LMFDB - Embedded newform 4761.2.a.bq.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
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 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.830830$ 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)

$f(q)$	$=$	$q+2.20362 q^{2} +2.85592 q^{4} +0.594351 q^{5} -2.88612 q^{7} +1.88612 q^{8} +O(q^{10})$	$q+2.20362 q^{2} +2.85592 q^{4} +0.594351 q^{5} -2.88612 q^{7} +1.88612 q^{8} +1.30972 q^{10} -4.39788 q^{11} +5.44724 q^{13} -6.35991 q^{14} -1.55555 q^{16} -3.81712 q^{17} +0.115460 q^{19} +1.69742 q^{20} -9.69123 q^{22} -4.64675 q^{25} +12.0036 q^{26} -8.24254 q^{28} +5.49621 q^{29} -6.70697 q^{31} -7.20009 q^{32} -8.41148 q^{34} -1.71537 q^{35} +9.20077 q^{37} +0.254429 q^{38} +1.12102 q^{40} -4.26315 q^{41} -3.14953 q^{43} -12.5600 q^{44} -3.51213 q^{47} +1.32970 q^{49} -10.2396 q^{50} +15.5569 q^{52} -9.81750 q^{53} -2.61388 q^{55} -5.44358 q^{56} +12.1115 q^{58} +4.05750 q^{59} -11.8283 q^{61} -14.7796 q^{62} -12.7551 q^{64} +3.23757 q^{65} -7.19097 q^{67} -10.9014 q^{68} -3.78002 q^{70} -8.06679 q^{71} -3.81795 q^{73} +20.2750 q^{74} +0.329744 q^{76} +12.6928 q^{77} -0.391365 q^{79} -0.924545 q^{80} -9.39435 q^{82} -0.955750 q^{83} -2.26871 q^{85} -6.94036 q^{86} -8.29494 q^{88} -6.00732 q^{89} -15.7214 q^{91} -7.73938 q^{94} +0.0686236 q^{95} +15.7125 q^{97} +2.93016 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 4 q^{7} - 9 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 4 * q^7 - 9 * q^8	$ 5 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 4 q^{7} - 9 q^{8} + q^{10} - 13 q^{11} + 7 q^{13} - 16 q^{14} + 6 q^{16} - q^{17} - 5 q^{19} + 2 q^{20} - 3 q^{22} - 10 q^{25} + 27 q^{26} + 12 q^{28} + 7 q^{29} + 8 q^{31} - 6 q^{32} - 18 q^{34} - 9 q^{35} + 7 q^{37} + 9 q^{38} + 12 q^{40} + 7 q^{41} - 10 q^{43} - 17 q^{44} - 13 q^{47} - q^{49} - 4 q^{50} - q^{52} - 17 q^{53} + 10 q^{55} - 38 q^{56} - 17 q^{58} + 10 q^{59} + 18 q^{61} + q^{62} - 21 q^{64} - 24 q^{65} + 9 q^{67} - 25 q^{68} + 3 q^{70} - 37 q^{71} - 2 q^{73} + 5 q^{74} - 4 q^{76} - 6 q^{77} - 32 q^{79} - 8 q^{80} - 17 q^{82} - 13 q^{83} + 5 q^{85} - 26 q^{86} + 19 q^{88} - 30 q^{89} - 23 q^{91} + 8 q^{94} + 14 q^{95} + 20 q^{97} - 51 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 4 * q^7 - 9 * q^8 + q^10 - 13 * q^11 + 7 * q^13 - 16 * q^14 + 6 * q^16 - q^17 - 5 * q^19 + 2 * q^20 - 3 * q^22 - 10 * q^25 + 27 * q^26 + 12 * q^28 + 7 * q^29 + 8 * q^31 - 6 * q^32 - 18 * q^34 - 9 * q^35 + 7 * q^37 + 9 * q^38 + 12 * q^40 + 7 * q^41 - 10 * q^43 - 17 * q^44 - 13 * q^47 - q^49 - 4 * q^50 - q^52 - 17 * q^53 + 10 * q^55 - 38 * q^56 - 17 * q^58 + 10 * q^59 + 18 * q^61 + q^62 - 21 * q^64 - 24 * q^65 + 9 * q^67 - 25 * q^68 + 3 * q^70 - 37 * q^71 - 2 * q^73 + 5 * q^74 - 4 * q^76 - 6 * q^77 - 32 * q^79 - 8 * q^80 - 17 * q^82 - 13 * q^83 + 5 * q^85 - 26 * q^86 + 19 * q^88 - 30 * q^89 - 23 * q^91 + 8 * q^94 + 14 * q^95 + 20 * q^97 - 51 * 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 $

$2$	2.20362	1.55819	0.779096	−	0.626905i	$-0.215679\pi$
$2$	2.20362	1.55819	0.779096	+	0.626905i	$0.215679\pi$
$3$	0	0
$3$	0	0
$4$	2.85592	1.42796
$5$	0.594351	0.265802	0.132901	−	0.991129i	$-0.457571\pi$
$5$	0.594351	0.265802	0.132901	+	0.991129i	$0.457571\pi$
$6$	0	0
$7$	−2.88612	−1.09085	−0.545426	−	0.838159i	$-0.683632\pi$
$7$	−2.88612	−1.09085	−0.545426	+	0.838159i	$0.683632\pi$
$8$	1.88612	0.666845
$9$	0	0
$10$	1.30972	0.414170
$11$	−4.39788	−1.32601	−0.663005	−	0.748615i	$-0.730719\pi$
$11$	−4.39788	−1.32601	−0.663005	+	0.748615i	$0.730719\pi$
$12$	0	0
$13$	5.44724	1.51079	0.755396	−	0.655269i	$-0.227445\pi$
$13$	5.44724	1.51079	0.755396	+	0.655269i	$0.227445\pi$
$14$	−6.35991	−1.69976
$15$	0	0
$16$	−1.55555	−0.388889
$17$	−3.81712	−0.925789	−0.462894	−	0.886413i	$-0.653189\pi$
$17$	−3.81712	−0.925789	−0.462894	+	0.886413i	$0.653189\pi$
$18$	0	0
$19$	0.115460	0.0264883	0.0132441	−	0.999912i	$-0.495784\pi$
$19$	0.115460	0.0264883	0.0132441	+	0.999912i	$0.495784\pi$
$20$	1.69742	0.379555
$21$	0	0
$22$	−9.69123	−2.06618
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.64675	−0.929349
$26$	12.0036	2.35410
$27$	0	0
$28$	−8.24254	−1.55769
$29$	5.49621	1.02062	0.510310	−	0.859990i	$-0.329530\pi$
$29$	5.49621	1.02062	0.510310	+	0.859990i	$0.329530\pi$
$30$	0	0
$31$	−6.70697	−1.20461	−0.602304	−	0.798267i	$-0.705750\pi$
$31$	−6.70697	−1.20461	−0.602304	+	0.798267i	$0.705750\pi$
$32$	−7.20009	−1.27281
$33$	0	0
$34$	−8.41148	−1.44256
$35$	−1.71537	−0.289951
$36$	0	0
$37$	9.20077	1.51260	0.756299	−	0.654226i	$-0.227006\pi$
$37$	9.20077	1.51260	0.756299	+	0.654226i	$0.227006\pi$
$38$	0.254429	0.0412738
$39$	0	0
$40$	1.12102	0.177249
$41$	−4.26315	−0.665793	−0.332896	−	0.942963i	$-0.608026\pi$
$41$	−4.26315	−0.665793	−0.332896	+	0.942963i	$0.608026\pi$
$42$	0	0
$43$	−3.14953	−0.480299	−0.240149	−	0.970736i	$-0.577196\pi$
$43$	−3.14953	−0.480299	−0.240149	+	0.970736i	$0.577196\pi$
$44$	−12.5600	−1.89349
$45$	0	0
$46$	0	0
$47$	−3.51213	−0.512297	−0.256148	−	0.966637i	$-0.582454\pi$
$47$	−3.51213	−0.512297	−0.256148	+	0.966637i	$0.582454\pi$
$48$	0	0
$49$	1.32970	0.189958
$50$	−10.2396	−1.44810
$51$	0	0
$52$	15.5569	2.15735
