LMFDB - Embedded newform 2178.4.a.ce.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 331x^{4} + 48x^{3} + 23386x^{2} - 36820x - 100804 $ x^6 - x^5 - 331x^4 + 48x^3 + 23386x^2 - 36820x - 100804
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 198)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-13.2590$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -4.57651 q^{5} -7.03954 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} -4.57651 q^{5} -7.03954 q^{7} -8.00000 q^{8} +9.15301 q^{10} -89.4824 q^{13} +14.0791 q^{14} +16.0000 q^{16} -103.166 q^{17} -21.0830 q^{19} -18.3060 q^{20} -89.9033 q^{23} -104.056 q^{25} +178.965 q^{26} -28.1581 q^{28} -157.130 q^{29} -226.077 q^{31} -32.0000 q^{32} +206.332 q^{34} +32.2165 q^{35} +174.764 q^{37} +42.1660 q^{38} +36.6120 q^{40} -91.8579 q^{41} -11.8603 q^{43} +179.807 q^{46} +294.448 q^{47} -293.445 q^{49} +208.111 q^{50} -357.929 q^{52} +502.358 q^{53} +56.3163 q^{56} +314.260 q^{58} -466.926 q^{59} +122.984 q^{61} +452.153 q^{62} +64.0000 q^{64} +409.517 q^{65} -874.971 q^{67} -412.665 q^{68} -64.4330 q^{70} -111.259 q^{71} -813.804 q^{73} -349.527 q^{74} -84.3320 q^{76} +435.924 q^{79} -73.2241 q^{80} +183.716 q^{82} +663.714 q^{83} +472.141 q^{85} +23.7206 q^{86} +1288.31 q^{89} +629.914 q^{91} -359.613 q^{92} -588.896 q^{94} +96.4864 q^{95} -204.724 q^{97} +586.890 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 24 q^{4} + 17 q^{5} - 7 q^{7} - 48 q^{8}+O(q^{10}) $ 6 * q - 12 * q^2 + 24 * q^4 + 17 * q^5 - 7 * q^7 - 48 * q^8	$ 6 q - 12 q^{2} + 24 q^{4} + 17 q^{5} - 7 q^{7} - 48 q^{8} - 34 q^{10} - 68 q^{13} + 14 q^{14} + 96 q^{16} - 42 q^{17} - 98 q^{19} + 68 q^{20} + 210 q^{23} + 47 q^{25} + 136 q^{26} - 28 q^{28} - 13 q^{29} - 125 q^{31} - 192 q^{32} + 84 q^{34} - 534 q^{35} + 282 q^{37} + 196 q^{38} - 136 q^{40} + 170 q^{41} - 868 q^{43} - 420 q^{46} + 782 q^{47} - 439 q^{49} - 94 q^{50} - 272 q^{52} + 645 q^{53} + 56 q^{56} + 26 q^{58} + 507 q^{59} - 1772 q^{61} + 250 q^{62} + 384 q^{64} + 1856 q^{65} + 686 q^{67} - 168 q^{68} + 1068 q^{70} + 2782 q^{71} - 335 q^{73} - 564 q^{74} - 392 q^{76} - 127 q^{79} + 272 q^{80} - 340 q^{82} + 9 q^{83} - 370 q^{85} + 1736 q^{86} + 2526 q^{89} + 296 q^{91} + 840 q^{92} - 1564 q^{94} + 1194 q^{95} + 89 q^{97} + 878 q^{98}+O(q^{100}) $ 6 * q - 12 * q^2 + 24 * q^4 + 17 * q^5 - 7 * q^7 - 48 * q^8 - 34 * q^10 - 68 * q^13 + 14 * q^14 + 96 * q^16 - 42 * q^17 - 98 * q^19 + 68 * q^20 + 210 * q^23 + 47 * q^25 + 136 * q^26 - 28 * q^28 - 13 * q^29 - 125 * q^31 - 192 * q^32 + 84 * q^34 - 534 * q^35 + 282 * q^37 + 196 * q^38 - 136 * q^40 + 170 * q^41 - 868 * q^43 - 420 * q^46 + 782 * q^47 - 439 * q^49 - 94 * q^50 - 272 * q^52 + 645 * q^53 + 56 * q^56 + 26 * q^58 + 507 * q^59 - 1772 * q^61 + 250 * q^62 + 384 * q^64 + 1856 * q^65 + 686 * q^67 - 168 * q^68 + 1068 * q^70 + 2782 * q^71 - 335 * q^73 - 564 * q^74 - 392 * q^76 - 127 * q^79 + 272 * q^80 - 340 * q^82 + 9 * q^83 - 370 * q^85 + 1736 * q^86 + 2526 * q^89 + 296 * q^91 + 840 * q^92 - 1564 * q^94 + 1194 * q^95 + 89 * q^97 + 878 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−4.57651	−0.409335	−0.204668	−	0.978832i	$-0.565611\pi$
$5$	−4.57651	−0.409335	−0.204668	+	0.978832i	$0.565611\pi$
$6$	0	0
$7$	−7.03954	−0.380099	−0.190050	−	0.981774i	$-0.560865\pi$
$7$	−7.03954	−0.380099	−0.190050	+	0.981774i	$0.560865\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	9.15301	0.289444
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−89.4824	−1.90907	−0.954536	−	0.298095i	$-0.903649\pi$
$13$	−89.4824	−1.90907	−0.954536	+	0.298095i	$0.903649\pi$
$14$	14.0791	0.268771
$15$	0	0
$16$	16.0000	0.250000
$17$	−103.166	−1.47185	−0.735926	−	0.677062i	$-0.763253\pi$
$17$	−103.166	−1.47185	−0.735926	+	0.677062i	$0.763253\pi$
$18$	0	0
$19$	−21.0830	−0.254567	−0.127283	−	0.991866i	$-0.540626\pi$
$19$	−21.0830	−0.254567	−0.127283	+	0.991866i	$0.540626\pi$
$20$	−18.3060	−0.204668
$21$	0	0
$22$	0	0
$23$	−89.9033	−0.815050	−0.407525	−	0.913194i	$-0.633608\pi$
$23$	−89.9033	−0.815050	−0.407525	+	0.913194i	$0.633608\pi$
$24$	0	0
$25$	−104.056	−0.832445
$26$	178.965	1.34992
$27$	0	0
$28$	−28.1581	−0.190050
$29$	−157.130	−1.00615	−0.503074	−	0.864243i	$-0.667798\pi$
$29$	−157.130	−1.00615	−0.503074	+	0.864243i	$0.667798\pi$
$30$	0	0
$31$	−226.077	−1.30983	−0.654913	−	0.755705i	$-0.727295\pi$
$31$	−226.077	−1.30983	−0.654913	+	0.755705i	$0.727295\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	206.332	1.04076
$35$	32.2165	0.155588
$36$	0	0
$37$	174.764	0.776513	0.388256	−	0.921551i	$-0.373077\pi$
$37$	174.764	0.776513	0.388256	+	0.921551i	$0.373077\pi$
$38$	42.1660	0.180006
$39$	0	0
$40$	36.6120	0.144722
$41$	−91.8579	−0.349898	−0.174949	−	0.984578i	$-0.555976\pi$
$41$	−91.8579	−0.349898	−0.174949	+	0.984578i	$0.555976\pi$
$42$	0	0
$43$	−11.8603	−0.0420622	−0.0210311	−	0.999779i	$-0.506695\pi$
$43$	−11.8603	−0.0420622	−0.0210311	+	0.999779i	$0.506695\pi$
$44$	0	0
$45$	0	0
$46$	179.807	0.576327
$47$	294.448	0.913823	0.456911	−	0.889512i	$-0.348956\pi$
$47$	294.448	0.913823	0.456911	+	0.889512i	$0.348956\pi$
$48$	0	0
$49$	−293.445	−0.855525
