LMFDB - Embedded newform 2850.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(1,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, 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("2850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.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:	$22.7573645761$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -1.00000 q^{38} +4.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +1.00000 q^{44} -5.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{51} -4.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} +1.00000 q^{57} +3.00000 q^{58} -11.0000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -1.00000 q^{66} +1.00000 q^{67} -4.00000 q^{68} +5.00000 q^{69} +2.00000 q^{71} +1.00000 q^{72} +3.00000 q^{73} +6.00000 q^{74} -1.00000 q^{76} +4.00000 q^{78} +17.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} -3.00000 q^{83} -4.00000 q^{86} -3.00000 q^{87} +1.00000 q^{88} +7.00000 q^{89} -5.00000 q^{92} +5.00000 q^{93} -1.00000 q^{96} -10.0000 q^{97} -7.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.00000	−0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	1.00000	0.213201
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−1.00000	−0.162221
$39$	4.00000	0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−5.00000	−0.737210
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	4.00000	0.560112
$52$	−4.00000	−0.554700
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	3.00000	0.393919
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	−4.00000	−0.485071
$69$	5.00000	0.601929
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	1.00000	0.117851
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	4.00000	0.452911
$79$	17.0000	1.91265	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	17.0000	1.91265	0.956325	+	0.292306i	$0.0944227\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	−3.00000	−0.321634
$88$	1.00000	0.106600
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	0	0
$91$	0	0
$92$	−5.00000	−0.521286
$93$	5.00000	0.518476
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−7.00000	−0.707107
$99$	1.00000	0.100504
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	4.00000	0.396059
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−9.00000	−0.874157
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	3.00000	0.278543
$117$	−4.00000	−0.369800
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−11.0000	−0.995893
$123$	2.00000	0.180334
$124$	−5.00000	−0.449013
$125$	0	0
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	1.00000	0.0863868
$135$	0	0
$136$	−4.00000	−0.342997
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	5.00000	0.425628
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	2.00000	0.167836
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	3.00000	0.248282
$147$	7.00000	0.577350
$148$	6.00000	0.493197
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	−1.00000	−0.0811107
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	17.0000	1.35245
$159$	9.00000	0.713746
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−3.00000	−0.232845
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−4.00000	−0.304997
$173$	−7.00000	−0.532200	−0.266100	−	0.963945i	$-0.585735\pi$
$173$	−7.00000	−0.532200	−0.266100	+	0.963945i	$0.585735\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	7.00000	0.524672
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	11.0000	0.813143
$184$	−5.00000	−0.368605
$185$	0	0
$186$	5.00000	0.366618
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.0000	0.795932	0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000	0.795932	0.397966	+	0.917400i	$0.369716\pi$
$192$	−1.00000	−0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−7.00000	−0.500000
$197$	20.0000	1.42494	0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000	1.42494	0.712470	+	0.701702i	$0.247576\pi$
$198$	1.00000	0.0710669
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	−8.00000	−0.562878
$203$	0	0
$204$	4.00000	0.280056
$205$	0	0
$206$	−13.0000	−0.905753
$207$	−5.00000	−0.347524
$208$	−4.00000	−0.277350
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−9.00000	−0.618123
$213$	−2.00000	−0.137038
$214$	−10.0000	−0.683586
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	6.00000	0.406371
$219$	−3.00000	−0.202721
$220$	0	0
$221$	16.0000	1.07628
$222$	−6.00000	−0.402694
