LMFDB - Embedded newform 825.6.a.u.1.10

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + \cdots - 7036608 $ x^10 - x^9 - 271x^8 + 309x^7 + 24456x^6 - 33410x^5 - 822204x^4 + 1367872x^3 + 7443872x^2 - 12856224x - 7036608
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$11.0151$ 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)

$f(q)$	$=$	$q+11.0151 q^{2} -9.00000 q^{3} +89.3332 q^{4} -99.1362 q^{6} +62.7690 q^{7} +631.533 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+11.0151 q^{2} -9.00000 q^{3} +89.3332 q^{4} -99.1362 q^{6} +62.7690 q^{7} +631.533 q^{8} +81.0000 q^{9} +121.000 q^{11} -803.999 q^{12} +986.561 q^{13} +691.409 q^{14} +4097.76 q^{16} +577.936 q^{17} +892.226 q^{18} -234.681 q^{19} -564.921 q^{21} +1332.83 q^{22} -2024.23 q^{23} -5683.79 q^{24} +10867.1 q^{26} -729.000 q^{27} +5607.35 q^{28} -6877.68 q^{29} -689.224 q^{31} +24928.3 q^{32} -1089.00 q^{33} +6366.04 q^{34} +7235.99 q^{36} +12437.0 q^{37} -2585.04 q^{38} -8879.05 q^{39} -5372.90 q^{41} -6222.68 q^{42} +13722.5 q^{43} +10809.3 q^{44} -22297.2 q^{46} -22652.2 q^{47} -36879.8 q^{48} -12867.1 q^{49} -5201.42 q^{51} +88132.7 q^{52} -5321.80 q^{53} -8030.03 q^{54} +39640.7 q^{56} +2112.12 q^{57} -75758.6 q^{58} +23792.4 q^{59} +45926.5 q^{61} -7591.90 q^{62} +5084.29 q^{63} +143460. q^{64} -11995.5 q^{66} +50509.5 q^{67} +51628.9 q^{68} +18218.1 q^{69} -29316.4 q^{71} +51154.1 q^{72} -42433.0 q^{73} +136995. q^{74} -20964.8 q^{76} +7595.05 q^{77} -97804.0 q^{78} +63806.8 q^{79} +6561.00 q^{81} -59183.2 q^{82} -57952.6 q^{83} -50466.2 q^{84} +151155. q^{86} +61899.1 q^{87} +76415.5 q^{88} +75013.4 q^{89} +61925.5 q^{91} -180831. q^{92} +6203.02 q^{93} -249517. q^{94} -224355. q^{96} -10869.1 q^{97} -141732. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) $ 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9 + 1210 * q^11 - 2007 * q^12 + 1102 * q^13 + 684 * q^14 + 7019 * q^16 - 1106 * q^17 + 81 * q^18 + 2586 * q^19 + 1692 * q^21 + 121 * q^22 - 2206 * q^23 + 1593 * q^24 + 12001 * q^26 - 7290 * q^27 - 14452 * q^28 + 5824 * q^29 - 4586 * q^31 + 10627 * q^32 - 10890 * q^33 + 7426 * q^34 + 18063 * q^36 + 18362 * q^37 - 44001 * q^38 - 9918 * q^39 - 5474 * q^41 - 6156 * q^42 - 20496 * q^43 + 26983 * q^44 + 19981 * q^46 + 14970 * q^47 - 63171 * q^48 + 68582 * q^49 + 9954 * q^51 + 58603 * q^52 - 61980 * q^53 - 729 * q^54 + 7132 * q^56 - 23274 * q^57 - 7161 * q^58 + 61190 * q^59 + 8230 * q^61 + 4509 * q^62 - 15228 * q^63 + 152223 * q^64 - 1089 * q^66 + 11930 * q^67 - 105598 * q^68 + 19854 * q^69 + 59822 * q^71 - 14337 * q^72 + 20680 * q^73 + 132564 * q^74 + 68165 * q^76 - 22748 * q^77 - 108009 * q^78 + 234494 * q^79 + 65610 * q^81 - 151948 * q^82 - 185478 * q^83 + 130068 * q^84 - 17825 * q^86 - 52416 * q^87 - 21417 * q^88 + 181834 * q^89 + 206274 * q^91 - 98373 * q^92 + 41274 * q^93 - 64998 * q^94 - 95643 * q^96 - 304358 * q^97 - 153453 * q^98 + 98010 * 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 $

$2$	11.0151	1.94722	0.973610	−	0.228220i	$-0.0732906\pi$
$2$	11.0151	1.94722	0.973610	+	0.228220i	$0.0732906\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	89.3332	2.79166
$5$	0	0
$5$	0	0
$6$	−99.1362	−1.12423
$7$	62.7690	0.484173	0.242086	−	0.970255i	$-0.422168\pi$
$7$	62.7690	0.484173	0.242086	+	0.970255i	$0.422168\pi$
$8$	631.533	3.48876
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−803.999	−1.61177
$13$	986.561	1.61907	0.809535	−	0.587071i	$-0.199719\pi$
$13$	986.561	1.61907	0.809535	+	0.587071i	$0.199719\pi$
$14$	691.409	0.942790
$15$	0	0
$16$	4097.76	4.00171
$17$	577.936	0.485018	0.242509	−	0.970149i	$-0.422030\pi$
$17$	577.936	0.485018	0.242509	+	0.970149i	$0.422030\pi$
$18$	892.226	0.649073
$19$	−234.681	−0.149140	−0.0745698	−	0.997216i	$-0.523758\pi$
$19$	−234.681	−0.149140	−0.0745698	+	0.997216i	$0.523758\pi$
$20$	0	0
$21$	−564.921	−0.279537
$22$	1332.83	0.587109
$23$	−2024.23	−0.797885	−0.398943	−	0.916976i	$-0.630623\pi$
$23$	−2024.23	−0.797885	−0.398943	+	0.916976i	$0.630623\pi$
$24$	−5683.79	−2.01424
$25$	0	0
$26$	10867.1	3.15268
$27$	−729.000	−0.192450
$28$	5607.35	1.35165
$29$	−6877.68	−1.51861	−0.759306	−	0.650734i	$-0.774461\pi$
$29$	−6877.68	−1.51861	−0.759306	+	0.650734i	$0.774461\pi$
$30$	0	0
$31$	−689.224	−0.128812	−0.0644060	−	0.997924i	$-0.520515\pi$
$31$	−689.224	−0.128812	−0.0644060	+	0.997924i	$0.520515\pi$
$32$	24928.3	4.30346
$33$	−1089.00	−0.174078
$34$	6366.04	0.944436
$35$	0	0
$36$	7235.99	0.930554
$37$	12437.0	1.49352	0.746761	−	0.665092i	$-0.231608\pi$
$37$	12437.0	1.49352	0.746761	+	0.665092i	$0.231608\pi$
$38$	−2585.04	−0.290408
$39$	−8879.05	−0.934771
$40$	0	0
$41$	−5372.90	−0.499170	−0.249585	−	0.968353i	$-0.580294\pi$
$41$	−5372.90	−0.499170	−0.249585	+	0.968353i	$0.580294\pi$
$42$	−6222.68	−0.544320
$43$	13722.5	1.13178	0.565889	−	0.824482i	$-0.308533\pi$
$43$	13722.5	1.13178	0.565889	+	0.824482i	$0.308533\pi$
$44$	10809.3	0.841718
$45$	0	0
$46$	−22297.2	−1.55366
$47$	−22652.2	−1.49577	−0.747887	−	0.663826i	$-0.768932\pi$
$47$	−22652.2	−1.49577	−0.747887	+	0.663826i	$0.768932\pi$
$48$	−36879.8	−2.31039
$49$	−12867.1	−0.765577
$50$	0	0
$51$	−5201.42	−0.280025
$52$	88132.7	4.51990
$53$	−5321.80	−0.260237	−0.130118	−	0.991498i	$-0.541536\pi$
$53$	−5321.80	−0.260237	−0.130118	+	0.991498i	$0.541536\pi$
$54$	−8030.03	−0.374742
$55$	0	0
$56$	39640.7	1.68916
$57$	2112.12	0.0861058
$58$	−75758.6	−2.95707
$59$	23792.4	0.889833	0.444916	−	0.895572i	$-0.353233\pi$
$59$	23792.4	0.889833	0.444916	+	0.895572i	$0.353233\pi$
$60$	0	0
