LMFDB - Embedded newform 3696.2.a.bh.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1848)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +6.68466 q^{13} -3.56155 q^{15} -5.12311 q^{17} -5.56155 q^{19} +1.00000 q^{21} +3.12311 q^{23} +7.68466 q^{25} +1.00000 q^{27} -4.43845 q^{29} -4.00000 q^{31} -1.00000 q^{33} -3.56155 q^{35} -4.43845 q^{37} +6.68466 q^{39} +9.12311 q^{41} +7.12311 q^{43} -3.56155 q^{45} -4.68466 q^{47} +1.00000 q^{49} -5.12311 q^{51} -10.0000 q^{53} +3.56155 q^{55} -5.56155 q^{57} +1.56155 q^{59} -8.24621 q^{61} +1.00000 q^{63} -23.8078 q^{65} +9.56155 q^{67} +3.12311 q^{69} -6.24621 q^{71} -10.6847 q^{73} +7.68466 q^{75} -1.00000 q^{77} +14.2462 q^{79} +1.00000 q^{81} -8.00000 q^{83} +18.2462 q^{85} -4.43845 q^{87} +0.246211 q^{89} +6.68466 q^{91} -4.00000 q^{93} +19.8078 q^{95} +2.00000 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 2 q^{17} - 7 q^{19} + 2 q^{21} - 2 q^{23} + 3 q^{25} + 2 q^{27} - 13 q^{29} - 8 q^{31} - 2 q^{33} - 3 q^{35} - 13 q^{37} + q^{39} + 10 q^{41} + 6 q^{43} - 3 q^{45} + 3 q^{47} + 2 q^{49} - 2 q^{51} - 20 q^{53} + 3 q^{55} - 7 q^{57} - q^{59} + 2 q^{63} - 27 q^{65} + 15 q^{67} - 2 q^{69} + 4 q^{71} - 9 q^{73} + 3 q^{75} - 2 q^{77} + 12 q^{79} + 2 q^{81} - 16 q^{83} + 20 q^{85} - 13 q^{87} - 16 q^{89} + q^{91} - 8 q^{93} + 19 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 2 * q^17 - 7 * q^19 + 2 * q^21 - 2 * q^23 + 3 * q^25 + 2 * q^27 - 13 * q^29 - 8 * q^31 - 2 * q^33 - 3 * q^35 - 13 * q^37 + q^39 + 10 * q^41 + 6 * q^43 - 3 * q^45 + 3 * q^47 + 2 * q^49 - 2 * q^51 - 20 * q^53 + 3 * q^55 - 7 * q^57 - q^59 + 2 * q^63 - 27 * q^65 + 15 * q^67 - 2 * q^69 + 4 * q^71 - 9 * q^73 + 3 * q^75 - 2 * q^77 + 12 * q^79 + 2 * q^81 - 16 * q^83 + 20 * q^85 - 13 * q^87 - 16 * q^89 + q^91 - 8 * q^93 + 19 * q^95 + 4 * 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
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	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
$13$	6.68466	1.85399	0.926995	−	0.375073i	$-0.122382\pi$
$13$	6.68466	1.85399	0.926995	+	0.375073i	$0.122382\pi$
$14$	0	0
$15$	−3.56155	−0.919589
$16$	0	0
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	0	0
$19$	−5.56155	−1.27591	−0.637954	−	0.770075i	$-0.720219\pi$
$19$	−5.56155	−1.27591	−0.637954	+	0.770075i	$0.720219\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.43845	−0.824199	−0.412099	−	0.911139i	$-0.635204\pi$
$29$	−4.43845	−0.824199	−0.412099	+	0.911139i	$0.635204\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−3.56155	−0.602012
$36$	0	0
$37$	−4.43845	−0.729676	−0.364838	−	0.931071i	$-0.618876\pi$
$37$	−4.43845	−0.729676	−0.364838	+	0.931071i	$0.618876\pi$
$38$	0	0
$39$	6.68466	1.07040
$40$	0	0
$41$	9.12311	1.42479	0.712395	−	0.701779i	$-0.247611\pi$
$41$	9.12311	1.42479	0.712395	+	0.701779i	$0.247611\pi$
$42$	0	0
$43$	7.12311	1.08626	0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311	1.08626	0.543132	+	0.839648i	$0.317238\pi$
$44$	0	0
$45$	−3.56155	−0.530925
$46$	0	0
$47$	−4.68466	−0.683328	−0.341664	−	0.939822i	$-0.610990\pi$
$47$	−4.68466	−0.683328	−0.341664	+	0.939822i	$0.610990\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.12311	−0.717378
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	3.56155	0.480240
$56$	0	0
$57$	−5.56155	−0.736646
$58$	0	0
$59$	1.56155	0.203297	0.101648	−	0.994820i	$-0.467588\pi$
$59$	1.56155	0.203297	0.101648	+	0.994820i	$0.467588\pi$
$60$	0	0
$61$	−8.24621	−1.05582	−0.527910	−	0.849301i	$-0.677024\pi$
$61$	−8.24621	−1.05582	−0.527910	+	0.849301i	$0.677024\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−23.8078	−2.95299
$66$	0	0
$67$	9.56155	1.16813	0.584065	−	0.811707i	$-0.301461\pi$
$67$	9.56155	1.16813	0.584065	+	0.811707i	$0.301461\pi$
$68$	0	0
$69$	3.12311	0.375978
$70$	0	0
$71$	−6.24621	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$71$	−6.24621	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$72$	0	0
$73$	−10.6847	−1.25054	−0.625272	−	0.780407i	$-0.715012\pi$
$73$	−10.6847	−1.25054	−0.625272	+	0.780407i	$0.715012\pi$
$74$	0	0
$75$	7.68466	0.887348
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	14.2462	1.60282	0.801412	−	0.598113i	$-0.204082\pi$
$79$	14.2462	1.60282	0.801412	+	0.598113i	$0.204082\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	18.2462	1.97908
$86$	0	0
$87$	−4.43845	−0.475851
$88$	0	0
$89$	0.246211	0.0260983	0.0130492	−	0.999915i	$-0.495846\pi$
