LMFDB - Embedded newform 4598.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, 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("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.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:	$36.7152148494$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.772866$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +0.772866 q^{3} +1.00000 q^{4} +1.22713 q^{5} -0.772866 q^{6} -3.40268 q^{7} -1.00000 q^{8} -2.40268 q^{9} -1.22713 q^{10} +0.772866 q^{12} +1.40268 q^{13} +3.40268 q^{14} +0.948410 q^{15} +1.00000 q^{16} +4.80536 q^{17} +2.40268 q^{18} +1.00000 q^{19} +1.22713 q^{20} -2.62981 q^{21} +6.80536 q^{23} -0.772866 q^{24} -3.49414 q^{25} -1.40268 q^{26} -4.17554 q^{27} -3.40268 q^{28} -8.03249 q^{29} -0.948410 q^{30} -4.94841 q^{31} -1.00000 q^{32} -4.80536 q^{34} -4.17554 q^{35} -2.40268 q^{36} +2.35109 q^{37} -1.00000 q^{38} +1.08408 q^{39} -1.22713 q^{40} -2.94841 q^{41} +2.62981 q^{42} +9.12395 q^{43} -2.94841 q^{45} -6.80536 q^{46} +5.25963 q^{47} +0.772866 q^{48} +4.57822 q^{49} +3.49414 q^{50} +3.71390 q^{51} +1.40268 q^{52} -4.45427 q^{53} +4.17554 q^{54} +3.40268 q^{56} +0.772866 q^{57} +8.03249 q^{58} -14.5193 q^{59} +0.948410 q^{60} -2.00000 q^{61} +4.94841 q^{62} +8.17554 q^{63} +1.00000 q^{64} +1.72128 q^{65} +3.40268 q^{67} +4.80536 q^{68} +5.25963 q^{69} +4.17554 q^{70} -7.57822 q^{71} +2.40268 q^{72} +10.0650 q^{73} -2.35109 q^{74} -2.70050 q^{75} +1.00000 q^{76} -1.08408 q^{78} +3.71390 q^{79} +1.22713 q^{80} +3.98090 q^{81} +2.94841 q^{82} -10.6697 q^{83} -2.62981 q^{84} +5.89682 q^{85} -9.12395 q^{86} -6.20804 q^{87} +14.9883 q^{89} +2.94841 q^{90} -4.77287 q^{91} +6.80536 q^{92} -3.82446 q^{93} -5.25963 q^{94} +1.22713 q^{95} -0.772866 q^{96} +2.35109 q^{97} -4.57822 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} - q^{6} - q^{7} - 3 q^{8} + 2 q^{9} - 5 q^{10} + q^{12} - 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} - 4 q^{17} - 2 q^{18} + 3 q^{19} + 5 q^{20} + 2 q^{23} - q^{24}+ \cdots + 6 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 - q^6 - q^7 - 3 * q^8 + 2 * q^9 - 5 * q^10 + q^12 - 5 * q^13 + q^14 - 9 * q^15 + 3 * q^16 - 4 * q^17 - 2 * q^18 + 3 * q^19 + 5 * q^20 + 2 * q^23 - q^24 + 4 * q^25 + 5 * q^26 - 2 * q^27 - q^28 - 7 * q^29 + 9 * q^30 - 3 * q^31 - 3 * q^32 + 4 * q^34 - 2 * q^35 + 2 * q^36 - 14 * q^37 - 3 * q^38 - 2 * q^39 - 5 * q^40 + 3 * q^41 + 5 * q^43 + 3 * q^45 - 2 * q^46 + q^48 - 6 * q^49 - 4 * q^50 - 2 * q^51 - 5 * q^52 - 16 * q^53 + 2 * q^54 + q^56 + q^57 + 7 * q^58 - 12 * q^59 - 9 * q^60 - 6 * q^61 + 3 * q^62 + 14 * q^63 + 3 * q^64 - 8 * q^65 + q^67 - 4 * q^68 + 2 * q^70 - 3 * q^71 - 2 * q^72 - 4 * q^73 + 14 * q^74 - 41 * q^75 + 3 * q^76 + 2 * q^78 - 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 - 7 * q^83 - 6 * q^85 - 5 * q^86 + 9 * q^87 + 16 * q^89 - 3 * q^90 - 13 * q^91 + 2 * q^92 - 22 * q^93 + 5 * q^95 - q^96 - 14 * q^97 + 6 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.772866	0.446214	0.223107	−	0.974794i	$-0.428380\pi$
$3$	0.772866	0.446214	0.223107	+	0.974794i	$0.428380\pi$
$4$	1.00000	0.500000
$5$	1.22713	0.548791	0.274396	−	0.961617i	$-0.411522\pi$
$5$	1.22713	0.548791	0.274396	+	0.961617i	$0.411522\pi$
$6$	−0.772866	−0.315521
$7$	−3.40268	−1.28609	−0.643046	−	0.765828i	$-0.722330\pi$
$7$	−3.40268	−1.28609	−0.643046	+	0.765828i	$0.722330\pi$
$8$	−1.00000	−0.353553
$9$	−2.40268	−0.800893
$10$	−1.22713	−0.388054
$11$	0	0
$11$	0	0
$12$	0.772866	0.223107
$13$	1.40268	0.389033	0.194517	−	0.980899i	$-0.437686\pi$
$13$	1.40268	0.389033	0.194517	+	0.980899i	$0.437686\pi$
$14$	3.40268	0.909404
$15$	0.948410	0.244878
$16$	1.00000	0.250000
$17$	4.80536	1.16547	0.582735	−	0.812662i	$-0.301982\pi$
$17$	4.80536	1.16547	0.582735	+	0.812662i	$0.301982\pi$
$18$	2.40268	0.566317
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.22713	0.274396
$21$	−2.62981	−0.573872
$22$	0	0
$23$	6.80536	1.41902	0.709508	−	0.704698i	$-0.248917\pi$
$23$	6.80536	1.41902	0.709508	+	0.704698i	$0.248917\pi$
$24$	−0.772866	−0.157761
$25$	−3.49414	−0.698828
$26$	−1.40268	−0.275088
$27$	−4.17554	−0.803584
$28$	−3.40268	−0.643046
$29$	−8.03249	−1.49160	−0.745798	−	0.666172i	$-0.767932\pi$
$29$	−8.03249	−1.49160	−0.745798	+	0.666172i	$0.767932\pi$
$30$	−0.948410	−0.173155
$31$	−4.94841	−0.888761	−0.444380	−	0.895838i	$-0.646576\pi$
$31$	−4.94841	−0.888761	−0.444380	+	0.895838i	$0.646576\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.80536	−0.824112
$35$	−4.17554	−0.705796
$36$	−2.40268	−0.400446
$37$	2.35109	0.386517	0.193258	−	0.981148i	$-0.438094\pi$
$37$	2.35109	0.386517	0.193258	+	0.981148i	$0.438094\pi$
$38$	−1.00000	−0.162221
$39$	1.08408	0.173592
$40$	−1.22713	−0.194027
$41$	−2.94841	−0.460464	−0.230232	−	0.973136i	$-0.573949\pi$
$41$	−2.94841	−0.460464	−0.230232	+	0.973136i	$0.573949\pi$
$42$	2.62981	0.405789
$43$	9.12395	1.39139	0.695695	−	0.718337i	$-0.255097\pi$
$43$	9.12395	1.39139	0.695695	+	0.718337i	$0.255097\pi$
$44$	0	0
$45$	−2.94841	−0.439523
$46$	−6.80536	−1.00340
$47$	5.25963	0.767195	0.383598	−	0.923500i	$-0.374685\pi$
$47$	5.25963	0.767195	0.383598	+	0.923500i	$0.374685\pi$
$48$	0.772866	0.111554
$49$	4.57822	0.654032
$50$	3.49414	0.494146
$51$	3.71390	0.520049
$52$	1.40268	0.194517
$53$	−4.45427	−0.611841	−0.305920	−	0.952057i	$-0.598964\pi$
$53$	−4.45427	−0.611841	−0.305920	+	0.952057i	$0.598964\pi$
$54$	4.17554	0.568220
$55$	0	0
$56$	3.40268	0.454702
$57$	0.772866	0.102369
$58$	8.03249	1.05472
$59$	−14.5193	−1.89025	−0.945123	−	0.326715i	$-0.894058\pi$
