LMFDB - Embedded newform 4600.2.e.r.4049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.24681024.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 9 $ x^6 + 12x^4 + 36x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-2.66908i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.r.4049.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66908i q^{3} +3.66908i q^{7} -4.12398 q^{9} +O(q^{10})$	$q-2.66908i q^{3} +3.66908i q^{7} -4.12398 q^{9} -1.21417 q^{11} -2.21417i q^{13} +1.21417i q^{17} +2.57889 q^{19} +9.79306 q^{21} -1.00000i q^{23} +3.00000i q^{27} -1.45490 q^{29} -6.46214 q^{31} +3.24073i q^{33} +4.00000i q^{37} -5.90981 q^{39} -10.9170 q^{41} -6.90981i q^{43} +5.45490i q^{47} -6.46214 q^{49} +3.24073 q^{51} -3.81962i q^{53} -6.88325i q^{57} +4.24797 q^{59} +6.78583 q^{61} -15.1312i q^{63} +12.8567i q^{67} -2.66908 q^{69} -6.91705 q^{71} +15.2214i q^{73} -4.45490i q^{77} -4.36471 q^{81} +15.5861i q^{83} +3.88325i q^{87} +10.9098 q^{89} +8.12398 q^{91} +17.2480i q^{93} +1.69563i q^{97} +5.00724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 6 * q^11 - 6 * q^19 + 24 * q^21 - 6 * q^29 + 12 * q^31 - 30 * q^39 - 12 * q^41 + 12 * q^49 + 30 * q^51 - 12 * q^59 + 54 * q^61 + 12 * q^71 - 18 * q^81 + 60 * q^89 + 30 * q^91 - 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 2.66908i	− 1.54099i	−0.637444	−	0.770497i	$-0.720008\pi$
$3$	− 2.66908i	− 1.54099i	0.637444	−	0.770497i	$-0.279992\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.66908i	1.38678i	0.720562	+	0.693391i	$0.243884\pi$
$7$	3.66908i	1.38678i	−0.720562	+	0.693391i	$0.756116\pi$
$8$	0	0
$9$	−4.12398	−1.37466
$10$	0	0
$11$	−1.21417	−0.366088	−0.183044	−	0.983105i	$-0.558595\pi$
$11$	−1.21417	−0.366088	−0.183044	+	0.983105i	$0.558595\pi$
$12$	0	0
$13$	− 2.21417i	− 0.614102i	−0.951693	−	0.307051i	$-0.900658\pi$
$13$	− 2.21417i	− 0.614102i	0.951693	−	0.307051i	$-0.0993422\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.21417i	0.294481i	0.989101	+	0.147240i	$0.0470391\pi$
$17$	1.21417i	0.294481i	−0.989101	+	0.147240i	$0.952961\pi$
$18$	0	0
$19$	2.57889	0.591637	0.295819	−	0.955244i	$-0.404408\pi$
$19$	2.57889	0.591637	0.295819	+	0.955244i	$0.404408\pi$
$20$	0	0
$21$	9.79306	2.13702
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.00000i	0.577350i
$28$	0	0
$29$	−1.45490	−0.270169	−0.135084	−	0.990834i	$-0.543131\pi$
$29$	−1.45490	−0.270169	−0.135084	+	0.990834i	$0.543131\pi$
$30$	0	0
$31$	−6.46214	−1.16063	−0.580317	−	0.814390i	$-0.697072\pi$
$31$	−6.46214	−1.16063	−0.580317	+	0.814390i	$0.697072\pi$
$32$	0	0
$33$	3.24073i	0.564139i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	0	0
$39$	−5.90981	−0.946327
$40$	0	0
$41$	−10.9170	−1.70496	−0.852478	−	0.522763i	$-0.824901\pi$
$41$	−10.9170	−1.70496	−0.852478	+	0.522763i	$0.824901\pi$
$42$	0	0
$43$	− 6.90981i	− 1.05374i	−0.849947	−	0.526868i	$-0.823366\pi$
$43$	− 6.90981i	− 1.05374i	0.849947	−	0.526868i	$-0.176634\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.45490i	0.795680i	0.917455	+	0.397840i	$0.130240\pi$
$47$	5.45490i	0.795680i	−0.917455	+	0.397840i	$0.869760\pi$
$48$	0	0
$49$	−6.46214	−0.923163
$50$	0	0
$51$	3.24073	0.453793
$52$	0	0
$53$	− 3.81962i	− 0.524665i	−0.964978	−	0.262332i	$-0.915508\pi$
$53$	− 3.81962i	− 0.524665i	0.964978	−	0.262332i	$-0.0844917\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.88325i	− 0.911709i
$58$	0	0
$59$	4.24797	0.553038	0.276519	−	0.961008i	$-0.410819\pi$
$59$	4.24797	0.553038	0.276519	+	0.961008i	$0.410819\pi$
$60$	0	0
$61$	6.78583	0.868836	0.434418	−	0.900711i	$-0.356954\pi$
$61$	6.78583	0.868836	0.434418	+	0.900711i	$0.356954\pi$
$62$	0	0
$63$	− 15.1312i	− 1.90635i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.8567i	1.57070i	0.619055	+	0.785348i	$0.287516\pi$
$67$	12.8567i	1.57070i	−0.619055	+	0.785348i	$0.712484\pi$
$68$	0	0
$69$	−2.66908	−0.321319
$70$	0	0
$71$	−6.91705	−0.820902	−0.410451	−	0.911883i	$-0.634629\pi$
$71$	−6.91705	−0.820902	−0.410451	+	0.911883i	$0.634629\pi$
$72$	0	0
$73$	15.2214i	1.78153i	0.454463	+	0.890766i	$0.349831\pi$
$73$	15.2214i	1.78153i	−0.454463	+	0.890766i	$0.650169\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.45490i	− 0.507683i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−4.36471	−0.484968
$82$	0	0
$83$	15.5861i	1.71080i	0.517968	+	0.855400i	$0.326688\pi$
$83$	15.5861i	1.71080i	−0.517968	+	0.855400i	$0.673312\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.88325i	0.416329i
$88$	0	0
$89$	10.9098	1.15644	0.578219	−	0.815882i	$-0.303748\pi$
