LMFDB - Embedded newform 4761.2.a.bu.1.1

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:	$10$
Coefficient field:	10.10.5791333887977.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 12x^{8} + 22x^{7} + 49x^{6} - 84x^{5} - 73x^{4} + 132x^{3} + 17x^{2} - 74x + 23 $ x^10 - 2x^9 - 12x^8 + 22x^7 + 49x^6 - 84x^5 - 73x^4 + 132x^3 + 17x^2 - 74*x + 23
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$-2.10158$ 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)

$f(q)$	$=$	$q-2.10158 q^{2} +2.41664 q^{4} +3.44976 q^{5} -1.58175 q^{7} -0.875613 q^{8} +O(q^{10})$	$q-2.10158 q^{2} +2.41664 q^{4} +3.44976 q^{5} -1.58175 q^{7} -0.875613 q^{8} -7.24995 q^{10} +1.37979 q^{11} -4.55358 q^{13} +3.32418 q^{14} -2.99312 q^{16} -4.04550 q^{17} -6.10747 q^{19} +8.33685 q^{20} -2.89974 q^{22} +6.90085 q^{25} +9.56973 q^{26} -3.82253 q^{28} +10.0916 q^{29} +1.13727 q^{31} +8.04151 q^{32} +8.50196 q^{34} -5.45667 q^{35} +8.26429 q^{37} +12.8353 q^{38} -3.02065 q^{40} -4.90225 q^{41} +2.87439 q^{43} +3.33446 q^{44} -4.66656 q^{47} -4.49806 q^{49} -14.5027 q^{50} -11.0044 q^{52} +6.53652 q^{53} +4.75994 q^{55} +1.38500 q^{56} -21.2084 q^{58} -0.947273 q^{59} -0.762386 q^{61} -2.39007 q^{62} -10.9136 q^{64} -15.7088 q^{65} +2.79290 q^{67} -9.77655 q^{68} +11.4676 q^{70} +5.67469 q^{71} +0.910042 q^{73} -17.3681 q^{74} -14.7596 q^{76} -2.18249 q^{77} -9.74717 q^{79} -10.3255 q^{80} +10.3025 q^{82} -4.85439 q^{83} -13.9560 q^{85} -6.04077 q^{86} -1.20816 q^{88} -3.66643 q^{89} +7.20264 q^{91} +9.80716 q^{94} -21.0693 q^{95} +1.70609 q^{97} +9.45304 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8}+O(q^{10}) $ 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8	$ 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 13 q^{10} + 3 q^{11} - 4 q^{13} - 4 q^{16} + 11 q^{17} - 22 q^{19} + q^{20} - 13 q^{22} - 2 q^{25} - 4 q^{26} - 26 q^{28} + 5 q^{29} - 7 q^{31} + 34 q^{32} - 4 q^{34} - 9 q^{35} - 35 q^{37} - 9 q^{38} - 21 q^{40} - 28 q^{43} - 7 q^{44} - 9 q^{47} + 17 q^{49} - 52 q^{52} + 34 q^{53} - 14 q^{55} - 30 q^{56} - 24 q^{58} + 2 q^{59} - 49 q^{61} + 28 q^{62} + 10 q^{64} + 2 q^{65} - 26 q^{67} - 6 q^{68} + 16 q^{70} - 15 q^{71} + 14 q^{73} - 25 q^{74} - 19 q^{76} + 33 q^{77} - 43 q^{79} - 49 q^{80} + 24 q^{82} - 15 q^{83} - 21 q^{85} - 49 q^{86} - 15 q^{88} - 15 q^{89} - 4 q^{91} - 28 q^{94} - 28 q^{95} - 22 q^{97} - 15 q^{98}+O(q^{100}) $ 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8 - 13 * q^10 + 3 * q^11 - 4 * q^13 - 4 * q^16 + 11 * q^17 - 22 * q^19 + q^20 - 13 * q^22 - 2 * q^25 - 4 * q^26 - 26 * q^28 + 5 * q^29 - 7 * q^31 + 34 * q^32 - 4 * q^34 - 9 * q^35 - 35 * q^37 - 9 * q^38 - 21 * q^40 - 28 * q^43 - 7 * q^44 - 9 * q^47 + 17 * q^49 - 52 * q^52 + 34 * q^53 - 14 * q^55 - 30 * q^56 - 24 * q^58 + 2 * q^59 - 49 * q^61 + 28 * q^62 + 10 * q^64 + 2 * q^65 - 26 * q^67 - 6 * q^68 + 16 * q^70 - 15 * q^71 + 14 * q^73 - 25 * q^74 - 19 * q^76 + 33 * q^77 - 43 * q^79 - 49 * q^80 + 24 * q^82 - 15 * q^83 - 21 * q^85 - 49 * q^86 - 15 * q^88 - 15 * q^89 - 4 * q^91 - 28 * q^94 - 28 * q^95 - 22 * q^97 - 15 * 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 $

$2$	−2.10158	−1.48604	−0.743021	−	0.669268i	$-0.766608\pi$
$2$	−2.10158	−1.48604	−0.743021	+	0.669268i	$0.766608\pi$
$3$	0	0
$3$	0	0
$4$	2.41664	1.20832
$5$	3.44976	1.54278	0.771390	−	0.636363i	$-0.219562\pi$
$5$	3.44976	1.54278	0.771390	+	0.636363i	$0.219562\pi$
$6$	0	0
$7$	−1.58175	−0.597846	−0.298923	−	0.954277i	$-0.596627\pi$
$7$	−1.58175	−0.597846	−0.298923	+	0.954277i	$0.596627\pi$
$8$	−0.875613	−0.309576
$9$	0	0
$10$	−7.24995	−2.29264
$11$	1.37979	0.416022	0.208011	−	0.978126i	$-0.433301\pi$
$11$	1.37979	0.416022	0.208011	+	0.978126i	$0.433301\pi$
$12$	0	0
$13$	−4.55358	−1.26294	−0.631468	−	0.775402i	$-0.717547\pi$
$13$	−4.55358	−1.26294	−0.631468	+	0.775402i	$0.717547\pi$
$14$	3.32418	0.888425
$15$	0	0
$16$	−2.99312	−0.748279
$17$	−4.04550	−0.981179	−0.490589	−	0.871391i	$-0.663218\pi$
$17$	−4.04550	−0.981179	−0.490589	+	0.871391i	$0.663218\pi$
$18$	0	0
$19$	−6.10747	−1.40115	−0.700574	−	0.713579i	$-0.747073\pi$
$19$	−6.10747	−1.40115	−0.700574	+	0.713579i	$0.747073\pi$
$20$	8.33685	1.86418
$21$	0	0
$22$	−2.89974	−0.618227
$23$	0	0
$23$	0	0
$24$	0	0
$25$	6.90085	1.38017
$26$	9.56973	1.87678
$27$	0	0
$28$	−3.82253	−0.722391
$29$	10.0916	1.87397	0.936983	−	0.349374i	$-0.113606\pi$
$29$	10.0916	1.87397	0.936983	+	0.349374i	$0.113606\pi$
$30$	0	0
$31$	1.13727	0.204260	0.102130	−	0.994771i	$-0.467434\pi$
$31$	1.13727	0.204260	0.102130	+	0.994771i	$0.467434\pi$
$32$	8.04151	1.42155
$33$	0	0
$34$	8.50196	1.45807
$35$	−5.45667	−0.922345
$36$	0	0
$37$	8.26429	1.35864	0.679320	−	0.733842i	$-0.262275\pi$
$37$	8.26429	1.35864	0.679320	+	0.733842i	$0.262275\pi$
$38$	12.8353	2.08217
$39$	0	0
$40$	−3.02065	−0.477607
$41$	−4.90225	−0.765603	−0.382802	−	0.923831i	$-0.625041\pi$
$41$	−4.90225	−0.765603	−0.382802	+	0.923831i	$0.625041\pi$
$42$	0	0
$43$	2.87439	0.438340	0.219170	−	0.975687i	$-0.429665\pi$
$43$	2.87439	0.438340	0.219170	+	0.975687i	$0.429665\pi$
$44$	3.33446	0.502689
$45$	0	0
$46$	0	0
$47$	−4.66656	−0.680688	−0.340344	−	0.940301i	$-0.610544\pi$
