LMFDB - Embedded newform 3328.2.a.r.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); 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, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,2,0,-4] 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:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.82843 q^{5} -1.82843 q^{7} -2.00000 q^{9} -0.828427 q^{11} -1.00000 q^{13} +1.82843 q^{15} -4.65685 q^{17} -4.00000 q^{19} +1.82843 q^{21} -8.82843 q^{23} -1.65685 q^{25} +5.00000 q^{27} +8.82843 q^{29} +0.828427 q^{33} +3.34315 q^{35} -1.82843 q^{37} +1.00000 q^{39} -2.82843 q^{41} -3.00000 q^{43} +3.65685 q^{45} +5.48528 q^{47} -3.65685 q^{49} +4.65685 q^{51} +8.82843 q^{53} +1.51472 q^{55} +4.00000 q^{57} -10.8284 q^{59} +5.17157 q^{61} +3.65685 q^{63} +1.82843 q^{65} -7.65685 q^{67} +8.82843 q^{69} -15.8284 q^{71} +10.8284 q^{73} +1.65685 q^{75} +1.51472 q^{77} +3.65685 q^{79} +1.00000 q^{81} +6.00000 q^{83} +8.51472 q^{85} -8.82843 q^{87} -6.34315 q^{89} +1.82843 q^{91} +7.31371 q^{95} +2.48528 q^{97} +1.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} - 2 q^{21} - 12 q^{23} + 8 q^{25} + 10 q^{27} + 12 q^{29} - 4 q^{33} + 18 q^{35} + 2 q^{37} + 2 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 - 4 * q^9 + 4 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 - 2 * q^21 - 12 * q^23 + 8 * q^25 + 10 * q^27 + 12 * q^29 - 4 * q^33 + 18 * q^35 + 2 * q^37 + 2 * q^39 - 6 * q^43 - 4 * q^45 - 6 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 + 20 * q^55 + 8 * q^57 - 16 * q^59 + 16 * q^61 - 4 * q^63 - 2 * q^65 - 4 * q^67 + 12 * q^69 - 26 * q^71 + 16 * q^73 - 8 * q^75 + 20 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^83 + 34 * q^85 - 12 * q^87 - 24 * q^89 - 2 * q^91 - 8 * q^95 - 12 * q^97 - 8 * q^99

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.82843	−0.817697	−0.408849	−	0.912602i	$-0.634070\pi$
$5$	−1.82843	−0.817697	−0.408849	+	0.912602i	$0.634070\pi$
$6$	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
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	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
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	1.82843	0.398996
$22$	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
$27$	5.00000	0.962250
$28$	0	0
$29$	8.82843	1.63940	0.819699	−	0.572795i	$-0.194141\pi$
$29$	8.82843	1.63940	0.819699	+	0.572795i	$0.194141\pi$
$30$	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
$35$	3.34315	0.565095
$36$	0	0
$37$	−1.82843	−0.300592	−0.150296	−	0.988641i	$-0.548023\pi$
$37$	−1.82843	−0.300592	−0.150296	+	0.988641i	$0.548023\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−2.82843	−0.441726	−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843	−0.441726	−0.220863	+	0.975305i	$0.570887\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	0	0
$45$	3.65685	0.545132
$46$	0	0
$47$	5.48528	0.800111	0.400055	−	0.916491i	$-0.368991\pi$
$47$	5.48528	0.800111	0.400055	+	0.916491i	$0.368991\pi$
$48$	0	0
$49$	−3.65685	−0.522408
$50$	0	0
$51$	4.65685	0.652090
$52$	0	0
$53$	8.82843	1.21268	0.606339	−	0.795206i	$-0.292638\pi$
$53$	8.82843	1.21268	0.606339	+	0.795206i	$0.292638\pi$
$54$	0	0
$55$	1.51472	0.204245
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−10.8284	−1.40974	−0.704871	−	0.709336i	$-0.748995\pi$
$59$	−10.8284	−1.40974	−0.704871	+	0.709336i	$0.748995\pi$
$60$	0	0
$61$	5.17157	0.662152	0.331076	−	0.943604i	$-0.392588\pi$
$61$	5.17157	0.662152	0.331076	+	0.943604i	$0.392588\pi$
$62$	0	0
$63$	3.65685	0.460720
$64$	0	0
$65$	1.82843	0.226788
$66$	0	0
$67$	−7.65685	−0.935434	−0.467717	−	0.883878i	$-0.654923\pi$
$67$	−7.65685	−0.935434	−0.467717	+	0.883878i	$0.654923\pi$
$68$	0	0
$69$	8.82843	1.06282
$70$	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	$-0.281541\pi$
