LMFDB - Embedded newform 49.6.a.c.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,-4,0] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{39}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 39 $ x^2 - 39
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.24500$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +24.9800 q^{3} -28.0000 q^{4} +74.9400 q^{5} -49.9600 q^{6} +120.000 q^{8} +381.000 q^{9} -149.880 q^{10} -284.000 q^{11} -699.440 q^{12} +524.580 q^{13} +1872.00 q^{15} +656.000 q^{16} -149.880 q^{17} -762.000 q^{18} +2173.26 q^{19} -2098.32 q^{20} +568.000 q^{22} +1496.00 q^{23} +2997.60 q^{24} +2491.00 q^{25} -1049.16 q^{26} +3447.24 q^{27} -4366.00 q^{29} -3744.00 q^{30} -6444.84 q^{31} -5152.00 q^{32} -7094.32 q^{33} +299.760 q^{34} -10668.0 q^{36} -12630.0 q^{37} -4346.52 q^{38} +13104.0 q^{39} +8992.80 q^{40} -9442.44 q^{41} -1356.00 q^{43} +7952.00 q^{44} +28552.1 q^{45} -2992.00 q^{46} +10042.0 q^{47} +16386.9 q^{48} -4982.00 q^{50} -3744.00 q^{51} -14688.2 q^{52} +14150.0 q^{53} -6894.48 q^{54} -21283.0 q^{55} +54288.0 q^{57} +8732.00 q^{58} -37395.0 q^{59} -52416.0 q^{60} -35596.5 q^{61} +12889.7 q^{62} -10688.0 q^{64} +39312.0 q^{65} +14188.6 q^{66} -3644.00 q^{67} +4196.64 q^{68} +37370.1 q^{69} +35632.0 q^{71} +45720.0 q^{72} +40767.3 q^{73} +25260.0 q^{74} +62225.2 q^{75} -60851.3 q^{76} -26208.0 q^{78} -54616.0 q^{79} +49160.6 q^{80} -6471.00 q^{81} +18884.9 q^{82} -524.580 q^{83} -11232.0 q^{85} +2712.00 q^{86} -109063. q^{87} -34080.0 q^{88} -20383.7 q^{89} -57104.3 q^{90} -41888.0 q^{92} -160992. q^{93} -20083.9 q^{94} +162864. q^{95} -128697. q^{96} +183603. q^{97} -108204. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 56 q^{4} + 240 q^{8} + 762 q^{9} - 568 q^{11} + 3744 q^{15} + 1312 q^{16} - 1524 q^{18} + 1136 q^{22} + 2992 q^{23} + 4982 q^{25} - 8732 q^{29} - 7488 q^{30} - 10304 q^{32} - 21336 q^{36}+ \cdots - 216408 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 - 56 * q^4 + 240 * q^8 + 762 * q^9 - 568 * q^11 + 3744 * q^15 + 1312 * q^16 - 1524 * q^18 + 1136 * q^22 + 2992 * q^23 + 4982 * q^25 - 8732 * q^29 - 7488 * q^30 - 10304 * q^32 - 21336 * q^36 - 25260 * q^37 + 26208 * q^39 - 2712 * q^43 + 15904 * q^44 - 5984 * q^46 - 9964 * q^50 - 7488 * q^51 + 28300 * q^53 + 108576 * q^57 + 17464 * q^58 - 104832 * q^60 - 21376 * q^64 + 78624 * q^65 - 7288 * q^67 + 71264 * q^71 + 91440 * q^72 + 50520 * q^74 - 52416 * q^78 - 109232 * q^79 - 12942 * q^81 - 22464 * q^85 + 5424 * q^86 - 68160 * q^88 - 83776 * q^92 - 321984 * q^93 + 325728 * q^95 - 216408 * 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 $

$2$	−2.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−2.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	24.9800	1.60247	0.801234	−	0.598352i	$-0.204177\pi$
$3$	24.9800	1.60247	0.801234	+	0.598352i	$0.204177\pi$
$4$	−28.0000	−0.875000
$5$	74.9400	1.34057	0.670284	−	0.742105i	$-0.266172\pi$
$5$	74.9400	1.34057	0.670284	+	0.742105i	$0.266172\pi$
$6$	−49.9600	−0.566558
$7$	0	0
$7$	0	0
$8$	120.000	0.662913
$9$	381.000	1.56790
$10$	−149.880	−0.473962
$11$	−284.000	−0.707680	−0.353840	−	0.935306i	$-0.615124\pi$
$11$	−284.000	−0.707680	−0.353840	+	0.935306i	$0.615124\pi$
$12$	−699.440	−1.40216
$13$	524.580	0.860901	0.430450	−	0.902614i	$-0.358355\pi$
$13$	524.580	0.860901	0.430450	+	0.902614i	$0.358355\pi$
$14$	0	0
$15$	1872.00	2.14821
$16$	656.000	0.640625
$17$	−149.880	−0.125783	−0.0628914	−	0.998020i	$-0.520032\pi$
$17$	−149.880	−0.125783	−0.0628914	+	0.998020i	$0.520032\pi$
$18$	−762.000	−0.554337
$19$	2173.26	1.38111	0.690554	−	0.723281i	$-0.257367\pi$
$19$	2173.26	1.38111	0.690554	+	0.723281i	$0.257367\pi$
$20$	−2098.32	−1.17300
$21$	0	0
$22$	568.000	0.250202
$23$	1496.00	0.589674	0.294837	−	0.955548i	$-0.404735\pi$
$23$	1496.00	0.589674	0.294837	+	0.955548i	$0.404735\pi$
$24$	2997.60	1.06230
$25$	2491.00	0.797120
$26$	−1049.16	−0.304374
$27$	3447.24	0.910043
$28$	0	0
$29$	−4366.00	−0.964026	−0.482013	−	0.876164i	$-0.660094\pi$
$29$	−4366.00	−0.964026	−0.482013	+	0.876164i	$0.660094\pi$
$30$	−3744.00	−0.759509
$31$	−6444.84	−1.20450	−0.602251	−	0.798307i	$-0.705729\pi$
$31$	−6444.84	−1.20450	−0.602251	+	0.798307i	$0.705729\pi$
$32$	−5152.00	−0.889408
$33$	−7094.32	−1.13403
$34$	299.760	0.0444709
$35$	0	0
$36$	−10668.0	−1.37191
$37$	−12630.0	−1.51670	−0.758349	−	0.651849i	$-0.773994\pi$
$37$	−12630.0	−1.51670	−0.758349	+	0.651849i	$0.773994\pi$
$38$	−4346.52	−0.488295
$39$	13104.0	1.37957
$40$	8992.80	0.888679
$41$	−9442.44	−0.877252	−0.438626	−	0.898670i	$-0.644535\pi$
$41$	−9442.44	−0.877252	−0.438626	+	0.898670i	$0.644535\pi$
$42$	0	0
$43$	−1356.00	−0.111838	−0.0559189	−	0.998435i	$-0.517809\pi$
$43$	−1356.00	−0.111838	−0.0559189	+	0.998435i	$0.517809\pi$
$44$	7952.00	0.619220
$45$	28552.1	2.10188
$46$	−2992.00	−0.208481
$47$	10042.0	0.663092	0.331546	−	0.943439i	$-0.392430\pi$
$47$	10042.0	0.663092	0.331546	+	0.943439i	$0.392430\pi$
$48$	16386.9	1.02658
$49$	0	0
$50$	−4982.00	−0.281824
$51$	−3744.00	−0.201563
$52$	−14688.2	−0.753288
$53$	14150.0	0.691937	0.345969	−	0.938246i	$-0.387550\pi$
$53$	14150.0	0.691937	0.345969	+	0.938246i	$0.387550\pi$
$54$	−6894.48	−0.321749
$55$	−21283.0	−0.948692
$56$	0	0
$57$	54288.0	2.21318
$58$	8732.00	0.340835
$59$	−37395.0	−1.39857	−0.699285	−	0.714843i	$-0.746498\pi$
$59$	−37395.0	−1.39857	−0.699285	+	0.714843i	$0.746498\pi$
$60$	−52416.0	−1.87969
$61$	−35596.5	−1.22485	−0.612425	−	0.790529i	$-0.709806\pi$
$61$	−35596.5	−1.22485	−0.612425	+	0.790529i	$0.709806\pi$
$62$	12889.7	0.425856
$63$	0	0
$64$	−10688.0	−0.326172
$65$	39312.0	1.15410
$66$	14188.6	0.400941
