LMFDB - Embedded newform 8624.2.a.ce.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{3} +2.00000 q^{5} +7.47214 q^{9} +O(q^{10})$	$q+3.23607 q^{3} +2.00000 q^{5} +7.47214 q^{9} +1.00000 q^{11} +1.23607 q^{13} +6.47214 q^{15} -1.23607 q^{17} -2.47214 q^{19} +6.47214 q^{23} -1.00000 q^{25} +14.4721 q^{27} -0.472136 q^{29} -7.23607 q^{31} +3.23607 q^{33} +0.472136 q^{37} +4.00000 q^{39} +6.76393 q^{41} -8.00000 q^{43} +14.9443 q^{45} +7.23607 q^{47} -4.00000 q^{51} +8.47214 q^{53} +2.00000 q^{55} -8.00000 q^{57} +3.23607 q^{59} +2.76393 q^{61} +2.47214 q^{65} -5.52786 q^{67} +20.9443 q^{69} +1.52786 q^{71} +5.23607 q^{73} -3.23607 q^{75} -8.94427 q^{79} +24.4164 q^{81} +15.4164 q^{83} -2.47214 q^{85} -1.52786 q^{87} -2.00000 q^{89} -23.4164 q^{93} -4.94427 q^{95} +9.41641 q^{97} +7.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 + 6 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} + 6 q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{23} - 2 q^{25} + 20 q^{27} + 8 q^{29} - 10 q^{31} + 2 q^{33} - 8 q^{37} + 8 q^{39} + 18 q^{41} - 16 q^{43} + 12 q^{45} + 10 q^{47} - 8 q^{51} + 8 q^{53} + 4 q^{55} - 16 q^{57} + 2 q^{59} + 10 q^{61} - 4 q^{65} - 20 q^{67} + 24 q^{69} + 12 q^{71} + 6 q^{73} - 2 q^{75} + 22 q^{81} + 4 q^{83} + 4 q^{85} - 12 q^{87} - 4 q^{89} - 20 q^{93} + 8 q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 6 * q^9 + 2 * q^11 - 2 * q^13 + 4 * q^15 + 2 * q^17 + 4 * q^19 + 4 * q^23 - 2 * q^25 + 20 * q^27 + 8 * q^29 - 10 * q^31 + 2 * q^33 - 8 * q^37 + 8 * q^39 + 18 * q^41 - 16 * q^43 + 12 * q^45 + 10 * q^47 - 8 * q^51 + 8 * q^53 + 4 * q^55 - 16 * q^57 + 2 * q^59 + 10 * q^61 - 4 * q^65 - 20 * q^67 + 24 * q^69 + 12 * q^71 + 6 * q^73 - 2 * q^75 + 22 * q^81 + 4 * q^83 + 4 * q^85 - 12 * q^87 - 4 * q^89 - 20 * q^93 + 8 * q^95 - 8 * q^97 + 6 * 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
$3$	3.23607	1.86834	0.934172	−	0.356822i	$-0.116140\pi$
$3$	3.23607	1.86834	0.934172	+	0.356822i	$0.116140\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	6.47214	1.67110
$16$	0	0
$17$	−1.23607	−0.299791	−0.149895	−	0.988702i	$-0.547894\pi$
$17$	−1.23607	−0.299791	−0.149895	+	0.988702i	$0.547894\pi$
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.47214	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	14.4721	2.78516
$28$	0	0
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	0	0
$31$	−7.23607	−1.29964	−0.649818	−	0.760090i	$-0.725155\pi$
$31$	−7.23607	−1.29964	−0.649818	+	0.760090i	$0.725155\pi$
$32$	0	0
$33$	3.23607	0.563327
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.472136	0.0776187	0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136	0.0776187	0.0388093	+	0.999247i	$0.487644\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	6.76393	1.05635	0.528174	−	0.849136i	$-0.322877\pi$
$41$	6.76393	1.05635	0.528174	+	0.849136i	$0.322877\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	14.9443	2.22776
$46$	0	0
$47$	7.23607	1.05549	0.527744	−	0.849403i	$-0.323038\pi$
$47$	7.23607	1.05549	0.527744	+	0.849403i	$0.323038\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	3.23607	0.421300	0.210650	−	0.977562i	$-0.432442\pi$
$59$	3.23607	0.421300	0.210650	+	0.977562i	$0.432442\pi$
$60$	0	0
$61$	2.76393	0.353885	0.176943	−	0.984221i	$-0.443379\pi$
$61$	2.76393	0.353885	0.176943	+	0.984221i	$0.443379\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.47214	0.306631
$66$	0	0
$67$	−5.52786	−0.675336	−0.337668	−	0.941265i	$-0.609638\pi$
$67$	−5.52786	−0.675336	−0.337668	+	0.941265i	$0.609638\pi$
$68$	0	0
$69$	20.9443	2.52139
$70$	0	0
$71$	1.52786	0.181324	0.0906621	−	0.995882i	$-0.471102\pi$
$71$	1.52786	0.181324	0.0906621	+	0.995882i	$0.471102\pi$
$72$	0	0
$73$	5.23607	0.612835	0.306418	−	0.951897i	$-0.400870\pi$
$73$	5.23607	0.612835	0.306418	+	0.951897i	$0.400870\pi$
$74$	0	0
$75$	−3.23607	−0.373669
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	15.4164	1.69217	0.846085	−	0.533048i	$-0.178953\pi$
$83$	15.4164	1.69217	0.846085	+	0.533048i	$0.178953\pi$
$84$	0	0
$85$	−2.47214	−0.268141
$86$	0	0
$87$	−1.52786	−0.163804
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−23.4164	−2.42817
$94$	0	0
$95$	−4.94427	−0.507272
$96$	0	0
$97$	9.41641	0.956091	0.478046	−	0.878335i	$-0.341345\pi$
$97$	9.41641	0.956091	0.478046	+	0.878335i	$0.341345\pi$
$98$	0	0