$47$	−4.66656	−0.680688	−0.340344	+	0.940301i	$0.610544\pi$
$48$	0	0
$49$	−4.49806	−0.642580
$50$	−14.5027	−2.05099
$51$	0	0
$52$	−11.0044	−1.52603
$53$	6.53652	0.897859	0.448930	−	0.893567i	$-0.351805\pi$
$53$	6.53652	0.897859	0.448930	+	0.893567i	$0.351805\pi$
$54$	0	0
$55$	4.75994	0.641831
$56$	1.38500	0.185079
$57$	0	0
$58$	−21.2084	−2.78479
$59$	−0.947273	−0.123324	−0.0616622	−	0.998097i	$-0.519640\pi$
$59$	−0.947273	−0.123324	−0.0616622	+	0.998097i	$0.519640\pi$
$60$	0	0
$61$	−0.762386	−0.0976135	−0.0488068	−	0.998808i	$-0.515542\pi$
$61$	−0.762386	−0.0976135	−0.0488068	+	0.998808i	$0.515542\pi$
$62$	−2.39007	−0.303539
$63$	0	0
$64$	−10.9136	−1.36421
$65$	−15.7088	−1.94843
$66$	0	0
$67$	2.79290	0.341207	0.170603	−	0.985340i	$-0.445428\pi$
$67$	2.79290	0.341207	0.170603	+	0.985340i	$0.445428\pi$
$68$	−9.77655	−1.18558
$69$	0	0
$70$	11.4676	1.37064
$71$	5.67469	0.673462	0.336731	−	0.941601i	$-0.390679\pi$
$71$	5.67469	0.673462	0.336731	+	0.941601i	$0.390679\pi$
$72$	0	0
$73$	0.910042	0.106512	0.0532562	−	0.998581i	$-0.483040\pi$
$73$	0.910042	0.106512	0.0532562	+	0.998581i	$0.483040\pi$
$74$	−17.3681	−2.01900
$75$	0	0
$76$	−14.7596	−1.69304
$77$	−2.18249	−0.248717
$78$	0	0
$79$	−9.74717	−1.09664	−0.548321	−	0.836268i	$-0.684733\pi$
$79$	−9.74717	−1.09664	−0.548321	+	0.836268i	$0.684733\pi$
$80$	−10.3255	−1.15443
$81$	0	0
$82$	10.3025	1.13772
$83$	−4.85439	−0.532838	−0.266419	−	0.963857i	$-0.585841\pi$
$83$	−4.85439	−0.532838	−0.266419	+	0.963857i	$0.585841\pi$
$84$	0	0
$85$	−13.9560	−1.51374
$86$	−6.04077	−0.651392
$87$	0	0
$88$	−1.20816	−0.128790
$89$	−3.66643	−0.388641	−0.194321	−	0.980938i	$-0.562250\pi$
$89$	−3.66643	−0.388641	−0.194321	+	0.980938i	$0.562250\pi$
$90$	0	0
$91$	7.20264	0.755042
$92$	0	0
$93$	0	0
$94$	9.80716	1.01153
$95$	−21.0693	−2.16166
$96$	0	0
$97$	1.70609	0.173227	0.0866137	−	0.996242i	$-0.472395\pi$
$97$	1.70609	0.173227	0.0866137	+	0.996242i	$0.472395\pi$
$98$	9.45304	0.954901
$99$	0	0
$100$	16.6769	1.66769
$101$	−6.88534	−0.685117	−0.342558	−	0.939497i	$-0.611293\pi$
$101$	−6.88534	−0.685117	−0.342558	+	0.939497i	$0.611293\pi$
$102$	0	0
$103$	−0.287590	−0.0283371	−0.0141685	−	0.999900i	$-0.504510\pi$
$103$	−0.287590	−0.0283371	−0.0141685	+	0.999900i	$0.504510\pi$
$104$	3.98717	0.390975
$105$	0	0
$106$	−13.7370	−1.33426
$107$	8.29060	0.801483	0.400741	−	0.916191i	$-0.368753\pi$
$107$	8.29060	0.801483	0.400741	+	0.916191i	$0.368753\pi$
$108$	0	0
$109$	−20.5483	−1.96817	−0.984085	−	0.177697i	$-0.943135\pi$
$109$	−20.5483	−1.96817	−0.984085	+	0.177697i	$0.943135\pi$
$110$	−10.0034	−0.953788
$111$	0	0
$112$	4.73437	0.447356
$113$	−1.62281	−0.152661	−0.0763307	−	0.997083i	$-0.524320\pi$
$113$	−1.62281	−0.152661	−0.0763307	+	0.997083i	$0.524320\pi$
$114$	0	0
$115$	0	0
$116$	24.3879	2.26436
$117$	0	0
$118$	1.99077	0.183265
$119$	6.39899	0.586594
$120$	0	0
$121$	−9.09618	−0.826926
$122$	1.60222	0.145058
$123$	0	0
$124$	2.74838	0.246812
$125$	6.55748	0.586519
$126$	0	0
$127$	9.69077	0.859917	0.429958	−	0.902849i	$-0.358528\pi$
$127$	9.69077	0.859917	0.429958	+	0.902849i	$0.358528\pi$
$128$	6.85290	0.605717
$129$	0	0
$130$	33.0133	2.89545
$131$	−11.4566	−1.00097	−0.500486	−	0.865745i	$-0.666845\pi$
$131$	−11.4566	−1.00097	−0.500486	+	0.865745i	$0.666845\pi$
$132$	0	0
$133$	9.66050	0.837672
$134$	−5.86950	−0.507047
$135$	0	0
$136$	3.54229	0.303749
$137$	1.64372	0.140433	0.0702163	−	0.997532i	$-0.477631\pi$
$137$	1.64372	0.140433	0.0702163	+	0.997532i	$0.477631\pi$
$138$	0	0
$139$	−5.57603	−0.472952	−0.236476	−	0.971637i	$-0.575993\pi$
$139$	−5.57603	−0.472952	−0.236476	+	0.971637i	$0.575993\pi$
$140$	−13.1868	−1.11449
$141$	0	0
$142$	−11.9258	−1.00079
$143$	−6.28298	−0.525410
$144$	0	0
$145$	34.8137	2.89112
$146$	−1.91253	−0.158282
$147$	0	0
$148$	19.9718	1.64168
$149$	6.08189	0.498248	0.249124	−	0.968472i	$-0.419857\pi$
$149$	6.08189	0.498248	0.249124	+	0.968472i	$0.419857\pi$
$150$	0	0
$151$	−23.6673	−1.92601	−0.963007	−	0.269476i	$-0.913149\pi$
$151$	−23.6673	−1.92601	−0.963007	+	0.269476i	$0.913149\pi$
$152$	5.34777	0.433762
$153$	0	0
$154$	4.58667	0.369604
$155$	3.92331	0.315128
$156$	0	0
$157$	−15.5790	−1.24334	−0.621672	−	0.783278i	$-0.713546\pi$
$157$	−15.5790	−1.24334	−0.621672	+	0.783278i	$0.713546\pi$
$158$	20.4845	1.62966
$159$	0	0
$160$	27.7413	2.19314
$161$	0	0
$162$	0	0
$163$	−6.16185	−0.482633	−0.241317	−	0.970446i	$-0.577579\pi$
$163$	−6.16185	−0.482633	−0.241317	+	0.970446i	$0.577579\pi$
$164$	−11.8470	−0.925096
$165$	0	0
$166$	10.2019	0.791820
$167$	−9.44372	−0.730777	−0.365388	−	0.930855i	$-0.619064\pi$
$167$	−9.44372	−0.730777	−0.365388	+	0.930855i	$0.619064\pi$
$168$	0	0
$169$	7.73512	0.595009
$170$	29.3297	2.24949
$171$	0	0
$172$	6.94638	0.529656
$173$	−7.09393	−0.539341	−0.269671	−	0.962953i	$-0.586915\pi$
$173$	−7.09393	−0.539341	−0.269671	+	0.962953i	$0.586915\pi$
$174$	0	0
$175$	−10.9154	−0.825130
$176$	−4.12987	−0.311301
$177$	0	0
$178$	7.70531	0.577537
$179$	−8.02689	−0.599958	−0.299979	−	0.953946i	$-0.596980\pi$
$179$	−8.02689	−0.599958	−0.299979	+	0.953946i	$0.596980\pi$
$180$	0	0
$181$	11.7891	0.876274	0.438137	−	0.898908i	$-0.355638\pi$
$181$	11.7891	0.876274	0.438137	+	0.898908i	$0.355638\pi$
$182$	−15.1369	−1.12202