$59$	−14.5193	−1.89025	−0.945123	+	0.326715i	$0.894058\pi$
$60$	0.948410	0.122439
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.94841	0.628449
$63$	8.17554	1.03002
$64$	1.00000	0.125000
$65$	1.72128	0.213498
$66$	0	0
$67$	3.40268	0.415703	0.207852	−	0.978160i	$-0.433353\pi$
$67$	3.40268	0.415703	0.207852	+	0.978160i	$0.433353\pi$
$68$	4.80536	0.582735
$69$	5.25963	0.633185
$70$	4.17554	0.499073
$71$	−7.57822	−0.899370	−0.449685	−	0.893187i	$-0.648464\pi$
$71$	−7.57822	−0.899370	−0.449685	+	0.893187i	$0.648464\pi$
$72$	2.40268	0.283158
$73$	10.0650	1.17802	0.589009	−	0.808127i	$-0.299518\pi$
$73$	10.0650	1.17802	0.589009	+	0.808127i	$0.299518\pi$
$74$	−2.35109	−0.273309
$75$	−2.70050	−0.311827
$76$	1.00000	0.114708
$77$	0	0
$78$	−1.08408	−0.122748
$79$	3.71390	0.417846	0.208923	−	0.977932i	$-0.433004\pi$
$79$	3.71390	0.417846	0.208923	+	0.977932i	$0.433004\pi$
$80$	1.22713	0.137198
$81$	3.98090	0.442322
$82$	2.94841	0.325597
$83$	−10.6697	−1.17115	−0.585575	−	0.810618i	$-0.699131\pi$
$83$	−10.6697	−1.17115	−0.585575	+	0.810618i	$0.699131\pi$
$84$	−2.62981	−0.286936
$85$	5.89682	0.639600
$86$	−9.12395	−0.983861
$87$	−6.20804	−0.665571
$88$	0	0
$89$	14.9883	1.58875	0.794377	−	0.607425i	$-0.207797\pi$
$89$	14.9883	1.58875	0.794377	+	0.607425i	$0.207797\pi$
$90$	2.94841	0.310790
$91$	−4.77287	−0.500332
$92$	6.80536	0.709508
$93$	−3.82446	−0.396578
$94$	−5.25963	−0.542489
$95$	1.22713	0.125901
$96$	−0.772866	−0.0788803
$97$	2.35109	0.238717	0.119358	−	0.992851i	$-0.461916\pi$
$97$	2.35109	0.238717	0.119358	+	0.992851i	$0.461916\pi$
$98$	−4.57822	−0.462470
$99$	0	0
$100$	−3.49414	−0.349414
$101$	−14.0650	−1.39952	−0.699759	−	0.714379i	$-0.746709\pi$
$101$	−14.0650	−1.39952	−0.699759	+	0.714379i	$0.746709\pi$
$102$	−3.71390	−0.367730
$103$	−15.5782	−1.53497	−0.767484	−	0.641068i	$-0.778492\pi$
$103$	−15.5782	−1.53497	−0.767484	+	0.641068i	$0.778492\pi$
$104$	−1.40268	−0.137544
$105$	−3.22713	−0.314936
$106$	4.45427	0.432637
$107$	4.35109	0.420636	0.210318	−	0.977633i	$-0.432550\pi$
$107$	4.35109	0.420636	0.210318	+	0.977633i	$0.432550\pi$
$108$	−4.17554	−0.401792
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	1.81708	0.172469
$112$	−3.40268	−0.321523
$113$	−19.9618	−1.87785	−0.938924	−	0.344124i	$-0.888176\pi$
$113$	−19.9618	−1.87785	−0.938924	+	0.344124i	$0.888176\pi$
$114$	−0.772866	−0.0723855
$115$	8.35109	0.778743
$116$	−8.03249	−0.745798
$117$	−3.37019	−0.311574
$118$	14.5193	1.33661
$119$	−16.3511	−1.49890
$120$	−0.948410	−0.0865776
$121$	0	0
$122$	2.00000	0.181071
$123$	−2.27872	−0.205466
$124$	−4.94841	−0.444380
$125$	−10.4235	−0.932302
$126$	−8.17554	−0.728335
$127$	−11.7139	−1.03944	−0.519720	−	0.854337i	$-0.673964\pi$
$127$	−11.7139	−1.03944	−0.519720	+	0.854337i	$0.673964\pi$
$128$	−1.00000	−0.0883883
$129$	7.05159	0.620858
$130$	−1.72128	−0.150966
$131$	−1.85695	−0.162242	−0.0811211	−	0.996704i	$-0.525850\pi$
$131$	−1.85695	−0.162242	−0.0811211	+	0.996704i	$0.525850\pi$
$132$	0	0
$133$	−3.40268	−0.295050
$134$	−3.40268	−0.293947
$135$	−5.12395	−0.441000
$136$	−4.80536	−0.412056
$137$	10.5973	0.905390	0.452695	−	0.891665i	$-0.350463\pi$
$137$	10.5973	0.905390	0.452695	+	0.891665i	$0.350463\pi$
$138$	−5.25963	−0.447729
$139$	−14.7347	−1.24978	−0.624889	−	0.780713i	$-0.714856\pi$
$139$	−14.7347	−1.24978	−0.624889	+	0.780713i	$0.714856\pi$
$140$	−4.17554	−0.352898
$141$	4.06498	0.342333
$142$	7.57822	0.635950
$143$	0	0
$144$	−2.40268	−0.200223
$145$	−9.85695	−0.818575
$146$	−10.0650	−0.832984
$147$	3.53835	0.291838
$148$	2.35109	0.193258
$149$	−18.3511	−1.50338	−0.751690	−	0.659517i	$-0.770761\pi$
$149$	−18.3511	−1.50338	−0.751690	+	0.659517i	$0.770761\pi$
$150$	2.70050	0.220495
$151$	7.64891	0.622460	0.311230	−	0.950335i	$-0.399259\pi$
$151$	7.64891	0.622460	0.311230	+	0.950335i	$0.399259\pi$
$152$	−1.00000	−0.0811107
$153$	−11.5457	−0.933417
$154$	0	0
$155$	−6.07236	−0.487744
$156$	1.08408	0.0867960
$157$	8.38358	0.669083	0.334541	−	0.942381i	$-0.391419\pi$
$157$	8.38358	0.669083	0.334541	+	0.942381i	$0.391419\pi$
$158$	−3.71390	−0.295462
$159$	−3.44255	−0.273012
$160$	−1.22713	−0.0970135
$161$	−23.1564	−1.82498
$162$	−3.98090	−0.312769
$163$	−14.2479	−1.11598	−0.557991	−	0.829847i	$-0.688428\pi$
$163$	−14.2479	−1.11598	−0.557991	+	0.829847i	$0.688428\pi$
$164$	−2.94841	−0.230232
$165$	0	0
$166$	10.6697	0.828128
$167$	10.8703	0.841172	0.420586	−	0.907253i	$-0.361824\pi$
$167$	10.8703	0.841172	0.420586	+	0.907253i	$0.361824\pi$
$168$	2.62981	0.202894
$169$	−11.0325	−0.848653
$170$	−5.89682	−0.452265
$171$	−2.40268	−0.183737
$172$	9.12395	0.695695
$173$	−3.39530	−0.258140	−0.129070	−	0.991635i	$-0.541199\pi$
$173$	−3.39530	−0.258140	−0.129070	+	0.991635i	$0.541199\pi$
$174$	6.20804	0.470630
$175$	11.8894	0.898757
$176$	0	0
$177$	−11.2214	−0.843454
$178$	−14.9883	−1.12342
$179$	−0.311217	−0.0232614	−0.0116307	−	0.999932i	$-0.503702\pi$
$179$	−0.311217	−0.0232614	−0.0116307	+	0.999932i	$0.503702\pi$
$180$	−2.94841	−0.219762
$181$	−15.3246	−1.13907	−0.569535	−	0.821967i	$-0.692877\pi$
$181$	−15.3246	−1.13907	−0.569535	+	0.821967i	$0.692877\pi$
$182$	4.77287	0.353788
$183$	−1.54573	−0.114264
$184$	−6.80536	−0.501698
$185$	2.88510	0.212117
$186$	3.82446	0.280423
$187$	0	0
$188$	5.25963	0.383598
$189$	14.2080	1.03348
$190$	−1.22713	−0.0890257
$191$	26.8703	1.94427	0.972135	−	0.234422i	$-0.0753199\pi$
$191$	26.8703	1.94427	0.972135	+	0.234422i	$0.0753199\pi$
$192$	0.772866	0.0557768
$193$	−18.5665	−1.33645	−0.668223	−	0.743961i	$-0.732945\pi$
$193$	−18.5665	−1.33645	−0.668223	+	0.743961i	$0.732945\pi$
$194$	−2.35109	−0.168798
$195$	1.33031	0.0952658
