LMFDB - Embedded newform 525.6.d.p.274.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	525.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$84.2015054018$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 322x^{10} + 38621x^{8} + 2116025x^{6} + 52188850x^{4} + 497085625x^{2} + 1556302500 $ x^12 + 322x^10 + 38621x^8 + 2116025x^6 + 52188850x^4 + 497085625*x^2 + 1556302500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.8
Root			$2.98787i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.6.d.p.274.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.98787i q^{2} -9.00000i q^{3} +16.0969 q^{4} +35.8908 q^{6} -49.0000i q^{7} +191.804i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+3.98787i q^{2} -9.00000i q^{3} +16.0969 q^{4} +35.8908 q^{6} -49.0000i q^{7} +191.804i q^{8} -81.0000 q^{9} -608.104 q^{11} -144.872i q^{12} +183.868i q^{13} +195.405 q^{14} -249.788 q^{16} +316.692i q^{17} -323.017i q^{18} -396.758 q^{19} -441.000 q^{21} -2425.04i q^{22} -1889.71i q^{23} +1726.24 q^{24} -733.241 q^{26} +729.000i q^{27} -788.749i q^{28} +2617.09 q^{29} +9212.97 q^{31} +5141.61i q^{32} +5472.94i q^{33} -1262.93 q^{34} -1303.85 q^{36} +636.023i q^{37} -1582.22i q^{38} +1654.81 q^{39} +15763.8 q^{41} -1758.65i q^{42} -21352.6i q^{43} -9788.60 q^{44} +7535.92 q^{46} -20381.3i q^{47} +2248.09i q^{48} -2401.00 q^{49} +2850.23 q^{51} +2959.71i q^{52} +27449.4i q^{53} -2907.15 q^{54} +9398.40 q^{56} +3570.82i q^{57} +10436.6i q^{58} -6699.75 q^{59} +45353.9 q^{61} +36740.1i q^{62} +3969.00i q^{63} -28497.3 q^{64} -21825.3 q^{66} -53614.0i q^{67} +5097.76i q^{68} -17007.4 q^{69} -39980.7 q^{71} -15536.1i q^{72} -46871.5i q^{73} -2536.38 q^{74} -6386.59 q^{76} +29797.1i q^{77} +6599.17i q^{78} +4716.74 q^{79} +6561.00 q^{81} +62863.8i q^{82} -88790.7i q^{83} -7098.74 q^{84} +85151.2 q^{86} -23553.8i q^{87} -116637. i q^{88} -70517.0 q^{89} +9009.53 q^{91} -30418.5i q^{92} -82916.8i q^{93} +81278.0 q^{94} +46274.5 q^{96} -16773.0i q^{97} -9574.87i q^{98} +49256.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9}+O(q^{10}) $ 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9	$ 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9} + 392 q^{14} + 3568 q^{16} - 520 q^{19} - 5292 q^{21} - 5454 q^{24} - 6534 q^{26} + 4244 q^{29} + 4524 q^{31} - 24778 q^{34} + 21384 q^{36} + 3816 q^{39} + 16248 q^{41} - 62322 q^{44} + 58720 q^{46} - 28812 q^{49} + 67320 q^{51} - 5832 q^{54} - 29694 q^{56} - 51316 q^{59} + 82376 q^{61} - 131894 q^{64} + 124218 q^{66} + 33336 q^{69} + 35160 q^{71} - 7942 q^{74} + 507092 q^{76} + 19472 q^{79} + 78732 q^{81} + 116424 q^{84} + 287632 q^{86} + 660720 q^{89} + 20776 q^{91} + 747360 q^{94} + 491220 q^{96}+O(q^{100}) $ 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9 + 392 * q^14 + 3568 * q^16 - 520 * q^19 - 5292 * q^21 - 5454 * q^24 - 6534 * q^26 + 4244 * q^29 + 4524 * q^31 - 24778 * q^34 + 21384 * q^36 + 3816 * q^39 + 16248 * q^41 - 62322 * q^44 + 58720 * q^46 - 28812 * q^49 + 67320 * q^51 - 5832 * q^54 - 29694 * q^56 - 51316 * q^59 + 82376 * q^61 - 131894 * q^64 + 124218 * q^66 + 33336 * q^69 + 35160 * q^71 - 7942 * q^74 + 507092 * q^76 + 19472 * q^79 + 78732 * q^81 + 116424 * q^84 + 287632 * q^86 + 660720 * q^89 + 20776 * q^91 + 747360 * q^94 + 491220 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	3.98787i	0.704962i	0.935819	+	0.352481i	$0.114662\pi$
$2$	3.98787i	0.704962i	−0.935819	+	0.352481i	$0.885338\pi$
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	16.0969	0.503029
$5$	0	0
$5$	0	0
$6$	35.8908	0.407010
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	191.804i	1.05958i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−608.104	−1.51529	−0.757646	−	0.652666i	$-0.773651\pi$
$11$	−608.104	−1.51529	−0.757646	+	0.652666i	$0.773651\pi$
$12$	− 144.872i	− 0.290424i
$13$	183.868i	0.301750i	0.988553	+	0.150875i	$0.0482091\pi$
$13$	183.868i	0.301750i	−0.988553	+	0.150875i	$0.951791\pi$
$14$	195.405	0.266451
$15$	0	0
$16$	−249.788	−0.243933
$17$	316.692i	0.265775i	0.991131	+	0.132888i	$0.0424249\pi$
$17$	316.692i	0.265775i	−0.991131	+	0.132888i	$0.957575\pi$
$18$	− 323.017i	− 0.234987i
$19$	−396.758	−0.252140	−0.126070	−	0.992021i	$-0.540236\pi$
$19$	−396.758	−0.252140	−0.126070	+	0.992021i	$0.540236\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	− 2425.04i	− 1.06822i
$23$	− 1889.71i	− 0.744862i	−0.928060	−	0.372431i	$-0.878524\pi$
$23$	− 1889.71i	− 0.744862i	0.928060	−	0.372431i	$-0.121476\pi$
$24$	1726.24	0.611748
$25$	0	0
$26$	−733.241	−0.212722
$27$	729.000i	0.192450i
$28$	− 788.749i	− 0.190127i
$29$	2617.09	0.577862	0.288931	−	0.957350i	$-0.406700\pi$
$29$	2617.09	0.577862	0.288931	+	0.957350i	$0.406700\pi$
$30$	0	0
$31$	9212.97	1.72185	0.860925	−	0.508731i	$-0.169885\pi$
$31$	9212.97	1.72185	0.860925	+	0.508731i	$0.169885\pi$
$32$	5141.61i	0.887614i
$33$	5472.94i	0.874854i
$34$	−1262.93	−0.187361
$35$	0	0
$36$	−1303.85	−0.167676
$37$	636.023i	0.0763781i	0.999271	+	0.0381890i	$0.0121589\pi$
$37$	636.023i	0.0763781i	−0.999271	+	0.0381890i	$0.987841\pi$
$38$	− 1582.22i	− 0.177749i
$39$	1654.81	0.174216
$40$	0	0
$41$	15763.8	1.46454	0.732269	−	0.681016i	$-0.238461\pi$
$41$	15763.8	1.46454	0.732269	+	0.681016i	$0.238461\pi$
$42$	− 1758.65i	− 0.153835i
$43$	− 21352.6i	− 1.76108i	−0.473973	−	0.880539i	$-0.657181\pi$
$43$	− 21352.6i	− 1.76108i	0.473973	−	0.880539i	$-0.342819\pi$
$44$	−9788.60	−0.762235
$45$	0	0
$46$	7535.92	0.525099
$47$	− 20381.3i	− 1.34582i	−0.739723	−	0.672911i	$-0.765043\pi$
$47$	− 20381.3i	− 1.34582i	0.739723	−	0.672911i	$-0.234957\pi$
$48$	2248.09i	0.140835i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	2850.23	0.153445
$52$	2959.71i	0.151789i
$53$	27449.4i	1.34228i	0.741331	+	0.671140i	$0.234195\pi$
$53$	27449.4i	1.34228i	−0.741331	+	0.671140i	$0.765805\pi$
