LMFDB - Embedded newform 490.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,4,Mod(1,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("490.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 490 = 2 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	490.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:	$28.9109359028$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.78233$ of defining polynomial
Character	$\chi$	$=$	490.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -7.78233 q^{3} +4.00000 q^{4} +5.00000 q^{5} +15.5647 q^{6} -8.00000 q^{8} +33.5647 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -7.78233 q^{3} +4.00000 q^{4} +5.00000 q^{5} +15.5647 q^{6} -8.00000 q^{8} +33.5647 q^{9} -10.0000 q^{10} -47.5647 q^{11} -31.1293 q^{12} +1.12932 q^{13} -38.9116 q^{15} +16.0000 q^{16} +136.259 q^{17} -67.1293 q^{18} -88.8707 q^{19} +20.0000 q^{20} +95.1293 q^{22} +2.73495 q^{23} +62.2586 q^{24} +25.0000 q^{25} -2.25864 q^{26} -51.0884 q^{27} -131.082 q^{29} +77.8233 q^{30} +301.470 q^{31} -32.0000 q^{32} +370.164 q^{33} -272.517 q^{34} +134.259 q^{36} +161.388 q^{37} +177.741 q^{38} -8.78874 q^{39} -40.0000 q^{40} +288.729 q^{41} -70.5582 q^{43} -190.259 q^{44} +167.823 q^{45} -5.46990 q^{46} +136.259 q^{47} -124.517 q^{48} -50.0000 q^{50} -1060.41 q^{51} +4.51728 q^{52} -34.2586 q^{53} +102.177 q^{54} -237.823 q^{55} +691.621 q^{57} +262.164 q^{58} -745.362 q^{59} -155.647 q^{60} -683.845 q^{61} -602.940 q^{62} +64.0000 q^{64} +5.64660 q^{65} -740.328 q^{66} -72.5839 q^{67} +545.035 q^{68} -21.2843 q^{69} -687.293 q^{71} -268.517 q^{72} +84.4482 q^{73} -322.776 q^{74} -194.558 q^{75} -355.483 q^{76} +17.5775 q^{78} -722.069 q^{79} +80.0000 q^{80} -508.659 q^{81} -577.457 q^{82} -1010.36 q^{83} +681.293 q^{85} +141.116 q^{86} +1020.12 q^{87} +380.517 q^{88} +424.457 q^{89} -335.647 q^{90} +10.9398 q^{92} -2346.14 q^{93} -272.517 q^{94} -444.353 q^{95} +249.035 q^{96} +120.448 q^{97} -1596.49 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 10 q^{5} + 4 q^{6} - 16 q^{8} + 40 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 + 10 * q^5 + 4 * q^6 - 16 * q^8 + 40 * q^9	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 10 q^{5} + 4 q^{6} - 16 q^{8} + 40 q^{9} - 20 q^{10} - 68 q^{11} - 8 q^{12} - 52 q^{13} - 10 q^{15} + 32 q^{16} + 164 q^{17} - 80 q^{18} - 232 q^{19} + 40 q^{20} + 136 q^{22} - 198 q^{23} + 16 q^{24} + 50 q^{25} + 104 q^{26} - 170 q^{27} - 18 q^{29} + 20 q^{30} + 196 q^{31} - 64 q^{32} + 252 q^{33} - 328 q^{34} + 160 q^{36} + 160 q^{37} + 464 q^{38} - 316 q^{39} - 80 q^{40} + 62 q^{41} + 198 q^{43} - 272 q^{44} + 200 q^{45} + 396 q^{46} + 164 q^{47} - 32 q^{48} - 100 q^{50} - 900 q^{51} - 208 q^{52} + 40 q^{53} + 340 q^{54} - 340 q^{55} - 136 q^{57} + 36 q^{58} - 80 q^{59} - 40 q^{60} - 174 q^{61} - 392 q^{62} + 128 q^{64} - 260 q^{65} - 504 q^{66} - 1054 q^{67} + 656 q^{68} - 1182 q^{69} - 832 q^{71} - 320 q^{72} + 820 q^{73} - 320 q^{74} - 50 q^{75} - 928 q^{76} + 632 q^{78} - 576 q^{79} + 160 q^{80} - 1370 q^{81} - 124 q^{82} - 298 q^{83} + 820 q^{85} - 396 q^{86} + 1674 q^{87} + 544 q^{88} - 182 q^{89} - 400 q^{90} - 792 q^{92} - 2956 q^{93} - 328 q^{94} - 1160 q^{95} + 64 q^{96} + 892 q^{97} - 1728 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 + 10 * q^5 + 4 * q^6 - 16 * q^8 + 40 * q^9 - 20 * q^10 - 68 * q^11 - 8 * q^12 - 52 * q^13 - 10 * q^15 + 32 * q^16 + 164 * q^17 - 80 * q^18 - 232 * q^19 + 40 * q^20 + 136 * q^22 - 198 * q^23 + 16 * q^24 + 50 * q^25 + 104 * q^26 - 170 * q^27 - 18 * q^29 + 20 * q^30 + 196 * q^31 - 64 * q^32 + 252 * q^33 - 328 * q^34 + 160 * q^36 + 160 * q^37 + 464 * q^38 - 316 * q^39 - 80 * q^40 + 62 * q^41 + 198 * q^43 - 272 * q^44 + 200 * q^45 + 396 * q^46 + 164 * q^47 - 32 * q^48 - 100 * q^50 - 900 * q^51 - 208 * q^52 + 40 * q^53 + 340 * q^54 - 340 * q^55 - 136 * q^57 + 36 * q^58 - 80 * q^59 - 40 * q^60 - 174 * q^61 - 392 * q^62 + 128 * q^64 - 260 * q^65 - 504 * q^66 - 1054 * q^67 + 656 * q^68 - 1182 * q^69 - 832 * q^71 - 320 * q^72 + 820 * q^73 - 320 * q^74 - 50 * q^75 - 928 * q^76 + 632 * q^78 - 576 * q^79 + 160 * q^80 - 1370 * q^81 - 124 * q^82 - 298 * q^83 + 820 * q^85 - 396 * q^86 + 1674 * q^87 + 544 * q^88 - 182 * q^89 - 400 * q^90 - 792 * q^92 - 2956 * q^93 - 328 * q^94 - 1160 * q^95 + 64 * q^96 + 892 * q^97 - 1728 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−7.78233	−1.49771	−0.748855	−	0.662734i	$-0.769396\pi$
$3$	−7.78233	−1.49771	−0.748855	+	0.662734i	$0.769396\pi$
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	15.5647	1.05904
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	33.5647	1.24314
$10$	−10.0000	−0.316228
$11$	−47.5647	−1.30375	−0.651877	−	0.758325i	$-0.726018\pi$
$11$	−47.5647	−1.30375	−0.651877	+	0.758325i	$0.726018\pi$
$12$	−31.1293	−0.748855
$13$	1.12932	0.0240936	0.0120468	−	0.999927i	$-0.496165\pi$
$13$	1.12932	0.0240936	0.0120468	+	0.999927i	$0.496165\pi$
$14$	0	0
$15$	−38.9116	−0.669796
$16$	16.0000	0.250000
$17$	136.259	1.94397	0.971987	−	0.235033i	$-0.0755197\pi$
$17$	136.259	1.94397	0.971987	+	0.235033i	$0.0755197\pi$
$18$	−67.1293	−0.879030
$19$	−88.8707	−1.07307	−0.536535	−	0.843878i	$-0.680267\pi$
$19$	−88.8707	−1.07307	−0.536535	+	0.843878i	$0.680267\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	95.1293	0.921893
$23$	2.73495	0.0247946	0.0123973	−	0.999923i	$-0.496054\pi$
$23$	2.73495	0.0247946	0.0123973	+	0.999923i	$0.496054\pi$
$24$	62.2586	0.529520
$25$	25.0000	0.200000
$26$	−2.25864	−0.0170368
$27$	−51.0884	−0.364147
$28$	0	0
$29$	−131.082	−0.839355	−0.419678	−	0.907673i	$-0.637857\pi$
$29$	−131.082	−0.839355	−0.419678	+	0.907673i	$0.637857\pi$
$30$	77.8233	0.473618
$31$	301.470	1.74663	0.873316	−	0.487154i	$-0.161965\pi$
$31$	301.470	1.74663	0.873316	+	0.487154i	$0.161965\pi$
$32$	−32.0000	−0.176777
$33$	370.164	1.95264
$34$	−272.517	−1.37460
$35$	0	0
$36$	134.259	0.621568
$37$	161.388	0.717082	0.358541	−	0.933514i	$-0.383274\pi$