$196$	4.57822	0.327016
$197$	−14.9085	−1.06219	−0.531095	−	0.847312i	$-0.678219\pi$
$197$	−14.9085	−1.06219	−0.531095	+	0.847312i	$0.678219\pi$
$198$	0	0
$199$	15.1564	1.07441	0.537206	−	0.843451i	$-0.319480\pi$
$199$	15.1564	1.07441	0.537206	+	0.843451i	$0.319480\pi$
$200$	3.49414	0.247073
$201$	2.62981	0.185493
$202$	14.0650	0.989609
$203$	27.3320	1.91833
$204$	3.71390	0.260025
$205$	−3.61810	−0.252699
$206$	15.5782	1.08539
$207$	−16.3511	−1.13648
$208$	1.40268	0.0972583
$209$	0	0
$210$	3.22713	0.222693
$211$	−11.1564	−0.768041	−0.384021	−	0.923324i	$-0.625461\pi$
$211$	−11.1564	−0.768041	−0.384021	+	0.923324i	$0.625461\pi$
$212$	−4.45427	−0.305920
$213$	−5.85695	−0.401311
$214$	−4.35109	−0.297434
$215$	11.1963	0.763583
$216$	4.17554	0.284110
$217$	16.8378	1.14303
$218$	6.00000	0.406371
$219$	7.77888	0.525648
$220$	0	0
$221$	6.74037	0.453407
$222$	−1.81708	−0.121954
$223$	1.61072	0.107861	0.0539307	−	0.998545i	$-0.482825\pi$
$223$	1.61072	0.107861	0.0539307	+	0.998545i	$0.482825\pi$
$224$	3.40268	0.227351
$225$	8.39530	0.559687
$226$	19.9618	1.32784
$227$	16.7672	1.11288	0.556438	−	0.830889i	$-0.312168\pi$
$227$	16.7672	1.11288	0.556438	+	0.830889i	$0.312168\pi$
$228$	0.772866	0.0511843
$229$	−8.55913	−0.565603	−0.282801	−	0.959178i	$-0.591264\pi$
$229$	−8.55913	−0.565603	−0.282801	+	0.959178i	$0.591264\pi$
$230$	−8.35109	−0.550654
$231$	0	0
$232$	8.03249	0.527359
$233$	−5.44255	−0.356553	−0.178277	−	0.983980i	$-0.557052\pi$
$233$	−5.44255	−0.356553	−0.178277	+	0.983980i	$0.557052\pi$
$234$	3.37019	0.220316
$235$	6.45427	0.421030
$236$	−14.5193	−0.945123
$237$	2.87034	0.186449
$238$	16.3511	1.05988
$239$	−11.2921	−0.730426	−0.365213	−	0.930924i	$-0.619004\pi$
$239$	−11.2921	−0.730426	−0.365213	+	0.930924i	$0.619004\pi$
$240$	0.948410	0.0612196
$241$	−11.4101	−0.734987	−0.367493	−	0.930026i	$-0.619784\pi$
$241$	−11.4101	−0.734987	−0.367493	+	0.930026i	$0.619784\pi$
$242$	0	0
$243$	15.6033	1.00095
$244$	−2.00000	−0.128037
$245$	5.61810	0.358927
$246$	2.27872	0.145286
$247$	1.40268	0.0892503
$248$	4.94841	0.314224
$249$	−8.24623	−0.522584
$250$	10.4235	0.659237
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	8.17554	0.515011
$253$	0	0
$254$	11.7139	0.734995
$255$	4.55745	0.285399
$256$	1.00000	0.0625000
$257$	24.2331	1.51162	0.755811	−	0.654790i	$-0.227243\pi$
$257$	24.2331	1.51162	0.755811	+	0.654790i	$0.227243\pi$
$258$	−7.05159	−0.439013
$259$	−8.00000	−0.497096
$260$	1.72128	0.106749
$261$	19.2995	1.19461
$262$	1.85695	0.114723
$263$	−20.5665	−1.26819	−0.634093	−	0.773257i	$-0.718626\pi$
$263$	−20.5665	−1.26819	−0.634093	+	0.773257i	$0.718626\pi$
$264$	0	0
$265$	−5.46599	−0.335773
$266$	3.40268	0.208632
$267$	11.5839	0.708925
$268$	3.40268	0.207852
$269$	−1.36281	−0.0830918	−0.0415459	−	0.999137i	$-0.513228\pi$
$269$	−1.36281	−0.0830918	−0.0415459	+	0.999137i	$0.513228\pi$
$270$	5.12395	0.311834
$271$	23.1963	1.40908	0.704538	−	0.709666i	$-0.251154\pi$
$271$	23.1963	1.40908	0.704538	+	0.709666i	$0.251154\pi$
$272$	4.80536	0.291368
$273$	−3.68878	−0.223255
$274$	−10.5973	−0.640208
$275$	0	0
$276$	5.25963	0.316592
$277$	−31.0385	−1.86492	−0.932462	−	0.361269i	$-0.882344\pi$
$277$	−31.0385	−1.86492	−0.932462	+	0.361269i	$0.882344\pi$
$278$	14.7347	0.883727
$279$	11.8894	0.711802
$280$	4.17554	0.249537
$281$	2.77287	0.165415	0.0827076	−	0.996574i	$-0.473643\pi$
$281$	2.77287	0.165415	0.0827076	+	0.996574i	$0.473643\pi$
$282$	−4.06498	−0.242066
$283$	9.23451	0.548935	0.274467	−	0.961596i	$-0.411498\pi$
$283$	9.23451	0.548935	0.274467	+	0.961596i	$0.411498\pi$
$284$	−7.57822	−0.449685
$285$	0.948410	0.0561790
$286$	0	0
$287$	10.0325	0.592199
$288$	2.40268	0.141579
$289$	6.09146	0.358321
$290$	9.85695	0.578820
$291$	1.81708	0.106519
$292$	10.0650	0.589009
$293$	−17.7390	−1.03632	−0.518162	−	0.855283i	$-0.673384\pi$
$293$	−17.7390	−1.03632	−0.518162	+	0.855283i	$0.673384\pi$
$294$	−3.53835	−0.206361
$295$	−17.8171	−1.03735
$296$	−2.35109	−0.136654
$297$	0	0
$298$	18.3511	1.06305
$299$	9.54573	0.552044
$300$	−2.70050	−0.155914
$301$	−31.0459	−1.78946
$302$	−7.64891	−0.440145
$303$	−10.8703	−0.624485
$304$	1.00000	0.0573539
$305$	−2.45427	−0.140531
$306$	11.5457	0.660026
$307$	−20.9735	−1.19702	−0.598511	−	0.801115i	$-0.704241\pi$
$307$	−20.9735	−1.19702	−0.598511	+	0.801115i	$0.704241\pi$
$308$	0	0
$309$	−12.0399	−0.684924
$310$	6.07236	0.344887
$311$	−3.36281	−0.190687	−0.0953436	−	0.995444i	$-0.530395\pi$
$311$	−3.36281	−0.190687	−0.0953436	+	0.995444i	$0.530395\pi$
$312$	−1.08408	−0.0613741
$313$	−4.94103	−0.279284	−0.139642	−	0.990202i	$-0.544595\pi$
$313$	−4.94103	−0.279284	−0.139642	+	0.990202i	$0.544595\pi$
$314$	−8.38358	−0.473113
$315$	10.0325	0.565267
$316$	3.71390	0.208923
$317$	14.2861	0.802388	0.401194	−	0.915993i	$-0.368595\pi$
$317$	14.2861	0.802388	0.401194	+	0.915993i	$0.368595\pi$
$318$	3.44255	0.193049
$319$	0	0
$320$	1.22713	0.0685989
$321$	3.36281	0.187694
$322$	23.1564	1.29046
$323$	4.80536	0.267377
$324$	3.98090	0.221161
$325$	−4.90116	−0.271867
$326$	14.2479	0.789119
$327$	−4.63719	−0.256437
$328$	2.94841	0.162799
$329$	−17.8968	−0.986684
$330$	0	0
$331$	−31.9293	−1.75499	−0.877497	−	0.479582i	$-0.840788\pi$
$331$	−31.9293	−1.75499	−0.877497	+	0.479582i	$0.840788\pi$
$332$	−10.6697	−0.585575
$333$	−5.64891	−0.309558
$334$	−10.8703	−0.594799
$335$	4.17554	0.228134
$336$	−2.62981	−0.143468
$337$	8.84523	0.481830	0.240915	−	0.970546i	$-0.422552\pi$
$337$	8.84523	0.481830	0.240915	+	0.970546i	$0.422552\pi$
$338$	11.0325	0.600088
$339$	−15.4278	−0.837923
$340$	5.89682	0.319800
$341$	0	0
