LMFDB - Embedded newform 9075.2.a.dk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.9444552.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 7x^{2} + 9x - 6 $ x^5 - x^4 - 8x^3 + 7x^2 + 9*x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.39056$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.39056 q^{2} -1.00000 q^{3} -0.0663358 q^{4} +1.39056 q^{6} +4.73026 q^{7} +2.87337 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.39056 q^{2} -1.00000 q^{3} -0.0663358 q^{4} +1.39056 q^{6} +4.73026 q^{7} +2.87337 q^{8} +1.00000 q^{9} +0.0663358 q^{12} +3.26393 q^{13} -6.57772 q^{14} -3.86293 q^{16} +3.79659 q^{17} -1.39056 q^{18} +1.59900 q^{19} -4.73026 q^{21} +0.467340 q^{23} -2.87337 q^{24} -4.53870 q^{26} -1.00000 q^{27} -0.313785 q^{28} +4.44505 q^{29} -9.29251 q^{31} -0.375096 q^{32} -5.27940 q^{34} -0.0663358 q^{36} -0.117205 q^{37} -2.22350 q^{38} -3.26393 q^{39} +9.17773 q^{41} +6.57772 q^{42} +3.44505 q^{43} -0.649865 q^{46} -10.3743 q^{47} +3.86293 q^{48} +15.3753 q^{49} -3.79659 q^{51} -0.216516 q^{52} +5.92785 q^{53} +1.39056 q^{54} +13.5918 q^{56} -1.59900 q^{57} -6.18111 q^{58} +4.23434 q^{59} +3.54914 q^{61} +12.9218 q^{62} +4.73026 q^{63} +8.24745 q^{64} -10.0440 q^{67} -0.251850 q^{68} -0.467340 q^{69} +6.72924 q^{71} +2.87337 q^{72} +16.4194 q^{73} +0.162980 q^{74} -0.106071 q^{76} +4.53870 q^{78} -5.57772 q^{79} +1.00000 q^{81} -12.7622 q^{82} +16.9932 q^{83} +0.313785 q^{84} -4.79055 q^{86} -4.44505 q^{87} +9.17773 q^{89} +15.4392 q^{91} -0.0310014 q^{92} +9.29251 q^{93} +14.4261 q^{94} +0.375096 q^{96} +10.5564 q^{97} -21.3804 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} - 5 q^{3} + 7 q^{4} + q^{6} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - q^2 - 5 * q^3 + 7 * q^4 + q^6 + 4 * q^7 + 5 * q^9	$ 5 q - q^{2} - 5 q^{3} + 7 q^{4} + q^{6} + 4 q^{7} + 5 q^{9} - 7 q^{12} - 4 q^{13} + 6 q^{14} + 15 q^{16} - 8 q^{17} - q^{18} - 6 q^{19} - 4 q^{21} + 9 q^{23} + 13 q^{26} - 5 q^{27} + 17 q^{28} - 2 q^{29} - 3 q^{31} - 11 q^{32} + 9 q^{34} + 7 q^{36} - q^{37} - 3 q^{38} + 4 q^{39} - q^{41} - 6 q^{42} - 7 q^{43} + 3 q^{46} + 14 q^{47} - 15 q^{48} + 17 q^{49} + 8 q^{51} - 20 q^{52} - 3 q^{53} + q^{54} + 23 q^{56} + 6 q^{57} - 27 q^{58} + 18 q^{59} + 2 q^{61} + 61 q^{62} + 4 q^{63} + 30 q^{64} - 12 q^{67} - 39 q^{68} - 9 q^{69} + 8 q^{71} - 16 q^{73} + 40 q^{74} - 61 q^{76} - 13 q^{78} + 11 q^{79} + 5 q^{81} - 20 q^{82} + 39 q^{83} - 17 q^{84} - 26 q^{86} + 2 q^{87} - q^{89} + 13 q^{91} + 26 q^{92} + 3 q^{93} + 16 q^{94} + 11 q^{96} - 11 q^{97} - 12 q^{98}+O(q^{100}) $ 5 * q - q^2 - 5 * q^3 + 7 * q^4 + q^6 + 4 * q^7 + 5 * q^9 - 7 * q^12 - 4 * q^13 + 6 * q^14 + 15 * q^16 - 8 * q^17 - q^18 - 6 * q^19 - 4 * q^21 + 9 * q^23 + 13 * q^26 - 5 * q^27 + 17 * q^28 - 2 * q^29 - 3 * q^31 - 11 * q^32 + 9 * q^34 + 7 * q^36 - q^37 - 3 * q^38 + 4 * q^39 - q^41 - 6 * q^42 - 7 * q^43 + 3 * q^46 + 14 * q^47 - 15 * q^48 + 17 * q^49 + 8 * q^51 - 20 * q^52 - 3 * q^53 + q^54 + 23 * q^56 + 6 * q^57 - 27 * q^58 + 18 * q^59 + 2 * q^61 + 61 * q^62 + 4 * q^63 + 30 * q^64 - 12 * q^67 - 39 * q^68 - 9 * q^69 + 8 * q^71 - 16 * q^73 + 40 * q^74 - 61 * q^76 - 13 * q^78 + 11 * q^79 + 5 * q^81 - 20 * q^82 + 39 * q^83 - 17 * q^84 - 26 * q^86 + 2 * q^87 - q^89 + 13 * q^91 + 26 * q^92 + 3 * q^93 + 16 * q^94 + 11 * q^96 - 11 * q^97 - 12 * 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 $

$2$	−1.39056	−0.983276	−0.491638	−	0.870800i	$-0.663602\pi$
$2$	−1.39056	−0.983276	−0.491638	+	0.870800i	$0.663602\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.0663358	−0.0331679
$5$	0	0
$5$	0	0
$6$	1.39056	0.567695
$7$	4.73026	1.78787	0.893934	−	0.448198i	$-0.147934\pi$
$7$	4.73026	1.78787	0.893934	+	0.448198i	$0.147934\pi$
$8$	2.87337	1.01589
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.0663358	0.0191495
$13$	3.26393	0.905252	0.452626	−	0.891701i	$-0.350487\pi$
$13$	3.26393	0.905252	0.452626	+	0.891701i	$0.350487\pi$
$14$	−6.57772	−1.75797
$15$	0	0
$16$	−3.86293	−0.965732
$17$	3.79659	0.920809	0.460404	−	0.887709i	$-0.347704\pi$
$17$	3.79659	0.920809	0.460404	+	0.887709i	$0.347704\pi$
$18$	−1.39056	−0.327759
$19$	1.59900	0.366835	0.183417	−	0.983035i	$-0.441284\pi$
$19$	1.59900	0.366835	0.183417	+	0.983035i	$0.441284\pi$
$20$	0	0
$21$	−4.73026	−1.03223
$22$	0	0
$23$	0.467340	0.0974471	0.0487235	−	0.998812i	$-0.484485\pi$
$23$	0.467340	0.0974471	0.0487235	+	0.998812i	$0.484485\pi$
$24$	−2.87337	−0.586524
$25$	0	0
$26$	−4.53870	−0.890113
$27$	−1.00000	−0.192450
$28$	−0.313785	−0.0592999
$29$	4.44505	0.825424	0.412712	−	0.910862i	$-0.364582\pi$
$29$	4.44505	0.825424	0.412712	+	0.910862i	$0.364582\pi$
$30$	0	0
$31$	−9.29251	−1.66898	−0.834492	−	0.551020i	$-0.814239\pi$
$31$	−9.29251	−1.66898	−0.834492	+	0.551020i	$0.814239\pi$
$32$	−0.375096	−0.0663081
$33$	0	0
$34$	−5.27940	−0.905409
$35$	0	0
$36$	−0.0663358	−0.0110560
$37$	−0.117205	−0.0192683	−0.00963416	−	0.999954i	$-0.503067\pi$
$37$	−0.117205	−0.0192683	−0.00963416	+	0.999954i	$0.503067\pi$
$38$	−2.22350	−0.360700
$39$	−3.26393	−0.522647
$40$	0	0
$41$	9.17773	1.43332	0.716660	−	0.697423i	$-0.245670\pi$
$41$	9.17773	1.43332	0.716660	+	0.697423i	$0.245670\pi$
$42$	6.57772	1.01496
$43$	3.44505	0.525365	0.262682	−	0.964882i	$-0.415393\pi$
$43$	3.44505	0.525365	0.262682	+	0.964882i	$0.415393\pi$
$44$	0	0
$45$	0	0
$46$	−0.649865	−0.0958174
$47$	−10.3743	−1.51325	−0.756624	−	0.653850i	$-0.773153\pi$
