LMFDB - Embedded newform 2178.4.a.ce.1.1

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			1.1
Root			$7.99688$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -11.5573 q^{5} +15.9100 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} -11.5573 q^{5} +15.9100 q^{7} -8.00000 q^{8} +23.1145 q^{10} -84.2442 q^{13} -31.8200 q^{14} +16.0000 q^{16} +73.2907 q^{17} +93.4600 q^{19} -46.2290 q^{20} +188.856 q^{23} +8.57014 q^{25} +168.488 q^{26} +63.6400 q^{28} +162.863 q^{29} -153.989 q^{31} -32.0000 q^{32} -146.581 q^{34} -183.876 q^{35} -339.714 q^{37} -186.920 q^{38} +92.4580 q^{40} -96.2922 q^{41} -479.178 q^{43} -377.711 q^{46} +296.697 q^{47} -89.8715 q^{49} -17.1403 q^{50} -336.977 q^{52} -220.675 q^{53} -127.280 q^{56} -325.726 q^{58} -92.4275 q^{59} +242.746 q^{61} +307.979 q^{62} +64.0000 q^{64} +973.631 q^{65} +307.078 q^{67} +293.163 q^{68} +367.752 q^{70} +937.734 q^{71} +644.387 q^{73} +679.427 q^{74} +373.840 q^{76} -550.032 q^{79} -184.916 q^{80} +192.584 q^{82} -341.571 q^{83} -847.039 q^{85} +958.356 q^{86} +68.5702 q^{89} -1340.33 q^{91} +755.422 q^{92} -593.393 q^{94} -1080.14 q^{95} +1546.74 q^{97} +179.743 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−11.5573	−1.03371	−0.516856	−	0.856072i	$-0.672898\pi$
$5$	−11.5573	−1.03371	−0.516856	+	0.856072i	$0.672898\pi$
$6$	0	0
$7$	15.9100	0.859060	0.429530	−	0.903053i	$-0.358679\pi$
$7$	15.9100	0.859060	0.429530	+	0.903053i	$0.358679\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	23.1145	0.730945
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−84.2442	−1.79732	−0.898659	−	0.438648i	$-0.855457\pi$
$13$	−84.2442	−1.79732	−0.898659	+	0.438648i	$0.855457\pi$
$14$	−31.8200	−0.607447
$15$	0	0
$16$	16.0000	0.250000
$17$	73.2907	1.04562	0.522812	−	0.852448i	$-0.324883\pi$
$17$	73.2907	1.04562	0.522812	+	0.852448i	$0.324883\pi$
$18$	0	0
$19$	93.4600	1.12848	0.564242	−	0.825609i	$-0.309168\pi$
$19$	93.4600	1.12848	0.564242	+	0.825609i	$0.309168\pi$
$20$	−46.2290	−0.516856
$21$	0	0
$22$	0	0
$23$	188.856	1.71213	0.856067	−	0.516864i	$-0.172901\pi$
$23$	188.856	1.71213	0.856067	+	0.516864i	$0.172901\pi$
$24$	0	0
$25$	8.57014	0.0685611
$26$	168.488	1.27090
$27$	0	0
$28$	63.6400	0.429530
$29$	162.863	1.04286	0.521429	−	0.853295i	$-0.325399\pi$
$29$	162.863	1.04286	0.521429	+	0.853295i	$0.325399\pi$
$30$	0	0
$31$	−153.989	−0.892171	−0.446085	−	0.894990i	$-0.647182\pi$
$31$	−153.989	−0.892171	−0.446085	+	0.894990i	$0.647182\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−146.581	−0.739368
$35$	−183.876	−0.888021
$36$	0	0
$37$	−339.714	−1.50942	−0.754710	−	0.656058i	$-0.772223\pi$
$37$	−339.714	−1.50942	−0.754710	+	0.656058i	$0.772223\pi$
$38$	−186.920	−0.797959
$39$	0	0
$40$	92.4580	0.365472
$41$	−96.2922	−0.366788	−0.183394	−	0.983039i	$-0.558708\pi$
$41$	−96.2922	−0.366788	−0.183394	+	0.983039i	$0.558708\pi$
$42$	0	0
$43$	−479.178	−1.69939	−0.849697	−	0.527271i	$-0.823215\pi$
$43$	−479.178	−1.69939	−0.849697	+	0.527271i	$0.823215\pi$
$44$	0	0
$45$	0	0
$46$	−377.711	−1.21066
$47$	296.697	0.920801	0.460401	−	0.887711i	$-0.347706\pi$
$47$	296.697	0.920801	0.460401	+	0.887711i	$0.347706\pi$
$48$	0	0
$49$	−89.8715	−0.262016
$50$	−17.1403	−0.0484800
$51$	0	0
$52$	−336.977	−0.898659
$53$	−220.675	−0.571926	−0.285963	−	0.958241i	$-0.592313\pi$
$53$	−220.675	−0.571926	−0.285963	+	0.958241i	$0.592313\pi$
$54$	0	0
$55$	0	0
$56$	−127.280	−0.303724
$57$	0	0
$58$	−325.726	−0.737412
$59$	−92.4275	−0.203950	−0.101975	−	0.994787i	$-0.532516\pi$
$59$	−92.4275	−0.203950	−0.101975	+	0.994787i	$0.532516\pi$
$60$	0	0
$61$	242.746	0.509516	0.254758	−	0.967005i	$-0.418004\pi$
$61$	242.746	0.509516	0.254758	+	0.967005i	$0.418004\pi$
$62$	307.979	0.630860
$63$	0	0
$64$	64.0000	0.125000
$65$	973.631	1.85791
$66$	0	0
$67$	307.078	0.559934	0.279967	−	0.960010i	$-0.409676\pi$
$67$	307.078	0.559934	0.279967	+	0.960010i	$0.409676\pi$
$68$	293.163	0.522812
$69$	0	0
$70$	367.752	0.627925
$71$	937.734	1.56745	0.783723	−	0.621111i	$-0.213318\pi$
$71$	937.734	1.56745	0.783723	+	0.621111i	$0.213318\pi$
$72$	0	0
$73$	644.387	1.03315	0.516574	−	0.856243i	$-0.327207\pi$
$73$	644.387	1.03315	0.516574	+	0.856243i	$0.327207\pi$
$74$	679.427	1.06732
$75$	0	0
$76$	373.840	0.564242
$77$	0	0
$78$	0	0
$79$	−550.032	−0.783334	−0.391667	−	0.920107i	$-0.628101\pi$
$79$	−550.032	−0.783334	−0.391667	+	0.920107i	$0.628101\pi$
$80$	−184.916	−0.258428
$81$	0	0
$82$	192.584	0.259358
$83$	−341.571	−0.451714	−0.225857	−	0.974160i	$-0.572518\pi$
$83$	−341.571	−0.451714	−0.225857	+	0.974160i	$0.572518\pi$
$84$	0	0
$85$	−847.039	−1.08087
$86$	958.356	1.20165
$87$	0	0
$88$	0	0
$89$	68.5702	0.0816677	0.0408338	−	0.999166i	$-0.486999\pi$