$89$	10.9098	1.15644	0.578219	+	0.815882i	$0.303748\pi$
$90$	0	0
$91$	8.12398	0.851625
$92$	0	0
$93$	17.2480i	1.78853i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.69563i	0.172165i	0.996288	+	0.0860827i	$0.0274349\pi$
$97$	1.69563i	0.172165i	−0.996288	+	0.0860827i	$0.972565\pi$
$98$	0	0
$99$	5.00724	0.503246
$100$	0	0
$101$	−2.24797	−0.223681	−0.111841	−	0.993726i	$-0.535675\pi$
$101$	−2.24797	−0.223681	−0.111841	+	0.993726i	$0.535675\pi$
$102$	0	0
$103$	2.78583i	0.274495i	0.990537	+	0.137248i	$0.0438256\pi$
$103$	2.78583i	0.274495i	−0.990537	+	0.137248i	$0.956174\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.1578i	1.27201i	0.771685	+	0.636005i	$0.219414\pi$
$107$	13.1578i	1.27201i	−0.771685	+	0.636005i	$0.780586\pi$
$108$	0	0
$109$	−3.42111	−0.327683	−0.163842	−	0.986487i	$-0.552389\pi$
$109$	−3.42111	−0.327683	−0.163842	+	0.986487i	$0.552389\pi$
$110$	0	0
$111$	10.6763	1.01335
$112$	0	0
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.13122i	0.844182i
$118$	0	0
$119$	−4.45490	−0.408380
$120$	0	0
$121$	−9.52578	−0.865980
$122$	0	0
$123$	29.1385i	2.62733i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.9508i	1.77035i	0.465258	+	0.885175i	$0.345962\pi$
$127$	19.9508i	1.77035i	−0.465258	+	0.885175i	$0.654038\pi$
$128$	0	0
$129$	−18.4428	−1.62380
$130$	0	0
$131$	−9.70287	−0.847744	−0.423872	−	0.905722i	$-0.639329\pi$
$131$	−9.70287	−0.847744	−0.423872	+	0.905722i	$0.639329\pi$
$132$	0	0
$133$	9.46214i	0.820472i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.82685i	0.241514i	0.992682	+	0.120757i	$0.0385322\pi$
$137$	2.82685i	0.241514i	−0.992682	+	0.120757i	$0.961468\pi$
$138$	0	0
$139$	−2.36471	−0.200572	−0.100286	−	0.994959i	$-0.531976\pi$
$139$	−2.36471	−0.200572	−0.100286	+	0.994959i	$0.531976\pi$
$140$	0	0
$141$	14.5596	1.22614
$142$	0	0
$143$	2.68840i	0.224815i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.2480i	1.42259i
$148$	0	0
$149$	11.9170	0.976282	0.488141	−	0.872765i	$-0.337675\pi$
$149$	11.9170	0.976282	0.488141	+	0.872765i	$0.337675\pi$
$150$	0	0
$151$	9.57889	0.779519	0.389759	−	0.920917i	$-0.372558\pi$
$151$	9.57889	0.779519	0.389759	+	0.920917i	$0.372558\pi$
$152$	0	0
$153$	− 5.00724i	− 0.404811i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.7439i	− 1.49593i	−0.663740	−	0.747963i	$-0.731032\pi$
$157$	− 18.7439i	− 1.49593i	0.663740	−	0.747963i	$-0.268968\pi$
$158$	0	0
$159$	−10.1949	−0.808505
$160$	0	0
$161$	3.66908	0.289164
$162$	0	0
$163$	19.3188i	1.51317i	0.653896	+	0.756584i	$0.273133\pi$
$163$	19.3188i	1.51317i	−0.653896	+	0.756584i	$0.726867\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.81962i	− 0.140806i	−0.997519	−	0.0704031i	$-0.977571\pi$
$167$	− 1.81962i	− 0.140806i	0.997519	−	0.0704031i	$-0.0224285\pi$
$168$	0	0
$169$	8.09743	0.622879
$170$	0	0
$171$	−10.6353	−0.813301
$172$	0	0
$173$	− 15.9581i	− 1.21327i	−0.794981	−	0.606635i	$-0.792519\pi$
$173$	− 15.9581i	− 1.21327i	0.794981	−	0.606635i	$-0.207481\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 11.3382i	− 0.852228i
$178$	0	0
$179$	−17.9508	−1.34171	−0.670854	−	0.741589i	$-0.734072\pi$
$179$	−17.9508	−1.34171	−0.670854	+	0.741589i	$0.734072\pi$
$180$	0	0
$181$	6.33092	0.470574	0.235287	−	0.971926i	$-0.424397\pi$
$181$	6.33092	0.470574	0.235287	+	0.971926i	$0.424397\pi$
$182$	0	0
$183$	− 18.1119i	− 1.33887i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.47422i	− 0.107806i
$188$	0	0
$189$	−11.0072	−0.800659
$190$	0	0
$191$	−15.1578	−1.09678	−0.548389	−	0.836223i	$-0.684759\pi$
$191$	−15.1578	−1.09678	−0.548389	+	0.836223i	$0.684759\pi$
$192$	0	0
$193$	− 18.9879i	− 1.36678i	−0.730053	−	0.683390i	$-0.760505\pi$
$193$	− 18.9879i	− 1.36678i	0.730053	−	0.683390i	$-0.239495\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19.1650i	− 1.36545i	−0.730675	−	0.682725i	$-0.760795\pi$
$197$	− 19.1650i	− 1.36545i	0.730675	−	0.682725i	$-0.239205\pi$
$198$	0	0
$199$	5.76651	0.408777	0.204388	−	0.978890i	$-0.434479\pi$
$199$	5.76651	0.408777	0.204388	+	0.978890i	$0.434479\pi$
$200$	0	0
$201$	34.3155	2.42043
$202$	0	0
$203$	− 5.33816i	− 0.374665i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.12398i	0.286637i
$208$	0	0
$209$	−3.13122	−0.216591
$210$	0	0
$211$	−17.4057	−1.19826	−0.599130	−	0.800652i	$-0.704487\pi$
$211$	−17.4057	−1.19826	−0.599130	+	0.800652i	$0.704487\pi$
$212$	0	0
$213$	18.4621i	1.26501i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 23.7101i	− 1.60955i
$218$	0	0
$219$	40.6272	2.74533
$220$	0	0
$221$	2.68840	0.180841
$222$	0	0
$223$	14.6763i	0.982799i	0.870934	+	0.491399i	$0.163514\pi$