$47$	−10.3743	−1.51325	−0.756624	+	0.653850i	$0.773153\pi$
$48$	3.86293	0.557566
$49$	15.3753	2.19647
$50$	0	0
$51$	−3.79659	−0.531629
$52$	−0.216516	−0.0300253
$53$	5.92785	0.814253	0.407127	−	0.913372i	$-0.366531\pi$
$53$	5.92785	0.814253	0.407127	+	0.913372i	$0.366531\pi$
$54$	1.39056	0.189232
$55$	0	0
$56$	13.5918	1.81628
$57$	−1.59900	−0.211792
$58$	−6.18111	−0.811620
$59$	4.23434	0.551264	0.275632	−	0.961263i	$-0.411113\pi$
$59$	4.23434	0.551264	0.275632	+	0.961263i	$0.411113\pi$
$60$	0	0
$61$	3.54914	0.454421	0.227211	−	0.973846i	$-0.427039\pi$
$61$	3.54914	0.454421	0.227211	+	0.973846i	$0.427039\pi$
$62$	12.9218	1.64107
$63$	4.73026	0.595956
$64$	8.24745	1.03093
$65$	0	0
$66$	0	0
$67$	−10.0440	−1.22707	−0.613537	−	0.789666i	$-0.710254\pi$
$67$	−10.0440	−1.22707	−0.613537	+	0.789666i	$0.710254\pi$
$68$	−0.251850	−0.0305413
$69$	−0.467340	−0.0562611
$70$	0	0
$71$	6.72924	0.798614	0.399307	−	0.916817i	$-0.369251\pi$
$71$	6.72924	0.798614	0.399307	+	0.916817i	$0.369251\pi$
$72$	2.87337	0.338630
$73$	16.4194	1.92174	0.960871	−	0.276997i	$-0.0893393\pi$
$73$	16.4194	1.92174	0.960871	+	0.276997i	$0.0893393\pi$
$74$	0.162980	0.0189461
$75$	0	0
$76$	−0.106071	−0.0121671
$77$	0	0
$78$	4.53870	0.513907
$79$	−5.57772	−0.627542	−0.313771	−	0.949499i	$-0.601592\pi$
$79$	−5.57772	−0.627542	−0.313771	+	0.949499i	$0.601592\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−12.7622	−1.40935
$83$	16.9932	1.86524	0.932622	−	0.360856i	$-0.117515\pi$
$83$	16.9932	1.86524	0.932622	+	0.360856i	$0.117515\pi$
$84$	0.313785	0.0342368
$85$	0	0
$86$	−4.79055	−0.516578
$87$	−4.44505	−0.476559
$88$	0	0
$89$	9.17773	0.972837	0.486419	−	0.873726i	$-0.338303\pi$
$89$	9.17773	0.972837	0.486419	+	0.873726i	$0.338303\pi$
$90$	0	0
$91$	15.4392	1.61847
$92$	−0.0310014	−0.00323212
$93$	9.29251	0.963588
$94$	14.4261	1.48794
$95$	0	0
$96$	0.375096	0.0382830
$97$	10.5564	1.07184	0.535922	−	0.844268i	$-0.319964\pi$
$97$	10.5564	1.07184	0.535922	+	0.844268i	$0.319964\pi$
$98$	−21.3804	−2.15974
$99$	0	0
$100$	0	0
$101$	9.35743	0.931099	0.465550	−	0.885022i	$-0.345857\pi$
$101$	9.35743	0.931099	0.465550	+	0.885022i	$0.345857\pi$
$102$	5.27940	0.522738
$103$	−15.8402	−1.56078	−0.780392	−	0.625290i	$-0.784981\pi$
$103$	−15.8402	−1.56078	−0.780392	+	0.625290i	$0.784981\pi$
$104$	9.37848	0.919636
$105$	0	0
$106$	−8.24305	−0.800636
$107$	−18.6073	−1.79884	−0.899418	−	0.437089i	$-0.856009\pi$
$107$	−18.6073	−1.79884	−0.899418	+	0.437089i	$0.856009\pi$
$108$	0.0663358	0.00638317
$109$	0.549142	0.0525983	0.0262992	−	0.999654i	$-0.491628\pi$
$109$	0.549142	0.0525983	0.0262992	+	0.999654i	$0.491628\pi$
$110$	0	0
$111$	0.117205	0.0111246
$112$	−18.2726	−1.72660
$113$	−8.62960	−0.811805	−0.405902	−	0.913916i	$-0.633043\pi$
$113$	−8.62960	−0.811805	−0.405902	+	0.913916i	$0.633043\pi$
$114$	2.22350	0.208250
$115$	0	0
$116$	−0.294866	−0.0273776
$117$	3.26393	0.301751
$118$	−5.88812	−0.542045
$119$	17.9589	1.64629
$120$	0	0
$121$	0	0
$122$	−4.93530	−0.446821
$123$	−9.17773	−0.827528
$124$	0.616426	0.0553567
$125$	0	0
$126$	−6.57772	−0.585990
$127$	−17.2228	−1.52828	−0.764138	−	0.645053i	$-0.776835\pi$
$127$	−17.2228	−1.52828	−0.764138	+	0.645053i	$0.776835\pi$
$128$	−10.7184	−0.947382
$129$	−3.44505	−0.303319
$130$	0	0
$131$	20.3520	1.77816	0.889082	−	0.457748i	$-0.151344\pi$
$131$	20.3520	1.77816	0.889082	+	0.457748i	$0.151344\pi$
$132$	0	0
$133$	7.56366	0.655853
$134$	13.9669	1.20655
$135$	0	0
$136$	10.9090	0.935440
$137$	14.6368	1.25051	0.625254	−	0.780421i	$-0.284995\pi$
$137$	14.6368	1.25051	0.625254	+	0.780421i	$0.284995\pi$
$138$	0.649865	0.0553202
$139$	−18.4465	−1.56461	−0.782304	−	0.622897i	$-0.785956\pi$
$139$	−18.4465	−1.56461	−0.782304	+	0.622897i	$0.785956\pi$
$140$	0	0
$141$	10.3743	0.873674
$142$	−9.35743	−0.785258
$143$	0	0
$144$	−3.86293	−0.321911
$145$	0	0
$146$	−22.8322	−1.88960
$147$	−15.3753	−1.26814
$148$	0.00777487	0.000639090 0
$149$	−6.57631	−0.538752	−0.269376	−	0.963035i	$-0.586817\pi$
$149$	−6.57631	−0.538752	−0.269376	+	0.963035i	$0.586817\pi$
$150$	0	0
$151$	−2.14673	−0.174698	−0.0873491	−	0.996178i	$-0.527840\pi$
$151$	−2.14673	−0.174698	−0.0873491	+	0.996178i	$0.527840\pi$
$152$	4.59451	0.372664
$153$	3.79659	0.306936
$154$	0	0
$155$	0	0
$156$	0.216516	0.0173351
$157$	−7.03060	−0.561103	−0.280552	−	0.959839i	$-0.590517\pi$
$157$	−7.03060	−0.561103	−0.280552	+	0.959839i	$0.590517\pi$
$158$	7.75616	0.617047
$159$	−5.92785	−0.470109
$160$	0	0
$161$	2.21064	0.174223
$162$	−1.39056	−0.109253
$163$	−2.16801	−0.169811	−0.0849057	−	0.996389i	$-0.527059\pi$
$163$	−2.16801	−0.169811	−0.0849057	+	0.996389i	$0.527059\pi$
$164$	−0.608812	−0.0475402
$165$	0	0
$166$	−23.6301	−1.83405
$167$	21.9070	1.69521	0.847606	−	0.530626i	$-0.178043\pi$
$167$	21.9070	1.69521	0.847606	+	0.530626i	$0.178043\pi$
$168$	−13.5918	−1.04863
$169$	−2.34675	−0.180519
$170$	0	0
$171$	1.59900	0.122278
$172$	−0.228530	−0.0174252
$173$	−8.70757	−0.662024	−0.331012	−	0.943627i	$-0.607390\pi$
$173$	−8.70757	−0.662024	−0.331012	+	0.943627i	$0.607390\pi$
$174$	6.18111	0.468589
$175$	0	0
$176$	0	0
$177$	−4.23434	−0.318273
$178$	−12.7622	−0.956568
$179$	−13.4107	−1.00236	−0.501180	−	0.865343i	$-0.667100\pi$
$179$	−13.4107	−1.00236	−0.501180	+	0.865343i	$0.667100\pi$
$180$	0	0
$181$	14.0461	1.04404	0.522018	−	0.852934i	$-0.325179\pi$
$181$	14.0461	1.04404	0.522018	+	0.852934i	$0.325179\pi$
$182$	−21.4692	−1.59140