$99$	7.47214	0.750978
$100$	0	0
$101$	9.23607	0.919023	0.459512	−	0.888172i	$-0.348024\pi$
$101$	9.23607	0.919023	0.459512	+	0.888172i	$0.348024\pi$
$102$	0	0
$103$	−5.70820	−0.562446	−0.281223	−	0.959642i	$-0.590740\pi$
$103$	−5.70820	−0.562446	−0.281223	+	0.959642i	$0.590740\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	4.47214	0.428353	0.214176	−	0.976795i	$-0.431293\pi$
$109$	4.47214	0.428353	0.214176	+	0.976795i	$0.431293\pi$
$110$	0	0
$111$	1.52786	0.145018
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	12.9443	1.20706
$116$	0	0
$117$	9.23607	0.853875
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	21.8885	1.97362
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−20.9443	−1.85850	−0.929252	−	0.369447i	$-0.879547\pi$
$127$	−20.9443	−1.85850	−0.929252	+	0.369447i	$0.879547\pi$
$128$	0	0
$129$	−25.8885	−2.27936
$130$	0	0
$131$	−13.8885	−1.21345	−0.606724	−	0.794913i	$-0.707517\pi$
$131$	−13.8885	−1.21345	−0.606724	+	0.794913i	$0.707517\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	28.9443	2.49113
$136$	0	0
$137$	7.52786	0.643149	0.321574	−	0.946884i	$-0.395788\pi$
$137$	7.52786	0.643149	0.321574	+	0.946884i	$0.395788\pi$
$138$	0	0
$139$	−10.4721	−0.888235	−0.444117	−	0.895969i	$-0.646483\pi$
$139$	−10.4721	−0.888235	−0.444117	+	0.895969i	$0.646483\pi$
$140$	0	0
$141$	23.4164	1.97202
$142$	0	0
$143$	1.23607	0.103365
$144$	0	0
$145$	−0.944272	−0.0784175
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	0	0
$153$	−9.23607	−0.746692
$154$	0	0
$155$	−14.4721	−1.16243
$156$	0	0
$157$	6.94427	0.554213	0.277107	−	0.960839i	$-0.410624\pi$
$157$	6.94427	0.554213	0.277107	+	0.960839i	$0.410624\pi$
$158$	0	0
$159$	27.4164	2.17426
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−23.4164	−1.83411	−0.917057	−	0.398755i	$-0.869442\pi$
$163$	−23.4164	−1.83411	−0.917057	+	0.398755i	$0.869442\pi$
$164$	0	0
$165$	6.47214	0.503855
$166$	0	0
$167$	−12.9443	−1.00166	−0.500829	−	0.865546i	$-0.666971\pi$
$167$	−12.9443	−1.00166	−0.500829	+	0.865546i	$0.666971\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	−18.4721	−1.41260
$172$	0	0
$173$	17.2361	1.31043	0.655217	−	0.755441i	$-0.272577\pi$
$173$	17.2361	1.31043	0.655217	+	0.755441i	$0.272577\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.4721	0.787134
$178$	0	0
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	−1.41641	−0.105281	−0.0526404	−	0.998614i	$-0.516764\pi$
$181$	−1.41641	−0.105281	−0.0526404	+	0.998614i	$0.516764\pi$
$182$	0	0
$183$	8.94427	0.661180
$184$	0	0
$185$	0.944272	0.0694243
$186$	0	0
$187$	−1.23607	−0.0903902
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.9443	1.51547	0.757737	−	0.652560i	$-0.226305\pi$
$191$	20.9443	1.51547	0.757737	+	0.652560i	$0.226305\pi$
$192$	0	0
$193$	−23.8885	−1.71954	−0.859768	−	0.510686i	$-0.829392\pi$
$193$	−23.8885	−1.71954	−0.859768	+	0.510686i	$0.829392\pi$
$194$	0	0
$195$	8.00000	0.572892
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	20.1803	1.43055	0.715273	−	0.698845i	$-0.246302\pi$
$199$	20.1803	1.43055	0.715273	+	0.698845i	$0.246302\pi$
$200$	0	0
$201$	−17.8885	−1.26176
$202$	0	0
$203$	0	0
$204$	0	0
$205$	13.5279	0.944827
$206$	0	0
$207$	48.3607	3.36130
$208$	0	0
$209$	−2.47214	−0.171001
$210$	0	0
$211$	−21.8885	−1.50687	−0.753435	−	0.657523i	$-0.771604\pi$
$211$	−21.8885	−1.50687	−0.753435	+	0.657523i	$0.771604\pi$
$212$	0	0
$213$	4.94427	0.338776
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	0	0
$218$	0	0
$219$	16.9443	1.14499
$220$	0	0
$221$	−1.52786	−0.102775
$222$	0	0
$223$	12.1803	0.815656	0.407828	−	0.913059i	$-0.366286\pi$
$223$	12.1803	0.815656	0.407828	+	0.913059i	$0.366286\pi$
$224$	0	0
$225$	−7.47214	−0.498142
$226$	0	0
$227$	−29.8885	−1.98377	−0.991886	−	0.127129i	$-0.959424\pi$
$227$	−29.8885	−1.98377	−0.991886	+	0.127129i	$0.959424\pi$
$228$	0	0
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.4164	−1.14099	−0.570493	−	0.821302i	$-0.693248\pi$
$233$	−17.4164	−1.14099	−0.570493	+	0.821302i	$0.693248\pi$
$234$	0	0
$235$	14.4721	0.944058
$236$	0	0
$237$	−28.9443	−1.88013
$238$	0	0
$239$	−25.8885	−1.67459	−0.837295	−	0.546751i	$-0.815864\pi$
$239$	−25.8885	−1.67459	−0.837295	+	0.546751i	$0.815864\pi$
$240$	0	0
$241$	−27.1246	−1.74725	−0.873625	−	0.486600i	$-0.838237\pi$
$241$	−27.1246	−1.74725	−0.873625	+	0.486600i	$0.838237\pi$
$242$	0	0
