LMFDB - Embedded newform 5950.2.a.cc.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5950,2,Mod(1,5950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5950.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5950.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$47.5109892027$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 12x^{5} + 17x^{4} + 40x^{3} - 32x^{2} - 40x + 4 $ x^7 - 2x^6 - 12x^5 + 17x^4 + 40x^3 - 32x^2 - 40x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1190)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.54179$ of defining polynomial
Character	$\chi$	$=$	5950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.54179 q^{3} +1.00000 q^{4} +2.54179 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.46070 q^{9} -5.72650 q^{11} +2.54179 q^{12} -3.48296 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +3.46070 q^{18} -4.55477 q^{19} -2.54179 q^{21} -5.72650 q^{22} -8.26551 q^{23} +2.54179 q^{24} -3.48296 q^{26} +1.17101 q^{27} -1.00000 q^{28} -0.990385 q^{29} -3.99132 q^{31} +1.00000 q^{32} -14.5556 q^{33} -1.00000 q^{34} +3.46070 q^{36} +1.96580 q^{37} -4.55477 q^{38} -8.85296 q^{39} +7.88545 q^{41} -2.54179 q^{42} +12.2804 q^{43} -5.72650 q^{44} -8.26551 q^{46} +8.49457 q^{47} +2.54179 q^{48} +1.00000 q^{49} -2.54179 q^{51} -3.48296 q^{52} -10.3317 q^{53} +1.17101 q^{54} -1.00000 q^{56} -11.5773 q^{57} -0.990385 q^{58} -10.1278 q^{59} -0.952226 q^{61} -3.99132 q^{62} -3.46070 q^{63} +1.00000 q^{64} -14.5556 q^{66} -3.87612 q^{67} -1.00000 q^{68} -21.0092 q^{69} -6.23006 q^{71} +3.46070 q^{72} -7.07022 q^{73} +1.96580 q^{74} -4.55477 q^{76} +5.72650 q^{77} -8.85296 q^{78} +13.5580 q^{79} -7.40565 q^{81} +7.88545 q^{82} -15.3925 q^{83} -2.54179 q^{84} +12.2804 q^{86} -2.51735 q^{87} -5.72650 q^{88} +14.4517 q^{89} +3.48296 q^{91} -8.26551 q^{92} -10.1451 q^{93} +8.49457 q^{94} +2.54179 q^{96} +17.5462 q^{97} +1.00000 q^{98} -19.8177 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} - 2 q^{3} + 7 q^{4} - 2 q^{6} - 7 q^{7} + 7 q^{8} + 7 q^{9} - 7 q^{11} - 2 q^{12} + q^{13} - 7 q^{14} + 7 q^{16} - 7 q^{17} + 7 q^{18} - 2 q^{19} + 2 q^{21} - 7 q^{22} - 27 q^{23} - 2 q^{24}+ \cdots - 44 q^{99}+O(q^{100}) $ 7 * q + 7 * q^2 - 2 * q^3 + 7 * q^4 - 2 * q^6 - 7 * q^7 + 7 * q^8 + 7 * q^9 - 7 * q^11 - 2 * q^12 + q^13 - 7 * q^14 + 7 * q^16 - 7 * q^17 + 7 * q^18 - 2 * q^19 + 2 * q^21 - 7 * q^22 - 27 * q^23 - 2 * q^24 + q^26 - 17 * q^27 - 7 * q^28 - 10 * q^29 - 18 * q^31 + 7 * q^32 - 8 * q^33 - 7 * q^34 + 7 * q^36 - 12 * q^37 - 2 * q^38 - 25 * q^39 + 17 * q^41 + 2 * q^42 - 7 * q^43 - 7 * q^44 - 27 * q^46 - 8 * q^47 - 2 * q^48 + 7 * q^49 + 2 * q^51 + q^52 - 20 * q^53 - 17 * q^54 - 7 * q^56 - 25 * q^57 - 10 * q^58 + q^61 - 18 * q^62 - 7 * q^63 + 7 * q^64 - 8 * q^66 - 7 * q^67 - 7 * q^68 + 3 * q^69 - 11 * q^71 + 7 * q^72 + 7 * q^73 - 12 * q^74 - 2 * q^76 + 7 * q^77 - 25 * q^78 - 26 * q^79 + 11 * q^81 + 17 * q^82 - 22 * q^83 + 2 * q^84 - 7 * q^86 - 20 * q^87 - 7 * q^88 + 21 * q^89 - q^91 - 27 * q^92 - 25 * q^93 - 8 * q^94 - 2 * q^96 + 19 * q^97 + 7 * q^98 - 44 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.54179	1.46750	0.733752	−	0.679417i	$-0.237767\pi$
$3$	2.54179	1.46750	0.733752	+	0.679417i	$0.237767\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.54179	1.03768
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	3.46070	1.15357
$10$	0	0
$11$	−5.72650	−1.72660	−0.863302	−	0.504688i	$-0.831607\pi$
$11$	−5.72650	−1.72660	−0.863302	+	0.504688i	$0.831607\pi$
$12$	2.54179	0.733752
$13$	−3.48296	−0.966000	−0.483000	−	0.875620i	$-0.660453\pi$
$13$	−3.48296	−0.966000	−0.483000	+	0.875620i	$0.660453\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	3.46070	0.815695
$19$	−4.55477	−1.04494	−0.522468	−	0.852659i	$-0.674989\pi$
$19$	−4.55477	−1.04494	−0.522468	+	0.852659i	$0.674989\pi$
$20$	0	0
$21$	−2.54179	−0.554664
$22$	−5.72650	−1.22089
$23$	−8.26551	−1.72348	−0.861739	−	0.507352i	$-0.830625\pi$
$23$	−8.26551	−1.72348	−0.861739	+	0.507352i	$0.830625\pi$
$24$	2.54179	0.518841
$25$	0	0
$26$	−3.48296	−0.683065
$27$	1.17101	0.225361
$28$	−1.00000	−0.188982
$29$	−0.990385	−0.183910	−0.0919550	−	0.995763i	$-0.529312\pi$
$29$	−0.990385	−0.183910	−0.0919550	+	0.995763i	$0.529312\pi$
$30$	0	0
$31$	−3.99132	−0.716862	−0.358431	−	0.933556i	$-0.616688\pi$
$31$	−3.99132	−0.716862	−0.358431	+	0.933556i	$0.616688\pi$
$32$	1.00000	0.176777
$33$	−14.5556	−2.53380
$34$	−1.00000	−0.171499
$35$	0	0
$36$	3.46070	0.576784
$37$	1.96580	0.323176	0.161588	−	0.986858i	$-0.448338\pi$
$37$	1.96580	0.323176	0.161588	+	0.986858i	$0.448338\pi$
$38$	−4.55477	−0.738881
$39$	−8.85296	−1.41761
$40$	0	0
$41$	7.88545	1.23150	0.615750	−	0.787941i	$-0.288853\pi$
$41$	7.88545	1.23150	0.615750	+	0.787941i	$0.288853\pi$
$42$	−2.54179	−0.392207
$43$	12.2804	1.87274	0.936371	−	0.351011i	$-0.114162\pi$
$43$	12.2804	1.87274	0.936371	+	0.351011i	$0.114162\pi$
$44$	−5.72650	−0.863302
$45$	0	0
$46$	−8.26551	−1.21868
$47$	8.49457	1.23906	0.619530	−	0.784973i	$-0.287323\pi$
$47$	8.49457	1.23906	0.619530	+	0.784973i	$0.287323\pi$
$48$	2.54179	0.366876
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.54179	−0.355922
$52$	−3.48296	−0.483000
$53$	−10.3317	−1.41917	−0.709585	−	0.704619i	$-0.751118\pi$
$53$	−10.3317	−1.41917	−0.709585	+	0.704619i	$0.751118\pi$
$54$	1.17101	0.159354
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−11.5773	−1.53345
$58$	−0.990385	−0.130044
