LMFDB - Embedded newform 8550.2.a.cp.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:	$3$
Coefficient field:	3.3.993.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 3 $ x^3 - x^2 - 6*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.480031$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -4.76957 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -4.76957 q^{7} +1.00000 q^{8} -0.960061 q^{11} -2.24960 q^{13} -4.76957 q^{14} +1.00000 q^{16} -0.249601 q^{17} +1.00000 q^{19} -0.960061 q^{22} -9.01917 q^{23} -2.24960 q^{26} -4.76957 q^{28} -6.24960 q^{29} +2.96006 q^{31} +1.00000 q^{32} -0.249601 q^{34} +0.0399387 q^{37} +1.00000 q^{38} +4.96006 q^{43} -0.960061 q^{44} -9.01917 q^{46} +9.49920 q^{47} +15.7488 q^{49} -2.24960 q^{52} -6.84945 q^{53} -4.76957 q^{56} -6.24960 q^{58} +14.5583 q^{59} +7.53914 q^{61} +2.96006 q^{62} +1.00000 q^{64} -5.72963 q^{67} -0.249601 q^{68} +9.61902 q^{71} +12.0591 q^{73} +0.0399387 q^{74} +1.00000 q^{76} +4.57908 q^{77} +6.07988 q^{79} +7.45926 q^{83} +4.96006 q^{86} -0.960061 q^{88} -4.07988 q^{89} +10.7296 q^{91} -9.01917 q^{92} +9.49920 q^{94} +18.4193 q^{97} +15.7488 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 6 q^{13} - 2 q^{14} + 3 q^{16} + 12 q^{17} + 3 q^{19} - 2 q^{22} - 2 q^{23} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 8 q^{31} + 3 q^{32} + 12 q^{34} + q^{37} + 3 q^{38} + 14 q^{43} - 2 q^{44} - 2 q^{46} + 3 q^{47} + 9 q^{49} + 6 q^{52} - 10 q^{53} - 2 q^{56} - 6 q^{58} - 6 q^{59} - 2 q^{61} + 8 q^{62} + 3 q^{64} - 4 q^{67} + 12 q^{68} + 6 q^{71} + 12 q^{73} + q^{74} + 3 q^{76} - 10 q^{77} + 20 q^{79} - 4 q^{83} + 14 q^{86} - 2 q^{88} - 14 q^{89} + 19 q^{91} - 2 q^{92} + 3 q^{94} + 28 q^{97} + 9 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 6 * q^13 - 2 * q^14 + 3 * q^16 + 12 * q^17 + 3 * q^19 - 2 * q^22 - 2 * q^23 + 6 * q^26 - 2 * q^28 - 6 * q^29 + 8 * q^31 + 3 * q^32 + 12 * q^34 + q^37 + 3 * q^38 + 14 * q^43 - 2 * q^44 - 2 * q^46 + 3 * q^47 + 9 * q^49 + 6 * q^52 - 10 * q^53 - 2 * q^56 - 6 * q^58 - 6 * q^59 - 2 * q^61 + 8 * q^62 + 3 * q^64 - 4 * q^67 + 12 * q^68 + 6 * q^71 + 12 * q^73 + q^74 + 3 * q^76 - 10 * q^77 + 20 * q^79 - 4 * q^83 + 14 * q^86 - 2 * q^88 - 14 * q^89 + 19 * q^91 - 2 * q^92 + 3 * q^94 + 28 * q^97 + 9 * 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
$7$	−4.76957	−1.80273	−0.901364	−	0.433062i	$-0.857433\pi$
$7$	−4.76957	−1.80273	−0.901364	+	0.433062i	$0.857433\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−0.960061	−0.289469	−0.144735	−	0.989471i	$-0.546233\pi$
$11$	−0.960061	−0.289469	−0.144735	+	0.989471i	$0.546233\pi$
$12$	0	0
$13$	−2.24960	−0.623927	−0.311964	−	0.950094i	$-0.600987\pi$
$13$	−2.24960	−0.623927	−0.311964	+	0.950094i	$0.600987\pi$
$14$	−4.76957	−1.27472
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.249601	−0.0605372	−0.0302686	−	0.999542i	$-0.509636\pi$
$17$	−0.249601	−0.0605372	−0.0302686	+	0.999542i	$0.509636\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−0.960061	−0.204686
$23$	−9.01917	−1.88063	−0.940314	−	0.340309i	$-0.889468\pi$
$23$	−9.01917	−1.88063	−0.940314	+	0.340309i	$0.889468\pi$
$24$	0	0
$25$	0	0
$26$	−2.24960	−0.441183
$27$	0	0
$28$	−4.76957	−0.901364
$29$	−6.24960	−1.16052	−0.580261	−	0.814431i	$-0.697049\pi$
$29$	−6.24960	−1.16052	−0.580261	+	0.814431i	$0.697049\pi$
$30$	0	0
$31$	2.96006	0.531643	0.265821	−	0.964022i	$-0.414357\pi$
$31$	2.96006	0.531643	0.265821	+	0.964022i	$0.414357\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.249601	−0.0428063
$35$	0	0
$36$	0	0
$37$	0.0399387	0.00656589	0.00328294	−	0.999995i	$-0.498955\pi$
$37$	0.0399387	0.00656589	0.00328294	+	0.999995i	$0.498955\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.96006	0.756402	0.378201	−	0.925723i	$-0.376543\pi$
$43$	4.96006	0.756402	0.378201	+	0.925723i	$0.376543\pi$
$44$	−0.960061	−0.144735
$45$	0	0
$46$	−9.01917	−1.32980
$47$	9.49920	1.38560	0.692801	−	0.721129i	$-0.256377\pi$
$47$	9.49920	1.38560	0.692801	+	0.721129i	$0.256377\pi$
$48$	0	0
$49$	15.7488	2.24983
$50$	0	0
$51$	0	0
$52$	−2.24960	−0.311964
$53$	−6.84945	−0.940844	−0.470422	−	0.882442i	$-0.655898\pi$
$53$	−6.84945	−0.940844	−0.470422	+	0.882442i	$0.655898\pi$
$54$	0	0
$55$	0	0
$56$	−4.76957	−0.637361
$57$	0	0
$58$	−6.24960	−0.820613
$59$	14.5583	1.89533	0.947665	−	0.319265i	$-0.103436\pi$
$59$	14.5583	1.89533	0.947665	+	0.319265i	$0.103436\pi$
$60$	0	0
$61$	7.53914	0.965288	0.482644	−	0.875817i	$-0.339676\pi$
$61$	7.53914	0.965288	0.482644	+	0.875817i	$0.339676\pi$
$62$	2.96006	0.375928
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−5.72963	−0.699986	−0.349993	−	0.936752i	$-0.613816\pi$
$67$	−5.72963	−0.699986	−0.349993	+	0.936752i	$0.613816\pi$
$68$	−0.249601	−0.0302686
