LMFDB - Embedded newform 825.6.a.y.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:	$0$
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			$1.88656$ 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)

$f(q)$	$=$	$q-0.886559 q^{2} +9.00000 q^{3} -31.2140 q^{4} -7.97903 q^{6} +224.774 q^{7} +56.0430 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-0.886559 q^{2} +9.00000 q^{3} -31.2140 q^{4} -7.97903 q^{6} +224.774 q^{7} +56.0430 q^{8} +81.0000 q^{9} +121.000 q^{11} -280.926 q^{12} +1065.64 q^{13} -199.275 q^{14} +949.163 q^{16} -1347.75 q^{17} -71.8113 q^{18} +691.252 q^{19} +2022.96 q^{21} -107.274 q^{22} +3396.13 q^{23} +504.387 q^{24} -944.749 q^{26} +729.000 q^{27} -7016.09 q^{28} +8603.02 q^{29} -320.509 q^{31} -2634.86 q^{32} +1089.00 q^{33} +1194.86 q^{34} -2528.34 q^{36} -1906.23 q^{37} -612.836 q^{38} +9590.72 q^{39} +5821.85 q^{41} -1793.48 q^{42} +1421.42 q^{43} -3776.90 q^{44} -3010.87 q^{46} -6454.51 q^{47} +8542.47 q^{48} +33716.2 q^{49} -12129.7 q^{51} -33262.8 q^{52} +9109.26 q^{53} -646.302 q^{54} +12597.0 q^{56} +6221.27 q^{57} -7627.08 q^{58} +10213.6 q^{59} -34276.3 q^{61} +284.150 q^{62} +18206.7 q^{63} -28037.3 q^{64} -965.463 q^{66} -68436.0 q^{67} +42068.6 q^{68} +30565.2 q^{69} +1883.92 q^{71} +4539.48 q^{72} -87733.7 q^{73} +1689.99 q^{74} -21576.8 q^{76} +27197.6 q^{77} -8502.74 q^{78} -52408.8 q^{79} +6561.00 q^{81} -5161.42 q^{82} +58665.8 q^{83} -63144.8 q^{84} -1260.17 q^{86} +77427.1 q^{87} +6781.20 q^{88} -113843. q^{89} +239527. q^{91} -106007. q^{92} -2884.58 q^{93} +5722.30 q^{94} -23713.8 q^{96} -115329. q^{97} -29891.4 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.886559	−0.156723	−0.0783615	−	0.996925i	$-0.524969\pi$
$2$	−0.886559	−0.156723	−0.0783615	+	0.996925i	$0.524969\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.2140	−0.975438
$5$	0	0
$5$	0	0
$6$	−7.97903	−0.0904841
$7$	224.774	1.73380	0.866902	−	0.498478i	$-0.166108\pi$
$7$	224.774	1.73380	0.866902	+	0.498478i	$0.166108\pi$
$8$	56.0430	0.309597
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−280.926	−0.563169
$13$	1065.64	1.74884	0.874420	−	0.485169i	$-0.161242\pi$
$13$	1065.64	1.74884	0.874420	+	0.485169i	$0.161242\pi$
$14$	−199.275	−0.271727
$15$	0	0
$16$	949.163	0.926917
$17$	−1347.75	−1.13106	−0.565531	−	0.824727i	$-0.691329\pi$
$17$	−1347.75	−1.13106	−0.565531	+	0.824727i	$0.691329\pi$
$18$	−71.8113	−0.0522410
$19$	691.252	0.439291	0.219646	−	0.975580i	$-0.429510\pi$
$19$	691.252	0.439291	0.219646	+	0.975580i	$0.429510\pi$
$20$	0	0
$21$	2022.96	1.00101
$22$	−107.274	−0.0472538
$23$	3396.13	1.33864	0.669321	−	0.742973i	$-0.266585\pi$
$23$	3396.13	1.33864	0.669321	+	0.742973i	$0.266585\pi$
$24$	504.387	0.178746
$25$	0	0
$26$	−944.749	−0.274084
$27$	729.000	0.192450
$28$	−7016.09	−1.69122
$29$	8603.02	1.89957	0.949786	−	0.312900i	$-0.101301\pi$
$29$	8603.02	1.89957	0.949786	+	0.312900i	$0.101301\pi$
$30$	0	0
$31$	−320.509	−0.0599013	−0.0299507	−	0.999551i	$-0.509535\pi$
$31$	−320.509	−0.0599013	−0.0299507	+	0.999551i	$0.509535\pi$
$32$	−2634.86	−0.454866
$33$	1089.00	0.174078
$34$	1194.86	0.177263
$35$	0	0
$36$	−2528.34	−0.325146
$37$	−1906.23	−0.228913	−0.114457	−	0.993428i	$-0.536513\pi$
$37$	−1906.23	−0.228913	−0.114457	+	0.993428i	$0.536513\pi$
$38$	−612.836	−0.0688471
$39$	9590.72	1.00969
$40$	0	0
$41$	5821.85	0.540881	0.270440	−	0.962737i	$-0.412831\pi$
$41$	5821.85	0.540881	0.270440	+	0.962737i	$0.412831\pi$
$42$	−1793.48	−0.156882
$43$	1421.42	0.117234	0.0586168	−	0.998281i	$-0.481331\pi$
$43$	1421.42	0.117234	0.0586168	+	0.998281i	$0.481331\pi$
$44$	−3776.90	−0.294106
$45$	0	0
$46$	−3010.87	−0.209796
$47$	−6454.51	−0.426205	−0.213102	−	0.977030i	$-0.568357\pi$
$47$	−6454.51	−0.426205	−0.213102	+	0.977030i	$0.568357\pi$
$48$	8542.47	0.535156
$49$	33716.2	2.00608
$50$	0	0
$51$	−12129.7	−0.653019
$52$	−33262.8	−1.70589
$53$	9109.26	0.445444	0.222722	−	0.974882i	$-0.428506\pi$
$53$	9109.26	0.445444	0.222722	+	0.974882i	$0.428506\pi$
$54$	−646.302	−0.0301614
$55$	0	0
$56$	12597.0	0.536780
$57$	6221.27	0.253625
$58$	−7627.08	−0.297707
$59$	10213.6	0.381989	0.190995	−	0.981591i	$-0.438829\pi$
$59$	10213.6	0.381989	0.190995	+	0.981591i	$0.438829\pi$
$60$	0	0
$61$	−34276.3	−1.17942	−0.589712	−	0.807614i	$-0.700759\pi$
$61$	−34276.3	−1.17942	−0.589712	+	0.807614i	$0.700759\pi$
$62$	284.150	0.00938791
$63$	18206.7	0.577935
$64$	−28037.3	−0.855629
$65$	0	0
$66$	−965.463	−0.0272820
$67$	−68436.0	−1.86251	−0.931253	−	0.364374i	$-0.881283\pi$
$67$	−68436.0	−1.86251	−0.931253	+	0.364374i	$0.881283\pi$
$68$	42068.6	1.10328
$69$	30565.2	0.772866
$70$	0	0
$71$	1883.92	0.0443524	0.0221762	−	0.999754i	$-0.492941\pi$
$71$	1883.92	0.0443524	0.0221762	+	0.999754i	$0.492941\pi$
$72$	4539.48	0.103199
$73$	−87733.7	−1.92690	−0.963451	−	0.267885i	$-0.913675\pi$
$73$	−87733.7	−1.92690	−0.963451	+	0.267885i	$0.913675\pi$
$74$	1689.99	0.0358760
$75$	0	0
$76$	−21576.8	−0.428501
$77$	27197.6	0.522762
$78$	−8502.74	−0.158242
$79$	−52408.8	−0.944793	−0.472397	−	0.881386i	$-0.656611\pi$
$79$	−52408.8	−0.944793	−0.472397	+	0.881386i	$0.656611\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−5161.42	−0.0847685
$83$	58665.8	0.934737	0.467369	−	0.884062i	$-0.345202\pi$
