LMFDB - Embedded newform 3696.2.a.be.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.47214 q^{13} -1.00000 q^{15} +5.23607 q^{17} +6.70820 q^{19} +1.00000 q^{21} -5.70820 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} +5.23607 q^{31} +1.00000 q^{33} -1.00000 q^{35} -7.00000 q^{37} -3.47214 q^{39} -2.47214 q^{41} -5.70820 q^{43} +1.00000 q^{45} -0.236068 q^{47} +1.00000 q^{49} -5.23607 q^{51} -12.1803 q^{53} -1.00000 q^{55} -6.70820 q^{57} +11.1803 q^{59} +2.00000 q^{61} -1.00000 q^{63} +3.47214 q^{65} +9.76393 q^{67} +5.70820 q^{69} +2.47214 q^{71} +4.52786 q^{73} +4.00000 q^{75} +1.00000 q^{77} +14.4721 q^{79} +1.00000 q^{81} -6.76393 q^{83} +5.23607 q^{85} -5.00000 q^{87} -4.47214 q^{89} -3.47214 q^{91} -5.23607 q^{93} +6.70820 q^{95} +9.70820 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	3.47214	0.962997	0.481499	−	0.876447i	$-0.340093\pi$
$13$	3.47214	0.962997	0.481499	+	0.876447i	$0.340093\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−5.70820	−1.19024	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	−5.70820	−1.19024	−0.595121	+	0.803636i	$0.702896\pi$
$24$	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
$31$	5.23607	0.940426	0.470213	−	0.882553i	$-0.344177\pi$
$31$	5.23607	0.940426	0.470213	+	0.882553i	$0.344177\pi$
$32$	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
$39$	−3.47214	−0.555987
$40$	0	0
$41$	−2.47214	−0.386083	−0.193041	−	0.981191i	$-0.561835\pi$
$41$	−2.47214	−0.386083	−0.193041	+	0.981191i	$0.561835\pi$
$42$	0	0
$43$	−5.70820	−0.870493	−0.435246	−	0.900311i	$-0.643339\pi$
$43$	−5.70820	−0.870493	−0.435246	+	0.900311i	$0.643339\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−0.236068	−0.0344341	−0.0172170	−	0.999852i	$-0.505481\pi$
$47$	−0.236068	−0.0344341	−0.0172170	+	0.999852i	$0.505481\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.23607	−0.733196
$52$	0	0
$53$	−12.1803	−1.67310	−0.836549	−	0.547892i	$-0.815431\pi$
$53$	−12.1803	−1.67310	−0.836549	+	0.547892i	$0.815431\pi$
$54$	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	$-0.240553\pi$
$59$	11.1803	1.45556	0.727778	+	0.685813i	$0.240553\pi$
$60$	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
$65$	3.47214	0.430665
$66$	0	0
$67$	9.76393	1.19285	0.596427	−	0.802667i	$-0.296586\pi$
$67$	9.76393	1.19285	0.596427	+	0.802667i	$0.296586\pi$
$68$	0	0
$69$	5.70820	0.687187
$70$	0	0
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	4.52786	0.529946	0.264973	−	0.964256i	$-0.414637\pi$
$73$	4.52786	0.529946	0.264973	+	0.964256i	$0.414637\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	14.4721	1.62824	0.814121	−	0.580695i	$-0.197219\pi$
$79$	14.4721	1.62824	0.814121	+	0.580695i	$0.197219\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.76393	−0.742438	−0.371219	−	0.928545i	$-0.621060\pi$
$83$	−6.76393	−0.742438	−0.371219	+	0.928545i	$0.621060\pi$
$84$	0	0
$85$	5.23607	0.567931
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	−4.47214	−0.474045	−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214	−0.474045	−0.237023	+	0.971504i	$0.576172\pi$
$90$	0	0
$91$	−3.47214	−0.363979
$92$	0	0
$93$	−5.23607	−0.542955
$94$	0	0
$95$	6.70820	0.688247
$96$	0	0
$97$	9.70820	0.985719	0.492859	−	0.870109i	$-0.335952\pi$
$97$	9.70820	0.985719	0.492859	+	0.870109i	$0.335952\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	18.1803	1.80901	0.904506	−	0.426461i	$-0.140240\pi$
$101$	18.1803	1.80901	0.904506	+	0.426461i	$0.140240\pi$
$102$	0	0
$103$	−17.4164	−1.71609	−0.858045	−	0.513575i	$-0.828321\pi$
$103$	−17.4164	−1.71609	−0.858045	+	0.513575i	$0.828321\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	4.23607	0.409516	0.204758	−	0.978813i	$-0.434359\pi$
$107$	4.23607	0.409516	0.204758	+	0.978813i	$0.434359\pi$
$108$	0	0
$109$	2.76393	0.264737	0.132368	−	0.991201i	$-0.457742\pi$
$109$	2.76393	0.264737	0.132368	+	0.991201i	$0.457742\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	−0.472136	−0.0444148	−0.0222074	−	0.999753i	$-0.507069\pi$