$183$	0	0
$184$	0	0
$185$	28.5098	2.09608
$186$	0	0
$187$	−5.58194	−0.408192
$188$	−11.2774	−0.822491
$189$	0	0
$190$	44.2788	3.21233
$191$	−26.4497	−1.91384	−0.956918	−	0.290358i	$-0.906225\pi$
$191$	−26.4497	−1.91384	−0.956918	+	0.290358i	$0.906225\pi$
$192$	0	0
$193$	18.8490	1.35678	0.678389	−	0.734703i	$-0.262679\pi$
$193$	18.8490	1.35678	0.678389	+	0.734703i	$0.262679\pi$
$194$	−3.58549	−0.257423
$195$	0	0
$196$	−10.8702	−0.776443
$197$	−8.81816	−0.628268	−0.314134	−	0.949379i	$-0.601714\pi$
$197$	−8.81816	−0.628268	−0.314134	+	0.949379i	$0.601714\pi$
$198$	0	0
$199$	−15.0061	−1.06375	−0.531876	−	0.846822i	$-0.678513\pi$
$199$	−15.0061	−1.06375	−0.531876	+	0.846822i	$0.678513\pi$
$200$	−6.04247	−0.427267
$201$	0	0
$202$	14.4701	1.01811
$203$	−15.9624	−1.12034
$204$	0	0
$205$	−16.9116	−1.18116
$206$	0.604394	0.0421101
$207$	0	0
$208$	13.6294	0.945030
$209$	−8.42702	−0.582909
$210$	0	0
$211$	−26.7256	−1.83987	−0.919933	−	0.392075i	$-0.871757\pi$
$211$	−26.7256	−1.83987	−0.919933	+	0.392075i	$0.871757\pi$
$212$	15.7964	1.08490
$213$	0	0
$214$	−17.4234	−1.19104
$215$	9.91596	0.676263
$216$	0	0
$217$	−1.79888	−0.122116
$218$	43.1839	2.92479
$219$	0	0
$220$	11.5031	0.775538
$221$	18.4215	1.23917
$222$	0	0
$223$	12.2946	0.823310	0.411655	−	0.911340i	$-0.364951\pi$
$223$	12.2946	0.823310	0.411655	+	0.911340i	$0.364951\pi$
$224$	−12.7197	−0.849869
$225$	0	0
$226$	3.41047	0.226861
$227$	−16.4827	−1.09400	−0.546998	−	0.837134i	$-0.684229\pi$
$227$	−16.4827	−1.09400	−0.546998	+	0.837134i	$0.684229\pi$
$228$	0	0
$229$	0.636413	0.0420554	0.0210277	−	0.999779i	$-0.493306\pi$
$229$	0.636413	0.0420554	0.0210277	+	0.999779i	$0.493306\pi$
$230$	0	0
$231$	0	0
$232$	−8.83635	−0.580135
$233$	−0.643582	−0.0421625	−0.0210812	−	0.999778i	$-0.506711\pi$
$233$	−0.643582	−0.0421625	−0.0210812	+	0.999778i	$0.506711\pi$
$234$	0	0
$235$	−16.0985	−1.05015
$236$	−2.28922	−0.149016
$237$	0	0
$238$	−13.4480	−0.871704
$239$	9.62895	0.622845	0.311422	−	0.950272i	$-0.399195\pi$
$239$	9.62895	0.622845	0.311422	+	0.950272i	$0.399195\pi$
$240$	0	0
$241$	5.96496	0.384237	0.192119	−	0.981372i	$-0.438464\pi$
$241$	5.96496	0.384237	0.192119	+	0.981372i	$0.438464\pi$
$242$	19.1164	1.22885
$243$	0	0
$244$	−1.84242	−0.117949
$245$	−15.5172	−0.991359
$246$	0	0
$247$	27.8109	1.76956
$248$	−0.995808	−0.0632339
$249$	0	0
$250$	−13.7811	−0.871592
$251$	−3.85471	−0.243307	−0.121654	−	0.992573i	$-0.538820\pi$
$251$	−3.85471	−0.243307	−0.121654	+	0.992573i	$0.538820\pi$
$252$	0	0
$253$	0	0
$254$	−20.3659	−1.27787
$255$	0	0
$256$	7.42536	0.464085
$257$	11.8109	0.736741	0.368371	−	0.929679i	$-0.379916\pi$
$257$	11.8109	0.736741	0.368371	+	0.929679i	$0.379916\pi$
$258$	0	0
$259$	−13.0721	−0.812258
$260$	−37.9625	−2.35434
$261$	0	0
$262$	24.0771	1.48749
$263$	14.6534	0.903569	0.451784	−	0.892127i	$-0.350788\pi$
$263$	14.6534	0.903569	0.451784	+	0.892127i	$0.350788\pi$
$264$	0	0
$265$	22.5494	1.38520
$266$	−20.3023	−1.24482
$267$	0	0
$268$	6.74944	0.412288
$269$	27.2170	1.65945	0.829724	−	0.558174i	$-0.188498\pi$
$269$	27.2170	1.65945	0.829724	+	0.558174i	$0.188498\pi$
$270$	0	0
$271$	−4.28396	−0.260232	−0.130116	−	0.991499i	$-0.541535\pi$
$271$	−4.28396	−0.260232	−0.130116	+	0.991499i	$0.541535\pi$
$272$	12.1087	0.734196
$273$	0	0
$274$	−3.45442	−0.208689
$275$	9.52172	0.574181
$276$	0	0
$277$	6.86176	0.412283	0.206142	−	0.978522i	$-0.433909\pi$
$277$	6.86176	0.412283	0.206142	+	0.978522i	$0.433909\pi$
$278$	11.7185	0.702827
$279$	0	0
$280$	4.77793	0.285536
$281$	−13.3058	−0.793755	−0.396878	−	0.917871i	$-0.629906\pi$
$281$	−13.3058	−0.793755	−0.396878	+	0.917871i	$0.629906\pi$
$282$	0	0
$283$	−19.2217	−1.14261	−0.571307	−	0.820736i	$-0.693563\pi$
$283$	−19.2217	−1.14261	−0.571307	+	0.820736i	$0.693563\pi$
$284$	13.7137	0.813759
$285$	0	0
$286$	13.2042	0.780781
$287$	7.75415	0.457713
$288$	0	0
$289$	−0.633899	−0.0372882
$290$	−73.1638	−4.29633
$291$	0	0
$292$	2.19925	0.128701
$293$	13.6447	0.797133	0.398566	−	0.917139i	$-0.369508\pi$
$293$	13.6447	0.797133	0.398566	+	0.917139i	$0.369508\pi$
$294$	0	0
$295$	−3.26786	−0.190262
$296$	−7.23631	−0.420602
$297$	0	0
$298$	−12.7816	−0.740417
$299$	0	0
$300$	0	0
$301$	−4.54657	−0.262060
$302$	49.7387	2.86214
$303$	0	0
$304$	18.2804	1.04845
$305$	−2.63005	−0.150596
$306$	0	0
$307$	−25.4313	−1.45144	−0.725721	−	0.687989i	$-0.758494\pi$
$307$	−25.4313	−1.45144	−0.725721	+	0.687989i	$0.758494\pi$
$308$	−5.27429	−0.300531
$309$	0	0
$310$	−8.24516	−0.468294
$311$	31.4782	1.78496	0.892481	−	0.451085i	$-0.148963\pi$
$311$	31.4782	1.78496	0.892481	+	0.451085i	$0.148963\pi$
$312$	0	0
$313$	−13.4694	−0.761338	−0.380669	−	0.924711i	$-0.624306\pi$
$313$	−13.4694	−0.761338	−0.380669	+	0.924711i	$0.624306\pi$
$314$	32.7406	1.84766
$315$	0	0
$316$	−23.5554	−1.32510
$317$	3.84870	0.216165	0.108082	−	0.994142i	$-0.465529\pi$
$317$	3.84870	0.216165	0.108082	+	0.994142i	$0.465529\pi$
$318$	0	0
$319$	13.9243	0.779612
$320$	−37.6495	−2.10467
$321$	0	0
$322$	0	0
$323$	24.7078	1.37478
$324$	0	0
$325$	−31.4236	−1.74307
$326$	12.9496	0.717214
$327$	0	0
$328$	4.29247	0.237012
$329$	7.38135	0.406947
$330$	0	0
$331$	−20.2085	−1.11076	−0.555381	−	0.831596i	$-0.687427\pi$
