LMFDB - Embedded newform 9576.2.a.bg.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-3.23607 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-3.23607 q^{5} -1.00000 q^{7} +0.763932 q^{11} -5.23607 q^{13} -2.76393 q^{17} +1.00000 q^{19} -1.23607 q^{23} +5.47214 q^{25} +0.472136 q^{29} -8.47214 q^{31} +3.23607 q^{35} -8.94427 q^{37} -2.00000 q^{41} -4.00000 q^{43} -2.47214 q^{47} +1.00000 q^{49} -8.47214 q^{53} -2.47214 q^{55} +8.00000 q^{59} -0.472136 q^{61} +16.9443 q^{65} +5.23607 q^{67} +10.4721 q^{71} -4.47214 q^{73} -0.763932 q^{77} +2.29180 q^{79} -2.00000 q^{83} +8.94427 q^{85} -6.94427 q^{89} +5.23607 q^{91} -3.23607 q^{95} -3.23607 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	−5.23607	−1.45222	−0.726112	−	0.687576i	$-0.758675\pi$
$13$	−5.23607	−1.45222	−0.726112	+	0.687576i	$0.758675\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.76393	−0.670352	−0.335176	−	0.942156i	$-0.608796\pi$
$17$	−2.76393	−0.670352	−0.335176	+	0.942156i	$0.608796\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	−8.47214	−1.52164	−0.760820	−	0.648963i	$-0.775203\pi$
$31$	−8.47214	−1.52164	−0.760820	+	0.648963i	$0.775203\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.23607	0.546995
$36$	0	0
$37$	−8.94427	−1.47043	−0.735215	−	0.677834i	$-0.762919\pi$
$37$	−8.94427	−1.47043	−0.735215	+	0.677834i	$0.762919\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	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
$47$	−2.47214	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−0.472136	−0.0604508	−0.0302254	−	0.999543i	$-0.509623\pi$
$61$	−0.472136	−0.0604508	−0.0302254	+	0.999543i	$0.509623\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.9443	2.10168
$66$	0	0
$67$	5.23607	0.639688	0.319844	−	0.947470i	$-0.396370\pi$
$67$	5.23607	0.639688	0.319844	+	0.947470i	$0.396370\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.4721	1.24281	0.621407	−	0.783488i	$-0.286561\pi$
$71$	10.4721	1.24281	0.621407	+	0.783488i	$0.286561\pi$
$72$	0	0
$73$	−4.47214	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$73$	−4.47214	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.763932	−0.0870581
$78$	0	0
$79$	2.29180	0.257847	0.128924	−	0.991655i	$-0.458848\pi$
$79$	2.29180	0.257847	0.128924	+	0.991655i	$0.458848\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	8.94427	0.970143
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.94427	−0.736091	−0.368046	−	0.929808i	$-0.619973\pi$
$89$	−6.94427	−0.736091	−0.368046	+	0.929808i	$0.619973\pi$
$90$	0	0
$91$	5.23607	0.548889
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.23607	−0.332014
$96$	0	0
$97$	−3.23607	−0.328573	−0.164286	−	0.986413i	$-0.552532\pi$
$97$	−3.23607	−0.328573	−0.164286	+	0.986413i	$0.552532\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.18034	−0.415959	−0.207980	−	0.978133i	$-0.566689\pi$
$101$	−4.18034	−0.415959	−0.207980	+	0.978133i	$0.566689\pi$
$102$	0	0
$103$	−16.4721	−1.62305	−0.811524	−	0.584319i	$-0.801362\pi$
$103$	−16.4721	−1.62305	−0.811524	+	0.584319i	$0.801362\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.4164	1.49036	0.745180	−	0.666863i	$-0.232363\pi$
$107$	15.4164	1.49036	0.745180	+	0.666863i	$0.232363\pi$
$108$	0	0
$109$	−8.94427	−0.856706	−0.428353	−	0.903612i	$-0.640906\pi$
$109$	−8.94427	−0.856706	−0.428353	+	0.903612i	$0.640906\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.9443	−1.02955	−0.514775	−	0.857325i	$-0.672125\pi$
$113$	−10.9443	−1.02955	−0.514775	+	0.857325i	$0.672125\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.76393	0.253369
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−11.2361	−0.997040	−0.498520	−	0.866878i	$-0.666123\pi$