$342$	2.40268	0.129922
$343$	8.24053	0.444947
$344$	−9.12395	−0.491931
$345$	6.45427	0.347486
$346$	3.39530	0.182532
$347$	−5.27439	−0.283144	−0.141572	−	0.989928i	$-0.545216\pi$
$347$	−5.27439	−0.283144	−0.141572	+	0.989928i	$0.545216\pi$
$348$	−6.20804	−0.332786
$349$	10.9085	0.583921	0.291960	−	0.956430i	$-0.405692\pi$
$349$	10.9085	0.583921	0.291960	+	0.956430i	$0.405692\pi$
$350$	−11.8894	−0.635517
$351$	−5.85695	−0.312621
$352$	0	0
$353$	−22.1300	−1.17786	−0.588930	−	0.808184i	$-0.700451\pi$
$353$	−22.1300	−1.17786	−0.588930	+	0.808184i	$0.700451\pi$
$354$	11.2214	0.596412
$355$	−9.29950	−0.493566
$356$	14.9883	0.794377
$357$	−12.6372	−0.668831
$358$	0.311217	0.0164483
$359$	16.3910	0.865082	0.432541	−	0.901614i	$-0.357617\pi$
$359$	16.3910	0.865082	0.432541	+	0.901614i	$0.357617\pi$
$360$	2.94841	0.155395
$361$	1.00000	0.0526316
$362$	15.3246	0.805444
$363$	0	0
$364$	−4.77287	−0.250166
$365$	12.3511	0.646486
$366$	1.54573	0.0807967
$367$	−32.3511	−1.68871	−0.844357	−	0.535782i	$-0.820017\pi$
$367$	−32.3511	−1.68871	−0.844357	+	0.535782i	$0.820017\pi$
$368$	6.80536	0.354754
$369$	7.08408	0.368783
$370$	−2.88510	−0.149989
$371$	15.1564	0.786883
$372$	−3.82446	−0.198289
$373$	−17.5782	−0.910166	−0.455083	−	0.890449i	$-0.650390\pi$
$373$	−17.5782	−0.910166	−0.455083	+	0.890449i	$0.650390\pi$
$374$	0	0
$375$	−8.05593	−0.416006
$376$	−5.25963	−0.271245
$377$	−11.2670	−0.580280
$378$	−14.2080	−0.730783
$379$	8.93365	0.458891	0.229445	−	0.973322i	$-0.426309\pi$
$379$	8.93365	0.458891	0.229445	+	0.973322i	$0.426309\pi$
$380$	1.22713	0.0629507
$381$	−9.05327	−0.463813
$382$	−26.8703	−1.37481
$383$	−16.0399	−0.819599	−0.409800	−	0.912176i	$-0.634401\pi$
$383$	−16.0399	−0.819599	−0.409800	+	0.912176i	$0.634401\pi$
$384$	−0.772866	−0.0394401
$385$	0	0
$386$	18.5665	0.945010
$387$	−21.9219	−1.11435
$388$	2.35109	0.119358
$389$	17.2271	0.873450	0.436725	−	0.899595i	$-0.356138\pi$
$389$	17.2271	0.873450	0.436725	+	0.899595i	$0.356138\pi$
$390$	−1.33031	−0.0673631
$391$	32.7022	1.65382
$392$	−4.57822	−0.231235
$393$	−1.43517	−0.0723948
$394$	14.9085	0.751081
$395$	4.55745	0.229310
$396$	0	0
$397$	15.0784	0.756762	0.378381	−	0.925650i	$-0.376481\pi$
$397$	15.0784	0.756762	0.378381	+	0.925650i	$0.376481\pi$
$398$	−15.1564	−0.759724
$399$	−2.62981	−0.131655
$400$	−3.49414	−0.174707
$401$	−13.1564	−0.657002	−0.328501	−	0.944504i	$-0.606543\pi$
$401$	−13.1564	−0.657002	−0.328501	+	0.944504i	$0.606543\pi$
$402$	−2.62981	−0.131163
$403$	−6.94103	−0.345757
$404$	−14.0650	−0.699759
$405$	4.88510	0.242743
$406$	−27.3320	−1.35646
$407$	0	0
$408$	−3.71390	−0.183865
$409$	16.3836	0.810116	0.405058	−	0.914291i	$-0.367251\pi$
$409$	16.3836	0.810116	0.405058	+	0.914291i	$0.367251\pi$
$410$	3.61810	0.178685
$411$	8.19030	0.403998
$412$	−15.5782	−0.767484
$413$	49.4044	2.43103
$414$	16.3511	0.803612
$415$	−13.0931	−0.642717
$416$	−1.40268	−0.0687720
$417$	−11.3879	−0.557669
$418$	0	0
$419$	1.12966	0.0551874	0.0275937	−	0.999619i	$-0.491216\pi$
$419$	1.12966	0.0551874	0.0275937	+	0.999619i	$0.491216\pi$
$420$	−3.22713	−0.157468
$421$	−2.63719	−0.128529	−0.0642645	−	0.997933i	$-0.520470\pi$
$421$	−2.63719	−0.128529	−0.0642645	+	0.997933i	$0.520470\pi$
$422$	11.1564	0.543087
$423$	−12.6372	−0.614441
$424$	4.45427	0.216318
$425$	−16.7906	−0.814464
$426$	5.85695	0.283770
$427$	6.80536	0.329334
$428$	4.35109	0.210318
$429$	0	0
$430$	−11.1963	−0.539934
$431$	28.4958	1.37260	0.686298	−	0.727321i	$-0.259235\pi$
$431$	28.4958	1.37260	0.686298	+	0.727321i	$0.259235\pi$
$432$	−4.17554	−0.200896
$433$	−37.5075	−1.80250	−0.901249	−	0.433302i	$-0.857348\pi$
$433$	−37.5075	−1.80250	−0.901249	+	0.433302i	$0.857348\pi$
$434$	−16.8378	−0.808243
$435$	−7.61810	−0.365260
$436$	−6.00000	−0.287348
$437$	6.80536	0.325544
$438$	−7.77888	−0.371689
$439$	32.1300	1.53348	0.766740	−	0.641958i	$-0.221878\pi$
$439$	32.1300	1.53348	0.766740	+	0.641958i	$0.221878\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	−6.74037	−0.320607
$443$	33.0385	1.56971	0.784853	−	0.619681i	$-0.212738\pi$
$443$	33.0385	1.56971	0.784853	+	0.619681i	$0.212738\pi$
$444$	1.81708	0.0862346
$445$	18.3926	0.871895
$446$	−1.61072	−0.0762696
$447$	−14.1829	−0.670829
$448$	−3.40268	−0.160761
$449$	7.88206	0.371977	0.185989	−	0.982552i	$-0.440451\pi$
$449$	7.88206	0.371977	0.185989	+	0.982552i	$0.440451\pi$
$450$	−8.39530	−0.395758
$451$	0	0
$452$	−19.9618	−0.938924
$453$	5.91158	0.277750
$454$	−16.7672	−0.786922
$455$	−5.85695	−0.274578
$456$	−0.772866	−0.0361927
$457$	−24.9353	−1.16643	−0.583213	−	0.812320i	$-0.698205\pi$
$457$	−24.9353	−1.16643	−0.583213	+	0.812320i	$0.698205\pi$
$458$	8.55913	0.399942
$459$	−20.0650	−0.936553
$460$	8.35109	0.389372
$461$	26.6874	1.24296	0.621478	−	0.783431i	$-0.286532\pi$
$461$	26.6874	1.24296	0.621478	+	0.783431i	$0.286532\pi$
$462$	0	0
$463$	−32.7022	−1.51980	−0.759900	−	0.650041i	$-0.774752\pi$
$463$	−32.7022	−1.51980	−0.759900	+	0.650041i	$0.774752\pi$
$464$	−8.03249	−0.372899
$465$	−4.69312	−0.217638
$466$	5.44255	0.252121
$467$	11.3776	0.526491	0.263246	−	0.964729i	$-0.415207\pi$
$467$	11.3776	0.526491	0.263246	+	0.964729i	$0.415207\pi$
$468$	−3.37019	−0.155787
$469$	−11.5782	−0.534633
$470$	−6.45427	−0.297713
$471$	6.47938	0.298554
$472$	14.5193	0.668303
$473$	0	0
$474$	−2.87034	−0.131839
$475$	−3.49414	−0.160322
$476$	−16.3511	−0.749451
$477$	10.7022	0.490019
$478$	11.2921	0.516489
$479$	12.5973	0.575586	0.287793	−	0.957693i	$-0.407078\pi$
$479$	12.5973	0.575586	0.287793	+	0.957693i	$0.407078\pi$
$480$	−0.948410	−0.0432888
$481$	3.29782	0.150368
$482$	11.4101	0.519714
