LMFDB - Embedded newform 3328.2.b.y.1665.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1665,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3328.1665"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,2,0,-6,0,0,0,0,0,-14,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1665.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	3328.1665
Dual form			3328.2.b.y.1665.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} +3.56155i q^{5} -1.56155 q^{7} +0.561553 q^{9} +3.12311i q^{11} -1.00000i q^{13} -5.56155 q^{15} -6.68466 q^{17} +3.12311i q^{19} -2.43845i q^{21} +8.00000 q^{23} -7.68466 q^{25} +5.56155i q^{27} +2.00000i q^{29} +4.00000 q^{31} -4.87689 q^{33} -5.56155i q^{35} -2.68466i q^{37} +1.56155 q^{39} -5.12311 q^{41} +9.56155i q^{43} +2.00000i q^{45} -12.6847 q^{47} -4.56155 q^{49} -10.4384i q^{51} -5.12311i q^{53} -11.1231 q^{55} -4.87689 q^{57} -3.12311i q^{59} -2.87689i q^{61} -0.876894 q^{63} +3.56155 q^{65} -3.12311i q^{67} +12.4924i q^{69} -4.68466 q^{71} +6.00000 q^{73} -12.0000i q^{75} -4.87689i q^{77} +8.00000 q^{79} -7.00000 q^{81} +14.2462i q^{83} -23.8078i q^{85} -3.12311 q^{87} -10.0000 q^{89} +1.56155i q^{91} +6.24621i q^{93} -11.1231 q^{95} +8.24621 q^{97} +1.75379i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7} - 6 q^{9} - 14 q^{15} - 2 q^{17} + 32 q^{23} - 6 q^{25} + 16 q^{31} - 36 q^{33} - 2 q^{39} - 4 q^{41} - 26 q^{47} - 10 q^{49} - 28 q^{55} - 36 q^{57} - 20 q^{63} + 6 q^{65} + 6 q^{71} + 24 q^{73}+ \cdots - 28 q^{95}+O(q^{100}) $ 4 * q + 2 * q^7 - 6 * q^9 - 14 * q^15 - 2 * q^17 + 32 * q^23 - 6 * q^25 + 16 * q^31 - 36 * q^33 - 2 * q^39 - 4 * q^41 - 26 * q^47 - 10 * q^49 - 28 * q^55 - 36 * q^57 - 20 * q^63 + 6 * q^65 + 6 * q^71 + 24 * q^73 + 32 * q^79 - 28 * q^81 + 4 * q^87 - 40 * q^89 - 28 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	3.56155i	1.59277i	0.604787	+	0.796387i	$0.293258\pi$
$5$	3.56155i	1.59277i	−0.604787	+	0.796387i	$0.706742\pi$
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	3.12311i	0.941652i	0.882226	+	0.470826i	$0.156044\pi$
$11$	3.12311i	0.941652i	−0.882226	+	0.470826i	$0.843956\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−5.56155	−1.43599
$16$	0	0
$17$	−6.68466	−1.62127	−0.810634	−	0.585553i	$-0.800877\pi$
$17$	−6.68466	−1.62127	−0.810634	+	0.585553i	$0.800877\pi$
$18$	0	0
$19$	3.12311i	0.716490i	0.933628	+	0.358245i	$0.116625\pi$
$19$	3.12311i	0.716490i	−0.933628	+	0.358245i	$0.883375\pi$
$20$	0	0
$21$	− 2.43845i	− 0.532113i
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	−7.68466	−1.53693
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−4.87689	−0.848958
$34$	0	0
$35$	− 5.56155i	− 0.940074i
$36$	0	0
$37$	− 2.68466i	− 0.441355i	−0.975347	−	0.220678i	$-0.929173\pi$
$37$	− 2.68466i	− 0.441355i	0.975347	−	0.220678i	$-0.0708268\pi$
$38$	0	0
$39$	1.56155	0.250049
$40$	0	0
$41$	−5.12311	−0.800095	−0.400047	−	0.916494i	$-0.631006\pi$
$41$	−5.12311	−0.800095	−0.400047	+	0.916494i	$0.631006\pi$
$42$	0	0
$43$	9.56155i	1.45812i	0.684448	+	0.729062i	$0.260043\pi$
$43$	9.56155i	1.45812i	−0.684448	+	0.729062i	$0.739957\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	−12.6847	−1.85025	−0.925124	−	0.379666i	$-0.876039\pi$
$47$	−12.6847	−1.85025	−0.925124	+	0.379666i	$0.876039\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	− 10.4384i	− 1.46167i
$52$	0	0
$53$	− 5.12311i	− 0.703713i	−0.936054	−	0.351856i	$-0.885551\pi$
$53$	− 5.12311i	− 0.703713i	0.936054	−	0.351856i	$-0.114449\pi$
$54$	0	0
$55$	−11.1231	−1.49984
$56$	0	0
$57$	−4.87689	−0.645960
$58$	0	0
$59$	− 3.12311i	− 0.406594i	−0.979117	−	0.203297i	$-0.934834\pi$
$59$	− 3.12311i	− 0.406594i	0.979117	−	0.203297i	$-0.0651656\pi$
$60$	0	0
$61$	− 2.87689i	− 0.368349i	−0.982894	−	0.184174i	$-0.941039\pi$
$61$	− 2.87689i	− 0.368349i	0.982894	−	0.184174i	$-0.0589611\pi$
$62$	0	0
$63$	−0.876894	−0.110478
$64$	0	0
$65$	3.56155	0.441756
$66$	0	0
$67$	− 3.12311i	− 0.381548i	−0.981634	−	0.190774i	$-0.938900\pi$
$67$	− 3.12311i	− 0.381548i	0.981634	−	0.190774i	$-0.0610998\pi$
$68$	0	0
$69$	12.4924i	1.50391i
$70$	0	0
$71$	−4.68466	−0.555967	−0.277983	−	0.960586i	$-0.589666\pi$
$71$	−4.68466	−0.555967	−0.277983	+	0.960586i	$0.589666\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	− 12.0000i	− 1.38564i
$76$	0	0
$77$	− 4.87689i	− 0.555774i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	14.2462i	1.56372i	0.623451	+	0.781862i	$0.285730\pi$
$83$	14.2462i	1.56372i	−0.623451	+	0.781862i	$0.714270\pi$
$84$	0	0
$85$	− 23.8078i	− 2.58231i
$86$	0	0
$87$	−3.12311	−0.334832
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	1.56155i	0.163695i
$92$	0	0
$93$	6.24621i	0.647702i
$94$	0	0
$95$	−11.1231	−1.14121
$96$	0	0
$97$	8.24621	0.837276	0.418638	−	0.908153i	$-0.362508\pi$
