LMFDB - Embedded newform 3696.2.a.bl.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(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
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			$2.79129$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} -1.58258 q^{17} -2.58258 q^{19} -1.00000 q^{21} -3.58258 q^{23} +4.00000 q^{25} +1.00000 q^{27} +10.1652 q^{29} +5.58258 q^{31} +1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} +1.00000 q^{39} +7.16515 q^{41} +7.58258 q^{43} +3.00000 q^{45} -10.5826 q^{47} +1.00000 q^{49} -1.58258 q^{51} -0.417424 q^{53} +3.00000 q^{55} -2.58258 q^{57} +4.58258 q^{59} +10.0000 q^{61} -1.00000 q^{63} +3.00000 q^{65} +0.582576 q^{67} -3.58258 q^{69} +7.16515 q^{71} +7.00000 q^{73} +4.00000 q^{75} -1.00000 q^{77} +11.1652 q^{79} +1.00000 q^{81} +2.41742 q^{83} -4.74773 q^{85} +10.1652 q^{87} -9.16515 q^{89} -1.00000 q^{91} +5.58258 q^{93} -7.74773 q^{95} -11.5826 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} + 6 q^{43} + 6 q^{45} - 12 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 6 q^{55} + 4 q^{57} + 20 q^{61} - 2 q^{63} + 6 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} + 14 q^{73} + 8 q^{75} - 2 q^{77} + 4 q^{79} + 2 q^{81} + 14 q^{83} + 18 q^{85} + 2 q^{87} - 2 q^{91} + 2 q^{93} + 12 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 6 * q^15 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 + 2 * q^27 + 2 * q^29 + 2 * q^31 + 2 * q^33 - 6 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 + 6 * q^43 + 6 * q^45 - 12 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 6 * q^55 + 4 * q^57 + 20 * q^61 - 2 * q^63 + 6 * q^65 - 8 * q^67 + 2 * q^69 - 4 * q^71 + 14 * q^73 + 8 * q^75 - 2 * q^77 + 4 * q^79 + 2 * q^81 + 14 * q^83 + 18 * q^85 + 2 * q^87 - 2 * q^91 + 2 * q^93 + 12 * q^95 - 14 * 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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	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
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−1.58258	−0.383831	−0.191915	−	0.981411i	$-0.561470\pi$
$17$	−1.58258	−0.383831	−0.191915	+	0.981411i	$0.561470\pi$
$18$	0	0
$19$	−2.58258	−0.592483	−0.296242	−	0.955113i	$-0.595733\pi$
$19$	−2.58258	−0.592483	−0.296242	+	0.955113i	$0.595733\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−3.58258	−0.747019	−0.373509	−	0.927626i	$-0.621846\pi$
$23$	−3.58258	−0.747019	−0.373509	+	0.927626i	$0.621846\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	10.1652	1.88762	0.943811	−	0.330487i	$-0.107213\pi$
$29$	10.1652	1.88762	0.943811	+	0.330487i	$0.107213\pi$
$30$	0	0
$31$	5.58258	1.00266	0.501330	−	0.865256i	$-0.332844\pi$
$31$	5.58258	1.00266	0.501330	+	0.865256i	$0.332844\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.16515	1.11901	0.559504	−	0.828827i	$-0.310992\pi$
$41$	7.16515	1.11901	0.559504	+	0.828827i	$0.310992\pi$
$42$	0	0
$43$	7.58258	1.15633	0.578166	−	0.815919i	$-0.303769\pi$
$43$	7.58258	1.15633	0.578166	+	0.815919i	$0.303769\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	−10.5826	−1.54363	−0.771814	−	0.635849i	$-0.780650\pi$
$47$	−10.5826	−1.54363	−0.771814	+	0.635849i	$0.780650\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.58258	−0.221605
$52$	0	0
$53$	−0.417424	−0.0573376	−0.0286688	−	0.999589i	$-0.509127\pi$
$53$	−0.417424	−0.0573376	−0.0286688	+	0.999589i	$0.509127\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	−2.58258	−0.342071
$58$	0	0
$59$	4.58258	0.596601	0.298300	−	0.954472i	$-0.403580\pi$
$59$	4.58258	0.596601	0.298300	+	0.954472i	$0.403580\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	0.582576	0.0711729	0.0355865	−	0.999367i	$-0.488670\pi$
$67$	0.582576	0.0711729	0.0355865	+	0.999367i	$0.488670\pi$
$68$	0	0
$69$	−3.58258	−0.431291
$70$	0	0
$71$	7.16515	0.850347	0.425174	−	0.905112i	$-0.360213\pi$
$71$	7.16515	0.850347	0.425174	+	0.905112i	$0.360213\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	11.1652	1.25618	0.628089	−	0.778142i	$-0.283837\pi$
$79$	11.1652	1.25618	0.628089	+	0.778142i	$0.283837\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.41742	0.265347	0.132673	−	0.991160i	$-0.457644\pi$
$83$	2.41742	0.265347	0.132673	+	0.991160i	$0.457644\pi$
$84$	0	0
$85$	−4.74773	−0.514963