$89$	68.5702	0.0816677	0.0408338	+	0.999166i	$0.486999\pi$
$90$	0	0
$91$	−1340.33	−1.54400
$92$	755.422	0.856067
$93$	0	0
$94$	−593.393	−0.651105
$95$	−1080.14	−1.16653
$96$	0	0
$97$	1546.74	1.61905	0.809524	−	0.587087i	$-0.199725\pi$
$97$	1546.74	1.61905	0.809524	+	0.587087i	$0.199725\pi$
$98$	179.743	0.185273
$99$	0	0
$100$	34.2806	0.0342806
$101$	−1136.09	−1.11926	−0.559628	−	0.828744i	$-0.689056\pi$
$101$	−1136.09	−1.11926	−0.559628	+	0.828744i	$0.689056\pi$
$102$	0	0
$103$	−239.788	−0.229389	−0.114694	−	0.993401i	$-0.536589\pi$
$103$	−239.788	−0.229389	−0.114694	+	0.993401i	$0.536589\pi$
$104$	673.953	0.635448
$105$	0	0
$106$	441.350	0.404413
$107$	−182.482	−0.164871	−0.0824356	−	0.996596i	$-0.526270\pi$
$107$	−182.482	−0.164871	−0.0824356	+	0.996596i	$0.526270\pi$
$108$	0	0
$109$	−711.769	−0.625460	−0.312730	−	0.949842i	$-0.601243\pi$
$109$	−711.769	−0.625460	−0.312730	+	0.949842i	$0.601243\pi$
$110$	0	0
$111$	0	0
$112$	254.560	0.214765
$113$	−352.239	−0.293238	−0.146619	−	0.989193i	$-0.546839\pi$
$113$	−352.239	−0.293238	−0.146619	+	0.989193i	$0.546839\pi$
$114$	0	0
$115$	−2182.65	−1.76985
$116$	651.451	0.521429
$117$	0	0
$118$	184.855	0.144214
$119$	1166.06	0.898253
$120$	0	0
$121$	0	0
$122$	−485.493	−0.360283
$123$	0	0
$124$	−615.957	−0.446085
$125$	1345.61	0.962840
$126$	0	0
$127$	1060.27	0.740819	0.370410	−	0.928869i	$-0.379217\pi$
$127$	1060.27	0.740819	0.370410	+	0.928869i	$0.379217\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−1947.26	−1.31374
$131$	−1131.22	−0.754469	−0.377235	−	0.926118i	$-0.623125\pi$
$131$	−1131.22	−0.754469	−0.377235	+	0.926118i	$0.623125\pi$
$132$	0	0
$133$	1486.95	0.969436
$134$	−614.157	−0.395933
$135$	0	0
$136$	−586.325	−0.369684
$137$	−576.486	−0.359507	−0.179754	−	0.983712i	$-0.557530\pi$
$137$	−576.486	−0.359507	−0.179754	+	0.983712i	$0.557530\pi$
$138$	0	0
$139$	−977.212	−0.596302	−0.298151	−	0.954519i	$-0.596370\pi$
$139$	−977.212	−0.596302	−0.298151	+	0.954519i	$0.596370\pi$
$140$	−735.504	−0.444010
$141$	0	0
$142$	−1875.47	−1.10835
$143$	0	0
$144$	0	0
$145$	−1882.25	−1.07801
$146$	−1288.77	−0.730546
$147$	0	0
$148$	−1358.85	−0.754710
$149$	−2819.68	−1.55032	−0.775158	−	0.631768i	$-0.782330\pi$
$149$	−2819.68	−1.55032	−0.775158	+	0.631768i	$0.782330\pi$
$150$	0	0
$151$	1474.00	0.794387	0.397193	−	0.917735i	$-0.369984\pi$
$151$	1474.00	0.794387	0.397193	+	0.917735i	$0.369984\pi$
$152$	−747.680	−0.398979
$153$	0	0
$154$	0	0
$155$	1779.69	0.922248
$156$	0	0
$157$	2941.67	1.49536	0.747679	−	0.664061i	$-0.231168\pi$
$157$	2941.67	1.49536	0.747679	+	0.664061i	$0.231168\pi$
$158$	1100.06	0.553901
$159$	0	0
$160$	369.832	0.182736
$161$	3004.69	1.47083
$162$	0	0
$163$	−1203.67	−0.578395	−0.289198	−	0.957269i	$-0.593388\pi$
$163$	−1203.67	−0.578395	−0.289198	+	0.957269i	$0.593388\pi$
$164$	−385.169	−0.183394
$165$	0	0
$166$	683.142	0.319410
$167$	−3898.16	−1.80628	−0.903139	−	0.429347i	$-0.858744\pi$
$167$	−3898.16	−1.80628	−0.903139	+	0.429347i	$0.858744\pi$
$168$	0	0
$169$	4900.08	2.23035
$170$	1694.08	0.764293
$171$	0	0
$172$	−1916.71	−0.849697
$173$	−3135.07	−1.37777	−0.688887	−	0.724868i	$-0.741901\pi$
$173$	−3135.07	−1.37777	−0.688887	+	0.724868i	$0.741901\pi$
$174$	0	0
$175$	136.351	0.0588981
$176$	0	0
$177$	0	0
$178$	−137.140	−0.0577478
$179$	−232.089	−0.0969115	−0.0484557	−	0.998825i	$-0.515430\pi$
$179$	−232.089	−0.0969115	−0.0484557	+	0.998825i	$0.515430\pi$
$180$	0	0
$181$	2699.17	1.10844	0.554220	−	0.832370i	$-0.313016\pi$
$181$	2699.17	1.10844	0.554220	+	0.832370i	$0.313016\pi$
$182$	2680.65	1.09178
$183$	0	0
$184$	−1510.84	−0.605331
$185$	3926.16	1.56031
$186$	0	0
$187$	0	0
$188$	1186.79	0.460401
$189$	0	0
$190$	2160.28	0.824860
$191$	1988.61	0.753354	0.376677	−	0.926345i	$-0.377067\pi$
$191$	1988.61	0.753354	0.376677	+	0.926345i	$0.377067\pi$
$192$	0	0
$193$	−1193.96	−0.445301	−0.222650	−	0.974898i	$-0.571471\pi$
$193$	−1193.96	−0.445301	−0.222650	+	0.974898i	$0.571471\pi$
$194$	−3093.48	−1.14484
$195$	0	0
$196$	−359.486	−0.131008
$197$	4808.95	1.73920	0.869602	−	0.493753i	$-0.164375\pi$
$197$	4808.95	1.73920	0.869602	+	0.493753i	$0.164375\pi$
$198$	0	0
$199$	2681.69	0.955277	0.477639	−	0.878556i	$-0.341493\pi$
$199$	2681.69	0.955277	0.477639	+	0.878556i	$0.341493\pi$
$200$	−68.5611	−0.0242400
$201$	0	0
$202$	2272.17	0.791434
$203$	2591.15	0.895877
$204$	0	0
$205$	1112.87	0.379153
$206$	479.577	0.162202
$207$	0	0
$208$	−1347.91	−0.449329
$209$	0	0
$210$	0	0
$211$	−2561.80	−0.835836	−0.417918	−	0.908485i	$-0.637240\pi$
$211$	−2561.80	−0.835836	−0.417918	+	0.908485i	$0.637240\pi$
$212$	−882.701	−0.285963
$213$	0	0
$214$	364.964	0.116582
$215$	5537.98	1.75668
$216$	0	0
$217$	−2449.97	−0.766428
