LMFDB - Embedded newform 4050.2.a.bw.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4050,2,Mod(1,4050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4050.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4050 = 2 \cdot 3^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4050.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,0,0,-1,2,0,0,-3,0,2,-1,0,2,9,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 8$ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	4050.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.37228 q^{7} +1.00000 q^{8} +1.37228 q^{11} -4.74456 q^{13} +2.37228 q^{14} +1.00000 q^{16} +7.37228 q^{17} +3.37228 q^{19} +1.37228 q^{22} +4.37228 q^{23} -4.74456 q^{26} +2.37228 q^{28} -4.37228 q^{29} -6.74456 q^{31} +1.00000 q^{32} +7.37228 q^{34} +4.00000 q^{37} +3.37228 q^{38} -3.00000 q^{41} +11.3723 q^{43} +1.37228 q^{44} +4.37228 q^{46} +1.62772 q^{47} -1.37228 q^{49} -4.74456 q^{52} -11.4891 q^{53} +2.37228 q^{56} -4.37228 q^{58} -1.37228 q^{59} +9.11684 q^{61} -6.74456 q^{62} +1.00000 q^{64} +7.00000 q^{67} +7.37228 q^{68} -6.00000 q^{71} +14.1168 q^{73} +4.00000 q^{74} +3.37228 q^{76} +3.25544 q^{77} +2.00000 q^{79} -3.00000 q^{82} -1.62772 q^{83} +11.3723 q^{86} +1.37228 q^{88} -1.11684 q^{89} -11.2554 q^{91} +4.37228 q^{92} +1.62772 q^{94} +2.62772 q^{97} -1.37228 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} + 2 q^{4} - q^{7} + 2 q^{8} - 3 q^{11} + 2 q^{13} - q^{14} + 2 q^{16} + 9 q^{17} + q^{19} - 3 q^{22} + 3 q^{23} + 2 q^{26} - q^{28} - 3 q^{29} - 2 q^{31} + 2 q^{32} + 9 q^{34} + 8 q^{37}+ \cdots + 3 q^{98}+O(q^{100})$ 2 * q + 2 * q^2 + 2 * q^4 - q^7 + 2 * q^8 - 3 * q^11 + 2 * q^13 - q^14 + 2 * q^16 + 9 * q^17 + q^19 - 3 * q^22 + 3 * q^23 + 2 * q^26 - q^28 - 3 * q^29 - 2 * q^31 + 2 * q^32 + 9 * q^34 + 8 * q^37 + q^38 - 6 * q^41 + 17 * q^43 - 3 * q^44 + 3 * q^46 + 9 * q^47 + 3 * q^49 + 2 * q^52 - q^56 - 3 * q^58 + 3 * q^59 + q^61 - 2 * q^62 + 2 * q^64 + 14 * q^67 + 9 * q^68 - 12 * q^71 + 11 * q^73 + 8 * q^74 + q^76 + 18 * q^77 + 4 * q^79 - 6 * q^82 - 9 * q^83 + 17 * q^86 - 3 * q^88 + 15 * q^89 - 34 * q^91 + 3 * q^92 + 9 * q^94 + 11 * q^97 + 3 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	1.37228	0.413758	0.206879	−	0.978366i	$-0.433669\pi$
$11$	1.37228	0.413758	0.206879	+	0.978366i	$0.433669\pi$
$12$	0	0
$13$	−4.74456	−1.31590	−0.657952	−	0.753059i	$-0.728577\pi$
$13$	−4.74456	−1.31590	−0.657952	+	0.753059i	$0.728577\pi$
$14$	2.37228	0.634019
$15$	0	0
$16$	1.00000	0.250000
$17$	7.37228	1.78804	0.894020	−	0.448026i	$-0.147873\pi$
$17$	7.37228	1.78804	0.894020	+	0.448026i	$0.147873\pi$
$18$	0	0
$19$	3.37228	0.773654	0.386827	−	0.922152i	$-0.373571\pi$
$19$	3.37228	0.773654	0.386827	+	0.922152i	$0.373571\pi$
$20$	0	0
$21$	0	0
$22$	1.37228	0.292571
$23$	4.37228	0.911684	0.455842	−	0.890061i	$-0.349338\pi$
$23$	4.37228	0.911684	0.455842	+	0.890061i	$0.349338\pi$
$24$	0	0
$25$	0	0
$26$	−4.74456	−0.930485
$27$	0	0
$28$	2.37228	0.448319
$29$	−4.37228	−0.811912	−0.405956	−	0.913893i	$-0.633061\pi$
$29$	−4.37228	−0.811912	−0.405956	+	0.913893i	$0.633061\pi$
$30$	0	0
$31$	−6.74456	−1.21136	−0.605680	−	0.795709i	$-0.707099\pi$
$31$	−6.74456	−1.21136	−0.605680	+	0.795709i	$0.707099\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.37228	1.26434
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	3.37228	0.547056
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	11.3723	1.73426	0.867128	−	0.498085i	$-0.165963\pi$
$43$	11.3723	1.73426	0.867128	+	0.498085i	$0.165963\pi$
$44$	1.37228	0.206879
$45$	0	0
$46$	4.37228	0.644658
$47$	1.62772	0.237427	0.118714	−	0.992929i	$-0.462123\pi$
$47$	1.62772	0.237427	0.118714	+	0.992929i	$0.462123\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	0	0
$52$	−4.74456	−0.657952
$53$	−11.4891	−1.57815	−0.789076	−	0.614295i	$-0.789440\pi$
$53$	−11.4891	−1.57815	−0.789076	+	0.614295i	$0.789440\pi$
$54$	0	0
$55$	0	0
$56$	2.37228	0.317009
$57$	0	0
$58$	−4.37228	−0.574109
$59$	−1.37228	−0.178656	−0.0893279	−	0.996002i	$-0.528472\pi$
$59$	−1.37228	−0.178656	−0.0893279	+	0.996002i	$0.528472\pi$
$60$	0	0
$61$	9.11684	1.16729	0.583646	−	0.812008i	$-0.301626\pi$
$61$	9.11684	1.16729	0.583646	+	0.812008i	$0.301626\pi$
$62$	−6.74456	−0.856560
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	7.37228	0.894020
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	14.1168	1.65225	0.826126	−	0.563486i	$-0.190540\pi$
$73$	14.1168	1.65225	0.826126	+	0.563486i	$0.190540\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	3.37228	0.386827
$77$	3.25544	0.370992
