LMFDB - Embedded newform 1352.2.f.c.337.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1352,2,Mod(337,1352)] 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("1352.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,0,0,0,0,6,0,0,0,0,0,0,0,2] 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:	$10.7957743533$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	1352.337
Dual form			1352.2.f.c.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{3} -3.56155i q^{5} +1.56155i q^{7} -0.561553 q^{9} +3.12311i q^{11} +5.56155i q^{15} +6.68466 q^{17} +3.12311i q^{19} -2.43845i q^{21} +8.00000 q^{23} -7.68466 q^{25} +5.56155 q^{27} -2.00000 q^{29} -4.00000i q^{31} -4.87689i q^{33} +5.56155 q^{35} -2.68466i q^{37} -5.12311i q^{41} -9.56155 q^{43} +2.00000i q^{45} -12.6847i q^{47} +4.56155 q^{49} -10.4384 q^{51} -5.12311 q^{53} +11.1231 q^{55} -4.87689i q^{57} -3.12311i q^{59} +2.87689 q^{61} -0.876894i q^{63} -3.12311i q^{67} -12.4924 q^{69} -4.68466i q^{71} -6.00000i q^{73} +12.0000 q^{75} -4.87689 q^{77} +8.00000 q^{79} -7.00000 q^{81} +14.2462i q^{83} -23.8078i q^{85} +3.12311 q^{87} +10.0000i q^{89} +6.24621i q^{93} +11.1231 q^{95} -8.24621i q^{97} -1.75379i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{9} + 2 q^{17} + 32 q^{23} - 6 q^{25} + 14 q^{27} - 8 q^{29} + 14 q^{35} - 30 q^{43} + 10 q^{49} - 50 q^{51} - 4 q^{53} + 28 q^{55} + 28 q^{61} + 16 q^{69} + 48 q^{75} - 36 q^{77} + 32 q^{79}+ \cdots + 28 q^{95}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^9 + 2 * q^17 + 32 * q^23 - 6 * q^25 + 14 * q^27 - 8 * q^29 + 14 * q^35 - 30 * q^43 + 10 * q^49 - 50 * q^51 - 4 * q^53 + 28 * q^55 + 28 * q^61 + 16 * q^69 + 48 * q^75 - 36 * q^77 + 32 * q^79 - 28 * q^81 - 4 * q^87 + 28 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1185$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	− 3.56155i	− 1.59277i	−0.604787	−	0.796387i	$-0.706742\pi$
$5$	− 3.56155i	− 1.59277i	0.604787	−	0.796387i	$-0.293258\pi$
$6$	0	0
$7$	1.56155i	0.590211i	0.955465	+	0.295106i	$0.0953549\pi$
$7$	1.56155i	0.590211i	−0.955465	+	0.295106i	$0.904645\pi$
$8$	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
$13$	0	0
$13$	0	0
$14$	0	0
$15$	5.56155i	1.43599i
$16$	0	0
$17$	6.68466	1.62127	0.810634	−	0.585553i	$-0.199123\pi$
$17$	6.68466	1.62127	0.810634	+	0.585553i	$0.199123\pi$
$18$	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
$27$	5.56155	1.07032
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	0	0
$33$	− 4.87689i	− 0.848958i
$34$	0	0
$35$	5.56155	0.940074
$36$	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
$39$	0	0
$40$	0	0
$41$	− 5.12311i	− 0.800095i	−0.916494	−	0.400047i	$-0.868994\pi$
$41$	− 5.12311i	− 0.800095i	0.916494	−	0.400047i	$-0.131006\pi$
$42$	0	0
$43$	−9.56155	−1.45812	−0.729062	−	0.684448i	$-0.760043\pi$
$43$	−9.56155	−1.45812	−0.729062	+	0.684448i	$0.760043\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	− 12.6847i	− 1.85025i	−0.379666	−	0.925124i	$-0.623961\pi$
$47$	− 12.6847i	− 1.85025i	0.379666	−	0.925124i	$-0.376039\pi$
$48$	0	0
$49$	4.56155	0.651650
$50$	0	0
$51$	−10.4384	−1.46167
$52$	0	0
$53$	−5.12311	−0.703713	−0.351856	−	0.936054i	$-0.614449\pi$
$53$	−5.12311	−0.703713	−0.351856	+	0.936054i	$0.614449\pi$
$54$	0	0
$55$	11.1231	1.49984
$56$	0	0
$57$	− 4.87689i	− 0.645960i
$58$	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
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	0	0
$63$	− 0.876894i	− 0.110478i
$64$	0	0
$65$	0	0
$66$	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
$69$	−12.4924	−1.50391
$70$	0	0
$71$	− 4.68466i	− 0.555967i	−0.960586	−	0.277983i	$-0.910334\pi$
$71$	− 4.68466i	− 0.555967i	0.960586	−	0.277983i	$-0.0896660\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	0	0
$77$	−4.87689	−0.555774
$78$	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
$89$	10.0000i	1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$	10.0000i	1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.24621i	0.647702i
$94$	0	0
$95$	11.1231	1.14121
$96$	0	0
