LMFDB - Embedded newform 3696.2.a.bp.1.3

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:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*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			1.3
Root			$-0.167449$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.80451 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.80451 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.80451 q^{13} +3.80451 q^{15} +0.334898 q^{17} -8.13941 q^{19} +1.00000 q^{21} +1.66510 q^{23} +9.47431 q^{25} +1.00000 q^{27} +0.195488 q^{29} +9.94392 q^{31} -1.00000 q^{33} +3.80451 q^{35} -4.47431 q^{37} +3.80451 q^{39} -6.27882 q^{41} -2.33490 q^{43} +3.80451 q^{45} +12.1394 q^{47} +1.00000 q^{49} +0.334898 q^{51} +7.94392 q^{53} -3.80451 q^{55} -8.13941 q^{57} -3.74843 q^{59} +6.00000 q^{61} +1.00000 q^{63} +14.4743 q^{65} +0.139410 q^{67} +1.66510 q^{69} -4.66980 q^{71} +4.19549 q^{73} +9.47431 q^{75} -1.00000 q^{77} -3.33020 q^{79} +1.00000 q^{81} +13.9439 q^{83} +1.27412 q^{85} +0.195488 q^{87} -9.88784 q^{89} +3.80451 q^{91} +9.94392 q^{93} -30.9665 q^{95} +0.0560785 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 3 q^{11} - 12 q^{19} + 3 q^{21} + 6 q^{23} + 15 q^{25} + 3 q^{27} + 12 q^{29} + 6 q^{31} - 3 q^{33} + 6 q^{41} - 6 q^{43} + 24 q^{47} + 3 q^{49} - 12 q^{57} + 24 q^{59} + 18 q^{61} + 3 q^{63} + 30 q^{65} - 12 q^{67} + 6 q^{69} - 12 q^{71} + 24 q^{73} + 15 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 18 q^{83} - 18 q^{85} + 12 q^{87} + 18 q^{89} + 6 q^{93} - 12 q^{95} + 24 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 3 * q^11 - 12 * q^19 + 3 * q^21 + 6 * q^23 + 15 * q^25 + 3 * q^27 + 12 * q^29 + 6 * q^31 - 3 * q^33 + 6 * q^41 - 6 * q^43 + 24 * q^47 + 3 * q^49 - 12 * q^57 + 24 * q^59 + 18 * q^61 + 3 * q^63 + 30 * q^65 - 12 * q^67 + 6 * q^69 - 12 * q^71 + 24 * q^73 + 15 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 18 * q^83 - 18 * q^85 + 12 * q^87 + 18 * q^89 + 6 * q^93 - 12 * q^95 + 24 * q^97 - 3 * 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
$5$	3.80451	1.70143	0.850715	−	0.525628i	$-0.176170\pi$
$5$	3.80451	1.70143	0.850715	+	0.525628i	$0.176170\pi$
$6$	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
$13$	3.80451	1.05518	0.527591	−	0.849499i	$-0.323095\pi$
$13$	3.80451	1.05518	0.527591	+	0.849499i	$0.323095\pi$
$14$	0	0
$15$	3.80451	0.982321
$16$	0	0
$17$	0.334898	0.0812248	0.0406124	−	0.999175i	$-0.487069\pi$
$17$	0.334898	0.0812248	0.0406124	+	0.999175i	$0.487069\pi$
$18$	0	0
$19$	−8.13941	−1.86731	−0.933654	−	0.358175i	$-0.883399\pi$
$19$	−8.13941	−1.86731	−0.933654	+	0.358175i	$0.883399\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.66510	0.347198	0.173599	−	0.984816i	$-0.444460\pi$
$23$	1.66510	0.347198	0.173599	+	0.984816i	$0.444460\pi$
$24$	0	0
$25$	9.47431	1.89486
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.195488	0.0363013	0.0181506	−	0.999835i	$-0.494222\pi$
$29$	0.195488	0.0363013	0.0181506	+	0.999835i	$0.494222\pi$
$30$	0	0
$31$	9.94392	1.78598	0.892991	−	0.450075i	$-0.148603\pi$
$31$	9.94392	1.78598	0.892991	+	0.450075i	$0.148603\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	3.80451	0.643080
$36$	0	0
$37$	−4.47431	−0.735572	−0.367786	−	0.929911i	$-0.619884\pi$
$37$	−4.47431	−0.735572	−0.367786	+	0.929911i	$0.619884\pi$
$38$	0	0
$39$	3.80451	0.609209
$40$	0	0
$41$	−6.27882	−0.980587	−0.490293	−	0.871557i	$-0.663110\pi$
$41$	−6.27882	−0.980587	−0.490293	+	0.871557i	$0.663110\pi$
$42$	0	0
$43$	−2.33490	−0.356069	−0.178034	−	0.984024i	$-0.556974\pi$
$43$	−2.33490	−0.356069	−0.178034	+	0.984024i	$0.556974\pi$
$44$	0	0
$45$	3.80451	0.567143
$46$	0	0
$47$	12.1394	1.77071	0.885357	−	0.464911i	$-0.153914\pi$
$47$	12.1394	1.77071	0.885357	+	0.464911i	$0.153914\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.334898	0.0468952
$52$	0	0
$53$	7.94392	1.09118	0.545591	−	0.838052i	$-0.316305\pi$
$53$	7.94392	1.09118	0.545591	+	0.838052i	$0.316305\pi$
$54$	0	0
$55$	−3.80451	−0.513000
$56$	0	0
$57$	−8.13941	−1.07809
$58$	0	0
$59$	−3.74843	−0.488004	−0.244002	−	0.969775i	$-0.578460\pi$
$59$	−3.74843	−0.488004	−0.244002	+	0.969775i	$0.578460\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	14.4743	1.79532
$66$	0	0
$67$	0.139410	0.0170316	0.00851582	−	0.999964i	$-0.497289\pi$
$67$	0.139410	0.0170316	0.00851582	+	0.999964i	$0.497289\pi$
$68$	0	0
$69$	1.66510	0.200455
$70$	0	0
$71$	−4.66980	−0.554203	−0.277101	−	0.960841i	$-0.589374\pi$
$71$	−4.66980	−0.554203	−0.277101	+	0.960841i	$0.589374\pi$
$72$	0	0
$73$	4.19549	0.491045	0.245522	−	0.969391i	$-0.421041\pi$
$73$	4.19549	0.491045	0.245522	+	0.969391i	$0.421041\pi$