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	0	0
$82$	−3.00000	−0.331295
$83$	−1.62772	−0.178665	−0.0893327	−	0.996002i	$-0.528473\pi$
$83$	−1.62772	−0.178665	−0.0893327	+	0.996002i	$0.528473\pi$
$84$	0	0
$85$	0	0
$86$	11.3723	1.22630
$87$	0	0
$88$	1.37228	0.146286
$89$	−1.11684	−0.118385	−0.0591926	−	0.998247i	$-0.518853\pi$
$89$	−1.11684	−0.118385	−0.0591926	+	0.998247i	$0.518853\pi$
$90$	0	0
$91$	−11.2554	−1.17989
$92$	4.37228	0.455842
$93$	0	0
$94$	1.62772	0.167886
$95$	0	0
$96$	0	0
$97$	2.62772	0.266804	0.133402	−	0.991062i	$-0.457410\pi$
$97$	2.62772	0.266804	0.133402	+	0.991062i	$0.457410\pi$
$98$	−1.37228	−0.138621
$99$	0	0
$100$	0	0
$101$	−8.74456	−0.870117	−0.435058	−	0.900402i	$-0.643272\pi$
$101$	−8.74456	−0.870117	−0.435058	+	0.900402i	$0.643272\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−4.74456	−0.465243
$105$	0	0
$106$	−11.4891	−1.11592
$107$	14.4891	1.40072	0.700358	−	0.713791i	$-0.253024\pi$
$107$	14.4891	1.40072	0.700358	+	0.713791i	$0.253024\pi$
$108$	0	0
$109$	9.62772	0.922168	0.461084	−	0.887356i	$-0.347461\pi$
$109$	9.62772	0.922168	0.461084	+	0.887356i	$0.347461\pi$
$110$	0	0
$111$	0	0
$112$	2.37228	0.224160
$113$	−14.7446	−1.38705	−0.693526	−	0.720432i	$-0.743944\pi$
$113$	−14.7446	−1.38705	−0.693526	+	0.720432i	$0.743944\pi$
$114$	0	0
$115$	0	0
$116$	−4.37228	−0.405956
$117$	0	0
$118$	−1.37228	−0.126329
$119$	17.4891	1.60323
$120$	0	0
$121$	−9.11684	−0.828804
$122$	9.11684	0.825400
$123$	0	0
$124$	−6.74456	−0.605680
$125$	0	0
$126$	0	0
$127$	−9.11684	−0.808989	−0.404495	−	0.914540i	$-0.632553\pi$
$127$	−9.11684	−0.808989	−0.404495	+	0.914540i	$0.632553\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	8.74456	0.764016	0.382008	−	0.924159i	$-0.375233\pi$
$131$	8.74456	0.764016	0.382008	+	0.924159i	$0.375233\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	7.00000	0.604708
$135$	0	0
$136$	7.37228	0.632168
$137$	1.88316	0.160889	0.0804444	−	0.996759i	$-0.474366\pi$
$137$	1.88316	0.160889	0.0804444	+	0.996759i	$0.474366\pi$
$138$	0	0
$139$	18.1168	1.53665	0.768325	−	0.640060i	$-0.221090\pi$
$139$	18.1168	1.53665	0.768325	+	0.640060i	$0.221090\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−6.51087	−0.544467
$144$	0	0
$145$	0	0
$146$	14.1168	1.16832
$147$	0	0
$148$	4.00000	0.328798
$149$	19.1168	1.56611	0.783056	−	0.621951i	$-0.213660\pi$
$149$	19.1168	1.56611	0.783056	+	0.621951i	$0.213660\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	3.37228	0.273528
$153$	0	0
$154$	3.25544	0.262331
$155$	0	0
$156$	0	0
$157$	−4.74456	−0.378657	−0.189329	−	0.981914i	$-0.560631\pi$
$157$	−4.74456	−0.378657	−0.189329	+	0.981914i	$0.560631\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	0	0
$161$	10.3723	0.817450
$162$	0	0
$163$	−1.48913	−0.116637	−0.0583186	−	0.998298i	$-0.518574\pi$
$163$	−1.48913	−0.116637	−0.0583186	+	0.998298i	$0.518574\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−1.62772	−0.126335
$167$	−7.62772	−0.590251	−0.295125	−	0.955459i	$-0.595361\pi$
$167$	−7.62772	−0.590251	−0.295125	+	0.955459i	$0.595361\pi$
$168$	0	0
$169$	9.51087	0.731606
$170$	0	0
$171$	0	0
$172$	11.3723	0.867128
$173$	−9.25544	−0.703678	−0.351839	−	0.936061i	$-0.614444\pi$
$173$	−9.25544	−0.703678	−0.351839	+	0.936061i	$0.614444\pi$
$174$	0	0
$175$	0	0
$176$	1.37228	0.103440
$177$	0	0
$178$	−1.11684	−0.0837110
$179$	−3.25544	−0.243323	−0.121661	−	0.992572i	$-0.538822\pi$
$179$	−3.25544	−0.243323	−0.121661	+	0.992572i	$0.538822\pi$
$180$	0	0
$181$	−7.86141	−0.584334	−0.292167	−	0.956367i	$-0.594376\pi$
$181$	−7.86141	−0.584334	−0.292167	+	0.956367i	$0.594376\pi$
$182$	−11.2554	−0.834309
$183$	0	0
$184$	4.37228	0.322329
$185$	0	0
$186$	0	0
$187$	10.1168	0.739817
$188$	1.62772	0.118714
$189$	0	0
$190$	0	0
$191$	−5.48913	−0.397179	−0.198590	−	0.980083i	$-0.563636\pi$
$191$	−5.48913	−0.397179	−0.198590	+	0.980083i	$0.563636\pi$
$192$	0	0
$193$	−3.88316	−0.279516	−0.139758	−	0.990186i	$-0.544632\pi$
$193$	−3.88316	−0.279516	−0.139758	+	0.990186i	$0.544632\pi$
$194$	2.62772	0.188659
$195$	0	0
$196$	−1.37228	−0.0980201
$197$	−17.4891	−1.24605	−0.623024	−	0.782202i	$-0.714096\pi$
$197$	−17.4891	−1.24605	−0.623024	+	0.782202i	$0.714096\pi$
$198$	0	0
$199$	−9.48913	−0.672666	−0.336333	−	0.941743i	$-0.609187\pi$
$199$	−9.48913	−0.672666	−0.336333	+	0.941743i	$0.609187\pi$
$200$	0	0
$201$	0	0
$202$	−8.74456	−0.615265
$203$	−10.3723	−0.727991
$204$	0	0
$205$	0	0
$206$	16.0000	1.11477
$207$	0	0
$208$	−4.74456	−0.328976
$209$	4.62772	0.320106
$210$	0	0
$211$	−7.25544	−0.499485	−0.249742	−	0.968312i	$-0.580346\pi$