$50$	208.111	0.588627
$51$	0	0
$52$	−357.929	−0.954536
$53$	502.358	1.30196	0.650982	−	0.759093i	$-0.274357\pi$
$53$	502.358	1.30196	0.650982	+	0.759093i	$0.274357\pi$
$54$	0	0
$55$	0	0
$56$	56.3163	0.134385
$57$	0	0
$58$	314.260	0.711454
$59$	−466.926	−1.03032	−0.515158	−	0.857095i	$-0.672267\pi$
$59$	−466.926	−1.03032	−0.515158	+	0.857095i	$0.672267\pi$
$60$	0	0
$61$	122.984	0.258140	0.129070	−	0.991635i	$-0.458801\pi$
$61$	122.984	0.258140	0.129070	+	0.991635i	$0.458801\pi$
$62$	452.153	0.926186
$63$	0	0
$64$	64.0000	0.125000
$65$	409.517	0.781450
$66$	0	0
$67$	−874.971	−1.59544	−0.797722	−	0.603026i	$-0.793962\pi$
$67$	−874.971	−1.59544	−0.797722	+	0.603026i	$0.793962\pi$
$68$	−412.665	−0.735926
$69$	0	0
$70$	−64.4330	−0.110017
$71$	−111.259	−0.185972	−0.0929858	−	0.995667i	$-0.529641\pi$
$71$	−111.259	−0.185972	−0.0929858	+	0.995667i	$0.529641\pi$
$72$	0	0
$73$	−813.804	−1.30478	−0.652388	−	0.757885i	$-0.726233\pi$
$73$	−813.804	−1.30478	−0.652388	+	0.757885i	$0.726233\pi$
$74$	−349.527	−0.549078
$75$	0	0
$76$	−84.3320	−0.127283
$77$	0	0
$78$	0	0
$79$	435.924	0.620826	0.310413	−	0.950602i	$-0.399533\pi$
$79$	435.924	0.620826	0.310413	+	0.950602i	$0.399533\pi$
$80$	−73.2241	−0.102334
$81$	0	0
$82$	183.716	0.247415
$83$	663.714	0.877736	0.438868	−	0.898552i	$-0.355380\pi$
$83$	663.714	0.877736	0.438868	+	0.898552i	$0.355380\pi$
$84$	0	0
$85$	472.141	0.602481
$86$	23.7206	0.0297425
$87$	0	0
$88$	0	0
$89$	1288.31	1.53439	0.767195	−	0.641414i	$-0.221652\pi$
$89$	1288.31	1.53439	0.767195	+	0.641414i	$0.221652\pi$
$90$	0	0
$91$	629.914	0.725637
$92$	−359.613	−0.407525
$93$	0	0
$94$	−588.896	−0.646170
$95$	96.4864	0.104203
$96$	0	0
$97$	−204.724	−0.214294	−0.107147	−	0.994243i	$-0.534172\pi$
$97$	−204.724	−0.214294	−0.107147	+	0.994243i	$0.534172\pi$
$98$	586.890	0.604947
$99$	0	0
$100$	−416.222	−0.416222
$101$	105.866	0.104298	0.0521488	−	0.998639i	$-0.483393\pi$
$101$	105.866	0.104298	0.0521488	+	0.998639i	$0.483393\pi$
$102$	0	0
$103$	−482.170	−0.461258	−0.230629	−	0.973042i	$-0.574078\pi$
$103$	−482.170	−0.461258	−0.230629	+	0.973042i	$0.574078\pi$
$104$	715.859	0.674959
$105$	0	0
$106$	−1004.72	−0.920628
$107$	−449.733	−0.406330	−0.203165	−	0.979144i	$-0.565123\pi$
$107$	−449.733	−0.406330	−0.203165	+	0.979144i	$0.565123\pi$
$108$	0	0
$109$	−1214.86	−1.06754	−0.533772	−	0.845629i	$-0.679226\pi$
$109$	−1214.86	−1.06754	−0.533772	+	0.845629i	$0.679226\pi$
$110$	0	0
$111$	0	0
$112$	−112.633	−0.0950248
$113$	1564.83	1.30271	0.651357	−	0.758771i	$-0.274200\pi$
$113$	1564.83	1.30271	0.651357	+	0.758771i	$0.274200\pi$
$114$	0	0
$115$	411.443	0.333628
$116$	−628.519	−0.503074
$117$	0	0
$118$	933.852	0.728543
$119$	726.242	0.559450
$120$	0	0
$121$	0	0
$122$	−245.969	−0.182533
$123$	0	0
$124$	−904.307	−0.654913
$125$	1048.27	0.750084
$126$	0	0
$127$	−2451.96	−1.71320	−0.856598	−	0.515984i	$-0.827426\pi$
$127$	−2451.96	−1.71320	−0.856598	+	0.515984i	$0.827426\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−819.033	−0.552569
$131$	−2460.67	−1.64114	−0.820572	−	0.571543i	$-0.806345\pi$
$131$	−2460.67	−1.64114	−0.820572	+	0.571543i	$0.806345\pi$
$132$	0	0
$133$	148.414	0.0967606
$134$	1749.94	1.12815
$135$	0	0
$136$	825.330	0.520378
$137$	625.448	0.390041	0.195021	−	0.980799i	$-0.437523\pi$
$137$	625.448	0.390041	0.195021	+	0.980799i	$0.437523\pi$
$138$	0	0
$139$	1170.41	0.714193	0.357096	−	0.934068i	$-0.383767\pi$
$139$	1170.41	0.714193	0.357096	+	0.934068i	$0.383767\pi$
$140$	128.866	0.0777940
$141$	0	0
$142$	222.517	0.131502
$143$	0	0
$144$	0	0
$145$	719.106	0.411852
$146$	1627.61	0.922616
$147$	0	0
$148$	699.055	0.388256
$149$	−1980.42	−1.08888	−0.544438	−	0.838801i	$-0.683257\pi$
$149$	−1980.42	−1.08888	−0.544438	+	0.838801i	$0.683257\pi$
$150$	0	0
$151$	−2551.13	−1.37489	−0.687444	−	0.726238i	$-0.741267\pi$
$151$	−2551.13	−1.37489	−0.687444	+	0.726238i	$0.741267\pi$
$152$	168.664	0.0900030
$153$	0	0
$154$	0	0
$155$	1034.64	0.536157
$156$	0	0
$157$	2263.82	1.15078	0.575390	−	0.817880i	$-0.304850\pi$
$157$	2263.82	1.15078	0.575390	+	0.817880i	$0.304850\pi$
$158$	−871.848	−0.438990
$159$	0	0
$160$	146.448	0.0723609
$161$	632.878	0.309800
$162$	0	0
$163$	−493.276	−0.237033	−0.118516	−	0.992952i	$-0.537814\pi$
$163$	−493.276	−0.237033	−0.118516	+	0.992952i	$0.537814\pi$
$164$	−367.432	−0.174949
$165$	0	0
$166$	−1327.43	−0.620653
$167$	2346.52	1.08730	0.543650	−	0.839312i	$-0.317042\pi$
$167$	2346.52	1.08730	0.543650	+	0.839312i	$0.317042\pi$
$168$	0	0
$169$	5810.09	2.64456
$170$	−944.282	−0.426018
$171$	0	0
$172$	−47.4411	−0.0210311
$173$	−2961.95	−1.30169	−0.650847	−	0.759209i	$-0.725586\pi$
$173$	−2961.95	−1.30169	−0.650847	+	0.759209i	$0.725586\pi$
$174$	0	0
$175$	732.503	0.316412
$176$	0	0
$177$	0	0
$178$	−2576.62	−1.08498
$179$	−1492.52	−0.623220	−0.311610	−	0.950210i	$-0.600868\pi$
$179$	−1492.52	−0.623220	−0.311610	+	0.950210i	$0.600868\pi$
$180$	0	0
$181$	−1545.85	−0.634820	−0.317410	−	0.948288i	$-0.602813\pi$
$181$	−1545.85	−0.634820	−0.317410	+	0.948288i	$0.602813\pi$
$182$	−1259.83	−0.513103
$183$	0	0
$184$	719.227	0.288164
$185$	−799.807	−0.317854
$186$	0	0
$187$	0	0
$188$	1177.79	0.456911