$113$	−0.472136	−0.0444148	−0.0222074	+	0.999753i	$0.507069\pi$
$114$	0	0
$115$	−5.70820	−0.532293
$116$	0	0
$117$	3.47214	0.320999
$118$	0	0
$119$	−5.23607	−0.479990
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.47214	0.222905
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	12.6525	1.12273	0.561363	−	0.827570i	$-0.310277\pi$
$127$	12.6525	1.12273	0.561363	+	0.827570i	$0.310277\pi$
$128$	0	0
$129$	5.70820	0.502579
$130$	0	0
$131$	−0.944272	−0.0825014	−0.0412507	−	0.999149i	$-0.513134\pi$
$131$	−0.944272	−0.0825014	−0.0412507	+	0.999149i	$0.513134\pi$
$132$	0	0
$133$	−6.70820	−0.581675
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	19.7082	1.68379	0.841893	−	0.539645i	$-0.181441\pi$
$137$	19.7082	1.68379	0.841893	+	0.539645i	$0.181441\pi$
$138$	0	0
$139$	−14.4721	−1.22751	−0.613755	−	0.789496i	$-0.710342\pi$
$139$	−14.4721	−1.22751	−0.613755	+	0.789496i	$0.710342\pi$
$140$	0	0
$141$	0.236068	0.0198805
$142$	0	0
$143$	−3.47214	−0.290355
$144$	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
$151$	14.1803	1.15398	0.576990	−	0.816751i	$-0.304227\pi$
$151$	14.1803	1.15398	0.576990	+	0.816751i	$0.304227\pi$
$152$	0	0
$153$	5.23607	0.423311
$154$	0	0
$155$	5.23607	0.420571
$156$	0	0
$157$	−15.4164	−1.23036	−0.615182	−	0.788385i	$-0.710917\pi$
$157$	−15.4164	−1.23036	−0.615182	+	0.788385i	$0.710917\pi$
$158$	0	0
$159$	12.1803	0.965964
$160$	0	0
$161$	5.70820	0.449869
$162$	0	0
$163$	22.7082	1.77864	0.889322	−	0.457282i	$-0.151177\pi$
$163$	22.7082	1.77864	0.889322	+	0.457282i	$0.151177\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	22.6525	1.75290	0.876451	−	0.481492i	$-0.159905\pi$
$167$	22.6525	1.75290	0.876451	+	0.481492i	$0.159905\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	6.70820	0.512989
$172$	0	0
$173$	−1.52786	−0.116161	−0.0580807	−	0.998312i	$-0.518498\pi$
$173$	−1.52786	−0.116161	−0.0580807	+	0.998312i	$0.518498\pi$
$174$	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	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	−0.763932	−0.0567826	−0.0283913	−	0.999597i	$-0.509038\pi$
$181$	−0.763932	−0.0567826	−0.0283913	+	0.999597i	$0.509038\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	−5.23607	−0.382899
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	10.7639	0.778851	0.389425	−	0.921058i	$-0.372674\pi$
$191$	10.7639	0.778851	0.389425	+	0.921058i	$0.372674\pi$
$192$	0	0
$193$	14.6525	1.05471	0.527354	−	0.849646i	$-0.323184\pi$
$193$	14.6525	1.05471	0.527354	+	0.849646i	$0.323184\pi$
$194$	0	0
$195$	−3.47214	−0.248645
$196$	0	0
$197$	−16.4721	−1.17359	−0.586796	−	0.809735i	$-0.699611\pi$
$197$	−16.4721	−1.17359	−0.586796	+	0.809735i	$0.699611\pi$
$198$	0	0
$199$	3.81966	0.270769	0.135384	−	0.990793i	$-0.456773\pi$
$199$	3.81966	0.270769	0.135384	+	0.990793i	$0.456773\pi$
$200$	0	0
$201$	−9.76393	−0.688695
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	−2.47214	−0.172661
$206$	0	0
$207$	−5.70820	−0.396748
$208$	0	0
$209$	−6.70820	−0.464016
$210$	0	0
$211$	−5.41641	−0.372881	−0.186440	−	0.982466i	$-0.559695\pi$
$211$	−5.41641	−0.372881	−0.186440	+	0.982466i	$0.559695\pi$
$212$	0	0
$213$	−2.47214	−0.169388
$214$	0	0
$215$	−5.70820	−0.389296
$216$	0	0
$217$	−5.23607	−0.355447
$218$	0	0
$219$	−4.52786	−0.305965
$220$	0	0
$221$	18.1803	1.22294
$222$	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
$229$	7.23607	0.478173	0.239086	−	0.970998i	$-0.423152\pi$
$229$	7.23607	0.478173	0.239086	+	0.970998i	$0.423152\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	−0.236068	−0.0153994
$236$	0	0
$237$	−14.4721	−0.940066
$238$	0	0
$239$	−10.1246	−0.654907	−0.327453	−	0.944867i	$-0.606190\pi$
$239$	−10.1246	−0.654907	−0.327453	+	0.944867i	$0.606190\pi$
$240$	0	0
$241$	25.9443	1.67122	0.835609	−	0.549325i	$-0.185115\pi$
$241$	25.9443	1.67122	0.835609	+	0.549325i	$0.185115\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	23.2918	1.48202
$248$	0	0
$249$	6.76393	0.428647
$250$	0	0