$73$	10.8284	1.26737	0.633686	+	0.773591i	$0.281541\pi$
$74$	0	0
$75$	1.65685	0.191317
$76$	0	0
$77$	1.51472	0.172618
$78$	0	0
$79$	3.65685	0.411428	0.205714	−	0.978612i	$-0.434048\pi$
$79$	3.65685	0.411428	0.205714	+	0.978612i	$0.434048\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	8.51472	0.923551
$86$	0	0
$87$	−8.82843	−0.946507
$88$	0	0
$89$	−6.34315	−0.672372	−0.336186	−	0.941796i	$-0.609137\pi$
$89$	−6.34315	−0.672372	−0.336186	+	0.941796i	$0.609137\pi$
$90$	0	0
$91$	1.82843	0.191671
$92$	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
$99$	1.65685	0.166520
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	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
$107$	7.31371	0.707043	0.353521	−	0.935426i	$-0.384984\pi$
$107$	7.31371	0.707043	0.353521	+	0.935426i	$0.384984\pi$
$108$	0	0
$109$	−1.82843	−0.175132	−0.0875658	−	0.996159i	$-0.527909\pi$
$109$	−1.82843	−0.175132	−0.0875658	+	0.996159i	$0.527909\pi$
$110$	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
$115$	16.1421	1.50526
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	8.51472	0.780543
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	2.82843	0.255031
$124$	0	0
$125$	12.1716	1.08866
$126$	0	0
$127$	1.51472	0.134410	0.0672048	−	0.997739i	$-0.478592\pi$
$127$	1.51472	0.134410	0.0672048	+	0.997739i	$0.478592\pi$
$128$	0	0
$129$	3.00000	0.264135
$130$	0	0
$131$	−8.65685	−0.756353	−0.378176	−	0.925734i	$-0.623449\pi$
$131$	−8.65685	−0.756353	−0.378176	+	0.925734i	$0.623449\pi$
$132$	0	0
$133$	7.31371	0.634179
$134$	0	0
$135$	−9.14214	−0.786830
$136$	0	0
$137$	8.14214	0.695630	0.347815	−	0.937563i	$-0.386924\pi$
$137$	8.14214	0.695630	0.347815	+	0.937563i	$0.386924\pi$
$138$	0	0
$139$	7.34315	0.622837	0.311419	−	0.950273i	$-0.399196\pi$
$139$	7.34315	0.622837	0.311419	+	0.950273i	$0.399196\pi$
$140$	0	0
$141$	−5.48528	−0.461944
$142$	0	0
$143$	0.828427	0.0692766
$144$	0	0
$145$	−16.1421	−1.34053
$146$	0	0
$147$	3.65685	0.301612
$148$	0	0
$149$	21.3137	1.74609	0.873044	−	0.487642i	$-0.162143\pi$
$149$	21.3137	1.74609	0.873044	+	0.487642i	$0.162143\pi$
$150$	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
$157$	12.4853	0.996434	0.498217	−	0.867052i	$-0.333988\pi$
$157$	12.4853	0.996434	0.498217	+	0.867052i	$0.333988\pi$
$158$	0	0
$159$	−8.82843	−0.700140
$160$	0	0
$161$	16.1421	1.27218
$162$	0	0
$163$	−7.17157	−0.561721	−0.280860	−	0.959749i	$-0.590620\pi$
$163$	−7.17157	−0.561721	−0.280860	+	0.959749i	$0.590620\pi$
$164$	0	0
$165$	−1.51472	−0.117921
$166$	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
$171$	8.00000	0.611775
$172$	0	0
$173$	10.9706	0.834076	0.417038	−	0.908889i	$-0.363068\pi$
$173$	10.9706	0.834076	0.417038	+	0.908889i	$0.363068\pi$
$174$	0	0
$175$	3.02944	0.229004
$176$	0	0
$177$	10.8284	0.813914
$178$	0	0
$179$	−14.3137	−1.06986	−0.534928	−	0.844897i	$-0.679661\pi$
$179$	−14.3137	−1.06986	−0.534928	+	0.844897i	$0.679661\pi$
$180$	0	0
$181$	−10.9706	−0.815436	−0.407718	−	0.913108i	$-0.633675\pi$
$181$	−10.9706	−0.815436	−0.407718	+	0.913108i	$0.633675\pi$
$182$	0	0
$183$	−5.17157	−0.382294
$184$	0	0
$185$	3.34315	0.245793
$186$	0	0
$187$	3.85786	0.282115
$188$	0	0
$189$	−9.14214	−0.664993
$190$	0	0
$191$	8.82843	0.638803	0.319401	−	0.947620i	$-0.396518\pi$
$191$	8.82843	0.638803	0.319401	+	0.947620i	$0.396518\pi$
$192$	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
$195$	−1.82843	−0.130936
$196$	0	0
$197$	12.1716	0.867189	0.433594	−	0.901108i	$-0.357245\pi$
$197$	12.1716	0.867189	0.433594	+	0.901108i	$0.357245\pi$
$198$	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
$203$	−16.1421	−1.13296
$204$	0	0
$205$	5.17157	0.361198
$206$	0	0
$207$	17.6569	1.22724
$208$	0	0
$209$	3.31371	0.229214
$210$	0	0
$211$	−12.6569	−0.871334	−0.435667	−	0.900108i	$-0.643487\pi$
