LMFDB - Embedded newform 1664.2.b.i.833.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] 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("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.i.833.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.82843i q^{5} -1.82843 q^{7} +2.00000 q^{9} -0.828427i q^{11} -1.00000i q^{13} -1.82843 q^{15} -4.65685 q^{17} +4.00000i q^{19} -1.82843i q^{21} -8.82843 q^{23} +1.65685 q^{25} +5.00000i q^{27} +8.82843i q^{29} +0.828427 q^{33} -3.34315i q^{35} +1.82843i q^{37} +1.00000 q^{39} +2.82843 q^{41} -3.00000i q^{43} +3.65685i q^{45} -5.48528 q^{47} -3.65685 q^{49} -4.65685i q^{51} -8.82843i q^{53} +1.51472 q^{55} -4.00000 q^{57} -10.8284i q^{59} +5.17157i q^{61} -3.65685 q^{63} +1.82843 q^{65} +7.65685i q^{67} -8.82843i q^{69} -15.8284 q^{71} -10.8284 q^{73} +1.65685i q^{75} +1.51472i q^{77} -3.65685 q^{79} +1.00000 q^{81} -6.00000i q^{83} -8.51472i q^{85} -8.82843 q^{87} +6.34315 q^{89} +1.82843i q^{91} -7.31371 q^{95} +2.48528 q^{97} -1.65685i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} + 8 q^{9} + 4 q^{15} + 4 q^{17} - 24 q^{23} - 16 q^{25} - 8 q^{33} + 4 q^{39} + 12 q^{47} + 8 q^{49} + 40 q^{55} - 16 q^{57} + 8 q^{63} - 4 q^{65} - 52 q^{71} - 32 q^{73} + 8 q^{79} + 4 q^{81}+ \cdots - 24 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 8 * q^9 + 4 * q^15 + 4 * q^17 - 24 * q^23 - 16 * q^25 - 8 * q^33 + 4 * q^39 + 12 * q^47 + 8 * q^49 + 40 * q^55 - 16 * q^57 + 8 * q^63 - 4 * q^65 - 52 * q^71 - 32 * q^73 + 8 * q^79 + 4 * q^81 - 24 * q^87 + 48 * q^89 + 16 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

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

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	1.82843i	0.817697i	0.912602	+	0.408849i	$0.134070\pi$
$5$	1.82843i	0.817697i	−0.912602	+	0.408849i	$0.865930\pi$
$6$	0	0
$7$	−1.82843	−0.691080	−0.345540	−	0.938404i	$-0.612304\pi$
$7$	−1.82843	−0.691080	−0.345540	+	0.938404i	$0.612304\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	− 0.828427i	− 0.249780i	−0.992171	−	0.124890i	$-0.960142\pi$
$11$	− 0.828427i	− 0.249780i	0.992171	−	0.124890i	$-0.0398578\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−1.82843	−0.472098
$16$	0	0
$17$	−4.65685	−1.12945	−0.564727	−	0.825278i	$-0.691018\pi$
$17$	−4.65685	−1.12945	−0.564727	+	0.825278i	$0.691018\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	− 1.82843i	− 0.398996i
$22$	0	0
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	0	0
$25$	1.65685	0.331371
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	8.82843i	1.63940i	0.572795	+	0.819699i	$0.305859\pi$
$29$	8.82843i	1.63940i	−0.572795	+	0.819699i	$0.694141\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0.828427	0.144211
$34$	0	0
$35$	− 3.34315i	− 0.565095i
$36$	0	0
$37$	1.82843i	0.300592i	0.988641	+	0.150296i	$0.0480226\pi$
$37$	1.82843i	0.300592i	−0.988641	+	0.150296i	$0.951977\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	− 3.00000i	− 0.457496i	−0.973486	−	0.228748i	$-0.926537\pi$
$43$	− 3.00000i	− 0.457496i	0.973486	−	0.228748i	$-0.0734631\pi$
$44$	0	0
$45$	3.65685i	0.545132i
$46$	0	0
$47$	−5.48528	−0.800111	−0.400055	−	0.916491i	$-0.631009\pi$
$47$	−5.48528	−0.800111	−0.400055	+	0.916491i	$0.631009\pi$
$48$	0	0
$49$	−3.65685	−0.522408
$50$	0	0
$51$	− 4.65685i	− 0.652090i
$52$	0	0
$53$	− 8.82843i	− 1.21268i	−0.795206	−	0.606339i	$-0.792638\pi$
$53$	− 8.82843i	− 1.21268i	0.795206	−	0.606339i	$-0.207362\pi$
$54$	0	0
$55$	1.51472	0.204245
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 10.8284i	− 1.40974i	−0.709336	−	0.704871i	$-0.751005\pi$
$59$	− 10.8284i	− 1.40974i	0.709336	−	0.704871i	$-0.248995\pi$
$60$	0	0
$61$	5.17157i	0.662152i	0.943604	+	0.331076i	$0.107412\pi$
$61$	5.17157i	0.662152i	−0.943604	+	0.331076i	$0.892588\pi$
$62$	0	0
$63$	−3.65685	−0.460720
$64$	0	0
$65$	1.82843	0.226788
$66$	0	0
$67$	7.65685i	0.935434i	0.883878	+	0.467717i	$0.154923\pi$
$67$	7.65685i	0.935434i	−0.883878	+	0.467717i	$0.845077\pi$
$68$	0	0
$69$	− 8.82843i	− 1.06282i
$70$	0	0
$71$	−15.8284	−1.87849	−0.939244	−	0.343249i	$-0.888472\pi$
$71$	−15.8284	−1.87849	−0.939244	+	0.343249i	$0.888472\pi$
$72$	0	0
$73$	−10.8284	−1.26737	−0.633686	−	0.773591i	$-0.718459\pi$
$73$	−10.8284	−1.26737	−0.633686	+	0.773591i	$0.718459\pi$
$74$	0	0
$75$	1.65685i	0.191317i
$76$	0	0
$77$	1.51472i	0.172618i
$78$	0	0
$79$	−3.65685	−0.411428	−0.205714	−	0.978612i	$-0.565952\pi$
$79$	−3.65685	−0.411428	−0.205714	+	0.978612i	$0.565952\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	− 8.51472i	− 0.923551i