$218$	1423.54	0.442267
$219$	0	0
$220$	0	0
$221$	−6174.31	−1.87932
$222$	0	0
$223$	−950.148	−0.285321	−0.142661	−	0.989772i	$-0.545566\pi$
$223$	−950.148	−0.285321	−0.142661	+	0.989772i	$0.545566\pi$
$224$	−509.120	−0.151862
$225$	0	0
$226$	704.478	0.207350
$227$	827.522	0.241958	0.120979	−	0.992655i	$-0.461397\pi$
$227$	827.522	0.241958	0.120979	+	0.992655i	$0.461397\pi$
$228$	0	0
$229$	542.375	0.156512	0.0782558	−	0.996933i	$-0.475065\pi$
$229$	542.375	0.156512	0.0782558	+	0.996933i	$0.475065\pi$
$230$	4365.30	1.25148
$231$	0	0
$232$	−1302.90	−0.368706
$233$	5198.72	1.46172	0.730858	−	0.682530i	$-0.239120\pi$
$233$	5198.72	1.46172	0.730858	+	0.682530i	$0.239120\pi$
$234$	0	0
$235$	−3429.00	−0.951844
$236$	−369.710	−0.101975
$237$	0	0
$238$	−2332.11	−0.635161
$239$	−2481.26	−0.671544	−0.335772	−	0.941943i	$-0.608997\pi$
$239$	−2481.26	−0.671544	−0.335772	+	0.941943i	$0.608997\pi$
$240$	0	0
$241$	−50.4887	−0.0134949	−0.00674744	−	0.999977i	$-0.502148\pi$
$241$	−50.4887	−0.0134949	−0.00674744	+	0.999977i	$0.502148\pi$
$242$	0	0
$243$	0	0
$244$	970.986	0.254758
$245$	1038.67	0.270849
$246$	0	0
$247$	−7873.46	−2.02824
$248$	1231.91	0.315430
$249$	0	0
$250$	−2691.22	−0.680831
$251$	1111.12	0.279416	0.139708	−	0.990193i	$-0.455384\pi$
$251$	1111.12	0.279416	0.139708	+	0.990193i	$0.455384\pi$
$252$	0	0
$253$	0	0
$254$	−2120.55	−0.523838
$255$	0	0
$256$	256.000	0.0625000
$257$	7435.62	1.80475	0.902375	−	0.430951i	$-0.141822\pi$
$257$	7435.62	1.80475	0.902375	+	0.430951i	$0.141822\pi$
$258$	0	0
$259$	−5404.85	−1.29668
$260$	3894.53	0.928955
$261$	0	0
$262$	2262.45	0.533490
$263$	6483.45	1.52010	0.760051	−	0.649864i	$-0.225174\pi$
$263$	6483.45	1.52010	0.760051	+	0.649864i	$0.225174\pi$
$264$	0	0
$265$	2550.40	0.591207
$266$	−2973.90	−0.685494
$267$	0	0
$268$	1228.31	0.279967
$269$	3683.18	0.834823	0.417412	−	0.908717i	$-0.362937\pi$
$269$	3683.18	0.834823	0.417412	+	0.908717i	$0.362937\pi$
$270$	0	0
$271$	3380.09	0.757659	0.378829	−	0.925467i	$-0.376327\pi$
$271$	3380.09	0.757659	0.378829	+	0.925467i	$0.376327\pi$
$272$	1172.65	0.261406
$273$	0	0
$274$	1152.97	0.254210
$275$	0	0
$276$	0	0
$277$	8522.34	1.84858	0.924292	−	0.381687i	$-0.124657\pi$
$277$	8522.34	1.84858	0.924292	+	0.381687i	$0.124657\pi$
$278$	1954.42	0.421649
$279$	0	0
$280$	1471.01	0.313963
$281$	2776.04	0.589340	0.294670	−	0.955599i	$-0.404790\pi$
$281$	2776.04	0.589340	0.294670	+	0.955599i	$0.404790\pi$
$282$	0	0
$283$	3545.89	0.744810	0.372405	−	0.928070i	$-0.378533\pi$
$283$	3545.89	0.744810	0.372405	+	0.928070i	$0.378533\pi$
$284$	3750.94	0.783723
$285$	0	0
$286$	0	0
$287$	−1532.01	−0.315093
$288$	0	0
$289$	458.524	0.0933286
$290$	3764.49	0.762271
$291$	0	0
$292$	2577.55	0.516574
$293$	−4139.70	−0.825405	−0.412703	−	0.910866i	$-0.635415\pi$
$293$	−4139.70	−0.825405	−0.412703	+	0.910866i	$0.635415\pi$
$294$	0	0
$295$	1068.21	0.210825
$296$	2717.71	0.533661
$297$	0	0
$298$	5639.35	1.09624
$299$	−15910.0	−3.07725
$300$	0	0
$301$	−7623.73	−1.45988
$302$	−2948.00	−0.561716
$303$	0	0
$304$	1495.36	0.282121
$305$	−2805.48	−0.526693
$306$	0	0
$307$	3752.14	0.697545	0.348772	−	0.937208i	$-0.386599\pi$
$307$	3752.14	0.697545	0.348772	+	0.937208i	$0.386599\pi$
$308$	0	0
$309$	0	0
$310$	−3559.39	−0.652128
$311$	1705.92	0.311041	0.155521	−	0.987833i	$-0.450295\pi$
$311$	1705.92	0.311041	0.155521	+	0.987833i	$0.450295\pi$
$312$	0	0
$313$	−3555.97	−0.642157	−0.321078	−	0.947053i	$-0.604045\pi$
$313$	−3555.97	−0.642157	−0.321078	+	0.947053i	$0.604045\pi$
$314$	−5883.35	−1.05738
$315$	0	0
$316$	−2200.13	−0.391667
$317$	10299.5	1.82485	0.912423	−	0.409249i	$-0.134209\pi$
$317$	10299.5	1.82485	0.912423	+	0.409249i	$0.134209\pi$
$318$	0	0
$319$	0	0
$320$	−739.664	−0.129214
$321$	0	0
$322$	−6009.39	−1.04003
$323$	6849.75	1.17997
$324$	0	0
$325$	−721.984	−0.123226
$326$	2407.33	0.408987
$327$	0	0
$328$	770.338	0.129679
$329$	4720.45	0.791023
$330$	0	0
$331$	−6577.14	−1.09218	−0.546091	−	0.837726i	$-0.683885\pi$
$331$	−6577.14	−1.09218	−0.546091	+	0.837726i	$0.683885\pi$
$332$	−1366.28	−0.225857
$333$	0	0
$334$	7796.32	1.27723
$335$	−3548.98	−0.578811
$336$	0	0
$337$	657.062	0.106209	0.0531045	−	0.998589i	$-0.483088\pi$
$337$	657.062	0.106209	0.0531045	+	0.998589i	$0.483088\pi$
$338$	−9800.16	−1.57710
$339$	0	0
$340$	−3388.16	−0.540437
$341$	0	0
$342$	0	0
$343$	−6886.99	−1.08415
$344$	3833.42	0.600827
$345$	0	0
$346$	6270.14	0.974234
$347$	10937.5	1.69209	0.846044	−	0.533113i	$-0.178978\pi$
$347$	10937.5	1.69209	0.846044	+	0.533113i	$0.178978\pi$
$348$	0	0
$349$	5370.78	0.823758	0.411879	−	0.911239i	$-0.364873\pi$
$349$	5370.78	0.823758	0.411879	+	0.911239i	$0.364873\pi$
$350$	−272.702	−0.0416473
$351$	0	0
$352$	0	0