$211$	−7.25544	−0.499485	−0.249742	+	0.968312i	$0.580346\pi$
$212$	−11.4891	−0.789076
$213$	0	0
$214$	14.4891	0.990456
$215$	0	0
$216$	0	0
$217$	−16.0000	−1.08615
$218$	9.62772	0.652071
$219$	0	0
$220$	0	0
$221$	−34.9783	−2.35289
$222$	0	0
$223$	−12.3723	−0.828509	−0.414255	−	0.910161i	$-0.635958\pi$
$223$	−12.3723	−0.828509	−0.414255	+	0.910161i	$0.635958\pi$
$224$	2.37228	0.158505
$225$	0	0
$226$	−14.7446	−0.980794
$227$	−1.88316	−0.124989	−0.0624947	−	0.998045i	$-0.519906\pi$
$227$	−1.88316	−0.124989	−0.0624947	+	0.998045i	$0.519906\pi$
$228$	0	0
$229$	18.3723	1.21407	0.607037	−	0.794673i	$-0.292358\pi$
$229$	18.3723	1.21407	0.607037	+	0.794673i	$0.292358\pi$
$230$	0	0
$231$	0	0
$232$	−4.37228	−0.287054
$233$	10.1168	0.662776	0.331388	−	0.943494i	$-0.392483\pi$
$233$	10.1168	0.662776	0.331388	+	0.943494i	$0.392483\pi$
$234$	0	0
$235$	0	0
$236$	−1.37228	−0.0893279
$237$	0	0
$238$	17.4891	1.13365
$239$	−14.7446	−0.953746	−0.476873	−	0.878972i	$-0.658230\pi$
$239$	−14.7446	−0.953746	−0.476873	+	0.878972i	$0.658230\pi$
$240$	0	0
$241$	10.4891	0.675664	0.337832	−	0.941206i	$-0.390306\pi$
$241$	10.4891	0.675664	0.337832	+	0.941206i	$0.390306\pi$
$242$	−9.11684	−0.586053
$243$	0	0
$244$	9.11684	0.583646
$245$	0	0
$246$	0	0
$247$	−16.0000	−1.01806
$248$	−6.74456	−0.428280
$249$	0	0
$250$	0	0
$251$	15.6060	0.985040	0.492520	−	0.870301i	$-0.336076\pi$
$251$	15.6060	0.985040	0.492520	+	0.870301i	$0.336076\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−9.11684	−0.572042
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.37228	0.0856006	0.0428003	−	0.999084i	$-0.486372\pi$
$257$	1.37228	0.0856006	0.0428003	+	0.999084i	$0.486372\pi$
$258$	0	0
$259$	9.48913	0.589626
$260$	0	0
$261$	0	0
$262$	8.74456	0.540241
$263$	−5.48913	−0.338474	−0.169237	−	0.985575i	$-0.554130\pi$
$263$	−5.48913	−0.338474	−0.169237	+	0.985575i	$0.554130\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	7.00000	0.427593
$269$	4.37228	0.266583	0.133291	−	0.991077i	$-0.457445\pi$
$269$	4.37228	0.266583	0.133291	+	0.991077i	$0.457445\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	7.37228	0.447010
$273$	0	0
$274$	1.88316	0.113766
$275$	0	0
$276$	0	0
$277$	−5.25544	−0.315769	−0.157884	−	0.987458i	$-0.550467\pi$
$277$	−5.25544	−0.315769	−0.157884	+	0.987458i	$0.550467\pi$
$278$	18.1168	1.08658
$279$	0	0
$280$	0	0
$281$	4.37228	0.260828	0.130414	−	0.991460i	$-0.458369\pi$
$281$	4.37228	0.260828	0.130414	+	0.991460i	$0.458369\pi$
$282$	0	0
$283$	31.8614	1.89396	0.946982	−	0.321287i	$-0.104115\pi$
$283$	31.8614	1.89396	0.946982	+	0.321287i	$0.104115\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−6.51087	−0.384996
$287$	−7.11684	−0.420094
$288$	0	0
$289$	37.3505	2.19709
$290$	0	0
$291$	0	0
$292$	14.1168	0.826126
$293$	8.23369	0.481017	0.240509	−	0.970647i	$-0.422686\pi$
$293$	8.23369	0.481017	0.240509	+	0.970647i	$0.422686\pi$
$294$	0	0
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	19.1168	1.10741
$299$	−20.7446	−1.19969
$300$	0	0
$301$	26.9783	1.55500
$302$	−10.0000	−0.575435
$303$	0	0
$304$	3.37228	0.193414
$305$	0	0
$306$	0	0
$307$	33.2337	1.89675	0.948373	−	0.317156i	$-0.102728\pi$
$307$	33.2337	1.89675	0.948373	+	0.317156i	$0.102728\pi$
$308$	3.25544	0.185496
$309$	0	0
$310$	0	0
$311$	−9.25544	−0.524828	−0.262414	−	0.964955i	$-0.584519\pi$
$311$	−9.25544	−0.524828	−0.262414	+	0.964955i	$0.584519\pi$
$312$	0	0
$313$	−9.37228	−0.529753	−0.264876	−	0.964282i	$-0.585331\pi$
$313$	−9.37228	−0.529753	−0.264876	+	0.964282i	$0.585331\pi$
$314$	−4.74456	−0.267751
$315$	0	0
$316$	2.00000	0.112509
$317$	8.74456	0.491144	0.245572	−	0.969378i	$-0.421024\pi$
$317$	8.74456	0.491144	0.245572	+	0.969378i	$0.421024\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	0	0
$322$	10.3723	0.578025
$323$	24.8614	1.38333
$324$	0	0
$325$	0	0
$326$	−1.48913	−0.0824750
$327$	0	0
$328$	−3.00000	−0.165647
$329$	3.86141	0.212886
$330$	0	0
$331$	−18.2337	−1.00221	−0.501107	−	0.865385i	$-0.667074\pi$
$331$	−18.2337	−1.00221	−0.501107	+	0.865385i	$0.667074\pi$
$332$	−1.62772	−0.0893327
$333$	0	0
$334$	−7.62772	−0.417370
$335$	0	0
$336$	0	0
$337$	−3.37228	−0.183700	−0.0918499	−	0.995773i	$-0.529278\pi$
$337$	−3.37228	−0.183700	−0.0918499	+	0.995773i	$0.529278\pi$
$338$	9.51087	0.517323
$339$	0	0
$340$	0	0
$341$	−9.25544	−0.501210
$342$	0	0
$343$	−19.8614	−1.07242
$344$	11.3723	0.613152
$345$	0	0
$346$	−9.25544	−0.497575
$347$	22.1168	1.18729	0.593647	−	0.804725i	$-0.297687\pi$
$347$	22.1168	1.18729	0.593647	+	0.804725i	$0.297687\pi$
$348$	0	0
$349$	0.883156	0.0472743	0.0236371	−	0.999721i	$-0.492475\pi$