$223$	15.0000	1.00447	0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000	1.00447	0.502237	+	0.864730i	$0.332510\pi$
$224$	0	0
$225$	0	0
$226$	−5.00000	−0.332595
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	1.00000	0.0662266
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	0	0
$237$	−17.0000	−1.10427
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	−11.0000	−0.704203
$245$	0	0
$246$	2.00000	0.127515
$247$	4.00000	0.254514
$248$	−5.00000	−0.317500
$249$	3.00000	0.190117
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	5.00000	0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	15.0000	0.926703
$263$	11.0000	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$263$	11.0000	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	0	0
$267$	−7.00000	−0.428393
$268$	1.00000	0.0610847
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	4.00000	0.241649
$275$	0	0
$276$	5.00000	0.300965
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−2.00000	−0.119952
$279$	−5.00000	−0.299342
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	10.0000	0.586210
$292$	3.00000	0.175562
$293$	−11.0000	−0.642627	−0.321313	−	0.946973i	$-0.604124\pi$
$293$	−11.0000	−0.642627	−0.321313	+	0.946973i	$0.604124\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	6.00000	0.348743
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	20.0000	1.15663
$300$	0	0
$301$	0	0
$302$	−24.0000	−1.38104
$303$	8.00000	0.459588
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−4.00000	−0.228665
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	0	0
$309$	13.0000	0.739544
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	4.00000	0.226455
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	17.0000	0.956325
$317$	13.0000	0.730153	0.365076	−	0.930978i	$-0.381043\pi$
$317$	13.0000	0.730153	0.365076	+	0.930978i	$0.381043\pi$
$318$	9.00000	0.504695
$319$	3.00000	0.167968
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	−10.0000	−0.553849
$327$	−6.00000	−0.331801
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	−3.00000	−0.164646
$333$	6.00000	0.328798
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	3.00000	0.163178
$339$	5.00000	0.271563
$340$	0	0
$341$	−5.00000	−0.270765
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−7.00000	−0.376322
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−3.00000	−0.160817
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	1.00000	0.0533002
$353$	−28.0000	−1.49029	−0.745145	−	0.666903i	$-0.767620\pi$
$353$	−28.0000	−1.49029	−0.745145	+	0.666903i	$0.767620\pi$
$354$	0	0
$355$	0	0
$356$	7.00000	0.370999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	0	0
$366$	11.0000	0.574979
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−5.00000	−0.260643
$369$	−2.00000	−0.104116
$370$	0	0
$371$	0	0
$372$	5.00000	0.259238
$373$	−28.0000	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$373$	−28.0000	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−5.00000	−0.256158
$382$	11.0000	0.562809
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−4.00000	−0.203331
$388$	−10.0000	−0.507673
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	−7.00000	−0.353553
$393$	−15.0000	−0.756650
$394$	20.0000	1.00759
$395$	0	0
$396$	1.00000	0.0502519
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	6.00000	0.300753
$399$	0	0
$400$	0	0
$401$	−17.0000	−0.848939	−0.424470	−	0.905442i	$-0.639539\pi$
$401$	−17.0000	−0.848939	−0.424470	+	0.905442i	$0.639539\pi$
$402$	−1.00000	−0.0498755
$403$	20.0000	0.996271
$404$	−8.00000	−0.398015
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	4.00000	0.198030
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	−13.0000	−0.640464
$413$	0	0
$414$	−5.00000	−0.245737
$415$	0	0
$416$	−4.00000	−0.196116
$417$	2.00000	0.0979404
$418$	−1.00000	−0.0489116
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	−13.0000	−0.632830
$423$	0	0
$424$	−9.00000	−0.437079
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	0	0
$428$	−10.0000	−0.483368
$429$	4.00000	0.193122
$430$	0	0
$431$	38.0000	1.83040	0.915198	−	0.403005i	$-0.132034\pi$
$431$	38.0000	1.83040	0.915198	+	0.403005i	$0.132034\pi$
$432$	−1.00000	−0.0481125
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	0	0
$436$	6.00000	0.287348
$437$	5.00000	0.239182
$438$	−3.00000	−0.143346
$439$	21.0000	1.00228	0.501138	−	0.865368i	$-0.332915\pi$