$86$	0	0
$87$	10.1652	1.08982
$88$	0	0
$89$	−9.16515	−0.971504	−0.485752	−	0.874097i	$-0.661454\pi$
$89$	−9.16515	−0.971504	−0.485752	+	0.874097i	$0.661454\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	5.58258	0.578886
$94$	0	0
$95$	−7.74773	−0.794900
$96$	0	0
$97$	−11.5826	−1.17603	−0.588016	−	0.808849i	$-0.700091\pi$
$97$	−11.5826	−1.17603	−0.588016	+	0.808849i	$0.700091\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	2.41742	0.240543	0.120271	−	0.992741i	$-0.461624\pi$
$101$	2.41742	0.240543	0.120271	+	0.992741i	$0.461624\pi$
$102$	0	0
$103$	−17.1652	−1.69133	−0.845666	−	0.533712i	$-0.820797\pi$
$103$	−17.1652	−1.69133	−0.845666	+	0.533712i	$0.820797\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	3.41742	0.330375	0.165187	−	0.986262i	$-0.447177\pi$
$107$	3.41742	0.330375	0.165187	+	0.986262i	$0.447177\pi$
$108$	0	0
$109$	−5.58258	−0.534714	−0.267357	−	0.963598i	$-0.586150\pi$
$109$	−5.58258	−0.534714	−0.267357	+	0.963598i	$0.586150\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	−9.16515	−0.862185	−0.431092	−	0.902308i	$-0.641872\pi$
$113$	−9.16515	−0.862185	−0.431092	+	0.902308i	$0.641872\pi$
$114$	0	0
$115$	−10.7477	−1.00223
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	1.58258	0.145074
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.16515	0.646060
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	2.41742	0.214512	0.107256	−	0.994231i	$-0.465794\pi$
$127$	2.41742	0.214512	0.107256	+	0.994231i	$0.465794\pi$
$128$	0	0
$129$	7.58258	0.667609
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	2.58258	0.223938
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	−2.41742	−0.206534	−0.103267	−	0.994654i	$-0.532930\pi$
$137$	−2.41742	−0.206534	−0.103267	+	0.994654i	$0.532930\pi$
$138$	0	0
$139$	7.16515	0.607740	0.303870	−	0.952713i	$-0.401721\pi$
$139$	7.16515	0.607740	0.303870	+	0.952713i	$0.401721\pi$
$140$	0	0
$141$	−10.5826	−0.891214
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	30.4955	2.53251
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−12.1652	−0.996608	−0.498304	−	0.867002i	$-0.666044\pi$
$149$	−12.1652	−0.996608	−0.498304	+	0.867002i	$0.666044\pi$
$150$	0	0
$151$	5.58258	0.454304	0.227152	−	0.973859i	$-0.427059\pi$
$151$	5.58258	0.454304	0.227152	+	0.973859i	$0.427059\pi$
$152$	0	0
$153$	−1.58258	−0.127944
$154$	0	0
$155$	16.7477	1.34521
$156$	0	0
$157$	0.834849	0.0666282	0.0333141	−	0.999445i	$-0.489394\pi$
$157$	0.834849	0.0666282	0.0333141	+	0.999445i	$0.489394\pi$
$158$	0	0
$159$	−0.417424	−0.0331039
$160$	0	0
$161$	3.58258	0.282347
$162$	0	0
$163$	0.582576	0.0456309	0.0228154	−	0.999740i	$-0.492737\pi$
$163$	0.582576	0.0456309	0.0228154	+	0.999740i	$0.492737\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	−22.7477	−1.76027	−0.880136	−	0.474722i	$-0.842549\pi$
$167$	−22.7477	−1.76027	−0.880136	+	0.474722i	$0.842549\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−2.58258	−0.197494
$172$	0	0
$173$	11.1652	0.848871	0.424435	−	0.905458i	$-0.360473\pi$
$173$	11.1652	0.848871	0.424435	+	0.905458i	$0.360473\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	4.58258	0.344447
$178$	0	0
$179$	22.3303	1.66905	0.834523	−	0.550974i	$-0.185744\pi$
$179$	22.3303	1.66905	0.834523	+	0.550974i	$0.185744\pi$
$180$	0	0
$181$	3.58258	0.266291	0.133145	−	0.991097i	$-0.457492\pi$
$181$	3.58258	0.266291	0.133145	+	0.991097i	$0.457492\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	−1.58258	−0.115729
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−2.41742	−0.174919	−0.0874593	−	0.996168i	$-0.527875\pi$
$191$	−2.41742	−0.174919	−0.0874593	+	0.996168i	$0.527875\pi$
$192$	0	0
$193$	−11.5826	−0.833732	−0.416866	−	0.908968i	$-0.636872\pi$
$193$	−11.5826	−0.833732	−0.416866	+	0.908968i	$0.636872\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	0	0
$197$	−13.1652	−0.937978	−0.468989	−	0.883204i	$-0.655382\pi$
$197$	−13.1652	−0.937978	−0.468989	+	0.883204i	$0.655382\pi$
$198$	0	0
$199$	0.417424	0.0295904	0.0147952	−	0.999891i	$-0.495290\pi$
$199$	0.417424	0.0295904	0.0147952	+	0.999891i	$0.495290\pi$
$200$	0	0
$201$	0.582576	0.0410917
$202$	0	0
$203$	−10.1652	−0.713454
$204$	0	0
$205$	21.4955	1.50131
$206$	0	0
$207$	−3.58258	−0.249006
$208$	0	0
$209$	−2.58258	−0.178640
