LMFDB - Embedded newform 675.4.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("675.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.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:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 21x^{4} + 5x^{3} + 101x^{2} + 29x + 2 $ x^6 - x^5 - 21x^4 + 5x^3 + 101x^2 + 29x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.179777$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.92742 q^{2} +16.2795 q^{4} +34.7816 q^{7} -40.7965 q^{8} +O(q^{10})$	$q-4.92742 q^{2} +16.2795 q^{4} +34.7816 q^{7} -40.7965 q^{8} -12.4586 q^{11} +37.1137 q^{13} -171.384 q^{14} +70.7858 q^{16} +102.680 q^{17} +63.7858 q^{19} +61.3890 q^{22} +100.592 q^{23} -182.875 q^{26} +566.227 q^{28} +138.980 q^{29} +24.8911 q^{31} -22.4195 q^{32} -505.947 q^{34} +186.668 q^{37} -314.300 q^{38} -255.169 q^{41} -108.471 q^{43} -202.820 q^{44} -495.659 q^{46} -223.781 q^{47} +866.760 q^{49} +604.192 q^{52} -40.5151 q^{53} -1418.97 q^{56} -684.812 q^{58} +131.550 q^{59} -336.171 q^{61} -122.649 q^{62} -455.817 q^{64} +17.8284 q^{67} +1671.58 q^{68} -992.473 q^{71} -769.455 q^{73} -919.792 q^{74} +1038.40 q^{76} -433.331 q^{77} -252.252 q^{79} +1257.33 q^{82} +234.606 q^{83} +534.484 q^{86} +508.269 q^{88} +405.641 q^{89} +1290.87 q^{91} +1637.58 q^{92} +1102.67 q^{94} -1067.80 q^{97} -4270.89 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 30 q^{4} + 36 q^{7}+O(q^{10}) $ 6 * q + 30 * q^4 + 36 * q^7	$ 6 q + 30 q^{4} + 36 q^{7} + 162 q^{13} + 42 q^{16} + 450 q^{22} + 828 q^{28} + 126 q^{31} - 534 q^{34} + 1008 q^{37} + 558 q^{43} - 834 q^{46} + 1434 q^{49} + 2610 q^{52} - 270 q^{58} + 396 q^{61} - 1134 q^{64} + 2268 q^{67} + 144 q^{73} + 912 q^{76} - 1098 q^{79} + 5544 q^{82} + 702 q^{88} - 1692 q^{91} + 2574 q^{94} + 4104 q^{97}+O(q^{100}) $ 6 * q + 30 * q^4 + 36 * q^7 + 162 * q^13 + 42 * q^16 + 450 * q^22 + 828 * q^28 + 126 * q^31 - 534 * q^34 + 1008 * q^37 + 558 * q^43 - 834 * q^46 + 1434 * q^49 + 2610 * q^52 - 270 * q^58 + 396 * q^61 - 1134 * q^64 + 2268 * q^67 + 144 * q^73 + 912 * q^76 - 1098 * q^79 + 5544 * q^82 + 702 * q^88 - 1692 * q^91 + 2574 * q^94 + 4104 * q^97

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 $

$2$	−4.92742	−1.74211	−0.871053	−	0.491188i	$-0.836563\pi$
$2$	−4.92742	−1.74211	−0.871053	+	0.491188i	$0.836563\pi$
$3$	0	0
$3$	0	0
$4$	16.2795	2.03494
$5$	0	0
$5$	0	0
$6$	0	0
$7$	34.7816	1.87803	0.939015	−	0.343876i	$-0.111740\pi$
$7$	34.7816	1.87803	0.939015	+	0.343876i	$0.111740\pi$
$8$	−40.7965	−1.80297
$9$	0	0
$10$	0	0
$11$	−12.4586	−0.341493	−0.170746	−	0.985315i	$-0.554618\pi$
$11$	−12.4586	−0.341493	−0.170746	+	0.985315i	$0.554618\pi$
$12$	0	0
$13$	37.1137	0.791807	0.395904	−	0.918292i	$-0.370431\pi$
$13$	37.1137	0.791807	0.395904	+	0.918292i	$0.370431\pi$
$14$	−171.384	−3.27173
$15$	0	0
$16$	70.7858	1.10603
$17$	102.680	1.46491	0.732457	−	0.680813i	$-0.238373\pi$
$17$	102.680	1.46491	0.732457	+	0.680813i	$0.238373\pi$
$18$	0	0
$19$	63.7858	0.770183	0.385092	−	0.922878i	$-0.374170\pi$
$19$	63.7858	0.770183	0.385092	+	0.922878i	$0.374170\pi$
$20$	0	0
$21$	0	0
$22$	61.3890	0.594917
$23$	100.592	0.911951	0.455975	−	0.889992i	$-0.349291\pi$
$23$	100.592	0.911951	0.455975	+	0.889992i	$0.349291\pi$
$24$	0	0
$25$	0	0
$26$	−182.875	−1.37941
$27$	0	0
$28$	566.227	3.82167
$29$	138.980	0.889927	0.444964	−	0.895549i	$-0.353217\pi$
$29$	138.980	0.889927	0.444964	+	0.895549i	$0.353217\pi$
$30$	0	0
$31$	24.8911	0.144212	0.0721060	−	0.997397i	$-0.477028\pi$
$31$	24.8911	0.144212	0.0721060	+	0.997397i	$0.477028\pi$
$32$	−22.4195	−0.123851
$33$	0	0
$34$	−505.947	−2.55204
$35$	0	0
$36$	0	0
$37$	186.668	0.829406	0.414703	−	0.909957i	$-0.363886\pi$
$37$	186.668	0.829406	0.414703	+	0.909957i	$0.363886\pi$
$38$	−314.300	−1.34174
$39$	0	0
$40$	0	0
$41$	−255.169	−0.971970	−0.485985	−	0.873967i	$-0.661539\pi$
$41$	−255.169	−0.971970	−0.485985	+	0.873967i	$0.661539\pi$
$42$	0	0
$43$	−108.471	−0.384691	−0.192345	−	0.981327i	$-0.561609\pi$
$43$	−108.471	−0.384691	−0.192345	+	0.981327i	$0.561609\pi$
$44$	−202.820	−0.694916
$45$	0	0
$46$	−495.659	−1.58872
$47$	−223.781	−0.694508	−0.347254	−	0.937771i	$-0.612886\pi$
$47$	−223.781	−0.694508	−0.347254	+	0.937771i	$0.612886\pi$
$48$	0	0
$49$	866.760	2.52700
$50$	0	0
$51$	0	0
$52$	604.192	1.61128
$53$	−40.5151	−0.105003	−0.0525017	−	0.998621i	$-0.516720\pi$
$53$	−40.5151	−0.105003	−0.0525017	+	0.998621i	$0.516720\pi$
$54$	0	0
$55$	0	0
$56$	−1418.97	−3.38603
$57$	0	0
$58$	−684.812	−1.55035
$59$	131.550	0.290278	0.145139	−	0.989411i	$-0.453637\pi$
$59$	131.550	0.290278	0.145139	+	0.989411i	$0.453637\pi$
$60$	0	0
$61$	−336.171	−0.705610	−0.352805	−	0.935697i	$-0.614772\pi$
$61$	−336.171	−0.705610	−0.352805	+	0.935697i	$0.614772\pi$