$439$	21.0000	1.00228	0.501138	+	0.865368i	$0.332915\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	16.0000	0.761042
$443$	13.0000	0.617649	0.308824	−	0.951119i	$-0.400064\pi$
$443$	13.0000	0.617649	0.308824	+	0.951119i	$0.400064\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	15.0000	0.710271
$447$	0	0
$448$	0	0
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	−5.00000	−0.235180
$453$	24.0000	1.12762
$454$	−2.00000	−0.0938647
$455$	0	0
$456$	1.00000	0.0468293
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	5.00000	0.233635
$459$	4.00000	0.186704
$460$	0	0
$461$	4.00000	0.186299	0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	24.0000	1.11178
$467$	11.0000	0.509019	0.254510	−	0.967070i	$-0.418086\pi$
$467$	11.0000	0.509019	0.254510	+	0.967070i	$0.418086\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−4.00000	−0.183920
$474$	−17.0000	−0.780836
$475$	0	0
$476$	0	0
$477$	−9.00000	−0.412082
$478$	0	0
$479$	−37.0000	−1.69057	−0.845287	−	0.534313i	$-0.820570\pi$
$479$	−37.0000	−1.69057	−0.845287	+	0.534313i	$0.820570\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	−4.00000	−0.182195
$483$	0	0
$484$	−10.0000	−0.454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−11.0000	−0.497947
$489$	10.0000	0.452216
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	2.00000	0.0901670
$493$	−12.0000	−0.540453
$494$	4.00000	0.179969
$495$	0	0
$496$	−5.00000	−0.224507
$497$	0	0
$498$	3.00000	0.134433
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	16.0000	0.714115
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	0	0
$506$	−5.00000	−0.222277
$507$	−3.00000	−0.133235
$508$	5.00000	0.221839
$509$	−27.0000	−1.19675	−0.598377	−	0.801215i	$-0.704187\pi$
$509$	−27.0000	−1.19675	−0.598377	+	0.801215i	$0.704187\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	−7.00000	−0.308757
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	7.00000	0.307266
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	3.00000	0.131306
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	11.0000	0.479623
$527$	20.0000	0.871214
$528$	−1.00000	−0.0435194
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.00000	0.346518
$534$	−7.00000	−0.302920
$535$	0	0
$536$	1.00000	0.0431934
$537$	4.00000	0.172613
$538$	−2.00000	−0.0862261
$539$	−7.00000	−0.301511
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	−4.00000	−0.171499
$545$	0	0
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	4.00000	0.170872
$549$	−11.0000	−0.469469
$550$	0	0
$551$	−3.00000	−0.127804
$552$	5.00000	0.212814
$553$	0	0
$554$	1.00000	0.0424859
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	−5.00000	−0.211667
$559$	16.0000	0.676728
$560$	0	0
$561$	4.00000	0.168880
$562$	−3.00000	−0.126547
$563$	−26.0000	−1.09577	−0.547885	−	0.836554i	$-0.684567\pi$
$563$	−26.0000	−1.09577	−0.547885	+	0.836554i	$0.684567\pi$
$564$	0	0
$565$	0	0
$566$	8.00000	0.336265
$567$	0	0
$568$	2.00000	0.0839181
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	−4.00000	−0.167248
$573$	−11.0000	−0.459532
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	27.0000	1.12402	0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000	1.12402	0.562012	+	0.827129i	$0.310027\pi$
$578$	−1.00000	−0.0415945
$579$	10.0000	0.415586
$580$	0	0
$581$	0	0
$582$	10.0000	0.414513
$583$	−9.00000	−0.372742
$584$	3.00000	0.124141
$585$	0	0
$586$	−11.0000	−0.454406
$587$	17.0000	0.701665	0.350833	−	0.936438i	$-0.385899\pi$
$587$	17.0000	0.701665	0.350833	+	0.936438i	$0.385899\pi$
$588$	7.00000	0.288675
$589$	5.00000	0.206021
$590$	0	0
$591$	−20.0000	−0.822690
$592$	6.00000	0.246598
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	0	0
$597$	−6.00000	−0.245564
$598$	20.0000	0.817861
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	−24.0000	−0.976546
$605$	0	0
$606$	8.00000	0.324978
$607$	25.0000	1.01472	0.507359	−	0.861735i	$-0.330622\pi$
$607$	25.0000	1.01472	0.507359	+	0.861735i	$0.330622\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−4.00000	−0.161690
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	25.0000	1.00892
$615$	0	0
$616$	0	0
$617$	46.0000	1.85189	0.925945	−	0.377658i	$-0.123271\pi$
$617$	46.0000	1.85189	0.925945	+	0.377658i	$0.123271\pi$
$618$	13.0000	0.522937
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	32.0000	1.28308
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	−21.0000	−0.839329
$627$	1.00000	0.0399362
$628$	−14.0000	−0.558661
$629$	−24.0000	−0.956943