$54$	−2907.15	−0.135670
$55$	0	0
$56$	9398.40	0.400483
$57$	3570.82i	0.145573i
$58$	10436.6i	0.407370i
$59$	−6699.75	−0.250570	−0.125285	−	0.992121i	$-0.539984\pi$
$59$	−6699.75	−0.250570	−0.125285	+	0.992121i	$0.539984\pi$
$60$	0	0
$61$	45353.9	1.56060	0.780298	−	0.625408i	$-0.215067\pi$
$61$	45353.9	1.56060	0.780298	+	0.625408i	$0.215067\pi$
$62$	36740.1i	1.21384i
$63$	3969.00i	0.125988i
$64$	−28497.3	−0.869668
$65$	0	0
$66$	−21825.3	−0.616739
$67$	− 53614.0i	− 1.45912i	−0.683917	−	0.729560i	$-0.739725\pi$
$67$	− 53614.0i	− 1.45912i	0.683917	−	0.729560i	$-0.260275\pi$
$68$	5097.76i	0.133693i
$69$	−17007.4	−0.430046
$70$	0	0
$71$	−39980.7	−0.941249	−0.470624	−	0.882334i	$-0.655971\pi$
$71$	−39980.7	−0.941249	−0.470624	+	0.882334i	$0.655971\pi$
$72$	− 15536.1i	− 0.353193i
$73$	− 46871.5i	− 1.02944i	−0.857358	−	0.514721i	$-0.827896\pi$
$73$	− 46871.5i	− 1.02944i	0.857358	−	0.514721i	$-0.172104\pi$
$74$	−2536.38	−0.0538436
$75$	0	0
$76$	−6386.59	−0.126834
$77$	29797.1i	0.572726i
$78$	6599.17i	0.122815i
$79$	4716.74	0.0850304	0.0425152	−	0.999096i	$-0.486463\pi$
$79$	4716.74	0.0850304	0.0425152	+	0.999096i	$0.486463\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	62863.8i	1.03244i
$83$	− 88790.7i	− 1.41473i	−0.706851	−	0.707363i	$-0.749885\pi$
$83$	− 88790.7i	− 1.41473i	0.706851	−	0.707363i	$-0.250115\pi$
$84$	−7098.74	−0.109770
$85$	0	0
$86$	85151.2	1.24149
$87$	− 23553.8i	− 0.333629i
$88$	− 116637.i	− 1.60557i
$89$	−70517.0	−0.943668	−0.471834	−	0.881687i	$-0.656408\pi$
$89$	−70517.0	−0.943668	−0.471834	+	0.881687i	$0.656408\pi$
$90$	0	0
$91$	9009.53	0.114051
$92$	− 30418.5i	− 0.374687i
$93$	− 82916.8i	− 0.994111i
$94$	81278.0	0.948754
$95$	0	0
$96$	46274.5	0.512464
$97$	− 16773.0i	− 0.181001i	−0.995896	−	0.0905007i	$-0.971153\pi$
$97$	− 16773.0i	− 0.181001i	0.995896	−	0.0905007i	$-0.0288467\pi$
$98$	− 9574.87i	− 0.100709i
$99$	49256.4	0.505097
$100$	0	0
$101$	3061.82	0.0298659	0.0149330	−	0.999888i	$-0.495247\pi$
$101$	3061.82	0.0298659	0.0149330	+	0.999888i	$0.495247\pi$
$102$	11366.3i	0.108173i
$103$	33280.8i	0.309101i	0.987985	+	0.154551i	$0.0493930\pi$
$103$	33280.8i	0.309101i	−0.987985	+	0.154551i	$0.950607\pi$
$104$	−35266.6	−0.319728
$105$	0	0
$106$	−109465.	−0.946256
$107$	34602.3i	0.292177i	0.989272	+	0.146088i	$0.0466684\pi$
$107$	34602.3i	0.292177i	−0.989272	+	0.146088i	$0.953332\pi$
$108$	11734.7i	0.0968079i
$109$	159201.	1.28345	0.641725	−	0.766935i	$-0.278219\pi$
$109$	159201.	1.28345	0.641725	+	0.766935i	$0.278219\pi$
$110$	0	0
$111$	5724.21	0.0440969
$112$	12239.6i	0.0921982i
$113$	− 242126.i	− 1.78379i	−0.452238	−	0.891897i	$-0.649374\pi$
$113$	− 242126.i	− 1.78379i	0.452238	−	0.891897i	$-0.350626\pi$
$114$	−14240.0	−0.102624
$115$	0	0
$116$	42127.1	0.290681
$117$	− 14893.3i	− 0.100583i
$118$	− 26717.7i	− 0.176642i
$119$	15517.9	0.100454
$120$	0	0
$121$	208740.	1.29611
$122$	180865.i	1.10016i
$123$	− 141874.i	− 0.845551i
$124$	148300.	0.866140
$125$	0	0
$126$	−15827.8	−0.0888169
$127$	− 42609.6i	− 0.234422i	−0.993107	−	0.117211i	$-0.962605\pi$
$127$	− 42609.6i	− 0.234422i	0.993107	−	0.117211i	$-0.0373954\pi$
$128$	50888.2i	0.274532i
$129$	−192173.	−1.01676
$130$	0	0
$131$	127020.	0.646687	0.323344	−	0.946282i	$-0.395193\pi$
$131$	127020.	0.646687	0.323344	+	0.946282i	$0.395193\pi$
$132$	88097.4i	0.440077i
$133$	19441.2i	0.0953000i
$134$	213805.	1.02862
$135$	0	0
$136$	−60742.8	−0.281610
$137$	− 50488.0i	− 0.229820i	−0.993376	−	0.114910i	$-0.963342\pi$
$137$	− 50488.0i	− 0.229820i	0.993376	−	0.114910i	$-0.0366579\pi$
$138$	− 67823.3i	− 0.303166i
$139$	−224534.	−0.985702	−0.492851	−	0.870114i	$-0.664045\pi$
$139$	−224534.	−0.985702	−0.492851	+	0.870114i	$0.664045\pi$
$140$	0	0
$141$	−183432.	−0.777011
$142$	− 159438.i	− 0.663544i
$143$	− 111811.i	− 0.457240i
$144$	20232.8	0.0813111
$145$	0	0
$146$	186917.	0.725717
$147$	21609.0i	0.0824786i
$148$	10238.0i	0.0384204i
$149$	122520.	0.452107	0.226053	−	0.974115i	$-0.427418\pi$
$149$	122520.	0.452107	0.226053	+	0.974115i	$0.427418\pi$
$150$	0	0
$151$	65850.2	0.235026	0.117513	−	0.993071i	$-0.462508\pi$
$151$	65850.2	0.235026	0.117513	+	0.993071i	$0.462508\pi$
$152$	− 76099.9i	− 0.267162i
$153$	− 25652.0i	− 0.0885918i
$154$	−118827.	−0.403750
$155$	0	0
$156$	26637.4	0.0876354
$157$	117854.i	0.381589i	0.981630	+	0.190795i	$0.0611064\pi$
$157$	117854.i	0.381589i	−0.981630	+	0.190795i	$0.938894\pi$
$158$	18809.7i	0.0599432i
$159$	247045.	0.774966
$160$	0	0
$161$	−92595.9	−0.281531
$162$	26164.4i	0.0783291i
$163$	− 467286.i	− 1.37757i	−0.724966	−	0.688785i	$-0.758144\pi$
$163$	− 467286.i	− 1.37757i	0.724966	−	0.688785i	$-0.241856\pi$
$164$	253748.	0.736705
$165$	0	0
$166$	354085.	0.997328
$167$	503349.i	1.39662i	0.715796	+	0.698309i	$0.246064\pi$
$167$	503349.i	1.39662i	−0.715796	+	0.698309i	$0.753936\pi$
$168$	− 84585.6i	− 0.231219i
$169$	337486.	0.908947
$170$	0	0
$171$	32137.4	0.0840467
$172$	− 343710.i	− 0.885873i
$173$	− 117102.i	− 0.297474i	−0.988877	−	0.148737i	$-0.952479\pi$
$173$	− 117102.i	− 0.297474i	0.988877	−	0.148737i	$-0.0475207\pi$
$174$	93929.5	0.235195
$175$	0	0
$176$	151897.	0.369630
$177$	60297.7i	0.144666i
$178$	− 281213.i	− 0.665250i
$179$	721172.	1.68231	0.841155	−	0.540794i	$-0.181876\pi$
$179$	721172.	1.68231	0.841155	+	0.540794i	$0.181876\pi$
$180$	0	0
$181$	−298084.	−0.676305	−0.338153	−	0.941091i	$-0.609802\pi$
$181$	−298084.	−0.676305	−0.338153	+	0.941091i	$0.609802\pi$
$182$	35928.8i	0.0804015i
$183$	− 408185.i	− 0.901010i
$184$	362454.	0.789240
$185$	0	0
$186$	330661.	0.700810
$187$	− 192582.i	− 0.402727i
$188$	− 328077.i	− 0.676988i
$189$	35721.0	0.0727393
$190$	0	0
$191$	36894.7	0.0731780	0.0365890	−	0.999330i	$-0.488351\pi$
$191$	36894.7	0.0731780	0.0365890	+	0.999330i	$0.488351\pi$
$192$	256475.i	0.502103i
$193$	620771.i	1.19960i	0.800148	+	0.599802i	$0.204754\pi$