$62$	−122.649	−0.251233
$63$	0	0
$64$	−455.817	−0.890267
$65$	0	0
$66$	0	0
$67$	17.8284	0.0325087	0.0162543	−	0.999868i	$-0.494826\pi$
$67$	17.8284	0.0325087	0.0162543	+	0.999868i	$0.494826\pi$
$68$	1671.58	2.98101
$69$	0	0
$70$	0	0
$71$	−992.473	−1.65894	−0.829471	−	0.558549i	$-0.811358\pi$
$71$	−992.473	−1.65894	−0.829471	+	0.558549i	$0.811358\pi$
$72$	0	0
$73$	−769.455	−1.23367	−0.616835	−	0.787092i	$-0.711585\pi$
$73$	−769.455	−1.23367	−0.616835	+	0.787092i	$0.711585\pi$
$74$	−919.792	−1.44491
$75$	0	0
$76$	1038.40	1.56727
$77$	−433.331	−0.641334
$78$	0	0
$79$	−252.252	−0.359248	−0.179624	−	0.983735i	$-0.557488\pi$
$79$	−252.252	−0.359248	−0.179624	+	0.983735i	$0.557488\pi$
$80$	0	0
$81$	0	0
$82$	1257.33	1.69328
$83$	234.606	0.310258	0.155129	−	0.987894i	$-0.450421\pi$
$83$	234.606	0.310258	0.155129	+	0.987894i	$0.450421\pi$
$84$	0	0
$85$	0	0
$86$	534.484	0.670173
$87$	0	0
$88$	508.269	0.615701
$89$	405.641	0.483121	0.241561	−	0.970386i	$-0.422341\pi$
$89$	405.641	0.483121	0.241561	+	0.970386i	$0.422341\pi$
$90$	0	0
$91$	1290.87	1.48704
$92$	1637.58	1.85576
$93$	0	0
$94$	1102.67	1.20991
$95$	0	0
$96$	0	0
$97$	−1067.80	−1.11772	−0.558860	−	0.829262i	$-0.688761\pi$
$97$	−1067.80	−1.11772	−0.558860	+	0.829262i	$0.688761\pi$
$98$	−4270.89	−4.40230
$99$	0	0
$100$	0	0
$101$	1460.41	1.43877	0.719386	−	0.694611i	$-0.244423\pi$
$101$	1460.41	1.43877	0.719386	+	0.694611i	$0.244423\pi$
$102$	0	0
$103$	1503.55	1.43834	0.719170	−	0.694835i	$-0.244522\pi$
$103$	1503.55	1.43834	0.719170	+	0.694835i	$0.244522\pi$
$104$	−1514.11	−1.42760
$105$	0	0
$106$	199.635	0.182927
$107$	−1862.54	−1.68279	−0.841396	−	0.540418i	$-0.818266\pi$
$107$	−1862.54	−1.68279	−0.841396	+	0.540418i	$0.818266\pi$
$108$	0	0
$109$	−1489.05	−1.30848	−0.654242	−	0.756286i	$-0.727012\pi$
$109$	−1489.05	−1.30848	−0.654242	+	0.756286i	$0.727012\pi$
$110$	0	0
$111$	0	0
$112$	2462.05	2.07716
$113$	936.618	0.779731	0.389865	−	0.920872i	$-0.372522\pi$
$113$	936.618	0.779731	0.389865	+	0.920872i	$0.372522\pi$
$114$	0	0
$115$	0	0
$116$	2262.52	1.81094
$117$	0	0
$118$	−648.205	−0.505696
$119$	3571.37	2.75115
$120$	0	0
$121$	−1175.78	−0.883383
$122$	1656.45	1.22925
$123$	0	0
$124$	405.214	0.293462
$125$	0	0
$126$	0	0
$127$	285.249	0.199305	0.0996525	−	0.995022i	$-0.468227\pi$
$127$	285.249	0.199305	0.0996525	+	0.995022i	$0.468227\pi$
$128$	2425.36	1.67479
$129$	0	0
$130$	0	0
$131$	697.411	0.465138	0.232569	−	0.972580i	$-0.425287\pi$
$131$	697.411	0.465138	0.232569	+	0.972580i	$0.425287\pi$
$132$	0	0
$133$	2218.57	1.44643
$134$	−87.8479	−0.0566336
$135$	0	0
$136$	−4188.99	−2.64120
$137$	1557.87	0.971514	0.485757	−	0.874094i	$-0.338544\pi$
$137$	1557.87	0.971514	0.485757	+	0.874094i	$0.338544\pi$
$138$	0	0
$139$	−315.325	−0.192414	−0.0962068	−	0.995361i	$-0.530671\pi$
$139$	−315.325	−0.192414	−0.0962068	+	0.995361i	$0.530671\pi$
$140$	0	0
$141$	0	0
$142$	4890.34	2.89006
$143$	−462.386	−0.270396
$144$	0	0
$145$	0	0
$146$	3791.43	2.14919
$147$	0	0
$148$	3038.86	1.68779
$149$	1983.67	1.09066	0.545331	−	0.838221i	$-0.316404\pi$
$149$	1983.67	1.09066	0.545331	+	0.838221i	$0.316404\pi$
$150$	0	0
$151$	1057.25	0.569787	0.284894	−	0.958559i	$-0.408042\pi$
$151$	1057.25	0.569787	0.284894	+	0.958559i	$0.408042\pi$
$152$	−2602.24	−1.38862
$153$	0	0
$154$	2135.21	1.11727
$155$	0	0
$156$	0	0
$157$	2171.68	1.10394	0.551972	−	0.833862i	$-0.313875\pi$
$157$	2171.68	1.10394	0.551972	+	0.833862i	$0.313875\pi$
$158$	1242.95	0.625848
$159$	0	0
$160$	0	0
$161$	3498.75	1.71267
$162$	0	0
$163$	1219.04	0.585781	0.292890	−	0.956146i	$-0.405383\pi$
$163$	1219.04	0.585781	0.292890	+	0.956146i	$0.405383\pi$
$164$	−4154.03	−1.97790
$165$	0	0
$166$	−1156.01	−0.540502
$167$	3577.08	1.65750	0.828750	−	0.559619i	$-0.189052\pi$
$167$	3577.08	1.65750	0.828750	+	0.559619i	$0.189052\pi$
$168$	0	0
$169$	−819.572	−0.373041
$170$	0	0
$171$	0	0
$172$	−1765.86	−0.782822
$173$	−2372.95	−1.04284	−0.521422	−	0.853299i	$-0.674598\pi$
$173$	−2372.95	−1.04284	−0.521422	+	0.853299i	$0.674598\pi$
$174$	0	0
$175$	0	0
$176$	−881.895	−0.377701
$177$	0	0
$178$	−1998.76	−0.841649
$179$	774.792	0.323524	0.161762	−	0.986830i	$-0.448282\pi$
$179$	774.792	0.323524	0.161762	+	0.986830i	$0.448282\pi$
$180$	0	0
$181$	−1250.05	−0.513344	−0.256672	−	0.966499i	$-0.582626\pi$
$181$	−1250.05	−0.513344	−0.256672	+	0.966499i	$0.582626\pi$
$182$	−6360.68	−2.59058
$183$	0	0
$184$	−4103.80	−1.64422
$185$	0	0
$186$	0	0
$187$	−1279.25	−0.500258
$188$	−3643.05	−1.41328
$189$	0	0
$190$	0	0
$191$	−2141.03	−0.811097	−0.405548	−	0.914074i	$-0.632919\pi$
$191$	−2141.03	−0.811097	−0.405548	+	0.914074i	$0.632919\pi$
$192$	0	0
$193$	1829.86	0.682466	0.341233	−	0.939979i	$-0.389156\pi$
$193$	1829.86	0.682466	0.341233	+	0.939979i	$0.389156\pi$
