LMFDB - Embedded newform 8550.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.56155 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.56155 q^{7} -1.00000 q^{8} -4.00000 q^{11} +6.68466 q^{13} +1.56155 q^{14} +1.00000 q^{16} +7.56155 q^{17} -1.00000 q^{19} +4.00000 q^{22} -4.68466 q^{23} -6.68466 q^{26} -1.56155 q^{28} -6.68466 q^{29} +3.12311 q^{31} -1.00000 q^{32} -7.56155 q^{34} +6.00000 q^{37} +1.00000 q^{38} +4.24621 q^{41} +11.1231 q^{43} -4.00000 q^{44} +4.68466 q^{46} -10.2462 q^{47} -4.56155 q^{49} +6.68466 q^{52} -0.438447 q^{53} +1.56155 q^{56} +6.68466 q^{58} +1.56155 q^{59} +2.87689 q^{61} -3.12311 q^{62} +1.00000 q^{64} -1.56155 q^{67} +7.56155 q^{68} +6.24621 q^{71} -10.6847 q^{73} -6.00000 q^{74} -1.00000 q^{76} +6.24621 q^{77} +3.12311 q^{79} -4.24621 q^{82} +11.1231 q^{83} -11.1231 q^{86} +4.00000 q^{88} -2.00000 q^{89} -10.4384 q^{91} -4.68466 q^{92} +10.2462 q^{94} -6.00000 q^{97} +4.56155 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8} - 8 q^{11} + q^{13} - q^{14} + 2 q^{16} + 11 q^{17} - 2 q^{19} + 8 q^{22} + 3 q^{23} - q^{26} + q^{28} - q^{29} - 2 q^{31} - 2 q^{32} - 11 q^{34} + 12 q^{37} + 2 q^{38} - 8 q^{41} + 14 q^{43} - 8 q^{44} - 3 q^{46} - 4 q^{47} - 5 q^{49} + q^{52} - 5 q^{53} - q^{56} + q^{58} - q^{59} + 14 q^{61} + 2 q^{62} + 2 q^{64} + q^{67} + 11 q^{68} - 4 q^{71} - 9 q^{73} - 12 q^{74} - 2 q^{76} - 4 q^{77} - 2 q^{79} + 8 q^{82} + 14 q^{83} - 14 q^{86} + 8 q^{88} - 4 q^{89} - 25 q^{91} + 3 q^{92} + 4 q^{94} - 12 q^{97} + 5 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8 - 8 * q^11 + q^13 - q^14 + 2 * q^16 + 11 * q^17 - 2 * q^19 + 8 * q^22 + 3 * q^23 - q^26 + q^28 - q^29 - 2 * q^31 - 2 * q^32 - 11 * q^34 + 12 * q^37 + 2 * q^38 - 8 * q^41 + 14 * q^43 - 8 * q^44 - 3 * q^46 - 4 * q^47 - 5 * q^49 + q^52 - 5 * q^53 - q^56 + q^58 - q^59 + 14 * q^61 + 2 * q^62 + 2 * q^64 + q^67 + 11 * q^68 - 4 * q^71 - 9 * q^73 - 12 * q^74 - 2 * q^76 - 4 * q^77 - 2 * q^79 + 8 * q^82 + 14 * q^83 - 14 * q^86 + 8 * q^88 - 4 * q^89 - 25 * q^91 + 3 * q^92 + 4 * q^94 - 12 * q^97 + 5 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	6.68466	1.85399	0.926995	−	0.375073i	$-0.122382\pi$
$13$	6.68466	1.85399	0.926995	+	0.375073i	$0.122382\pi$
$14$	1.56155	0.417343
$15$	0	0
$16$	1.00000	0.250000
$17$	7.56155	1.83395	0.916973	−	0.398949i	$-0.130625\pi$
$17$	7.56155	1.83395	0.916973	+	0.398949i	$0.130625\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	−4.68466	−0.976819	−0.488409	−	0.872615i	$-0.662423\pi$
$23$	−4.68466	−0.976819	−0.488409	+	0.872615i	$0.662423\pi$
$24$	0	0
$25$	0	0
$26$	−6.68466	−1.31097
$27$	0	0
$28$	−1.56155	−0.295106
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	3.12311	0.560926	0.280463	−	0.959865i	$-0.409512\pi$
$31$	3.12311	0.560926	0.280463	+	0.959865i	$0.409512\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.56155	−1.29680
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	4.24621	0.663147	0.331573	−	0.943429i	$-0.392421\pi$
$41$	4.24621	0.663147	0.331573	+	0.943429i	$0.392421\pi$
$42$	0	0
$43$	11.1231	1.69626	0.848129	−	0.529790i	$-0.177729\pi$
$43$	11.1231	1.69626	0.848129	+	0.529790i	$0.177729\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	4.68466	0.690715
$47$	−10.2462	−1.49456	−0.747282	−	0.664507i	$-0.768641\pi$
$47$	−10.2462	−1.49456	−0.747282	+	0.664507i	$0.768641\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	0	0
$52$	6.68466	0.926995
$53$	−0.438447	−0.0602254	−0.0301127	−	0.999547i	$-0.509587\pi$
$53$	−0.438447	−0.0602254	−0.0301127	+	0.999547i	$0.509587\pi$
$54$	0	0
$55$	0	0
$56$	1.56155	0.208671
$57$	0	0
$58$	6.68466	0.877739
$59$	1.56155	0.203297	0.101648	−	0.994820i	$-0.467588\pi$
$59$	1.56155	0.203297	0.101648	+	0.994820i	$0.467588\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	−3.12311	−0.396635
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−1.56155	−0.190774	−0.0953870	−	0.995440i	$-0.530409\pi$
$67$	−1.56155	−0.190774	−0.0953870	+	0.995440i	$0.530409\pi$
$68$	7.56155	0.916973
$69$	0	0
$70$	0	0
$71$	6.24621	0.741289	0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621	0.741289	0.370644	+	0.928775i	$0.379137\pi$
$72$	0	0
$73$	−10.6847	−1.25054	−0.625272	−	0.780407i	$-0.715012\pi$
$73$	−10.6847	−1.25054	−0.625272	+	0.780407i	$0.715012\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−1.00000	−0.114708
$77$	6.24621	0.711822
$78$	0	0
$79$	3.12311	0.351377	0.175688	−	0.984446i	$-0.443785\pi$
$79$	3.12311	0.351377	0.175688	+	0.984446i	$0.443785\pi$
$80$	0	0
$81$	0	0
$82$	−4.24621	−0.468916