$59$	−10.1278	−1.31852	−0.659262	−	0.751913i	$-0.729131\pi$
$59$	−10.1278	−1.31852	−0.659262	+	0.751913i	$0.729131\pi$
$60$	0	0
$61$	−0.952226	−0.121920	−0.0609600	−	0.998140i	$-0.519416\pi$
$61$	−0.952226	−0.121920	−0.0609600	+	0.998140i	$0.519416\pi$
$62$	−3.99132	−0.506898
$63$	−3.46070	−0.436008
$64$	1.00000	0.125000
$65$	0	0
$66$	−14.5556	−1.79167
$67$	−3.87612	−0.473543	−0.236772	−	0.971565i	$-0.576089\pi$
$67$	−3.87612	−0.473543	−0.236772	+	0.971565i	$0.576089\pi$
$68$	−1.00000	−0.121268
$69$	−21.0092	−2.52921
$70$	0	0
$71$	−6.23006	−0.739372	−0.369686	−	0.929157i	$-0.620535\pi$
$71$	−6.23006	−0.739372	−0.369686	+	0.929157i	$0.620535\pi$
$72$	3.46070	0.407848
$73$	−7.07022	−0.827507	−0.413753	−	0.910389i	$-0.635782\pi$
$73$	−7.07022	−0.827507	−0.413753	+	0.910389i	$0.635782\pi$
$74$	1.96580	0.228520
$75$	0	0
$76$	−4.55477	−0.522468
$77$	5.72650	0.652595
$78$	−8.85296	−1.00240
$79$	13.5580	1.52539	0.762695	−	0.646759i	$-0.223876\pi$
$79$	13.5580	1.52539	0.762695	+	0.646759i	$0.223876\pi$
$80$	0	0
$81$	−7.40565	−0.822850
$82$	7.88545	0.870802
$83$	−15.3925	−1.68955	−0.844775	−	0.535121i	$-0.820266\pi$
$83$	−15.3925	−1.68955	−0.844775	+	0.535121i	$0.820266\pi$
$84$	−2.54179	−0.277332
$85$	0	0
$86$	12.2804	1.32423
$87$	−2.51735	−0.269889
$88$	−5.72650	−0.610446
$89$	14.4517	1.53188	0.765938	−	0.642915i	$-0.222275\pi$
$89$	14.4517	1.53188	0.765938	+	0.642915i	$0.222275\pi$
$90$	0	0
$91$	3.48296	0.365114
$92$	−8.26551	−0.861739
$93$	−10.1451	−1.05200
$94$	8.49457	0.876148
$95$	0	0
$96$	2.54179	0.259420
$97$	17.5462	1.78154	0.890771	−	0.454452i	$-0.150165\pi$
$97$	17.5462	1.78154	0.890771	+	0.454452i	$0.150165\pi$
$98$	1.00000	0.101015
$99$	−19.8177	−1.99175
$100$	0	0
$101$	−3.17217	−0.315643	−0.157821	−	0.987468i	$-0.550447\pi$
$101$	−3.17217	−0.315643	−0.157821	+	0.987468i	$0.550447\pi$
$102$	−2.54179	−0.251675
$103$	−1.94049	−0.191202	−0.0956009	−	0.995420i	$-0.530477\pi$
$103$	−1.94049	−0.191202	−0.0956009	+	0.995420i	$0.530477\pi$
$104$	−3.48296	−0.341532
$105$	0	0
$106$	−10.3317	−1.00351
$107$	−2.05039	−0.198218	−0.0991091	−	0.995077i	$-0.531599\pi$
$107$	−2.05039	−0.198218	−0.0991091	+	0.995077i	$0.531599\pi$
$108$	1.17101	0.112680
$109$	−19.3457	−1.85298	−0.926490	−	0.376320i	$-0.877189\pi$
$109$	−19.3457	−1.85298	−0.926490	+	0.376320i	$0.877189\pi$
$110$	0	0
$111$	4.99666	0.474262
$112$	−1.00000	−0.0944911
$113$	−6.29785	−0.592452	−0.296226	−	0.955118i	$-0.595728\pi$
$113$	−6.29785	−0.592452	−0.296226	+	0.955118i	$0.595728\pi$
$114$	−11.5773	−1.08431
$115$	0	0
$116$	−0.990385	−0.0919550
$117$	−12.0535	−1.11435
$118$	−10.1278	−0.932338
$119$	1.00000	0.0916698
$120$	0	0
$121$	21.7928	1.98116
$122$	−0.952226	−0.0862105
$123$	20.0432	1.80723
$124$	−3.99132	−0.358431
$125$	0	0
$126$	−3.46070	−0.308304
$127$	−3.92166	−0.347991	−0.173996	−	0.984746i	$-0.555668\pi$
$127$	−3.92166	−0.347991	−0.173996	+	0.984746i	$0.555668\pi$
$128$	1.00000	0.0883883
$129$	31.2142	2.74826
$130$	0	0
$131$	−9.94843	−0.869198	−0.434599	−	0.900624i	$-0.643110\pi$
$131$	−9.94843	−0.869198	−0.434599	+	0.900624i	$0.643110\pi$
$132$	−14.5556	−1.26690
$133$	4.55477	0.394949
$134$	−3.87612	−0.334846
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	21.0684	1.80000	0.899998	−	0.435895i	$-0.143568\pi$
$137$	21.0684	1.80000	0.899998	+	0.435895i	$0.143568\pi$
$138$	−21.0092	−1.78842
$139$	1.43988	0.122129	0.0610643	−	0.998134i	$-0.480551\pi$
$139$	1.43988	0.122129	0.0610643	+	0.998134i	$0.480551\pi$
$140$	0	0
$141$	21.5914	1.81833
$142$	−6.23006	−0.522815
$143$	19.9452	1.66790
$144$	3.46070	0.288392
$145$	0	0
$146$	−7.07022	−0.585136
$147$	2.54179	0.209643
$148$	1.96580	0.161588
$149$	22.3068	1.82744	0.913722	−	0.406339i	$-0.133195\pi$
$149$	22.3068	1.82744	0.913722	+	0.406339i	$0.133195\pi$
$150$	0	0
$151$	−8.43690	−0.686585	−0.343293	−	0.939228i	$-0.611542\pi$
$151$	−8.43690	−0.686585	−0.343293	+	0.939228i	$0.611542\pi$
$152$	−4.55477	−0.369441
$153$	−3.46070	−0.279781
$154$	5.72650	0.461454
$155$	0	0
$156$	−8.85296	−0.708804
$157$	19.5520	1.56042	0.780212	−	0.625516i	$-0.215111\pi$
$157$	19.5520	1.56042	0.780212	+	0.625516i	$0.215111\pi$
$158$	13.5580	1.07861
$159$	−26.2611	−2.08264
$160$	0	0
$161$	8.26551	0.651414
$162$	−7.40565	−0.581843
$163$	15.2346	1.19326	0.596632	−	0.802515i	$-0.296505\pi$
$163$	15.2346	1.19326	0.596632	+	0.802515i	$0.296505\pi$
$164$	7.88545	0.615750
$165$	0	0
$166$	−15.3925	−1.19469
$167$	−1.44115	−0.111520	−0.0557598	−	0.998444i	$-0.517758\pi$
$167$	−1.44115	−0.111520	−0.0557598	+	0.998444i	$0.517758\pi$
$168$	−2.54179	−0.196103
$169$	−0.868981	−0.0668447
$170$	0	0
$171$	−15.7627	−1.20540
$172$	12.2804	0.936371
$173$	−0.0905351	−0.00688326	−0.00344163	−	0.999994i	$-0.501096\pi$
$173$	−0.0905351	−0.00688326	−0.00344163	+	0.999994i	$0.501096\pi$
$174$	−2.51735	−0.190840
$175$	0	0
$176$	−5.72650	−0.431651
$177$	−25.7427	−1.93494
$178$	14.4517	1.08320
$179$	−6.72039	−0.502306	−0.251153	−	0.967947i	$-0.580810\pi$
$179$	−6.72039	−0.502306	−0.251153	+	0.967947i	$0.580810\pi$
$180$	0	0
$181$	−20.3547	−1.51295	−0.756476	−	0.654021i	$-0.773081\pi$
$181$	−20.3547	−1.51295	−0.756476	+	0.654021i	$0.773081\pi$
$182$	3.48296	0.258174
$183$	−2.42036	−0.178918
$184$	−8.26551	−0.609342
$185$	0	0
$186$	−10.1451	−0.743874
$187$	5.72650	0.418763
$188$	8.49457	0.619530
$189$	−1.17101	−0.0851784
$190$	0	0
$191$	−5.38668	−0.389766	−0.194883	−	0.980826i	$-0.562433\pi$
$191$	−5.38668	−0.389766	−0.194883	+	0.980826i	$0.562433\pi$