$194$	5261.51	1.94719
$195$	0	0
$196$	14110.4	5.14228
$197$	1142.23	0.413099	0.206550	−	0.978436i	$-0.433777\pi$
$197$	1142.23	0.413099	0.206550	+	0.978436i	$0.433777\pi$
$198$	0	0
$199$	284.078	0.101195	0.0505973	−	0.998719i	$-0.483887\pi$
$199$	284.078	0.101195	0.0505973	+	0.998719i	$0.483887\pi$
$200$	0	0
$201$	0	0
$202$	−7196.04	−2.50649
$203$	4833.94	1.67131
$204$	0	0
$205$	0	0
$206$	−7408.61	−2.50574
$207$	0	0
$208$	2627.13	0.875762
$209$	−794.685	−0.263012
$210$	0	0
$211$	813.075	0.265281	0.132641	−	0.991164i	$-0.457654\pi$
$211$	813.075	0.265281	0.132641	+	0.991164i	$0.457654\pi$
$212$	−659.566	−0.213675
$213$	0	0
$214$	9177.54	2.93160
$215$	0	0
$216$	0	0
$217$	865.752	0.270834
$218$	7337.16	2.27952
$219$	0	0
$220$	0	0
$221$	3810.83	1.15993
$222$	0	0
$223$	−4936.43	−1.48237	−0.741183	−	0.671303i	$-0.765735\pi$
$223$	−4936.43	−1.48237	−0.741183	+	0.671303i	$0.765735\pi$
$224$	−779.785	−0.232596
$225$	0	0
$226$	−4615.11	−1.35837
$227$	−3589.88	−1.04964	−0.524820	−	0.851213i	$-0.675867\pi$
$227$	−3589.88	−1.04964	−0.524820	+	0.851213i	$0.675867\pi$
$228$	0	0
$229$	−3869.82	−1.11670	−0.558352	−	0.829604i	$-0.688566\pi$
$229$	−3869.82	−1.11670	−0.558352	+	0.829604i	$0.688566\pi$
$230$	0	0
$231$	0	0
$232$	−5669.89	−1.60451
$233$	−2983.29	−0.838805	−0.419403	−	0.907800i	$-0.637760\pi$
$233$	−2983.29	−0.838805	−0.419403	+	0.907800i	$0.637760\pi$
$234$	0	0
$235$	0	0
$236$	2141.57	0.590698
$237$	0	0
$238$	−17597.7	−4.79280
$239$	2627.57	0.711145	0.355572	−	0.934649i	$-0.384286\pi$
$239$	2627.57	0.711145	0.355572	+	0.934649i	$0.384286\pi$
$240$	0	0
$241$	−4236.43	−1.13233	−0.566167	−	0.824291i	$-0.691574\pi$
$241$	−4236.43	−1.13233	−0.566167	+	0.824291i	$0.691574\pi$
$242$	5793.58	1.53895
$243$	0	0
$244$	−5472.68	−1.43587
$245$	0	0
$246$	0	0
$247$	2367.33	0.609836
$248$	−1015.47	−0.260010
$249$	0	0
$250$	0	0
$251$	6298.35	1.58386	0.791929	−	0.610614i	$-0.209077\pi$
$251$	6298.35	1.58386	0.791929	+	0.610614i	$0.209077\pi$
$252$	0	0
$253$	−1253.24	−0.311425
$254$	−1405.54	−0.347211
$255$	0	0
$256$	−8304.22	−2.02740
$257$	3034.13	0.736434	0.368217	−	0.929740i	$-0.379968\pi$
$257$	3034.13	0.736434	0.368217	+	0.929740i	$0.379968\pi$
$258$	0	0
$259$	6492.61	1.55765
$260$	0	0
$261$	0	0
$262$	−3436.44	−0.810320
$263$	−5212.09	−1.22202	−0.611010	−	0.791623i	$-0.709237\pi$
$263$	−5212.09	−1.22202	−0.611010	+	0.791623i	$0.709237\pi$
$264$	0	0
$265$	0	0
$266$	−10931.9	−2.51983
$267$	0	0
$268$	290.237	0.0661531
$269$	1851.11	0.419571	0.209785	−	0.977747i	$-0.432724\pi$
$269$	1851.11	0.419571	0.209785	+	0.977747i	$0.432724\pi$
$270$	0	0
$271$	922.751	0.206838	0.103419	−	0.994638i	$-0.467022\pi$
$271$	922.751	0.206838	0.103419	+	0.994638i	$0.467022\pi$
$272$	7268.29	1.62024
$273$	0	0
$274$	−7676.26	−1.69248
$275$	0	0
$276$	0	0
$277$	1932.26	0.419126	0.209563	−	0.977795i	$-0.432796\pi$
$277$	1932.26	0.419126	0.209563	+	0.977795i	$0.432796\pi$
$278$	1553.74	0.335205
$279$	0	0
$280$	0	0
$281$	−7020.02	−1.49032	−0.745159	−	0.666887i	$-0.767626\pi$
$281$	−7020.02	−1.49032	−0.745159	+	0.666887i	$0.767626\pi$
$282$	0	0
$283$	1497.15	0.314475	0.157237	−	0.987561i	$-0.449741\pi$
$283$	1497.15	0.314475	0.157237	+	0.987561i	$0.449741\pi$
$284$	−16157.0	−3.37584
$285$	0	0
$286$	2278.37	0.471059
$287$	−8875.21	−1.82539
$288$	0	0
$289$	5630.17	1.14597
$290$	0	0
$291$	0	0
$292$	−12526.3	−2.51044
$293$	−7003.84	−1.39648	−0.698240	−	0.715863i	$-0.746033\pi$
$293$	−7003.84	−1.39648	−0.698240	+	0.715863i	$0.746033\pi$
$294$	0	0
$295$	0	0
$296$	−7615.40	−1.49539
$297$	0	0
$298$	−9774.38	−1.90005
$299$	3733.34	0.722089
$300$	0	0
$301$	−3772.80	−0.722461
$302$	−5209.52	−0.992630
$303$	0	0
$304$	4515.14	0.851845
$305$	0	0
$306$	0	0
$307$	1297.09	0.241137	0.120568	−	0.992705i	$-0.461528\pi$
$307$	1297.09	0.241137	0.120568	+	0.992705i	$0.461528\pi$
$308$	−7054.41	−1.30507
$309$	0	0
$310$	0	0
$311$	−3073.25	−0.560348	−0.280174	−	0.959949i	$-0.590392\pi$
$311$	−3073.25	−0.560348	−0.280174	+	0.959949i	$0.590392\pi$
$312$	0	0
$313$	8290.40	1.49713	0.748564	−	0.663062i	$-0.230744\pi$
$313$	8290.40	1.49713	0.748564	+	0.663062i	$0.230744\pi$
$314$	−10700.8	−1.92319
$315$	0	0
$316$	−4106.53	−0.731046
$317$	2882.26	0.510675	0.255338	−	0.966852i	$-0.417813\pi$
$317$	2882.26	0.510675	0.255338	+	0.966852i	$0.417813\pi$
$318$	0	0
$319$	−1731.50	−0.303904
$320$	0	0
$321$	0	0
$322$	−17239.8	−2.98366
$323$	6549.53	1.12825
$324$	0	0
$325$	0	0
$326$	−6006.71	−1.02049
$327$	0	0
$328$	10410.0	1.75243
$329$	−7783.48	−1.30431
$330$	0	0
$331$	9505.77	1.57850	0.789251	−	0.614071i	$-0.210469\pi$
$331$	9505.77	1.57850	0.789251	+	0.614071i	$0.210469\pi$
$332$	3819.27	0.631355
$333$	0	0
$334$	−17625.8	−2.88754
$335$	0	0
$336$	0	0
$337$	5929.45	0.958451	0.479225	−	0.877692i	$-0.340918\pi$