$89$	0.246211	0.0260983	0.0130492	+	0.999915i	$0.495846\pi$
$90$	0	0
$91$	6.68466	0.700743
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	19.8078	2.03223
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−13.1231	−1.30580	−0.652899	−	0.757445i	$-0.726447\pi$
$101$	−13.1231	−1.30580	−0.652899	+	0.757445i	$0.726447\pi$
$102$	0	0
$103$	0.876894	0.0864030	0.0432015	−	0.999066i	$-0.486244\pi$
$103$	0.876894	0.0864030	0.0432015	+	0.999066i	$0.486244\pi$
$104$	0	0
$105$	−3.56155	−0.347572
$106$	0	0
$107$	9.56155	0.924350	0.462175	−	0.886789i	$-0.347069\pi$
$107$	9.56155	0.924350	0.462175	+	0.886789i	$0.347069\pi$
$108$	0	0
$109$	−13.1231	−1.25697	−0.628483	−	0.777824i	$-0.716324\pi$
$109$	−13.1231	−1.25697	−0.628483	+	0.777824i	$0.716324\pi$
$110$	0	0
$111$	−4.43845	−0.421279
$112$	0	0
$113$	−20.2462	−1.90460	−0.952302	−	0.305158i	$-0.901291\pi$
$113$	−20.2462	−1.90460	−0.952302	+	0.305158i	$0.901291\pi$
$114$	0	0
$115$	−11.1231	−1.03723
$116$	0	0
$117$	6.68466	0.617997
$118$	0	0
$119$	−5.12311	−0.469634
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	9.12311	0.822603
$124$	0	0
$125$	−9.56155	−0.855211
$126$	0	0
$127$	−20.4924	−1.81841	−0.909204	−	0.416350i	$-0.863309\pi$
$127$	−20.4924	−1.81841	−0.909204	+	0.416350i	$0.863309\pi$
$128$	0	0
$129$	7.12311	0.627154
$130$	0	0
$131$	−12.4924	−1.09147	−0.545734	−	0.837958i	$-0.683749\pi$
$131$	−12.4924	−1.09147	−0.545734	+	0.837958i	$0.683749\pi$
$132$	0	0
$133$	−5.56155	−0.482248
$134$	0	0
$135$	−3.56155	−0.306530
$136$	0	0
$137$	11.3693	0.971346	0.485673	−	0.874140i	$-0.338575\pi$
$137$	11.3693	0.971346	0.485673	+	0.874140i	$0.338575\pi$
$138$	0	0
$139$	−1.75379	−0.148754	−0.0743772	−	0.997230i	$-0.523697\pi$
$139$	−1.75379	−0.148754	−0.0743772	+	0.997230i	$0.523697\pi$
$140$	0	0
$141$	−4.68466	−0.394519
$142$	0	0
$143$	−6.68466	−0.558999
$144$	0	0
$145$	15.8078	1.31276
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−18.6847	−1.53071	−0.765353	−	0.643610i	$-0.777436\pi$
$149$	−18.6847	−1.53071	−0.765353	+	0.643610i	$0.777436\pi$
$150$	0	0
$151$	−20.4924	−1.66765	−0.833825	−	0.552029i	$-0.813854\pi$
$151$	−20.4924	−1.66765	−0.833825	+	0.552029i	$0.813854\pi$
$152$	0	0
$153$	−5.12311	−0.414179
$154$	0	0
$155$	14.2462	1.14428
$156$	0	0
$157$	−10.8769	−0.868071	−0.434035	−	0.900896i	$-0.642911\pi$
$157$	−10.8769	−0.868071	−0.434035	+	0.900896i	$0.642911\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	3.12311	0.246135
$162$	0	0
$163$	−10.9309	−0.856172	−0.428086	−	0.903738i	$-0.640812\pi$
$163$	−10.9309	−0.856172	−0.428086	+	0.903738i	$0.640812\pi$
$164$	0	0
$165$	3.56155	0.277267
$166$	0	0
$167$	17.3693	1.34408	0.672039	−	0.740516i	$-0.265419\pi$
$167$	17.3693	1.34408	0.672039	+	0.740516i	$0.265419\pi$
$168$	0	0
$169$	31.6847	2.43728
$170$	0	0
$171$	−5.56155	−0.425303
$172$	0	0
$173$	−19.3693	−1.47262	−0.736311	−	0.676643i	$-0.763434\pi$
$173$	−19.3693	−1.47262	−0.736311	+	0.676643i	$0.763434\pi$
$174$	0	0
$175$	7.68466	0.580906
$176$	0	0
$177$	1.56155	0.117373
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−18.4924	−1.37453	−0.687265	−	0.726406i	$-0.741189\pi$
$181$	−18.4924	−1.37453	−0.687265	+	0.726406i	$0.741189\pi$
$182$	0	0
$183$	−8.24621	−0.609577
$184$	0	0
$185$	15.8078	1.16221
$186$	0	0
$187$	5.12311	0.374639
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−10.4924	−0.755261	−0.377631	−	0.925956i	$-0.623261\pi$
$193$	−10.4924	−0.755261	−0.377631	+	0.925956i	$0.623261\pi$
$194$	0	0
$195$	−23.8078	−1.70491
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	7.12311	0.504944	0.252472	−	0.967604i	$-0.418757\pi$
$199$	7.12311	0.504944	0.252472	+	0.967604i	$0.418757\pi$
$200$	0	0
$201$	9.56155	0.674420
$202$	0	0
$203$	−4.43845	−0.311518
$204$	0	0
$205$	−32.4924	−2.26937
$206$	0	0
$207$	3.12311	0.217071
$208$	0	0
$209$	5.56155	0.384701
$210$	0	0
$211$	15.1231	1.04112	0.520559	−	0.853826i	$-0.325724\pi$
$211$	15.1231	1.04112	0.520559	+	0.853826i	$0.325724\pi$
$212$	0	0
$213$	−6.24621	−0.427983
$214$	0	0
$215$	−25.3693	−1.73017
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−10.6847	−0.722002
$220$	0	0
$221$	−34.2462	−2.30365
$222$	0	0
$223$	−8.87689	−0.594441	−0.297220	−	0.954809i	$-0.596060\pi$
$223$	−8.87689	−0.594441	−0.297220	+	0.954809i	$0.596060\pi$
$224$	0	0
$225$	7.68466	0.512311
$226$	0	0
$227$	−23.6155	−1.56742	−0.783709	−	0.621128i	$-0.786675\pi$
