LMFDB - Embedded newform 693.4.a.l.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.20317$ 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)

$f(q)$	$=$	\(q+4.89098 q^{2} +15.9217 q^{4} -15.5514 q^{5} -7.00000 q^{7} +38.7449 q^{8} -76.0614 q^{10} +11.0000 q^{11} +74.3459 q^{13} -34.2369 q^{14} +62.1271 q^{16} +94.0836 q^{17} +135.682 q^{19} -247.604 q^{20} +53.8008 q^{22} -81.1793 q^{23} +116.845 q^{25} +363.625 q^{26} -111.452 q^{28} +53.4259 q^{29} -9.50536 q^{31} -6.09673 q^{32} +460.161 q^{34} +108.860 q^{35} -9.14224 q^{37} +663.620 q^{38} -602.536 q^{40} +339.461 q^{41} +433.078 q^{43} +175.139 q^{44} -397.046 q^{46} +54.4784 q^{47} +49.0000 q^{49} +571.486 q^{50} +1183.71 q^{52} -123.830 q^{53} -171.065 q^{55} -271.215 q^{56} +261.305 q^{58} +534.396 q^{59} -358.624 q^{61} -46.4905 q^{62} -526.836 q^{64} -1156.18 q^{65} -694.318 q^{67} +1497.97 q^{68} +532.430 q^{70} +278.330 q^{71} -886.688 q^{73} -44.7145 q^{74} +2160.29 q^{76} -77.0000 q^{77} -185.631 q^{79} -966.162 q^{80} +1660.30 q^{82} +122.624 q^{83} -1463.13 q^{85} +2118.18 q^{86} +426.194 q^{88} +847.086 q^{89} -520.422 q^{91} -1292.51 q^{92} +266.453 q^{94} -2110.04 q^{95} +1002.49 q^{97} +239.658 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 26 * q^4 - 10 * q^5 - 28 * q^7 + 18 * q^8 - 2 * q^10 + 44 * q^11 + 58 * q^13 - 14 * q^14 + 2 * q^16 - 4 * q^17 + 258 * q^19 - 182 * q^20 + 22 * q^22 - 8 * q^23 + 80 * q^25 + 482 * q^26 - 182 * q^28 + 396 * q^29 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 + 84 * q^37 + 942 * q^38 - 1026 * q^40 - 52 * q^41 + 408 * q^43 + 286 * q^44 + 368 * q^46 - 8 * q^47 + 196 * q^49 + 1642 * q^50 + 2030 * q^52 - 624 * q^53 - 110 * q^55 - 126 * q^56 + 864 * q^58 + 238 * q^59 - 162 * q^61 - 688 * q^62 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 14 * q^70 - 1788 * q^71 + 1456 * q^73 - 996 * q^74 + 3042 * q^76 - 308 * q^77 - 1324 * q^79 - 2342 * q^80 + 1984 * q^82 - 450 * q^83 - 1736 * q^85 + 4380 * q^86 + 198 * q^88 + 3072 * q^89 - 406 * q^91 - 544 * q^92 - 1696 * q^94 - 24 * q^95 - 652 * 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 $

$2$	4.89098	1.72922	0.864612	−	0.502441i	$-0.167564\pi$
$2$	4.89098	1.72922	0.864612	+	0.502441i	$0.167564\pi$
$3$	0	0
$3$	0	0
$4$	15.9217	1.99021
$5$	−15.5514	−1.39096	−0.695478	−	0.718547i	$-0.744807\pi$
$5$	−15.5514	−1.39096	−0.695478	+	0.718547i	$0.744807\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	38.7449	1.71230
$9$	0	0
$10$	−76.0614	−2.40527
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	74.3459	1.58614	0.793071	−	0.609129i	$-0.208481\pi$
$13$	74.3459	1.58614	0.793071	+	0.609129i	$0.208481\pi$
$14$	−34.2369	−0.653585
$15$	0	0
$16$	62.1271	0.970737
$17$	94.0836	1.34227	0.671136	−	0.741334i	$-0.265807\pi$
$17$	94.0836	1.34227	0.671136	+	0.741334i	$0.265807\pi$
$18$	0	0
$19$	135.682	1.63830	0.819149	−	0.573581i	$-0.194446\pi$
$19$	135.682	1.63830	0.819149	+	0.573581i	$0.194446\pi$
$20$	−247.604	−2.76830
$21$	0	0
$22$	53.8008	0.521380
$23$	−81.1793	−0.735959	−0.367979	−	0.929834i	$-0.619950\pi$
$23$	−81.1793	−0.735959	−0.367979	+	0.929834i	$0.619950\pi$
$24$	0	0
$25$	116.845	0.934758
$26$	363.625	2.74279
$27$	0	0
$28$	−111.452	−0.752230
$29$	53.4259	0.342102	0.171051	−	0.985262i	$-0.445284\pi$
$29$	53.4259	0.342102	0.171051	+	0.985262i	$0.445284\pi$
$30$	0	0
$31$	−9.50536	−0.0550714	−0.0275357	−	0.999621i	$-0.508766\pi$
$31$	−9.50536	−0.0550714	−0.0275357	+	0.999621i	$0.508766\pi$
$32$	−6.09673	−0.0336800
$33$	0	0
$34$	460.161	2.32109
$35$	108.860	0.525732
$36$	0	0
$37$	−9.14224	−0.0406209	−0.0203105	−	0.999794i	$-0.506465\pi$
$37$	−9.14224	−0.0406209	−0.0203105	+	0.999794i	$0.506465\pi$
$38$	663.620	2.83298
$39$	0	0
$40$	−602.536	−2.38173
$41$	339.461	1.29305	0.646523	−	0.762894i	$-0.276222\pi$
$41$	339.461	1.29305	0.646523	+	0.762894i	$0.276222\pi$
$42$	0	0
$43$	433.078	1.53590	0.767950	−	0.640509i	$-0.221277\pi$
$43$	433.078	1.53590	0.767950	+	0.640509i	$0.221277\pi$
$44$	175.139	0.600072
$45$	0	0
$46$	−397.046	−1.27264
$47$	54.4784	0.169074	0.0845371	−	0.996420i	$-0.473059\pi$
$47$	54.4784	0.169074	0.0845371	+	0.996420i	$0.473059\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	571.486	1.61641
$51$	0	0
$52$	1183.71	3.15676
$53$	−123.830	−0.320932	−0.160466	−	0.987041i	$-0.551300\pi$
$53$	−123.830	−0.320932	−0.160466	+	0.987041i	$0.551300\pi$
$54$	0	0
$55$	−171.065	−0.419389
$56$	−271.215	−0.647189
$57$	0	0
$58$	261.305	0.591570
$59$	534.396	1.17919	0.589596	−	0.807698i	$-0.299287\pi$
$59$	534.396	1.17919	0.589596	+	0.807698i	$0.299287\pi$
$60$	0	0
$61$	−358.624	−0.752740	−0.376370	−	0.926469i	$-0.622828\pi$
$61$	−358.624	−0.752740	−0.376370	+	0.926469i	$0.622828\pi$
$62$	−46.4905	−0.0952307
$63$	0	0
$64$	−526.836	−1.02898
$65$	−1156.18	−2.20625
$66$	0	0
$67$	−694.318	−1.26604	−0.633019	−	0.774137i	$-0.718184\pi$
$67$	−694.318	−1.26604	−0.633019	+	0.774137i	$0.718184\pi$
$68$	1497.97	2.67141
$69$	0	0
$70$	532.430	0.909108
$71$	278.330	0.465236	0.232618	−	0.972568i	$-0.425271\pi$
