LMFDB - Embedded newform 693.4.a.s.1.8

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			1.8
Root			$-4.93411$ 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)

$f(q)$	$=$	\(q+4.93411 q^{2} +16.3455 q^{4} -14.7743 q^{5} -7.00000 q^{7} +41.1774 q^{8} -72.8980 q^{10} +11.0000 q^{11} -32.3144 q^{13} -34.5388 q^{14} +72.4103 q^{16} +45.2603 q^{17} -146.544 q^{19} -241.493 q^{20} +54.2752 q^{22} -153.307 q^{23} +93.2798 q^{25} -159.443 q^{26} -114.418 q^{28} -78.6971 q^{29} +106.083 q^{31} +27.8612 q^{32} +223.320 q^{34} +103.420 q^{35} +94.0807 q^{37} -723.064 q^{38} -608.367 q^{40} -417.245 q^{41} +60.3987 q^{43} +179.800 q^{44} -756.431 q^{46} -253.648 q^{47} +49.0000 q^{49} +460.253 q^{50} -528.194 q^{52} +647.312 q^{53} -162.517 q^{55} -288.242 q^{56} -388.300 q^{58} -559.388 q^{59} -602.774 q^{61} +523.425 q^{62} -441.812 q^{64} +477.423 q^{65} +343.967 q^{67} +739.801 q^{68} +510.286 q^{70} +224.903 q^{71} +1051.84 q^{73} +464.205 q^{74} -2395.33 q^{76} -77.0000 q^{77} -102.782 q^{79} -1069.81 q^{80} -2058.73 q^{82} -730.867 q^{83} -668.690 q^{85} +298.014 q^{86} +452.952 q^{88} +43.7903 q^{89} +226.201 q^{91} -2505.86 q^{92} -1251.53 q^{94} +2165.08 q^{95} +8.01384 q^{97} +241.771 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.93411	1.74447	0.872236	−	0.489085i	$-0.162669\pi$
$2$	4.93411	1.74447	0.872236	+	0.489085i	$0.162669\pi$
$3$	0	0
$3$	0	0
$4$	16.3455	2.04318
$5$	−14.7743	−1.32145	−0.660727	−	0.750627i	$-0.729752\pi$
$5$	−14.7743	−1.32145	−0.660727	+	0.750627i	$0.729752\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	41.1774	1.81980
$9$	0	0
$10$	−72.8980	−2.30524
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−32.3144	−0.689417	−0.344708	−	0.938710i	$-0.612022\pi$
$13$	−32.3144	−0.689417	−0.344708	+	0.938710i	$0.612022\pi$
$14$	−34.5388	−0.659348
$15$	0	0
$16$	72.4103	1.13141
$17$	45.2603	0.645720	0.322860	−	0.946447i	$-0.395356\pi$
$17$	45.2603	0.645720	0.322860	+	0.946447i	$0.395356\pi$
$18$	0	0
$19$	−146.544	−1.76945	−0.884723	−	0.466118i	$-0.845652\pi$
$19$	−146.544	−1.76945	−0.884723	+	0.466118i	$0.845652\pi$
$20$	−241.493	−2.69997
$21$	0	0
$22$	54.2752	0.525978
$23$	−153.307	−1.38985	−0.694926	−	0.719081i	$-0.744563\pi$
$23$	−153.307	−1.38985	−0.694926	+	0.719081i	$0.744563\pi$
$24$	0	0
$25$	93.2798	0.746239
$26$	−159.443	−1.20267
$27$	0	0
$28$	−114.418	−0.772250
$29$	−78.6971	−0.503920	−0.251960	−	0.967738i	$-0.581075\pi$
$29$	−78.6971	−0.503920	−0.251960	+	0.967738i	$0.581075\pi$
$30$	0	0
$31$	106.083	0.614614	0.307307	−	0.951610i	$-0.400572\pi$
$31$	106.083	0.614614	0.307307	+	0.951610i	$0.400572\pi$
$32$	27.8612	0.153913
$33$	0	0
$34$	223.320	1.12644
$35$	103.420	0.499462
$36$	0	0
$37$	94.0807	0.418021	0.209010	−	0.977913i	$-0.432976\pi$
$37$	94.0807	0.418021	0.209010	+	0.977913i	$0.432976\pi$
$38$	−723.064	−3.08675
$39$	0	0
$40$	−608.367	−2.40478
$41$	−417.245	−1.58933	−0.794667	−	0.607045i	$-0.792355\pi$
$41$	−417.245	−1.58933	−0.794667	+	0.607045i	$0.792355\pi$
$42$	0	0
$43$	60.3987	0.214203	0.107101	−	0.994248i	$-0.465843\pi$
$43$	60.3987	0.214203	0.107101	+	0.994248i	$0.465843\pi$
$44$	179.800	0.616043
$45$	0	0
$46$	−756.431	−2.42456
$47$	−253.648	−0.787200	−0.393600	−	0.919282i	$-0.628771\pi$
$47$	−253.648	−0.787200	−0.393600	+	0.919282i	$0.628771\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	460.253	1.30179
$51$	0	0
$52$	−528.194	−1.40860
$53$	647.312	1.67765	0.838823	−	0.544405i	$-0.183244\pi$
$53$	647.312	1.67765	0.838823	+	0.544405i	$0.183244\pi$
$54$	0	0
$55$	−162.517	−0.398433
$56$	−288.242	−0.687820
$57$	0	0
$58$	−388.300	−0.879075
$59$	−559.388	−1.23434	−0.617171	−	0.786829i	$-0.711721\pi$
$59$	−559.388	−1.23434	−0.617171	+	0.786829i	$0.711721\pi$
$60$	0	0
$61$	−602.774	−1.26520	−0.632601	−	0.774478i	$-0.718013\pi$
$61$	−602.774	−1.26520	−0.632601	+	0.774478i	$0.718013\pi$
$62$	523.425	1.07218
$63$	0	0
$64$	−441.812	−0.862914
$65$	477.423	0.911032
$66$	0	0
$67$	343.967	0.627197	0.313599	−	0.949556i	$-0.398465\pi$
$67$	343.967	0.627197	0.313599	+	0.949556i	$0.398465\pi$
$68$	739.801	1.31932
$69$	0	0
$70$	510.286	0.871298
$71$	224.903	0.375930	0.187965	−	0.982176i	$-0.439811\pi$
$71$	224.903	0.375930	0.187965	+	0.982176i	$0.439811\pi$
$72$	0	0
$73$	1051.84	1.68642	0.843209	−	0.537586i	$-0.180664\pi$
$73$	1051.84	1.68642	0.843209	+	0.537586i	$0.180664\pi$
$74$	464.205	0.729225
$75$	0	0
$76$	−2395.33	−3.61530
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−102.782	−0.146378	−0.0731890	−	0.997318i	$-0.523318\pi$
$79$	−102.782	−0.146378	−0.0731890	+	0.997318i	$0.523318\pi$
$80$	−1069.81	−1.49511
$81$	0	0
$82$	−2058.73	−2.77255
$83$	−730.867	−0.966542	−0.483271	−	0.875471i	$-0.660552\pi$
$83$	−730.867	−0.966542	−0.483271	+	0.875471i	$0.660552\pi$
