LMFDB - Embedded newform 49.6.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [49,6,Mod(1,49)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("49.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	49.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.85880717084$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.54138$ of defining polynomial
Character	$\chi$	$=$	49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+7.08276 q^{2} -10.0828 q^{3} +18.1655 q^{4} -79.8276 q^{5} -71.4138 q^{6} -97.9863 q^{8} -141.338 q^{9} -565.400 q^{10} +351.904 q^{11} -183.159 q^{12} -291.683 q^{13} +804.883 q^{15} -1275.31 q^{16} -370.075 q^{17} -1001.06 q^{18} +1504.93 q^{19} -1450.11 q^{20} +2492.45 q^{22} -425.711 q^{23} +987.973 q^{24} +3247.45 q^{25} -2065.92 q^{26} +3875.19 q^{27} -7783.93 q^{29} +5700.80 q^{30} -2575.18 q^{31} -5897.16 q^{32} -3548.16 q^{33} -2621.16 q^{34} -2567.48 q^{36} +739.618 q^{37} +10659.0 q^{38} +2940.97 q^{39} +7822.01 q^{40} +7029.84 q^{41} +1835.23 q^{43} +6392.51 q^{44} +11282.7 q^{45} -3015.21 q^{46} -1532.68 q^{47} +12858.7 q^{48} +23000.9 q^{50} +3731.38 q^{51} -5298.57 q^{52} -9537.46 q^{53} +27447.0 q^{54} -28091.6 q^{55} -15173.8 q^{57} -55131.8 q^{58} -29674.1 q^{59} +14621.1 q^{60} -46510.8 q^{61} -18239.4 q^{62} -958.246 q^{64} +23284.3 q^{65} -25130.8 q^{66} +26746.1 q^{67} -6722.61 q^{68} +4292.34 q^{69} -14388.8 q^{71} +13849.2 q^{72} -70095.1 q^{73} +5238.54 q^{74} -32743.3 q^{75} +27337.8 q^{76} +20830.2 q^{78} -27085.8 q^{79} +101805. q^{80} -4727.49 q^{81} +49790.7 q^{82} -79755.4 q^{83} +29542.2 q^{85} +12998.5 q^{86} +78483.6 q^{87} -34481.7 q^{88} +43577.3 q^{89} +79912.5 q^{90} -7733.26 q^{92} +25964.9 q^{93} -10855.6 q^{94} -120135. q^{95} +59459.7 q^{96} +103374. q^{97} -49737.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 8 q^{3} + 12 q^{4} - 38 q^{5} - 82 q^{6} + 96 q^{8} - 380 q^{9} - 778 q^{10} + 424 q^{11} - 196 q^{12} - 924 q^{13} + 892 q^{15} - 2064 q^{16} - 2346 q^{17} + 212 q^{18} - 360 q^{19} - 1708 q^{20}+ \cdots - 66944 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 8 * q^3 + 12 * q^4 - 38 * q^5 - 82 * q^6 + 96 * q^8 - 380 * q^9 - 778 * q^10 + 424 * q^11 - 196 * q^12 - 924 * q^13 + 892 * q^15 - 2064 * q^16 - 2346 * q^17 + 212 * q^18 - 360 * q^19 - 1708 * q^20 + 2126 * q^22 - 12 * q^23 + 1392 * q^24 + 1872 * q^25 + 1148 * q^26 + 2872 * q^27 - 7052 * q^29 + 5258 * q^30 + 3548 * q^31 - 8096 * q^32 - 3398 * q^33 + 7422 * q^34 - 1096 * q^36 + 11090 * q^37 + 20138 * q^38 + 1624 * q^39 + 15936 * q^40 + 3500 * q^41 - 12680 * q^43 + 5948 * q^44 + 1300 * q^45 - 5118 * q^46 - 22956 * q^47 + 11216 * q^48 + 29992 * q^50 - 384 * q^51 - 1400 * q^52 + 3042 * q^53 + 32546 * q^54 - 25076 * q^55 - 19058 * q^57 - 58852 * q^58 - 65808 * q^59 + 14084 * q^60 - 42486 * q^61 - 49362 * q^62 + 35456 * q^64 - 3164 * q^65 - 25894 * q^66 + 42312 * q^67 + 5460 * q^68 + 5154 * q^69 - 2208 * q^71 - 32448 * q^72 - 50506 * q^73 - 47370 * q^74 - 35608 * q^75 + 38836 * q^76 + 27524 * q^78 + 9004 * q^79 + 68816 * q^80 + 51178 * q^81 + 67732 * q^82 - 104328 * q^83 - 53106 * q^85 + 86776 * q^86 + 80008 * q^87 - 20496 * q^88 - 26666 * q^89 + 130652 * q^90 - 10284 * q^92 + 38718 * q^93 + 98034 * q^94 - 198140 * q^95 + 54880 * q^96 + 209132 * q^97 - 66944 * q^99

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 $

$2$	7.08276	1.25207	0.626034	−	0.779796i	$-0.284677\pi$
$2$	7.08276	1.25207	0.626034	+	0.779796i	$0.284677\pi$
$3$	−10.0828	−0.646810	−0.323405	−	0.946261i	$-0.604828\pi$
$3$	−10.0828	−0.646810	−0.323405	+	0.946261i	$0.604828\pi$
$4$	18.1655	0.567673
$5$	−79.8276	−1.42800	−0.714000	−	0.700146i	$-0.753118\pi$
$5$	−79.8276	−1.42800	−0.714000	+	0.700146i	$0.753118\pi$
$6$	−71.4138	−0.809849
$7$	0	0
$7$	0	0
$8$	−97.9863	−0.541303
$9$	−141.338	−0.581637
$10$	−565.400	−1.78795
$11$	351.904	0.876884	0.438442	−	0.898760i	$-0.355531\pi$
$11$	351.904	0.876884	0.438442	+	0.898760i	$0.355531\pi$
$12$	−183.159	−0.367176
$13$	−291.683	−0.478688	−0.239344	−	0.970935i	$-0.576932\pi$
$13$	−291.683	−0.478688	−0.239344	+	0.970935i	$0.576932\pi$
$14$	0	0
$15$	804.883	0.923644
$16$	−1275.31	−1.24542
$17$	−370.075	−0.310576	−0.155288	−	0.987869i	$-0.549631\pi$
$17$	−370.075	−0.310576	−0.155288	+	0.987869i	$0.549631\pi$
$18$	−1001.06	−0.728249
$19$	1504.93	0.956381	0.478190	−	0.878256i	$-0.341293\pi$
$19$	1504.93	0.956381	0.478190	+	0.878256i	$0.341293\pi$
$20$	−1450.11	−0.810637
$21$	0	0
$22$	2492.45	1.09792
$23$	−425.711	−0.167801	−0.0839006	−	0.996474i	$-0.526738\pi$
$23$	−425.711	−0.167801	−0.0839006	+	0.996474i	$0.526738\pi$
$24$	987.973	0.350120
$25$	3247.45	1.03918
$26$	−2065.92	−0.599349
$27$	3875.19	1.02302
$28$	0	0
$29$	−7783.93	−1.71872	−0.859358	−	0.511374i	$-0.829137\pi$
$29$	−7783.93	−1.71872	−0.859358	+	0.511374i	$0.829137\pi$
$30$	5700.80	1.15646
$31$	−2575.18	−0.481285	−0.240643	−	0.970614i	$-0.577358\pi$
$31$	−2575.18	−0.481285	−0.240643	+	0.970614i	$0.577358\pi$
$32$	−5897.16	−1.01805
$33$	−3548.16	−0.567177
$34$	−2621.16	−0.388862
$35$	0	0
$36$	−2567.48	−0.330180
$37$	739.618	0.0888184	0.0444092	−	0.999013i	$-0.485859\pi$
$37$	739.618	0.0888184	0.0444092	+	0.999013i	$0.485859\pi$
$38$	10659.0	1.19745
$39$	2940.97	0.309620
$40$	7822.01	0.772981
$41$	7029.84	0.653109	0.326554	−	0.945178i	$-0.394112\pi$
$41$	7029.84	0.653109	0.326554	+	0.945178i	$0.394112\pi$
$42$	0	0
$43$	1835.23	0.151363	0.0756816	−	0.997132i	$-0.475887\pi$
$43$	1835.23	0.151363	0.0756816	+	0.997132i	$0.475887\pi$
$44$	6392.51	0.497783
$45$	11282.7	0.830578
$46$	−3015.21	−0.210098
$47$	−1532.68	−0.101206	−0.0506032	−	0.998719i	$-0.516114\pi$
$47$	−1532.68	−0.101206	−0.0506032	+	0.998719i	$0.516114\pi$
$48$	12858.7	0.805550
$49$	0	0
$50$	23000.9	1.30113
$51$	3731.38	0.200883
$52$	−5298.57	−0.271738
$53$	−9537.46	−0.466383	−0.233192	−	0.972431i	$-0.574917\pi$