$251$	−12.1246	−0.765299	−0.382649	−	0.923894i	$-0.624988\pi$
$251$	−12.1246	−0.765299	−0.382649	+	0.923894i	$0.624988\pi$
$252$	0	0
$253$	5.70820	0.358872
$254$	0	0
$255$	−5.23607	−0.327895
$256$	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
$263$	26.1246	1.61091	0.805456	−	0.592655i	$-0.201920\pi$
$263$	26.1246	1.61091	0.805456	+	0.592655i	$0.201920\pi$
$264$	0	0
$265$	−12.1803	−0.748232
$266$	0	0
$267$	4.47214	0.273690
$268$	0	0
$269$	1.05573	0.0643689	0.0321844	−	0.999482i	$-0.489754\pi$
$269$	1.05573	0.0643689	0.0321844	+	0.999482i	$0.489754\pi$
$270$	0	0
$271$	−5.29180	−0.321454	−0.160727	−	0.986999i	$-0.551384\pi$
$271$	−5.29180	−0.321454	−0.160727	+	0.986999i	$0.551384\pi$
$272$	0	0
$273$	3.47214	0.210143
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−6.47214	−0.388873	−0.194436	−	0.980915i	$-0.562288\pi$
$277$	−6.47214	−0.388873	−0.194436	+	0.980915i	$0.562288\pi$
$278$	0	0
$279$	5.23607	0.313475
$280$	0	0
$281$	11.4721	0.684370	0.342185	−	0.939633i	$-0.388833\pi$
$281$	11.4721	0.684370	0.342185	+	0.939633i	$0.388833\pi$
$282$	0	0
$283$	13.7639	0.818181	0.409090	−	0.912494i	$-0.365846\pi$
$283$	13.7639	0.818181	0.409090	+	0.912494i	$0.365846\pi$
$284$	0	0
$285$	−6.70820	−0.397360
$286$	0	0
$287$	2.47214	0.145926
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	−9.70820	−0.569105
$292$	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
$299$	−19.8197	−1.14620
$300$	0	0
$301$	5.70820	0.329015
$302$	0	0
$303$	−18.1803	−1.04443
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	20.9443	1.19535	0.597676	−	0.801737i	$-0.296091\pi$
$307$	20.9443	1.19535	0.597676	+	0.801737i	$0.296091\pi$
$308$	0	0
$309$	17.4164	0.990785
$310$	0	0
$311$	−9.88854	−0.560728	−0.280364	−	0.959894i	$-0.590455\pi$
$311$	−9.88854	−0.560728	−0.280364	+	0.959894i	$0.590455\pi$
$312$	0	0
$313$	24.6525	1.39344	0.696720	−	0.717343i	$-0.254642\pi$
$313$	24.6525	1.39344	0.696720	+	0.717343i	$0.254642\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	24.1803	1.35810	0.679052	−	0.734091i	$-0.262391\pi$
$317$	24.1803	1.35810	0.679052	+	0.734091i	$0.262391\pi$
$318$	0	0
$319$	−5.00000	−0.279946
$320$	0	0
$321$	−4.23607	−0.236434
$322$	0	0
$323$	35.1246	1.95439
$324$	0	0
$325$	−13.8885	−0.770398
$326$	0	0
$327$	−2.76393	−0.152846
$328$	0	0
$329$	0.236068	0.0130148
$330$	0	0
$331$	11.4164	0.627503	0.313751	−	0.949505i	$-0.398414\pi$
$331$	11.4164	0.627503	0.313751	+	0.949505i	$0.398414\pi$
$332$	0	0
$333$	−7.00000	−0.383598
$334$	0	0
$335$	9.76393	0.533461
$336$	0	0
$337$	−8.18034	−0.445612	−0.222806	−	0.974863i	$-0.571522\pi$
$337$	−8.18034	−0.445612	−0.222806	+	0.974863i	$0.571522\pi$
$338$	0	0
$339$	0.472136	0.0256429
$340$	0	0
$341$	−5.23607	−0.283549
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.70820	0.307319
$346$	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
$349$	1.58359	0.0847677	0.0423839	−	0.999101i	$-0.486505\pi$
$349$	1.58359	0.0847677	0.0423839	+	0.999101i	$0.486505\pi$
$350$	0	0
$351$	−3.47214	−0.185329
$352$	0	0
$353$	24.5279	1.30549	0.652743	−	0.757579i	$-0.273618\pi$
$353$	24.5279	1.30549	0.652743	+	0.757579i	$0.273618\pi$
$354$	0	0
$355$	2.47214	0.131207
$356$	0	0
$357$	5.23607	0.277122
$358$	0	0
$359$	−23.4164	−1.23587	−0.617935	−	0.786229i	$-0.712031\pi$
$359$	−23.4164	−1.23587	−0.617935	+	0.786229i	$0.712031\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	4.52786	0.236999
$366$	0	0
$367$	19.8885	1.03817	0.519087	−	0.854722i	$-0.326272\pi$
$367$	19.8885	1.03817	0.519087	+	0.854722i	$0.326272\pi$
$368$	0	0
$369$	−2.47214	−0.128694
$370$	0	0
$371$	12.1803	0.632372
$372$	0	0
$373$	4.65248	0.240896	0.120448	−	0.992720i	$-0.461567\pi$
$373$	4.65248	0.240896	0.120448	+	0.992720i	$0.461567\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	17.3607	0.894120
$378$	0	0
$379$	31.1803	1.60163	0.800813	−	0.598914i	$-0.204401\pi$
$379$	31.1803	1.60163	0.800813	+	0.598914i	$0.204401\pi$
$380$	0	0
$381$	−12.6525	−0.648206
$382$	0	0
