LMFDB - Embedded newform 3696.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3696,2,Mod(1,3696)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3696, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3696.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3696.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.5127085871$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3696.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -5.47214 q^{13} -1.00000 q^{15} +0.763932 q^{17} -6.70820 q^{19} +1.00000 q^{21} +7.70820 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} +0.763932 q^{31} +1.00000 q^{33} -1.00000 q^{35} -7.00000 q^{37} +5.47214 q^{39} +6.47214 q^{41} +7.70820 q^{43} +1.00000 q^{45} +4.23607 q^{47} +1.00000 q^{49} -0.763932 q^{51} +10.1803 q^{53} -1.00000 q^{55} +6.70820 q^{57} -11.1803 q^{59} +2.00000 q^{61} -1.00000 q^{63} -5.47214 q^{65} +14.2361 q^{67} -7.70820 q^{69} -6.47214 q^{71} +13.4721 q^{73} +4.00000 q^{75} +1.00000 q^{77} +5.52786 q^{79} +1.00000 q^{81} -11.2361 q^{83} +0.763932 q^{85} -5.00000 q^{87} +4.47214 q^{89} +5.47214 q^{91} -0.763932 q^{93} -6.70820 q^{95} -3.70820 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} + 2 q^{21} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 10 q^{29} + 6 q^{31} + 2 q^{33} - 2 q^{35} - 14 q^{37} + 2 q^{39} + 4 q^{41} + 2 q^{43} + 2 q^{45} + 4 q^{47} + 2 q^{49} - 6 q^{51} - 2 q^{53} - 2 q^{55} + 4 q^{61} - 2 q^{63} - 2 q^{65} + 24 q^{67} - 2 q^{69} - 4 q^{71} + 18 q^{73} + 8 q^{75} + 2 q^{77} + 20 q^{79} + 2 q^{81} - 18 q^{83} + 6 q^{85} - 10 q^{87} + 2 q^{91} - 6 q^{93} + 6 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 - 2 * q^15 + 6 * q^17 + 2 * q^21 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 10 * q^29 + 6 * q^31 + 2 * q^33 - 2 * q^35 - 14 * q^37 + 2 * q^39 + 4 * q^41 + 2 * q^43 + 2 * q^45 + 4 * q^47 + 2 * q^49 - 6 * q^51 - 2 * q^53 - 2 * q^55 + 4 * q^61 - 2 * q^63 - 2 * q^65 + 24 * q^67 - 2 * q^69 - 4 * q^71 + 18 * q^73 + 8 * q^75 + 2 * q^77 + 20 * q^79 + 2 * q^81 - 18 * q^83 + 6 * q^85 - 10 * q^87 + 2 * q^91 - 6 * q^93 + 6 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	0	0
$19$	−6.70820	−1.53897	−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820	−1.53897	−0.769484	+	0.638666i	$0.779486\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	7.70820	1.60727	0.803636	−	0.595121i	$-0.202896\pi$
$23$	7.70820	1.60727	0.803636	+	0.595121i	$0.202896\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	0.763932	0.137206	0.0686031	−	0.997644i	$-0.478146\pi$
$31$	0.763932	0.137206	0.0686031	+	0.997644i	$0.478146\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	5.47214	0.876243
$40$	0	0
$41$	6.47214	1.01078	0.505389	−	0.862892i	$-0.331349\pi$
$41$	6.47214	1.01078	0.505389	+	0.862892i	$0.331349\pi$
$42$	0	0
$43$	7.70820	1.17549	0.587745	−	0.809046i	$-0.300016\pi$
$43$	7.70820	1.17549	0.587745	+	0.809046i	$0.300016\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	4.23607	0.617894	0.308947	−	0.951079i	$-0.400023\pi$
$47$	4.23607	0.617894	0.308947	+	0.951079i	$0.400023\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.763932	−0.106972
$52$	0	0
$53$	10.1803	1.39838	0.699189	−	0.714937i	$-0.253545\pi$
$53$	10.1803	1.39838	0.699189	+	0.714937i	$0.253545\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	6.70820	0.888523
$58$	0	0
$59$	−11.1803	−1.45556	−0.727778	−	0.685813i	$-0.759447\pi$
$59$	−11.1803	−1.45556	−0.727778	+	0.685813i	$0.759447\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−5.47214	−0.678735
$66$	0	0
$67$	14.2361	1.73921	0.869606	−	0.493746i	$-0.164373\pi$
$67$	14.2361	1.73921	0.869606	+	0.493746i	$0.164373\pi$
$68$	0	0
$69$	−7.70820	−0.927959
$70$	0	0
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	13.4721	1.57679	0.788397	−	0.615167i	$-0.210911\pi$
$73$	13.4721	1.57679	0.788397	+	0.615167i	$0.210911\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	5.52786	0.621933	0.310967	−	0.950421i	$-0.399347\pi$
$79$	5.52786	0.621933	0.310967	+	0.950421i	$0.399347\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.2361	−1.23332	−0.616659	−	0.787230i	$-0.711514\pi$
$83$	−11.2361	−1.23332	−0.616659	+	0.787230i	$0.711514\pi$
$84$	0	0
$85$	0.763932	0.0828601
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	4.47214	0.474045	0.237023	−	0.971504i	$-0.423828\pi$
$89$	4.47214	0.474045	0.237023	+	0.971504i	$0.423828\pi$
