LMFDB - Embedded newform 9576.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.23607 q^{5} -1.00000 q^{7} -5.23607 q^{11} -0.763932 q^{13} +7.23607 q^{17} +1.00000 q^{19} -3.23607 q^{23} -3.47214 q^{25} +8.47214 q^{29} +0.472136 q^{31} +1.23607 q^{35} +8.94427 q^{37} +2.00000 q^{41} -4.00000 q^{43} -6.47214 q^{47} +1.00000 q^{49} -0.472136 q^{53} +6.47214 q^{55} -8.00000 q^{59} +8.47214 q^{61} +0.944272 q^{65} +0.763932 q^{67} -1.52786 q^{71} +4.47214 q^{73} +5.23607 q^{77} +15.7082 q^{79} +2.00000 q^{83} -8.94427 q^{85} -10.9443 q^{89} +0.763932 q^{91} -1.23607 q^{95} +1.23607 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{5} - 2 q^{7} - 6 q^{11} - 6 q^{13} + 10 q^{17} + 2 q^{19} - 2 q^{23} + 2 q^{25} + 8 q^{29} - 8 q^{31} - 2 q^{35} + 4 q^{41} - 8 q^{43} - 4 q^{47} + 2 q^{49} + 8 q^{53} + 4 q^{55} - 16 q^{59} + 8 q^{61} - 16 q^{65} + 6 q^{67} - 12 q^{71} + 6 q^{77} + 18 q^{79} + 4 q^{83} - 4 q^{89} + 6 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 6 * q^11 - 6 * q^13 + 10 * q^17 + 2 * q^19 - 2 * q^23 + 2 * q^25 + 8 * q^29 - 8 * q^31 - 2 * q^35 + 4 * q^41 - 8 * q^43 - 4 * q^47 + 2 * q^49 + 8 * q^53 + 4 * q^55 - 16 * q^59 + 8 * q^61 - 16 * q^65 + 6 * q^67 - 12 * q^71 + 6 * q^77 + 18 * q^79 + 4 * q^83 - 4 * q^89 + 6 * q^91 + 2 * q^95 - 2 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	0	0
$13$	−0.763932	−0.211877	−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932	−0.211877	−0.105938	+	0.994373i	$0.533785\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.23607	1.75500	0.877502	−	0.479573i	$-0.159208\pi$
$17$	7.23607	1.75500	0.877502	+	0.479573i	$0.159208\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.23607	−0.674767	−0.337383	−	0.941367i	$-0.609542\pi$
$23$	−3.23607	−0.674767	−0.337383	+	0.941367i	$0.609542\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	0.472136	0.0847981	0.0423991	−	0.999101i	$-0.486500\pi$
$31$	0.472136	0.0847981	0.0423991	+	0.999101i	$0.486500\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.23607	0.208934
$36$	0	0
$37$	8.94427	1.47043	0.735215	−	0.677834i	$-0.237081\pi$
$37$	8.94427	1.47043	0.735215	+	0.677834i	$0.237081\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.47214	−0.944058	−0.472029	−	0.881583i	$-0.656478\pi$
$47$	−6.47214	−0.944058	−0.472029	+	0.881583i	$0.656478\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	6.47214	0.872703
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	8.47214	1.08475	0.542373	−	0.840138i	$-0.317526\pi$
$61$	8.47214	1.08475	0.542373	+	0.840138i	$0.317526\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.944272	0.117123
$66$	0	0
$67$	0.763932	0.0933292	0.0466646	−	0.998911i	$-0.485141\pi$
$67$	0.763932	0.0933292	0.0466646	+	0.998911i	$0.485141\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.52786	−0.181324	−0.0906621	−	0.995882i	$-0.528898\pi$
$71$	−1.52786	−0.181324	−0.0906621	+	0.995882i	$0.528898\pi$
$72$	0	0
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.23607	0.596705
$78$	0	0
$79$	15.7082	1.76731	0.883656	−	0.468138i	$-0.155075\pi$
$79$	15.7082	1.76731	0.883656	+	0.468138i	$0.155075\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−8.94427	−0.970143
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.9443	−1.16009	−0.580045	−	0.814584i	$-0.696965\pi$
$89$	−10.9443	−1.16009	−0.580045	+	0.814584i	$0.696965\pi$
$90$	0	0
$91$	0.763932	0.0800818
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.23607	−0.126818
$96$	0	0
$97$	1.23607	0.125504	0.0627518	−	0.998029i	$-0.480012\pi$
$97$	1.23607	0.125504	0.0627518	+	0.998029i	$0.480012\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.1803	−1.80901	−0.904506	−	0.426461i	$-0.859760\pi$
$101$	−18.1803	−1.80901	−0.904506	+	0.426461i	$0.859760\pi$
$102$	0	0
$103$	−7.52786	−0.741742	−0.370871	−	0.928684i	$-0.620941\pi$
$103$	−7.52786	−0.741742	−0.370871	+	0.928684i	$0.620941\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.4164	1.10367	0.551833	−	0.833955i	$-0.313929\pi$
