LMFDB - Embedded newform 4805.2.a.bb.1.21

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4805,2,Mod(1,4805)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4805.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4805, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4805 = 5 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4805.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,0,0,32,-24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.3681181712$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	4805.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.55674 q^{2} -2.41709 q^{3} +4.53694 q^{4} -1.00000 q^{5} -6.17988 q^{6} +3.41271 q^{7} +6.48631 q^{8} +2.84233 q^{9} -2.55674 q^{10} -5.46610 q^{11} -10.9662 q^{12} +5.38349 q^{13} +8.72542 q^{14} +2.41709 q^{15} +7.50995 q^{16} -2.83128 q^{17} +7.26711 q^{18} -1.20252 q^{19} -4.53694 q^{20} -8.24883 q^{21} -13.9754 q^{22} +7.33676 q^{23} -15.6780 q^{24} +1.00000 q^{25} +13.7642 q^{26} +0.381101 q^{27} +15.4833 q^{28} +2.64099 q^{29} +6.17988 q^{30} +6.22840 q^{32} +13.2121 q^{33} -7.23885 q^{34} -3.41271 q^{35} +12.8955 q^{36} +2.54608 q^{37} -3.07455 q^{38} -13.0124 q^{39} -6.48631 q^{40} -4.95941 q^{41} -21.0901 q^{42} +1.90043 q^{43} -24.7994 q^{44} -2.84233 q^{45} +18.7582 q^{46} +3.88911 q^{47} -18.1522 q^{48} +4.64659 q^{49} +2.55674 q^{50} +6.84346 q^{51} +24.4246 q^{52} +3.36913 q^{53} +0.974379 q^{54} +5.46610 q^{55} +22.1359 q^{56} +2.90661 q^{57} +6.75233 q^{58} +3.61973 q^{59} +10.9662 q^{60} +7.45118 q^{61} +9.70005 q^{63} +0.904535 q^{64} -5.38349 q^{65} +33.7799 q^{66} +12.0235 q^{67} -12.8453 q^{68} -17.7336 q^{69} -8.72542 q^{70} -3.34899 q^{71} +18.4362 q^{72} +15.4323 q^{73} +6.50968 q^{74} -2.41709 q^{75} -5.45578 q^{76} -18.6542 q^{77} -33.2693 q^{78} -3.97731 q^{79} -7.50995 q^{80} -9.44815 q^{81} -12.6799 q^{82} -1.32742 q^{83} -37.4245 q^{84} +2.83128 q^{85} +4.85892 q^{86} -6.38351 q^{87} -35.4548 q^{88} -18.0557 q^{89} -7.26711 q^{90} +18.3723 q^{91} +33.2865 q^{92} +9.94345 q^{94} +1.20252 q^{95} -15.0546 q^{96} +6.87122 q^{97} +11.8801 q^{98} -15.5365 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} - 24 q^{5} + 8 q^{7} + 40 q^{9} + 8 q^{14} + 88 q^{16} - 64 q^{18} + 40 q^{19} - 32 q^{20} + 24 q^{25} + 72 q^{28} + 56 q^{33} - 8 q^{35} + 88 q^{36} - 72 q^{38} + 64 q^{39} - 56 q^{41}+ \cdots - 8 q^{98}+O(q^{100}) $ 24 * q + 32 * q^4 - 24 * q^5 + 8 * q^7 + 40 * q^9 + 8 * q^14 + 88 * q^16 - 64 * q^18 + 40 * q^19 - 32 * q^20 + 24 * q^25 + 72 * q^28 + 56 * q^33 - 8 * q^35 + 88 * q^36 - 72 * q^38 + 64 * q^39 - 56 * q^41 - 40 * q^45 - 32 * q^47 + 56 * q^49 + 88 * q^51 + 104 * q^56 + 8 * q^59 + 56 * q^63 + 160 * q^64 - 8 * q^66 + 96 * q^67 + 8 * q^69 - 8 * q^70 - 64 * q^71 - 120 * q^72 + 128 * q^76 - 128 * q^78 - 88 * q^80 + 72 * q^81 + 32 * q^82 - 32 * q^87 + 64 * q^90 + 56 * q^94 - 40 * q^95 + 48 * q^97 - 8 * q^98

Coefficient data

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

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

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

$2$	2.55674	1.80789	0.903946	−	0.427648i	$-0.140658\pi$
$2$	2.55674	1.80789	0.903946	+	0.427648i	$0.140658\pi$
$3$	−2.41709	−1.39551	−0.697754	−	0.716337i	$-0.745817\pi$
$3$	−2.41709	−1.39551	−0.697754	+	0.716337i	$0.745817\pi$
$4$	4.53694	2.26847
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−6.17988	−2.52293
$7$	3.41271	1.28988	0.644941	−	0.764232i	$-0.276882\pi$
$7$	3.41271	1.28988	0.644941	+	0.764232i	$0.276882\pi$
$8$	6.48631	2.29326
$9$	2.84233	0.947444
$10$	−2.55674	−0.808513
$11$	−5.46610	−1.64809	−0.824046	−	0.566523i	$-0.808288\pi$
$11$	−5.46610	−1.64809	−0.824046	+	0.566523i	$0.808288\pi$
$12$	−10.9662	−3.16567
$13$	5.38349	1.49311	0.746555	−	0.665323i	$-0.231706\pi$
$13$	5.38349	1.49311	0.746555	+	0.665323i	$0.231706\pi$
$14$	8.72542	2.33197
$15$	2.41709	0.624090
$16$	7.50995	1.87749
$17$	−2.83128	−0.686686	−0.343343	−	0.939210i	$-0.611559\pi$
$17$	−2.83128	−0.686686	−0.343343	+	0.939210i	$0.611559\pi$
$18$	7.26711	1.71287
$19$	−1.20252	−0.275878	−0.137939	−	0.990441i	$-0.544048\pi$
$19$	−1.20252	−0.275878	−0.137939	+	0.990441i	$0.544048\pi$
$20$	−4.53694	−1.01449
$21$	−8.24883	−1.80004
$22$	−13.9754	−2.97957
$23$	7.33676	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$23$	7.33676	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$24$	−15.6780	−3.20026
$25$	1.00000	0.200000
$26$	13.7642	2.69938
$27$	0.381101	0.0733430
$28$	15.4833	2.92606
$29$	2.64099	0.490419	0.245210	−	0.969470i	$-0.421143\pi$
$29$	2.64099	0.490419	0.245210	+	0.969470i	$0.421143\pi$
$30$	6.17988	1.12829
$31$	0	0
$31$	0	0
$32$	6.22840	1.10104
$33$	13.2121	2.29993
$34$	−7.23885	−1.24145
$35$	−3.41271	−0.576853
$36$	12.8955	2.14925
$37$	2.54608	0.418573	0.209287	−	0.977854i	$-0.432886\pi$
$37$	2.54608	0.418573	0.209287	+	0.977854i	$0.432886\pi$
$38$	−3.07455	−0.498757
$39$	−13.0124	−2.08365
$40$	−6.48631	−1.02558
$41$	−4.95941	−0.774529	−0.387265	−	0.921969i	$-0.626580\pi$
$41$	−4.95941	−0.774529	−0.387265	+	0.921969i	$0.626580\pi$
$42$	−21.0901	−3.25428
$43$	1.90043	0.289813	0.144907	−	0.989445i	$-0.453712\pi$
$43$	1.90043	0.289813	0.144907	+	0.989445i	$0.453712\pi$
$44$	−24.7994	−3.73865
$45$	−2.84233	−0.423710
$46$	18.7582	2.76575
$47$	3.88911	0.567285	0.283642	−	0.958930i	$-0.408457\pi$
$47$	3.88911	0.567285	0.283642	+	0.958930i	$0.408457\pi$
$48$	−18.1522	−2.62005
$49$	4.64659	0.663798
$50$	2.55674	0.361578
$51$	6.84346	0.958276
$52$	24.4246	3.38708
$53$	3.36913	0.462785	0.231393	−	0.972860i	$-0.425672\pi$
$53$	3.36913	0.462785	0.231393	+	0.972860i	$0.425672\pi$
$54$	0.974379	0.132596
