LMFDB - Embedded newform 1075.6.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$1$
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} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$9.86547$ of defining polynomial
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.86547 q^{2} -1.50169 q^{3} +46.5966 q^{4} -13.3132 q^{6} +124.747 q^{7} +129.406 q^{8} -240.745 q^{9} +O(q^{10})$	\(q+8.86547 q^{2} -1.50169 q^{3} +46.5966 q^{4} -13.3132 q^{6} +124.747 q^{7} +129.406 q^{8} -240.745 q^{9} +590.079 q^{11} -69.9735 q^{12} -434.774 q^{13} +1105.94 q^{14} -343.849 q^{16} -1925.58 q^{17} -2134.32 q^{18} +654.129 q^{19} -187.332 q^{21} +5231.33 q^{22} -2805.03 q^{23} -194.327 q^{24} -3854.47 q^{26} +726.434 q^{27} +5812.80 q^{28} -1456.23 q^{29} +4419.85 q^{31} -7189.36 q^{32} -886.115 q^{33} -17071.2 q^{34} -11217.9 q^{36} -3753.09 q^{37} +5799.17 q^{38} +652.894 q^{39} +1972.85 q^{41} -1660.78 q^{42} -1849.00 q^{43} +27495.7 q^{44} -24867.9 q^{46} -2204.84 q^{47} +516.354 q^{48} -1245.12 q^{49} +2891.62 q^{51} -20259.0 q^{52} -24984.2 q^{53} +6440.18 q^{54} +16143.0 q^{56} -982.298 q^{57} -12910.1 q^{58} -42756.2 q^{59} -21022.8 q^{61} +39184.0 q^{62} -30032.3 q^{63} -52733.9 q^{64} -7855.83 q^{66} +25272.1 q^{67} -89725.4 q^{68} +4212.28 q^{69} +48082.8 q^{71} -31153.7 q^{72} -58801.2 q^{73} -33272.9 q^{74} +30480.2 q^{76} +73610.8 q^{77} +5788.22 q^{78} +92704.7 q^{79} +57410.1 q^{81} +17490.3 q^{82} +1849.64 q^{83} -8729.01 q^{84} -16392.3 q^{86} +2186.80 q^{87} +76359.5 q^{88} -70380.3 q^{89} -54236.8 q^{91} -130705. q^{92} -6637.23 q^{93} -19546.9 q^{94} +10796.2 q^{96} +88842.1 q^{97} -11038.6 q^{98} -142059. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) $ 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * 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 $

$2$	8.86547	1.56721	0.783604	−	0.621260i	$-0.213379\pi$
$2$	8.86547	1.56721	0.783604	+	0.621260i	$0.213379\pi$
$3$	−1.50169	−0.0963333	−0.0481667	−	0.998839i	$-0.515338\pi$
$3$	−1.50169	−0.0963333	−0.0481667	+	0.998839i	$0.515338\pi$
$4$	46.5966	1.45614
$5$	0	0
$5$	0	0
$6$	−13.3132	−0.150974
$7$	124.747	0.962246	0.481123	−	0.876653i	$-0.340229\pi$
$7$	124.747	0.962246	0.481123	+	0.876653i	$0.340229\pi$
$8$	129.406	0.714872
$9$	−240.745	−0.990720
$10$	0	0
$11$	590.079	1.47038	0.735188	−	0.677863i	$-0.237094\pi$
$11$	590.079	1.47038	0.735188	+	0.677863i	$0.237094\pi$
$12$	−69.9735	−0.140275
$13$	−434.774	−0.713518	−0.356759	−	0.934196i	$-0.616118\pi$
$13$	−434.774	−0.713518	−0.356759	+	0.934196i	$0.616118\pi$
$14$	1105.94	1.50804
$15$	0	0
$16$	−343.849	−0.335790
$17$	−1925.58	−1.61599	−0.807995	−	0.589189i	$-0.799447\pi$
$17$	−1925.58	−1.61599	−0.807995	+	0.589189i	$0.799447\pi$
$18$	−2134.32	−1.55266
$19$	654.129	0.415700	0.207850	−	0.978161i	$-0.433353\pi$
$19$	654.129	0.415700	0.207850	+	0.978161i	$0.433353\pi$
$20$	0	0
$21$	−187.332	−0.0926963
$22$	5231.33	2.30439
$23$	−2805.03	−1.10565	−0.552825	−	0.833297i	$-0.686450\pi$
$23$	−2805.03	−1.10565	−0.552825	+	0.833297i	$0.686450\pi$
$24$	−194.327	−0.0688660
$25$	0	0
$26$	−3854.47	−1.11823
$27$	726.434	0.191773
$28$	5812.80	1.40117
$29$	−1456.23	−0.321540	−0.160770	−	0.986992i	$-0.551398\pi$
$29$	−1456.23	−0.321540	−0.160770	+	0.986992i	$0.551398\pi$
$30$	0	0
$31$	4419.85	0.826044	0.413022	−	0.910721i	$-0.364473\pi$
$31$	4419.85	0.826044	0.413022	+	0.910721i	$0.364473\pi$
$32$	−7189.36	−1.24112
$33$	−886.115	−0.141646
$34$	−17071.2	−2.53259
$35$	0	0
$36$	−11217.9	−1.44263
$37$	−3753.09	−0.450697	−0.225348	−	0.974278i	$-0.572352\pi$
$37$	−3753.09	−0.450697	−0.225348	+	0.974278i	$0.572352\pi$
$38$	5799.17	0.651488
$39$	652.894	0.0687356
$40$	0	0
$41$	1972.85	0.183288	0.0916442	−	0.995792i	$-0.470788\pi$
$41$	1972.85	0.183288	0.0916442	+	0.995792i	$0.470788\pi$
$42$	−1660.78	−0.145275
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	27495.7	2.14108
$45$	0	0
$46$	−24867.9	−1.73279
$47$	−2204.84	−0.145590	−0.0727950	−	0.997347i	$-0.523192\pi$
$47$	−2204.84	−0.145590	−0.0727950	+	0.997347i	$0.523192\pi$
$48$	516.354	0.0323478
$49$	−1245.12	−0.0740834
$50$	0	0
$51$	2891.62	0.155674
$52$	−20259.0	−1.03898
$53$	−24984.2	−1.22173	−0.610865	−	0.791735i	$-0.709178\pi$
$53$	−24984.2	−1.22173	−0.610865	+	0.791735i	$0.709178\pi$
$54$	6440.18	0.300548
$55$	0	0
$56$	16143.0	0.687882
$57$	−982.298	−0.0400457
$58$	−12910.1	−0.503920
$59$	−42756.2	−1.59907	−0.799537	−	0.600616i	$-0.794922\pi$
$59$	−42756.2	−1.59907	−0.799537	+	0.600616i	$0.794922\pi$
$60$	0	0
$61$	−21022.8	−0.723379	−0.361689	−	0.932299i	$-0.617800\pi$
$61$	−21022.8	−0.723379	−0.361689	+	0.932299i	$0.617800\pi$
$62$	39184.0	1.29458
$63$	−30032.3	−0.953316
$64$	−52733.9	−1.60931
$65$	0	0
$66$	−7855.83	−0.221989
$67$	25272.1	0.687788	0.343894	−	0.939008i	$-0.388254\pi$
$67$	25272.1	0.687788	0.343894	+	0.939008i	$0.388254\pi$
$68$	−89725.4	−2.35311
$69$	4212.28	0.106511
$70$	0	0
$71$	48082.8	1.13199	0.565997	−	0.824407i	$-0.308491\pi$
$71$	48082.8	1.13199	0.565997	+	0.824407i	$0.308491\pi$
$72$	−31153.7	−0.708237
$73$	−58801.2	−1.29145	−0.645727	−	0.763568i	$-0.723446\pi$
$73$	−58801.2	−1.29145	−0.645727	+	0.763568i	$0.723446\pi$
$74$	−33272.9	−0.706336
$75$	0	0
$76$	30480.2	0.605318
$77$	73610.8	1.41486
$78$	5788.22	0.107723
$79$	92704.7	1.67122	0.835611	−	0.549322i	$-0.185114\pi$
$79$	92704.7	1.67122	0.835611	+	0.549322i	$0.185114\pi$
$80$	0	0
$81$	57410.1	0.972246
$82$	17490.3	0.287251
$83$	1849.64	0.0294708	0.0147354	−	0.999891i	$-0.495309\pi$
$83$	1849.64	0.0294708	0.0147354	+	0.999891i	$0.495309\pi$
$84$	−8729.01	−0.134979