$107$	11.4164	1.10367	0.551833	+	0.833955i	$0.313929\pi$
$108$	0	0
$109$	8.94427	0.856706	0.428353	−	0.903612i	$-0.359094\pi$
$109$	8.94427	0.856706	0.428353	+	0.903612i	$0.359094\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.94427	−0.653262	−0.326631	−	0.945152i	$-0.605913\pi$
$113$	−6.94427	−0.653262	−0.326631	+	0.945152i	$0.605913\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.23607	−0.663329
$120$	0	0
$121$	16.4164	1.49240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	−6.76393	−0.600202	−0.300101	−	0.953907i	$-0.597020\pi$
$127$	−6.76393	−0.600202	−0.300101	+	0.953907i	$0.597020\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.472136	−0.0412507	−0.0206254	−	0.999787i	$-0.506566\pi$
$131$	−0.472136	−0.0412507	−0.0206254	+	0.999787i	$0.506566\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.47214	−0.552952	−0.276476	−	0.961021i	$-0.589167\pi$
$137$	−6.47214	−0.552952	−0.276476	+	0.961021i	$0.589167\pi$
$138$	0	0
$139$	−8.94427	−0.758643	−0.379322	−	0.925265i	$-0.623843\pi$
$139$	−8.94427	−0.758643	−0.379322	+	0.925265i	$0.623843\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	−10.4721	−0.869664
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.4721	−1.34945	−0.674725	−	0.738069i	$-0.735738\pi$
$149$	−16.4721	−1.34945	−0.674725	+	0.738069i	$0.735738\pi$
$150$	0	0
$151$	−0.291796	−0.0237460	−0.0118730	−	0.999930i	$-0.503779\pi$
$151$	−0.291796	−0.0237460	−0.0118730	+	0.999930i	$0.503779\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.583592	−0.0468752
$156$	0	0
$157$	−16.4721	−1.31462	−0.657310	−	0.753620i	$-0.728306\pi$
$157$	−16.4721	−1.31462	−0.657310	+	0.753620i	$0.728306\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.23607	0.255038
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.9443	−1.31119	−0.655594	−	0.755114i	$-0.727582\pi$
$167$	−16.9443	−1.31119	−0.655594	+	0.755114i	$0.727582\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.8885	1.51210	0.756049	−	0.654515i	$-0.227127\pi$
$173$	19.8885	1.51210	0.756049	+	0.654515i	$0.227127\pi$
$174$	0	0
$175$	3.47214	0.262469
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	12.7639	0.948736	0.474368	−	0.880327i	$-0.342677\pi$
$181$	12.7639	0.948736	0.474368	+	0.880327i	$0.342677\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.0557	−0.812833
$186$	0	0
$187$	−37.8885	−2.77068
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.6525	0.770786	0.385393	−	0.922753i	$-0.374066\pi$
$191$	10.6525	0.770786	0.385393	+	0.922753i	$0.374066\pi$
$192$	0	0
$193$	18.9443	1.36364	0.681819	−	0.731521i	$-0.261189\pi$
$193$	18.9443	1.36364	0.681819	+	0.731521i	$0.261189\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.4164	−1.24087	−0.620434	−	0.784259i	$-0.713043\pi$
$197$	−17.4164	−1.24087	−0.620434	+	0.784259i	$0.713043\pi$
$198$	0	0
$199$	−14.4721	−1.02590	−0.512951	−	0.858418i	$-0.671448\pi$
$199$	−14.4721	−1.02590	−0.512951	+	0.858418i	$0.671448\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.47214	−0.594627
$204$	0	0
$205$	−2.47214	−0.172661
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.23607	−0.362186
$210$	0	0
$211$	−16.1803	−1.11390	−0.556950	−	0.830546i	$-0.688029\pi$
$211$	−16.1803	−1.11390	−0.556950	+	0.830546i	$0.688029\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.94427	0.337197
$216$	0	0
$217$	−0.472136	−0.0320507
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.52786	−0.371844
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.4164	1.02322	0.511611	−	0.859217i	$-0.329049\pi$
$227$	15.4164	1.02322	0.511611	+	0.859217i	$0.329049\pi$
$228$	0	0
$229$	−1.05573	−0.0697645	−0.0348822	−	0.999391i	$-0.511106\pi$
$229$	−1.05573	−0.0697645	−0.0348822	+	0.999391i	$0.511106\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.3607	−1.59592	−0.797961	−	0.602710i	$-0.794088\pi$