$227$	−23.6155	−1.56742	−0.783709	+	0.621128i	$0.786675\pi$
$228$	0	0
$229$	0.246211	0.0162701	0.00813505	−	0.999967i	$-0.497411\pi$
$229$	0.246211	0.0162701	0.00813505	+	0.999967i	$0.497411\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	16.6847	1.08839
$236$	0	0
$237$	14.2462	0.925391
$238$	0	0
$239$	29.1771	1.88731	0.943654	−	0.330933i	$-0.107363\pi$
$239$	29.1771	1.88731	0.943654	+	0.330933i	$0.107363\pi$
$240$	0	0
$241$	8.43845	0.543568	0.271784	−	0.962358i	$-0.412386\pi$
$241$	8.43845	0.543568	0.271784	+	0.962358i	$0.412386\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.56155	−0.227539
$246$	0	0
$247$	−37.1771	−2.36552
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−9.56155	−0.603520	−0.301760	−	0.953384i	$-0.597574\pi$
$251$	−9.56155	−0.603520	−0.301760	+	0.953384i	$0.597574\pi$
$252$	0	0
$253$	−3.12311	−0.196348
$254$	0	0
$255$	18.2462	1.14262
$256$	0	0
$257$	−5.31534	−0.331562	−0.165781	−	0.986163i	$-0.553014\pi$
$257$	−5.31534	−0.331562	−0.165781	+	0.986163i	$0.553014\pi$
$258$	0	0
$259$	−4.43845	−0.275792
$260$	0	0
$261$	−4.43845	−0.274733
$262$	0	0
$263$	−4.19224	−0.258504	−0.129252	−	0.991612i	$-0.541258\pi$
$263$	−4.19224	−0.258504	−0.129252	+	0.991612i	$0.541258\pi$
$264$	0	0
$265$	35.6155	2.18784
$266$	0	0
$267$	0.246211	0.0150679
$268$	0	0
$269$	−4.24621	−0.258896	−0.129448	−	0.991586i	$-0.541321\pi$
$269$	−4.24621	−0.258896	−0.129448	+	0.991586i	$0.541321\pi$
$270$	0	0
$271$	22.9309	1.39295	0.696476	−	0.717581i	$-0.254750\pi$
$271$	22.9309	1.39295	0.696476	+	0.717581i	$0.254750\pi$
$272$	0	0
$273$	6.68466	0.404574
$274$	0	0
$275$	−7.68466	−0.463402
$276$	0	0
$277$	−17.6155	−1.05841	−0.529207	−	0.848493i	$-0.677511\pi$
$277$	−17.6155	−1.05841	−0.529207	+	0.848493i	$0.677511\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−0.438447	−0.0261556	−0.0130778	−	0.999914i	$-0.504163\pi$
$281$	−0.438447	−0.0261556	−0.0130778	+	0.999914i	$0.504163\pi$
$282$	0	0
$283$	−21.5616	−1.28170	−0.640851	−	0.767666i	$-0.721418\pi$
$283$	−21.5616	−1.28170	−0.640851	+	0.767666i	$0.721418\pi$
$284$	0	0
$285$	19.8078	1.17331
$286$	0	0
$287$	9.12311	0.538520
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−8.63068	−0.504210	−0.252105	−	0.967700i	$-0.581123\pi$
$293$	−8.63068	−0.504210	−0.252105	+	0.967700i	$0.581123\pi$
$294$	0	0
$295$	−5.56155	−0.323806
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	20.8769	1.20734
$300$	0	0
$301$	7.12311	0.410569
$302$	0	0
$303$	−13.1231	−0.753903
$304$	0	0
$305$	29.3693	1.68168
$306$	0	0
$307$	22.2462	1.26966	0.634829	−	0.772653i	$-0.281070\pi$
$307$	22.2462	1.26966	0.634829	+	0.772653i	$0.281070\pi$
$308$	0	0
$309$	0.876894	0.0498848
$310$	0	0
$311$	10.2462	0.581009	0.290505	−	0.956874i	$-0.406177\pi$
$311$	10.2462	0.581009	0.290505	+	0.956874i	$0.406177\pi$
$312$	0	0
$313$	−21.6155	−1.22178	−0.610891	−	0.791715i	$-0.709189\pi$
$313$	−21.6155	−1.22178	−0.610891	+	0.791715i	$0.709189\pi$
$314$	0	0
$315$	−3.56155	−0.200671
$316$	0	0
$317$	32.7386	1.83878	0.919392	−	0.393342i	$-0.128681\pi$
$317$	32.7386	1.83878	0.919392	+	0.393342i	$0.128681\pi$
$318$	0	0
$319$	4.43845	0.248505
$320$	0	0
$321$	9.56155	0.533674
$322$	0	0
$323$	28.4924	1.58536
$324$	0	0
$325$	51.3693	2.84946
$326$	0	0
$327$	−13.1231	−0.725709
$328$	0	0
$329$	−4.68466	−0.258274
$330$	0	0
$331$	16.4924	0.906506	0.453253	−	0.891382i	$-0.350263\pi$
$331$	16.4924	0.906506	0.453253	+	0.891382i	$0.350263\pi$
$332$	0	0
$333$	−4.43845	−0.243225
$334$	0	0
$335$	−34.0540	−1.86057
$336$	0	0
$337$	22.4924	1.22524	0.612620	−	0.790377i	$-0.290116\pi$
$337$	22.4924	1.22524	0.612620	+	0.790377i	$0.290116\pi$
$338$	0	0
$339$	−20.2462	−1.09962
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−11.1231	−0.598848
$346$	0	0
$347$	18.2462	0.979508	0.489754	−	0.871861i	$-0.337087\pi$
$347$	18.2462	0.979508	0.489754	+	0.871861i	$0.337087\pi$
$348$	0	0
$349$	24.4384	1.30816	0.654080	−	0.756425i	$-0.273056\pi$
$349$	24.4384	1.30816	0.654080	+	0.756425i	$0.273056\pi$
$350$	0	0
$351$	6.68466	0.356801
$352$	0	0
$353$	15.5616	0.828258	0.414129	−	0.910218i	$-0.364086\pi$
$353$	15.5616	0.828258	0.414129	+	0.910218i	$0.364086\pi$
$354$	0	0
$355$	22.2462	1.18071
$356$	0	0
$357$	−5.12311	−0.271144
$358$	0	0
$359$	20.4924	1.08155	0.540774	−	0.841168i	$-0.318131\pi$
$359$	20.4924	1.08155	0.540774	+	0.841168i	$0.318131\pi$
$360$	0	0
$361$	11.9309	0.627941
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	38.0540	1.99184