$67$	−3644.00	−0.0991725	−0.0495863	−	0.998770i	$-0.515790\pi$
$67$	−3644.00	−0.0991725	−0.0495863	+	0.998770i	$0.515790\pi$
$68$	4196.64	0.110060
$69$	37370.1	0.944933
$70$	0	0
$71$	35632.0	0.838869	0.419435	−	0.907786i	$-0.362228\pi$
$71$	35632.0	0.838869	0.419435	+	0.907786i	$0.362228\pi$
$72$	45720.0	1.03938
$73$	40767.3	0.895376	0.447688	−	0.894190i	$-0.352248\pi$
$73$	40767.3	0.895376	0.447688	+	0.894190i	$0.352248\pi$
$74$	25260.0	0.536234
$75$	62225.2	1.27736
$76$	−60851.3	−1.20847
$77$	0	0
$78$	−26208.0	−0.487750
$79$	−54616.0	−0.984583	−0.492291	−	0.870431i	$-0.663841\pi$
$79$	−54616.0	−0.984583	−0.492291	+	0.870431i	$0.663841\pi$
$80$	49160.6	0.858801
$81$	−6471.00	−0.109587
$82$	18884.9	0.310155
$83$	−524.580	−0.00835827	−0.00417913	−	0.999991i	$-0.501330\pi$
$83$	−524.580	−0.00835827	−0.00417913	+	0.999991i	$0.501330\pi$
$84$	0	0
$85$	−11232.0	−0.168620
$86$	2712.00	0.0395406
$87$	−109063.	−1.54482
$88$	−34080.0	−0.469130
$89$	−20383.7	−0.272777	−0.136388	−	0.990655i	$-0.543550\pi$
$89$	−20383.7	−0.272777	−0.136388	+	0.990655i	$0.543550\pi$
$90$	−57104.3	−0.743126
$91$	0	0
$92$	−41888.0	−0.515965
$93$	−160992.	−1.93018
$94$	−20083.9	−0.234438
$95$	162864.	1.85147
$96$	−128697.	−1.42525
$97$	183603.	1.98130	0.990650	−	0.136427i	$-0.0435619\pi$
$97$	183603.	1.98130	0.990650	+	0.136427i	$0.0435619\pi$
$98$	0	0
$99$	−108204.	−1.10957
$100$	−69748.0	−0.697480
$101$	75914.2	0.740491	0.370245	−	0.928934i	$-0.379274\pi$
$101$	75914.2	0.740491	0.370245	+	0.928934i	$0.379274\pi$
$102$	7488.00	0.0712632
$103$	−10941.2	−0.101619	−0.0508093	−	0.998708i	$-0.516180\pi$
$103$	−10941.2	−0.101619	−0.0508093	+	0.998708i	$0.516180\pi$
$104$	62949.6	0.570702
$105$	0	0
$106$	−28300.0	−0.244637
$107$	218188.	1.84235	0.921173	−	0.389152i	$-0.127232\pi$
$107$	218188.	1.84235	0.921173	+	0.389152i	$0.127232\pi$
$108$	−96522.7	−0.796288
$109$	−96030.0	−0.774178	−0.387089	−	0.922042i	$-0.626519\pi$
$109$	−96030.0	−0.774178	−0.387089	+	0.922042i	$0.626519\pi$
$110$	42565.9	0.335413
$111$	−315497.	−2.43046
$112$	0	0
$113$	−137422.	−1.01242	−0.506209	−	0.862411i	$-0.668954\pi$
$113$	−137422.	−1.01242	−0.506209	+	0.862411i	$0.668954\pi$
$114$	−108576.	−0.782477
$115$	112110.	0.790498
$116$	122248.	0.843523
$117$	199865.	1.34981
$118$	74790.1	0.494469
$119$	0	0
$120$	224640.	1.42408
$121$	−80395.0	−0.499190
$122$	71193.0	0.433050
$123$	−235872.	−1.40577
$124$	180455.	1.05394
$125$	−47511.9	−0.271974
$126$	0	0
$127$	170368.	0.937300	0.468650	−	0.883384i	$-0.344741\pi$
$127$	170368.	0.937300	0.468650	+	0.883384i	$0.344741\pi$
$128$	186240.	1.00473
$129$	−33872.9	−0.179216
$130$	−78624.0	−0.408034
$131$	−348246.	−1.77300	−0.886498	−	0.462732i	$-0.846869\pi$
$131$	−348246.	−1.77300	−0.886498	+	0.462732i	$0.846869\pi$
$132$	198641.	0.992279
$133$	0	0
$134$	7288.00	0.0350628
$135$	258336.	1.21997
$136$	−17985.6	−0.0833830
$137$	−75562.0	−0.343955	−0.171978	−	0.985101i	$-0.555016\pi$
$137$	−75562.0	−0.343955	−0.171978	+	0.985101i	$0.555016\pi$
$138$	−74740.1	−0.334084
$139$	97047.3	0.426036	0.213018	−	0.977048i	$-0.431671\pi$
$139$	97047.3	0.426036	0.213018	+	0.977048i	$0.431671\pi$
$140$	0	0
$141$	250848.	1.06258
$142$	−71264.0	−0.296585
$143$	−148981.	−0.609242
$144$	249936.	1.00444
$145$	−327188.	−1.29234
$146$	−81534.7	−0.316563
$147$	0	0
$148$	353640.	1.32711
$149$	361030.	1.33223	0.666113	−	0.745851i	$-0.267957\pi$
$149$	361030.	1.33223	0.666113	+	0.745851i	$0.267957\pi$
$150$	−124450.	−0.451614
$151$	32280.0	0.115210	0.0576051	−	0.998339i	$-0.481654\pi$
$151$	32280.0	0.115210	0.0576051	+	0.998339i	$0.481654\pi$
$152$	260791.	0.915554
$153$	−57104.3	−0.197215
$154$	0	0
$155$	−482976.	−1.61472
$156$	−366912.	−1.20712
$157$	−132869.	−0.430203	−0.215101	−	0.976592i	$-0.569008\pi$
$157$	−132869.	−0.430203	−0.215101	+	0.976592i	$0.569008\pi$
$158$	109232.	0.348103
$159$	353467.	1.10881
$160$	−386091.	−1.19231
$161$	0	0
$162$	12942.0	0.0387448
$163$	−61364.0	−0.180903	−0.0904513	−	0.995901i	$-0.528831\pi$
$163$	−61364.0	−0.180903	−0.0904513	+	0.995901i	$0.528831\pi$
$164$	264388.	0.767596
$165$	−531648.	−1.52025
$166$	1049.16	0.00295509
$167$	380845.	1.05671	0.528356	−	0.849023i	$-0.322808\pi$
$167$	380845.	1.05671	0.528356	+	0.849023i	$0.322808\pi$
$168$	0	0
$169$	−96109.0	−0.258849
$170$	22464.0	0.0596163
$171$	828012.	2.16544
$172$	37968.0	0.0978581
$173$	517311.	1.31412	0.657062	−	0.753837i	$-0.271799\pi$
$173$	517311.	1.31412	0.657062	+	0.753837i	$0.271799\pi$
$174$	218125.	0.546176
$175$	0	0
$176$	−186304.	−0.453357
$177$	−934128.	−2.24116
$178$	40767.3	0.0964412
$179$	610564.	1.42429	0.712145	−	0.702032i	$-0.247724\pi$
$179$	610564.	1.42429	0.712145	+	0.702032i	$0.247724\pi$
$180$	−799460.	−1.83914
$181$	−433828.	−0.984285	−0.492142	−	0.870515i	$-0.663786\pi$
$181$	−433828.	−0.984285	−0.492142	+	0.870515i	$0.663786\pi$
$182$	0	0
$183$	−889200.	−1.96278
$184$	179520.	0.390902
$185$	−946492.	−2.03324
$186$	321984.	0.682420
$187$	42565.9	0.0890139
$188$	−281175.	−0.580205
$189$	0	0
$190$	−325728.	−0.654593
$191$	341192.	0.676730	0.338365	−	0.941015i	$-0.390126\pi$
$191$	341192.	0.676730	0.338365	+	0.941015i	$0.390126\pi$
$192$	−266986.	−0.522680
$193$	−616158.	−1.19069	−0.595345	−	0.803470i	$-0.702985\pi$
$193$	−616158.	−1.19069	−0.595345	+	0.803470i	$0.702985\pi$
$194$	−367206.	−0.700495
$195$	982013.	1.84940
$196$	0	0
$197$	231478.	0.424956	0.212478	−	0.977166i	$-0.431847\pi$
$197$	231478.	0.424956	0.212478	+	0.977166i	$0.431847\pi$
$198$	216408.	0.392293
$199$	−405126.	−0.725199	−0.362599	−	0.931945i	$-0.618111\pi$
$199$	−405126.	−0.725199	−0.362599	+	0.931945i	$0.618111\pi$
$200$	298920.	0.528421