$53$	−9.81750	−1.34854	−0.674269	−	0.738486i	$-0.735541\pi$
$53$	−9.81750	−1.34854	−0.674269	+	0.738486i	$0.735541\pi$
$54$	0	0
$55$	−2.61388	−0.352456
$56$	−5.44358	−0.727429
$57$	0	0
$58$	12.1115	1.59032
$59$	4.05750	0.528242	0.264121	−	0.964490i	$-0.414918\pi$
$59$	4.05750	0.528242	0.264121	+	0.964490i	$0.414918\pi$
$60$	0	0
$61$	−11.8283	−1.51445	−0.757227	−	0.653152i	$-0.773446\pi$
$61$	−11.8283	−1.51445	−0.757227	+	0.653152i	$0.773446\pi$
$62$	−14.7796	−1.87701
$63$	0	0
$64$	−12.7551	−1.59439
$65$	3.23757	0.401571
$66$	0	0
$67$	−7.19097	−0.878517	−0.439258	−	0.898361i	$-0.644759\pi$
$67$	−7.19097	−0.878517	−0.439258	+	0.898361i	$0.644759\pi$
$68$	−10.9014	−1.32199
$69$	0	0
$70$	−3.78002	−0.451798
$71$	−8.06679	−0.957351	−0.478676	−	0.877992i	$-0.658883\pi$
$71$	−8.06679	−0.957351	−0.478676	+	0.877992i	$0.658883\pi$
$72$	0	0
$73$	−3.81795	−0.446857	−0.223429	−	0.974720i	$-0.571725\pi$
$73$	−3.81795	−0.446857	−0.223429	+	0.974720i	$0.571725\pi$
$74$	20.2750	2.35692
$75$	0	0
$76$	0.329744	0.0378242
$77$	12.6928	1.44648
$78$	0	0
$79$	−0.391365	−0.0440320	−0.0220160	−	0.999758i	$-0.507008\pi$
$79$	−0.391365	−0.0440320	−0.0220160	+	0.999758i	$0.507008\pi$
$80$	−0.924545	−0.103367
$81$	0	0
$82$	−9.39435	−1.03743
$83$	−0.955750	−0.104907	−0.0524536	−	0.998623i	$-0.516704\pi$
$83$	−0.955750	−0.104907	−0.0524536	+	0.998623i	$0.516704\pi$
$84$	0	0
$85$	−2.26871	−0.246076
$86$	−6.94036	−0.748398
$87$	0	0
$88$	−8.29494	−0.884243
$89$	−6.00732	−0.636775	−0.318387	−	0.947961i	$-0.603141\pi$
$89$	−6.00732	−0.636775	−0.318387	+	0.947961i	$0.603141\pi$
$90$	0	0
$91$	−15.7214	−1.64805
$92$	0	0
$93$	0	0
$94$	−7.73938	−0.798256
$95$	0.0686236	0.00704063
$96$	0	0
$97$	15.7125	1.59536	0.797679	−	0.603082i	$-0.206061\pi$
$97$	15.7125	1.59536	0.797679	+	0.603082i	$0.206061\pi$
$98$	2.93016	0.295991
$99$	0	0
$100$	−13.2707	−1.32707
$101$	−1.15415	−0.114842	−0.0574211	−	0.998350i	$-0.518288\pi$
$101$	−1.15415	−0.114842	−0.0574211	+	0.998350i	$0.518288\pi$
$102$	0	0
$103$	−0.542406	−0.0534448	−0.0267224	−	0.999643i	$-0.508507\pi$
$103$	−0.542406	−0.0534448	−0.0267224	+	0.999643i	$0.508507\pi$
$104$	10.2742	1.00746
$105$	0	0
$106$	−21.6340	−2.10128
$107$	−11.4001	−1.10209	−0.551044	−	0.834476i	$-0.685770\pi$
$107$	−11.4001	−1.10209	−0.551044	+	0.834476i	$0.685770\pi$
$108$	0	0
$109$	5.62658	0.538929	0.269465	−	0.963010i	$-0.413153\pi$
$109$	5.62658	0.538929	0.269465	+	0.963010i	$0.413153\pi$
$110$	−5.75999	−0.549194
$111$	0	0
$112$	4.48952	0.424220
$113$	4.84196	0.455493	0.227747	−	0.973720i	$-0.426864\pi$
$113$	4.84196	0.455493	0.227747	+	0.973720i	$0.426864\pi$
$114$	0	0
$115$	0	0
$116$	15.6968	1.45741
$117$	0	0
$118$	8.94118	0.823102
$119$	11.0167	1.00990
$120$	0	0
$121$	8.34133	0.758302
$122$	−26.0649	−2.35981
$123$	0	0
$124$	−19.1546	−1.72013
$125$	−5.73355	−0.512825
$126$	0	0
$127$	15.8827	1.40936	0.704679	−	0.709527i	$-0.251091\pi$
$127$	15.8827	1.40936	0.704679	+	0.709527i	$0.251091\pi$
$128$	−13.7072	−1.21156
$129$	0	0
$130$	7.13436	0.625725
$131$	15.9211	1.39104	0.695518	−	0.718509i	$-0.255175\pi$
$131$	15.9211	1.39104	0.695518	+	0.718509i	$0.255175\pi$
$132$	0	0
$133$	−0.333231	−0.0288948
$134$	−15.8461	−1.36890
$135$	0	0
$136$	−7.19957	−0.617358
$137$	−19.3249	−1.65104	−0.825519	−	0.564374i	$-0.809117\pi$
$137$	−19.3249	−1.65104	−0.825519	+	0.564374i	$0.809117\pi$
$138$	0	0
$139$	0.855237	0.0725402	0.0362701	−	0.999342i	$-0.488452\pi$
$139$	0.855237	0.0725402	0.0362701	+	0.999342i	$0.488452\pi$
$140$	−4.89896	−0.414038
$141$	0	0
$142$	−17.7761	−1.49174
$143$	−23.9563	−2.00332
$144$	0	0
$145$	3.26668	0.271283
$146$	−8.41329	−0.696289
$147$	0	0
$148$	26.2767	2.15993
$149$	5.37514	0.440349	0.220174	−	0.975461i	$-0.429337\pi$
$149$	5.37514	0.440349	0.220174	+	0.975461i	$0.429337\pi$
$150$	0	0
$151$	11.7670	0.957587	0.478794	−	0.877927i	$-0.341074\pi$
$151$	11.7670	0.957587	0.478794	+	0.877927i	$0.341074\pi$
$152$	0.217771	0.0176636
$153$	0	0
$154$	27.9701	2.25389
$155$	−3.98629	−0.320187
$156$	0	0
$157$	18.4446	1.47204	0.736020	−	0.676960i	$-0.236703\pi$
$157$	18.4446	1.47204	0.736020	+	0.676960i	$0.236703\pi$
$158$	−0.862418	−0.0686103
$159$	0	0
$160$	−4.27938	−0.338315
$161$	0	0
$162$	0	0
$163$	−20.8490	−1.63302	−0.816508	−	0.577334i	$-0.804093\pi$
$163$	−20.8490	−1.63302	−0.816508	+	0.577334i	$0.804093\pi$
$164$	−12.1752	−0.950726
$165$	0	0
$166$	−2.10611	−0.163466
$167$	−6.10692	−0.472568	−0.236284	−	0.971684i	$-0.575930\pi$
$167$	−6.10692	−0.472568	−0.236284	+	0.971684i	$0.575930\pi$
$168$	0	0
$169$	16.6724	1.28249
$170$	−4.99937	−0.383434
$171$	0	0
$172$	−8.99481	−0.685848
$173$	14.3920	1.09420	0.547102	−	0.837066i	$-0.315731\pi$
$173$	14.3920	1.09420	0.547102	+	0.837066i	$0.315731\pi$
$174$	0	0
$175$	13.4111	1.01378
$176$	6.84114	0.515670
$177$	0	0
$178$	−13.2378	−0.992217
$179$	−18.9043	−1.41297	−0.706486	−	0.707727i	$-0.749721\pi$
$179$	−18.9043	−1.41297	−0.706486	+	0.707727i	$0.749721\pi$
$180$	0	0
$181$	2.08204	0.154757	0.0773785	−	0.997002i	$-0.475345\pi$
$181$	2.08204	0.154757	0.0773785	+	0.997002i	$0.475345\pi$
$182$	−34.6439	−2.56798
$183$	0	0
$184$	0	0
$185$	5.46849	0.402051
$186$	0	0
$187$	16.7872	1.22761
$188$	−10.0304	−0.731540
$189$	0	0