$71$	278.330	0.465236	0.232618	+	0.972568i	$0.425271\pi$
$72$	0	0
$73$	−886.688	−1.42163	−0.710815	−	0.703379i	$-0.751674\pi$
$73$	−886.688	−1.42163	−0.710815	+	0.703379i	$0.751674\pi$
$74$	−44.7145	−0.0702427
$75$	0	0
$76$	2160.29	3.26056
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−185.631	−0.264369	−0.132184	−	0.991225i	$-0.542199\pi$
$79$	−185.631	−0.264369	−0.132184	+	0.991225i	$0.542199\pi$
$80$	−966.162	−1.35025
$81$	0	0
$82$	1660.30	2.23597
$83$	122.624	0.162166	0.0810830	−	0.996707i	$-0.474162\pi$
$83$	122.624	0.162166	0.0810830	+	0.996707i	$0.474162\pi$
$84$	0	0
$85$	−1463.13	−1.86704
$86$	2118.18	2.65592
$87$	0	0
$88$	426.194	0.516278
$89$	847.086	1.00889	0.504443	−	0.863445i	$-0.331698\pi$
$89$	847.086	1.00889	0.504443	+	0.863445i	$0.331698\pi$
$90$	0	0
$91$	−520.422	−0.599506
$92$	−1292.51	−1.46471
$93$	0	0
$94$	266.453	0.292367
$95$	−2110.04	−2.27880
$96$	0	0
$97$	1002.49	1.04935	0.524676	−	0.851302i	$-0.324186\pi$
$97$	1002.49	1.04935	0.524676	+	0.851302i	$0.324186\pi$
$98$	239.658	0.247032
$99$	0	0
$100$	1860.37	1.86037
$101$	−1124.79	−1.10812	−0.554062	−	0.832476i	$-0.686923\pi$
$101$	−1124.79	−1.10812	−0.554062	+	0.832476i	$0.686923\pi$
$102$	0	0
$103$	966.118	0.924218	0.462109	−	0.886823i	$-0.347093\pi$
$103$	966.118	0.924218	0.462109	+	0.886823i	$0.347093\pi$
$104$	2880.53	2.71595
$105$	0	0
$106$	−605.652	−0.554963
$107$	144.202	0.130286	0.0651428	−	0.997876i	$-0.479250\pi$
$107$	144.202	0.130286	0.0651428	+	0.997876i	$0.479250\pi$
$108$	0	0
$109$	−1875.32	−1.64792	−0.823961	−	0.566647i	$-0.808240\pi$
$109$	−1875.32	−1.64792	−0.823961	+	0.566647i	$0.808240\pi$
$110$	−836.676	−0.725217
$111$	0	0
$112$	−434.890	−0.366904
$113$	−1207.46	−1.00521	−0.502603	−	0.864517i	$-0.667624\pi$
$113$	−1207.46	−1.00521	−0.502603	+	0.864517i	$0.667624\pi$
$114$	0	0
$115$	1262.45	1.02369
$116$	850.632	0.680855
$117$	0	0
$118$	2613.72	2.03909
$119$	−658.585	−0.507331
$120$	0	0
$121$	121.000	0.0909091
$122$	−1754.03	−1.30166
$123$	0	0
$124$	−151.342	−0.109604
$125$	126.825	0.0907483
$126$	0	0
$127$	1143.56	0.799009	0.399504	−	0.916731i	$-0.369182\pi$
$127$	1143.56	0.799009	0.399504	+	0.916731i	$0.369182\pi$
$128$	−2527.97	−1.74565
$129$	0	0
$130$	−5654.86	−3.81511
$131$	−2478.40	−1.65297	−0.826484	−	0.562961i	$-0.809662\pi$
$131$	−2478.40	−1.65297	−0.826484	+	0.562961i	$0.809662\pi$
$132$	0	0
$133$	−949.776	−0.619218
$134$	−3395.90	−2.18926
$135$	0	0
$136$	3645.26	2.29837
$137$	835.661	0.521134	0.260567	−	0.965456i	$-0.416091\pi$
$137$	835.661	0.521134	0.260567	+	0.965456i	$0.416091\pi$
$138$	0	0
$139$	−1726.99	−1.05382	−0.526912	−	0.849920i	$-0.676650\pi$
$139$	−1726.99	−1.05382	−0.526912	+	0.849920i	$0.676650\pi$
$140$	1733.23	1.04632
$141$	0	0
$142$	1361.31	0.804497
$143$	817.805	0.478240
$144$	0	0
$145$	−830.846	−0.475848
$146$	−4336.78	−2.45832
$147$	0	0
$148$	−145.560	−0.0808443
$149$	3454.95	1.89960	0.949799	−	0.312859i	$-0.101287\pi$
$149$	3454.95	1.89960	0.949799	+	0.312859i	$0.101287\pi$
$150$	0	0
$151$	−2468.17	−1.33018	−0.665089	−	0.746764i	$-0.731606\pi$
$151$	−2468.17	−1.33018	−0.665089	+	0.746764i	$0.731606\pi$
$152$	5257.00	2.80526
$153$	0	0
$154$	−376.606	−0.197063
$155$	147.821	0.0766018
$156$	0	0
$157$	−1561.27	−0.793649	−0.396824	−	0.917895i	$-0.629888\pi$
$157$	−1561.27	−0.793649	−0.396824	+	0.917895i	$0.629888\pi$
$158$	−907.919	−0.457153
$159$	0	0
$160$	94.8125	0.0468474
$161$	568.255	0.278166
$162$	0	0
$163$	3458.59	1.66195	0.830973	−	0.556312i	$-0.187784\pi$
$163$	3458.59	1.66195	0.830973	+	0.556312i	$0.187784\pi$
$164$	5404.80	2.57344
$165$	0	0
$166$	599.754	0.280421
$167$	972.527	0.450637	0.225319	−	0.974285i	$-0.427658\pi$
$167$	972.527	0.450637	0.225319	+	0.974285i	$0.427658\pi$
$168$	0	0
$169$	3330.32	1.51585
$170$	−7156.13	−3.22853
$171$	0	0
$172$	6895.34	3.05677
$173$	1154.18	0.507230	0.253615	−	0.967305i	$-0.418380\pi$
$173$	1154.18	0.507230	0.253615	+	0.967305i	$0.418380\pi$
$174$	0	0
$175$	−817.914	−0.353305
$176$	683.399	0.292688
$177$	0	0
$178$	4143.08	1.74459
$179$	−259.234	−0.108246	−0.0541231	−	0.998534i	$-0.517236\pi$
$179$	−259.234	−0.108246	−0.0541231	+	0.998534i	$0.517236\pi$
$180$	0	0
$181$	−2121.49	−0.871209	−0.435604	−	0.900138i	$-0.643465\pi$
$181$	−2121.49	−0.871209	−0.435604	+	0.900138i	$0.643465\pi$
$182$	−2545.37	−1.03668
$183$	0	0
$184$	−3145.29	−1.26018
$185$	142.174	0.0565019
$186$	0	0
$187$	1034.92	0.404710
$188$	867.389	0.336494
$189$	0	0
$190$	−10320.2	−3.94055
$191$	−2918.27	−1.10554	−0.552772	−	0.833332i	$-0.686430\pi$
$191$	−2918.27	−1.10554	−0.552772	+	0.833332i	$0.686430\pi$
$192$	0	0
$193$	3757.48	1.40139	0.700697	−	0.713459i	$-0.252872\pi$
$193$	3757.48	1.40139	0.700697	+	0.713459i	$0.252872\pi$
$194$	4903.15	1.81456
$195$	0	0
$196$	780.164	0.284316
$197$	−1608.39	−0.581689	−0.290845	−	0.956770i	$-0.593936\pi$
$197$	−1608.39	−0.581689	−0.290845	+	0.956770i	$0.593936\pi$
$198$	0	0
$199$	2865.53	1.02076	0.510382	−	0.859948i	$-0.329504\pi$