$211$	−12.6569	−0.871334	−0.435667	+	0.900108i	$0.643487\pi$
$212$	0	0
$213$	15.8284	1.08455
$214$	0	0
$215$	5.48528	0.374093
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−10.8284	−0.731717
$220$	0	0
$221$	4.65685	0.313254
$222$	0	0
$223$	−15.8284	−1.05995	−0.529975	−	0.848013i	$-0.677799\pi$
$223$	−15.8284	−1.05995	−0.529975	+	0.848013i	$0.677799\pi$
$224$	0	0
$225$	3.31371	0.220914
$226$	0	0
$227$	14.8284	0.984197	0.492099	−	0.870539i	$-0.336230\pi$
$227$	14.8284	0.984197	0.492099	+	0.870539i	$0.336230\pi$
$228$	0	0
$229$	−15.8284	−1.04597	−0.522986	−	0.852341i	$-0.675182\pi$
$229$	−15.8284	−1.04597	−0.522986	+	0.852341i	$0.675182\pi$
$230$	0	0
$231$	−1.51472	−0.0996612
$232$	0	0
$233$	−2.31371	−0.151576	−0.0757880	−	0.997124i	$-0.524147\pi$
$233$	−2.31371	−0.151576	−0.0757880	+	0.997124i	$0.524147\pi$
$234$	0	0
$235$	−10.0294	−0.654248
$236$	0	0
$237$	−3.65685	−0.237538
$238$	0	0
$239$	−9.14214	−0.591356	−0.295678	−	0.955288i	$-0.595545\pi$
$239$	−9.14214	−0.591356	−0.295678	+	0.955288i	$0.595545\pi$
$240$	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
$243$	−16.0000	−1.02640
$244$	0	0
$245$	6.68629	0.427171
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−18.6274	−1.17575	−0.587876	−	0.808951i	$-0.700036\pi$
$251$	−18.6274	−1.17575	−0.587876	+	0.808951i	$0.700036\pi$
$252$	0	0
$253$	7.31371	0.459809
$254$	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
$259$	3.34315	0.207733
$260$	0	0
$261$	−17.6569	−1.09293
$262$	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
$267$	6.34315	0.388194
$268$	0	0
$269$	19.7990	1.20717	0.603583	−	0.797300i	$-0.293739\pi$
$269$	19.7990	1.20717	0.603583	+	0.797300i	$0.293739\pi$
$270$	0	0
$271$	−19.4853	−1.18365	−0.591823	−	0.806068i	$-0.701592\pi$
$271$	−19.4853	−1.18365	−0.591823	+	0.806068i	$0.701592\pi$
$272$	0	0
$273$	−1.82843	−0.110661
$274$	0	0
$275$	1.37258	0.0827699
$276$	0	0
$277$	12.4853	0.750168	0.375084	−	0.926991i	$-0.377614\pi$
$277$	12.4853	0.750168	0.375084	+	0.926991i	$0.377614\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	28.9706	1.72824	0.864119	−	0.503287i	$-0.167876\pi$
$281$	28.9706	1.72824	0.864119	+	0.503287i	$0.167876\pi$
$282$	0	0
$283$	30.6274	1.82061	0.910305	−	0.413937i	$-0.135847\pi$
$283$	30.6274	1.82061	0.910305	+	0.413937i	$0.135847\pi$
$284$	0	0
$285$	−7.31371	−0.433227
$286$	0	0
$287$	5.17157	0.305268
$288$	0	0
$289$	4.68629	0.275664
$290$	0	0
$291$	−2.48528	−0.145690
$292$	0	0
$293$	−26.7990	−1.56561	−0.782807	−	0.622265i	$-0.786213\pi$
$293$	−26.7990	−1.56561	−0.782807	+	0.622265i	$0.786213\pi$
$294$	0	0
$295$	19.7990	1.15274
$296$	0	0
$297$	−4.14214	−0.240351
$298$	0	0
$299$	8.82843	0.510561
$300$	0	0
$301$	5.48528	0.316166
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.45584	−0.541440
$306$	0	0
$307$	18.8284	1.07460	0.537298	−	0.843393i	$-0.319445\pi$
$307$	18.8284	1.07460	0.537298	+	0.843393i	$0.319445\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	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	$-0.629197\pi$
$313$	−13.9706	−0.789663	−0.394831	+	0.918754i	$0.629197\pi$
$314$	0	0
$315$	−6.68629	−0.376730
$316$	0	0
$317$	6.68629	0.375540	0.187770	−	0.982213i	$-0.439874\pi$
$317$	6.68629	0.375540	0.187770	+	0.982213i	$0.439874\pi$
$318$	0	0
$319$	−7.31371	−0.409489
$320$	0	0
$321$	−7.31371	−0.408211
$322$	0	0
$323$	18.6274	1.03646
$324$	0	0
$325$	1.65685	0.0919057
$326$	0	0
$327$	1.82843	0.101112
$328$	0	0
$329$	−10.0294	−0.552941
$330$	0	0
$331$	−22.9706	−1.26258	−0.631288	−	0.775548i	$-0.717473\pi$
$331$	−22.9706	−1.26258	−0.631288	+	0.775548i	$0.717473\pi$
$332$	0	0
$333$	3.65685	0.200394
$334$	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
$339$	−7.65685	−0.415863
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.4853	1.05211
$344$	0	0
$345$	−16.1421	−0.869063