$37$	161.388	0.717082	0.358541	+	0.933514i	$0.383274\pi$
$38$	177.741	0.758775
$39$	−8.78874	−0.0360853
$40$	−40.0000	−0.158114
$41$	288.729	1.09980	0.549900	−	0.835230i	$-0.314666\pi$
$41$	288.729	1.09980	0.549900	+	0.835230i	$0.314666\pi$
$42$	0	0
$43$	−70.5582	−0.250233	−0.125117	−	0.992142i	$-0.539931\pi$
$43$	−70.5582	−0.250233	−0.125117	+	0.992142i	$0.539931\pi$
$44$	−190.259	−0.651877
$45$	167.823	0.555947
$46$	−5.46990	−0.0175324
$47$	136.259	0.422880	0.211440	−	0.977391i	$-0.432185\pi$
$47$	136.259	0.422880	0.211440	+	0.977391i	$0.432185\pi$
$48$	−124.517	−0.374428
$49$	0	0
$50$	−50.0000	−0.141421
$51$	−1060.41	−2.91151
$52$	4.51728	0.0120468
$53$	−34.2586	−0.0887884	−0.0443942	−	0.999014i	$-0.514136\pi$
$53$	−34.2586	−0.0887884	−0.0443942	+	0.999014i	$0.514136\pi$
$54$	102.177	0.257491
$55$	−237.823	−0.583056
$56$	0	0
$57$	691.621	1.60715
$58$	262.164	0.593514
$59$	−745.362	−1.64471	−0.822355	−	0.568975i	$-0.807340\pi$
$59$	−745.362	−1.64471	−0.822355	+	0.568975i	$0.807340\pi$
$60$	−155.647	−0.334898
$61$	−683.845	−1.43537	−0.717683	−	0.696369i	$-0.754798\pi$
$61$	−683.845	−1.43537	−0.717683	+	0.696369i	$0.754798\pi$
$62$	−602.940	−1.23506
$63$	0	0
$64$	64.0000	0.125000
$65$	5.64660	0.0107750
$66$	−740.328	−1.38073
$67$	−72.5839	−0.132351	−0.0661756	−	0.997808i	$-0.521080\pi$
$67$	−72.5839	−0.132351	−0.0661756	+	0.997808i	$0.521080\pi$
$68$	545.035	0.971987
$69$	−21.2843	−0.0371352
$70$	0	0
$71$	−687.293	−1.14883	−0.574413	−	0.818565i	$-0.694770\pi$
$71$	−687.293	−1.14883	−0.574413	+	0.818565i	$0.694770\pi$
$72$	−268.517	−0.439515
$73$	84.4482	0.135396	0.0676980	−	0.997706i	$-0.478435\pi$
$73$	84.4482	0.135396	0.0676980	+	0.997706i	$0.478435\pi$
$74$	−322.776	−0.507053
$75$	−194.558	−0.299542
$76$	−355.483	−0.536535
$77$	0	0
$78$	17.5775	0.0255161
$79$	−722.069	−1.02834	−0.514172	−	0.857687i	$-0.671901\pi$
$79$	−722.069	−1.02834	−0.514172	+	0.857687i	$0.671901\pi$
$80$	80.0000	0.111803
$81$	−508.659	−0.697750
$82$	−577.457	−0.777676
$83$	−1010.36	−1.33616	−0.668078	−	0.744091i	$-0.732883\pi$
$83$	−1010.36	−1.33616	−0.668078	+	0.744091i	$0.732883\pi$
$84$	0	0
$85$	681.293	0.869372
$86$	141.116	0.176942
$87$	1020.12	1.25711
$88$	380.517	0.460946
$89$	424.457	0.505532	0.252766	−	0.967527i	$-0.418660\pi$
$89$	424.457	0.505532	0.252766	+	0.967527i	$0.418660\pi$
$90$	−335.647	−0.393114
$91$	0	0
$92$	10.9398	0.0123973
$93$	−2346.14	−2.61595
$94$	−272.517	−0.299021
$95$	−444.353	−0.479892
$96$	249.035	0.264760
$97$	120.448	0.126079	0.0630395	−	0.998011i	$-0.479921\pi$
$97$	120.448	0.126079	0.0630395	+	0.998011i	$0.479921\pi$
$98$	0	0
$99$	−1596.49	−1.62074
$100$	100.000	0.100000
$101$	−244.590	−0.240967	−0.120483	−	0.992715i	$-0.538444\pi$
$101$	−244.590	−0.240967	−0.120483	+	0.992715i	$0.538444\pi$
$102$	2120.82	2.05875
$103$	829.688	0.793704	0.396852	−	0.917883i	$-0.370103\pi$
$103$	829.688	0.793704	0.396852	+	0.917883i	$0.370103\pi$
$104$	−9.03456	−0.00851838
$105$	0	0
$106$	68.5173	0.0627829
$107$	786.330	0.710443	0.355222	−	0.934782i	$-0.384405\pi$
$107$	786.330	0.710443	0.355222	+	0.934782i	$0.384405\pi$
$108$	−204.353	−0.182073
$109$	−1312.97	−1.15376	−0.576881	−	0.816828i	$-0.695731\pi$
$109$	−1312.97	−1.15376	−0.576881	+	0.816828i	$0.695731\pi$
$110$	475.647	0.412283
$111$	−1255.97	−1.07398
$112$	0	0
$113$	−1970.23	−1.64021	−0.820106	−	0.572212i	$-0.806086\pi$
$113$	−1970.23	−1.64021	−0.820106	+	0.572212i	$0.806086\pi$
$114$	−1383.24	−1.13643
$115$	13.6747	0.0110885
$116$	−524.328	−0.419678
$117$	37.9052	0.0299516
$118$	1490.72	1.16299
$119$	0	0
$120$	311.293	0.236809
$121$	931.397	0.699772
$122$	1367.69	1.01496
$123$	−2246.98	−1.64718
$124$	1205.88	0.873316
$125$	125.000	0.0894427
$126$	0	0
$127$	222.466	0.155438	0.0777192	−	0.996975i	$-0.475236\pi$
$127$	222.466	0.155438	0.0777192	+	0.996975i	$0.475236\pi$
$128$	−128.000	−0.0883883
$129$	549.108	0.374777
$130$	−11.2932	−0.00761907
$131$	−1466.80	−0.978283	−0.489141	−	0.872204i	$-0.662690\pi$
$131$	−1466.80	−0.978283	−0.489141	+	0.872204i	$0.662690\pi$
$132$	1480.66	0.976322
$133$	0	0
$134$	145.168	0.0935865
$135$	−255.442	−0.162851
$136$	−1090.07	−0.687299
$137$	−2818.86	−1.75790	−0.878948	−	0.476918i	$-0.841754\pi$
$137$	−2818.86	−1.75790	−0.878948	+	0.476918i	$0.841754\pi$
$138$	42.5686	0.0262585
$139$	2026.67	1.23669	0.618344	−	0.785907i	$-0.287804\pi$
$139$	2026.67	1.23669	0.618344	+	0.785907i	$0.287804\pi$
$140$	0	0
$141$	−1060.41	−0.633352
$142$	1374.59	0.812343
$143$	−53.7157	−0.0314121
$144$	537.035	0.310784
$145$	−655.410	−0.375371
$146$	−168.896	−0.0957394
$147$	0	0
$148$	645.552	0.358541
$149$	2917.86	1.60430	0.802150	−	0.597123i	$-0.203690\pi$
$149$	2917.86	1.60430	0.802150	+	0.597123i	$0.203690\pi$
$150$	389.116	0.211808
$151$	2910.69	1.56867	0.784334	−	0.620339i	$-0.213005\pi$
$151$	2910.69	1.56867	0.784334	+	0.620339i	$0.213005\pi$
$152$	710.965	0.379388
$153$	4573.47	2.41662
$154$	0	0
$155$	1507.35	0.781118
$156$	−35.1550	−0.0180426
$157$	2075.12	1.05486	0.527429	−	0.849599i	$-0.323156\pi$
$157$	2075.12	1.05486	0.527429	+	0.849599i	$0.323156\pi$
$158$	1444.14	0.727149
$159$	266.612	0.132979
$160$	−160.000	−0.0790569
$161$	0	0
$162$	1017.32	0.493383
$163$	−108.535	−0.0521541	−0.0260771	−	0.999660i	$-0.508302\pi$
$163$	−108.535	−0.0521541	−0.0260771	+	0.999660i	$0.508302\pi$
$164$	1154.91	0.549900
$165$	1850.82	0.873249
$166$	2020.71	0.944805
$167$	−2904.58	−1.34589	−0.672943	−	0.739695i	$-0.734970\pi$
$167$	−2904.58	−1.34589	−0.672943	+	0.739695i	$0.734970\pi$
$168$	0	0
$169$	−2195.72	−0.999419
$170$	−1362.59	−0.614739
$171$	−2982.91	−1.33397
$172$	−282.233	−0.125117
$173$	−1110.89	−0.488204	−0.244102	−	0.969750i	$-0.578493\pi$
$173$	−1110.89	−0.488204	−0.244102	+	0.969750i	$0.578493\pi$
$174$	−2040.25	−0.888912
$175$	0	0
$176$	−761.035	−0.325938
$177$	5800.66	2.46330
$178$	−848.914	−0.357465
$179$	−627.426	−0.261989	−0.130995	−	0.991383i	$-0.541817\pi$
$179$	−627.426	−0.261989	−0.130995	+	0.991383i	$0.541817\pi$
