LMFDB - Embedded newform 4600.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24698 q^{3} -4.49396 q^{7} +2.04892 q^{9} +3.38404 q^{11} -3.04892 q^{13} -4.49396 q^{17} +7.20775 q^{19} +10.0978 q^{21} +1.00000 q^{23} +2.13706 q^{27} -5.51573 q^{29} -1.29590 q^{31} -7.60388 q^{33} +5.82371 q^{37} +6.85086 q^{39} +3.63102 q^{41} +10.7138 q^{43} -2.06100 q^{47} +13.1957 q^{49} +10.0978 q^{51} +2.98792 q^{53} -16.1957 q^{57} -9.31767 q^{59} -13.3056 q^{61} -9.20775 q^{63} +8.19567 q^{67} -2.24698 q^{69} +3.48858 q^{71} +8.72348 q^{73} -15.2078 q^{77} -9.92154 q^{79} -10.9487 q^{81} +15.7560 q^{83} +12.3937 q^{87} -0.121998 q^{89} +13.7017 q^{91} +2.91185 q^{93} -15.6039 q^{97} +6.93362 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 4 q^{7} - 3 q^{9} - 4 q^{17} + 4 q^{19} + 12 q^{21} + 3 q^{23} + q^{27} - 4 q^{29} + 10 q^{31} - 14 q^{33} + 10 q^{37} + 7 q^{39} - 4 q^{41} + 24 q^{43} - 16 q^{47} + 3 q^{49} + 12 q^{51}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 4 * q^7 - 3 * q^9 - 4 * q^17 + 4 * q^19 + 12 * q^21 + 3 * q^23 + q^27 - 4 * q^29 + 10 * q^31 - 14 * q^33 + 10 * q^37 + 7 * q^39 - 4 * q^41 + 24 * q^43 - 16 * q^47 + 3 * q^49 + 12 * q^51 - 10 * q^53 - 12 * q^57 - 11 * q^59 - 4 * q^61 - 10 * q^63 - 12 * q^67 - 2 * q^69 + 4 * q^71 - 4 * q^73 - 28 * q^77 - 4 * q^79 - q^81 + 8 * q^83 + 5 * q^87 - 20 * q^89 + 14 * q^91 + 5 * q^93 - 38 * q^97 + 14 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.24698	−1.29729	−0.648647	−	0.761089i	$-0.724665\pi$
$3$	−2.24698	−1.29729	−0.648647	+	0.761089i	$0.724665\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.49396	−1.69856	−0.849278	−	0.527945i	$-0.822963\pi$
$7$	−4.49396	−1.69856	−0.849278	+	0.527945i	$0.822963\pi$
$8$	0	0
$9$	2.04892	0.682972
$10$	0	0
$11$	3.38404	1.02033	0.510164	−	0.860077i	$-0.329585\pi$
$11$	3.38404	1.02033	0.510164	+	0.860077i	$0.329585\pi$
$12$	0	0
$13$	−3.04892	−0.845618	−0.422809	−	0.906219i	$-0.638956\pi$
$13$	−3.04892	−0.845618	−0.422809	+	0.906219i	$0.638956\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.49396	−1.08995	−0.544973	−	0.838454i	$-0.683460\pi$
$17$	−4.49396	−1.08995	−0.544973	+	0.838454i	$0.683460\pi$
$18$	0	0
$19$	7.20775	1.65357	0.826786	−	0.562517i	$-0.190167\pi$
$19$	7.20775	1.65357	0.826786	+	0.562517i	$0.190167\pi$
$20$	0	0
$21$	10.0978	2.20353
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.13706	0.411278
$28$	0	0
$29$	−5.51573	−1.02425	−0.512123	−	0.858912i	$-0.671141\pi$
$29$	−5.51573	−1.02425	−0.512123	+	0.858912i	$0.671141\pi$
$30$	0	0
$31$	−1.29590	−0.232750	−0.116375	−	0.993205i	$-0.537127\pi$
$31$	−1.29590	−0.232750	−0.116375	+	0.993205i	$0.537127\pi$
$32$	0	0
$33$	−7.60388	−1.32366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.82371	0.957412	0.478706	−	0.877975i	$-0.341106\pi$
$37$	5.82371	0.957412	0.478706	+	0.877975i	$0.341106\pi$
$38$	0	0
$39$	6.85086	1.09701
$40$	0	0
$41$	3.63102	0.567070	0.283535	−	0.958962i	$-0.408493\pi$
$41$	3.63102	0.567070	0.283535	+	0.958962i	$0.408493\pi$
$42$	0	0
$43$	10.7138	1.63384	0.816919	−	0.576752i	$-0.195680\pi$
