LMFDB - Embedded newform 4761.2.a.bn.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4761,2,Mod(1,4761)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4761.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4761, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,0,4,-7,0,8,9,0,5,-13,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.284630$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.236479 q^{2} -1.94408 q^{4} +1.51334 q^{5} +2.54620 q^{7} +0.932691 q^{8} -0.357872 q^{10} -3.37279 q^{11} +3.29177 q^{13} -0.602123 q^{14} +3.66759 q^{16} -6.27686 q^{17} +4.23092 q^{19} -2.94204 q^{20} +0.797593 q^{22} -2.70981 q^{25} -0.778434 q^{26} -4.95001 q^{28} -6.28621 q^{29} -4.74204 q^{31} -2.73269 q^{32} +1.48434 q^{34} +3.85326 q^{35} -0.0379723 q^{37} -1.00052 q^{38} +1.41148 q^{40} -3.24666 q^{41} -1.28887 q^{43} +6.55696 q^{44} -3.41741 q^{47} -0.516864 q^{49} +0.640813 q^{50} -6.39946 q^{52} -6.72578 q^{53} -5.10416 q^{55} +2.37482 q^{56} +1.48656 q^{58} +10.4186 q^{59} -3.59980 q^{61} +1.12139 q^{62} -6.68896 q^{64} +4.98156 q^{65} +6.81939 q^{67} +12.2027 q^{68} -0.911214 q^{70} -4.08974 q^{71} +4.36149 q^{73} +0.00897966 q^{74} -8.22524 q^{76} -8.58779 q^{77} -2.40198 q^{79} +5.55031 q^{80} +0.767766 q^{82} +2.95715 q^{83} -9.49900 q^{85} +0.304792 q^{86} -3.14577 q^{88} +12.3607 q^{89} +8.38151 q^{91} +0.808145 q^{94} +6.40281 q^{95} +10.6165 q^{97} +0.122227 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} - 7 q^{5} + 8 q^{7} + 9 q^{8} + 5 q^{10} - 13 q^{11} + 4 q^{13} - 12 q^{14} + 6 q^{16} - 16 q^{17} + 10 q^{19} - 10 q^{20} + 3 q^{22} - 2 q^{25} - 6 q^{26} - 9 q^{28} - 7 q^{29}+ \cdots - 3 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 - 7 * q^5 + 8 * q^7 + 9 * q^8 + 5 * q^10 - 13 * q^11 + 4 * q^13 - 12 * q^14 + 6 * q^16 - 16 * q^17 + 10 * q^19 - 10 * q^20 + 3 * q^22 - 2 * q^25 - 6 * q^26 - 9 * q^28 - 7 * q^29 + 5 * q^31 + 6 * q^32 - 9 * q^34 - 9 * q^35 - 7 * q^37 - 4 * q^38 - 17 * q^40 - 9 * q^41 - 17 * q^44 + 9 * q^47 - 9 * q^49 - 8 * q^50 + q^52 - 2 * q^53 + 16 * q^55 - q^56 - 6 * q^58 + 16 * q^59 + 4 * q^61 - 2 * q^62 - 21 * q^64 + q^65 - 17 * q^67 - 15 * q^68 + 19 * q^70 + 7 * q^71 - 18 * q^73 + 5 * q^74 + 30 * q^76 - 12 * q^77 + 13 * q^79 + 7 * q^80 - 3 * q^82 - 13 * q^83 + 7 * q^85 + 11 * q^86 - 30 * q^88 - 15 * q^89 + 2 * q^91 - 19 * q^94 + 8 * q^95 - 16 * q^97 - 3 * q^98

Coefficient data

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

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

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

$2$	−0.236479	−0.167216	−0.0836079	−	0.996499i	$-0.526644\pi$
$2$	−0.236479	−0.167216	−0.0836079	+	0.996499i	$0.526644\pi$
$3$	0	0
$3$	0	0
$4$	−1.94408	−0.972039
$5$	1.51334	0.676785	0.338392	−	0.941005i	$-0.390117\pi$
$5$	1.51334	0.676785	0.338392	+	0.941005i	$0.390117\pi$
$6$	0	0
$7$	2.54620	0.962373	0.481187	−	0.876618i	$-0.340206\pi$
$7$	2.54620	0.962373	0.481187	+	0.876618i	$0.340206\pi$
$8$	0.932691	0.329756
$9$	0	0
$10$	−0.357872	−0.113169
$11$	−3.37279	−1.01693	−0.508467	−	0.861082i	$-0.669787\pi$
$11$	−3.37279	−1.01693	−0.508467	+	0.861082i	$0.669787\pi$
$12$	0	0
$13$	3.29177	0.912973	0.456487	−	0.889730i	$-0.349108\pi$
$13$	3.29177	0.912973	0.456487	+	0.889730i	$0.349108\pi$
$14$	−0.602123	−0.160924
$15$	0	0
$16$	3.66759	0.916898
$17$	−6.27686	−1.52236	−0.761181	−	0.648540i	$-0.775380\pi$
$17$	−6.27686	−1.52236	−0.761181	+	0.648540i	$0.775380\pi$
$18$	0	0
$19$	4.23092	0.970639	0.485320	−	0.874337i	$-0.338703\pi$
$19$	4.23092	0.970639	0.485320	+	0.874337i	$0.338703\pi$
$20$	−2.94204	−0.657861
$21$	0	0
$22$	0.797593	0.170047
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.70981	−0.541962
$26$	−0.778434	−0.152664
$27$	0	0
$28$	−4.95001	−0.935464
$29$	−6.28621	−1.16732	−0.583660	−	0.811998i	$-0.698380\pi$
$29$	−6.28621	−1.16732	−0.583660	+	0.811998i	$0.698380\pi$
$30$	0	0
$31$	−4.74204	−0.851696	−0.425848	−	0.904795i	$-0.640024\pi$
$31$	−4.74204	−0.851696	−0.425848	+	0.904795i	$0.640024\pi$
$32$	−2.73269	−0.483076
$33$	0	0
$34$	1.48434	0.254563
$35$	3.85326	0.651320
$36$	0	0
$37$	−0.0379723	−0.00624261	−0.00312131	−	0.999995i	$-0.500994\pi$
$37$	−0.0379723	−0.00624261	−0.00312131	+	0.999995i	$0.500994\pi$
$38$	−1.00052	−0.162306
$39$	0	0
$40$	1.41148	0.223174
$41$	−3.24666	−0.507043	−0.253521	−	0.967330i	$-0.581589\pi$
$41$	−3.24666	−0.507043	−0.253521	+	0.967330i	$0.581589\pi$
$42$	0	0
$43$	−1.28887	−0.196552	−0.0982758	−	0.995159i	$-0.531333\pi$
$43$	−1.28887	−0.196552	−0.0982758	+	0.995159i	$0.531333\pi$
$44$	6.55696	0.988499
$45$	0	0
$46$	0	0
$47$	−3.41741	−0.498480	−0.249240	−	0.968442i	$-0.580181\pi$
$47$	−3.41741	−0.498480	−0.249240	+	0.968442i	$0.580181\pi$
$48$	0	0
$49$	−0.516864	−0.0738377
$50$	0.640813	0.0906246
$51$	0	0
$52$	−6.39946	−0.887445
$53$	−6.72578	−0.923857	−0.461929	−	0.886917i	$-0.652842\pi$
$53$	−6.72578	−0.923857	−0.461929	+	0.886917i	$0.652842\pi$
$54$	0	0
$55$	−5.10416	−0.688245
$56$	2.37482	0.317348
$57$	0	0
$58$	1.48656	0.195194
$59$	10.4186	1.35639	0.678194	−	0.734883i	$-0.262763\pi$
$59$	10.4186	1.35639	0.678194	+	0.734883i	$0.262763\pi$
$60$	0	0
$61$	−3.59980	−0.460908	−0.230454	−	0.973083i	$-0.574021\pi$
$61$	−3.59980	−0.460908	−0.230454	+	0.973083i	$0.574021\pi$
$62$	1.12139	0.142417
$63$	0	0