$210$	0	0
$211$	−5.16515	−0.355584	−0.177792	−	0.984068i	$-0.556895\pi$
$211$	−5.16515	−0.355584	−0.177792	+	0.984068i	$0.556895\pi$
$212$	0	0
$213$	7.16515	0.490948
$214$	0	0
$215$	22.7477	1.55138
$216$	0	0
$217$	−5.58258	−0.378970
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	−1.58258	−0.106456
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	−26.7477	−1.76754	−0.883770	−	0.467922i	$-0.845003\pi$
$229$	−26.7477	−1.76754	−0.883770	+	0.467922i	$0.845003\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	−31.7477	−2.07099
$236$	0	0
$237$	11.1652	0.725255
$238$	0	0
$239$	−7.41742	−0.479793	−0.239897	−	0.970798i	$-0.577114\pi$
$239$	−7.41742	−0.479793	−0.239897	+	0.970798i	$0.577114\pi$
$240$	0	0
$241$	8.16515	0.525964	0.262982	−	0.964801i	$-0.415294\pi$
$241$	8.16515	0.525964	0.262982	+	0.964801i	$0.415294\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	−2.58258	−0.164325
$248$	0	0
$249$	2.41742	0.153198
$250$	0	0
$251$	−16.5826	−1.04668	−0.523341	−	0.852123i	$-0.675315\pi$
$251$	−16.5826	−1.04668	−0.523341	+	0.852123i	$0.675315\pi$
$252$	0	0
$253$	−3.58258	−0.225235
$254$	0	0
$255$	−4.74773	−0.297314
$256$	0	0
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	10.1652	0.629207
$262$	0	0
$263$	−22.9129	−1.41287	−0.706434	−	0.707779i	$-0.749697\pi$
$263$	−22.9129	−1.41287	−0.706434	+	0.707779i	$0.749697\pi$
$264$	0	0
$265$	−1.25227	−0.0769265
$266$	0	0
$267$	−9.16515	−0.560898
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−14.5826	−0.885828	−0.442914	−	0.896564i	$-0.646055\pi$
$271$	−14.5826	−0.885828	−0.442914	+	0.896564i	$0.646055\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−0.834849	−0.0501612	−0.0250806	−	0.999685i	$-0.507984\pi$
$277$	−0.834849	−0.0501612	−0.0250806	+	0.999685i	$0.507984\pi$
$278$	0	0
$279$	5.58258	0.334220
$280$	0	0
$281$	9.33030	0.556599	0.278300	−	0.960494i	$-0.410229\pi$
$281$	9.33030	0.556599	0.278300	+	0.960494i	$0.410229\pi$
$282$	0	0
$283$	−0.252273	−0.0149961	−0.00749803	−	0.999972i	$-0.502387\pi$
$283$	−0.252273	−0.0149961	−0.00749803	+	0.999972i	$0.502387\pi$
$284$	0	0
$285$	−7.74773	−0.458936
$286$	0	0
$287$	−7.16515	−0.422946
$288$	0	0
$289$	−14.4955	−0.852674
$290$	0	0
$291$	−11.5826	−0.678983
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	13.7477	0.800424
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−3.58258	−0.207186
$300$	0	0
$301$	−7.58258	−0.437052
$302$	0	0
$303$	2.41742	0.138877
$304$	0	0
$305$	30.0000	1.71780
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−17.1652	−0.976491
$310$	0	0
$311$	22.3303	1.26624	0.633118	−	0.774056i	$-0.281775\pi$
$311$	22.3303	1.26624	0.633118	+	0.774056i	$0.281775\pi$
$312$	0	0
$313$	10.4174	0.588828	0.294414	−	0.955678i	$-0.404876\pi$
$313$	10.4174	0.588828	0.294414	+	0.955678i	$0.404876\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	−31.5826	−1.77385	−0.886927	−	0.461909i	$-0.847165\pi$
$317$	−31.5826	−1.77385	−0.886927	+	0.461909i	$0.847165\pi$
$318$	0	0
$319$	10.1652	0.569139
$320$	0	0
$321$	3.41742	0.190742
$322$	0	0
$323$	4.08712	0.227414
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−5.58258	−0.308717
$328$	0	0
$329$	10.5826	0.583436
$330$	0	0
$331$	−15.1652	−0.833552	−0.416776	−	0.909009i	$-0.636840\pi$
$331$	−15.1652	−0.833552	−0.416776	+	0.909009i	$0.636840\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	1.74773	0.0954885
$336$	0	0
$337$	8.41742	0.458526	0.229263	−	0.973364i	$-0.426368\pi$
$337$	8.41742	0.458526	0.229263	+	0.973364i	$0.426368\pi$
$338$	0	0
$339$	−9.16515	−0.497783
$340$	0	0
$341$	5.58258	0.302313
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−10.7477	−0.578638
$346$	0	0
$347$	10.3303	0.554560	0.277280	−	0.960789i	$-0.410567\pi$
$347$	10.3303	0.554560	0.277280	+	0.960789i	$0.410567\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	5.83485	0.310558	0.155279	−	0.987871i	$-0.450372\pi$
$353$	5.83485	0.310558	0.155279	+	0.987871i	$0.450372\pi$
$354$	0	0
$355$	21.4955	1.14086
$356$	0	0
$357$	1.58258	0.0837588
$358$	0	0
$359$	27.1652	1.43372	0.716861	−	0.697216i	$-0.245578\pi$
$359$	27.1652	1.43372	0.716861	+	0.697216i	$0.245578\pi$
$360$	0	0
$361$	−12.3303	−0.648963
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	21.0000	1.09919
