LMFDB - Embedded newform 693.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 $ x^8 - 2x^7 - 45x^6 + 77x^5 + 540x^4 - 915x^3 - 1452x^2 + 2660*x - 672
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.62357$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.62357 q^{2} +23.6246 q^{4} -7.12019 q^{5} -7.00000 q^{7} -87.8658 q^{8} +40.0409 q^{10} +11.0000 q^{11} -28.1697 q^{13} +39.3650 q^{14} +305.123 q^{16} -53.4303 q^{17} +154.054 q^{19} -168.211 q^{20} -61.8593 q^{22} +44.4801 q^{23} -74.3030 q^{25} +158.414 q^{26} -165.372 q^{28} +154.435 q^{29} +63.2765 q^{31} -1012.95 q^{32} +300.469 q^{34} +49.8413 q^{35} -137.541 q^{37} -866.336 q^{38} +625.621 q^{40} -442.195 q^{41} -48.2577 q^{43} +259.870 q^{44} -250.137 q^{46} +122.856 q^{47} +49.0000 q^{49} +417.848 q^{50} -665.496 q^{52} +298.159 q^{53} -78.3220 q^{55} +615.060 q^{56} -868.475 q^{58} +694.313 q^{59} -7.85696 q^{61} -355.840 q^{62} +3255.44 q^{64} +200.573 q^{65} +467.586 q^{67} -1262.27 q^{68} -280.286 q^{70} +984.767 q^{71} -128.695 q^{73} +773.474 q^{74} +3639.47 q^{76} -77.0000 q^{77} -1333.44 q^{79} -2172.53 q^{80} +2486.72 q^{82} +984.007 q^{83} +380.434 q^{85} +271.381 q^{86} -966.524 q^{88} -918.790 q^{89} +197.188 q^{91} +1050.82 q^{92} -690.889 q^{94} -1096.90 q^{95} -1539.96 q^{97} -275.555 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26}+ \cdots - 98 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8 - 13 * q^10 + 88 * q^11 - 148 * q^13 + 14 * q^14 + 266 * q^16 - 114 * q^17 + 58 * q^19 - 291 * q^20 - 22 * q^22 - 246 * q^23 + 244 * q^25 - 305 * q^26 - 210 * q^28 + 72 * q^29 + 252 * q^31 - 1272 * q^32 + 630 * q^34 + 70 * q^35 - 80 * q^37 - 1885 * q^38 - 342 * q^40 - 682 * q^41 - 106 * q^43 + 330 * q^44 + 120 * q^46 - 828 * q^47 + 392 * q^49 - 801 * q^50 - 1681 * q^52 - 462 * q^53 - 110 * q^55 + 105 * q^56 - 1087 * q^58 - 626 * q^59 - 854 * q^61 - 1350 * q^62 + 2997 * q^64 - 22 * q^65 + 130 * q^67 - 2202 * q^68 + 91 * q^70 - 326 * q^71 - 390 * q^73 + 359 * q^74 + 2041 * q^76 - 616 * q^77 - 508 * q^79 - 4391 * q^80 + 1528 * q^82 - 1596 * q^83 - 880 * q^85 + 414 * q^86 - 165 * q^88 - 4324 * q^89 + 1036 * q^91 - 2092 * q^92 - 1685 * q^94 - 1076 * q^95 - 964 * q^97 - 98 * q^98

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 $

$2$	−5.62357	−1.98823	−0.994116	−	0.108318i	$-0.965454\pi$
$2$	−5.62357	−1.98823	−0.994116	+	0.108318i	$0.965454\pi$
$3$	0	0
$3$	0	0
$4$	23.6246	2.95307
$5$	−7.12019	−0.636849	−0.318424	−	0.947948i	$-0.603154\pi$
$5$	−7.12019	−0.636849	−0.318424	+	0.947948i	$0.603154\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−87.8658	−3.88316
$9$	0	0
$10$	40.0409	1.26620
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−28.1697	−0.600989	−0.300495	−	0.953783i	$-0.597152\pi$
$13$	−28.1697	−0.600989	−0.300495	+	0.953783i	$0.597152\pi$
$14$	39.3650	0.751481
$15$	0	0
$16$	305.123	4.76755
$17$	−53.4303	−0.762280	−0.381140	−	0.924517i	$-0.624468\pi$
$17$	−53.4303	−0.762280	−0.381140	+	0.924517i	$0.624468\pi$
$18$	0	0
$19$	154.054	1.86013	0.930066	−	0.367391i	$-0.119749\pi$
$19$	154.054	1.86013	0.930066	+	0.367391i	$0.119749\pi$
$20$	−168.211	−1.88066
$21$	0	0
$22$	−61.8593	−0.599475
$23$	44.4801	0.403250	0.201625	−	0.979463i	$-0.435378\pi$
$23$	44.4801	0.403250	0.201625	+	0.979463i	$0.435378\pi$
$24$	0	0
$25$	−74.3030	−0.594424
$26$	158.414	1.19491
$27$	0	0
$28$	−165.372	−1.11616
$29$	154.435	0.988890	0.494445	−	0.869209i	$-0.335371\pi$
$29$	154.435	0.988890	0.494445	+	0.869209i	$0.335371\pi$
$30$	0	0
$31$	63.2765	0.366606	0.183303	−	0.983056i	$-0.441321\pi$
$31$	63.2765	0.366606	0.183303	+	0.983056i	$0.441321\pi$
$32$	−1012.95	−5.59584
$33$	0	0
$34$	300.469	1.51559
$35$	49.8413	0.240706
$36$	0	0
$37$	−137.541	−0.611126	−0.305563	−	0.952172i	$-0.598845\pi$
$37$	−137.541	−0.611126	−0.305563	+	0.952172i	$0.598845\pi$
$38$	−866.336	−3.69838
$39$	0	0
$40$	625.621	2.47298
$41$	−442.195	−1.68437	−0.842186	−	0.539187i	$-0.818732\pi$
$41$	−442.195	−1.68437	−0.842186	+	0.539187i	$0.818732\pi$
$42$	0	0
$43$	−48.2577	−0.171145	−0.0855724	−	0.996332i	$-0.527272\pi$
$43$	−48.2577	−0.171145	−0.0855724	+	0.996332i	$0.527272\pi$
$44$	259.870	0.890384
$45$	0	0
$46$	−250.137	−0.801755
$47$	122.856	0.381285	0.190642	−	0.981660i	$-0.438943\pi$
$47$	122.856	0.381285	0.190642	+	0.981660i	$0.438943\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	417.848	1.18185
$51$	0	0
$52$	−665.496	−1.77476
$53$	298.159	0.772741	0.386370	−	0.922344i	$-0.373729\pi$
$53$	298.159	0.772741	0.386370	+	0.922344i	$0.373729\pi$
$54$	0	0
$55$	−78.3220	−0.192017
$56$	615.060	1.46769
$57$	0	0
$58$	−868.475	−1.96614
$59$	694.313	1.53207	0.766033	−	0.642801i	$-0.222228\pi$
$59$	694.313	1.53207	0.766033	+	0.642801i	$0.222228\pi$
$60$	0	0
$61$	−7.85696	−0.0164915	−0.00824575	−	0.999966i	$-0.502625\pi$
$61$	−7.85696	−0.0164915	−0.00824575	+	0.999966i	$0.502625\pi$
$62$	−355.840	−0.728899
$63$	0	0
$64$	3255.44	6.35828
$65$	200.573	0.382739
$66$	0	0