$366$	0	0
$367$	13.3693	0.697873	0.348936	−	0.937146i	$-0.386543\pi$
$367$	13.3693	0.697873	0.348936	+	0.937146i	$0.386543\pi$
$368$	0	0
$369$	9.12311	0.474930
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	−27.3693	−1.41713	−0.708565	−	0.705646i	$-0.750657\pi$
$373$	−27.3693	−1.41713	−0.708565	+	0.705646i	$0.750657\pi$
$374$	0	0
$375$	−9.56155	−0.493756
$376$	0	0
$377$	−29.6695	−1.52806
$378$	0	0
$379$	23.4233	1.20317	0.601587	−	0.798807i	$-0.294535\pi$
$379$	23.4233	1.20317	0.601587	+	0.798807i	$0.294535\pi$
$380$	0	0
$381$	−20.4924	−1.04986
$382$	0	0
$383$	30.7386	1.57067	0.785335	−	0.619071i	$-0.212491\pi$
$383$	30.7386	1.57067	0.785335	+	0.619071i	$0.212491\pi$
$384$	0	0
$385$	3.56155	0.181514
$386$	0	0
$387$	7.12311	0.362088
$388$	0	0
$389$	9.12311	0.462560	0.231280	−	0.972887i	$-0.425709\pi$
$389$	9.12311	0.462560	0.231280	+	0.972887i	$0.425709\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−12.4924	−0.630159
$394$	0	0
$395$	−50.7386	−2.55294
$396$	0	0
$397$	−26.8769	−1.34891	−0.674456	−	0.738315i	$-0.735622\pi$
$397$	−26.8769	−1.34891	−0.674456	+	0.738315i	$0.735622\pi$
$398$	0	0
$399$	−5.56155	−0.278426
$400$	0	0
$401$	24.2462	1.21080	0.605399	−	0.795922i	$-0.293014\pi$
$401$	24.2462	1.21080	0.605399	+	0.795922i	$0.293014\pi$
$402$	0	0
$403$	−26.7386	−1.33195
$404$	0	0
$405$	−3.56155	−0.176975
$406$	0	0
$407$	4.43845	0.220006
$408$	0	0
$409$	−0.246211	−0.0121744	−0.00608718	−	0.999981i	$-0.501938\pi$
$409$	−0.246211	−0.0121744	−0.00608718	+	0.999981i	$0.501938\pi$
$410$	0	0
$411$	11.3693	0.560807
$412$	0	0
$413$	1.56155	0.0768390
$414$	0	0
$415$	28.4924	1.39864
$416$	0	0
$417$	−1.75379	−0.0858834
$418$	0	0
$419$	−15.8078	−0.772260	−0.386130	−	0.922444i	$-0.626188\pi$
$419$	−15.8078	−0.772260	−0.386130	+	0.922444i	$0.626188\pi$
$420$	0	0
$421$	30.6847	1.49548	0.747739	−	0.663992i	$-0.231139\pi$
$421$	30.6847	1.49548	0.747739	+	0.663992i	$0.231139\pi$
$422$	0	0
$423$	−4.68466	−0.227776
$424$	0	0
$425$	−39.3693	−1.90969
$426$	0	0
$427$	−8.24621	−0.399062
$428$	0	0
$429$	−6.68466	−0.322738
$430$	0	0
$431$	−5.17708	−0.249371	−0.124686	−	0.992196i	$-0.539792\pi$
$431$	−5.17708	−0.249371	−0.124686	+	0.992196i	$0.539792\pi$
$432$	0	0
$433$	0.630683	0.0303087	0.0151543	−	0.999885i	$-0.495176\pi$
$433$	0.630683	0.0303087	0.0151543	+	0.999885i	$0.495176\pi$
$434$	0	0
$435$	15.8078	0.757924
$436$	0	0
$437$	−17.3693	−0.830887
$438$	0	0
$439$	−13.5616	−0.647258	−0.323629	−	0.946184i	$-0.604903\pi$
$439$	−13.5616	−0.647258	−0.323629	+	0.946184i	$0.604903\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−0.492423	−0.0233957	−0.0116978	−	0.999932i	$-0.503724\pi$
$443$	−0.492423	−0.0233957	−0.0116978	+	0.999932i	$0.503724\pi$
$444$	0	0
$445$	−0.876894	−0.0415688
$446$	0	0
$447$	−18.6847	−0.883754
$448$	0	0
$449$	0.630683	0.0297638	0.0148819	−	0.999889i	$-0.495263\pi$
$449$	0.630683	0.0297638	0.0148819	+	0.999889i	$0.495263\pi$
$450$	0	0
$451$	−9.12311	−0.429590
$452$	0	0
$453$	−20.4924	−0.962818
$454$	0	0
$455$	−23.8078	−1.11613
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	−5.12311	−0.239126
$460$	0	0
$461$	10.8769	0.506587	0.253294	−	0.967389i	$-0.418486\pi$
$461$	10.8769	0.506587	0.253294	+	0.967389i	$0.418486\pi$
$462$	0	0
$463$	−19.4233	−0.902677	−0.451338	−	0.892353i	$-0.649053\pi$
$463$	−19.4233	−0.902677	−0.451338	+	0.892353i	$0.649053\pi$
$464$	0	0
$465$	14.2462	0.660652
$466$	0	0
$467$	14.0540	0.650340	0.325170	−	0.945656i	$-0.394578\pi$
$467$	14.0540	0.650340	0.325170	+	0.945656i	$0.394578\pi$
$468$	0	0
$469$	9.56155	0.441511
$470$	0	0
$471$	−10.8769	−0.501181
$472$	0	0
$473$	−7.12311	−0.327521
$474$	0	0
$475$	−42.7386	−1.96098
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	18.7386	0.856190	0.428095	−	0.903734i	$-0.359185\pi$
$479$	18.7386	0.856190	0.428095	+	0.903734i	$0.359185\pi$
$480$	0	0
$481$	−29.6695	−1.35281
$482$	0	0
$483$	3.12311	0.142106
$484$	0	0
$485$	−7.12311	−0.323444
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	−10.9309	−0.494311
$490$	0	0
$491$	−10.9309	−0.493303	−0.246652	−	0.969104i	$-0.579330\pi$
$491$	−10.9309	−0.493303	−0.246652	+	0.969104i	$0.579330\pi$
$492$	0	0
$493$	22.7386	1.02410
$494$	0	0
$495$	3.56155	0.160080
$496$	0	0
$497$	−6.24621	−0.280181
$498$	0	0
$499$	−5.06913	−0.226925	−0.113463	−	0.993542i	$-0.536194\pi$
$499$	−5.06913	−0.226925	−0.113463	+	0.993542i	$0.536194\pi$
$500$	0	0