$64$	−6.68896	−0.836120
$65$	4.98156	0.617886
$66$	0	0
$67$	6.81939	0.833121	0.416561	−	0.909108i	$-0.363235\pi$
$67$	6.81939	0.833121	0.416561	+	0.909108i	$0.363235\pi$
$68$	12.2027	1.47979
$69$	0	0
$70$	−0.911214	−0.108911
$71$	−4.08974	−0.485363	−0.242681	−	0.970106i	$-0.578027\pi$
$71$	−4.08974	−0.485363	−0.242681	+	0.970106i	$0.578027\pi$
$72$	0	0
$73$	4.36149	0.510473	0.255237	−	0.966879i	$-0.417847\pi$
$73$	4.36149	0.510473	0.255237	+	0.966879i	$0.417847\pi$
$74$	0.00897966	0.00104386
$75$	0	0
$76$	−8.22524	−0.943499
$77$	−8.58779	−0.978669
$78$	0	0
$79$	−2.40198	−0.270244	−0.135122	−	0.990829i	$-0.543143\pi$
$79$	−2.40198	−0.270244	−0.135122	+	0.990829i	$0.543143\pi$
$80$	5.55031	0.620543
$81$	0	0
$82$	0.767766	0.0847856
$83$	2.95715	0.324590	0.162295	−	0.986742i	$-0.448110\pi$
$83$	2.95715	0.324590	0.162295	+	0.986742i	$0.448110\pi$
$84$	0	0
$85$	−9.49900	−1.03031
$86$	0.304792	0.0328665
$87$	0	0
$88$	−3.14577	−0.335340
$89$	12.3607	1.31023	0.655117	−	0.755528i	$-0.272619\pi$
$89$	12.3607	1.31023	0.655117	+	0.755528i	$0.272619\pi$
$90$	0	0
$91$	8.38151	0.878621
$92$	0	0
$93$	0	0
$94$	0.808145	0.0833538
$95$	6.40281	0.656914
$96$	0	0
$97$	10.6165	1.07794	0.538971	−	0.842324i	$-0.318813\pi$
$97$	10.6165	1.07794	0.538971	+	0.842324i	$0.318813\pi$
$98$	0.122227	0.0123468
$99$	0	0
$100$	5.26808	0.526808
$101$	−9.19901	−0.915336	−0.457668	−	0.889123i	$-0.651315\pi$
$101$	−9.19901	−0.915336	−0.457668	+	0.889123i	$0.651315\pi$
$102$	0	0
$103$	−14.7689	−1.45522	−0.727612	−	0.685989i	$-0.759370\pi$
$103$	−14.7689	−1.45522	−0.727612	+	0.685989i	$0.759370\pi$
$104$	3.07021	0.301058
$105$	0	0
$106$	1.59051	0.154484
$107$	−16.2066	−1.56675	−0.783376	−	0.621549i	$-0.786504\pi$
$107$	−16.2066	−1.56675	−0.783376	+	0.621549i	$0.786504\pi$
$108$	0	0
$109$	−15.0296	−1.43958	−0.719788	−	0.694194i	$-0.755761\pi$
$109$	−15.0296	−1.43958	−0.719788	+	0.694194i	$0.755761\pi$
$110$	1.20703	0.115085
$111$	0	0
$112$	9.33843	0.882399
$113$	−1.15546	−0.108697	−0.0543485	−	0.998522i	$-0.517308\pi$
$113$	−1.15546	−0.108697	−0.0543485	+	0.998522i	$0.517308\pi$
$114$	0	0
$115$	0	0
$116$	12.2209	1.13468
$117$	0	0
$118$	−2.46378	−0.226810
$119$	−15.9821	−1.46508
$120$	0	0
$121$	0.375683	0.0341530
$122$	0.851278	0.0770711
$123$	0	0
$124$	9.21890	0.827882
$125$	−11.6675	−1.04358
$126$	0	0
$127$	10.4435	0.926711	0.463356	−	0.886172i	$-0.346645\pi$
$127$	10.4435	0.926711	0.463356	+	0.886172i	$0.346645\pi$
$128$	7.04718	0.622889
$129$	0	0
$130$	−1.17803	−0.103320
$131$	15.1945	1.32755	0.663776	−	0.747931i	$-0.268953\pi$
$131$	15.1945	1.32755	0.663776	+	0.747931i	$0.268953\pi$
$132$	0	0
$133$	10.7728	0.934117
$134$	−1.61264	−0.139311
$135$	0	0
$136$	−5.85437	−0.502008
$137$	1.22243	0.104439	0.0522196	−	0.998636i	$-0.483370\pi$
$137$	1.22243	0.104439	0.0522196	+	0.998636i	$0.483370\pi$
$138$	0	0
$139$	2.18447	0.185284	0.0926420	−	0.995699i	$-0.470469\pi$
$139$	2.18447	0.185284	0.0926420	+	0.995699i	$0.470469\pi$
$140$	−7.49104	−0.633108
$141$	0	0
$142$	0.967137	0.0811603
$143$	−11.1024	−0.928433
$144$	0	0
$145$	−9.51316	−0.790025
$146$	−1.03140	−0.0853592
$147$	0	0
$148$	0.0738212	0.00606806
$149$	−4.13987	−0.339151	−0.169576	−	0.985517i	$-0.554240\pi$
$149$	−4.13987	−0.339151	−0.169576	+	0.985517i	$0.554240\pi$
$150$	0	0
$151$	−6.79599	−0.553050	−0.276525	−	0.961007i	$-0.589183\pi$
$151$	−6.79599	−0.553050	−0.276525	+	0.961007i	$0.589183\pi$
$152$	3.94614	0.320074
$153$	0	0
$154$	2.03083	0.163649
$155$	−7.17631	−0.576415
$156$	0	0
$157$	−9.73735	−0.777125	−0.388563	−	0.921422i	$-0.627028\pi$
$157$	−9.73735	−0.777125	−0.388563	+	0.921422i	$0.627028\pi$
$158$	0.568018	0.0451891
$159$	0	0
$160$	−4.13548	−0.326939
$161$	0	0
$162$	0	0
$163$	−6.02878	−0.472210	−0.236105	−	0.971727i	$-0.575871\pi$
$163$	−6.02878	−0.472210	−0.236105	+	0.971727i	$0.575871\pi$
$164$	6.31175	0.492865
$165$	0	0
$166$	−0.699304	−0.0542765
$167$	−1.75962	−0.136164	−0.0680818	−	0.997680i	$-0.521688\pi$
$167$	−1.75962	−0.136164	−0.0680818	+	0.997680i	$0.521688\pi$
$168$	0	0
$169$	−2.16424	−0.166480
$170$	2.24631	0.172284
$171$	0	0
$172$	2.50567	0.191056
$173$	10.2117	0.776378	0.388189	−	0.921580i	$-0.373101\pi$
$173$	10.2117	0.776378	0.388189	+	0.921580i	$0.373101\pi$
$174$	0	0
$175$	−6.89972	−0.521570
$176$	−12.3700	−0.932424
$177$	0	0
$178$	−2.92305	−0.219092
$179$	10.3055	0.770266	0.385133	−	0.922861i	$-0.374156\pi$
$179$	10.3055	0.770266	0.385133	+	0.922861i	$0.374156\pi$
$180$	0	0
$181$	−9.29148	−0.690630	−0.345315	−	0.938487i	$-0.612228\pi$
$181$	−9.29148	−0.690630	−0.345315	+	0.938487i	$0.612228\pi$
$182$	−1.98205	−0.146919
$183$	0	0
$184$	0	0
$185$	−0.0574649	−0.00422491
$186$	0	0
$187$	21.1705	1.54814
$188$	6.64371	0.484542
$189$	0	0
$190$	−1.51413	−0.109846
$191$	−20.5302	−1.48551	−0.742756	−	0.669562i	$-0.766482\pi$
$191$	−20.5302	−1.48551	−0.742756	+	0.669562i	$0.766482\pi$
$192$	0	0
$193$	−13.7365	−0.988778	−0.494389	−	0.869241i	$-0.664608\pi$
$193$	−13.7365	−0.988778	−0.494389	+	0.869241i	$0.664608\pi$
$194$	−2.51058	−0.180249
$195$	0	0
$196$	1.00482	0.0717731
$197$	−2.29944	−0.163828	−0.0819140	−	0.996639i	$-0.526103\pi$
$197$	−2.29944	−0.163828	−0.0819140	+	0.996639i	$0.526103\pi$
$198$	0	0
$199$	−1.71064	−0.121264	−0.0606319	−	0.998160i	$-0.519312\pi$
$199$	−1.71064	−0.121264	−0.0606319	+	0.998160i	$0.519312\pi$