$43$	10.7138	1.63384	0.816919	+	0.576752i	$0.195680\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.06100	−0.300628	−0.150314	−	0.988638i	$-0.548028\pi$
$47$	−2.06100	−0.300628	−0.150314	+	0.988638i	$0.548028\pi$
$48$	0	0
$49$	13.1957	1.88510
$50$	0	0
$51$	10.0978	1.41398
$52$	0	0
$53$	2.98792	0.410422	0.205211	−	0.978718i	$-0.434212\pi$
$53$	2.98792	0.410422	0.205211	+	0.978718i	$0.434212\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−16.1957	−2.14517
$58$	0	0
$59$	−9.31767	−1.21306	−0.606528	−	0.795062i	$-0.707438\pi$
$59$	−9.31767	−1.21306	−0.606528	+	0.795062i	$0.707438\pi$
$60$	0	0
$61$	−13.3056	−1.70361	−0.851803	−	0.523863i	$-0.824491\pi$
$61$	−13.3056	−1.70361	−0.851803	+	0.523863i	$0.824491\pi$
$62$	0	0
$63$	−9.20775	−1.16007
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.19567	1.00126	0.500630	−	0.865661i	$-0.333102\pi$
$67$	8.19567	1.00126	0.500630	+	0.865661i	$0.333102\pi$
$68$	0	0
$69$	−2.24698	−0.270505
$70$	0	0
$71$	3.48858	0.414019	0.207009	−	0.978339i	$-0.433627\pi$
$71$	3.48858	0.414019	0.207009	+	0.978339i	$0.433627\pi$
$72$	0	0
$73$	8.72348	1.02101	0.510503	−	0.859876i	$-0.329459\pi$
$73$	8.72348	1.02101	0.510503	+	0.859876i	$0.329459\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.2078	−1.73308
$78$	0	0
$79$	−9.92154	−1.11626	−0.558130	−	0.829753i	$-0.688481\pi$
$79$	−9.92154	−1.11626	−0.558130	+	0.829753i	$0.688481\pi$
$80$	0	0
$81$	−10.9487	−1.21652
$82$	0	0
$83$	15.7560	1.72945	0.864723	−	0.502249i	$-0.167494\pi$
$83$	15.7560	1.72945	0.864723	+	0.502249i	$0.167494\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.3937	1.32875
$88$	0	0
$89$	−0.121998	−0.0129317	−0.00646587	−	0.999979i	$-0.502058\pi$
$89$	−0.121998	−0.0129317	−0.00646587	+	0.999979i	$0.502058\pi$
$90$	0	0
$91$	13.7017	1.43633
$92$	0	0
$93$	2.91185	0.301945
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.6039	−1.58433	−0.792167	−	0.610305i	$-0.791047\pi$
$97$	−15.6039	−1.58433	−0.792167	+	0.610305i	$0.791047\pi$
$98$	0	0
$99$	6.93362	0.696855
$100$	0	0
$101$	17.4155	1.73291	0.866454	−	0.499258i	$-0.166394\pi$
$101$	17.4155	1.73291	0.866454	+	0.499258i	$0.166394\pi$
$102$	0	0
$103$	17.5797	1.73218	0.866090	−	0.499888i	$-0.166625\pi$
$103$	17.5797	1.73218	0.866090	+	0.499888i	$0.166625\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.7138	−1.22909	−0.614544	−	0.788882i	$-0.710660\pi$
$107$	−12.7138	−1.22909	−0.614544	+	0.788882i	$0.710660\pi$
$108$	0	0
$109$	−5.32975	−0.510497	−0.255249	−	0.966875i	$-0.582157\pi$
$109$	−5.32975	−0.510497	−0.255249	+	0.966875i	$0.582157\pi$
$110$	0	0
$111$	−13.0858	−1.24204
$112$	0	0
$113$	−6.21983	−0.585113	−0.292556	−	0.956248i	$-0.594506\pi$
$113$	−6.21983	−0.585113	−0.292556	+	0.956248i	$0.594506\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.24698	−0.577533
$118$	0	0
$119$	20.1957	1.85133
$120$	0	0
$121$	0.451747	0.0410679
$122$	0	0
$123$	−8.15883	−0.735657
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.08815	−0.185293	−0.0926465	−	0.995699i	$-0.529533\pi$
$127$	−2.08815	−0.185293	−0.0926465	+	0.995699i	$0.529533\pi$
$128$	0	0
$129$	−24.0737	−2.11957
$130$	0	0
$131$	10.7952	0.943184	0.471592	−	0.881817i	$-0.343680\pi$