$353$	6918.39	1.04314	0.521571	−	0.853208i	$-0.325346\pi$
$353$	6918.39	1.04314	0.521571	+	0.853208i	$0.325346\pi$
$354$	0	0
$355$	−10837.6	−1.62029
$356$	274.281	0.0408338
$357$	0	0
$358$	464.178	0.0685268
$359$	−1386.23	−0.203795	−0.101897	−	0.994795i	$-0.532491\pi$
$359$	−1386.23	−0.203795	−0.101897	+	0.994795i	$0.532491\pi$
$360$	0	0
$361$	1875.78	0.273477
$362$	−5398.34	−0.783786
$363$	0	0
$364$	−5361.30	−0.772002
$365$	−7447.35	−1.06798
$366$	0	0
$367$	−3773.44	−0.536709	−0.268354	−	0.963320i	$-0.586480\pi$
$367$	−3773.44	−0.536709	−0.268354	+	0.963320i	$0.586480\pi$
$368$	3021.69	0.428034
$369$	0	0
$370$	−7852.31	−1.10330
$371$	−3510.94	−0.491319
$372$	0	0
$373$	1267.21	0.175908	0.0879541	−	0.996125i	$-0.471967\pi$
$373$	1267.21	0.175908	0.0879541	+	0.996125i	$0.471967\pi$
$374$	0	0
$375$	0	0
$376$	−2373.57	−0.325552
$377$	−13720.2	−1.87435
$378$	0	0
$379$	12827.1	1.73849	0.869243	−	0.494386i	$-0.164607\pi$
$379$	12827.1	1.73849	0.869243	+	0.494386i	$0.164607\pi$
$380$	−4320.57	−0.583264
$381$	0	0
$382$	−3977.22	−0.532702
$383$	1795.77	0.239582	0.119791	−	0.992799i	$-0.461778\pi$
$383$	1795.77	0.239582	0.119791	+	0.992799i	$0.461778\pi$
$384$	0	0
$385$	0	0
$386$	2387.92	0.314875
$387$	0	0
$388$	6186.96	0.809524
$389$	5052.14	0.658493	0.329246	−	0.944244i	$-0.393205\pi$
$389$	5052.14	0.658493	0.329246	+	0.944244i	$0.393205\pi$
$390$	0	0
$391$	13841.3	1.79025
$392$	718.972	0.0926367
$393$	0	0
$394$	−9617.89	−1.22980
$395$	6356.85	0.809742
$396$	0	0
$397$	−9769.86	−1.23510	−0.617550	−	0.786531i	$-0.711875\pi$
$397$	−9769.86	−1.23510	−0.617550	+	0.786531i	$0.711875\pi$
$398$	−5363.39	−0.675483
$399$	0	0
$400$	137.122	0.0171403
$401$	14793.8	1.84232	0.921158	−	0.389188i	$-0.127244\pi$
$401$	14793.8	1.84232	0.921158	+	0.389188i	$0.127244\pi$
$402$	0	0
$403$	12972.7	1.60351
$404$	−4544.35	−0.559628
$405$	0	0
$406$	−5182.30	−0.633481
$407$	0	0
$408$	0	0
$409$	−1893.67	−0.228938	−0.114469	−	0.993427i	$-0.536517\pi$
$409$	−1893.67	−0.228938	−0.114469	+	0.993427i	$0.536517\pi$
$410$	−2225.75	−0.268102
$411$	0	0
$412$	−959.154	−0.114694
$413$	−1470.52	−0.175205
$414$	0	0
$415$	3947.62	0.466942
$416$	2695.81	0.317724
$417$	0	0
$418$	0	0
$419$	1889.23	0.220274	0.110137	−	0.993916i	$-0.464871\pi$
$419$	1889.23	0.220274	0.110137	+	0.993916i	$0.464871\pi$
$420$	0	0
$421$	6209.48	0.718840	0.359420	−	0.933176i	$-0.382975\pi$
$421$	6209.48	0.718840	0.359420	+	0.933176i	$0.382975\pi$
$422$	5123.59	0.591025
$423$	0	0
$424$	1765.40	0.202206
$425$	628.111	0.0716891
$426$	0	0
$427$	3862.10	0.437705
$428$	−729.929	−0.0824356
$429$	0	0
$430$	−11076.0	−1.24216
$431$	1260.07	0.140825	0.0704126	−	0.997518i	$-0.477568\pi$
$431$	1260.07	0.140825	0.0704126	+	0.997518i	$0.477568\pi$
$432$	0	0
$433$	−9455.77	−1.04946	−0.524729	−	0.851269i	$-0.675833\pi$
$433$	−9455.77	−1.04946	−0.524729	+	0.851269i	$0.675833\pi$
$434$	4899.94	0.541946
$435$	0	0
$436$	−2847.08	−0.312730
$437$	17650.4	1.93212
$438$	0	0
$439$	5992.08	0.651449	0.325725	−	0.945465i	$-0.394392\pi$
$439$	5992.08	0.651449	0.325725	+	0.945465i	$0.394392\pi$
$440$	0	0
$441$	0	0
$442$	12348.6	1.32888
$443$	−11457.2	−1.22877	−0.614386	−	0.789006i	$-0.710596\pi$
$443$	−11457.2	−1.22877	−0.614386	+	0.789006i	$0.710596\pi$
$444$	0	0
$445$	−792.483	−0.0844209
$446$	1900.30	0.201753
$447$	0	0
$448$	1018.24	0.107382
$449$	9615.18	1.01062	0.505310	−	0.862938i	$-0.331378\pi$
$449$	9615.18	1.01062	0.505310	+	0.862938i	$0.331378\pi$
$450$	0	0
$451$	0	0
$452$	−1408.96	−0.146619
$453$	0	0
$454$	−1655.04	−0.171090
$455$	15490.5	1.59606
$456$	0	0
$457$	−6925.66	−0.708903	−0.354452	−	0.935074i	$-0.615332\pi$
$457$	−6925.66	−0.708903	−0.354452	+	0.935074i	$0.615332\pi$
$458$	−1084.75	−0.110670
$459$	0	0
$460$	−8730.61	−0.884927
$461$	4385.77	0.443092	0.221546	−	0.975150i	$-0.428890\pi$
$461$	4385.77	0.443092	0.221546	+	0.975150i	$0.428890\pi$
$462$	0	0
$463$	644.748	0.0647171	0.0323585	−	0.999476i	$-0.489698\pi$
$463$	644.748	0.0647171	0.0323585	+	0.999476i	$0.489698\pi$
$464$	2605.80	0.260714
$465$	0	0
$466$	−10397.4	−1.03359
$467$	12724.6	1.26087	0.630434	−	0.776243i	$-0.282877\pi$
$467$	12724.6	1.26087	0.630434	+	0.776243i	$0.282877\pi$
$468$	0	0
$469$	4885.62	0.481017
$470$	6858.00	0.673055
$471$	0	0
$472$	739.420	0.0721071
$473$	0	0
$474$	0	0
$475$	800.966	0.0773702
$476$	4664.22	0.449127
$477$	0	0
$478$	4962.51	0.474854
$479$	−8809.29	−0.840306	−0.420153	−	0.907453i	$-0.638024\pi$
$479$	−8809.29	−0.840306	−0.420153	+	0.907453i	$0.638024\pi$
$480$	0	0
$481$	28618.9	2.71291
$482$	100.977	0.00954232
$483$	0	0
$484$	0	0
$485$	−17876.1	−1.67363
$486$	0	0
$487$	14185.3	1.31991	0.659957	−	0.751303i	$-0.270574\pi$