$223$	14.6763i	0.982799i	−0.870934	+	0.491399i	$0.836486\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.4283i	1.09039i	0.838310	+	0.545194i	$0.183544\pi$
$227$	16.4283i	1.09039i	−0.838310	+	0.545194i	$0.816456\pi$
$228$	0	0
$229$	−19.3526	−1.27886	−0.639429	−	0.768850i	$-0.720829\pi$
$229$	−19.3526	−1.27886	−0.639429	+	0.768850i	$0.720829\pi$
$230$	0	0
$231$	−11.8905	−0.782337
$232$	0	0
$233$	19.4549i	1.27453i	0.770643	+	0.637267i	$0.219935\pi$
$233$	19.4549i	1.27453i	−0.770643	+	0.637267i	$0.780065\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.7029	−1.53321	−0.766606	−	0.642118i	$-0.778056\pi$
$239$	−23.7029	−1.53321	−0.766606	+	0.642118i	$0.778056\pi$
$240$	0	0
$241$	0.233492	0.0150405	0.00752027	−	0.999972i	$-0.497606\pi$
$241$	0.233492	0.0150405	0.00752027	+	0.999972i	$0.497606\pi$
$242$	0	0
$243$	20.6498i	1.32468i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.71011i	− 0.363325i
$248$	0	0
$249$	41.6006	2.63633
$250$	0	0
$251$	21.7511	1.37292	0.686460	−	0.727168i	$-0.259164\pi$
$251$	21.7511	1.37292	0.686460	+	0.727168i	$0.259164\pi$
$252$	0	0
$253$	1.21417i	0.0763345i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.3116i	− 1.01749i	−0.860917	−	0.508745i	$-0.830110\pi$
$257$	− 16.3116i	− 1.01749i	0.860917	−	0.508745i	$-0.169890\pi$
$258$	0	0
$259$	−14.6763	−0.911942
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	2.53786i	0.156491i	0.996934	+	0.0782455i	$0.0249318\pi$
$263$	2.53786i	0.156491i	−0.996934	+	0.0782455i	$0.975068\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 29.1191i	− 1.78206i
$268$	0	0
$269$	2.54510	0.155177	0.0775886	−	0.996985i	$-0.475278\pi$
$269$	2.54510	0.155177	0.0775886	+	0.996985i	$0.475278\pi$
$270$	0	0
$271$	24.1795	1.46880	0.734400	−	0.678717i	$-0.237464\pi$
$271$	24.1795	1.46880	0.734400	+	0.678717i	$0.237464\pi$
$272$	0	0
$273$	− 21.6836i	− 1.31235i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 21.2359i	− 1.27594i	−0.770061	−	0.637970i	$-0.779774\pi$
$277$	− 21.2359i	− 1.27594i	0.770061	−	0.637970i	$-0.220226\pi$
$278$	0	0
$279$	26.6498	1.59548
$280$	0	0
$281$	−13.8872	−0.828441	−0.414220	−	0.910177i	$-0.635946\pi$
$281$	−13.8872	−0.828441	−0.414220	+	0.910177i	$0.635946\pi$
$282$	0	0
$283$	29.5330i	1.75556i	0.479068	+	0.877778i	$0.340975\pi$
$283$	29.5330i	1.75556i	−0.479068	+	0.877778i	$0.659025\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 40.0555i	− 2.36440i
$288$	0	0
$289$	15.5258	0.913281
$290$	0	0
$291$	4.52578	0.265306
$292$	0	0
$293$	24.0821i	1.40689i	0.710750	+	0.703444i	$0.248356\pi$
$293$	24.0821i	1.40689i	−0.710750	+	0.703444i	$0.751644\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.64252i	− 0.211361i
$298$	0	0
$299$	−2.21417	−0.128049
$300$	0	0
$301$	25.3526	1.46130
$302$	0	0
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.9436i	1.13824i	0.822254	+	0.569121i	$0.192716\pi$
$307$	19.9436i	1.13824i	−0.822254	+	0.569121i	$0.807284\pi$
$308$	0	0
$309$	7.43559	0.422996
$310$	0	0
$311$	−2.97345	−0.168609	−0.0843043	−	0.996440i	$-0.526867\pi$
$311$	−2.97345	−0.168609	−0.0843043	+	0.996440i	$0.526867\pi$
$312$	0	0
$313$	− 15.4887i	− 0.875473i	−0.899103	−	0.437736i	$-0.855780\pi$
$313$	− 15.4887i	− 0.875473i	0.899103	−	0.437736i	$-0.144220\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.82685i	0.495765i	0.968790	+	0.247883i	$0.0797348\pi$
$317$	8.82685i	0.495765i	−0.968790	+	0.247883i	$0.920265\pi$
$318$	0	0
$319$	1.76651	0.0989055
$320$	0	0
$321$	35.1191	1.96016
$322$	0	0
$323$	3.13122i	0.174226i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.13122i	0.504958i
$328$	0	0
$329$	−20.0145	−1.10343
$330$	0	0
$331$	17.4018	0.956489	0.478245	−	0.878227i	$-0.341273\pi$
$331$	17.4018	0.956489	0.478245	+	0.878227i	$0.341273\pi$
$332$	0	0
$333$	− 16.4959i	− 0.903972i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.9541i	1.46828i	0.678995	+	0.734142i	$0.262416\pi$
$337$	26.9541i	1.46828i	−0.678995	+	0.734142i	$0.737584\pi$
$338$	0	0
$339$	26.6908	1.44964
$340$	0	0
$341$	7.84617	0.424894
$342$	0	0
$343$	1.97345i	0.106556i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10.1240i	− 0.543484i	−0.962370	−	0.271742i	$-0.912400\pi$
$347$	− 10.1240i	− 0.543484i	0.962370	−	0.271742i	$-0.0875997\pi$
$348$	0	0
$349$	−7.38732	−0.395434	−0.197717	−	0.980259i	$-0.563353\pi$
$349$	−7.38732	−0.395434	−0.197717	+	0.980259i	$0.563353\pi$
$350$	0	0
$351$	6.64252	0.354552
$352$	0	0
$353$	− 13.8833i	− 0.738931i	−0.929244	−	0.369466i	$-0.879541\pi$