$55$	5.46610	0.737049
$56$	22.1359	2.95803
$57$	2.90661	0.384990
$58$	6.75233	0.886625
$59$	3.61973	0.471248	0.235624	−	0.971844i	$-0.424287\pi$
$59$	3.61973	0.471248	0.235624	+	0.971844i	$0.424287\pi$
$60$	10.9662	1.41573
$61$	7.45118	0.954026	0.477013	−	0.878896i	$-0.341719\pi$
$61$	7.45118	0.954026	0.477013	+	0.878896i	$0.341719\pi$
$62$	0	0
$63$	9.70005	1.22209
$64$	0.904535	0.113067
$65$	−5.38349	−0.667739
$66$	33.7799	4.15801
$67$	12.0235	1.46891	0.734453	−	0.678659i	$-0.237439\pi$
$67$	12.0235	1.46891	0.734453	+	0.678659i	$0.237439\pi$
$68$	−12.8453	−1.55773
$69$	−17.7336	−2.13488
$70$	−8.72542	−1.04289
$71$	−3.34899	−0.397452	−0.198726	−	0.980055i	$-0.563680\pi$
$71$	−3.34899	−0.397452	−0.198726	+	0.980055i	$0.563680\pi$
$72$	18.4362	2.17273
$73$	15.4323	1.80622	0.903110	−	0.429410i	$-0.141278\pi$
$73$	15.4323	1.80622	0.903110	+	0.429410i	$0.141278\pi$
$74$	6.50968	0.756735
$75$	−2.41709	−0.279102
$76$	−5.45578	−0.625821
$77$	−18.6542	−2.12584
$78$	−33.2693	−3.76701
$79$	−3.97731	−0.447482	−0.223741	−	0.974649i	$-0.571827\pi$
$79$	−3.97731	−0.447482	−0.223741	+	0.974649i	$0.571827\pi$
$80$	−7.50995	−0.839638
$81$	−9.44815	−1.04979
$82$	−12.6799	−1.40026
$83$	−1.32742	−0.145704	−0.0728518	−	0.997343i	$-0.523210\pi$
$83$	−1.32742	−0.145704	−0.0728518	+	0.997343i	$0.523210\pi$
$84$	−37.4245	−4.08334
$85$	2.83128	0.307095
$86$	4.85892	0.523951
$87$	−6.38351	−0.684384
$88$	−35.4548	−3.77950
$89$	−18.0557	−1.91390	−0.956951	−	0.290250i	$-0.906261\pi$
$89$	−18.0557	−1.91390	−0.956951	+	0.290250i	$0.906261\pi$
$90$	−7.26711	−0.766021
$91$	18.3723	1.92594
$92$	33.2865	3.47035
$93$	0	0
$94$	9.94345	1.02559
$95$	1.20252	0.123376
$96$	−15.0546	−1.53651
$97$	6.87122	0.697667	0.348833	−	0.937185i	$-0.386578\pi$
$97$	6.87122	0.697667	0.348833	+	0.937185i	$0.386578\pi$
$98$	11.8801	1.20007
$99$	−15.5365	−1.56147
$100$	4.53694	0.453694
$101$	−13.2625	−1.31967	−0.659833	−	0.751413i	$-0.729373\pi$
$101$	−13.2625	−1.31967	−0.659833	+	0.751413i	$0.729373\pi$
$102$	17.4970	1.73246
$103$	−5.18607	−0.510999	−0.255499	−	0.966809i	$-0.582240\pi$
$103$	−5.18607	−0.510999	−0.255499	+	0.966809i	$0.582240\pi$
$104$	34.9190	3.42409
$105$	8.24883	0.805003
$106$	8.61400	0.836665
$107$	13.5434	1.30929	0.654644	−	0.755937i	$-0.272819\pi$
$107$	13.5434	1.30929	0.654644	+	0.755937i	$0.272819\pi$
$108$	1.72903	0.166376
$109$	5.53736	0.530383	0.265192	−	0.964196i	$-0.414565\pi$
$109$	5.53736	0.530383	0.265192	+	0.964196i	$0.414565\pi$
$110$	13.9754	1.33250
$111$	−6.15411	−0.584123
$112$	25.6293	2.42174
$113$	5.14611	0.484105	0.242053	−	0.970263i	$-0.422179\pi$
$113$	5.14611	0.484105	0.242053	+	0.970263i	$0.422179\pi$
$114$	7.43146	0.696020
$115$	−7.33676	−0.684157
$116$	11.9820	1.11250
$117$	15.3017	1.41464
$118$	9.25472	0.851966
$119$	−9.66233	−0.885744
$120$	15.6780	1.43120
$121$	18.8783	1.71620
$122$	19.0508	1.72478
$123$	11.9873	1.08086
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	24.8005	2.20941
$127$	11.2632	0.999452	0.499726	−	0.866184i	$-0.333434\pi$
$127$	11.2632	0.999452	0.499726	+	0.866184i	$0.333434\pi$
$128$	−10.1441	−0.896624
$129$	−4.59352	−0.404437
$130$	−13.7642	−1.20720
$131$	12.9343	1.13008	0.565038	−	0.825065i	$-0.308862\pi$
$131$	12.9343	1.13008	0.565038	+	0.825065i	$0.308862\pi$
$132$	59.9424	5.21731
$133$	−4.10387	−0.355850
$134$	30.7411	2.65562
$135$	−0.381101	−0.0328000
$136$	−18.3645	−1.57475
$137$	5.27414	0.450600	0.225300	−	0.974289i	$-0.427664\pi$
$137$	5.27414	0.450600	0.225300	+	0.974289i	$0.427664\pi$
$138$	−45.3404	−3.85963
$139$	3.51245	0.297922	0.148961	−	0.988843i	$-0.452407\pi$
$139$	3.51245	0.297922	0.148961	+	0.988843i	$0.452407\pi$
$140$	−15.4833	−1.30857
$141$	−9.40033	−0.791650
$142$	−8.56251	−0.718549
$143$	−29.4267	−2.46078
$144$	21.3458	1.77881
$145$	−2.64099	−0.219322
$146$	39.4566	3.26545
$147$	−11.2312	−0.926336
$148$	11.5514	0.949521
$149$	6.85604	0.561669	0.280834	−	0.959756i	$-0.409389\pi$
$149$	6.85604	0.561669	0.280834	+	0.959756i	$0.409389\pi$
$150$	−6.17988	−0.504585
$151$	−13.1910	−1.07347	−0.536735	−	0.843751i	$-0.680342\pi$
$151$	−13.1910	−1.07347	−0.536735	+	0.843751i	$0.680342\pi$
$152$	−7.79995	−0.632659
$153$	−8.04743	−0.650596
$154$	−47.6940	−3.84330
$155$	0	0
$156$	−59.0364	−4.72669
$157$	−13.9741	−1.11526	−0.557629	−	0.830090i	$-0.688289\pi$
$157$	−13.9741	−1.11526	−0.557629	+	0.830090i	$0.688289\pi$
$158$	−10.1690	−0.808999
$159$	−8.14349	−0.645821
$160$	−6.22840	−0.492399
$161$	25.0382	1.97329
$162$	−24.1565	−1.89791
$163$	16.3515	1.28075	0.640374	−	0.768063i	$-0.278779\pi$
$163$	16.3515	1.28075	0.640374	+	0.768063i	$0.278779\pi$
$164$	−22.5005	−1.75700
$165$	−13.2121	−1.02856
$166$	−3.39388	−0.263416
$167$	−8.53209	−0.660233	−0.330116	−	0.943940i	$-0.607088\pi$
$167$	−8.53209	−0.660233	−0.330116	+	0.943940i	$0.607088\pi$
$168$	−53.5045	−4.12796
$169$	15.9819	1.22938
$170$	7.23885	0.555195
$171$	−3.41797	−0.261379
$172$	8.62215	0.657433
$173$	22.9386	1.74399	0.871994	−	0.489517i	$-0.162827\pi$
$173$	22.9386	1.74399	0.871994	+	0.489517i	$0.162827\pi$
$174$	−16.3210	−1.23729
$175$	3.41271	0.257977
$176$	−41.0501	−3.09427
$177$	−8.74921	−0.657631
$178$	−46.1638	−3.46013
$179$	14.9896	1.12038	0.560188	−	0.828366i	$-0.310729\pi$
$179$	14.9896	1.12038	0.560188	+	0.828366i	$0.310729\pi$
$180$	−12.8955	−0.961173
$181$	2.09879	0.156002	0.0780009	−	0.996953i	$-0.475146\pi$
$181$	2.09879	0.156002	0.0780009	+	0.996953i	$0.475146\pi$
$182$	46.9732	3.48189
$183$	−18.0102	−1.33135
$184$	47.5885	3.50827
$185$	−2.54608	−0.187192
$186$	0	0
$187$	15.4760	1.13172
$188$	17.6446	1.28687
$189$	1.30059	0.0946039
$190$	3.07455	0.223051
$191$	15.1699	1.09765	0.548827	−	0.835936i	$-0.315075\pi$