$201$	−91027.1	−0.158921
$202$	−151828.	−0.261803
$203$	0	0
$204$	104832.	0.176367
$205$	−707616.	−1.17602
$206$	21882.5	0.0359276
$207$	569976.	0.924551
$208$	344124.	0.551515
$209$	−617206.	−0.977382
$210$	0	0
$211$	776820.	1.20120	0.600599	−	0.799551i	$-0.294929\pi$
$211$	776820.	1.20120	0.600599	+	0.799551i	$0.294929\pi$
$212$	−396200.	−0.605445
$213$	890087.	1.34426
$214$	−436376.	−0.651368
$215$	−101619.	−0.149926
$216$	413669.	0.603279
$217$	0	0
$218$	192060.	0.273713
$219$	1.01837e6	1.43481
$220$	595923.	0.830105
$221$	−78624.0	−0.108287
$222$	630995.	0.859297
$223$	81834.5	0.110198	0.0550990	−	0.998481i	$-0.482453\pi$
$223$	81834.5	0.110198	0.0550990	+	0.998481i	$0.482453\pi$
$224$	0	0
$225$	949071.	1.24981
$226$	274844.	0.357944
$227$	753671.	0.970772	0.485386	−	0.874300i	$-0.338679\pi$
$227$	753671.	0.970772	0.485386	+	0.874300i	$0.338679\pi$
$228$	−1.52006e6	−1.93653
$229$	26303.9	0.0331461	0.0165730	−	0.999863i	$-0.494724\pi$
$229$	26303.9	0.0331461	0.0165730	+	0.999863i	$0.494724\pi$
$230$	−224220.	−0.279483
$231$	0	0
$232$	−523920.	−0.639065
$233$	47542.0	0.0573704	0.0286852	−	0.999588i	$-0.490868\pi$
$233$	47542.0	0.0573704	0.0286852	+	0.999588i	$0.490868\pi$
$234$	−399730.	−0.477229
$235$	752544.	0.888919
$236$	1.04706e6	1.22375
$237$	−1.36431e6	−1.57776
$238$	0	0
$239$	1.08899e6	1.23319	0.616595	−	0.787281i	$-0.288512\pi$
$239$	1.08899e6	1.23319	0.616595	+	0.787281i	$0.288512\pi$
$240$	1.22803e6	1.37620
$241$	−1.34937e6	−1.49654	−0.748270	−	0.663395i	$-0.769115\pi$
$241$	−1.34937e6	−1.49654	−0.748270	+	0.663395i	$0.769115\pi$
$242$	160790.	0.176490
$243$	−999325.	−1.08565
$244$	996702.	1.07174
$245$	0	0
$246$	471744.	0.497014
$247$	1.14005e6	1.18900
$248$	−773381.	−0.798480
$249$	−13104.0	−0.0133939
$250$	95023.9	0.0961574
$251$	630020.	0.631205	0.315602	−	0.948892i	$-0.397793\pi$
$251$	630020.	0.631205	0.315602	+	0.948892i	$0.397793\pi$
$252$	0	0
$253$	−424864.	−0.417300
$254$	−340736.	−0.331386
$255$	−280575.	−0.270208
$256$	−30464.0	−0.0290527
$257$	−1.93405e6	−1.82656	−0.913282	−	0.407327i	$-0.866461\pi$
$257$	−1.93405e6	−1.82656	−0.913282	+	0.407327i	$0.866461\pi$
$258$	67745.7	0.0633626
$259$	0	0
$260$	−1.10074e6	−1.00983
$261$	−1.66345e6	−1.51150
$262$	696492.	0.626849
$263$	−1.11712e6	−0.995888	−0.497944	−	0.867209i	$-0.665911\pi$
$263$	−1.11712e6	−0.995888	−0.497944	+	0.867209i	$0.665911\pi$
$264$	−851318.	−0.751765
$265$	1.06040e6	0.927588
$266$	0	0
$267$	−509184.	−0.437116
$268$	102032.	0.0867760
$269$	1.52061e6	1.28126	0.640629	−	0.767851i	$-0.278674\pi$
$269$	1.52061e6	1.28126	0.640629	+	0.767851i	$0.278674\pi$
$270$	−516672.	−0.431326
$271$	1.02038e6	0.843995	0.421997	−	0.906597i	$-0.361329\pi$
$271$	1.02038e6	0.843995	0.421997	+	0.906597i	$0.361329\pi$
$272$	−98321.2	−0.0805796
$273$	0	0
$274$	151124.	0.121607
$275$	−707444.	−0.564105
$276$	−1.04636e6	−0.826817
$277$	1.89642e6	1.48503	0.742516	−	0.669829i	$-0.233633\pi$
$277$	1.89642e6	1.48503	0.742516	+	0.669829i	$0.233633\pi$
$278$	−194095.	−0.150626
$279$	−2.45548e6	−1.88854
$280$	0	0
$281$	1.31911e6	0.996587	0.498293	−	0.867008i	$-0.333960\pi$
$281$	1.31911e6	0.996587	0.498293	+	0.867008i	$0.333960\pi$
$282$	−501696.	−0.375680
$283$	478792.	0.355370	0.177685	−	0.984087i	$-0.443139\pi$
$283$	478792.	0.355370	0.177685	+	0.984087i	$0.443139\pi$
$284$	−997696.	−0.734011
$285$	4.06834e6	2.96692
$286$	297961.	0.215400
$287$	0	0
$288$	−1.96291e6	−1.39450
$289$	−1.39739e6	−0.984179
$290$	654376.	0.456912
$291$	4.58640e6	3.17497
$292$	−1.14149e6	−0.783454
$293$	1.50187e6	1.02203	0.511015	−	0.859572i	$-0.329270\pi$
$293$	1.50187e6	1.02203	0.511015	+	0.859572i	$0.329270\pi$
$294$	0	0
$295$	−2.80238e6	−1.87488
$296$	−1.51560e6	−1.00544
$297$	−979016.	−0.644019
$298$	−722060.	−0.471013
$299$	784771.	0.507651
$300$	−1.74230e6	−1.11769
$301$	0	0
$302$	−64560.0	−0.0407330
$303$	1.89634e6	1.18661
$304$	1.42566e6	0.884772
$305$	−2.66760e6	−1.64199
$306$	114209.	0.0697260
$307$	−1.19657e6	−0.724588	−0.362294	−	0.932064i	$-0.618006\pi$
$307$	−1.19657e6	−0.724588	−0.362294	+	0.932064i	$0.618006\pi$
$308$	0	0
$309$	−273312.	−0.162841
$310$	965952.	0.570889
$311$	−1.31475e6	−0.770799	−0.385400	−	0.922750i	$-0.625936\pi$
$311$	−1.31475e6	−0.770799	−0.385400	+	0.922750i	$0.625936\pi$
$312$	1.57248e6	0.914531
$313$	−2.65273e6	−1.53049	−0.765247	−	0.643737i	$-0.777383\pi$
$313$	−2.65273e6	−1.53049	−0.765247	+	0.643737i	$0.777383\pi$
$314$	265737.	0.152100
$315$	0	0
$316$	1.52925e6	0.861510
$317$	−1.59992e6	−0.894233	−0.447116	−	0.894476i	$-0.647549\pi$
$317$	−1.59992e6	−0.894233	−0.447116	+	0.894476i	$0.647549\pi$
$318$	−706934.	−0.392022
$319$	1.23994e6	0.682221
$320$	−800958.	−0.437255
$321$	5.45033e6	2.95230
$322$	0	0
$323$	−325728.	−0.173720
$324$	181188.	0.0958886
$325$	1.30673e6	0.686241
$326$	122728.	0.0639587
$327$	−2.39883e6	−1.24059
$328$	−1.13309e6	−0.581542
$329$	0	0
$330$	1.06330e6	0.537489
$331$	−111708.	−0.0560421	−0.0280210	−	0.999607i	$-0.508921\pi$
$331$	−111708.	−0.0560421	−0.0280210	+	0.999607i	$0.508921\pi$
$332$	14688.2	0.00731349
$333$	−4.81203e6	−2.37803
$334$	−761690.	−0.373604
$335$	−273081.	−0.132947
$336$	0	0
$337$	1.59301e6	0.764087	0.382043	−	0.924144i	$-0.375220\pi$
$337$	1.59301e6	0.764087	0.382043	+	0.924144i	$0.375220\pi$
$338$	192218.	0.0915171
$339$	−3.43280e6	−1.62237
$340$	314496.	0.147543
$341$	1.83033e6	0.852402
$342$	−1.65602e6	−0.765599
$343$	0	0
$344$	−162720.	−0.0741387
$345$	2.80051e6	1.26675
$346$	−1.03462e6	−0.464613
$347$	−3.33676e6	−1.48765	−0.743827	−	0.668372i	$-0.766991\pi$
$347$	−3.33676e6	−1.48765	−0.743827	+	0.668372i	$0.766991\pi$
$348$	3.05375e6	1.35172
