LMFDB - Embedded newform 4598.2.a.by.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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 $ x^8 - 16x^6 - 4x^5 + 75x^4 + 32x^3 - 90x^2 - 28x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.115899$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} -2.11129 q^{3} +1.00000 q^{4} -2.75070 q^{5} +2.11129 q^{6} -3.66381 q^{7} -1.00000 q^{8} +1.45756 q^{9} +2.75070 q^{10} -2.11129 q^{12} +3.81790 q^{13} +3.66381 q^{14} +5.80753 q^{15} +1.00000 q^{16} +4.31816 q^{17} -1.45756 q^{18} +1.00000 q^{19} -2.75070 q^{20} +7.73537 q^{21} -0.695478 q^{23} +2.11129 q^{24} +2.56634 q^{25} -3.81790 q^{26} +3.25654 q^{27} -3.66381 q^{28} -7.77089 q^{29} -5.80753 q^{30} +0.340036 q^{31} -1.00000 q^{32} -4.31816 q^{34} +10.0780 q^{35} +1.45756 q^{36} -0.692199 q^{37} -1.00000 q^{38} -8.06072 q^{39} +2.75070 q^{40} -9.21705 q^{41} -7.73537 q^{42} -2.01495 q^{43} -4.00932 q^{45} +0.695478 q^{46} -9.80308 q^{47} -2.11129 q^{48} +6.42347 q^{49} -2.56634 q^{50} -9.11690 q^{51} +3.81790 q^{52} -5.54956 q^{53} -3.25654 q^{54} +3.66381 q^{56} -2.11129 q^{57} +7.77089 q^{58} -8.22759 q^{59} +5.80753 q^{60} +0.344547 q^{61} -0.340036 q^{62} -5.34023 q^{63} +1.00000 q^{64} -10.5019 q^{65} -5.04040 q^{67} +4.31816 q^{68} +1.46836 q^{69} -10.0780 q^{70} -3.22075 q^{71} -1.45756 q^{72} -14.6020 q^{73} +0.692199 q^{74} -5.41830 q^{75} +1.00000 q^{76} +8.06072 q^{78} +4.44082 q^{79} -2.75070 q^{80} -11.2482 q^{81} +9.21705 q^{82} -12.0464 q^{83} +7.73537 q^{84} -11.8779 q^{85} +2.01495 q^{86} +16.4066 q^{87} -0.966434 q^{89} +4.00932 q^{90} -13.9881 q^{91} -0.695478 q^{92} -0.717915 q^{93} +9.80308 q^{94} -2.75070 q^{95} +2.11129 q^{96} +16.3170 q^{97} -6.42347 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25}+ \cdots - 34 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^14 + 4 * q^15 + 8 * q^16 + 4 * q^17 - 22 * q^18 + 8 * q^19 + 20 * q^21 + 14 * q^23 - 8 * q^24 + 36 * q^25 - 12 * q^26 + 32 * q^27 - 4 * q^28 + 2 * q^29 - 4 * q^30 - 8 * q^32 - 4 * q^34 - 36 * q^35 + 22 * q^36 + 24 * q^37 - 8 * q^38 - 16 * q^39 - 8 * q^41 - 20 * q^42 - 8 * q^43 + 16 * q^45 - 14 * q^46 - 16 * q^47 + 8 * q^48 + 34 * q^49 - 36 * q^50 - 18 * q^51 + 12 * q^52 + 36 * q^53 - 32 * q^54 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 24 * q^59 + 4 * q^60 - 12 * q^61 - 24 * q^63 + 8 * q^64 - 16 * q^65 + 16 * q^67 + 4 * q^68 + 4 * q^69 + 36 * q^70 + 4 * q^71 - 22 * q^72 + 20 * q^73 - 24 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^78 + 12 * q^79 + 40 * q^81 + 8 * q^82 - 20 * q^83 + 20 * q^84 - 12 * q^85 + 8 * q^86 + 36 * q^87 + 8 * q^89 - 16 * q^90 - 24 * q^91 + 14 * q^92 + 12 * q^93 + 16 * q^94 - 8 * q^96 + 4 * q^97 - 34 * 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
$3$	−2.11129	−1.21896	−0.609478	−	0.792803i	$-0.708621\pi$
$3$	−2.11129	−1.21896	−0.609478	+	0.792803i	$0.708621\pi$
$4$	1.00000	0.500000
$5$	−2.75070	−1.23015	−0.615075	−	0.788469i	$-0.710874\pi$
$5$	−2.75070	−1.23015	−0.615075	+	0.788469i	$0.710874\pi$
$6$	2.11129	0.861932
$7$	−3.66381	−1.38479	−0.692394	−	0.721519i	$-0.743444\pi$
$7$	−3.66381	−1.38479	−0.692394	+	0.721519i	$0.743444\pi$
$8$	−1.00000	−0.353553
$9$	1.45756	0.485855
$10$	2.75070	0.869847
$11$	0	0
$11$	0	0
$12$	−2.11129	−0.609478
$13$	3.81790	1.05890	0.529448	−	0.848342i	$-0.322399\pi$
$13$	3.81790	1.05890	0.529448	+	0.848342i	$0.322399\pi$
$14$	3.66381	0.979193
$15$	5.80753	1.49950
$16$	1.00000	0.250000
$17$	4.31816	1.04731	0.523653	−	0.851931i	$-0.324569\pi$
$17$	4.31816	1.04731	0.523653	+	0.851931i	$0.324569\pi$
$18$	−1.45756	−0.343551
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.75070	−0.615075
$21$	7.73537	1.68800
$22$	0	0
$23$	−0.695478	−0.145017	−0.0725086	−	0.997368i	$-0.523100\pi$
$23$	−0.695478	−0.145017	−0.0725086	+	0.997368i	$0.523100\pi$
$24$	2.11129	0.430966
$25$	2.56634	0.513268
$26$	−3.81790	−0.748753
$27$	3.25654	0.626721
$28$	−3.66381	−0.692394
$29$	−7.77089	−1.44302	−0.721509	−	0.692405i	$-0.756551\pi$
$29$	−7.77089	−1.44302	−0.721509	+	0.692405i	$0.756551\pi$
$30$	−5.80753	−1.06031
$31$	0.340036	0.0610722	0.0305361	−	0.999534i	$-0.490279\pi$
$31$	0.340036	0.0610722	0.0305361	+	0.999534i	$0.490279\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.31816	−0.740558
$35$	10.0780	1.70350
$36$	1.45756	0.242927
$37$	−0.692199	−0.113797	−0.0568984	−	0.998380i	$-0.518121\pi$
$37$	−0.692199	−0.113797	−0.0568984	+	0.998380i	$0.518121\pi$
$38$	−1.00000	−0.162221
$39$	−8.06072	−1.29075
$40$	2.75070	0.434924
$41$	−9.21705	−1.43946	−0.719731	−	0.694253i	$-0.755735\pi$
$41$	−9.21705	−1.43946	−0.719731	+	0.694253i	$0.755735\pi$
$42$	−7.73537	−1.19359
$43$	−2.01495	−0.307277	−0.153638	−	0.988127i	$-0.549099\pi$
$43$	−2.01495	−0.307277	−0.153638	+	0.988127i	$0.549099\pi$
$44$	0	0
$45$	−4.00932	−0.597674
$46$	0.695478	0.102543
$47$	−9.80308	−1.42993	−0.714963	−	0.699162i	$-0.753557\pi$
$47$	−9.80308	−1.42993	−0.714963	+	0.699162i	$0.753557\pi$
$48$	−2.11129	−0.304739
$49$	6.42347	0.917639
$50$	−2.56634	−0.362935
$51$	−9.11690	−1.27662
$52$	3.81790	0.529448
$53$	−5.54956	−0.762290	−0.381145	−	0.924515i	$-0.624470\pi$
$53$	−5.54956	−0.762290	−0.381145	+	0.924515i	$0.624470\pi$
$54$	−3.25654	−0.443158
$55$	0	0
$56$	3.66381	0.489597
$57$	−2.11129	−0.279648
$58$	7.77089	1.02037
$59$	−8.22759	−1.07114	−0.535570	−	0.844491i	$-0.679903\pi$
$59$	−8.22759	−1.07114	−0.535570	+	0.844491i	$0.679903\pi$
$60$	5.80753	0.749749
$61$	0.344547	0.0441148	0.0220574	−	0.999757i	$-0.492978\pi$
$61$	0.344547	0.0441148	0.0220574	+	0.999757i	$0.492978\pi$