$191$	15.1699	1.09765	0.548827	+	0.835936i	$0.315075\pi$
$192$	−2.18634	−0.157786
$193$	−8.33660	−0.600082	−0.300041	−	0.953926i	$-0.597000\pi$
$193$	−8.33660	−0.600082	−0.300041	+	0.953926i	$0.597000\pi$
$194$	17.5680	1.26131
$195$	13.0124	0.931836
$196$	21.0813	1.50581
$197$	18.9006	1.34661	0.673305	−	0.739365i	$-0.264874\pi$
$197$	18.9006	1.34661	0.673305	+	0.739365i	$0.264874\pi$
$198$	−39.7228	−2.82297
$199$	−7.51729	−0.532887	−0.266443	−	0.963851i	$-0.585849\pi$
$199$	−7.51729	−0.532887	−0.266443	+	0.963851i	$0.585849\pi$
$200$	6.48631	0.458651
$201$	−29.0619	−2.04987
$202$	−33.9087	−2.38581
$203$	9.01293	0.632584
$204$	31.0484	2.17382
$205$	4.95941	0.346380
$206$	−13.2595	−0.923830
$207$	20.8535	1.44942
$208$	40.4297	2.80330
$209$	6.57312	0.454672
$210$	21.0901	1.45536
$211$	18.7278	1.28928	0.644638	−	0.764488i	$-0.277008\pi$
$211$	18.7278	1.28928	0.644638	+	0.764488i	$0.277008\pi$
$212$	15.2855	1.04981
$213$	8.09481	0.554647
$214$	34.6270	2.36705
$215$	−1.90043	−0.129608
$216$	2.47194	0.168194
$217$	0	0
$218$	14.1576	0.958875
$219$	−37.3014	−2.52059
$220$	24.7994	1.67197
$221$	−15.2421	−1.02530
$222$	−15.7345	−1.05603
$223$	13.2247	0.885592	0.442796	−	0.896622i	$-0.353987\pi$
$223$	13.2247	0.885592	0.442796	+	0.896622i	$0.353987\pi$
$224$	21.2557	1.42021
$225$	2.84233	0.189489
$226$	13.1573	0.875209
$227$	−23.1084	−1.53376	−0.766880	−	0.641790i	$-0.778192\pi$
$227$	−23.1084	−1.53376	−0.766880	+	0.641790i	$0.778192\pi$
$228$	13.1871	0.873339
$229$	−9.26446	−0.612213	−0.306106	−	0.951997i	$-0.599026\pi$
$229$	−9.26446	−0.612213	−0.306106	+	0.951997i	$0.599026\pi$
$230$	−18.7582	−1.23688
$231$	45.0889	2.96663
$232$	17.1303	1.12466
$233$	−13.0053	−0.852008	−0.426004	−	0.904721i	$-0.640079\pi$
$233$	−13.0053	−0.852008	−0.426004	+	0.904721i	$0.640079\pi$
$234$	39.1224	2.55751
$235$	−3.88911	−0.253697
$236$	16.4225	1.06901
$237$	9.61351	0.624465
$238$	−24.7041	−1.60133
$239$	2.90520	0.187921	0.0939607	−	0.995576i	$-0.470047\pi$
$239$	2.90520	0.187921	0.0939607	+	0.995576i	$0.470047\pi$
$240$	18.1522	1.17172
$241$	−19.3655	−1.24744	−0.623720	−	0.781648i	$-0.714379\pi$
$241$	−19.3655	−1.24744	−0.623720	+	0.781648i	$0.714379\pi$
$242$	48.2669	3.10271
$243$	21.6937	1.39165
$244$	33.8056	2.16418
$245$	−4.64659	−0.296859
$246$	30.6486	1.95408
$247$	−6.47378	−0.411916
$248$	0	0
$249$	3.20850	0.203331
$250$	−2.55674	−0.161703
$251$	−13.2700	−0.837593	−0.418797	−	0.908080i	$-0.637548\pi$
$251$	−13.2700	−0.837593	−0.418797	+	0.908080i	$0.637548\pi$
$252$	44.0085	2.77228
$253$	−40.1035	−2.52128
$254$	28.7972	1.80690
$255$	−6.84346	−0.428554
$256$	−27.7450	−1.73407
$257$	−3.18176	−0.198472	−0.0992362	−	0.995064i	$-0.531640\pi$
$257$	−3.18176	−0.198472	−0.0992362	+	0.995064i	$0.531640\pi$
$258$	−11.7445	−0.731178
$259$	8.68904	0.539911
$260$	−24.4246	−1.51475
$261$	7.50657	0.464645
$262$	33.0697	2.04305
$263$	−17.6702	−1.08959	−0.544795	−	0.838569i	$-0.683392\pi$
$263$	−17.6702	−1.08959	−0.544795	+	0.838569i	$0.683392\pi$
$264$	85.6975	5.27432
$265$	−3.36913	−0.206964
$266$	−10.4925	−0.643339
$267$	43.6423	2.67087
$268$	54.5500	3.33217
$269$	−1.34708	−0.0821331	−0.0410666	−	0.999156i	$-0.513076\pi$
$269$	−1.34708	−0.0821331	−0.0410666	+	0.999156i	$0.513076\pi$
$270$	−0.974379	−0.0592988
$271$	3.66547	0.222662	0.111331	−	0.993783i	$-0.464489\pi$
$271$	3.66547	0.222662	0.111331	+	0.993783i	$0.464489\pi$
$272$	−21.2628	−1.28924
$273$	−44.4075	−2.68766
$274$	13.4846	0.814637
$275$	−5.46610	−0.329618
$276$	−80.4564	−4.84291
$277$	−20.2583	−1.21720	−0.608601	−	0.793476i	$-0.708269\pi$
$277$	−20.2583	−1.21720	−0.608601	+	0.793476i	$0.708269\pi$
$278$	8.98044	0.538611
$279$	0	0
$280$	−22.1359	−1.32287
$281$	−2.76581	−0.164994	−0.0824972	−	0.996591i	$-0.526290\pi$
$281$	−2.76581	−0.164994	−0.0824972	+	0.996591i	$0.526290\pi$
$282$	−24.0342	−1.43122
$283$	30.3025	1.80130	0.900648	−	0.434549i	$-0.143092\pi$
$283$	30.3025	1.80130	0.900648	+	0.434549i	$0.143092\pi$
$284$	−15.1942	−0.901608
$285$	−2.90661	−0.172173
$286$	−75.2365	−4.44883
$287$	−16.9250	−0.999052
$288$	17.7032	1.04317
$289$	−8.98387	−0.528463
$290$	−6.75233	−0.396511
$291$	−16.6084	−0.973600
$292$	70.0156	4.09736
$293$	−26.0050	−1.51923	−0.759616	−	0.650372i	$-0.774613\pi$
$293$	−26.0050	−1.51923	−0.759616	+	0.650372i	$0.774613\pi$
$294$	−28.7154	−1.67471
$295$	−3.61973	−0.210749
$296$	16.5147	0.959896
$297$	−2.08314	−0.120876
$298$	17.5291	1.01544
$299$	39.4974	2.28419
$300$	−10.9662	−0.633134
$301$	6.48563	0.373825
$302$	−33.7260	−1.94072
$303$	32.0566	1.84160
$304$	−9.03090	−0.517958
$305$	−7.45118	−0.426653
$306$	−20.5752	−1.17621
$307$	−17.1009	−0.976002	−0.488001	−	0.872843i	$-0.662274\pi$
$307$	−17.1009	−0.976002	−0.488001	+	0.872843i	$0.662274\pi$
$308$	−84.6331	−4.82242
$309$	12.5352	0.713103
$310$	0	0
$311$	−8.36576	−0.474379	−0.237189	−	0.971463i	$-0.576226\pi$
$311$	−8.36576	−0.474379	−0.237189	+	0.971463i	$0.576226\pi$
$312$	−84.4023	−4.77834
$313$	−16.9903	−0.960350	−0.480175	−	0.877173i	$-0.659427\pi$
$313$	−16.9903	−0.960350	−0.480175	+	0.877173i	$0.659427\pi$
$314$	−35.7283	−2.01627
$315$	−9.70005	−0.546536
$316$	−18.0448	−1.01510
$317$	21.8888	1.22940	0.614699	−	0.788761i	$-0.289277\pi$
$317$	21.8888	1.22940	0.614699	+	0.788761i	$0.289277\pi$
$318$	−20.8208	−1.16757
$319$	−14.4359	−0.808256
$320$	−0.904535	−0.0505650
$321$	−32.7356	−1.82712
$322$	64.0164	3.56749
$323$	3.40468	0.189442
$324$	−42.8657	−2.38143
$325$	5.38349	0.298622
$326$	41.8066	2.31545
$327$	−13.3843	−0.740154
$328$	−32.1682	−1.77619
$329$	13.2724	0.731731
$330$	−33.7799	−1.85952
$331$	20.6052	1.13257	0.566283	−	0.824211i	$-0.308381\pi$