$243$	35.5967	2.28353
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.05573	−0.194431
$248$	0	0
$249$	49.8885	3.16156
$250$	0	0
$251$	17.7082	1.11773	0.558866	−	0.829258i	$-0.311237\pi$
$251$	17.7082	1.11773	0.558866	+	0.829258i	$0.311237\pi$
$252$	0	0
$253$	6.47214	0.406900
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.52786	−0.218369
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	16.9443	1.04088
$266$	0	0
$267$	−6.47214	−0.396088
$268$	0	0
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	0	0
$271$	−1.52786	−0.0928111	−0.0464056	−	0.998923i	$-0.514777\pi$
$271$	−1.52786	−0.0928111	−0.0464056	+	0.998923i	$0.514777\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	15.8885	0.954650	0.477325	−	0.878727i	$-0.341606\pi$
$277$	15.8885	0.954650	0.477325	+	0.878727i	$0.341606\pi$
$278$	0	0
$279$	−54.0689	−3.23702
$280$	0	0
$281$	−12.4721	−0.744025	−0.372013	−	0.928228i	$-0.621332\pi$
$281$	−12.4721	−0.744025	−0.372013	+	0.928228i	$0.621332\pi$
$282$	0	0
$283$	−5.88854	−0.350038	−0.175019	−	0.984565i	$-0.555999\pi$
$283$	−5.88854	−0.350038	−0.175019	+	0.984565i	$0.555999\pi$
$284$	0	0
$285$	−16.0000	−0.947758
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.4721	−0.910126
$290$	0	0
$291$	30.4721	1.78631
$292$	0	0
$293$	−15.1246	−0.883589	−0.441795	−	0.897116i	$-0.645658\pi$
$293$	−15.1246	−0.883589	−0.441795	+	0.897116i	$0.645658\pi$
$294$	0	0
$295$	6.47214	0.376822
$296$	0	0
$297$	14.4721	0.839759
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	0	0
$303$	29.8885	1.71705
$304$	0	0
$305$	5.52786	0.316525
$306$	0	0
$307$	8.94427	0.510477	0.255238	−	0.966878i	$-0.417846\pi$
$307$	8.94427	0.510477	0.255238	+	0.966878i	$0.417846\pi$
$308$	0	0
$309$	−18.4721	−1.05084
$310$	0	0
$311$	−21.7082	−1.23096	−0.615480	−	0.788153i	$-0.711038\pi$
$311$	−21.7082	−1.23096	−0.615480	+	0.788153i	$0.711038\pi$
$312$	0	0
$313$	2.94427	0.166420	0.0832100	−	0.996532i	$-0.473483\pi$
$313$	2.94427	0.166420	0.0832100	+	0.996532i	$0.473483\pi$
$314$	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
$319$	−0.472136	−0.0264345
$320$	0	0
$321$	12.9443	0.722479
$322$	0	0
$323$	3.05573	0.170025
$324$	0	0
$325$	−1.23607	−0.0685647
$326$	0	0
$327$	14.4721	0.800311
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−21.8885	−1.20310	−0.601552	−	0.798834i	$-0.705451\pi$
$331$	−21.8885	−1.20310	−0.601552	+	0.798834i	$0.705451\pi$
$332$	0	0
$333$	3.52786	0.193326
$334$	0	0
$335$	−11.0557	−0.604039
$336$	0	0
$337$	−20.4721	−1.11519	−0.557594	−	0.830114i	$-0.688275\pi$
$337$	−20.4721	−1.11519	−0.557594	+	0.830114i	$0.688275\pi$
$338$	0	0
$339$	6.47214	0.351518
$340$	0	0
$341$	−7.23607	−0.391855
$342$	0	0
$343$	0	0
$344$	0	0
$345$	41.8885	2.25520
$346$	0	0
$347$	−3.05573	−0.164040	−0.0820200	−	0.996631i	$-0.526137\pi$
$347$	−3.05573	−0.164040	−0.0820200	+	0.996631i	$0.526137\pi$
$348$	0	0
$349$	2.76393	0.147950	0.0739749	−	0.997260i	$-0.476432\pi$
$349$	2.76393	0.147950	0.0739749	+	0.997260i	$0.476432\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	0	0
$353$	15.8885	0.845662	0.422831	−	0.906209i	$-0.361036\pi$
$353$	15.8885	0.845662	0.422831	+	0.906209i	$0.361036\pi$
$354$	0	0
$355$	3.05573	0.162181
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.05573	0.372387	0.186194	−	0.982513i	$-0.440385\pi$
$359$	7.05573	0.372387	0.186194	+	0.982513i	$0.440385\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	3.23607	0.169850
$364$	0	0
$365$	10.4721	0.548137
$366$	0	0
$367$	17.1246	0.893897	0.446949	−	0.894560i	$-0.352511\pi$
$367$	17.1246	0.893897	0.446949	+	0.894560i	$0.352511\pi$
$368$	0	0
$369$	50.5410	2.63106
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	−38.8328	−2.00532
$376$	0	0
$377$	−0.583592	−0.0300565
$378$	0	0
$379$	25.3050	1.29983	0.649914	−	0.760008i	$-0.274805\pi$
$379$	25.3050	1.29983	0.649914	+	0.760008i	$0.274805\pi$
$380$	0	0
$381$	−67.7771	−3.47233
$382$	0	0
$383$	26.6525	1.36188	0.680939	−	0.732340i	$-0.261572\pi$
$383$	26.6525	1.36188	0.680939	+	0.732340i	$0.261572\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−59.7771	−3.03864
$388$	0	0
$389$	−19.8885	−1.00839	−0.504195	−	0.863590i	$-0.668211\pi$
$389$	−19.8885	−1.00839	−0.504195	+	0.863590i	$0.668211\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	−44.9443	−2.26714
$394$	0	0
$395$	−17.8885	−0.900070
$396$	0	0
