LMFDB - Embedded newform 825.6.a.v.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + \cdots + 522579400 $ x^13 - 306x^11 - 206x^10 + 34574x^9 + 39928x^8 - 1788312x^7 - 2591628x^6 + 42852537x^5 + 63733360x^4 - 448113518x^3 - 549984598x^2 + 1518551280*x + 522579400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-3.97394$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.97394 q^{2} -9.00000 q^{3} -7.25990 q^{4} +44.7655 q^{6} +155.412 q^{7} +195.276 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-4.97394 q^{2} -9.00000 q^{3} -7.25990 q^{4} +44.7655 q^{6} +155.412 q^{7} +195.276 q^{8} +81.0000 q^{9} +121.000 q^{11} +65.3391 q^{12} +1173.30 q^{13} -773.008 q^{14} -738.977 q^{16} +898.118 q^{17} -402.889 q^{18} -2407.95 q^{19} -1398.70 q^{21} -601.847 q^{22} -4121.12 q^{23} -1757.49 q^{24} -5835.95 q^{26} -729.000 q^{27} -1128.27 q^{28} +75.4479 q^{29} +3595.47 q^{31} -2573.22 q^{32} -1089.00 q^{33} -4467.18 q^{34} -588.052 q^{36} -16063.1 q^{37} +11977.0 q^{38} -10559.7 q^{39} +15062.8 q^{41} +6957.07 q^{42} -12867.8 q^{43} -878.448 q^{44} +20498.2 q^{46} -6698.84 q^{47} +6650.79 q^{48} +7345.75 q^{49} -8083.06 q^{51} -8518.07 q^{52} +22850.8 q^{53} +3626.00 q^{54} +30348.2 q^{56} +21671.6 q^{57} -375.274 q^{58} -13021.3 q^{59} +882.310 q^{61} -17883.6 q^{62} +12588.3 q^{63} +36446.3 q^{64} +5416.62 q^{66} -14191.7 q^{67} -6520.24 q^{68} +37090.1 q^{69} -78746.0 q^{71} +15817.4 q^{72} +34227.3 q^{73} +79896.8 q^{74} +17481.5 q^{76} +18804.8 q^{77} +52523.5 q^{78} -48694.6 q^{79} +6561.00 q^{81} -74921.3 q^{82} -95905.3 q^{83} +10154.5 q^{84} +64003.6 q^{86} -679.031 q^{87} +23628.5 q^{88} +1342.55 q^{89} +182345. q^{91} +29918.9 q^{92} -32359.2 q^{93} +33319.6 q^{94} +23159.0 q^{96} +80938.9 q^{97} -36537.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * q^99

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 $

$2$	−4.97394	−0.879277	−0.439639	−	0.898175i	$-0.644893\pi$
$2$	−4.97394	−0.879277	−0.439639	+	0.898175i	$0.644893\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−7.25990	−0.226872
$5$	0	0
$5$	0	0
$6$	44.7655	0.507651
$7$	155.412	1.19878	0.599388	−	0.800459i	$-0.295411\pi$
$7$	155.412	1.19878	0.599388	+	0.800459i	$0.295411\pi$
$8$	195.276	1.07876
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	65.3391	0.130985
$13$	1173.30	1.92554	0.962770	−	0.270323i	$-0.0871306\pi$
$13$	1173.30	1.92554	0.962770	+	0.270323i	$0.0871306\pi$
$14$	−773.008	−1.05406
$15$	0	0
$16$	−738.977	−0.721657
$17$	898.118	0.753721	0.376861	−	0.926270i	$-0.377004\pi$
$17$	898.118	0.753721	0.376861	+	0.926270i	$0.377004\pi$
$18$	−402.889	−0.293092
$19$	−2407.95	−1.53026	−0.765128	−	0.643879i	$-0.777324\pi$
$19$	−2407.95	−1.53026	−0.765128	+	0.643879i	$0.777324\pi$
$20$	0	0
$21$	−1398.70	−0.692114
$22$	−601.847	−0.265112
$23$	−4121.12	−1.62441	−0.812205	−	0.583372i	$-0.801733\pi$
$23$	−4121.12	−1.62441	−0.812205	+	0.583372i	$0.801733\pi$
$24$	−1757.49	−0.622823
$25$	0	0
$26$	−5835.95	−1.69308
$27$	−729.000	−0.192450
$28$	−1128.27	−0.271969
$29$	75.4479	0.0166591	0.00832957	−	0.999965i	$-0.497349\pi$
$29$	75.4479	0.0166591	0.00832957	+	0.999965i	$0.497349\pi$
$30$	0	0
$31$	3595.47	0.671972	0.335986	−	0.941867i	$-0.390931\pi$
$31$	3595.47	0.671972	0.335986	+	0.941867i	$0.390931\pi$
$32$	−2573.22	−0.444224
$33$	−1089.00	−0.174078
$34$	−4467.18	−0.662730
$35$	0	0
$36$	−588.052	−0.0756240
$37$	−16063.1	−1.92897	−0.964483	−	0.264145i	$-0.914910\pi$
$37$	−16063.1	−1.92897	−0.964483	+	0.264145i	$0.914910\pi$
$38$	11977.0	1.34552
$39$	−10559.7	−1.11171
$40$	0	0
$41$	15062.8	1.39941	0.699705	−	0.714432i	$-0.253315\pi$
$41$	15062.8	1.39941	0.699705	+	0.714432i	$0.253315\pi$
$42$	6957.07	0.608560
$43$	−12867.8	−1.06129	−0.530643	−	0.847595i	$-0.678050\pi$
$43$	−12867.8	−1.06129	−0.530643	+	0.847595i	$0.678050\pi$
$44$	−878.448	−0.0684045
$45$	0	0
$46$	20498.2	1.42831
$47$	−6698.84	−0.442339	−0.221169	−	0.975235i	$-0.570987\pi$
$47$	−6698.84	−0.442339	−0.221169	+	0.975235i	$0.570987\pi$
$48$	6650.79	0.416649
$49$	7345.75	0.437065
$50$	0	0
$51$	−8083.06	−0.435161
$52$	−8518.07	−0.436851
$53$	22850.8	1.11741	0.558705	−	0.829367i	$-0.311298\pi$
$53$	22850.8	1.11741	0.558705	+	0.829367i	$0.311298\pi$
$54$	3626.00	0.169217
$55$	0	0
$56$	30348.2	1.29319
$57$	21671.6	0.883493
$58$	−375.274	−0.0146480
$59$	−13021.3	−0.486995	−0.243498	−	0.969901i	$-0.578295\pi$
$59$	−13021.3	−0.486995	−0.243498	+	0.969901i	$0.578295\pi$
$60$	0	0
$61$	882.310	0.0303596	0.0151798	−	0.999885i	$-0.495168\pi$
$61$	882.310	0.0303596	0.0151798	+	0.999885i	$0.495168\pi$
$62$	−17883.6	−0.590849
$63$	12588.3	0.399592
$64$	36446.3	1.11225
$65$	0	0
$66$	5416.62	0.153062
$67$	−14191.7	−0.386231	−0.193115	−	0.981176i	$-0.561859\pi$
$67$	−14191.7	−0.386231	−0.193115	+	0.981176i	$0.561859\pi$
$68$	−6520.24	−0.170998
$69$	37090.1	0.937854
$70$	0	0
$71$	−78746.0	−1.85388	−0.926942	−	0.375205i	$-0.877572\pi$
$71$	−78746.0	−1.85388	−0.926942	+	0.375205i	$0.877572\pi$
$72$	15817.4	0.359587
$73$	34227.3	0.751736	0.375868	−	0.926673i	$-0.377345\pi$
$73$	34227.3	0.751736	0.375868	+	0.926673i	$0.377345\pi$
$74$	79896.8	1.69610
$75$	0	0
$76$	17481.5	0.347172
$77$	18804.8	0.361445
$78$	52523.5	0.977502
$79$	−48694.6	−0.877835	−0.438917	−	0.898527i	$-0.644638\pi$
