LMFDB - Embedded newform 4761.2.a.bx.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:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 30 x^{18} + 376 x^{16} - 2566 x^{14} + 10441 x^{12} - 26158 x^{10} + 40383 x^{8} - 37458 x^{6} + \cdots + 529 $ x^20 - 30x^18 + 376x^16 - 2566x^14 + 10441x^12 - 26158x^10 + 40383x^8 - 37458x^6 + 19703x^4 - 5244*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 207)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.58900$ 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)

$f(q)$	$=$	$q-1.58900 q^{2} +0.524921 q^{4} +1.06419 q^{5} -3.24210 q^{7} +2.34390 q^{8} +O(q^{10})$	$q-1.58900 q^{2} +0.524921 q^{4} +1.06419 q^{5} -3.24210 q^{7} +2.34390 q^{8} -1.69100 q^{10} -3.21821 q^{11} -2.29511 q^{13} +5.15170 q^{14} -4.77430 q^{16} -3.37951 q^{17} -0.226545 q^{19} +0.558618 q^{20} +5.11373 q^{22} -3.86749 q^{25} +3.64694 q^{26} -1.70185 q^{28} +1.67832 q^{29} -5.68493 q^{31} +2.89856 q^{32} +5.37005 q^{34} -3.45023 q^{35} +6.77374 q^{37} +0.359980 q^{38} +2.49436 q^{40} -2.75278 q^{41} +0.337080 q^{43} -1.68930 q^{44} -13.3242 q^{47} +3.51123 q^{49} +6.14544 q^{50} -1.20475 q^{52} -12.0397 q^{53} -3.42480 q^{55} -7.59917 q^{56} -2.66684 q^{58} -8.32790 q^{59} +10.6084 q^{61} +9.03335 q^{62} +4.94278 q^{64} -2.44245 q^{65} +12.2850 q^{67} -1.77398 q^{68} +5.48241 q^{70} +15.6491 q^{71} -5.12570 q^{73} -10.7635 q^{74} -0.118918 q^{76} +10.4338 q^{77} +8.14772 q^{79} -5.08078 q^{80} +4.37416 q^{82} +7.74513 q^{83} -3.59646 q^{85} -0.535620 q^{86} -7.54315 q^{88} +8.48928 q^{89} +7.44100 q^{91} +21.1722 q^{94} -0.241088 q^{95} -12.9690 q^{97} -5.57935 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4} + 18 q^{7}+O(q^{10}) $ 20 * q + 20 * q^4 + 18 * q^7	$ 20 q + 20 q^{4} + 18 q^{7} + 22 q^{10} + 16 q^{16} + 40 q^{19} + 14 q^{22} + 20 q^{25} + 32 q^{28} - 22 q^{31} + 60 q^{34} + 18 q^{37} + 74 q^{40} + 32 q^{43} + 2 q^{49} - 52 q^{55} - 24 q^{58} + 70 q^{61} + 36 q^{64} + 64 q^{67} + 48 q^{70} + 40 q^{73} + 82 q^{76} + 106 q^{79} - 36 q^{82} + 2 q^{85} + 58 q^{88} + 88 q^{91} + 40 q^{94} + 24 q^{97}+O(q^{100}) $ 20 * q + 20 * q^4 + 18 * q^7 + 22 * q^10 + 16 * q^16 + 40 * q^19 + 14 * q^22 + 20 * q^25 + 32 * q^28 - 22 * q^31 + 60 * q^34 + 18 * q^37 + 74 * q^40 + 32 * q^43 + 2 * q^49 - 52 * q^55 - 24 * q^58 + 70 * q^61 + 36 * q^64 + 64 * q^67 + 48 * q^70 + 40 * q^73 + 82 * q^76 + 106 * q^79 - 36 * q^82 + 2 * q^85 + 58 * q^88 + 88 * q^91 + 40 * q^94 + 24 * q^97

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 $

$2$	−1.58900	−1.12359	−0.561796	−	0.827276i	$-0.689890\pi$
$2$	−1.58900	−1.12359	−0.561796	+	0.827276i	$0.689890\pi$
$3$	0	0
$3$	0	0
$4$	0.524921	0.262461
$5$	1.06419	0.475922	0.237961	−	0.971275i	$-0.423521\pi$
$5$	1.06419	0.475922	0.237961	+	0.971275i	$0.423521\pi$
$6$	0	0
$7$	−3.24210	−1.22540	−0.612700	−	0.790316i	$-0.709917\pi$
$7$	−3.24210	−1.22540	−0.612700	+	0.790316i	$0.709917\pi$
$8$	2.34390	0.828694
$9$	0	0
$10$	−1.69100	−0.534743
$11$	−3.21821	−0.970325	−0.485163	−	0.874424i	$-0.661240\pi$
$11$	−3.21821	−0.970325	−0.485163	+	0.874424i	$0.661240\pi$
$12$	0	0
$13$	−2.29511	−0.636550	−0.318275	−	0.947998i	$-0.603104\pi$
$13$	−2.29511	−0.636550	−0.318275	+	0.947998i	$0.603104\pi$
$14$	5.15170	1.37685
$15$	0	0
$16$	−4.77430	−1.19358
$17$	−3.37951	−0.819653	−0.409826	−	0.912164i	$-0.634411\pi$
$17$	−3.37951	−0.819653	−0.409826	+	0.912164i	$0.634411\pi$
$18$	0	0
$19$	−0.226545	−0.0519730	−0.0259865	−	0.999662i	$-0.508273\pi$
$19$	−0.226545	−0.0519730	−0.0259865	+	0.999662i	$0.508273\pi$
$20$	0.558618	0.124911
$21$	0	0
$22$	5.11373	1.09025
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.86749	−0.773498
$26$	3.64694	0.715223
$27$	0	0
$28$	−1.70185	−0.321619
$29$	1.67832	0.311655	0.155828	−	0.987784i	$-0.450196\pi$
$29$	1.67832	0.311655	0.155828	+	0.987784i	$0.450196\pi$
$30$	0	0
$31$	−5.68493	−1.02104	−0.510521	−	0.859865i	$-0.670548\pi$
$31$	−5.68493	−1.02104	−0.510521	+	0.859865i	$0.670548\pi$
$32$	2.89856	0.512398
$33$	0	0
$34$	5.37005	0.920956
$35$	−3.45023	−0.583195
$36$	0	0
$37$	6.77374	1.11360	0.556798	−	0.830648i	$-0.312030\pi$
$37$	6.77374	1.11360	0.556798	+	0.830648i	$0.312030\pi$
$38$	0.359980	0.0583965
$39$	0	0
$40$	2.49436	0.394394
$41$	−2.75278	−0.429912	−0.214956	−	0.976624i	$-0.568961\pi$
$41$	−2.75278	−0.429912	−0.214956	+	0.976624i	$0.568961\pi$
$42$	0	0
$43$	0.337080	0.0514042	0.0257021	−	0.999670i	$-0.491818\pi$
$43$	0.337080	0.0514042	0.0257021	+	0.999670i	$0.491818\pi$
$44$	−1.68930	−0.254672
$45$	0	0
$46$	0	0
$47$	−13.3242	−1.94354	−0.971769	−	0.235936i	$-0.924184\pi$
$47$	−13.3242	−1.94354	−0.971769	+	0.235936i	$0.924184\pi$
$48$	0	0
$49$	3.51123	0.501605
$50$	6.14544	0.869097
$51$	0	0
$52$	−1.20475	−0.167069
$53$	−12.0397	−1.65378	−0.826889	−	0.562364i	$-0.809892\pi$
$53$	−12.0397	−1.65378	−0.826889	+	0.562364i	$0.809892\pi$
$54$	0	0
$55$	−3.42480	−0.461799
$56$	−7.59917	−1.01548
$57$	0	0
$58$	−2.66684	−0.350174
$59$	−8.32790	−1.08420	−0.542100	−	0.840314i	$-0.682371\pi$
$59$	−8.32790	−1.08420	−0.542100	+	0.840314i	$0.682371\pi$
