LMFDB - Embedded newform 5700.2.f.p.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5700,2,Mod(3649,5700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5700.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5700.f (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:	$45.5147291521$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.37161216.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 15x^{4} + 51x^{2} + 1 $ x^6 + 15x^4 + 51x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$2.27307i$ of defining polynomial
Character	$\chi$	$=$	5700.3649
Dual form			5700.2.f.p.3649.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +0.166860i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +0.166860i q^{7} -1.00000 q^{9} +4.71301 q^{11} +1.83314i q^{13} +4.54615i q^{17} -1.00000 q^{19} +0.166860 q^{21} -4.54615i q^{23} +1.00000i q^{27} -6.71301 q^{29} -2.54615 q^{31} -4.71301i q^{33} -2.16686i q^{37} +1.83314 q^{39} +10.7130 q^{41} +9.25915i q^{43} -0.546146i q^{47} +6.97216 q^{49} +4.54615 q^{51} +10.5461i q^{53} +1.00000i q^{57} +9.42601 q^{59} +12.8799 q^{61} -0.166860i q^{63} -4.54615 q^{69} -5.42601 q^{71} +15.0923i q^{73} +0.786413i q^{77} -13.7597 q^{79} +1.00000 q^{81} -13.9722i q^{83} +6.71301i q^{87} -6.04557 q^{89} -0.305878 q^{91} +2.54615i q^{93} +2.16686i q^{97} -4.71301 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} - 4 q^{11} - 6 q^{19} - 8 q^{29} + 16 q^{31} + 12 q^{39} + 32 q^{41} - 54 q^{49} - 4 q^{51} - 8 q^{59} + 44 q^{61} + 4 q^{69} + 32 q^{71} - 16 q^{79} + 6 q^{81} - 8 q^{89} + 96 q^{91} + 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 - 4 * q^11 - 6 * q^19 - 8 * q^29 + 16 * q^31 + 12 * q^39 + 32 * q^41 - 54 * q^49 - 4 * q^51 - 8 * q^59 + 44 * q^61 + 4 * q^69 + 32 * q^71 - 16 * q^79 + 6 * q^81 - 8 * q^89 + 96 * q^91 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times$.

$n$	$1901$	$2851$	$3877$	$4201$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.166860i	0.0630672i	0.999503	+	0.0315336i	$0.0100391\pi$
$7$	0.166860i	0.0630672i	−0.999503	+	0.0315336i	$0.989961\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.71301	1.42102	0.710512	−	0.703685i	$-0.248463\pi$
$11$	4.71301	1.42102	0.710512	+	0.703685i	$0.248463\pi$
$12$	0	0
$13$	1.83314i	0.508421i	0.967149	+	0.254211i	$0.0818157\pi$
$13$	1.83314i	0.508421i	−0.967149	+	0.254211i	$0.918184\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.54615i	1.10260i	0.834306	+	0.551301i	$0.185868\pi$
$17$	4.54615i	1.10260i	−0.834306	+	0.551301i	$0.814132\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.166860	0.0364119
$22$	0	0
$23$	− 4.54615i	− 0.947937i	−0.880542	−	0.473968i	$-0.842821\pi$
$23$	− 4.54615i	− 0.947937i	0.880542	−	0.473968i	$-0.157179\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−6.71301	−1.24657	−0.623287	−	0.781993i	$-0.714203\pi$
$29$	−6.71301	−1.24657	−0.623287	+	0.781993i	$0.714203\pi$
$30$	0	0
$31$	−2.54615	−0.457301	−0.228651	−	0.973509i	$-0.573431\pi$
$31$	−2.54615	−0.457301	−0.228651	+	0.973509i	$0.573431\pi$
$32$	0	0
$33$	− 4.71301i	− 0.820429i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.16686i	− 0.356230i	−0.984010	−	0.178115i	$-0.943000\pi$
$37$	− 2.16686i	− 0.356230i	0.984010	−	0.178115i	$-0.0569998\pi$
$38$	0	0
$39$	1.83314	0.293537
$40$	0	0
$41$	10.7130	1.67309	0.836545	−	0.547898i	$-0.184572\pi$
$41$	10.7130	1.67309	0.836545	+	0.547898i	$0.184572\pi$
$42$	0	0
$43$	9.25915i	1.41201i	0.708208	+	0.706004i	$0.249504\pi$
$43$	9.25915i	1.41201i	−0.708208	+	0.706004i	$0.750496\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.546146i	− 0.0796635i	−0.999206	−	0.0398318i	$-0.987318\pi$
$47$	− 0.546146i	− 0.0796635i	0.999206	−	0.0398318i	$-0.0126822\pi$
$48$	0	0
$49$	6.97216	0.996023
$50$	0	0
$51$	4.54615	0.636588
$52$	0	0
$53$	10.5461i	1.44862i	0.689472	+	0.724312i	$0.257843\pi$
$53$	10.5461i	1.44862i	−0.689472	+	0.724312i	$0.742157\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	9.42601	1.22716	0.613581	−	0.789632i	$-0.289728\pi$
$59$	9.42601	1.22716	0.613581	+	0.789632i	$0.289728\pi$
$60$	0	0
$61$	12.8799	1.64910	0.824549	−	0.565791i	$-0.191429\pi$
$61$	12.8799	1.64910	0.824549	+	0.565791i	$0.191429\pi$
$62$	0	0
$63$	− 0.166860i	− 0.0210224i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−4.54615	−0.547292
$70$	0	0
$71$	−5.42601	−0.643949	−0.321975	−	0.946748i	$-0.604347\pi$
$71$	−5.42601	−0.643949	−0.321975	+	0.946748i	$0.604347\pi$
$72$	0	0
$73$	15.0923i	1.76642i	0.468979	+	0.883210i	$0.344622\pi$
$73$	15.0923i	1.76642i	−0.468979	+	0.883210i	$0.655378\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.786413i	0.0896201i