$180$	671.293	0.277974
$181$	−1961.52	−0.805518	−0.402759	−	0.915306i	$-0.631949\pi$
$181$	−1961.52	−0.805518	−0.402759	+	0.915306i	$0.631949\pi$
$182$	0	0
$183$	5321.91	2.14976
$184$	−21.8796	−0.00876622
$185$	806.940	0.320689
$186$	4692.28	1.84975
$187$	−6481.10	−2.53446
$188$	545.035	0.211440
$189$	0	0
$190$	888.707	0.339335
$191$	−3100.90	−1.17473	−0.587365	−	0.809322i	$-0.699835\pi$
$191$	−3100.90	−1.17473	−0.587365	+	0.809322i	$0.699835\pi$
$192$	−498.069	−0.187214
$193$	−784.207	−0.292479	−0.146240	−	0.989249i	$-0.546717\pi$
$193$	−784.207	−0.292479	−0.146240	+	0.989249i	$0.546717\pi$
$194$	−240.896	−0.0891513
$195$	−43.9437	−0.0161378
$196$	0	0
$197$	−2315.34	−0.837365	−0.418683	−	0.908133i	$-0.637508\pi$
$197$	−2315.34	−0.837365	−0.418683	+	0.908133i	$0.637508\pi$
$198$	3192.98	1.14604
$199$	−2776.41	−0.989019	−0.494510	−	0.869172i	$-0.664652\pi$
$199$	−2776.41	−0.989019	−0.494510	+	0.869172i	$0.664652\pi$
$200$	−200.000	−0.0707107
$201$	564.872	0.198224
$202$	489.181	0.170389
$203$	0	0
$204$	−4241.64	−1.45576
$205$	1443.64	0.491846
$206$	−1659.38	−0.561234
$207$	91.7977	0.0308231
$208$	18.0691	0.00602340
$209$	4227.10	1.39902
$210$	0	0
$211$	745.035	0.243082	0.121541	−	0.992586i	$-0.461216\pi$
$211$	745.035	0.243082	0.121541	+	0.992586i	$0.461216\pi$
$212$	−137.035	−0.0443942
$213$	5348.74	1.72061
$214$	−1572.66	−0.502359
$215$	−352.791	−0.111908
$216$	408.707	0.128745
$217$	0	0
$218$	2625.95	0.815833
$219$	−657.203	−0.202784
$220$	−951.293	−0.291528
$221$	153.880	0.0468374
$222$	2511.95	0.759419
$223$	−4379.59	−1.31515	−0.657576	−	0.753388i	$-0.728418\pi$
$223$	−4379.59	−1.31515	−0.657576	+	0.753388i	$0.728418\pi$
$224$	0	0
$225$	839.116	0.248627
$226$	3940.47	1.15980
$227$	882.456	0.258021	0.129010	−	0.991643i	$-0.458820\pi$
$227$	882.456	0.258021	0.129010	+	0.991643i	$0.458820\pi$
$228$	2766.48	0.803574
$229$	−3917.95	−1.13059	−0.565295	−	0.824888i	$-0.691238\pi$
$229$	−3917.95	−1.13059	−0.565295	+	0.824888i	$0.691238\pi$
$230$	−27.3495	−0.00784075
$231$	0	0
$232$	1048.66	0.296757
$233$	−1367.64	−0.384536	−0.192268	−	0.981342i	$-0.561584\pi$
$233$	−1367.64	−0.384536	−0.192268	+	0.981342i	$0.561584\pi$
$234$	−75.8105	−0.0211790
$235$	681.293	0.189118
$236$	−2981.45	−0.822355
$237$	5619.38	1.54016
$238$	0	0
$239$	5080.94	1.37514	0.687570	−	0.726118i	$-0.258677\pi$
$239$	5080.94	1.37514	0.687570	+	0.726118i	$0.258677\pi$
$240$	−622.586	−0.167449
$241$	−6491.74	−1.73514	−0.867572	−	0.497311i	$-0.834321\pi$
$241$	−6491.74	−1.73514	−0.867572	+	0.497311i	$0.834321\pi$
$242$	−1862.79	−0.494814
$243$	5337.94	1.40917
$244$	−2735.38	−0.717683
$245$	0	0
$246$	4493.96	1.16473
$247$	−100.363	−0.0258541
$248$	−2411.76	−0.617528
$249$	7862.92	2.00117
$250$	−250.000	−0.0632456
$251$	−2112.81	−0.531313	−0.265657	−	0.964068i	$-0.585589\pi$
$251$	−2112.81	−0.531313	−0.265657	+	0.964068i	$0.585589\pi$
$252$	0	0
$253$	−130.087	−0.0323261
$254$	−444.932	−0.109911
$255$	−5302.05	−1.30207
$256$	256.000	0.0625000
$257$	−804.758	−0.195329	−0.0976643	−	0.995219i	$-0.531137\pi$
$257$	−804.758	−0.195329	−0.0976643	+	0.995219i	$0.531137\pi$
$258$	−1098.22	−0.265007
$259$	0	0
$260$	22.5864	0.00538750
$261$	−4399.72	−1.04343
$262$	2933.60	0.691750
$263$	−3975.62	−0.932120	−0.466060	−	0.884753i	$-0.654327\pi$
$263$	−3975.62	−0.932120	−0.466060	+	0.884753i	$0.654327\pi$
$264$	−2961.31	−0.690364
$265$	−171.293	−0.0397074
$266$	0	0
$267$	−3303.27	−0.757141
$268$	−290.336	−0.0661756
$269$	948.880	0.215071	0.107536	−	0.994201i	$-0.465704\pi$
$269$	948.880	0.215071	0.107536	+	0.994201i	$0.465704\pi$
$270$	510.884	0.115153
$271$	128.190	0.0287342	0.0143671	−	0.999897i	$-0.495427\pi$
$271$	128.190	0.0287342	0.0143671	+	0.999897i	$0.495427\pi$
$272$	2180.14	0.485994
$273$	0	0
$274$	5637.73	1.24302
$275$	−1189.12	−0.260751
$276$	−85.1371	−0.0185676
$277$	3754.86	0.814467	0.407234	−	0.913324i	$-0.366494\pi$
$277$	3754.86	0.814467	0.407234	+	0.913324i	$0.366494\pi$
$278$	−4053.34	−0.874471
$279$	10118.7	2.17130
$280$	0	0
$281$	6172.10	1.31031	0.655155	−	0.755495i	$-0.272603\pi$
$281$	6172.10	1.31031	0.655155	+	0.755495i	$0.272603\pi$
$282$	2120.82	0.447847
$283$	3351.06	0.703886	0.351943	−	0.936021i	$-0.385521\pi$
$283$	3351.06	0.703886	0.351943	+	0.936021i	$0.385521\pi$
$284$	−2749.17	−0.574413
$285$	3458.10	0.718738
$286$	107.431	0.0222117
$287$	0	0
$288$	−1074.07	−0.219757
$289$	13653.4	2.77904
$290$	1310.82	0.265427
$291$	−937.367	−0.188830
$292$	337.793	0.0676980
$293$	681.163	0.135816	0.0679078	−	0.997692i	$-0.478368\pi$
$293$	681.163	0.135816	0.0679078	+	0.997692i	$0.478368\pi$
$294$	0	0
$295$	−3726.81	−0.735537
$296$	−1291.10	−0.253527
$297$	2430.00	0.474757
$298$	−5835.73	−1.13441
$299$	3.08863	0.000597392 0
$300$	−778.233	−0.149771
$301$	0	0
$302$	−5821.39	−1.10922
$303$	1903.48	0.360898
$304$	−1421.93	−0.268268
$305$	−3419.23	−0.641916
$306$	−9146.95	−1.70881
$307$	6246.87	1.16133	0.580664	−	0.814143i	$-0.302793\pi$
$307$	6246.87	1.16133	0.580664	+	0.814143i	$0.302793\pi$
$308$	0	0
$309$	−6456.90	−1.18874
$310$	−3014.70	−0.552334
$311$	−6065.15	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$311$	−6065.15	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$312$	70.3099	0.0127581
$313$	8285.23	1.49620	0.748098	−	0.663589i	$-0.230967\pi$
$313$	8285.23	1.49620	0.748098	+	0.663589i	$0.230967\pi$
$314$	−4150.24	−0.745897
$315$	0	0
$316$	−2888.28	−0.514172
$317$	−1191.04	−0.211028	−0.105514	−	0.994418i	$-0.533649\pi$
$317$	−1191.04	−0.211028	−0.105514	+	0.994418i	$0.533649\pi$
$318$	−533.224	−0.0940306
$319$	6234.87	1.09431
$320$	320.000	0.0559017
$321$	−6119.48	−1.06404
$322$	0	0
$323$	−12109.4	−2.08602
$324$	−2034.64	−0.348875
$325$	28.2330	0.00481872
$326$	217.070	0.0368786
$327$	10218.0	1.72800
$328$	−2309.83	−0.388838
$329$	0	0
$330$	−3701.64	−0.617480
$331$	−4789.90	−0.795398	−0.397699	−	0.917516i	$-0.630191\pi$
$331$	−4789.90	−0.795398	−0.397699	+	0.917516i	$0.630191\pi$
$332$	−4041.42	−0.668078
$333$	5416.93	0.891430