$62$	−0.340036	−0.0431846
$63$	−5.34023	−0.672806
$64$	1.00000	0.125000
$65$	−10.5019	−1.30260
$66$	0	0
$67$	−5.04040	−0.615783	−0.307892	−	0.951421i	$-0.599623\pi$
$67$	−5.04040	−0.615783	−0.307892	+	0.951421i	$0.599623\pi$
$68$	4.31816	0.523653
$69$	1.46836	0.176770
$70$	−10.0780	−1.20455
$71$	−3.22075	−0.382233	−0.191117	−	0.981567i	$-0.561211\pi$
$71$	−3.22075	−0.382233	−0.191117	+	0.981567i	$0.561211\pi$
$72$	−1.45756	−0.171776
$73$	−14.6020	−1.70903	−0.854517	−	0.519423i	$-0.826147\pi$
$73$	−14.6020	−1.70903	−0.854517	+	0.519423i	$0.826147\pi$
$74$	0.692199	0.0804666
$75$	−5.41830	−0.625651
$76$	1.00000	0.114708
$77$	0	0
$78$	8.06072	0.912697
$79$	4.44082	0.499631	0.249816	−	0.968293i	$-0.419630\pi$
$79$	4.44082	0.499631	0.249816	+	0.968293i	$0.419630\pi$
$80$	−2.75070	−0.307537
$81$	−11.2482	−1.24980
$82$	9.21705	1.01785
$83$	−12.0464	−1.32227	−0.661133	−	0.750269i	$-0.729924\pi$
$83$	−12.0464	−1.32227	−0.661133	+	0.750269i	$0.729924\pi$
$84$	7.73537	0.843998
$85$	−11.8779	−1.28834
$86$	2.01495	0.217277
$87$	16.4066	1.75898
$88$	0	0
$89$	−0.966434	−0.102442	−0.0512209	−	0.998687i	$-0.516311\pi$
$89$	−0.966434	−0.102442	−0.0512209	+	0.998687i	$0.516311\pi$
$90$	4.00932	0.422619
$91$	−13.9881	−1.46635
$92$	−0.695478	−0.0725086
$93$	−0.717915	−0.0744443
$94$	9.80308	1.01111
$95$	−2.75070	−0.282216
$96$	2.11129	0.215483
$97$	16.3170	1.65674	0.828369	−	0.560183i	$-0.189269\pi$
$97$	16.3170	1.65674	0.828369	+	0.560183i	$0.189269\pi$
$98$	−6.42347	−0.648869
$99$	0	0
$100$	2.56634	0.256634
$101$	14.1398	1.40696	0.703479	−	0.710716i	$-0.251629\pi$
$101$	14.1398	1.40696	0.703479	+	0.710716i	$0.251629\pi$
$102$	9.11690	0.902708
$103$	0.880175	0.0867262	0.0433631	−	0.999059i	$-0.486193\pi$
$103$	0.880175	0.0867262	0.0433631	+	0.999059i	$0.486193\pi$
$104$	−3.81790	−0.374376
$105$	−21.2777	−2.07649
$106$	5.54956	0.539020
$107$	1.76713	0.170835	0.0854174	−	0.996345i	$-0.472778\pi$
$107$	1.76713	0.170835	0.0854174	+	0.996345i	$0.472778\pi$
$108$	3.25654	0.313360
$109$	−11.6553	−1.11638	−0.558190	−	0.829713i	$-0.688504\pi$
$109$	−11.6553	−1.11638	−0.558190	+	0.829713i	$0.688504\pi$
$110$	0	0
$111$	1.46144	0.138713
$112$	−3.66381	−0.346197
$113$	19.9467	1.87643	0.938215	−	0.346054i	$-0.112478\pi$
$113$	19.9467	1.87643	0.938215	+	0.346054i	$0.112478\pi$
$114$	2.11129	0.197741
$115$	1.91305	0.178393
$116$	−7.77089	−0.721509
$117$	5.56484	0.514470
$118$	8.22759	0.757411
$119$	−15.8209	−1.45030
$120$	−5.80753	−0.530153
$121$	0	0
$122$	−0.344547	−0.0311938
$123$	19.4599	1.75464
$124$	0.340036	0.0305361
$125$	6.69426	0.598753
$126$	5.34023	0.475746
$127$	0.568592	0.0504544	0.0252272	−	0.999682i	$-0.491969\pi$
$127$	0.568592	0.0504544	0.0252272	+	0.999682i	$0.491969\pi$
$128$	−1.00000	−0.0883883
$129$	4.25415	0.374557
$130$	10.5019	0.921078
$131$	−11.8544	−1.03572	−0.517862	−	0.855464i	$-0.673272\pi$
$131$	−11.8544	−1.03572	−0.517862	+	0.855464i	$0.673272\pi$
$132$	0	0
$133$	−3.66381	−0.317692
$134$	5.04040	0.435424
$135$	−8.95775	−0.770960
$136$	−4.31816	−0.370279
$137$	11.2562	0.961679	0.480840	−	0.876809i	$-0.340332\pi$
$137$	11.2562	0.961679	0.480840	+	0.876809i	$0.340332\pi$
$138$	−1.46836	−0.124995
$139$	−1.69259	−0.143564	−0.0717820	−	0.997420i	$-0.522869\pi$
$139$	−1.69259	−0.143564	−0.0717820	+	0.997420i	$0.522869\pi$
$140$	10.0780	0.851749
$141$	20.6972	1.74302
$142$	3.22075	0.270280
$143$	0	0
$144$	1.45756	0.121464
$145$	21.3754	1.77513
$146$	14.6020	1.20847
$147$	−13.5618	−1.11856
$148$	−0.692199	−0.0568984
$149$	21.5429	1.76487	0.882433	−	0.470438i	$-0.155904\pi$
$149$	21.5429	1.76487	0.882433	+	0.470438i	$0.155904\pi$
$150$	5.41830	0.442402
$151$	−19.9995	−1.62754	−0.813768	−	0.581190i	$-0.802587\pi$
$151$	−19.9995	−1.62754	−0.813768	+	0.581190i	$0.802587\pi$
$152$	−1.00000	−0.0811107
$153$	6.29399	0.508839
$154$	0	0
$155$	−0.935335	−0.0751279
$156$	−8.06072	−0.645374
$157$	6.94839	0.554542	0.277271	−	0.960792i	$-0.410570\pi$
$157$	6.94839	0.554542	0.277271	+	0.960792i	$0.410570\pi$
$158$	−4.44082	−0.353293
$159$	11.7167	0.929198
$160$	2.75070	0.217462
$161$	2.54810	0.200818
$162$	11.2482	0.883742
$163$	16.0104	1.25403	0.627014	−	0.779008i	$-0.284277\pi$
$163$	16.0104	1.25403	0.627014	+	0.779008i	$0.284277\pi$
$164$	−9.21705	−0.719731
$165$	0	0
$166$	12.0464	0.934983
$167$	−3.68048	−0.284804	−0.142402	−	0.989809i	$-0.545483\pi$
$167$	−3.68048	−0.284804	−0.142402	+	0.989809i	$0.545483\pi$
$168$	−7.73537	−0.596797
$169$	1.57639	0.121261
$170$	11.8779	0.910997
$171$	1.45756	0.111463
$172$	−2.01495	−0.153638
$173$	12.9528	0.984782	0.492391	−	0.870374i	$-0.336123\pi$
$173$	12.9528	0.984782	0.492391	+	0.870374i	$0.336123\pi$
$174$	−16.4066	−1.24378
$175$	−9.40257	−0.710768
$176$	0	0
$177$	17.3709	1.30567
$178$	0.966434	0.0724373
$179$	−13.7864	−1.03044	−0.515222	−	0.857056i	$-0.672291\pi$
$179$	−13.7864	−1.03044	−0.515222	+	0.857056i	$0.672291\pi$
$180$	−4.00932	−0.298837
$181$	−15.2785	−1.13564	−0.567820	−	0.823153i	$-0.692213\pi$
$181$	−15.2785	−1.13564	−0.567820	+	0.823153i	$0.692213\pi$
$182$	13.9881	1.03686
$183$	−0.727441	−0.0537740
$184$	0.695478	0.0512713
$185$	1.90403	0.139987
$186$	0.717915	0.0526401
$187$	0	0
$188$	−9.80308	−0.714963
$189$	−11.9313	−0.867876
$190$	2.75070	0.199557
$191$	23.9456	1.73265	0.866323	−	0.499485i	$-0.166477\pi$
$191$	23.9456	1.73265	0.866323	+	0.499485i	$0.166477\pi$
$192$	−2.11129	−0.152370
$193$	−27.1307	−1.95291	−0.976456	−	0.215715i	$-0.930792\pi$
$193$	−27.1307	−1.95291	−0.976456	+	0.215715i	$0.930792\pi$