$189$	0	0
$190$	−192.973	−0.0736827
$191$	−1161.25	−0.439922	−0.219961	−	0.975509i	$-0.570593\pi$
$191$	−1161.25	−0.439922	−0.219961	+	0.975509i	$0.570593\pi$
$192$	0	0
$193$	−717.225	−0.267497	−0.133749	−	0.991015i	$-0.542702\pi$
$193$	−717.225	−0.267497	−0.133749	+	0.991015i	$0.542702\pi$
$194$	409.447	0.151529
$195$	0	0
$196$	−1173.78	−0.427762
$197$	3628.45	1.31227	0.656133	−	0.754645i	$-0.272191\pi$
$197$	3628.45	1.31227	0.656133	+	0.754645i	$0.272191\pi$
$198$	0	0
$199$	−1276.68	−0.454782	−0.227391	−	0.973804i	$-0.573020\pi$
$199$	−1276.68	−0.454782	−0.227391	+	0.973804i	$0.573020\pi$
$200$	832.445	0.294314
$201$	0	0
$202$	−211.732	−0.0737496
$203$	1106.12	0.382436
$204$	0	0
$205$	420.388	0.143225
$206$	964.339	0.326159
$207$	0	0
$208$	−1431.72	−0.477268
$209$	0	0
$210$	0	0
$211$	3229.47	1.05368	0.526839	−	0.849965i	$-0.323377\pi$
$211$	3229.47	1.05368	0.526839	+	0.849965i	$0.323377\pi$
$212$	2009.43	0.650982
$213$	0	0
$214$	899.466	0.287319
$215$	54.2787	0.0172176
$216$	0	0
$217$	1591.48	0.497863
$218$	2429.71	0.754867
$219$	0	0
$220$	0	0
$221$	9231.56	2.80987
$222$	0	0
$223$	−3287.01	−0.987060	−0.493530	−	0.869729i	$-0.664294\pi$
$223$	−3287.01	−0.987060	−0.493530	+	0.869729i	$0.664294\pi$
$224$	225.265	0.0671927
$225$	0	0
$226$	−3129.66	−0.921158
$227$	6290.58	1.83930	0.919649	−	0.392741i	$-0.128473\pi$
$227$	6290.58	1.83930	0.919649	+	0.392741i	$0.128473\pi$
$228$	0	0
$229$	−1066.43	−0.307737	−0.153869	−	0.988091i	$-0.549173\pi$
$229$	−1066.43	−0.307737	−0.153869	+	0.988091i	$0.549173\pi$
$230$	−822.886	−0.235911
$231$	0	0
$232$	1257.04	0.355727
$233$	3689.53	1.03738	0.518689	−	0.854963i	$-0.326420\pi$
$233$	3689.53	1.03738	0.518689	+	0.854963i	$0.326420\pi$
$234$	0	0
$235$	−1347.54	−0.374060
$236$	−1867.70	−0.515158
$237$	0	0
$238$	−1452.48	−0.395591
$239$	−7131.13	−1.93002	−0.965009	−	0.262218i	$-0.915546\pi$
$239$	−7131.13	−1.93002	−0.965009	+	0.262218i	$0.915546\pi$
$240$	0	0
$241$	−3210.60	−0.858146	−0.429073	−	0.903270i	$-0.641160\pi$
$241$	−3210.60	−0.858146	−0.429073	+	0.903270i	$0.641160\pi$
$242$	0	0
$243$	0	0
$244$	491.938	0.129070
$245$	1342.95	0.350196
$246$	0	0
$247$	1886.56	0.485987
$248$	1808.61	0.463093
$249$	0	0
$250$	−2096.55	−0.530389
$251$	6834.90	1.71878	0.859392	−	0.511317i	$-0.170842\pi$
$251$	6834.90	1.71878	0.859392	+	0.511317i	$0.170842\pi$
$252$	0	0
$253$	0	0
$254$	4903.91	1.21141
$255$	0	0
$256$	256.000	0.0625000
$257$	−5527.16	−1.34154	−0.670768	−	0.741667i	$-0.734035\pi$
$257$	−5527.16	−1.34154	−0.670768	+	0.741667i	$0.734035\pi$
$258$	0	0
$259$	−1230.26	−0.295152
$260$	1638.07	0.390725
$261$	0	0
$262$	4921.35	1.16046
$263$	−3472.45	−0.814147	−0.407073	−	0.913395i	$-0.633451\pi$
$263$	−3472.45	−0.814147	−0.407073	+	0.913395i	$0.633451\pi$
$264$	0	0
$265$	−2299.04	−0.532940
$266$	−296.829	−0.0684201
$267$	0	0
$268$	−3499.88	−0.797722
$269$	−4395.18	−0.996204	−0.498102	−	0.867118i	$-0.665970\pi$
$269$	−4395.18	−0.996204	−0.498102	+	0.867118i	$0.665970\pi$
$270$	0	0
$271$	−5904.86	−1.32360	−0.661798	−	0.749682i	$-0.730206\pi$
$271$	−5904.86	−1.32360	−0.661798	+	0.749682i	$0.730206\pi$
$272$	−1650.66	−0.367963
$273$	0	0
$274$	−1250.90	−0.275801
$275$	0	0
$276$	0	0
$277$	−229.840	−0.0498547	−0.0249273	−	0.999689i	$-0.507935\pi$
$277$	−229.840	−0.0498547	−0.0249273	+	0.999689i	$0.507935\pi$
$278$	−2340.82	−0.505010
$279$	0	0
$280$	−257.732	−0.0550086
$281$	−3990.15	−0.847090	−0.423545	−	0.905875i	$-0.639215\pi$
$281$	−3990.15	−0.847090	−0.423545	+	0.905875i	$0.639215\pi$
$282$	0	0
$283$	2027.72	0.425919	0.212960	−	0.977061i	$-0.431690\pi$
$283$	2027.72	0.425919	0.212960	+	0.977061i	$0.431690\pi$
$284$	−445.035	−0.0929858
$285$	0	0
$286$	0	0
$287$	646.637	0.132996
$288$	0	0
$289$	5730.27	1.16635
$290$	−1438.21	−0.291223
$291$	0	0
$292$	−3255.22	−0.652388
$293$	2875.03	0.573247	0.286623	−	0.958043i	$-0.407467\pi$
$293$	2875.03	0.573247	0.286623	+	0.958043i	$0.407467\pi$
$294$	0	0
$295$	2136.89	0.421744
$296$	−1398.11	−0.274539
$297$	0	0
$298$	3960.84	0.769951
$299$	8044.76	1.55599
$300$	0	0
$301$	83.4909	0.0159878
$302$	5102.26	0.972192
$303$	0	0
$304$	−337.328	−0.0636417
$305$	−562.839	−0.105666
$306$	0	0
$307$	841.144	0.156373	0.0781867	−	0.996939i	$-0.475087\pi$
$307$	841.144	0.156373	0.0781867	+	0.996939i	$0.475087\pi$
$308$	0	0
$309$	0	0
$310$	−2069.28	−0.379121
$311$	9174.55	1.67280	0.836400	−	0.548120i	$-0.184656\pi$
$311$	9174.55	1.67280	0.836400	+	0.548120i	$0.184656\pi$
$312$	0	0
$313$	−2912.34	−0.525926	−0.262963	−	0.964806i	$-0.584700\pi$
$313$	−2912.34	−0.525926	−0.262963	+	0.964806i	$0.584700\pi$
$314$	−4527.63	−0.813724
$315$	0	0
$316$	1743.70	0.310413
$317$	1828.97	0.324054	0.162027	−	0.986786i	$-0.448197\pi$
$317$	1828.97	0.324054	0.162027	+	0.986786i	$0.448197\pi$
$318$	0	0
$319$	0	0
$320$	−292.896	−0.0511669
$321$	0	0
$322$	−1265.76	−0.219061
$323$	2175.05	0.374685
$324$	0	0
$325$	9311.14	1.58920
$326$	986.551	0.167607
$327$	0	0
$328$	734.863	0.123707
$329$	−2072.78	−0.347343
$330$	0	0
$331$	−6235.51	−1.03545	−0.517726	−	0.855547i	$-0.673221\pi$
$331$	−6235.51	−1.03545	−0.517726	+	0.855547i	$0.673221\pi$
$332$	2654.86	0.438868