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	7.16515	0.373003
$370$	0	0
$371$	0.417424	0.0216716
$372$	0	0
$373$	−7.25227	−0.375508	−0.187754	−	0.982216i	$-0.560121\pi$
$373$	−7.25227	−0.375508	−0.187754	+	0.982216i	$0.560121\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	10.1652	0.523532
$378$	0	0
$379$	3.41742	0.175541	0.0877706	−	0.996141i	$-0.472026\pi$
$379$	3.41742	0.175541	0.0877706	+	0.996141i	$0.472026\pi$
$380$	0	0
$381$	2.41742	0.123848
$382$	0	0
$383$	26.3303	1.34542	0.672708	−	0.739908i	$-0.265131\pi$
$383$	26.3303	1.34542	0.672708	+	0.739908i	$0.265131\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	7.58258	0.385444
$388$	0	0
$389$	10.3303	0.523767	0.261884	−	0.965099i	$-0.415656\pi$
$389$	10.3303	0.523767	0.261884	+	0.965099i	$0.415656\pi$
$390$	0	0
$391$	5.66970	0.286729
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	33.4955	1.68534
$396$	0	0
$397$	−22.4174	−1.12510	−0.562549	−	0.826764i	$-0.690179\pi$
$397$	−22.4174	−1.12510	−0.562549	+	0.826764i	$0.690179\pi$
$398$	0	0
$399$	2.58258	0.129290
$400$	0	0
$401$	−13.9129	−0.694776	−0.347388	−	0.937721i	$-0.612931\pi$
$401$	−13.9129	−0.694776	−0.347388	+	0.937721i	$0.612931\pi$
$402$	0	0
$403$	5.58258	0.278088
$404$	0	0
$405$	3.00000	0.149071
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	−28.3303	−1.40084	−0.700422	−	0.713729i	$-0.747005\pi$
$409$	−28.3303	−1.40084	−0.700422	+	0.713729i	$0.747005\pi$
$410$	0	0
$411$	−2.41742	−0.119243
$412$	0	0
$413$	−4.58258	−0.225494
$414$	0	0
$415$	7.25227	0.356000
$416$	0	0
$417$	7.16515	0.350879
$418$	0	0
$419$	6.58258	0.321580	0.160790	−	0.986989i	$-0.448596\pi$
$419$	6.58258	0.321580	0.160790	+	0.986989i	$0.448596\pi$
$420$	0	0
$421$	39.6606	1.93294	0.966470	−	0.256780i	$-0.0826617\pi$
$421$	39.6606	1.93294	0.966470	+	0.256780i	$0.0826617\pi$
$422$	0	0
$423$	−10.5826	−0.514542
$424$	0	0
$425$	−6.33030	−0.307065
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	9.74773	0.469531	0.234766	−	0.972052i	$-0.424568\pi$
$431$	9.74773	0.469531	0.234766	+	0.972052i	$0.424568\pi$
$432$	0	0
$433$	7.16515	0.344335	0.172168	−	0.985068i	$-0.444923\pi$
$433$	7.16515	0.344335	0.172168	+	0.985068i	$0.444923\pi$
$434$	0	0
$435$	30.4955	1.46215
$436$	0	0
$437$	9.25227	0.442596
$438$	0	0
$439$	−26.5826	−1.26872	−0.634359	−	0.773039i	$-0.718736\pi$
$439$	−26.5826	−1.26872	−0.634359	+	0.773039i	$0.718736\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.83485	0.229711	0.114855	−	0.993382i	$-0.463360\pi$
$443$	4.83485	0.229711	0.114855	+	0.993382i	$0.463360\pi$
$444$	0	0
$445$	−27.4955	−1.30341
$446$	0	0
$447$	−12.1652	−0.575392
$448$	0	0
$449$	18.3303	0.865060	0.432530	−	0.901619i	$-0.357621\pi$
$449$	18.3303	0.865060	0.432530	+	0.901619i	$0.357621\pi$
$450$	0	0
$451$	7.16515	0.337394
$452$	0	0
$453$	5.58258	0.262292
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	25.9129	1.21215	0.606077	−	0.795406i	$-0.292742\pi$
$457$	25.9129	1.21215	0.606077	+	0.795406i	$0.292742\pi$
$458$	0	0
$459$	−1.58258	−0.0738683
$460$	0	0
$461$	−18.3303	−0.853727	−0.426864	−	0.904316i	$-0.640382\pi$
$461$	−18.3303	−0.853727	−0.426864	+	0.904316i	$0.640382\pi$
$462$	0	0
$463$	0.582576	0.0270746	0.0135373	−	0.999908i	$-0.495691\pi$
$463$	0.582576	0.0270746	0.0135373	+	0.999908i	$0.495691\pi$
$464$	0	0
$465$	16.7477	0.776657
$466$	0	0
$467$	29.4174	1.36128	0.680638	−	0.732620i	$-0.261703\pi$
$467$	29.4174	1.36128	0.680638	+	0.732620i	$0.261703\pi$
$468$	0	0
$469$	−0.582576	−0.0269008
$470$	0	0
$471$	0.834849	0.0384678
$472$	0	0
$473$	7.58258	0.348647
$474$	0	0
$475$	−10.3303	−0.473987
$476$	0	0
$477$	−0.417424	−0.0191125
$478$	0	0
$479$	6.41742	0.293220	0.146610	−	0.989194i	$-0.453164\pi$
$479$	6.41742	0.293220	0.146610	+	0.989194i	$0.453164\pi$
$480$	0	0
$481$	1.00000	0.0455961
$482$	0	0
$483$	3.58258	0.163013
$484$	0	0
$485$	−34.7477	−1.57781
$486$	0	0
$487$	26.3303	1.19314	0.596570	−	0.802561i	$-0.296530\pi$
$487$	26.3303	1.19314	0.596570	+	0.802561i	$0.296530\pi$
$488$	0	0
$489$	0.582576	0.0263450
$490$	0	0
$491$	−22.9129	−1.03404	−0.517022	−	0.855972i	$-0.672959\pi$
$491$	−22.9129	−1.03404	−0.517022	+	0.855972i	$0.672959\pi$
$492$	0	0
$493$	−16.0871	−0.724528
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	−7.16515	−0.321401
$498$	0	0