$200$	−2.52742	−0.178715
$201$	0	0
$202$	2.17537	0.153059
$203$	−16.0060	−1.12340
$204$	0	0
$205$	−4.91329	−0.343159
$206$	3.49254	0.243337
$207$	0	0
$208$	12.0729	0.837104
$209$	−14.2700	−0.987075
$210$	0	0
$211$	−8.00293	−0.550944	−0.275472	−	0.961309i	$-0.588834\pi$
$211$	−8.00293	−0.550944	−0.275472	+	0.961309i	$0.588834\pi$
$212$	13.0754	0.898025
$213$	0	0
$214$	3.83252	0.261986
$215$	−1.95050	−0.133023
$216$	0	0
$217$	−12.0742	−0.819650
$218$	3.55418	0.240720
$219$	0	0
$220$	9.92289	0.669001
$221$	−20.6620	−1.38988
$222$	0	0
$223$	28.5231	1.91005	0.955024	−	0.296528i	$-0.0958289\pi$
$223$	28.5231	1.91005	0.955024	+	0.296528i	$0.0958289\pi$
$224$	−6.95798	−0.464899
$225$	0	0
$226$	0.273243	0.0181759
$227$	−9.38310	−0.622778	−0.311389	−	0.950283i	$-0.600794\pi$
$227$	−9.38310	−0.622778	−0.311389	+	0.950283i	$0.600794\pi$
$228$	0	0
$229$	6.65292	0.439637	0.219819	−	0.975541i	$-0.429453\pi$
$229$	6.65292	0.439637	0.219819	+	0.975541i	$0.429453\pi$
$230$	0	0
$231$	0	0
$232$	−5.86309	−0.384931
$233$	25.6392	1.67968	0.839841	−	0.542833i	$-0.182648\pi$
$233$	25.6392	1.67968	0.839841	+	0.542833i	$0.182648\pi$
$234$	0	0
$235$	−5.17169	−0.337364
$236$	−20.2546	−1.31846
$237$	0	0
$238$	3.77944	0.244985
$239$	1.53210	0.0991030	0.0495515	−	0.998772i	$-0.484221\pi$
$239$	1.53210	0.0991030	0.0495515	+	0.998772i	$0.484221\pi$
$240$	0	0
$241$	27.9057	1.79756	0.898782	−	0.438396i	$-0.144453\pi$
$241$	27.9057	1.79756	0.898782	+	0.438396i	$0.144453\pi$
$242$	−0.0888410	−0.00571092
$243$	0	0
$244$	6.99830	0.448020
$245$	−0.782189	−0.0499722
$246$	0	0
$247$	13.9272	0.886168
$248$	−4.42286	−0.280852
$249$	0	0
$250$	2.75913	0.174503
$251$	−11.2672	−0.711182	−0.355591	−	0.934642i	$-0.615720\pi$
$251$	−11.2672	−0.711182	−0.355591	+	0.934642i	$0.615720\pi$
$252$	0	0
$253$	0	0
$254$	−2.46967	−0.154961
$255$	0	0
$256$	11.7114	0.731964
$257$	16.4410	1.02556	0.512780	−	0.858520i	$-0.328616\pi$
$257$	16.4410	1.02556	0.512780	+	0.858520i	$0.328616\pi$
$258$	0	0
$259$	−0.0966852	−0.00600772
$260$	−9.68454	−0.600610
$261$	0	0
$262$	−3.59319	−0.221988
$263$	−23.4981	−1.44896	−0.724479	−	0.689297i	$-0.757919\pi$
$263$	−23.4981	−1.44896	−0.724479	+	0.689297i	$0.757919\pi$
$264$	0	0
$265$	−10.1784	−0.625253
$266$	−2.54753	−0.156199
$267$	0	0
$268$	−13.2574	−0.809826
$269$	−4.44189	−0.270827	−0.135413	−	0.990789i	$-0.543236\pi$
$269$	−4.44189	−0.270827	−0.135413	+	0.990789i	$0.543236\pi$
$270$	0	0
$271$	−19.7009	−1.19675	−0.598373	−	0.801218i	$-0.704186\pi$
$271$	−19.7009	−1.19675	−0.598373	+	0.801218i	$0.704186\pi$
$272$	−23.0210	−1.39585
$273$	0	0
$274$	−0.289079	−0.0174639
$275$	9.13961	0.551139
$276$	0	0
$277$	−18.8580	−1.13307	−0.566535	−	0.824038i	$-0.691716\pi$
$277$	−18.8580	−1.13307	−0.566535	+	0.824038i	$0.691716\pi$
$278$	−0.516580	−0.0309824
$279$	0	0
$280$	3.59390	0.214777
$281$	19.1925	1.14493	0.572464	−	0.819930i	$-0.305988\pi$
$281$	19.1925	1.14493	0.572464	+	0.819930i	$0.305988\pi$
$282$	0	0
$283$	−1.09006	−0.0647972	−0.0323986	−	0.999475i	$-0.510315\pi$
$283$	−1.09006	−0.0647972	−0.0323986	+	0.999475i	$0.510315\pi$
$284$	7.95077	0.471791
$285$	0	0
$286$	2.62549	0.155249
$287$	−8.26664	−0.487964
$288$	0	0
$289$	22.3989	1.31759
$290$	2.24966	0.132105
$291$	0	0
$292$	−8.47907	−0.496200
$293$	−13.9393	−0.814340	−0.407170	−	0.913352i	$-0.633484\pi$
$293$	−13.9393	−0.814340	−0.407170	+	0.913352i	$0.633484\pi$
$294$	0	0
$295$	15.7669	0.917983
$296$	−0.0354165	−0.00205854
$297$	0	0
$298$	0.978991	0.0567114
$299$	0	0
$300$	0	0
$301$	−3.28173	−0.189156
$302$	1.60711	0.0924787
$303$	0	0
$304$	15.5173	0.889978
$305$	−5.44772	−0.311935
$306$	0	0
$307$	−25.6303	−1.46280	−0.731398	−	0.681951i	$-0.761132\pi$
$307$	−25.6303	−1.46280	−0.731398	+	0.681951i	$0.761132\pi$
$308$	16.6953	0.951305
$309$	0	0
$310$	1.69705	0.0963857
$311$	3.71365	0.210582	0.105291	−	0.994441i	$-0.466423\pi$
$311$	3.71365	0.210582	0.105291	+	0.994441i	$0.466423\pi$
$312$	0	0
$313$	−15.7287	−0.889041	−0.444521	−	0.895769i	$-0.646626\pi$
$313$	−15.7287	−0.889041	−0.444521	+	0.895769i	$0.646626\pi$
$314$	2.30268	0.129948
$315$	0	0
$316$	4.66964	0.262688
$317$	3.30492	0.185623	0.0928114	−	0.995684i	$-0.470415\pi$
$317$	3.30492	0.185623	0.0928114	+	0.995684i	$0.470415\pi$
$318$	0	0
$319$	21.2020	1.18709
$320$	−10.1227	−0.565874
$321$	0	0
$322$	0	0
$323$	−26.5569	−1.47766
$324$	0	0
$325$	−8.92008	−0.494797
$326$	1.42568	0.0789611
$327$	0	0
$328$	−3.02813	−0.167200
$329$	−8.70141	−0.479724
$330$	0	0
$331$	8.00066	0.439756	0.219878	−	0.975527i	$-0.429434\pi$
$331$	8.00066	0.439756	0.219878	+	0.975527i	$0.429434\pi$
$332$	−5.74894	−0.315514
$333$	0	0
$334$	0.416113	0.0227687
$335$	10.3200	0.563844
$336$	0	0
$337$	−1.39225	−0.0758409	−0.0379205	−	0.999281i	$-0.512073\pi$
$337$	−1.39225	−0.0758409	−0.0379205	+	0.999281i	$0.512073\pi$
$338$	0.511797	0.0278381
$339$	0	0
$340$	18.4668	1.00150
$341$	15.9939	0.866118
$342$	0	0
$343$	−19.1394	−1.03343
$344$	−1.20212	−0.0648141
$345$	0	0
$346$	−2.41484	−0.129823
$347$	−16.8286	−0.903405	−0.451702	−	0.892169i	$-0.649183\pi$
$347$	−16.8286	−0.903405	−0.451702	+	0.892169i	$0.649183\pi$
$348$	0	0
$349$	−19.1277	−1.02388	−0.511942	−	0.859020i	$-0.671074\pi$
$349$	−19.1277	−1.02388	−0.511942	+	0.859020i	$0.671074\pi$
$350$	1.63164	0.0872147
$351$	0	0
