LMFDB - Embedded newform 49.4.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,4,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	49.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$2.89109359028$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 35x^{2} + 36x + 194 $ x^4 - 2x^3 - 35x^2 + 36*x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.94534$ 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)

$f(q)$	$=$	$q+4.53113 q^{2} +3.57956 q^{3} +12.5311 q^{4} -13.4791 q^{5} +16.2194 q^{6} +20.5311 q^{8} -14.1868 q^{9} +O(q^{10})$	\(q+4.53113 q^{2} +3.57956 q^{3} +12.5311 q^{4} -13.4791 q^{5} +16.2194 q^{6} +20.5311 q^{8} -14.1868 q^{9} -61.0753 q^{10} +0.813227 q^{11} +44.8559 q^{12} +34.9564 q^{13} -48.2490 q^{15} -7.21984 q^{16} +117.732 q^{17} -64.2821 q^{18} -93.2913 q^{19} -168.908 q^{20} +3.68484 q^{22} +120.249 q^{23} +73.4923 q^{24} +56.6848 q^{25} +158.392 q^{26} -147.430 q^{27} +8.56420 q^{29} -218.623 q^{30} -82.1070 q^{31} -196.963 q^{32} +2.91099 q^{33} +533.458 q^{34} -177.776 q^{36} +28.8132 q^{37} -422.715 q^{38} +125.128 q^{39} -276.740 q^{40} -70.5291 q^{41} +417.179 q^{43} +10.1906 q^{44} +191.224 q^{45} +544.864 q^{46} +338.261 q^{47} -25.8438 q^{48} +256.846 q^{50} +421.428 q^{51} +438.043 q^{52} +149.121 q^{53} -668.026 q^{54} -10.9615 q^{55} -333.942 q^{57} +38.8055 q^{58} -94.1828 q^{59} -604.615 q^{60} -120.525 q^{61} -372.037 q^{62} -834.706 q^{64} -471.179 q^{65} +13.1901 q^{66} -792.366 q^{67} +1475.31 q^{68} +430.438 q^{69} +449.128 q^{71} -291.270 q^{72} -469.420 q^{73} +130.556 q^{74} +202.907 q^{75} -1169.05 q^{76} +566.973 q^{78} -1019.85 q^{79} +97.3166 q^{80} -144.693 q^{81} -319.576 q^{82} -104.253 q^{83} -1586.91 q^{85} +1890.29 q^{86} +30.6560 q^{87} +16.6965 q^{88} -1572.92 q^{89} +866.462 q^{90} +1506.86 q^{92} -293.906 q^{93} +1532.70 q^{94} +1257.48 q^{95} -705.040 q^{96} +550.057 q^{97} -11.5371 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9	$ 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9} + 100 q^{11} - 64 q^{15} - 174 q^{16} - 370 q^{18} - 340 q^{22} + 352 q^{23} - 128 q^{25} + 260 q^{29} - 552 q^{30} - 30 q^{32} - 50 q^{36} + 212 q^{37} + 952 q^{39} + 540 q^{43} + 460 q^{44} + 696 q^{46} + 1366 q^{50} + 428 q^{51} + 16 q^{53} - 1884 q^{57} - 780 q^{58} - 1064 q^{60} - 1678 q^{64} - 756 q^{65} - 1944 q^{67} + 2248 q^{71} + 270 q^{72} - 284 q^{74} - 1344 q^{78} - 1048 q^{79} - 1256 q^{81} - 3284 q^{85} + 4820 q^{86} + 1260 q^{88} + 3512 q^{92} - 5368 q^{93} + 2192 q^{95} + 3340 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9 + 100 * q^11 - 64 * q^15 - 174 * q^16 - 370 * q^18 - 340 * q^22 + 352 * q^23 - 128 * q^25 + 260 * q^29 - 552 * q^30 - 30 * q^32 - 50 * q^36 + 212 * q^37 + 952 * q^39 + 540 * q^43 + 460 * q^44 + 696 * q^46 + 1366 * q^50 + 428 * q^51 + 16 * q^53 - 1884 * q^57 - 780 * q^58 - 1064 * q^60 - 1678 * q^64 - 756 * q^65 - 1944 * q^67 + 2248 * q^71 + 270 * q^72 - 284 * q^74 - 1344 * q^78 - 1048 * q^79 - 1256 * q^81 - 3284 * q^85 + 4820 * q^86 + 1260 * q^88 + 3512 * q^92 - 5368 * q^93 + 2192 * q^95 + 3340 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.53113	1.60200	0.800998	−	0.598667i	$-0.204303\pi$
$2$	4.53113	1.60200	0.800998	+	0.598667i	$0.204303\pi$
$3$	3.57956	0.688886	0.344443	−	0.938807i	$-0.388068\pi$
$3$	3.57956	0.688886	0.344443	+	0.938807i	$0.388068\pi$
$4$	12.5311	1.56639
$5$	−13.4791	−1.20560	−0.602802	−	0.797891i	$-0.705949\pi$
$5$	−13.4791	−1.20560	−0.602802	+	0.797891i	$0.705949\pi$
$6$	16.2194	1.10359
$7$	0	0
$7$	0	0
$8$	20.5311	0.907356
$9$	−14.1868	−0.525436
$10$	−61.0753	−1.93137
$11$	0.813227	0.0222906	0.0111453	−	0.999938i	$-0.496452\pi$
$11$	0.813227	0.0222906	0.0111453	+	0.999938i	$0.496452\pi$
$12$	44.8559	1.07906
$13$	34.9564	0.745781	0.372891	−	0.927875i	$-0.378367\pi$
$13$	34.9564	0.745781	0.372891	+	0.927875i	$0.378367\pi$
$14$	0	0
$15$	−48.2490	−0.830523
$16$	−7.21984	−0.112810
$17$	117.732	1.67966	0.839829	−	0.542851i	$-0.182655\pi$
$17$	117.732	1.67966	0.839829	+	0.542851i	$0.182655\pi$
$18$	−64.2821	−0.841746
$19$	−93.2913	−1.12645	−0.563224	−	0.826304i	$-0.690439\pi$
$19$	−93.2913	−1.12645	−0.563224	+	0.826304i	$0.690439\pi$
$20$	−168.908	−1.88845
$21$	0	0
$22$	3.68484	0.0357095
$23$	120.249	1.09016	0.545079	−	0.838384i	$-0.316499\pi$
$23$	120.249	1.09016	0.545079	+	0.838384i	$0.316499\pi$
$24$	73.4923	0.625065
$25$	56.6848	0.453479
$26$	158.392	1.19474
$27$	−147.430	−1.05085
$28$	0	0
$29$	8.56420	0.0548390	0.0274195	−	0.999624i	$-0.491271\pi$
$29$	8.56420	0.0548390	0.0274195	+	0.999624i	$0.491271\pi$
$30$	−218.623	−1.33049
$31$	−82.1070	−0.475705	−0.237852	−	0.971301i	$-0.576443\pi$
$31$	−82.1070	−0.475705	−0.237852	+	0.971301i	$0.576443\pi$
$32$	−196.963	−1.08808
$33$	2.91099	0.0153557
$34$	533.458	2.69080
$35$	0	0
$36$	−177.776	−0.823038
$37$	28.8132	0.128023	0.0640117	−	0.997949i	$-0.479611\pi$
$37$	28.8132	0.128023	0.0640117	+	0.997949i	$0.479611\pi$
$38$	−422.715	−1.80456
$39$	125.128	0.513758
$40$	−276.740	−1.09391
$41$	−70.5291	−0.268654	−0.134327	−	0.990937i	$-0.542887\pi$
$41$	−70.5291	−0.268654	−0.134327	+	0.990937i	$0.542887\pi$
$42$	0	0
$43$	417.179	1.47952	0.739758	−	0.672873i	$-0.234940\pi$
$43$	417.179	1.47952	0.739758	+	0.672873i	$0.234940\pi$
$44$	10.1906	0.0349159
$45$	191.224	0.633467
$46$	544.864	1.74643
$47$	338.261	1.04980	0.524899	−	0.851165i	$-0.324103\pi$
$47$	338.261	1.04980	0.524899	+	0.851165i	$0.324103\pi$
$48$	−25.8438	−0.0777132
$49$	0	0
$50$	256.846	0.726471
$51$	421.428	1.15709
$52$	438.043	1.16819
$53$	149.121	0.386477	0.193239	−	0.981152i	$-0.438101\pi$
$53$	149.121	0.386477	0.193239	+	0.981152i	$0.438101\pi$