$192$	2.54179	0.183438
$193$	0.617801	0.0444703	0.0222351	−	0.999753i	$-0.492922\pi$
$193$	0.617801	0.0444703	0.0222351	+	0.999753i	$0.492922\pi$
$194$	17.5462	1.25974
$195$	0	0
$196$	1.00000	0.0714286
$197$	−8.90420	−0.634398	−0.317199	−	0.948359i	$-0.602742\pi$
$197$	−8.90420	−0.634398	−0.317199	+	0.948359i	$0.602742\pi$
$198$	−19.8177	−1.40838
$199$	2.74906	0.194876	0.0974380	−	0.995242i	$-0.468935\pi$
$199$	2.74906	0.194876	0.0974380	+	0.995242i	$0.468935\pi$
$200$	0	0
$201$	−9.85228	−0.694926
$202$	−3.17217	−0.223193
$203$	0.990385	0.0695114
$204$	−2.54179	−0.177961
$205$	0	0
$206$	−1.94049	−0.135200
$207$	−28.6045	−1.98815
$208$	−3.48296	−0.241500
$209$	26.0829	1.80419
$210$	0	0
$211$	−9.35373	−0.643937	−0.321969	−	0.946750i	$-0.604345\pi$
$211$	−9.35373	−0.643937	−0.321969	+	0.946750i	$0.604345\pi$
$212$	−10.3317	−0.709585
$213$	−15.8355	−1.08503
$214$	−2.05039	−0.140161
$215$	0	0
$216$	1.17101	0.0796771
$217$	3.99132	0.270948
$218$	−19.3457	−1.31025
$219$	−17.9710	−1.21437
$220$	0	0
$221$	3.48296	0.234289
$222$	4.99666	0.335354
$223$	−0.946770	−0.0634004	−0.0317002	−	0.999497i	$-0.510092\pi$
$223$	−0.946770	−0.0634004	−0.0317002	+	0.999497i	$0.510092\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−6.29785	−0.418927
$227$	−12.3408	−0.819085	−0.409543	−	0.912291i	$-0.634312\pi$
$227$	−12.3408	−0.819085	−0.409543	+	0.912291i	$0.634312\pi$
$228$	−11.5773	−0.766724
$229$	−13.3470	−0.881993	−0.440996	−	0.897509i	$-0.645375\pi$
$229$	−13.3470	−0.881993	−0.440996	+	0.897509i	$0.645375\pi$
$230$	0	0
$231$	14.5556	0.957685
$232$	−0.990385	−0.0650220
$233$	23.2229	1.52138	0.760691	−	0.649114i	$-0.224860\pi$
$233$	23.2229	1.52138	0.760691	+	0.649114i	$0.224860\pi$
$234$	−12.0535	−0.787961
$235$	0	0
$236$	−10.1278	−0.659262
$237$	34.4615	2.23851
$238$	1.00000	0.0648204
$239$	10.8663	0.702880	0.351440	−	0.936210i	$-0.385692\pi$
$239$	10.8663	0.702880	0.351440	+	0.936210i	$0.385692\pi$
$240$	0	0
$241$	9.61298	0.619226	0.309613	−	0.950863i	$-0.399800\pi$
$241$	9.61298	0.619226	0.309613	+	0.950863i	$0.399800\pi$
$242$	21.7928	1.40089
$243$	−22.3366	−1.43290
$244$	−0.952226	−0.0609600
$245$	0	0
$246$	20.0432	1.27791
$247$	15.8641	1.00941
$248$	−3.99132	−0.253449
$249$	−39.1246	−2.47942
$250$	0	0
$251$	13.6959	0.864477	0.432239	−	0.901759i	$-0.357724\pi$
$251$	13.6959	0.864477	0.432239	+	0.901759i	$0.357724\pi$
$252$	−3.46070	−0.218004
$253$	47.3324	2.97576
$254$	−3.92166	−0.246067
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.2766	−1.20244	−0.601222	−	0.799082i	$-0.705319\pi$
$257$	−19.2766	−1.20244	−0.601222	+	0.799082i	$0.705319\pi$
$258$	31.2142	1.94331
$259$	−1.96580	−0.122149
$260$	0	0
$261$	−3.42743	−0.212153
$262$	−9.94843	−0.614616
$263$	−15.4341	−0.951706	−0.475853	−	0.879525i	$-0.657861\pi$
$263$	−15.4341	−0.951706	−0.475853	+	0.879525i	$0.657861\pi$
$264$	−14.5556	−0.895833
$265$	0	0
$266$	4.55477	0.279271
$267$	36.7332	2.24803
$268$	−3.87612	−0.236772
$269$	18.7045	1.14044	0.570218	−	0.821494i	$-0.306859\pi$
$269$	18.7045	1.14044	0.570218	+	0.821494i	$0.306859\pi$
$270$	0	0
$271$	27.6242	1.67805	0.839026	−	0.544092i	$-0.183126\pi$
$271$	27.6242	1.67805	0.839026	+	0.544092i	$0.183126\pi$
$272$	−1.00000	−0.0606339
$273$	8.85296	0.535806
$274$	21.0684	1.27279
$275$	0	0
$276$	−21.0092	−1.26461
$277$	2.15313	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$277$	2.15313	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$278$	1.43988	0.0863580
$279$	−13.8128	−0.826948
$280$	0	0
$281$	3.87390	0.231098	0.115549	−	0.993302i	$-0.463137\pi$
$281$	3.87390	0.231098	0.115549	+	0.993302i	$0.463137\pi$
$282$	21.5914	1.28575
$283$	−5.48685	−0.326160	−0.163080	−	0.986613i	$-0.552143\pi$
$283$	−5.48685	−0.326160	−0.163080	+	0.986613i	$0.552143\pi$
$284$	−6.23006	−0.369686
$285$	0	0
$286$	19.9452	1.17938
$287$	−7.88545	−0.465463
$288$	3.46070	0.203924
$289$	1.00000	0.0588235
$290$	0	0
$291$	44.5987	2.61442
$292$	−7.07022	−0.413753
$293$	−21.7657	−1.27156	−0.635782	−	0.771869i	$-0.719322\pi$
$293$	−21.7657	−1.27156	−0.635782	+	0.771869i	$0.719322\pi$
$294$	2.54179	0.148240
$295$	0	0
$296$	1.96580	0.114260
$297$	−6.70578	−0.389109
$298$	22.3068	1.29220
$299$	28.7885	1.66488
$300$	0	0
$301$	−12.2804	−0.707830
$302$	−8.43690	−0.485489
$303$	−8.06299	−0.463207
$304$	−4.55477	−0.261234
$305$	0	0
$306$	−3.46070	−0.197835
$307$	−13.1008	−0.747704	−0.373852	−	0.927488i	$-0.621963\pi$
$307$	−13.1008	−0.747704	−0.373852	+	0.927488i	$0.621963\pi$
$308$	5.72650	0.326297
$309$	−4.93231	−0.280589
$310$	0	0
$311$	−4.20193	−0.238270	−0.119135	−	0.992878i	$-0.538012\pi$
$311$	−4.20193	−0.238270	−0.119135	+	0.992878i	$0.538012\pi$
$312$	−8.85296	−0.501200
$313$	−11.6520	−0.658612	−0.329306	−	0.944223i	$-0.606815\pi$
$313$	−11.6520	−0.658612	−0.329306	+	0.944223i	$0.606815\pi$
$314$	19.5520	1.10339
$315$	0	0
$316$	13.5580	0.762695
$317$	−3.45462	−0.194031	−0.0970155	−	0.995283i	$-0.530930\pi$
$317$	−3.45462	−0.194031	−0.0970155	+	0.995283i	$0.530930\pi$
$318$	−26.2611	−1.47265
$319$	5.67144	0.317540
$320$	0	0
$321$	−5.21165	−0.290886
$322$	8.26551	0.460619
$323$	4.55477	0.253434
$324$	−7.40565	−0.411425
$325$	0	0
$326$	15.2346	0.843766
$327$	−49.1727	−2.71925
$328$	7.88545	0.435401
$329$	−8.49457	−0.468321
$330$	0	0
$331$	12.4973	0.686911	0.343456	−	0.939169i	$-0.388403\pi$
$331$	12.4973	0.686911	0.343456	+	0.939169i	$0.388403\pi$
$332$	−15.3925	−0.844775