$349$	−1.60259e6	−0.704303	−0.352151	−	0.935943i	$-0.614550\pi$
$349$	−1.60259e6	−0.704303	−0.352151	+	0.935943i	$0.614550\pi$
$350$	0	0
$351$	1.80835e6	0.783457
$352$	1.46317e6	0.629416
$353$	1.80965e6	0.772962	0.386481	−	0.922297i	$-0.373691\pi$
$353$	1.80965e6	0.772962	0.386481	+	0.922297i	$0.373691\pi$
$354$	1.86826e6	0.792370
$355$	2.67026e6	1.12456
$356$	570743.	0.238680
$357$	0	0
$358$	−1.22113e6	−0.503563
$359$	920792.	0.377073	0.188536	−	0.982066i	$-0.439626\pi$
$359$	920792.	0.377073	0.188536	+	0.982066i	$0.439626\pi$
$360$	3.42626e6	1.39336
$361$	2.24696e6	0.907458
$362$	867655.	0.347997
$363$	−2.00827e6	−0.799935
$364$	0	0
$365$	3.05510e6	1.20031
$366$	1.77840e6	0.693948
$367$	1.34802e6	0.522434	0.261217	−	0.965280i	$-0.415876\pi$
$367$	1.34802e6	0.522434	0.261217	+	0.965280i	$0.415876\pi$
$368$	981376.	0.377760
$369$	−3.59757e6	−1.37544
$370$	1.89298e6	0.718857
$371$	0	0
$372$	4.50778e6	1.68890
$373$	3.13773e6	1.16773	0.583867	−	0.811849i	$-0.301539\pi$
$373$	3.13773e6	1.16773	0.583867	+	0.811849i	$0.301539\pi$
$374$	−85131.8	−0.0314712
$375$	−1.18685e6	−0.435830
$376$	1.20503e6	0.439572
$377$	−2.29032e6	−0.829931
$378$	0	0
$379$	−1.83188e6	−0.655088	−0.327544	−	0.944836i	$-0.606221\pi$
$379$	−1.83188e6	−0.655088	−0.327544	+	0.944836i	$0.606221\pi$
$380$	−4.56019e6	−1.62003
$381$	4.25579e6	1.50199
$382$	−682384.	−0.239260
$383$	−27727.8	−0.00965869	−0.00482935	−	0.999988i	$-0.501537\pi$
$383$	−27727.8	−0.00965869	−0.00482935	+	0.999988i	$0.501537\pi$
$384$	4.65227e6	1.61004
$385$	0	0
$386$	1.23232e6	0.420973
$387$	−516636.	−0.175351
$388$	−5.14088e6	−1.73364
$389$	−548342.	−0.183729	−0.0918645	−	0.995772i	$-0.529283\pi$
$389$	−548342.	−0.183729	−0.0918645	+	0.995772i	$0.529283\pi$
$390$	−1.96403e6	−0.653862
$391$	−224220.	−0.0741708
$392$	0	0
$393$	−8.69918e6	−2.84117
$394$	−462956.	−0.150245
$395$	−4.09292e6	−1.31990
$396$	3.02971e6	0.970875
$397$	−403852.	−0.128601	−0.0643007	−	0.997931i	$-0.520482\pi$
$397$	−403852.	−0.128601	−0.0643007	+	0.997931i	$0.520482\pi$
$398$	810251.	0.256396
$399$	0	0
$400$	1.63410e6	0.510655
$401$	−4.16427e6	−1.29324	−0.646619	−	0.762813i	$-0.723818\pi$
$401$	−4.16427e6	−1.29324	−0.646619	+	0.762813i	$0.723818\pi$
$402$	182054.	0.0561870
$403$	−3.38083e6	−1.03696
$404$	−2.12560e6	−0.647929
$405$	−484937.	−0.146909
$406$	0	0
$407$	3.58692e6	1.07334
$408$	−449280.	−0.133619
$409$	−2.54121e6	−0.751161	−0.375581	−	0.926790i	$-0.622557\pi$
$409$	−2.54121e6	−0.751161	−0.375581	+	0.926790i	$0.622557\pi$
$410$	1.41523e6	0.415784
$411$	−1.88754e6	−0.551177
$412$	306355.	0.0889163
$413$	0	0
$414$	−1.13995e6	−0.326878
$415$	−39312.0	−0.0112048
$416$	−2.70264e6	−0.765692
$417$	2.42424e6	0.682709
$418$	1.23441e6	0.345557
$419$	−2.30133e6	−0.640389	−0.320195	−	0.947352i	$-0.603748\pi$
$419$	−2.30133e6	−0.640389	−0.320195	+	0.947352i	$0.603748\pi$
$420$	0	0
$421$	−2.79991e6	−0.769909	−0.384955	−	0.922936i	$-0.625783\pi$
$421$	−2.79991e6	−0.769909	−0.384955	+	0.922936i	$0.625783\pi$
$422$	−1.55364e6	−0.424687
$423$	3.82599e6	1.03966
$424$	1.69800e6	0.458694
$425$	−373351.	−0.100264
$426$	−1.78017e6	−0.475268
$427$	0	0
$428$	−6.10926e6	−1.61205
$429$	−3.72154e6	−0.976290
$430$	203237.	0.0530069
$431$	954320.	0.247458	0.123729	−	0.992316i	$-0.460515\pi$
$431$	954320.	0.247458	0.123729	+	0.992316i	$0.460515\pi$
$432$	2.26139e6	0.582996
$433$	519334.	0.133115	0.0665575	−	0.997783i	$-0.478798\pi$
$433$	519334.	0.133115	0.0665575	+	0.997783i	$0.478798\pi$
$434$	0	0
$435$	−8.17315e6	−2.07093
$436$	2.68884e6	0.677406
$437$	3.25120e6	0.814403
$438$	−2.03674e6	−0.507282
$439$	5.98081e6	1.48115	0.740574	−	0.671974i	$-0.234554\pi$
$439$	5.98081e6	1.48115	0.740574	+	0.671974i	$0.234554\pi$
$440$	−2.55395e6	−0.628900
$441$	0	0
$442$	157248.	0.0382851
$443$	3.74820e6	0.907432	0.453716	−	0.891146i	$-0.350098\pi$
$443$	3.74820e6	0.907432	0.453716	+	0.891146i	$0.350098\pi$
$444$	8.83392e6	2.12665
$445$	−1.52755e6	−0.365676
$446$	−163669.	−0.0389609
$447$	9.01853e6	2.13485
$448$	0	0
$449$	99458.0	0.0232822	0.0116411	−	0.999932i	$-0.496294\pi$
$449$	99458.0	0.0232822	0.0116411	+	0.999932i	$0.496294\pi$
$450$	−1.89814e6	−0.441873
$451$	2.68165e6	0.620813
$452$	3.84782e6	0.885866
$453$	806354.	0.184621
$454$	−1.50734e6	−0.343220
$455$	0	0
$456$	6.51456e6	1.46714
$457$	−161814.	−0.0362431	−0.0181216	−	0.999836i	$-0.505769\pi$
$457$	−161814.	−0.0362431	−0.0181216	+	0.999836i	$0.505769\pi$
$458$	−52607.9	−0.0117189
$459$	−516672.	−0.114468
$460$	−3.13909e6	−0.691685
$461$	−4.49198e6	−0.984431	−0.492215	−	0.870473i	$-0.663813\pi$
$461$	−4.49198e6	−0.984431	−0.492215	+	0.870473i	$0.663813\pi$
$462$	0	0
$463$	3.59382e6	0.779118	0.389559	−	0.921001i	$-0.372627\pi$
$463$	3.59382e6	0.779118	0.389559	+	0.921001i	$0.372627\pi$
$464$	−2.86410e6	−0.617579
$465$	−1.20647e7	−2.58753
$466$	−95084.0	−0.0202835
$467$	2.05223e6	0.435446	0.217723	−	0.976011i	$-0.430137\pi$
$467$	2.05223e6	0.435446	0.217723	+	0.976011i	$0.430137\pi$
$468$	−5.59622e6	−1.18108
$469$	0	0
$470$	−1.50509e6	−0.314280
$471$	−3.31906e6	−0.689386
$472$	−4.48741e6	−0.927129
$473$	385104.	0.0791453
$474$	2.72861e6	0.557823
$475$	5.41359e6	1.10091
$476$	0	0
$477$	5.39115e6	1.08489
$478$	−2.17798e6	−0.435998
$479$	−1.99985e6	−0.398252	−0.199126	−	0.979974i	$-0.563810\pi$
$479$	−1.99985e6	−0.398252	−0.199126	+	0.979974i	$0.563810\pi$
$480$	−9.64454e6	−1.91064
$481$	−6.62544e6	−1.30573
$482$	2.69874e6	0.529107
$483$	0	0
$484$	2.25106e6	0.436791
$485$	1.37592e7	2.65607
$486$	1.99865e6	0.383836
$487$	2.17126e6	0.414848	0.207424	−	0.978251i	$-0.433492\pi$
$487$	2.17126e6	0.414848	0.207424	+	0.978251i	$0.433492\pi$
$488$	−4.27158e6	−0.811968