$84$	0	0
$85$	−668.690	−0.853289
$86$	298.014	0.373670
$87$	0	0
$88$	452.952	0.548691
$89$	43.7903	0.0521546	0.0260773	−	0.999660i	$-0.491698\pi$
$89$	43.7903	0.0521546	0.0260773	+	0.999660i	$0.491698\pi$
$90$	0	0
$91$	226.201	0.260575
$92$	−2505.86	−2.83972
$93$	0	0
$94$	−1251.53	−1.37325
$95$	2165.08	2.33824
$96$	0	0
$97$	8.01384	0.00838847	0.00419424	−	0.999991i	$-0.498665\pi$
$97$	8.01384	0.00838847	0.00419424	+	0.999991i	$0.498665\pi$
$98$	241.771	0.249210
$99$	0	0
$100$	1524.70	1.52470
$101$	−369.969	−0.364488	−0.182244	−	0.983253i	$-0.558336\pi$
$101$	−369.969	−0.364488	−0.182244	+	0.983253i	$0.558336\pi$
$102$	0	0
$103$	1350.49	1.29192	0.645958	−	0.763373i	$-0.276458\pi$
$103$	1350.49	1.29192	0.645958	+	0.763373i	$0.276458\pi$
$104$	−1330.63	−1.25460
$105$	0	0
$106$	3193.91	2.92661
$107$	−1590.45	−1.43696	−0.718479	−	0.695549i	$-0.755161\pi$
$107$	−1590.45	−1.43696	−0.718479	+	0.695549i	$0.755161\pi$
$108$	0	0
$109$	2230.29	1.95984	0.979922	−	0.199382i	$-0.0638933\pi$
$109$	2230.29	1.95984	0.979922	+	0.199382i	$0.0638933\pi$
$110$	−801.878	−0.695055
$111$	0	0
$112$	−506.872	−0.427633
$113$	1145.75	0.953830	0.476915	−	0.878949i	$-0.341755\pi$
$113$	1145.75	0.953830	0.476915	+	0.878949i	$0.341755\pi$
$114$	0	0
$115$	2265.00	1.83663
$116$	−1286.34	−1.02960
$117$	0	0
$118$	−2760.08	−2.15327
$119$	−316.822	−0.244059
$120$	0	0
$121$	121.000	0.0909091
$122$	−2974.15	−2.20711
$123$	0	0
$124$	1733.97	1.25577
$125$	468.643	0.335334
$126$	0	0
$127$	916.800	0.640574	0.320287	−	0.947321i	$-0.396221\pi$
$127$	916.800	0.640574	0.320287	+	0.947321i	$0.396221\pi$
$128$	−2402.84	−1.65924
$129$	0	0
$130$	2355.66	1.58927
$131$	488.489	0.325798	0.162899	−	0.986643i	$-0.447916\pi$
$131$	488.489	0.325798	0.162899	+	0.986643i	$0.447916\pi$
$132$	0	0
$133$	1025.81	0.668787
$134$	1697.17	1.09413
$135$	0	0
$136$	1863.70	1.17508
$137$	433.883	0.270578	0.135289	−	0.990806i	$-0.456804\pi$
$137$	433.883	0.270578	0.135289	+	0.990806i	$0.456804\pi$
$138$	0	0
$139$	−657.839	−0.401418	−0.200709	−	0.979651i	$-0.564325\pi$
$139$	−657.839	−0.401418	−0.200709	+	0.979651i	$0.564325\pi$
$140$	1690.45	1.02049
$141$	0	0
$142$	1109.70	0.655800
$143$	−355.459	−0.207867
$144$	0	0
$145$	1162.69	0.665907
$146$	5189.89	2.94191
$147$	0	0
$148$	1537.79	0.854093
$149$	−1672.94	−0.919815	−0.459907	−	0.887967i	$-0.652117\pi$
$149$	−1672.94	−0.919815	−0.459907	+	0.887967i	$0.652117\pi$
$150$	0	0
$151$	−1730.15	−0.932432	−0.466216	−	0.884671i	$-0.654383\pi$
$151$	−1730.15	−0.932432	−0.466216	+	0.884671i	$0.654383\pi$
$152$	−6034.30	−3.22004
$153$	0	0
$154$	−379.927	−0.198801
$155$	−1567.30	−0.812184
$156$	0	0
$157$	1606.88	0.816832	0.408416	−	0.912796i	$-0.366081\pi$
$157$	1606.88	0.816832	0.408416	+	0.912796i	$0.366081\pi$
$158$	−507.137	−0.255352
$159$	0	0
$160$	−411.630	−0.203389
$161$	1073.15	0.525315
$162$	0	0
$163$	1439.38	0.691664	0.345832	−	0.938296i	$-0.387597\pi$
$163$	1439.38	0.691664	0.345832	+	0.938296i	$0.387597\pi$
$164$	−6820.06	−3.24730
$165$	0	0
$166$	−3606.18	−1.68611
$167$	1875.59	0.869087	0.434544	−	0.900651i	$-0.356910\pi$
$167$	1875.59	0.869087	0.434544	+	0.900651i	$0.356910\pi$
$168$	0	0
$169$	−1152.78	−0.524705
$170$	−3299.39	−1.48854
$171$	0	0
$172$	987.244	0.437655
$173$	−576.664	−0.253428	−0.126714	−	0.991939i	$-0.540443\pi$
$173$	−576.664	−0.253428	−0.126714	+	0.991939i	$0.540443\pi$
$174$	0	0
$175$	−652.959	−0.282052
$176$	796.513	0.341133
$177$	0	0
$178$	216.066	0.0909822
$179$	1365.58	0.570215	0.285107	−	0.958496i	$-0.407971\pi$
$179$	1365.58	0.570215	0.285107	+	0.958496i	$0.407971\pi$
$180$	0	0
$181$	−1409.35	−0.578765	−0.289383	−	0.957214i	$-0.593450\pi$
$181$	−1409.35	−0.578765	−0.289383	+	0.957214i	$0.593450\pi$
$182$	1116.10	0.454566
$183$	0	0
$184$	−6312.77	−2.52926
$185$	−1389.98	−0.552395
$186$	0	0
$187$	497.864	0.194692
$188$	−4146.00	−1.60839
$189$	0	0
$190$	10682.8	4.07899
$191$	−5210.94	−1.97409	−0.987043	−	0.160457i	$-0.948703\pi$
$191$	−5210.94	−1.97409	−0.987043	+	0.160457i	$0.948703\pi$
$192$	0	0
$193$	−4236.23	−1.57995	−0.789976	−	0.613138i	$-0.789907\pi$
$193$	−4236.23	−1.57995	−0.789976	+	0.613138i	$0.789907\pi$
$194$	39.5412	0.0146335
$195$	0	0
$196$	800.927	0.291883
$197$	−2534.40	−0.916592	−0.458296	−	0.888800i	$-0.651540\pi$
$197$	−2534.40	−0.916592	−0.458296	+	0.888800i	$0.651540\pi$
$198$	0	0
$199$	−1486.78	−0.529622	−0.264811	−	0.964300i	$-0.585310\pi$
$199$	−1486.78	−0.529622	−0.264811	+	0.964300i	$0.585310\pi$
$200$	3841.02	1.35801
$201$	0	0
$202$	−1825.47	−0.635840
$203$	550.880	0.190464
$204$	0	0
$205$	6164.50	2.10023
$206$	6663.45	2.25371
$207$	0	0
$208$	−2339.90	−0.780014
$209$	−1611.98	−0.533508
$210$	0	0
$211$	4445.93	1.45057	0.725285	−	0.688449i	$-0.241708\pi$