$127$	−11.2361	−0.997040	−0.498520	+	0.866878i	$0.666123\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.47214	−0.740214	−0.370107	−	0.928989i	$-0.620679\pi$
$131$	−8.47214	−0.740214	−0.370107	+	0.928989i	$0.620679\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.47214	−0.211209	−0.105604	−	0.994408i	$-0.533678\pi$
$137$	−2.47214	−0.211209	−0.105604	+	0.994408i	$0.533678\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−1.52786	−0.126882
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.52786	0.616707	0.308353	−	0.951272i	$-0.400222\pi$
$149$	7.52786	0.616707	0.308353	+	0.951272i	$0.400222\pi$
$150$	0	0
$151$	−13.7082	−1.11556	−0.557779	−	0.829990i	$-0.688346\pi$
$151$	−13.7082	−1.11556	−0.557779	+	0.829990i	$0.688346\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	27.4164	2.20214
$156$	0	0
$157$	−7.52786	−0.600789	−0.300394	−	0.953815i	$-0.597118\pi$
$157$	−7.52786	−0.600789	−0.300394	+	0.953815i	$0.597118\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.23607	0.0974158
$162$	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
$167$	−0.944272	−0.0730700	−0.0365350	−	0.999332i	$-0.511632\pi$
$167$	−0.944272	−0.0730700	−0.0365350	+	0.999332i	$0.511632\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.8885	1.20798	0.603992	−	0.796991i	$-0.293576\pi$
$173$	15.8885	1.20798	0.603992	+	0.796991i	$0.293576\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	0	0
$177$	0	0
$178$	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
$181$	17.2361	1.28115	0.640573	−	0.767897i	$-0.278697\pi$
$181$	17.2361	1.28115	0.640573	+	0.767897i	$0.278697\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	28.9443	2.12803
$186$	0	0
$187$	−2.11146	−0.154405
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.6525	1.49436	0.747180	−	0.664621i	$-0.231407\pi$
$191$	20.6525	1.49436	0.747180	+	0.664621i	$0.231407\pi$
$192$	0	0
$193$	1.05573	0.0759930	0.0379965	−	0.999278i	$-0.487902\pi$
$193$	1.05573	0.0759930	0.0379965	+	0.999278i	$0.487902\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.41641	−0.670891	−0.335446	−	0.942060i	$-0.608887\pi$
$197$	−9.41641	−0.670891	−0.335446	+	0.942060i	$0.608887\pi$
$198$	0	0
$199$	−5.52786	−0.391860	−0.195930	−	0.980618i	$-0.562773\pi$
$199$	−5.52786	−0.391860	−0.195930	+	0.980618i	$0.562773\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.472136	−0.0331374
$204$	0	0
$205$	6.47214	0.452034
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.763932	0.0528423
$210$	0	0
$211$	6.18034	0.425472	0.212736	−	0.977110i	$-0.431763\pi$
$211$	6.18034	0.425472	0.212736	+	0.977110i	$0.431763\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.9443	0.882792
$216$	0	0
$217$	8.47214	0.575126
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.4721	0.973501
$222$	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
$227$	11.4164	0.757734	0.378867	−	0.925451i	$-0.376314\pi$
$227$	11.4164	0.757734	0.378867	+	0.925451i	$0.376314\pi$
$228$	0	0
$229$	−18.9443	−1.25187	−0.625936	−	0.779874i	$-0.715283\pi$
$229$	−18.9443	−1.25187	−0.625936	+	0.779874i	$0.715283\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.3607	−1.33387	−0.666936	−	0.745115i	$-0.732395\pi$
$233$	−20.3607	−1.33387	−0.666936	+	0.745115i	$0.732395\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.7639	1.47248	0.736238	−	0.676723i	$-0.236600\pi$
$239$	22.7639	1.47248	0.736238	+	0.676723i	$0.236600\pi$
$240$	0	0
$241$	7.23607	0.466116	0.233058	−	0.972463i	$-0.425127\pi$
$241$	7.23607	0.466116	0.233058	+	0.972463i	$0.425127\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	−5.23607	−0.333163
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.472136	0.0298010	0.0149005	−	0.999889i	$-0.495257\pi$
