6155	1.75533	0.877664	+	0.479276i	$0.159101\pi$
$182$	− 7.12311i	− 0.528000i
$183$	0	0
$184$	−12.4924	−0.920954
$185$	0	0
$186$	0	0
$187$	11.6847i	0.854467i
$188$	− 1.61553i	− 0.117824i
$189$	0	0
$190$	0	0
$191$	9.43845	0.682942	0.341471	−	0.939892i	$-0.389075\pi$
$191$	9.43845	0.682942	0.341471	+	0.939892i	$0.389075\pi$
$192$	0	0
$193$	− 5.36932i	− 0.386492i	−0.981150	−	0.193246i	$-0.938098\pi$
$193$	− 5.36932i	− 0.386492i	0.981150	−	0.193246i	$-0.0619015\pi$
$194$	−23.1231	−1.66014
$195$	0	0
$196$	0.438447	0.0313177
$197$	− 7.12311i	− 0.507500i	−0.967270	−	0.253750i	$-0.918336\pi$
$197$	− 7.12311i	− 0.507500i	0.967270	−	0.253750i	$-0.0816641\pi$
$198$	0	0
$199$	18.2462	1.29344	0.646720	−	0.762728i	$-0.276140\pi$
$199$	18.2462	1.29344	0.646720	+	0.762728i	$0.276140\pi$
$200$	0	0
$201$	0	0
$202$	− 0.384472i	− 0.0270513i
$203$	− 5.68466i	− 0.398985i
$204$	0	0
$205$	0	0
$206$	−2.24621	−0.156501
$207$	0	0
$208$	− 21.3693i	− 1.48170i
$209$	2.87689	0.198999
$210$	0	0
$211$	−23.0540	−1.58710	−0.793551	−	0.608504i	$-0.791770\pi$
$211$	−23.0540	−1.58710	−0.793551	+	0.608504i	$0.791770\pi$
$212$	1.36932i	0.0940451i
$213$	0	0
$214$	17.7538	1.21362
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 27.6155i	− 1.87036i
$219$	0	0
$220$	0	0
$221$	20.8078	1.39968
$222$	0	0
$223$	− 6.56155i	− 0.439394i	−0.975568	−	0.219697i	$-0.929493\pi$
$223$	− 6.56155i	− 0.439394i	0.975568	−	0.219697i	$-0.0705069\pi$
$224$	2.43845	0.162926
$225$	0	0
$226$	−21.8617	−1.45422
$227$	23.6847i	1.57201i	0.618223	+	0.786003i	$0.287853\pi$
$227$	23.6847i	1.57201i	−0.618223	+	0.786003i	$0.712147\pi$
$228$	0	0
$229$	−19.1231	−1.26369	−0.631845	−	0.775095i	$-0.717702\pi$
$229$	−19.1231	−1.26369	−0.631845	+	0.775095i	$0.717702\pi$
$230$	0	0
$231$	0	0
$232$	− 13.8617i	− 0.910068i
$233$	3.12311i	0.204601i	0.994754	+	0.102301i	$0.0326204\pi$
$233$	3.12311i	0.204601i	−0.994754	+	0.102301i	$0.967380\pi$
$234$	0	0
$235$	0	0
$236$	1.75379	0.114162
$237$	0	0
$238$	7.12311i	0.461722i
$239$	−0.807764	−0.0522499	−0.0261250	−	0.999659i	$-0.508317\pi$
$239$	−0.807764	−0.0522499	−0.0261250	+	0.999659i	$0.508317\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	− 6.93087i	− 0.445533i
$243$	0	0
$244$	4.10795	0.262985
$245$	0	0
$246$	0	0
$247$	− 5.12311i	− 0.325975i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.1231	1.08080	0.540400	−	0.841408i	$-0.318273\pi$
$251$	17.1231	1.08080	0.540400	+	0.841408i	$0.318273\pi$
$252$	0	0
$253$	− 13.1231i	− 0.825043i
$254$	16.0000	1.00393
$255$	0	0
$256$	10.0540	0.628373
$257$	22.4924i	1.40304i	0.712650	+	0.701519i	$0.247495\pi$
$257$	22.4924i	1.40304i	−0.712650	+	0.701519i	$0.752505\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	14.2462i	0.880134i
$263$	21.1231i	1.30251i	0.758860	+	0.651253i	$0.225756\pi$
$263$	21.1231i	1.30251i	−0.758860	+	0.651253i	$0.774244\pi$
$264$	0	0
$265$	0	0
$266$	1.75379	0.107532
$267$	0	0
$268$	− 2.73863i	− 0.167289i
$269$	28.7386	1.75223	0.876113	−	0.482106i	$-0.160128\pi$
$269$	28.7386	1.75223	0.876113	+	0.482106i	$0.160128\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	21.3693i	1.29571i
$273$	0	0
$274$	13.8617	0.837418
$275$	0	0
$276$	0	0
$277$	− 16.2462i	− 0.976140i	−0.872804	−	0.488070i	$-0.837701\pi$
$277$	− 16.2462i	− 0.976140i	0.872804	−	0.488070i	$-0.162299\pi$
$278$	10.7386i	0.644060i
$279$	0	0
$280$	0	0
$281$	−16.5616	−0.987979	−0.493990	−	0.869468i	$-0.664462\pi$
$281$	−16.5616	−0.987979	−0.493990	+	0.869468i	$0.664462\pi$
$282$	0	0
$283$	− 23.6847i	− 1.40791i	−0.710246	−	0.703953i	$-0.751416\pi$
$283$	− 23.6847i	− 1.40791i	0.710246	−	0.703953i	$-0.248584\pi$
$284$	3.50758	0.208136
$285$	0	0
$286$	18.2462	1.07892
$287$	3.12311i	0.184351i
$288$	0	0
$289$	−3.80776	−0.223986
$290$	0	0
$291$	0	0