$97$	8.24621	0.837276	0.418638	+	0.908153i	$0.362508\pi$
$98$	0	0
$99$	1.75379i	0.176262i
$100$	0	0
$101$	1.12311i	0.111753i	0.998438	+	0.0558766i	$0.0177953\pi$
$101$	1.12311i	0.111753i	−0.998438	+	0.0558766i	$0.982205\pi$
$102$	0	0
$103$	14.2462	1.40372	0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462	1.40372	0.701860	+	0.712314i	$0.252353\pi$
$104$	0	0
$105$	8.68466	0.847536
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	− 17.8078i	− 1.70567i	−0.522177	−	0.852837i	$-0.674880\pi$
$109$	− 17.8078i	− 1.70567i	0.522177	−	0.852837i	$-0.325120\pi$
$110$	0	0
$111$	4.19224	0.397909
$112$	0	0
$113$	14.4924	1.36333	0.681666	−	0.731663i	$-0.261256\pi$
$113$	14.4924	1.36333	0.681666	+	0.731663i	$0.261256\pi$
$114$	0	0
$115$	28.4924i	2.65693i
$116$	0	0
$117$	− 0.561553i	− 0.0519156i
$118$	0	0
$119$	10.4384	0.956891
$120$	0	0
$121$	1.24621	0.113292
$122$	0	0
$123$	− 8.00000i	− 0.721336i
$124$	0	0
$125$	− 9.56155i	− 0.855211i
$126$	0	0
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	0	0
$129$	−14.9309	−1.31459
$130$	0	0
$131$	− 15.8078i	− 1.38113i	−0.723270	−	0.690565i	$-0.757362\pi$
$131$	− 15.8078i	− 1.38113i	0.723270	−	0.690565i	$-0.242638\pi$
$132$	0	0
$133$	− 4.87689i	− 0.422880i
$134$	0	0
$135$	−19.8078	−1.70478
$136$	0	0
$137$	−5.12311	−0.437696	−0.218848	−	0.975759i	$-0.570230\pi$
$137$	−5.12311	−0.437696	−0.218848	+	0.975759i	$0.570230\pi$
$138$	0	0
$139$	− 4.68466i	− 0.397348i	−0.980066	−	0.198674i	$-0.936337\pi$
$139$	− 4.68466i	− 0.397348i	0.980066	−	0.198674i	$-0.0636634\pi$
$140$	0	0
$141$	− 19.8078i	− 1.66811i
$142$	0	0
$143$	3.12311	0.261167
$144$	0	0
$145$	−7.12311	−0.591542
$146$	0	0
$147$	− 7.12311i	− 0.587504i
$148$	0	0
$149$	− 0.246211i	− 0.0201704i	−0.999949	−	0.0100852i	$-0.996790\pi$
$149$	− 0.246211i	− 0.0201704i	0.999949	−	0.0100852i	$-0.00321028\pi$
$150$	0	0
$151$	−6.43845	−0.523953	−0.261977	−	0.965074i	$-0.584374\pi$
$151$	−6.43845	−0.523953	−0.261977	+	0.965074i	$0.584374\pi$
$152$	0	0
$153$	−3.75379	−0.303476
$154$	0	0
$155$	14.2462i	1.14428i
$156$	0	0
$157$	− 10.4924i	− 0.837386i	−0.908128	−	0.418693i	$-0.862488\pi$
$157$	− 10.4924i	− 0.837386i	0.908128	−	0.418693i	$-0.137512\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	−12.4924	−0.984541
$162$	0	0
$163$	− 12.4924i	− 0.978482i	−0.872149	−	0.489241i	$-0.837274\pi$
$163$	− 12.4924i	− 0.978482i	0.872149	−	0.489241i	$-0.162726\pi$
$164$	0	0
$165$	− 17.3693i	− 1.35220i
$166$	0	0
$167$	2.24621	0.173817	0.0869085	−	0.996216i	$-0.472301\pi$
$167$	2.24621	0.173817	0.0869085	+	0.996216i	$0.472301\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	1.75379i	0.134116i
$172$	0	0
$173$	13.1231i	0.997731i	0.866679	+	0.498866i	$0.166250\pi$
$173$	13.1231i	0.997731i	−0.866679	+	0.498866i	$0.833750\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	4.87689	0.366570
$178$	0	0
$179$	4.68466i	0.350148i	0.984555	+	0.175074i	$0.0560164\pi$
$179$	4.68466i	0.350148i	−0.984555	+	0.175074i	$0.943984\pi$
$180$	0	0
$181$	− 14.8769i	− 1.10579i	−0.833251	−	0.552895i	$-0.813523\pi$
$181$	− 14.8769i	− 1.10579i	0.833251	−	0.552895i	$-0.186477\pi$
$182$	0	0
$183$	4.49242	0.332089
$184$	0	0
$185$	9.56155	0.702979
$186$	0	0
$187$	− 20.8769i	− 1.52667i
$188$	0	0
$189$	− 8.68466i	− 0.631716i
$190$	0	0
$191$	−9.36932	−0.677940	−0.338970	−	0.940797i	$-0.610079\pi$
$191$	−9.36932	−0.677940	−0.338970	+	0.940797i	$0.610079\pi$
$192$	0	0
$193$	3.36932	0.242529	0.121264	−	0.992620i	$-0.461305\pi$
$193$	3.36932	0.242529	0.121264	+	0.992620i	$0.461305\pi$
$194$	0	0
$195$	5.56155i	0.398271i
$196$	0	0
$197$	14.6847i	1.04624i	0.852259	+	0.523119i	$0.175232\pi$
$197$	14.6847i	1.04624i	−0.852259	+	0.523119i	$0.824768\pi$
$198$	0	0
$199$	3.12311	0.221391	0.110696	−	0.993854i	$-0.464692\pi$
$199$	3.12311	0.221391	0.110696	+	0.993854i	$0.464692\pi$
$200$	0	0
$201$	4.87689	0.343990
$202$	0	0
$203$	− 3.12311i	− 0.219199i
$204$	0	0
$205$	− 18.2462i	− 1.27437i
$206$	0	0
$207$	4.49242	0.312245
$208$	0	0
$209$	−9.75379	−0.674684
$210$	0	0
$211$	− 3.31534i	− 0.228238i	−0.993467	−	0.114119i	$-0.963596\pi$
$211$	− 3.31534i	− 0.228238i	0.993467	−	0.114119i	$-0.0364044\pi$
$212$	0	0
$213$	− 7.31534i	− 0.501239i
$214$	0	0
$215$	−34.0540	−2.32246
$216$	0	0
$217$	−6.24621	−0.424020
$218$	0	0
$219$	9.36932i	0.633120i
$220$	0	0
$221$	6.68466i	0.449659i
$222$	0	0
$223$	−3.31534	−0.222012	−0.111006	−	0.993820i	$-0.535407\pi$
$223$	−3.31534	−0.222012	−0.111006	+	0.993820i	$0.535407\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	0	0
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	0	0
$229$	− 13.8078i	− 0.912443i	−0.889866	−	0.456221i	$-0.849203\pi$
$229$	− 13.8078i	− 0.912443i	0.889866	−	0.456221i	$-0.150797\pi$
$230$	0	0
$231$	7.61553	0.501065
$232$	0	0