$352$	9.21678	0.491256
$353$	−33.6499	−1.79100	−0.895502	−	0.445058i	$-0.853183\pi$
$353$	−33.6499	−1.79100	−0.895502	+	0.445058i	$0.853183\pi$
$354$	0	0
$355$	−6.18915	−0.328486
$356$	−24.0302	−1.27360
$357$	0	0
$358$	−2.43702	−0.128801
$359$	−19.6315	−1.03611	−0.518056	−	0.855347i	$-0.673344\pi$
$359$	−19.6315	−1.03611	−0.518056	+	0.855347i	$0.673344\pi$
$360$	0	0
$361$	−1.09932	−0.0578590
$362$	2.19724	0.115484
$363$	0	0
$364$	−16.2943	−0.854054
$365$	6.60040	0.345481
$366$	0	0
$367$	−6.73062	−0.351336	−0.175668	−	0.984449i	$-0.556208\pi$
$367$	−6.73062	−0.351336	−0.175668	+	0.984449i	$0.556208\pi$
$368$	0	0
$369$	0	0
$370$	0.0135892	0.000706471 0
$371$	−17.1252	−0.889096
$372$	0	0
$373$	−25.3280	−1.31144	−0.655718	−	0.755006i	$-0.727634\pi$
$373$	−25.3280	−1.31144	−0.655718	+	0.755006i	$0.727634\pi$
$374$	−5.00638	−0.258874
$375$	0	0
$376$	−3.18739	−0.164377
$377$	−20.6928	−1.06573
$378$	0	0
$379$	−28.8573	−1.48230	−0.741149	−	0.671340i	$-0.765719\pi$
$379$	−28.8573	−1.48230	−0.741149	+	0.671340i	$0.765719\pi$
$380$	−12.4476	−0.638546
$381$	0	0
$382$	4.85495	0.248401
$383$	−12.8954	−0.658926	−0.329463	−	0.944168i	$-0.606868\pi$
$383$	−12.8954	−0.658926	−0.329463	+	0.944168i	$0.606868\pi$
$384$	0	0
$385$	−12.9962	−0.662349
$386$	3.24840	0.165339
$387$	0	0
$388$	−20.6393	−1.04780
$389$	2.09646	0.106295	0.0531475	−	0.998587i	$-0.483075\pi$
$389$	2.09646	0.106295	0.0531475	+	0.998587i	$0.483075\pi$
$390$	0	0
$391$	0	0
$392$	−0.482074	−0.0243484
$393$	0	0
$394$	0.543768	0.0273946
$395$	−3.63501	−0.182897
$396$	0	0
$397$	1.87901	0.0943046	0.0471523	−	0.998888i	$-0.484985\pi$
$397$	1.87901	0.0943046	0.0471523	+	0.998888i	$0.484985\pi$
$398$	0.404529	0.0202772
$399$	0	0
$400$	−9.93849	−0.496924
$401$	16.8547	0.841681	0.420841	−	0.907135i	$-0.361735\pi$
$401$	16.8547	0.841681	0.420841	+	0.907135i	$0.361735\pi$
$402$	0	0
$403$	−15.6097	−0.777576
$404$	17.8836	0.889742
$405$	0	0
$406$	3.78507	0.187850
$407$	0.128073	0.00634832
$408$	0	0
$409$	13.0096	0.643284	0.321642	−	0.946861i	$-0.395765\pi$
$409$	13.0096	0.643284	0.321642	+	0.946861i	$0.395765\pi$
$410$	1.16189	0.0573816
$411$	0	0
$412$	28.7119	1.41453
$413$	26.5279	1.30535
$414$	0	0
$415$	4.47517	0.219677
$416$	−8.99539	−0.441035
$417$	0	0
$418$	3.37455	0.165055
$419$	−29.4107	−1.43680	−0.718402	−	0.695628i	$-0.755126\pi$
$419$	−29.4107	−1.43680	−0.718402	+	0.695628i	$0.755126\pi$
$420$	0	0
$421$	−22.2973	−1.08670	−0.543351	−	0.839506i	$-0.682845\pi$
$421$	−22.2973	−1.08670	−0.543351	+	0.839506i	$0.682845\pi$
$422$	1.89252	0.0921266
$423$	0	0
$424$	−6.27308	−0.304648
$425$	17.0091	0.825062
$426$	0	0
$427$	−9.16582	−0.443565
$428$	31.5069	1.52294
$429$	0	0
$430$	0.461252	0.0222436
$431$	−1.76154	−0.0848504	−0.0424252	−	0.999100i	$-0.513508\pi$
$431$	−1.76154	−0.0848504	−0.0424252	+	0.999100i	$0.513508\pi$
$432$	0	0
$433$	−13.7988	−0.663126	−0.331563	−	0.943433i	$-0.607576\pi$
$433$	−13.7988	−0.663126	−0.331563	+	0.943433i	$0.607576\pi$
$434$	2.85529	0.137058
$435$	0	0
$436$	29.2187	1.39932
$437$	0	0
$438$	0	0
$439$	7.18849	0.343088	0.171544	−	0.985176i	$-0.445124\pi$
$439$	7.18849	0.343088	0.171544	+	0.985176i	$0.445124\pi$
$440$	−4.76061	−0.226953
$441$	0	0
$442$	4.88612	0.232409
$443$	23.3915	1.11136	0.555682	−	0.831395i	$-0.312457\pi$
$443$	23.3915	1.11136	0.555682	+	0.831395i	$0.312457\pi$
$444$	0	0
$445$	18.7059	0.886746
$446$	−6.74511	−0.319390
$447$	0	0
$448$	−17.0314	−0.804660
$449$	−6.75255	−0.318673	−0.159336	−	0.987224i	$-0.550935\pi$
$449$	−6.75255	−0.318673	−0.159336	+	0.987224i	$0.550935\pi$
$450$	0	0
$451$	10.9503	0.515629
$452$	2.24631	0.105658
$453$	0	0
$454$	2.21890	0.104138
$455$	12.6840	0.594637
$456$	0	0
$457$	−7.22487	−0.337965	−0.168983	−	0.985619i	$-0.554048\pi$
$457$	−7.22487	−0.337965	−0.168983	+	0.985619i	$0.554048\pi$
$458$	−1.57327	−0.0735143
$459$	0	0
$460$	0	0
$461$	32.1800	1.49877	0.749385	−	0.662134i	$-0.230349\pi$
$461$	32.1800	1.49877	0.749385	+	0.662134i	$0.230349\pi$
$462$	0	0
$463$	−18.7742	−0.872510	−0.436255	−	0.899823i	$-0.643695\pi$
$463$	−18.7742	−0.872510	−0.436255	+	0.899823i	$0.643695\pi$
$464$	−23.0553	−1.07031
$465$	0	0
$466$	−6.06313	−0.280869
$467$	28.4750	1.31767	0.658833	−	0.752289i	$-0.271050\pi$
$467$	28.4750	1.31767	0.658833	+	0.752289i	$0.271050\pi$
$468$	0	0
$469$	17.3635	0.801774
$470$	1.22300	0.0564126
$471$	0	0
$472$	9.71735	0.447277
$473$	4.34710	0.199880
$474$	0	0
$475$	−11.4650	−0.526050
$476$	31.0705	1.42411
$477$	0	0
$478$	−0.362308	−0.0165716
$479$	4.72745	0.216003	0.108001	−	0.994151i	$-0.465555\pi$
$479$	4.72745	0.216003	0.108001	+	0.994151i	$0.465555\pi$
$480$	0	0
$481$	−0.124996	−0.00569934
$482$	−6.59911	−0.300581
$483$	0	0
$484$	−0.730356	−0.0331980
$485$	16.0663	0.729535
$486$	0	0
$487$	31.6764	1.43540	0.717698	−	0.696354i	$-0.245196\pi$
$487$	31.6764	1.43540	0.717698	+	0.696354i	$0.245196\pi$
$488$	−3.35751	−0.151987
$489$	0	0
$490$	0.184971	0.00835615
$491$	7.19879	0.324877	0.162439	−	0.986719i	$-0.448064\pi$
$491$	7.19879	0.324877	0.162439	+	0.986719i	$0.448064\pi$
$492$	0	0
$493$	39.4577	1.77708
$494$	−3.29349	−0.148181
$495$	0	0
$496$	−17.3919	−0.780919
$497$	−10.4133	−0.467100
$498$	0	0
$499$	−22.7933	−1.02037	−0.510184	−	0.860065i	$-0.670423\pi$
$499$	−22.7933	−1.02037	−0.510184	+	0.860065i	$0.670423\pi$