$78$	0	0
$79$	−13.7597	−1.54809	−0.774045	−	0.633130i	$-0.781770\pi$
$79$	−13.7597	−1.54809	−0.774045	+	0.633130i	$0.781770\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 13.9722i	− 1.53364i	−0.641860	−	0.766822i	$-0.721837\pi$
$83$	− 13.9722i	− 1.53364i	0.641860	−	0.766822i	$-0.278163\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.71301i	0.719710i
$88$	0	0
$89$	−6.04557	−0.640829	−0.320414	−	0.947278i	$-0.603822\pi$
$89$	−6.04557	−0.640829	−0.320414	+	0.947278i	$0.603822\pi$
$90$	0	0
$91$	−0.305878	−0.0320647
$92$	0	0
$93$	2.54615i	0.264023i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.16686i	0.220011i	0.993931	+	0.110006i	$0.0350869\pi$
$97$	2.16686i	0.220011i	−0.993931	+	0.110006i	$0.964913\pi$
$98$	0	0
$99$	−4.71301	−0.473675
$100$	0	0
$101$	−16.5183	−1.64363	−0.821816	−	0.569753i	$-0.807039\pi$
$101$	−16.5183	−1.64363	−0.821816	+	0.569753i	$0.807039\pi$
$102$	0	0
$103$	− 5.09229i	− 0.501758i	−0.968018	−	0.250879i	$-0.919280\pi$
$103$	− 5.09229i	− 0.501758i	0.968018	−	0.250879i	$-0.0807197\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	0	0
$109$	−15.4260	−1.47754	−0.738772	−	0.673955i	$-0.764594\pi$
$109$	−15.4260	−1.47754	−0.738772	+	0.673955i	$0.764594\pi$
$110$	0	0
$111$	−2.16686	−0.205669
$112$	0	0
$113$	14.8799i	1.39978i	0.714251	+	0.699890i	$0.246768\pi$
$113$	14.8799i	1.39978i	−0.714251	+	0.699890i	$0.753232\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.83314i	− 0.169474i
$118$	0	0
$119$	−0.758571	−0.0695381
$120$	0	0
$121$	11.2124	1.01931
$122$	0	0
$123$	− 10.7130i	− 0.965959i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	0	0
$129$	9.25915	0.815223
$130$	0	0
$131$	14.1390	1.23533	0.617666	−	0.786441i	$-0.288078\pi$
$131$	14.1390	1.23533	0.617666	+	0.786441i	$0.288078\pi$
$132$	0	0
$133$	− 0.166860i	− 0.0144686i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−9.42601	−0.799504	−0.399752	−	0.916623i	$-0.630904\pi$
$139$	−9.42601	−0.799504	−0.399752	+	0.916623i	$0.630904\pi$
$140$	0	0
$141$	−0.546146	−0.0459938
$142$	0	0
$143$	8.63960i	0.722480i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.97216i	− 0.575054i
$148$	0	0
$149$	15.0923	1.23641	0.618204	−	0.786017i	$-0.287860\pi$
$149$	15.0923	1.23641	0.618204	+	0.786017i	$0.287860\pi$
$150$	0	0
$151$	11.2136	0.912549	0.456274	−	0.889839i	$-0.349184\pi$
$151$	11.2136	0.912549	0.456274	+	0.889839i	$0.349184\pi$
$152$	0	0
$153$	− 4.54615i	− 0.367534i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.7586i	1.17786i	0.808183	+	0.588931i	$0.200451\pi$
$157$	14.7586i	1.17786i	−0.808183	+	0.588931i	$0.799549\pi$
$158$	0	0
$159$	10.5461	0.836364
$160$	0	0
$161$	0.758571	0.0597838
$162$	0	0
$163$	− 3.49942i	− 0.274096i	−0.990564	−	0.137048i	$-0.956239\pi$
$163$	− 3.49942i	− 0.274096i	0.990564	−	0.137048i	$-0.0437614\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.21243i	− 0.480732i	−0.970682	−	0.240366i	$-0.922733\pi$
$167$	− 6.21243i	− 0.480732i	0.970682	−	0.240366i	$-0.0772674\pi$
$168$	0	0
$169$	9.63960	0.741508
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	− 1.12013i	− 0.0851622i	−0.999093	−	0.0425811i	$-0.986442\pi$
$173$	− 1.12013i	− 0.0851622i	0.999093	−	0.0425811i	$-0.0135581\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 9.42601i	− 0.708502i
$178$	0	0
$179$	11.6663	0.871979	0.435989	−	0.899952i	$-0.356399\pi$
$179$	11.6663	0.871979	0.435989	+	0.899952i	$0.356399\pi$
$180$	0	0
$181$	9.66628	0.718489	0.359244	−	0.933243i	$-0.383034\pi$
$181$	9.66628	0.718489	0.359244	+	0.933243i	$0.383034\pi$
$182$	0	0
$183$	− 12.8799i	− 0.952107i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	21.4260i	1.56683i
$188$	0	0
$189$	−0.166860	−0.0121373
$190$	0	0
$191$	1.04673	0.0757385	0.0378692	−	0.999283i	$-0.487943\pi$
$191$	1.04673	0.0757385	0.0378692	+	0.999283i	$0.487943\pi$
$192$	0	0
$193$	− 10.9254i	− 0.786430i	−0.919447	−	0.393215i	$-0.871363\pi$
$193$	− 10.9254i	− 0.786430i	0.919447	−	0.393215i	$-0.128637\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.4539i	0.816053i	0.912970	+	0.408027i	$0.133783\pi$
$197$	11.4539i	0.816053i	−0.912970	+	0.408027i	$0.866217\pi$
$198$	0	0
$199$	24.7586	1.75509	0.877544	−	0.479496i	$-0.159180\pi$
$199$	24.7586	1.75509	0.877544	+	0.479496i	$0.159180\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.12013i	− 0.0786180i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.54615i	0.315979i
$208$	0	0
$209$	−4.71301	−0.326005
$210$	0	0
$211$	16.4249	1.13073	0.565367	−	0.824840i	$-0.308735\pi$
