LMFDB - Embedded newform 2178.4.a.x.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{273}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 68 $ x^2 - x - 68
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 726)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.76136$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +13.5227 q^{5} +13.5227 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} +13.5227 q^{5} +13.5227 q^{7} -8.00000 q^{8} -27.0454 q^{10} -40.4773 q^{13} -27.0454 q^{14} +16.0000 q^{16} -111.136 q^{17} +36.0908 q^{19} +54.0908 q^{20} +193.614 q^{23} +57.8637 q^{25} +80.9546 q^{26} +54.0908 q^{28} -204.273 q^{29} -58.8637 q^{31} -32.0000 q^{32} +222.273 q^{34} +182.864 q^{35} -317.409 q^{37} -72.1817 q^{38} -108.182 q^{40} +393.409 q^{41} +44.8637 q^{43} -387.227 q^{46} +22.5681 q^{47} -160.136 q^{49} -115.727 q^{50} -161.909 q^{52} -94.8407 q^{53} -108.182 q^{56} +408.545 q^{58} -9.31796 q^{59} -104.068 q^{61} +117.727 q^{62} +64.0000 q^{64} -547.363 q^{65} -524.000 q^{67} -444.545 q^{68} -365.727 q^{70} +229.614 q^{71} -1152.91 q^{73} +634.818 q^{74} +144.363 q^{76} -1075.80 q^{79} +216.363 q^{80} -786.818 q^{82} -1401.14 q^{83} -1502.86 q^{85} -89.7275 q^{86} +1116.09 q^{89} -547.363 q^{91} +774.454 q^{92} -45.1363 q^{94} +488.046 q^{95} -1395.04 q^{97} +320.273 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 12 q^{10} - 114 q^{13} + 12 q^{14} + 32 q^{16} - 24 q^{17} - 60 q^{19} - 24 q^{20} + 222 q^{23} + 314 q^{25} + 228 q^{26} - 24 q^{28} - 12 q^{29} - 316 q^{31} - 64 q^{32} + 48 q^{34} + 564 q^{35} - 40 q^{37} + 120 q^{38} + 48 q^{40} + 192 q^{41} + 288 q^{43} - 444 q^{46} - 54 q^{47} - 122 q^{49} - 628 q^{50} - 456 q^{52} + 306 q^{53} + 48 q^{56} + 24 q^{58} + 444 q^{59} + 618 q^{61} + 632 q^{62} + 128 q^{64} + 888 q^{65} - 1048 q^{67} - 96 q^{68} - 1128 q^{70} + 294 q^{71} - 984 q^{73} + 80 q^{74} - 240 q^{76} - 1722 q^{79} - 96 q^{80} - 384 q^{82} - 2604 q^{83} - 3204 q^{85} - 576 q^{86} + 2100 q^{89} + 888 q^{91} + 888 q^{92} + 108 q^{94} + 2364 q^{95} + 184 q^{97} + 244 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 - 16 * q^8 + 12 * q^10 - 114 * q^13 + 12 * q^14 + 32 * q^16 - 24 * q^17 - 60 * q^19 - 24 * q^20 + 222 * q^23 + 314 * q^25 + 228 * q^26 - 24 * q^28 - 12 * q^29 - 316 * q^31 - 64 * q^32 + 48 * q^34 + 564 * q^35 - 40 * q^37 + 120 * q^38 + 48 * q^40 + 192 * q^41 + 288 * q^43 - 444 * q^46 - 54 * q^47 - 122 * q^49 - 628 * q^50 - 456 * q^52 + 306 * q^53 + 48 * q^56 + 24 * q^58 + 444 * q^59 + 618 * q^61 + 632 * q^62 + 128 * q^64 + 888 * q^65 - 1048 * q^67 - 96 * q^68 - 1128 * q^70 + 294 * q^71 - 984 * q^73 + 80 * q^74 - 240 * q^76 - 1722 * q^79 - 96 * q^80 - 384 * q^82 - 2604 * q^83 - 3204 * q^85 - 576 * q^86 + 2100 * q^89 + 888 * q^91 + 888 * q^92 + 108 * q^94 + 2364 * q^95 + 184 * q^97 + 244 * 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
$5$	13.5227	1.20951	0.604754	−	0.796412i	$-0.293271\pi$
$5$	13.5227	1.20951	0.604754	+	0.796412i	$0.293271\pi$
$6$	0	0
$7$	13.5227	0.730158	0.365079	−	0.930977i	$-0.381042\pi$
$7$	13.5227	0.730158	0.365079	+	0.930977i	$0.381042\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−27.0454	−0.855251
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−40.4773	−0.863568	−0.431784	−	0.901977i	$-0.642116\pi$
$13$	−40.4773	−0.863568	−0.431784	+	0.901977i	$0.642116\pi$
$14$	−27.0454	−0.516300
$15$	0	0
$16$	16.0000	0.250000
$17$	−111.136	−1.58556	−0.792780	−	0.609508i	$-0.791367\pi$
$17$	−111.136	−1.58556	−0.792780	+	0.609508i	$0.791367\pi$
$18$	0	0
$19$	36.0908	0.435779	0.217890	−	0.975973i	$-0.430083\pi$
$19$	36.0908	0.435779	0.217890	+	0.975973i	$0.430083\pi$
$20$	54.0908	0.604754
$21$	0	0
$22$	0	0
$23$	193.614	1.75527	0.877635	−	0.479329i	$-0.159120\pi$
$23$	193.614	1.75527	0.877635	+	0.479329i	$0.159120\pi$
$24$	0	0
$25$	57.8637	0.462910
$26$	80.9546	0.610635
$27$	0	0
$28$	54.0908	0.365079
$29$	−204.273	−1.30802	−0.654008	−	0.756488i	$-0.726914\pi$
$29$	−204.273	−1.30802	−0.654008	+	0.756488i	$0.726914\pi$
$30$	0	0
$31$	−58.8637	−0.341040	−0.170520	−	0.985354i	$-0.554545\pi$
$31$	−58.8637	−0.341040	−0.170520	+	0.985354i	$0.554545\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	222.273	1.12116
$35$	182.864	0.883132
$36$	0	0
$37$	−317.409	−1.41032	−0.705158	−	0.709050i	$-0.749124\pi$
$37$	−317.409	−1.41032	−0.705158	+	0.709050i	$0.749124\pi$
$38$	−72.1817	−0.308143
$39$	0	0
$40$	−108.182	−0.427626
$41$	393.409	1.49854	0.749270	−	0.662265i	$-0.230405\pi$
$41$	393.409	1.49854	0.749270	+	0.662265i	$0.230405\pi$
$42$	0	0
$43$	44.8637	0.159108	0.0795541	−	0.996831i	$-0.474650\pi$
$43$	44.8637	0.159108	0.0795541	+	0.996831i	$0.474650\pi$
$44$	0	0
$45$	0	0
$46$	−387.227	−1.24116
$47$	22.5681	0.0700404	0.0350202	−	0.999387i	$-0.488850\pi$
$47$	22.5681	0.0700404	0.0350202	+	0.999387i	$0.488850\pi$
$48$	0	0
$49$	−160.136	−0.466870
$50$	−115.727	−0.327327
$51$	0	0
$52$	−161.909	−0.431784
$53$	−94.8407	−0.245799	−0.122900	−	0.992419i	$-0.539219\pi$
$53$	−94.8407	−0.245799	−0.122900	+	0.992419i	$0.539219\pi$