$346$	0	0
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	0	0
$349$	8.51472	0.455782	0.227891	−	0.973687i	$-0.426817\pi$
$349$	8.51472	0.455782	0.227891	+	0.973687i	$0.426817\pi$
$350$	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
$355$	28.9411	1.53604
$356$	0	0
$357$	−8.51472	−0.450647
$358$	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
$363$	10.3137	0.541329
$364$	0	0
$365$	−19.7990	−1.03633
$366$	0	0
$367$	−35.9411	−1.87611	−0.938056	−	0.346484i	$-0.887375\pi$
$367$	−35.9411	−1.87611	−0.938056	+	0.346484i	$0.887375\pi$
$368$	0	0
$369$	5.65685	0.294484
$370$	0	0
$371$	−16.1421	−0.838058
$372$	0	0
$373$	24.9706	1.29293	0.646463	−	0.762945i	$-0.276247\pi$
$373$	24.9706	1.29293	0.646463	+	0.762945i	$0.276247\pi$
$374$	0	0
$375$	−12.1716	−0.628537
$376$	0	0
$377$	−8.82843	−0.454687
$378$	0	0
$379$	27.4558	1.41031	0.705156	−	0.709052i	$-0.250877\pi$
$379$	27.4558	1.41031	0.705156	+	0.709052i	$0.250877\pi$
$380$	0	0
$381$	−1.51472	−0.0776014
$382$	0	0
$383$	12.1716	0.621938	0.310969	−	0.950420i	$-0.399346\pi$
$383$	12.1716	0.621938	0.310969	+	0.950420i	$0.399346\pi$
$384$	0	0
$385$	−2.76955	−0.141149
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	−2.14214	−0.108611	−0.0543053	−	0.998524i	$-0.517294\pi$
$389$	−2.14214	−0.108611	−0.0543053	+	0.998524i	$0.517294\pi$
$390$	0	0
$391$	41.1127	2.07916
$392$	0	0
$393$	8.65685	0.436681
$394$	0	0
$395$	−6.68629	−0.336424
$396$	0	0
$397$	−21.3137	−1.06970	−0.534852	−	0.844946i	$-0.679633\pi$
$397$	−21.3137	−1.06970	−0.534852	+	0.844946i	$0.679633\pi$
$398$	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
$405$	−1.82843	−0.0908553
$406$	0	0
$407$	1.51472	0.0750818
$408$	0	0
$409$	−18.4853	−0.914038	−0.457019	−	0.889457i	$-0.651083\pi$
$409$	−18.4853	−0.914038	−0.457019	+	0.889457i	$0.651083\pi$
$410$	0	0
$411$	−8.14214	−0.401622
$412$	0	0
$413$	19.7990	0.974245
$414$	0	0
$415$	−10.9706	−0.538524
$416$	0	0
$417$	−7.34315	−0.359595
$418$	0	0
$419$	−30.6569	−1.49769	−0.748843	−	0.662748i	$-0.769390\pi$
$419$	−30.6569	−1.49769	−0.748843	+	0.662748i	$0.769390\pi$
$420$	0	0
$421$	−37.1421	−1.81020	−0.905098	−	0.425202i	$-0.860203\pi$
$421$	−37.1421	−1.81020	−0.905098	+	0.425202i	$0.860203\pi$
$422$	0	0
$423$	−10.9706	−0.533407
$424$	0	0
$425$	7.71573	0.374268
$426$	0	0
$427$	−9.45584	−0.457600
$428$	0	0
$429$	−0.828427	−0.0399968
$430$	0	0
$431$	−12.1716	−0.586284	−0.293142	−	0.956069i	$-0.594701\pi$
$431$	−12.1716	−0.586284	−0.293142	+	0.956069i	$0.594701\pi$
$432$	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
$435$	16.1421	0.773956
$436$	0	0
$437$	35.3137	1.68928
$438$	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
$443$	22.3137	1.06016	0.530078	−	0.847949i	$-0.322163\pi$
$443$	22.3137	1.06016	0.530078	+	0.847949i	$0.322163\pi$
$444$	0	0
$445$	11.5980	0.549797
$446$	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
$451$	2.34315	0.110334
$452$	0	0
$453$	−19.4853	−0.915498
$454$	0	0
$455$	−3.34315	−0.156729
$456$	0	0
$457$	12.9706	0.606737	0.303369	−	0.952873i	$-0.401889\pi$
$457$	12.9706	0.606737	0.303369	+	0.952873i	$0.401889\pi$
$458$	0	0
$459$	−23.2843	−1.08682
$460$	0	0
$461$	5.48528	0.255475	0.127738	−	0.991808i	$-0.459228\pi$
$461$	5.48528	0.255475	0.127738	+	0.991808i	$0.459228\pi$
$462$	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
$467$	−41.6569	−1.92765	−0.963825	−	0.266537i	$-0.914121\pi$
$467$	−41.6569	−1.92765	−0.963825	+	0.266537i	$0.914121\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	−12.4853	−0.575291
$472$	0	0
$473$	2.48528	0.114273
$474$	0	0
$475$	6.62742	0.304087
$476$	0	0
$477$	−17.6569	−0.808452
$478$	0	0
$479$	−9.14214	−0.417715	−0.208857	−	0.977946i	$-0.566974\pi$
$479$	−9.14214	−0.417715	−0.208857	+	0.977946i	$0.566974\pi$
$480$	0	0
$481$	1.82843	0.0833691
$482$	0	0