$487$	14185.3	1.31991	0.659957	+	0.751303i	$0.270574\pi$
$488$	−1941.97	−0.180141
$489$	0	0
$490$	−2077.34	−0.191519
$491$	12321.1	1.13247	0.566236	−	0.824243i	$-0.308399\pi$
$491$	12321.1	1.13247	0.566236	+	0.824243i	$0.308399\pi$
$492$	0	0
$493$	11936.3	1.09044
$494$	15746.9	1.43419
$495$	0	0
$496$	−2463.83	−0.223043
$497$	14919.4	1.34653
$498$	0	0
$499$	−2399.23	−0.215239	−0.107620	−	0.994192i	$-0.534323\pi$
$499$	−2399.23	−0.215239	−0.107620	+	0.994192i	$0.534323\pi$
$500$	5382.44	0.481420
$501$	0	0
$502$	−2222.24	−0.197577
$503$	−5807.58	−0.514805	−0.257403	−	0.966304i	$-0.582867\pi$
$503$	−5807.58	−0.514805	−0.257403	+	0.966304i	$0.582867\pi$
$504$	0	0
$505$	13130.0	1.15699
$506$	0	0
$507$	0	0
$508$	4241.09	0.370410
$509$	14430.6	1.25663	0.628316	−	0.777958i	$-0.283744\pi$
$509$	14430.6	1.25663	0.628316	+	0.777958i	$0.283744\pi$
$510$	0	0
$511$	10252.2	0.887536
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−14871.2	−1.27615
$515$	2771.30	0.237122
$516$	0	0
$517$	0	0
$518$	10809.7	0.916893
$519$	0	0
$520$	−7789.05	−0.656870
$521$	−2176.19	−0.182995	−0.0914975	−	0.995805i	$-0.529165\pi$
$521$	−2176.19	−0.182995	−0.0914975	+	0.995805i	$0.529165\pi$
$522$	0	0
$523$	−4330.18	−0.362038	−0.181019	−	0.983480i	$-0.557939\pi$
$523$	−4330.18	−0.362038	−0.181019	+	0.983480i	$0.557939\pi$
$524$	−4524.89	−0.377235
$525$	0	0
$526$	−12966.9	−1.07487
$527$	−11286.0	−0.932875
$528$	0	0
$529$	23499.4	1.93141
$530$	−5100.80	−0.418046
$531$	0	0
$532$	5947.80	0.484718
$533$	8112.06	0.659235
$534$	0	0
$535$	2108.99	0.170429
$536$	−2456.63	−0.197967
$537$	0	0
$538$	−7366.36	−0.590309
$539$	0	0
$540$	0	0
$541$	9665.06	0.768084	0.384042	−	0.923316i	$-0.374532\pi$
$541$	9665.06	0.768084	0.384042	+	0.923316i	$0.374532\pi$
$542$	−6760.17	−0.535746
$543$	0	0
$544$	−2345.30	−0.184842
$545$	8226.09	0.646545
$546$	0	0
$547$	−5417.54	−0.423468	−0.211734	−	0.977327i	$-0.567911\pi$
$547$	−5417.54	−0.423468	−0.211734	+	0.977327i	$0.567911\pi$
$548$	−2305.94	−0.179754
$549$	0	0
$550$	0	0
$551$	15221.2	1.17685
$552$	0	0
$553$	−8751.01	−0.672931
$554$	−17044.7	−1.30715
$555$	0	0
$556$	−3908.85	−0.298151
$557$	−7475.43	−0.568661	−0.284330	−	0.958726i	$-0.591771\pi$
$557$	−7475.43	−0.568661	−0.284330	+	0.958726i	$0.591771\pi$
$558$	0	0
$559$	40367.9	3.05435
$560$	−2942.02	−0.222005
$561$	0	0
$562$	−5552.07	−0.416726
$563$	−9189.09	−0.687876	−0.343938	−	0.938992i	$-0.611761\pi$
$563$	−9189.09	−0.687876	−0.343938	+	0.938992i	$0.611761\pi$
$564$	0	0
$565$	4070.92	0.303123
$566$	−7091.78	−0.526661
$567$	0	0
$568$	−7501.87	−0.554175
$569$	−12451.3	−0.917374	−0.458687	−	0.888598i	$-0.651680\pi$
$569$	−12451.3	−0.917374	−0.458687	+	0.888598i	$0.651680\pi$
$570$	0	0
$571$	20045.5	1.46914	0.734570	−	0.678532i	$-0.237384\pi$
$571$	20045.5	1.46914	0.734570	+	0.678532i	$0.237384\pi$
$572$	0	0
$573$	0	0
$574$	3064.02	0.222804
$575$	1618.52	0.117386
$576$	0	0
$577$	−9292.91	−0.670483	−0.335242	−	0.942132i	$-0.608818\pi$
$577$	−9292.91	−0.670483	−0.335242	+	0.942132i	$0.608818\pi$
$578$	−917.047	−0.0659933
$579$	0	0
$580$	−7528.99	−0.539007
$581$	−5434.40	−0.388049
$582$	0	0
$583$	0	0
$584$	−5155.10	−0.365273
$585$	0	0
$586$	8279.40	0.583650
$587$	17675.2	1.24281	0.621407	−	0.783488i	$-0.286561\pi$
$587$	17675.2	1.24281	0.621407	+	0.783488i	$0.286561\pi$
$588$	0	0
$589$	−14391.8	−1.00680
$590$	−2136.42	−0.149076
$591$	0	0
$592$	−5435.42	−0.377355
$593$	11433.4	0.791760	0.395880	−	0.918302i	$-0.370440\pi$
$593$	11433.4	0.791760	0.395880	+	0.918302i	$0.370440\pi$
$594$	0	0
$595$	−13476.4	−0.928535
$596$	−11278.7	−0.775158
$597$	0	0
$598$	31820.0	2.17594
$599$	−25721.1	−1.75449	−0.877243	−	0.480047i	$-0.840620\pi$
$599$	−25721.1	−1.75449	−0.877243	+	0.480047i	$0.840620\pi$
$600$	0	0
$601$	−26821.1	−1.82039	−0.910197	−	0.414175i	$-0.864070\pi$
$601$	−26821.1	−1.82039	−0.910197	+	0.414175i	$0.864070\pi$
$602$	15247.5	1.03229
$603$	0	0
$604$	5896.00	0.397193
$605$	0	0
$606$	0	0
$607$	25766.3	1.72294	0.861469	−	0.507809i	$-0.169545\pi$
$607$	25766.3	1.72294	0.861469	+	0.507809i	$0.169545\pi$
$608$	−2990.72	−0.199490
$609$	0	0
$610$	5610.97	0.372428
$611$	−24995.0	−1.65497
$612$	0	0
$613$	−4463.49	−0.294092	−0.147046	−	0.989130i	$-0.546977\pi$
$613$	−4463.49	−0.294092	−0.147046	+	0.989130i	$0.546977\pi$
$614$	−7504.29	−0.493238
$615$	0	0
$616$	0	0
$617$	3826.92	0.249702	0.124851	−	0.992176i	$-0.460155\pi$
$617$	3826.92	0.249702	0.124851	+	0.992176i	$0.460155\pi$
$618$	0	0
$619$	16747.3	1.08745	0.543724	−	0.839264i	$-0.317014\pi$
$619$	16747.3	1.08745	0.543724	+	0.839264i	$0.317014\pi$
$620$	7118.77	0.461124
$621$	0	0
$622$	−3411.84	−0.219939
$623$	1090.95	0.0701574
$624$	0	0
$625$	−16622.8	−1.06386
$626$	7111.93	0.454073
$627$	0	0