$333$	6.80305	0.372805
$334$	−1.44115	−0.0788562
$335$	0	0
$336$	−2.54179	−0.138666
$337$	−8.93416	−0.486675	−0.243337	−	0.969942i	$-0.578242\pi$
$337$	−8.93416	−0.486675	−0.243337	+	0.969942i	$0.578242\pi$
$338$	−0.868981	−0.0472663
$339$	−16.0078	−0.869426
$340$	0	0
$341$	22.8563	1.23774
$342$	−15.7627	−0.852350
$343$	−1.00000	−0.0539949
$344$	12.2804	0.662115
$345$	0	0
$346$	−0.0905351	−0.00486720
$347$	−19.0850	−1.02454	−0.512269	−	0.858825i	$-0.671195\pi$
$347$	−19.0850	−1.02454	−0.512269	+	0.858825i	$0.671195\pi$
$348$	−2.51735	−0.134944
$349$	15.3497	0.821652	0.410826	−	0.911714i	$-0.365240\pi$
$349$	15.3497	0.821652	0.410826	+	0.911714i	$0.365240\pi$
$350$	0	0
$351$	−4.07858	−0.217698
$352$	−5.72650	−0.305223
$353$	−3.33185	−0.177337	−0.0886684	−	0.996061i	$-0.528261\pi$
$353$	−3.33185	−0.177337	−0.0886684	+	0.996061i	$0.528261\pi$
$354$	−25.7427	−1.36821
$355$	0	0
$356$	14.4517	0.765938
$357$	2.54179	0.134526
$358$	−6.72039	−0.355184
$359$	−10.1746	−0.536995	−0.268498	−	0.963280i	$-0.586527\pi$
$359$	−10.1746	−0.536995	−0.268498	+	0.963280i	$0.586527\pi$
$360$	0	0
$361$	1.74594	0.0918915
$362$	−20.3547	−1.06982
$363$	55.3926	2.90736
$364$	3.48296	0.182557
$365$	0	0
$366$	−2.42036	−0.126514
$367$	11.2202	0.585688	0.292844	−	0.956160i	$-0.405398\pi$
$367$	11.2202	0.585688	0.292844	+	0.956160i	$0.405398\pi$
$368$	−8.26551	−0.430870
$369$	27.2892	1.42062
$370$	0	0
$371$	10.3317	0.536396
$372$	−10.1451	−0.525999
$373$	−19.7975	−1.02508	−0.512538	−	0.858665i	$-0.671295\pi$
$373$	−19.7975	−1.02508	−0.512538	+	0.858665i	$0.671295\pi$
$374$	5.72650	0.296110
$375$	0	0
$376$	8.49457	0.438074
$377$	3.44947	0.177657
$378$	−1.17101	−0.0602302
$379$	−33.0938	−1.69992	−0.849958	−	0.526851i	$-0.823373\pi$
$379$	−33.0938	−1.69992	−0.849958	+	0.526851i	$0.823373\pi$
$380$	0	0
$381$	−9.96804	−0.510678
$382$	−5.38668	−0.275606
$383$	−17.2749	−0.882706	−0.441353	−	0.897333i	$-0.645501\pi$
$383$	−17.2749	−0.882706	−0.441353	+	0.897333i	$0.645501\pi$
$384$	2.54179	0.129710
$385$	0	0
$386$	0.617801	0.0314452
$387$	42.4988	2.16034
$388$	17.5462	0.890771
$389$	5.88067	0.298162	0.149081	−	0.988825i	$-0.452369\pi$
$389$	5.88067	0.298162	0.149081	+	0.988825i	$0.452369\pi$
$390$	0	0
$391$	8.26551	0.418005
$392$	1.00000	0.0505076
$393$	−25.2868	−1.27555
$394$	−8.90420	−0.448587
$395$	0	0
$396$	−19.8177	−0.995877
$397$	−17.3518	−0.870859	−0.435430	−	0.900223i	$-0.643404\pi$
$397$	−17.3518	−0.870859	−0.435430	+	0.900223i	$0.643404\pi$
$398$	2.74906	0.137798
$399$	11.5773	0.579589
$400$	0	0
$401$	−19.9909	−0.998299	−0.499149	−	0.866516i	$-0.666354\pi$
$401$	−19.9909	−0.998299	−0.499149	+	0.866516i	$0.666354\pi$
$402$	−9.85228	−0.491387
$403$	13.9016	0.692488
$404$	−3.17217	−0.157821
$405$	0	0
$406$	0.990385	0.0491520
$407$	−11.2572	−0.557996
$408$	−2.54179	−0.125837
$409$	35.3383	1.74737	0.873683	−	0.486495i	$-0.161725\pi$
$409$	35.3383	1.74737	0.873683	+	0.486495i	$0.161725\pi$
$410$	0	0
$411$	53.5515	2.64150
$412$	−1.94049	−0.0956009
$413$	10.1278	0.498355
$414$	−28.6045	−1.40583
$415$	0	0
$416$	−3.48296	−0.170766
$417$	3.65986	0.179224
$418$	26.0829	1.27576
$419$	−27.0860	−1.32324	−0.661619	−	0.749841i	$-0.730130\pi$
$419$	−27.0860	−1.32324	−0.661619	+	0.749841i	$0.730130\pi$
$420$	0	0
$421$	12.6808	0.618025	0.309013	−	0.951058i	$-0.400001\pi$
$421$	12.6808	0.618025	0.309013	+	0.951058i	$0.400001\pi$
$422$	−9.35373	−0.455332
$423$	29.3972	1.42934
$424$	−10.3317	−0.501753
$425$	0	0
$426$	−15.8355	−0.767233
$427$	0.952226	0.0460814
$428$	−2.05039	−0.0991091
$429$	50.6964	2.44765
$430$	0	0
$431$	−2.22147	−0.107005	−0.0535023	−	0.998568i	$-0.517038\pi$
$431$	−2.22147	−0.107005	−0.0535023	+	0.998568i	$0.517038\pi$
$432$	1.17101	0.0563402
$433$	−29.8916	−1.43650	−0.718249	−	0.695786i	$-0.755056\pi$
$433$	−29.8916	−1.43650	−0.718249	+	0.695786i	$0.755056\pi$
$434$	3.99132	0.191589
$435$	0	0
$436$	−19.3457	−0.926490
$437$	37.6475	1.80092
$438$	−17.9710	−0.858689
$439$	−14.5552	−0.694683	−0.347342	−	0.937739i	$-0.612916\pi$
$439$	−14.5552	−0.694683	−0.347342	+	0.937739i	$0.612916\pi$
$440$	0	0
$441$	3.46070	0.164795
$442$	3.48296	0.165668
$443$	−6.99922	−0.332543	−0.166271	−	0.986080i	$-0.553173\pi$
$443$	−6.99922	−0.332543	−0.166271	+	0.986080i	$0.553173\pi$
$444$	4.99666	0.237131
$445$	0	0
$446$	−0.946770	−0.0448309
$447$	56.6992	2.68178
$448$	−1.00000	−0.0472456
$449$	−4.29546	−0.202715	−0.101358	−	0.994850i	$-0.532319\pi$
$449$	−4.29546	−0.202715	−0.101358	+	0.994850i	$0.532319\pi$
$450$	0	0
$451$	−45.1560	−2.12631
$452$	−6.29785	−0.296226
$453$	−21.4448	−1.00757
$454$	−12.3408	−0.579181
$455$	0	0
$456$	−11.5773	−0.542156
$457$	−0.936921	−0.0438273	−0.0219137	−	0.999760i	$-0.506976\pi$
$457$	−0.936921	−0.0438273	−0.0219137	+	0.999760i	$0.506976\pi$
$458$	−13.3470	−0.623663
$459$	−1.17101	−0.0546580
$460$	0	0
$461$	24.6279	1.14704	0.573518	−	0.819193i	$-0.305578\pi$
$461$	24.6279	1.14704	0.573518	+	0.819193i	$0.305578\pi$
$462$	14.5556	0.677186
$463$	−29.9356	−1.39122	−0.695612	−	0.718418i	$-0.744867\pi$
$463$	−29.9356	−1.39122	−0.695612	+	0.718418i	$0.744867\pi$
$464$	−0.990385	−0.0459775
$465$	0	0
$466$	23.2229	1.07578
$467$	−9.31665	−0.431123	−0.215562	−	0.976490i	$-0.569158\pi$
$467$	−9.31665	−0.431123	−0.215562	+	0.976490i	$0.569158\pi$
$468$	−12.0535	−0.557173
$469$	3.87612	0.178982
$470$	0	0
$471$	49.6972	2.28993
$472$	−10.1278	−0.466169
$473$	−70.3236	−3.23348
$474$	34.4615	1.58287
$475$	0	0
$476$	1.00000	0.0458349