$67$	467.586	0.852608	0.426304	−	0.904580i	$-0.359815\pi$
$67$	467.586	0.852608	0.426304	+	0.904580i	$0.359815\pi$
$68$	−1262.27	−2.25106
$69$	0	0
$70$	−280.286	−0.478580
$71$	984.767	1.64606	0.823031	−	0.567997i	$-0.192281\pi$
$71$	984.767	1.64606	0.823031	+	0.567997i	$0.192281\pi$
$72$	0	0
$73$	−128.695	−0.206337	−0.103169	−	0.994664i	$-0.532898\pi$
$73$	−128.695	−0.206337	−0.103169	+	0.994664i	$0.532898\pi$
$74$	773.474	1.21506
$75$	0	0
$76$	3639.47	5.49310
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−1333.44	−1.89904	−0.949520	−	0.313706i	$-0.898429\pi$
$79$	−1333.44	−1.89904	−0.949520	+	0.313706i	$0.898429\pi$
$80$	−2172.53	−3.03621
$81$	0	0
$82$	2486.72	3.34892
$83$	984.007	1.30131	0.650655	−	0.759373i	$-0.274494\pi$
$83$	984.007	1.30131	0.650655	+	0.759373i	$0.274494\pi$
$84$	0	0
$85$	380.434	0.485457
$86$	271.381	0.340276
$87$	0	0
$88$	−966.524	−1.17082
$89$	−918.790	−1.09429	−0.547143	−	0.837039i	$-0.684285\pi$
$89$	−918.790	−1.09429	−0.547143	+	0.837039i	$0.684285\pi$
$90$	0	0
$91$	197.188	0.227153
$92$	1050.82	1.19082
$93$	0	0
$94$	−690.889	−0.758083
$95$	−1096.90	−1.18462
$96$	0	0
$97$	−1539.96	−1.61195	−0.805975	−	0.591949i	$-0.798359\pi$
$97$	−1539.96	−1.61195	−0.805975	+	0.591949i	$0.798359\pi$
$98$	−275.555	−0.284033
$99$	0	0
$100$	−1755.37	−1.75537
$101$	−1094.19	−1.07798	−0.538988	−	0.842313i	$-0.681193\pi$
$101$	−1094.19	−1.07798	−0.538988	+	0.842313i	$0.681193\pi$
$102$	0	0
$103$	866.791	0.829199	0.414599	−	0.910004i	$-0.363922\pi$
$103$	866.791	0.829199	0.414599	+	0.910004i	$0.363922\pi$
$104$	2475.15	2.33374
$105$	0	0
$106$	−1676.72	−1.53639
$107$	−1232.70	−1.11373	−0.556865	−	0.830603i	$-0.687996\pi$
$107$	−1232.70	−1.11373	−0.556865	+	0.830603i	$0.687996\pi$
$108$	0	0
$109$	123.514	0.108537	0.0542685	−	0.998526i	$-0.482717\pi$
$109$	123.514	0.108537	0.0542685	+	0.998526i	$0.482717\pi$
$110$	440.450	0.381775
$111$	0	0
$112$	−2135.86	−1.80196
$113$	−30.4112	−0.0253172	−0.0126586	−	0.999920i	$-0.504029\pi$
$113$	−30.4112	−0.0253172	−0.0126586	+	0.999920i	$0.504029\pi$
$114$	0	0
$115$	−316.707	−0.256809
$116$	3648.45	2.92026
$117$	0	0
$118$	−3904.52	−3.04610
$119$	374.012	0.288115
$120$	0	0
$121$	121.000	0.0909091
$122$	44.1842	0.0327889
$123$	0	0
$124$	1494.88	1.08261
$125$	1419.07	1.01541
$126$	0	0
$127$	−701.964	−0.490466	−0.245233	−	0.969464i	$-0.578865\pi$
$127$	−701.964	−0.490466	−0.245233	+	0.969464i	$0.578865\pi$
$128$	−10203.6	−7.04590
$129$	0	0
$130$	−1127.94	−0.760975
$131$	216.835	0.144618	0.0723089	−	0.997382i	$-0.476963\pi$
$131$	216.835	0.144618	0.0723089	+	0.997382i	$0.476963\pi$
$132$	0	0
$133$	−1078.38	−0.703064
$134$	−2629.50	−1.69518
$135$	0	0
$136$	4694.70	2.96005
$137$	−1501.33	−0.936258	−0.468129	−	0.883660i	$-0.655072\pi$
$137$	−1501.33	−0.936258	−0.468129	+	0.883660i	$0.655072\pi$
$138$	0	0
$139$	1552.97	0.947633	0.473816	−	0.880624i	$-0.342876\pi$
$139$	1552.97	0.947633	0.473816	+	0.880624i	$0.342876\pi$
$140$	1177.48	0.710822
$141$	0	0
$142$	−5537.91	−3.27275
$143$	−309.866	−0.181205
$144$	0	0
$145$	−1099.60	−0.629774
$146$	723.727	0.410247
$147$	0	0
$148$	−3249.36	−1.80470
$149$	−318.363	−0.175043	−0.0875213	−	0.996163i	$-0.527895\pi$
$149$	−318.363	−0.175043	−0.0875213	+	0.996163i	$0.527895\pi$
$150$	0	0
$151$	−2841.58	−1.53142	−0.765709	−	0.643187i	$-0.777612\pi$
$151$	−2841.58	−1.53142	−0.765709	+	0.643187i	$0.777612\pi$
$152$	−13536.1	−7.22319
$153$	0	0
$154$	433.015	0.226580
$155$	−450.540	−0.233473
$156$	0	0
$157$	−2339.36	−1.18918	−0.594589	−	0.804030i	$-0.702685\pi$
$157$	−2339.36	−1.18918	−0.594589	+	0.804030i	$0.702685\pi$
$158$	7498.72	3.77573
$159$	0	0
$160$	7212.43	3.56370
$161$	−311.361	−0.152414
$162$	0	0
$163$	2398.25	1.15242	0.576212	−	0.817300i	$-0.304530\pi$
$163$	2398.25	1.15242	0.576212	+	0.817300i	$0.304530\pi$
$164$	−10446.7	−4.97407
$165$	0	0
$166$	−5533.63	−2.58731
$167$	−1855.95	−0.859984	−0.429992	−	0.902833i	$-0.641484\pi$
$167$	−1855.95	−0.859984	−0.429992	+	0.902833i	$0.641484\pi$
$168$	0	0
$169$	−1403.47	−0.638812
$170$	−2139.40	−0.965201
$171$	0	0
$172$	−1140.07	−0.505403
$173$	−992.101	−0.436000	−0.218000	−	0.975949i	$-0.569953\pi$
$173$	−992.101	−0.436000	−0.218000	+	0.975949i	$0.569953\pi$
$174$	0	0
$175$	520.121	0.224671
$176$	3356.35	1.43747
$177$	0	0
$178$	5166.88	2.17570
$179$	−1214.54	−0.507144	−0.253572	−	0.967316i	$-0.581606\pi$
$179$	−1214.54	−0.507144	−0.253572	+	0.967316i	$0.581606\pi$
$180$	0	0
$181$	−3278.81	−1.34648	−0.673238	−	0.739426i	$-0.735097\pi$
$181$	−3278.81	−1.34648	−0.673238	+	0.739426i	$0.735097\pi$
$182$	−1108.90	−0.451632
$183$	0	0
$184$	−3908.28	−1.56588
$185$	979.321	0.389195
$186$	0	0
$187$	−587.733	−0.229836
$188$	2902.42	1.12596
$189$	0	0
$190$	6168.48	2.35531
$191$	1497.12	0.567163	0.283582	−	0.958948i	$-0.408477\pi$
$191$	1497.12	0.567163	0.283582	+	0.958948i	$0.408477\pi$
$192$	0	0
$193$	2555.04	0.952931	0.476465	−	0.879193i	$-0.341918\pi$
$193$	2555.04	0.952931	0.476465	+	0.879193i	$0.341918\pi$
$194$	8660.07	3.20493
$195$	0	0