$383$	−32.9443	−1.68337	−0.841687	−	0.539966i	$-0.818437\pi$
$383$	−32.9443	−1.68337	−0.841687	+	0.539966i	$0.818437\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−5.70820	−0.290164
$388$	0	0
$389$	11.0557	0.560548	0.280274	−	0.959920i	$-0.409575\pi$
$389$	11.0557	0.560548	0.280274	+	0.959920i	$0.409575\pi$
$390$	0	0
$391$	−29.8885	−1.51153
$392$	0	0
$393$	0.944272	0.0476322
$394$	0	0
$395$	14.4721	0.728172
$396$	0	0
$397$	23.1246	1.16059	0.580295	−	0.814406i	$-0.302937\pi$
$397$	23.1246	1.16059	0.580295	+	0.814406i	$0.302937\pi$
$398$	0	0
$399$	6.70820	0.335830
$400$	0	0
$401$	−29.7082	−1.48356	−0.741778	−	0.670645i	$-0.766017\pi$
$401$	−29.7082	−1.48356	−0.741778	+	0.670645i	$0.766017\pi$
$402$	0	0
$403$	18.1803	0.905627
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	7.00000	0.346977
$408$	0	0
$409$	21.0557	1.04114	0.520569	−	0.853819i	$-0.325720\pi$
$409$	21.0557	1.04114	0.520569	+	0.853819i	$0.325720\pi$
$410$	0	0
$411$	−19.7082	−0.972134
$412$	0	0
$413$	−11.1803	−0.550149
$414$	0	0
$415$	−6.76393	−0.332028
$416$	0	0
$417$	14.4721	0.708704
$418$	0	0
$419$	1.18034	0.0576634	0.0288317	−	0.999584i	$-0.490821\pi$
$419$	1.18034	0.0576634	0.0288317	+	0.999584i	$0.490821\pi$
$420$	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
$423$	−0.236068	−0.0114780
$424$	0	0
$425$	−20.9443	−1.01595
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	3.47214	0.167636
$430$	0	0
$431$	−8.70820	−0.419459	−0.209730	−	0.977759i	$-0.567258\pi$
$431$	−8.70820	−0.419459	−0.209730	+	0.977759i	$0.567258\pi$
$432$	0	0
$433$	−10.4721	−0.503259	−0.251629	−	0.967824i	$-0.580966\pi$
$433$	−10.4721	−0.503259	−0.251629	+	0.967824i	$0.580966\pi$
$434$	0	0
$435$	−5.00000	−0.239732
$436$	0	0
$437$	−38.2918	−1.83175
$438$	0	0
$439$	−11.1803	−0.533609	−0.266804	−	0.963751i	$-0.585968\pi$
$439$	−11.1803	−0.533609	−0.266804	+	0.963751i	$0.585968\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−18.4721	−0.877638	−0.438819	−	0.898576i	$-0.644603\pi$
$443$	−18.4721	−0.877638	−0.438819	+	0.898576i	$0.644603\pi$
$444$	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
$451$	2.47214	0.116408
$452$	0	0
$453$	−14.1803	−0.666250
$454$	0	0
$455$	−3.47214	−0.162776
$456$	0	0
$457$	15.2361	0.712713	0.356357	−	0.934350i	$-0.384019\pi$
$457$	15.2361	0.712713	0.356357	+	0.934350i	$0.384019\pi$
$458$	0	0
$459$	−5.23607	−0.244399
$460$	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
$463$	4.81966	0.223989	0.111994	−	0.993709i	$-0.464276\pi$
$463$	4.81966	0.223989	0.111994	+	0.993709i	$0.464276\pi$
$464$	0	0
$465$	−5.23607	−0.242817
$466$	0	0
$467$	−29.1803	−1.35031	−0.675153	−	0.737678i	$-0.735922\pi$
$467$	−29.1803	−1.35031	−0.675153	+	0.737678i	$0.735922\pi$
$468$	0	0
$469$	−9.76393	−0.450856
$470$	0	0
$471$	15.4164	0.710351
$472$	0	0
$473$	5.70820	0.262463
$474$	0	0
$475$	−26.8328	−1.23117
$476$	0	0
$477$	−12.1803	−0.557699
$478$	0	0
$479$	−11.7082	−0.534961	−0.267481	−	0.963563i	$-0.586191\pi$
$479$	−11.7082	−0.534961	−0.267481	+	0.963563i	$0.586191\pi$
$480$	0	0
$481$	−24.3050	−1.10821
$482$	0	0
$483$	−5.70820	−0.259732
$484$	0	0
$485$	9.70820	0.440827
$486$	0	0
$487$	−16.9443	−0.767818	−0.383909	−	0.923371i	$-0.625422\pi$
$487$	−16.9443	−0.767818	−0.383909	+	0.923371i	$0.625422\pi$
$488$	0	0
$489$	−22.7082	−1.02690
$490$	0	0
$491$	18.1246	0.817952	0.408976	−	0.912545i	$-0.365886\pi$
$491$	18.1246	0.817952	0.408976	+	0.912545i	$0.365886\pi$
$492$	0	0
$493$	26.1803	1.17910
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	−2.47214	−0.110890
$498$	0	0
$499$	2.23607	0.100100	0.0500501	−	0.998747i	$-0.484062\pi$
$499$	2.23607	0.100100	0.0500501	+	0.998747i	$0.484062\pi$
$500$	0	0
$501$	−22.6525	−1.01204
$502$	0	0
$503$	−2.29180	−0.102186	−0.0510931	−	0.998694i	$-0.516271\pi$
$503$	−2.29180	−0.102186	−0.0510931	+	0.998694i	$0.516271\pi$
$504$	0	0
$505$	18.1803	0.809015
$506$	0	0
$507$	0.944272	0.0419366
$508$	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
$511$	−4.52786	−0.200301
$512$	0	0