$194$	−16.3170	−1.17149
$195$	22.1726	1.58781
$196$	6.42347	0.458820
$197$	−21.1960	−1.51015	−0.755076	−	0.655637i	$-0.772400\pi$
$197$	−21.1960	−1.51015	−0.755076	+	0.655637i	$0.772400\pi$
$198$	0	0
$199$	−3.03824	−0.215375	−0.107688	−	0.994185i	$-0.534345\pi$
$199$	−3.03824	−0.215375	−0.107688	+	0.994185i	$0.534345\pi$
$200$	−2.56634	−0.181468
$201$	10.6418	0.750613
$202$	−14.1398	−0.994870
$203$	28.4710	1.99827
$204$	−9.11690	−0.638311
$205$	25.3533	1.77075
$206$	−0.880175	−0.0613247
$207$	−1.01370	−0.0704572
$208$	3.81790	0.264724
$209$	0	0
$210$	21.2777	1.46830
$211$	−27.4130	−1.88719	−0.943593	−	0.331108i	$-0.892578\pi$
$211$	−27.4130	−1.88719	−0.943593	+	0.331108i	$0.892578\pi$
$212$	−5.54956	−0.381145
$213$	6.79996	0.465925
$214$	−1.76713	−0.120799
$215$	5.54251	0.377996
$216$	−3.25654	−0.221579
$217$	−1.24582	−0.0845721
$218$	11.6553	0.789399
$219$	30.8291	2.08324
$220$	0	0
$221$	16.4863	1.10899
$222$	−1.46144	−0.0980852
$223$	18.7060	1.25265	0.626323	−	0.779564i	$-0.284559\pi$
$223$	18.7060	1.25265	0.626323	+	0.779564i	$0.284559\pi$
$224$	3.66381	0.244798
$225$	3.74060	0.249374
$226$	−19.9467	−1.32684
$227$	−12.6689	−0.840862	−0.420431	−	0.907324i	$-0.638121\pi$
$227$	−12.6689	−0.840862	−0.420431	+	0.907324i	$0.638121\pi$
$228$	−2.11129	−0.139824
$229$	22.6558	1.49714	0.748568	−	0.663058i	$-0.230742\pi$
$229$	22.6558	1.49714	0.748568	+	0.663058i	$0.230742\pi$
$230$	−1.91305	−0.126143
$231$	0	0
$232$	7.77089	0.510184
$233$	−20.0819	−1.31561	−0.657805	−	0.753189i	$-0.728515\pi$
$233$	−20.0819	−1.31561	−0.657805	+	0.753189i	$0.728515\pi$
$234$	−5.56484	−0.363785
$235$	26.9653	1.75902
$236$	−8.22759	−0.535570
$237$	−9.37587	−0.609028
$238$	15.8209	1.02552
$239$	−19.7417	−1.27699	−0.638493	−	0.769627i	$-0.720442\pi$
$239$	−19.7417	−1.27699	−0.638493	+	0.769627i	$0.720442\pi$
$240$	5.80753	0.374875
$241$	20.0092	1.28890	0.644452	−	0.764645i	$-0.277085\pi$
$241$	20.0092	1.28890	0.644452	+	0.764645i	$0.277085\pi$
$242$	0	0
$243$	13.9787	0.896731
$244$	0.344547	0.0220574
$245$	−17.6690	−1.12883
$246$	−19.4599	−1.24072
$247$	3.81790	0.242927
$248$	−0.340036	−0.0215923
$249$	25.4335	1.61178
$250$	−6.69426	−0.423382
$251$	−28.0383	−1.76976	−0.884880	−	0.465818i	$-0.845760\pi$
$251$	−28.0383	−1.76976	−0.884880	+	0.465818i	$0.845760\pi$
$252$	−5.34023	−0.336403
$253$	0	0
$254$	−0.568592	−0.0356766
$255$	25.0778	1.57044
$256$	1.00000	0.0625000
$257$	−13.5564	−0.845628	−0.422814	−	0.906217i	$-0.638958\pi$
$257$	−13.5564	−0.845628	−0.422814	+	0.906217i	$0.638958\pi$
$258$	−4.25415	−0.264852
$259$	2.53608	0.157585
$260$	−10.5019	−0.651300
$261$	−11.3266	−0.701097
$262$	11.8544	0.732367
$263$	4.42685	0.272971	0.136486	−	0.990642i	$-0.456419\pi$
$263$	4.42685	0.272971	0.136486	+	0.990642i	$0.456419\pi$
$264$	0	0
$265$	15.2652	0.937731
$266$	3.66381	0.224642
$267$	2.04043	0.124872
$268$	−5.04040	−0.307892
$269$	−7.98617	−0.486926	−0.243463	−	0.969910i	$-0.578283\pi$
$269$	−7.98617	−0.486926	−0.243463	+	0.969910i	$0.578283\pi$
$270$	8.95775	0.545151
$271$	−17.9453	−1.09010	−0.545048	−	0.838405i	$-0.683489\pi$
$271$	−17.9453	−1.09010	−0.545048	+	0.838405i	$0.683489\pi$
$272$	4.31816	0.261827
$273$	29.5329	1.78741
$274$	−11.2562	−0.680010
$275$	0	0
$276$	1.46836	0.0883848
$277$	21.4997	1.29179	0.645896	−	0.763425i	$-0.276484\pi$
$277$	21.4997	1.29179	0.645896	+	0.763425i	$0.276484\pi$
$278$	1.69259	0.101515
$279$	0.495624	0.0296722
$280$	−10.0780	−0.602277
$281$	−19.5998	−1.16923	−0.584614	−	0.811311i	$-0.698754\pi$
$281$	−19.5998	−1.16923	−0.584614	+	0.811311i	$0.698754\pi$
$282$	−20.6972	−1.23250
$283$	−13.0294	−0.774518	−0.387259	−	0.921971i	$-0.626578\pi$
$283$	−13.0294	−0.774518	−0.387259	+	0.921971i	$0.626578\pi$
$284$	−3.22075	−0.191117
$285$	5.80753	0.344009
$286$	0	0
$287$	33.7695	1.99335
$288$	−1.45756	−0.0858878
$289$	1.64648	0.0968515
$290$	−21.3754	−1.25520
$291$	−34.4499	−2.01949
$292$	−14.6020	−0.854517
$293$	12.4424	0.726895	0.363448	−	0.931615i	$-0.381600\pi$
$293$	12.4424	0.726895	0.363448	+	0.931615i	$0.381600\pi$
$294$	13.5618	0.790943
$295$	22.6316	1.31766
$296$	0.692199	0.0402333
$297$	0	0
$298$	−21.5429	−1.24795
$299$	−2.65527	−0.153558
$300$	−5.41830	−0.312826
$301$	7.38237	0.425513
$302$	19.9995	1.15084
$303$	−29.8532	−1.71502
$304$	1.00000	0.0573539
$305$	−0.947746	−0.0542678
$306$	−6.29399	−0.359803
$307$	18.8124	1.07368	0.536840	−	0.843684i	$-0.319618\pi$
$307$	18.8124	1.07368	0.536840	+	0.843684i	$0.319618\pi$
$308$	0	0
$309$	−1.85831	−0.105715
$310$	0.935335	0.0531235
$311$	−13.2168	−0.749455	−0.374727	−	0.927135i	$-0.622264\pi$
$311$	−13.2168	−0.749455	−0.374727	+	0.927135i	$0.622264\pi$
$312$	8.06072	0.456348
$313$	−3.51233	−0.198529	−0.0992644	−	0.995061i	$-0.531649\pi$
$313$	−3.51233	−0.198529	−0.0992644	+	0.995061i	$0.531649\pi$
$314$	−6.94839	−0.392120
$315$	14.6894	0.827652
$316$	4.44082	0.249816
$317$	8.53754	0.479516	0.239758	−	0.970833i	$-0.422932\pi$
$317$	8.53754	0.479516	0.239758	+	0.970833i	$0.422932\pi$
$318$	−11.7167	−0.657042
$319$	0	0
$320$	−2.75070	−0.153769
$321$	−3.73093	−0.208240
$322$	−2.54810	−0.142000
$323$	4.31816	0.240269
$324$	−11.2482	−0.624900
$325$	9.79804	0.543497
$326$	−16.0104	−0.886732
$327$	24.6079	1.36082
$328$	9.21705	0.508927
$329$	35.9166	1.98015
$330$	0	0
$331$	−3.43770	−0.188953	−0.0944766	−	0.995527i	$-0.530118\pi$
$331$	−3.43770	−0.188953	−0.0944766	+	0.995527i	$0.530118\pi$
$332$	−12.0464	−0.661133
$333$	−1.00893	−0.0552888
$334$	3.68048	0.201387
$335$	13.8646	0.757505
$336$	7.73537	0.421999
$337$	−23.8752	−1.30057	−0.650283	−	0.759692i	$-0.725350\pi$