$331$	20.6052	1.13257	0.566283	+	0.824211i	$0.308381\pi$
$332$	−6.02244	−0.330524
$333$	7.23681	0.396575
$334$	−21.8144	−1.19363
$335$	−12.0235	−0.656915
$336$	−61.9483	−3.37956
$337$	−6.70887	−0.365455	−0.182728	−	0.983164i	$-0.558493\pi$
$337$	−6.70887	−0.365455	−0.182728	+	0.983164i	$0.558493\pi$
$338$	40.8617	2.22258
$339$	−12.4386	−0.675573
$340$	12.8453	0.696636
$341$	0	0
$342$	−8.73888	−0.472545
$343$	−8.03152	−0.433661
$344$	12.3268	0.664616
$345$	17.7336	0.954747
$346$	58.6481	3.15294
$347$	8.12022	0.435916	0.217958	−	0.975958i	$-0.430060\pi$
$347$	8.12022	0.435916	0.217958	+	0.975958i	$0.430060\pi$
$348$	−28.9616	−1.55251
$349$	−7.00132	−0.374772	−0.187386	−	0.982286i	$-0.560002\pi$
$349$	−7.00132	−0.374772	−0.187386	+	0.982286i	$0.560002\pi$
$350$	8.72542	0.466394
$351$	2.05165	0.109509
$352$	−34.0451	−1.81461
$353$	1.62311	0.0863897	0.0431948	−	0.999067i	$-0.486246\pi$
$353$	1.62311	0.0863897	0.0431948	+	0.999067i	$0.486246\pi$
$354$	−22.3695	−1.18893
$355$	3.34899	0.177746
$356$	−81.9177	−4.34163
$357$	23.3547	1.23606
$358$	38.3246	2.02552
$359$	4.12321	0.217615	0.108807	−	0.994063i	$-0.465297\pi$
$359$	4.12321	0.217615	0.108807	+	0.994063i	$0.465297\pi$
$360$	−18.4362	−0.971675
$361$	−17.5539	−0.923891
$362$	5.36607	0.282034
$363$	−45.6305	−2.39498
$364$	83.3539	4.36893
$365$	−15.4323	−0.807766
$366$	−46.0474	−2.40694
$367$	12.5426	0.654716	0.327358	−	0.944900i	$-0.393842\pi$
$367$	12.5426	0.654716	0.327358	+	0.944900i	$0.393842\pi$
$368$	55.0987	2.87222
$369$	−14.0963	−0.733823
$370$	−6.50968	−0.338422
$371$	11.4979	0.596939
$372$	0	0
$373$	32.5633	1.68607	0.843033	−	0.537862i	$-0.180768\pi$
$373$	32.5633	1.68607	0.843033	+	0.537862i	$0.180768\pi$
$374$	39.5683	2.04603
$375$	2.41709	0.124818
$376$	25.2259	1.30093
$377$	14.2177	0.732251
$378$	3.32527	0.171034
$379$	−21.3538	−1.09687	−0.548435	−	0.836193i	$-0.684776\pi$
$379$	−21.3538	−1.09687	−0.548435	+	0.836193i	$0.684776\pi$
$380$	5.45578	0.279876
$381$	−27.2243	−1.39474
$382$	38.7855	1.98444
$383$	−10.4260	−0.532744	−0.266372	−	0.963870i	$-0.585825\pi$
$383$	−10.4260	−0.532744	−0.266372	+	0.963870i	$0.585825\pi$
$384$	24.5193	1.25125
$385$	18.6542	0.950707
$386$	−21.3146	−1.08488
$387$	5.40166	0.274582
$388$	31.1743	1.58264
$389$	−32.9923	−1.67278	−0.836388	−	0.548138i	$-0.815337\pi$
$389$	−32.9923	−1.67278	−0.836388	+	0.548138i	$0.815337\pi$
$390$	33.2693	1.68466
$391$	−20.7724	−1.05051
$392$	30.1392	1.52226
$393$	−31.2634	−1.57703
$394$	48.3239	2.43452
$395$	3.97731	0.200120
$396$	−70.4880	−3.54216
$397$	−4.63109	−0.232428	−0.116214	−	0.993224i	$-0.537076\pi$
$397$	−4.63109	−0.232428	−0.116214	+	0.993224i	$0.537076\pi$
$398$	−19.2198	−0.963401
$399$	9.91942	0.496592
$400$	7.50995	0.375497
$401$	−19.1106	−0.954338	−0.477169	−	0.878812i	$-0.658337\pi$
$401$	−19.1106	−0.954338	−0.477169	+	0.878812i	$0.658337\pi$
$402$	−74.3040	−3.70594
$403$	0	0
$404$	−60.1710	−2.99362
$405$	9.44815	0.469482
$406$	23.0438	1.14364
$407$	−13.9171	−0.689847
$408$	44.3888	2.19757
$409$	−21.5580	−1.06597	−0.532987	−	0.846123i	$-0.678931\pi$
$409$	−21.5580	−1.06597	−0.532987	+	0.846123i	$0.678931\pi$
$410$	12.6799	0.626217
$411$	−12.7481	−0.628817
$412$	−23.5289	−1.15919
$413$	12.3531	0.607855
$414$	53.3171	2.62039
$415$	1.32742	0.0651606
$416$	33.5305	1.64397
$417$	−8.48992	−0.415753
$418$	16.8058	0.821998
$419$	−23.1121	−1.12910	−0.564551	−	0.825398i	$-0.690951\pi$
$419$	−23.1121	−1.12910	−0.564551	+	0.825398i	$0.690951\pi$
$420$	37.4245	1.82613
$421$	9.56282	0.466063	0.233032	−	0.972469i	$-0.425135\pi$
$421$	9.56282	0.466063	0.233032	+	0.972469i	$0.425135\pi$
$422$	47.8822	2.33087
$423$	11.0541	0.537470
$424$	21.8532	1.06129
$425$	−2.83128	−0.137337
$426$	20.6964	1.00274
$427$	25.4287	1.23058
$428$	61.4455	2.97008
$429$	71.1270	3.43404
$430$	−4.85892	−0.234318
$431$	12.5266	0.603385	0.301692	−	0.953405i	$-0.402448\pi$
$431$	12.5266	0.603385	0.301692	+	0.953405i	$0.402448\pi$
$432$	2.86205	0.137701
$433$	17.3509	0.833832	0.416916	−	0.908945i	$-0.363111\pi$
$433$	17.3509	0.833832	0.416916	+	0.908945i	$0.363111\pi$
$434$	0	0
$435$	6.38351	0.306066
$436$	25.1227	1.20316
$437$	−8.82264	−0.422044
$438$	−95.3701	−4.55696
$439$	−23.3256	−1.11327	−0.556636	−	0.830757i	$-0.687908\pi$
$439$	−23.3256	−1.11327	−0.556636	+	0.830757i	$0.687908\pi$
$440$	35.4548	1.69024
$441$	13.2071	0.628911
$442$	−38.9703	−1.85363
$443$	−28.1551	−1.33769	−0.668845	−	0.743402i	$-0.733211\pi$
$443$	−28.1551	−1.33769	−0.668845	+	0.743402i	$0.733211\pi$
$444$	−27.9209	−1.32507
$445$	18.0557	0.855923
$446$	33.8122	1.60105
$447$	−16.5717	−0.783814
$448$	3.08691	0.145843
$449$	5.06308	0.238941	0.119471	−	0.992838i	$-0.461880\pi$
$449$	5.06308	0.238941	0.119471	+	0.992838i	$0.461880\pi$
$450$	7.26711	0.342575
$451$	27.1086	1.27649
$452$	23.3476	1.09818
$453$	31.8839	1.49804
$454$	−59.0823	−2.77287
$455$	−18.3723	−0.861306
$456$	18.8532	0.882881
$457$	−29.4244	−1.37641	−0.688207	−	0.725515i	$-0.741602\pi$
$457$	−29.4244	−1.37641	−0.688207	+	0.725515i	$0.741602\pi$
$458$	−23.6869	−1.10681
$459$	−1.07900	−0.0503636
$460$	−33.2865	−1.55199
$461$	−7.80537	−0.363532	−0.181766	−	0.983342i	$-0.558181\pi$
$461$	−7.80537	−0.363532	−0.181766	+	0.983342i	$0.558181\pi$
$462$	115.281	5.36335
$463$	4.27397	0.198628	0.0993141	−	0.995056i	$-0.468335\pi$
$463$	4.27397	0.198628	0.0993141	+	0.995056i	$0.468335\pi$
$464$	19.8337	0.920756
$465$	0	0
$466$	−33.2513	−1.54034
$467$	−4.94347	−0.228757	−0.114378	−	0.993437i	$-0.536488\pi$
$467$	−4.94347	−0.228757	−0.114378	+	0.993437i	$0.536488\pi$
$468$	69.4227	3.20906
$469$	41.0328	1.89472
$470$	−9.94345	−0.458657
$471$	33.7768	1.55635
$472$	23.4787	1.08069