$334$	5809.15	0.951685
$335$	−362.919	−0.0591893
$336$	0	0
$337$	−948.778	−0.153363	−0.0766814	−	0.997056i	$-0.524432\pi$
$337$	−948.778	−0.153363	−0.0766814	+	0.997056i	$0.524432\pi$
$338$	4391.45	0.706696
$339$	15333.0	2.45656
$340$	2725.17	0.434686
$341$	−14339.3	−2.27718
$342$	5965.83	0.943260
$343$	0	0
$344$	564.466	0.0884708
$345$	−106.421	−0.0166074
$346$	2221.78	0.345212
$347$	−5663.76	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$347$	−5663.76	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$348$	4080.49	0.628555
$349$	−2369.73	−0.363463	−0.181732	−	0.983348i	$-0.558170\pi$
$349$	−2369.73	−0.363463	−0.181732	+	0.983348i	$0.558170\pi$
$350$	0	0
$351$	−57.6951	−0.00877361
$352$	1522.07	0.230473
$353$	−1416.15	−0.213525	−0.106762	−	0.994285i	$-0.534048\pi$
$353$	−1416.15	−0.213525	−0.106762	+	0.994285i	$0.534048\pi$
$354$	−11601.3	−1.74182
$355$	−3436.47	−0.513771
$356$	1697.83	0.252766
$357$	0	0
$358$	1254.85	0.185254
$359$	−7772.84	−1.14272	−0.571358	−	0.820701i	$-0.693583\pi$
$359$	−7772.84	−1.14272	−0.571358	+	0.820701i	$0.693583\pi$
$360$	−1342.59	−0.196557
$361$	1039.00	0.151479
$362$	3923.04	0.569587
$363$	−7248.44	−1.04806
$364$	0	0
$365$	422.241	0.0605509
$366$	−10643.8	−1.52011
$367$	−432.717	−0.0615467	−0.0307734	−	0.999526i	$-0.509797\pi$
$367$	−432.717	−0.0615467	−0.0307734	+	0.999526i	$0.509797\pi$
$368$	43.7592	0.00619866
$369$	9691.08	1.36720
$370$	−1613.88	−0.226761
$371$	0	0
$372$	−9384.55	−1.30797
$373$	6157.79	0.854795	0.427397	−	0.904064i	$-0.359431\pi$
$373$	6157.79	0.854795	0.427397	+	0.904064i	$0.359431\pi$
$374$	12962.2	1.79214
$375$	−972.791	−0.133959
$376$	−1090.07	−0.149511
$377$	−148.033	−0.0202231
$378$	0	0
$379$	10109.6	1.37017	0.685087	−	0.728462i	$-0.259764\pi$
$379$	10109.6	1.37017	0.685087	+	0.728462i	$0.259764\pi$
$380$	−1777.41	−0.239946
$381$	−1731.30	−0.232802
$382$	6201.80	0.830659
$383$	−8208.37	−1.09511	−0.547556	−	0.836769i	$-0.684442\pi$
$383$	−8208.37	−1.09511	−0.547556	+	0.836769i	$0.684442\pi$
$384$	996.138	0.132380
$385$	0	0
$386$	1568.41	0.206814
$387$	−2368.26	−0.311074
$388$	481.793	0.0630395
$389$	4918.31	0.641050	0.320525	−	0.947240i	$-0.396141\pi$
$389$	4918.31	0.641050	0.320525	+	0.947240i	$0.396141\pi$
$390$	87.8874	0.0114112
$391$	372.661	0.0482001
$392$	0	0
$393$	11415.1	1.46518
$394$	4630.67	0.592106
$395$	−3610.35	−0.459889
$396$	−6385.97	−0.810371
$397$	8138.95	1.02892	0.514461	−	0.857514i	$-0.327992\pi$
$397$	8138.95	1.02892	0.514461	+	0.857514i	$0.327992\pi$
$398$	5552.83	0.699342
$399$	0	0
$400$	400.000	0.0500000
$401$	−345.948	−0.0430818	−0.0215409	−	0.999768i	$-0.506857\pi$
$401$	−345.948	−0.0430818	−0.0215409	+	0.999768i	$0.506857\pi$
$402$	−1129.74	−0.140165
$403$	340.456	0.0420827
$404$	−978.361	−0.120483
$405$	−2543.30	−0.312043
$406$	0	0
$407$	−7676.36	−0.934897
$408$	8483.28	1.02937
$409$	−366.127	−0.0442635	−0.0221318	−	0.999755i	$-0.507045\pi$
$409$	−366.127	−0.0442635	−0.0221318	+	0.999755i	$0.507045\pi$
$410$	−2887.29	−0.347787
$411$	21937.3	2.63282
$412$	3318.75	0.396852
$413$	0	0
$414$	−183.595	−0.0217952
$415$	−5051.78	−0.597547
$416$	−36.1382	−0.00425919
$417$	−15772.2	−1.85220
$418$	−8454.21	−0.989255
$419$	−6915.52	−0.806313	−0.403157	−	0.915131i	$-0.632087\pi$
$419$	−6915.52	−0.806313	−0.403157	+	0.915131i	$0.632087\pi$
$420$	0	0
$421$	6467.62	0.748724	0.374362	−	0.927283i	$-0.377862\pi$
$421$	6467.62	0.748724	0.374362	+	0.927283i	$0.377862\pi$
$422$	−1490.07	−0.171885
$423$	4573.47	0.525697
$424$	274.069	0.0313914
$425$	3406.47	0.388795
$426$	−10697.5	−1.21665
$427$	0	0
$428$	3145.32	0.355222
$429$	418.033	0.0470463
$430$	705.582	0.0791307
$431$	−9470.21	−1.05839	−0.529193	−	0.848502i	$-0.677505\pi$
$431$	−9470.21	−1.05839	−0.529193	+	0.848502i	$0.677505\pi$
$432$	−817.414	−0.0910366
$433$	5097.47	0.565748	0.282874	−	0.959157i	$-0.408712\pi$
$433$	5097.47	0.565748	0.282874	+	0.959157i	$0.408712\pi$
$434$	0	0
$435$	5100.61	0.562197
$436$	−5251.90	−0.576881
$437$	−243.057	−0.0266064
$438$	1314.41	0.143390
$439$	−113.511	−0.0123407	−0.00617035	−	0.999981i	$-0.501964\pi$
$439$	−113.511	−0.0123407	−0.00617035	+	0.999981i	$0.501964\pi$
$440$	1902.59	0.206141
$441$	0	0
$442$	−307.759	−0.0331190
$443$	−17199.1	−1.84459	−0.922294	−	0.386488i	$-0.873688\pi$
$443$	−17199.1	−1.84459	−0.922294	+	0.386488i	$0.873688\pi$
$444$	−5023.90	−0.536990
$445$	2122.29	0.226081
$446$	8759.18	0.929953
$447$	−22707.8	−2.40278
$448$	0	0
$449$	2452.08	0.257730	0.128865	−	0.991662i	$-0.458867\pi$
$449$	2452.08	0.257730	0.128865	+	0.991662i	$0.458867\pi$
$450$	−1678.23	−0.175806
$451$	−13733.3	−1.43387
$452$	−7880.93	−0.820106
$453$	−22652.0	−2.34941
$454$	−1764.91	−0.182448
$455$	0	0
$456$	−5532.97	−0.568213
$457$	−13424.5	−1.37411	−0.687057	−	0.726604i	$-0.741098\pi$
$457$	−13424.5	−1.37411	−0.687057	+	0.726604i	$0.741098\pi$
$458$	7835.90	0.799449
$459$	−6961.23	−0.707892
$460$	54.6990	0.00554425
$461$	−3681.74	−0.371965	−0.185983	−	0.982553i	$-0.559547\pi$
$461$	−3681.74	−0.371965	−0.185983	+	0.982553i	$0.559547\pi$
$462$	0	0
$463$	1700.08	0.170647	0.0853236	−	0.996353i	$-0.472808\pi$
$463$	1700.08	0.170647	0.0853236	+	0.996353i	$0.472808\pi$
$464$	−2097.31	−0.209839
$465$	−11730.7	−1.16989
$466$	2735.28	0.271908
$467$	−7951.90	−0.787945	−0.393972	−	0.919122i	$-0.628899\pi$
$467$	−7951.90	−0.787945	−0.393972	+	0.919122i	$0.628899\pi$
$468$	151.621	0.0149758
$469$	0	0
$470$	−1362.59	−0.133726
$471$	−16149.3	−1.57987
$472$	5962.90	0.581493
$473$	3356.08	0.326242
$474$	−11238.8	−1.08906
$475$	−2221.77	−0.214614
$476$	0	0
$477$	−1149.88	−0.110376
$478$	−10161.9	−0.972371
$479$	12820.3	1.22291	0.611453	−	0.791280i	$-0.290585\pi$
$479$	12820.3	1.22291	0.611453	+	0.791280i	$0.290585\pi$
$480$	1245.17	0.118404
$481$	182.259	0.0172771
$482$	12983.5	1.22693
$483$	0	0
$484$	3725.59	0.349886
$485$	602.241	0.0563842
$486$	−10675.9	−0.996436
$487$	5543.73	0.515832	0.257916	−	0.966167i	$-0.416964\pi$
$487$	5543.73	0.515832	0.257916	+	0.966167i	$0.416964\pi$