$500$	22.6826	1.01440
$501$	0	0
$502$	2.66446	0.118921
$503$	−24.4294	−1.08925	−0.544626	−	0.838679i	$-0.683328\pi$
$503$	−24.4294	−1.08925	−0.544626	+	0.838679i	$0.683328\pi$
$504$	0	0
$505$	−13.9212	−0.619485
$506$	0	0
$507$	0	0
$508$	−20.3030	−0.900799
$509$	7.83025	0.347070	0.173535	−	0.984828i	$-0.444481\pi$
$509$	7.83025	0.347070	0.173535	+	0.984828i	$0.444481\pi$
$510$	0	0
$511$	11.1052	0.491266
$512$	−16.8639	−0.745284
$513$	0	0
$514$	−3.88795	−0.171490
$515$	−22.3503	−0.984874
$516$	0	0
$517$	11.5262	0.506921
$518$	0.0228640	0.00100459
$519$	0	0
$520$	4.64626	0.203752
$521$	3.03368	0.132908	0.0664539	−	0.997789i	$-0.478831\pi$
$521$	3.03368	0.132908	0.0664539	+	0.997789i	$0.478831\pi$
$522$	0	0
$523$	−29.6487	−1.29645	−0.648224	−	0.761449i	$-0.724488\pi$
$523$	−29.6487	−1.29645	−0.648224	+	0.761449i	$0.724488\pi$
$524$	−29.5393	−1.29043
$525$	0	0
$526$	5.55681	0.242289
$527$	29.7651	1.29659
$528$	0	0
$529$	0	0
$530$	2.40697	0.104552
$531$	0	0
$532$	−20.9431	−0.907999
$533$	−10.6873	−0.462916
$534$	0	0
$535$	−24.5261	−1.06035
$536$	6.36039	0.274727
$537$	0	0
$538$	1.05041	0.0452865
$539$	1.74327	0.0750880
$540$	0	0
$541$	23.7648	1.02173	0.510864	−	0.859662i	$-0.329326\pi$
$541$	23.7648	1.02173	0.510864	+	0.859662i	$0.329326\pi$
$542$	4.65885	0.200115
$543$	0	0
$544$	17.1527	0.735416
$545$	−22.7449	−0.974283
$546$	0	0
$547$	−19.1130	−0.817214	−0.408607	−	0.912710i	$-0.633985\pi$
$547$	−19.1130	−0.817214	−0.408607	+	0.912710i	$0.633985\pi$
$548$	−2.37650	−0.101519
$549$	0	0
$550$	−2.16132	−0.0921592
$551$	−26.5965	−1.13305
$552$	0	0
$553$	−6.11593	−0.260076
$554$	4.45953	0.189467
$555$	0	0
$556$	−4.24677	−0.180103
$557$	−19.6258	−0.831571	−0.415785	−	0.909463i	$-0.636493\pi$
$557$	−19.6258	−0.831571	−0.415785	+	0.909463i	$0.636493\pi$
$558$	0	0
$559$	−4.24268	−0.179446
$560$	14.1322	0.597194
$561$	0	0
$562$	−4.53862	−0.191450
$563$	44.6795	1.88302	0.941509	−	0.336989i	$-0.109408\pi$
$563$	44.6795	1.88302	0.941509	+	0.336989i	$0.109408\pi$
$564$	0	0
$565$	−1.74861	−0.0735645
$566$	0.257775	0.0108351
$567$	0	0
$568$	−3.81446	−0.160051
$569$	4.14523	0.173777	0.0868885	−	0.996218i	$-0.472308\pi$
$569$	4.14523	0.173777	0.0868885	+	0.996218i	$0.472308\pi$
$570$	0	0
$571$	13.3883	0.560281	0.280141	−	0.959959i	$-0.409619\pi$
$571$	13.3883	0.560281	0.280141	+	0.959959i	$0.409619\pi$
$572$	21.5840	0.902473
$573$	0	0
$574$	1.95489	0.0815954
$575$	0	0
$576$	0	0
$577$	23.4280	0.975322	0.487661	−	0.873033i	$-0.337850\pi$
$577$	23.4280	0.975322	0.487661	+	0.873033i	$0.337850\pi$
$578$	−5.29688	−0.220321
$579$	0	0
$580$	18.4943	0.767935
$581$	7.52950	0.312376
$582$	0	0
$583$	22.6846	0.939501
$584$	4.06792	0.168332
$585$	0	0
$586$	3.29634	0.136170
$587$	40.0658	1.65369	0.826847	−	0.562427i	$-0.190132\pi$
$587$	40.0658	1.65369	0.826847	+	0.562427i	$0.190132\pi$
$588$	0	0
$589$	−20.0632	−0.826690
$590$	−3.72853	−0.153501
$591$	0	0
$592$	−0.139267	−0.00572384
$593$	−5.84119	−0.239869	−0.119935	−	0.992782i	$-0.538268\pi$
$593$	−5.84119	−0.239869	−0.119935	+	0.992782i	$0.538268\pi$
$594$	0	0
$595$	−24.1864	−0.991544
$596$	8.04822	0.329668
$597$	0	0
$598$	0	0
$599$	5.01179	0.204776	0.102388	−	0.994745i	$-0.467352\pi$
$599$	5.01179	0.204776	0.102388	+	0.994745i	$0.467352\pi$
$600$	0	0
$601$	12.7427	0.519784	0.259892	−	0.965638i	$-0.416313\pi$
$601$	12.7427	0.519784	0.259892	+	0.965638i	$0.416313\pi$
$602$	0.776060	0.0316299
$603$	0	0
$604$	13.2119	0.537586
$605$	0.568535	0.0231142
$606$	0	0
$607$	−0.618065	−0.0250865	−0.0125432	−	0.999921i	$-0.503993\pi$
$607$	−0.618065	−0.0250865	−0.0125432	+	0.999921i	$0.503993\pi$
$608$	−11.5618	−0.468893
$609$	0	0
$610$	1.28827	0.0521605
$611$	−11.2493	−0.455099
$612$	0	0
$613$	−31.8070	−1.28467	−0.642336	−	0.766423i	$-0.722034\pi$
$613$	−31.8070	−1.28467	−0.642336	+	0.766423i	$0.722034\pi$
$614$	6.06101	0.244603
$615$	0	0
$616$	−8.00975	−0.322722
$617$	39.9754	1.60935	0.804675	−	0.593715i	$-0.202340\pi$
$617$	39.9754	1.60935	0.804675	+	0.593715i	$0.202340\pi$
$618$	0	0
$619$	35.1336	1.41214	0.706069	−	0.708143i	$-0.250467\pi$
$619$	35.1336	1.41214	0.706069	+	0.708143i	$0.250467\pi$
$620$	13.9513	0.560298
$621$	0	0
$622$	−0.878199	−0.0352126
$623$	31.4729	1.26093
$624$	0	0
$625$	−4.10787	−0.164315
$626$	3.71952	0.148662
$627$	0	0
$628$	18.9302	0.755396
$629$	0.238347	0.00950352
$630$	0	0
$631$	−47.2227	−1.87990	−0.939952	−	0.341306i	$-0.889131\pi$
$631$	−47.2227	−1.87990	−0.939952	+	0.341306i	$0.889131\pi$
$632$	−2.24031	−0.0891147
$633$	0	0
$634$	−0.781544	−0.0310391
$635$	15.8045	0.627184
$636$	0	0
$637$	−1.70140	−0.0674118
$638$	−5.01384	−0.198500
$639$	0	0
$640$	10.6648	0.421562
$641$	6.91023	0.272938	0.136469	−	0.990644i	$-0.456425\pi$
$641$	6.91023	0.272938	0.136469	+	0.990644i	$0.456425\pi$
$642$	0	0
$643$	−38.9219	−1.53493	−0.767465	−	0.641091i	$-0.778482\pi$
$643$	−38.9219	−1.53493	−0.767465	+	0.641091i	$0.778482\pi$
$644$	0	0
$645$	0	0
$646$	6.28014	0.247089
$647$	−11.6534	−0.458142	−0.229071	−	0.973410i	$-0.573569\pi$
$647$	−11.6534	−0.458142	−0.229071	+	0.973410i	$0.573569\pi$
$648$	0	0
$649$	−35.1398	−1.37936
$650$	2.10941	0.0827379
$651$	0	0
$652$	11.7204	0.459007
$653$	39.4525	1.54389	0.771947	−	0.635687i	$-0.219283\pi$
$653$	39.4525	1.54389	0.771947	+	0.635687i	$0.219283\pi$
$654$	0	0
$655$	22.9944	0.898467