$193$	620771.i	1.19960i	−0.800148	+	0.599802i	$0.795246\pi$
$194$	66888.6	0.127599
$195$	0	0
$196$	−38648.7	−0.0718612
$197$	336496.i	0.617752i	0.951102	+	0.308876i	$0.0999528\pi$
$197$	336496.i	0.617752i	−0.951102	+	0.308876i	$0.900047\pi$
$198$	196428.i	0.356074i
$199$	257845.	0.461557	0.230779	−	0.973006i	$-0.425873\pi$
$199$	257845.	0.461557	0.230779	+	0.973006i	$0.425873\pi$
$200$	0	0
$201$	−482526.	−0.842423
$202$	12210.1i	0.0210543i
$203$	− 128237.i	− 0.218411i
$204$	45879.9	0.0771875
$205$	0	0
$206$	−132719.	−0.217905
$207$	153067.i	0.248287i
$208$	− 45928.0i	− 0.0736070i
$209$	241270.	0.382066
$210$	0	0
$211$	631996.	0.977255	0.488628	−	0.872492i	$-0.337498\pi$
$211$	631996.	0.977255	0.488628	+	0.872492i	$0.337498\pi$
$212$	441851.i	0.675205i
$213$	359826.i	0.543430i
$214$	−137989.	−0.205974
$215$	0	0
$216$	−139825.	−0.203916
$217$	− 451436.i	− 0.650798i
$218$	634871.i	0.904783i
$219$	−421843.	−0.594348
$220$	0	0
$221$	−58229.5	−0.0801977
$222$	22827.4i	0.0310866i
$223$	− 1.01140e6i	− 1.36195i	−0.732307	−	0.680975i	$-0.761556\pi$
$223$	− 1.01140e6i	− 1.36195i	0.732307	−	0.680975i	$-0.238444\pi$
$224$	251939.	0.335487
$225$	0	0
$226$	965565.	1.25751
$227$	682099.i	0.878583i	0.898345	+	0.439292i	$0.144771\pi$
$227$	682099.i	0.878583i	−0.898345	+	0.439292i	$0.855229\pi$
$228$	57479.3i	0.0732275i
$229$	−56079.1	−0.0706663	−0.0353332	−	0.999376i	$-0.511249\pi$
$229$	−56079.1	−0.0706663	−0.0353332	+	0.999376i	$0.511249\pi$
$230$	0	0
$231$	268174.	0.330664
$232$	501969.i	0.612289i
$233$	− 21363.4i	− 0.0257799i	−0.999917	−	0.0128899i	$-0.995897\pi$
$233$	− 21363.4i	− 0.0257799i	0.999917	−	0.0128899i	$-0.00410310\pi$
$234$	59392.5	0.0709075
$235$	0	0
$236$	−107845.	−0.126044
$237$	− 42450.6i	− 0.0490923i
$238$	61883.3i	0.0708160i
$239$	407013.	0.460907	0.230453	−	0.973083i	$-0.425979\pi$
$239$	407013.	0.460907	0.230453	+	0.973083i	$0.425979\pi$
$240$	0	0
$241$	1.47891e6	1.64021	0.820104	−	0.572214i	$-0.193915\pi$
$241$	1.47891e6	1.64021	0.820104	+	0.572214i	$0.193915\pi$
$242$	832426.i	0.913707i
$243$	− 59049.0i	− 0.0641500i
$244$	730059.	0.785024
$245$	0	0
$246$	565775.	0.596081
$247$	− 72951.1i	− 0.0760833i
$248$	1.76709e6i	1.82444i
$249$	−799116.	−0.816792
$250$	0	0
$251$	−990670.	−0.992533	−0.496266	−	0.868170i	$-0.665296\pi$
$251$	−990670.	−0.992533	−0.496266	+	0.868170i	$0.665296\pi$
$252$	63888.7i	0.0633757i
$253$	1.14914e6i	1.12868i
$254$	169921.	0.165259
$255$	0	0
$256$	−1.11485e6	−1.06320
$257$	− 33118.7i	− 0.0312782i	−0.999878	−	0.0156391i	$-0.995022\pi$
$257$	− 33118.7i	− 0.0312782i	0.999878	−	0.0156391i	$-0.00497828\pi$
$258$	− 766360.i	− 0.716777i
$259$	31165.1	0.0288682
$260$	0	0
$261$	−211984.	−0.192621
$262$	506539.i	0.455890i
$263$	− 745885.i	− 0.664940i	−0.943114	−	0.332470i	$-0.892118\pi$
$263$	− 745885.i	− 0.664940i	0.943114	−	0.332470i	$-0.107882\pi$
$264$	−1.04973e6	−0.926976
$265$	0	0
$266$	−77528.7	−0.0671829
$267$	634653.i	0.544827i
$268$	− 863020.i	− 0.733979i
$269$	−933817.	−0.786831	−0.393415	−	0.919361i	$-0.628707\pi$
$269$	−933817.	−0.786831	−0.393415	+	0.919361i	$0.628707\pi$
$270$	0	0
$271$	1.33585e6	1.10493	0.552464	−	0.833537i	$-0.313688\pi$
$271$	1.33585e6	1.10493	0.552464	+	0.833537i	$0.313688\pi$
$272$	− 79105.8i	− 0.0648315i
$273$	− 81085.7i	− 0.0658473i
$274$	201340.	0.162014
$275$	0	0
$276$	−273767.	−0.216326
$277$	257580.i	0.201703i	0.994901	+	0.100852i	$0.0321567\pi$
$277$	257580.i	0.201703i	−0.994901	+	0.100852i	$0.967843\pi$
$278$	− 895412.i	− 0.694882i
$279$	−746251.	−0.573950
$280$	0	0
$281$	40780.4	0.0308096	0.0154048	−	0.999881i	$-0.495096\pi$
$281$	40780.4	0.0308096	0.0154048	+	0.999881i	$0.495096\pi$
$282$	− 731502.i	− 0.547763i
$283$	− 1.67482e6i	− 1.24309i	−0.783380	−	0.621543i	$-0.786506\pi$
$283$	− 1.67482e6i	− 1.24309i	0.783380	−	0.621543i	$-0.213494\pi$
$284$	−643566.	−0.473475
$285$	0	0
$286$	445887.	0.322336
$287$	− 772425.i	− 0.553543i
$288$	− 416470.i	− 0.295871i
$289$	1.31956e6	0.929363
$290$	0	0
$291$	−150957.	−0.104501
$292$	− 754486.i	− 0.517838i
$293$	− 1.05299e6i	− 0.716562i	−0.933614	−	0.358281i	$-0.883363\pi$
$293$	− 1.05299e6i	− 0.716562i	0.933614	−	0.358281i	$-0.116637\pi$
$294$	−86173.8	−0.0581443
$295$	0	0
$296$	−121992.	−0.0809285
$297$	− 443308.i	− 0.291618i
$298$	488593.i	0.318718i
$299$	347457.	0.224762
$300$	0	0
$301$	−1.04628e6	−0.665625
$302$	262602.i	0.165684i
$303$	− 27556.3i	− 0.0172431i
$304$	99105.4	0.0615054
$305$	0	0
$306$	102297.	0.0624538
$307$	− 785188.i	− 0.475475i	−0.971329	−	0.237738i	$-0.923594\pi$
$307$	− 785188.i	− 0.475475i	0.971329	−	0.237738i	$-0.0764058\pi$
$308$	479642.i	0.288098i
$309$	299527.	0.178460
$310$	0	0
$311$	−2.85107e6	−1.67150	−0.835752	−	0.549107i	$-0.814968\pi$
$311$	−2.85107e6	−1.67150	−0.835752	+	0.549107i	$0.814968\pi$
$312$	317400.i	0.184595i
$313$	1.00721e6i	0.581110i	0.956858	+	0.290555i	$0.0938399\pi$
$313$	1.00721e6i	0.581110i	−0.956858	+	0.290555i	$0.906160\pi$
$314$	−469987.	−0.269006
$315$	0	0
$316$	75925.0	0.0427727
$317$	− 50317.7i	− 0.0281237i	−0.999901	−	0.0140618i	$-0.995524\pi$
$317$	− 50317.7i	− 0.0281237i	0.999901	−	0.0140618i	$-0.00447617\pi$
$318$	985181.i	0.546321i
$319$	−1.59146e6	−0.875629
$320$	0	0
$321$	311421.	0.168688
$322$	− 369260.i	− 0.198469i
$323$	− 125650.i	− 0.0670126i
$324$	105612.	0.0558921
$325$	0	0
$326$	1.86347e6	0.971135
$327$	− 1.43281e6i	− 0.741000i
$328$	3.02356e6i	1.55179i
$329$	−998685.	−0.508673
$330$	0	0
$331$	−1.78739e6	−0.896706	−0.448353	−	0.893857i	$-0.647989\pi$
$331$	−1.78739e6	−0.896706	−0.448353	+	0.893857i	$0.647989\pi$
$332$	− 1.42926e6i	− 0.711648i
$333$	− 51517.9i	− 0.0254594i
$334$	−2.00729e6	−0.984562
$335$	0	0
$336$	110156.	0.0532306
$337$	− 3.64919e6i	− 1.75034i	−0.483817	−	0.875169i	$-0.660750\pi$
$337$	− 3.64919e6i	− 1.75034i	0.483817	−	0.875169i	$-0.339250\pi$
$338$	1.34585e6i	0.640773i