$61$	45926.5	1.58030	0.790149	−	0.612915i	$-0.210003\pi$
$61$	45926.5	1.58030	0.790149	+	0.612915i	$0.210003\pi$
$62$	−7591.90	−0.250825
$63$	5084.29	0.161391
$64$	143460.	4.37806
$65$	0	0
$66$	−11995.5	−0.338967
$67$	50509.5	1.37463	0.687315	−	0.726359i	$-0.258789\pi$
$67$	50509.5	1.37463	0.687315	+	0.726359i	$0.258789\pi$
$68$	51628.9	1.35401
$69$	18218.1	0.460659
$70$	0	0
$71$	−29316.4	−0.690184	−0.345092	−	0.938569i	$-0.612152\pi$
$71$	−29316.4	−0.690184	−0.345092	+	0.938569i	$0.612152\pi$
$72$	51154.1	1.16292
$73$	−42433.0	−0.931959	−0.465979	−	0.884796i	$-0.654298\pi$
$73$	−42433.0	−0.931959	−0.465979	+	0.884796i	$0.654298\pi$
$74$	136995.	2.90822
$75$	0	0
$76$	−20964.8	−0.416347
$77$	7595.05	0.145984
$78$	−97804.0	−1.82020
$79$	63806.8	1.15027	0.575134	−	0.818059i	$-0.304950\pi$
$79$	63806.8	1.15027	0.575134	+	0.818059i	$0.304950\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−59183.2	−0.971994
$83$	−57952.6	−0.923374	−0.461687	−	0.887043i	$-0.652756\pi$
$83$	−57952.6	−0.923374	−0.461687	+	0.887043i	$0.652756\pi$
$84$	−50466.2	−0.780373
$85$	0	0
$86$	151155.	2.20382
$87$	61899.1	0.876771
$88$	76415.5	1.05190
$89$	75013.4	1.00384	0.501919	−	0.864915i	$-0.332628\pi$
$89$	75013.4	1.00384	0.501919	+	0.864915i	$0.332628\pi$
$90$	0	0
$91$	61925.5	0.783909
$92$	−180831.	−2.22743
$93$	6203.02	0.0743696
$94$	−249517.	−2.91260
$95$	0	0
$96$	−224355.	−2.48460
$97$	−10869.1	−0.117291	−0.0586454	−	0.998279i	$-0.518678\pi$
$97$	−10869.1	−0.117291	−0.0586454	+	0.998279i	$0.518678\pi$
$98$	−141732.	−1.49075
$99$	9801.00	0.100504
$100$	0	0
$101$	112358.	1.09597	0.547987	−	0.836487i	$-0.315394\pi$
$101$	112358.	1.09597	0.547987	+	0.836487i	$0.315394\pi$
$102$	−57294.4	−0.545270
$103$	63393.0	0.588774	0.294387	−	0.955686i	$-0.404885\pi$
$103$	63393.0	0.588774	0.294387	+	0.955686i	$0.404885\pi$
$104$	623046.	5.64855
$105$	0	0
$106$	−58620.3	−0.506738
$107$	163095.	1.37715	0.688574	−	0.725167i	$-0.258237\pi$
$107$	163095.	1.37715	0.688574	+	0.725167i	$0.258237\pi$
$108$	−65123.9	−0.537256
$109$	−177552.	−1.43139	−0.715697	−	0.698411i	$-0.753891\pi$
$109$	−177552.	−1.43139	−0.715697	+	0.698411i	$0.753891\pi$
$110$	0	0
$111$	−111933.	−0.862286
$112$	257212.	1.93752
$113$	−3009.41	−0.0221710	−0.0110855	−	0.999939i	$-0.503529\pi$
$113$	−3009.41	−0.0221710	−0.0110855	+	0.999939i	$0.503529\pi$
$114$	23265.3	0.167667
$115$	0	0
$116$	−614405.	−4.23945
$117$	79911.5	0.539690
$118$	262077.	1.73270
$119$	36276.5	0.234832
$120$	0	0
$121$	14641.0	0.0909091
$122$	505887.	3.07719
$123$	48356.1	0.288196
$124$	−61570.6	−0.359599
$125$	0	0
$126$	56004.1	0.314263
$127$	151116.	0.831382	0.415691	−	0.909506i	$-0.363540\pi$
$127$	151116.	0.831382	0.415691	+	0.909506i	$0.363540\pi$
$128$	782528.	4.22158
$129$	−123502.	−0.653432
$130$	0	0
$131$	−45401.3	−0.231148	−0.115574	−	0.993299i	$-0.536871\pi$
$131$	−45401.3	−0.231148	−0.115574	+	0.993299i	$0.536871\pi$
$132$	−97283.8	−0.485966
$133$	−14730.7	−0.0722093
$134$	556369.	2.67671
$135$	0	0
$136$	364986.	1.69211
$137$	−297728.	−1.35525	−0.677624	−	0.735408i	$-0.736990\pi$
$137$	−297728.	−1.35525	−0.677624	+	0.735408i	$0.736990\pi$
$138$	200675.	0.897005
$139$	−214343.	−0.940962	−0.470481	−	0.882410i	$-0.655920\pi$
$139$	−214343.	−0.940962	−0.470481	+	0.882410i	$0.655920\pi$
$140$	0	0
$141$	203870.	0.863586
$142$	−322924.	−1.34394
$143$	119374.	0.488168
$144$	331918.	1.33390
$145$	0	0
$146$	−467405.	−1.81473
$147$	115803.	0.442006
$148$	1.11104e6	4.16941
$149$	234432.	0.865071	0.432536	−	0.901617i	$-0.357619\pi$
$149$	234432.	0.865071	0.432536	+	0.901617i	$0.357619\pi$
$150$	0	0
$151$	60872.6	0.217260	0.108630	−	0.994082i	$-0.465354\pi$
$151$	60872.6	0.217260	0.108630	+	0.994082i	$0.465354\pi$
$152$	−148208.	−0.520312
$153$	46812.8	0.161673
$154$	83660.5	0.284262
$155$	0	0
$156$	−793194.	−2.60956
$157$	112122.	0.363029	0.181514	−	0.983388i	$-0.441900\pi$
$157$	112122.	0.363029	0.181514	+	0.983388i	$0.441900\pi$
$158$	702840.	2.23982
$159$	47896.2	0.150248
$160$	0	0
$161$	−127059.	−0.386314
$162$	72270.3	0.216358
$163$	364063.	1.07327	0.536634	−	0.843815i	$-0.319696\pi$
$163$	364063.	1.07327	0.536634	+	0.843815i	$0.319696\pi$
$164$	−479978.	−1.39351
$165$	0	0
$166$	−638356.	−1.79801
$167$	−599451.	−1.66327	−0.831635	−	0.555323i	$-0.812595\pi$
$167$	−599451.	−1.66327	−0.831635	+	0.555323i	$0.812595\pi$
$168$	−356766.	−0.975237
$169$	602011.	1.62139
$170$	0	0
$171$	−19009.1	−0.0497132
$172$	1.22587e6	3.15954
$173$	−546315.	−1.38780	−0.693902	−	0.720070i	$-0.744110\pi$
$173$	−546315.	−1.38780	−0.693902	+	0.720070i	$0.744110\pi$
$174$	681827.	1.70727
$175$	0	0
$176$	495828.	1.20656
$177$	−214132.	−0.513745
$178$	826282.	1.95469
$179$	−482038.	−1.12447	−0.562236	−	0.826977i	$-0.690059\pi$
$179$	−482038.	−1.12447	−0.562236	+	0.826977i	$0.690059\pi$
$180$	0	0
$181$	−481935.	−1.09343	−0.546717	−	0.837318i	$-0.684123\pi$
$181$	−481935.	−1.09343	−0.546717	+	0.837318i	$0.684123\pi$
$182$	682117.	1.52644
$183$	−413339.	−0.912385
$184$	−1.27837e6	−2.78363
$185$	0	0
$186$	68327.1	0.144814
$187$	69930.3	0.146238
$188$	−2.02359e6	−4.17570
$189$	−45758.6	−0.0931790
$190$	0	0
$191$	−300485.	−0.595990	−0.297995	−	0.954567i	$-0.596318\pi$
$191$	−300485.	−0.595990	−0.297995	+	0.954567i	$0.596318\pi$
$192$	−1.29114e6	−2.52767
$193$	−860084.	−1.66206	−0.831032	−	0.556225i	$-0.812249\pi$
$193$	−860084.	−1.66206	−0.831032	+	0.556225i	$0.812249\pi$
$194$	−119724.	−0.228391
$195$	0	0
$196$	−1.14945e6	−2.13723
$197$	−623099.	−1.14391	−0.571954	−	0.820286i	$-0.693814\pi$