$501$	17.3693	0.776004
$502$	0	0
$503$	−23.6155	−1.05296	−0.526482	−	0.850186i	$-0.676489\pi$
$503$	−23.6155	−1.05296	−0.526482	+	0.850186i	$0.676489\pi$
$504$	0	0
$505$	46.7386	2.07984
$506$	0	0
$507$	31.6847	1.40717
$508$	0	0
$509$	−23.7538	−1.05287	−0.526434	−	0.850216i	$-0.676471\pi$
$509$	−23.7538	−1.05287	−0.526434	+	0.850216i	$0.676471\pi$
$510$	0	0
$511$	−10.6847	−0.472661
$512$	0	0
$513$	−5.56155	−0.245549
$514$	0	0
$515$	−3.12311	−0.137620
$516$	0	0
$517$	4.68466	0.206031
$518$	0	0
$519$	−19.3693	−0.850219
$520$	0	0
$521$	−3.56155	−0.156034	−0.0780172	−	0.996952i	$-0.524859\pi$
$521$	−3.56155	−0.156034	−0.0780172	+	0.996952i	$0.524859\pi$
$522$	0	0
$523$	−16.6847	−0.729569	−0.364785	−	0.931092i	$-0.618857\pi$
$523$	−16.6847	−0.729569	−0.364785	+	0.931092i	$0.618857\pi$
$524$	0	0
$525$	7.68466	0.335386
$526$	0	0
$527$	20.4924	0.892664
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	1.56155	0.0677656
$532$	0	0
$533$	60.9848	2.64155
$534$	0	0
$535$	−34.0540	−1.47228
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	25.1231	1.08013	0.540063	−	0.841624i	$-0.318400\pi$
$541$	25.1231	1.08013	0.540063	+	0.841624i	$0.318400\pi$
$542$	0	0
$543$	−18.4924	−0.793586
$544$	0	0
$545$	46.7386	2.00206
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	−8.24621	−0.351940
$550$	0	0
$551$	24.6847	1.05160
$552$	0	0
$553$	14.2462	0.605811
$554$	0	0
$555$	15.8078	0.671002
$556$	0	0
$557$	−36.4384	−1.54395	−0.771973	−	0.635655i	$-0.780730\pi$
$557$	−36.4384	−1.54395	−0.771973	+	0.635655i	$0.780730\pi$
$558$	0	0
$559$	47.6155	2.01392
$560$	0	0
$561$	5.12311	0.216298
$562$	0	0
$563$	−4.87689	−0.205537	−0.102768	−	0.994705i	$-0.532770\pi$
$563$	−4.87689	−0.205537	−0.102768	+	0.994705i	$0.532770\pi$
$564$	0	0
$565$	72.1080	3.03360
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	20.7386	0.869409	0.434704	−	0.900573i	$-0.356853\pi$
$569$	20.7386	0.869409	0.434704	+	0.900573i	$0.356853\pi$
$570$	0	0
$571$	10.2462	0.428791	0.214395	−	0.976747i	$-0.431222\pi$
$571$	10.2462	0.428791	0.214395	+	0.976747i	$0.431222\pi$
$572$	0	0
$573$	−16.0000	−0.668410
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	−13.6155	−0.566822	−0.283411	−	0.958999i	$-0.591466\pi$
$577$	−13.6155	−0.566822	−0.283411	+	0.958999i	$0.591466\pi$
$578$	0	0
$579$	−10.4924	−0.436050
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	10.0000	0.414158
$584$	0	0
$585$	−23.8078	−0.984330
$586$	0	0
$587$	25.1771	1.03917	0.519585	−	0.854419i	$-0.326087\pi$
$587$	25.1771	1.03917	0.519585	+	0.854419i	$0.326087\pi$
$588$	0	0
$589$	22.2462	0.916639
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−14.8769	−0.610921	−0.305460	−	0.952205i	$-0.598810\pi$
$593$	−14.8769	−0.610921	−0.305460	+	0.952205i	$0.598810\pi$
$594$	0	0
$595$	18.2462	0.748022
$596$	0	0
$597$	7.12311	0.291529
$598$	0	0
$599$	−9.75379	−0.398529	−0.199265	−	0.979946i	$-0.563855\pi$
$599$	−9.75379	−0.398529	−0.199265	+	0.979946i	$0.563855\pi$
$600$	0	0
$601$	−18.6847	−0.762163	−0.381082	−	0.924541i	$-0.624448\pi$
$601$	−18.6847	−0.762163	−0.381082	+	0.924541i	$0.624448\pi$
$602$	0	0
$603$	9.56155	0.389377
$604$	0	0
$605$	−3.56155	−0.144798
$606$	0	0
$607$	−3.80776	−0.154552	−0.0772762	−	0.997010i	$-0.524622\pi$
$607$	−3.80776	−0.154552	−0.0772762	+	0.997010i	$0.524622\pi$
$608$	0	0
$609$	−4.43845	−0.179855
$610$	0	0
$611$	−31.3153	−1.26688
$612$	0	0
$613$	48.3542	1.95301	0.976503	−	0.215503i	$-0.0691392\pi$
$613$	48.3542	1.95301	0.976503	+	0.215503i	$0.0691392\pi$
$614$	0	0
$615$	−32.4924	−1.31022
$616$	0	0
$617$	−40.7386	−1.64008	−0.820038	−	0.572309i	$-0.806048\pi$
$617$	−40.7386	−1.64008	−0.820038	+	0.572309i	$0.806048\pi$
$618$	0	0
$619$	−6.73863	−0.270849	−0.135424	−	0.990788i	$-0.543240\pi$
$619$	−6.73863	−0.270849	−0.135424	+	0.990788i	$0.543240\pi$
$620$	0	0
$621$	3.12311	0.125326
$622$	0	0
$623$	0.246211	0.00986425
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	5.56155	0.222107
$628$	0	0
$629$	22.7386	0.906649
$630$	0	0
$631$	−36.4924	−1.45274	−0.726370	−	0.687304i	$-0.758794\pi$
$631$	−36.4924	−1.45274	−0.726370	+	0.687304i	$0.758794\pi$
$632$	0	0
$633$	15.1231	0.601089
$634$	0	0
$635$	72.9848	2.89632
$636$	0	0
$637$	6.68466	0.264856
$638$	0	0
$639$	−6.24621	−0.247096
$640$	0	0
$641$	0.630683	0.0249105	0.0124552	−	0.999922i	$-0.496035\pi$
$641$	0.630683	0.0249105	0.0124552	+	0.999922i	$0.496035\pi$
$642$	0	0