$339$	−2.17913e6	−1.02987
$340$	0	0
$341$	−5.60245e6	−2.60911
$342$	128160.i	0.0592497i
$343$	117649.i	0.0539949i
$344$	4.09551e6	1.86600
$345$	0	0
$346$	466986.	0.209708
$347$	3.69234e6i	1.64618i	0.567909	+	0.823091i	$0.307752\pi$
$347$	3.69234e6i	1.64618i	−0.567909	+	0.823091i	$0.692248\pi$
$348$	− 379144.i	− 0.167825i
$349$	−3.78967e6	−1.66548	−0.832738	−	0.553668i	$-0.813228\pi$
$349$	−3.78967e6	−1.66548	−0.832738	+	0.553668i	$0.813228\pi$
$350$	0	0
$351$	−134040.	−0.0580718
$352$	− 3.12663e6i	− 1.34499i
$353$	2.10255e6i	0.898068i	0.893515	+	0.449034i	$0.148232\pi$
$353$	2.10255e6i	0.898068i	−0.893515	+	0.449034i	$0.851768\pi$
$354$	−240459.	−0.101984
$355$	0	0
$356$	−1.13511e6	−0.474692
$357$	− 139661.i	− 0.0579969i
$358$	2.87594e6i	1.18596i
$359$	1.18153e6	0.483848	0.241924	−	0.970295i	$-0.422222\pi$
$359$	1.18153e6	0.483848	0.241924	+	0.970295i	$0.422222\pi$
$360$	0	0
$361$	−2.31868e6	−0.936425
$362$	− 1.18872e6i	− 0.476769i
$363$	− 1.87866e6i	− 0.748309i
$364$	145026.	0.0573708
$365$	0	0
$366$	1.62779e6	0.635178
$367$	− 3.62660e6i	− 1.40551i	−0.711432	−	0.702755i	$-0.751953\pi$
$367$	− 3.62660e6i	− 1.40551i	0.711432	−	0.702755i	$-0.248047\pi$
$368$	472027.i	0.181697i
$369$	−1.27687e6	−0.488179
$370$	0	0
$371$	1.34502e6	0.507334
$372$	− 1.33470e6i	− 0.500066i
$373$	− 3.86121e6i	− 1.43698i	−0.695536	−	0.718492i	$-0.744833\pi$
$373$	− 3.86121e6i	− 1.43698i	0.695536	−	0.718492i	$-0.255167\pi$
$374$	767990.	0.283907
$375$	0	0
$376$	3.90922e6	1.42600
$377$	481199.i	0.174370i
$378$	142451.i	0.0512784i
$379$	−3.88391e6	−1.38890	−0.694450	−	0.719541i	$-0.744352\pi$
$379$	−3.88391e6	−1.38890	−0.694450	+	0.719541i	$0.744352\pi$
$380$	0	0
$381$	−383487.	−0.135344
$382$	147131.i	0.0515877i
$383$	4.99124e6i	1.73865i	0.494243	+	0.869324i	$0.335445\pi$
$383$	4.99124e6i	1.73865i	−0.494243	+	0.869324i	$0.664555\pi$
$384$	457994.	0.158501
$385$	0	0
$386$	−2.47555e6	−0.845675
$387$	1.72956e6i	0.587026i
$388$	− 269994.i	− 0.0910489i
$389$	−1.32373e6	−0.443534	−0.221767	−	0.975100i	$-0.571182\pi$
$389$	−1.32373e6	−0.443534	−0.221767	+	0.975100i	$0.571182\pi$
$390$	0	0
$391$	598456.	0.197966
$392$	− 460522.i	− 0.151368i
$393$	− 1.14318e6i	− 0.373365i
$394$	−1.34190e6	−0.435492
$395$	0	0
$396$	792877.	0.254078
$397$	2.37011e6i	0.754732i	0.926064	+	0.377366i	$0.123170\pi$
$397$	2.37011e6i	0.754732i	−0.926064	+	0.377366i	$0.876830\pi$
$398$	1.02825e6i	0.325380i
$399$	174970.	0.0550215
$400$	0	0
$401$	5.49213e6	1.70561	0.852806	−	0.522229i	$-0.174899\pi$
$401$	5.49213e6	1.70561	0.852806	+	0.522229i	$0.174899\pi$
$402$	− 1.92425e6i	− 0.593876i
$403$	1.69397e6i	0.519569i
$404$	49285.8	0.0150234
$405$	0	0
$406$	511394.	0.153972
$407$	− 386768.i	− 0.115735i
$408$	546685.i	0.162587i
$409$	3.19531e6	0.944506	0.472253	−	0.881463i	$-0.343441\pi$
$409$	3.19531e6	0.944506	0.472253	+	0.881463i	$0.343441\pi$
$410$	0	0
$411$	−454392.	−0.132686
$412$	535718.i	0.155487i
$413$	328288.i	0.0947064i
$414$	−610409.	−0.175033
$415$	0	0
$416$	−945377.	−0.267838
$417$	2.02081e6i	0.569095i
$418$	962154.i	0.269342i
$419$	4.51295e6	1.25581	0.627907	−	0.778288i	$-0.283912\pi$
$419$	4.51295e6	1.25581	0.627907	+	0.778288i	$0.283912\pi$
$420$	0	0
$421$	−2.10532e6	−0.578913	−0.289457	−	0.957191i	$-0.593475\pi$
$421$	−2.10532e6	−0.578913	−0.289457	+	0.957191i	$0.593475\pi$
$422$	2.52032e6i	0.688928i
$423$	1.65089e6i	0.448608i
$424$	−5.26491e6	−1.42225
$425$	0	0
$426$	−1.43494e6	−0.383098
$427$	− 2.22234e6i	− 0.589850i
$428$	556991.i	0.146973i
$429$	−1.00630e6	−0.263987
$430$	0	0
$431$	−6.41232e6	−1.66273	−0.831366	−	0.555726i	$-0.812440\pi$
$431$	−6.41232e6	−1.66273	−0.831366	+	0.555726i	$0.812440\pi$
$432$	− 182095.i	− 0.0469450i
$433$	− 2.54265e6i	− 0.651728i	−0.945417	−	0.325864i	$-0.894345\pi$
$433$	− 2.54265e6i	− 0.651728i	0.945417	−	0.325864i	$-0.105655\pi$
$434$	1.80027e6	0.458788
$435$	0	0
$436$	2.56264e6	0.645612
$437$	749759.i	0.187810i
$438$	− 1.68226e6i	− 0.418993i
$439$	−6.08589e6	−1.50717	−0.753586	−	0.657349i	$-0.771678\pi$
$439$	−6.08589e6	−1.50717	−0.753586	+	0.657349i	$0.771678\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	− 232211.i	− 0.0565364i
$443$	− 371353.i	− 0.0899037i	−0.998989	−	0.0449519i	$-0.985687\pi$
$443$	− 371353.i	− 0.0899037i	0.998989	−	0.0449519i	$-0.0143134\pi$
$444$	92142.1	0.0221820
$445$	0	0
$446$	4.03333e6	0.960122
$447$	− 1.10268e6i	− 0.261024i
$448$	1.39637e6i	0.328703i
$449$	4.57079e6	1.06998	0.534989	−	0.844859i	$-0.320316\pi$
$449$	4.57079e6	1.06998	0.534989	+	0.844859i	$0.320316\pi$
$450$	0	0
$451$	−9.58602e6	−2.21920
$452$	− 3.89748e6i	− 0.897300i
$453$	− 592652.i	− 0.135692i
$454$	−2.72012e6	−0.619368
$455$	0	0
$456$	−684899.	−0.154246
$457$	1.75669e6i	0.393464i	0.980457	+	0.196732i	$0.0630329\pi$
$457$	1.75669e6i	0.393464i	−0.980457	+	0.196732i	$0.936967\pi$
$458$	− 223636.i	− 0.0498171i
$459$	−230868.	−0.0511485
$460$	0	0
$461$	−1.08092e6	−0.236888	−0.118444	−	0.992961i	$-0.537791\pi$
$461$	−1.08092e6	−0.236888	−0.118444	+	0.992961i	$0.537791\pi$
$462$	1.06944e6i	0.233105i
$463$	− 2.03144e6i	− 0.440405i	−0.975454	−	0.220203i	$-0.929328\pi$
$463$	− 2.03144e6i	− 0.440405i	0.975454	−	0.220203i	$-0.0706719\pi$
$464$	−653718.	−0.140960
$465$	0	0
$466$	85194.4	0.0181738
$467$	3.30524e6i	0.701311i	0.936505	+	0.350655i	$0.114041\pi$
$467$	3.30524e6i	0.701311i	−0.936505	+	0.350655i	$0.885959\pi$
$468$	− 239736.i	− 0.0505963i
$469$	−2.62708e6	−0.551496
$470$	0	0
$471$	1.06069e6	0.220311
$472$	− 1.28504e6i	− 0.265498i
$473$	1.29846e7i	2.66855i
$474$	169288.	0.0346082
$475$	0	0
$476$	249790.	0.0505311
$477$	− 2.22340e6i	− 0.447427i
$478$	1.62311e6i	0.324922i
$479$	−3.09748e6	−0.616836	−0.308418	−	0.951251i	$-0.599799\pi$
$479$	−3.09748e6	−0.616836	−0.308418	+	0.951251i	$0.599799\pi$
$480$	0	0
$481$	−116944.	−0.0230471
$482$	5.89769e6i	1.15628i
