LMFDB - Embedded newform 8775.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8775,2,Mod(1,8775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8775 = 3^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8775.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:	$70.0687277737$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1755)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.785261$ of defining polynomial
Character	$\chi$	$=$	8775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.214739 q^{2} -1.95389 q^{4} +3.65683 q^{7} -0.849052 q^{8} +O(q^{10})$	$q+0.214739 q^{2} -1.95389 q^{4} +3.65683 q^{7} -0.849052 q^{8} -0.401897 q^{11} +1.00000 q^{13} +0.785261 q^{14} +3.72545 q^{16} -6.98901 q^{17} +3.87156 q^{19} -0.0863028 q^{22} -5.80185 q^{23} +0.214739 q^{26} -7.14503 q^{28} -7.63431 q^{29} -1.02251 q^{31} +2.49810 q^{32} -1.50081 q^{34} +11.5820 q^{37} +0.831374 q^{38} -10.1214 q^{41} +2.87156 q^{43} +0.785261 q^{44} -1.24588 q^{46} +11.4000 q^{47} +6.37237 q^{49} -1.95389 q^{52} +10.9067 q^{53} -3.10484 q^{56} -1.63938 q^{58} -1.68043 q^{59} +4.49218 q^{61} -0.219573 q^{62} -6.91446 q^{64} +7.59725 q^{67} +13.6557 q^{68} +7.86564 q^{71} -10.6560 q^{73} +2.48711 q^{74} -7.56460 q^{76} -1.46967 q^{77} +12.6820 q^{79} -2.17346 q^{82} +9.31474 q^{83} +0.616636 q^{86} +0.341231 q^{88} -3.29706 q^{89} +3.65683 q^{91} +11.3362 q^{92} +2.44801 q^{94} +10.2960 q^{97} +1.36839 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{2} + 3 q^{4} + q^{7} + 9 q^{8}+O(q^{10}) $ 4 * q + 3 * q^2 + 3 * q^4 + q^7 + 9 * q^8	$ 4 q + 3 q^{2} + 3 q^{4} + q^{7} + 9 q^{8} - 6 q^{11} + 4 q^{13} + q^{14} + 13 q^{16} + 2 q^{17} + 4 q^{19} + 9 q^{22} + 9 q^{23} + 3 q^{26} - 10 q^{28} - 16 q^{29} - 5 q^{31} + 26 q^{32} + 19 q^{34} + 13 q^{37} + 12 q^{38} - 12 q^{41} + q^{44} + 2 q^{46} + 9 q^{47} - 11 q^{49} + 3 q^{52} + 13 q^{53} - 14 q^{56} + 9 q^{58} - 3 q^{59} + 3 q^{61} - 25 q^{62} + 45 q^{64} - 6 q^{67} + 32 q^{68} - 11 q^{71} + 3 q^{73} - 4 q^{74} + 5 q^{76} + 10 q^{77} - 3 q^{79} - 22 q^{82} + 19 q^{83} + 9 q^{86} - 26 q^{88} - 16 q^{89} + q^{91} + 19 q^{92} + 25 q^{94} + 35 q^{97} - 21 q^{98}+O(q^{100}) $ 4 * q + 3 * q^2 + 3 * q^4 + q^7 + 9 * q^8 - 6 * q^11 + 4 * q^13 + q^14 + 13 * q^16 + 2 * q^17 + 4 * q^19 + 9 * q^22 + 9 * q^23 + 3 * q^26 - 10 * q^28 - 16 * q^29 - 5 * q^31 + 26 * q^32 + 19 * q^34 + 13 * q^37 + 12 * q^38 - 12 * q^41 + q^44 + 2 * q^46 + 9 * q^47 - 11 * q^49 + 3 * q^52 + 13 * q^53 - 14 * q^56 + 9 * q^58 - 3 * q^59 + 3 * q^61 - 25 * q^62 + 45 * q^64 - 6 * q^67 + 32 * q^68 - 11 * q^71 + 3 * q^73 - 4 * q^74 + 5 * q^76 + 10 * q^77 - 3 * q^79 - 22 * q^82 + 19 * q^83 + 9 * q^86 - 26 * q^88 - 16 * q^89 + q^91 + 19 * q^92 + 25 * q^94 + 35 * q^97 - 21 * q^98

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 $

$2$	0.214739	0.151843	0.0759215	−	0.997114i	$-0.475810\pi$
$2$	0.214739	0.151843	0.0759215	+	0.997114i	$0.475810\pi$
$3$	0	0
$3$	0	0
$4$	−1.95389	−0.976944
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.65683	1.38215	0.691075	−	0.722783i	$-0.257137\pi$
$7$	3.65683	1.38215	0.691075	+	0.722783i	$0.257137\pi$
$8$	−0.849052	−0.300185
$9$	0	0
$10$	0	0
$11$	−0.401897	−0.121177	−0.0605883	−	0.998163i	$-0.519298\pi$
$11$	−0.401897	−0.121177	−0.0605883	+	0.998163i	$0.519298\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.785261	0.209870
$15$	0	0
$16$	3.72545	0.931363
$17$	−6.98901	−1.69508	−0.847542	−	0.530728i	$-0.821918\pi$
$17$	−6.98901	−1.69508	−0.847542	+	0.530728i	$0.821918\pi$
$18$	0	0
$19$	3.87156	0.888198	0.444099	−	0.895978i	$-0.353524\pi$
$19$	3.87156	0.888198	0.444099	+	0.895978i	$0.353524\pi$
$20$	0	0
$21$	0	0
$22$	−0.0863028	−0.0183998
$23$	−5.80185	−1.20977	−0.604885	−	0.796313i	$-0.706781\pi$
$23$	−5.80185	−1.20977	−0.604885	+	0.796313i	$0.706781\pi$
$24$	0	0
$25$	0	0
$26$	0.214739	0.0421137
$27$	0	0
$28$	−7.14503	−1.35028
$29$	−7.63431	−1.41766	−0.708828	−	0.705381i	$-0.750776\pi$
$29$	−7.63431	−1.41766	−0.708828	+	0.705381i	$0.750776\pi$
$30$	0	0
$31$	−1.02251	−0.183649	−0.0918243	−	0.995775i	$-0.529270\pi$
$31$	−1.02251	−0.183649	−0.0918243	+	0.995775i	$0.529270\pi$
$32$	2.49810	0.441606
$33$	0	0
$34$	−1.50081	−0.257387
$35$	0	0
$36$	0	0
$37$	11.5820	1.90408	0.952038	−	0.305979i	$-0.0989838\pi$
$37$	11.5820	1.90408	0.952038	+	0.305979i	$0.0989838\pi$
$38$	0.831374	0.134867
$39$	0	0
$40$	0	0
$41$	−10.1214	−1.58070	−0.790351	−	0.612655i	$-0.790102\pi$
$41$	−10.1214	−1.58070	−0.790351	+	0.612655i	$0.790102\pi$
$42$	0	0
$43$	2.87156	0.437909	0.218955	−	0.975735i	$-0.429735\pi$
$43$	2.87156	0.437909	0.218955	+	0.975735i	$0.429735\pi$
$44$	0.785261	0.118383
$45$	0	0
$46$	−1.24588	−0.183695
$47$	11.4000	1.66285	0.831427	−	0.555634i	$-0.187524\pi$
$47$	11.4000	1.66285	0.831427	+	0.555634i	$0.187524\pi$
$48$	0	0
$49$	6.37237	0.910339
$50$	0	0
$51$	0	0
$52$	−1.95389	−0.270955
$53$	10.9067	1.49815	0.749074	−	0.662486i	$-0.230499\pi$