$83$	58665.8	0.934737	0.467369	+	0.884062i	$0.345202\pi$
$84$	−63144.8	−0.976426
$85$	0	0
$86$	−1260.17	−0.0183732
$87$	77427.1	1.09672
$88$	6781.20	0.0933469
$89$	−113843.	−1.52346	−0.761728	−	0.647896i	$-0.775649\pi$
$89$	−113843.	−1.52346	−0.761728	+	0.647896i	$0.775649\pi$
$90$	0	0
$91$	239527.	3.03215
$92$	−106007.	−1.30576
$93$	−2884.58	−0.0345840
$94$	5722.30	0.0667961
$95$	0	0
$96$	−23713.8	−0.262617
$97$	−115329.	−1.24454	−0.622271	−	0.782802i	$-0.713790\pi$
$97$	−115329.	−1.24454	−0.622271	+	0.782802i	$0.713790\pi$
$98$	−29891.4	−0.314399
$99$	9801.00	0.100504
$100$	0	0
$101$	39906.6	0.389261	0.194631	−	0.980877i	$-0.437649\pi$
$101$	39906.6	0.389261	0.194631	+	0.980877i	$0.437649\pi$
$102$	10753.7	0.102343
$103$	69320.0	0.643822	0.321911	−	0.946770i	$-0.395675\pi$
$103$	69320.0	0.643822	0.321911	+	0.946770i	$0.395675\pi$
$104$	59721.4	0.541435
$105$	0	0
$106$	−8075.90	−0.0698114
$107$	131149.	1.10740	0.553700	−	0.832717i	$-0.313216\pi$
$107$	131149.	1.10740	0.553700	+	0.832717i	$0.313216\pi$
$108$	−22755.0	−0.187723
$109$	146929.	1.18451	0.592257	−	0.805749i	$-0.298237\pi$
$109$	146929.	1.18451	0.592257	+	0.805749i	$0.298237\pi$
$110$	0	0
$111$	−17156.1	−0.132163
$112$	213347.	1.60709
$113$	−57346.0	−0.422481	−0.211240	−	0.977434i	$-0.567750\pi$
$113$	−57346.0	−0.422481	−0.211240	+	0.977434i	$0.567750\pi$
$114$	−5515.52	−0.0397489
$115$	0	0
$116$	−268535.	−1.85291
$117$	86316.5	0.582947
$118$	−9055.00	−0.0598665
$119$	−302938.	−1.96104
$120$	0	0
$121$	14641.0	0.0909091
$122$	30388.0	0.184843
$123$	52396.7	0.312278
$124$	10004.4	0.0584300
$125$	0	0
$126$	−16141.3	−0.0905757
$127$	172649.	0.949852	0.474926	−	0.880026i	$-0.342475\pi$
$127$	172649.	0.949852	0.474926	+	0.880026i	$0.342475\pi$
$128$	109172.	0.588963
$129$	12792.8	0.0676848
$130$	0	0
$131$	297723.	1.51577	0.757886	−	0.652387i	$-0.226233\pi$
$131$	297723.	1.51577	0.757886	+	0.652387i	$0.226233\pi$
$132$	−33992.1	−0.169802
$133$	155375.	0.761645
$134$	60672.6	0.291897
$135$	0	0
$136$	−75531.8	−0.350173
$137$	−45559.8	−0.207387	−0.103693	−	0.994609i	$-0.533066\pi$
$137$	−45559.8	−0.207387	−0.103693	+	0.994609i	$0.533066\pi$
$138$	−27097.8	−0.121126
$139$	−124306.	−0.545701	−0.272850	−	0.962056i	$-0.587966\pi$
$139$	−124306.	−0.545701	−0.272850	+	0.962056i	$0.587966\pi$
$140$	0	0
$141$	−58090.6	−0.246070
$142$	−1670.21	−0.00695104
$143$	128942.	0.527295
$144$	76882.2	0.308972
$145$	0	0
$146$	77781.2	0.301990
$147$	303445.	1.15821
$148$	59501.1	0.223291
$149$	−107578.	−0.396971	−0.198485	−	0.980104i	$-0.563602\pi$
$149$	−107578.	−0.396971	−0.198485	+	0.980104i	$0.563602\pi$
$150$	0	0
$151$	147513.	0.526487	0.263244	−	0.964729i	$-0.415208\pi$
$151$	147513.	0.526487	0.263244	+	0.964729i	$0.415208\pi$
$152$	38739.8	0.136003
$153$	−109168.	−0.377021
$154$	−24112.3	−0.0819288
$155$	0	0
$156$	−299365.	−0.984893
$157$	231901.	0.750849	0.375425	−	0.926853i	$-0.377497\pi$
$157$	231901.	0.750849	0.375425	+	0.926853i	$0.377497\pi$
$158$	46463.5	0.148071
$159$	81983.3	0.257177
$160$	0	0
$161$	763360.	2.32094
$162$	−5816.72	−0.0174137
$163$	−6862.35	−0.0202304	−0.0101152	−	0.999949i	$-0.503220\pi$
$163$	−6862.35	−0.0202304	−0.0101152	+	0.999949i	$0.503220\pi$
$164$	−181723.	−0.527596
$165$	0	0
$166$	−52010.7	−0.146495
$167$	−10230.0	−0.0283847	−0.0141923	−	0.999899i	$-0.504518\pi$
$167$	−10230.0	−0.0283847	−0.0141923	+	0.999899i	$0.504518\pi$
$168$	113373.	0.309910
$169$	764285.	2.05844
$170$	0	0
$171$	55991.4	0.146430
$172$	−44368.3	−0.114354
$173$	421259.	1.07012	0.535062	−	0.844813i	$-0.320288\pi$
$173$	421259.	1.07012	0.535062	+	0.844813i	$0.320288\pi$
$174$	−68643.8	−0.171881
$175$	0	0
$176$	114849.	0.279476
$177$	91922.8	0.220541
$178$	100928.	0.238761
$179$	330517.	0.771012	0.385506	−	0.922705i	$-0.374027\pi$
$179$	330517.	0.771012	0.385506	+	0.922705i	$0.374027\pi$
$180$	0	0
$181$	668805.	1.51741	0.758705	−	0.651434i	$-0.225832\pi$
$181$	668805.	1.51741	0.758705	+	0.651434i	$0.225832\pi$
$182$	−212355.	−0.475207
$183$	−308487.	−0.680940
$184$	190329.	0.414439
$185$	0	0
$186$	2557.35	0.00542012
$187$	−163078.	−0.341028
$188$	201471.	0.415736
$189$	163860.	0.333671
$190$	0	0
$191$	−961866.	−1.90779	−0.953896	−	0.300137i	$-0.902967\pi$
$191$	−961866.	−1.90779	−0.953896	+	0.300137i	$0.902967\pi$
$192$	−252335.	−0.493998
$193$	−87470.4	−0.169032	−0.0845158	−	0.996422i	$-0.526934\pi$
$193$	−87470.4	−0.169032	−0.0845158	+	0.996422i	$0.526934\pi$
$194$	102246.	0.195048
$195$	0	0
$196$	−1.05242e6	−1.95680
$197$	799656.	1.46804	0.734019	−	0.679129i	$-0.237642\pi$
$197$	799656.	1.46804	0.734019	+	0.679129i	$0.237642\pi$
$198$	−8689.17	−0.0157513
$199$	−635515.	−1.13761	−0.568804	−	0.822473i	$-0.692594\pi$
$199$	−635515.	−1.13761	−0.568804	+	0.822473i	$0.692594\pi$
$200$	0	0
$201$	−615924.	−1.07532
$202$	−35379.6	−0.0610062
$203$	1.93373e6	3.29349
$204$	378618.	0.636980
$205$	0	0
$206$	−61456.3	−0.100902
$207$	275086.	0.446214
$208$	1.01146e6	1.62103
$209$	83641.5	0.132451
$210$	0	0
$211$	−349594.	−0.540578	−0.270289	−	0.962779i	$-0.587119\pi$
$211$	−349594.	−0.540578	−0.270289	+	0.962779i	$0.587119\pi$
$212$	−284337.	−0.434503
$213$	16955.3	0.0256069
$214$	−116271.	−0.173555
$215$	0	0
$216$	40855.3	0.0595819
$217$	−72042.0	−0.103857
$218$	−130261.	−0.185641