$337$	−23.8752	−1.30057	−0.650283	+	0.759692i	$0.725350\pi$
$338$	−1.57639	−0.0857444
$339$	−42.1134	−2.28729
$340$	−11.8779	−0.644172
$341$	0	0
$342$	−1.45756	−0.0788160
$343$	2.11228	0.114052
$344$	2.01495	0.108639
$345$	−4.03901	−0.217453
$346$	−12.9528	−0.696346
$347$	−7.32500	−0.393227	−0.196613	−	0.980481i	$-0.562994\pi$
$347$	−7.32500	−0.393227	−0.196613	+	0.980481i	$0.562994\pi$
$348$	16.4066	0.879488
$349$	−8.11645	−0.434464	−0.217232	−	0.976120i	$-0.569703\pi$
$349$	−8.11645	−0.434464	−0.217232	+	0.976120i	$0.569703\pi$
$350$	9.40257	0.502589
$351$	12.4331	0.663632
$352$	0	0
$353$	−5.08395	−0.270591	−0.135296	−	0.990805i	$-0.543198\pi$
$353$	−5.08395	−0.270591	−0.135296	+	0.990805i	$0.543198\pi$
$354$	−17.3709	−0.923251
$355$	8.85932	0.470204
$356$	−0.966434	−0.0512209
$357$	33.4026	1.76785
$358$	13.7864	0.728635
$359$	−15.2156	−0.803050	−0.401525	−	0.915848i	$-0.631520\pi$
$359$	−15.2156	−0.803050	−0.401525	+	0.915848i	$0.631520\pi$
$360$	4.00932	0.211310
$361$	1.00000	0.0526316
$362$	15.2785	0.803019
$363$	0	0
$364$	−13.9881	−0.733174
$365$	40.1657	2.10237
$366$	0.727441	0.0380239
$367$	29.9084	1.56120	0.780602	−	0.625028i	$-0.214913\pi$
$367$	29.9084	1.56120	0.780602	+	0.625028i	$0.214913\pi$
$368$	−0.695478	−0.0362543
$369$	−13.4344	−0.699369
$370$	−1.90403	−0.0989859
$371$	20.3325	1.05561
$372$	−0.717915	−0.0372222
$373$	32.7204	1.69420	0.847100	−	0.531434i	$-0.178347\pi$
$373$	32.7204	1.69420	0.847100	+	0.531434i	$0.178347\pi$
$374$	0	0
$375$	−14.1336	−0.729854
$376$	9.80308	0.505555
$377$	−29.6685	−1.52801
$378$	11.9313	0.613681
$379$	21.4522	1.10193	0.550963	−	0.834530i	$-0.314261\pi$
$379$	21.4522	1.10193	0.550963	+	0.834530i	$0.314261\pi$
$380$	−2.75070	−0.141108
$381$	−1.20046	−0.0615017
$382$	−23.9456	−1.22517
$383$	−17.1641	−0.877044	−0.438522	−	0.898720i	$-0.644498\pi$
$383$	−17.1641	−0.877044	−0.438522	+	0.898720i	$0.644498\pi$
$384$	2.11129	0.107742
$385$	0	0
$386$	27.1307	1.38092
$387$	−2.93691	−0.149292
$388$	16.3170	0.828369
$389$	28.8568	1.46310	0.731550	−	0.681788i	$-0.238797\pi$
$389$	28.8568	1.46310	0.731550	+	0.681788i	$0.238797\pi$
$390$	−22.1726	−1.12275
$391$	−3.00318	−0.151877
$392$	−6.42347	−0.324434
$393$	25.0281	1.26250
$394$	21.1960	1.06784
$395$	−12.2153	−0.614621
$396$	0	0
$397$	−5.83003	−0.292601	−0.146300	−	0.989240i	$-0.546737\pi$
$397$	−5.83003	−0.292601	−0.146300	+	0.989240i	$0.546737\pi$
$398$	3.03824	0.152293
$399$	7.73537	0.387253
$400$	2.56634	0.128317
$401$	13.2388	0.661113	0.330556	−	0.943786i	$-0.392764\pi$
$401$	13.2388	0.661113	0.330556	+	0.943786i	$0.392764\pi$
$402$	−10.6418	−0.530763
$403$	1.29822	0.0646691
$404$	14.1398	0.703479
$405$	30.9404	1.53744
$406$	−28.4710	−1.41299
$407$	0	0
$408$	9.11690	0.451354
$409$	17.6372	0.872102	0.436051	−	0.899922i	$-0.356377\pi$
$409$	17.6372	0.872102	0.436051	+	0.899922i	$0.356377\pi$
$410$	−25.3533	−1.25211
$411$	−23.7651	−1.17224
$412$	0.880175	0.0433631
$413$	30.1443	1.48330
$414$	1.01370	0.0498208
$415$	33.1360	1.62658
$416$	−3.81790	−0.187188
$417$	3.57357	0.174998
$418$	0	0
$419$	−28.5638	−1.39543	−0.697715	−	0.716375i	$-0.745800\pi$
$419$	−28.5638	−1.39543	−0.697715	+	0.716375i	$0.745800\pi$
$420$	−21.2777	−1.03824
$421$	10.1449	0.494432	0.247216	−	0.968960i	$-0.420484\pi$
$421$	10.1449	0.494432	0.247216	+	0.968960i	$0.420484\pi$
$422$	27.4130	1.33444
$423$	−14.2886	−0.694737
$424$	5.54956	0.269510
$425$	11.0819	0.537549
$426$	−6.79996	−0.329459
$427$	−1.26235	−0.0610896
$428$	1.76713	0.0854174
$429$	0	0
$430$	−5.54251	−0.267284
$431$	−22.6879	−1.09284	−0.546418	−	0.837512i	$-0.684009\pi$
$431$	−22.6879	−1.09284	−0.546418	+	0.837512i	$0.684009\pi$
$432$	3.25654	0.156680
$433$	31.0778	1.49350	0.746751	−	0.665104i	$-0.231613\pi$
$433$	31.0778	1.49350	0.746751	+	0.665104i	$0.231613\pi$
$434$	1.24582	0.0598015
$435$	−45.1297	−2.16380
$436$	−11.6553	−0.558190
$437$	−0.695478	−0.0332692
$438$	−30.8291	−1.47307
$439$	30.3380	1.44795	0.723977	−	0.689824i	$-0.242312\pi$
$439$	30.3380	1.44795	0.723977	+	0.689824i	$0.242312\pi$
$440$	0	0
$441$	9.36263	0.445839
$442$	−16.4863	−0.784174
$443$	5.08267	0.241485	0.120742	−	0.992684i	$-0.461472\pi$
$443$	5.08267	0.241485	0.120742	+	0.992684i	$0.461472\pi$
$444$	1.46144	0.0693567
$445$	2.65837	0.126019
$446$	−18.7060	−0.885754
$447$	−45.4835	−2.15129
$448$	−3.66381	−0.173099
$449$	−34.4255	−1.62464	−0.812321	−	0.583211i	$-0.801796\pi$
$449$	−34.4255	−1.62464	−0.812321	+	0.583211i	$0.801796\pi$
$450$	−3.74060	−0.176334
$451$	0	0
$452$	19.9467	0.938215
$453$	42.2248	1.98390
$454$	12.6689	0.594579
$455$	38.4769	1.80383
$456$	2.11129	0.0988704
$457$	5.43151	0.254075	0.127038	−	0.991898i	$-0.459453\pi$
$457$	5.43151	0.254075	0.127038	+	0.991898i	$0.459453\pi$
$458$	−22.6558	−1.05864
$459$	14.0622	0.656369
$460$	1.91305	0.0891964
$461$	−0.796700	−0.0371060	−0.0185530	−	0.999828i	$-0.505906\pi$
$461$	−0.796700	−0.0371060	−0.0185530	+	0.999828i	$0.505906\pi$
$462$	0	0
$463$	24.1337	1.12159	0.560795	−	0.827955i	$-0.310496\pi$
$463$	24.1337	1.12159	0.560795	+	0.827955i	$0.310496\pi$
$464$	−7.77089	−0.360754
$465$	1.97477	0.0915777
$466$	20.0819	0.930276
$467$	8.96049	0.414642	0.207321	−	0.978273i	$-0.433526\pi$
$467$	8.96049	0.414642	0.207321	+	0.978273i	$0.433526\pi$
$468$	5.56484	0.257235
$469$	18.4671	0.852729
$470$	−26.9653	−1.24382
$471$	−14.6701	−0.675962
$472$	8.22759	0.378705
$473$	0	0
$474$	9.37587	0.430648
$475$	2.56634	0.117752
$476$	−15.8209	−0.725149
$477$	−8.08883	−0.370362
$478$	19.7417	0.902966