$499$	−14.2523	−0.638019	−0.319010	−	0.947751i	$-0.603350\pi$
$499$	−14.2523	−0.638019	−0.319010	+	0.947751i	$0.603350\pi$
$500$	0	0
$501$	−22.7477	−1.01629
$502$	0	0
$503$	−26.7477	−1.19262	−0.596311	−	0.802753i	$-0.703368\pi$
$503$	−26.7477	−1.19262	−0.596311	+	0.802753i	$0.703368\pi$
$504$	0	0
$505$	7.25227	0.322722
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−7.00000	−0.309662
$512$	0	0
$513$	−2.58258	−0.114024
$514$	0	0
$515$	−51.4955	−2.26916
$516$	0	0
$517$	−10.5826	−0.465421
$518$	0	0
$519$	11.1652	0.490096
$520$	0	0
$521$	34.1652	1.49680	0.748401	−	0.663246i	$-0.230822\pi$
$521$	34.1652	1.49680	0.748401	+	0.663246i	$0.230822\pi$
$522$	0	0
$523$	−24.5826	−1.07492	−0.537460	−	0.843289i	$-0.680616\pi$
$523$	−24.5826	−1.07492	−0.537460	+	0.843289i	$0.680616\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−8.83485	−0.384852
$528$	0	0
$529$	−10.1652	−0.441963
$530$	0	0
$531$	4.58258	0.198867
$532$	0	0
$533$	7.16515	0.310357
$534$	0	0
$535$	10.2523	0.443244
$536$	0	0
$537$	22.3303	0.963624
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−18.3303	−0.788081	−0.394041	−	0.919093i	$-0.628923\pi$
$541$	−18.3303	−0.788081	−0.394041	+	0.919093i	$0.628923\pi$
$542$	0	0
$543$	3.58258	0.153743
$544$	0	0
$545$	−16.7477	−0.717394
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−26.2523	−1.11838
$552$	0	0
$553$	−11.1652	−0.474791
$554$	0	0
$555$	3.00000	0.127343
$556$	0	0
$557$	−27.3303	−1.15802	−0.579011	−	0.815320i	$-0.696561\pi$
$557$	−27.3303	−1.15802	−0.579011	+	0.815320i	$0.696561\pi$
$558$	0	0
$559$	7.58258	0.320709
$560$	0	0
$561$	−1.58258	−0.0668164
$562$	0	0
$563$	28.4174	1.19765	0.598826	−	0.800879i	$-0.295634\pi$
$563$	28.4174	1.19765	0.598826	+	0.800879i	$0.295634\pi$
$564$	0	0
$565$	−27.4955	−1.15674
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	46.6606	1.95611	0.978057	−	0.208337i	$-0.0668050\pi$
$569$	46.6606	1.95611	0.978057	+	0.208337i	$0.0668050\pi$
$570$	0	0
$571$	−47.1652	−1.97380	−0.986900	−	0.161333i	$-0.948421\pi$
$571$	−47.1652	−1.97380	−0.986900	+	0.161333i	$0.948421\pi$
$572$	0	0
$573$	−2.41742	−0.100989
$574$	0	0
$575$	−14.3303	−0.597615
$576$	0	0
$577$	23.9129	0.995506	0.497753	−	0.867319i	$-0.334159\pi$
$577$	23.9129	0.995506	0.497753	+	0.867319i	$0.334159\pi$
$578$	0	0
$579$	−11.5826	−0.481355
$580$	0	0
$581$	−2.41742	−0.100292
$582$	0	0
$583$	−0.417424	−0.0172879
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	−10.2523	−0.423157	−0.211578	−	0.977361i	$-0.567860\pi$
$587$	−10.2523	−0.423157	−0.211578	+	0.977361i	$0.567860\pi$
$588$	0	0
$589$	−14.4174	−0.594060
$590$	0	0
$591$	−13.1652	−0.541542
$592$	0	0
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	0	0
$595$	4.74773	0.194638
$596$	0	0
$597$	0.417424	0.0170840
$598$	0	0
$599$	11.1652	0.456196	0.228098	−	0.973638i	$-0.426749\pi$
$599$	11.1652	0.456196	0.228098	+	0.973638i	$0.426749\pi$
$600$	0	0
$601$	−30.4955	−1.24394	−0.621968	−	0.783043i	$-0.713667\pi$
$601$	−30.4955	−1.24394	−0.621968	+	0.783043i	$0.713667\pi$
$602$	0	0
$603$	0.582576	0.0237243
$604$	0	0
$605$	3.00000	0.121967
$606$	0	0
$607$	−5.74773	−0.233293	−0.116647	−	0.993173i	$-0.537214\pi$
$607$	−5.74773	−0.233293	−0.116647	+	0.993173i	$0.537214\pi$
$608$	0	0
$609$	−10.1652	−0.411913
$610$	0	0
$611$	−10.5826	−0.428125
$612$	0	0
$613$	0.747727	0.0302004	0.0151002	−	0.999886i	$-0.495193\pi$
$613$	0.747727	0.0302004	0.0151002	+	0.999886i	$0.495193\pi$
$614$	0	0
$615$	21.4955	0.866780
$616$	0	0
$617$	21.1652	0.852077	0.426038	−	0.904705i	$-0.359909\pi$
$617$	21.1652	0.852077	0.426038	+	0.904705i	$0.359909\pi$
$618$	0	0
$619$	−35.0780	−1.40991	−0.704953	−	0.709254i	$-0.749032\pi$
$619$	−35.0780	−1.40991	−0.704953	+	0.709254i	$0.749032\pi$
$620$	0	0
$621$	−3.58258	−0.143764
$622$	0	0
$623$	9.16515	0.367194
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−2.58258	−0.103138
$628$	0	0
$629$	−1.58258	−0.0631014
$630$	0	0
$631$	−4.83485	−0.192472	−0.0962361	−	0.995359i	$-0.530680\pi$
$631$	−4.83485	−0.192472	−0.0962361	+	0.995359i	$0.530680\pi$
$632$	0	0
$633$	−5.16515	−0.205296
$634$	0	0
$635$	7.25227	0.287798
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	7.16515	0.283449
$640$	0	0
$641$	34.4174	1.35941	0.679703	−	0.733487i	$-0.262109\pi$
$641$	34.4174	1.35941	0.679703	+	0.733487i	$0.262109\pi$