$628$	11766.7	0.747679
$629$	−24897.8	−1.57829
$630$	0	0
$631$	−20048.7	−1.26486	−0.632430	−	0.774617i	$-0.717943\pi$
$631$	−20048.7	−1.26486	−0.632430	+	0.774617i	$0.717943\pi$
$632$	4400.25	0.276950
$633$	0	0
$634$	−20598.9	−1.29036
$635$	−12253.8	−0.765794
$636$	0	0
$637$	7571.15	0.470926
$638$	0	0
$639$	0	0
$640$	1479.33	0.0913681
$641$	15366.1	0.946837	0.473419	−	0.880838i	$-0.343020\pi$
$641$	15366.1	0.946837	0.473419	+	0.880838i	$0.343020\pi$
$642$	0	0
$643$	−23711.4	−1.45426	−0.727129	−	0.686501i	$-0.759146\pi$
$643$	−23711.4	−1.45426	−0.727129	+	0.686501i	$0.759146\pi$
$644$	12018.8	0.735413
$645$	0	0
$646$	−13699.5	−0.834365
$647$	9714.41	0.590283	0.295141	−	0.955454i	$-0.404633\pi$
$647$	9714.41	0.590283	0.295141	+	0.955454i	$0.404633\pi$
$648$	0	0
$649$	0	0
$650$	1443.97	0.0871340
$651$	0	0
$652$	−4814.66	−0.289198
$653$	−16789.2	−1.00615	−0.503073	−	0.864244i	$-0.667797\pi$
$653$	−16789.2	−1.00615	−0.503073	+	0.864244i	$0.667797\pi$
$654$	0	0
$655$	13073.8	0.779904
$656$	−1540.68	−0.0916970
$657$	0	0
$658$	−9440.90	−0.559338
$659$	−14946.6	−0.883517	−0.441759	−	0.897134i	$-0.645645\pi$
$659$	−14946.6	−0.883517	−0.441759	+	0.897134i	$0.645645\pi$
$660$	0	0
$661$	1035.71	0.0609448	0.0304724	−	0.999536i	$-0.490299\pi$
$661$	1035.71	0.0609448	0.0304724	+	0.999536i	$0.490299\pi$
$662$	13154.3	0.772289
$663$	0	0
$664$	2732.57	0.159705
$665$	−17185.1	−1.00212
$666$	0	0
$667$	30757.5	1.78551
$668$	−15592.6	−0.903139
$669$	0	0
$670$	7097.97	0.409281
$671$	0	0
$672$	0	0
$673$	7573.67	0.433794	0.216897	−	0.976194i	$-0.430406\pi$
$673$	7573.67	0.433794	0.216897	+	0.976194i	$0.430406\pi$
$674$	−1314.12	−0.0751011
$675$	0	0
$676$	19600.3	1.11518
$677$	28720.5	1.63046	0.815229	−	0.579138i	$-0.196611\pi$
$677$	28720.5	1.63046	0.815229	+	0.579138i	$0.196611\pi$
$678$	0	0
$679$	24608.7	1.39086
$680$	6776.31	0.382147
$681$	0	0
$682$	0	0
$683$	−10626.4	−0.595326	−0.297663	−	0.954671i	$-0.596207\pi$
$683$	−10626.4	−0.595326	−0.297663	+	0.954671i	$0.596207\pi$
$684$	0	0
$685$	6662.60	0.371627
$686$	13774.0	0.766608
$687$	0	0
$688$	−7666.85	−0.424849
$689$	18590.6	1.02793
$690$	0	0
$691$	−29261.2	−1.61093	−0.805463	−	0.592646i	$-0.798083\pi$
$691$	−29261.2	−1.61093	−0.805463	+	0.592646i	$0.798083\pi$
$692$	−12540.3	−0.688887
$693$	0	0
$694$	−21875.0	−1.19649
$695$	11293.9	0.616405
$696$	0	0
$697$	−7057.32	−0.383522
$698$	−10741.6	−0.582485
$699$	0	0
$700$	545.404	0.0294491
$701$	21055.3	1.13445	0.567223	−	0.823565i	$-0.308018\pi$
$701$	21055.3	1.13445	0.567223	+	0.823565i	$0.308018\pi$
$702$	0	0
$703$	−31749.6	−1.70336
$704$	0	0
$705$	0	0
$706$	−13836.8	−0.737612
$707$	−18075.2	−0.961508
$708$	0	0
$709$	442.581	0.0234436	0.0117218	−	0.999931i	$-0.496269\pi$
$709$	442.581	0.0234436	0.0117218	+	0.999931i	$0.496269\pi$
$710$	21675.3	1.14572
$711$	0	0
$712$	−548.561	−0.0288739
$713$	−29081.7	−1.52752
$714$	0	0
$715$	0	0
$716$	−928.356	−0.0484557
$717$	0	0
$718$	2772.46	0.144105
$719$	−27955.9	−1.45004	−0.725020	−	0.688727i	$-0.758170\pi$
$719$	−27955.9	−1.45004	−0.725020	+	0.688727i	$0.758170\pi$
$720$	0	0
$721$	−3815.04	−0.197059
$722$	−3751.56	−0.193377
$723$	0	0
$724$	10796.7	0.554220
$725$	1395.76	0.0714995
$726$	0	0
$727$	32275.7	1.64655	0.823274	−	0.567644i	$-0.192145\pi$
$727$	32275.7	1.64655	0.823274	+	0.567644i	$0.192145\pi$
$728$	10722.6	0.545888
$729$	0	0
$730$	14894.7	0.755174
$731$	−35119.3	−1.77693
$732$	0	0
$733$	−2040.36	−0.102813	−0.0514067	−	0.998678i	$-0.516370\pi$
$733$	−2040.36	−0.102813	−0.0514067	+	0.998678i	$0.516370\pi$
$734$	7546.88	0.379510
$735$	0	0
$736$	−6043.38	−0.302666
$737$	0	0
$738$	0	0
$739$	−11118.7	−0.553462	−0.276731	−	0.960947i	$-0.589251\pi$
$739$	−11118.7	−0.553462	−0.276731	+	0.960947i	$0.589251\pi$
$740$	15704.6	0.780153
$741$	0	0
$742$	7021.89	0.347415
$743$	−9426.04	−0.465421	−0.232710	−	0.972546i	$-0.574759\pi$
$743$	−9426.04	−0.465421	−0.232710	+	0.972546i	$0.574759\pi$
$744$	0	0
$745$	32587.7	1.60258
$746$	−2534.42	−0.124386
$747$	0	0
$748$	0	0
$749$	−2903.29	−0.141634
$750$	0	0
$751$	26665.8	1.29567	0.647836	−	0.761779i	$-0.275674\pi$
$751$	26665.8	1.29567	0.647836	+	0.761779i	$0.275674\pi$
$752$	4747.15	0.230200
$753$	0	0
$754$	27440.5	1.32536
$755$	−17035.4	−0.821168
$756$	0	0
$757$	29629.7	1.42260	0.711302	−	0.702886i	$-0.248106\pi$
$757$	29629.7	1.42260	0.711302	+	0.702886i	$0.248106\pi$
$758$	−25654.3	−1.22929
$759$	0	0
$760$	8641.13	0.412430
$761$	28940.1	1.37855	0.689275	−	0.724500i	$-0.257929\pi$
$761$	28940.1	1.37855	0.689275	+	0.724500i	$0.257929\pi$
$762$	0	0
$763$	−11324.2	−0.537307
$764$	7954.43	0.376677
$765$	0	0
$766$	−3591.55	−0.169410
$767$	7786.48	0.366563
$768$	0	0
$769$	−9579.01	−0.449191	−0.224596	−	0.974452i	$-0.572106\pi$