$69$	0	0
$70$	0	0
$71$	9.61902	1.14157	0.570784	−	0.821100i	$-0.306639\pi$
$71$	9.61902	1.14157	0.570784	+	0.821100i	$0.306639\pi$
$72$	0	0
$73$	12.0591	1.41141	0.705706	−	0.708505i	$-0.250630\pi$
$73$	12.0591	1.41141	0.705706	+	0.708505i	$0.250630\pi$
$74$	0.0399387	0.00464278
$75$	0	0
$76$	1.00000	0.114708
$77$	4.57908	0.521835
$78$	0	0
$79$	6.07988	0.684040	0.342020	−	0.939693i	$-0.388889\pi$
$79$	6.07988	0.684040	0.342020	+	0.939693i	$0.388889\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.45926	0.818761	0.409380	−	0.912364i	$-0.365745\pi$
$83$	7.45926	0.818761	0.409380	+	0.912364i	$0.365745\pi$
$84$	0	0
$85$	0	0
$86$	4.96006	0.534857
$87$	0	0
$88$	−0.960061	−0.102343
$89$	−4.07988	−0.432466	−0.216233	−	0.976342i	$-0.569377\pi$
$89$	−4.07988	−0.432466	−0.216233	+	0.976342i	$0.569377\pi$
$90$	0	0
$91$	10.7296	1.12477
$92$	−9.01917	−0.940314
$93$	0	0
$94$	9.49920	0.979768
$95$	0	0
$96$	0	0
$97$	18.4193	1.87020	0.935100	−	0.354385i	$-0.115310\pi$
$97$	18.4193	1.87020	0.935100	+	0.354385i	$0.115310\pi$
$98$	15.7488	1.59087
$99$	0	0
$100$	0	0
$101$	−5.53914	−0.551165	−0.275583	−	0.961277i	$-0.588871\pi$
$101$	−5.53914	−0.551165	−0.275583	+	0.961277i	$0.588871\pi$
$102$	0	0
$103$	−6.57908	−0.648256	−0.324128	−	0.946013i	$-0.605071\pi$
$103$	−6.57908	−0.648256	−0.324128	+	0.946013i	$0.605071\pi$
$104$	−2.24960	−0.220592
$105$	0	0
$106$	−6.84945	−0.665277
$107$	−14.9086	−1.44126	−0.720632	−	0.693317i	$-0.756148\pi$
$107$	−14.9086	−1.44126	−0.720632	+	0.693317i	$0.756148\pi$
$108$	0	0
$109$	4.76957	0.456842	0.228421	−	0.973562i	$-0.426644\pi$
$109$	4.76957	0.456842	0.228421	+	0.973562i	$0.426644\pi$
$110$	0	0
$111$	0	0
$112$	−4.76957	−0.450682
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	−6.24960	−0.580261
$117$	0	0
$118$	14.5583	1.34020
$119$	1.19049	0.109132
$120$	0	0
$121$	−10.0783	−0.916207
$122$	7.53914	0.682562
$123$	0	0
$124$	2.96006	0.265821
$125$	0	0
$126$	0	0
$127$	17.9585	1.59356	0.796778	−	0.604272i	$-0.206536\pi$
$127$	17.9585	1.59356	0.796778	+	0.604272i	$0.206536\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	6.96006	0.608103	0.304052	−	0.952656i	$-0.401660\pi$
$131$	6.96006	0.608103	0.304052	+	0.952656i	$0.401660\pi$
$132$	0	0
$133$	−4.76957	−0.413574
$134$	−5.72963	−0.494965
$135$	0	0
$136$	−0.249601	−0.0214031
$137$	−7.80951	−0.667211	−0.333606	−	0.942713i	$-0.608265\pi$
$137$	−7.80951	−0.667211	−0.333606	+	0.942713i	$0.608265\pi$
$138$	0	0
$139$	−16.0383	−1.36035	−0.680177	−	0.733048i	$-0.738097\pi$
$139$	−16.0383	−1.36035	−0.680177	+	0.733048i	$0.738097\pi$
$140$	0	0
$141$	0	0
$142$	9.61902	0.807210
$143$	2.15975	0.180608
$144$	0	0
$145$	0	0
$146$	12.0591	0.998019
$147$	0	0
$148$	0.0399387	0.00328294
$149$	8.41932	0.689738	0.344869	−	0.938651i	$-0.387923\pi$
$149$	8.41932	0.689738	0.344869	+	0.938651i	$0.387923\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	4.57908	0.368993
$155$	0	0
$156$	0	0
$157$	4.96006	0.395856	0.197928	−	0.980217i	$-0.436579\pi$
$157$	4.96006	0.395856	0.197928	+	0.980217i	$0.436579\pi$
$158$	6.07988	0.483689
$159$	0	0
$160$	0	0
$161$	43.0176	3.39026
$162$	0	0
$163$	−21.4593	−1.68082	−0.840410	−	0.541952i	$-0.817686\pi$
$163$	−21.4593	−1.68082	−0.840410	+	0.541952i	$0.817686\pi$
$164$	0	0
$165$	0	0
$166$	7.45926	0.578951
$167$	11.0399	0.854296	0.427148	−	0.904182i	$-0.359518\pi$
$167$	11.0399	0.854296	0.427148	+	0.904182i	$0.359518\pi$
$168$	0	0
$169$	−7.93929	−0.610715
$170$	0	0
$171$	0	0
$172$	4.96006	0.378201
$173$	−16.9585	−1.28933	−0.644664	−	0.764466i	$-0.723003\pi$
$173$	−16.9585	−1.28933	−0.644664	+	0.764466i	$0.723003\pi$
$174$	0	0
$175$	0	0
$176$	−0.960061	−0.0723673
$177$	0	0
$178$	−4.07988	−0.305800
$179$	−2.53914	−0.189784	−0.0948922	−	0.995488i	$-0.530251\pi$
$179$	−2.53914	−0.189784	−0.0948922	+	0.995488i	$0.530251\pi$
$180$	0	0
$181$	−3.88018	−0.288412	−0.144206	−	0.989548i	$-0.546063\pi$
$181$	−3.88018	−0.288412	−0.144206	+	0.989548i	$0.546063\pi$
$182$	10.7296	0.795333
$183$	0	0
$184$	−9.01917	−0.664902
$185$	0	0
$186$	0	0
$187$	0.239632	0.0175237
$188$	9.49920	0.692801
$189$	0	0
$190$	0	0
$191$	−2.05911	−0.148992	−0.0744960	−	0.997221i	$-0.523735\pi$
$191$	−2.05911	−0.148992	−0.0744960	+	0.997221i	$0.523735\pi$
$192$	0	0
$193$	−8.46086	−0.609026	−0.304513	−	0.952508i	$-0.598494\pi$
$193$	−8.46086	−0.609026	−0.304513	+	0.952508i	$0.598494\pi$
$194$	18.4193	1.32243
$195$	0	0
$196$	15.7488	1.12491
$197$	−13.9201	−0.991768	−0.495884	−	0.868389i	$-0.665156\pi$
$197$	−13.9201	−0.991768	−0.495884	+	0.868389i	$0.665156\pi$
$198$	0	0
$199$	−9.15055	−0.648665	−0.324333	−	0.945943i	$-0.605140\pi$
$199$	−9.15055	−0.648665	−0.324333	+	0.945943i	$0.605140\pi$
$200$	0	0
$201$	0	0
$202$	−5.53914	−0.389733