$183$	−3.54914	−0.262360
$184$	1.34284	0.0989954
$185$	0	0
$186$	−12.9218	−0.947473
$187$	0	0
$188$	0.688188	0.0501913
$189$	−4.73026	−0.344076
$190$	0	0
$191$	3.79518	0.274610	0.137305	−	0.990529i	$-0.456156\pi$
$191$	3.79518	0.274610	0.137305	+	0.990529i	$0.456156\pi$
$192$	−8.24745	−0.595208
$193$	15.6818	1.12880	0.564401	−	0.825501i	$-0.309107\pi$
$193$	15.6818	1.12880	0.564401	+	0.825501i	$0.309107\pi$
$194$	−14.6794	−1.05392
$195$	0	0
$196$	−1.01993	−0.0728525
$197$	4.81065	0.342744	0.171372	−	0.985206i	$-0.445180\pi$
$197$	4.81065	0.342744	0.171372	+	0.985206i	$0.445180\pi$
$198$	0	0
$199$	6.36459	0.451174	0.225587	−	0.974223i	$-0.427570\pi$
$199$	6.36459	0.451174	0.225587	+	0.974223i	$0.427570\pi$
$200$	0	0
$201$	10.0440	0.708452
$202$	−13.0121	−0.915528
$203$	21.0262	1.47575
$204$	0.251850	0.0176330
$205$	0	0
$206$	22.0268	1.53468
$207$	0.467340	0.0324824
$208$	−12.6083	−0.874231
$209$	0	0
$210$	0	0
$211$	−16.0688	−1.10622	−0.553112	−	0.833107i	$-0.686560\pi$
$211$	−16.0688	−1.10622	−0.553112	+	0.833107i	$0.686560\pi$
$212$	−0.393229	−0.0270071
$213$	−6.72924	−0.461080
$214$	25.8746	1.76875
$215$	0	0
$216$	−2.87337	−0.195508
$217$	−43.9559	−2.98392
$218$	−0.763617	−0.0517187
$219$	−16.4194	−1.10952
$220$	0	0
$221$	12.3918	0.833564
$222$	−0.162980	−0.0109385
$223$	−3.56806	−0.238935	−0.119468	−	0.992838i	$-0.538119\pi$
$223$	−3.56806	−0.238935	−0.119468	+	0.992838i	$0.538119\pi$
$224$	−1.77430	−0.118550
$225$	0	0
$226$	12.0000	0.798228
$227$	11.3825	0.755486	0.377743	−	0.925910i	$-0.376700\pi$
$227$	11.3825	0.755486	0.377743	+	0.925910i	$0.376700\pi$
$228$	0.106071	0.00702471
$229$	16.3811	1.08250	0.541248	−	0.840863i	$-0.317952\pi$
$229$	16.3811	1.08250	0.541248	+	0.840863i	$0.317952\pi$
$230$	0	0
$231$	0	0
$232$	12.7723	0.838540
$233$	−20.4557	−1.34010	−0.670049	−	0.742317i	$-0.733727\pi$
$233$	−20.4557	−1.34010	−0.670049	+	0.742317i	$0.733727\pi$
$234$	−4.53870	−0.296704
$235$	0	0
$236$	−0.280889	−0.0182843
$237$	5.57772	0.362312
$238$	−24.9729	−1.61875
$239$	10.7103	0.692793	0.346397	−	0.938088i	$-0.387405\pi$
$239$	10.7103	0.692793	0.346397	+	0.938088i	$0.387405\pi$
$240$	0	0
$241$	−6.09625	−0.392694	−0.196347	−	0.980534i	$-0.562908\pi$
$241$	−6.09625	−0.392694	−0.196347	+	0.980534i	$0.562908\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−0.235435	−0.0150722
$245$	0	0
$246$	12.7622	0.813688
$247$	5.21901	0.332078
$248$	−26.7008	−1.69550
$249$	−16.9932	−1.07690
$250$	0	0
$251$	−5.88671	−0.371566	−0.185783	−	0.982591i	$-0.559482\pi$
$251$	−5.88671	−0.371566	−0.185783	+	0.982591i	$0.559482\pi$
$252$	−0.313785	−0.0197666
$253$	0	0
$254$	23.9494	1.50272
$255$	0	0
$256$	−1.59029	−0.0993931
$257$	−20.3095	−1.26687	−0.633435	−	0.773796i	$-0.718355\pi$
$257$	−20.3095	−1.26687	−0.633435	+	0.773796i	$0.718355\pi$
$258$	4.79055	0.298247
$259$	−0.554408	−0.0344492
$260$	0	0
$261$	4.44505	0.275141
$262$	−28.3007	−1.74843
$263$	−16.6324	−1.02560	−0.512798	−	0.858510i	$-0.671391\pi$
$263$	−16.6324	−1.02560	−0.512798	+	0.858510i	$0.671391\pi$
$264$	0	0
$265$	0	0
$266$	−10.5177	−0.644884
$267$	−9.17773	−0.561668
$268$	0.666280	0.0406995
$269$	19.0169	1.15948	0.579740	−	0.814802i	$-0.303154\pi$
$269$	19.0169	1.15948	0.579740	+	0.814802i	$0.303154\pi$
$270$	0	0
$271$	−9.61272	−0.583931	−0.291966	−	0.956429i	$-0.594309\pi$
$271$	−9.61272	−0.583931	−0.291966	+	0.956429i	$0.594309\pi$
$272$	−14.6660	−0.889255
$273$	−15.4392	−0.934425
$274$	−20.3534	−1.22960
$275$	0	0
$276$	0.0310014	0.00186606
$277$	−10.6238	−0.638322	−0.319161	−	0.947701i	$-0.603401\pi$
$277$	−10.6238	−0.638322	−0.319161	+	0.947701i	$0.603401\pi$
$278$	25.6510	1.53844
$279$	−9.29251	−0.556328
$280$	0	0
$281$	20.1695	1.20321	0.601605	−	0.798793i	$-0.294528\pi$
$281$	20.1695	1.20321	0.601605	+	0.798793i	$0.294528\pi$
$282$	−14.4261	−0.859063
$283$	18.8548	1.12080	0.560399	−	0.828222i	$-0.310648\pi$
$283$	18.8548	1.12080	0.560399	+	0.828222i	$0.310648\pi$
$284$	−0.446390	−0.0264884
$285$	0	0
$286$	0	0
$287$	43.4130	2.56259
$288$	−0.375096	−0.0221027
$289$	−2.58589	−0.152111
$290$	0	0
$291$	−10.5564	−0.618829
$292$	−1.08919	−0.0637401
$293$	−2.57042	−0.150166	−0.0750828	−	0.997177i	$-0.523922\pi$
$293$	−2.57042	−0.150166	−0.0750828	+	0.997177i	$0.523922\pi$
$294$	21.3804	1.24693
$295$	0	0
$296$	−0.336772	−0.0195745
$297$	0	0
$298$	9.14476	0.529742
$299$	1.52536	0.0882141
$300$	0	0
$301$	16.2959	0.939283
$302$	2.98516	0.171777
$303$	−9.35743	−0.537570
$304$	−6.17681	−0.354264
$305$	0	0
$306$	−5.27940	−0.301803
$307$	14.8489	0.847470	0.423735	−	0.905786i	$-0.360719\pi$
$307$	14.8489	0.847470	0.423735	+	0.905786i	$0.360719\pi$
$308$	0	0
$309$	15.8402	0.901120
$310$	0	0
$311$	−2.64783	−0.150145	−0.0750725	−	0.997178i	$-0.523919\pi$
$311$	−2.64783	−0.150145	−0.0750725	+	0.997178i	$0.523919\pi$
$312$	−9.37848	−0.530952
$313$	34.8449	1.96955	0.984774	−	0.173838i	$-0.0556168\pi$
$313$	34.8449	1.96955	0.984774	+	0.173838i	$0.0556168\pi$
$314$	9.77650	0.551720
$315$	0	0
$316$	0.370002	0.0208143
$317$	−31.6314	−1.77660	−0.888299	−	0.459266i	$-0.848112\pi$
$317$	−31.6314	−1.77660	−0.888299	+	0.459266i	$0.848112\pi$
$318$	8.24305	0.462247
$319$	0	0
$320$	0	0
$321$	18.6073	1.03856
$322$	−3.07403	−0.171309
$323$	6.07074	0.337785
$324$	−0.0663358	−0.00368532
$325$	0	0
$326$	3.01475	0.166971
$327$	−0.549142	−0.0303676
$328$	26.3710	1.45610
$329$	−49.0731	−2.70549
$330$	0	0
$331$	−16.7230	−0.919181	−0.459591	−	0.888131i	$-0.652004\pi$