$53$	−9537.46	−0.466383	−0.233192	+	0.972431i	$0.574917\pi$
$54$	27447.0	1.28089
$55$	−28091.6	−1.25219
$56$	0	0
$57$	−15173.8	−0.618596
$58$	−55131.8	−2.15195
$59$	−29674.1	−1.10981	−0.554903	−	0.831915i	$-0.687245\pi$
$59$	−29674.1	−1.10981	−0.554903	+	0.831915i	$0.687245\pi$
$60$	14621.1	0.524327
$61$	−46510.8	−1.60040	−0.800201	−	0.599732i	$-0.795274\pi$
$61$	−46510.8	−1.60040	−0.800201	+	0.599732i	$0.795274\pi$
$62$	−18239.4	−0.602602
$63$	0	0
$64$	−958.246	−0.0292434
$65$	23284.3	0.683566
$66$	−25130.8	−0.710143
$67$	26746.1	0.727902	0.363951	−	0.931418i	$-0.381428\pi$
$67$	26746.1	0.727902	0.363951	+	0.931418i	$0.381428\pi$
$68$	−6722.61	−0.176305
$69$	4292.34	0.108535
$70$	0	0
$71$	−14388.8	−0.338748	−0.169374	−	0.985552i	$-0.554175\pi$
$71$	−14388.8	−0.338748	−0.169374	+	0.985552i	$0.554175\pi$
$72$	13849.2	0.314842
$73$	−70095.1	−1.53950	−0.769752	−	0.638343i	$-0.779620\pi$
$73$	−70095.1	−1.53950	−0.769752	+	0.638343i	$0.779620\pi$
$74$	5238.54	0.111207
$75$	−32743.3	−0.672154
$76$	27337.8	0.542911
$77$	0	0
$78$	20830.2	0.387665
$79$	−27085.8	−0.488285	−0.244143	−	0.969739i	$-0.578507\pi$
$79$	−27085.8	−0.488285	−0.244143	+	0.969739i	$0.578507\pi$
$80$	101805.	1.77846
$81$	−4727.49	−0.0800604
$82$	49790.7	0.817736
$83$	−79755.4	−1.27076	−0.635382	−	0.772198i	$-0.719157\pi$
$83$	−79755.4	−1.27076	−0.635382	+	0.772198i	$0.719157\pi$
$84$	0	0
$85$	29542.2	0.443502
$86$	12998.5	0.189517
$87$	78483.6	1.11168
$88$	−34481.7	−0.474660
$89$	43577.3	0.583157	0.291579	−	0.956547i	$-0.405820\pi$
$89$	43577.3	0.583157	0.291579	+	0.956547i	$0.405820\pi$
$90$	79912.5	1.03994
$91$	0	0
$92$	−7733.26	−0.0952561
$93$	25964.9	0.311300
$94$	−10855.6	−0.126717
$95$	−120135.	−1.36571
$96$	59459.7	0.658483
$97$	103374.	1.11553	0.557765	−	0.829999i	$-0.311659\pi$
$97$	103374.	1.11553	0.557765	+	0.829999i	$0.311659\pi$
$98$	0	0
$99$	−49737.3	−0.510028
$100$	58991.6	0.589916
$101$	28700.2	0.279951	0.139975	−	0.990155i	$-0.455298\pi$
$101$	28700.2	0.279951	0.139975	+	0.990155i	$0.455298\pi$
$102$	26428.5	0.251520
$103$	29227.9	0.271459	0.135730	−	0.990746i	$-0.456662\pi$
$103$	29227.9	0.271459	0.135730	+	0.990746i	$0.456662\pi$
$104$	28580.9	0.259115
$105$	0	0
$106$	−67551.6	−0.583944
$107$	87858.4	0.741863	0.370932	−	0.928660i	$-0.379038\pi$
$107$	87858.4	0.741863	0.370932	+	0.928660i	$0.379038\pi$
$108$	70394.8	0.580739
$109$	220628.	1.77867	0.889333	−	0.457260i	$-0.151169\pi$
$109$	220628.	1.77867	0.889333	+	0.457260i	$0.151169\pi$
$110$	−198966.	−1.56783
$111$	−7457.39	−0.0574486
$112$	0	0
$113$	39665.6	0.292225	0.146113	−	0.989268i	$-0.453324\pi$
$113$	39665.6	0.292225	0.146113	+	0.989268i	$0.453324\pi$
$114$	−107472.	−0.774524
$115$	33983.5	0.239620
$116$	−141399.	−0.975668
$117$	41225.8	0.278423
$118$	−210174.	−1.38955
$119$	0	0
$120$	−78867.5	−0.499971
$121$	−37214.9	−0.231075
$122$	−329425.	−2.00381
$123$	−70880.2	−0.422437
$124$	−46779.4	−0.273212
$125$	−9774.87	−0.0559546
$126$	0	0
$127$	51740.3	0.284655	0.142328	−	0.989820i	$-0.454541\pi$
$127$	51740.3	0.284655	0.142328	+	0.989820i	$0.454541\pi$
$128$	181922.	0.981433
$129$	−18504.2	−0.0979031
$130$	164917.	0.855871
$131$	−166674.	−0.848572	−0.424286	−	0.905528i	$-0.639475\pi$
$131$	−166674.	−0.848572	−0.424286	+	0.905528i	$0.639475\pi$
$132$	−64454.2	−0.321971
$133$	0	0
$134$	189436.	0.911382
$135$	−309347.	−1.46087
$136$	36262.3	0.168116
$137$	28259.4	0.128636	0.0643178	−	0.997929i	$-0.479513\pi$
$137$	28259.4	0.128636	0.0643178	+	0.997929i	$0.479513\pi$
$138$	30401.6	0.135894
$139$	336393.	1.47676	0.738380	−	0.674384i	$-0.235591\pi$
$139$	336393.	1.47676	0.738380	+	0.674384i	$0.235591\pi$
$140$	0	0
$141$	15453.7	0.0654612
$142$	−101912.	−0.424136
$143$	−102644.	−0.419753
$144$	180250.	0.724383
$145$	621373.	2.45433
$146$	−496467.	−1.92756
$147$	0	0
$148$	13435.5	0.0504198
$149$	−355381.	−1.31138	−0.655691	−	0.755030i	$-0.727622\pi$
$149$	−355381.	−1.31138	−0.655691	+	0.755030i	$0.727622\pi$
$150$	−231913.	−0.841582
$151$	−358797.	−1.28058	−0.640290	−	0.768133i	$-0.721186\pi$
$151$	−358797.	−1.28058	−0.640290	+	0.768133i	$0.721186\pi$
$152$	−147462.	−0.517692
$153$	52305.7	0.180643
$154$	0	0
$155$	205570.	0.687275
$156$	53424.2	0.175763
$157$	458911.	1.48586	0.742932	−	0.669367i	$-0.233435\pi$
$157$	458911.	1.48586	0.742932	+	0.669367i	$0.233435\pi$
$158$	−191842.	−0.611366
$159$	96164.0	0.301661
$160$	470756.	1.45377
$161$	0	0
$162$	−33483.7	−0.100241
$163$	502441.	1.48121	0.740603	−	0.671943i	$-0.234540\pi$
$163$	502441.	1.48121	0.740603	+	0.671943i	$0.234540\pi$
$164$	127701.	0.370752
$165$	283241.	0.809928
$166$	−564889.	−1.59108
$167$	−676652.	−1.87748	−0.938738	−	0.344632i	$-0.888004\pi$
$167$	−676652.	−1.87748	−0.938738	+	0.344632i	$0.888004\pi$
$168$	0	0
$169$	−286214.	−0.770858
$170$	209241.	0.555295
$171$	−212703.	−0.556267
$172$	33338.0	0.0859247
$173$	−249160.	−0.632941	−0.316470	−	0.948602i	$-0.602498\pi$
$173$	−249160.	−0.632941	−0.316470	+	0.948602i	$0.602498\pi$
$174$	555880.	1.39190
$175$	0	0
$176$	−448786.	−1.09209
$177$	299196.	0.717833
$178$	308648.	0.730152
$179$	139258.	0.324853	0.162427	−	0.986721i	$-0.448068\pi$
$179$	139258.	0.324853	0.162427	+	0.986721i	$0.448068\pi$
$180$	204956.	0.471497
$181$	306246.	0.694823	0.347412	−	0.937713i	$-0.387061\pi$
$181$	306246.	0.694823	0.347412	+	0.937713i	$0.387061\pi$
$182$	0	0
$183$	468957.	1.03516
$184$	41713.8	0.0908312
$185$	−59041.9	−0.126833
$186$	183903.	0.389768
$187$	−130231.	−0.272339
$188$	−27842.0	−0.0574521
$189$	0	0
$190$	−850885.	−1.70996
$191$	227494.	0.451218	0.225609	−	0.974218i	$-0.427563\pi$
$191$	227494.	0.451218	0.225609	+	0.974218i	$0.427563\pi$
$192$	9661.77	0.0189149
$193$	−672374.	−1.29933	−0.649663	−	0.760223i	$-0.725090\pi$
$193$	−672374.	−1.29933	−0.649663	+	0.760223i	$0.725090\pi$
$194$	732172.	1.39672
$195$	−234770.	−0.442137