$333$	0	0
$334$	−4693.04	−0.768837
$335$	4004.31	0.653071
$336$	0	0
$337$	986.658	0.159486	0.0797428	−	0.996815i	$-0.474590\pi$
$337$	986.658	0.159486	0.0797428	+	0.996815i	$0.474590\pi$
$338$	−11620.2	−1.86998
$339$	0	0
$340$	1888.56	0.301240
$341$	0	0
$342$	0	0
$343$	4480.28	0.705283
$344$	94.8823	0.0148712
$345$	0	0
$346$	5923.90	0.920436
$347$	−11482.2	−1.77636	−0.888178	−	0.459499i	$-0.848029\pi$
$347$	−11482.2	−1.77636	−0.888178	+	0.459499i	$0.848029\pi$
$348$	0	0
$349$	−3203.25	−0.491307	−0.245654	−	0.969358i	$-0.579003\pi$
$349$	−3203.25	−0.491307	−0.245654	+	0.969358i	$0.579003\pi$
$350$	−1465.01	−0.223737
$351$	0	0
$352$	0	0
$353$	8669.58	1.30718	0.653591	−	0.756848i	$-0.273262\pi$
$353$	8669.58	1.30718	0.653591	+	0.756848i	$0.273262\pi$
$354$	0	0
$355$	509.176	0.0761247
$356$	5153.25	0.767195
$357$	0	0
$358$	2985.05	0.440683
$359$	−4294.29	−0.631320	−0.315660	−	0.948872i	$-0.602226\pi$
$359$	−4294.29	−0.631320	−0.315660	+	0.948872i	$0.602226\pi$
$360$	0	0
$361$	−6414.51	−0.935196
$362$	3091.71	0.448885
$363$	0	0
$364$	2519.66	0.362818
$365$	3724.38	0.534090
$366$	0	0
$367$	9107.04	1.29532	0.647662	−	0.761928i	$-0.275747\pi$
$367$	9107.04	1.29532	0.647662	+	0.761928i	$0.275747\pi$
$368$	−1438.45	−0.203762
$369$	0	0
$370$	1599.61	0.224757
$371$	−3536.36	−0.494876
$372$	0	0
$373$	7648.04	1.06166	0.530831	−	0.847477i	$-0.321880\pi$
$373$	7648.04	1.06166	0.530831	+	0.847477i	$0.321880\pi$
$374$	0	0
$375$	0	0
$376$	−2355.59	−0.323085
$377$	14060.4	1.92081
$378$	0	0
$379$	−9302.19	−1.26074	−0.630371	−	0.776294i	$-0.717097\pi$
$379$	−9302.19	−1.26074	−0.630371	+	0.776294i	$0.717097\pi$
$380$	385.946	0.0521016
$381$	0	0
$382$	2322.50	0.311072
$383$	−395.475	−0.0527620	−0.0263810	−	0.999652i	$-0.508398\pi$
$383$	−395.475	−0.0527620	−0.0263810	+	0.999652i	$0.508398\pi$
$384$	0	0
$385$	0	0
$386$	1434.45	0.189149
$387$	0	0
$388$	−818.894	−0.107147
$389$	−2413.79	−0.314612	−0.157306	−	0.987550i	$-0.550281\pi$
$389$	−2413.79	−0.314612	−0.157306	+	0.987550i	$0.550281\pi$
$390$	0	0
$391$	9274.99	1.19963
$392$	2347.56	0.302474
$393$	0	0
$394$	−7256.91	−0.927913
$395$	−1995.01	−0.254126
$396$	0	0
$397$	4882.03	0.617185	0.308592	−	0.951194i	$-0.400142\pi$
$397$	4882.03	0.617185	0.308592	+	0.951194i	$0.400142\pi$
$398$	2553.37	0.321580
$399$	0	0
$400$	−1664.89	−0.208111
$401$	−1748.86	−0.217790	−0.108895	−	0.994053i	$-0.534731\pi$
$401$	−1748.86	−0.217790	−0.108895	+	0.994053i	$0.534731\pi$
$402$	0	0
$403$	20229.9	2.50055
$404$	423.464	0.0521488
$405$	0	0
$406$	−2212.24	−0.270423
$407$	0	0
$408$	0	0
$409$	4968.49	0.600674	0.300337	−	0.953833i	$-0.402901\pi$
$409$	4968.49	0.600674	0.300337	+	0.953833i	$0.402901\pi$
$410$	−840.777	−0.101276
$411$	0	0
$412$	−1928.68	−0.230629
$413$	3286.94	0.391622
$414$	0	0
$415$	−3037.49	−0.359288
$416$	2863.44	0.337480
$417$	0	0
$418$	0	0
$419$	11763.3	1.37154	0.685770	−	0.727819i	$-0.259466\pi$
$419$	11763.3	1.37154	0.685770	+	0.727819i	$0.259466\pi$
$420$	0	0
$421$	−4369.22	−0.505802	−0.252901	−	0.967492i	$-0.581385\pi$
$421$	−4369.22	−0.505802	−0.252901	+	0.967492i	$0.581385\pi$
$422$	−6458.94	−0.745063
$423$	0	0
$424$	−4018.86	−0.460314
$425$	10735.0	1.22524
$426$	0	0
$427$	−865.753	−0.0981188
$428$	−1798.93	−0.203165
$429$	0	0
$430$	−108.557	−0.0121746
$431$	3184.71	0.355921	0.177961	−	0.984038i	$-0.443050\pi$
$431$	3184.71	0.355921	0.177961	+	0.984038i	$0.443050\pi$
$432$	0	0
$433$	−5538.61	−0.614708	−0.307354	−	0.951595i	$-0.599444\pi$
$433$	−5538.61	−0.614708	−0.307354	+	0.951595i	$0.599444\pi$
$434$	−3182.95	−0.352043
$435$	0	0
$436$	−4859.43	−0.533772
$437$	1895.43	0.207485
$438$	0	0
$439$	8450.31	0.918704	0.459352	−	0.888254i	$-0.348082\pi$
$439$	8450.31	0.918704	0.459352	+	0.888254i	$0.348082\pi$
$440$	0	0
$441$	0	0
$442$	−18463.1	−1.98688
$443$	1586.14	0.170113	0.0850564	−	0.996376i	$-0.472893\pi$
$443$	1586.14	0.170113	0.0850564	+	0.996376i	$0.472893\pi$
$444$	0	0
$445$	−5895.96	−0.628080
$446$	6574.02	0.697957
$447$	0	0
$448$	−450.530	−0.0475124
$449$	14257.0	1.49851	0.749255	−	0.662282i	$-0.230412\pi$
$449$	14257.0	1.49851	0.749255	+	0.662282i	$0.230412\pi$
$450$	0	0
$451$	0	0
$452$	6259.31	0.651357
$453$	0	0
$454$	−12581.2	−1.30058
$455$	−2882.81	−0.297029
$456$	0	0
$457$	878.746	0.0899474	0.0449737	−	0.998988i	$-0.485680\pi$
$457$	878.746	0.0899474	0.0449737	+	0.998988i	$0.485680\pi$
$458$	2132.86	0.217603
$459$	0	0
$460$	1645.77	0.166814
$461$	4936.37	0.498719	0.249360	−	0.968411i	$-0.419780\pi$
$461$	4936.37	0.498719	0.249360	+	0.968411i	$0.419780\pi$
$462$	0	0
$463$	16955.6	1.70193	0.850964	−	0.525224i	$-0.176019\pi$
$463$	16955.6	1.70193	0.850964	+	0.525224i	$0.176019\pi$
$464$	−2514.08	−0.251537
$465$	0	0
$466$	−7379.06	−0.733538
$467$	9637.03	0.954922	0.477461	−	0.878653i	$-0.341557\pi$
$467$	9637.03	0.954922	0.477461	+	0.878653i	$0.341557\pi$
$468$	0	0
$469$	6159.39	0.606427
$470$	2695.09	0.264500
$471$	0	0
$472$	3735.41	0.364271
$473$	0	0
$474$	0	0
$475$	2193.80	0.211913
$476$	2904.97	0.279725
$477$	0	0
$478$	14262.3	1.36473
$479$	12521.3	1.19439	0.597193	−	0.802097i	$-0.296282\pi$
$479$	12521.3	1.19439	0.597193	+	0.802097i	$0.296282\pi$