$211$	4445.93	1.45057	0.725285	+	0.688449i	$0.241708\pi$
$212$	10580.6	3.42773
$213$	0	0
$214$	−7847.45	−2.50673
$215$	−892.348	−0.283059
$216$	0	0
$217$	−742.580	−0.232302
$218$	11004.5	3.41889
$219$	0	0
$220$	−2656.42	−0.814071
$221$	−1462.56	−0.445170
$222$	0	0
$223$	5051.11	1.51680	0.758402	−	0.651787i	$-0.225980\pi$
$223$	5051.11	1.51680	0.758402	+	0.651787i	$0.225980\pi$
$224$	−195.028	−0.0581736
$225$	0	0
$226$	5653.24	1.66393
$227$	−694.112	−0.202951	−0.101475	−	0.994838i	$-0.532356\pi$
$227$	−694.112	−0.202951	−0.101475	+	0.994838i	$0.532356\pi$
$228$	0	0
$229$	1791.51	0.516972	0.258486	−	0.966015i	$-0.416776\pi$
$229$	1791.51	0.516972	0.258486	+	0.966015i	$0.416776\pi$
$230$	11175.7	3.20394
$231$	0	0
$232$	−3240.54	−0.917035
$233$	−4166.66	−1.17153	−0.585766	−	0.810481i	$-0.699206\pi$
$233$	−4166.66	−1.17153	−0.585766	+	0.810481i	$0.699206\pi$
$234$	0	0
$235$	3747.48	1.04025
$236$	−9143.46	−2.52198
$237$	0	0
$238$	−1563.24	−0.425755
$239$	6789.46	1.83755	0.918774	−	0.394785i	$-0.129181\pi$
$239$	6789.46	1.83755	0.918774	+	0.394785i	$0.129181\pi$
$240$	0	0
$241$	−7170.37	−1.91653	−0.958265	−	0.285880i	$-0.907714\pi$
$241$	−7170.37	−1.91653	−0.958265	+	0.285880i	$0.907714\pi$
$242$	597.027	0.158588
$243$	0	0
$244$	−9852.62	−2.58504
$245$	−723.941	−0.188779
$246$	0	0
$247$	4735.48	1.21988
$248$	4368.22	1.11848
$249$	0	0
$250$	2312.34	0.584980
$251$	−4269.76	−1.07372	−0.536862	−	0.843670i	$-0.680391\pi$
$251$	−4269.76	−1.07372	−0.536862	+	0.843670i	$0.680391\pi$
$252$	0	0
$253$	−1686.37	−0.419056
$254$	4523.59	1.11746
$255$	0	0
$256$	−8321.38	−2.03159
$257$	1267.24	0.307581	0.153790	−	0.988103i	$-0.450852\pi$
$257$	1267.24	0.307581	0.153790	+	0.988103i	$0.450852\pi$
$258$	0	0
$259$	−658.565	−0.157997
$260$	7803.70	1.86140
$261$	0	0
$262$	2410.26	0.568345
$263$	−5297.20	−1.24197	−0.620987	−	0.783821i	$-0.713268\pi$
$263$	−5297.20	−1.24197	−0.620987	+	0.783821i	$0.713268\pi$
$264$	0	0
$265$	−9563.59	−2.21693
$266$	5061.44	1.16668
$267$	0	0
$268$	5622.29	1.28148
$269$	138.869	0.0314758	0.0157379	−	0.999876i	$-0.494990\pi$
$269$	138.869	0.0314758	0.0157379	+	0.999876i	$0.494990\pi$
$270$	0	0
$271$	6201.39	1.39006	0.695032	−	0.718979i	$-0.255390\pi$
$271$	6201.39	1.39006	0.695032	+	0.718979i	$0.255390\pi$
$272$	3277.31	0.730575
$273$	0	0
$274$	2140.83	0.472015
$275$	1026.08	0.224999
$276$	0	0
$277$	−1486.69	−0.322478	−0.161239	−	0.986915i	$-0.551549\pi$
$277$	−1486.69	−0.322478	−0.161239	+	0.986915i	$0.551549\pi$
$278$	−3245.85	−0.700263
$279$	0	0
$280$	4258.57	0.908923
$281$	4425.70	0.939555	0.469778	−	0.882785i	$-0.344334\pi$
$281$	4425.70	0.939555	0.469778	+	0.882785i	$0.344334\pi$
$282$	0	0
$283$	−1502.45	−0.315588	−0.157794	−	0.987472i	$-0.550438\pi$
$283$	−1502.45	−0.315588	−0.157794	+	0.987472i	$0.550438\pi$
$284$	3676.14	0.768094
$285$	0	0
$286$	−1753.87	−0.362618
$287$	2920.72	0.600712
$288$	0	0
$289$	−2864.50	−0.583045
$290$	5736.87	1.16166
$291$	0	0
$292$	17192.8	3.44566
$293$	−5606.97	−1.11796	−0.558980	−	0.829181i	$-0.688807\pi$
$293$	−5606.97	−1.11796	−0.558980	+	0.829181i	$0.688807\pi$
$294$	0	0
$295$	8264.57	1.63112
$296$	3874.00	0.760715
$297$	0	0
$298$	−8254.46	−1.60459
$299$	4954.01	0.958188
$300$	0	0
$301$	−422.791	−0.0809609
$302$	−8536.73	−1.62660
$303$	0	0
$304$	−10611.3	−2.00197
$305$	8905.56	1.67191
$306$	0	0
$307$	2660.03	0.494514	0.247257	−	0.968950i	$-0.420471\pi$
$307$	2660.03	0.494514	0.247257	+	0.968950i	$0.420471\pi$
$308$	−1258.60	−0.232842
$309$	0	0
$310$	−7733.23	−1.41683
$311$	−5233.58	−0.954241	−0.477121	−	0.878838i	$-0.658320\pi$
$311$	−5233.58	−0.954241	−0.477121	+	0.878838i	$0.658320\pi$
$312$	0	0
$313$	2462.59	0.444708	0.222354	−	0.974966i	$-0.428626\pi$
$313$	2462.59	0.444708	0.222354	+	0.974966i	$0.428626\pi$
$314$	7928.50	1.42494
$315$	0	0
$316$	−1680.02	−0.299077
$317$	−3920.02	−0.694543	−0.347272	−	0.937765i	$-0.612892\pi$
$317$	−3920.02	−0.694543	−0.347272	+	0.937765i	$0.612892\pi$
$318$	0	0
$319$	−865.669	−0.151938
$320$	6527.46	1.14030
$321$	0	0
$322$	5295.02	0.916397
$323$	−6632.62	−1.14257
$324$	0	0
$325$	−3014.29	−0.514469
$326$	7102.08	1.20659
$327$	0	0
$328$	−17181.1	−2.89227
$329$	1775.54	0.297534
$330$	0	0
$331$	−11647.6	−1.93417	−0.967086	−	0.254450i	$-0.918106\pi$
$331$	−11647.6	−1.93417	−0.967086	+	0.254450i	$0.918106\pi$
$332$	−11946.3	−1.97482
$333$	0	0
$334$	9254.37	1.51610
$335$	−5081.87	−0.828812
$336$	0	0
$337$	−8628.94	−1.39480	−0.697401	−	0.716681i	$-0.745660\pi$
$337$	−8628.94	−1.39480	−0.697401	+	0.716681i	$0.745660\pi$
$338$	−5687.93	−0.915333
$339$	0	0
$340$	−10930.0	−1.74343
$341$	1166.91	0.185313
$342$	0	0
$343$	−343.000	−0.0539949
$344$	2487.06	0.389806
$345$	0	0
$346$	−2845.33	−0.442097
$347$	−1753.00	−0.271199	−0.135600	−	0.990764i	$-0.543296\pi$