$54$	0	0
$55$	0	0
$56$	−108.182	−0.258150
$57$	0	0
$58$	408.545	0.924907
$59$	−9.31796	−0.0205609	−0.0102805	−	0.999947i	$-0.503272\pi$
$59$	−9.31796	−0.0205609	−0.0102805	+	0.999947i	$0.503272\pi$
$60$	0	0
$61$	−104.068	−0.218435	−0.109217	−	0.994018i	$-0.534834\pi$
$61$	−104.068	−0.218435	−0.109217	+	0.994018i	$0.534834\pi$
$62$	117.727	0.241152
$63$	0	0
$64$	64.0000	0.125000
$65$	−547.363	−1.04449
$66$	0	0
$67$	−524.000	−0.955474	−0.477737	−	0.878503i	$-0.658543\pi$
$67$	−524.000	−0.955474	−0.477737	+	0.878503i	$0.658543\pi$
$68$	−444.545	−0.792780
$69$	0	0
$70$	−365.727	−0.624468
$71$	229.614	0.383804	0.191902	−	0.981414i	$-0.438534\pi$
$71$	229.614	0.383804	0.191902	+	0.981414i	$0.438534\pi$
$72$	0	0
$73$	−1152.91	−1.84846	−0.924231	−	0.381834i	$-0.875293\pi$
$73$	−1152.91	−1.84846	−0.924231	+	0.381834i	$0.875293\pi$
$74$	634.818	0.997244
$75$	0	0
$76$	144.363	0.217890
$77$	0	0
$78$	0	0
$79$	−1075.80	−1.53211	−0.766053	−	0.642777i	$-0.777782\pi$
$79$	−1075.80	−1.53211	−0.766053	+	0.642777i	$0.777782\pi$
$80$	216.363	0.302377
$81$	0	0
$82$	−786.818	−1.05963
$83$	−1401.14	−1.85295	−0.926474	−	0.376359i	$-0.877176\pi$
$83$	−1401.14	−1.85295	−0.926474	+	0.376359i	$0.877176\pi$
$84$	0	0
$85$	−1502.86	−1.91775
$86$	−89.7275	−0.112507
$87$	0	0
$88$	0	0
$89$	1116.09	1.32927	0.664637	−	0.747166i	$-0.268586\pi$
$89$	1116.09	1.32927	0.664637	+	0.747166i	$0.268586\pi$
$90$	0	0
$91$	−547.363	−0.630541
$92$	774.454	0.877635
$93$	0	0
$94$	−45.1363	−0.0495261
$95$	488.046	0.527079
$96$	0	0
$97$	−1395.04	−1.46026	−0.730130	−	0.683308i	$-0.760541\pi$
$97$	−1395.04	−1.46026	−0.730130	+	0.683308i	$0.760541\pi$
$98$	320.273	0.330127
$99$	0	0
$100$	231.455	0.231455
$101$	27.1363	0.0267343	0.0133671	−	0.999911i	$-0.495745\pi$
$101$	27.1363	0.0267343	0.0133671	+	0.999911i	$0.495745\pi$
$102$	0	0
$103$	1475.95	1.41194	0.705971	−	0.708241i	$-0.250511\pi$
$103$	1475.95	1.41194	0.705971	+	0.708241i	$0.250511\pi$
$104$	323.818	0.305317
$105$	0	0
$106$	189.681	0.173806
$107$	1223.45	1.10538	0.552691	−	0.833386i	$-0.313601\pi$
$107$	1223.45	1.10538	0.552691	+	0.833386i	$0.313601\pi$
$108$	0	0
$109$	−1913.93	−1.68185	−0.840924	−	0.541154i	$-0.817988\pi$
$109$	−1913.93	−1.68185	−0.840924	+	0.541154i	$0.817988\pi$
$110$	0	0
$111$	0	0
$112$	216.363	0.182539
$113$	−703.817	−0.585925	−0.292963	−	0.956124i	$-0.594641\pi$
$113$	−703.817	−0.585925	−0.292963	+	0.956124i	$0.594641\pi$
$114$	0	0
$115$	2618.18	2.12301
$116$	−817.090	−0.654008
$117$	0	0
$118$	18.6359	0.0145388
$119$	−1502.86	−1.15771
$120$	0	0
$121$	0	0
$122$	208.136	0.154457
$123$	0	0
$124$	−235.455	−0.170520
$125$	−907.864	−0.649615
$126$	0	0
$127$	1014.75	0.709011	0.354505	−	0.935054i	$-0.384649\pi$
$127$	1014.75	0.709011	0.354505	+	0.935054i	$0.384649\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	1094.73	0.738567
$131$	1570.77	1.04763	0.523813	−	0.851833i	$-0.324509\pi$
$131$	1570.77	1.04763	0.523813	+	0.851833i	$0.324509\pi$
$132$	0	0
$133$	488.046	0.318188
$134$	1048.00	0.675622
$135$	0	0
$136$	889.090	0.560580
$137$	279.863	0.174528	0.0872639	−	0.996185i	$-0.472188\pi$
$137$	279.863	0.174528	0.0872639	+	0.996185i	$0.472188\pi$
$138$	0	0
$139$	1981.54	1.20915	0.604577	−	0.796547i	$-0.293342\pi$
$139$	1981.54	1.20915	0.604577	+	0.796547i	$0.293342\pi$
$140$	731.455	0.441566
$141$	0	0
$142$	−459.227	−0.271391
$143$	0	0
$144$	0	0
$145$	−2762.32	−1.58206
$146$	2305.82	1.30706
$147$	0	0
$148$	−1269.64	−0.705158
$149$	−1533.41	−0.843099	−0.421549	−	0.906805i	$-0.638514\pi$
$149$	−1533.41	−0.843099	−0.421549	+	0.906805i	$0.638514\pi$
$150$	0	0
$151$	−1886.07	−1.01646	−0.508232	−	0.861220i	$-0.669701\pi$
$151$	−1886.07	−1.01646	−0.508232	+	0.861220i	$0.669701\pi$
$152$	−288.727	−0.154071
$153$	0	0
$154$	0	0
$155$	−795.997	−0.412491
$156$	0	0
$157$	2796.18	1.42140	0.710699	−	0.703496i	$-0.248379\pi$
$157$	2796.18	1.42140	0.710699	+	0.703496i	$0.248379\pi$
$158$	2151.59	1.08336
$159$	0	0
$160$	−432.727	−0.213813
$161$	2618.18	1.28162
$162$	0	0
$163$	−152.637	−0.0733465	−0.0366732	−	0.999327i	$-0.511676\pi$
$163$	−152.637	−0.0733465	−0.0366732	+	0.999327i	$0.511676\pi$
$164$	1573.64	0.749270
$165$	0	0
$166$	2802.27	1.31023
$167$	838.772	0.388659	0.194330	−	0.980936i	$-0.437747\pi$
$167$	838.772	0.388659	0.194330	+	0.980936i	$0.437747\pi$
$168$	0	0
$169$	−558.589	−0.254251
$170$	3005.73	1.35605
$171$	0	0
$172$	179.455	0.0795541
$173$	−1789.36	−0.786374	−0.393187	−	0.919458i	$-0.628628\pi$
$173$	−1789.36	−0.786374	−0.393187	+	0.919458i	$0.628628\pi$
$174$	0	0
$175$	782.475	0.337997
$176$	0	0
$177$	0	0
$178$	−2232.18	−0.939939
$179$	4330.95	1.80844	0.904220	−	0.427068i	$-0.140453\pi$
$179$	4330.95	1.80844	0.904220	+	0.427068i	$0.140453\pi$
$180$	0	0
$181$	3649.86	1.49885	0.749426	−	0.662089i	$-0.230330\pi$
$181$	3649.86	1.49885	0.749426	+	0.662089i	$0.230330\pi$
$182$	1094.73	0.445860
$183$	0	0
$184$	−1548.91	−0.620582
$185$	−4292.23	−1.70579
$186$	0	0
$187$	0	0
$188$	90.2725	0.0350202
$189$	0	0
$190$	−976.092	−0.372701
$191$	−4878.16	−1.84802	−0.924008	−	0.382374i	$-0.875107\pi$