$131$	10.7952	0.943184	0.471592	+	0.881817i	$0.343680\pi$
$132$	0	0
$133$	−32.3913	−2.80869
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.39612	0.204715	0.102357	−	0.994748i	$-0.467361\pi$
$137$	2.39612	0.204715	0.102357	+	0.994748i	$0.467361\pi$
$138$	0	0
$139$	−10.0610	−0.853363	−0.426681	−	0.904402i	$-0.640317\pi$
$139$	−10.0610	−0.853363	−0.426681	+	0.904402i	$0.640317\pi$
$140$	0	0
$141$	4.63102	0.390002
$142$	0	0
$143$	−10.3177	−0.862807
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−29.6504	−2.44552
$148$	0	0
$149$	−0.846543	−0.0693515	−0.0346758	−	0.999399i	$-0.511040\pi$
$149$	−0.846543	−0.0693515	−0.0346758	+	0.999399i	$0.511040\pi$
$150$	0	0
$151$	−11.3666	−0.925000	−0.462500	−	0.886619i	$-0.653047\pi$
$151$	−11.3666	−0.925000	−0.462500	+	0.886619i	$0.653047\pi$
$152$	0	0
$153$	−9.20775	−0.744403
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−15.0858	−1.20397	−0.601987	−	0.798506i	$-0.705624\pi$
$157$	−15.0858	−1.20397	−0.601987	+	0.798506i	$0.705624\pi$
$158$	0	0
$159$	−6.71379	−0.532438
$160$	0	0
$161$	−4.49396	−0.354174
$162$	0	0
$163$	9.11960	0.714303	0.357151	−	0.934047i	$-0.383748\pi$
$163$	9.11960	0.714303	0.357151	+	0.934047i	$0.383748\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.4383	−0.962507	−0.481254	−	0.876581i	$-0.659818\pi$
$167$	−12.4383	−0.962507	−0.481254	+	0.876581i	$0.659818\pi$
$168$	0	0
$169$	−3.70410	−0.284931
$170$	0	0
$171$	14.7681	1.12934
$172$	0	0
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.9366	1.57369
$178$	0	0
$179$	8.30798	0.620967	0.310484	−	0.950579i	$-0.399509\pi$
$179$	8.30798	0.620967	0.310484	+	0.950579i	$0.399509\pi$
$180$	0	0
$181$	−16.7138	−1.24233	−0.621163	−	0.783681i	$-0.713340\pi$
$181$	−16.7138	−1.24233	−0.621163	+	0.783681i	$0.713340\pi$
$182$	0	0
$183$	29.8974	2.21008
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−15.2078	−1.11210
$188$	0	0
$189$	−9.60388	−0.698579
$190$	0	0
$191$	−12.9638	−0.938024	−0.469012	−	0.883192i	$-0.655390\pi$
$191$	−12.9638	−0.938024	−0.469012	+	0.883192i	$0.655390\pi$
$192$	0	0
$193$	−5.70948	−0.410977	−0.205489	−	0.978659i	$-0.565878\pi$
$193$	−5.70948	−0.410977	−0.205489	+	0.978659i	$0.565878\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.618941	0.0440977	0.0220489	−	0.999757i	$-0.492981\pi$
$197$	0.618941	0.0440977	0.0220489	+	0.999757i	$0.492981\pi$
$198$	0	0
$199$	−10.4155	−0.738335	−0.369168	−	0.929363i	$-0.620357\pi$
$199$	−10.4155	−0.738335	−0.369168	+	0.929363i	$0.620357\pi$
$200$	0	0
$201$	−18.4155	−1.29893
$202$	0	0
$203$	24.7875	1.73974
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.04892	0.142410
$208$	0	0
$209$	24.3913	1.68718
$210$	0	0
$211$	28.3056	1.94864	0.974318	−	0.225175i	$-0.0722952\pi$
$211$	28.3056	1.94864	0.974318	+	0.225175i	$0.0722952\pi$
$212$	0	0
$213$	−7.83877	−0.537104
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.82371	0.395339
$218$	0	0
$219$	−19.6015	−1.32455
$220$	0	0
$221$	13.7017	0.921677
$222$	0	0
$223$	−4.08575	−0.273602	−0.136801	−	0.990599i	$-0.543682\pi$