$353$	− 13.8833i	− 0.738931i	0.929244	−	0.369466i	$-0.120459\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11.8905i	0.629312i
$358$	0	0
$359$	−15.0902	−0.796430	−0.398215	−	0.917292i	$-0.630370\pi$
$359$	−15.0902	−0.796430	−0.398215	+	0.917292i	$0.630370\pi$
$360$	0	0
$361$	−12.3493	−0.649965
$362$	0	0
$363$	25.4251i	1.33447i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 24.0289i	− 1.25430i	−0.778898	−	0.627150i	$-0.784221\pi$
$367$	− 24.0289i	− 1.25430i	0.778898	−	0.627150i	$-0.215779\pi$
$368$	0	0
$369$	45.0217	2.34374
$370$	0	0
$371$	14.0145	0.727595
$372$	0	0
$373$	− 30.6908i	− 1.58911i	−0.607193	−	0.794554i	$-0.707704\pi$
$373$	− 30.6908i	− 1.58911i	0.607193	−	0.794554i	$-0.292296\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.22141i	0.165911i
$378$	0	0
$379$	−2.07087	−0.106374	−0.0531869	−	0.998585i	$-0.516938\pi$
$379$	−2.07087	−0.106374	−0.0531869	+	0.998585i	$0.516938\pi$
$380$	0	0
$381$	53.2504	2.72810
$382$	0	0
$383$	1.58612i	0.0810472i	0.999179	+	0.0405236i	$0.0129026\pi$
$383$	1.58612i	0.0810472i	−0.999179	+	0.0405236i	$0.987097\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	28.4959i	1.44853i
$388$	0	0
$389$	−21.2142	−1.07560	−0.537801	−	0.843072i	$-0.680745\pi$
$389$	−21.2142	−1.07560	−0.537801	+	0.843072i	$0.680745\pi$
$390$	0	0
$391$	1.21417	0.0614035
$392$	0	0
$393$	25.8977i	1.30637i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.93965i	− 0.0973485i	−0.998815	−	0.0486742i	$-0.984500\pi$
$397$	− 1.93965i	− 0.0973485i	0.998815	−	0.0486742i	$-0.0154996\pi$
$398$	0	0
$399$	25.2552	1.26434
$400$	0	0
$401$	−5.69893	−0.284591	−0.142295	−	0.989824i	$-0.545448\pi$
$401$	−5.69893	−0.284591	−0.142295	+	0.989824i	$0.545448\pi$
$402$	0	0
$403$	14.3083i	0.712748i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.85670i	− 0.240738i
$408$	0	0
$409$	−3.53786	−0.174936	−0.0874679	−	0.996167i	$-0.527878\pi$
$409$	−3.53786	−0.174936	−0.0874679	+	0.996167i	$0.527878\pi$
$410$	0	0
$411$	7.54510	0.372172
$412$	0	0
$413$	15.5861i	0.766943i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.31160i	0.309081i
$418$	0	0
$419$	−28.2624	−1.38071	−0.690355	−	0.723471i	$-0.742546\pi$
$419$	−28.2624	−1.38071	−0.690355	+	0.723471i	$0.742546\pi$
$420$	0	0
$421$	34.0974	1.66181	0.830904	−	0.556417i	$-0.187824\pi$
$421$	34.0974	1.66181	0.830904	+	0.556417i	$0.187824\pi$
$422$	0	0
$423$	− 22.4959i	− 1.09379i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	24.8977i	1.20489i
$428$	0	0
$429$	7.17554	0.346438
$430$	0	0
$431$	−13.2850	−0.639918	−0.319959	−	0.947431i	$-0.603669\pi$
$431$	−13.2850	−0.639918	−0.319959	+	0.947431i	$0.603669\pi$
$432$	0	0
$433$	− 11.0603i	− 0.531526i	−0.964038	−	0.265763i	$-0.914376\pi$
$433$	− 11.0603i	− 0.531526i	0.964038	−	0.265763i	$-0.0856239\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.57889i	− 0.123365i
$438$	0	0
$439$	19.1916	0.915963	0.457982	−	0.888962i	$-0.348573\pi$
$439$	19.1916	0.915963	0.457982	+	0.888962i	$0.348573\pi$
$440$	0	0
$441$	26.6498	1.26904
$442$	0	0
$443$	− 24.2818i	− 1.15366i	−0.816864	−	0.576831i	$-0.804289\pi$
$443$	− 24.2818i	− 1.15366i	0.816864	−	0.576831i	$-0.195711\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 31.8075i	− 1.50444i
$448$	0	0
$449$	−14.8413	−0.700406	−0.350203	−	0.936674i	$-0.613887\pi$
$449$	−14.8413	−0.700406	−0.350203	+	0.936674i	$0.613887\pi$
$450$	0	0
$451$	13.2552	0.624163
$452$	0	0
$453$	− 25.5668i	− 1.20123i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.5861i	0.729088i	0.931186	+	0.364544i	$0.118775\pi$
$457$	15.5861i	0.729088i	−0.931186	+	0.364544i	$0.881225\pi$
$458$	0	0
$459$	−3.64252	−0.170019
$460$	0	0
$461$	−14.1312	−0.658157	−0.329078	−	0.944303i	$-0.606738\pi$
$461$	−14.1312	−0.658157	−0.329078	+	0.944303i	$0.606738\pi$
$462$	0	0
$463$	− 26.7584i	− 1.24357i	−0.783189	−	0.621784i	$-0.786408\pi$
$463$	− 26.7584i	− 1.24357i	0.783189	−	0.621784i	$-0.213592\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.6908i	1.23510i	0.786531	+	0.617551i	$0.211875\pi$
$467$	26.6908i	1.23510i	−0.786531	+	0.617551i	$0.788125\pi$
$468$	0	0
$469$	−47.1722	−2.17821
$470$	0	0
$471$	−50.0289	−2.30521
$472$	0	0
$473$	8.38972i	0.385760i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	15.7520i	0.721236i
$478$	0	0
$479$	−4.90981	−0.224335	−0.112167	−	0.993689i	$-0.535779\pi$
$479$	−4.90981	−0.224335	−0.112167	+	0.993689i	$0.535779\pi$
$480$	0	0
$481$	8.85670	0.403831
$482$	0	0
$483$	− 9.79306i	− 0.445600i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.7786i	0.760310i	0.924923	+	0.380155i	$0.124129\pi$
$487$	16.7786i	0.760310i	−0.924923	+	0.380155i	$0.875871\pi$