$479$	13.5637	0.619743	0.309871	−	0.950778i	$-0.399714\pi$
$479$	13.5637	0.619743	0.309871	+	0.950778i	$0.399714\pi$
$480$	−5.80753	−0.265076
$481$	−2.64275	−0.120499
$482$	−20.0092	−0.911393
$483$	−5.37978	−0.244788
$484$	0	0
$485$	−44.8831	−2.03804
$486$	−13.9787	−0.634084
$487$	1.14086	0.0516975	0.0258488	−	0.999666i	$-0.491771\pi$
$487$	1.14086	0.0516975	0.0258488	+	0.999666i	$0.491771\pi$
$488$	−0.344547	−0.0155969
$489$	−33.8026	−1.52861
$490$	17.6690	0.798206
$491$	16.4537	0.742545	0.371272	−	0.928524i	$-0.378922\pi$
$491$	16.4537	0.742545	0.371272	+	0.928524i	$0.378922\pi$
$492$	19.4599	0.877320
$493$	−33.5559	−1.51128
$494$	−3.81790	−0.171776
$495$	0	0
$496$	0.340036	0.0152680
$497$	11.8002	0.529312
$498$	−25.4335	−1.13970
$499$	27.3412	1.22396	0.611980	−	0.790873i	$-0.290373\pi$
$499$	27.3412	1.22396	0.611980	+	0.790873i	$0.290373\pi$
$500$	6.69426	0.299377
$501$	7.77057	0.347163
$502$	28.0383	1.25141
$503$	−20.2121	−0.901213	−0.450607	−	0.892723i	$-0.648792\pi$
$503$	−20.2121	−0.901213	−0.450607	+	0.892723i	$0.648792\pi$
$504$	5.34023	0.237873
$505$	−38.8942	−1.73077
$506$	0	0
$507$	−3.32823	−0.147812
$508$	0.568592	0.0252272
$509$	−0.629459	−0.0279003	−0.0139502	−	0.999903i	$-0.504441\pi$
$509$	−0.629459	−0.0279003	−0.0139502	+	0.999903i	$0.504441\pi$
$510$	−25.0778	−1.11047
$511$	53.4989	2.36665
$512$	−1.00000	−0.0441942
$513$	3.25654	0.143780
$514$	13.5564	0.597949
$515$	−2.42109	−0.106686
$516$	4.25415	0.187278
$517$	0	0
$518$	−2.53608	−0.111429
$519$	−27.3471	−1.20041
$520$	10.5019	0.460539
$521$	−25.0746	−1.09854	−0.549270	−	0.835645i	$-0.685094\pi$
$521$	−25.0746	−1.09854	−0.549270	+	0.835645i	$0.685094\pi$
$522$	11.3266	0.495750
$523$	13.1638	0.575611	0.287806	−	0.957689i	$-0.407074\pi$
$523$	13.1638	0.575611	0.287806	+	0.957689i	$0.407074\pi$
$524$	−11.8544	−0.517862
$525$	19.8516	0.866395
$526$	−4.42685	−0.193020
$527$	1.46833	0.0639613
$528$	0	0
$529$	−22.5163	−0.978970
$530$	−15.2652	−0.663076
$531$	−11.9922	−0.520419
$532$	−3.66381	−0.158846
$533$	−35.1898	−1.52424
$534$	−2.04043	−0.0882979
$535$	−4.86084	−0.210152
$536$	5.04040	0.217712
$537$	29.1072	1.25607
$538$	7.98617	0.344308
$539$	0	0
$540$	−8.95775	−0.385480
$541$	−29.6788	−1.27599	−0.637995	−	0.770041i	$-0.720236\pi$
$541$	−29.6788	−1.27599	−0.637995	+	0.770041i	$0.720236\pi$
$542$	17.9453	0.770815
$543$	32.2574	1.38430
$544$	−4.31816	−0.185139
$545$	32.0603	1.37331
$546$	−29.5329	−1.26389
$547$	38.2994	1.63756	0.818781	−	0.574106i	$-0.194650\pi$
$547$	38.2994	1.63756	0.818781	+	0.574106i	$0.194650\pi$
$548$	11.2562	0.480840
$549$	0.502200	0.0214334
$550$	0	0
$551$	−7.77089	−0.331051
$552$	−1.46836	−0.0624975
$553$	−16.2703	−0.691883
$554$	−21.4997	−0.913435
$555$	−4.01997	−0.170638
$556$	−1.69259	−0.0717820
$557$	7.05263	0.298829	0.149415	−	0.988775i	$-0.452261\pi$
$557$	7.05263	0.298829	0.149415	+	0.988775i	$0.452261\pi$
$558$	−0.495624	−0.0209814
$559$	−7.69287	−0.325374
$560$	10.0780	0.425874
$561$	0	0
$562$	19.5998	0.826769
$563$	17.3112	0.729581	0.364790	−	0.931090i	$-0.381141\pi$
$563$	17.3112	0.729581	0.364790	+	0.931090i	$0.381141\pi$
$564$	20.6972	0.871509
$565$	−54.8674	−2.30829
$566$	13.0294	0.547667
$567$	41.2112	1.73071
$568$	3.22075	0.135140
$569$	15.7749	0.661318	0.330659	−	0.943750i	$-0.392729\pi$
$569$	15.7749	0.661318	0.330659	+	0.943750i	$0.392729\pi$
$570$	−5.80753	−0.243251
$571$	−24.8790	−1.04116	−0.520578	−	0.853814i	$-0.674283\pi$
$571$	−24.8790	−1.04116	−0.520578	+	0.853814i	$0.674283\pi$
$572$	0	0
$573$	−50.5563	−2.11202
$574$	−33.7695	−1.40951
$575$	−1.78483	−0.0744326
$576$	1.45756	0.0607318
$577$	12.6925	0.528395	0.264197	−	0.964469i	$-0.414893\pi$
$577$	12.6925	0.528395	0.264197	+	0.964469i	$0.414893\pi$
$578$	−1.64648	−0.0684844
$579$	57.2810	2.38052
$580$	21.3754	0.887564
$581$	44.1357	1.83106
$582$	34.4499	1.42800
$583$	0	0
$584$	14.6020	0.604235
$585$	−15.3072	−0.632875
$586$	−12.4424	−0.513993
$587$	−37.8091	−1.56055	−0.780275	−	0.625437i	$-0.784921\pi$
$587$	−37.8091	−1.56055	−0.780275	+	0.625437i	$0.784921\pi$
$588$	−13.5618	−0.559281
$589$	0.340036	0.0140109
$590$	−22.6316	−0.931729
$591$	44.7510	1.84081
$592$	−0.692199	−0.0284492
$593$	11.7942	0.484331	0.242166	−	0.970235i	$-0.422142\pi$
$593$	11.7942	0.484331	0.242166	+	0.970235i	$0.422142\pi$
$594$	0	0
$595$	43.5185	1.78408
$596$	21.5429	0.882433
$597$	6.41462	0.262533
$598$	2.65527	0.108582
$599$	−2.24280	−0.0916382	−0.0458191	−	0.998950i	$-0.514590\pi$
$599$	−2.24280	−0.0916382	−0.0458191	+	0.998950i	$0.514590\pi$
$600$	5.41830	0.221201
$601$	34.5443	1.40909	0.704545	−	0.709659i	$-0.251151\pi$
$601$	34.5443	1.40909	0.704545	+	0.709659i	$0.251151\pi$
$602$	−7.38237	−0.300883
$603$	−7.34671	−0.299181
$604$	−19.9995	−0.813768
$605$	0	0
$606$	29.8532	1.21270
$607$	0.623652	0.0253133	0.0126566	−	0.999920i	$-0.495971\pi$
$607$	0.623652	0.0253133	0.0126566	+	0.999920i	$0.495971\pi$
$608$	−1.00000	−0.0405554
$609$	−60.1107	−2.43581
$610$	0.947746	0.0383731
$611$	−37.4272	−1.51414
$612$	6.29399	0.254419
$613$	19.0977	0.771347	0.385673	−	0.922635i	$-0.373969\pi$
$613$	19.0977	0.771347	0.385673	+	0.922635i	$0.373969\pi$
$614$	−18.8124	−0.759206
$615$	−53.5283	−2.15847
$616$	0	0
$617$	−12.8189	−0.516070	−0.258035	−	0.966136i	$-0.583075\pi$
$617$	−12.8189	−0.516070	−0.258035	+	0.966136i	$0.583075\pi$
$618$	1.85831	0.0747521
$619$	18.3684	0.738288	0.369144	−	0.929372i	$-0.379651\pi$
$619$	18.3684	0.738288	0.369144	+	0.929372i	$0.379651\pi$
$620$	−0.935335	−0.0375640
$621$	−2.26485	−0.0908852
$622$	13.2168	0.529944
$623$	3.54083	0.141860