$197$	−623099.	−1.14391	−0.571954	+	0.820286i	$0.693814\pi$
$198$	107959.	0.195703
$199$	−417370.	−0.747116	−0.373558	−	0.927607i	$-0.621862\pi$
$199$	−417370.	−0.747116	−0.373558	+	0.927607i	$0.621862\pi$
$200$	0	0
$201$	−454585.	−0.793643
$202$	1.23764e6	2.13410
$203$	−431705.	−0.735270
$204$	−464660.	−0.781735
$205$	0	0
$206$	698283.	1.14647
$207$	−163963.	−0.265962
$208$	4.04269e6	6.47906
$209$	−28396.3	−0.0449673
$210$	0	0
$211$	1.24795e6	1.92970	0.964852	−	0.262795i	$-0.0846441\pi$
$211$	1.24795e6	1.92970	0.964852	+	0.262795i	$0.0846441\pi$
$212$	−475413.	−0.726493
$213$	263848.	0.398478
$214$	1.79651e6	2.68161
$215$	0	0
$216$	−460387.	−0.671412
$217$	−43261.9	−0.0623672
$218$	−1.95576e6	−2.78724
$219$	381897.	0.538067
$220$	0	0
$221$	570169.	0.785278
$222$	−1.23296e6	−1.67906
$223$	−399761.	−0.538317	−0.269159	−	0.963096i	$-0.586746\pi$
$223$	−399761.	−0.538317	−0.269159	+	0.963096i	$0.586746\pi$
$224$	1.56472e6	2.08362
$225$	0	0
$226$	−33149.1	−0.0431718
$227$	1.14478e6	1.47454	0.737269	−	0.675599i	$-0.236115\pi$
$227$	1.14478e6	1.47454	0.737269	+	0.675599i	$0.236115\pi$
$228$	188683.	0.240378
$229$	24928.6	0.0314130	0.0157065	−	0.999877i	$-0.495000\pi$
$229$	24928.6	0.0314130	0.0157065	+	0.999877i	$0.495000\pi$
$230$	0	0
$231$	−68355.4	−0.0842836
$232$	−4.34348e6	−5.29807
$233$	−626233.	−0.755694	−0.377847	−	0.925868i	$-0.623336\pi$
$233$	−626233.	−0.755694	−0.377847	+	0.925868i	$0.623336\pi$
$234$	880236.	1.05089
$235$	0	0
$236$	2.12545e6	2.48411
$237$	−574261.	−0.664108
$238$	399590.	0.457270
$239$	−1.05033e6	−1.18941	−0.594704	−	0.803945i	$-0.702731\pi$
$239$	−1.05033e6	−1.18941	−0.594704	+	0.803945i	$0.702731\pi$
$240$	0	0
$241$	251724.	0.279178	0.139589	−	0.990210i	$-0.455422\pi$
$241$	251724.	0.279178	0.139589	+	0.990210i	$0.455422\pi$
$242$	161273.	0.177020
$243$	−59049.0	−0.0641500
$244$	4.10276e6	4.41166
$245$	0	0
$246$	532649.	0.561181
$247$	−231527.	−0.241468
$248$	−435268.	−0.449394
$249$	521574.	0.533111
$250$	0	0
$251$	926617.	0.928359	0.464180	−	0.885741i	$-0.346349\pi$
$251$	926617.	0.928359	0.464180	+	0.885741i	$0.346349\pi$
$252$	454196.	0.450549
$253$	−244932.	−0.240571
$254$	1.66456e6	1.61888
$255$	0	0
$256$	4.02893e6	3.84228
$257$	1.37226e6	1.29600	0.647998	−	0.761642i	$-0.275607\pi$
$257$	1.37226e6	1.29600	0.647998	+	0.761642i	$0.275607\pi$
$258$	−1.36039e6	−1.27238
$259$	780659.	0.723123
$260$	0	0
$261$	−557092.	−0.506204
$262$	−500101.	−0.450095
$263$	1.81605e6	1.61896	0.809482	−	0.587144i	$-0.199748\pi$
$263$	1.81605e6	1.61896	0.809482	+	0.587144i	$0.199748\pi$
$264$	−687739.	−0.607315
$265$	0	0
$266$	−162260.	−0.140607
$267$	−675120.	−0.579566
$268$	4.51217e6	3.83750
$269$	1.27139e6	1.07127	0.535634	−	0.844450i	$-0.320073\pi$
$269$	1.27139e6	1.07127	0.535634	+	0.844450i	$0.320073\pi$
$270$	0	0
$271$	1.44540e6	1.19554	0.597769	−	0.801668i	$-0.296054\pi$
$271$	1.44540e6	1.19554	0.597769	+	0.801668i	$0.296054\pi$
$272$	2.36824e6	1.94090
$273$	−557329.	−0.452590
$274$	−3.27952e6	−2.63897
$275$	0	0
$276$	1.62748e6	1.28601
$277$	1.74706e6	1.36807	0.684034	−	0.729450i	$-0.260224\pi$
$277$	1.74706e6	1.36807	0.684034	+	0.729450i	$0.260224\pi$
$278$	−2.36102e6	−1.83226
$279$	−55827.1	−0.0429373
$280$	0	0
$281$	−1.84101e6	−1.39088	−0.695441	−	0.718583i	$-0.744791\pi$
$281$	−1.84101e6	−1.39088	−0.695441	+	0.718583i	$0.744791\pi$
$282$	2.24565e6	1.68159
$283$	−986043.	−0.731863	−0.365931	−	0.930642i	$-0.619249\pi$
$283$	−986043.	−0.731863	−0.365931	+	0.930642i	$0.619249\pi$
$284$	−2.61893e6	−1.92676
$285$	0	0
$286$	1.31492e6	0.950570
$287$	−337251.	−0.241685
$288$	2.01919e6	1.43449
$289$	−1.08585e6	−0.764758
$290$	0	0
$291$	97821.8	0.0677179
$292$	−3.79068e6	−2.60171
$293$	−1.98564e6	−1.35124	−0.675618	−	0.737252i	$-0.736123\pi$
$293$	−1.98564e6	−1.35124	−0.675618	+	0.737252i	$0.736123\pi$
$294$	1.27559e6	0.860683
$295$	0	0
$296$	7.85438e6	5.21054
$297$	−88209.0	−0.0580259
$298$	2.58230e6	1.68448
$299$	−1.99703e6	−1.29183
$300$	0	0
$301$	861346.	0.547976
$302$	670520.	0.423052
$303$	−1.01122e6	−0.632761
$304$	−961663.	−0.596814
$305$	0	0
$306$	515650.	0.314812
$307$	−1.13983e6	−0.690232	−0.345116	−	0.938560i	$-0.612160\pi$
$307$	−1.13983e6	−0.690232	−0.345116	+	0.938560i	$0.612160\pi$
$308$	678490.	0.407537
$309$	−570537.	−0.339929
$310$	0	0
$311$	−131904.	−0.0773319	−0.0386659	−	0.999252i	$-0.512311\pi$
$311$	−131904.	−0.0773319	−0.0386659	+	0.999252i	$0.512311\pi$
$312$	−5.60741e6	−3.26119
$313$	−2.07325e6	−1.19616	−0.598082	−	0.801435i	$-0.704070\pi$
$313$	−2.07325e6	−1.19616	−0.598082	+	0.801435i	$0.704070\pi$
$314$	1.23504e6	0.706896
$315$	0	0
$316$	5.70006e6	3.21116
$317$	95123.6	0.0531667	0.0265834	−	0.999647i	$-0.491537\pi$
$317$	95123.6	0.0531667	0.0265834	+	0.999647i	$0.491537\pi$
$318$	527583.	0.292565
$319$	−832199.	−0.457879
$320$	0	0
$321$	−1.46785e6	−0.795096
$322$	−1.39957e6	−0.752238
$323$	−135630.	−0.0723353
$324$	586115.	0.310185
$325$	0	0
$326$	4.01021e6	2.08989
$327$	1.59797e6	0.826415
$328$	−3.39316e6	−1.74148
$329$	−1.42186e6	−0.724213
$330$	0	0
$331$	−3.20153e6	−1.60616	−0.803078	−	0.595874i	$-0.796806\pi$
$331$	−3.20153e6	−1.60616	−0.803078	+	0.595874i	$0.796806\pi$
$332$	−5.17709e6	−2.57775
$333$	1.00740e6	0.497841
$334$	−6.60304e6	−3.23875
$335$	0	0
$336$	−2.31491e6	−1.11863
$337$	−2.16577e6	−1.03881	−0.519406	−	0.854527i	$-0.673847\pi$
$337$	−2.16577e6	−1.03881	−0.519406	+	0.854527i	$0.673847\pi$
$338$	6.63123e6	3.15720
$339$	27084.7	0.0128004
$340$	0	0
$341$	−83396.1	−0.0388383
$342$	−209388.	−0.0968025
$343$	−1.86261e6	−0.854844
$344$	8.66619e6	3.94850