$251$	0.472136	0.0298010	0.0149005	+	0.999889i	$0.495257\pi$
$252$	0	0
$253$	−0.944272	−0.0593659
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.5279	0.968602	0.484301	−	0.874902i	$-0.339074\pi$
$257$	15.5279	0.968602	0.484301	+	0.874902i	$0.339074\pi$
$258$	0	0
$259$	8.94427	0.555770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.1803	0.874397	0.437199	−	0.899365i	$-0.355971\pi$
$263$	14.1803	0.874397	0.437199	+	0.899365i	$0.355971\pi$
$264$	0	0
$265$	27.4164	1.68418
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.94427	−0.423400	−0.211700	−	0.977335i	$-0.567900\pi$
$269$	−6.94427	−0.423400	−0.211700	+	0.977335i	$0.567900\pi$
$270$	0	0
$271$	−7.05573	−0.428605	−0.214302	−	0.976767i	$-0.568748\pi$
$271$	−7.05573	−0.428605	−0.214302	+	0.976767i	$0.568748\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.18034	0.252084
$276$	0	0
$277$	23.8885	1.43532	0.717662	−	0.696392i	$-0.245212\pi$
$277$	23.8885	1.43532	0.717662	+	0.696392i	$0.245212\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.9443	1.60736	0.803680	−	0.595061i	$-0.202872\pi$
$281$	26.9443	1.60736	0.803680	+	0.595061i	$0.202872\pi$
$282$	0	0
$283$	−24.9443	−1.48278	−0.741392	−	0.671073i	$-0.765834\pi$
$283$	−24.9443	−1.48278	−0.741392	+	0.671073i	$0.765834\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−9.36068	−0.550628
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	−25.8885	−1.50729
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.47214	0.374293
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.52786	0.0874852
$306$	0	0
$307$	−24.3607	−1.39034	−0.695169	−	0.718847i	$-0.744670\pi$
$307$	−24.3607	−1.39034	−0.695169	+	0.718847i	$0.744670\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.94427	−0.280364	−0.140182	−	0.990126i	$-0.544769\pi$
$311$	−4.94427	−0.280364	−0.140182	+	0.990126i	$0.544769\pi$
$312$	0	0
$313$	5.05573	0.285767	0.142883	−	0.989740i	$-0.454363\pi$
$313$	5.05573	0.285767	0.142883	+	0.989740i	$0.454363\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	0	0
$319$	0.360680	0.0201942
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.76393	−0.153789
$324$	0	0
$325$	−28.6525	−1.58935
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.47214	0.136293
$330$	0	0
$331$	14.7639	0.811499	0.405750	−	0.913984i	$-0.367011\pi$
$331$	14.7639	0.811499	0.405750	+	0.913984i	$0.367011\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−16.9443	−0.925764
$336$	0	0
$337$	31.3050	1.70529	0.852645	−	0.522491i	$-0.174997\pi$
$337$	31.3050	1.70529	0.852645	+	0.522491i	$0.174997\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.47214	−0.350486
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.70820	−0.521164	−0.260582	−	0.965452i	$-0.583914\pi$
$347$	−9.70820	−0.521164	−0.260582	+	0.965452i	$0.583914\pi$
$348$	0	0
$349$	28.8328	1.54339	0.771693	−	0.635996i	$-0.219410\pi$
$349$	28.8328	1.54339	0.771693	+	0.635996i	$0.219410\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.70820	−0.197368	−0.0986839	−	0.995119i	$-0.531463\pi$
$353$	−3.70820	−0.197368	−0.0986839	+	0.995119i	$0.531463\pi$
$354$	0	0
$355$	−33.8885	−1.79862
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.70820	−0.406823	−0.203412	−	0.979093i	$-0.565203\pi$
$359$	−7.70820	−0.406823	−0.203412	+	0.979093i	$0.565203\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.4721	0.757506
$366$	0	0
$367$	−0.944272	−0.0492906	−0.0246453	−	0.999696i	$-0.507846\pi$
$367$	−0.944272	−0.0492906	−0.0246453	+	0.999696i	$0.507846\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.47214	0.439851
$372$	0	0
$373$	16.3607	0.847124	0.423562	−	0.905867i	$-0.360780\pi$
$373$	16.3607	0.847124	0.423562	+	0.905867i	$0.360780\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.47214	−0.127321