$90$	0	0
$91$	5.47214	0.573636
$92$	0	0
$93$	−0.763932	−0.0792161
$94$	0	0
$95$	−6.70820	−0.688247
$96$	0	0
$97$	−3.70820	−0.376511	−0.188256	−	0.982120i	$-0.560283\pi$
$97$	−3.70820	−0.376511	−0.188256	+	0.982120i	$0.560283\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−4.18034	−0.415959	−0.207980	−	0.978133i	$-0.566689\pi$
$101$	−4.18034	−0.415959	−0.207980	+	0.978133i	$0.566689\pi$
$102$	0	0
$103$	9.41641	0.927826	0.463913	−	0.885881i	$-0.346445\pi$
$103$	9.41641	0.927826	0.463913	+	0.885881i	$0.346445\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−0.236068	−0.0228216	−0.0114108	−	0.999935i	$-0.503632\pi$
$107$	−0.236068	−0.0228216	−0.0114108	+	0.999935i	$0.503632\pi$
$108$	0	0
$109$	7.23607	0.693090	0.346545	−	0.938033i	$-0.387355\pi$
$109$	7.23607	0.693090	0.346545	+	0.938033i	$0.387355\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	8.47214	0.796992	0.398496	−	0.917170i	$-0.369532\pi$
$113$	8.47214	0.796992	0.398496	+	0.917170i	$0.369532\pi$
$114$	0	0
$115$	7.70820	0.718794
$116$	0	0
$117$	−5.47214	−0.505899
$118$	0	0
$119$	−0.763932	−0.0700295
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.47214	−0.583573
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−18.6525	−1.65514	−0.827570	−	0.561363i	$-0.810277\pi$
$127$	−18.6525	−1.65514	−0.827570	+	0.561363i	$0.810277\pi$
$128$	0	0
$129$	−7.70820	−0.678670
$130$	0	0
$131$	16.9443	1.48043	0.740214	−	0.672371i	$-0.234724\pi$
$131$	16.9443	1.48043	0.740214	+	0.672371i	$0.234724\pi$
$132$	0	0
$133$	6.70820	0.581675
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	6.29180	0.537544	0.268772	−	0.963204i	$-0.413382\pi$
$137$	6.29180	0.537544	0.268772	+	0.963204i	$0.413382\pi$
$138$	0	0
$139$	−5.52786	−0.468867	−0.234434	−	0.972132i	$-0.575324\pi$
$139$	−5.52786	−0.468867	−0.234434	+	0.972132i	$0.575324\pi$
$140$	0	0
$141$	−4.23607	−0.356741
$142$	0	0
$143$	5.47214	0.457603
$144$	0	0
$145$	5.00000	0.415227
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	−8.18034	−0.665707	−0.332853	−	0.942979i	$-0.608011\pi$
$151$	−8.18034	−0.665707	−0.332853	+	0.942979i	$0.608011\pi$
$152$	0	0
$153$	0.763932	0.0617602
$154$	0	0
$155$	0.763932	0.0613605
$156$	0	0
$157$	11.4164	0.911129	0.455564	−	0.890203i	$-0.349438\pi$
$157$	11.4164	0.911129	0.455564	+	0.890203i	$0.349438\pi$
$158$	0	0
$159$	−10.1803	−0.807353
$160$	0	0
$161$	−7.70820	−0.607492
$162$	0	0
$163$	9.29180	0.727790	0.363895	−	0.931440i	$-0.381447\pi$
$163$	9.29180	0.727790	0.363895	+	0.931440i	$0.381447\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−8.65248	−0.669549	−0.334774	−	0.942298i	$-0.608660\pi$
$167$	−8.65248	−0.669549	−0.334774	+	0.942298i	$0.608660\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	−6.70820	−0.512989
$172$	0	0
$173$	−10.4721	−0.796182	−0.398091	−	0.917346i	$-0.630327\pi$
$173$	−10.4721	−0.796182	−0.398091	+	0.917346i	$0.630327\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	11.1803	0.840366
$178$	0	0
$179$	8.94427	0.668526	0.334263	−	0.942480i	$-0.391513\pi$
$179$	8.94427	0.668526	0.334263	+	0.942480i	$0.391513\pi$
$180$	0	0
$181$	−5.23607	−0.389194	−0.194597	−	0.980883i	$-0.562340\pi$
$181$	−5.23607	−0.389194	−0.194597	+	0.980883i	$0.562340\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	−0.763932	−0.0558642
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	15.2361	1.10244	0.551222	−	0.834359i	$-0.314162\pi$
$191$	15.2361	1.10244	0.551222	+	0.834359i	$0.314162\pi$
$192$	0	0
$193$	−16.6525	−1.19867	−0.599336	−	0.800498i	$-0.704569\pi$
$193$	−16.6525	−1.19867	−0.599336	+	0.800498i	$0.704569\pi$
$194$	0	0
$195$	5.47214	0.391868
$196$	0	0
$197$	−7.52786	−0.536338	−0.268169	−	0.963372i	$-0.586419\pi$
$197$	−7.52786	−0.536338	−0.268169	+	0.963372i	$0.586419\pi$
$198$	0	0
$199$	26.1803	1.85588	0.927938	−	0.372736i	$-0.121580\pi$
$199$	26.1803	1.85588	0.927938	+	0.372736i	$0.121580\pi$
$200$	0	0
$201$	−14.2361	−1.00413
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	6.47214	0.452034
$206$	0	0
$207$	7.70820	0.535757
$208$	0	0
$209$	6.70820	0.464016
$210$	0	0
$211$	21.4164	1.47437	0.737183	−	0.675693i	$-0.236155\pi$
$211$	21.4164	1.47437	0.737183	+	0.675693i	$0.236155\pi$
$212$	0	0
$213$	6.47214	0.443463
$214$	0	0
$215$	7.70820	0.525695
$216$	0	0
$217$	−0.763932	−0.0518591
$218$	0	0
$219$	−13.4721	−0.910363
$220$	0	0
$221$	−4.18034	−0.281200