$488$	0	0
$489$	51.5635	2.33178
$490$	0	0
$491$	41.4839	1.87214	0.936070	−	0.351814i	$-0.114435\pi$
$491$	41.4839	1.87214	0.936070	+	0.351814i	$0.114435\pi$
$492$	0	0
$493$	− 1.76651i	− 0.0795595i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 25.3792i	− 1.13841i
$498$	0	0
$499$	6.47751	0.289973	0.144987	−	0.989434i	$-0.453686\pi$
$499$	6.47751	0.289973	0.144987	+	0.989434i	$0.453686\pi$
$500$	0	0
$501$	−4.85670	−0.216981
$502$	0	0
$503$	− 37.3825i	− 1.66680i	−0.552669	−	0.833401i	$-0.686390\pi$
$503$	− 37.3825i	− 1.66680i	0.552669	−	0.833401i	$-0.313610\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 21.6127i	− 0.959853i
$508$	0	0
$509$	15.0410	0.666682	0.333341	−	0.942806i	$-0.391824\pi$
$509$	15.0410	0.666682	0.333341	+	0.942806i	$0.391824\pi$
$510$	0	0
$511$	−55.8486	−2.47060
$512$	0	0
$513$	7.73666i	0.341582i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.62321i	− 0.291288i
$518$	0	0
$519$	−42.5934	−1.86964
$520$	0	0
$521$	15.9469	0.698646	0.349323	−	0.937002i	$-0.386412\pi$
$521$	15.9469	0.698646	0.349323	+	0.937002i	$0.386412\pi$
$522$	0	0
$523$	− 10.8567i	− 0.474730i	−0.971420	−	0.237365i	$-0.923716\pi$
$523$	− 10.8567i	− 0.474730i	0.971420	−	0.237365i	$-0.0762838\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 7.84617i	− 0.341785i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−17.5185	−0.760240
$532$	0	0
$533$	24.1722i	1.04702i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	47.9122i	2.06756i
$538$	0	0
$539$	7.84617	0.337958
$540$	0	0
$541$	−30.3792	−1.30610	−0.653052	−	0.757313i	$-0.726512\pi$
$541$	−30.3792	−1.30610	−0.653052	+	0.757313i	$0.726512\pi$
$542$	0	0
$543$	− 16.8977i	− 0.725151i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.4090i	1.21468i	0.794441	+	0.607341i	$0.207764\pi$
$547$	28.4090i	1.21468i	−0.794441	+	0.607341i	$0.792236\pi$
$548$	0	0
$549$	−27.9846	−1.19435
$550$	0	0
$551$	−3.75203	−0.159842
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.7584i	1.55750i	0.627333	+	0.778751i	$0.284147\pi$
$557$	36.7584i	1.55750i	−0.627333	+	0.778751i	$0.715853\pi$
$558$	0	0
$559$	−15.2995	−0.647101
$560$	0	0
$561$	−3.93481	−0.166128
$562$	0	0
$563$	3.03708i	0.127998i	0.997950	+	0.0639989i	$0.0203854\pi$
$563$	3.03708i	0.127998i	−0.997950	+	0.0639989i	$0.979615\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 16.0145i	− 0.672545i
$568$	0	0
$569$	32.2093	1.35029	0.675143	−	0.737687i	$-0.264082\pi$
$569$	32.2093	1.35029	0.675143	+	0.737687i	$0.264082\pi$
$570$	0	0
$571$	27.7777	1.16246	0.581230	−	0.813739i	$-0.302572\pi$
$571$	27.7777	1.16246	0.581230	+	0.813739i	$0.302572\pi$
$572$	0	0
$573$	40.4573i	1.69013i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.2890i	0.469967i	0.971999	+	0.234984i	$0.0755036\pi$
$577$	11.2890i	0.469967i	−0.971999	+	0.234984i	$0.924496\pi$
$578$	0	0
$579$	−50.6803	−2.10620
$580$	0	0
$581$	−57.1867	−2.37251
$582$	0	0
$583$	4.63768i	0.192073i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.6609i	0.976592i	0.872678	+	0.488296i	$0.162381\pi$
$587$	23.6609i	0.976592i	−0.872678	+	0.488296i	$0.837619\pi$
$588$	0	0
$589$	−16.6651	−0.686675
$590$	0	0
$591$	−51.1529	−2.10415
$592$	0	0
$593$	17.8341i	0.732358i	0.930544	+	0.366179i	$0.119334\pi$
$593$	17.8341i	0.732358i	−0.930544	+	0.366179i	$0.880666\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 15.3913i	− 0.629923i
$598$	0	0
$599$	23.8308	0.973700	0.486850	−	0.873486i	$-0.338146\pi$
$599$	23.8308	0.973700	0.486850	+	0.873486i	$0.338146\pi$
$600$	0	0
$601$	12.5297	0.511098	0.255549	−	0.966796i	$-0.417744\pi$
$601$	12.5297	0.511098	0.255549	+	0.966796i	$0.417744\pi$
$602$	0	0
$603$	− 53.0208i	− 2.15917i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 23.1722i	− 0.940533i	−0.882525	−	0.470266i	$-0.844158\pi$
$607$	− 23.1722i	− 0.940533i	0.882525	−	0.470266i	$-0.155842\pi$
$608$	0	0
$609$	−14.2480	−0.577357
$610$	0	0
$611$	12.0781	0.488628
$612$	0	0
$613$	44.5780i	1.80049i	0.435386	+	0.900244i	$0.356612\pi$
$613$	44.5780i	1.80049i	−0.435386	+	0.900244i	$0.643388\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 32.1385i	− 1.29385i	−0.762556	−	0.646923i	$-0.776056\pi$
$617$	− 32.1385i	− 1.29385i	0.762556	−	0.646923i	$-0.223944\pi$
$618$	0	0
$619$	−31.5708	−1.26894	−0.634468	−	0.772949i	$-0.718781\pi$
$619$	−31.5708	−1.26894	−0.634468	+	0.772949i	$0.718781\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	40.0289i	1.60373i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.35748i	0.333765i
$628$	0	0
$629$	−4.85670	−0.193649
$630$	0	0