$97$	− 8.24621i	− 0.837276i	−0.908153	−	0.418638i	$-0.862508\pi$
$97$	− 8.24621i	− 0.837276i	0.908153	−	0.418638i	$-0.137492\pi$
$98$	0	0
$99$	− 1.75379i	− 0.176262i
$100$	0	0
$101$	−1.12311	−0.111753	−0.0558766	−	0.998438i	$-0.517795\pi$
$101$	−1.12311	−0.111753	−0.0558766	+	0.998438i	$0.517795\pi$
$102$	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
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	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
$111$	4.19224i	0.397909i
$112$	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	0
$118$	0	0
$119$	10.4384i	0.956891i
$120$	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	$-0.589384\pi$
$127$	−6.24621	−0.554262	−0.277131	+	0.960832i	$0.589384\pi$
$128$	0	0
$129$	14.9309	1.31459
$130$	0	0
$131$	15.8078	1.38113	0.690565	−	0.723270i	$-0.257362\pi$
$131$	15.8078	1.38113	0.690565	+	0.723270i	$0.257362\pi$
$132$	0	0
$133$	−4.87689	−0.422880
$134$	0	0
$135$	− 19.8078i	− 1.70478i
$136$	0	0
$137$	5.12311i	0.437696i	0.975759	+	0.218848i	$0.0702300\pi$
$137$	5.12311i	0.437696i	−0.975759	+	0.218848i	$0.929770\pi$
$138$	0	0
$139$	−4.68466	−0.397348	−0.198674	−	0.980066i	$-0.563663\pi$
$139$	−4.68466	−0.397348	−0.198674	+	0.980066i	$0.563663\pi$
$140$	0	0
$141$	19.8078i	1.66811i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.12311i	0.591542i
$146$	0	0
$147$	−7.12311	−0.587504
$148$	0	0
$149$	0.246211i	0.0201704i	0.999949	+	0.0100852i	$0.00321028\pi$
$149$	0.246211i	0.0201704i	−0.999949	+	0.0100852i	$0.996790\pi$
$150$	0	0
$151$	6.43845i	0.523953i	0.965074	+	0.261977i	$0.0843744\pi$
$151$	6.43845i	0.523953i	−0.965074	+	0.261977i	$0.915626\pi$
$152$	0	0
$153$	−3.75379	−0.303476
$154$	0	0
$155$	−14.2462	−1.14428
$156$	0	0
$157$	10.4924	0.837386	0.418693	−	0.908128i	$-0.362488\pi$
$157$	10.4924	0.837386	0.418693	+	0.908128i	$0.362488\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	12.4924i	0.984541i
$162$	0	0
$163$	12.4924i	0.978482i	0.872149	+	0.489241i	$0.162726\pi$
$163$	12.4924i	0.978482i	−0.872149	+	0.489241i	$0.837274\pi$
$164$	0	0
$165$	−17.3693	−1.35220
$166$	0	0
$167$	− 2.24621i	− 0.173817i	−0.996216	−	0.0869085i	$-0.972301\pi$
$167$	− 2.24621i	− 0.173817i	0.996216	−	0.0869085i	$-0.0276988\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 1.75379i	− 0.134116i
$172$	0	0
$173$	13.1231	0.997731	0.498866	−	0.866679i	$-0.333750\pi$
$173$	13.1231	0.997731	0.498866	+	0.866679i	$0.333750\pi$
$174$	0	0
$175$	− 12.0000i	− 0.907115i
$176$	0	0
$177$	4.87689i	0.366570i
$178$	0	0
$179$	4.68466	0.350148	0.175074	−	0.984555i	$-0.443984\pi$
$179$	4.68466	0.350148	0.175074	+	0.984555i	$0.443984\pi$
$180$	0	0
$181$	14.8769	1.10579	0.552895	−	0.833251i	$-0.313523\pi$
$181$	14.8769	1.10579	0.552895	+	0.833251i	$0.313523\pi$
$182$	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
$193$	3.36932i	0.242529i	0.992620	+	0.121264i	$0.0386949\pi$
$193$	3.36932i	0.242529i	−0.992620	+	0.121264i	$0.961305\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.6847i	− 1.04624i	−0.852259	−	0.523119i	$-0.824768\pi$
$197$	− 14.6847i	− 1.04624i	0.852259	−	0.523119i	$-0.175232\pi$
$198$	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
$201$	4.87689i	0.343990i
$202$	0	0
$203$	− 3.12311i	− 0.219199i
$204$	0	0
$205$	−18.2462	−1.27437
$206$	0	0
$207$	−4.49242	−0.312245
$208$	0	0
$209$	−9.75379	−0.674684
$210$	0	0
$211$	3.31534	0.228238	0.114119	−	0.993467i	$-0.463596\pi$
$211$	3.31534	0.228238	0.114119	+	0.993467i	$0.463596\pi$
$212$	0	0
$213$	7.31534i	0.501239i
$214$	0	0
$215$	34.0540i	2.32246i
$216$	0	0
$217$	6.24621	0.424020
$218$	0	0
$219$	9.36932i	0.633120i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3.31534i	0.222012i	0.993820	+	0.111006i	$0.0354072\pi$
$223$	3.31534i	0.222012i	−0.993820	+	0.111006i	$0.964593\pi$
$224$	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
$235$	−45.1771	−2.94703
$236$	0	0
$237$	−12.4924	−0.811470
$238$	0	0
$239$	− 1.56155i	− 0.101008i	−0.998724	−	0.0505042i	$-0.983917\pi$
$239$	− 1.56155i	− 0.101008i	0.998724	−	0.0505042i	$-0.0160828\pi$
$240$	0	0