$203$	29.8079	2.09211
$204$	0	0
$205$	0	0
$206$	−6.57908	−0.458386
$207$	0	0
$208$	−2.24960	−0.155982
$209$	−0.960061	−0.0664088
$210$	0	0
$211$	16.5200	1.13728	0.568641	−	0.822586i	$-0.307469\pi$
$211$	16.5200	1.13728	0.568641	+	0.822586i	$0.307469\pi$
$212$	−6.84945	−0.470422
$213$	0	0
$214$	−14.9086	−1.01913
$215$	0	0
$216$	0	0
$217$	−14.1182	−0.958407
$218$	4.76957	0.323036
$219$	0	0
$220$	0	0
$221$	0.561503	0.0377708
$222$	0	0
$223$	9.03994	0.605359	0.302680	−	0.953092i	$-0.402119\pi$
$223$	9.03994	0.605359	0.302680	+	0.953092i	$0.402119\pi$
$224$	−4.76957	−0.318680
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−0.350246	−0.0232466	−0.0116233	−	0.999932i	$-0.503700\pi$
$227$	−0.350246	−0.0232466	−0.0116233	+	0.999932i	$0.503700\pi$
$228$	0	0
$229$	−8.07988	−0.533933	−0.266967	−	0.963706i	$-0.586021\pi$
$229$	−8.07988	−0.533933	−0.266967	+	0.963706i	$0.586021\pi$
$230$	0	0
$231$	0	0
$232$	−6.24960	−0.410306
$233$	4.92012	0.322328	0.161164	−	0.986928i	$-0.448475\pi$
$233$	4.92012	0.322328	0.161164	+	0.986928i	$0.448475\pi$
$234$	0	0
$235$	0	0
$236$	14.5583	0.947665
$237$	0	0
$238$	1.19049	0.0771680
$239$	−2.98083	−0.192814	−0.0964069	−	0.995342i	$-0.530735\pi$
$239$	−2.98083	−0.192814	−0.0964069	+	0.995342i	$0.530735\pi$
$240$	0	0
$241$	20.0383	1.29078	0.645392	−	0.763852i	$-0.276694\pi$
$241$	20.0383	1.29078	0.645392	+	0.763852i	$0.276694\pi$
$242$	−10.0783	−0.647857
$243$	0	0
$244$	7.53914	0.482644
$245$	0	0
$246$	0	0
$247$	−2.24960	−0.143139
$248$	2.96006	0.187964
$249$	0	0
$250$	0	0
$251$	14.8802	0.939229	0.469614	−	0.882872i	$-0.344393\pi$
$251$	14.8802	0.939229	0.469614	+	0.882872i	$0.344393\pi$
$252$	0	0
$253$	8.65896	0.544384
$254$	17.9585	1.12681
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.0399	0.688652	0.344326	−	0.938850i	$-0.388107\pi$
$257$	11.0399	0.688652	0.344326	+	0.938850i	$0.388107\pi$
$258$	0	0
$259$	−0.190491	−0.0118365
$260$	0	0
$261$	0	0
$262$	6.96006	0.429994
$263$	3.84025	0.236800	0.118400	−	0.992966i	$-0.462224\pi$
$263$	3.84025	0.236800	0.118400	+	0.992966i	$0.462224\pi$
$264$	0	0
$265$	0	0
$266$	−4.76957	−0.292441
$267$	0	0
$268$	−5.72963	−0.349993
$269$	−23.1981	−1.41441	−0.707207	−	0.707007i	$-0.750045\pi$
$269$	−23.1981	−1.41441	−0.707207	+	0.707007i	$0.750045\pi$
$270$	0	0
$271$	23.9792	1.45663	0.728317	−	0.685240i	$-0.240303\pi$
$271$	23.9792	1.45663	0.728317	+	0.685240i	$0.240303\pi$
$272$	−0.249601	−0.0151343
$273$	0	0
$274$	−7.80951	−0.471790
$275$	0	0
$276$	0	0
$277$	24.6590	1.48161	0.740807	−	0.671718i	$-0.234444\pi$
$277$	24.6590	1.48161	0.740807	+	0.671718i	$0.234444\pi$
$278$	−16.0383	−0.961916
$279$	0	0
$280$	0	0
$281$	18.9984	1.13335	0.566675	−	0.823941i	$-0.308230\pi$
$281$	18.9984	1.13335	0.566675	+	0.823941i	$0.308230\pi$
$282$	0	0
$283$	29.0367	1.72606	0.863028	−	0.505156i	$-0.168565\pi$
$283$	29.0367	1.72606	0.863028	+	0.505156i	$0.168565\pi$
$284$	9.61902	0.570784
$285$	0	0
$286$	2.15975	0.127709
$287$	0	0
$288$	0	0
$289$	−16.9377	−0.996335
$290$	0	0
$291$	0	0
$292$	12.0591	0.705706
$293$	−6.24960	−0.365106	−0.182553	−	0.983196i	$-0.558436\pi$
$293$	−6.24960	−0.365106	−0.182553	+	0.983196i	$0.558436\pi$
$294$	0	0
$295$	0	0
$296$	0.0399387	0.00232139
$297$	0	0
$298$	8.41932	0.487718
$299$	20.2895	1.17337
$300$	0	0
$301$	−23.6574	−1.36359
$302$	20.0000	1.15087
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	12.5391	0.715647	0.357823	−	0.933789i	$-0.383519\pi$
$307$	12.5391	0.715647	0.357823	+	0.933789i	$0.383519\pi$
$308$	4.57908	0.260917
$309$	0	0
$310$	0	0
$311$	−7.84785	−0.445011	−0.222505	−	0.974931i	$-0.571424\pi$
$311$	−7.84785	−0.445011	−0.222505	+	0.974931i	$0.571424\pi$
$312$	0	0
$313$	10.9884	0.621103	0.310552	−	0.950557i	$-0.399486\pi$
$313$	10.9884	0.621103	0.310552	+	0.950557i	$0.399486\pi$
$314$	4.96006	0.279912
$315$	0	0
$316$	6.07988	0.342020
$317$	21.5567	1.21075	0.605373	−	0.795942i	$-0.293024\pi$
$317$	21.5567	1.21075	0.605373	+	0.795942i	$0.293024\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	43.0176	2.39728
$323$	−0.249601	−0.0138882
$324$	0	0
$325$	0	0
$326$	−21.4593	−1.18852
$327$	0	0
$328$	0	0
$329$	−45.3071	−2.49786
$330$	0	0
$331$	33.4876	1.84065	0.920324	−	0.391158i	$-0.127925\pi$
$331$	33.4876	1.84065	0.920324	+	0.391158i	$0.127925\pi$
$332$	7.45926	0.409380
$333$	0	0
$334$	11.0399	0.604079
$335$	0	0
$336$	0	0
$337$	5.69890	0.310439	0.155219	−	0.987880i	$-0.450392\pi$
$337$	5.69890	0.310439	0.155219	+	0.987880i	$0.450392\pi$
$338$	−7.93929	−0.431841
$339$	0	0
$340$	0	0
$341$	−2.84184	−0.153894
$342$	0	0
$343$	−41.7280	−2.25310
$344$	4.96006	0.267429
$345$	0	0
$346$	−16.9585	−0.911693
$347$	0.239632	0.0128641	0.00643207	−	0.999979i	$-0.497953\pi$
$347$	0.239632	0.0128641	0.00643207	+	0.999979i	$0.497953\pi$