$199$	2865.53	1.02076	0.510382	+	0.859948i	$0.329504\pi$
$200$	4527.14	1.60059
$201$	0	0
$202$	−5501.31	−1.91619
$203$	−373.981	−0.129302
$204$	0	0
$205$	−5279.08	−1.79857
$206$	4725.27	1.59818
$207$	0	0
$208$	4618.90	1.53973
$209$	1492.50	0.493965
$210$	0	0
$211$	821.996	0.268192	0.134096	−	0.990968i	$-0.457187\pi$
$211$	821.996	0.268192	0.134096	+	0.990968i	$0.457187\pi$
$212$	−1971.59	−0.638723
$213$	0	0
$214$	705.290	0.225293
$215$	−6734.95	−2.13637
$216$	0	0
$217$	66.5375	0.0208150
$218$	−9172.18	−2.84963
$219$	0	0
$220$	−2723.65	−0.834674
$221$	6994.73	2.12903
$222$	0	0
$223$	109.532	0.0328916	0.0164458	−	0.999865i	$-0.494765\pi$
$223$	109.532	0.0328916	0.0164458	+	0.999865i	$0.494765\pi$
$224$	42.6771	0.0127298
$225$	0	0
$226$	−5905.67	−1.73823
$227$	−3023.03	−0.883900	−0.441950	−	0.897040i	$-0.645713\pi$
$227$	−3023.03	−0.883900	−0.441950	+	0.897040i	$0.645713\pi$
$228$	0	0
$229$	−2278.75	−0.657571	−0.328786	−	0.944405i	$-0.606639\pi$
$229$	−2278.75	−0.657571	−0.328786	+	0.944405i	$0.606639\pi$
$230$	6174.61	1.77018
$231$	0	0
$232$	2069.98	0.585781
$233$	1864.08	0.524121	0.262061	−	0.965051i	$-0.415598\pi$
$233$	1864.08	0.524121	0.262061	+	0.965051i	$0.415598\pi$
$234$	0	0
$235$	−847.213	−0.235175
$236$	8508.49	2.34685
$237$	0	0
$238$	−3221.13	−0.877289
$239$	−1404.69	−0.380174	−0.190087	−	0.981767i	$-0.560877\pi$
$239$	−1404.69	−0.380174	−0.190087	+	0.981767i	$0.560877\pi$
$240$	0	0
$241$	3879.32	1.03688	0.518442	−	0.855113i	$-0.326512\pi$
$241$	3879.32	1.03688	0.518442	+	0.855113i	$0.326512\pi$
$242$	591.809	0.157202
$243$	0	0
$244$	−5709.91	−1.49811
$245$	−762.017	−0.198708
$246$	0	0
$247$	10087.4	2.59857
$248$	−368.284	−0.0942987
$249$	0	0
$250$	620.297	0.156924
$251$	2143.54	0.539040	0.269520	−	0.962995i	$-0.413135\pi$
$251$	2143.54	0.539040	0.269520	+	0.962995i	$0.413135\pi$
$252$	0	0
$253$	−892.972	−0.221900
$254$	5593.11	1.38166
$255$	0	0
$256$	−8149.58	−1.98964
$257$	−7288.62	−1.76907	−0.884537	−	0.466471i	$-0.845525\pi$
$257$	−7288.62	−1.76907	−0.884537	+	0.466471i	$0.845525\pi$
$258$	0	0
$259$	63.9957	0.0153533
$260$	−18408.4	−4.39092
$261$	0	0
$262$	−12121.8	−2.85835
$263$	−2670.52	−0.626127	−0.313064	−	0.949732i	$-0.601355\pi$
$263$	−2670.52	−0.626127	−0.313064	+	0.949732i	$0.601355\pi$
$264$	0	0
$265$	1925.73	0.446402
$266$	−4645.34	−1.07077
$267$	0	0
$268$	−11054.7	−2.51968
$269$	−1073.80	−0.243387	−0.121693	−	0.992568i	$-0.538832\pi$
$269$	−1073.80	−0.243387	−0.121693	+	0.992568i	$0.538832\pi$
$270$	0	0
$271$	3624.88	0.812530	0.406265	−	0.913755i	$-0.366831\pi$
$271$	3624.88	0.812530	0.406265	+	0.913755i	$0.366831\pi$
$272$	5845.14	1.30299
$273$	0	0
$274$	4087.20	0.901157
$275$	1285.29	0.281840
$276$	0	0
$277$	−4905.35	−1.06402	−0.532010	−	0.846738i	$-0.678563\pi$
$277$	−4905.35	−1.06402	−0.532010	+	0.846738i	$0.678563\pi$
$278$	−8446.68	−1.82230
$279$	0	0
$280$	4217.76	0.900211
$281$	2661.90	0.565109	0.282555	−	0.959251i	$-0.408818\pi$
$281$	2661.90	0.565109	0.282555	+	0.959251i	$0.408818\pi$
$282$	0	0
$283$	−4367.36	−0.917359	−0.458680	−	0.888602i	$-0.651677\pi$
$283$	−4367.36	−0.917359	−0.458680	+	0.888602i	$0.651677\pi$
$284$	4431.50	0.925919
$285$	0	0
$286$	3999.87	0.826984
$287$	−2376.23	−0.488726
$288$	0	0
$289$	3938.72	0.801693
$290$	−4063.65	−0.822848
$291$	0	0
$292$	−14117.6	−2.82935
$293$	−1992.30	−0.397240	−0.198620	−	0.980077i	$-0.563646\pi$
$293$	−1992.30	−0.397240	−0.198620	+	0.980077i	$0.563646\pi$
$294$	0	0
$295$	−8310.58	−1.64021
$296$	−354.215	−0.0695553
$297$	0	0
$298$	16898.1	3.28483
$299$	−6035.35	−1.16734
$300$	0	0
$301$	−3031.54	−0.580516
$302$	−12071.8	−2.30018
$303$	0	0
$304$	8429.55	1.59036
$305$	5577.10	1.04703
$306$	0	0
$307$	7633.53	1.41912	0.709558	−	0.704647i	$-0.248895\pi$
$307$	7633.53	1.41912	0.709558	+	0.704647i	$0.248895\pi$
$308$	−1225.97	−0.226806
$309$	0	0
$310$	722.991	0.132462
$311$	−1453.52	−0.265020	−0.132510	−	0.991182i	$-0.542304\pi$
$311$	−1453.52	−0.265020	−0.132510	+	0.991182i	$0.542304\pi$
$312$	0	0
$313$	1936.27	0.349663	0.174831	−	0.984598i	$-0.444062\pi$
$313$	1936.27	0.349663	0.174831	+	0.984598i	$0.444062\pi$
$314$	−7636.14	−1.37240
$315$	0	0
$316$	−2955.56	−0.526150
$317$	1534.36	0.271856	0.135928	−	0.990719i	$-0.456598\pi$
$317$	1534.36	0.271856	0.135928	+	0.990719i	$0.456598\pi$
$318$	0	0
$319$	587.685	0.103148
$320$	8193.02	1.43126
$321$	0	0
$322$	2779.32	0.481012
$323$	12765.5	2.19904
$324$	0	0
$325$	8686.94	1.48266
$326$	16915.9	2.87388
$327$	0	0
$328$	13152.4	2.21408
$329$	−381.349	−0.0639041
$330$	0	0
$331$	3807.42	0.632251	0.316125	−	0.948717i	$-0.397618\pi$
$331$	3807.42	0.632251	0.316125	+	0.948717i	$0.397618\pi$
$332$	1952.39	0.322745
$333$	0	0
$334$	4756.61	0.779252
$335$	10797.6	1.76100
$336$	0	0
$337$	−4724.09	−0.763614	−0.381807	−	0.924242i	$-0.624698\pi$
$337$	−4724.09	−0.763614	−0.381807	+	0.924242i	$0.624698\pi$
$338$	16288.5	2.62124
$339$	0	0
$340$	−23295.5	−3.71581
$341$	−104.559	−0.0166046
$342$	0	0
$343$	−343.000	−0.0539949
$344$	16779.6	2.62992