$79$	−48694.6	−0.877835	−0.438917	+	0.898527i	$0.644638\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−74921.3	−1.23047
$83$	−95905.3	−1.52809	−0.764043	−	0.645166i	$-0.776788\pi$
$83$	−95905.3	−1.52809	−0.764043	+	0.645166i	$0.776788\pi$
$84$	10154.5	0.157021
$85$	0	0
$86$	64003.6	0.933165
$87$	−679.031	−0.00961815
$88$	23628.5	0.325258
$89$	1342.55	0.0179662	0.00898308	−	0.999960i	$-0.497141\pi$
$89$	1342.55	0.0179662	0.00898308	+	0.999960i	$0.497141\pi$
$90$	0	0
$91$	182345.	2.30829
$92$	29918.9	0.368533
$93$	−32359.2	−0.387963
$94$	33319.6	0.388938
$95$	0	0
$96$	23159.0	0.256473
$97$	80938.9	0.873430	0.436715	−	0.899600i	$-0.356142\pi$
$97$	80938.9	0.873430	0.436715	+	0.899600i	$0.356142\pi$
$98$	−36537.3	−0.384301
$99$	9801.00	0.100504
$100$	0	0
$101$	19225.2	0.187529	0.0937644	−	0.995594i	$-0.470110\pi$
$101$	19225.2	0.187529	0.0937644	+	0.995594i	$0.470110\pi$
$102$	40204.7	0.382627
$103$	−126651.	−1.17630	−0.588148	−	0.808753i	$-0.700143\pi$
$103$	−126651.	−1.17630	−0.588148	+	0.808753i	$0.700143\pi$
$104$	229119.	2.07720
$105$	0	0
$106$	−113659.	−0.982513
$107$	77773.3	0.656706	0.328353	−	0.944555i	$-0.393506\pi$
$107$	77773.3	0.656706	0.328353	+	0.944555i	$0.393506\pi$
$108$	5292.47	0.0436615
$109$	68872.6	0.555240	0.277620	−	0.960691i	$-0.410454\pi$
$109$	68872.6	0.555240	0.277620	+	0.960691i	$0.410454\pi$
$110$	0	0
$111$	144568.	1.11369
$112$	−114846.	−0.865106
$113$	−43794.9	−0.322647	−0.161323	−	0.986902i	$-0.551576\pi$
$113$	−43794.9	−0.322647	−0.161323	+	0.986902i	$0.551576\pi$
$114$	−107793.	−0.776835
$115$	0	0
$116$	−547.745	−0.00377949
$117$	95037.7	0.641846
$118$	64767.3	0.428204
$119$	139578.	0.903543
$120$	0	0
$121$	14641.0	0.0909091
$122$	−4388.56	−0.0266945
$123$	−135565.	−0.807949
$124$	−26102.7	−0.152452
$125$	0	0
$126$	−62613.6	−0.351352
$127$	−209089.	−1.15033	−0.575165	−	0.818037i	$-0.695062\pi$
$127$	−209089.	−1.15033	−0.575165	+	0.818037i	$0.695062\pi$
$128$	−98938.8	−0.533755
$129$	115810.	0.612734
$130$	0	0
$131$	−85677.3	−0.436202	−0.218101	−	0.975926i	$-0.569986\pi$
$131$	−85677.3	−0.436202	−0.218101	+	0.975926i	$0.569986\pi$
$132$	7906.03	0.0394933
$133$	−374224.	−1.83443
$134$	70588.6	0.339604
$135$	0	0
$136$	175381.	0.813085
$137$	115510.	0.525795	0.262898	−	0.964824i	$-0.415322\pi$
$137$	115510.	0.525795	0.262898	+	0.964824i	$0.415322\pi$
$138$	−184484.	−0.824633
$139$	−120621.	−0.529523	−0.264761	−	0.964314i	$-0.585293\pi$
$139$	−120621.	−0.529523	−0.264761	+	0.964314i	$0.585293\pi$
$140$	0	0
$141$	60289.5	0.255384
$142$	391678.	1.63008
$143$	141970.	0.580572
$144$	−59857.1	−0.240552
$145$	0	0
$146$	−170245.	−0.660984
$147$	−66111.7	−0.252339
$148$	116616.	0.437628
$149$	49190.8	0.181517	0.0907587	−	0.995873i	$-0.471071\pi$
$149$	49190.8	0.181517	0.0907587	+	0.995873i	$0.471071\pi$
$150$	0	0
$151$	−490808.	−1.75174	−0.875870	−	0.482547i	$-0.839712\pi$
$151$	−490808.	−1.75174	−0.875870	+	0.482547i	$0.839712\pi$
$152$	−470216.	−1.65078
$153$	72747.5	0.251240
$154$	−93534.0	−0.317810
$155$	0	0
$156$	76662.7	0.252216
$157$	−273703.	−0.886199	−0.443099	−	0.896473i	$-0.646121\pi$
$157$	−273703.	−0.886199	−0.443099	+	0.896473i	$0.646121\pi$
$158$	242204.	0.771860
$159$	−205657.	−0.645137
$160$	0	0
$161$	−640470.	−1.94730
$162$	−32634.0	−0.0976974
$163$	64881.0	0.191271	0.0956354	−	0.995416i	$-0.469512\pi$
$163$	64881.0	0.191271	0.0956354	+	0.995416i	$0.469512\pi$
$164$	−109354.	−0.317487
$165$	0	0
$166$	477028.	1.34361
$167$	379513.	1.05302	0.526508	−	0.850170i	$-0.323501\pi$
$167$	379513.	1.05302	0.526508	+	0.850170i	$0.323501\pi$
$168$	−273134.	−0.746625
$169$	1.00535e6	2.70770
$170$	0	0
$171$	−195044.	−0.510085
$172$	93418.8	0.240776
$173$	−509354.	−1.29391	−0.646955	−	0.762528i	$-0.723958\pi$
$173$	−509354.	−1.29391	−0.646955	+	0.762528i	$0.723958\pi$
$174$	3377.46	0.00845702
$175$	0	0
$176$	−89416.2	−0.217588
$177$	117192.	0.281167
$178$	−6677.76	−0.0157972
$179$	706267.	1.64754	0.823771	−	0.566923i	$-0.191866\pi$
$179$	706267.	1.64754	0.823771	+	0.566923i	$0.191866\pi$
$180$	0	0
$181$	250646.	0.568674	0.284337	−	0.958724i	$-0.408227\pi$
$181$	250646.	0.568674	0.284337	+	0.958724i	$0.408227\pi$
$182$	−906974.	−2.02963
$183$	−7940.79	−0.0175281
$184$	−804758.	−1.75235
$185$	0	0
$186$	160953.	0.341127
$187$	108672.	0.227256
$188$	48632.9	0.100354
$189$	−113295.	−0.230705
$190$	0	0
$191$	−592850.	−1.17588	−0.587938	−	0.808906i	$-0.700060\pi$
$191$	−592850.	−1.17588	−0.587938	+	0.808906i	$0.700060\pi$
$192$	−328017.	−0.642160
$193$	−965415.	−1.86561	−0.932804	−	0.360383i	$-0.882646\pi$
$193$	−965415.	−1.86561	−0.932804	+	0.360383i	$0.882646\pi$
$194$	−402585.	−0.767986
$195$	0	0
$196$	−53329.4	−0.0991577
$197$	−175247.	−0.321725	−0.160862	−	0.986977i	$-0.551427\pi$
$197$	−175247.	−0.321725	−0.160862	+	0.986977i	$0.551427\pi$
$198$	−48749.6	−0.0883707
$199$	−48708.5	−0.0871911	−0.0435955	−	0.999049i	$-0.513881\pi$
$199$	−48708.5	−0.0871911	−0.0435955	+	0.999049i	$0.513881\pi$
$200$	0	0
$201$	127725.	0.222990
$202$	−95625.1	−0.164890
$203$	11725.5	0.0199706
$204$	58682.2	0.0987259
$205$	0	0
$206$	629956.	1.03429
$207$	−333811.	−0.541470
$208$	−867045.	−1.38958
$209$	−291362.	−0.461389
$210$	0	0
$211$	662206.	1.02397	0.511985	−	0.858995i	$-0.328910\pi$
$211$	662206.	1.02397	0.511985	+	0.858995i	$0.328910\pi$