$196$	1157.60	0.421867
$197$	3509.36	1.26919	0.634597	−	0.772843i	$-0.281166\pi$
$197$	3509.36	1.26919	0.634597	+	0.772843i	$0.281166\pi$
$198$	0	0
$199$	−4744.61	−1.69013	−0.845066	−	0.534662i	$-0.820439\pi$
$199$	−4744.61	−1.69013	−0.845066	+	0.534662i	$0.820439\pi$
$200$	6528.69	2.30824
$201$	0	0
$202$	6153.23	2.14327
$203$	−1081.04	−0.373765
$204$	0	0
$205$	3148.51	1.07269
$206$	−4874.46	−1.64864
$207$	0	0
$208$	−8595.22	−2.86525
$209$	1694.60	0.560851
$210$	0	0
$211$	−1771.64	−0.578032	−0.289016	−	0.957324i	$-0.593328\pi$
$211$	−1771.64	−0.578032	−0.289016	+	0.957324i	$0.593328\pi$
$212$	7043.87	2.28196
$213$	0	0
$214$	6932.15	2.21436
$215$	343.604	0.108993
$216$	0	0
$217$	−442.936	−0.138564
$218$	−694.592	−0.215797
$219$	0	0
$220$	−1850.32	−0.567040
$221$	1505.11	0.458122
$222$	0	0
$223$	4664.90	1.40083	0.700415	−	0.713736i	$-0.252998\pi$
$223$	4664.90	1.40083	0.700415	+	0.713736i	$0.252998\pi$
$224$	7090.68	2.11503
$225$	0	0
$226$	171.020	0.0503365
$227$	−3817.04	−1.11606	−0.558031	−	0.829820i	$-0.688443\pi$
$227$	−3817.04	−1.11606	−0.558031	+	0.829820i	$0.688443\pi$
$228$	0	0
$229$	−3450.88	−0.995810	−0.497905	−	0.867232i	$-0.665897\pi$
$229$	−3450.88	−0.995810	−0.497905	+	0.867232i	$0.665897\pi$
$230$	1781.02	0.510596
$231$	0	0
$232$	−13569.5	−3.84002
$233$	600.780	0.168920	0.0844601	−	0.996427i	$-0.473083\pi$
$233$	600.780	0.168920	0.0844601	+	0.996427i	$0.473083\pi$
$234$	0	0
$235$	−874.757	−0.242821
$236$	16402.8	4.52430
$237$	0	0
$238$	−2103.28	−0.572839
$239$	2809.33	0.760336	0.380168	−	0.924917i	$-0.375866\pi$
$239$	2809.33	0.760336	0.380168	+	0.924917i	$0.375866\pi$
$240$	0	0
$241$	−6213.34	−1.66073	−0.830366	−	0.557218i	$-0.811869\pi$
$241$	−6213.34	−1.66073	−0.830366	+	0.557218i	$0.811869\pi$
$242$	−680.452	−0.180748
$243$	0	0
$244$	−185.617	−0.0487005
$245$	−348.889	−0.0909784
$246$	0	0
$247$	−4339.66	−1.11792
$248$	−5559.84	−1.42359
$249$	0	0
$250$	−7980.26	−2.01886
$251$	−4102.72	−1.03172	−0.515860	−	0.856673i	$-0.672527\pi$
$251$	−4102.72	−1.03172	−0.515860	+	0.856673i	$0.672527\pi$
$252$	0	0
$253$	489.281	0.121584
$254$	3947.54	0.975161
$255$	0	0
$256$	31336.9	7.65061
$257$	−3125.14	−0.758525	−0.379263	−	0.925289i	$-0.623822\pi$
$257$	−3125.14	−0.758525	−0.379263	+	0.925289i	$0.623822\pi$
$258$	0	0
$259$	962.790	0.230984
$260$	4738.45	1.13026
$261$	0	0
$262$	−1219.39	−0.287534
$263$	−5041.12	−1.18193	−0.590967	−	0.806695i	$-0.701254\pi$
$263$	−5041.12	−1.18193	−0.590967	+	0.806695i	$0.701254\pi$
$264$	0	0
$265$	−2122.95	−0.492119
$266$	6064.35	1.39786
$267$	0	0
$268$	11046.5	2.51781
$269$	−3461.31	−0.784535	−0.392267	−	0.919851i	$-0.628309\pi$
$269$	−3461.31	−0.784535	−0.392267	+	0.919851i	$0.628309\pi$
$270$	0	0
$271$	−5994.89	−1.34378	−0.671889	−	0.740652i	$-0.734517\pi$
$271$	−5994.89	−1.34378	−0.671889	+	0.740652i	$0.734517\pi$
$272$	−16302.8	−3.63421
$273$	0	0
$274$	8442.84	1.86150
$275$	−817.333	−0.179225
$276$	0	0
$277$	−6382.09	−1.38434	−0.692171	−	0.721734i	$-0.743345\pi$
$277$	−6382.09	−1.38434	−0.692171	+	0.721734i	$0.743345\pi$
$278$	−8733.22	−1.88411
$279$	0	0
$280$	−4379.34	−0.934700
$281$	−1290.85	−0.274042	−0.137021	−	0.990568i	$-0.543753\pi$
$281$	−1290.85	−0.274042	−0.137021	+	0.990568i	$0.543753\pi$
$282$	0	0
$283$	4551.37	0.956009	0.478005	−	0.878357i	$-0.341360\pi$
$283$	4551.37	0.956009	0.478005	+	0.878357i	$0.341360\pi$
$284$	23264.7	4.86093
$285$	0	0
$286$	1742.56	0.360278
$287$	3095.37	0.636633
$288$	0	0
$289$	−2058.20	−0.418930
$290$	6183.70	1.25214
$291$	0	0
$292$	−3040.37	−0.609329
$293$	5990.36	1.19440	0.597202	−	0.802091i	$-0.296279\pi$
$293$	5990.36	1.19440	0.597202	+	0.802091i	$0.296279\pi$
$294$	0	0
$295$	−4943.64	−0.975694
$296$	12085.2	2.37310
$297$	0	0
$298$	1790.34	0.348025
$299$	−1252.99	−0.242349
$300$	0	0
$301$	337.804	0.0646867
$302$	15979.8	3.04482
$303$	0	0
$304$	47005.6	8.86827
$305$	55.9430	0.0105026
$306$	0	0
$307$	−3165.24	−0.588435	−0.294218	−	0.955738i	$-0.595059\pi$
$307$	−3165.24	−0.588435	−0.294218	+	0.955738i	$0.595059\pi$
$308$	−1819.09	−0.336533
$309$	0	0
$310$	2533.65	0.464198
$311$	−7904.12	−1.44116	−0.720582	−	0.693370i	$-0.756125\pi$
$311$	−7904.12	−1.44116	−0.720582	+	0.693370i	$0.756125\pi$
$312$	0	0
$313$	−906.983	−0.163788	−0.0818941	−	0.996641i	$-0.526097\pi$
$313$	−906.983	−0.163788	−0.0818941	+	0.996641i	$0.526097\pi$
$314$	13155.5	2.36436
$315$	0	0
$316$	−31502.0	−5.60800
$317$	3437.04	0.608970	0.304485	−	0.952517i	$-0.401516\pi$
$317$	3437.04	0.608970	0.304485	+	0.952517i	$0.401516\pi$
$318$	0	0
$319$	1698.78	0.298162
$320$	−23179.3	−4.04926
$321$	0	0
$322$	1750.96	0.303035
$323$	−8231.18	−1.41794
$324$	0	0
$325$	2093.09	0.357242
$326$	−13486.7	−2.29129
$327$	0	0
$328$	38853.8	6.54068
$329$	−859.992	−0.144112
$330$	0	0
$331$	6468.68	1.07417	0.537086	−	0.843528i	$-0.319525\pi$
$331$	6468.68	1.07417	0.537086	+	0.843528i	$0.319525\pi$
$332$	23246.7	3.84286
$333$	0	0
$334$	10437.0	1.70985
$335$	−3329.30	−0.542982
$336$	0	0