$292$	− 1.86174i	− 0.108950i
$293$	− 9.68466i	− 0.565784i	−0.959152	−	0.282892i	$-0.908706\pi$
$293$	− 9.68466i	− 0.565784i	0.959152	−	0.282892i	$-0.0912938\pi$
$294$	0	0
$295$	0	0
$296$	14.6307	0.850391
$297$	0	0
$298$	− 6.63068i	− 0.384105i
$299$	−23.3693	−1.35148
$300$	0	0
$301$	−9.12311	−0.525847
$302$	34.2462i	1.97065i
$303$	0	0
$304$	5.26137	0.301760
$305$	0	0
$306$	0	0
$307$	31.6847i	1.80834i	0.427174	+	0.904169i	$0.359509\pi$
$307$	31.6847i	1.80834i	−0.427174	+	0.904169i	$0.640491\pi$
$308$	1.12311i	0.0639949i
$309$	0	0
$310$	0	0
$311$	9.61553	0.545247	0.272623	−	0.962121i	$-0.412109\pi$
$311$	9.61553	0.545247	0.272623	+	0.962121i	$0.412109\pi$
$312$	0	0
$313$	31.3002i	1.76919i	0.466359	+	0.884596i	$0.345566\pi$
$313$	31.3002i	1.76919i	−0.466359	+	0.884596i	$0.654434\pi$
$314$	5.86174	0.330797
$315$	0	0
$316$	−2.87689	−0.161838
$317$	− 22.4924i	− 1.26330i	−0.775254	−	0.631650i	$-0.782378\pi$
$317$	− 22.4924i	− 1.26330i	0.775254	−	0.631650i	$-0.217622\pi$
$318$	0	0
$319$	14.5616	0.815290
$320$	0	0
$321$	0	0
$322$	− 8.00000i	− 0.445823i
$323$	5.12311i	0.285057i
$324$	0	0
$325$	0	0
$326$	−1.75379	−0.0971334
$327$	0	0
$328$	7.61553i	0.420497i
$329$	−3.68466	−0.203142
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	1.75379i	0.0962517i
$333$	0	0
$334$	−34.2462	−1.87387
$335$	0	0
$336$	0	0
$337$	34.4924i	1.87892i	0.342656	+	0.939461i	$0.388674\pi$
$337$	34.4924i	1.87892i	−0.342656	+	0.939461i	$0.611326\pi$
$338$	− 12.1922i	− 0.663170i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	−22.2462	−1.19944
$345$	0	0
$346$	−13.3693	−0.718739
$347$	− 1.12311i	− 0.0602915i	−0.999546	−	0.0301457i	$-0.990403\pi$
$347$	− 1.12311i	− 0.0602915i	0.999546	−	0.0301457i	$-0.00959714\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	0	0
$352$	6.24621i	0.332924i
$353$	14.8078i	0.788138i	0.919081	+	0.394069i	$0.128933\pi$
$353$	14.8078i	0.788138i	−0.919081	+	0.394069i	$0.871067\pi$
$354$	0	0
$355$	0	0
$356$	−3.12311	−0.165524
$357$	0	0
$358$	31.2311i	1.65061i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	36.8769i	1.93821i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	− 3.68466i	− 0.192338i	−0.995365	−	0.0961688i	$-0.969341\pi$
$367$	− 3.68466i	− 0.192338i	0.995365	−	0.0961688i	$-0.0306589\pi$
$368$	− 24.0000i	− 1.25109i
$369$	0	0
$370$	0	0
$371$	3.12311	0.162144
$372$	0	0
$373$	29.3693i	1.52069i	0.649522	+	0.760343i	$0.274969\pi$
$373$	29.3693i	1.52069i	−0.649522	+	0.760343i	$0.725031\pi$
$374$	−18.2462	−0.943489
$375$	0	0
$376$	−8.98485	−0.463358
$377$	− 25.9309i	− 1.33551i
$378$	0	0
$379$	−16.4924	−0.847159	−0.423579	−	0.905859i	$-0.639227\pi$
$379$	−16.4924	−0.847159	−0.423579	+	0.905859i	$0.639227\pi$
$380$	0	0
$381$	0	0
$382$	14.7386i	0.754094i
$383$	10.2462i	0.523557i	0.965128	+	0.261778i	$0.0843090\pi$
$383$	10.2462i	0.523557i	−0.965128	+	0.261778i	$0.915691\pi$
$384$	0	0
$385$	0	0
$386$	8.38447	0.426758
$387$	0	0
$388$	− 6.49242i	− 0.329603i
$389$	3.93087	0.199303	0.0996515	−	0.995022i	$-0.468227\pi$
$389$	3.93087	0.199303	0.0996515	+	0.995022i	$0.468227\pi$
$390$	0	0
$391$	23.3693	1.18184
$392$	− 2.43845i	− 0.123160i
$393$	0	0
$394$	11.1231	0.560374
$395$	0	0
$396$	0	0
$397$	− 23.4384i	− 1.17634i	−0.808737	−	0.588171i	$-0.799848\pi$
$397$	− 23.4384i	− 1.17634i	0.808737	−	0.588171i	$-0.200152\pi$
$398$	28.4924i	1.42820i
$399$	0	0
$400$	0	0
$401$	−27.4384	−1.37021	−0.685105	−	0.728444i	$-0.740244\pi$
$401$	−27.4384	−1.37021	−0.685105	+	0.728444i	$0.740244\pi$
$402$	0	0
$403$	0	0
$404$	0.107951	0.00537074
$405$	0	0
$406$	8.87689	0.440553
$407$	15.3693i	0.761829i
$408$	0	0
$409$	26.4924	1.30997	0.654983	−	0.755644i	$-0.272676\pi$
$409$	26.4924	1.30997	0.654983	+	0.755644i	$0.272676\pi$
$410$	0	0
$411$	0	0
$412$	− 0.630683i	− 0.0310715i
