LMFDB - Embedded newform 693.4.a.s.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:	$1$
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			1.4
Root			$1.22364$ 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)

$f(q)$	$=$	$q-1.22364 q^{2} -6.50270 q^{4} +11.7152 q^{5} -7.00000 q^{7} +17.7461 q^{8} +O(q^{10})$	\(q-1.22364 q^{2} -6.50270 q^{4} +11.7152 q^{5} -7.00000 q^{7} +17.7461 q^{8} -14.3353 q^{10} +11.0000 q^{11} -25.5594 q^{13} +8.56550 q^{14} +30.3067 q^{16} -16.3295 q^{17} -126.020 q^{19} -76.1807 q^{20} -13.4601 q^{22} +94.6529 q^{23} +12.2469 q^{25} +31.2756 q^{26} +45.5189 q^{28} +160.092 q^{29} -12.1160 q^{31} -179.054 q^{32} +19.9815 q^{34} -82.0067 q^{35} +436.954 q^{37} +154.204 q^{38} +207.900 q^{40} -264.689 q^{41} -171.820 q^{43} -71.5297 q^{44} -115.821 q^{46} -535.747 q^{47} +49.0000 q^{49} -14.9859 q^{50} +166.205 q^{52} +514.087 q^{53} +128.868 q^{55} -124.223 q^{56} -195.895 q^{58} -607.038 q^{59} +47.8997 q^{61} +14.8256 q^{62} -23.3557 q^{64} -299.435 q^{65} +1054.56 q^{67} +106.186 q^{68} +100.347 q^{70} -783.445 q^{71} -1103.40 q^{73} -534.676 q^{74} +819.473 q^{76} -77.0000 q^{77} -194.840 q^{79} +355.050 q^{80} +323.885 q^{82} -489.127 q^{83} -191.305 q^{85} +210.247 q^{86} +195.207 q^{88} -466.942 q^{89} +178.916 q^{91} -615.499 q^{92} +655.563 q^{94} -1476.36 q^{95} -429.704 q^{97} -59.9585 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.22364	−0.432623	−0.216312	−	0.976324i	$-0.569403\pi$
$2$	−1.22364	−0.432623	−0.216312	+	0.976324i	$0.569403\pi$
$3$	0	0
$3$	0	0
$4$	−6.50270	−0.812837
$5$	11.7152	1.04784	0.523922	−	0.851766i	$-0.324468\pi$
$5$	11.7152	1.04784	0.523922	+	0.851766i	$0.324468\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	17.7461	0.784275
$9$	0	0
$10$	−14.3353	−0.453321
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−25.5594	−0.545301	−0.272650	−	0.962113i	$-0.587900\pi$
$13$	−25.5594	−0.545301	−0.272650	+	0.962113i	$0.587900\pi$
$14$	8.56550	0.163516
$15$	0	0
$16$	30.3067	0.473542
$17$	−16.3295	−0.232970	−0.116485	−	0.993192i	$-0.537163\pi$
$17$	−16.3295	−0.232970	−0.116485	+	0.993192i	$0.537163\pi$
$18$	0	0
$19$	−126.020	−1.52164	−0.760818	−	0.648965i	$-0.775202\pi$
$19$	−126.020	−1.52164	−0.760818	+	0.648965i	$0.775202\pi$
$20$	−76.1807	−0.851726
$21$	0	0
$22$	−13.4601	−0.130441
$23$	94.6529	0.858108	0.429054	−	0.903279i	$-0.358847\pi$
$23$	94.6529	0.858108	0.429054	+	0.903279i	$0.358847\pi$
$24$	0	0
$25$	12.2469	0.0979756
$26$	31.2756	0.235910
$27$	0	0
$28$	45.5189	0.307224
$29$	160.092	1.02511	0.512557	−	0.858653i	$-0.328698\pi$
$29$	160.092	1.02511	0.512557	+	0.858653i	$0.328698\pi$
$30$	0	0
$31$	−12.1160	−0.0701966	−0.0350983	−	0.999384i	$-0.511174\pi$
$31$	−12.1160	−0.0701966	−0.0350983	+	0.999384i	$0.511174\pi$
$32$	−179.054	−0.989140
$33$	0	0
$34$	19.9815	0.100788
$35$	−82.0067	−0.396048
$36$	0	0
$37$	436.954	1.94148	0.970742	−	0.240127i	$-0.0771890\pi$
$37$	436.954	1.94148	0.970742	+	0.240127i	$0.0771890\pi$
$38$	154.204	0.658295
$39$	0	0
$40$	207.900	0.821798
$41$	−264.689	−1.00823	−0.504116	−	0.863636i	$-0.668182\pi$
$41$	−264.689	−1.00823	−0.504116	+	0.863636i	$0.668182\pi$
$42$	0	0
$43$	−171.820	−0.609357	−0.304679	−	0.952455i	$-0.598549\pi$
$43$	−171.820	−0.609357	−0.304679	+	0.952455i	$0.598549\pi$
$44$	−71.5297	−0.245080
$45$	0	0
$46$	−115.821	−0.371237
$47$	−535.747	−1.66270	−0.831348	−	0.555752i	$-0.812431\pi$
$47$	−535.747	−1.66270	−0.831348	+	0.555752i	$0.812431\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−14.9859	−0.0423865
$51$	0	0
$52$	166.205	0.443241
$53$	514.087	1.33236	0.666182	−	0.745789i	$-0.267927\pi$
$53$	514.087	1.33236	0.666182	+	0.745789i	$0.267927\pi$
$54$	0	0
$55$	128.868	0.315937
$56$	−124.223	−0.296428
$57$	0	0
$58$	−195.895	−0.443488
$59$	−607.038	−1.33948	−0.669742	−	0.742594i	$-0.733595\pi$
$59$	−607.038	−1.33948	−0.669742	+	0.742594i	$0.733595\pi$
$60$	0	0
$61$	47.8997	0.100540	0.0502699	−	0.998736i	$-0.483992\pi$
$61$	47.8997	0.100540	0.0502699	+	0.998736i	$0.483992\pi$
$62$	14.8256	0.0303687
$63$	0	0
$64$	−23.3557	−0.0456167
$65$	−299.435	−0.571390
$66$	0	0
$67$	1054.56	1.92291	0.961456	−	0.274958i	$-0.0886640\pi$
$67$	1054.56	1.92291	0.961456	+	0.274958i	$0.0886640\pi$
$68$	106.186	0.189367
$69$	0	0
$70$	100.347	0.171339
$71$	−783.445	−1.30955	−0.654774	−	0.755825i	$-0.727236\pi$
$71$	−783.445	−1.30955	−0.654774	+	0.755825i	$0.727236\pi$
$72$	0	0
$73$	−1103.40	−1.76908	−0.884541	−	0.466462i	$-0.845529\pi$
$73$	−1103.40	−1.76908	−0.884541	+	0.466462i	$0.845529\pi$
$74$	−534.676	−0.839930
$75$	0	0
$76$	819.473	1.23684
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−194.840	−0.277483	−0.138742	−	0.990329i	$-0.544306\pi$
$79$	−194.840	−0.277483	−0.138742	+	0.990329i	$0.544306\pi$
$80$	355.050	0.496197
$81$	0	0
$82$	323.885	0.436185
$83$	−489.127	−0.646851	−0.323425	−	0.946254i	$-0.604834\pi$