$54$	−668.026	−1.68346
$55$	−10.9615	−0.0268737
$56$	0	0
$57$	−333.942	−0.775994
$58$	38.8055	0.0878519
$59$	−94.1828	−0.207823	−0.103911	−	0.994587i	$-0.533136\pi$
$59$	−94.1828	−0.207823	−0.103911	+	0.994587i	$0.533136\pi$
$60$	−604.615	−1.30092
$61$	−120.525	−0.252977	−0.126488	−	0.991968i	$-0.540371\pi$
$61$	−120.525	−0.252977	−0.126488	+	0.991968i	$0.540371\pi$
$62$	−372.037	−0.762077
$63$	0	0
$64$	−834.706	−1.63029
$65$	−471.179	−0.899116
$66$	13.1901	0.0245998
$67$	−792.366	−1.44482	−0.722410	−	0.691465i	$-0.756965\pi$
$67$	−792.366	−1.44482	−0.722410	+	0.691465i	$0.756965\pi$
$68$	1475.31	2.63100
$69$	430.438	0.750995
$70$	0	0
$71$	449.128	0.750729	0.375364	−	0.926877i	$-0.377518\pi$
$71$	449.128	0.750729	0.375364	+	0.926877i	$0.377518\pi$
$72$	−291.270	−0.476758
$73$	−469.420	−0.752623	−0.376311	−	0.926493i	$-0.622808\pi$
$73$	−469.420	−0.752623	−0.376311	+	0.926493i	$0.622808\pi$
$74$	130.556	0.205093
$75$	202.907	0.312395
$76$	−1169.05	−1.76446
$77$	0	0
$78$	566.973	0.823039
$79$	−1019.85	−1.45243	−0.726217	−	0.687465i	$-0.758723\pi$
$79$	−1019.85	−1.45243	−0.726217	+	0.687465i	$0.758723\pi$
$80$	97.3166	0.136004
$81$	−144.693	−0.198481
$82$	−319.576	−0.430382
$83$	−104.253	−0.137870	−0.0689352	−	0.997621i	$-0.521960\pi$
$83$	−104.253	−0.137870	−0.0689352	+	0.997621i	$0.521960\pi$
$84$	0	0
$85$	−1586.91	−2.02500
$86$	1890.29	2.37018
$87$	30.6560	0.0377778
$88$	16.6965	0.0202256
$89$	−1572.92	−1.87336	−0.936680	−	0.350185i	$-0.886119\pi$
$89$	−1572.92	−1.87336	−0.936680	+	0.350185i	$0.886119\pi$
$90$	866.462	1.01481
$91$	0	0
$92$	1506.86	1.70762
$93$	−293.906	−0.327706
$94$	1532.70	1.68177
$95$	1257.48	1.35805
$96$	−705.040	−0.749561
$97$	550.057	0.575772	0.287886	−	0.957665i	$-0.407048\pi$
$97$	550.057	0.575772	0.287886	+	0.957665i	$0.407048\pi$
$98$	0	0
$99$	−11.5371	−0.0117123
$100$	710.325	0.710325
$101$	−65.7169	−0.0647433	−0.0323717	−	0.999476i	$-0.510306\pi$
$101$	−65.7169	−0.0647433	−0.0323717	+	0.999476i	$0.510306\pi$
$102$	1909.54	1.85366
$103$	1829.35	1.75001	0.875005	−	0.484113i	$-0.160858\pi$
$103$	1829.35	1.75001	0.875005	+	0.484113i	$0.160858\pi$
$104$	717.694	0.676689
$105$	0	0
$106$	675.685	0.619135
$107$	861.377	0.778248	0.389124	−	0.921185i	$-0.372778\pi$
$107$	861.377	0.778248	0.389124	+	0.921185i	$0.372778\pi$
$108$	−1847.47	−1.64604
$109$	−1620.52	−1.42401	−0.712007	−	0.702173i	$-0.752213\pi$
$109$	−1620.52	−1.42401	−0.712007	+	0.702173i	$0.752213\pi$
$110$	−49.6681	−0.0430515
$111$	103.139	0.0881935
$112$	0	0
$113$	380.409	0.316689	0.158344	−	0.987384i	$-0.449384\pi$
$113$	380.409	0.316689	0.158344	+	0.987384i	$0.449384\pi$
$114$	−1513.13	−1.24314
$115$	−1620.84	−1.31430
$116$	107.319	0.0858993
$117$	−495.918	−0.391860
$118$	−426.754	−0.332931
$119$	0	0
$120$	−990.607	−0.753580
$121$	−1330.34	−0.999503
$122$	−546.112	−0.405268
$123$	−252.463	−0.185072
$124$	−1028.89	−0.745140
$125$	920.824	0.658888
$126$	0	0
$127$	958.358	0.669610	0.334805	−	0.942287i	$-0.391329\pi$
$127$	958.358	0.669610	0.334805	+	0.942287i	$0.391329\pi$
$128$	−2206.46	−1.52363
$129$	1493.32	1.01922
$130$	−2134.97	−1.44038
$131$	1152.16	0.768431	0.384216	−	0.923243i	$-0.374472\pi$
$131$	1152.16	0.768431	0.384216	+	0.923243i	$0.374472\pi$
$132$	36.4780	0.0240531
$133$	0	0
$134$	−3590.31	−2.31459
$135$	1987.22	1.26691
$136$	2417.17	1.52405
$137$	357.377	0.222867	0.111434	−	0.993772i	$-0.464456\pi$
$137$	357.377	0.222867	0.111434	+	0.993772i	$0.464456\pi$
$138$	1950.37	1.20309
$139$	−2736.29	−1.66970	−0.834852	−	0.550475i	$-0.814447\pi$
$139$	−2736.29	−1.66970	−0.834852	+	0.550475i	$0.814447\pi$
$140$	0	0
$141$	1210.83	0.723191
$142$	2035.06	1.20266
$143$	28.4275	0.0166239
$144$	102.426	0.0592744
$145$	−115.437	−0.0661141
$146$	−2127.00	−1.20570
$147$	0	0
$148$	361.062	0.200535
$149$	1409.94	0.775212	0.387606	−	0.921825i	$-0.373302\pi$
$149$	1409.94	0.775212	0.387606	+	0.921825i	$0.373302\pi$
$150$	919.396	0.500456
$151$	2352.35	1.26776	0.633879	−	0.773432i	$-0.281462\pi$
$151$	2352.35	1.26776	0.633879	+	0.773432i	$0.281462\pi$
$152$	−1915.38	−1.02209
$153$	−1670.24	−0.882553
$154$	0	0
$155$	1106.72	0.573511
$156$	1568.00	0.804747
$157$	1213.82	0.617029	0.308514	−	0.951220i	$-0.400168\pi$
$157$	1213.82	0.617029	0.308514	+	0.951220i	$0.400168\pi$
$158$	−4621.08	−2.32679
$159$	533.786	0.266239
$160$	2654.88	1.31179
$161$	0	0
$162$	−655.621	−0.317966
$163$	−722.774	−0.347313	−0.173657	−	0.984806i	$-0.555558\pi$
$163$	−722.774	−0.347313	−0.173657	+	0.984806i	$0.555558\pi$
$164$	−883.809	−0.420817
$165$	−39.2374	−0.0185129
$166$	−472.383	−0.220868
$167$	753.016	0.348923	0.174462	−	0.984664i	$-0.444182\pi$
$167$	753.016	0.348923	0.174462	+	0.984664i	$0.444182\pi$
$168$	0	0
$169$	−975.051	−0.443810
$170$	−7190.51	−3.24404
$171$	1323.50	0.591876
$172$	5227.72	2.31750
$173$	−1859.14	−0.817038	−0.408519	−	0.912750i	$-0.633955\pi$
$173$	−1859.14	−0.817038	−0.408519	+	0.912750i	$0.633955\pi$
$174$	138.906	0.0605199
$175$	0	0
$176$	−5.87137	−0.00251461
$177$	−337.133	−0.143166
$178$	−7127.10	−3.00112
$179$	−522.825	−0.218312	−0.109156	−	0.994025i	$-0.534815\pi$
$179$	−522.825	−0.218312	−0.109156	+	0.994025i	$0.534815\pi$
$180$	2396.26	0.992258
$181$	2901.38	1.19148	0.595740	−	0.803177i	$-0.296859\pi$
$181$	2901.38	1.19148	0.595740	+	0.803177i	$0.296859\pi$
$182$	0	0
$183$	−431.424	−0.174272
$184$	2468.85	0.989163
$185$	−388.375	−0.154345
$186$	−1331.73	−0.524984
$187$	95.7427	0.0374407
$188$	4238.79	1.64439
$189$	0	0
$190$	5697.80	2.17559
$191$	2604.35	0.986619	0.493309	−	0.869854i	$-0.335787\pi$
$191$	2604.35	0.986619	0.493309	+	0.869854i	$0.335787\pi$
$192$	−2987.88	−1.12308