$337$	−771.467	−0.124702	−0.0623509	−	0.998054i	$-0.519860\pi$
$337$	−771.467	−0.124702	−0.0623509	+	0.998054i	$0.519860\pi$
$338$	7892.51	1.27011
$339$	0	0
$340$	8987.58	1.43359
$341$	696.042	0.110536
$342$	0	0
$343$	−343.000	−0.0539949
$344$	4240.20	0.664582
$345$	0	0
$346$	5579.15	0.866870
$347$	599.893	0.0928067	0.0464033	−	0.998923i	$-0.485224\pi$
$347$	599.893	0.0928067	0.0464033	+	0.998923i	$0.485224\pi$
$348$	0	0
$349$	4214.06	0.646342	0.323171	−	0.946341i	$-0.395251\pi$
$349$	4214.06	0.646342	0.323171	+	0.946341i	$0.395251\pi$
$350$	−2924.94	−0.446698
$351$	0	0
$352$	−11142.5	−1.68721
$353$	−8652.67	−1.30463	−0.652316	−	0.757947i	$-0.726203\pi$
$353$	−8652.67	−1.30463	−0.652316	+	0.757947i	$0.726203\pi$
$354$	0	0
$355$	−7011.72	−1.04829
$356$	−21706.0	−3.23150
$357$	0	0
$358$	6830.04	1.00832
$359$	−7963.79	−1.17079	−0.585394	−	0.810749i	$-0.699060\pi$
$359$	−7963.79	−1.17079	−0.585394	+	0.810749i	$0.699060\pi$
$360$	0	0
$361$	16873.8	2.46009
$362$	18438.6	2.67711
$363$	0	0
$364$	4658.47	0.670797
$365$	916.334	0.131406
$366$	0	0
$367$	10230.5	1.45511	0.727557	−	0.686047i	$-0.240655\pi$
$367$	10230.5	1.45511	0.727557	+	0.686047i	$0.240655\pi$
$368$	13571.9	1.92251
$369$	0	0
$370$	−5507.28	−0.773810
$371$	−2087.11	−0.292069
$372$	0	0
$373$	−8392.49	−1.16500	−0.582502	−	0.812829i	$-0.697926\pi$
$373$	−8392.49	−1.16500	−0.582502	+	0.812829i	$0.697926\pi$
$374$	3305.16	0.456967
$375$	0	0
$376$	−10794.8	−1.48059
$377$	−4350.38	−0.594313
$378$	0	0
$379$	−668.982	−0.0906684	−0.0453342	−	0.998972i	$-0.514435\pi$
$379$	−668.982	−0.0906684	−0.0453342	+	0.998972i	$0.514435\pi$
$380$	−25913.7	−3.49827
$381$	0	0
$382$	−8419.19	−1.12765
$383$	−7477.42	−0.997594	−0.498797	−	0.866719i	$-0.666225\pi$
$383$	−7477.42	−0.997594	−0.498797	+	0.866719i	$0.666225\pi$
$384$	0	0
$385$	548.254	0.0725756
$386$	−14368.4	−1.89465
$387$	0	0
$388$	−36380.9	−4.76020
$389$	−11824.2	−1.54116	−0.770582	−	0.637341i	$-0.780034\pi$
$389$	−11824.2	−1.54116	−0.770582	+	0.637341i	$0.780034\pi$
$390$	0	0
$391$	−2376.59	−0.307389
$392$	−4305.42	−0.554737
$393$	0	0
$394$	−19735.1	−2.52345
$395$	9494.37	1.20940
$396$	0	0
$397$	2724.20	0.344392	0.172196	−	0.985063i	$-0.444914\pi$
$397$	2724.20	0.344392	0.172196	+	0.985063i	$0.444914\pi$
$398$	26681.6	3.36037
$399$	0	0
$400$	−22671.5	−2.83394
$401$	−1287.20	−0.160298	−0.0801492	−	0.996783i	$-0.525540\pi$
$401$	−1287.20	−0.160298	−0.0801492	+	0.996783i	$0.525540\pi$
$402$	0	0
$403$	−1782.48	−0.220327
$404$	−25849.7	−3.18334
$405$	0	0
$406$	6079.33	0.743133
$407$	−1512.96	−0.184262
$408$	0	0
$409$	−3246.35	−0.392473	−0.196237	−	0.980557i	$-0.562872\pi$
$409$	−3246.35	−0.392473	−0.196237	+	0.980557i	$0.562872\pi$
$410$	−17705.9	−2.13276
$411$	0	0
$412$	20477.5	2.44868
$413$	−4860.19	−0.579067
$414$	0	0
$415$	−7006.31	−0.828738
$416$	28534.6	3.36304
$417$	0	0
$418$	−9529.70	−1.11510
$419$	6853.05	0.799030	0.399515	−	0.916727i	$-0.369179\pi$
$419$	6853.05	0.799030	0.399515	+	0.916727i	$0.369179\pi$
$420$	0	0
$421$	8735.11	1.01122	0.505610	−	0.862762i	$-0.331268\pi$
$421$	8735.11	1.01122	0.505610	+	0.862762i	$0.331268\pi$
$422$	9962.95	1.14926
$423$	0	0
$424$	−26198.0	−3.00067
$425$	3970.03	0.453117
$426$	0	0
$427$	54.9988	0.00623320
$428$	−29121.9	−3.28892
$429$	0	0
$430$	−1932.28	−0.216704
$431$	12397.2	1.38550	0.692750	−	0.721178i	$-0.256399\pi$
$431$	12397.2	1.38550	0.692750	+	0.721178i	$0.256399\pi$
$432$	0	0
$433$	6950.88	0.771451	0.385725	−	0.922614i	$-0.373951\pi$
$433$	6950.88	0.771451	0.385725	+	0.922614i	$0.373951\pi$
$434$	2490.88	0.275498
$435$	0	0
$436$	2917.97	0.320517
$437$	6852.36	0.750098
$438$	0	0
$439$	10205.6	1.10954	0.554769	−	0.832005i	$-0.312807\pi$
$439$	10205.6	1.10954	0.554769	+	0.832005i	$0.312807\pi$
$440$	6881.83	0.745632
$441$	0	0
$442$	−8464.12	−0.910853
$443$	−919.299	−0.0985942	−0.0492971	−	0.998784i	$-0.515698\pi$
$443$	−919.299	−0.0985942	−0.0492971	+	0.998784i	$0.515698\pi$
$444$	0	0
$445$	6541.95	0.696895
$446$	−26233.4	−2.78518
$447$	0	0
$448$	−22788.1	−2.40320
$449$	1588.39	0.166951	0.0834753	−	0.996510i	$-0.473398\pi$
$449$	1588.39	0.166951	0.0834753	+	0.996510i	$0.473398\pi$
$450$	0	0
$451$	−4864.15	−0.507857
$452$	−718.451	−0.0747635
$453$	0	0
$454$	21465.4	2.21899
$455$	−1404.01	−0.144662
$456$	0	0
$457$	−19146.7	−1.95984	−0.979918	−	0.199399i	$-0.936101\pi$
$457$	−19146.7	−1.95984	−0.979918	+	0.199399i	$0.936101\pi$
$458$	19406.3	1.97990
$459$	0	0
$460$	−7482.05	−0.758375
$461$	−4034.54	−0.407608	−0.203804	−	0.979012i	$-0.565331\pi$
$461$	−4034.54	−0.407608	−0.203804	+	0.979012i	$0.565331\pi$
$462$	0	0
$463$	12370.4	1.24169	0.620844	−	0.783934i	$-0.286790\pi$
$463$	12370.4	1.24169	0.620844	+	0.783934i	$0.286790\pi$
$464$	47121.6	4.71458
$465$	0	0
$466$	−3378.53	−0.335852
$467$	−4367.91	−0.432811	−0.216405	−	0.976304i	$-0.569433\pi$
$467$	−4367.91	−0.432811	−0.216405	+	0.976304i	$0.569433\pi$
$468$	0	0
$469$	−3273.10	−0.322255
$470$	4919.26	0.482784
$471$	0	0
$472$	−61006.4	−5.94925
$473$	−530.835	−0.0516021
$474$	0	0