$83$	11.1231	1.22092	0.610460	−	0.792047i	$-0.290985\pi$
$83$	11.1231	1.22092	0.610460	+	0.792047i	$0.290985\pi$
$84$	0	0
$85$	0	0
$86$	−11.1231	−1.19944
$87$	0	0
$88$	4.00000	0.426401
$89$	−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
$91$	−10.4384	−1.09425
$92$	−4.68466	−0.488409
$93$	0	0
$94$	10.2462	1.05682
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	4.56155	0.460786
$99$	0	0
$100$	0	0
$101$	−7.36932	−0.733274	−0.366637	−	0.930364i	$-0.619491\pi$
$101$	−7.36932	−0.733274	−0.366637	+	0.930364i	$0.619491\pi$
$102$	0	0
$103$	14.2462	1.40372	0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462	1.40372	0.701860	+	0.712314i	$0.252353\pi$
$104$	−6.68466	−0.655485
$105$	0	0
$106$	0.438447	0.0425858
$107$	−9.56155	−0.924350	−0.462175	−	0.886789i	$-0.652931\pi$
$107$	−9.56155	−0.924350	−0.462175	+	0.886789i	$0.652931\pi$
$108$	0	0
$109$	3.56155	0.341135	0.170567	−	0.985346i	$-0.445440\pi$
$109$	3.56155	0.341135	0.170567	+	0.985346i	$0.445440\pi$
$110$	0	0
$111$	0	0
$112$	−1.56155	−0.147553
$113$	17.1231	1.61081	0.805403	−	0.592727i	$-0.201949\pi$
$113$	17.1231	1.61081	0.805403	+	0.592727i	$0.201949\pi$
$114$	0	0
$115$	0	0
$116$	−6.68466	−0.620655
$117$	0	0
$118$	−1.56155	−0.143753
$119$	−11.8078	−1.08242
$120$	0	0
$121$	5.00000	0.454545
$122$	−2.87689	−0.260462
$123$	0	0
$124$	3.12311	0.280463
$125$	0	0
$126$	0	0
$127$	−4.87689	−0.432754	−0.216377	−	0.976310i	$-0.569424\pi$
$127$	−4.87689	−0.432754	−0.216377	+	0.976310i	$0.569424\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−16.4924	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$131$	−16.4924	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$132$	0	0
$133$	1.56155	0.135404
$134$	1.56155	0.134898
$135$	0	0
$136$	−7.56155	−0.648398
$137$	5.80776	0.496191	0.248095	−	0.968736i	$-0.420195\pi$
$137$	5.80776	0.496191	0.248095	+	0.968736i	$0.420195\pi$
$138$	0	0
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	0	0
$141$	0	0
$142$	−6.24621	−0.524170
$143$	−26.7386	−2.23600
$144$	0	0
$145$	0	0
$146$	10.6847	0.884269
$147$	0	0
$148$	6.00000	0.493197
$149$	11.3693	0.931411	0.465705	−	0.884940i	$-0.345801\pi$
$149$	11.3693	0.931411	0.465705	+	0.884940i	$0.345801\pi$
$150$	0	0
$151$	−3.12311	−0.254155	−0.127077	−	0.991893i	$-0.540560\pi$
$151$	−3.12311	−0.254155	−0.127077	+	0.991893i	$0.540560\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−6.24621	−0.503334
$155$	0	0
$156$	0	0
$157$	3.75379	0.299585	0.149792	−	0.988717i	$-0.452139\pi$
$157$	3.75379	0.299585	0.149792	+	0.988717i	$0.452139\pi$
$158$	−3.12311	−0.248461
$159$	0	0
$160$	0	0
$161$	7.31534	0.576530
$162$	0	0
$163$	−9.36932	−0.733862	−0.366931	−	0.930248i	$-0.619591\pi$
$163$	−9.36932	−0.733862	−0.366931	+	0.930248i	$0.619591\pi$
$164$	4.24621	0.331573
$165$	0	0
$166$	−11.1231	−0.863320
$167$	17.3693	1.34408	0.672039	−	0.740516i	$-0.265419\pi$
$167$	17.3693	1.34408	0.672039	+	0.740516i	$0.265419\pi$
$168$	0	0
$169$	31.6847	2.43728
$170$	0	0
$171$	0	0
$172$	11.1231	0.848129
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	2.00000	0.149906
$179$	5.75379	0.430058	0.215029	−	0.976608i	$-0.431015\pi$
$179$	5.75379	0.430058	0.215029	+	0.976608i	$0.431015\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	10.4384	0.773749
$183$	0	0
$184$	4.68466	0.345358
$185$	0	0
$186$	0	0
$187$	−30.2462	−2.21182
$188$	−10.2462	−0.747282
$189$	0	0
$190$	0	0
$191$	−8.68466	−0.628400	−0.314200	−	0.949357i	$-0.601736\pi$
$191$	−8.68466	−0.628400	−0.314200	+	0.949357i	$0.601736\pi$
$192$	0	0
$193$	−18.4924	−1.33111	−0.665557	−	0.746347i	$-0.731806\pi$
$193$	−18.4924	−1.33111	−0.665557	+	0.746347i	$0.731806\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−4.56155	−0.325825
$197$	−3.75379	−0.267446	−0.133723	−	0.991019i	$-0.542693\pi$
$197$	−3.75379	−0.267446	−0.133723	+	0.991019i	$0.542693\pi$
$198$	0	0
$199$	−3.80776	−0.269925	−0.134963	−	0.990851i	$-0.543091\pi$
$199$	−3.80776	−0.269925	−0.134963	+	0.990851i	$0.543091\pi$
$200$	0	0
$201$	0	0
$202$	7.36932	0.518503
$203$	10.4384	0.732635
$204$	0	0
$205$	0	0
$206$	−14.2462	−0.992581
$207$	0	0
$208$	6.68466	0.463498
$209$	4.00000	0.276686
$210$	0	0
$211$	20.6847	1.42399	0.711995	−	0.702184i	$-0.247792\pi$
$211$	20.6847	1.42399	0.711995	+	0.702184i	$0.247792\pi$
$212$	−0.438447	−0.0301127
$213$	0	0
$214$	9.56155	0.653614
$215$	0	0
$216$	0	0
$217$	−4.87689	−0.331065
$218$	−3.56155	−0.241219
$219$	0	0
$220$	0	0
$221$	50.5464	3.40012
$222$	0	0
$223$	−1.36932	−0.0916962	−0.0458481	−	0.998948i	$-0.514599\pi$