$85$	0	0
$86$	−16392.3	−0.238997
$87$	2186.80	0.0309750
$88$	76359.5	1.05113
$89$	−70380.3	−0.941838	−0.470919	−	0.882176i	$-0.656078\pi$
$89$	−70380.3	−0.941838	−0.470919	+	0.882176i	$0.656078\pi$
$90$	0	0
$91$	−54236.8	−0.686579
$92$	−130705.	−1.60999
$93$	−6637.23	−0.0795756
$94$	−19546.9	−0.228170
$95$	0	0
$96$	10796.2	0.119562
$97$	88842.1	0.958714	0.479357	−	0.877620i	$-0.340870\pi$
$97$	88842.1	0.958714	0.479357	+	0.877620i	$0.340870\pi$
$98$	−11038.6	−0.116104
$99$	−142059.	−1.45673
$100$	0	0
$101$	179221.	1.74818	0.874090	−	0.485763i	$-0.161458\pi$
$101$	179221.	1.74818	0.874090	+	0.485763i	$0.161458\pi$
$102$	25635.6	0.243973
$103$	−123443.	−1.14650	−0.573248	−	0.819382i	$-0.694317\pi$
$103$	−123443.	−1.14650	−0.573248	+	0.819382i	$0.694317\pi$
$104$	−56262.1	−0.510074
$105$	0	0
$106$	−221496.	−1.91471
$107$	−98991.7	−0.835871	−0.417936	−	0.908477i	$-0.637246\pi$
$107$	−98991.7	−0.835871	−0.417936	+	0.908477i	$0.637246\pi$
$108$	33849.3	0.279249
$109$	46356.0	0.373715	0.186857	−	0.982387i	$-0.440170\pi$
$109$	46356.0	0.373715	0.186857	+	0.982387i	$0.440170\pi$
$110$	0	0
$111$	5635.97	0.0434171
$112$	−42894.2	−0.323113
$113$	−185076.	−1.36350	−0.681750	−	0.731586i	$-0.738781\pi$
$113$	−185076.	−1.36350	−0.681750	+	0.731586i	$0.738781\pi$
$114$	−8708.54	−0.0627600
$115$	0	0
$116$	−67855.3	−0.468208
$117$	104670.	0.706896
$118$	−379054.	−2.50608
$119$	−240211.	−1.55498
$120$	0	0
$121$	187142.	1.16201
$122$	−186377.	−1.13369
$123$	−2962.61	−0.0176568
$124$	205950.	1.20284
$125$	0	0
$126$	−266250.	−1.49404
$127$	237332.	1.30571	0.652856	−	0.757482i	$-0.273571\pi$
$127$	237332.	1.30571	0.652856	+	0.757482i	$0.273571\pi$
$128$	−237451.	−1.28100
$129$	2776.62	0.0146907
$130$	0	0
$131$	−283496.	−1.44334	−0.721671	−	0.692236i	$-0.756626\pi$
$131$	−283496.	−1.44334	−0.721671	+	0.692236i	$0.756626\pi$
$132$	−41289.9	−0.206257
$133$	81600.9	0.400005
$134$	224049.	1.07791
$135$	0	0
$136$	−249181.	−1.15523
$137$	−330992.	−1.50666	−0.753331	−	0.657642i	$-0.771554\pi$
$137$	−330992.	−1.50666	−0.753331	+	0.657642i	$0.771554\pi$
$138$	37343.9	0.166925
$139$	105975.	0.465230	0.232615	−	0.972569i	$-0.425272\pi$
$139$	105975.	0.465230	0.232615	+	0.972569i	$0.425272\pi$
$140$	0	0
$141$	3310.98	0.0140252
$142$	426277.	1.77407
$143$	−256551.	−1.04914
$144$	82780.0	0.332674
$145$	0	0
$146$	−521300.	−2.02398
$147$	1869.78	0.00713670
$148$	−174881.	−0.656279
$149$	−196700.	−0.725836	−0.362918	−	0.931821i	$-0.618220\pi$
$149$	−196700.	−0.725836	−0.362918	+	0.931821i	$0.618220\pi$
$150$	0	0
$151$	−412513.	−1.47230	−0.736148	−	0.676820i	$-0.763357\pi$
$151$	−412513.	−1.47230	−0.736148	+	0.676820i	$0.763357\pi$
$152$	84648.0	0.297172
$153$	463573.	1.60099
$154$	652594.	2.21739
$155$	0	0
$156$	30422.7	0.100089
$157$	−410971.	−1.33064	−0.665322	−	0.746557i	$-0.731706\pi$
$157$	−410971.	−1.33064	−0.665322	+	0.746557i	$0.731706\pi$
$158$	821871.	2.61915
$159$	37518.4	0.117693
$160$	0	0
$161$	−349920.	−1.06391
$162$	508968.	1.52371
$163$	82488.7	0.243179	0.121589	−	0.992580i	$-0.461201\pi$
$163$	82488.7	0.243179	0.121589	+	0.992580i	$0.461201\pi$
$164$	91928.2	0.266894
$165$	0	0
$166$	16397.9	0.0461868
$167$	380336.	1.05530	0.527650	−	0.849462i	$-0.323073\pi$
$167$	380336.	1.05530	0.527650	+	0.849462i	$0.323073\pi$
$168$	−24241.7	−0.0662660
$169$	−182265.	−0.490892
$170$	0	0
$171$	−157478.	−0.411842
$172$	−86157.1	−0.222060
$173$	−91482.2	−0.232392	−0.116196	−	0.993226i	$-0.537070\pi$
$173$	−91482.2	−0.232392	−0.116196	+	0.993226i	$0.537070\pi$
$174$	19387.0	0.0485443
$175$	0	0
$176$	−202898.	−0.493738
$177$	64206.4	0.154044
$178$	−623955.	−1.47606
$179$	−607572.	−1.41731	−0.708656	−	0.705554i	$-0.750698\pi$
$179$	−607572.	−1.41731	−0.708656	+	0.705554i	$0.750698\pi$
$180$	0	0
$181$	351519.	0.797540	0.398770	−	0.917051i	$-0.369437\pi$
$181$	351519.	0.797540	0.398770	+	0.917051i	$0.369437\pi$
$182$	−480835.	−1.07601
$183$	31569.7	0.0696855
$184$	−362987.	−0.790398
$185$	0	0
$186$	−58842.2	−0.124712
$187$	−1.13624e6	−2.37611
$188$	−102738.	−0.212000
$189$	90620.7	0.184532
$190$	0	0
$191$	−123308.	−0.244573	−0.122286	−	0.992495i	$-0.539023\pi$
$191$	−123308.	−0.244573	−0.122286	+	0.992495i	$0.539023\pi$
$192$	79189.9	0.155030
$193$	163481.	0.315917	0.157959	−	0.987446i	$-0.449509\pi$
$193$	163481.	0.315917	0.157959	+	0.987446i	$0.449509\pi$
$194$	787627.	1.50251
$195$	0	0
$196$	−58018.3	−0.107876
$197$	−581124.	−1.06685	−0.533425	−	0.845848i	$-0.679095\pi$
$197$	−581124.	−1.06685	−0.533425	+	0.845848i	$0.679095\pi$
$198$	−1.25942e6	−2.28300
$199$	351526.	0.629253	0.314627	−	0.949216i	$-0.398121\pi$
$199$	351526.	0.629253	0.314627	+	0.949216i	$0.398121\pi$
$200$	0	0
$201$	−37950.9	−0.0662570
$202$	1.58888e6	2.73976
$203$	−181660.	−0.309400
$204$	134740.	0.226683
$205$	0	0
$206$	−1.09438e6	−1.79680
$207$	675297.	1.09539
$208$	149497.	0.239592
$209$	385988.	0.611235
$210$	0	0
$211$	327480.	0.506383	0.253192	−	0.967416i	$-0.418520\pi$
$211$	327480.	0.506383	0.253192	+	0.967416i	$0.418520\pi$
$212$	−1.16418e6	−1.77901
$213$	−72205.4	−0.109049
$214$	−877608.	−1.30998
$215$	0	0
$216$	94004.6	0.137093
$217$	551364.	0.794857
$218$	410968.	0.585689
$219$	88301.0	0.124410
$220$	0	0
$221$	837191.	1.15304
$222$	49965.5	0.0680437
$223$	−619602.	−0.834355	−0.417177	−	0.908825i	$-0.636981\pi$
$223$	−619602.	−0.834355	−0.417177	+	0.908825i	$0.636981\pi$
$224$	−896853.	−1.19427
$225$	0	0
$226$	−1.64079e6	−2.13689
$227$	757446.	0.975635	0.487817	−	0.872946i	$-0.337793\pi$
$227$	757446.	0.975635	0.487817	+	0.872946i	$0.337793\pi$
$228$	−45771.8	−0.0583123