$349$	0.883156	0.0472743	0.0236371	+	0.999721i	$0.492475\pi$
$350$	0	0
$351$	0	0
$352$	1.37228	0.0731428
$353$	−30.3505	−1.61540	−0.807698	−	0.589597i	$-0.799287\pi$
$353$	−30.3505	−1.61540	−0.807698	+	0.589597i	$0.799287\pi$
$354$	0	0
$355$	0	0
$356$	−1.11684	−0.0591926
$357$	0	0
$358$	−3.25544	−0.172055
$359$	5.48913	0.289705	0.144852	−	0.989453i	$-0.453729\pi$
$359$	5.48913	0.289705	0.144852	+	0.989453i	$0.453729\pi$
$360$	0	0
$361$	−7.62772	−0.401459
$362$	−7.86141	−0.413186
$363$	0	0
$364$	−11.2554	−0.589945
$365$	0	0
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	4.37228	0.227921
$369$	0	0
$370$	0	0
$371$	−27.2554	−1.41503
$372$	0	0
$373$	−7.48913	−0.387772	−0.193886	−	0.981024i	$-0.562109\pi$
$373$	−7.48913	−0.387772	−0.193886	+	0.981024i	$0.562109\pi$
$374$	10.1168	0.523130
$375$	0	0
$376$	1.62772	0.0839432
$377$	20.7446	1.06840
$378$	0	0
$379$	−10.8614	−0.557913	−0.278956	−	0.960304i	$-0.589989\pi$
$379$	−10.8614	−0.557913	−0.278956	+	0.960304i	$0.589989\pi$
$380$	0	0
$381$	0	0
$382$	−5.48913	−0.280848
$383$	−22.9783	−1.17413	−0.587067	−	0.809538i	$-0.699717\pi$
$383$	−22.9783	−1.17413	−0.587067	+	0.809538i	$0.699717\pi$
$384$	0	0
$385$	0	0
$386$	−3.88316	−0.197647
$387$	0	0
$388$	2.62772	0.133402
$389$	−10.3723	−0.525896	−0.262948	−	0.964810i	$-0.584695\pi$
$389$	−10.3723	−0.525896	−0.262948	+	0.964810i	$0.584695\pi$
$390$	0	0
$391$	32.2337	1.63013
$392$	−1.37228	−0.0693107
$393$	0	0
$394$	−17.4891	−0.881089
$395$	0	0
$396$	0	0
$397$	−11.2554	−0.564894	−0.282447	−	0.959283i	$-0.591146\pi$
$397$	−11.2554	−0.564894	−0.282447	+	0.959283i	$0.591146\pi$
$398$	−9.48913	−0.475647
$399$	0	0
$400$	0	0
$401$	16.1168	0.804837	0.402418	−	0.915456i	$-0.368170\pi$
$401$	16.1168	0.804837	0.402418	+	0.915456i	$0.368170\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	−8.74456	−0.435058
$405$	0	0
$406$	−10.3723	−0.514768
$407$	5.48913	0.272086
$408$	0	0
$409$	−10.8614	−0.537062	−0.268531	−	0.963271i	$-0.586538\pi$
$409$	−10.8614	−0.537062	−0.268531	+	0.963271i	$0.586538\pi$
$410$	0	0
$411$	0	0
$412$	16.0000	0.788263
$413$	−3.25544	−0.160190
$414$	0	0
$415$	0	0
$416$	−4.74456	−0.232621
$417$	0	0
$418$	4.62772	0.226349
$419$	−31.7228	−1.54976	−0.774880	−	0.632108i	$-0.782190\pi$
$419$	−31.7228	−1.54976	−0.774880	+	0.632108i	$0.782190\pi$
$420$	0	0
$421$	−38.4674	−1.87479	−0.937393	−	0.348274i	$-0.886768\pi$
$421$	−38.4674	−1.87479	−0.937393	+	0.348274i	$0.886768\pi$
$422$	−7.25544	−0.353189
$423$	0	0
$424$	−11.4891	−0.557961
$425$	0	0
$426$	0	0
$427$	21.6277	1.04664
$428$	14.4891	0.700358
$429$	0	0
$430$	0	0
$431$	−26.2337	−1.26363	−0.631816	−	0.775118i	$-0.717690\pi$
$431$	−26.2337	−1.26363	−0.631816	+	0.775118i	$0.717690\pi$
$432$	0	0
$433$	−0.627719	−0.0301662	−0.0150831	−	0.999886i	$-0.504801\pi$
$433$	−0.627719	−0.0301662	−0.0150831	+	0.999886i	$0.504801\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	9.62772	0.461084
$437$	14.7446	0.705328
$438$	0	0
$439$	16.2337	0.774792	0.387396	−	0.921913i	$-0.373375\pi$
$439$	16.2337	0.774792	0.387396	+	0.921913i	$0.373375\pi$
$440$	0	0
$441$	0	0
$442$	−34.9783	−1.66375
$443$	26.4891	1.25854	0.629268	−	0.777188i	$-0.283355\pi$
$443$	26.4891	1.25854	0.629268	+	0.777188i	$0.283355\pi$
$444$	0	0
$445$	0	0
$446$	−12.3723	−0.585845
$447$	0	0
$448$	2.37228	0.112080
$449$	−18.8614	−0.890125	−0.445062	−	0.895500i	$-0.646819\pi$
$449$	−18.8614	−0.890125	−0.445062	+	0.895500i	$0.646819\pi$
$450$	0	0
$451$	−4.11684	−0.193855
$452$	−14.7446	−0.693526
$453$	0	0
$454$	−1.88316	−0.0883809
$455$	0	0
$456$	0	0
$457$	−30.1168	−1.40881	−0.704403	−	0.709800i	$-0.748785\pi$
$457$	−30.1168	−1.40881	−0.704403	+	0.709800i	$0.748785\pi$
$458$	18.3723	0.858480
$459$	0	0
$460$	0	0
$461$	19.1168	0.890360	0.445180	−	0.895441i	$-0.353140\pi$
$461$	19.1168	0.890360	0.445180	+	0.895441i	$0.353140\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	−4.37228	−0.202978
$465$	0	0
$466$	10.1168	0.468654
$467$	−25.8832	−1.19773	−0.598865	−	0.800850i	$-0.704381\pi$
$467$	−25.8832	−1.19773	−0.598865	+	0.800850i	$0.704381\pi$
$468$	0	0
$469$	16.6060	0.766792
$470$	0	0
$471$	0	0
$472$	−1.37228	−0.0631644
$473$	15.6060	0.717563
$474$	0	0
$475$	0	0
$476$	17.4891	0.801613
$477$	0	0
$478$	−14.7446	−0.674401
$479$	−23.4891	−1.07325	−0.536623	−	0.843822i	$-0.680300\pi$
$479$	−23.4891	−1.07325	−0.536623	+	0.843822i	$0.680300\pi$
$480$	0	0
$481$	−18.9783	−0.865334
$482$	10.4891	0.477767
$483$	0	0
$484$	−9.11684	−0.414402
$485$	0	0
$486$	0	0
$487$	1.25544	0.0568893	0.0284446	−	0.999595i	$-0.490945\pi$