$345$	0	0
$346$	−6.01773e6	−2.70236
$347$	2.76730e6	1.23377	0.616883	−	0.787055i	$-0.288395\pi$
$347$	2.76730e6	1.23377	0.616883	+	0.787055i	$0.288395\pi$
$348$	5.52964e6	2.44765
$349$	3.44895e6	1.51574	0.757868	−	0.652408i	$-0.226241\pi$
$349$	3.44895e6	1.51574	0.757868	+	0.652408i	$0.226241\pi$
$350$	0	0
$351$	−719203.	−0.311590
$352$	3.01632e6	1.29754
$353$	−4.34110e6	−1.85423	−0.927113	−	0.374781i	$-0.877718\pi$
$353$	−4.34110e6	−1.85423	−0.927113	+	0.374781i	$0.877718\pi$
$354$	−2.35869e6	−1.00037
$355$	0	0
$356$	6.70118e6	2.80238
$357$	−326488.	−0.135580
$358$	−5.30971e6	−2.18959
$359$	2.70091e6	1.10605	0.553024	−	0.833166i	$-0.313474\pi$
$359$	2.70091e6	1.10605	0.553024	+	0.833166i	$0.313474\pi$
$360$	0	0
$361$	−2.42102e6	−0.977757
$362$	−5.30858e6	−2.12915
$363$	−131769.	−0.0524864
$364$	5.53200e6	2.18841
$365$	0	0
$366$	−4.55298e6	−1.77661
$367$	−3.59856e6	−1.39465	−0.697323	−	0.716757i	$-0.745626\pi$
$367$	−3.59856e6	−1.39465	−0.697323	+	0.716757i	$0.745626\pi$
$368$	−8.29480e6	−3.19291
$369$	−435205.	−0.166390
$370$	0	0
$371$	−334044.	−0.126000
$372$	554135.	0.207615
$373$	4.71222e6	1.75369	0.876846	−	0.480772i	$-0.159643\pi$
$373$	4.71222e6	1.75369	0.876846	+	0.480772i	$0.159643\pi$
$374$	770291.	0.284758
$375$	0	0
$376$	−1.43056e7	−5.21839
$377$	−6.78525e6	−2.45874
$378$	−504037.	−0.181440
$379$	2.89500e6	1.03526	0.517631	−	0.855604i	$-0.326814\pi$
$379$	2.89500e6	1.03526	0.517631	+	0.855604i	$0.326814\pi$
$380$	0	0
$381$	−1.36004e6	−0.479999
$382$	−3.30988e6	−1.16052
$383$	−691654.	−0.240931	−0.120465	−	0.992718i	$-0.538439\pi$
$383$	−691654.	−0.240931	−0.120465	+	0.992718i	$0.538439\pi$
$384$	−7.04275e6	−2.43733
$385$	0	0
$386$	−9.47394e6	−3.23640
$387$	1.11152e6	0.377259
$388$	−970970.	−0.327436
$389$	−210206.	−0.0704323	−0.0352161	−	0.999380i	$-0.511212\pi$
$389$	−210206.	−0.0704323	−0.0352161	+	0.999380i	$0.511212\pi$
$390$	0	0
$391$	−1.16988e6	−0.386988
$392$	−8.12596e6	−2.67091
$393$	408611.	0.133453
$394$	−6.86352e6	−2.22744
$395$	0	0
$396$	875555.	0.280573
$397$	−2.04826e6	−0.652242	−0.326121	−	0.945328i	$-0.605742\pi$
$397$	−2.04826e6	−0.652242	−0.326121	+	0.945328i	$0.605742\pi$
$398$	−4.59738e6	−1.45480
$399$	132576.	0.0416901
$400$	0	0
$401$	1.26605e6	0.393179	0.196589	−	0.980486i	$-0.437013\pi$
$401$	1.26605e6	0.393179	0.196589	+	0.980486i	$0.437013\pi$
$402$	−5.00732e6	−1.54540
$403$	−679962.	−0.208556
$404$	1.00373e7	3.05959
$405$	0	0
$406$	−4.75529e6	−1.43173
$407$	1.50488e6	0.450314
$408$	−3.28487e6	−0.976940
$409$	2.04510e6	0.604514	0.302257	−	0.953226i	$-0.402260\pi$
$409$	2.04510e6	0.604514	0.302257	+	0.953226i	$0.402260\pi$
$410$	0	0
$411$	2.67956e6	0.782453
$412$	5.66310e6	1.64366
$413$	1.49343e6	0.430832
$414$	−1.80607e6	−0.517886
$415$	0	0
$416$	2.45933e7	6.96760
$417$	1.92909e6	0.543265
$418$	−312790.	−0.0875612
$419$	−1.00407e6	−0.279401	−0.139701	−	0.990194i	$-0.544614\pi$
$419$	−1.00407e6	−0.279401	−0.139701	+	0.990194i	$0.544614\pi$
$420$	0	0
$421$	−4.99725e6	−1.37412	−0.687061	−	0.726599i	$-0.741100\pi$
$421$	−4.99725e6	−1.37412	−0.687061	+	0.726599i	$0.741100\pi$
$422$	1.37463e7	3.75756
$423$	−1.83483e6	−0.498591
$424$	−3.36089e6	−0.907903
$425$	0	0
$426$	2.90632e6	0.775924
$427$	2.88276e6	0.765137
$428$	1.45698e7	3.84453
$429$	−1.07437e6	−0.281844
$430$	0	0
$431$	−397569.	−0.103091	−0.0515453	−	0.998671i	$-0.516415\pi$
$431$	−397569.	−0.103091	−0.0515453	+	0.998671i	$0.516415\pi$
$432$	−2.98726e6	−0.770130
$433$	−5.89453e6	−1.51088	−0.755439	−	0.655219i	$-0.772576\pi$
$433$	−5.89453e6	−1.51088	−0.755439	+	0.655219i	$0.772576\pi$
$434$	−476536.	−0.121443
$435$	0	0
$436$	−1.58613e7	−3.99597
$437$	475048.	0.118996
$438$	4.20665e6	1.04773
$439$	−2.26759e6	−0.561570	−0.280785	−	0.959771i	$-0.590595\pi$
$439$	−2.26759e6	−0.561570	−0.280785	+	0.959771i	$0.590595\pi$
$440$	0	0
$441$	−1.04223e6	−0.255192
$442$	6.28049e6	1.52911
$443$	−1.42871e6	−0.345888	−0.172944	−	0.984932i	$-0.555328\pi$
$443$	−1.42871e6	−0.345888	−0.172944	+	0.984932i	$0.555328\pi$
$444$	−9.99934e6	−2.40721
$445$	0	0
$446$	−4.40342e6	−1.04822
$447$	−2.10989e6	−0.499449
$448$	9.00485e6	2.11974
$449$	4.49834e6	1.05302	0.526510	−	0.850169i	$-0.323500\pi$
$449$	4.49834e6	1.05302	0.526510	+	0.850169i	$0.323500\pi$
$450$	0	0
$451$	−650120.	−0.150506
$452$	−268840.	−0.0618940
$453$	−547853.	−0.125435
$454$	1.26099e7	2.87125
$455$	0	0
$456$	1.33388e6	0.300402
$457$	−5.64865e6	−1.26518	−0.632592	−	0.774485i	$-0.718009\pi$
$457$	−5.64865e6	−1.26518	−0.632592	+	0.774485i	$0.718009\pi$
$458$	274592.	0.0611680
$459$	−421315.	−0.0933417
$460$	0	0
$461$	−8.44590e6	−1.85094	−0.925472	−	0.378815i	$-0.876332\pi$
$461$	−8.44590e6	−1.85094	−0.925472	+	0.378815i	$0.876332\pi$
$462$	−752944.	−0.164119
$463$	−4.07836e6	−0.884165	−0.442082	−	0.896974i	$-0.645760\pi$
$463$	−4.07836e6	−0.884165	−0.442082	+	0.896974i	$0.645760\pi$
$464$	−2.81830e7	−6.07705
$465$	0	0
$466$	−6.89804e6	−1.47150
$467$	−3.43895e6	−0.729682	−0.364841	−	0.931070i	$-0.618877\pi$
$467$	−3.43895e6	−0.729682	−0.364841	+	0.931070i	$0.618877\pi$
$468$	7.13875e6	1.50663
$469$	3.17043e6	0.665558
$470$	0	0
$471$	−1.00910e6	−0.209595
$472$	1.50257e7	3.10441
$473$	1.66042e6	0.341244
$474$	−6.32556e6	−1.29316
$475$	0	0
$476$	3.24069e6	0.655572
$477$	−431066.	−0.0867456
$478$	−1.15695e7	−2.31604
$479$	3.52502e6	0.701978	0.350989	−	0.936380i	$-0.385845\pi$
$479$	3.52502e6	0.701978	0.350989	+	0.936380i	$0.385845\pi$
$480$	0	0
$481$	1.22699e7	2.41812
$482$	2.77277e6	0.543621
$483$	1.14353e6	0.223039
$484$	1.30793e6	0.253787
$485$	0	0