$233$	−24.3607	−1.59592	−0.797961	+	0.602710i	$0.794088\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−27.2361	−1.76175	−0.880877	−	0.473345i	$-0.843047\pi$
$239$	−27.2361	−1.76175	−0.880877	+	0.473345i	$0.843047\pi$
$240$	0	0
$241$	2.76393	0.178041	0.0890203	−	0.996030i	$-0.471626\pi$
$241$	2.76393	0.178041	0.0890203	+	0.996030i	$0.471626\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.23607	−0.0789695
$246$	0	0
$247$	−0.763932	−0.0486078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.47214	0.534756	0.267378	−	0.963592i	$-0.413843\pi$
$251$	8.47214	0.534756	0.267378	+	0.963592i	$0.413843\pi$
$252$	0	0
$253$	16.9443	1.06528
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.4721	−1.52653	−0.763265	−	0.646086i	$-0.776405\pi$
$257$	−24.4721	−1.52653	−0.763265	+	0.646086i	$0.776405\pi$
$258$	0	0
$259$	−8.94427	−0.555770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.18034	0.504421	0.252211	−	0.967672i	$-0.418842\pi$
$263$	8.18034	0.504421	0.252211	+	0.967672i	$0.418842\pi$
$264$	0	0
$265$	0.583592	0.0358498
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.9443	−0.667284	−0.333642	−	0.942700i	$-0.608278\pi$
$269$	−10.9443	−0.667284	−0.333642	+	0.942700i	$0.608278\pi$
$270$	0	0
$271$	−24.9443	−1.51526	−0.757628	−	0.652686i	$-0.773642\pi$
$271$	−24.9443	−1.51526	−0.757628	+	0.652686i	$0.773642\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.1803	1.09632
$276$	0	0
$277$	−11.8885	−0.714313	−0.357157	−	0.934044i	$-0.616254\pi$
$277$	−11.8885	−0.714313	−0.357157	+	0.934044i	$0.616254\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.05573	−0.540219	−0.270110	−	0.962830i	$-0.587060\pi$
$281$	−9.05573	−0.540219	−0.270110	+	0.962830i	$0.587060\pi$
$282$	0	0
$283$	−7.05573	−0.419419	−0.209710	−	0.977764i	$-0.567252\pi$
$283$	−7.05573	−0.419419	−0.209710	+	0.977764i	$0.567252\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	35.3607	2.08004
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	9.88854	0.575733
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.47214	0.142967
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.4721	−0.599633
$306$	0	0
$307$	20.3607	1.16205	0.581023	−	0.813887i	$-0.302653\pi$
$307$	20.3607	1.16205	0.581023	+	0.813887i	$0.302653\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.9443	−0.734002	−0.367001	−	0.930220i	$-0.619615\pi$
$311$	−12.9443	−0.734002	−0.367001	+	0.930220i	$0.619615\pi$
$312$	0	0
$313$	22.9443	1.29689	0.648443	−	0.761263i	$-0.275420\pi$
$313$	22.9443	1.29689	0.648443	+	0.761263i	$0.275420\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−44.3607	−2.48372
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.23607	0.402626
$324$	0	0
$325$	2.65248	0.147133
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.47214	0.356820
$330$	0	0
$331$	19.2361	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$331$	19.2361	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.944272	−0.0515911
$336$	0	0
$337$	−31.3050	−1.70529	−0.852645	−	0.522491i	$-0.825003\pi$
$337$	−31.3050	−1.70529	−0.852645	+	0.522491i	$0.825003\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.47214	−0.133874
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.70820	−0.199067	−0.0995334	−	0.995034i	$-0.531735\pi$
$347$	−3.70820	−0.199067	−0.0995334	+	0.995034i	$0.531735\pi$
$348$	0	0
$349$	−24.8328	−1.32927	−0.664635	−	0.747168i	$-0.731413\pi$
$349$	−24.8328	−1.32927	−0.664635	+	0.747168i	$0.731413\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.70820	−0.516716	−0.258358	−	0.966049i	$-0.583181\pi$
$353$	−9.70820	−0.516716	−0.258358	+	0.966049i	$0.583181\pi$
$354$	0	0
$355$	1.88854	0.100233
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.70820	−0.301267	−0.150634	−	0.988590i	$-0.548131\pi$
$359$	−5.70820	−0.301267	−0.150634	+	0.988590i	$0.548131\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.52786	−0.289342
$366$	0	0
$367$	16.9443	0.884484	0.442242	−	0.896896i	$-0.354183\pi$