$483$	−17.8968	−0.814334
$484$	0	0
$485$	2.88510	0.131006
$486$	−15.6033	−0.707782
$487$	9.69616	0.439375	0.219688	−	0.975570i	$-0.429496\pi$
$487$	9.69616	0.439375	0.219688	+	0.975570i	$0.429496\pi$
$488$	2.00000	0.0905357
$489$	−11.0117	−0.497967
$490$	−5.61810	−0.253800
$491$	11.7538	0.530440	0.265220	−	0.964188i	$-0.414555\pi$
$491$	11.7538	0.530440	0.265220	+	0.964188i	$0.414555\pi$
$492$	−2.27872	−0.102733
$493$	−38.5990	−1.73841
$494$	−1.40268	−0.0631095
$495$	0	0
$496$	−4.94841	−0.222190
$497$	25.7863	1.15667
$498$	8.24623	0.369523
$499$	26.6640	1.19364	0.596822	−	0.802374i	$-0.296430\pi$
$499$	26.6640	1.19364	0.596822	+	0.802374i	$0.296430\pi$
$500$	−10.4235	−0.466151
$501$	8.40131	0.375343
$502$	12.0000	0.535586
$503$	−11.1622	−0.497696	−0.248848	−	0.968543i	$-0.580052\pi$
$503$	−11.1622	−0.497696	−0.248848	+	0.968543i	$0.580052\pi$
$504$	−8.17554	−0.364168
$505$	−17.2596	−0.768043
$506$	0	0
$507$	−8.52663	−0.378681
$508$	−11.7139	−0.519720
$509$	−35.9618	−1.59398	−0.796989	−	0.603993i	$-0.793575\pi$
$509$	−35.9618	−1.59398	−0.796989	+	0.603993i	$0.793575\pi$
$510$	−4.55745	−0.201807
$511$	−34.2479	−1.51504
$512$	−1.00000	−0.0441942
$513$	−4.17554	−0.184355
$514$	−24.2331	−1.06888
$515$	−19.1166	−0.842377
$516$	7.05159	0.310429
$517$	0	0
$518$	8.00000	0.351500
$519$	−2.62411	−0.115186
$520$	−1.72128	−0.0754829
$521$	10.1300	0.443802	0.221901	−	0.975069i	$-0.428774\pi$
$521$	10.1300	0.443802	0.221901	+	0.975069i	$0.428774\pi$
$522$	−19.2995	−0.844716
$523$	7.93502	0.346974	0.173487	−	0.984836i	$-0.444497\pi$
$523$	7.93502	0.346974	0.173487	+	0.984836i	$0.444497\pi$
$524$	−1.85695	−0.0811211
$525$	9.18894	0.401038
$526$	20.5665	0.896742
$527$	−23.7789	−1.03582
$528$	0	0
$529$	23.3129	1.01360
$530$	5.46599	0.237427
$531$	34.8851	1.51388
$532$	−3.40268	−0.147525
$533$	−4.13567	−0.179136
$534$	−11.5839	−0.501286
$535$	5.33937	0.230841
$536$	−3.40268	−0.146973
$537$	−0.240529	−0.0103796
$538$	1.36281	0.0587548
$539$	0	0
$540$	−5.12395	−0.220500
$541$	−10.8436	−0.466201	−0.233100	−	0.972453i	$-0.574887\pi$
$541$	−10.8436	−0.466201	−0.233100	+	0.972453i	$0.574887\pi$
$542$	−23.1963	−0.996367
$543$	−11.8439	−0.508269
$544$	−4.80536	−0.206028
$545$	−7.36281	−0.315388
$546$	3.68878	0.157865
$547$	10.3129	0.440947	0.220474	−	0.975393i	$-0.429240\pi$
$547$	10.3129	0.440947	0.220474	+	0.975393i	$0.429240\pi$
$548$	10.5973	0.452695
$549$	4.80536	0.205088
$550$	0	0
$551$	−8.03249	−0.342196
$552$	−5.25963	−0.223865
$553$	−12.6372	−0.537388
$554$	31.0385	1.31870
$555$	2.22980	0.0946496
$556$	−14.7347	−0.624889
$557$	14.4811	0.613582	0.306791	−	0.951777i	$-0.400745\pi$
$557$	14.4811	0.613582	0.306791	+	0.951777i	$0.400745\pi$
$558$	−11.8894	−0.503320
$559$	12.7980	0.541297
$560$	−4.17554	−0.176449
$561$	0	0
$562$	−2.77287	−0.116966
$563$	−10.5340	−0.443956	−0.221978	−	0.975052i	$-0.571251\pi$
$563$	−10.5340	−0.443956	−0.221978	+	0.975052i	$0.571251\pi$
$564$	4.06498	0.171167
$565$	−24.4958	−1.03055
$566$	−9.23451	−0.388156
$567$	−13.5457	−0.568867
$568$	7.57822	0.317975
$569$	−10.3762	−0.434993	−0.217496	−	0.976061i	$-0.569789\pi$
$569$	−10.3762	−0.434993	−0.217496	+	0.976061i	$0.569789\pi$
$570$	−0.948410	−0.0397245
$571$	−33.2847	−1.39292	−0.696461	−	0.717594i	$-0.745243\pi$
$571$	−33.2847	−1.39292	−0.696461	+	0.717594i	$0.745243\pi$
$572$	0	0
$573$	20.7672	0.867561
$574$	−10.0325	−0.418748
$575$	−23.7789	−0.991648
$576$	−2.40268	−0.100112
$577$	−2.93365	−0.122129	−0.0610647	−	0.998134i	$-0.519450\pi$
$577$	−2.93365	−0.122129	−0.0610647	+	0.998134i	$0.519450\pi$
$578$	−6.09146	−0.253371
$579$	−14.3494	−0.596341
$580$	−9.85695	−0.409287
$581$	36.3055	1.50621
$582$	−1.81708	−0.0753202
$583$	0	0
$584$	−10.0650	−0.416492
$585$	−4.13567	−0.170989
$586$	17.7390	0.732792
$587$	8.90854	0.367695	0.183847	−	0.982955i	$-0.441145\pi$
$587$	8.90854	0.367695	0.183847	+	0.982955i	$0.441145\pi$
$588$	3.53835	0.145919
$589$	−4.94841	−0.203896
$590$	17.8171	0.733517
$591$	−11.5223	−0.473964
$592$	2.35109	0.0966292
$593$	−16.3129	−0.669890	−0.334945	−	0.942238i	$-0.608718\pi$
$593$	−16.3129	−0.669890	−0.334945	+	0.942238i	$0.608718\pi$
$594$	0	0
$595$	−20.0650	−0.822584
$596$	−18.3511	−0.751690
$597$	11.7139	0.479418
$598$	−9.54573	−0.390354
$599$	40.9826	1.67450	0.837251	−	0.546818i	$-0.184161\pi$
$599$	40.9826	1.67450	0.837251	+	0.546818i	$0.184161\pi$
$600$	2.70050	0.110248
$601$	−0.349413	−0.0142528	−0.00712642	−	0.999975i	$-0.502268\pi$
$601$	−0.349413	−0.0142528	−0.00712642	+	0.999975i	$0.502268\pi$
$602$	31.0459	1.26534
$603$	−8.17554	−0.332934
$604$	7.64891	0.311230
$605$	0	0
$606$	10.8703	0.441577
$607$	−5.56049	−0.225693	−0.112847	−	0.993612i	$-0.535997\pi$
$607$	−5.56049	−0.225693	−0.112847	+	0.993612i	$0.535997\pi$
$608$	−1.00000	−0.0405554
$609$	21.1240	0.855986
$610$	2.45427	0.0993704
$611$	7.37757	0.298464
$612$	−11.5457	−0.466709
$613$	19.7407	0.797319	0.398659	−	0.917099i	$-0.369476\pi$
$613$	19.7407	0.797319	0.398659	+	0.917099i	$0.369476\pi$
$614$	20.9735	0.846422
$615$	−2.79630	−0.112758
$616$	0	0
$617$	10.5973	0.426632	0.213316	−	0.976983i	$-0.431574\pi$
$617$	10.5973	0.426632	0.213316	+	0.976983i	$0.431574\pi$
$618$	12.0399	0.484315
$619$	4.22112	0.169661	0.0848306	−	0.996395i	$-0.472965\pi$
$619$	4.22112	0.169661	0.0848306	+	0.996395i	$0.472965\pi$
$620$	−6.07236	−0.243872
$621$	−28.4161	−1.14030
$622$	3.36281	0.134836
$623$	−51.0003	−2.04328
$624$	1.08408	0.0433980
$625$	4.67973	0.187189
$626$	4.94103	0.197483
$627$	0	0
$628$	8.38358	0.334541
$629$	11.2978	0.450474
$630$	−10.0325	−0.399704
$631$	13.6905	0.545009	0.272504	−	0.962155i	$-0.412148\pi$