$473$	−10.3880	−0.477639
$474$	24.5793	1.12896
$475$	−1.20252	−0.0551756
$476$	−43.8374	−2.00928
$477$	9.57617	0.438463
$478$	7.42784	0.339742
$479$	13.3220	0.608700	0.304350	−	0.952560i	$-0.401561\pi$
$479$	13.3220	0.608700	0.304350	+	0.952560i	$0.401561\pi$
$480$	15.0546	0.687146
$481$	13.7068	0.624976
$482$	−49.5125	−2.25523
$483$	−60.5197	−2.75374
$484$	85.6495	3.89316
$485$	−6.87122	−0.312006
$486$	55.4653	2.51596
$487$	42.0019	1.90329	0.951644	−	0.307204i	$-0.0993935\pi$
$487$	42.0019	1.90329	0.951644	+	0.307204i	$0.0993935\pi$
$488$	48.3307	2.18783
$489$	−39.5231	−1.78729
$490$	−11.8801	−0.536690
$491$	−39.5937	−1.78684	−0.893420	−	0.449222i	$-0.851701\pi$
$491$	−39.5937	−1.78684	−0.893420	+	0.449222i	$0.851701\pi$
$492$	54.3858	2.45190
$493$	−7.47738	−0.336764
$494$	−16.5518	−0.744700
$495$	15.5365	0.698312
$496$	0	0
$497$	−11.4291	−0.512666
$498$	8.20332	0.367599
$499$	−9.09941	−0.407346	−0.203673	−	0.979039i	$-0.565288\pi$
$499$	−9.09941	−0.407346	−0.203673	+	0.979039i	$0.565288\pi$
$500$	−4.53694	−0.202898
$501$	20.6228	0.921360
$502$	−33.9279	−1.51428
$503$	−15.9995	−0.713383	−0.356692	−	0.934222i	$-0.616095\pi$
$503$	−15.9995	−0.713383	−0.356692	+	0.934222i	$0.616095\pi$
$504$	62.9175	2.80257
$505$	13.2625	0.590172
$506$	−102.534	−4.55821
$507$	−38.6298	−1.71561
$508$	51.1007	2.26723
$509$	26.2944	1.16548	0.582739	−	0.812660i	$-0.301981\pi$
$509$	26.2944	1.16548	0.582739	+	0.812660i	$0.301981\pi$
$510$	−17.4970	−0.774779
$511$	52.6661	2.32981
$512$	−50.6487	−2.23838
$513$	−0.458284	−0.0202337
$514$	−8.13494	−0.358817
$515$	5.18607	0.228526
$516$	−20.8405	−0.917453
$517$	−21.2582	−0.934937
$518$	22.2157	0.976100
$519$	−55.4446	−2.43375
$520$	−34.9190	−1.53130
$521$	−20.6956	−0.906690	−0.453345	−	0.891335i	$-0.649769\pi$
$521$	−20.6956	−0.906690	−0.453345	+	0.891335i	$0.649769\pi$
$522$	19.1924	0.840027
$523$	−22.0517	−0.964255	−0.482127	−	0.876101i	$-0.660136\pi$
$523$	−22.0517	−0.964255	−0.482127	+	0.876101i	$0.660136\pi$
$524$	58.6822	2.56354
$525$	−8.24883	−0.360008
$526$	−45.1781	−1.96986
$527$	0	0
$528$	99.2219	4.31808
$529$	30.8281	1.34035
$530$	−8.61400	−0.374168
$531$	10.2885	0.446481
$532$	−18.6190	−0.807236
$533$	−26.6989	−1.15646
$534$	111.582	4.82863
$535$	−13.5434	−0.585531
$536$	77.9882	3.36858
$537$	−36.2313	−1.56349
$538$	−3.44415	−0.148488
$539$	−25.3987	−1.09400
$540$	−1.72903	−0.0744058
$541$	1.24621	0.0535786	0.0267893	−	0.999641i	$-0.491472\pi$
$541$	1.24621	0.0535786	0.0267893	+	0.999641i	$0.491472\pi$
$542$	9.37167	0.402548
$543$	−5.07297	−0.217702
$544$	−17.6343	−0.756066
$545$	−5.53736	−0.237195
$546$	−113.539	−4.85900
$547$	6.27178	0.268162	0.134081	−	0.990970i	$-0.457192\pi$
$547$	6.27178	0.268162	0.134081	+	0.990970i	$0.457192\pi$
$548$	23.9285	1.02217
$549$	21.1787	0.903886
$550$	−13.9754	−0.595914
$551$	−3.17585	−0.135296
$552$	−115.026	−4.89582
$553$	−13.5734	−0.577199
$554$	−51.7953	−2.20057
$555$	6.15411	0.261228
$556$	15.9358	0.675828
$557$	14.8349	0.628575	0.314288	−	0.949328i	$-0.398234\pi$
$557$	14.8349	0.628575	0.314288	+	0.949328i	$0.398234\pi$
$558$	0	0
$559$	10.2310	0.432724
$560$	−25.6293	−1.08303
$561$	−37.4070	−1.57933
$562$	−7.07147	−0.298292
$563$	−38.7429	−1.63282	−0.816410	−	0.577473i	$-0.804039\pi$
$563$	−38.7429	−1.63282	−0.816410	+	0.577473i	$0.804039\pi$
$564$	−42.6487	−1.79584
$565$	−5.14611	−0.216498
$566$	77.4757	3.25655
$567$	−32.2438	−1.35411
$568$	−21.7226	−0.911459
$569$	−12.5825	−0.527485	−0.263743	−	0.964593i	$-0.584957\pi$
$569$	−12.5825	−0.527485	−0.263743	+	0.964593i	$0.584957\pi$
$570$	−7.43146	−0.311270
$571$	−11.8059	−0.494063	−0.247032	−	0.969007i	$-0.579455\pi$
$571$	−11.8059	−0.494063	−0.247032	+	0.969007i	$0.579455\pi$
$572$	−133.507	−5.58221
$573$	−36.6670	−1.53178
$574$	−43.2729	−1.80618
$575$	7.33676	0.305964
$576$	2.57099	0.107124
$577$	−11.4457	−0.476491	−0.238246	−	0.971205i	$-0.576572\pi$
$577$	−11.4457	−0.476491	−0.238246	+	0.971205i	$0.576572\pi$
$578$	−22.9694	−0.955403
$579$	20.1503	0.837419
$580$	−11.9820	−0.497526
$581$	−4.53011	−0.187941
$582$	−42.4634	−1.76016
$583$	−18.4160	−0.762712
$584$	100.099	4.14212
$585$	−15.3017	−0.632645
$586$	−66.4883	−2.74660
$587$	38.0604	1.57092	0.785459	−	0.618913i	$-0.212427\pi$
$587$	38.0604	1.57092	0.785459	+	0.618913i	$0.212427\pi$
$588$	−50.9554	−2.10136
$589$	0	0
$590$	−9.25472	−0.381011
$591$	−45.6844	−1.87920
$592$	19.1210	0.785866
$593$	−21.6817	−0.890360	−0.445180	−	0.895441i	$-0.646860\pi$
$593$	−21.6817	−0.890360	−0.445180	+	0.895441i	$0.646860\pi$
$594$	−5.32605	−0.218531
$595$	9.66233	0.396117
$596$	31.1055	1.27413
$597$	18.1700	0.743648
$598$	100.985	4.12957
$599$	−11.6740	−0.476985	−0.238492	−	0.971144i	$-0.576653\pi$
$599$	−11.6740	−0.476985	−0.238492	+	0.971144i	$0.576653\pi$
$600$	−15.6780	−0.640052
$601$	4.73860	0.193291	0.0966457	−	0.995319i	$-0.469189\pi$
$601$	4.73860	0.193291	0.0966457	+	0.995319i	$0.469189\pi$
$602$	16.5821	0.675836
$603$	34.1748	1.39171
$604$	−59.8468	−2.43513
$605$	−18.8783	−0.767510
$606$	81.9605	3.32942
$607$	−27.4282	−1.11328	−0.556638	−	0.830755i	$-0.687909\pi$
$607$	−27.4282	−1.11328	−0.556638	+	0.830755i	$0.687909\pi$
$608$	−7.48981	−0.303752
$609$	−21.7851	−0.882776
$610$	−19.0508	−0.771343
$611$	20.9370	0.847019
$612$	−36.5107	−1.47586
$613$	−7.68442	−0.310371	−0.155185	−	0.987885i	$-0.549597\pi$
$613$	−7.68442	−0.310371	−0.155185	+	0.987885i	$0.549597\pi$
$614$	−43.7227	−1.76451
$615$	−11.9873	−0.483376
$616$	−120.997	−4.87511
$617$	28.5096	1.14775	0.573876	−	0.818942i	$-0.305439\pi$
$617$	28.5096	1.14775	0.573876	+	0.818942i	$0.305439\pi$
$618$	32.0493	1.28921
$619$	11.9420	0.479988	0.239994	−	0.970774i	$-0.422855\pi$