$345$	0	0
$346$	5645.08	0.877113
$347$	−6122.41	−0.947171	−0.473586	−	0.880748i	$-0.657041\pi$
$347$	−6122.41	−0.947171	−0.473586	+	0.880748i	$0.657041\pi$
$348$	0	0
$349$	6778.93	1.03974	0.519868	−	0.854247i	$-0.325981\pi$
$349$	6778.93	1.03974	0.519868	+	0.854247i	$0.325981\pi$
$350$	−4000.40	−0.610944
$351$	0	0
$352$	−67.0640	−0.0101549
$353$	3486.98	0.525760	0.262880	−	0.964829i	$-0.415328\pi$
$353$	3486.98	0.525760	0.262880	+	0.964829i	$0.415328\pi$
$354$	0	0
$355$	−4328.42	−0.647123
$356$	13487.1	2.00790
$357$	0	0
$358$	−1267.91	−0.187182
$359$	−3230.16	−0.474878	−0.237439	−	0.971402i	$-0.576308\pi$
$359$	−3230.16	−0.474878	−0.237439	+	0.971402i	$0.576308\pi$
$360$	0	0
$361$	11550.7	1.68402
$362$	−10376.2	−1.50651
$363$	0	0
$364$	−8286.00	−1.19314
$365$	13789.2	1.97742
$366$	0	0
$367$	−8190.16	−1.16491	−0.582456	−	0.812862i	$-0.697908\pi$
$367$	−8190.16	−1.16491	−0.582456	+	0.812862i	$0.697908\pi$
$368$	−5043.44	−0.714422
$369$	0	0
$370$	695.372	0.0977045
$371$	866.812	0.121301
$372$	0	0
$373$	5008.53	0.695260	0.347630	−	0.937632i	$-0.386987\pi$
$373$	5008.53	0.695260	0.347630	+	0.937632i	$0.386987\pi$
$374$	5061.77	0.699834
$375$	0	0
$376$	2110.76	0.289506
$377$	3972.00	0.542622
$378$	0	0
$379$	1522.26	0.206314	0.103157	−	0.994665i	$-0.467106\pi$
$379$	1522.26	0.206314	0.103157	+	0.994665i	$0.467106\pi$
$380$	−33595.5	−4.53530
$381$	0	0
$382$	−14273.2	−1.91173
$383$	−8520.08	−1.13670	−0.568349	−	0.822787i	$-0.692418\pi$
$383$	−8520.08	−1.13670	−0.568349	+	0.822787i	$0.692418\pi$
$384$	0	0
$385$	1197.45	0.158514
$386$	18377.8	2.42332
$387$	0	0
$388$	15961.3	2.08844
$389$	−1588.83	−0.207088	−0.103544	−	0.994625i	$-0.533018\pi$
$389$	−1588.83	−0.207088	−0.103544	+	0.994625i	$0.533018\pi$
$390$	0	0
$391$	−7637.64	−0.987856
$392$	1898.50	0.244614
$393$	0	0
$394$	−7866.59	−1.00587
$395$	2886.82	0.367725
$396$	0	0
$397$	−1483.46	−0.187539	−0.0937694	−	0.995594i	$-0.529892\pi$
$397$	−1483.46	−0.187539	−0.0937694	+	0.995594i	$0.529892\pi$
$398$	14015.3	1.76513
$399$	0	0
$400$	7259.23	0.907404
$401$	−10189.5	−1.26892	−0.634460	−	0.772956i	$-0.718778\pi$
$401$	−10189.5	−1.26892	−0.634460	+	0.772956i	$0.718778\pi$
$402$	0	0
$403$	−706.685	−0.0873510
$404$	−17908.5	−2.20540
$405$	0	0
$406$	−1829.14	−0.223592
$407$	−100.565	−0.0122477
$408$	0	0
$409$	3465.96	0.419024	0.209512	−	0.977806i	$-0.432812\pi$
$409$	3465.96	0.419024	0.209512	+	0.977806i	$0.432812\pi$
$410$	−25819.9	−3.11013
$411$	0	0
$412$	15382.2	1.83939
$413$	−3740.77	−0.445693
$414$	0	0
$415$	−1906.98	−0.225566
$416$	−453.267	−0.0534213
$417$	0	0
$418$	7299.81	0.854176
$419$	−2754.83	−0.321199	−0.160599	−	0.987020i	$-0.551343\pi$
$419$	−2754.83	−0.321199	−0.160599	+	0.987020i	$0.551343\pi$
$420$	0	0
$421$	−3140.19	−0.363524	−0.181762	−	0.983343i	$-0.558180\pi$
$421$	−3140.19	−0.363524	−0.181762	+	0.983343i	$0.558180\pi$
$422$	4020.37	0.463764
$423$	0	0
$424$	−4797.80	−0.549532
$425$	10993.2	1.25470
$426$	0	0
$427$	2510.37	0.284509
$428$	2295.94	0.259296
$429$	0	0
$430$	−32940.5	−3.69426
$431$	4476.22	0.500260	0.250130	−	0.968212i	$-0.419527\pi$
$431$	4476.22	0.500260	0.250130	+	0.968212i	$0.419527\pi$
$432$	0	0
$433$	1646.00	0.182683	0.0913416	−	0.995820i	$-0.470884\pi$
$433$	1646.00	0.182683	0.0913416	+	0.995820i	$0.470884\pi$
$434$	325.434	0.0359938
$435$	0	0
$436$	−29858.4	−3.27972
$437$	−11014.6	−1.20572
$438$	0	0
$439$	−7241.05	−0.787236	−0.393618	−	0.919274i	$-0.628777\pi$
$439$	−7241.05	−0.787236	−0.393618	+	0.919274i	$0.628777\pi$
$440$	−6627.90	−0.718120
$441$	0	0
$442$	34211.1	3.68158
$443$	−5887.27	−0.631405	−0.315703	−	0.948858i	$-0.602240\pi$
$443$	−5887.27	−0.631405	−0.315703	+	0.948858i	$0.602240\pi$
$444$	0	0
$445$	−13173.3	−1.40332
$446$	535.721	0.0568770
$447$	0	0
$448$	3687.85	0.388917
$449$	−6378.42	−0.670415	−0.335207	−	0.942144i	$-0.608806\pi$
$449$	−6378.42	−0.670415	−0.335207	+	0.942144i	$0.608806\pi$
$450$	0	0
$451$	3734.07	0.389868
$452$	−19224.8	−2.00058
$453$	0	0
$454$	−14785.6	−1.52846
$455$	8093.26	0.833886
$456$	0	0
$457$	−18368.0	−1.88013	−0.940064	−	0.340998i	$-0.889235\pi$
$457$	−18368.0	−1.88013	−0.940064	+	0.340998i	$0.889235\pi$
$458$	−11145.3	−1.13709
$459$	0	0
$460$	20100.3	2.03735
$461$	−17510.6	−1.76909	−0.884546	−	0.466453i	$-0.845532\pi$
$461$	−17510.6	−1.76909	−0.884546	+	0.466453i	$0.845532\pi$
$462$	0	0
$463$	−12732.5	−1.27803	−0.639016	−	0.769193i	$-0.720658\pi$
$463$	−12732.5	−1.27803	−0.639016	+	0.769193i	$0.720658\pi$
$464$	3319.20	0.332091
$465$	0	0
$466$	9117.20	0.906323
$467$	4997.23	0.495170	0.247585	−	0.968866i	$-0.420363\pi$
$467$	4997.23	0.495170	0.247585	+	0.968866i	$0.420363\pi$
$468$	0	0
$469$	4860.23	0.478517
$470$	−4143.70	−0.406670
$471$	0	0
$472$	20705.1	2.01913
$473$	4763.85	0.463091
$474$	0	0
$475$	15853.8	1.53141
$476$	−10485.8	−1.00970
$477$	0	0
$478$	−6870.30	−0.657406
$479$	9075.53	0.865703	0.432851	−	0.901465i	$-0.357507\pi$
$479$	9075.53	0.865703	0.432851	+	0.901465i	$0.357507\pi$
$480$	0	0
$481$	−679.688	−0.0644306