$223$	−1.36932	−0.0916962	−0.0458481	+	0.998948i	$0.514599\pi$
$224$	1.56155	0.104336
$225$	0	0
$226$	−17.1231	−1.13901
$227$	−2.93087	−0.194529	−0.0972643	−	0.995259i	$-0.531009\pi$
$227$	−2.93087	−0.194529	−0.0972643	+	0.995259i	$0.531009\pi$
$228$	0	0
$229$	18.4924	1.22201	0.611007	−	0.791625i	$-0.290765\pi$
$229$	18.4924	1.22201	0.611007	+	0.791625i	$0.290765\pi$
$230$	0	0
$231$	0	0
$232$	6.68466	0.438869
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	1.56155	0.101648
$237$	0	0
$238$	11.8078	0.765384
$239$	5.56155	0.359747	0.179873	−	0.983690i	$-0.442431\pi$
$239$	5.56155	0.359747	0.179873	+	0.983690i	$0.442431\pi$
$240$	0	0
$241$	14.8769	0.958305	0.479153	−	0.877732i	$-0.340944\pi$
$241$	14.8769	0.958305	0.479153	+	0.877732i	$0.340944\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	2.87689	0.184174
$245$	0	0
$246$	0	0
$247$	−6.68466	−0.425335
$248$	−3.12311	−0.198317
$249$	0	0
$250$	0	0
$251$	10.2462	0.646735	0.323368	−	0.946273i	$-0.395185\pi$
$251$	10.2462	0.646735	0.323368	+	0.946273i	$0.395185\pi$
$252$	0	0
$253$	18.7386	1.17809
$254$	4.87689	0.306004
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	−9.36932	−0.582181
$260$	0	0
$261$	0	0
$262$	16.4924	1.01891
$263$	−5.75379	−0.354794	−0.177397	−	0.984139i	$-0.556768\pi$
$263$	−5.75379	−0.354794	−0.177397	+	0.984139i	$0.556768\pi$
$264$	0	0
$265$	0	0
$266$	−1.56155	−0.0957449
$267$	0	0
$268$	−1.56155	−0.0953870
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	6.93087	0.421020	0.210510	−	0.977592i	$-0.432487\pi$
$271$	6.93087	0.421020	0.210510	+	0.977592i	$0.432487\pi$
$272$	7.56155	0.458486
$273$	0	0
$274$	−5.80776	−0.350860
$275$	0	0
$276$	0	0
$277$	−9.12311	−0.548154	−0.274077	−	0.961708i	$-0.588372\pi$
$277$	−9.12311	−0.548154	−0.274077	+	0.961708i	$0.588372\pi$
$278$	16.4924	0.989150
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	12.8769	0.765452	0.382726	−	0.923862i	$-0.374985\pi$
$283$	12.8769	0.765452	0.382726	+	0.923862i	$0.374985\pi$
$284$	6.24621	0.370644
$285$	0	0
$286$	26.7386	1.58109
$287$	−6.63068	−0.391397
$288$	0	0
$289$	40.1771	2.36336
$290$	0	0
$291$	0	0
$292$	−10.6847	−0.625272
$293$	23.1771	1.35402	0.677010	−	0.735974i	$-0.263275\pi$
$293$	23.1771	1.35402	0.677010	+	0.735974i	$0.263275\pi$
$294$	0	0
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−11.3693	−0.658607
$299$	−31.3153	−1.81101
$300$	0	0
$301$	−17.3693	−1.00115
$302$	3.12311	0.179715
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	0.492423	0.0281040	0.0140520	−	0.999901i	$-0.495527\pi$
$307$	0.492423	0.0281040	0.0140520	+	0.999901i	$0.495527\pi$
$308$	6.24621	0.355911
$309$	0	0
$310$	0	0
$311$	−8.68466	−0.492462	−0.246231	−	0.969211i	$-0.579192\pi$
$311$	−8.68466	−0.492462	−0.246231	+	0.969211i	$0.579192\pi$
$312$	0	0
$313$	32.0540	1.81180	0.905899	−	0.423494i	$-0.139197\pi$
$313$	32.0540	1.81180	0.905899	+	0.423494i	$0.139197\pi$
$314$	−3.75379	−0.211839
$315$	0	0
$316$	3.12311	0.175688
$317$	−24.0540	−1.35101	−0.675503	−	0.737357i	$-0.736073\pi$
$317$	−24.0540	−1.35101	−0.675503	+	0.737357i	$0.736073\pi$
$318$	0	0
$319$	26.7386	1.49708
$320$	0	0
$321$	0	0
$322$	−7.31534	−0.407668
$323$	−7.56155	−0.420736
$324$	0	0
$325$	0	0
$326$	9.36932	0.518918
$327$	0	0
$328$	−4.24621	−0.234458
$329$	16.0000	0.882109
$330$	0	0
$331$	1.56155	0.0858307	0.0429154	−	0.999079i	$-0.486335\pi$
$331$	1.56155	0.0858307	0.0429154	+	0.999079i	$0.486335\pi$
$332$	11.1231	0.610460
$333$	0	0
$334$	−17.3693	−0.950407
$335$	0	0
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−31.6847	−1.72342
$339$	0	0
$340$	0	0
$341$	−12.4924	−0.676503
$342$	0	0
$343$	18.0540	0.974823
$344$	−11.1231	−0.599718
$345$	0	0
$346$	−3.75379	−0.201805
$347$	−33.3693	−1.79136	−0.895679	−	0.444700i	$-0.853310\pi$
$347$	−33.3693	−1.79136	−0.895679	+	0.444700i	$0.853310\pi$
$348$	0	0
$349$	20.2462	1.08375	0.541877	−	0.840458i	$-0.317714\pi$
$349$	20.2462	1.08375	0.541877	+	0.840458i	$0.317714\pi$
$350$	0	0
$351$	0	0
$352$	4.00000	0.213201
$353$	24.9309	1.32694	0.663468	−	0.748205i	$-0.269084\pi$
$353$	24.9309	1.32694	0.663468	+	0.748205i	$0.269084\pi$
$354$	0	0
$355$	0	0
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−5.75379	−0.304097
$359$	−5.56155	−0.293528	−0.146764	−	0.989172i	$-0.546886\pi$
$359$	−5.56155	−0.293528	−0.146764	+	0.989172i	$0.546886\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.0000	0.946059
$363$	0	0
$364$	−10.4384	−0.547123
$365$	0	0
$366$	0	0
$367$	10.2462	0.534848	0.267424	−	0.963579i	$-0.413828\pi$
$367$	10.2462	0.534848	0.267424	+	0.963579i	$0.413828\pi$