$488$	5470.76	0.507479
$489$	844.656	0.0781118
$490$	0	0
$491$	−16382.1	−1.50573	−0.752865	−	0.658175i	$-0.771329\pi$
$491$	−16382.1	−1.50573	−0.752865	+	0.658175i	$0.771329\pi$
$492$	−8987.92	−0.823591
$493$	−17861.0	−1.63169
$494$	200.727	0.0182816
$495$	−7982.46	−0.724818
$496$	4823.52	0.436658
$497$	0	0
$498$	−15725.8	−1.41504
$499$	11537.6	1.03506	0.517529	−	0.855666i	$-0.326852\pi$
$499$	11537.6	1.03506	0.517529	+	0.855666i	$0.326852\pi$
$500$	500.000	0.0447214
$501$	22604.4	2.01575
$502$	4225.63	0.375695
$503$	−13364.5	−1.18468	−0.592338	−	0.805690i	$-0.701795\pi$
$503$	−13364.5	−1.18468	−0.592338	+	0.805690i	$0.701795\pi$
$504$	0	0
$505$	−1222.95	−0.107764
$506$	260.174	0.0228580
$507$	17087.9	1.49684
$508$	889.864	0.0777192
$509$	−6874.82	−0.598666	−0.299333	−	0.954149i	$-0.596764\pi$
$509$	−6874.82	−0.598666	−0.299333	+	0.954149i	$0.596764\pi$
$510$	10604.1	0.920701
$511$	0	0
$512$	−512.000	−0.0441942
$513$	4540.26	0.390755
$514$	1609.52	0.138118
$515$	4148.44	0.354955
$516$	2196.43	0.187388
$517$	−6481.10	−0.551331
$518$	0	0
$519$	8645.30	0.731188
$520$	−45.1728	−0.00380954
$521$	15396.4	1.29468	0.647342	−	0.762200i	$-0.275881\pi$
$521$	15396.4	1.29468	0.647342	+	0.762200i	$0.275881\pi$
$522$	8799.44	0.737818
$523$	21761.3	1.81941	0.909707	−	0.415252i	$-0.136306\pi$
$523$	21761.3	1.81941	0.909707	+	0.415252i	$0.136306\pi$
$524$	−5867.21	−0.489141
$525$	0	0
$526$	7951.25	0.659108
$527$	41077.9	3.39541
$528$	5922.62	0.488161
$529$	−12159.5	−0.999385
$530$	342.586	0.0280774
$531$	−25017.8	−2.04460
$532$	0	0
$533$	326.067	0.0264982
$534$	6606.53	0.535379
$535$	3931.65	0.317720
$536$	580.671	0.0467932
$537$	4882.84	0.392384
$538$	−1897.76	−0.152078
$539$	0	0
$540$	−1021.77	−0.0814256
$541$	7050.81	0.560329	0.280165	−	0.959952i	$-0.409611\pi$
$541$	7050.81	0.560329	0.280165	+	0.959952i	$0.409611\pi$
$542$	−256.379	−0.0203181
$543$	15265.2	1.20643
$544$	−4360.28	−0.343649
$545$	−6564.87	−0.515978
$546$	0	0
$547$	7083.09	0.553658	0.276829	−	0.960919i	$-0.410716\pi$
$547$	7083.09	0.553658	0.276829	+	0.960919i	$0.410716\pi$
$548$	−11275.5	−0.878948
$549$	−22953.0	−1.78436
$550$	2378.23	0.184379
$551$	11649.3	0.900687
$552$	170.274	0.0131293
$553$	0	0
$554$	−7509.71	−0.575915
$555$	−6279.87	−0.480299
$556$	8106.67	0.618344
$557$	−11654.7	−0.886581	−0.443291	−	0.896378i	$-0.646189\pi$
$557$	−11654.7	−0.886581	−0.443291	+	0.896378i	$0.646189\pi$
$558$	−20237.5	−1.53534
$559$	−79.6828	−0.00602902
$560$	0	0
$561$	50438.0	3.79589
$562$	−12344.2	−0.926529
$563$	5910.23	0.442427	0.221214	−	0.975225i	$-0.428998\pi$
$563$	5910.23	0.442427	0.221214	+	0.975225i	$0.428998\pi$
$564$	−4241.64	−0.316676
$565$	−9851.16	−0.733525
$566$	−6702.12	−0.497723
$567$	0	0
$568$	5498.35	0.406172
$569$	−3632.95	−0.267665	−0.133832	−	0.991004i	$-0.542728\pi$
$569$	−3632.95	−0.267665	−0.133832	+	0.991004i	$0.542728\pi$
$570$	−6916.21	−0.508225
$571$	9704.22	0.711224	0.355612	−	0.934634i	$-0.384272\pi$
$571$	9704.22	0.711224	0.355612	+	0.934634i	$0.384272\pi$
$572$	−214.863	−0.0157061
$573$	24132.2	1.75940
$574$	0	0
$575$	68.3737	0.00495893
$576$	2148.14	0.155392
$577$	−5559.91	−0.401147	−0.200574	−	0.979679i	$-0.564281\pi$
$577$	−5559.91	−0.401147	−0.200574	+	0.979679i	$0.564281\pi$
$578$	−27306.8	−1.96508
$579$	6102.96	0.438049
$580$	−2621.64	−0.187686
$581$	0	0
$582$	1874.73	0.133523
$583$	1629.50	0.115758
$584$	−675.585	−0.0478697
$585$	189.526	0.0133948
$586$	−1362.33	−0.0960361
$587$	22559.1	1.58622	0.793111	−	0.609077i	$-0.208460\pi$
$587$	22559.1	1.58622	0.793111	+	0.609077i	$0.208460\pi$
$588$	0	0
$589$	−26791.8	−1.87426
$590$	7453.62	0.520103
$591$	18018.7	1.25413
$592$	2582.21	0.179270
$593$	1487.62	0.103017	0.0515085	−	0.998673i	$-0.483597\pi$
$593$	1487.62	0.103017	0.0515085	+	0.998673i	$0.483597\pi$
$594$	−4860.00	−0.335704
$595$	0	0
$596$	11671.5	0.802150
$597$	21607.0	1.48126
$598$	−6.17727	−0.000422420 0
$599$	−3908.33	−0.266594	−0.133297	−	0.991076i	$-0.542556\pi$
$599$	−3908.33	−0.266594	−0.133297	+	0.991076i	$0.542556\pi$
$600$	1556.47	0.105904
$601$	14849.4	1.00785	0.503927	−	0.863747i	$-0.331888\pi$
$601$	14849.4	1.00785	0.503927	+	0.863747i	$0.331888\pi$
$602$	0	0
$603$	−2436.25	−0.164531
$604$	11642.8	0.784334
$605$	4656.98	0.312948
$606$	−3806.96	−0.255194
$607$	−1923.28	−0.128606	−0.0643028	−	0.997930i	$-0.520482\pi$
$607$	−1923.28	−0.128606	−0.0643028	+	0.997930i	$0.520482\pi$
$608$	2843.86	0.189694
$609$	0	0
$610$	6838.45	0.453903
$611$	153.880	0.0101887
$612$	18293.9	1.20831
$613$	6887.96	0.453837	0.226919	−	0.973914i	$-0.427135\pi$
$613$	6887.96	0.453837	0.226919	+	0.973914i	$0.427135\pi$
$614$	−12493.7	−0.821183
$615$	−11234.9	−0.736642
$616$	0	0
$617$	4032.07	0.263088	0.131544	−	0.991310i	$-0.458007\pi$
$617$	4032.07	0.263088	0.131544	+	0.991310i	$0.458007\pi$
$618$	12913.8	0.840565
$619$	12643.8	0.820996	0.410498	−	0.911861i	$-0.365355\pi$
$619$	12643.8	0.820996	0.410498	+	0.911861i	$0.365355\pi$
$620$	6029.40	0.390559
$621$	−139.724	−0.00902888
$622$	12130.3	0.781962
$623$	0	0
$624$	−140.620	−0.00902131
$625$	625.000	0.0400000
$626$	−16570.5	−1.05797
$627$	−32896.7	−2.09532
$628$	8300.49	0.527429
$629$	21990.5	1.39399
$630$	0	0
$631$	−19282.6	−1.21652	−0.608262	−	0.793736i	$-0.708133\pi$
$631$	−19282.6	−1.21652	−0.608262	+	0.793736i	$0.708133\pi$
$632$	5776.55	0.363574
$633$	−5798.10	−0.364066
$634$	2382.09	0.149219
$635$	1112.33	0.0695141
$636$	1066.45	0.0664897
$637$	0	0
$638$	−12469.7	−0.773796
$639$	−23068.8	−1.42815
$640$	−640.000	−0.0395285
$641$	4821.65	0.297104	0.148552	−	0.988905i	$-0.452539\pi$
$641$	4821.65	0.297104	0.148552	+	0.988905i	$0.452539\pi$
$642$	12239.0	0.752388
$643$	31195.9	1.91329	0.956645	−	0.291255i	$-0.0940729\pi$
$643$	31195.9	1.91329	0.956645	+	0.291255i	$0.0940729\pi$
$644$	0	0
$645$	2745.54	0.167605
$646$	24218.8	1.47504
$647$	20524.7	1.24715	0.623577	−	0.781762i	$-0.285679\pi$
$647$	20524.7	1.24715	0.623577	+	0.781762i	$0.285679\pi$