$193$	676.245	0.252214	0.126107	−	0.992017i	$-0.459752\pi$
$193$	676.245	0.252214	0.126107	+	0.992017i	$0.459752\pi$
$194$	2492.38	0.922384
$195$	−1686.61	−0.619389
$196$	0	0
$197$	−3685.99	−1.33308	−0.666538	−	0.745471i	$-0.732225\pi$
$197$	−3685.99	−1.33308	−0.666538	+	0.745471i	$0.732225\pi$
$198$	−52.2759	−0.0187631
$199$	−799.801	−0.284907	−0.142453	−	0.989802i	$-0.545499\pi$
$199$	−799.801	−0.284907	−0.142453	+	0.989802i	$0.545499\pi$
$200$	1163.80	0.411467
$201$	−2836.32	−0.995316
$202$	−297.772	−0.103719
$203$	0	0
$204$	5280.97	1.81246
$205$	950.665	0.323890
$206$	8289.01	2.80351
$207$	−1705.95	−0.572809
$208$	−252.380	−0.0841316
$209$	−75.8670	−0.0251092
$210$	0	0
$211$	−667.385	−0.217747	−0.108874	−	0.994056i	$-0.534724\pi$
$211$	−667.385	−0.217747	−0.108874	+	0.994056i	$0.534724\pi$
$212$	1868.65	0.605375
$213$	1607.68	0.517166
$214$	3903.01	1.24675
$215$	−5623.18	−1.78371
$216$	−3026.91	−0.953497
$217$	0	0
$218$	−7342.77	−2.28126
$219$	−1680.32	−0.518471
$220$	−137.360	−0.0420947
$221$	4115.48	1.25266
$222$	467.334	0.141286
$223$	2646.82	0.794818	0.397409	−	0.917642i	$-0.369909\pi$
$223$	2646.82	0.794818	0.397409	+	0.917642i	$0.369909\pi$
$224$	0	0
$225$	−804.175	−0.238274
$226$	1723.68	0.507334
$227$	4121.11	1.20497	0.602485	−	0.798131i	$-0.294178\pi$
$227$	4121.11	1.20497	0.602485	+	0.798131i	$0.294178\pi$
$228$	−4184.67	−1.21551
$229$	4066.92	1.17358	0.586790	−	0.809739i	$-0.300391\pi$
$229$	4066.92	1.17358	0.586790	+	0.809739i	$0.300391\pi$
$230$	−7344.25	−2.10550
$231$	0	0
$232$	175.833	0.0497585
$233$	−3904.67	−1.09787	−0.548934	−	0.835865i	$-0.684966\pi$
$233$	−3904.67	−1.09787	−0.548934	+	0.835865i	$0.684966\pi$
$234$	−2247.07	−0.627759
$235$	−4559.44	−1.26564
$236$	−1180.22	−0.325532
$237$	−3650.62	−1.00056
$238$	0	0
$239$	5425.12	1.46829	0.734146	−	0.678991i	$-0.237583\pi$
$239$	5425.12	1.46829	0.734146	+	0.678991i	$0.237583\pi$
$240$	348.350	0.0936913
$241$	−1602.89	−0.428429	−0.214215	−	0.976787i	$-0.568719\pi$
$241$	−1602.89	−0.428429	−0.214215	+	0.976787i	$0.568719\pi$
$242$	−6027.94	−1.60120
$243$	3462.69	0.914121
$244$	−1510.31	−0.396261
$245$	0	0
$246$	−1143.94	−0.296484
$247$	−3261.13	−0.840084
$248$	−1685.75	−0.431634
$249$	−373.179	−0.0949769
$250$	4172.37	1.05554
$251$	3805.93	0.957085	0.478542	−	0.878064i	$-0.341165\pi$
$251$	3805.93	0.957085	0.478542	+	0.878064i	$0.341165\pi$
$252$	0	0
$253$	97.7897	0.0243003
$254$	4342.44	1.07271
$255$	−5680.45	−1.39499
$256$	−3320.09	−0.810569
$257$	4589.34	1.11391	0.556956	−	0.830542i	$-0.311969\pi$
$257$	4589.34	1.11391	0.556956	+	0.830542i	$0.311969\pi$
$258$	6766.41	1.63278
$259$	0	0
$260$	−5904.41	−1.40837
$261$	−121.498	−0.0288144
$262$	5220.58	1.23102
$263$	877.175	0.205661	0.102831	−	0.994699i	$-0.467210\pi$
$263$	877.175	0.205661	0.102831	+	0.994699i	$0.467210\pi$
$264$	59.7659	0.0139331
$265$	−2010.00	−0.465938
$266$	0	0
$267$	−5630.35	−1.29053
$268$	−9929.24	−2.26315
$269$	−6123.55	−1.38795	−0.693977	−	0.719997i	$-0.744143\pi$
$269$	−6123.55	−1.38795	−0.693977	+	0.719997i	$0.744143\pi$
$270$	9004.36	2.02958
$271$	−3489.76	−0.782243	−0.391122	−	0.920339i	$-0.627913\pi$
$271$	−3489.76	−0.782243	−0.391122	+	0.920339i	$0.627913\pi$
$272$	−850.006	−0.189482
$273$	0	0
$274$	1619.32	0.357032
$275$	46.0976	0.0101083
$276$	5393.88	1.17635
$277$	−4891.70	−1.06106	−0.530530	−	0.847666i	$-0.678007\pi$
$277$	−4891.70	−1.06106	−0.530530	+	0.847666i	$0.678007\pi$
$278$	−12398.5	−2.67486
$279$	1164.83	0.249952
$280$	0	0
$281$	6914.46	1.46791	0.733954	−	0.679199i	$-0.237673\pi$
$281$	6914.46	1.46791	0.733954	+	0.679199i	$0.237673\pi$
$282$	5486.40	1.15855
$283$	3559.85	0.747742	0.373871	−	0.927481i	$-0.378030\pi$
$283$	3559.85	0.747742	0.373871	+	0.927481i	$0.378030\pi$
$284$	5628.09	1.17593
$285$	4501.22	0.935541
$286$	128.809	0.0266315
$287$	0	0
$288$	2794.27	0.571715
$289$	8947.80	1.82125
$290$	−523.061	−0.105914
$291$	1968.96	0.396641
$292$	−5882.36	−1.17890
$293$	3285.11	0.655011	0.327505	−	0.944849i	$-0.393792\pi$
$293$	3285.11	0.655011	0.327505	+	0.944849i	$0.393792\pi$
$294$	0	0
$295$	1269.49	0.250552
$296$	591.568	0.116163
$297$	−119.894	−0.0234242
$298$	6388.61	1.24189
$299$	4203.47	0.813020
$300$	2542.65	0.489333
$301$	0	0
$302$	10658.8	2.03094
$303$	−235.237	−0.0446008
$304$	673.548	0.127075
$305$	1624.56	0.304990
$306$	−7568.05	−1.41385
$307$	−9094.65	−1.69075	−0.845373	−	0.534176i	$-0.820622\pi$
$307$	−9094.65	−1.69075	−0.845373	+	0.534176i	$0.820622\pi$
$308$	0	0
$309$	6548.26	1.20556
$310$	5014.71	0.918762
$311$	−8163.06	−1.48838	−0.744188	−	0.667971i	$-0.767163\pi$
$311$	−8163.06	−1.48838	−0.744188	+	0.667971i	$0.767163\pi$
$312$	2569.03	0.466162
$313$	2979.62	0.538076	0.269038	−	0.963130i	$-0.413294\pi$
$313$	2979.62	0.538076	0.269038	+	0.963130i	$0.413294\pi$
$314$	5499.98	0.988478
$315$	0	0
$316$	−12779.9	−2.27508
$317$	−3888.11	−0.688889	−0.344445	−	0.938807i	$-0.611933\pi$
$317$	−3888.11	−0.688889	−0.344445	+	0.938807i	$0.611933\pi$
$318$	2418.65	0.426513
$319$	6.96463	0.00122240
$320$	11251.0	1.96548
$321$	3083.35	0.536124
$322$	0	0
$323$	−10983.4	−1.89205
$324$	−1813.16	−0.310899
$325$	1981.50	0.338196
$326$	−3274.98	−0.556394
$327$	−5800.74	−0.980983
$328$	−1448.04	−0.243764
$329$	0	0
$330$	−177.790	−0.0296576
$331$	−4893.03	−0.812524	−0.406262	−	0.913757i	$-0.633168\pi$
$331$	−4893.03	−0.812524	−0.406262	+	0.913757i	$0.633168\pi$
$332$	−1306.41	−0.215959
$333$	−408.767	−0.0672681
$334$	3412.01	0.558973
$335$	10680.3	1.74188
$336$	0	0
$337$	−1722.10	−0.278364	−0.139182	−	0.990267i	$-0.544447\pi$
$337$	−1722.10	−0.278364	−0.139182	+	0.990267i	$0.544447\pi$
$338$	−4418.08	−0.710982
$339$	1361.69	0.218163