$196$	0	0
$197$	−1282.76	−0.00235493	−0.00117747	−	0.999999i	$-0.500375\pi$
$197$	−1282.76	−0.00235493	−0.00117747	+	0.999999i	$0.500375\pi$
$198$	−352278.	−0.638590
$199$	368898.	0.660349	0.330175	−	0.943920i	$-0.392892\pi$
$199$	368898.	0.660349	0.330175	+	0.943920i	$0.392892\pi$
$200$	−318206.	−0.562513
$201$	−269674.	−0.470814
$202$	203277.	0.350517
$203$	0	0
$204$	67782.5	0.114036
$205$	−561175.	−0.932640
$206$	207014.	0.339885
$207$	60169.0	0.0975994
$208$	371986.	0.596167
$209$	529589.	0.838635
$210$	0	0
$211$	502168.	0.776503	0.388251	−	0.921553i	$-0.373079\pi$
$211$	502168.	0.776503	0.388251	+	0.921553i	$0.373079\pi$
$212$	−173253.	−0.264753
$213$	145078.	0.219106
$214$	622280.	0.928863
$215$	−146502.	−0.216147
$216$	−379715.	−0.553763
$217$	0	0
$218$	1.56266e6	2.22701
$219$	706753.	0.995765
$220$	−510299.	−0.710834
$221$	107945.	0.148669
$222$	−52818.9	−0.0719295
$223$	−1.17328e6	−1.57993	−0.789967	−	0.613149i	$-0.789902\pi$
$223$	−1.17328e6	−1.57993	−0.789967	+	0.613149i	$0.789902\pi$
$224$	0	0
$225$	−458988.	−0.604428
$226$	280942.	0.365886
$227$	−910159.	−1.17234	−0.586168	−	0.810189i	$-0.699364\pi$
$227$	−910159.	−1.17234	−0.586168	+	0.810189i	$0.699364\pi$
$228$	−275640.	−0.351160
$229$	521924.	0.657686	0.328843	−	0.944385i	$-0.393341\pi$
$229$	521924.	0.657686	0.328843	+	0.944385i	$0.393341\pi$
$230$	240697.	0.300020
$231$	0	0
$232$	762719.	0.930346
$233$	−1.04279e6	−1.25836	−0.629182	−	0.777258i	$-0.716610\pi$
$233$	−1.04279e6	−1.25836	−0.629182	+	0.777258i	$0.716610\pi$
$234$	291993.	0.348604
$235$	122350.	0.144523
$236$	−539045.	−0.630006
$237$	273100.	0.315828
$238$	0	0
$239$	−1.53447e6	−1.73766	−0.868830	−	0.495110i	$-0.835128\pi$
$239$	−1.53447e6	−1.73766	−0.868830	+	0.495110i	$0.835128\pi$
$240$	−1.02648e6	−1.15033
$241$	−1.00758e6	−1.11747	−0.558735	−	0.829346i	$-0.688713\pi$
$241$	−1.00758e6	−1.11747	−0.558735	+	0.829346i	$0.688713\pi$
$242$	−263584.	−0.289322
$243$	−894004.	−0.971234
$244$	−844893.	−0.908505
$245$	0	0
$246$	−502028.	−0.528920
$247$	−438961.	−0.457808
$248$	252332.	0.260521
$249$	804155.	0.821942
$250$	−69233.1	−0.0700590
$251$	8511.89	0.00852789	0.00426394	−	0.999991i	$-0.498643\pi$
$251$	8511.89	0.00852789	0.00426394	+	0.999991i	$0.498643\pi$
$252$	0	0
$253$	−149809.	−0.147142
$254$	366464.	0.356408
$255$	−297867.	−0.286862
$256$	1.31917e6	1.25806
$257$	−527532.	−0.498214	−0.249107	−	0.968476i	$-0.580137\pi$
$257$	−527532.	−0.498214	−0.249107	+	0.968476i	$0.580137\pi$
$258$	−131061.	−0.122581
$259$	0	0
$260$	422972.	0.388042
$261$	1.10016e6	0.999670
$262$	−1.18051e6	−1.06247
$263$	−352085.	−0.313876	−0.156938	−	0.987608i	$-0.550162\pi$
$263$	−352085.	−0.313876	−0.156938	+	0.987608i	$0.550162\pi$
$264$	347671.	0.307014
$265$	761353.	0.665996
$266$	0	0
$267$	−439380.	−0.377192
$268$	485856.	0.413210
$269$	479540.	0.404058	0.202029	−	0.979380i	$-0.435246\pi$
$269$	479540.	0.404058	0.202029	+	0.979380i	$0.435246\pi$
$270$	−2.19103e6	−1.82911
$271$	−977611.	−0.808617	−0.404308	−	0.914623i	$-0.632488\pi$
$271$	−977611.	−0.808617	−0.404308	+	0.914623i	$0.632488\pi$
$272$	471961.	0.386798
$273$	0	0
$274$	200155.	0.161061
$275$	1.14279e6	0.911243
$276$	77972.6	0.0616126
$277$	968723.	0.758578	0.379289	−	0.925278i	$-0.376169\pi$
$277$	968723.	0.758578	0.379289	+	0.925278i	$0.376169\pi$
$278$	2.38259e6	1.84900
$279$	363970.	0.279934
$280$	0	0
$281$	−318333.	−0.240501	−0.120250	−	0.992744i	$-0.538370\pi$
$281$	−318333.	−0.240501	−0.120250	+	0.992744i	$0.538370\pi$
$282$	109455.	0.0819619
$283$	1.77210e6	1.31529	0.657646	−	0.753327i	$-0.271552\pi$
$283$	1.77210e6	1.31529	0.657646	+	0.753327i	$0.271552\pi$
$284$	−261379.	−0.192298
$285$	1.21129e6	0.883356
$286$	−727004.	−0.525559
$287$	0	0
$288$	833492.	0.592134
$289$	−1.28290e6	−0.903543
$290$	4.40104e6	3.07298
$291$	−1.04229e6	−0.721535
$292$	−1.27331e6	−0.873934
$293$	1.64148e6	1.11703	0.558516	−	0.829494i	$-0.311371\pi$
$293$	1.64148e6	1.11703	0.558516	+	0.829494i	$0.311371\pi$
$294$	0	0
$295$	2.36881e6	1.58480
$296$	−72472.4	−0.0480777
$297$	1.36369e6	0.897068
$298$	−2.51708e6	−1.64194
$299$	124172.	0.0803243
$300$	−594799.	−0.381563
$301$	0	0
$302$	−2.54128e6	−1.60337
$303$	−289377.	−0.181075
$304$	−1.91925e6	−1.19110
$305$	3.71285e6	2.28537
$306$	370469.	0.226177
$307$	−466930.	−0.282752	−0.141376	−	0.989956i	$-0.545153\pi$
$307$	−466930.	−0.282752	−0.141376	+	0.989956i	$0.545153\pi$
$308$	0	0
$309$	−294698.	−0.175582
$310$	1.45600e6	0.860515
$311$	−2.43796e6	−1.42931	−0.714654	−	0.699478i	$-0.753416\pi$
$311$	−2.43796e6	−1.42931	−0.714654	+	0.699478i	$0.753416\pi$
$312$	−288174.	−0.167598
$313$	2.42094e6	1.39676	0.698381	−	0.715726i	$-0.253904\pi$
$313$	2.42094e6	1.39676	0.698381	+	0.715726i	$0.253904\pi$
$314$	3.25035e6	1.86040
$315$	0	0
$316$	−492028.	−0.277186
$317$	1.87611e6	1.04860	0.524301	−	0.851533i	$-0.324327\pi$
$317$	1.87611e6	1.04860	0.524301	+	0.851533i	$0.324327\pi$
$318$	681107.	0.377700
$319$	−2.73919e6	−1.50711
$320$	76494.5	0.0417595
$321$	−885855.	−0.479844
$322$	0	0
$323$	−556936.	−0.297029
$324$	−85877.3	−0.0454481
$325$	−947225.	−0.497445
$326$	3.55867e6	1.85457
$327$	−2.22454e6	−1.15046
$328$	−688828.	−0.353530
$329$	0	0
$330$	2.00613e6	1.01408
$331$	−1.08310e6	−0.543373	−0.271686	−	0.962386i	$-0.587581\pi$
$331$	−1.08310e6	−0.543373	−0.271686	+	0.962386i	$0.587581\pi$
$332$	−1.44880e6	−0.721378
$333$	−104536.	−0.0516601
$334$	−4.79257e6	−2.35073
$335$	−2.13507e6	−1.03944
$336$	0	0
$337$	−2.59465e6	−1.24453	−0.622263	−	0.782809i	$-0.713786\pi$
$337$	−2.59465e6	−1.24453	−0.622263	+	0.782809i	$0.713786\pi$
$338$	−2.02719e6	−0.965166
$339$	−399939.	−0.189014
$340$	536650.	0.251764
$341$	−906213.	−0.422031
$342$	−1.50652e6	−0.696484
$343$	0	0
$344$	−179828.	−0.0819333
$345$	−342647.	−0.154989
$346$	−1.76474e6	−0.792485
$347$	−1.87051e6	−0.833943	−0.416972	−	0.908920i	$-0.636909\pi$