$86$	0	0
$87$	−8.82843	−0.946507
$88$	0	0
$89$	6.34315	0.672372	0.336186	−	0.941796i	$-0.390863\pi$
$89$	6.34315	0.672372	0.336186	+	0.941796i	$0.390863\pi$
$90$	0	0
$91$	1.82843i	0.191671i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.31371	−0.750371
$96$	0	0
$97$	2.48528	0.252342	0.126171	−	0.992009i	$-0.459731\pi$
$97$	2.48528	0.252342	0.126171	+	0.992009i	$0.459731\pi$
$98$	0	0
$99$	− 1.65685i	− 0.166520i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	3.34315	0.326258
$106$	0	0
$107$	7.31371i	0.707043i	0.935426	+	0.353521i	$0.115016\pi$
$107$	7.31371i	0.707043i	−0.935426	+	0.353521i	$0.884984\pi$
$108$	0	0
$109$	− 1.82843i	− 0.175132i	−0.996159	−	0.0875658i	$-0.972091\pi$
$109$	− 1.82843i	− 0.175132i	0.996159	−	0.0875658i	$-0.0279088\pi$
$110$	0	0
$111$	−1.82843	−0.173547
$112$	0	0
$113$	7.65685	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$114$	0	0
$115$	− 16.1421i	− 1.50526i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	8.51472	0.780543
$120$	0	0
$121$	10.3137	0.937610
$122$	0	0
$123$	2.82843i	0.255031i
$124$	0	0
$125$	12.1716i	1.08866i
$126$	0	0
$127$	−1.51472	−0.134410	−0.0672048	−	0.997739i	$-0.521408\pi$
$127$	−1.51472	−0.134410	−0.0672048	+	0.997739i	$0.521408\pi$
$128$	0	0
$129$	3.00000	0.264135
$130$	0	0
$131$	8.65685i	0.756353i	0.925734	+	0.378176i	$0.123449\pi$
$131$	8.65685i	0.756353i	−0.925734	+	0.378176i	$0.876551\pi$
$132$	0	0
$133$	− 7.31371i	− 0.634179i
$134$	0	0
$135$	−9.14214	−0.786830
$136$	0	0
$137$	−8.14214	−0.695630	−0.347815	−	0.937563i	$-0.613076\pi$
$137$	−8.14214	−0.695630	−0.347815	+	0.937563i	$0.613076\pi$
$138$	0	0
$139$	7.34315i	0.622837i	0.950273	+	0.311419i	$0.100804\pi$
$139$	7.34315i	0.622837i	−0.950273	+	0.311419i	$0.899196\pi$
$140$	0	0
$141$	− 5.48528i	− 0.461944i
$142$	0	0
$143$	−0.828427	−0.0692766
$144$	0	0
$145$	−16.1421	−1.34053
$146$	0	0
$147$	− 3.65685i	− 0.301612i
$148$	0	0
$149$	− 21.3137i	− 1.74609i	−0.487642	−	0.873044i	$-0.662143\pi$
$149$	− 21.3137i	− 1.74609i	0.487642	−	0.873044i	$-0.337857\pi$
$150$	0	0
$151$	19.4853	1.58569	0.792845	−	0.609424i	$-0.208599\pi$
$151$	19.4853	1.58569	0.792845	+	0.609424i	$0.208599\pi$
$152$	0	0
$153$	−9.31371	−0.752969
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.4853i	0.996434i	0.867052	+	0.498217i	$0.166012\pi$
$157$	12.4853i	0.996434i	−0.867052	+	0.498217i	$0.833988\pi$
$158$	0	0
$159$	8.82843	0.700140
$160$	0	0
$161$	16.1421	1.27218
$162$	0	0
$163$	7.17157i	0.561721i	0.959749	+	0.280860i	$0.0906198\pi$
$163$	7.17157i	0.561721i	−0.959749	+	0.280860i	$0.909380\pi$
$164$	0	0
$165$	1.51472i	0.117921i
$166$	0	0
$167$	10.3431	0.800377	0.400188	−	0.916433i	$-0.368945\pi$
$167$	10.3431	0.800377	0.400188	+	0.916433i	$0.368945\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	8.00000i	0.611775i
$172$	0	0
$173$	10.9706i	0.834076i	0.908889	+	0.417038i	$0.136932\pi$
$173$	10.9706i	0.834076i	−0.908889	+	0.417038i	$0.863068\pi$
$174$	0	0
$175$	−3.02944	−0.229004
$176$	0	0
$177$	10.8284	0.813914
$178$	0	0
$179$	14.3137i	1.06986i	0.844897	+	0.534928i	$0.179661\pi$
$179$	14.3137i	1.06986i	−0.844897	+	0.534928i	$0.820339\pi$
$180$	0	0
$181$	10.9706i	0.815436i	0.913108	+	0.407718i	$0.133675\pi$
$181$	10.9706i	0.815436i	−0.913108	+	0.407718i	$0.866325\pi$
$182$	0	0
$183$	−5.17157	−0.382294
$184$	0	0
$185$	−3.34315	−0.245793
$186$	0	0
$187$	3.85786i	0.282115i
$188$	0	0
$189$	− 9.14214i	− 0.664993i
$190$	0	0
$191$	−8.82843	−0.638803	−0.319401	−	0.947620i	$-0.603482\pi$
$191$	−8.82843	−0.638803	−0.319401	+	0.947620i	$0.603482\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	1.82843i	0.130936i
$196$	0	0
$197$	− 12.1716i	− 0.867189i	−0.901108	−	0.433594i	$-0.857245\pi$
$197$	− 12.1716i	− 0.867189i	0.901108	−	0.433594i	$-0.142755\pi$
$198$	0	0
$199$	10.9706	0.777683	0.388841	−	0.921305i	$-0.372875\pi$
$199$	10.9706	0.777683	0.388841	+	0.921305i	$0.372875\pi$
$200$	0	0
$201$	−7.65685	−0.540073
$202$	0	0
$203$	− 16.1421i	− 1.13296i
$204$	0	0
$205$	5.17157i	0.361198i
$206$	0	0
$207$	−17.6569	−1.22724
$208$	0	0
$209$	3.31371	0.229214
$210$	0	0
$211$	12.6569i	0.871334i	0.900108	+	0.435667i	$0.143487\pi$
$211$	12.6569i	0.871334i	−0.900108	+	0.435667i	$0.856513\pi$
$212$	0	0
$213$	− 15.8284i	− 1.08455i
$214$	0	0
$215$	5.48528	0.374093
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 10.8284i	− 0.731717i
$220$	0	0
$221$	4.65685i	0.313254i
$222$	0	0
$223$	15.8284	1.05995	0.529975	−	0.848013i	$-0.322201\pi$
$223$	15.8284	1.05995	0.529975	+	0.848013i	$0.322201\pi$
$224$	0	0
$225$	3.31371	0.220914
$226$	0	0