$60$	0	0
$61$	10.6084	1.35826	0.679132	−	0.734016i	$-0.262356\pi$
$61$	10.6084	1.35826	0.679132	+	0.734016i	$0.262356\pi$
$62$	9.03335	1.14724
$63$	0	0
$64$	4.94278	0.617848
$65$	−2.44245	−0.302948
$66$	0	0
$67$	12.2850	1.50086	0.750428	−	0.660952i	$-0.229847\pi$
$67$	12.2850	1.50086	0.750428	+	0.660952i	$0.229847\pi$
$68$	−1.77398	−0.215127
$69$	0	0
$70$	5.48241	0.655273
$71$	15.6491	1.85720	0.928601	−	0.371079i	$-0.121012\pi$
$71$	15.6491	1.85720	0.928601	+	0.371079i	$0.121012\pi$
$72$	0	0
$73$	−5.12570	−0.599918	−0.299959	−	0.953952i	$-0.596973\pi$
$73$	−5.12570	−0.599918	−0.299959	+	0.953952i	$0.596973\pi$
$74$	−10.7635	−1.25123
$75$	0	0
$76$	−0.118918	−0.0136409
$77$	10.4338	1.18904
$78$	0	0
$79$	8.14772	0.916690	0.458345	−	0.888774i	$-0.348442\pi$
$79$	8.14772	0.916690	0.458345	+	0.888774i	$0.348442\pi$
$80$	−5.08078	−0.568049
$81$	0	0
$82$	4.37416	0.483046
$83$	7.74513	0.850138	0.425069	−	0.905161i	$-0.360250\pi$
$83$	7.74513	0.850138	0.425069	+	0.905161i	$0.360250\pi$
$84$	0	0
$85$	−3.59646	−0.390091
$86$	−0.535620	−0.0577574
$87$	0	0
$88$	−7.54315	−0.804103
$89$	8.48928	0.899862	0.449931	−	0.893063i	$-0.351449\pi$
$89$	8.48928	0.899862	0.449931	+	0.893063i	$0.351449\pi$
$90$	0	0
$91$	7.44100	0.780028
$92$	0	0
$93$	0	0
$94$	21.1722	2.18374
$95$	−0.241088	−0.0247351
$96$	0	0
$97$	−12.9690	−1.31680	−0.658402	−	0.752666i	$-0.728767\pi$
$97$	−12.9690	−1.31680	−0.658402	+	0.752666i	$0.728767\pi$
$98$	−5.57935	−0.563599
$99$	0	0
$100$	−2.03013	−0.203013
$101$	−17.2651	−1.71794	−0.858970	−	0.512026i	$-0.828895\pi$
$101$	−17.2651	−1.71794	−0.858970	+	0.512026i	$0.828895\pi$
$102$	0	0
$103$	−0.232626	−0.0229213	−0.0114606	−	0.999934i	$-0.503648\pi$
$103$	−0.232626	−0.0229213	−0.0114606	+	0.999934i	$0.503648\pi$
$104$	−5.37952	−0.527505
$105$	0	0
$106$	19.1311	1.85817
$107$	0.0846912	0.00818740	0.00409370	−	0.999992i	$-0.498697\pi$
$107$	0.0846912	0.00818740	0.00409370	+	0.999992i	$0.498697\pi$
$108$	0	0
$109$	−9.60336	−0.919835	−0.459918	−	0.887962i	$-0.652121\pi$
$109$	−9.60336	−0.919835	−0.459918	+	0.887962i	$0.652121\pi$
$110$	5.44200	0.518874
$111$	0	0
$112$	15.4788	1.46261
$113$	10.3147	0.970321	0.485160	−	0.874425i	$-0.338761\pi$
$113$	10.3147	0.970321	0.485160	+	0.874425i	$0.338761\pi$
$114$	0	0
$115$	0	0
$116$	0.880984	0.0817973
$117$	0	0
$118$	13.2330	1.21820
$119$	10.9567	1.00440
$120$	0	0
$121$	−0.643154	−0.0584685
$122$	−16.8567	−1.52614
$123$	0	0
$124$	−2.98414	−0.267984
$125$	−9.43673	−0.844047
$126$	0	0
$127$	7.64752	0.678608	0.339304	−	0.940677i	$-0.389808\pi$
$127$	7.64752	0.678608	0.339304	+	0.940677i	$0.389808\pi$
$128$	−13.6512	−1.20661
$129$	0	0
$130$	3.88105	0.340390
$131$	2.96149	0.258747	0.129374	−	0.991596i	$-0.458703\pi$
$131$	2.96149	0.258747	0.129374	+	0.991596i	$0.458703\pi$
$132$	0	0
$133$	0.734483	0.0636877
$134$	−19.5209	−1.68635
$135$	0	0
$136$	−7.92124	−0.679241
$137$	−13.8811	−1.18594	−0.592971	−	0.805224i	$-0.702045\pi$
$137$	−13.8811	−1.18594	−0.592971	+	0.805224i	$0.702045\pi$
$138$	0	0
$139$	14.8326	1.25809	0.629044	−	0.777370i	$-0.283447\pi$
$139$	14.8326	1.25809	0.629044	+	0.777370i	$0.283447\pi$
$140$	−1.81110	−0.153066
$141$	0	0
$142$	−24.8664	−2.08674
$143$	7.38615	0.617661
$144$	0	0
$145$	1.78605	0.148324
$146$	8.14474	0.674064
$147$	0	0
$148$	3.55568	0.292275
$149$	−18.9771	−1.55467	−0.777334	−	0.629088i	$-0.783429\pi$
$149$	−18.9771	−1.55467	−0.777334	+	0.629088i	$0.783429\pi$
$150$	0	0
$151$	11.4480	0.931629	0.465814	−	0.884882i	$-0.345761\pi$
$151$	11.4480	0.931629	0.465814	+	0.884882i	$0.345761\pi$
$152$	−0.530999	−0.0430697
$153$	0	0
$154$	−16.5792	−1.33599
$155$	−6.04986	−0.485937
$156$	0	0
$157$	22.5585	1.80036	0.900181	−	0.435516i	$-0.143434\pi$
$157$	22.5585	1.80036	0.900181	+	0.435516i	$0.143434\pi$
$158$	−12.9467	−1.02999
$159$	0	0
$160$	3.08463	0.243862
$161$	0	0
$162$	0	0
$163$	18.9327	1.48293	0.741463	−	0.670994i	$-0.234132\pi$
$163$	18.9327	1.48293	0.741463	+	0.670994i	$0.234132\pi$
$164$	−1.44499	−0.112835
$165$	0	0
$166$	−12.3070	−0.955209
$167$	6.60475	0.511091	0.255546	−	0.966797i	$-0.417745\pi$
$167$	6.60475	0.511091	0.255546	+	0.966797i	$0.417745\pi$
$168$	0	0
$169$	−7.73245	−0.594804
$170$	5.71477	0.438303
$171$	0	0
$172$	0.176940	0.0134916
$173$	−16.6509	−1.26594	−0.632971	−	0.774175i	$-0.718165\pi$
$173$	−16.6509	−1.26594	−0.632971	+	0.774175i	$0.718165\pi$
$174$	0	0
$175$	12.5388	0.947844
$176$	15.3647	1.15816
$177$	0	0
$178$	−13.4895	−1.01108
$179$	−4.19799	−0.313773	−0.156886	−	0.987617i	$-0.550146\pi$
$179$	−4.19799	−0.313773	−0.156886	+	0.987617i	$0.550146\pi$
$180$	0	0
$181$	4.29701	0.319394	0.159697	−	0.987166i	$-0.448948\pi$
$181$	4.29701	0.319394	0.159697	+	0.987166i	$0.448948\pi$
$182$	−11.8237	−0.876434
$183$	0	0
$184$	0	0
$185$	7.20857	0.529985
$186$	0	0
$187$	10.8760	0.795330
$188$	−6.99417	−0.510102
$189$	0	0
$190$	0.383089	0.0277922
$191$	1.28733	0.0931480	0.0465740	−	0.998915i	$-0.485170\pi$
$191$	1.28733	0.0931480	0.0465740	+	0.998915i	$0.485170\pi$
$192$	0	0
$193$	−21.7878	−1.56832	−0.784160	−	0.620559i	$-0.786906\pi$