$211$	16.4249	1.13073	0.565367	+	0.824840i	$0.308735\pi$
$212$	0	0
$213$	5.42601i	0.371784i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.424850i	− 0.0288407i
$218$	0	0
$219$	15.0923	1.01984
$220$	0	0
$221$	−8.33372	−0.560587
$222$	0	0
$223$	− 5.09229i	− 0.341005i	−0.985357	−	0.170503i	$-0.945461\pi$
$223$	− 5.09229i	− 0.341005i	0.985357	−	0.170503i	$-0.0545391\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.6396i	1.63539i	0.575653	+	0.817694i	$0.304748\pi$
$227$	24.6396i	1.63539i	−0.575653	+	0.817694i	$0.695252\pi$
$228$	0	0
$229$	7.78757	0.514617	0.257309	−	0.966329i	$-0.417164\pi$
$229$	7.78757	0.514617	0.257309	+	0.966329i	$0.417164\pi$
$230$	0	0
$231$	0.786413	0.0517422
$232$	0	0
$233$	− 25.9443i	− 1.69967i	−0.527050	−	0.849834i	$-0.676702\pi$
$233$	− 25.9443i	− 1.69967i	0.527050	−	0.849834i	$-0.323298\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.7597i	0.893791i
$238$	0	0
$239$	16.3793	1.05949	0.529744	−	0.848158i	$-0.322288\pi$
$239$	16.3793	1.05949	0.529744	+	0.848158i	$0.322288\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.83314i	− 0.116640i
$248$	0	0
$249$	−13.9722	−0.885450
$250$	0	0
$251$	−21.4716	−1.35527	−0.677637	−	0.735397i	$-0.736996\pi$
$251$	−21.4716	−1.35527	−0.677637	+	0.735397i	$0.736996\pi$
$252$	0	0
$253$	− 21.4260i	− 1.34704i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.9722i	0.996316i	0.867086	+	0.498158i	$0.165990\pi$
$257$	15.9722i	0.996316i	−0.867086	+	0.498158i	$0.834010\pi$
$258$	0	0
$259$	0.361563	0.0224664
$260$	0	0
$261$	6.71301	0.415525
$262$	0	0
$263$	17.3047i	1.06705i	0.845783	+	0.533527i	$0.179134\pi$
$263$	17.3047i	1.06705i	−0.845783	+	0.533527i	$0.820866\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.04557i	0.369983i
$268$	0	0
$269$	7.04673	0.429646	0.214823	−	0.976653i	$-0.431082\pi$
$269$	7.04673	0.429646	0.214823	+	0.976653i	$0.431082\pi$
$270$	0	0
$271$	11.6663	0.708676	0.354338	−	0.935117i	$-0.384706\pi$
$271$	11.6663	0.708676	0.354338	+	0.935117i	$0.384706\pi$
$272$	0	0
$273$	0.305878i	0.0185126i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 29.9443i	− 1.79918i	−0.436736	−	0.899590i	$-0.643866\pi$
$277$	− 29.9443i	− 1.79918i	0.436736	−	0.899590i	$-0.356134\pi$
$278$	0	0
$279$	2.54615	0.152434
$280$	0	0
$281$	24.1390	1.44001	0.720007	−	0.693967i	$-0.244139\pi$
$281$	24.1390	1.44001	0.720007	+	0.693967i	$0.244139\pi$
$282$	0	0
$283$	4.92543i	0.292786i	0.989226	+	0.146393i	$0.0467665\pi$
$283$	4.92543i	0.292786i	−0.989226	+	0.146393i	$0.953234\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.78757i	0.105517i
$288$	0	0
$289$	−3.66744	−0.215732
$290$	0	0
$291$	2.16686	0.127024
$292$	0	0
$293$	9.12013i	0.532804i	0.963862	+	0.266402i	$0.0858349\pi$
$293$	9.12013i	0.532804i	−0.963862	+	0.266402i	$0.914165\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.71301i	0.273476i
$298$	0	0
$299$	8.33372	0.481951
$300$	0	0
$301$	−1.54498	−0.0890514
$302$	0	0
$303$	16.5183i	0.948952i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.75857i	0.499878i	0.968262	+	0.249939i	$0.0804106\pi$
$307$	8.75857i	0.499878i	−0.968262	+	0.249939i	$0.919589\pi$
$308$	0	0
$309$	−5.09229	−0.289690
$310$	0	0
$311$	1.04673	0.0593544	0.0296772	−	0.999560i	$-0.490552\pi$
$311$	1.04673	0.0593544	0.0296772	+	0.999560i	$0.490552\pi$
$312$	0	0
$313$	− 21.6663i	− 1.22465i	−0.790606	−	0.612325i	$-0.790234\pi$
$313$	− 21.6663i	− 1.22465i	0.790606	−	0.612325i	$-0.209766\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.5461i	0.816993i	0.912760	+	0.408496i	$0.133947\pi$
$317$	14.5461i	0.816993i	−0.912760	+	0.408496i	$0.866053\pi$
$318$	0	0
$319$	−31.6384	−1.77141
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	− 4.54615i	− 0.252954i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.4260i	0.853060i
$328$	0	0
$329$	0.0911300	0.00502416
$330$	0	0
$331$	7.21359	0.396495	0.198247	−	0.980152i	$-0.436475\pi$
$331$	7.21359	0.396495	0.198247	+	0.980152i	$0.436475\pi$
$332$	0	0
$333$	2.16686i	0.118743i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.59171i	0.141179i	0.997505	+	0.0705897i	$0.0224881\pi$
$337$	2.59171i	0.141179i	−0.997505	+	0.0705897i	$0.977512\pi$
$338$	0	0
$339$	14.8799	0.808163
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	2.33140i	0.125884i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.54498i	0.190305i	0.995463	+	0.0951524i	$0.0303338\pi$
$347$	3.54498i	0.190305i	−0.995463	+	0.0951524i	$0.969666\pi$
$348$	0	0
$349$	28.8520	1.54441	0.772207	−	0.635371i	$-0.219153\pi$
$349$	28.8520	1.54441	0.772207	+	0.635371i	$0.219153\pi$
$350$	0	0
$351$	−1.83314	−0.0978458