$480$	0	0
$481$	−15638.3	−1.48242
$482$	6421.21	0.606801
$483$	0	0
$484$	0	0
$485$	936.919	0.0877181
$486$	0	0
$487$	12575.7	1.17014	0.585072	−	0.810982i	$-0.301066\pi$
$487$	12575.7	1.17014	0.585072	+	0.810982i	$0.301066\pi$
$488$	−983.875	−0.0912663
$489$	0	0
$490$	−2685.90	−0.247626
$491$	−1971.22	−0.181181	−0.0905904	−	0.995888i	$-0.528875\pi$
$491$	−1971.22	−0.181181	−0.0905904	+	0.995888i	$0.528875\pi$
$492$	0	0
$493$	16210.5	1.48090
$494$	−3773.11	−0.343644
$495$	0	0
$496$	−3617.23	−0.327456
$497$	783.209	0.0706876
$498$	0	0
$499$	−7374.80	−0.661606	−0.330803	−	0.943700i	$-0.607320\pi$
$499$	−7374.80	−0.661606	−0.330803	+	0.943700i	$0.607320\pi$
$500$	4193.10	0.375042
$501$	0	0
$502$	−13669.8	−1.21536
$503$	−12188.8	−1.08046	−0.540229	−	0.841518i	$-0.681662\pi$
$503$	−12188.8	−1.08046	−0.540229	+	0.841518i	$0.681662\pi$
$504$	0	0
$505$	−484.497	−0.0426927
$506$	0	0
$507$	0	0
$508$	−9807.82	−0.856598
$509$	−7388.58	−0.643405	−0.321702	−	0.946841i	$-0.604255\pi$
$509$	−7388.58	−0.643405	−0.321702	+	0.946841i	$0.604255\pi$
$510$	0	0
$511$	5728.81	0.495944
$512$	−512.000	−0.0441942
$513$	0	0
$514$	11054.3	0.948610
$515$	2206.65	0.188809
$516$	0	0
$517$	0	0
$518$	2460.51	0.208704
$519$	0	0
$520$	−3276.13	−0.276284
$521$	−13529.2	−1.13767	−0.568836	−	0.822451i	$-0.692606\pi$
$521$	−13529.2	−1.13767	−0.568836	+	0.822451i	$0.692606\pi$
$522$	0	0
$523$	18923.5	1.58215	0.791077	−	0.611716i	$-0.209521\pi$
$523$	18923.5	1.58215	0.791077	+	0.611716i	$0.209521\pi$
$524$	−9842.69	−0.820572
$525$	0	0
$526$	6944.90	0.575689
$527$	23323.5	1.92787
$528$	0	0
$529$	−4084.39	−0.335694
$530$	4598.08	0.376845
$531$	0	0
$532$	593.658	0.0483803
$533$	8219.67	0.667980
$534$	0	0
$535$	2058.21	0.166325
$536$	6999.77	0.564075
$537$	0	0
$538$	8790.36	0.704423
$539$	0	0
$540$	0	0
$541$	12655.2	1.00571	0.502854	−	0.864371i	$-0.332283\pi$
$541$	12655.2	1.00571	0.502854	+	0.864371i	$0.332283\pi$
$542$	11809.7	0.935924
$543$	0	0
$544$	3301.32	0.260189
$545$	5559.80	0.436983
$546$	0	0
$547$	23429.8	1.83142	0.915708	−	0.401844i	$-0.131631\pi$
$547$	23429.8	1.83142	0.915708	+	0.401844i	$0.131631\pi$
$548$	2501.79	0.195021
$549$	0	0
$550$	0	0
$551$	3312.77	0.256132
$552$	0	0
$553$	−3068.70	−0.235976
$554$	459.680	0.0352526
$555$	0	0
$556$	4681.63	0.357096
$557$	22742.9	1.73007	0.865034	−	0.501712i	$-0.167296\pi$
$557$	22742.9	1.73007	0.865034	+	0.501712i	$0.167296\pi$
$558$	0	0
$559$	1061.29	0.0802999
$560$	515.464	0.0388970
$561$	0	0
$562$	7980.30	0.598983
$563$	2397.29	0.179456	0.0897280	−	0.995966i	$-0.471400\pi$
$563$	2397.29	0.179456	0.0897280	+	0.995966i	$0.471400\pi$
$564$	0	0
$565$	−7161.45	−0.533247
$566$	−4055.43	−0.301171
$567$	0	0
$568$	890.070	0.0657509
$569$	−12443.6	−0.916808	−0.458404	−	0.888744i	$-0.651579\pi$
$569$	−12443.6	−0.916808	−0.458404	+	0.888744i	$0.651579\pi$
$570$	0	0
$571$	−25337.2	−1.85697	−0.928485	−	0.371370i	$-0.878888\pi$
$571$	−25337.2	−1.85697	−0.928485	+	0.371370i	$0.878888\pi$
$572$	0	0
$573$	0	0
$574$	−1293.27	−0.0940422
$575$	9354.95	0.678484
$576$	0	0
$577$	−16450.5	−1.18690	−0.593450	−	0.804871i	$-0.702235\pi$
$577$	−16450.5	−1.18690	−0.593450	+	0.804871i	$0.702235\pi$
$578$	−11460.5	−0.824733
$579$	0	0
$580$	2876.42	0.205926
$581$	−4672.24	−0.333627
$582$	0	0
$583$	0	0
$584$	6510.44	0.461308
$585$	0	0
$586$	−5750.07	−0.405347
$587$	15910.1	1.11870	0.559352	−	0.828931i	$-0.311050\pi$
$587$	15910.1	1.11870	0.559352	+	0.828931i	$0.311050\pi$
$588$	0	0
$589$	4766.37	0.333438
$590$	−4273.78	−0.298218
$591$	0	0
$592$	2796.22	0.194128
$593$	−26361.5	−1.82552	−0.912762	−	0.408492i	$-0.866055\pi$
$593$	−26361.5	−1.82552	−0.912762	+	0.408492i	$0.866055\pi$
$594$	0	0
$595$	−3323.65	−0.229002
$596$	−7921.68	−0.544438
$597$	0	0
$598$	−16089.5	−1.10025
$599$	−15578.4	−1.06263	−0.531315	−	0.847174i	$-0.678302\pi$
$599$	−15578.4	−1.06263	−0.531315	+	0.847174i	$0.678302\pi$
$600$	0	0
$601$	2647.38	0.179682	0.0898408	−	0.995956i	$-0.471364\pi$
$601$	2647.38	0.179682	0.0898408	+	0.995956i	$0.471364\pi$
$602$	−166.982	−0.0113051
$603$	0	0
$604$	−10204.5	−0.687444
$605$	0	0
$606$	0	0
$607$	17591.7	1.17632	0.588160	−	0.808745i	$-0.299852\pi$
$607$	17591.7	1.17632	0.588160	+	0.808745i	$0.299852\pi$
$608$	674.656	0.0450015
$609$	0	0
$610$	1125.68	0.0747170
$611$	−26347.9	−1.74455
$612$	0	0
$613$	−13244.5	−0.872658	−0.436329	−	0.899787i	$-0.643722\pi$
$613$	−13244.5	−0.872658	−0.436329	+	0.899787i	$0.643722\pi$
$614$	−1682.29	−0.110573
$615$	0	0
$616$	0	0
$617$	−20902.5	−1.36386	−0.681930	−	0.731417i	$-0.738859\pi$
$617$	−20902.5	−1.36386	−0.681930	+	0.731417i	$0.738859\pi$
$618$	0	0
$619$	6969.51	0.452550	0.226275	−	0.974063i	$-0.427345\pi$
$619$	6969.51	0.452550	0.226275	+	0.974063i	$0.427345\pi$
$620$	4138.57	0.268079
$621$	0	0
$622$	−18349.1	−1.18285
$623$	−9069.11	−0.583221
$624$	0	0
$625$	8209.52	0.525409
$626$	5824.67	0.371886
$627$	0	0
$628$	9055.27	0.575390
$629$	−18029.7	−1.14291
$630$	0	0
$631$	30496.5	1.92400	0.962002	−	0.273043i	$-0.0880300\pi$
$631$	30496.5	1.92400	0.962002	+	0.273043i	$0.0880300\pi$
$632$	−3487.39	−0.219495
$633$	0	0
$634$	−3657.94	−0.229141