$489$	−1.53287e6	−0.289890
$490$	0	0
$491$	3.04555e6	0.570114	0.285057	−	0.958511i	$-0.407987\pi$
$491$	3.04555e6	0.570114	0.285057	+	0.958511i	$0.407987\pi$
$492$	6.60442e6	1.23005
$493$	654376.	0.121258
$494$	−2.28010e6	−0.420374
$495$	−8.10881e6	−1.48746
$496$	−4.22781e6	−0.771635
$497$	0	0
$498$	26208.0	0.00473544
$499$	−7.99225e6	−1.43687	−0.718436	−	0.695594i	$-0.755141\pi$
$499$	−7.99225e6	−1.43687	−0.718436	+	0.695594i	$0.755141\pi$
$500$	1.33033e6	0.237977
$501$	9.51350e6	1.69335
$502$	−1.26004e6	−0.223165
$503$	9.47811e6	1.67033	0.835164	−	0.550001i	$-0.185373\pi$
$503$	9.47811e6	1.67033	0.835164	+	0.550001i	$0.185373\pi$
$504$	0	0
$505$	5.68901e6	0.992677
$506$	849728.	0.147538
$507$	−2.40080e6	−0.414798
$508$	−4.77030e6	−0.820138
$509$	1.01003e7	1.72799	0.863995	−	0.503500i	$-0.167955\pi$
$509$	1.01003e7	1.72799	0.863995	+	0.503500i	$0.167955\pi$
$510$	561151.	0.0955331
$511$	0	0
$512$	−5.89875e6	−0.994455
$513$	7.49174e6	1.25687
$514$	3.86810e6	0.645788
$515$	−819936.	−0.136227
$516$	948440.	0.156814
$517$	−2.85192e6	−0.469257
$518$	0	0
$519$	1.29224e7	2.10584
$520$	4.71744e6	0.765064
$521$	5.11975e6	0.826332	0.413166	−	0.910656i	$-0.364423\pi$
$521$	5.11975e6	0.826332	0.413166	+	0.910656i	$0.364423\pi$
$522$	3.32689e6	0.534395
$523$	3.51476e6	0.561877	0.280939	−	0.959726i	$-0.409354\pi$
$523$	3.51476e6	0.561877	0.280939	+	0.959726i	$0.409354\pi$
$524$	9.75089e6	1.55137
$525$	0	0
$526$	2.23424e6	0.352100
$527$	965952.	0.151506
$528$	−4.65387e6	−0.726490
$529$	−4.19833e6	−0.652285
$530$	−2.12080e6	−0.327952
$531$	−1.42475e7	−2.19282
$532$	0	0
$533$	−4.95331e6	−0.755227
$534$	1.01837e6	0.154544
$535$	1.63510e7	2.46979
$536$	−437280.	−0.0657427
$537$	1.52519e7	2.28238
$538$	−3.04121e6	−0.452993
$539$	0	0
$540$	−7.23341e6	−1.06748
$541$	−3.47547e6	−0.510530	−0.255265	−	0.966871i	$-0.582163\pi$
$541$	−3.47547e6	−0.510530	−0.255265	+	0.966871i	$0.582163\pi$
$542$	−2.04077e6	−0.298397
$543$	−1.08370e7	−1.57728
$544$	772182.	0.111872
$545$	−7.19649e6	−1.03784
$546$	0	0
$547$	−7.85765e6	−1.12286	−0.561429	−	0.827525i	$-0.689748\pi$
$547$	−7.85765e6	−1.12286	−0.561429	+	0.827525i	$0.689748\pi$
$548$	2.11574e6	0.300961
$549$	−1.35623e7	−1.92044
$550$	1.41489e6	0.199441
$551$	−9.48845e6	−1.33142
$552$	4.48441e6	0.626408
$553$	0	0
$554$	−3.79284e6	−0.525038
$555$	−2.36434e7	−3.25819
$556$	−2.71732e6	−0.372782
$557$	9.06537e6	1.23808	0.619039	−	0.785361i	$-0.287522\pi$
$557$	9.06537e6	1.23808	0.619039	+	0.785361i	$0.287522\pi$
$558$	4.91097e6	0.667700
$559$	−711330.	−0.0962813
$560$	0	0
$561$	1.06330e6	0.142642
$562$	−2.63822e6	−0.352347
$563$	5.80957e6	0.772455	0.386227	−	0.922404i	$-0.373778\pi$
$563$	5.80957e6	0.772455	0.386227	+	0.922404i	$0.373778\pi$
$564$	−7.02374e6	−0.929760
$565$	−1.02984e7	−1.35722
$566$	−957583.	−0.125642
$567$	0	0
$568$	4.27584e6	0.556097
$569$	1.05559e7	1.36684	0.683418	−	0.730027i	$-0.260493\pi$
$569$	1.05559e7	1.36684	0.683418	+	0.730027i	$0.260493\pi$
$570$	−8.13668e6	−1.04896
$571$	7.99584e6	1.02630	0.513150	−	0.858299i	$-0.328479\pi$
$571$	7.99584e6	1.02630	0.513150	+	0.858299i	$0.328479\pi$
$572$	4.17146e6	0.533087
$573$	8.52297e6	1.08444
$574$	0	0
$575$	3.72654e6	0.470041
$576$	−4.07213e6	−0.511405
$577$	−2.60941e6	−0.326289	−0.163145	−	0.986602i	$-0.552164\pi$
$577$	−2.60941e6	−0.326289	−0.163145	+	0.986602i	$0.552164\pi$
$578$	2.79479e6	0.347960
$579$	−1.53916e7	−1.90804
$580$	9.16126e6	1.13080
$581$	0	0
$582$	−9.17280e6	−1.12252
$583$	−4.01860e6	−0.489670
$584$	4.89208e6	0.593556
$585$	1.49779e7	1.80951
$586$	−3.00374e6	−0.361342
$587$	4.41749e6	0.529151	0.264576	−	0.964365i	$-0.414768\pi$
$587$	4.41749e6	0.529151	0.264576	+	0.964365i	$0.414768\pi$
$588$	0	0
$589$	−1.40063e7	−1.66355
$590$	5.60477e6	0.662869
$591$	5.78232e6	0.680978
$592$	−8.28528e6	−0.971634
$593$	−5.54106e6	−0.647077	−0.323539	−	0.946215i	$-0.604873\pi$
$593$	−5.54106e6	−0.647077	−0.323539	+	0.946215i	$0.604873\pi$
$594$	1.95803e6	0.227695
$595$	0	0
$596$	−1.01088e7	−1.16570
$597$	−1.01200e7	−1.16211
$598$	−1.56954e6	−0.179482
$599$	5.08611e6	0.579187	0.289594	−	0.957150i	$-0.406480\pi$
$599$	5.08611e6	0.579187	0.289594	+	0.957150i	$0.406480\pi$
$600$	7.46702e6	0.846777
$601$	2.41307e6	0.272511	0.136255	−	0.990674i	$-0.456493\pi$
$601$	2.41307e6	0.272511	0.136255	+	0.990674i	$0.456493\pi$
$602$	0	0
$603$	−1.38836e6	−0.155493
$604$	−903840.	−0.100809
$605$	−6.02480e6	−0.669197
$606$	−3.79267e6	−0.419531
$607$	7.62170e6	0.839614	0.419807	−	0.907613i	$-0.362098\pi$
$607$	7.62170e6	0.839614	0.419807	+	0.907613i	$0.362098\pi$
$608$	−1.11966e7	−1.22837
$609$	0	0
$610$	5.33520e6	0.580532
$611$	5.26781e6	0.570856
$612$	1.59892e6	0.172563
$613$	3.60126e6	0.387082	0.193541	−	0.981092i	$-0.438003\pi$
$613$	3.60126e6	0.387082	0.193541	+	0.981092i	$0.438003\pi$
$614$	2.39313e6	0.256180
$615$	−1.76762e7	−1.88453
$616$	0	0
$617$	7.22901e6	0.764480	0.382240	−	0.924063i	$-0.375153\pi$
$617$	7.22901e6	0.764480	0.382240	+	0.924063i	$0.375153\pi$
$618$	546624.	0.0575728
$619$	−1.25832e7	−1.31998	−0.659988	−	0.751276i	$-0.729439\pi$
$619$	−1.25832e7	−1.31998	−0.659988	+	0.751276i	$0.729439\pi$
$620$	1.35233e7	1.41288
$621$	5.15707e6	0.536629
$622$	2.62949e6	0.272519
$623$	0	0
$624$	8.59622e6	0.883784
$625$	−1.13449e7	−1.16172
$626$	5.30545e6	0.541111
$627$	−1.54178e7	−1.56622
$628$	3.72032e6	0.376427
$629$	1.89298e6	0.190774
$630$	0	0
$631$	5.98350e6	0.598249	0.299125	−	0.954214i	$-0.403305\pi$
$631$	5.98350e6	0.598249	0.299125	+	0.954214i	$0.403305\pi$
$632$	−6.55392e6	−0.652692
$633$	1.94050e7	1.92488
$634$	3.19984e6	0.316159
$635$	1.27674e7	1.25651
$636$	−9.89707e6	−0.970206
$637$	0	0