$241$	− 20.2462i	− 1.30417i	−0.758145	−	0.652087i	$-0.773894\pi$
$241$	− 20.2462i	− 1.30417i	0.758145	−	0.652087i	$-0.226106\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	− 16.2462i	− 1.03793i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 22.2462i	− 1.40980i
$250$	0	0
$251$	−5.75379	−0.363176	−0.181588	−	0.983375i	$-0.558124\pi$
$251$	−5.75379	−0.363176	−0.181588	+	0.983375i	$0.558124\pi$
$252$	0	0
$253$	24.9848i	1.57078i
$254$	0	0
$255$	37.1771i	2.32812i
$256$	0	0
$257$	−15.5616	−0.970703	−0.485351	−	0.874319i	$-0.661308\pi$
$257$	−15.5616	−0.970703	−0.485351	+	0.874319i	$0.661308\pi$
$258$	0	0
$259$	4.19224	0.260493
$260$	0	0
$261$	1.12311	0.0695185
$262$	0	0
$263$	−9.75379	−0.601444	−0.300722	−	0.953712i	$-0.597228\pi$
$263$	−9.75379	−0.601444	−0.300722	+	0.953712i	$0.597228\pi$
$264$	0	0
$265$	18.2462i	1.12086i
$266$	0	0
$267$	− 15.6155i	− 0.955655i
$268$	0	0
$269$	−13.1231	−0.800130	−0.400065	−	0.916487i	$-0.631012\pi$
$269$	−13.1231	−0.800130	−0.400065	+	0.916487i	$0.631012\pi$
$270$	0	0
$271$	− 0.192236i	− 0.0116775i	−0.999983	−	0.00583875i	$-0.998141\pi$
$271$	− 0.192236i	− 0.0116775i	0.999983	−	0.00583875i	$-0.00185854\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 24.0000i	− 1.44725i
$276$	0	0
$277$	−4.63068	−0.278231	−0.139115	−	0.990276i	$-0.544426\pi$
$277$	−4.63068	−0.278231	−0.139115	+	0.990276i	$0.544426\pi$
$278$	0	0
$279$	2.24621i	0.134477i
$280$	0	0
$281$	10.0000i	0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$	10.0000i	0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	−24.4924	−1.45592	−0.727962	−	0.685618i	$-0.759532\pi$
$283$	−24.4924	−1.45592	−0.727962	+	0.685618i	$0.759532\pi$
$284$	0	0
$285$	−17.3693	−1.02887
$286$	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
$297$	17.3693i	1.00787i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 14.9309i	− 0.860601i
$302$	0	0
$303$	1.75379	0.100753
$304$	0	0
$305$	− 10.2462i	− 0.586696i
$306$	0	0
$307$	4.87689i	0.278339i	0.990269	+	0.139170i	$0.0444433\pi$
$307$	4.87689i	0.278339i	−0.990269	+	0.139170i	$0.955557\pi$
$308$	0	0
$309$	−22.2462	−1.26554
$310$	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	$-0.444007\pi$
$313$	6.19224	0.350006	0.175003	+	0.984568i	$0.444007\pi$
$314$	0	0
$315$	−3.12311	−0.175967
$316$	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
$319$	− 6.24621i	− 0.349721i
$320$	0	0
$321$	−6.24621	−0.348630
$322$	0	0
$323$	20.8769i	1.16162i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.8078i	1.53777i
$328$	0	0
$329$	19.8078	1.09204
$330$	0	0
$331$	− 28.4924i	− 1.56609i	−0.621968	−	0.783043i	$-0.713667\pi$
$331$	− 28.4924i	− 1.56609i	0.621968	−	0.783043i	$-0.286333\pi$
$332$	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	$-0.127526\pi$
$337$	33.8078	1.84163	0.920813	+	0.390004i	$0.127526\pi$
$338$	0	0
$339$	−22.6307	−1.22913
$340$	0	0
$341$	12.4924	0.676503
$342$	0	0
$343$	18.0540i	0.974823i
$344$	0	0
$345$	44.4924i	2.39539i
$346$	0	0
$347$	−4.68466	−0.251486	−0.125743	−	0.992063i	$-0.540131\pi$
$347$	−4.68466	−0.251486	−0.125743	+	0.992063i	$0.540131\pi$
$348$	0	0
$349$	− 16.9309i	− 0.906289i	−0.891437	−	0.453144i	$-0.850302\pi$
$349$	− 16.9309i	− 0.906289i	0.891437	−	0.453144i	$-0.149698\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.3693i	0.818026i	0.912529	+	0.409013i	$0.134127\pi$
$353$	15.3693i	0.818026i	−0.912529	+	0.409013i	$0.865873\pi$
$354$	0	0
$355$	−16.6847	−0.885530
$356$	0	0
$357$	− 16.3002i	− 0.862697i
$358$	0	0
$359$	− 2.24621i	− 0.118550i	−0.998242	−	0.0592752i	$-0.981121\pi$
$359$	− 2.24621i	− 0.118550i	0.998242	−	0.0592752i	$-0.0188790\pi$
$360$	0	0
$361$	9.24621	0.486643
$362$	0	0
$363$	−1.94602	−0.102140
$364$	0	0
$365$	−21.3693	−1.11852
$366$	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
$369$	2.87689i	0.149765i
$370$	0	0
$371$	− 8.00000i	− 0.415339i
$372$	0	0
$373$	29.6155	1.53343	0.766717	−	0.641985i	$-0.221889\pi$
$373$	29.6155	1.53343	0.766717	+	0.641985i	$0.221889\pi$
$374$	0	0
$375$	− 14.9309i	− 0.771027i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 22.2462i	− 1.14271i	−0.820703	−	0.571356i	$-0.806418\pi$