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	2.76393	0.182646	0.0913229	−	0.995821i	$-0.470890\pi$
$229$	2.76393	0.182646	0.0913229	+	0.995821i	$0.470890\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	2.94427	0.192886	0.0964428	−	0.995339i	$-0.469254\pi$
$233$	2.94427	0.192886	0.0964428	+	0.995339i	$0.469254\pi$
$234$	0	0
$235$	4.23607	0.276331
$236$	0	0
$237$	−5.52786	−0.359073
$238$	0	0
$239$	30.1246	1.94860	0.974300	−	0.225256i	$-0.0723219\pi$
$239$	30.1246	1.94860	0.974300	+	0.225256i	$0.0723219\pi$
$240$	0	0
$241$	8.05573	0.518915	0.259458	−	0.965755i	$-0.416456\pi$
$241$	8.05573	0.518915	0.259458	+	0.965755i	$0.416456\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	36.7082	2.33569
$248$	0	0
$249$	11.2361	0.712057
$250$	0	0
$251$	28.1246	1.77521	0.887605	−	0.460606i	$-0.152368\pi$
$251$	28.1246	1.77521	0.887605	+	0.460606i	$0.152368\pi$
$252$	0	0
$253$	−7.70820	−0.484611
$254$	0	0
$255$	−0.763932	−0.0478393
$256$	0	0
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	5.00000	0.309492
$262$	0	0
$263$	−14.1246	−0.870961	−0.435480	−	0.900198i	$-0.643421\pi$
$263$	−14.1246	−0.870961	−0.435480	+	0.900198i	$0.643421\pi$
$264$	0	0
$265$	10.1803	0.625373
$266$	0	0
$267$	−4.47214	−0.273690
$268$	0	0
$269$	18.9443	1.15505	0.577526	−	0.816372i	$-0.304018\pi$
$269$	18.9443	1.15505	0.577526	+	0.816372i	$0.304018\pi$
$270$	0	0
$271$	−18.7082	−1.13644	−0.568221	−	0.822876i	$-0.692368\pi$
$271$	−18.7082	−1.13644	−0.568221	+	0.822876i	$0.692368\pi$
$272$	0	0
$273$	−5.47214	−0.331189
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	2.47214	0.148536	0.0742681	−	0.997238i	$-0.476338\pi$
$277$	2.47214	0.148536	0.0742681	+	0.997238i	$0.476338\pi$
$278$	0	0
$279$	0.763932	0.0457354
$280$	0	0
$281$	2.52786	0.150800	0.0753999	−	0.997153i	$-0.475977\pi$
$281$	2.52786	0.150800	0.0753999	+	0.997153i	$0.475977\pi$
$282$	0	0
$283$	18.2361	1.08402	0.542011	−	0.840371i	$-0.317663\pi$
$283$	18.2361	1.08402	0.542011	+	0.840371i	$0.317663\pi$
$284$	0	0
$285$	6.70820	0.397360
$286$	0	0
$287$	−6.47214	−0.382038
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	3.70820	0.217379
$292$	0	0
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	−11.1803	−0.650945
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−42.1803	−2.43935
$300$	0	0
$301$	−7.70820	−0.444293
$302$	0	0
$303$	4.18034	0.240154
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	3.05573	0.174400	0.0871998	−	0.996191i	$-0.472208\pi$
$307$	3.05573	0.174400	0.0871998	+	0.996191i	$0.472208\pi$
$308$	0	0
$309$	−9.41641	−0.535681
$310$	0	0
$311$	25.8885	1.46800	0.734002	−	0.679147i	$-0.237650\pi$
$311$	25.8885	1.46800	0.734002	+	0.679147i	$0.237650\pi$
$312$	0	0
$313$	−6.65248	−0.376020	−0.188010	−	0.982167i	$-0.560204\pi$
$313$	−6.65248	−0.376020	−0.188010	+	0.982167i	$0.560204\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	1.81966	0.102202	0.0511011	−	0.998693i	$-0.483727\pi$
$317$	1.81966	0.102202	0.0511011	+	0.998693i	$0.483727\pi$
$318$	0	0
$319$	−5.00000	−0.279946
$320$	0	0
$321$	0.236068	0.0131760
$322$	0	0
$323$	−5.12461	−0.285141
$324$	0	0
$325$	21.8885	1.21416
$326$	0	0
$327$	−7.23607	−0.400155
$328$	0	0
$329$	−4.23607	−0.233542
$330$	0	0
$331$	−15.4164	−0.847362	−0.423681	−	0.905811i	$-0.639262\pi$
$331$	−15.4164	−0.847362	−0.423681	+	0.905811i	$0.639262\pi$
$332$	0	0
$333$	−7.00000	−0.383598
$334$	0	0
$335$	14.2361	0.777799
$336$	0	0
$337$	14.1803	0.772452	0.386226	−	0.922404i	$-0.373778\pi$
$337$	14.1803	0.772452	0.386226	+	0.922404i	$0.373778\pi$
$338$	0	0
$339$	−8.47214	−0.460143
$340$	0	0
$341$	−0.763932	−0.0413692
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.70820	−0.414996
$346$	0	0
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	28.4164	1.52110	0.760548	−	0.649282i	$-0.224931\pi$
$349$	28.4164	1.52110	0.760548	+	0.649282i	$0.224931\pi$
$350$	0	0
$351$	5.47214	0.292081
$352$	0	0
$353$	33.4721	1.78154	0.890771	−	0.454452i	$-0.150165\pi$
$353$	33.4721	1.78154	0.890771	+	0.454452i	$0.150165\pi$
$354$	0	0
$355$	−6.47214	−0.343505
$356$	0	0
$357$	0.763932	0.0404316
$358$	0	0
$359$	3.41641	0.180311	0.0901556	−	0.995928i	$-0.471264\pi$
$359$	3.41641	0.180311	0.0901556	+	0.995928i	$0.471264\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	13.4721	0.705164
$366$	0	0
$367$	−15.8885	−0.829375	−0.414688	−	0.909964i	$-0.636109\pi$