$229$	1.21739e6	1.53405	0.767025	−	0.641617i	$-0.221736\pi$
$229$	1.21739e6	1.53405	0.767025	+	0.641617i	$0.221736\pi$
$230$	0	0
$231$	−110540.	−0.136299
$232$	−188444.	−0.229860
$233$	−379715.	−0.458213	−0.229107	−	0.973401i	$-0.573580\pi$
$233$	−379715.	−0.458213	−0.229107	+	0.973401i	$0.573580\pi$
$234$	927945.	1.10785
$235$	0	0
$236$	−1.99229e6	−2.32848
$237$	−139214.	−0.160994
$238$	−2.12958e6	−2.43698
$239$	281756.	0.319064	0.159532	−	0.987193i	$-0.449001\pi$
$239$	281756.	0.319064	0.159532	+	0.987193i	$0.449001\pi$
$240$	0	0
$241$	−1.12524e6	−1.24796	−0.623981	−	0.781439i	$-0.714486\pi$
$241$	−1.12524e6	−1.24796	−0.623981	+	0.781439i	$0.714486\pi$
$242$	1.65911e6	1.82111
$243$	−262736.	−0.285432
$244$	−979590.	−1.05334
$245$	0	0
$246$	−26264.9	−0.0276719
$247$	−284398.	−0.296609
$248$	571953.	0.590515
$249$	−2777.58	−0.00283902
$250$	0	0
$251$	−675567.	−0.676837	−0.338419	−	0.940996i	$-0.609892\pi$
$251$	−675567.	−0.676837	−0.338419	+	0.940996i	$0.609892\pi$
$252$	−1.39940e6	−1.38816
$253$	−1.65519e6	−1.62572
$254$	2.10406e6	2.04632
$255$	0	0
$256$	−417633.	−0.398286
$257$	−1.85623e6	−1.75306	−0.876532	−	0.481343i	$-0.840149\pi$
$257$	−1.85623e6	−1.75306	−0.876532	+	0.481343i	$0.840149\pi$
$258$	24616.1	0.0230234
$259$	−468188.	−0.433681
$260$	0	0
$261$	350580.	0.318556
$262$	−2.51333e6	−2.26202
$263$	1.54160e6	1.37430	0.687149	−	0.726516i	$-0.258862\pi$
$263$	1.54160e6	1.37430	0.687149	+	0.726516i	$0.258862\pi$
$264$	−114668.	−0.101259
$265$	0	0
$266$	723430.	0.626892
$267$	105689.	0.0907304
$268$	1.17759e6	1.00152
$269$	1.16775e6	0.983945	0.491973	−	0.870611i	$-0.336276\pi$
$269$	1.16775e6	0.983945	0.491973	+	0.870611i	$0.336276\pi$
$270$	0	0
$271$	−1.20256e6	−0.994684	−0.497342	−	0.867555i	$-0.665691\pi$
$271$	−1.20256e6	−0.994684	−0.497342	+	0.867555i	$0.665691\pi$
$272$	662109.	0.542634
$273$	81446.8	0.0661405
$274$	−2.93440e6	−2.36125
$275$	0	0
$276$	196278.	0.155095
$277$	504463.	0.395030	0.197515	−	0.980300i	$-0.436713\pi$
$277$	504463.	0.395030	0.197515	+	0.980300i	$0.436713\pi$
$278$	939521.	0.729112
$279$	−1.06406e6	−0.818378
$280$	0	0
$281$	1.52712e6	1.15374	0.576870	−	0.816836i	$-0.304274\pi$
$281$	1.52712e6	1.15374	0.576870	+	0.816836i	$0.304274\pi$
$282$	29353.4	0.0219804
$283$	−1.81916e6	−1.35022	−0.675110	−	0.737717i	$-0.735904\pi$
$283$	−1.81916e6	−1.35022	−0.675110	+	0.737717i	$0.735904\pi$
$284$	2.24049e6	1.64834
$285$	0	0
$286$	−2.27444e6	−1.64422
$287$	246108.	0.176369
$288$	1.73080e6	1.22961
$289$	2.28799e6	1.61143
$290$	0	0
$291$	−133413.	−0.0923562
$292$	−2.73993e6	−1.88054
$293$	2.35975e6	1.60582	0.802911	−	0.596099i	$-0.203283\pi$
$293$	2.35975e6	1.60582	0.802911	+	0.596099i	$0.203283\pi$
$294$	16576.5	0.0111847
$295$	0	0
$296$	−485671.	−0.322190
$297$	428654.	0.281978
$298$	−1.74384e6	−1.13754
$299$	1.21955e6	0.788901
$300$	0	0
$301$	−230658.	−0.146741
$302$	−3.65712e6	−2.30740
$303$	−269135.	−0.168408
$304$	−224922.	−0.139588
$305$	0	0
$306$	4.10979e6	2.50909
$307$	2.59790e6	1.57317	0.786587	−	0.617480i	$-0.211846\pi$
$307$	2.59790e6	1.57317	0.786587	+	0.617480i	$0.211846\pi$
$308$	3.43001e6	2.06024
$309$	185373.	0.110446
$310$	0	0
$311$	−2.21471e6	−1.29842	−0.649212	−	0.760607i	$-0.724901\pi$
$311$	−2.21471e6	−1.29842	−0.649212	+	0.760607i	$0.724901\pi$
$312$	84488.2	0.0491371
$313$	2.43630e6	1.40563	0.702813	−	0.711374i	$-0.251927\pi$
$313$	2.43630e6	1.40563	0.702813	+	0.711374i	$0.251927\pi$
$314$	−3.64345e6	−2.08540
$315$	0	0
$316$	4.31972e6	2.43354
$317$	2.59108e6	1.44821	0.724106	−	0.689689i	$-0.242253\pi$
$317$	2.59108e6	1.44821	0.724106	+	0.689689i	$0.242253\pi$
$318$	332619.	0.184450
$319$	−859290.	−0.472784
$320$	0	0
$321$	148655.	0.0805223
$322$	−3.10220e6	−1.66737
$323$	−1.25958e6	−0.671767
$324$	2.67512e6	1.41573
$325$	0	0
$326$	731301.	0.381112
$327$	−69612.3	−0.0360012
$328$	255298.	0.131028
$329$	−275047.	−0.140093
$330$	0	0
$331$	−2.49362e6	−1.25101	−0.625503	−	0.780221i	$-0.715106\pi$
$331$	−2.49362e6	−1.25101	−0.625503	+	0.780221i	$0.715106\pi$
$332$	86186.7	0.0429136
$333$	903537.	0.446514
$334$	3.37186e6	1.65388
$335$	0	0
$336$	64413.8	0.0311265
$337$	159199.	0.0763600	0.0381800	−	0.999271i	$-0.487844\pi$
$337$	159199.	0.0763600	0.0381800	+	0.999271i	$0.487844\pi$
$338$	−1.61586e6	−0.769331
$339$	277927.	0.131350
$340$	0	0
$341$	2.60806e6	1.21460
$342$	−1.39612e6	−0.645442
$343$	−2.25195e6	−1.03353
$344$	−239271.	−0.109017
$345$	0	0
$346$	−811033.	−0.364207
$347$	−4.35264e6	−1.94057	−0.970285	−	0.241966i	$-0.922208\pi$
$347$	−4.35264e6	−1.94057	−0.970285	+	0.241966i	$0.922208\pi$
$348$	101897.	0.0451040
$349$	−502730.	−0.220938	−0.110469	−	0.993880i	$-0.535235\pi$
$349$	−502730.	−0.220938	−0.110469	+	0.993880i	$0.535235\pi$
$350$	0	0
$351$	−315834.	−0.136833
$352$	−4.24229e6	−1.82492
$353$	990932.	0.423260	0.211630	−	0.977350i	$-0.432123\pi$
$353$	990932.	0.423260	0.211630	+	0.977350i	$0.432123\pi$
$354$	569220.	0.241419
$355$	0	0
$356$	−3.27948e6	−1.37145
$357$	360721.	0.149796
$358$	−5.38641e6	−2.22122
$359$	−1.78805e6	−0.732223	−0.366112	−	0.930571i	$-0.619311\pi$
$359$	−1.78805e6	−0.732223	−0.366112	+	0.930571i	$0.619311\pi$
$360$	0	0
$361$	−2.04821e6	−0.827194
$362$	3.11638e6	1.24991
$363$	−281029.	−0.111940
$364$	−2.52725e6	−0.999758
$365$	0	0
$366$	279880.	0.109212
$367$	68021.3	0.0263621	0.0131810	−	0.999913i	$-0.495804\pi$
$367$	68021.3	0.0263621	0.0131810	+	0.999913i	$0.495804\pi$
$368$	964507.	0.371267
$369$	−474954.	−0.181588
$370$	0	0
$371$	−3.11671e6	−1.17560
$372$	−309272.	−0.115873
$373$	949477.	0.353356	0.176678	−	0.984269i	$-0.443465\pi$