$379$	− 22.2462i	− 1.14271i	0.820703	−	0.571356i	$-0.193582\pi$
$380$	0	0
$381$	9.75379	0.499702
$382$	0	0
$383$	31.8078i	1.62530i	0.582752	+	0.812650i	$0.301976\pi$
$383$	31.8078i	1.62530i	−0.582752	+	0.812650i	$0.698024\pi$
$384$	0	0
$385$	17.3693i	0.885222i
$386$	0	0
$387$	5.36932	0.272938
$388$	0	0
$389$	28.7386	1.45711	0.728553	−	0.684989i	$-0.240193\pi$
$389$	28.7386	1.45711	0.728553	+	0.684989i	$0.240193\pi$
$390$	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	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	7.61553	0.381253
$400$	0	0
$401$	− 37.6155i	− 1.87843i	−0.343330	−	0.939215i	$-0.611555\pi$
$401$	− 37.6155i	− 1.87843i	0.343330	−	0.939215i	$-0.388445\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	24.9309i	1.23882i
$406$	0	0
$407$	8.38447	0.415603
$408$	0	0
$409$	− 21.1231i	− 1.04447i	−0.852802	−	0.522235i	$-0.825098\pi$
$409$	− 21.1231i	− 1.04447i	0.852802	−	0.522235i	$-0.174902\pi$
$410$	0	0
$411$	− 8.00000i	− 0.394611i
$412$	0	0
$413$	4.87689	0.239976
$414$	0	0
$415$	50.7386	2.49066
$416$	0	0
$417$	7.31534	0.358234
$418$	0	0
$419$	26.9309	1.31566	0.657830	−	0.753167i	$-0.271475\pi$
$419$	26.9309	1.31566	0.657830	+	0.753167i	$0.271475\pi$
$420$	0	0
$421$	18.6847i	0.910635i	0.890329	+	0.455317i	$0.150474\pi$
$421$	18.6847i	0.910635i	−0.890329	+	0.455317i	$0.849526\pi$
$422$	0	0
$423$	7.12311i	0.346337i
$424$	0	0
$425$	−51.3693	−2.49178
$426$	0	0
$427$	4.49242i	0.217404i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.4384i	0.695476i	0.937592	+	0.347738i	$0.113050\pi$
$431$	14.4384i	0.695476i	−0.937592	+	0.347738i	$0.886950\pi$
$432$	0	0
$433$	−39.1771	−1.88273	−0.941365	−	0.337389i	$-0.890456\pi$
$433$	−39.1771	−1.88273	−0.941365	+	0.337389i	$0.890456\pi$
$434$	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
$443$	0.192236	0.00913340	0.00456670	−	0.999990i	$-0.498546\pi$
$443$	0.192236	0.00913340	0.00456670	+	0.999990i	$0.498546\pi$
$444$	0	0
$445$	35.6155	1.68834
$446$	0	0
$447$	− 0.384472i	− 0.0181849i
$448$	0	0
$449$	− 16.7386i	− 0.789945i	−0.918693	−	0.394972i	$-0.870754\pi$
$449$	− 16.7386i	− 0.789945i	0.918693	−	0.394972i	$-0.129246\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	− 10.0540i	− 0.472377i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.7386i	1.15722i	0.815603	+	0.578612i	$0.196406\pi$
$457$	24.7386i	1.15722i	−0.815603	+	0.578612i	$0.803594\pi$
$458$	0	0
$459$	37.1771	1.73528
$460$	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
$463$	21.7538i	1.01098i	0.862831	+	0.505492i	$0.168689\pi$
$463$	21.7538i	1.01098i	−0.862831	+	0.505492i	$0.831311\pi$
$464$	0	0
$465$	22.2462	1.03164
$466$	0	0
$467$	−7.50758	−0.347409	−0.173705	−	0.984798i	$-0.555574\pi$
$467$	−7.50758	−0.347409	−0.173705	+	0.984798i	$0.555574\pi$
$468$	0	0
$469$	4.87689	0.225194
$470$	0	0
$471$	−16.3845	−0.754957
$472$	0	0
$473$	− 29.8617i	− 1.37304i
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	0	0
$477$	2.87689	0.131724
$478$	0	0
$479$	31.8078i	1.45333i	0.686990	+	0.726667i	$0.258932\pi$
$479$	31.8078i	1.45333i	−0.686990	+	0.726667i	$0.741068\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 19.5076i	− 0.887626i
$484$	0	0
$485$	−29.3693	−1.33359
$486$	0	0
$487$	− 18.2462i	− 0.826815i	−0.910546	−	0.413407i	$-0.864339\pi$
$487$	− 18.2462i	− 0.826815i	0.910546	−	0.413407i	$-0.135661\pi$
$488$	0	0
$489$	− 19.5076i	− 0.882163i
$490$	0	0
$491$	−38.0540	−1.71735	−0.858676	−	0.512519i	$-0.828712\pi$
$491$	−38.0540	−1.71735	−0.858676	+	0.512519i	$0.828712\pi$
$492$	0	0
$493$	−13.3693	−0.602124
$494$	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	$-0.373432\pi$
$503$	17.3693	0.774460	0.387230	+	0.921983i	$0.373432\pi$
$504$	0	0
$505$	4.00000i	0.177998i
$506$	0	0
$507$	0	0
$508$	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
$513$	17.3693i	0.766874i
$514$	0	0
$515$	− 50.7386i	− 2.23581i
$516$	0	0
$517$	39.6155	1.74229
$518$	0	0
$519$	−20.4924	−0.899518
$520$	0	0
$521$	2.68466	0.117617	0.0588085	−	0.998269i	$-0.481270\pi$