$352$	0	0
$353$	20.8520i	1.10984i	0.831903	+	0.554921i	$0.187251\pi$
$353$	20.8520i	1.10984i	−0.831903	+	0.554921i	$0.812749\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.758571i	0.0401478i
$358$	0	0
$359$	10.4727	0.552730	0.276365	−	0.961053i	$-0.410870\pi$
$359$	10.4727	0.552730	0.276365	+	0.961053i	$0.410870\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 11.2124i	− 0.588500i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.40713i	− 0.125651i	−0.998025	−	0.0628255i	$-0.979989\pi$
$367$	− 2.40713i	− 0.125651i	0.998025	−	0.0628255i	$-0.0200112\pi$
$368$	0	0
$369$	−10.7130	−0.557697
$370$	0	0
$371$	−1.75973	−0.0913608
$372$	0	0
$373$	− 31.2592i	− 1.61854i	−0.587439	−	0.809269i	$-0.699864\pi$
$373$	− 31.2592i	− 1.61854i	0.587439	−	0.809269i	$-0.300136\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.3059i	− 0.633785i
$378$	0	0
$379$	11.6384	0.597826	0.298913	−	0.954280i	$-0.403376\pi$
$379$	11.6384	0.597826	0.298913	+	0.954280i	$0.403376\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	25.0923i	1.28216i	0.767476	+	0.641078i	$0.221513\pi$
$383$	25.0923i	1.28216i	−0.767476	+	0.641078i	$0.778487\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 9.25915i	− 0.470669i
$388$	0	0
$389$	−11.7597	−0.596242	−0.298121	−	0.954528i	$-0.596360\pi$
$389$	−11.7597	−0.596242	−0.298121	+	0.954528i	$0.596360\pi$
$390$	0	0
$391$	20.6674	1.04520
$392$	0	0
$393$	− 14.1390i	− 0.713219i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 25.5195i	− 1.28079i	−0.768048	−	0.640393i	$-0.778772\pi$
$397$	− 25.5195i	− 1.28079i	0.768048	−	0.640393i	$-0.221228\pi$
$398$	0	0
$399$	−0.166860	−0.00835346
$400$	0	0
$401$	0.194703	0.00972298	0.00486149	−	0.999988i	$-0.498453\pi$
$401$	0.194703	0.00972298	0.00486149	+	0.999988i	$0.498453\pi$
$402$	0	0
$403$	− 4.66744i	− 0.232502i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10.2124i	− 0.506211i
$408$	0	0
$409$	−29.6106	−1.46415	−0.732075	−	0.681224i	$-0.761448\pi$
$409$	−29.6106	−1.46415	−0.732075	+	0.681224i	$0.761448\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	1.57283i	0.0773937i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.42601i	0.461594i
$418$	0	0
$419$	3.95443	0.193187	0.0965934	−	0.995324i	$-0.469205\pi$
$419$	3.95443	0.193187	0.0965934	+	0.995324i	$0.469205\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0.546146i	0.0265545i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.14914i	0.104004i
$428$	0	0
$429$	8.63960	0.417124
$430$	0	0
$431$	−29.8509	−1.43787	−0.718933	−	0.695080i	$-0.755369\pi$
$431$	−29.8509	−1.43787	−0.718933	+	0.695080i	$0.755369\pi$
$432$	0	0
$433$	5.07457i	0.243868i	0.992538	+	0.121934i	$0.0389097\pi$
$433$	5.07457i	0.243868i	−0.992538	+	0.121934i	$0.961090\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.54615i	0.217472i
$438$	0	0
$439$	34.8520	1.66340	0.831698	−	0.555228i	$-0.187369\pi$
$439$	34.8520	1.66340	0.831698	+	0.555228i	$0.187369\pi$
$440$	0	0
$441$	−6.97216	−0.332008
$442$	0	0
$443$	15.8787i	0.754420i	0.926128	+	0.377210i	$0.123117\pi$
$443$	15.8787i	0.754420i	−0.926128	+	0.377210i	$0.876883\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 15.0923i	− 0.713841i
$448$	0	0
$449$	−11.8053	−0.557126	−0.278563	−	0.960418i	$-0.589858\pi$
$449$	−11.8053	−0.557126	−0.278563	+	0.960418i	$0.589858\pi$
$450$	0	0
$451$	50.4905	2.37750
$452$	0	0
$453$	− 11.2136i	− 0.526860i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.4260i	1.65716i	0.559871	+	0.828579i	$0.310851\pi$
$457$	35.4260i	1.65716i	−0.559871	+	0.828579i	$0.689149\pi$
$458$	0	0
$459$	−4.54615	−0.212196
$460$	0	0
$461$	−10.6674	−0.496832	−0.248416	−	0.968653i	$-0.579910\pi$
$461$	−10.6674	−0.496832	−0.248416	+	0.968653i	$0.579910\pi$
$462$	0	0
$463$	13.9266i	0.647224i	0.946190	+	0.323612i	$0.104897\pi$
$463$	13.9266i	0.647224i	−0.946190	+	0.323612i	$0.895103\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.30588i	− 0.106703i	−0.998576	−	0.0533517i	$-0.983010\pi$
$467$	− 2.30588i	− 0.106703i	0.998576	−	0.0533517i	$-0.0169904\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.7586	0.680039
$472$	0	0
$473$	43.6384i	2.00650i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.5461i	− 0.482875i
$478$	0	0
$479$	−2.13902	−0.0977342	−0.0488671	−	0.998805i	$-0.515561\pi$
$479$	−2.13902	−0.0977342	−0.0488671	+	0.998805i	$0.515561\pi$
$480$	0	0
$481$	3.97216	0.181115
$482$	0	0
$483$	− 0.758571i	− 0.0345162i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.09113i	0.366644i	0.983053	+	0.183322i	$0.0586851\pi$
$487$	8.09113i	0.366644i	−0.983053	+	0.183322i	$0.941315\pi$