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	11.1231	0.545355
$417$	0	0
$418$	4.49242i	0.219732i
$419$	9.75379	0.476504	0.238252	−	0.971203i	$-0.423426\pi$
$419$	9.75379	0.476504	0.238252	+	0.971203i	$0.423426\pi$
$420$	0	0
$421$	9.68466	0.472001	0.236001	−	0.971753i	$-0.424163\pi$
$421$	9.68466	0.472001	0.236001	+	0.971753i	$0.424163\pi$
$422$	− 36.0000i	− 1.75245i
$423$	0	0
$424$	7.61553	0.369843
$425$	0	0
$426$	0	0
$427$	− 9.36932i	− 0.453413i
$428$	4.98485i	0.240952i
$429$	0	0
$430$	0	0
$431$	−0.807764	−0.0389086	−0.0194543	−	0.999811i	$-0.506193\pi$
$431$	−0.807764	−0.0389086	−0.0194543	+	0.999811i	$0.506193\pi$
$432$	0	0
$433$	− 8.24621i	− 0.396288i	−0.980173	−	0.198144i	$-0.936509\pi$
$433$	− 8.24621i	− 0.396288i	0.980173	−	0.198144i	$-0.0634913\pi$
$434$	0	0
$435$	0	0
$436$	7.75379	0.371339
$437$	− 5.75379i	− 0.275241i
$438$	0	0
$439$	15.3693	0.733537	0.366769	−	0.930312i	$-0.380464\pi$
$439$	15.3693	0.733537	0.366769	+	0.930312i	$0.380464\pi$
$440$	0	0
$441$	0	0
$442$	32.4924i	1.54551i
$443$	27.3693i	1.30036i	0.759782	+	0.650178i	$0.225306\pi$
$443$	27.3693i	1.30036i	−0.759782	+	0.650178i	$0.774694\pi$
$444$	0	0
$445$	0	0
$446$	10.2462	0.485172
$447$	0	0
$448$	− 5.56155i	− 0.262759i
$449$	18.8078	0.887593	0.443797	−	0.896128i	$-0.353631\pi$
$449$	18.8078	0.887593	0.443797	+	0.896128i	$0.353631\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	− 6.13826i	− 0.288719i
$453$	0	0
$454$	−36.9848	−1.73578
$455$	0	0
$456$	0	0
$457$	8.87689i	0.415244i	0.978209	+	0.207622i	$0.0665723\pi$
$457$	8.87689i	0.415244i	−0.978209	+	0.207622i	$0.933428\pi$
$458$	− 29.8617i	− 1.39535i
$459$	0	0
$460$	0	0
$461$	4.87689	0.227140	0.113570	−	0.993530i	$-0.463771\pi$
$461$	4.87689	0.227140	0.113570	+	0.993530i	$0.463771\pi$
$462$	0	0
$463$	− 20.4924i	− 0.952364i	−0.879347	−	0.476182i	$-0.842020\pi$
$463$	− 20.4924i	− 0.952364i	0.879347	−	0.476182i	$-0.157980\pi$
$464$	26.6307	1.23630
$465$	0	0
$466$	−4.87689	−0.225918
$467$	26.5616i	1.22912i	0.788869	+	0.614561i	$0.210667\pi$
$467$	26.5616i	1.22912i	−0.788869	+	0.614561i	$0.789333\pi$
$468$	0	0
$469$	−6.24621	−0.288423
$470$	0	0
$471$	0	0
$472$	− 9.75379i	− 0.448955i
$473$	− 23.3693i	− 1.07452i
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	− 1.26137i	− 0.0576935i
$479$	13.1231	0.599610	0.299805	−	0.954001i	$-0.403078\pi$
$479$	13.1231	0.599610	0.299805	+	0.954001i	$0.403078\pi$
$480$	0	0
$481$	27.3693	1.24793
$482$	19.1231i	0.871034i
$483$	0	0
$484$	1.94602	0.0884557
$485$	0	0
$486$	0	0
$487$	− 5.12311i	− 0.232150i	−0.993240	−	0.116075i	$-0.962969\pi$
$487$	− 5.12311i	− 0.232150i	0.993240	−	0.116075i	$-0.0370313\pi$
$488$	− 22.8466i	− 1.03422i
$489$	0	0
$490$	0	0
$491$	−4.17708	−0.188509	−0.0942545	−	0.995548i	$-0.530047\pi$
$491$	−4.17708	−0.188509	−0.0942545	+	0.995548i	$0.530047\pi$
$492$	0	0
$493$	25.9309i	1.16787i
$494$	8.00000	0.359937
$495$	0	0
$496$	0	0
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	4.17708	0.186992	0.0934959	−	0.995620i	$-0.470196\pi$
$499$	4.17708	0.186992	0.0934959	+	0.995620i	$0.470196\pi$
$500$	0	0
$501$	0	0
$502$	26.7386i	1.19340i
$503$	− 10.0691i	− 0.448960i	−0.974479	−	0.224480i	$-0.927932\pi$
$503$	− 10.0691i	− 0.448960i	0.974479	−	0.224480i	$-0.0720684\pi$
$504$	0	0
$505$	0	0
$506$	20.4924	0.910999
$507$	0	0
$508$	4.49242i	0.199319i
$509$	−28.2462	−1.25199	−0.625996	−	0.779827i	$-0.715307\pi$
$509$	−28.2462	−1.25199	−0.625996	+	0.779827i	$0.715307\pi$
$510$	0	0
$511$	−4.24621	−0.187841
$512$	− 11.4233i	− 0.504843i
$513$	0	0
$514$	−35.1231	−1.54921
$515$	0	0
$516$	0	0
$517$	− 9.43845i	− 0.415102i
$518$	9.36932i	0.411664i
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	7.50758i	0.328283i	0.986437	+	0.164142i	$0.0524854\pi$