$368$	−4.68466	−0.244205
$369$	0	0
$370$	0	0
$371$	0.684658	0.0355457
$372$	0	0
$373$	27.5616	1.42708	0.713542	−	0.700613i	$-0.247090\pi$
$373$	27.5616	1.42708	0.713542	+	0.700613i	$0.247090\pi$
$374$	30.2462	1.56399
$375$	0	0
$376$	10.2462	0.528408
$377$	−44.6847	−2.30138
$378$	0	0
$379$	6.43845	0.330721	0.165360	−	0.986233i	$-0.447121\pi$
$379$	6.43845	0.330721	0.165360	+	0.986233i	$0.447121\pi$
$380$	0	0
$381$	0	0
$382$	8.68466	0.444346
$383$	30.2462	1.54551	0.772755	−	0.634705i	$-0.218878\pi$
$383$	30.2462	1.54551	0.772755	+	0.634705i	$0.218878\pi$
$384$	0	0
$385$	0	0
$386$	18.4924	0.941240
$387$	0	0
$388$	−6.00000	−0.304604
$389$	−1.12311	−0.0569437	−0.0284719	−	0.999595i	$-0.509064\pi$
$389$	−1.12311	−0.0569437	−0.0284719	+	0.999595i	$0.509064\pi$
$390$	0	0
$391$	−35.4233	−1.79143
$392$	4.56155	0.230393
$393$	0	0
$394$	3.75379	0.189113
$395$	0	0
$396$	0	0
$397$	−1.12311	−0.0563671	−0.0281835	−	0.999603i	$-0.508972\pi$
$397$	−1.12311	−0.0563671	−0.0281835	+	0.999603i	$0.508972\pi$
$398$	3.80776	0.190866
$399$	0	0
$400$	0	0
$401$	20.2462	1.01105	0.505524	−	0.862813i	$-0.331299\pi$
$401$	20.2462	1.01105	0.505524	+	0.862813i	$0.331299\pi$
$402$	0	0
$403$	20.8769	1.03995
$404$	−7.36932	−0.366637
$405$	0	0
$406$	−10.4384	−0.518051
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−24.7386	−1.22325	−0.611623	−	0.791149i	$-0.709483\pi$
$409$	−24.7386	−1.22325	−0.611623	+	0.791149i	$0.709483\pi$
$410$	0	0
$411$	0	0
$412$	14.2462	0.701860
$413$	−2.43845	−0.119988
$414$	0	0
$415$	0	0
$416$	−6.68466	−0.327742
$417$	0	0
$418$	−4.00000	−0.195646
$419$	33.8617	1.65425	0.827127	−	0.562015i	$-0.189974\pi$
$419$	33.8617	1.65425	0.827127	+	0.562015i	$0.189974\pi$
$420$	0	0
$421$	4.93087	0.240316	0.120158	−	0.992755i	$-0.461660\pi$
$421$	4.93087	0.240316	0.120158	+	0.992755i	$0.461660\pi$
$422$	−20.6847	−1.00691
$423$	0	0
$424$	0.438447	0.0212929
$425$	0	0
$426$	0	0
$427$	−4.49242	−0.217404
$428$	−9.56155	−0.462175
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−39.3693	−1.89197	−0.945984	−	0.324212i	$-0.894901\pi$
$433$	−39.3693	−1.89197	−0.945984	+	0.324212i	$0.894901\pi$
$434$	4.87689	0.234098
$435$	0	0
$436$	3.56155	0.170567
$437$	4.68466	0.224098
$438$	0	0
$439$	−4.87689	−0.232761	−0.116381	−	0.993205i	$-0.537129\pi$
$439$	−4.87689	−0.232761	−0.116381	+	0.993205i	$0.537129\pi$
$440$	0	0
$441$	0	0
$442$	−50.5464	−2.40425
$443$	−14.2462	−0.676858	−0.338429	−	0.940992i	$-0.609895\pi$
$443$	−14.2462	−0.676858	−0.338429	+	0.940992i	$0.609895\pi$
$444$	0	0
$445$	0	0
$446$	1.36932	0.0648390
$447$	0	0
$448$	−1.56155	−0.0737764
$449$	−20.7386	−0.978717	−0.489358	−	0.872083i	$-0.662769\pi$
$449$	−20.7386	−0.978717	−0.489358	+	0.872083i	$0.662769\pi$
$450$	0	0
$451$	−16.9848	−0.799785
$452$	17.1231	0.805403
$453$	0	0
$454$	2.93087	0.137553
$455$	0	0
$456$	0	0
$457$	−18.6847	−0.874031	−0.437016	−	0.899454i	$-0.643965\pi$
$457$	−18.6847	−0.874031	−0.437016	+	0.899454i	$0.643965\pi$
$458$	−18.4924	−0.864094
$459$	0	0
$460$	0	0
$461$	−20.2462	−0.942960	−0.471480	−	0.881877i	$-0.656280\pi$
$461$	−20.2462	−0.942960	−0.471480	+	0.881877i	$0.656280\pi$
$462$	0	0
$463$	13.7538	0.639193	0.319596	−	0.947554i	$-0.396453\pi$
$463$	13.7538	0.639193	0.319596	+	0.947554i	$0.396453\pi$
$464$	−6.68466	−0.310327
$465$	0	0
$466$	−10.0000	−0.463241
$467$	1.75379	0.0811557	0.0405778	−	0.999176i	$-0.487080\pi$
$467$	1.75379	0.0811557	0.0405778	+	0.999176i	$0.487080\pi$
$468$	0	0
$469$	2.43845	0.112597
$470$	0	0
$471$	0	0
$472$	−1.56155	−0.0718763
$473$	−44.4924	−2.04576
$474$	0	0
$475$	0	0
$476$	−11.8078	−0.541208
$477$	0	0
$478$	−5.56155	−0.254380
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	40.1080	1.82877
$482$	−14.8769	−0.677624
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	0	0
$487$	−23.6155	−1.07012	−0.535061	−	0.844814i	$-0.679711\pi$
$487$	−23.6155	−1.07012	−0.535061	+	0.844814i	$0.679711\pi$
$488$	−2.87689	−0.130231
$489$	0	0
$490$	0	0
$491$	−7.12311	−0.321461	−0.160731	−	0.986998i	$-0.551385\pi$
$491$	−7.12311	−0.321461	−0.160731	+	0.986998i	$0.551385\pi$
$492$	0	0
$493$	−50.5464	−2.27650
$494$	6.68466	0.300757
$495$	0	0
$496$	3.12311	0.140232
$497$	−9.75379	−0.437517
$498$	0	0
$499$	−32.1080	−1.43735	−0.718675	−	0.695347i	$-0.755251\pi$
$499$	−32.1080	−1.43735	−0.718675	+	0.695347i	$0.755251\pi$
$500$	0	0
$501$	0	0
$502$	−10.2462	−0.457311
$503$	30.0540	1.34004	0.670020	−	0.742343i	$-0.266285\pi$
$503$	30.0540	1.34004	0.670020	+	0.742343i	$0.266285\pi$
$504$	0	0
$505$	0	0
$506$	−18.7386	−0.833034
$507$	0	0