$191$	−4878.16	−1.84802	−0.924008	+	0.382374i	$0.875107\pi$
$192$	0	0
$193$	−3414.86	−1.27361	−0.636806	−	0.771024i	$-0.719745\pi$
$193$	−3414.86	−1.27361	−0.636806	+	0.771024i	$0.719745\pi$
$194$	2790.09	1.03256
$195$	0	0
$196$	−640.545	−0.233435
$197$	−2930.59	−1.05988	−0.529939	−	0.848036i	$-0.677785\pi$
$197$	−2930.59	−1.05988	−0.529939	+	0.848036i	$0.677785\pi$
$198$	0	0
$199$	−1628.55	−0.580123	−0.290062	−	0.957008i	$-0.593676\pi$
$199$	−1628.55	−0.580123	−0.290062	+	0.957008i	$0.593676\pi$
$200$	−462.910	−0.163663
$201$	0	0
$202$	−54.2725	−0.0189040
$203$	−2762.32	−0.955058
$204$	0	0
$205$	5319.95	1.81250
$206$	−2951.91	−0.998394
$207$	0	0
$208$	−647.637	−0.215892
$209$	0	0
$210$	0	0
$211$	−1879.59	−0.613253	−0.306627	−	0.951830i	$-0.599200\pi$
$211$	−1879.59	−0.613253	−0.306627	+	0.951830i	$0.599200\pi$
$212$	−379.363	−0.122900
$213$	0	0
$214$	−2446.91	−0.781623
$215$	606.679	0.192443
$216$	0	0
$217$	−795.997	−0.249013
$218$	3827.86	1.18925
$219$	0	0
$220$	0	0
$221$	4498.49	1.36924
$222$	0	0
$223$	−3268.27	−0.981432	−0.490716	−	0.871319i	$-0.663265\pi$
$223$	−3268.27	−0.981432	−0.490716	+	0.871319i	$0.663265\pi$
$224$	−432.727	−0.129075
$225$	0	0
$226$	1407.63	0.414312
$227$	5375.18	1.57164	0.785822	−	0.618453i	$-0.212240\pi$
$227$	5375.18	1.57164	0.785822	+	0.618453i	$0.212240\pi$
$228$	0	0
$229$	2522.09	0.727793	0.363896	−	0.931439i	$-0.381446\pi$
$229$	2522.09	0.727793	0.363896	+	0.931439i	$0.381446\pi$
$230$	−5236.36	−1.50120
$231$	0	0
$232$	1634.18	0.462453
$233$	−2342.59	−0.658662	−0.329331	−	0.944215i	$-0.606823\pi$
$233$	−2342.59	−0.658662	−0.329331	+	0.944215i	$0.606823\pi$
$234$	0	0
$235$	305.182	0.0847145
$236$	−37.2719	−0.0102805
$237$	0	0
$238$	3005.73	0.818623
$239$	−3174.68	−0.859217	−0.429609	−	0.903015i	$-0.641348\pi$
$239$	−3174.68	−0.859217	−0.429609	+	0.903015i	$0.641348\pi$
$240$	0	0
$241$	−1070.14	−0.286031	−0.143016	−	0.989720i	$-0.545680\pi$
$241$	−1070.14	−0.286031	−0.143016	+	0.989720i	$0.545680\pi$
$242$	0	0
$243$	0	0
$244$	−416.271	−0.109217
$245$	−2165.48	−0.564683
$246$	0	0
$247$	−1460.86	−0.376325
$248$	470.910	0.120576
$249$	0	0
$250$	1815.73	0.459347
$251$	544.957	0.137041	0.0685207	−	0.997650i	$-0.478172\pi$
$251$	544.957	0.137041	0.0685207	+	0.997650i	$0.478172\pi$
$252$	0	0
$253$	0	0
$254$	−2029.50	−0.501346
$255$	0	0
$256$	256.000	0.0625000
$257$	5117.72	1.24216	0.621079	−	0.783748i	$-0.286694\pi$
$257$	5117.72	1.24216	0.621079	+	0.783748i	$0.286694\pi$
$258$	0	0
$259$	−4292.23	−1.02975
$260$	−2189.45	−0.522246
$261$	0	0
$262$	−3141.54	−0.740783
$263$	1238.59	0.290398	0.145199	−	0.989403i	$-0.453618\pi$
$263$	1238.59	0.290398	0.145199	+	0.989403i	$0.453618\pi$
$264$	0	0
$265$	−1282.50	−0.297296
$266$	−976.092	−0.224993
$267$	0	0
$268$	−2096.00	−0.477737
$269$	−507.478	−0.115024	−0.0575120	−	0.998345i	$-0.518317\pi$
$269$	−507.478	−0.115024	−0.0575120	+	0.998345i	$0.518317\pi$
$270$	0	0
$271$	6169.52	1.38292	0.691461	−	0.722414i	$-0.256967\pi$
$271$	6169.52	1.38292	0.691461	+	0.722414i	$0.256967\pi$
$272$	−1778.18	−0.396390
$273$	0	0
$274$	−559.726	−0.123410
$275$	0	0
$276$	0	0
$277$	−67.7044	−0.0146858	−0.00734289	−	0.999973i	$-0.502337\pi$
$277$	−67.7044	−0.0146858	−0.00734289	+	0.999973i	$0.502337\pi$
$278$	−3963.09	−0.855001
$279$	0	0
$280$	−1462.91	−0.312234
$281$	2915.87	0.619025	0.309512	−	0.950895i	$-0.399834\pi$
$281$	2915.87	0.619025	0.309512	+	0.950895i	$0.399834\pi$
$282$	0	0
$283$	9034.22	1.89763	0.948814	−	0.315835i	$-0.102285\pi$
$283$	9034.22	1.89763	0.948814	+	0.315835i	$0.102285\pi$
$284$	918.454	0.191902
$285$	0	0
$286$	0	0
$287$	5319.95	1.09417
$288$	0	0
$289$	7438.27	1.51400
$290$	5524.64	1.11868
$291$	0	0
$292$	−4611.63	−0.924231
$293$	−5309.72	−1.05869	−0.529347	−	0.848405i	$-0.677563\pi$
$293$	−5309.72	−1.05869	−0.529347	+	0.848405i	$0.677563\pi$
$294$	0	0
$295$	−126.004	−0.0248686
$296$	2539.27	0.498622
$297$	0	0
$298$	3066.82	0.596161
$299$	−7836.95	−1.51579
$300$	0	0
$301$	606.679	0.116174
$302$	3772.14	0.718748
$303$	0	0
$304$	577.454	0.108945
$305$	−1407.28	−0.264199
$306$	0	0
$307$	−6791.45	−1.26257	−0.631284	−	0.775551i	$-0.717472\pi$
$307$	−6791.45	−1.26257	−0.631284	+	0.775551i	$0.717472\pi$
$308$	0	0
$309$	0	0
$310$	1591.99	0.291675
$311$	−1519.02	−0.276964	−0.138482	−	0.990365i	$-0.544222\pi$
$311$	−1519.02	−0.276964	−0.138482	+	0.990365i	$0.544222\pi$
$312$	0	0
$313$	−2041.72	−0.368706	−0.184353	−	0.982860i	$-0.559019\pi$
$313$	−2041.72	−0.368706	−0.184353	+	0.982860i	$0.559019\pi$
$314$	−5592.36	−1.00508
$315$	0	0
$316$	−4303.18	−0.766053
$317$	−4982.07	−0.882715	−0.441358	−	0.897331i	$-0.645503\pi$
$317$	−4982.07	−0.882715	−0.441358	+	0.897331i	$0.645503\pi$
$318$	0	0
$319$	0	0
$320$	865.454	0.151189
$321$	0	0
$322$	−5236.36	−0.906245
$323$	−4011.00	−0.690954
$324$	0	0
$325$	−2342.17	−0.399754
$326$	305.275	0.0518638
$327$	0	0
$328$	−3147.27	−0.529814
$329$	305.182	0.0511406
$330$	0	0
$331$	−5242.54	−0.870562	−0.435281	−	0.900295i	$-0.643351\pi$
$331$	−5242.54	−0.870562	−0.435281	+	0.900295i	$0.643351\pi$