$523$	7.50758i	0.328283i	−0.986437	+	0.164142i	$0.947515\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−32.9848	−1.43821
$527$	0	0
$528$	0	0
$529$	−3.24621	−0.141140
$530$	0	0
$531$	0	0
$532$	0.492423i	0.0213492i
$533$	14.2462i	0.617072i
$534$	0	0
$535$	0	0
$536$	−15.2311	−0.657881
$537$	0	0
$538$	44.8769i	1.93478i
$539$	2.56155	0.110334
$540$	0	0
$541$	−17.1922	−0.739152	−0.369576	−	0.929201i	$-0.620497\pi$
$541$	−17.1922	−0.739152	−0.369576	+	0.929201i	$0.620497\pi$
$542$	− 24.9848i	− 1.07319i
$543$	0	0
$544$	−11.1231	−0.476899
$545$	0	0
$546$	0	0
$547$	− 14.2462i	− 0.609124i	−0.952493	−	0.304562i	$-0.901490\pi$
$547$	− 14.2462i	− 0.609124i	0.952493	−	0.304562i	$-0.0985101\pi$
$548$	3.89205i	0.166260i
$549$	0	0
$550$	0	0
$551$	6.38447	0.271988
$552$	0	0
$553$	6.56155i	0.279026i
$554$	25.3693	1.07784
$555$	0	0
$556$	−3.01515	−0.127871
$557$	− 4.87689i	− 0.206641i	−0.994648	−	0.103320i	$-0.967053\pi$
$557$	− 4.87689i	− 0.206641i	0.994648	−	0.103320i	$-0.0329467\pi$
$558$	0	0
$559$	−41.6155	−1.76015
$560$	0	0
$561$	0	0
$562$	− 25.8617i	− 1.09091i
$563$	28.0000i	1.18006i	0.807382	+	0.590030i	$0.200884\pi$
$563$	28.0000i	1.18006i	−0.807382	+	0.590030i	$0.799116\pi$
$564$	0	0
$565$	0	0
$566$	36.9848	1.55459
$567$	0	0
$568$	− 19.5076i	− 0.818520i
$569$	34.9848	1.46664	0.733320	−	0.679883i	$-0.237969\pi$
$569$	34.9848	1.46664	0.733320	+	0.679883i	$0.237969\pi$
$570$	0	0
$571$	7.50758	0.314182	0.157091	−	0.987584i	$-0.449788\pi$
$571$	7.50758	0.314182	0.157091	+	0.987584i	$0.449788\pi$
$572$	5.12311i	0.214208i
$573$	0	0
$574$	−4.87689	−0.203558
$575$	0	0
$576$	0	0
$577$	− 13.0540i	− 0.543444i	−0.962376	−	0.271722i	$-0.912407\pi$
$577$	− 13.0540i	− 0.543444i	0.962376	−	0.271722i	$-0.0875931\pi$
$578$	− 5.94602i	− 0.247322i
$579$	0	0
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	8.00000i	0.331326i
$584$	−10.3542	−0.428458
$585$	0	0
$586$	15.1231	0.624730
$587$	− 9.75379i	− 0.402582i	−0.979531	−	0.201291i	$-0.935486\pi$
$587$	− 9.75379i	− 0.402582i	0.979531	−	0.201291i	$-0.0645137\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	28.1080i	1.15523i
$593$	23.4384i	0.962502i	0.876583	+	0.481251i	$0.159817\pi$
$593$	23.4384i	0.962502i	−0.876583	+	0.481251i	$0.840183\pi$
$594$	0	0
$595$	0	0
$596$	1.86174	0.0762598
$597$	0	0
$598$	− 36.4924i	− 1.49229i
$599$	8.80776	0.359875	0.179938	−	0.983678i	$-0.442410\pi$
$599$	8.80776	0.359875	0.179938	+	0.983678i	$0.442410\pi$
$600$	0	0
$601$	−26.4924	−1.08065	−0.540324	−	0.841457i	$-0.681698\pi$
$601$	−26.4924	−1.08065	−0.540324	+	0.841457i	$0.681698\pi$
$602$	− 14.2462i	− 0.580632i
$603$	0	0
$604$	−9.61553	−0.391250
$605$	0	0
$606$	0	0
$607$	4.94602i	0.200753i	0.994950	+	0.100376i	$0.0320047\pi$
$607$	4.94602i	0.200753i	−0.994950	+	0.100376i	$0.967995\pi$
$608$	2.73863i	0.111066i
$609$	0	0
$610$	0	0
$611$	−16.8078	−0.679969
$612$	0	0
$613$	− 8.73863i	− 0.352950i	−0.984305	−	0.176475i	$-0.943531\pi$
$613$	− 8.73863i	− 0.352950i	0.984305	−	0.176475i	$-0.0564695\pi$
$614$	−49.4773	−1.99674
$615$	0	0
$616$	6.24621	0.251667
$617$	15.7538i	0.634224i	0.948388	+	0.317112i	$0.102713\pi$
$617$	15.7538i	0.634224i	−0.948388	+	0.317112i	$0.897287\pi$
$618$	0	0
$619$	42.1080	1.69246	0.846231	−	0.532817i	$-0.178866\pi$
$619$	42.1080	1.69246	0.846231	+	0.532817i	$0.178866\pi$
$620$	0	0
$621$	0	0
$622$	15.0152i	0.602053i
$623$	7.12311i	0.285381i
$624$	0	0
$625$	0	0
$626$	−48.8769	−1.95351
$627$	0	0
$628$	1.64584i	0.0656761i
$629$	−27.3693	−1.09129
$630$	0	0
$631$	8.80776	0.350632	0.175316	−	0.984512i	$-0.443905\pi$
$631$	8.80776	0.350632	0.175316	+	0.984512i	$0.443905\pi$
$632$	16.0000i	0.636446i
$633$	0	0
$634$	35.1231	1.39492
$635$	0	0
$636$	0	0
$637$	− 4.56155i	− 0.180735i
$638$	22.7386i	0.900231i