$227$	− 14.8284i	− 0.984197i	−0.870539	−	0.492099i	$-0.836230\pi$
$227$	− 14.8284i	− 0.984197i	0.870539	−	0.492099i	$-0.163770\pi$
$228$	0	0
$229$	15.8284i	1.04597i	0.852341	+	0.522986i	$0.175182\pi$
$229$	15.8284i	1.04597i	−0.852341	+	0.522986i	$0.824818\pi$
$230$	0	0
$231$	−1.51472	−0.0996612
$232$	0	0
$233$	2.31371	0.151576	0.0757880	−	0.997124i	$-0.475853\pi$
$233$	2.31371	0.151576	0.0757880	+	0.997124i	$0.475853\pi$
$234$	0	0
$235$	− 10.0294i	− 0.654248i
$236$	0	0
$237$	− 3.65685i	− 0.237538i
$238$	0	0
$239$	9.14214	0.591356	0.295678	−	0.955288i	$-0.404455\pi$
$239$	9.14214	0.591356	0.295678	+	0.955288i	$0.404455\pi$
$240$	0	0
$241$	−19.3137	−1.24411	−0.622053	−	0.782975i	$-0.713701\pi$
$241$	−19.3137	−1.24411	−0.622053	+	0.782975i	$0.713701\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	− 6.68629i	− 0.427171i
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	− 18.6274i	− 1.17575i	−0.808951	−	0.587876i	$-0.799964\pi$
$251$	− 18.6274i	− 1.17575i	0.808951	−	0.587876i	$-0.200036\pi$
$252$	0	0
$253$	7.31371i	0.459809i
$254$	0	0
$255$	8.51472	0.533212
$256$	0	0
$257$	−12.6569	−0.789513	−0.394756	−	0.918786i	$-0.629171\pi$
$257$	−12.6569	−0.789513	−0.394756	+	0.918786i	$0.629171\pi$
$258$	0	0
$259$	− 3.34315i	− 0.207733i
$260$	0	0
$261$	17.6569i	1.09293i
$262$	0	0
$263$	−2.14214	−0.132090	−0.0660449	−	0.997817i	$-0.521038\pi$
$263$	−2.14214	−0.132090	−0.0660449	+	0.997817i	$0.521038\pi$
$264$	0	0
$265$	16.1421	0.991604
$266$	0	0
$267$	6.34315i	0.388194i
$268$	0	0
$269$	19.7990i	1.20717i	0.797300	+	0.603583i	$0.206261\pi$
$269$	19.7990i	1.20717i	−0.797300	+	0.603583i	$0.793739\pi$
$270$	0	0
$271$	19.4853	1.18365	0.591823	−	0.806068i	$-0.298408\pi$
$271$	19.4853	1.18365	0.591823	+	0.806068i	$0.298408\pi$
$272$	0	0
$273$	−1.82843	−0.110661
$274$	0	0
$275$	− 1.37258i	− 0.0827699i
$276$	0	0
$277$	− 12.4853i	− 0.750168i	−0.926991	−	0.375084i	$-0.877614\pi$
$277$	− 12.4853i	− 0.750168i	0.926991	−	0.375084i	$-0.122386\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−28.9706	−1.72824	−0.864119	−	0.503287i	$-0.832124\pi$
$281$	−28.9706	−1.72824	−0.864119	+	0.503287i	$0.832124\pi$
$282$	0	0
$283$	30.6274i	1.82061i	0.413937	+	0.910305i	$0.364153\pi$
$283$	30.6274i	1.82061i	−0.413937	+	0.910305i	$0.635847\pi$
$284$	0	0
$285$	− 7.31371i	− 0.433227i
$286$	0	0
$287$	−5.17157	−0.305268
$288$	0	0
$289$	4.68629	0.275664
$290$	0	0
$291$	2.48528i	0.145690i
$292$	0	0
$293$	26.7990i	1.56561i	0.622265	+	0.782807i	$0.286213\pi$
$293$	26.7990i	1.56561i	−0.622265	+	0.782807i	$0.713787\pi$
$294$	0	0
$295$	19.7990	1.15274
$296$	0	0
$297$	4.14214	0.240351
$298$	0	0
$299$	8.82843i	0.510561i
$300$	0	0
$301$	5.48528i	0.316166i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.45584	−0.541440
$306$	0	0
$307$	− 18.8284i	− 1.07460i	−0.843393	−	0.537298i	$-0.819445\pi$
$307$	− 18.8284i	− 1.07460i	0.843393	−	0.537298i	$-0.180555\pi$
$308$	0	0
$309$	− 14.0000i	− 0.796432i
$310$	0	0
$311$	32.2843	1.83067	0.915337	−	0.402690i	$-0.131925\pi$
$311$	32.2843	1.83067	0.915337	+	0.402690i	$0.131925\pi$
$312$	0	0
$313$	13.9706	0.789663	0.394831	−	0.918754i	$-0.370803\pi$
$313$	13.9706	0.789663	0.394831	+	0.918754i	$0.370803\pi$
$314$	0	0
$315$	− 6.68629i	− 0.376730i
$316$	0	0
$317$	6.68629i	0.375540i	0.982213	+	0.187770i	$0.0601259\pi$
$317$	6.68629i	0.375540i	−0.982213	+	0.187770i	$0.939874\pi$
$318$	0	0
$319$	7.31371	0.409489
$320$	0	0
$321$	−7.31371	−0.408211
$322$	0	0
$323$	− 18.6274i	− 1.03646i
$324$	0	0
$325$	− 1.65685i	− 0.0919057i
$326$	0	0
$327$	1.82843	0.101112
$328$	0	0
$329$	10.0294	0.552941
$330$	0	0
$331$	− 22.9706i	− 1.26258i	−0.775548	−	0.631288i	$-0.782527\pi$
$331$	− 22.9706i	− 1.26258i	0.775548	−	0.631288i	$-0.217473\pi$
$332$	0	0
$333$	3.65685i	0.200394i
$334$	0	0
$335$	−14.0000	−0.764902
$336$	0	0
$337$	31.9706	1.74155	0.870774	−	0.491684i	$-0.163618\pi$
$337$	31.9706	1.74155	0.870774	+	0.491684i	$0.163618\pi$
$338$	0	0
$339$	7.65685i	0.415863i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.4853	1.05211
$344$	0	0
$345$	16.1421	0.869063
$346$	0	0
$347$	− 15.0000i	− 0.805242i	−0.915367	−	0.402621i	$-0.868099\pi$
$347$	− 15.0000i	− 0.805242i	0.915367	−	0.402621i	$-0.131901\pi$
$348$	0	0
$349$	8.51472i	0.455782i	0.973687	+	0.227891i	$0.0731831\pi$
$349$	8.51472i	0.455782i	−0.973687	+	0.227891i	$0.926817\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	5.31371	0.282820	0.141410	−	0.989951i	$-0.454836\pi$
$353$	5.31371	0.282820	0.141410	+	0.989951i	$0.454836\pi$
$354$	0	0
$355$	− 28.9411i	− 1.53604i
$356$	0	0