$347$	−1753.00	−0.271199	−0.135600	+	0.990764i	$0.543296\pi$
$348$	0	0
$349$	−11437.8	−1.75430	−0.877152	−	0.480213i	$-0.840559\pi$
$349$	−11437.8	−1.75430	−0.877152	+	0.480213i	$0.840559\pi$
$350$	−3221.77	−0.492031
$351$	0	0
$352$	306.473	0.0464065
$353$	363.466	0.0548027	0.0274014	−	0.999625i	$-0.491277\pi$
$353$	363.466	0.0548027	0.0274014	+	0.999625i	$0.491277\pi$
$354$	0	0
$355$	−3322.78	−0.496774
$356$	715.772	0.106561
$357$	0	0
$358$	6737.94	0.994724
$359$	5084.34	0.747469	0.373734	−	0.927536i	$-0.378077\pi$
$359$	5084.34	0.747469	0.373734	+	0.927536i	$0.378077\pi$
$360$	0	0
$361$	14616.1	2.13094
$362$	−6953.91	−1.00964
$363$	0	0
$364$	3697.36	0.532402
$365$	−15540.2	−2.22852
$366$	0	0
$367$	3672.38	0.522334	0.261167	−	0.965294i	$-0.415893\pi$
$367$	3672.38	0.522334	0.261167	+	0.965294i	$0.415893\pi$
$368$	−11101.0	−1.57249
$369$	0	0
$370$	−6858.30	−0.963637
$371$	−4531.19	−0.634090
$372$	0	0
$373$	9877.17	1.37110	0.685550	−	0.728025i	$-0.259562\pi$
$373$	9877.17	1.37110	0.685550	+	0.728025i	$0.259562\pi$
$374$	2456.52	0.339635
$375$	0	0
$376$	−10444.6	−1.43255
$377$	2543.05	0.347411
$378$	0	0
$379$	−806.035	−0.109243	−0.0546217	−	0.998507i	$-0.517395\pi$
$379$	−806.035	−0.109243	−0.0546217	+	0.998507i	$0.517395\pi$
$380$	35389.3	4.77745
$381$	0	0
$382$	−25711.4	−3.44374
$383$	−10298.5	−1.37397	−0.686986	−	0.726671i	$-0.741067\pi$
$383$	−10298.5	−1.37397	−0.686986	+	0.726671i	$0.741067\pi$
$384$	0	0
$385$	1137.62	0.150594
$386$	−20902.0	−2.75618
$387$	0	0
$388$	130.990	0.0171392
$389$	−261.011	−0.0340200	−0.0170100	−	0.999855i	$-0.505415\pi$
$389$	−261.011	−0.0340200	−0.0170100	+	0.999855i	$0.505415\pi$
$390$	0	0
$391$	−6938.70	−0.897456
$392$	2017.69	0.259972
$393$	0	0
$394$	−12505.0	−1.59897
$395$	1518.53	0.193432
$396$	0	0
$397$	5217.48	0.659591	0.329796	−	0.944052i	$-0.393020\pi$
$397$	5217.48	0.659591	0.329796	+	0.944052i	$0.393020\pi$
$398$	−7335.92	−0.923911
$399$	0	0
$400$	6754.42	0.844303
$401$	−9584.00	−1.19352	−0.596761	−	0.802419i	$-0.703546\pi$
$401$	−9584.00	−1.19352	−0.596761	+	0.802419i	$0.703546\pi$
$402$	0	0
$403$	−3428.01	−0.423725
$404$	−6047.32	−0.744716
$405$	0	0
$406$	2718.10	0.332259
$407$	1034.89	0.126038
$408$	0	0
$409$	8828.72	1.06737	0.533683	−	0.845685i	$-0.320808\pi$
$409$	8828.72	1.06737	0.533683	+	0.845685i	$0.320808\pi$
$410$	30416.3	3.66379
$411$	0	0
$412$	22074.3	2.63962
$413$	3915.72	0.466537
$414$	0	0
$415$	10798.0	1.27724
$416$	−900.320	−0.106110
$417$	0	0
$418$	−7953.70	−0.930689
$419$	8400.12	0.979410	0.489705	−	0.871888i	$-0.337105\pi$
$419$	8400.12	0.979410	0.489705	+	0.871888i	$0.337105\pi$
$420$	0	0
$421$	1479.36	0.171258	0.0856289	−	0.996327i	$-0.472710\pi$
$421$	1479.36	0.171258	0.0856289	+	0.996327i	$0.472710\pi$
$422$	21936.7	2.53048
$423$	0	0
$424$	26654.7	3.05298
$425$	4221.88	0.481861
$426$	0	0
$427$	4219.42	0.478201
$428$	−25996.6	−2.93597
$429$	0	0
$430$	−4402.94	−0.493788
$431$	−5864.95	−0.655463	−0.327731	−	0.944771i	$-0.606284\pi$
$431$	−5864.95	−0.655463	−0.327731	+	0.944771i	$0.606284\pi$
$432$	0	0
$433$	−4186.93	−0.464690	−0.232345	−	0.972633i	$-0.574640\pi$
$433$	−4186.93	−0.464690	−0.232345	+	0.972633i	$0.574640\pi$
$434$	−3663.97	−0.405245
$435$	0	0
$436$	36455.1	4.00432
$437$	22466.1	2.45927
$438$	0	0
$439$	4744.75	0.515842	0.257921	−	0.966166i	$-0.416963\pi$
$439$	4744.75	0.515842	0.257921	+	0.966166i	$0.416963\pi$
$440$	−6692.04	−0.725069
$441$	0	0
$442$	−7216.45	−0.776587
$443$	−6661.14	−0.714403	−0.357201	−	0.934027i	$-0.616269\pi$
$443$	−6661.14	−0.714403	−0.357201	+	0.934027i	$0.616269\pi$
$444$	0	0
$445$	−646.970	−0.0689199
$446$	24922.7	2.64602
$447$	0	0
$448$	3092.68	0.326151
$449$	18086.5	1.90101	0.950506	−	0.310707i	$-0.100566\pi$
$449$	18086.5	1.90101	0.950506	+	0.310707i	$0.100566\pi$
$450$	0	0
$451$	−4589.70	−0.479202
$452$	18727.8	1.94885
$453$	0	0
$454$	−3424.83	−0.354042
$455$	−3341.96	−0.344338
$456$	0	0
$457$	403.581	0.0413101	0.0206550	−	0.999787i	$-0.493425\pi$
$457$	403.581	0.0413101	0.0206550	+	0.999787i	$0.493425\pi$
$458$	8839.53	0.901843
$459$	0	0
$460$	37022.4	3.75256
$461$	−12833.0	−1.29651	−0.648256	−	0.761422i	$-0.724501\pi$
$461$	−12833.0	−1.29651	−0.648256	+	0.761422i	$0.724501\pi$
$462$	0	0
$463$	−6730.20	−0.675549	−0.337774	−	0.941227i	$-0.609674\pi$
$463$	−6730.20	−0.675549	−0.337774	+	0.941227i	$0.609674\pi$
$464$	−5698.48	−0.570141
$465$	0	0
$466$	−20558.7	−2.04370
$467$	−11787.7	−1.16803	−0.584015	−	0.811743i	$-0.698519\pi$
$467$	−11787.7	−1.16803	−0.584015	+	0.811743i	$0.698519\pi$
$468$	0	0
$469$	−2407.77	−0.237058
$470$	18490.5	1.81468
$471$	0	0
$472$	−23034.2	−2.24626
$473$	664.385	0.0645845
$474$	0	0
$475$	−13669.6	−1.32043
$476$	−5178.61	−0.498658
$477$	0	0