$83$	−489.127	−0.646851	−0.323425	+	0.946254i	$0.604834\pi$
$84$	0	0
$85$	−191.305	−0.244116
$86$	210.247	0.263622
$87$	0	0
$88$	195.207	0.236468
$89$	−466.942	−0.556131	−0.278066	−	0.960562i	$-0.589693\pi$
$89$	−466.942	−0.556131	−0.278066	+	0.960562i	$0.589693\pi$
$90$	0	0
$91$	178.916	0.206104
$92$	−615.499	−0.697502
$93$	0	0
$94$	655.563	0.719321
$95$	−1476.36	−1.59444
$96$	0	0
$97$	−429.704	−0.449793	−0.224896	−	0.974383i	$-0.572204\pi$
$97$	−429.704	−0.449793	−0.224896	+	0.974383i	$0.572204\pi$
$98$	−59.9585	−0.0618033
$99$	0	0
$100$	−79.6382	−0.0796382
$101$	−163.645	−0.161221	−0.0806104	−	0.996746i	$-0.525687\pi$
$101$	−163.645	−0.161221	−0.0806104	+	0.996746i	$0.525687\pi$
$102$	0	0
$103$	−1695.99	−1.62244	−0.811220	−	0.584741i	$-0.801196\pi$
$103$	−1695.99	−1.62244	−0.811220	+	0.584741i	$0.801196\pi$
$104$	−453.581	−0.427666
$105$	0	0
$106$	−629.059	−0.576411
$107$	482.153	0.435622	0.217811	−	0.975991i	$-0.430108\pi$
$107$	482.153	0.435622	0.217811	+	0.975991i	$0.430108\pi$
$108$	0	0
$109$	−1387.58	−1.21932	−0.609660	−	0.792663i	$-0.708694\pi$
$109$	−1387.58	−1.21932	−0.609660	+	0.792663i	$0.708694\pi$
$110$	−157.688	−0.136681
$111$	0	0
$112$	−212.147	−0.178982
$113$	−1881.83	−1.56662	−0.783309	−	0.621632i	$-0.786470\pi$
$113$	−1881.83	−1.56662	−0.783309	+	0.621632i	$0.786470\pi$
$114$	0	0
$115$	1108.88	0.899163
$116$	−1041.03	−0.833250
$117$	0	0
$118$	742.797	0.579492
$119$	114.307	0.0880545
$120$	0	0
$121$	121.000	0.0909091
$122$	−58.6121	−0.0434958
$123$	0	0
$124$	78.7866	0.0570584
$125$	−1320.93	−0.945180
$126$	0	0
$127$	−2591.48	−1.81068	−0.905342	−	0.424684i	$-0.860385\pi$
$127$	−2591.48	−1.81068	−0.905342	+	0.424684i	$0.860385\pi$
$128$	1461.01	1.00888
$129$	0	0
$130$	366.401	0.247196
$131$	2833.91	1.89008	0.945039	−	0.326958i	$-0.106023\pi$
$131$	2833.91	1.89008	0.945039	+	0.326958i	$0.106023\pi$
$132$	0	0
$133$	882.143	0.575124
$134$	−1290.41	−0.831896
$135$	0	0
$136$	−289.786	−0.182713
$137$	2480.55	1.54692	0.773458	−	0.633847i	$-0.218525\pi$
$137$	2480.55	1.54692	0.773458	+	0.633847i	$0.218525\pi$
$138$	0	0
$139$	1471.83	0.898119	0.449060	−	0.893502i	$-0.351759\pi$
$139$	1471.83	0.898119	0.449060	+	0.893502i	$0.351759\pi$
$140$	533.265	0.321922
$141$	0	0
$142$	958.657	0.566540
$143$	−281.154	−0.164414
$144$	0	0
$145$	1875.51	1.07416
$146$	1350.17	0.765346
$147$	0	0
$148$	−2841.38	−1.57811
$149$	−2925.89	−1.60871	−0.804356	−	0.594148i	$-0.797489\pi$
$149$	−2925.89	−1.60871	−0.804356	+	0.594148i	$0.797489\pi$
$150$	0	0
$151$	−1713.98	−0.923721	−0.461860	−	0.886953i	$-0.652818\pi$
$151$	−1713.98	−0.923721	−0.461860	+	0.886953i	$0.652818\pi$
$152$	−2236.38	−1.19338
$153$	0	0
$154$	94.2205	0.0493020
$155$	−141.942	−0.0735550
$156$	0	0
$157$	−1419.24	−0.721448	−0.360724	−	0.932673i	$-0.617470\pi$
$157$	−1419.24	−0.721448	−0.360724	+	0.932673i	$0.617470\pi$
$158$	238.414	0.120046
$159$	0	0
$160$	−2097.66	−1.03646
$161$	−662.570	−0.324334
$162$	0	0
$163$	−256.658	−0.123331	−0.0616656	−	0.998097i	$-0.519641\pi$
$163$	−256.658	−0.123331	−0.0616656	+	0.998097i	$0.519641\pi$
$164$	1721.19	0.819529
$165$	0	0
$166$	598.516	0.279843
$167$	−2122.63	−0.983555	−0.491778	−	0.870721i	$-0.663653\pi$
$167$	−2122.63	−0.983555	−0.491778	+	0.870721i	$0.663653\pi$
$168$	0	0
$169$	−1543.72	−0.702647
$170$	234.089	0.105610
$171$	0	0
$172$	1117.30	0.495308
$173$	−3533.40	−1.55283	−0.776414	−	0.630223i	$-0.782964\pi$
$173$	−3533.40	−1.55283	−0.776414	+	0.630223i	$0.782964\pi$
$174$	0	0
$175$	−85.7286	−0.0370313
$176$	333.373	0.142778
$177$	0	0
$178$	571.370	0.240595
$179$	−2331.73	−0.973642	−0.486821	−	0.873502i	$-0.661844\pi$
$179$	−2331.73	−0.973642	−0.486821	+	0.873502i	$0.661844\pi$
$180$	0	0
$181$	1102.06	0.452574	0.226287	−	0.974061i	$-0.427341\pi$
$181$	1102.06	0.452574	0.226287	+	0.974061i	$0.427341\pi$
$182$	−218.929	−0.0891655
$183$	0	0
$184$	1679.72	0.672993
$185$	5119.03	2.03437
$186$	0	0
$187$	−179.625	−0.0702432
$188$	3483.80	1.35150
$189$	0	0
$190$	1806.54	0.689790
$191$	2221.20	0.841468	0.420734	−	0.907184i	$-0.361773\pi$
$191$	2221.20	0.841468	0.420734	+	0.907184i	$0.361773\pi$
$192$	0	0
$193$	3176.73	1.18480	0.592399	−	0.805645i	$-0.298181\pi$
$193$	3176.73	1.18480	0.592399	+	0.805645i	$0.298181\pi$
$194$	525.805	0.194591
$195$	0	0
$196$	−318.632	−0.116120
$197$	−728.736	−0.263555	−0.131777	−	0.991279i	$-0.542068\pi$
$197$	−728.736	−0.263555	−0.131777	+	0.991279i	$0.542068\pi$
$198$	0	0
$199$	−3316.74	−1.18149	−0.590747	−	0.806857i	$-0.701167\pi$
$199$	−3316.74	−1.18149	−0.590747	+	0.806857i	$0.701167\pi$
$200$	217.336	0.0768398
$201$	0	0
$202$	200.243	0.0697478
$203$	−1120.64	−0.387456
$204$	0	0
$205$	−3100.90	−1.05647
$206$	2075.29	0.701905
$207$	0	0
$208$	−774.621	−0.258223
$209$	−1386.23	−0.458790
$210$	0	0
$211$	1005.81	0.328165	0.164083	−	0.986447i	$-0.447534\pi$
$211$	1005.81	0.328165	0.164083	+	0.986447i	$0.447534\pi$
$212$	−3342.95	−1.08299
$213$	0	0