$53$	10.9067	1.49815	0.749074	+	0.662486i	$0.230499\pi$
$54$	0	0
$55$	0	0
$56$	−3.10484	−0.414901
$57$	0	0
$58$	−1.63938	−0.215261
$59$	−1.68043	−0.218773	−0.109386	−	0.993999i	$-0.534889\pi$
$59$	−1.68043	−0.218773	−0.109386	+	0.993999i	$0.534889\pi$
$60$	0	0
$61$	4.49218	0.575165	0.287582	−	0.957756i	$-0.407149\pi$
$61$	4.49218	0.575165	0.287582	+	0.957756i	$0.407149\pi$
$62$	−0.219573	−0.0278858
$63$	0	0
$64$	−6.91446	−0.864308
$65$	0	0
$66$	0	0
$67$	7.59725	0.928152	0.464076	−	0.885795i	$-0.346387\pi$
$67$	7.59725	0.928152	0.464076	+	0.885795i	$0.346387\pi$
$68$	13.6557	1.65600
$69$	0	0
$70$	0	0
$71$	7.86564	0.933480	0.466740	−	0.884395i	$-0.345428\pi$
$71$	7.86564	0.933480	0.466740	+	0.884395i	$0.345428\pi$
$72$	0	0
$73$	−10.6560	−1.24719	−0.623594	−	0.781749i	$-0.714328\pi$
$73$	−10.6560	−1.24719	−0.623594	+	0.781749i	$0.714328\pi$
$74$	2.48711	0.289121
$75$	0	0
$76$	−7.56460	−0.867719
$77$	−1.46967	−0.167484
$78$	0	0
$79$	12.6820	1.42684	0.713421	−	0.700736i	$-0.247145\pi$
$79$	12.6820	1.42684	0.713421	+	0.700736i	$0.247145\pi$
$80$	0	0
$81$	0	0
$82$	−2.17346	−0.240019
$83$	9.31474	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$83$	9.31474	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$84$	0	0
$85$	0	0
$86$	0.616636	0.0664935
$87$	0	0
$88$	0.341231	0.0363754
$89$	−3.29706	−0.349488	−0.174744	−	0.984614i	$-0.555910\pi$
$89$	−3.29706	−0.349488	−0.174744	+	0.984614i	$0.555910\pi$
$90$	0	0
$91$	3.65683	0.383339
$92$	11.3362	1.18188
$93$	0	0
$94$	2.44801	0.252493
$95$	0	0
$96$	0	0
$97$	10.2960	1.04540	0.522699	−	0.852517i	$-0.324925\pi$
$97$	10.2960	1.04540	0.522699	+	0.852517i	$0.324925\pi$
$98$	1.36839	0.138229
$99$	0	0
$100$	0	0
$101$	−14.1663	−1.40960	−0.704798	−	0.709408i	$-0.748962\pi$
$101$	−14.1663	−1.40960	−0.704798	+	0.709408i	$0.748962\pi$
$102$	0	0
$103$	−13.3984	−1.32019	−0.660094	−	0.751183i	$-0.729484\pi$
$103$	−13.3984	−1.32019	−0.660094	+	0.751183i	$0.729484\pi$
$104$	−0.849052	−0.0832564
$105$	0	0
$106$	2.34209	0.227484
$107$	12.2490	1.18416	0.592078	−	0.805881i	$-0.298308\pi$
$107$	12.2490	1.18416	0.592078	+	0.805881i	$0.298308\pi$
$108$	0	0
$109$	−4.64584	−0.444990	−0.222495	−	0.974934i	$-0.571420\pi$
$109$	−4.64584	−0.444990	−0.222495	+	0.974934i	$0.571420\pi$
$110$	0	0
$111$	0	0
$112$	13.6233	1.28728
$113$	−11.7381	−1.10422	−0.552112	−	0.833770i	$-0.686178\pi$
$113$	−11.7381	−1.10422	−0.552112	+	0.833770i	$0.686178\pi$
$114$	0	0
$115$	0	0
$116$	14.9166	1.38497
$117$	0	0
$118$	−0.360852	−0.0332191
$119$	−25.5576	−2.34286
$120$	0	0
$121$	−10.8385	−0.985316
$122$	0.964644	0.0873348
$123$	0	0
$124$	1.99787	0.179414
$125$	0	0
$126$	0	0
$127$	−4.76057	−0.422433	−0.211216	−	0.977439i	$-0.567742\pi$
$127$	−4.76057	−0.422433	−0.211216	+	0.977439i	$0.567742\pi$
$128$	−6.48101	−0.572845
$129$	0	0
$130$	0	0
$131$	19.4576	1.70002	0.850009	−	0.526769i	$-0.176597\pi$
$131$	19.4576	1.70002	0.850009	+	0.526769i	$0.176597\pi$
$132$	0	0
$133$	14.1576	1.22762
$134$	1.63142	0.140933
$135$	0	0
$136$	5.93403	0.508839
$137$	−4.70854	−0.402278	−0.201139	−	0.979563i	$-0.564464\pi$
$137$	−4.70854	−0.402278	−0.201139	+	0.979563i	$0.564464\pi$
$138$	0	0
$139$	−17.1917	−1.45818	−0.729089	−	0.684419i	$-0.760056\pi$
$139$	−17.1917	−1.45818	−0.729089	+	0.684419i	$0.760056\pi$
$140$	0	0
$141$	0	0
$142$	1.68906	0.141742
$143$	−0.401897	−0.0336083
$144$	0	0
$145$	0	0
$146$	−2.28825	−0.189377
$147$	0	0
$148$	−22.6300	−1.86018
$149$	−9.94526	−0.814747	−0.407374	−	0.913262i	$-0.633555\pi$
$149$	−9.94526	−0.814747	−0.407374	+	0.913262i	$0.633555\pi$
$150$	0	0
$151$	1.47668	0.120170	0.0600852	−	0.998193i	$-0.480863\pi$
$151$	1.47668	0.120170	0.0600852	+	0.998193i	$0.480863\pi$
$152$	−3.28716	−0.266624
$153$	0	0
$154$	−0.315594	−0.0254313
$155$	0	0
$156$	0	0
$157$	16.6804	1.33124	0.665621	−	0.746290i	$-0.268167\pi$
$157$	16.6804	1.33124	0.665621	+	0.746290i	$0.268167\pi$
$158$	2.72332	0.216656
$159$	0	0
$160$	0	0
$161$	−21.2164	−1.67208
$162$	0	0
$163$	−1.94991	−0.152729	−0.0763643	−	0.997080i	$-0.524331\pi$
$163$	−1.94991	−0.152729	−0.0763643	+	0.997080i	$0.524331\pi$
$164$	19.7761	1.54426
$165$	0	0
$166$	2.00023	0.155248
$167$	6.61555	0.511926	0.255963	−	0.966687i	$-0.417607\pi$
$167$	6.61555	0.511926	0.255963	+	0.966687i	$0.417607\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−5.61071	−0.427813
$173$	16.2128	1.23264	0.616318	−	0.787497i	$-0.288624\pi$
$173$	16.2128	1.23264	0.616318	+	0.787497i	$0.288624\pi$
$174$	0	0
$175$	0	0
$176$	−1.49725	−0.112859
$177$	0	0
$178$	−0.708006	−0.0530673
$179$	11.2040	0.837425	0.418712	−	0.908119i	$-0.362482\pi$
$179$	11.2040	0.837425	0.418712	+	0.908119i	$0.362482\pi$
$180$	0	0
$181$	−13.5595	−1.00787	−0.503936	−	0.863741i	$-0.668115\pi$
$181$	−13.5595	−1.00787	−0.503936	+	0.863741i	$0.668115\pi$
$182$	0.785261	0.0582075
$183$	0	0
$184$	4.92607	0.363155
$185$	0	0
$186$	0	0
$187$	2.80886	0.205404