$478$	33500.0	3.20555
$479$	8240.02	0.786005	0.393002	−	0.919537i	$-0.371436\pi$
$479$	8240.02	0.786005	0.393002	+	0.919537i	$0.371436\pi$
$480$	0	0
$481$	−3040.17	−0.288190
$482$	−35379.4	−3.34333
$483$	0	0
$484$	1977.80	0.185744
$485$	−118.399	−0.0110850
$486$	0	0
$487$	14711.3	1.36886	0.684429	−	0.729080i	$-0.260052\pi$
$487$	14711.3	1.36886	0.684429	+	0.729080i	$0.260052\pi$
$488$	−24820.7	−2.30242
$489$	0	0
$490$	−3572.00	−0.329320
$491$	−17891.4	−1.64445	−0.822227	−	0.569160i	$-0.807268\pi$
$491$	−17891.4	−1.64445	−0.822227	+	0.569160i	$0.807268\pi$
$492$	0	0
$493$	−3561.86	−0.325392
$494$	23365.4	2.12805
$495$	0	0
$496$	7681.49	0.695381
$497$	−1574.32	−0.142088
$498$	0	0
$499$	−6616.76	−0.593601	−0.296801	−	0.954939i	$-0.595920\pi$
$499$	−6616.76	−0.593601	−0.296801	+	0.954939i	$0.595920\pi$
$500$	7660.18	0.685148
$501$	0	0
$502$	−21067.5	−1.87308
$503$	12112.3	1.07368	0.536841	−	0.843683i	$-0.319617\pi$
$503$	12112.3	1.07368	0.536841	+	0.843683i	$0.319617\pi$
$504$	0	0
$505$	5466.04	0.481654
$506$	−8320.75	−0.731032
$507$	0	0
$508$	14985.5	1.30881
$509$	9955.07	0.866897	0.433449	−	0.901178i	$-0.357297\pi$
$509$	9955.07	0.866897	0.433449	+	0.901178i	$0.357297\pi$
$510$	0	0
$511$	−7362.88	−0.637406
$512$	−21835.9	−1.88481
$513$	0	0
$514$	6252.70	0.536566
$515$	−19952.5	−1.70721
$516$	0	0
$517$	−2790.13	−0.237350
$518$	−3249.43	−0.275621
$519$	0	0
$520$	19659.1	1.65790
$521$	10077.7	0.847434	0.423717	−	0.905795i	$-0.360725\pi$
$521$	10077.7	0.847434	0.423717	+	0.905795i	$0.360725\pi$
$522$	0	0
$523$	1287.74	0.107666	0.0538328	−	0.998550i	$-0.482856\pi$
$523$	1287.74	0.107666	0.0538328	+	0.998550i	$0.482856\pi$
$524$	7984.58	0.665664
$525$	0	0
$526$	−26137.0	−2.16659
$527$	4801.35	0.396869
$528$	0	0
$529$	11335.9	0.931691
$530$	−47187.8	−3.86737
$531$	0	0
$532$	16767.3	1.36645
$533$	13483.0	1.09571
$534$	0	0
$535$	23497.8	1.89887
$536$	14163.7	1.14138
$537$	0	0
$538$	685.194	0.0549086
$539$	539.000	0.0430730
$540$	0	0
$541$	21639.6	1.71970	0.859850	−	0.510546i	$-0.170557\pi$
$541$	21639.6	1.71970	0.859850	+	0.510546i	$0.170557\pi$
$542$	30598.3	2.42493
$543$	0	0
$544$	1261.01	0.0993847
$545$	−32951.0	−2.58984
$546$	0	0
$547$	7156.42	0.559390	0.279695	−	0.960089i	$-0.409767\pi$
$547$	7156.42	0.559390	0.279695	+	0.960089i	$0.409767\pi$
$548$	7092.02	0.552840
$549$	0	0
$550$	5062.78	0.392505
$551$	11532.6	0.891660
$552$	0	0
$553$	719.473	0.0553257
$554$	−7335.47	−0.562553
$555$	0	0
$556$	−10752.7	−0.820171
$557$	−14142.9	−1.07586	−0.537929	−	0.842990i	$-0.680793\pi$
$557$	−14142.9	−1.07586	−0.537929	+	0.842990i	$0.680793\pi$
$558$	0	0
$559$	−1951.75	−0.147675
$560$	7488.68	0.565097
$561$	0	0
$562$	21836.9	1.63903
$563$	−23097.9	−1.72906	−0.864529	−	0.502583i	$-0.832383\pi$
$563$	−23097.9	−1.72906	−0.864529	+	0.502583i	$0.832383\pi$
$564$	0	0
$565$	−16927.6	−1.26044
$566$	−7413.25	−0.550534
$567$	0	0
$568$	9260.92	0.684119
$569$	6152.18	0.453274	0.226637	−	0.973979i	$-0.427227\pi$
$569$	6152.18	0.453274	0.226637	+	0.973979i	$0.427227\pi$
$570$	0	0
$571$	−22342.5	−1.63748	−0.818741	−	0.574162i	$-0.805328\pi$
$571$	−22342.5	−1.63748	−0.818741	+	0.574162i	$0.805328\pi$
$572$	−5810.14	−0.424710
$573$	0	0
$574$	14411.1	1.04793
$575$	−14300.4	−1.03716
$576$	0	0
$577$	−23692.0	−1.70938	−0.854689	−	0.519140i	$-0.826252\pi$
$577$	−23692.0	−1.70938	−0.854689	+	0.519140i	$0.826252\pi$
$578$	−14133.8	−1.01711
$579$	0	0
$580$	19004.8	1.36057
$581$	5116.07	0.365319
$582$	0	0
$583$	7120.44	0.505829
$584$	43312.0	3.06895
$585$	0	0
$586$	−27665.4	−1.95025
$587$	3747.23	0.263483	0.131742	−	0.991284i	$-0.457943\pi$
$587$	3747.23	0.263483	0.131742	+	0.991284i	$0.457943\pi$
$588$	0	0
$589$	−15545.8	−1.08753
$590$	40778.3	2.84545
$591$	0	0
$592$	6812.41	0.472953
$593$	14175.4	0.981639	0.490820	−	0.871261i	$-0.336697\pi$
$593$	14175.4	0.981639	0.490820	+	0.871261i	$0.336697\pi$
$594$	0	0
$595$	4680.83	0.322513
$596$	−27344.9	−1.87935
$597$	0	0
$598$	24443.7	1.67153
$599$	−14398.5	−0.982151	−0.491075	−	0.871117i	$-0.663396\pi$
$599$	−14398.5	−0.982151	−0.491075	+	0.871117i	$0.663396\pi$
$600$	0	0
$601$	14943.3	1.01423	0.507114	−	0.861879i	$-0.330712\pi$
$601$	14943.3	1.01423	0.507114	+	0.861879i	$0.330712\pi$
$602$	−2086.10	−0.141234
$603$	0	0
$604$	−28280.0	−1.90513
$605$	−1787.69	−0.120132
$606$	0	0
$607$	16994.3	1.13637	0.568185	−	0.822901i	$-0.307646\pi$
$607$	16994.3	1.13637	0.568185	+	0.822901i	$0.307646\pi$
$608$	−4082.89	−0.272340
$609$	0	0
$610$	43941.0	2.91659
$611$	8196.50	0.542709
$612$	0	0
$613$	−19859.4	−1.30851	−0.654253	−	0.756276i	$-0.727017\pi$
$613$	−19859.4	−1.30851	−0.654253	+	0.756276i	$0.727017\pi$
$614$	13124.9	0.862666
$615$	0	0
$616$	−3170.66	−0.207386
$617$	−15541.0	−1.01403	−0.507016	−	0.861937i	$-0.669251\pi$