$233$	−13.8078	−0.904577	−0.452288	−	0.891872i	$-0.649392\pi$
$233$	−13.8078	−0.904577	−0.452288	+	0.891872i	$0.649392\pi$
$234$	0	0
$235$	− 45.1771i	− 2.94703i
$236$	0	0
$237$	12.4924i	0.811470i
$238$	0	0
$239$	1.56155	0.101008	0.0505042	−	0.998724i	$-0.483917\pi$
$239$	1.56155	0.101008	0.0505042	+	0.998724i	$0.483917\pi$
$240$	0	0
$241$	−20.2462	−1.30417	−0.652087	−	0.758145i	$-0.726106\pi$
$241$	−20.2462	−1.30417	−0.652087	+	0.758145i	$0.726106\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	− 16.2462i	− 1.03793i
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	−22.2462	−1.40980
$250$	0	0
$251$	5.75379i	0.363176i	0.983375	+	0.181588i	$0.0581237\pi$
$251$	5.75379i	0.363176i	−0.983375	+	0.181588i	$0.941876\pi$
$252$	0	0
$253$	24.9848i	1.57078i
$254$	0	0
$255$	37.1771	2.32812
$256$	0	0
$257$	15.5616	0.970703	0.485351	−	0.874319i	$-0.338692\pi$
$257$	15.5616	0.970703	0.485351	+	0.874319i	$0.338692\pi$
$258$	0	0
$259$	4.19224i	0.260493i
$260$	0	0
$261$	1.12311i	0.0695185i
$262$	0	0
$263$	9.75379	0.601444	0.300722	−	0.953712i	$-0.402772\pi$
$263$	9.75379	0.601444	0.300722	+	0.953712i	$0.402772\pi$
$264$	0	0
$265$	18.2462	1.12086
$266$	0	0
$267$	− 15.6155i	− 0.955655i
$268$	0	0
$269$	13.1231i	0.800130i	0.916487	+	0.400065i	$0.131012\pi$
$269$	13.1231i	0.800130i	−0.916487	+	0.400065i	$0.868988\pi$
$270$	0	0
$271$	−0.192236	−0.0116775	−0.00583875	−	0.999983i	$-0.501859\pi$
$271$	−0.192236	−0.0116775	−0.00583875	+	0.999983i	$0.501859\pi$
$272$	0	0
$273$	−2.43845	−0.147582
$274$	0	0
$275$	− 24.0000i	− 1.44725i
$276$	0	0
$277$	4.63068i	0.278231i	0.990276	+	0.139115i	$0.0444259\pi$
$277$	4.63068i	0.278231i	−0.990276	+	0.139115i	$0.955574\pi$
$278$	0	0
$279$	2.24621	0.134477
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	24.4924i	1.45592i	0.685618	+	0.727962i	$0.259532\pi$
$283$	24.4924i	1.45592i	−0.685618	+	0.727962i	$0.740468\pi$
$284$	0	0
$285$	− 17.3693i	− 1.02887i
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	27.6847	1.62851
$290$	0	0
$291$	12.8769i	0.754857i
$292$	0	0
$293$	19.5616i	1.14280i	0.820672	+	0.571399i	$0.193599\pi$
$293$	19.5616i	1.14280i	−0.820672	+	0.571399i	$0.806401\pi$
$294$	0	0
$295$	11.1231	0.647612
$296$	0	0
$297$	−17.3693	−1.00787
$298$	0	0
$299$	− 8.00000i	− 0.462652i
$300$	0	0
$301$	− 14.9309i	− 0.860601i
$302$	0	0
$303$	−1.75379	−0.100753
$304$	0	0
$305$	10.2462	0.586696
$306$	0	0
$307$	− 4.87689i	− 0.278339i	−0.990269	−	0.139170i	$-0.955557\pi$
$307$	− 4.87689i	− 0.278339i	0.990269	−	0.139170i	$-0.0444433\pi$
$308$	0	0
$309$	22.2462i	1.26554i
$310$	0	0
$311$	19.1231	1.08437	0.542186	−	0.840259i	$-0.317597\pi$
$311$	19.1231	1.08437	0.542186	+	0.840259i	$0.317597\pi$
$312$	0	0
$313$	−6.19224	−0.350006	−0.175003	−	0.984568i	$-0.555993\pi$
$313$	−6.19224	−0.350006	−0.175003	+	0.984568i	$0.555993\pi$
$314$	0	0
$315$	− 3.12311i	− 0.175967i
$316$	0	0
$317$	− 4.24621i	− 0.238491i	−0.992865	−	0.119245i	$-0.961952\pi$
$317$	− 4.24621i	− 0.238491i	0.992865	−	0.119245i	$-0.0380476\pi$
$318$	0	0
$319$	−6.24621	−0.349721
$320$	0	0
$321$	−6.24621	−0.348630
$322$	0	0
$323$	− 20.8769i	− 1.16162i
$324$	0	0
$325$	7.68466i	0.426268i
$326$	0	0
$327$	27.8078	1.53777
$328$	0	0
$329$	19.8078	1.09204
$330$	0	0
$331$	28.4924i	1.56609i	0.621968	+	0.783043i	$0.286333\pi$
$331$	28.4924i	1.56609i	−0.621968	+	0.783043i	$0.713667\pi$
$332$	0	0
$333$	− 1.50758i	− 0.0826147i
$334$	0	0
$335$	11.1231	0.607720
$336$	0	0
$337$	−33.8078	−1.84163	−0.920813	−	0.390004i	$-0.872474\pi$
$337$	−33.8078	−1.84163	−0.920813	+	0.390004i	$0.872474\pi$
$338$	0	0
$339$	22.6307i	1.22913i
$340$	0	0
$341$	12.4924i	0.676503i
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	−44.4924	−2.39539
$346$	0	0
$347$	− 4.68466i	− 0.251486i	−0.992063	−	0.125743i	$-0.959869\pi$
$347$	− 4.68466i	− 0.251486i	0.992063	−	0.125743i	$-0.0401314\pi$
$348$	0	0
$349$	16.9309i	0.906289i	0.891437	+	0.453144i	$0.149698\pi$
$349$	16.9309i	0.906289i	−0.891437	+	0.453144i	$0.850302\pi$
$350$	0	0
$351$	5.56155	0.296854
$352$	0	0
$353$	−15.3693	−0.818026	−0.409013	−	0.912529i	$-0.634127\pi$
$353$	−15.3693	−0.818026	−0.409013	+	0.912529i	$0.634127\pi$
$354$	0	0
$355$	− 16.6847i	− 0.885530i
$356$	0	0
$357$	16.3002i	0.862697i
$358$	0	0
$359$	2.24621	0.118550	0.0592752	−	0.998242i	$-0.481121\pi$
$359$	2.24621	0.118550	0.0592752	+	0.998242i	$0.481121\pi$
$360$	0	0
$361$	9.24621	0.486643
$362$	0	0
$363$	1.94602i	0.102140i
$364$	0	0
$365$	21.3693i	1.11852i
$366$	0	0
$367$	−31.6155	−1.65032	−0.825159	−	0.564901i	$-0.808914\pi$
$367$	−31.6155	−1.65032	−0.825159	+	0.564901i	$0.808914\pi$
$368$	0	0
$369$	−2.87689	−0.149765
$370$	0	0
$371$	8.00000i	0.415339i
$372$	0	0
$373$	29.6155i	1.53343i	0.641985	+	0.766717i	$0.278111\pi$
$373$	29.6155i	1.53343i	−0.641985	+	0.766717i	$0.721889\pi$