$190$	0.151220	0.0109707
$191$	−0.879742	−0.0636559	−0.0318280	−	0.999493i	$-0.510133\pi$
$191$	−0.879742	−0.0636559	−0.0318280	+	0.999493i	$0.510133\pi$
$192$	0	0
$193$	−2.71571	−0.195481	−0.0977407	−	0.995212i	$-0.531162\pi$
$193$	−2.71571	−0.195481	−0.0977407	+	0.995212i	$0.531162\pi$
$194$	34.6242	2.48587
$195$	0	0
$196$	3.79753	0.271252
$197$	1.93562	0.137907	0.0689536	−	0.997620i	$-0.478034\pi$
$197$	1.93562	0.137907	0.0689536	+	0.997620i	$0.478034\pi$
$198$	0	0
$199$	21.6714	1.53625	0.768123	−	0.640302i	$-0.221191\pi$
$199$	21.6714	1.53625	0.768123	+	0.640302i	$0.221191\pi$
$200$	−8.76433	−0.619732
$201$	0	0
$202$	−2.54330	−0.178946
$203$	−15.8627	−1.11335
$204$	0	0
$205$	−2.53381	−0.176969
$206$	−1.19525	−0.0832773
$207$	0	0
$208$	−8.47347	−0.587530
$209$	−0.507778	−0.0351237
$210$	0	0
$211$	7.90247	0.544028	0.272014	−	0.962293i	$-0.412310\pi$
$211$	7.90247	0.544028	0.272014	+	0.962293i	$0.412310\pi$
$212$	−28.0380	−1.92566
$213$	0	0
$214$	−25.1214	−1.71726
$215$	−1.87193	−0.127664
$216$	0	0
$217$	19.3571	1.31405
$218$	12.3988	0.839755
$219$	0	0
$220$	−7.46505	−0.503293
$221$	−20.7928	−1.39867
$222$	0	0
$223$	−12.0165	−0.804685	−0.402343	−	0.915489i	$-0.631804\pi$
$223$	−12.0165	−0.804685	−0.402343	+	0.915489i	$0.631804\pi$
$224$	20.7803	1.38844
$225$	0	0
$226$	10.6698	0.709746
$227$	0.134329	0.00891576	0.00445788	−	0.999990i	$-0.498581\pi$
$227$	0.134329	0.00891576	0.00445788	+	0.999990i	$0.498581\pi$
$228$	0	0
$229$	9.04483	0.597699	0.298850	−	0.954300i	$-0.403397\pi$
$229$	9.04483	0.597699	0.298850	+	0.954300i	$0.403397\pi$
$230$	0	0
$231$	0	0
$232$	10.3665	0.680596
$233$	−24.9283	−1.63311	−0.816553	−	0.577271i	$-0.804118\pi$
$233$	−24.9283	−1.63311	−0.816553	+	0.577271i	$0.804118\pi$
$234$	0	0
$235$	−2.08744	−0.136169
$236$	11.5879	0.754309
$237$	0	0
$238$	24.2766	1.57362
$239$	4.40749	0.285097	0.142548	−	0.989788i	$-0.454470\pi$
$239$	4.40749	0.285097	0.142548	+	0.989788i	$0.454470\pi$
$240$	0	0
$241$	16.3457	1.05292	0.526461	−	0.850199i	$-0.323519\pi$
$241$	16.3457	1.05292	0.526461	+	0.850199i	$0.323519\pi$
$242$	18.3811	1.18158
$243$	0	0
$244$	−33.7806	−2.16258
$245$	0.790311	0.0504911
$246$	0	0
$247$	0.628936	0.0400183
$248$	−12.6502	−0.803286
$249$	0	0
$250$	−12.6346	−0.799079
$251$	1.33105	0.0840152	0.0420076	−	0.999117i	$-0.486625\pi$
$251$	1.33105	0.0840152	0.0420076	+	0.999117i	$0.486625\pi$
$252$	0	0
$253$	0	0
$254$	34.9993	2.19605
$255$	0	0
$256$	−4.69517	−0.293448
$257$	18.3892	1.14709	0.573544	−	0.819175i	$-0.305568\pi$
$257$	18.3892	1.14709	0.573544	+	0.819175i	$0.305568\pi$
$258$	0	0
$259$	−26.5546	−1.65002
$260$	9.24625	0.573428
$261$	0	0
$262$	35.0841	2.16750
$263$	4.69044	0.289225	0.144612	−	0.989488i	$-0.453806\pi$
$263$	4.69044	0.289225	0.144612	+	0.989488i	$0.453806\pi$
$264$	0	0
$265$	−5.83504	−0.358444
$266$	−0.734313	−0.0450236
$267$	0	0
$268$	−20.5368	−1.25449
$269$	−1.68931	−0.102999	−0.0514995	−	0.998673i	$-0.516400\pi$
$269$	−1.68931	−0.102999	−0.0514995	+	0.998673i	$0.516400\pi$
$270$	0	0
$271$	1.72953	0.105061	0.0525306	−	0.998619i	$-0.483271\pi$
$271$	1.72953	0.105061	0.0525306	+	0.998619i	$0.483271\pi$
$272$	5.93774	0.360029
$273$	0	0
$274$	−42.5847	−2.57263
$275$	20.4358	1.23233
$276$	0	0
$277$	20.8841	1.25481	0.627403	−	0.778695i	$-0.284118\pi$
$277$	20.8841	1.25481	0.627403	+	0.778695i	$0.284118\pi$
$278$	1.88461	0.113032
$279$	0	0
$280$	−3.23540	−0.193352
$281$	9.64608	0.575437	0.287719	−	0.957715i	$-0.407103\pi$
$281$	9.64608	0.575437	0.287719	+	0.957715i	$0.407103\pi$
$282$	0	0
$283$	−10.6869	−0.635273	−0.317637	−	0.948213i	$-0.602889\pi$
$283$	−10.6869	−0.635273	−0.317637	+	0.948213i	$0.602889\pi$
$284$	−23.0381	−1.36706
$285$	0	0
$286$	−52.7904	−3.12156
$287$	12.3040	0.726281
$288$	0	0
$289$	−2.42956	−0.142915
$290$	7.19851	0.422711
$291$	0	0
$292$	−10.9038	−0.638095
$293$	−19.0622	−1.11363	−0.556814	−	0.830637i	$-0.687976\pi$
$293$	−19.0622	−1.11363	−0.556814	+	0.830637i	$0.687976\pi$
$294$	0	0
$295$	2.41158	0.140408
$296$	17.3538	1.00867
$297$	0	0
$298$	11.8447	0.686148
$299$	0	0
$300$	0	0
$301$	9.08993	0.523935
$302$	25.9300	1.49210
$303$	0	0
$304$	−0.179604	−0.0103010
$305$	−7.03014	−0.402545
$306$	0	0
$307$	−16.8244	−0.960217	−0.480108	−	0.877209i	$-0.659403\pi$
$307$	−16.8244	−0.960217	−0.480108	+	0.877209i	$0.659403\pi$
$308$	36.2497	2.06552
$309$	0	0
$310$	−8.78426	−0.498913
$311$	−28.2368	−1.60116	−0.800580	−	0.599226i	$-0.795475\pi$
$311$	−28.2368	−1.60116	−0.800580	+	0.599226i	$0.795475\pi$
$312$	0	0
$313$	9.83722	0.556033	0.278016	−	0.960576i	$-0.410323\pi$
$313$	9.83722	0.556033	0.278016	+	0.960576i	$0.410323\pi$
$314$	40.6448	2.29372
$315$	0	0
$316$	−1.11771	−0.0628760
$317$	−2.22149	−0.124771	−0.0623856	−	0.998052i	$-0.519871\pi$
$317$	−2.22149	−0.124771	−0.0623856	+	0.998052i	$0.519871\pi$
$318$	0	0
$319$	−24.1717	−1.35335
$320$	−7.58102	−0.423792
$321$	0	0
$322$	0	0
$323$	−0.440724	−0.0245225
$324$	0	0
$325$	−25.3119	−1.40405
$326$	−45.9431	−2.54455
$327$	0	0
$328$	−8.04083	−0.443981
$329$	10.1364	0.558840
$330$	0	0
$331$	−20.2108	−1.11089	−0.555444	−	0.831554i	$-0.687452\pi$
$331$	−20.2108	−1.11089	−0.555444	+	0.831554i	$0.687452\pi$
$332$	−2.72955	−0.149803
$333$	0	0