$373$	949477.	0.353356	0.176678	+	0.984269i	$0.443465\pi$
$374$	−1.00733e7	−3.72387
$375$	0	0
$376$	−285318.	−0.104078
$377$	633130.	0.229424
$378$	803395.	0.289201
$379$	2.53069e6	0.904982	0.452491	−	0.891769i	$-0.350535\pi$
$379$	2.53069e6	0.904982	0.452491	+	0.891769i	$0.350535\pi$
$380$	0	0
$381$	−356399.	−0.125784
$382$	−1.09318e6	−0.383297
$383$	−500631.	−0.174390	−0.0871948	−	0.996191i	$-0.527790\pi$
$383$	−500631.	−0.174390	−0.0871948	+	0.996191i	$0.527790\pi$
$384$	356578.	0.123403
$385$	0	0
$386$	1.44933e6	0.495108
$387$	445137.	0.151083
$388$	4.13974e6	1.39603
$389$	−4.08546e6	−1.36889	−0.684443	−	0.729066i	$-0.739955\pi$
$389$	−4.08546e6	−1.36889	−0.684443	+	0.729066i	$0.739955\pi$
$390$	0	0
$391$	5.40130e6	1.78672
$392$	−161125.	−0.0529601
$393$	425723.	0.139042
$394$	−5.15194e6	−1.67198
$395$	0	0
$396$	−6.61944e6	−2.12121
$397$	−953651.	−0.303678	−0.151839	−	0.988405i	$-0.548520\pi$
$397$	−953651.	−0.303678	−0.151839	+	0.988405i	$0.548520\pi$
$398$	3.11645e6	0.986171
$399$	−122539.	−0.0385338
$400$	0	0
$401$	4.81115e6	1.49413	0.747064	−	0.664752i	$-0.231463\pi$
$401$	4.81115e6	1.49413	0.747064	+	0.664752i	$0.231463\pi$
$402$	−336452.	−0.103838
$403$	−1.92163e6	−0.589397
$404$	8.35110e6	2.54560
$405$	0	0
$406$	−1.61051e6	−0.484894
$407$	−2.21462e6	−0.662694
$408$	374192.	0.111287
$409$	−1.42551e6	−0.421370	−0.210685	−	0.977554i	$-0.567569\pi$
$409$	−1.42551e6	−0.421370	−0.210685	+	0.977554i	$0.567569\pi$
$410$	0	0
$411$	497046.	0.145142
$412$	−5.75201e6	−1.66946
$413$	−5.33372e6	−1.53870
$414$	5.98682e6	1.71670
$415$	0	0
$416$	3.12575e6	0.885565
$417$	−159142.	−0.0448171
$418$	3.42197e6	0.957933
$419$	2.91631e6	0.811520	0.405760	−	0.913980i	$-0.367007\pi$
$419$	2.91631e6	0.811520	0.405760	+	0.913980i	$0.367007\pi$
$420$	0	0
$421$	6.91223e6	1.90070	0.950349	−	0.311186i	$-0.100726\pi$
$421$	6.91223e6	1.90070	0.950349	+	0.311186i	$0.100726\pi$
$422$	2.90327e6	0.793608
$423$	530803.	0.144239
$424$	−3.23309e6	−0.873380
$425$	0	0
$426$	−640135.	−0.170902
$427$	−2.62254e6	−0.696068
$428$	−4.61268e6	−1.21715
$429$	385259.	0.101067
$430$	0	0
$431$	2.58575e6	0.670493	0.335246	−	0.942131i	$-0.391180\pi$
$431$	2.58575e6	0.670493	0.335246	+	0.942131i	$0.391180\pi$
$432$	−249784.	−0.0643954
$433$	3.71200e6	0.951455	0.475727	−	0.879593i	$-0.342185\pi$
$433$	3.71200e6	0.951455	0.475727	+	0.879593i	$0.342185\pi$
$434$	4.88810e6	1.24571
$435$	0	0
$436$	2.16003e6	0.544182
$437$	−1.83485e6	−0.459619
$438$	782830.	0.194977
$439$	−415772.	−0.102966	−0.0514830	−	0.998674i	$-0.516395\pi$
$439$	−415772.	−0.102966	−0.0514830	+	0.998674i	$0.516395\pi$
$440$	0	0
$441$	299756.	0.0733959
$442$	7.42209e6	1.80705
$443$	602736.	0.145921	0.0729605	−	0.997335i	$-0.476755\pi$
$443$	602736.	0.145921	0.0729605	+	0.997335i	$0.476755\pi$
$444$	262617.	0.0632216
$445$	0	0
$446$	−5.49307e6	−1.30761
$447$	295382.	0.0699223
$448$	−6.57841e6	−1.54855
$449$	1.84907e6	0.432849	0.216424	−	0.976299i	$-0.430560\pi$
$449$	1.84907e6	0.432849	0.216424	+	0.976299i	$0.430560\pi$
$450$	0	0
$451$	1.16414e6	0.269503
$452$	−8.62393e6	−1.98545
$453$	619466.	0.141831
$454$	6.71512e6	1.52902
$455$	0	0
$456$	−127115.	−0.0286276
$457$	7.35472e6	1.64731	0.823655	−	0.567091i	$-0.191931\pi$
$457$	7.35472e6	1.64731	0.823655	+	0.567091i	$0.191931\pi$
$458$	1.07927e7	2.40418
$459$	−1.39881e6	−0.309903
$460$	0	0
$461$	−3.51032e6	−0.769298	−0.384649	−	0.923063i	$-0.625678\pi$
$461$	−3.51032e6	−0.769298	−0.384649	+	0.923063i	$0.625678\pi$
$462$	−979993.	−0.213608
$463$	6.86607e6	1.48852	0.744262	−	0.667888i	$-0.232801\pi$
$463$	6.86607e6	1.48852	0.744262	+	0.667888i	$0.232801\pi$
$464$	500723.	0.107970
$465$	0	0
$466$	−3.36635e6	−0.718116
$467$	−5.66490e6	−1.20199	−0.600993	−	0.799254i	$-0.705228\pi$
$467$	−5.66490e6	−1.20199	−0.600993	+	0.799254i	$0.705228\pi$
$468$	4.87724e6	1.02934
$469$	3.15263e6	0.661821
$470$	0	0
$471$	617150.	0.128185
$472$	−5.53289e6	−1.14313
$473$	−1.09106e6	−0.224230
$474$	−1.23419e6	−0.252312
$475$	0	0
$476$	−1.11930e7	−2.26427
$477$	6.01481e6	1.21039
$478$	2.49790e6	0.500041
$479$	4.55961e6	0.908006	0.454003	−	0.891000i	$-0.349996\pi$
$479$	4.55961e6	0.908006	0.454003	+	0.891000i	$0.349996\pi$
$480$	0	0
$481$	1.63174e6	0.321580
$482$	−9.97576e6	−1.95582
$483$	525470.	0.102490
$484$	8.72019e6	1.69205
$485$	0	0
$486$	−2.32928e6	−0.447332
$487$	9.17550e6	1.75310	0.876552	−	0.481308i	$-0.159838\pi$
$487$	9.17550e6	1.75310	0.876552	+	0.481308i	$0.159838\pi$
$488$	−2.72047e6	−0.517123
$489$	−123872.	−0.0234262
$490$	0	0
$491$	3.41449e6	0.639179	0.319589	−	0.947556i	$-0.396455\pi$
$491$	3.41449e6	0.639179	0.319589	+	0.947556i	$0.396455\pi$
$492$	−138048.	−0.0257108
$493$	2.80408e6	0.519605
$494$	−2.52132e6	−0.464848
$495$	0	0
$496$	−1.51976e6	−0.277377
$497$	5.99820e6	1.08926
$498$	−24624.5	−0.00444933
$499$	−1.99364e6	−0.358423	−0.179211	−	0.983811i	$-0.557355\pi$
$499$	−1.99364e6	−0.358423	−0.179211	+	0.983811i	$0.557355\pi$
$500$	0	0
$501$	−571146.	−0.101661
$502$	−5.98922e6	−1.06074
$503$	−9.15685e6	−1.61371	−0.806856	−	0.590748i	$-0.798833\pi$
$503$	−9.15685e6	−1.61371	−0.806856	+	0.590748i	$0.798833\pi$
$504$	−3.88634e6	−0.681498
$505$	0	0
$506$	−1.46740e7	−2.54785
$507$	273705.	0.0472893
$508$	1.10589e7	1.90130
$509$	6.56697e6	1.12349	0.561746	−	0.827309i	$-0.310130\pi$
$509$	6.56697e6	1.12349	0.561746	+	0.827309i	$0.310130\pi$
$510$	0	0
$511$	−7.33529e6	−1.24270
$512$	3.89593e6	0.656805
$513$	475182.	0.0797199
$514$	−1.64563e7	−2.74742
$515$	0	0
$516$	129381.	0.0213918
$517$	−1.30103e6	−0.214072
$518$	−4.15070e6	−0.679669
$519$	137378.	0.0223871
$520$	0	0