$214$	−589.984	−0.188460
$215$	−2012.92	−0.638511
$216$	0	0
$217$	84.8119	0.0265318
$218$	1697.90	0.527506
$219$	0	0
$220$	−837.988	−0.256805
$221$	417.374	0.127039
$222$	0	0
$223$	4528.06	1.35974	0.679869	−	0.733334i	$-0.262037\pi$
$223$	4528.06	1.35974	0.679869	+	0.733334i	$0.262037\pi$
$224$	1253.37	0.373860
$225$	0	0
$226$	2302.69	0.677755
$227$	2221.82	0.649635	0.324818	−	0.945777i	$-0.394697\pi$
$227$	2221.82	0.649635	0.324818	+	0.945777i	$0.394697\pi$
$228$	0	0
$229$	2923.23	0.843548	0.421774	−	0.906701i	$-0.361408\pi$
$229$	2923.23	0.843548	0.421774	+	0.906701i	$0.361408\pi$
$230$	−1356.88	−0.388999
$231$	0	0
$232$	2841.01	0.803971
$233$	3905.86	1.09820	0.549102	−	0.835755i	$-0.314970\pi$
$233$	3905.86	1.09820	0.549102	+	0.835755i	$0.314970\pi$
$234$	0	0
$235$	−6276.41	−1.74225
$236$	3947.38	1.08878
$237$	0	0
$238$	−139.871	−0.0380944
$239$	−1684.06	−0.455785	−0.227893	−	0.973686i	$-0.573184\pi$
$239$	−1684.06	−0.455785	−0.227893	+	0.973686i	$0.573184\pi$
$240$	0	0
$241$	2753.98	0.736096	0.368048	−	0.929807i	$-0.380026\pi$
$241$	2753.98	0.736096	0.368048	+	0.929807i	$0.380026\pi$
$242$	−148.061	−0.0393294
$243$	0	0
$244$	−311.477	−0.0817225
$245$	574.047	0.149692
$246$	0	0
$247$	3221.01	0.829749
$248$	−215.012	−0.0550534
$249$	0	0
$250$	1616.35	0.408907
$251$	−5315.23	−1.33663	−0.668316	−	0.743878i	$-0.732984\pi$
$251$	−5315.23	−1.33663	−0.668316	+	0.743878i	$0.732984\pi$
$252$	0	0
$253$	1041.18	0.258729
$254$	3171.05	0.783343
$255$	0	0
$256$	−1600.91	−0.390846
$257$	−2536.26	−0.615594	−0.307797	−	0.951452i	$-0.599592\pi$
$257$	−2536.26	−0.615594	−0.307797	+	0.951452i	$0.599592\pi$
$258$	0	0
$259$	−3058.68	−0.733812
$260$	1947.13	0.464447
$261$	0	0
$262$	−3467.70	−0.817691
$263$	−1555.04	−0.364593	−0.182297	−	0.983244i	$-0.558353\pi$
$263$	−1555.04	−0.364593	−0.182297	+	0.983244i	$0.558353\pi$
$264$	0	0
$265$	6022.65	1.39611
$266$	−1079.43	−0.248812
$267$	0	0
$268$	−6857.49	−1.56301
$269$	848.803	0.192388	0.0961941	−	0.995363i	$-0.469333\pi$
$269$	848.803	0.192388	0.0961941	+	0.995363i	$0.469333\pi$
$270$	0	0
$271$	−3319.30	−0.744034	−0.372017	−	0.928226i	$-0.621334\pi$
$271$	−3319.30	−0.744034	−0.372017	+	0.928226i	$0.621334\pi$
$272$	−494.894	−0.110321
$273$	0	0
$274$	−3035.31	−0.669232
$275$	134.716	0.0295407
$276$	0	0
$277$	1783.43	0.386843	0.193422	−	0.981116i	$-0.438041\pi$
$277$	1783.43	0.386843	0.193422	+	0.981116i	$0.438041\pi$
$278$	−1800.99	−0.388547
$279$	0	0
$280$	−1455.30	−0.310610
$281$	72.4269	0.0153759	0.00768795	−	0.999970i	$-0.497553\pi$
$281$	72.4269	0.0153759	0.00768795	+	0.999970i	$0.497553\pi$
$282$	0	0
$283$	7993.56	1.67904	0.839519	−	0.543330i	$-0.182837\pi$
$283$	7993.56	1.67904	0.839519	+	0.543330i	$0.182837\pi$
$284$	5094.51	1.06445
$285$	0	0
$286$	344.032	0.0711294
$287$	1852.82	0.381076
$288$	0	0
$289$	−4646.35	−0.945725
$290$	−2294.96	−0.464706
$291$	0	0
$292$	7175.07	1.43798
$293$	−1165.36	−0.232358	−0.116179	−	0.993228i	$-0.537065\pi$
$293$	−1165.36	−0.232358	−0.116179	+	0.993228i	$0.537065\pi$
$294$	0	0
$295$	−7111.59	−1.40357
$296$	7754.25	1.52266
$297$	0	0
$298$	3580.24	0.695966
$299$	−2419.27	−0.467927
$300$	0	0
$301$	1202.74	0.230315
$302$	2097.30	0.399623
$303$	0	0
$304$	−3819.26	−0.720558
$305$	561.157	0.105350
$306$	0	0
$307$	−459.180	−0.0853640	−0.0426820	−	0.999089i	$-0.513590\pi$
$307$	−459.180	−0.0853640	−0.0426820	+	0.999089i	$0.513590\pi$
$308$	500.708	0.0926314
$309$	0	0
$310$	173.686	0.0318216
$311$	3247.07	0.592040	0.296020	−	0.955182i	$-0.404340\pi$
$311$	3247.07	0.592040	0.296020	+	0.955182i	$0.404340\pi$
$312$	0	0
$313$	−3263.89	−0.589412	−0.294706	−	0.955588i	$-0.595222\pi$
$313$	−3263.89	−0.589412	−0.294706	+	0.955588i	$0.595222\pi$
$314$	1736.64	0.312115
$315$	0	0
$316$	1266.98	0.225549
$317$	6820.40	1.20843	0.604214	−	0.796822i	$-0.293487\pi$
$317$	6820.40	1.20843	0.604214	+	0.796822i	$0.293487\pi$
$318$	0	0
$319$	1761.01	0.309083
$320$	−273.618	−0.0477991
$321$	0	0
$322$	810.749	0.140315
$323$	2057.86	0.354496
$324$	0	0
$325$	−313.025	−0.0534261
$326$	314.057	0.0533559
$327$	0	0
$328$	−4697.21	−0.790732
$329$	3750.23	0.628440
$330$	0	0
$331$	25.5296	0.00423938	0.00211969	−	0.999998i	$-0.499325\pi$
$331$	25.5296	0.00423938	0.00211969	+	0.999998i	$0.499325\pi$
$332$	3180.64	0.525784
$333$	0	0
$334$	2597.34	0.425509
$335$	12354.4	2.01491
$336$	0	0
$337$	847.626	0.137012	0.0685061	−	0.997651i	$-0.478177\pi$
$337$	847.626	0.137012	0.0685061	+	0.997651i	$0.478177\pi$
$338$	1888.96	0.303981
$339$	0	0
$340$	1244.00	0.198427
$341$	−133.276	−0.0211651
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−3049.15	−0.477904
$345$	0	0
$346$	4323.62	0.671790
$347$	5389.78	0.833828	0.416914	−	0.908946i	$-0.363112\pi$
$347$	5389.78	0.833828	0.416914	+	0.908946i	$0.363112\pi$
$348$	0	0
$349$	5693.81	0.873302	0.436651	−	0.899631i	$-0.356164\pi$
$349$	5693.81	0.873302	0.436651	+	0.899631i	$0.356164\pi$