$340$	−19885.8	−3.17194
$341$	−66.7716	−0.0106038
$342$	5996.96	0.948183
$343$	0	0
$344$	8565.16	1.34245
$345$	−5801.90	−0.905402
$346$	−8423.99	−1.30889
$347$	−238.058	−0.0368289	−0.0184145	−	0.999830i	$-0.505862\pi$
$347$	−238.058	−0.0368289	−0.0184145	+	0.999830i	$0.505862\pi$
$348$	384.155	0.0591749
$349$	−10053.1	−1.54192	−0.770959	−	0.636884i	$-0.780223\pi$
$349$	−10053.1	−1.54192	−0.770959	+	0.636884i	$0.780223\pi$
$350$	0	0
$351$	−5153.63	−0.783706
$352$	−160.176	−0.0242539
$353$	−3470.16	−0.523224	−0.261612	−	0.965173i	$-0.584254\pi$
$353$	−3470.16	−0.523224	−0.261612	+	0.965173i	$0.584254\pi$
$354$	−1527.59	−0.229352
$355$	−6053.82	−0.905081
$356$	−19710.5	−2.93442
$357$	0	0
$358$	−2368.99	−0.349734
$359$	1407.54	0.206928	0.103464	−	0.994633i	$-0.467007\pi$
$359$	1407.54	0.206928	0.103464	+	0.994633i	$0.467007\pi$
$360$	3926.05	0.574781
$361$	1844.27	0.268884
$362$	13146.5	1.90875
$363$	−4762.02	−0.688544
$364$	0	0
$365$	6327.34	0.907364
$366$	−1954.84	−0.279183
$367$	11133.0	1.58348	0.791742	−	0.610855i	$-0.209174\pi$
$367$	11133.0	1.58348	0.791742	+	0.610855i	$0.209174\pi$
$368$	−868.179	−0.122981
$369$	1000.58	0.141160
$370$	−1759.78	−0.247261
$371$	0	0
$372$	−3682.98	−0.513316
$373$	9025.94	1.25294	0.626468	−	0.779447i	$-0.284500\pi$
$373$	9025.94	1.25294	0.626468	+	0.779447i	$0.284500\pi$
$374$	433.823	0.0599798
$375$	3296.14	0.453899
$376$	6944.88	0.952540
$377$	299.373	0.0408979
$378$	0	0
$379$	−5855.75	−0.793640	−0.396820	−	0.917896i	$-0.629886\pi$
$379$	−5855.75	−0.793640	−0.396820	+	0.917896i	$0.629886\pi$
$380$	15757.6	2.12723
$381$	3430.50	0.461285
$382$	11800.6	1.58056
$383$	−7788.03	−1.03903	−0.519517	−	0.854460i	$-0.673888\pi$
$383$	−7788.03	−1.03903	−0.519517	+	0.854460i	$0.673888\pi$
$384$	−7898.14	−1.04961
$385$	0	0
$386$	3064.16	0.404045
$387$	−5918.42	−0.777391
$388$	6892.84	0.901884
$389$	−3814.05	−0.497121	−0.248560	−	0.968616i	$-0.579957\pi$
$389$	−3814.05	−0.497121	−0.248560	+	0.968616i	$0.579957\pi$
$390$	−7642.26	−0.992258
$391$	14157.1	1.83109
$392$	0	0
$393$	4124.21	0.529362
$394$	−16701.7	−2.13558
$395$	13746.6	1.75106
$396$	−144.572	−0.0183461
$397$	−10165.3	−1.28509	−0.642545	−	0.766248i	$-0.722121\pi$
$397$	−10165.3	−1.28509	−0.642545	+	0.766248i	$0.722121\pi$
$398$	−3624.00	−0.456419
$399$	0	0
$400$	−409.255	−0.0511569
$401$	11502.5	1.43244	0.716222	−	0.697873i	$-0.245870\pi$
$401$	11502.5	1.43244	0.716222	+	0.697873i	$0.245870\pi$
$402$	−12851.7	−1.59449
$403$	−2870.16	−0.354772
$404$	−823.507	−0.101413
$405$	1950.32	0.239289
$406$	0	0
$407$	23.4317	0.00285372
$408$	8652.39	1.04990
$409$	3266.27	0.394882	0.197441	−	0.980315i	$-0.436737\pi$
$409$	3266.27	0.394882	0.197441	+	0.980315i	$0.436737\pi$
$410$	4307.59	0.518870
$411$	1279.25	0.153530
$412$	22923.8	2.74120
$413$	0	0
$414$	−7729.86	−0.917637
$415$	1405.23	0.166217
$416$	−6885.12	−0.811468
$417$	−9794.69	−1.15024
$418$	−343.763	−0.0402249
$419$	−6822.93	−0.795518	−0.397759	−	0.917490i	$-0.630212\pi$
$419$	−6822.93	−0.795518	−0.397759	+	0.917490i	$0.630212\pi$
$420$	0	0
$421$	1431.63	0.165733	0.0828665	−	0.996561i	$-0.473592\pi$
$421$	1431.63	0.165733	0.0828665	+	0.996561i	$0.473592\pi$
$422$	−3024.01	−0.348830
$423$	−4798.83	−0.551601
$424$	3061.62	0.350673
$425$	6673.61	0.761689
$426$	7284.61	0.828499
$427$	0	0
$428$	10794.0	1.21904
$429$	101.758	0.0114520
$430$	−25479.3	−2.85750
$431$	15142.2	1.69228	0.846141	−	0.532959i	$-0.178920\pi$
$431$	15142.2	1.69228	0.846141	+	0.532959i	$0.178920\pi$
$432$	1064.42	0.118547
$433$	−5475.65	−0.607721	−0.303860	−	0.952717i	$-0.598276\pi$
$433$	−5475.65	−0.607721	−0.303860	+	0.952717i	$0.598276\pi$
$434$	0	0
$435$	−413.214	−0.0455451
$436$	−20306.9	−2.23056
$437$	−11218.2	−1.22801
$438$	−7613.72	−0.830589
$439$	1780.54	0.193578	0.0967890	−	0.995305i	$-0.469143\pi$
$439$	1780.54	0.193578	0.0967890	+	0.995305i	$0.469143\pi$
$440$	−225.052	−0.0243840
$441$	0	0
$442$	18647.8	2.00675
$443$	−3259.64	−0.349594	−0.174797	−	0.984605i	$-0.555927\pi$
$443$	−3259.64	−0.349594	−0.174797	+	0.984605i	$0.555927\pi$
$444$	1292.44	0.138146
$445$	21201.5	2.25853
$446$	11993.1	1.27330
$447$	5046.95	0.534033
$448$	0	0
$449$	−6826.19	−0.717478	−0.358739	−	0.933438i	$-0.616793\pi$
$449$	−6826.19	−0.717478	−0.358739	+	0.933438i	$0.616793\pi$
$450$	−3643.82	−0.381714
$451$	−57.3562	−0.00598846
$452$	4766.95	0.496059
$453$	8420.37	0.873341
$454$	18673.3	1.93036
$455$	0	0
$456$	−6856.20	−0.704103
$457$	3700.03	0.378731	0.189365	−	0.981907i	$-0.439357\pi$
$457$	3700.03	0.378731	0.189365	+	0.981907i	$0.439357\pi$
$458$	18427.8	1.88007
$459$	−17357.3	−1.76507
$460$	−20311.0	−2.05871
$461$	−9400.80	−0.949759	−0.474880	−	0.880051i	$-0.657508\pi$
$461$	−9400.80	−0.949759	−0.474880	+	0.880051i	$0.657508\pi$
$462$	0	0
$463$	15483.9	1.55420	0.777102	−	0.629374i	$-0.216689\pi$
$463$	15483.9	1.55420	0.777102	+	0.629374i	$0.216689\pi$
$464$	−61.8321	−0.00618639
$465$	3961.58	0.395084
$466$	−17692.6	−1.75878
$467$	−2205.62	−0.218552	−0.109276	−	0.994011i	$-0.534853\pi$
$467$	−2205.62	−0.218552	−0.109276	+	0.994011i	$0.534853\pi$
$468$	−6214.42	−0.613807
$469$	0	0
$470$	−20659.4	−2.02755
$471$	4344.94	0.425062
$472$	−1933.68	−0.188569
$473$	339.261	0.0329794
$474$	−16541.4	−1.60290
$475$	−5288.20	−0.510820
$476$	0	0
$477$	−2115.54	−0.203069
$478$	24581.9	2.35220
$479$	2349.32	0.224098	0.112049	−	0.993703i	$-0.464259\pi$
$479$	2349.32	0.224098	0.112049	+	0.993703i	$0.464259\pi$
$480$	9503.28	0.903673
$481$	1007.21	0.0954775
$482$	−7262.91	−0.686342
$483$	0	0
$484$	−16670.6	−1.56561
$485$	−7414.25	−0.694152
$486$	15689.9	1.46442
$487$	10394.3	0.967167	0.483583	−	0.875298i	$-0.339335\pi$