$367$	−15.8885	−0.829375	−0.414688	+	0.909964i	$0.636109\pi$
$368$	0	0
$369$	6.47214	0.336926
$370$	0	0
$371$	−10.1803	−0.528537
$372$	0	0
$373$	−26.6525	−1.38001	−0.690006	−	0.723803i	$-0.742392\pi$
$373$	−26.6525	−1.38001	−0.690006	+	0.723803i	$0.742392\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	−27.3607	−1.40915
$378$	0	0
$379$	8.81966	0.453036	0.226518	−	0.974007i	$-0.427266\pi$
$379$	8.81966	0.453036	0.226518	+	0.974007i	$0.427266\pi$
$380$	0	0
$381$	18.6525	0.955595
$382$	0	0
$383$	−15.0557	−0.769312	−0.384656	−	0.923060i	$-0.625680\pi$
$383$	−15.0557	−0.769312	−0.384656	+	0.923060i	$0.625680\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	7.70820	0.391830
$388$	0	0
$389$	28.9443	1.46753	0.733766	−	0.679402i	$-0.237761\pi$
$389$	28.9443	1.46753	0.733766	+	0.679402i	$0.237761\pi$
$390$	0	0
$391$	5.88854	0.297796
$392$	0	0
$393$	−16.9443	−0.854725
$394$	0	0
$395$	5.52786	0.278137
$396$	0	0
$397$	−17.1246	−0.859460	−0.429730	−	0.902958i	$-0.641391\pi$
$397$	−17.1246	−0.859460	−0.429730	+	0.902958i	$0.641391\pi$
$398$	0	0
$399$	−6.70820	−0.335830
$400$	0	0
$401$	−16.2918	−0.813573	−0.406787	−	0.913523i	$-0.633351\pi$
$401$	−16.2918	−0.813573	−0.406787	+	0.913523i	$0.633351\pi$
$402$	0	0
$403$	−4.18034	−0.208238
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	7.00000	0.346977
$408$	0	0
$409$	38.9443	1.92567	0.962835	−	0.270090i	$-0.0870534\pi$
$409$	38.9443	1.92567	0.962835	+	0.270090i	$0.0870534\pi$
$410$	0	0
$411$	−6.29180	−0.310351
$412$	0	0
$413$	11.1803	0.550149
$414$	0	0
$415$	−11.2361	−0.551557
$416$	0	0
$417$	5.52786	0.270701
$418$	0	0
$419$	−21.1803	−1.03473	−0.517364	−	0.855766i	$-0.673087\pi$
$419$	−21.1803	−1.03473	−0.517364	+	0.855766i	$0.673087\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	0	0
$423$	4.23607	0.205965
$424$	0	0
$425$	−3.05573	−0.148225
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−5.47214	−0.264197
$430$	0	0
$431$	4.70820	0.226786	0.113393	−	0.993550i	$-0.463828\pi$
$431$	4.70820	0.226786	0.113393	+	0.993550i	$0.463828\pi$
$432$	0	0
$433$	−1.52786	−0.0734245	−0.0367122	−	0.999326i	$-0.511688\pi$
$433$	−1.52786	−0.0734245	−0.0367122	+	0.999326i	$0.511688\pi$
$434$	0	0
$435$	−5.00000	−0.239732
$436$	0	0
$437$	−51.7082	−2.47354
$438$	0	0
$439$	11.1803	0.533609	0.266804	−	0.963751i	$-0.414032\pi$
$439$	11.1803	0.533609	0.266804	+	0.963751i	$0.414032\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−9.52786	−0.452682	−0.226341	−	0.974048i	$-0.572676\pi$
$443$	−9.52786	−0.452682	−0.226341	+	0.974048i	$0.572676\pi$
$444$	0	0
$445$	4.47214	0.212000
$446$	0	0
$447$	−5.00000	−0.236492
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−6.47214	−0.304761
$452$	0	0
$453$	8.18034	0.384346
$454$	0	0
$455$	5.47214	0.256538
$456$	0	0
$457$	10.7639	0.503516	0.251758	−	0.967790i	$-0.418991\pi$
$457$	10.7639	0.503516	0.251758	+	0.967790i	$0.418991\pi$
$458$	0	0
$459$	−0.763932	−0.0356573
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	27.1803	1.26318	0.631589	−	0.775304i	$-0.282403\pi$
$463$	27.1803	1.26318	0.631589	+	0.775304i	$0.282403\pi$
$464$	0	0
$465$	−0.763932	−0.0354265
$466$	0	0
$467$	−6.81966	−0.315576	−0.157788	−	0.987473i	$-0.550436\pi$
$467$	−6.81966	−0.315576	−0.157788	+	0.987473i	$0.550436\pi$
$468$	0	0
$469$	−14.2361	−0.657361
$470$	0	0
$471$	−11.4164	−0.526040
$472$	0	0
$473$	−7.70820	−0.354424
$474$	0	0
$475$	26.8328	1.23117
$476$	0	0
$477$	10.1803	0.466126
$478$	0	0
$479$	1.70820	0.0780498	0.0390249	−	0.999238i	$-0.487575\pi$
$479$	1.70820	0.0780498	0.0390249	+	0.999238i	$0.487575\pi$
$480$	0	0
$481$	38.3050	1.74656
$482$	0	0
$483$	7.70820	0.350735
$484$	0	0
$485$	−3.70820	−0.168381
$486$	0	0
$487$	0.944272	0.0427890	0.0213945	−	0.999771i	$-0.493189\pi$
$487$	0.944272	0.0427890	0.0213945	+	0.999771i	$0.493189\pi$
$488$	0	0
$489$	−9.29180	−0.420190
$490$	0	0
$491$	−22.1246	−0.998470	−0.499235	−	0.866467i	$-0.666386\pi$
$491$	−22.1246	−0.998470	−0.499235	+	0.866467i	$0.666386\pi$
$492$	0	0
$493$	3.81966	0.172029
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	6.47214	0.290315
$498$	0	0
$499$	−2.23607	−0.100100	−0.0500501	−	0.998747i	$-0.515938\pi$
$499$	−2.23607	−0.100100	−0.0500501	+	0.998747i	$0.515938\pi$
$500$	0	0
$501$	8.65248	0.386564
$502$	0	0
$503$	−15.7082	−0.700394	−0.350197	−	0.936676i	$-0.613885\pi$
$503$	−15.7082	−0.700394	−0.350197	+	0.936676i	$0.613885\pi$