$350$	104.901	0.0160206
$351$	0	0
$352$	−1969.59	−0.298237
$353$	5227.14	0.788138	0.394069	−	0.919081i	$-0.371067\pi$
$353$	5227.14	0.788138	0.394069	+	0.919081i	$0.371067\pi$
$354$	0	0
$355$	−9178.25	−1.37220
$356$	3036.38	0.452044
$357$	0	0
$358$	2853.21	0.421220
$359$	−3789.54	−0.557115	−0.278557	−	0.960420i	$-0.589856\pi$
$359$	−3789.54	−0.557115	−0.278557	+	0.960420i	$0.589856\pi$
$360$	0	0
$361$	9022.16	1.31538
$362$	−1348.53	−0.195794
$363$	0	0
$364$	−1163.44	−0.167529
$365$	−12926.6	−1.85372
$366$	0	0
$367$	1680.96	0.239089	0.119544	−	0.992829i	$-0.461857\pi$
$367$	1680.96	0.239089	0.119544	+	0.992829i	$0.461857\pi$
$368$	2868.61	0.406350
$369$	0	0
$370$	−6263.86	−0.880116
$371$	−3598.61	−0.503586
$372$	0	0
$373$	−9682.93	−1.34414	−0.672068	−	0.740489i	$-0.734594\pi$
$373$	−9682.93	−1.34414	−0.672068	+	0.740489i	$0.734594\pi$
$374$	219.797	0.0303888
$375$	0	0
$376$	−9507.43	−1.30401
$377$	−4091.85	−0.558995
$378$	0	0
$379$	10765.6	1.45908	0.729539	−	0.683939i	$-0.239735\pi$
$379$	10765.6	1.45908	0.729539	+	0.683939i	$0.239735\pi$
$380$	9600.33	1.29602
$381$	0	0
$382$	−2717.95	−0.364038
$383$	−4404.20	−0.587582	−0.293791	−	0.955870i	$-0.594917\pi$
$383$	−4404.20	−0.587582	−0.293791	+	0.955870i	$0.594917\pi$
$384$	0	0
$385$	−902.074	−0.119413
$386$	−3887.18	−0.512571
$387$	0	0
$388$	2794.24	0.365608
$389$	13109.8	1.70872	0.854359	−	0.519684i	$-0.173950\pi$
$389$	13109.8	1.70872	0.854359	+	0.519684i	$0.173950\pi$
$390$	0	0
$391$	−1545.64	−0.199914
$392$	869.560	0.112039
$393$	0	0
$394$	891.712	0.114020
$395$	−2282.59	−0.290759
$396$	0	0
$397$	9725.94	1.22955	0.614775	−	0.788703i	$-0.289247\pi$
$397$	9725.94	1.22955	0.614775	+	0.788703i	$0.289247\pi$
$398$	4058.50	0.511141
$399$	0	0
$400$	371.164	0.0463955
$401$	−171.626	−0.0213731	−0.0106865	−	0.999943i	$-0.503402\pi$
$401$	−171.626	−0.0213731	−0.0106865	+	0.999943i	$0.503402\pi$
$402$	0	0
$403$	309.677	0.0382782
$404$	1064.13	0.131046
$405$	0	0
$406$	1371.27	0.167623
$407$	4806.50	0.585379
$408$	0	0
$409$	6111.86	0.738905	0.369453	−	0.929250i	$-0.379545\pi$
$409$	6111.86	0.738905	0.369453	+	0.929250i	$0.379545\pi$
$410$	3794.39	0.457053
$411$	0	0
$412$	11028.5	1.31878
$413$	4249.26	0.506277
$414$	0	0
$415$	−5730.24	−0.677798
$416$	4576.50	0.539379
$417$	0	0
$418$	1696.24	0.198483
$419$	−1032.33	−0.120365	−0.0601824	−	0.998187i	$-0.519168\pi$
$419$	−1032.33	−0.120365	−0.0601824	+	0.998187i	$0.519168\pi$
$420$	0	0
$421$	11648.9	1.34853	0.674266	−	0.738489i	$-0.264460\pi$
$421$	11648.9	1.34853	0.674266	+	0.738489i	$0.264460\pi$
$422$	−1230.75	−0.141972
$423$	0	0
$424$	9123.05	1.04494
$425$	−199.987	−0.0228254
$426$	0	0
$427$	−335.298	−0.0380005
$428$	−3135.30	−0.354090
$429$	0	0
$430$	2463.09	0.276235
$431$	3012.71	0.336699	0.168349	−	0.985727i	$-0.446156\pi$
$431$	3012.71	0.336699	0.168349	+	0.985727i	$0.446156\pi$
$432$	0	0
$433$	−2094.06	−0.232411	−0.116206	−	0.993225i	$-0.537073\pi$
$433$	−2094.06	−0.232411	−0.116206	+	0.993225i	$0.537073\pi$
$434$	−103.779	−0.0114783
$435$	0	0
$436$	9023.00	0.991109
$437$	−11928.2	−1.30573
$438$	0	0
$439$	−8438.17	−0.917385	−0.458692	−	0.888595i	$-0.651682\pi$
$439$	−8438.17	−0.917385	−0.458692	+	0.888595i	$0.651682\pi$
$440$	2286.90	0.247781
$441$	0	0
$442$	−510.716	−0.0549600
$443$	−1796.37	−0.192660	−0.0963298	−	0.995349i	$-0.530710\pi$
$443$	−1796.37	−0.192660	−0.0963298	+	0.995349i	$0.530710\pi$
$444$	0	0
$445$	−5470.33	−0.582739
$446$	−5540.73	−0.588254
$447$	0	0
$448$	163.490	0.0172415
$449$	6300.70	0.662245	0.331123	−	0.943588i	$-0.392573\pi$
$449$	6300.70	0.662245	0.331123	+	0.943588i	$0.392573\pi$
$450$	0	0
$451$	−2911.58	−0.303993
$452$	12237.0	1.27341
$453$	0	0
$454$	−2718.71	−0.281047
$455$	2096.04	0.215965
$456$	0	0
$457$	5760.46	0.589634	0.294817	−	0.955554i	$-0.404741\pi$
$457$	5760.46	0.589634	0.294817	+	0.955554i	$0.404741\pi$
$458$	−3576.99	−0.364939
$459$	0	0
$460$	−7210.72	−0.730873
$461$	−74.6854	−0.00754543	−0.00377272	−	0.999993i	$-0.501201\pi$
$461$	−74.6854	−0.00754543	−0.00377272	+	0.999993i	$0.501201\pi$
$462$	0	0
$463$	−3485.88	−0.349898	−0.174949	−	0.984578i	$-0.555976\pi$
$463$	−3485.88	−0.349898	−0.174949	+	0.984578i	$0.555976\pi$
$464$	4851.85	0.485434
$465$	0	0
$466$	−4779.38	−0.475109
$467$	−19514.4	−1.93366	−0.966828	−	0.255427i	$-0.917784\pi$
$467$	−19514.4	−1.93366	−0.966828	+	0.255427i	$0.917784\pi$
$468$	0	0
$469$	−7381.93	−0.726793
$470$	7680.08	0.753736
$471$	0	0
$472$	−10772.6	−1.05052
$473$	−1890.02	−0.183728
$474$	0	0
$475$	−1543.37	−0.149083
$476$	−743.303	−0.0715740
$477$	0	0
$478$	2060.69	0.197183
$479$	−17061.5	−1.62748	−0.813738	−	0.581231i	$-0.802571\pi$
$479$	−17061.5	−1.62748	−0.813738	+	0.581231i	$0.802571\pi$
$480$	0	0
$481$	−11168.3	−1.05869
$482$	−3369.88	−0.318452
$483$	0	0
$484$	−786.826	−0.0738943
$485$	−5034.09	−0.471312
$486$	0	0
$487$	−9161.83	−0.852489	−0.426245	−	0.904608i	$-0.640164\pi$