$642$	0	0
$643$	44.2432	1.74478	0.872390	−	0.488810i	$-0.162569\pi$
$643$	44.2432	1.74478	0.872390	+	0.488810i	$0.162569\pi$
$644$	0	0
$645$	22.7477	0.895691
$646$	0	0
$647$	−34.9129	−1.37257	−0.686283	−	0.727334i	$-0.740759\pi$
$647$	−34.9129	−1.37257	−0.686283	+	0.727334i	$0.740759\pi$
$648$	0	0
$649$	4.58258	0.179882
$650$	0	0
$651$	−5.58258	−0.218798
$652$	0	0
$653$	6.33030	0.247724	0.123862	−	0.992299i	$-0.460472\pi$
$653$	6.33030	0.247724	0.123862	+	0.992299i	$0.460472\pi$
$654$	0	0
$655$	48.0000	1.87552
$656$	0	0
$657$	7.00000	0.273096
$658$	0	0
$659$	19.4174	0.756395	0.378198	−	0.925725i	$-0.376544\pi$
$659$	19.4174	0.756395	0.378198	+	0.925725i	$0.376544\pi$
$660$	0	0
$661$	25.0780	0.975422	0.487711	−	0.873005i	$-0.337832\pi$
$661$	25.0780	0.975422	0.487711	+	0.873005i	$0.337832\pi$
$662$	0	0
$663$	−1.58258	−0.0614621
$664$	0	0
$665$	7.74773	0.300444
$666$	0	0
$667$	−36.4174	−1.41009
$668$	0	0
$669$	−6.00000	−0.231973
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	38.7477	1.49362	0.746808	−	0.665040i	$-0.231586\pi$
$673$	38.7477	1.49362	0.746808	+	0.665040i	$0.231586\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−26.8348	−1.03135	−0.515674	−	0.856785i	$-0.672458\pi$
$677$	−26.8348	−1.03135	−0.515674	+	0.856785i	$0.672458\pi$
$678$	0	0
$679$	11.5826	0.444498
$680$	0	0
$681$	22.0000	0.843042
$682$	0	0
$683$	−31.0780	−1.18917	−0.594584	−	0.804034i	$-0.702683\pi$
$683$	−31.0780	−1.18917	−0.594584	+	0.804034i	$0.702683\pi$
$684$	0	0
$685$	−7.25227	−0.277095
$686$	0	0
$687$	−26.7477	−1.02049
$688$	0	0
$689$	−0.417424	−0.0159026
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	21.4955	0.815369
$696$	0	0
$697$	−11.3394	−0.429510
$698$	0	0
$699$	−14.0000	−0.529529
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	−2.58258	−0.0974037
$704$	0	0
$705$	−31.7477	−1.19569
$706$	0	0
$707$	−2.41742	−0.0909166
$708$	0	0
$709$	−45.6606	−1.71482	−0.857410	−	0.514634i	$-0.827928\pi$
$709$	−45.6606	−1.71482	−0.857410	+	0.514634i	$0.827928\pi$
$710$	0	0
$711$	11.1652	0.418726
$712$	0	0
$713$	−20.0000	−0.749006
$714$	0	0
$715$	3.00000	0.112194
$716$	0	0
$717$	−7.41742	−0.277009
$718$	0	0
$719$	−50.0780	−1.86760	−0.933798	−	0.357801i	$-0.883526\pi$
$719$	−50.0780	−1.86760	−0.933798	+	0.357801i	$0.883526\pi$
$720$	0	0
$721$	17.1652	0.639264
$722$	0	0
$723$	8.16515	0.303665
$724$	0	0
$725$	40.6606	1.51010
$726$	0	0
$727$	−29.9129	−1.10941	−0.554704	−	0.832048i	$-0.687168\pi$
$727$	−29.9129	−1.10941	−0.554704	+	0.832048i	$0.687168\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	0.582576	0.0214595
$738$	0	0
$739$	−13.9129	−0.511794	−0.255897	−	0.966704i	$-0.582371\pi$
$739$	−13.9129	−0.511794	−0.255897	+	0.966704i	$0.582371\pi$
$740$	0	0
$741$	−2.58258	−0.0948733
$742$	0	0
$743$	−29.2432	−1.07283	−0.536414	−	0.843955i	$-0.680221\pi$
$743$	−29.2432	−1.07283	−0.536414	+	0.843955i	$0.680221\pi$
$744$	0	0
$745$	−36.4955	−1.33709
$746$	0	0
$747$	2.41742	0.0884489
$748$	0	0
$749$	−3.41742	−0.124870
$750$	0	0
$751$	−36.9129	−1.34697	−0.673485	−	0.739201i	$-0.735203\pi$
$751$	−36.9129	−1.34697	−0.673485	+	0.739201i	$0.735203\pi$
$752$	0	0
$753$	−16.5826	−0.604303
$754$	0	0
$755$	16.7477	0.609512
$756$	0	0
$757$	−9.33030	−0.339116	−0.169558	−	0.985520i	$-0.554234\pi$
$757$	−9.33030	−0.339116	−0.169558	+	0.985520i	$0.554234\pi$
$758$	0	0
$759$	−3.58258	−0.130039
$760$	0	0
$761$	5.66970	0.205526	0.102763	−	0.994706i	$-0.467232\pi$
$761$	5.66970	0.205526	0.102763	+	0.994706i	$0.467232\pi$
$762$	0	0
$763$	5.58258	0.202103
$764$	0	0
$765$	−4.74773	−0.171654
$766$	0	0
$767$	4.58258	0.165467
$768$	0	0
$769$	−48.4955	−1.74879	−0.874395	−	0.485214i	$-0.838742\pi$
$769$	−48.4955	−1.74879	−0.874395	+	0.485214i	$0.838742\pi$
$770$	0	0
$771$	19.0000	0.684268
$772$	0	0
$773$	−12.1652	−0.437550	−0.218775	−	0.975775i	$-0.570206\pi$
$773$	−12.1652	−0.437550	−0.218775	+	0.975775i	$0.570206\pi$
$774$	0	0
$775$	22.3303	0.802128
$776$	0	0
$777$	−1.00000	−0.0358748
$778$	0	0
$779$	−18.5045	−0.662994
$780$	0	0
$781$	7.16515	0.256389
$782$	0	0
$783$	10.1652	0.363273
$784$	0	0
$785$	2.50455	0.0893911
$786$	0	0
$787$	−29.4174	−1.04862	−0.524309	−	0.851528i	$-0.675676\pi$
$787$	−29.4174	−1.04862	−0.524309	+	0.851528i	$0.675676\pi$
$788$	0	0