$483$	−16.1421	−0.734493
$484$	0	0
$485$	−4.54416	−0.206339
$486$	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
$491$	40.3137	1.81933	0.909666	−	0.415340i	$-0.136338\pi$
$491$	40.3137	1.81933	0.909666	+	0.415340i	$0.136338\pi$
$492$	0	0
$493$	−41.1127	−1.85162
$494$	0	0
$495$	−3.02944	−0.136163
$496$	0	0
$497$	28.9411	1.29819
$498$	0	0
$499$	10.1421	0.454024	0.227012	−	0.973892i	$-0.427104\pi$
$499$	10.1421	0.454024	0.227012	+	0.973892i	$0.427104\pi$
$500$	0	0
$501$	−10.3431	−0.462098
$502$	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
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−18.2843	−0.810436	−0.405218	−	0.914220i	$-0.632804\pi$
$509$	−18.2843	−0.810436	−0.405218	+	0.914220i	$0.632804\pi$
$510$	0	0
$511$	−19.7990	−0.875856
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	25.5980	1.12798
$516$	0	0
$517$	−4.54416	−0.199852
$518$	0	0
$519$	−10.9706	−0.481554
$520$	0	0
$521$	−8.31371	−0.364230	−0.182115	−	0.983277i	$-0.558294\pi$
$521$	−8.31371	−0.364230	−0.182115	+	0.983277i	$0.558294\pi$
$522$	0	0
$523$	−1.65685	−0.0724492	−0.0362246	−	0.999344i	$-0.511533\pi$
$523$	−1.65685	−0.0724492	−0.0362246	+	0.999344i	$0.511533\pi$
$524$	0	0
$525$	−3.02944	−0.132215
$526$	0	0
$527$	0	0
$528$	0	0
$529$	54.9411	2.38874
$530$	0	0
$531$	21.6569	0.939827
$532$	0	0
$533$	2.82843	0.122513
$534$	0	0
$535$	−13.3726	−0.578147
$536$	0	0
$537$	14.3137	0.617682
$538$	0	0
$539$	3.02944	0.130487
$540$	0	0
$541$	−33.4853	−1.43964	−0.719822	−	0.694158i	$-0.755777\pi$
$541$	−33.4853	−1.43964	−0.719822	+	0.694158i	$0.755777\pi$
$542$	0	0
$543$	10.9706	0.470792
$544$	0	0
$545$	3.34315	0.143205
$546$	0	0
$547$	−39.6274	−1.69435	−0.847173	−	0.531317i	$-0.821697\pi$
$547$	−39.6274	−1.69435	−0.847173	+	0.531317i	$0.821697\pi$
$548$	0	0
$549$	−10.3431	−0.441435
$550$	0	0
$551$	−35.3137	−1.50441
$552$	0	0
$553$	−6.68629	−0.284330
$554$	0	0
$555$	−3.34315	−0.141909
$556$	0	0
$557$	−23.1421	−0.980564	−0.490282	−	0.871564i	$-0.663106\pi$
$557$	−23.1421	−0.980564	−0.490282	+	0.871564i	$0.663106\pi$
$558$	0	0
$559$	3.00000	0.126886
$560$	0	0
$561$	−3.85786	−0.162879
$562$	0	0
$563$	40.5980	1.71100	0.855500	−	0.517802i	$-0.173250\pi$
$563$	40.5980	1.71100	0.855500	+	0.517802i	$0.173250\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	0	0
$567$	−1.82843	−0.0767867
$568$	0	0
$569$	31.9706	1.34028	0.670138	−	0.742237i	$-0.266235\pi$
$569$	31.9706	1.34028	0.670138	+	0.742237i	$0.266235\pi$
$570$	0	0
$571$	−11.3431	−0.474696	−0.237348	−	0.971425i	$-0.576278\pi$
$571$	−11.3431	−0.474696	−0.237348	+	0.971425i	$0.576278\pi$
$572$	0	0
$573$	−8.82843	−0.368813
$574$	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
$579$	6.00000	0.249351
$580$	0	0
$581$	−10.9706	−0.455136
$582$	0	0
$583$	−7.31371	−0.302903
$584$	0	0
$585$	−3.65685	−0.151192
$586$	0	0
$587$	37.1127	1.53180	0.765902	−	0.642957i	$-0.222293\pi$
$587$	37.1127	1.53180	0.765902	+	0.642957i	$0.222293\pi$
$588$	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
$595$	−15.5685	−0.638248
$596$	0	0
$597$	−10.9706	−0.448995
$598$	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	$-0.334842\pi$
$601$	24.3137	0.991777	0.495888	+	0.868386i	$0.334842\pi$
$602$	0	0
$603$	15.3137	0.623622
$604$	0	0
$605$	18.8579	0.766681
$606$	0	0
$607$	−5.17157	−0.209908	−0.104954	−	0.994477i	$-0.533469\pi$
$607$	−5.17157	−0.209908	−0.104954	+	0.994477i	$0.533469\pi$
$608$	0	0
$609$	16.1421	0.654112
$610$	0	0
$611$	−5.48528	−0.221911
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	0	0
$615$	−5.17157	−0.208538
$616$	0	0
$617$	−23.3137	−0.938575	−0.469287	−	0.883046i	$-0.655489\pi$
$617$	−23.3137	−0.938575	−0.469287	+	0.883046i	$0.655489\pi$
$618$	0	0
$619$	−10.1421	−0.407647	−0.203823	−	0.979008i	$-0.565337\pi$
$619$	−10.1421	−0.407647	−0.203823	+	0.979008i	$0.565337\pi$