$487$	−9161.83	−0.852489	−0.426245	+	0.904608i	$0.640164\pi$
$488$	850.034	0.0788509
$489$	0	0
$490$	−702.429	−0.0647602
$491$	−16122.5	−1.48187	−0.740936	−	0.671575i	$-0.765618\pi$
$491$	−16122.5	−1.48187	−0.740936	+	0.671575i	$0.765618\pi$
$492$	0	0
$493$	−2614.23	−0.238821
$494$	−3941.37	−0.358969
$495$	0	0
$496$	−367.195	−0.0332410
$497$	5484.12	0.494962
$498$	0	0
$499$	−7773.19	−0.697346	−0.348673	−	0.937244i	$-0.613368\pi$
$499$	−7773.19	−0.697346	−0.348673	+	0.937244i	$0.613368\pi$
$500$	8589.61	0.768278
$501$	0	0
$502$	6503.95	0.578258
$503$	−11111.5	−0.984966	−0.492483	−	0.870322i	$-0.663911\pi$
$503$	−11111.5	−0.984966	−0.492483	+	0.870322i	$0.663911\pi$
$504$	0	0
$505$	−1917.14	−0.168934
$506$	−1274.03	−0.111932
$507$	0	0
$508$	16851.6	1.47179
$509$	−15251.2	−1.32809	−0.664045	−	0.747692i	$-0.731162\pi$
$509$	−15251.2	−1.32809	−0.664045	+	0.747692i	$0.731162\pi$
$510$	0	0
$511$	7723.79	0.668650
$512$	−9729.12	−0.839786
$513$	0	0
$514$	3103.48	0.266320
$515$	−19869.0	−1.70006
$516$	0	0
$517$	−5893.22	−0.501322
$518$	3742.73	0.317464
$519$	0	0
$520$	−5313.81	−0.448127
$521$	8258.80	0.694481	0.347241	−	0.937776i	$-0.387119\pi$
$521$	8258.80	0.694481	0.347241	+	0.937776i	$0.387119\pi$
$522$	0	0
$523$	−14442.7	−1.20752	−0.603762	−	0.797165i	$-0.706332\pi$
$523$	−14442.7	−1.20752	−0.603762	+	0.797165i	$0.706332\pi$
$524$	−18428.1	−1.53633
$525$	0	0
$526$	1902.82	0.157731
$527$	197.848	0.0163537
$528$	0	0
$529$	−3207.83	−0.263650
$530$	−7369.58	−0.603989
$531$	0	0
$532$	−5736.31	−0.467482
$533$	6765.30	0.549790
$534$	0	0
$535$	5648.54	0.456463
$536$	18714.4	1.50809
$537$	0	0
$538$	−1038.63	−0.0832316
$539$	539.000	0.0430730
$540$	0	0
$541$	−3078.16	−0.244622	−0.122311	−	0.992492i	$-0.539031\pi$
$541$	−3078.16	−0.244622	−0.122311	+	0.992492i	$0.539031\pi$
$542$	4061.64	0.321886
$543$	0	0
$544$	2923.86	0.230440
$545$	−16255.8	−1.27766
$546$	0	0
$547$	525.331	0.0410631	0.0205316	−	0.999789i	$-0.493464\pi$
$547$	525.331	0.0410631	0.0205316	+	0.999789i	$0.493464\pi$
$548$	−16130.3	−1.25739
$549$	0	0
$550$	−164.845	−0.0127800
$551$	−20174.8	−1.55985
$552$	0	0
$553$	1363.88	0.104879
$554$	−2182.28	−0.167357
$555$	0	0
$556$	−9570.83	−0.730025
$557$	19741.0	1.50171	0.750855	−	0.660467i	$-0.229642\pi$
$557$	19741.0	1.50171	0.750855	+	0.660467i	$0.229642\pi$
$558$	0	0
$559$	4391.63	0.332283
$560$	−2485.35	−0.187545
$561$	0	0
$562$	−88.6247	−0.00665197
$563$	11977.6	0.896621	0.448310	−	0.893878i	$-0.352026\pi$
$563$	11977.6	0.896621	0.448310	+	0.893878i	$0.352026\pi$
$564$	0	0
$565$	−22046.1	−1.64157
$566$	−9781.26	−0.726391
$567$	0	0
$568$	−13903.1	−1.02705
$569$	−6601.66	−0.486390	−0.243195	−	0.969977i	$-0.578195\pi$
$569$	−6601.66	−0.486390	−0.243195	+	0.969977i	$0.578195\pi$
$570$	0	0
$571$	−7394.46	−0.541942	−0.270971	−	0.962588i	$-0.587345\pi$
$571$	−7394.46	−0.541942	−0.270971	+	0.962588i	$0.587345\pi$
$572$	1828.26	0.133642
$573$	0	0
$574$	−2267.20	−0.164862
$575$	1159.21	0.0840736
$576$	0	0
$577$	3298.51	0.237988	0.118994	−	0.992895i	$-0.462033\pi$
$577$	3298.51	0.237988	0.118994	+	0.992895i	$0.462033\pi$
$578$	5685.47	0.409142
$579$	0	0
$580$	−12195.9	−0.873116
$581$	3423.89	0.244487
$582$	0	0
$583$	5654.96	0.401723
$584$	−19581.0	−1.38745
$585$	0	0
$586$	1425.98	0.100523
$587$	2250.87	0.158268	0.0791339	−	0.996864i	$-0.474785\pi$
$587$	2250.87	0.158268	0.0791339	+	0.996864i	$0.474785\pi$
$588$	0	0
$589$	1526.86	0.106814
$590$	8702.05	0.607217
$591$	0	0
$592$	13242.6	0.919373
$593$	−25955.6	−1.79742	−0.898710	−	0.438544i	$-0.855494\pi$
$593$	−25955.6	−1.79742	−0.898710	+	0.438544i	$0.855494\pi$
$594$	0	0
$595$	1339.13	0.0922673
$596$	19026.2	1.30762
$597$	0	0
$598$	2960.33	0.202436
$599$	10200.8	0.695819	0.347909	−	0.937528i	$-0.386892\pi$
$599$	10200.8	0.695819	0.347909	+	0.937528i	$0.386892\pi$
$600$	0	0
$601$	5563.68	0.377616	0.188808	−	0.982014i	$-0.439538\pi$
$601$	5563.68	0.377616	0.188808	+	0.982014i	$0.439538\pi$
$602$	−1471.73	−0.0996398
$603$	0	0
$604$	11145.5	0.750835
$605$	1417.54	0.0952585
$606$	0	0
$607$	12109.8	0.809758	0.404879	−	0.914370i	$-0.367314\pi$
$607$	12109.8	0.809758	0.404879	+	0.914370i	$0.367314\pi$
$608$	22564.4	1.50511
$609$	0	0
$610$	−686.655	−0.0455768
$611$	13693.4	0.906669
$612$	0	0
$613$	−2026.27	−0.133508	−0.0667540	−	0.997769i	$-0.521264\pi$
$613$	−2026.27	−0.133508	−0.0667540	+	0.997769i	$0.521264\pi$
$614$	561.872	0.0369305
$615$	0	0
$616$	−1366.45	−0.0893765
$617$	−9471.18	−0.617983	−0.308991	−	0.951065i	$-0.599991\pi$
$617$	−9471.18	−0.617983	−0.308991	+	0.951065i	$0.599991\pi$
$618$	0	0
$619$	21610.6	1.40324	0.701619	−	0.712552i	$-0.252461\pi$
$619$	21610.6	1.40324	0.701619	+	0.712552i	$0.252461\pi$
$620$	923.004	0.0597883
$621$	0	0
$622$	−3973.26	−0.256130
$623$	3268.59	0.210198
$624$	0	0
$625$	−17005.9	−1.08838
$626$	3993.84	0.254993
$627$	0	0
$628$	9228.86	0.586420
$629$	−7135.27	−0.452308
$630$	0	0