$508$	−4.87689	−0.216377
$509$	30.4924	1.35155	0.675776	−	0.737107i	$-0.263808\pi$
$509$	30.4924	1.35155	0.675776	+	0.737107i	$0.263808\pi$
$510$	0	0
$511$	16.6847	0.738086
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−14.0000	−0.617514
$515$	0	0
$516$	0	0
$517$	40.9848	1.80251
$518$	9.36932	0.411664
$519$	0	0
$520$	0	0
$521$	−5.12311	−0.224447	−0.112224	−	0.993683i	$-0.535797\pi$
$521$	−5.12311	−0.224447	−0.112224	+	0.993683i	$0.535797\pi$
$522$	0	0
$523$	19.3153	0.844601	0.422300	−	0.906456i	$-0.361223\pi$
$523$	19.3153	0.844601	0.422300	+	0.906456i	$0.361223\pi$
$524$	−16.4924	−0.720475
$525$	0	0
$526$	5.75379	0.250877
$527$	23.6155	1.02871
$528$	0	0
$529$	−1.05398	−0.0458250
$530$	0	0
$531$	0	0
$532$	1.56155	0.0677019
$533$	28.3845	1.22947
$534$	0	0
$535$	0	0
$536$	1.56155	0.0674488
$537$	0	0
$538$	−26.0000	−1.12094
$539$	18.2462	0.785920
$540$	0	0
$541$	−41.6155	−1.78919	−0.894596	−	0.446877i	$-0.852536\pi$
$541$	−41.6155	−1.78919	−0.894596	+	0.446877i	$0.852536\pi$
$542$	−6.93087	−0.297706
$543$	0	0
$544$	−7.56155	−0.324199
$545$	0	0
$546$	0	0
$547$	16.4924	0.705165	0.352583	−	0.935781i	$-0.385304\pi$
$547$	16.4924	0.705165	0.352583	+	0.935781i	$0.385304\pi$
$548$	5.80776	0.248095
$549$	0	0
$550$	0	0
$551$	6.68466	0.284776
$552$	0	0
$553$	−4.87689	−0.207387
$554$	9.12311	0.387604
$555$	0	0
$556$	−16.4924	−0.699435
$557$	−1.61553	−0.0684521	−0.0342261	−	0.999414i	$-0.510897\pi$
$557$	−1.61553	−0.0684521	−0.0342261	+	0.999414i	$0.510897\pi$
$558$	0	0
$559$	74.3542	3.14485
$560$	0	0
$561$	0	0
$562$	2.00000	0.0843649
$563$	24.4924	1.03223	0.516116	−	0.856519i	$-0.327377\pi$
$563$	24.4924	1.03223	0.516116	+	0.856519i	$0.327377\pi$
$564$	0	0
$565$	0	0
$566$	−12.8769	−0.541256
$567$	0	0
$568$	−6.24621	−0.262085
$569$	−5.12311	−0.214772	−0.107386	−	0.994217i	$-0.534248\pi$
$569$	−5.12311	−0.214772	−0.107386	+	0.994217i	$0.534248\pi$
$570$	0	0
$571$	−19.6155	−0.820884	−0.410442	−	0.911887i	$-0.634626\pi$
$571$	−19.6155	−0.820884	−0.410442	+	0.911887i	$0.634626\pi$
$572$	−26.7386	−1.11800
$573$	0	0
$574$	6.63068	0.276759
$575$	0	0
$576$	0	0
$577$	22.6847	0.944375	0.472187	−	0.881498i	$-0.343465\pi$
$577$	22.6847	0.944375	0.472187	+	0.881498i	$0.343465\pi$
$578$	−40.1771	−1.67115
$579$	0	0
$580$	0	0
$581$	−17.3693	−0.720601
$582$	0	0
$583$	1.75379	0.0726345
$584$	10.6847	0.442134
$585$	0	0
$586$	−23.1771	−0.957436
$587$	17.3693	0.716908	0.358454	−	0.933547i	$-0.383304\pi$
$587$	17.3693	0.716908	0.358454	+	0.933547i	$0.383304\pi$
$588$	0	0
$589$	−3.12311	−0.128685
$590$	0	0
$591$	0	0
$592$	6.00000	0.246598
$593$	24.2462	0.995673	0.497836	−	0.867271i	$-0.334128\pi$
$593$	24.2462	0.995673	0.497836	+	0.867271i	$0.334128\pi$
$594$	0	0
$595$	0	0
$596$	11.3693	0.465705
$597$	0	0
$598$	31.3153	1.28058
$599$	−45.8617	−1.87386	−0.936930	−	0.349517i	$-0.886346\pi$
$599$	−45.8617	−1.87386	−0.936930	+	0.349517i	$0.886346\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	17.3693	0.707921
$603$	0	0
$604$	−3.12311	−0.127077
$605$	0	0
$606$	0	0
$607$	12.8769	0.522657	0.261329	−	0.965250i	$-0.415839\pi$
$607$	12.8769	0.522657	0.261329	+	0.965250i	$0.415839\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−68.4924	−2.77091
$612$	0	0
$613$	19.3693	0.782319	0.391160	−	0.920323i	$-0.372074\pi$
$613$	19.3693	0.782319	0.391160	+	0.920323i	$0.372074\pi$
$614$	−0.492423	−0.0198726
$615$	0	0
$616$	−6.24621	−0.251667
$617$	−4.24621	−0.170946	−0.0854730	−	0.996340i	$-0.527240\pi$
$617$	−4.24621	−0.170946	−0.0854730	+	0.996340i	$0.527240\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	8.68466	0.348223
$623$	3.12311	0.125125
$624$	0	0
$625$	0	0
$626$	−32.0540	−1.28113
$627$	0	0
$628$	3.75379	0.149792
$629$	45.3693	1.80899
$630$	0	0
$631$	12.4924	0.497315	0.248658	−	0.968591i	$-0.420011\pi$
$631$	12.4924	0.497315	0.248658	+	0.968591i	$0.420011\pi$
$632$	−3.12311	−0.124230
$633$	0	0
$634$	24.0540	0.955305
$635$	0	0
$636$	0	0
$637$	−30.4924	−1.20815
$638$	−26.7386	−1.05859
$639$	0	0
$640$	0	0
$641$	35.8617	1.41645	0.708227	−	0.705985i	$-0.249495\pi$
$641$	35.8617	1.41645	0.708227	+	0.705985i	$0.249495\pi$
$642$	0	0
$643$	−7.61553	−0.300327	−0.150164	−	0.988661i	$-0.547980\pi$
$643$	−7.61553	−0.300327	−0.150164	+	0.988661i	$0.547980\pi$
$644$	7.31534	0.288265
$645$	0	0
$646$	7.56155	0.297505
$647$	9.56155	0.375903	0.187952	−	0.982178i	$-0.439815\pi$
$647$	9.56155	0.375903	0.187952	+	0.982178i	$0.439815\pi$
$648$	0	0
$649$	−6.24621	−0.245185
$650$	0	0
$651$	0	0
$652$	−9.36932	−0.366931
$653$	1.12311	0.0439505	0.0219753	−	0.999759i	$-0.493004\pi$