$374$	0	0
$375$	14.9309	0.771027
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	22.2462i	1.14271i	0.820703	+	0.571356i	$0.193582\pi$
$379$	22.2462i	1.14271i	−0.820703	+	0.571356i	$0.806418\pi$
$380$	0	0
$381$	9.75379i	0.499702i
$382$	0	0
$383$	−31.8078	−1.62530	−0.812650	−	0.582752i	$-0.801976\pi$
$383$	−31.8078	−1.62530	−0.812650	+	0.582752i	$0.801976\pi$
$384$	0	0
$385$	17.3693	0.885222
$386$	0	0
$387$	5.36932i	0.272938i
$388$	0	0
$389$	− 28.7386i	− 1.45711i	−0.684989	−	0.728553i	$-0.740193\pi$
$389$	− 28.7386i	− 1.45711i	0.684989	−	0.728553i	$-0.259807\pi$
$390$	0	0
$391$	−53.4773	−2.70446
$392$	0	0
$393$	24.6847	1.24518
$394$	0	0
$395$	28.4924i	1.43361i
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	7.61553	0.381253
$400$	0	0
$401$	−37.6155	−1.87843	−0.939215	−	0.343330i	$-0.888445\pi$
$401$	−37.6155	−1.87843	−0.939215	+	0.343330i	$0.888445\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	0	0
$405$	− 24.9309i	− 1.23882i
$406$	0	0
$407$	8.38447	0.415603
$408$	0	0
$409$	−21.1231	−1.04447	−0.522235	−	0.852802i	$-0.674902\pi$
$409$	−21.1231	−1.04447	−0.522235	+	0.852802i	$0.674902\pi$
$410$	0	0
$411$	− 8.00000i	− 0.394611i
$412$	0	0
$413$	4.87689i	0.239976i
$414$	0	0
$415$	−50.7386	−2.49066
$416$	0	0
$417$	7.31534	0.358234
$418$	0	0
$419$	− 26.9309i	− 1.31566i	−0.753167	−	0.657830i	$-0.771475\pi$
$419$	− 26.9309i	− 1.31566i	0.753167	−	0.657830i	$-0.228525\pi$
$420$	0	0
$421$	− 18.6847i	− 0.910635i	−0.890329	−	0.455317i	$-0.849526\pi$
$421$	− 18.6847i	− 0.910635i	0.890329	−	0.455317i	$-0.150474\pi$
$422$	0	0
$423$	−7.12311	−0.346337
$424$	0	0
$425$	51.3693	2.49178
$426$	0	0
$427$	4.49242i	0.217404i
$428$	0	0
$429$	4.87689i	0.235459i
$430$	0	0
$431$	−14.4384	−0.695476	−0.347738	−	0.937592i	$-0.613050\pi$
$431$	−14.4384	−0.695476	−0.347738	+	0.937592i	$0.613050\pi$
$432$	0	0
$433$	39.1771	1.88273	0.941365	−	0.337389i	$-0.109544\pi$
$433$	39.1771	1.88273	0.941365	+	0.337389i	$0.109544\pi$
$434$	0	0
$435$	− 11.1231i	− 0.533312i
$436$	0	0
$437$	24.9848i	1.19519i
$438$	0	0
$439$	−12.8769	−0.614581	−0.307290	−	0.951616i	$-0.599422\pi$
$439$	−12.8769	−0.614581	−0.307290	+	0.951616i	$0.599422\pi$
$440$	0	0
$441$	−2.56155	−0.121979
$442$	0	0
$443$	0.192236i	0.00913340i	0.999990	+	0.00456670i	$0.00145363\pi$
$443$	0.192236i	0.00913340i	−0.999990	+	0.00456670i	$0.998546\pi$
$444$	0	0
$445$	− 35.6155i	− 1.68834i
$446$	0	0
$447$	0.384472	0.0181849
$448$	0	0
$449$	−16.7386	−0.789945	−0.394972	−	0.918693i	$-0.629246\pi$
$449$	−16.7386	−0.789945	−0.394972	+	0.918693i	$0.629246\pi$
$450$	0	0
$451$	− 16.0000i	− 0.753411i
$452$	0	0
$453$	− 10.0540i	− 0.472377i
$454$	0	0
$455$	−5.56155	−0.260730
$456$	0	0
$457$	24.7386	1.15722	0.578612	−	0.815603i	$-0.303594\pi$
$457$	24.7386	1.15722	0.578612	+	0.815603i	$0.303594\pi$
$458$	0	0
$459$	− 37.1771i	− 1.73528i
$460$	0	0
$461$	14.1922i	0.660998i	0.943806	+	0.330499i	$0.107217\pi$
$461$	14.1922i	0.660998i	−0.943806	+	0.330499i	$0.892783\pi$
$462$	0	0
$463$	21.7538	1.01098	0.505492	−	0.862831i	$-0.331311\pi$
$463$	21.7538	1.01098	0.505492	+	0.862831i	$0.331311\pi$
$464$	0	0
$465$	−22.2462	−1.03164
$466$	0	0
$467$	− 7.50758i	− 0.347409i	−0.984798	−	0.173705i	$-0.944426\pi$
$467$	− 7.50758i	− 0.347409i	0.984798	−	0.173705i	$-0.0555738\pi$
$468$	0	0
$469$	4.87689i	0.225194i
$470$	0	0
$471$	16.3845	0.754957
$472$	0	0
$473$	−29.8617	−1.37304
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	0	0
$477$	− 2.87689i	− 0.131724i
$478$	0	0
$479$	31.8078	1.45333	0.726667	−	0.686990i	$-0.241068\pi$
$479$	31.8078	1.45333	0.726667	+	0.686990i	$0.241068\pi$
$480$	0	0
$481$	−2.68466	−0.122410
$482$	0	0
$483$	− 19.5076i	− 0.887626i
$484$	0	0
$485$	29.3693i	1.33359i
$486$	0	0
$487$	−18.2462	−0.826815	−0.413407	−	0.910546i	$-0.635661\pi$
$487$	−18.2462	−0.826815	−0.413407	+	0.910546i	$0.635661\pi$
$488$	0	0
$489$	19.5076	0.882163
$490$	0	0
$491$	38.0540i	1.71735i	0.512519	+	0.858676i	$0.328712\pi$
$491$	38.0540i	1.71735i	−0.512519	+	0.858676i	$0.671288\pi$
$492$	0	0
$493$	− 13.3693i	− 0.602124i
$494$	0	0
$495$	−6.24621	−0.280746
$496$	0	0
$497$	7.31534	0.328138
$498$	0	0
$499$	4.49242i	0.201108i	0.994932	+	0.100554i	$0.0320616\pi$
$499$	4.49242i	0.201108i	−0.994932	+	0.100554i	$0.967938\pi$
$500$	0	0
$501$	3.50758i	0.156707i
$502$	0	0
$503$	−17.3693	−0.774460	−0.387230	−	0.921983i	$-0.626568\pi$
$503$	−17.3693	−0.774460	−0.387230	+	0.921983i	$0.626568\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	− 1.56155i	− 0.0693510i
$508$	0	0
$509$	18.0000i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$	18.0000i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$	0	0
$511$	−9.36932	−0.414474
$512$	0	0
$513$	−17.3693	−0.766874
$514$	0	0
$515$	50.7386i	2.23581i
$516$	0	0