$631$	13.6905	0.545009	0.272504	+	0.962155i	$0.412148\pi$
$632$	−3.71390	−0.147731
$633$	−8.62243	−0.342711
$634$	−14.2861	−0.567374
$635$	−14.3745	−0.570436
$636$	−3.44255	−0.136506
$637$	6.42178	0.254440
$638$	0	0
$639$	18.2080	0.720299
$640$	−1.22713	−0.0485067
$641$	−42.9883	−1.69794	−0.848968	−	0.528445i	$-0.822775\pi$
$641$	−42.9883	−1.69794	−0.848968	+	0.528445i	$0.822775\pi$
$642$	−3.36281	−0.132719
$643$	16.3658	0.645406	0.322703	−	0.946500i	$-0.395408\pi$
$643$	16.3658	0.645406	0.322703	+	0.946500i	$0.395408\pi$
$644$	−23.1564	−0.912492
$645$	8.65325	0.340721
$646$	−4.80536	−0.189064
$647$	39.9886	1.57211	0.786057	−	0.618154i	$-0.212119\pi$
$647$	39.9886	1.57211	0.786057	+	0.618154i	$0.212119\pi$
$648$	−3.98090	−0.156385
$649$	0	0
$650$	4.90116	0.192239
$651$	13.0134	0.510035
$652$	−14.2479	−0.557991
$653$	−37.5976	−1.47131	−0.735655	−	0.677357i	$-0.763125\pi$
$653$	−37.5976	−1.47131	−0.735655	+	0.677357i	$0.763125\pi$
$654$	4.63719	0.181329
$655$	−2.27872	−0.0890371
$656$	−2.94841	−0.115116
$657$	−24.1829	−0.943466
$658$	17.8968	0.697691
$659$	11.5075	0.448270	0.224135	−	0.974558i	$-0.428044\pi$
$659$	11.5075	0.448270	0.224135	+	0.974558i	$0.428044\pi$
$660$	0	0
$661$	−29.2362	−1.13716	−0.568578	−	0.822629i	$-0.692506\pi$
$661$	−29.2362	−1.13716	−0.568578	+	0.822629i	$0.692506\pi$
$662$	31.9293	1.24097
$663$	5.20940	0.202316
$664$	10.6697	0.414064
$665$	−4.17554	−0.161921
$666$	5.64891	0.218891
$667$	−54.6640	−2.11660
$668$	10.8703	0.420586
$669$	1.24487	0.0481293
$670$	−4.17554	−0.161315
$671$	0	0
$672$	2.62981	0.101447
$673$	47.0784	1.81474	0.907369	−	0.420335i	$-0.138087\pi$
$673$	47.0784	1.81474	0.907369	+	0.420335i	$0.138087\pi$
$674$	−8.84523	−0.340706
$675$	14.5899	0.561567
$676$	−11.0325	−0.424327
$677$	−43.2539	−1.66238	−0.831192	−	0.555986i	$-0.812341\pi$
$677$	−43.2539	−1.66238	−0.831192	+	0.555986i	$0.812341\pi$
$678$	15.4278	0.592501
$679$	−8.00000	−0.307012
$680$	−5.89682	−0.226133
$681$	12.9588	0.496581
$682$	0	0
$683$	34.8556	1.33371	0.666856	−	0.745187i	$-0.267640\pi$
$683$	34.8556	1.33371	0.666856	+	0.745187i	$0.267640\pi$
$684$	−2.40268	−0.0918687
$685$	13.0043	0.496870
$686$	−8.24053	−0.314625
$687$	−6.61505	−0.252380
$688$	9.12395	0.347847
$689$	−6.24791	−0.238026
$690$	−6.45427	−0.245710
$691$	28.4811	1.08347	0.541735	−	0.840549i	$-0.317768\pi$
$691$	28.4811	1.08347	0.541735	+	0.840549i	$0.317768\pi$
$692$	−3.39530	−0.129070
$693$	0	0
$694$	5.27439	0.200213
$695$	−18.0814	−0.685867
$696$	6.20804	0.235315
$697$	−14.1682	−0.536657
$698$	−10.9085	−0.412894
$699$	−4.20636	−0.159099
$700$	11.8894	0.449379
$701$	−12.8703	−0.486106	−0.243053	−	0.970013i	$-0.578149\pi$
$701$	−12.8703	−0.486106	−0.243053	+	0.970013i	$0.578149\pi$
$702$	5.85695	0.221056
$703$	2.35109	0.0886730
$704$	0	0
$705$	4.98828	0.187870
$706$	22.1300	0.832872
$707$	47.8586	1.79991
$708$	−11.2214	−0.421727
$709$	1.41744	0.0532330	0.0266165	−	0.999646i	$-0.491527\pi$
$709$	1.41744	0.0532330	0.0266165	+	0.999646i	$0.491527\pi$
$710$	9.29950	0.349004
$711$	−8.92330	−0.334650
$712$	−14.9883	−0.561710
$713$	−33.6757	−1.26116
$714$	12.6372	0.472935
$715$	0	0
$716$	−0.311217	−0.0116307
$717$	−8.72729	−0.325927
$718$	−16.3910	−0.611705
$719$	−31.1564	−1.16194	−0.580970	−	0.813925i	$-0.697327\pi$
$719$	−31.1564	−1.16194	−0.580970	+	0.813925i	$0.697327\pi$
$720$	−2.94841	−0.109881
$721$	53.0077	1.97411
$722$	−1.00000	−0.0372161
$723$	−8.81844	−0.327961
$724$	−15.3246	−0.569535
$725$	28.0667	1.04237
$726$	0	0
$727$	−8.57221	−0.317926	−0.158963	−	0.987285i	$-0.550815\pi$
$727$	−8.57221	−0.317926	−0.158963	+	0.987285i	$0.550815\pi$
$728$	4.77287	0.176894
$729$	0.116574	0.00431757
$730$	−12.3511	−0.457134
$731$	43.8439	1.62162
$732$	−1.54573	−0.0571319
$733$	−44.6787	−1.65025	−0.825123	−	0.564952i	$-0.808895\pi$
$733$	−44.6787	−1.65025	−0.825123	+	0.564952i	$0.808895\pi$
$734$	32.3511	1.19410
$735$	4.34203	0.160158
$736$	−6.80536	−0.250849
$737$	0	0
$738$	−7.08408	−0.260769
$739$	−29.4898	−1.08480	−0.542400	−	0.840120i	$-0.682484\pi$
$739$	−29.4898	−1.08480	−0.542400	+	0.840120i	$0.682484\pi$
$740$	2.88510	0.106058
$741$	1.08408	0.0398248
$742$	−15.1564	−0.556411
$743$	13.2596	0.486449	0.243224	−	0.969970i	$-0.421795\pi$
$743$	13.2596	0.486449	0.243224	+	0.969970i	$0.421795\pi$
$744$	3.82446	0.140211
$745$	−22.5193	−0.825042
$746$	17.5782	0.643584
$747$	25.6358	0.937966
$748$	0	0
$749$	−14.8054	−0.540976
$750$	8.05593	0.294161
$751$	−10.1829	−0.371580	−0.185790	−	0.982589i	$-0.559484\pi$
$751$	−10.1829	−0.371580	−0.185790	+	0.982589i	$0.559484\pi$
$752$	5.25963	0.191799
$753$	−9.27439	−0.337977
$754$	11.2670	0.410320
$755$	9.38624	0.341600
$756$	14.2080	0.516741
$757$	40.1122	1.45790	0.728952	−	0.684565i	$-0.240008\pi$
$757$	40.1122	1.45790	0.728952	+	0.684565i	$0.240008\pi$
$758$	−8.93365	−0.324485
$759$	0	0
$760$	−1.22713	−0.0445128
$761$	38.7819	1.40584	0.702922	−	0.711267i	$-0.251878\pi$
$761$	38.7819	1.40584	0.702922	+	0.711267i	$0.251878\pi$
$762$	9.05327	0.327965
$763$	20.4161	0.739111
$764$	26.8703	0.972135
$765$	−14.1682	−0.512251
$766$	16.0399	0.579544
$767$	−20.3658	−0.735368
$768$	0.772866	0.0278884
$769$	−8.15340	−0.294019	−0.147010	−	0.989135i	$-0.546965\pi$
$769$	−8.15340	−0.294019	−0.147010	+	0.989135i	$0.546965\pi$
$770$	0	0
$771$	18.7290	0.674507
$772$	−18.5665	−0.668223
$773$	−24.3893	−0.877222	−0.438611	−	0.898677i	$-0.644529\pi$
$773$	−24.3893	−0.877222	−0.438611	+	0.898677i	$0.644529\pi$
$774$	21.9219	0.787968
$775$	17.2904	0.621091
$776$	−2.35109	−0.0843992
$777$	−6.18292	−0.221811