$475$	−11446.7	−1.10571
$476$	8835.87	0.850822
$477$	0	0
$478$	−15798.5	−1.51172
$479$	3679.31	0.350964	0.175482	−	0.984483i	$-0.443852\pi$
$479$	3679.31	0.350964	0.175482	+	0.984483i	$0.443852\pi$
$480$	0	0
$481$	3874.50	0.367280
$482$	34941.2	3.30192
$483$	0	0
$484$	2858.57	0.268461
$485$	10964.8	1.02657
$486$	0	0
$487$	8515.80	0.792378	0.396189	−	0.918169i	$-0.370333\pi$
$487$	8515.80	0.792378	0.396189	+	0.918169i	$0.370333\pi$
$488$	690.358	0.0640390
$489$	0	0
$490$	1962.00	0.180886
$491$	−11839.3	−1.08818	−0.544092	−	0.839025i	$-0.683126\pi$
$491$	−11839.3	−1.08818	−0.544092	+	0.839025i	$0.683126\pi$
$492$	0	0
$493$	−8251.50	−0.753811
$494$	24404.4	2.22269
$495$	0	0
$496$	19307.1	1.74781
$497$	−6893.37	−0.622153
$498$	0	0
$499$	12015.4	1.07792	0.538962	−	0.842330i	$-0.318817\pi$
$499$	12015.4	1.07792	0.538962	+	0.842330i	$0.318817\pi$
$500$	33525.0	2.99857
$501$	0	0
$502$	23071.9	2.05130
$503$	−18060.7	−1.60097	−0.800485	−	0.599352i	$-0.795425\pi$
$503$	−18060.7	−1.60097	−0.800485	+	0.599352i	$0.795425\pi$
$504$	0	0
$505$	7790.81	0.686508
$506$	−2751.51	−0.241738
$507$	0	0
$508$	−16583.6	−1.44838
$509$	16258.9	1.41584	0.707920	−	0.706293i	$-0.249634\pi$
$509$	16258.9	1.41584	0.707920	+	0.706293i	$0.249634\pi$
$510$	0	0
$511$	900.866	0.0779882
$512$	−94596.9	−8.16530
$513$	0	0
$514$	17574.5	1.50812
$515$	−6171.71	−0.528074
$516$	0	0
$517$	1351.42	0.114962
$518$	−5414.32	−0.459250
$519$	0	0
$520$	−17623.5	−1.48624
$521$	−15726.2	−1.32241	−0.661206	−	0.750204i	$-0.729955\pi$
$521$	−15726.2	−1.32241	−0.661206	+	0.750204i	$0.729955\pi$
$522$	0	0
$523$	18910.5	1.58107	0.790534	−	0.612418i	$-0.209803\pi$
$523$	18910.5	1.58107	0.790534	+	0.612418i	$0.209803\pi$
$524$	5122.62	0.427066
$525$	0	0
$526$	28349.1	2.34996
$527$	−3380.88	−0.279457
$528$	0	0
$529$	−10188.5	−0.837390
$530$	11938.5	0.978447
$531$	0	0
$532$	−25476.3	−2.07620
$533$	12456.5	1.01229
$534$	0	0
$535$	8777.02	0.709278
$536$	−41084.8	−3.31081
$537$	0	0
$538$	19464.9	1.55984
$539$	539.000	0.0430730
$540$	0	0
$541$	5082.93	0.403942	0.201971	−	0.979392i	$-0.435265\pi$
$541$	5082.93	0.403942	0.201971	+	0.979392i	$0.435265\pi$
$542$	33712.7	2.67174
$543$	0	0
$544$	54122.5	4.26559
$545$	−879.445	−0.0691216
$546$	0	0
$547$	−4228.16	−0.330500	−0.165250	−	0.986252i	$-0.552843\pi$
$547$	−4228.16	−0.330500	−0.165250	+	0.986252i	$0.552843\pi$
$548$	−35468.3	−2.76484
$549$	0	0
$550$	4596.33	0.356342
$551$	23791.4	1.83947
$552$	0	0
$553$	9334.11	0.717770
$554$	35890.1	2.75239
$555$	0	0
$556$	36688.1	2.79842
$557$	4684.87	0.356381	0.178191	−	0.983996i	$-0.442976\pi$
$557$	4684.87	0.356381	0.178191	+	0.983996i	$0.442976\pi$
$558$	0	0
$559$	1359.40	0.102856
$560$	15207.7	1.14758
$561$	0	0
$562$	7259.20	0.544859
$563$	3563.28	0.266740	0.133370	−	0.991066i	$-0.457420\pi$
$563$	3563.28	0.266740	0.133370	+	0.991066i	$0.457420\pi$
$564$	0	0
$565$	216.533	0.0161232
$566$	−25594.9	−1.90077
$567$	0	0
$568$	−86527.3	−6.39191
$569$	−10462.9	−0.770875	−0.385437	−	0.922734i	$-0.625949\pi$
$569$	−10462.9	−0.770875	−0.385437	+	0.922734i	$0.625949\pi$
$570$	0	0
$571$	12141.9	0.889884	0.444942	−	0.895559i	$-0.353224\pi$
$571$	12141.9	0.889884	0.444942	+	0.895559i	$0.353224\pi$
$572$	−7320.46	−0.535111
$573$	0	0
$574$	−17407.0	−1.26577
$575$	−3305.00	−0.239701
$576$	0	0
$577$	−10570.3	−0.762649	−0.381324	−	0.924441i	$-0.624532\pi$
$577$	−10570.3	−0.762649	−0.381324	+	0.924441i	$0.624532\pi$
$578$	11574.4	0.832930
$579$	0	0
$580$	−25977.7	−1.85976
$581$	−6888.05	−0.491849
$582$	0	0
$583$	3279.75	0.232990
$584$	11307.9	0.801240
$585$	0	0
$586$	−33687.2	−2.37475
$587$	−10148.2	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$587$	−10148.2	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$588$	0	0
$589$	9748.03	0.681936
$590$	27800.9	1.93991
$591$	0	0
$592$	−41967.1	−2.91357
$593$	−8782.28	−0.608170	−0.304085	−	0.952645i	$-0.598351\pi$
$593$	−8782.28	−0.608170	−0.304085	+	0.952645i	$0.598351\pi$
$594$	0	0
$595$	−2663.04	−0.183485
$596$	−7521.19	−0.516913
$597$	0	0
$598$	7046.28	0.481846
$599$	−4017.42	−0.274035	−0.137018	−	0.990569i	$-0.543752\pi$
$599$	−4017.42	−0.274035	−0.137018	+	0.990569i	$0.543752\pi$
$600$	0	0
$601$	−4540.55	−0.308175	−0.154087	−	0.988057i	$-0.549244\pi$
$601$	−4540.55	−0.308175	−0.154087	+	0.988057i	$0.549244\pi$
$602$	−1899.66	−0.128612
$603$	0	0
$604$	−67131.0	−4.52239
$605$	−861.542	−0.0578953
$606$	0	0
$607$	22149.3	1.48107	0.740537	−	0.672015i	$-0.234571\pi$
$607$	22149.3	1.48107	0.740537	+	0.672015i	$0.234571\pi$
$608$	−156050.	−10.4090
$609$	0	0
$610$	−314.600	−0.0208816
$611$	−3460.81	−0.229148
$612$	0	0
$613$	5443.93	0.358692	0.179346	−	0.983786i	$-0.442602\pi$
$613$	5443.93	0.358692	0.179346	+	0.983786i	$0.442602\pi$
$614$	17799.9	1.16995
$615$	0	0
$616$	6765.67	0.442527
$617$	−15820.9	−1.03229	−0.516147	−	0.856500i	$-0.672634\pi$
$617$	−15820.9	−1.03229	−0.516147	+	0.856500i	$0.672634\pi$
$618$	0	0
$619$	−16423.8	−1.06644	−0.533221	−	0.845976i	$-0.679019\pi$