$504$	0	0
$505$	−4.18034	−0.186023
$506$	0	0
$507$	−16.9443	−0.752522
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−13.4721	−0.595972
$512$	0	0
$513$	6.70820	0.296174
$514$	0	0
$515$	9.41641	0.414937
$516$	0	0
$517$	−4.23607	−0.186302
$518$	0	0
$519$	10.4721	0.459676
$520$	0	0
$521$	−24.3050	−1.06482	−0.532410	−	0.846487i	$-0.678713\pi$
$521$	−24.3050	−1.06482	−0.532410	+	0.846487i	$0.678713\pi$
$522$	0	0
$523$	−9.65248	−0.422073	−0.211037	−	0.977478i	$-0.567684\pi$
$523$	−9.65248	−0.422073	−0.211037	+	0.977478i	$0.567684\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	0.583592	0.0254217
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	−11.1803	−0.485185
$532$	0	0
$533$	−35.4164	−1.53405
$534$	0	0
$535$	−0.236068	−0.0102061
$536$	0	0
$537$	−8.94427	−0.385974
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−19.0557	−0.819270	−0.409635	−	0.912250i	$-0.634344\pi$
$541$	−19.0557	−0.819270	−0.409635	+	0.912250i	$0.634344\pi$
$542$	0	0
$543$	5.23607	0.224701
$544$	0	0
$545$	7.23607	0.309959
$546$	0	0
$547$	38.8328	1.66037	0.830186	−	0.557487i	$-0.188234\pi$
$547$	38.8328	1.66037	0.830186	+	0.557487i	$0.188234\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−33.5410	−1.42890
$552$	0	0
$553$	−5.52786	−0.235069
$554$	0	0
$555$	7.00000	0.297133
$556$	0	0
$557$	−21.4721	−0.909804	−0.454902	−	0.890542i	$-0.650326\pi$
$557$	−21.4721	−0.909804	−0.454902	+	0.890542i	$0.650326\pi$
$558$	0	0
$559$	−42.1803	−1.78404
$560$	0	0
$561$	0.763932	0.0322532
$562$	0	0
$563$	−3.34752	−0.141081	−0.0705407	−	0.997509i	$-0.522472\pi$
$563$	−3.34752	−0.141081	−0.0705407	+	0.997509i	$0.522472\pi$
$564$	0	0
$565$	8.47214	0.356425
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−26.4721	−1.10782	−0.553912	−	0.832575i	$-0.686866\pi$
$571$	−26.4721	−1.10782	−0.553912	+	0.832575i	$0.686866\pi$
$572$	0	0
$573$	−15.2361	−0.636496
$574$	0	0
$575$	−30.8328	−1.28582
$576$	0	0
$577$	38.6525	1.60912	0.804562	−	0.593869i	$-0.202400\pi$
$577$	38.6525	1.60912	0.804562	+	0.593869i	$0.202400\pi$
$578$	0	0
$579$	16.6525	0.692053
$580$	0	0
$581$	11.2361	0.466151
$582$	0	0
$583$	−10.1803	−0.421627
$584$	0	0
$585$	−5.47214	−0.226245
$586$	0	0
$587$	30.0132	1.23878	0.619388	−	0.785085i	$-0.287381\pi$
$587$	30.0132	1.23878	0.619388	+	0.785085i	$0.287381\pi$
$588$	0	0
$589$	−5.12461	−0.211156
$590$	0	0
$591$	7.52786	0.309655
$592$	0	0
$593$	30.8328	1.26615	0.633076	−	0.774090i	$-0.281792\pi$
$593$	30.8328	1.26615	0.633076	+	0.774090i	$0.281792\pi$
$594$	0	0
$595$	−0.763932	−0.0313182
$596$	0	0
$597$	−26.1803	−1.07149
$598$	0	0
$599$	34.4721	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$599$	34.4721	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$600$	0	0
$601$	27.0000	1.10135	0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000	1.10135	0.550676	+	0.834719i	$0.314370\pi$
$602$	0	0
$603$	14.2361	0.579738
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−29.1803	−1.18439	−0.592197	−	0.805793i	$-0.701739\pi$
$607$	−29.1803	−1.18439	−0.592197	+	0.805793i	$0.701739\pi$
$608$	0	0
$609$	5.00000	0.202610
$610$	0	0
$611$	−23.1803	−0.937776
$612$	0	0
$613$	−35.5967	−1.43774	−0.718870	−	0.695145i	$-0.755340\pi$
$613$	−35.5967	−1.43774	−0.718870	+	0.695145i	$0.755340\pi$
$614$	0	0
$615$	−6.47214	−0.260982
$616$	0	0
$617$	23.5279	0.947196	0.473598	−	0.880741i	$-0.342955\pi$
$617$	23.5279	0.947196	0.473598	+	0.880741i	$0.342955\pi$
$618$	0	0
$619$	−14.0689	−0.565476	−0.282738	−	0.959197i	$-0.591243\pi$
$619$	−14.0689	−0.565476	−0.282738	+	0.959197i	$0.591243\pi$
$620$	0	0
$621$	−7.70820	−0.309320
$622$	0	0
$623$	−4.47214	−0.179172
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−6.70820	−0.267900
$628$	0	0
$629$	−5.34752	−0.213220
$630$	0	0
$631$	0.360680	0.0143584	0.00717922	−	0.999974i	$-0.497715\pi$
$631$	0.360680	0.0143584	0.00717922	+	0.999974i	$0.497715\pi$
$632$	0	0
$633$	−21.4164	−0.851226
$634$	0	0
$635$	−18.6525	−0.740201
$636$	0	0
$637$	−5.47214	−0.216814
$638$	0	0
$639$	−6.47214	−0.256034
$640$	0	0
$641$	20.5410	0.811321	0.405661	−	0.914024i	$-0.367041\pi$
$641$	20.5410	0.811321	0.405661	+	0.914024i	$0.367041\pi$
$642$	0	0
$643$	−45.9574	−1.81238	−0.906192	−	0.422866i	$-0.861024\pi$
$643$	−45.9574	−1.81238	−0.906192	+	0.422866i	$0.861024\pi$
$644$	0	0
$645$	−7.70820	−0.303510
$646$	0	0
$647$	−43.6525	−1.71616	−0.858078	−	0.513519i	$-0.828341\pi$