$348$	0	0
$349$	−17.2380	−0.922731	−0.461365	−	0.887210i	$-0.652640\pi$
$349$	−17.2380	−0.922731	−0.461365	+	0.887210i	$0.652640\pi$
$350$	0	0
$351$	0	0
$352$	−0.960061	−0.0511714
$353$	34.7280	1.84839	0.924193	−	0.381925i	$-0.124739\pi$
$353$	34.7280	1.84839	0.924193	+	0.381925i	$0.124739\pi$
$354$	0	0
$355$	0	0
$356$	−4.07988	−0.216233
$357$	0	0
$358$	−2.53914	−0.134198
$359$	−26.6689	−1.40753	−0.703766	−	0.710432i	$-0.748500\pi$
$359$	−26.6689	−1.40753	−0.703766	+	0.710432i	$0.748500\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−3.88018	−0.203938
$363$	0	0
$364$	10.7296	0.562386
$365$	0	0
$366$	0	0
$367$	−15.2380	−0.795419	−0.397710	−	0.917511i	$-0.630195\pi$
$367$	−15.2380	−0.795419	−0.397710	+	0.917511i	$0.630195\pi$
$368$	−9.01917	−0.470157
$369$	0	0
$370$	0	0
$371$	32.6689	1.69609
$372$	0	0
$373$	−26.6382	−1.37927	−0.689637	−	0.724156i	$-0.742230\pi$
$373$	−26.6382	−1.37927	−0.689637	+	0.724156i	$0.742230\pi$
$374$	0.239632	0.0123911
$375$	0	0
$376$	9.49920	0.489884
$377$	14.0591	0.724081
$378$	0	0
$379$	11.9693	0.614820	0.307410	−	0.951577i	$-0.400538\pi$
$379$	11.9693	0.614820	0.307410	+	0.951577i	$0.400538\pi$
$380$	0	0
$381$	0	0
$382$	−2.05911	−0.105353
$383$	9.84025	0.502813	0.251407	−	0.967882i	$-0.419107\pi$
$383$	9.84025	0.502813	0.251407	+	0.967882i	$0.419107\pi$
$384$	0	0
$385$	0	0
$386$	−8.46086	−0.430646
$387$	0	0
$388$	18.4193	0.935100
$389$	8.65896	0.439027	0.219513	−	0.975610i	$-0.429553\pi$
$389$	8.65896	0.439027	0.219513	+	0.975610i	$0.429553\pi$
$390$	0	0
$391$	2.25120	0.113848
$392$	15.7488	0.795435
$393$	0	0
$394$	−13.9201	−0.701286
$395$	0	0
$396$	0	0
$397$	28.9984	1.45539	0.727694	−	0.685902i	$-0.240592\pi$
$397$	28.9984	1.45539	0.727694	+	0.685902i	$0.240592\pi$
$398$	−9.15055	−0.458676
$399$	0	0
$400$	0	0
$401$	−22.6174	−1.12946	−0.564730	−	0.825276i	$-0.691020\pi$
$401$	−22.6174	−1.12946	−0.564730	+	0.825276i	$0.691020\pi$
$402$	0	0
$403$	−6.65896	−0.331706
$404$	−5.53914	−0.275583
$405$	0	0
$406$	29.8079	1.47934
$407$	−0.0383436	−0.00190062
$408$	0	0
$409$	5.34104	0.264098	0.132049	−	0.991243i	$-0.457844\pi$
$409$	5.34104	0.264098	0.132049	+	0.991243i	$0.457844\pi$
$410$	0	0
$411$	0	0
$412$	−6.57908	−0.324128
$413$	−69.4369	−3.41677
$414$	0	0
$415$	0	0
$416$	−2.24960	−0.110296
$417$	0	0
$418$	−0.960061	−0.0469581
$419$	22.1566	1.08242	0.541210	−	0.840888i	$-0.317967\pi$
$419$	22.1566	1.08242	0.541210	+	0.840888i	$0.317967\pi$
$420$	0	0
$421$	29.2787	1.42696	0.713479	−	0.700676i	$-0.247118\pi$
$421$	29.2787	1.42696	0.713479	+	0.700676i	$0.247118\pi$
$422$	16.5200	0.804180
$423$	0	0
$424$	−6.84945	−0.332639
$425$	0	0
$426$	0	0
$427$	−35.9585	−1.74015
$428$	−14.9086	−0.720632
$429$	0	0
$430$	0	0
$431$	−21.3794	−1.02981	−0.514904	−	0.857248i	$-0.672173\pi$
$431$	−21.3794	−1.02981	−0.514904	+	0.857248i	$0.672173\pi$
$432$	0	0
$433$	−36.1182	−1.73573	−0.867865	−	0.496799i	$-0.834509\pi$
$433$	−36.1182	−1.73573	−0.867865	+	0.496799i	$0.834509\pi$
$434$	−14.1182	−0.677696
$435$	0	0
$436$	4.76957	0.228421
$437$	−9.01917	−0.431445
$438$	0	0
$439$	20.9984	1.00220	0.501100	−	0.865390i	$-0.332929\pi$
$439$	20.9984	1.00220	0.501100	+	0.865390i	$0.332929\pi$
$440$	0	0
$441$	0	0
$442$	0.561503	0.0267080
$443$	31.4976	1.49650	0.748248	−	0.663419i	$-0.230895\pi$
$443$	31.4976	1.49650	0.748248	+	0.663419i	$0.230895\pi$
$444$	0	0
$445$	0	0
$446$	9.03994	0.428054
$447$	0	0
$448$	−4.76957	−0.225341
$449$	−18.9601	−0.894781	−0.447390	−	0.894339i	$-0.647647\pi$
$449$	−18.9601	−0.894781	−0.447390	+	0.894339i	$0.647647\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−0.350246	−0.0164378
$455$	0	0
$456$	0	0
$457$	−6.32948	−0.296081	−0.148040	−	0.988981i	$-0.547297\pi$
$457$	−6.32948	−0.296081	−0.148040	+	0.988981i	$0.547297\pi$
$458$	−8.07988	−0.377548
$459$	0	0
$460$	0	0
$461$	6.49920	0.302698	0.151349	−	0.988480i	$-0.451638\pi$
$461$	6.49920	0.302698	0.151349	+	0.988480i	$0.451638\pi$
$462$	0	0
$463$	28.4177	1.32068	0.660342	−	0.750965i	$-0.270411\pi$
$463$	28.4177	1.32068	0.660342	+	0.750965i	$0.270411\pi$
$464$	−6.24960	−0.290130
$465$	0	0
$466$	4.92012	0.227920
$467$	−17.5775	−0.813389	−0.406694	−	0.913564i	$-0.633319\pi$
$467$	−17.5775	−0.813389	−0.406694	+	0.913564i	$0.633319\pi$
$468$	0	0
$469$	27.3279	1.26188
$470$	0	0
$471$	0	0
$472$	14.5583	0.670101
$473$	−4.76196	−0.218955
$474$	0	0
$475$	0	0
$476$	1.19049	0.0545660
$477$	0	0
$478$	−2.98083	−0.136340
$479$	−22.7372	−1.03889	−0.519446	−	0.854504i	$-0.673861\pi$
$479$	−22.7372	−1.03889	−0.519446	+	0.854504i	$0.673861\pi$
$480$	0	0
$481$	−0.0898462	−0.00409664
$482$	20.0383	0.912722
$483$	0	0
$484$	−10.0783	−0.458104
$485$	0	0
$486$	0	0
$487$	−22.1981	−1.00589	−0.502946	−	0.864318i	$-0.667751\pi$