$188$	−22.2742	−1.62451
$189$	0	0
$190$	0	0
$191$	−0.118300	−0.00855988	−0.00427994	−	0.999991i	$-0.501362\pi$
$191$	−0.118300	−0.00855988	−0.00427994	+	0.999991i	$0.501362\pi$
$192$	0	0
$193$	−21.1045	−1.51914	−0.759568	−	0.650428i	$-0.774590\pi$
$193$	−21.1045	−1.51914	−0.759568	+	0.650428i	$0.774590\pi$
$194$	2.21094	0.158736
$195$	0	0
$196$	−12.4509	−0.889350
$197$	18.6509	1.32882	0.664411	−	0.747367i	$-0.268683\pi$
$197$	18.6509	1.32882	0.664411	+	0.747367i	$0.268683\pi$
$198$	0	0
$199$	15.9102	1.12785	0.563924	−	0.825827i	$-0.309291\pi$
$199$	15.9102	1.12785	0.563924	+	0.825827i	$0.309291\pi$
$200$	0	0
$201$	0	0
$202$	−3.04204	−0.214037
$203$	−27.9174	−1.95941
$204$	0	0
$205$	0	0
$206$	−2.87716	−0.200461
$207$	0	0
$208$	3.72545	0.258314
$209$	−1.55597	−0.107629
$210$	0	0
$211$	−4.86564	−0.334965	−0.167482	−	0.985875i	$-0.553564\pi$
$211$	−4.86564	−0.334965	−0.167482	+	0.985875i	$0.553564\pi$
$212$	−21.3104	−1.46361
$213$	0	0
$214$	2.63033	0.179806
$215$	0	0
$216$	0	0
$217$	−3.73915	−0.253830
$218$	−0.997640	−0.0675687
$219$	0	0
$220$	0	0
$221$	−6.98901	−0.470132
$222$	0	0
$223$	13.9067	0.931261	0.465630	−	0.884979i	$-0.345828\pi$
$223$	13.9067	0.931261	0.465630	+	0.884979i	$0.345828\pi$
$224$	9.13512	0.610366
$225$	0	0
$226$	−2.52061	−0.167669
$227$	−10.9005	−0.723494	−0.361747	−	0.932276i	$-0.617820\pi$
$227$	−10.9005	−0.723494	−0.361747	+	0.932276i	$0.617820\pi$
$228$	0	0
$229$	19.8311	1.31047	0.655236	−	0.755424i	$-0.272569\pi$
$229$	19.8311	1.31047	0.655236	+	0.755424i	$0.272569\pi$
$230$	0	0
$231$	0	0
$232$	6.48193	0.425560
$233$	2.59616	0.170080	0.0850401	−	0.996378i	$-0.472898\pi$
$233$	2.59616	0.170080	0.0850401	+	0.996378i	$0.472898\pi$
$234$	0	0
$235$	0	0
$236$	3.28336	0.213729
$237$	0	0
$238$	−5.48820	−0.355747
$239$	5.46062	0.353218	0.176609	−	0.984281i	$-0.443487\pi$
$239$	5.46062	0.353218	0.176609	+	0.984281i	$0.443487\pi$
$240$	0	0
$241$	−4.56104	−0.293802	−0.146901	−	0.989151i	$-0.546930\pi$
$241$	−4.56104	−0.293802	−0.146901	+	0.989151i	$0.546930\pi$
$242$	−2.32744	−0.149613
$243$	0	0
$244$	−8.77721	−0.561903
$245$	0	0
$246$	0	0
$247$	3.87156	0.246342
$248$	0.868166	0.0551286
$249$	0	0
$250$	0	0
$251$	28.1146	1.77458	0.887290	−	0.461211i	$-0.152585\pi$
$251$	28.1146	1.77458	0.887290	+	0.461211i	$0.152585\pi$
$252$	0	0
$253$	2.33175	0.146596
$254$	−1.02228	−0.0641435
$255$	0	0
$256$	12.4372	0.777325
$257$	12.3305	0.769154	0.384577	−	0.923093i	$-0.374347\pi$
$257$	12.3305	0.769154	0.384577	+	0.923093i	$0.374347\pi$
$258$	0	0
$259$	42.3535	2.63172
$260$	0	0
$261$	0	0
$262$	4.17829	0.258136
$263$	26.6174	1.64130	0.820650	−	0.571432i	$-0.193612\pi$
$263$	26.6174	1.64130	0.820650	+	0.571432i	$0.193612\pi$
$264$	0	0
$265$	0	0
$266$	3.04019	0.186406
$267$	0	0
$268$	−14.8442	−0.906752
$269$	5.31129	0.323835	0.161918	−	0.986804i	$-0.448232\pi$
$269$	5.31129	0.323835	0.161918	+	0.986804i	$0.448232\pi$
$270$	0	0
$271$	−5.00507	−0.304036	−0.152018	−	0.988378i	$-0.548577\pi$
$271$	−5.00507	−0.304036	−0.152018	+	0.988378i	$0.548577\pi$
$272$	−26.0372	−1.57874
$273$	0	0
$274$	−1.01110	−0.0610831
$275$	0	0
$276$	0	0
$277$	−1.16138	−0.0697806	−0.0348903	−	0.999391i	$-0.511108\pi$
$277$	−1.16138	−0.0697806	−0.0348903	+	0.999391i	$0.511108\pi$
$278$	−3.69171	−0.221414
$279$	0	0
$280$	0	0
$281$	−1.58290	−0.0944279	−0.0472139	−	0.998885i	$-0.515034\pi$
$281$	−1.58290	−0.0944279	−0.0472139	+	0.998885i	$0.515034\pi$
$282$	0	0
$283$	24.2142	1.43938	0.719692	−	0.694294i	$-0.244283\pi$
$283$	24.2142	1.43938	0.719692	+	0.694294i	$0.244283\pi$
$284$	−15.3686	−0.911957
$285$	0	0
$286$	−0.0863028	−0.00510319
$287$	−37.0123	−2.18477
$288$	0	0
$289$	31.8463	1.87331
$290$	0	0
$291$	0	0
$292$	20.8206	1.21843
$293$	31.2619	1.82634	0.913170	−	0.407578i	$-0.133627\pi$
$293$	31.2619	1.82634	0.913170	+	0.407578i	$0.133627\pi$
$294$	0	0
$295$	0	0
$296$	−9.83376	−0.571576
$297$	0	0
$298$	−2.13563	−0.123714
$299$	−5.80185	−0.335530
$300$	0	0
$301$	10.5008	0.605257
$302$	0.317100	0.0182470
$303$	0	0
$304$	14.4233	0.827234
$305$	0	0
$306$	0	0
$307$	−11.0267	−0.629328	−0.314664	−	0.949203i	$-0.601892\pi$
$307$	−11.0267	−0.629328	−0.314664	+	0.949203i	$0.601892\pi$
$308$	2.87156	0.163623
$309$	0	0
$310$	0	0
$311$	−3.02564	−0.171568	−0.0857841	−	0.996314i	$-0.527340\pi$
$311$	−3.02564	−0.171568	−0.0857841	+	0.996314i	$0.527340\pi$
$312$	0	0
$313$	21.6275	1.22246	0.611230	−	0.791453i	$-0.290675\pi$
$313$	21.6275	1.22246	0.611230	+	0.791453i	$0.290675\pi$
$314$	3.58193	0.202140
$315$	0	0
$316$	−24.7793	−1.39394
$317$	26.4410	1.48507	0.742537	−	0.669805i	$-0.233622\pi$
$317$	26.4410	1.48507	0.742537	+	0.669805i	$0.233622\pi$
$318$	0	0
$319$	3.06821	0.171787
$320$	0	0
$321$	0	0
$322$	−4.55597	−0.253894
$323$	−27.0584	−1.50557
$324$	0	0
$325$	0	0
$326$	−0.418720	−0.0231908
$327$	0	0
$328$	8.59362	0.474503
$329$	41.6876	2.29831
$330$	0	0
$331$	15.8519	0.871302	0.435651	−	0.900116i	$-0.356518\pi$