$212$	−165895.	−0.253509
$213$	708714.	1.07034
$214$	−386840.	−0.577427
$215$	0	0
$216$	−142357.	−0.207608
$217$	558777.	0.805544
$218$	−342568.	−0.488209
$219$	−308046.	−0.434015
$220$	0	0
$221$	1.05377e6	1.45132
$222$	−719071.	−0.979241
$223$	−44174.7	−0.0594855	−0.0297428	−	0.999558i	$-0.509469\pi$
$223$	−44174.7	−0.0594855	−0.0297428	+	0.999558i	$0.509469\pi$
$224$	−399908.	−0.532525
$225$	0	0
$226$	217833.	0.283696
$227$	585184.	0.753750	0.376875	−	0.926264i	$-0.376999\pi$
$227$	585184.	0.753750	0.376875	+	0.926264i	$0.376999\pi$
$228$	−157333.	−0.200440
$229$	914300.	1.15213	0.576063	−	0.817405i	$-0.304588\pi$
$229$	914300.	1.15213	0.576063	+	0.817405i	$0.304588\pi$
$230$	0	0
$231$	−169243.	−0.208680
$232$	14733.2	0.0179712
$233$	702641.	0.847898	0.423949	−	0.905686i	$-0.360644\pi$
$233$	702641.	0.847898	0.423949	+	0.905686i	$0.360644\pi$
$234$	−472712.	−0.564361
$235$	0	0
$236$	94533.5	0.110486
$237$	438251.	0.506818
$238$	−694252.	−0.794465
$239$	572014.	0.647757	0.323879	−	0.946099i	$-0.395013\pi$
$239$	572014.	0.647757	0.323879	+	0.946099i	$0.395013\pi$
$240$	0	0
$241$	−485324.	−0.538256	−0.269128	−	0.963104i	$-0.586736\pi$
$241$	−485324.	−0.538256	−0.269128	+	0.963104i	$0.586736\pi$
$242$	−72823.5	−0.0799343
$243$	−59049.0	−0.0641500
$244$	−6405.48	−0.00688775
$245$	0	0
$246$	674291.	0.710411
$247$	−2.82526e6	−2.94657
$248$	702110.	0.724897
$249$	863148.	0.882240
$250$	0	0
$251$	25755.4	0.0258039	0.0129019	−	0.999917i	$-0.495893\pi$
$251$	25755.4	0.0258039	0.0129019	+	0.999917i	$0.495893\pi$
$252$	−91390.1	−0.0906562
$253$	−498656.	−0.489778
$254$	1.04000e6	1.01146
$255$	0	0
$256$	−674166.	−0.642935
$257$	243229.	0.229711	0.114856	−	0.993382i	$-0.463359\pi$
$257$	243229.	0.229711	0.114856	+	0.993382i	$0.463359\pi$
$258$	−576032.	−0.538763
$259$	−2.49639e6	−2.31240
$260$	0	0
$261$	6111.28	0.00555304
$262$	426154.	0.383542
$263$	46278.0	0.0412558	0.0206279	−	0.999787i	$-0.493433\pi$
$263$	46278.0	0.0412558	0.0206279	+	0.999787i	$0.493433\pi$
$264$	−212656.	−0.187788
$265$	0	0
$266$	1.86137e6	1.61298
$267$	−12082.9	−0.0103728
$268$	103030.	0.0876249
$269$	174634.	0.147146	0.0735729	−	0.997290i	$-0.476560\pi$
$269$	174634.	0.147146	0.0735729	+	0.997290i	$0.476560\pi$
$270$	0	0
$271$	−1.78096e6	−1.47309	−0.736546	−	0.676387i	$-0.763545\pi$
$271$	−1.78096e6	−1.47309	−0.736546	+	0.676387i	$0.763545\pi$
$272$	−663688.	−0.543929
$273$	−1.64111e6	−1.33269
$274$	−574538.	−0.462320
$275$	0	0
$276$	−269270.	−0.212773
$277$	545088.	0.426842	0.213421	−	0.976960i	$-0.431539\pi$
$277$	545088.	0.426842	0.213421	+	0.976960i	$0.431539\pi$
$278$	599960.	0.465597
$279$	291233.	0.223991
$280$	0	0
$281$	85469.5	0.0645722	0.0322861	−	0.999479i	$-0.489721\pi$
$281$	85469.5	0.0645722	0.0322861	+	0.999479i	$0.489721\pi$
$282$	−299877.	−0.224554
$283$	−1.51069e6	−1.12127	−0.560634	−	0.828064i	$-0.689442\pi$
$283$	−1.51069e6	−1.12127	−0.560634	+	0.828064i	$0.689442\pi$
$284$	571688.	0.420594
$285$	0	0
$286$	−706150.	−0.510484
$287$	2.34093e6	1.67758
$288$	−208431.	−0.148075
$289$	−613242.	−0.431904
$290$	0	0
$291$	−728450.	−0.504275
$292$	−248487.	−0.170548
$293$	−1.05179e6	−0.715746	−0.357873	−	0.933770i	$-0.616498\pi$
$293$	−1.05179e6	−0.715746	−0.357873	+	0.933770i	$0.616498\pi$
$294$	328836.	0.221876
$295$	0	0
$296$	−3.13674e6	−2.08089
$297$	−88209.0	−0.0580259
$298$	−244672.	−0.159604
$299$	−4.83533e6	−3.12787
$300$	0	0
$301$	−1.99980e6	−1.27224
$302$	2.44125e6	1.54026
$303$	−173027.	−0.108270
$304$	1.77942e6	1.10432
$305$	0	0
$306$	−361842.	−0.220910
$307$	−1.68373e6	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$307$	−1.68373e6	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$308$	−136521.	−0.0820016
$309$	1.13986e6	0.679135
$310$	0	0
$311$	−2.72412e6	−1.59707	−0.798536	−	0.601947i	$-0.794392\pi$
$311$	−2.72412e6	−1.59707	−0.798536	+	0.601947i	$0.794392\pi$
$312$	−2.06207e6	−1.19927
$313$	1.69158e6	0.975959	0.487979	−	0.872855i	$-0.337734\pi$
$313$	1.69158e6	0.975959	0.487979	+	0.872855i	$0.337734\pi$
$314$	1.36138e6	0.779214
$315$	0	0
$316$	353518.	0.199156
$317$	−371370.	−0.207567	−0.103784	−	0.994600i	$-0.533095\pi$
$317$	−371370.	−0.207567	−0.103784	+	0.994600i	$0.533095\pi$
$318$	1.02293e6	0.567254
$319$	9129.20	0.00502292
$320$	0	0
$321$	−699960.	−0.379149
$322$	3.18566e6	1.71222
$323$	−2.16262e6	−1.15339
$324$	−47632.2	−0.0252080
$325$	0	0
$326$	−322714.	−0.168180
$327$	−619854.	−0.320568
$328$	2.94140e6	1.50963
$329$	−1.04108e6	−0.530265
$330$	0	0
$331$	1.54525e6	0.775229	0.387614	−	0.921822i	$-0.373299\pi$
$331$	1.54525e6	0.775229	0.387614	+	0.921822i	$0.373299\pi$
$332$	696263.	0.346680
$333$	−1.30111e6	−0.642989
$334$	−1.88767e6	−0.925893
$335$	0	0
$336$	1.03361e6	0.499469
$337$	−1.56563e6	−0.750956	−0.375478	−	0.926831i	$-0.622521\pi$
$337$	−1.56563e6	−0.750956	−0.375478	+	0.926831i	$0.622521\pi$
$338$	−5.00055e6	−2.38082
$339$	394154.	0.186280
$340$	0	0
$341$	435052.	0.202607
$342$	970138.	0.448506
$343$	−1.47039e6	−0.674833
$344$	−2.51277e6	−1.14487
$345$	0	0
$346$	2.53350e6	1.13771
$347$	2.21769e6	0.988727	0.494364	−	0.869255i	$-0.335401\pi$
$347$	2.21769e6	0.988727	0.494364	+	0.869255i	$0.335401\pi$
$348$	4929.70	0.00218209
$349$	−3.10101e6	−1.36283	−0.681413	−	0.731899i	$-0.738634\pi$