$769$	−9579.01	−0.449191	−0.224596	+	0.974452i	$0.572106\pi$
$770$	0	0
$771$	0	0
$772$	−4775.83	−0.222650
$773$	29405.4	1.36822	0.684112	−	0.729377i	$-0.260190\pi$
$773$	29405.4	1.36822	0.684112	+	0.729377i	$0.260190\pi$
$774$	0	0
$775$	−1319.71	−0.0611682
$776$	−12373.9	−0.572420
$777$	0	0
$778$	−10104.3	−0.465625
$779$	−8999.47	−0.413915
$780$	0	0
$781$	0	0
$782$	−27682.7	−1.26590
$783$	0	0
$784$	−1437.94	−0.0655040
$785$	−33997.7	−1.54577
$786$	0	0
$787$	−18319.1	−0.829738	−0.414869	−	0.909881i	$-0.636173\pi$
$787$	−18319.1	−0.829738	−0.414869	+	0.909881i	$0.636173\pi$
$788$	19235.8	0.869602
$789$	0	0
$790$	−12713.7	−0.572574
$791$	−5604.13	−0.251909
$792$	0	0
$793$	−20450.0	−0.915763
$794$	19539.7	0.873348
$795$	0	0
$796$	10726.8	0.477639
$797$	31532.7	1.40144	0.700719	−	0.713437i	$-0.252862\pi$
$797$	31532.7	1.40144	0.700719	+	0.713437i	$0.252862\pi$
$798$	0	0
$799$	21745.1	0.962812
$800$	−274.244	−0.0121200
$801$	0	0
$802$	−29587.7	−1.30271
$803$	0	0
$804$	0	0
$805$	−34726.0	−1.52041
$806$	−25945.4	−1.13386
$807$	0	0
$808$	9088.70	0.395717
$809$	−38389.6	−1.66837	−0.834183	−	0.551488i	$-0.814060\pi$
$809$	−38389.6	−1.66837	−0.834183	+	0.551488i	$0.814060\pi$
$810$	0	0
$811$	1614.36	0.0698987	0.0349493	−	0.999389i	$-0.488873\pi$
$811$	1614.36	0.0698987	0.0349493	+	0.999389i	$0.488873\pi$
$812$	10364.6	0.447938
$813$	0	0
$814$	0	0
$815$	13911.1	0.597894
$816$	0	0
$817$	−44784.0	−1.91774
$818$	3787.33	0.161884
$819$	0	0
$820$	4451.49	0.189577
$821$	−26620.4	−1.13162	−0.565810	−	0.824536i	$-0.691436\pi$
$821$	−26620.4	−1.13162	−0.565810	+	0.824536i	$0.691436\pi$
$822$	0	0
$823$	6790.44	0.287606	0.143803	−	0.989606i	$-0.454067\pi$
$823$	6790.44	0.287606	0.143803	+	0.989606i	$0.454067\pi$
$824$	1918.31	0.0811012
$825$	0	0
$826$	2941.05	0.123889
$827$	25641.8	1.07818	0.539089	−	0.842249i	$-0.318769\pi$
$827$	25641.8	1.07818	0.539089	+	0.842249i	$0.318769\pi$
$828$	0	0
$829$	−4864.08	−0.203784	−0.101892	−	0.994795i	$-0.532490\pi$
$829$	−4864.08	−0.203784	−0.101892	+	0.994795i	$0.532490\pi$
$830$	−7895.24	−0.330178
$831$	0	0
$832$	−5391.63	−0.224665
$833$	−6586.75	−0.273970
$834$	0	0
$835$	45052.0	1.86717
$836$	0	0
$837$	0	0
$838$	−3778.45	−0.155757
$839$	15915.0	0.654884	0.327442	−	0.944871i	$-0.393813\pi$
$839$	15915.0	0.654884	0.327442	+	0.944871i	$0.393813\pi$
$840$	0	0
$841$	2135.30	0.0875516
$842$	−12419.0	−0.508296
$843$	0	0
$844$	−10247.2	−0.417918
$845$	−56631.5	−2.30554
$846$	0	0
$847$	0	0
$848$	−3530.80	−0.142981
$849$	0	0
$850$	−1256.22	−0.0506919
$851$	−64156.8	−2.58433
$852$	0	0
$853$	38675.2	1.55242	0.776209	−	0.630476i	$-0.217140\pi$
$853$	38675.2	1.55242	0.776209	+	0.630476i	$0.217140\pi$
$854$	−7724.20	−0.309504
$855$	0	0
$856$	1459.86	0.0582908
$857$	19067.8	0.760029	0.380015	−	0.924981i	$-0.375919\pi$
$857$	19067.8	0.760029	0.380015	+	0.924981i	$0.375919\pi$
$858$	0	0
$859$	−6779.06	−0.269265	−0.134632	−	0.990896i	$-0.542985\pi$
$859$	−6779.06	−0.269265	−0.134632	+	0.990896i	$0.542985\pi$
$860$	22151.9	0.878342
$861$	0	0
$862$	−2520.15	−0.0995784
$863$	14744.3	0.581579	0.290789	−	0.956787i	$-0.406082\pi$
$863$	14744.3	0.581579	0.290789	+	0.956787i	$0.406082\pi$
$864$	0	0
$865$	36232.8	1.42422
$866$	18911.5	0.742079
$867$	0	0
$868$	−9799.88	−0.383214
$869$	0	0
$870$	0	0
$871$	−25869.6	−1.00638
$872$	5694.15	0.221133
$873$	0	0
$874$	−35300.9	−1.36621
$875$	21408.7	0.827137
$876$	0	0
$877$	−18934.6	−0.729048	−0.364524	−	0.931194i	$-0.618768\pi$
$877$	−18934.6	−0.729048	−0.364524	+	0.931194i	$0.618768\pi$
$878$	−11984.2	−0.460644
$879$	0	0
$880$	0	0
$881$	6518.05	0.249261	0.124630	−	0.992203i	$-0.460225\pi$
$881$	6518.05	0.249261	0.124630	+	0.992203i	$0.460225\pi$
$882$	0	0
$883$	−14983.5	−0.571047	−0.285524	−	0.958372i	$-0.592168\pi$
$883$	−14983.5	−0.571047	−0.285524	+	0.958372i	$0.592168\pi$
$884$	−24697.3	−0.939659
$885$	0	0
$886$	22914.3	0.868873
$887$	7170.17	0.271422	0.135711	−	0.990748i	$-0.456668\pi$
$887$	7170.17	0.271422	0.135711	+	0.990748i	$0.456668\pi$
$888$	0	0
$889$	16869.0	0.636408
$890$	1584.97	0.0596946
$891$	0	0
$892$	−3800.59	−0.142661
$893$	27729.3	1.03911
$894$	0	0
$895$	2682.31	0.100179
$896$	−2036.48	−0.0759309
$897$	0	0
$898$	−19230.4	−0.714616
$899$	−25079.1	−0.930407
$900$	0	0
$901$	−16173.4	−0.598019
$902$	0	0
$903$	0	0
$904$	2817.91	0.103675
$905$	−31195.0	−1.14581
$906$	0	0
$907$	−16666.7	−0.610152	−0.305076	−	0.952328i	$-0.598682\pi$
$907$	−16666.7	−0.610152	−0.305076	+	0.952328i	$0.598682\pi$
$908$	3310.09	0.120979
$909$	0	0
$910$	−30981.0	−1.12858
$911$	−991.707	−0.0360667	−0.0180333	−	0.999837i	$-0.505740\pi$
$911$	−991.707	−0.0360667	−0.0180333	+	0.999837i	$0.505740\pi$