$638$	−2.47989e6	−0.241202
$639$	1.35758e7	1.31526
$640$	1.39568e7	1.34690
$641$	−7.74000e6	−0.744040	−0.372020	−	0.928225i	$-0.621335\pi$
$641$	−7.74000e6	−0.744040	−0.372020	+	0.928225i	$0.621335\pi$
$642$	−1.09007e7	−1.04380
$643$	6.81377e6	0.649920	0.324960	−	0.945728i	$-0.394649\pi$
$643$	6.81377e6	0.649920	0.324960	+	0.945728i	$0.394649\pi$
$644$	0	0
$645$	−2.53843e6	−0.240252
$646$	651456.	0.0614191
$647$	−1.07121e7	−1.00603	−0.503017	−	0.864276i	$-0.667777\pi$
$647$	−1.07121e7	−1.00603	−0.503017	+	0.864276i	$0.667777\pi$
$648$	−776520.	−0.0726466
$649$	1.06202e7	0.989739
$650$	−2.61346e6	−0.242623
$651$	0	0
$652$	1.71819e6	0.158290
$653$	1.34167e7	1.23129	0.615647	−	0.788022i	$-0.288895\pi$
$653$	1.34167e7	1.23129	0.615647	+	0.788022i	$0.288895\pi$
$654$	4.79766e6	0.438616
$655$	−2.60976e7	−2.37682
$656$	−6.19424e6	−0.561990
$657$	1.55324e7	1.40386
$658$	0	0
$659$	1.38574e7	1.24299	0.621494	−	0.783419i	$-0.286526\pi$
$659$	1.38574e7	1.24299	0.621494	+	0.783419i	$0.286526\pi$
$660$	1.48861e7	1.33022
$661$	−1.20734e7	−1.07479	−0.537396	−	0.843330i	$-0.680592\pi$
$661$	−1.20734e7	−1.07479	−0.537396	+	0.843330i	$0.680592\pi$
$662$	223416.	0.0198139
$663$	−1.96403e6	−0.173526
$664$	−62949.6	−0.00554080
$665$	0	0
$666$	9.62406e6	0.840761
$667$	−6.53154e6	−0.568461
$668$	−1.06637e7	−0.924624
$669$	2.04422e6	0.176589
$670$	546163.	0.0470040
$671$	1.01094e7	0.866801
$672$	0	0
$673$	−5.32490e6	−0.453183	−0.226592	−	0.973990i	$-0.572758\pi$
$673$	−5.32490e6	−0.453183	−0.226592	+	0.973990i	$0.572758\pi$
$674$	−3.18601e6	−0.270145
$675$	8.58707e6	0.725414
$676$	2.69105e6	0.226493
$677$	−2.34518e7	−1.96655	−0.983274	−	0.182135i	$-0.941699\pi$
$677$	−2.34518e7	−1.96655	−0.983274	+	0.182135i	$0.941699\pi$
$678$	6.86560e6	0.573594
$679$	0	0
$680$	−1.34784e6	−0.111781
$681$	1.88267e7	1.55563
$682$	−3.66067e6	−0.301370
$683$	1.82270e7	1.49508	0.747538	−	0.664219i	$-0.231236\pi$
$683$	1.82270e7	1.49508	0.747538	+	0.664219i	$0.231236\pi$
$684$	−2.31843e7	−1.89476
$685$	−5.66261e6	−0.461095
$686$	0	0
$687$	657072.	0.0531155
$688$	−889536.	−0.0716461
$689$	7.42280e6	0.595690
$690$	−5.60102e6	−0.447863
$691$	7.60858e6	0.606190	0.303095	−	0.952960i	$-0.401980\pi$
$691$	7.60858e6	0.606190	0.303095	+	0.952960i	$0.401980\pi$
$692$	−1.44847e7	−1.14986
$693$	0	0
$694$	6.67353e6	0.525965
$695$	7.27272e6	0.571130
$696$	−1.30875e7	−1.02408
$697$	1.41523e6	0.110343
$698$	3.20518e6	0.249009
$699$	1.18760e6	0.0919341
$700$	0	0
$701$	314162.	0.0241467	0.0120734	−	0.999927i	$-0.496157\pi$
$701$	314162.	0.0241467	0.0120734	+	0.999927i	$0.496157\pi$
$702$	−3.61670e6	−0.276994
$703$	−2.74483e7	−2.09472
$704$	3.03539e6	0.230825
$705$	1.87985e7	1.42446
$706$	−3.61930e6	−0.273283
$707$	0	0
$708$	2.61556e7	1.96102
$709$	2.48285e7	1.85496	0.927482	−	0.373869i	$-0.121969\pi$
$709$	2.48285e7	1.85496	0.927482	+	0.373869i	$0.121969\pi$
$710$	−5.34052e6	−0.397592
$711$	−2.08087e7	−1.54373
$712$	−2.44604e6	−0.180827
$713$	−9.64148e6	−0.710264
$714$	0	0
$715$	−1.11646e7	−0.816730
$716$	−1.70958e7	−1.24625
$717$	2.72030e7	1.97615
$718$	−1.84158e6	−0.133315
$719$	1.71932e7	1.24032	0.620160	−	0.784475i	$-0.287068\pi$
$719$	1.71932e7	1.24032	0.620160	+	0.784475i	$0.287068\pi$
$720$	1.87302e7	1.34651
$721$	0	0
$722$	−4.49391e6	−0.320835
$723$	−3.37072e7	−2.39816
$724$	1.21472e7	0.861249
$725$	−1.08757e7	−0.768444
$726$	4.01653e6	0.282820
$727$	−2.12927e7	−1.49415	−0.747076	−	0.664739i	$-0.768543\pi$
$727$	−2.12927e7	−1.49415	−0.747076	+	0.664739i	$0.768543\pi$
$728$	0	0
$729$	−2.33907e7	−1.63014
$730$	−6.11021e6	−0.424374
$731$	203237.	0.0140673
$732$	2.48976e7	1.71743
$733$	−1.96024e7	−1.34757	−0.673783	−	0.738930i	$-0.735332\pi$
$733$	−1.96024e7	−1.34757	−0.673783	+	0.738930i	$0.735332\pi$
$734$	−2.69604e6	−0.184708
$735$	0	0
$736$	−7.70739e6	−0.524461
$737$	1.03490e6	0.0701824
$738$	7.19514e6	0.486293
$739$	1.44181e7	0.971173	0.485587	−	0.874189i	$-0.338606\pi$
$739$	1.44181e7	0.971173	0.485587	+	0.874189i	$0.338606\pi$
$740$	2.65018e7	1.77908
$741$	2.84784e7	1.90533
$742$	0	0
$743$	5.57521e6	0.370501	0.185250	−	0.982691i	$-0.440690\pi$
$743$	5.57521e6	0.370501	0.185250	+	0.982691i	$0.440690\pi$
$744$	−1.93190e7	−1.27954
$745$	2.70556e7	1.78594
$746$	−6.27547e6	−0.412856
$747$	−199865.	−0.0131049
$748$	−1.19185e6	−0.0778872
$749$	0	0
$750$	2.37370e6	0.154089
$751$	508800.	0.0329190	0.0164595	−	0.999865i	$-0.494761\pi$
$751$	508800.	0.0329190	0.0164595	+	0.999865i	$0.494761\pi$
$752$	6.58752e6	0.424793
$753$	1.57379e7	1.01149
$754$	4.58063e6	0.293425
$755$	2.41906e6	0.154447
$756$	0	0
$757$	1.10466e7	0.700631	0.350316	−	0.936632i	$-0.386074\pi$
$757$	1.10466e7	0.700631	0.350316	+	0.936632i	$0.386074\pi$
$758$	3.66377e6	0.231609
$759$	−1.06131e7	−0.668710
$760$	1.95437e7	1.22736
$761$	−6.77173e6	−0.423875	−0.211937	−	0.977283i	$-0.567977\pi$
$761$	−6.77173e6	−0.423875	−0.211937	+	0.977283i	$0.567977\pi$
$762$	−8.51158e6	−0.531035
$763$	0	0
$764$	−9.55338e6	−0.592139
$765$	−4.27939e6	−0.264380
$766$	55455.6	0.00341486
$767$	−1.96167e7	−1.20403
$768$	−760990.	−0.0465561
$769$	−2.48053e7	−1.51261	−0.756307	−	0.654216i	$-0.772999\pi$
$769$	−2.48053e7	−1.51261	−0.756307	+	0.654216i	$0.772999\pi$
$770$	0	0
$771$	−4.83126e7	−2.92701
$772$	1.72524e7	1.04185
$773$	−1.62952e7	−0.980867	−0.490434	−	0.871479i	$-0.663162\pi$
$773$	−1.62952e7	−0.980867	−0.490434	+	0.871479i	$0.663162\pi$
$774$	1.03327e6	0.0619958
$775$	−1.60541e7	−0.960133
$776$	2.20324e7	1.31343
$777$	0	0
$778$	1.09668e6	0.0649580
$779$	−2.05209e7	−1.21158
$780$	−2.74964e7	−1.61823
$781$	−1.01195e7	−0.593651
$782$	448441.	0.0262234
$783$	−1.50506e7	−0.877305
$784$	0	0