$647$	−43.6525	−1.71616	−0.858078	+	0.513519i	$0.828341\pi$
$648$	0	0
$649$	11.1803	0.438867
$650$	0	0
$651$	0.763932	0.0299409
$652$	0	0
$653$	−27.0557	−1.05877	−0.529386	−	0.848381i	$-0.677578\pi$
$653$	−27.0557	−1.05877	−0.529386	+	0.848381i	$0.677578\pi$
$654$	0	0
$655$	16.9443	0.662067
$656$	0	0
$657$	13.4721	0.525598
$658$	0	0
$659$	43.5410	1.69612	0.848059	−	0.529902i	$-0.177771\pi$
$659$	43.5410	1.69612	0.848059	+	0.529902i	$0.177771\pi$
$660$	0	0
$661$	−26.5410	−1.03233	−0.516163	−	0.856490i	$-0.672640\pi$
$661$	−26.5410	−1.03233	−0.516163	+	0.856490i	$0.672640\pi$
$662$	0	0
$663$	4.18034	0.162351
$664$	0	0
$665$	6.70820	0.260133
$666$	0	0
$667$	38.5410	1.49231
$668$	0	0
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−45.5967	−1.75763	−0.878813	−	0.477167i	$-0.841664\pi$
$673$	−45.5967	−1.75763	−0.878813	+	0.477167i	$0.841664\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−43.3050	−1.66434	−0.832172	−	0.554517i	$-0.812903\pi$
$677$	−43.3050	−1.66434	−0.832172	+	0.554517i	$0.812903\pi$
$678$	0	0
$679$	3.70820	0.142308
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	0	0
$683$	39.0132	1.49280	0.746398	−	0.665499i	$-0.231781\pi$
$683$	39.0132	1.49280	0.746398	+	0.665499i	$0.231781\pi$
$684$	0	0
$685$	6.29180	0.240397
$686$	0	0
$687$	−2.76393	−0.105451
$688$	0	0
$689$	−55.7082	−2.12231
$690$	0	0
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	−5.52786	−0.209684
$696$	0	0
$697$	4.94427	0.187278
$698$	0	0
$699$	−2.94427	−0.111363
$700$	0	0
$701$	39.8885	1.50657	0.753285	−	0.657695i	$-0.228468\pi$
$701$	39.8885	1.50657	0.753285	+	0.657695i	$0.228468\pi$
$702$	0	0
$703$	46.9574	1.77103
$704$	0	0
$705$	−4.23607	−0.159540
$706$	0	0
$707$	4.18034	0.157218
$708$	0	0
$709$	39.7214	1.49177	0.745883	−	0.666076i	$-0.232028\pi$
$709$	39.7214	1.49177	0.745883	+	0.666076i	$0.232028\pi$
$710$	0	0
$711$	5.52786	0.207311
$712$	0	0
$713$	5.88854	0.220528
$714$	0	0
$715$	5.47214	0.204646
$716$	0	0
$717$	−30.1246	−1.12502
$718$	0	0
$719$	−12.2361	−0.456328	−0.228164	−	0.973623i	$-0.573272\pi$
$719$	−12.2361	−0.456328	−0.228164	+	0.973623i	$0.573272\pi$
$720$	0	0
$721$	−9.41641	−0.350685
$722$	0	0
$723$	−8.05573	−0.299596
$724$	0	0
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−1.81966	−0.0674875	−0.0337437	−	0.999431i	$-0.510743\pi$
$727$	−1.81966	−0.0674875	−0.0337437	+	0.999431i	$0.510743\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.88854	0.217796
$732$	0	0
$733$	−43.8885	−1.62106	−0.810530	−	0.585697i	$-0.800821\pi$
$733$	−43.8885	−1.62106	−0.810530	+	0.585697i	$0.800821\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	−14.2361	−0.524392
$738$	0	0
$739$	24.0689	0.885388	0.442694	−	0.896673i	$-0.354023\pi$
$739$	24.0689	0.885388	0.442694	+	0.896673i	$0.354023\pi$
$740$	0	0
$741$	−36.7082	−1.34851
$742$	0	0
$743$	−25.1803	−0.923777	−0.461889	−	0.886938i	$-0.652828\pi$
$743$	−25.1803	−0.923777	−0.461889	+	0.886938i	$0.652828\pi$
$744$	0	0
$745$	5.00000	0.183186
$746$	0	0
$747$	−11.2361	−0.411106
$748$	0	0
$749$	0.236068	0.00862574
$750$	0	0
$751$	−44.2361	−1.61420	−0.807099	−	0.590417i	$-0.798963\pi$
$751$	−44.2361	−1.61420	−0.807099	+	0.590417i	$0.798963\pi$
$752$	0	0
$753$	−28.1246	−1.02492
$754$	0	0
$755$	−8.18034	−0.297713
$756$	0	0
$757$	37.7214	1.37101	0.685503	−	0.728070i	$-0.259582\pi$
$757$	37.7214	1.37101	0.685503	+	0.728070i	$0.259582\pi$
$758$	0	0
$759$	7.70820	0.279790
$760$	0	0
$761$	−43.7771	−1.58692	−0.793459	−	0.608624i	$-0.791722\pi$
$761$	−43.7771	−1.58692	−0.793459	+	0.608624i	$0.791722\pi$
$762$	0	0
$763$	−7.23607	−0.261963
$764$	0	0
$765$	0.763932	0.0276200
$766$	0	0
$767$	61.1803	2.20909
$768$	0	0
$769$	−3.94427	−0.142234	−0.0711170	−	0.997468i	$-0.522656\pi$
$769$	−3.94427	−0.142234	−0.0711170	+	0.997468i	$0.522656\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	0	0
$773$	3.47214	0.124884	0.0624420	−	0.998049i	$-0.480111\pi$
$773$	3.47214	0.124884	0.0624420	+	0.998049i	$0.480111\pi$
$774$	0	0
$775$	−3.05573	−0.109765
$776$	0	0
$777$	−7.00000	−0.251124
$778$	0	0
$779$	−43.4164	−1.55555
$780$	0	0
$781$	6.47214	0.231591
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	11.4164	0.407469
$786$	0	0
$787$	27.6525	0.985704	0.492852	−	0.870113i	$-0.335954\pi$
$787$	27.6525	0.985704	0.492852	+	0.870113i	$0.335954\pi$
$788$	0	0
$789$	14.1246	0.502849
$790$	0	0
$791$	−8.47214	−0.301234