$486$	−650433.	−0.124914
$487$	−1.06326e6	−0.203151	−0.101575	−	0.994828i	$-0.532388\pi$
$487$	−1.06326e6	−0.203151	−0.101575	+	0.994828i	$0.532388\pi$
$488$	2.90041e7	5.51328
$489$	−3.27657e6	−0.619651
$490$	0	0
$491$	9.16347e6	1.71536	0.857682	−	0.514180i	$-0.171904\pi$
$491$	9.16347e6	1.71536	0.857682	+	0.514180i	$0.171904\pi$
$492$	4.31980e6	0.804546
$493$	−3.97486e6	−0.736554
$494$	−2.55030e6	−0.470190
$495$	0	0
$496$	−2.82427e6	−0.515469
$497$	−1.84016e6	−0.334168
$498$	5.74520e6	1.03808
$499$	−4.49589e6	−0.808285	−0.404142	−	0.914696i	$-0.632430\pi$
$499$	−4.49589e6	−0.808285	−0.404142	+	0.914696i	$0.632430\pi$
$500$	0	0
$501$	5.39506e6	0.960289
$502$	1.02068e7	1.80772
$503$	−8.70281e6	−1.53370	−0.766848	−	0.641828i	$-0.778176\pi$
$503$	−8.70281e6	−1.53370	−0.766848	+	0.641828i	$0.778176\pi$
$504$	3.21089e6	0.563054
$505$	0	0
$506$	−2.69796e6	−0.468445
$507$	−5.41809e6	−0.936110
$508$	1.34997e7	2.32094
$509$	5.96436e6	1.02040	0.510199	−	0.860057i	$-0.329572\pi$
$509$	5.96436e6	1.02040	0.510199	+	0.860057i	$0.329572\pi$
$510$	0	0
$511$	−2.66348e6	−0.451229
$512$	1.93383e7	3.26019
$513$	171082.	0.0287019
$514$	1.51156e7	2.52359
$515$	0	0
$516$	−1.10328e7	−1.82416
$517$	−2.74092e6	−0.450993
$518$	8.59906e6	1.40808
$519$	4.91683e6	0.801248
$520$	0	0
$521$	−2.15902e6	−0.348468	−0.174234	−	0.984704i	$-0.555745\pi$
$521$	−2.15902e6	−0.348468	−0.174234	+	0.984704i	$0.555745\pi$
$522$	−6.13644e6	−0.985690
$523$	768246.	0.122814	0.0614068	−	0.998113i	$-0.480441\pi$
$523$	768246.	0.122814	0.0614068	+	0.998113i	$0.480441\pi$
$524$	−4.05584e6	−0.645286
$525$	0	0
$526$	2.00040e7	3.15248
$527$	−398327.	−0.0624761
$528$	−4.46246e6	−0.696609
$529$	−2.33883e6	−0.363379
$530$	0	0
$531$	1.92718e6	0.296611
$532$	−1.31594e6	−0.201584
$533$	−5.30069e6	−0.808192
$534$	−7.43654e6	−1.12854
$535$	0	0
$536$	3.18984e7	4.79575
$537$	4.33834e6	0.649214
$538$	1.40045e7	2.08599
$539$	−1.55691e6	−0.230830
$540$	0	0
$541$	−518319.	−0.0761385	−0.0380693	−	0.999275i	$-0.512121\pi$
$541$	−518319.	−0.0761385	−0.0380693	+	0.999275i	$0.512121\pi$
$542$	1.59212e7	2.32797
$543$	4.33742e6	0.631294
$544$	1.44070e7	2.08725
$545$	0	0
$546$	−6.13906e6	−0.881292
$547$	−4.96112e6	−0.708944	−0.354472	−	0.935067i	$-0.615339\pi$
$547$	−4.96112e6	−0.708944	−0.354472	+	0.935067i	$0.615339\pi$
$548$	−2.65970e7	−3.78340
$549$	3.72005e6	0.526766
$550$	0	0
$551$	1.61406e6	0.226485
$552$	1.15053e7	1.60713
$553$	4.00509e6	0.556928
$554$	1.92441e7	2.66393
$555$	0	0
$556$	−1.91479e7	−2.62685
$557$	9.82979e6	1.34248	0.671238	−	0.741242i	$-0.265763\pi$
$557$	9.82979e6	1.34248	0.671238	+	0.741242i	$0.265763\pi$
$558$	−614944.	−0.0836084
$559$	1.35381e7	1.83243
$560$	0	0
$561$	−629372.	−0.0844307
$562$	−2.02790e7	−2.70835
$563$	−7.66483e6	−1.01913	−0.509567	−	0.860431i	$-0.670195\pi$
$563$	−7.66483e6	−1.01913	−0.509567	+	0.860431i	$0.670195\pi$
$564$	1.82124e7	2.41084
$565$	0	0
$566$	−1.08614e7	−1.42510
$567$	411827.	0.0537969
$568$	−1.85143e7	−2.40788
$569$	−62102.5	−0.00804134	−0.00402067	−	0.999992i	$-0.501280\pi$
$569$	−62102.5	−0.00804134	−0.00402067	+	0.999992i	$0.501280\pi$
$570$	0	0
$571$	−3.48638e6	−0.447491	−0.223745	−	0.974648i	$-0.571828\pi$
$571$	−3.48638e6	−0.447491	−0.223745	+	0.974648i	$0.571828\pi$
$572$	1.06641e7	1.36280
$573$	2.70436e6	0.344095
$574$	−3.71487e6	−0.470613
$575$	0	0
$576$	1.16203e7	1.45935
$577$	9.55413e6	1.19468	0.597340	−	0.801988i	$-0.296224\pi$
$577$	9.55413e6	1.19468	0.597340	+	0.801988i	$0.296224\pi$
$578$	−1.19607e7	−1.48915
$579$	7.74076e6	0.959593
$580$	0	0
$581$	−3.63763e6	−0.447073
$582$	1.07752e6	0.131861
$583$	−643938.	−0.0784644
$584$	−2.67978e7	−3.25138
$585$	0	0
$586$	−2.18721e7	−2.63115
$587$	5.56273e6	0.666335	0.333168	−	0.942868i	$-0.391882\pi$
$587$	5.56273e6	0.666335	0.333168	+	0.942868i	$0.391882\pi$
$588$	1.03451e7	1.23393
$589$	161747.	0.0192110
$590$	0	0
$591$	5.60789e6	0.660436
$592$	5.09639e7	5.97665
$593$	−1.10409e7	−1.28934	−0.644669	−	0.764462i	$-0.723005\pi$
$593$	−1.10409e7	−1.28934	−0.644669	+	0.764462i	$0.723005\pi$
$594$	−971634.	−0.112989
$595$	0	0
$596$	2.09426e7	2.41499
$597$	3.75633e6	0.431348
$598$	−2.19975e7	−2.51548
$599$	−1.47678e6	−0.168170	−0.0840852	−	0.996459i	$-0.526797\pi$
$599$	−1.47678e6	−0.168170	−0.0840852	+	0.996459i	$0.526797\pi$
$600$	0	0
$601$	−7.70993e6	−0.870691	−0.435346	−	0.900263i	$-0.643374\pi$
$601$	−7.70993e6	−0.870691	−0.435346	+	0.900263i	$0.643374\pi$
$602$	9.48784e6	1.06703
$603$	4.09127e6	0.458210
$604$	5.43794e6	0.606516
$605$	0	0
$606$	−1.11387e7	−1.23212
$607$	−1.55229e7	−1.71002	−0.855009	−	0.518613i	$-0.826449\pi$
$607$	−1.55229e7	−1.71002	−0.855009	+	0.518613i	$0.826449\pi$
$608$	−5.85018e6	−0.641816
$609$	3.88535e6	0.424509
$610$	0	0
$611$	−2.23478e7	−2.42176
$612$	4.18194e6	0.451335
$613$	−7.98615e6	−0.858394	−0.429197	−	0.903211i	$-0.641203\pi$
$613$	−7.98615e6	−0.858394	−0.429197	+	0.903211i	$0.641203\pi$
$614$	−1.25554e7	−1.34403
$615$	0	0
$616$	4.79652e6	0.509301
$617$	8.91087e6	0.942339	0.471170	−	0.882043i	$-0.343832\pi$
$617$	8.91087e6	0.942339	0.471170	+	0.882043i	$0.343832\pi$
$618$	−6.28454e6	−0.661915
$619$	−4.54740e6	−0.477019	−0.238510	−	0.971140i	$-0.576659\pi$
$619$	−4.54740e6	−0.477019	−0.238510	+	0.971140i	$0.576659\pi$
$620$	0	0
$621$	1.47566e6	0.153553
$622$	−1.45295e6	−0.150582
$623$	4.70851e6	0.486031
$624$	−3.63842e7	−3.74069
$625$	0	0
$626$	−2.28371e7	−2.32919
$627$	255567.	0.0259619
$628$	1.00162e7	1.01345
$629$	7.18780e6	0.724385
$630$	0	0
$631$	1.26569e7	1.26548	0.632740	−	0.774365i	$-0.281930\pi$