$483$	833363.i	0.162542i
$484$	3.36006e6	0.651980
$485$	0	0
$486$	235480.	0.0452233
$487$	151309.i	0.0289096i	0.999896	+	0.0144548i	$0.00460126\pi$
$487$	151309.i	0.0289096i	−0.999896	+	0.0144548i	$0.995399\pi$
$488$	8.69907e6i	1.65357i
$489$	−4.20557e6	−0.795341
$490$	0	0
$491$	436245.	0.0816632	0.0408316	−	0.999166i	$-0.486999\pi$
$491$	436245.	0.0816632	0.0408316	+	0.999166i	$0.486999\pi$
$492$	− 2.28373e6i	− 0.425337i
$493$	828812.i	0.153581i
$494$	290919.	0.0536359
$495$	0	0
$496$	−2.30129e6	−0.420017
$497$	1.95905e6i	0.355759i
$498$	− 3.18677e6i	− 0.575807i
$499$	5.33341e6	0.958856	0.479428	−	0.877581i	$-0.340844\pi$
$499$	5.33341e6	0.958856	0.479428	+	0.877581i	$0.340844\pi$
$500$	0	0
$501$	4.53014e6	0.806338
$502$	− 3.95066e6i	− 0.699698i
$503$	− 605474.i	− 0.106703i	−0.998576	−	0.0533513i	$-0.983010\pi$
$503$	− 605474.i	− 0.106703i	0.998576	−	0.0533513i	$-0.0169903\pi$
$504$	−761270.	−0.133494
$505$	0	0
$506$	−4.58262e6	−0.795679
$507$	− 3.03737e6i	− 0.524781i
$508$	− 685884.i	− 0.117921i
$509$	4.72311e6	0.808041	0.404020	−	0.914750i	$-0.367612\pi$
$509$	4.72311e6	0.808041	0.404020	+	0.914750i	$0.367612\pi$
$510$	0	0
$511$	−2.29670e6	−0.389092
$512$	− 2.81744e6i	− 0.474985i
$513$	− 289237.i	− 0.0485244i
$514$	132073.	0.0220499
$515$	0	0
$516$	−3.09339e6	−0.511459
$517$	1.23940e7i	2.03931i
$518$	124282.i	0.0203510i
$519$	−1.05392e6	−0.171746
$520$	0	0
$521$	−8.37792e6	−1.35220	−0.676102	−	0.736808i	$-0.736332\pi$
$521$	−8.37792e6	−1.35220	−0.676102	+	0.736808i	$0.736332\pi$
$522$	− 845365.i	− 0.135790i
$523$	7.55960e6i	1.20849i	0.796797	+	0.604247i	$0.206526\pi$
$523$	7.55960e6i	1.20849i	−0.796797	+	0.604247i	$0.793474\pi$
$524$	2.04463e6	0.325302
$525$	0	0
$526$	2.97449e6	0.468757
$527$	2.91767e6i	0.457625i
$528$	− 1.36707e6i	− 0.213406i
$529$	2.86533e6	0.445180
$530$	0	0
$531$	542680.	0.0835232
$532$	312943.i	0.0479386i
$533$	2.89845e6i	0.441925i
$534$	−2.53091e6	−0.384082
$535$	0	0
$536$	1.02834e7	1.54605
$537$	− 6.49054e6i	− 0.971282i
$538$	− 3.72394e6i	− 0.554686i
$539$	1.46006e6	0.216470
$540$	0	0
$541$	6.86308e6	1.00815	0.504076	−	0.863659i	$-0.331833\pi$
$541$	6.86308e6	1.00815	0.504076	+	0.863659i	$0.331833\pi$
$542$	5.32719e6i	0.778932i
$543$	2.68276e6i	0.390465i
$544$	−1.62831e6	−0.235906
$545$	0	0
$546$	323359.	0.0464198
$547$	7.50773e6i	1.07285i	0.843947	+	0.536426i	$0.180226\pi$
$547$	7.50773e6i	1.07285i	−0.843947	+	0.536426i	$0.819774\pi$
$548$	− 812702.i	− 0.115606i
$549$	−3.67367e6	−0.520199
$550$	0	0
$551$	−1.03835e6	−0.145702
$552$	− 3.26209e6i	− 0.455668i
$553$	− 231120.i	− 0.0321385i
$554$	−1.02719e6	−0.142193
$555$	0	0
$556$	−3.61431e6	−0.495836
$557$	1.08263e7i	1.47857i	0.673392	+	0.739286i	$0.264837\pi$
$557$	1.08263e7i	1.47857i	−0.673392	+	0.739286i	$0.735163\pi$
$558$	− 2.97595e6i	− 0.404613i
$559$	3.92605e6	0.531406
$560$	0	0
$561$	−1.73323e6	−0.232515
$562$	162627.i	0.0217196i
$563$	− 1.61894e6i	− 0.215259i	−0.994191	−	0.107629i	$-0.965674\pi$
$563$	− 1.61894e6i	− 0.215259i	0.994191	−	0.107629i	$-0.0343260\pi$
$564$	−2.95269e6	−0.390859
$565$	0	0
$566$	6.67895e6	0.876329
$567$	− 321489.i	− 0.0419961i
$568$	− 7.66846e6i	− 0.997326i
$569$	1.47472e7	1.90954	0.954770	−	0.297344i	$-0.0961009\pi$
$569$	1.47472e7	1.90954	0.954770	+	0.297344i	$0.0961009\pi$
$570$	0	0
$571$	5.37082e6	0.689367	0.344683	−	0.938719i	$-0.387986\pi$
$571$	5.37082e6	0.689367	0.344683	+	0.938719i	$0.387986\pi$
$572$	− 1.79981e6i	− 0.230005i
$573$	− 332052.i	− 0.0422493i
$574$	3.08033e6	0.390227
$575$	0	0
$576$	2.30828e6	0.289889
$577$	3.96034e6i	0.495214i	0.968860	+	0.247607i	$0.0796443\pi$
$577$	3.96034e6i	0.495214i	−0.968860	+	0.247607i	$0.920356\pi$
$578$	5.26224e6i	0.655166i
$579$	5.58694e6	0.692592
$580$	0	0
$581$	−4.35074e6	−0.534716
$582$	− 601997.i	− 0.0736694i
$583$	− 1.66921e7i	− 2.03395i
$584$	8.99014e6	1.09077
$585$	0	0
$586$	4.19917e6	0.505149
$587$	− 6.79201e6i	− 0.813586i	−0.913520	−	0.406793i	$-0.866647\pi$
$587$	− 6.79201e6i	− 0.813586i	0.913520	−	0.406793i	$-0.133353\pi$
$588$	347838.i	0.0414891i
$589$	−3.65532e6	−0.434148
$590$	0	0
$591$	3.02846e6	0.356659
$592$	− 158871.i	− 0.0186312i
$593$	− 1.98637e6i	− 0.231965i	−0.993251	−	0.115983i	$-0.962998\pi$
$593$	− 1.98637e6i	− 0.231965i	0.993251	−	0.115983i	$-0.0370017\pi$
$594$	1.76785e6	0.205580
$595$	0	0
$596$	1.97219e6	0.227423
$597$	− 2.32060e6i	− 0.266480i
$598$	1.38561e6i	0.158449i
$599$	3.97648e6	0.452827	0.226413	−	0.974031i	$-0.427300\pi$
$599$	3.97648e6	0.452827	0.226413	+	0.974031i	$0.427300\pi$
$600$	0	0
$601$	−7.35239e6	−0.830314	−0.415157	−	0.909750i	$-0.636273\pi$
$601$	−7.35239e6	−0.830314	−0.415157	+	0.909750i	$0.636273\pi$
$602$	− 4.17241e6i	− 0.469240i
$603$	4.34273e6i	0.486373i
$604$	1.05999e6	0.118225
$605$	0	0
$606$	109891.	0.0121557
$607$	1.43297e7i	1.57858i	0.614020	+	0.789290i	$0.289551\pi$
$607$	1.43297e7i	1.57858i	−0.614020	+	0.789290i	$0.710449\pi$
$608$	− 2.03998e6i	− 0.223803i
$609$	−1.15414e6	−0.126100
$610$	0	0
$611$	3.74747e6	0.406102
$612$	− 412919.i	− 0.0445642i
$613$	− 2.07696e6i	− 0.223243i	−0.993751	−	0.111621i	$-0.964396\pi$
$613$	− 2.07696e6i	− 0.223243i	0.993751	−	0.111621i	$-0.0356044\pi$
$614$	3.13123e6	0.335192
$615$	0	0
$616$	−5.71521e6	−0.606848
$617$	3.26565e6i	0.345348i	0.984979	+	0.172674i	$0.0552407\pi$
$617$	3.26565e6i	0.345348i	−0.984979	+	0.172674i	$0.944759\pi$
$618$	1.19447e6i	0.125807i
$619$	−1.78136e7	−1.86864	−0.934319	−	0.356439i	$-0.883991\pi$
$619$	−1.78136e7	−1.86864	−0.934319	+	0.356439i	$0.883991\pi$
$620$	0	0
$621$	1.37760e6	0.143349
$622$	− 1.13697e7i	− 1.17835i
$623$	3.45534e6i	0.356673i
$624$	−413352.	−0.0424970
$625$	0	0
$626$	−4.01661e6	−0.409660
$627$	− 2.17143e6i	− 0.220586i
$628$	1.89709e6i	0.191950i
$629$	−201423.	−0.0202994
$630$	0	0
$631$	6.73211e6	0.673097	0.336548	−	0.941666i	$-0.390740\pi$