$334$	−13.4573	−0.736351
$335$	−4.27396	−0.233511
$336$	0	0
$337$	20.8038	1.13326	0.566628	−	0.823974i	$-0.308248\pi$
$337$	20.8038	1.13326	0.566628	+	0.823974i	$0.308248\pi$
$338$	36.7395	1.99837
$339$	0	0
$340$	−6.47927	−0.351388
$341$	29.4964	1.59732
$342$	0	0
$343$	16.3652	0.883636
$344$	−5.94040	−0.320285
$345$	0	0
$346$	31.7144	1.70498
$347$	−10.4268	−0.559743	−0.279871	−	0.960038i	$-0.590292\pi$
$347$	−10.4268	−0.559743	−0.279871	+	0.960038i	$0.590292\pi$
$348$	0	0
$349$	23.3151	1.24803	0.624014	−	0.781413i	$-0.285501\pi$
$349$	23.3151	1.24803	0.624014	+	0.781413i	$0.285501\pi$
$350$	29.5529	1.57967
$351$	0	0
$352$	31.6651	1.68776
$353$	−8.18396	−0.435588	−0.217794	−	0.975995i	$-0.569886\pi$
$353$	−8.18396	−0.435588	−0.217794	+	0.975995i	$0.569886\pi$
$354$	0	0
$355$	−4.79450	−0.254466
$356$	−17.1564	−0.909289
$357$	0	0
$358$	−41.6577	−2.20168
$359$	26.7308	1.41080	0.705399	−	0.708810i	$-0.250768\pi$
$359$	26.7308	1.41080	0.705399	+	0.708810i	$0.250768\pi$
$360$	0	0
$361$	−18.9867	−0.999298
$362$	4.58802	0.241141
$363$	0	0
$364$	−44.8991	−2.35335
$365$	−2.26920	−0.118775
$366$	0	0
$367$	6.47075	0.337770	0.168885	−	0.985636i	$-0.445983\pi$
$367$	6.47075	0.337770	0.168885	+	0.985636i	$0.445983\pi$
$368$	0	0
$369$	0	0
$370$	12.0505	0.626473
$371$	28.3345	1.47105
$372$	0	0
$373$	10.4802	0.542645	0.271322	−	0.962489i	$-0.412539\pi$
$373$	10.4802	0.542645	0.271322	+	0.962489i	$0.412539\pi$
$374$	36.9926	1.91284
$375$	0	0
$376$	−6.62431	−0.341622
$377$	29.9392	1.54195
$378$	0	0
$379$	25.0266	1.28553	0.642765	−	0.766063i	$-0.277787\pi$
$379$	25.0266	1.28553	0.642765	+	0.766063i	$0.277787\pi$
$380$	0.195984	0.0100537
$381$	0	0
$382$	−1.93861	−0.0991881
$383$	34.5285	1.76432	0.882161	−	0.470947i	$-0.156088\pi$
$383$	34.5285	1.76432	0.882161	+	0.470947i	$0.156088\pi$
$384$	0	0
$385$	7.54399	0.384477
$386$	−5.98439	−0.304597
$387$	0	0
$388$	44.8735	2.27811
$389$	0.635442	0.0322182	0.0161091	−	0.999870i	$-0.494872\pi$
$389$	0.635442	0.0322182	0.0161091	+	0.999870i	$0.494872\pi$
$390$	0	0
$391$	0	0
$392$	2.50799	0.126672
$393$	0	0
$394$	4.26537	0.214886
$395$	−0.232608	−0.0117038
$396$	0	0
$397$	25.8693	1.29834	0.649171	−	0.760642i	$-0.275116\pi$
$397$	25.8693	1.29834	0.649171	+	0.760642i	$0.275116\pi$
$398$	47.7555	2.39377
$399$	0	0
$400$	7.22827	0.361413
$401$	4.07335	0.203413	0.101707	−	0.994814i	$-0.467570\pi$
$401$	4.07335	0.203413	0.101707	+	0.994814i	$0.467570\pi$
$402$	0	0
$403$	−36.5344	−1.81991
$404$	−3.29616	−0.163990
$405$	0	0
$406$	−34.9554	−1.73481
$407$	−40.4639	−2.00572
$408$	0	0
$409$	4.72671	0.233721	0.116860	−	0.993148i	$-0.462717\pi$
$409$	4.72671	0.233721	0.116860	+	0.993148i	$0.462717\pi$
$410$	−5.58354	−0.275752
$411$	0	0
$412$	−1.54907	−0.0763172
$413$	−11.7105	−0.576234
$414$	0	0
$415$	−0.568051	−0.0278845
$416$	−39.2206	−1.92295
$417$	0	0
$418$	−1.11895	−0.0547295
$419$	−29.9545	−1.46338	−0.731688	−	0.681640i	$-0.761267\pi$
$419$	−29.9545	−1.46338	−0.731688	+	0.681640i	$0.761267\pi$
$420$	0	0
$421$	−35.1645	−1.71381	−0.856906	−	0.515472i	$-0.827616\pi$
$421$	−35.1645	−1.71381	−0.856906	+	0.515472i	$0.827616\pi$
$422$	17.4140	0.847701
$423$	0	0
$424$	−18.5170	−0.899265
$425$	17.7372	0.860381
$426$	0	0
$427$	34.1378	1.65204
$428$	−32.5578	−1.57374
$429$	0	0
$430$	−4.12501	−0.198926
$431$	−2.21082	−0.106491	−0.0532457	−	0.998581i	$-0.516957\pi$
$431$	−2.21082	−0.106491	−0.0532457	+	0.998581i	$0.516957\pi$
$432$	0	0
$433$	−22.7495	−1.09327	−0.546636	−	0.837370i	$-0.684092\pi$
$433$	−22.7495	−1.09327	−0.546636	+	0.837370i	$0.684092\pi$
$434$	42.6557	2.04754
$435$	0	0
$436$	16.0691	0.769570
$437$	0	0
$438$	0	0
$439$	25.5621	1.22001	0.610005	−	0.792397i	$-0.291167\pi$
$439$	25.5621	1.22001	0.610005	+	0.792397i	$0.291167\pi$
$440$	−4.93010	−0.235034
$441$	0	0
$442$	−45.8193	−2.17940
$443$	−1.60459	−0.0762361	−0.0381181	−	0.999273i	$-0.512136\pi$
$443$	−1.60459	−0.0762361	−0.0381181	+	0.999273i	$0.512136\pi$
$444$	0	0
$445$	−3.57046	−0.169256
$446$	−26.4798	−1.25385
$447$	0	0
$448$	36.8128	1.73924
$449$	10.5887	0.499712	0.249856	−	0.968283i	$-0.419617\pi$
$449$	10.5887	0.499712	0.249856	+	0.968283i	$0.419617\pi$
$450$	0	0
$451$	18.7488	0.882848
$452$	13.8283	0.650427
$453$	0	0
$454$	0.296010	0.0138925
$455$	−9.34403	−0.438055
$456$	0	0
$457$	−0.473302	−0.0221401	−0.0110701	−	0.999939i	$-0.503524\pi$
$457$	−0.473302	−0.0221401	−0.0110701	+	0.999939i	$0.503524\pi$
$458$	19.9313	0.931330
$459$	0	0
$460$	0	0
$461$	−20.7268	−0.965342	−0.482671	−	0.875802i	$-0.660333\pi$
$461$	−20.7268	−0.965342	−0.482671	+	0.875802i	$0.660333\pi$
$462$	0	0
$463$	−20.8560	−0.969262	−0.484631	−	0.874719i	$-0.661046\pi$
$463$	−20.8560	−0.969262	−0.484631	+	0.874719i	$0.661046\pi$
$464$	−8.54966	−0.396908
$465$	0	0
$466$	−54.9323	−2.54469
$467$	−34.9474	−1.61717	−0.808586	−	0.588378i	$-0.799767\pi$
$467$	−34.9474	−1.61717	−0.808586	+	0.588378i	$0.799767\pi$
$468$	0	0
$469$	20.7540	0.958332
$470$	−4.59991	−0.212178
$471$	0	0
$472$	7.65295	0.352256
$473$	13.8513	0.636881
$474$	0	0
$475$	−0.536512	−0.0246169
$476$	31.4628	1.44210
$477$	0	0
$478$	9.71241	0.444235
$479$	−8.41854	−0.384653	−0.192326	−	0.981331i	$-0.561603\pi$