$219$	−789604.	−1.11250
$220$	0	0
$221$	−1.43621e6	−1.97805
$222$	15209.9	0.0207130
$223$	−309538.	−0.416824	−0.208412	−	0.978041i	$-0.566829\pi$
$223$	−309538.	−0.416824	−0.208412	+	0.978041i	$0.566829\pi$
$224$	−592248.	−0.788648
$225$	0	0
$226$	50840.6	0.0662124
$227$	−509733.	−0.656566	−0.328283	−	0.944579i	$-0.606470\pi$
$227$	−509733.	−0.656566	−0.328283	+	0.944579i	$0.606470\pi$
$228$	−194191.	−0.247395
$229$	−1.28351e6	−1.61737	−0.808684	−	0.588243i	$-0.799820\pi$
$229$	−1.28351e6	−1.61737	−0.808684	+	0.588243i	$0.799820\pi$
$230$	0	0
$231$	244778.	0.301817
$232$	482139.	0.588101
$233$	312877.	0.377558	0.188779	−	0.982020i	$-0.439547\pi$
$233$	312877.	0.377558	0.188779	+	0.982020i	$0.439547\pi$
$234$	−76524.7	−0.0913612
$235$	0	0
$236$	−318809.	−0.372607
$237$	−471679.	−0.545477
$238$	268573.	0.307340
$239$	−357254.	−0.404559	−0.202280	−	0.979328i	$-0.564835\pi$
$239$	−357254.	−0.404559	−0.202280	+	0.979328i	$0.564835\pi$
$240$	0	0
$241$	−680168.	−0.754351	−0.377176	−	0.926142i	$-0.623105\pi$
$241$	−680168.	−0.754351	−0.377176	+	0.926142i	$0.623105\pi$
$242$	−12980.1	−0.0142475
$243$	59049.0	0.0641500
$244$	1.06990e6	1.15045
$245$	0	0
$246$	−46452.8	−0.0489411
$247$	736623.	0.768250
$248$	−17962.3	−0.0185452
$249$	527992.	0.539671
$250$	0	0
$251$	157817.	0.158114	0.0790568	−	0.996870i	$-0.474809\pi$
$251$	157817.	0.158114	0.0790568	+	0.996870i	$0.474809\pi$
$252$	−568303.	−0.563740
$253$	410932.	0.403616
$254$	−153064.	−0.148864
$255$	0	0
$256$	800404.	0.763325
$257$	557296.	0.526324	0.263162	−	0.964752i	$-0.415235\pi$
$257$	557296.	0.526324	0.263162	+	0.964752i	$0.415235\pi$
$258$	−11341.6	−0.0106078
$259$	−428470.	−0.396891
$260$	0	0
$261$	696844.	0.633191
$262$	−263949.	−0.237556
$263$	−645647.	−0.575580	−0.287790	−	0.957693i	$-0.592921\pi$
$263$	−645647.	−0.575580	−0.287790	+	0.957693i	$0.592921\pi$
$264$	61030.8	0.0538939
$265$	0	0
$266$	−137749.	−0.119367
$267$	−1.02458e6	−0.879568
$268$	2.13616e6	1.81676
$269$	−219755.	−0.185165	−0.0925825	−	0.995705i	$-0.529512\pi$
$269$	−219755.	−0.185165	−0.0925825	+	0.995705i	$0.529512\pi$
$270$	0	0
$271$	1.54792e6	1.28034	0.640171	−	0.768233i	$-0.278864\pi$
$271$	1.54792e6	1.28034	0.640171	+	0.768233i	$0.278864\pi$
$272$	−1.27923e6	−1.04840
$273$	2.15574e6	1.75061
$274$	40391.5	0.0325023
$275$	0	0
$276$	−954061.	−0.753882
$277$	−980724.	−0.767976	−0.383988	−	0.923338i	$-0.625450\pi$
$277$	−980724.	−0.767976	−0.383988	+	0.923338i	$0.625450\pi$
$278$	110205.	0.0855239
$279$	−25961.2	−0.0199671
$280$	0	0
$281$	2.46283e6	1.86067	0.930334	−	0.366713i	$-0.119517\pi$
$281$	2.46283e6	1.86067	0.930334	+	0.366713i	$0.119517\pi$
$282$	51500.7	0.0385648
$283$	−838610.	−0.622435	−0.311217	−	0.950339i	$-0.600737\pi$
$283$	−838610.	−0.622435	−0.311217	+	0.950339i	$0.600737\pi$
$284$	−58804.8	−0.0432630
$285$	0	0
$286$	−114315.	−0.0826393
$287$	1.30860e6	0.937782
$288$	−213424.	−0.151622
$289$	396569.	0.279302
$290$	0	0
$291$	−1.03796e6	−0.718536
$292$	2.73852e6	1.87957
$293$	1.32689e6	0.902955	0.451478	−	0.892282i	$-0.350897\pi$
$293$	1.32689e6	0.902955	0.451478	+	0.892282i	$0.350897\pi$
$294$	−269022.	−0.181518
$295$	0	0
$296$	−106831.	−0.0708708
$297$	88209.0	0.0580259
$298$	95374.4	0.0622144
$299$	3.61904e6	2.34107
$300$	0	0
$301$	319498.	0.203260
$302$	−130779.	−0.0825126
$303$	359159.	0.224740
$304$	656111.	0.407187
$305$	0	0
$306$	96783.6	0.0590878
$307$	1.04842e6	0.634878	0.317439	−	0.948279i	$-0.397177\pi$
$307$	1.04842e6	0.634878	0.317439	+	0.948279i	$0.397177\pi$
$308$	−848946.	−0.509922
$309$	623880.	0.371711
$310$	0	0
$311$	−1.53786e6	−0.901601	−0.450801	−	0.892625i	$-0.648861\pi$
$311$	−1.53786e6	−0.901601	−0.450801	+	0.892625i	$0.648861\pi$
$312$	537492.	0.312598
$313$	−2.58473e6	−1.49126	−0.745631	−	0.666359i	$-0.767852\pi$
$313$	−2.58473e6	−1.49126	−0.745631	+	0.666359i	$0.767852\pi$
$314$	−205594.	−0.117675
$315$	0	0
$316$	1.63589e6	0.921587
$317$	−1.29146e6	−0.721827	−0.360914	−	0.932599i	$-0.617535\pi$
$317$	−1.29146e6	−0.721827	−0.360914	+	0.932599i	$0.617535\pi$
$318$	−72683.1	−0.0403056
$319$	1.04096e6	0.572742
$320$	0	0
$321$	1.18034e6	0.639357
$322$	−676764.	−0.363745
$323$	−931634.	−0.496866
$324$	−204795.	−0.108382
$325$	0	0
$326$	6083.88	0.00317057
$327$	1.32236e6	0.683880
$328$	326274.	0.167455
$329$	−1.45080e6	−0.738956
$330$	0	0
$331$	1.77716e6	0.891573	0.445787	−	0.895139i	$-0.352924\pi$
$331$	1.77716e6	0.891573	0.445787	+	0.895139i	$0.352924\pi$
$332$	−1.83119e6	−0.911778
$333$	−154405.	−0.0763045
$334$	9069.48	0.00444853
$335$	0	0
$336$	1.92012e6	0.927856
$337$	1.99819e6	0.958435	0.479217	−	0.877696i	$-0.340921\pi$
$337$	1.99819e6	0.958435	0.479217	+	0.877696i	$0.340921\pi$
$338$	−677584.	−0.322605
$339$	−516114.	−0.243919
$340$	0	0
$341$	−38781.6	−0.0180609
$342$	−49639.7	−0.0229490
$343$	3.80073e6	1.74434
$344$	79660.7	0.0362951
$345$	0	0
$346$	−373471.	−0.167713
$347$	2.94954e6	1.31501	0.657507	−	0.753449i	$-0.271611\pi$
$347$	2.94954e6	1.31501	0.657507	+	0.753449i	$0.271611\pi$
$348$	−2.41681e6	−1.06978
$349$	−1.15473e6	−0.507476	−0.253738	−	0.967273i	$-0.581660\pi$
$349$	−1.15473e6	−0.507476	−0.253738	+	0.967273i	$0.581660\pi$
$350$	0	0
$351$	776848.	0.336565
$352$	−318819.	−0.137147
$353$	−829780.	−0.354427	−0.177213	−	0.984172i	$-0.556708\pi$