$331$	−16.7230	−0.919181	−0.459591	+	0.888131i	$0.652004\pi$
$332$	−1.12726	−0.0618662
$333$	−0.117205	−0.00642277
$334$	−30.4630	−1.66686
$335$	0	0
$336$	18.2726	0.996854
$337$	−20.5046	−1.11695	−0.558477	−	0.829520i	$-0.688614\pi$
$337$	−20.5046	−1.11695	−0.558477	+	0.829520i	$0.688614\pi$
$338$	3.26330	0.177500
$339$	8.62960	0.468696
$340$	0	0
$341$	0	0
$342$	−2.22350	−0.120233
$343$	39.6174	2.13914
$344$	9.89889	0.533712
$345$	0	0
$346$	12.1084	0.650952
$347$	−22.3757	−1.20119	−0.600596	−	0.799553i	$-0.705070\pi$
$347$	−22.3757	−1.20119	−0.600596	+	0.799553i	$0.705070\pi$
$348$	0.294866	0.0158065
$349$	−18.1923	−0.973809	−0.486904	−	0.873455i	$-0.661874\pi$
$349$	−18.1923	−0.973809	−0.486904	+	0.873455i	$0.661874\pi$
$350$	0	0
$351$	−3.26393	−0.174216
$352$	0	0
$353$	−11.6334	−0.619186	−0.309593	−	0.950869i	$-0.600193\pi$
$353$	−11.6334	−0.619186	−0.309593	+	0.950869i	$0.600193\pi$
$354$	5.88812	0.312950
$355$	0	0
$356$	−0.608812	−0.0322670
$357$	−17.9589	−0.950483
$358$	18.6484	0.985596
$359$	−5.08627	−0.268443	−0.134222	−	0.990951i	$-0.542853\pi$
$359$	−5.08627	−0.268443	−0.134222	+	0.990951i	$0.542853\pi$
$360$	0	0
$361$	−16.4432	−0.865432
$362$	−19.5319	−1.02658
$363$	0	0
$364$	−1.02417	−0.0536813
$365$	0	0
$366$	4.93530	0.257973
$367$	−20.3278	−1.06110	−0.530552	−	0.847652i	$-0.678015\pi$
$367$	−20.3278	−1.06110	−0.530552	+	0.847652i	$0.678015\pi$
$368$	−1.80530	−0.0941077
$369$	9.17773	0.477773
$370$	0	0
$371$	28.0403	1.45578
$372$	−0.616426	−0.0319602
$373$	5.40971	0.280104	0.140052	−	0.990144i	$-0.455273\pi$
$373$	5.40971	0.280104	0.140052	+	0.990144i	$0.455273\pi$
$374$	0	0
$375$	0	0
$376$	−29.8092	−1.53729
$377$	14.5083	0.747217
$378$	6.57772	0.338321
$379$	−8.33373	−0.428075	−0.214037	−	0.976825i	$-0.568661\pi$
$379$	−8.33373	−0.428075	−0.214037	+	0.976825i	$0.568661\pi$
$380$	0	0
$381$	17.2228	0.882350
$382$	−5.27744	−0.270017
$383$	1.16840	0.0597026	0.0298513	−	0.999554i	$-0.490497\pi$
$383$	1.16840	0.0597026	0.0298513	+	0.999554i	$0.490497\pi$
$384$	10.7184	0.546971
$385$	0	0
$386$	−21.8065	−1.10992
$387$	3.44505	0.175122
$388$	−0.700270	−0.0355508
$389$	22.0121	1.11606	0.558028	−	0.829822i	$-0.311558\pi$
$389$	22.0121	1.11606	0.558028	+	0.829822i	$0.311558\pi$
$390$	0	0
$391$	1.77430	0.0897301
$392$	44.1790	2.23138
$393$	−20.3520	−1.02662
$394$	−6.68951	−0.337012
$395$	0	0
$396$	0	0
$397$	−31.7355	−1.59276	−0.796380	−	0.604797i	$-0.793254\pi$
$397$	−31.7355	−1.59276	−0.796380	+	0.604797i	$0.793254\pi$
$398$	−8.85036	−0.443628
$399$	−7.56366	−0.378657
$400$	0	0
$401$	−18.1802	−0.907874	−0.453937	−	0.891034i	$-0.649981\pi$
$401$	−18.1802	−0.907874	−0.453937	+	0.891034i	$0.649981\pi$
$402$	−13.9669	−0.696604
$403$	−30.3301	−1.51085
$404$	−0.620733	−0.0308826
$405$	0	0
$406$	−29.2383	−1.45107
$407$	0	0
$408$	−10.9090	−0.540077
$409$	−5.26974	−0.260572	−0.130286	−	0.991476i	$-0.541590\pi$
$409$	−5.26974	−0.260572	−0.130286	+	0.991476i	$0.541590\pi$
$410$	0	0
$411$	−14.6368	−0.721982
$412$	1.05078	0.0517680
$413$	20.0295	0.985588
$414$	−0.649865	−0.0319391
$415$	0	0
$416$	−1.22429	−0.0600256
$417$	18.4465	0.903327
$418$	0	0
$419$	−10.8388	−0.529511	−0.264756	−	0.964316i	$-0.585291\pi$
$419$	−10.8388	−0.529511	−0.264756	+	0.964316i	$0.585291\pi$
$420$	0	0
$421$	−2.30075	−0.112132	−0.0560658	−	0.998427i	$-0.517856\pi$
$421$	−2.30075	−0.112132	−0.0560658	+	0.998427i	$0.517856\pi$
$422$	22.3447	1.08772
$423$	−10.3743	−0.504416
$424$	17.0329	0.827191
$425$	0	0
$426$	9.35743	0.453369
$427$	16.7884	0.812445
$428$	1.23433	0.0596636
$429$	0	0
$430$	0	0
$431$	17.7662	0.855767	0.427883	−	0.903834i	$-0.359259\pi$
$431$	17.7662	0.855767	0.427883	+	0.903834i	$0.359259\pi$
$432$	3.86293	0.185855
$433$	4.22719	0.203146	0.101573	−	0.994828i	$-0.467613\pi$
$433$	4.22719	0.203146	0.101573	+	0.994828i	$0.467613\pi$
$434$	61.1235	2.93402
$435$	0	0
$436$	−0.0364278	−0.00174458
$437$	0.747274	0.0357470
$438$	22.8322	1.09096
$439$	6.99216	0.333717	0.166859	−	0.985981i	$-0.446638\pi$
$439$	6.99216	0.333717	0.166859	+	0.985981i	$0.446638\pi$
$440$	0	0
$441$	15.3753	0.732158
$442$	−17.2316	−0.819623
$443$	−15.6431	−0.743226	−0.371613	−	0.928388i	$-0.621195\pi$
$443$	−15.6431	−0.743226	−0.371613	+	0.928388i	$0.621195\pi$
$444$	−0.00777487	−0.000368979 0
$445$	0	0
$446$	4.96161	0.234939
$447$	6.57631	0.311049
$448$	39.0126	1.84317
$449$	10.7084	0.505359	0.252679	−	0.967550i	$-0.418688\pi$
$449$	10.7084	0.505359	0.252679	+	0.967550i	$0.418688\pi$
$450$	0	0
$451$	0	0
$452$	0.572452	0.0269259
$453$	2.14673	0.100862
$454$	−15.8281	−0.742852
$455$	0	0
$456$	−4.59451	−0.215157
$457$	−21.3070	−0.996701	−0.498350	−	0.866976i	$-0.666061\pi$
$457$	−21.3070	−0.996701	−0.498350	+	0.866976i	$0.666061\pi$
$458$	−22.7790	−1.06439
$459$	−3.79659	−0.177210
$460$	0	0
$461$	37.3681	1.74041	0.870203	−	0.492693i	$-0.163988\pi$
$461$	37.3681	1.74041	0.870203	+	0.492693i	$0.163988\pi$
$462$	0	0
$463$	27.5909	1.28226	0.641129	−	0.767433i	$-0.278467\pi$
$463$	27.5909	1.28226	0.641129	+	0.767433i	$0.278467\pi$
$464$	−17.1709	−0.797139
$465$	0	0
$466$	28.4450	1.31769
$467$	−37.9840	−1.75769	−0.878846	−	0.477105i	$-0.841686\pi$
$467$	−37.9840	−1.75769	−0.878846	+	0.477105i	$0.841686\pi$
$468$	−0.216516	−0.0100084
$469$	−47.5109	−2.19385
$470$	0	0
$471$	7.03060	0.323953
$472$	12.1668	0.560024
$473$	0	0
$474$	−7.75616	−0.356252
$475$	0	0
$476$	−1.19132	−0.0546038
$477$	5.92785	0.271418