$487$	10394.3	0.967167	0.483583	+	0.875298i	$0.339335\pi$
$488$	−2474.50	−0.229540
$489$	−2587.21	−0.239259
$490$	0	0
$491$	12586.7	1.15689	0.578444	−	0.815722i	$-0.303660\pi$
$491$	12586.7	1.15689	0.578444	+	0.815722i	$0.303660\pi$
$492$	−3163.65	−0.289895
$493$	1008.28	0.0921108
$494$	−14776.6	−1.34581
$495$	155.509	0.0141204
$496$	592.799	0.0536642
$497$	0	0
$498$	−1690.92	−0.152153
$499$	10627.9	0.953450	0.476725	−	0.879053i	$-0.341824\pi$
$499$	10627.9	0.953450	0.476725	+	0.879053i	$0.341824\pi$
$500$	11539.0	1.03208
$501$	2695.46	0.240368
$502$	17245.2	1.53325
$503$	−6719.02	−0.595599	−0.297800	−	0.954628i	$-0.596253\pi$
$503$	−6719.02	−0.595599	−0.297800	+	0.954628i	$0.596253\pi$
$504$	0	0
$505$	885.802	0.0780548
$506$	443.098	0.0389291
$507$	−3490.25	−0.305735
$508$	12009.3	1.04887
$509$	3904.33	0.339993	0.169997	−	0.985445i	$-0.445624\pi$
$509$	3904.33	0.339993	0.169997	+	0.985445i	$0.445624\pi$
$510$	−25738.9	−2.23478
$511$	0	0
$512$	2607.89	0.225105
$513$	13754.0	1.18373
$514$	20794.9	1.78448
$515$	−24657.9	−2.10982
$516$	18712.9	1.59649
$517$	275.083	0.0234007
$518$	0	0
$519$	−6654.89	−0.562846
$520$	−9673.84	−0.815819
$521$	−15699.7	−1.32018	−0.660092	−	0.751184i	$-0.729483\pi$
$521$	−15699.7	−1.32018	−0.660092	+	0.751184i	$0.729483\pi$
$522$	−550.524	−0.0461605
$523$	10152.1	0.848794	0.424397	−	0.905476i	$-0.360486\pi$
$523$	10152.1	0.848794	0.424397	+	0.905476i	$0.360486\pi$
$524$	14437.8	1.20366
$525$	0	0
$526$	3974.59	0.329469
$527$	−9666.61	−0.799021
$528$	−21.0169	−0.00173228
$529$	2292.83	0.188447
$530$	−9107.59	−0.746431
$531$	1336.15	0.109198
$532$	0	0
$533$	−2465.44	−0.200357
$534$	−25511.9	−2.06743
$535$	−11610.6	−0.938258
$536$	−16268.2	−1.31097
$537$	−1871.48	−0.150392
$538$	−27746.6	−2.22350
$539$	0	0
$540$	24902.1	1.98448
$541$	−19846.6	−1.57722	−0.788608	−	0.614896i	$-0.789198\pi$
$541$	−19846.6	−1.57722	−0.788608	+	0.614896i	$0.789198\pi$
$542$	−15812.6	−1.25315
$543$	10385.7	0.820794
$544$	−23188.8	−1.82760
$545$	21843.0	1.71679
$546$	0	0
$547$	−22798.9	−1.78210	−0.891052	−	0.453901i	$-0.850032\pi$
$547$	−22798.9	−1.78210	−0.891052	+	0.453901i	$0.850032\pi$
$548$	4478.34	0.349097
$549$	1709.85	0.132923
$550$	208.874	0.0161935
$551$	−798.965	−0.0617733
$552$	8837.38	0.681420
$553$	0	0
$554$	−22164.9	−1.69981
$555$	−1390.21	−0.106326
$556$	−34288.8	−2.61541
$557$	−17998.3	−1.36914	−0.684570	−	0.728947i	$-0.740010\pi$
$557$	−17998.3	−1.36914	−0.684570	+	0.728947i	$0.740010\pi$
$558$	5278.01	0.400423
$559$	14583.1	1.10340
$560$	0	0
$561$	342.717	0.0257923
$562$	31330.3	2.35158
$563$	195.636	0.0146449	0.00732246	−	0.999973i	$-0.497669\pi$
$563$	195.636	0.0146449	0.00732246	+	0.999973i	$0.497669\pi$
$564$	15173.0	1.13280
$565$	−5127.55	−0.381801
$566$	16130.1	1.19788
$567$	0	0
$568$	9221.11	0.681178
$569$	−19660.4	−1.44852	−0.724260	−	0.689527i	$-0.757818\pi$
$569$	−19660.4	−1.44852	−0.724260	+	0.689527i	$0.757818\pi$
$570$	20395.6	1.49873
$571$	−15764.5	−1.15538	−0.577691	−	0.816255i	$-0.696046\pi$
$571$	−15764.5	−1.15538	−0.577691	+	0.816255i	$0.696046\pi$
$572$	356.228	0.0260396
$573$	9322.42	0.679668
$574$	0	0
$575$	6816.30	0.494364
$576$	11841.8	0.856611
$577$	22306.4	1.60941	0.804704	−	0.593676i	$-0.202324\pi$
$577$	22306.4	1.60941	0.804704	+	0.593676i	$0.202324\pi$
$578$	40543.6	2.91764
$579$	2420.66	0.173746
$580$	−1446.56	−0.103560
$581$	0	0
$582$	8921.62	0.635418
$583$	121.269	0.00861483
$584$	−9637.72	−0.682897
$585$	6684.51	0.472428
$586$	14885.3	1.04932
$587$	−15953.2	−1.12173	−0.560866	−	0.827906i	$-0.689532\pi$
$587$	−15953.2	−1.12173	−0.560866	+	0.827906i	$0.689532\pi$
$588$	0	0
$589$	7659.87	0.535856
$590$	5752.24	0.401383
$591$	−13194.2	−0.918338
$592$	−208.027	−0.0144423
$593$	−3155.68	−0.218530	−0.109265	−	0.994013i	$-0.534850\pi$
$593$	−3155.68	−0.218530	−0.109265	+	0.994013i	$0.534850\pi$
$594$	−543.257	−0.0375254
$595$	0	0
$596$	17668.1	1.21429
$597$	−2862.93	−0.196268
$598$	19046.5	1.30246
$599$	25456.3	1.73642	0.868212	−	0.496194i	$-0.165270\pi$
$599$	25456.3	1.73642	0.868212	+	0.496194i	$0.165270\pi$
$600$	4165.90	0.283454
$601$	5580.96	0.378789	0.189395	−	0.981901i	$-0.439347\pi$
$601$	5580.96	0.378789	0.189395	+	0.981901i	$0.439347\pi$
$602$	0	0
$603$	11241.1	0.759160
$604$	29477.6	1.98581
$605$	17931.7	1.20500
$606$	−1065.89	−0.0714503
$607$	381.133	0.0254855	0.0127427	−	0.999919i	$-0.495944\pi$
$607$	381.133	0.0254855	0.0127427	+	0.999919i	$0.495944\pi$
$608$	18374.9	1.22566
$609$	0	0
$610$	7361.07	0.488592
$611$	11824.4	0.782919
$612$	−20929.9	−1.38242
$613$	−8235.98	−0.542656	−0.271328	−	0.962487i	$-0.587463\pi$
$613$	−8235.98	−0.542656	−0.271328	+	0.962487i	$0.587463\pi$
$614$	−41209.0	−2.70857
$615$	3402.96	0.223123
$616$	0	0
$617$	−27419.8	−1.78911	−0.894555	−	0.446958i	$-0.852507\pi$
$617$	−27419.8	−1.78911	−0.894555	+	0.446958i	$0.852507\pi$
$618$	29671.0	1.93130
$619$	−16373.4	−1.06317	−0.531585	−	0.847005i	$-0.678403\pi$
$619$	−16373.4	−1.06317	−0.531585	+	0.847005i	$0.678403\pi$
$620$	13868.5	0.898342
$621$	−17728.4	−1.14560
$622$	−36987.9	−2.38437
$623$	0	0
$624$	−903.407	−0.0579571
$625$	−19497.4	−1.24784
$626$	13501.0	0.861996
$627$	−271.570	−0.0172974
$628$	15210.6	0.966508
$629$	3392.24	0.215035
$630$	0	0
$631$	4059.60	0.256118	0.128059	−	0.991767i	$-0.459125\pi$
$631$	4059.60	0.256118	0.128059	+	0.991767i	$0.459125\pi$
$632$	−20938.7	−1.31788
$633$	−2388.94	−0.150003
$634$	−17617.5	−1.10360
$635$	−12917.8	−0.807284
$636$	6688.94	0.417034
$637$	0	0
$638$	31.5577	0.00195827
$639$	−6371.68	−0.394460
$640$	29741.0	1.83690
$641$	−6388.63	−0.393660	−0.196830	−	0.980438i	$-0.563065\pi$
$641$	−6388.63	−0.393660	−0.196830	+	0.980438i	$0.563065\pi$