$331$	15.8519	0.871302	0.435651	+	0.900116i	$0.356518\pi$
$332$	−18.2000	−0.998852
$333$	0	0
$334$	1.42061	0.0777325
$335$	0	0
$336$	0	0
$337$	13.8519	0.754563	0.377282	−	0.926099i	$-0.376859\pi$
$337$	13.8519	0.754563	0.377282	+	0.926099i	$0.376859\pi$
$338$	0.214739	0.0116802
$339$	0	0
$340$	0	0
$341$	0.410945	0.0222539
$342$	0	0
$343$	−2.29512	−0.123925
$344$	−2.43811	−0.131454
$345$	0	0
$346$	3.48151	0.187167
$347$	9.26934	0.497604	0.248802	−	0.968554i	$-0.419963\pi$
$347$	9.26934	0.497604	0.248802	+	0.968554i	$0.419963\pi$
$348$	0	0
$349$	1.82071	0.0974602	0.0487301	−	0.998812i	$-0.484483\pi$
$349$	1.82071	0.0974602	0.0487301	+	0.998812i	$0.484483\pi$
$350$	0	0
$351$	0	0
$352$	−1.00398	−0.0535123
$353$	−19.7437	−1.05085	−0.525424	−	0.850840i	$-0.676093\pi$
$353$	−19.7437	−1.05085	−0.525424	+	0.850840i	$0.676093\pi$
$354$	0	0
$355$	0	0
$356$	6.44209	0.341430
$357$	0	0
$358$	2.40593	0.127157
$359$	10.4010	0.548946	0.274473	−	0.961595i	$-0.411497\pi$
$359$	10.4010	0.548946	0.274473	+	0.961595i	$0.411497\pi$
$360$	0	0
$361$	−4.01099	−0.211105
$362$	−2.91175	−0.153038
$363$	0	0
$364$	−7.14503	−0.374501
$365$	0	0
$366$	0	0
$367$	16.4338	0.857838	0.428919	−	0.903343i	$-0.358895\pi$
$367$	16.4338	0.857838	0.428919	+	0.903343i	$0.358895\pi$
$368$	−21.6145	−1.12673
$369$	0	0
$370$	0	0
$371$	39.8839	2.07067
$372$	0	0
$373$	−8.96800	−0.464346	−0.232173	−	0.972675i	$-0.574583\pi$
$373$	−8.96800	−0.464346	−0.232173	+	0.972675i	$0.574583\pi$
$374$	0.603171	0.0311892
$375$	0	0
$376$	−9.67915	−0.499164
$377$	−7.63431	−0.393187
$378$	0	0
$379$	−4.22517	−0.217033	−0.108516	−	0.994095i	$-0.534610\pi$
$379$	−4.22517	−0.217033	−0.108516	+	0.994095i	$0.534610\pi$
$380$	0	0
$381$	0	0
$382$	−0.0254036	−0.00129976
$383$	31.2421	1.59640	0.798199	−	0.602394i	$-0.205787\pi$
$383$	31.2421	1.59640	0.798199	+	0.602394i	$0.205787\pi$
$384$	0	0
$385$	0	0
$386$	−4.53195	−0.230670
$387$	0	0
$388$	−20.1172	−1.02129
$389$	8.18177	0.414832	0.207416	−	0.978253i	$-0.433495\pi$
$389$	8.18177	0.414832	0.207416	+	0.978253i	$0.433495\pi$
$390$	0	0
$391$	40.5492	2.05066
$392$	−5.41048	−0.273270
$393$	0	0
$394$	4.00507	0.201772
$395$	0	0
$396$	0	0
$397$	11.3404	0.569157	0.284579	−	0.958653i	$-0.408146\pi$
$397$	11.3404	0.569157	0.284579	+	0.958653i	$0.408146\pi$
$398$	3.41654	0.171256
$399$	0	0
$400$	0	0
$401$	−15.7434	−0.786186	−0.393093	−	0.919499i	$-0.628595\pi$
$401$	−15.7434	−0.786186	−0.393093	+	0.919499i	$0.628595\pi$
$402$	0	0
$403$	−1.02251	−0.0509350
$404$	27.6793	1.37710
$405$	0	0
$406$	−5.99493	−0.297523
$407$	−4.65479	−0.230729
$408$	0	0
$409$	15.2025	0.751714	0.375857	−	0.926678i	$-0.377348\pi$
$409$	15.2025	0.751714	0.375857	+	0.926678i	$0.377348\pi$
$410$	0	0
$411$	0	0
$412$	26.1791	1.28975
$413$	−6.14503	−0.302377
$414$	0	0
$415$	0	0
$416$	2.49810	0.122480
$417$	0	0
$418$	−0.334127	−0.0163427
$419$	−20.4322	−0.998178	−0.499089	−	0.866551i	$-0.666332\pi$
$419$	−20.4322	−0.998178	−0.499089	+	0.866551i	$0.666332\pi$
$420$	0	0
$421$	9.88744	0.481884	0.240942	−	0.970539i	$-0.422544\pi$
$421$	9.88744	0.481884	0.240942	+	0.970539i	$0.422544\pi$
$422$	−1.04484	−0.0508621
$423$	0	0
$424$	−9.26034	−0.449722
$425$	0	0
$426$	0	0
$427$	16.4271	0.794964
$428$	−23.9332	−1.15685
$429$	0	0
$430$	0	0
$431$	−11.1804	−0.538540	−0.269270	−	0.963065i	$-0.586782\pi$
$431$	−11.1804	−0.538540	−0.269270	+	0.963065i	$0.586782\pi$
$432$	0	0
$433$	−36.9033	−1.77346	−0.886730	−	0.462287i	$-0.847029\pi$
$433$	−36.9033	−1.77346	−0.886730	+	0.462287i	$0.847029\pi$
$434$	−0.802939	−0.0385423
$435$	0	0
$436$	9.07744	0.434730
$437$	−22.4622	−1.07451
$438$	0	0
$439$	−4.11238	−0.196273	−0.0981365	−	0.995173i	$-0.531288\pi$
$439$	−4.11238	−0.196273	−0.0981365	+	0.995173i	$0.531288\pi$
$440$	0	0
$441$	0	0
$442$	−1.50081	−0.0713862
$443$	−6.43605	−0.305786	−0.152893	−	0.988243i	$-0.548859\pi$
$443$	−6.43605	−0.305786	−0.152893	+	0.988243i	$0.548859\pi$
$444$	0	0
$445$	0	0
$446$	2.98630	0.141405
$447$	0	0
$448$	−25.2850	−1.19460
$449$	−28.2890	−1.33504	−0.667519	−	0.744592i	$-0.732644\pi$
$449$	−28.2890	−1.33504	−0.667519	+	0.744592i	$0.732644\pi$
$450$	0	0
$451$	4.06777	0.191544
$452$	22.9348	1.07876
$453$	0	0
$454$	−2.34076	−0.109858
$455$	0	0
$456$	0	0
$457$	−35.1411	−1.64383	−0.821917	−	0.569608i	$-0.807095\pi$
$457$	−35.1411	−1.64383	−0.821917	+	0.569608i	$0.807095\pi$
$458$	4.25849	0.198986
$459$	0	0
$460$	0	0
$461$	−21.1261	−0.983939	−0.491970	−	0.870612i	$-0.663723\pi$
$461$	−21.1261	−0.983939	−0.491970	+	0.870612i	$0.663723\pi$
$462$	0	0
$463$	−6.65326	−0.309203	−0.154602	−	0.987977i	$-0.549409\pi$
$463$	−6.65326	−0.309203	−0.154602	+	0.987977i	$0.549409\pi$
$464$	−28.4413	−1.32035
$465$	0	0
$466$	0.557496	0.0258255
$467$	38.0920	1.76269	0.881344	−	0.472475i	$-0.156639\pi$
$467$	38.0920	1.76269	0.881344	+	0.472475i	$0.156639\pi$
$468$	0	0
$469$	27.7818	1.28284
$470$	0	0