$487$	1.25544	0.0568893	0.0284446	+	0.999595i	$0.490945\pi$
$488$	9.11684	0.412700
$489$	0	0
$490$	0	0
$491$	3.60597	0.162735	0.0813676	−	0.996684i	$-0.474071\pi$
$491$	3.60597	0.162735	0.0813676	+	0.996684i	$0.474071\pi$
$492$	0	0
$493$	−32.2337	−1.45173
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−6.74456	−0.302840
$497$	−14.2337	−0.638468
$498$	0	0
$499$	−2.11684	−0.0947630	−0.0473815	−	0.998877i	$-0.515088\pi$
$499$	−2.11684	−0.0947630	−0.0473815	+	0.998877i	$0.515088\pi$
$500$	0	0
$501$	0	0
$502$	15.6060	0.696528
$503$	−21.8614	−0.974752	−0.487376	−	0.873192i	$-0.662046\pi$
$503$	−21.8614	−0.974752	−0.487376	+	0.873192i	$0.662046\pi$
$504$	0	0
$505$	0	0
$506$	6.00000	0.266733
$507$	0	0
$508$	−9.11684	−0.404495
$509$	9.35053	0.414455	0.207228	−	0.978293i	$-0.433556\pi$
$509$	9.35053	0.414455	0.207228	+	0.978293i	$0.433556\pi$
$510$	0	0
$511$	33.4891	1.48147
$512$	1.00000	0.0441942
$513$	0	0
$514$	1.37228	0.0605287
$515$	0	0
$516$	0	0
$517$	2.23369	0.0982375
$518$	9.48913	0.416928
$519$	0	0
$520$	0	0
$521$	−41.2337	−1.80648	−0.903240	−	0.429135i	$-0.858818\pi$
$521$	−41.2337	−1.80648	−0.903240	+	0.429135i	$0.858818\pi$
$522$	0	0
$523$	11.1168	0.486106	0.243053	−	0.970013i	$-0.421851\pi$
$523$	11.1168	0.486106	0.243053	+	0.970013i	$0.421851\pi$
$524$	8.74456	0.382008
$525$	0	0
$526$	−5.48913	−0.239337
$527$	−49.7228	−2.16596
$528$	0	0
$529$	−3.88316	−0.168833
$530$	0	0
$531$	0	0
$532$	8.00000	0.346844
$533$	14.2337	0.616529
$534$	0	0
$535$	0	0
$536$	7.00000	0.302354
$537$	0	0
$538$	4.37228	0.188502
$539$	−1.88316	−0.0811133
$540$	0	0
$541$	21.6277	0.929848	0.464924	−	0.885351i	$-0.346082\pi$
$541$	21.6277	0.929848	0.464924	+	0.885351i	$0.346082\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	7.37228	0.316084
$545$	0	0
$546$	0	0
$547$	−39.4674	−1.68750	−0.843752	−	0.536734i	$-0.819658\pi$
$547$	−39.4674	−1.68750	−0.843752	+	0.536734i	$0.819658\pi$
$548$	1.88316	0.0804444
$549$	0	0
$550$	0	0
$551$	−14.7446	−0.628139
$552$	0	0
$553$	4.74456	0.201759
$554$	−5.25544	−0.223282
$555$	0	0
$556$	18.1168	0.768325
$557$	9.76631	0.413812	0.206906	−	0.978361i	$-0.433661\pi$
$557$	9.76631	0.413812	0.206906	+	0.978361i	$0.433661\pi$
$558$	0	0
$559$	−53.9565	−2.28212
$560$	0	0
$561$	0	0
$562$	4.37228	0.184434
$563$	−16.7228	−0.704783	−0.352391	−	0.935853i	$-0.614631\pi$
$563$	−16.7228	−0.704783	−0.352391	+	0.935853i	$0.614631\pi$
$564$	0	0
$565$	0	0
$566$	31.8614	1.33923
$567$	0	0
$568$	−6.00000	−0.251754
$569$	9.60597	0.402703	0.201352	−	0.979519i	$-0.435467\pi$
$569$	9.60597	0.402703	0.201352	+	0.979519i	$0.435467\pi$
$570$	0	0
$571$	−31.6060	−1.32267	−0.661334	−	0.750091i	$-0.730010\pi$
$571$	−31.6060	−1.32267	−0.661334	+	0.750091i	$0.730010\pi$
$572$	−6.51087	−0.272233
$573$	0	0
$574$	−7.11684	−0.297051
$575$	0	0
$576$	0	0
$577$	23.8832	0.994269	0.497134	−	0.867674i	$-0.334386\pi$
$577$	23.8832	0.994269	0.497134	+	0.867674i	$0.334386\pi$
$578$	37.3505	1.55358
$579$	0	0
$580$	0	0
$581$	−3.86141	−0.160198
$582$	0	0
$583$	−15.7663	−0.652974
$584$	14.1168	0.584159
$585$	0	0
$586$	8.23369	0.340131
$587$	−27.0000	−1.11441	−0.557205	−	0.830375i	$-0.688126\pi$
$587$	−27.0000	−1.11441	−0.557205	+	0.830375i	$0.688126\pi$
$588$	0	0
$589$	−22.7446	−0.937173
$590$	0	0
$591$	0	0
$592$	4.00000	0.164399
$593$	−37.7228	−1.54909	−0.774545	−	0.632519i	$-0.782021\pi$
$593$	−37.7228	−1.54909	−0.774545	+	0.632519i	$0.782021\pi$
$594$	0	0
$595$	0	0
$596$	19.1168	0.783056
$597$	0	0
$598$	−20.7446	−0.848308
$599$	−38.2337	−1.56219	−0.781093	−	0.624415i	$-0.785338\pi$
$599$	−38.2337	−1.56219	−0.781093	+	0.624415i	$0.785338\pi$
$600$	0	0
$601$	26.8614	1.09570	0.547850	−	0.836577i	$-0.315447\pi$
$601$	26.8614	1.09570	0.547850	+	0.836577i	$0.315447\pi$
$602$	26.9783	1.09955
$603$	0	0
$604$	−10.0000	−0.406894
$605$	0	0
$606$	0	0
$607$	−0.883156	−0.0358462	−0.0179231	−	0.999839i	$-0.505705\pi$
$607$	−0.883156	−0.0358462	−0.0179231	+	0.999839i	$0.505705\pi$
$608$	3.37228	0.136764
$609$	0	0
$610$	0	0
$611$	−7.72281	−0.312432
$612$	0	0
$613$	0.233688	0.00943857	0.00471928	−	0.999989i	$-0.498498\pi$
$613$	0.233688	0.00943857	0.00471928	+	0.999989i	$0.498498\pi$
$614$	33.2337	1.34120
$615$	0	0
$616$	3.25544	0.131165
$617$	22.1168	0.890391	0.445195	−	0.895433i	$-0.353134\pi$
$617$	22.1168	0.890391	0.445195	+	0.895433i	$0.353134\pi$
$618$	0	0
$619$	−38.1168	−1.53205	−0.766023	−	0.642814i	$-0.777767\pi$
$619$	−38.1168	−1.53205	−0.766023	+	0.642814i	$0.777767\pi$
$620$	0	0
$621$	0	0
$622$	−9.25544	−0.371109
$623$	−2.64947	−0.106149
$624$	0	0
$625$	0	0
$626$	−9.37228	−0.374592
$627$	0	0
$628$	−4.74456	−0.189329