$619$	−16423.8	−1.06644	−0.533221	+	0.845976i	$0.679019\pi$
$620$	−10643.8	−0.689461
$621$	0	0
$622$	44449.4	2.86537
$623$	6431.53	0.413602
$624$	0	0
$625$	−816.200	−0.0522368
$626$	5100.48	0.325649
$627$	0	0
$628$	−55266.2	−3.51172
$629$	7348.88	0.465849
$630$	0	0
$631$	11789.7	0.743803	0.371901	−	0.928272i	$-0.378706\pi$
$631$	11789.7	0.743803	0.371901	+	0.928272i	$0.378706\pi$
$632$	117164.	7.37427
$633$	0	0
$634$	−19328.5	−1.21077
$635$	4998.11	0.312353
$636$	0	0
$637$	−1380.31	−0.0858556
$638$	−9553.23	−0.592815
$639$	0	0
$640$	72651.2	4.48717
$641$	−1837.09	−0.113199	−0.0565995	−	0.998397i	$-0.518026\pi$
$641$	−1837.09	−0.113199	−0.0565995	+	0.998397i	$0.518026\pi$
$642$	0	0
$643$	−764.066	−0.0468613	−0.0234306	−	0.999725i	$-0.507459\pi$
$643$	−764.066	−0.0468613	−0.0234306	+	0.999725i	$0.507459\pi$
$644$	−7355.76	−0.450089
$645$	0	0
$646$	46288.6	2.81920
$647$	7504.93	0.456027	0.228013	−	0.973658i	$-0.426777\pi$
$647$	7504.93	0.456027	0.228013	+	0.973658i	$0.426777\pi$
$648$	0	0
$649$	7637.45	0.461935
$650$	−11770.6	−0.710281
$651$	0	0
$652$	56657.5	3.40319
$653$	28325.2	1.69747	0.848736	−	0.528817i	$-0.177364\pi$
$653$	28325.2	1.69747	0.848736	+	0.528817i	$0.177364\pi$
$654$	0	0
$655$	−1543.90	−0.0920997
$656$	−134924.	−8.03033
$657$	0	0
$658$	4836.22	0.286528
$659$	20098.9	1.18808	0.594038	−	0.804437i	$-0.297533\pi$
$659$	20098.9	1.18808	0.594038	+	0.804437i	$0.297533\pi$
$660$	0	0
$661$	−8508.04	−0.500642	−0.250321	−	0.968163i	$-0.580536\pi$
$661$	−8508.04	−0.500642	−0.250321	+	0.968163i	$0.580536\pi$
$662$	−36377.1	−2.13570
$663$	0	0
$664$	−86460.6	−5.05319
$665$	7678.28	0.447746
$666$	0	0
$667$	6869.28	0.398770
$668$	−43845.9	−2.53959
$669$	0	0
$670$	18722.5	1.07957
$671$	−86.4266	−0.00497237
$672$	0	0
$673$	−2658.98	−0.152297	−0.0761486	−	0.997096i	$-0.524262\pi$
$673$	−2658.98	−0.152297	−0.0761486	+	0.997096i	$0.524262\pi$
$674$	4338.40	0.247936
$675$	0	0
$676$	−33156.3	−1.88646
$677$	−12080.0	−0.685780	−0.342890	−	0.939376i	$-0.611406\pi$
$677$	−12080.0	−0.685780	−0.342890	+	0.939376i	$0.611406\pi$
$678$	0	0
$679$	10779.7	0.609260
$680$	−33427.1	−1.88510
$681$	0	0
$682$	−3914.24	−0.219771
$683$	−35043.6	−1.96326	−0.981629	−	0.190798i	$-0.938893\pi$
$683$	−35043.6	−1.96326	−0.981629	+	0.190798i	$0.938893\pi$
$684$	0	0
$685$	10689.8	0.596255
$686$	1928.88	0.107354
$687$	0	0
$688$	−14724.5	−0.815941
$689$	−8399.04	−0.464409
$690$	0	0
$691$	−17131.1	−0.943123	−0.471561	−	0.881833i	$-0.656309\pi$
$691$	−17131.1	−0.943123	−0.471561	+	0.881833i	$0.656309\pi$
$692$	−23438.0	−1.28754
$693$	0	0
$694$	−3373.54	−0.184521
$695$	−11057.4	−0.603499
$696$	0	0
$697$	23626.6	1.28396
$698$	−23698.0	−1.28508
$699$	0	0
$700$	12287.6	0.663469
$701$	4893.69	0.263669	0.131835	−	0.991272i	$-0.457913\pi$
$701$	4893.69	0.263669	0.131835	+	0.991272i	$0.457913\pi$
$702$	0	0
$703$	−21188.9	−1.13678
$704$	35809.8	1.91709
$705$	0	0
$706$	48658.9	2.59391
$707$	7659.30	0.407437
$708$	0	0
$709$	21196.6	1.12279	0.561393	−	0.827549i	$-0.310266\pi$
$709$	21196.6	1.12279	0.561393	+	0.827549i	$0.310266\pi$
$710$	39430.9	2.08425
$711$	0	0
$712$	80730.2	4.24929
$713$	2814.55	0.147834
$714$	0	0
$715$	2206.31	0.115400
$716$	−28692.9	−1.49763
$717$	0	0
$718$	44784.9	2.32780
$719$	35267.5	1.82929	0.914643	−	0.404262i	$-0.132472\pi$
$719$	35267.5	1.82929	0.914643	+	0.404262i	$0.132472\pi$
$720$	0	0
$721$	−6067.54	−0.313408
$722$	−94890.9	−4.89124
$723$	0	0
$724$	−77460.4	−3.97623
$725$	−11475.0	−0.587820
$726$	0	0
$727$	27284.4	1.39191	0.695957	−	0.718084i	$-0.254981\pi$
$727$	27284.4	1.39191	0.695957	+	0.718084i	$0.254981\pi$
$728$	−17326.1	−0.882069
$729$	0	0
$730$	−5153.07	−0.261265
$731$	2578.42	0.130460
$732$	0	0
$733$	−10568.0	−0.532521	−0.266261	−	0.963901i	$-0.585788\pi$
$733$	−10568.0	−0.532521	−0.266261	+	0.963901i	$0.585788\pi$
$734$	−57531.9	−2.89311
$735$	0	0
$736$	−45056.4	−2.25652
$737$	5143.44	0.257071
$738$	0	0
$739$	37217.1	1.85257	0.926287	−	0.376820i	$-0.122982\pi$
$739$	37217.1	1.85257	0.926287	+	0.376820i	$0.122982\pi$
$740$	23136.0	1.14932
$741$	0	0
$742$	11737.0	0.580700
$743$	18389.8	0.908017	0.454009	−	0.890997i	$-0.349994\pi$
$743$	18389.8	0.908017	0.454009	+	0.890997i	$0.349994\pi$
$744$	0	0
$745$	2266.81	0.111476
$746$	47195.7	2.31630
$747$	0	0
$748$	−13884.9	−0.678722
$749$	8628.87	0.420951
$750$	0	0
$751$	−23345.7	−1.13435	−0.567175	−	0.823597i	$-0.691964\pi$
$751$	−23345.7	−1.13435	−0.567175	+	0.823597i	$0.691964\pi$
$752$	37486.2	1.81779
$753$	0	0
$754$	24464.7	1.18163
$755$	20232.6	0.975282
$756$	0	0
$757$	21304.9	1.02291	0.511454	−	0.859311i	$-0.329107\pi$
$757$	21304.9	1.02291	0.511454	+	0.859311i	$0.329107\pi$
$758$	3762.07	0.180270
$759$	0	0
$760$	96379.7	4.60008
$761$	29063.1	1.38441	0.692205	−	0.721701i	$-0.256639\pi$
$761$	29063.1	1.38441	0.692205	+	0.721701i	$0.256639\pi$
$762$	0	0
$763$	−864.600	−0.0410231
$764$	35368.9	1.67487
$765$	0	0
$766$	42049.8	1.98345
$767$	−19558.6	−0.920756
$768$	0	0
$769$	31525.6	1.47834	0.739168	−	0.673521i	$-0.235219\pi$