$631$	6.73211e6	0.673097	0.336548	+	0.941666i	$0.390740\pi$
$632$	904690.i	0.0900963i
$633$	− 5.68796e6i	− 0.564219i
$634$	200660.	0.0198261
$635$	0	0
$636$	3.97666e6	0.389830
$637$	− 441467.i	− 0.0431072i
$638$	− 6.34655e6i	− 0.617285i
$639$	3.23844e6	0.313750
$640$	0	0
$641$	1.68085e7	1.61579	0.807893	−	0.589329i	$-0.200608\pi$
$641$	1.68085e7	1.61579	0.807893	+	0.589329i	$0.200608\pi$
$642$	1.24191e6i	0.118919i
$643$	− 1.97564e7i	− 1.88443i	−0.335013	−	0.942214i	$-0.608741\pi$
$643$	− 1.97564e7i	− 1.88443i	0.335013	−	0.942214i	$-0.391259\pi$
$644$	−1.49051e6	−0.141618
$645$	0	0
$646$	501076.	0.0472413
$647$	7.46809e6i	0.701373i	0.936493	+	0.350686i	$0.114052\pi$
$647$	7.46809e6i	0.701373i	−0.936493	+	0.350686i	$0.885948\pi$
$648$	1.25843e6i	0.117731i
$649$	4.07414e6	0.379686
$650$	0	0
$651$	−4.06292e6	−0.375739
$652$	− 7.52186e6i	− 0.692957i
$653$	− 7.80886e6i	− 0.716646i	−0.933598	−	0.358323i	$-0.883349\pi$
$653$	− 7.80886e6i	− 0.716646i	0.933598	−	0.358323i	$-0.116651\pi$
$654$	5.71384e6	0.522377
$655$	0	0
$656$	−3.93760e6	−0.357250
$657$	3.79659e6i	0.343147i
$658$	− 3.98262e6i	− 0.358595i
$659$	4.36507e6	0.391541	0.195771	−	0.980650i	$-0.437279\pi$
$659$	4.36507e6	0.391541	0.195771	+	0.980650i	$0.437279\pi$
$660$	0	0
$661$	1.56375e6	0.139208	0.0696038	−	0.997575i	$-0.477826\pi$
$661$	1.56375e6	0.139208	0.0696038	+	0.997575i	$0.477826\pi$
$662$	− 7.12789e6i	− 0.632144i
$663$	524065.i	0.0463022i
$664$	1.70304e7	1.49901
$665$	0	0
$666$	205446.	0.0179479
$667$	− 4.94555e6i	− 0.430427i
$668$	8.10236e6i	0.702539i
$669$	−9.10260e6	−0.786322
$670$	0	0
$671$	−2.75799e7	−2.36476
$672$	− 2.26745e6i	− 0.193693i
$673$	− 1.56674e7i	− 1.33340i	−0.745327	−	0.666700i	$-0.767706\pi$
$673$	− 1.56674e7i	− 1.33340i	0.745327	−	0.666700i	$-0.232294\pi$
$674$	1.45525e7	1.23392
$675$	0	0
$676$	5.43248e6	0.457226
$677$	1.01619e6i	0.0852126i	0.999092	+	0.0426063i	$0.0135661\pi$
$677$	1.01619e6i	0.0852126i	−0.999092	+	0.0426063i	$0.986434\pi$
$678$	− 8.69009e6i	− 0.726022i
$679$	−821878.	−0.0684121
$680$	0	0
$681$	6.13889e6	0.507250
$682$	− 2.23418e7i	− 1.83932i
$683$	55688.2i	0.00456784i	0.999997	+	0.00228392i	$0.000726995\pi$
$683$	55688.2i	0.00456784i	−0.999997	+	0.00228392i	$0.999273\pi$
$684$	517313.	0.0422779
$685$	0	0
$686$	−469169.	−0.0380644
$687$	504712.i	0.0407992i
$688$	5.33361e6i	0.429586i
$689$	−5.04706e6	−0.405033
$690$	0	0
$691$	2.55441e6	0.203515	0.101757	−	0.994809i	$-0.467553\pi$
$691$	2.55441e6	0.203515	0.101757	+	0.994809i	$0.467553\pi$
$692$	− 1.88498e6i	− 0.149638i
$693$	− 2.41357e6i	− 0.190909i
$694$	−1.47246e7	−1.16050
$695$	0	0
$696$	4.51772e6	0.353505
$697$	4.99226e6i	0.389238i
$698$	− 1.51127e7i	− 1.17410i
$699$	−192271.	−0.0148840
$700$	0	0
$701$	454132.	0.0349050	0.0174525	−	0.999848i	$-0.494444\pi$
$701$	454132.	0.0349050	0.0174525	+	0.999848i	$0.494444\pi$
$702$	− 534532.i	− 0.0409384i
$703$	− 252347.i	− 0.0192580i
$704$	1.73293e7	1.31780
$705$	0	0
$706$	−8.38468e6	−0.633103
$707$	− 150029.i	− 0.0112883i
$708$	970608.i	0.0727714i
$709$	−2.26775e7	−1.69426	−0.847128	−	0.531388i	$-0.821670\pi$
$709$	−2.26775e7	−1.69426	−0.847128	+	0.531388i	$0.821670\pi$
$710$	0	0
$711$	−382056.	−0.0283435
$712$	− 1.35255e7i	− 0.999889i
$713$	− 1.74099e7i	− 1.28254i
$714$	556950.	0.0408856
$715$	0	0
$716$	1.16086e7	0.846250
$717$	− 3.66312e6i	− 0.266105i
$718$	4.71179e6i	0.341094i
$719$	−6.28298e6	−0.453256	−0.226628	−	0.973981i	$-0.572770\pi$
$719$	−6.28298e6	−0.453256	−0.226628	+	0.973981i	$0.572770\pi$
$720$	0	0
$721$	1.63076e6	0.116829
$722$	− 9.24659e6i	− 0.660144i
$723$	− 1.33102e7i	− 0.946975i
$724$	−4.79824e6	−0.340201
$725$	0	0
$726$	7.49183e6	0.527529
$727$	9.17981e6i	0.644166i	0.946711	+	0.322083i	$0.104383\pi$
$727$	9.17981e6i	0.644166i	−0.946711	+	0.322083i	$0.895617\pi$
$728$	1.72806e6i	0.120846i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	6.76218e6	0.468051
$732$	− 6.57053e6i	− 0.453234i
$733$	− 1.90756e6i	− 0.131135i	−0.997848	−	0.0655674i	$-0.979114\pi$
$733$	− 1.90756e6i	− 0.131135i	0.997848	−	0.0655674i	$-0.0208857\pi$
$734$	1.44624e7	0.990832
$735$	0	0
$736$	9.71616e6	0.661150
$737$	3.26029e7i	2.21099i
$738$	− 5.09197e6i	− 0.344148i
$739$	−1.74486e7	−1.17530	−0.587650	−	0.809115i	$-0.699947\pi$
$739$	−1.74486e7	−1.17530	−0.587650	+	0.809115i	$0.699947\pi$
$740$	0	0
$741$	−656560.	−0.0439267
$742$	5.36376e6i	0.357651i
$743$	− 1.96676e6i	− 0.130701i	−0.997862	−	0.0653504i	$-0.979183\pi$
$743$	− 1.96676e6i	− 0.130701i	0.997862	−	0.0653504i	$-0.0208165\pi$
$744$	1.59038e7	1.05334
$745$	0	0
$746$	1.53980e7	1.01302
$747$	7.19205e6i	0.471575i
$748$	− 3.09997e6i	− 0.202583i
$749$	1.69551e6	0.110432
$750$	0	0
$751$	3.85900e6	0.249675	0.124838	−	0.992177i	$-0.460159\pi$
$751$	3.85900e6	0.249675	0.124838	+	0.992177i	$0.460159\pi$
$752$	5.09101e6i	0.328291i
$753$	8.91603e6i	0.573039i
$754$	−1.91896e6	−0.122924
$755$	0	0
$756$	574998.	0.0365900
$757$	2.69828e7i	1.71138i	0.517485	+	0.855692i	$0.326868\pi$
$757$	2.69828e7i	1.71138i	−0.517485	+	0.855692i	$0.673132\pi$
$758$	− 1.54885e7i	− 0.979121i
$759$	1.03423e7	0.651646
$760$	0	0
$761$	2.88746e6	0.180740	0.0903700	−	0.995908i	$-0.471195\pi$
$761$	2.88746e6	0.180740	0.0903700	+	0.995908i	$0.471195\pi$
$762$	− 1.52929e6i	− 0.0954121i
$763$	− 7.80084e6i	− 0.485098i
$764$	593891.	0.0368106
$765$	0	0
$766$	−1.99044e7	−1.22568
$767$	− 1.23187e6i	− 0.0756094i
$768$	1.00336e7i	0.613840i
$769$	1.77458e7	1.08213	0.541065	−	0.840981i	$-0.318021\pi$
$769$	1.77458e7	1.08213	0.541065	+	0.840981i	$0.318021\pi$
$770$	0	0
$771$	−298069.	−0.0180585
$772$	9.99250e6i	0.603435i
$773$	− 1.76434e7i	− 1.06202i	−0.847365	−	0.531011i	$-0.821812\pi$
$773$	− 1.76434e7i	− 1.06202i	0.847365	−	0.531011i	$-0.178188\pi$
$774$	−6.89724e6	−0.413831
$775$	0	0
$776$	3.21713e6	0.191785
$777$	− 280486.i	− 0.0166671i
$778$	− 5.27887e6i	− 0.312674i
$779$	−6.25441e6	−0.369269