$482$	18973.7	1.79300
$483$	0	0
$484$	1926.53	0.180929
$485$	−15590.0	−1.45960
$486$	0	0
$487$	−3867.63	−0.359875	−0.179937	−	0.983678i	$-0.557590\pi$
$487$	−3867.63	−0.359875	−0.179937	+	0.983678i	$0.557590\pi$
$488$	−13894.9	−1.28892
$489$	0	0
$490$	−3727.01	−0.343611
$491$	−7334.24	−0.674113	−0.337057	−	0.941484i	$-0.609431\pi$
$491$	−7334.24	−0.674113	−0.337057	+	0.941484i	$0.609431\pi$
$492$	0	0
$493$	5026.50	0.459193
$494$	49337.4	4.49351
$495$	0	0
$496$	−590.541	−0.0534598
$497$	−1948.31	−0.175843
$498$	0	0
$499$	−4365.93	−0.391675	−0.195838	−	0.980636i	$-0.562743\pi$
$499$	−4365.93	−0.391675	−0.195838	+	0.980636i	$0.562743\pi$
$500$	2019.26	0.180609
$501$	0	0
$502$	10484.0	0.932120
$503$	16352.6	1.44956	0.724779	−	0.688981i	$-0.241942\pi$
$503$	16352.6	1.44956	0.724779	+	0.688981i	$0.241942\pi$
$504$	0	0
$505$	17492.0	1.54135
$506$	−4367.51	−0.383714
$507$	0	0
$508$	18207.4	1.59020
$509$	12108.2	1.05440	0.527199	−	0.849742i	$-0.323242\pi$
$509$	12108.2	1.05440	0.527199	+	0.849742i	$0.323242\pi$
$510$	0	0
$511$	6206.82	0.537326
$512$	−19635.7	−1.69489
$513$	0	0
$514$	−35648.5	−3.05912
$515$	−15024.4	−1.28555
$516$	0	0
$517$	599.262	0.0509778
$518$	313.002	0.0265492
$519$	0	0
$520$	−44796.1	−3.77777
$521$	5625.67	0.473062	0.236531	−	0.971624i	$-0.423990\pi$
$521$	5625.67	0.473062	0.236531	+	0.971624i	$0.423990\pi$
$522$	0	0
$523$	−3280.32	−0.274261	−0.137130	−	0.990553i	$-0.543788\pi$
$523$	−3280.32	−0.274261	−0.137130	+	0.990553i	$0.543788\pi$
$524$	−39460.3	−3.28976
$525$	0	0
$526$	−13061.5	−1.08271
$527$	−894.298	−0.0739207
$528$	0	0
$529$	−5576.93	−0.458365
$530$	9418.70	0.771929
$531$	0	0
$532$	−15122.1	−1.23238
$533$	25237.6	2.05096
$534$	0	0
$535$	−2242.54	−0.181221
$536$	−26901.3	−2.16784
$537$	0	0
$538$	−5251.96	−0.420870
$539$	539.000	0.0430730
$540$	0	0
$541$	−7772.10	−0.617650	−0.308825	−	0.951119i	$-0.599936\pi$
$541$	−7772.10	−0.617650	−0.308825	+	0.951119i	$0.599936\pi$
$542$	17729.2	1.40505
$543$	0	0
$544$	−573.602	−0.0452077
$545$	29163.8	2.29219
$546$	0	0
$547$	10022.2	0.783398	0.391699	−	0.920094i	$-0.371888\pi$
$547$	10022.2	0.783398	0.391699	+	0.920094i	$0.371888\pi$
$548$	13305.1	1.03717
$549$	0	0
$550$	6286.34	0.487365
$551$	7248.95	0.560464
$552$	0	0
$553$	1299.42	0.0999220
$554$	−23992.0	−1.83993
$555$	0	0
$556$	−27496.7	−2.09733
$557$	10849.3	0.825310	0.412655	−	0.910887i	$-0.364601\pi$
$557$	10849.3	0.825310	0.412655	+	0.910887i	$0.364601\pi$
$558$	0	0
$559$	32197.6	2.43616
$560$	6763.13	0.510347
$561$	0	0
$562$	13019.3	0.977200
$563$	−22019.9	−1.64836	−0.824180	−	0.566328i	$-0.808363\pi$
$563$	−22019.9	−1.64836	−0.824180	+	0.566328i	$0.808363\pi$
$564$	0	0
$565$	18777.7	1.39820
$566$	−21360.7	−1.58632
$567$	0	0
$568$	10783.9	0.796624
$569$	−13742.9	−1.01253	−0.506267	−	0.862377i	$-0.668975\pi$
$569$	−13742.9	−1.01253	−0.506267	+	0.862377i	$0.668975\pi$
$570$	0	0
$571$	19382.4	1.42054	0.710269	−	0.703931i	$-0.248573\pi$
$571$	19382.4	1.42054	0.710269	+	0.703931i	$0.248573\pi$
$572$	13020.9	0.951800
$573$	0	0
$574$	−11622.1	−0.845116
$575$	−9485.37	−0.687943
$576$	0	0
$577$	−565.348	−0.0407899	−0.0203949	−	0.999792i	$-0.506492\pi$
$577$	−565.348	−0.0407899	−0.0203949	+	0.999792i	$0.506492\pi$
$578$	19264.2	1.38631
$579$	0	0
$580$	−13228.5	−0.947040
$581$	−858.371	−0.0612930
$582$	0	0
$583$	−1362.13	−0.0967646
$584$	−34354.7	−2.43426
$585$	0	0
$586$	−9744.30	−0.686917
$587$	20727.4	1.45743	0.728714	−	0.684818i	$-0.240118\pi$
$587$	20727.4	1.45743	0.728714	+	0.684818i	$0.240118\pi$
$588$	0	0
$589$	−1289.71	−0.0902233
$590$	−40646.9	−2.83628
$591$	0	0
$592$	−567.981	−0.0394322
$593$	2787.10	0.193006	0.0965030	−	0.995333i	$-0.469234\pi$
$593$	2787.10	0.193006	0.0965030	+	0.995333i	$0.469234\pi$
$594$	0	0
$595$	10241.9	0.705675
$596$	55008.7	3.78061
$597$	0	0
$598$	−29518.8	−2.01858
$599$	−18935.5	−1.29162	−0.645812	−	0.763497i	$-0.723481\pi$
$599$	−18935.5	−1.29162	−0.645812	+	0.763497i	$0.723481\pi$
$600$	0	0
$601$	15821.3	1.07382	0.536908	−	0.843641i	$-0.319592\pi$
$601$	15821.3	1.07382	0.536908	+	0.843641i	$0.319592\pi$
$602$	−14827.2	−1.00384
$603$	0	0
$604$	−39297.5	−2.64734
$605$	−1881.71	−0.126451
$606$	0	0
$607$	−2084.63	−0.139394	−0.0696972	−	0.997568i	$-0.522203\pi$
$607$	−2084.63	−0.139394	−0.0696972	+	0.997568i	$0.522203\pi$
$608$	−827.218	−0.0551778
$609$	0	0
$610$	27277.5	1.81055
$611$	4050.25	0.268176
$612$	0	0
$613$	−4395.15	−0.289590	−0.144795	−	0.989462i	$-0.546252\pi$
$613$	−4395.15	−0.289590	−0.144795	+	0.989462i	$0.546252\pi$
$614$	37335.5	2.45397
$615$	0	0
$616$	−2983.36	−0.195135
$617$	98.5856	0.00643259	0.00321629	−	0.999995i	$-0.498976\pi$
$617$	98.5856	0.00643259	0.00321629	+	0.999995i	$0.498976\pi$
$618$	0	0
$619$	−16533.9	−1.07359	−0.536796	−	0.843712i	$-0.680366\pi$
$619$	−16533.9	−1.07359	−0.536796	+	0.843712i	$0.680366\pi$
$620$	2353.57	0.152454
$621$	0	0
$622$	−7109.12	−0.458280
$623$	−5929.60	−0.381323
$624$	0	0
$625$	−16577.9	−1.06099
$626$	9470.27	0.604645
$627$	0	0
$628$	−24858.1	−1.57953