$477$	−35.7550	−1.63711
$478$	10.8663	0.497011
$479$	12.2197	0.558333	0.279167	−	0.960243i	$-0.409942\pi$
$479$	12.2197	0.558333	0.279167	+	0.960243i	$0.409942\pi$
$480$	0	0
$481$	−6.84681	−0.312188
$482$	9.61298	0.437859
$483$	21.0092	0.955952
$484$	21.7928	0.990580
$485$	0	0
$486$	−22.3366	−1.01321
$487$	−2.11355	−0.0957739	−0.0478870	−	0.998853i	$-0.515249\pi$
$487$	−2.11355	−0.0957739	−0.0478870	+	0.998853i	$0.515249\pi$
$488$	−0.952226	−0.0431052
$489$	38.7231	1.75112
$490$	0	0
$491$	−6.35587	−0.286836	−0.143418	−	0.989662i	$-0.545809\pi$
$491$	−6.35587	−0.286836	−0.143418	+	0.989662i	$0.545809\pi$
$492$	20.0432	0.903616
$493$	0.990385	0.0446047
$494$	15.8641	0.713759
$495$	0	0
$496$	−3.99132	−0.179215
$497$	6.23006	0.279456
$498$	−39.1246	−1.75322
$499$	42.4519	1.90041	0.950204	−	0.311627i	$-0.100874\pi$
$499$	42.4519	1.90041	0.950204	+	0.311627i	$0.100874\pi$
$500$	0	0
$501$	−3.66310	−0.163655
$502$	13.6959	0.611278
$503$	−20.9628	−0.934683	−0.467342	−	0.884077i	$-0.654788\pi$
$503$	−20.9628	−0.934683	−0.467342	+	0.884077i	$0.654788\pi$
$504$	−3.46070	−0.154152
$505$	0	0
$506$	47.3324	2.10418
$507$	−2.20877	−0.0980948
$508$	−3.92166	−0.173996
$509$	−3.66956	−0.162650	−0.0813251	−	0.996688i	$-0.525915\pi$
$509$	−3.66956	−0.162650	−0.0813251	+	0.996688i	$0.525915\pi$
$510$	0	0
$511$	7.07022	0.312768
$512$	1.00000	0.0441942
$513$	−5.33368	−0.235488
$514$	−19.2766	−0.850256
$515$	0	0
$516$	31.2142	1.37413
$517$	−48.6441	−2.13937
$518$	−1.96580	−0.0863723
$519$	−0.230121	−0.0101012
$520$	0	0
$521$	21.8630	0.957836	0.478918	−	0.877860i	$-0.341029\pi$
$521$	21.8630	0.957836	0.478918	+	0.877860i	$0.341029\pi$
$522$	−3.42743	−0.150014
$523$	−2.31826	−0.101370	−0.0506852	−	0.998715i	$-0.516141\pi$
$523$	−2.31826	−0.101370	−0.0506852	+	0.998715i	$0.516141\pi$
$524$	−9.94843	−0.434599
$525$	0	0
$526$	−15.4341	−0.672958
$527$	3.99132	0.173864
$528$	−14.5556	−0.633449
$529$	45.3187	1.97038
$530$	0	0
$531$	−35.0492	−1.52101
$532$	4.55477	0.197474
$533$	−27.4647	−1.18963
$534$	36.7332	1.58960
$535$	0	0
$536$	−3.87612	−0.167423
$537$	−17.0818	−0.737136
$538$	18.7045	0.806409
$539$	−5.72650	−0.246658
$540$	0	0
$541$	−26.3538	−1.13304	−0.566520	−	0.824048i	$-0.691711\pi$
$541$	−26.3538	−1.13304	−0.566520	+	0.824048i	$0.691711\pi$
$542$	27.6242	1.18656
$543$	−51.7374	−2.22026
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	8.85296	0.378872
$547$	20.5567	0.878943	0.439471	−	0.898257i	$-0.355166\pi$
$547$	20.5567	0.878943	0.439471	+	0.898257i	$0.355166\pi$
$548$	21.0684	0.899998
$549$	−3.29537	−0.140643
$550$	0	0
$551$	4.51098	0.192174
$552$	−21.0092	−0.894211
$553$	−13.5580	−0.576543
$554$	2.15313	0.0914778
$555$	0	0
$556$	1.43988	0.0610643
$557$	−30.1050	−1.27559	−0.637794	−	0.770207i	$-0.720153\pi$
$557$	−30.1050	−1.27559	−0.637794	+	0.770207i	$0.720153\pi$
$558$	−13.8128	−0.584741
$559$	−42.7721	−1.80907
$560$	0	0
$561$	14.5556	0.614536
$562$	3.87390	0.163411
$563$	−13.4551	−0.567065	−0.283532	−	0.958963i	$-0.591506\pi$
$563$	−13.4551	−0.567065	−0.283532	+	0.958963i	$0.591506\pi$
$564$	21.5914	0.909163
$565$	0	0
$566$	−5.48685	−0.230630
$567$	7.40565	0.311008
$568$	−6.23006	−0.261407
$569$	14.1962	0.595135	0.297567	−	0.954701i	$-0.403825\pi$
$569$	14.1962	0.595135	0.297567	+	0.954701i	$0.403825\pi$
$570$	0	0
$571$	4.06927	0.170294	0.0851469	−	0.996368i	$-0.472864\pi$
$571$	4.06927	0.170294	0.0851469	+	0.996368i	$0.472864\pi$
$572$	19.9452	0.833949
$573$	−13.6918	−0.571984
$574$	−7.88545	−0.329132
$575$	0	0
$576$	3.46070	0.144196
$577$	23.0464	0.959433	0.479716	−	0.877424i	$-0.340740\pi$
$577$	23.0464	0.959433	0.479716	+	0.877424i	$0.340740\pi$
$578$	1.00000	0.0415945
$579$	1.57032	0.0652603
$580$	0	0
$581$	15.3925	0.638590
$582$	44.5987	1.84867
$583$	59.1645	2.45035
$584$	−7.07022	−0.292568
$585$	0	0
$586$	−21.7657	−0.899131
$587$	9.46700	0.390745	0.195372	−	0.980729i	$-0.437408\pi$
$587$	9.46700	0.390745	0.195372	+	0.980729i	$0.437408\pi$
$588$	2.54179	0.104822
$589$	18.1795	0.749075
$590$	0	0
$591$	−22.6326	−0.930981
$592$	1.96580	0.0807939
$593$	0.297927	0.0122344	0.00611719	−	0.999981i	$-0.498053\pi$
$593$	0.297927	0.0122344	0.00611719	+	0.999981i	$0.498053\pi$
$594$	−6.70578	−0.275141
$595$	0	0
$596$	22.3068	0.913722
$597$	6.98754	0.285981
$598$	28.7885	1.17725
$599$	30.4804	1.24539	0.622697	−	0.782463i	$-0.286037\pi$
$599$	30.4804	1.24539	0.622697	+	0.782463i	$0.286037\pi$
$600$	0	0
$601$	−11.4524	−0.467154	−0.233577	−	0.972338i	$-0.575043\pi$
$601$	−11.4524	−0.467154	−0.233577	+	0.972338i	$0.575043\pi$
$602$	−12.2804	−0.500512
$603$	−13.4141	−0.546264
$604$	−8.43690	−0.343293
$605$	0	0
$606$	−8.06299	−0.327537
$607$	−16.7894	−0.681461	−0.340730	−	0.940161i	$-0.610674\pi$
$607$	−16.7894	−0.681461	−0.340730	+	0.940161i	$0.610674\pi$
$608$	−4.55477	−0.184720
$609$	2.51735	0.102008
$610$	0	0
$611$	−29.5863	−1.19693
$612$	−3.46070	−0.139891
$613$	4.05568	0.163808	0.0819038	−	0.996640i	$-0.473900\pi$
$613$	4.05568	0.163808	0.0819038	+	0.996640i	$0.473900\pi$
$614$	−13.1008	−0.528707
$615$	0	0
$616$	5.72650	0.230727
$617$	31.7150	1.27680	0.638399	−	0.769706i	$-0.279597\pi$
$617$	31.7150	1.27680	0.638399	+	0.769706i	$0.279597\pi$
$618$	−4.93231	−0.198407
$619$	−10.3777	−0.417114	−0.208557	−	0.978010i	$-0.566877\pi$
$619$	−10.3777	−0.417114	−0.208557	+	0.978010i	$0.566877\pi$
$620$	0	0
$621$	−9.67899	−0.388405
$622$	−4.20193	−0.168482
$623$	−14.4517	−0.578994