$624$	−8.06072	−0.322687
$625$	−31.2456	−1.24982
$626$	3.51233	0.140381
$627$	0	0
$628$	6.94839	0.277271
$629$	−2.98903	−0.119180
$630$	−14.6894	−0.585238
$631$	5.04337	0.200774	0.100387	−	0.994948i	$-0.467992\pi$
$631$	5.04337	0.200774	0.100387	+	0.994948i	$0.467992\pi$
$632$	−4.44082	−0.176646
$633$	57.8768	2.30040
$634$	−8.53754	−0.339069
$635$	−1.56402	−0.0620664
$636$	11.7167	0.464599
$637$	24.5242	0.971685
$638$	0	0
$639$	−4.69445	−0.185710
$640$	2.75070	0.108731
$641$	46.3321	1.83001	0.915003	−	0.403446i	$-0.132188\pi$
$641$	46.3321	1.83001	0.915003	+	0.403446i	$0.132188\pi$
$642$	3.73093	0.147248
$643$	−25.3374	−0.999210	−0.499605	−	0.866253i	$-0.666522\pi$
$643$	−25.3374	−0.999210	−0.499605	+	0.866253i	$0.666522\pi$
$644$	2.54810	0.100409
$645$	−11.7019	−0.460761
$646$	−4.31816	−0.169896
$647$	−21.2531	−0.835545	−0.417773	−	0.908552i	$-0.637189\pi$
$647$	−21.2531	−0.835545	−0.417773	+	0.908552i	$0.637189\pi$
$648$	11.2482	0.441871
$649$	0	0
$650$	−9.79804	−0.384311
$651$	2.63030	0.103090
$652$	16.0104	0.627014
$653$	6.92969	0.271180	0.135590	−	0.990765i	$-0.456707\pi$
$653$	6.92969	0.271180	0.135590	+	0.990765i	$0.456707\pi$
$654$	−24.6079	−0.962243
$655$	32.6079	1.27409
$656$	−9.21705	−0.359865
$657$	−21.2834	−0.830343
$658$	−35.9166	−1.40017
$659$	34.0324	1.32571	0.662857	−	0.748746i	$-0.269344\pi$
$659$	34.0324	1.32571	0.662857	+	0.748746i	$0.269344\pi$
$660$	0	0
$661$	12.8692	0.500552	0.250276	−	0.968175i	$-0.419479\pi$
$661$	12.8692	0.500552	0.250276	+	0.968175i	$0.419479\pi$
$662$	3.43770	0.133610
$663$	−34.8074	−1.35181
$664$	12.0464	0.467491
$665$	10.0780	0.390809
$666$	1.00893	0.0390951
$667$	5.40448	0.209262
$668$	−3.68048	−0.142402
$669$	−39.4938	−1.52692
$670$	−13.8646	−0.535637
$671$	0	0
$672$	−7.73537	−0.298398
$673$	32.0632	1.23595	0.617973	−	0.786199i	$-0.287954\pi$
$673$	32.0632	1.23595	0.617973	+	0.786199i	$0.287954\pi$
$674$	23.8752	0.919638
$675$	8.35738	0.321676
$676$	1.57639	0.0606304
$677$	34.6314	1.33099	0.665496	−	0.746402i	$-0.268220\pi$
$677$	34.6314	1.33099	0.665496	+	0.746402i	$0.268220\pi$
$678$	42.1134	1.61736
$679$	−59.7823	−2.29423
$680$	11.8779	0.455498
$681$	26.7477	1.02497
$682$	0	0
$683$	−11.1244	−0.425663	−0.212831	−	0.977089i	$-0.568269\pi$
$683$	−11.1244	−0.425663	−0.212831	+	0.977089i	$0.568269\pi$
$684$	1.45756	0.0557314
$685$	−30.9623	−1.18301
$686$	−2.11228	−0.0806471
$687$	−47.8330	−1.82494
$688$	−2.01495	−0.0768191
$689$	−21.1877	−0.807186
$690$	4.03901	0.153762
$691$	30.1726	1.14782	0.573910	−	0.818919i	$-0.305426\pi$
$691$	30.1726	1.14782	0.573910	+	0.818919i	$0.305426\pi$
$692$	12.9528	0.492391
$693$	0	0
$694$	7.32500	0.278053
$695$	4.65582	0.176605
$696$	−16.4066	−0.621892
$697$	−39.8007	−1.50756
$698$	8.11645	0.307212
$699$	42.3988	1.60367
$700$	−9.40257	−0.355384
$701$	20.1483	0.760991	0.380495	−	0.924783i	$-0.375753\pi$
$701$	20.1483	0.760991	0.380495	+	0.924783i	$0.375753\pi$
$702$	−12.4331	−0.469259
$703$	−0.692199	−0.0261068
$704$	0	0
$705$	−56.9317	−2.14417
$706$	5.08395	0.191337
$707$	−51.8053	−1.94834
$708$	17.3709	0.652837
$709$	−1.20513	−0.0452597	−0.0226298	−	0.999744i	$-0.507204\pi$
$709$	−1.20513	−0.0452597	−0.0226298	+	0.999744i	$0.507204\pi$
$710$	−8.85932	−0.332484
$711$	6.47278	0.242748
$712$	0.966434	0.0362186
$713$	−0.236487	−0.00885651
$714$	−33.4026	−1.25006
$715$	0	0
$716$	−13.7864	−0.515222
$717$	41.6806	1.55659
$718$	15.2156	0.567842
$719$	−49.8350	−1.85853	−0.929267	−	0.369409i	$-0.879560\pi$
$719$	−49.8350	−1.85853	−0.929267	+	0.369409i	$0.879560\pi$
$720$	−4.00932	−0.149418
$721$	−3.22479	−0.120097
$722$	−1.00000	−0.0372161
$723$	−42.2453	−1.57112
$724$	−15.2785	−0.567820
$725$	−19.9427	−0.740655
$726$	0	0
$727$	6.51901	0.241777	0.120888	−	0.992666i	$-0.461426\pi$
$727$	6.51901	0.241777	0.120888	+	0.992666i	$0.461426\pi$
$728$	13.9881	0.518432
$729$	4.23155	0.156724
$730$	−40.1657	−1.48660
$731$	−8.70086	−0.321813
$732$	−0.727441	−0.0268870
$733$	1.19268	0.0440526	0.0220263	−	0.999757i	$-0.492988\pi$
$733$	1.19268	0.0440526	0.0220263	+	0.999757i	$0.492988\pi$
$734$	−29.9084	−1.10394
$735$	37.3045	1.37600
$736$	0.695478	0.0256356
$737$	0	0
$738$	13.4344	0.494529
$739$	−21.4841	−0.790305	−0.395152	−	0.918616i	$-0.629308\pi$
$739$	−21.4841	−0.790305	−0.395152	+	0.918616i	$0.629308\pi$
$740$	1.90403	0.0699936
$741$	−8.06072	−0.296118
$742$	−20.3325	−0.746429
$743$	33.2777	1.22084	0.610421	−	0.792077i	$-0.291000\pi$
$743$	33.2777	1.22084	0.610421	+	0.792077i	$0.291000\pi$
$744$	0.717915	0.0263201
$745$	−59.2581	−2.17105
$746$	−32.7204	−1.19798
$747$	−17.5584	−0.642429
$748$	0	0
$749$	−6.47442	−0.236570
$750$	14.1336	0.516085
$751$	23.0796	0.842185	0.421092	−	0.907018i	$-0.361647\pi$
$751$	23.0796	0.842185	0.421092	+	0.907018i	$0.361647\pi$
$752$	−9.80308	−0.357482
$753$	59.1971	2.15726
$754$	29.6685	1.08046
$755$	55.0126	2.00211
$756$	−11.9313	−0.433938
$757$	25.7018	0.934149	0.467074	−	0.884218i	$-0.345308\pi$
$757$	25.7018	0.934149	0.467074	+	0.884218i	$0.345308\pi$
$758$	−21.4522	−0.779179
$759$	0	0
$760$	2.75070	0.0997783
$761$	−45.5538	−1.65132	−0.825661	−	0.564166i	$-0.809198\pi$
$761$	−45.5538	−1.65132	−0.825661	+	0.564166i	$0.809198\pi$
$762$	1.20046	0.0434882
$763$	42.7029	1.54595
$764$	23.9456	0.866323
$765$	−17.3129	−0.625948
$766$	17.1641	0.620164
$767$	−31.4121	−1.13423
$768$	−2.11129	−0.0761848
$769$	−50.7820	−1.83125	−0.915623	−	0.402038i	$-0.868302\pi$
$769$	−50.7820	−1.83125	−0.915623	+	0.402038i	$0.868302\pi$
$770$	0	0
$771$	28.6216	1.03078
$772$	−27.1307	−0.976456