$337$	5929.45	0.958451	0.479225	+	0.877692i	$0.340918\pi$
$338$	4038.38	0.649878
$339$	0	0
$340$	0	0
$341$	−310.109	−0.0492473
$342$	0	0
$343$	18217.2	2.86775
$344$	4425.25	0.693586
$345$	0	0
$346$	11692.5	1.81675
$347$	4199.08	0.649621	0.324811	−	0.945779i	$-0.394699\pi$
$347$	4199.08	0.649621	0.324811	+	0.945779i	$0.394699\pi$
$348$	0	0
$349$	−4490.59	−0.688756	−0.344378	−	0.938831i	$-0.611910\pi$
$349$	−4490.59	−0.688756	−0.344378	+	0.938831i	$0.611910\pi$
$350$	0	0
$351$	0	0
$352$	279.316	0.0422943
$353$	3992.72	0.602014	0.301007	−	0.953622i	$-0.402677\pi$
$353$	3992.72	0.602014	0.301007	+	0.953622i	$0.402677\pi$
$354$	0	0
$355$	0	0
$356$	6603.62	0.983121
$357$	0	0
$358$	−3817.73	−0.563613
$359$	1630.18	0.239659	0.119830	−	0.992794i	$-0.461765\pi$
$359$	1630.18	0.239659	0.119830	+	0.992794i	$0.461765\pi$
$360$	0	0
$361$	−2790.37	−0.406818
$362$	6159.51	0.894301
$363$	0	0
$364$	21014.8	3.02603
$365$	0	0
$366$	0	0
$367$	−1402.36	−0.199462	−0.0997312	−	0.995014i	$-0.531798\pi$
$367$	−1402.36	−0.199462	−0.0997312	+	0.995014i	$0.531798\pi$
$368$	7120.48	1.00864
$369$	0	0
$370$	0	0
$371$	−1409.18	−0.197200
$372$	0	0
$373$	4988.43	0.692469	0.346235	−	0.938148i	$-0.387460\pi$
$373$	4988.43	0.692469	0.346235	+	0.938148i	$0.387460\pi$
$374$	6303.41	0.871502
$375$	0	0
$376$	9129.51	1.25218
$377$	5158.05	0.704651
$378$	0	0
$379$	9475.33	1.28421	0.642104	−	0.766617i	$-0.278062\pi$
$379$	9475.33	1.28421	0.642104	+	0.766617i	$0.278062\pi$
$380$	0	0
$381$	0	0
$382$	10549.8	1.41302
$383$	7172.68	0.956937	0.478469	−	0.878105i	$-0.341192\pi$
$383$	7172.68	0.956937	0.478469	+	0.878105i	$0.341192\pi$
$384$	0	0
$385$	0	0
$386$	−9016.47	−1.18893
$387$	0	0
$388$	−17383.3	−2.27449
$389$	64.9700	0.00846815	0.00423407	−	0.999991i	$-0.498652\pi$
$389$	64.9700	0.00846815	0.00423407	+	0.999991i	$0.498652\pi$
$390$	0	0
$391$	10328.8	1.33593
$392$	−35360.8	−4.55610
$393$	0	0
$394$	−5628.25	−0.719663
$395$	0	0
$396$	0	0
$397$	−9638.18	−1.21845	−0.609227	−	0.792996i	$-0.708520\pi$
$397$	−9638.18	−1.21845	−0.609227	+	0.792996i	$0.708520\pi$
$398$	−1399.77	−0.176292
$399$	0	0
$400$	0	0
$401$	7130.33	0.887959	0.443979	−	0.896037i	$-0.353566\pi$
$401$	7130.33	0.887959	0.443979	+	0.896037i	$0.353566\pi$
$402$	0	0
$403$	923.801	0.114188
$404$	23774.7	2.92781
$405$	0	0
$406$	−23818.9	−2.91160
$407$	−2325.63	−0.283236
$408$	0	0
$409$	−950.932	−0.114965	−0.0574824	−	0.998347i	$-0.518307\pi$
$409$	−950.932	−0.114965	−0.0574824	+	0.998347i	$0.518307\pi$
$410$	0	0
$411$	0	0
$412$	24477.0	2.92693
$413$	4575.54	0.545151
$414$	0	0
$415$	0	0
$416$	−832.069	−0.0980662
$417$	0	0
$418$	3915.75	0.458195
$419$	16381.7	1.91002	0.955008	−	0.296579i	$-0.0958457\pi$
$419$	16381.7	1.91002	0.955008	+	0.296579i	$0.0958457\pi$
$420$	0	0
$421$	−408.506	−0.0472907	−0.0236453	−	0.999720i	$-0.507527\pi$
$421$	−408.506	−0.0472907	−0.0236453	+	0.999720i	$0.507527\pi$
$422$	−4006.36	−0.462148
$423$	0	0
$424$	1652.88	0.189318
$425$	0	0
$426$	0	0
$427$	−11692.6	−1.32516
$428$	−30321.3	−3.42438
$429$	0	0
$430$	0	0
$431$	−8202.47	−0.916703	−0.458352	−	0.888771i	$-0.651560\pi$
$431$	−8202.47	−0.916703	−0.458352	+	0.888771i	$0.651560\pi$
$432$	0	0
$433$	1435.71	0.159343	0.0796716	−	0.996821i	$-0.474613\pi$
$433$	1435.71	0.159343	0.0796716	+	0.996821i	$0.474613\pi$
$434$	−4265.93	−0.471823
$435$	0	0
$436$	−24240.9	−2.66268
$437$	6416.34	0.702369
$438$	0	0
$439$	−13629.3	−1.48176	−0.740880	−	0.671637i	$-0.765591\pi$
$439$	−13629.3	−1.48176	−0.740880	+	0.671637i	$0.765591\pi$
$440$	0	0
$441$	0	0
$442$	−18777.6	−2.02072
$443$	−8832.42	−0.947271	−0.473636	−	0.880721i	$-0.657059\pi$
$443$	−8832.42	−0.947271	−0.473636	+	0.880721i	$0.657059\pi$
$444$	0	0
$445$	0	0
$446$	24323.9	2.58244
$447$	0	0
$448$	−15854.0	−1.67195
$449$	−11128.7	−1.16970	−0.584848	−	0.811143i	$-0.698846\pi$
$449$	−11128.7	−1.16970	−0.584848	+	0.811143i	$0.698846\pi$
$450$	0	0
$451$	3179.06	0.331921
$452$	15247.7	1.58670
$453$	0	0
$454$	17688.8	1.82859
$455$	0	0
$456$	0	0
$457$	16984.0	1.73846	0.869231	−	0.494405i	$-0.164614\pi$
$457$	16984.0	1.73846	0.869231	+	0.494405i	$0.164614\pi$
$458$	19068.3	1.94542
$459$	0	0
$460$	0	0
$461$	15698.3	1.58600	0.792999	−	0.609223i	$-0.208519\pi$
$461$	15698.3	1.58600	0.792999	+	0.609223i	$0.208519\pi$
$462$	0	0
$463$	−7996.32	−0.802636	−0.401318	−	0.915939i	$-0.631448\pi$
$463$	−7996.32	−0.802636	−0.401318	+	0.915939i	$0.631448\pi$
$464$	9837.80	0.984285
$465$	0	0
$466$	14699.9	1.46129
$467$	6347.66	0.628982	0.314491	−	0.949260i	$-0.398166\pi$
$467$	6347.66	0.628982	0.314491	+	0.949260i	$0.398166\pi$
$468$	0	0
$469$	620.099	0.0610523
$470$	0	0
$471$	0	0
$472$	−5366.80	−0.523363
$473$	1351.40	0.131369
$474$	0	0
$475$	0	0
$476$	58140.1	5.59842
$477$	0	0
$478$	−12947.2	−1.23889