$347$	−1.87051e6	−0.833943	−0.416972	+	0.908920i	$0.636909\pi$
$348$	1.42569e6	0.631071
$349$	−1.61685e6	−0.710568	−0.355284	−	0.934758i	$-0.615616\pi$
$349$	−1.61685e6	−0.710568	−0.355284	+	0.934758i	$0.615616\pi$
$350$	0	0
$351$	−1.13033e6	−0.489706
$352$	−2.07523e6	−0.892709
$353$	−578305.	−0.247013	−0.123507	−	0.992344i	$-0.539414\pi$
$353$	−578305.	−0.247013	−0.123507	+	0.992344i	$0.539414\pi$
$354$	2.11914e6	0.898775
$355$	1.14862e6	0.483733
$356$	791605.	0.331042
$357$	0	0
$358$	986330.	0.406738
$359$	−1.96818e6	−0.805988	−0.402994	−	0.915203i	$-0.632030\pi$
$359$	−1.96818e6	−0.805988	−0.402994	+	0.915203i	$0.632030\pi$
$360$	−1.10555e6	−0.449594
$361$	−211299.	−0.0853355
$362$	2.16907e6	0.869965
$363$	375229.	0.149462
$364$	0	0
$365$	5.59553e6	2.19841
$366$	3.32151e6	1.29608
$367$	2.17452e6	0.842749	0.421375	−	0.906887i	$-0.361548\pi$
$367$	2.17452e6	0.842749	0.421375	+	0.906887i	$0.361548\pi$
$368$	542913.	0.208983
$369$	−993583.	−0.379873
$370$	−418180.	−0.158803
$371$	0	0
$372$	471666.	0.176716
$373$	1.38476e6	0.515349	0.257675	−	0.966232i	$-0.417044\pi$
$373$	1.38476e6	0.515349	0.257675	+	0.966232i	$0.417044\pi$
$374$	−922394.	−0.340987
$375$	98557.7	0.0361920
$376$	150182.	0.0547833
$377$	2.27044e6	0.822728
$378$	0	0
$379$	3.37190e6	1.20580	0.602902	−	0.797815i	$-0.294011\pi$
$379$	3.37190e6	1.20580	0.602902	+	0.797815i	$0.294011\pi$
$380$	−2.18231e6	−0.775277
$381$	−521685.	−0.184118
$382$	1.61128e6	0.564955
$383$	3.28060e6	1.14276	0.571382	−	0.820685i	$-0.306408\pi$
$383$	3.28060e6	1.14276	0.571382	+	0.820685i	$0.306408\pi$
$384$	−1.83428e6	−0.634800
$385$	0	0
$386$	−4.76227e6	−1.62684
$387$	−259388.	−0.0880385
$388$	1.87784e6	0.633256
$389$	−2.94810e6	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$389$	−2.94810e6	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$390$	−1.66282e6	−0.553585
$391$	157545.	0.0521150
$392$	0	0
$393$	1.68053e6	0.548865
$394$	−9085.46	−0.00294854
$395$	2.16219e6	0.697271
$396$	−903504.	−0.289529
$397$	−69270.2	−0.0220582	−0.0110291	−	0.999939i	$-0.503511\pi$
$397$	−69270.2	−0.0220582	−0.0110291	+	0.999939i	$0.503511\pi$
$398$	2.61282e6	0.826802
$399$	0	0
$400$	−4.14151e6	−1.29422
$401$	−3.34786e6	−1.03970	−0.519848	−	0.854259i	$-0.674012\pi$
$401$	−3.34786e6	−1.03970	−0.519848	+	0.854259i	$0.674012\pi$
$402$	−1.91004e6	−0.589491
$403$	751134.	0.230385
$404$	521354.	0.158920
$405$	377384.	0.114326
$406$	0	0
$407$	260274.	0.0778834
$408$	−365624.	−0.108739
$409$	2.91217e6	0.860812	0.430406	−	0.902636i	$-0.358370\pi$
$409$	2.91217e6	0.860812	0.430406	+	0.902636i	$0.358370\pi$
$410$	−3.97467e6	−1.16773
$411$	−284933.	−0.0832028
$412$	530940.	0.154100
$413$	0	0
$414$	426163.	0.122201
$415$	6.36669e6	1.81465
$416$	1.72010e6	0.487327
$417$	−3.39177e6	−0.955183
$418$	3.75095e6	1.05003
$419$	−4.62361e6	−1.28661	−0.643304	−	0.765611i	$-0.722437\pi$
$419$	−4.62361e6	−1.28661	−0.643304	+	0.765611i	$0.722437\pi$
$420$	0	0
$421$	−2.63042e6	−0.723303	−0.361652	−	0.932313i	$-0.617787\pi$
$421$	−2.63042e6	−0.723303	−0.361652	+	0.932313i	$0.617787\pi$
$422$	3.55674e6	0.972234
$423$	216626.	0.0588654
$424$	934541.	0.252455
$425$	−1.20180e6	−0.322746
$426$	1.02756e6	0.274335
$427$	0	0
$428$	1.59599e6	0.421135
$429$	1.03494e6	0.271500
$430$	−1.03764e6	−0.270630
$431$	7.54128e6	1.95547	0.977736	−	0.209837i	$-0.0672932\pi$
$431$	7.54128e6	1.95547	0.977736	+	0.209837i	$0.0672932\pi$
$432$	−4.94207e6	−1.27409
$433$	5.83558e6	1.49577	0.747883	−	0.663830i	$-0.231070\pi$
$433$	5.83558e6	1.49577	0.747883	+	0.663830i	$0.231070\pi$
$434$	0	0
$435$	−6.26516e6	−1.58748
$436$	4.00782e6	1.00970
$437$	−640663.	−0.160482
$438$	5.00576e6	1.24677
$439$	168104.	0.0416310	0.0208155	−	0.999783i	$-0.493374\pi$
$439$	168104.	0.0416310	0.0208155	+	0.999783i	$0.493374\pi$
$440$	2.75259e6	0.677814
$441$	0	0
$442$	764546.	0.186143
$443$	−2.84151e6	−0.687924	−0.343962	−	0.938984i	$-0.611769\pi$
$443$	−2.84151e6	−0.687924	−0.343962	+	0.938984i	$0.611769\pi$
$444$	−135467.	−0.0326120
$445$	−3.47867e6	−0.832748
$446$	−8.31005e6	−1.97818
$447$	3.58323e6	0.848214
$448$	0	0
$449$	−1.41567e6	−0.331396	−0.165698	−	0.986177i	$-0.552988\pi$
$449$	−1.41567e6	−0.331396	−0.165698	+	0.986177i	$0.552988\pi$
$450$	−3.25090e6	−0.756785
$451$	2.47382e6	0.572701
$452$	720547.	0.165888
$453$	3.61767e6	0.828291
$454$	−6.44644e6	−1.46784
$455$	0	0
$456$	1.48682e6	0.334848
$457$	1.55727e6	0.348799	0.174399	−	0.984675i	$-0.444202\pi$
$457$	1.55727e6	0.348799	0.174399	+	0.984675i	$0.444202\pi$
$458$	3.69667e6	0.823468
$459$	−1.43411e6	−0.317725
$460$	617328.	0.136026
$461$	−4.45345e6	−0.975987	−0.487994	−	0.872847i	$-0.662271\pi$
$461$	−4.45345e6	−0.975987	−0.487994	+	0.872847i	$0.662271\pi$
$462$	0	0
$463$	4.92263e6	1.06720	0.533599	−	0.845738i	$-0.320839\pi$
$463$	4.92263e6	1.06720	0.533599	+	0.845738i	$0.320839\pi$
$464$	9.92693e6	2.14052
$465$	−2.07271e6	−0.444536
$466$	−7.38582e6	−1.57556
$467$	5.09090e6	1.08020	0.540098	−	0.841602i	$-0.318387\pi$
$467$	5.09090e6	1.08020	0.540098	+	0.841602i	$0.318387\pi$
$468$	748889.	0.158053
$469$	0	0
$470$	866579.	0.180952
$471$	−4.62709e6	−0.961071
$472$	2.90765e6	0.600741
$473$	645825.	0.132728
$474$	1.93430e6	0.395437
$475$	4.88717e6	0.993856
$476$	0	0
$477$	1.34800e6	0.271266
$478$	−1.08683e7	−2.17567
$479$	−8.30085e6	−1.65304	−0.826521	−	0.562907i	$-0.809683\pi$
$479$	−8.30085e6	−1.65304	−0.826521	+	0.562907i	$0.809683\pi$
$480$	−4.74652e6	−0.940313
$481$	−215734.	−0.0425163
$482$	−7.13644e6	−1.39915
$483$	0	0
$484$	−676028.	−0.131175
$485$	−8.25208e6	−1.59298
$486$	−6.33202e6	−1.21605
$487$	8.63401e6	1.64964	0.824822	−	0.565392i	$-0.191275\pi$
$487$	8.63401e6	1.64964	0.824822	+	0.565392i	$0.191275\pi$
$488$	4.55742e6	0.866303
$489$	−5.06599e6	−0.958059
$490$	0	0
$491$	95039.5	0.0177910	0.00889550	−	0.999960i	$-0.497168\pi$