$619$	11.9420	0.479988	0.239994	+	0.970774i	$0.422855\pi$
$620$	0	0
$621$	2.79605	0.112202
$622$	−21.3891	−0.857626
$623$	−61.6189	−2.46871
$624$	−97.7223	−3.91202
$625$	1.00000	0.0400000
$626$	−43.4399	−1.73621
$627$	−15.8878	−0.634499
$628$	−63.3999	−2.52993
$629$	−7.20867	−0.287428
$630$	−24.8005	−0.988077
$631$	30.8241	1.22709	0.613543	−	0.789661i	$-0.289744\pi$
$631$	30.8241	1.22709	0.613543	+	0.789661i	$0.289744\pi$
$632$	−25.7980	−1.02619
$633$	−45.2668	−1.79919
$634$	55.9641	2.22262
$635$	−11.2632	−0.446968
$636$	−36.9465	−1.46502
$637$	25.0148	0.991124
$638$	−36.9089	−1.46124
$639$	−9.51893	−0.376563
$640$	10.1441	0.400982
$641$	−29.2867	−1.15675	−0.578377	−	0.815770i	$-0.696314\pi$
$641$	−29.2867	−1.15675	−0.578377	+	0.815770i	$0.696314\pi$
$642$	−83.6965	−3.30324
$643$	28.7834	1.13511	0.567553	−	0.823337i	$-0.307890\pi$
$643$	28.7834	1.13511	0.567553	+	0.823337i	$0.307890\pi$
$644$	113.597	4.47635
$645$	4.59352	0.180870
$646$	8.70490	0.342490
$647$	−6.73042	−0.264600	−0.132300	−	0.991210i	$-0.542236\pi$
$647$	−6.73042	−0.264600	−0.132300	+	0.991210i	$0.542236\pi$
$648$	−61.2836	−2.40745
$649$	−19.7858	−0.776660
$650$	13.7642	0.539876
$651$	0	0
$652$	74.1858	2.90534
$653$	9.93549	0.388806	0.194403	−	0.980922i	$-0.437723\pi$
$653$	9.93549	0.388806	0.194403	+	0.980922i	$0.437723\pi$
$654$	−34.2203	−1.33812
$655$	−12.9343	−0.505385
$656$	−37.2449	−1.45417
$657$	43.8638	1.71129
$658$	33.9341	1.32289
$659$	10.3088	0.401572	0.200786	−	0.979635i	$-0.435650\pi$
$659$	10.3088	0.401572	0.200786	+	0.979635i	$0.435650\pi$
$660$	−59.9424	−2.33325
$661$	−31.9769	−1.24376	−0.621878	−	0.783114i	$-0.713630\pi$
$661$	−31.9769	−1.24376	−0.621878	+	0.783114i	$0.713630\pi$
$662$	52.6823	2.04755
$663$	36.8417	1.43081
$664$	−8.61007	−0.334136
$665$	4.10387	0.159141
$666$	18.5027	0.716964
$667$	19.3763	0.750254
$668$	−38.7096	−1.49772
$669$	−31.9653	−1.23585
$670$	−30.7411	−1.18763
$671$	−40.7289	−1.57232
$672$	−51.3770	−1.98191
$673$	−36.0633	−1.39014	−0.695068	−	0.718944i	$-0.744626\pi$
$673$	−36.0633	−1.39014	−0.695068	+	0.718944i	$0.744626\pi$
$674$	−17.1529	−0.660703
$675$	0.381101	0.0146686
$676$	72.5091	2.78881
$677$	45.9548	1.76619	0.883093	−	0.469198i	$-0.155457\pi$
$677$	45.9548	1.76619	0.883093	+	0.469198i	$0.155457\pi$
$678$	−31.8023	−1.22136
$679$	23.4495	0.899909
$680$	18.3645	0.704248
$681$	55.8552	2.14038
$682$	0	0
$683$	−36.9842	−1.41516	−0.707581	−	0.706632i	$-0.750213\pi$
$683$	−36.9842	−1.41516	−0.707581	+	0.706632i	$0.750213\pi$
$684$	−15.5071	−0.592930
$685$	−5.27414	−0.201515
$686$	−20.5345	−0.784012
$687$	22.3930	0.854348
$688$	14.2722	0.544121
$689$	18.1377	0.690990
$690$	45.3404	1.72608
$691$	8.79157	0.334447	0.167223	−	0.985919i	$-0.446520\pi$
$691$	8.79157	0.334447	0.167223	+	0.985919i	$0.446520\pi$
$692$	104.071	3.95618
$693$	−53.0214	−2.01412
$694$	20.7613	0.788089
$695$	−3.51245	−0.133235
$696$	−41.4054	−1.56947
$697$	14.0415	0.531858
$698$	−17.9006	−0.677547
$699$	31.4351	1.18898
$700$	15.4833	0.585212
$701$	17.7710	0.671201	0.335601	−	0.942004i	$-0.391061\pi$
$701$	17.7710	0.671201	0.335601	+	0.942004i	$0.391061\pi$
$702$	5.24556	0.197981
$703$	−3.06173	−0.115475
$704$	−4.94428	−0.186345
$705$	9.40033	0.354037
$706$	4.14989	0.156183
$707$	−45.2610	−1.70221
$708$	−39.6947	−1.49182
$709$	−7.01800	−0.263566	−0.131783	−	0.991279i	$-0.542070\pi$
$709$	−7.01800	−0.263566	−0.131783	+	0.991279i	$0.542070\pi$
$710$	8.56251	0.321345
$711$	−11.3048	−0.423964
$712$	−117.115	−4.38907
$713$	0	0
$714$	59.7121	2.23467
$715$	29.4267	1.10050
$716$	68.0070	2.54154
$717$	−7.02212	−0.262246
$718$	10.5420	0.393423
$719$	20.1553	0.751666	0.375833	−	0.926687i	$-0.377357\pi$
$719$	20.1553	0.751666	0.375833	+	0.926687i	$0.377357\pi$
$720$	−21.3458	−0.795510
$721$	−17.6985	−0.659128
$722$	−44.8809	−1.67029
$723$	46.8081	1.74081
$724$	9.52208	0.353885
$725$	2.64099	0.0980839
$726$	−116.665	−4.32986
$727$	30.1784	1.11926	0.559628	−	0.828744i	$-0.310944\pi$
$727$	30.1784	1.11926	0.559628	+	0.828744i	$0.310944\pi$
$728$	119.168	4.41667
$729$	−24.0913	−0.892270
$730$	−39.4566	−1.46035
$731$	−5.38066	−0.199011
$732$	−81.7112	−3.02013
$733$	9.26840	0.342336	0.171168	−	0.985242i	$-0.445246\pi$
$733$	9.26840	0.342336	0.171168	+	0.985242i	$0.445246\pi$
$734$	32.0681	1.18365
$735$	11.2312	0.414270
$736$	45.6963	1.68439
$737$	−65.7218	−2.42089
$738$	−36.0406	−1.32667
$739$	−43.7100	−1.60790	−0.803949	−	0.594698i	$-0.797272\pi$
$739$	−43.7100	−1.60790	−0.803949	+	0.594698i	$0.797272\pi$
$740$	−11.5514	−0.424639
$741$	15.6477	0.574833
$742$	29.3971	1.07920
$743$	−5.98061	−0.219407	−0.109704	−	0.993964i	$-0.534990\pi$
$743$	−5.98061	−0.219407	−0.109704	+	0.993964i	$0.534990\pi$
$744$	0	0
$745$	−6.85604	−0.251186
$746$	83.2561	3.04822
$747$	−3.77297	−0.138046
$748$	70.2139	2.56728
$749$	46.2196	1.68883
$750$	6.17988	0.225657
$751$	−40.0362	−1.46094	−0.730471	−	0.682944i	$-0.760699\pi$
$751$	−40.0362	−1.46094	−0.730471	+	0.682944i	$0.760699\pi$
$752$	29.2070	1.06507
$753$	32.0747	1.16887
$754$	36.3511	1.32383
$755$	13.1910	0.480070
$756$	5.90069	0.214606
$757$	−32.5127	−1.18169	−0.590847	−	0.806783i	$-0.701206\pi$
$757$	−32.5127	−1.18169	−0.590847	+	0.806783i	$0.701206\pi$
$758$	−54.5962	−1.98302
$759$	96.9338	3.51847
$760$	7.79995	0.282934
$761$	9.05050	0.328080	0.164040	−	0.986454i	$-0.447547\pi$
$761$	9.05050	0.328080	0.164040	+	0.986454i	$0.447547\pi$
$762$	−69.6056	−2.52154
$763$	18.8974	0.684132
$764$	68.8248	2.48999
$765$	8.04743	0.290955
$766$	−26.6566	−0.963143
$767$	19.4868	0.703626
$768$	67.0623	2.41990
$769$	−39.6295	−1.42908	−0.714539	−	0.699596i	$-0.753363\pi$