$487$	−22.1981	−1.00589	−0.502946	+	0.864318i	$0.667751\pi$
$488$	7.53914	0.341281
$489$	0	0
$490$	0	0
$491$	24.9984	1.12816	0.564081	−	0.825719i	$-0.309231\pi$
$491$	24.9984	1.12816	0.564081	+	0.825719i	$0.309231\pi$
$492$	0	0
$493$	1.55991	0.0702547
$494$	−2.24960	−0.101214
$495$	0	0
$496$	2.96006	0.132911
$497$	−45.8786	−2.05794
$498$	0	0
$499$	11.6574	0.521855	0.260928	−	0.965358i	$-0.415972\pi$
$499$	11.6574	0.521855	0.260928	+	0.965358i	$0.415972\pi$
$500$	0	0
$501$	0	0
$502$	14.8802	0.664135
$503$	−11.7504	−0.523924	−0.261962	−	0.965078i	$-0.584370\pi$
$503$	−11.7504	−0.523924	−0.261962	+	0.965078i	$0.584370\pi$
$504$	0	0
$505$	0	0
$506$	8.65896	0.384938
$507$	0	0
$508$	17.9585	0.796778
$509$	−29.9569	−1.32781	−0.663907	−	0.747815i	$-0.731103\pi$
$509$	−29.9569	−1.32781	−0.663907	+	0.747815i	$0.731103\pi$
$510$	0	0
$511$	−57.5168	−2.54439
$512$	1.00000	0.0441942
$513$	0	0
$514$	11.0399	0.486951
$515$	0	0
$516$	0	0
$517$	−9.11982	−0.401089
$518$	−0.190491	−0.00836968
$519$	0	0
$520$	0	0
$521$	15.1198	0.662411	0.331206	−	0.943559i	$-0.392545\pi$
$521$	15.1198	0.662411	0.331206	+	0.943559i	$0.392545\pi$
$522$	0	0
$523$	15.6498	0.684316	0.342158	−	0.939642i	$-0.388842\pi$
$523$	15.6498	0.684316	0.342158	+	0.939642i	$0.388842\pi$
$524$	6.96006	0.304052
$525$	0	0
$526$	3.84025	0.167443
$527$	−0.738835	−0.0321842
$528$	0	0
$529$	58.3455	2.53676
$530$	0	0
$531$	0	0
$532$	−4.76957	−0.206787
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−5.72963	−0.247482
$537$	0	0
$538$	−23.1981	−1.00014
$539$	−15.1198	−0.651257
$540$	0	0
$541$	−27.2995	−1.17370	−0.586849	−	0.809697i	$-0.699632\pi$
$541$	−27.2995	−1.17370	−0.586849	+	0.809697i	$0.699632\pi$
$542$	23.9792	1.03000
$543$	0	0
$544$	−0.249601	−0.0107016
$545$	0	0
$546$	0	0
$547$	−33.9968	−1.45360	−0.726799	−	0.686850i	$-0.758993\pi$
$547$	−33.9968	−1.45360	−0.726799	+	0.686850i	$0.758993\pi$
$548$	−7.80951	−0.333606
$549$	0	0
$550$	0	0
$551$	−6.24960	−0.266242
$552$	0	0
$553$	−28.9984	−1.23314
$554$	24.6590	1.04766
$555$	0	0
$556$	−16.0383	−0.680177
$557$	−33.8786	−1.43548	−0.717741	−	0.696310i	$-0.754824\pi$
$557$	−33.8786	−1.43548	−0.717741	+	0.696310i	$0.754824\pi$
$558$	0	0
$559$	−11.1582	−0.471940
$560$	0	0
$561$	0	0
$562$	18.9984	0.801399
$563$	−32.2995	−1.36126	−0.680631	−	0.732626i	$-0.738294\pi$
$563$	−32.2995	−1.36126	−0.680631	+	0.732626i	$0.738294\pi$
$564$	0	0
$565$	0	0
$566$	29.0367	1.22051
$567$	0	0
$568$	9.61902	0.403605
$569$	6.73883	0.282507	0.141253	−	0.989973i	$-0.454887\pi$
$569$	6.73883	0.282507	0.141253	+	0.989973i	$0.454887\pi$
$570$	0	0
$571$	34.4193	1.44040	0.720202	−	0.693764i	$-0.244049\pi$
$571$	34.4193	1.44040	0.720202	+	0.693764i	$0.244049\pi$
$572$	2.15975	0.0903039
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.1773	1.17304	0.586519	−	0.809936i	$-0.300498\pi$
$577$	28.1773	1.17304	0.586519	+	0.809936i	$0.300498\pi$
$578$	−16.9377	−0.704515
$579$	0	0
$580$	0	0
$581$	−35.5775	−1.47600
$582$	0	0
$583$	6.57589	0.272346
$584$	12.0591	0.499010
$585$	0	0
$586$	−6.24960	−0.258169
$587$	34.6174	1.42881	0.714407	−	0.699730i	$-0.246697\pi$
$587$	34.6174	1.42881	0.714407	+	0.699730i	$0.246697\pi$
$588$	0	0
$589$	2.96006	0.121967
$590$	0	0
$591$	0	0
$592$	0.0399387	0.00164147
$593$	3.99840	0.164195	0.0820974	−	0.996624i	$-0.473838\pi$
$593$	3.99840	0.164195	0.0820974	+	0.996624i	$0.473838\pi$
$594$	0	0
$595$	0	0
$596$	8.41932	0.344869
$597$	0	0
$598$	20.2895	0.829701
$599$	42.5759	1.73960	0.869802	−	0.493401i	$-0.164247\pi$
$599$	42.5759	1.73960	0.869802	+	0.493401i	$0.164247\pi$
$600$	0	0
$601$	−18.4577	−0.752904	−0.376452	−	0.926436i	$-0.622856\pi$
$601$	−18.4577	−0.752904	−0.376452	+	0.926436i	$0.622856\pi$
$602$	−23.6574	−0.964202
$603$	0	0
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	−12.5791	−0.510569	−0.255285	−	0.966866i	$-0.582169\pi$
$607$	−12.5791	−0.510569	−0.255285	+	0.966866i	$0.582169\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−21.3694	−0.864514
$612$	0	0
$613$	−26.5759	−1.07339	−0.536695	−	0.843776i	$-0.680327\pi$
$613$	−26.5759	−1.07339	−0.536695	+	0.843776i	$0.680327\pi$
$614$	12.5391	0.506039
$615$	0	0
$616$	4.57908	0.184496
$617$	37.8386	1.52333	0.761663	−	0.647973i	$-0.224383\pi$
$617$	37.8386	1.52333	0.761663	+	0.647973i	$0.224383\pi$
$618$	0	0
$619$	30.1566	1.21209	0.606047	−	0.795429i	$-0.292754\pi$
$619$	30.1566	1.21209	0.606047	+	0.795429i	$0.292754\pi$
$620$	0	0
$621$	0	0
$622$	−7.84785	−0.314670
$623$	19.4593	0.779619
$624$	0	0
$625$	0	0
$626$	10.9884	0.439186
$627$	0	0
$628$	4.96006	0.197928
$629$	−0.00996876	−0.000397480 0
$630$	0	0
$631$	−25.4992	−1.01511	−0.507554	−	0.861620i	$-0.669450\pi$
$631$	−25.4992	−1.01511	−0.507554	+	0.861620i	$0.669450\pi$
$632$	6.07988	0.241845