$488$	0	0
$489$	−3.49942	−0.158249
$490$	0	0
$491$	4.37929	0.197634	0.0988172	−	0.995106i	$-0.468494\pi$
$491$	4.37929	0.197634	0.0988172	+	0.995106i	$0.468494\pi$
$492$	0	0
$493$	− 30.5183i	− 1.37448i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.905386i	− 0.0406121i
$498$	0	0
$499$	10.0935	0.451845	0.225923	−	0.974145i	$-0.427460\pi$
$499$	10.0935	0.451845	0.225923	+	0.974145i	$0.427460\pi$
$500$	0	0
$501$	−6.21243	−0.277551
$502$	0	0
$503$	− 8.21243i	− 0.366174i	−0.983097	−	0.183087i	$-0.941391\pi$
$503$	− 8.21243i	− 0.366174i	0.983097	−	0.183087i	$-0.0586090\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.63960i	− 0.428110i
$508$	0	0
$509$	−32.8065	−1.45412	−0.727060	−	0.686574i	$-0.759114\pi$
$509$	−32.8065	−1.45412	−0.727060	+	0.686574i	$0.759114\pi$
$510$	0	0
$511$	−2.51830	−0.111403
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.57399i	− 0.113204i
$518$	0	0
$519$	−1.12013	−0.0491684
$520$	0	0
$521$	−9.62071	−0.421491	−0.210746	−	0.977541i	$-0.567589\pi$
$521$	−9.62071	−0.421491	−0.210746	+	0.977541i	$0.567589\pi$
$522$	0	0
$523$	− 31.6106i	− 1.38223i	−0.722743	−	0.691117i	$-0.757119\pi$
$523$	− 31.6106i	− 1.38223i	0.722743	−	0.691117i	$-0.242881\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 11.5751i	− 0.504221i
$528$	0	0
$529$	2.33256	0.101416
$530$	0	0
$531$	−9.42601	−0.409054
$532$	0	0
$533$	19.6384i	0.850635i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 11.6663i	− 0.503437i
$538$	0	0
$539$	32.8598	1.41537
$540$	0	0
$541$	−32.1846	−1.38372	−0.691862	−	0.722030i	$-0.743209\pi$
$541$	−32.1846	−1.38372	−0.691862	+	0.722030i	$0.743209\pi$
$542$	0	0
$543$	− 9.66628i	− 0.414820i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 33.4260i	− 1.42919i	−0.699537	−	0.714597i	$-0.746610\pi$
$547$	− 33.4260i	− 1.42919i	0.699537	−	0.714597i	$-0.253390\pi$
$548$	0	0
$549$	−12.8799	−0.549699
$550$	0	0
$551$	6.71301	0.285984
$552$	0	0
$553$	− 2.29595i	− 0.0976338i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 25.2769i	− 1.07102i	−0.844530	−	0.535508i	$-0.820120\pi$
$557$	− 25.2769i	− 1.07102i	0.844530	−	0.535508i	$-0.179880\pi$
$558$	0	0
$559$	−16.9733	−0.717895
$560$	0	0
$561$	21.4260	0.904607
$562$	0	0
$563$	− 32.6396i	− 1.37560i	−0.725903	−	0.687798i	$-0.758578\pi$
$563$	− 32.6396i	− 1.37560i	0.725903	−	0.687798i	$-0.241422\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.166860i	0.00700747i
$568$	0	0
$569$	−25.6207	−1.07408	−0.537038	−	0.843558i	$-0.680457\pi$
$569$	−25.6207	−1.07408	−0.537038	+	0.843558i	$0.680457\pi$
$570$	0	0
$571$	−27.1857	−1.13769	−0.568844	−	0.822445i	$-0.692609\pi$
$571$	−27.1857	−1.13769	−0.568844	+	0.822445i	$0.692609\pi$
$572$	0	0
$573$	− 1.04673i	− 0.0437276i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 4.51830i	− 0.188099i	−0.995568	−	0.0940497i	$-0.970019\pi$
$577$	− 4.51830i	− 0.188099i	0.995568	−	0.0940497i	$-0.0299813\pi$
$578$	0	0
$579$	−10.9254	−0.454045
$580$	0	0
$581$	2.33140	0.0967227
$582$	0	0
$583$	49.7040i	2.05853i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 26.9733i	− 1.11331i	−0.830744	−	0.556654i	$-0.812085\pi$
$587$	− 26.9733i	− 1.11331i	0.830744	−	0.556654i	$-0.187915\pi$
$588$	0	0
$589$	2.54615	0.104912
$590$	0	0
$591$	11.4539	0.471149
$592$	0	0
$593$	− 7.09229i	− 0.291246i	−0.989340	−	0.145623i	$-0.953481\pi$
$593$	− 7.09229i	− 0.291246i	0.989340	−	0.145623i	$-0.0465186\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 24.7586i	− 1.01330i
$598$	0	0
$599$	6.85202	0.279966	0.139983	−	0.990154i	$-0.455295\pi$
$599$	6.85202	0.279966	0.139983	+	0.990154i	$0.455295\pi$
$600$	0	0
$601$	−31.8509	−1.29922	−0.649612	−	0.760266i	$-0.725069\pi$
$601$	−31.8509	−1.29922	−0.649612	+	0.760266i	$0.725069\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 41.3703i	− 1.67917i	−0.543229	−	0.839585i	$-0.682798\pi$
$607$	− 41.3703i	− 1.67917i	0.543229	−	0.839585i	$-0.317202\pi$
$608$	0	0
$609$	−1.12013	−0.0453901
$610$	0	0
$611$	1.00116	0.0405027
$612$	0	0
$613$	5.66628i	0.228859i	0.993431	+	0.114429i	$0.0365040\pi$
$613$	5.66628i	0.228859i	−0.993431	+	0.114429i	$0.963496\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.7307i	1.23717i	0.785717	+	0.618586i	$0.212294\pi$
$617$	30.7307i	1.23717i	−0.785717	+	0.618586i	$0.787706\pi$
$618$	0	0
$619$	−2.99884	−0.120533	−0.0602667	−	0.998182i	$-0.519195\pi$
$619$	−2.99884	−0.120533	−0.0602667	+	0.998182i	$0.519195\pi$
$620$	0	0
$621$	4.54615	0.182431
$622$	0	0
$623$	− 1.00876i	− 0.0404153i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.71301i	0.188219i
$628$	0	0
$629$	9.85086	0.392780