$778$	−17.2271	−0.617623
$779$	−2.94841	−0.105638
$780$	1.33031	0.0476329
$781$	0	0
$782$	−32.7022	−1.16943
$783$	33.5400	1.19862
$784$	4.57822	0.163508
$785$	10.2878	0.367187
$786$	1.43517	0.0511909
$787$	46.3779	1.65319	0.826596	−	0.562795i	$-0.190274\pi$
$787$	46.3779	1.65319	0.826596	+	0.562795i	$0.190274\pi$
$788$	−14.9085	−0.531095
$789$	−15.8951	−0.565882
$790$	−4.55745	−0.162147
$791$	67.9236	2.41509
$792$	0	0
$793$	−2.80536	−0.0996212
$794$	−15.0784	−0.535112
$795$	−4.22447	−0.149827
$796$	15.1564	0.537206
$797$	3.89682	0.138032	0.0690162	−	0.997616i	$-0.478014\pi$
$797$	3.89682	0.138032	0.0690162	+	0.997616i	$0.478014\pi$
$798$	2.62981	0.0930944
$799$	25.2744	0.894144
$800$	3.49414	0.123537
$801$	−36.0120	−1.27242
$802$	13.1564	0.464570
$803$	0	0
$804$	2.62981	0.0927464
$805$	−28.4161	−1.00153
$806$	6.94103	0.244487
$807$	−1.05327	−0.0370767
$808$	14.0650	0.494804
$809$	43.3896	1.52550	0.762748	−	0.646695i	$-0.223849\pi$
$809$	43.3896	1.52550	0.762748	+	0.646695i	$0.223849\pi$
$810$	−4.88510	−0.171645
$811$	12.9233	0.453798	0.226899	−	0.973918i	$-0.427141\pi$
$811$	12.9233	0.453798	0.226899	+	0.973918i	$0.427141\pi$
$812$	27.3320	0.959165
$813$	17.9276	0.628750
$814$	0	0
$815$	−17.4841	−0.612441
$816$	3.71390	0.130012
$817$	9.12395	0.319207
$818$	−16.3836	−0.572838
$819$	11.4677	0.400713
$820$	−3.61810	−0.126349
$821$	36.5842	1.27680	0.638399	−	0.769705i	$-0.279597\pi$
$821$	36.5842	1.27680	0.638399	+	0.769705i	$0.279597\pi$
$822$	−8.19030	−0.285670
$823$	−21.6107	−0.753302	−0.376651	−	0.926355i	$-0.622924\pi$
$823$	−21.6107	−0.753302	−0.376651	+	0.926355i	$0.622924\pi$
$824$	15.5782	0.542693
$825$	0	0
$826$	−49.4044	−1.71900
$827$	34.8851	1.21307	0.606537	−	0.795055i	$-0.292558\pi$
$827$	34.8851	1.21307	0.606537	+	0.795055i	$0.292558\pi$
$828$	−16.3511	−0.568240
$829$	31.7407	1.10240	0.551200	−	0.834373i	$-0.314170\pi$
$829$	31.7407	1.10240	0.551200	+	0.834373i	$0.314170\pi$
$830$	13.0931	0.454469
$831$	−23.9886	−0.832155
$832$	1.40268	0.0486291
$833$	22.0000	0.762255
$834$	11.3879	0.394331
$835$	13.3394	0.461628
$836$	0	0
$837$	20.6623	0.714194
$838$	−1.12966	−0.0390234
$839$	13.4603	0.464701	0.232350	−	0.972632i	$-0.425358\pi$
$839$	13.4603	0.464701	0.232350	+	0.972632i	$0.425358\pi$
$840$	3.22713	0.111347
$841$	35.5209	1.22486
$842$	2.63719	0.0908837
$843$	2.14305	0.0738106
$844$	−11.1564	−0.384021
$845$	−13.5384	−0.465733
$846$	12.6372	0.434476
$847$	0	0
$848$	−4.45427	−0.152960
$849$	7.13704	0.244943
$850$	16.7906	0.575913
$851$	16.0000	0.548473
$852$	−5.85695	−0.200656
$853$	−36.2981	−1.24282	−0.621412	−	0.783484i	$-0.713441\pi$
$853$	−36.2981	−1.24282	−0.621412	+	0.783484i	$0.713441\pi$
$854$	−6.80536	−0.232875
$855$	−2.94841	−0.100833
$856$	−4.35109	−0.148717
$857$	43.8569	1.49812	0.749062	−	0.662499i	$-0.230504\pi$
$857$	43.8569	1.49812	0.749062	+	0.662499i	$0.230504\pi$
$858$	0	0
$859$	−41.1980	−1.40566	−0.702829	−	0.711359i	$-0.748080\pi$
$859$	−41.1980	−1.40566	−0.702829	+	0.711359i	$0.748080\pi$
$860$	11.1963	0.381791
$861$	7.75377	0.264248
$862$	−28.4958	−0.970571
$863$	−24.8720	−0.846653	−0.423327	−	0.905977i	$-0.639138\pi$
$863$	−24.8720	−0.846653	−0.423327	+	0.905977i	$0.639138\pi$
$864$	4.17554	0.142055
$865$	−4.16649	−0.141665
$866$	37.5075	1.27456
$867$	4.70788	0.159888
$868$	16.8378	0.571514
$869$	0	0
$870$	7.61810	0.258278
$871$	4.77287	0.161722
$872$	6.00000	0.203186
$873$	−5.64891	−0.191187
$874$	−6.80536	−0.230195
$875$	35.4677	1.19903
$876$	7.77888	0.262824
$877$	−39.8911	−1.34703	−0.673514	−	0.739175i	$-0.735216\pi$
$877$	−39.8911	−1.34703	−0.673514	+	0.739175i	$0.735216\pi$
$878$	−32.1300	−1.08433
$879$	−13.7099	−0.462422
$880$	0	0
$881$	−2.88343	−0.0971451	−0.0485725	−	0.998820i	$-0.515467\pi$
$881$	−2.88343	−0.0971451	−0.0485725	+	0.998820i	$0.515467\pi$
$882$	11.0000	0.370389
$883$	−4.27134	−0.143742	−0.0718711	−	0.997414i	$-0.522897\pi$
$883$	−4.27134	−0.143742	−0.0718711	+	0.997414i	$0.522897\pi$
$884$	6.74037	0.226703
$885$	−13.7702	−0.462880
$886$	−33.0385	−1.10995
$887$	−22.3129	−0.749194	−0.374597	−	0.927188i	$-0.622219\pi$
$887$	−22.3129	−0.749194	−0.374597	+	0.927188i	$0.622219\pi$
$888$	−1.81708	−0.0609771
$889$	39.8586	1.33682
$890$	−18.3926	−0.616523
$891$	0	0
$892$	1.61072	0.0539307
$893$	5.25963	0.176007
$894$	14.1829	0.474348
$895$	−0.381905	−0.0127657
$896$	3.40268	0.113676
$897$	7.37757	0.246330
$898$	−7.88206	−0.263028
$899$	39.7481	1.32567
$900$	8.39530	0.279843
$901$	−21.4044	−0.713082
$902$	0	0
$903$	−23.9943	−0.798480
$904$	19.9618	0.663920
$905$	−18.8054	−0.625111
$906$	−5.91158	−0.196399
$907$	−35.5578	−1.18068	−0.590338	−	0.807156i	$-0.701006\pi$
$907$	−35.5578	−1.18068	−0.590338	+	0.807156i	$0.701006\pi$
$908$	16.7672	0.556438
$909$	33.7936	1.12086
$910$	5.85695	0.194156
$911$	−49.1980	−1.63000	−0.815001	−	0.579459i	$-0.803264\pi$
$911$	−49.1980	−1.63000	−0.815001	+	0.579459i	$0.803264\pi$
$912$	0.772866	0.0255921
$913$	0	0
$914$	24.9353	0.824787
$915$	−1.89682	−0.0627069
$916$	−8.55913	−0.282801
$917$	6.31860	0.208658
$918$	20.0650	0.662243
$919$	8.47200	0.279466	0.139733	−	0.990189i	$-0.455376\pi$
$919$	8.47200	0.279466	0.139733	+	0.990189i	$0.455376\pi$
$920$	−8.35109	−0.275327
$921$	−16.2097	−0.534128
$922$	−26.6874	−0.878903
$923$	−10.6298	−0.349885
$924$	0	0
$925$	−8.21504	−0.270109
$926$	32.7022	1.07466
$927$	37.4295	1.22934
$928$	8.03249	0.263679
$929$	−16.0017	−0.524998	−0.262499	−	0.964932i	$-0.584547\pi$
$929$	−16.0017	−0.524998	−0.262499	+	0.964932i	$0.584547\pi$
$930$	4.69312	0.153894
$931$	4.57822	0.150045
$932$	−5.44255	−0.178277
$933$	−2.59900	−0.0850874