$332$	−5604.55	−0.926474
$333$	0	0
$334$	−1677.54	−0.274824
$335$	−7085.90	−1.15565
$336$	0	0
$337$	7291.18	1.17856	0.589282	−	0.807928i	$-0.299411\pi$
$337$	7291.18	1.17856	0.589282	+	0.807928i	$0.299411\pi$
$338$	1117.18	0.179783
$339$	0	0
$340$	−6011.45	−0.958873
$341$	0	0
$342$	0	0
$343$	−6803.77	−1.07105
$344$	−358.910	−0.0562533
$345$	0	0
$346$	3578.73	0.556050
$347$	2785.23	0.430890	0.215445	−	0.976516i	$-0.430880\pi$
$347$	2785.23	0.430890	0.215445	+	0.976516i	$0.430880\pi$
$348$	0	0
$349$	3821.33	0.586107	0.293053	−	0.956096i	$-0.405329\pi$
$349$	3821.33	0.586107	0.293053	+	0.956096i	$0.405329\pi$
$350$	−1564.95	−0.239000
$351$	0	0
$352$	0	0
$353$	−10538.0	−1.58890	−0.794449	−	0.607331i	$-0.792240\pi$
$353$	−10538.0	−1.58890	−0.794449	+	0.607331i	$0.792240\pi$
$354$	0	0
$355$	3105.00	0.464215
$356$	4464.36	0.664637
$357$	0	0
$358$	−8661.91	−1.27876
$359$	−731.455	−0.107534	−0.0537670	−	0.998554i	$-0.517123\pi$
$359$	−731.455	−0.107534	−0.0537670	+	0.998554i	$0.517123\pi$
$360$	0	0
$361$	−5556.45	−0.810096
$362$	−7299.72	−1.05985
$363$	0	0
$364$	−2189.45	−0.315270
$365$	−15590.4	−2.23573
$366$	0	0
$367$	−4802.36	−0.683056	−0.341528	−	0.939872i	$-0.610944\pi$
$367$	−4802.36	−0.683056	−0.341528	+	0.939872i	$0.610944\pi$
$368$	3097.82	0.438818
$369$	0	0
$370$	8584.46	1.20617
$371$	−1282.50	−0.179472
$372$	0	0
$373$	−4271.48	−0.592946	−0.296473	−	0.955041i	$-0.595810\pi$
$373$	−4271.48	−0.592946	−0.296473	+	0.955041i	$0.595810\pi$
$374$	0	0
$375$	0	0
$376$	−180.545	−0.0247630
$377$	8268.40	1.12956
$378$	0	0
$379$	−10570.5	−1.43264	−0.716322	−	0.697770i	$-0.754176\pi$
$379$	−10570.5	−1.43264	−0.716322	+	0.697770i	$0.754176\pi$
$380$	1952.18	0.263539
$381$	0	0
$382$	9756.31	1.30674
$383$	12537.0	1.67261	0.836304	−	0.548265i	$-0.184712\pi$
$383$	12537.0	1.67261	0.836304	+	0.548265i	$0.184712\pi$
$384$	0	0
$385$	0	0
$386$	6829.72	0.900580
$387$	0	0
$388$	−5580.18	−0.730130
$389$	−11303.6	−1.47330	−0.736649	−	0.676275i	$-0.763593\pi$
$389$	−11303.6	−1.47330	−0.736649	+	0.676275i	$0.763593\pi$
$390$	0	0
$391$	−21517.5	−2.78308
$392$	1281.09	0.165063
$393$	0	0
$394$	5861.18	0.749447
$395$	−14547.7	−1.85310
$396$	0	0
$397$	1640.40	0.207379	0.103689	−	0.994610i	$-0.466935\pi$
$397$	1640.40	0.207379	0.103689	+	0.994610i	$0.466935\pi$
$398$	3257.09	0.410209
$399$	0	0
$400$	925.820	0.115727
$401$	735.950	0.0916498	0.0458249	−	0.998949i	$-0.485408\pi$
$401$	735.950	0.0916498	0.0458249	+	0.998949i	$0.485408\pi$
$402$	0	0
$403$	2382.64	0.294511
$404$	108.545	0.0133671
$405$	0	0
$406$	5524.64	0.675328
$407$	0	0
$408$	0	0
$409$	−8704.32	−1.05232	−0.526162	−	0.850384i	$-0.676370\pi$
$409$	−8704.32	−1.05232	−0.526162	+	0.850384i	$0.676370\pi$
$410$	−10639.9	−1.28163
$411$	0	0
$412$	5903.82	0.705971
$413$	−126.004	−0.0150127
$414$	0	0
$415$	−18947.2	−2.24116
$416$	1295.27	0.152659
$417$	0	0
$418$	0	0
$419$	−8022.68	−0.935402	−0.467701	−	0.883887i	$-0.654918\pi$
$419$	−8022.68	−0.935402	−0.467701	+	0.883887i	$0.654918\pi$
$420$	0	0
$421$	14871.3	1.72157	0.860787	−	0.508965i	$-0.169972\pi$
$421$	14871.3	1.72157	0.860787	+	0.508965i	$0.169972\pi$
$422$	3759.18	0.433636
$423$	0	0
$424$	758.725	0.0869032
$425$	−6430.76	−0.733971
$426$	0	0
$427$	−1407.28	−0.159492
$428$	4893.82	0.552691
$429$	0	0
$430$	−1213.36	−0.136078
$431$	3787.22	0.423258	0.211629	−	0.977350i	$-0.432123\pi$
$431$	3787.22	0.423258	0.211629	+	0.977350i	$0.432123\pi$
$432$	0	0
$433$	4758.08	0.528081	0.264040	−	0.964512i	$-0.414945\pi$
$433$	4758.08	0.528081	0.264040	+	0.964512i	$0.414945\pi$
$434$	1591.99	0.176079
$435$	0	0
$436$	−7655.72	−0.840924
$437$	6987.68	0.764911
$438$	0	0
$439$	3896.38	0.423609	0.211804	−	0.977312i	$-0.432066\pi$
$439$	3896.38	0.423609	0.211804	+	0.977312i	$0.432066\pi$
$440$	0	0
$441$	0	0
$442$	−8996.99	−0.968197
$443$	−4326.31	−0.463993	−0.231997	−	0.972717i	$-0.574526\pi$
$443$	−4326.31	−0.463993	−0.231997	+	0.972717i	$0.574526\pi$
$444$	0	0
$445$	15092.6	1.60777
$446$	6536.54	0.693978
$447$	0	0
$448$	865.454	0.0912697
$449$	−4423.68	−0.464958	−0.232479	−	0.972601i	$-0.574684\pi$
$449$	−4423.68	−0.464958	−0.232479	+	0.972601i	$0.574684\pi$
$450$	0	0
$451$	0	0
$452$	−2815.27	−0.292963
$453$	0	0
$454$	−10750.4	−1.11132
$455$	−7401.83	−0.762644
$456$	0	0
$457$	−4347.41	−0.444996	−0.222498	−	0.974933i	$-0.571421\pi$
$457$	−4347.41	−0.444996	−0.222498	+	0.974933i	$0.571421\pi$
$458$	−5044.18	−0.514627
$459$	0	0
$460$	10472.7	1.06151
$461$	−3268.63	−0.330229	−0.165114	−	0.986274i	$-0.552799\pi$
$461$	−3268.63	−0.330229	−0.165114	+	0.986274i	$0.552799\pi$
$462$	0	0
$463$	−14091.6	−1.41445	−0.707226	−	0.706987i	$-0.750054\pi$
$463$	−14091.6	−1.41445	−0.707226	+	0.706987i	$0.750054\pi$
$464$	−3268.36	−0.327004
$465$	0	0
$466$	4685.18	0.465744
$467$	3662.81	0.362944	0.181472	−	0.983396i	$-0.441914\pi$
$467$	3662.81	0.362944	0.181472	+	0.983396i	$0.441914\pi$
$468$	0	0
$469$	−7085.90	−0.697647
$470$	−610.365	−0.0599022
$471$	0	0
$472$	74.5437	0.00726939
$473$	0	0
$474$	0	0
$475$	2088.35	0.201727
$476$	−6011.45	−0.578854