$642$	13971.1	0.858868
$643$	−18308.0	−1.12286	−0.561428	−	0.827525i	$-0.689748\pi$
$643$	−18308.0	−1.12286	−0.561428	+	0.827525i	$0.689748\pi$
$644$	0	0
$645$	−20128.5	−1.22877
$646$	−49767.0	−3.03105
$647$	3303.90	0.200757	0.100379	−	0.994949i	$-0.467995\pi$
$647$	3303.90	0.200757	0.100379	+	0.994949i	$0.467995\pi$
$648$	−2970.70	−0.180093
$649$	−76.5919	−0.00463251
$650$	8978.42	0.541789
$651$	0	0
$652$	−9057.18	−0.544028
$653$	4371.27	0.261961	0.130981	−	0.991385i	$-0.458187\pi$
$653$	4371.27	0.261961	0.130981	+	0.991385i	$0.458187\pi$
$654$	−26283.9	−1.57153
$655$	−15530.0	−0.926423
$656$	509.209	0.0303068
$657$	6659.55	0.395455
$658$	0	0
$659$	6259.75	0.370023	0.185012	−	0.982736i	$-0.440768\pi$
$659$	6259.75	0.370023	0.185012	+	0.982736i	$0.440768\pi$
$660$	−491.689	−0.0289984
$661$	−14845.7	−0.873574	−0.436787	−	0.899565i	$-0.643884\pi$
$661$	−14845.7	−0.873574	−0.436787	+	0.899565i	$0.643884\pi$
$662$	−22171.0	−1.30166
$663$	14731.6	0.862938
$664$	−2140.43	−0.125098
$665$	0	0
$666$	−1852.17	−0.107763
$667$	1029.84	0.0597832
$668$	9436.14	0.546550
$669$	9474.46	0.547539
$670$	48394.0	2.79048
$671$	−98.0138	−0.00563902
$672$	0	0
$673$	9409.13	0.538923	0.269462	−	0.963011i	$-0.413154\pi$
$673$	9409.13	0.538923	0.269462	+	0.963011i	$0.413154\pi$
$674$	−7803.05	−0.445938
$675$	−8357.07	−0.476539
$676$	−12218.5	−0.695180
$677$	2950.63	0.167507	0.0837533	−	0.996487i	$-0.473309\pi$
$677$	2950.63	0.167507	0.0837533	+	0.996487i	$0.473309\pi$
$678$	6170.01	0.349495
$679$	0	0
$680$	−32581.1	−1.83740
$681$	14751.8	0.830086
$682$	−302.551	−0.0169872
$683$	6280.85	0.351874	0.175937	−	0.984401i	$-0.443704\pi$
$683$	6280.85	0.351874	0.175937	+	0.984401i	$0.443704\pi$
$684$	16585.0	0.927109
$685$	−4817.11	−0.268689
$686$	0	0
$687$	14557.8	0.808463
$688$	−3011.97	−0.166904
$689$	5212.72	0.288228
$690$	−26289.2	−1.45045
$691$	32763.2	1.80372	0.901861	−	0.432027i	$-0.142202\pi$
$691$	32763.2	1.80372	0.901861	+	0.432027i	$0.142202\pi$
$692$	−23297.1	−1.27980
$693$	0	0
$694$	−1078.67	−0.0589998
$695$	36882.5	2.01300
$696$	629.403	0.0342779
$697$	−8303.53	−0.451246
$698$	−45551.9	−2.47015
$699$	−13977.0	−0.756306
$700$	0	0
$701$	−1775.97	−0.0956883	−0.0478442	−	0.998855i	$-0.515235\pi$
$701$	−1775.97	−0.0956883	−0.0478442	+	0.998855i	$0.515235\pi$
$702$	−23351.8	−1.25549
$703$	−2688.02	−0.144212
$704$	−678.805	−0.0363401
$705$	−16320.8	−0.871881
$706$	−15723.7	−0.838203
$707$	0	0
$708$	−4224.65	−0.224254
$709$	8862.43	0.469444	0.234722	−	0.972063i	$-0.424582\pi$
$709$	8862.43	0.469444	0.234722	+	0.972063i	$0.424582\pi$
$710$	−27430.7	−1.44994
$711$	14468.4	0.763162
$712$	−32293.8	−1.69981
$713$	−9873.28	−0.518594
$714$	0	0
$715$	−383.175	−0.0200419
$716$	−6551.59	−0.341961
$717$	19419.5	1.01149
$718$	6377.75	0.331498
$719$	27499.2	1.42635	0.713177	−	0.700984i	$-0.247256\pi$
$719$	27499.2	1.42635	0.713177	+	0.700984i	$0.247256\pi$
$720$	−1380.61	−0.0714614
$721$	0	0
$722$	8356.64	0.430750
$723$	−5737.65	−0.295139
$724$	36357.6	1.86632
$725$	485.460	0.0248683
$726$	−21577.3	−1.10304
$727$	25434.9	1.29756	0.648781	−	0.760975i	$-0.275279\pi$
$727$	25434.9	1.29756	0.648781	+	0.760975i	$0.275279\pi$
$728$	0	0
$729$	16301.6	0.828206
$730$	28670.0	1.45359
$731$	49115.3	2.48508
$732$	−5406.23	−0.272978
$733$	−24155.6	−1.21720	−0.608600	−	0.793477i	$-0.708269\pi$
$733$	−24155.6	−1.21720	−0.608600	+	0.793477i	$0.708269\pi$
$734$	50445.2	2.53674
$735$	0	0
$736$	−23684.6	−1.18618
$737$	−644.373	−0.0322060
$738$	4533.76	0.226138
$739$	27512.9	1.36952	0.684762	−	0.728767i	$-0.259906\pi$
$739$	27512.9	1.36952	0.684762	+	0.728767i	$0.259906\pi$
$740$	−4866.78	−0.241765
$741$	−11673.4	−0.578722
$742$	0	0
$743$	5995.09	0.296014	0.148007	−	0.988986i	$-0.452714\pi$
$743$	5995.09	0.296014	0.148007	+	0.988986i	$0.452714\pi$
$744$	−6034.23	−0.297346
$745$	−19004.6	−0.934598
$746$	40897.7	2.00720
$747$	1479.01	0.0724420
$748$	1199.76	0.0586467
$749$	0	0
$750$	14935.2	0.727144
$751$	1545.09	0.0750747	0.0375373	−	0.999295i	$-0.488049\pi$
$751$	1545.09	0.0750747	0.0375373	+	0.999295i	$0.488049\pi$
$752$	−2442.19	−0.118428
$753$	13623.6	0.659322
$754$	1356.50	0.0655183
$755$	−31707.5	−1.52841
$756$	0	0
$757$	−5157.82	−0.247641	−0.123820	−	0.992305i	$-0.539515\pi$
$757$	−5157.82	−0.247641	−0.123820	+	0.992305i	$0.539515\pi$
$758$	−26533.1	−1.27141
$759$	350.044	0.0167402
$760$	25817.5	1.23223
$761$	−3289.96	−0.156716	−0.0783581	−	0.996925i	$-0.524968\pi$
$761$	−3289.96	−0.156716	−0.0783581	+	0.996925i	$0.524968\pi$
$762$	15544.0	0.738977
$763$	0	0
$764$	32635.4	1.54543
$765$	22513.2	1.06401
$766$	−35288.6	−1.66453
$767$	−3292.29	−0.154990
$768$	−11884.5	−0.558390
$769$	−11146.5	−0.522697	−0.261348	−	0.965245i	$-0.584167\pi$
$769$	−11146.5	−0.522697	−0.261348	+	0.965245i	$0.584167\pi$
$770$	0	0
$771$	16427.8	0.767358
$772$	8474.12	0.395065
$773$	−15830.2	−0.736576	−0.368288	−	0.929712i	$-0.620056\pi$
$773$	−15830.2	−0.736576	−0.368288	+	0.929712i	$0.620056\pi$
$774$	−26817.1	−1.24538
$775$	−4654.22	−0.215722
$776$	11293.3	0.522430
$777$	0	0
$778$	−17282.0	−0.796386
$779$	6579.75	0.302624
$780$	−21135.2	−0.970205
$781$	365.243	0.0167342
$782$	64147.9	2.93341
$783$	−1262.62	−0.0576277
$784$	0	0
$785$	−16361.2	−0.743892
$786$	18687.3	0.848035
$787$	−15163.4	−0.686809	−0.343404	−	0.939188i	$-0.611580\pi$
$787$	−15163.4	−0.686809	−0.343404	+	0.939188i	$0.611580\pi$
$788$	−46189.6	−2.08812
$789$	3139.90	0.141677
$790$	62287.8	2.80519
$791$	0	0
$792$	−236.869	−0.0106272
$793$	−4213.10	−0.188665
$794$	−46060.2	−2.05871
$795$	−7194.93	−0.320978
$796$	−10022.4	−0.446275
$797$	29398.3	1.30658	0.653289	−	0.757109i	$-0.273389\pi$