$769$	31525.6	1.47834	0.739168	+	0.673521i	$0.235219\pi$
$770$	−3083.15	−0.144297
$771$	0	0
$772$	60361.6	2.81407
$773$	−31560.1	−1.46848	−0.734242	−	0.678887i	$-0.762462\pi$
$773$	−31560.1	−1.46848	−0.734242	+	0.678887i	$0.762462\pi$
$774$	0	0
$775$	−4701.63	−0.217919
$776$	135310.	6.25946
$777$	0	0
$778$	66494.5	3.06419
$779$	−68122.1	−3.13316
$780$	0	0
$781$	10832.4	0.496306
$782$	13364.9	0.611161
$783$	0	0
$784$	14951.0	0.681078
$785$	16656.6	0.757326
$786$	0	0
$787$	−23463.6	−1.06275	−0.531377	−	0.847135i	$-0.678325\pi$
$787$	−23463.6	−1.06275	−0.531377	+	0.847135i	$0.678325\pi$
$788$	82907.0	3.74802
$789$	0	0
$790$	−53392.3	−2.40457
$791$	212.878	0.00956901
$792$	0	0
$793$	221.328	0.00991121
$794$	−15319.7	−0.684732
$795$	0	0
$796$	−112089.	−4.99108
$797$	−15372.1	−0.683195	−0.341597	−	0.939846i	$-0.610968\pi$
$797$	−15372.1	−0.683195	−0.341597	+	0.939846i	$0.610968\pi$
$798$	0	0
$799$	−6564.23	−0.290646
$800$	75265.6	3.32630
$801$	0	0
$802$	7238.66	0.318711
$803$	−1415.65	−0.0622131
$804$	0	0
$805$	2216.95	0.0970647
$806$	10023.9	0.438060
$807$	0	0
$808$	96141.5	4.18595
$809$	36115.6	1.56954	0.784769	−	0.619789i	$-0.212782\pi$
$809$	36115.6	1.56954	0.784769	+	0.619789i	$0.212782\pi$
$810$	0	0
$811$	−11277.4	−0.488291	−0.244146	−	0.969739i	$-0.578508\pi$
$811$	−11277.4	−0.488291	−0.244146	+	0.969739i	$0.578508\pi$
$812$	−25539.2	−1.10376
$813$	0	0
$814$	8508.21	0.366355
$815$	−17076.0	−0.733920
$816$	0	0
$817$	−7434.31	−0.318352
$818$	18256.1	0.780328
$819$	0	0
$820$	74382.2	3.16773
$821$	622.509	0.0264625	0.0132313	−	0.999912i	$-0.495788\pi$
$821$	622.509	0.0264625	0.0132313	+	0.999912i	$0.495788\pi$
$822$	0	0
$823$	−17003.4	−0.720170	−0.360085	−	0.932919i	$-0.617252\pi$
$823$	−17003.4	−0.720170	−0.360085	+	0.932919i	$0.617252\pi$
$824$	−76161.3	−3.21991
$825$	0	0
$826$	27331.6	1.15132
$827$	−15759.0	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$827$	−15759.0	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$828$	0	0
$829$	−26620.5	−1.11528	−0.557640	−	0.830083i	$-0.688293\pi$
$829$	−26620.5	−1.11528	−0.557640	+	0.830083i	$0.688293\pi$
$830$	39400.5	1.64772
$831$	0	0
$832$	−91704.7	−3.82126
$833$	−2618.09	−0.108897
$834$	0	0
$835$	13214.7	0.547680
$836$	40034.2	1.65623
$837$	0	0
$838$	−38538.6	−1.58866
$839$	−21158.9	−0.870664	−0.435332	−	0.900270i	$-0.643369\pi$
$839$	−21158.9	−0.870664	−0.435332	+	0.900270i	$0.643369\pi$
$840$	0	0
$841$	−538.896	−0.0220959
$842$	−49122.5	−2.01054
$843$	0	0
$844$	−41854.2	−1.70697
$845$	9992.96	0.406826
$846$	0	0
$847$	−847.000	−0.0343604
$848$	90975.1	3.68408
$849$	0	0
$850$	−22325.8	−0.900902
$851$	−6117.86	−0.246437
$852$	0	0
$853$	1655.46	0.0664499	0.0332250	−	0.999448i	$-0.489422\pi$
$853$	1655.46	0.0664499	0.0332250	+	0.999448i	$0.489422\pi$
$854$	−309.289	−0.0123931
$855$	0	0
$856$	108312.	4.32479
$857$	41597.7	1.65805	0.829027	−	0.559209i	$-0.188895\pi$
$857$	41597.7	1.65805	0.829027	+	0.559209i	$0.188895\pi$
$858$	0	0
$859$	−18953.4	−0.752832	−0.376416	−	0.926451i	$-0.622844\pi$
$859$	−18953.4	−0.752832	−0.376416	+	0.926451i	$0.622844\pi$
$860$	8117.48	0.321865
$861$	0	0
$862$	−69716.4	−2.75470
$863$	−9701.17	−0.382655	−0.191328	−	0.981526i	$-0.561279\pi$
$863$	−9701.17	−0.382655	−0.191328	+	0.981526i	$0.561279\pi$
$864$	0	0
$865$	7063.95	0.277666
$866$	−39088.8	−1.53382
$867$	0	0
$868$	−10464.2	−0.409190
$869$	−14667.9	−0.572582
$870$	0	0
$871$	−13171.7	−0.512408
$872$	−10852.7	−0.421466
$873$	0	0
$874$	−38534.7	−1.49137
$875$	−9933.52	−0.383788
$876$	0	0
$877$	17359.2	0.668390	0.334195	−	0.942504i	$-0.391536\pi$
$877$	17359.2	0.668390	0.334195	+	0.942504i	$0.391536\pi$
$878$	−57391.9	−2.20602
$879$	0	0
$880$	−23897.9	−0.915451
$881$	47705.2	1.82432	0.912162	−	0.409830i	$-0.134412\pi$
$881$	47705.2	1.82432	0.912162	+	0.409830i	$0.134412\pi$
$882$	0	0
$883$	−49223.3	−1.87598	−0.937992	−	0.346656i	$-0.887317\pi$
$883$	−49223.3	−1.87598	−0.937992	+	0.346656i	$0.887317\pi$
$884$	35557.7	1.35287
$885$	0	0
$886$	5169.75	0.196028
$887$	−14818.3	−0.560934	−0.280467	−	0.959864i	$-0.590489\pi$
$887$	−14818.3	−0.560934	−0.280467	+	0.959864i	$0.590489\pi$
$888$	0	0
$889$	4913.75	0.185379
$890$	−36789.1	−1.38559
$891$	0	0
$892$	110206.	4.13675
$893$	18926.5	0.709240
$894$	0	0
$895$	8647.73	0.322974
$896$	71424.9	2.66310
$897$	0	0
$898$	−8932.43	−0.331937
$899$	9772.09	0.362533
$900$	0	0
$901$	−15930.7	−0.589045
$902$	27353.9	1.00974
$903$	0	0
$904$	2672.10	0.0983107
$905$	23345.7	0.857501
$906$	0	0
$907$	7365.25	0.269635	0.134818	−	0.990870i	$-0.456955\pi$
$907$	7365.25	0.269635	0.134818	+	0.990870i	$0.456955\pi$
$908$	−90176.0	−3.29581
$909$	0	0
$910$	7895.57	0.287621
$911$	−18318.8	−0.666223	−0.333111	−	0.942888i	$-0.608098\pi$
$911$	−18318.8	−0.666223	−0.333111	+	0.942888i	$0.608098\pi$
$912$	0	0
$913$	10824.1	0.392360
$914$	107673.	3.89661
$915$	0	0
$916$	−81525.5	−2.94070
$917$	−1517.84	−0.0546604
$918$	0	0
$919$	−8776.63	−0.315032	−0.157516	−	0.987516i	$-0.550349\pi$