$620$	0	0
$621$	−44.1421	−1.77136
$622$	0	0
$623$	11.5980	0.464663
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	−3.31371	−0.132337
$628$	0	0
$629$	8.51472	0.339504
$630$	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
$635$	−2.76955	−0.109906
$636$	0	0
$637$	3.65685	0.144890
$638$	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
$643$	−3.37258	−0.133002	−0.0665008	−	0.997786i	$-0.521184\pi$
$643$	−3.37258	−0.133002	−0.0665008	+	0.997786i	$0.521184\pi$
$644$	0	0
$645$	−5.48528	−0.215983
$646$	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
$653$	−7.31371	−0.286208	−0.143104	−	0.989708i	$-0.545708\pi$
$653$	−7.31371	−0.286208	−0.143104	+	0.989708i	$0.545708\pi$
$654$	0	0
$655$	15.8284	0.618468
$656$	0	0
$657$	−21.6569	−0.844914
$658$	0	0
$659$	28.9706	1.12853	0.564266	−	0.825593i	$-0.309159\pi$
$659$	28.9706	1.12853	0.564266	+	0.825593i	$0.309159\pi$
$660$	0	0
$661$	−21.3137	−0.829007	−0.414504	−	0.910048i	$-0.636045\pi$
$661$	−21.3137	−0.829007	−0.414504	+	0.910048i	$0.636045\pi$
$662$	0	0
$663$	−4.65685	−0.180857
$664$	0	0
$665$	−13.3726	−0.518567
$666$	0	0
$667$	−77.9411	−3.01789
$668$	0	0
$669$	15.8284	0.611962
$670$	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
$675$	−8.28427	−0.318862
$676$	0	0
$677$	−6.68629	−0.256975	−0.128488	−	0.991711i	$-0.541012\pi$
$677$	−6.68629	−0.256975	−0.128488	+	0.991711i	$0.541012\pi$
$678$	0	0
$679$	−4.54416	−0.174389
$680$	0	0
$681$	−14.8284	−0.568227
$682$	0	0
$683$	−28.3431	−1.08452	−0.542260	−	0.840211i	$-0.682431\pi$
$683$	−28.3431	−1.08452	−0.542260	+	0.840211i	$0.682431\pi$
$684$	0	0
$685$	−14.8873	−0.568815
$686$	0	0
$687$	15.8284	0.603892
$688$	0	0
$689$	−8.82843	−0.336336
$690$	0	0
$691$	−40.9706	−1.55859	−0.779297	−	0.626655i	$-0.784424\pi$
$691$	−40.9706	−1.55859	−0.779297	+	0.626655i	$0.784424\pi$
$692$	0	0
$693$	−3.02944	−0.115079
$694$	0	0
$695$	−13.4264	−0.509293
$696$	0	0
$697$	13.1716	0.498909
$698$	0	0
$699$	2.31371	0.0875125
$700$	0	0
$701$	16.1421	0.609680	0.304840	−	0.952404i	$-0.401397\pi$
$701$	16.1421	0.609680	0.304840	+	0.952404i	$0.401397\pi$
$702$	0	0
$703$	7.31371	0.275842
$704$	0	0
$705$	10.0294	0.377730
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−3.65685	−0.137336	−0.0686680	−	0.997640i	$-0.521875\pi$
$709$	−3.65685	−0.137336	−0.0686680	+	0.997640i	$0.521875\pi$
$710$	0	0
$711$	−7.31371	−0.274285
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−1.51472	−0.0566473
$716$	0	0
$717$	9.14214	0.341419
$718$	0	0
$719$	10.9706	0.409133	0.204566	−	0.978853i	$-0.434422\pi$
$719$	10.9706	0.409133	0.204566	+	0.978853i	$0.434422\pi$
$720$	0	0
$721$	25.5980	0.953319
$722$	0	0
$723$	19.3137	0.718285
$724$	0	0
$725$	−14.6274	−0.543249
$726$	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
$731$	13.9706	0.516720
$732$	0	0
$733$	−37.7696	−1.39505	−0.697525	−	0.716560i	$-0.745715\pi$
$733$	−37.7696	−1.39505	−0.697525	+	0.716560i	$0.745715\pi$
$734$	0	0
$735$	−6.68629	−0.246628
$736$	0	0
$737$	6.34315	0.233653
$738$	0	0
$739$	48.4853	1.78356	0.891780	−	0.452469i	$-0.149457\pi$
$739$	48.4853	1.78356	0.891780	+	0.452469i	$0.149457\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	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
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−13.3726	−0.488624
$750$	0	0
$751$	−40.4853	−1.47733	−0.738664	−	0.674073i	$-0.764543\pi$
$751$	−40.4853	−1.47733	−0.738664	+	0.674073i	$0.764543\pi$
$752$	0	0
$753$	18.6274	0.678821
$754$	0	0
$755$	−35.6274	−1.29661
$756$	0	0
$757$	36.8284	1.33855	0.669276	−	0.743014i	$-0.266604\pi$
$757$	36.8284	1.33855	0.669276	+	0.743014i	$0.266604\pi$
$758$	0	0
$759$	−7.31371	−0.265471
$760$	0	0
$761$	−38.8284	−1.40753	−0.703765	−	0.710433i	$-0.748499\pi$
$761$	−38.8284	−1.40753	−0.703765	+	0.710433i	$0.748499\pi$
$762$	0	0