$74$	0	0
$75$	9.47431	1.09400
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−3.33020	−0.374677	−0.187339	−	0.982295i	$-0.559986\pi$
$79$	−3.33020	−0.374677	−0.187339	+	0.982295i	$0.559986\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.9439	1.53054	0.765272	−	0.643707i	$-0.222604\pi$
$83$	13.9439	1.53054	0.765272	+	0.643707i	$0.222604\pi$
$84$	0	0
$85$	1.27412	0.138198
$86$	0	0
$87$	0.195488	0.0209586
$88$	0	0
$89$	−9.88784	−1.04811	−0.524055	−	0.851685i	$-0.675581\pi$
$89$	−9.88784	−1.04811	−0.524055	+	0.851685i	$0.675581\pi$
$90$	0	0
$91$	3.80451	0.398821
$92$	0	0
$93$	9.94392	1.03114
$94$	0	0
$95$	−30.9665	−3.17709
$96$	0	0
$97$	0.0560785	0.00569391	0.00284695	−	0.999996i	$-0.499094\pi$
$97$	0.0560785	0.00569391	0.00284695	+	0.999996i	$0.499094\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−18.8831	−1.87894	−0.939472	−	0.342626i	$-0.888683\pi$
$101$	−18.8831	−1.87894	−0.939472	+	0.342626i	$0.888683\pi$
$102$	0	0
$103$	8.27882	0.815736	0.407868	−	0.913041i	$-0.366272\pi$
$103$	8.27882	0.815736	0.407868	+	0.913041i	$0.366272\pi$
$104$	0	0
$105$	3.80451	0.371282
$106$	0	0
$107$	8.13941	0.786866	0.393433	−	0.919353i	$-0.371287\pi$
$107$	8.13941	0.786866	0.393433	+	0.919353i	$0.371287\pi$
$108$	0	0
$109$	11.5529	1.10657	0.553286	−	0.832992i	$-0.313374\pi$
$109$	11.5529	1.10657	0.553286	+	0.832992i	$0.313374\pi$
$110$	0	0
$111$	−4.47431	−0.424683
$112$	0	0
$113$	−1.33020	−0.125135	−0.0625675	−	0.998041i	$-0.519929\pi$
$113$	−1.33020	−0.125135	−0.0625675	+	0.998041i	$0.519929\pi$
$114$	0	0
$115$	6.33490	0.590732
$116$	0	0
$117$	3.80451	0.351727
$118$	0	0
$119$	0.334898	0.0307001
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.27882	−0.566142
$124$	0	0
$125$	17.0226	1.52254
$126$	0	0
$127$	−14.6137	−1.29676	−0.648379	−	0.761318i	$-0.724553\pi$
$127$	−14.6137	−1.29676	−0.648379	+	0.761318i	$0.724553\pi$
$128$	0	0
$129$	−2.33490	−0.205576
$130$	0	0
$131$	−20.5576	−1.79613	−0.898065	−	0.439863i	$-0.855027\pi$
$131$	−20.5576	−1.79613	−0.898065	+	0.439863i	$0.855027\pi$
$132$	0	0
$133$	−8.13941	−0.705776
$134$	0	0
$135$	3.80451	0.327440
$136$	0	0
$137$	−16.2227	−1.38600	−0.693001	−	0.720936i	$-0.743712\pi$
$137$	−16.2227	−1.38600	−0.693001	+	0.720936i	$0.743712\pi$
$138$	0	0
$139$	−22.5482	−1.91252	−0.956259	−	0.292522i	$-0.905506\pi$
$139$	−22.5482	−1.91252	−0.956259	+	0.292522i	$0.905506\pi$
$140$	0	0
$141$	12.1394	1.02232
$142$	0	0
$143$	−3.80451	−0.318149
$144$	0	0
$145$	0.743738	0.0617641
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	8.19549	0.671401	0.335700	−	0.941969i	$-0.391027\pi$
$149$	8.19549	0.671401	0.335700	+	0.941969i	$0.391027\pi$
$150$	0	0
$151$	13.2741	1.08023	0.540116	−	0.841590i	$-0.318380\pi$
$151$	13.2741	1.08023	0.540116	+	0.841590i	$0.318380\pi$
$152$	0	0
$153$	0.334898	0.0270749
$154$	0	0
$155$	37.8318	3.03872
$156$	0	0
$157$	−16.9392	−1.35190	−0.675949	−	0.736949i	$-0.736266\pi$
$157$	−16.9392	−1.35190	−0.675949	+	0.736949i	$0.736266\pi$
$158$	0	0
$159$	7.94392	0.629994
$160$	0	0
$161$	1.66510	0.131228
$162$	0	0
$163$	6.79982	0.532603	0.266301	−	0.963890i	$-0.414198\pi$
$163$	6.79982	0.532603	0.266301	+	0.963890i	$0.414198\pi$
$164$	0	0
$165$	−3.80451	−0.296181
$166$	0	0
$167$	18.2227	1.41012	0.705059	−	0.709149i	$-0.250920\pi$
$167$	18.2227	1.41012	0.705059	+	0.709149i	$0.250920\pi$
$168$	0	0
$169$	1.47431	0.113408
$170$	0	0
$171$	−8.13941	−0.622436
$172$	0	0
$173$	−1.72118	−0.130859	−0.0654294	−	0.997857i	$-0.520842\pi$
$173$	−1.72118	−0.130859	−0.0654294	+	0.997857i	$0.520842\pi$
$174$	0	0
$175$	9.47431	0.716190
$176$	0	0
$177$	−3.74843	−0.281749
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	0.725875	0.0539539	0.0269769	−	0.999636i	$-0.491412\pi$
$181$	0.725875	0.0539539	0.0269769	+	0.999636i	$0.491412\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−17.0226	−1.25152
$186$	0	0
$187$	−0.334898	−0.0244902
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−5.27412	−0.381622	−0.190811	−	0.981627i	$-0.561112\pi$
$191$	−5.27412	−0.381622	−0.190811	+	0.981627i	$0.561112\pi$
$192$	0	0
$193$	−19.8318	−1.42752	−0.713761	−	0.700390i	$-0.753010\pi$
$193$	−19.8318	−1.42752	−0.713761	+	0.700390i	$0.753010\pi$
$194$	0	0
$195$	14.4743	1.03653
$196$	0	0
$197$	2.66980	0.190215	0.0951076	−	0.995467i	$-0.469680\pi$
$197$	2.66980	0.190215	0.0951076	+	0.995467i	$0.469680\pi$
$198$	0	0
$199$	−13.5529	−0.960743	−0.480371	−	0.877065i	$-0.659498\pi$
$199$	−13.5529	−0.960743	−0.480371	+	0.877065i	$0.659498\pi$
$200$	0	0
$201$	0.139410	0.00983322
$202$	0	0
$203$	0.195488	0.0137206
$204$	0	0
$205$	−23.8878	−1.66840