$624$	−8.85296	−0.354402
$625$	0	0
$626$	−11.6520	−0.465709
$627$	66.2972	2.64766
$628$	19.5520	0.780212
$629$	−1.96580	−0.0783816
$630$	0	0
$631$	13.3731	0.532373	0.266187	−	0.963922i	$-0.414236\pi$
$631$	13.3731	0.532373	0.266187	+	0.963922i	$0.414236\pi$
$632$	13.5580	0.539307
$633$	−23.7752	−0.944980
$634$	−3.45462	−0.137201
$635$	0	0
$636$	−26.2611	−1.04132
$637$	−3.48296	−0.138000
$638$	5.67144	0.224534
$639$	−21.5604	−0.852915
$640$	0	0
$641$	−44.6667	−1.76423	−0.882113	−	0.471037i	$-0.843880\pi$
$641$	−44.6667	−1.76423	−0.882113	+	0.471037i	$0.843880\pi$
$642$	−5.21165	−0.205688
$643$	3.96886	0.156517	0.0782583	−	0.996933i	$-0.475064\pi$
$643$	3.96886	0.156517	0.0782583	+	0.996933i	$0.475064\pi$
$644$	8.26551	0.325707
$645$	0	0
$646$	4.55477	0.179205
$647$	−8.10463	−0.318626	−0.159313	−	0.987228i	$-0.550928\pi$
$647$	−8.10463	−0.318626	−0.159313	+	0.987228i	$0.550928\pi$
$648$	−7.40565	−0.290921
$649$	57.9967	2.27657
$650$	0	0
$651$	10.1451	0.397618
$652$	15.2346	0.596632
$653$	−14.6228	−0.572235	−0.286118	−	0.958194i	$-0.592365\pi$
$653$	−14.6228	−0.572235	−0.286118	+	0.958194i	$0.592365\pi$
$654$	−49.1727	−1.92280
$655$	0	0
$656$	7.88545	0.307875
$657$	−24.4679	−0.954585
$658$	−8.49457	−0.331153
$659$	10.9571	0.426829	0.213414	−	0.976962i	$-0.431542\pi$
$659$	10.9571	0.426829	0.213414	+	0.976962i	$0.431542\pi$
$660$	0	0
$661$	41.7506	1.62391	0.811955	−	0.583720i	$-0.198404\pi$
$661$	41.7506	1.62391	0.811955	+	0.583720i	$0.198404\pi$
$662$	12.4973	0.485720
$663$	8.85296	0.343820
$664$	−15.3925	−0.597346
$665$	0	0
$666$	6.80305	0.263613
$667$	8.18604	0.316965
$668$	−1.44115	−0.0557598
$669$	−2.40649	−0.0930403
$670$	0	0
$671$	5.45292	0.210508
$672$	−2.54179	−0.0980517
$673$	−7.26123	−0.279900	−0.139950	−	0.990159i	$-0.544694\pi$
$673$	−7.26123	−0.279900	−0.139950	+	0.990159i	$0.544694\pi$
$674$	−8.93416	−0.344131
$675$	0	0
$676$	−0.868981	−0.0334223
$677$	−32.7711	−1.25950	−0.629748	−	0.776799i	$-0.716842\pi$
$677$	−32.7711	−1.25950	−0.629748	+	0.776799i	$0.716842\pi$
$678$	−16.0078	−0.614777
$679$	−17.5462	−0.673360
$680$	0	0
$681$	−31.3676	−1.20201
$682$	22.8563	0.875211
$683$	−1.54929	−0.0592818	−0.0296409	−	0.999561i	$-0.509436\pi$
$683$	−1.54929	−0.0592818	−0.0296409	+	0.999561i	$0.509436\pi$
$684$	−15.7627	−0.602702
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−33.9252	−1.29433
$688$	12.2804	0.468186
$689$	35.9850	1.37092
$690$	0	0
$691$	−31.3105	−1.19111	−0.595554	−	0.803315i	$-0.703068\pi$
$691$	−31.3105	−1.19111	−0.595554	+	0.803315i	$0.703068\pi$
$692$	−0.0905351	−0.00344163
$693$	19.8177	0.752812
$694$	−19.0850	−0.724457
$695$	0	0
$696$	−2.51735	−0.0954200
$697$	−7.88545	−0.298683
$698$	15.3497	0.580996
$699$	59.0277	2.23263
$700$	0	0
$701$	−26.7200	−1.00920	−0.504601	−	0.863353i	$-0.668360\pi$
$701$	−26.7200	−1.00920	−0.504601	+	0.863353i	$0.668360\pi$
$702$	−4.07858	−0.153936
$703$	−8.95377	−0.337698
$704$	−5.72650	−0.215825
$705$	0	0
$706$	−3.33185	−0.125396
$707$	3.17217	0.119302
$708$	−25.7427	−0.967470
$709$	−18.5720	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$709$	−18.5720	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$710$	0	0
$711$	46.9200	1.75964
$712$	14.4517	0.541600
$713$	32.9903	1.23550
$714$	2.54179	0.0951241
$715$	0	0
$716$	−6.72039	−0.251153
$717$	27.6198	1.03148
$718$	−10.1746	−0.379713
$719$	6.42124	0.239472	0.119736	−	0.992806i	$-0.461795\pi$
$719$	6.42124	0.239472	0.119736	+	0.992806i	$0.461795\pi$
$720$	0	0
$721$	1.94049	0.0722675
$722$	1.74594	0.0649771
$723$	24.4342	0.908717
$724$	−20.3547	−0.756476
$725$	0	0
$726$	55.3926	2.05581
$727$	−41.2275	−1.52904	−0.764522	−	0.644598i	$-0.777025\pi$
$727$	−41.2275	−1.52904	−0.764522	+	0.644598i	$0.777025\pi$
$728$	3.48296	0.129087
$729$	−34.5581	−1.27993
$730$	0	0
$731$	−12.2804	−0.454207
$732$	−2.42036	−0.0894590
$733$	−16.0832	−0.594048	−0.297024	−	0.954870i	$-0.595994\pi$
$733$	−16.0832	−0.594048	−0.297024	+	0.954870i	$0.595994\pi$
$734$	11.2202	0.414144
$735$	0	0
$736$	−8.26551	−0.304671
$737$	22.1966	0.817621
$738$	27.2892	1.00453
$739$	24.2742	0.892941	0.446470	−	0.894798i	$-0.352681\pi$
$739$	24.2742	0.892941	0.446470	+	0.894798i	$0.352681\pi$
$740$	0	0
$741$	40.3232	1.48131
$742$	10.3317	0.379289
$743$	13.4662	0.494027	0.247013	−	0.969012i	$-0.420551\pi$
$743$	13.4662	0.494027	0.247013	+	0.969012i	$0.420551\pi$
$744$	−10.1451	−0.371937
$745$	0	0
$746$	−19.7975	−0.724838
$747$	−53.2690	−1.94901
$748$	5.72650	0.209381
$749$	2.05039	0.0749195
$750$	0	0
$751$	38.8084	1.41614	0.708069	−	0.706144i	$-0.249567\pi$
$751$	38.8084	1.41614	0.708069	+	0.706144i	$0.249567\pi$
$752$	8.49457	0.309765
$753$	34.8121	1.26862
$754$	3.44947	0.125622
$755$	0	0
$756$	−1.17101	−0.0425892
$757$	−35.9312	−1.30594	−0.652970	−	0.757384i	$-0.726477\pi$
$757$	−35.9312	−1.30594	−0.652970	+	0.757384i	$0.726477\pi$
$758$	−33.0938	−1.20202
$759$	120.309	4.36694
$760$	0	0
$761$	44.4288	1.61054	0.805272	−	0.592906i	$-0.202020\pi$
$761$	44.4288	1.61054	0.805272	+	0.592906i	$0.202020\pi$
$762$	−9.96804	−0.361104
$763$	19.3457	0.700360
$764$	−5.38668	−0.194883
$765$	0	0
$766$	−17.2749	−0.624168
$767$	35.2747	1.27369
$768$	2.54179	0.0917190
$769$	0.248795	0.00897177	0.00448589	−	0.999990i	$-0.498572\pi$
$769$	0.248795	0.00897177	0.00448589	+	0.999990i	$0.498572\pi$
$770$	0	0
$771$	−48.9972	−1.76459
$772$	0.617801	0.0222351
$773$	13.7137	0.493248	0.246624	−	0.969111i	$-0.420679\pi$