$521$	6.45805e6	1.04234	0.521168	−	0.853454i	$-0.325497\pi$
$521$	6.45805e6	1.04234	0.521168	+	0.853454i	$0.325497\pi$
$522$	3.10805e6	0.499243
$523$	−184405.	−0.0294794	−0.0147397	−	0.999891i	$-0.504692\pi$
$523$	−184405.	−0.0294794	−0.0147397	+	0.999891i	$0.504692\pi$
$524$	−1.32100e7	−2.10171
$525$	0	0
$526$	1.36670e7	2.15381
$527$	−8.51076e6	−1.33488
$528$	304690.	0.0475634
$529$	1.43185e6	0.222463
$530$	0	0
$531$	1.02933e7	1.58424
$532$	3.80232e6	0.582465
$533$	−857745.	−0.130780
$534$	936986.	0.142194
$535$	0	0
$536$	3.27035e6	0.491680
$537$	912384.	0.136534
$538$	1.03527e7	1.54205
$539$	−734719.	−0.108931
$540$	0	0
$541$	−7.10369e6	−1.04350	−0.521748	−	0.853100i	$-0.674720\pi$
$541$	−7.10369e6	−1.04350	−0.521748	+	0.853100i	$0.674720\pi$
$542$	−1.06613e7	−1.55888
$543$	−527872.	−0.0768297
$544$	1.38437e7	2.00565
$545$	0	0
$546$	722064.	0.103656
$547$	9.75210e6	1.39357	0.696787	−	0.717279i	$-0.254612\pi$
$547$	9.75210e6	1.39357	0.696787	+	0.717279i	$0.254612\pi$
$548$	−1.54231e7	−2.19392
$549$	5.06113e6	0.716666
$550$	0	0
$551$	−952562.	−0.133664
$552$	545093.	0.0761417
$553$	1.15647e7	1.60813
$554$	4.47230e6	0.619094
$555$	0	0
$556$	4.93809e6	0.677441
$557$	2.28586e6	0.312185	0.156092	−	0.987742i	$-0.450110\pi$
$557$	2.28586e6	0.312185	0.156092	+	0.987742i	$0.450110\pi$
$558$	−9.43336e6	−1.28257
$559$	803896.	0.108810
$560$	0	0
$561$	1.70628e6	0.228899
$562$	1.35386e7	1.80815
$563$	−2.31099e6	−0.307275	−0.153638	−	0.988127i	$-0.549099\pi$
$563$	−2.31099e6	−0.307275	−0.153638	+	0.988127i	$0.549099\pi$
$564$	154280.	0.0204227
$565$	0	0
$566$	−1.61277e7	−2.11608
$567$	7.16176e6	0.935539
$568$	6.22218e6	0.809230
$569$	1.17226e7	1.51790	0.758949	−	0.651151i	$-0.225713\pi$
$569$	1.17226e7	1.51790	0.758949	+	0.651151i	$0.225713\pi$
$570$	0	0
$571$	−5.10057e6	−0.654679	−0.327340	−	0.944907i	$-0.606152\pi$
$571$	−5.10057e6	−0.654679	−0.327340	+	0.944907i	$0.606152\pi$
$572$	−1.19544e7	−1.52770
$573$	185170.	0.0235605
$574$	2.18186e6	0.276406
$575$	0	0
$576$	1.26954e7	1.59438
$577$	−599429.	−0.0749546	−0.0374773	−	0.999297i	$-0.511932\pi$
$577$	−599429.	−0.0749546	−0.0374773	+	0.999297i	$0.511932\pi$
$578$	2.02841e7	2.52544
$579$	−245497.	−0.0304334
$580$	0	0
$581$	230737.	0.0283581
$582$	−1.18277e6	−0.144741
$583$	−1.47426e7	−1.79640
$584$	−7.60920e6	−0.923224
$585$	0	0
$586$	2.09203e7	2.51666
$587$	−4.32755e6	−0.518379	−0.259189	−	0.965826i	$-0.583455\pi$
$587$	−4.32755e6	−0.518379	−0.259189	+	0.965826i	$0.583455\pi$
$588$	87125.5	0.0103921
$589$	2.89115e6	0.343386
$590$	0	0
$591$	872667.	0.102773
$592$	1.29050e6	0.151340
$593$	1.59099e7	1.85794	0.928968	−	0.370160i	$-0.120697\pi$
$593$	1.59099e7	1.85794	0.928968	+	0.370160i	$0.120697\pi$
$594$	3.80022e6	0.441918
$595$	0	0
$596$	−9.16555e6	−1.05692
$597$	−527883.	−0.0606181
$598$	1.08119e7	1.23637
$599$	73202.1	0.00833598	0.00416799	−	0.999991i	$-0.498673\pi$
$599$	73202.1	0.00833598	0.00416799	+	0.999991i	$0.498673\pi$
$600$	0	0
$601$	−3.77328e6	−0.426121	−0.213060	−	0.977039i	$-0.568343\pi$
$601$	−3.77328e6	−0.426121	−0.213060	+	0.977039i	$0.568343\pi$
$602$	−2.04489e6	−0.229974
$603$	−6.08414e6	−0.681406
$604$	−1.92217e7	−2.14387
$605$	0	0
$606$	−2.38601e6	−0.263931
$607$	−3.39314e6	−0.373792	−0.186896	−	0.982380i	$-0.559843\pi$
$607$	−3.39314e6	−0.373792	−0.186896	+	0.982380i	$0.559843\pi$
$608$	−4.70277e6	−0.515935
$609$	272797.	0.0298055
$610$	0	0
$611$	958605.	0.103881
$612$	2.16009e7	2.33128
$613$	−5.93491e6	−0.637915	−0.318957	−	0.947769i	$-0.603333\pi$
$613$	−5.93491e6	−0.637915	−0.318957	+	0.947769i	$0.603333\pi$
$614$	2.30316e7	2.46549
$615$	0	0
$616$	9.52564e6	1.01145
$617$	1.07871e7	1.14076	0.570379	−	0.821382i	$-0.306796\pi$
$617$	1.07871e7	1.14076	0.570379	+	0.821382i	$0.306796\pi$
$618$	1.64341e6	0.173092
$619$	9.97743e6	1.04663	0.523313	−	0.852140i	$-0.324696\pi$
$619$	9.97743e6	1.04663	0.523313	+	0.852140i	$0.324696\pi$
$620$	0	0
$621$	−2.03767e6	−0.212034
$622$	−1.96345e7	−2.03490
$623$	−8.77976e6	−0.906280
$624$	−224497.	−0.0230807
$625$	0	0
$626$	2.15990e7	2.20291
$627$	−579634.	−0.0588823
$628$	−1.91498e7	−1.93761
$629$	7.22687e6	0.728322
$630$	0	0
$631$	−3.42994e6	−0.342936	−0.171468	−	0.985190i	$-0.554851\pi$
$631$	−3.42994e6	−0.342936	−0.171468	+	0.985190i	$0.554851\pi$
$632$	1.19965e7	1.19471
$633$	−491774.	−0.0487816
$634$	2.29711e7	2.26965
$635$	0	0
$636$	1.74823e6	0.171378
$637$	541345.	0.0528598
$638$	−7.61801e6	−0.740952
$639$	−1.15757e7	−1.12149
$640$	0	0
$641$	−1.41725e7	−1.36239	−0.681193	−	0.732104i	$-0.738538\pi$
$641$	−1.41725e7	−1.36239	−0.681193	+	0.732104i	$0.738538\pi$
$642$	1.31789e6	0.126195
$643$	−4.26802e6	−0.407098	−0.203549	−	0.979065i	$-0.565248\pi$
$643$	−4.26802e6	−0.407098	−0.203549	+	0.979065i	$0.565248\pi$
$644$	−1.63051e7	−1.54920
$645$	0	0
$646$	−1.11667e7	−1.05280
$647$	−1.26038e7	−1.18370	−0.591849	−	0.806049i	$-0.701602\pi$
$647$	−1.26038e7	−1.18370	−0.591849	+	0.806049i	$0.701602\pi$
$648$	7.42919e6	0.695031
$649$	−2.52295e7	−2.35124
$650$	0	0
$651$	−827977.	−0.0765712
$652$	3.84369e6	0.354103
$653$	−1.67724e7	−1.53926	−0.769631	−	0.638489i	$-0.779560\pi$
$653$	−1.67724e7	−1.53926	−0.769631	+	0.638489i	$0.779560\pi$
$654$	−617146.	−0.0564214
$655$	0	0
$656$	−678364.	−0.0615465
$657$	1.41561e7	1.27947
$658$	−2.43842e6	−0.219556
$659$	−1.72113e7	−1.54383	−0.771915	−	0.635725i	$-0.780701\pi$
$659$	−1.72113e7	−1.54383	−0.771915	+	0.635725i	$0.780701\pi$
$660$	0	0
$661$	5.67723e6	0.505397	0.252699	−	0.967545i	$-0.418682\pi$
$661$	5.67723e6	0.505397	0.252699	+	0.967545i	$0.418682\pi$
$662$	−2.21071e7	−1.96059