$643$	13.3693	0.527234	0.263617	−	0.964627i	$-0.415084\pi$
$643$	13.3693	0.527234	0.263617	+	0.964627i	$0.415084\pi$
$644$	0	0
$645$	−25.3693	−0.998916
$646$	0	0
$647$	9.17708	0.360788	0.180394	−	0.983594i	$-0.442263\pi$
$647$	9.17708	0.360788	0.180394	+	0.983594i	$0.442263\pi$
$648$	0	0
$649$	−1.56155	−0.0612963
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	35.8617	1.40338	0.701689	−	0.712483i	$-0.252430\pi$
$653$	35.8617	1.40338	0.701689	+	0.712483i	$0.252430\pi$
$654$	0	0
$655$	44.4924	1.73846
$656$	0	0
$657$	−10.6847	−0.416848
$658$	0	0
$659$	−4.68466	−0.182488	−0.0912442	−	0.995829i	$-0.529084\pi$
$659$	−4.68466	−0.182488	−0.0912442	+	0.995829i	$0.529084\pi$
$660$	0	0
$661$	0.246211	0.00957651	0.00478825	−	0.999989i	$-0.498476\pi$
$661$	0.246211	0.00957651	0.00478825	+	0.999989i	$0.498476\pi$
$662$	0	0
$663$	−34.2462	−1.33001
$664$	0	0
$665$	19.8078	0.768112
$666$	0	0
$667$	−13.8617	−0.536729
$668$	0	0
$669$	−8.87689	−0.343201
$670$	0	0
$671$	8.24621	0.318341
$672$	0	0
$673$	19.7538	0.761453	0.380726	−	0.924688i	$-0.375674\pi$
$673$	19.7538	0.761453	0.380726	+	0.924688i	$0.375674\pi$
$674$	0	0
$675$	7.68466	0.295783
$676$	0	0
$677$	−16.2462	−0.624393	−0.312196	−	0.950018i	$-0.601065\pi$
$677$	−16.2462	−0.624393	−0.312196	+	0.950018i	$0.601065\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−23.6155	−0.904949
$682$	0	0
$683$	−32.4924	−1.24329	−0.621644	−	0.783300i	$-0.713535\pi$
$683$	−32.4924	−1.24329	−0.621644	+	0.783300i	$0.713535\pi$
$684$	0	0
$685$	−40.4924	−1.54714
$686$	0	0
$687$	0.246211	0.00939355
$688$	0	0
$689$	−66.8466	−2.54665
$690$	0	0
$691$	3.61553	0.137541	0.0687706	−	0.997633i	$-0.478092\pi$
$691$	3.61553	0.137541	0.0687706	+	0.997633i	$0.478092\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	6.24621	0.236932
$696$	0	0
$697$	−46.7386	−1.77035
$698$	0	0
$699$	2.00000	0.0756469
$700$	0	0
$701$	−36.7386	−1.38760	−0.693800	−	0.720168i	$-0.744065\pi$
$701$	−36.7386	−1.38760	−0.693800	+	0.720168i	$0.744065\pi$
$702$	0	0
$703$	24.6847	0.931000
$704$	0	0
$705$	16.6847	0.628381
$706$	0	0
$707$	−13.1231	−0.493545
$708$	0	0
$709$	−47.5616	−1.78621	−0.893106	−	0.449847i	$-0.851479\pi$
$709$	−47.5616	−1.78621	−0.893106	+	0.449847i	$0.851479\pi$
$710$	0	0
$711$	14.2462	0.534275
$712$	0	0
$713$	−12.4924	−0.467845
$714$	0	0
$715$	23.8078	0.890360
$716$	0	0
$717$	29.1771	1.08964
$718$	0	0
$719$	−36.3002	−1.35377	−0.676884	−	0.736089i	$-0.736670\pi$
$719$	−36.3002	−1.35377	−0.676884	+	0.736089i	$0.736670\pi$
$720$	0	0
$721$	0.876894	0.0326573
$722$	0	0
$723$	8.43845	0.313829
$724$	0	0
$725$	−34.1080	−1.26674
$726$	0	0
$727$	2.63068	0.0975666	0.0487833	−	0.998809i	$-0.484466\pi$
$727$	2.63068	0.0975666	0.0487833	+	0.998809i	$0.484466\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.4924	−1.34972
$732$	0	0
$733$	−33.2311	−1.22742	−0.613709	−	0.789533i	$-0.710323\pi$
$733$	−33.2311	−1.22742	−0.613709	+	0.789533i	$0.710323\pi$
$734$	0	0
$735$	−3.56155	−0.131370
$736$	0	0
$737$	−9.56155	−0.352204
$738$	0	0
$739$	32.8769	1.20940	0.604698	−	0.796455i	$-0.293294\pi$
$739$	32.8769	1.20940	0.604698	+	0.796455i	$0.293294\pi$
$740$	0	0
$741$	−37.1771	−1.36573
$742$	0	0
$743$	32.3002	1.18498	0.592489	−	0.805578i	$-0.298145\pi$
$743$	32.3002	1.18498	0.592489	+	0.805578i	$0.298145\pi$
$744$	0	0
$745$	66.5464	2.43807
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	9.56155	0.349372
$750$	0	0
$751$	7.31534	0.266941	0.133470	−	0.991053i	$-0.457388\pi$
$751$	7.31534	0.266941	0.133470	+	0.991053i	$0.457388\pi$
$752$	0	0
$753$	−9.56155	−0.348442
$754$	0	0
$755$	72.9848	2.65619
$756$	0	0
$757$	−41.9157	−1.52345	−0.761726	−	0.647899i	$-0.775648\pi$
$757$	−41.9157	−1.52345	−0.761726	+	0.647899i	$0.775648\pi$
$758$	0	0
$759$	−3.12311	−0.113362
$760$	0	0
$761$	51.8617	1.87999	0.939993	−	0.341193i	$-0.110831\pi$
$761$	51.8617	1.87999	0.939993	+	0.341193i	$0.110831\pi$
$762$	0	0
$763$	−13.1231	−0.475088
$764$	0	0
$765$	18.2462	0.659693
$766$	0	0
$767$	10.4384	0.376910
$768$	0	0
$769$	12.9309	0.466299	0.233150	−	0.972441i	$-0.425097\pi$
$769$	12.9309	0.466299	0.233150	+	0.972441i	$0.425097\pi$
$770$	0	0
$771$	−5.31534	−0.191427
$772$	0	0
$773$	36.4384	1.31060	0.655300	−	0.755369i	$-0.272542\pi$
$773$	36.4384	1.31060	0.655300	+	0.755369i	$0.272542\pi$
$774$	0	0
$775$	−30.7386	−1.10416
$776$	0	0
$777$	−4.43845	−0.159228
$778$	0	0
$779$	−50.7386	−1.81790
$780$	0	0
$781$	6.24621	0.223507
$782$	0	0