$630$	0	0
$631$	25.5171	1.01582	0.507911	−	0.861410i	$-0.330418\pi$
$631$	25.5171	1.01582	0.507911	+	0.861410i	$0.330418\pi$
$632$	0	0
$633$	− 16.4249i	− 0.652829i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.7809i	0.506399i
$638$	0	0
$639$	5.42601	0.214650
$640$	0	0
$641$	−4.13902	−0.163481	−0.0817407	−	0.996654i	$-0.526048\pi$
$641$	−4.13902	−0.163481	−0.0817407	+	0.996654i	$0.526048\pi$
$642$	0	0
$643$	26.2603i	1.03561i	0.855500	+	0.517803i	$0.173250\pi$
$643$	26.2603i	1.03561i	−0.855500	+	0.517803i	$0.826750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.3970i	0.880517i	0.897871	+	0.440259i	$0.145113\pi$
$647$	22.3970i	0.880517i	−0.897871	+	0.440259i	$0.854887\pi$
$648$	0	0
$649$	44.4249	1.74383
$650$	0	0
$651$	−0.424850	−0.0166512
$652$	0	0
$653$	13.6384i	0.533713i	0.963736	+	0.266857i	$0.0859850\pi$
$653$	13.6384i	0.533713i	−0.963736	+	0.266857i	$0.914015\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 15.0923i	− 0.588806i
$658$	0	0
$659$	13.0923	0.510003	0.255002	−	0.966941i	$-0.417924\pi$
$659$	13.0923	0.510003	0.255002	+	0.966941i	$0.417924\pi$
$660$	0	0
$661$	−20.5183	−0.798070	−0.399035	−	0.916936i	$-0.630655\pi$
$661$	−20.5183	−0.798070	−0.399035	+	0.916936i	$0.630655\pi$
$662$	0	0
$663$	8.33372i	0.323655i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30.5183i	1.18167i
$668$	0	0
$669$	−5.09229	−0.196879
$670$	0	0
$671$	60.7029	2.34341
$672$	0	0
$673$	− 2.16686i	− 0.0835263i	−0.999128	−	0.0417632i	$-0.986703\pi$
$673$	− 2.16686i	− 0.0835263i	0.999128	−	0.0417632i	$-0.0132975\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.3036i	0.857195i	0.903495	+	0.428598i	$0.140992\pi$
$677$	22.3036i	0.857195i	−0.903495	+	0.428598i	$0.859008\pi$
$678$	0	0
$679$	−0.361563	−0.0138755
$680$	0	0
$681$	24.6396	0.944191
$682$	0	0
$683$	7.33256i	0.280573i	0.990111	+	0.140286i	$0.0448023\pi$
$683$	7.33256i	0.280573i	−0.990111	+	0.140286i	$0.955198\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.78757i	− 0.297115i
$688$	0	0
$689$	−19.3326	−0.736512
$690$	0	0
$691$	0.333720	0.0126953	0.00634766	−	0.999980i	$-0.497979\pi$
$691$	0.333720	0.0126953	0.00634766	+	0.999980i	$0.497979\pi$
$692$	0	0
$693$	− 0.786413i	− 0.0298734i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	48.7029i	1.84475i
$698$	0	0
$699$	−25.9443	−0.981304
$700$	0	0
$701$	19.7597	0.746315	0.373157	−	0.927768i	$-0.378275\pi$
$701$	19.7597	0.746315	0.373157	+	0.927768i	$0.378275\pi$
$702$	0	0
$703$	2.16686i	0.0817247i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.75625i	− 0.103659i
$708$	0	0
$709$	−30.3970	−1.14158	−0.570792	−	0.821095i	$-0.693364\pi$
$709$	−30.3970	−1.14158	−0.570792	+	0.821095i	$0.693364\pi$
$710$	0	0
$711$	13.7597	0.516030
$712$	0	0
$713$	11.5751i	0.433493i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16.3793i	− 0.611696i
$718$	0	0
$719$	34.1390	1.27317	0.636585	−	0.771206i	$-0.280346\pi$
$719$	34.1390	1.27317	0.636585	+	0.771206i	$0.280346\pi$
$720$	0	0
$721$	0.849701	0.0316445
$722$	0	0
$723$	2.00000i	0.0743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 41.2592i	− 1.53022i	−0.643901	−	0.765109i	$-0.722685\pi$
$727$	− 41.2592i	− 1.53022i	0.643901	−	0.765109i	$-0.277315\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−42.0935	−1.55688
$732$	0	0
$733$	3.09229i	0.114216i	0.998368	+	0.0571082i	$0.0181880\pi$
$733$	3.09229i	0.114216i	−0.998368	+	0.0571082i	$0.981812\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	15.5195	0.570893	0.285446	−	0.958395i	$-0.407858\pi$
$739$	15.5195	0.570893	0.285446	+	0.958395i	$0.407858\pi$
$740$	0	0
$741$	−1.83314	−0.0673421
$742$	0	0
$743$	26.4272i	0.969519i	0.874647	+	0.484759i	$0.161093\pi$
$743$	26.4272i	0.969519i	−0.874647	+	0.484759i	$0.838907\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.9722i	0.511215i
$748$	0	0
$749$	−2.66976	−0.0975510
$750$	0	0
$751$	40.3059	1.47078	0.735391	−	0.677643i	$-0.236998\pi$
$751$	40.3059	1.47078	0.735391	+	0.677643i	$0.236998\pi$
$752$	0	0
$753$	21.4716i	0.782468i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 0.573988i	− 0.0208620i	−0.999946	−	0.0104310i	$-0.996680\pi$
$757$	− 0.573988i	− 0.0208620i	0.999946	−	0.0104310i	$-0.00332034\pi$
$758$	0	0
$759$	−21.4260	−0.777715
$760$	0	0
$761$	−43.7597	−1.58629	−0.793145	−	0.609033i	$-0.791558\pi$
$761$	−43.7597	−1.58629	−0.793145	+	0.609033i	$0.791558\pi$
$762$	0	0
$763$	− 2.57399i	− 0.0931846i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.2792i	0.623916i
$768$	0	0
$769$	−13.3326	−0.480784	−0.240392	−	0.970676i	$-0.577276\pi$
$769$	−13.3326	−0.480784	−0.240392	+	0.970676i	$0.577276\pi$