$478$	−14.8934	−0.681207
$479$	37.6624	1.72084	0.860420	−	0.509586i	$-0.170201\pi$
$479$	37.6624	1.72084	0.860420	+	0.509586i	$0.170201\pi$
$480$	0	0
$481$	−0.382548	−0.0174427
$482$	8.47722	0.386127
$483$	−2.21064	−0.100587
$484$	0	0
$485$	0	0
$486$	1.39056	0.0630772
$487$	0.200023	0.00906389	0.00453194	−	0.999990i	$-0.498557\pi$
$487$	0.200023	0.00906389	0.00453194	+	0.999990i	$0.498557\pi$
$488$	10.1980	0.461642
$489$	2.16801	0.0980406
$490$	0	0
$491$	−36.7708	−1.65945	−0.829723	−	0.558176i	$-0.811501\pi$
$491$	−36.7708	−1.65945	−0.829723	+	0.558176i	$0.811501\pi$
$492$	0.608812	0.0274474
$493$	16.8760	0.760058
$494$	−7.25737	−0.326524
$495$	0	0
$496$	35.8963	1.61179
$497$	31.8310	1.42782
$498$	23.6301	1.05889
$499$	25.7139	1.15111	0.575556	−	0.817762i	$-0.304786\pi$
$499$	25.7139	1.15111	0.575556	+	0.817762i	$0.304786\pi$
$500$	0	0
$501$	−21.9070	−0.978731
$502$	8.18583	0.365352
$503$	−15.3971	−0.686522	−0.343261	−	0.939240i	$-0.611531\pi$
$503$	−15.3971	−0.686522	−0.343261	+	0.939240i	$0.611531\pi$
$504$	13.5918	0.605426
$505$	0	0
$506$	0	0
$507$	2.34675	0.104223
$508$	1.14249	0.0506897
$509$	−0.696277	−0.0308619	−0.0154310	−	0.999881i	$-0.504912\pi$
$509$	−0.696277	−0.0308619	−0.0154310	+	0.999881i	$0.504912\pi$
$510$	0	0
$511$	77.6678	3.43582
$512$	23.6482	1.04511
$513$	−1.59900	−0.0705974
$514$	28.2416	1.24568
$515$	0	0
$516$	0.228530	0.0100605
$517$	0	0
$518$	0.770939	0.0338731
$519$	8.70757	0.382220
$520$	0	0
$521$	−23.6712	−1.03705	−0.518527	−	0.855061i	$-0.673520\pi$
$521$	−23.6712	−1.03705	−0.518527	+	0.855061i	$0.673520\pi$
$522$	−6.18111	−0.270540
$523$	10.1444	0.443582	0.221791	−	0.975094i	$-0.428810\pi$
$523$	10.1444	0.443582	0.221791	+	0.975094i	$0.428810\pi$
$524$	−1.35007	−0.0589780
$525$	0	0
$526$	23.1283	1.00844
$527$	−35.2799	−1.53681
$528$	0	0
$529$	−22.7816	−0.990504
$530$	0	0
$531$	4.23434	0.183755
$532$	−0.501742	−0.0217533
$533$	29.9555	1.29752
$534$	12.7622	0.552275
$535$	0	0
$536$	−28.8602	−1.24657
$537$	13.4107	0.578713
$538$	−26.4442	−1.14009
$539$	0	0
$540$	0	0
$541$	9.64980	0.414877	0.207439	−	0.978248i	$-0.433487\pi$
$541$	9.64980	0.414877	0.207439	+	0.978248i	$0.433487\pi$
$542$	13.3671	0.574166
$543$	−14.0461	−0.602775
$544$	−1.42408	−0.0610571
$545$	0	0
$546$	21.4692	0.918798
$547$	21.1345	0.903645	0.451823	−	0.892108i	$-0.350774\pi$
$547$	21.1345	0.903645	0.451823	+	0.892108i	$0.350774\pi$
$548$	−0.970946	−0.0414768
$549$	3.54914	0.151474
$550$	0	0
$551$	7.10761	0.302794
$552$	−1.34284	−0.0571550
$553$	−26.3840	−1.12196
$554$	14.7730	0.627646
$555$	0	0
$556$	1.22366	0.0518948
$557$	22.1535	0.938672	0.469336	−	0.883020i	$-0.344493\pi$
$557$	22.1535	0.938672	0.469336	+	0.883020i	$0.344493\pi$
$558$	12.9218	0.547024
$559$	11.2444	0.475587
$560$	0	0
$561$	0	0
$562$	−28.0469	−1.18309
$563$	24.5811	1.03597	0.517985	−	0.855390i	$-0.326682\pi$
$563$	24.5811	1.03597	0.517985	+	0.855390i	$0.326682\pi$
$564$	−0.688188	−0.0289780
$565$	0	0
$566$	−26.2187	−1.10205
$567$	4.73026	0.198652
$568$	19.3356	0.811304
$569$	2.85186	0.119556	0.0597781	−	0.998212i	$-0.480961\pi$
$569$	2.85186	0.119556	0.0597781	+	0.998212i	$0.480961\pi$
$570$	0	0
$571$	41.1394	1.72163	0.860816	−	0.508917i	$-0.169954\pi$
$571$	41.1394	1.72163	0.860816	+	0.508917i	$0.169954\pi$
$572$	0	0
$573$	−3.79518	−0.158546
$574$	−60.3685	−2.51973
$575$	0	0
$576$	8.24745	0.343644
$577$	28.3036	1.17830	0.589148	−	0.808025i	$-0.299463\pi$
$577$	28.3036	1.17830	0.589148	+	0.808025i	$0.299463\pi$
$578$	3.59584	0.149567
$579$	−15.6818	−0.651714
$580$	0	0
$581$	80.3821	3.33481
$582$	14.6794	0.608480
$583$	0	0
$584$	47.1789	1.95228
$585$	0	0
$586$	3.57433	0.147654
$587$	33.1070	1.36647	0.683236	−	0.730198i	$-0.260572\pi$
$587$	33.1070	1.36647	0.683236	+	0.730198i	$0.260572\pi$
$588$	1.01993	0.0420614
$589$	−14.8587	−0.612241
$590$	0	0
$591$	−4.81065	−0.197884
$592$	0.452753	0.0186080
$593$	18.6126	0.764327	0.382163	−	0.924095i	$-0.375179\pi$
$593$	18.6126	0.764327	0.382163	+	0.924095i	$0.375179\pi$
$594$	0	0
$595$	0	0
$596$	0.436245	0.0178693
$597$	−6.36459	−0.260485
$598$	−2.12111	−0.0867388
$599$	−14.0348	−0.573446	−0.286723	−	0.958014i	$-0.592566\pi$
$599$	−14.0348	−0.573446	−0.286723	+	0.958014i	$0.592566\pi$
$600$	0	0
$601$	−28.5597	−1.16497	−0.582486	−	0.812840i	$-0.697920\pi$
$601$	−28.5597	−1.16497	−0.582486	+	0.812840i	$0.697920\pi$
$602$	−22.6605	−0.923574
$603$	−10.0440	−0.409025
$604$	0.142405	0.00579437
$605$	0	0
$606$	13.0121	0.528580
$607$	−2.94480	−0.119526	−0.0597628	−	0.998213i	$-0.519034\pi$
$607$	−2.94480	−0.119526	−0.0597628	+	0.998213i	$0.519034\pi$
$608$	−0.599776	−0.0243241
$609$	−21.0262	−0.852025
$610$	0	0
$611$	−33.8610	−1.36987
$612$	−0.251850	−0.0101804
$613$	28.2974	1.14292	0.571461	−	0.820629i	$-0.306377\pi$
$613$	28.2974	1.14292	0.571461	+	0.820629i	$0.306377\pi$
$614$	−20.6483	−0.833297
$615$	0	0
$616$	0	0
$617$	−42.7336	−1.72039	−0.860194	−	0.509967i	$-0.829658\pi$
$617$	−42.7336	−1.72039	−0.860194	+	0.509967i	$0.829658\pi$
$618$	−22.0268	−0.886049
$619$	−6.57859	−0.264416	−0.132208	−	0.991222i	$-0.542207\pi$
$619$	−6.57859	−0.264416	−0.132208	+	0.991222i	$0.542207\pi$
$620$	0	0
$621$	−0.467340	−0.0187537
$622$	3.68198	0.147634
$623$	43.4130	1.73931
$624$	12.6083	0.504737
$625$	0	0
$626$	−48.4540	−1.93661
$627$	0	0
$628$	0.466381	0.0186106
$629$	−0.444978	−0.0177424
$630$	0	0
$631$	−1.29204	−0.0514354	−0.0257177	−	0.999669i	$-0.508187\pi$