$471$	0	0
$472$	1.42677	0.0656724
$473$	−1.15407	−0.0530643
$474$	0	0
$475$	0	0
$476$	49.9367	2.28884
$477$	0	0
$478$	1.17261	0.0536337
$479$	19.5852	0.894869	0.447435	−	0.894317i	$-0.352338\pi$
$479$	19.5852	0.894869	0.447435	+	0.894317i	$0.352338\pi$
$480$	0	0
$481$	11.5820	0.528096
$482$	−0.979431	−0.0446118
$483$	0	0
$484$	21.1772	0.962598
$485$	0	0
$486$	0	0
$487$	8.64734	0.391848	0.195924	−	0.980619i	$-0.437229\pi$
$487$	8.64734	0.391848	0.195924	+	0.980619i	$0.437229\pi$
$488$	−3.81409	−0.172656
$489$	0	0
$490$	0	0
$491$	17.3471	0.782862	0.391431	−	0.920207i	$-0.371980\pi$
$491$	17.3471	0.782862	0.391431	+	0.920207i	$0.371980\pi$
$492$	0	0
$493$	53.3563	2.40305
$494$	0.831374	0.0374053
$495$	0	0
$496$	−3.80932	−0.171043
$497$	28.7633	1.29021
$498$	0	0
$499$	−32.1800	−1.44057	−0.720287	−	0.693677i	$-0.755990\pi$
$499$	−32.1800	−1.44057	−0.720287	+	0.693677i	$0.755990\pi$
$500$	0	0
$501$	0	0
$502$	6.03730	0.269458
$503$	−5.61911	−0.250544	−0.125272	−	0.992122i	$-0.539980\pi$
$503$	−5.61911	−0.250544	−0.125272	+	0.992122i	$0.539980\pi$
$504$	0	0
$505$	0	0
$506$	0.500716	0.0222595
$507$	0	0
$508$	9.30162	0.412693
$509$	31.8432	1.41143	0.705713	−	0.708497i	$-0.250627\pi$
$509$	31.8432	1.41143	0.705713	+	0.708497i	$0.250627\pi$
$510$	0	0
$511$	−38.9670	−1.72380
$512$	15.6328	0.690877
$513$	0	0
$514$	2.64783	0.116791
$515$	0	0
$516$	0	0
$517$	−4.58161	−0.201499
$518$	9.09493	0.399608
$519$	0	0
$520$	0	0
$521$	−7.63255	−0.334388	−0.167194	−	0.985924i	$-0.553471\pi$
$521$	−7.63255	−0.334388	−0.167194	+	0.985924i	$0.553471\pi$
$522$	0	0
$523$	28.6541	1.25296	0.626479	−	0.779438i	$-0.284495\pi$
$523$	28.6541	1.25296	0.626479	+	0.779438i	$0.284495\pi$
$524$	−38.0179	−1.66082
$525$	0	0
$526$	5.71578	0.249220
$527$	7.14635	0.311300
$528$	0	0
$529$	10.6615	0.463542
$530$	0	0
$531$	0	0
$532$	−27.6624	−1.19932
$533$	−10.1214	−0.438408
$534$	0	0
$535$	0	0
$536$	−6.45046	−0.278617
$537$	0	0
$538$	1.14054	0.0491721
$539$	−2.56104	−0.110312
$540$	0	0
$541$	−15.7258	−0.676105	−0.338052	−	0.941127i	$-0.609768\pi$
$541$	−15.7258	−0.676105	−0.338052	+	0.941127i	$0.609768\pi$
$542$	−1.07478	−0.0461658
$543$	0	0
$544$	−17.4593	−0.748560
$545$	0	0
$546$	0	0
$547$	10.2745	0.439308	0.219654	−	0.975578i	$-0.429507\pi$
$547$	10.2745	0.439308	0.219654	+	0.975578i	$0.429507\pi$
$548$	9.19995	0.393002
$549$	0	0
$550$	0	0
$551$	−29.5567	−1.25916
$552$	0	0
$553$	46.3760	1.97211
$554$	−0.249393	−0.0105957
$555$	0	0
$556$	33.5906	1.42456
$557$	−24.2175	−1.02613	−0.513065	−	0.858350i	$-0.671490\pi$
$557$	−24.2175	−1.02613	−0.513065	+	0.858350i	$0.671490\pi$
$558$	0	0
$559$	2.87156	0.121454
$560$	0	0
$561$	0	0
$562$	−0.339909	−0.0143382
$563$	−32.0806	−1.35204	−0.676018	−	0.736885i	$-0.736296\pi$
$563$	−32.0806	−1.35204	−0.676018	+	0.736885i	$0.736296\pi$
$564$	0	0
$565$	0	0
$566$	5.19972	0.218560
$567$	0	0
$568$	−6.67834	−0.280217
$569$	−19.7319	−0.827204	−0.413602	−	0.910458i	$-0.635730\pi$
$569$	−19.7319	−0.827204	−0.413602	+	0.910458i	$0.635730\pi$
$570$	0	0
$571$	43.3171	1.81277	0.906383	−	0.422458i	$-0.138833\pi$
$571$	43.3171	1.81277	0.906383	+	0.422458i	$0.138833\pi$
$572$	0.785261	0.0328334
$573$	0	0
$574$	−7.94796	−0.331742
$575$	0	0
$576$	0	0
$577$	−6.57233	−0.273610	−0.136805	−	0.990598i	$-0.543683\pi$
$577$	−6.57233	−0.273610	−0.136805	+	0.990598i	$0.543683\pi$
$578$	6.83862	0.284449
$579$	0	0
$580$	0	0
$581$	34.0624	1.41315
$582$	0	0
$583$	−4.38336	−0.181540
$584$	9.04747	0.374387
$585$	0	0
$586$	6.71314	0.277317
$587$	−23.2200	−0.958394	−0.479197	−	0.877707i	$-0.659072\pi$
$587$	−23.2200	−0.958394	−0.479197	+	0.877707i	$0.659072\pi$
$588$	0	0
$589$	−3.95872	−0.163116
$590$	0	0
$591$	0	0
$592$	43.1483	1.77339
$593$	10.9561	0.449914	0.224957	−	0.974369i	$-0.427776\pi$
$593$	10.9561	0.449914	0.224957	+	0.974369i	$0.427776\pi$
$594$	0	0
$595$	0	0
$596$	19.4319	0.795962
$597$	0	0
$598$	−1.24588	−0.0509479
$599$	−5.27279	−0.215440	−0.107720	−	0.994181i	$-0.534355\pi$
$599$	−5.27279	−0.215440	−0.107720	+	0.994181i	$0.534355\pi$
$600$	0	0
$601$	27.6639	1.12844	0.564218	−	0.825626i	$-0.309178\pi$
$601$	27.6639	1.12844	0.564218	+	0.825626i	$0.309178\pi$
$602$	2.25493	0.0919040
$603$	0	0
$604$	−2.88526	−0.117400
$605$	0	0
$606$	0	0
$607$	6.02964	0.244735	0.122368	−	0.992485i	$-0.460951\pi$
$607$	6.02964	0.244735	0.122368	+	0.992485i	$0.460951\pi$
$608$	9.67156	0.392234
$609$	0	0
$610$	0	0
$611$	11.4000	0.461193
$612$	0	0
$613$	−5.86455	−0.236867	−0.118434	−	0.992962i	$-0.537787\pi$
$613$	−5.86455	−0.236867	−0.118434	+	0.992962i	$0.537787\pi$
$614$	−2.36786	−0.0955592
$615$	0	0
$616$	1.24782	0.0502763
$617$	5.79938	0.233474	0.116737	−	0.993163i	$-0.462757\pi$
$617$	5.79938	0.233474	0.116737	+	0.993163i	$0.462757\pi$
$618$	0	0
$619$	−9.82557	−0.394923	−0.197461	−	0.980311i	$-0.563270\pi$
$619$	−9.82557	−0.394923	−0.197461	+	0.980311i	$0.563270\pi$
$620$	0	0
$621$	0	0
$622$	−0.649721	−0.0260514