$513$	−6.70820	−0.296174
$514$	0	0
$515$	−17.4164	−0.767459
$516$	0	0
$517$	0.236068	0.0103823
$518$	0	0
$519$	1.52786	0.0670658
$520$	0	0
$521$	38.3050	1.67817	0.839085	−	0.544000i	$-0.183091\pi$
$521$	38.3050	1.67817	0.839085	+	0.544000i	$0.183091\pi$
$522$	0	0
$523$	21.6525	0.946797	0.473398	−	0.880848i	$-0.343027\pi$
$523$	21.6525	0.946797	0.473398	+	0.880848i	$0.343027\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	27.4164	1.19428
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	11.1803	0.485185
$532$	0	0
$533$	−8.58359	−0.371797
$534$	0	0
$535$	4.23607	0.183141
$536$	0	0
$537$	8.94427	0.385974
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−36.9443	−1.58836	−0.794179	−	0.607684i	$-0.792099\pi$
$541$	−36.9443	−1.58836	−0.794179	+	0.607684i	$0.792099\pi$
$542$	0	0
$543$	0.763932	0.0327835
$544$	0	0
$545$	2.76393	0.118394
$546$	0	0
$547$	−14.8328	−0.634205	−0.317103	−	0.948391i	$-0.602710\pi$
$547$	−14.8328	−0.634205	−0.317103	+	0.948391i	$0.602710\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	33.5410	1.42890
$552$	0	0
$553$	−14.4721	−0.615418
$554$	0	0
$555$	7.00000	0.297133
$556$	0	0
$557$	−12.5279	−0.530823	−0.265411	−	0.964135i	$-0.585508\pi$
$557$	−12.5279	−0.530823	−0.265411	+	0.964135i	$0.585508\pi$
$558$	0	0
$559$	−19.8197	−0.838282
$560$	0	0
$561$	5.23607	0.221067
$562$	0	0
$563$	−34.6525	−1.46043	−0.730214	−	0.683219i	$-0.760580\pi$
$563$	−34.6525	−1.46043	−0.730214	+	0.683219i	$0.760580\pi$
$564$	0	0
$565$	−0.472136	−0.0198629
$566$	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
$571$	−17.5279	−0.733518	−0.366759	−	0.930316i	$-0.619533\pi$
$571$	−17.5279	−0.733518	−0.366759	+	0.930316i	$0.619533\pi$
$572$	0	0
$573$	−10.7639	−0.449670
$574$	0	0
$575$	22.8328	0.952194
$576$	0	0
$577$	7.34752	0.305881	0.152941	−	0.988235i	$-0.451126\pi$
$577$	7.34752	0.305881	0.152941	+	0.988235i	$0.451126\pi$
$578$	0	0
$579$	−14.6525	−0.608936
$580$	0	0
$581$	6.76393	0.280615
$582$	0	0
$583$	12.1803	0.504458
$584$	0	0
$585$	3.47214	0.143555
$586$	0	0
$587$	−46.0132	−1.89917	−0.949583	−	0.313515i	$-0.898493\pi$
$587$	−46.0132	−1.89917	−0.949583	+	0.313515i	$0.898493\pi$
$588$	0	0
$589$	35.1246	1.44728
$590$	0	0
$591$	16.4721	0.677573
$592$	0	0
$593$	−22.8328	−0.937631	−0.468816	−	0.883296i	$-0.655319\pi$
$593$	−22.8328	−0.937631	−0.468816	+	0.883296i	$0.655319\pi$
$594$	0	0
$595$	−5.23607	−0.214658
$596$	0	0
$597$	−3.81966	−0.156328
$598$	0	0
$599$	25.5279	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$599$	25.5279	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$600$	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
$603$	9.76393	0.397618
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−6.81966	−0.276801	−0.138401	−	0.990376i	$-0.544196\pi$
$607$	−6.81966	−0.276801	−0.138401	+	0.990376i	$0.544196\pi$
$608$	0	0
$609$	5.00000	0.202610
$610$	0	0
$611$	−0.819660	−0.0331599
$612$	0	0
$613$	13.5967	0.549167	0.274584	−	0.961563i	$-0.411460\pi$
$613$	13.5967	0.549167	0.274584	+	0.961563i	$0.411460\pi$
$614$	0	0
$615$	2.47214	0.0996861
$616$	0	0
$617$	32.4721	1.30728	0.653639	−	0.756806i	$-0.273241\pi$
$617$	32.4721	1.30728	0.653639	+	0.756806i	$0.273241\pi$
$618$	0	0
$619$	44.0689	1.77128	0.885639	−	0.464374i	$-0.153721\pi$
$619$	44.0689	1.77128	0.885639	+	0.464374i	$0.153721\pi$
$620$	0	0
$621$	5.70820	0.229062
$622$	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
$629$	−36.6525	−1.46143
$630$	0	0
$631$	−44.3607	−1.76597	−0.882985	−	0.469400i	$-0.844470\pi$
$631$	−44.3607	−1.76597	−0.882985	+	0.469400i	$0.844470\pi$
$632$	0	0
$633$	5.41641	0.215283
$634$	0	0
$635$	12.6525	0.502098
$636$	0	0
$637$	3.47214	0.137571
$638$	0	0
$639$	2.47214	0.0977962
$640$	0	0
$641$	−46.5410	−1.83826	−0.919130	−	0.393955i	$-0.871107\pi$
$641$	−46.5410	−1.83826	−0.919130	+	0.393955i	$0.871107\pi$
$642$	0	0
$643$	47.9574	1.89126	0.945628	−	0.325250i	$-0.105448\pi$
$643$	47.9574	1.89126	0.945628	+	0.325250i	$0.105448\pi$
$644$	0	0