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 2.56155i	− 0.101018i	−0.998724	−	0.0505089i	$-0.983916\pi$
$643$	− 2.56155i	− 0.101018i	0.998724	−	0.0505089i	$-0.0160843\pi$
$644$	2.24621	0.0885131
$645$	0	0
$646$	−8.00000	−0.314756
$647$	3.50758i	0.137897i	0.997620	+	0.0689486i	$0.0219644\pi$
$647$	3.50758i	0.137897i	−0.997620	+	0.0689486i	$0.978036\pi$
$648$	0	0
$649$	10.2462	0.402199
$650$	0	0
$651$	0	0
$652$	− 0.492423i	− 0.0192848i
$653$	− 49.2311i	− 1.92656i	−0.268499	−	0.963280i	$-0.586527\pi$
$653$	− 49.2311i	− 1.92656i	0.268499	−	0.963280i	$-0.413473\pi$
$654$	0	0
$655$	0	0
$656$	−14.6307	−0.571232
$657$	0	0
$658$	− 5.75379i	− 0.224306i
$659$	−36.1771	−1.40926	−0.704629	−	0.709575i	$-0.748887\pi$
$659$	−36.1771	−1.40926	−0.704629	+	0.709575i	$0.748887\pi$
$660$	0	0
$661$	3.12311	0.121475	0.0607374	−	0.998154i	$-0.480655\pi$
$661$	3.12311	0.121475	0.0607374	+	0.998154i	$0.480655\pi$
$662$	18.7386i	0.728298i
$663$	0	0
$664$	9.75379	0.378520
$665$	0	0
$666$	0	0
$667$	− 29.1231i	− 1.12765i
$668$	− 9.61553i	− 0.372036i
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	− 25.8617i	− 0.996897i	−0.866919	−	0.498448i	$-0.833903\pi$
$673$	− 25.8617i	− 0.996897i	0.866919	−	0.498448i	$-0.166097\pi$
$674$	−53.8617	−2.07468
$675$	0	0
$676$	3.42329	0.131665
$677$	− 23.9309i	− 0.919738i	−0.887987	−	0.459869i	$-0.847896\pi$
$677$	− 23.9309i	− 0.919738i	0.887987	−	0.459869i	$-0.152104\pi$
$678$	0	0
$679$	−14.8078	−0.568270
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 42.7386i	− 1.63535i	−0.575681	−	0.817674i	$-0.695263\pi$
$683$	− 42.7386i	− 1.63535i	0.575681	−	0.817674i	$-0.304737\pi$
$684$	0	0
$685$	0	0
$686$	1.56155	0.0596204
$687$	0	0
$688$	− 42.7386i	− 1.62940i
$689$	14.2462	0.542737
$690$	0	0
$691$	8.49242	0.323067	0.161533	−	0.986867i	$-0.448356\pi$
$691$	8.49242	0.323067	0.161533	+	0.986867i	$0.448356\pi$
$692$	− 3.75379i	− 0.142698i
$693$	0	0
$694$	1.75379	0.0665729
$695$	0	0
$696$	0	0
$697$	− 14.2462i	− 0.539614i
$698$	35.1231i	1.32943i
$699$	0	0
$700$	0	0
$701$	−0.0691303	−0.00261102	−0.00130551	−	0.999999i	$-0.500416\pi$
$701$	−0.0691303	−0.00261102	−0.00130551	+	0.999999i	$0.500416\pi$
$702$	0	0
$703$	6.73863i	0.254152i
$704$	14.2462	0.536924
$705$	0	0
$706$	−23.1231	−0.870250
$707$	− 0.246211i	− 0.00925973i
$708$	0	0
$709$	18.1771	0.682655	0.341327	−	0.939945i	$-0.389124\pi$
$709$	18.1771	0.682655	0.341327	+	0.939945i	$0.389124\pi$
$710$	0	0
$711$	0	0
$712$	17.3693i	0.650943i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−8.76894	−0.327711
$717$	0	0
$718$	12.4924i	0.466213i
$719$	49.6155	1.85035	0.925173	−	0.379544i	$-0.123919\pi$
$719$	49.6155	1.85035	0.925173	+	0.379544i	$0.123919\pi$
$720$	0	0
$721$	−1.43845	−0.0535706
$722$	− 27.6998i	− 1.03088i
$723$	0	0
$724$	−10.3542	−0.384809
$725$	0	0
$726$	0	0
$727$	− 19.5076i	− 0.723496i	−0.932276	−	0.361748i	$-0.882180\pi$
$727$	− 19.5076i	− 0.723496i	0.932276	−	0.361748i	$-0.117820\pi$
$728$	− 11.1231i	− 0.412250i
$729$	0	0
$730$	0	0
$731$	41.6155	1.53921
$732$	0	0
$733$	− 5.68466i	− 0.209968i	−0.994474	−	0.104984i	$-0.966521\pi$
$733$	− 5.68466i	− 0.209968i	0.994474	−	0.104984i	$-0.0334791\pi$
$734$	5.75379	0.212376
$735$	0	0
$736$	12.4924	0.460477
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	−6.06913	−0.223257	−0.111628	−	0.993750i	$-0.535607\pi$
$739$	−6.06913	−0.223257	−0.111628	+	0.993750i	$0.535607\pi$
$740$	0	0
$741$	0	0
$742$	4.87689i	0.179036i
$743$	− 32.9848i	− 1.21010i	−0.796189	−	0.605048i	$-0.793154\pi$
$743$	− 32.9848i	− 1.21010i	0.796189	−	0.605048i	$-0.206846\pi$
$744$	0	0
$745$	0	0
$746$	−45.8617	−1.67912
$747$	0	0
$748$	− 5.12311i	− 0.187319i
$749$	11.3693	0.415426
$750$	0	0
$751$	45.9309	1.67604	0.838021	−	0.545639i	$-0.183713\pi$
$751$	45.9309	1.67604	0.838021	+	0.545639i	$0.183713\pi$