$763$	3.34315	0.121030
$764$	0	0
$765$	−17.0294	−0.615701
$766$	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
$771$	12.6569	0.455825
$772$	0	0
$773$	12.7990	0.460348	0.230174	−	0.973150i	$-0.426071\pi$
$773$	12.7990	0.460348	0.230174	+	0.973150i	$0.426071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.34315	−0.119935
$778$	0	0
$779$	11.3137	0.405356
$780$	0	0
$781$	13.1127	0.469209
$782$	0	0
$783$	44.1421	1.57751
$784$	0	0
$785$	−22.8284	−0.814782
$786$	0	0
$787$	−28.4853	−1.01539	−0.507695	−	0.861537i	$-0.669502\pi$
$787$	−28.4853	−1.01539	−0.507695	+	0.861537i	$0.669502\pi$
$788$	0	0
$789$	2.14214	0.0762620
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−5.17157	−0.183648
$794$	0	0
$795$	16.1421	0.572503
$796$	0	0
$797$	45.6569	1.61725	0.808624	−	0.588325i	$-0.200213\pi$
$797$	45.6569	1.61725	0.808624	+	0.588325i	$0.200213\pi$
$798$	0	0
$799$	−25.5442	−0.903687
$800$	0	0
$801$	12.6863	0.448248
$802$	0	0
$803$	−8.97056	−0.316564
$804$	0	0
$805$	−29.5147	−1.04026
$806$	0	0
$807$	−19.7990	−0.696957
$808$	0	0
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	0	0
$811$	−4.97056	−0.174540	−0.0872700	−	0.996185i	$-0.527814\pi$
$811$	−4.97056	−0.174540	−0.0872700	+	0.996185i	$0.527814\pi$
$812$	0	0
$813$	19.4853	0.683379
$814$	0	0
$815$	13.1127	0.459318
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	−3.65685	−0.127781
$820$	0	0
$821$	5.48528	0.191438	0.0957188	−	0.995408i	$-0.469485\pi$
$821$	5.48528	0.191438	0.0957188	+	0.995408i	$0.469485\pi$
$822$	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
$827$	−14.6274	−0.508645	−0.254323	−	0.967119i	$-0.581852\pi$
$827$	−14.6274	−0.508645	−0.254323	+	0.967119i	$0.581852\pi$
$828$	0	0
$829$	10.9706	0.381023	0.190512	−	0.981685i	$-0.438985\pi$
$829$	10.9706	0.381023	0.190512	+	0.981685i	$0.438985\pi$
$830$	0	0
$831$	−12.4853	−0.433110
$832$	0	0
$833$	17.0294	0.590035
$834$	0	0
$835$	−18.9117	−0.654466
$836$	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
$843$	−28.9706	−0.997799
$844$	0	0
$845$	−1.82843	−0.0628998
$846$	0	0
$847$	18.8579	0.647964
$848$	0	0
$849$	−30.6274	−1.05113
$850$	0	0
$851$	16.1421	0.553345
$852$	0	0
$853$	−37.7696	−1.29320	−0.646602	−	0.762827i	$-0.723811\pi$
$853$	−37.7696	−1.29320	−0.646602	+	0.762827i	$0.723811\pi$
$854$	0	0
$855$	−14.6274	−0.500247
$856$	0	0
$857$	18.6863	0.638312	0.319156	−	0.947702i	$-0.396601\pi$
$857$	18.6863	0.638312	0.319156	+	0.947702i	$0.396601\pi$
$858$	0	0
$859$	28.9706	0.988463	0.494231	−	0.869330i	$-0.335450\pi$
$859$	28.9706	0.988463	0.494231	+	0.869330i	$0.335450\pi$
$860$	0	0
$861$	−5.17157	−0.176247
$862$	0	0
$863$	48.1127	1.63778	0.818888	−	0.573954i	$-0.194591\pi$
$863$	48.1127	1.63778	0.818888	+	0.573954i	$0.194591\pi$
$864$	0	0
$865$	−20.0589	−0.682022
$866$	0	0
$867$	−4.68629	−0.159155
$868$	0	0
$869$	−3.02944	−0.102767
$870$	0	0
$871$	7.65685	0.259443
$872$	0	0
$873$	−4.97056	−0.168228
$874$	0	0
$875$	−22.2548	−0.752351
$876$	0	0
$877$	−33.4853	−1.13072	−0.565359	−	0.824845i	$-0.691262\pi$
$877$	−33.4853	−1.13072	−0.565359	+	0.824845i	$0.691262\pi$
$878$	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
$883$	31.9706	1.07590	0.537948	−	0.842978i	$-0.319200\pi$
$883$	31.9706	1.07590	0.537948	+	0.842978i	$0.319200\pi$
$884$	0	0
$885$	−19.7990	−0.665536
$886$	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
$891$	−0.828427	−0.0277534
$892$	0	0
$893$	−21.9411	−0.734232
$894$	0	0
$895$	26.1716	0.874819
$896$	0	0
$897$	−8.82843	−0.294773
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−41.1127	−1.36966
$902$	0	0
$903$	−5.48528	−0.182539
$904$	0	0
$905$	20.0589	0.666780
$906$	0	0
$907$	42.9411	1.42584	0.712918	−	0.701247i	$-0.247373\pi$
$907$	42.9411	1.42584	0.712918	+	0.701247i	$0.247373\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.54416	0.150555	0.0752773	−	0.997163i	$-0.476016\pi$