$349$	−3.10101e6	−1.36283	−0.681413	+	0.731899i	$0.738634\pi$
$350$	0	0
$351$	−855339.	−0.370570
$352$	−311359.	−0.133938
$353$	−2.34837e6	−1.00307	−0.501534	−	0.865138i	$-0.667231\pi$
$353$	−2.34837e6	−1.00307	−0.501534	+	0.865138i	$0.667231\pi$
$354$	−582905.	−0.247224
$355$	0	0
$356$	−9746.77	−0.00407602
$357$	−1.25620e6	−0.521661
$358$	−3.51293e6	−1.44865
$359$	1.76347e6	0.722158	0.361079	−	0.932535i	$-0.382408\pi$
$359$	1.76347e6	0.722158	0.361079	+	0.932535i	$0.382408\pi$
$360$	0	0
$361$	3.32213e6	1.34168
$362$	−1.24670e6	−0.500022
$363$	−131769.	−0.0524864
$364$	−1.32381e6	−0.523686
$365$	0	0
$366$	39497.0	0.0154121
$367$	−2.99827e6	−1.16200	−0.580999	−	0.813904i	$-0.697338\pi$
$367$	−2.99827e6	−1.16200	−0.580999	+	0.813904i	$0.697338\pi$
$368$	3.04541e6	1.17227
$369$	1.22008e6	0.466470
$370$	0	0
$371$	3.55128e6	1.33952
$372$	234925.	0.0880179
$373$	−2.01705e6	−0.750663	−0.375331	−	0.926891i	$-0.622471\pi$
$373$	−2.01705e6	−0.750663	−0.375331	+	0.926891i	$0.622471\pi$
$374$	−540529.	−0.199821
$375$	0	0
$376$	−1.30813e6	−0.477177
$377$	88523.4	0.0320778
$378$	563523.	0.202853
$379$	3.78909e6	1.35499	0.677497	−	0.735526i	$-0.263065\pi$
$379$	3.78909e6	1.35499	0.677497	+	0.735526i	$0.263065\pi$
$380$	0	0
$381$	1.88180e6	0.664144
$382$	2.94880e6	1.03392
$383$	−2.72827e6	−0.950366	−0.475183	−	0.879887i	$-0.657618\pi$
$383$	−2.72827e6	−0.950366	−0.475183	+	0.879887i	$0.657618\pi$
$384$	890449.	0.308163
$385$	0	0
$386$	4.80192e6	1.64039
$387$	−1.04229e6	−0.353762
$388$	−587608.	−0.198157
$389$	432266.	0.144836	0.0724181	−	0.997374i	$-0.476928\pi$
$389$	432266.	0.144836	0.0724181	+	0.997374i	$0.476928\pi$
$390$	0	0
$391$	−3.70125e6	−1.22435
$392$	1.43445e6	0.471488
$393$	771095.	0.251841
$394$	871667.	0.282885
$395$	0	0
$396$	−71154.3	−0.0228015
$397$	−3.90462e6	−1.24338	−0.621688	−	0.783265i	$-0.713553\pi$
$397$	−3.90462e6	−1.24338	−0.621688	+	0.783265i	$0.713553\pi$
$398$	242273.	0.0766651
$399$	3.36801e6	1.05911
$400$	0	0
$401$	4.60863e6	1.43124	0.715618	−	0.698492i	$-0.246145\pi$
$401$	4.60863e6	1.43124	0.715618	+	0.698492i	$0.246145\pi$
$402$	−635297.	−0.196070
$403$	4.21858e6	1.29391
$404$	−139573.	−0.0425450
$405$	0	0
$406$	−58321.9	−0.0175597
$407$	−1.94363e6	−0.581605
$408$	−1.57843e6	−0.469435
$409$	−3.86604e6	−1.14277	−0.571384	−	0.820683i	$-0.693593\pi$
$409$	−3.86604e6	−1.14277	−0.571384	+	0.820683i	$0.693593\pi$
$410$	0	0
$411$	−1.03959e6	−0.303568
$412$	919476.	0.266868
$413$	−2.02366e6	−0.583799
$414$	1.66036e6	0.476102
$415$	0	0
$416$	−3.01917e6	−0.855370
$417$	1.08559e6	0.305720
$418$	1.44922e6	0.405689
$419$	3.60488e6	1.00313	0.501563	−	0.865121i	$-0.332759\pi$
$419$	3.60488e6	1.00313	0.501563	+	0.865121i	$0.332759\pi$
$420$	0	0
$421$	2.69664e6	0.741510	0.370755	−	0.928731i	$-0.379099\pi$
$421$	2.69664e6	0.741510	0.370755	+	0.928731i	$0.379099\pi$
$422$	−3.29377e6	−0.900353
$423$	−542606.	−0.147446
$424$	4.46223e6	1.20542
$425$	0	0
$426$	−3.52510e6	−0.941126
$427$	137121.	0.0363944
$428$	−564626.	−0.148988
$429$	−1.27773e6	−0.335193
$430$	0	0
$431$	−337978.	−0.0876385	−0.0438193	−	0.999039i	$-0.513953\pi$
$431$	−337978.	−0.0876385	−0.0438193	+	0.999039i	$0.513953\pi$
$432$	538714.	0.138883
$433$	5.43565e6	1.39326	0.696629	−	0.717431i	$-0.254682\pi$
$433$	5.43565e6	1.39326	0.696629	+	0.717431i	$0.254682\pi$
$434$	−2.77932e6	−0.708296
$435$	0	0
$436$	−500008.	−0.125968
$437$	9.92346e6	2.48576
$438$	1.53220e6	0.381620
$439$	−170878.	−0.0423180	−0.0211590	−	0.999776i	$-0.506736\pi$
$439$	−170878.	−0.0423180	−0.0211590	+	0.999776i	$0.506736\pi$
$440$	0	0
$441$	595006.	0.145688
$442$	−5.24137e6	−1.27611
$443$	6.94673e6	1.68179	0.840893	−	0.541201i	$-0.182030\pi$
$443$	6.94673e6	1.68179	0.840893	+	0.541201i	$0.182030\pi$
$444$	−1.04955e6	−0.252665
$445$	0	0
$446$	219722.	0.0523043
$447$	−442717.	−0.104799
$448$	5.66418e6	1.33334
$449$	−1.91825e6	−0.449045	−0.224522	−	0.974469i	$-0.572082\pi$
$449$	−1.91825e6	−0.449045	−0.224522	+	0.974469i	$0.572082\pi$
$450$	0	0
$451$	1.82259e6	0.421938
$452$	317946.	0.0731995
$453$	4.41728e6	1.01137
$454$	−2.91067e6	−0.662755
$455$	0	0
$456$	4.23195e6	0.953077
$457$	−1.14248e6	−0.255892	−0.127946	−	0.991781i	$-0.540838\pi$
$457$	−1.14248e6	−0.255892	−0.127946	+	0.991781i	$0.540838\pi$
$458$	−4.54768e6	−1.01304
$459$	−654728.	−0.145054
$460$	0	0
$461$	3.63295e6	0.796172	0.398086	−	0.917348i	$-0.369675\pi$
$461$	3.63295e6	0.796172	0.398086	+	0.917348i	$0.369675\pi$
$462$	841806.	0.183488
$463$	−6.29651e6	−1.36505	−0.682523	−	0.730864i	$-0.739117\pi$
$463$	−6.29651e6	−1.36505	−0.682523	+	0.730864i	$0.739117\pi$
$464$	−55754.3	−0.0120222
$465$	0	0
$466$	−3.49490e6	−0.745537
$467$	46216.3	0.00980624	0.00490312	−	0.999988i	$-0.498439\pi$
$467$	46216.3	0.00980624	0.00490312	+	0.999988i	$0.498439\pi$
$468$	−689964.	−0.145617
$469$	−2.20555e6	−0.463004
$470$	0	0
$471$	2.46333e6	0.511647
$472$	−2.54276e6	−0.525351
$473$	−1.55700e6	−0.319990
$474$	−2.17983e6	−0.445633
$475$	0	0
$476$	−1.01332e6	−0.204989
$477$	1.85092e6	0.372470
$478$	−2.84517e6	−0.569558
$479$	1.87454e6	0.373299	0.186649	−	0.982427i	$-0.440237\pi$
$479$	1.87454e6	0.373299	0.186649	+	0.982427i	$0.440237\pi$
$480$	0	0
$481$	−1.88469e7	−3.71430
$482$	2.41397e6	0.473276
$483$	5.76423e6	1.12428
$484$	−106292.	−0.0206247