$934$	−11.3776	−0.372285
$935$	0	0
$936$	3.37019	0.110158
$937$	15.9766	0.521932	0.260966	−	0.965348i	$-0.415959\pi$
$937$	15.9766	0.521932	0.260966	+	0.965348i	$0.415959\pi$
$938$	11.5782	0.378042
$939$	−3.81875	−0.124620
$940$	6.45427	0.210515
$941$	21.0915	0.687562	0.343781	−	0.939050i	$-0.388292\pi$
$941$	21.0915	0.687562	0.343781	+	0.939050i	$0.388292\pi$
$942$	−6.47938	−0.211110
$943$	−20.0650	−0.653406
$944$	−14.5193	−0.472561
$945$	17.4352	0.567166
$946$	0	0
$947$	20.9735	0.681548	0.340774	−	0.940145i	$-0.389311\pi$
$947$	20.9735	0.681548	0.340774	+	0.940145i	$0.389311\pi$
$948$	2.87034	0.0932244
$949$	14.1179	0.458288
$950$	3.49414	0.113365
$951$	11.0412	0.358037
$952$	16.3511	0.529942
$953$	−3.27439	−0.106068	−0.0530339	−	0.998593i	$-0.516889\pi$
$953$	−3.27439	−0.106068	−0.0530339	+	0.998593i	$0.516889\pi$
$954$	−10.7022	−0.346496
$955$	32.9735	1.06700
$956$	−11.2921	−0.365213
$957$	0	0
$958$	−12.5973	−0.407001
$959$	−36.0593	−1.16441
$960$	0.948410	0.0306098
$961$	−6.51324	−0.210104
$962$	−3.29782	−0.106326
$963$	−10.4543	−0.336884
$964$	−11.4101	−0.367493
$965$	−22.7836	−0.733430
$966$	17.8968	0.575821
$967$	29.2744	0.941401	0.470700	−	0.882293i	$-0.344001\pi$
$967$	29.2744	0.941401	0.470700	+	0.882293i	$0.344001\pi$
$968$	0	0
$969$	3.71390	0.119308
$970$	−2.88510	−0.0926350
$971$	−38.3836	−1.23179	−0.615894	−	0.787829i	$-0.711205\pi$
$971$	−38.3836	−1.23179	−0.615894	+	0.787829i	$0.711205\pi$
$972$	15.6033	0.500477
$973$	50.1373	1.60733
$974$	−9.69616	−0.310685
$975$	−3.78794	−0.121311
$976$	−2.00000	−0.0640184
$977$	10.9233	0.349467	0.174734	−	0.984616i	$-0.444094\pi$
$977$	10.9233	0.349467	0.174734	+	0.984616i	$0.444094\pi$
$978$	11.0117	0.352116
$979$	0	0
$980$	5.61810	0.179463
$981$	14.4161	0.460270
$982$	−11.7538	−0.375078
$983$	−32.1196	−1.02446	−0.512228	−	0.858849i	$-0.671180\pi$
$983$	−32.1196	−1.02446	−0.512228	+	0.858849i	$0.671180\pi$
$984$	2.27872	0.0726431
$985$	−18.2948	−0.582920
$986$	38.5990	1.22924
$987$	−13.8318	−0.440272
$988$	1.40268	0.0446252
$989$	62.0918	1.97440
$990$	0	0
$991$	4.37620	0.139015	0.0695073	−	0.997581i	$-0.477857\pi$
$991$	4.37620	0.139015	0.0695073	+	0.997581i	$0.477857\pi$
$992$	4.94841	0.157112
$993$	−24.6771	−0.783103
$994$	−25.7863	−0.817890
$995$	18.5990	0.589628
$996$	−8.24623	−0.261292
$997$	8.37788	0.265330	0.132665	−	0.991161i	$-0.457647\pi$
$997$	8.37788	0.265330	0.132665	+	0.991161i	$0.457647\pi$
$998$	−26.6640	−0.844034
$999$	−9.81708	−0.310599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bm.1.2		3
11.10	odd	2		418.2.a.h.1.2	✓	3
33.32	even	2		3762.2.a.bd.1.2		3
44.43	even	2		3344.2.a.p.1.2		3
209.208	even	2		7942.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.h.1.2	✓	3	11.10	odd	2
3344.2.a.p.1.2		3	44.43	even	2
3762.2.a.bd.1.2		3	33.32	even	2
4598.2.a.bm.1.2		3	1.1	even	1	trivial
7942.2.a.bc.1.2		3	209.208	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.772866 q^{3} +1.00000 q^{4} +1.22713 q^{5} -0.772866 q^{6} -3.40268 q^{7} -1.00000 q^{8} -2.40268 q^{9} -1.22713 q^{10} +0.772866 q^{12} +1.40268 q^{13} +3.40268 q^{14} +0.948410 q^{15} +1.00000 q^{16} +4.80536 q^{17} +2.40268 q^{18} +1.00000 q^{19} +1.22713 q^{20} -2.62981 q^{21} +6.80536 q^{23} -0.772866 q^{24} -3.49414 q^{25} -1.40268 q^{26} -4.17554 q^{27} -3.40268 q^{28} -8.03249 q^{29} -0.948410 q^{30} -4.94841 q^{31} -1.00000 q^{32} -4.80536 q^{34} -4.17554 q^{35} -2.40268 q^{36} +2.35109 q^{37} -1.00000 q^{38} +1.08408 q^{39} -1.22713 q^{40} -2.94841 q^{41} +2.62981 q^{42} +9.12395 q^{43} -2.94841 q^{45} -6.80536 q^{46} +5.25963 q^{47} +0.772866 q^{48} +4.57822 q^{49} +3.49414 q^{50} +3.71390 q^{51} +1.40268 q^{52} -4.45427 q^{53} +4.17554 q^{54} +3.40268 q^{56} +0.772866 q^{57} +8.03249 q^{58} -14.5193 q^{59} +0.948410 q^{60} -2.00000 q^{61} +4.94841 q^{62} +8.17554 q^{63} +1.00000 q^{64} +1.72128 q^{65} +3.40268 q^{67} +4.80536 q^{68} +5.25963 q^{69} +4.17554 q^{70} -7.57822 q^{71} +2.40268 q^{72} +10.0650 q^{73} -2.35109 q^{74} -2.70050 q^{75} +1.00000 q^{76} -1.08408 q^{78} +3.71390 q^{79} +1.22713 q^{80} +3.98090 q^{81} +2.94841 q^{82} -10.6697 q^{83} -2.62981 q^{84} +5.89682 q^{85} -9.12395 q^{86} -6.20804 q^{87} +14.9883 q^{89} +2.94841 q^{90} -4.77287 q^{91} +6.80536 q^{92} -3.82446 q^{93} -5.25963 q^{94} +1.22713 q^{95} -0.772866 q^{96} +2.35109 q^{97} -4.57822 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} - q^{6} - q^{7} - 3 q^{8} + 2 q^{9} - 5 q^{10} + q^{12} - 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} - 4 q^{17} - 2 q^{18} + 3 q^{19} + 5 q^{20} + 2 q^{23} - q^{24}+ \cdots + 6 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 - q^6 - q^7 - 3 * q^8 + 2 * q^9 - 5 * q^10 + q^12 - 5 * q^13 + q^14 - 9 * q^15 + 3 * q^16 - 4 * q^17 - 2 * q^18 + 3 * q^19 + 5 * q^20 + 2 * q^23 - q^24 + 4 * q^25 + 5 * q^26 - 2 * q^27 - q^28 - 7 * q^29 + 9 * q^30 - 3 * q^31 - 3 * q^32 + 4 * q^34 - 2 * q^35 + 2 * q^36 - 14 * q^37 - 3 * q^38 - 2 * q^39 - 5 * q^40 + 3 * q^41 + 5 * q^43 + 3 * q^45 - 2 * q^46 + q^48 - 6 * q^49 - 4 * q^50 - 2 * q^51 - 5 * q^52 - 16 * q^53 + 2 * q^54 + q^56 + q^57 + 7 * q^58 - 12 * q^59 - 9 * q^60 - 6 * q^61 + 3 * q^62 + 14 * q^63 + 3 * q^64 - 8 * q^65 + q^67 - 4 * q^68 + 2 * q^70 - 3 * q^71 - 2 * q^72 - 4 * q^73 + 14 * q^74 - 41 * q^75 + 3 * q^76 + 2 * q^78 - 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 - 7 * q^83 - 6 * q^85 - 5 * q^86 + 9 * q^87 + 16 * q^89 - 3 * q^90 - 13 * q^91 + 2 * q^92 - 22 * q^93 + 5 * q^95 - q^96 - 14 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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