$789$	−22.9129	−0.815720
$790$	0	0
$791$	9.16515	0.325875
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	−1.25227	−0.0444135
$796$	0	0
$797$	2.49545	0.0883935	0.0441968	−	0.999023i	$-0.485927\pi$
$797$	2.49545	0.0883935	0.0441968	+	0.999023i	$0.485927\pi$
$798$	0	0
$799$	16.7477	0.592492
$800$	0	0
$801$	−9.16515	−0.323835
$802$	0	0
$803$	7.00000	0.247025
$804$	0	0
$805$	10.7477	0.378808
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	−27.3303	−0.960882	−0.480441	−	0.877027i	$-0.659523\pi$
$809$	−27.3303	−0.960882	−0.480441	+	0.877027i	$0.659523\pi$
$810$	0	0
$811$	29.7477	1.04458	0.522292	−	0.852767i	$-0.325077\pi$
$811$	29.7477	1.04458	0.522292	+	0.852767i	$0.325077\pi$
$812$	0	0
$813$	−14.5826	−0.511433
$814$	0	0
$815$	1.74773	0.0612202
$816$	0	0
$817$	−19.5826	−0.685108
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−47.0000	−1.64031	−0.820156	−	0.572140i	$-0.806113\pi$
$821$	−47.0000	−1.64031	−0.820156	+	0.572140i	$0.806113\pi$
$822$	0	0
$823$	21.4174	0.746564	0.373282	−	0.927718i	$-0.378232\pi$
$823$	21.4174	0.746564	0.373282	+	0.927718i	$0.378232\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	36.9129	1.28359	0.641793	−	0.766878i	$-0.278191\pi$
$827$	36.9129	1.28359	0.641793	+	0.766878i	$0.278191\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	−0.834849	−0.0289606
$832$	0	0
$833$	−1.58258	−0.0548330
$834$	0	0
$835$	−68.2432	−2.36165
$836$	0	0
$837$	5.58258	0.192962
$838$	0	0
$839$	−52.9129	−1.82676	−0.913378	−	0.407113i	$-0.866535\pi$
$839$	−52.9129	−1.82676	−0.913378	+	0.407113i	$0.866535\pi$
$840$	0	0
$841$	74.3303	2.56311
$842$	0	0
$843$	9.33030	0.321353
$844$	0	0
$845$	−36.0000	−1.23844
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−0.252273	−0.00865798
$850$	0	0
$851$	−3.58258	−0.122809
$852$	0	0
$853$	−1.16515	−0.0398940	−0.0199470	−	0.999801i	$-0.506350\pi$
$853$	−1.16515	−0.0398940	−0.0199470	+	0.999801i	$0.506350\pi$
$854$	0	0
$855$	−7.74773	−0.264967
$856$	0	0
$857$	−44.3303	−1.51429	−0.757147	−	0.653244i	$-0.773407\pi$
$857$	−44.3303	−1.51429	−0.757147	+	0.653244i	$0.773407\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	0	0
$861$	−7.16515	−0.244188
$862$	0	0
$863$	−23.5826	−0.802760	−0.401380	−	0.915912i	$-0.631469\pi$
$863$	−23.5826	−0.802760	−0.401380	+	0.915912i	$0.631469\pi$
$864$	0	0
$865$	33.4955	1.13888
$866$	0	0
$867$	−14.4955	−0.492291
$868$	0	0
$869$	11.1652	0.378752
$870$	0	0
$871$	0.582576	0.0197398
$872$	0	0
$873$	−11.5826	−0.392011
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−45.4955	−1.53627	−0.768136	−	0.640287i	$-0.778816\pi$
$877$	−45.4955	−1.53627	−0.768136	+	0.640287i	$0.778816\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35.6606	1.20144	0.600718	−	0.799461i	$-0.294881\pi$
$881$	35.6606	1.20144	0.600718	+	0.799461i	$0.294881\pi$
$882$	0	0
$883$	28.2523	0.950765	0.475382	−	0.879779i	$-0.342310\pi$
$883$	28.2523	0.950765	0.475382	+	0.879779i	$0.342310\pi$
$884$	0	0
$885$	13.7477	0.462125
$886$	0	0
$887$	−11.2523	−0.377814	−0.188907	−	0.981995i	$-0.560495\pi$
$887$	−11.2523	−0.377814	−0.188907	+	0.981995i	$0.560495\pi$
$888$	0	0
$889$	−2.41742	−0.0810778
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	27.3303	0.914574
$894$	0	0
$895$	66.9909	2.23926
$896$	0	0
$897$	−3.58258	−0.119619
$898$	0	0
$899$	56.7477	1.89264
$900$	0	0
$901$	0.660606	0.0220080
$902$	0	0
$903$	−7.58258	−0.252332
$904$	0	0
$905$	10.7477	0.357267
$906$	0	0
$907$	−5.66970	−0.188259	−0.0941296	−	0.995560i	$-0.530007\pi$
$907$	−5.66970	−0.188259	−0.0941296	+	0.995560i	$0.530007\pi$
$908$	0	0
$909$	2.41742	0.0801809
$910$	0	0
$911$	−51.4955	−1.70612	−0.853060	−	0.521812i	$-0.825256\pi$
$911$	−51.4955	−1.70612	−0.853060	+	0.521812i	$0.825256\pi$
$912$	0	0
$913$	2.41742	0.0800051
$914$	0	0
$915$	30.0000	0.991769
$916$	0	0
$917$	−16.0000	−0.528367
$918$	0	0
$919$	53.9129	1.77842	0.889211	−	0.457498i	$-0.151254\pi$
$919$	53.9129	1.77842	0.889211	+	0.457498i	$0.151254\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.16515	0.235844
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	−17.1652	−0.563778
$928$	0	0
$929$	15.3303	0.502971	0.251485	−	0.967861i	$-0.419081\pi$
$929$	15.3303	0.502971	0.251485	+	0.967861i	$0.419081\pi$
$930$	0	0
$931$	−2.58258	−0.0846405
$932$	0	0