$517$	− 39.6155i	− 1.74229i
$518$	0	0
$519$	−20.4924	−0.899518
$520$	0	0
$521$	−2.68466	−0.117617	−0.0588085	−	0.998269i	$-0.518730\pi$
$521$	−2.68466	−0.117617	−0.0588085	+	0.998269i	$0.518730\pi$
$522$	0	0
$523$	32.4924i	1.42079i	0.703801	+	0.710397i	$0.251485\pi$
$523$	32.4924i	1.42079i	−0.703801	+	0.710397i	$0.748515\pi$
$524$	0	0
$525$	18.7386i	0.817821i
$526$	0	0
$527$	−26.7386	−1.16475
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	− 1.75379i	− 0.0761079i
$532$	0	0
$533$	5.12311i	0.221906i
$534$	0	0
$535$	−14.2462	−0.615917
$536$	0	0
$537$	−7.31534	−0.315680
$538$	0	0
$539$	− 14.2462i	− 0.613628i
$540$	0	0
$541$	− 6.68466i	− 0.287396i	−0.989622	−	0.143698i	$-0.954101\pi$
$541$	− 6.68466i	− 0.287396i	0.989622	−	0.143698i	$-0.0458994\pi$
$542$	0	0
$543$	23.2311	0.996940
$544$	0	0
$545$	63.4233	2.71676
$546$	0	0
$547$	33.5616i	1.43499i	0.696564	+	0.717494i	$0.254711\pi$
$547$	33.5616i	1.43499i	−0.696564	+	0.717494i	$0.745289\pi$
$548$	0	0
$549$	− 1.61553i	− 0.0689491i
$550$	0	0
$551$	−6.24621	−0.266098
$552$	0	0
$553$	−12.4924	−0.531232
$554$	0	0
$555$	14.9309i	0.633780i
$556$	0	0
$557$	− 3.17708i	− 0.134617i	−0.997732	−	0.0673086i	$-0.978559\pi$
$557$	− 3.17708i	− 0.134617i	0.997732	−	0.0673086i	$-0.0214412\pi$
$558$	0	0
$559$	9.56155	0.404411
$560$	0	0
$561$	32.6004	1.37639
$562$	0	0
$563$	− 30.0540i	− 1.26662i	−0.773897	−	0.633312i	$-0.781695\pi$
$563$	− 30.0540i	− 1.26662i	0.773897	−	0.633312i	$-0.218305\pi$
$564$	0	0
$565$	51.6155i	2.17148i
$566$	0	0
$567$	10.9309	0.459053
$568$	0	0
$569$	29.3153	1.22896	0.614482	−	0.788931i	$-0.289365\pi$
$569$	29.3153	1.22896	0.614482	+	0.788931i	$0.289365\pi$
$570$	0	0
$571$	8.19224i	0.342834i	0.985199	+	0.171417i	$0.0548346\pi$
$571$	8.19224i	0.342834i	−0.985199	+	0.171417i	$0.945165\pi$
$572$	0	0
$573$	− 14.6307i	− 0.611206i
$574$	0	0
$575$	−61.4773	−2.56378
$576$	0	0
$577$	−7.75379	−0.322794	−0.161397	−	0.986890i	$-0.551600\pi$
$577$	−7.75379	−0.322794	−0.161397	+	0.986890i	$0.551600\pi$
$578$	0	0
$579$	5.26137i	0.218655i
$580$	0	0
$581$	− 22.2462i	− 0.922928i
$582$	0	0
$583$	16.0000	0.662652
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	40.9848i	1.69163i	0.533480	+	0.845813i	$0.320884\pi$
$587$	40.9848i	1.69163i	−0.533480	+	0.845813i	$0.679116\pi$
$588$	0	0
$589$	12.4924i	0.514741i
$590$	0	0
$591$	−22.9309	−0.943250
$592$	0	0
$593$	5.50758	0.226169	0.113085	−	0.993585i	$-0.463927\pi$
$593$	5.50758	0.226169	0.113085	+	0.993585i	$0.463927\pi$
$594$	0	0
$595$	37.1771i	1.52411i
$596$	0	0
$597$	4.87689i	0.199598i
$598$	0	0
$599$	1.36932	0.0559488	0.0279744	−	0.999609i	$-0.491094\pi$
$599$	1.36932	0.0559488	0.0279744	+	0.999609i	$0.491094\pi$
$600$	0	0
$601$	11.1771	0.455923	0.227961	−	0.973670i	$-0.426794\pi$
$601$	11.1771	0.455923	0.227961	+	0.973670i	$0.426794\pi$
$602$	0	0
$603$	− 1.75379i	− 0.0714198i
$604$	0	0
$605$	4.43845i	0.180449i
$606$	0	0
$607$	−9.36932	−0.380289	−0.190144	−	0.981756i	$-0.560896\pi$
$607$	−9.36932	−0.380289	−0.190144	+	0.981756i	$0.560896\pi$
$608$	0	0
$609$	4.87689	0.197622
$610$	0	0
$611$	12.6847i	0.513166i
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	0	0
$615$	28.4924	1.14893
$616$	0	0
$617$	35.8617	1.44374	0.721870	−	0.692029i	$-0.243283\pi$
$617$	35.8617	1.44374	0.721870	+	0.692029i	$0.243283\pi$
$618$	0	0
$619$	− 46.2462i	− 1.85879i	−0.369084	−	0.929396i	$-0.620328\pi$
$619$	− 46.2462i	− 1.85879i	0.369084	−	0.929396i	$-0.379672\pi$
$620$	0	0
$621$	44.4924i	1.78542i
$622$	0	0
$623$	15.6155	0.625623
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	− 15.2311i	− 0.608270i
$628$	0	0
$629$	17.9460i	0.715555i
$630$	0	0
$631$	35.3153	1.40588	0.702941	−	0.711248i	$-0.251870\pi$
$631$	35.3153	1.40588	0.702941	+	0.711248i	$0.251870\pi$
$632$	0	0
$633$	5.17708	0.205770
$634$	0	0
$635$	22.2462i	0.882814i
$636$	0	0
$637$	4.56155i	0.180735i
$638$	0	0
$639$	−2.63068	−0.104068
$640$	0	0
$641$	8.24621	0.325706	0.162853	−	0.986650i	$-0.447930\pi$
$641$	8.24621	0.325706	0.162853	+	0.986650i	$0.447930\pi$
$642$	0	0
$643$	21.8617i	0.862143i	0.902318	+	0.431071i	$0.141864\pi$
$643$	21.8617i	0.862143i	−0.902318	+	0.431071i	$0.858136\pi$
$644$	0	0
$645$	− 53.1771i	− 2.09385i
$646$	0	0
$647$	3.12311	0.122782	0.0613910	−	0.998114i	$-0.480446\pi$
$647$	3.12311	0.122782	0.0613910	+	0.998114i	$0.480446\pi$
$648$	0	0
$649$	9.75379	0.382870
$650$	0	0
$651$	− 9.75379i	− 0.382281i
$652$	0	0
$653$	13.1231i	0.513547i	0.966472	+	0.256773i	$0.0826594\pi$
$653$	13.1231i	0.513547i	−0.966472	+	0.256773i	$0.917341\pi$
$654$	0	0
$655$	56.3002	2.19983
$656$	0	0
$657$	3.36932	0.131450
$658$	0	0
$659$	16.4924i	0.642454i	0.947002	+	0.321227i	$0.104095\pi$
$659$	16.4924i	0.642454i	−0.947002	+	0.321227i	$0.895905\pi$
$660$	0	0