$629$	−860.135	−0.0545243
$630$	0	0
$631$	22032.9	1.39004	0.695022	−	0.718989i	$-0.255395\pi$
$631$	22032.9	1.39004	0.695022	+	0.718989i	$0.255395\pi$
$632$	−7192.27	−0.452679
$633$	0	0
$634$	7504.54	0.470100
$635$	−17783.8	−1.11139
$636$	0	0
$637$	3642.95	0.226592
$638$	2874.36	0.178365
$639$	0	0
$640$	39313.4	2.42812
$641$	−434.281	−0.0267598	−0.0133799	−	0.999910i	$-0.504259\pi$
$641$	−434.281	−0.0267598	−0.0133799	+	0.999910i	$0.504259\pi$
$642$	0	0
$643$	10963.1	0.672381	0.336190	−	0.941794i	$-0.390861\pi$
$643$	10963.1	0.672381	0.336190	+	0.941794i	$0.390861\pi$
$644$	9047.59	0.553610
$645$	0	0
$646$	62435.7	3.80263
$647$	452.083	0.0274702	0.0137351	−	0.999906i	$-0.495628\pi$
$647$	452.083	0.0274702	0.0137351	+	0.999906i	$0.495628\pi$
$648$	0	0
$649$	5878.35	0.355540
$650$	42487.6	2.56385
$651$	0	0
$652$	55066.6	3.30763
$653$	10438.7	0.625572	0.312786	−	0.949824i	$-0.398738\pi$
$653$	10438.7	0.625572	0.312786	+	0.949824i	$0.398738\pi$
$654$	0	0
$655$	38542.5	2.29920
$656$	21089.8	1.25521
$657$	0	0
$658$	−1865.17	−0.110504
$659$	8738.79	0.516563	0.258281	−	0.966070i	$-0.416844\pi$
$659$	8738.79	0.516563	0.258281	+	0.966070i	$0.416844\pi$
$660$	0	0
$661$	12849.4	0.756102	0.378051	−	0.925785i	$-0.376594\pi$
$661$	12849.4	0.756102	0.378051	+	0.925785i	$0.376594\pi$
$662$	18622.0	1.09330
$663$	0	0
$664$	4751.08	0.277677
$665$	14770.3	0.861305
$666$	0	0
$667$	−4337.08	−0.251773
$668$	15484.3	0.896864
$669$	0	0
$670$	52810.8	3.04517
$671$	−3944.87	−0.226960
$672$	0	0
$673$	−22926.5	−1.31315	−0.656577	−	0.754259i	$-0.727996\pi$
$673$	−22926.5	−1.31315	−0.656577	+	0.754259i	$0.727996\pi$
$674$	−23105.5	−1.32046
$675$	0	0
$676$	53024.4	3.01686
$677$	−13633.4	−0.773965	−0.386983	−	0.922087i	$-0.626483\pi$
$677$	−13633.4	−0.773965	−0.386983	+	0.922087i	$0.626483\pi$
$678$	0	0
$679$	−7017.41	−0.396618
$680$	−56688.8	−3.19694
$681$	0	0
$682$	−511.396	−0.0287131
$683$	−33307.5	−1.86600	−0.932998	−	0.359882i	$-0.882817\pi$
$683$	−33307.5	−1.86600	−0.932998	+	0.359882i	$0.882817\pi$
$684$	0	0
$685$	−12995.7	−0.724874
$686$	−1677.61	−0.0933693
$687$	0	0
$688$	26905.9	1.49096
$689$	−9206.28	−0.509044
$690$	0	0
$691$	−10077.3	−0.554786	−0.277393	−	0.960757i	$-0.589470\pi$
$691$	−10077.3	−0.554786	−0.277393	+	0.960757i	$0.589470\pi$
$692$	18376.5	1.00950
$693$	0	0
$694$	−29944.6	−1.63787
$695$	26857.1	1.46582
$696$	0	0
$697$	31937.7	1.73562
$698$	33155.6	1.79794
$699$	0	0
$700$	−13022.6	−0.703153
$701$	4621.74	0.249017	0.124508	−	0.992219i	$-0.460265\pi$
$701$	4621.74	0.249017	0.124508	+	0.992219i	$0.460265\pi$
$702$	0	0
$703$	−1240.44	−0.0665492
$704$	−5795.20	−0.310248
$705$	0	0
$706$	17054.8	0.909157
$707$	7873.51	0.418831
$708$	0	0
$709$	17746.6	0.940038	0.470019	−	0.882656i	$-0.344247\pi$
$709$	17746.6	0.940038	0.470019	+	0.882656i	$0.344247\pi$
$710$	−21170.2	−1.11902
$711$	0	0
$712$	32820.3	1.72752
$713$	771.638	0.0405302
$714$	0	0
$715$	−12718.0	−0.665211
$716$	−4127.45	−0.215433
$717$	0	0
$718$	−15798.6	−0.821170
$719$	31652.1	1.64176	0.820879	−	0.571103i	$-0.193484\pi$
$719$	31652.1	1.64176	0.820879	+	0.571103i	$0.193484\pi$
$720$	0	0
$721$	−6762.83	−0.349322
$722$	56494.1	2.91204
$723$	0	0
$724$	−33777.7	−1.73389
$725$	6242.54	0.319782
$726$	0	0
$727$	21610.8	1.10248	0.551239	−	0.834348i	$-0.314155\pi$
$727$	21610.8	1.10248	0.551239	+	0.834348i	$0.314155\pi$
$728$	−20163.7	−1.02653
$729$	0	0
$730$	67442.8	3.41941
$731$	40745.5	2.06160
$732$	0	0
$733$	−7665.43	−0.386261	−0.193130	−	0.981173i	$-0.561864\pi$
$733$	−7665.43	−0.386261	−0.193130	+	0.981173i	$0.561864\pi$
$734$	−40057.9	−2.01439
$735$	0	0
$736$	494.928	0.0247871
$737$	−7637.50	−0.381725
$738$	0	0
$739$	−13841.5	−0.688996	−0.344498	−	0.938787i	$-0.611951\pi$
$739$	−13841.5	−0.688996	−0.344498	+	0.938787i	$0.611951\pi$
$740$	2263.66	0.112451
$741$	0	0
$742$	4239.56	0.209756
$743$	12136.3	0.599243	0.299621	−	0.954058i	$-0.403140\pi$
$743$	12136.3	0.599243	0.299621	+	0.954058i	$0.403140\pi$
$744$	0	0
$745$	−53729.1	−2.64226
$746$	24496.6	1.20226
$747$	0	0
$748$	16477.7	0.805460
$749$	−1009.42	−0.0492433
$750$	0	0
$751$	28866.7	1.40261	0.701305	−	0.712861i	$-0.252601\pi$
$751$	28866.7	1.40261	0.701305	+	0.712861i	$0.252601\pi$
$752$	3384.59	0.164127
$753$	0	0
$754$	19427.0	0.938314
$755$	38383.4	1.85022
$756$	0	0
$757$	8370.48	0.401889	0.200945	−	0.979603i	$-0.435599\pi$
$757$	8370.48	0.401889	0.200945	+	0.979603i	$0.435599\pi$
$758$	7445.33	0.356763
$759$	0	0
$760$	−81753.5	−3.90199
$761$	−30458.1	−1.45086	−0.725430	−	0.688296i	$-0.758359\pi$
$761$	−30458.1	−1.45086	−0.725430	+	0.688296i	$0.758359\pi$
$762$	0	0
$763$	13127.3	0.622856
$764$	−46463.9	−2.20027
$765$	0	0
$766$	−41671.5	−1.96561
$767$	39730.1	1.87037
$768$	0	0
$769$	−32239.5	−1.51182	−0.755908	−	0.654678i	$-0.772804\pi$
$769$	−32239.5	−1.51182	−0.755908	+	0.654678i	$0.772804\pi$
$770$	5856.73	0.274106
$771$	0	0
$772$	59825.5	2.78907
$773$	7524.47	0.350112	0.175056	−	0.984559i	$-0.443989\pi$
$773$	7524.47	0.350112	0.175056	+	0.984559i	$0.443989\pi$