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	17.0000	0.676224
$633$	13.0000	0.516704
$634$	13.0000	0.516296
$635$	0	0
$636$	9.00000	0.356873
$637$	28.0000	1.10940
$638$	3.00000	0.118771
$639$	2.00000	0.0791188
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	10.0000	0.394669
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	21.0000	0.825595	0.412798	−	0.910823i	$-0.364552\pi$
$647$	21.0000	0.825595	0.412798	+	0.910823i	$0.364552\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−10.0000	−0.391630
$653$	48.0000	1.87839	0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000	1.87839	0.939193	+	0.343391i	$0.111576\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	3.00000	0.117041
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	13.0000	0.505259
$663$	−16.0000	−0.621389
$664$	−3.00000	−0.116423
$665$	0	0
$666$	6.00000	0.232495
$667$	−15.0000	−0.580802
$668$	−18.0000	−0.696441
$669$	−15.0000	−0.579934
$670$	0	0
$671$	−11.0000	−0.424650
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	3.00000	0.115385
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	5.00000	0.192024
$679$	0	0
$680$	0	0
$681$	2.00000	0.0766402
$682$	−5.00000	−0.191460
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	0	0
$687$	−5.00000	−0.190762
$688$	−4.00000	−0.152499
$689$	36.0000	1.37149
$690$	0	0
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	−7.00000	−0.266100
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	−3.00000	−0.113715
$697$	8.00000	0.303022
$698$	−1.00000	−0.0378506
$699$	−24.0000	−0.907763
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	4.00000	0.150970
$703$	−6.00000	−0.226294
$704$	1.00000	0.0376889
$705$	0	0
$706$	−28.0000	−1.05379
$707$	0	0
$708$	0	0
$709$	−43.0000	−1.61490	−0.807449	−	0.589937i	$-0.799153\pi$
$709$	−43.0000	−1.61490	−0.807449	+	0.589937i	$0.799153\pi$
$710$	0	0
$711$	17.0000	0.637550
$712$	7.00000	0.262336
$713$	25.0000	0.936257
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	24.0000	0.895672
$719$	−29.0000	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$719$	−29.0000	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$720$	0	0
$721$	0	0
$722$	1.00000	0.0372161
$723$	4.00000	0.148762
$724$	0	0
$725$	0	0
$726$	10.0000	0.371135
$727$	10.0000	0.370879	0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000	0.370879	0.185440	+	0.982656i	$0.440629\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	11.0000	0.406572
$733$	−41.0000	−1.51437	−0.757185	−	0.653201i	$-0.773426\pi$
$733$	−41.0000	−1.51437	−0.757185	+	0.653201i	$0.773426\pi$
$734$	18.0000	0.664392
$735$	0	0
$736$	−5.00000	−0.184302
$737$	1.00000	0.0368355
$738$	−2.00000	−0.0736210
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	5.00000	0.183309
$745$	0	0
$746$	−28.0000	−1.02515
$747$	−3.00000	−0.109764
$748$	−4.00000	−0.146254
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	−8.00000	−0.290573
$759$	5.00000	0.181489
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−5.00000	−0.181131
$763$	0	0
$764$	11.0000	0.397966
$765$	0	0
$766$	6.00000	0.216789
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	−10.0000	−0.359908
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−10.0000	−0.358979
$777$	0	0
$778$	26.0000	0.932145
$779$	2.00000	0.0716574
$780$	0	0
$781$	2.00000	0.0715656
$782$	20.0000	0.715199
$783$	−3.00000	−0.107211
$784$	−7.00000	−0.250000
$785$	0	0
$786$	−15.0000	−0.535032
$787$	−37.0000	−1.31891	−0.659454	−	0.751745i	$-0.729212\pi$
$787$	−37.0000	−1.31891	−0.659454	+	0.751745i	$0.729212\pi$
$788$	20.0000	0.712470
$789$	−11.0000	−0.391610
$790$	0	0
$791$	0	0
$792$	1.00000	0.0355335
$793$	44.0000	1.56249
$794$	3.00000	0.106466
$795$	0	0
$796$	6.00000	0.212664
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	7.00000	0.247333
$802$	−17.0000	−0.600291
$803$	3.00000	0.105868
$804$	−1.00000	−0.0352673
$805$	0	0
$806$	20.0000	0.704470
$807$	2.00000	0.0704033
$808$	−8.00000	−0.281439
$809$	−44.0000	−1.54696	−0.773479	−	0.633822i	$-0.781485\pi$
$809$	−44.0000	−1.54696	−0.773479	+	0.633822i	$0.781485\pi$
$810$	0	0
$811$	−35.0000	−1.22902	−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000	−1.22902	−0.614508	+	0.788911i	$0.710645\pi$
$812$	0	0
$813$	−2.00000	−0.0701431
$814$	6.00000	0.210300
$815$	0	0
$816$	4.00000	0.140028
$817$	4.00000	0.139942
$818$	6.00000	0.209785
$819$	0	0
$820$	0	0
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	−4.00000	−0.139516