$656$	−11.9074	−0.464907
$657$	0	0
$658$	2.05770	0.0802175
$659$	−0.746115	−0.0290645	−0.0145323	−	0.999894i	$-0.504626\pi$
$659$	−0.746115	−0.0290645	−0.0145323	+	0.999894i	$0.504626\pi$
$660$	0	0
$661$	29.2025	1.13585	0.567923	−	0.823082i	$-0.307747\pi$
$661$	29.2025	1.13585	0.567923	+	0.823082i	$0.307747\pi$
$662$	−1.89199	−0.0735342
$663$	0	0
$664$	2.75811	0.107035
$665$	16.3028	0.632197
$666$	0	0
$667$	0	0
$668$	3.42084	0.132356
$669$	0	0
$670$	−2.44047	−0.0942836
$671$	12.1414	0.468712
$672$	0	0
$673$	16.7351	0.645089	0.322545	−	0.946554i	$-0.395462\pi$
$673$	16.7351	0.645089	0.322545	+	0.946554i	$0.395462\pi$
$674$	0.329239	0.0126818
$675$	0	0
$676$	4.20745	0.161825
$677$	45.2291	1.73829	0.869147	−	0.494554i	$-0.164669\pi$
$677$	45.2291	1.73829	0.869147	+	0.494554i	$0.164669\pi$
$678$	0	0
$679$	27.0317	1.03738
$680$	−8.85963	−0.339751
$681$	0	0
$682$	−3.78222	−0.144829
$683$	−34.8041	−1.33174	−0.665870	−	0.746068i	$-0.731940\pi$
$683$	−34.8041	−1.33174	−0.665870	+	0.746068i	$0.731940\pi$
$684$	0	0
$685$	1.84995	0.0706829
$686$	4.52607	0.172806
$687$	0	0
$688$	−4.72707	−0.180218
$689$	−22.1397	−0.843457
$690$	0	0
$691$	10.7550	0.409140	0.204570	−	0.978852i	$-0.434420\pi$
$691$	10.7550	0.409140	0.204570	+	0.978852i	$0.434420\pi$
$692$	−19.8523	−0.754669
$693$	0	0
$694$	3.97960	0.151064
$695$	3.30583	0.125397
$696$	0	0
$697$	20.3788	0.771902
$698$	4.52331	0.171210
$699$	0	0
$700$	13.4136	0.506986
$701$	−20.2600	−0.765210	−0.382605	−	0.923912i	$-0.624973\pi$
$701$	−20.2600	−0.765210	−0.382605	+	0.923912i	$0.624973\pi$
$702$	0	0
$703$	−0.160658	−0.00605933
$704$	22.5604	0.850279
$705$	0	0
$706$	7.95749	0.299484
$707$	−23.4225	−0.880895
$708$	0	0
$709$	13.1519	0.493929	0.246965	−	0.969024i	$-0.420567\pi$
$709$	13.1519	0.493929	0.246965	+	0.969024i	$0.420567\pi$
$710$	1.46360	0.0549281
$711$	0	0
$712$	11.5287	0.432057
$713$	0	0
$714$	0	0
$715$	−16.8017	−0.628349
$716$	−20.0346	−0.748729
$717$	0	0
$718$	4.64244	0.173254
$719$	28.7406	1.07184	0.535922	−	0.844268i	$-0.319964\pi$
$719$	28.7406	1.07184	0.535922	+	0.844268i	$0.319964\pi$
$720$	0	0
$721$	−37.6046	−1.40047
$722$	0.259966	0.00967494
$723$	0	0
$724$	18.0634	0.671319
$725$	17.0344	0.632643
$726$	0	0
$727$	27.8488	1.03285	0.516427	−	0.856331i	$-0.327262\pi$
$727$	27.8488	1.03285	0.516427	+	0.856331i	$0.327262\pi$
$728$	7.81736	0.289731
$729$	0	0
$730$	−1.56086	−0.0577698
$731$	8.09008	0.299222
$732$	0	0
$733$	25.0147	0.923939	0.461970	−	0.886896i	$-0.347143\pi$
$733$	25.0147	0.923939	0.461970	+	0.886896i	$0.347143\pi$
$734$	1.59165	0.0587489
$735$	0	0
$736$	0	0
$737$	−23.0003	−0.847229
$738$	0	0
$739$	6.38838	0.235000	0.117500	−	0.993073i	$-0.462512\pi$
$739$	6.38838	0.235000	0.117500	+	0.993073i	$0.462512\pi$
$740$	0.111716	0.00410677
$741$	0	0
$742$	4.04975	0.148671
$743$	−28.9196	−1.06096	−0.530479	−	0.847698i	$-0.677988\pi$
$743$	−28.9196	−1.06096	−0.530479	+	0.847698i	$0.677988\pi$
$744$	0	0
$745$	−6.26501	−0.229532
$746$	5.98954	0.219293
$747$	0	0
$748$	−41.1571	−1.50485
$749$	−41.2653	−1.50780
$750$	0	0
$751$	50.2692	1.83435	0.917174	−	0.398487i	$-0.130465\pi$
$751$	50.2692	1.83435	0.917174	+	0.398487i	$0.130465\pi$
$752$	−12.5337	−0.457056
$753$	0	0
$754$	4.89340	0.178207
$755$	−10.2846	−0.374296
$756$	0	0
$757$	−35.1746	−1.27844	−0.639222	−	0.769023i	$-0.720743\pi$
$757$	−35.1746	−1.27844	−0.639222	+	0.769023i	$0.720743\pi$
$758$	6.82414	0.247864
$759$	0	0
$760$	5.97184	0.216621
$761$	35.3214	1.28040	0.640200	−	0.768208i	$-0.278851\pi$
$761$	35.3214	1.28040	0.640200	+	0.768208i	$0.278851\pi$
$762$	0	0
$763$	−38.2684	−1.38541
$764$	39.9123	1.44397
$765$	0	0
$766$	3.04950	0.110183
$767$	34.2957	1.23835
$768$	0	0
$769$	49.3025	1.77789	0.888946	−	0.458012i	$-0.151438\pi$
$769$	49.3025	1.77789	0.888946	+	0.458012i	$0.151438\pi$
$770$	3.07333	0.110755
$771$	0	0
$772$	26.7049	0.961130
$773$	3.74473	0.134689	0.0673443	−	0.997730i	$-0.478547\pi$
$773$	3.74473	0.134689	0.0673443	+	0.997730i	$0.478547\pi$
$774$	0	0
$775$	12.8500	0.461587
$776$	9.90191	0.355458
$777$	0	0
$778$	−0.495769	−0.0177742
$779$	−13.7363	−0.492156
$780$	0	0
$781$	13.7938	0.493581
$782$	0	0
$783$	0	0
$784$	−1.89565	−0.0677017
$785$	−14.7359	−0.525947
$786$	0	0
$787$	−11.6368	−0.414806	−0.207403	−	0.978256i	$-0.566501\pi$
$787$	−11.6368	−0.414806	−0.207403	+	0.978256i	$0.566501\pi$
$788$	4.47028	0.159247
$789$	0	0
$790$	0.859603	0.0305833
$791$	−2.94204	−0.104607
$792$	0	0
$793$	−11.8497	−0.420796
$794$	−0.444345	−0.0157692
$795$	0	0
$796$	3.32561	0.117873
$797$	50.0561	1.77308	0.886540	−	0.462653i	$-0.153102\pi$
$797$	50.0561	1.77308	0.886540	+	0.462653i	$0.153102\pi$
$798$	0	0
$799$	21.4506	0.758868
$800$	7.40507	0.261809
$801$	0	0
$802$	−3.98577	−0.140742
$803$	−14.7104	−0.519117
$804$	0	0
$805$	0	0
$806$	3.69137	0.130023
$807$	0	0
$808$	−8.57983	−0.301837
$809$	32.0554	1.12701	0.563504	−	0.826113i	$-0.309453\pi$
$809$	32.0554	1.12701	0.563504	+	0.826113i	$0.309453\pi$
$810$	0	0
$811$	48.3682	1.69844	0.849218	−	0.528042i	$-0.177074\pi$
$811$	48.3682	1.69844	0.849218	+	0.528042i	$0.177074\pi$
$812$	31.1168	1.09199
$813$	0	0
$814$	−0.0302865	−0.00106154
$815$	−9.12358	−0.319585
$816$	0	0
$817$	−5.45312	−0.190781
$818$	−3.07650	−0.107567
$819$	0	0
$820$	9.55181	0.333564
$821$	10.5245	0.367307	0.183653	−	0.982991i	$-0.441208\pi$