$661$	8.73863i	0.339893i	0.985453	+	0.169947i	$0.0543596\pi$
$661$	8.73863i	0.339893i	−0.985453	+	0.169947i	$0.945640\pi$
$662$	0	0
$663$	−10.4384	−0.405396
$664$	0	0
$665$	17.3693	0.673553
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	− 5.17708i	− 0.200158i
$670$	0	0
$671$	8.98485	0.346856
$672$	0	0
$673$	34.3002	1.32218	0.661088	−	0.750309i	$-0.270095\pi$
$673$	34.3002	1.32218	0.661088	+	0.750309i	$0.270095\pi$
$674$	0	0
$675$	− 42.7386i	− 1.64501i
$676$	0	0
$677$	23.3693i	0.898156i	0.893493	+	0.449078i	$0.148247\pi$
$677$	23.3693i	0.898156i	−0.893493	+	0.449078i	$0.851753\pi$
$678$	0	0
$679$	−12.8769	−0.494170
$680$	0	0
$681$	−12.4924	−0.478711
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	− 18.2462i	− 0.697152i
$686$	0	0
$687$	21.5616	0.822625
$688$	0	0
$689$	−5.12311	−0.195175
$690$	0	0
$691$	48.9848i	1.86347i	0.363137	+	0.931736i	$0.381706\pi$
$691$	48.9848i	1.86347i	−0.363137	+	0.931736i	$0.618294\pi$
$692$	0	0
$693$	− 2.73863i	− 0.104032i
$694$	0	0
$695$	16.6847	0.632885
$696$	0	0
$697$	34.2462	1.29717
$698$	0	0
$699$	− 21.5616i	− 0.815533i
$700$	0	0
$701$	− 47.3693i	− 1.78911i	−0.446953	−	0.894557i	$-0.647491\pi$
$701$	− 47.3693i	− 1.78911i	0.446953	−	0.894557i	$-0.352509\pi$
$702$	0	0
$703$	8.38447	0.316226
$704$	0	0
$705$	70.5464	2.65693
$706$	0	0
$707$	− 1.75379i	− 0.0659580i
$708$	0	0
$709$	24.7386i	0.929079i	0.885552	+	0.464539i	$0.153780\pi$
$709$	24.7386i	0.929079i	−0.885552	+	0.464539i	$0.846220\pi$
$710$	0	0
$711$	4.49242	0.168479
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	11.1231i	0.415981i
$716$	0	0
$717$	2.43845i	0.0910655i
$718$	0	0
$719$	−4.87689	−0.181877	−0.0909387	−	0.995856i	$-0.528987\pi$
$719$	−4.87689	−0.181877	−0.0909387	+	0.995856i	$0.528987\pi$
$720$	0	0
$721$	−22.2462	−0.828492
$722$	0	0
$723$	− 31.6155i	− 1.17579i
$724$	0	0
$725$	− 15.3693i	− 0.570802i
$726$	0	0
$727$	−28.1080	−1.04247	−0.521233	−	0.853414i	$-0.674528\pi$
$727$	−28.1080	−1.04247	−0.521233	+	0.853414i	$0.674528\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	− 63.9157i	− 2.36401i
$732$	0	0
$733$	31.5616i	1.16575i	0.812561	+	0.582876i	$0.198073\pi$
$733$	31.5616i	1.16575i	−0.812561	+	0.582876i	$0.801927\pi$
$734$	0	0
$735$	25.3693	0.935761
$736$	0	0
$737$	9.75379	0.359285
$738$	0	0
$739$	− 6.24621i	− 0.229771i	−0.993379	−	0.114885i	$-0.963350\pi$
$739$	− 6.24621i	− 0.229771i	0.993379	−	0.114885i	$-0.0366501\pi$
$740$	0	0
$741$	4.87689i	0.179157i
$742$	0	0
$743$	43.9157	1.61111	0.805556	−	0.592520i	$-0.201867\pi$
$743$	43.9157	1.61111	0.805556	+	0.592520i	$0.201867\pi$
$744$	0	0
$745$	0.876894	0.0321269
$746$	0	0
$747$	8.00000i	0.292705i
$748$	0	0
$749$	− 6.24621i	− 0.228232i
$750$	0	0
$751$	−40.9848	−1.49556	−0.747779	−	0.663948i	$-0.768880\pi$
$751$	−40.9848	−1.49556	−0.747779	+	0.663948i	$0.768880\pi$
$752$	0	0
$753$	−8.98485	−0.327426
$754$	0	0
$755$	− 22.9309i	− 0.834540i
$756$	0	0
$757$	− 5.12311i	− 0.186202i	−0.995657	−	0.0931012i	$-0.970322\pi$
$757$	− 5.12311i	− 0.186202i	0.995657	−	0.0931012i	$-0.0296780\pi$
$758$	0	0
$759$	−39.0152	−1.41616
$760$	0	0
$761$	−13.5076	−0.489649	−0.244825	−	0.969567i	$-0.578730\pi$
$761$	−13.5076	−0.489649	−0.244825	+	0.969567i	$0.578730\pi$
$762$	0	0
$763$	27.8078i	1.00671i
$764$	0	0
$765$	− 13.3693i	− 0.483369i
$766$	0	0
$767$	−3.12311	−0.112769
$768$	0	0
$769$	−43.8617	−1.58169	−0.790847	−	0.612013i	$-0.790360\pi$
$769$	−43.8617	−1.58169	−0.790847	+	0.612013i	$0.790360\pi$
$770$	0	0
$771$	24.3002i	0.875150i
$772$	0	0
$773$	0.822919i	0.0295983i	0.999890	+	0.0147992i	$0.00471089\pi$
$773$	0.822919i	0.0295983i	−0.999890	+	0.0147992i	$0.995289\pi$
$774$	0	0
$775$	−30.7386	−1.10416
$776$	0	0
$777$	−6.54640	−0.234851
$778$	0	0
$779$	− 16.0000i	− 0.573259i
$780$	0	0
$781$	− 14.6307i	− 0.523527i
$782$	0	0
$783$	−11.1231	−0.397507
$784$	0	0
$785$	37.3693	1.33377
$786$	0	0
$787$	16.0000i	0.570338i	0.958477	+	0.285169i	$0.0920498\pi$
$787$	16.0000i	0.570338i	−0.958477	+	0.285169i	$0.907950\pi$
$788$	0	0
$789$	15.2311i	0.542240i
$790$	0	0
$791$	−22.6307	−0.804654
$792$	0	0
$793$	−2.87689	−0.102162
$794$	0	0
$795$	28.4924i	1.01052i
$796$	0	0
$797$	46.4924i	1.64685i	0.567428	+	0.823423i	$0.307939\pi$
$797$	46.4924i	1.64685i	−0.567428	+	0.823423i	$0.692061\pi$
$798$	0	0
$799$	84.7926	2.99975
$800$	0	0
$801$	−5.61553	−0.198415
$802$	0	0
$803$	18.7386i	0.661272i
$804$	0	0
$805$	− 44.4924i	− 1.56815i
$806$	0	0
$807$	−20.4924	−0.721367
$808$	0	0
$809$	−15.1771	−0.533598	−0.266799	−	0.963752i	$-0.585966\pi$
$809$	−15.1771	−0.533598	−0.266799	+	0.963752i	$0.585966\pi$
$810$	0	0
$811$	2.73863i	0.0961664i	0.998843	+	0.0480832i	$0.0153113\pi$
$811$	2.73863i	0.0961664i	−0.998843	+	0.0480832i	$0.984689\pi$
$812$	0	0
$813$	− 0.300187i	− 0.0105280i
$814$	0	0
$815$	44.4924	1.55850
$816$	0	0