$617$	−15541.0	−1.01403	−0.507016	+	0.861937i	$0.669251\pi$
$618$	0	0
$619$	−11192.3	−0.726746	−0.363373	−	0.931644i	$-0.618375\pi$
$619$	−11192.3	−0.726746	−0.363373	+	0.931644i	$0.618375\pi$
$620$	−25618.2	−1.65944
$621$	0	0
$622$	−25823.1	−1.66465
$623$	−306.532	−0.0197126
$624$	0	0
$625$	−18583.9	−1.18937
$626$	12150.7	0.775781
$627$	0	0
$628$	26265.1	1.66894
$629$	4258.12	0.269924
$630$	0	0
$631$	−21967.4	−1.38591	−0.692955	−	0.720981i	$-0.743692\pi$
$631$	−21967.4	−1.38591	−0.692955	+	0.720981i	$0.743692\pi$
$632$	−4232.29	−0.266379
$633$	0	0
$634$	−19341.8	−1.21161
$635$	−13545.1	−0.846488
$636$	0	0
$637$	−1583.41	−0.0984881
$638$	−4271.31	−0.265051
$639$	0	0
$640$	35500.3	2.19261
$641$	5862.10	0.361215	0.180608	−	0.983555i	$-0.442194\pi$
$641$	5862.10	0.361215	0.180608	+	0.983555i	$0.442194\pi$
$642$	0	0
$643$	−23698.2	−1.45344	−0.726722	−	0.686932i	$-0.758957\pi$
$643$	−23698.2	−1.45344	−0.726722	+	0.686932i	$0.758957\pi$
$644$	17541.1	1.07331
$645$	0	0
$646$	−32726.1	−1.99318
$647$	13171.6	0.800355	0.400177	−	0.916438i	$-0.368949\pi$
$647$	13171.6	0.800355	0.400177	+	0.916438i	$0.368949\pi$
$648$	0	0
$649$	−6153.27	−0.372168
$650$	−14872.8	−0.897477
$651$	0	0
$652$	23527.4	1.41320
$653$	−13475.4	−0.807555	−0.403778	−	0.914857i	$-0.632303\pi$
$653$	−13475.4	−0.807555	−0.403778	+	0.914857i	$0.632303\pi$
$654$	0	0
$655$	−7217.08	−0.430526
$656$	−30212.8	−1.79819
$657$	0	0
$658$	8760.70	0.519039
$659$	−15669.7	−0.926261	−0.463131	−	0.886290i	$-0.653274\pi$
$659$	−15669.7	−0.926261	−0.463131	+	0.886290i	$0.653274\pi$
$660$	0	0
$661$	24770.3	1.45757	0.728784	−	0.684743i	$-0.240086\pi$
$661$	24770.3	1.45757	0.728784	+	0.684743i	$0.240086\pi$
$662$	−57470.6	−3.37411
$663$	0	0
$664$	−30095.2	−1.75892
$665$	−15155.6	−0.883771
$666$	0	0
$667$	12064.8	0.700375
$668$	30657.4	1.77570
$669$	0	0
$670$	−25074.5	−1.44584
$671$	−6630.51	−0.381473
$672$	0	0
$673$	−16808.7	−0.962748	−0.481374	−	0.876515i	$-0.659862\pi$
$673$	−16808.7	−0.962748	−0.481374	+	0.876515i	$0.659862\pi$
$674$	−42576.2	−2.43319
$675$	0	0
$676$	−18842.7	−1.07207
$677$	−8739.56	−0.496142	−0.248071	−	0.968742i	$-0.579797\pi$
$677$	−8739.56	−0.496142	−0.248071	+	0.968742i	$0.579797\pi$
$678$	0	0
$679$	−56.0969	−0.00317055
$680$	−27534.9	−1.55282
$681$	0	0
$682$	5757.67	0.323274
$683$	−14225.9	−0.796980	−0.398490	−	0.917173i	$-0.630466\pi$
$683$	−14225.9	−0.796980	−0.398490	+	0.917173i	$0.630466\pi$
$684$	0	0
$685$	−6410.32	−0.357556
$686$	−1692.40	−0.0941926
$687$	0	0
$688$	4373.49	0.242351
$689$	−20917.5	−1.15660
$690$	0	0
$691$	11640.4	0.640842	0.320421	−	0.947275i	$-0.396176\pi$
$691$	11640.4	0.640842	0.320421	+	0.947275i	$0.396176\pi$
$692$	−9425.84	−0.517799
$693$	0	0
$694$	−8649.52	−0.473100
$695$	9719.11	0.530456
$696$	0	0
$697$	−18884.6	−1.02627
$698$	−56435.5	−3.06033
$699$	0	0
$700$	−10672.9	−0.576283
$701$	6420.66	0.345942	0.172971	−	0.984927i	$-0.444663\pi$
$701$	6420.66	0.345942	0.172971	+	0.984927i	$0.444663\pi$
$702$	0	0
$703$	−13786.9	−0.739665
$704$	−4859.93	−0.260178
$705$	0	0
$706$	1793.38	0.0956018
$707$	2589.78	0.137764
$708$	0	0
$709$	4597.37	0.243523	0.121762	−	0.992559i	$-0.461146\pi$
$709$	4597.37	0.243523	0.121762	+	0.992559i	$0.461146\pi$
$710$	−16395.0	−0.866609
$711$	0	0
$712$	1803.17	0.0949110
$713$	−16263.2	−0.854223
$714$	0	0
$715$	5251.66	0.274686
$716$	22321.1	1.16505
$717$	0	0
$718$	25086.7	1.30394
$719$	−21320.2	−1.10585	−0.552926	−	0.833230i	$-0.686489\pi$
$719$	−21320.2	−1.10585	−0.552926	+	0.833230i	$0.686489\pi$
$720$	0	0
$721$	−9453.40	−0.488298
$722$	72117.4	3.71736
$723$	0	0
$724$	−23036.5	−1.18252
$725$	−7340.86	−0.376045
$726$	0	0
$727$	−30010.9	−1.53101	−0.765504	−	0.643431i	$-0.777510\pi$
$727$	−30010.9	−1.53101	−0.765504	+	0.643431i	$0.777510\pi$
$728$	9314.38	0.474195
$729$	0	0
$730$	−76677.0	−3.88759
$731$	2733.66	0.138315
$732$	0	0
$733$	−33304.1	−1.67819	−0.839097	−	0.543982i	$-0.816916\pi$
$733$	−33304.1	−1.67819	−0.839097	+	0.543982i	$0.816916\pi$
$734$	18119.9	0.911197
$735$	0	0
$736$	−4271.30	−0.213916
$737$	3783.63	0.189107
$738$	0	0
$739$	13024.7	0.648338	0.324169	−	0.945999i	$-0.394915\pi$
$739$	13024.7	0.648338	0.324169	+	0.945999i	$0.394915\pi$
$740$	−22719.8	−1.12864
$741$	0	0
$742$	−22357.4	−1.10615
$743$	27910.3	1.37810	0.689051	−	0.724713i	$-0.258028\pi$
$743$	27910.3	1.37810	0.689051	+	0.724713i	$0.258028\pi$
$744$	0	0
$745$	24716.5	1.21549
$746$	48735.1	2.39185
$747$	0	0
$748$	8137.81	0.397791
$749$	11133.1	0.543119
$750$	0	0
$751$	3762.49	0.182816	0.0914082	−	0.995814i	$-0.470863\pi$
$751$	3762.49	0.182816	0.0914082	+	0.995814i	$0.470863\pi$
$752$	−18366.8	−0.890647
$753$	0	0
$754$	12547.7	0.606049
$755$	25561.7	1.23217
$756$	0	0