$752$	− 17.2614i	− 0.629457i
$753$	0	0
$754$	40.4924	1.47465
$755$	0	0
$756$	0	0
$757$	− 14.6307i	− 0.531761i	−0.964006	−	0.265881i	$-0.914337\pi$
$757$	− 14.6307i	− 0.531761i	0.964006	−	0.265881i	$-0.0856627\pi$
$758$	− 25.7538i	− 0.935420i
$759$	0	0
$760$	0	0
$761$	−31.7538	−1.15107	−0.575537	−	0.817776i	$-0.695207\pi$
$761$	−31.7538	−1.15107	−0.575537	+	0.817776i	$0.695207\pi$
$762$	0	0
$763$	− 17.6847i	− 0.640228i
$764$	−4.13826	−0.149717
$765$	0	0
$766$	−16.0000	−0.578103
$767$	− 18.2462i	− 0.658833i
$768$	0	0
$769$	9.50758	0.342852	0.171426	−	0.985197i	$-0.445163\pi$
$769$	9.50758	0.342852	0.171426	+	0.985197i	$0.445163\pi$
$770$	0	0
$771$	0	0
$772$	2.35416i	0.0847281i
$773$	− 8.06913i	− 0.290226i	−0.989415	−	0.145113i	$-0.953645\pi$
$773$	− 8.06913i	− 0.290226i	0.989415	−	0.145113i	$-0.0463546\pi$
$774$	0	0
$775$	0	0
$776$	−36.1080	−1.29620
$777$	0	0
$778$	6.13826i	0.220067i
$779$	−3.50758	−0.125672
$780$	0	0
$781$	20.4924	0.733277
$782$	36.4924i	1.30497i
$783$	0	0
$784$	4.68466	0.167309
$785$	0	0
$786$	0	0
$787$	3.82292i	0.136272i	0.997676	+	0.0681362i	$0.0217052\pi$
$787$	3.82292i	0.136272i	−0.997676	+	0.0681362i	$0.978295\pi$
$788$	3.12311i	0.111256i
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	− 42.7386i	− 1.51769i
$794$	36.6004	1.29890
$795$	0	0
$796$	−8.00000	−0.283552
$797$	− 13.0540i	− 0.462396i	−0.972907	−	0.231198i	$-0.925736\pi$
$797$	− 13.0540i	− 0.462396i	0.972907	−	0.231198i	$-0.0742644\pi$
$798$	0	0
$799$	16.8078	0.594616
$800$	0	0
$801$	0	0
$802$	− 42.8466i	− 1.51297i
$803$	− 10.8769i	− 0.383837i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 0.600373i	− 0.0211211i
$809$	−53.5464	−1.88259	−0.941296	−	0.337584i	$-0.890390\pi$
$809$	−53.5464	−1.88259	−0.941296	+	0.337584i	$0.890390\pi$
$810$	0	0
$811$	−21.6155	−0.759024	−0.379512	−	0.925187i	$-0.623908\pi$
$811$	−21.6155	−0.759024	−0.379512	+	0.925187i	$0.623908\pi$
$812$	2.49242i	0.0874669i
$813$	0	0
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	− 10.2462i	− 0.358470i
$818$	41.3693i	1.44644i
$819$	0	0
$820$	0	0
$821$	−40.4233	−1.41078	−0.705391	−	0.708818i	$-0.749229\pi$
$821$	−40.4233	−1.41078	−0.705391	+	0.708818i	$0.749229\pi$
$822$	0	0
$823$	− 3.50758i	− 0.122266i	−0.998130	−	0.0611332i	$-0.980529\pi$
$823$	− 3.50758i	− 0.122266i	0.998130	−	0.0611332i	$-0.0194715\pi$
$824$	−3.50758	−0.122192
$825$	0	0
$826$	6.24621	0.217333
$827$	19.3693i	0.673537i	0.941587	+	0.336769i	$0.109334\pi$
$827$	19.3693i	0.673537i	−0.941587	+	0.336769i	$0.890666\pi$
$828$	0	0
$829$	−43.1231	−1.49773	−0.748864	−	0.662724i	$-0.769400\pi$
$829$	−43.1231	−1.49773	−0.748864	+	0.662724i	$0.769400\pi$
$830$	0	0
$831$	0	0
$832$	− 25.3693i	− 0.879523i
$833$	4.56155i	0.158048i
$834$	0	0
$835$	0	0
$836$	−1.26137	−0.0436253
$837$	0	0
$838$	15.2311i	0.526148i
$839$	−37.1231	−1.28163	−0.640816	−	0.767695i	$-0.721404\pi$
$839$	−37.1231	−1.28163	−0.640816	+	0.767695i	$0.721404\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	15.1231i	0.521177i
$843$	0	0
$844$	10.1080	0.347930
$845$	0	0
$846$	0	0
$847$	− 4.43845i	− 0.152507i
$848$	14.6307i	0.502420i
$849$	0	0
$850$	0	0
$851$	30.7386	1.05371
$852$	0	0
$853$	− 56.7386i	− 1.94269i	−0.237666	−	0.971347i	$-0.576382\pi$
$853$	− 56.7386i	− 1.94269i	0.237666	−	0.971347i	$-0.423618\pi$
$854$	14.6307	0.500652
$855$	0	0
$856$	27.7235	0.947569
$857$	− 32.2462i	− 1.10151i	−0.834667	−	0.550755i	$-0.814340\pi$
$857$	− 32.2462i	− 1.10151i	0.834667	−	0.550755i	$-0.185660\pi$
$858$	0	0
$859$	−16.4924	−0.562714	−0.281357	−	0.959603i	$-0.590785\pi$
$859$	−16.4924	−0.562714	−0.281357	+	0.959603i	$0.590785\pi$
$860$	0	0
$861$	0	0
$862$	− 1.26137i	− 0.0429623i
$863$	42.2462i	1.43808i	0.694970	+	0.719039i	$0.255418\pi$
$863$	42.2462i	1.43808i	−0.694970	+	0.719039i	$0.744582\pi$