$206$	0	0
$207$	1.66510	0.115733
$208$	0	0
$209$	8.13941	0.563015
$210$	0	0
$211$	4.27882	0.294566	0.147283	−	0.989094i	$-0.452947\pi$
$211$	4.27882	0.294566	0.147283	+	0.989094i	$0.452947\pi$
$212$	0	0
$213$	−4.66980	−0.319969
$214$	0	0
$215$	−8.88315	−0.605826
$216$	0	0
$217$	9.94392	0.675037
$218$	0	0
$219$	4.19549	0.283505
$220$	0	0
$221$	1.27412	0.0857069
$222$	0	0
$223$	10.2694	0.687692	0.343846	−	0.939026i	$-0.388270\pi$
$223$	10.2694	0.687692	0.343846	+	0.939026i	$0.388270\pi$
$224$	0	0
$225$	9.47431	0.631621
$226$	0	0
$227$	0.390977	0.0259500	0.0129750	−	0.999916i	$-0.495870\pi$
$227$	0.390977	0.0259500	0.0129750	+	0.999916i	$0.495870\pi$
$228$	0	0
$229$	7.94392	0.524949	0.262475	−	0.964939i	$-0.415461\pi$
$229$	7.94392	0.524949	0.262475	+	0.964939i	$0.415461\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	26.5576	1.73985	0.869924	−	0.493185i	$-0.164167\pi$
$233$	26.5576	1.73985	0.869924	+	0.493185i	$0.164167\pi$
$234$	0	0
$235$	46.1845	3.01275
$236$	0	0
$237$	−3.33020	−0.216320
$238$	0	0
$239$	−10.7998	−0.698582	−0.349291	−	0.937014i	$-0.613578\pi$
$239$	−10.7998	−0.698582	−0.349291	+	0.937014i	$0.613578\pi$
$240$	0	0
$241$	−12.0833	−0.778356	−0.389178	−	0.921163i	$-0.627241\pi$
$241$	−12.0833	−0.778356	−0.389178	+	0.921163i	$0.627241\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.80451	0.243061
$246$	0	0
$247$	−30.9665	−1.97035
$248$	0	0
$249$	13.9439	0.883660
$250$	0	0
$251$	−4.80921	−0.303554	−0.151777	−	0.988415i	$-0.548500\pi$
$251$	−4.80921	−0.303554	−0.151777	+	0.988415i	$0.548500\pi$
$252$	0	0
$253$	−1.66510	−0.104684
$254$	0	0
$255$	1.27412	0.0797888
$256$	0	0
$257$	−16.7531	−1.04503	−0.522516	−	0.852630i	$-0.675006\pi$
$257$	−16.7531	−1.04503	−0.522516	+	0.852630i	$0.675006\pi$
$258$	0	0
$259$	−4.47431	−0.278020
$260$	0	0
$261$	0.195488	0.0121004
$262$	0	0
$263$	12.1394	0.748548	0.374274	−	0.927318i	$-0.377892\pi$
$263$	12.1394	0.748548	0.374274	+	0.927318i	$0.377892\pi$
$264$	0	0
$265$	30.2227	1.85657
$266$	0	0
$267$	−9.88784	−0.605126
$268$	0	0
$269$	−25.2180	−1.53757	−0.768786	−	0.639506i	$-0.779139\pi$
$269$	−25.2180	−1.53757	−0.768786	+	0.639506i	$0.779139\pi$
$270$	0	0
$271$	−4.13941	−0.251451	−0.125726	−	0.992065i	$-0.540126\pi$
$271$	−4.13941	−0.251451	−0.125726	+	0.992065i	$0.540126\pi$
$272$	0	0
$273$	3.80451	0.230260
$274$	0	0
$275$	−9.47431	−0.571322
$276$	0	0
$277$	−18.2788	−1.09827	−0.549134	−	0.835734i	$-0.685042\pi$
$277$	−18.2788	−1.09827	−0.549134	+	0.835734i	$0.685042\pi$
$278$	0	0
$279$	9.94392	0.595327
$280$	0	0
$281$	6.74374	0.402298	0.201149	−	0.979561i	$-0.435533\pi$
$281$	6.74374	0.402298	0.201149	+	0.979561i	$0.435533\pi$
$282$	0	0
$283$	−23.3575	−1.38846	−0.694228	−	0.719755i	$-0.744254\pi$
$283$	−23.3575	−1.38846	−0.694228	+	0.719755i	$0.744254\pi$
$284$	0	0
$285$	−30.9665	−1.83430
$286$	0	0
$287$	−6.27882	−0.370627
$288$	0	0
$289$	−16.8878	−0.993403
$290$	0	0
$291$	0.0560785	0.00328738
$292$	0	0
$293$	−14.1667	−0.827625	−0.413813	−	0.910362i	$-0.635803\pi$
$293$	−14.1667	−0.827625	−0.413813	+	0.910362i	$0.635803\pi$
$294$	0	0
$295$	−14.2610	−0.830305
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	6.33490	0.366357
$300$	0	0
$301$	−2.33490	−0.134581
$302$	0	0
$303$	−18.8831	−1.08481
$304$	0	0
$305$	22.8271	1.30707
$306$	0	0
$307$	12.5576	0.716702	0.358351	−	0.933587i	$-0.383339\pi$
$307$	12.5576	0.716702	0.358351	+	0.933587i	$0.383339\pi$
$308$	0	0
$309$	8.27882	0.470966
$310$	0	0
$311$	−9.33959	−0.529600	−0.264800	−	0.964303i	$-0.585306\pi$
$311$	−9.33959	−0.529600	−0.264800	+	0.964303i	$0.585306\pi$
$312$	0	0
$313$	−2.99530	−0.169305	−0.0846523	−	0.996411i	$-0.526978\pi$
$313$	−2.99530	−0.169305	−0.0846523	+	0.996411i	$0.526978\pi$
$314$	0	0
$315$	3.80451	0.214360
$316$	0	0
$317$	9.93453	0.557979	0.278989	−	0.960294i	$-0.410001\pi$
$317$	9.93453	0.557979	0.278989	+	0.960294i	$0.410001\pi$
$318$	0	0
$319$	−0.195488	−0.0109453
$320$	0	0
$321$	8.13941	0.454298
$322$	0	0
$323$	−2.72588	−0.151672
$324$	0	0
$325$	36.0451	1.99942
$326$	0	0
$327$	11.5529	0.638879
$328$	0	0
$329$	12.1394	0.669267
$330$	0	0
$331$	−22.5482	−1.23936	−0.619682	−	0.784853i	$-0.712738\pi$
$331$	−22.5482	−1.23936	−0.619682	+	0.784853i	$0.712738\pi$
$332$	0	0
$333$	−4.47431	−0.245191
$334$	0	0
$335$	0.530387	0.0289781
$336$	0	0
$337$	13.2835	0.723599	0.361800	−	0.932256i	$-0.382162\pi$
$337$	13.2835	0.723599	0.361800	+	0.932256i	$0.382162\pi$
$338$	0	0
$339$	−1.33020	−0.0722467
$340$	0	0
$341$	−9.94392	−0.538494
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	6.33490	0.341059
$346$	0	0
$347$	27.7757	1.49108	0.745538	−	0.666463i	$-0.232192\pi$