$631$	−1.29204	−0.0514354	−0.0257177	+	0.999669i	$0.508187\pi$
$632$	−16.0268	−0.637513
$633$	16.0688	0.638679
$634$	43.9855	1.74689
$635$	0	0
$636$	0.393229	0.0155925
$637$	50.1840	1.98836
$638$	0	0
$639$	6.72924	0.266205
$640$	0	0
$641$	22.1083	0.873225	0.436612	−	0.899650i	$-0.356178\pi$
$641$	22.1083	0.873225	0.436612	+	0.899650i	$0.356178\pi$
$642$	−25.8746	−1.02119
$643$	−44.2576	−1.74535	−0.872674	−	0.488303i	$-0.837616\pi$
$643$	−44.2576	−1.74535	−0.872674	+	0.488303i	$0.837616\pi$
$644$	−0.146644	−0.00577860
$645$	0	0
$646$	−8.44174	−0.332136
$647$	36.6939	1.44259	0.721293	−	0.692630i	$-0.243548\pi$
$647$	36.6939	1.44259	0.721293	+	0.692630i	$0.243548\pi$
$648$	2.87337	0.112877
$649$	0	0
$650$	0	0
$651$	43.9559	1.72277
$652$	0.143816	0.00563229
$653$	22.8928	0.895866	0.447933	−	0.894067i	$-0.352160\pi$
$653$	22.8928	0.895866	0.447933	+	0.894067i	$0.352160\pi$
$654$	0.763617	0.0298598
$655$	0	0
$656$	−35.4529	−1.38420
$657$	16.4194	0.640580
$658$	68.2393	2.66024
$659$	11.8873	0.463062	0.231531	−	0.972828i	$-0.425627\pi$
$659$	11.8873	0.463062	0.231531	+	0.972828i	$0.425627\pi$
$660$	0	0
$661$	10.6632	0.414752	0.207376	−	0.978261i	$-0.433508\pi$
$661$	10.6632	0.414752	0.207376	+	0.978261i	$0.433508\pi$
$662$	23.2544	0.903809
$663$	−12.3918	−0.481258
$664$	48.8277	1.89488
$665$	0	0
$666$	0.162980	0.00631536
$667$	2.07735	0.0804352
$668$	−1.45322	−0.0562266
$669$	3.56806	0.137949
$670$	0	0
$671$	0	0
$672$	1.77430	0.0684450
$673$	−11.1838	−0.431104	−0.215552	−	0.976492i	$-0.569155\pi$
$673$	−11.1838	−0.431104	−0.215552	+	0.976492i	$0.569155\pi$
$674$	28.5129	1.09827
$675$	0	0
$676$	0.155674	0.00598744
$677$	−5.19861	−0.199799	−0.0998994	−	0.994998i	$-0.531852\pi$
$677$	−5.19861	−0.199799	−0.0998994	+	0.994998i	$0.531852\pi$
$678$	−12.0000	−0.460857
$679$	49.9347	1.91632
$680$	0	0
$681$	−11.3825	−0.436180
$682$	0	0
$683$	27.9250	1.06852	0.534260	−	0.845320i	$-0.320590\pi$
$683$	27.9250	1.06852	0.534260	+	0.845320i	$0.320590\pi$
$684$	−0.106071	−0.00405572
$685$	0	0
$686$	−55.0905	−2.10337
$687$	−16.3811	−0.624979
$688$	−13.3080	−0.507361
$689$	19.3481	0.737104
$690$	0	0
$691$	−18.0596	−0.687019	−0.343509	−	0.939149i	$-0.611616\pi$
$691$	−18.0596	−0.687019	−0.343509	+	0.939149i	$0.611616\pi$
$692$	0.577624	0.0219580
$693$	0	0
$694$	31.1148	1.18110
$695$	0	0
$696$	−12.7723	−0.484131
$697$	34.8441	1.31981
$698$	25.2975	0.957523
$699$	20.4557	0.773706
$700$	0	0
$701$	13.4135	0.506620	0.253310	−	0.967385i	$-0.418481\pi$
$701$	13.4135	0.506620	0.253310	+	0.967385i	$0.418481\pi$
$702$	4.53870	0.171302
$703$	−0.187410	−0.00706829
$704$	0	0
$705$	0	0
$706$	16.1770	0.608831
$707$	44.2630	1.66468
$708$	0.280889	0.0105564
$709$	33.8987	1.27309	0.636546	−	0.771239i	$-0.280363\pi$
$709$	33.8987	1.27309	0.636546	+	0.771239i	$0.280363\pi$
$710$	0	0
$711$	−5.57772	−0.209181
$712$	26.3710	0.988295
$713$	−4.34276	−0.162638
$714$	24.9729	0.934588
$715$	0	0
$716$	0.889607	0.0332462
$717$	−10.7103	−0.399984
$718$	7.07278	0.263954
$719$	40.1999	1.49920	0.749601	−	0.661890i	$-0.230245\pi$
$719$	40.1999	1.49920	0.749601	+	0.661890i	$0.230245\pi$
$720$	0	0
$721$	−74.9284	−2.79048
$722$	22.8653	0.850959
$723$	6.09625	0.226722
$724$	−0.931758	−0.0346285
$725$	0	0
$726$	0	0
$727$	−17.6302	−0.653869	−0.326934	−	0.945047i	$-0.606016\pi$
$727$	−17.6302	−0.653869	−0.326934	+	0.945047i	$0.606016\pi$
$728$	44.3626	1.64419
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.0794	0.483760
$732$	0.235435	0.00870194
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	28.2671	1.04336
$735$	0	0
$736$	−0.175297	−0.00646153
$737$	0	0
$738$	−12.7622	−0.469783
$739$	30.5466	1.12367	0.561837	−	0.827248i	$-0.310095\pi$
$739$	30.5466	1.12367	0.561837	+	0.827248i	$0.310095\pi$
$740$	0	0
$741$	−5.21901	−0.191725
$742$	−38.9917	−1.43143
$743$	−9.13517	−0.335137	−0.167568	−	0.985860i	$-0.553592\pi$
$743$	−9.13517	−0.335137	−0.167568	+	0.985860i	$0.553592\pi$
$744$	26.7008	0.978899
$745$	0	0
$746$	−7.52254	−0.275420
$747$	16.9932	0.621748
$748$	0	0
$749$	−88.0173	−3.21608
$750$	0	0
$751$	−21.0490	−0.768088	−0.384044	−	0.923315i	$-0.625469\pi$
$751$	−21.0490	−0.768088	−0.384044	+	0.923315i	$0.625469\pi$
$752$	40.0752	1.46139
$753$	5.88671	0.214523
$754$	−20.1747	−0.734720
$755$	0	0
$756$	0.313785	0.0114123
$757$	−28.2854	−1.02805	−0.514024	−	0.857776i	$-0.671846\pi$
$757$	−28.2854	−1.02805	−0.514024	+	0.857776i	$0.671846\pi$
$758$	11.5886	0.420916
$759$	0	0
$760$	0	0
$761$	−14.0194	−0.508202	−0.254101	−	0.967178i	$-0.581780\pi$
$761$	−14.0194	−0.508202	−0.254101	+	0.967178i	$0.581780\pi$
$762$	−23.9494	−0.867594
$763$	2.59758	0.0940389
$764$	−0.251756	−0.00910823
$765$	0	0
$766$	−1.62474	−0.0587041
$767$	13.8206	0.499033
$768$	1.59029	0.0573846
$769$	12.8224	0.462388	0.231194	−	0.972908i	$-0.425737\pi$
$769$	12.8224	0.462388	0.231194	+	0.972908i	$0.425737\pi$
$770$	0	0
$771$	20.3095	0.731427
$772$	−1.04027	−0.0374400
$773$	−23.2818	−0.837390	−0.418695	−	0.908127i	$-0.637512\pi$
$773$	−23.2818	−0.837390	−0.418695	+	0.908127i	$0.637512\pi$
$774$	−4.79055	−0.172193
$775$	0	0
$776$	30.3325	1.08887
$777$	0.554408	0.0198893
$778$	−30.6092	−1.09739
$779$	14.6752	0.525792
$780$	0	0
$781$	0	0
$782$	−2.46727	−0.0882295
$783$	−4.44505	−0.158853
$784$	−59.3938	−2.12121
$785$	0	0
$786$	28.3007	1.00945
$787$	17.8314	0.635622	0.317811	−	0.948154i	$-0.397052\pi$
$787$	17.8314	0.635622	0.317811	+	0.948154i	$0.397052\pi$