$631$	24.0145	0.956001	0.478001	−	0.878360i	$-0.341362\pi$
$631$	24.0145	0.956001	0.478001	+	0.878360i	$0.341362\pi$
$632$	0	0
$633$	46.4573i	1.84651i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.3083i	0.566916i
$638$	0	0
$639$	28.5258	1.12846
$640$	0	0
$641$	−10.4428	−0.412467	−0.206233	−	0.978503i	$-0.566121\pi$
$641$	−10.4428	−0.412467	−0.206233	+	0.978503i	$0.566121\pi$
$642$	0	0
$643$	− 4.39940i	− 0.173495i	−0.996230	−	0.0867477i	$-0.972353\pi$
$643$	− 4.39940i	− 0.173495i	0.996230	−	0.0867477i	$-0.0276474\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 23.6498i	− 0.929768i	−0.885372	−	0.464884i	$-0.846096\pi$
$647$	− 23.6498i	− 0.929768i	0.885372	−	0.464884i	$-0.153904\pi$
$648$	0	0
$649$	−5.15777	−0.202460
$650$	0	0
$651$	−63.2842	−2.48030
$652$	0	0
$653$	− 40.9194i	− 1.60130i	−0.599131	−	0.800651i	$-0.704487\pi$
$653$	− 40.9194i	− 1.60130i	0.599131	−	0.800651i	$-0.295513\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 62.7728i	− 2.44900i
$658$	0	0
$659$	−48.8567	−1.90319	−0.951593	−	0.307360i	$-0.900555\pi$
$659$	−48.8567	−1.90319	−0.951593	+	0.307360i	$0.900555\pi$
$660$	0	0
$661$	−12.1529	−0.472694	−0.236347	−	0.971669i	$-0.575950\pi$
$661$	−12.1529	−0.472694	−0.236347	+	0.971669i	$0.575950\pi$
$662$	0	0
$663$	− 7.17554i	− 0.278675i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.45490i	0.0563341i
$668$	0	0
$669$	39.1722	1.51449
$670$	0	0
$671$	−8.23918	−0.318070
$672$	0	0
$673$	− 9.20694i	− 0.354901i	−0.984130	−	0.177451i	$-0.943215\pi$
$673$	− 9.20694i	− 0.354901i	0.984130	−	0.177451i	$-0.0567850\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 39.8196i	− 1.53039i	−0.643797	−	0.765196i	$-0.722642\pi$
$677$	− 39.8196i	− 1.53039i	0.643797	−	0.765196i	$-0.277358\pi$
$678$	0	0
$679$	−6.22141	−0.238756
$680$	0	0
$681$	43.8486	1.68028
$682$	0	0
$683$	5.60544i	0.214486i	0.994233	+	0.107243i	$0.0342023\pi$
$683$	5.60544i	0.214486i	−0.994233	+	0.107243i	$0.965798\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	51.6537i	1.97071i
$688$	0	0
$689$	−8.45730	−0.322197
$690$	0	0
$691$	−2.84223	−0.108123	−0.0540617	−	0.998538i	$-0.517217\pi$
$691$	−2.84223	−0.108123	−0.0540617	+	0.998538i	$0.517217\pi$
$692$	0	0
$693$	18.3719i	0.697893i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 13.2552i	− 0.502077i
$698$	0	0
$699$	51.9267	1.96405
$700$	0	0
$701$	−24.4130	−0.922065	−0.461033	−	0.887383i	$-0.652521\pi$
$701$	−24.4130	−0.922065	−0.461033	+	0.887383i	$0.652521\pi$
$702$	0	0
$703$	10.3155i	0.389058i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.24797i	− 0.310197i
$708$	0	0
$709$	−0.123983	−0.00465629	−0.00232814	−	0.999997i	$-0.500741\pi$
$709$	−0.123983	−0.00465629	−0.00232814	+	0.999997i	$0.500741\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.46214i	0.242009i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	63.2648i	2.36267i
$718$	0	0
$719$	−47.1569	−1.75865	−0.879327	−	0.476218i	$-0.842007\pi$
$719$	−47.1569	−1.75865	−0.879327	+	0.476218i	$0.842007\pi$
$720$	0	0
$721$	−10.2214	−0.380665
$722$	0	0
$723$	− 0.623208i	− 0.0231774i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	43.2962i	1.60577i	0.596135	+	0.802884i	$0.296702\pi$
$727$	43.2962i	1.60577i	−0.596135	+	0.802884i	$0.703298\pi$
$728$	0	0
$729$	42.0217	1.55636
$730$	0	0
$731$	8.38972	0.310305
$732$	0	0
$733$	− 11.5861i	− 0.427943i	−0.976840	−	0.213972i	$-0.931360\pi$
$733$	− 11.5861i	− 0.427943i	0.976840	−	0.213972i	$-0.0686400\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 15.6103i	− 0.575012i
$738$	0	0
$739$	30.3792	1.11752	0.558758	−	0.829331i	$-0.311278\pi$
$739$	30.3792	1.11752	0.558758	+	0.829331i	$0.311278\pi$
$740$	0	0
$741$	−15.2407	−0.559882
$742$	0	0
$743$	− 35.5708i	− 1.30496i	−0.757804	−	0.652482i	$-0.773728\pi$
$743$	− 35.5708i	− 1.30496i	0.757804	−	0.652482i	$-0.226272\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 64.2769i	− 2.35177i
$748$	0	0
$749$	−48.2769	−1.76400
$750$	0	0
$751$	−24.9774	−0.911438	−0.455719	−	0.890124i	$-0.650618\pi$
$751$	−24.9774	−0.911438	−0.455719	+	0.890124i	$0.650618\pi$
$752$	0	0
$753$	− 58.0555i	− 2.11566i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.35263i	0.339927i	0.985450	+	0.169964i	$0.0543650\pi$
$757$	9.35263i	0.339927i	−0.985450	+	0.169964i	$0.945635\pi$
$758$	0	0
$759$	3.24073	0.117631
$760$	0	0
$761$	−18.1466	−0.657813	−0.328907	−	0.944362i	$-0.606680\pi$
$761$	−18.1466	−0.657813	−0.328907	+	0.944362i	$0.606680\pi$
$762$	0	0
$763$	− 12.5523i	− 0.454425i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 9.40574i	− 0.339622i