$357$	8.51472i	0.450647i
$358$	0	0
$359$	−10.3431	−0.545890	−0.272945	−	0.962030i	$-0.587998\pi$
$359$	−10.3431	−0.545890	−0.272945	+	0.962030i	$0.587998\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	10.3137i	0.541329i
$364$	0	0
$365$	− 19.7990i	− 1.03633i
$366$	0	0
$367$	35.9411	1.87611	0.938056	−	0.346484i	$-0.112625\pi$
$367$	35.9411	1.87611	0.938056	+	0.346484i	$0.112625\pi$
$368$	0	0
$369$	5.65685	0.294484
$370$	0	0
$371$	16.1421i	0.838058i
$372$	0	0
$373$	− 24.9706i	− 1.29293i	−0.762945	−	0.646463i	$-0.776247\pi$
$373$	− 24.9706i	− 1.29293i	0.762945	−	0.646463i	$-0.223753\pi$
$374$	0	0
$375$	−12.1716	−0.628537
$376$	0	0
$377$	8.82843	0.454687
$378$	0	0
$379$	27.4558i	1.41031i	0.709052	+	0.705156i	$0.249123\pi$
$379$	27.4558i	1.41031i	−0.709052	+	0.705156i	$0.750877\pi$
$380$	0	0
$381$	− 1.51472i	− 0.0776014i
$382$	0	0
$383$	−12.1716	−0.621938	−0.310969	−	0.950420i	$-0.600654\pi$
$383$	−12.1716	−0.621938	−0.310969	+	0.950420i	$0.600654\pi$
$384$	0	0
$385$	−2.76955	−0.141149
$386$	0	0
$387$	− 6.00000i	− 0.304997i
$388$	0	0
$389$	2.14214i	0.108611i	0.998524	+	0.0543053i	$0.0172944\pi$
$389$	2.14214i	0.108611i	−0.998524	+	0.0543053i	$0.982706\pi$
$390$	0	0
$391$	41.1127	2.07916
$392$	0	0
$393$	−8.65685	−0.436681
$394$	0	0
$395$	− 6.68629i	− 0.336424i
$396$	0	0
$397$	− 21.3137i	− 1.06970i	−0.844946	−	0.534852i	$-0.820367\pi$
$397$	− 21.3137i	− 1.06970i	0.844946	−	0.534852i	$-0.179633\pi$
$398$	0	0
$399$	7.31371	0.366143
$400$	0	0
$401$	−16.8284	−0.840372	−0.420186	−	0.907438i	$-0.638035\pi$
$401$	−16.8284	−0.840372	−0.420186	+	0.907438i	$0.638035\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.82843i	0.0908553i
$406$	0	0
$407$	1.51472	0.0750818
$408$	0	0
$409$	18.4853	0.914038	0.457019	−	0.889457i	$-0.348917\pi$
$409$	18.4853	0.914038	0.457019	+	0.889457i	$0.348917\pi$
$410$	0	0
$411$	− 8.14214i	− 0.401622i
$412$	0	0
$413$	19.7990i	0.974245i
$414$	0	0
$415$	10.9706	0.538524
$416$	0	0
$417$	−7.34315	−0.359595
$418$	0	0
$419$	30.6569i	1.49769i	0.662748	+	0.748843i	$0.269390\pi$
$419$	30.6569i	1.49769i	−0.662748	+	0.748843i	$0.730610\pi$
$420$	0	0
$421$	37.1421i	1.81020i	0.425202	+	0.905098i	$0.360203\pi$
$421$	37.1421i	1.81020i	−0.425202	+	0.905098i	$0.639797\pi$
$422$	0	0
$423$	−10.9706	−0.533407
$424$	0	0
$425$	−7.71573	−0.374268
$426$	0	0
$427$	− 9.45584i	− 0.457600i
$428$	0	0
$429$	− 0.828427i	− 0.0399968i
$430$	0	0
$431$	12.1716	0.586284	0.293142	−	0.956069i	$-0.405299\pi$
$431$	12.1716	0.586284	0.293142	+	0.956069i	$0.405299\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	− 16.1421i	− 0.773956i
$436$	0	0
$437$	− 35.3137i	− 1.68928i
$438$	0	0
$439$	12.4853	0.595890	0.297945	−	0.954583i	$-0.403699\pi$
$439$	12.4853	0.595890	0.297945	+	0.954583i	$0.403699\pi$
$440$	0	0
$441$	−7.31371	−0.348272
$442$	0	0
$443$	22.3137i	1.06016i	0.847949	+	0.530078i	$0.177837\pi$
$443$	22.3137i	1.06016i	−0.847949	+	0.530078i	$0.822163\pi$
$444$	0	0
$445$	11.5980i	0.549797i
$446$	0	0
$447$	21.3137	1.00810
$448$	0	0
$449$	−16.1421	−0.761794	−0.380897	−	0.924617i	$-0.624385\pi$
$449$	−16.1421	−0.761794	−0.380897	+	0.924617i	$0.624385\pi$
$450$	0	0
$451$	− 2.34315i	− 0.110334i
$452$	0	0
$453$	19.4853i	0.915498i
$454$	0	0
$455$	−3.34315	−0.156729
$456$	0	0
$457$	−12.9706	−0.606737	−0.303369	−	0.952873i	$-0.598111\pi$
$457$	−12.9706	−0.606737	−0.303369	+	0.952873i	$0.598111\pi$
$458$	0	0
$459$	− 23.2843i	− 1.08682i
$460$	0	0
$461$	5.48528i	0.255475i	0.991808	+	0.127738i	$0.0407715\pi$
$461$	5.48528i	0.255475i	−0.991808	+	0.127738i	$0.959228\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.6569i	1.92765i	0.266537	+	0.963825i	$0.414121\pi$
$467$	41.6569i	1.92765i	−0.266537	+	0.963825i	$0.585879\pi$
$468$	0	0
$469$	− 14.0000i	− 0.646460i
$470$	0	0
$471$	−12.4853	−0.575291
$472$	0	0
$473$	−2.48528	−0.114273
$474$	0	0
$475$	6.62742i	0.304087i
$476$	0	0
$477$	− 17.6569i	− 0.808452i
$478$	0	0
$479$	9.14214	0.417715	0.208857	−	0.977946i	$-0.433026\pi$
$479$	9.14214	0.417715	0.208857	+	0.977946i	$0.433026\pi$
$480$	0	0
$481$	1.82843	0.0833691
$482$	0	0
$483$	16.1421i	0.734493i
$484$	0	0
$485$	4.54416i	0.206339i
$486$	0	0
$487$	24.9706	1.13152	0.565762	−	0.824569i	$-0.308582\pi$
$487$	24.9706	1.13152	0.565762	+	0.824569i	$0.308582\pi$
$488$	0	0
$489$	−7.17157	−0.324310
$490$	0	0
$491$	40.3137i	1.81933i	0.415340	+	0.909666i	$0.363662\pi$
$491$	40.3137i	1.81933i	−0.415340	+	0.909666i	$0.636338\pi$
$492$	0	0
$493$	− 41.1127i	− 1.85162i
$494$	0	0
$495$	3.02944	0.136163
$496$	0	0
$497$	28.9411	1.29819
$498$	0	0