$353$	−829780.	−0.354427	−0.177213	+	0.984172i	$0.556708\pi$
$354$	−81495.0	−0.0345639
$355$	0	0
$356$	3.55349e6	1.48604
$357$	−2.72644e6	−1.13221
$358$	−293023.	−0.120835
$359$	3.60644e6	1.47687	0.738435	−	0.674325i	$-0.235565\pi$
$359$	3.60644e6	1.47687	0.738435	+	0.674325i	$0.235565\pi$
$360$	0	0
$361$	−1.99827e6	−0.807023
$362$	−592935.	−0.237813
$363$	131769.	0.0524864
$364$	−7.47659e6	−2.95767
$365$	0	0
$366$	273492.	0.106719
$367$	−403294.	−0.156299	−0.0781496	−	0.996942i	$-0.524901\pi$
$367$	−403294.	−0.156299	−0.0781496	+	0.996942i	$0.524901\pi$
$368$	3.22348e6	1.24081
$369$	471570.	0.180294
$370$	0	0
$371$	2.04752e6	0.772313
$372$	90039.4	0.0337346
$373$	3.48177e6	1.29577	0.647884	−	0.761739i	$-0.275654\pi$
$373$	3.48177e6	1.29577	0.647884	+	0.761739i	$0.275654\pi$
$374$	144578.	0.0534470
$375$	0	0
$376$	−361730.	−0.131952
$377$	9.16768e6	3.32205
$378$	−145272.	−0.0522939
$379$	−704863.	−0.252062	−0.126031	−	0.992026i	$-0.540224\pi$
$379$	−704863.	−0.252062	−0.126031	+	0.992026i	$0.540224\pi$
$380$	0	0
$381$	1.55384e6	0.548397
$382$	852751.	0.298995
$383$	−2.16463e6	−0.754028	−0.377014	−	0.926208i	$-0.623049\pi$
$383$	−2.16463e6	−0.754028	−0.377014	+	0.926208i	$0.623049\pi$
$384$	982551.	0.340038
$385$	0	0
$386$	77547.7	0.0264911
$387$	115135.	0.0390778
$388$	3.59988e6	1.21397
$389$	2.50750e6	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$389$	2.50750e6	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$390$	0	0
$391$	−4.57713e6	−1.51409
$392$	1.88955e6	0.621075
$393$	2.67950e6	0.875131
$394$	−708942.	−0.230075
$395$	0	0
$396$	−305929.	−0.0980352
$397$	−2.75852e6	−0.878417	−0.439208	−	0.898385i	$-0.644741\pi$
$397$	−2.75852e6	−0.878417	−0.439208	+	0.898385i	$0.644741\pi$
$398$	563422.	0.178289
$399$	1.39838e6	0.439736
$400$	0	0
$401$	−6.11243e6	−1.89825	−0.949124	−	0.314901i	$-0.898029\pi$
$401$	−6.11243e6	−1.89825	−0.949124	+	0.314901i	$0.898029\pi$
$402$	546053.	0.168527
$403$	−341546.	−0.104758
$404$	−1.24565e6	−0.379700
$405$	0	0
$406$	−1.71437e6	−0.516165
$407$	−230654.	−0.0690200
$408$	−679786.	−0.202172
$409$	4.96574e6	1.46783	0.733915	−	0.679241i	$-0.237691\pi$
$409$	4.96574e6	1.46783	0.733915	+	0.679241i	$0.237691\pi$
$410$	0	0
$411$	−410039.	−0.119735
$412$	−2.16376e6	−0.628008
$413$	2.29576e6	0.662294
$414$	−243880.	−0.0699320
$415$	0	0
$416$	−2.80780e6	−0.795488
$417$	−1.11875e6	−0.315061
$418$	−74153.2	−0.0207582
$419$	−3.30467e6	−0.919589	−0.459794	−	0.888025i	$-0.652077\pi$
$419$	−3.30467e6	−0.919589	−0.459794	+	0.888025i	$0.652077\pi$
$420$	0	0
$421$	5.04577e6	1.38747	0.693733	−	0.720232i	$-0.255965\pi$
$421$	5.04577e6	1.38747	0.693733	+	0.720232i	$0.255965\pi$
$422$	309936.	0.0847209
$423$	−522815.	−0.142068
$424$	510510.	0.137908
$425$	0	0
$426$	−15031.9	−0.00401318
$427$	−7.70441e6	−2.04489
$428$	−4.09367e6	−1.08020
$429$	1.16048e6	0.304434
$430$	0	0
$431$	889002.	0.230520	0.115260	−	0.993335i	$-0.463230\pi$
$431$	889002.	0.230520	0.115260	+	0.993335i	$0.463230\pi$
$432$	691940.	0.178385
$433$	406627.	0.104226	0.0521131	−	0.998641i	$-0.483404\pi$
$433$	406627.	0.104226	0.0521131	+	0.998641i	$0.483404\pi$
$434$	63869.5	0.0162768
$435$	0	0
$436$	−4.58623e6	−1.15542
$437$	2.34758e6	0.588054
$438$	700031.	0.174354
$439$	−5.04798e6	−1.25013	−0.625067	−	0.780571i	$-0.714928\pi$
$439$	−5.04798e6	−1.25013	−0.625067	+	0.780571i	$0.714928\pi$
$440$	0	0
$441$	2.73101e6	0.668693
$442$	1.27328e6	0.310006
$443$	−888258.	−0.215045	−0.107523	−	0.994203i	$-0.534292\pi$
$443$	−888258.	−0.215045	−0.107523	+	0.994203i	$0.534292\pi$
$444$	535510.	0.128917
$445$	0	0
$446$	274424.	0.0653259
$447$	−968203.	−0.229191
$448$	−6.30203e6	−1.48349
$449$	3.82230e6	0.894765	0.447383	−	0.894343i	$-0.352356\pi$
$449$	3.82230e6	0.894765	0.447383	+	0.894343i	$0.352356\pi$
$450$	0	0
$451$	704444.	0.163082
$452$	1.79000e6	0.412104
$453$	1.32762e6	0.303967
$454$	451909.	0.102899
$455$	0	0
$456$	348658.	0.0785214
$457$	2.10600e6	0.471703	0.235852	−	0.971789i	$-0.424212\pi$
$457$	2.10600e6	0.471703	0.235852	+	0.971789i	$0.424212\pi$
$458$	1.13790e6	0.253479
$459$	−982509.	−0.217673
$460$	0	0
$461$	−2.43217e6	−0.533017	−0.266508	−	0.963833i	$-0.585870\pi$
$461$	−2.43217e6	−0.533017	−0.266508	+	0.963833i	$0.585870\pi$
$462$	−217011.	−0.0473016
$463$	7.53506e6	1.63356	0.816779	−	0.576951i	$-0.195758\pi$
$463$	7.53506e6	1.63356	0.816779	+	0.576951i	$0.195758\pi$
$464$	8.16566e6	1.76075
$465$	0	0
$466$	−277384.	−0.0591720
$467$	−3.17287e6	−0.673225	−0.336612	−	0.941643i	$-0.609281\pi$
$467$	−3.17287e6	−0.673225	−0.336612	+	0.941643i	$0.609281\pi$
$468$	−2.69428e6	−0.568628
$469$	−1.53826e7	−3.22922
$470$	0	0
$471$	2.08711e6	0.433503
$472$	572403.	0.118262
$473$	171992.	0.0353472
$474$	418172.	0.0854887
$475$	0	0
$476$	9.45592e6	1.91287
$477$	737850.	0.148481
$478$	316727.	0.0634038
$479$	3.20281e6	0.637811	0.318905	−	0.947787i	$-0.396685\pi$
$479$	3.20281e6	0.637811	0.318905	+	0.947787i	$0.396685\pi$
$480$	0	0
$481$	−2.03135e6	−0.400333
$482$	603009.	0.118224
$483$	6.87024e6	1.34000
$484$	−457004.	−0.0886762
$485$	0	0
$486$	−52350.4	−0.0100538
$487$	−4.45541e6	−0.851266	−0.425633	−	0.904896i	$-0.639949\pi$
$487$	−4.45541e6	−0.851266	−0.425633	+	0.904896i	$0.639949\pi$