$933$	22.3303	0.731061
$934$	0	0
$935$	−4.74773	−0.155267
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	10.4174	0.339960
$940$	0	0
$941$	16.8348	0.548800	0.274400	−	0.961616i	$-0.411521\pi$
$941$	16.8348	0.548800	0.274400	+	0.961616i	$0.411521\pi$
$942$	0	0
$943$	−25.6697	−0.835920
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	−26.8348	−0.872015	−0.436008	−	0.899943i	$-0.643608\pi$
$947$	−26.8348	−0.872015	−0.436008	+	0.899943i	$0.643608\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	0	0
$951$	−31.5826	−1.02414
$952$	0	0
$953$	−9.83485	−0.318582	−0.159291	−	0.987232i	$-0.550921\pi$
$953$	−9.83485	−0.318582	−0.159291	+	0.987232i	$0.550921\pi$
$954$	0	0
$955$	−7.25227	−0.234678
$956$	0	0
$957$	10.1652	0.328593
$958$	0	0
$959$	2.41742	0.0780627
$960$	0	0
$961$	0.165151	0.00532746
$962$	0	0
$963$	3.41742	0.110125
$964$	0	0
$965$	−34.7477	−1.11857
$966$	0	0
$967$	−5.16515	−0.166100	−0.0830500	−	0.996545i	$-0.526466\pi$
$967$	−5.16515	−0.166100	−0.0830500	+	0.996545i	$0.526466\pi$
$968$	0	0
$969$	4.08712	0.131297
$970$	0	0
$971$	3.41742	0.109670	0.0548352	−	0.998495i	$-0.482537\pi$
$971$	3.41742	0.109670	0.0548352	+	0.998495i	$0.482537\pi$
$972$	0	0
$973$	−7.16515	−0.229704
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	50.7477	1.62356	0.811782	−	0.583961i	$-0.198498\pi$
$977$	50.7477	1.62356	0.811782	+	0.583961i	$0.198498\pi$
$978$	0	0
$979$	−9.16515	−0.292920
$980$	0	0
$981$	−5.58258	−0.178238
$982$	0	0
$983$	23.1652	0.738854	0.369427	−	0.929260i	$-0.379554\pi$
$983$	23.1652	0.738854	0.369427	+	0.929260i	$0.379554\pi$
$984$	0	0
$985$	−39.4955	−1.25843
$986$	0	0
$987$	10.5826	0.336847
$988$	0	0
$989$	−27.1652	−0.863802
$990$	0	0
$991$	47.7477	1.51676	0.758378	−	0.651815i	$-0.225992\pi$
$991$	47.7477	1.51676	0.758378	+	0.651815i	$0.225992\pi$
$992$	0	0
$993$	−15.1652	−0.481252
$994$	0	0
$995$	1.25227	0.0396997
$996$	0	0
$997$	−62.6606	−1.98448	−0.992241	−	0.124332i	$-0.960321\pi$
$997$	−62.6606	−1.98448	−0.992241	+	0.124332i	$0.960321\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bl.1.1		2
4.3	odd	2		231.2.a.b.1.1	✓	2
12.11	even	2		693.2.a.j.1.2		2
20.19	odd	2		5775.2.a.bn.1.2		2
28.27	even	2		1617.2.a.o.1.1		2
44.43	even	2		2541.2.a.z.1.2		2
84.83	odd	2		4851.2.a.ba.1.2		2
132.131	odd	2		7623.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.b.1.1	✓	2	4.3	odd	2
693.2.a.j.1.2		2	12.11	even	2
1617.2.a.o.1.1		2	28.27	even	2
2541.2.a.z.1.2		2	44.43	even	2
3696.2.a.bl.1.1		2	1.1	even	1	trivial
4851.2.a.ba.1.2		2	84.83	odd	2
5775.2.a.bn.1.2		2	20.19	odd	2
7623.2.a.bf.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} -1.58258 q^{17} -2.58258 q^{19} -1.00000 q^{21} -3.58258 q^{23} +4.00000 q^{25} +1.00000 q^{27} +10.1652 q^{29} +5.58258 q^{31} +1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} +1.00000 q^{39} +7.16515 q^{41} +7.58258 q^{43} +3.00000 q^{45} -10.5826 q^{47} +1.00000 q^{49} -1.58258 q^{51} -0.417424 q^{53} +3.00000 q^{55} -2.58258 q^{57} +4.58258 q^{59} +10.0000 q^{61} -1.00000 q^{63} +3.00000 q^{65} +0.582576 q^{67} -3.58258 q^{69} +7.16515 q^{71} +7.00000 q^{73} +4.00000 q^{75} -1.00000 q^{77} +11.1652 q^{79} +1.00000 q^{81} +2.41742 q^{83} -4.74773 q^{85} +10.1652 q^{87} -9.16515 q^{89} -1.00000 q^{91} +5.58258 q^{93} -7.74773 q^{95} -11.5826 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} + 6 q^{43} + 6 q^{45} - 12 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 6 q^{55} + 4 q^{57} + 20 q^{61} - 2 q^{63} + 6 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} + 14 q^{73} + 8 q^{75} - 2 q^{77} + 4 q^{79} + 2 q^{81} + 14 q^{83} + 18 q^{85} + 2 q^{87} - 2 q^{91} + 2 q^{93} + 12 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 6 * q^15 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 + 2 * q^27 + 2 * q^29 + 2 * q^31 + 2 * q^33 - 6 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 + 6 * q^43 + 6 * q^45 - 12 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 6 * q^55 + 4 * q^57 + 20 * q^61 - 2 * q^63 + 6 * q^65 - 8 * q^67 + 2 * q^69 - 4 * q^71 + 14 * q^73 + 8 * q^75 - 2 * q^77 + 4 * q^79 + 2 * q^81 + 14 * q^83 + 18 * q^85 + 2 * q^87 - 2 * q^91 + 2 * q^93 + 12 * q^95 - 14 * q^97 + 2 * q^99

Introduction

L-functions

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