$479$	−8.41854	−0.384653	−0.192326	+	0.981331i	$0.561603\pi$
$480$	0	0
$481$	50.1188	2.28522
$482$	36.0197	1.64065
$483$	0	0
$484$	23.8222	1.08283
$485$	9.33871	0.424049
$486$	0	0
$487$	−14.4772	−0.656026	−0.328013	−	0.944673i	$-0.606379\pi$
$487$	−14.4772	−0.656026	−0.328013	+	0.944673i	$0.606379\pi$
$488$	−22.3095	−1.00991
$489$	0	0
$490$	1.74154	0.0786749
$491$	17.0833	0.770959	0.385480	−	0.922716i	$-0.374036\pi$
$491$	17.0833	0.770959	0.385480	+	0.922716i	$0.374036\pi$
$492$	0	0
$493$	−20.9797	−0.944879
$494$	1.38593	0.0623561
$495$	0	0
$496$	10.4331	0.468458
$497$	23.2817	1.04433
$498$	0	0
$499$	−0.884464	−0.0395941	−0.0197970	−	0.999804i	$-0.506302\pi$
$499$	−0.884464	−0.0395941	−0.0197970	+	0.999804i	$0.506302\pi$
$500$	−16.3746	−0.732294
$501$	0	0
$502$	2.93313	0.130912
$503$	12.3377	0.550111	0.275056	−	0.961428i	$-0.411304\pi$
$503$	12.3377	0.550111	0.275056	+	0.961428i	$0.411304\pi$
$504$	0	0
$505$	−0.685970	−0.0305253
$506$	0	0
$507$	0	0
$508$	45.3596	2.01251
$509$	−16.8614	−0.747369	−0.373684	−	0.927556i	$-0.621906\pi$
$509$	−16.8614	−0.747369	−0.373684	+	0.927556i	$0.621906\pi$
$510$	0	0
$511$	11.0191	0.487455
$512$	17.0681	0.754309
$513$	0	0
$514$	40.5228	1.78738
$515$	−0.322380	−0.0142057
$516$	0	0
$517$	15.4459	0.679310
$518$	−58.5160	−2.57105
$519$	0	0
$520$	6.10646	0.267786
$521$	−30.7748	−1.34827	−0.674135	−	0.738609i	$-0.735483\pi$
$521$	−30.7748	−1.34827	−0.674135	+	0.738609i	$0.735483\pi$
$522$	0	0
$523$	−33.9892	−1.48624	−0.743121	−	0.669157i	$-0.766655\pi$
$523$	−33.9892	−1.48624	−0.743121	+	0.669157i	$0.766655\pi$
$524$	45.4695	1.98635
$525$	0	0
$526$	10.3359	0.450668
$527$	25.6013	1.11521
$528$	0	0
$529$	0	0
$530$	−12.8582	−0.558524
$531$	0	0
$532$	−0.951681	−0.0412606
$533$	−23.2224	−1.00587
$534$	0	0
$535$	−6.77566	−0.292937
$536$	−13.5631	−0.585835
$537$	0	0
$538$	−3.72258	−0.160492
$539$	−5.84788	−0.251886
$540$	0	0
$541$	−13.5814	−0.583911	−0.291955	−	0.956432i	$-0.594306\pi$
$541$	−13.5814	−0.583911	−0.291955	+	0.956432i	$0.594306\pi$
$542$	3.81121	0.163705
$543$	0	0
$544$	27.4836	1.17835
$545$	3.34417	0.143248
$546$	0	0
$547$	−31.4163	−1.34326	−0.671632	−	0.740885i	$-0.734406\pi$
$547$	−31.4163	−1.34326	−0.671632	+	0.740885i	$0.734406\pi$
$548$	−55.1904	−2.35762
$549$	0	0
$550$	45.0327	1.92020
$551$	0.634591	0.0270345
$552$	0	0
$553$	1.12953	0.0480324
$554$	46.0206	1.95523
$555$	0	0
$556$	2.44249	0.103585
$557$	22.9669	0.973141	0.486570	−	0.873641i	$-0.338248\pi$
$557$	22.9669	0.973141	0.486570	+	0.873641i	$0.338248\pi$
$558$	0	0
$559$	−17.1562	−0.725632
$560$	2.66835	0.112758
$561$	0	0
$562$	21.2563	0.896641
$563$	−30.9592	−1.30478	−0.652388	−	0.757885i	$-0.726233\pi$
$563$	−30.9592	−1.30478	−0.652388	+	0.757885i	$0.726233\pi$
$564$	0	0
$565$	2.87783	0.121071
$566$	−23.5499	−0.989877
$567$	0	0
$568$	−15.2149	−0.638405
$569$	24.5490	1.02915	0.514573	−	0.857446i	$-0.327950\pi$
$569$	24.5490	1.02915	0.514573	+	0.857446i	$0.327950\pi$
$570$	0	0
$571$	3.39701	0.142160	0.0710802	−	0.997471i	$-0.477355\pi$
$571$	3.39701	0.142160	0.0710802	+	0.997471i	$0.477355\pi$
$572$	−68.4173	−2.86067
$573$	0	0
$574$	27.1132	1.13169
$575$	0	0
$576$	0	0
$577$	−38.4527	−1.60081	−0.800403	−	0.599463i	$-0.795381\pi$
$577$	−38.4527	−1.60081	−0.800403	+	0.599463i	$0.795381\pi$
$578$	−5.35381	−0.222689
$579$	0	0
$580$	9.32938	0.387382
$581$	2.75841	0.114438
$582$	0	0
$583$	43.1762	1.78817
$584$	−7.20112	−0.297985
$585$	0	0
$586$	−42.0059	−1.73525
$587$	13.9378	0.575276	0.287638	−	0.957739i	$-0.407130\pi$
$587$	13.9378	0.575276	0.287638	+	0.957739i	$0.407130\pi$
$588$	0	0
$589$	−0.774385	−0.0319080
$590$	5.31420	0.218782
$591$	0	0
$592$	−14.3123	−0.588232
$593$	28.5530	1.17253	0.586266	−	0.810119i	$-0.300597\pi$
$593$	28.5530	1.17253	0.586266	+	0.810119i	$0.300597\pi$
$594$	0	0
$595$	6.54778	0.268433
$596$	15.3510	0.628801
$597$	0	0
$598$	0	0
$599$	1.09226	0.0446286	0.0223143	−	0.999751i	$-0.492897\pi$
$599$	1.09226	0.0446286	0.0223143	+	0.999751i	$0.492897\pi$
$600$	0	0
$601$	−8.14533	−0.332255	−0.166127	−	0.986104i	$-0.553126\pi$
$601$	−8.14533	−0.332255	−0.166127	+	0.986104i	$0.553126\pi$
$602$	20.0307	0.816391
$603$	0	0
$604$	33.6057	1.36740
$605$	4.95768	0.201558
$606$	0	0
$607$	29.7621	1.20801	0.604003	−	0.796982i	$-0.293571\pi$
$607$	29.7621	1.20801	0.604003	+	0.796982i	$0.293571\pi$
$608$	−0.831320	−0.0337145
$609$	0	0
$610$	−15.4917	−0.627241
$611$	−19.1314	−0.773973
$612$	0	0
$613$	24.7835	1.00099	0.500497	−	0.865738i	$-0.333151\pi$
$613$	24.7835	1.00099	0.500497	+	0.865738i	$0.333151\pi$
$614$	−37.0744	−1.49620
$615$	0	0
$616$	23.9402	0.964578
$617$	−30.1318	−1.21306	−0.606531	−	0.795060i	$-0.707440\pi$
$617$	−30.1318	−1.21306	−0.606531	+	0.795060i	$0.707440\pi$
$618$	0	0
$619$	−15.0511	−0.604956	−0.302478	−	0.953156i	$-0.597814\pi$
$619$	−15.0511	−0.604956	−0.302478	+	0.953156i	$0.597814\pi$
$620$	−11.3845	−0.457214
$621$	0	0
$622$	−62.2230	−2.49491
$623$	17.3379	0.694627
$624$	0	0
$625$	19.8260	0.793040
$626$	21.6775	0.866405
$627$	0	0
$628$	52.6764	2.10202
$629$	−35.1205	−1.40035
$630$	0	0
$631$	8.35583	0.332640	0.166320	−	0.986072i	$-0.446811\pi$
$631$	8.35583	0.332640	0.166320	+	0.986072i	$0.446811\pi$
$632$	−0.738163	−0.0293625
$633$	0	0