$347$	27.7757	1.49108	0.745538	+	0.666463i	$0.232192\pi$
$348$	0	0
$349$	−2.85589	−0.152873	−0.0764363	−	0.997074i	$-0.524354\pi$
$349$	−2.85589	−0.152873	−0.0764363	+	0.997074i	$0.524354\pi$
$350$	0	0
$351$	3.80451	0.203070
$352$	0	0
$353$	24.6410	1.31151	0.655753	−	0.754975i	$-0.272351\pi$
$353$	24.6410	1.31151	0.655753	+	0.754975i	$0.272351\pi$
$354$	0	0
$355$	−17.7663	−0.942937
$356$	0	0
$357$	0.334898	0.0177247
$358$	0	0
$359$	11.8878	0.627416	0.313708	−	0.949519i	$-0.398429\pi$
$359$	11.8878	0.627416	0.313708	+	0.949519i	$0.398429\pi$
$360$	0	0
$361$	47.2500	2.48684
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	15.9618	0.835478
$366$	0	0
$367$	−3.60902	−0.188389	−0.0941947	−	0.995554i	$-0.530028\pi$
$367$	−3.60902	−0.188389	−0.0941947	+	0.995554i	$0.530028\pi$
$368$	0	0
$369$	−6.27882	−0.326862
$370$	0	0
$371$	7.94392	0.412428
$372$	0	0
$373$	12.3349	0.638677	0.319338	−	0.947641i	$-0.396539\pi$
$373$	12.3349	0.638677	0.319338	+	0.947641i	$0.396539\pi$
$374$	0	0
$375$	17.0226	0.879041
$376$	0	0
$377$	0.743738	0.0383045
$378$	0	0
$379$	−23.3575	−1.19979	−0.599896	−	0.800078i	$-0.704791\pi$
$379$	−23.3575	−1.19979	−0.599896	+	0.800078i	$0.704791\pi$
$380$	0	0
$381$	−14.6137	−0.748683
$382$	0	0
$383$	15.2180	0.777606	0.388803	−	0.921321i	$-0.372889\pi$
$383$	15.2180	0.777606	0.388803	+	0.921321i	$0.372889\pi$
$384$	0	0
$385$	−3.80451	−0.193896
$386$	0	0
$387$	−2.33490	−0.118690
$388$	0	0
$389$	3.73057	0.189147	0.0945737	−	0.995518i	$-0.469851\pi$
$389$	3.73057	0.189147	0.0945737	+	0.995518i	$0.469851\pi$
$390$	0	0
$391$	0.557640	0.0282011
$392$	0	0
$393$	−20.5576	−1.03700
$394$	0	0
$395$	−12.6698	−0.637487
$396$	0	0
$397$	20.8925	1.04857	0.524283	−	0.851544i	$-0.324333\pi$
$397$	20.8925	1.04857	0.524283	+	0.851544i	$0.324333\pi$
$398$	0	0
$399$	−8.13941	−0.407480
$400$	0	0
$401$	23.2741	1.16225	0.581127	−	0.813813i	$-0.302612\pi$
$401$	23.2741	1.16225	0.581127	+	0.813813i	$0.302612\pi$
$402$	0	0
$403$	37.8318	1.88453
$404$	0	0
$405$	3.80451	0.189048
$406$	0	0
$407$	4.47431	0.221783
$408$	0	0
$409$	−23.8972	−1.18164	−0.590821	−	0.806803i	$-0.701196\pi$
$409$	−23.8972	−1.18164	−0.590821	+	0.806803i	$0.701196\pi$
$410$	0	0
$411$	−16.2227	−0.800209
$412$	0	0
$413$	−3.74843	−0.184448
$414$	0	0
$415$	53.0498	2.60411
$416$	0	0
$417$	−22.5482	−1.10419
$418$	0	0
$419$	30.2967	1.48009	0.740045	−	0.672557i	$-0.234804\pi$
$419$	30.2967	1.48009	0.740045	+	0.672557i	$0.234804\pi$
$420$	0	0
$421$	−15.5257	−0.756676	−0.378338	−	0.925668i	$-0.623504\pi$
$421$	−15.5257	−0.756676	−0.378338	+	0.925668i	$0.623504\pi$
$422$	0	0
$423$	12.1394	0.590238
$424$	0	0
$425$	3.17293	0.153910
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	−3.80451	−0.183684
$430$	0	0
$431$	−3.07864	−0.148293	−0.0741463	−	0.997247i	$-0.523623\pi$
$431$	−3.07864	−0.148293	−0.0741463	+	0.997247i	$0.523623\pi$
$432$	0	0
$433$	16.9392	0.814047	0.407024	−	0.913418i	$-0.366567\pi$
$433$	16.9392	0.814047	0.407024	+	0.913418i	$0.366567\pi$
$434$	0	0
$435$	0.743738	0.0356595
$436$	0	0
$437$	−13.5529	−0.648325
$438$	0	0
$439$	11.0786	0.528754	0.264377	−	0.964419i	$-0.414834\pi$
$439$	11.0786	0.528754	0.264377	+	0.964419i	$0.414834\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	14.5482	0.691208	0.345604	−	0.938380i	$-0.387674\pi$
$443$	14.5482	0.691208	0.345604	+	0.938380i	$0.387674\pi$
$444$	0	0
$445$	−37.6184	−1.78328
$446$	0	0
$447$	8.19549	0.387633
$448$	0	0
$449$	27.4875	1.29721	0.648607	−	0.761123i	$-0.275352\pi$
$449$	27.4875	1.29721	0.648607	+	0.761123i	$0.275352\pi$
$450$	0	0
$451$	6.27882	0.295658
$452$	0	0
$453$	13.2741	0.623673
$454$	0	0
$455$	14.4743	0.678566
$456$	0	0
$457$	−7.94392	−0.371601	−0.185800	−	0.982587i	$-0.559488\pi$
$457$	−7.94392	−0.371601	−0.185800	+	0.982587i	$0.559488\pi$
$458$	0	0
$459$	0.334898	0.0156317
$460$	0	0
$461$	8.26943	0.385146	0.192573	−	0.981283i	$-0.438317\pi$
$461$	8.26943	0.385146	0.192573	+	0.981283i	$0.438317\pi$
$462$	0	0
$463$	−9.73904	−0.452612	−0.226306	−	0.974056i	$-0.572665\pi$
$463$	−9.73904	−0.452612	−0.226306	+	0.974056i	$0.572665\pi$
$464$	0	0
$465$	37.8318	1.75441
$466$	0	0
$467$	40.6970	1.88323	0.941617	−	0.336685i	$-0.109306\pi$
$467$	40.6970	1.88323	0.941617	+	0.336685i	$0.109306\pi$
$468$	0	0
$469$	0.139410	0.00643735
$470$	0	0
$471$	−16.9392	−0.780518
$472$	0	0
$473$	2.33490	0.107359
$474$	0	0
$475$	−77.1153	−3.53829
$476$	0	0
$477$	7.94392	0.363727
$478$	0	0
$479$	−37.8318	−1.72858	−0.864289	−	0.502996i	$-0.832231\pi$
$479$	−37.8318	−1.72858	−0.864289	+	0.502996i	$0.832231\pi$
$480$	0	0
$481$	−17.0226	−0.776162
$482$	0	0