$499$	− 10.1421i	− 0.454024i	−0.973892	−	0.227012i	$-0.927104\pi$
$499$	− 10.1421i	− 0.454024i	0.973892	−	0.227012i	$-0.0728957\pi$
$500$	0	0
$501$	10.3431i	0.462098i
$502$	0	0
$503$	−7.31371	−0.326102	−0.163051	−	0.986618i	$-0.552134\pi$
$503$	−7.31371	−0.326102	−0.163051	+	0.986618i	$0.552134\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	0	0
$509$	− 18.2843i	− 0.810436i	−0.914220	−	0.405218i	$-0.867196\pi$
$509$	− 18.2843i	− 0.810436i	0.914220	−	0.405218i	$-0.132804\pi$
$510$	0	0
$511$	19.7990	0.875856
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	− 25.5980i	− 1.12798i
$516$	0	0
$517$	4.54416i	0.199852i
$518$	0	0
$519$	−10.9706	−0.481554
$520$	0	0
$521$	8.31371	0.364230	0.182115	−	0.983277i	$-0.441706\pi$
$521$	8.31371	0.364230	0.182115	+	0.983277i	$0.441706\pi$
$522$	0	0
$523$	− 1.65685i	− 0.0724492i	−0.999344	−	0.0362246i	$-0.988467\pi$
$523$	− 1.65685i	− 0.0724492i	0.999344	−	0.0362246i	$-0.0115332\pi$
$524$	0	0
$525$	− 3.02944i	− 0.132215i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	54.9411	2.38874
$530$	0	0
$531$	− 21.6569i	− 0.939827i
$532$	0	0
$533$	− 2.82843i	− 0.122513i
$534$	0	0
$535$	−13.3726	−0.578147
$536$	0	0
$537$	−14.3137	−0.617682
$538$	0	0
$539$	3.02944i	0.130487i
$540$	0	0
$541$	− 33.4853i	− 1.43964i	−0.694158	−	0.719822i	$-0.744223\pi$
$541$	− 33.4853i	− 1.43964i	0.694158	−	0.719822i	$-0.255777\pi$
$542$	0	0
$543$	−10.9706	−0.470792
$544$	0	0
$545$	3.34315	0.143205
$546$	0	0
$547$	39.6274i	1.69435i	0.531317	+	0.847173i	$0.321697\pi$
$547$	39.6274i	1.69435i	−0.531317	+	0.847173i	$0.678303\pi$
$548$	0	0
$549$	10.3431i	0.441435i
$550$	0	0
$551$	−35.3137	−1.50441
$552$	0	0
$553$	6.68629	0.284330
$554$	0	0
$555$	− 3.34315i	− 0.141909i
$556$	0	0
$557$	− 23.1421i	− 0.980564i	−0.871564	−	0.490282i	$-0.836894\pi$
$557$	− 23.1421i	− 0.980564i	0.871564	−	0.490282i	$-0.163106\pi$
$558$	0	0
$559$	−3.00000	−0.126886
$560$	0	0
$561$	−3.85786	−0.162879
$562$	0	0
$563$	− 40.5980i	− 1.71100i	−0.517802	−	0.855500i	$-0.673250\pi$
$563$	− 40.5980i	− 1.71100i	0.517802	−	0.855500i	$-0.326750\pi$
$564$	0	0
$565$	14.0000i	0.588984i
$566$	0	0
$567$	−1.82843	−0.0767867
$568$	0	0
$569$	−31.9706	−1.34028	−0.670138	−	0.742237i	$-0.733765\pi$
$569$	−31.9706	−1.34028	−0.670138	+	0.742237i	$0.733765\pi$
$570$	0	0
$571$	− 11.3431i	− 0.474696i	−0.971425	−	0.237348i	$-0.923722\pi$
$571$	− 11.3431i	− 0.474696i	0.971425	−	0.237348i	$-0.0762781\pi$
$572$	0	0
$573$	− 8.82843i	− 0.368813i
$574$	0	0
$575$	−14.6274	−0.610005
$576$	0	0
$577$	−17.4558	−0.726696	−0.363348	−	0.931653i	$-0.618366\pi$
$577$	−17.4558	−0.726696	−0.363348	+	0.931653i	$0.618366\pi$
$578$	0	0
$579$	− 6.00000i	− 0.249351i
$580$	0	0
$581$	10.9706i	0.455136i
$582$	0	0
$583$	−7.31371	−0.302903
$584$	0	0
$585$	3.65685	0.151192
$586$	0	0
$587$	37.1127i	1.53180i	0.642957	+	0.765902i	$0.277707\pi$
$587$	37.1127i	1.53180i	−0.642957	+	0.765902i	$0.722293\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.1716	0.500672
$592$	0	0
$593$	20.1421	0.827138	0.413569	−	0.910473i	$-0.364282\pi$
$593$	20.1421	0.827138	0.413569	+	0.910473i	$0.364282\pi$
$594$	0	0
$595$	15.5685i	0.638248i
$596$	0	0
$597$	10.9706i	0.448995i
$598$	0	0
$599$	−30.1421	−1.23157	−0.615787	−	0.787913i	$-0.711162\pi$
$599$	−30.1421	−1.23157	−0.615787	+	0.787913i	$0.711162\pi$
$600$	0	0
$601$	−24.3137	−0.991777	−0.495888	−	0.868386i	$-0.665158\pi$
$601$	−24.3137	−0.991777	−0.495888	+	0.868386i	$0.665158\pi$
$602$	0	0
$603$	15.3137i	0.623622i
$604$	0	0
$605$	18.8579i	0.766681i
$606$	0	0
$607$	5.17157	0.209908	0.104954	−	0.994477i	$-0.466531\pi$
$607$	5.17157	0.209908	0.104954	+	0.994477i	$0.466531\pi$
$608$	0	0
$609$	16.1421	0.654112
$610$	0	0
$611$	5.48528i	0.221911i
$612$	0	0
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	0	0
$615$	−5.17157	−0.208538
$616$	0	0
$617$	23.3137	0.938575	0.469287	−	0.883046i	$-0.344511\pi$
$617$	23.3137	0.938575	0.469287	+	0.883046i	$0.344511\pi$
$618$	0	0
$619$	− 10.1421i	− 0.407647i	−0.979008	−	0.203823i	$-0.934663\pi$
$619$	− 10.1421i	− 0.407647i	0.979008	−	0.203823i	$-0.0653368\pi$
$620$	0	0
$621$	− 44.1421i	− 1.77136i
$622$	0	0
$623$	−11.5980	−0.464663
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	3.31371i	0.132337i
$628$	0	0
$629$	− 8.51472i	− 0.339504i
$630$	0	0
$631$	26.7990	1.06685	0.533425	−	0.845847i	$-0.320904\pi$
$631$	26.7990	1.06685	0.533425	+	0.845847i	$0.320904\pi$
$632$	0	0
$633$	−12.6569	−0.503065
$634$	0	0
$635$	− 2.76955i	− 0.109906i
$636$	0	0
$637$	3.65685i	0.144890i
$638$	0	0
$639$	−31.6569	−1.25233
$640$	0	0
$641$	10.9706	0.433311	0.216656	−	0.976248i	$-0.430485\pi$