$773$	33.7448	1.21372	0.606858	−	0.794810i	$-0.292430\pi$
$773$	33.7448	1.21372	0.606858	+	0.794810i	$0.292430\pi$
$774$	2.93691	0.105565
$775$	0.872647	0.0313464
$776$	−16.3170	−0.585745
$777$	−5.35442	−0.192089
$778$	−28.8568	−1.03457
$779$	−9.21705	−0.330235
$780$	22.1726	0.793907
$781$	0	0
$782$	3.00318	0.107394
$783$	−25.3062	−0.904369
$784$	6.42347	0.229410
$785$	−19.1129	−0.682169
$786$	−25.0281	−0.892723
$787$	−21.9550	−0.782611	−0.391306	−	0.920261i	$-0.627976\pi$
$787$	−21.9550	−0.782611	−0.391306	+	0.920261i	$0.627976\pi$
$788$	−21.1960	−0.755076
$789$	−9.34639	−0.332740
$790$	12.2153	0.434603
$791$	−73.0809	−2.59846
$792$	0	0
$793$	1.31545	0.0467129
$794$	5.83003	0.206900
$795$	−32.2292	−1.14305
$796$	−3.03824	−0.107688
$797$	32.3449	1.14571	0.572857	−	0.819655i	$-0.305835\pi$
$797$	32.3449	1.14571	0.572857	+	0.819655i	$0.305835\pi$
$798$	−7.73537	−0.273829
$799$	−42.3313	−1.49757
$800$	−2.56634	−0.0907338
$801$	−1.40864	−0.0497718
$802$	−13.2388	−0.467477
$803$	0	0
$804$	10.6418	0.375306
$805$	−7.00904	−0.247036
$806$	−1.29822	−0.0457280
$807$	16.8612	0.593541
$808$	−14.1398	−0.497435
$809$	9.49040	0.333665	0.166832	−	0.985985i	$-0.446646\pi$
$809$	9.49040	0.333665	0.166832	+	0.985985i	$0.446646\pi$
$810$	−30.9404	−1.08713
$811$	−35.3671	−1.24191	−0.620953	−	0.783847i	$-0.713254\pi$
$811$	−35.3671	−1.24191	−0.620953	+	0.783847i	$0.713254\pi$
$812$	28.4710	0.999137
$813$	37.8877	1.32878
$814$	0	0
$815$	−44.0397	−1.54264
$816$	−9.11690	−0.319155
$817$	−2.01495	−0.0704941
$818$	−17.6372	−0.616669
$819$	−20.3885	−0.712432
$820$	25.3533	0.885377
$821$	14.6841	0.512478	0.256239	−	0.966613i	$-0.417517\pi$
$821$	14.6841	0.512478	0.256239	+	0.966613i	$0.417517\pi$
$822$	23.7651	0.828902
$823$	28.4915	0.993152	0.496576	−	0.867993i	$-0.334590\pi$
$823$	28.4915	0.993152	0.496576	+	0.867993i	$0.334590\pi$
$824$	−0.880175	−0.0306623
$825$	0	0
$826$	−30.1443	−1.04885
$827$	20.6508	0.718098	0.359049	−	0.933319i	$-0.383101\pi$
$827$	20.6508	0.718098	0.359049	+	0.933319i	$0.383101\pi$
$828$	−1.01370	−0.0352286
$829$	10.7947	0.374916	0.187458	−	0.982273i	$-0.439975\pi$
$829$	10.7947	0.374916	0.187458	+	0.982273i	$0.439975\pi$
$830$	−33.1360	−1.15017
$831$	−45.3922	−1.57464
$832$	3.81790	0.132362
$833$	27.7376	0.961050
$834$	−3.57357	−0.123742
$835$	10.1239	0.350351
$836$	0	0
$837$	1.10734	0.0382752
$838$	28.5638	0.986719
$839$	−26.2123	−0.904950	−0.452475	−	0.891777i	$-0.649459\pi$
$839$	−26.2123	−0.904950	−0.452475	+	0.891777i	$0.649459\pi$
$840$	21.2777	0.734150
$841$	31.3867	1.08230
$842$	−10.1449	−0.349616
$843$	41.3810	1.42524
$844$	−27.4130	−0.943593
$845$	−4.33618	−0.149169
$846$	14.2886	0.491253
$847$	0	0
$848$	−5.54956	−0.190573
$849$	27.5089	0.944104
$850$	−11.0819	−0.380105
$851$	0.481409	0.0165025
$852$	6.79996	0.232963
$853$	7.01542	0.240203	0.120102	−	0.992762i	$-0.461678\pi$
$853$	7.01542	0.240203	0.120102	+	0.992762i	$0.461678\pi$
$854$	1.26235	0.0431969
$855$	−4.00932	−0.137116
$856$	−1.76713	−0.0603993
$857$	−22.6511	−0.773746	−0.386873	−	0.922133i	$-0.626445\pi$
$857$	−22.6511	−0.773746	−0.386873	+	0.922133i	$0.626445\pi$
$858$	0	0
$859$	36.4532	1.24377	0.621883	−	0.783110i	$-0.286368\pi$
$859$	36.4532	1.24377	0.621883	+	0.783110i	$0.286368\pi$
$860$	5.54251	0.188998
$861$	−71.2973	−2.42981
$862$	22.6879	0.772752
$863$	31.8436	1.08397	0.541984	−	0.840389i	$-0.317673\pi$
$863$	31.8436	1.08397	0.541984	+	0.840389i	$0.317673\pi$
$864$	−3.25654	−0.110790
$865$	−35.6292	−1.21143
$866$	−31.0778	−1.05607
$867$	−3.47620	−0.118058
$868$	−1.24582	−0.0422860
$869$	0	0
$870$	45.1297	1.53004
$871$	−19.2438	−0.652050
$872$	11.6553	0.394700
$873$	23.7830	0.804934
$874$	0.695478	0.0235249
$875$	−24.5265	−0.829147
$876$	30.8291	1.04162
$877$	32.0478	1.08218	0.541089	−	0.840966i	$-0.318012\pi$
$877$	32.0478	1.08218	0.541089	+	0.840966i	$0.318012\pi$
$878$	−30.3380	−1.02386
$879$	−26.2697	−0.886054
$880$	0	0
$881$	−34.2368	−1.15347	−0.576733	−	0.816932i	$-0.695673\pi$
$881$	−34.2368	−1.15347	−0.576733	+	0.816932i	$0.695673\pi$
$882$	−9.36263	−0.315256
$883$	38.2142	1.28601	0.643005	−	0.765862i	$-0.277687\pi$
$883$	38.2142	1.28601	0.643005	+	0.765862i	$0.277687\pi$
$884$	16.4863	0.554495
$885$	−47.7820	−1.60617
$886$	−5.08267	−0.170756
$887$	−38.7150	−1.29992	−0.649961	−	0.759968i	$-0.725215\pi$
$887$	−38.7150	−1.29992	−0.649961	+	0.759968i	$0.725215\pi$
$888$	−1.46144	−0.0490426
$889$	−2.08321	−0.0698686
$890$	−2.65837	−0.0891087
$891$	0	0
$892$	18.7060	0.626323
$893$	−9.80308	−0.328048
$894$	45.4835	1.52120
$895$	37.9223	1.26760
$896$	3.66381	0.122399
$897$	5.60605	0.187181
$898$	34.4255	1.14880
$899$	−2.64238	−0.0881282
$900$	3.74060	0.124687
$901$	−23.9638	−0.798352
$902$	0	0
$903$	−15.5864	−0.518682
$904$	−19.9467	−0.663418
$905$	42.0265	1.39701
$906$	−42.2248	−1.40283
$907$	−30.2793	−1.00541	−0.502704	−	0.864458i	$-0.667662\pi$
$907$	−30.2793	−1.00541	−0.502704	+	0.864458i	$0.667662\pi$
$908$	−12.6689	−0.420431
$909$	20.6096	0.683577
$910$	−38.4769	−1.27550
$911$	−38.8426	−1.28691	−0.643456	−	0.765483i	$-0.722500\pi$
$911$	−38.8426	−1.28691	−0.643456	+	0.765483i	$0.722500\pi$
$912$	−2.11129	−0.0699119
$913$	0	0
$914$	−5.43151	−0.179658
$915$	2.00097	0.0661500
$916$	22.6558	0.748568
$917$	43.4322	1.43426
$918$	−14.0622	−0.464123
$919$	40.6344	1.34041	0.670203	−	0.742178i	$-0.266207\pi$
$919$	40.6344	1.34041	0.670203	+	0.742178i	$0.266207\pi$
$920$	−1.91305	−0.0630714
$921$	−39.7185	−1.30877
$922$	0.796700	0.0262379
$923$	−12.2965	−0.404745
$924$	0	0
$925$	−1.77642	−0.0584083