$629$	29.4891	1.17581
$630$	0	0
$631$	33.7228	1.34248	0.671242	−	0.741238i	$-0.265761\pi$
$631$	33.7228	1.34248	0.671242	+	0.741238i	$0.265761\pi$
$632$	2.00000	0.0795557
$633$	0	0
$634$	8.74456	0.347291
$635$	0	0
$636$	0	0
$637$	6.51087	0.257970
$638$	−6.00000	−0.237542
$639$	0	0
$640$	0	0
$641$	−39.0000	−1.54041	−0.770204	−	0.637798i	$-0.779845\pi$
$641$	−39.0000	−1.54041	−0.770204	+	0.637798i	$0.779845\pi$
$642$	0	0
$643$	−11.0000	−0.433798	−0.216899	−	0.976194i	$-0.569594\pi$
$643$	−11.0000	−0.433798	−0.216899	+	0.976194i	$0.569594\pi$
$644$	10.3723	0.408725
$645$	0	0
$646$	24.8614	0.978159
$647$	−24.0951	−0.947276	−0.473638	−	0.880720i	$-0.657059\pi$
$647$	−24.0951	−0.947276	−0.473638	+	0.880720i	$0.657059\pi$
$648$	0	0
$649$	−1.88316	−0.0739203
$650$	0	0
$651$	0	0
$652$	−1.48913	−0.0583186
$653$	37.7228	1.47621	0.738104	−	0.674687i	$-0.235721\pi$
$653$	37.7228	1.47621	0.738104	+	0.674687i	$0.235721\pi$
$654$	0	0
$655$	0	0
$656$	−3.00000	−0.117130
$657$	0	0
$658$	3.86141	0.150533
$659$	−5.48913	−0.213826	−0.106913	−	0.994268i	$-0.534097\pi$
$659$	−5.48913	−0.213826	−0.106913	+	0.994268i	$0.534097\pi$
$660$	0	0
$661$	22.2337	0.864790	0.432395	−	0.901684i	$-0.357669\pi$
$661$	22.2337	0.864790	0.432395	+	0.901684i	$0.357669\pi$
$662$	−18.2337	−0.708672
$663$	0	0
$664$	−1.62772	−0.0631677
$665$	0	0
$666$	0	0
$667$	−19.1168	−0.740207
$668$	−7.62772	−0.295125
$669$	0	0
$670$	0	0
$671$	12.5109	0.482977
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	−3.37228	−0.129895
$675$	0	0
$676$	9.51087	0.365803
$677$	43.7228	1.68040	0.840202	−	0.542273i	$-0.182436\pi$
$677$	43.7228	1.68040	0.840202	+	0.542273i	$0.182436\pi$
$678$	0	0
$679$	6.23369	0.239227
$680$	0	0
$681$	0	0
$682$	−9.25544	−0.354409
$683$	−33.0951	−1.26635	−0.633174	−	0.774009i	$-0.718248\pi$
$683$	−33.0951	−1.26635	−0.633174	+	0.774009i	$0.718248\pi$
$684$	0	0
$685$	0	0
$686$	−19.8614	−0.758312
$687$	0	0
$688$	11.3723	0.433564
$689$	54.5109	2.07670
$690$	0	0
$691$	−1.76631	−0.0671937	−0.0335968	−	0.999435i	$-0.510696\pi$
$691$	−1.76631	−0.0671937	−0.0335968	+	0.999435i	$0.510696\pi$
$692$	−9.25544	−0.351839
$693$	0	0
$694$	22.1168	0.839544
$695$	0	0
$696$	0	0
$697$	−22.1168	−0.837735
$698$	0.883156	0.0334279
$699$	0	0
$700$	0	0
$701$	14.1386	0.534007	0.267004	−	0.963696i	$-0.413966\pi$
$701$	14.1386	0.534007	0.267004	+	0.963696i	$0.413966\pi$
$702$	0	0
$703$	13.4891	0.508752
$704$	1.37228	0.0517198
$705$	0	0
$706$	−30.3505	−1.14226
$707$	−20.7446	−0.780180
$708$	0	0
$709$	−25.8614	−0.971246	−0.485623	−	0.874168i	$-0.661407\pi$
$709$	−25.8614	−0.971246	−0.485623	+	0.874168i	$0.661407\pi$
$710$	0	0
$711$	0	0
$712$	−1.11684	−0.0418555
$713$	−29.4891	−1.10438
$714$	0	0
$715$	0	0
$716$	−3.25544	−0.121661
$717$	0	0
$718$	5.48913	0.204852
$719$	−38.2337	−1.42588	−0.712938	−	0.701227i	$-0.752636\pi$
$719$	−38.2337	−1.42588	−0.712938	+	0.701227i	$0.752636\pi$
$720$	0	0
$721$	37.9565	1.41357
$722$	−7.62772	−0.283874
$723$	0	0
$724$	−7.86141	−0.292167
$725$	0	0
$726$	0	0
$727$	−0.883156	−0.0327544	−0.0163772	−	0.999866i	$-0.505213\pi$
$727$	−0.883156	−0.0327544	−0.0163772	+	0.999866i	$0.505213\pi$
$728$	−11.2554	−0.417154
$729$	0	0
$730$	0	0
$731$	83.8397	3.10092
$732$	0	0
$733$	−34.2337	−1.26445	−0.632225	−	0.774785i	$-0.717858\pi$
$733$	−34.2337	−1.26445	−0.632225	+	0.774785i	$0.717858\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	4.37228	0.161164
$737$	9.60597	0.353840
$738$	0	0
$739$	−23.8832	−0.878556	−0.439278	−	0.898351i	$-0.644766\pi$
$739$	−23.8832	−0.878556	−0.439278	+	0.898351i	$0.644766\pi$
$740$	0	0
$741$	0	0
$742$	−27.2554	−1.00058
$743$	−25.1168	−0.921448	−0.460724	−	0.887544i	$-0.652410\pi$
$743$	−25.1168	−0.921448	−0.460724	+	0.887544i	$0.652410\pi$
$744$	0	0
$745$	0	0
$746$	−7.48913	−0.274196
$747$	0	0
$748$	10.1168	0.369908
$749$	34.3723	1.25594
$750$	0	0
$751$	−18.2337	−0.665357	−0.332678	−	0.943040i	$-0.607952\pi$
$751$	−18.2337	−0.665357	−0.332678	+	0.943040i	$0.607952\pi$
$752$	1.62772	0.0593568
$753$	0	0
$754$	20.7446	0.755472
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	−10.8614	−0.394504
$759$	0	0
$760$	0	0
$761$	12.0951	0.438447	0.219223	−	0.975675i	$-0.429648\pi$
$761$	12.0951	0.438447	0.219223	+	0.975675i	$0.429648\pi$
$762$	0	0
$763$	22.8397	0.826851
$764$	−5.48913	−0.198590
$765$	0	0
$766$	−22.9783	−0.830238
$767$	6.51087	0.235094
$768$	0	0
$769$	−18.1386	−0.654094	−0.327047	−	0.945008i	$-0.606054\pi$
$769$	−18.1386	−0.654094	−0.327047	+	0.945008i	$0.606054\pi$
$770$	0	0
$771$	0	0
$772$	−3.88316	−0.139758