$821$	10.5245	0.367307	0.183653	+	0.982991i	$0.441208\pi$
$822$	0	0
$823$	47.5631	1.65795	0.828973	−	0.559288i	$-0.188926\pi$
$823$	47.5631	1.65795	0.828973	+	0.559288i	$0.188926\pi$
$824$	−13.7748	−0.479869
$825$	0	0
$826$	−6.27329	−0.218276
$827$	−52.9294	−1.84053	−0.920267	−	0.391291i	$-0.872029\pi$
$827$	−52.9294	−1.84053	−0.920267	+	0.391291i	$0.872029\pi$
$828$	0	0
$829$	12.4245	0.431522	0.215761	−	0.976446i	$-0.430777\pi$
$829$	12.4245	0.431522	0.215761	+	0.976446i	$0.430777\pi$
$830$	−1.05828	−0.0367335
$831$	0	0
$832$	−22.0185	−0.763356
$833$	3.24428	0.112408
$834$	0	0
$835$	−2.66290	−0.0921534
$836$	27.7420	0.959476
$837$	0	0
$838$	6.95500	0.240256
$839$	−43.6840	−1.50814	−0.754070	−	0.656794i	$-0.771912\pi$
$839$	−43.6840	−1.50814	−0.754070	+	0.656794i	$0.771912\pi$
$840$	0	0
$841$	10.5165	0.362637
$842$	5.27283	0.181714
$843$	0	0
$844$	15.5583	0.535539
$845$	−3.27522	−0.112671
$846$	0	0
$847$	0.956563	0.0328679
$848$	−24.6674	−0.847083
$849$	0	0
$850$	−4.02229	−0.137963
$851$	0	0
$852$	0	0
$853$	29.8296	1.02135	0.510673	−	0.859775i	$-0.329396\pi$
$853$	29.8296	1.02135	0.510673	+	0.859775i	$0.329396\pi$
$854$	2.16752	0.0741711
$855$	0	0
$856$	−15.1158	−0.516646
$857$	6.57662	0.224653	0.112326	−	0.993671i	$-0.464170\pi$
$857$	6.57662	0.224653	0.112326	+	0.993671i	$0.464170\pi$
$858$	0	0
$859$	18.4762	0.630401	0.315200	−	0.949025i	$-0.397928\pi$
$859$	18.4762	0.630401	0.315200	+	0.949025i	$0.397928\pi$
$860$	3.79193	0.129304
$861$	0	0
$862$	0.416567	0.0141883
$863$	−29.5347	−1.00537	−0.502686	−	0.864469i	$-0.667655\pi$
$863$	−29.5347	−1.00537	−0.502686	+	0.864469i	$0.667655\pi$
$864$	0	0
$865$	15.4537	0.525441
$866$	3.26312	0.110885
$867$	0	0
$868$	23.4732	0.796731
$869$	8.10137	0.274820
$870$	0	0
$871$	22.4479	0.760617
$872$	−14.0180	−0.474709
$873$	0	0
$874$	0	0
$875$	−29.7079	−1.00431
$876$	0	0
$877$	8.60818	0.290678	0.145339	−	0.989382i	$-0.453573\pi$
$877$	8.60818	0.290678	0.145339	+	0.989382i	$0.453573\pi$
$878$	−1.69993	−0.0573697
$879$	0	0
$880$	−18.7200	−0.631051
$881$	−5.03012	−0.169469	−0.0847344	−	0.996404i	$-0.527004\pi$
$881$	−5.03012	−0.169469	−0.0847344	+	0.996404i	$0.527004\pi$
$882$	0	0
$883$	−21.6359	−0.728108	−0.364054	−	0.931378i	$-0.618608\pi$
$883$	−21.6359	−0.728108	−0.364054	+	0.931378i	$0.618608\pi$
$884$	40.1685	1.35101
$885$	0	0
$886$	−5.53160	−0.185838
$887$	−27.5141	−0.923834	−0.461917	−	0.886923i	$-0.652838\pi$
$887$	−27.5141	−0.923834	−0.461917	+	0.886923i	$0.652838\pi$
$888$	0	0
$889$	26.5913	0.891842
$890$	−4.42356	−0.148278
$891$	0	0
$892$	−55.4511	−1.85664
$893$	−14.4588	−0.483845
$894$	0	0
$895$	15.5956	0.521305
$896$	17.9435	0.599451
$897$	0	0
$898$	1.59684	0.0532871
$899$	29.8095	0.994202
$900$	0	0
$901$	42.2168	1.40645
$902$	−2.58951	−0.0862212
$903$	0	0
$904$	−1.07769	−0.0358435
$905$	−14.0611	−0.467408
$906$	0	0
$907$	−1.90638	−0.0633002	−0.0316501	−	0.999499i	$-0.510076\pi$
$907$	−1.90638	−0.0633002	−0.0316501	+	0.999499i	$0.510076\pi$
$908$	18.2415	0.605364
$909$	0	0
$910$	−2.99951	−0.0994328
$911$	−37.7053	−1.24923	−0.624616	−	0.780932i	$-0.714745\pi$
$911$	−37.7053	−1.24923	−0.624616	+	0.780932i	$0.714745\pi$
$912$	0	0
$913$	−9.97384	−0.330086
$914$	1.70853	0.0565131
$915$	0	0
$916$	−12.9338	−0.427344
$917$	38.6883	1.27760
$918$	0	0
$919$	1.89744	0.0625909	0.0312955	−	0.999510i	$-0.490037\pi$
$919$	1.89744	0.0625909	0.0312955	+	0.999510i	$0.490037\pi$
$920$	0	0
$921$	0	0
$922$	−7.60988	−0.250618
$923$	−13.4625	−0.443123
$924$	0	0
$925$	0.102898	0.00338326
$926$	4.43970	0.145898
$927$	0	0
$928$	17.1783	0.563904
$929$	27.8481	0.913666	0.456833	−	0.889553i	$-0.348984\pi$
$929$	27.8481	0.913666	0.456833	+	0.889553i	$0.348984\pi$
$930$	0	0
$931$	−2.18681	−0.0716698
$932$	−49.8446	−1.63272
$933$	0	0
$934$	−6.73374	−0.220335
$935$	32.0381	1.04776
$936$	0	0
$937$	−14.3279	−0.468072	−0.234036	−	0.972228i	$-0.575193\pi$
$937$	−14.3279	−0.468072	−0.234036	+	0.972228i	$0.575193\pi$
$938$	−4.10611	−0.134069
$939$	0	0
$940$	10.0542	0.327931
$941$	−35.0559	−1.14279	−0.571395	−	0.820676i	$-0.693597\pi$
$941$	−35.0559	−1.14279	−0.571395	+	0.820676i	$0.693597\pi$
$942$	0	0
$943$	0	0
$944$	38.2113	1.24367
$945$	0	0
$946$	−1.02800	−0.0334231
$947$	7.91512	0.257207	0.128603	−	0.991696i	$-0.458951\pi$
$947$	7.91512	0.257207	0.128603	+	0.991696i	$0.458951\pi$
$948$	0	0
$949$	14.3570	0.466049
$950$	2.71123	0.0879639
$951$	0	0
$952$	−14.9064	−0.483119
$953$	17.6442	0.571552	0.285776	−	0.958296i	$-0.407749\pi$
$953$	17.6442	0.571552	0.285776	+	0.958296i	$0.407749\pi$
$954$	0	0
$955$	−31.0691	−1.00537
$956$	−2.97851	−0.0963320
$957$	0	0
$958$	−1.11794	−0.0361191
$959$	3.11255	0.100510
$960$	0	0
$961$	−8.51301	−0.274613
$962$	0.0295590	0.000953020 0
$963$	0	0
$964$	−54.2508	−1.74730
$965$	−20.7880	−0.669190
$966$	0	0
$967$	−18.1226	−0.582785	−0.291392	−	0.956604i	$-0.594119\pi$
$967$	−18.1226	−0.582785	−0.291392	+	0.956604i	$0.594119\pi$
$968$	0.350396	0.0112622
$969$	0	0
$970$	−3.79935	−0.121990
$971$	−10.6050	−0.340331	−0.170166	−	0.985415i	$-0.554430\pi$
$971$	−10.6050	−0.340331	−0.170166	+	0.985415i	$0.554430\pi$
$972$	0	0
$973$	5.56209	0.178312
$974$	−7.49081	−0.240021
$975$	0	0
$976$	−13.2026	−0.422606
$977$	6.57503	0.210354	0.105177	−	0.994454i	$-0.466459\pi$
$977$	6.57503	0.210354	0.105177	+	0.994454i	$0.466459\pi$