$631$	−13483.4	−0.850659	−0.425330	−	0.905038i	$-0.639842\pi$
$631$	−13483.4	−0.850659	−0.425330	+	0.905038i	$0.639842\pi$
$632$	−3457.65	−0.217623
$633$	0	0
$634$	−8345.73	−0.522794
$635$	−30359.8	−1.89731
$636$	0	0
$637$	−1252.41	−0.0779001
$638$	−2154.85	−0.133717
$639$	0	0
$640$	17116.1	1.05714
$641$	6064.55	0.373690	0.186845	−	0.982389i	$-0.440174\pi$
$641$	6064.55	0.373690	0.186845	+	0.982389i	$0.440174\pi$
$642$	0	0
$643$	9981.12	0.612157	0.306079	−	0.952006i	$-0.400983\pi$
$643$	9981.12	0.612157	0.306079	+	0.952006i	$0.400983\pi$
$644$	4308.49	0.263631
$645$	0	0
$646$	−2518.08	−0.153363
$647$	−19235.0	−1.16879	−0.584395	−	0.811469i	$-0.698668\pi$
$647$	−19235.0	−1.16879	−0.584395	+	0.811469i	$0.698668\pi$
$648$	0	0
$649$	−6677.41	−0.403870
$650$	383.031	0.0231134
$651$	0	0
$652$	1668.97	0.100248
$653$	−7626.96	−0.457069	−0.228534	−	0.973536i	$-0.573393\pi$
$653$	−7626.96	−0.457069	−0.228534	+	0.973536i	$0.573393\pi$
$654$	0	0
$655$	33200.0	1.98051
$656$	−8021.85	−0.477440
$657$	0	0
$658$	−4588.94	−0.271878
$659$	−17527.8	−1.03609	−0.518046	−	0.855353i	$-0.673341\pi$
$659$	−17527.8	−1.03609	−0.518046	+	0.855353i	$0.673341\pi$
$660$	0	0
$661$	33122.0	1.94901	0.974507	−	0.224357i	$-0.0720281\pi$
$661$	33122.0	1.94901	0.974507	+	0.224357i	$0.0720281\pi$
$662$	−31.2391	−0.00183405
$663$	0	0
$664$	−8680.10	−0.507309
$665$	10334.5	0.602640
$666$	0	0
$667$	15153.1	0.879658
$668$	13802.8	0.799470
$669$	0	0
$670$	−15117.4	−0.871697
$671$	526.897	0.0303139
$672$	0	0
$673$	18913.2	1.08328	0.541641	−	0.840610i	$-0.317803\pi$
$673$	18913.2	1.08328	0.541641	+	0.840610i	$0.317803\pi$
$674$	−1037.19	−0.0592746
$675$	0	0
$676$	10038.3	0.571138
$677$	12615.3	0.716168	0.358084	−	0.933689i	$-0.383430\pi$
$677$	12615.3	0.716168	0.358084	+	0.933689i	$0.383430\pi$
$678$	0	0
$679$	3007.93	0.170006
$680$	−3394.92	−0.191455
$681$	0	0
$682$	163.082	0.00915650
$683$	−22131.1	−1.23986	−0.619929	−	0.784658i	$-0.712839\pi$
$683$	−22131.1	−1.23986	−0.619929	+	0.784658i	$0.712839\pi$
$684$	0	0
$685$	29060.2	1.62093
$686$	419.710	0.0233595
$687$	0	0
$688$	−5207.30	−0.288556
$689$	−13139.8	−0.726539
$690$	0	0
$691$	29230.2	1.60921	0.804607	−	0.593807i	$-0.202376\pi$
$691$	29230.2	1.60921	0.804607	+	0.593807i	$0.202376\pi$
$692$	22976.6	1.26220
$693$	0	0
$694$	−6595.16	−0.360733
$695$	17242.8	0.941088
$696$	0	0
$697$	4322.26	0.234888
$698$	−6967.19	−0.377811
$699$	0	0
$700$	557.467	0.0301004
$701$	−26085.9	−1.40549	−0.702746	−	0.711441i	$-0.748043\pi$
$701$	−26085.9	−1.40549	−0.702746	+	0.711441i	$0.748043\pi$
$702$	0	0
$703$	−55065.2	−2.95423
$704$	−256.913	−0.0137539
$705$	0	0
$706$	−6396.16	−0.340967
$707$	1145.52	0.0609357
$708$	0	0
$709$	14446.3	0.765223	0.382612	−	0.923909i	$-0.375025\pi$
$709$	14446.3	0.765223	0.382612	+	0.923909i	$0.375025\pi$
$710$	11230.9	0.593646
$711$	0	0
$712$	−8286.40	−0.436160
$713$	−1146.81	−0.0602363
$714$	0	0
$715$	−3293.78	−0.172280
$716$	15162.6	0.791412
$717$	0	0
$718$	4637.04	0.241021
$719$	−17315.1	−0.898116	−0.449058	−	0.893502i	$-0.648240\pi$
$719$	−17315.1	−0.898116	−0.449058	+	0.893502i	$0.648240\pi$
$720$	0	0
$721$	11872.0	0.613225
$722$	−11039.9	−0.569062
$723$	0	0
$724$	−7166.40	−0.367869
$725$	1960.63	0.100436
$726$	0	0
$727$	4864.75	0.248176	0.124088	−	0.992271i	$-0.460400\pi$
$727$	4864.75	0.248176	0.124088	+	0.992271i	$0.460400\pi$
$728$	3175.06	0.161642
$729$	0	0
$730$	15817.5	0.801963
$731$	2805.75	0.141962
$732$	0	0
$733$	26051.3	1.31272	0.656362	−	0.754447i	$-0.272095\pi$
$733$	26051.3	1.31272	0.656362	+	0.754447i	$0.272095\pi$
$734$	−2056.90	−0.103435
$735$	0	0
$736$	−16947.9	−0.848790
$737$	11600.2	0.579780
$738$	0	0
$739$	−29680.2	−1.47741	−0.738703	−	0.674031i	$-0.764562\pi$
$739$	−29680.2	−1.47741	−0.738703	+	0.674031i	$0.764562\pi$
$740$	−33287.5	−1.65361
$741$	0	0
$742$	4403.41	0.217863
$743$	6957.44	0.343531	0.171766	−	0.985138i	$-0.445053\pi$
$743$	6957.44	0.343531	0.171766	+	0.985138i	$0.445053\pi$
$744$	0	0
$745$	−34277.5	−1.68568
$746$	11848.4	0.581505
$747$	0	0
$748$	1168.05	0.0570963
$749$	−3375.07	−0.164650
$750$	0	0
$751$	9769.17	0.474676	0.237338	−	0.971427i	$-0.423725\pi$
$751$	9769.17	0.474676	0.237338	+	0.971427i	$0.423725\pi$
$752$	−16236.7	−0.787356
$753$	0	0
$754$	5006.97	0.241834
$755$	−20079.7	−0.967915
$756$	0	0
$757$	−7310.67	−0.351005	−0.175503	−	0.984479i	$-0.556155\pi$
$757$	−7310.67	−0.351005	−0.175503	+	0.984479i	$0.556155\pi$
$758$	−13173.2	−0.631231
$759$	0	0
$760$	−26199.7	−1.25048
$761$	33724.1	1.60643	0.803217	−	0.595686i	$-0.203120\pi$
$761$	33724.1	1.60643	0.803217	+	0.595686i	$0.203120\pi$
$762$	0	0
$763$	9713.05	0.460860
$764$	−14443.8	−0.683976
$765$	0	0
$766$	5389.17	0.254202
$767$	15515.5	0.730421
$768$	0	0
$769$	−6968.29	−0.326766	−0.163383	−	0.986563i	$-0.552241\pi$
$769$	−6968.29	−0.326766	−0.163383	+	0.986563i	$0.552241\pi$
$770$	1103.82	0.0516607
$771$	0	0
$772$	−20657.3	−0.963047