$634$	−4.89531	−0.194418
$635$	9.43987	0.374610
$636$	0	0
$637$	7.24321	0.286987
$638$	−53.2651	−2.10878
$639$	0	0
$640$	−8.14689	−0.322034
$641$	27.8767	1.10106	0.550532	−	0.834814i	$-0.314425\pi$
$641$	27.8767	1.10106	0.550532	+	0.834814i	$0.314425\pi$
$642$	0	0
$643$	−23.4722	−0.925651	−0.462826	−	0.886449i	$-0.653164\pi$
$643$	−23.4722	−0.925651	−0.462826	+	0.886449i	$0.653164\pi$
$644$	0	0
$645$	0	0
$646$	−0.971187	−0.0382108
$647$	−22.7373	−0.893895	−0.446948	−	0.894560i	$-0.647489\pi$
$647$	−22.7373	−0.893895	−0.446948	+	0.894560i	$0.647489\pi$
$648$	0	0
$649$	−17.8444	−0.700454
$650$	−55.7778	−2.18778
$651$	0	0
$652$	−59.5430	−2.33188
$653$	26.3235	1.03012	0.515060	−	0.857154i	$-0.327770\pi$
$653$	26.3235	1.03012	0.515060	+	0.857154i	$0.327770\pi$
$654$	0	0
$655$	9.46275	0.369740
$656$	6.63157	0.258919
$657$	0	0
$658$	22.3368	0.870779
$659$	−40.5222	−1.57852	−0.789260	−	0.614060i	$-0.789535\pi$
$659$	−40.5222	−1.57852	−0.789260	+	0.614060i	$0.789535\pi$
$660$	0	0
$661$	13.1323	0.510786	0.255393	−	0.966837i	$-0.417795\pi$
$661$	13.1323	0.510786	0.255393	+	0.966837i	$0.417795\pi$
$662$	−44.5369	−1.73098
$663$	0	0
$664$	−1.80266	−0.0699569
$665$	−0.198056	−0.00768029
$666$	0	0
$667$	0	0
$668$	−17.4409	−0.674808
$669$	0	0
$670$	−9.41817	−0.363856
$671$	52.0192	2.00818
$672$	0	0
$673$	−9.30350	−0.358624	−0.179312	−	0.983792i	$-0.557387\pi$
$673$	−9.30350	−0.358624	−0.179312	+	0.983792i	$0.557387\pi$
$674$	45.8436	1.76583
$675$	0	0
$676$	47.6150	1.83135
$677$	−51.7655	−1.98951	−0.994755	−	0.102282i	$-0.967386\pi$
$677$	−51.7655	−1.98951	−0.994755	+	0.102282i	$0.967386\pi$
$678$	0	0
$679$	−45.3481	−1.74030
$680$	−4.27907	−0.164095
$681$	0	0
$682$	64.9988	2.48893
$683$	−34.6238	−1.32484	−0.662422	−	0.749131i	$-0.730471\pi$
$683$	−34.6238	−1.32484	−0.662422	+	0.749131i	$0.730471\pi$
$684$	0	0
$685$	−11.4858	−0.438849
$686$	36.0625	1.37687
$687$	0	0
$688$	4.89927	0.186783
$689$	−53.4782	−2.03736
$690$	0	0
$691$	−36.6389	−1.39381	−0.696905	−	0.717163i	$-0.745440\pi$
$691$	−36.6389	−1.39381	−0.696905	+	0.717163i	$0.745440\pi$
$692$	41.1024	1.56248
$693$	0	0
$694$	−22.9768	−0.872186
$695$	0.508311	0.0192813
$696$	0	0
$697$	16.2730	0.616383
$698$	51.3775	1.94467
$699$	0	0
$700$	38.3010	1.44764
$701$	−29.4378	−1.11185	−0.555925	−	0.831233i	$-0.687636\pi$
$701$	−29.4378	−1.11185	−0.555925	+	0.831233i	$0.687636\pi$
$702$	0	0
$703$	1.06232	0.0400661
$704$	56.0955	2.11418
$705$	0	0
$706$	−18.0343	−0.678730
$707$	3.33102	0.125276
$708$	0	0
$709$	17.4506	0.655370	0.327685	−	0.944787i	$-0.393731\pi$
$709$	17.4506	0.655370	0.327685	+	0.944787i	$0.393731\pi$
$710$	−10.5652	−0.396506
$711$	0	0
$712$	−11.3305	−0.424630
$713$	0	0
$714$	0	0
$715$	−14.2384	−0.532488
$716$	−53.9891	−2.01767
$717$	0	0
$718$	58.9044	2.19829
$719$	−9.26469	−0.345515	−0.172757	−	0.984964i	$-0.555268\pi$
$719$	−9.26469	−0.345515	−0.172757	+	0.984964i	$0.555268\pi$
$720$	0	0
$721$	1.56545	0.0583004
$722$	−41.8393	−1.55710
$723$	0	0
$724$	5.94615	0.220987
$725$	−25.5395	−0.948513
$726$	0	0
$727$	−22.2769	−0.826205	−0.413103	−	0.910684i	$-0.635555\pi$
$727$	−22.2769	−0.826205	−0.413103	+	0.910684i	$0.635555\pi$
$728$	−29.6525	−1.09899
$729$	0	0
$730$	−5.00045	−0.185075
$731$	12.0222	0.444655
$732$	0	0
$733$	24.1452	0.891824	0.445912	−	0.895077i	$-0.352879\pi$
$733$	24.1452	0.891824	0.445912	+	0.895077i	$0.352879\pi$
$734$	14.2590	0.526311
$735$	0	0
$736$	0	0
$737$	31.6250	1.16492
$738$	0	0
$739$	−0.408839	−0.0150394	−0.00751969	−	0.999972i	$-0.502394\pi$
$739$	−0.408839	−0.0150394	−0.00751969	+	0.999972i	$0.502394\pi$
$740$	15.6176	0.574114
$741$	0	0
$742$	62.4384	2.29218
$743$	−33.8578	−1.24212	−0.621062	−	0.783762i	$-0.713298\pi$
$743$	−33.8578	−1.24212	−0.621062	+	0.783762i	$0.713298\pi$
$744$	0	0
$745$	3.19472	0.117046
$746$	23.0943	0.845544
$747$	0	0
$748$	47.9431	1.75297
$749$	32.9021	1.20221
$750$	0	0
$751$	−13.3715	−0.487934	−0.243967	−	0.969784i	$-0.578449\pi$
$751$	−13.3715	−0.487934	−0.243967	+	0.969784i	$0.578449\pi$
$752$	5.46331	0.199226
$753$	0	0
$754$	65.9744	2.40265
$755$	6.99375	0.254529
$756$	0	0
$757$	8.38136	0.304626	0.152313	−	0.988332i	$-0.451328\pi$
$757$	8.38136	0.304626	0.152313	+	0.988332i	$0.451328\pi$
$758$	55.1490	2.00310
$759$	0	0
$760$	0.129433	0.00469501
$761$	−10.2677	−0.372205	−0.186102	−	0.982530i	$-0.559586\pi$
$761$	−10.2677	−0.372205	−0.186102	+	0.982530i	$0.559586\pi$
$762$	0	0
$763$	−16.2390	−0.587892
$764$	−2.51247	−0.0908981
$765$	0	0
$766$	76.0875	2.74915
$767$	22.1022	0.798064
$768$	0	0
$769$	17.5193	0.631761	0.315881	−	0.948799i	$-0.397700\pi$
$769$	17.5193	0.631761	0.315881	+	0.948799i	$0.397700\pi$
$770$	16.6241	0.599089
$771$	0	0
$772$	−7.75587	−0.279140
$773$	39.8673	1.43393	0.716964	−	0.697110i	$-0.245531\pi$
$773$	39.8673	1.43393	0.716964	+	0.697110i	$0.245531\pi$
$774$	0	0
$775$	31.1656	1.11950
$776$	29.6356	1.06386
$777$	0	0
$778$	1.40027	0.0502021
$779$	−0.492222	−0.0176357
$780$	0	0
$781$	35.4767	1.26946
$782$	0	0
$783$	0	0
$784$	−2.06843	−0.0738724
$785$	10.9626	0.391271
$786$	0	0
$787$	−8.74536	−0.311738	−0.155869	−	0.987778i	$-0.549818\pi$
$787$	−8.74536	−0.311738	−0.155869	+	0.987778i	$0.549818\pi$
$788$	5.52798	0.196926