$769$	−39.6295	−1.42908	−0.714539	+	0.699596i	$0.753363\pi$
$770$	47.6940	1.71877
$771$	7.69059	0.276970
$772$	−37.8227	−1.36127
$773$	−12.2401	−0.440244	−0.220122	−	0.975472i	$-0.570646\pi$
$773$	−12.2401	−0.440244	−0.220122	+	0.975472i	$0.570646\pi$
$774$	13.8107	0.496414
$775$	0	0
$776$	44.5689	1.59993
$777$	−21.0022	−0.753450
$778$	−84.3528	−3.02420
$779$	5.96381	0.213676
$780$	59.0364	2.11384
$781$	18.3059	0.655037
$782$	−53.1098	−1.89920
$783$	1.00648	0.0359688
$784$	34.8956	1.24627
$785$	13.9741	0.498759
$786$	−79.9325	−2.85110
$787$	0.383970	0.0136871	0.00684353	−	0.999977i	$-0.497822\pi$
$787$	0.383970	0.0136871	0.00684353	+	0.999977i	$0.497822\pi$
$788$	85.7507	3.05474
$789$	42.7104	1.52053
$790$	10.1690	0.361795
$791$	17.5622	0.624439
$792$	−100.774	−3.58086
$793$	40.1133	1.42447
$794$	−11.8405	−0.420204
$795$	8.14349	0.288820
$796$	−34.1055	−1.20884
$797$	−29.9745	−1.06175	−0.530875	−	0.847450i	$-0.678137\pi$
$797$	−29.9745	−1.06175	−0.530875	+	0.847450i	$0.678137\pi$
$798$	25.3614	0.897785
$799$	−11.0111	−0.389546
$800$	6.22840	0.220207
$801$	−51.3203	−1.81331
$802$	−48.8609	−1.72534
$803$	−84.3548	−2.97681
$804$	−131.852	−4.65007
$805$	−25.0382	−0.882482
$806$	0	0
$807$	3.25602	0.114617
$808$	−86.0245	−3.02633
$809$	53.9576	1.89705	0.948524	−	0.316706i	$-0.102577\pi$
$809$	53.9576	1.89705	0.948524	+	0.316706i	$0.102577\pi$
$810$	24.1565	0.848773
$811$	−11.3034	−0.396917	−0.198458	−	0.980109i	$-0.563593\pi$
$811$	−11.3034	−0.396917	−0.198458	+	0.980109i	$0.563593\pi$
$812$	40.8911	1.43500
$813$	−8.85978	−0.310726
$814$	−35.5826	−1.24717
$815$	−16.3515	−0.572768
$816$	51.3940	1.79915
$817$	−2.28532	−0.0799532
$818$	−55.1183	−1.92716
$819$	52.2201	1.82472
$820$	22.5005	0.785753
$821$	42.7826	1.49312	0.746562	−	0.665316i	$-0.231703\pi$
$821$	42.7826	1.49312	0.746562	+	0.665316i	$0.231703\pi$
$822$	−32.5936	−1.13683
$823$	−24.9511	−0.869740	−0.434870	−	0.900493i	$-0.643206\pi$
$823$	−24.9511	−0.869740	−0.434870	+	0.900493i	$0.643206\pi$
$824$	−33.6384	−1.17185
$825$	13.2121	0.459985
$826$	31.5837	1.09894
$827$	−31.5430	−1.09686	−0.548429	−	0.836197i	$-0.684774\pi$
$827$	−31.5430	−1.09686	−0.548429	+	0.836197i	$0.684774\pi$
$828$	94.6111	3.28796
$829$	−19.2045	−0.667000	−0.333500	−	0.942750i	$-0.608230\pi$
$829$	−19.2045	−0.667000	−0.333500	+	0.942750i	$0.608230\pi$
$830$	3.39388	0.117803
$831$	48.9661	1.69862
$832$	4.86955	0.168821
$833$	−13.1558	−0.455821
$834$	−21.7065	−0.751636
$835$	8.53209	0.295265
$836$	29.8219	1.03141
$837$	0	0
$838$	−59.0918	−2.04129
$839$	18.7264	0.646506	0.323253	−	0.946313i	$-0.395223\pi$
$839$	18.7264	0.646506	0.323253	+	0.946313i	$0.395223\pi$
$840$	53.5045	1.84608
$841$	−22.0252	−0.759489
$842$	24.4497	0.842592
$843$	6.68522	0.230251
$844$	84.9669	2.92468
$845$	−15.9819	−0.549795
$846$	28.2626	0.971687
$847$	64.4260	2.21370
$848$	25.3020	0.868873
$849$	−73.2439	−2.51372
$850$	−7.23885	−0.248291
$851$	18.6800	0.640342
$852$	36.7257	1.25820
$853$	56.2107	1.92462	0.962310	−	0.271956i	$-0.0876705\pi$
$853$	56.2107	1.92462	0.962310	+	0.271956i	$0.0876705\pi$
$854$	65.0147	2.22476
$855$	3.41797	0.116892
$856$	87.8466	3.00253
$857$	−18.5419	−0.633378	−0.316689	−	0.948529i	$-0.602571\pi$
$857$	−18.5419	−0.633378	−0.316689	+	0.948529i	$0.602571\pi$
$858$	181.853	6.20838
$859$	−3.57211	−0.121879	−0.0609395	−	0.998141i	$-0.519410\pi$
$859$	−3.57211	−0.121879	−0.0609395	+	0.998141i	$0.519410\pi$
$860$	−8.62215	−0.294013
$861$	40.9093	1.39419
$862$	32.0273	1.09085
$863$	38.0561	1.29544	0.647722	−	0.761877i	$-0.275722\pi$
$863$	38.0561	1.29544	0.647722	+	0.761877i	$0.275722\pi$
$864$	2.37365	0.0807533
$865$	−22.9386	−0.779935
$866$	44.3618	1.50748
$867$	21.7148	0.737474
$868$	0	0
$869$	21.7404	0.737491
$870$	16.3210	0.553334
$871$	64.7285	2.19324
$872$	35.9170	1.21630
$873$	19.5303	0.661000
$874$	−22.5572	−0.763010
$875$	−3.41271	−0.115371
$876$	−169.234	−5.71789
$877$	−24.6139	−0.831152	−0.415576	−	0.909559i	$-0.636420\pi$
$877$	−24.6139	−0.831152	−0.415576	+	0.909559i	$0.636420\pi$
$878$	−59.6376	−2.01267
$879$	62.8566	2.12010
$880$	41.0501	1.38380
$881$	−4.56011	−0.153634	−0.0768171	−	0.997045i	$-0.524476\pi$
$881$	−4.56011	−0.153634	−0.0768171	+	0.997045i	$0.524476\pi$
$882$	33.7673	1.13700
$883$	−4.50147	−0.151486	−0.0757432	−	0.997127i	$-0.524133\pi$
$883$	−4.50147	−0.151486	−0.0757432	+	0.997127i	$0.524133\pi$
$884$	−69.1527	−2.32586
$885$	8.74921	0.294102
$886$	−71.9855	−2.41840
$887$	−56.1143	−1.88413	−0.942067	−	0.335425i	$-0.891120\pi$
$887$	−56.1143	−1.88413	−0.942067	+	0.335425i	$0.891120\pi$
$888$	−39.9175	−1.33954
$889$	38.4382	1.28918
$890$	46.1638	1.54742
$891$	51.6445	1.73016
$892$	59.9997	2.00894
$893$	−4.67675	−0.156501
$894$	−42.3695	−1.41705
$895$	−14.9896	−0.501047
$896$	−34.6190	−1.15654
$897$	−95.4688	−3.18761
$898$	12.9450	0.431980
$899$	0	0
$900$	12.8955	0.429850
$901$	−9.53893	−0.317788
$902$	69.3098	2.30776
$903$	−15.6764	−0.521676
$904$	33.3792	1.11018
$905$	−2.09879	−0.0697661
$906$	81.5189	2.70829
$907$	55.6829	1.84892	0.924460	−	0.381279i	$-0.124516\pi$
$907$	55.6829	1.84892	0.924460	+	0.381279i	$0.124516\pi$
$908$	−104.842	−3.47929
$909$	−37.6963	−1.25031
$910$	−46.9732	−1.55715
$911$	17.3307	0.574190	0.287095	−	0.957902i	$-0.407310\pi$
$911$	17.3307	0.574190	0.287095	+	0.957902i	$0.407310\pi$
$912$	21.8285	0.722814
$913$	7.25582	0.240133
$914$	−75.2306	−2.48841
$915$	18.0102	0.595398
$916$	−42.0323	−1.38879
$917$	44.1410	1.45767
$918$	−2.75874	−0.0910519
$919$	34.3449	1.13293	0.566466	−	0.824085i	$-0.308310\pi$
$919$	34.3449	1.13293	0.566466	+	0.824085i	$0.308310\pi$
$920$	−47.5885	−1.56895
$921$	41.3345	1.36202
$922$	−19.9563	−0.657227