$773$	20407.7	0.949567	0.474783	−	0.880103i	$-0.342526\pi$
$773$	20407.7	0.949567	0.474783	+	0.880103i	$0.342526\pi$
$774$	0	0
$775$	−148.384	−0.00687755
$776$	−7625.59	−0.352761
$777$	0	0
$778$	−16041.7	−0.739231
$779$	33356.3	1.53416
$780$	0	0
$781$	−8617.90	−0.394843
$782$	1891.31	0.0864873
$783$	0	0
$784$	1485.03	0.0676488
$785$	−16626.7	−0.755964
$786$	0	0
$787$	13319.7	0.603297	0.301648	−	0.953419i	$-0.402463\pi$
$787$	13319.7	0.603297	0.301648	+	0.953419i	$0.402463\pi$
$788$	4738.75	0.214227
$789$	0	0
$790$	2793.08	0.125789
$791$	13172.8	0.592126
$792$	0	0
$793$	−1224.29	−0.0548244
$794$	−11901.1	−0.531931
$795$	0	0
$796$	21567.7	0.960362
$797$	−11122.9	−0.494344	−0.247172	−	0.968972i	$-0.579501\pi$
$797$	−11122.9	−0.494344	−0.247172	+	0.968972i	$0.579501\pi$
$798$	0	0
$799$	8748.51	0.387359
$800$	−2192.86	−0.0969116
$801$	0	0
$802$	210.009	0.00924649
$803$	−12137.4	−0.533399
$804$	0	0
$805$	−7762.17	−0.339852
$806$	−378.935	−0.0165600
$807$	0	0
$808$	−2904.07	−0.126441
$809$	−35906.0	−1.56043	−0.780215	−	0.625511i	$-0.784891\pi$
$809$	−35906.0	−1.56043	−0.780215	+	0.625511i	$0.784891\pi$
$810$	0	0
$811$	−4084.08	−0.176833	−0.0884164	−	0.996084i	$-0.528181\pi$
$811$	−4084.08	−0.176833	−0.0884164	+	0.996084i	$0.528181\pi$
$812$	7287.20	0.314939
$813$	0	0
$814$	−5881.44	−0.253249
$815$	−3006.81	−0.129232
$816$	0	0
$817$	21652.9	0.927220
$818$	−7478.74	−0.319668
$819$	0	0
$820$	20164.2	0.858738
$821$	29147.1	1.23903	0.619514	−	0.784986i	$-0.287330\pi$
$821$	29147.1	1.23903	0.619514	+	0.784986i	$0.287330\pi$
$822$	0	0
$823$	−14387.7	−0.609387	−0.304693	−	0.952450i	$-0.598554\pi$
$823$	−14387.7	−0.609387	−0.304693	+	0.952450i	$0.598554\pi$
$824$	−30097.3	−1.27244
$825$	0	0
$826$	−5199.58	−0.219027
$827$	−6056.48	−0.254661	−0.127330	−	0.991860i	$-0.540641\pi$
$827$	−6056.48	−0.254661	−0.127330	+	0.991860i	$0.540641\pi$
$828$	0	0
$829$	−3167.61	−0.132709	−0.0663545	−	0.997796i	$-0.521137\pi$
$829$	−3167.61	−0.132709	−0.0663545	+	0.997796i	$0.521137\pi$
$830$	7011.76	0.293231
$831$	0	0
$832$	596.959	0.0248748
$833$	−800.148	−0.0332815
$834$	0	0
$835$	−24867.1	−1.03061
$836$	9014.20	0.372922
$837$	0	0
$838$	1263.21	0.0520726
$839$	24152.2	0.993833	0.496916	−	0.867798i	$-0.334466\pi$
$839$	24152.2	0.993833	0.496916	+	0.867798i	$0.334466\pi$
$840$	0	0
$841$	1240.37	0.0508576
$842$	−14254.1	−0.583406
$843$	0	0
$844$	−6540.48	−0.266745
$845$	−18085.0	−0.736264
$846$	0	0
$847$	−847.000	−0.0343604
$848$	15580.3	0.630930
$849$	0	0
$850$	244.713	0.00987480
$851$	41359.0	1.66600
$852$	0	0
$853$	11497.2	0.461496	0.230748	−	0.973013i	$-0.425883\pi$
$853$	11497.2	0.461496	0.230748	+	0.973013i	$0.425883\pi$
$854$	410.285	0.0164399
$855$	0	0
$856$	8556.35	0.341647
$857$	25002.9	0.996595	0.498298	−	0.867006i	$-0.333959\pi$
$857$	25002.9	0.996595	0.498298	+	0.867006i	$0.333959\pi$
$858$	0	0
$859$	36363.8	1.44437	0.722187	−	0.691698i	$-0.243137\pi$
$859$	36363.8	1.44437	0.722187	+	0.691698i	$0.243137\pi$
$860$	13089.4	0.519006
$861$	0	0
$862$	−3686.48	−0.145664
$863$	28922.4	1.14082	0.570410	−	0.821360i	$-0.306784\pi$
$863$	28922.4	1.14082	0.570410	+	0.821360i	$0.306784\pi$
$864$	0	0
$865$	−41394.6	−1.62712
$866$	2562.38	0.100547
$867$	0	0
$868$	−551.506	−0.0215660
$869$	−2143.24	−0.0836644
$870$	0	0
$871$	−26954.0	−1.04857
$872$	−24624.1	−0.956283
$873$	0	0
$874$	14595.9	0.564888
$875$	9246.51	0.357245
$876$	0	0
$877$	−8749.24	−0.336877	−0.168438	−	0.985712i	$-0.553872\pi$
$877$	−8749.24	−0.336877	−0.168438	+	0.985712i	$0.553872\pi$
$878$	10325.3	0.396882
$879$	0	0
$880$	3905.55	0.149609
$881$	−32477.1	−1.24198	−0.620988	−	0.783820i	$-0.713269\pi$
$881$	−32477.1	−1.24198	−0.620988	+	0.783820i	$0.713269\pi$
$882$	0	0
$883$	41807.7	1.59337	0.796683	−	0.604397i	$-0.206586\pi$
$883$	41807.7	1.59337	0.796683	+	0.604397i	$0.206586\pi$
$884$	−2714.06	−0.103262
$885$	0	0
$886$	2198.12	0.0833490
$887$	−17267.0	−0.653631	−0.326815	−	0.945088i	$-0.605976\pi$
$887$	−17267.0	−0.653631	−0.326815	+	0.945088i	$0.605976\pi$
$888$	0	0
$889$	18140.4	0.684374
$890$	6693.74	0.252106
$891$	0	0
$892$	−29444.6	−1.10525
$893$	67515.1	2.53002
$894$	0	0
$895$	−27316.8	−1.02022
$896$	−10227.1	−0.381319
$897$	0	0
$898$	−7709.80	−0.286503
$899$	−1939.67	−0.0719595
$900$	0	0
$901$	−8394.81	−0.310401
$902$	3562.74	0.131515
$903$	0	0
$904$	−33395.2	−1.22866
$905$	12911.0	0.474226
$906$	0	0
$907$	24132.0	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$907$	24132.0	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$908$	−14447.8	−0.528048
$909$	0	0
$910$	−2564.81	−0.0934314
$911$	9391.82	0.341564	0.170782	−	0.985309i	$-0.445371\pi$
$911$	9391.82	0.341564	0.170782	+	0.985309i	$0.445371\pi$
$912$	0	0
$913$	−5380.39	−0.195033
$914$	−7048.74	−0.255089
$915$	0	0
$916$	−19008.9	−0.685668
$917$	−19837.4	−0.714382
$918$	0	0
$919$	46124.1	1.65560	0.827798	−	0.561026i	$-0.189593\pi$
$919$	46124.1	1.65560	0.827798	+	0.561026i	$0.189593\pi$