$397$	0.111456	0.00559383	0.00279691	−	0.999996i	$-0.499110\pi$
$397$	0.111456	0.00559383	0.00279691	+	0.999996i	$0.499110\pi$
$398$	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
$403$	−8.94427	−0.445546
$404$	0	0
$405$	48.8328	2.42652
$406$	0	0
$407$	0.472136	0.0234029
$408$	0	0
$409$	31.1246	1.53901	0.769507	−	0.638639i	$-0.220502\pi$
$409$	31.1246	1.53901	0.769507	+	0.638639i	$0.220502\pi$
$410$	0	0
$411$	24.3607	1.20162
$412$	0	0
$413$	0	0
$414$	0	0
$415$	30.8328	1.51352
$416$	0	0
$417$	−33.8885	−1.65953
$418$	0	0
$419$	−6.65248	−0.324995	−0.162497	−	0.986709i	$-0.551955\pi$
$419$	−6.65248	−0.324995	−0.162497	+	0.986709i	$0.551955\pi$
$420$	0	0
$421$	−22.3607	−1.08979	−0.544896	−	0.838503i	$-0.683431\pi$
$421$	−22.3607	−1.08979	−0.544896	+	0.838503i	$0.683431\pi$
$422$	0	0
$423$	54.0689	2.62892
$424$	0	0
$425$	1.23607	0.0599581
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−0.472136	−0.0226894	−0.0113447	−	0.999936i	$-0.503611\pi$
$433$	−0.472136	−0.0226894	−0.0113447	+	0.999936i	$0.503611\pi$
$434$	0	0
$435$	−3.05573	−0.146511
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	1.52786	0.0729210	0.0364605	−	0.999335i	$-0.488392\pi$
$439$	1.52786	0.0729210	0.0364605	+	0.999335i	$0.488392\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.05573	0.335228	0.167614	−	0.985853i	$-0.446394\pi$
$443$	7.05573	0.335228	0.167614	+	0.985853i	$0.446394\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	0	0
$447$	45.3050	2.14285
$448$	0	0
$449$	−19.5279	−0.921577	−0.460788	−	0.887510i	$-0.652433\pi$
$449$	−19.5279	−0.921577	−0.460788	+	0.887510i	$0.652433\pi$
$450$	0	0
$451$	6.76393	0.318501
$452$	0	0
$453$	28.9443	1.35992
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.8328	1.16163	0.580815	−	0.814036i	$-0.302734\pi$
$457$	24.8328	1.16163	0.580815	+	0.814036i	$0.302734\pi$
$458$	0	0
$459$	−17.8885	−0.834966
$460$	0	0
$461$	−10.1803	−0.474146	−0.237073	−	0.971492i	$-0.576188\pi$
$461$	−10.1803	−0.474146	−0.237073	+	0.971492i	$0.576188\pi$
$462$	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
$465$	−46.8328	−2.17182
$466$	0	0
$467$	−34.0689	−1.57652	−0.788260	−	0.615342i	$-0.789018\pi$
$467$	−34.0689	−1.57652	−0.788260	+	0.615342i	$0.789018\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	22.4721	1.03546
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	2.47214	0.113429
$476$	0	0
$477$	63.3050	2.89853
$478$	0	0
$479$	22.4721	1.02678	0.513389	−	0.858156i	$-0.328390\pi$
$479$	22.4721	1.02678	0.513389	+	0.858156i	$0.328390\pi$
$480$	0	0
$481$	0.583592	0.0266095
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.8328	0.855154
$486$	0	0
$487$	−8.36068	−0.378859	−0.189429	−	0.981894i	$-0.560664\pi$
$487$	−8.36068	−0.378859	−0.189429	+	0.981894i	$0.560664\pi$
$488$	0	0
$489$	−75.7771	−3.42676
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0.583592	0.0262837
$494$	0	0
$495$	14.9443	0.671695
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−10.4721	−0.468797	−0.234399	−	0.972141i	$-0.575312\pi$
$499$	−10.4721	−0.468797	−0.234399	+	0.972141i	$0.575312\pi$
$500$	0	0
$501$	−41.8885	−1.87144
$502$	0	0
$503$	−3.41641	−0.152330	−0.0761650	−	0.997095i	$-0.524268\pi$
$503$	−3.41641	−0.152330	−0.0761650	+	0.997095i	$0.524268\pi$
$504$	0	0
$505$	18.4721	0.821999
$506$	0	0
$507$	−37.1246	−1.64876
$508$	0	0
$509$	−31.5279	−1.39745	−0.698724	−	0.715391i	$-0.746248\pi$
$509$	−31.5279	−1.39745	−0.698724	+	0.715391i	$0.746248\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−35.7771	−1.57960
$514$	0	0
$515$	−11.4164	−0.503067
$516$	0	0
$517$	7.23607	0.318242
$518$	0	0
$519$	55.7771	2.44834
$520$	0	0
$521$	14.3607	0.629153	0.314576	−	0.949232i	$-0.398138\pi$
$521$	14.3607	0.629153	0.314576	+	0.949232i	$0.398138\pi$
$522$	0	0
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.94427	0.389619
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	24.1803	1.04934
$532$	0	0
$533$	8.36068	0.362141
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	−28.9443	−1.24904
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−32.8328	−1.41159	−0.705797	−	0.708415i	$-0.749411\pi$
$541$	−32.8328	−1.41159	−0.705797	+	0.708415i	$0.749411\pi$
$542$	0	0
$543$	−4.58359	−0.196701
$544$	0	0
$545$	8.94427	0.383131
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	20.6525	0.881426
$550$	0	0
$551$	1.16718	0.0497237