$912$	0	0
$913$	0	0
$914$	13851.3	0.501270
$915$	0	0
$916$	2169.50	0.0782558
$917$	−17997.8	−0.648134
$918$	0	0
$919$	−12832.7	−0.460621	−0.230310	−	0.973117i	$-0.573974\pi$
$919$	−12832.7	−0.460621	−0.230310	+	0.973117i	$0.573974\pi$
$920$	17461.2	0.625738
$921$	0	0
$922$	−8771.54	−0.313314
$923$	−78998.7	−2.81720
$924$	0	0
$925$	−2911.39	−0.103488
$926$	−1289.50	−0.0457619
$927$	0	0
$928$	−5211.61	−0.184353
$929$	−11925.8	−0.421178	−0.210589	−	0.977575i	$-0.567538\pi$
$929$	−11925.8	−0.421178	−0.210589	+	0.977575i	$0.567538\pi$
$930$	0	0
$931$	−8399.40	−0.295681
$932$	20794.9	0.730858
$933$	0	0
$934$	−25449.2	−0.891568
$935$	0	0
$936$	0	0
$937$	−31308.0	−1.09156	−0.545779	−	0.837929i	$-0.683766\pi$
$937$	−31308.0	−1.09156	−0.545779	+	0.837929i	$0.683766\pi$
$938$	−9771.24	−0.340131
$939$	0	0
$940$	−13716.0	−0.475922
$941$	−40.8796	−0.00141619	−0.000708096	−	1.00000i	$-0.500225\pi$
$941$	−40.8796	−0.00141619	−0.000708096		1.00000i	$0.500225\pi$
$942$	0	0
$943$	−18185.3	−0.627991
$944$	−1478.84	−0.0509874
$945$	0	0
$946$	0	0
$947$	8016.53	0.275081	0.137541	−	0.990496i	$-0.456080\pi$
$947$	8016.53	0.275081	0.137541	+	0.990496i	$0.456080\pi$
$948$	0	0
$949$	−54285.9	−1.85690
$950$	−1601.93	−0.0547090
$951$	0	0
$952$	−9328.44	−0.317580
$953$	−25916.2	−0.880909	−0.440455	−	0.897775i	$-0.645183\pi$
$953$	−25916.2	−0.880909	−0.440455	+	0.897775i	$0.645183\pi$
$954$	0	0
$955$	−22982.9	−0.778751
$956$	−9925.02	−0.335772
$957$	0	0
$958$	17618.6	0.594186
$959$	−9171.90	−0.308838
$960$	0	0
$961$	−6078.31	−0.204032
$962$	−57237.8	−1.91832
$963$	0	0
$964$	−201.955	−0.00674744
$965$	13798.9	0.460313
$966$	0	0
$967$	−4975.56	−0.165463	−0.0827317	−	0.996572i	$-0.526364\pi$
$967$	−4975.56	−0.165463	−0.0827317	+	0.996572i	$0.526364\pi$
$968$	0	0
$969$	0	0
$970$	35752.1	1.18344
$971$	−2916.76	−0.0963990	−0.0481995	−	0.998838i	$-0.515348\pi$
$971$	−2916.76	−0.0963990	−0.0481995	+	0.998838i	$0.515348\pi$
$972$	0	0
$973$	−15547.4	−0.512259
$974$	−28370.6	−0.933320
$975$	0	0
$976$	3883.94	0.127379
$977$	25480.7	0.834389	0.417195	−	0.908817i	$-0.363013\pi$
$977$	25480.7	0.834389	0.417195	+	0.908817i	$0.363013\pi$
$978$	0	0
$979$	0	0
$980$	4154.67	0.135425
$981$	0	0
$982$	−24642.2	−0.800778
$983$	25588.9	0.830275	0.415137	−	0.909759i	$-0.363734\pi$
$983$	25588.9	0.830275	0.415137	+	0.909759i	$0.363734\pi$
$984$	0	0
$985$	−55578.2	−1.79784
$986$	−23872.7	−0.771055
$987$	0	0
$988$	−31493.9	−1.01412
$989$	−90495.4	−2.90959
$990$	0	0
$991$	48863.5	1.56630	0.783148	−	0.621836i	$-0.213613\pi$
$991$	48863.5	1.56630	0.783148	+	0.621836i	$0.213613\pi$
$992$	4927.66	0.157715
$993$	0	0
$994$	−29838.7	−0.952140
$995$	−30993.0	−0.987482
$996$	0	0
$997$	56868.9	1.80648	0.903238	−	0.429141i	$-0.141184\pi$
$997$	56868.9	1.80648	0.903238	+	0.429141i	$0.141184\pi$
$998$	4798.47	0.152197
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.ce.1.1		6
3.2	odd	2		2178.4.a.cf.1.6		6
11.3	even	5		198.4.f.h.163.1	yes	12
11.4	even	5		198.4.f.h.181.1	yes	12
11.10	odd	2		2178.4.a.cg.1.1		6
33.14	odd	10		198.4.f.g.163.3	✓	12
33.26	odd	10		198.4.f.g.181.3	yes	12
33.32	even	2		2178.4.a.cd.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.4.f.g.163.3	✓	12	33.14	odd	10
198.4.f.g.181.3	yes	12	33.26	odd	10
198.4.f.h.163.1	yes	12	11.3	even	5
198.4.f.h.181.1	yes	12	11.4	even	5
2178.4.a.cd.1.6		6	33.32	even	2
2178.4.a.ce.1.1		6	1.1	even	1	trivial
2178.4.a.cf.1.6		6	3.2	odd	2
2178.4.a.cg.1.1		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -11.5573 q^{5} +15.9100 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -11.5573 q^{5} +15.9100 q^{7} -8.00000 q^{8} +23.1145 q^{10} -84.2442 q^{13} -31.8200 q^{14} +16.0000 q^{16} +73.2907 q^{17} +93.4600 q^{19} -46.2290 q^{20} +188.856 q^{23} +8.57014 q^{25} +168.488 q^{26} +63.6400 q^{28} +162.863 q^{29} -153.989 q^{31} -32.0000 q^{32} -146.581 q^{34} -183.876 q^{35} -339.714 q^{37} -186.920 q^{38} +92.4580 q^{40} -96.2922 q^{41} -479.178 q^{43} -377.711 q^{46} +296.697 q^{47} -89.8715 q^{49} -17.1403 q^{50} -336.977 q^{52} -220.675 q^{53} -127.280 q^{56} -325.726 q^{58} -92.4275 q^{59} +242.746 q^{61} +307.979 q^{62} +64.0000 q^{64} +973.631 q^{65} +307.078 q^{67} +293.163 q^{68} +367.752 q^{70} +937.734 q^{71} +644.387 q^{73} +679.427 q^{74} +373.840 q^{76} -550.032 q^{79} -184.916 q^{80} +192.584 q^{82} -341.571 q^{83} -847.039 q^{85} +958.356 q^{86} +68.5702 q^{89} -1340.33 q^{91} +755.422 q^{92} -593.393 q^{94} -1080.14 q^{95} +1546.74 q^{97} +179.743 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 24 q^{4} + 17 q^{5} - 7 q^{7} - 48 q^{8}+O(q^{10}) \) 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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