$483$	1.66510	0.0757647
$484$	0	0
$485$	0.213351	0.00968778
$486$	0	0
$487$	−40.5576	−1.83784	−0.918921	−	0.394442i	$-0.870938\pi$
$487$	−40.5576	−1.83784	−0.918921	+	0.394442i	$0.870938\pi$
$488$	0	0
$489$	6.79982	0.307498
$490$	0	0
$491$	−31.0786	−1.40256	−0.701280	−	0.712886i	$-0.747388\pi$
$491$	−31.0786	−1.40256	−0.701280	+	0.712886i	$0.747388\pi$
$492$	0	0
$493$	0.0654688	0.00294856
$494$	0	0
$495$	−3.80451	−0.171000
$496$	0	0
$497$	−4.66980	−0.209469
$498$	0	0
$499$	−15.3575	−0.687494	−0.343747	−	0.939062i	$-0.611696\pi$
$499$	−15.3575	−0.687494	−0.343747	+	0.939062i	$0.611696\pi$
$500$	0	0
$501$	18.2227	0.814132
$502$	0	0
$503$	−14.8925	−0.664025	−0.332013	−	0.943275i	$-0.607728\pi$
$503$	−14.8925	−0.664025	−0.332013	+	0.943275i	$0.607728\pi$
$504$	0	0
$505$	−71.8412	−3.19689
$506$	0	0
$507$	1.47431	0.0654763
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	4.19549	0.185597
$512$	0	0
$513$	−8.13941	−0.359364
$514$	0	0
$515$	31.4969	1.38792
$516$	0	0
$517$	−12.1394	−0.533891
$518$	0	0
$519$	−1.72118	−0.0755514
$520$	0	0
$521$	27.6924	1.21322	0.606612	−	0.794998i	$-0.292528\pi$
$521$	27.6924	1.21322	0.606612	+	0.794998i	$0.292528\pi$
$522$	0	0
$523$	−18.4088	−0.804962	−0.402481	−	0.915428i	$-0.631852\pi$
$523$	−18.4088	−0.804962	−0.402481	+	0.915428i	$0.631852\pi$
$524$	0	0
$525$	9.47431	0.413493
$526$	0	0
$527$	3.33020	0.145066
$528$	0	0
$529$	−20.2274	−0.879454
$530$	0	0
$531$	−3.74843	−0.162668
$532$	0	0
$533$	−23.8878	−1.03470
$534$	0	0
$535$	30.9665	1.33880
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−4.26943	−0.183557	−0.0917786	−	0.995779i	$-0.529255\pi$
$541$	−4.26943	−0.183557	−0.0917786	+	0.995779i	$0.529255\pi$
$542$	0	0
$543$	0.725875	0.0311503
$544$	0	0
$545$	43.9533	1.88275
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−1.59116	−0.0677857
$552$	0	0
$553$	−3.33020	−0.141615
$554$	0	0
$555$	−17.0226	−0.722567
$556$	0	0
$557$	−12.4743	−0.528553	−0.264277	−	0.964447i	$-0.585133\pi$
$557$	−12.4743	−0.528553	−0.264277	+	0.964447i	$0.585133\pi$
$558$	0	0
$559$	−8.88315	−0.375717
$560$	0	0
$561$	−0.334898	−0.0141394
$562$	0	0
$563$	32.7710	1.38113	0.690566	−	0.723269i	$-0.257361\pi$
$563$	32.7710	1.38113	0.690566	+	0.723269i	$0.257361\pi$
$564$	0	0
$565$	−5.06077	−0.212908
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−29.2180	−1.22488	−0.612442	−	0.790515i	$-0.709813\pi$
$569$	−29.2180	−1.22488	−0.612442	+	0.790515i	$0.709813\pi$
$570$	0	0
$571$	−32.4455	−1.35780	−0.678901	−	0.734230i	$-0.737543\pi$
$571$	−32.4455	−1.35780	−0.678901	+	0.734230i	$0.737543\pi$
$572$	0	0
$573$	−5.27412	−0.220330
$574$	0	0
$575$	15.7757	0.657892
$576$	0	0
$577$	−14.8831	−0.619594	−0.309797	−	0.950803i	$-0.600261\pi$
$577$	−14.8831	−0.619594	−0.309797	+	0.950803i	$0.600261\pi$
$578$	0	0
$579$	−19.8318	−0.824180
$580$	0	0
$581$	13.9439	0.578491
$582$	0	0
$583$	−7.94392	−0.329004
$584$	0	0
$585$	14.4743	0.598439
$586$	0	0
$587$	4.53039	0.186989	0.0934945	−	0.995620i	$-0.470196\pi$
$587$	4.53039	0.186989	0.0934945	+	0.995620i	$0.470196\pi$
$588$	0	0
$589$	−80.9377	−3.33498
$590$	0	0
$591$	2.66980	0.109821
$592$	0	0
$593$	8.28821	0.340356	0.170178	−	0.985413i	$-0.445566\pi$
$593$	8.28821	0.340356	0.170178	+	0.985413i	$0.445566\pi$
$594$	0	0
$595$	1.27412	0.0522340
$596$	0	0
$597$	−13.5529	−0.554685
$598$	0	0
$599$	−1.99061	−0.0813341	−0.0406671	−	0.999173i	$-0.512948\pi$
$599$	−1.99061	−0.0813341	−0.0406671	+	0.999173i	$0.512948\pi$
$600$	0	0
$601$	−8.47431	−0.345674	−0.172837	−	0.984950i	$-0.555293\pi$
$601$	−8.47431	−0.345674	−0.172837	+	0.984950i	$0.555293\pi$
$602$	0	0
$603$	0.139410	0.00567721
$604$	0	0
$605$	3.80451	0.154675
$606$	0	0
$607$	22.9665	0.932181	0.466090	−	0.884737i	$-0.345662\pi$
$607$	22.9665	0.932181	0.466090	+	0.884737i	$0.345662\pi$
$608$	0	0
$609$	0.195488	0.00792159
$610$	0	0
$611$	46.1845	1.86843
$612$	0	0
$613$	3.55294	0.143502	0.0717510	−	0.997423i	$-0.477141\pi$
$613$	3.55294	0.143502	0.0717510	+	0.997423i	$0.477141\pi$
$614$	0	0
$615$	−23.8878	−0.963251
$616$	0	0
$617$	22.6698	0.912652	0.456326	−	0.889813i	$-0.349165\pi$
$617$	22.6698	0.912652	0.456326	+	0.889813i	$0.349165\pi$
$618$	0	0
$619$	−8.71648	−0.350345	−0.175173	−	0.984538i	$-0.556048\pi$
$619$	−8.71648	−0.350345	−0.175173	+	0.984538i	$0.556048\pi$
$620$	0	0
$621$	1.66510	0.0668182
$622$	0	0
$623$	−9.88784	−0.396148
$624$	0	0
$625$	17.3910	0.695639
$626$	0	0
$627$	8.13941	0.325057
$628$	0	0
$629$	−1.49844	−0.0597467
$630$	0	0
$631$	11.8878	0.473248	0.236624	−	0.971601i	$-0.423959\pi$
$631$	11.8878	0.473248	0.236624	+	0.971601i	$0.423959\pi$
$632$	0	0
$633$	4.27882	0.170068
$634$	0	0