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.05573	0.129708
$556$	0	0
$557$	−21.0557	−0.892160	−0.446080	−	0.894993i	$-0.647180\pi$
$557$	−21.0557	−0.892160	−0.446080	+	0.894993i	$0.647180\pi$
$558$	0	0
$559$	−9.88854	−0.418241
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	39.4164	1.66120	0.830602	−	0.556867i	$-0.187997\pi$
$563$	39.4164	1.66120	0.830602	+	0.556867i	$0.187997\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.4721	0.690548	0.345274	−	0.938502i	$-0.387786\pi$
$569$	16.4721	0.690548	0.345274	+	0.938502i	$0.387786\pi$
$570$	0	0
$571$	32.9443	1.37867	0.689337	−	0.724440i	$-0.257902\pi$
$571$	32.9443	1.37867	0.689337	+	0.724440i	$0.257902\pi$
$572$	0	0
$573$	67.7771	2.83143
$574$	0	0
$575$	−6.47214	−0.269907
$576$	0	0
$577$	28.4721	1.18531	0.592655	−	0.805456i	$-0.298080\pi$
$577$	28.4721	1.18531	0.592655	+	0.805456i	$0.298080\pi$
$578$	0	0
$579$	−77.3050	−3.21268
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.47214	0.350880
$584$	0	0
$585$	18.4721	0.763729
$586$	0	0
$587$	−13.1246	−0.541711	−0.270855	−	0.962620i	$-0.587307\pi$
$587$	−13.1246	−0.541711	−0.270855	+	0.962620i	$0.587307\pi$
$588$	0	0
$589$	17.8885	0.737085
$590$	0	0
$591$	−6.47214	−0.266228
$592$	0	0
$593$	32.2918	1.32607	0.663033	−	0.748591i	$-0.269269\pi$
$593$	32.2918	1.32607	0.663033	+	0.748591i	$0.269269\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	65.3050	2.67275
$598$	0	0
$599$	−3.41641	−0.139591	−0.0697953	−	0.997561i	$-0.522235\pi$
$599$	−3.41641	−0.139591	−0.0697953	+	0.997561i	$0.522235\pi$
$600$	0	0
$601$	−3.12461	−0.127456	−0.0637278	−	0.997967i	$-0.520299\pi$
$601$	−3.12461	−0.127456	−0.0637278	+	0.997967i	$0.520299\pi$
$602$	0	0
$603$	−41.3050	−1.68207
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	−4.94427	−0.200682	−0.100341	−	0.994953i	$-0.531993\pi$
$607$	−4.94427	−0.200682	−0.100341	+	0.994953i	$0.531993\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.94427	0.361847
$612$	0	0
$613$	−47.3050	−1.91063	−0.955315	−	0.295591i	$-0.904483\pi$
$613$	−47.3050	−1.91063	−0.955315	+	0.295591i	$0.904483\pi$
$614$	0	0
$615$	43.7771	1.76526
$616$	0	0
$617$	33.4164	1.34529	0.672647	−	0.739964i	$-0.265157\pi$
$617$	33.4164	1.34529	0.672647	+	0.739964i	$0.265157\pi$
$618$	0	0
$619$	−29.1246	−1.17062	−0.585308	−	0.810811i	$-0.699027\pi$
$619$	−29.1246	−1.17062	−0.585308	+	0.810811i	$0.699027\pi$
$620$	0	0
$621$	93.6656	3.75867
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	−0.583592	−0.0232693
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	−70.8328	−2.81535
$634$	0	0
$635$	−41.8885	−1.66230
$636$	0	0
$637$	0	0
$638$	0	0
$639$	11.4164	0.451626
$640$	0	0
$641$	−24.4721	−0.966591	−0.483296	−	0.875457i	$-0.660560\pi$
$641$	−24.4721	−0.966591	−0.483296	+	0.875457i	$0.660560\pi$
$642$	0	0
$643$	−29.1246	−1.14856	−0.574281	−	0.818658i	$-0.694718\pi$
$643$	−29.1246	−1.14856	−0.574281	+	0.818658i	$0.694718\pi$
$644$	0	0
$645$	−51.7771	−2.03872
$646$	0	0
$647$	−22.0689	−0.867617	−0.433809	−	0.901005i	$-0.642831\pi$
$647$	−22.0689	−0.867617	−0.433809	+	0.901005i	$0.642831\pi$
$648$	0	0
$649$	3.23607	0.127027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.9443	1.68054	0.840270	−	0.542169i	$-0.182397\pi$
$653$	42.9443	1.68054	0.840270	+	0.542169i	$0.182397\pi$
$654$	0	0
$655$	−27.7771	−1.08534
$656$	0	0
$657$	39.1246	1.52640
$658$	0	0
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	−12.8328	−0.499139	−0.249569	−	0.968357i	$-0.580289\pi$
$661$	−12.8328	−0.499139	−0.249569	+	0.968357i	$0.580289\pi$
$662$	0	0
$663$	−4.94427	−0.192020
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.05573	−0.118318
$668$	0	0
$669$	39.4164	1.52393
$670$	0	0
$671$	2.76393	0.106700
$672$	0	0
$673$	5.41641	0.208787	0.104394	−	0.994536i	$-0.466710\pi$
$673$	5.41641	0.208787	0.104394	+	0.994536i	$0.466710\pi$
$674$	0	0
$675$	−14.4721	−0.557033
$676$	0	0
$677$	−3.70820	−0.142518	−0.0712589	−	0.997458i	$-0.522702\pi$
$677$	−3.70820	−0.142518	−0.0712589	+	0.997458i	$0.522702\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−96.7214	−3.70637
$682$	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
$685$	15.0557	0.575250
$686$	0	0
$687$	−14.4721	−0.552146
$688$	0	0
$689$	10.4721	0.398957
$690$	0	0
$691$	−48.5410	−1.84659	−0.923294	−	0.384095i	$-0.874514\pi$