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	−5.00000	−0.173762
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	−1.00000	−0.0346896
$832$	−4.00000	−0.138675
$833$	28.0000	0.970143
$834$	2.00000	0.0692543
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	5.00000	0.172825
$838$	−12.0000	−0.414533
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−34.0000	−1.17172
$843$	3.00000	0.103325
$844$	−13.0000	−0.447478
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−9.00000	−0.309061
$849$	−8.00000	−0.274559
$850$	0	0
$851$	−30.0000	−1.02839
$852$	−2.00000	−0.0685189
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	0	0
$856$	−10.0000	−0.341793
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	4.00000	0.136558
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	0	0
$861$	0	0
$862$	38.0000	1.29429
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	22.0000	0.747590
$867$	1.00000	0.0339618
$868$	0	0
$869$	17.0000	0.576686
$870$	0	0
$871$	−4.00000	−0.135535
$872$	6.00000	0.203186
$873$	−10.0000	−0.338449
$874$	5.00000	0.169128
$875$	0	0
$876$	−3.00000	−0.101361
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	21.0000	0.708716
$879$	11.0000	0.371021
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	−7.00000	−0.235702
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	13.0000	0.436744
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	−6.00000	−0.201347
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	15.0000	0.502237
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−20.0000	−0.667781
$898$	−27.0000	−0.901002
$899$	−15.0000	−0.500278
$900$	0	0
$901$	36.0000	1.19933
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	−5.00000	−0.166298
$905$	0	0
$906$	24.0000	0.797347
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−2.00000	−0.0663723
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	1.00000	0.0331133
$913$	−3.00000	−0.0992855
$914$	18.0000	0.595387
$915$	0	0
$916$	5.00000	0.165205
$917$	0	0
$918$	4.00000	0.132020
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	0	0
$921$	−25.0000	−0.823778
$922$	4.00000	0.131733
$923$	−8.00000	−0.263323
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	−13.0000	−0.426976
$928$	3.00000	0.0984798
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	24.0000	0.786146
$933$	−32.0000	−1.04763
$934$	11.0000	0.359931
$935$	0	0
$936$	−4.00000	−0.130744
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	21.0000	0.685309
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	14.0000	0.456145
$943$	10.0000	0.325645
$944$	0	0
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	−17.0000	−0.552134
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−13.0000	−0.421554
$952$	0	0
$953$	−3.00000	−0.0971795	−0.0485898	−	0.998819i	$-0.515473\pi$
$953$	−3.00000	−0.0971795	−0.0485898	+	0.998819i	$0.515473\pi$
$954$	−9.00000	−0.291386
$955$	0	0
$956$	0	0
$957$	−3.00000	−0.0969762
$958$	−37.0000	−1.19542
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−24.0000	−0.773791
$963$	−10.0000	−0.322245
$964$	−4.00000	−0.128831
$965$	0	0
$966$	0	0
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	−10.0000	−0.321412
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−11.0000	−0.352101
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	10.0000	0.319765
$979$	7.00000	0.223721
$980$	0	0
$981$	6.00000	0.191565
$982$	36.0000	1.14881
$983$	−34.0000	−1.08443	−0.542216	−	0.840239i	$-0.682414\pi$
$983$	−34.0000	−1.08443	−0.542216	+	0.840239i	$0.682414\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	4.00000	0.127257
$989$	20.0000	0.635963
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	−5.00000	−0.158750
$993$	−13.0000	−0.412543
$994$	0	0
$995$	0	0
$996$	3.00000	0.0950586
$997$	−37.0000	−1.17180	−0.585901	−	0.810383i	$-0.699259\pi$
$997$	−37.0000	−1.17180	−0.585901	+	0.810383i	$0.699259\pi$
$998$	−24.0000	−0.759707
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.a.u.1.1	yes	1
3.2	odd	2		8550.2.a.h.1.1		1
5.2	odd	4		2850.2.d.g.799.2		2
5.3	odd	4		2850.2.d.g.799.1		2
5.4	even	2		2850.2.a.l.1.1	✓	1
15.14	odd	2		8550.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.l.1.1	✓	1	5.4	even	2
2850.2.a.u.1.1	yes	1	1.1	even	1	trivial
2850.2.d.g.799.1		2	5.3	odd	4
2850.2.d.g.799.2		2	5.2	odd	4
8550.2.a.h.1.1		1	3.2	odd	2
8550.2.a.z.1.1		1	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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