$477$	0	0
$478$	6349.36	0.607558
$479$	−10763.3	−1.02670	−0.513349	−	0.858180i	$-0.671595\pi$
$479$	−10763.3	−1.02670	−0.513349	+	0.858180i	$0.671595\pi$
$480$	0	0
$481$	12847.8	1.21790
$482$	2140.27	0.202255
$483$	0	0
$484$	0	0
$485$	−18864.8	−1.76620
$486$	0	0
$487$	−15299.5	−1.42359	−0.711795	−	0.702387i	$-0.752118\pi$
$487$	−15299.5	−1.42359	−0.711795	+	0.702387i	$0.752118\pi$
$488$	832.542	0.0772283
$489$	0	0
$490$	4330.95	0.399291
$491$	5183.18	0.476402	0.238201	−	0.971216i	$-0.423442\pi$
$491$	5183.18	0.476402	0.238201	+	0.971216i	$0.423442\pi$
$492$	0	0
$493$	22702.1	2.07394
$494$	2921.72	0.266102
$495$	0	0
$496$	−941.820	−0.0852600
$497$	3105.00	0.280238
$498$	0	0
$499$	−430.717	−0.0386404	−0.0193202	−	0.999813i	$-0.506150\pi$
$499$	−430.717	−0.0386404	−0.0193202	+	0.999813i	$0.506150\pi$
$500$	−3631.46	−0.324807
$501$	0	0
$502$	−1089.91	−0.0969029
$503$	−16928.6	−1.50061	−0.750306	−	0.661090i	$-0.770094\pi$
$503$	−16928.6	−1.50061	−0.750306	+	0.661090i	$0.770094\pi$
$504$	0	0
$505$	366.956	0.0323353
$506$	0	0
$507$	0	0
$508$	4058.99	0.354505
$509$	13359.2	1.16334	0.581668	−	0.813426i	$-0.302400\pi$
$509$	13359.2	1.16334	0.581668	+	0.813426i	$0.302400\pi$
$510$	0	0
$511$	−15590.4	−1.34967
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−10235.4	−0.878339
$515$	19958.9	1.70776
$516$	0	0
$517$	0	0
$518$	8584.46	0.728145
$519$	0	0
$520$	4378.90	0.369284
$521$	6963.77	0.585582	0.292791	−	0.956176i	$-0.405416\pi$
$521$	6963.77	0.585582	0.292791	+	0.956176i	$0.405416\pi$
$522$	0	0
$523$	15002.5	1.25433	0.627163	−	0.778888i	$-0.284216\pi$
$523$	15002.5	1.25433	0.627163	+	0.778888i	$0.284216\pi$
$524$	6283.09	0.523813
$525$	0	0
$526$	−2477.17	−0.205342
$527$	6541.90	0.540739
$528$	0	0
$529$	25319.2	2.08097
$530$	2565.01	0.210220
$531$	0	0
$532$	1952.18	0.159094
$533$	−15924.1	−1.29409
$534$	0	0
$535$	16544.4	1.33697
$536$	4192.00	0.337811
$537$	0	0
$538$	1014.96	0.0813343
$539$	0	0
$540$	0	0
$541$	−20270.2	−1.61088	−0.805438	−	0.592680i	$-0.798070\pi$
$541$	−20270.2	−1.61088	−0.805438	+	0.592680i	$0.798070\pi$
$542$	−12339.0	−0.977873
$543$	0	0
$544$	3556.36	0.280290
$545$	−25881.5	−2.03421
$546$	0	0
$547$	−4200.46	−0.328334	−0.164167	−	0.986433i	$-0.552494\pi$
$547$	−4200.46	−0.328334	−0.164167	+	0.986433i	$0.552494\pi$
$548$	1119.45	0.0872639
$549$	0	0
$550$	0	0
$551$	−7372.37	−0.570006
$552$	0	0
$553$	−14547.7	−1.11868
$554$	135.409	0.0103844
$555$	0	0
$556$	7926.18	0.604577
$557$	17898.5	1.36155	0.680777	−	0.732491i	$-0.261642\pi$
$557$	17898.5	1.36155	0.680777	+	0.732491i	$0.261642\pi$
$558$	0	0
$559$	−1815.96	−0.137401
$560$	2925.82	0.220783
$561$	0	0
$562$	−5831.73	−0.437717
$563$	98.7171	0.00738975	0.00369488	−	0.999993i	$-0.498824\pi$
$563$	98.7171	0.00738975	0.00369488	+	0.999993i	$0.498824\pi$
$564$	0	0
$565$	−9517.51	−0.708681
$566$	−18068.4	−1.34183
$567$	0	0
$568$	−1836.91	−0.135695
$569$	4802.60	0.353841	0.176920	−	0.984225i	$-0.443386\pi$
$569$	4802.60	0.353841	0.176920	+	0.984225i	$0.443386\pi$
$570$	0	0
$571$	−4181.00	−0.306426	−0.153213	−	0.988193i	$-0.548962\pi$
$571$	−4181.00	−0.306426	−0.153213	+	0.988193i	$0.548962\pi$
$572$	0	0
$573$	0	0
$574$	−10639.9	−0.773695
$575$	11203.2	0.812532
$576$	0	0
$577$	−11618.9	−0.838307	−0.419153	−	0.907915i	$-0.637673\pi$
$577$	−11618.9	−0.838307	−0.419153	+	0.907915i	$0.637673\pi$
$578$	−14876.5	−1.07056
$579$	0	0
$580$	−11049.3	−0.791028
$581$	−18947.2	−1.35294
$582$	0	0
$583$	0	0
$584$	9223.27	0.653530
$585$	0	0
$586$	10619.4	0.748610
$587$	−17966.9	−1.26332	−0.631662	−	0.775244i	$-0.717627\pi$
$587$	−17966.9	−1.26332	−0.631662	+	0.775244i	$0.717627\pi$
$588$	0	0
$589$	−2124.44	−0.148618
$590$	252.008	0.0175848
$591$	0	0
$592$	−5078.54	−0.352579
$593$	8341.22	0.577627	0.288813	−	0.957385i	$-0.406739\pi$
$593$	8341.22	0.577627	0.288813	+	0.957385i	$0.406739\pi$
$594$	0	0
$595$	−20322.8	−1.40026
$596$	−6133.64	−0.421549
$597$	0	0
$598$	15673.9	1.07183
$599$	12708.1	0.866841	0.433420	−	0.901192i	$-0.357307\pi$
$599$	12708.1	0.866841	0.433420	+	0.901192i	$0.357307\pi$
$600$	0	0
$601$	−737.182	−0.0500338	−0.0250169	−	0.999687i	$-0.507964\pi$
$601$	−737.182	−0.0500338	−0.0250169	+	0.999687i	$0.507964\pi$
$602$	−1213.36	−0.0821475
$603$	0	0
$604$	−7544.27	−0.508232
$605$	0	0
$606$	0	0
$607$	13543.4	0.905615	0.452808	−	0.891608i	$-0.350422\pi$
$607$	13543.4	0.905615	0.452808	+	0.891608i	$0.350422\pi$
$608$	−1154.91	−0.0770356
$609$	0	0
$610$	2814.56	0.186817
$611$	−913.497	−0.0604847
$612$	0	0
$613$	−18531.5	−1.22101	−0.610506	−	0.792012i	$-0.709034\pi$
$613$	−18531.5	−1.22101	−0.610506	+	0.792012i	$0.709034\pi$
$614$	13582.9	0.892771
$615$	0	0
$616$	0	0
$617$	−19033.1	−1.24189	−0.620943	−	0.783856i	$-0.713250\pi$
$617$	−19033.1	−1.24189	−0.620943	+	0.783856i	$0.713250\pi$
$618$	0	0
$619$	3085.74	0.200365	0.100183	−	0.994969i	$-0.468057\pi$
$619$	3085.74	0.200365	0.100183	+	0.994969i	$0.468057\pi$
$620$	−3183.99	−0.206245
$621$	0	0
$622$	3038.04	0.195843
$623$	15092.6	0.970580
$624$	0	0
$625$	−19509.8	−1.24862
$626$	4083.45	0.260715