$635$	11221.4	0.701271
$636$	0	0
$637$	26258.1	1.63326
$638$	0	0
$639$	0	0
$640$	585.793	0.0361805
$641$	−14561.2	−0.897244	−0.448622	−	0.893722i	$-0.648085\pi$
$641$	−14561.2	−0.897244	−0.448622	+	0.893722i	$0.648085\pi$
$642$	0	0
$643$	−19398.1	−1.18971	−0.594856	−	0.803832i	$-0.702791\pi$
$643$	−19398.1	−1.18971	−0.594856	+	0.803832i	$0.702791\pi$
$644$	2531.51	0.154900
$645$	0	0
$646$	−4350.10	−0.264942
$647$	−29892.7	−1.81639	−0.908195	−	0.418548i	$-0.862539\pi$
$647$	−29892.7	−1.81639	−0.908195	+	0.418548i	$0.862539\pi$
$648$	0	0
$649$	0	0
$650$	−18622.3	−1.12373
$651$	0	0
$652$	−1973.10	−0.118516
$653$	22590.7	1.35382	0.676909	−	0.736067i	$-0.263319\pi$
$653$	22590.7	1.35382	0.676909	+	0.736067i	$0.263319\pi$
$654$	0	0
$655$	11261.3	0.671778
$656$	−1469.73	−0.0874744
$657$	0	0
$658$	4145.56	0.245609
$659$	−12824.3	−0.758061	−0.379030	−	0.925384i	$-0.623742\pi$
$659$	−12824.3	−0.758061	−0.379030	+	0.925384i	$0.623742\pi$
$660$	0	0
$661$	−28829.4	−1.69642	−0.848209	−	0.529662i	$-0.822319\pi$
$661$	−28829.4	−1.69642	−0.848209	+	0.529662i	$0.822319\pi$
$662$	12471.0	0.732175
$663$	0	0
$664$	−5309.71	−0.310326
$665$	−679.220	−0.0396075
$666$	0	0
$667$	14126.5	0.820060
$668$	9386.07	0.543650
$669$	0	0
$670$	−8008.62	−0.461791
$671$	0	0
$672$	0	0
$673$	10133.3	0.580403	0.290202	−	0.956966i	$-0.406278\pi$
$673$	10133.3	0.580403	0.290202	+	0.956966i	$0.406278\pi$
$674$	−1973.32	−0.112773
$675$	0	0
$676$	23240.4	1.32228
$677$	5051.06	0.286748	0.143374	−	0.989669i	$-0.454205\pi$
$677$	5051.06	0.286748	0.143374	+	0.989669i	$0.454205\pi$
$678$	0	0
$679$	1441.16	0.0814530
$680$	−3777.13	−0.213009
$681$	0	0
$682$	0	0
$683$	−5966.92	−0.334287	−0.167144	−	0.985933i	$-0.553454\pi$
$683$	−5966.92	−0.334287	−0.167144	+	0.985933i	$0.553454\pi$
$684$	0	0
$685$	−2862.37	−0.159658
$686$	−8960.55	−0.498711
$687$	0	0
$688$	−189.765	−0.0105156
$689$	−44952.1	−2.48554
$690$	0	0
$691$	6026.67	0.331788	0.165894	−	0.986144i	$-0.446949\pi$
$691$	6026.67	0.331788	0.165894	+	0.986144i	$0.446949\pi$
$692$	−11847.8	−0.650847
$693$	0	0
$694$	22964.4	1.25607
$695$	−5356.38	−0.292344
$696$	0	0
$697$	9476.64	0.514997
$698$	6406.51	0.347407
$699$	0	0
$700$	2930.01	0.158206
$701$	−9568.15	−0.515526	−0.257763	−	0.966208i	$-0.582985\pi$
$701$	−9568.15	−0.515526	−0.257763	+	0.966208i	$0.582985\pi$
$702$	0	0
$703$	−3684.54	−0.197674
$704$	0	0
$705$	0	0
$706$	−17339.2	−0.924318
$707$	−745.248	−0.0396435
$708$	0	0
$709$	−5158.01	−0.273220	−0.136610	−	0.990625i	$-0.543621\pi$
$709$	−5158.01	−0.273220	−0.136610	+	0.990625i	$0.543621\pi$
$710$	−1018.35	−0.0538283
$711$	0	0
$712$	−10306.5	−0.542489
$713$	20325.1	1.06757
$714$	0	0
$715$	0	0
$716$	−5970.09	−0.311610
$717$	0	0
$718$	8588.58	0.446411
$719$	−7027.41	−0.364504	−0.182252	−	0.983252i	$-0.558339\pi$
$719$	−7027.41	−0.364504	−0.182252	+	0.983252i	$0.558339\pi$
$720$	0	0
$721$	3394.25	0.175324
$722$	12829.0	0.661283
$723$	0	0
$724$	−6183.41	−0.317410
$725$	16350.2	0.837562
$726$	0	0
$727$	9911.22	0.505621	0.252811	−	0.967516i	$-0.418645\pi$
$727$	9911.22	0.505621	0.252811	+	0.967516i	$0.418645\pi$
$728$	−5039.31	−0.256551
$729$	0	0
$730$	−7448.76	−0.377659
$731$	1223.58	0.0619094
$732$	0	0
$733$	14769.5	0.744235	0.372118	−	0.928186i	$-0.378632\pi$
$733$	14769.5	0.744235	0.372118	+	0.928186i	$0.378632\pi$
$734$	−18214.1	−0.915932
$735$	0	0
$736$	2876.91	0.144082
$737$	0	0
$738$	0	0
$739$	−30674.8	−1.52691	−0.763457	−	0.645859i	$-0.776499\pi$
$739$	−30674.8	−1.52691	−0.763457	+	0.645859i	$0.776499\pi$
$740$	−3199.23	−0.158927
$741$	0	0
$742$	7072.73	0.349930
$743$	5993.05	0.295913	0.147957	−	0.988994i	$-0.452730\pi$
$743$	5993.05	0.295913	0.147957	+	0.988994i	$0.452730\pi$
$744$	0	0
$745$	9063.41	0.445715
$746$	−15296.1	−0.750709
$747$	0	0
$748$	0	0
$749$	3165.91	0.154446
$750$	0	0
$751$	−2650.42	−0.128782	−0.0643910	−	0.997925i	$-0.520510\pi$
$751$	−2650.42	−0.128782	−0.0643910	+	0.997925i	$0.520510\pi$
$752$	4711.17	0.228456
$753$	0	0
$754$	−28120.7	−1.35822
$755$	11675.3	0.562790
$756$	0	0
$757$	−29830.3	−1.43224	−0.716118	−	0.697980i	$-0.754083\pi$
$757$	−29830.3	−1.43224	−0.716118	+	0.697980i	$0.754083\pi$
$758$	18604.4	0.891479
$759$	0	0
$760$	−771.891	−0.0368414
$761$	20738.0	0.987848	0.493924	−	0.869505i	$-0.335562\pi$
$761$	20738.0	0.987848	0.493924	+	0.869505i	$0.335562\pi$
$762$	0	0
$763$	8552.03	0.405772
$764$	−4645.00	−0.219961
$765$	0	0
$766$	790.950	0.0373083
$767$	41781.7	1.96695
$768$	0	0
$769$	20601.9	0.966092	0.483046	−	0.875595i	$-0.339530\pi$
$769$	20601.9	0.966092	0.483046	+	0.875595i	$0.339530\pi$
$770$	0	0
$771$	0	0
$772$	−2868.90	−0.133749
$773$	−28146.9	−1.30967	−0.654835	−	0.755772i	$-0.727262\pi$
$773$	−28146.9	−1.30967	−0.654835	+	0.755772i	$0.727262\pi$
$774$	0	0
$775$	23524.5	1.09036
$776$	1637.79	0.0757644
$777$	0	0
$778$	4827.59	0.222465
$779$	1936.64	0.0890723
$780$	0	0
$781$	0	0
$782$	−18550.0	−0.848268
$783$	0	0
$784$	−4695.12	−0.213881
$785$	−10360.4	−0.471054
$786$	0	0
$787$	−5175.21	−0.234404	−0.117202	−	0.993108i	$-0.537393\pi$
$787$	−5175.21	−0.234404	−0.117202	+	0.993108i	$0.537393\pi$
$788$	14513.8	0.656133
$789$	0	0
$790$	3990.02	0.179694