$193$	−21.7878	−1.56832	−0.784160	+	0.620559i	$0.786906\pi$
$194$	20.6078	1.47955
$195$	0	0
$196$	1.84312	0.131651
$197$	12.9591	0.923295	0.461647	−	0.887064i	$-0.347259\pi$
$197$	12.9591	0.923295	0.461647	+	0.887064i	$0.347259\pi$
$198$	0	0
$199$	11.1679	0.791673	0.395836	−	0.918321i	$-0.370455\pi$
$199$	11.1679	0.791673	0.395836	+	0.918321i	$0.370455\pi$
$200$	−9.06501	−0.640993
$201$	0	0
$202$	27.4342	1.93027
$203$	−5.44127	−0.381903
$204$	0	0
$205$	−2.92949	−0.204604
$206$	0.369642	0.0257542
$207$	0	0
$208$	10.9576	0.759770
$209$	0.729069	0.0504308
$210$	0	0
$211$	11.6429	0.801528	0.400764	−	0.916181i	$-0.368745\pi$
$211$	11.6429	0.801528	0.400764	+	0.916181i	$0.368745\pi$
$212$	−6.31989	−0.434052
$213$	0	0
$214$	−0.134574	−0.00919931
$215$	0.358718	0.0244644
$216$	0	0
$217$	18.4311	1.25119
$218$	15.2597	1.03352
$219$	0	0
$220$	−1.79775	−0.121204
$221$	7.75637	0.521750
$222$	0	0
$223$	−19.9793	−1.33791	−0.668957	−	0.743301i	$-0.733259\pi$
$223$	−19.9793	−1.33791	−0.668957	+	0.743301i	$0.733259\pi$
$224$	−9.39744	−0.627893
$225$	0	0
$226$	−16.3900	−1.09025
$227$	11.1104	0.737422	0.368711	−	0.929544i	$-0.379799\pi$
$227$	11.1104	0.737422	0.368711	+	0.929544i	$0.379799\pi$
$228$	0	0
$229$	3.11850	0.206076	0.103038	−	0.994677i	$-0.467144\pi$
$229$	3.11850	0.206076	0.103038	+	0.994677i	$0.467144\pi$
$230$	0	0
$231$	0	0
$232$	3.93381	0.258267
$233$	−0.126734	−0.00830262	−0.00415131	−	0.999991i	$-0.501321\pi$
$233$	−0.126734	−0.00830262	−0.00415131	+	0.999991i	$0.501321\pi$
$234$	0	0
$235$	−14.1796	−0.924972
$236$	−4.37149	−0.284560
$237$	0	0
$238$	−17.4103	−1.12854
$239$	−0.993127	−0.0642400	−0.0321200	−	0.999484i	$-0.510226\pi$
$239$	−0.993127	−0.0642400	−0.0321200	+	0.999484i	$0.510226\pi$
$240$	0	0
$241$	10.6154	0.683797	0.341899	−	0.939737i	$-0.388930\pi$
$241$	10.6154	0.683797	0.341899	+	0.939737i	$0.388930\pi$
$242$	1.02197	0.0656948
$243$	0	0
$244$	5.56856	0.356491
$245$	3.73663	0.238725
$246$	0	0
$247$	0.519947	0.0330834
$248$	−13.3249	−0.846132
$249$	0	0
$250$	14.9950	0.948365
$251$	−3.35228	−0.211594	−0.105797	−	0.994388i	$-0.533739\pi$
$251$	−3.35228	−0.211594	−0.105797	+	0.994388i	$0.533739\pi$
$252$	0	0
$253$	0	0
$254$	−12.1519	−0.762479
$255$	0	0
$256$	11.8062	0.737888
$257$	5.75926	0.359253	0.179626	−	0.983735i	$-0.442511\pi$
$257$	5.75926	0.359253	0.179626	+	0.983735i	$0.442511\pi$
$258$	0	0
$259$	−21.9611	−1.36460
$260$	−1.28209	−0.0795120
$261$	0	0
$262$	−4.70582	−0.290726
$263$	−20.9585	−1.29236	−0.646178	−	0.763186i	$-0.723634\pi$
$263$	−20.9585	−1.29236	−0.646178	+	0.763186i	$0.723634\pi$
$264$	0	0
$265$	−12.8126	−0.787070
$266$	−1.16709	−0.0715591
$267$	0	0
$268$	6.44868	0.393916
$269$	−1.08452	−0.0661242	−0.0330621	−	0.999453i	$-0.510526\pi$
$269$	−1.08452	−0.0661242	−0.0330621	+	0.999453i	$0.510526\pi$
$270$	0	0
$271$	−27.2208	−1.65354	−0.826772	−	0.562537i	$-0.809825\pi$
$271$	−27.2208	−1.65354	−0.826772	+	0.562537i	$0.809825\pi$
$272$	16.1348	0.978317
$273$	0	0
$274$	22.0570	1.33252
$275$	12.4464	0.750545
$276$	0	0
$277$	−6.26139	−0.376211	−0.188105	−	0.982149i	$-0.560235\pi$
$277$	−6.26139	−0.376211	−0.188105	+	0.982149i	$0.560235\pi$
$278$	−23.5691	−1.41358
$279$	0	0
$280$	−8.08699	−0.483290
$281$	−0.779985	−0.0465300	−0.0232650	−	0.999729i	$-0.507406\pi$
$281$	−0.779985	−0.0465300	−0.0232650	+	0.999729i	$0.507406\pi$
$282$	0	0
$283$	−11.4791	−0.682360	−0.341180	−	0.939998i	$-0.610827\pi$
$283$	−11.4791	−0.682360	−0.341180	+	0.939998i	$0.610827\pi$
$284$	8.21453	0.487443
$285$	0	0
$286$	−11.7366	−0.693999
$287$	8.92479	0.526814
$288$	0	0
$289$	−5.57888	−0.328170
$290$	−2.83804	−0.166655
$291$	0	0
$292$	−2.69059	−0.157455
$293$	34.0640	1.99004	0.995019	−	0.0996822i	$-0.0317826\pi$
$293$	34.0640	1.99004	0.995019	+	0.0996822i	$0.0317826\pi$
$294$	0	0
$295$	−8.86250	−0.515995
$296$	15.8770	0.922829
$297$	0	0
$298$	30.1547	1.74681
$299$	0	0
$300$	0	0
$301$	−1.09285	−0.0629907
$302$	−18.1909	−1.04677
$303$	0	0
$304$	1.08159	0.0620337
$305$	11.2894	0.646428
$306$	0	0
$307$	14.9241	0.851763	0.425882	−	0.904779i	$-0.359964\pi$
$307$	14.9241	0.851763	0.425882	+	0.904779i	$0.359964\pi$
$308$	5.47690	0.312075
$309$	0	0
$310$	9.61323	0.545995
$311$	19.3778	1.09881	0.549407	−	0.835555i	$-0.314853\pi$
$311$	19.3778	1.09881	0.549407	+	0.835555i	$0.314853\pi$
$312$	0	0
$313$	28.1762	1.59261	0.796305	−	0.604895i	$-0.206785\pi$
$313$	28.1762	1.59261	0.796305	+	0.604895i	$0.206785\pi$
$314$	−35.8454	−2.02287
$315$	0	0
$316$	4.27691	0.240595
$317$	−7.79443	−0.437779	−0.218889	−	0.975750i	$-0.570243\pi$
$317$	−7.79443	−0.437779	−0.218889	+	0.975750i	$0.570243\pi$
$318$	0	0
$319$	−5.40117	−0.302407
$320$	5.26008	0.294047
$321$	0	0
$322$	0	0
$323$	0.765613	0.0425998
$324$	0	0
$325$	8.87633	0.492370
$326$	−30.0841	−1.66621
$327$	0	0
$328$	−6.45224	−0.356265
$329$	43.1985	2.38161
$330$	0	0
$331$	12.5557	0.690122	0.345061	−	0.938580i	$-0.387858\pi$
$331$	12.5557	0.690122	0.345061	+	0.938580i	$0.387858\pi$
$332$	4.06558	0.223128
$333$	0	0
$334$	−10.4950	−0.574258
$335$	13.0737	0.714290