$768$	0	0
$769$	−44.3445	−1.59910	−0.799552	−	0.600597i	$-0.794930\pi$
$769$	−44.3445	−1.59910	−0.799552	+	0.600597i	$0.794930\pi$
$770$	0	0
$771$	−43.5370	−1.56795
$772$	0	0
$773$	10.8036i	0.388578i	0.980944	+	0.194289i	$0.0622400\pi$
$773$	10.8036i	0.388578i	−0.980944	+	0.194289i	$0.937760\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	39.1722i	1.40530i
$778$	0	0
$779$	−28.1538	−1.00872
$780$	0	0
$781$	8.39850	0.300522
$782$	0	0
$783$	− 4.36471i	− 0.155982i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 31.7810i	− 1.13287i	−0.824107	−	0.566435i	$-0.808322\pi$
$787$	− 31.7810i	− 1.13287i	0.824107	−	0.566435i	$-0.191678\pi$
$788$	0	0
$789$	6.77375	0.241152
$790$	0	0
$791$	−36.6908	−1.30457
$792$	0	0
$793$	− 15.0250i	− 0.533554i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.0594i	1.24187i	0.783862	+	0.620935i	$0.213247\pi$
$797$	35.0594i	1.24187i	−0.783862	+	0.620935i	$0.786753\pi$
$798$	0	0
$799$	−6.62321	−0.234312
$800$	0	0
$801$	−44.9919	−1.58971
$802$	0	0
$803$	− 18.4815i	− 0.652196i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.79306i	− 0.239127i
$808$	0	0
$809$	51.4911	1.81033	0.905165	−	0.425060i	$-0.139747\pi$
$809$	51.4911	1.81033	0.905165	+	0.425060i	$0.139747\pi$
$810$	0	0
$811$	−19.7705	−0.694235	−0.347117	−	0.937822i	$-0.612839\pi$
$811$	−19.7705	−0.694235	−0.347117	+	0.937822i	$0.612839\pi$
$812$	0	0
$813$	− 64.5370i	− 2.26341i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 17.8196i	− 0.623429i
$818$	0	0
$819$	−33.5032	−1.17070
$820$	0	0
$821$	31.8486	1.11152	0.555761	−	0.831342i	$-0.312427\pi$
$821$	31.8486	1.11152	0.555761	+	0.831342i	$0.312427\pi$
$822$	0	0
$823$	− 9.34210i	− 0.325645i	−0.986655	−	0.162823i	$-0.947940\pi$
$823$	− 9.34210i	− 0.325645i	0.986655	−	0.162823i	$-0.0520598\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.9243i	− 1.07534i	−0.843155	−	0.537671i	$-0.819304\pi$
$827$	− 30.9243i	− 1.07534i	0.843155	−	0.537671i	$-0.180696\pi$
$828$	0	0
$829$	32.5701	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$829$	32.5701	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$830$	0	0
$831$	−56.6803	−1.96622
$832$	0	0
$833$	− 7.84617i	− 0.271854i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 19.3864i	− 0.670093i
$838$	0	0
$839$	−48.5635	−1.67660	−0.838299	−	0.545210i	$-0.816450\pi$
$839$	−48.5635	−1.67660	−0.838299	+	0.545210i	$0.816450\pi$
$840$	0	0
$841$	−26.8833	−0.927009
$842$	0	0
$843$	37.0660i	1.27662i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 34.9508i	− 1.20092i
$848$	0	0
$849$	78.8260	2.70530
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	33.6546i	1.15231i	0.817340	+	0.576156i	$0.195448\pi$
$853$	33.6546i	1.15231i	−0.817340	+	0.576156i	$0.804552\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.4081i	0.833766i	0.908960	+	0.416883i	$0.136878\pi$
$857$	24.4081i	0.833766i	−0.908960	+	0.416883i	$0.863122\pi$
$858$	0	0
$859$	11.1394	0.380070	0.190035	−	0.981777i	$-0.439140\pi$
$859$	11.1394	0.380070	0.190035	+	0.981777i	$0.439140\pi$
$860$	0	0
$861$	−106.911	−3.64353
$862$	0	0
$863$	15.7173i	0.535025i	0.963554	+	0.267512i	$0.0862016\pi$
$863$	15.7173i	0.535025i	−0.963554	+	0.267512i	$0.913798\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 41.4395i	− 1.40736i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	28.4670	0.964567
$872$	0	0
$873$	− 6.99276i	− 0.236669i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 33.2818i	− 1.12385i	−0.827190	−	0.561923i	$-0.810062\pi$
$877$	− 33.2818i	− 1.12385i	0.827190	−	0.561923i	$-0.189938\pi$
$878$	0	0
$879$	64.2769	2.16801
$880$	0	0
$881$	8.99187	0.302944	0.151472	−	0.988462i	$-0.451599\pi$
$881$	8.99187	0.302944	0.151472	+	0.988462i	$0.451599\pi$
$882$	0	0
$883$	− 58.5258i	− 1.96955i	−0.173836	−	0.984775i	$-0.555616\pi$
$883$	− 58.5258i	− 1.96955i	0.173836	−	0.984775i	$-0.444384\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.8220i	1.10206i	0.834487	+	0.551028i	$0.185764\pi$
$887$	32.8220i	1.10206i	−0.834487	+	0.551028i	$0.814236\pi$
$888$	0	0
$889$	−73.2012	−2.45509
$890$	0	0
$891$	5.29952	0.177541
$892$	0	0
$893$	14.0676i	0.470754i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.90981i	0.197323i
$898$	0	0
$899$	9.40180	0.313567
$900$	0	0
$901$	4.63768	0.154504
$902$	0	0
$903$	− 67.6682i	− 2.25186i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.3300i	0.940683i	0.882484	+	0.470341i	$0.155869\pi$
$907$	28.3300i	0.940683i	−0.882484	+	0.470341i	$0.844131\pi$
$908$	0	0
$909$	9.27058	0.307486
$910$	0	0
$911$	15.3382	0.508176	0.254088	−	0.967181i	$-0.418225\pi$