$923$	−18.0292	−0.593439
$924$	204.566	6.72972
$925$	2.54608	0.0837147
$926$	10.9274	0.359098
$927$	−14.7405	−0.484142
$928$	16.4491	0.539970
$929$	−15.9851	−0.524455	−0.262228	−	0.965006i	$-0.584457\pi$
$929$	−15.9851	−0.524455	−0.262228	+	0.965006i	$0.584457\pi$
$930$	0	0
$931$	−5.58763	−0.183127
$932$	−59.0044	−1.93275
$933$	20.2208	0.662000
$934$	−12.6392	−0.413567
$935$	−15.4760	−0.506121
$936$	99.2512	3.24413
$937$	−0.797551	−0.0260549	−0.0130274	−	0.999915i	$-0.504147\pi$
$937$	−0.797551	−0.0260549	−0.0130274	+	0.999915i	$0.504147\pi$
$938$	104.910	3.42544
$939$	41.0672	1.34018
$940$	−17.6446	−0.575505
$941$	21.1320	0.688883	0.344442	−	0.938808i	$-0.388068\pi$
$941$	21.1320	0.688883	0.344442	+	0.938808i	$0.388068\pi$
$942$	86.3586	2.81371
$943$	−36.3860	−1.18489
$944$	27.1840	0.884763
$945$	−1.30059	−0.0423081
$946$	−26.5594	−0.863519
$947$	−49.9648	−1.62364	−0.811819	−	0.583909i	$-0.801522\pi$
$947$	−49.9648	−1.62364	−0.811819	+	0.583909i	$0.801522\pi$
$948$	43.6159	1.41658
$949$	83.0798	2.69689
$950$	−3.07455	−0.0997515
$951$	−52.9073	−1.71564
$952$	−62.6728	−2.03124
$953$	−8.50612	−0.275540	−0.137770	−	0.990464i	$-0.543994\pi$
$953$	−8.50612	−0.275540	−0.137770	+	0.990464i	$0.543994\pi$
$954$	24.4838	0.792693
$955$	−15.1699	−0.490886
$956$	13.1807	0.426294
$957$	34.8929	1.12793
$958$	34.0611	1.10046
$959$	17.9991	0.581222
$960$	2.18634	0.0705639
$961$	0	0
$962$	35.0448	1.12989
$963$	38.4948	1.24048
$964$	−87.8600	−2.82978
$965$	8.33660	0.268365
$966$	−154.733	−4.97847
$967$	−15.0769	−0.484842	−0.242421	−	0.970171i	$-0.577941\pi$
$967$	−15.0769	−0.484842	−0.242421	+	0.970171i	$0.577941\pi$
$968$	122.450	3.93570
$969$	−8.22943	−0.264367
$970$	−17.5680	−0.564073
$971$	45.7540	1.46832	0.734158	−	0.678978i	$-0.237577\pi$
$971$	45.7540	1.46832	0.734158	+	0.678978i	$0.237577\pi$
$972$	98.4232	3.15693
$973$	11.9870	0.384285
$974$	107.388	3.44094
$975$	−13.0124	−0.416730
$976$	55.9580	1.79117
$977$	46.2643	1.48013	0.740063	−	0.672537i	$-0.234795\pi$
$977$	46.2643	1.48013	0.740063	+	0.672537i	$0.234795\pi$
$978$	−101.050	−3.23123
$979$	98.6943	3.15428
$980$	−21.0813	−0.673417
$981$	15.7390	0.502508
$982$	−101.231	−3.23041
$983$	−49.1527	−1.56773	−0.783864	−	0.620932i	$-0.786754\pi$
$983$	−49.1527	−1.56773	−0.783864	+	0.620932i	$0.786754\pi$
$984$	77.7536	2.47869
$985$	−18.9006	−0.602222
$986$	−19.1177	−0.608833
$987$	−32.0806	−1.02114
$988$	−29.3711	−0.934420
$989$	13.9430	0.443363
$990$	39.7228	1.26247
$991$	37.9195	1.20455	0.602276	−	0.798288i	$-0.294261\pi$
$991$	37.9195	1.20455	0.602276	+	0.798288i	$0.294261\pi$
$992$	0	0
$993$	−49.8047	−1.58050
$994$	−29.2213	−0.926845
$995$	7.51729	0.238314
$996$	14.5568	0.461249
$997$	−26.5898	−0.842107	−0.421054	−	0.907036i	$-0.638340\pi$
$997$	−26.5898	−0.842107	−0.421054	+	0.907036i	$0.638340\pi$
$998$	−23.2649	−0.736437
$999$	0.970316	0.0306994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4805.2.a.bb.1.21	✓	24
31.30	odd	2	inner	4805.2.a.bb.1.22	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4805.2.a.bb.1.21	✓	24	1.1	even	1	trivial
4805.2.a.bb.1.22	yes	24	31.30	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4805 = 5 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4805.a (trivial)

\(f(q)\)	\(=\)	\(q+2.55674 q^{2} -2.41709 q^{3} +4.53694 q^{4} -1.00000 q^{5} -6.17988 q^{6} +3.41271 q^{7} +6.48631 q^{8} +2.84233 q^{9} -2.55674 q^{10} -5.46610 q^{11} -10.9662 q^{12} +5.38349 q^{13} +8.72542 q^{14} +2.41709 q^{15} +7.50995 q^{16} -2.83128 q^{17} +7.26711 q^{18} -1.20252 q^{19} -4.53694 q^{20} -8.24883 q^{21} -13.9754 q^{22} +7.33676 q^{23} -15.6780 q^{24} +1.00000 q^{25} +13.7642 q^{26} +0.381101 q^{27} +15.4833 q^{28} +2.64099 q^{29} +6.17988 q^{30} +6.22840 q^{32} +13.2121 q^{33} -7.23885 q^{34} -3.41271 q^{35} +12.8955 q^{36} +2.54608 q^{37} -3.07455 q^{38} -13.0124 q^{39} -6.48631 q^{40} -4.95941 q^{41} -21.0901 q^{42} +1.90043 q^{43} -24.7994 q^{44} -2.84233 q^{45} +18.7582 q^{46} +3.88911 q^{47} -18.1522 q^{48} +4.64659 q^{49} +2.55674 q^{50} +6.84346 q^{51} +24.4246 q^{52} +3.36913 q^{53} +0.974379 q^{54} +5.46610 q^{55} +22.1359 q^{56} +2.90661 q^{57} +6.75233 q^{58} +3.61973 q^{59} +10.9662 q^{60} +7.45118 q^{61} +9.70005 q^{63} +0.904535 q^{64} -5.38349 q^{65} +33.7799 q^{66} +12.0235 q^{67} -12.8453 q^{68} -17.7336 q^{69} -8.72542 q^{70} -3.34899 q^{71} +18.4362 q^{72} +15.4323 q^{73} +6.50968 q^{74} -2.41709 q^{75} -5.45578 q^{76} -18.6542 q^{77} -33.2693 q^{78} -3.97731 q^{79} -7.50995 q^{80} -9.44815 q^{81} -12.6799 q^{82} -1.32742 q^{83} -37.4245 q^{84} +2.83128 q^{85} +4.85892 q^{86} -6.38351 q^{87} -35.4548 q^{88} -18.0557 q^{89} -7.26711 q^{90} +18.3723 q^{91} +33.2865 q^{92} +9.94345 q^{94} +1.20252 q^{95} -15.0546 q^{96} +6.87122 q^{97} +11.8801 q^{98} -15.5365 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} - 24 q^{5} + 8 q^{7} + 40 q^{9} + 8 q^{14} + 88 q^{16} - 64 q^{18} + 40 q^{19} - 32 q^{20} + 24 q^{25} + 72 q^{28} + 56 q^{33} - 8 q^{35} + 88 q^{36} - 72 q^{38} + 64 q^{39} - 56 q^{41}+ \cdots - 8 q^{98}+O(q^{100}) \) 24 * q + 32 * q^4 - 24 * q^5 + 8 * q^7 + 40 * q^9 + 8 * q^14 + 88 * q^16 - 64 * q^18 + 40 * q^19 - 32 * q^20 + 24 * q^25 + 72 * q^28 + 56 * q^33 - 8 * q^35 + 88 * q^36 - 72 * q^38 + 64 * q^39 - 56 * q^41 - 40 * q^45 - 32 * q^47 + 56 * q^49 + 88 * q^51 + 104 * q^56 + 8 * q^59 + 56 * q^63 + 160 * q^64 - 8 * q^66 + 96 * q^67 + 8 * q^69 - 8 * q^70 - 64 * q^71 - 120 * q^72 + 128 * q^76 - 128 * q^78 - 88 * q^80 + 72 * q^81 + 32 * q^82 - 32 * q^87 + 64 * q^90 + 56 * q^94 - 40 * q^95 + 48 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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