$521$	2.68466	0.117617	0.0588085	+	0.998269i	$0.481270\pi$
$522$	0	0
$523$	32.4924	1.42079	0.710397	−	0.703801i	$-0.248515\pi$
$523$	32.4924	1.42079	0.710397	+	0.703801i	$0.248515\pi$
$524$	0	0
$525$	18.7386i	0.817821i
$526$	0	0
$527$	− 26.7386i	− 1.16475i
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	1.75379i	0.0761079i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 14.2462i	− 0.615917i
$536$	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	$0.0458994\pi$
$541$	6.68466i	0.287396i	−0.989622	+	0.143698i	$0.954101\pi$
$542$	0	0
$543$	−23.2311	−0.996940
$544$	0	0
$545$	−63.4233	−2.71676
$546$	0	0
$547$	−33.5616	−1.43499	−0.717494	−	0.696564i	$-0.754711\pi$
$547$	−33.5616	−1.43499	−0.717494	+	0.696564i	$0.754711\pi$
$548$	0	0
$549$	−1.61553	−0.0689491
$550$	0	0
$551$	− 6.24621i	− 0.266098i
$552$	0	0
$553$	12.4924i	0.531232i
$554$	0	0
$555$	14.9309	0.633780
$556$	0	0
$557$	3.17708i	0.134617i	0.997732	+	0.0673086i	$0.0214412\pi$
$557$	3.17708i	0.134617i	−0.997732	+	0.0673086i	$0.978559\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 32.6004i	− 1.37639i
$562$	0	0
$563$	−30.0540	−1.26662	−0.633312	−	0.773897i	$-0.718305\pi$
$563$	−30.0540	−1.26662	−0.633312	+	0.773897i	$0.718305\pi$
$564$	0	0
$565$	− 51.6155i	− 2.17148i
$566$	0	0
$567$	− 10.9309i	− 0.459053i
$568$	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
$571$	−8.19224	−0.342834	−0.171417	−	0.985199i	$-0.554835\pi$
$571$	−8.19224	−0.342834	−0.171417	+	0.985199i	$0.554835\pi$
$572$	0	0
$573$	14.6307	0.611206
$574$	0	0
$575$	−61.4773	−2.56378
$576$	0	0
$577$	7.75379i	0.322794i	0.986890	+	0.161397i	$0.0516000\pi$
$577$	7.75379i	0.322794i	−0.986890	+	0.161397i	$0.948400\pi$
$578$	0	0
$579$	− 5.26137i	− 0.218655i
$580$	0	0
$581$	−22.2462	−0.922928
$582$	0	0
$583$	− 16.0000i	− 0.662652i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 40.9848i	− 1.69163i	−0.533480	−	0.845813i	$-0.679116\pi$
$587$	− 40.9848i	− 1.69163i	0.533480	−	0.845813i	$-0.320884\pi$
$588$	0	0
$589$	12.4924	0.514741
$590$	0	0
$591$	22.9309i	0.943250i
$592$	0	0
$593$	5.50758i	0.226169i	0.993585	+	0.113085i	$0.0360731\pi$
$593$	5.50758i	0.226169i	−0.993585	+	0.113085i	$0.963927\pi$
$594$	0	0
$595$	37.1771	1.52411
$596$	0	0
$597$	−4.87689	−0.199598
$598$	0	0
$599$	−1.36932	−0.0559488	−0.0279744	−	0.999609i	$-0.508906\pi$
$599$	−1.36932	−0.0559488	−0.0279744	+	0.999609i	$0.508906\pi$
$600$	0	0
$601$	−11.1771	−0.455923	−0.227961	−	0.973670i	$-0.573206\pi$
$601$	−11.1771	−0.455923	−0.227961	+	0.973670i	$0.573206\pi$
$602$	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
$609$	4.87689i	0.197622i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	0	0
$615$	28.4924	1.14893
$616$	0	0
$617$	35.8617i	1.44374i	0.692029	+	0.721870i	$0.256717\pi$
$617$	35.8617i	1.44374i	−0.692029	+	0.721870i	$0.743283\pi$
$618$	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
$621$	44.4924	1.78542
$622$	0	0
$623$	−15.6155	−0.625623
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	15.2311	0.608270
$628$	0	0
$629$	− 17.9460i	− 0.715555i
$630$	0	0
$631$	− 35.3153i	− 1.40588i	−0.711248	−	0.702941i	$-0.751870\pi$
$631$	− 35.3153i	− 1.40588i	0.711248	−	0.702941i	$-0.248130\pi$
$632$	0	0
$633$	−5.17708	−0.205770
$634$	0	0
$635$	22.2462i	0.882814i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.63068i	0.104068i
$640$	0	0
$641$	−8.24621	−0.325706	−0.162853	−	0.986650i	$-0.552070\pi$
$641$	−8.24621	−0.325706	−0.162853	+	0.986650i	$0.552070\pi$
$642$	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
$651$	−9.75379	−0.382281
$652$	0	0
$653$	−13.1231	−0.513547	−0.256773	−	0.966472i	$-0.582659\pi$
$653$	−13.1231	−0.513547	−0.256773	+	0.966472i	$0.582659\pi$
$654$	0	0
$655$	− 56.3002i	− 2.19983i
$656$	0	0
$657$	3.36932i	0.131450i
$658$	0	0
$659$	−16.4924	−0.642454	−0.321227	−	0.947002i	$-0.604095\pi$
$659$	−16.4924	−0.642454	−0.321227	+	0.947002i	$0.604095\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	17.3693i	0.673553i
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	− 5.17708i	− 0.200158i