$367$	16.9443	0.884484	0.442242	+	0.896896i	$0.354183\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.472136	0.0245121
$372$	0	0
$373$	−28.3607	−1.46846	−0.734230	−	0.678901i	$-0.762457\pi$
$373$	−28.3607	−1.46846	−0.734230	+	0.678901i	$0.762457\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.47214	−0.333332
$378$	0	0
$379$	32.1803	1.65299	0.826497	−	0.562942i	$-0.190331\pi$
$379$	32.1803	1.65299	0.826497	+	0.562942i	$0.190331\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.05573	−0.156140	−0.0780702	−	0.996948i	$-0.524876\pi$
$383$	−3.05573	−0.156140	−0.0780702	+	0.996948i	$0.524876\pi$
$384$	0	0
$385$	−6.47214	−0.329851
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−23.4164	−1.18422
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−19.4164	−0.976946
$396$	0	0
$397$	−34.3607	−1.72451	−0.862257	−	0.506472i	$-0.830949\pi$
$397$	−34.3607	−1.72451	−0.862257	+	0.506472i	$0.830949\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.05573	0.252471	0.126236	−	0.992000i	$-0.459710\pi$
$401$	5.05573	0.252471	0.126236	+	0.992000i	$0.459710\pi$
$402$	0	0
$403$	−0.360680	−0.0179667
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−46.8328	−2.32142
$408$	0	0
$409$	17.2361	0.852269	0.426134	−	0.904660i	$-0.359875\pi$
$409$	17.2361	0.852269	0.426134	+	0.904660i	$0.359875\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−2.47214	−0.121352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.4164	−0.850847	−0.425424	−	0.904994i	$-0.639875\pi$
$419$	−17.4164	−0.850847	−0.425424	+	0.904994i	$0.639875\pi$
$420$	0	0
$421$	17.5279	0.854256	0.427128	−	0.904191i	$-0.359525\pi$
$421$	17.5279	0.854256	0.427128	+	0.904191i	$0.359525\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−25.1246	−1.21872
$426$	0	0
$427$	−8.47214	−0.409995
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.0557	0.532536	0.266268	−	0.963899i	$-0.414209\pi$
$431$	11.0557	0.532536	0.266268	+	0.963899i	$0.414209\pi$
$432$	0	0
$433$	21.2361	1.02054	0.510270	−	0.860014i	$-0.329545\pi$
$433$	21.2361	1.02054	0.510270	+	0.860014i	$0.329545\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.23607	−0.154802
$438$	0	0
$439$	2.58359	0.123308	0.0616541	−	0.998098i	$-0.480362\pi$
$439$	2.58359	0.123308	0.0616541	+	0.998098i	$0.480362\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.1803	−0.863774	−0.431887	−	0.901928i	$-0.642152\pi$
$443$	−18.1803	−0.863774	−0.431887	+	0.901928i	$0.642152\pi$
$444$	0	0
$445$	13.5279	0.641282
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.0557	−0.804910	−0.402455	−	0.915440i	$-0.631843\pi$
$449$	−17.0557	−0.804910	−0.402455	+	0.915440i	$0.631843\pi$
$450$	0	0
$451$	−10.4721	−0.493114
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.944272	−0.0442681
$456$	0	0
$457$	9.41641	0.440481	0.220240	−	0.975446i	$-0.429316\pi$
$457$	9.41641	0.440481	0.220240	+	0.975446i	$0.429316\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.76393	0.315028	0.157514	−	0.987517i	$-0.449652\pi$
$461$	6.76393	0.315028	0.157514	+	0.987517i	$0.449652\pi$
$462$	0	0
$463$	5.52786	0.256902	0.128451	−	0.991716i	$-0.459000\pi$
$463$	5.52786	0.256902	0.128451	+	0.991716i	$0.459000\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.8885	1.29053	0.645264	−	0.763960i	$-0.276747\pi$
$467$	27.8885	1.29053	0.645264	+	0.763960i	$0.276747\pi$
$468$	0	0
$469$	−0.763932	−0.0352751
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.9443	0.963019
$474$	0	0
$475$	−3.47214	−0.159313
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−6.83282	−0.311550
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.52786	−0.0693767
$486$	0	0
$487$	−23.7082	−1.07432	−0.537161	−	0.843480i	$-0.680503\pi$
$487$	−23.7082	−1.07432	−0.537161	+	0.843480i	$0.680503\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−0.875388	−0.0395057	−0.0197529	−	0.999805i	$-0.506288\pi$
$491$	−0.875388	−0.0395057	−0.0197529	+	0.999805i	$0.506288\pi$
$492$	0	0
$493$	61.3050	2.76104
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.52786	0.0685341