$631$	1.26569e7	1.26548	0.632740	+	0.774365i	$0.281930\pi$
$632$	4.02961e7	4.01301
$633$	−1.12315e7	−1.11411
$634$	1.04780e6	0.103527
$635$	0	0
$636$	4.27872e6	0.419441
$637$	−1.26941e7	−1.23952
$638$	−9.16679e6	−0.891590
$639$	−2.37463e6	−0.230061
$640$	0	0
$641$	3.39088e6	0.325963	0.162981	−	0.986629i	$-0.447889\pi$
$641$	3.39088e6	0.325963	0.162981	+	0.986629i	$0.447889\pi$
$642$	−1.61686e7	−1.54823
$643$	−5.58626e6	−0.532837	−0.266418	−	0.963858i	$-0.585840\pi$
$643$	−5.58626e6	−0.532837	−0.266418	+	0.963858i	$0.585840\pi$
$644$	−1.13506e7	−1.07846
$645$	0	0
$646$	−1.49399e6	−0.140853
$647$	−1.49548e7	−1.40450	−0.702248	−	0.711932i	$-0.747820\pi$
$647$	−1.49548e7	−1.40450	−0.702248	+	0.711932i	$0.747820\pi$
$648$	4.14349e6	0.387640
$649$	2.87888e6	0.268295
$650$	0	0
$651$	389357.	0.0360077
$652$	3.25229e7	2.99620
$653$	−3.85494e6	−0.353782	−0.176891	−	0.984230i	$-0.556604\pi$
$653$	−3.85494e6	−0.353782	−0.176891	+	0.984230i	$0.556604\pi$
$654$	1.76018e7	1.60921
$655$	0	0
$656$	−2.20168e7	−1.99754
$657$	−3.43707e6	−0.310653
$658$	−1.56619e7	−1.41020
$659$	−8.60888e6	−0.772206	−0.386103	−	0.922456i	$-0.626179\pi$
$659$	−8.60888e6	−0.772206	−0.386103	+	0.922456i	$0.626179\pi$
$660$	0	0
$661$	3.93015e6	0.349869	0.174934	−	0.984580i	$-0.444029\pi$
$661$	3.93015e6	0.349869	0.174934	+	0.984580i	$0.444029\pi$
$662$	−3.52653e7	−3.12754
$663$	−5.13153e6	−0.453380
$664$	−3.65990e7	−3.22143
$665$	0	0
$666$	1.10966e7	0.969405
$667$	1.39220e7	1.21168
$668$	−5.35509e7	−4.64329
$669$	3.59785e6	0.310798
$670$	0	0
$671$	5.55711e6	0.476478
$672$	−1.40825e7	−1.20298
$673$	1.74231e7	1.48282	0.741408	−	0.671054i	$-0.234158\pi$
$673$	1.74231e7	1.48282	0.741408	+	0.671054i	$0.234158\pi$
$674$	−2.38562e7	−2.02280
$675$	0	0
$676$	5.37795e7	4.52637
$677$	1.74041e7	1.45942	0.729711	−	0.683756i	$-0.239655\pi$
$677$	1.74041e7	1.45942	0.729711	+	0.683756i	$0.239655\pi$
$678$	298342.	0.0249253
$679$	−682242.	−0.0567890
$680$	0	0
$681$	−1.03030e7	−0.851325
$682$	−918619.	−0.0756266
$683$	−6.21136e6	−0.509489	−0.254745	−	0.967008i	$-0.581991\pi$
$683$	−6.21136e6	−0.509489	−0.254745	+	0.967008i	$0.581991\pi$
$684$	−1.69815e6	−0.138782
$685$	0	0
$686$	−2.05169e7	−1.66457
$687$	−224358.	−0.0181363
$688$	5.62313e7	4.52905
$689$	−5.25028e6	−0.421342
$690$	0	0
$691$	−3.49150e6	−0.278174	−0.139087	−	0.990280i	$-0.544417\pi$
$691$	−3.49150e6	−0.278174	−0.139087	+	0.990280i	$0.544417\pi$
$692$	−4.88041e7	−3.87428
$693$	615199.	0.0486612
$694$	3.04822e7	2.40241
$695$	0	0
$696$	3.90913e7	3.05884
$697$	−3.10519e6	−0.242106
$698$	3.79907e7	2.95147
$699$	5.63610e6	0.436300
$700$	0	0
$701$	9.84192e6	0.756458	0.378229	−	0.925712i	$-0.376533\pi$
$701$	9.84192e6	0.756458	0.378229	+	0.925712i	$0.376533\pi$
$702$	−7.92212e6	−0.606734
$703$	−2.91873e6	−0.222743
$704$	1.73587e7	1.32003
$705$	0	0
$706$	−4.78178e7	−3.61059
$707$	7.05260e6	0.530641
$708$	−1.91291e7	−1.43420
$709$	2.04292e7	1.52629	0.763144	−	0.646228i	$-0.223655\pi$
$709$	2.04292e7	1.52629	0.763144	+	0.646228i	$0.223655\pi$
$710$	0	0
$711$	5.16835e6	0.383423
$712$	4.73734e7	3.50215
$713$	1.39515e6	0.102777
$714$	−3.59631e6	−0.264005
$715$	0	0
$716$	−4.30620e7	−3.13915
$717$	9.45296e6	0.686705
$718$	2.97509e7	2.15372
$719$	1.70520e7	1.23014	0.615069	−	0.788474i	$-0.289128\pi$
$719$	1.70520e7	1.23014	0.615069	+	0.788474i	$0.289128\pi$
$720$	0	0
$721$	3.97912e6	0.285068
$722$	−2.66679e7	−1.90391
$723$	−2.26551e6	−0.161184
$724$	−4.30528e7	−3.05250
$725$	0	0
$726$	−1.45145e6	−0.102202
$727$	−6.52920e6	−0.458167	−0.229084	−	0.973407i	$-0.573573\pi$
$727$	−6.52920e6	−0.458167	−0.229084	+	0.973407i	$0.573573\pi$
$728$	3.91080e7	2.73487
$729$	531441.	0.0370370
$730$	0	0
$731$	7.93071e6	0.548932
$732$	−3.69248e7	−2.54707
$733$	−1.96819e7	−1.35303	−0.676513	−	0.736430i	$-0.736510\pi$
$733$	−1.96819e7	−1.35303	−0.676513	+	0.736430i	$0.736510\pi$
$734$	−3.96386e7	−2.71568
$735$	0	0
$736$	−5.04606e7	−3.43366
$737$	6.11165e6	0.414467
$738$	−4.79384e6	−0.323998
$739$	5.50059e6	0.370509	0.185254	−	0.982691i	$-0.440689\pi$
$739$	5.50059e6	0.370509	0.185254	+	0.982691i	$0.440689\pi$
$740$	0	0
$741$	2.08374e6	0.139411
$742$	−3.67954e6	−0.245349
$743$	6.56551e6	0.436311	0.218156	−	0.975914i	$-0.429996\pi$
$743$	6.56551e6	0.436311	0.218156	+	0.975914i	$0.429996\pi$
$744$	3.91741e6	0.259458
$745$	0	0
$746$	5.19057e7	3.41482
$747$	−4.69416e6	−0.307791
$748$	6.24709e6	0.408248
$749$	1.02373e7	0.666777
$750$	0	0
$751$	2.04233e7	1.32137	0.660687	−	0.750662i	$-0.270265\pi$
$751$	2.04233e7	1.32137	0.660687	+	0.750662i	$0.270265\pi$
$752$	−9.28232e7	−5.98566
$753$	−8.33956e6	−0.535989
$754$	−7.47405e7	−4.78771
$755$	0	0
$756$	−4.08776e6	−0.260124
$757$	1.28683e7	0.816174	0.408087	−	0.912943i	$-0.366196\pi$
$757$	1.28683e7	0.816174	0.408087	+	0.912943i	$0.366196\pi$
$758$	3.18888e7	2.01588
$759$	2.20439e6	0.138894
$760$	0	0
$761$	1.16269e7	0.727782	0.363891	−	0.931442i	$-0.381448\pi$
$761$	1.16269e7	0.727782	0.363891	+	0.931442i	$0.381448\pi$
$762$	−1.49811e7	−0.934663
$763$	−1.11448e7	−0.693041
$764$	−2.68432e7	−1.66380
$765$	0	0
$766$	−7.61866e6	−0.469144
$767$	2.34727e7	1.44070
$768$	−3.62603e7	−2.21834
$769$	2.67451e7	1.63091	0.815453	−	0.578823i	$-0.196488\pi$
$769$	2.67451e7	1.63091	0.815453	+	0.578823i	$0.196488\pi$
$770$	0	0
$771$	−1.23503e7	−0.748243
$772$	−7.68340e7	−4.63992
$773$	−1.62196e7	−0.976320	−0.488160	−	0.872754i	$-0.662332\pi$
$773$	−1.62196e7	−0.976320	−0.488160	+	0.872754i	$0.662332\pi$
$774$	1.22435e7	0.734606
$775$	0	0
$776$	−6.86419e6	−0.409199
$777$	−7.02593e6	−0.417495