$653$	1.12311	0.0439505	0.0219753	+	0.999759i	$0.493004\pi$
$654$	0	0
$655$	0	0
$656$	4.24621	0.165787
$657$	0	0
$658$	−16.0000	−0.623745
$659$	42.9309	1.67235	0.836175	−	0.548463i	$-0.184787\pi$
$659$	42.9309	1.67235	0.836175	+	0.548463i	$0.184787\pi$
$660$	0	0
$661$	9.80776	0.381478	0.190739	−	0.981641i	$-0.438912\pi$
$661$	9.80776	0.381478	0.190739	+	0.981641i	$0.438912\pi$
$662$	−1.56155	−0.0606915
$663$	0	0
$664$	−11.1231	−0.431660
$665$	0	0
$666$	0	0
$667$	31.3153	1.21253
$668$	17.3693	0.672039
$669$	0	0
$670$	0	0
$671$	−11.5076	−0.444245
$672$	0	0
$673$	3.36932	0.129878	0.0649388	−	0.997889i	$-0.479315\pi$
$673$	3.36932	0.129878	0.0649388	+	0.997889i	$0.479315\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	31.6847	1.21864
$677$	36.4384	1.40044	0.700222	−	0.713926i	$-0.253085\pi$
$677$	36.4384	1.40044	0.700222	+	0.713926i	$0.253085\pi$
$678$	0	0
$679$	9.36932	0.359561
$680$	0	0
$681$	0	0
$682$	12.4924	0.478360
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	0	0
$686$	−18.0540	−0.689304
$687$	0	0
$688$	11.1231	0.424064
$689$	−2.93087	−0.111657
$690$	0	0
$691$	−8.87689	−0.337693	−0.168846	−	0.985642i	$-0.554004\pi$
$691$	−8.87689	−0.337693	−0.168846	+	0.985642i	$0.554004\pi$
$692$	3.75379	0.142698
$693$	0	0
$694$	33.3693	1.26668
$695$	0	0
$696$	0	0
$697$	32.1080	1.21618
$698$	−20.2462	−0.766330
$699$	0	0
$700$	0	0
$701$	29.1231	1.09996	0.549982	−	0.835176i	$-0.314634\pi$
$701$	29.1231	1.09996	0.549982	+	0.835176i	$0.314634\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−24.9309	−0.938286
$707$	11.5076	0.432787
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	2.00000	0.0749532
$713$	−14.6307	−0.547923
$714$	0	0
$715$	0	0
$716$	5.75379	0.215029
$717$	0	0
$718$	5.56155	0.207555
$719$	29.5616	1.10246	0.551230	−	0.834353i	$-0.314159\pi$
$719$	29.5616	1.10246	0.551230	+	0.834353i	$0.314159\pi$
$720$	0	0
$721$	−22.2462	−0.828492
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−18.0000	−0.668965
$725$	0	0
$726$	0	0
$727$	36.6847	1.36056	0.680279	−	0.732953i	$-0.261858\pi$
$727$	36.6847	1.36056	0.680279	+	0.732953i	$0.261858\pi$
$728$	10.4384	0.386875
$729$	0	0
$730$	0	0
$731$	84.1080	3.11084
$732$	0	0
$733$	45.1231	1.66666	0.833330	−	0.552776i	$-0.186431\pi$
$733$	45.1231	1.66666	0.833330	+	0.552776i	$0.186431\pi$
$734$	−10.2462	−0.378195
$735$	0	0
$736$	4.68466	0.172679
$737$	6.24621	0.230082
$738$	0	0
$739$	16.8769	0.620827	0.310413	−	0.950602i	$-0.399533\pi$
$739$	16.8769	0.620827	0.310413	+	0.950602i	$0.399533\pi$
$740$	0	0
$741$	0	0
$742$	−0.684658	−0.0251346
$743$	27.1231	0.995050	0.497525	−	0.867450i	$-0.334242\pi$
$743$	27.1231	0.995050	0.497525	+	0.867450i	$0.334242\pi$
$744$	0	0
$745$	0	0
$746$	−27.5616	−1.00910
$747$	0	0
$748$	−30.2462	−1.10591
$749$	14.9309	0.545562
$750$	0	0
$751$	−43.1231	−1.57358	−0.786792	−	0.617218i	$-0.788260\pi$
$751$	−43.1231	−1.57358	−0.786792	+	0.617218i	$0.788260\pi$
$752$	−10.2462	−0.373641
$753$	0	0
$754$	44.6847	1.62732
$755$	0	0
$756$	0	0
$757$	−9.50758	−0.345559	−0.172779	−	0.984961i	$-0.555275\pi$
$757$	−9.50758	−0.345559	−0.172779	+	0.984961i	$0.555275\pi$
$758$	−6.43845	−0.233855
$759$	0	0
$760$	0	0
$761$	28.5464	1.03481	0.517403	−	0.855742i	$-0.326899\pi$
$761$	28.5464	1.03481	0.517403	+	0.855742i	$0.326899\pi$
$762$	0	0
$763$	−5.56155	−0.201342
$764$	−8.68466	−0.314200
$765$	0	0
$766$	−30.2462	−1.09284
$767$	10.4384	0.376910
$768$	0	0
$769$	31.5616	1.13814	0.569069	−	0.822290i	$-0.307304\pi$
$769$	31.5616	1.13814	0.569069	+	0.822290i	$0.307304\pi$
$770$	0	0
$771$	0	0
$772$	−18.4924	−0.665557
$773$	−9.80776	−0.352761	−0.176380	−	0.984322i	$-0.556439\pi$
$773$	−9.80776	−0.352761	−0.176380	+	0.984322i	$0.556439\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	1.12311	0.0402653
$779$	−4.24621	−0.152136
$780$	0	0
$781$	−24.9848	−0.894028
$782$	35.4233	1.26673
$783$	0	0
$784$	−4.56155	−0.162913
$785$	0	0
$786$	0	0
$787$	−31.8078	−1.13382	−0.566912	−	0.823778i	$-0.691862\pi$
$787$	−31.8078	−1.13382	−0.566912	+	0.823778i	$0.691862\pi$
$788$	−3.75379	−0.133723
$789$	0	0
$790$	0	0
$791$	−26.7386	−0.950716
$792$	0	0
$793$	19.2311	0.682915
$794$	1.12311	0.0398575
$795$	0	0
$796$	−3.80776	−0.134963
$797$	42.3002	1.49835	0.749175	−	0.662372i	$-0.230450\pi$
$797$	42.3002	1.49835	0.749175	+	0.662372i	$0.230450\pi$
$798$	0	0
$799$	−77.4773	−2.74095
$800$	0	0
$801$	0	0
$802$	−20.2462	−0.714919
$803$	42.7386	1.50821
$804$	0	0
$805$	0	0
$806$	−20.8769	−0.735357
$807$	0	0
$808$	7.36932	0.259252
$809$	8.43845	0.296680	0.148340	−	0.988936i	$-0.452607\pi$