$645$	5.70820	0.224760
$646$	0	0
$647$	−12.3475	−0.485431	−0.242716	−	0.970097i	$-0.578038\pi$
$647$	−12.3475	−0.485431	−0.242716	+	0.970097i	$0.578038\pi$
$648$	0	0
$649$	−11.1803	−0.438867
$650$	0	0
$651$	5.23607	0.205218
$652$	0	0
$653$	−44.9443	−1.75881	−0.879403	−	0.476079i	$-0.842058\pi$
$653$	−44.9443	−1.75881	−0.879403	+	0.476079i	$0.842058\pi$
$654$	0	0
$655$	−0.944272	−0.0368958
$656$	0	0
$657$	4.52786	0.176649
$658$	0	0
$659$	−23.5410	−0.917028	−0.458514	−	0.888687i	$-0.651618\pi$
$659$	−23.5410	−0.917028	−0.458514	+	0.888687i	$0.651618\pi$
$660$	0	0
$661$	40.5410	1.57686	0.788431	−	0.615123i	$-0.210894\pi$
$661$	40.5410	1.57686	0.788431	+	0.615123i	$0.210894\pi$
$662$	0	0
$663$	−18.1803	−0.706066
$664$	0	0
$665$	−6.70820	−0.260133
$666$	0	0
$667$	−28.5410	−1.10511
$668$	0	0
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	3.59675	0.138644	0.0693222	−	0.997594i	$-0.477916\pi$
$673$	3.59675	0.138644	0.0693222	+	0.997594i	$0.477916\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	19.3050	0.741950	0.370975	−	0.928643i	$-0.379024\pi$
$677$	19.3050	0.741950	0.370975	+	0.928643i	$0.379024\pi$
$678$	0	0
$679$	−9.70820	−0.372567
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	0	0
$683$	−37.0132	−1.41627	−0.708135	−	0.706078i	$-0.750463\pi$
$683$	−37.0132	−1.41627	−0.708135	+	0.706078i	$0.750463\pi$
$684$	0	0
$685$	19.7082	0.753012
$686$	0	0
$687$	−7.23607	−0.276073
$688$	0	0
$689$	−42.2918	−1.61119
$690$	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
$695$	−14.4721	−0.548959
$696$	0	0
$697$	−12.9443	−0.490299
$698$	0	0
$699$	14.9443	0.565244
$700$	0	0
$701$	4.11146	0.155288	0.0776438	−	0.996981i	$-0.475260\pi$
$701$	4.11146	0.155288	0.0776438	+	0.996981i	$0.475260\pi$
$702$	0	0
$703$	−46.9574	−1.77103
$704$	0	0
$705$	0.236068	0.00889083
$706$	0	0
$707$	−18.1803	−0.683742
$708$	0	0
$709$	−49.7214	−1.86732	−0.933662	−	0.358154i	$-0.883406\pi$
$709$	−49.7214	−1.86732	−0.933662	+	0.358154i	$0.883406\pi$
$710$	0	0
$711$	14.4721	0.542748
$712$	0	0
$713$	−29.8885	−1.11933
$714$	0	0
$715$	−3.47214	−0.129851
$716$	0	0
$717$	10.1246	0.378111
$718$	0	0
$719$	−7.76393	−0.289546	−0.144773	−	0.989465i	$-0.546245\pi$
$719$	−7.76393	−0.289546	−0.144773	+	0.989465i	$0.546245\pi$
$720$	0	0
$721$	17.4164	0.648621
$722$	0	0
$723$	−25.9443	−0.964878
$724$	0	0
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−24.1803	−0.896799	−0.448400	−	0.893833i	$-0.648006\pi$
$727$	−24.1803	−0.896799	−0.448400	+	0.893833i	$0.648006\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−29.8885	−1.10547
$732$	0	0
$733$	−8.11146	−0.299603	−0.149802	−	0.988716i	$-0.547864\pi$
$733$	−8.11146	−0.299603	−0.149802	+	0.988716i	$0.547864\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	−9.76393	−0.359659
$738$	0	0
$739$	−34.0689	−1.25324	−0.626622	−	0.779323i	$-0.715563\pi$
$739$	−34.0689	−1.25324	−0.626622	+	0.779323i	$0.715563\pi$
$740$	0	0
$741$	−23.2918	−0.855646
$742$	0	0
$743$	−2.81966	−0.103443	−0.0517216	−	0.998662i	$-0.516471\pi$
$743$	−2.81966	−0.103443	−0.0517216	+	0.998662i	$0.516471\pi$
$744$	0	0
$745$	5.00000	0.183186
$746$	0	0
$747$	−6.76393	−0.247479
$748$	0	0
$749$	−4.23607	−0.154783
$750$	0	0
$751$	−39.7639	−1.45101	−0.725503	−	0.688219i	$-0.758393\pi$
$751$	−39.7639	−1.45101	−0.725503	+	0.688219i	$0.758393\pi$
$752$	0	0
$753$	12.1246	0.441845
$754$	0	0
$755$	14.1803	0.516075
$756$	0	0
$757$	−51.7214	−1.87984	−0.939922	−	0.341388i	$-0.889103\pi$
$757$	−51.7214	−1.87984	−0.939922	+	0.341388i	$0.889103\pi$
$758$	0	0
$759$	−5.70820	−0.207195
$760$	0	0
$761$	27.7771	1.00692	0.503459	−	0.864019i	$-0.332060\pi$
$761$	27.7771	1.00692	0.503459	+	0.864019i	$0.332060\pi$
$762$	0	0
$763$	−2.76393	−0.100061
$764$	0	0
$765$	5.23607	0.189310
$766$	0	0
$767$	38.8197	1.40170
$768$	0	0
$769$	13.9443	0.502843	0.251422	−	0.967878i	$-0.419102\pi$
$769$	13.9443	0.502843	0.251422	+	0.967878i	$0.419102\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	0	0
$773$	−5.47214	−0.196819	−0.0984095	−	0.995146i	$-0.531376\pi$