$378$	0	0
$379$	9.81966	0.504402	0.252201	−	0.967675i	$-0.418846\pi$
$379$	9.81966	0.504402	0.252201	+	0.967675i	$0.418846\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.9443	1.07020	0.535101	−	0.844788i	$-0.320273\pi$
$383$	20.9443	1.07020	0.535101	+	0.844788i	$0.320273\pi$
$384$	0	0
$385$	2.47214	0.125992
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	3.41641	0.172775
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.41641	−0.373160
$396$	0	0
$397$	10.3607	0.519988	0.259994	−	0.965610i	$-0.416279\pi$
$397$	10.3607	0.519988	0.259994	+	0.965610i	$0.416279\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.9443	−1.14578	−0.572891	−	0.819631i	$-0.694178\pi$
$401$	−22.9443	−1.14578	−0.572891	+	0.819631i	$0.694178\pi$
$402$	0	0
$403$	44.3607	2.20976
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.83282	−0.338690
$408$	0	0
$409$	12.7639	0.631136	0.315568	−	0.948903i	$-0.397805\pi$
$409$	12.7639	0.631136	0.315568	+	0.948903i	$0.397805\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	6.47214	0.317705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.41641	−0.460022	−0.230011	−	0.973188i	$-0.573876\pi$
$419$	−9.41641	−0.460022	−0.230011	+	0.973188i	$0.573876\pi$
$420$	0	0
$421$	26.4721	1.29017	0.645086	−	0.764110i	$-0.276821\pi$
$421$	26.4721	1.29017	0.645086	+	0.764110i	$0.276821\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.1246	−0.733651
$426$	0	0
$427$	0.472136	0.0228483
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.9443	−1.39420	−0.697098	−	0.716976i	$-0.745526\pi$
$431$	−28.9443	−1.39420	−0.697098	+	0.716976i	$0.745526\pi$
$432$	0	0
$433$	16.7639	0.805623	0.402812	−	0.915283i	$-0.368033\pi$
$433$	16.7639	0.805623	0.402812	+	0.915283i	$0.368033\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.23607	−0.0591292
$438$	0	0
$439$	29.4164	1.40397	0.701984	−	0.712192i	$-0.252298\pi$
$439$	29.4164	1.40397	0.701984	+	0.712192i	$0.252298\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.18034	−0.198614	−0.0993070	−	0.995057i	$-0.531663\pi$
$443$	−4.18034	−0.198614	−0.0993070	+	0.995057i	$0.531663\pi$
$444$	0	0
$445$	22.4721	1.06528
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.9443	1.64912	0.824561	−	0.565773i	$-0.191422\pi$
$449$	34.9443	1.64912	0.824561	+	0.565773i	$0.191422\pi$
$450$	0	0
$451$	−1.52786	−0.0719443
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16.9443	−0.794360
$456$	0	0
$457$	−17.4164	−0.814705	−0.407353	−	0.913271i	$-0.633548\pi$
$457$	−17.4164	−0.814705	−0.407353	+	0.913271i	$0.633548\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.2361	−0.523316	−0.261658	−	0.965161i	$-0.584269\pi$
$461$	−11.2361	−0.523316	−0.261658	+	0.965161i	$0.584269\pi$
$462$	0	0
$463$	14.4721	0.672577	0.336289	−	0.941759i	$-0.390828\pi$
$463$	14.4721	0.672577	0.336289	+	0.941759i	$0.390828\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.88854	0.365038	0.182519	−	0.983202i	$-0.441575\pi$
$467$	7.88854	0.365038	0.182519	+	0.983202i	$0.441575\pi$
$468$	0	0
$469$	−5.23607	−0.241779
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.05573	−0.140503
$474$	0	0
$475$	5.47214	0.251079
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	46.8328	2.13539
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.4721	0.475515
$486$	0	0
$487$	−10.2918	−0.466366	−0.233183	−	0.972433i	$-0.574914\pi$
$487$	−10.2918	−0.466366	−0.233183	+	0.972433i	$0.574914\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	41.1246	1.85593	0.927964	−	0.372670i	$-0.121558\pi$
$491$	41.1246	1.85593	0.927964	+	0.372670i	$0.121558\pi$
$492$	0	0
$493$	−1.30495	−0.0587721
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.4721	−0.469739