$670$	0	0
$671$	8.98485i	0.346856i
$672$	0	0
$673$	−34.3002	−1.32218	−0.661088	−	0.750309i	$-0.729905\pi$
$673$	−34.3002	−1.32218	−0.661088	+	0.750309i	$0.729905\pi$
$674$	0	0
$675$	−42.7386	−1.64501
$676$	0	0
$677$	23.3693	0.898156	0.449078	−	0.893493i	$-0.351753\pi$
$677$	23.3693	0.898156	0.449078	+	0.893493i	$0.351753\pi$
$678$	0	0
$679$	12.8769	0.494170
$680$	0	0
$681$	− 12.4924i	− 0.478711i
$682$	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
$685$	18.2462	0.697152
$686$	0	0
$687$	21.5616i	0.822625i
$688$	0	0
$689$	0	0
$690$	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
$693$	2.73863	0.104032
$694$	0	0
$695$	16.6847i	0.632885i
$696$	0	0
$697$	− 34.2462i	− 1.29717i
$698$	0	0
$699$	21.5616	0.815533
$700$	0	0
$701$	−47.3693	−1.78911	−0.894557	−	0.446953i	$-0.852509\pi$
$701$	−47.3693	−1.78911	−0.894557	+	0.446953i	$0.852509\pi$
$702$	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
$713$	− 32.0000i	− 1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.43845i	0.0910655i
$718$	0	0
$719$	4.87689	0.181877	0.0909387	−	0.995856i	$-0.471013\pi$
$719$	4.87689	0.181877	0.0909387	+	0.995856i	$0.471013\pi$
$720$	0	0
$721$	22.2462i	0.828492i
$722$	0	0
$723$	31.6155i	1.17579i
$724$	0	0
$725$	15.3693	0.570802
$726$	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
$731$	−63.9157	−2.36401
$732$	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
$735$	25.3693i	0.935761i
$736$	0	0
$737$	9.75379	0.359285
$738$	0	0
$739$	6.24621i	0.229771i	0.993379	+	0.114885i	$0.0366501\pi$
$739$	6.24621i	0.229771i	−0.993379	+	0.114885i	$0.963350\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	43.9157i	1.61111i	0.592520	+	0.805556i	$0.298133\pi$
$743$	43.9157i	1.61111i	−0.592520	+	0.805556i	$0.701867\pi$
$744$	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	$-0.231120\pi$
$751$	40.9848	1.49556	0.747779	+	0.663948i	$0.231120\pi$
$752$	0	0
$753$	8.98485	0.327426
$754$	0	0
$755$	22.9309	0.834540
$756$	0	0
$757$	−5.12311	−0.186202	−0.0931012	−	0.995657i	$-0.529678\pi$
$757$	−5.12311	−0.186202	−0.0931012	+	0.995657i	$0.529678\pi$
$758$	0	0
$759$	− 39.0152i	− 1.41616i
$760$	0	0
$761$	13.5076i	0.489649i	0.969567	+	0.244825i	$0.0787304\pi$
$761$	13.5076i	0.489649i	−0.969567	+	0.244825i	$0.921270\pi$
$762$	0	0
$763$	27.8078	1.00671
$764$	0	0
$765$	13.3693i	0.483369i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	43.8617i	1.58169i	0.612013	+	0.790847i	$0.290360\pi$
$769$	43.8617i	1.58169i	−0.612013	+	0.790847i	$0.709640\pi$
$770$	0	0
$771$	24.3002	0.875150
$772$	0	0
$773$	− 0.822919i	− 0.0295983i	−0.999890	−	0.0147992i	$-0.995289\pi$
$773$	− 0.822919i	− 0.0295983i	0.999890	−	0.0147992i	$-0.00471089\pi$
$774$	0	0
$775$	30.7386i	1.10416i
$776$	0	0
$777$	−6.54640	−0.234851
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	14.6307	0.523527
$782$	0	0
$783$	−11.1231	−0.397507
$784$	0	0
$785$	− 37.3693i	− 1.33377i
$786$	0	0
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	0	0
$789$	15.2311	0.542240
$790$	0	0
$791$	22.6307i	0.804654i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 28.4924i	− 1.01052i
$796$	0	0
$797$	46.4924	1.64685	0.823423	−	0.567428i	$-0.192061\pi$
$797$	46.4924	1.64685	0.823423	+	0.567428i	$0.192061\pi$
$798$	0	0
$799$	− 84.7926i	− 2.99975i
$800$	0	0
$801$	− 5.61553i	− 0.198415i
$802$	0	0
$803$	18.7386	0.661272
$804$	0	0
$805$	44.4924	1.56815
$806$	0	0
$807$	20.4924	0.721367
$808$	0	0
$809$	15.1771	0.533598	0.266799	−	0.963752i	$-0.414034\pi$
$809$	15.1771	0.533598	0.266799	+	0.963752i	$0.414034\pi$
$810$	0	0
$811$	− 2.73863i	− 0.0961664i	−0.998843	−	0.0480832i	$-0.984689\pi$
$811$	− 2.73863i	− 0.0961664i	0.998843	−	0.0480832i	$-0.0153113\pi$
$812$	0	0
$813$	0.300187i	0.0105280i
$814$	0	0
$815$	44.4924	1.55850
$816$	0	0
$817$	− 29.8617i	− 1.04473i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 19.5616i	− 0.682703i	−0.939936	−	0.341351i	$-0.889115\pi$
$821$	− 19.5616i	− 0.682703i	0.939936	−	0.341351i	$-0.110885\pi$
$822$	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
$825$	37.4773i	1.30479i