$479$	−5794.01	−0.552683	−0.276341	−	0.961060i	$-0.589122\pi$
$479$	−5794.01	−0.552683	−0.276341	+	0.961060i	$0.589122\pi$
$480$	0	0
$481$	6927.94	0.656729
$482$	20874.7	1.97265
$483$	0	0
$484$	−19141.1	−1.79763
$485$	0	0
$486$	0	0
$487$	10363.7	0.964323	0.482162	−	0.876082i	$-0.339852\pi$
$487$	10363.7	0.964323	0.482162	+	0.876082i	$0.339852\pi$
$488$	13714.6	1.27219
$489$	0	0
$490$	0	0
$491$	−21420.5	−1.96882	−0.984412	−	0.175878i	$-0.943723\pi$
$491$	−21420.5	−1.96882	−0.984412	+	0.175878i	$0.943723\pi$
$492$	0	0
$493$	14270.4	1.30367
$494$	−11664.8	−1.06240
$495$	0	0
$496$	1761.94	0.159503
$497$	−34519.8	−3.11554
$498$	0	0
$499$	1661.78	0.149081	0.0745404	−	0.997218i	$-0.476251\pi$
$499$	1661.78	0.149081	0.0745404	+	0.997218i	$0.476251\pi$
$500$	0	0
$501$	0	0
$502$	−31034.6	−2.75925
$503$	13256.5	1.17510	0.587551	−	0.809187i	$-0.300092\pi$
$503$	13256.5	1.17510	0.587551	+	0.809187i	$0.300092\pi$
$504$	0	0
$505$	0	0
$506$	6175.23	0.542535
$507$	0	0
$508$	4643.70	0.405573
$509$	−13818.0	−1.20329	−0.601643	−	0.798765i	$-0.705487\pi$
$509$	−13818.0	−1.20329	−0.601643	+	0.798765i	$0.705487\pi$
$510$	0	0
$511$	−26762.9	−2.31687
$512$	21515.6	1.85715
$513$	0	0
$514$	−14950.4	−1.28295
$515$	0	0
$516$	0	0
$517$	2788.01	0.237170
$518$	−31991.8	−2.71359
$519$	0	0
$520$	0	0
$521$	−11817.3	−0.993718	−0.496859	−	0.867831i	$-0.665513\pi$
$521$	−11817.3	−0.993718	−0.496859	+	0.867831i	$0.665513\pi$
$522$	0	0
$523$	−3358.57	−0.280803	−0.140401	−	0.990095i	$-0.544839\pi$
$523$	−3358.57	−0.280803	−0.140401	+	0.990095i	$0.544839\pi$
$524$	11353.5	0.946526
$525$	0	0
$526$	25682.2	2.12889
$527$	2555.81	0.211258
$528$	0	0
$529$	−2048.27	−0.168346
$530$	0	0
$531$	0	0
$532$	36117.3	2.94339
$533$	−9470.29	−0.769613
$534$	0	0
$535$	0	0
$536$	−727.335	−0.0586121
$537$	0	0
$538$	−9121.23	−0.730937
$539$	−10798.7	−0.862952
$540$	0	0
$541$	−1111.42	−0.0883243	−0.0441622	−	0.999024i	$-0.514062\pi$
$541$	−1111.42	−0.0883243	−0.0441622	+	0.999024i	$0.514062\pi$
$542$	−4546.78	−0.360334
$543$	0	0
$544$	−2302.03	−0.181431
$545$	0	0
$546$	0	0
$547$	−4216.48	−0.329586	−0.164793	−	0.986328i	$-0.552696\pi$
$547$	−4216.48	−0.329586	−0.164793	+	0.986328i	$0.552696\pi$
$548$	25361.2	1.97697
$549$	0	0
$550$	0	0
$551$	8864.94	0.685407
$552$	0	0
$553$	−8773.73	−0.674678
$554$	−9521.04	−0.730163
$555$	0	0
$556$	−5133.33	−0.391549
$557$	1403.06	0.106732	0.0533658	−	0.998575i	$-0.483005\pi$
$557$	1403.06	0.106732	0.0533658	+	0.998575i	$0.483005\pi$
$558$	0	0
$559$	−4025.77	−0.304601
$560$	0	0
$561$	0	0
$562$	34590.6	2.59629
$563$	−5088.40	−0.380907	−0.190453	−	0.981696i	$-0.560996\pi$
$563$	−5088.40	−0.380907	−0.190453	+	0.981696i	$0.560996\pi$
$564$	0	0
$565$	0	0
$566$	−7377.10	−0.547849
$567$	0	0
$568$	40489.5	2.99102
$569$	8310.91	0.612323	0.306161	−	0.951980i	$-0.400955\pi$
$569$	8310.91	0.612323	0.306161	+	0.951980i	$0.400955\pi$
$570$	0	0
$571$	−24264.2	−1.77833	−0.889166	−	0.457585i	$-0.848714\pi$
$571$	−24264.2	−1.77833	−0.889166	+	0.457585i	$0.848714\pi$
$572$	−7527.41	−0.550239
$573$	0	0
$574$	43731.9	3.18002
$575$	0	0
$576$	0	0
$577$	2740.03	0.197693	0.0988464	−	0.995103i	$-0.468485\pi$
$577$	2740.03	0.197693	0.0988464	+	0.995103i	$0.468485\pi$
$578$	−27742.2	−1.99641
$579$	0	0
$580$	0	0
$581$	8159.99	0.582674
$582$	0	0
$583$	504.763	0.0358579
$584$	31391.1	2.22427
$585$	0	0
$586$	34510.9	2.43282
$587$	−23433.9	−1.64773	−0.823867	−	0.566784i	$-0.808188\pi$
$587$	−23433.9	−1.64773	−0.823867	+	0.566784i	$0.808188\pi$
$588$	0	0
$589$	1587.70	0.111070
$590$	0	0
$591$	0	0
$592$	13213.4	0.917347
$593$	−18399.2	−1.27414	−0.637068	−	0.770807i	$-0.719853\pi$
$593$	−18399.2	−1.27414	−0.637068	+	0.770807i	$0.719853\pi$
$594$	0	0
$595$	0	0
$596$	32293.1	2.21943
$597$	0	0
$598$	−18395.7	−1.25796
$599$	−20464.1	−1.39589	−0.697946	−	0.716151i	$-0.745902\pi$
$599$	−20464.1	−1.39589	−0.697946	+	0.716151i	$0.745902\pi$
$600$	0	0
$601$	20627.9	1.40005	0.700023	−	0.714120i	$-0.253173\pi$
$601$	20627.9	1.40005	0.700023	+	0.714120i	$0.253173\pi$
$602$	18590.2	1.25860
$603$	0	0
$604$	17211.5	1.15948
$605$	0	0
$606$	0	0
$607$	−4733.11	−0.316492	−0.158246	−	0.987400i	$-0.550584\pi$
$607$	−4733.11	−0.316492	−0.158246	+	0.987400i	$0.550584\pi$
$608$	−1430.04	−0.0953881
$609$	0	0
$610$	0	0
$611$	−8305.36	−0.549917
$612$	0	0
$613$	−13741.6	−0.905414	−0.452707	−	0.891659i	$-0.649542\pi$
$613$	−13741.6	−0.905414	−0.452707	+	0.891659i	$0.649542\pi$
$614$	−6391.32	−0.420086
$615$	0	0
$616$	17678.4	1.15631
$617$	18762.1	1.22420	0.612100	−	0.790780i	$-0.290325\pi$
$617$	18762.1	1.22420	0.612100	+	0.790780i	$0.290325\pi$
$618$	0	0
$619$	8550.03	0.555177	0.277589	−	0.960700i	$-0.410465\pi$
$619$	8550.03	0.555177	0.277589	+	0.960700i	$0.410465\pi$
$620$	0	0
$621$	0	0
$622$	15143.2	0.976186