$331$	−20.2085	−1.11076	−0.555381	+	0.831596i	$0.687427\pi$
$332$	−11.7313	−0.643840
$333$	0	0
$334$	19.8467	1.08597
$335$	9.63482	0.526407
$336$	0	0
$337$	−11.0123	−0.599877	−0.299939	−	0.953959i	$-0.596966\pi$
$337$	−11.0123	−0.599877	−0.299939	+	0.953959i	$0.596966\pi$
$338$	−16.2560	−0.884209
$339$	0	0
$340$	−33.7267	−1.82909
$341$	1.56919	0.0849766
$342$	0	0
$343$	18.1871	0.982010
$344$	−2.51685	−0.135700
$345$	0	0
$346$	14.9085	0.801484
$347$	26.3629	1.41523	0.707617	−	0.706596i	$-0.249770\pi$
$347$	26.3629	1.41523	0.707617	+	0.706596i	$0.249770\pi$
$348$	0	0
$349$	18.6963	1.00079	0.500394	−	0.865798i	$-0.333188\pi$
$349$	18.6963	1.00079	0.500394	+	0.865798i	$0.333188\pi$
$350$	22.9397	1.22618
$351$	0	0
$352$	11.0956	0.591397
$353$	−23.5876	−1.25544	−0.627722	−	0.778438i	$-0.716012\pi$
$353$	−23.5876	−1.25544	−0.627722	+	0.778438i	$0.716012\pi$
$354$	0	0
$355$	19.5763	1.03900
$356$	−8.86047	−0.469604
$357$	0	0
$358$	16.8692	0.891563
$359$	−15.0554	−0.794594	−0.397297	−	0.917690i	$-0.630052\pi$
$359$	−15.0554	−0.794594	−0.397297	+	0.917690i	$0.630052\pi$
$360$	0	0
$361$	18.3011	0.963218
$362$	−24.7757	−1.30218
$363$	0	0
$364$	17.4062	0.912334
$365$	3.13943	0.164325
$366$	0	0
$367$	−33.1504	−1.73043	−0.865217	−	0.501397i	$-0.832820\pi$
$367$	−33.1504	−1.73043	−0.865217	+	0.501397i	$0.832820\pi$
$368$	0	0
$369$	0	0
$370$	−59.9157	−3.11487
$371$	−10.3392	−0.536782
$372$	0	0
$373$	−32.9380	−1.70546	−0.852732	−	0.522349i	$-0.825056\pi$
$373$	−32.9380	−1.70546	−0.852732	+	0.522349i	$0.825056\pi$
$374$	11.7309	0.606591
$375$	0	0
$376$	4.08610	0.210725
$377$	−45.9530	−2.36670
$378$	0	0
$379$	−22.0053	−1.13034	−0.565168	−	0.824976i	$-0.691189\pi$
$379$	−22.0053	−1.13034	−0.565168	+	0.824976i	$0.691189\pi$
$380$	−50.9170	−2.61199
$381$	0	0
$382$	55.5863	2.84404
$383$	4.68252	0.239266	0.119633	−	0.992818i	$-0.461828\pi$
$383$	4.68252	0.239266	0.119633	+	0.992818i	$0.461828\pi$
$384$	0	0
$385$	−7.52905	−0.383716
$386$	−39.6126	−2.01623
$387$	0	0
$388$	4.12302	0.209315
$389$	−14.0305	−0.711373	−0.355687	−	0.934605i	$-0.615753\pi$
$389$	−14.0305	−0.711373	−0.355687	+	0.934605i	$0.615753\pi$
$390$	0	0
$391$	0	0
$392$	3.93856	0.198927
$393$	0	0
$394$	18.5321	0.933633
$395$	−33.6254	−1.69188
$396$	0	0
$397$	11.2613	0.565189	0.282595	−	0.959239i	$-0.408805\pi$
$397$	11.2613	0.565189	0.282595	+	0.959239i	$0.408805\pi$
$398$	31.5365	1.58078
$399$	0	0
$400$	−20.6551	−1.03275
$401$	−17.1233	−0.855097	−0.427548	−	0.903992i	$-0.640623\pi$
$401$	−17.1233	−0.855097	−0.427548	+	0.903992i	$0.640623\pi$
$402$	0	0
$403$	−5.17866	−0.257967
$404$	−16.6394	−0.827842
$405$	0	0
$406$	33.5464	1.66488
$407$	11.4030	0.565224
$408$	0	0
$409$	−8.98166	−0.444115	−0.222057	−	0.975034i	$-0.571277\pi$
$409$	−8.98166	−0.444115	−0.222057	+	0.975034i	$0.571277\pi$
$410$	35.5411	1.75525
$411$	0	0
$412$	−0.695003	−0.0342403
$413$	1.49835	0.0737290
$414$	0	0
$415$	−16.7465	−0.822052
$416$	−36.6177	−1.79533
$417$	0	0
$418$	17.7101	0.866227
$419$	−7.04191	−0.344020	−0.172010	−	0.985095i	$-0.555026\pi$
$419$	−7.04191	−0.344020	−0.172010	+	0.985095i	$0.555026\pi$
$420$	0	0
$421$	−12.2973	−0.599332	−0.299666	−	0.954044i	$-0.596875\pi$
$421$	−12.2973	−0.599332	−0.299666	+	0.954044i	$0.596875\pi$
$422$	56.1660	2.73412
$423$	0	0
$424$	−5.72346	−0.277956
$425$	−27.9174	−1.35419
$426$	0	0
$427$	1.20591	0.0583579
$428$	20.0354	0.968450
$429$	0	0
$430$	−20.8392	−1.00496
$431$	5.04310	0.242917	0.121459	−	0.992596i	$-0.461243\pi$
$431$	5.04310	0.242917	0.121459	+	0.992596i	$0.461243\pi$
$432$	0	0
$433$	0.889545	0.0427488	0.0213744	−	0.999772i	$-0.493196\pi$
$433$	0.889545	0.0427488	0.0213744	+	0.999772i	$0.493196\pi$
$434$	3.78049	0.181470
$435$	0	0
$436$	−49.6580	−2.37818
$437$	0	0
$438$	0	0
$439$	39.8837	1.90354	0.951772	−	0.306807i	$-0.0992607\pi$
$439$	39.8837	1.90354	0.951772	+	0.306807i	$0.0992607\pi$
$440$	−4.16787	−0.198695
$441$	0	0
$442$	−38.7144	−1.84145
$443$	−12.8103	−0.608636	−0.304318	−	0.952570i	$-0.598429\pi$
$443$	−12.8103	−0.608636	−0.304318	+	0.952570i	$0.598429\pi$
$444$	0	0
$445$	−12.6483	−0.599588
$446$	−25.8382	−1.22347
$447$	0	0
$448$	17.2627	0.815585
$449$	13.3440	0.629744	0.314872	−	0.949134i	$-0.398038\pi$
$449$	13.3440	0.629744	0.314872	+	0.949134i	$0.398038\pi$
$450$	0	0
$451$	−6.76408	−0.318508
$452$	−3.92176	−0.184464
$453$	0	0
$454$	34.6398	1.62572
$455$	24.8474	1.16486
$456$	0	0
$457$	−23.0625	−1.07882	−0.539409	−	0.842044i	$-0.681352\pi$
$457$	−23.0625	−1.07882	−0.539409	+	0.842044i	$0.681352\pi$
$458$	−1.33747	−0.0624961
$459$	0	0
$460$	0	0
$461$	7.59985	0.353960	0.176980	−	0.984214i	$-0.443367\pi$
$461$	7.59985	0.353960	0.176980	+	0.984214i	$0.443367\pi$
$462$	0	0
$463$	10.1675	0.472522	0.236261	−	0.971690i	$-0.424078\pi$
$463$	10.1675	0.472522	0.236261	+	0.971690i	$0.424078\pi$
$464$	−30.2054	−1.40225
$465$	0	0
$466$	1.35254	0.0626552
$467$	−10.3330	−0.478154	−0.239077	−	0.971001i	$-0.576845\pi$
$467$	−10.3330	−0.478154	−0.239077	+	0.971001i	$0.576845\pi$
$468$	0	0
$469$	−4.41767	−0.203989
$470$	33.8324	1.56057
$471$	0	0
$472$	0.829444	0.0381783
$473$	3.96605	0.182359
$474$	0	0
$475$	−42.1467	−1.93382
$476$	15.4641	0.708795
$477$	0	0
$478$	−20.2360	−0.925574