$864$	0	0
$865$	0	0
$866$	12.8769	0.437575
$867$	0	0
$868$	0	0
$869$	−16.8078	−0.570164
$870$	0	0
$871$	−28.4924	−0.965429
$872$	− 43.1231i	− 1.46033i
$873$	0	0
$874$	8.98485	0.303917
$875$	0	0
$876$	0	0
$877$	23.7538i	0.802108i	0.916054	+	0.401054i	$0.131356\pi$
$877$	23.7538i	0.802108i	−0.916054	+	0.401054i	$0.868644\pi$
$878$	24.0000i	0.809961i
$879$	0	0
$880$	0	0
$881$	−45.8617	−1.54512	−0.772561	−	0.634941i	$-0.781024\pi$
$881$	−45.8617	−1.54512	−0.772561	+	0.634941i	$0.781024\pi$
$882$	0	0
$883$	24.4924i	0.824236i	0.911131	+	0.412118i	$0.135211\pi$
$883$	24.4924i	0.824236i	−0.911131	+	0.412118i	$0.864789\pi$
$884$	−9.12311	−0.306843
$885$	0	0
$886$	−42.7386	−1.43583
$887$	− 12.4924i	− 0.419454i	−0.977760	−	0.209727i	$-0.932742\pi$
$887$	− 12.4924i	− 0.419454i	0.977760	−	0.209727i	$-0.0672576\pi$
$888$	0	0
$889$	10.2462	0.343647
$890$	0	0
$891$	0	0
$892$	2.87689i	0.0963255i
$893$	− 4.13826i	− 0.138482i
$894$	0	0
$895$	0	0
$896$	13.5616	0.453060
$897$	0	0
$898$	29.3693i	0.980067i
$899$	0	0
$900$	0	0
$901$	−14.2462	−0.474610
$902$	− 12.4924i	− 0.415952i
$903$	0	0
$904$	−34.1383	−1.13542
$905$	0	0
$906$	0	0
$907$	− 50.1080i	− 1.66381i	−0.554920	−	0.831904i	$-0.687251\pi$
$907$	− 50.1080i	− 1.66381i	0.554920	−	0.831904i	$-0.312749\pi$
$908$	− 10.3845i	− 0.344621i
$909$	0	0
$910$	0	0
$911$	−4.49242	−0.148841	−0.0744203	−	0.997227i	$-0.523711\pi$
$911$	−4.49242	−0.148841	−0.0744203	+	0.997227i	$0.523711\pi$
$912$	0	0
$913$	10.2462i	0.339100i
$914$	−13.8617	−0.458506
$915$	0	0
$916$	8.38447	0.277031
$917$	9.12311i	0.301271i
$918$	0	0
$919$	13.3002	0.438733	0.219366	−	0.975643i	$-0.429601\pi$
$919$	13.3002	0.438733	0.219366	+	0.975643i	$0.429601\pi$
$920$	0	0
$921$	0	0
$922$	7.61553i	0.250804i
$923$	− 36.4924i	− 1.20116i
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	0	0
$928$	13.8617i	0.455034i
$929$	−52.1080	−1.70961	−0.854803	−	0.518952i	$-0.826322\pi$
$929$	−52.1080	−1.70961	−0.854803	+	0.518952i	$0.826322\pi$
$930$	0	0
$931$	1.12311	0.0368083
$932$	− 1.36932i	− 0.0448535i
$933$	0	0
$934$	−41.4773	−1.35718
$935$	0	0
$936$	0	0
$937$	− 22.6695i	− 0.740580i	−0.928916	−	0.370290i	$-0.879258\pi$
$937$	− 22.6695i	− 0.740580i	0.928916	−	0.370290i	$-0.120742\pi$
$938$	− 9.75379i	− 0.318472i
$939$	0	0
$940$	0	0
$941$	13.8617	0.451880	0.225940	−	0.974141i	$-0.427455\pi$
$941$	13.8617	0.451880	0.225940	+	0.974141i	$0.427455\pi$
$942$	0	0
$943$	16.0000i	0.521032i
$944$	18.7386	0.609891
$945$	0	0
$946$	36.4924	1.18647
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	−19.3693	−0.628755
$950$	0	0
$951$	0	0
$952$	11.1231i	0.360502i
$953$	24.8769i	0.805842i	0.915235	+	0.402921i	$0.132005\pi$
$953$	24.8769i	0.805842i	−0.915235	+	0.402921i	$0.867995\pi$
$954$	0	0
$955$	0	0
$956$	0.354162	0.0114544
$957$	0	0
$958$	20.4924i	0.662080i
$959$	8.87689	0.286650
$960$	0	0
$961$	−31.0000	−1.00000
$962$	42.7386i	1.37795i
$963$	0	0
$964$	−5.36932	−0.172934
$965$	0	0
$966$	0	0
$967$	26.8769i	0.864303i	0.901801	+	0.432151i	$0.142245\pi$
$967$	26.8769i	0.864303i	−0.901801	+	0.432151i	$0.857755\pi$
$968$	− 10.8229i	− 0.347862i
$969$	0	0
$970$	0	0
$971$	49.4773	1.58780	0.793901	−	0.608048i	$-0.208047\pi$
$971$	49.4773	1.58780	0.793901	+	0.608048i	$0.208047\pi$
$972$	0	0
$973$	6.87689i	0.220463i
$974$	8.00000	0.256337
$975$	0	0
$976$	43.8920	1.40495
$977$	− 49.2311i	− 1.57504i	−0.616288	−	0.787521i	$-0.711364\pi$
$977$	− 49.2311i	− 1.57504i	0.616288	−	0.787521i	$-0.288636\pi$
$978$	0	0
$979$	−18.2462	−0.583151
$980$	0	0
$981$	0	0
$982$	− 6.52273i	− 0.208149i
$983$	10.4233i	0.332451i	0.986088	+	0.166226i	$0.0531580\pi$
$983$	10.4233i	0.332451i	−0.986088	+	0.166226i	$0.946842\pi$
$984$	0	0
$985$	0	0
$986$	−40.4924	−1.28954
$987$	0	0
$988$	2.24621i	0.0714615i