$691$	−48.5410	−1.84659	−0.923294	+	0.384095i	$0.874514\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.9443	−0.794462
$696$	0	0
$697$	−8.36068	−0.316683
$698$	0	0
$699$	−56.3607	−2.13176
$700$	0	0
$701$	−24.4721	−0.924300	−0.462150	−	0.886802i	$-0.652922\pi$
$701$	−24.4721	−0.924300	−0.462150	+	0.886802i	$0.652922\pi$
$702$	0	0
$703$	−1.16718	−0.0440212
$704$	0	0
$705$	46.8328	1.76383
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.94427	0.110574	0.0552872	−	0.998470i	$-0.482393\pi$
$709$	2.94427	0.110574	0.0552872	+	0.998470i	$0.482393\pi$
$710$	0	0
$711$	−66.8328	−2.50643
$712$	0	0
$713$	−46.8328	−1.75390
$714$	0	0
$715$	2.47214	0.0924526
$716$	0	0
$717$	−83.7771	−3.12871
$718$	0	0
$719$	33.4853	1.24879	0.624395	−	0.781108i	$-0.285345\pi$
$719$	33.4853	1.24879	0.624395	+	0.781108i	$0.285345\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−87.7771	−3.26447
$724$	0	0
$725$	0.472136	0.0175347
$726$	0	0
$727$	−51.0132	−1.89197	−0.945987	−	0.324206i	$-0.894903\pi$
$727$	−51.0132	−1.89197	−0.945987	+	0.324206i	$0.894903\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	9.88854	0.365741
$732$	0	0
$733$	−13.2361	−0.488885	−0.244443	−	0.969664i	$-0.578605\pi$
$733$	−13.2361	−0.488885	−0.244443	+	0.969664i	$0.578605\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.52786	−0.203621
$738$	0	0
$739$	−7.05573	−0.259549	−0.129775	−	0.991544i	$-0.541425\pi$
$739$	−7.05573	−0.259549	−0.129775	+	0.991544i	$0.541425\pi$
$740$	0	0
$741$	−9.88854	−0.363265
$742$	0	0
$743$	33.8885	1.24325	0.621625	−	0.783315i	$-0.286473\pi$
$743$	33.8885	1.24325	0.621625	+	0.783315i	$0.286473\pi$
$744$	0	0
$745$	28.0000	1.02584
$746$	0	0
$747$	115.193	4.21471
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−38.4721	−1.40387	−0.701934	−	0.712242i	$-0.747680\pi$
$751$	−38.4721	−1.40387	−0.701934	+	0.712242i	$0.747680\pi$
$752$	0	0
$753$	57.3050	2.08831
$754$	0	0
$755$	17.8885	0.651031
$756$	0	0
$757$	−19.8885	−0.722861	−0.361431	−	0.932399i	$-0.617712\pi$
$757$	−19.8885	−0.722861	−0.361431	+	0.932399i	$0.617712\pi$
$758$	0	0
$759$	20.9443	0.760229
$760$	0	0
$761$	−17.5967	−0.637882	−0.318941	−	0.947775i	$-0.603327\pi$
$761$	−17.5967	−0.637882	−0.318941	+	0.947775i	$0.603327\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−18.4721	−0.667861
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	−31.7082	−1.14343	−0.571714	−	0.820453i	$-0.693721\pi$
$769$	−31.7082	−1.14343	−0.571714	+	0.820453i	$0.693721\pi$
$770$	0	0
$771$	19.4164	0.699265
$772$	0	0
$773$	−6.36068	−0.228778	−0.114389	−	0.993436i	$-0.536491\pi$
$773$	−6.36068	−0.228778	−0.114389	+	0.993436i	$0.536491\pi$
$774$	0	0
$775$	7.23607	0.259927
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.7214	−0.599105
$780$	0	0
$781$	1.52786	0.0546713
$782$	0	0
$783$	−6.83282	−0.244185
$784$	0	0
$785$	13.8885	0.495703
$786$	0	0
$787$	−16.5836	−0.591141	−0.295571	−	0.955321i	$-0.595510\pi$
$787$	−16.5836	−0.591141	−0.295571	+	0.955321i	$0.595510\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.41641	0.121320
$794$	0	0
$795$	54.8328	1.94472
$796$	0	0
$797$	−2.94427	−0.104291	−0.0521457	−	0.998639i	$-0.516606\pi$
$797$	−2.94427	−0.104291	−0.0521457	+	0.998639i	$0.516606\pi$
$798$	0	0
$799$	−8.94427	−0.316426
$800$	0	0
$801$	−14.9443	−0.528030
$802$	0	0
$803$	5.23607	0.184777
$804$	0	0
$805$	0	0
$806$	0	0
$807$	43.4164	1.52833
$808$	0	0
$809$	38.9443	1.36921	0.684604	−	0.728915i	$-0.259975\pi$
$809$	38.9443	1.36921	0.684604	+	0.728915i	$0.259975\pi$
$810$	0	0
$811$	18.8328	0.661310	0.330655	−	0.943752i	$-0.392730\pi$
$811$	18.8328	0.661310	0.330655	+	0.943752i	$0.392730\pi$
$812$	0	0
$813$	−4.94427	−0.173403
$814$	0	0
$815$	−46.8328	−1.64048
$816$	0	0
$817$	19.7771	0.691913
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.83282	−0.308267	−0.154134	−	0.988050i	$-0.549259\pi$
$821$	−8.83282	−0.308267	−0.154134	+	0.988050i	$0.549259\pi$
$822$	0	0
$823$	49.8885	1.73901	0.869503	−	0.493928i	$-0.164439\pi$
$823$	49.8885	1.73901	0.869503	+	0.493928i	$0.164439\pi$
$824$	0	0
$825$	−3.23607	−0.112665
$826$	0	0
$827$	−4.94427	−0.171929	−0.0859646	−	0.996298i	$-0.527397\pi$
$827$	−4.94427	−0.171929	−0.0859646	+	0.996298i	$0.527397\pi$
$828$	0	0
$829$	16.8328	0.584628	0.292314	−	0.956322i	$-0.405575\pi$
$829$	16.8328	0.584628	0.292314	+	0.956322i	$0.405575\pi$
$830$	0	0
$831$	51.4164	1.78362