$498$	0	0
$499$	−5.52786	−0.247461	−0.123731	−	0.992316i	$-0.539486\pi$
$499$	−5.52786	−0.247461	−0.123731	+	0.992316i	$0.539486\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−39.7771	−1.77357	−0.886786	−	0.462180i	$-0.847068\pi$
$503$	−39.7771	−1.77357	−0.886786	+	0.462180i	$0.847068\pi$
$504$	0	0
$505$	22.4721	0.999997
$506$	0	0
$507$	0	0
$508$	0	0
$509$	43.3050	1.91946	0.959729	−	0.280927i	$-0.0906419\pi$
$509$	43.3050	1.91946	0.959729	+	0.280927i	$0.0906419\pi$
$510$	0	0
$511$	−4.47214	−0.197836
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.30495	0.410025
$516$	0	0
$517$	33.8885	1.49042
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.4164	0.587784	0.293892	−	0.955839i	$-0.405049\pi$
$521$	13.4164	0.587784	0.293892	+	0.955839i	$0.405049\pi$
$522$	0	0
$523$	29.3050	1.28142	0.640708	−	0.767785i	$-0.278641\pi$
$523$	29.3050	1.28142	0.640708	+	0.767785i	$0.278641\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.41641	0.148821
$528$	0	0
$529$	−12.5279	−0.544690
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.52786	−0.0661791
$534$	0	0
$535$	−14.1115	−0.610091
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.23607	−0.225533
$540$	0	0
$541$	2.94427	0.126584	0.0632921	−	0.997995i	$-0.479840\pi$
$541$	2.94427	0.126584	0.0632921	+	0.997995i	$0.479840\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.0557	−0.473575
$546$	0	0
$547$	9.70820	0.415093	0.207546	−	0.978225i	$-0.433452\pi$
$547$	9.70820	0.415093	0.207546	+	0.978225i	$0.433452\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.47214	0.360925
$552$	0	0
$553$	−15.7082	−0.667981
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.58359	−0.278956	−0.139478	−	0.990225i	$-0.544542\pi$
$557$	−6.58359	−0.278956	−0.139478	+	0.990225i	$0.544542\pi$
$558$	0	0
$559$	3.05573	0.129244
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	8.58359	0.361114
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.8328	−0.705668	−0.352834	−	0.935686i	$-0.614782\pi$
$569$	−16.8328	−0.705668	−0.352834	+	0.935686i	$0.614782\pi$
$570$	0	0
$571$	34.4721	1.44261	0.721307	−	0.692615i	$-0.243542\pi$
$571$	34.4721	1.44261	0.721307	+	0.692615i	$0.243542\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.2361	0.468576
$576$	0	0
$577$	−37.4164	−1.55767	−0.778833	−	0.627232i	$-0.784188\pi$
$577$	−37.4164	−1.55767	−0.778833	+	0.627232i	$0.784188\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	2.47214	0.102385
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.0000	0.577842	0.288921	−	0.957353i	$-0.406704\pi$
$587$	14.0000	0.577842	0.288921	+	0.957353i	$0.406704\pi$
$588$	0	0
$589$	0.472136	0.0194540
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−41.1246	−1.68879	−0.844393	−	0.535725i	$-0.820038\pi$
$593$	−41.1246	−1.68879	−0.844393	+	0.535725i	$0.820038\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.4721	−0.918187	−0.459093	−	0.888388i	$-0.651826\pi$
$599$	−22.4721	−0.918187	−0.459093	+	0.888388i	$0.651826\pi$
$600$	0	0
$601$	−4.87539	−0.198871	−0.0994356	−	0.995044i	$-0.531704\pi$
$601$	−4.87539	−0.198871	−0.0994356	+	0.995044i	$0.531704\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20.2918	−0.824979
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.94427	0.200024
$612$	0	0
$613$	−23.8885	−0.964849	−0.482425	−	0.875938i	$-0.660244\pi$
$613$	−23.8885	−0.964849	−0.482425	+	0.875938i	$0.660244\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.4721	0.743660	0.371830	−	0.928301i	$-0.378730\pi$
$617$	18.4721	0.743660	0.371830	+	0.928301i	$0.378730\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.9443	0.438473
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	64.7214	2.58061
$630$	0	0
$631$	−12.3607	−0.492071	−0.246035	−	0.969261i	$-0.579128\pi$
$631$	−12.3607	−0.492071	−0.246035	+	0.969261i	$0.579128\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.36068	0.331783
$636$	0	0
$637$	−0.763932	−0.0302681