$809$	8.43845	0.296680	0.148340	+	0.988936i	$0.452607\pi$
$810$	0	0
$811$	−0.192236	−0.00675032	−0.00337516	−	0.999994i	$-0.501074\pi$
$811$	−0.192236	−0.00675032	−0.00337516	+	0.999994i	$0.501074\pi$
$812$	10.4384	0.366318
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	−11.1231	−0.389148
$818$	24.7386	0.864966
$819$	0	0
$820$	0	0
$821$	−7.36932	−0.257191	−0.128595	−	0.991697i	$-0.541047\pi$
$821$	−7.36932	−0.257191	−0.128595	+	0.991697i	$0.541047\pi$
$822$	0	0
$823$	33.5616	1.16988	0.584941	−	0.811076i	$-0.301118\pi$
$823$	33.5616	1.16988	0.584941	+	0.811076i	$0.301118\pi$
$824$	−14.2462	−0.496290
$825$	0	0
$826$	2.43845	0.0848444
$827$	−6.43845	−0.223887	−0.111943	−	0.993715i	$-0.535708\pi$
$827$	−6.43845	−0.223887	−0.111943	+	0.993715i	$0.535708\pi$
$828$	0	0
$829$	16.0540	0.557578	0.278789	−	0.960352i	$-0.410067\pi$
$829$	16.0540	0.557578	0.278789	+	0.960352i	$0.410067\pi$
$830$	0	0
$831$	0	0
$832$	6.68466	0.231749
$833$	−34.4924	−1.19509
$834$	0	0
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	−33.8617	−1.16973
$839$	12.4924	0.431286	0.215643	−	0.976472i	$-0.430815\pi$
$839$	12.4924	0.431286	0.215643	+	0.976472i	$0.430815\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	−4.93087	−0.169929
$843$	0	0
$844$	20.6847	0.711995
$845$	0	0
$846$	0	0
$847$	−7.80776	−0.268278
$848$	−0.438447	−0.0150563
$849$	0	0
$850$	0	0
$851$	−28.1080	−0.963528
$852$	0	0
$853$	−24.7386	−0.847035	−0.423517	−	0.905888i	$-0.639205\pi$
$853$	−24.7386	−0.847035	−0.423517	+	0.905888i	$0.639205\pi$
$854$	4.49242	0.153728
$855$	0	0
$856$	9.56155	0.326807
$857$	39.3693	1.34483	0.672415	−	0.740174i	$-0.265257\pi$
$857$	39.3693	1.34483	0.672415	+	0.740174i	$0.265257\pi$
$858$	0	0
$859$	12.9848	0.443037	0.221519	−	0.975156i	$-0.428899\pi$
$859$	12.9848	0.443037	0.221519	+	0.975156i	$0.428899\pi$
$860$	0	0
$861$	0	0
$862$	−16.0000	−0.544962
$863$	−14.2462	−0.484947	−0.242473	−	0.970158i	$-0.577959\pi$
$863$	−14.2462	−0.484947	−0.242473	+	0.970158i	$0.577959\pi$
$864$	0	0
$865$	0	0
$866$	39.3693	1.33782
$867$	0	0
$868$	−4.87689	−0.165533
$869$	−12.4924	−0.423776
$870$	0	0
$871$	−10.4384	−0.353693
$872$	−3.56155	−0.120609
$873$	0	0
$874$	−4.68466	−0.158461
$875$	0	0
$876$	0	0
$877$	−24.9309	−0.841856	−0.420928	−	0.907094i	$-0.638295\pi$
$877$	−24.9309	−0.841856	−0.420928	+	0.907094i	$0.638295\pi$
$878$	4.87689	0.164587
$879$	0	0
$880$	0	0
$881$	22.9848	0.774379	0.387190	−	0.922000i	$-0.373446\pi$
$881$	22.9848	0.774379	0.387190	+	0.922000i	$0.373446\pi$
$882$	0	0
$883$	47.6155	1.60239	0.801195	−	0.598403i	$-0.204198\pi$
$883$	47.6155	1.60239	0.801195	+	0.598403i	$0.204198\pi$
$884$	50.5464	1.70006
$885$	0	0
$886$	14.2462	0.478611
$887$	−4.49242	−0.150841	−0.0754204	−	0.997152i	$-0.524030\pi$
$887$	−4.49242	−0.150841	−0.0754204	+	0.997152i	$0.524030\pi$
$888$	0	0
$889$	7.61553	0.255417
$890$	0	0
$891$	0	0
$892$	−1.36932	−0.0458481
$893$	10.2462	0.342876
$894$	0	0
$895$	0	0
$896$	1.56155	0.0521678
$897$	0	0
$898$	20.7386	0.692057
$899$	−20.8769	−0.696283
$900$	0	0
$901$	−3.31534	−0.110450
$902$	16.9848	0.565533
$903$	0	0
$904$	−17.1231	−0.569506
$905$	0	0
$906$	0	0
$907$	−25.1771	−0.835991	−0.417996	−	0.908449i	$-0.637267\pi$
$907$	−25.1771	−0.835991	−0.417996	+	0.908449i	$0.637267\pi$
$908$	−2.93087	−0.0972643
$909$	0	0
$910$	0	0
$911$	28.4924	0.943996	0.471998	−	0.881600i	$-0.343533\pi$
$911$	28.4924	0.943996	0.471998	+	0.881600i	$0.343533\pi$
$912$	0	0
$913$	−44.4924	−1.47248
$914$	18.6847	0.618034
$915$	0	0
$916$	18.4924	0.611007
$917$	25.7538	0.850465
$918$	0	0
$919$	−30.9309	−1.02032	−0.510158	−	0.860081i	$-0.670413\pi$
$919$	−30.9309	−1.02032	−0.510158	+	0.860081i	$0.670413\pi$
$920$	0	0
$921$	0	0
$922$	20.2462	0.666773
$923$	41.7538	1.37434
$924$	0	0
$925$	0	0
$926$	−13.7538	−0.451978
$927$	0	0
$928$	6.68466	0.219435
$929$	−34.3002	−1.12535	−0.562676	−	0.826677i	$-0.690228\pi$
$929$	−34.3002	−1.12535	−0.562676	+	0.826677i	$0.690228\pi$
$930$	0	0
$931$	4.56155	0.149499
$932$	10.0000	0.327561
$933$	0	0
$934$	−1.75379	−0.0573857
$935$	0	0
$936$	0	0
$937$	−36.4384	−1.19039	−0.595196	−	0.803580i	$-0.702926\pi$
$937$	−36.4384	−1.19039	−0.595196	+	0.803580i	$0.702926\pi$
$938$	−2.43845	−0.0796181
$939$	0	0
$940$	0	0
$941$	−34.1922	−1.11464	−0.557318	−	0.830299i	$-0.688169\pi$
$941$	−34.1922	−1.11464	−0.557318	+	0.830299i	$0.688169\pi$
$942$	0	0
$943$	−19.8920	−0.647774
$944$	1.56155	0.0508242
$945$	0	0
$946$	44.4924	1.44657
$947$	−17.7538	−0.576921	−0.288460	−	0.957492i	$-0.593143\pi$