$223$	−4.08575	−0.273602	−0.136801	+	0.990599i	$0.543682\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.7681	−1.51117	−0.755585	−	0.655050i	$-0.772648\pi$
$227$	−22.7681	−1.51117	−0.755585	+	0.655050i	$0.772648\pi$
$228$	0	0
$229$	3.74525	0.247493	0.123747	−	0.992314i	$-0.460509\pi$
$229$	3.74525	0.247493	0.123747	+	0.992314i	$0.460509\pi$
$230$	0	0
$231$	34.1715	2.24832
$232$	0	0
$233$	−1.62133	−0.106217	−0.0531086	−	0.998589i	$-0.516913\pi$
$233$	−1.62133	−0.106217	−0.0531086	+	0.998589i	$0.516913\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	22.2935	1.44812
$238$	0	0
$239$	17.1021	1.10625	0.553123	−	0.833100i	$-0.313436\pi$
$239$	17.1021	1.10625	0.553123	+	0.833100i	$0.313436\pi$
$240$	0	0
$241$	0.165538	0.0106633	0.00533163	−	0.999986i	$-0.498303\pi$
$241$	0.165538	0.0106633	0.00533163	+	0.999986i	$0.498303\pi$
$242$	0	0
$243$	18.1903	1.16691
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−21.9758	−1.39829
$248$	0	0
$249$	−35.4034	−2.24360
$250$	0	0
$251$	−15.6775	−0.989558	−0.494779	−	0.869019i	$-0.664751\pi$
$251$	−15.6775	−0.989558	−0.494779	+	0.869019i	$0.664751\pi$
$252$	0	0
$253$	3.38404	0.212753
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.9922	1.12232	0.561162	−	0.827706i	$-0.310354\pi$
$257$	17.9922	1.12232	0.561162	+	0.827706i	$0.310354\pi$
$258$	0	0
$259$	−26.1715	−1.62622
$260$	0	0
$261$	−11.3013	−0.699531
$262$	0	0
$263$	−9.70171	−0.598233	−0.299117	−	0.954217i	$-0.596692\pi$
$263$	−9.70171	−0.598233	−0.299117	+	0.954217i	$0.596692\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.274127	0.0167763
$268$	0	0
$269$	−21.4644	−1.30871	−0.654354	−	0.756188i	$-0.727060\pi$
$269$	−21.4644	−1.30871	−0.654354	+	0.756188i	$0.727060\pi$
$270$	0	0
$271$	1.89008	0.114814	0.0574072	−	0.998351i	$-0.481717\pi$
$271$	1.89008	0.114814	0.0574072	+	0.998351i	$0.481717\pi$
$272$	0	0
$273$	−30.7875	−1.86334
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−12.5646	−0.754936	−0.377468	−	0.926023i	$-0.623205\pi$
$277$	−12.5646	−0.754936	−0.377468	+	0.926023i	$0.623205\pi$
$278$	0	0
$279$	−2.65519	−0.158962
$280$	0	0
$281$	22.1521	1.32149	0.660743	−	0.750613i	$-0.270241\pi$
$281$	22.1521	1.32149	0.660743	+	0.750613i	$0.270241\pi$
$282$	0	0
$283$	−26.1172	−1.55251	−0.776254	−	0.630421i	$-0.782882\pi$
$283$	−26.1172	−1.55251	−0.776254	+	0.630421i	$0.782882\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−16.3177	−0.963201
$288$	0	0
$289$	3.19567	0.187981
$290$	0	0
$291$	35.0616	2.05535
$292$	0	0
$293$	−32.2500	−1.88406	−0.942031	−	0.335524i	$-0.891086\pi$
$293$	−32.2500	−1.88406	−0.942031	+	0.335524i	$0.891086\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.23191	0.419638
$298$	0	0
$299$	−3.04892	−0.176323
$300$	0	0
$301$	−48.1473	−2.77517
$302$	0	0
$303$	−39.1323	−2.24809
$304$	0	0
$305$	0	0
$306$	0	0
$307$	11.2078	0.639660	0.319830	−	0.947475i	$-0.396374\pi$
$307$	11.2078	0.639660	0.319830	+	0.947475i	$0.396374\pi$
$308$	0	0
$309$	−39.5013	−2.24715
$310$	0	0
$311$	−23.8200	−1.35071	−0.675354	−	0.737494i	$-0.736009\pi$
$311$	−23.8200	−1.35071	−0.675354	+	0.737494i	$0.736009\pi$
$312$	0	0
$313$	1.43834	0.0812996	0.0406498	−	0.999173i	$-0.487057\pi$