$783$	−4.43845	−0.158617
$784$	0	0
$785$	38.7386	1.38264
$786$	0	0
$787$	−22.9309	−0.817397	−0.408699	−	0.912669i	$-0.634017\pi$
$787$	−22.9309	−0.817397	−0.408699	+	0.912669i	$0.634017\pi$
$788$	0	0
$789$	−4.19224	−0.149248
$790$	0	0
$791$	−20.2462	−0.719872
$792$	0	0
$793$	−55.1231	−1.95748
$794$	0	0
$795$	35.6155	1.26315
$796$	0	0
$797$	−18.1922	−0.644402	−0.322201	−	0.946671i	$-0.604423\pi$
$797$	−18.1922	−0.644402	−0.322201	+	0.946671i	$0.604423\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	0.246211	0.00869945
$802$	0	0
$803$	10.6847	0.377053
$804$	0	0
$805$	−11.1231	−0.392038
$806$	0	0
$807$	−4.24621	−0.149474
$808$	0	0
$809$	29.8078	1.04799	0.523993	−	0.851723i	$-0.324442\pi$
$809$	29.8078	1.04799	0.523993	+	0.851723i	$0.324442\pi$
$810$	0	0
$811$	−48.6847	−1.70955	−0.854775	−	0.518999i	$-0.826305\pi$
$811$	−48.6847	−1.70955	−0.854775	+	0.518999i	$0.826305\pi$
$812$	0	0
$813$	22.9309	0.804221
$814$	0	0
$815$	38.9309	1.36369
$816$	0	0
$817$	−39.6155	−1.38597
$818$	0	0
$819$	6.68466	0.233581
$820$	0	0
$821$	11.1771	0.390083	0.195041	−	0.980795i	$-0.437516\pi$
$821$	11.1771	0.390083	0.195041	+	0.980795i	$0.437516\pi$
$822$	0	0
$823$	56.3002	1.96250	0.981251	−	0.192736i	$-0.0617362\pi$
$823$	56.3002	1.96250	0.981251	+	0.192736i	$0.0617362\pi$
$824$	0	0
$825$	−7.68466	−0.267545
$826$	0	0
$827$	27.3153	0.949847	0.474924	−	0.880027i	$-0.342476\pi$
$827$	27.3153	0.949847	0.474924	+	0.880027i	$0.342476\pi$
$828$	0	0
$829$	−18.8769	−0.655622	−0.327811	−	0.944743i	$-0.606311\pi$
$829$	−18.8769	−0.655622	−0.327811	+	0.944743i	$0.606311\pi$
$830$	0	0
$831$	−17.6155	−0.611076
$832$	0	0
$833$	−5.12311	−0.177505
$834$	0	0
$835$	−61.8617	−2.14081
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−17.1771	−0.593019	−0.296509	−	0.955030i	$-0.595823\pi$
$839$	−17.1771	−0.593019	−0.296509	+	0.955030i	$0.595823\pi$
$840$	0	0
$841$	−9.30019	−0.320696
$842$	0	0
$843$	−0.438447	−0.0151009
$844$	0	0
$845$	−112.847	−3.88204
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−21.5616	−0.739991
$850$	0	0
$851$	−13.8617	−0.475174
$852$	0	0
$853$	−0.246211	−0.00843011	−0.00421506	−	0.999991i	$-0.501342\pi$
$853$	−0.246211	−0.00843011	−0.00421506	+	0.999991i	$0.501342\pi$
$854$	0	0
$855$	19.8078	0.677411
$856$	0	0
$857$	34.4924	1.17824	0.589119	−	0.808046i	$-0.299475\pi$
$857$	34.4924	1.17824	0.589119	+	0.808046i	$0.299475\pi$
$858$	0	0
$859$	−2.24621	−0.0766397	−0.0383199	−	0.999266i	$-0.512201\pi$
$859$	−2.24621	−0.0766397	−0.0383199	+	0.999266i	$0.512201\pi$
$860$	0	0
$861$	9.12311	0.310915
$862$	0	0
$863$	−1.36932	−0.0466121	−0.0233060	−	0.999728i	$-0.507419\pi$
$863$	−1.36932	−0.0466121	−0.0233060	+	0.999728i	$0.507419\pi$
$864$	0	0
$865$	68.9848	2.34556
$866$	0	0
$867$	9.24621	0.314018
$868$	0	0
$869$	−14.2462	−0.483270
$870$	0	0
$871$	63.9157	2.16570
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	−9.56155	−0.323239
$876$	0	0
$877$	0.138261	0.00466873	0.00233436	−	0.999997i	$-0.499257\pi$
$877$	0.138261	0.00466873	0.00233436	+	0.999997i	$0.499257\pi$
$878$	0	0
$879$	−8.63068	−0.291106
$880$	0	0
$881$	15.1771	0.511329	0.255664	−	0.966766i	$-0.417706\pi$
$881$	15.1771	0.511329	0.255664	+	0.966766i	$0.417706\pi$
$882$	0	0
$883$	3.69981	0.124509	0.0622543	−	0.998060i	$-0.480171\pi$
$883$	3.69981	0.124509	0.0622543	+	0.998060i	$0.480171\pi$
$884$	0	0
$885$	−5.56155	−0.186950
$886$	0	0
$887$	−48.9848	−1.64475	−0.822375	−	0.568946i	$-0.807351\pi$
$887$	−48.9848	−1.64475	−0.822375	+	0.568946i	$0.807351\pi$
$888$	0	0
$889$	−20.4924	−0.687294
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	26.0540	0.871863
$894$	0	0
$895$	−14.2462	−0.476198
$896$	0	0
$897$	20.8769	0.697059
$898$	0	0
$899$	17.7538	0.592122
$900$	0	0
$901$	51.2311	1.70675
$902$	0	0
$903$	7.12311	0.237042
$904$	0	0
$905$	65.8617	2.18932
$906$	0	0
$907$	−24.4924	−0.813258	−0.406629	−	0.913593i	$-0.633296\pi$
$907$	−24.4924	−0.813258	−0.406629	+	0.913593i	$0.633296\pi$
$908$	0	0
$909$	−13.1231	−0.435266
$910$	0	0
$911$	−23.6155	−0.782417	−0.391209	−	0.920302i	$-0.627943\pi$
$911$	−23.6155	−0.782417	−0.391209	+	0.920302i	$0.627943\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	29.3693	0.970920
$916$	0	0
$917$	−12.4924	−0.412536
$918$	0	0
$919$	19.1231	0.630813	0.315407	−	0.948957i	$-0.397859\pi$
$919$	19.1231	0.630813	0.315407	+	0.948957i	$0.397859\pi$
$920$	0	0
$921$	22.2462	0.733038
$922$	0	0
$923$	−41.7538	−1.37434