$498$	0	0
$499$	−14.4721	−0.647862	−0.323931	−	0.946081i	$-0.605005\pi$
$499$	−14.4721	−0.647862	−0.323931	+	0.946081i	$0.605005\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.7771	−1.41687	−0.708435	−	0.705776i	$-0.750599\pi$
$503$	−31.7771	−1.41687	−0.708435	+	0.705776i	$0.750599\pi$
$504$	0	0
$505$	13.5279	0.601982
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.3050	0.855677	0.427838	−	0.903855i	$-0.359275\pi$
$509$	19.3050	0.855677	0.427838	+	0.903855i	$0.359275\pi$
$510$	0	0
$511$	4.47214	0.197836
$512$	0	0
$513$	0	0
$514$	0	0
$515$	53.3050	2.34890
$516$	0	0
$517$	−1.88854	−0.0830581
$518$	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
$523$	−33.3050	−1.45632	−0.728162	−	0.685405i	$-0.759625\pi$
$523$	−33.3050	−1.45632	−0.728162	+	0.685405i	$0.759625\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.4164	1.02003
$528$	0	0
$529$	−21.4721	−0.933571
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.4721	0.453599
$534$	0	0
$535$	−49.8885	−2.15687
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.763932	0.0329049
$540$	0	0
$541$	−14.9443	−0.642504	−0.321252	−	0.946994i	$-0.604104\pi$
$541$	−14.9443	−0.642504	−0.321252	+	0.946994i	$0.604104\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	28.9443	1.23984
$546$	0	0
$547$	−3.70820	−0.158551	−0.0792757	−	0.996853i	$-0.525261\pi$
$547$	−3.70820	−0.158551	−0.0792757	+	0.996853i	$0.525261\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.472136	0.0201137
$552$	0	0
$553$	−2.29180	−0.0974571
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.4164	1.41590	0.707949	−	0.706263i	$-0.249620\pi$
$557$	33.4164	1.41590	0.707949	+	0.706263i	$0.249620\pi$
$558$	0	0
$559$	20.9443	0.885848
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	35.4164	1.48998
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.8328	−1.54411	−0.772056	−	0.635555i	$-0.780772\pi$
$569$	−36.8328	−1.54411	−0.772056	+	0.635555i	$0.780772\pi$
$570$	0	0
$571$	25.5279	1.06831	0.534154	−	0.845387i	$-0.320630\pi$
$571$	25.5279	1.06831	0.534154	+	0.845387i	$0.320630\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.76393	−0.282075
$576$	0	0
$577$	−10.5836	−0.440601	−0.220300	−	0.975432i	$-0.570704\pi$
$577$	−10.5836	−0.440601	−0.220300	+	0.975432i	$0.570704\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.00000	0.0829740
$582$	0	0
$583$	−6.47214	−0.268048
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	−8.47214	−0.349088
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.875388	0.0359479	0.0179739	−	0.999838i	$-0.494278\pi$
$593$	0.875388	0.0359479	0.0179739	+	0.999838i	$0.494278\pi$
$594$	0	0
$595$	−8.94427	−0.366679
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.5279	0.552734	0.276367	−	0.961052i	$-0.410870\pi$
$599$	13.5279	0.552734	0.276367	+	0.961052i	$0.410870\pi$
$600$	0	0
$601$	−45.1246	−1.84067	−0.920336	−	0.391129i	$-0.872084\pi$
$601$	−45.1246	−1.84067	−0.920336	+	0.391129i	$0.872084\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	33.7082	1.37043
$606$	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
$611$	12.9443	0.523669
$612$	0	0
$613$	11.8885	0.480174	0.240087	−	0.970751i	$-0.422824\pi$
$613$	11.8885	0.480174	0.240087	+	0.970751i	$0.422824\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.52786	−0.383577	−0.191789	−	0.981436i	$-0.561429\pi$
$617$	−9.52786	−0.383577	−0.191789	+	0.981436i	$0.561429\pi$
$618$	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
$623$	6.94427	0.278216
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.7214	0.985705
$630$	0	0
$631$	32.3607	1.28826	0.644129	−	0.764917i	$-0.277220\pi$
$631$	32.3607	1.28826	0.644129	+	0.764917i	$0.277220\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	36.3607	1.44293