$641$	10.9706	0.433311	0.216656	+	0.976248i	$0.430485\pi$
$642$	0	0
$643$	3.37258i	0.133002i	0.997786	+	0.0665008i	$0.0211835\pi$
$643$	3.37258i	0.133002i	−0.997786	+	0.0665008i	$0.978816\pi$
$644$	0	0
$645$	5.48528i	0.215983i
$646$	0	0
$647$	22.8284	0.897478	0.448739	−	0.893663i	$-0.351873\pi$
$647$	22.8284	0.897478	0.448739	+	0.893663i	$0.351873\pi$
$648$	0	0
$649$	−8.97056	−0.352125
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 7.31371i	− 0.286208i	−0.989708	−	0.143104i	$-0.954292\pi$
$653$	− 7.31371i	− 0.286208i	0.989708	−	0.143104i	$-0.0457083\pi$
$654$	0	0
$655$	−15.8284	−0.618468
$656$	0	0
$657$	−21.6569	−0.844914
$658$	0	0
$659$	− 28.9706i	− 1.12853i	−0.825593	−	0.564266i	$-0.809159\pi$
$659$	− 28.9706i	− 1.12853i	0.825593	−	0.564266i	$-0.190841\pi$
$660$	0	0
$661$	21.3137i	0.829007i	0.910048	+	0.414504i	$0.136045\pi$
$661$	21.3137i	0.829007i	−0.910048	+	0.414504i	$0.863955\pi$
$662$	0	0
$663$	−4.65685	−0.180857
$664$	0	0
$665$	13.3726	0.518567
$666$	0	0
$667$	− 77.9411i	− 3.01789i
$668$	0	0
$669$	15.8284i	0.611962i
$670$	0	0
$671$	4.28427	0.165392
$672$	0	0
$673$	−10.3137	−0.397564	−0.198782	−	0.980044i	$-0.563699\pi$
$673$	−10.3137	−0.397564	−0.198782	+	0.980044i	$0.563699\pi$
$674$	0	0
$675$	8.28427i	0.318862i
$676$	0	0
$677$	6.68629i	0.256975i	0.991711	+	0.128488i	$0.0410122\pi$
$677$	6.68629i	0.256975i	−0.991711	+	0.128488i	$0.958988\pi$
$678$	0	0
$679$	−4.54416	−0.174389
$680$	0	0
$681$	14.8284	0.568227
$682$	0	0
$683$	− 28.3431i	− 1.08452i	−0.840211	−	0.542260i	$-0.817569\pi$
$683$	− 28.3431i	− 1.08452i	0.840211	−	0.542260i	$-0.182431\pi$
$684$	0	0
$685$	− 14.8873i	− 0.568815i
$686$	0	0
$687$	−15.8284	−0.603892
$688$	0	0
$689$	−8.82843	−0.336336
$690$	0	0
$691$	40.9706i	1.55859i	0.626655	+	0.779297i	$0.284424\pi$
$691$	40.9706i	1.55859i	−0.626655	+	0.779297i	$0.715576\pi$
$692$	0	0
$693$	3.02944i	0.115079i
$694$	0	0
$695$	−13.4264	−0.509293
$696$	0	0
$697$	−13.1716	−0.498909
$698$	0	0
$699$	2.31371i	0.0875125i
$700$	0	0
$701$	16.1421i	0.609680i	0.952404	+	0.304840i	$0.0986030\pi$
$701$	16.1421i	0.609680i	−0.952404	+	0.304840i	$0.901397\pi$
$702$	0	0
$703$	−7.31371	−0.275842
$704$	0	0
$705$	10.0294	0.377730
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.65685i	0.137336i	0.997640	+	0.0686680i	$0.0218749\pi$
$709$	3.65685i	0.137336i	−0.997640	+	0.0686680i	$0.978125\pi$
$710$	0	0
$711$	−7.31371	−0.274285
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 1.51472i	− 0.0566473i
$716$	0	0
$717$	9.14214i	0.341419i
$718$	0	0
$719$	−10.9706	−0.409133	−0.204566	−	0.978853i	$-0.565578\pi$
$719$	−10.9706	−0.409133	−0.204566	+	0.978853i	$0.565578\pi$
$720$	0	0
$721$	25.5980	0.953319
$722$	0	0
$723$	− 19.3137i	− 0.718285i
$724$	0	0
$725$	14.6274i	0.543249i
$726$	0	0
$727$	−1.51472	−0.0561778	−0.0280889	−	0.999605i	$-0.508942\pi$
$727$	−1.51472	−0.0561778	−0.0280889	+	0.999605i	$0.508942\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	13.9706i	0.516720i
$732$	0	0
$733$	− 37.7696i	− 1.39505i	−0.716560	−	0.697525i	$-0.754285\pi$
$733$	− 37.7696i	− 1.39505i	0.716560	−	0.697525i	$-0.245715\pi$
$734$	0	0
$735$	6.68629	0.246628
$736$	0	0
$737$	6.34315	0.233653
$738$	0	0
$739$	− 48.4853i	− 1.78356i	−0.452469	−	0.891780i	$-0.649457\pi$
$739$	− 48.4853i	− 1.78356i	0.452469	−	0.891780i	$-0.350543\pi$
$740$	0	0
$741$	4.00000i	0.146944i
$742$	0	0
$743$	1.20101	0.0440608	0.0220304	−	0.999757i	$-0.492987\pi$
$743$	1.20101	0.0440608	0.0220304	+	0.999757i	$0.492987\pi$
$744$	0	0
$745$	38.9706	1.42777
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	− 13.3726i	− 0.488624i
$750$	0	0
$751$	40.4853	1.47733	0.738664	−	0.674073i	$-0.235457\pi$
$751$	40.4853	1.47733	0.738664	+	0.674073i	$0.235457\pi$
$752$	0	0
$753$	18.6274	0.678821
$754$	0	0
$755$	35.6274i	1.29661i
$756$	0	0
$757$	− 36.8284i	− 1.33855i	−0.743014	−	0.669276i	$-0.766604\pi$
$757$	− 36.8284i	− 1.33855i	0.743014	−	0.669276i	$-0.233396\pi$
$758$	0	0
$759$	−7.31371	−0.265471
$760$	0	0
$761$	38.8284	1.40753	0.703765	−	0.710433i	$-0.251501\pi$
$761$	38.8284	1.40753	0.703765	+	0.710433i	$0.251501\pi$
$762$	0	0
$763$	3.34315i	0.121030i
$764$	0	0
$765$	− 17.0294i	− 0.615701i
$766$	0	0
$767$	−10.8284	−0.390992
$768$	0	0
$769$	−2.34315	−0.0844960	−0.0422480	−	0.999107i	$-0.513452\pi$
$769$	−2.34315	−0.0844960	−0.0422480	+	0.999107i	$0.513452\pi$
$770$	0	0
$771$	− 12.6569i	− 0.455825i
$772$	0	0
$773$	− 12.7990i	− 0.460348i	−0.973150	−	0.230174i	$-0.926071\pi$
$773$	− 12.7990i	− 0.460348i	0.973150	−	0.230174i	$-0.0739295\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.34315	0.119935