$336$	0	0
$337$	−5.65280	−0.307927	−0.153964	−	0.988077i	$-0.549204\pi$
$337$	−5.65280	−0.307927	−0.153964	+	0.988077i	$0.549204\pi$
$338$	12.2869	0.668317
$339$	0	0
$340$	−1.88786	−0.102383
$341$	18.2953	0.990744
$342$	0	0
$343$	11.3109	0.610734
$344$	0.790082	0.0425983
$345$	0	0
$346$	26.4582	1.42240
$347$	15.5070	0.832459	0.416229	−	0.909260i	$-0.363351\pi$
$347$	15.5070	0.832459	0.416229	+	0.909260i	$0.363351\pi$
$348$	0	0
$349$	15.9779	0.855277	0.427639	−	0.903950i	$-0.359346\pi$
$349$	15.9779	0.855277	0.427639	+	0.903950i	$0.359346\pi$
$350$	−19.9242	−1.06499
$351$	0	0
$352$	−9.32817	−0.497193
$353$	−29.5466	−1.57261	−0.786303	−	0.617842i	$-0.788007\pi$
$353$	−29.5466	−1.57261	−0.786303	+	0.617842i	$0.788007\pi$
$354$	0	0
$355$	16.6536	0.883884
$356$	4.45621	0.236178
$357$	0	0
$358$	6.67061	0.352553
$359$	−19.2942	−1.01831	−0.509153	−	0.860676i	$-0.670041\pi$
$359$	−19.2942	−1.01831	−0.509153	+	0.860676i	$0.670041\pi$
$360$	0	0
$361$	−18.9487	−0.997299
$362$	−6.82794	−0.358869
$363$	0	0
$364$	3.90594	0.204727
$365$	−5.45474	−0.285514
$366$	0	0
$367$	20.2508	1.05708	0.528541	−	0.848908i	$-0.322739\pi$
$367$	20.2508	1.05708	0.528541	+	0.848908i	$0.322739\pi$
$368$	0	0
$369$	0	0
$370$	−11.4544	−0.595487
$371$	39.0339	2.02654
$372$	0	0
$373$	−10.5273	−0.545084	−0.272542	−	0.962144i	$-0.587864\pi$
$373$	−10.5273	−0.545084	−0.272542	+	0.962144i	$0.587864\pi$
$374$	−17.2819	−0.893627
$375$	0	0
$376$	−31.2306	−1.61060
$377$	−3.85193	−0.198384
$378$	0	0
$379$	2.88548	0.148217	0.0741086	−	0.997250i	$-0.476389\pi$
$379$	2.88548	0.148217	0.0741086	+	0.997250i	$0.476389\pi$
$380$	−0.126552	−0.00649199
$381$	0	0
$382$	−2.04557	−0.104660
$383$	−2.64488	−0.135147	−0.0675736	−	0.997714i	$-0.521526\pi$
$383$	−2.64488	−0.135147	−0.0675736	+	0.997714i	$0.521526\pi$
$384$	0	0
$385$	11.1035	0.565889
$386$	34.6208	1.76215
$387$	0	0
$388$	−6.80772	−0.345609
$389$	2.71119	0.137463	0.0687315	−	0.997635i	$-0.478105\pi$
$389$	2.71119	0.137463	0.0687315	+	0.997635i	$0.478105\pi$
$390$	0	0
$391$	0	0
$392$	8.22998	0.415677
$393$	0	0
$394$	−20.5919	−1.03741
$395$	8.67076	0.436273
$396$	0	0
$397$	25.1242	1.26095	0.630473	−	0.776211i	$-0.282861\pi$
$397$	25.1242	1.26095	0.630473	+	0.776211i	$0.282861\pi$
$398$	−17.7458	−0.889518
$399$	0	0
$400$	18.4646	0.923228
$401$	24.2934	1.21316	0.606578	−	0.795024i	$-0.292542\pi$
$401$	24.2934	1.21316	0.606578	+	0.795024i	$0.292542\pi$
$402$	0	0
$403$	13.0476	0.649945
$404$	−9.06281	−0.450892
$405$	0	0
$406$	8.64619	0.429103
$407$	−21.7993	−1.08055
$408$	0	0
$409$	17.1694	0.848972	0.424486	−	0.905435i	$-0.360455\pi$
$409$	17.1694	0.848972	0.424486	+	0.905435i	$0.360455\pi$
$410$	4.65496	0.229892
$411$	0	0
$412$	−0.122110	−0.00601593
$413$	26.9999	1.32858
$414$	0	0
$415$	8.24232	0.404599
$416$	−6.65253	−0.326167
$417$	0	0
$418$	−1.15849	−0.0566636
$419$	34.3160	1.67645	0.838223	−	0.545327i	$-0.183595\pi$
$419$	34.3160	1.67645	0.838223	+	0.545327i	$0.183595\pi$
$420$	0	0
$421$	11.1697	0.544380	0.272190	−	0.962244i	$-0.412252\pi$
$421$	11.1697	0.544380	0.272190	+	0.962244i	$0.412252\pi$
$422$	−18.5005	−0.900591
$423$	0	0
$424$	−28.2198	−1.37048
$425$	13.0702	0.634000
$426$	0	0
$427$	−34.3935	−1.66442
$428$	0.0444562	0.00214887
$429$	0	0
$430$	−0.570004	−0.0274880
$431$	6.55066	0.315534	0.157767	−	0.987476i	$-0.449570\pi$
$431$	6.55066	0.315534	0.157767	+	0.987476i	$0.449570\pi$
$432$	0	0
$433$	3.81311	0.183246	0.0916231	−	0.995794i	$-0.470794\pi$
$433$	3.81311	0.183246	0.0916231	+	0.995794i	$0.470794\pi$
$434$	−29.2870	−1.40582
$435$	0	0
$436$	−5.04101	−0.241421
$437$	0	0
$438$	0	0
$439$	13.5274	0.645629	0.322815	−	0.946462i	$-0.395371\pi$
$439$	13.5274	0.645629	0.322815	+	0.946462i	$0.395371\pi$
$440$	−8.02738	−0.382690
$441$	0	0
$442$	−12.3249	−0.586235
$443$	15.1980	0.722080	0.361040	−	0.932550i	$-0.382422\pi$
$443$	15.1980	0.722080	0.361040	+	0.932550i	$0.382422\pi$
$444$	0	0
$445$	9.03424	0.428264
$446$	31.7471	1.50327
$447$	0	0
$448$	−16.0250	−0.757111
$449$	7.38046	0.348305	0.174153	−	0.984719i	$-0.444281\pi$
$449$	7.38046	0.348305	0.174153	+	0.984719i	$0.444281\pi$
$450$	0	0
$451$	8.85901	0.417154
$452$	5.41438	0.254671
$453$	0	0
$454$	−17.6544	−0.828562
$455$	7.91867	0.371233
$456$	0	0
$457$	−32.6319	−1.52646	−0.763229	−	0.646128i	$-0.776387\pi$
$457$	−32.6319	−1.52646	−0.763229	+	0.646128i	$0.776387\pi$
$458$	−4.95529	−0.231546
$459$	0	0
$460$	0	0
$461$	32.0815	1.49418	0.747092	−	0.664721i	$-0.231449\pi$
$461$	32.0815	1.49418	0.747092	+	0.664721i	$0.231449\pi$
$462$	0	0
$463$	−5.08956	−0.236532	−0.118266	−	0.992982i	$-0.537734\pi$
$463$	−5.08956	−0.236532	−0.118266	+	0.992982i	$0.537734\pi$
$464$	−8.01279	−0.371984
$465$	0	0
$466$	0.201380	0.00932877
$467$	28.5662	1.32188	0.660942	−	0.750437i	$-0.270157\pi$
$467$	28.5662	1.32188	0.660942	+	0.750437i	$0.270157\pi$
$468$	0	0
$469$	−39.8293	−1.83915
$470$	22.5313	1.03929
$471$	0	0
$472$	−19.5198	−0.898470
$473$	−1.08479	−0.0498788
$474$	0	0
$475$	0.876161	0.0402010
$476$	5.75142	0.263616
$477$	0	0
$478$	1.57808	0.0721796