$770$	0	0
$771$	15.9722	0.575223
$772$	0	0
$773$	− 31.2136i	− 1.12267i	−0.827587	−	0.561337i	$-0.810287\pi$
$773$	− 31.2136i	− 1.12267i	0.827587	−	0.561337i	$-0.189713\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.361563i	− 0.0129710i
$778$	0	0
$779$	−10.7130	−0.383833
$780$	0	0
$781$	−25.5728	−0.915068
$782$	0	0
$783$	− 6.71301i	− 0.239903i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	48.2780i	1.72093i	0.509513	+	0.860463i	$0.329826\pi$
$787$	48.2780i	1.72093i	−0.509513	+	0.860463i	$0.670174\pi$
$788$	0	0
$789$	17.3047	0.616064
$790$	0	0
$791$	−2.48286	−0.0882803
$792$	0	0
$793$	23.6106i	0.838437i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 17.0644i	− 0.604454i	−0.953236	−	0.302227i	$-0.902270\pi$
$797$	− 17.0644i	− 0.604454i	0.953236	−	0.302227i	$-0.0977300\pi$
$798$	0	0
$799$	2.48286	0.0878372
$800$	0	0
$801$	6.04557	0.213610
$802$	0	0
$803$	71.1301i	2.51013i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.04673i	− 0.248057i
$808$	0	0
$809$	3.75973	0.132185	0.0660926	−	0.997813i	$-0.478947\pi$
$809$	3.75973	0.132185	0.0660926	+	0.997813i	$0.478947\pi$
$810$	0	0
$811$	−5.33488	−0.187333	−0.0936665	−	0.995604i	$-0.529859\pi$
$811$	−5.33488	−0.187333	−0.0936665	+	0.995604i	$0.529859\pi$
$812$	0	0
$813$	− 11.6663i	− 0.409154i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.25915i	− 0.323937i
$818$	0	0
$819$	0.305878	0.0106882
$820$	0	0
$821$	7.42601	0.259170	0.129585	−	0.991568i	$-0.458636\pi$
$821$	7.42601	0.259170	0.129585	+	0.991568i	$0.458636\pi$
$822$	0	0
$823$	− 30.6852i	− 1.06962i	−0.844973	−	0.534809i	$-0.820384\pi$
$823$	− 30.6852i	− 1.06962i	0.844973	−	0.534809i	$-0.179616\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.8242i	− 1.07186i	−0.844262	−	0.535931i	$-0.819961\pi$
$827$	− 30.8242i	− 1.07186i	0.844262	−	0.535931i	$-0.180039\pi$
$828$	0	0
$829$	−20.5183	−0.712630	−0.356315	−	0.934366i	$-0.615967\pi$
$829$	−20.5183	−0.712630	−0.356315	+	0.934366i	$0.615967\pi$
$830$	0	0
$831$	−29.9443	−1.03876
$832$	0	0
$833$	31.6964i	1.09822i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.54615i	− 0.0880077i
$838$	0	0
$839$	−30.9432	−1.06828	−0.534138	−	0.845397i	$-0.679364\pi$
$839$	−30.9432	−1.06828	−0.534138	+	0.845397i	$0.679364\pi$
$840$	0	0
$841$	16.0644	0.553947
$842$	0	0
$843$	− 24.1390i	− 0.831392i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.87091i	0.0642852i
$848$	0	0
$849$	4.92543	0.169040
$850$	0	0
$851$	−9.85086	−0.337683
$852$	0	0
$853$	− 31.4260i	− 1.07601i	−0.842943	−	0.538003i	$-0.819179\pi$
$853$	− 31.4260i	− 1.07601i	0.842943	−	0.538003i	$-0.180821\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.6941i	0.399464i	0.979851	+	0.199732i	$0.0640071\pi$
$857$	11.6941i	0.399464i	−0.979851	+	0.199732i	$0.935993\pi$
$858$	0	0
$859$	38.1289	1.30094	0.650471	−	0.759531i	$-0.274572\pi$
$859$	38.1289	1.30094	0.650471	+	0.759531i	$0.274572\pi$
$860$	0	0
$861$	1.78757	0.0609204
$862$	0	0
$863$	44.0000i	1.49778i	0.662696	+	0.748889i	$0.269412\pi$
$863$	44.0000i	1.49778i	−0.662696	+	0.748889i	$0.730588\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.66744i	0.124553i
$868$	0	0
$869$	−64.8497	−2.19988
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.16686i	− 0.0733371i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 22.8697i	− 0.772256i	−0.922445	−	0.386128i	$-0.873812\pi$
$877$	− 22.8697i	− 0.772256i	0.922445	−	0.386128i	$-0.126188\pi$
$878$	0	0
$879$	9.12013	0.307614
$880$	0	0
$881$	−58.2780	−1.96344	−0.981718	−	0.190339i	$-0.939041\pi$
$881$	−58.2780	−1.96344	−0.981718	+	0.190339i	$0.939041\pi$
$882$	0	0
$883$	− 1.31484i	− 0.0442478i	−0.999755	−	0.0221239i	$-0.992957\pi$
$883$	− 1.31484i	− 0.0442478i	0.999755	−	0.0221239i	$-0.00704283\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 10.1846i	− 0.341965i	−0.985274	−	0.170982i	$-0.945306\pi$
$887$	− 10.1846i	− 0.341965i	0.985274	−	0.170982i	$-0.0546941\pi$
$888$	0	0
$889$	0.667441	0.0223853
$890$	0	0
$891$	4.71301	0.157892
$892$	0	0
$893$	0.546146i	0.0182761i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.33372i	− 0.278255i
$898$	0	0
$899$	17.0923	0.570060
$900$	0	0
$901$	−47.9443	−1.59726
$902$	0	0
$903$	1.54498i	0.0514139i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	44.0354i	1.46217i	0.682285	+	0.731086i	$0.260986\pi$
$907$	44.0354i	1.46217i	−0.682285	+	0.731086i	$0.739014\pi$
$908$	0	0
$909$	16.5183	0.547878
$910$	0	0
$911$	−36.4249	−1.20681	−0.603405	−	0.797435i	$-0.706190\pi$
$911$	−36.4249	−1.20681	−0.603405	+	0.797435i	$0.706190\pi$
$912$	0	0