$485$	0	0
$486$	293706.	0.0564056
$487$	−2.93849e6	−0.561439	−0.280719	−	0.959790i	$-0.590573\pi$
$487$	−2.93849e6	−0.561439	−0.280719	+	0.959790i	$0.590573\pi$
$488$	172294.	0.0327508
$489$	−583929.	−0.110430
$490$	0	0
$491$	7.56455e6	1.41605	0.708026	−	0.706186i	$-0.249586\pi$
$491$	7.56455e6	1.41605	0.708026	+	0.706186i	$0.249586\pi$
$492$	984187.	0.183301
$493$	67761.1	0.0125563
$494$	1.40527e7	2.59085
$495$	0	0
$496$	−2.65697e6	−0.484933
$497$	−1.22380e7	−2.22239
$498$	−4.29325e6	−0.775734
$499$	−3.96894e6	−0.713548	−0.356774	−	0.934191i	$-0.616123\pi$
$499$	−3.96894e6	−0.713548	−0.356774	+	0.934191i	$0.616123\pi$
$500$	0	0
$501$	−3.41561e6	−0.607959
$502$	−128106.	−0.0226887
$503$	7.67415e6	1.35242	0.676208	−	0.736711i	$-0.263622\pi$
$503$	7.67415e6	1.35242	0.676208	+	0.736711i	$0.263622\pi$
$504$	2.45821e6	0.431064
$505$	0	0
$506$	2.48028e6	0.430651
$507$	−9.04815e6	−1.56329
$508$	1.51797e6	0.260978
$509$	−2.34465e6	−0.401129	−0.200564	−	0.979681i	$-0.564278\pi$
$509$	−2.34465e6	−0.401129	−0.200564	+	0.979681i	$0.564278\pi$
$510$	0	0
$511$	5.31932e6	0.901164
$512$	6.51930e6	1.09907
$513$	1.75540e6	0.294498
$514$	−1.20981e6	−0.201980
$515$	0	0
$516$	−840769.	−0.139012
$517$	−810559.	−0.133370
$518$	1.24169e7	2.03324
$519$	4.58418e6	0.747040
$520$	0	0
$521$	−4.60694e6	−0.743563	−0.371782	−	0.928320i	$-0.621253\pi$
$521$	−4.60694e6	−0.743563	−0.371782	+	0.928320i	$0.621253\pi$
$522$	−30397.2	−0.00488266
$523$	−3.79276e6	−0.606319	−0.303159	−	0.952940i	$-0.598041\pi$
$523$	−3.79276e6	−0.606319	−0.303159	+	0.952940i	$0.598041\pi$
$524$	622008.	0.0989619
$525$	0	0
$526$	−230184.	−0.0362753
$527$	3.22915e6	0.506480
$528$	804746.	0.125624
$529$	1.05473e7	1.63871
$530$	0	0
$531$	−1.05473e6	−0.162332
$532$	2.71683e6	0.416181
$533$	1.76732e7	2.69462
$534$	60099.9	0.00912053
$535$	0	0
$536$	−2.77130e6	−0.416650
$537$	−6.35640e6	−0.951208
$538$	−868619.	−0.129382
$539$	888835.	0.131780
$540$	0	0
$541$	−8.95976e6	−1.31614	−0.658072	−	0.752955i	$-0.728628\pi$
$541$	−8.95976e6	−1.31614	−0.658072	+	0.752955i	$0.728628\pi$
$542$	8.85837e6	1.29526
$543$	−2.25581e6	−0.328324
$544$	−2.31105e6	−0.334821
$545$	0	0
$546$	8.16276e6	1.17181
$547$	2.79065e6	0.398783	0.199391	−	0.979920i	$-0.436104\pi$
$547$	2.79065e6	0.398783	0.199391	+	0.979920i	$0.436104\pi$
$548$	−838588.	−0.119288
$549$	71467.1	0.0101199
$550$	0	0
$551$	−181675.	−0.0254927
$552$	7.24282e6	1.01172
$553$	−7.56770e6	−1.05233
$554$	−2.71124e6	−0.375313
$555$	0	0
$556$	875694.	0.120134
$557$	1.22903e7	1.67851	0.839256	−	0.543736i	$-0.182991\pi$
$557$	1.22903e7	1.67851	0.839256	+	0.543736i	$0.182991\pi$
$558$	−1.44858e6	−0.196950
$559$	−1.50978e7	−2.04355
$560$	0	0
$561$	−978050.	−0.131206
$562$	−425120.	−0.0567768
$563$	1.13900e7	1.51444	0.757220	−	0.653160i	$-0.226557\pi$
$563$	1.13900e7	1.51444	0.757220	+	0.653160i	$0.226557\pi$
$564$	−437696.	−0.0579395
$565$	0	0
$566$	7.51408e6	0.985905
$567$	1.01966e6	0.133197
$568$	−1.53772e7	−1.99990
$569$	4.71367e6	0.610350	0.305175	−	0.952296i	$-0.401285\pi$
$569$	4.71367e6	0.610350	0.305175	+	0.952296i	$0.401285\pi$
$570$	0	0
$571$	7.49421e6	0.961913	0.480957	−	0.876744i	$-0.340289\pi$
$571$	7.49421e6	0.961913	0.480957	+	0.876744i	$0.340289\pi$
$572$	−1.03069e6	−0.131715
$573$	5.33565e6	0.678893
$574$	−1.16436e7	−1.47506
$575$	0	0
$576$	2.95215e6	0.370751
$577$	−5.77654e6	−0.722318	−0.361159	−	0.932504i	$-0.617619\pi$
$577$	−5.77654e6	−0.722318	−0.361159	+	0.932504i	$0.617619\pi$
$578$	3.05023e6	0.379763
$579$	8.68873e6	1.07711
$580$	0	0
$581$	−1.49048e7	−1.83183
$582$	3.62327e6	0.443397
$583$	2.76495e6	0.336912
$584$	6.68379e6	0.810943
$585$	0	0
$586$	5.23153e6	0.629339
$587$	−1.19273e6	−0.142872	−0.0714358	−	0.997445i	$-0.522758\pi$
$587$	−1.19273e6	−0.142872	−0.0714358	+	0.997445i	$0.522758\pi$
$588$	479965.	0.0572487
$589$	−8.65771e6	−1.02829
$590$	0	0
$591$	1.57722e6	0.185748
$592$	1.18702e7	1.39205
$593$	−1.14801e7	−1.34064	−0.670318	−	0.742074i	$-0.733842\pi$
$593$	−1.14801e7	−1.34064	−0.670318	+	0.742074i	$0.733842\pi$
$594$	438746.	0.0510208
$595$	0	0
$596$	−357120.	−0.0411812
$597$	438376.	0.0503398
$598$	2.40506e7	2.75026
$599$	1.07989e7	1.22974	0.614869	−	0.788630i	$-0.289209\pi$
$599$	1.07989e7	1.22974	0.614869	+	0.788630i	$0.289209\pi$
$600$	0	0
$601$	6.43742e6	0.726986	0.363493	−	0.931597i	$-0.381584\pi$
$601$	6.43742e6	0.726986	0.363493	+	0.931597i	$0.381584\pi$
$602$	9.94690e6	1.11866
$603$	−1.14953e6	−0.128744
$604$	3.56322e6	0.397421
$605$	0	0
$606$	860626.	0.0951991
$607$	2.78009e6	0.306258	0.153129	−	0.988206i	$-0.451065\pi$
$607$	2.78009e6	0.306258	0.153129	+	0.988206i	$0.451065\pi$
$608$	6.19619e6	0.679776
$609$	−105529.	−0.0115300
$610$	0	0
$611$	−7.85978e6	−0.851740
$612$	−528140.	−0.0569994
$613$	−1.25907e7	−1.35331	−0.676657	−	0.736299i	$-0.736572\pi$
$613$	−1.25907e7	−1.35331	−0.676657	+	0.736299i	$0.736572\pi$
$614$	8.37477e6	0.896504
$615$	0	0
$616$	3.67213e6	0.389912
$617$	1.20724e6	0.127668	0.0638339	−	0.997961i	$-0.479667\pi$
$617$	1.20724e6	0.127668	0.0638339	+	0.997961i	$0.479667\pi$
$618$	−5.66960e6	−0.597147
$619$	1.03935e7	1.09028	0.545138	−	0.838346i	$-0.316477\pi$
$619$	1.03935e7	1.09028	0.545138	+	0.838346i	$0.316477\pi$
$620$	0	0
$621$	3.00430e6	0.312618
$622$	1.35496e7	1.40427