$989$	−46.7386	−1.48620
$990$	0	0
$991$	−20.4924	−0.650963	−0.325482	−	0.945548i	$-0.605526\pi$
$991$	−20.4924	−0.650963	−0.325482	+	0.945548i	$0.605526\pi$
$992$	0	0
$993$	0	0
$994$	12.4924	0.396236
$995$	0	0
$996$	0	0
$997$	− 9.68466i	− 0.306716i	−0.988171	−	0.153358i	$-0.950991\pi$
$997$	− 9.68466i	− 0.306716i	0.988171	−	0.153358i	$-0.0490088\pi$
$998$	6.52273i	0.206473i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.e.1324.3		4
3.2	odd	2		175.2.b.b.99.2		4
5.2	odd	4		315.2.a.e.1.1		2
5.3	odd	4		1575.2.a.p.1.2		2
5.4	even	2	inner	1575.2.d.e.1324.2		4
12.11	even	2		2800.2.g.t.449.4		4
15.2	even	4		35.2.a.b.1.2	✓	2
15.8	even	4		175.2.a.f.1.1		2
15.14	odd	2		175.2.b.b.99.3		4
20.7	even	4		5040.2.a.bt.1.2		2
21.20	even	2		1225.2.b.f.99.2		4
35.27	even	4		2205.2.a.x.1.1		2
60.23	odd	4		2800.2.a.bi.1.1		2
60.47	odd	4		560.2.a.i.1.2		2
60.59	even	2		2800.2.g.t.449.1		4
105.2	even	12		245.2.e.i.116.1		4
105.17	odd	12		245.2.e.h.226.1		4
105.32	even	12		245.2.e.i.226.1		4
105.47	odd	12		245.2.e.h.116.1		4
105.62	odd	4		245.2.a.d.1.2		2
105.83	odd	4		1225.2.a.s.1.1		2
105.104	even	2		1225.2.b.f.99.3		4
120.77	even	4		2240.2.a.bh.1.2		2
120.107	odd	4		2240.2.a.bd.1.1		2
165.32	odd	4		4235.2.a.m.1.1		2
195.77	even	4		5915.2.a.l.1.1		2
420.167	even	4		3920.2.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.2	✓	2	15.2	even	4
175.2.a.f.1.1		2	15.8	even	4
175.2.b.b.99.2		4	3.2	odd	2
175.2.b.b.99.3		4	15.14	odd	2
245.2.a.d.1.2		2	105.62	odd	4
245.2.e.h.116.1		4	105.47	odd	12
245.2.e.h.226.1		4	105.17	odd	12
245.2.e.i.116.1		4	105.2	even	12
245.2.e.i.226.1		4	105.32	even	12
315.2.a.e.1.1		2	5.2	odd	4
560.2.a.i.1.2		2	60.47	odd	4
1225.2.a.s.1.1		2	105.83	odd	4
1225.2.b.f.99.2		4	21.20	even	2
1225.2.b.f.99.3		4	105.104	even	2
1575.2.a.p.1.2		2	5.3	odd	4
1575.2.d.e.1324.2		4	5.4	even	2	inner
1575.2.d.e.1324.3		4	1.1	even	1	trivial
2205.2.a.x.1.1		2	35.27	even	4
2240.2.a.bd.1.1		2	120.107	odd	4
2240.2.a.bh.1.2		2	120.77	even	4
2800.2.a.bi.1.1		2	60.23	odd	4
2800.2.g.t.449.1		4	60.59	even	2
2800.2.g.t.449.4		4	12.11	even	2
3920.2.a.bs.1.1		2	420.167	even	4
4235.2.a.m.1.1		2	165.32	odd	4
5040.2.a.bt.1.2		2	20.7	even	4
5915.2.a.l.1.1		2	195.77	even	4

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{2} -0.438447 q^{4} +1.00000i q^{7} +2.43845i q^{8} -2.56155 q^{11} +4.56155i q^{13} -1.56155 q^{14} -4.68466 q^{16} -4.56155i q^{17} -1.12311 q^{19} -4.00000i q^{22} +5.12311i q^{23} -7.12311 q^{26} -0.438447i q^{28} -5.68466 q^{29} -2.43845i q^{32} +7.12311 q^{34} -6.00000i q^{37} -1.75379i q^{38} +3.12311 q^{41} +9.12311i q^{43} +1.12311 q^{44} -8.00000 q^{46} +3.68466i q^{47} -1.00000 q^{49} -2.00000i q^{52} -3.12311i q^{53} -2.43845 q^{56} -8.87689i q^{58} -4.00000 q^{59} -9.36932 q^{61} -5.56155 q^{64} +6.24621i q^{67} +2.00000i q^{68} -8.00000 q^{71} +4.24621i q^{73} +9.36932 q^{74} +0.492423 q^{76} -2.56155i q^{77} +6.56155 q^{79} +4.87689i q^{82} -4.00000i q^{83} -14.2462 q^{86} -6.24621i q^{88} +7.12311 q^{89} -4.56155 q^{91} -2.24621i q^{92} -5.75379 q^{94} +14.8078i q^{97} -1.56155i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4} - 2 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{19} - 12 q^{26} + 2 q^{29} + 12 q^{34} - 4 q^{41} - 12 q^{44} - 32 q^{46} - 4 q^{49} - 18 q^{56} - 16 q^{59} + 12 q^{61} - 14 q^{64} - 32 q^{71} - 12 q^{74}+ \cdots - 56 q^{94}+O(q^{100}) \) 4 * q - 10 * q^4 - 2 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^19 - 12 * q^26 + 2 * q^29 + 12 * q^34 - 4 * q^41 - 12 * q^44 - 32 * q^46 - 4 * q^49 - 18 * q^56 - 16 * q^59 + 12 * q^61 - 14 * q^64 - 32 * q^71 - 12 * q^74 - 64 * q^76 + 18 * q^79 - 24 * q^86 + 12 * q^89 - 10 * q^91 - 56 * q^94

Introduction

L-functions

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