$623$	14108.8	0.907317
$624$	0	0
$625$	0	0
$626$	−40850.3	−2.60816
$627$	0	0
$628$	35353.9	2.24646
$629$	19167.0	1.21501
$630$	0	0
$631$	−11772.7	−0.742730	−0.371365	−	0.928487i	$-0.621110\pi$
$631$	−11772.7	−0.742730	−0.371365	+	0.928487i	$0.621110\pi$
$632$	10291.0	0.647713
$633$	0	0
$634$	−14202.1	−0.889651
$635$	0	0
$636$	0	0
$637$	32168.7	2.00090
$638$	8531.82	0.529433
$639$	0	0
$640$	0	0
$641$	−13219.7	−0.814585	−0.407292	−	0.913298i	$-0.633527\pi$
$641$	−13219.7	−0.814585	−0.407292	+	0.913298i	$0.633527\pi$
$642$	0	0
$643$	29569.6	1.81355	0.906774	−	0.421617i	$-0.138537\pi$
$643$	29569.6	1.81355	0.906774	+	0.421617i	$0.138537\pi$
$644$	56957.8	3.48518
$645$	0	0
$646$	−32272.3	−1.96554
$647$	−16887.3	−1.02613	−0.513066	−	0.858349i	$-0.671490\pi$
$647$	−16887.3	−1.02613	−0.513066	+	0.858349i	$0.671490\pi$
$648$	0	0
$649$	−1638.94	−0.0991279
$650$	0	0
$651$	0	0
$652$	19845.3	1.19203
$653$	20847.4	1.24934	0.624671	−	0.780888i	$-0.285233\pi$
$653$	20847.4	1.24934	0.624671	+	0.780888i	$0.285233\pi$
$654$	0	0
$655$	0	0
$656$	−18062.4	−1.07503
$657$	0	0
$658$	38352.5	2.27224
$659$	−21697.2	−1.28255	−0.641277	−	0.767309i	$-0.721595\pi$
$659$	−21697.2	−1.28255	−0.641277	+	0.767309i	$0.721595\pi$
$660$	0	0
$661$	−2987.43	−0.175790	−0.0878952	−	0.996130i	$-0.528014\pi$
$661$	−2987.43	−0.175790	−0.0878952	+	0.996130i	$0.528014\pi$
$662$	−46838.9	−2.74992
$663$	0	0
$664$	−9571.13	−0.559385
$665$	0	0
$666$	0	0
$667$	13980.2	0.811570
$668$	58233.0	3.37291
$669$	0	0
$670$	0	0
$671$	4188.23	0.240961
$672$	0	0
$673$	23532.4	1.34785	0.673927	−	0.738798i	$-0.264606\pi$
$673$	23532.4	1.34785	0.673927	+	0.738798i	$0.264606\pi$
$674$	−29216.9	−1.66972
$675$	0	0
$676$	−13342.2	−0.759116
$677$	18598.3	1.05582	0.527911	−	0.849300i	$-0.322975\pi$
$677$	18598.3	1.05582	0.527911	+	0.849300i	$0.322975\pi$
$678$	0	0
$679$	−37139.9	−2.09911
$680$	0	0
$681$	0	0
$682$	1528.04	0.0857941
$683$	−474.740	−0.0265965	−0.0132983	−	0.999912i	$-0.504233\pi$
$683$	−474.740	−0.0265965	−0.0132983	+	0.999912i	$0.504233\pi$
$684$	0	0
$685$	0	0
$686$	−89764.0	−4.99592
$687$	0	0
$688$	−7678.23	−0.425479
$689$	−1503.67	−0.0831425
$690$	0	0
$691$	1586.50	0.0873418	0.0436709	−	0.999046i	$-0.486095\pi$
$691$	1586.50	0.0873418	0.0436709	+	0.999046i	$0.486095\pi$
$692$	−38630.4	−2.12212
$693$	0	0
$694$	−20690.6	−1.13171
$695$	0	0
$696$	0	0
$697$	−26200.8	−1.42385
$698$	22127.0	1.19989
$699$	0	0
$700$	0	0
$701$	−12342.4	−0.664999	−0.332499	−	0.943103i	$-0.607892\pi$
$701$	−12342.4	−0.664999	−0.332499	+	0.943103i	$0.607892\pi$
$702$	0	0
$703$	11906.8	0.638794
$704$	5678.85	0.304020
$705$	0	0
$706$	−19673.8	−1.04877
$707$	50795.3	2.70206
$708$	0	0
$709$	27714.9	1.46806	0.734031	−	0.679116i	$-0.237637\pi$
$709$	27714.9	1.46806	0.734031	+	0.679116i	$0.237637\pi$
$710$	0	0
$711$	0	0
$712$	−16548.7	−0.871053
$713$	2503.84	0.131514
$714$	0	0
$715$	0	0
$716$	12613.2	0.658350
$717$	0	0
$718$	−8032.60	−0.417512
$719$	−29126.8	−1.51077	−0.755387	−	0.655279i	$-0.772551\pi$
$719$	−29126.8	−1.51077	−0.755387	+	0.655279i	$0.772551\pi$
$720$	0	0
$721$	52295.8	2.70124
$722$	13749.3	0.708721
$723$	0	0
$724$	−20350.1	−1.04462
$725$	0	0
$726$	0	0
$727$	−32102.8	−1.63772	−0.818862	−	0.573990i	$-0.805395\pi$
$727$	−32102.8	−1.63772	−0.818862	+	0.573990i	$0.805395\pi$
$728$	−52663.2	−2.68108
$729$	0	0
$730$	0	0
$731$	−11137.8	−0.563539
$732$	0	0
$733$	18658.3	0.940190	0.470095	−	0.882616i	$-0.344220\pi$
$733$	18658.3	0.940190	0.470095	+	0.882616i	$0.344220\pi$
$734$	6910.03	0.347485
$735$	0	0
$736$	−2255.22	−0.112946
$737$	−222.117	−0.0111015
$738$	0	0
$739$	19105.8	0.951040	0.475520	−	0.879705i	$-0.342260\pi$
$739$	19105.8	0.951040	0.475520	+	0.879705i	$0.342260\pi$
$740$	0	0
$741$	0	0
$742$	6943.63	0.343543
$743$	−34177.1	−1.68753	−0.843765	−	0.536712i	$-0.819666\pi$
$743$	−34177.1	−1.68753	−0.843765	+	0.536712i	$0.819666\pi$
$744$	0	0
$745$	0	0
$746$	−24580.1	−1.20636
$747$	0	0
$748$	−20825.6	−1.01799
$749$	−64782.2	−3.16034
$750$	0	0
$751$	−25906.0	−1.25875	−0.629376	−	0.777101i	$-0.716690\pi$
$751$	−25906.0	−1.25875	−0.629376	+	0.777101i	$0.716690\pi$
$752$	−15840.6	−0.768146
$753$	0	0
$754$	−25415.9	−1.22758
$755$	0	0
$756$	0	0
$757$	12734.0	0.611391	0.305696	−	0.952129i	$-0.401111\pi$
$757$	12734.0	0.611391	0.305696	+	0.952129i	$0.401111\pi$
$758$	−46689.0	−2.23723
$759$	0	0
$760$	0	0
$761$	−26680.6	−1.27092	−0.635461	−	0.772133i	$-0.719190\pi$
$761$	−26680.6	−1.27092	−0.635461	+	0.772133i	$0.719190\pi$
$762$	0	0
$763$	−51791.4	−2.45737
$764$	−34854.9	−1.65053
$765$	0	0
$766$	−35342.8	−1.66709
$767$	4882.33	0.229844
$768$	0	0
$769$	−15123.8	−0.709203	−0.354602	−	0.935017i	$-0.615384\pi$
$769$	−15123.8	−0.709203	−0.354602	+	0.935017i	$0.615384\pi$
$770$	0	0
$771$	0	0