$636$	0	0
$637$	−5.23607	−0.207461
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	−17.8885	−0.705455	−0.352728	−	0.935726i	$-0.614746\pi$
$643$	−17.8885	−0.705455	−0.352728	+	0.935726i	$0.614746\pi$
$644$	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
$649$	6.11146	0.239896
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.8885	0.465235	0.232617	−	0.972568i	$-0.425271\pi$
$653$	11.8885	0.465235	0.232617	+	0.972568i	$0.425271\pi$
$654$	0	0
$655$	27.4164	1.07125
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23.4164	0.912174	0.456087	−	0.889935i	$-0.349251\pi$
$659$	23.4164	0.912174	0.456087	+	0.889935i	$0.349251\pi$
$660$	0	0
$661$	−3.70820	−0.144232	−0.0721162	−	0.997396i	$-0.522975\pi$
$661$	−3.70820	−0.144232	−0.0721162	+	0.997396i	$0.522975\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.23607	0.125489
$666$	0	0
$667$	−0.583592	−0.0225968
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.360680	−0.0139239
$672$	0	0
$673$	29.4164	1.13392	0.566960	−	0.823746i	$-0.308120\pi$
$673$	29.4164	1.13392	0.566960	+	0.823746i	$0.308120\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.4164	−1.28430	−0.642148	−	0.766580i	$-0.721957\pi$
$677$	−33.4164	−1.28430	−0.642148	+	0.766580i	$0.721957\pi$
$678$	0	0
$679$	3.23607	0.124189
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29.8885	−1.14365	−0.571827	−	0.820374i	$-0.693765\pi$
$683$	−29.8885	−1.14365	−0.571827	+	0.820374i	$0.693765\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	0	0
$687$	0	0
$688$	0	0
$689$	44.3607	1.69001
$690$	0	0
$691$	8.94427	0.340256	0.170128	−	0.985422i	$-0.445582\pi$
$691$	8.94427	0.340256	0.170128	+	0.985422i	$0.445582\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.9443	−1.09792
$696$	0	0
$697$	5.52786	0.209383
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32.8328	−1.24008	−0.620039	−	0.784571i	$-0.712883\pi$
$701$	−32.8328	−1.24008	−0.620039	+	0.784571i	$0.712883\pi$
$702$	0	0
$703$	−8.94427	−0.337340
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.18034	0.157218
$708$	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
$713$	10.4721	0.392185
$714$	0	0
$715$	12.9443	0.484088
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	16.4721	0.613454
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.58359	0.0959522
$726$	0	0
$727$	17.8885	0.663449	0.331725	−	0.943376i	$-0.392369\pi$
$727$	17.8885	0.663449	0.331725	+	0.943376i	$0.392369\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.0557	0.408911
$732$	0	0
$733$	1.05573	0.0389942	0.0194971	−	0.999810i	$-0.493793\pi$
$733$	1.05573	0.0389942	0.0194971	+	0.999810i	$0.493793\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−7.63932	−0.281017	−0.140508	−	0.990079i	$-0.544874\pi$
$739$	−7.63932	−0.281017	−0.140508	+	0.990079i	$0.544874\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.3050	−0.634857	−0.317429	−	0.948282i	$-0.602819\pi$
$743$	−17.3050	−0.634857	−0.317429	+	0.948282i	$0.602819\pi$
$744$	0	0
$745$	−24.3607	−0.892506
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15.4164	−0.563303
$750$	0	0
$751$	−37.1246	−1.35470	−0.677348	−	0.735663i	$-0.736871\pi$
$751$	−37.1246	−1.35470	−0.677348	+	0.735663i	$0.736871\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	44.3607	1.61445
$756$	0	0
$757$	−15.8885	−0.577479	−0.288739	−	0.957408i	$-0.593236\pi$
$757$	−15.8885	−0.577479	−0.288739	+	0.957408i	$0.593236\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	49.5967	1.79788	0.898940	−	0.438071i	$-0.144338\pi$
$761$	49.5967	1.79788	0.898940	+	0.438071i	$0.144338\pi$
$762$	0	0
$763$	8.94427	0.323804
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−41.8885	−1.51251