$757$	−34217.1	−1.64286	−0.821429	−	0.570311i	$-0.806823\pi$
$757$	−34217.1	−1.64286	−0.821429	+	0.570311i	$0.806823\pi$
$758$	−3977.07	−0.190572
$759$	0	0
$760$	89152.5	4.25513
$761$	−7946.81	−0.378544	−0.189272	−	0.981925i	$-0.560613\pi$
$761$	−7946.81	−0.378544	−0.189272	+	0.981925i	$0.560613\pi$
$762$	0	0
$763$	−15612.0	−0.740751
$764$	−85175.2	−4.03342
$765$	0	0
$766$	−50814.2	−2.39686
$767$	18076.3	0.850975
$768$	0	0
$769$	21908.8	1.02738	0.513688	−	0.857977i	$-0.328279\pi$
$769$	21908.8	1.02738	0.513688	+	0.857977i	$0.328279\pi$
$770$	5613.15	0.262706
$771$	0	0
$772$	−69243.1	−3.22813
$773$	1544.58	0.0718688	0.0359344	−	0.999354i	$-0.488559\pi$
$773$	1544.58	0.0718688	0.0359344	+	0.999354i	$0.488559\pi$
$774$	0	0
$775$	9895.39	0.458649
$776$	329.989	0.0152654
$777$	0	0
$778$	−1287.86	−0.0593470
$779$	61144.7	2.81224
$780$	0	0
$781$	2473.93	0.113347
$782$	−34236.3	−1.56559
$783$	0	0
$784$	3548.11	0.161630
$785$	−23740.5	−1.07941
$786$	0	0
$787$	−18156.3	−0.822366	−0.411183	−	0.911553i	$-0.634884\pi$
$787$	−18156.3	−0.822366	−0.411183	+	0.911553i	$0.634884\pi$
$788$	−41425.9	−1.87276
$789$	0	0
$790$	7492.59	0.337436
$791$	−8020.23	−0.360514
$792$	0	0
$793$	19478.3	0.872251
$794$	25743.6	1.15064
$795$	0	0
$796$	−24302.0	−1.08211
$797$	16596.1	0.737598	0.368799	−	0.929509i	$-0.379769\pi$
$797$	16596.1	0.737598	0.368799	+	0.929509i	$0.379769\pi$
$798$	0	0
$799$	−11480.2	−0.508311
$800$	2598.89	0.114856
$801$	0	0
$802$	−47288.5	−2.08207
$803$	11570.2	0.508474
$804$	0	0
$805$	−15855.0	−0.694179
$806$	−16914.2	−0.739177
$807$	0	0
$808$	−15234.4	−0.663296
$809$	30953.6	1.34520	0.672602	−	0.740005i	$-0.265177\pi$
$809$	30953.6	1.34520	0.672602	+	0.740005i	$0.265177\pi$
$810$	0	0
$811$	−12176.8	−0.527232	−0.263616	−	0.964628i	$-0.584915\pi$
$811$	−12176.8	−0.527232	−0.263616	+	0.964628i	$0.584915\pi$
$812$	9004.39	0.389153
$813$	0	0
$814$	5106.25	0.219870
$815$	−21265.9	−0.914002
$816$	0	0
$817$	−8851.05	−0.379020
$818$	43561.9	1.86199
$819$	0	0
$820$	100762.	4.29116
$821$	44485.3	1.89105	0.945523	−	0.325556i	$-0.105551\pi$
$821$	44485.3	1.89105	0.945523	+	0.325556i	$0.105551\pi$
$822$	0	0
$823$	−21772.2	−0.922151	−0.461075	−	0.887361i	$-0.652536\pi$
$823$	−21772.2	−0.922151	−0.461075	+	0.887361i	$0.652536\pi$
$824$	55609.5	2.35103
$825$	0	0
$826$	19320.6	0.813861
$827$	33239.7	1.39765	0.698826	−	0.715292i	$-0.253706\pi$
$827$	33239.7	1.39765	0.698826	+	0.715292i	$0.253706\pi$
$828$	0	0
$829$	33230.5	1.39221	0.696105	−	0.717940i	$-0.254915\pi$
$829$	33230.5	1.39221	0.696105	+	0.717940i	$0.254915\pi$
$830$	53278.7	2.22811
$831$	0	0
$832$	14276.9	0.594907
$833$	2217.76	0.0922457
$834$	0	0
$835$	−27710.5	−1.14846
$836$	−26348.6	−1.09005
$837$	0	0
$838$	41447.1	1.70855
$839$	25659.1	1.05584	0.527919	−	0.849294i	$-0.322972\pi$
$839$	25659.1	1.05584	0.527919	+	0.849294i	$0.322972\pi$
$840$	0	0
$841$	−18195.8	−0.746064
$842$	7299.32	0.298754
$843$	0	0
$844$	72670.7	2.96378
$845$	17031.5	0.693373
$846$	0	0
$847$	−847.000	−0.0343604
$848$	46872.1	1.89811
$849$	0	0
$850$	20831.2	0.840594
$851$	−14423.2	−0.580987
$852$	0	0
$853$	44687.0	1.79373	0.896866	−	0.442303i	$-0.145838\pi$
$853$	44687.0	1.79373	0.896866	+	0.442303i	$0.145838\pi$
$854$	20819.1	0.834209
$855$	0	0
$856$	−65490.6	−2.61498
$857$	14194.9	0.565798	0.282899	−	0.959150i	$-0.408704\pi$
$857$	14194.9	0.565798	0.282899	+	0.959150i	$0.408704\pi$
$858$	0	0
$859$	−1277.85	−0.0507561	−0.0253781	−	0.999678i	$-0.508079\pi$
$859$	−1277.85	−0.0507561	−0.0253781	+	0.999678i	$0.508079\pi$
$860$	−14585.8	−0.578340
$861$	0	0
$862$	−28938.3	−1.14344
$863$	−46845.2	−1.84778	−0.923888	−	0.382664i	$-0.875007\pi$
$863$	−46845.2	−1.84778	−0.923888	+	0.382664i	$0.875007\pi$
$864$	0	0
$865$	8519.81	0.334893
$866$	−20658.8	−0.810639
$867$	0	0
$868$	−12137.8	−0.474636
$869$	−1130.60	−0.0441346
$870$	0	0
$871$	−11115.1	−0.432400
$872$	91837.6	3.56653
$873$	0	0
$874$	110850.	4.29012
$875$	−3280.50	−0.126744
$876$	0	0
$877$	16116.7	0.620551	0.310275	−	0.950647i	$-0.399579\pi$
$877$	16116.7	0.620551	0.310275	+	0.950647i	$0.399579\pi$
$878$	23411.1	0.899872
$879$	0	0
$880$	−11767.9	−0.450792
$881$	12707.2	0.485944	0.242972	−	0.970033i	$-0.421878\pi$
$881$	12707.2	0.485944	0.242972	+	0.970033i	$0.421878\pi$
$882$	0	0
$883$	−7591.93	−0.289342	−0.144671	−	0.989480i	$-0.546212\pi$
$883$	−7591.93	−0.289342	−0.144671	+	0.989480i	$0.546212\pi$
$884$	−23906.3	−0.909564
$885$	0	0
$886$	−32866.8	−1.24626
$887$	7953.37	0.301069	0.150534	−	0.988605i	$-0.451901\pi$
$887$	7953.37	0.301069	0.150534	+	0.988605i	$0.451901\pi$
$888$	0	0
$889$	−6417.60	−0.242114
$890$	−3192.22	−0.120229
$891$	0	0
$892$	82562.7	3.09911
$893$	37170.6	1.39291
$894$	0	0
$895$	−20175.5	−0.753512
$896$	16819.9	0.627135
$897$	0	0
$898$	89240.8	3.31626