$947$	−17.7538	−0.576921	−0.288460	+	0.957492i	$0.593143\pi$
$948$	0	0
$949$	−71.4233	−2.31850
$950$	0	0
$951$	0	0
$952$	11.8078	0.382692
$953$	−30.1080	−0.975292	−0.487646	−	0.873041i	$-0.662144\pi$
$953$	−30.1080	−0.975292	−0.487646	+	0.873041i	$0.662144\pi$
$954$	0	0
$955$	0	0
$956$	5.56155	0.179873
$957$	0	0
$958$	32.0000	1.03387
$959$	−9.06913	−0.292857
$960$	0	0
$961$	−21.2462	−0.685362
$962$	−40.1080	−1.29313
$963$	0	0
$964$	14.8769	0.479153
$965$	0	0
$966$	0	0
$967$	48.4924	1.55941	0.779706	−	0.626146i	$-0.215369\pi$
$967$	48.4924	1.55941	0.779706	+	0.626146i	$0.215369\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	0	0
$971$	23.5076	0.754394	0.377197	−	0.926133i	$-0.376888\pi$
$971$	23.5076	0.754394	0.377197	+	0.926133i	$0.376888\pi$
$972$	0	0
$973$	25.7538	0.825629
$974$	23.6155	0.756690
$975$	0	0
$976$	2.87689	0.0920871
$977$	−20.7386	−0.663488	−0.331744	−	0.943370i	$-0.607637\pi$
$977$	−20.7386	−0.663488	−0.331744	+	0.943370i	$0.607637\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	0	0
$982$	7.12311	0.227307
$983$	27.1231	0.865093	0.432546	−	0.901612i	$-0.357615\pi$
$983$	27.1231	0.865093	0.432546	+	0.901612i	$0.357615\pi$
$984$	0	0
$985$	0	0
$986$	50.5464	1.60973
$987$	0	0
$988$	−6.68466	−0.212667
$989$	−52.1080	−1.65694
$990$	0	0
$991$	11.1231	0.353337	0.176669	−	0.984270i	$-0.443468\pi$
$991$	11.1231	0.353337	0.176669	+	0.984270i	$0.443468\pi$
$992$	−3.12311	−0.0991587
$993$	0	0
$994$	9.75379	0.309371
$995$	0	0
$996$	0	0
$997$	−32.7386	−1.03684	−0.518421	−	0.855125i	$-0.673480\pi$
$997$	−32.7386	−1.03684	−0.518421	+	0.855125i	$0.673480\pi$
$998$	32.1080	1.01636
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.br.1.1		2
3.2	odd	2		950.2.a.h.1.1		2
5.4	even	2		1710.2.a.w.1.2		2
12.11	even	2		7600.2.a.y.1.2		2
15.2	even	4		950.2.b.f.799.4		4
15.8	even	4		950.2.b.f.799.1		4
15.14	odd	2		190.2.a.d.1.2	✓	2
60.59	even	2		1520.2.a.n.1.1		2
105.104	even	2		9310.2.a.bc.1.1		2
120.29	odd	2		6080.2.a.bh.1.1		2
120.59	even	2		6080.2.a.bb.1.2		2
285.284	even	2		3610.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.d.1.2	✓	2	15.14	odd	2
950.2.a.h.1.1		2	3.2	odd	2
950.2.b.f.799.1		4	15.8	even	4
950.2.b.f.799.4		4	15.2	even	4
1520.2.a.n.1.1		2	60.59	even	2
1710.2.a.w.1.2		2	5.4	even	2
3610.2.a.t.1.1		2	285.284	even	2
6080.2.a.bb.1.2		2	120.59	even	2
6080.2.a.bh.1.1		2	120.29	odd	2
7600.2.a.y.1.2		2	12.11	even	2
8550.2.a.br.1.1		2	1.1	even	1	trivial
9310.2.a.bc.1.1		2	105.104	even	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.56155 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.56155 q^{7} -1.00000 q^{8} -4.00000 q^{11} +6.68466 q^{13} +1.56155 q^{14} +1.00000 q^{16} +7.56155 q^{17} -1.00000 q^{19} +4.00000 q^{22} -4.68466 q^{23} -6.68466 q^{26} -1.56155 q^{28} -6.68466 q^{29} +3.12311 q^{31} -1.00000 q^{32} -7.56155 q^{34} +6.00000 q^{37} +1.00000 q^{38} +4.24621 q^{41} +11.1231 q^{43} -4.00000 q^{44} +4.68466 q^{46} -10.2462 q^{47} -4.56155 q^{49} +6.68466 q^{52} -0.438447 q^{53} +1.56155 q^{56} +6.68466 q^{58} +1.56155 q^{59} +2.87689 q^{61} -3.12311 q^{62} +1.00000 q^{64} -1.56155 q^{67} +7.56155 q^{68} +6.24621 q^{71} -10.6847 q^{73} -6.00000 q^{74} -1.00000 q^{76} +6.24621 q^{77} +3.12311 q^{79} -4.24621 q^{82} +11.1231 q^{83} -11.1231 q^{86} +4.00000 q^{88} -2.00000 q^{89} -10.4384 q^{91} -4.68466 q^{92} +10.2462 q^{94} -6.00000 q^{97} +4.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8} - 8 q^{11} + q^{13} - q^{14} + 2 q^{16} + 11 q^{17} - 2 q^{19} + 8 q^{22} + 3 q^{23} - q^{26} + q^{28} - q^{29} - 2 q^{31} - 2 q^{32} - 11 q^{34} + 12 q^{37} + 2 q^{38} - 8 q^{41} + 14 q^{43} - 8 q^{44} - 3 q^{46} - 4 q^{47} - 5 q^{49} + q^{52} - 5 q^{53} - q^{56} + q^{58} - q^{59} + 14 q^{61} + 2 q^{62} + 2 q^{64} + q^{67} + 11 q^{68} - 4 q^{71} - 9 q^{73} - 12 q^{74} - 2 q^{76} - 4 q^{77} - 2 q^{79} + 8 q^{82} + 14 q^{83} - 14 q^{86} + 8 q^{88} - 4 q^{89} - 25 q^{91} + 3 q^{92} + 4 q^{94} - 12 q^{97} + 5 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8 - 8 * q^11 + q^13 - q^14 + 2 * q^16 + 11 * q^17 - 2 * q^19 + 8 * q^22 + 3 * q^23 - q^26 + q^28 - q^29 - 2 * q^31 - 2 * q^32 - 11 * q^34 + 12 * q^37 + 2 * q^38 - 8 * q^41 + 14 * q^43 - 8 * q^44 - 3 * q^46 - 4 * q^47 - 5 * q^49 + q^52 - 5 * q^53 - q^56 + q^58 - q^59 + 14 * q^61 + 2 * q^62 + 2 * q^64 + q^67 + 11 * q^68 - 4 * q^71 - 9 * q^73 - 12 * q^74 - 2 * q^76 - 4 * q^77 - 2 * q^79 + 8 * q^82 + 14 * q^83 - 14 * q^86 + 8 * q^88 - 4 * q^89 - 25 * q^91 + 3 * q^92 + 4 * q^94 - 12 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more