$768$	0	0
$769$	−25.0557	−0.903533	−0.451766	−	0.892136i	$-0.649206\pi$
$769$	−25.0557	−0.903533	−0.451766	+	0.892136i	$0.649206\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35.3050	1.26983	0.634915	−	0.772582i	$-0.281035\pi$
$773$	35.3050	1.26983	0.634915	+	0.772582i	$0.281035\pi$
$774$	0	0
$775$	−46.3607	−1.66532
$776$	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
$785$	24.3607	0.869470
$786$	0	0
$787$	19.4164	0.692120	0.346060	−	0.938212i	$-0.387519\pi$
$787$	19.4164	0.692120	0.346060	+	0.938212i	$0.387519\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.9443	0.389134
$792$	0	0
$793$	2.47214	0.0877881
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.3050	0.967191	0.483596	−	0.875292i	$-0.339331\pi$
$797$	27.3050	0.967191	0.483596	+	0.875292i	$0.339331\pi$
$798$	0	0
$799$	6.83282	0.241728
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.41641	−0.120562
$804$	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	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	26.2492	0.921735	0.460867	−	0.887469i	$-0.347538\pi$
$811$	26.2492	0.921735	0.460867	+	0.887469i	$0.347538\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−38.8328	−1.36025
$816$	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
$823$	−20.5836	−0.717499	−0.358749	−	0.933434i	$-0.616797\pi$
$823$	−20.5836	−0.717499	−0.358749	+	0.933434i	$0.616797\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.0557	1.07991	0.539957	−	0.841693i	$-0.318441\pi$
$827$	31.0557	1.07991	0.539957	+	0.841693i	$0.318441\pi$
$828$	0	0
$829$	−32.0689	−1.11380	−0.556899	−	0.830580i	$-0.688009\pi$
$829$	−32.0689	−1.11380	−0.556899	+	0.830580i	$0.688009\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.76393	−0.0957646
$834$	0	0
$835$	3.05573	0.105748
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.8328	−0.650181	−0.325091	−	0.945683i	$-0.605395\pi$
$839$	−18.8328	−0.650181	−0.325091	+	0.945683i	$0.605395\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−46.6525	−1.60489
$846$	0	0
$847$	10.4164	0.357912
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.0557	0.378985
$852$	0	0
$853$	27.8885	0.954886	0.477443	−	0.878663i	$-0.341564\pi$
$853$	27.8885	0.954886	0.477443	+	0.878663i	$0.341564\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.8328	−1.66810	−0.834049	−	0.551691i	$-0.813983\pi$
$857$	−48.8328	−1.66810	−0.834049	+	0.551691i	$0.813983\pi$
$858$	0	0
$859$	−19.7771	−0.674786	−0.337393	−	0.941364i	$-0.609545\pi$
$859$	−19.7771	−0.674786	−0.337393	+	0.941364i	$0.609545\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.3607	−0.420762	−0.210381	−	0.977619i	$-0.567471\pi$
$863$	−12.3607	−0.420762	−0.210381	+	0.977619i	$0.567471\pi$
$864$	0	0
$865$	−51.4164	−1.74821
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.75078	0.0593910
$870$	0	0
$871$	−27.4164	−0.928970
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.52786	0.0516512
$876$	0	0
$877$	−23.0557	−0.778537	−0.389268	−	0.921124i	$-0.627272\pi$
$877$	−23.0557	−0.778537	−0.389268	+	0.921124i	$0.627272\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	57.9574	1.95264	0.976318	−	0.216342i	$-0.0694126\pi$
$881$	57.9574	1.95264	0.976318	+	0.216342i	$0.0694126\pi$
$882$	0	0
$883$	−1.52786	−0.0514167	−0.0257084	−	0.999669i	$-0.508184\pi$
$883$	−1.52786	−0.0514167	−0.0257084	+	0.999669i	$0.508184\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.88854	0.332025	0.166012	−	0.986124i	$-0.446911\pi$
$887$	9.88854	0.332025	0.166012	+	0.986124i	$0.446911\pi$
$888$	0	0
$889$	11.2361	0.376846
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.47214	−0.0827269
$894$	0	0
$895$	−12.9443	−0.432679
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	23.4164	0.780114