$627$	0	0
$628$	11184.7	0.710699
$629$	35275.6	2.23614
$630$	0	0
$631$	−2923.31	−0.184430	−0.0922148	−	0.995739i	$-0.529395\pi$
$631$	−2923.31	−0.184430	−0.0922148	+	0.995739i	$0.529395\pi$
$632$	8606.36	0.541681
$633$	0	0
$634$	9964.14	0.624174
$635$	13722.2	0.857554
$636$	0	0
$637$	6481.88	0.403173
$638$	0	0
$639$	0	0
$640$	−1730.91	−0.106906
$641$	12788.7	0.788023	0.394011	−	0.919106i	$-0.371087\pi$
$641$	12788.7	0.788023	0.394011	+	0.919106i	$0.371087\pi$
$642$	0	0
$643$	−25527.7	−1.56565	−0.782826	−	0.622240i	$-0.786223\pi$
$643$	−25527.7	−1.56565	−0.782826	+	0.622240i	$0.786223\pi$
$644$	10472.7	0.640812
$645$	0	0
$646$	8022.00	0.488578
$647$	−17467.2	−1.06137	−0.530685	−	0.847569i	$-0.678065\pi$
$647$	−17467.2	−1.06137	−0.530685	+	0.847569i	$0.678065\pi$
$648$	0	0
$649$	0	0
$650$	4684.33	0.282669
$651$	0	0
$652$	−610.549	−0.0366732
$653$	−3177.42	−0.190417	−0.0952084	−	0.995457i	$-0.530352\pi$
$653$	−3177.42	−0.190417	−0.0952084	+	0.995457i	$0.530352\pi$
$654$	0	0
$655$	21241.1	1.26711
$656$	6294.54	0.374635
$657$	0	0
$658$	−610.365	−0.0361618
$659$	26099.5	1.54278	0.771389	−	0.636364i	$-0.219562\pi$
$659$	26099.5	1.54278	0.771389	+	0.636364i	$0.219562\pi$
$660$	0	0
$661$	17187.2	1.01135	0.505676	−	0.862723i	$-0.331243\pi$
$661$	17187.2	1.01135	0.505676	+	0.862723i	$0.331243\pi$
$662$	10485.1	0.615580
$663$	0	0
$664$	11209.1	0.655116
$665$	6599.71	0.384851
$666$	0	0
$667$	−39549.9	−2.29592
$668$	3355.09	0.194330
$669$	0	0
$670$	14171.8	0.817171
$671$	0	0
$672$	0	0
$673$	13317.2	0.762766	0.381383	−	0.924417i	$-0.375448\pi$
$673$	13317.2	0.762766	0.381383	+	0.924417i	$0.375448\pi$
$674$	−14582.4	−0.833370
$675$	0	0
$676$	−2234.36	−0.127125
$677$	20030.0	1.13710	0.568550	−	0.822649i	$-0.307505\pi$
$677$	20030.0	1.13710	0.568550	+	0.822649i	$0.307505\pi$
$678$	0	0
$679$	−18864.8	−1.06622
$680$	12022.9	0.678026
$681$	0	0
$682$	0	0
$683$	−10170.9	−0.569810	−0.284905	−	0.958556i	$-0.591962\pi$
$683$	−10170.9	−0.569810	−0.284905	+	0.958556i	$0.591962\pi$
$684$	0	0
$685$	3784.51	0.211093
$686$	13607.5	0.757344
$687$	0	0
$688$	717.820	0.0397771
$689$	3838.89	0.212264
$690$	0	0
$691$	−17817.2	−0.980893	−0.490447	−	0.871471i	$-0.663166\pi$
$691$	−17817.2	−0.980893	−0.490447	+	0.871471i	$0.663166\pi$
$692$	−7157.45	−0.393187
$693$	0	0
$694$	−5570.46	−0.304685
$695$	26795.9	1.46248
$696$	0	0
$697$	−43722.0	−2.37602
$698$	−7642.67	−0.414440
$699$	0	0
$700$	3129.90	0.168999
$701$	10453.6	0.563236	0.281618	−	0.959527i	$-0.409129\pi$
$701$	10453.6	0.563236	0.281618	+	0.959527i	$0.409129\pi$
$702$	0	0
$703$	−11455.6	−0.614587
$704$	0	0
$705$	0	0
$706$	21076.0	1.12352
$707$	366.956	0.0195202
$708$	0	0
$709$	25472.9	1.34930	0.674651	−	0.738137i	$-0.264294\pi$
$709$	25472.9	1.34930	0.674651	+	0.738137i	$0.264294\pi$
$710$	−6210.00	−0.328249
$711$	0	0
$712$	−8928.73	−0.469969
$713$	−11396.8	−0.598617
$714$	0	0
$715$	0	0
$716$	17323.8	0.904220
$717$	0	0
$718$	1462.91	0.0760380
$719$	3858.42	0.200132	0.100066	−	0.994981i	$-0.468095\pi$
$719$	3858.42	0.200132	0.100066	+	0.994981i	$0.468095\pi$
$720$	0	0
$721$	19958.9	1.03094
$722$	11112.9	0.572825
$723$	0	0
$724$	14599.4	0.749426
$725$	−11820.0	−0.605493
$726$	0	0
$727$	−14285.8	−0.728792	−0.364396	−	0.931244i	$-0.618725\pi$
$727$	−14285.8	−0.728792	−0.364396	+	0.931244i	$0.618725\pi$
$728$	4378.90	0.222930
$729$	0	0
$730$	31180.9	1.58090
$731$	−4985.99	−0.252276
$732$	0	0
$733$	5145.60	0.259287	0.129643	−	0.991561i	$-0.458617\pi$
$733$	5145.60	0.259287	0.129643	+	0.991561i	$0.458617\pi$
$734$	9604.73	0.482993
$735$	0	0
$736$	−6195.63	−0.310291
$737$	0	0
$738$	0	0
$739$	−23913.4	−1.19035	−0.595175	−	0.803596i	$-0.702917\pi$
$739$	−23913.4	−1.19035	−0.595175	+	0.803596i	$0.702917\pi$
$740$	−17168.9	−0.852894
$741$	0	0
$742$	2565.01	0.126906
$743$	26428.9	1.30496	0.652478	−	0.757808i	$-0.273730\pi$
$743$	26428.9	1.30496	0.652478	+	0.757808i	$0.273730\pi$
$744$	0	0
$745$	−20735.8	−1.01974
$746$	8542.95	0.419276
$747$	0	0
$748$	0	0
$749$	16544.4	0.807103
$750$	0	0
$751$	1736.24	0.0843625	0.0421813	−	0.999110i	$-0.486569\pi$
$751$	1736.24	0.0843625	0.0421813	+	0.999110i	$0.486569\pi$
$752$	361.090	0.0175101
$753$	0	0
$754$	−16536.8	−0.798720
$755$	−25504.8	−1.22942
$756$	0	0
$757$	−12106.6	−0.581270	−0.290635	−	0.956834i	$-0.593867\pi$
$757$	−12106.6	−0.581270	−0.290635	+	0.956834i	$0.593867\pi$
$758$	21141.1	1.01303
$759$	0	0
$760$	−3904.37	−0.186350
$761$	−12935.3	−0.616169	−0.308084	−	0.951359i	$-0.599688\pi$
$761$	−12935.3	−0.616169	−0.308084	+	0.951359i	$0.599688\pi$
$762$	0	0
$763$	−25881.5	−1.22801
$764$	−19512.6	−0.924008
$765$	0	0
$766$	−25073.9	−1.18271
$767$	377.166	0.0177558
$768$	0	0
$769$	876.225	0.0410891	0.0205445	−	0.999789i	$-0.493460\pi$
$769$	876.225	0.0410891	0.0205445	+	0.999789i	$0.493460\pi$
$770$	0	0
$771$	0	0
$772$	−13659.4	−0.636806
$773$	−16063.0	−0.747408	−0.373704	−	0.927548i	$-0.621912\pi$
$773$	−16063.0	−0.747408	−0.373704	+	0.927548i	$0.621912\pi$
$774$	0	0
$775$	−3406.07	−0.157871
$776$	11160.4	0.516280
$777$	0	0
$778$	22607.1	1.04178