$791$	−11015.7	−0.495160
$792$	0	0
$793$	−11004.9	−0.492808
$794$	−9764.07	−0.436415
$795$	0	0
$796$	−5106.73	−0.227391
$797$	−2195.96	−0.0975973	−0.0487986	−	0.998809i	$-0.515539\pi$
$797$	−2195.96	−0.0975973	−0.0487986	+	0.998809i	$0.515539\pi$
$798$	0	0
$799$	−30377.1	−1.34501
$800$	3329.78	0.147157
$801$	0	0
$802$	3497.72	0.154001
$803$	0	0
$804$	0	0
$805$	−2896.37	−0.126812
$806$	−40459.8	−1.76816
$807$	0	0
$808$	−846.928	−0.0368748
$809$	30979.9	1.34635	0.673174	−	0.739484i	$-0.264930\pi$
$809$	30979.9	1.34635	0.673174	+	0.739484i	$0.264930\pi$
$810$	0	0
$811$	−10160.9	−0.439949	−0.219974	−	0.975506i	$-0.570597\pi$
$811$	−10160.9	−0.439949	−0.219974	+	0.975506i	$0.570597\pi$
$812$	4424.48	0.191218
$813$	0	0
$814$	0	0
$815$	2257.48	0.0970258
$816$	0	0
$817$	250.050	0.0107077
$818$	−9936.97	−0.424741
$819$	0	0
$820$	1681.55	0.0716127
$821$	2149.26	0.0913638	0.0456819	−	0.998956i	$-0.485454\pi$
$821$	2149.26	0.0913638	0.0456819	+	0.998956i	$0.485454\pi$
$822$	0	0
$823$	−44421.2	−1.88144	−0.940721	−	0.339181i	$-0.889850\pi$
$823$	−44421.2	−1.88144	−0.940721	+	0.339181i	$0.889850\pi$
$824$	3857.36	0.163079
$825$	0	0
$826$	−6573.89	−0.276919
$827$	−40483.6	−1.70224	−0.851121	−	0.524970i	$-0.824077\pi$
$827$	−40483.6	−1.70224	−0.851121	+	0.524970i	$0.824077\pi$
$828$	0	0
$829$	21766.1	0.911903	0.455952	−	0.890005i	$-0.349299\pi$
$829$	21766.1	0.911903	0.455952	+	0.890005i	$0.349299\pi$
$830$	6074.98	0.254055
$831$	0	0
$832$	−5726.87	−0.238634
$833$	30273.6	1.25921
$834$	0	0
$835$	−10738.9	−0.445070
$836$	0	0
$837$	0	0
$838$	−23526.6	−0.969825
$839$	−25521.6	−1.05018	−0.525092	−	0.851046i	$-0.675969\pi$
$839$	−25521.6	−1.05018	−0.525092	+	0.851046i	$0.675969\pi$
$840$	0	0
$841$	300.790	0.0123330
$842$	8738.43	0.357656
$843$	0	0
$844$	12917.9	0.526839
$845$	−26589.9	−1.08251
$846$	0	0
$847$	0	0
$848$	8037.72	0.325491
$849$	0	0
$850$	−21470.0	−0.866372
$851$	−15711.8	−0.632897
$852$	0	0
$853$	−20892.4	−0.838618	−0.419309	−	0.907844i	$-0.637728\pi$
$853$	−20892.4	−0.838618	−0.419309	+	0.907844i	$0.637728\pi$
$854$	1731.51	0.0693805
$855$	0	0
$856$	3597.87	0.143659
$857$	4786.08	0.190769	0.0953847	−	0.995440i	$-0.469592\pi$
$857$	4786.08	0.190769	0.0953847	+	0.995440i	$0.469592\pi$
$858$	0	0
$859$	−13013.8	−0.516911	−0.258455	−	0.966023i	$-0.583214\pi$
$859$	−13013.8	−0.516911	−0.258455	+	0.966023i	$0.583214\pi$
$860$	217.115	0.00860878
$861$	0	0
$862$	−6369.42	−0.251674
$863$	−49593.1	−1.95616	−0.978082	−	0.208220i	$-0.933233\pi$
$863$	−49593.1	−1.95616	−0.978082	+	0.208220i	$0.933233\pi$
$864$	0	0
$865$	13555.4	0.532829
$866$	11077.2	0.434664
$867$	0	0
$868$	6365.90	0.248932
$869$	0	0
$870$	0	0
$871$	78294.5	3.04582
$872$	9718.86	0.377433
$873$	0	0
$874$	−3790.86	−0.146714
$875$	−7379.36	−0.285106
$876$	0	0
$877$	−24170.2	−0.930638	−0.465319	−	0.885143i	$-0.654060\pi$
$877$	−24170.2	−0.930638	−0.465319	+	0.885143i	$0.654060\pi$
$878$	−16900.6	−0.649622
$879$	0	0
$880$	0	0
$881$	−31630.8	−1.20961	−0.604806	−	0.796373i	$-0.706749\pi$
$881$	−31630.8	−1.20961	−0.604806	+	0.796373i	$0.706749\pi$
$882$	0	0
$883$	22389.6	0.853305	0.426653	−	0.904416i	$-0.359693\pi$
$883$	22389.6	0.853305	0.426653	+	0.904416i	$0.359693\pi$
$884$	36926.2	1.40494
$885$	0	0
$886$	−3172.29	−0.120288
$887$	−16044.8	−0.607362	−0.303681	−	0.952774i	$-0.598216\pi$
$887$	−16044.8	−0.607362	−0.303681	+	0.952774i	$0.598216\pi$
$888$	0	0
$889$	17260.6	0.651185
$890$	11791.9	0.444120
$891$	0	0
$892$	−13148.0	−0.493530
$893$	−6207.85	−0.232629
$894$	0	0
$895$	6830.54	0.255106
$896$	901.061	0.0335963
$897$	0	0
$898$	−28514.0	−1.05961
$899$	35523.4	1.31788
$900$	0	0
$901$	−51826.3	−1.91630
$902$	0	0
$903$	0	0
$904$	−12518.6	−0.460579
$905$	7074.61	0.259854
$906$	0	0
$907$	−50950.8	−1.86526	−0.932632	−	0.360829i	$-0.882494\pi$
$907$	−50950.8	−1.86526	−0.932632	+	0.360829i	$0.882494\pi$
$908$	25162.3	0.919649
$909$	0	0
$910$	5765.61	0.210031
$911$	−7826.10	−0.284621	−0.142311	−	0.989822i	$-0.545453\pi$
$911$	−7826.10	−0.284621	−0.142311	+	0.989822i	$0.545453\pi$
$912$	0	0
$913$	0	0
$914$	−1757.49	−0.0636024
$915$	0	0
$916$	−4265.73	−0.153869
$917$	17322.0	0.623798
$918$	0	0
$919$	11790.8	0.423222	0.211611	−	0.977354i	$-0.432129\pi$
$919$	11790.8	0.423222	0.211611	+	0.977354i	$0.432129\pi$
$920$	−3291.55	−0.117955
$921$	0	0
$922$	−9872.73	−0.352648
$923$	9955.69	0.355033
$924$	0	0
$925$	−18185.1	−0.646404
$926$	−33911.2	−1.20344
$927$	0	0
$928$	5028.16	0.177863
$929$	28489.6	1.00615	0.503076	−	0.864242i	$-0.332202\pi$
$929$	28489.6	1.00615	0.503076	+	0.864242i	$0.332202\pi$
$930$	0	0
$931$	6186.70	0.217788
$932$	14758.1	0.518689
$933$	0	0
$934$	−19274.1	−0.675232
$935$	0	0
$936$	0	0
$937$	19992.7	0.697047	0.348523	−	0.937300i	$-0.386683\pi$
$937$	19992.7	0.697047	0.348523	+	0.937300i	$0.386683\pi$
$938$	−12318.8	−0.428809
$939$	0	0
$940$	−5390.17	−0.187030
$941$	13579.4	0.470432	0.235216	−	0.971943i	$-0.424420\pi$
$941$	13579.4	0.470432	0.235216	+	0.971943i	$0.424420\pi$
$942$	0	0
$943$	8258.34	0.285184
$944$	−7470.82	−0.257579
$945$	0	0
$946$	0	0
$947$	−54932.1	−1.88496	−0.942478	−	0.334269i	$-0.891511\pi$