$491$	95039.5	0.0177910	0.00889550	+	0.999960i	$0.497168\pi$
$492$	−1.28758e6	−0.239806
$493$	2.88064e6	0.533792
$494$	−3.10905e6	−0.573206
$495$	3.97041e6	0.728320
$496$	3.28415e6	0.599402
$497$	0	0
$498$	5.69564e6	1.02913
$499$	2.14203e6	0.385101	0.192551	−	0.981287i	$-0.438324\pi$
$499$	2.14203e6	0.385101	0.192551	+	0.981287i	$0.438324\pi$
$500$	−177566.	−0.0317639
$501$	6.82252e6	1.21437
$502$	60287.7	0.0106775
$503$	5.24794e6	0.924844	0.462422	−	0.886660i	$-0.346981\pi$
$503$	5.24794e6	0.924844	0.462422	+	0.886660i	$0.346981\pi$
$504$	0	0
$505$	−2.29107e6	−0.399769
$506$	−1.06106e6	−0.184232
$507$	2.88583e6	0.498598
$508$	939889.	0.161591
$509$	−1.05891e7	−1.81160	−0.905802	−	0.423702i	$-0.860730\pi$
$509$	−1.05891e7	−1.81160	−0.905802	+	0.423702i	$0.860730\pi$
$510$	−2.10972e6	−0.359170
$511$	0	0
$512$	3.52190e6	0.593747
$513$	5.83187e6	0.978395
$514$	−3.73639e6	−0.623798
$515$	−2.33319e6	−0.387644
$516$	−336139.	−0.0555769
$517$	−539357.	−0.0887462
$518$	0	0
$519$	2.51222e6	0.409392
$520$	−2.28155e6	−0.370016
$521$	−4.54465e6	−0.733510	−0.366755	−	0.930318i	$-0.619531\pi$
$521$	−4.54465e6	−0.733510	−0.366755	+	0.930318i	$0.619531\pi$
$522$	7.79221e6	1.25165
$523$	−5.27197e6	−0.842789	−0.421394	−	0.906877i	$-0.638459\pi$
$523$	−5.27197e6	−0.842789	−0.421394	+	0.906877i	$0.638459\pi$
$524$	−3.02772e6	−0.481711
$525$	0	0
$526$	−2.49373e6	−0.392993
$527$	953009.	0.149476
$528$	4.52501e6	0.706373
$529$	−6.25511e6	−0.971843
$530$	5.39248e6	0.833871
$531$	4.19407e6	0.645504
$532$	0	0
$533$	−2.05048e6	−0.312635
$534$	−3.11202e6	−0.472269
$535$	−7.01353e6	−1.05938
$536$	−2.62075e6	−0.394015
$537$	−1.40410e6	−0.210118
$538$	3.39647e6	0.505908
$539$	0	0
$540$	−5.61945e6	−0.829296
$541$	5.93445e6	0.871741	0.435871	−	0.900009i	$-0.356441\pi$
$541$	5.93445e6	0.871741	0.435871	+	0.900009i	$0.356441\pi$
$542$	−6.92418e6	−1.01244
$543$	−3.08781e6	−0.449418
$544$	2.18239e6	0.316181
$545$	−1.76122e7	−2.53994
$546$	0	0
$547$	−8.82017e6	−1.26040	−0.630200	−	0.776433i	$-0.717027\pi$
$547$	−8.82017e6	−1.26040	−0.630200	+	0.776433i	$0.717027\pi$
$548$	513347.	0.0730230
$549$	6.57374e6	0.930854
$550$	8.09410e6	1.14094
$551$	−1.17142e7	−1.64375
$552$	−420590.	−0.0587505
$553$	0	0
$554$	6.86124e6	0.949791
$555$	595306.	0.0820366
$556$	6.11076e6	0.838317
$557$	−1.18224e6	−0.161461	−0.0807304	−	0.996736i	$-0.525725\pi$
$557$	−1.18224e6	−0.161461	−0.0807304	+	0.996736i	$0.525725\pi$
$558$	2.57791e6	0.350496
$559$	−535306.	−0.0724556
$560$	0	0
$561$	1.31309e6	0.176151
$562$	−2.25468e6	−0.301123
$563$	4.07741e6	0.542142	0.271071	−	0.962559i	$-0.412622\pi$
$563$	4.07741e6	0.542142	0.271071	+	0.962559i	$0.412622\pi$
$564$	280724.	0.0371606
$565$	−3.16641e6	−0.417298
$566$	1.25514e7	1.64684
$567$	0	0
$568$	1.40990e6	0.183366
$569$	8.17615e6	1.05869	0.529344	−	0.848407i	$-0.322438\pi$
$569$	8.17615e6	1.05869	0.529344	+	0.848407i	$0.322438\pi$
$570$	8.57927e6	1.10602
$571$	3.30615e6	0.424357	0.212179	−	0.977231i	$-0.431944\pi$
$571$	3.30615e6	0.424357	0.212179	+	0.977231i	$0.431944\pi$
$572$	−1.86458e6	−0.238282
$573$	−2.29377e6	−0.291852
$574$	0	0
$575$	−1.38247e6	−0.174376
$576$	135437.	0.0170090
$577$	−7.14994e6	−0.894052	−0.447026	−	0.894521i	$-0.647517\pi$
$577$	−7.14994e6	−0.894052	−0.447026	+	0.894521i	$0.647517\pi$
$578$	−9.08648e6	−1.13130
$579$	6.77939e6	0.840416
$580$	1.12876e7	1.39325
$581$	0	0
$582$	−7.38232e6	−0.903411
$583$	−3.35627e6	−0.408964
$584$	6.86836e6	0.833338
$585$	−3.29096e6	−0.397588
$586$	1.16262e7	1.39860
$587$	9.69191e6	1.16095	0.580476	−	0.814277i	$-0.302867\pi$
$587$	9.69191e6	1.16095	0.580476	+	0.814277i	$0.302867\pi$
$588$	0	0
$589$	−3.87545e6	−0.460292
$590$	1.67777e7	1.98428
$591$	12933.7	0.00152319
$592$	−943242.	−0.110616
$593$	6.63960e6	0.775363	0.387682	−	0.921793i	$-0.373276\pi$
$593$	6.63960e6	0.775363	0.387682	+	0.921793i	$0.373276\pi$
$594$	9.65871e6	1.12319
$595$	0	0
$596$	−6.45569e6	−0.744435
$597$	−3.71951e6	−0.427120
$598$	879484.	0.100571
$599$	−3.24191e6	−0.369177	−0.184588	−	0.982816i	$-0.559095\pi$
$599$	−3.24191e6	−0.369177	−0.184588	+	0.982816i	$0.559095\pi$
$600$	3.20839e6	0.363839
$601$	−5.65076e6	−0.638147	−0.319074	−	0.947730i	$-0.603372\pi$
$601$	−5.65076e6	−0.638147	−0.319074	+	0.947730i	$0.603372\pi$
$602$	0	0
$603$	−3.78023e6	−0.423375
$604$	−6.51774e6	−0.726950
$605$	2.97078e6	0.329975
$606$	−2.04959e6	−0.226718
$607$	235674.	0.0259621	0.0129811	−	0.999916i	$-0.495868\pi$
$607$	235674.	0.0259621	0.0129811	+	0.999916i	$0.495868\pi$
$608$	−8.87478e6	−0.973641
$609$	0	0
$610$	2.62972e7	2.86144
$611$	447057.	0.0484462
$612$	950160.	0.102546
$613$	788877.	0.0847926	0.0423963	−	0.999101i	$-0.486501\pi$
$613$	788877.	0.0847926	0.0423963	+	0.999101i	$0.486501\pi$
$614$	−3.30716e6	−0.354025
$615$	5.65820e6	0.603240
$616$	0	0
$617$	1.67739e7	1.77387	0.886935	−	0.461894i	$-0.152830\pi$
$617$	1.67739e7	1.77387	0.886935	+	0.461894i	$0.152830\pi$
$618$	−2.08728e6	−0.219841
$619$	8.22300e6	0.862588	0.431294	−	0.902211i	$-0.358057\pi$
$619$	8.22300e6	0.862588	0.431294	+	0.902211i	$0.358057\pi$
$620$	3.73429e6	0.390147
$621$	−1.64971e6	−0.171664
$622$	−1.72675e7	−1.78959
$623$	0	0
$624$	−3.75065e6	−0.385607
$625$	−9.36798e6	−0.959281
$626$	1.71469e7	1.74884
$627$	−5.33972e6	−0.542437
$628$	8.33635e6	0.843484
$629$	−273714.	−0.0275849
$630$	0	0
$631$	−5.94507e6	−0.594406	−0.297203	−	0.954814i	$-0.596054\pi$
$631$	−5.94507e6	−0.594406	−0.297203	+	0.954814i	$0.596054\pi$
$632$	2.65404e6	0.264310
$633$	−5.06324e6	−0.502249
$634$	1.32881e7	1.31292
$635$	−4.13030e6	−0.406488
$636$	1.74687e6	0.171245
$637$	0	0
$638$	−1.94011e7	−1.88701
$639$	2.03368e6	0.197029
$640$	−1.45224e7	−1.40149
$641$	−1.06761e7	−1.02628	−0.513141	−	0.858304i	$-0.671518\pi$
$641$	−1.06761e7	−1.02628	−0.513141	+	0.858304i	$0.671518\pi$
$642$	−6.27430e6	−0.600797
$643$	−3.13159e6	−0.298701	−0.149351	−	0.988784i	$-0.547718\pi$