$788$	−0.319118	−0.0113681
$789$	16.6324	0.592128
$790$	0	0
$791$	−40.8202	−1.45140
$792$	0	0
$793$	11.5842	0.411366
$794$	44.1302	1.56612
$795$	0	0
$796$	−0.422200	−0.0149645
$797$	10.3839	0.367817	0.183908	−	0.982943i	$-0.441125\pi$
$797$	10.3839	0.367817	0.183908	+	0.982943i	$0.441125\pi$
$798$	10.5177	0.372324
$799$	−39.3870	−1.39341
$800$	0	0
$801$	9.17773	0.324279
$802$	25.2806	0.892691
$803$	0	0
$804$	−0.666280	−0.0234979
$805$	0	0
$806$	42.1759	1.48558
$807$	−19.0169	−0.669426
$808$	26.8874	0.945894
$809$	34.2998	1.20592	0.602958	−	0.797773i	$-0.293989\pi$
$809$	34.2998	1.20592	0.602958	+	0.797773i	$0.293989\pi$
$810$	0	0
$811$	−37.2747	−1.30889	−0.654447	−	0.756108i	$-0.727098\pi$
$811$	−37.2747	−1.30889	−0.654447	+	0.756108i	$0.727098\pi$
$812$	−1.39479	−0.0489476
$813$	9.61272	0.337133
$814$	0	0
$815$	0	0
$816$	14.6660	0.513411
$817$	5.50861	0.192722
$818$	7.32791	0.256214
$819$	15.4392	0.539490
$820$	0	0
$821$	20.5531	0.717309	0.358655	−	0.933470i	$-0.383236\pi$
$821$	20.5531	0.717309	0.358655	+	0.933470i	$0.383236\pi$
$822$	20.3534	0.709907
$823$	−35.9672	−1.25374	−0.626869	−	0.779124i	$-0.715664\pi$
$823$	−35.9672	−1.25374	−0.626869	+	0.779124i	$0.715664\pi$
$824$	−45.5148	−1.58558
$825$	0	0
$826$	−27.8523	−0.969106
$827$	33.6155	1.16892	0.584462	−	0.811421i	$-0.301305\pi$
$827$	33.6155	1.16892	0.584462	+	0.811421i	$0.301305\pi$
$828$	−0.0310014	−0.00107737
$829$	4.07809	0.141638	0.0708189	−	0.997489i	$-0.477439\pi$
$829$	4.07809	0.141638	0.0708189	+	0.997489i	$0.477439\pi$
$830$	0	0
$831$	10.6238	0.368535
$832$	26.9191	0.933252
$833$	58.3738	2.02253
$834$	−25.6510	−0.888220
$835$	0	0
$836$	0	0
$837$	9.29251	0.321196
$838$	15.0721	0.520656
$839$	−22.5177	−0.777396	−0.388698	−	0.921365i	$-0.627075\pi$
$839$	−22.5177	−0.777396	−0.388698	+	0.921365i	$0.627075\pi$
$840$	0	0
$841$	−9.24157	−0.318675
$842$	3.19934	0.110256
$843$	−20.1695	−0.694674
$844$	1.06594	0.0366911
$845$	0	0
$846$	14.4261	0.495980
$847$	0	0
$848$	−22.8989	−0.786350
$849$	−18.8548	−0.647094
$850$	0	0
$851$	−0.0547744	−0.00187764
$852$	0.446390	0.0152931
$853$	−28.8843	−0.988979	−0.494489	−	0.869184i	$-0.664645\pi$
$853$	−28.8843	−0.988979	−0.494489	+	0.869184i	$0.664645\pi$
$854$	−23.3453	−0.798858
$855$	0	0
$856$	−53.4657	−1.82742
$857$	−4.35431	−0.148741	−0.0743703	−	0.997231i	$-0.523695\pi$
$857$	−4.35431	−0.148741	−0.0743703	+	0.997231i	$0.523695\pi$
$858$	0	0
$859$	12.6726	0.432382	0.216191	−	0.976351i	$-0.430637\pi$
$859$	12.6726	0.432382	0.216191	+	0.976351i	$0.430637\pi$
$860$	0	0
$861$	−43.4130	−1.47951
$862$	−24.7050	−0.841455
$863$	56.9196	1.93757	0.968783	−	0.247912i	$-0.0797443\pi$
$863$	56.9196	1.93757	0.968783	+	0.247912i	$0.0797443\pi$
$864$	0.375096	0.0127610
$865$	0	0
$866$	−5.87817	−0.199748
$867$	2.58589	0.0878214
$868$	2.91585	0.0989705
$869$	0	0
$870$	0	0
$871$	−32.7831	−1.11081
$872$	1.57789	0.0534341
$873$	10.5564	0.357281
$874$	−1.03913	−0.0351492
$875$	0	0
$876$	1.08919	0.0368004
$877$	−13.2024	−0.445813	−0.222907	−	0.974840i	$-0.571554\pi$
$877$	−13.2024	−0.445813	−0.222907	+	0.974840i	$0.571554\pi$
$878$	−9.72303	−0.328136
$879$	2.57042	0.0866982
$880$	0	0
$881$	−5.94805	−0.200395	−0.100197	−	0.994968i	$-0.531947\pi$
$881$	−5.94805	−0.200395	−0.100197	+	0.994968i	$0.531947\pi$
$882$	−21.3804	−0.719914
$883$	−4.50950	−0.151757	−0.0758784	−	0.997117i	$-0.524176\pi$
$883$	−4.50950	−0.151757	−0.0758784	+	0.997117i	$0.524176\pi$
$884$	−0.822021	−0.0276476
$885$	0	0
$886$	21.7527	0.730797
$887$	−40.5545	−1.36169	−0.680843	−	0.732429i	$-0.738386\pi$
$887$	−40.5545	−1.36169	−0.680843	+	0.732429i	$0.738386\pi$
$888$	0.336772	0.0113013
$889$	−81.4682	−2.73236
$890$	0	0
$891$	0	0
$892$	0.236690	0.00792498
$893$	−16.5885	−0.555112
$894$	−9.14476	−0.305847
$895$	0	0
$896$	−50.7008	−1.69379
$897$	−1.52536	−0.0509304
$898$	−14.8906	−0.496907
$899$	−41.3056	−1.37762
$900$	0	0
$901$	22.5056	0.749771
$902$	0	0
$903$	−16.2959	−0.542295
$904$	−24.7960	−0.824704
$905$	0	0
$906$	−2.98516	−0.0991752
$907$	−37.3902	−1.24152	−0.620761	−	0.784000i	$-0.713176\pi$
$907$	−37.3902	−1.24152	−0.620761	+	0.784000i	$0.713176\pi$
$908$	−0.755071	−0.0250579
$909$	9.35743	0.310366
$910$	0	0
$911$	38.6363	1.28008	0.640039	−	0.768343i	$-0.278918\pi$
$911$	38.6363	1.28008	0.640039	+	0.768343i	$0.278918\pi$
$912$	6.17681	0.204535
$913$	0	0
$914$	29.6288	0.980032
$915$	0	0
$916$	−1.08666	−0.0359041
$917$	96.2702	3.17912
$918$	5.27940	0.174246
$919$	48.3704	1.59559	0.797796	−	0.602928i	$-0.205999\pi$
$919$	48.3704	1.59559	0.797796	+	0.602928i	$0.205999\pi$
$920$	0	0
$921$	−14.8489	−0.489287
$922$	−51.9627	−1.71130
$923$	21.9638	0.722947
$924$	0	0
$925$	0	0
$926$	−38.3669	−1.26081
$927$	−15.8402	−0.520262
$928$	−1.66732	−0.0547324
$929$	−14.5149	−0.476218	−0.238109	−	0.971238i	$-0.576528\pi$
$929$	−14.5149	−0.476218	−0.238109	+	0.971238i	$0.576528\pi$
$930$	0	0
$931$	24.5851	0.805744
$932$	1.35695	0.0444483
$933$	2.64783	0.0866862
$934$	52.8192	1.72830
$935$	0	0
$936$	9.37848	0.306545
$937$	−46.0328	−1.50383	−0.751913	−	0.659262i	$-0.770869\pi$
$937$	−46.0328	−1.50383	−0.751913	+	0.659262i	$0.770869\pi$
$938$	66.0669	2.15716
$939$	−34.8449	−1.13712
$940$	0	0
$941$	39.1113	1.27499	0.637496	−	0.770454i	$-0.279970\pi$
$941$	39.1113	1.27499	0.637496	+	0.770454i	$0.279970\pi$
$942$	−9.77650	−0.318535
$943$	4.28912	0.139673
$944$	−16.3570	−0.532374
$945$	0	0
$946$	0	0
$947$	35.4165	1.15088	0.575442	−	0.817843i	$-0.304830\pi$