$826$	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
$829$	14.4924	0.503343	0.251671	−	0.967813i	$-0.419020\pi$
$829$	14.4924	0.503343	0.251671	+	0.967813i	$0.419020\pi$
$830$	0	0
$831$	7.23106	0.250843
$832$	0	0
$833$	30.4924	1.05650
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	− 22.2462i	− 0.768942i
$838$	0	0
$839$	38.7386i	1.33741i	0.743530	+	0.668703i	$0.233150\pi$
$839$	38.7386i	1.33741i	−0.743530	+	0.668703i	$0.766850\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	− 15.6155i	− 0.537827i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.94602i	0.0668662i
$848$	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
$859$	−52.9848	−1.80782	−0.903910	−	0.427723i	$-0.859316\pi$
$859$	−52.9848	−1.80782	−0.903910	+	0.427723i	$0.859316\pi$
$860$	0	0
$861$	−12.4924	−0.425741
$862$	0	0
$863$	− 4.30019i	− 0.146380i	−0.997318	−	0.0731900i	$-0.976682\pi$
$863$	− 4.30019i	− 0.146380i	0.997318	−	0.0731900i	$-0.0233180\pi$
$864$	0	0
$865$	− 46.7386i	− 1.58916i
$866$	0	0
$867$	−43.2311	−1.46820
$868$	0	0
$869$	24.9848i	0.847553i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4.63068i	0.156725i
$874$	0	0
$875$	−14.9309	−0.504756
$876$	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
$879$	− 30.5464i	− 1.03030i
$880$	0	0
$881$	10.1922	0.343385	0.171693	−	0.985151i	$-0.445076\pi$
$881$	10.1922	0.343385	0.171693	+	0.985151i	$0.445076\pi$
$882$	0	0
$883$	−12.6847	−0.426873	−0.213436	−	0.976957i	$-0.568466\pi$
$883$	−12.6847	−0.426873	−0.213436	+	0.976957i	$0.568466\pi$
$884$	0	0
$885$	17.3693	0.583863
$886$	0	0
$887$	53.4773	1.79559	0.897795	−	0.440413i	$-0.145168\pi$
$887$	53.4773	1.79559	0.897795	+	0.440413i	$0.145168\pi$
$888$	0	0
$889$	− 9.75379i	− 0.327132i
$890$	0	0
$891$	− 21.8617i	− 0.732396i
$892$	0	0
$893$	39.6155	1.32568
$894$	0	0
$895$	− 16.6847i	− 0.557707i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000i	0.266815i
$900$	0	0
$901$	−34.2462	−1.14091
$902$	0	0
$903$	23.3153i	0.775886i
$904$	0	0
$905$	− 52.9848i	− 1.76128i
$906$	0	0
$907$	0.192236	0.00638309	0.00319154	−	0.999995i	$-0.498984\pi$
$907$	0.192236	0.00638309	0.00319154	+	0.999995i	$0.498984\pi$
$908$	0	0
$909$	0.630683	0.0209184
$910$	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	$-0.736281\pi$
$919$	−40.9848	−1.35197	−0.675983	+	0.736918i	$0.736281\pi$
$920$	0	0
$921$	− 7.61553i	− 0.250940i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	20.6307i	0.678333i
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	− 47.8617i	− 1.57029i	−0.619310	−	0.785146i	$-0.712588\pi$
$929$	− 47.8617i	− 1.57029i	0.619310	−	0.785146i	$-0.287412\pi$
$930$	0	0
$931$	14.2462i	0.466901i
$932$	0	0
$933$	−29.8617	−0.977629
$934$	0	0
$935$	74.3542	2.43164
$936$	0	0
$937$	12.7386	0.416153	0.208077	−	0.978113i	$-0.433280\pi$
$937$	12.7386	0.416153	0.208077	+	0.978113i	$0.433280\pi$
$938$	0	0
$939$	−9.66950	−0.315552
$940$	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
$943$	− 40.9848i	− 1.33465i
$944$	0	0
$945$	30.9309	1.00618
$946$	0	0
$947$	− 17.3693i	− 0.564427i	−0.959352	−	0.282213i	$-0.908931\pi$
$947$	− 17.3693i	− 0.564427i	0.959352	−	0.282213i	$-0.0910686\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	6.63068i	0.215015i
$952$	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
$963$	−2.24621	−0.0723831
$964$	0	0
$965$	12.0000	0.386294
$966$	0	0
$967$	1.56155i	0.0502162i	0.999685	+	0.0251081i	$0.00799299\pi$
$967$	1.56155i	0.0502162i	−0.999685	+	0.0251081i	$0.992007\pi$
$968$	0	0
$969$	− 32.6004i	− 1.04727i
$970$	0	0
$971$	−19.3153	−0.619859	−0.309929	−	0.950760i	$-0.600305\pi$
$971$	−19.3153	−0.619859	−0.309929	+	0.950760i	$0.600305\pi$
$972$	0	0
$973$	− 7.31534i	− 0.234519i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 21.5076i	− 0.688088i	−0.938953	−	0.344044i	$-0.888203\pi$
$977$	− 21.5076i	− 0.688088i	0.938953	−	0.344044i	$-0.111797\pi$
$978$	0	0
$979$	−31.2311	−0.998149
$980$	0	0
$981$	10.0000i	0.319275i
$982$	0	0
$983$	− 10.9309i	− 0.348641i	−0.984689	−	0.174320i	$-0.944227\pi$
$983$	− 10.9309i	− 0.348641i	0.984689	−	0.174320i	$-0.0557728\pi$