$792$	0	0
$793$	−10.9443	−0.388642
$794$	0	0
$795$	−10.1803	−0.361059
$796$	0	0
$797$	−11.4721	−0.406364	−0.203182	−	0.979141i	$-0.565128\pi$
$797$	−11.4721	−0.406364	−0.203182	+	0.979141i	$0.565128\pi$
$798$	0	0
$799$	3.23607	0.114484
$800$	0	0
$801$	4.47214	0.158015
$802$	0	0
$803$	−13.4721	−0.475421
$804$	0	0
$805$	−7.70820	−0.271678
$806$	0	0
$807$	−18.9443	−0.666870
$808$	0	0
$809$	6.30495	0.221670	0.110835	−	0.993839i	$-0.464647\pi$
$809$	6.30495	0.221670	0.110835	+	0.993839i	$0.464647\pi$
$810$	0	0
$811$	−28.7082	−1.00808	−0.504041	−	0.863680i	$-0.668154\pi$
$811$	−28.7082	−1.00808	−0.504041	+	0.863680i	$0.668154\pi$
$812$	0	0
$813$	18.7082	0.656125
$814$	0	0
$815$	9.29180	0.325477
$816$	0	0
$817$	−51.7082	−1.80904
$818$	0	0
$819$	5.47214	0.191212
$820$	0	0
$821$	1.47214	0.0513779	0.0256889	−	0.999670i	$-0.491822\pi$
$821$	1.47214	0.0513779	0.0256889	+	0.999670i	$0.491822\pi$
$822$	0	0
$823$	−5.18034	−0.180575	−0.0902876	−	0.995916i	$-0.528779\pi$
$823$	−5.18034	−0.180575	−0.0902876	+	0.995916i	$0.528779\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−43.6525	−1.51795	−0.758973	−	0.651122i	$-0.774298\pi$
$827$	−43.6525	−1.51795	−0.758973	+	0.651122i	$0.774298\pi$
$828$	0	0
$829$	−35.7771	−1.24259	−0.621295	−	0.783577i	$-0.713393\pi$
$829$	−35.7771	−1.24259	−0.621295	+	0.783577i	$0.713393\pi$
$830$	0	0
$831$	−2.47214	−0.0857574
$832$	0	0
$833$	0.763932	0.0264687
$834$	0	0
$835$	−8.65248	−0.299431
$836$	0	0
$837$	−0.763932	−0.0264054
$838$	0	0
$839$	−43.5410	−1.50320	−0.751601	−	0.659617i	$-0.770718\pi$
$839$	−43.5410	−1.50320	−0.751601	+	0.659617i	$0.770718\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−2.52786	−0.0870643
$844$	0	0
$845$	16.9443	0.582901
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−18.2361	−0.625860
$850$	0	0
$851$	−53.9574	−1.84964
$852$	0	0
$853$	−2.58359	−0.0884605	−0.0442303	−	0.999021i	$-0.514084\pi$
$853$	−2.58359	−0.0884605	−0.0442303	+	0.999021i	$0.514084\pi$
$854$	0	0
$855$	−6.70820	−0.229416
$856$	0	0
$857$	35.8885	1.22593	0.612965	−	0.790110i	$-0.289977\pi$
$857$	35.8885	1.22593	0.612965	+	0.790110i	$0.289977\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	6.47214	0.220570
$862$	0	0
$863$	38.7639	1.31954	0.659770	−	0.751468i	$-0.270654\pi$
$863$	38.7639	1.31954	0.659770	+	0.751468i	$0.270654\pi$
$864$	0	0
$865$	−10.4721	−0.356063
$866$	0	0
$867$	16.4164	0.557530
$868$	0	0
$869$	−5.52786	−0.187520
$870$	0	0
$871$	−77.9017	−2.63960
$872$	0	0
$873$	−3.70820	−0.125504
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	31.4164	1.06086	0.530428	−	0.847730i	$-0.322031\pi$
$877$	31.4164	1.06086	0.530428	+	0.847730i	$0.322031\pi$
$878$	0	0
$879$	16.0000	0.539667
$880$	0	0
$881$	19.1115	0.643881	0.321941	−	0.946760i	$-0.395665\pi$
$881$	19.1115	0.643881	0.321941	+	0.946760i	$0.395665\pi$
$882$	0	0
$883$	37.1803	1.25122	0.625609	−	0.780137i	$-0.284851\pi$
$883$	37.1803	1.25122	0.625609	+	0.780137i	$0.284851\pi$
$884$	0	0
$885$	11.1803	0.375823
$886$	0	0
$887$	−15.2361	−0.511577	−0.255789	−	0.966733i	$-0.582335\pi$
$887$	−15.2361	−0.511577	−0.255789	+	0.966733i	$0.582335\pi$
$888$	0	0
$889$	18.6525	0.625584
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−28.4164	−0.950919
$894$	0	0
$895$	8.94427	0.298974
$896$	0	0
$897$	42.1803	1.40836
$898$	0	0
$899$	3.81966	0.127393
$900$	0	0
$901$	7.77709	0.259092
$902$	0	0
$903$	7.70820	0.256513
$904$	0	0
$905$	−5.23607	−0.174053
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−4.18034	−0.138653
$910$	0	0
$911$	41.4164	1.37219	0.686093	−	0.727513i	$-0.259324\pi$
$911$	41.4164	1.37219	0.686093	+	0.727513i	$0.259324\pi$
$912$	0	0
$913$	11.2361	0.371860
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	0	0
$917$	−16.9443	−0.559549
$918$	0	0
$919$	9.59675	0.316567	0.158284	−	0.987394i	$-0.449404\pi$
$919$	9.59675	0.316567	0.158284	+	0.987394i	$0.449404\pi$
$920$	0	0
$921$	−3.05573	−0.100690
$922$	0	0
$923$	35.4164	1.16575
$924$	0	0
$925$	28.0000	0.920634
$926$	0	0
$927$	9.41641	0.309275
$928$	0	0
$929$	−22.8885	−0.750949	−0.375474	−	0.926833i	$-0.622520\pi$
$929$	−22.8885	−0.750949	−0.375474	+	0.926833i	$0.622520\pi$
$930$	0	0
$931$	−6.70820	−0.219853
$932$	0	0
$933$	−25.8885	−0.847553
$934$	0	0
$935$	−0.763932	−0.0249832
$936$	0	0
$937$	−4.11146	−0.134315	−0.0671577	−	0.997742i	$-0.521393\pi$
$937$	−4.11146	−0.134315	−0.0671577	+	0.997742i	$0.521393\pi$