$479$	−8.71133	−0.398031	−0.199015	−	0.979996i	$-0.563774\pi$
$479$	−8.71133	−0.398031	−0.199015	+	0.979996i	$0.563774\pi$
$480$	0	0
$481$	−37.6321	−1.71588
$482$	−12.5359	−0.570993
$483$	0	0
$484$	−21.9822	−0.999193
$485$	5.88561	0.267252
$486$	0	0
$487$	22.7674	1.03169	0.515844	−	0.856682i	$-0.327478\pi$
$487$	22.7674	1.03169	0.515844	+	0.856682i	$0.327478\pi$
$488$	0.667555	0.0302188
$489$	0	0
$490$	32.6107	1.47320
$491$	−28.0432	−1.26557	−0.632785	−	0.774328i	$-0.718088\pi$
$491$	−28.0432	−1.26557	−0.632785	+	0.774328i	$0.718088\pi$
$492$	0	0
$493$	−40.8257	−1.83870
$494$	−58.4468	−2.62964
$495$	0	0
$496$	−3.40398	−0.152843
$497$	−8.97596	−0.402627
$498$	0	0
$499$	−14.6488	−0.655771	−0.327885	−	0.944718i	$-0.606336\pi$
$499$	−14.6488	−0.655771	−0.327885	+	0.944718i	$0.606336\pi$
$500$	15.8471	0.708704
$501$	0	0
$502$	8.10099	0.361565
$503$	−1.61306	−0.0719230	−0.0359615	−	0.999353i	$-0.511449\pi$
$503$	−1.61306	−0.0719230	−0.0359615	+	0.999353i	$0.511449\pi$
$504$	0	0
$505$	−23.7528	−1.05698
$506$	0	0
$507$	0	0
$508$	23.4191	1.03906
$509$	24.4829	1.08518	0.542592	−	0.839997i	$-0.317443\pi$
$509$	24.4829	1.08518	0.542592	+	0.839997i	$0.317443\pi$
$510$	0	0
$511$	−1.43946	−0.0636781
$512$	−29.3108	−1.29537
$513$	0	0
$514$	−24.8215	−1.09483
$515$	−0.992117	−0.0437179
$516$	0	0
$517$	−6.43887	−0.283181
$518$	27.4720	1.20705
$519$	0	0
$520$	13.7548	0.603188
$521$	−36.3502	−1.59253	−0.796265	−	0.604949i	$-0.793194\pi$
$521$	−36.3502	−1.59253	−0.796265	+	0.604949i	$0.793194\pi$
$522$	0	0
$523$	18.1125	0.792005	0.396002	−	0.918249i	$-0.370397\pi$
$523$	18.1125	0.792005	0.396002	+	0.918249i	$0.370397\pi$
$524$	−27.6866	−1.20950
$525$	0	0
$526$	−30.7954	−1.34274
$527$	−4.60083	−0.200415
$528$	0	0
$529$	0	0
$530$	−47.3894	−2.05847
$531$	0	0
$532$	23.3460	1.01218
$533$	22.3228	0.966908
$534$	0	0
$535$	28.6006	1.23651
$536$	−2.44550	−0.105629
$537$	0	0
$538$	−57.1987	−2.46601
$539$	−6.20637	−0.267327
$540$	0	0
$541$	36.2904	1.56025	0.780123	−	0.625627i	$-0.215157\pi$
$541$	36.2904	1.56025	0.780123	+	0.625627i	$0.215157\pi$
$542$	9.00308	0.386716
$543$	0	0
$544$	−32.5319	−1.39480
$545$	−70.8867	−3.03645
$546$	0	0
$547$	−11.5431	−0.493548	−0.246774	−	0.969073i	$-0.579371\pi$
$547$	−11.5431	−0.493548	−0.246774	+	0.969073i	$0.579371\pi$
$548$	3.97229	0.169688
$549$	0	0
$550$	−20.0107	−0.853258
$551$	−61.6342	−2.62571
$552$	0	0
$553$	15.4176	0.655623
$554$	−14.4205	−0.612670
$555$	0	0
$556$	−13.4753	−0.571479
$557$	16.9829	0.719588	0.359794	−	0.933032i	$-0.382847\pi$
$557$	16.9829	0.719588	0.359794	+	0.933032i	$0.382847\pi$
$558$	0	0
$559$	−13.0888	−0.553596
$560$	16.3325	0.690172
$561$	0	0
$562$	27.9632	1.17955
$563$	16.4137	0.691753	0.345877	−	0.938280i	$-0.387582\pi$
$563$	16.4137	0.691753	0.345877	+	0.938280i	$0.387582\pi$
$564$	0	0
$565$	−5.59832	−0.235523
$566$	40.3961	1.69797
$567$	0	0
$568$	−4.96883	−0.208488
$569$	−0.755568	−0.0316751	−0.0158375	−	0.999875i	$-0.505041\pi$
$569$	−0.755568	−0.0316751	−0.0158375	+	0.999875i	$0.505041\pi$
$570$	0	0
$571$	−27.0996	−1.13408	−0.567040	−	0.823690i	$-0.691912\pi$
$571$	−27.0996	−1.13408	−0.567040	+	0.823690i	$0.691912\pi$
$572$	−15.1837	−0.634864
$573$	0	0
$574$	−16.2960	−0.680181
$575$	0	0
$576$	0	0
$577$	9.51857	0.396263	0.198132	−	0.980175i	$-0.436513\pi$
$577$	9.51857	0.396263	0.198132	+	0.980175i	$0.436513\pi$
$578$	1.33219	0.0554118
$579$	0	0
$580$	84.1323	3.49340
$581$	7.67844	0.318555
$582$	0	0
$583$	9.01901	0.373529
$584$	−0.796845	−0.0329737
$585$	0	0
$586$	−28.6755	−1.18457
$587$	−42.7401	−1.76407	−0.882036	−	0.471182i	$-0.843828\pi$
$587$	−42.7401	−1.76407	−0.882036	+	0.471182i	$0.843828\pi$
$588$	0	0
$589$	−6.94584	−0.286198
$590$	6.86768	0.282738
$591$	0	0
$592$	−24.7360	−1.01664
$593$	12.7912	0.525272	0.262636	−	0.964895i	$-0.415408\pi$
$593$	12.7912	0.525272	0.262636	+	0.964895i	$0.415408\pi$
$594$	0	0
$595$	22.0750	0.904986
$596$	14.6978	0.602044
$597$	0	0
$598$	0	0
$599$	−27.4971	−1.12350	−0.561751	−	0.827306i	$-0.689872\pi$
$599$	−27.4971	−1.12350	−0.561751	+	0.827306i	$0.689872\pi$
$600$	0	0
$601$	34.1537	1.39316	0.696579	−	0.717480i	$-0.254705\pi$
$601$	34.1537	1.39316	0.696579	+	0.717480i	$0.254705\pi$
$602$	9.55500	0.389433
$603$	0	0
$604$	−57.1953	−2.32725
$605$	−31.3797	−1.27576
$606$	0	0
$607$	−29.9922	−1.21735	−0.608673	−	0.793421i	$-0.708298\pi$
$607$	−29.9922	−1.21735	−0.608673	+	0.793421i	$0.708298\pi$
$608$	−49.1132	−1.99180
$609$	0	0
$610$	5.52726	0.223792
$611$	21.2496	0.859666
$612$	0	0
$613$	−6.66283	−0.269109	−0.134555	−	0.990906i	$-0.542960\pi$
$613$	−6.66283	−0.269109	−0.134555	+	0.990906i	$0.542960\pi$
$614$	53.4460	2.15691
$615$	0	0
$616$	1.91101	0.0769969
$617$	28.5932	1.15112	0.575560	−	0.817759i	$-0.304784\pi$
$617$	28.5932	1.15112	0.575560	+	0.817759i	$0.304784\pi$
$618$	0	0
$619$	3.01580	0.121215	0.0606076	−	0.998162i	$-0.480696\pi$
$619$	3.01580	0.121215	0.0606076	+	0.998162i	$0.480696\pi$
$620$	9.48125	0.380776
$621$	0	0
$622$	−66.1539	−2.65253
$623$	5.79939	0.232348
$624$	0	0
$625$	−11.8825	−0.475300
$626$	28.3071	1.13138
$627$	0	0
$628$	−37.6490	−1.50236
$629$	−33.4332	−1.33307
$630$	0	0
$631$	26.3615	1.04943	0.524717	−	0.851277i	$-0.324171\pi$