$488$	−1.92095e6	−0.365145
$489$	−61761.2	−0.0116800
$490$	0	0
$491$	150957.	0.0282586	0.0141293	−	0.999900i	$-0.495502\pi$
$491$	150957.	0.0282586	0.0141293	+	0.999900i	$0.495502\pi$
$492$	−1.63551e6	−0.304608
$493$	−1.15947e7	−2.14853
$494$	−653060.	−0.120403
$495$	0	0
$496$	−304216.	−0.0555235
$497$	423456.	0.0768984
$498$	−468096.	−0.0845789
$499$	−4.56196e6	−0.820163	−0.410081	−	0.912049i	$-0.634500\pi$
$499$	−4.56196e6	−0.820163	−0.410081	+	0.912049i	$0.634500\pi$
$500$	0	0
$501$	−92069.8	−0.0163879
$502$	−139914.	−0.0247800
$503$	−5.18773e6	−0.914234	−0.457117	−	0.889407i	$-0.651118\pi$
$503$	−5.18773e6	−0.914234	−0.457117	+	0.889407i	$0.651118\pi$
$504$	1.02036e6	0.178927
$505$	0	0
$506$	−364315.	−0.0632559
$507$	6.87857e6	1.18844
$508$	−5.38908e6	−0.926521
$509$	6.85927e6	1.17350	0.586750	−	0.809768i	$-0.300407\pi$
$509$	6.85927e6	1.17350	0.586750	+	0.809768i	$0.300407\pi$
$510$	0	0
$511$	−1.97202e7	−3.34087
$512$	−4.20312e6	−0.708593
$513$	503923.	0.0845416
$514$	−494076.	−0.0824871
$515$	0	0
$516$	−399314.	−0.0660223
$517$	−780995.	−0.128506
$518$	379864.	0.0622020
$519$	3.79133e6	0.617836
$520$	0	0
$521$	−3.91860e6	−0.632466	−0.316233	−	0.948682i	$-0.602418\pi$
$521$	−3.91860e6	−0.632466	−0.316233	+	0.948682i	$0.602418\pi$
$522$	−617794.	−0.0992356
$523$	−3.64804e6	−0.583183	−0.291592	−	0.956543i	$-0.594185\pi$
$523$	−3.64804e6	−0.583183	−0.291592	+	0.956543i	$0.594185\pi$
$524$	−9.29312e6	−1.47854
$525$	0	0
$526$	572405.	0.0902067
$527$	431966.	0.0677521
$528$	1.03364e6	0.161356
$529$	5.09735e6	0.791964
$530$	0	0
$531$	827305.	0.127330
$532$	−4.84988e6	−0.742938
$533$	6.20397e6	0.945914
$534$	908355.	0.137849
$535$	0	0
$536$	−3.83536e6	−0.576625
$537$	2.97465e6	0.445144
$538$	194826.	0.0290196
$539$	4.07966e6	0.604855
$540$	0	0
$541$	−4.23861e6	−0.622630	−0.311315	−	0.950307i	$-0.600769\pi$
$541$	−4.23861e6	−0.622630	−0.311315	+	0.950307i	$0.600769\pi$
$542$	−1.37232e6	−0.200659
$543$	6.01924e6	0.876077
$544$	3.55113e6	0.514482
$545$	0	0
$546$	−1.91119e6	−0.274361
$547$	−513913.	−0.0734380	−0.0367190	−	0.999326i	$-0.511691\pi$
$547$	−513913.	−0.0734380	−0.0367190	+	0.999326i	$0.511691\pi$
$548$	1.42211e6	0.202293
$549$	−2.77638e6	−0.393141
$550$	0	0
$551$	5.94685e6	0.834465
$552$	1.71296e6	0.239277
$553$	−1.17801e7	−1.63809
$554$	869470.	0.120359
$555$	0	0
$556$	3.88009e6	0.532297
$557$	−1.04431e7	−1.42624	−0.713121	−	0.701041i	$-0.752719\pi$
$557$	−1.04431e7	−1.42624	−0.713121	+	0.701041i	$0.752719\pi$
$558$	23016.2	0.00312930
$559$	1.51472e6	0.205023
$560$	0	0
$561$	−1.46770e6	−0.196893
$562$	−2.18345e6	−0.291610
$563$	5.42113e6	0.720806	0.360403	−	0.932797i	$-0.382639\pi$
$563$	5.42113e6	0.720806	0.360403	+	0.932797i	$0.382639\pi$
$564$	1.81324e6	0.240026
$565$	0	0
$566$	743477.	0.0975499
$567$	1.47474e6	0.192645
$568$	105581.	0.0137313
$569$	−4.65152e6	−0.602301	−0.301151	−	0.953577i	$-0.597371\pi$
$569$	−4.65152e6	−0.602301	−0.301151	+	0.953577i	$0.597371\pi$
$570$	0	0
$571$	836920.	0.107422	0.0537111	−	0.998557i	$-0.482895\pi$
$571$	836920.	0.107422	0.0537111	+	0.998557i	$0.482895\pi$
$572$	−4.02479e6	−0.514344
$573$	−8.65679e6	−1.10146
$574$	−1.16015e6	−0.146972
$575$	0	0
$576$	−2.27102e6	−0.285210
$577$	−1.22020e7	−1.52578	−0.762892	−	0.646526i	$-0.776221\pi$
$577$	−1.22020e7	−1.52578	−0.762892	+	0.646526i	$0.776221\pi$
$578$	−351582.	−0.0437730
$579$	−787233.	−0.0975904
$580$	0	0
$581$	1.31865e7	1.62065
$582$	920214.	0.112611
$583$	1.10222e6	0.134307
$584$	−4.91686e6	−0.596562
$585$	0	0
$586$	−1.17637e6	−0.141514
$587$	−4.64629e6	−0.556559	−0.278279	−	0.960500i	$-0.589764\pi$
$587$	−4.64629e6	−0.556559	−0.278279	+	0.960500i	$0.589764\pi$
$588$	−9.47175e6	−1.12976
$589$	−221553.	−0.0263141
$590$	0	0
$591$	7.19690e6	0.847572
$592$	−1.80932e6	−0.212184
$593$	−3.76104e6	−0.439209	−0.219605	−	0.975589i	$-0.570477\pi$
$593$	−3.76104e6	−0.439209	−0.219605	+	0.975589i	$0.570477\pi$
$594$	−78202.5	−0.00909399
$595$	0	0
$596$	3.35794e6	0.387220
$597$	−5.71963e6	−0.656799
$598$	−3.20849e6	−0.366900
$599$	−2.65769e6	−0.302648	−0.151324	−	0.988484i	$-0.548354\pi$
$599$	−2.65769e6	−0.302648	−0.151324	+	0.988484i	$0.548354\pi$
$600$	0	0
$601$	5.84003e6	0.659521	0.329761	−	0.944065i	$-0.393032\pi$
$601$	5.84003e6	0.659521	0.329761	+	0.944065i	$0.393032\pi$
$602$	−283254.	−0.0318555
$603$	−5.54331e6	−0.620835
$604$	−4.60447e6	−0.513555
$605$	0	0
$606$	−318416.	−0.0352220
$607$	−7.10119e6	−0.782274	−0.391137	−	0.920332i	$-0.627918\pi$
$607$	−7.10119e6	−0.782274	−0.391137	+	0.920332i	$0.627918\pi$
$608$	−1.82136e6	−0.199819
$609$	1.74036e7	1.90150
$610$	0	0
$611$	−6.87815e6	−0.745364
$612$	3.40756e6	0.367760
$613$	−1.10978e7	−1.19285	−0.596423	−	0.802670i	$-0.703412\pi$
$613$	−1.10978e7	−1.19285	−0.596423	+	0.802670i	$0.703412\pi$
$614$	−929488.	−0.0995000
$615$	0	0
$616$	1.52423e6	0.161845
$617$	1.34866e7	1.42623	0.713114	−	0.701049i	$-0.247284\pi$
$617$	1.34866e7	1.42623	0.713114	+	0.701049i	$0.247284\pi$
$618$	−553107.	−0.0582556
$619$	−1.19228e7	−1.25069	−0.625346	−	0.780347i	$-0.715042\pi$
$619$	−1.19228e7	−1.25069	−0.625346	+	0.780347i	$0.715042\pi$
$620$	0	0
$621$	2.47578e6	0.257622
$622$	1.36340e6	0.141302
$623$	−2.55888e7	−2.64138
$624$	9.10315e6	0.935902
$625$	0	0
$626$	2.29151e6	0.233715