$789$	0	0
$790$	−0.512579	−0.0182368
$791$	−13.9745	−0.496876
$792$	0	0
$793$	−64.4313	−2.28802
$794$	57.0060	2.02307
$795$	0	0
$796$	61.8919	2.19370
$797$	−27.9625	−0.990484	−0.495242	−	0.868755i	$-0.664921\pi$
$797$	−27.9625	−0.990484	−0.495242	+	0.868755i	$0.664921\pi$
$798$	0	0
$799$	13.4062	0.474278
$800$	33.4570	1.18288
$801$	0	0
$802$	8.97610	0.316957
$803$	16.7909	0.592537
$804$	0	0
$805$	0	0
$806$	−80.5079	−2.83577
$807$	0	0
$808$	−2.17687	−0.0765820
$809$	−31.7004	−1.11453	−0.557263	−	0.830336i	$-0.688149\pi$
$809$	−31.7004	−1.11453	−0.557263	+	0.830336i	$0.688149\pi$
$810$	0	0
$811$	−3.01197	−0.105765	−0.0528824	−	0.998601i	$-0.516841\pi$
$811$	−3.01197	−0.105765	−0.0528824	+	0.998601i	$0.516841\pi$
$812$	−45.3027	−1.58981
$813$	0	0
$814$	−89.1668	−3.12530
$815$	−12.3916	−0.434059
$816$	0	0
$817$	−0.363644	−0.0127223
$818$	10.4159	0.364182
$819$	0	0
$820$	−7.23636	−0.252705
$821$	−48.9500	−1.70837	−0.854183	−	0.519973i	$-0.825942\pi$
$821$	−48.9500	−1.70837	−0.854183	+	0.519973i	$0.825942\pi$
$822$	0	0
$823$	29.3561	1.02329	0.511645	−	0.859197i	$-0.329036\pi$
$823$	29.3561	1.02329	0.511645	+	0.859197i	$0.329036\pi$
$824$	−1.02304	−0.0356394
$825$	0	0
$826$	−25.8053	−0.897883
$827$	1.94586	0.0676642	0.0338321	−	0.999428i	$-0.489229\pi$
$827$	1.94586	0.0676642	0.0338321	+	0.999428i	$0.489229\pi$
$828$	0	0
$829$	20.0800	0.697406	0.348703	−	0.937233i	$-0.386622\pi$
$829$	20.0800	0.697406	0.348703	+	0.937233i	$0.386622\pi$
$830$	−1.25177	−0.0434494
$831$	0	0
$832$	−69.4802	−2.40879
$833$	−5.07565	−0.175861
$834$	0	0
$835$	−3.62965	−0.125609
$836$	−1.45017	−0.0501553
$837$	0	0
$838$	−66.0083	−2.28022
$839$	45.2643	1.56270	0.781349	−	0.624095i	$-0.214532\pi$
$839$	45.2643	1.56270	0.781349	+	0.624095i	$0.214532\pi$
$840$	0	0
$841$	1.20834	0.0416670
$842$	−77.4890	−2.67045
$843$	0	0
$844$	22.5688	0.776851
$845$	9.90925	0.340889
$846$	0	0
$847$	−24.0741	−0.827195
$848$	15.2717	0.524431
$849$	0	0
$850$	39.0860	1.34064
$851$	0	0
$852$	0	0
$853$	8.33955	0.285541	0.142770	−	0.989756i	$-0.454399\pi$
$853$	8.33955	0.285541	0.142770	+	0.989756i	$0.454399\pi$
$854$	75.2266	2.57420
$855$	0	0
$856$	−21.5020	−0.734922
$857$	−44.1542	−1.50828	−0.754140	−	0.656714i	$-0.771946\pi$
$857$	−44.1542	−1.50828	−0.754140	+	0.656714i	$0.771946\pi$
$858$	0	0
$859$	−4.98381	−0.170045	−0.0850227	−	0.996379i	$-0.527096\pi$
$859$	−4.98381	−0.170045	−0.0850227	+	0.996379i	$0.527096\pi$
$860$	−5.34608	−0.182300
$861$	0	0
$862$	−4.87180	−0.165934
$863$	−46.8167	−1.59366	−0.796829	−	0.604204i	$-0.793491\pi$
$863$	−46.8167	−1.59366	−0.796829	+	0.604204i	$0.793491\pi$
$864$	0	0
$865$	8.55390	0.290841
$866$	−50.1312	−1.70353
$867$	0	0
$868$	55.2825	1.87641
$869$	1.72118	0.0583869
$870$	0	0
$871$	−39.1709	−1.32726
$872$	10.6124	0.359382
$873$	0	0
$874$	0	0
$875$	16.5477	0.559416
$876$	0	0
$877$	32.7521	1.10596	0.552979	−	0.833195i	$-0.313491\pi$
$877$	32.7521	1.10596	0.552979	+	0.833195i	$0.313491\pi$
$878$	56.3289	1.90101
$879$	0	0
$880$	4.06604	0.137066
$881$	20.7147	0.697895	0.348947	−	0.937142i	$-0.386539\pi$
$881$	20.7147	0.697895	0.348947	+	0.937142i	$0.386539\pi$
$882$	0	0
$883$	19.6356	0.660790	0.330395	−	0.943843i	$-0.392818\pi$
$883$	19.6356	0.660790	0.330395	+	0.943843i	$0.392818\pi$
$884$	−59.3826	−1.99725
$885$	0	0
$886$	−3.53589	−0.118791
$887$	4.01145	0.134691	0.0673456	−	0.997730i	$-0.478547\pi$
$887$	4.01145	0.134691	0.0673456	+	0.997730i	$0.478547\pi$
$888$	0	0
$889$	−45.8393	−1.53740
$890$	−7.86792	−0.263733
$891$	0	0
$892$	−34.3182	−1.14906
$893$	−0.405509	−0.0135699
$894$	0	0
$895$	−11.2358	−0.375570
$896$	39.5607	1.32163
$897$	0	0
$898$	23.3335	0.778648
$899$	−36.8629	−1.22945
$900$	0	0
$901$	37.4746	1.24846
$902$	41.3152	1.37565
$903$	0	0
$904$	9.13253	0.303744
$905$	1.23746	0.0411347
$906$	0	0
$907$	22.7616	0.755787	0.377893	−	0.925849i	$-0.376649\pi$
$907$	22.7616	0.755787	0.377893	+	0.925849i	$0.376649\pi$
$908$	0.383634	0.0127314
$909$	0	0
$910$	−20.5906	−0.682573
$911$	21.5042	0.712466	0.356233	−	0.934397i	$-0.384061\pi$
$911$	21.5042	0.712466	0.356233	+	0.934397i	$0.384061\pi$
$912$	0	0
$913$	4.20327	0.139108
$914$	−1.04297	−0.0344985
$915$	0	0
$916$	25.8313	0.853491
$917$	−45.9504	−1.51741
$918$	0	0
$919$	−21.5131	−0.709653	−0.354826	−	0.934932i	$-0.615460\pi$
$919$	−21.5131	−0.709653	−0.354826	+	0.934932i	$0.615460\pi$
$920$	0	0
$921$	0	0
$922$	−45.6739	−1.50419
$923$	−43.9417	−1.44636
$924$	0	0
$925$	−42.7537	−1.40573
$926$	−45.9587	−1.51030
$927$	0	0
$928$	−39.5732	−1.29905
$929$	9.43124	0.309429	0.154715	−	0.987959i	$-0.450554\pi$
$929$	9.43124	0.309429	0.154715	+	0.987959i	$0.450554\pi$
$930$	0	0
$931$	0.153527	0.00503165
$932$	−71.1932	−2.33201
$933$	0	0
$934$	−77.0106	−2.51986
$935$	9.97752	0.326300
$936$	0	0
$937$	−30.5180	−0.996981	−0.498490	−	0.866895i	$-0.666112\pi$
$937$	−30.5180	−0.996981	−0.498490	+	0.866895i	$0.666112\pi$
$938$	45.7339	1.49326
$939$	0	0
$940$	−5.96156	−0.194445
$941$	−17.1830	−0.560151	−0.280075	−	0.959978i	$-0.590359\pi$
$941$	−17.1830	−0.560151	−0.280075	+	0.959978i	$0.590359\pi$
$942$	0	0
$943$	0	0
$944$	−6.31167	−0.205427
$945$	0	0
$946$	30.5228	0.992383
$947$	22.8933	0.743933	0.371967	−	0.928246i	$-0.378684\pi$