$635$	−55.5981	−2.20634
$636$	0	0
$637$	3.80451	0.150740
$638$	0	0
$639$	−4.66980	−0.184734
$640$	0	0
$641$	−4.89254	−0.193244	−0.0966218	−	0.995321i	$-0.530804\pi$
$641$	−4.89254	−0.193244	−0.0966218	+	0.995321i	$0.530804\pi$
$642$	0	0
$643$	−22.7804	−0.898371	−0.449185	−	0.893439i	$-0.648286\pi$
$643$	−22.7804	−0.898371	−0.449185	+	0.893439i	$0.648286\pi$
$644$	0	0
$645$	−8.88315	−0.349774
$646$	0	0
$647$	−31.2453	−1.22838	−0.614190	−	0.789158i	$-0.710517\pi$
$647$	−31.2453	−1.22838	−0.614190	+	0.789158i	$0.710517\pi$
$648$	0	0
$649$	3.74843	0.147139
$650$	0	0
$651$	9.94392	0.389733
$652$	0	0
$653$	28.2882	1.10700	0.553502	−	0.832848i	$-0.313291\pi$
$653$	28.2882	1.10700	0.553502	+	0.832848i	$0.313291\pi$
$654$	0	0
$655$	−78.2118	−3.05599
$656$	0	0
$657$	4.19549	0.163682
$658$	0	0
$659$	−30.2967	−1.18019	−0.590096	−	0.807333i	$-0.700910\pi$
$659$	−30.2967	−1.18019	−0.590096	+	0.807333i	$0.700910\pi$
$660$	0	0
$661$	−35.7196	−1.38933	−0.694666	−	0.719333i	$-0.744448\pi$
$661$	−35.7196	−1.38933	−0.694666	+	0.719333i	$0.744448\pi$
$662$	0	0
$663$	1.27412	0.0494829
$664$	0	0
$665$	−30.9665	−1.20083
$666$	0	0
$667$	0.325508	0.0126037
$668$	0	0
$669$	10.2694	0.397039
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	34.9377	1.34675	0.673374	−	0.739302i	$-0.264844\pi$
$673$	34.9377	1.34675	0.673374	+	0.739302i	$0.264844\pi$
$674$	0	0
$675$	9.47431	0.364666
$676$	0	0
$677$	20.0094	0.769023	0.384512	−	0.923120i	$-0.374370\pi$
$677$	20.0094	0.769023	0.384512	+	0.923120i	$0.374370\pi$
$678$	0	0
$679$	0.0560785	0.00215209
$680$	0	0
$681$	0.390977	0.0149823
$682$	0	0
$683$	−25.0498	−0.958504	−0.479252	−	0.877677i	$-0.659092\pi$
$683$	−25.0498	−0.958504	−0.479252	+	0.877677i	$0.659092\pi$
$684$	0	0
$685$	−61.7196	−2.35818
$686$	0	0
$687$	7.94392	0.303080
$688$	0	0
$689$	30.2227	1.15139
$690$	0	0
$691$	38.8271	1.47705	0.738526	−	0.674225i	$-0.235522\pi$
$691$	38.8271	1.47705	0.738526	+	0.674225i	$0.235522\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−85.7851	−3.25401
$696$	0	0
$697$	−2.10277	−0.0796480
$698$	0	0
$699$	26.5576	1.00450
$700$	0	0
$701$	−41.7757	−1.57785	−0.788923	−	0.614492i	$-0.789361\pi$
$701$	−41.7757	−1.57785	−0.788923	+	0.614492i	$0.789361\pi$
$702$	0	0
$703$	36.4182	1.37354
$704$	0	0
$705$	46.1845	1.73941
$706$	0	0
$707$	−18.8831	−0.710174
$708$	0	0
$709$	−13.4041	−0.503403	−0.251702	−	0.967805i	$-0.580990\pi$
$709$	−13.4041	−0.503403	−0.251702	+	0.967805i	$0.580990\pi$
$710$	0	0
$711$	−3.33020	−0.124892
$712$	0	0
$713$	16.5576	0.620088
$714$	0	0
$715$	−14.4743	−0.541308
$716$	0	0
$717$	−10.7998	−0.403327
$718$	0	0
$719$	−5.47900	−0.204332	−0.102166	−	0.994767i	$-0.532577\pi$
$719$	−5.47900	−0.204332	−0.102166	+	0.994767i	$0.532577\pi$
$720$	0	0
$721$	8.27882	0.308319
$722$	0	0
$723$	−12.0833	−0.449384
$724$	0	0
$725$	1.85212	0.0687859
$726$	0	0
$727$	32.8831	1.21957	0.609784	−	0.792567i	$-0.291256\pi$
$727$	32.8831	1.21957	0.609784	+	0.792567i	$0.291256\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.781954	−0.0289216
$732$	0	0
$733$	−35.8972	−1.32589	−0.662947	−	0.748666i	$-0.730695\pi$
$733$	−35.8972	−1.32589	−0.662947	+	0.748666i	$0.730695\pi$
$734$	0	0
$735$	3.80451	0.140332
$736$	0	0
$737$	−0.139410	−0.00513523
$738$	0	0
$739$	−29.6651	−1.09125	−0.545624	−	0.838030i	$-0.683707\pi$
$739$	−29.6651	−1.09125	−0.545624	+	0.838030i	$0.683707\pi$
$740$	0	0
$741$	−30.9665	−1.13758
$742$	0	0
$743$	18.7998	0.689698	0.344849	−	0.938658i	$-0.387930\pi$
$743$	18.7998	0.689698	0.344849	+	0.938658i	$0.387930\pi$
$744$	0	0
$745$	31.1798	1.14234
$746$	0	0
$747$	13.9439	0.510181
$748$	0	0
$749$	8.13941	0.297408
$750$	0	0
$751$	9.59116	0.349986	0.174993	−	0.984570i	$-0.444010\pi$
$751$	9.59116	0.349986	0.174993	+	0.984570i	$0.444010\pi$
$752$	0	0
$753$	−4.80921	−0.175257
$754$	0	0
$755$	50.5016	1.83794
$756$	0	0
$757$	2.74374	0.0997229	0.0498614	−	0.998756i	$-0.484122\pi$
$757$	2.74374	0.0997229	0.0498614	+	0.998756i	$0.484122\pi$
$758$	0	0
$759$	−1.66510	−0.0604394
$760$	0	0
$761$	6.39098	0.231673	0.115836	−	0.993268i	$-0.463045\pi$
$761$	6.39098	0.231673	0.115836	+	0.993268i	$0.463045\pi$
$762$	0	0
$763$	11.5529	0.418245
$764$	0	0
$765$	1.27412	0.0460661
$766$	0	0
$767$	−14.2610	−0.514933
$768$	0	0
$769$	17.7951	0.641708	0.320854	−	0.947129i	$-0.396030\pi$
$769$	17.7951	0.641708	0.320854	+	0.947129i	$0.396030\pi$
$770$	0	0
$771$	−16.7531	−0.603349
$772$	0	0
$773$	−31.5257	−1.13390	−0.566950	−	0.823752i	$-0.691877\pi$
$773$	−31.5257	−1.13390	−0.566950	+	0.823752i	$0.691877\pi$
$774$	0	0
$775$	94.2118	3.38419
$776$	0	0
$777$	−4.47431	−0.160515
$778$	0	0