$926$	−24.1337	−0.793083
$927$	1.28291	0.0421363
$928$	7.77089	0.255092
$929$	−24.9807	−0.819591	−0.409796	−	0.912177i	$-0.634400\pi$
$929$	−24.9807	−0.819591	−0.409796	+	0.912177i	$0.634400\pi$
$930$	−1.97477	−0.0647552
$931$	6.42347	0.210521
$932$	−20.0819	−0.657805
$933$	27.9045	0.913552
$934$	−8.96049	−0.293196
$935$	0	0
$936$	−5.56484	−0.181892
$937$	−9.69911	−0.316856	−0.158428	−	0.987371i	$-0.550643\pi$
$937$	−9.69911	−0.316856	−0.158428	+	0.987371i	$0.550643\pi$
$938$	−18.4671	−0.602971
$939$	7.41557	0.241998
$940$	26.9653	0.879512
$941$	20.4974	0.668195	0.334098	−	0.942539i	$-0.391569\pi$
$941$	20.4974	0.668195	0.334098	+	0.942539i	$0.391569\pi$
$942$	14.6701	0.477978
$943$	6.41025	0.208747
$944$	−8.22759	−0.267785
$945$	32.8195	1.06762
$946$	0	0
$947$	52.0636	1.69184	0.845920	−	0.533309i	$-0.179052\pi$
$947$	52.0636	1.69184	0.845920	+	0.533309i	$0.179052\pi$
$948$	−9.37587	−0.304514
$949$	−55.7490	−1.80969
$950$	−2.56634	−0.0832631
$951$	−18.0253	−0.584509
$952$	15.8209	0.512758
$953$	2.51516	0.0814739	0.0407370	−	0.999170i	$-0.487029\pi$
$953$	2.51516	0.0814739	0.0407370	+	0.999170i	$0.487029\pi$
$954$	8.08883	0.261886
$955$	−65.8672	−2.13141
$956$	−19.7417	−0.638493
$957$	0	0
$958$	−13.5637	−0.438224
$959$	−41.2404	−1.33172
$960$	5.80753	0.187437
$961$	−30.8844	−0.996270
$962$	2.64275	0.0852057
$963$	2.57570	0.0830009
$964$	20.0092	0.644452
$965$	74.6284	2.40237
$966$	5.37978	0.173092
$967$	−16.5427	−0.531979	−0.265989	−	0.963976i	$-0.585699\pi$
$967$	−16.5427	−0.531979	−0.265989	+	0.963976i	$0.585699\pi$
$968$	0	0
$969$	−9.11690	−0.292877
$970$	44.8831	1.44111
$971$	40.7237	1.30688	0.653442	−	0.756976i	$-0.273324\pi$
$971$	40.7237	1.30688	0.653442	+	0.756976i	$0.273324\pi$
$972$	13.9787	0.448365
$973$	6.20134	0.198806
$974$	−1.14086	−0.0365557
$975$	−20.6865	−0.662500
$976$	0.344547	0.0110287
$977$	0.602998	0.0192916	0.00964580	−	0.999953i	$-0.496930\pi$
$977$	0.602998	0.0192916	0.00964580	+	0.999953i	$0.496930\pi$
$978$	33.8026	1.08089
$979$	0	0
$980$	−17.6690	−0.564417
$981$	−16.9884	−0.542398
$982$	−16.4537	−0.525058
$983$	23.4122	0.746734	0.373367	−	0.927684i	$-0.378203\pi$
$983$	23.4122	0.746734	0.373367	+	0.927684i	$0.378203\pi$
$984$	−19.4599	−0.620359
$985$	58.3038	1.85771
$986$	33.5559	1.06864
$987$	−75.8305	−2.41371
$988$	3.81790	0.121464
$989$	1.40135	0.0445603
$990$	0	0
$991$	−34.2320	−1.08742	−0.543708	−	0.839275i	$-0.682980\pi$
$991$	−34.2320	−1.08742	−0.543708	+	0.839275i	$0.682980\pi$
$992$	−0.340036	−0.0107961
$993$	7.25800	0.230326
$994$	−11.8002	−0.374280
$995$	8.35728	0.264944
$996$	25.4335	0.805892
$997$	−21.6379	−0.685280	−0.342640	−	0.939467i	$-0.611321\pi$
$997$	−21.6379	−0.685280	−0.342640	+	0.939467i	$0.611321\pi$
$998$	−27.3412	−0.865471
$999$	−2.25417	−0.0713189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.by.1.2	✓	8
11.10	odd	2		4598.2.a.cb.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.by.1.2	✓	8	1.1	even	1	trivial
4598.2.a.cb.1.2	yes	8	11.10	odd	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.11129 q^{3} +1.00000 q^{4} -2.75070 q^{5} +2.11129 q^{6} -3.66381 q^{7} -1.00000 q^{8} +1.45756 q^{9} +2.75070 q^{10} -2.11129 q^{12} +3.81790 q^{13} +3.66381 q^{14} +5.80753 q^{15} +1.00000 q^{16} +4.31816 q^{17} -1.45756 q^{18} +1.00000 q^{19} -2.75070 q^{20} +7.73537 q^{21} -0.695478 q^{23} +2.11129 q^{24} +2.56634 q^{25} -3.81790 q^{26} +3.25654 q^{27} -3.66381 q^{28} -7.77089 q^{29} -5.80753 q^{30} +0.340036 q^{31} -1.00000 q^{32} -4.31816 q^{34} +10.0780 q^{35} +1.45756 q^{36} -0.692199 q^{37} -1.00000 q^{38} -8.06072 q^{39} +2.75070 q^{40} -9.21705 q^{41} -7.73537 q^{42} -2.01495 q^{43} -4.00932 q^{45} +0.695478 q^{46} -9.80308 q^{47} -2.11129 q^{48} +6.42347 q^{49} -2.56634 q^{50} -9.11690 q^{51} +3.81790 q^{52} -5.54956 q^{53} -3.25654 q^{54} +3.66381 q^{56} -2.11129 q^{57} +7.77089 q^{58} -8.22759 q^{59} +5.80753 q^{60} +0.344547 q^{61} -0.340036 q^{62} -5.34023 q^{63} +1.00000 q^{64} -10.5019 q^{65} -5.04040 q^{67} +4.31816 q^{68} +1.46836 q^{69} -10.0780 q^{70} -3.22075 q^{71} -1.45756 q^{72} -14.6020 q^{73} +0.692199 q^{74} -5.41830 q^{75} +1.00000 q^{76} +8.06072 q^{78} +4.44082 q^{79} -2.75070 q^{80} -11.2482 q^{81} +9.21705 q^{82} -12.0464 q^{83} +7.73537 q^{84} -11.8779 q^{85} +2.01495 q^{86} +16.4066 q^{87} -0.966434 q^{89} +4.00932 q^{90} -13.9881 q^{91} -0.695478 q^{92} -0.717915 q^{93} +9.80308 q^{94} -2.75070 q^{95} +2.11129 q^{96} +16.3170 q^{97} -6.42347 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25}+ \cdots - 34 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^14 + 4 * q^15 + 8 * q^16 + 4 * q^17 - 22 * q^18 + 8 * q^19 + 20 * q^21 + 14 * q^23 - 8 * q^24 + 36 * q^25 - 12 * q^26 + 32 * q^27 - 4 * q^28 + 2 * q^29 - 4 * q^30 - 8 * q^32 - 4 * q^34 - 36 * q^35 + 22 * q^36 + 24 * q^37 - 8 * q^38 - 16 * q^39 - 8 * q^41 - 20 * q^42 - 8 * q^43 + 16 * q^45 - 14 * q^46 - 16 * q^47 + 8 * q^48 + 34 * q^49 - 36 * q^50 - 18 * q^51 + 12 * q^52 + 36 * q^53 - 32 * q^54 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 24 * q^59 + 4 * q^60 - 12 * q^61 - 24 * q^63 + 8 * q^64 - 16 * q^65 + 16 * q^67 + 4 * q^68 + 4 * q^69 + 36 * q^70 + 4 * q^71 - 22 * q^72 + 20 * q^73 - 24 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^78 + 12 * q^79 + 40 * q^81 + 8 * q^82 - 20 * q^83 + 20 * q^84 - 12 * q^85 + 8 * q^86 + 36 * q^87 + 8 * q^89 - 16 * q^90 - 24 * q^91 + 14 * q^92 + 12 * q^93 + 16 * q^94 - 8 * q^96 + 4 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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