$648$	4069.28	0.246692
$649$	35452.9	2.14430
$650$	−56.4660	−0.00340735
$651$	0	0
$652$	−434.140	−0.0260771
$653$	1562.03	0.0936095	0.0468048	−	0.998904i	$-0.485096\pi$
$653$	1562.03	0.0936095	0.0468048	+	0.998904i	$0.485096\pi$
$654$	−20436.0	−1.22188
$655$	−7334.01	−0.437501
$656$	4619.66	0.274950
$657$	2834.47	0.168316
$658$	0	0
$659$	−18041.4	−1.06646	−0.533228	−	0.845971i	$-0.679021\pi$
$659$	−18041.4	−1.06646	−0.533228	+	0.845971i	$0.679021\pi$
$660$	7403.28	0.436625
$661$	−25703.4	−1.51247	−0.756236	−	0.654298i	$-0.772964\pi$
$661$	−25703.4	−1.51247	−0.756236	+	0.654298i	$0.772964\pi$
$662$	9579.80	0.562431
$663$	−1197.54	−0.0701488
$664$	8082.85	0.472403
$665$	0	0
$666$	−10833.9	−0.630336
$667$	−358.503	−0.0208115
$668$	−11618.3	−0.672943
$669$	34083.4	1.96972
$670$	725.839	0.0418531
$671$	32526.9	1.87136
$672$	0	0
$673$	−21106.5	−1.20891	−0.604454	−	0.796640i	$-0.706609\pi$
$673$	−21106.5	−1.20891	−0.604454	+	0.796640i	$0.706609\pi$
$674$	1897.56	0.108444
$675$	−1277.21	−0.0728293
$676$	−8782.90	−0.499710
$677$	−26770.9	−1.51978	−0.759889	−	0.650053i	$-0.774747\pi$
$677$	−26770.9	−1.51978	−0.759889	+	0.650053i	$0.774747\pi$
$678$	−30666.0	−1.73705
$679$	0	0
$680$	−5450.35	−0.307369
$681$	−6867.56	−0.386440
$682$	28678.6	1.61021
$683$	4157.06	0.232892	0.116446	−	0.993197i	$-0.462850\pi$
$683$	4157.06	0.232892	0.116446	+	0.993197i	$0.462850\pi$
$684$	−11931.7	−0.666986
$685$	−14094.3	−0.786155
$686$	0	0
$687$	30490.8	1.69330
$688$	−1128.93	−0.0625583
$689$	−38.6890	−0.00213923
$690$	212.843	0.0117432
$691$	−8598.82	−0.473393	−0.236696	−	0.971584i	$-0.576065\pi$
$691$	−8598.82	−0.473393	−0.236696	+	0.971584i	$0.576065\pi$
$692$	−4443.55	−0.244102
$693$	0	0
$694$	11327.5	0.619578
$695$	10133.3	0.553064
$696$	−8160.98	−0.444456
$697$	39341.8	2.13798
$698$	4739.46	0.257007
$699$	10643.4	0.575924
$700$	0	0
$701$	−8786.01	−0.473385	−0.236693	−	0.971585i	$-0.576063\pi$
$701$	−8786.01	−0.473385	−0.236693	+	0.971585i	$0.576063\pi$
$702$	115.390	0.00620388
$703$	−14342.7	−0.769479
$704$	−3044.14	−0.162969
$705$	−5302.05	−0.283244
$706$	2832.31	0.150985
$707$	0	0
$708$	23202.6	1.23165
$709$	−22322.6	−1.18243	−0.591214	−	0.806514i	$-0.701351\pi$
$709$	−22322.6	−1.18243	−0.591214	+	0.806514i	$0.701351\pi$
$710$	6872.93	0.363291
$711$	−24236.0	−1.27837
$712$	−3395.66	−0.178733
$713$	824.505	0.0433071
$714$	0	0
$715$	−268.579	−0.0140479
$716$	−2509.71	−0.130995
$717$	−39541.5	−2.05956
$718$	15545.7	0.808022
$719$	−7668.32	−0.397747	−0.198874	−	0.980025i	$-0.563728\pi$
$719$	−7668.32	−0.397747	−0.198874	+	0.980025i	$0.563728\pi$
$720$	2685.17	0.138987
$721$	0	0
$722$	−2078.00	−0.107112
$723$	50520.9	2.59874
$724$	−7846.09	−0.402759
$725$	−3277.05	−0.167871
$726$	14496.9	0.741087
$727$	−15189.8	−0.774907	−0.387453	−	0.921889i	$-0.626645\pi$
$727$	−15189.8	−0.774907	−0.387453	+	0.921889i	$0.626645\pi$
$728$	0	0
$729$	−27807.8	−1.41278
$730$	−844.482	−0.0428160
$731$	−9614.17	−0.486447
$732$	21287.6	1.07488
$733$	−10061.0	−0.506976	−0.253488	−	0.967339i	$-0.581578\pi$
$733$	−10061.0	−0.506976	−0.253488	+	0.967339i	$0.581578\pi$
$734$	865.434	0.0435201
$735$	0	0
$736$	−87.5184	−0.00438311
$737$	3452.43	0.172553
$738$	−19382.2	−0.966757
$739$	−5137.10	−0.255712	−0.127856	−	0.991793i	$-0.540810\pi$
$739$	−5137.10	−0.255712	−0.127856	+	0.991793i	$0.540810\pi$
$740$	3227.76	0.160344
$741$	781.061	0.0387220
$742$	0	0
$743$	−32710.9	−1.61514	−0.807569	−	0.589773i	$-0.799217\pi$
$743$	−32710.9	−1.61514	−0.807569	+	0.589773i	$0.799217\pi$
$744$	18769.1	0.924877
$745$	14589.3	0.717465
$746$	−12315.6	−0.604431
$747$	−33912.3	−1.66102
$748$	−25924.4	−1.26723
$749$	0	0
$750$	1945.58	0.0947235
$751$	−19293.9	−0.937474	−0.468737	−	0.883338i	$-0.655291\pi$
$751$	−19293.9	−0.937474	−0.468737	+	0.883338i	$0.655291\pi$
$752$	2180.14	0.105720
$753$	16442.6	0.795753
$754$	296.067	0.0142999
$755$	14553.5	0.701530
$756$	0	0
$757$	2631.70	0.126355	0.0631775	−	0.998002i	$-0.479877\pi$
$757$	2631.70	0.126355	0.0631775	+	0.998002i	$0.479877\pi$
$758$	−20219.2	−0.968859
$759$	1012.38	0.0484151
$760$	3554.83	0.169667
$761$	−12339.7	−0.587796	−0.293898	−	0.955837i	$-0.594953\pi$
$761$	−12339.7	−0.587796	−0.293898	+	0.955837i	$0.594953\pi$
$762$	3462.61	0.164616
$763$	0	0
$764$	−12403.6	−0.587365
$765$	22867.4	1.08075
$766$	16416.7	0.774361
$767$	−841.753	−0.0396270
$768$	−1992.28	−0.0936069
$769$	1477.03	0.0692627	0.0346313	−	0.999400i	$-0.488974\pi$
$769$	1477.03	0.0692627	0.0346313	+	0.999400i	$0.488974\pi$
$770$	0	0
$771$	6262.89	0.292546
$772$	−3136.83	−0.146240
$773$	10308.3	0.479641	0.239820	−	0.970817i	$-0.422911\pi$
$773$	10308.3	0.479641	0.239820	+	0.970817i	$0.422911\pi$
$774$	4736.53	0.219962
$775$	7536.75	0.349326
$776$	−963.585	−0.0445756
$777$	0	0
$778$	−9836.62	−0.453291
$779$	−25659.5	−1.18016
$780$	−175.775	−0.00806891
$781$	32690.9	1.49779
$782$	−745.321	−0.0340826
$783$	6696.76	0.305648
$784$	0	0
$785$	10375.6	0.471747
$786$	−22830.3	−1.03604
$787$	10430.1	0.472418	0.236209	−	0.971702i	$-0.424095\pi$
$787$	10430.1	0.472418	0.236209	+	0.971702i	$0.424095\pi$
$788$	−9261.35	−0.418683
$789$	30939.6	1.39604
$790$	7220.69	0.325191
$791$	0	0
$792$	12771.9	0.573019
$793$	−772.280	−0.0345832
$794$	−16277.9	−0.727558
$795$	1333.06	0.0594702
$796$	−11105.7	−0.494510
$797$	10521.2	0.467602	0.233801	−	0.972285i	$-0.424884\pi$
$797$	10521.2	0.467602	0.233801	+	0.972285i	$0.424884\pi$
$798$	0	0
$799$	18566.4	0.822068
$800$	−800.000	−0.0353553
$801$	14246.8	0.628445
$802$	691.895	0.0304634
$803$	−4016.75	−0.176523
$804$	2259.49	0.0991119
$805$	0	0
$806$	−680.912	−0.0297569
$807$	−7384.49	−0.322115
$808$	1956.72	0.0851946
$809$	−25637.6	−1.11418	−0.557088	−	0.830453i	$-0.688081\pi$
$809$	−25637.6	−1.11418	−0.557088	+	0.830453i	$0.688081\pi$
$810$	5086.59	0.220648
$811$	21946.1	0.950225	0.475112	−	0.879925i	$-0.342407\pi$
$811$	21946.1	0.950225	0.475112	+	0.879925i	$0.342407\pi$
$812$	0	0
$813$	−997.613	−0.0430354
$814$	15352.7	0.661072