$778$	−2.31545e6	−0.137147
$779$	1.26091e6	0.0744461
$780$	0	0
$781$	−3.54728e6	−0.208098
$782$	−1.28863e7	−0.753551
$783$	5.01383e6	0.292257
$784$	−5.27260e7	−3.06362
$785$	0	0
$786$	4.50091e6	0.259863
$787$	3.30833e7	1.90402	0.952011	−	0.306063i	$-0.0990119\pi$
$787$	3.30833e7	1.90402	0.952011	+	0.306063i	$0.0990119\pi$
$788$	−5.56634e7	−3.19341
$789$	−1.63444e7	−0.934710
$790$	0	0
$791$	−188898.	−0.0107346
$792$	6.18965e6	0.350633
$793$	4.53093e7	2.55861
$794$	−2.25619e7	−1.27006
$795$	0	0
$796$	−3.72850e7	−2.08570
$797$	6.59997e6	0.368041	0.184020	−	0.982922i	$-0.441089\pi$
$797$	6.59997e6	0.368041	0.184020	+	0.982922i	$0.441089\pi$
$798$	1.46034e6	0.0811797
$799$	−1.30915e7	−0.725477
$800$	0	0
$801$	6.07608e6	0.334613
$802$	1.39457e7	0.765605
$803$	−5.13439e6	−0.280996
$804$	−4.06096e7	−2.21558
$805$	0	0
$806$	−7.48987e6	−0.406104
$807$	−1.14425e7	−0.618497
$808$	7.09577e7	3.82359
$809$	−2.14723e7	−1.15347	−0.576736	−	0.816931i	$-0.695674\pi$
$809$	−2.14723e7	−1.15347	−0.576736	+	0.816931i	$0.695674\pi$
$810$	0	0
$811$	−1.13634e7	−0.606678	−0.303339	−	0.952883i	$-0.598101\pi$
$811$	−1.13634e7	−0.606678	−0.303339	+	0.952883i	$0.598101\pi$
$812$	−3.85656e7	−2.05263
$813$	−1.30086e7	−0.690244
$814$	1.65764e7	0.876860
$815$	0	0
$816$	−2.13142e7	−1.12058
$817$	−3.22040e6	−0.168793
$818$	2.25271e7	1.17712
$819$	5.01596e6	0.261303
$820$	0	0
$821$	−6.33658e6	−0.328093	−0.164047	−	0.986453i	$-0.552455\pi$
$821$	−6.33658e6	−0.328093	−0.164047	+	0.986453i	$0.552455\pi$
$822$	2.95157e7	1.52361
$823$	−3.27073e7	−1.68324	−0.841619	−	0.540071i	$-0.818397\pi$
$823$	−3.27073e7	−1.68324	−0.841619	+	0.540071i	$0.818397\pi$
$824$	4.00348e7	2.05409
$825$	0	0
$826$	1.64503e7	0.838925
$827$	3.83082e6	0.194773	0.0973863	−	0.995247i	$-0.468952\pi$
$827$	3.83082e6	0.194773	0.0973863	+	0.995247i	$0.468952\pi$
$828$	−1.46473e7	−0.742475
$829$	−2.45814e7	−1.24228	−0.621142	−	0.783698i	$-0.713331\pi$
$829$	−2.45814e7	−1.24228	−0.621142	+	0.783698i	$0.713331\pi$
$830$	0	0
$831$	−1.57235e7	−0.789854
$832$	1.41532e8	7.08838
$833$	−7.43633e6	−0.371318
$834$	2.12491e7	1.05786
$835$	0	0
$836$	−2.53674e6	−0.125533
$837$	502444.	0.0247899
$838$	−1.10599e7	−0.544055
$839$	8.44908e6	0.414386	0.207193	−	0.978300i	$-0.433567\pi$
$839$	8.44908e6	0.414386	0.207193	+	0.978300i	$0.433567\pi$
$840$	0	0
$841$	2.67913e7	1.30618
$842$	−5.50453e7	−2.67572
$843$	1.65691e7	0.803026
$844$	1.11483e8	5.38708
$845$	0	0
$846$	−2.02109e7	−0.970867
$847$	919001.	0.0440157
$848$	−2.18074e7	−1.04139
$849$	8.87438e6	0.422541
$850$	0	0
$851$	−2.51754e7	−1.19166
$852$	2.35703e7	1.11242
$853$	1.03931e7	0.489072	0.244536	−	0.969640i	$-0.421364\pi$
$853$	1.03931e7	0.489072	0.244536	+	0.969640i	$0.421364\pi$
$854$	3.17540e7	1.48989
$855$	0	0
$856$	1.03000e8	4.80453
$857$	−1.78951e7	−0.832302	−0.416151	−	0.909295i	$-0.636621\pi$
$857$	−1.78951e7	−0.832302	−0.416151	+	0.909295i	$0.636621\pi$
$858$	−1.18343e7	−0.548812
$859$	−1.57750e7	−0.729435	−0.364717	−	0.931118i	$-0.618834\pi$
$859$	−1.57750e7	−0.729435	−0.364717	+	0.931118i	$0.618834\pi$
$860$	0	0
$861$	3.03526e6	0.139537
$862$	−4.37928e6	−0.200740
$863$	3.34930e7	1.53083	0.765416	−	0.643536i	$-0.222533\pi$
$863$	3.34930e7	1.53083	0.765416	+	0.643536i	$0.222533\pi$
$864$	−1.81727e7	−0.828201
$865$	0	0
$866$	−6.49290e7	−2.94201
$867$	9.77262e6	0.441533
$868$	−3.86472e6	−0.174108
$869$	7.72062e6	0.346819
$870$	0	0
$871$	4.98307e7	2.22562
$872$	−1.12130e8	−4.99379
$873$	−880396.	−0.0390969
$874$	5.23271e6	0.231712
$875$	0	0
$876$	3.41161e7	1.50210
$877$	−3.05516e7	−1.34133	−0.670665	−	0.741760i	$-0.733991\pi$
$877$	−3.05516e7	−1.34133	−0.670665	+	0.741760i	$0.733991\pi$
$878$	−2.49778e7	−1.09350
$879$	1.78708e7	0.780137
$880$	0	0
$881$	−6.31867e6	−0.274275	−0.137137	−	0.990552i	$-0.543790\pi$
$881$	−6.31867e6	−0.274275	−0.137137	+	0.990552i	$0.543790\pi$
$882$	−1.14803e7	−0.496915
$883$	−2.91551e7	−1.25838	−0.629191	−	0.777251i	$-0.716614\pi$
$883$	−2.91551e7	−1.25838	−0.629191	+	0.777251i	$0.716614\pi$
$884$	5.09351e7	2.19223
$885$	0	0
$886$	−1.57375e7	−0.673519
$887$	−1.26078e7	−0.538061	−0.269030	−	0.963132i	$-0.586703\pi$
$887$	−1.26078e7	−0.538061	−0.269030	+	0.963132i	$0.586703\pi$
$888$	−7.06894e7	−3.00831
$889$	9.48539e6	0.402532
$890$	0	0
$891$	793881.	0.0335013
$892$	−3.57119e7	−1.50280
$893$	5.31603e6	0.223079
$894$	−2.32407e7	−0.972537
$895$	0	0
$896$	4.91185e7	2.04397
$897$	1.79733e7	0.745840
$898$	4.95499e7	2.05046
$899$	4.74026e6	0.195615
$900$	0	0
$901$	−3.07566e6	−0.126219
$902$	−7.16116e6	−0.293067
$903$	−7.75211e6	−0.316374
$904$	−1.90054e6	−0.0773493
$905$	0	0
$906$	−6.03468e6	−0.244249
$907$	−2.75312e7	−1.11124	−0.555618	−	0.831437i	$-0.687518\pi$
$907$	−2.75312e7	−1.11124	−0.555618	+	0.831437i	$0.687518\pi$
$908$	1.02267e8	4.11641
$909$	9.10099e6	0.365325
$910$	0	0
$911$	2.45616e7	0.980531	0.490265	−	0.871573i	$-0.336900\pi$
$911$	2.45616e7	0.980531	0.490265	+	0.871573i	$0.336900\pi$
$912$	8.65497e6	0.344571
$913$	−7.01227e6	−0.278408
$914$	−6.22206e7	−2.46359
$915$	0	0
$916$	2.22695e6	0.0876945
$917$	−2.84979e6	−0.111915
$918$	−4.64085e6	−0.181757
$919$	1.79061e7	0.699379	0.349690	−	0.936866i	$-0.386287\pi$
$919$	1.79061e7	0.699379	0.349690	+	0.936866i	$0.386287\pi$
$920$	0	0
$921$	1.02585e7	0.398505
$922$	−9.30327e7	−3.60419
$923$	−2.89224e7	−1.11746
$924$	−6.10641e6	−0.235291
$925$	0	0
$926$	−4.49237e7	−1.72166
$927$	5.13483e6	0.196258
$928$	−1.71449e8	−6.53528
$929$	−1.63508e7	−0.621583	−0.310791	−	0.950478i	$-0.600594\pi$
$929$	−1.63508e7	−0.621583	−0.310791	+	0.950478i	$0.600594\pi$