$947$	22.8933	0.743933	0.371967	+	0.928246i	$0.378684\pi$
$948$	0	0
$949$	−20.7973	−0.675108
$950$	−1.18227	−0.0383578
$951$	0	0
$952$	20.7788	0.673446
$953$	−17.6887	−0.572995	−0.286497	−	0.958081i	$-0.592491\pi$
$953$	−17.6887	−0.572995	−0.286497	+	0.958081i	$0.592491\pi$
$954$	0	0
$955$	−0.522876	−0.0169199
$956$	12.5874	0.407107
$957$	0	0
$958$	−18.5512	−0.599363
$959$	55.7740	1.80104
$960$	0	0
$961$	13.9834	0.451079
$962$	110.443	3.56081
$963$	0	0
$964$	46.6822	1.50353
$965$	−1.61409	−0.0519593
$966$	0	0
$967$	15.3851	0.494752	0.247376	−	0.968920i	$-0.420432\pi$
$967$	15.3851	0.494752	0.247376	+	0.968920i	$0.420432\pi$
$968$	15.7328	0.505670
$969$	0	0
$970$	20.5789	0.660750
$971$	−2.24847	−0.0721568	−0.0360784	−	0.999349i	$-0.511487\pi$
$971$	−2.24847	−0.0721568	−0.0360784	+	0.999349i	$0.511487\pi$
$972$	0	0
$973$	−2.46832	−0.0791306
$974$	−31.9023	−1.02221
$975$	0	0
$976$	18.3995	0.588953
$977$	44.1125	1.41129	0.705643	−	0.708568i	$-0.250658\pi$
$977$	44.1125	1.41129	0.705643	+	0.708568i	$0.250658\pi$
$978$	0	0
$979$	26.4195	0.844370
$980$	2.25707	0.0720994
$981$	0	0
$982$	37.6451	1.20130
$983$	45.7047	1.45776	0.728878	−	0.684644i	$-0.240042\pi$
$983$	45.7047	1.45776	0.728878	+	0.684644i	$0.240042\pi$
$984$	0	0
$985$	1.15044	0.0366560
$986$	−46.2313	−1.47230
$987$	0	0
$988$	1.79619	0.0571445
$989$	0	0
$990$	0	0
$991$	12.1385	0.385593	0.192796	−	0.981239i	$-0.438244\pi$
$991$	12.1385	0.385593	0.192796	+	0.981239i	$0.438244\pi$
$992$	48.2908	1.53323
$993$	0	0
$994$	51.3040	1.62726
$995$	12.8804	0.408337
$996$	0	0
$997$	−24.7942	−0.785241	−0.392620	−	0.919701i	$-0.628431\pi$
$997$	−24.7942	−0.785241	−0.392620	+	0.919701i	$0.628431\pi$
$998$	−1.94902	−0.0616951
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bq.1.5		5
3.2	odd	2		1587.2.a.p.1.1		5
23.17	odd	22		207.2.i.b.82.1		10
23.19	odd	22		207.2.i.b.154.1		10
23.22	odd	2		4761.2.a.br.1.5		5
69.17	even	22		69.2.e.a.13.1	✓	10
69.65	even	22		69.2.e.a.16.1	yes	10
69.68	even	2		1587.2.a.o.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.a.13.1	✓	10	69.17	even	22
69.2.e.a.16.1	yes	10	69.65	even	22
207.2.i.b.82.1		10	23.17	odd	22
207.2.i.b.154.1		10	23.19	odd	22
1587.2.a.o.1.1		5	69.68	even	2
1587.2.a.p.1.1		5	3.2	odd	2
4761.2.a.bq.1.5		5	1.1	even	1	trivial
4761.2.a.br.1.5		5	23.22	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:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20362 q^{2} +2.85592 q^{4} +0.594351 q^{5} -2.88612 q^{7} +1.88612 q^{8} +O(q^{10})\)	\(q+2.20362 q^{2} +2.85592 q^{4} +0.594351 q^{5} -2.88612 q^{7} +1.88612 q^{8} +1.30972 q^{10} -4.39788 q^{11} +5.44724 q^{13} -6.35991 q^{14} -1.55555 q^{16} -3.81712 q^{17} +0.115460 q^{19} +1.69742 q^{20} -9.69123 q^{22} -4.64675 q^{25} +12.0036 q^{26} -8.24254 q^{28} +5.49621 q^{29} -6.70697 q^{31} -7.20009 q^{32} -8.41148 q^{34} -1.71537 q^{35} +9.20077 q^{37} +0.254429 q^{38} +1.12102 q^{40} -4.26315 q^{41} -3.14953 q^{43} -12.5600 q^{44} -3.51213 q^{47} +1.32970 q^{49} -10.2396 q^{50} +15.5569 q^{52} -9.81750 q^{53} -2.61388 q^{55} -5.44358 q^{56} +12.1115 q^{58} +4.05750 q^{59} -11.8283 q^{61} -14.7796 q^{62} -12.7551 q^{64} +3.23757 q^{65} -7.19097 q^{67} -10.9014 q^{68} -3.78002 q^{70} -8.06679 q^{71} -3.81795 q^{73} +20.2750 q^{74} +0.329744 q^{76} +12.6928 q^{77} -0.391365 q^{79} -0.924545 q^{80} -9.39435 q^{82} -0.955750 q^{83} -2.26871 q^{85} -6.94036 q^{86} -8.29494 q^{88} -6.00732 q^{89} -15.7214 q^{91} -7.73938 q^{94} +0.0686236 q^{95} +15.7125 q^{97} +2.93016 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 4 q^{7} - 9 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 4 * q^7 - 9 * q^8	\( 5 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 4 q^{7} - 9 q^{8} + q^{10} - 13 q^{11} + 7 q^{13} - 16 q^{14} + 6 q^{16} - q^{17} - 5 q^{19} + 2 q^{20} - 3 q^{22} - 10 q^{25} + 27 q^{26} + 12 q^{28} + 7 q^{29} + 8 q^{31} - 6 q^{32} - 18 q^{34} - 9 q^{35} + 7 q^{37} + 9 q^{38} + 12 q^{40} + 7 q^{41} - 10 q^{43} - 17 q^{44} - 13 q^{47} - q^{49} - 4 q^{50} - q^{52} - 17 q^{53} + 10 q^{55} - 38 q^{56} - 17 q^{58} + 10 q^{59} + 18 q^{61} + q^{62} - 21 q^{64} - 24 q^{65} + 9 q^{67} - 25 q^{68} + 3 q^{70} - 37 q^{71} - 2 q^{73} + 5 q^{74} - 4 q^{76} - 6 q^{77} - 32 q^{79} - 8 q^{80} - 17 q^{82} - 13 q^{83} + 5 q^{85} - 26 q^{86} + 19 q^{88} - 30 q^{89} - 23 q^{91} + 8 q^{94} + 14 q^{95} + 20 q^{97} - 51 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 4 * q^7 - 9 * q^8 + q^10 - 13 * q^11 + 7 * q^13 - 16 * q^14 + 6 * q^16 - q^17 - 5 * q^19 + 2 * q^20 - 3 * q^22 - 10 * q^25 + 27 * q^26 + 12 * q^28 + 7 * q^29 + 8 * q^31 - 6 * q^32 - 18 * q^34 - 9 * q^35 + 7 * q^37 + 9 * q^38 + 12 * q^40 + 7 * q^41 - 10 * q^43 - 17 * q^44 - 13 * q^47 - q^49 - 4 * q^50 - q^52 - 17 * q^53 + 10 * q^55 - 38 * q^56 - 17 * q^58 + 10 * q^59 + 18 * q^61 + q^62 - 21 * q^64 - 24 * q^65 + 9 * q^67 - 25 * q^68 + 3 * q^70 - 37 * q^71 - 2 * q^73 + 5 * q^74 - 4 * q^76 - 6 * q^77 - 32 * q^79 - 8 * q^80 - 17 * q^82 - 13 * q^83 + 5 * q^85 - 26 * q^86 + 19 * q^88 - 30 * q^89 - 23 * q^91 + 8 * q^94 + 14 * q^95 + 20 * q^97 - 51 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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