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	17.8885	0.705455	0.352728	−	0.935726i	$-0.385254\pi$
$643$	17.8885	0.705455	0.352728	+	0.935726i	$0.385254\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.94427	−0.351636	−0.175818	−	0.984423i	$-0.556257\pi$
$647$	−8.94427	−0.351636	−0.175818	+	0.984423i	$0.556257\pi$
$648$	0	0
$649$	41.8885	1.64427
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.8885	0.934831	0.467415	−	0.884038i	$-0.345185\pi$
$653$	23.8885	0.934831	0.467415	+	0.884038i	$0.345185\pi$
$654$	0	0
$655$	0.583592	0.0228028
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.41641	0.133084	0.0665422	−	0.997784i	$-0.478803\pi$
$659$	3.41641	0.133084	0.0665422	+	0.997784i	$0.478803\pi$
$660$	0	0
$661$	9.70820	0.377605	0.188803	−	0.982015i	$-0.439539\pi$
$661$	9.70820	0.377605	0.188803	+	0.982015i	$0.439539\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.23607	0.0479327
$666$	0	0
$667$	−27.4164	−1.06157
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−44.3607	−1.71253
$672$	0	0
$673$	2.58359	0.0995902	0.0497951	−	0.998759i	$-0.484143\pi$
$673$	2.58359	0.0995902	0.0497951	+	0.998759i	$0.484143\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.58359	0.253028	0.126514	−	0.991965i	$-0.459621\pi$
$677$	6.58359	0.253028	0.126514	+	0.991965i	$0.459621\pi$
$678$	0	0
$679$	−1.23607	−0.0474359
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.88854	−0.225319	−0.112659	−	0.993634i	$-0.535937\pi$
$683$	−5.88854	−0.225319	−0.112659	+	0.993634i	$0.535937\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.360680	0.0137408
$690$	0	0
$691$	−8.94427	−0.340256	−0.170128	−	0.985422i	$-0.554418\pi$
$691$	−8.94427	−0.340256	−0.170128	+	0.985422i	$0.554418\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.0557	0.419368
$696$	0	0
$697$	14.4721	0.548171
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−20.8328	−0.786845	−0.393422	−	0.919358i	$-0.628709\pi$
$701$	−20.8328	−0.786845	−0.393422	+	0.919358i	$0.628709\pi$
$702$	0	0
$703$	8.94427	0.337340
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.1803	0.683742
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.52786	−0.0572190
$714$	0	0
$715$	−4.94427	−0.184905
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	7.52786	0.280352
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−29.4164	−1.09250
$726$	0	0
$727$	−17.8885	−0.663449	−0.331725	−	0.943376i	$-0.607631\pi$
$727$	−17.8885	−0.663449	−0.331725	+	0.943376i	$0.607631\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−28.9443	−1.07054
$732$	0	0
$733$	18.9443	0.699723	0.349861	−	0.936802i	$-0.386229\pi$
$733$	18.9443	0.699723	0.349861	+	0.936802i	$0.386229\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−52.3607	−1.92612	−0.963059	−	0.269289i	$-0.913211\pi$
$739$	−52.3607	−1.92612	−0.963059	+	0.269289i	$0.913211\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.3050	−1.66208	−0.831039	−	0.556215i	$-0.812253\pi$
$743$	−45.3050	−1.66208	−0.831039	+	0.556215i	$0.812253\pi$
$744$	0	0
$745$	20.3607	0.745958
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11.4164	−0.417146
$750$	0	0
$751$	3.12461	0.114019	0.0570094	−	0.998374i	$-0.481844\pi$
$751$	3.12461	0.114019	0.0570094	+	0.998374i	$0.481844\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.360680	0.0131265
$756$	0	0
$757$	19.8885	0.722861	0.361431	−	0.932399i	$-0.382288\pi$
$757$	19.8885	0.722861	0.361431	+	0.932399i	$0.382288\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.403252	−0.0146179	−0.00730894	−	0.999973i	$-0.502327\pi$
$761$	−0.403252	−0.0146179	−0.00730894	+	0.999973i	$0.502327\pi$
$762$	0	0
$763$	−8.94427	−0.323804
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.11146	0.220672
$768$	0	0
$769$	−42.9443	−1.54861	−0.774305	−	0.632813i	$-0.781900\pi$
$769$	−42.9443	−1.54861	−0.774305	+	0.632813i	$0.781900\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.3050	0.982091	0.491045	−	0.871134i	$-0.336615\pi$