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−25.8885	−0.895910
$836$	0	0
$837$	−104.721	−3.61970
$838$	0	0
$839$	−14.0689	−0.485712	−0.242856	−	0.970062i	$-0.578084\pi$
$839$	−14.0689	−0.485712	−0.242856	+	0.970062i	$0.578084\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	−40.3607	−1.39010
$844$	0	0
$845$	−22.9443	−0.789307
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−19.0557	−0.653991
$850$	0	0
$851$	3.05573	0.104749
$852$	0	0
$853$	−0.652476	−0.0223403	−0.0111702	−	0.999938i	$-0.503556\pi$
$853$	−0.652476	−0.0223403	−0.0111702	+	0.999938i	$0.503556\pi$
$854$	0	0
$855$	−36.9443	−1.26347
$856$	0	0
$857$	−10.7639	−0.367689	−0.183844	−	0.982955i	$-0.558854\pi$
$857$	−10.7639	−0.367689	−0.183844	+	0.982955i	$0.558854\pi$
$858$	0	0
$859$	40.5410	1.38324	0.691621	−	0.722261i	$-0.256897\pi$
$859$	40.5410	1.38324	0.691621	+	0.722261i	$0.256897\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.9443	0.712951	0.356476	−	0.934305i	$-0.383978\pi$
$863$	20.9443	0.712951	0.356476	+	0.934305i	$0.383978\pi$
$864$	0	0
$865$	34.4721	1.17209
$866$	0	0
$867$	−50.0689	−1.70043
$868$	0	0
$869$	−8.94427	−0.303414
$870$	0	0
$871$	−6.83282	−0.231521
$872$	0	0
$873$	70.3607	2.38135
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.4164	1.39853	0.699266	−	0.714861i	$-0.253510\pi$
$877$	41.4164	1.39853	0.699266	+	0.714861i	$0.253510\pi$
$878$	0	0
$879$	−48.9443	−1.65085
$880$	0	0
$881$	−29.4164	−0.991064	−0.495532	−	0.868590i	$-0.665027\pi$
$881$	−29.4164	−0.991064	−0.495532	+	0.868590i	$0.665027\pi$
$882$	0	0
$883$	−8.94427	−0.300999	−0.150499	−	0.988610i	$-0.548088\pi$
$883$	−8.94427	−0.300999	−0.150499	+	0.988610i	$0.548088\pi$
$884$	0	0
$885$	20.9443	0.704034
$886$	0	0
$887$	40.3607	1.35518	0.677589	−	0.735440i	$-0.263025\pi$
$887$	40.3607	1.35518	0.677589	+	0.735440i	$0.263025\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	24.4164	0.817980
$892$	0	0
$893$	−17.8885	−0.598617
$894$	0	0
$895$	−17.8885	−0.597948
$896$	0	0
$897$	25.8885	0.864393
$898$	0	0
$899$	3.41641	0.113944
$900$	0	0
$901$	−10.4721	−0.348877
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.83282	−0.0941660
$906$	0	0
$907$	−13.5279	−0.449185	−0.224593	−	0.974453i	$-0.572105\pi$
$907$	−13.5279	−0.449185	−0.224593	+	0.974453i	$0.572105\pi$
$908$	0	0
$909$	69.0132	2.28902
$910$	0	0
$911$	−33.5279	−1.11083	−0.555414	−	0.831574i	$-0.687440\pi$
$911$	−33.5279	−1.11083	−0.555414	+	0.831574i	$0.687440\pi$
$912$	0	0
$913$	15.4164	0.510209
$914$	0	0
$915$	17.8885	0.591377
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−6.11146	−0.201598	−0.100799	−	0.994907i	$-0.532140\pi$
$919$	−6.11146	−0.201598	−0.100799	+	0.994907i	$0.532140\pi$
$920$	0	0
$921$	28.9443	0.953746
$922$	0	0
$923$	1.88854	0.0621622
$924$	0	0
$925$	−0.472136	−0.0155237
$926$	0	0
$927$	−42.6525	−1.40089
$928$	0	0
$929$	−28.2492	−0.926827	−0.463413	−	0.886142i	$-0.653376\pi$
$929$	−28.2492	−0.926827	−0.463413	+	0.886142i	$0.653376\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−70.2492	−2.29986
$934$	0	0
$935$	−2.47214	−0.0808475
$936$	0	0
$937$	−20.6525	−0.674687	−0.337343	−	0.941382i	$-0.609528\pi$
$937$	−20.6525	−0.674687	−0.337343	+	0.941382i	$0.609528\pi$
$938$	0	0
$939$	9.52786	0.310930
$940$	0	0
$941$	41.5967	1.35602	0.678008	−	0.735055i	$-0.262844\pi$
$941$	41.5967	1.35602	0.678008	+	0.735055i	$0.262844\pi$
$942$	0	0
$943$	43.7771	1.42558
$944$	0	0
$945$	0	0
$946$	0	0
$947$	58.8328	1.91181	0.955905	−	0.293677i	$-0.0948789\pi$
$947$	58.8328	1.91181	0.955905	+	0.293677i	$0.0948789\pi$
$948$	0	0
$949$	6.47214	0.210094
$950$	0	0
$951$	45.3050	1.46911
$952$	0	0
$953$	5.05573	0.163771	0.0818855	−	0.996642i	$-0.473906\pi$
$953$	5.05573	0.163771	0.0818855	+	0.996642i	$0.473906\pi$
$954$	0	0
$955$	41.8885	1.35548
$956$	0	0
$957$	−1.52786	−0.0493888
$958$	0	0
$959$	0	0
$960$	0	0
$961$	21.3607	0.689054
$962$	0	0
$963$	29.8885	0.963145
$964$	0	0
$965$	−47.7771	−1.53800
$966$	0	0
$967$	−21.8885	−0.703888	−0.351944	−	0.936021i	$-0.614479\pi$
$967$	−21.8885	−0.703888	−0.351944	+	0.936021i	$0.614479\pi$
$968$	0	0
$969$	9.88854	0.317666
$970$	0	0
$971$	−29.1246	−0.934653	−0.467327	−	0.884085i	$-0.654783\pi$
$971$	−29.1246	−0.934653	−0.467327	+	0.884085i	$0.654783\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	5.05573	0.161747	0.0808735	−	0.996724i	$-0.474229\pi$