$779$	51.1059	1.83106
$780$	0	0
$781$	4.66980	0.167098
$782$	0	0
$783$	0.195488	0.00698619
$784$	0	0
$785$	−64.4455	−2.30016
$786$	0	0
$787$	−31.8606	−1.13571	−0.567854	−	0.823130i	$-0.692226\pi$
$787$	−31.8606	−1.13571	−0.567854	+	0.823130i	$0.692226\pi$
$788$	0	0
$789$	12.1394	0.432174
$790$	0	0
$791$	−1.33020	−0.0472966
$792$	0	0
$793$	22.8271	0.810613
$794$	0	0
$795$	30.2227	1.07189
$796$	0	0
$797$	43.8590	1.55357	0.776783	−	0.629768i	$-0.216850\pi$
$797$	43.8590	1.55357	0.776783	+	0.629768i	$0.216850\pi$
$798$	0	0
$799$	4.06547	0.143826
$800$	0	0
$801$	−9.88784	−0.349370
$802$	0	0
$803$	−4.19549	−0.148056
$804$	0	0
$805$	6.33490	0.223276
$806$	0	0
$807$	−25.2180	−0.887717
$808$	0	0
$809$	6.74374	0.237097	0.118549	−	0.992948i	$-0.462176\pi$
$809$	6.74374	0.237097	0.118549	+	0.992948i	$0.462176\pi$
$810$	0	0
$811$	−3.97275	−0.139502	−0.0697510	−	0.997564i	$-0.522220\pi$
$811$	−3.97275	−0.139502	−0.0697510	+	0.997564i	$0.522220\pi$
$812$	0	0
$813$	−4.13941	−0.145175
$814$	0	0
$815$	25.8700	0.906186
$816$	0	0
$817$	19.0047	0.664890
$818$	0	0
$819$	3.80451	0.132940
$820$	0	0
$821$	49.5896	1.73069	0.865344	−	0.501178i	$-0.167100\pi$
$821$	49.5896	1.73069	0.865344	+	0.501178i	$0.167100\pi$
$822$	0	0
$823$	−23.1908	−0.808380	−0.404190	−	0.914675i	$-0.632447\pi$
$823$	−23.1908	−0.808380	−0.404190	+	0.914675i	$0.632447\pi$
$824$	0	0
$825$	−9.47431	−0.329853
$826$	0	0
$827$	−27.7484	−0.964908	−0.482454	−	0.875921i	$-0.660254\pi$
$827$	−27.7484	−0.964908	−0.482454	+	0.875921i	$0.660254\pi$
$828$	0	0
$829$	0.269430	0.00935768	0.00467884	−	0.999989i	$-0.498511\pi$
$829$	0.269430	0.00935768	0.00467884	+	0.999989i	$0.498511\pi$
$830$	0	0
$831$	−18.2788	−0.634085
$832$	0	0
$833$	0.334898	0.0116035
$834$	0	0
$835$	69.3286	2.39922
$836$	0	0
$837$	9.94392	0.343712
$838$	0	0
$839$	27.3575	0.944484	0.472242	−	0.881469i	$-0.343445\pi$
$839$	27.3575	0.944484	0.472242	+	0.881469i	$0.343445\pi$
$840$	0	0
$841$	−28.9618	−0.998682
$842$	0	0
$843$	6.74374	0.232267
$844$	0	0
$845$	5.60902	0.192956
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−23.3575	−0.801626
$850$	0	0
$851$	−7.45018	−0.255389
$852$	0	0
$853$	−28.5482	−0.977473	−0.488737	−	0.872431i	$-0.662542\pi$
$853$	−28.5482	−0.977473	−0.488737	+	0.872431i	$0.662542\pi$
$854$	0	0
$855$	−30.9665	−1.05903
$856$	0	0
$857$	−39.8972	−1.36286	−0.681432	−	0.731882i	$-0.738642\pi$
$857$	−39.8972	−1.36286	−0.681432	+	0.731882i	$0.738642\pi$
$858$	0	0
$859$	−20.5576	−0.701418	−0.350709	−	0.936485i	$-0.614059\pi$
$859$	−20.5576	−0.701418	−0.350709	+	0.936485i	$0.614059\pi$
$860$	0	0
$861$	−6.27882	−0.213982
$862$	0	0
$863$	9.66510	0.329004	0.164502	−	0.986377i	$-0.447398\pi$
$863$	9.66510	0.329004	0.164502	+	0.986377i	$0.447398\pi$
$864$	0	0
$865$	−6.54825	−0.222647
$866$	0	0
$867$	−16.8878	−0.573541
$868$	0	0
$869$	3.33020	0.112969
$870$	0	0
$871$	0.530387	0.0179715
$872$	0	0
$873$	0.0560785	0.00189797
$874$	0	0
$875$	17.0226	0.575467
$876$	0	0
$877$	−25.7212	−0.868543	−0.434271	−	0.900782i	$-0.642994\pi$
$877$	−25.7212	−0.868543	−0.434271	+	0.900782i	$0.642994\pi$
$878$	0	0
$879$	−14.1667	−0.477830
$880$	0	0
$881$	−14.6316	−0.492950	−0.246475	−	0.969149i	$-0.579272\pi$
$881$	−14.6316	−0.492950	−0.246475	+	0.969149i	$0.579272\pi$
$882$	0	0
$883$	−18.6877	−0.628890	−0.314445	−	0.949276i	$-0.601818\pi$
$883$	−18.6877	−0.628890	−0.314445	+	0.949276i	$0.601818\pi$
$884$	0	0
$885$	−14.2610	−0.479377
$886$	0	0
$887$	−38.6137	−1.29652	−0.648261	−	0.761418i	$-0.724503\pi$
$887$	−38.6137	−1.29652	−0.648261	+	0.761418i	$0.724503\pi$
$888$	0	0
$889$	−14.6137	−0.490128
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−98.8076	−3.30647
$894$	0	0
$895$	45.6541	1.52605
$896$	0	0
$897$	6.33490	0.211516
$898$	0	0
$899$	1.94392	0.0648334
$900$	0	0
$901$	2.66041	0.0886310
$902$	0	0
$903$	−2.33490	−0.0777006
$904$	0	0
$905$	2.76160	0.0917987
$906$	0	0
$907$	−49.0965	−1.63022	−0.815111	−	0.579304i	$-0.803324\pi$
$907$	−49.0965	−1.63022	−0.815111	+	0.579304i	$0.803324\pi$
$908$	0	0
$909$	−18.8831	−0.626314
$910$	0	0
$911$	−29.3753	−0.973248	−0.486624	−	0.873612i	$-0.661772\pi$
$911$	−29.3753	−0.973248	−0.486624	+	0.873612i	$0.661772\pi$
$912$	0	0
$913$	−13.9439	−0.461476
$914$	0	0
$915$	22.8271	0.754640
$916$	0	0
$917$	−20.5576	−0.678873
$918$	0	0
$919$	27.0047	0.890803	0.445401	−	0.895331i	$-0.353061\pi$
$919$	27.0047	0.890803	0.445401	+	0.895331i	$0.353061\pi$
$920$	0	0
$921$	12.5576	0.413788
$922$	0	0
$923$	−17.7663	−0.584785
$924$	0	0
$925$	−42.3910	−1.39381
$926$	0	0
$927$	8.27882	0.271912
$928$	0	0
$929$	57.8496	1.89798	0.948992	−	0.315299i	$-0.102105\pi$