$633$	0	0
$634$	21.5567	0.856127
$635$	0	0
$636$	0	0
$637$	−35.4285	−1.40373
$638$	6.00000	0.237542
$639$	0	0
$640$	0	0
$641$	−39.1965	−1.54817	−0.774084	−	0.633082i	$-0.781789\pi$
$641$	−39.1965	−1.54817	−0.774084	+	0.633082i	$0.781789\pi$
$642$	0	0
$643$	−36.3778	−1.43460	−0.717300	−	0.696764i	$-0.754622\pi$
$643$	−36.3778	−1.43460	−0.717300	+	0.696764i	$0.754622\pi$
$644$	43.0176	1.69513
$645$	0	0
$646$	−0.249601	−0.00982043
$647$	10.6897	0.420255	0.210128	−	0.977674i	$-0.432612\pi$
$647$	10.6897	0.420255	0.210128	+	0.977674i	$0.432612\pi$
$648$	0	0
$649$	−13.9769	−0.548640
$650$	0	0
$651$	0	0
$652$	−21.4593	−0.840410
$653$	−36.0383	−1.41029	−0.705145	−	0.709063i	$-0.749118\pi$
$653$	−36.0383	−1.41029	−0.705145	+	0.709063i	$0.749118\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−45.3071	−1.76626
$659$	−17.1889	−0.669584	−0.334792	−	0.942292i	$-0.608666\pi$
$659$	−17.1889	−0.669584	−0.334792	+	0.942292i	$0.608666\pi$
$660$	0	0
$661$	48.9054	1.90220	0.951099	−	0.308886i	$-0.0999560\pi$
$661$	48.9054	1.90220	0.951099	+	0.308886i	$0.0999560\pi$
$662$	33.4876	1.30153
$663$	0	0
$664$	7.45926	0.289476
$665$	0	0
$666$	0	0
$667$	56.3662	2.18251
$668$	11.0399	0.427148
$669$	0	0
$670$	0	0
$671$	−7.23804	−0.279421
$672$	0	0
$673$	−6.57908	−0.253605	−0.126802	−	0.991928i	$-0.540471\pi$
$673$	−6.57908	−0.253605	−0.126802	+	0.991928i	$0.540471\pi$
$674$	5.69890	0.219513
$675$	0	0
$676$	−7.93929	−0.305357
$677$	−38.3087	−1.47232	−0.736162	−	0.676806i	$-0.763364\pi$
$677$	−38.3087	−1.47232	−0.736162	+	0.676806i	$0.763364\pi$
$678$	0	0
$679$	−87.8523	−3.37146
$680$	0	0
$681$	0	0
$682$	−2.84184	−0.108820
$683$	7.54074	0.288538	0.144269	−	0.989538i	$-0.453917\pi$
$683$	7.54074	0.288538	0.144269	+	0.989538i	$0.453917\pi$
$684$	0	0
$685$	0	0
$686$	−41.7280	−1.59318
$687$	0	0
$688$	4.96006	0.189101
$689$	15.4085	0.587018
$690$	0	0
$691$	28.4193	1.08112	0.540561	−	0.841305i	$-0.318212\pi$
$691$	28.4193	1.08112	0.540561	+	0.841305i	$0.318212\pi$
$692$	−16.9585	−0.644664
$693$	0	0
$694$	0.239632	0.00909632
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−17.2380	−0.652469
$699$	0	0
$700$	0	0
$701$	24.2596	0.916271	0.458136	−	0.888882i	$-0.348517\pi$
$701$	24.2596	0.916271	0.458136	+	0.888882i	$0.348517\pi$
$702$	0	0
$703$	0.0399387	0.00150632
$704$	−0.960061	−0.0361837
$705$	0	0
$706$	34.7280	1.30701
$707$	26.4193	0.993601
$708$	0	0
$709$	21.7372	0.816359	0.408180	−	0.912902i	$-0.366164\pi$
$709$	21.7372	0.816359	0.408180	+	0.912902i	$0.366164\pi$
$710$	0	0
$711$	0	0
$712$	−4.07988	−0.152900
$713$	−26.6973	−0.999822
$714$	0	0
$715$	0	0
$716$	−2.53914	−0.0948922
$717$	0	0
$718$	−26.6689	−0.995275
$719$	48.9361	1.82501	0.912504	−	0.409067i	$-0.134146\pi$
$719$	48.9361	1.82501	0.912504	+	0.409067i	$0.134146\pi$
$720$	0	0
$721$	31.3794	1.16863
$722$	1.00000	0.0372161
$723$	0	0
$724$	−3.88018	−0.144206
$725$	0	0
$726$	0	0
$727$	−4.29874	−0.159432	−0.0797158	−	0.996818i	$-0.525401\pi$
$727$	−4.29874	−0.159432	−0.0797158	+	0.996818i	$0.525401\pi$
$728$	10.7296	0.397667
$729$	0	0
$730$	0	0
$731$	−1.23804	−0.0457905
$732$	0	0
$733$	−12.0415	−0.444764	−0.222382	−	0.974960i	$-0.571383\pi$
$733$	−12.0415	−0.444764	−0.222382	+	0.974960i	$0.571383\pi$
$734$	−15.2380	−0.562446
$735$	0	0
$736$	−9.01917	−0.332451
$737$	5.50080	0.202624
$738$	0	0
$739$	6.61742	0.243426	0.121713	−	0.992565i	$-0.461161\pi$
$739$	6.61742	0.243426	0.121713	+	0.992565i	$0.461161\pi$
$740$	0	0
$741$	0	0
$742$	32.6689	1.19931
$743$	47.6158	1.74686	0.873428	−	0.486954i	$-0.161892\pi$
$743$	47.6158	1.74686	0.873428	+	0.486954i	$0.161892\pi$
$744$	0	0
$745$	0	0
$746$	−26.6382	−0.975293
$747$	0	0
$748$	0.239632	0.00876183
$749$	71.1074	2.59821
$750$	0	0
$751$	−25.6574	−0.936250	−0.468125	−	0.883662i	$-0.655070\pi$
$751$	−25.6574	−0.936250	−0.468125	+	0.883662i	$0.655070\pi$
$752$	9.49920	0.346400
$753$	0	0
$754$	14.0591	0.512003
$755$	0	0
$756$	0	0
$757$	8.83865	0.321246	0.160623	−	0.987016i	$-0.448650\pi$
$757$	8.83865	0.321246	0.160623	+	0.987016i	$0.448650\pi$
$758$	11.9693	0.434743
$759$	0	0
$760$	0	0
$761$	9.40776	0.341031	0.170516	−	0.985355i	$-0.445457\pi$
$761$	9.40776	0.341031	0.170516	+	0.985355i	$0.445457\pi$
$762$	0	0
$763$	−22.7488	−0.823562
$764$	−2.05911	−0.0744960
$765$	0	0
$766$	9.84025	0.355543
$767$	−32.7504	−1.18255
$768$	0	0
$769$	7.32788	0.264250	0.132125	−	0.991233i	$-0.457820\pi$
$769$	7.32788	0.264250	0.132125	+	0.991233i	$0.457820\pi$
$770$	0	0
$771$	0	0
$772$	−8.46086	−0.304513
$773$	−54.4369	−1.95796	−0.978980	−	0.203958i	$-0.934619\pi$
$773$	−54.4369	−1.95796	−0.978980	+	0.203958i	$0.934619\pi$
$774$	0	0
$775$	0	0
$776$	18.4193	0.661215
$777$	0	0
$778$	8.65896	0.310439
$779$	0	0
$780$	0	0
$781$	−9.23485	−0.330449