$663$	−1.25720e6	−0.111076
$664$	239353.	0.0210678
$665$	0	0
$666$	8.01028e6	0.699781
$667$	4.08476e6	0.355510
$668$	1.77223e7	1.53667
$669$	930449.	0.0803762
$670$	0	0
$671$	−1.24051e7	−1.06364
$672$	1.34679e6	0.115048
$673$	−4.92989e6	−0.419565	−0.209783	−	0.977748i	$-0.567276\pi$
$673$	−4.92989e6	−0.419565	−0.209783	+	0.977748i	$0.567276\pi$
$674$	1.41137e6	0.119672
$675$	0	0
$676$	−8.49292e6	−0.714809
$677$	1.25498e7	1.05236	0.526180	−	0.850373i	$-0.323624\pi$
$677$	1.25498e7	1.05236	0.526180	+	0.850373i	$0.323624\pi$
$678$	2.46395e6	0.205854
$679$	1.10828e7	0.922519
$680$	0	0
$681$	−1.13745e6	−0.0939861
$682$	2.31217e7	1.90352
$683$	7.08503e6	0.581152	0.290576	−	0.956852i	$-0.406153\pi$
$683$	7.08503e6	0.581152	0.290576	+	0.956852i	$0.406153\pi$
$684$	−7.33795e6	−0.599701
$685$	0	0
$686$	−1.99646e7	−1.61976
$687$	−1.82813e6	−0.147780
$688$	635777.	0.0512075
$689$	1.08625e7	0.871726
$690$	0	0
$691$	−9.40053e6	−0.748958	−0.374479	−	0.927235i	$-0.622178\pi$
$691$	−9.40053e6	−0.748958	−0.374479	+	0.927235i	$0.622178\pi$
$692$	−4.26276e6	−0.338396
$693$	−1.77214e7	−1.40173
$694$	−3.85882e7	−3.04128
$695$	0	0
$696$	282984.	0.0221431
$697$	−3.79888e6	−0.296192
$698$	−4.45694e6	−0.346256
$699$	570213.	0.0441412
$700$	0	0
$701$	2.14131e7	1.64583	0.822914	−	0.568167i	$-0.192347\pi$
$701$	2.14131e7	1.64583	0.822914	+	0.568167i	$0.192347\pi$
$702$	−2.80002e6	−0.214446
$703$	−2.45501e6	−0.187355
$704$	−3.11172e7	−2.36629
$705$	0	0
$706$	8.78508e6	0.663337
$707$	2.23574e7	1.68218
$708$	2.99180e6	0.224310
$709$	1.70494e7	1.27377	0.636887	−	0.770957i	$-0.280222\pi$
$709$	1.70494e7	1.27377	0.636887	+	0.770957i	$0.280222\pi$
$710$	0	0
$711$	−2.23182e7	−1.65571
$712$	−9.10761e6	−0.673293
$713$	−1.23978e7	−0.913316
$714$	3.19797e6	0.234762
$715$	0	0
$716$	−2.83108e7	−2.06381
$717$	−423110.	−0.0307365
$718$	−1.58519e7	−1.14755
$719$	−1.59376e7	−1.14974	−0.574872	−	0.818243i	$-0.694948\pi$
$719$	−1.59376e7	−1.14974	−0.574872	+	0.818243i	$0.694948\pi$
$720$	0	0
$721$	−1.53991e7	−1.10321
$722$	−1.81584e7	−1.29639
$723$	1.68976e6	0.120220
$724$	1.63796e7	1.16133
$725$	0	0
$726$	−2.49146e6	−0.175433
$727$	−8.26202e6	−0.579763	−0.289881	−	0.957063i	$-0.593616\pi$
$727$	−8.26202e6	−0.579763	−0.289881	+	0.957063i	$0.593616\pi$
$728$	−7.01855e6	−0.490816
$729$	−1.35561e7	−0.944749
$730$	0	0
$731$	3.56039e6	0.246436
$732$	1.47104e6	0.101472
$733$	−6.20168e6	−0.426334	−0.213167	−	0.977016i	$-0.568378\pi$
$733$	−6.20168e6	−0.426334	−0.213167	+	0.977016i	$0.568378\pi$
$734$	603041.	0.0413149
$735$	0	0
$736$	2.01664e7	1.37225
$737$	1.49126e7	1.01131
$738$	−4.21069e6	−0.284586
$739$	−1.19003e7	−0.801578	−0.400789	−	0.916170i	$-0.631264\pi$
$739$	−1.19003e7	−0.801578	−0.400789	+	0.916170i	$0.631264\pi$
$740$	0	0
$741$	427077.	0.0285734
$742$	−2.76311e7	−1.84242
$743$	−7.41722e6	−0.492912	−0.246456	−	0.969154i	$-0.579266\pi$
$743$	−7.41722e6	−0.492912	−0.246456	+	0.969154i	$0.579266\pi$
$744$	−858895.	−0.0568863
$745$	0	0
$746$	8.41756e6	0.553782
$747$	−445291.	−0.0291973
$748$	−5.29451e7	−3.45996
$749$	−1.23489e7	−0.804314
$750$	0	0
$751$	1.46691e7	0.949084	0.474542	−	0.880233i	$-0.342614\pi$
$751$	1.46691e7	0.949084	0.474542	+	0.880233i	$0.342614\pi$
$752$	758131.	0.0488877
$753$	1.01449e6	0.0652020
$754$	5.61299e6	0.359556
$755$	0	0
$756$	4.22261e6	0.268706
$757$	−2.41775e6	−0.153346	−0.0766729	−	0.997056i	$-0.524430\pi$
$757$	−2.41775e6	−0.153346	−0.0766729	+	0.997056i	$0.524430\pi$
$758$	2.24357e7	1.41830
$759$	2.48558e6	0.156611
$760$	0	0
$761$	2.32422e7	1.45484	0.727422	−	0.686191i	$-0.240718\pi$
$761$	2.32422e7	1.45484	0.727422	+	0.686191i	$0.240718\pi$
$762$	−3.15964e6	−0.197129
$763$	5.78279e6	0.359605
$764$	−5.74574e6	−0.356133
$765$	0	0
$766$	−4.43833e6	−0.273305
$767$	1.85893e7	1.14097
$768$	627155.	0.0383682
$769$	−2.55432e7	−1.55761	−0.778806	−	0.627265i	$-0.784174\pi$
$769$	−2.55432e7	−1.55761	−0.778806	+	0.627265i	$0.784174\pi$
$770$	0	0
$771$	2.78747e6	0.168879
$772$	7.61765e6	0.460021
$773$	−1.72159e7	−1.03629	−0.518146	−	0.855292i	$-0.673378\pi$
$773$	−1.72159e7	−1.03629	−0.518146	+	0.855292i	$0.673378\pi$
$774$	3.94635e6	0.236779
$775$	0	0
$776$	1.14967e7	0.685358
$777$	703072.	0.0417780
$778$	−3.62196e7	−2.14533
$779$	1.29050e6	0.0761930
$780$	0	0
$781$	2.83727e7	1.66446
$782$	4.78851e7	2.80016
$783$	−1.05785e6	−0.0616625
$784$	428134.	0.0248765
$785$	0	0
$786$	3.77424e6	0.217908
$787$	1.48812e7	0.856449	0.428225	−	0.903672i	$-0.359139\pi$
$787$	1.48812e7	0.856449	0.428225	+	0.903672i	$0.359139\pi$
$788$	−2.70784e7	−1.55349
$789$	−2.31500e6	−0.132391
$790$	0	0
$791$	−2.30878e7	−1.31202
$792$	−1.83832e7	−1.04138
$793$	9.14016e6	0.516144
$794$	−8.45456e6	−0.475927
$795$	0	0
$796$	1.63799e7	0.916283
$797$	−5.81135e6	−0.324064	−0.162032	−	0.986785i	$-0.551805\pi$
$797$	−5.81135e6	−0.324064	−0.162032	+	0.986785i	$0.551805\pi$
$798$	−1.08637e6	−0.0603906
$799$	4.24559e6	0.235272
$800$	0	0
$801$	1.69437e7	0.933098
$802$	4.26531e7	2.34161
$803$	−3.46973e7	−1.89892
$804$	−1.76838e6	−0.0964796
$805$	0	0
$806$	−1.70362e7	−0.923708
$807$	−1.75360e6	−0.0947867
$808$	2.31922e7	1.24972
$809$	−2.41530e7	−1.29748	−0.648740	−	0.761010i	$-0.724704\pi$
$809$	−2.41530e7	−1.29748	−0.648740	+	0.761010i	$0.724704\pi$
$810$	0	0
$811$	2.14467e7	1.14501	0.572505	−	0.819901i	$-0.305972\pi$
$811$	2.14467e7	1.14501	0.572505	+	0.819901i	$0.305972\pi$
$812$	−8.46476e6	−0.450531
$813$	1.80588e6	0.0958212
$814$	−1.96336e7	−1.03858
$815$	0	0
$816$	−994281.	−0.0522737
$817$	−1.20949e6	−0.0633936
$818$	−1.26379e7	−0.660375
$819$	1.30572e7	0.680208
$820$	0	0