$920$	19678.4	0.705191
$921$	0	0
$922$	91.3882	0.00326433
$923$	20024.4	0.714097
$924$	0	0
$925$	5351.36	0.190218
$926$	4265.47	0.151374
$927$	0	0
$928$	−28665.0	−1.01398
$929$	−10411.9	−0.367709	−0.183855	−	0.982953i	$-0.558858\pi$
$929$	−10411.9	−0.367709	−0.183855	+	0.982953i	$0.558858\pi$
$930$	0	0
$931$	−6175.00	−0.217377
$932$	−25398.7	−0.892662
$933$	0	0
$934$	23878.6	0.836545
$935$	−2104.35	−0.0736039
$936$	0	0
$937$	11415.2	0.397992	0.198996	−	0.980000i	$-0.436232\pi$
$937$	11415.2	0.397992	0.198996	+	0.980000i	$0.436232\pi$
$938$	9032.84	0.314427
$939$	0	0
$940$	40813.6	1.41616
$941$	36890.1	1.27798	0.638992	−	0.769213i	$-0.279351\pi$
$941$	36890.1	1.27798	0.638992	+	0.769213i	$0.279351\pi$
$942$	0	0
$943$	−25053.6	−0.865172
$944$	−18397.3	−0.634302
$945$	0	0
$946$	2312.72	0.0794851
$947$	−1002.01	−0.0343832	−0.0171916	−	0.999852i	$-0.505473\pi$
$947$	−1002.01	−0.0343832	−0.0171916	+	0.999852i	$0.505473\pi$
$948$	0	0
$949$	28202.2	0.964682
$950$	1888.53	0.0644968
$951$	0	0
$952$	2028.50	0.0690590
$953$	33037.6	1.12297	0.561486	−	0.827486i	$-0.310230\pi$
$953$	33037.6	1.12297	0.561486	+	0.827486i	$0.310230\pi$
$954$	0	0
$955$	26021.9	0.881726
$956$	10950.9	0.370479
$957$	0	0
$958$	20877.2	0.704084
$959$	−17363.8	−0.584679
$960$	0	0
$961$	−29644.2	−0.995072
$962$	13666.0	0.458015
$963$	0	0
$964$	−17908.3	−0.598326
$965$	37216.1	1.24148
$966$	0	0
$967$	−37338.7	−1.24171	−0.620855	−	0.783926i	$-0.713214\pi$
$967$	−37338.7	−1.24171	−0.620855	+	0.783926i	$0.713214\pi$
$968$	2147.28	0.0712978
$969$	0	0
$970$	6159.93	0.203901
$971$	3258.01	0.107677	0.0538386	−	0.998550i	$-0.482854\pi$
$971$	3258.01	0.107677	0.0538386	+	0.998550i	$0.482854\pi$
$972$	0	0
$973$	−10302.8	−0.339457
$974$	11210.8	0.368807
$975$	0	0
$976$	1451.68	0.0476098
$977$	−2513.84	−0.0823183	−0.0411591	−	0.999153i	$-0.513105\pi$
$977$	−2513.84	−0.0823183	−0.0411591	+	0.999153i	$0.513105\pi$
$978$	0	0
$979$	−5136.36	−0.167680
$980$	−3732.85	−0.121675
$981$	0	0
$982$	19728.2	0.641092
$983$	50258.4	1.63072	0.815358	−	0.578957i	$-0.196540\pi$
$983$	50258.4	1.63072	0.815358	+	0.578957i	$0.196540\pi$
$984$	0	0
$985$	−8537.32	−0.276164
$986$	3198.88	0.103320
$987$	0	0
$988$	−20945.3	−0.674451
$989$	−16263.3	−0.522895
$990$	0	0
$991$	23304.6	0.747019	0.373510	−	0.927626i	$-0.378154\pi$
$991$	23304.6	0.747019	0.373510	+	0.927626i	$0.378154\pi$
$992$	2169.41	0.0694343
$993$	0	0
$994$	−6710.60	−0.214132
$995$	−38856.4	−1.23802
$996$	0	0
$997$	−36576.0	−1.16186	−0.580929	−	0.813954i	$-0.697311\pi$
$997$	−36576.0	−1.16186	−0.580929	+	0.813954i	$0.697311\pi$
$998$	9511.60	0.301688
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.s.1.4	✓	8
3.2	odd	2		693.4.a.t.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.4	✓	8	1.1	even	1	trivial
693.4.a.t.1.5	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)

\(f(q)\)	\(=\)	\(q-1.22364 q^{2} -6.50270 q^{4} +11.7152 q^{5} -7.00000 q^{7} +17.7461 q^{8} +O(q^{10})\)	\(q-1.22364 q^{2} -6.50270 q^{4} +11.7152 q^{5} -7.00000 q^{7} +17.7461 q^{8} -14.3353 q^{10} +11.0000 q^{11} -25.5594 q^{13} +8.56550 q^{14} +30.3067 q^{16} -16.3295 q^{17} -126.020 q^{19} -76.1807 q^{20} -13.4601 q^{22} +94.6529 q^{23} +12.2469 q^{25} +31.2756 q^{26} +45.5189 q^{28} +160.092 q^{29} -12.1160 q^{31} -179.054 q^{32} +19.9815 q^{34} -82.0067 q^{35} +436.954 q^{37} +154.204 q^{38} +207.900 q^{40} -264.689 q^{41} -171.820 q^{43} -71.5297 q^{44} -115.821 q^{46} -535.747 q^{47} +49.0000 q^{49} -14.9859 q^{50} +166.205 q^{52} +514.087 q^{53} +128.868 q^{55} -124.223 q^{56} -195.895 q^{58} -607.038 q^{59} +47.8997 q^{61} +14.8256 q^{62} -23.3557 q^{64} -299.435 q^{65} +1054.56 q^{67} +106.186 q^{68} +100.347 q^{70} -783.445 q^{71} -1103.40 q^{73} -534.676 q^{74} +819.473 q^{76} -77.0000 q^{77} -194.840 q^{79} +355.050 q^{80} +323.885 q^{82} -489.127 q^{83} -191.305 q^{85} +210.247 q^{86} +195.207 q^{88} -466.942 q^{89} +178.916 q^{91} -615.499 q^{92} +655.563 q^{94} -1476.36 q^{95} -429.704 q^{97} -59.9585 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8	\( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26} - 210 q^{28} + 72 q^{29} + 252 q^{31} - 1272 q^{32} + 630 q^{34} + 70 q^{35} - 80 q^{37} - 1885 q^{38} - 342 q^{40} - 682 q^{41} - 106 q^{43} + 330 q^{44} + 120 q^{46} - 828 q^{47} + 392 q^{49} - 801 q^{50} - 1681 q^{52} - 462 q^{53} - 110 q^{55} + 105 q^{56} - 1087 q^{58} - 626 q^{59} - 854 q^{61} - 1350 q^{62} + 2997 q^{64} - 22 q^{65} + 130 q^{67} - 2202 q^{68} + 91 q^{70} - 326 q^{71} - 390 q^{73} + 359 q^{74} + 2041 q^{76} - 616 q^{77} - 508 q^{79} - 4391 q^{80} + 1528 q^{82} - 1596 q^{83} - 880 q^{85} + 414 q^{86} - 165 q^{88} - 4324 q^{89} + 1036 q^{91} - 2092 q^{92} - 1685 q^{94} - 1076 q^{95} - 964 q^{97} - 98 q^{98}+O(q^{100}) \) 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

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