$977$	5.05573	0.161747	0.0808735	+	0.996724i	$0.474229\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	33.4164	1.06690
$982$	0	0
$983$	44.1803	1.40913	0.704567	−	0.709637i	$-0.251141\pi$
$983$	44.1803	1.40913	0.704567	+	0.709637i	$0.251141\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−51.7771	−1.64642
$990$	0	0
$991$	26.2492	0.833834	0.416917	−	0.908945i	$-0.363111\pi$
$991$	26.2492	0.833834	0.416917	+	0.908945i	$0.363111\pi$
$992$	0	0
$993$	−70.8328	−2.24781
$994$	0	0
$995$	40.3607	1.27952
$996$	0	0
$997$	−32.6525	−1.03411	−0.517057	−	0.855951i	$-0.672973\pi$
$997$	−32.6525	−1.03411	−0.517057	+	0.855951i	$0.672973\pi$
$998$	0	0
$999$	6.83282	0.216181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.ce.1.2		2
4.3	odd	2		539.2.a.f.1.1		2
7.6	odd	2		1232.2.a.m.1.1		2
12.11	even	2		4851.2.a.y.1.2		2
28.3	even	6		539.2.e.i.177.2		4
28.11	odd	6		539.2.e.j.177.2		4
28.19	even	6		539.2.e.i.67.2		4
28.23	odd	6		539.2.e.j.67.2		4
28.27	even	2		77.2.a.d.1.1	✓	2
44.43	even	2		5929.2.a.m.1.2		2
56.13	odd	2		4928.2.a.bv.1.2		2
56.27	even	2		4928.2.a.bm.1.1		2
84.83	odd	2		693.2.a.h.1.2		2
140.27	odd	4		1925.2.b.h.1849.1		4
140.83	odd	4		1925.2.b.h.1849.4		4
140.139	even	2		1925.2.a.r.1.2		2
308.27	even	10		847.2.f.n.729.1		4
308.83	odd	10		847.2.f.b.729.1		4
308.139	odd	10		847.2.f.m.148.1		4
308.167	odd	10		847.2.f.b.323.1		4
308.195	odd	10		847.2.f.m.372.1		4
308.223	even	10		847.2.f.a.372.1		4
308.251	even	10		847.2.f.n.323.1		4
308.279	even	10		847.2.f.a.148.1		4
308.307	odd	2		847.2.a.f.1.2		2
924.923	even	2		7623.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.d.1.1	✓	2	28.27	even	2
539.2.a.f.1.1		2	4.3	odd	2
539.2.e.i.67.2		4	28.19	even	6
539.2.e.i.177.2		4	28.3	even	6
539.2.e.j.67.2		4	28.23	odd	6
539.2.e.j.177.2		4	28.11	odd	6
693.2.a.h.1.2		2	84.83	odd	2
847.2.a.f.1.2		2	308.307	odd	2
847.2.f.a.148.1		4	308.279	even	10
847.2.f.a.372.1		4	308.223	even	10
847.2.f.b.323.1		4	308.167	odd	10
847.2.f.b.729.1		4	308.83	odd	10
847.2.f.m.148.1		4	308.139	odd	10
847.2.f.m.372.1		4	308.195	odd	10
847.2.f.n.323.1		4	308.251	even	10
847.2.f.n.729.1		4	308.27	even	10
1232.2.a.m.1.1		2	7.6	odd	2
1925.2.a.r.1.2		2	140.139	even	2
1925.2.b.h.1849.1		4	140.27	odd	4
1925.2.b.h.1849.4		4	140.83	odd	4
4851.2.a.y.1.2		2	12.11	even	2
4928.2.a.bm.1.1		2	56.27	even	2
4928.2.a.bv.1.2		2	56.13	odd	2
5929.2.a.m.1.2		2	44.43	even	2
7623.2.a.bl.1.1		2	924.923	even	2
8624.2.a.ce.1.2		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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23607 q^{3} +2.00000 q^{5} +7.47214 q^{9} +O(q^{10})\)	\(q+3.23607 q^{3} +2.00000 q^{5} +7.47214 q^{9} +1.00000 q^{11} +1.23607 q^{13} +6.47214 q^{15} -1.23607 q^{17} -2.47214 q^{19} +6.47214 q^{23} -1.00000 q^{25} +14.4721 q^{27} -0.472136 q^{29} -7.23607 q^{31} +3.23607 q^{33} +0.472136 q^{37} +4.00000 q^{39} +6.76393 q^{41} -8.00000 q^{43} +14.9443 q^{45} +7.23607 q^{47} -4.00000 q^{51} +8.47214 q^{53} +2.00000 q^{55} -8.00000 q^{57} +3.23607 q^{59} +2.76393 q^{61} +2.47214 q^{65} -5.52786 q^{67} +20.9443 q^{69} +1.52786 q^{71} +5.23607 q^{73} -3.23607 q^{75} -8.94427 q^{79} +24.4164 q^{81} +15.4164 q^{83} -2.47214 q^{85} -1.52786 q^{87} -2.00000 q^{89} -23.4164 q^{93} -4.94427 q^{95} +9.41641 q^{97} +7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 + 6 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} + 6 q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{23} - 2 q^{25} + 20 q^{27} + 8 q^{29} - 10 q^{31} + 2 q^{33} - 8 q^{37} + 8 q^{39} + 18 q^{41} - 16 q^{43} + 12 q^{45} + 10 q^{47} - 8 q^{51} + 8 q^{53} + 4 q^{55} - 16 q^{57} + 2 q^{59} + 10 q^{61} - 4 q^{65} - 20 q^{67} + 24 q^{69} + 12 q^{71} + 6 q^{73} - 2 q^{75} + 22 q^{81} + 4 q^{83} + 4 q^{85} - 12 q^{87} - 4 q^{89} - 20 q^{93} + 8 q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 6 * q^9 + 2 * q^11 - 2 * q^13 + 4 * q^15 + 2 * q^17 + 4 * q^19 + 4 * q^23 - 2 * q^25 + 20 * q^27 + 8 * q^29 - 10 * q^31 + 2 * q^33 - 8 * q^37 + 8 * q^39 + 18 * q^41 - 16 * q^43 + 12 * q^45 + 10 * q^47 - 8 * q^51 + 8 * q^53 + 4 * q^55 - 16 * q^57 + 2 * q^59 + 10 * q^61 - 4 * q^65 - 20 * q^67 + 24 * q^69 + 12 * q^71 + 6 * q^73 - 2 * q^75 + 22 * q^81 + 4 * q^83 + 4 * q^85 - 12 * q^87 - 4 * q^89 - 20 * q^93 + 8 * q^95 - 8 * q^97 + 6 * q^99

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