$821$	−1.06313e7	−0.550465	−0.275232	−	0.961378i	$-0.588755\pi$
$821$	−1.06313e7	−0.550465	−0.275232	+	0.961378i	$0.588755\pi$
$822$	4.40655e6	0.227467
$823$	871518.	0.0448515	0.0224257	−	0.999749i	$-0.492861\pi$
$823$	871518.	0.0448515	0.0224257	+	0.999749i	$0.492861\pi$
$824$	−1.59742e7	−0.819597
$825$	0	0
$826$	−4.72859e7	−2.41147
$827$	1.70354e7	0.866141	0.433071	−	0.901360i	$-0.357430\pi$
$827$	1.70354e7	0.866141	0.433071	+	0.901360i	$0.357430\pi$
$828$	3.14665e7	1.59504
$829$	5.65731e6	0.285906	0.142953	−	0.989729i	$-0.454340\pi$
$829$	5.65731e6	0.285906	0.142953	+	0.989729i	$0.454340\pi$
$830$	0	0
$831$	−757546.	−0.0380546
$832$	2.29273e7	1.14827
$833$	2.39758e6	0.119718
$834$	−1.41087e6	−0.0702378
$835$	0	0
$836$	1.79857e7	0.890046
$837$	3.21073e6	0.158413
$838$	2.58545e7	1.27182
$839$	3.42452e7	1.67956	0.839779	−	0.542929i	$-0.182685\pi$
$839$	3.42452e7	1.67956	0.839779	+	0.542929i	$0.182685\pi$
$840$	0	0
$841$	−1.83905e7	−0.896612
$842$	6.12802e7	2.97879
$843$	−2.29326e6	−0.111144
$844$	1.52595e7	0.737366
$845$	0	0
$846$	4.70582e6	0.226053
$847$	2.33455e7	1.11814
$848$	8.59078e6	0.410245
$849$	2.73181e6	0.130071
$850$	0	0
$851$	1.05275e7	0.498313
$852$	−3.36452e6	−0.158791
$853$	2.21968e7	1.04452	0.522261	−	0.852785i	$-0.325089\pi$
$853$	2.21968e7	1.04452	0.522261	+	0.852785i	$0.325089\pi$
$854$	−2.32500e7	−1.09088
$855$	0	0
$856$	−1.28101e7	−0.597541
$857$	−3.15803e7	−1.46881	−0.734403	−	0.678714i	$-0.762538\pi$
$857$	−3.15803e7	−1.46881	−0.734403	+	0.678714i	$0.762538\pi$
$858$	3.41551e6	0.158393
$859$	−1.22037e6	−0.0564297	−0.0282148	−	0.999602i	$-0.508982\pi$
$859$	−1.22037e6	−0.0564297	−0.0282148	+	0.999602i	$0.508982\pi$
$860$	0	0
$861$	−369578.	−0.0169902
$862$	2.29239e7	1.05080
$863$	2.43543e7	1.11314	0.556570	−	0.830801i	$-0.312117\pi$
$863$	2.43543e7	1.11314	0.556570	+	0.830801i	$0.312117\pi$
$864$	−5.22260e6	−0.238014
$865$	0	0
$866$	3.29086e7	1.49113
$867$	−3.43585e6	−0.155234
$868$	2.56917e7	1.15743
$869$	5.47031e7	2.45732
$870$	0	0
$871$	−1.09877e7	−0.490749
$872$	5.99873e6	0.267158
$873$	−2.13883e7	−0.949817
$874$	−1.62668e7	−0.720318
$875$	0	0
$876$	4.11453e6	0.181159
$877$	−2.52016e6	−0.110644	−0.0553222	−	0.998469i	$-0.517619\pi$
$877$	−2.52016e6	−0.110644	−0.0553222	+	0.998469i	$0.517619\pi$
$878$	−3.68601e6	−0.161369
$879$	−3.54361e6	−0.154694
$880$	0	0
$881$	1.04311e6	0.0452785	0.0226393	−	0.999744i	$-0.492793\pi$
$881$	1.04311e6	0.0452785	0.0226393	+	0.999744i	$0.492793\pi$
$882$	2.65748e6	0.115027
$883$	3.57753e7	1.54412	0.772061	−	0.635548i	$-0.219226\pi$
$883$	3.57753e7	1.54412	0.772061	+	0.635548i	$0.219226\pi$
$884$	3.90102e7	1.67899
$885$	0	0
$886$	5.34354e6	0.228689
$887$	1.66616e7	0.711061	0.355531	−	0.934665i	$-0.384300\pi$
$887$	1.66616e7	0.711061	0.355531	+	0.934665i	$0.384300\pi$
$888$	729326.	0.0310377
$889$	2.96065e7	1.25642
$890$	0	0
$891$	3.38765e7	1.42957
$892$	−2.88713e7	−1.21494
$893$	−1.44225e6	−0.0605218
$894$	2.61870e6	0.109583
$895$	0	0
$896$	−2.96214e7	−1.23264
$897$	−1.83139e6	−0.0759975
$898$	1.63928e7	0.678365
$899$	−6.43631e6	−0.265606
$900$	0	0
$901$	4.81090e7	1.97430
$902$	1.03206e7	0.422368
$903$	346376.	0.0141361
$904$	−2.39499e7	−0.974727
$905$	0	0
$906$	5.49186e6	0.222279
$907$	−3.59470e7	−1.45092	−0.725462	−	0.688262i	$-0.758374\pi$
$907$	−3.59470e7	−1.45092	−0.725462	+	0.688262i	$0.758374\pi$
$908$	3.52944e7	1.42066
$909$	−4.31466e7	−1.73196
$910$	0	0
$911$	7.56404e6	0.301966	0.150983	−	0.988536i	$-0.451756\pi$
$911$	7.56404e6	0.301966	0.150983	+	0.988536i	$0.451756\pi$
$912$	337763.	0.0134470
$913$	1.09143e6	0.0433331
$914$	6.52030e7	2.58168
$915$	0	0
$916$	5.67260e7	2.23380
$917$	−3.53654e7	−1.38885
$918$	−1.24011e7	−0.485683
$919$	−2.22167e7	−0.867741	−0.433870	−	0.900975i	$-0.642852\pi$
$919$	−2.22167e7	−0.867741	−0.433870	+	0.900975i	$0.642852\pi$
$920$	0	0
$921$	−3.90124e6	−0.151549
$922$	−3.11207e7	−1.20565
$923$	−2.09051e7	−0.807698
$924$	−5.15081e6	−0.198470
$925$	0	0
$926$	6.08709e7	2.33283
$927$	2.97182e7	1.13586
$928$	1.04694e7	0.399071
$929$	9.18809e6	0.349290	0.174645	−	0.984631i	$-0.444122\pi$
$929$	9.18809e6	0.349290	0.174645	+	0.984631i	$0.444122\pi$
$930$	0	0
$931$	−814470.	−0.0307965
$932$	−1.76934e7	−0.667224
$933$	3.32581e6	0.125082
$934$	−5.02220e7	−1.88376
$935$	0	0
$936$	1.35448e7	0.505340
$937$	−4.06654e7	−1.51313	−0.756565	−	0.653919i	$-0.773124\pi$
$937$	−4.06654e7	−1.51313	−0.756565	+	0.653919i	$0.773124\pi$
$938$	2.79495e7	1.03721
$939$	−3.65856e6	−0.135409
$940$	0	0
$941$	−2.60105e7	−0.957580	−0.478790	−	0.877930i	$-0.658924\pi$
$941$	−2.60105e7	−0.957580	−0.478790	+	0.877930i	$0.658924\pi$
$942$	5.47132e6	0.200893
$943$	−5.53391e6	−0.202653
$944$	1.47017e7	0.536954
$945$	0	0
$946$	−9.67273e6	−0.351416
$947$	2.03967e7	0.739067	0.369534	−	0.929217i	$-0.379517\pi$
$947$	2.03967e7	0.739067	0.369534	+	0.929217i	$0.379517\pi$
$948$	−6.48688e6	−0.234431
$949$	2.55652e7	0.921475
$950$	0	0
$951$	−3.89099e6	−0.139511
$952$	−3.10846e7	−1.11161
$953$	4.87462e6	0.173864	0.0869318	−	0.996214i	$-0.472294\pi$
$953$	4.87462e6	0.173864	0.0869318	+	0.996214i	$0.472294\pi$
$954$	5.33241e7	1.89694
$955$	0	0
$956$	1.31289e7	0.464603
$957$	1.29039e6	0.0455449
$958$	4.04231e7	1.42304
$959$	−4.12903e7	−1.44978
$960$	0	0
$961$	−9.09409e6	−0.317651
$962$	1.44662e7	0.503984
$963$	2.38318e7	0.828114
$964$	−5.24322e7	−1.81721
$965$	0	0
$966$	4.65854e6	0.160623
$967$	1.26016e7	0.433371	0.216686	−	0.976241i	$-0.430475\pi$
$967$	1.26016e7	0.433371	0.216686	+	0.976241i	$0.430475\pi$
$968$	2.42173e7	0.830686
$969$	1.89149e6	0.0647135
$970$	0	0