$313$	1.43834	0.0812996	0.0406498	+	0.999173i	$0.487057\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.3163	0.747920	0.373960	−	0.927445i	$-0.378000\pi$
$317$	13.3163	0.747920	0.373960	+	0.927445i	$0.378000\pi$
$318$	0	0
$319$	−18.6655	−1.04507
$320$	0	0
$321$	28.5676	1.59449
$322$	0	0
$323$	−32.3913	−1.80230
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.9758	0.662265
$328$	0	0
$329$	9.26205	0.510633
$330$	0	0
$331$	7.09352	0.389895	0.194948	−	0.980814i	$-0.437546\pi$
$331$	7.09352	0.389895	0.194948	+	0.980814i	$0.437546\pi$
$332$	0	0
$333$	11.9323	0.653886
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.9202	1.68433	0.842166	−	0.539219i	$-0.181280\pi$
$337$	30.9202	1.68433	0.842166	+	0.539219i	$0.181280\pi$
$338$	0	0
$339$	13.9758	0.759063
$340$	0	0
$341$	−4.38537	−0.237481
$342$	0	0
$343$	−27.8431	−1.50339
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.6340	−1.10769	−0.553846	−	0.832619i	$-0.686840\pi$
$347$	−20.6340	−1.10769	−0.553846	+	0.832619i	$0.686840\pi$
$348$	0	0
$349$	−11.8412	−0.633843	−0.316922	−	0.948452i	$-0.602649\pi$
$349$	−11.8412	−0.633843	−0.316922	+	0.948452i	$0.602649\pi$
$350$	0	0
$351$	−6.51573	−0.347784
$352$	0	0
$353$	−10.3448	−0.550599	−0.275299	−	0.961359i	$-0.588777\pi$
$353$	−10.3448	−0.550599	−0.275299	+	0.961359i	$0.588777\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−45.3793	−2.40172
$358$	0	0
$359$	5.70171	0.300925	0.150462	−	0.988616i	$-0.451924\pi$
$359$	5.70171	0.300925	0.150462	+	0.988616i	$0.451924\pi$
$360$	0	0
$361$	32.9517	1.73430
$362$	0	0
$363$	−1.01507	−0.0532771
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.6789	−1.49703	−0.748513	−	0.663121i	$-0.769232\pi$
$367$	−28.6789	−1.49703	−0.748513	+	0.663121i	$0.769232\pi$
$368$	0	0
$369$	7.43967	0.387293
$370$	0	0
$371$	−13.4276	−0.697125
$372$	0	0
$373$	−2.72455	−0.141072	−0.0705358	−	0.997509i	$-0.522471\pi$
$373$	−2.72455	−0.141072	−0.0705358	+	0.997509i	$0.522471\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.8170	0.866120
$378$	0	0
$379$	−26.0737	−1.33931	−0.669657	−	0.742670i	$-0.733559\pi$
$379$	−26.0737	−1.33931	−0.669657	+	0.742670i	$0.733559\pi$
$380$	0	0
$381$	4.69202	0.240380
$382$	0	0
$383$	30.8745	1.57762	0.788808	−	0.614640i	$-0.210699\pi$
$383$	30.8745	1.57762	0.788808	+	0.614640i	$0.210699\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	21.9517	1.11587
$388$	0	0
$389$	2.76809	0.140348	0.0701738	−	0.997535i	$-0.477645\pi$
$389$	2.76809	0.140348	0.0701738	+	0.997535i	$0.477645\pi$
$390$	0	0
$391$	−4.49396	−0.227269
$392$	0	0
$393$	−24.2567	−1.22359
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.2790	1.51966	0.759830	−	0.650121i	$-0.225282\pi$
$397$	30.2790	1.51966	0.759830	+	0.650121i	$0.225282\pi$
$398$	0	0
$399$	72.7827	3.64369
$400$	0	0
$401$	11.4276	0.570666	0.285333	−	0.958428i	$-0.407896\pi$
$401$	11.4276	0.570666	0.285333	+	0.958428i	$0.407896\pi$
$402$	0	0
$403$	3.95108	0.196817
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.7077	0.976873
$408$	0	0
$409$	3.68532	0.182227	0.0911136	−	0.995841i	$-0.470957\pi$
$409$	3.68532	0.182227	0.0911136	+	0.995841i	$0.470957\pi$