$773$	−14.7446	−0.530325	−0.265163	−	0.964204i	$-0.585426\pi$
$773$	−14.7446	−0.530325	−0.265163	+	0.964204i	$0.585426\pi$
$774$	0	0
$775$	0	0
$776$	2.62772	0.0943296
$777$	0	0
$778$	−10.3723	−0.371864
$779$	−10.1168	−0.362474
$780$	0	0
$781$	−8.23369	−0.294625
$782$	32.2337	1.15267
$783$	0	0
$784$	−1.37228	−0.0490100
$785$	0	0
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−17.4891	−0.623024
$789$	0	0
$790$	0	0
$791$	−34.9783	−1.24368
$792$	0	0
$793$	−43.2554	−1.53605
$794$	−11.2554	−0.399441
$795$	0	0
$796$	−9.48913	−0.336333
$797$	−3.25544	−0.115314	−0.0576568	−	0.998336i	$-0.518363\pi$
$797$	−3.25544	−0.115314	−0.0576568	+	0.998336i	$0.518363\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	0	0
$802$	16.1168	0.569106
$803$	19.3723	0.683633
$804$	0	0
$805$	0	0
$806$	32.0000	1.12715
$807$	0	0
$808$	−8.74456	−0.307633
$809$	−12.3505	−0.434222	−0.217111	−	0.976147i	$-0.569663\pi$
$809$	−12.3505	−0.434222	−0.217111	+	0.976147i	$0.569663\pi$
$810$	0	0
$811$	9.37228	0.329105	0.164553	−	0.986368i	$-0.447382\pi$
$811$	9.37228	0.329105	0.164553	+	0.986368i	$0.447382\pi$
$812$	−10.3723	−0.363996
$813$	0	0
$814$	5.48913	0.192394
$815$	0	0
$816$	0	0
$817$	38.3505	1.34172
$818$	−10.8614	−0.379760
$819$	0	0
$820$	0	0
$821$	50.8397	1.77432	0.887158	−	0.461466i	$-0.152676\pi$
$821$	50.8397	1.77432	0.887158	+	0.461466i	$0.152676\pi$
$822$	0	0
$823$	−38.0951	−1.32791	−0.663956	−	0.747772i	$-0.731124\pi$
$823$	−38.0951	−1.32791	−0.663956	+	0.747772i	$0.731124\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−3.25544	−0.113271
$827$	−8.13859	−0.283007	−0.141503	−	0.989938i	$-0.545194\pi$
$827$	−8.13859	−0.283007	−0.141503	+	0.989938i	$0.545194\pi$
$828$	0	0
$829$	−32.8832	−1.14208	−0.571040	−	0.820923i	$-0.693460\pi$
$829$	−32.8832	−1.14208	−0.571040	+	0.820923i	$0.693460\pi$
$830$	0	0
$831$	0	0
$832$	−4.74456	−0.164488
$833$	−10.1168	−0.350528
$834$	0	0
$835$	0	0
$836$	4.62772	0.160053
$837$	0	0
$838$	−31.7228	−1.09585
$839$	44.2337	1.52712	0.763558	−	0.645739i	$-0.223451\pi$
$839$	44.2337	1.52712	0.763558	+	0.645739i	$0.223451\pi$
$840$	0	0
$841$	−9.88316	−0.340798
$842$	−38.4674	−1.32567
$843$	0	0
$844$	−7.25544	−0.249742
$845$	0	0
$846$	0	0
$847$	−21.6277	−0.743137
$848$	−11.4891	−0.394538
$849$	0	0
$850$	0	0
$851$	17.4891	0.599519
$852$	0	0
$853$	1.76631	0.0604774	0.0302387	−	0.999543i	$-0.490373\pi$
$853$	1.76631	0.0604774	0.0302387	+	0.999543i	$0.490373\pi$
$854$	21.6277	0.740085
$855$	0	0
$856$	14.4891	0.495228
$857$	−23.4891	−0.802373	−0.401187	−	0.915996i	$-0.631402\pi$
$857$	−23.4891	−0.802373	−0.401187	+	0.915996i	$0.631402\pi$
$858$	0	0
$859$	0.116844	0.00398666	0.00199333	−	0.999998i	$-0.499366\pi$
$859$	0.116844	0.00398666	0.00199333	+	0.999998i	$0.499366\pi$
$860$	0	0
$861$	0	0
$862$	−26.2337	−0.893523
$863$	42.6060	1.45032	0.725162	−	0.688578i	$-0.241765\pi$
$863$	42.6060	1.45032	0.725162	+	0.688578i	$0.241765\pi$
$864$	0	0
$865$	0	0
$866$	−0.627719	−0.0213307
$867$	0	0
$868$	−16.0000	−0.543075
$869$	2.74456	0.0931029
$870$	0	0
$871$	−33.2119	−1.12534
$872$	9.62772	0.326036
$873$	0	0
$874$	14.7446	0.498742
$875$	0	0
$876$	0	0
$877$	55.9565	1.88952	0.944758	−	0.327768i	$-0.106296\pi$
$877$	55.9565	1.88952	0.944758	+	0.327768i	$0.106296\pi$
$878$	16.2337	0.547860
$879$	0	0
$880$	0	0
$881$	27.3505	0.921463	0.460731	−	0.887540i	$-0.347587\pi$
$881$	27.3505	0.921463	0.460731	+	0.887540i	$0.347587\pi$
$882$	0	0
$883$	−12.7228	−0.428157	−0.214078	−	0.976816i	$-0.568675\pi$
$883$	−12.7228	−0.428157	−0.214078	+	0.976816i	$0.568675\pi$
$884$	−34.9783	−1.17645
$885$	0	0
$886$	26.4891	0.889920
$887$	37.7228	1.26661	0.633304	−	0.773903i	$-0.281698\pi$
$887$	37.7228	1.26661	0.633304	+	0.773903i	$0.281698\pi$
$888$	0	0
$889$	−21.6277	−0.725370
$890$	0	0
$891$	0	0
$892$	−12.3723	−0.414255
$893$	5.48913	0.183687
$894$	0	0
$895$	0	0
$896$	2.37228	0.0792524
$897$	0	0
$898$	−18.8614	−0.629413
$899$	29.4891	0.983517
$900$	0	0
$901$	−84.7011	−2.82180
$902$	−4.11684	−0.137076
$903$	0	0
$904$	−14.7446	−0.490397
$905$	0	0
$906$	0	0
$907$	7.00000	0.232431	0.116216	−	0.993224i	$-0.462924\pi$
$907$	7.00000	0.232431	0.116216	+	0.993224i	$0.462924\pi$
$908$	−1.88316	−0.0624947
$909$	0	0
$910$	0	0
$911$	−42.0000	−1.39152	−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000	−1.39152	−0.695761	+	0.718273i	$0.744933\pi$
$912$	0	0
$913$	−2.23369	−0.0739243
$914$	−30.1168	−0.996177
$915$	0	0
$916$	18.3723	0.607037
$917$	20.7446	0.685046
$918$	0	0
$919$	42.4674	1.40087	0.700435	−	0.713716i	$-0.252989\pi$
$919$	42.4674	1.40087	0.700435	+	0.713716i	$0.252989\pi$
$920$	0	0
$921$	0	0