$782$	2.25120	0.0805026
$783$	0	0
$784$	15.7488	0.562457
$785$	0	0
$786$	0	0
$787$	−27.6705	−0.986348	−0.493174	−	0.869931i	$-0.664163\pi$
$787$	−27.6705	−0.986348	−0.493174	+	0.869931i	$0.664163\pi$
$788$	−13.9201	−0.495884
$789$	0	0
$790$	0	0
$791$	28.6174	1.01752
$792$	0	0
$793$	−16.9601	−0.602269
$794$	28.9984	1.02911
$795$	0	0
$796$	−9.15055	−0.324333
$797$	−21.5906	−0.764780	−0.382390	−	0.924001i	$-0.624899\pi$
$797$	−21.5906	−0.764780	−0.382390	+	0.924001i	$0.624899\pi$
$798$	0	0
$799$	−2.37101	−0.0838804
$800$	0	0
$801$	0	0
$802$	−22.6174	−0.798649
$803$	−11.5775	−0.408561
$804$	0	0
$805$	0	0
$806$	−6.65896	−0.234552
$807$	0	0
$808$	−5.53914	−0.194866
$809$	−9.72963	−0.342076	−0.171038	−	0.985264i	$-0.554712\pi$
$809$	−9.72963	−0.342076	−0.171038	+	0.985264i	$0.554712\pi$
$810$	0	0
$811$	−38.7280	−1.35993	−0.679963	−	0.733247i	$-0.738004\pi$
$811$	−38.7280	−1.35993	−0.679963	+	0.733247i	$0.738004\pi$
$812$	29.8079	1.04605
$813$	0	0
$814$	−0.0383436	−0.00134394
$815$	0	0
$816$	0	0
$817$	4.96006	0.173531
$818$	5.34104	0.186745
$819$	0	0
$820$	0	0
$821$	30.2780	1.05671	0.528354	−	0.849024i	$-0.322809\pi$
$821$	30.2780	1.05671	0.528354	+	0.849024i	$0.322809\pi$
$822$	0	0
$823$	−7.93770	−0.276691	−0.138345	−	0.990384i	$-0.544178\pi$
$823$	−7.93770	−0.276691	−0.138345	+	0.990384i	$0.544178\pi$
$824$	−6.57908	−0.229193
$825$	0	0
$826$	−69.4369	−2.41602
$827$	−30.2496	−1.05188	−0.525941	−	0.850521i	$-0.676287\pi$
$827$	−30.2496	−1.05188	−0.525941	+	0.850521i	$0.676287\pi$
$828$	0	0
$829$	−40.6765	−1.41275	−0.706377	−	0.707836i	$-0.749672\pi$
$829$	−40.6765	−1.41275	−0.706377	+	0.707836i	$0.749672\pi$
$830$	0	0
$831$	0	0
$832$	−2.24960	−0.0779909
$833$	−3.93092	−0.136198
$834$	0	0
$835$	0	0
$836$	−0.960061	−0.0332044
$837$	0	0
$838$	22.1566	0.765386
$839$	−45.9553	−1.58655	−0.793276	−	0.608862i	$-0.791626\pi$
$839$	−45.9553	−1.58655	−0.793276	+	0.608862i	$0.791626\pi$
$840$	0	0
$841$	10.0575	0.346811
$842$	29.2787	1.00901
$843$	0	0
$844$	16.5200	0.568641
$845$	0	0
$846$	0	0
$847$	48.0691	1.65167
$848$	−6.84945	−0.235211
$849$	0	0
$850$	0	0
$851$	−0.360214	−0.0123480
$852$	0	0
$853$	17.2196	0.589589	0.294794	−	0.955561i	$-0.404749\pi$
$853$	17.2196	0.589589	0.294794	+	0.955561i	$0.404749\pi$
$854$	−35.9585	−1.23047
$855$	0	0
$856$	−14.9086	−0.509564
$857$	−17.2995	−0.590940	−0.295470	−	0.955352i	$-0.595476\pi$
$857$	−17.2995	−0.590940	−0.295470	+	0.955352i	$0.595476\pi$
$858$	0	0
$859$	17.6190	0.601153	0.300577	−	0.953758i	$-0.402821\pi$
$859$	17.6190	0.601153	0.300577	+	0.953758i	$0.402821\pi$
$860$	0	0
$861$	0	0
$862$	−21.3794	−0.728185
$863$	4.07988	0.138881	0.0694403	−	0.997586i	$-0.477879\pi$
$863$	4.07988	0.138881	0.0694403	+	0.997586i	$0.477879\pi$
$864$	0	0
$865$	0	0
$866$	−36.1182	−1.22735
$867$	0	0
$868$	−14.1182	−0.479204
$869$	−5.83705	−0.198009
$870$	0	0
$871$	12.8894	0.436740
$872$	4.76957	0.161518
$873$	0	0
$874$	−9.01917	−0.305078
$875$	0	0
$876$	0	0
$877$	25.2787	0.853602	0.426801	−	0.904345i	$-0.359640\pi$
$877$	25.2787	0.853602	0.426801	+	0.904345i	$0.359640\pi$
$878$	20.9984	0.708662
$879$	0	0
$880$	0	0
$881$	0.0814726	0.00274488	0.00137244	−	0.999999i	$-0.499563\pi$
$881$	0.0814726	0.00274488	0.00137244	+	0.999999i	$0.499563\pi$
$882$	0	0
$883$	19.3410	0.650878	0.325439	−	0.945563i	$-0.394488\pi$
$883$	19.3410	0.650878	0.325439	+	0.945563i	$0.394488\pi$
$884$	0.561503	0.0188854
$885$	0	0
$886$	31.4976	1.05818
$887$	−5.53914	−0.185986	−0.0929931	−	0.995667i	$-0.529643\pi$
$887$	−5.53914	−0.185986	−0.0929931	+	0.995667i	$0.529643\pi$
$888$	0	0
$889$	−85.6542	−2.87275
$890$	0	0
$891$	0	0
$892$	9.03994	0.302680
$893$	9.49920	0.317879
$894$	0	0
$895$	0	0
$896$	−4.76957	−0.159340
$897$	0	0
$898$	−18.9601	−0.632705
$899$	−18.4992	−0.616983
$900$	0	0
$901$	1.70963	0.0569561
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	10.8111	0.358977	0.179488	−	0.983760i	$-0.442556\pi$
$907$	10.8111	0.358977	0.179488	+	0.983760i	$0.442556\pi$
$908$	−0.350246	−0.0116233
$909$	0	0
$910$	0	0
$911$	−35.1166	−1.16347	−0.581733	−	0.813380i	$-0.697625\pi$
$911$	−35.1166	−1.16347	−0.581733	+	0.813380i	$0.697625\pi$
$912$	0	0
$913$	−7.16135	−0.237006
$914$	−6.32948	−0.209361
$915$	0	0
$916$	−8.07988	−0.266967
$917$	−33.1965	−1.09625
$918$	0	0
$919$	30.8287	1.01694	0.508472	−	0.861078i	$-0.330210\pi$
$919$	30.8287	1.01694	0.508472	+	0.861078i	$0.330210\pi$
$920$	0	0
$921$	0	0
$922$	6.49920	0.214040
$923$	−21.6390	−0.712255
$924$	0	0
$925$	0	0
$926$	28.4177	0.933865
$927$	0	0
$928$	−6.24960	−0.205153
$929$	45.0575	1.47829	0.739145	−	0.673547i	$-0.235230\pi$
$929$	45.0575	1.47829	0.739145	+	0.673547i	$0.235230\pi$
$930$	0	0
$931$	15.7488	0.516146
$932$	4.92012	0.161164
$933$	0	0
$934$	−17.5775	−0.575153