$778$	0	0
$779$	11.3137i	0.405356i
$780$	0	0
$781$	13.1127i	0.469209i
$782$	0	0
$783$	−44.1421	−1.57751
$784$	0	0
$785$	−22.8284	−0.814782
$786$	0	0
$787$	28.4853i	1.01539i	0.861537	+	0.507695i	$0.169502\pi$
$787$	28.4853i	1.01539i	−0.861537	+	0.507695i	$0.830498\pi$
$788$	0	0
$789$	− 2.14214i	− 0.0762620i
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	5.17157	0.183648
$794$	0	0
$795$	16.1421i	0.572503i
$796$	0	0
$797$	45.6569i	1.61725i	0.588325	+	0.808624i	$0.299787\pi$
$797$	45.6569i	1.61725i	−0.588325	+	0.808624i	$0.700213\pi$
$798$	0	0
$799$	25.5442	0.903687
$800$	0	0
$801$	12.6863	0.448248
$802$	0	0
$803$	8.97056i	0.316564i
$804$	0	0
$805$	29.5147i	1.04026i
$806$	0	0
$807$	−19.7990	−0.696957
$808$	0	0
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	− 4.97056i	− 0.174540i	−0.996185	−	0.0872700i	$-0.972186\pi$
$811$	− 4.97056i	− 0.174540i	0.996185	−	0.0872700i	$-0.0278143\pi$
$812$	0	0
$813$	19.4853i	0.683379i
$814$	0	0
$815$	−13.1127	−0.459318
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	3.65685i	0.127781i
$820$	0	0
$821$	− 5.48528i	− 0.191438i	−0.995408	−	0.0957188i	$-0.969485\pi$
$821$	− 5.48528i	− 0.191438i	0.995408	−	0.0957188i	$-0.0305150\pi$
$822$	0	0
$823$	−44.7696	−1.56057	−0.780284	−	0.625425i	$-0.784926\pi$
$823$	−44.7696	−1.56057	−0.780284	+	0.625425i	$0.784926\pi$
$824$	0	0
$825$	1.37258	0.0477872
$826$	0	0
$827$	− 14.6274i	− 0.508645i	−0.967119	−	0.254323i	$-0.918148\pi$
$827$	− 14.6274i	− 0.508645i	0.967119	−	0.254323i	$-0.0818525\pi$
$828$	0	0
$829$	10.9706i	0.381023i	0.981685	+	0.190512i	$0.0610147\pi$
$829$	10.9706i	0.381023i	−0.981685	+	0.190512i	$0.938985\pi$
$830$	0	0
$831$	12.4853	0.433110
$832$	0	0
$833$	17.0294	0.590035
$834$	0	0
$835$	18.9117i	0.654466i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−35.3137	−1.21916	−0.609582	−	0.792723i	$-0.708663\pi$
$839$	−35.3137	−1.21916	−0.609582	+	0.792723i	$0.708663\pi$
$840$	0	0
$841$	−48.9411	−1.68763
$842$	0	0
$843$	− 28.9706i	− 0.997799i
$844$	0	0
$845$	− 1.82843i	− 0.0628998i
$846$	0	0
$847$	−18.8579	−0.647964
$848$	0	0
$849$	−30.6274	−1.05113
$850$	0	0
$851$	− 16.1421i	− 0.553345i
$852$	0	0
$853$	37.7696i	1.29320i	0.762827	+	0.646602i	$0.223811\pi$
$853$	37.7696i	1.29320i	−0.762827	+	0.646602i	$0.776189\pi$
$854$	0	0
$855$	−14.6274	−0.500247
$856$	0	0
$857$	−18.6863	−0.638312	−0.319156	−	0.947702i	$-0.603399\pi$
$857$	−18.6863	−0.638312	−0.319156	+	0.947702i	$0.603399\pi$
$858$	0	0
$859$	28.9706i	0.988463i	0.869330	+	0.494231i	$0.164550\pi$
$859$	28.9706i	0.988463i	−0.869330	+	0.494231i	$0.835450\pi$
$860$	0	0
$861$	− 5.17157i	− 0.176247i
$862$	0	0
$863$	−48.1127	−1.63778	−0.818888	−	0.573954i	$-0.805409\pi$
$863$	−48.1127	−1.63778	−0.818888	+	0.573954i	$0.805409\pi$
$864$	0	0
$865$	−20.0589	−0.682022
$866$	0	0
$867$	4.68629i	0.159155i
$868$	0	0
$869$	3.02944i	0.102767i
$870$	0	0
$871$	7.65685	0.259443
$872$	0	0
$873$	4.97056	0.168228
$874$	0	0
$875$	− 22.2548i	− 0.752351i
$876$	0	0
$877$	− 33.4853i	− 1.13072i	−0.824845	−	0.565359i	$-0.808738\pi$
$877$	− 33.4853i	− 1.13072i	0.824845	−	0.565359i	$-0.191262\pi$
$878$	0	0
$879$	−26.7990	−0.903907
$880$	0	0
$881$	17.6863	0.595866	0.297933	−	0.954587i	$-0.403703\pi$
$881$	17.6863	0.595866	0.297933	+	0.954587i	$0.403703\pi$
$882$	0	0
$883$	− 31.9706i	− 1.07590i	−0.842978	−	0.537948i	$-0.819200\pi$
$883$	− 31.9706i	− 1.07590i	0.842978	−	0.537948i	$-0.180800\pi$
$884$	0	0
$885$	19.7990i	0.665536i
$886$	0	0
$887$	2.14214	0.0719259	0.0359629	−	0.999353i	$-0.488550\pi$
$887$	2.14214	0.0719259	0.0359629	+	0.999353i	$0.488550\pi$
$888$	0	0
$889$	2.76955	0.0928878
$890$	0	0
$891$	− 0.828427i	− 0.0277534i
$892$	0	0
$893$	− 21.9411i	− 0.734232i
$894$	0	0
$895$	−26.1716	−0.874819
$896$	0	0
$897$	−8.82843	−0.294773
$898$	0	0
$899$	0	0
$900$	0	0
$901$	41.1127i	1.36966i
$902$	0	0
$903$	−5.48528	−0.182539
$904$	0	0
$905$	−20.0589	−0.666780
$906$	0	0
$907$	42.9411i	1.42584i	0.701247	+	0.712918i	$0.252627\pi$
$907$	42.9411i	1.42584i	−0.701247	+	0.712918i	$0.747373\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.54416	−0.150555	−0.0752773	−	0.997163i	$-0.523984\pi$
$911$	−4.54416	−0.150555	−0.0752773	+	0.997163i	$0.523984\pi$
$912$	0	0
$913$	−4.97056	−0.164502
$914$	0	0
$915$	− 9.45584i	− 0.312601i
$916$	0	0
$917$	− 15.8284i	− 0.522701i
$918$	0	0
$919$	−7.31371	−0.241257	−0.120628	−	0.992698i	$-0.538491\pi$
$919$	−7.31371	−0.241257	−0.120628	+	0.992698i	$0.538491\pi$
$920$	0	0
$921$	18.8284	0.620418
$922$	0	0
$923$	15.8284i	0.520999i