$780$	0	0
$781$	2.43124e7	1.42627
$782$	2.38656e6i	0.139558i
$783$	1.90786e6i	0.111210i
$784$	599741.	0.0348476
$785$	0	0
$786$	4.55885e6	0.263208
$787$	− 4.67447e6i	− 0.269027i	−0.990912	−	0.134513i	$-0.957053\pi$
$787$	− 4.67447e6i	− 0.269027i	0.990912	−	0.134513i	$-0.0429472\pi$
$788$	5.41655e6i	0.310747i
$789$	−6.71297e6	−0.383903
$790$	0	0
$791$	−1.18642e7	−0.674211
$792$	9.44759e6i	0.535190i
$793$	8.33913e6i	0.470910i
$794$	−9.45169e6	−0.532057
$795$	0	0
$796$	4.15051e6	0.232177
$797$	− 1.38221e7i	− 0.770778i	−0.922754	−	0.385389i	$-0.874067\pi$
$797$	− 1.38221e7i	− 0.770778i	0.922754	−	0.385389i	$-0.125933\pi$
$798$	697759.i	0.0387881i
$799$	6.45460e6	0.357687
$800$	0	0
$801$	5.71188e6	0.314556
$802$	2.19019e7i	1.20239i
$803$	2.85027e7i	1.55990i
$804$	−7.76718e6	−0.423763
$805$	0	0
$806$	−6.75533e6	−0.366276
$807$	8.40436e6i	0.454277i
$808$	587269.i	0.0316453i
$809$	−3.66736e7	−1.97007	−0.985037	−	0.172345i	$-0.944866\pi$
$809$	−3.66736e7	−1.97007	−0.985037	+	0.172345i	$0.944866\pi$
$810$	0	0
$811$	1.43912e7	0.768327	0.384163	−	0.923265i	$-0.374490\pi$
$811$	1.43912e7	0.768327	0.384163	+	0.923265i	$0.374490\pi$
$812$	− 2.06423e6i	− 0.109867i
$813$	− 1.20226e7i	− 0.637930i
$814$	1.54238e6	0.0815888
$815$	0	0
$816$	−711952.	−0.0374305
$817$	8.47180e6i	0.444039i
$818$	1.27425e7i	0.665841i
$819$	−729772.	−0.0380169
$820$	0	0
$821$	−3.79866e6	−0.196685	−0.0983427	−	0.995153i	$-0.531354\pi$
$821$	−3.79866e6	−0.196685	−0.0983427	+	0.995153i	$0.531354\pi$
$822$	− 1.81206e6i	− 0.0935388i
$823$	1.11281e7i	0.572691i	0.958126	+	0.286345i	$0.0924405\pi$
$823$	1.11281e7i	0.572691i	−0.958126	+	0.286345i	$0.907559\pi$
$824$	−6.38340e6	−0.327517
$825$	0	0
$826$	−1.30917e6	−0.0667644
$827$	− 2.47766e7i	− 1.25973i	−0.776704	−	0.629866i	$-0.783110\pi$
$827$	− 2.47766e7i	− 1.25973i	0.776704	−	0.629866i	$-0.216890\pi$
$828$	2.46390e6i	0.124896i
$829$	1.95608e7	0.988555	0.494278	−	0.869304i	$-0.335433\pi$
$829$	1.95608e7	0.988555	0.494278	+	0.869304i	$0.335433\pi$
$830$	0	0
$831$	2.31822e6	0.116453
$832$	− 5.23973e6i	− 0.262422i
$833$	− 760377.i	− 0.0379679i
$834$	−8.05871e6	−0.401190
$835$	0	0
$836$	3.88371e6	0.192190
$837$	6.71626e6i	0.331370i
$838$	1.79971e7i	0.885302i
$839$	9.56061e6	0.468900	0.234450	−	0.972128i	$-0.424671\pi$
$839$	9.56061e6	0.468900	0.234450	+	0.972128i	$0.424671\pi$
$840$	0	0
$841$	−1.36620e7	−0.666076
$842$	− 8.39574e6i	− 0.408112i
$843$	− 367023.i	− 0.0177879i
$844$	1.01732e7	0.491588
$845$	0	0
$846$	−6.58352e6	−0.316251
$847$	− 1.02282e7i	− 0.489883i
$848$	− 6.85653e6i	− 0.327427i
$849$	−1.50734e7	−0.717696
$850$	0	0
$851$	1.20190e6	0.0568911
$852$	5.79209e6i	0.273361i
$853$	1.75892e7i	0.827701i	0.910345	+	0.413851i	$0.135816\pi$
$853$	1.75892e7i	0.827701i	−0.910345	+	0.413851i	$0.864184\pi$
$854$	8.86241e6	0.415822
$855$	0	0
$856$	−6.63687e6	−0.309584
$857$	− 2.95318e7i	− 1.37353i	−0.726880	−	0.686764i	$-0.759031\pi$
$857$	− 2.95318e7i	− 1.37353i	0.726880	−	0.686764i	$-0.240969\pi$
$858$	− 4.01298e6i	− 0.186101i
$859$	−2.55915e7	−1.18335	−0.591674	−	0.806177i	$-0.701533\pi$
$859$	−2.55915e7	−1.18335	−0.591674	+	0.806177i	$0.701533\pi$
$860$	0	0
$861$	−6.95183e6	−0.319588
$862$	− 2.55715e7i	− 1.17216i
$863$	− 1.21161e7i	− 0.553776i	−0.960902	−	0.276888i	$-0.910697\pi$
$863$	− 1.21161e7i	− 0.553776i	0.960902	−	0.276888i	$-0.0893031\pi$
$864$	−3.74823e6	−0.170821
$865$	0	0
$866$	1.01397e7	0.459444
$867$	− 1.18761e7i	− 0.536568i
$868$	− 7.26672e6i	− 0.327370i
$869$	−2.86827e6	−0.128846
$870$	0	0
$871$	9.85789e6	0.440290
$872$	3.05354e7i	1.35991i
$873$	1.35861e6i	0.0603338i
$874$	−2.98994e6	−0.132399
$875$	0	0
$876$	−6.79038e6	−0.298974
$877$	1.90755e7i	0.837483i	0.908105	+	0.418742i	$0.137529\pi$
$877$	1.90755e7i	0.837483i	−0.908105	+	0.418742i	$0.862471\pi$
$878$	− 2.42697e7i	− 1.06250i
$879$	−9.47688e6	−0.413707
$880$	0	0
$881$	2.18358e7	0.947830	0.473915	−	0.880571i	$-0.342840\pi$
$881$	2.18358e7	0.947830	0.473915	+	0.880571i	$0.342840\pi$
$882$	775564.i	0.0335696i
$883$	− 4.10178e7i	− 1.77040i	−0.465214	−	0.885198i	$-0.654023\pi$
$883$	− 4.10178e7i	− 1.77040i	0.465214	−	0.885198i	$-0.345977\pi$
$884$	−937315.	−0.0403418
$885$	0	0
$886$	1.48091e6	0.0633787
$887$	3.64575e7i	1.55589i	0.628335	+	0.777943i	$0.283737\pi$
$887$	3.64575e7i	1.55589i	−0.628335	+	0.777943i	$0.716263\pi$
$888$	1.09793e6i	0.0467241i
$889$	−2.08787e6	−0.0886032
$890$	0	0
$891$	−3.98977e6	−0.168366
$892$	− 1.62804e7i	− 0.685100i
$893$	8.08646e6i	0.339336i
$894$	4.39734e6	0.184012
$895$	0	0
$896$	2.49352e6	0.103763
$897$	− 3.12712e6i	− 0.129767i
$898$	1.82277e7i	0.754294i
$899$	2.41112e7	0.994992
$900$	0	0
$901$	−8.69300e6	−0.356745
$902$	− 3.82278e7i	− 1.56445i
$903$	9.41648e6i	0.384299i
$904$	4.64407e7	1.89007
$905$	0	0
$906$	2.36342e6	0.0956577
$907$	3.68833e7i	1.48871i	0.667782	+	0.744357i	$0.267244\pi$
$907$	3.68833e7i	1.48871i	−0.667782	+	0.744357i	$0.732756\pi$
$908$	1.09797e7i	0.441953i
$909$	−248007.	−0.00995530
$910$	0	0
$911$	−3.65024e7	−1.45722	−0.728611	−	0.684927i	$-0.759834\pi$
$911$	−3.65024e7	−1.45722	−0.728611	+	0.684927i	$0.759834\pi$
$912$	− 891949.i	− 0.0355102i
$913$	5.39940e7i	2.14372i
$914$	−7.00545e6	−0.277377
$915$	0	0
$916$	−902701.	−0.0355472
$917$	− 6.22399e6i	− 0.244425i
$918$	− 920672.i	− 0.0360577i
$919$	2.16771e7	0.846668	0.423334	−	0.905974i	$-0.360860\pi$
$919$	2.16771e7	0.846668	0.423334	+	0.905974i	$0.360860\pi$
$920$	0	0
$921$	−7.06669e6	−0.274516
$922$	− 4.31058e6i	− 0.166997i
$923$	− 7.35116e6i	− 0.284022i
$924$	4.31677e6	0.166333
$925$	0	0
$926$	8.10113e6	0.310469
$927$	− 2.69575e6i	− 0.103034i
$928$	1.34561e7i	0.512918i
$929$	−2.13217e7	−0.810557	−0.405278	−	0.914193i	$-0.632825\pi$
$929$	−2.13217e7	−0.810557	−0.405278	+	0.914193i	$0.632825\pi$
$930$	0	0
$931$	952617.	0.0360200
$932$	− 343885.i	− 0.0129680i
$933$	2.56597e7i	0.965043i