$815$	−542.676	−0.0233240
$816$	−16966.6	−0.727878
$817$	6270.56	0.268518
$818$	732.253	0.0312991
$819$	0	0
$820$	5774.57	0.245923
$821$	24586.1	1.04514	0.522572	−	0.852595i	$-0.324973\pi$
$821$	24586.1	1.04514	0.522572	+	0.852595i	$0.324973\pi$
$822$	−43874.6	−1.86168
$823$	−22966.4	−0.972731	−0.486366	−	0.873755i	$-0.661678\pi$
$823$	−22966.4	−0.972731	−0.486366	+	0.873755i	$0.661678\pi$
$824$	−6637.50	−0.280617
$825$	9254.10	0.390529
$826$	0	0
$827$	38028.0	1.59899	0.799494	−	0.600674i	$-0.205101\pi$
$827$	38028.0	1.59899	0.799494	+	0.600674i	$0.205101\pi$
$828$	367.191	0.0154115
$829$	−27697.2	−1.16039	−0.580194	−	0.814478i	$-0.697023\pi$
$829$	−27697.2	−1.16039	−0.580194	+	0.814478i	$0.697023\pi$
$830$	10103.6	0.422530
$831$	−29221.5	−1.21984
$832$	72.2765	0.00301170
$833$	0	0
$834$	31544.4	1.30970
$835$	−14522.9	−0.601898
$836$	16908.4	0.699509
$837$	−15401.6	−0.636030
$838$	13831.0	0.570149
$839$	5304.21	0.218262	0.109131	−	0.994027i	$-0.465193\pi$
$839$	5304.21	0.218262	0.109131	+	0.994027i	$0.465193\pi$
$840$	0	0
$841$	−7206.53	−0.295483
$842$	−12935.2	−0.529427
$843$	−48033.4	−1.96246
$844$	2980.14	0.121541
$845$	−10978.6	−0.446954
$846$	−9146.95	−0.371724
$847$	0	0
$848$	−548.138	−0.0221971
$849$	−26079.1	−1.05422
$850$	−6812.93	−0.274920
$851$	441.388	0.0177798
$852$	21395.0	0.860305
$853$	−14972.1	−0.600980	−0.300490	−	0.953785i	$-0.597150\pi$
$853$	−14972.1	−0.600980	−0.300490	+	0.953785i	$0.597150\pi$
$854$	0	0
$855$	−14914.6	−0.596570
$856$	−6290.64	−0.251180
$857$	−8884.53	−0.354130	−0.177065	−	0.984199i	$-0.556660\pi$
$857$	−8884.53	−0.354130	−0.177065	+	0.984199i	$0.556660\pi$
$858$	−836.067	−0.0332667
$859$	16654.0	0.661499	0.330749	−	0.943719i	$-0.392699\pi$
$859$	16654.0	0.661499	0.330749	+	0.943719i	$0.392699\pi$
$860$	−1411.16	−0.0559539
$861$	0	0
$862$	18940.4	0.748391
$863$	−24985.4	−0.985531	−0.492766	−	0.870162i	$-0.664014\pi$
$863$	−24985.4	−0.985531	−0.492766	+	0.870162i	$0.664014\pi$
$864$	1634.83	0.0643726
$865$	−5554.44	−0.218331
$866$	−10194.9	−0.400044
$867$	−106255.	−4.16219
$868$	0	0
$869$	34345.0	1.34071
$870$	−10201.2	−0.397533
$871$	−81.9704	−0.00318882
$872$	10503.8	0.407917
$873$	4042.80	0.156733
$874$	486.114	0.0188135
$875$	0	0
$876$	−2628.81	−0.101392
$877$	17465.3	0.672474	0.336237	−	0.941777i	$-0.390846\pi$
$877$	17465.3	0.672474	0.336237	+	0.941777i	$0.390846\pi$
$878$	227.021	0.00872619
$879$	−5301.03	−0.203412
$880$	−3805.17	−0.145764
$881$	15147.3	0.579259	0.289629	−	0.957139i	$-0.406468\pi$
$881$	15147.3	0.579259	0.289629	+	0.957139i	$0.406468\pi$
$882$	0	0
$883$	−9446.54	−0.360024	−0.180012	−	0.983664i	$-0.557614\pi$
$883$	−9446.54	−0.360024	−0.180012	+	0.983664i	$0.557614\pi$
$884$	615.518	0.0234187
$885$	29003.3	1.10162
$886$	34398.2	1.30432
$887$	16388.8	0.620384	0.310192	−	0.950674i	$-0.399607\pi$
$887$	16388.8	0.620384	0.310192	+	0.950674i	$0.399607\pi$
$888$	10047.8	0.379709
$889$	0	0
$890$	−4244.57	−0.159863
$891$	24194.2	0.909693
$892$	−17518.4	−0.657576
$893$	−12109.4	−0.453780
$894$	45415.5	1.69902
$895$	−3137.13	−0.117165
$896$	0	0
$897$	−24.0368	−0.000894720 0
$898$	−4904.15	−0.182242
$899$	−39517.3	−1.46604
$900$	3356.47	0.124314
$901$	−4668.04	−0.172602
$902$	27466.5	1.01390
$903$	0	0
$904$	15761.9	0.579902
$905$	−9807.61	−0.360239
$906$	45304.0	1.66128
$907$	−18465.3	−0.675997	−0.337998	−	0.941147i	$-0.609750\pi$
$907$	−18465.3	−0.675997	−0.337998	+	0.941147i	$0.609750\pi$
$908$	3529.82	0.129010
$909$	−8209.59	−0.299554
$910$	0	0
$911$	−29356.9	−1.06766	−0.533830	−	0.845592i	$-0.679248\pi$
$911$	−29356.9	−1.06766	−0.533830	+	0.845592i	$0.679248\pi$
$912$	11065.9	0.401787
$913$	48057.2	1.74202
$914$	26848.9	0.971645
$915$	26609.5	0.961404
$916$	−15671.8	−0.565295
$917$	0	0
$918$	13922.5	0.500555
$919$	911.579	0.0327206	0.0163603	−	0.999866i	$-0.494792\pi$
$919$	911.579	0.0327206	0.0163603	+	0.999866i	$0.494792\pi$
$920$	−109.398	−0.00392037
$921$	−48615.2	−1.73933
$922$	7363.49	0.263019
$923$	−776.174	−0.0276794
$924$	0	0
$925$	4034.70	0.143416
$926$	−3400.17	−0.120666
$927$	27848.2	0.986682
$928$	4194.62	0.148378
$929$	−47788.8	−1.68773	−0.843864	−	0.536557i	$-0.819725\pi$
$929$	−47788.8	−1.68773	−0.843864	+	0.536557i	$0.819725\pi$
$930$	23461.4	0.827236
$931$	0	0
$932$	−5470.56	−0.192268
$933$	47201.0	1.65626
$934$	15903.8	0.557161
$935$	−32405.5	−1.13345
$936$	−303.242	−0.0105895
$937$	−41650.7	−1.45216	−0.726078	−	0.687612i	$-0.758659\pi$
$937$	−41650.7	−1.45216	−0.726078	+	0.687612i	$0.758659\pi$
$938$	0	0
$939$	−64478.4	−2.24087
$940$	2725.17	0.0945589
$941$	51606.4	1.78780	0.893901	−	0.448264i	$-0.147958\pi$
$941$	51606.4	1.78780	0.893901	+	0.448264i	$0.147958\pi$
$942$	32298.6	1.11714
$943$	789.658	0.0272691
$944$	−11925.8	−0.411178
$945$	0	0
$946$	−6712.16	−0.230688
$947$	−28910.3	−0.992038	−0.496019	−	0.868312i	$-0.665205\pi$
$947$	−28910.3	−0.992038	−0.496019	+	0.868312i	$0.665205\pi$
$948$	22477.5	0.770080
$949$	95.3690	0.00326218
$950$	4443.53	0.151755
$951$	9269.10	0.316058
$952$	0	0
$953$	37077.6	1.26029	0.630147	−	0.776476i	$-0.282995\pi$
$953$	37077.6	1.26029	0.630147	+	0.776476i	$0.282995\pi$
$954$	2299.76	0.0780476
$955$	−15504.5	−0.525355
$956$	20323.7	0.687570
$957$	−48521.8	−1.63896
$958$	−25640.5	−0.864726
$959$	0	0
$960$	−2490.35	−0.0837245
$961$	61093.1	2.05072
$962$	−364.517	−0.0122167
$963$	26392.9	0.883177
$964$	−25967.0	−0.867572
$965$	−3921.04	−0.130801
$966$	0	0
$967$	36422.8	1.21125	0.605625	−	0.795750i	$-0.292923\pi$
$967$	36422.8	1.21125	0.605625	+	0.795750i	$0.292923\pi$
$968$	−7451.18	−0.247407
$969$	94239.3	3.12426
$970$	−1204.48	−0.0398697
$971$	13119.2	0.433590	0.216795	−	0.976217i	$-0.430440\pi$
$971$	13119.2	0.433590	0.216795	+	0.976217i	$0.430440\pi$
$972$	21351.8	0.704587
$973$	0	0
$974$	−11087.5	−0.364748
$975$	−219.719	−0.00721705
$976$	−10941.5	−0.358842
$977$	11606.2	0.380057	0.190028	−	0.981779i	$-0.439142\pi$
$977$	11606.2	0.380057	0.190028	+	0.981779i	$0.439142\pi$