$938$	0	0
$939$	6.65248	0.217095
$940$	0	0
$941$	−15.6393	−0.509827	−0.254914	−	0.966964i	$-0.582047\pi$
$941$	−15.6393	−0.509827	−0.254914	+	0.966964i	$0.582047\pi$
$942$	0	0
$943$	49.8885	1.62459
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−21.4164	−0.695940	−0.347970	−	0.937506i	$-0.613129\pi$
$947$	−21.4164	−0.695940	−0.347970	+	0.937506i	$0.613129\pi$
$948$	0	0
$949$	−73.7214	−2.39310
$950$	0	0
$951$	−1.81966	−0.0590065
$952$	0	0
$953$	−26.7771	−0.867395	−0.433697	−	0.901059i	$-0.642791\pi$
$953$	−26.7771	−0.867395	−0.433697	+	0.901059i	$0.642791\pi$
$954$	0	0
$955$	15.2361	0.493028
$956$	0	0
$957$	5.00000	0.161627
$958$	0	0
$959$	−6.29180	−0.203173
$960$	0	0
$961$	−30.4164	−0.981174
$962$	0	0
$963$	−0.236068	−0.00760718
$964$	0	0
$965$	−16.6525	−0.536062
$966$	0	0
$967$	−37.1935	−1.19606	−0.598031	−	0.801473i	$-0.704050\pi$
$967$	−37.1935	−1.19606	−0.598031	+	0.801473i	$0.704050\pi$
$968$	0	0
$969$	5.12461	0.164626
$970$	0	0
$971$	8.12461	0.260731	0.130366	−	0.991466i	$-0.458385\pi$
$971$	8.12461	0.260731	0.130366	+	0.991466i	$0.458385\pi$
$972$	0	0
$973$	5.52786	0.177215
$974$	0	0
$975$	−21.8885	−0.700994
$976$	0	0
$977$	−18.1803	−0.581641	−0.290820	−	0.956778i	$-0.593928\pi$
$977$	−18.1803	−0.581641	−0.290820	+	0.956778i	$0.593928\pi$
$978$	0	0
$979$	−4.47214	−0.142930
$980$	0	0
$981$	7.23607	0.231030
$982$	0	0
$983$	−0.583592	−0.0186137	−0.00930685	−	0.999957i	$-0.502963\pi$
$983$	−0.583592	−0.0186137	−0.00930685	+	0.999957i	$0.502963\pi$
$984$	0	0
$985$	−7.52786	−0.239858
$986$	0	0
$987$	4.23607	0.134836
$988$	0	0
$989$	59.4164	1.88933
$990$	0	0
$991$	6.81966	0.216634	0.108317	−	0.994116i	$-0.465454\pi$
$991$	6.81966	0.216634	0.108317	+	0.994116i	$0.465454\pi$
$992$	0	0
$993$	15.4164	0.489225
$994$	0	0
$995$	26.1803	0.829973
$996$	0	0
$997$	9.05573	0.286798	0.143399	−	0.989665i	$-0.454197\pi$
$997$	9.05573	0.286798	0.143399	+	0.989665i	$0.454197\pi$
$998$	0	0
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.be.1.1		2
4.3	odd	2		231.2.a.c.1.2	✓	2
12.11	even	2		693.2.a.f.1.1		2
20.19	odd	2		5775.2.a.be.1.1		2
28.27	even	2		1617.2.a.p.1.2		2
44.43	even	2		2541.2.a.t.1.1		2
84.83	odd	2		4851.2.a.w.1.1		2
132.131	odd	2		7623.2.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.c.1.2	✓	2	4.3	odd	2
693.2.a.f.1.1		2	12.11	even	2
1617.2.a.p.1.2		2	28.27	even	2
2541.2.a.t.1.1		2	44.43	even	2
3696.2.a.be.1.1		2	1.1	even	1	trivial
4851.2.a.w.1.1		2	84.83	odd	2
5775.2.a.be.1.1		2	20.19	odd	2
7623.2.a.bm.1.2		2	132.131	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3696.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -5.47214 q^{13} -1.00000 q^{15} +0.763932 q^{17} -6.70820 q^{19} +1.00000 q^{21} +7.70820 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} +0.763932 q^{31} +1.00000 q^{33} -1.00000 q^{35} -7.00000 q^{37} +5.47214 q^{39} +6.47214 q^{41} +7.70820 q^{43} +1.00000 q^{45} +4.23607 q^{47} +1.00000 q^{49} -0.763932 q^{51} +10.1803 q^{53} -1.00000 q^{55} +6.70820 q^{57} -11.1803 q^{59} +2.00000 q^{61} -1.00000 q^{63} -5.47214 q^{65} +14.2361 q^{67} -7.70820 q^{69} -6.47214 q^{71} +13.4721 q^{73} +4.00000 q^{75} +1.00000 q^{77} +5.52786 q^{79} +1.00000 q^{81} -11.2361 q^{83} +0.763932 q^{85} -5.00000 q^{87} +4.47214 q^{89} +5.47214 q^{91} -0.763932 q^{93} -6.70820 q^{95} -3.70820 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} + 2 q^{21} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 10 q^{29} + 6 q^{31} + 2 q^{33} - 2 q^{35} - 14 q^{37} + 2 q^{39} + 4 q^{41} + 2 q^{43} + 2 q^{45} + 4 q^{47} + 2 q^{49} - 6 q^{51} - 2 q^{53} - 2 q^{55} + 4 q^{61} - 2 q^{63} - 2 q^{65} + 24 q^{67} - 2 q^{69} - 4 q^{71} + 18 q^{73} + 8 q^{75} + 2 q^{77} + 20 q^{79} + 2 q^{81} - 18 q^{83} + 6 q^{85} - 10 q^{87} + 2 q^{91} - 6 q^{93} + 6 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 - 2 * q^15 + 6 * q^17 + 2 * q^21 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 10 * q^29 + 6 * q^31 + 2 * q^33 - 2 * q^35 - 14 * q^37 + 2 * q^39 + 4 * q^41 + 2 * q^43 + 2 * q^45 + 4 * q^47 + 2 * q^49 - 6 * q^51 - 2 * q^53 - 2 * q^55 + 4 * q^61 - 2 * q^63 - 2 * q^65 + 24 * q^67 - 2 * q^69 - 4 * q^71 + 18 * q^73 + 8 * q^75 + 2 * q^77 + 20 * q^79 + 2 * q^81 - 18 * q^83 + 6 * q^85 - 10 * q^87 + 2 * q^91 - 6 * q^93 + 6 * q^97 - 2 * q^99

Introduction

L-functions

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