$772$	29789.1	1.38877
$773$	−32394.6	−1.50731	−0.753656	−	0.657269i	$-0.771712\pi$
$773$	−32394.6	−1.50731	−0.753656	+	0.657269i	$0.771712\pi$
$774$	0	0
$775$	0	0
$776$	43562.6	2.01522
$777$	0	0
$778$	−320.135	−0.0147524
$779$	−16276.2	−0.748595
$780$	0	0
$781$	12364.9	0.566517
$782$	−50894.2	−2.32733
$783$	0	0
$784$	61354.4	2.79493
$785$	0	0
$786$	0	0
$787$	−13328.5	−0.603699	−0.301849	−	0.953356i	$-0.597604\pi$
$787$	−13328.5	−0.603699	−0.301849	+	0.953356i	$0.597604\pi$
$788$	18594.9	0.840630
$789$	0	0
$790$	0	0
$791$	32577.1	1.46436
$792$	0	0
$793$	−12476.5	−0.558707
$794$	47491.4	2.12268
$795$	0	0
$796$	4624.64	0.205925
$797$	3904.61	0.173536	0.0867681	−	0.996229i	$-0.472346\pi$
$797$	3904.61	0.173536	0.0867681	+	0.996229i	$0.472346\pi$
$798$	0	0
$799$	−22977.9	−1.01739
$800$	0	0
$801$	0	0
$802$	−35134.1	−1.54692
$803$	9586.36	0.421289
$804$	0	0
$805$	0	0
$806$	−4551.96	−0.198928
$807$	0	0
$808$	−59579.6	−2.59406
$809$	27021.7	1.17433	0.587165	−	0.809467i	$-0.300244\pi$
$809$	27021.7	1.17433	0.587165	+	0.809467i	$0.300244\pi$
$810$	0	0
$811$	13873.1	0.600679	0.300340	−	0.953832i	$-0.402900\pi$
$811$	13873.1	0.600679	0.300340	+	0.953832i	$0.402900\pi$
$812$	78694.0	3.40101
$813$	0	0
$814$	11459.3	0.493427
$815$	0	0
$816$	0	0
$817$	−6918.93	−0.296282
$818$	4685.65	0.200281
$819$	0	0
$820$	0	0
$821$	11284.2	0.479684	0.239842	−	0.970812i	$-0.422904\pi$
$821$	11284.2	0.479684	0.239842	+	0.970812i	$0.422904\pi$
$822$	0	0
$823$	−30236.7	−1.28066	−0.640332	−	0.768098i	$-0.721203\pi$
$823$	−30236.7	−1.28066	−0.640332	+	0.768098i	$0.721203\pi$
$824$	−61339.5	−2.59328
$825$	0	0
$826$	−22545.6	−0.949712
$827$	2610.71	0.109774	0.0548871	−	0.998493i	$-0.482520\pi$
$827$	2610.71	0.109774	0.0548871	+	0.998493i	$0.482520\pi$
$828$	0	0
$829$	43070.0	1.80444	0.902222	−	0.431272i	$-0.141935\pi$
$829$	43070.0	1.80444	0.902222	+	0.431272i	$0.141935\pi$
$830$	0	0
$831$	0	0
$832$	−16917.0	−0.704920
$833$	88998.9	3.70183
$834$	0	0
$835$	0	0
$836$	−12937.1	−0.535212
$837$	0	0
$838$	−80719.4	−3.32745
$839$	19767.0	0.813388	0.406694	−	0.913564i	$-0.366681\pi$
$839$	19767.0	0.813388	0.406694	+	0.913564i	$0.366681\pi$
$840$	0	0
$841$	−5073.64	−0.208030
$842$	2012.88	0.0823854
$843$	0	0
$844$	13236.4	0.539831
$845$	0	0
$846$	0	0
$847$	−40895.6	−1.65902
$848$	−2867.90	−0.116137
$849$	0	0
$850$	0	0
$851$	18777.3	0.756377
$852$	0	0
$853$	28437.0	1.14146	0.570729	−	0.821139i	$-0.306661\pi$
$853$	28437.0	1.14146	0.570729	+	0.821139i	$0.306661\pi$
$854$	57614.1	2.30857
$855$	0	0
$856$	75985.3	3.03402
$857$	−45202.3	−1.80173	−0.900864	−	0.434101i	$-0.857066\pi$
$857$	−45202.3	−1.80173	−0.900864	+	0.434101i	$0.857066\pi$
$858$	0	0
$859$	39304.1	1.56116	0.780582	−	0.625054i	$-0.214923\pi$
$859$	39304.1	1.56116	0.780582	+	0.625054i	$0.214923\pi$
$860$	0	0
$861$	0	0
$862$	40417.0	1.59699
$863$	−41102.8	−1.62127	−0.810635	−	0.585552i	$-0.800878\pi$
$863$	−41102.8	−1.62127	−0.810635	+	0.585552i	$0.800878\pi$
$864$	0	0
$865$	0	0
$866$	−7074.33	−0.277593
$867$	0	0
$868$	14094.0	0.551131
$869$	3142.72	0.122680
$870$	0	0
$871$	661.677	0.0257406
$872$	60747.9	2.35916
$873$	0	0
$874$	−31616.0	−1.22360
$875$	0	0
$876$	0	0
$877$	48649.4	1.87317	0.936587	−	0.350436i	$-0.113967\pi$
$877$	48649.4	1.87317	0.936587	+	0.350436i	$0.113967\pi$
$878$	67157.5	2.58139
$879$	0	0
$880$	0	0
$881$	−1470.17	−0.0562217	−0.0281108	−	0.999605i	$-0.508949\pi$
$881$	−1470.17	−0.0562217	−0.0281108	+	0.999605i	$0.508949\pi$
$882$	0	0
$883$	103.543	0.00394622	0.00197311	−	0.999998i	$-0.499372\pi$
$883$	103.543	0.00394622	0.00197311	+	0.999998i	$0.499372\pi$
$884$	62038.4	2.36038
$885$	0	0
$886$	43521.1	1.65025
$887$	21162.1	0.801074	0.400537	−	0.916281i	$-0.368824\pi$
$887$	21162.1	0.801074	0.400537	+	0.916281i	$0.368824\pi$
$888$	0	0
$889$	9921.41	0.374301
$890$	0	0
$891$	0	0
$892$	−80362.5	−3.01652
$893$	−14274.1	−0.534898
$894$	0	0
$895$	0	0
$896$	84357.8	3.14531
$897$	0	0
$898$	54835.6	2.03774
$899$	3459.36	0.128338
$900$	0	0
$901$	−4160.09	−0.153821
$902$	−15664.6	−0.578242
$903$	0	0
$904$	−38210.8	−1.40583
$905$	0	0
$906$	0	0
$907$	−31376.9	−1.14868	−0.574340	−	0.818617i	$-0.694741\pi$
$907$	−31376.9	−1.14868	−0.574340	+	0.818617i	$0.694741\pi$
$908$	−58441.3	−2.13595
$909$	0	0
$910$	0	0
$911$	−9271.17	−0.337176	−0.168588	−	0.985687i	$-0.553921\pi$
$911$	−9271.17	−0.337176	−0.168588	+	0.985687i	$0.553921\pi$
$912$	0	0
$913$	−2922.88	−0.105951
$914$	−83687.3	−3.02859
$915$	0	0
$916$	−62998.8	−2.27242
$917$	24257.1	0.873544
$918$	0	0
$919$	−39558.3	−1.41992	−0.709962	−	0.704240i	$-0.751288\pi$
$919$	−39558.3	−1.41992	−0.709962	+	0.704240i	$0.751288\pi$
$920$	0	0
$921$	0	0
$922$	−77352.4	−2.76298
$923$	−36834.4	−1.31356
$924$	0	0
$925$	0	0
$926$	39401.2	1.39828
$927$	0	0