$971$	−1.39978e6	−0.0476443	−0.0238221	−	0.999716i	$-0.507584\pi$
$971$	−1.39978e6	−0.0476443	−0.0238221	+	0.999716i	$0.507584\pi$
$972$	−1.22426e7	−0.415630
$973$	1.32201e7	0.447665
$974$	8.13452e7	2.74748
$975$	0	0
$976$	7.22867e6	0.242904
$977$	−1.43010e6	−0.0479325	−0.0239662	−	0.999713i	$-0.507629\pi$
$977$	−1.43010e6	−0.0479325	−0.0239662	+	0.999713i	$0.507629\pi$
$978$	−1.09819e6	−0.0367138
$979$	−4.15300e7	−1.38486
$980$	0	0
$981$	−1.11600e7	−0.370247
$982$	3.02711e7	1.00173
$983$	−4.26255e7	−1.40697	−0.703487	−	0.710708i	$-0.748375\pi$
$983$	−4.26255e7	−1.40697	−0.703487	+	0.710708i	$0.748375\pi$
$984$	−383378.	−0.0126223
$985$	0	0
$986$	2.48595e7	0.814329
$987$	413035.	0.0134957
$988$	−1.32520e7	−0.431905
$989$	5.18650e6	0.168610
$990$	0	0
$991$	1.88273e7	0.608980	0.304490	−	0.952516i	$-0.401514\pi$
$991$	1.88273e7	0.608980	0.304490	+	0.952516i	$0.401514\pi$
$992$	−3.17759e7	−1.02522
$993$	3.74463e6	0.120514
$994$	5.31769e7	1.70709
$995$	0	0
$996$	−129426.	−0.00413401
$997$	−2.10574e7	−0.670915	−0.335458	−	0.942055i	$-0.608891\pi$
$997$	−2.10574e7	−0.670915	−0.335458	+	0.942055i	$0.608891\pi$
$998$	−1.76746e7	−0.561724
$999$	−2.72637e6	−0.0864314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.b.1.9		10
5.4	even	2		43.6.a.b.1.2	✓	10
15.14	odd	2		387.6.a.e.1.9		10
20.19	odd	2		688.6.a.h.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.2	✓	10	5.4	even	2
387.6.a.e.1.9		10	15.14	odd	2
688.6.a.h.1.6		10	20.19	odd	2
1075.6.a.b.1.9		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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q+8.86547 q^{2} -1.50169 q^{3} +46.5966 q^{4} -13.3132 q^{6} +124.747 q^{7} +129.406 q^{8} -240.745 q^{9} +O(q^{10})\)	\(q+8.86547 q^{2} -1.50169 q^{3} +46.5966 q^{4} -13.3132 q^{6} +124.747 q^{7} +129.406 q^{8} -240.745 q^{9} +590.079 q^{11} -69.9735 q^{12} -434.774 q^{13} +1105.94 q^{14} -343.849 q^{16} -1925.58 q^{17} -2134.32 q^{18} +654.129 q^{19} -187.332 q^{21} +5231.33 q^{22} -2805.03 q^{23} -194.327 q^{24} -3854.47 q^{26} +726.434 q^{27} +5812.80 q^{28} -1456.23 q^{29} +4419.85 q^{31} -7189.36 q^{32} -886.115 q^{33} -17071.2 q^{34} -11217.9 q^{36} -3753.09 q^{37} +5799.17 q^{38} +652.894 q^{39} +1972.85 q^{41} -1660.78 q^{42} -1849.00 q^{43} +27495.7 q^{44} -24867.9 q^{46} -2204.84 q^{47} +516.354 q^{48} -1245.12 q^{49} +2891.62 q^{51} -20259.0 q^{52} -24984.2 q^{53} +6440.18 q^{54} +16143.0 q^{56} -982.298 q^{57} -12910.1 q^{58} -42756.2 q^{59} -21022.8 q^{61} +39184.0 q^{62} -30032.3 q^{63} -52733.9 q^{64} -7855.83 q^{66} +25272.1 q^{67} -89725.4 q^{68} +4212.28 q^{69} +48082.8 q^{71} -31153.7 q^{72} -58801.2 q^{73} -33272.9 q^{74} +30480.2 q^{76} +73610.8 q^{77} +5788.22 q^{78} +92704.7 q^{79} +57410.1 q^{81} +17490.3 q^{82} +1849.64 q^{83} -8729.01 q^{84} -16392.3 q^{86} +2186.80 q^{87} +76359.5 q^{88} -70380.3 q^{89} -54236.8 q^{91} -130705. q^{92} -6637.23 q^{93} -19546.9 q^{94} +10796.2 q^{96} +88842.1 q^{97} -11038.6 q^{98} -142059. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9}+O(q^{10}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9	\( 10 q - 8 q^{2} - 28 q^{3} + 202 q^{4} + 75 q^{6} - 60 q^{7} - 294 q^{8} + 1356 q^{9} + 745 q^{11} - 4627 q^{12} - 1917 q^{13} + 1936 q^{14} + 5354 q^{16} - 4017 q^{17} + 2725 q^{18} - 2404 q^{19} - 228 q^{21} + 5836 q^{22} - 1733 q^{23} - 10711 q^{24} - 1484 q^{26} + 2324 q^{27} + 15028 q^{28} + 6996 q^{29} - 4899 q^{31} + 7554 q^{32} + 15734 q^{33} - 27033 q^{34} + 4433 q^{36} - 1466 q^{37} - 13905 q^{38} - 26542 q^{39} + 10297 q^{41} + 37642 q^{42} - 18490 q^{43} - 36140 q^{44} + 17991 q^{46} - 48592 q^{47} - 83607 q^{48} + 29458 q^{49} + 92972 q^{51} - 14232 q^{52} - 127165 q^{53} - 92002 q^{54} - 7780 q^{56} - 34060 q^{57} + 10305 q^{58} + 99372 q^{59} + 17408 q^{61} - 28265 q^{62} - 2244 q^{63} + 47202 q^{64} - 150292 q^{66} + 2021 q^{67} - 192151 q^{68} + 1654 q^{69} + 11286 q^{71} + 298365 q^{72} - 49892 q^{73} - 125431 q^{74} - 249803 q^{76} - 98144 q^{77} + 28494 q^{78} - 91524 q^{79} - 26450 q^{81} + 158909 q^{82} + 105203 q^{83} - 357682 q^{84} + 14792 q^{86} - 181200 q^{87} + 461824 q^{88} - 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 238430 q^{93} + 7259 q^{94} - 162399 q^{96} - 108383 q^{97} - 354656 q^{98} - 270499 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 28 * q^3 + 202 * q^4 + 75 * q^6 - 60 * q^7 - 294 * q^8 + 1356 * q^9 + 745 * q^11 - 4627 * q^12 - 1917 * q^13 + 1936 * q^14 + 5354 * q^16 - 4017 * q^17 + 2725 * q^18 - 2404 * q^19 - 228 * q^21 + 5836 * q^22 - 1733 * q^23 - 10711 * q^24 - 1484 * q^26 + 2324 * q^27 + 15028 * q^28 + 6996 * q^29 - 4899 * q^31 + 7554 * q^32 + 15734 * q^33 - 27033 * q^34 + 4433 * q^36 - 1466 * q^37 - 13905 * q^38 - 26542 * q^39 + 10297 * q^41 + 37642 * q^42 - 18490 * q^43 - 36140 * q^44 + 17991 * q^46 - 48592 * q^47 - 83607 * q^48 + 29458 * q^49 + 92972 * q^51 - 14232 * q^52 - 127165 * q^53 - 92002 * q^54 - 7780 * q^56 - 34060 * q^57 + 10305 * q^58 + 99372 * q^59 + 17408 * q^61 - 28265 * q^62 - 2244 * q^63 + 47202 * q^64 - 150292 * q^66 + 2021 * q^67 - 192151 * q^68 + 1654 * q^69 + 11286 * q^71 + 298365 * q^72 - 49892 * q^73 - 125431 * q^74 - 249803 * q^76 - 98144 * q^77 + 28494 * q^78 - 91524 * q^79 - 26450 * q^81 + 158909 * q^82 + 105203 * q^83 - 357682 * q^84 + 14792 * q^86 - 181200 * q^87 + 461824 * q^88 - 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 238430 * q^93 + 7259 * q^94 - 162399 * q^96 - 108383 * q^97 - 354656 * q^98 - 270499 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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