$410$	0	0
$411$	−5.38404	−0.265575
$412$	0	0
$413$	41.8732	2.06045
$414$	0	0
$415$	0	0
$416$	0	0
$417$	22.6069	1.10706
$418$	0	0
$419$	−39.5749	−1.93336	−0.966681	−	0.255985i	$-0.917600\pi$
$419$	−39.5749	−1.93336	−0.966681	+	0.255985i	$0.917600\pi$
$420$	0	0
$421$	−14.9202	−0.727167	−0.363583	−	0.931562i	$-0.618447\pi$
$421$	−14.9202	−0.727167	−0.363583	+	0.931562i	$0.618447\pi$
$422$	0	0
$423$	−4.22282	−0.205320
$424$	0	0
$425$	0	0
$426$	0	0
$427$	59.7948	2.89367
$428$	0	0
$429$	23.1836	1.11931
$430$	0	0
$431$	25.2271	1.21515	0.607574	−	0.794263i	$-0.292143\pi$
$431$	25.2271	1.21515	0.607574	+	0.794263i	$0.292143\pi$
$432$	0	0
$433$	−29.4577	−1.41565	−0.707824	−	0.706389i	$-0.750323\pi$
$433$	−29.4577	−1.41565	−0.707824	+	0.706389i	$0.750323\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.20775	0.344793
$438$	0	0
$439$	31.3763	1.49751	0.748754	−	0.662848i	$-0.230652\pi$
$439$	31.3763	1.49751	0.748754	+	0.662848i	$0.230652\pi$
$440$	0	0
$441$	27.0368	1.28747
$442$	0	0
$443$	−35.8799	−1.70471	−0.852353	−	0.522966i	$-0.824825\pi$
$443$	−35.8799	−1.70471	−0.852353	+	0.522966i	$0.824825\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.90217	0.0899693
$448$	0	0
$449$	−9.74392	−0.459844	−0.229922	−	0.973209i	$-0.573847\pi$
$449$	−9.74392	−0.459844	−0.229922	+	0.973209i	$0.573847\pi$
$450$	0	0
$451$	12.2875	0.578597
$452$	0	0
$453$	25.5405	1.20000
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.2935	−1.13640	−0.568201	−	0.822890i	$-0.692360\pi$
$457$	−24.2935	−1.13640	−0.568201	+	0.822890i	$0.692360\pi$
$458$	0	0
$459$	−9.60388	−0.448271
$460$	0	0
$461$	−37.2687	−1.73578	−0.867889	−	0.496758i	$-0.834524\pi$
$461$	−37.2687	−1.73578	−0.867889	+	0.496758i	$0.834524\pi$
$462$	0	0
$463$	11.7332	0.545287	0.272643	−	0.962115i	$-0.412102\pi$
$463$	11.7332	0.545287	0.272643	+	0.962115i	$0.412102\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.7861	−1.93363	−0.966816	−	0.255474i	$-0.917768\pi$
$467$	−41.7861	−1.93363	−0.966816	+	0.255474i	$0.917768\pi$
$468$	0	0
$469$	−36.8310	−1.70070
$470$	0	0
$471$	33.8974	1.56191
$472$	0	0
$473$	36.2559	1.66705
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.12200	0.280307
$478$	0	0
$479$	3.34050	0.152631	0.0763157	−	0.997084i	$-0.475684\pi$
$479$	3.34050	0.152631	0.0763157	+	0.997084i	$0.475684\pi$
$480$	0	0
$481$	−17.7560	−0.809604
$482$	0	0
$483$	10.0978	0.459467
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.3653	1.01347	0.506733	−	0.862103i	$-0.330853\pi$
$487$	22.3653	1.01347	0.506733	+	0.862103i	$0.330853\pi$
$488$	0	0
$489$	−20.4916	−0.926661
$490$	0	0
$491$	−7.64550	−0.345036	−0.172518	−	0.985006i	$-0.555190\pi$
$491$	−7.64550	−0.345036	−0.172518	+	0.985006i	$0.555190\pi$
$492$	0	0
$493$	24.7875	1.11637
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.6775	−0.703234
$498$	0	0
$499$	−22.9366	−1.02678	−0.513392	−	0.858154i	$-0.671611\pi$
$499$	−22.9366	−1.02678	−0.513392	+	0.858154i	$0.671611\pi$
$500$	0	0
$501$	27.9487	1.24866
$502$	0	0
$503$	6.37196	0.284112	0.142056	−	0.989859i	$-0.454629\pi$