$978$	0	0
$979$	−41.6901	−1.33242
$980$	1.52064	0.0485750
$981$	0	0
$982$	−1.70236	−0.0543246
$983$	43.7116	1.39418	0.697091	−	0.716983i	$-0.254477\pi$
$983$	43.7116	1.39418	0.697091	+	0.716983i	$0.254477\pi$
$984$	0	0
$985$	−3.47982	−0.110876
$986$	−9.33090	−0.297157
$987$	0	0
$988$	−27.0756	−0.861390
$989$	0	0
$990$	0	0
$991$	54.2307	1.72270	0.861348	−	0.508016i	$-0.169621\pi$
$991$	54.2307	1.72270	0.861348	+	0.508016i	$0.169621\pi$
$992$	12.9585	0.411434
$993$	0	0
$994$	2.46252	0.0781065
$995$	−2.58877	−0.0820695
$996$	0	0
$997$	0.687738	0.0217809	0.0108904	−	0.999941i	$-0.496533\pi$
$997$	0.687738	0.0217809	0.0108904	+	0.999941i	$0.496533\pi$
$998$	5.39013	0.170622
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bn.1.4		5
3.2	odd	2		529.2.a.j.1.2		5
12.11	even	2		8464.2.a.bt.1.3		5
23.15	odd	22		207.2.i.c.64.1		10
23.20	odd	22		207.2.i.c.55.1		10
23.22	odd	2		4761.2.a.bo.1.4		5
69.2	odd	22		529.2.c.h.487.1		10
69.5	even	22		529.2.c.b.255.1		10
69.8	odd	22		529.2.c.a.501.1		10
69.11	even	22		529.2.c.i.466.1		10
69.14	even	22		529.2.c.b.334.1		10
69.17	even	22		529.2.c.g.266.1		10
69.20	even	22		23.2.c.a.9.1	✓	10
69.26	odd	22		529.2.c.a.170.1		10
69.29	odd	22		529.2.c.f.266.1		10
69.32	odd	22		529.2.c.c.334.1		10
69.35	odd	22		529.2.c.h.466.1		10
69.38	even	22		23.2.c.a.18.1	yes	10
69.41	odd	22		529.2.c.c.255.1		10
69.44	even	22		529.2.c.i.487.1		10
69.50	odd	22		529.2.c.f.177.1		10
69.53	even	22		529.2.c.d.118.1		10
69.56	even	22		529.2.c.d.399.1		10
69.59	odd	22		529.2.c.e.399.1		10
69.62	odd	22		529.2.c.e.118.1		10
69.65	even	22		529.2.c.g.177.1		10
69.68	even	2		529.2.a.i.1.2		5
276.107	odd	22		368.2.m.c.225.1		10
276.227	odd	22		368.2.m.c.193.1		10
276.275	odd	2		8464.2.a.bs.1.3		5
345.38	odd	44		575.2.p.b.524.1		20
345.89	even	22		575.2.k.b.101.1		10
345.107	odd	44		575.2.p.b.524.2		20
345.158	odd	44		575.2.p.b.124.2		20
345.227	odd	44		575.2.p.b.124.1		20
345.314	even	22		575.2.k.b.501.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.2.c.a.9.1	✓	10	69.20	even	22
23.2.c.a.18.1	yes	10	69.38	even	22
207.2.i.c.55.1		10	23.20	odd	22
207.2.i.c.64.1		10	23.15	odd	22
368.2.m.c.193.1		10	276.227	odd	22
368.2.m.c.225.1		10	276.107	odd	22
529.2.a.i.1.2		5	69.68	even	2
529.2.a.j.1.2		5	3.2	odd	2
529.2.c.a.170.1		10	69.26	odd	22
529.2.c.a.501.1		10	69.8	odd	22
529.2.c.b.255.1		10	69.5	even	22
529.2.c.b.334.1		10	69.14	even	22
529.2.c.c.255.1		10	69.41	odd	22
529.2.c.c.334.1		10	69.32	odd	22
529.2.c.d.118.1		10	69.53	even	22
529.2.c.d.399.1		10	69.56	even	22
529.2.c.e.118.1		10	69.62	odd	22
529.2.c.e.399.1		10	69.59	odd	22
529.2.c.f.177.1		10	69.50	odd	22
529.2.c.f.266.1		10	69.29	odd	22
529.2.c.g.177.1		10	69.65	even	22
529.2.c.g.266.1		10	69.17	even	22
529.2.c.h.466.1		10	69.35	odd	22
529.2.c.h.487.1		10	69.2	odd	22
529.2.c.i.466.1		10	69.11	even	22
529.2.c.i.487.1		10	69.44	even	22
575.2.k.b.101.1		10	345.89	even	22
575.2.k.b.501.1		10	345.314	even	22
575.2.p.b.124.1		20	345.227	odd	44
575.2.p.b.124.2		20	345.158	odd	44
575.2.p.b.524.1		20	345.38	odd	44
575.2.p.b.524.2		20	345.107	odd	44
4761.2.a.bn.1.4		5	1.1	even	1	trivial
4761.2.a.bo.1.4		5	23.22	odd	2
8464.2.a.bs.1.3		5	276.275	odd	2
8464.2.a.bt.1.3		5	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-0.236479 q^{2} -1.94408 q^{4} +1.51334 q^{5} +2.54620 q^{7} +0.932691 q^{8} -0.357872 q^{10} -3.37279 q^{11} +3.29177 q^{13} -0.602123 q^{14} +3.66759 q^{16} -6.27686 q^{17} +4.23092 q^{19} -2.94204 q^{20} +0.797593 q^{22} -2.70981 q^{25} -0.778434 q^{26} -4.95001 q^{28} -6.28621 q^{29} -4.74204 q^{31} -2.73269 q^{32} +1.48434 q^{34} +3.85326 q^{35} -0.0379723 q^{37} -1.00052 q^{38} +1.41148 q^{40} -3.24666 q^{41} -1.28887 q^{43} +6.55696 q^{44} -3.41741 q^{47} -0.516864 q^{49} +0.640813 q^{50} -6.39946 q^{52} -6.72578 q^{53} -5.10416 q^{55} +2.37482 q^{56} +1.48656 q^{58} +10.4186 q^{59} -3.59980 q^{61} +1.12139 q^{62} -6.68896 q^{64} +4.98156 q^{65} +6.81939 q^{67} +12.2027 q^{68} -0.911214 q^{70} -4.08974 q^{71} +4.36149 q^{73} +0.00897966 q^{74} -8.22524 q^{76} -8.58779 q^{77} -2.40198 q^{79} +5.55031 q^{80} +0.767766 q^{82} +2.95715 q^{83} -9.49900 q^{85} +0.304792 q^{86} -3.14577 q^{88} +12.3607 q^{89} +8.38151 q^{91} +0.808145 q^{94} +6.40281 q^{95} +10.6165 q^{97} +0.122227 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} - 7 q^{5} + 8 q^{7} + 9 q^{8} + 5 q^{10} - 13 q^{11} + 4 q^{13} - 12 q^{14} + 6 q^{16} - 16 q^{17} + 10 q^{19} - 10 q^{20} + 3 q^{22} - 2 q^{25} - 6 q^{26} - 9 q^{28} - 7 q^{29}+ \cdots - 3 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 - 7 * q^5 + 8 * q^7 + 9 * q^8 + 5 * q^10 - 13 * q^11 + 4 * q^13 - 12 * q^14 + 6 * q^16 - 16 * q^17 + 10 * q^19 - 10 * q^20 + 3 * q^22 - 2 * q^25 - 6 * q^26 - 9 * q^28 - 7 * q^29 + 5 * q^31 + 6 * q^32 - 9 * q^34 - 9 * q^35 - 7 * q^37 - 4 * q^38 - 17 * q^40 - 9 * q^41 - 17 * q^44 + 9 * q^47 - 9 * q^49 - 8 * q^50 + q^52 - 2 * q^53 + 16 * q^55 - q^56 - 6 * q^58 + 16 * q^59 + 4 * q^61 - 2 * q^62 - 21 * q^64 + q^65 - 17 * q^67 - 15 * q^68 + 19 * q^70 + 7 * q^71 - 18 * q^73 + 5 * q^74 + 30 * q^76 - 12 * q^77 + 13 * q^79 + 7 * q^80 - 3 * q^82 - 13 * q^83 + 7 * q^85 + 11 * q^86 - 30 * q^88 - 15 * q^89 + 2 * q^91 - 19 * q^94 + 8 * q^95 - 16 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more