$773$	−5.47214	−0.196819	−0.0984095	+	0.995146i	$0.531376\pi$
$774$	0	0
$775$	−20.9443	−0.752340
$776$	0	0
$777$	−7.00000	−0.251124
$778$	0	0
$779$	−16.5836	−0.594169
$780$	0	0
$781$	−2.47214	−0.0884600
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	−15.4164	−0.550235
$786$	0	0
$787$	−3.65248	−0.130197	−0.0650984	−	0.997879i	$-0.520736\pi$
$787$	−3.65248	−0.130197	−0.0650984	+	0.997879i	$0.520736\pi$
$788$	0	0
$789$	−26.1246	−0.930061
$790$	0	0
$791$	0.472136	0.0167872
$792$	0	0
$793$	6.94427	0.246598
$794$	0	0
$795$	12.1803	0.431992
$796$	0	0
$797$	−2.52786	−0.0895415	−0.0447708	−	0.998997i	$-0.514256\pi$
$797$	−2.52786	−0.0895415	−0.0447708	+	0.998997i	$0.514256\pi$
$798$	0	0
$799$	−1.23607	−0.0437289
$800$	0	0
$801$	−4.47214	−0.158015
$802$	0	0
$803$	−4.52786	−0.159785
$804$	0	0
$805$	5.70820	0.201188
$806$	0	0
$807$	−1.05573	−0.0371634
$808$	0	0
$809$	−56.3050	−1.97958	−0.989788	−	0.142545i	$-0.954471\pi$
$809$	−56.3050	−1.97958	−0.989788	+	0.142545i	$0.954471\pi$
$810$	0	0
$811$	−15.2918	−0.536968	−0.268484	−	0.963284i	$-0.586523\pi$
$811$	−15.2918	−0.536968	−0.268484	+	0.963284i	$0.586523\pi$
$812$	0	0
$813$	5.29180	0.185591
$814$	0	0
$815$	22.7082	0.795434
$816$	0	0
$817$	−38.2918	−1.33966
$818$	0	0
$819$	−3.47214	−0.121326
$820$	0	0
$821$	−7.47214	−0.260779	−0.130390	−	0.991463i	$-0.541623\pi$
$821$	−7.47214	−0.260779	−0.130390	+	0.991463i	$0.541623\pi$
$822$	0	0
$823$	17.1803	0.598869	0.299435	−	0.954117i	$-0.403202\pi$
$823$	17.1803	0.598869	0.299435	+	0.954117i	$0.403202\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−12.3475	−0.429365	−0.214683	−	0.976684i	$-0.568872\pi$
$827$	−12.3475	−0.429365	−0.214683	+	0.976684i	$0.568872\pi$
$828$	0	0
$829$	35.7771	1.24259	0.621295	−	0.783577i	$-0.286607\pi$
$829$	35.7771	1.24259	0.621295	+	0.783577i	$0.286607\pi$
$830$	0	0
$831$	6.47214	0.224516
$832$	0	0
$833$	5.23607	0.181419
$834$	0	0
$835$	22.6525	0.783921
$836$	0	0
$837$	−5.23607	−0.180985
$838$	0	0
$839$	23.5410	0.812726	0.406363	−	0.913712i	$-0.366797\pi$
$839$	23.5410	0.812726	0.406363	+	0.913712i	$0.366797\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−11.4721	−0.395121
$844$	0	0
$845$	−0.944272	−0.0324839
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−13.7639	−0.472377
$850$	0	0
$851$	39.9574	1.36972
$852$	0	0
$853$	−29.4164	−1.00720	−0.503599	−	0.863937i	$-0.667991\pi$
$853$	−29.4164	−1.00720	−0.503599	+	0.863937i	$0.667991\pi$
$854$	0	0
$855$	6.70820	0.229416
$856$	0	0
$857$	0.111456	0.00380727	0.00190364	−	0.999998i	$-0.499394\pi$
$857$	0.111456	0.00380727	0.00190364	+	0.999998i	$0.499394\pi$
$858$	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
$861$	−2.47214	−0.0842502
$862$	0	0
$863$	43.2361	1.47177	0.735886	−	0.677105i	$-0.236766\pi$
$863$	43.2361	1.47177	0.735886	+	0.677105i	$0.236766\pi$
$864$	0	0
$865$	−1.52786	−0.0519489
$866$	0	0
$867$	−10.4164	−0.353760
$868$	0	0
$869$	−14.4721	−0.490934
$870$	0	0
$871$	33.9017	1.14872
$872$	0	0
$873$	9.70820	0.328573
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	4.58359	0.154777	0.0773885	−	0.997001i	$-0.475342\pi$
$877$	4.58359	0.154777	0.0773885	+	0.997001i	$0.475342\pi$
$878$	0	0
$879$	16.0000	0.539667
$880$	0	0
$881$	54.8885	1.84924	0.924621	−	0.380888i	$-0.124382\pi$
$881$	54.8885	1.84924	0.924621	+	0.380888i	$0.124382\pi$
$882$	0	0
$883$	14.8197	0.498721	0.249361	−	0.968411i	$-0.419780\pi$
$883$	14.8197	0.498721	0.249361	+	0.968411i	$0.419780\pi$
$884$	0	0
$885$	−11.1803	−0.375823
$886$	0	0
$887$	−10.7639	−0.361417	−0.180709	−	0.983537i	$-0.557839\pi$
$887$	−10.7639	−0.361417	−0.180709	+	0.983537i	$0.557839\pi$
$888$	0	0
$889$	−12.6525	−0.424350
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−1.58359	−0.0529929
$894$	0	0
$895$	−8.94427	−0.298974
$896$	0	0
$897$	19.8197	0.661759
$898$	0	0
$899$	26.1803	0.873163
$900$	0	0
$901$	−63.7771	−2.12472
$902$	0	0
$903$	−5.70820	−0.189957
$904$	0	0
$905$	−0.763932	−0.0253940