$627$	752774.	0.0764708
$628$	−7.23855e6	−0.732407
$629$	2.56912e6	0.258915
$630$	0	0
$631$	−5.56411e6	−0.556317	−0.278159	−	0.960535i	$-0.589724\pi$
$631$	−5.56411e6	−0.556317	−0.278159	+	0.960535i	$0.589724\pi$
$632$	−2.93715e6	−0.292505
$633$	−3.14635e6	−0.312103
$634$	1.14496e6	0.113127
$635$	0	0
$636$	−2.55903e6	−0.250861
$637$	3.59291e7	3.50831
$638$	−922877.	−0.0897619
$639$	152598.	0.0147841
$640$	0	0
$641$	7.43898e6	0.715103	0.357551	−	0.933894i	$-0.383612\pi$
$641$	7.43898e6	0.715103	0.357551	+	0.933894i	$0.383612\pi$
$642$	−1.04644e6	−0.100202
$643$	8.56732e6	0.817180	0.408590	−	0.912718i	$-0.366021\pi$
$643$	8.56732e6	0.817180	0.408590	+	0.912718i	$0.366021\pi$
$644$	−2.38275e7	−2.26394
$645$	0	0
$646$	825949.	0.0778703
$647$	−1.65852e7	−1.55762	−0.778808	−	0.627263i	$-0.784175\pi$
$647$	−1.65852e7	−1.55762	−0.778808	+	0.627263i	$0.784175\pi$
$648$	367698.	0.0343996
$649$	1.23585e6	0.115174
$650$	0	0
$651$	−648378.	−0.0599620
$652$	214202.	0.0197335
$653$	−7.12943e6	−0.654292	−0.327146	−	0.944974i	$-0.606087\pi$
$653$	−7.12943e6	−0.654292	−0.327146	+	0.944974i	$0.606087\pi$
$654$	−1.17235e6	−0.107180
$655$	0	0
$656$	5.52589e6	0.501352
$657$	−7.10643e6	−0.642301
$658$	1.28622e6	0.115811
$659$	1.00795e7	0.904116	0.452058	−	0.891988i	$-0.350690\pi$
$659$	1.00795e7	0.904116	0.452058	+	0.891988i	$0.350690\pi$
$660$	0	0
$661$	1.12712e7	1.00338	0.501689	−	0.865048i	$-0.332712\pi$
$661$	1.12712e7	1.00338	0.501689	+	0.865048i	$0.332712\pi$
$662$	−1.57556e6	−0.139730
$663$	−1.29259e7	−1.14203
$664$	3.28780e6	0.289392
$665$	0	0
$666$	136889.	0.0119587
$667$	2.92170e7	2.54285
$668$	319319.	0.0276875
$669$	−2.78585e6	−0.240653
$670$	0	0
$671$	−4.14743e6	−0.355609
$672$	−5.33023e6	−0.455326
$673$	−2.71329e6	−0.230918	−0.115459	−	0.993312i	$-0.536834\pi$
$673$	−2.71329e6	−0.230918	−0.115459	+	0.993312i	$0.536834\pi$
$674$	−1.77152e6	−0.150209
$675$	0	0
$676$	−2.38564e7	−2.00788
$677$	−1.70935e6	−0.143337	−0.0716686	−	0.997429i	$-0.522832\pi$
$677$	−1.70935e6	−0.143337	−0.0716686	+	0.997429i	$0.522832\pi$
$678$	457565.	0.0382278
$679$	−2.59229e7	−2.15779
$680$	0	0
$681$	−4.58760e6	−0.379069
$682$	34382.2	0.00283056
$683$	2.06905e7	1.69715	0.848573	−	0.529079i	$-0.177462\pi$
$683$	2.06905e7	1.69715	0.848573	+	0.529079i	$0.177462\pi$
$684$	−1.74772e6	−0.142834
$685$	0	0
$686$	−3.36957e6	−0.273379
$687$	−1.15516e7	−0.933788
$688$	1.34916e6	0.108666
$689$	9.70715e6	0.779011
$690$	0	0
$691$	3.42722e6	0.273053	0.136526	−	0.990636i	$-0.456406\pi$
$691$	3.42722e6	0.273053	0.136526	+	0.990636i	$0.456406\pi$
$692$	−1.31492e7	−1.04384
$693$	2.20301e6	0.174254
$694$	−2.61494e6	−0.206093
$695$	0	0
$696$	4.33925e6	0.339540
$697$	−7.84639e6	−0.611770
$698$	1.02373e6	0.0795331
$699$	2.81589e6	0.217983
$700$	0	0
$701$	1.05204e7	0.808608	0.404304	−	0.914625i	$-0.367514\pi$
$701$	1.05204e7	0.808608	0.404304	+	0.914625i	$0.367514\pi$
$702$	−688722.	−0.0527474
$703$	−1.31769e6	−0.100560
$704$	−3.39251e6	−0.257982
$705$	0	0
$706$	735650.	0.0555468
$707$	8.96995e6	0.674903
$708$	−2.86928e6	−0.215124
$709$	−5.27905e6	−0.394403	−0.197202	−	0.980363i	$-0.563185\pi$
$709$	−5.27905e6	−0.394403	−0.197202	+	0.980363i	$0.563185\pi$
$710$	0	0
$711$	−4.24512e6	−0.314931
$712$	−6.38008e6	−0.471657
$713$	−1.08849e6	−0.0801864
$714$	2.41715e6	0.177443
$715$	0	0
$716$	−1.03168e7	−0.752074
$717$	−3.21529e6	−0.233572
$718$	−3.19732e6	−0.231459
$719$	−5.38506e6	−0.388480	−0.194240	−	0.980954i	$-0.562224\pi$
$719$	−5.38506e6	−0.388480	−0.194240	+	0.980954i	$0.562224\pi$
$720$	0	0
$721$	1.55813e7	1.11626
$722$	1.77158e6	0.126479
$723$	−6.12151e6	−0.435525
$724$	−2.08761e7	−1.48014
$725$	0	0
$726$	−116821.	−0.00822583
$727$	−1.05731e6	−0.0741934	−0.0370967	−	0.999312i	$-0.511811\pi$
$727$	−1.05731e6	−0.0741934	−0.0370967	+	0.999312i	$0.511811\pi$
$728$	1.34238e7	0.938743
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.91572e6	−0.132598
$732$	9.62911e6	0.664215
$733$	1.02639e6	0.0705589	0.0352794	−	0.999377i	$-0.488768\pi$
$733$	1.02639e6	0.0705589	0.0352794	+	0.999377i	$0.488768\pi$
$734$	357544.	0.0244957
$735$	0	0
$736$	−8.94834e6	−0.608903
$737$	−8.28075e6	−0.561566
$738$	−418075.	−0.0282562
$739$	−3.33815e6	−0.224851	−0.112426	−	0.993660i	$-0.535862\pi$
$739$	−3.33815e6	−0.224851	−0.112426	+	0.993660i	$0.535862\pi$
$740$	0	0
$741$	6.62960e6	0.443550
$742$	−1.81525e6	−0.121039
$743$	−2.15478e7	−1.43196	−0.715981	−	0.698120i	$-0.754020\pi$
$743$	−2.15478e7	−1.43196	−0.715981	+	0.698120i	$0.754020\pi$
$744$	−161661.	−0.0107071
$745$	0	0
$746$	−3.08679e6	−0.203077
$747$	4.75193e6	0.311579
$748$	5.09030e6	0.332652
$749$	2.94787e7	1.92001
$750$	0	0
$751$	−2.02621e7	−1.31095	−0.655474	−	0.755218i	$-0.727531\pi$
$751$	−2.02621e7	−1.31095	−0.655474	+	0.755218i	$0.727531\pi$
$752$	−6.12638e6	−0.395057
$753$	1.42035e6	0.0912869
$754$	−8.12769e6	−0.520641
$755$	0	0
$756$	−5.11473e6	−0.325475
$757$	1.23455e7	0.783012	0.391506	−	0.920176i	$-0.371954\pi$
$757$	1.23455e7	0.783012	0.391506	+	0.920176i	$0.371954\pi$
$758$	624903.	0.0395039
$759$	3.69838e6	0.233028
$760$	0	0
$761$	9.77671e6	0.611971	0.305986	−	0.952036i	$-0.401014\pi$
$761$	9.77671e6	0.611971	0.305986	+	0.952036i	$0.401014\pi$
$762$	−1.37758e6	−0.0859465
$763$	3.30257e7	2.05372