$643$	−3.13159e6	−0.298701	−0.149351	+	0.988784i	$0.547718\pi$
$644$	0	0
$645$	1.47715e6	0.139806
$646$	−3.94464e6	−0.371900
$647$	−4.93457e6	−0.463435	−0.231717	−	0.972783i	$-0.574434\pi$
$647$	−4.93457e6	−0.463435	−0.231717	+	0.972783i	$0.574434\pi$
$648$	463229.	0.0433369
$649$	−1.04424e7	−0.973170
$650$	−6.70897e6	−0.622834
$651$	0	0
$652$	9.12710e6	0.840841
$653$	5.72224e6	0.525150	0.262575	−	0.964912i	$-0.415428\pi$
$653$	5.72224e6	0.525150	0.262575	+	0.964912i	$0.415428\pi$
$654$	−1.57559e7	−1.44045
$655$	1.33052e7	1.21176
$656$	−8.96523e6	−0.813395
$657$	9.90710e6	0.895433
$658$	0	0
$659$	362477.	0.0325137	0.0162569	−	0.999868i	$-0.494825\pi$
$659$	362477.	0.0325137	0.0162569	+	0.999868i	$0.494825\pi$
$660$	5.14522e6	0.459774
$661$	−1.91211e7	−1.70219	−0.851096	−	0.525011i	$-0.824061\pi$
$661$	−1.91211e7	−1.70219	−0.851096	+	0.525011i	$0.824061\pi$
$662$	−7.67133e6	−0.680339
$663$	−1.08838e6	−0.0961604
$664$	7.81494e6	0.687868
$665$	0	0
$666$	−740404.	−0.0646819
$667$	3.31370e6	0.288403
$668$	−1.22917e7	−1.06579
$669$	1.18299e7	1.02192
$670$	−1.51222e7	−1.30145
$671$	−1.63673e7	−1.40337
$672$	0	0
$673$	−573374.	−0.0487978	−0.0243989	−	0.999702i	$-0.507767\pi$
$673$	−573374.	−0.0487978	−0.0243989	+	0.999702i	$0.507767\pi$
$674$	−1.83773e7	−1.55823
$675$	1.25845e7	1.06310
$676$	−5.19923e6	−0.437595
$677$	1.16903e7	0.980291	0.490146	−	0.871641i	$-0.336944\pi$
$677$	1.16903e7	0.980291	0.490146	+	0.871641i	$0.336944\pi$
$678$	−2.83267e6	−0.236659
$679$	0	0
$680$	−2.89473e6	−0.240069
$681$	9.17691e6	0.758279
$682$	−6.41849e6	−0.528411
$683$	1.83674e7	1.50659	0.753297	−	0.657681i	$-0.228462\pi$
$683$	1.83674e7	1.50659	0.753297	+	0.657681i	$0.228462\pi$
$684$	−3.86386e6	−0.315778
$685$	−2.25588e6	−0.183692
$686$	0	0
$687$	−5.26244e6	−0.425398
$688$	−2.34049e6	−0.188511
$689$	2.78191e6	0.223252
$690$	−2.42689e6	−0.194056
$691$	2.35611e7	1.87716	0.938579	−	0.345066i	$-0.112143\pi$
$691$	2.35611e7	1.87716	0.938579	+	0.345066i	$0.112143\pi$
$692$	−4.52612e6	−0.359303
$693$	0	0
$694$	−1.32484e7	−1.04415
$695$	−2.68535e7	−2.10881
$696$	−7.69031e6	−0.601757
$697$	−2.60157e6	−0.202840
$698$	−1.14517e7	−0.889679
$699$	1.05142e7	0.813921
$700$	0	0
$701$	1.32980e7	1.02210	0.511048	−	0.859552i	$-0.329257\pi$
$701$	1.32980e7	1.02210	0.511048	+	0.859552i	$0.329257\pi$
$702$	−8.00582e6	−0.613145
$703$	1.11307e6	0.0849442
$704$	−337210.	−0.0256430
$705$	−1.23363e6	−0.0934786
$706$	−4.09600e6	−0.309277
$707$	0	0
$708$	5.43506e6	0.407494
$709$	−6.85353e6	−0.512034	−0.256017	−	0.966672i	$-0.582410\pi$
$709$	−6.85353e6	−0.512034	−0.256017	+	0.966672i	$0.582410\pi$
$710$	8.13540e6	0.605666
$711$	3.82825e6	0.284005
$712$	−4.26998e6	−0.315665
$713$	1.09628e6	0.0807602
$714$	0	0
$715$	8.19384e6	0.599408
$716$	2.52969e6	0.184410
$717$	1.54717e7	1.12394
$718$	−1.39402e7	−1.00915
$719$	2.65729e7	1.91698	0.958490	−	0.285127i	$-0.0920357\pi$
$719$	2.65729e7	1.91698	0.958490	+	0.285127i	$0.0920357\pi$
$720$	−1.43889e7	−1.03442
$721$	0	0
$722$	−1.49658e6	−0.106846
$723$	1.01592e7	0.722791
$724$	5.56312e6	0.394432
$725$	−2.52779e7	−1.78606
$726$	2.65766e6	0.187136
$727$	2.16991e6	0.152267	0.0761335	−	0.997098i	$-0.475742\pi$
$727$	2.16991e6	0.152267	0.0761335	+	0.997098i	$0.475742\pi$
$728$	0	0
$729$	1.01628e7	0.708264
$730$	3.96318e7	2.75256
$731$	−679174.	−0.0470097
$732$	8.51886e6	0.587629
$733$	−1.74653e7	−1.20065	−0.600324	−	0.799757i	$-0.704962\pi$
$733$	−1.74653e7	−1.20065	−0.600324	+	0.799757i	$0.704962\pi$
$734$	1.54016e7	1.05518
$735$	0	0
$736$	2.51048e6	0.170829
$737$	9.41203e6	0.638285
$738$	−7.03731e6	−0.475626
$739$	−1.36461e7	−0.919172	−0.459586	−	0.888133i	$-0.652002\pi$
$739$	−1.36461e7	−0.919172	−0.459586	+	0.888133i	$0.652002\pi$
$740$	−1.07253e6	−0.0719994
$741$	4.42594e6	0.296114
$742$	0	0
$743$	1.48965e7	0.989944	0.494972	−	0.868909i	$-0.335178\pi$
$743$	1.48965e7	0.989944	0.494972	+	0.868909i	$0.335178\pi$
$744$	−2.54420e6	−0.168508
$745$	2.83692e7	1.87265
$746$	9.80791e6	0.645252
$747$	1.12725e7	0.739124
$748$	−2.36571e6	−0.154599
$749$	0	0
$750$	698061.	0.0453148
$751$	−2.53463e7	−1.63989	−0.819944	−	0.572443i	$-0.805996\pi$
$751$	−2.53463e7	−1.63989	−0.819944	+	0.572443i	$0.805996\pi$
$752$	1.95465e6	0.126044
$753$	−85823.3	−0.00551592
$754$	1.60810e7	1.03011
$755$	2.86419e7	1.82867
$756$	0	0
$757$	−2.66725e7	−1.69170	−0.845852	−	0.533417i	$-0.820908\pi$
$757$	−2.66725e7	−1.69170	−0.845852	+	0.533417i	$0.820908\pi$
$758$	2.38824e7	1.50975
$759$	1.51049e6	0.0951729
$760$	1.17715e7	0.739264
$761$	579829.	0.0362943	0.0181471	−	0.999835i	$-0.494223\pi$
$761$	579829.	0.0362943	0.0181471	+	0.999835i	$0.494223\pi$
$762$	−3.69497e6	−0.230528
$763$	0	0
$764$	4.13255e6	0.256144
$765$	−4.17544e6	−0.257958
$766$	2.32357e7	1.43082
$767$	8.65541e6	0.531250
$768$	−1.33009e7	−0.813727
$769$	1.52438e7	0.929562	0.464781	−	0.885426i	$-0.346133\pi$
$769$	1.52438e7	0.929562	0.464781	+	0.885426i	$0.346133\pi$
$770$	0	0
$771$	5.31898e6	0.322250
$772$	−1.22140e7	−0.737591
$773$	−1.94926e7	−1.17333	−0.586665	−	0.809830i	$-0.699559\pi$
$773$	−1.94926e7	−1.17333	−0.586665	+	0.809830i	$0.699559\pi$
$774$	−1.83718e6	−0.110230
$775$	−8.36275e6	−0.500144
$776$	−1.01292e7	−0.603839
$777$	0	0
$778$	−2.08807e7	−1.23679
$779$	1.05794e7	0.624621
$780$	−4.26473e6	−0.250989
$781$	−5.06345e6	−0.297043
$782$	1.11585e6	0.0652515
$783$	−3.01642e7	−1.75828
$784$	0	0
$785$	−3.66337e7	−2.12181
$786$	1.19028e7	0.687215
$787$	1.36277e7	0.784305	0.392153	−	0.919900i	$-0.371730\pi$
$787$	1.36277e7	0.784305	0.392153	+	0.919900i	$0.371730\pi$
$788$	−23302.0	−0.00133683
$789$	3.54999e6	0.203018
$790$	1.53143e7	0.873031
$791$	0	0
$792$	4.87357e6	0.276080
$793$	1.35664e7	0.766093
$794$	−490624.	−0.0276184
$795$	−7.67654e6	−0.430772
$796$	6.70123e6	0.374862
$797$	3.62853e6	0.202341	0.101171	−	0.994869i	$-0.467741\pi$