$899$	−8348.42	−0.309717
$900$	0	0
$901$	29297.6	1.08329
$902$	−22646.1	−0.835955
$903$	0	0
$904$	47178.9	1.73578
$905$	20822.2	0.764811
$906$	0	0
$907$	−2325.47	−0.0851335	−0.0425668	−	0.999094i	$-0.513554\pi$
$907$	−2325.47	−0.0851335	−0.0425668	+	0.999094i	$0.513554\pi$
$908$	−11345.6	−0.414666
$909$	0	0
$910$	−16489.6	−0.600687
$911$	12179.3	0.442939	0.221469	−	0.975167i	$-0.428915\pi$
$911$	12179.3	0.442939	0.221469	+	0.975167i	$0.428915\pi$
$912$	0	0
$913$	−8039.53	−0.291424
$914$	1991.31	0.0720643
$915$	0	0
$916$	29283.1	1.05627
$917$	−3419.42	−0.123140
$918$	0	0
$919$	−13972.9	−0.501550	−0.250775	−	0.968045i	$-0.580685\pi$
$919$	−13972.9	−0.501550	−0.250775	+	0.968045i	$0.580685\pi$
$920$	93266.7	3.34229
$921$	0	0
$922$	−63319.4	−2.26173
$923$	−7267.61	−0.259173
$924$	0	0
$925$	8775.83	0.311943
$926$	−33207.6	−1.17848
$927$	0	0
$928$	−2192.60	−0.0775599
$929$	11612.8	0.410123	0.205062	−	0.978749i	$-0.434261\pi$
$929$	11612.8	0.410123	0.205062	+	0.978749i	$0.434261\pi$
$930$	0	0
$931$	−7180.65	−0.252778
$932$	−68105.9	−2.39365
$933$	0	0
$934$	−58161.9	−2.03760
$935$	−7355.59	−0.257276
$936$	0	0
$937$	−5811.23	−0.202609	−0.101305	−	0.994855i	$-0.532302\pi$
$937$	−5811.23	−0.202609	−0.101305	+	0.994855i	$0.532302\pi$
$938$	−11880.2	−0.413542
$939$	0	0
$940$	61254.2	2.12542
$941$	−35047.0	−1.21413	−0.607067	−	0.794651i	$-0.707654\pi$
$941$	−35047.0	−1.21413	−0.607067	+	0.794651i	$0.707654\pi$
$942$	0	0
$943$	63966.4	2.20894
$944$	−40505.5	−1.39655
$945$	0	0
$946$	3278.15	0.112666
$947$	24845.3	0.852550	0.426275	−	0.904594i	$-0.359826\pi$
$947$	24845.3	0.852550	0.426275	+	0.904594i	$0.359826\pi$
$948$	0	0
$949$	−33989.6	−1.16264
$950$	−67447.3	−2.30345
$951$	0	0
$952$	−13045.9	−0.444140
$953$	−26556.9	−0.902688	−0.451344	−	0.892350i	$-0.649055\pi$
$953$	−26556.9	−0.902688	−0.451344	+	0.892350i	$0.649055\pi$
$954$	0	0
$955$	76988.0	2.60866
$956$	110977.	3.75444
$957$	0	0
$958$	40657.2	1.37116
$959$	−3037.18	−0.102269
$960$	0	0
$961$	−18537.4	−0.622249
$962$	−15000.5	−0.502740
$963$	0	0
$964$	−117203.	−3.91582
$965$	62587.3	2.08783
$966$	0	0
$967$	−10813.2	−0.359596	−0.179798	−	0.983704i	$-0.557544\pi$
$967$	−10813.2	−0.359596	−0.179798	+	0.983704i	$0.557544\pi$
$968$	4982.47	0.165437
$969$	0	0
$970$	−584.193	−0.0193374
$971$	−44816.6	−1.48119	−0.740595	−	0.671952i	$-0.765456\pi$
$971$	−44816.6	−1.48119	−0.740595	+	0.671952i	$0.765456\pi$
$972$	0	0
$973$	4604.87	0.151722
$974$	72587.3	2.38793
$975$	0	0
$976$	−43647.1	−1.43146
$977$	−23952.8	−0.784359	−0.392180	−	0.919889i	$-0.628279\pi$
$977$	−23952.8	−0.784359	−0.392180	+	0.919889i	$0.628279\pi$
$978$	0	0
$979$	481.693	0.0157252
$980$	−11833.1	−0.385710
$981$	0	0
$982$	−88278.1	−2.86870
$983$	−35723.9	−1.15912	−0.579560	−	0.814930i	$-0.696776\pi$
$983$	−35723.9	−1.15912	−0.579560	+	0.814930i	$0.696776\pi$
$984$	0	0
$985$	37444.0	1.21123
$986$	−17574.6	−0.567637
$987$	0	0
$988$	77403.6	2.49245
$989$	−9259.51	−0.297710
$990$	0	0
$991$	23977.0	0.768572	0.384286	−	0.923214i	$-0.374448\pi$
$991$	23977.0	0.768572	0.384286	+	0.923214i	$0.374448\pi$
$992$	2955.60	0.0945971
$993$	0	0
$994$	−7767.87	−0.247869
$995$	21966.1	0.699871
$996$	0	0
$997$	−43975.2	−1.39690	−0.698450	−	0.715659i	$-0.746126\pi$
$997$	−43975.2	−1.39690	−0.698450	+	0.715659i	$0.746126\pi$
$998$	−32647.8	−1.03552
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.8	✓	8	1.1	even	1	trivial
693.4.a.t.1.1	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)

\(f(q)\)	\(=\)	\(q+4.93411 q^{2} +16.3455 q^{4} -14.7743 q^{5} -7.00000 q^{7} +41.1774 q^{8} -72.8980 q^{10} +11.0000 q^{11} -32.3144 q^{13} -34.5388 q^{14} +72.4103 q^{16} +45.2603 q^{17} -146.544 q^{19} -241.493 q^{20} +54.2752 q^{22} -153.307 q^{23} +93.2798 q^{25} -159.443 q^{26} -114.418 q^{28} -78.6971 q^{29} +106.083 q^{31} +27.8612 q^{32} +223.320 q^{34} +103.420 q^{35} +94.0807 q^{37} -723.064 q^{38} -608.367 q^{40} -417.245 q^{41} +60.3987 q^{43} +179.800 q^{44} -756.431 q^{46} -253.648 q^{47} +49.0000 q^{49} +460.253 q^{50} -528.194 q^{52} +647.312 q^{53} -162.517 q^{55} -288.242 q^{56} -388.300 q^{58} -559.388 q^{59} -602.774 q^{61} +523.425 q^{62} -441.812 q^{64} +477.423 q^{65} +343.967 q^{67} +739.801 q^{68} +510.286 q^{70} +224.903 q^{71} +1051.84 q^{73} +464.205 q^{74} -2395.33 q^{76} -77.0000 q^{77} -102.782 q^{79} -1069.81 q^{80} -2058.73 q^{82} -730.867 q^{83} -668.690 q^{85} +298.014 q^{86} +452.952 q^{88} +43.7903 q^{89} +226.201 q^{91} -2505.86 q^{92} -1251.53 q^{94} +2165.08 q^{95} +8.01384 q^{97} +241.771 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26}+ \cdots - 98 q^{98}+O(q^{100}) \) 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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