$911$	15.3382	0.508176	0.254088	+	0.967181i	$0.418225\pi$
$912$	0	0
$913$	− 18.9243i	− 0.626302i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 35.6006i	− 1.17564i
$918$	0	0
$919$	38.8036	1.28001	0.640006	−	0.768370i	$-0.278932\pi$
$919$	38.8036	1.28001	0.640006	+	0.768370i	$0.278932\pi$
$920$	0	0
$921$	53.2310	1.75402
$922$	0	0
$923$	15.3155i	0.504117i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 11.4887i	− 0.377338i
$928$	0	0
$929$	−4.01053	−0.131581	−0.0657906	−	0.997833i	$-0.520957\pi$
$929$	−4.01053	−0.131581	−0.0657906	+	0.997833i	$0.520957\pi$
$930$	0	0
$931$	−16.6651	−0.546178
$932$	0	0
$933$	7.93636i	0.259825i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.6836i	1.42708i	0.700615	+	0.713540i	$0.252909\pi$
$937$	43.6836i	1.42708i	−0.700615	+	0.713540i	$0.747091\pi$
$938$	0	0
$939$	−41.3406	−1.34910
$940$	0	0
$941$	−24.8534	−0.810198	−0.405099	−	0.914273i	$-0.632763\pi$
$941$	−24.8534	−0.810198	−0.405099	+	0.914273i	$0.632763\pi$
$942$	0	0
$943$	10.9170i	0.355508i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22.1690i	− 0.720394i	−0.932876	−	0.360197i	$-0.882709\pi$
$947$	− 22.1690i	− 0.720394i	0.932876	−	0.360197i	$-0.117291\pi$
$948$	0	0
$949$	33.7029	1.09404
$950$	0	0
$951$	23.5596	0.763971
$952$	0	0
$953$	− 15.9050i	− 0.515212i	−0.966250	−	0.257606i	$-0.917066\pi$
$953$	− 15.9050i	− 0.515212i	0.966250	−	0.257606i	$-0.0829337\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 4.71495i	− 0.152413i
$958$	0	0
$959$	−10.3719	−0.334928
$960$	0	0
$961$	10.7593	0.347073
$962$	0	0
$963$	− 54.2624i	− 1.74858i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	50.6272i	1.62806i	0.580823	+	0.814030i	$0.302731\pi$
$967$	50.6272i	1.62806i	−0.580823	+	0.814030i	$0.697269\pi$
$968$	0	0
$969$	8.35748	0.268481
$970$	0	0
$971$	−52.4009	−1.68162	−0.840812	−	0.541327i	$-0.817922\pi$
$971$	−52.4009	−1.68162	−0.840812	+	0.541327i	$0.817922\pi$
$972$	0	0
$973$	− 8.67632i	− 0.278150i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.5258i	0.336750i	0.985723	+	0.168375i	$0.0538519\pi$
$977$	10.5258i	0.336750i	−0.985723	+	0.168375i	$0.946148\pi$
$978$	0	0
$979$	−13.2464	−0.423357
$980$	0	0
$981$	14.1086	0.450453
$982$	0	0
$983$	− 9.44767i	− 0.301334i	−0.988585	−	0.150667i	$-0.951858\pi$
$983$	− 9.44767i	− 0.301334i	0.988585	−	0.150667i	$-0.0481421\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	53.4202i	1.70038i
$988$	0	0
$989$	−6.90981	−0.219719
$990$	0	0
$991$	33.3252	1.05861	0.529305	−	0.848432i	$-0.322453\pi$
$991$	33.3252	1.05861	0.529305	+	0.848432i	$0.322453\pi$
$992$	0	0
$993$	− 46.4468i	− 1.47394i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	54.0289i	1.71111i	0.517709	+	0.855557i	$0.326785\pi$
$997$	54.0289i	1.71111i	−0.517709	+	0.855557i	$0.673215\pi$
$998$	0	0
$999$	−12.0000	−0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.r.4049.1		6
5.2	odd	4		4600.2.a.y.1.1		3
5.3	odd	4		920.2.a.g.1.3	✓	3
5.4	even	2	inner	4600.2.e.r.4049.6		6
15.8	even	4		8280.2.a.bo.1.3		3
20.3	even	4		1840.2.a.t.1.1		3
20.7	even	4		9200.2.a.cd.1.3		3
40.3	even	4		7360.2.a.ca.1.3		3
40.13	odd	4		7360.2.a.cb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.g.1.3	✓	3	5.3	odd	4
1840.2.a.t.1.1		3	20.3	even	4
4600.2.a.y.1.1		3	5.2	odd	4
4600.2.e.r.4049.1		6	1.1	even	1	trivial
4600.2.e.r.4049.6		6	5.4	even	2	inner
7360.2.a.ca.1.3		3	40.3	even	4
7360.2.a.cb.1.1		3	40.13	odd	4
8280.2.a.bo.1.3		3	15.8	even	4
9200.2.a.cd.1.3		3	20.7	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.66908i q^{3} +3.66908i q^{7} -4.12398 q^{9} +O(q^{10})\)	\(q-2.66908i q^{3} +3.66908i q^{7} -4.12398 q^{9} -1.21417 q^{11} -2.21417i q^{13} +1.21417i q^{17} +2.57889 q^{19} +9.79306 q^{21} -1.00000i q^{23} +3.00000i q^{27} -1.45490 q^{29} -6.46214 q^{31} +3.24073i q^{33} +4.00000i q^{37} -5.90981 q^{39} -10.9170 q^{41} -6.90981i q^{43} +5.45490i q^{47} -6.46214 q^{49} +3.24073 q^{51} -3.81962i q^{53} -6.88325i q^{57} +4.24797 q^{59} +6.78583 q^{61} -15.1312i q^{63} +12.8567i q^{67} -2.66908 q^{69} -6.91705 q^{71} +15.2214i q^{73} -4.45490i q^{77} -4.36471 q^{81} +15.5861i q^{83} +3.88325i q^{87} +10.9098 q^{89} +8.12398 q^{91} +17.2480i q^{93} +1.69563i q^{97} +5.00724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 6 * q^11 - 6 * q^19 + 24 * q^21 - 6 * q^29 + 12 * q^31 - 30 * q^39 - 12 * q^41 + 12 * q^49 + 30 * q^51 - 12 * q^59 + 54 * q^61 + 12 * q^71 - 18 * q^81 + 60 * q^89 + 30 * q^91 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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