$773$	27.3050	0.982091	0.491045	+	0.871134i	$0.336615\pi$
$774$	0	0
$775$	−1.63932	−0.0588861
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.3607	0.726704
$786$	0	0
$787$	−7.41641	−0.264366	−0.132183	−	0.991225i	$-0.542199\pi$
$787$	−7.41641	−0.264366	−0.132183	+	0.991225i	$0.542199\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.94427	0.246910
$792$	0	0
$793$	−6.47214	−0.229832
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.3050	1.25057	0.625283	−	0.780398i	$-0.284984\pi$
$797$	35.3050	1.25057	0.625283	+	0.780398i	$0.284984\pi$
$798$	0	0
$799$	−46.8328	−1.65683
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.4164	−0.826347
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	−54.2492	−1.90495	−0.952474	−	0.304620i	$-0.901471\pi$
$811$	−54.2492	−1.90495	−0.952474	+	0.304620i	$0.901471\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.8328	−0.519571
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.3607	−0.780393	−0.390197	−	0.920732i	$-0.627593\pi$
$821$	−22.3607	−0.780393	−0.390197	+	0.920732i	$0.627593\pi$
$822$	0	0
$823$	−47.4164	−1.65283	−0.826416	−	0.563060i	$-0.809624\pi$
$823$	−47.4164	−1.65283	−0.826416	+	0.563060i	$0.809624\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.9443	−1.70196	−0.850980	−	0.525199i	$-0.823991\pi$
$827$	−48.9443	−1.70196	−0.850980	+	0.525199i	$0.823991\pi$
$828$	0	0
$829$	26.0689	0.905410	0.452705	−	0.891660i	$-0.350459\pi$
$829$	26.0689	0.905410	0.452705	+	0.891660i	$0.350459\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.23607	0.250715
$834$	0	0
$835$	20.9443	0.724806
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.8328	−1.20256	−0.601281	−	0.799038i	$-0.705343\pi$
$839$	−34.8328	−1.20256	−0.601281	+	0.799038i	$0.705343\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.3475	0.527971
$846$	0	0
$847$	−16.4164	−0.564074
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−28.9443	−0.992197
$852$	0	0
$853$	−7.88854	−0.270099	−0.135049	−	0.990839i	$-0.543119\pi$
$853$	−7.88854	−0.270099	−0.135049	+	0.990839i	$0.543119\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.83282	−0.165086	−0.0825429	−	0.996588i	$-0.526304\pi$
$857$	−4.83282	−0.165086	−0.0825429	+	0.996588i	$0.526304\pi$
$858$	0	0
$859$	51.7771	1.76661	0.883306	−	0.468797i	$-0.155313\pi$
$859$	51.7771	1.76661	0.883306	+	0.468797i	$0.155313\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.3607	−1.10157	−0.550785	−	0.834647i	$-0.685672\pi$
$863$	−32.3607	−1.10157	−0.550785	+	0.834647i	$0.685672\pi$
$864$	0	0
$865$	−24.5836	−0.835867
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−82.2492	−2.79011
$870$	0	0
$871$	−0.583592	−0.0197743
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.4721	−0.354023
$876$	0	0
$877$	−40.9443	−1.38259	−0.691295	−	0.722573i	$-0.742959\pi$
$877$	−40.9443	−1.38259	−0.691295	+	0.722573i	$0.742959\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35.9574	1.21144	0.605718	−	0.795679i	$-0.292886\pi$
$881$	35.9574	1.21144	0.605718	+	0.795679i	$0.292886\pi$
$882$	0	0
$883$	−10.4721	−0.352415	−0.176208	−	0.984353i	$-0.556383\pi$
$883$	−10.4721	−0.352415	−0.176208	+	0.984353i	$0.556383\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.8885	0.869252	0.434626	−	0.900611i	$-0.356881\pi$
$887$	25.8885	0.869252	0.434626	+	0.900611i	$0.356881\pi$
$888$	0	0
$889$	6.76393	0.226855
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.47214	−0.216582
$894$	0	0
$895$	4.94427	0.165269
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.00000	0.133407
$900$	0	0
$901$	−3.41641	−0.113817
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.7771	−0.524448
$906$	0	0
$907$	−37.7082	−1.25208	−0.626040	−	0.779791i	$-0.715325\pi$
$907$	−37.7082	−1.25208	−0.626040	+	0.779791i	$0.715325\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.4164	0.643294	0.321647	−	0.946860i	$-0.395764\pi$
$911$	19.4164	0.643294	0.321647	+	0.946860i	$0.395764\pi$