$779$	14198.5	0.653033
$780$	0	0
$781$	0	0
$782$	43035.0	1.96794
$783$	0	0
$784$	−2562.18	−0.116717
$785$	37811.9	1.71919
$786$	0	0
$787$	16485.1	0.746669	0.373335	−	0.927697i	$-0.378214\pi$
$787$	16485.1	0.746669	0.373335	+	0.927697i	$0.378214\pi$
$788$	−11722.4	−0.529939
$789$	0	0
$790$	29095.3	1.31034
$791$	−9517.51	−0.427818
$792$	0	0
$793$	4212.38	0.188633
$794$	−3280.81	−0.146639
$795$	0	0
$796$	−6514.18	−0.290062
$797$	9380.17	0.416892	0.208446	−	0.978034i	$-0.433160\pi$
$797$	9380.17	0.416892	0.208446	+	0.978034i	$0.433160\pi$
$798$	0	0
$799$	−2508.14	−0.111053
$800$	−1851.64	−0.0818317
$801$	0	0
$802$	−1471.90	−0.0648062
$803$	0	0
$804$	0	0
$805$	35404.9	1.55013
$806$	−4765.29	−0.208251
$807$	0	0
$808$	−217.090	−0.00945199
$809$	−12139.8	−0.527579	−0.263789	−	0.964580i	$-0.584972\pi$
$809$	−12139.8	−0.527579	−0.263789	+	0.964580i	$0.584972\pi$
$810$	0	0
$811$	30960.4	1.34053	0.670263	−	0.742124i	$-0.266181\pi$
$811$	30960.4	1.34053	0.670263	+	0.742124i	$0.266181\pi$
$812$	−11049.3	−0.477529
$813$	0	0
$814$	0	0
$815$	−2064.07	−0.0887132
$816$	0	0
$817$	1619.17	0.0693361
$818$	17408.6	0.744106
$819$	0	0
$820$	21279.8	0.906248
$821$	2352.14	0.0999882	0.0499941	−	0.998750i	$-0.484080\pi$
$821$	2352.14	0.0999882	0.0499941	+	0.998750i	$0.484080\pi$
$822$	0	0
$823$	5139.35	0.217675	0.108837	−	0.994060i	$-0.465287\pi$
$823$	5139.35	0.217675	0.108837	+	0.994060i	$0.465287\pi$
$824$	−11807.6	−0.499197
$825$	0	0
$826$	252.008	0.0106156
$827$	−34658.3	−1.45730	−0.728650	−	0.684887i	$-0.759852\pi$
$827$	−34658.3	−1.45730	−0.728650	+	0.684887i	$0.759852\pi$
$828$	0	0
$829$	−13319.6	−0.558034	−0.279017	−	0.960286i	$-0.590009\pi$
$829$	−13319.6	−0.558034	−0.279017	+	0.960286i	$0.590009\pi$
$830$	37894.3	1.58474
$831$	0	0
$832$	−2590.55	−0.107946
$833$	17796.9	0.740249
$834$	0	0
$835$	11342.5	0.470086
$836$	0	0
$837$	0	0
$838$	16045.4	0.661429
$839$	−1485.41	−0.0611228	−0.0305614	−	0.999533i	$-0.509730\pi$
$839$	−1485.41	−0.0611228	−0.0305614	+	0.999533i	$0.509730\pi$
$840$	0	0
$841$	17338.3	0.710905
$842$	−29742.6	−1.21734
$843$	0	0
$844$	−7518.37	−0.306627
$845$	−7553.64	−0.307518
$846$	0	0
$847$	0	0
$848$	−1517.45	−0.0614499
$849$	0	0
$850$	12861.5	0.518996
$851$	−61454.6	−2.47549
$852$	0	0
$853$	18720.0	0.751420	0.375710	−	0.926737i	$-0.377399\pi$
$853$	18720.0	0.751420	0.375710	+	0.926737i	$0.377399\pi$
$854$	2814.56	0.112778
$855$	0	0
$856$	−9787.64	−0.390811
$857$	8898.12	0.354672	0.177336	−	0.984150i	$-0.443252\pi$
$857$	8898.12	0.354672	0.177336	+	0.984150i	$0.443252\pi$
$858$	0	0
$859$	10820.0	0.429771	0.214886	−	0.976639i	$-0.431062\pi$
$859$	10820.0	0.429771	0.214886	+	0.976639i	$0.431062\pi$
$860$	2426.72	0.0962214
$861$	0	0
$862$	−7574.45	−0.299289
$863$	41524.5	1.63791	0.818953	−	0.573861i	$-0.194555\pi$
$863$	41524.5	1.63791	0.818953	+	0.573861i	$0.194555\pi$
$864$	0	0
$865$	−24197.0	−0.951126
$866$	−9516.17	−0.373409
$867$	0	0
$868$	−3183.99	−0.124506
$869$	0	0
$870$	0	0
$871$	21210.1	0.825117
$872$	15311.4	0.594623
$873$	0	0
$874$	−13975.4	−0.540873
$875$	−12276.8	−0.474321
$876$	0	0
$877$	42734.1	1.64542	0.822708	−	0.568465i	$-0.192462\pi$
$877$	42734.1	1.64542	0.822708	+	0.568465i	$0.192462\pi$
$878$	−7792.77	−0.299537
$879$	0	0
$880$	0	0
$881$	−52155.1	−1.99449	−0.997247	−	0.0741559i	$-0.976374\pi$
$881$	−52155.1	−1.99449	−0.997247	+	0.0741559i	$0.976374\pi$
$882$	0	0
$883$	−814.457	−0.0310404	−0.0155202	−	0.999880i	$-0.504940\pi$
$883$	−814.457	−0.0310404	−0.0155202	+	0.999880i	$0.504940\pi$
$884$	17994.0	0.684619
$885$	0	0
$886$	8652.62	0.328093
$887$	−42246.8	−1.59922	−0.799610	−	0.600520i	$-0.794960\pi$
$887$	−42246.8	−1.59922	−0.799610	+	0.600520i	$0.794960\pi$
$888$	0	0
$889$	13722.2	0.517690
$890$	−30185.1	−1.13686
$891$	0	0
$892$	−13073.1	−0.490716
$893$	814.503	0.0305222
$894$	0	0
$895$	58566.2	2.18732
$896$	−1730.91	−0.0645374
$897$	0	0
$898$	8847.36	0.328775
$899$	12024.2	0.446086
$900$	0	0
$901$	10540.2	0.389729
$902$	0	0
$903$	0	0
$904$	5630.54	0.207156
$905$	49356.0	1.81287
$906$	0	0
$907$	11328.9	0.414741	0.207370	−	0.978263i	$-0.433509\pi$
$907$	11328.9	0.414741	0.207370	+	0.978263i	$0.433509\pi$
$908$	21500.7	0.785822
$909$	0	0
$910$	14803.7	0.539271
$911$	1390.44	0.0505678	0.0252839	−	0.999680i	$-0.491951\pi$
$911$	1390.44	0.0505678	0.0252839	+	0.999680i	$0.491951\pi$
$912$	0	0
$913$	0	0
$914$	8694.82	0.314660
$915$	0	0
$916$	10088.4	0.363896
$917$	21241.1	0.764932
$918$	0	0
$919$	23289.6	0.835966	0.417983	−	0.908455i	$-0.362737\pi$
$919$	23289.6	0.835966	0.417983	+	0.908455i	$0.362737\pi$
$920$	−20945.4	−0.750599
$921$	0	0
$922$	6537.27	0.233507
$923$	−9294.13	−0.331441
$924$	0	0
$925$	−18366.5	−0.652849
$926$	28183.2	1.00017
$927$	0	0
$928$	6536.72	0.231227
$929$	−4263.14	−0.150559	−0.0752795	−	0.997162i	$-0.523985\pi$
$929$	−4263.14	−0.150559	−0.0752795	+	0.997162i	$0.523985\pi$
$930$	0	0
$931$	−5779.45	−0.203452
$932$	−9370.36	−0.329331
$933$	0	0
$934$	−7325.62	−0.256640
$935$	0	0
$936$	0	0
$937$	47180.1	1.64494	0.822469	−	0.568810i	$-0.192596\pi$