$797$	3.62853e6	0.202341	0.101171	+	0.994869i	$0.467741\pi$
$798$	0	0
$799$	567208.	0.0314323
$800$	−1.91507e7	−1.05794
$801$	−6.15913e6	−0.339186
$802$	−2.37121e7	−1.30177
$803$	−2.46667e7	−1.34997
$804$	−4.89877e6	−0.267268
$805$	0	0
$806$	5.32010e6	0.288458
$807$	−4.83509e6	−0.261349
$808$	−2.81223e6	−0.151538
$809$	2.98780e7	1.60502	0.802509	−	0.596640i	$-0.203498\pi$
$809$	2.98780e7	1.60502	0.802509	+	0.596640i	$0.203498\pi$
$810$	2.67292e6	0.143144
$811$	−1.02643e6	−0.0547995	−0.0273998	−	0.999625i	$-0.508723\pi$
$811$	−1.02643e6	−0.0547995	−0.0273998	+	0.999625i	$0.508723\pi$
$812$	0	0
$813$	9.85702e6	0.523021
$814$	1.84346e6	0.0975153
$815$	−4.01086e7	−2.11516
$816$	−4.75867e6	−0.250184
$817$	2.76189e6	0.144761
$818$	2.06262e7	1.07779
$819$	0	0
$820$	−1.01940e7	−0.529434
$821$	−1.15062e7	−0.595766	−0.297883	−	0.954602i	$-0.596281\pi$
$821$	−1.15062e7	−0.595766	−0.297883	+	0.954602i	$0.596281\pi$
$822$	−2.01811e6	−0.104175
$823$	−2.51210e7	−1.29282	−0.646408	−	0.762992i	$-0.723730\pi$
$823$	−2.51210e7	−1.29282	−0.646408	+	0.762992i	$0.723730\pi$
$824$	−2.86393e6	−0.146942
$825$	−1.15225e7	−0.589401
$826$	0	0
$827$	−2.14447e6	−0.109032	−0.0545162	−	0.998513i	$-0.517362\pi$
$827$	−2.14447e6	−0.109032	−0.0545162	+	0.998513i	$0.517362\pi$
$828$	1.09300e6	0.0554045
$829$	818065.	0.0413430	0.0206715	−	0.999786i	$-0.493420\pi$
$829$	818065.	0.0413430	0.0206715	+	0.999786i	$0.493420\pi$
$830$	4.50937e7	2.27207
$831$	−9.76741e6	−0.490656
$832$	279504.	0.0139984
$833$	0	0
$834$	−2.40231e7	−1.19595
$835$	5.40155e7	2.68104
$836$	9.62025e6	0.476070
$837$	−9.97929e6	−0.492364
$838$	−3.27480e7	−1.61092
$839$	−2.11279e7	−1.03622	−0.518110	−	0.855314i	$-0.673364\pi$
$839$	−2.11279e7	−1.03622	−0.518110	+	0.855314i	$0.673364\pi$
$840$	0	0
$841$	4.00785e7	1.95398
$842$	−1.86307e7	−0.905624
$843$	3.20968e6	0.155558
$844$	9.12215e6	0.440799
$845$	2.28478e7	1.10079
$846$	1.53431e6	0.0737035
$847$	0	0
$848$	1.21632e7	0.580844
$849$	−1.78677e7	−0.850744
$850$	−8.51207e6	−0.404099
$851$	−314863.	−0.0149038
$852$	2.63543e6	0.124380
$853$	−1.89000e7	−0.889386	−0.444693	−	0.895683i	$-0.646687\pi$
$853$	−1.89000e7	−0.889386	−0.444693	+	0.895683i	$0.646687\pi$
$854$	0	0
$855$	1.69796e7	0.794349
$856$	−8.60892e6	−0.401573
$857$	3.72286e7	1.73151	0.865753	−	0.500471i	$-0.166840\pi$
$857$	3.72286e7	1.73151	0.865753	+	0.500471i	$0.166840\pi$
$858$	7.33021e6	0.339937
$859$	3.02064e6	0.139674	0.0698371	−	0.997558i	$-0.477752\pi$
$859$	3.02064e6	0.139674	0.0698371	+	0.997558i	$0.477752\pi$
$860$	−2.66129e6	−0.122700
$861$	0	0
$862$	5.34131e7	2.44838
$863$	−2.71843e7	−1.24248	−0.621242	−	0.783619i	$-0.713371\pi$
$863$	−2.71843e7	−1.24248	−0.621242	+	0.783619i	$0.713371\pi$
$864$	−2.28526e7	−1.04148
$865$	1.98899e7	0.903839
$866$	4.13320e7	1.87280
$867$	1.29352e7	0.584420
$868$	0	0
$869$	−9.53158e6	−0.428169
$870$	−4.43746e7	−1.98763
$871$	−7.80136e6	−0.348438
$872$	−2.16185e7	−0.962797
$873$	−1.46106e7	−0.648834
$874$	−4.53766e6	−0.200934
$875$	0	0
$876$	1.28385e7	0.565269
$877$	1.73868e7	0.763344	0.381672	−	0.924298i	$-0.375348\pi$
$877$	1.73868e7	0.763344	0.381672	+	0.924298i	$0.375348\pi$
$878$	1.19064e6	0.0521248
$879$	−1.65506e7	−0.722507
$880$	3.58255e7	1.55950
$881$	−8.14472e6	−0.353538	−0.176769	−	0.984252i	$-0.556565\pi$
$881$	−8.14472e6	−0.353538	−0.176769	+	0.984252i	$0.556565\pi$
$882$	0	0
$883$	−3.10298e7	−1.33930	−0.669649	−	0.742678i	$-0.733555\pi$
$883$	−3.10298e7	−1.33930	−0.669649	+	0.742678i	$0.733555\pi$
$884$	1.96087e6	0.0843953
$885$	−2.38841e7	−1.02507
$886$	−2.01258e7	−0.861327
$887$	−1.47028e7	−0.627465	−0.313733	−	0.949511i	$-0.601580\pi$
$887$	−1.47028e7	−0.627465	−0.313733	+	0.949511i	$0.601580\pi$
$888$	730722.	0.0310971
$889$	0	0
$890$	−2.46386e7	−1.04266
$891$	−1.66362e6	−0.0702037
$892$	−2.13132e7	−0.896885
$893$	−2.30657e6	−0.0967918
$894$	2.53791e7	1.06202
$895$	−1.11166e7	−0.463890
$896$	0	0
$897$	−1.25200e6	−0.0519545
$898$	−1.00269e7	−0.414930
$899$	2.00450e7	0.827193
$900$	−8.33775e6	−0.343117
$901$	3.52958e6	0.144848
$902$	1.75215e7	0.717060
$903$	0	0
$904$	−3.88669e6	−0.158183
$905$	−2.44469e7	−0.992208
$906$	2.56231e7	1.03708
$907$	1.30940e7	0.528512	0.264256	−	0.964453i	$-0.414874\pi$
$907$	1.30940e7	0.528512	0.264256	+	0.964453i	$0.414874\pi$
$908$	−1.65335e7	−0.665504
$909$	−4.05643e6	−0.162830
$910$	0	0
$911$	2.68695e7	1.07266	0.536332	−	0.844007i	$-0.319809\pi$
$911$	2.68695e7	1.07266	0.536332	+	0.844007i	$0.319809\pi$
$912$	1.93513e7	0.770412
$913$	−2.80662e7	−1.11431
$914$	1.10298e7	0.436719
$915$	−3.74357e7	−1.47820
$916$	9.48103e6	0.373351
$917$	0	0
$918$	−1.01575e7	−0.397813
$919$	1.27317e6	0.0497277	0.0248638	−	0.999691i	$-0.492085\pi$
$919$	1.27317e6	0.0497277	0.0248638	+	0.999691i	$0.492085\pi$
$920$	−3.32991e6	−0.129707
$921$	4.70795e6	0.182887
$922$	−3.15427e7	−1.22200
$923$	4.19695e6	0.162155
$924$	0	0
$925$	2.40187e6	0.0922986
$926$	3.48658e7	1.33620
$927$	−4.13101e6	−0.157891
$928$	4.59031e7	1.74973
$929$	−9.31705e6	−0.354192	−0.177096	−	0.984194i	$-0.556670\pi$
$929$	−9.31705e6	−0.354192	−0.177096	+	0.984194i	$0.556670\pi$
$930$	−1.46805e7	−0.556589
$931$	0	0
$932$	−1.89428e7	−0.714338
$933$	2.45814e7	0.924490
$934$	3.60577e7	1.35248
$935$	1.03960e7	0.388900
$936$	−4.03956e6	−0.150711
$937$	1.18158e7	0.439657	0.219829	−	0.975539i	$-0.429450\pi$
$937$	1.18158e7	0.439657	0.219829	+	0.975539i	$0.429450\pi$
$938$	0	0
$939$	−2.44097e7	−0.903439
$940$	2.22256e6	0.0820416
$941$	−2.53529e7	−0.933371	−0.466685	−	0.884423i	$-0.654552\pi$
$941$	−2.53529e7	−0.933371	−0.466685	+	0.884423i	$0.654552\pi$
$942$	−3.27725e7	−1.20333
$943$	−2.99268e6	−0.109592
$944$	3.78436e7	1.38217
$945$	0	0
$946$	4.57422e6	0.166184
$947$	2.64941e7	0.960008	0.480004	−	0.877266i	$-0.340635\pi$
$947$	2.64941e7	0.960008	0.480004	+	0.877266i	$0.340635\pi$
$948$	4.96100e6	0.179287