$984$	0	0
$985$	−52.3002	−1.66642
$986$	0	0
$987$	−30.9309	−0.984540
$988$	0	0
$989$	−76.4924	−2.43232
$990$	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
$993$	44.4924i	1.41192i
$994$	0	0
$995$	− 11.1231i	− 0.352626i
$996$	0	0
$997$	−16.2462	−0.514523	−0.257261	−	0.966342i	$-0.582820\pi$
$997$	−16.2462	−0.514523	−0.257261	+	0.966342i	$0.582820\pi$
$998$	0	0
$999$	− 14.9309i	− 0.472392i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.f.c.337.1		4
4.3	odd	2		2704.2.f.k.337.3		4
13.2	odd	12		1352.2.i.d.529.2		4
13.3	even	3		1352.2.o.d.1161.3		8
13.4	even	6		1352.2.o.d.361.4		8
13.5	odd	4		1352.2.a.g.1.1		2
13.6	odd	12		1352.2.i.d.1329.2		4
13.7	odd	12		1352.2.i.f.1329.2		4
13.8	odd	4		104.2.a.b.1.1	✓	2
13.9	even	3		1352.2.o.d.361.3		8
13.10	even	6		1352.2.o.d.1161.4		8
13.11	odd	12		1352.2.i.f.529.2		4
13.12	even	2	inner	1352.2.f.c.337.2		4
39.8	even	4		936.2.a.j.1.1		2
52.31	even	4		2704.2.a.p.1.2		2
52.47	even	4		208.2.a.e.1.2		2
52.51	odd	2		2704.2.f.k.337.4		4
65.8	even	4		2600.2.d.k.1249.2		4
65.34	odd	4		2600.2.a.p.1.2		2
65.47	even	4		2600.2.d.k.1249.3		4
91.34	even	4		5096.2.a.m.1.2		2
104.21	odd	4		832.2.a.k.1.2		2
104.99	even	4		832.2.a.n.1.1		2
156.47	odd	4		1872.2.a.u.1.1		2
208.21	odd	4		3328.2.b.y.1665.3		4
208.99	even	4		3328.2.b.w.1665.3		4
208.125	odd	4		3328.2.b.y.1665.2		4
208.203	even	4		3328.2.b.w.1665.2		4
260.99	even	4		5200.2.a.bw.1.1		2
312.125	even	4		7488.2.a.cu.1.2		2
312.203	odd	4		7488.2.a.cv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.a.b.1.1	✓	2	13.8	odd	4
208.2.a.e.1.2		2	52.47	even	4
832.2.a.k.1.2		2	104.21	odd	4
832.2.a.n.1.1		2	104.99	even	4
936.2.a.j.1.1		2	39.8	even	4
1352.2.a.g.1.1		2	13.5	odd	4
1352.2.f.c.337.1		4	1.1	even	1	trivial
1352.2.f.c.337.2		4	13.12	even	2	inner
1352.2.i.d.529.2		4	13.2	odd	12
1352.2.i.d.1329.2		4	13.6	odd	12
1352.2.i.f.529.2		4	13.11	odd	12
1352.2.i.f.1329.2		4	13.7	odd	12
1352.2.o.d.361.3		8	13.9	even	3
1352.2.o.d.361.4		8	13.4	even	6
1352.2.o.d.1161.3		8	13.3	even	3
1352.2.o.d.1161.4		8	13.10	even	6
1872.2.a.u.1.1		2	156.47	odd	4
2600.2.a.p.1.2		2	65.34	odd	4
2600.2.d.k.1249.2		4	65.8	even	4
2600.2.d.k.1249.3		4	65.47	even	4
2704.2.a.p.1.2		2	52.31	even	4
2704.2.f.k.337.3		4	4.3	odd	2
2704.2.f.k.337.4		4	52.51	odd	2
3328.2.b.w.1665.2		4	208.203	even	4
3328.2.b.w.1665.3		4	208.99	even	4
3328.2.b.y.1665.2		4	208.125	odd	4
3328.2.b.y.1665.3		4	208.21	odd	4
5096.2.a.m.1.2		2	91.34	even	4
5200.2.a.bw.1.1		2	260.99	even	4
7488.2.a.cu.1.2		2	312.125	even	4
7488.2.a.cv.1.2		2	312.203	odd	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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} -3.56155i q^{5} +1.56155i q^{7} -0.561553 q^{9} +3.12311i q^{11} +5.56155i q^{15} +6.68466 q^{17} +3.12311i q^{19} -2.43845i q^{21} +8.00000 q^{23} -7.68466 q^{25} +5.56155 q^{27} -2.00000 q^{29} -4.00000i q^{31} -4.87689i q^{33} +5.56155 q^{35} -2.68466i q^{37} -5.12311i q^{41} -9.56155 q^{43} +2.00000i q^{45} -12.6847i q^{47} +4.56155 q^{49} -10.4384 q^{51} -5.12311 q^{53} +11.1231 q^{55} -4.87689i q^{57} -3.12311i q^{59} +2.87689 q^{61} -0.876894i q^{63} -3.12311i q^{67} -12.4924 q^{69} -4.68466i q^{71} -6.00000i q^{73} +12.0000 q^{75} -4.87689 q^{77} +8.00000 q^{79} -7.00000 q^{81} +14.2462i q^{83} -23.8078i q^{85} +3.12311 q^{87} +10.0000i q^{89} +6.24621i q^{93} +11.1231 q^{95} -8.24621i q^{97} -1.75379i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{9} + 2 q^{17} + 32 q^{23} - 6 q^{25} + 14 q^{27} - 8 q^{29} + 14 q^{35} - 30 q^{43} + 10 q^{49} - 50 q^{51} - 4 q^{53} + 28 q^{55} + 28 q^{61} + 16 q^{69} + 48 q^{75} - 36 q^{77} + 32 q^{79}+ \cdots + 28 q^{95}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^9 + 2 * q^17 + 32 * q^23 - 6 * q^25 + 14 * q^27 - 8 * q^29 + 14 * q^35 - 30 * q^43 + 10 * q^49 - 50 * q^51 - 4 * q^53 + 28 * q^55 + 28 * q^61 + 16 * q^69 + 48 * q^75 - 36 * q^77 + 32 * q^79 - 28 * q^81 - 4 * q^87 + 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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