$503$	6.37196	0.284112	0.142056	+	0.989859i	$0.454629\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.32304	0.369639
$508$	0	0
$509$	19.8810	0.881209	0.440605	−	0.897701i	$-0.354764\pi$
$509$	19.8810	0.881209	0.440605	+	0.897701i	$0.354764\pi$
$510$	0	0
$511$	−39.2030	−1.73424
$512$	0	0
$513$	15.4034	0.680078
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.97451	−0.306739
$518$	0	0
$519$	2.24698	0.0986315
$520$	0	0
$521$	−31.9409	−1.39936	−0.699679	−	0.714458i	$-0.746673\pi$
$521$	−31.9409	−1.39936	−0.699679	+	0.714458i	$0.746673\pi$
$522$	0	0
$523$	−13.3599	−0.584187	−0.292093	−	0.956390i	$-0.594352\pi$
$523$	−13.3599	−0.584187	−0.292093	+	0.956390i	$0.594352\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.82371	0.253685
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−19.0911	−0.828484
$532$	0	0
$533$	−11.0707	−0.479525
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−18.6679	−0.805578
$538$	0	0
$539$	44.6547	1.92341
$540$	0	0
$541$	−5.65519	−0.243135	−0.121568	−	0.992583i	$-0.538792\pi$
$541$	−5.65519	−0.243135	−0.121568	+	0.992583i	$0.538792\pi$
$542$	0	0
$543$	37.5555	1.61166
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24.9028	−1.06476	−0.532382	−	0.846504i	$-0.678703\pi$
$547$	−24.9028	−1.06476	−0.532382	+	0.846504i	$0.678703\pi$
$548$	0	0
$549$	−27.2620	−1.16352
$550$	0	0
$551$	−39.7560	−1.69366
$552$	0	0
$553$	44.5870	1.89603
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.38537	0.355300	0.177650	−	0.984094i	$-0.443151\pi$
$557$	8.38537	0.355300	0.177650	+	0.984094i	$0.443151\pi$
$558$	0	0
$559$	−32.6655	−1.38160
$560$	0	0
$561$	34.1715	1.44272
$562$	0	0
$563$	−2.17629	−0.0917198	−0.0458599	−	0.998948i	$-0.514603\pi$
$563$	−2.17629	−0.0917198	−0.0458599	+	0.998948i	$0.514603\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	49.2030	2.06633
$568$	0	0
$569$	1.25129	0.0524569	0.0262284	−	0.999656i	$-0.491650\pi$
$569$	1.25129	0.0524569	0.0262284	+	0.999656i	$0.491650\pi$
$570$	0	0
$571$	30.8068	1.28923	0.644613	−	0.764509i	$-0.277018\pi$
$571$	30.8068	1.28923	0.644613	+	0.764509i	$0.277018\pi$
$572$	0	0
$573$	29.1293	1.21689
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.1879	−0.424128	−0.212064	−	0.977256i	$-0.568018\pi$
$577$	−10.1879	−0.424128	−0.212064	+	0.977256i	$0.568018\pi$
$578$	0	0
$579$	12.8291	0.533159
$580$	0	0
$581$	−70.8068	−2.93756
$582$	0	0
$583$	10.1112	0.418765
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.0489	−0.786233	−0.393116	−	0.919489i	$-0.628603\pi$
$587$	−19.0489	−0.786233	−0.393116	+	0.919489i	$0.628603\pi$
$588$	0	0
$589$	−9.34050	−0.384869
$590$	0	0
$591$	−1.39075	−0.0572077
$592$	0	0
$593$	3.07500	0.126275	0.0631375	−	0.998005i	$-0.479889\pi$
$593$	3.07500	0.126275	0.0631375	+	0.998005i	$0.479889\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.4034	0.957838
$598$	0	0
$599$	40.8538	1.66924	0.834621	−	0.550824i	$-0.185687\pi$
$599$	40.8538	1.66924	0.834621	+	0.550824i	$0.185687\pi$
$600$	0	0
$601$	14.2403	0.580873	0.290436	−	0.956894i	$-0.406200\pi$
$601$	14.2403	0.580873	0.290436	+	0.956894i	$0.406200\pi$
$602$	0	0
$603$	16.7922	0.683833