$935$	0	0
$936$	0	0
$937$	−22.2464	−0.726759	−0.363379	−	0.931641i	$-0.618377\pi$
$937$	−22.2464	−0.726759	−0.363379	+	0.931641i	$0.618377\pi$
$938$	27.3279	0.892287
$939$	0	0
$940$	0	0
$941$	53.9277	1.75799	0.878997	−	0.476828i	$-0.158213\pi$
$941$	53.9277	1.75799	0.878997	+	0.476828i	$0.158213\pi$
$942$	0	0
$943$	0	0
$944$	14.5583	0.473833
$945$	0	0
$946$	−4.76196	−0.154825
$947$	−4.76196	−0.154743	−0.0773715	−	0.997002i	$-0.524653\pi$
$947$	−4.76196	−0.154743	−0.0773715	+	0.997002i	$0.524653\pi$
$948$	0	0
$949$	−27.1282	−0.880618
$950$	0	0
$951$	0	0
$952$	1.19049	0.0385840
$953$	4.37779	0.141811	0.0709053	−	0.997483i	$-0.477411\pi$
$953$	4.37779	0.141811	0.0709053	+	0.997483i	$0.477411\pi$
$954$	0	0
$955$	0	0
$956$	−2.98083	−0.0964069
$957$	0	0
$958$	−22.7372	−0.734607
$959$	37.2480	1.20280
$960$	0	0
$961$	−22.2380	−0.717356
$962$	−0.0898462	−0.00289676
$963$	0	0
$964$	20.0383	0.645392
$965$	0	0
$966$	0	0
$967$	13.1582	0.423138	0.211569	−	0.977363i	$-0.432143\pi$
$967$	13.1582	0.423138	0.211569	+	0.977363i	$0.432143\pi$
$968$	−10.0783	−0.323928
$969$	0	0
$970$	0	0
$971$	−25.9201	−0.831816	−0.415908	−	0.909407i	$-0.636536\pi$
$971$	−25.9201	−0.831816	−0.415908	+	0.909407i	$0.636536\pi$
$972$	0	0
$973$	76.4960	2.45235
$974$	−22.1981	−0.711273
$975$	0	0
$976$	7.53914	0.241322
$977$	31.2764	1.00062	0.500310	−	0.865846i	$-0.333219\pi$
$977$	31.2764	1.00062	0.500310	+	0.865846i	$0.333219\pi$
$978$	0	0
$979$	3.91693	0.125186
$980$	0	0
$981$	0	0
$982$	24.9984	0.797731
$983$	26.2364	0.836813	0.418406	−	0.908260i	$-0.362589\pi$
$983$	26.2364	0.836813	0.418406	+	0.908260i	$0.362589\pi$
$984$	0	0
$985$	0	0
$986$	1.55991	0.0496776
$987$	0	0
$988$	−2.24960	−0.0715693
$989$	−44.7356	−1.42251
$990$	0	0
$991$	39.2764	1.24766	0.623828	−	0.781562i	$-0.285577\pi$
$991$	39.2764	1.24766	0.623828	+	0.781562i	$0.285577\pi$
$992$	2.96006	0.0939820
$993$	0	0
$994$	−45.8786	−1.45518
$995$	0	0
$996$	0	0
$997$	−14.4609	−0.457980	−0.228990	−	0.973429i	$-0.573542\pi$
$997$	−14.4609	−0.457980	−0.228990	+	0.973429i	$0.573542\pi$
$998$	11.6574	0.369007
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cp.1.1		3
3.2	odd	2		950.2.a.j.1.2	✓	3
5.4	even	2		8550.2.a.ci.1.3		3
12.11	even	2		7600.2.a.bz.1.2		3
15.2	even	4		950.2.b.h.799.2		6
15.8	even	4		950.2.b.h.799.5		6
15.14	odd	2		950.2.a.l.1.2	yes	3
60.59	even	2		7600.2.a.bk.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.j.1.2	✓	3	3.2	odd	2
950.2.a.l.1.2	yes	3	15.14	odd	2
950.2.b.h.799.2		6	15.2	even	4
950.2.b.h.799.5		6	15.8	even	4
7600.2.a.bk.1.2		3	60.59	even	2
7600.2.a.bz.1.2		3	12.11	even	2
8550.2.a.ci.1.3		3	5.4	even	2
8550.2.a.cp.1.1		3	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.76957 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.76957 q^{7} +1.00000 q^{8} -0.960061 q^{11} -2.24960 q^{13} -4.76957 q^{14} +1.00000 q^{16} -0.249601 q^{17} +1.00000 q^{19} -0.960061 q^{22} -9.01917 q^{23} -2.24960 q^{26} -4.76957 q^{28} -6.24960 q^{29} +2.96006 q^{31} +1.00000 q^{32} -0.249601 q^{34} +0.0399387 q^{37} +1.00000 q^{38} +4.96006 q^{43} -0.960061 q^{44} -9.01917 q^{46} +9.49920 q^{47} +15.7488 q^{49} -2.24960 q^{52} -6.84945 q^{53} -4.76957 q^{56} -6.24960 q^{58} +14.5583 q^{59} +7.53914 q^{61} +2.96006 q^{62} +1.00000 q^{64} -5.72963 q^{67} -0.249601 q^{68} +9.61902 q^{71} +12.0591 q^{73} +0.0399387 q^{74} +1.00000 q^{76} +4.57908 q^{77} +6.07988 q^{79} +7.45926 q^{83} +4.96006 q^{86} -0.960061 q^{88} -4.07988 q^{89} +10.7296 q^{91} -9.01917 q^{92} +9.49920 q^{94} +18.4193 q^{97} +15.7488 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 6 q^{13} - 2 q^{14} + 3 q^{16} + 12 q^{17} + 3 q^{19} - 2 q^{22} - 2 q^{23} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 8 q^{31} + 3 q^{32} + 12 q^{34} + q^{37} + 3 q^{38} + 14 q^{43} - 2 q^{44} - 2 q^{46} + 3 q^{47} + 9 q^{49} + 6 q^{52} - 10 q^{53} - 2 q^{56} - 6 q^{58} - 6 q^{59} - 2 q^{61} + 8 q^{62} + 3 q^{64} - 4 q^{67} + 12 q^{68} + 6 q^{71} + 12 q^{73} + q^{74} + 3 q^{76} - 10 q^{77} + 20 q^{79} - 4 q^{83} + 14 q^{86} - 2 q^{88} - 14 q^{89} + 19 q^{91} - 2 q^{92} + 3 q^{94} + 28 q^{97} + 9 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 6 * q^13 - 2 * q^14 + 3 * q^16 + 12 * q^17 + 3 * q^19 - 2 * q^22 - 2 * q^23 + 6 * q^26 - 2 * q^28 - 6 * q^29 + 8 * q^31 + 3 * q^32 + 12 * q^34 + q^37 + 3 * q^38 + 14 * q^43 - 2 * q^44 - 2 * q^46 + 3 * q^47 + 9 * q^49 + 6 * q^52 - 10 * q^53 - 2 * q^56 - 6 * q^58 - 6 * q^59 - 2 * q^61 + 8 * q^62 + 3 * q^64 - 4 * q^67 + 12 * q^68 + 6 * q^71 + 12 * q^73 + q^74 + 3 * q^76 - 10 * q^77 + 20 * q^79 - 4 * q^83 + 14 * q^86 - 2 * q^88 - 14 * q^89 + 19 * q^91 - 2 * q^92 + 3 * q^94 + 28 * q^97 + 9 * q^98

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