$937$	47180.1	1.64494	0.822469	+	0.568810i	$0.192596\pi$
$938$	14171.8	0.493311
$939$	0	0
$940$	1220.73	0.0423572
$941$	26919.7	0.932579	0.466289	−	0.884632i	$-0.345591\pi$
$941$	26919.7	0.932579	0.466289	+	0.884632i	$0.345591\pi$
$942$	0	0
$943$	76169.3	2.63034
$944$	−149.087	−0.00514023
$945$	0	0
$946$	0	0
$947$	42195.7	1.44792	0.723958	−	0.689844i	$-0.242321\pi$
$947$	42195.7	1.44792	0.723958	+	0.689844i	$0.242321\pi$
$948$	0	0
$949$	46666.6	1.59627
$950$	−4176.70	−0.142642
$951$	0	0
$952$	12022.9	0.409312
$953$	−38133.4	−1.29618	−0.648091	−	0.761563i	$-0.724432\pi$
$953$	−38133.4	−1.29618	−0.648091	+	0.761563i	$0.724432\pi$
$954$	0	0
$955$	−65965.9	−2.23519
$956$	−12698.7	−0.429609
$957$	0	0
$958$	21526.6	0.725985
$959$	3784.51	0.127433
$960$	0	0
$961$	−26326.1	−0.883692
$962$	−25695.7	−0.861188
$963$	0	0
$964$	−4280.55	−0.143016
$965$	−46178.2	−1.54044
$966$	0	0
$967$	43514.8	1.44710	0.723549	−	0.690274i	$-0.242510\pi$
$967$	43514.8	1.44710	0.723549	+	0.690274i	$0.242510\pi$
$968$	0	0
$969$	0	0
$970$	37729.6	1.24889
$971$	27857.7	0.920698	0.460349	−	0.887738i	$-0.347724\pi$
$971$	27857.7	0.920698	0.460349	+	0.887738i	$0.347724\pi$
$972$	0	0
$973$	26795.9	0.882873
$974$	30599.1	1.00663
$975$	0	0
$976$	−1665.08	−0.0546087
$977$	−37269.8	−1.22044	−0.610218	−	0.792234i	$-0.708918\pi$
$977$	−37269.8	−1.22044	−0.610218	+	0.792234i	$0.708918\pi$
$978$	0	0
$979$	0	0
$980$	−8661.91	−0.282341
$981$	0	0
$982$	−10366.4	−0.336867
$983$	−46669.7	−1.51427	−0.757137	−	0.653256i	$-0.773402\pi$
$983$	−46669.7	−1.51427	−0.757137	+	0.653256i	$0.773402\pi$
$984$	0	0
$985$	−39629.5	−1.28193
$986$	−45404.2	−1.46649
$987$	0	0
$988$	−5843.44	−0.188162
$989$	8686.23	0.279278
$990$	0	0
$991$	27744.6	0.889340	0.444670	−	0.895695i	$-0.353321\pi$
$991$	27744.6	0.889340	0.444670	+	0.895695i	$0.353321\pi$
$992$	1883.64	0.0602879
$993$	0	0
$994$	−6210.00	−0.198158
$995$	−22022.3	−0.701664
$996$	0	0
$997$	−38572.3	−1.22527	−0.612636	−	0.790365i	$-0.709891\pi$
$997$	−38572.3	−1.22527	−0.612636	+	0.790365i	$0.709891\pi$
$998$	861.434	0.0273229
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.x.1.2		2
3.2	odd	2		726.4.a.s.1.1	yes	2
11.10	odd	2		2178.4.a.bg.1.2		2
33.32	even	2		726.4.a.m.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
726.4.a.m.1.1	✓	2	33.32	even	2
726.4.a.s.1.1	yes	2	3.2	odd	2
2178.4.a.x.1.2		2	1.1	even	1	trivial
2178.4.a.bg.1.2		2	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +13.5227 q^{5} +13.5227 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +13.5227 q^{5} +13.5227 q^{7} -8.00000 q^{8} -27.0454 q^{10} -40.4773 q^{13} -27.0454 q^{14} +16.0000 q^{16} -111.136 q^{17} +36.0908 q^{19} +54.0908 q^{20} +193.614 q^{23} +57.8637 q^{25} +80.9546 q^{26} +54.0908 q^{28} -204.273 q^{29} -58.8637 q^{31} -32.0000 q^{32} +222.273 q^{34} +182.864 q^{35} -317.409 q^{37} -72.1817 q^{38} -108.182 q^{40} +393.409 q^{41} +44.8637 q^{43} -387.227 q^{46} +22.5681 q^{47} -160.136 q^{49} -115.727 q^{50} -161.909 q^{52} -94.8407 q^{53} -108.182 q^{56} +408.545 q^{58} -9.31796 q^{59} -104.068 q^{61} +117.727 q^{62} +64.0000 q^{64} -547.363 q^{65} -524.000 q^{67} -444.545 q^{68} -365.727 q^{70} +229.614 q^{71} -1152.91 q^{73} +634.818 q^{74} +144.363 q^{76} -1075.80 q^{79} +216.363 q^{80} -786.818 q^{82} -1401.14 q^{83} -1502.86 q^{85} -89.7275 q^{86} +1116.09 q^{89} -547.363 q^{91} +774.454 q^{92} -45.1363 q^{94} +488.046 q^{95} -1395.04 q^{97} +320.273 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 12 q^{10} - 114 q^{13} + 12 q^{14} + 32 q^{16} - 24 q^{17} - 60 q^{19} - 24 q^{20} + 222 q^{23} + 314 q^{25} + 228 q^{26} - 24 q^{28} - 12 q^{29} - 316 q^{31} - 64 q^{32} + 48 q^{34} + 564 q^{35} - 40 q^{37} + 120 q^{38} + 48 q^{40} + 192 q^{41} + 288 q^{43} - 444 q^{46} - 54 q^{47} - 122 q^{49} - 628 q^{50} - 456 q^{52} + 306 q^{53} + 48 q^{56} + 24 q^{58} + 444 q^{59} + 618 q^{61} + 632 q^{62} + 128 q^{64} + 888 q^{65} - 1048 q^{67} - 96 q^{68} - 1128 q^{70} + 294 q^{71} - 984 q^{73} + 80 q^{74} - 240 q^{76} - 1722 q^{79} - 96 q^{80} - 384 q^{82} - 2604 q^{83} - 3204 q^{85} - 576 q^{86} + 2100 q^{89} + 888 q^{91} + 888 q^{92} + 108 q^{94} + 2364 q^{95} + 184 q^{97} + 244 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 - 16 * q^8 + 12 * q^10 - 114 * q^13 + 12 * q^14 + 32 * q^16 - 24 * q^17 - 60 * q^19 - 24 * q^20 + 222 * q^23 + 314 * q^25 + 228 * q^26 - 24 * q^28 - 12 * q^29 - 316 * q^31 - 64 * q^32 + 48 * q^34 + 564 * q^35 - 40 * q^37 + 120 * q^38 + 48 * q^40 + 192 * q^41 + 288 * q^43 - 444 * q^46 - 54 * q^47 - 122 * q^49 - 628 * q^50 - 456 * q^52 + 306 * q^53 + 48 * q^56 + 24 * q^58 + 444 * q^59 + 618 * q^61 + 632 * q^62 + 128 * q^64 + 888 * q^65 - 1048 * q^67 - 96 * q^68 - 1128 * q^70 + 294 * q^71 - 984 * q^73 + 80 * q^74 - 240 * q^76 - 1722 * q^79 - 96 * q^80 - 384 * q^82 - 2604 * q^83 - 3204 * q^85 - 576 * q^86 + 2100 * q^89 + 888 * q^91 + 888 * q^92 + 108 * q^94 + 2364 * q^95 + 184 * q^97 + 244 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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