$911$	4.54416	0.150555	0.0752773	+	0.997163i	$0.476016\pi$
$912$	0	0
$913$	−4.97056	−0.164502
$914$	0	0
$915$	9.45584	0.312601
$916$	0	0
$917$	15.8284	0.522701
$918$	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
$923$	15.8284	0.520999
$924$	0	0
$925$	3.02944	0.0996073
$926$	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
$931$	14.6274	0.479394
$932$	0	0
$933$	−32.2843	−1.05694
$934$	0	0
$935$	−7.05382	−0.230685
$936$	0	0
$937$	−4.34315	−0.141884	−0.0709422	−	0.997480i	$-0.522601\pi$
$937$	−4.34315	−0.141884	−0.0709422	+	0.997480i	$0.522601\pi$
$938$	0	0
$939$	13.9706	0.455912
$940$	0	0
$941$	−1.20101	−0.0391518	−0.0195759	−	0.999808i	$-0.506232\pi$
$941$	−1.20101	−0.0391518	−0.0195759	+	0.999808i	$0.506232\pi$
$942$	0	0
$943$	24.9706	0.813153
$944$	0	0
$945$	16.7157	0.543763
$946$	0	0
$947$	25.1127	0.816053	0.408027	−	0.912970i	$-0.366217\pi$
$947$	25.1127	0.816053	0.408027	+	0.912970i	$0.366217\pi$
$948$	0	0
$949$	−10.8284	−0.351506
$950$	0	0
$951$	−6.68629	−0.216818
$952$	0	0
$953$	−28.3137	−0.917171	−0.458585	−	0.888650i	$-0.651644\pi$
$953$	−28.3137	−0.917171	−0.458585	+	0.888650i	$0.651644\pi$
$954$	0	0
$955$	−16.1421	−0.522347
$956$	0	0
$957$	7.31371	0.236419
$958$	0	0
$959$	−14.8873	−0.480736
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−14.6274	−0.471362
$964$	0	0
$965$	10.9706	0.353155
$966$	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
$971$	42.5980	1.36703	0.683517	−	0.729934i	$-0.260449\pi$
$971$	42.5980	1.36703	0.683517	+	0.729934i	$0.260449\pi$
$972$	0	0
$973$	−13.4264	−0.430431
$974$	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
$979$	5.25483	0.167945
$980$	0	0
$981$	3.65685	0.116754
$982$	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
$987$	10.0294	0.319241
$988$	0	0
$989$	26.4853	0.842183
$990$	0	0
$991$	52.9706	1.68267	0.841333	−	0.540518i	$-0.181772\pi$
$991$	52.9706	1.68267	0.841333	+	0.540518i	$0.181772\pi$
$992$	0	0
$993$	22.9706	0.728949
$994$	0	0
$995$	−20.0589	−0.635909
$996$	0	0
$997$	−23.4558	−0.742854	−0.371427	−	0.928462i	$-0.621131\pi$
$997$	−23.4558	−0.742854	−0.371427	+	0.928462i	$0.621131\pi$
$998$	0	0
$999$	−9.14214	−0.289244

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.82843 q^{5} -1.82843 q^{7} -2.00000 q^{9} -0.828427 q^{11} -1.00000 q^{13} +1.82843 q^{15} -4.65685 q^{17} -4.00000 q^{19} +1.82843 q^{21} -8.82843 q^{23} -1.65685 q^{25} +5.00000 q^{27} +8.82843 q^{29} +0.828427 q^{33} +3.34315 q^{35} -1.82843 q^{37} +1.00000 q^{39} -2.82843 q^{41} -3.00000 q^{43} +3.65685 q^{45} +5.48528 q^{47} -3.65685 q^{49} +4.65685 q^{51} +8.82843 q^{53} +1.51472 q^{55} +4.00000 q^{57} -10.8284 q^{59} +5.17157 q^{61} +3.65685 q^{63} +1.82843 q^{65} -7.65685 q^{67} +8.82843 q^{69} -15.8284 q^{71} +10.8284 q^{73} +1.65685 q^{75} +1.51472 q^{77} +3.65685 q^{79} +1.00000 q^{81} +6.00000 q^{83} +8.51472 q^{85} -8.82843 q^{87} -6.34315 q^{89} +1.82843 q^{91} +7.31371 q^{95} +2.48528 q^{97} +1.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} - 2 q^{21} - 12 q^{23} + 8 q^{25} + 10 q^{27} + 12 q^{29} - 4 q^{33} + 18 q^{35} + 2 q^{37} + 2 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 - 4 * q^9 + 4 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 - 2 * q^21 - 12 * q^23 + 8 * q^25 + 10 * q^27 + 12 * q^29 - 4 * q^33 + 18 * q^35 + 2 * q^37 + 2 * q^39 - 6 * q^43 - 4 * q^45 - 6 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 + 20 * q^55 + 8 * q^57 - 16 * q^59 + 16 * q^61 - 4 * q^63 - 2 * q^65 - 4 * q^67 + 12 * q^69 - 26 * q^71 + 16 * q^73 - 8 * q^75 + 20 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^83 + 34 * q^85 - 12 * q^87 - 24 * q^89 - 2 * q^91 - 8 * q^95 - 12 * q^97 - 8 * q^99

Introduction

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