$623$	208648.	0.0215374
$624$	7.80341e6	0.802274
$625$	0	0
$626$	−8.41381e6	−0.858138
$627$	2.62226e6	0.266383
$628$	1.98706e6	0.201054
$629$	−1.44265e7	−1.45390
$630$	0	0
$631$	389320.	0.0389254	0.0194627	−	0.999811i	$-0.493804\pi$
$631$	389320.	0.0389254	0.0194627	+	0.999811i	$0.493804\pi$
$632$	−9.50890e6	−0.946973
$633$	−5.95985e6	−0.591189
$634$	1.84717e6	0.182509
$635$	0	0
$636$	1.49305e6	0.146363
$637$	8.61880e6	0.841585
$638$	−45408.1	−0.00441654
$639$	−6.37843e6	−0.617961
$640$	0	0
$641$	9.85176e6	0.947041	0.473520	−	0.880783i	$-0.342983\pi$
$641$	9.85176e6	0.947041	0.473520	+	0.880783i	$0.342983\pi$
$642$	3.48156e6	0.333377
$643$	9.41361e6	0.897901	0.448951	−	0.893557i	$-0.351798\pi$
$643$	9.41361e6	0.897901	0.448951	+	0.893557i	$0.351798\pi$
$644$	4.64975e6	0.441789
$645$	0	0
$646$	1.07568e7	1.01415
$647$	−8.37846e6	−0.786871	−0.393435	−	0.919352i	$-0.628713\pi$
$647$	−8.37846e6	−0.786871	−0.393435	+	0.919352i	$0.628713\pi$
$648$	1.28121e6	0.119862
$649$	−1.57558e6	−0.146835
$650$	0	0
$651$	−5.02899e6	−0.465081
$652$	−471030.	−0.0433940
$653$	−1.87196e7	−1.71796	−0.858979	−	0.512011i	$-0.828901\pi$
$653$	−1.87196e7	−1.71796	−0.858979	+	0.512011i	$0.828901\pi$
$654$	3.08312e6	0.281868
$655$	0	0
$656$	−1.11310e7	−1.00989
$657$	2.77241e6	0.250579
$658$	5.17826e6	0.466250
$659$	−7.53086e6	−0.675509	−0.337755	−	0.941234i	$-0.609667\pi$
$659$	−7.53086e6	−0.675509	−0.337755	+	0.941234i	$0.609667\pi$
$660$	0	0
$661$	−1.12673e7	−1.00304	−0.501518	−	0.865147i	$-0.667225\pi$
$661$	−1.12673e7	−1.00304	−0.501518	+	0.865147i	$0.667225\pi$
$662$	−7.68600e6	−0.681641
$663$	−9.48389e6	−0.837920
$664$	−1.87281e7	−1.64844
$665$	0	0
$666$	6.47164e6	0.565365
$667$	−310930.	−0.0270613
$668$	−2.75522e6	−0.238900
$669$	397572.	0.0343440
$670$	0	0
$671$	106759.	0.00915377
$672$	3.59917e6	0.307453
$673$	747965.	0.0636566	0.0318283	−	0.999493i	$-0.489867\pi$
$673$	747965.	0.0636566	0.0318283	+	0.999493i	$0.489867\pi$
$674$	7.78735e6	0.660298
$675$	0	0
$676$	−7.29874e6	−0.614301
$677$	−1.52419e7	−1.27811	−0.639056	−	0.769160i	$-0.720675\pi$
$677$	−1.52419e7	−1.27811	−0.639056	+	0.769160i	$0.720675\pi$
$678$	−1.96050e6	−0.163792
$679$	1.25788e7	1.04705
$680$	0	0
$681$	−5.26665e6	−0.435178
$682$	−2.16392e6	−0.178148
$683$	891262.	0.0731061	0.0365531	−	0.999332i	$-0.488362\pi$
$683$	891262.	0.0731061	0.0365531	+	0.999332i	$0.488362\pi$
$684$	1.41600e6	0.115724
$685$	0	0
$686$	7.31362e6	0.593366
$687$	−8.22870e6	−0.665180
$688$	9.50899e6	0.765885
$689$	2.68110e7	2.15162
$690$	0	0
$691$	−2.23692e6	−0.178220	−0.0891100	−	0.996022i	$-0.528402\pi$
$691$	−2.23692e6	−0.178220	−0.0891100	+	0.996022i	$0.528402\pi$
$692$	3.69786e6	0.293552
$693$	1.52319e6	0.120482
$694$	−1.10306e7	−0.869365
$695$	0	0
$696$	−132599.	−0.0103757
$697$	1.35281e7	1.05476
$698$	1.54243e7	1.19830
$699$	−6.32377e6	−0.489534
$700$	0	0
$701$	−9.23251e6	−0.709618	−0.354809	−	0.934939i	$-0.615454\pi$
$701$	−9.23251e6	−0.709618	−0.354809	+	0.934939i	$0.615454\pi$
$702$	4.25441e6	0.325834
$703$	3.86791e7	2.95181
$704$	4.41000e6	0.335357
$705$	0	0
$706$	1.16807e7	0.881975
$707$	2.98782e6	0.224805
$708$	−850801.	−0.0637889
$709$	−1.91122e7	−1.42789	−0.713945	−	0.700201i	$-0.753094\pi$
$709$	−1.91122e7	−1.42789	−0.713945	+	0.700201i	$0.753094\pi$
$710$	0	0
$711$	−3.94426e6	−0.292612
$712$	262168.	0.0193812
$713$	−1.48174e7	−1.09156
$714$	6.24827e6	0.458685
$715$	0	0
$716$	−5.12743e6	−0.373781
$717$	−5.14813e6	−0.373983
$718$	−8.77140e6	−0.634977
$719$	3.98132e6	0.287213	0.143607	−	0.989635i	$-0.454130\pi$
$719$	3.98132e6	0.287213	0.143607	+	0.989635i	$0.454130\pi$
$720$	0	0
$721$	−1.96831e7	−1.41012
$722$	−1.65241e7	−1.17971
$723$	4.36792e6	0.310762
$724$	−1.81966e6	−0.129016
$725$	0	0
$726$	655411.	0.0461501
$727$	−2.31098e6	−0.162166	−0.0810832	−	0.996707i	$-0.525838\pi$
$727$	−2.31098e6	−0.162166	−0.0810832	+	0.996707i	$0.525838\pi$
$728$	3.56077e7	2.49009
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.15568e7	−0.799914
$732$	57649.3	0.00397664
$733$	1.76034e7	1.21015	0.605073	−	0.796170i	$-0.293144\pi$
$733$	1.76034e7	1.21015	0.605073	+	0.796170i	$0.293144\pi$
$734$	1.49132e7	1.02172
$735$	0	0
$736$	1.06045e7	0.721601
$737$	−1.71719e6	−0.116453
$738$	−6.06862e6	−0.410156
$739$	−8.91956e6	−0.600803	−0.300402	−	0.953813i	$-0.597121\pi$
$739$	−8.91956e6	−0.600803	−0.300402	+	0.953813i	$0.597121\pi$
$740$	0	0
$741$	2.54273e7	1.70120
$742$	−1.76639e7	−1.17781
$743$	3.22802e6	0.214519	0.107259	−	0.994231i	$-0.465793\pi$
$743$	3.22802e6	0.214519	0.107259	+	0.994231i	$0.465793\pi$
$744$	−6.31899e6	−0.418519
$745$	0	0
$746$	1.00327e7	0.660040
$747$	−7.76833e6	−0.509362
$748$	−788950.	−0.0515579
$749$	1.20869e7	0.787244
$750$	0	0
$751$	−2.04930e6	−0.132588	−0.0662942	−	0.997800i	$-0.521118\pi$
$751$	−2.04930e6	−0.132588	−0.0662942	+	0.997800i	$0.521118\pi$
$752$	4.95029e6	0.319217
$753$	−231799.	−0.0148979
$754$	−440310.	−0.0282053
$755$	0	0
$756$	822511.	0.0523404
$757$	2.62464e6	0.166468	0.0832340	−	0.996530i	$-0.473475\pi$
$757$	2.62464e6	0.166468	0.0832340	+	0.996530i	$0.473475\pi$
$758$	−1.88467e7	−1.19141
$759$	4.48790e6	0.282773
$760$	0	0
$761$	1.90240e7	1.19080	0.595401	−	0.803428i	$-0.296993\pi$
$761$	1.90240e7	1.19080	0.595401	+	0.803428i	$0.296993\pi$