$919$	−8776.63	−0.315032	−0.157516	+	0.987516i	$0.550349\pi$
$920$	27827.7	0.997230
$921$	0	0
$922$	22688.5	0.810419
$923$	−27740.6	−0.989265
$924$	0	0
$925$	10219.7	0.363268
$926$	−69565.9	−2.46876
$927$	0	0
$928$	−156435.	−5.53367
$929$	−23450.2	−0.828176	−0.414088	−	0.910237i	$-0.635899\pi$
$929$	−23450.2	−0.828176	−0.414088	+	0.910237i	$0.635899\pi$
$930$	0	0
$931$	7548.67	0.265733
$932$	14193.2	0.498833
$933$	0	0
$934$	24563.2	0.860529
$935$	4184.77	0.146371
$936$	0	0
$937$	−30682.7	−1.06975	−0.534877	−	0.844930i	$-0.679642\pi$
$937$	−30682.7	−1.06975	−0.534877	+	0.844930i	$0.679642\pi$
$938$	18406.5	0.640719
$939$	0	0
$940$	−20665.7	−0.717066
$941$	−3051.86	−0.105726	−0.0528628	−	0.998602i	$-0.516835\pi$
$941$	−3051.86	−0.105726	−0.0528628	+	0.998602i	$0.516835\pi$
$942$	0	0
$943$	−19668.9	−0.679223
$944$	211851.	7.30420
$945$	0	0
$946$	2985.19	0.102597
$947$	15859.5	0.544209	0.272104	−	0.962268i	$-0.412280\pi$
$947$	15859.5	0.544209	0.272104	+	0.962268i	$0.412280\pi$
$948$	0	0
$949$	3625.30	0.124007
$950$	64371.4	2.19840
$951$	0	0
$952$	−32862.9	−1.11879
$953$	−13989.8	−0.475524	−0.237762	−	0.971323i	$-0.576414\pi$
$953$	−13989.8	−0.475524	−0.237762	+	0.971323i	$0.576414\pi$
$954$	0	0
$955$	−10659.8	−0.361197
$956$	66369.1	2.24532
$957$	0	0
$958$	−20690.8	−0.697798
$959$	10509.3	0.353872
$960$	0	0
$961$	−25787.1	−0.865600
$962$	−21788.5	−0.730239
$963$	0	0
$964$	−146787.	−4.90426
$965$	−18192.3	−0.606873
$966$	0	0
$967$	55411.1	1.84271	0.921355	−	0.388723i	$-0.127084\pi$
$967$	55411.1	1.84271	0.921355	+	0.388723i	$0.127084\pi$
$968$	−10631.8	−0.353014
$969$	0	0
$970$	−61661.3	−2.04106
$971$	9133.50	0.301862	0.150931	−	0.988544i	$-0.451773\pi$
$971$	9133.50	0.301862	0.150931	+	0.988544i	$0.451773\pi$
$972$	0	0
$973$	−10870.8	−0.358171
$974$	−47889.2	−1.57543
$975$	0	0
$976$	−2397.34	−0.0786240
$977$	−45379.9	−1.48601	−0.743005	−	0.669286i	$-0.766600\pi$
$977$	−45379.9	−1.48601	−0.743005	+	0.669286i	$0.766600\pi$
$978$	0	0
$979$	−10106.7	−0.329940
$980$	−8242.35	−0.268665
$981$	0	0
$982$	66579.0	2.16356
$983$	14931.3	0.484469	0.242235	−	0.970218i	$-0.422120\pi$
$983$	14931.3	0.484469	0.242235	+	0.970218i	$0.422120\pi$
$984$	0	0
$985$	−24987.3	−0.808285
$986$	46402.9	1.49875
$987$	0	0
$988$	−102523.	−3.30130
$989$	−2146.51	−0.0690141
$990$	0	0
$991$	33062.1	1.05979	0.529895	−	0.848063i	$-0.322231\pi$
$991$	33062.1	1.05979	0.529895	+	0.848063i	$0.322231\pi$
$992$	−64096.3	−2.05147
$993$	0	0
$994$	38765.4	1.23698
$995$	33782.5	1.07636
$996$	0	0
$997$	535.694	0.0170167	0.00850833	−	0.999964i	$-0.497292\pi$
$997$	535.694	0.0170167	0.00850833	+	0.999964i	$0.497292\pi$
$998$	−67569.5	−2.14316
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.s.1.1	✓	8
3.2	odd	2		693.4.a.t.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.1	✓	8	1.1	even	1	trivial
693.4.a.t.1.8	yes	8	3.2	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-5.62357 q^{2} +23.6246 q^{4} -7.12019 q^{5} -7.00000 q^{7} -87.8658 q^{8} +40.0409 q^{10} +11.0000 q^{11} -28.1697 q^{13} +39.3650 q^{14} +305.123 q^{16} -53.4303 q^{17} +154.054 q^{19} -168.211 q^{20} -61.8593 q^{22} +44.4801 q^{23} -74.3030 q^{25} +158.414 q^{26} -165.372 q^{28} +154.435 q^{29} +63.2765 q^{31} -1012.95 q^{32} +300.469 q^{34} +49.8413 q^{35} -137.541 q^{37} -866.336 q^{38} +625.621 q^{40} -442.195 q^{41} -48.2577 q^{43} +259.870 q^{44} -250.137 q^{46} +122.856 q^{47} +49.0000 q^{49} +417.848 q^{50} -665.496 q^{52} +298.159 q^{53} -78.3220 q^{55} +615.060 q^{56} -868.475 q^{58} +694.313 q^{59} -7.85696 q^{61} -355.840 q^{62} +3255.44 q^{64} +200.573 q^{65} +467.586 q^{67} -1262.27 q^{68} -280.286 q^{70} +984.767 q^{71} -128.695 q^{73} +773.474 q^{74} +3639.47 q^{76} -77.0000 q^{77} -1333.44 q^{79} -2172.53 q^{80} +2486.72 q^{82} +984.007 q^{83} +380.434 q^{85} +271.381 q^{86} -966.524 q^{88} -918.790 q^{89} +197.188 q^{91} +1050.82 q^{92} -690.889 q^{94} -1096.90 q^{95} -1539.96 q^{97} -275.555 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26}+ \cdots - 98 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8 - 13 * q^10 + 88 * q^11 - 148 * q^13 + 14 * q^14 + 266 * q^16 - 114 * q^17 + 58 * q^19 - 291 * q^20 - 22 * q^22 - 246 * q^23 + 244 * q^25 - 305 * q^26 - 210 * q^28 + 72 * q^29 + 252 * q^31 - 1272 * q^32 + 630 * q^34 + 70 * q^35 - 80 * q^37 - 1885 * q^38 - 342 * q^40 - 682 * q^41 - 106 * q^43 + 330 * q^44 + 120 * q^46 - 828 * q^47 + 392 * q^49 - 801 * q^50 - 1681 * q^52 - 462 * q^53 - 110 * q^55 + 105 * q^56 - 1087 * q^58 - 626 * q^59 - 854 * q^61 - 1350 * q^62 + 2997 * q^64 - 22 * q^65 + 130 * q^67 - 2202 * q^68 + 91 * q^70 - 326 * q^71 - 390 * q^73 + 359 * q^74 + 2041 * q^76 - 616 * q^77 - 508 * q^79 - 4391 * q^80 + 1528 * q^82 - 1596 * q^83 - 880 * q^85 + 414 * q^86 - 165 * q^88 - 4324 * q^89 + 1036 * q^91 - 2092 * q^92 - 1685 * q^94 - 1076 * q^95 - 964 * q^97 - 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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