$934$	−1.31808e7	−0.494397
$935$	0	0
$936$	2.85660e6	0.106576
$937$	− 2.29451e7i	− 0.853768i	−0.904306	−	0.426884i	$-0.859611\pi$
$937$	− 2.29451e7i	− 0.853768i	0.904306	−	0.426884i	$-0.140389\pi$
$938$	− 1.04765e7i	− 0.388783i
$939$	9.06487e6	0.335504
$940$	0	0
$941$	−5.02665e7	−1.85057	−0.925283	−	0.379278i	$-0.876172\pi$
$941$	−5.02665e7	−1.85057	−0.925283	+	0.379278i	$0.876172\pi$
$942$	4.22988e6i	0.155311i
$943$	− 2.97890e7i	− 1.09088i
$944$	1.67352e6	0.0611223
$945$	0	0
$946$	−5.17808e7	−1.88122
$947$	2.39762e7i	0.868772i	0.900727	+	0.434386i	$0.143035\pi$
$947$	2.39762e7i	0.868772i	−0.900727	+	0.434386i	$0.856965\pi$
$948$	− 683325.i	− 0.0246948i
$949$	8.61816e6	0.310634
$950$	0	0
$951$	−452859.	−0.0162372
$952$	2.97640e6i	0.106438i
$953$	2.76130e7i	0.984875i	0.870348	+	0.492438i	$0.163894\pi$
$953$	2.76130e7i	0.984875i	−0.870348	+	0.492438i	$0.836106\pi$
$954$	8.86663e6	0.315419
$955$	0	0
$956$	6.55165e6	0.231849
$957$	1.43232e7i	0.505545i
$958$	− 1.23523e7i	− 0.434846i
$959$	−2.47391e6	−0.0868636
$960$	0	0
$961$	5.62497e7	1.96477
$962$	− 466358.i	− 0.0162473i
$963$	− 2.80279e6i	− 0.0973923i
$964$	2.38059e7	0.825072
$965$	0	0
$966$	−3.32334e6	−0.114586
$967$	− 5.30618e7i	− 1.82480i	−0.409298	−	0.912401i	$-0.634227\pi$
$967$	− 5.30618e7i	− 1.82480i	0.409298	−	0.912401i	$-0.365773\pi$
$968$	4.00371e7i	1.37333i
$969$	−1.13085e6	−0.0386898
$970$	0	0
$971$	2.93111e7	0.997664	0.498832	−	0.866699i	$-0.333763\pi$
$971$	2.93111e7	0.997664	0.498832	+	0.866699i	$0.333763\pi$
$972$	− 950507.i	− 0.0322693i
$973$	1.10022e7i	0.372560i
$974$	−603400.	−0.0203802
$975$	0	0
$976$	−1.13289e7	−0.380681
$977$	− 5.32670e7i	− 1.78534i	−0.450707	−	0.892672i	$-0.648828\pi$
$977$	− 5.32670e7i	− 1.78534i	0.450707	−	0.892672i	$-0.351172\pi$
$978$	− 1.67713e7i	− 0.560685i
$979$	4.28817e7	1.42993
$980$	0	0
$981$	−1.28953e7	−0.427817
$982$	1.73969e6i	0.0575695i
$983$	− 3.88419e7i	− 1.28208i	−0.767506	−	0.641042i	$-0.778503\pi$
$983$	− 3.88419e7i	− 1.28208i	0.767506	−	0.641042i	$-0.221497\pi$
$984$	2.72120e7	0.895928
$985$	0	0
$986$	−3.30519e6	−0.108269
$987$	8.98817e6i	0.293683i
$988$	− 1.17429e6i	− 0.0382721i
$989$	−4.03502e7	−1.31176
$990$	0	0
$991$	6.16595e7	1.99442	0.997208	−	0.0746757i	$-0.0237922\pi$
$991$	6.16595e7	1.99442	0.997208	+	0.0746757i	$0.0237922\pi$
$992$	4.73695e7i	1.52834i
$993$	1.60865e7i	0.517714i
$994$	−7.81244e6	−0.250796
$995$	0	0
$996$	−1.28633e7	−0.410870
$997$	− 9.78217e6i	− 0.311672i	−0.987783	−	0.155836i	$-0.950193\pi$
$997$	− 9.78217e6i	− 0.311672i	0.987783	−	0.155836i	$-0.0498071\pi$
$998$	2.12689e7i	0.675957i
$999$	−463661.	−0.0146990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.6.d.p.274.8		12
5.2	odd	4		525.6.a.r.1.3	✓	6
5.3	odd	4		525.6.a.s.1.4	yes	6
5.4	even	2	inner	525.6.d.p.274.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.6.a.r.1.3	✓	6	5.2	odd	4
525.6.a.s.1.4	yes	6	5.3	odd	4
525.6.d.p.274.5		12	5.4	even	2	inner
525.6.d.p.274.8		12	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.98787i q^{2} -9.00000i q^{3} +16.0969 q^{4} +35.8908 q^{6} -49.0000i q^{7} +191.804i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+3.98787i q^{2} -9.00000i q^{3} +16.0969 q^{4} +35.8908 q^{6} -49.0000i q^{7} +191.804i q^{8} -81.0000 q^{9} -608.104 q^{11} -144.872i q^{12} +183.868i q^{13} +195.405 q^{14} -249.788 q^{16} +316.692i q^{17} -323.017i q^{18} -396.758 q^{19} -441.000 q^{21} -2425.04i q^{22} -1889.71i q^{23} +1726.24 q^{24} -733.241 q^{26} +729.000i q^{27} -788.749i q^{28} +2617.09 q^{29} +9212.97 q^{31} +5141.61i q^{32} +5472.94i q^{33} -1262.93 q^{34} -1303.85 q^{36} +636.023i q^{37} -1582.22i q^{38} +1654.81 q^{39} +15763.8 q^{41} -1758.65i q^{42} -21352.6i q^{43} -9788.60 q^{44} +7535.92 q^{46} -20381.3i q^{47} +2248.09i q^{48} -2401.00 q^{49} +2850.23 q^{51} +2959.71i q^{52} +27449.4i q^{53} -2907.15 q^{54} +9398.40 q^{56} +3570.82i q^{57} +10436.6i q^{58} -6699.75 q^{59} +45353.9 q^{61} +36740.1i q^{62} +3969.00i q^{63} -28497.3 q^{64} -21825.3 q^{66} -53614.0i q^{67} +5097.76i q^{68} -17007.4 q^{69} -39980.7 q^{71} -15536.1i q^{72} -46871.5i q^{73} -2536.38 q^{74} -6386.59 q^{76} +29797.1i q^{77} +6599.17i q^{78} +4716.74 q^{79} +6561.00 q^{81} +62863.8i q^{82} -88790.7i q^{83} -7098.74 q^{84} +85151.2 q^{86} -23553.8i q^{87} -116637. i q^{88} -70517.0 q^{89} +9009.53 q^{91} -30418.5i q^{92} -82916.8i q^{93} +81278.0 q^{94} +46274.5 q^{96} -16773.0i q^{97} -9574.87i q^{98} +49256.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9}+O(q^{10}) \) 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9	\( 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9} + 392 q^{14} + 3568 q^{16} - 520 q^{19} - 5292 q^{21} - 5454 q^{24} - 6534 q^{26} + 4244 q^{29} + 4524 q^{31} - 24778 q^{34} + 21384 q^{36} + 3816 q^{39} + 16248 q^{41} - 62322 q^{44} + 58720 q^{46} - 28812 q^{49} + 67320 q^{51} - 5832 q^{54} - 29694 q^{56} - 51316 q^{59} + 82376 q^{61} - 131894 q^{64} + 124218 q^{66} + 33336 q^{69} + 35160 q^{71} - 7942 q^{74} + 507092 q^{76} + 19472 q^{79} + 78732 q^{81} + 116424 q^{84} + 287632 q^{86} + 660720 q^{89} + 20776 q^{91} + 747360 q^{94} + 491220 q^{96}+O(q^{100}) \) 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9 + 392 * q^14 + 3568 * q^16 - 520 * q^19 - 5292 * q^21 - 5454 * q^24 - 6534 * q^26 + 4244 * q^29 + 4524 * q^31 - 24778 * q^34 + 21384 * q^36 + 3816 * q^39 + 16248 * q^41 - 62322 * q^44 + 58720 * q^46 - 28812 * q^49 + 67320 * q^51 - 5832 * q^54 - 29694 * q^56 - 51316 * q^59 + 82376 * q^61 - 131894 * q^64 + 124218 * q^66 + 33336 * q^69 + 35160 * q^71 - 7942 * q^74 + 507092 * q^76 + 19472 * q^79 + 78732 * q^81 + 116424 * q^84 + 287632 * q^86 + 660720 * q^89 + 20776 * q^91 + 747360 * q^94 + 491220 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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