$623$	−12.0568	−0.483045
$624$	0	0
$625$	0	0
$626$	4.64427	0.185622
$627$	0	0
$628$	−32.5917	−1.30055
$629$	−80.9470	−3.22757
$630$	0	0
$631$	36.4526	1.45116	0.725578	−	0.688140i	$-0.241572\pi$
$631$	36.4526	1.45116	0.725578	+	0.688140i	$0.241572\pi$
$632$	−10.7677	−0.428317
$633$	0	0
$634$	5.67790	0.225498
$635$	0	0
$636$	0	0
$637$	6.37237	0.252483
$638$	0.658862	0.0260846
$639$	0	0
$640$	0	0
$641$	9.15752	0.361700	0.180850	−	0.983511i	$-0.442115\pi$
$641$	9.15752	0.361700	0.180850	+	0.983511i	$0.442115\pi$
$642$	0	0
$643$	−1.19836	−0.0472586	−0.0236293	−	0.999721i	$-0.507522\pi$
$643$	−1.19836	−0.0472586	−0.0236293	+	0.999721i	$0.507522\pi$
$644$	41.4544	1.63353
$645$	0	0
$646$	−5.81048	−0.228610
$647$	−4.05601	−0.159458	−0.0797292	−	0.996817i	$-0.525406\pi$
$647$	−4.05601	−0.159458	−0.0797292	+	0.996817i	$0.525406\pi$
$648$	0	0
$649$	0.675358	0.0265101
$650$	0	0
$651$	0	0
$652$	3.80990	0.149207
$653$	−6.78598	−0.265556	−0.132778	−	0.991146i	$-0.542390\pi$
$653$	−6.78598	−0.265556	−0.132778	+	0.991146i	$0.542390\pi$
$654$	0	0
$655$	0	0
$656$	−37.7069	−1.47221
$657$	0	0
$658$	8.95194	0.348983
$659$	29.8513	1.16284	0.581421	−	0.813603i	$-0.302497\pi$
$659$	29.8513	1.16284	0.581421	+	0.813603i	$0.302497\pi$
$660$	0	0
$661$	12.1299	0.471800	0.235900	−	0.971777i	$-0.424196\pi$
$661$	12.1299	0.471800	0.235900	+	0.971777i	$0.424196\pi$
$662$	3.40402	0.132301
$663$	0	0
$664$	−7.90870	−0.306917
$665$	0	0
$666$	0	0
$667$	44.2932	1.71504
$668$	−12.9260	−0.500123
$669$	0	0
$670$	0	0
$671$	−1.80539	−0.0696964
$672$	0	0
$673$	−37.8753	−1.45999	−0.729993	−	0.683454i	$-0.760477\pi$
$673$	−37.8753	−1.45999	−0.729993	+	0.683454i	$0.760477\pi$
$674$	2.97455	0.114575
$675$	0	0
$676$	−1.95389	−0.0751495
$677$	−26.0801	−1.00234	−0.501169	−	0.865349i	$-0.667097\pi$
$677$	−26.0801	−1.00234	−0.501169	+	0.865349i	$0.667097\pi$
$678$	0	0
$679$	37.6506	1.44490
$680$	0	0
$681$	0	0
$682$	0.0882456	0.00337910
$683$	44.5861	1.70604	0.853019	−	0.521879i	$-0.174769\pi$
$683$	44.5861	1.70604	0.853019	+	0.521879i	$0.174769\pi$
$684$	0	0
$685$	0	0
$686$	−0.492850	−0.0188171
$687$	0	0
$688$	10.6979	0.407852
$689$	10.9067	0.415512
$690$	0	0
$691$	37.3562	1.42110	0.710548	−	0.703649i	$-0.248447\pi$
$691$	37.3562	1.42110	0.710548	+	0.703649i	$0.248447\pi$
$692$	−31.6780	−1.20422
$693$	0	0
$694$	1.99049	0.0755578
$695$	0	0
$696$	0	0
$697$	70.7387	2.67942
$698$	0.390976	0.0147987
$699$	0	0
$700$	0	0
$701$	3.18371	0.120247	0.0601235	−	0.998191i	$-0.480851\pi$
$701$	3.18371	0.120247	0.0601235	+	0.998191i	$0.480851\pi$
$702$	0	0
$703$	44.8406	1.69120
$704$	2.77890	0.104734
$705$	0	0
$706$	−4.23972	−0.159564
$707$	−51.8036	−1.94827
$708$	0	0
$709$	52.7634	1.98157	0.990785	−	0.135444i	$-0.0432459\pi$
$709$	52.7634	1.98157	0.990785	+	0.135444i	$0.0432459\pi$
$710$	0	0
$711$	0	0
$712$	2.79938	0.104911
$713$	5.93246	0.222173
$714$	0	0
$715$	0	0
$716$	−21.8913	−0.818117
$717$	0	0
$718$	2.23350	0.0833537
$719$	14.0517	0.524040	0.262020	−	0.965062i	$-0.415611\pi$
$719$	14.0517	0.524040	0.262020	+	0.965062i	$0.415611\pi$
$720$	0	0
$721$	−48.9958	−1.82470
$722$	−0.861314	−0.0320548
$723$	0	0
$724$	26.4938	0.984634
$725$	0	0
$726$	0	0
$727$	−10.4482	−0.387504	−0.193752	−	0.981051i	$-0.562066\pi$
$727$	−10.4482	−0.387504	−0.193752	+	0.981051i	$0.562066\pi$
$728$	−3.10484	−0.115073
$729$	0	0
$730$	0	0
$731$	−20.0694	−0.742293
$732$	0	0
$733$	22.3609	0.825920	0.412960	−	0.910749i	$-0.364495\pi$
$733$	22.3609	0.825920	0.412960	+	0.910749i	$0.364495\pi$
$734$	3.52897	0.130257
$735$	0	0
$736$	−14.4936	−0.534242
$737$	−3.05331	−0.112470
$738$	0	0
$739$	−27.5953	−1.01511	−0.507555	−	0.861619i	$-0.669451\pi$
$739$	−27.5953	−1.01511	−0.507555	+	0.861619i	$0.669451\pi$
$740$	0	0
$741$	0	0
$742$	8.56460	0.314416
$743$	−29.9663	−1.09936	−0.549678	−	0.835376i	$-0.685250\pi$
$743$	−29.9663	−1.09936	−0.549678	+	0.835376i	$0.685250\pi$
$744$	0	0
$745$	0	0
$746$	−1.92578	−0.0705077
$747$	0	0
$748$	−5.48820	−0.200668
$749$	44.7925	1.63668
$750$	0	0
$751$	50.0220	1.82533	0.912664	−	0.408710i	$-0.134021\pi$
$751$	50.0220	1.82533	0.912664	+	0.408710i	$0.134021\pi$
$752$	42.4700	1.54872
$753$	0	0
$754$	−1.63938	−0.0597028
$755$	0	0
$756$	0	0
$757$	−30.5218	−1.10934	−0.554668	−	0.832072i	$-0.687155\pi$
$757$	−30.5218	−1.10934	−0.554668	+	0.832072i	$0.687155\pi$
$758$	−0.907307	−0.0329549
$759$	0	0
$760$	0	0
$761$	−12.0427	−0.436546	−0.218273	−	0.975888i	$-0.570042\pi$
$761$	−12.0427	−0.436546	−0.218273	+	0.975888i	$0.570042\pi$
$762$	0	0
$763$	−16.9890	−0.615043
$764$	0.231145	0.00836252
$765$	0	0
$766$	6.70889	0.242402
$767$	−1.68043	−0.0606767
$768$	0	0
$769$	−12.8081	−0.461873	−0.230936	−	0.972969i	$-0.574179\pi$
$769$	−12.8081	−0.461873	−0.230936	+	0.972969i	$0.574179\pi$
$770$	0	0
$771$	0	0
$772$	41.2358	1.48411
$773$	36.0955	1.29826	0.649132	−	0.760676i	$-0.275132\pi$
$773$	36.0955	1.29826	0.649132	+	0.760676i	$0.275132\pi$