$797$	29398.3	1.30658	0.653289	+	0.757109i	$0.273389\pi$
$798$	0	0
$799$	39824.1	1.76330
$800$	−11164.8	−0.493420
$801$	22314.6	0.984331
$802$	52119.5	2.29477
$803$	−381.745	−0.0167764
$804$	−35542.3	−1.55905
$805$	0	0
$806$	−13005.1	−0.568343
$807$	−21919.6	−0.956142
$808$	−1349.24	−0.0587453
$809$	−20712.9	−0.900155	−0.450078	−	0.892989i	$-0.648604\pi$
$809$	−20712.9	−0.900155	−0.450078	+	0.892989i	$0.648604\pi$
$810$	8837.15	0.383340
$811$	27369.9	1.18506	0.592532	−	0.805547i	$-0.298128\pi$
$811$	27369.9	1.18506	0.592532	+	0.805547i	$0.298128\pi$
$812$	0	0
$813$	−12491.8	−0.538876
$814$	106.172	0.00457165
$815$	9742.31	0.418722
$816$	−3042.64	−0.130532
$817$	−38919.2	−1.66660
$818$	14799.9	0.632600
$819$	0	0
$820$	11912.9	0.507338
$821$	−35362.7	−1.50325	−0.751623	−	0.659592i	$-0.770729\pi$
$821$	−35362.7	−1.50325	−0.751623	+	0.659592i	$0.770729\pi$
$822$	5796.46	0.245955
$823$	−29190.4	−1.23635	−0.618174	−	0.786042i	$-0.712127\pi$
$823$	−29190.4	−1.23635	−0.618174	+	0.786042i	$0.712127\pi$
$824$	37558.6	1.58788
$825$	165.009	0.00696349
$826$	0	0
$827$	−7302.08	−0.307035	−0.153518	−	0.988146i	$-0.549060\pi$
$827$	−7302.08	−0.307035	−0.153518	+	0.988146i	$0.549060\pi$
$828$	−21377.4	−0.897243
$829$	4250.77	0.178088	0.0890442	−	0.996028i	$-0.471619\pi$
$829$	4250.77	0.178088	0.0890442	+	0.996028i	$0.471619\pi$
$830$	6367.28	0.266279
$831$	−17510.1	−0.730949
$832$	−29178.3	−1.21584
$833$	0	0
$834$	−44381.0	−1.84267
$835$	−10149.9	−0.420663
$836$	−950.699	−0.0393309
$837$	12105.1	0.499895
$838$	−30915.6	−1.27442
$839$	−39527.7	−1.62652	−0.813258	−	0.581903i	$-0.802308\pi$
$839$	−39527.7	−1.62652	−0.813258	+	0.581903i	$0.802308\pi$
$840$	0	0
$841$	−24315.7	−0.996993
$842$	6486.92	0.265504
$843$	24750.7	1.01122
$844$	−8363.09	−0.341078
$845$	13142.8	0.535059
$846$	−21744.1	−0.883663
$847$	0	0
$848$	−1076.63	−0.0435985
$849$	12742.7	0.515109
$850$	30239.0	1.22022
$851$	3464.76	0.139566
$852$	20146.1	0.810085
$853$	31656.1	1.27067	0.635337	−	0.772235i	$-0.280861\pi$
$853$	31656.1	1.27067	0.635337	+	0.772235i	$0.280861\pi$
$854$	0	0
$855$	−17839.6	−0.713568
$856$	17685.1	0.706148
$857$	1193.16	0.0475583	0.0237792	−	0.999717i	$-0.492430\pi$
$857$	1193.16	0.0475583	0.0237792	+	0.999717i	$0.492430\pi$
$858$	461.078	0.0183461
$859$	29060.0	1.15427	0.577134	−	0.816650i	$-0.304171\pi$
$859$	29060.0	1.15427	0.577134	+	0.816650i	$0.304171\pi$
$860$	−70464.8	−2.79399
$861$	0	0
$862$	68611.2	2.71103
$863$	23063.0	0.909702	0.454851	−	0.890567i	$-0.349692\pi$
$863$	23063.0	0.909702	0.454851	+	0.890567i	$0.349692\pi$
$864$	29038.3	1.14341
$865$	25059.4	0.985024
$866$	−24810.9	−0.973566
$867$	32029.2	1.25463
$868$	0	0
$869$	−829.371	−0.0323757
$870$	−1872.33	−0.0729630
$871$	−27698.2	−1.07752
$872$	−33271.1	−1.29209
$873$	−7803.54	−0.302531
$874$	−50831.1	−1.96726
$875$	0	0
$876$	−21056.2	−0.812129
$877$	33871.0	1.30415	0.652077	−	0.758153i	$-0.273898\pi$
$877$	33871.0	1.30415	0.652077	+	0.758153i	$0.273898\pi$
$878$	8067.88	0.310111
$879$	11759.2	0.451228
$880$	79.1405	0.00303162
$881$	−43331.1	−1.65705	−0.828525	−	0.559953i	$-0.810819\pi$
$881$	−43331.1	−1.65705	−0.828525	+	0.559953i	$0.810819\pi$
$882$	0	0
$883$	−40897.3	−1.55867	−0.779334	−	0.626609i	$-0.784442\pi$
$883$	−40897.3	−1.55867	−0.779334	+	0.626609i	$0.784442\pi$
$884$	51571.6	1.96215
$885$	4544.23	0.172602
$886$	−14769.8	−0.560047
$887$	−45065.8	−1.70593	−0.852965	−	0.521968i	$-0.825198\pi$
$887$	−45065.8	−1.70593	−0.852965	+	0.521968i	$0.825198\pi$
$888$	2117.55	0.0800229
$889$	0	0
$890$	96066.5	3.61816
$891$	−117.668	−0.00442427
$892$	33167.7	1.24500
$893$	−31556.8	−1.18254
$894$	22868.4	0.855518
$895$	7047.19	0.263197
$896$	0	0
$897$	15046.6	0.560078
$898$	−30930.3	−1.14940
$899$	−703.180	−0.0260872
$900$	−10077.2	−0.373230
$901$	17556.3	0.649150
$902$	−259.888	−0.00959349
$903$	0	0
$904$	7810.22	0.287350
$905$	−39107.9	−1.43645
$906$	38153.8	1.39909
$907$	25282.5	0.925570	0.462785	−	0.886471i	$-0.346850\pi$
$907$	25282.5	0.925570	0.462785	+	0.886471i	$0.346850\pi$
$908$	51642.2	1.88745
$909$	932.311	0.0340185
$910$	0	0
$911$	−41646.1	−1.51460	−0.757298	−	0.653070i	$-0.773481\pi$
$911$	−41646.1	−1.51460	−0.757298	+	0.653070i	$0.773481\pi$
$912$	2411.01	0.0875399
$913$	−84.7812	−0.00307322
$914$	16765.3	0.606725
$915$	5815.19	0.210103
$916$	50963.1	1.83829
$917$	0	0
$918$	−78648.0	−2.82764
$919$	−26112.5	−0.937292	−0.468646	−	0.883386i	$-0.655258\pi$
$919$	−26112.5	−0.937292	−0.468646	+	0.883386i	$0.655258\pi$
$920$	−33277.7	−1.19254
$921$	−32554.8	−1.16473
$922$	−42596.3	−1.52151
$923$	15699.9	0.559879
$924$	0	0
$925$	1633.27	0.0580559
$926$	70159.4	2.48983
$927$	−25952.6	−0.919519
$928$	−1686.83	−0.0596691
$929$	32357.0	1.14273	0.571366	−	0.820695i	$-0.306414\pi$
$929$	32357.0	1.14273	0.571366	+	0.820695i	$0.306414\pi$
$930$	17950.4	0.632922
$931$	0	0
$932$	−48929.9	−1.71969
$933$	−29220.1	−1.02532
$934$	−9993.95	−0.350120
$935$	−1290.52	−0.0451386
$936$	−10181.8	−0.355557
$937$	32947.0	1.14870	0.574350	−	0.818610i	$-0.305255\pi$
$937$	32947.0	1.14870	0.574350	+	0.818610i	$0.305255\pi$
$938$	0	0
$939$	10665.7	0.370673
$940$	−57134.9	−1.98249
$941$	−501.602	−0.0173770	−0.00868849	−	0.999962i	$-0.502766\pi$
$941$	−501.602	−0.0173770	−0.00868849	+	0.999962i	$0.502766\pi$
$942$	19687.5	0.680948
$943$	−8481.06	−0.292875
$944$	679.984	0.0234445
$945$	0	0
$946$	1537.24	0.0528328
$947$	−6436.90	−0.220878	−0.110439	−	0.993883i	$-0.535226\pi$
$947$	−6436.90	−0.220878	−0.110439	+	0.993883i	$0.535226\pi$
$948$	−45746.4	−1.56727
$949$	−16409.2	−0.561292
$950$	−23961.5	−0.818331
$951$	−13917.7	−0.474566
$952$	0	0