$631$	26.3615	1.04943	0.524717	+	0.851277i	$0.324171\pi$
$632$	8.53474	0.339494
$633$	0	0
$634$	−8.08836	−0.321230
$635$	33.4308	1.32666
$636$	0	0
$637$	20.4823	0.811538
$638$	−29.2631	−1.15854
$639$	0	0
$640$	23.6409	0.934488
$641$	15.5390	0.613753	0.306877	−	0.951749i	$-0.400716\pi$
$641$	15.5390	0.613753	0.306877	+	0.951749i	$0.400716\pi$
$642$	0	0
$643$	40.0898	1.58099	0.790493	−	0.612471i	$-0.209824\pi$
$643$	40.0898	1.58099	0.790493	+	0.612471i	$0.209824\pi$
$644$	0	0
$645$	0	0
$646$	−51.9254	−2.04298
$647$	18.6564	0.733459	0.366730	−	0.930328i	$-0.380477\pi$
$647$	18.6564	0.733459	0.366730	+	0.930328i	$0.380477\pi$
$648$	0	0
$649$	−1.30704	−0.0513057
$650$	66.0393	2.59027
$651$	0	0
$652$	−14.8910	−0.583177
$653$	46.6877	1.82703	0.913516	−	0.406803i	$-0.133356\pi$
$653$	46.6877	1.82703	0.913516	+	0.406803i	$0.133356\pi$
$654$	0	0
$655$	−39.5227	−1.54428
$656$	14.6730	0.572885
$657$	0	0
$658$	−15.5125	−0.604740
$659$	−12.3106	−0.479552	−0.239776	−	0.970828i	$-0.577074\pi$
$659$	−12.3106	−0.479552	−0.239776	+	0.970828i	$0.577074\pi$
$660$	0	0
$661$	−35.3668	−1.37561	−0.687805	−	0.725895i	$-0.741426\pi$
$661$	−35.3668	−1.37561	−0.687805	+	0.725895i	$0.741426\pi$
$662$	42.4699	1.65064
$663$	0	0
$664$	4.25056	0.164954
$665$	33.3264	1.29234
$666$	0	0
$667$	0	0
$668$	−22.8221	−0.883014
$669$	0	0
$670$	−20.2484	−0.782263
$671$	−1.05193	−0.0406094
$672$	0	0
$673$	−0.700020	−0.0269838	−0.0134919	−	0.999909i	$-0.504295\pi$
$673$	−0.700020	−0.0269838	−0.0134919	+	0.999909i	$0.504295\pi$
$674$	23.1432	0.891443
$675$	0	0
$676$	18.6930	0.718963
$677$	−11.4154	−0.438730	−0.219365	−	0.975643i	$-0.570399\pi$
$677$	−11.4154	−0.438730	−0.219365	+	0.975643i	$0.570399\pi$
$678$	0	0
$679$	−2.69862	−0.103563
$680$	12.2201	0.468618
$681$	0	0
$682$	−3.29779	−0.126279
$683$	−24.6583	−0.943524	−0.471762	−	0.881726i	$-0.656382\pi$
$683$	−24.6583	−0.943524	−0.471762	+	0.881726i	$0.656382\pi$
$684$	0	0
$685$	5.67045	0.216657
$686$	−38.2216	−1.45931
$687$	0	0
$688$	−8.60339	−0.328001
$689$	−29.7646	−1.13394
$690$	0	0
$691$	16.2668	0.618819	0.309410	−	0.950929i	$-0.399869\pi$
$691$	16.2668	0.618819	0.309410	+	0.950929i	$0.399869\pi$
$692$	−17.1435	−0.651698
$693$	0	0
$694$	−55.4038	−2.10310
$695$	−19.2360	−0.729661
$696$	0	0
$697$	19.8321	0.751194
$698$	−39.2917	−1.48721
$699$	0	0
$700$	−26.3787	−0.997023
$701$	24.2283	0.915090	0.457545	−	0.889186i	$-0.348729\pi$
$701$	24.2283	0.915090	0.457545	+	0.889186i	$0.348729\pi$
$702$	0	0
$703$	−50.4738	−1.90366
$704$	−15.0585	−0.567540
$705$	0	0
$706$	49.5714	1.86564
$707$	10.8909	0.409594
$708$	0	0
$709$	20.3858	0.765604	0.382802	−	0.923830i	$-0.374959\pi$
$709$	20.3858	0.765604	0.382802	+	0.923830i	$0.374959\pi$
$710$	−41.1413	−1.54400
$711$	0	0
$712$	3.21038	0.120314
$713$	0	0
$714$	0	0
$715$	−21.6748	−0.810591
$716$	−19.3981	−0.724943
$717$	0	0
$718$	31.6402	1.18080
$719$	23.4769	0.875540	0.437770	−	0.899087i	$-0.355768\pi$
$719$	23.4769	0.875540	0.437770	+	0.899087i	$0.355768\pi$
$720$	0	0
$721$	0.454896	0.0169412
$722$	−38.4613	−1.43138
$723$	0	0
$724$	28.4900	1.05882
$725$	69.6408	2.58639
$726$	0	0
$727$	−21.6990	−0.804771	−0.402385	−	0.915470i	$-0.631819\pi$
$727$	−21.6990	−0.804771	−0.402385	+	0.915470i	$0.631819\pi$
$728$	−6.30672	−0.233743
$729$	0	0
$730$	−6.59777	−0.244194
$731$	−11.6284	−0.430090
$732$	0	0
$733$	−32.3971	−1.19661	−0.598307	−	0.801267i	$-0.704160\pi$
$733$	−32.3971	−1.19661	−0.598307	+	0.801267i	$0.704160\pi$
$734$	69.6682	2.57150
$735$	0	0
$736$	0	0
$737$	3.85361	0.141949
$738$	0	0
$739$	36.5732	1.34537	0.672683	−	0.739931i	$-0.265142\pi$
$739$	36.5732	1.34537	0.672683	+	0.739931i	$0.265142\pi$
$740$	68.8981	2.53274
$741$	0	0
$742$	21.7286	0.797681
$743$	27.8032	1.02000	0.510001	−	0.860174i	$-0.329645\pi$
$743$	27.8032	1.02000	0.510001	+	0.860174i	$0.329645\pi$
$744$	0	0
$745$	20.9811	0.768687
$746$	69.2218	2.53439
$747$	0	0
$748$	−13.4896	−0.493228
$749$	−13.1137	−0.479164
$750$	0	0
$751$	24.5221	0.894824	0.447412	−	0.894328i	$-0.352346\pi$
$751$	24.5221	0.894824	0.447412	+	0.894328i	$0.352346\pi$
$752$	13.9676	0.509345
$753$	0	0
$754$	96.5740	3.51702
$755$	−81.6464	−2.97142
$756$	0	0
$757$	49.9792	1.81652	0.908262	−	0.418402i	$-0.137410\pi$
$757$	49.9792	1.81652	0.908262	+	0.418402i	$0.137410\pi$
$758$	46.2460	1.67973
$759$	0	0
$760$	18.4485	0.669199
$761$	0.903993	0.0327697	0.0163849	−	0.999866i	$-0.494784\pi$
$761$	0.903993	0.0327697	0.0163849	+	0.999866i	$0.494784\pi$
$762$	0	0
$763$	32.5023	1.17666
$764$	−63.9196	−2.31253
$765$	0	0
$766$	−9.84070	−0.355559
$767$	4.31349	0.155751
$768$	0	0
$769$	−46.9777	−1.69406	−0.847030	−	0.531546i	$-0.821611\pi$
$769$	−46.9777	−1.69406	−0.847030	+	0.531546i	$0.821611\pi$
$770$	15.8229	0.570218
$771$	0	0
$772$	45.5512	1.63942
$773$	42.2237	1.51868	0.759340	−	0.650694i	$-0.225522\pi$
$773$	42.2237	1.51868	0.759340	+	0.650694i	$0.225522\pi$
$774$	0	0
$775$	7.84813	0.281913
$776$	−1.49388	−0.0536270
$777$	0	0
$778$	29.4862	1.05713
$779$	29.9403	1.07272
$780$	0	0
$781$	7.82988	0.280175
$782$	0	0
$783$	0	0
$784$	13.4632	0.480829
$785$	−53.7440	−1.91820
$786$	0	0
$787$	−18.7438	−0.668144	−0.334072	−	0.942548i	$-0.608423\pi$