$922$	19.1168	0.629580
$923$	28.4674	0.937015
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	0	0
$928$	−4.37228	−0.143527
$929$	−39.9565	−1.31093	−0.655465	−	0.755225i	$-0.727527\pi$
$929$	−39.9565	−1.31093	−0.655465	+	0.755225i	$0.727527\pi$
$930$	0	0
$931$	−4.62772	−0.151667
$932$	10.1168	0.331388
$933$	0	0
$934$	−25.8832	−0.846923
$935$	0	0
$936$	0	0
$937$	17.7228	0.578979	0.289490	−	0.957181i	$-0.406514\pi$
$937$	17.7228	0.578979	0.289490	+	0.957181i	$0.406514\pi$
$938$	16.6060	0.542204
$939$	0	0
$940$	0	0
$941$	37.6277	1.22663	0.613314	−	0.789839i	$-0.289836\pi$
$941$	37.6277	1.22663	0.613314	+	0.789839i	$0.289836\pi$
$942$	0	0
$943$	−13.1168	−0.427143
$944$	−1.37228	−0.0446640
$945$	0	0
$946$	15.6060	0.507394
$947$	−22.7228	−0.738392	−0.369196	−	0.929352i	$-0.620367\pi$
$947$	−22.7228	−0.738392	−0.369196	+	0.929352i	$0.620367\pi$
$948$	0	0
$949$	−66.9783	−2.17421
$950$	0	0
$951$	0	0
$952$	17.4891	0.566826
$953$	−30.8614	−0.999699	−0.499850	−	0.866112i	$-0.666611\pi$
$953$	−30.8614	−0.999699	−0.499850	+	0.866112i	$0.666611\pi$
$954$	0	0
$955$	0	0
$956$	−14.7446	−0.476873
$957$	0	0
$958$	−23.4891	−0.758899
$959$	4.46738	0.144259
$960$	0	0
$961$	14.4891	0.467391
$962$	−18.9783	−0.611883
$963$	0	0
$964$	10.4891	0.337832
$965$	0	0
$966$	0	0
$967$	−18.8832	−0.607241	−0.303621	−	0.952793i	$-0.598196\pi$
$967$	−18.8832	−0.607241	−0.303621	+	0.952793i	$0.598196\pi$
$968$	−9.11684	−0.293026
$969$	0	0
$970$	0	0
$971$	22.9783	0.737407	0.368704	−	0.929547i	$-0.379802\pi$
$971$	22.9783	0.737407	0.368704	+	0.929547i	$0.379802\pi$
$972$	0	0
$973$	42.9783	1.37782
$974$	1.25544	0.0402268
$975$	0	0
$976$	9.11684	0.291823
$977$	40.1168	1.28345	0.641726	−	0.766934i	$-0.278219\pi$
$977$	40.1168	1.28345	0.641726	+	0.766934i	$0.278219\pi$
$978$	0	0
$979$	−1.53262	−0.0489829
$980$	0	0
$981$	0	0
$982$	3.60597	0.115071
$983$	−15.8614	−0.505900	−0.252950	−	0.967479i	$-0.581401\pi$
$983$	−15.8614	−0.505900	−0.252950	+	0.967479i	$0.581401\pi$
$984$	0	0
$985$	0	0
$986$	−32.2337	−1.02653
$987$	0	0
$988$	−16.0000	−0.509028
$989$	49.7228	1.58109
$990$	0	0
$991$	−18.2337	−0.579212	−0.289606	−	0.957146i	$-0.593524\pi$
$991$	−18.2337	−0.579212	−0.289606	+	0.957146i	$0.593524\pi$
$992$	−6.74456	−0.214140
$993$	0	0
$994$	−14.2337	−0.451465
$995$	0	0
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−2.11684	−0.0670075
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.a.bw.1.2		2
3.2	odd	2		4050.2.a.bo.1.2		2
5.2	odd	4		4050.2.c.v.649.4		4
5.3	odd	4		4050.2.c.v.649.1		4
5.4	even	2		810.2.a.i.1.1		2
9.2	odd	6		1350.2.e.l.901.1		4
9.4	even	3		450.2.e.j.151.1		4
9.5	odd	6		1350.2.e.l.451.1		4
9.7	even	3		450.2.e.j.301.2		4
15.2	even	4		4050.2.c.ba.649.2		4
15.8	even	4		4050.2.c.ba.649.3		4
15.14	odd	2		810.2.a.k.1.1		2
20.19	odd	2		6480.2.a.be.1.2		2
45.2	even	12		1350.2.j.f.199.2		8
45.4	even	6		90.2.e.c.61.2	yes	4
45.7	odd	12		450.2.j.g.49.4		8
45.13	odd	12		450.2.j.g.349.4		8
45.14	odd	6		270.2.e.c.181.2		4
45.22	odd	12		450.2.j.g.349.1		8
45.23	even	12		1350.2.j.f.1099.2		8
45.29	odd	6		270.2.e.c.91.2		4
45.32	even	12		1350.2.j.f.1099.3		8
45.34	even	6		90.2.e.c.31.1	✓	4
45.38	even	12		1350.2.j.f.199.3		8
45.43	odd	12		450.2.j.g.49.1		8
60.59	even	2		6480.2.a.bn.1.2		2
180.59	even	6		2160.2.q.f.721.1		4
180.79	odd	6		720.2.q.f.481.2		4
180.119	even	6		2160.2.q.f.1441.1		4
180.139	odd	6		720.2.q.f.241.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.c.31.1	✓	4	45.34	even	6
90.2.e.c.61.2	yes	4	45.4	even	6
270.2.e.c.91.2		4	45.29	odd	6
270.2.e.c.181.2		4	45.14	odd	6
450.2.e.j.151.1		4	9.4	even	3
450.2.e.j.301.2		4	9.7	even	3
450.2.j.g.49.1		8	45.43	odd	12
450.2.j.g.49.4		8	45.7	odd	12
450.2.j.g.349.1		8	45.22	odd	12
450.2.j.g.349.4		8	45.13	odd	12
720.2.q.f.241.1		4	180.139	odd	6
720.2.q.f.481.2		4	180.79	odd	6
810.2.a.i.1.1		2	5.4	even	2
810.2.a.k.1.1		2	15.14	odd	2
1350.2.e.l.451.1		4	9.5	odd	6
1350.2.e.l.901.1		4	9.2	odd	6
1350.2.j.f.199.2		8	45.2	even	12
1350.2.j.f.199.3		8	45.38	even	12
1350.2.j.f.1099.2		8	45.23	even	12
1350.2.j.f.1099.3		8	45.32	even	12
2160.2.q.f.721.1		4	180.59	even	6
2160.2.q.f.1441.1		4	180.119	even	6
4050.2.a.bo.1.2		2	3.2	odd	2
4050.2.a.bw.1.2		2	1.1	even	1	trivial
4050.2.c.v.649.1		4	5.3	odd	4
4050.2.c.v.649.4		4	5.2	odd	4
4050.2.c.ba.649.2		4	15.2	even	4
4050.2.c.ba.649.3		4	15.8	even	4
6480.2.a.be.1.2		2	20.19	odd	2
6480.2.a.bn.1.2		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4050.2.a.bw.1.2

Level	$4050$
Weight	$2$
Character	4050.1
Self dual	yes
Analytic conductor	$32.339$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$