$953$	47511.2	1.61494	0.807470	−	0.589908i	$-0.200836\pi$
$953$	47511.2	1.61494	0.807470	+	0.589908i	$0.200836\pi$
$954$	−9585.79	−0.325316
$955$	−35104.2	−1.18947
$956$	67982.9	2.29992
$957$	24.9303	0.000842092 0
$958$	10645.1	0.359004
$959$	0	0
$960$	40273.8	1.35399
$961$	−23049.4	−0.773705
$962$	4563.78	0.152955
$963$	−12220.2	−0.408919
$964$	−20086.1	−0.671087
$965$	−9115.15	−0.304069
$966$	0	0
$967$	7817.32	0.259967	0.129984	−	0.991516i	$-0.458508\pi$
$967$	7817.32	0.259967	0.129984	+	0.991516i	$0.458508\pi$
$968$	−27313.4	−0.906905
$969$	−39315.6	−1.30340
$970$	−33594.9	−1.11203
$971$	1503.50	0.0496905	0.0248453	−	0.999691i	$-0.492091\pi$
$971$	1503.50	0.0496905	0.0248453	+	0.999691i	$0.492091\pi$
$972$	43391.4	1.43187
$973$	0	0
$974$	47097.9	1.54940
$975$	7092.88	0.232978
$976$	870.168	0.0285383
$977$	33389.1	1.09336	0.546680	−	0.837342i	$-0.315892\pi$
$977$	33389.1	1.09336	0.546680	+	0.837342i	$0.315892\pi$
$978$	−11723.0	−0.383292
$979$	−1279.14	−0.0417584
$980$	0	0
$981$	22989.9	0.748228
$982$	57032.2	1.85333
$983$	−5451.02	−0.176867	−0.0884337	−	0.996082i	$-0.528186\pi$
$983$	−5451.02	−0.176867	−0.0884337	+	0.996082i	$0.528186\pi$
$984$	−5183.35	−0.167926
$985$	49683.7	1.60716
$986$	4568.64	0.147561
$987$	0	0
$988$	−40865.6	−1.31590
$989$	50165.4	1.61291
$990$	704.630	0.0226208
$991$	−46530.0	−1.49150	−0.745750	−	0.666226i	$-0.767908\pi$
$991$	−46530.0	−1.49150	−0.745750	+	0.666226i	$0.767908\pi$
$992$	16172.0	0.517603
$993$	−17514.9	−0.559736
$994$	0	0
$995$	10780.6	0.343484
$996$	−4676.35	−0.148771
$997$	11410.0	0.362444	0.181222	−	0.983442i	$-0.441995\pi$
$997$	11410.0	0.362444	0.181222	+	0.983442i	$0.441995\pi$
$998$	48156.5	1.52742
$999$	−4247.95	−0.134534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.a.e.1.4	yes	4
3.2	odd	2		441.4.a.u.1.2		4
4.3	odd	2		784.4.a.bf.1.2		4
5.4	even	2		1225.4.a.bb.1.1		4
7.2	even	3		49.4.c.e.18.1		8
7.3	odd	6		49.4.c.e.30.2		8
7.4	even	3		49.4.c.e.30.1		8
7.5	odd	6		49.4.c.e.18.2		8
7.6	odd	2	inner	49.4.a.e.1.3	✓	4
21.2	odd	6		441.4.e.y.361.3		8
21.5	even	6		441.4.e.y.361.4		8
21.11	odd	6		441.4.e.y.226.3		8
21.17	even	6		441.4.e.y.226.4		8
21.20	even	2		441.4.a.u.1.1		4
28.27	even	2		784.4.a.bf.1.3		4
35.34	odd	2		1225.4.a.bb.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.a.e.1.3	✓	4	7.6	odd	2	inner
49.4.a.e.1.4	yes	4	1.1	even	1	trivial
49.4.c.e.18.1		8	7.2	even	3
49.4.c.e.18.2		8	7.5	odd	6
49.4.c.e.30.1		8	7.4	even	3
49.4.c.e.30.2		8	7.3	odd	6
441.4.a.u.1.1		4	21.20	even	2
441.4.a.u.1.2		4	3.2	odd	2
441.4.e.y.226.3		8	21.11	odd	6
441.4.e.y.226.4		8	21.17	even	6
441.4.e.y.361.3		8	21.2	odd	6
441.4.e.y.361.4		8	21.5	even	6
784.4.a.bf.1.2		4	4.3	odd	2
784.4.a.bf.1.3		4	28.27	even	2
1225.4.a.bb.1.1		4	5.4	even	2
1225.4.a.bb.1.2		4	35.34	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)

\(f(q)\)	\(=\)	\(q+4.53113 q^{2} +3.57956 q^{3} +12.5311 q^{4} -13.4791 q^{5} +16.2194 q^{6} +20.5311 q^{8} -14.1868 q^{9} +O(q^{10})\)	\(q+4.53113 q^{2} +3.57956 q^{3} +12.5311 q^{4} -13.4791 q^{5} +16.2194 q^{6} +20.5311 q^{8} -14.1868 q^{9} -61.0753 q^{10} +0.813227 q^{11} +44.8559 q^{12} +34.9564 q^{13} -48.2490 q^{15} -7.21984 q^{16} +117.732 q^{17} -64.2821 q^{18} -93.2913 q^{19} -168.908 q^{20} +3.68484 q^{22} +120.249 q^{23} +73.4923 q^{24} +56.6848 q^{25} +158.392 q^{26} -147.430 q^{27} +8.56420 q^{29} -218.623 q^{30} -82.1070 q^{31} -196.963 q^{32} +2.91099 q^{33} +533.458 q^{34} -177.776 q^{36} +28.8132 q^{37} -422.715 q^{38} +125.128 q^{39} -276.740 q^{40} -70.5291 q^{41} +417.179 q^{43} +10.1906 q^{44} +191.224 q^{45} +544.864 q^{46} +338.261 q^{47} -25.8438 q^{48} +256.846 q^{50} +421.428 q^{51} +438.043 q^{52} +149.121 q^{53} -668.026 q^{54} -10.9615 q^{55} -333.942 q^{57} +38.8055 q^{58} -94.1828 q^{59} -604.615 q^{60} -120.525 q^{61} -372.037 q^{62} -834.706 q^{64} -471.179 q^{65} +13.1901 q^{66} -792.366 q^{67} +1475.31 q^{68} +430.438 q^{69} +449.128 q^{71} -291.270 q^{72} -469.420 q^{73} +130.556 q^{74} +202.907 q^{75} -1169.05 q^{76} +566.973 q^{78} -1019.85 q^{79} +97.3166 q^{80} -144.693 q^{81} -319.576 q^{82} -104.253 q^{83} -1586.91 q^{85} +1890.29 q^{86} +30.6560 q^{87} +16.6965 q^{88} -1572.92 q^{89} +866.462 q^{90} +1506.86 q^{92} -293.906 q^{93} +1532.70 q^{94} +1257.48 q^{95} -705.040 q^{96} +550.057 q^{97} -11.5371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9	\( 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9} + 100 q^{11} - 64 q^{15} - 174 q^{16} - 370 q^{18} - 340 q^{22} + 352 q^{23} - 128 q^{25} + 260 q^{29} - 552 q^{30} - 30 q^{32} - 50 q^{36} + 212 q^{37} + 952 q^{39} + 540 q^{43} + 460 q^{44} + 696 q^{46} + 1366 q^{50} + 428 q^{51} + 16 q^{53} - 1884 q^{57} - 780 q^{58} - 1064 q^{60} - 1678 q^{64} - 756 q^{65} - 1944 q^{67} + 2248 q^{71} + 270 q^{72} - 284 q^{74} - 1344 q^{78} - 1048 q^{79} - 1256 q^{81} - 3284 q^{85} + 4820 q^{86} + 1260 q^{88} + 3512 q^{92} - 5368 q^{93} + 2192 q^{95} + 3340 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9 + 100 * q^11 - 64 * q^15 - 174 * q^16 - 370 * q^18 - 340 * q^22 + 352 * q^23 - 128 * q^25 + 260 * q^29 - 552 * q^30 - 30 * q^32 - 50 * q^36 + 212 * q^37 + 952 * q^39 + 540 * q^43 + 460 * q^44 + 696 * q^46 + 1366 * q^50 + 428 * q^51 + 16 * q^53 - 1884 * q^57 - 780 * q^58 - 1064 * q^60 - 1678 * q^64 - 756 * q^65 - 1944 * q^67 + 2248 * q^71 + 270 * q^72 - 284 * q^74 - 1344 * q^78 - 1048 * q^79 - 1256 * q^81 - 3284 * q^85 + 4820 * q^86 + 1260 * q^88 + 3512 * q^92 - 5368 * q^93 + 2192 * q^95 + 3340 * q^99

Introduction

L-functions

Modular forms

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Database

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