$787$	−18.7438	−0.668144	−0.334072	+	0.942548i	$0.608423\pi$
$788$	−21.3104	−0.759151
$789$	0	0
$790$	70.6665	2.51420
$791$	2.56689	0.0912680
$792$	0	0
$793$	3.47159	0.123280
$794$	−23.6666	−0.839896
$795$	0	0
$796$	−36.2643	−1.28536
$797$	−48.2282	−1.70833	−0.854165	−	0.520002i	$-0.825931\pi$
$797$	−48.2282	−1.70833	−0.854165	+	0.520002i	$0.825931\pi$
$798$	0	0
$799$	18.8786	0.667877
$800$	55.4932	1.96198
$801$	0	0
$802$	35.9860	1.27071
$803$	1.25567	0.0443115
$804$	0	0
$805$	0	0
$806$	10.8834	0.383350
$807$	0	0
$808$	6.02889	0.212096
$809$	53.2038	1.87055	0.935273	−	0.353928i	$-0.115154\pi$
$809$	53.2038	1.87055	0.935273	+	0.353928i	$0.115154\pi$
$810$	0	0
$811$	3.68024	0.129231	0.0646153	−	0.997910i	$-0.479418\pi$
$811$	3.68024	0.129231	0.0646153	+	0.997910i	$0.479418\pi$
$812$	−38.5756	−1.35374
$813$	0	0
$814$	−23.9643	−0.839947
$815$	−21.2569	−0.744597
$816$	0	0
$817$	−17.5552	−0.614180
$818$	18.8757	0.659973
$819$	0	0
$820$	−40.8693	−1.42722
$821$	−2.44781	−0.0854291	−0.0427146	−	0.999087i	$-0.513601\pi$
$821$	−2.44781	−0.0854291	−0.0427146	+	0.999087i	$0.513601\pi$
$822$	0	0
$823$	0.887969	0.0309527	0.0154763	−	0.999880i	$-0.495074\pi$
$823$	0.887969	0.0309527	0.0154763	+	0.999880i	$0.495074\pi$
$824$	0.251818	0.00877248
$825$	0	0
$826$	−3.14891	−0.109564
$827$	25.9780	0.903343	0.451671	−	0.892184i	$-0.350828\pi$
$827$	25.9780	0.903343	0.451671	+	0.892184i	$0.350828\pi$
$828$	0	0
$829$	−18.5613	−0.644662	−0.322331	−	0.946627i	$-0.604466\pi$
$829$	−18.5613	−0.644662	−0.322331	+	0.946627i	$0.604466\pi$
$830$	35.1941	1.22160
$831$	0	0
$832$	49.6962	1.72291
$833$	18.1969	0.630486
$834$	0	0
$835$	−32.5786	−1.12743
$836$	−20.3651	−0.704342
$837$	0	0
$838$	14.7991	0.511228
$839$	−17.0133	−0.587366	−0.293683	−	0.955903i	$-0.594881\pi$
$839$	−17.0133	−0.587366	−0.293683	+	0.955903i	$0.594881\pi$
$840$	0	0
$841$	72.8408	2.51175
$842$	25.8437	0.890633
$843$	0	0
$844$	−64.5863	−2.22315
$845$	26.6843	0.917968
$846$	0	0
$847$	14.3879	0.494374
$848$	−19.5646	−0.671850
$849$	0	0
$850$	58.6707	2.01239
$851$	0	0
$852$	0	0
$853$	33.0854	1.13282	0.566411	−	0.824123i	$-0.308331\pi$
$853$	33.0854	1.13282	0.566411	+	0.824123i	$0.308331\pi$
$854$	−2.53431	−0.0867223
$855$	0	0
$856$	−7.25936	−0.248120
$857$	10.0151	0.342110	0.171055	−	0.985262i	$-0.445282\pi$
$857$	10.0151	0.342110	0.171055	+	0.985262i	$0.445282\pi$
$858$	0	0
$859$	37.7898	1.28937	0.644686	−	0.764447i	$-0.276988\pi$
$859$	37.7898	1.28937	0.644686	+	0.764447i	$0.276988\pi$
$860$	23.9634	0.817143
$861$	0	0
$862$	−10.5985	−0.360986
$863$	−38.7898	−1.32042	−0.660210	−	0.751081i	$-0.729533\pi$
$863$	−38.7898	−1.32042	−0.660210	+	0.751081i	$0.729533\pi$
$864$	0	0
$865$	−24.4724	−0.832085
$866$	−1.86945	−0.0635266
$867$	0	0
$868$	−4.34726	−0.147555
$869$	−13.4490	−0.456227
$870$	0	0
$871$	−12.7177	−0.430922
$872$	17.9924	0.609298
$873$	0	0
$874$	0	0
$875$	−10.3723	−0.350648
$876$	0	0
$877$	−23.7055	−0.800477	−0.400238	−	0.916411i	$-0.631073\pi$
$877$	−23.7055	−0.800477	−0.400238	+	0.916411i	$0.631073\pi$
$878$	−83.8188	−2.82875
$879$	0	0
$880$	−14.2471	−0.480269
$881$	19.7637	0.665857	0.332929	−	0.942952i	$-0.391963\pi$
$881$	19.7637	0.665857	0.332929	+	0.942952i	$0.391963\pi$
$882$	0	0
$883$	26.5396	0.893130	0.446565	−	0.894751i	$-0.352647\pi$
$883$	26.5396	0.893130	0.446565	+	0.894751i	$0.352647\pi$
$884$	44.5183	1.49731
$885$	0	0
$886$	26.9219	0.904459
$887$	12.8547	0.431620	0.215810	−	0.976435i	$-0.430761\pi$
$887$	12.8547	0.431620	0.215810	+	0.976435i	$0.430761\pi$
$888$	0	0
$889$	−15.3284	−0.514098
$890$	26.5815	0.891013
$891$	0	0
$892$	29.7118	0.994824
$893$	28.5009	0.953745
$894$	0	0
$895$	−27.6909	−0.925604
$896$	−10.8396	−0.362125
$897$	0	0
$898$	−28.0436	−0.935827
$899$	11.4769	0.382776
$900$	0	0
$901$	−26.4435	−0.880961
$902$	14.2153	0.473316
$903$	0	0
$904$	1.42096	0.0472603
$905$	40.6695	1.35190
$906$	0	0
$907$	40.0575	1.33009	0.665044	−	0.746804i	$-0.268413\pi$
$907$	40.0575	1.33009	0.665044	+	0.746804i	$0.268413\pi$
$908$	−39.8329	−1.32190
$909$	0	0
$910$	−52.2188	−1.73104
$911$	−45.6764	−1.51333	−0.756663	−	0.653806i	$-0.773171\pi$
$911$	−45.6764	−1.51333	−0.756663	+	0.653806i	$0.773171\pi$
$912$	0	0
$913$	−6.69803	−0.221672
$914$	48.4677	1.60317
$915$	0	0
$916$	1.53799	0.0508165
$917$	18.1216	0.598427
$918$	0	0
$919$	−9.12733	−0.301083	−0.150541	−	0.988604i	$-0.548102\pi$
$919$	−9.12733	−0.301083	−0.150541	+	0.988604i	$0.548102\pi$
$920$	0	0
$921$	0	0
$922$	−15.9717	−0.526000
$923$	−25.8402	−0.850540
$924$	0	0
$925$	57.0306	1.87515
$926$	−21.3677	−0.702187
$927$	0	0
$928$	81.1518	2.66394
$929$	−0.583776	−0.0191531	−0.00957653	−	0.999954i	$-0.503048\pi$
$929$	−0.583776	−0.0191531	−0.00957653	+	0.999954i	$0.503048\pi$
$930$	0	0
$931$	27.4717	0.900350
$932$	−1.55531	−0.0509458
$933$	0	0
$934$	21.7156	0.710557
$935$	−19.2564	−0.629751
$936$	0	0
$937$	52.5112	1.71547	0.857733	−	0.514096i	$-0.171873\pi$
$937$	52.5112	1.71547	0.857733	+	0.514096i	$0.171873\pi$
$938$	9.28410	0.303136
$939$	0	0
$940$	−38.9044	−1.26892
$941$	−53.3424	−1.73891	−0.869456	−	0.494010i	$-0.835531\pi$
$941$	−53.3424	−1.73891	−0.869456	+	0.494010i	$0.835531\pi$
$942$	0	0
$943$	0	0
$944$	2.83530	0.0922811
$945$	0	0
$946$	−8.33498	−0.270994