$785$	−9.95717e6	−0.576716
$786$	1.73984e7	1.00451
$787$	−1.44214e7	−0.829984	−0.414992	−	0.909825i	$-0.636216\pi$
$787$	−1.44214e7	−0.829984	−0.414992	+	0.909825i	$0.636216\pi$
$788$	−6.48138e6	−0.371837
$789$	−2.79056e7	−1.59588
$790$	8.18584e6	0.466655
$791$	0	0
$792$	−1.29845e7	−0.735549
$793$	−1.86732e7	−1.05447
$794$	807703.	0.0454674
$795$	2.64888e7	1.48643
$796$	1.13435e7	0.634549
$797$	−1.10109e6	−0.0614014	−0.0307007	−	0.999529i	$-0.509774\pi$
$797$	−1.10109e6	−0.0614014	−0.0307007	+	0.999529i	$0.509774\pi$
$798$	0	0
$799$	−1.50509e6	−0.0834055
$800$	−1.28336e7	−0.708965
$801$	−7.76618e6	−0.427687
$802$	8.32855e6	0.457229
$803$	−1.15779e7	−0.633639
$804$	2.54876e6	0.139056
$805$	0	0
$806$	6.76166e6	0.366620
$807$	3.79848e7	2.05317
$808$	9.10970e6	0.490881
$809$	−2.45146e7	−1.31690	−0.658450	−	0.752625i	$-0.728787\pi$
$809$	−2.45146e7	−1.31690	−0.658450	+	0.752625i	$0.728787\pi$
$810$	969873.	0.0519401
$811$	3.03580e7	1.62077	0.810383	−	0.585900i	$-0.199259\pi$
$811$	3.03580e7	1.62077	0.810383	+	0.585900i	$0.199259\pi$
$812$	0	0
$813$	2.54892e7	1.35247
$814$	−7.17384e6	−0.379482
$815$	−4.59862e6	−0.242512
$816$	−2.45606e6	−0.129126
$817$	−2.94694e6	−0.154460
$818$	5.08243e6	0.265576
$819$	0	0
$820$	1.98132e7	1.02901
$821$	−2.54599e7	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$821$	−2.54599e7	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$822$	3.77508e6	0.194871
$823$	5.95232e6	0.306328	0.153164	−	0.988201i	$-0.451054\pi$
$823$	5.95232e6	0.306328	0.153164	+	0.988201i	$0.451054\pi$
$824$	−1.31295e6	−0.0673643
$825$	−1.76719e7	−0.903961
$826$	0	0
$827$	−7.85900e6	−0.399580	−0.199790	−	0.979839i	$-0.564026\pi$
$827$	−7.85900e6	−0.399580	−0.199790	+	0.979839i	$0.564026\pi$
$828$	−1.59593e7	−0.808982
$829$	1.12642e7	0.569262	0.284631	−	0.958637i	$-0.408129\pi$
$829$	1.12642e7	0.569262	0.284631	+	0.958637i	$0.408129\pi$
$830$	78624.0	0.00396150
$831$	4.73726e7	2.37971
$832$	−5.60671e6	−0.280802
$833$	0	0
$834$	−4.84848e6	−0.241374
$835$	2.85405e7	1.41659
$836$	1.72818e7	0.855209
$837$	−2.22169e7	−1.09615
$838$	4.60266e6	0.226412
$839$	−7.80470e6	−0.382782	−0.191391	−	0.981514i	$-0.561300\pi$
$839$	−7.80470e6	−0.382782	−0.191391	+	0.981514i	$0.561300\pi$
$840$	0	0
$841$	−1.44919e6	−0.0706539
$842$	5.59983e6	0.272204
$843$	3.29514e7	1.59700
$844$	−2.17510e7	−1.05105
$845$	−7.20241e6	−0.347005
$846$	−7.65197e6	−0.367576
$847$	0	0
$848$	9.28240e6	0.443272
$849$	1.19602e7	0.569468
$850$	746702.	0.0354487
$851$	−1.88945e7	−0.894357
$852$	−2.49224e7	−1.17623
$853$	1.50581e7	0.708593	0.354296	−	0.935133i	$-0.384720\pi$
$853$	1.50581e7	0.708593	0.354296	+	0.935133i	$0.384720\pi$
$854$	0	0
$855$	6.20512e7	2.90292
$856$	2.61826e7	1.22132
$857$	2.28736e7	1.06386	0.531928	−	0.846789i	$-0.321468\pi$
$857$	2.28736e7	1.06386	0.531928	+	0.846789i	$0.321468\pi$
$858$	7.44307e6	0.345171
$859$	8.22863e6	0.380491	0.190246	−	0.981737i	$-0.439072\pi$
$859$	8.22863e6	0.380491	0.190246	+	0.981737i	$0.439072\pi$
$860$	2.84532e6	0.131185
$861$	0	0
$862$	−1.90864e6	−0.0874895
$863$	4.09902e6	0.187350	0.0936748	−	0.995603i	$-0.470139\pi$
$863$	4.09902e6	0.187350	0.0936748	+	0.995603i	$0.470139\pi$
$864$	−1.77602e7	−0.809399
$865$	3.87672e7	1.76167
$866$	−1.03867e6	−0.0470633
$867$	−3.49069e7	−1.57711
$868$	0	0
$869$	1.55109e7	0.696769
$870$	1.63463e7	0.732186
$871$	−1.91157e6	−0.0853777
$872$	−1.15236e7	−0.513212
$873$	6.99527e7	3.10648
$874$	−6.50239e6	−0.287935
$875$	0	0
$876$	−2.85143e7	−1.25546
$877$	787778.	0.0345864	0.0172932	−	0.999850i	$-0.494495\pi$
$877$	787778.	0.0345864	0.0172932	+	0.999850i	$0.494495\pi$
$878$	−1.19616e7	−0.523665
$879$	3.75168e7	1.63777
$880$	−1.39616e7	−0.607756
$881$	1.73321e7	0.752336	0.376168	−	0.926551i	$-0.377242\pi$
$881$	1.73321e7	0.752336	0.376168	+	0.926551i	$0.377242\pi$
$882$	0	0
$883$	1.23991e7	0.535164	0.267582	−	0.963535i	$-0.413775\pi$
$883$	1.23991e7	0.535164	0.267582	+	0.963535i	$0.413775\pi$
$884$	2.20147e6	0.0947507
$885$	−7.00035e7	−3.00443
$886$	−7.49641e6	−0.320826
$887$	−1.67555e7	−0.715071	−0.357535	−	0.933900i	$-0.616383\pi$
$887$	−1.67555e7	−0.715071	−0.357535	+	0.933900i	$0.616383\pi$
$888$	−3.78597e7	−1.61118
$889$	0	0
$890$	3.05510e6	0.129286
$891$	1.83776e6	0.0775524
$892$	−2.29136e6	−0.0964233
$893$	2.18238e7	0.915801
$894$	−1.80371e7	−0.754782
$895$	4.57557e7	1.90936
$896$	0	0
$897$	1.96036e7	0.813494
$898$	−198916.	−0.00823150
$899$	2.81382e7	1.16117
$900$	−2.65740e7	−1.09358
$901$	−2.12080e6	−0.0870338
$902$	−5.36330e6	−0.219491
$903$	0	0
$904$	−1.64906e7	−0.671145
$905$	−3.25110e7	−1.31950
$906$	−1.61271e6	−0.0652733
$907$	−3.96543e7	−1.60056	−0.800280	−	0.599627i	$-0.795316\pi$
$907$	−3.96543e7	−1.60056	−0.800280	+	0.599627i	$0.795316\pi$
$908$	−2.11028e7	−0.849426
$909$	2.89233e7	1.16102
$910$	0	0
$911$	2.99138e7	1.19419	0.597097	−	0.802169i	$-0.296321\pi$
$911$	2.99138e7	1.19419	0.597097	+	0.802169i	$0.296321\pi$
$912$	3.56129e7	1.41782
$913$	148981.	0.00591498
$914$	323628.	0.0128139
$915$	−6.66366e7	−2.63124
$916$	−736510.	−0.0290028
$917$	0	0
$918$	1.03334e6	0.0404705
$919$	1.98997e7	0.777245	0.388622	−	0.921397i	$-0.372951\pi$
$919$	1.98997e7	0.777245	0.388622	+	0.921397i	$0.372951\pi$
$920$	1.34532e7	0.524031
$921$	−2.98902e7	−1.16113
$922$	8.98395e6	0.348049
$923$	1.86918e7	0.722183
$924$	0	0
$925$	−3.14613e7	−1.20899
$926$	−7.18763e6	−0.275460
$927$	−4.16861e6	−0.159328
$928$	2.24936e7	0.857412
$929$	1.19663e7	0.454904	0.227452	−	0.973789i	$-0.426961\pi$
$929$	1.19663e7	0.454904	0.227452	+	0.973789i	$0.426961\pi$
$930$	2.41295e7	0.914830
$931$	0	0
$932$	−1.33118e6	−0.0501991
$933$	−3.28424e7	−1.23518