$906$	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
$909$	18.1803	0.603004
$910$	0	0
$911$	14.5836	0.483176	0.241588	−	0.970379i	$-0.422332\pi$
$911$	14.5836	0.483176	0.241588	+	0.970379i	$0.422332\pi$
$912$	0	0
$913$	6.76393	0.223853
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	0	0
$917$	0.944272	0.0311826
$918$	0	0
$919$	−39.5967	−1.30618	−0.653088	−	0.757282i	$-0.726527\pi$
$919$	−39.5967	−1.30618	−0.653088	+	0.757282i	$0.726527\pi$
$920$	0	0
$921$	−20.9443	−0.690137
$922$	0	0
$923$	8.58359	0.282532
$924$	0	0
$925$	28.0000	0.920634
$926$	0	0
$927$	−17.4164	−0.572030
$928$	0	0
$929$	12.8885	0.422859	0.211430	−	0.977393i	$-0.432188\pi$
$929$	12.8885	0.422859	0.211430	+	0.977393i	$0.432188\pi$
$930$	0	0
$931$	6.70820	0.219853
$932$	0	0
$933$	9.88854	0.323736
$934$	0	0
$935$	−5.23607	−0.171238
$936$	0	0
$937$	−39.8885	−1.30310	−0.651551	−	0.758605i	$-0.725881\pi$
$937$	−39.8885	−1.30310	−0.651551	+	0.758605i	$0.725881\pi$
$938$	0	0
$939$	−24.6525	−0.804503
$940$	0	0
$941$	−60.3607	−1.96770	−0.983851	−	0.178990i	$-0.942717\pi$
$941$	−60.3607	−1.96770	−0.983851	+	0.178990i	$0.942717\pi$
$942$	0	0
$943$	14.1115	0.459532
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	5.41641	0.176010	0.0880048	−	0.996120i	$-0.471951\pi$
$947$	5.41641	0.176010	0.0880048	+	0.996120i	$0.471951\pi$
$948$	0	0
$949$	15.7214	0.510337
$950$	0	0
$951$	−24.1803	−0.784101
$952$	0	0
$953$	44.7771	1.45047	0.725236	−	0.688500i	$-0.241731\pi$
$953$	44.7771	1.45047	0.725236	+	0.688500i	$0.241731\pi$
$954$	0	0
$955$	10.7639	0.348313
$956$	0	0
$957$	5.00000	0.161627
$958$	0	0
$959$	−19.7082	−0.636411
$960$	0	0
$961$	−3.58359	−0.115600
$962$	0	0
$963$	4.23607	0.136505
$964$	0	0
$965$	14.6525	0.471680
$966$	0	0
$967$	61.1935	1.96785	0.983925	−	0.178582i	$-0.0571509\pi$
$967$	61.1935	1.96785	0.983925	+	0.178582i	$0.0571509\pi$
$968$	0	0
$969$	−35.1246	−1.12837
$970$	0	0
$971$	−32.1246	−1.03093	−0.515464	−	0.856911i	$-0.672380\pi$
$971$	−32.1246	−1.03093	−0.515464	+	0.856911i	$0.672380\pi$
$972$	0	0
$973$	14.4721	0.463955
$974$	0	0
$975$	13.8885	0.444789
$976$	0	0
$977$	4.18034	0.133741	0.0668705	−	0.997762i	$-0.478699\pi$
$977$	4.18034	0.133741	0.0668705	+	0.997762i	$0.478699\pi$
$978$	0	0
$979$	4.47214	0.142930
$980$	0	0
$981$	2.76393	0.0882456
$982$	0	0
$983$	−27.4164	−0.874448	−0.437224	−	0.899353i	$-0.644038\pi$
$983$	−27.4164	−0.874448	−0.437224	+	0.899353i	$0.644038\pi$
$984$	0	0
$985$	−16.4721	−0.524846
$986$	0	0
$987$	−0.236068	−0.00751413
$988$	0	0
$989$	32.5836	1.03610
$990$	0	0
$991$	29.1803	0.926944	0.463472	−	0.886112i	$-0.346603\pi$
$991$	29.1803	0.926944	0.463472	+	0.886112i	$0.346603\pi$
$992$	0	0
$993$	−11.4164	−0.362289
$994$	0	0
$995$	3.81966	0.121091
$996$	0	0
$997$	26.9443	0.853334	0.426667	−	0.904409i	$-0.359688\pi$
$997$	26.9443	0.853334	0.426667	+	0.904409i	$0.359688\pi$
$998$	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.1.2		2
4.3	odd	2		231.2.a.c.1.1	✓	2
12.11	even	2		693.2.a.f.1.2		2
20.19	odd	2		5775.2.a.be.1.2		2
28.27	even	2		1617.2.a.p.1.1		2
44.43	even	2		2541.2.a.t.1.2		2
84.83	odd	2		4851.2.a.w.1.2		2
132.131	odd	2		7623.2.a.bm.1.1		2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.47214 q^{13} -1.00000 q^{15} +5.23607 q^{17} +6.70820 q^{19} +1.00000 q^{21} -5.70820 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} +5.23607 q^{31} +1.00000 q^{33} -1.00000 q^{35} -7.00000 q^{37} -3.47214 q^{39} -2.47214 q^{41} -5.70820 q^{43} +1.00000 q^{45} -0.236068 q^{47} +1.00000 q^{49} -5.23607 q^{51} -12.1803 q^{53} -1.00000 q^{55} -6.70820 q^{57} +11.1803 q^{59} +2.00000 q^{61} -1.00000 q^{63} +3.47214 q^{65} +9.76393 q^{67} +5.70820 q^{69} +2.47214 q^{71} +4.52786 q^{73} +4.00000 q^{75} +1.00000 q^{77} +14.4721 q^{79} +1.00000 q^{81} -6.76393 q^{83} +5.23607 q^{85} -5.00000 q^{87} -4.47214 q^{89} -3.47214 q^{91} -5.23607 q^{93} +6.70820 q^{95} +9.70820 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

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