$930$	0	0
$931$	3.01965e6	0.114178
$932$	−5.59434e7	−2.10964
$933$	1.18714e6	0.0446476
$934$	−3.78805e7	−1.42085
$935$	0	0
$936$	5.04667e7	1.88285
$937$	5.61949e6	0.209097	0.104548	−	0.994520i	$-0.466660\pi$
$937$	5.61949e6	0.209097	0.104548	+	0.994520i	$0.466660\pi$
$938$	3.49227e7	1.29599
$939$	1.86592e7	0.690605
$940$	0	0
$941$	5.03132e6	0.185228	0.0926142	−	0.995702i	$-0.470478\pi$
$941$	5.03132e6	0.185228	0.0926142	+	0.995702i	$0.470478\pi$
$942$	−1.11153e7	−0.408127
$943$	1.08760e7	0.398281
$944$	9.74955e7	3.56086
$945$	0	0
$946$	1.82897e7	0.664476
$947$	−1.16336e7	−0.421540	−0.210770	−	0.977536i	$-0.567597\pi$
$947$	−1.16336e7	−0.421540	−0.210770	+	0.977536i	$0.567597\pi$
$948$	−5.13006e7	−1.85396
$949$	−4.18628e7	−1.50891
$950$	0	0
$951$	−856112.	−0.0306958
$952$	2.29098e7	0.819273
$953$	1.43027e7	0.510135	0.255068	−	0.966923i	$-0.417902\pi$
$953$	1.43027e7	0.510135	0.255068	+	0.966923i	$0.417902\pi$
$954$	−4.74825e6	−0.168913
$955$	0	0
$956$	−9.38293e7	−3.32042
$957$	7.48979e6	0.264356
$958$	3.88286e7	1.36690
$959$	−1.86881e7	−0.656174
$960$	0	0
$961$	−2.81541e7	−0.983407
$962$	1.35154e8	4.70861
$963$	1.32107e7	0.459049
$964$	2.24873e7	0.779372
$965$	0	0
$966$	1.25961e7	0.434305
$967$	1.87953e7	0.646372	0.323186	−	0.946335i	$-0.395246\pi$
$967$	1.87953e7	0.646372	0.323186	+	0.946335i	$0.395246\pi$
$968$	9.24627e6	0.317160
$969$	1.22067e6	0.0417628
$970$	0	0
$971$	3.65105e7	1.24271	0.621355	−	0.783529i	$-0.286582\pi$
$971$	3.65105e7	1.24271	0.621355	+	0.783529i	$0.286582\pi$
$972$	−5.27504e6	−0.179085
$973$	−1.34541e7	−0.455588
$974$	−1.17120e7	−0.395579
$975$	0	0
$976$	1.88196e8	6.32390
$977$	2.62418e7	0.879542	0.439771	−	0.898110i	$-0.355060\pi$
$977$	2.62418e7	0.879542	0.439771	+	0.898110i	$0.355060\pi$
$978$	−3.60919e7	−1.20660
$979$	9.07662e6	0.302669
$980$	0	0
$981$	−1.43817e7	−0.477131
$982$	1.00937e8	3.34019
$983$	−4.19211e7	−1.38372	−0.691861	−	0.722030i	$-0.743209\pi$
$983$	−4.19211e7	−1.38372	−0.691861	+	0.722030i	$0.743209\pi$
$984$	3.05384e7	1.00545
$985$	0	0
$986$	−4.37836e7	−1.43423
$987$	1.27967e7	0.418124
$988$	−2.06830e7	−0.674096
$989$	−2.77774e7	−0.903029
$990$	0	0
$991$	3.68229e7	1.19106	0.595530	−	0.803333i	$-0.296942\pi$
$991$	3.68229e7	1.19106	0.595530	+	0.803333i	$0.296942\pi$
$992$	−1.71812e7	−0.554337
$993$	2.88138e7	0.927315
$994$	−2.02696e7	−0.650698
$995$	0	0
$996$	4.65938e7	1.48826
$997$	5.96749e6	0.190132	0.0950658	−	0.995471i	$-0.469694\pi$
$997$	5.96749e6	0.190132	0.0950658	+	0.995471i	$0.469694\pi$
$998$	−4.95228e7	−1.57391
$999$	−9.06658e6	−0.287429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.u.1.10	yes	10
5.4	even	2		825.6.a.t.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.t.1.1	✓	10	5.4	even	2
825.6.a.u.1.10	yes	10	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)

\(f(q)\)	\(=\)	\(q+11.0151 q^{2} -9.00000 q^{3} +89.3332 q^{4} -99.1362 q^{6} +62.7690 q^{7} +631.533 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+11.0151 q^{2} -9.00000 q^{3} +89.3332 q^{4} -99.1362 q^{6} +62.7690 q^{7} +631.533 q^{8} +81.0000 q^{9} +121.000 q^{11} -803.999 q^{12} +986.561 q^{13} +691.409 q^{14} +4097.76 q^{16} +577.936 q^{17} +892.226 q^{18} -234.681 q^{19} -564.921 q^{21} +1332.83 q^{22} -2024.23 q^{23} -5683.79 q^{24} +10867.1 q^{26} -729.000 q^{27} +5607.35 q^{28} -6877.68 q^{29} -689.224 q^{31} +24928.3 q^{32} -1089.00 q^{33} +6366.04 q^{34} +7235.99 q^{36} +12437.0 q^{37} -2585.04 q^{38} -8879.05 q^{39} -5372.90 q^{41} -6222.68 q^{42} +13722.5 q^{43} +10809.3 q^{44} -22297.2 q^{46} -22652.2 q^{47} -36879.8 q^{48} -12867.1 q^{49} -5201.42 q^{51} +88132.7 q^{52} -5321.80 q^{53} -8030.03 q^{54} +39640.7 q^{56} +2112.12 q^{57} -75758.6 q^{58} +23792.4 q^{59} +45926.5 q^{61} -7591.90 q^{62} +5084.29 q^{63} +143460. q^{64} -11995.5 q^{66} +50509.5 q^{67} +51628.9 q^{68} +18218.1 q^{69} -29316.4 q^{71} +51154.1 q^{72} -42433.0 q^{73} +136995. q^{74} -20964.8 q^{76} +7595.05 q^{77} -97804.0 q^{78} +63806.8 q^{79} +6561.00 q^{81} -59183.2 q^{82} -57952.6 q^{83} -50466.2 q^{84} +151155. q^{86} +61899.1 q^{87} +76415.5 q^{88} +75013.4 q^{89} +61925.5 q^{91} -180831. q^{92} +6203.02 q^{93} -249517. q^{94} -224355. q^{96} -10869.1 q^{97} -141732. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q + q^2 - 90 * q^3 + 223 * q^4 - 9 * q^6 - 188 * q^7 - 177 * q^8 + 810 * q^9 + 1210 * q^11 - 2007 * q^12 + 1102 * q^13 + 684 * q^14 + 7019 * q^16 - 1106 * q^17 + 81 * q^18 + 2586 * q^19 + 1692 * q^21 + 121 * q^22 - 2206 * q^23 + 1593 * q^24 + 12001 * q^26 - 7290 * q^27 - 14452 * q^28 + 5824 * q^29 - 4586 * q^31 + 10627 * q^32 - 10890 * q^33 + 7426 * q^34 + 18063 * q^36 + 18362 * q^37 - 44001 * q^38 - 9918 * q^39 - 5474 * q^41 - 6156 * q^42 - 20496 * q^43 + 26983 * q^44 + 19981 * q^46 + 14970 * q^47 - 63171 * q^48 + 68582 * q^49 + 9954 * q^51 + 58603 * q^52 - 61980 * q^53 - 729 * q^54 + 7132 * q^56 - 23274 * q^57 - 7161 * q^58 + 61190 * q^59 + 8230 * q^61 + 4509 * q^62 - 15228 * q^63 + 152223 * q^64 - 1089 * q^66 + 11930 * q^67 - 105598 * q^68 + 19854 * q^69 + 59822 * q^71 - 14337 * q^72 + 20680 * q^73 + 132564 * q^74 + 68165 * q^76 - 22748 * q^77 - 108009 * q^78 + 234494 * q^79 + 65610 * q^81 - 151948 * q^82 - 185478 * q^83 + 130068 * q^84 - 17825 * q^86 - 52416 * q^87 - 21417 * q^88 + 181834 * q^89 + 206274 * q^91 - 98373 * q^92 + 41274 * q^93 - 64998 * q^94 - 95643 * q^96 - 304358 * q^97 - 153453 * q^98 + 98010 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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