$774$	0	0
$775$	0	0
$776$	−8.74182	−0.313813
$777$	0	0
$778$	1.75694	0.0629894
$779$	−39.1857	−1.40398
$780$	0	0
$781$	−3.16118	−0.113116
$782$	8.70747	0.311379
$783$	0	0
$784$	23.7400	0.847856
$785$	0	0
$786$	0	0
$787$	6.23265	0.222170	0.111085	−	0.993811i	$-0.464567\pi$
$787$	6.23265	0.222170	0.111085	+	0.993811i	$0.464567\pi$
$788$	−36.4418	−1.29818
$789$	0	0
$790$	0	0
$791$	−42.9240	−1.52620
$792$	0	0
$793$	4.49218	0.159522
$794$	2.43522	0.0864226
$795$	0	0
$796$	−31.0868	−1.10184
$797$	12.2040	0.432289	0.216145	−	0.976361i	$-0.430652\pi$
$797$	12.2040	0.432289	0.216145	+	0.976361i	$0.430652\pi$
$798$	0	0
$799$	−79.6744	−2.81868
$800$	0	0
$801$	0	0
$802$	−3.38071	−0.119377
$803$	4.28260	0.151130
$804$	0	0
$805$	0	0
$806$	−0.219573	−0.00773412
$807$	0	0
$808$	12.0279	0.423140
$809$	−11.8278	−0.415844	−0.207922	−	0.978145i	$-0.566670\pi$
$809$	−11.8278	−0.415844	−0.207922	+	0.978145i	$0.566670\pi$
$810$	0	0
$811$	24.9577	0.876384	0.438192	−	0.898881i	$-0.355619\pi$
$811$	24.9577	0.876384	0.438192	+	0.898881i	$0.355619\pi$
$812$	54.5474	1.91424
$813$	0	0
$814$	−0.999563	−0.0350347
$815$	0	0
$816$	0	0
$817$	11.1174	0.388950
$818$	3.26456	0.114143
$819$	0	0
$820$	0	0
$821$	0.809160	0.0282399	0.0141199	−	0.999900i	$-0.495505\pi$
$821$	0.809160	0.0282399	0.0141199	+	0.999900i	$0.495505\pi$
$822$	0	0
$823$	−12.6701	−0.441650	−0.220825	−	0.975313i	$-0.570875\pi$
$823$	−12.6701	−0.441650	−0.220825	+	0.975313i	$0.570875\pi$
$824$	11.3760	0.396301
$825$	0	0
$826$	−1.31957	−0.0459138
$827$	−22.7641	−0.791587	−0.395793	−	0.918340i	$-0.629530\pi$
$827$	−22.7641	−0.791587	−0.395793	+	0.918340i	$0.629530\pi$
$828$	0	0
$829$	8.82934	0.306656	0.153328	−	0.988175i	$-0.451001\pi$
$829$	8.82934	0.306656	0.153328	+	0.988175i	$0.451001\pi$
$830$	0	0
$831$	0	0
$832$	−6.91446	−0.239716
$833$	−44.5366	−1.54310
$834$	0	0
$835$	0	0
$836$	3.04019	0.105147
$837$	0	0
$838$	−4.38758	−0.151566
$839$	22.5191	0.777447	0.388724	−	0.921354i	$-0.372916\pi$
$839$	22.5191	0.777447	0.388724	+	0.921354i	$0.372916\pi$
$840$	0	0
$841$	29.2827	1.00975
$842$	2.12321	0.0731708
$843$	0	0
$844$	9.50692	0.327242
$845$	0	0
$846$	0	0
$847$	−39.6344	−1.36186
$848$	40.6323	1.39532
$849$	0	0
$850$	0	0
$851$	−67.1973	−2.30349
$852$	0	0
$853$	−45.7438	−1.56624	−0.783118	−	0.621873i	$-0.786372\pi$
$853$	−45.7438	−1.56624	−0.783118	+	0.621873i	$0.786372\pi$
$854$	3.52754	0.120710
$855$	0	0
$856$	−10.4000	−0.355466
$857$	−26.5404	−0.906602	−0.453301	−	0.891357i	$-0.649754\pi$
$857$	−26.5404	−0.906602	−0.453301	+	0.891357i	$0.649754\pi$
$858$	0	0
$859$	20.2798	0.691939	0.345969	−	0.938246i	$-0.387550\pi$
$859$	20.2798	0.691939	0.345969	+	0.938246i	$0.387550\pi$
$860$	0	0
$861$	0	0
$862$	−2.40086	−0.0817736
$863$	56.1533	1.91148	0.955741	−	0.294210i	$-0.0950565\pi$
$863$	56.1533	1.91148	0.955741	+	0.294210i	$0.0950565\pi$
$864$	0	0
$865$	0	0
$866$	−7.92457	−0.269288
$867$	0	0
$868$	7.30588	0.247978
$869$	−5.09688	−0.172900
$870$	0	0
$871$	7.59725	0.257423
$872$	3.94456	0.133580
$873$	0	0
$874$	−4.82351	−0.163158
$875$	0	0
$876$	0	0
$877$	−20.8647	−0.704550	−0.352275	−	0.935897i	$-0.614592\pi$
$877$	−20.8647	−0.704550	−0.352275	+	0.935897i	$0.614592\pi$
$878$	−0.883086	−0.0298027
$879$	0	0
$880$	0	0
$881$	21.0864	0.710419	0.355210	−	0.934787i	$-0.384409\pi$
$881$	21.0864	0.710419	0.355210	+	0.934787i	$0.384409\pi$
$882$	0	0
$883$	−53.2824	−1.79310	−0.896548	−	0.442947i	$-0.853933\pi$
$883$	−53.2824	−1.79310	−0.896548	+	0.442947i	$0.853933\pi$
$884$	13.6557	0.459292
$885$	0	0
$886$	−1.38207	−0.0464315
$887$	28.7341	0.964798	0.482399	−	0.875952i	$-0.339766\pi$
$887$	28.7341	0.964798	0.482399	+	0.875952i	$0.339766\pi$
$888$	0	0
$889$	−17.4086	−0.583865
$890$	0	0
$891$	0	0
$892$	−27.1721	−0.909789
$893$	44.1357	1.47694
$894$	0	0
$895$	0	0
$896$	−23.6999	−0.791758
$897$	0	0
$898$	−6.07473	−0.202716
$899$	7.80618	0.260351
$900$	0	0
$901$	−76.2269	−2.53949
$902$	0.873507	0.0290846
$903$	0	0
$904$	9.96622	0.331472
$905$	0	0
$906$	0	0
$907$	15.8650	0.526788	0.263394	−	0.964688i	$-0.415158\pi$
$907$	15.8650	0.526788	0.263394	+	0.964688i	$0.415158\pi$
$908$	21.2984	0.706812
$909$	0	0
$910$	0	0
$911$	39.6668	1.31422	0.657111	−	0.753794i	$-0.271778\pi$
$911$	39.6668	1.31422	0.657111	+	0.753794i	$0.271778\pi$
$912$	0	0
$913$	−3.74357	−0.123894
$914$	−7.54616	−0.249605
$915$	0	0
$916$	−38.7476	−1.28026
$917$	71.1530	2.34968
$918$	0	0
$919$	12.9526	0.427267	0.213633	−	0.976914i	$-0.431470\pi$
$919$	12.9526	0.427267	0.213633	+	0.976914i	$0.431470\pi$
$920$	0	0
$921$	0	0
$922$	−4.53658	−0.149404
$923$	7.86564	0.258901
$924$	0	0
$925$	0	0
$926$	−1.42871	−0.0469504
$927$	0	0
$928$	−19.0713	−0.626046
$929$	−45.1625	−1.48173	−0.740867	−	0.671652i	$-0.765585\pi$
$929$	−45.1625	−1.48173	−0.740867	+	0.671652i	$0.765585\pi$
$930$	0	0
$931$	24.6711	0.808561
$932$	−5.07260	−0.166159
$933$	0	0
$934$	8.17982	0.267652