$817$	−29.8617	−1.04473
$818$	0	0
$819$	0.876894i	0.0306412i
$820$	0	0
$821$	19.5616i	0.682703i	0.939936	+	0.341351i	$0.110885\pi$
$821$	19.5616i	0.682703i	−0.939936	+	0.341351i	$0.889115\pi$
$822$	0	0
$823$	−31.6155	−1.10205	−0.551024	−	0.834489i	$-0.685763\pi$
$823$	−31.6155	−1.10205	−0.551024	+	0.834489i	$0.685763\pi$
$824$	0	0
$825$	37.4773	1.30479
$826$	0	0
$827$	21.8617i	0.760207i	0.924944	+	0.380104i	$0.124112\pi$
$827$	21.8617i	0.760207i	−0.924944	+	0.380104i	$0.875888\pi$
$828$	0	0
$829$	14.4924i	0.503343i	0.967813	+	0.251671i	$0.0809802\pi$
$829$	14.4924i	0.503343i	−0.967813	+	0.251671i	$0.919020\pi$
$830$	0	0
$831$	−7.23106	−0.250843
$832$	0	0
$833$	30.4924	1.05650
$834$	0	0
$835$	8.00000i	0.276851i
$836$	0	0
$837$	22.2462i	0.768942i
$838$	0	0
$839$	−38.7386	−1.33741	−0.668703	−	0.743530i	$-0.733150\pi$
$839$	−38.7386	−1.33741	−0.668703	+	0.743530i	$0.733150\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 15.6155i	− 0.537827i
$844$	0	0
$845$	− 3.56155i	− 0.122521i
$846$	0	0
$847$	−1.94602	−0.0668662
$848$	0	0
$849$	−38.2462	−1.31261
$850$	0	0
$851$	− 21.4773i	− 0.736231i
$852$	0	0
$853$	− 17.3153i	− 0.592866i	−0.955054	−	0.296433i	$-0.904203\pi$
$853$	− 17.3153i	− 0.592866i	0.955054	−	0.296433i	$-0.0957971\pi$
$854$	0	0
$855$	−6.24621	−0.213616
$856$	0	0
$857$	8.73863	0.298506	0.149253	−	0.988799i	$-0.452313\pi$
$857$	8.73863	0.298506	0.149253	+	0.988799i	$0.452313\pi$
$858$	0	0
$859$	− 52.9848i	− 1.80782i	−0.427723	−	0.903910i	$-0.640684\pi$
$859$	− 52.9848i	− 1.80782i	0.427723	−	0.903910i	$-0.359316\pi$
$860$	0	0
$861$	12.4924i	0.425741i
$862$	0	0
$863$	4.30019	0.146380	0.0731900	−	0.997318i	$-0.476682\pi$
$863$	4.30019	0.146380	0.0731900	+	0.997318i	$0.476682\pi$
$864$	0	0
$865$	−46.7386	−1.58916
$866$	0	0
$867$	43.2311i	1.46820i
$868$	0	0
$869$	24.9848i	0.847553i
$870$	0	0
$871$	−3.12311	−0.105822
$872$	0	0
$873$	4.63068	0.156725
$874$	0	0
$875$	14.9309i	0.504756i
$876$	0	0
$877$	25.3153i	0.854838i	0.904054	+	0.427419i	$0.140577\pi$
$877$	25.3153i	0.854838i	−0.904054	+	0.427419i	$0.859423\pi$
$878$	0	0
$879$	−30.5464	−1.03030
$880$	0	0
$881$	−10.1922	−0.343385	−0.171693	−	0.985151i	$-0.554924\pi$
$881$	−10.1922	−0.343385	−0.171693	+	0.985151i	$0.554924\pi$
$882$	0	0
$883$	− 12.6847i	− 0.426873i	−0.976957	−	0.213436i	$-0.931534\pi$
$883$	− 12.6847i	− 0.426873i	0.976957	−	0.213436i	$-0.0684656\pi$
$884$	0	0
$885$	17.3693i	0.583863i
$886$	0	0
$887$	−53.4773	−1.79559	−0.897795	−	0.440413i	$-0.854832\pi$
$887$	−53.4773	−1.79559	−0.897795	+	0.440413i	$0.854832\pi$
$888$	0	0
$889$	−9.75379	−0.327132
$890$	0	0
$891$	− 21.8617i	− 0.732396i
$892$	0	0
$893$	− 39.6155i	− 1.32568i
$894$	0	0
$895$	−16.6847	−0.557707
$896$	0	0
$897$	12.4924	0.417110
$898$	0	0
$899$	8.00000i	0.266815i
$900$	0	0
$901$	34.2462i	1.14091i
$902$	0	0
$903$	23.3153	0.775886
$904$	0	0
$905$	52.9848	1.76128
$906$	0	0
$907$	− 0.192236i	− 0.00638309i	−0.999995	−	0.00319154i	$-0.998984\pi$
$907$	− 0.192236i	− 0.00638309i	0.999995	−	0.00319154i	$-0.00101590\pi$
$908$	0	0
$909$	0.630683i	0.0209184i
$910$	0	0
$911$	−9.36932	−0.310419	−0.155210	−	0.987882i	$-0.549605\pi$
$911$	−9.36932	−0.310419	−0.155210	+	0.987882i	$0.549605\pi$
$912$	0	0
$913$	−44.4924	−1.47248
$914$	0	0
$915$	16.0000i	0.528944i
$916$	0	0
$917$	24.6847i	0.815159i
$918$	0	0
$919$	40.9848	1.35197	0.675983	−	0.736918i	$-0.263719\pi$
$919$	40.9848	1.35197	0.675983	+	0.736918i	$0.263719\pi$
$920$	0	0
$921$	7.61553	0.250940
$922$	0	0
$923$	4.68466i	0.154197i
$924$	0	0
$925$	20.6307i	0.678333i
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	47.8617	1.57029	0.785146	−	0.619310i	$-0.212588\pi$
$929$	47.8617	1.57029	0.785146	+	0.619310i	$0.212588\pi$
$930$	0	0
$931$	− 14.2462i	− 0.466901i
$932$	0	0
$933$	29.8617i	0.977629i
$934$	0	0
$935$	74.3542	2.43164
$936$	0	0
$937$	−12.7386	−0.416153	−0.208077	−	0.978113i	$-0.566720\pi$
$937$	−12.7386	−0.416153	−0.208077	+	0.978113i	$0.566720\pi$
$938$	0	0
$939$	− 9.66950i	− 0.315552i
$940$	0	0
$941$	30.7926i	1.00381i	0.864923	+	0.501905i	$0.167367\pi$
$941$	30.7926i	1.00381i	−0.864923	+	0.501905i	$0.832633\pi$
$942$	0	0
$943$	−40.9848	−1.33465
$944$	0	0
$945$	30.9309	1.00618
$946$	0	0
$947$	17.3693i	0.564427i	0.959352	+	0.282213i	$0.0910686\pi$
$947$	17.3693i	0.564427i	−0.959352	+	0.282213i	$0.908931\pi$
$948$	0	0
$949$	− 6.00000i	− 0.194768i
$950$	0	0
$951$	6.63068	0.215015
$952$	0	0
$953$	−59.6695	−1.93288	−0.966442	−	0.256883i	$-0.917304\pi$
$953$	−59.6695	−1.93288	−0.966442	+	0.256883i	$0.917304\pi$
$954$	0	0
$955$	− 33.3693i	− 1.07981i
$956$	0	0
$957$	− 9.75379i	− 0.315295i
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	2.24621i	0.0723831i
$964$	0	0
$965$	12.0000i	0.386294i
$966$	0	0
$967$	1.56155	0.0502162	0.0251081	−	0.999685i	$-0.492007\pi$