$479$	24.9098	1.13816	0.569079	−	0.822283i	$-0.307300\pi$
$479$	24.9098	1.13816	0.569079	+	0.822283i	$0.307300\pi$
$480$	0	0
$481$	−15.5465	−0.708859
$482$	−16.8679	−0.768310
$483$	0	0
$484$	−0.337605	−0.0153457
$485$	−13.8016	−0.626696
$486$	0	0
$487$	−26.5308	−1.20223	−0.601113	−	0.799164i	$-0.705276\pi$
$487$	−26.5308	−1.20223	−0.601113	+	0.799164i	$0.705276\pi$
$488$	24.8650	1.12558
$489$	0	0
$490$	−5.93751	−0.268229
$491$	−26.1840	−1.18167	−0.590834	−	0.806793i	$-0.701201\pi$
$491$	−26.1840	−1.18167	−0.590834	+	0.806793i	$0.701201\pi$
$492$	0	0
$493$	−5.67189	−0.255449
$494$	−0.826196	−0.0371723
$495$	0	0
$496$	27.1415	1.21869
$497$	−50.7359	−2.27582
$498$	0	0
$499$	1.28052	0.0573241	0.0286620	−	0.999589i	$-0.490875\pi$
$499$	1.28052	0.0573241	0.0286620	+	0.999589i	$0.490875\pi$
$500$	−4.95354	−0.221529
$501$	0	0
$502$	5.32678	0.237746
$503$	−22.7477	−1.01427	−0.507135	−	0.861866i	$-0.669296\pi$
$503$	−22.7477	−1.01427	−0.507135	+	0.861866i	$0.669296\pi$
$504$	0	0
$505$	−18.3734	−0.817606
$506$	0	0
$507$	0	0
$508$	4.01435	0.178108
$509$	−26.9938	−1.19648	−0.598240	−	0.801317i	$-0.704133\pi$
$509$	−26.9938	−1.19648	−0.598240	+	0.801317i	$0.704133\pi$
$510$	0	0
$511$	16.6181	0.735140
$512$	8.54235	0.377522
$513$	0	0
$514$	−9.15147	−0.403654
$515$	−0.247559	−0.0109087
$516$	0	0
$517$	42.8801	1.88586
$518$	34.8963	1.53325
$519$	0	0
$520$	−5.72485	−0.251051
$521$	17.0187	0.745605	0.372802	−	0.927911i	$-0.378397\pi$
$521$	17.0187	0.745605	0.372802	+	0.927911i	$0.378397\pi$
$522$	0	0
$523$	17.6789	0.773045	0.386523	−	0.922280i	$-0.373676\pi$
$523$	17.6789	0.773045	0.386523	+	0.922280i	$0.373676\pi$
$524$	1.55455	0.0679109
$525$	0	0
$526$	33.3031	1.45208
$527$	19.2123	0.836900
$528$	0	0
$529$	0	0
$530$	20.3592	0.884346
$531$	0	0
$532$	0.385546	0.0167155
$533$	6.31794	0.273660
$534$	0	0
$535$	0.0901278	0.00389657
$536$	28.7949	1.24375
$537$	0	0
$538$	1.72330	0.0742967
$539$	−11.2999	−0.486720
$540$	0	0
$541$	27.3744	1.17692	0.588459	−	0.808527i	$-0.299735\pi$
$541$	27.3744	1.17692	0.588459	+	0.808527i	$0.299735\pi$
$542$	43.2538	1.85791
$543$	0	0
$544$	−9.79574	−0.419989
$545$	−10.2198	−0.437770
$546$	0	0
$547$	−13.7779	−0.589100	−0.294550	−	0.955636i	$-0.595170\pi$
$547$	−13.7779	−0.589100	−0.294550	+	0.955636i	$0.595170\pi$
$548$	−7.28648	−0.311263
$549$	0	0
$550$	−19.7773	−0.843307
$551$	−0.380214	−0.0161977
$552$	0	0
$553$	−26.4158	−1.12331
$554$	9.94935	0.422708
$555$	0	0
$556$	7.78597	0.330198
$557$	26.7278	1.13249	0.566246	−	0.824236i	$-0.308395\pi$
$557$	26.7278	1.13249	0.566246	+	0.824236i	$0.308395\pi$
$558$	0	0
$559$	−0.773637	−0.0327214
$560$	16.4724	0.696087
$561$	0	0
$562$	1.23940	0.0522808
$563$	−1.10451	−0.0465495	−0.0232747	−	0.999729i	$-0.507409\pi$
$563$	−1.10451	−0.0465495	−0.0232747	+	0.999729i	$0.507409\pi$
$564$	0	0
$565$	10.9768	0.461797
$566$	18.2403	0.766695
$567$	0	0
$568$	36.6799	1.53905
$569$	−29.0780	−1.21901	−0.609506	−	0.792781i	$-0.708632\pi$
$569$	−29.0780	−1.21901	−0.609506	+	0.792781i	$0.708632\pi$
$570$	0	0
$571$	0.0114549	0.000479371 0	0.000239686	−	1.00000i	$-0.499924\pi$
$571$	0.0114549	0.000479371 0	0.000239686		1.00000i	$0.499924\pi$
$572$	3.87715	0.162112
$573$	0	0
$574$	−14.1815	−0.591924
$575$	0	0
$576$	0	0
$577$	−14.4281	−0.600650	−0.300325	−	0.953837i	$-0.597095\pi$
$577$	−14.4281	−0.600650	−0.300325	+	0.953837i	$0.597095\pi$
$578$	8.86485	0.368729
$579$	0	0
$580$	0.937538	0.0389291
$581$	−25.1105	−1.04176
$582$	0	0
$583$	38.7462	1.60470
$584$	−12.0141	−0.497148
$585$	0	0
$586$	−54.1277	−2.23599
$587$	28.9164	1.19351	0.596754	−	0.802424i	$-0.296457\pi$
$587$	28.9164	1.19351	0.596754	+	0.802424i	$0.296457\pi$
$588$	0	0
$589$	1.28789	0.0530667
$590$	14.0825	0.579768
$591$	0	0
$592$	−32.3398	−1.32916
$593$	0.225899	0.00927658	0.00463829	−	0.999989i	$-0.498524\pi$
$593$	0.225899	0.00927658	0.00463829	+	0.999989i	$0.498524\pi$
$594$	0	0
$595$	11.6601	0.478017
$596$	−9.96151	−0.408039
$597$	0	0
$598$	0	0
$599$	8.69708	0.355353	0.177677	−	0.984089i	$-0.443142\pi$
$599$	8.69708	0.355353	0.177677	+	0.984089i	$0.443142\pi$
$600$	0	0
$601$	38.0553	1.55231	0.776155	−	0.630543i	$-0.217168\pi$
$601$	38.0553	1.55231	0.776155	+	0.630543i	$0.217168\pi$
$602$	1.73654	0.0707759
$603$	0	0
$604$	6.00932	0.244516
$605$	−0.684440	−0.0278265
$606$	0	0
$607$	35.0891	1.42422	0.712112	−	0.702066i	$-0.247739\pi$
$607$	35.0891	1.42422	0.712112	+	0.702066i	$0.247739\pi$
$608$	−0.656656	−0.0266309
$609$	0	0
$610$	−17.9388	−0.726321
$611$	30.5806	1.23716
$612$	0	0
$613$	19.2651	0.778111	0.389055	−	0.921214i	$-0.372802\pi$
$613$	19.2651	0.778111	0.389055	+	0.921214i	$0.372802\pi$
$614$	−23.7144	−0.957035
$615$	0	0
$616$	24.4557	0.985347
$617$	2.26084	0.0910179	0.0455090	−	0.998964i	$-0.485509\pi$
$617$	2.26084	0.0910179	0.0455090	+	0.998964i	$0.485509\pi$
$618$	0	0
$619$	28.5845	1.14891	0.574455	−	0.818536i	$-0.305214\pi$
$619$	28.5845	1.14891	0.574455	+	0.818536i	$0.305214\pi$
$620$	−3.17570	−0.127539
$621$	0	0
$622$	−30.7913	−1.23462
$623$	−27.5231	−1.10269
$624$	0	0
$625$	9.29494	0.371798
$626$	−44.7719	−1.78945