$947$	35.4165	1.15088	0.575442	+	0.817843i	$0.304830\pi$
$948$	−0.370002	−0.0120171
$949$	53.5917	1.73966
$950$	0	0
$951$	31.6314	1.02572
$952$	51.6024	1.67244
$953$	−0.110378	−0.00357548	−0.00178774	−	0.999998i	$-0.500569\pi$
$953$	−0.110378	−0.00357548	−0.00178774	+	0.999998i	$0.500569\pi$
$954$	−8.24305	−0.266879
$955$	0	0
$956$	−0.710478	−0.0229785
$957$	0	0
$958$	−52.3719	−1.69206
$959$	69.2360	2.23575
$960$	0	0
$961$	55.3507	1.78551
$962$	0.531957	0.0171510
$963$	−18.6073	−0.599612
$964$	0.404400	0.0130248
$965$	0	0
$966$	3.07403	0.0989052
$967$	48.5424	1.56102	0.780510	−	0.625143i	$-0.214960\pi$
$967$	48.5424	1.56102	0.780510	+	0.625143i	$0.214960\pi$
$968$	0	0
$969$	−6.07074	−0.195020
$970$	0	0
$971$	−53.1531	−1.70577	−0.852883	−	0.522103i	$-0.825148\pi$
$971$	−53.1531	−1.70577	−0.852883	+	0.522103i	$0.825148\pi$
$972$	0.0663358	0.00212772
$973$	−87.2565	−2.79731
$974$	−0.278144	−0.00891231
$975$	0	0
$976$	−13.7101	−0.438849
$977$	−56.2770	−1.80046	−0.900230	−	0.435416i	$-0.856601\pi$
$977$	−56.2770	−1.80046	−0.900230	+	0.435416i	$0.856601\pi$
$978$	−3.01475	−0.0964010
$979$	0	0
$980$	0	0
$981$	0.549142	0.0175328
$982$	51.1322	1.63169
$983$	35.6247	1.13625	0.568126	−	0.822942i	$-0.307669\pi$
$983$	35.6247	1.13625	0.568126	+	0.822942i	$0.307669\pi$
$984$	−26.3710	−0.840677
$985$	0	0
$986$	−23.4672	−0.747347
$987$	49.0731	1.56202
$988$	−0.346208	−0.0110143
$989$	1.61001	0.0511952
$990$	0	0
$991$	−36.2187	−1.15053	−0.575263	−	0.817969i	$-0.695100\pi$
$991$	−36.2187	−1.15053	−0.575263	+	0.817969i	$0.695100\pi$
$992$	3.48558	0.110667
$993$	16.7230	0.530689
$994$	−44.2630	−1.40394
$995$	0	0
$996$	1.12726	0.0357185
$997$	−18.1791	−0.575737	−0.287869	−	0.957670i	$-0.592947\pi$
$997$	−18.1791	−0.575737	−0.287869	+	0.957670i	$0.592947\pi$
$998$	−35.7568	−1.13186
$999$	0.117205	0.00370819

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dk.1.2	✓	5
5.4	even	2		9075.2.a.dn.1.4	yes	5
11.10	odd	2		9075.2.a.dm.1.4	yes	5
55.54	odd	2		9075.2.a.dl.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.dk.1.2	✓	5	1.1	even	1	trivial
9075.2.a.dl.1.2	yes	5	55.54	odd	2
9075.2.a.dm.1.4	yes	5	11.10	odd	2
9075.2.a.dn.1.4	yes	5	5.4	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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-1.39056 q^{2} -1.00000 q^{3} -0.0663358 q^{4} +1.39056 q^{6} +4.73026 q^{7} +2.87337 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.39056 q^{2} -1.00000 q^{3} -0.0663358 q^{4} +1.39056 q^{6} +4.73026 q^{7} +2.87337 q^{8} +1.00000 q^{9} +0.0663358 q^{12} +3.26393 q^{13} -6.57772 q^{14} -3.86293 q^{16} +3.79659 q^{17} -1.39056 q^{18} +1.59900 q^{19} -4.73026 q^{21} +0.467340 q^{23} -2.87337 q^{24} -4.53870 q^{26} -1.00000 q^{27} -0.313785 q^{28} +4.44505 q^{29} -9.29251 q^{31} -0.375096 q^{32} -5.27940 q^{34} -0.0663358 q^{36} -0.117205 q^{37} -2.22350 q^{38} -3.26393 q^{39} +9.17773 q^{41} +6.57772 q^{42} +3.44505 q^{43} -0.649865 q^{46} -10.3743 q^{47} +3.86293 q^{48} +15.3753 q^{49} -3.79659 q^{51} -0.216516 q^{52} +5.92785 q^{53} +1.39056 q^{54} +13.5918 q^{56} -1.59900 q^{57} -6.18111 q^{58} +4.23434 q^{59} +3.54914 q^{61} +12.9218 q^{62} +4.73026 q^{63} +8.24745 q^{64} -10.0440 q^{67} -0.251850 q^{68} -0.467340 q^{69} +6.72924 q^{71} +2.87337 q^{72} +16.4194 q^{73} +0.162980 q^{74} -0.106071 q^{76} +4.53870 q^{78} -5.57772 q^{79} +1.00000 q^{81} -12.7622 q^{82} +16.9932 q^{83} +0.313785 q^{84} -4.79055 q^{86} -4.44505 q^{87} +9.17773 q^{89} +15.4392 q^{91} -0.0310014 q^{92} +9.29251 q^{93} +14.4261 q^{94} +0.375096 q^{96} +10.5564 q^{97} -21.3804 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 5 q^{3} + 7 q^{4} + q^{6} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - q^2 - 5 * q^3 + 7 * q^4 + q^6 + 4 * q^7 + 5 * q^9	\( 5 q - q^{2} - 5 q^{3} + 7 q^{4} + q^{6} + 4 q^{7} + 5 q^{9} - 7 q^{12} - 4 q^{13} + 6 q^{14} + 15 q^{16} - 8 q^{17} - q^{18} - 6 q^{19} - 4 q^{21} + 9 q^{23} + 13 q^{26} - 5 q^{27} + 17 q^{28} - 2 q^{29} - 3 q^{31} - 11 q^{32} + 9 q^{34} + 7 q^{36} - q^{37} - 3 q^{38} + 4 q^{39} - q^{41} - 6 q^{42} - 7 q^{43} + 3 q^{46} + 14 q^{47} - 15 q^{48} + 17 q^{49} + 8 q^{51} - 20 q^{52} - 3 q^{53} + q^{54} + 23 q^{56} + 6 q^{57} - 27 q^{58} + 18 q^{59} + 2 q^{61} + 61 q^{62} + 4 q^{63} + 30 q^{64} - 12 q^{67} - 39 q^{68} - 9 q^{69} + 8 q^{71} - 16 q^{73} + 40 q^{74} - 61 q^{76} - 13 q^{78} + 11 q^{79} + 5 q^{81} - 20 q^{82} + 39 q^{83} - 17 q^{84} - 26 q^{86} + 2 q^{87} - q^{89} + 13 q^{91} + 26 q^{92} + 3 q^{93} + 16 q^{94} + 11 q^{96} - 11 q^{97} - 12 q^{98}+O(q^{100}) \) 5 * q - q^2 - 5 * q^3 + 7 * q^4 + q^6 + 4 * q^7 + 5 * q^9 - 7 * q^12 - 4 * q^13 + 6 * q^14 + 15 * q^16 - 8 * q^17 - q^18 - 6 * q^19 - 4 * q^21 + 9 * q^23 + 13 * q^26 - 5 * q^27 + 17 * q^28 - 2 * q^29 - 3 * q^31 - 11 * q^32 + 9 * q^34 + 7 * q^36 - q^37 - 3 * q^38 + 4 * q^39 - q^41 - 6 * q^42 - 7 * q^43 + 3 * q^46 + 14 * q^47 - 15 * q^48 + 17 * q^49 + 8 * q^51 - 20 * q^52 - 3 * q^53 + q^54 + 23 * q^56 + 6 * q^57 - 27 * q^58 + 18 * q^59 + 2 * q^61 + 61 * q^62 + 4 * q^63 + 30 * q^64 - 12 * q^67 - 39 * q^68 - 9 * q^69 + 8 * q^71 - 16 * q^73 + 40 * q^74 - 61 * q^76 - 13 * q^78 + 11 * q^79 + 5 * q^81 - 20 * q^82 + 39 * q^83 - 17 * q^84 - 26 * q^86 + 2 * q^87 - q^89 + 13 * q^91 + 26 * q^92 + 3 * q^93 + 16 * q^94 + 11 * q^96 - 11 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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