$928$	−3115.85	−0.110219
$929$	37330.3	1.31837	0.659186	−	0.751980i	$-0.270901\pi$
$929$	37330.3	1.31837	0.659186	+	0.751980i	$0.270901\pi$
$930$	0	0
$931$	55287.0	1.94625
$932$	−48566.4	−1.70692
$933$	0	0
$934$	−31277.6	−1.09575
$935$	0	0
$936$	0	0
$937$	32301.0	1.12618	0.563088	−	0.826397i	$-0.309613\pi$
$937$	32301.0	1.12618	0.563088	+	0.826397i	$0.309613\pi$
$938$	−3055.49	−0.106360
$939$	0	0
$940$	0	0
$941$	11473.3	0.397468	0.198734	−	0.980053i	$-0.436317\pi$
$941$	11473.3	0.397468	0.198734	+	0.980053i	$0.436317\pi$
$942$	0	0
$943$	−25668.0	−0.886389
$944$	9311.91	0.321056
$945$	0	0
$946$	−6658.94	−0.228859
$947$	−34000.9	−1.16672	−0.583359	−	0.812215i	$-0.698262\pi$
$947$	−34000.9	−1.16672	−0.583359	+	0.812215i	$0.698262\pi$
$948$	0	0
$949$	−28557.3	−0.976829
$950$	0	0
$951$	0	0
$952$	−145700.	−4.96024
$953$	−2018.64	−0.0686149	−0.0343075	−	0.999411i	$-0.510923\pi$
$953$	−2018.64	−0.0686149	−0.0343075	+	0.999411i	$0.510923\pi$
$954$	0	0
$955$	0	0
$956$	42775.6	1.44713
$957$	0	0
$958$	28549.5	0.962833
$959$	54185.1	1.82453
$960$	0	0
$961$	−29171.4	−0.979203
$962$	−34136.9	−1.14409
$963$	0	0
$964$	−68966.9	−2.30423
$965$	0	0
$966$	0	0
$967$	18525.1	0.616056	0.308028	−	0.951377i	$-0.400331\pi$
$967$	18525.1	0.616056	0.308028	+	0.951377i	$0.400331\pi$
$968$	47967.8	1.59271
$969$	0	0
$970$	0	0
$971$	48076.1	1.58891	0.794457	−	0.607320i	$-0.207755\pi$
$971$	48076.1	1.58891	0.794457	+	0.607320i	$0.207755\pi$
$972$	0	0
$973$	−10967.5	−0.361359
$974$	−51066.5	−1.67995
$975$	0	0
$976$	−23796.1	−0.780425
$977$	−47135.6	−1.54350	−0.771751	−	0.635925i	$-0.780619\pi$
$977$	−47135.6	−1.54350	−0.771751	+	0.635925i	$0.780619\pi$
$978$	0	0
$979$	−5053.73	−0.164982
$980$	0	0
$981$	0	0
$982$	105548.	3.42990
$983$	−20180.8	−0.654799	−0.327399	−	0.944886i	$-0.606172\pi$
$983$	−20180.8	−0.654799	−0.327399	+	0.944886i	$0.606172\pi$
$984$	0	0
$985$	0	0
$986$	−70316.4	−2.27113
$987$	0	0
$988$	38538.9	1.24098
$989$	−10911.3	−0.350819
$990$	0	0
$991$	−21110.9	−0.676701	−0.338351	−	0.941020i	$-0.609869\pi$
$991$	−21110.9	−0.676701	−0.338351	+	0.941020i	$0.609869\pi$
$992$	−558.045	−0.0178608
$993$	0	0
$994$	170094.	5.42761
$995$	0	0
$996$	0	0
$997$	−11304.0	−0.359079	−0.179540	−	0.983751i	$-0.557461\pi$
$997$	−11304.0	−0.359079	−0.179540	+	0.983751i	$0.557461\pi$
$998$	−8188.28	−0.259715
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.bc.1.1		6
3.2	odd	2	inner	675.4.a.bc.1.6		6
5.2	odd	4		135.4.b.c.109.1	✓	12
5.3	odd	4		135.4.b.c.109.11	yes	12
5.4	even	2		675.4.a.bb.1.6		6
15.2	even	4		135.4.b.c.109.12	yes	12
15.8	even	4		135.4.b.c.109.2	yes	12
15.14	odd	2		675.4.a.bb.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.c.109.1	✓	12	5.2	odd	4
135.4.b.c.109.2	yes	12	15.8	even	4
135.4.b.c.109.11	yes	12	5.3	odd	4
135.4.b.c.109.12	yes	12	15.2	even	4
675.4.a.bb.1.1		6	15.14	odd	2
675.4.a.bb.1.6		6	5.4	even	2
675.4.a.bc.1.1		6	1.1	even	1	trivial
675.4.a.bc.1.6		6	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-4.92742 q^{2} +16.2795 q^{4} +34.7816 q^{7} -40.7965 q^{8} +O(q^{10})\)	\(q-4.92742 q^{2} +16.2795 q^{4} +34.7816 q^{7} -40.7965 q^{8} -12.4586 q^{11} +37.1137 q^{13} -171.384 q^{14} +70.7858 q^{16} +102.680 q^{17} +63.7858 q^{19} +61.3890 q^{22} +100.592 q^{23} -182.875 q^{26} +566.227 q^{28} +138.980 q^{29} +24.8911 q^{31} -22.4195 q^{32} -505.947 q^{34} +186.668 q^{37} -314.300 q^{38} -255.169 q^{41} -108.471 q^{43} -202.820 q^{44} -495.659 q^{46} -223.781 q^{47} +866.760 q^{49} +604.192 q^{52} -40.5151 q^{53} -1418.97 q^{56} -684.812 q^{58} +131.550 q^{59} -336.171 q^{61} -122.649 q^{62} -455.817 q^{64} +17.8284 q^{67} +1671.58 q^{68} -992.473 q^{71} -769.455 q^{73} -919.792 q^{74} +1038.40 q^{76} -433.331 q^{77} -252.252 q^{79} +1257.33 q^{82} +234.606 q^{83} +534.484 q^{86} +508.269 q^{88} +405.641 q^{89} +1290.87 q^{91} +1637.58 q^{92} +1102.67 q^{94} -1067.80 q^{97} -4270.89 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 30 q^{4} + 36 q^{7}+O(q^{10}) \) 6 * q + 30 * q^4 + 36 * q^7	\( 6 q + 30 q^{4} + 36 q^{7} + 162 q^{13} + 42 q^{16} + 450 q^{22} + 828 q^{28} + 126 q^{31} - 534 q^{34} + 1008 q^{37} + 558 q^{43} - 834 q^{46} + 1434 q^{49} + 2610 q^{52} - 270 q^{58} + 396 q^{61} - 1134 q^{64} + 2268 q^{67} + 144 q^{73} + 912 q^{76} - 1098 q^{79} + 5544 q^{82} + 702 q^{88} - 1692 q^{91} + 2574 q^{94} + 4104 q^{97}+O(q^{100}) \) 6 * q + 30 * q^4 + 36 * q^7 + 162 * q^13 + 42 * q^16 + 450 * q^22 + 828 * q^28 + 126 * q^31 - 534 * q^34 + 1008 * q^37 + 558 * q^43 - 834 * q^46 + 1434 * q^49 + 2610 * q^52 - 270 * q^58 + 396 * q^61 - 1134 * q^64 + 2268 * q^67 + 144 * q^73 + 912 * q^76 - 1098 * q^79 + 5544 * q^82 + 702 * q^88 - 1692 * q^91 + 2574 * q^94 + 4104 * q^97

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