$947$	12.7808	0.415319	0.207659	−	0.978201i	$-0.433415\pi$
$947$	12.7808	0.415319	0.207659	+	0.978201i	$0.433415\pi$
$948$	0	0
$949$	−4.14395	−0.134518
$950$	88.5748	2.87374
$951$	0	0
$952$	−5.60303	−0.181595
$953$	−26.8697	−0.870394	−0.435197	−	0.900335i	$-0.643321\pi$
$953$	−26.8697	−0.870394	−0.435197	+	0.900335i	$0.643321\pi$
$954$	0	0
$955$	−91.2453	−2.95263
$956$	23.2697	0.752597
$957$	0	0
$958$	18.3076	0.591491
$959$	−2.59996	−0.0839571
$960$	0	0
$961$	−29.7066	−0.958278
$962$	79.0869	2.54987
$963$	0	0
$964$	14.4152	0.464282
$965$	65.0244	2.09321
$966$	0	0
$967$	12.5704	0.404236	0.202118	−	0.979361i	$-0.435218\pi$
$967$	12.5704	0.404236	0.202118	+	0.979361i	$0.435218\pi$
$968$	7.96473	0.255996
$969$	0	0
$970$	−12.3691	−0.397148
$971$	−19.9924	−0.641585	−0.320793	−	0.947149i	$-0.603949\pi$
$971$	−19.9924	−0.641585	−0.320793	+	0.947149i	$0.603949\pi$
$972$	0	0
$973$	8.81989	0.282753
$974$	−47.8475	−1.53313
$975$	0	0
$976$	2.28191	0.0730422
$977$	−27.6322	−0.884031	−0.442015	−	0.897007i	$-0.645736\pi$
$977$	−27.6322	−0.884031	−0.442015	+	0.897007i	$0.645736\pi$
$978$	0	0
$979$	−5.05891	−0.161683
$980$	−37.4996	−1.19788
$981$	0	0
$982$	58.9350	1.88069
$983$	23.5308	0.750515	0.375258	−	0.926921i	$-0.377554\pi$
$983$	23.5308	0.750515	0.375258	+	0.926921i	$0.377554\pi$
$984$	0	0
$985$	−30.4206	−0.969280
$986$	85.7985	2.73238
$987$	0	0
$988$	67.2089	2.13820
$989$	0	0
$990$	0	0
$991$	42.8907	1.36247	0.681234	−	0.732066i	$-0.261444\pi$
$991$	42.8907	1.36247	0.681234	+	0.732066i	$0.261444\pi$
$992$	9.14537	0.290366
$993$	0	0
$994$	18.8637	0.598321
$995$	−51.7674	−1.64114
$996$	0	0
$997$	−38.8923	−1.23173	−0.615866	−	0.787851i	$-0.711194\pi$
$997$	−38.8923	−1.23173	−0.615866	+	0.787851i	$0.711194\pi$
$998$	30.7857	0.974503
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bu.1.1		10
3.2	odd	2		1587.2.a.t.1.10		10
23.17	odd	22		207.2.i.d.82.2		20
23.19	odd	22		207.2.i.d.154.2		20
23.22	odd	2		4761.2.a.bt.1.1		10
69.17	even	22		69.2.e.c.13.1	✓	20
69.65	even	22		69.2.e.c.16.1	yes	20
69.68	even	2		1587.2.a.u.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.c.13.1	✓	20	69.17	even	22
69.2.e.c.16.1	yes	20	69.65	even	22
207.2.i.d.82.2		20	23.17	odd	22
207.2.i.d.154.2		20	23.19	odd	22
1587.2.a.t.1.10		10	3.2	odd	2
1587.2.a.u.1.10		10	69.68	even	2
4761.2.a.bt.1.1		10	23.22	odd	2
4761.2.a.bu.1.1		10	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.10158 q^{2} +2.41664 q^{4} +3.44976 q^{5} -1.58175 q^{7} -0.875613 q^{8} +O(q^{10})\)	\(q-2.10158 q^{2} +2.41664 q^{4} +3.44976 q^{5} -1.58175 q^{7} -0.875613 q^{8} -7.24995 q^{10} +1.37979 q^{11} -4.55358 q^{13} +3.32418 q^{14} -2.99312 q^{16} -4.04550 q^{17} -6.10747 q^{19} +8.33685 q^{20} -2.89974 q^{22} +6.90085 q^{25} +9.56973 q^{26} -3.82253 q^{28} +10.0916 q^{29} +1.13727 q^{31} +8.04151 q^{32} +8.50196 q^{34} -5.45667 q^{35} +8.26429 q^{37} +12.8353 q^{38} -3.02065 q^{40} -4.90225 q^{41} +2.87439 q^{43} +3.33446 q^{44} -4.66656 q^{47} -4.49806 q^{49} -14.5027 q^{50} -11.0044 q^{52} +6.53652 q^{53} +4.75994 q^{55} +1.38500 q^{56} -21.2084 q^{58} -0.947273 q^{59} -0.762386 q^{61} -2.39007 q^{62} -10.9136 q^{64} -15.7088 q^{65} +2.79290 q^{67} -9.77655 q^{68} +11.4676 q^{70} +5.67469 q^{71} +0.910042 q^{73} -17.3681 q^{74} -14.7596 q^{76} -2.18249 q^{77} -9.74717 q^{79} -10.3255 q^{80} +10.3025 q^{82} -4.85439 q^{83} -13.9560 q^{85} -6.04077 q^{86} -1.20816 q^{88} -3.66643 q^{89} +7.20264 q^{91} +9.80716 q^{94} -21.0693 q^{95} +1.70609 q^{97} +9.45304 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8}+O(q^{10}) \) 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8	\( 10 q + 2 q^{2} + 8 q^{4} + 8 q^{5} - 19 q^{7} + 6 q^{8} - 13 q^{10} + 3 q^{11} - 4 q^{13} - 4 q^{16} + 11 q^{17} - 22 q^{19} + q^{20} - 13 q^{22} - 2 q^{25} - 4 q^{26} - 26 q^{28} + 5 q^{29} - 7 q^{31} + 34 q^{32} - 4 q^{34} - 9 q^{35} - 35 q^{37} - 9 q^{38} - 21 q^{40} - 28 q^{43} - 7 q^{44} - 9 q^{47} + 17 q^{49} - 52 q^{52} + 34 q^{53} - 14 q^{55} - 30 q^{56} - 24 q^{58} + 2 q^{59} - 49 q^{61} + 28 q^{62} + 10 q^{64} + 2 q^{65} - 26 q^{67} - 6 q^{68} + 16 q^{70} - 15 q^{71} + 14 q^{73} - 25 q^{74} - 19 q^{76} + 33 q^{77} - 43 q^{79} - 49 q^{80} + 24 q^{82} - 15 q^{83} - 21 q^{85} - 49 q^{86} - 15 q^{88} - 15 q^{89} - 4 q^{91} - 28 q^{94} - 28 q^{95} - 22 q^{97} - 15 q^{98}+O(q^{100}) \) 10 * q + 2 * q^2 + 8 * q^4 + 8 * q^5 - 19 * q^7 + 6 * q^8 - 13 * q^10 + 3 * q^11 - 4 * q^13 - 4 * q^16 + 11 * q^17 - 22 * q^19 + q^20 - 13 * q^22 - 2 * q^25 - 4 * q^26 - 26 * q^28 + 5 * q^29 - 7 * q^31 + 34 * q^32 - 4 * q^34 - 9 * q^35 - 35 * q^37 - 9 * q^38 - 21 * q^40 - 28 * q^43 - 7 * q^44 - 9 * q^47 + 17 * q^49 - 52 * q^52 + 34 * q^53 - 14 * q^55 - 30 * q^56 - 24 * q^58 + 2 * q^59 - 49 * q^61 + 28 * q^62 + 10 * q^64 + 2 * q^65 - 26 * q^67 - 6 * q^68 + 16 * q^70 - 15 * q^71 + 14 * q^73 - 25 * q^74 - 19 * q^76 + 33 * q^77 - 43 * q^79 - 49 * q^80 + 24 * q^82 - 15 * q^83 - 21 * q^85 - 49 * q^86 - 15 * q^88 - 15 * q^89 - 4 * q^91 - 28 * q^94 - 28 * q^95 - 22 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

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Database

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