$764$	3.00237e7	1.86093
$765$	0	0
$766$	1.91908e6	0.118174
$767$	1.08840e7	0.668038
$768$	7.20364e6	0.440706
$769$	−4.31458e6	−0.263101	−0.131551	−	0.991309i	$-0.541996\pi$
$769$	−4.31458e6	−0.263101	−0.131551	+	0.991309i	$0.541996\pi$
$770$	0	0
$771$	5.01567e6	0.303873
$772$	2.73030e6	0.164880
$773$	3.28419e7	1.97688	0.988439	−	0.151620i	$-0.0484490\pi$
$773$	3.28419e7	1.97688	0.988439	+	0.151620i	$0.0484490\pi$
$774$	−102074.	−0.00612440
$775$	0	0
$776$	−6.46338e6	−0.385306
$777$	−3.85623e6	−0.229145
$778$	−2.22304e6	−0.131674
$779$	4.02437e6	0.237604
$780$	0	0
$781$	227955.	0.0133727
$782$	4.05790e6	0.237292
$783$	6.27160e6	0.365573
$784$	3.20021e7	1.85947
$785$	0	0
$786$	−2.37554e6	−0.137153
$787$	1.45886e7	0.839608	0.419804	−	0.907615i	$-0.362099\pi$
$787$	1.45886e7	0.839608	0.419804	+	0.907615i	$0.362099\pi$
$788$	−2.49605e7	−1.43198
$789$	−5.81083e6	−0.332312
$790$	0	0
$791$	−1.28899e7	−0.732499
$792$	549277.	0.0311156
$793$	−3.65261e7	−2.06262
$794$	2.44560e6	0.137668
$795$	0	0
$796$	1.98370e7	1.10967
$797$	4.63451e6	0.258439	0.129220	−	0.991616i	$-0.458753\pi$
$797$	4.63451e6	0.258439	0.129220	+	0.991616i	$0.458753\pi$
$798$	−1.23974e6	−0.0689168
$799$	8.69905e6	0.482064
$800$	0	0
$801$	−9.22126e6	−0.507819
$802$	5.41903e6	0.297499
$803$	−1.06158e7	−0.580983
$804$	1.92255e7	1.04891
$805$	0	0
$806$	302801.	0.0164180
$807$	−1.97780e6	−0.106905
$808$	2.23648e6	0.120514
$809$	1.28933e6	0.0692618	0.0346309	−	0.999400i	$-0.488974\pi$
$809$	1.28933e6	0.0692618	0.0346309	+	0.999400i	$0.488974\pi$
$810$	0	0
$811$	6.47343e6	0.345607	0.172803	−	0.984956i	$-0.444718\pi$
$811$	6.47343e6	0.345607	0.172803	+	0.984956i	$0.444718\pi$
$812$	−6.03595e7	−3.21259
$813$	1.39313e7	0.739205
$814$	204488.	0.0108170
$815$	0	0
$816$	−1.15131e7	−0.605294
$817$	982561.	0.0514997
$818$	−4.40242e6	−0.230043
$819$	1.94017e7	1.01072
$820$	0	0
$821$	2.48796e7	1.28820	0.644102	−	0.764939i	$-0.277231\pi$
$821$	2.48796e7	1.28820	0.644102	+	0.764939i	$0.277231\pi$
$822$	363524.	0.0187652
$823$	2.73317e7	1.40659	0.703295	−	0.710898i	$-0.251711\pi$
$823$	2.73317e7	1.40659	0.703295	+	0.710898i	$0.251711\pi$
$824$	3.88490e6	0.199325
$825$	0	0
$826$	−2.03533e6	−0.103797
$827$	3.20369e7	1.62887	0.814437	−	0.580252i	$-0.197046\pi$
$827$	3.20369e7	1.62887	0.814437	+	0.580252i	$0.197046\pi$
$828$	−8.58655e6	−0.435254
$829$	−2.54756e7	−1.28747	−0.643736	−	0.765247i	$-0.722617\pi$
$829$	−2.54756e7	−1.28747	−0.643736	+	0.765247i	$0.722617\pi$
$830$	0	0
$831$	−8.82652e6	−0.443391
$832$	−2.98775e7	−1.49636
$833$	−4.54409e7	−2.26900
$834$	991841.	0.0493772
$835$	0	0
$836$	−2.61079e6	−0.129198
$837$	−233651.	−0.0115280
$838$	2.92979e6	0.144121
$839$	−2.56179e7	−1.25643	−0.628216	−	0.778039i	$-0.716215\pi$
$839$	−2.56179e7	−1.25643	−0.628216	+	0.778039i	$0.716215\pi$
$840$	0	0
$841$	5.35007e7	2.60837
$842$	−4.47338e6	−0.217448
$843$	2.21655e7	1.07426
$844$	1.09122e7	0.527300
$845$	0	0
$846$	463507.	0.0222654
$847$	3.29091e6	0.157619
$848$	8.64617e6	0.412890
$849$	−7.54749e6	−0.359363
$850$	0	0
$851$	−6.47381e6	−0.306433
$852$	−529243.	−0.0249779
$853$	8.49389e6	0.399700	0.199850	−	0.979827i	$-0.435955\pi$
$853$	8.49389e6	0.399700	0.199850	+	0.979827i	$0.435955\pi$
$854$	6.83042e6	0.320481
$855$	0	0
$856$	7.34996e6	0.342847
$857$	2.29100e7	1.06555	0.532773	−	0.846258i	$-0.321150\pi$
$857$	2.29100e7	1.06555	0.532773	+	0.846258i	$0.321150\pi$
$858$	−1.02883e6	−0.0477118
$859$	1.05172e7	0.486315	0.243158	−	0.969987i	$-0.421817\pi$
$859$	1.05172e7	0.486315	0.243158	+	0.969987i	$0.421817\pi$
$860$	0	0
$861$	1.17774e7	0.541429
$862$	−788153.	−0.0361279
$863$	2.93900e7	1.34330	0.671649	−	0.740870i	$-0.265587\pi$
$863$	2.93900e7	1.34330	0.671649	+	0.740870i	$0.265587\pi$
$864$	−1.92082e6	−0.0875390
$865$	0	0
$866$	−360499.	−0.0163346
$867$	3.56912e6	0.161255
$868$	2.24872e6	0.101306
$869$	−6.34147e6	−0.284866
$870$	0	0
$871$	−7.29278e7	−3.25722
$872$	8.23432e6	0.366722
$873$	−9.34165e6	−0.414847
$874$	−2.08127e6	−0.0921616
$875$	0	0
$876$	2.46467e7	1.08517
$877$	−3.81404e7	−1.67451	−0.837253	−	0.546816i	$-0.815840\pi$
$877$	−3.81404e7	−1.67451	−0.837253	+	0.546816i	$0.815840\pi$
$878$	4.47533e6	0.195925
$879$	1.19420e7	0.521321
$880$	0	0
$881$	2.39208e7	1.03833	0.519165	−	0.854674i	$-0.326243\pi$
$881$	2.39208e7	1.03833	0.519165	+	0.854674i	$0.326243\pi$
$882$	−2.42120e6	−0.104800
$883$	−1.95650e6	−0.0844457	−0.0422229	−	0.999108i	$-0.513444\pi$
$883$	−1.95650e6	−0.0844457	−0.0422229	+	0.999108i	$0.513444\pi$
$884$	4.48298e7	1.92946
$885$	0	0
$886$	787493.	0.0337025
$887$	−2.80192e7	−1.19577	−0.597884	−	0.801583i	$-0.703992\pi$
$887$	−2.80192e7	−1.19577	−0.597884	+	0.801583i	$0.703992\pi$
$888$	−961478.	−0.0409173
$889$	3.88070e7	1.64686
$890$	0	0
$891$	793881.	0.0335013
$892$	9.66194e6	0.406586
$893$	−4.46169e6	−0.187228
$894$	858369.	0.0359195
$895$	0	0
$896$	2.45391e7	1.02115
$897$	3.25713e7	1.35162
$898$	−3.38870e6	−0.140230
$899$	−2.75735e6	−0.113787
$900$	0	0
$901$	−1.22770e7	−0.503825
$902$	−624532.	−0.0255587
$903$	2.87548e6	0.117352
$904$	−3.21384e6	−0.130799
$905$	0	0
$906$	−1.17701e6	−0.0476387
$907$	−5.99454e6	−0.241957	−0.120978	−	0.992655i	$-0.538603\pi$
$907$	−5.99454e6	−0.241957	−0.120978	+	0.992655i	$0.538603\pi$
$908$	1.59108e7	0.640439