$929$	57.8496	1.89798	0.948992	+	0.315299i	$0.102105\pi$
$930$	0	0
$931$	−8.13941	−0.266758
$932$	0	0
$933$	−9.33959	−0.305765
$934$	0	0
$935$	−1.27412	−0.0416683
$936$	0	0
$937$	25.2180	0.823838	0.411919	−	0.911221i	$-0.364859\pi$
$937$	25.2180	0.823838	0.411919	+	0.911221i	$0.364859\pi$
$938$	0	0
$939$	−2.99530	−0.0977481
$940$	0	0
$941$	−34.2788	−1.11746	−0.558729	−	0.829350i	$-0.688711\pi$
$941$	−34.2788	−1.11746	−0.558729	+	0.829350i	$0.688711\pi$
$942$	0	0
$943$	−10.4549	−0.340458
$944$	0	0
$945$	3.80451	0.123761
$946$	0	0
$947$	18.1573	0.590032	0.295016	−	0.955492i	$-0.404675\pi$
$947$	18.1573	0.590032	0.295016	+	0.955492i	$0.404675\pi$
$948$	0	0
$949$	15.9618	0.518141
$950$	0	0
$951$	9.93453	0.322149
$952$	0	0
$953$	−19.8045	−0.641531	−0.320766	−	0.947159i	$-0.603940\pi$
$953$	−19.8045	−0.641531	−0.320766	+	0.947159i	$0.603940\pi$
$954$	0	0
$955$	−20.0655	−0.649303
$956$	0	0
$957$	−0.195488	−0.00631924
$958$	0	0
$959$	−16.2227	−0.523860
$960$	0	0
$961$	67.8816	2.18973
$962$	0	0
$963$	8.13941	0.262289
$964$	0	0
$965$	−75.4502	−2.42883
$966$	0	0
$967$	0.278820	0.00896624	0.00448312	−	0.999990i	$-0.498573\pi$
$967$	0.278820	0.00896624	0.00448312	+	0.999990i	$0.498573\pi$
$968$	0	0
$969$	−2.72588	−0.0875677
$970$	0	0
$971$	−25.3481	−0.813458	−0.406729	−	0.913549i	$-0.633331\pi$
$971$	−25.3481	−0.813458	−0.406729	+	0.913549i	$0.633331\pi$
$972$	0	0
$973$	−22.5482	−0.722864
$974$	0	0
$975$	36.0451	1.15437
$976$	0	0
$977$	6.71648	0.214879	0.107440	−	0.994212i	$-0.465735\pi$
$977$	6.71648	0.214879	0.107440	+	0.994212i	$0.465735\pi$
$978$	0	0
$979$	9.88784	0.316017
$980$	0	0
$981$	11.5529	0.368857
$982$	0	0
$983$	−9.99061	−0.318651	−0.159325	−	0.987226i	$-0.550932\pi$
$983$	−9.99061	−0.318651	−0.159325	+	0.987226i	$0.550932\pi$
$984$	0	0
$985$	10.1573	0.323638
$986$	0	0
$987$	12.1394	0.386402
$988$	0	0
$989$	−3.88784	−0.123626
$990$	0	0
$991$	26.7064	0.848358	0.424179	−	0.905578i	$-0.360563\pi$
$991$	26.7064	0.848358	0.424179	+	0.905578i	$0.360563\pi$
$992$	0	0
$993$	−22.5482	−0.715547
$994$	0	0
$995$	−51.5623	−1.63464
$996$	0	0
$997$	54.3333	1.72075	0.860377	−	0.509658i	$-0.170228\pi$
$997$	54.3333	1.72075	0.860377	+	0.509658i	$0.170228\pi$
$998$	0	0
$999$	−4.47431	−0.141561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bp.1.3		3
4.3	odd	2		231.2.a.d.1.2	✓	3
12.11	even	2		693.2.a.m.1.2		3
20.19	odd	2		5775.2.a.bw.1.2		3
28.27	even	2		1617.2.a.s.1.2		3
44.43	even	2		2541.2.a.bi.1.2		3
84.83	odd	2		4851.2.a.bp.1.2		3
132.131	odd	2		7623.2.a.cb.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.2	✓	3	4.3	odd	2
693.2.a.m.1.2		3	12.11	even	2
1617.2.a.s.1.2		3	28.27	even	2
2541.2.a.bi.1.2		3	44.43	even	2
3696.2.a.bp.1.3		3	1.1	even	1	trivial
4851.2.a.bp.1.2		3	84.83	odd	2
5775.2.a.bw.1.2		3	20.19	odd	2
7623.2.a.cb.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.80451 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.80451 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.80451 q^{13} +3.80451 q^{15} +0.334898 q^{17} -8.13941 q^{19} +1.00000 q^{21} +1.66510 q^{23} +9.47431 q^{25} +1.00000 q^{27} +0.195488 q^{29} +9.94392 q^{31} -1.00000 q^{33} +3.80451 q^{35} -4.47431 q^{37} +3.80451 q^{39} -6.27882 q^{41} -2.33490 q^{43} +3.80451 q^{45} +12.1394 q^{47} +1.00000 q^{49} +0.334898 q^{51} +7.94392 q^{53} -3.80451 q^{55} -8.13941 q^{57} -3.74843 q^{59} +6.00000 q^{61} +1.00000 q^{63} +14.4743 q^{65} +0.139410 q^{67} +1.66510 q^{69} -4.66980 q^{71} +4.19549 q^{73} +9.47431 q^{75} -1.00000 q^{77} -3.33020 q^{79} +1.00000 q^{81} +13.9439 q^{83} +1.27412 q^{85} +0.195488 q^{87} -9.88784 q^{89} +3.80451 q^{91} +9.94392 q^{93} -30.9665 q^{95} +0.0560785 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 3 q^{11} - 12 q^{19} + 3 q^{21} + 6 q^{23} + 15 q^{25} + 3 q^{27} + 12 q^{29} + 6 q^{31} - 3 q^{33} + 6 q^{41} - 6 q^{43} + 24 q^{47} + 3 q^{49} - 12 q^{57} + 24 q^{59} + 18 q^{61} + 3 q^{63} + 30 q^{65} - 12 q^{67} + 6 q^{69} - 12 q^{71} + 24 q^{73} + 15 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 18 q^{83} - 18 q^{85} + 12 q^{87} + 18 q^{89} + 6 q^{93} - 12 q^{95} + 24 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 3 * q^11 - 12 * q^19 + 3 * q^21 + 6 * q^23 + 15 * q^25 + 3 * q^27 + 12 * q^29 + 6 * q^31 - 3 * q^33 + 6 * q^41 - 6 * q^43 + 24 * q^47 + 3 * q^49 - 12 * q^57 + 24 * q^59 + 18 * q^61 + 3 * q^63 + 30 * q^65 - 12 * q^67 + 6 * q^69 - 12 * q^71 + 24 * q^73 + 15 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 18 * q^83 - 18 * q^85 + 12 * q^87 + 18 * q^89 + 6 * q^93 - 12 * q^95 + 24 * q^97 - 3 * q^99

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