$913$	− 65.8509i	− 2.17935i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.35924i	0.0779090i
$918$	0	0
$919$	5.75973	0.189996	0.0949980	−	0.995477i	$-0.469716\pi$
$919$	5.75973	0.189996	0.0949980	+	0.995477i	$0.469716\pi$
$920$	0	0
$921$	8.75857	0.288605
$922$	0	0
$923$	− 9.94664i	− 0.327398i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.09229i	0.167253i
$928$	0	0
$929$	−50.7586	−1.66533	−0.832667	−	0.553774i	$-0.813187\pi$
$929$	−50.7586	−1.66533	−0.832667	+	0.553774i	$0.813187\pi$
$930$	0	0
$931$	−6.97216	−0.228503
$932$	0	0
$933$	− 1.04673i	− 0.0342683i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.2780i	1.25049i	0.780429	+	0.625244i	$0.215001\pi$
$937$	38.2780i	1.25049i	−0.780429	+	0.625244i	$0.784999\pi$
$938$	0	0
$939$	−21.6663	−0.707052
$940$	0	0
$941$	30.6573	0.999400	0.499700	−	0.866199i	$-0.333444\pi$
$941$	30.6573	0.999400	0.499700	+	0.866199i	$0.333444\pi$
$942$	0	0
$943$	− 48.7029i	− 1.58598i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 11.1201i	− 0.361356i	−0.983542	−	0.180678i	$-0.942171\pi$
$947$	− 11.1201i	− 0.361356i	0.983542	−	0.180678i	$-0.0578292\pi$
$948$	0	0
$949$	−27.6663	−0.898085
$950$	0	0
$951$	14.5461	0.471691
$952$	0	0
$953$	− 30.6373i	− 0.992439i	−0.868197	−	0.496219i	$-0.834721\pi$
$953$	− 30.6373i	− 0.992439i	0.868197	−	0.496219i	$-0.165279\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	31.6384i	1.02273i
$958$	0	0
$959$	−1.00116	−0.0323292
$960$	0	0
$961$	−24.5171	−0.790876
$962$	0	0
$963$	− 16.0000i	− 0.515593i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 11.6863i	− 0.375807i	−0.982188	−	0.187903i	$-0.939831\pi$
$967$	− 11.6863i	− 0.375807i	0.982188	−	0.187903i	$-0.0601692\pi$
$968$	0	0
$969$	−4.54615	−0.146043
$970$	0	0
$971$	15.2769	0.490258	0.245129	−	0.969490i	$-0.421170\pi$
$971$	15.2769	0.490258	0.245129	+	0.969490i	$0.421170\pi$
$972$	0	0
$973$	− 1.57283i	− 0.0504225i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.97332i	0.159111i	0.996830	+	0.0795553i	$0.0253500\pi$
$977$	4.97332i	0.159111i	−0.996830	+	0.0795553i	$0.974650\pi$
$978$	0	0
$979$	−28.4928	−0.910633
$980$	0	0
$981$	15.4260	0.492515
$982$	0	0
$983$	27.9722i	0.892173i	0.894990	+	0.446087i	$0.147183\pi$
$983$	27.9722i	0.892173i	−0.894990	+	0.446087i	$0.852817\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.0911300i	− 0.00290070i
$988$	0	0
$989$	42.0935	1.33849
$990$	0	0
$991$	−34.8520	−1.10711	−0.553556	−	0.832812i	$-0.686729\pi$
$991$	−34.8520	−1.10711	−0.553556	+	0.832812i	$0.686729\pi$
$992$	0	0
$993$	− 7.21359i	− 0.228916i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 37.6106i	− 1.19114i	−0.803304	−	0.595570i	$-0.796926\pi$
$997$	− 37.6106i	− 1.19114i	0.803304	−	0.595570i	$-0.203074\pi$
$998$	0	0
$999$	2.16686	0.0685564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5700.2.f.p.3649.2		6
5.2	odd	4		1140.2.a.f.1.2	✓	3
5.3	odd	4		5700.2.a.z.1.2		3
5.4	even	2	inner	5700.2.f.p.3649.5		6
15.2	even	4		3420.2.a.k.1.2		3
20.7	even	4		4560.2.a.bu.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.f.1.2	✓	3	5.2	odd	4
3420.2.a.k.1.2		3	15.2	even	4
4560.2.a.bu.1.2		3	20.7	even	4
5700.2.a.z.1.2		3	5.3	odd	4
5700.2.f.p.3649.2		6	1.1	even	1	trivial
5700.2.f.p.3649.5		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5700.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +0.166860i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +0.166860i q^{7} -1.00000 q^{9} +4.71301 q^{11} +1.83314i q^{13} +4.54615i q^{17} -1.00000 q^{19} +0.166860 q^{21} -4.54615i q^{23} +1.00000i q^{27} -6.71301 q^{29} -2.54615 q^{31} -4.71301i q^{33} -2.16686i q^{37} +1.83314 q^{39} +10.7130 q^{41} +9.25915i q^{43} -0.546146i q^{47} +6.97216 q^{49} +4.54615 q^{51} +10.5461i q^{53} +1.00000i q^{57} +9.42601 q^{59} +12.8799 q^{61} -0.166860i q^{63} -4.54615 q^{69} -5.42601 q^{71} +15.0923i q^{73} +0.786413i q^{77} -13.7597 q^{79} +1.00000 q^{81} -13.9722i q^{83} +6.71301i q^{87} -6.04557 q^{89} -0.305878 q^{91} +2.54615i q^{93} +2.16686i q^{97} -4.71301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} - 4 q^{11} - 6 q^{19} - 8 q^{29} + 16 q^{31} + 12 q^{39} + 32 q^{41} - 54 q^{49} - 4 q^{51} - 8 q^{59} + 44 q^{61} + 4 q^{69} + 32 q^{71} - 16 q^{79} + 6 q^{81} - 8 q^{89} + 96 q^{91} + 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 - 4 * q^11 - 6 * q^19 - 8 * q^29 + 16 * q^31 + 12 * q^39 + 32 * q^41 - 54 * q^49 - 4 * q^51 - 8 * q^59 + 44 * q^61 + 4 * q^69 + 32 * q^71 - 16 * q^79 + 6 * q^81 - 8 * q^89 + 96 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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