$934$	−4.10446e6	−0.153953
$935$	3.18989e6	0.119329
$936$	2.39838e7	0.894805
$937$	−6.18165e6	−0.230015	−0.115007	−	0.993365i	$-0.536689\pi$
$937$	−6.18165e6	−0.230015	−0.115007	+	0.993365i	$0.536689\pi$
$938$	0	0
$939$	−6.62651e7	−2.45257
$940$	−2.10712e7	−0.777804
$941$	−2.99426e7	−1.10234	−0.551171	−	0.834393i	$-0.685819\pi$
$941$	−2.99426e7	−1.10234	−0.551171	+	0.834393i	$0.685819\pi$
$942$	6.63811e6	0.243735
$943$	−1.41259e7	−0.517293
$944$	−2.45312e7	−0.895959
$945$	0	0
$946$	−770208.	−0.0279821
$947$	−4.84279e7	−1.75477	−0.877386	−	0.479785i	$-0.840714\pi$
$947$	−4.84279e7	−1.75477	−0.877386	+	0.479785i	$0.840714\pi$
$948$	3.82006e7	1.38054
$949$	2.13857e7	0.770830
$950$	−1.08272e7	−0.389230
$951$	−3.99660e7	−1.43298
$952$	0	0
$953$	2.26780e7	0.808860	0.404430	−	0.914569i	$-0.367470\pi$
$953$	2.26780e7	0.808860	0.404430	+	0.914569i	$0.367470\pi$
$954$	−1.07823e7	−0.383566
$955$	2.55689e7	0.907202
$956$	−3.04918e7	−1.07904
$957$	3.09738e7	1.09324
$958$	3.99970e6	0.140803
$959$	0	0
$960$	−2.00079e7	−0.700687
$961$	1.29068e7	0.450827
$962$	1.32509e7	0.461644
$963$	8.31296e7	2.88862
$964$	3.77823e7	1.30947
$965$	−4.61749e7	−1.59620
$966$	0	0
$967$	−3.60431e6	−0.123953	−0.0619764	−	0.998078i	$-0.519740\pi$
$967$	−3.60431e6	−0.123953	−0.0619764	+	0.998078i	$0.519740\pi$
$968$	−9.64740e6	−0.330919
$969$	−8.13668e6	−0.278380
$970$	−2.75184e7	−0.939061
$971$	2.90807e7	0.989821	0.494910	−	0.868944i	$-0.335201\pi$
$971$	2.90807e7	0.989821	0.494910	+	0.868944i	$0.335201\pi$
$972$	2.79811e7	0.949946
$973$	0	0
$974$	−4.34251e6	−0.146671
$975$	3.26421e7	1.09968
$976$	−2.33513e7	−0.784669
$977$	−1.81960e7	−0.609873	−0.304936	−	0.952373i	$-0.598635\pi$
$977$	−1.81960e7	−0.609873	−0.304936	+	0.952373i	$0.598635\pi$
$978$	3.06574e6	0.102492
$979$	5.78896e6	0.193039
$980$	0	0
$981$	−3.65874e7	−1.21383
$982$	−6.09110e6	−0.201566
$983$	−3.32808e6	−0.109853	−0.0549263	−	0.998490i	$-0.517492\pi$
$983$	−3.32808e6	−0.109853	−0.0549263	+	0.998490i	$0.517492\pi$
$984$	−2.83046e7	−0.931901
$985$	1.73470e7	0.569682
$986$	−1.30875e6	−0.0428711
$987$	0	0
$988$	−3.19213e7	−1.04037
$989$	−2.02858e6	−0.0659478
$990$	1.62176e7	0.525895
$991$	5.25420e7	1.69951	0.849753	−	0.527181i	$-0.176751\pi$
$991$	5.25420e7	1.69951	0.849753	+	0.527181i	$0.176751\pi$
$992$	3.32038e7	1.07129
$993$	−2.79046e6	−0.0898056
$994$	0	0
$995$	−3.03601e7	−0.972177
$996$	366912.	0.0117196
$997$	1.22999e6	0.0391889	0.0195945	−	0.999808i	$-0.493762\pi$
$997$	1.22999e6	0.0391889	0.0195945	+	0.999808i	$0.493762\pi$
$998$	1.59845e7	0.508011
$999$	−4.35386e7	−1.38026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.a.c.1.2	yes	2
3.2	odd	2		441.6.a.u.1.1		2
4.3	odd	2		784.6.a.z.1.1		2
7.2	even	3		49.6.c.g.18.1		4
7.3	odd	6		49.6.c.g.30.2		4
7.4	even	3		49.6.c.g.30.1		4
7.5	odd	6		49.6.c.g.18.2		4
7.6	odd	2	inner	49.6.a.c.1.1	✓	2
21.20	even	2		441.6.a.u.1.2		2
28.27	even	2		784.6.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.a.c.1.1	✓	2	7.6	odd	2	inner
49.6.a.c.1.2	yes	2	1.1	even	1	trivial
49.6.c.g.18.1		4	7.2	even	3
49.6.c.g.18.2		4	7.5	odd	6
49.6.c.g.30.1		4	7.4	even	3
49.6.c.g.30.2		4	7.3	odd	6
441.6.a.u.1.1		2	3.2	odd	2
441.6.a.u.1.2		2	21.20	even	2
784.6.a.z.1.1		2	4.3	odd	2
784.6.a.z.1.2		2	28.27	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +24.9800 q^{3} -28.0000 q^{4} +74.9400 q^{5} -49.9600 q^{6} +120.000 q^{8} +381.000 q^{9} -149.880 q^{10} -284.000 q^{11} -699.440 q^{12} +524.580 q^{13} +1872.00 q^{15} +656.000 q^{16} -149.880 q^{17} -762.000 q^{18} +2173.26 q^{19} -2098.32 q^{20} +568.000 q^{22} +1496.00 q^{23} +2997.60 q^{24} +2491.00 q^{25} -1049.16 q^{26} +3447.24 q^{27} -4366.00 q^{29} -3744.00 q^{30} -6444.84 q^{31} -5152.00 q^{32} -7094.32 q^{33} +299.760 q^{34} -10668.0 q^{36} -12630.0 q^{37} -4346.52 q^{38} +13104.0 q^{39} +8992.80 q^{40} -9442.44 q^{41} -1356.00 q^{43} +7952.00 q^{44} +28552.1 q^{45} -2992.00 q^{46} +10042.0 q^{47} +16386.9 q^{48} -4982.00 q^{50} -3744.00 q^{51} -14688.2 q^{52} +14150.0 q^{53} -6894.48 q^{54} -21283.0 q^{55} +54288.0 q^{57} +8732.00 q^{58} -37395.0 q^{59} -52416.0 q^{60} -35596.5 q^{61} +12889.7 q^{62} -10688.0 q^{64} +39312.0 q^{65} +14188.6 q^{66} -3644.00 q^{67} +4196.64 q^{68} +37370.1 q^{69} +35632.0 q^{71} +45720.0 q^{72} +40767.3 q^{73} +25260.0 q^{74} +62225.2 q^{75} -60851.3 q^{76} -26208.0 q^{78} -54616.0 q^{79} +49160.6 q^{80} -6471.00 q^{81} +18884.9 q^{82} -524.580 q^{83} -11232.0 q^{85} +2712.00 q^{86} -109063. q^{87} -34080.0 q^{88} -20383.7 q^{89} -57104.3 q^{90} -41888.0 q^{92} -160992. q^{93} -20083.9 q^{94} +162864. q^{95} -128697. q^{96} +183603. q^{97} -108204. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 56 q^{4} + 240 q^{8} + 762 q^{9} - 568 q^{11} + 3744 q^{15} + 1312 q^{16} - 1524 q^{18} + 1136 q^{22} + 2992 q^{23} + 4982 q^{25} - 8732 q^{29} - 7488 q^{30} - 10304 q^{32} - 21336 q^{36}+ \cdots - 216408 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 56 * q^4 + 240 * q^8 + 762 * q^9 - 568 * q^11 + 3744 * q^15 + 1312 * q^16 - 1524 * q^18 + 1136 * q^22 + 2992 * q^23 + 4982 * q^25 - 8732 * q^29 - 7488 * q^30 - 10304 * q^32 - 21336 * q^36 - 25260 * q^37 + 26208 * q^39 - 2712 * q^43 + 15904 * q^44 - 5984 * q^46 - 9964 * q^50 - 7488 * q^51 + 28300 * q^53 + 108576 * q^57 + 17464 * q^58 - 104832 * q^60 - 21376 * q^64 + 78624 * q^65 - 7288 * q^67 + 71264 * q^71 + 91440 * q^72 + 50520 * q^74 - 52416 * q^78 - 109232 * q^79 - 12942 * q^81 - 22464 * q^85 + 5424 * q^86 - 68160 * q^88 - 83776 * q^92 - 321984 * q^93 + 325728 * q^95 - 216408 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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