$774$	0	0
$775$	−1110.65	−0.0514784
$776$	38841.3	1.79681
$777$	0	0
$778$	−7770.96	−0.358101
$779$	46058.8	2.11839
$780$	0	0
$781$	3061.63	0.140274
$782$	−37355.5	−1.70822
$783$	0	0
$784$	3044.23	0.138677
$785$	24279.9	1.10393
$786$	0	0
$787$	33196.6	1.50360	0.751798	−	0.659394i	$-0.229187\pi$
$787$	33196.6	1.50360	0.751798	+	0.659394i	$0.229187\pi$
$788$	−25608.3	−1.15769
$789$	0	0
$790$	14119.4	0.635879
$791$	8452.23	0.379932
$792$	0	0
$793$	−26662.3	−1.19395
$794$	−7255.59	−0.324296
$795$	0	0
$796$	45624.1	2.03154
$797$	−8817.06	−0.391865	−0.195932	−	0.980617i	$-0.562773\pi$
$797$	−8817.06	−0.391865	−0.195932	+	0.980617i	$0.562773\pi$
$798$	0	0
$799$	5125.52	0.226944
$800$	−712.371	−0.0314827
$801$	0	0
$802$	−49836.4	−2.19425
$803$	−9753.57	−0.428638
$804$	0	0
$805$	−8837.14	−0.386917
$806$	−3456.38	−0.151049
$807$	0	0
$808$	−43579.8	−1.89744
$809$	28930.3	1.25727	0.628637	−	0.777699i	$-0.283613\pi$
$809$	28930.3	1.25727	0.628637	+	0.777699i	$0.283613\pi$
$810$	0	0
$811$	−45923.3	−1.98839	−0.994195	−	0.107594i	$-0.965685\pi$
$811$	−45923.3	−1.98839	−0.994195	+	0.107594i	$0.965685\pi$
$812$	−5954.42	−0.257339
$813$	0	0
$814$	−491.860	−0.0211790
$815$	−53785.7	−2.31169
$816$	0	0
$817$	58761.0	2.51626
$818$	16952.0	0.724586
$819$	0	0
$820$	−84052.0	−3.57954
$821$	−16382.8	−0.696423	−0.348211	−	0.937416i	$-0.613211\pi$
$821$	−16382.8	−0.696423	−0.348211	+	0.937416i	$0.613211\pi$
$822$	0	0
$823$	−32395.3	−1.37209	−0.686045	−	0.727559i	$-0.740655\pi$
$823$	−32395.3	−1.37209	−0.686045	+	0.727559i	$0.740655\pi$
$824$	37432.2	1.58254
$825$	0	0
$826$	−18296.0	−0.770703
$827$	14329.2	0.602508	0.301254	−	0.953544i	$-0.402595\pi$
$827$	14329.2	0.602508	0.301254	+	0.953544i	$0.402595\pi$
$828$	0	0
$829$	16445.8	0.689005	0.344503	−	0.938785i	$-0.388048\pi$
$829$	16445.8	0.689005	0.344503	+	0.938785i	$0.388048\pi$
$830$	−9326.99	−0.390054
$831$	0	0
$832$	−39168.1	−1.63210
$833$	4610.10	0.191753
$834$	0	0
$835$	−15124.1	−0.626816
$836$	23763.2	0.983096
$837$	0	0
$838$	−13473.8	−0.555424
$839$	−40514.7	−1.66713	−0.833565	−	0.552422i	$-0.813704\pi$
$839$	−40514.7	−1.66713	−0.833565	+	0.552422i	$0.813704\pi$
$840$	0	0
$841$	−21534.7	−0.882967
$842$	−15358.6	−0.628614
$843$	0	0
$844$	13087.6	0.533760
$845$	−51791.0	−2.10848
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−7693.22	−0.311540
$849$	0	0
$850$	53767.4	2.16966
$851$	742.160	0.0298953
$852$	0	0
$853$	−27962.7	−1.12242	−0.561211	−	0.827673i	$-0.689664\pi$
$853$	−27962.7	−1.12242	−0.561211	+	0.827673i	$0.689664\pi$
$854$	12278.2	0.491980
$855$	0	0
$856$	5587.10	0.223088
$857$	4639.52	0.184928	0.0924639	−	0.995716i	$-0.470526\pi$
$857$	4639.52	0.184928	0.0924639	+	0.995716i	$0.470526\pi$
$858$	0	0
$859$	18521.2	0.735663	0.367831	−	0.929892i	$-0.380100\pi$
$859$	18521.2	0.735663	0.367831	+	0.929892i	$0.380100\pi$
$860$	−107232.	−4.25183
$861$	0	0
$862$	21893.1	0.865061
$863$	−40491.4	−1.59715	−0.798577	−	0.601893i	$-0.794413\pi$
$863$	−40491.4	−1.59715	−0.798577	+	0.601893i	$0.794413\pi$
$864$	0	0
$865$	−17949.1	−0.705534
$866$	8050.57	0.315900
$867$	0	0
$868$	1059.39	0.0414263
$869$	−2041.94	−0.0797102
$870$	0	0
$871$	−51619.8	−2.00812
$872$	−72659.3	−2.82174
$873$	0	0
$874$	−53872.1	−2.08496
$875$	−887.772	−0.0342996
$876$	0	0
$877$	−48530.3	−1.86859	−0.934295	−	0.356502i	$-0.883969\pi$
$877$	−48530.3	−1.86859	−0.934295	+	0.356502i	$0.883969\pi$
$878$	−35415.9	−1.36131
$879$	0	0
$880$	−10627.8	−0.407116
$881$	11590.0	0.443219	0.221609	−	0.975135i	$-0.428869\pi$
$881$	11590.0	0.443219	0.221609	+	0.975135i	$0.428869\pi$
$882$	0	0
$883$	41900.7	1.59691	0.798455	−	0.602054i	$-0.205651\pi$
$883$	41900.7	1.59691	0.798455	+	0.602054i	$0.205651\pi$
$884$	111368.	4.23723
$885$	0	0
$886$	−28794.5	−1.09184
$887$	17136.8	0.648700	0.324350	−	0.945937i	$-0.394854\pi$
$887$	17136.8	0.648700	0.324350	+	0.945937i	$0.394854\pi$
$888$	0	0
$889$	−8004.89	−0.301997
$890$	−64430.6	−2.42665
$891$	0	0
$892$	1743.94	0.0654614
$893$	7391.75	0.276994
$894$	0	0
$895$	4031.44	0.150566
$896$	17695.8	0.659794
$897$	0	0
$898$	−31196.7	−1.15930
$899$	−507.832	−0.0188400
$900$	0	0
$901$	−11650.4	−0.430778
$902$	18263.3	0.674169
$903$	0	0
$904$	−46783.0	−1.72122
$905$	32992.0	1.21181
$906$	0	0
$907$	40350.7	1.47720	0.738601	−	0.674143i	$-0.235487\pi$
$907$	40350.7	1.47720	0.738601	+	0.674143i	$0.235487\pi$
$908$	−48131.8	−1.75915
$909$	0	0
$910$	39584.0	1.44197
$911$	32322.7	1.17552	0.587760	−	0.809035i	$-0.300010\pi$
$911$	32322.7	1.17552	0.587760	+	0.809035i	$0.300010\pi$
$912$	0	0
$913$	1348.87	0.0488949
$914$	−89837.5	−3.25116
$915$	0	0
$916$	−36281.5	−1.30871
$917$	17348.8	0.624763
$918$	0	0
$919$	2431.28	0.0872694	0.0436347	−	0.999048i	$-0.486106\pi$
$919$	2431.28	0.0872694	0.0436347	+	0.999048i	$0.486106\pi$
$920$	48913.5	1.75286
$921$	0	0
$922$	−85644.2	−3.05916
$923$	20692.7	0.737930
$924$	0	0
$925$	−1068.22	−0.0379708
$926$	−62274.4	−2.21000
$927$	0	0
$928$	−325.724	−0.0115220