$924$	0	0
$925$	−34.1080	−1.12146
$926$	0	0
$927$	0.876894	0.0288010
$928$	0	0
$929$	5.80776	0.190547	0.0952733	−	0.995451i	$-0.469628\pi$
$929$	5.80776	0.190547	0.0952733	+	0.995451i	$0.469628\pi$
$930$	0	0
$931$	−5.56155	−0.182273
$932$	0	0
$933$	10.2462	0.335446
$934$	0	0
$935$	−18.2462	−0.596715
$936$	0	0
$937$	1.50758	0.0492504	0.0246252	−	0.999697i	$-0.492161\pi$
$937$	1.50758	0.0492504	0.0246252	+	0.999697i	$0.492161\pi$
$938$	0	0
$939$	−21.6155	−0.705396
$940$	0	0
$941$	18.8769	0.615369	0.307685	−	0.951488i	$-0.400446\pi$
$941$	18.8769	0.615369	0.307685	+	0.951488i	$0.400446\pi$
$942$	0	0
$943$	28.4924	0.927841
$944$	0	0
$945$	−3.56155	−0.115857
$946$	0	0
$947$	29.3693	0.954375	0.477187	−	0.878802i	$-0.341656\pi$
$947$	29.3693	0.954375	0.477187	+	0.878802i	$0.341656\pi$
$948$	0	0
$949$	−71.4233	−2.31850
$950$	0	0
$951$	32.7386	1.06162
$952$	0	0
$953$	−4.93087	−0.159727	−0.0798633	−	0.996806i	$-0.525448\pi$
$953$	−4.93087	−0.159727	−0.0798633	+	0.996806i	$0.525448\pi$
$954$	0	0
$955$	56.9848	1.84399
$956$	0	0
$957$	4.43845	0.143475
$958$	0	0
$959$	11.3693	0.367134
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	9.56155	0.308117
$964$	0	0
$965$	37.3693	1.20296
$966$	0	0
$967$	47.6155	1.53121	0.765606	−	0.643310i	$-0.222439\pi$
$967$	47.6155	1.53121	0.765606	+	0.643310i	$0.222439\pi$
$968$	0	0
$969$	28.4924	0.915308
$970$	0	0
$971$	33.1771	1.06470	0.532352	−	0.846523i	$-0.321308\pi$
$971$	33.1771	1.06470	0.532352	+	0.846523i	$0.321308\pi$
$972$	0	0
$973$	−1.75379	−0.0562239
$974$	0	0
$975$	51.3693	1.64513
$976$	0	0
$977$	−58.4924	−1.87134	−0.935669	−	0.352878i	$-0.885203\pi$
$977$	−58.4924	−1.87134	−0.935669	+	0.352878i	$0.885203\pi$
$978$	0	0
$979$	−0.246211	−0.00786895
$980$	0	0
$981$	−13.1231	−0.418989
$982$	0	0
$983$	−12.9848	−0.414152	−0.207076	−	0.978325i	$-0.566395\pi$
$983$	−12.9848	−0.414152	−0.207076	+	0.978325i	$0.566395\pi$
$984$	0	0
$985$	−21.3693	−0.680883
$986$	0	0
$987$	−4.68466	−0.149114
$988$	0	0
$989$	22.2462	0.707388
$990$	0	0
$991$	31.3153	0.994765	0.497382	−	0.867531i	$-0.334295\pi$
$991$	31.3153	0.994765	0.497382	+	0.867531i	$0.334295\pi$
$992$	0	0
$993$	16.4924	0.523371
$994$	0	0
$995$	−25.3693	−0.804261
$996$	0	0
$997$	26.4924	0.839023	0.419512	−	0.907750i	$-0.362201\pi$
$997$	26.4924	0.839023	0.419512	+	0.907750i	$0.362201\pi$
$998$	0	0
$999$	−4.43845	−0.140426

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bh.1.1		2
4.3	odd	2		1848.2.a.n.1.1	✓	2
12.11	even	2		5544.2.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.a.n.1.1	✓	2	4.3	odd	2
3696.2.a.bh.1.1		2	1.1	even	1	trivial
5544.2.a.bg.1.2		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +6.68466 q^{13} -3.56155 q^{15} -5.12311 q^{17} -5.56155 q^{19} +1.00000 q^{21} +3.12311 q^{23} +7.68466 q^{25} +1.00000 q^{27} -4.43845 q^{29} -4.00000 q^{31} -1.00000 q^{33} -3.56155 q^{35} -4.43845 q^{37} +6.68466 q^{39} +9.12311 q^{41} +7.12311 q^{43} -3.56155 q^{45} -4.68466 q^{47} +1.00000 q^{49} -5.12311 q^{51} -10.0000 q^{53} +3.56155 q^{55} -5.56155 q^{57} +1.56155 q^{59} -8.24621 q^{61} +1.00000 q^{63} -23.8078 q^{65} +9.56155 q^{67} +3.12311 q^{69} -6.24621 q^{71} -10.6847 q^{73} +7.68466 q^{75} -1.00000 q^{77} +14.2462 q^{79} +1.00000 q^{81} -8.00000 q^{83} +18.2462 q^{85} -4.43845 q^{87} +0.246211 q^{89} +6.68466 q^{91} -4.00000 q^{93} +19.8078 q^{95} +2.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 2 q^{17} - 7 q^{19} + 2 q^{21} - 2 q^{23} + 3 q^{25} + 2 q^{27} - 13 q^{29} - 8 q^{31} - 2 q^{33} - 3 q^{35} - 13 q^{37} + q^{39} + 10 q^{41} + 6 q^{43} - 3 q^{45} + 3 q^{47} + 2 q^{49} - 2 q^{51} - 20 q^{53} + 3 q^{55} - 7 q^{57} - q^{59} + 2 q^{63} - 27 q^{65} + 15 q^{67} - 2 q^{69} + 4 q^{71} - 9 q^{73} + 3 q^{75} - 2 q^{77} + 12 q^{79} + 2 q^{81} - 16 q^{83} + 20 q^{85} - 13 q^{87} - 16 q^{89} + q^{91} - 8 q^{93} + 19 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 2 * q^17 - 7 * q^19 + 2 * q^21 - 2 * q^23 + 3 * q^25 + 2 * q^27 - 13 * q^29 - 8 * q^31 - 2 * q^33 - 3 * q^35 - 13 * q^37 + q^39 + 10 * q^41 + 6 * q^43 - 3 * q^45 + 3 * q^47 + 2 * q^49 - 2 * q^51 - 20 * q^53 + 3 * q^55 - 7 * q^57 - q^59 + 2 * q^63 - 27 * q^65 + 15 * q^67 - 2 * q^69 + 4 * q^71 - 9 * q^73 + 3 * q^75 - 2 * q^77 + 12 * q^79 + 2 * q^81 - 16 * q^83 + 20 * q^85 - 13 * q^87 - 16 * q^89 + q^91 - 8 * q^93 + 19 * q^95 + 4 * q^97 - 2 * q^99

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