$773$	13.7137	0.493248	0.246624	+	0.969111i	$0.420679\pi$
$774$	42.4988	1.52759
$775$	0	0
$776$	17.5462	0.629870
$777$	−4.99666	−0.179254
$778$	5.88067	0.210832
$779$	−35.9164	−1.28684
$780$	0	0
$781$	35.6764	1.27660
$782$	8.26551	0.295574
$783$	−1.15975	−0.0414461
$784$	1.00000	0.0357143
$785$	0	0
$786$	−25.2868	−0.901951
$787$	39.3773	1.40365	0.701824	−	0.712350i	$-0.252369\pi$
$787$	39.3773	1.40365	0.701824	+	0.712350i	$0.252369\pi$
$788$	−8.90420	−0.317199
$789$	−39.2302	−1.39663
$790$	0	0
$791$	6.29785	0.223926
$792$	−19.8177	−0.704191
$793$	3.31657	0.117775
$794$	−17.3518	−0.615791
$795$	0	0
$796$	2.74906	0.0974380
$797$	−28.4245	−1.00685	−0.503424	−	0.864039i	$-0.667927\pi$
$797$	−28.4245	−1.00685	−0.503424	+	0.864039i	$0.667927\pi$
$798$	11.5773	0.409831
$799$	−8.49457	−0.300516
$800$	0	0
$801$	50.0130	1.76712
$802$	−19.9909	−0.705904
$803$	40.4876	1.42878
$804$	−9.85228	−0.347463
$805$	0	0
$806$	13.9016	0.489663
$807$	47.5430	1.67359
$808$	−3.17217	−0.111597
$809$	−36.4046	−1.27992	−0.639958	−	0.768410i	$-0.721048\pi$
$809$	−36.4046	−1.27992	−0.639958	+	0.768410i	$0.721048\pi$
$810$	0	0
$811$	29.5590	1.03796	0.518978	−	0.854787i	$-0.326312\pi$
$811$	29.5590	1.03796	0.518978	+	0.854787i	$0.326312\pi$
$812$	0.990385	0.0347557
$813$	70.2150	2.46255
$814$	−11.2572	−0.394563
$815$	0	0
$816$	−2.54179	−0.0889805
$817$	−55.9344	−1.95690
$818$	35.3383	1.23557
$819$	12.0535	0.421183
$820$	0	0
$821$	−7.70862	−0.269033	−0.134516	−	0.990911i	$-0.542948\pi$
$821$	−7.70862	−0.269033	−0.134516	+	0.990911i	$0.542948\pi$
$822$	53.5515	1.86782
$823$	−28.4869	−0.992990	−0.496495	−	0.868040i	$-0.665380\pi$
$823$	−28.4869	−0.992990	−0.496495	+	0.868040i	$0.665380\pi$
$824$	−1.94049	−0.0676000
$825$	0	0
$826$	10.1278	0.352390
$827$	−34.5262	−1.20060	−0.600298	−	0.799777i	$-0.704951\pi$
$827$	−34.5262	−1.20060	−0.600298	+	0.799777i	$0.704951\pi$
$828$	−28.6045	−0.994074
$829$	−19.5009	−0.677293	−0.338647	−	0.940914i	$-0.609969\pi$
$829$	−19.5009	−0.677293	−0.338647	+	0.940914i	$0.609969\pi$
$830$	0	0
$831$	5.47281	0.189850
$832$	−3.48296	−0.120750
$833$	−1.00000	−0.0346479
$834$	3.65986	0.126731
$835$	0	0
$836$	26.0829	0.902095
$837$	−4.67387	−0.161553
$838$	−27.0860	−0.935670
$839$	−33.4690	−1.15548	−0.577738	−	0.816222i	$-0.696065\pi$
$839$	−33.4690	−1.15548	−0.577738	+	0.816222i	$0.696065\pi$
$840$	0	0
$841$	−28.0191	−0.966177
$842$	12.6808	0.437010
$843$	9.84665	0.339137
$844$	−9.35373	−0.321969
$845$	0	0
$846$	29.3972	1.01070
$847$	−21.7928	−0.748808
$848$	−10.3317	−0.354793
$849$	−13.9464	−0.478641
$850$	0	0
$851$	−16.2484	−0.556986
$852$	−15.8355	−0.542516
$853$	−13.7146	−0.469580	−0.234790	−	0.972046i	$-0.575440\pi$
$853$	−13.7146	−0.469580	−0.234790	+	0.972046i	$0.575440\pi$
$854$	0.952226	0.0325845
$855$	0	0
$856$	−2.05039	−0.0700807
$857$	42.1511	1.43986	0.719928	−	0.694049i	$-0.244175\pi$
$857$	42.1511	1.43986	0.719928	+	0.694049i	$0.244175\pi$
$858$	50.6964	1.73075
$859$	45.9832	1.56893	0.784463	−	0.620176i	$-0.212939\pi$
$859$	45.9832	1.56893	0.784463	+	0.620176i	$0.212939\pi$
$860$	0	0
$861$	−20.0432	−0.683069
$862$	−2.22147	−0.0756637
$863$	21.0234	0.715644	0.357822	−	0.933790i	$-0.383519\pi$
$863$	21.0234	0.715644	0.357822	+	0.933790i	$0.383519\pi$
$864$	1.17101	0.0398385
$865$	0	0
$866$	−29.8916	−1.01576
$867$	2.54179	0.0863238
$868$	3.99132	0.135474
$869$	−77.6396	−2.63374
$870$	0	0
$871$	13.5004	0.457443
$872$	−19.3457	−0.655127
$873$	60.7220	2.05513
$874$	37.6475	1.27345
$875$	0	0
$876$	−17.9710	−0.607185
$877$	−3.93131	−0.132751	−0.0663754	−	0.997795i	$-0.521143\pi$
$877$	−3.93131	−0.132751	−0.0663754	+	0.997795i	$0.521143\pi$
$878$	−14.5552	−0.491215
$879$	−55.3238	−1.86602
$880$	0	0
$881$	5.41729	0.182513	0.0912566	−	0.995827i	$-0.470912\pi$
$881$	5.41729	0.182513	0.0912566	+	0.995827i	$0.470912\pi$
$882$	3.46070	0.116528
$883$	33.5752	1.12990	0.564948	−	0.825127i	$-0.308896\pi$
$883$	33.5752	1.12990	0.564948	+	0.825127i	$0.308896\pi$
$884$	3.48296	0.117145
$885$	0	0
$886$	−6.99922	−0.235143
$887$	−2.92415	−0.0981834	−0.0490917	−	0.998794i	$-0.515633\pi$
$887$	−2.92415	−0.0981834	−0.0490917	+	0.998794i	$0.515633\pi$
$888$	4.99666	0.167677
$889$	3.92166	0.131528
$890$	0	0
$891$	42.4084	1.42073
$892$	−0.946770	−0.0317002
$893$	−38.6908	−1.29474
$894$	56.6992	1.89631
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	73.1742	2.44322
$898$	−4.29546	−0.143341
$899$	3.95294	0.131838
$900$	0	0
$901$	10.3317	0.344199
$902$	−45.1560	−1.50353
$903$	−31.2142	−1.03874
$904$	−6.29785	−0.209464
$905$	0	0
$906$	−21.4448	−0.712457
$907$	27.5005	0.913138	0.456569	−	0.889688i	$-0.349078\pi$
$907$	27.5005	0.913138	0.456569	+	0.889688i	$0.349078\pi$
$908$	−12.3408	−0.409543
$909$	−10.9779	−0.364115
$910$	0	0
$911$	−5.75095	−0.190537	−0.0952687	−	0.995452i	$-0.530371\pi$
$911$	−5.75095	−0.190537	−0.0952687	+	0.995452i	$0.530371\pi$
$912$	−11.5773	−0.383362
$913$	88.1453	2.91718
$914$	−0.936921	−0.0309906
$915$	0	0
$916$	−13.3470	−0.440996
$917$	9.94843	0.328526
$918$	−1.17101	−0.0386491
$919$	−6.18604	−0.204059	−0.102029	−	0.994781i	$-0.532534\pi$
$919$	−6.18604	−0.204059	−0.102029	+	0.994781i	$0.532534\pi$
$920$	0	0
$921$	−33.2996	−1.09726
$922$	24.6279	0.811077
$923$	21.6991	0.714233
$924$	14.5556	0.478843
$925$	0	0
$926$	−29.9356	−0.983744
$927$	−6.71544	−0.220564
$928$	−0.990385	−0.0325110
$929$	44.0088	1.44388	0.721941	−	0.691955i	$-0.243250\pi$
$929$	44.0088	1.44388	0.721941	+	0.691955i	$0.243250\pi$