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−55.7771	−1.85409
$906$	0	0
$907$	−24.2918	−0.806596	−0.403298	−	0.915069i	$-0.632136\pi$
$907$	−24.2918	−0.806596	−0.403298	+	0.915069i	$0.632136\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.41641	0.245717	0.122858	−	0.992424i	$-0.460794\pi$
$911$	7.41641	0.245717	0.122858	+	0.992424i	$0.460794\pi$
$912$	0	0
$913$	−1.52786	−0.0505649
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.47214	0.279775
$918$	0	0
$919$	−14.4721	−0.477392	−0.238696	−	0.971094i	$-0.576720\pi$
$919$	−14.4721	−0.477392	−0.238696	+	0.971094i	$0.576720\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−54.8328	−1.80484
$924$	0	0
$925$	−48.9443	−1.60928
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40.6525	−1.33376	−0.666882	−	0.745163i	$-0.732372\pi$
$929$	−40.6525	−1.33376	−0.666882	+	0.745163i	$0.732372\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.83282	0.223457
$936$	0	0
$937$	52.8328	1.72597	0.862986	−	0.505227i	$-0.168591\pi$
$937$	52.8328	1.72597	0.862986	+	0.505227i	$0.168591\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	35.3050	1.15091	0.575454	−	0.817834i	$-0.304825\pi$
$941$	35.3050	1.15091	0.575454	+	0.817834i	$0.304825\pi$
$942$	0	0
$943$	2.47214	0.0805038
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.8197	−0.904017	−0.452009	−	0.892014i	$-0.649292\pi$
$947$	−27.8197	−0.904017	−0.452009	+	0.892014i	$0.649292\pi$
$948$	0	0
$949$	23.4164	0.760129
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	−66.8328	−2.16266
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.47214	0.0798294
$960$	0	0
$961$	40.7771	1.31539
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.41641	−0.109978
$966$	0	0
$967$	−1.16718	−0.0375341	−0.0187671	−	0.999824i	$-0.505974\pi$
$967$	−1.16718	−0.0375341	−0.0187671	+	0.999824i	$0.505974\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−58.2492	−1.86931	−0.934653	−	0.355560i	$-0.884290\pi$
$971$	−58.2492	−1.86931	−0.934653	+	0.355560i	$0.884290\pi$
$972$	0	0
$973$	−8.94427	−0.286740
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.8328	1.43433	0.717164	−	0.696904i	$-0.245440\pi$
$977$	44.8328	1.43433	0.717164	+	0.696904i	$0.245440\pi$
$978$	0	0
$979$	−5.30495	−0.169547
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.7771	1.26869	0.634346	−	0.773049i	$-0.281269\pi$
$983$	39.7771	1.26869	0.634346	+	0.773049i	$0.281269\pi$
$984$	0	0
$985$	30.4721	0.970923
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.94427	0.157219
$990$	0	0
$991$	−44.5410	−1.41489	−0.707446	−	0.706767i	$-0.750153\pi$
$991$	−44.5410	−1.41489	−0.707446	+	0.706767i	$0.750153\pi$
$992$	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	$-0.522561\pi$
$997$	−4.47214	−0.141634	−0.0708170	+	0.997489i	$0.522561\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bg.1.1	✓	2	1.1	even	1	trivial
9576.2.a.bp.1.2	yes	2	3.2	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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-3.23607 q^{5} -1.00000 q^{7} +0.763932 q^{11} -5.23607 q^{13} -2.76393 q^{17} +1.00000 q^{19} -1.23607 q^{23} +5.47214 q^{25} +0.472136 q^{29} -8.47214 q^{31} +3.23607 q^{35} -8.94427 q^{37} -2.00000 q^{41} -4.00000 q^{43} -2.47214 q^{47} +1.00000 q^{49} -8.47214 q^{53} -2.47214 q^{55} +8.00000 q^{59} -0.472136 q^{61} +16.9443 q^{65} +5.23607 q^{67} +10.4721 q^{71} -4.47214 q^{73} -0.763932 q^{77} +2.29180 q^{79} -2.00000 q^{83} +8.94427 q^{85} -6.94427 q^{89} +5.23607 q^{91} -3.23607 q^{95} -3.23607 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 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

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