$627$	0	0
$628$	11.8414	0.472524
$629$	−22.8919	−0.912761
$630$	0	0
$631$	36.3182	1.44580	0.722902	−	0.690950i	$-0.242808\pi$
$631$	36.3182	1.44580	0.722902	+	0.690950i	$0.242808\pi$
$632$	19.0974	0.759656
$633$	0	0
$634$	12.3853	0.491885
$635$	8.13845	0.322965
$636$	0	0
$637$	−8.05868	−0.319296
$638$	8.58245	0.339783
$639$	0	0
$640$	−14.5275	−0.574251
$641$	16.2001	0.639865	0.319933	−	0.947440i	$-0.396340\pi$
$641$	16.2001	0.639865	0.319933	+	0.947440i	$0.396340\pi$
$642$	0	0
$643$	−39.4841	−1.55710	−0.778551	−	0.627582i	$-0.784045\pi$
$643$	−39.4841	−1.55710	−0.778551	+	0.627582i	$0.784045\pi$
$644$	0	0
$645$	0	0
$646$	−1.21656	−0.0478649
$647$	−33.2548	−1.30738	−0.653691	−	0.756761i	$-0.726780\pi$
$647$	−33.2548	−1.30738	−0.653691	+	0.756761i	$0.726780\pi$
$648$	0	0
$649$	26.8009	1.05203
$650$	−14.1045	−0.553224
$651$	0	0
$652$	9.93820	0.389210
$653$	2.47077	0.0966888	0.0483444	−	0.998831i	$-0.484605\pi$
$653$	2.47077	0.0966888	0.0483444	+	0.998831i	$0.484605\pi$
$654$	0	0
$655$	3.15161	0.123143
$656$	13.1426	0.513132
$657$	0	0
$658$	−68.6424	−2.67596
$659$	−32.7084	−1.27414	−0.637069	−	0.770807i	$-0.719853\pi$
$659$	−32.7084	−1.27414	−0.637069	+	0.770807i	$0.719853\pi$
$660$	0	0
$661$	14.2771	0.555314	0.277657	−	0.960680i	$-0.410442\pi$
$661$	14.2771	0.555314	0.277657	+	0.960680i	$0.410442\pi$
$662$	−19.9510	−0.775416
$663$	0	0
$664$	18.1538	0.704504
$665$	0.781632	0.0303104
$666$	0	0
$667$	0	0
$668$	3.46698	0.134141
$669$	0	0
$670$	−20.7741	−0.802572
$671$	−34.1399	−1.31796
$672$	0	0
$673$	14.2503	0.549308	0.274654	−	0.961543i	$-0.411437\pi$
$673$	14.2503	0.549308	0.274654	+	0.961543i	$0.411437\pi$
$674$	8.98229	0.345985
$675$	0	0
$676$	−4.05893	−0.156113
$677$	−25.8847	−0.994830	−0.497415	−	0.867513i	$-0.665717\pi$
$677$	−25.8847	−0.994830	−0.497415	+	0.867513i	$0.665717\pi$
$678$	0	0
$679$	42.0469	1.61361
$680$	−8.42974	−0.323266
$681$	0	0
$682$	−29.0712	−1.11319
$683$	−26.3751	−1.00921	−0.504607	−	0.863349i	$-0.668363\pi$
$683$	−26.3751	−1.00921	−0.504607	+	0.863349i	$0.668363\pi$
$684$	0	0
$685$	−14.7722	−0.564416
$686$	−17.9731	−0.686216
$687$	0	0
$688$	−1.60932	−0.0613548
$689$	27.6325	1.05271
$690$	0	0
$691$	−2.77076	−0.105405	−0.0527023	−	0.998610i	$-0.516783\pi$
$691$	−2.77076	−0.105405	−0.0527023	+	0.998610i	$0.516783\pi$
$692$	−8.74040	−0.332260
$693$	0	0
$694$	−24.6406	−0.935345
$695$	15.7848	0.598752
$696$	0	0
$697$	9.30305	0.352378
$698$	−25.3889	−0.960984
$699$	0	0
$700$	6.58189	0.248772
$701$	38.8149	1.46602	0.733008	−	0.680220i	$-0.238116\pi$
$701$	38.8149	1.46602	0.733008	+	0.680220i	$0.238116\pi$
$702$	0	0
$703$	−1.53456	−0.0578769
$704$	−15.9069	−0.599514
$705$	0	0
$706$	46.9495	1.76697
$707$	55.9752	2.10516
$708$	0	0
$709$	19.3696	0.727440	0.363720	−	0.931508i	$-0.381507\pi$
$709$	19.3696	0.727440	0.363720	+	0.931508i	$0.381507\pi$
$710$	−26.4626	−0.993126
$711$	0	0
$712$	19.8980	0.745710
$713$	0	0
$714$	0	0
$715$	7.86030	0.293958
$716$	−2.20362	−0.0823530
$717$	0	0
$718$	30.6584	1.14416
$719$	−8.67434	−0.323498	−0.161749	−	0.986832i	$-0.551714\pi$
$719$	−8.67434	−0.323498	−0.161749	+	0.986832i	$0.551714\pi$
$720$	0	0
$721$	0.754196	0.0280877
$722$	30.1095	1.12056
$723$	0	0
$724$	2.25559	0.0838283
$725$	−6.49087	−0.241065
$726$	0	0
$727$	−21.3440	−0.791606	−0.395803	−	0.918335i	$-0.629534\pi$
$727$	−21.3440	−0.791606	−0.395803	+	0.918335i	$0.629534\pi$
$728$	17.4410	0.646405
$729$	0	0
$730$	8.66759	0.320802
$731$	−1.13917	−0.0421336
$732$	0	0
$733$	−7.90576	−0.292006	−0.146003	−	0.989284i	$-0.546641\pi$
$733$	−7.90576	−0.292006	−0.146003	+	0.989284i	$0.546641\pi$
$734$	−32.1785	−1.18773
$735$	0	0
$736$	0	0
$737$	−39.5358	−1.45632
$738$	0	0
$739$	−10.7259	−0.394558	−0.197279	−	0.980347i	$-0.563211\pi$
$739$	−10.7259	−0.394558	−0.197279	+	0.980347i	$0.563211\pi$
$740$	3.78393	0.139100
$741$	0	0
$742$	−62.0249	−2.27701
$743$	23.4563	0.860529	0.430265	−	0.902703i	$-0.358420\pi$
$743$	23.4563	0.860529	0.430265	+	0.902703i	$0.358420\pi$
$744$	0	0
$745$	−20.1954	−0.739901
$746$	16.7279	0.612452
$747$	0	0
$748$	5.70903	0.208743
$749$	−0.274577	−0.0100328
$750$	0	0
$751$	0.531262	0.0193860	0.00969302	−	0.999953i	$-0.496915\pi$
$751$	0.531262	0.0193860	0.00969302	+	0.999953i	$0.496915\pi$
$752$	63.6138	2.31976
$753$	0	0
$754$	6.12071	0.222903
$755$	12.1829	0.443383
$756$	0	0
$757$	−19.8302	−0.720742	−0.360371	−	0.932809i	$-0.617350\pi$
$757$	−19.8302	−0.720742	−0.360371	+	0.932809i	$0.617350\pi$
$758$	−4.58503	−0.166536
$759$	0	0
$760$	−0.565086	−0.0204978
$761$	39.8979	1.44630	0.723149	−	0.690692i	$-0.242694\pi$
$761$	39.8979	1.44630	0.723149	+	0.690692i	$0.242694\pi$
$762$	0	0
$763$	31.1351	1.12717
$764$	0.675748	0.0244477
$765$	0	0
$766$	4.20272	0.151850
$767$	19.1135	0.690148
$768$	0	0
$769$	−9.74004	−0.351235	−0.175617	−	0.984458i	$-0.556192\pi$
$769$	−9.74004	−0.351235	−0.175617	+	0.984458i	$0.556192\pi$
$770$	−17.6435	−0.635829
$771$	0	0
$772$	−11.4369	−0.411622
$773$	3.68154	0.132416	0.0662079	−	0.997806i	$-0.478910\pi$
$773$	3.68154	0.132416	0.0662079	+	0.997806i	$0.478910\pi$