$967$	1.56155	0.0502162	0.0251081	+	0.999685i	$0.492007\pi$
$968$	0	0
$969$	32.6004	1.04727
$970$	0	0
$971$	− 19.3153i	− 0.619859i	−0.950760	−	0.309929i	$-0.899695\pi$
$971$	− 19.3153i	− 0.619859i	0.950760	−	0.309929i	$-0.100305\pi$
$972$	0	0
$973$	7.31534i	0.234519i
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	21.5076	0.688088	0.344044	−	0.938953i	$-0.388203\pi$
$977$	21.5076	0.688088	0.344044	+	0.938953i	$0.388203\pi$
$978$	0	0
$979$	− 31.2311i	− 0.998149i
$980$	0	0
$981$	− 10.0000i	− 0.319275i
$982$	0	0
$983$	10.9309	0.348641	0.174320	−	0.984689i	$-0.444227\pi$
$983$	10.9309	0.348641	0.174320	+	0.984689i	$0.444227\pi$
$984$	0	0
$985$	−52.3002	−1.66642
$986$	0	0
$987$	30.9309i	0.984540i
$988$	0	0
$989$	76.4924i	2.43232i
$990$	0	0
$991$	21.8617	0.694461	0.347231	−	0.937780i	$-0.387122\pi$
$991$	21.8617	0.694461	0.347231	+	0.937780i	$0.387122\pi$
$992$	0	0
$993$	−44.4924	−1.41192
$994$	0	0
$995$	11.1231i	0.352626i
$996$	0	0
$997$	− 16.2462i	− 0.514523i	−0.966342	−	0.257261i	$-0.917180\pi$
$997$	− 16.2462i	− 0.514523i	0.966342	−	0.257261i	$-0.0828201\pi$
$998$	0	0
$999$	14.9309	0.472392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.b.y.1665.3		4
4.3	odd	2		3328.2.b.w.1665.2		4
8.3	odd	2		3328.2.b.w.1665.3		4
8.5	even	2	inner	3328.2.b.y.1665.2		4
16.3	odd	4		208.2.a.e.1.2		2
16.5	even	4		832.2.a.k.1.2		2
16.11	odd	4		832.2.a.n.1.1		2
16.13	even	4		104.2.a.b.1.1	✓	2
48.5	odd	4		7488.2.a.cu.1.2		2
48.11	even	4		7488.2.a.cv.1.2		2
48.29	odd	4		936.2.a.j.1.1		2
48.35	even	4		1872.2.a.u.1.1		2
80.13	odd	4		2600.2.d.k.1249.2		4
80.19	odd	4		5200.2.a.bw.1.1		2
80.29	even	4		2600.2.a.p.1.2		2
80.77	odd	4		2600.2.d.k.1249.3		4
112.13	odd	4		5096.2.a.m.1.2		2
208.29	even	12		1352.2.i.f.529.2		4
208.45	odd	12		1352.2.o.d.361.3		8
208.51	odd	4		2704.2.a.p.1.2		2
208.61	even	12		1352.2.i.f.1329.2		4
208.77	even	4		1352.2.a.g.1.1		2
208.83	even	4		2704.2.f.k.337.3		4
208.93	odd	12		1352.2.o.d.1161.3		8
208.99	even	4		2704.2.f.k.337.4		4
208.109	odd	4		1352.2.f.c.337.1		4
208.125	odd	4		1352.2.f.c.337.2		4
208.141	odd	12		1352.2.o.d.1161.4		8
208.173	even	12		1352.2.i.d.1329.2		4
208.189	odd	12		1352.2.o.d.361.4		8
208.205	even	12		1352.2.i.d.529.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.a.b.1.1	✓	2	16.13	even	4
208.2.a.e.1.2		2	16.3	odd	4
832.2.a.k.1.2		2	16.5	even	4
832.2.a.n.1.1		2	16.11	odd	4
936.2.a.j.1.1		2	48.29	odd	4
1352.2.a.g.1.1		2	208.77	even	4
1352.2.f.c.337.1		4	208.109	odd	4
1352.2.f.c.337.2		4	208.125	odd	4
1352.2.i.d.529.2		4	208.205	even	12
1352.2.i.d.1329.2		4	208.173	even	12
1352.2.i.f.529.2		4	208.29	even	12
1352.2.i.f.1329.2		4	208.61	even	12
1352.2.o.d.361.3		8	208.45	odd	12
1352.2.o.d.361.4		8	208.189	odd	12
1352.2.o.d.1161.3		8	208.93	odd	12
1352.2.o.d.1161.4		8	208.141	odd	12
1872.2.a.u.1.1		2	48.35	even	4
2600.2.a.p.1.2		2	80.29	even	4
2600.2.d.k.1249.2		4	80.13	odd	4
2600.2.d.k.1249.3		4	80.77	odd	4
2704.2.a.p.1.2		2	208.51	odd	4
2704.2.f.k.337.3		4	208.83	even	4
2704.2.f.k.337.4		4	208.99	even	4
3328.2.b.w.1665.2		4	4.3	odd	2
3328.2.b.w.1665.3		4	8.3	odd	2
3328.2.b.y.1665.2		4	8.5	even	2	inner
3328.2.b.y.1665.3		4	1.1	even	1	trivial
5096.2.a.m.1.2		2	112.13	odd	4
5200.2.a.bw.1.1		2	80.19	odd	4
7488.2.a.cu.1.2		2	48.5	odd	4
7488.2.a.cv.1.2		2	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} +3.56155i q^{5} -1.56155 q^{7} +0.561553 q^{9} +3.12311i q^{11} -1.00000i q^{13} -5.56155 q^{15} -6.68466 q^{17} +3.12311i q^{19} -2.43845i q^{21} +8.00000 q^{23} -7.68466 q^{25} +5.56155i q^{27} +2.00000i q^{29} +4.00000 q^{31} -4.87689 q^{33} -5.56155i q^{35} -2.68466i q^{37} +1.56155 q^{39} -5.12311 q^{41} +9.56155i q^{43} +2.00000i q^{45} -12.6847 q^{47} -4.56155 q^{49} -10.4384i q^{51} -5.12311i q^{53} -11.1231 q^{55} -4.87689 q^{57} -3.12311i q^{59} -2.87689i q^{61} -0.876894 q^{63} +3.56155 q^{65} -3.12311i q^{67} +12.4924i q^{69} -4.68466 q^{71} +6.00000 q^{73} -12.0000i q^{75} -4.87689i q^{77} +8.00000 q^{79} -7.00000 q^{81} +14.2462i q^{83} -23.8078i q^{85} -3.12311 q^{87} -10.0000 q^{89} +1.56155i q^{91} +6.24621i q^{93} -11.1231 q^{95} +8.24621 q^{97} +1.75379i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7} - 6 q^{9} - 14 q^{15} - 2 q^{17} + 32 q^{23} - 6 q^{25} + 16 q^{31} - 36 q^{33} - 2 q^{39} - 4 q^{41} - 26 q^{47} - 10 q^{49} - 28 q^{55} - 36 q^{57} - 20 q^{63} + 6 q^{65} + 6 q^{71} + 24 q^{73}+ \cdots - 28 q^{95}+O(q^{100}) \) 4 * q + 2 * q^7 - 6 * q^9 - 14 * q^15 - 2 * q^17 + 32 * q^23 - 6 * q^25 + 16 * q^31 - 36 * q^33 - 2 * q^39 - 4 * q^41 - 26 * q^47 - 10 * q^49 - 28 * q^55 - 36 * q^57 - 20 * q^63 + 6 * q^65 + 6 * q^71 + 24 * q^73 + 32 * q^79 - 28 * q^81 + 4 * q^87 - 40 * q^89 - 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more