$924$	0	0
$925$	3.02944i	0.0996073i
$926$	0	0
$927$	−28.0000	−0.919641
$928$	0	0
$929$	−51.3137	−1.68355	−0.841774	−	0.539830i	$-0.818489\pi$
$929$	−51.3137	−1.68355	−0.841774	+	0.539830i	$0.818489\pi$
$930$	0	0
$931$	− 14.6274i	− 0.479394i
$932$	0	0
$933$	32.2843i	1.05694i
$934$	0	0
$935$	−7.05382	−0.230685
$936$	0	0
$937$	4.34315	0.141884	0.0709422	−	0.997480i	$-0.477399\pi$
$937$	4.34315	0.141884	0.0709422	+	0.997480i	$0.477399\pi$
$938$	0	0
$939$	13.9706i	0.455912i
$940$	0	0
$941$	− 1.20101i	− 0.0391518i	−0.999808	−	0.0195759i	$-0.993768\pi$
$941$	− 1.20101i	− 0.0391518i	0.999808	−	0.0195759i	$-0.00623160\pi$
$942$	0	0
$943$	−24.9706	−0.813153
$944$	0	0
$945$	16.7157	0.543763
$946$	0	0
$947$	− 25.1127i	− 0.816053i	−0.912970	−	0.408027i	$-0.866217\pi$
$947$	− 25.1127i	− 0.816053i	0.912970	−	0.408027i	$-0.133783\pi$
$948$	0	0
$949$	10.8284i	0.351506i
$950$	0	0
$951$	−6.68629	−0.216818
$952$	0	0
$953$	28.3137	0.917171	0.458585	−	0.888650i	$-0.348356\pi$
$953$	28.3137	0.917171	0.458585	+	0.888650i	$0.348356\pi$
$954$	0	0
$955$	− 16.1421i	− 0.522347i
$956$	0	0
$957$	7.31371i	0.236419i
$958$	0	0
$959$	14.8873	0.480736
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	14.6274i	0.471362i
$964$	0	0
$965$	− 10.9706i	− 0.353155i
$966$	0	0
$967$	54.7990	1.76222	0.881108	−	0.472914i	$-0.156798\pi$
$967$	54.7990	1.76222	0.881108	+	0.472914i	$0.156798\pi$
$968$	0	0
$969$	18.6274	0.598399
$970$	0	0
$971$	42.5980i	1.36703i	0.729934	+	0.683517i	$0.239551\pi$
$971$	42.5980i	1.36703i	−0.729934	+	0.683517i	$0.760449\pi$
$972$	0	0
$973$	− 13.4264i	− 0.430431i
$974$	0	0
$975$	1.65685	0.0530618
$976$	0	0
$977$	−54.2843	−1.73671	−0.868354	−	0.495945i	$-0.834822\pi$
$977$	−54.2843	−1.73671	−0.868354	+	0.495945i	$0.834822\pi$
$978$	0	0
$979$	− 5.25483i	− 0.167945i
$980$	0	0
$981$	− 3.65685i	− 0.116754i
$982$	0	0
$983$	−51.7696	−1.65119	−0.825596	−	0.564261i	$-0.809161\pi$
$983$	−51.7696	−1.65119	−0.825596	+	0.564261i	$0.809161\pi$
$984$	0	0
$985$	22.2548	0.709098
$986$	0	0
$987$	10.0294i	0.319241i
$988$	0	0
$989$	26.4853i	0.842183i
$990$	0	0
$991$	−52.9706	−1.68267	−0.841333	−	0.540518i	$-0.818228\pi$
$991$	−52.9706	−1.68267	−0.841333	+	0.540518i	$0.818228\pi$
$992$	0	0
$993$	22.9706	0.728949
$994$	0	0
$995$	20.0589i	0.635909i
$996$	0	0
$997$	23.4558i	0.742854i	0.928462	+	0.371427i	$0.121131\pi$
$997$	23.4558i	0.742854i	−0.928462	+	0.371427i	$0.878869\pi$
$998$	0	0
$999$	−9.14214	−0.289244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.i.833.4	yes	4
4.3	odd	2		1664.2.b.e.833.2	✓	4
8.3	odd	2		1664.2.b.e.833.3	yes	4
8.5	even	2	inner	1664.2.b.i.833.1	yes	4
16.3	odd	4		3328.2.a.z.1.2		2
16.5	even	4		3328.2.a.ba.1.1		2
16.11	odd	4		3328.2.a.r.1.1		2
16.13	even	4		3328.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.e.833.2	✓	4	4.3	odd	2
1664.2.b.e.833.3	yes	4	8.3	odd	2
1664.2.b.i.833.1	yes	4	8.5	even	2	inner
1664.2.b.i.833.4	yes	4	1.1	even	1	trivial
3328.2.a.o.1.2		2	16.13	even	4
3328.2.a.r.1.1		2	16.11	odd	4
3328.2.a.z.1.2		2	16.3	odd	4
3328.2.a.ba.1.1		2	16.5	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:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.82843i q^{5} -1.82843 q^{7} +2.00000 q^{9} -0.828427i q^{11} -1.00000i q^{13} -1.82843 q^{15} -4.65685 q^{17} +4.00000i q^{19} -1.82843i q^{21} -8.82843 q^{23} +1.65685 q^{25} +5.00000i q^{27} +8.82843i q^{29} +0.828427 q^{33} -3.34315i q^{35} +1.82843i q^{37} +1.00000 q^{39} +2.82843 q^{41} -3.00000i q^{43} +3.65685i q^{45} -5.48528 q^{47} -3.65685 q^{49} -4.65685i q^{51} -8.82843i q^{53} +1.51472 q^{55} -4.00000 q^{57} -10.8284i q^{59} +5.17157i q^{61} -3.65685 q^{63} +1.82843 q^{65} +7.65685i q^{67} -8.82843i q^{69} -15.8284 q^{71} -10.8284 q^{73} +1.65685i q^{75} +1.51472i q^{77} -3.65685 q^{79} +1.00000 q^{81} -6.00000i q^{83} -8.51472i q^{85} -8.82843 q^{87} +6.34315 q^{89} +1.82843i q^{91} -7.31371 q^{95} +2.48528 q^{97} -1.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} + 8 q^{9} + 4 q^{15} + 4 q^{17} - 24 q^{23} - 16 q^{25} - 8 q^{33} + 4 q^{39} + 12 q^{47} + 8 q^{49} + 40 q^{55} - 16 q^{57} + 8 q^{63} - 4 q^{65} - 52 q^{71} - 32 q^{73} + 8 q^{79} + 4 q^{81}+ \cdots - 24 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 8 * q^9 + 4 * q^15 + 4 * q^17 - 24 * q^23 - 16 * q^25 - 8 * q^33 + 4 * q^39 + 12 * q^47 + 8 * q^49 + 40 * q^55 - 16 * q^57 + 8 * q^63 - 4 * q^65 - 52 * q^71 - 32 * q^73 + 8 * q^79 + 4 * q^81 - 24 * q^87 + 48 * q^89 + 16 * q^95 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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