$774$	0	0
$775$	21.9864	0.789775
$776$	−30.3981	−1.09123
$777$	0	0
$778$	−4.30809	−0.154452
$779$	0.623629	0.0223438
$780$	0	0
$781$	−50.3619	−1.80209
$782$	0	0
$783$	0	0
$784$	−16.7637	−0.598703
$785$	24.0066	0.856832
$786$	0	0
$787$	13.3624	0.476317	0.238159	−	0.971226i	$-0.423456\pi$
$787$	13.3624	0.476317	0.238159	+	0.971226i	$0.423456\pi$
$788$	6.80249	0.242329
$789$	0	0
$790$	−13.7778	−0.490193
$791$	−33.4412	−1.18903
$792$	0	0
$793$	−24.3474	−0.864603
$794$	−39.9223	−1.41679
$795$	0	0
$796$	5.86228	0.207783
$797$	−25.4121	−0.900143	−0.450072	−	0.892992i	$-0.648602\pi$
$797$	−25.4121	−0.900143	−0.450072	+	0.892992i	$0.648602\pi$
$798$	0	0
$799$	45.0294	1.59303
$800$	−11.2102	−0.396339
$801$	0	0
$802$	−38.6023	−1.36309
$803$	16.4956	0.582116
$804$	0	0
$805$	0	0
$806$	−20.7326	−0.730273
$807$	0	0
$808$	−40.4676	−1.42365
$809$	−0.591285	−0.0207885	−0.0103942	−	0.999946i	$-0.503309\pi$
$809$	−0.591285	−0.0207885	−0.0103942	+	0.999946i	$0.503309\pi$
$810$	0	0
$811$	41.9133	1.47178	0.735888	−	0.677103i	$-0.236765\pi$
$811$	41.9133	1.47178	0.735888	+	0.677103i	$0.236765\pi$
$812$	−2.85624	−0.100234
$813$	0	0
$814$	34.6390	1.21410
$815$	20.1481	0.705757
$816$	0	0
$817$	−0.0763638	−0.00267163
$818$	−27.2822	−0.953899
$819$	0	0
$820$	−1.53775	−0.0537006
$821$	−26.5899	−0.927995	−0.463998	−	0.885836i	$-0.653585\pi$
$821$	−26.5899	−0.927995	−0.463998	+	0.885836i	$0.653585\pi$
$822$	0	0
$823$	−32.4588	−1.13144	−0.565721	−	0.824596i	$-0.691402\pi$
$823$	−32.4588	−1.13144	−0.565721	+	0.824596i	$0.691402\pi$
$824$	−0.545251	−0.0189947
$825$	0	0
$826$	−42.9029	−1.49278
$827$	−16.7672	−0.583054	−0.291527	−	0.956563i	$-0.594163\pi$
$827$	−16.7672	−0.583054	−0.291527	+	0.956563i	$0.594163\pi$
$828$	0	0
$829$	−30.5689	−1.06170	−0.530851	−	0.847465i	$-0.678128\pi$
$829$	−30.5689	−1.06170	−0.530851	+	0.847465i	$0.678128\pi$
$830$	−13.0970	−0.454605
$831$	0	0
$832$	−11.3443	−0.393291
$833$	−11.8663	−0.411142
$834$	0	0
$835$	7.02874	0.243240
$836$	0.382704	0.0132361
$837$	0	0
$838$	−54.5281	−1.88364
$839$	−2.65715	−0.0917348	−0.0458674	−	0.998948i	$-0.514605\pi$
$839$	−2.65715	−0.0917348	−0.0458674	+	0.998948i	$0.514605\pi$
$840$	0	0
$841$	−26.1833	−0.902871
$842$	−17.7487	−0.611661
$843$	0	0
$844$	6.11159	0.210370
$845$	−8.22883	−0.283080
$846$	0	0
$847$	2.08517	0.0716473
$848$	57.4811	1.97391
$849$	0	0
$850$	−20.7686	−0.712358
$851$	0	0
$852$	0	0
$853$	−23.5810	−0.807397	−0.403698	−	0.914892i	$-0.632275\pi$
$853$	−23.5810	−0.807397	−0.403698	+	0.914892i	$0.632275\pi$
$854$	54.6512	1.87013
$855$	0	0
$856$	0.198508	0.00678485
$857$	39.6297	1.35372	0.676862	−	0.736110i	$-0.263340\pi$
$857$	39.6297	1.35372	0.676862	+	0.736110i	$0.263340\pi$
$858$	0	0
$859$	26.2941	0.897141	0.448571	−	0.893747i	$-0.351933\pi$
$859$	26.2941	0.897141	0.448571	+	0.893747i	$0.351933\pi$
$860$	0.188299	0.00642094
$861$	0	0
$862$	−10.4090	−0.354532
$863$	−34.6794	−1.18050	−0.590250	−	0.807221i	$-0.700971\pi$
$863$	−34.6794	−1.18050	−0.590250	+	0.807221i	$0.700971\pi$
$864$	0	0
$865$	−17.7198	−0.602490
$866$	−6.05903	−0.205894
$867$	0	0
$868$	9.67488	0.328387
$869$	−26.2210	−0.889488
$870$	0	0
$871$	−28.1956	−0.955370
$872$	−22.5093	−0.762262
$873$	0	0
$874$	0	0
$875$	30.5949	1.03429
$876$	0	0
$877$	−51.2212	−1.72962	−0.864808	−	0.502102i	$-0.832560\pi$
$877$	−51.2212	−1.72962	−0.864808	+	0.502102i	$0.832560\pi$
$878$	−21.4951	−0.725424
$879$	0	0
$880$	16.3510	0.551192
$881$	−37.2799	−1.25599	−0.627997	−	0.778216i	$-0.716125\pi$
$881$	−37.2799	−1.25599	−0.627997	+	0.778216i	$0.716125\pi$
$882$	0	0
$883$	7.95835	0.267820	0.133910	−	0.990994i	$-0.457247\pi$
$883$	7.95835	0.267820	0.133910	+	0.990994i	$0.457247\pi$
$884$	4.07148	0.136939
$885$	0	0
$886$	−24.1497	−0.811324
$887$	−43.7278	−1.46824	−0.734119	−	0.679021i	$-0.762404\pi$
$887$	−43.7278	−1.46824	−0.734119	+	0.679021i	$0.762404\pi$
$888$	0	0
$889$	−24.7941	−0.831566
$890$	−14.3554	−0.481195
$891$	0	0
$892$	−10.4876	−0.351150
$893$	3.01854	0.101012
$894$	0	0
$895$	−4.46748	−0.149331
$896$	44.2586	1.47858
$897$	0	0
$898$	−11.7275	−0.391353
$899$	−9.54110	−0.318214
$900$	0	0
$901$	40.6883	1.35552
$902$	−14.0770	−0.468712
$903$	0	0
$904$	24.1765	0.804099
$905$	4.57285	0.152007
$906$	0	0
$907$	15.7166	0.521862	0.260931	−	0.965358i	$-0.415971\pi$
$907$	15.7166	0.521862	0.260931	+	0.965358i	$0.415971\pi$
$908$	5.83208	0.193544
$909$	0	0
$910$	−12.5828	−0.417114
$911$	28.0712	0.930040	0.465020	−	0.885300i	$-0.346047\pi$
$911$	28.0712	0.930040	0.465020	+	0.885300i	$0.346047\pi$
$912$	0	0
$913$	−24.9254	−0.824911
$914$	51.8521	1.71512
$915$	0	0
$916$	1.63697	0.0540869
$917$	−9.60147	−0.317069
$918$	0	0
$919$	40.9693	1.35145	0.675726	−	0.737153i	$-0.263830\pi$
$919$	40.9693	1.35145	0.675726	+	0.737153i	$0.263830\pi$
$920$	0	0
$921$	0	0
$922$	−50.9775	−1.67885
$923$	−35.9164	−1.18220
$924$	0	0
$925$	−26.1974	−0.861364
$926$	8.08730	0.265765
$927$	0	0
$928$	4.86471	0.159692
$929$	−14.0169	−0.459881	−0.229940	−	0.973205i	$-0.573853\pi$
$929$	−14.0169	−0.459881	−0.229940	+	0.973205i	$0.573853\pi$
$930$	0	0