$935$	0	0
$936$	0	0
$937$	−21.9191	−0.716065	−0.358032	−	0.933709i	$-0.616552\pi$
$937$	−21.9191	−0.716065	−0.358032	+	0.933709i	$0.616552\pi$
$938$	5.96583	0.194791
$939$	0	0
$940$	0	0
$941$	−17.7099	−0.577325	−0.288662	−	0.957431i	$-0.593211\pi$
$941$	−17.7099	−0.577325	−0.288662	+	0.957431i	$0.593211\pi$
$942$	0	0
$943$	58.7230	1.91228
$944$	−6.26034	−0.203757
$945$	0	0
$946$	−0.247824	−0.00805745
$947$	30.9677	1.00631	0.503157	−	0.864195i	$-0.332172\pi$
$947$	30.9677	1.00631	0.503157	+	0.864195i	$0.332172\pi$
$948$	0	0
$949$	−10.6560	−0.345907
$950$	0	0
$951$	0	0
$952$	21.6997	0.703292
$953$	−3.69613	−0.119729	−0.0598647	−	0.998206i	$-0.519067\pi$
$953$	−3.69613	−0.119729	−0.0598647	+	0.998206i	$0.519067\pi$
$954$	0	0
$955$	0	0
$956$	−10.6694	−0.345074
$957$	0	0
$958$	4.20569	0.135880
$959$	−17.2183	−0.556008
$960$	0	0
$961$	−29.9545	−0.966273
$962$	2.48711	0.0801877
$963$	0	0
$964$	8.91175	0.287028
$965$	0	0
$966$	0	0
$967$	10.5007	0.337680	0.168840	−	0.985644i	$-0.445998\pi$
$967$	10.5007	0.337680	0.168840	+	0.985644i	$0.445998\pi$
$968$	9.20243	0.295777
$969$	0	0
$970$	0	0
$971$	−35.6481	−1.14400	−0.572001	−	0.820253i	$-0.693833\pi$
$971$	−35.6481	−1.14400	−0.572001	+	0.820253i	$0.693833\pi$
$972$	0	0
$973$	−62.8669	−2.01542
$974$	1.85692	0.0594995
$975$	0	0
$976$	16.7354	0.535687
$977$	22.9839	0.735319	0.367659	−	0.929961i	$-0.380159\pi$
$977$	22.9839	0.735319	0.367659	+	0.929961i	$0.380159\pi$
$978$	0	0
$979$	1.32508	0.0423497
$980$	0	0
$981$	0	0
$982$	3.72508	0.118872
$983$	−4.63216	−0.147743	−0.0738715	−	0.997268i	$-0.523535\pi$
$983$	−4.63216	−0.147743	−0.0738715	+	0.997268i	$0.523535\pi$
$984$	0	0
$985$	0	0
$986$	11.4577	0.364886
$987$	0	0
$988$	−7.56460	−0.240662
$989$	−16.6604	−0.529770
$990$	0	0
$991$	−24.5457	−0.779718	−0.389859	−	0.920875i	$-0.627476\pi$
$991$	−24.5457	−0.779718	−0.389859	+	0.920875i	$0.627476\pi$
$992$	−2.55434	−0.0811004
$993$	0	0
$994$	6.17659	0.195909
$995$	0	0
$996$	0	0
$997$	−12.2159	−0.386881	−0.193441	−	0.981112i	$-0.561965\pi$
$997$	−12.2159	−0.386881	−0.193441	+	0.981112i	$0.561965\pi$
$998$	−6.91028	−0.218741
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8775.2.a.bs.1.2		4
3.2	odd	2		8775.2.a.bg.1.3		4
5.4	even	2		1755.2.a.n.1.3	✓	4
15.14	odd	2		1755.2.a.t.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.n.1.3	✓	4	5.4	even	2
1755.2.a.t.1.2	yes	4	15.14	odd	2
8775.2.a.bg.1.3		4	3.2	odd	2
8775.2.a.bs.1.2		4	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:	\( N \)	\(=\)	\( 8775 = 3^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8775.a (trivial)

\(f(q)\)	\(=\)	\(q+0.214739 q^{2} -1.95389 q^{4} +3.65683 q^{7} -0.849052 q^{8} +O(q^{10})\)	\(q+0.214739 q^{2} -1.95389 q^{4} +3.65683 q^{7} -0.849052 q^{8} -0.401897 q^{11} +1.00000 q^{13} +0.785261 q^{14} +3.72545 q^{16} -6.98901 q^{17} +3.87156 q^{19} -0.0863028 q^{22} -5.80185 q^{23} +0.214739 q^{26} -7.14503 q^{28} -7.63431 q^{29} -1.02251 q^{31} +2.49810 q^{32} -1.50081 q^{34} +11.5820 q^{37} +0.831374 q^{38} -10.1214 q^{41} +2.87156 q^{43} +0.785261 q^{44} -1.24588 q^{46} +11.4000 q^{47} +6.37237 q^{49} -1.95389 q^{52} +10.9067 q^{53} -3.10484 q^{56} -1.63938 q^{58} -1.68043 q^{59} +4.49218 q^{61} -0.219573 q^{62} -6.91446 q^{64} +7.59725 q^{67} +13.6557 q^{68} +7.86564 q^{71} -10.6560 q^{73} +2.48711 q^{74} -7.56460 q^{76} -1.46967 q^{77} +12.6820 q^{79} -2.17346 q^{82} +9.31474 q^{83} +0.616636 q^{86} +0.341231 q^{88} -3.29706 q^{89} +3.65683 q^{91} +11.3362 q^{92} +2.44801 q^{94} +10.2960 q^{97} +1.36839 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 3 q^{4} + q^{7} + 9 q^{8}+O(q^{10}) \) 4 * q + 3 * q^2 + 3 * q^4 + q^7 + 9 * q^8	\( 4 q + 3 q^{2} + 3 q^{4} + q^{7} + 9 q^{8} - 6 q^{11} + 4 q^{13} + q^{14} + 13 q^{16} + 2 q^{17} + 4 q^{19} + 9 q^{22} + 9 q^{23} + 3 q^{26} - 10 q^{28} - 16 q^{29} - 5 q^{31} + 26 q^{32} + 19 q^{34} + 13 q^{37} + 12 q^{38} - 12 q^{41} + q^{44} + 2 q^{46} + 9 q^{47} - 11 q^{49} + 3 q^{52} + 13 q^{53} - 14 q^{56} + 9 q^{58} - 3 q^{59} + 3 q^{61} - 25 q^{62} + 45 q^{64} - 6 q^{67} + 32 q^{68} - 11 q^{71} + 3 q^{73} - 4 q^{74} + 5 q^{76} + 10 q^{77} - 3 q^{79} - 22 q^{82} + 19 q^{83} + 9 q^{86} - 26 q^{88} - 16 q^{89} + q^{91} + 19 q^{92} + 25 q^{94} + 35 q^{97} - 21 q^{98}+O(q^{100}) \) 4 * q + 3 * q^2 + 3 * q^4 + q^7 + 9 * q^8 - 6 * q^11 + 4 * q^13 + q^14 + 13 * q^16 + 2 * q^17 + 4 * q^19 + 9 * q^22 + 9 * q^23 + 3 * q^26 - 10 * q^28 - 16 * q^29 - 5 * q^31 + 26 * q^32 + 19 * q^34 + 13 * q^37 + 12 * q^38 - 12 * q^41 + q^44 + 2 * q^46 + 9 * q^47 - 11 * q^49 + 3 * q^52 + 13 * q^53 - 14 * q^56 + 9 * q^58 - 3 * q^59 + 3 * q^61 - 25 * q^62 + 45 * q^64 - 6 * q^67 + 32 * q^68 - 11 * q^71 + 3 * q^73 - 4 * q^74 + 5 * q^76 + 10 * q^77 - 3 * q^79 - 22 * q^82 + 19 * q^83 + 9 * q^86 - 26 * q^88 - 16 * q^89 + q^91 + 19 * q^92 + 25 * q^94 + 35 * q^97 - 21 * q^98

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