$978$	−1689.31	−0.0552334
$979$	−20189.2	−0.659089
$980$	0	0
$981$	−44069.5	−1.43428
$982$	32764.2	1.06471
$983$	30458.7	0.988283	0.494142	−	0.869381i	$-0.335482\pi$
$983$	30458.7	0.988283	0.494142	+	0.869381i	$0.335482\pi$
$984$	17975.8	0.582367
$985$	−11576.7	−0.374481
$986$	35722.1	1.15378
$987$	0	0
$988$	−401.454	−0.0129271
$989$	−192.973	−0.00620444
$990$	15964.9	0.512524
$991$	−39763.4	−1.27460	−0.637299	−	0.770617i	$-0.719948\pi$
$991$	−39763.4	−1.27460	−0.637299	+	0.770617i	$0.719948\pi$
$992$	−9647.04	−0.308764
$993$	37276.6	1.19128
$994$	0	0
$995$	−13882.1	−0.442303
$996$	31451.7	1.00059
$997$	−40668.8	−1.29187	−0.645935	−	0.763393i	$-0.723532\pi$
$997$	−40668.8	−1.29187	−0.645935	+	0.763393i	$0.723532\pi$
$998$	−23075.2	−0.731896
$999$	−8245.04	−0.261123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.4.a.r.1.1		2
5.4	even	2		2450.4.a.bz.1.2		2
7.2	even	3		70.4.e.d.11.2	✓	4
7.3	odd	6		490.4.e.u.471.1		4
7.4	even	3		70.4.e.d.51.2	yes	4
7.5	odd	6		490.4.e.u.361.1		4
7.6	odd	2		490.4.a.t.1.2		2
21.2	odd	6		630.4.k.l.361.2		4
21.11	odd	6		630.4.k.l.541.2		4
28.11	odd	6		560.4.q.j.401.1		4
28.23	odd	6		560.4.q.j.81.1		4
35.2	odd	12		350.4.j.g.249.1		8
35.4	even	6		350.4.e.h.51.1		4
35.9	even	6		350.4.e.h.151.1		4
35.18	odd	12		350.4.j.g.149.1		8
35.23	odd	12		350.4.j.g.249.4		8
35.32	odd	12		350.4.j.g.149.4		8
35.34	odd	2		2450.4.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.e.d.11.2	✓	4	7.2	even	3
70.4.e.d.51.2	yes	4	7.4	even	3
350.4.e.h.51.1		4	35.4	even	6
350.4.e.h.151.1		4	35.9	even	6
350.4.j.g.149.1		8	35.18	odd	12
350.4.j.g.149.4		8	35.32	odd	12
350.4.j.g.249.1		8	35.2	odd	12
350.4.j.g.249.4		8	35.23	odd	12
490.4.a.r.1.1		2	1.1	even	1	trivial
490.4.a.t.1.2		2	7.6	odd	2
490.4.e.u.361.1		4	7.5	odd	6
490.4.e.u.471.1		4	7.3	odd	6
560.4.q.j.81.1		4	28.23	odd	6
560.4.q.j.401.1		4	28.11	odd	6
630.4.k.l.361.2		4	21.2	odd	6
630.4.k.l.541.2		4	21.11	odd	6
2450.4.a.bv.1.1		2	35.34	odd	2
2450.4.a.bz.1.2		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	490.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -7.78233 q^{3} +4.00000 q^{4} +5.00000 q^{5} +15.5647 q^{6} -8.00000 q^{8} +33.5647 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -7.78233 q^{3} +4.00000 q^{4} +5.00000 q^{5} +15.5647 q^{6} -8.00000 q^{8} +33.5647 q^{9} -10.0000 q^{10} -47.5647 q^{11} -31.1293 q^{12} +1.12932 q^{13} -38.9116 q^{15} +16.0000 q^{16} +136.259 q^{17} -67.1293 q^{18} -88.8707 q^{19} +20.0000 q^{20} +95.1293 q^{22} +2.73495 q^{23} +62.2586 q^{24} +25.0000 q^{25} -2.25864 q^{26} -51.0884 q^{27} -131.082 q^{29} +77.8233 q^{30} +301.470 q^{31} -32.0000 q^{32} +370.164 q^{33} -272.517 q^{34} +134.259 q^{36} +161.388 q^{37} +177.741 q^{38} -8.78874 q^{39} -40.0000 q^{40} +288.729 q^{41} -70.5582 q^{43} -190.259 q^{44} +167.823 q^{45} -5.46990 q^{46} +136.259 q^{47} -124.517 q^{48} -50.0000 q^{50} -1060.41 q^{51} +4.51728 q^{52} -34.2586 q^{53} +102.177 q^{54} -237.823 q^{55} +691.621 q^{57} +262.164 q^{58} -745.362 q^{59} -155.647 q^{60} -683.845 q^{61} -602.940 q^{62} +64.0000 q^{64} +5.64660 q^{65} -740.328 q^{66} -72.5839 q^{67} +545.035 q^{68} -21.2843 q^{69} -687.293 q^{71} -268.517 q^{72} +84.4482 q^{73} -322.776 q^{74} -194.558 q^{75} -355.483 q^{76} +17.5775 q^{78} -722.069 q^{79} +80.0000 q^{80} -508.659 q^{81} -577.457 q^{82} -1010.36 q^{83} +681.293 q^{85} +141.116 q^{86} +1020.12 q^{87} +380.517 q^{88} +424.457 q^{89} -335.647 q^{90} +10.9398 q^{92} -2346.14 q^{93} -272.517 q^{94} -444.353 q^{95} +249.035 q^{96} +120.448 q^{97} -1596.49 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 10 q^{5} + 4 q^{6} - 16 q^{8} + 40 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 + 10 * q^5 + 4 * q^6 - 16 * q^8 + 40 * q^9	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 10 q^{5} + 4 q^{6} - 16 q^{8} + 40 q^{9} - 20 q^{10} - 68 q^{11} - 8 q^{12} - 52 q^{13} - 10 q^{15} + 32 q^{16} + 164 q^{17} - 80 q^{18} - 232 q^{19} + 40 q^{20} + 136 q^{22} - 198 q^{23} + 16 q^{24} + 50 q^{25} + 104 q^{26} - 170 q^{27} - 18 q^{29} + 20 q^{30} + 196 q^{31} - 64 q^{32} + 252 q^{33} - 328 q^{34} + 160 q^{36} + 160 q^{37} + 464 q^{38} - 316 q^{39} - 80 q^{40} + 62 q^{41} + 198 q^{43} - 272 q^{44} + 200 q^{45} + 396 q^{46} + 164 q^{47} - 32 q^{48} - 100 q^{50} - 900 q^{51} - 208 q^{52} + 40 q^{53} + 340 q^{54} - 340 q^{55} - 136 q^{57} + 36 q^{58} - 80 q^{59} - 40 q^{60} - 174 q^{61} - 392 q^{62} + 128 q^{64} - 260 q^{65} - 504 q^{66} - 1054 q^{67} + 656 q^{68} - 1182 q^{69} - 832 q^{71} - 320 q^{72} + 820 q^{73} - 320 q^{74} - 50 q^{75} - 928 q^{76} + 632 q^{78} - 576 q^{79} + 160 q^{80} - 1370 q^{81} - 124 q^{82} - 298 q^{83} + 820 q^{85} - 396 q^{86} + 1674 q^{87} + 544 q^{88} - 182 q^{89} - 400 q^{90} - 792 q^{92} - 2956 q^{93} - 328 q^{94} - 1160 q^{95} + 64 q^{96} + 892 q^{97} - 1728 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 + 10 * q^5 + 4 * q^6 - 16 * q^8 + 40 * q^9 - 20 * q^10 - 68 * q^11 - 8 * q^12 - 52 * q^13 - 10 * q^15 + 32 * q^16 + 164 * q^17 - 80 * q^18 - 232 * q^19 + 40 * q^20 + 136 * q^22 - 198 * q^23 + 16 * q^24 + 50 * q^25 + 104 * q^26 - 170 * q^27 - 18 * q^29 + 20 * q^30 + 196 * q^31 - 64 * q^32 + 252 * q^33 - 328 * q^34 + 160 * q^36 + 160 * q^37 + 464 * q^38 - 316 * q^39 - 80 * q^40 + 62 * q^41 + 198 * q^43 - 272 * q^44 + 200 * q^45 + 396 * q^46 + 164 * q^47 - 32 * q^48 - 100 * q^50 - 900 * q^51 - 208 * q^52 + 40 * q^53 + 340 * q^54 - 340 * q^55 - 136 * q^57 + 36 * q^58 - 80 * q^59 - 40 * q^60 - 174 * q^61 - 392 * q^62 + 128 * q^64 - 260 * q^65 - 504 * q^66 - 1054 * q^67 + 656 * q^68 - 1182 * q^69 - 832 * q^71 - 320 * q^72 + 820 * q^73 - 320 * q^74 - 50 * q^75 - 928 * q^76 + 632 * q^78 - 576 * q^79 + 160 * q^80 - 1370 * q^81 - 124 * q^82 - 298 * q^83 + 820 * q^85 - 396 * q^86 + 1674 * q^87 + 544 * q^88 - 182 * q^89 - 400 * q^90 - 792 * q^92 - 2956 * q^93 - 328 * q^94 - 1160 * q^95 + 64 * q^96 + 892 * q^97 - 1728 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more