$931$	−0.795453	−0.0260699
$932$	−0.0665254	−0.00217911
$933$	0	0
$934$	−45.3917	−1.48526
$935$	11.5741	0.378515
$936$	0	0
$937$	−40.8051	−1.33305	−0.666523	−	0.745485i	$-0.732218\pi$
$937$	−40.8051	−1.33305	−0.666523	+	0.745485i	$0.732218\pi$
$938$	63.2888	2.06645
$939$	0	0
$940$	−7.44315	−0.242769
$941$	37.8641	1.23433	0.617167	−	0.786832i	$-0.288280\pi$
$941$	37.8641	1.23433	0.617167	+	0.786832i	$0.288280\pi$
$942$	0	0
$943$	0	0
$944$	39.7599	1.29407
$945$	0	0
$946$	1.72374	0.0560435
$947$	31.1378	1.01184	0.505921	−	0.862580i	$-0.331153\pi$
$947$	31.1378	1.01184	0.505921	+	0.862580i	$0.331153\pi$
$948$	0	0
$949$	11.7641	0.381878
$950$	−1.39222	−0.0451696
$951$	0	0
$952$	25.6815	0.832342
$953$	11.6948	0.378831	0.189415	−	0.981897i	$-0.439341\pi$
$953$	11.6948	0.378831	0.189415	+	0.981897i	$0.439341\pi$
$954$	0	0
$955$	1.36997	0.0443312
$956$	−0.521314	−0.0168605
$957$	0	0
$958$	−39.5817	−1.27883
$959$	45.0039	1.45325
$960$	0	0
$961$	1.31837	0.0425282
$962$	24.7034	0.796469
$963$	0	0
$964$	5.57225	0.179470
$965$	−23.1864	−0.746398
$966$	0	0
$967$	6.26021	0.201315	0.100657	−	0.994921i	$-0.467905\pi$
$967$	6.26021	0.201315	0.100657	+	0.994921i	$0.467905\pi$
$968$	−1.50749	−0.0484525
$969$	0	0
$970$	21.9307	0.704152
$971$	−19.7811	−0.634806	−0.317403	−	0.948291i	$-0.602811\pi$
$971$	−19.7811	−0.634806	−0.317403	+	0.948291i	$0.602811\pi$
$972$	0	0
$973$	−48.0889	−1.54166
$974$	42.1575	1.35081
$975$	0	0
$976$	−50.6476	−1.62119
$977$	−58.8723	−1.88349	−0.941746	−	0.336324i	$-0.890816\pi$
$977$	−58.8723	−1.88349	−0.941746	+	0.336324i	$0.890816\pi$
$978$	0	0
$979$	−27.3203	−0.873159
$980$	1.96144	0.0626559
$981$	0	0
$982$	41.6064	1.32771
$983$	−31.1170	−0.992478	−0.496239	−	0.868186i	$-0.665286\pi$
$983$	−31.1170	−0.992478	−0.496239	+	0.868186i	$0.665286\pi$
$984$	0	0
$985$	13.7910	0.439416
$986$	9.01264	0.287021
$987$	0	0
$988$	0.272931	0.00868310
$989$	0	0
$990$	0	0
$991$	−26.7980	−0.851268	−0.425634	−	0.904895i	$-0.639949\pi$
$991$	−26.7980	−0.851268	−0.425634	+	0.904895i	$0.639949\pi$
$992$	−16.4781	−0.523181
$993$	0	0
$994$	80.6193	2.55709
$995$	11.8848	0.376775
$996$	0	0
$997$	46.2134	1.46359	0.731797	−	0.681522i	$-0.238682\pi$
$997$	46.2134	1.46359	0.731797	+	0.681522i	$0.238682\pi$
$998$	−2.03475	−0.0644089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bx.1.5		20
3.2	odd	2	inner	4761.2.a.bx.1.16		20
23.7	odd	22		207.2.i.e.118.4	yes	40
23.10	odd	22		207.2.i.e.100.4	yes	40
23.22	odd	2		4761.2.a.bw.1.5		20
69.53	even	22		207.2.i.e.118.1	yes	40
69.56	even	22		207.2.i.e.100.1	✓	40
69.68	even	2		4761.2.a.bw.1.16		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.i.e.100.1	✓	40	69.56	even	22
207.2.i.e.100.4	yes	40	23.10	odd	22
207.2.i.e.118.1	yes	40	69.53	even	22
207.2.i.e.118.4	yes	40	23.7	odd	22
4761.2.a.bw.1.5		20	23.22	odd	2
4761.2.a.bw.1.16		20	69.68	even	2
4761.2.a.bx.1.5		20	1.1	even	1	trivial
4761.2.a.bx.1.16		20	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.58900 q^{2} +0.524921 q^{4} +1.06419 q^{5} -3.24210 q^{7} +2.34390 q^{8} +O(q^{10})\)	\(q-1.58900 q^{2} +0.524921 q^{4} +1.06419 q^{5} -3.24210 q^{7} +2.34390 q^{8} -1.69100 q^{10} -3.21821 q^{11} -2.29511 q^{13} +5.15170 q^{14} -4.77430 q^{16} -3.37951 q^{17} -0.226545 q^{19} +0.558618 q^{20} +5.11373 q^{22} -3.86749 q^{25} +3.64694 q^{26} -1.70185 q^{28} +1.67832 q^{29} -5.68493 q^{31} +2.89856 q^{32} +5.37005 q^{34} -3.45023 q^{35} +6.77374 q^{37} +0.359980 q^{38} +2.49436 q^{40} -2.75278 q^{41} +0.337080 q^{43} -1.68930 q^{44} -13.3242 q^{47} +3.51123 q^{49} +6.14544 q^{50} -1.20475 q^{52} -12.0397 q^{53} -3.42480 q^{55} -7.59917 q^{56} -2.66684 q^{58} -8.32790 q^{59} +10.6084 q^{61} +9.03335 q^{62} +4.94278 q^{64} -2.44245 q^{65} +12.2850 q^{67} -1.77398 q^{68} +5.48241 q^{70} +15.6491 q^{71} -5.12570 q^{73} -10.7635 q^{74} -0.118918 q^{76} +10.4338 q^{77} +8.14772 q^{79} -5.08078 q^{80} +4.37416 q^{82} +7.74513 q^{83} -3.59646 q^{85} -0.535620 q^{86} -7.54315 q^{88} +8.48928 q^{89} +7.44100 q^{91} +21.1722 q^{94} -0.241088 q^{95} -12.9690 q^{97} -5.57935 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} + 18 q^{7}+O(q^{10}) \) 20 * q + 20 * q^4 + 18 * q^7	\( 20 q + 20 q^{4} + 18 q^{7} + 22 q^{10} + 16 q^{16} + 40 q^{19} + 14 q^{22} + 20 q^{25} + 32 q^{28} - 22 q^{31} + 60 q^{34} + 18 q^{37} + 74 q^{40} + 32 q^{43} + 2 q^{49} - 52 q^{55} - 24 q^{58} + 70 q^{61} + 36 q^{64} + 64 q^{67} + 48 q^{70} + 40 q^{73} + 82 q^{76} + 106 q^{79} - 36 q^{82} + 2 q^{85} + 58 q^{88} + 88 q^{91} + 40 q^{94} + 24 q^{97}+O(q^{100}) \) 20 * q + 20 * q^4 + 18 * q^7 + 22 * q^10 + 16 * q^16 + 40 * q^19 + 14 * q^22 + 20 * q^25 + 32 * q^28 - 22 * q^31 + 60 * q^34 + 18 * q^37 + 74 * q^40 + 32 * q^43 + 2 * q^49 - 52 * q^55 - 24 * q^58 + 70 * q^61 + 36 * q^64 + 64 * q^67 + 48 * q^70 + 40 * q^73 + 82 * q^76 + 106 * q^79 - 36 * q^82 + 2 * q^85 + 58 * q^88 + 88 * q^91 + 40 * q^94 + 24 * q^97

Introduction

L-functions

Modular forms

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Database

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