$762$	−9.35999e6	−0.583966
$763$	1.07036e7	0.665608
$764$	4.30404e6	0.266773
$765$	0	0
$766$	1.35703e7	0.835635
$767$	−1.52780e7	−0.937729
$768$	6.06749e6	0.371198
$769$	−1.84565e7	−1.12547	−0.562735	−	0.826637i	$-0.690251\pi$
$769$	−1.84565e7	−1.12547	−0.562735	+	0.826637i	$0.690251\pi$
$770$	0	0
$771$	−2.18906e6	−0.132624
$772$	7.00881e6	0.423254
$773$	−6.98835e6	−0.420655	−0.210327	−	0.977631i	$-0.567453\pi$
$773$	−6.98835e6	−0.420655	−0.210327	+	0.977631i	$0.567453\pi$
$774$	5.18429e6	0.311055
$775$	0	0
$776$	1.58055e7	0.942221
$777$	2.24675e7	1.33506
$778$	−2.15007e6	−0.127351
$779$	−3.62704e7	−2.14145
$780$	0	0
$781$	−9.52826e6	−0.558967
$782$	1.84098e7	1.07655
$783$	−55001.5	−0.00320605
$784$	−5.42834e6	−0.315411
$785$	0	0
$786$	−3.83538e6	−0.221438
$787$	−1.02434e7	−0.589533	−0.294766	−	0.955569i	$-0.595242\pi$
$787$	−1.02434e7	−0.589533	−0.294766	+	0.955569i	$0.595242\pi$
$788$	1.27227e6	0.0729903
$789$	−416502.	−0.0238190
$790$	0	0
$791$	−6.80623e6	−0.386781
$792$	1.91390e6	0.108419
$793$	1.03522e6	0.0584587
$794$	1.94213e7	1.09327
$795$	0	0
$796$	353619.	0.0197812
$797$	1.36372e7	0.760466	0.380233	−	0.924891i	$-0.375844\pi$
$797$	1.36372e7	0.760466	0.380233	+	0.924891i	$0.375844\pi$
$798$	−1.67523e7	−0.931252
$799$	−6.01634e6	−0.333400
$800$	0	0
$801$	108746.	0.00598872
$802$	−2.29231e7	−1.25845
$803$	4.14150e6	0.226657
$804$	−927271.	−0.0505902
$805$	0	0
$806$	−2.09830e7	−1.13770
$807$	−1.57171e6	−0.0849547
$808$	3.75423e6	0.202299
$809$	−2.94019e7	−1.57944	−0.789722	−	0.613465i	$-0.789775\pi$
$809$	−2.94019e7	−1.57944	−0.789722	+	0.613465i	$0.789775\pi$
$810$	0	0
$811$	−1.42304e7	−0.759737	−0.379869	−	0.925040i	$-0.624031\pi$
$811$	−1.42304e7	−0.759737	−0.379869	+	0.925040i	$0.624031\pi$
$812$	−85125.8	−0.00453076
$813$	1.60286e7	0.850490
$814$	9.66752e6	0.511392
$815$	0	0
$816$	5.97319e6	0.314037
$817$	3.09850e7	1.62404
$818$	1.92294e7	1.00481
$819$	1.47699e7	0.769430
$820$	0	0
$821$	2.28257e7	1.18186	0.590931	−	0.806722i	$-0.298761\pi$
$821$	2.28257e7	1.18186	0.590931	+	0.806722i	$0.298761\pi$
$822$	5.17084e6	0.266920
$823$	−1.12391e6	−0.0578405	−0.0289202	−	0.999582i	$-0.509207\pi$
$823$	−1.12391e6	−0.0578405	−0.0289202	+	0.999582i	$0.509207\pi$
$824$	−2.47320e7	−1.26894
$825$	0	0
$826$	1.00656e7	0.513321
$827$	−2.69107e7	−1.36824	−0.684119	−	0.729370i	$-0.739813\pi$
$827$	−2.69107e7	−1.36824	−0.684119	+	0.729370i	$0.739813\pi$
$828$	2.42343e6	0.122844
$829$	−6.83042e6	−0.345192	−0.172596	−	0.984993i	$-0.555216\pi$
$829$	−6.83042e6	−0.345192	−0.172596	+	0.984993i	$0.555216\pi$
$830$	0	0
$831$	−4.90580e6	−0.246438
$832$	4.27626e7	2.14169
$833$	6.59734e6	0.329425
$834$	−5.39964e6	−0.268813
$835$	0	0
$836$	2.11526e6	0.104676
$837$	−2.62110e6	−0.129321
$838$	−1.79305e7	−0.882026
$839$	−1.37641e7	−0.675061	−0.337531	−	0.941315i	$-0.609592\pi$
$839$	−1.37641e7	−0.675061	−0.337531	+	0.941315i	$0.609592\pi$
$840$	0	0
$841$	−2.05055e7	−0.999722
$842$	−1.34129e7	−0.651993
$843$	−769225.	−0.0372808
$844$	−4.80755e6	−0.232310
$845$	0	0
$846$	2.69889e6	0.129646
$847$	2.27538e6	0.108980
$848$	−1.68862e7	−0.806387
$849$	1.35962e7	0.647364
$850$	0	0
$851$	6.61979e7	3.13343
$852$	−5.14519e6	−0.242830
$853$	−2.38141e7	−1.12063	−0.560313	−	0.828281i	$-0.689319\pi$
$853$	−2.38141e7	−1.12063	−0.560313	+	0.828281i	$0.689319\pi$
$854$	−682032.	−0.0320008
$855$	0	0
$856$	1.51873e7	0.708428
$857$	−1.96463e7	−0.913751	−0.456875	−	0.889531i	$-0.651031\pi$
$857$	−1.96463e7	−0.913751	−0.456875	+	0.889531i	$0.651031\pi$
$858$	6.35535e6	0.294728
$859$	−4.10315e7	−1.89729	−0.948646	−	0.316339i	$-0.897546\pi$
$859$	−4.10315e7	−1.89729	−0.948646	+	0.316339i	$0.897546\pi$
$860$	0	0
$861$	−2.10683e7	−0.968550
$862$	1.68108e6	0.0770585
$863$	−1.10401e7	−0.504599	−0.252300	−	0.967649i	$-0.581187\pi$
$863$	−1.10401e7	−0.504599	−0.252300	+	0.967649i	$0.581187\pi$
$864$	1.87588e6	0.0854909
$865$	0	0
$866$	−2.70366e7	−1.22506
$867$	5.51918e6	0.249360
$868$	−4.05667e6	−0.182755
$869$	−5.89204e6	−0.264677
$870$	0	0
$871$	−1.66512e7	−0.743702
$872$	1.34492e7	0.598970
$873$	6.55605e6	0.291143
$874$	−4.93587e7	−2.18567
$875$	0	0
$876$	2.23638e6	0.0984658
$877$	−2.06101e7	−0.904861	−0.452430	−	0.891800i	$-0.649443\pi$
$877$	−2.06101e7	−0.904861	−0.452430	+	0.891800i	$0.649443\pi$
$878$	849937.	0.0372092
$879$	9.46608e6	0.413236
$880$	0	0
$881$	−7.17874e6	−0.311608	−0.155804	−	0.987788i	$-0.549797\pi$
$881$	−7.17874e6	−0.311608	−0.155804	+	0.987788i	$0.549797\pi$
$882$	−2.95952e6	−0.128100
$883$	1.55190e6	0.0669824	0.0334912	−	0.999439i	$-0.489337\pi$
$883$	1.55190e6	0.0669824	0.0334912	+	0.999439i	$0.489337\pi$
$884$	−7.65023e6	−0.329264
$885$	0	0
$886$	−3.45526e7	−1.47876
$887$	−3.39646e7	−1.44950	−0.724748	−	0.689014i	$-0.758044\pi$
$887$	−3.39646e7	−1.44950	−0.724748	+	0.689014i	$0.758044\pi$
$888$	2.82307e7	1.20140
$889$	−3.24949e7	−1.37899
$890$	0	0
$891$	793881.	0.0335013
$892$	320704.	0.0134956
$893$	1.61305e7	0.676891
$894$	2.20205e6	0.0921474
$895$	0	0
$896$	−1.53762e7	−0.639853
$897$	4.35180e7	1.80587
$898$	9.54127e6	0.394835
$899$	271271.	0.0111945
$900$	0	0
$901$	2.05227e7	0.842216
$902$	−9.06547e6	−0.371000
$903$	1.79982e7	0.734531
$904$	−8.55211e6	−0.348058
$905$	0	0
$906$	−2.19713e7	−0.889272
$907$	−2.58319e7	−1.04265	−0.521324	−	0.853359i	$-0.674562\pi$