$930$	0	0
$931$	−4.55477	−0.149277
$932$	23.2229	0.760691
$933$	−10.6804	−0.349662
$934$	−9.31665	−0.304850
$935$	0	0
$936$	−12.0535	−0.393981
$937$	−27.3208	−0.892533	−0.446266	−	0.894900i	$-0.647247\pi$
$937$	−27.3208	−0.892533	−0.446266	+	0.894900i	$0.647247\pi$
$938$	3.87612	0.126560
$939$	−29.6170	−0.966515
$940$	0	0
$941$	44.8806	1.46306	0.731532	−	0.681807i	$-0.238806\pi$
$941$	44.8806	1.46306	0.731532	+	0.681807i	$0.238806\pi$
$942$	49.6972	1.61922
$943$	−65.1773	−2.12246
$944$	−10.1278	−0.329631
$945$	0	0
$946$	−70.3236	−2.28642
$947$	−32.0215	−1.04056	−0.520279	−	0.853996i	$-0.674172\pi$
$947$	−32.0215	−1.04056	−0.520279	+	0.853996i	$0.674172\pi$
$948$	34.4615	1.11926
$949$	24.6253	0.799371
$950$	0	0
$951$	−8.78093	−0.284741
$952$	1.00000	0.0324102
$953$	−6.66420	−0.215875	−0.107937	−	0.994158i	$-0.534425\pi$
$953$	−6.66420	−0.215875	−0.107937	+	0.994158i	$0.534425\pi$
$954$	−35.7550	−1.15761
$955$	0	0
$956$	10.8663	0.351440
$957$	14.4156	0.465991
$958$	12.2197	0.394801
$959$	−21.0684	−0.680334
$960$	0	0
$961$	−15.0694	−0.486109
$962$	−6.84681	−0.220750
$963$	−7.09577	−0.228658
$964$	9.61298	0.309613
$965$	0	0
$966$	21.0092	0.675960
$967$	−32.3370	−1.03989	−0.519944	−	0.854200i	$-0.674047\pi$
$967$	−32.3370	−1.03989	−0.519944	+	0.854200i	$0.674047\pi$
$968$	21.7928	0.700446
$969$	11.5773	0.371916
$970$	0	0
$971$	26.8528	0.861747	0.430873	−	0.902412i	$-0.358206\pi$
$971$	26.8528	0.861747	0.430873	+	0.902412i	$0.358206\pi$
$972$	−22.3366	−0.716448
$973$	−1.43988	−0.0461603
$974$	−2.11355	−0.0677224
$975$	0	0
$976$	−0.952226	−0.0304800
$977$	14.3089	0.457783	0.228892	−	0.973452i	$-0.426490\pi$
$977$	14.3089	0.457783	0.228892	+	0.973452i	$0.426490\pi$
$978$	38.7231	1.23823
$979$	−82.7575	−2.64494
$980$	0	0
$981$	−66.9496	−2.13754
$982$	−6.35587	−0.202824
$983$	−10.9246	−0.348442	−0.174221	−	0.984707i	$-0.555741\pi$
$983$	−10.9246	−0.348442	−0.174221	+	0.984707i	$0.555741\pi$
$984$	20.0432	0.638953
$985$	0	0
$986$	0.990385	0.0315403
$987$	−21.5914	−0.687262
$988$	15.8641	0.504704
$989$	−101.504	−3.22763
$990$	0	0
$991$	46.6895	1.48314	0.741570	−	0.670876i	$-0.234082\pi$
$991$	46.6895	1.48314	0.741570	+	0.670876i	$0.234082\pi$
$992$	−3.99132	−0.126724
$993$	31.7654	1.00804
$994$	6.23006	0.197605
$995$	0	0
$996$	−39.1246	−1.23971
$997$	−7.52339	−0.238268	−0.119134	−	0.992878i	$-0.538012\pi$
$997$	−7.52339	−0.238268	−0.119134	+	0.992878i	$0.538012\pi$
$998$	42.4519	1.34379
$999$	2.30197	0.0728312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5950.2.a.cc.1.7		7
5.2	odd	4		1190.2.e.g.239.8	yes	14
5.3	odd	4		1190.2.e.g.239.7	✓	14
5.4	even	2		5950.2.a.cb.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.e.g.239.7	✓	14	5.3	odd	4
1190.2.e.g.239.8	yes	14	5.2	odd	4
5950.2.a.cb.1.1		7	5.4	even	2
5950.2.a.cc.1.7		7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5950.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.54179 q^{3} +1.00000 q^{4} +2.54179 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.46070 q^{9} -5.72650 q^{11} +2.54179 q^{12} -3.48296 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +3.46070 q^{18} -4.55477 q^{19} -2.54179 q^{21} -5.72650 q^{22} -8.26551 q^{23} +2.54179 q^{24} -3.48296 q^{26} +1.17101 q^{27} -1.00000 q^{28} -0.990385 q^{29} -3.99132 q^{31} +1.00000 q^{32} -14.5556 q^{33} -1.00000 q^{34} +3.46070 q^{36} +1.96580 q^{37} -4.55477 q^{38} -8.85296 q^{39} +7.88545 q^{41} -2.54179 q^{42} +12.2804 q^{43} -5.72650 q^{44} -8.26551 q^{46} +8.49457 q^{47} +2.54179 q^{48} +1.00000 q^{49} -2.54179 q^{51} -3.48296 q^{52} -10.3317 q^{53} +1.17101 q^{54} -1.00000 q^{56} -11.5773 q^{57} -0.990385 q^{58} -10.1278 q^{59} -0.952226 q^{61} -3.99132 q^{62} -3.46070 q^{63} +1.00000 q^{64} -14.5556 q^{66} -3.87612 q^{67} -1.00000 q^{68} -21.0092 q^{69} -6.23006 q^{71} +3.46070 q^{72} -7.07022 q^{73} +1.96580 q^{74} -4.55477 q^{76} +5.72650 q^{77} -8.85296 q^{78} +13.5580 q^{79} -7.40565 q^{81} +7.88545 q^{82} -15.3925 q^{83} -2.54179 q^{84} +12.2804 q^{86} -2.51735 q^{87} -5.72650 q^{88} +14.4517 q^{89} +3.48296 q^{91} -8.26551 q^{92} -10.1451 q^{93} +8.49457 q^{94} +2.54179 q^{96} +17.5462 q^{97} +1.00000 q^{98} -19.8177 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} - 2 q^{3} + 7 q^{4} - 2 q^{6} - 7 q^{7} + 7 q^{8} + 7 q^{9} - 7 q^{11} - 2 q^{12} + q^{13} - 7 q^{14} + 7 q^{16} - 7 q^{17} + 7 q^{18} - 2 q^{19} + 2 q^{21} - 7 q^{22} - 27 q^{23} - 2 q^{24}+ \cdots - 44 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 - 2 * q^3 + 7 * q^4 - 2 * q^6 - 7 * q^7 + 7 * q^8 + 7 * q^9 - 7 * q^11 - 2 * q^12 + q^13 - 7 * q^14 + 7 * q^16 - 7 * q^17 + 7 * q^18 - 2 * q^19 + 2 * q^21 - 7 * q^22 - 27 * q^23 - 2 * q^24 + q^26 - 17 * q^27 - 7 * q^28 - 10 * q^29 - 18 * q^31 + 7 * q^32 - 8 * q^33 - 7 * q^34 + 7 * q^36 - 12 * q^37 - 2 * q^38 - 25 * q^39 + 17 * q^41 + 2 * q^42 - 7 * q^43 - 7 * q^44 - 27 * q^46 - 8 * q^47 - 2 * q^48 + 7 * q^49 + 2 * q^51 + q^52 - 20 * q^53 - 17 * q^54 - 7 * q^56 - 25 * q^57 - 10 * q^58 + q^61 - 18 * q^62 - 7 * q^63 + 7 * q^64 - 8 * q^66 - 7 * q^67 - 7 * q^68 + 3 * q^69 - 11 * q^71 + 7 * q^72 + 7 * q^73 - 12 * q^74 - 2 * q^76 + 7 * q^77 - 25 * q^78 - 26 * q^79 + 11 * q^81 + 17 * q^82 - 22 * q^83 + 2 * q^84 - 7 * q^86 - 20 * q^87 - 7 * q^88 + 21 * q^89 - q^91 - 27 * q^92 - 25 * q^93 - 8 * q^94 - 2 * q^96 + 19 * q^97 + 7 * q^98 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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