$912$	0	0
$913$	−10.4721	−0.346577
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.472136	0.0155913
$918$	0	0
$919$	−5.52786	−0.182347	−0.0911737	−	0.995835i	$-0.529062\pi$
$919$	−5.52786	−0.182347	−0.0911737	+	0.995835i	$0.529062\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.16718	0.0384183
$924$	0	0
$925$	−31.0557	−1.02111
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.34752	0.306682	0.153341	−	0.988173i	$-0.450997\pi$
$929$	9.34752	0.306682	0.153341	+	0.988173i	$0.450997\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	46.8328	1.53160
$936$	0	0
$937$	−0.832816	−0.0272069	−0.0136035	−	0.999907i	$-0.504330\pi$
$937$	−0.832816	−0.0272069	−0.0136035	+	0.999907i	$0.504330\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.3050	0.890116	0.445058	−	0.895502i	$-0.353183\pi$
$941$	27.3050	0.890116	0.445058	+	0.895502i	$0.353183\pi$
$942$	0	0
$943$	−6.47214	−0.210762
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.1803	1.63064	0.815321	−	0.579009i	$-0.196560\pi$
$947$	50.1803	1.63064	0.815321	+	0.579009i	$0.196560\pi$
$948$	0	0
$949$	−3.41641	−0.110901
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	−13.1672	−0.426080
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.47214	0.208996
$960$	0	0
$961$	−30.7771	−0.992809
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23.4164	−0.753801
$966$	0	0
$967$	−54.8328	−1.76330	−0.881652	−	0.471900i	$-0.843568\pi$
$967$	−54.8328	−1.76330	−0.881652	+	0.471900i	$0.843568\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.2492	−0.714012	−0.357006	−	0.934102i	$-0.616202\pi$
$971$	−22.2492	−0.714012	−0.357006	+	0.934102i	$0.616202\pi$
$972$	0	0
$973$	8.94427	0.286740
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.83282	0.282587	0.141293	−	0.989968i	$-0.454874\pi$
$977$	8.83282	0.282587	0.141293	+	0.989968i	$0.454874\pi$
$978$	0	0
$979$	57.3050	1.83147
$980$	0	0
$981$	0	0
$982$	0	0
$983$	31.7771	1.01353	0.506766	−	0.862084i	$-0.330841\pi$
$983$	31.7771	1.01353	0.506766	+	0.862084i	$0.330841\pi$
$984$	0	0
$985$	21.5279	0.685935
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.9443	0.411604
$990$	0	0
$991$	22.5410	0.716039	0.358020	−	0.933714i	$-0.383452\pi$
$991$	22.5410	0.716039	0.358020	+	0.933714i	$0.383452\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.8885	0.567105
$996$	0	0
$997$	4.47214	0.141634	0.0708170	−	0.997489i	$-0.477439\pi$
$997$	4.47214	0.141634	0.0708170	+	0.997489i	$0.477439\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bp.1.1	yes	2
3.2	odd	2		9576.2.a.bg.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bg.1.2	✓	2	3.2	odd	2
9576.2.a.bp.1.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.23607 q^{5} -1.00000 q^{7} -5.23607 q^{11} -0.763932 q^{13} +7.23607 q^{17} +1.00000 q^{19} -3.23607 q^{23} -3.47214 q^{25} +8.47214 q^{29} +0.472136 q^{31} +1.23607 q^{35} +8.94427 q^{37} +2.00000 q^{41} -4.00000 q^{43} -6.47214 q^{47} +1.00000 q^{49} -0.472136 q^{53} +6.47214 q^{55} -8.00000 q^{59} +8.47214 q^{61} +0.944272 q^{65} +0.763932 q^{67} -1.52786 q^{71} +4.47214 q^{73} +5.23607 q^{77} +15.7082 q^{79} +2.00000 q^{83} -8.94427 q^{85} -10.9443 q^{89} +0.763932 q^{91} -1.23607 q^{95} +1.23607 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{5} - 2 q^{7} - 6 q^{11} - 6 q^{13} + 10 q^{17} + 2 q^{19} - 2 q^{23} + 2 q^{25} + 8 q^{29} - 8 q^{31} - 2 q^{35} + 4 q^{41} - 8 q^{43} - 4 q^{47} + 2 q^{49} + 8 q^{53} + 4 q^{55} - 16 q^{59} + 8 q^{61} - 16 q^{65} + 6 q^{67} - 12 q^{71} + 6 q^{77} + 18 q^{79} + 4 q^{83} - 4 q^{89} + 6 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 6 * q^11 - 6 * q^13 + 10 * q^17 + 2 * q^19 - 2 * q^23 + 2 * q^25 + 8 * q^29 - 8 * q^31 - 2 * q^35 + 4 * q^41 - 8 * q^43 - 4 * q^47 + 2 * q^49 + 8 * q^53 + 4 * q^55 - 16 * q^59 + 8 * q^61 - 16 * q^65 + 6 * q^67 - 12 * q^71 + 6 * q^77 + 18 * q^79 + 4 * q^83 - 4 * q^89 + 6 * q^91 + 2 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more