$907$	−2.58319e7	−1.04265	−0.521324	+	0.853359i	$0.674562\pi$
$908$	−4.24837e6	−0.171005
$909$	1.55724e6	0.0625096
$910$	0	0
$911$	549431.	0.0219340	0.0109670	−	0.999940i	$-0.496509\pi$
$911$	549431.	0.0219340	0.0109670	+	0.999940i	$0.496509\pi$
$912$	−1.60148e7	−0.637579
$913$	−1.16045e7	−0.460735
$914$	5.68262e6	0.225000
$915$	0	0
$916$	−6.63773e6	−0.261385
$917$	−1.33152e7	−0.522908
$918$	3.25658e6	0.127542
$919$	3.84177e7	1.50052	0.750261	−	0.661142i	$-0.229928\pi$
$919$	3.84177e7	1.50052	0.750261	+	0.661142i	$0.229928\pi$
$920$	0	0
$921$	1.51536e7	0.588662
$922$	−1.80701e7	−0.700055
$923$	−9.23930e7	−3.56973
$924$	1.22869e6	0.0473437
$925$	0	0
$926$	3.13185e7	1.20025
$927$	−1.02588e7	−0.392099
$928$	−194144.	−0.00740038
$929$	1.50560e7	0.572360	0.286180	−	0.958176i	$-0.407614\pi$
$929$	1.50560e7	0.572360	0.286180	+	0.958176i	$0.407614\pi$
$930$	0	0
$931$	−1.76882e7	−0.668821
$932$	−5.10110e6	−0.192364
$933$	2.45170e7	0.922070
$934$	−229877.	−0.00862240
$935$	0	0
$936$	1.85586e7	0.692398
$937$	1.83320e7	0.682119	0.341059	−	0.940042i	$-0.389214\pi$
$937$	1.83320e7	0.682119	0.341059	+	0.940042i	$0.389214\pi$
$938$	1.09703e7	0.407109
$939$	−1.52242e7	−0.563470
$940$	0	0
$941$	−3.82211e7	−1.40711	−0.703557	−	0.710639i	$-0.748406\pi$
$941$	−3.82211e7	−1.40711	−0.703557	+	0.710639i	$0.748406\pi$
$942$	−1.22525e7	−0.449879
$943$	−6.20754e7	−2.27321
$944$	9.62245e6	0.351444
$945$	0	0
$946$	7.74444e6	0.281360
$947$	3.35057e7	1.21407	0.607035	−	0.794675i	$-0.292359\pi$
$947$	3.35057e7	1.21407	0.607035	+	0.794675i	$0.292359\pi$
$948$	−3.18166e6	−0.114983
$949$	4.01590e7	1.44750
$950$	0	0
$951$	3.34233e6	0.119839
$952$	2.72563e7	0.974707
$953$	−3.29329e7	−1.17462	−0.587310	−	0.809362i	$-0.699813\pi$
$953$	−3.29329e7	−1.17462	−0.587310	+	0.809362i	$0.699813\pi$
$954$	−9.20636e6	−0.327504
$955$	0	0
$956$	−4.15277e6	−0.146958
$957$	−82162.8	−0.00289998
$958$	−9.32386e6	−0.328233
$959$	1.79515e7	0.630311
$960$	0	0
$961$	−1.57018e7	−0.548454
$962$	9.37433e7	3.26590
$963$	6.29964e6	0.218902
$964$	3.52340e6	0.122115
$965$	0	0
$966$	−2.86709e7	−0.988551
$967$	2.24586e7	0.772353	0.386177	−	0.922425i	$-0.373796\pi$
$967$	2.24586e7	0.772353	0.386177	+	0.922425i	$0.373796\pi$
$968$	2.85904e6	0.0980691
$969$	1.94636e7	0.665908
$970$	0	0
$971$	3.76102e7	1.28014	0.640069	−	0.768317i	$-0.278906\pi$
$971$	3.76102e7	1.28014	0.640069	+	0.768317i	$0.278906\pi$
$972$	428690.	0.0145538
$973$	−1.87458e7	−0.634780
$974$	1.46159e7	0.493660
$975$	0	0
$976$	−652007.	−0.0219092
$977$	−1.90408e7	−0.638187	−0.319094	−	0.947723i	$-0.603378\pi$
$977$	−1.90408e7	−0.638187	−0.319094	+	0.947723i	$0.603378\pi$
$978$	2.90443e6	0.0970987
$979$	162448.	0.00541700
$980$	0	0
$981$	5.57868e6	0.185080
$982$	−3.76256e7	−1.24510
$983$	−3.59407e7	−1.18632	−0.593162	−	0.805083i	$-0.702120\pi$
$983$	−3.59407e7	−1.18632	−0.593162	+	0.805083i	$0.702120\pi$
$984$	−2.64726e7	−0.871584
$985$	0	0
$986$	−337040.	−0.0110405
$987$	9.36969e6	0.306149
$988$	2.05111e7	0.668493
$989$	5.30297e7	1.72396
$990$	0	0
$991$	2.52119e7	0.815495	0.407748	−	0.913095i	$-0.366314\pi$
$991$	2.52119e7	0.815495	0.407748	+	0.913095i	$0.366314\pi$
$992$	−9.25192e6	−0.298506
$993$	−1.39073e7	−0.447578
$994$	6.08713e7	1.95410
$995$	0	0
$996$	−6.26637e6	−0.200156
$997$	4.86935e6	0.155143	0.0775716	−	0.996987i	$-0.475283\pi$
$997$	4.86935e6	0.155143	0.0775716	+	0.996987i	$0.475283\pi$
$998$	1.97413e7	0.627406
$999$	1.17100e7	0.371230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.v.1.6		13
5.2	odd	4		165.6.c.b.34.9	✓	26
5.3	odd	4		165.6.c.b.34.18	yes	26
5.4	even	2		825.6.a.y.1.8		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.9	✓	26	5.2	odd	4
165.6.c.b.34.18	yes	26	5.3	odd	4
825.6.a.v.1.6		13	1.1	even	1	trivial
825.6.a.y.1.8		13	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-4.97394 q^{2} -9.00000 q^{3} -7.25990 q^{4} +44.7655 q^{6} +155.412 q^{7} +195.276 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-4.97394 q^{2} -9.00000 q^{3} -7.25990 q^{4} +44.7655 q^{6} +155.412 q^{7} +195.276 q^{8} +81.0000 q^{9} +121.000 q^{11} +65.3391 q^{12} +1173.30 q^{13} -773.008 q^{14} -738.977 q^{16} +898.118 q^{17} -402.889 q^{18} -2407.95 q^{19} -1398.70 q^{21} -601.847 q^{22} -4121.12 q^{23} -1757.49 q^{24} -5835.95 q^{26} -729.000 q^{27} -1128.27 q^{28} +75.4479 q^{29} +3595.47 q^{31} -2573.22 q^{32} -1089.00 q^{33} -4467.18 q^{34} -588.052 q^{36} -16063.1 q^{37} +11977.0 q^{38} -10559.7 q^{39} +15062.8 q^{41} +6957.07 q^{42} -12867.8 q^{43} -878.448 q^{44} +20498.2 q^{46} -6698.84 q^{47} +6650.79 q^{48} +7345.75 q^{49} -8083.06 q^{51} -8518.07 q^{52} +22850.8 q^{53} +3626.00 q^{54} +30348.2 q^{56} +21671.6 q^{57} -375.274 q^{58} -13021.3 q^{59} +882.310 q^{61} -17883.6 q^{62} +12588.3 q^{63} +36446.3 q^{64} +5416.62 q^{66} -14191.7 q^{67} -6520.24 q^{68} +37090.1 q^{69} -78746.0 q^{71} +15817.4 q^{72} +34227.3 q^{73} +79896.8 q^{74} +17481.5 q^{76} +18804.8 q^{77} +52523.5 q^{78} -48694.6 q^{79} +6561.00 q^{81} -74921.3 q^{82} -95905.3 q^{83} +10154.5 q^{84} +64003.6 q^{86} -679.031 q^{87} +23628.5 q^{88} +1342.55 q^{89} +182345. q^{91} +29918.9 q^{92} -32359.2 q^{93} +33319.6 q^{94} +23159.0 q^{96} +80938.9 q^{97} -36537.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more