$947$	−54932.1	−1.88496	−0.942478	+	0.334269i	$0.891511\pi$
$948$	0	0
$949$	72821.1	2.49091
$950$	−4387.61	−0.149845
$951$	0	0
$952$	−5809.94	−0.197795
$953$	−34099.1	−1.15905	−0.579527	−	0.814953i	$-0.696763\pi$
$953$	−34099.1	−1.15905	−0.579527	+	0.814953i	$0.696763\pi$
$954$	0	0
$955$	5314.47	0.180076
$956$	−28524.5	−0.965009
$957$	0	0
$958$	−25042.5	−0.844559
$959$	−4402.86	−0.148254
$960$	0	0
$961$	21319.7	0.715642
$962$	31276.5	1.04823
$963$	0	0
$964$	−12842.4	−0.429073
$965$	3282.39	0.109496
$966$	0	0
$967$	33962.8	1.12944	0.564721	−	0.825282i	$-0.308984\pi$
$967$	33962.8	1.12944	0.564721	+	0.825282i	$0.308984\pi$
$968$	0	0
$969$	0	0
$970$	−1873.84	−0.0620261
$971$	−28810.8	−0.952195	−0.476097	−	0.879393i	$-0.657949\pi$
$971$	−28810.8	−0.952195	−0.476097	+	0.879393i	$0.657949\pi$
$972$	0	0
$973$	−8239.13	−0.271464
$974$	−25151.4	−0.827416
$975$	0	0
$976$	1967.75	0.0645350
$977$	−42065.2	−1.37747	−0.688733	−	0.725015i	$-0.741833\pi$
$977$	−42065.2	−1.37747	−0.688733	+	0.725015i	$0.741833\pi$
$978$	0	0
$979$	0	0
$980$	5371.81	0.175098
$981$	0	0
$982$	3942.43	0.128114
$983$	21808.6	0.707615	0.353808	−	0.935318i	$-0.384887\pi$
$983$	21808.6	0.707615	0.353808	+	0.935318i	$0.384887\pi$
$984$	0	0
$985$	−16605.6	−0.537157
$986$	−32421.0	−1.04715
$987$	0	0
$988$	7546.22	0.242993
$989$	1066.28	0.0342828
$990$	0	0
$991$	11621.1	0.372508	0.186254	−	0.982502i	$-0.440365\pi$
$991$	11621.1	0.372508	0.186254	+	0.982502i	$0.440365\pi$
$992$	7234.46	0.231547
$993$	0	0
$994$	−1566.42	−0.0499837
$995$	5842.75	0.186158
$996$	0	0
$997$	−54897.6	−1.74385	−0.871927	−	0.489635i	$-0.837130\pi$
$997$	−54897.6	−1.74385	−0.871927	+	0.489635i	$0.837130\pi$
$998$	14749.6	0.467826
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.ce.1.2		6
3.2	odd	2		2178.4.a.cf.1.5		6
11.5	even	5		198.4.f.h.91.3	yes	12
11.9	even	5		198.4.f.h.37.3	yes	12
11.10	odd	2		2178.4.a.cg.1.2		6
33.5	odd	10		198.4.f.g.91.1	yes	12
33.20	odd	10		198.4.f.g.37.1	✓	12
33.32	even	2		2178.4.a.cd.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.4.f.g.37.1	✓	12	33.20	odd	10
198.4.f.g.91.1	yes	12	33.5	odd	10
198.4.f.h.37.3	yes	12	11.9	even	5
198.4.f.h.91.3	yes	12	11.5	even	5
2178.4.a.cd.1.5		6	33.32	even	2
2178.4.a.ce.1.2		6	1.1	even	1	trivial
2178.4.a.cf.1.5		6	3.2	odd	2
2178.4.a.cg.1.2		6	11.10	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -4.57651 q^{5} -7.03954 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -4.57651 q^{5} -7.03954 q^{7} -8.00000 q^{8} +9.15301 q^{10} -89.4824 q^{13} +14.0791 q^{14} +16.0000 q^{16} -103.166 q^{17} -21.0830 q^{19} -18.3060 q^{20} -89.9033 q^{23} -104.056 q^{25} +178.965 q^{26} -28.1581 q^{28} -157.130 q^{29} -226.077 q^{31} -32.0000 q^{32} +206.332 q^{34} +32.2165 q^{35} +174.764 q^{37} +42.1660 q^{38} +36.6120 q^{40} -91.8579 q^{41} -11.8603 q^{43} +179.807 q^{46} +294.448 q^{47} -293.445 q^{49} +208.111 q^{50} -357.929 q^{52} +502.358 q^{53} +56.3163 q^{56} +314.260 q^{58} -466.926 q^{59} +122.984 q^{61} +452.153 q^{62} +64.0000 q^{64} +409.517 q^{65} -874.971 q^{67} -412.665 q^{68} -64.4330 q^{70} -111.259 q^{71} -813.804 q^{73} -349.527 q^{74} -84.3320 q^{76} +435.924 q^{79} -73.2241 q^{80} +183.716 q^{82} +663.714 q^{83} +472.141 q^{85} +23.7206 q^{86} +1288.31 q^{89} +629.914 q^{91} -359.613 q^{92} -588.896 q^{94} +96.4864 q^{95} -204.724 q^{97} +586.890 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 24 q^{4} + 17 q^{5} - 7 q^{7} - 48 q^{8}+O(q^{10}) \) 6 * q - 12 * q^2 + 24 * q^4 + 17 * q^5 - 7 * q^7 - 48 * q^8	\( 6 q - 12 q^{2} + 24 q^{4} + 17 q^{5} - 7 q^{7} - 48 q^{8} - 34 q^{10} - 68 q^{13} + 14 q^{14} + 96 q^{16} - 42 q^{17} - 98 q^{19} + 68 q^{20} + 210 q^{23} + 47 q^{25} + 136 q^{26} - 28 q^{28} - 13 q^{29} - 125 q^{31} - 192 q^{32} + 84 q^{34} - 534 q^{35} + 282 q^{37} + 196 q^{38} - 136 q^{40} + 170 q^{41} - 868 q^{43} - 420 q^{46} + 782 q^{47} - 439 q^{49} - 94 q^{50} - 272 q^{52} + 645 q^{53} + 56 q^{56} + 26 q^{58} + 507 q^{59} - 1772 q^{61} + 250 q^{62} + 384 q^{64} + 1856 q^{65} + 686 q^{67} - 168 q^{68} + 1068 q^{70} + 2782 q^{71} - 335 q^{73} - 564 q^{74} - 392 q^{76} - 127 q^{79} + 272 q^{80} - 340 q^{82} + 9 q^{83} - 370 q^{85} + 1736 q^{86} + 2526 q^{89} + 296 q^{91} + 840 q^{92} - 1564 q^{94} + 1194 q^{95} + 89 q^{97} + 878 q^{98}+O(q^{100}) \) 6 * q - 12 * q^2 + 24 * q^4 + 17 * q^5 - 7 * q^7 - 48 * q^8 - 34 * q^10 - 68 * q^13 + 14 * q^14 + 96 * q^16 - 42 * q^17 - 98 * q^19 + 68 * q^20 + 210 * q^23 + 47 * q^25 + 136 * q^26 - 28 * q^28 - 13 * q^29 - 125 * q^31 - 192 * q^32 + 84 * q^34 - 534 * q^35 + 282 * q^37 + 196 * q^38 - 136 * q^40 + 170 * q^41 - 868 * q^43 - 420 * q^46 + 782 * q^47 - 439 * q^49 - 94 * q^50 - 272 * q^52 + 645 * q^53 + 56 * q^56 + 26 * q^58 + 507 * q^59 - 1772 * q^61 + 250 * q^62 + 384 * q^64 + 1856 * q^65 + 686 * q^67 - 168 * q^68 + 1068 * q^70 + 2782 * q^71 - 335 * q^73 - 564 * q^74 - 392 * q^76 - 127 * q^79 + 272 * q^80 - 340 * q^82 + 9 * q^83 - 370 * q^85 + 1736 * q^86 + 2526 * q^89 + 296 * q^91 + 840 * q^92 - 1564 * q^94 + 1194 * q^95 + 89 * q^97 + 878 * q^98

Introduction

L-functions

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