$949$	2.04455e7	0.736941
$950$	3.46147e7	1.24437
$951$	−1.89164e7	−0.678246
$952$	0	0
$953$	−1.88335e7	−0.671735	−0.335868	−	0.941909i	$-0.609030\pi$
$953$	−1.88335e7	−0.671735	−0.335868	+	0.941909i	$0.609030\pi$
$954$	9.54760e6	0.339643
$955$	−1.81603e7	−0.644339
$956$	−2.78745e7	−0.986423
$957$	2.76186e7	0.974816
$958$	−5.87929e7	−2.06972
$959$	0	0
$960$	−771276.	−0.0270105
$961$	−2.19976e7	−0.768365
$962$	−1.52799e6	−0.0532332
$963$	−1.24177e7	−0.431495
$964$	−1.83032e7	−0.634357
$965$	5.36740e7	1.85544
$966$	0	0
$967$	−3.14956e7	−1.08314	−0.541569	−	0.840656i	$-0.682169\pi$
$967$	−3.14956e7	−1.08314	−0.541569	+	0.840656i	$0.682169\pi$
$968$	3.64655e6	0.125082
$969$	5.61545e6	0.192121
$970$	−5.84475e7	−1.99451
$971$	−6.85669e6	−0.233381	−0.116691	−	0.993168i	$-0.537229\pi$
$971$	−6.85669e6	−0.233381	−0.116691	+	0.993168i	$0.537229\pi$
$972$	−1.62401e7	−0.551343
$973$	0	0
$974$	6.11527e7	2.06547
$975$	9.55064e6	0.321752
$976$	5.93157e7	1.99317
$977$	2.81471e7	0.943402	0.471701	−	0.881759i	$-0.343640\pi$
$977$	2.81471e7	0.943402	0.471701	+	0.881759i	$0.343640\pi$
$978$	−3.58812e7	−1.19955
$979$	1.53350e7	0.511361
$980$	0	0
$981$	−3.11831e7	−1.03454
$982$	673142.	0.0222755
$983$	2.34916e7	0.775406	0.387703	−	0.921784i	$-0.373269\pi$
$983$	2.34916e7	0.775406	0.387703	+	0.921784i	$0.373269\pi$
$984$	6.94529e6	0.228666
$985$	102399.	0.00336285
$986$	2.04029e7	0.668343
$987$	0	0
$988$	−7.97395e6	−0.259885
$989$	−781278.	−0.0253989
$990$	2.81215e7	0.911906
$991$	−2.14412e6	−0.0693530	−0.0346765	−	0.999399i	$-0.511040\pi$
$991$	−2.14412e6	−0.0693530	−0.0346765	+	0.999399i	$0.511040\pi$
$992$	1.51862e7	0.489971
$993$	1.09206e7	0.351459
$994$	0	0
$995$	−2.94483e7	−0.942979
$996$	1.46079e7	0.466594
$997$	2.50872e7	0.799307	0.399654	−	0.916666i	$-0.369130\pi$
$997$	2.50872e7	0.799307	0.399654	+	0.916666i	$0.369130\pi$
$998$	1.51715e7	0.482173
$999$	2.86616e6	0.0908628

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.a.d.1.2		2
3.2	odd	2		441.6.a.n.1.1		2
4.3	odd	2		784.6.a.ba.1.2		2
7.2	even	3		7.6.c.a.4.1	yes	4
7.3	odd	6		49.6.c.f.30.1		4
7.4	even	3		7.6.c.a.2.1	✓	4
7.5	odd	6		49.6.c.f.18.1		4
7.6	odd	2		49.6.a.e.1.2		2
21.2	odd	6		63.6.e.d.46.2		4
21.11	odd	6		63.6.e.d.37.2		4
21.20	even	2		441.6.a.m.1.1		2
28.11	odd	6		112.6.i.c.65.1		4
28.23	odd	6		112.6.i.c.81.1		4
28.27	even	2		784.6.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.c.a.2.1	✓	4	7.4	even	3
7.6.c.a.4.1	yes	4	7.2	even	3
49.6.a.d.1.2		2	1.1	even	1	trivial
49.6.a.e.1.2		2	7.6	odd	2
49.6.c.f.18.1		4	7.5	odd	6
49.6.c.f.30.1		4	7.3	odd	6
63.6.e.d.37.2		4	21.11	odd	6
63.6.e.d.46.2		4	21.2	odd	6
112.6.i.c.65.1		4	28.11	odd	6
112.6.i.c.81.1		4	28.23	odd	6
441.6.a.m.1.1		2	21.20	even	2
441.6.a.n.1.1		2	3.2	odd	2
784.6.a.t.1.1		2	28.27	even	2
784.6.a.ba.1.2		2	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)

\(f(q)\)	\(=\)	\(q+7.08276 q^{2} -10.0828 q^{3} +18.1655 q^{4} -79.8276 q^{5} -71.4138 q^{6} -97.9863 q^{8} -141.338 q^{9} -565.400 q^{10} +351.904 q^{11} -183.159 q^{12} -291.683 q^{13} +804.883 q^{15} -1275.31 q^{16} -370.075 q^{17} -1001.06 q^{18} +1504.93 q^{19} -1450.11 q^{20} +2492.45 q^{22} -425.711 q^{23} +987.973 q^{24} +3247.45 q^{25} -2065.92 q^{26} +3875.19 q^{27} -7783.93 q^{29} +5700.80 q^{30} -2575.18 q^{31} -5897.16 q^{32} -3548.16 q^{33} -2621.16 q^{34} -2567.48 q^{36} +739.618 q^{37} +10659.0 q^{38} +2940.97 q^{39} +7822.01 q^{40} +7029.84 q^{41} +1835.23 q^{43} +6392.51 q^{44} +11282.7 q^{45} -3015.21 q^{46} -1532.68 q^{47} +12858.7 q^{48} +23000.9 q^{50} +3731.38 q^{51} -5298.57 q^{52} -9537.46 q^{53} +27447.0 q^{54} -28091.6 q^{55} -15173.8 q^{57} -55131.8 q^{58} -29674.1 q^{59} +14621.1 q^{60} -46510.8 q^{61} -18239.4 q^{62} -958.246 q^{64} +23284.3 q^{65} -25130.8 q^{66} +26746.1 q^{67} -6722.61 q^{68} +4292.34 q^{69} -14388.8 q^{71} +13849.2 q^{72} -70095.1 q^{73} +5238.54 q^{74} -32743.3 q^{75} +27337.8 q^{76} +20830.2 q^{78} -27085.8 q^{79} +101805. q^{80} -4727.49 q^{81} +49790.7 q^{82} -79755.4 q^{83} +29542.2 q^{85} +12998.5 q^{86} +78483.6 q^{87} -34481.7 q^{88} +43577.3 q^{89} +79912.5 q^{90} -7733.26 q^{92} +25964.9 q^{93} -10855.6 q^{94} -120135. q^{95} +59459.7 q^{96} +103374. q^{97} -49737.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 8 q^{3} + 12 q^{4} - 38 q^{5} - 82 q^{6} + 96 q^{8} - 380 q^{9} - 778 q^{10} + 424 q^{11} - 196 q^{12} - 924 q^{13} + 892 q^{15} - 2064 q^{16} - 2346 q^{17} + 212 q^{18} - 360 q^{19} - 1708 q^{20}+ \cdots - 66944 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 8 * q^3 + 12 * q^4 - 38 * q^5 - 82 * q^6 + 96 * q^8 - 380 * q^9 - 778 * q^10 + 424 * q^11 - 196 * q^12 - 924 * q^13 + 892 * q^15 - 2064 * q^16 - 2346 * q^17 + 212 * q^18 - 360 * q^19 - 1708 * q^20 + 2126 * q^22 - 12 * q^23 + 1392 * q^24 + 1872 * q^25 + 1148 * q^26 + 2872 * q^27 - 7052 * q^29 + 5258 * q^30 + 3548 * q^31 - 8096 * q^32 - 3398 * q^33 + 7422 * q^34 - 1096 * q^36 + 11090 * q^37 + 20138 * q^38 + 1624 * q^39 + 15936 * q^40 + 3500 * q^41 - 12680 * q^43 + 5948 * q^44 + 1300 * q^45 - 5118 * q^46 - 22956 * q^47 + 11216 * q^48 + 29992 * q^50 - 384 * q^51 - 1400 * q^52 + 3042 * q^53 + 32546 * q^54 - 25076 * q^55 - 19058 * q^57 - 58852 * q^58 - 65808 * q^59 + 14084 * q^60 - 42486 * q^61 - 49362 * q^62 + 35456 * q^64 - 3164 * q^65 - 25894 * q^66 + 42312 * q^67 + 5460 * q^68 + 5154 * q^69 - 2208 * q^71 - 32448 * q^72 - 50506 * q^73 - 47370 * q^74 - 35608 * q^75 + 38836 * q^76 + 27524 * q^78 + 9004 * q^79 + 68816 * q^80 + 51178 * q^81 + 67732 * q^82 - 104328 * q^83 - 53106 * q^85 + 86776 * q^86 + 80008 * q^87 - 20496 * q^88 - 26666 * q^89 + 130652 * q^90 - 10284 * q^92 + 38718 * q^93 + 98034 * q^94 - 198140 * q^95 + 54880 * q^96 + 209132 * q^97 - 66944 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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