$909$	3.23243e6	0.129754
$910$	0	0
$911$	−3.83047e7	−1.52917	−0.764585	−	0.644523i	$-0.777056\pi$
$911$	−3.83047e7	−1.52917	−0.764585	+	0.644523i	$0.777056\pi$
$912$	5.90500e6	0.235089
$913$	7.09856e6	0.281834
$914$	−1.86710e6	−0.0739267
$915$	0	0
$916$	4.00634e7	1.57764
$917$	6.69202e7	2.62805
$918$	871052.	0.0341144
$919$	−4.41558e7	−1.72464	−0.862321	−	0.506362i	$-0.830990\pi$
$919$	−4.41558e7	−1.72464	−0.862321	+	0.506362i	$0.830990\pi$
$920$	0	0
$921$	9.43580e6	0.366547
$922$	2.15626e6	0.0835360
$923$	2.00757e6	0.0775652
$924$	−7.64052e6	−0.294403
$925$	0	0
$926$	−6.68028e6	−0.256016
$927$	5.61492e6	0.214607
$928$	−2.26678e7	−0.864050
$929$	1.00353e7	0.381499	0.190749	−	0.981639i	$-0.438908\pi$
$929$	1.00353e7	0.381499	0.190749	+	0.981639i	$0.438908\pi$
$930$	0	0
$931$	2.33064e7	0.881253
$932$	−9.76613e6	−0.368284
$933$	−1.38407e7	−0.520540
$934$	2.81294e6	0.105510
$935$	0	0
$936$	4.83743e6	0.180478
$937$	3.04913e7	1.13456	0.567279	−	0.823526i	$-0.307996\pi$
$937$	3.04913e7	1.13456	0.567279	+	0.823526i	$0.307996\pi$
$938$	1.36376e7	0.506093
$939$	−2.32626e7	−0.860981
$940$	0	0
$941$	−4.13865e7	−1.52365	−0.761824	−	0.647785i	$-0.775696\pi$
$941$	−4.13865e7	−1.52365	−0.761824	+	0.647785i	$0.775696\pi$
$942$	−1.85034e6	−0.0679399
$943$	1.97718e7	0.724046
$944$	9.69442e6	0.354072
$945$	0	0
$946$	−152481.	−0.00553973
$947$	−9.38834e6	−0.340184	−0.170092	−	0.985428i	$-0.554406\pi$
$947$	−9.38834e6	−0.340184	−0.170092	+	0.985428i	$0.554406\pi$
$948$	1.47230e7	0.532079
$949$	−9.34922e7	−3.36984
$950$	0	0
$951$	−1.16232e7	−0.416747
$952$	−1.69776e7	−0.607132
$953$	3.35369e7	1.19616	0.598081	−	0.801435i	$-0.295930\pi$
$953$	3.35369e7	1.19616	0.598081	+	0.801435i	$0.295930\pi$
$954$	−654148.	−0.0232705
$955$	0	0
$956$	1.11513e7	0.394623
$957$	9.36868e6	0.330673
$958$	−2.83948e6	−0.0999597
$959$	−1.02406e7	−0.359568
$960$	0	0
$961$	−2.85264e7	−0.996412
$962$	1.80091e6	0.0627414
$963$	1.06230e7	0.369133
$964$	2.12308e7	0.735823
$965$	0	0
$966$	−6.09088e6	−0.210009
$967$	−5.35398e7	−1.84124	−0.920620	−	0.390459i	$-0.872316\pi$
$967$	−5.35398e7	−1.84124	−0.920620	+	0.390459i	$0.872316\pi$
$968$	820525.	0.0281451
$969$	−8.38471e6	−0.286866
$970$	0	0
$971$	−3.17286e7	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$971$	−3.17286e7	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$972$	−1.84316e6	−0.0625744
$973$	−2.79407e7	−0.946139
$974$	3.94999e6	0.133413
$975$	0	0
$976$	−3.25338e7	−1.09323
$977$	5.28990e6	0.177301	0.0886505	−	0.996063i	$-0.471745\pi$
$977$	5.28990e6	0.177301	0.0886505	+	0.996063i	$0.471745\pi$
$978$	54754.9	0.00183053
$979$	−1.37750e7	−0.459340
$980$	0	0
$981$	1.19012e7	0.394838
$982$	−133832.	−0.00442877
$983$	−2.55796e7	−0.844327	−0.422164	−	0.906520i	$-0.638729\pi$
$983$	−2.55796e7	−0.844327	−0.422164	+	0.906520i	$0.638729\pi$
$984$	2.93647e6	0.0966801
$985$	0	0
$986$	1.02794e7	0.336725
$987$	−1.30572e7	−0.426637
$988$	−2.29930e7	−0.749380
$989$	4.82733e6	0.156934
$990$	0	0
$991$	4.14167e7	1.33965	0.669825	−	0.742519i	$-0.266369\pi$
$991$	4.14167e7	1.33965	0.669825	+	0.742519i	$0.266369\pi$
$992$	844498.	0.0272471
$993$	1.59945e7	0.514750
$994$	−375419.	−0.0120517
$995$	0	0
$996$	−1.64808e7	−0.526415
$997$	3.02210e7	0.962877	0.481438	−	0.876480i	$-0.340115\pi$
$997$	3.02210e7	0.962877	0.481438	+	0.876480i	$0.340115\pi$
$998$	4.04445e6	0.128538
$999$	−1.38964e6	−0.0440544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.y.1.6		13
5.2	odd	4		165.6.c.b.34.13	✓	26
5.3	odd	4		165.6.c.b.34.14	yes	26
5.4	even	2		825.6.a.v.1.8		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.13	✓	26	5.2	odd	4
165.6.c.b.34.14	yes	26	5.3	odd	4
825.6.a.v.1.8		13	5.4	even	2
825.6.a.y.1.6		13	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-0.886559 q^{2} +9.00000 q^{3} -31.2140 q^{4} -7.97903 q^{6} +224.774 q^{7} +56.0430 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-0.886559 q^{2} +9.00000 q^{3} -31.2140 q^{4} -7.97903 q^{6} +224.774 q^{7} +56.0430 q^{8} +81.0000 q^{9} +121.000 q^{11} -280.926 q^{12} +1065.64 q^{13} -199.275 q^{14} +949.163 q^{16} -1347.75 q^{17} -71.8113 q^{18} +691.252 q^{19} +2022.96 q^{21} -107.274 q^{22} +3396.13 q^{23} +504.387 q^{24} -944.749 q^{26} +729.000 q^{27} -7016.09 q^{28} +8603.02 q^{29} -320.509 q^{31} -2634.86 q^{32} +1089.00 q^{33} +1194.86 q^{34} -2528.34 q^{36} -1906.23 q^{37} -612.836 q^{38} +9590.72 q^{39} +5821.85 q^{41} -1793.48 q^{42} +1421.42 q^{43} -3776.90 q^{44} -3010.87 q^{46} -6454.51 q^{47} +8542.47 q^{48} +33716.2 q^{49} -12129.7 q^{51} -33262.8 q^{52} +9109.26 q^{53} -646.302 q^{54} +12597.0 q^{56} +6221.27 q^{57} -7627.08 q^{58} +10213.6 q^{59} -34276.3 q^{61} +284.150 q^{62} +18206.7 q^{63} -28037.3 q^{64} -965.463 q^{66} -68436.0 q^{67} +42068.6 q^{68} +30565.2 q^{69} +1883.92 q^{71} +4539.48 q^{72} -87733.7 q^{73} +1689.99 q^{74} -21576.8 q^{76} +27197.6 q^{77} -8502.74 q^{78} -52408.8 q^{79} +6561.00 q^{81} -5161.42 q^{82} +58665.8 q^{83} -63144.8 q^{84} -1260.17 q^{86} +77427.1 q^{87} +6781.20 q^{88} -113843. q^{89} +239527. q^{91} -106007. q^{92} -2884.58 q^{93} +5722.30 q^{94} -23713.8 q^{96} -115329. q^{97} -29891.4 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 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