$929$	−40221.0	−1.42046	−0.710230	−	0.703970i	$-0.751409\pi$
$929$	−40221.0	−1.42046	−0.710230	+	0.703970i	$0.751409\pi$
$930$	0	0
$931$	6648.43	0.234042
$932$	29679.4	1.04311
$933$	0	0
$934$	24441.4	0.856260
$935$	−16094.4	−0.562934
$936$	0	0
$937$	41503.4	1.44702	0.723510	−	0.690314i	$-0.242528\pi$
$937$	41503.4	1.44702	0.723510	+	0.690314i	$0.242528\pi$
$938$	23771.3	0.827463
$939$	0	0
$940$	−13489.1	−0.468048
$941$	9032.78	0.312923	0.156461	−	0.987684i	$-0.449991\pi$
$941$	9032.78	0.312923	0.156461	+	0.987684i	$0.449991\pi$
$942$	0	0
$943$	−27557.2	−0.951629
$944$	33200.5	1.14469
$945$	0	0
$946$	23299.9	0.800789
$947$	−5565.96	−0.190992	−0.0954960	−	0.995430i	$-0.530444\pi$
$947$	−5565.96	−0.190992	−0.0954960	+	0.995430i	$0.530444\pi$
$948$	0	0
$949$	−65921.7	−2.25491
$950$	77540.5	2.64815
$951$	0	0
$952$	−25516.8	−0.868703
$953$	−32657.4	−1.11005	−0.555024	−	0.831834i	$-0.687291\pi$
$953$	−32657.4	−1.11005	−0.555024	+	0.831834i	$0.687291\pi$
$954$	0	0
$955$	45383.1	1.53776
$956$	−22365.0	−0.756628
$957$	0	0
$958$	44388.3	1.49699
$959$	−5849.63	−0.196970
$960$	0	0
$961$	−29700.6	−0.996967
$962$	−3324.34	−0.111415
$963$	0	0
$964$	61765.4	2.06362
$965$	−58433.9	−1.94928
$966$	0	0
$967$	−610.079	−0.0202883	−0.0101442	−	0.999949i	$-0.503229\pi$
$967$	−610.079	−0.0202883	−0.0101442	+	0.999949i	$0.503229\pi$
$968$	4688.14	0.155664
$969$	0	0
$970$	−76250.6	−2.52398
$971$	45371.4	1.49952	0.749762	−	0.661708i	$-0.230168\pi$
$971$	45371.4	1.49952	0.749762	+	0.661708i	$0.230168\pi$
$972$	0	0
$973$	12088.9	0.398308
$974$	−18916.5	−0.622304
$975$	0	0
$976$	−22280.3	−0.730713
$977$	−27229.0	−0.891639	−0.445820	−	0.895123i	$-0.647088\pi$
$977$	−27229.0	−0.891639	−0.445820	+	0.895123i	$0.647088\pi$
$978$	0	0
$979$	9317.94	0.304191
$980$	−12132.6	−0.395471
$981$	0	0
$982$	−35871.6	−1.16569
$983$	−7458.49	−0.242003	−0.121002	−	0.992652i	$-0.538611\pi$
$983$	−7458.49	−0.242003	−0.121002	+	0.992652i	$0.538611\pi$
$984$	0	0
$985$	25012.6	0.809104
$986$	24584.5	0.794048
$987$	0	0
$988$	160609.	5.17171
$989$	−35156.9	−1.13036
$990$	0	0
$991$	−24357.8	−0.780778	−0.390389	−	0.920650i	$-0.627660\pi$
$991$	−24357.8	−0.780778	−0.390389	+	0.920650i	$0.627660\pi$
$992$	57.9516	0.00185480
$993$	0	0
$994$	−9529.17	−0.304071
$995$	−44562.9	−1.41984
$996$	0	0
$997$	19954.3	0.633862	0.316931	−	0.948449i	$-0.397348\pi$
$997$	19954.3	0.633862	0.316931	+	0.948449i	$0.397348\pi$
$998$	−21353.7	−0.677294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.l.1.4		4
3.2	odd	2		77.4.a.d.1.1	✓	4
12.11	even	2		1232.4.a.s.1.2		4
15.14	odd	2		1925.4.a.p.1.4		4
21.20	even	2		539.4.a.g.1.1		4
33.32	even	2		847.4.a.d.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.1	✓	4	3.2	odd	2
539.4.a.g.1.1		4	21.20	even	2
693.4.a.l.1.4		4	1.1	even	1	trivial
847.4.a.d.1.4		4	33.32	even	2
1232.4.a.s.1.2		4	12.11	even	2
1925.4.a.p.1.4		4	15.14	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)

\(f(q)\)	\(=\)	\(q+4.89098 q^{2} +15.9217 q^{4} -15.5514 q^{5} -7.00000 q^{7} +38.7449 q^{8} -76.0614 q^{10} +11.0000 q^{11} +74.3459 q^{13} -34.2369 q^{14} +62.1271 q^{16} +94.0836 q^{17} +135.682 q^{19} -247.604 q^{20} +53.8008 q^{22} -81.1793 q^{23} +116.845 q^{25} +363.625 q^{26} -111.452 q^{28} +53.4259 q^{29} -9.50536 q^{31} -6.09673 q^{32} +460.161 q^{34} +108.860 q^{35} -9.14224 q^{37} +663.620 q^{38} -602.536 q^{40} +339.461 q^{41} +433.078 q^{43} +175.139 q^{44} -397.046 q^{46} +54.4784 q^{47} +49.0000 q^{49} +571.486 q^{50} +1183.71 q^{52} -123.830 q^{53} -171.065 q^{55} -271.215 q^{56} +261.305 q^{58} +534.396 q^{59} -358.624 q^{61} -46.4905 q^{62} -526.836 q^{64} -1156.18 q^{65} -694.318 q^{67} +1497.97 q^{68} +532.430 q^{70} +278.330 q^{71} -886.688 q^{73} -44.7145 q^{74} +2160.29 q^{76} -77.0000 q^{77} -185.631 q^{79} -966.162 q^{80} +1660.30 q^{82} +122.624 q^{83} -1463.13 q^{85} +2118.18 q^{86} +426.194 q^{88} +847.086 q^{89} -520.422 q^{91} -1292.51 q^{92} +266.453 q^{94} -2110.04 q^{95} +1002.49 q^{97} +239.658 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 26 * q^4 - 10 * q^5 - 28 * q^7 + 18 * q^8 - 2 * q^10 + 44 * q^11 + 58 * q^13 - 14 * q^14 + 2 * q^16 - 4 * q^17 + 258 * q^19 - 182 * q^20 + 22 * q^22 - 8 * q^23 + 80 * q^25 + 482 * q^26 - 182 * q^28 + 396 * q^29 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 + 84 * q^37 + 942 * q^38 - 1026 * q^40 - 52 * q^41 + 408 * q^43 + 286 * q^44 + 368 * q^46 - 8 * q^47 + 196 * q^49 + 1642 * q^50 + 2030 * q^52 - 624 * q^53 - 110 * q^55 - 126 * q^56 + 864 * q^58 + 238 * q^59 - 162 * q^61 - 688 * q^62 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 14 * q^70 - 1788 * q^71 + 1456 * q^73 - 996 * q^74 + 3042 * q^76 - 308 * q^77 - 1324 * q^79 - 2342 * q^80 + 1984 * q^82 - 450 * q^83 - 1736 * q^85 + 4380 * q^86 + 198 * q^88 + 3072 * q^89 - 406 * q^91 - 544 * q^92 - 1696 * q^94 - 24 * q^95 - 652 * q^97 + 98 * q^98

Introduction

L-functions

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Database

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