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.10859	−0.329117	−0.164559	−	0.986367i	$-0.552620\pi$
$607$	−8.10859	−0.329117	−0.164559	+	0.986367i	$0.552620\pi$
$608$	0	0
$609$	−55.6969	−2.25695
$610$	0	0
$611$	6.28382	0.254216
$612$	0	0
$613$	−28.9047	−1.16745	−0.583724	−	0.811952i	$-0.698405\pi$
$613$	−28.9047	−1.16745	−0.583724	+	0.811952i	$0.698405\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.9517	−1.44736	−0.723680	−	0.690136i	$-0.757551\pi$
$617$	−35.9517	−1.44736	−0.723680	+	0.690136i	$0.757551\pi$
$618$	0	0
$619$	35.1728	1.41372	0.706858	−	0.707356i	$-0.250112\pi$
$619$	35.1728	1.41372	0.706858	+	0.707356i	$0.250112\pi$
$620$	0	0
$621$	2.13706	0.0857574
$622$	0	0
$623$	0.548253	0.0219653
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−54.8068	−2.18877
$628$	0	0
$629$	−26.1715	−1.04353
$630$	0	0
$631$	29.1943	1.16221	0.581104	−	0.813829i	$-0.302621\pi$
$631$	29.1943	1.16221	0.581104	+	0.813829i	$0.302621\pi$
$632$	0	0
$633$	−63.6021	−2.52796
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−40.2325	−1.59407
$638$	0	0
$639$	7.14782	0.282763
$640$	0	0
$641$	16.6160	0.656291	0.328145	−	0.944627i	$-0.393576\pi$
$641$	16.6160	0.656291	0.328145	+	0.944627i	$0.393576\pi$
$642$	0	0
$643$	45.8404	1.80777	0.903885	−	0.427775i	$-0.140703\pi$
$643$	45.8404	1.80777	0.903885	+	0.427775i	$0.140703\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.8635	−1.01680	−0.508400	−	0.861121i	$-0.669763\pi$
$647$	−25.8635	−1.01680	−0.508400	+	0.861121i	$0.669763\pi$
$648$	0	0
$649$	−31.5314	−1.23772
$650$	0	0
$651$	−13.0858	−0.512871
$652$	0	0
$653$	−5.95539	−0.233053	−0.116526	−	0.993188i	$-0.537176\pi$
$653$	−5.95539	−0.233053	−0.116526	+	0.993188i	$0.537176\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	17.8737	0.697319
$658$	0	0
$659$	43.1400	1.68050	0.840249	−	0.542201i	$-0.182409\pi$
$659$	43.1400	1.68050	0.840249	+	0.542201i	$0.182409\pi$
$660$	0	0
$661$	12.4349	0.483661	0.241830	−	0.970319i	$-0.422252\pi$
$661$	12.4349	0.483661	0.241830	+	0.970319i	$0.422252\pi$
$662$	0	0
$663$	−30.7875	−1.19569
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.51573	−0.213570
$668$	0	0
$669$	9.18060	0.354943
$670$	0	0
$671$	−45.0267	−1.73824
$672$	0	0
$673$	−4.15777	−0.160270	−0.0801351	−	0.996784i	$-0.525535\pi$
$673$	−4.15777	−0.160270	−0.0801351	+	0.996784i	$0.525535\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.97716	−0.345020	−0.172510	−	0.985008i	$-0.555188\pi$
$677$	−8.97716	−0.345020	−0.172510	+	0.985008i	$0.555188\pi$
$678$	0	0
$679$	70.1232	2.69108
$680$	0	0
$681$	51.1594	1.96043
$682$	0	0
$683$	−2.08815	−0.0799007	−0.0399503	−	0.999202i	$-0.512720\pi$
$683$	−2.08815	−0.0799007	−0.0399503	+	0.999202i	$0.512720\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.41550	−0.321071
$688$	0	0
$689$	−9.10992	−0.347060
$690$	0	0
$691$	−35.9047	−1.36588	−0.682939	−	0.730475i	$-0.739299\pi$
$691$	−35.9047	−1.36588	−0.682939	+	0.730475i	$0.739299\pi$
$692$	0	0
$693$	−31.1594	−1.18365
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−16.3177	−0.618076
$698$	0	0
$699$	3.64310	0.137795
$700$	0	0
$701$	−24.8853	−0.939905	−0.469952	−	0.882692i	$-0.655729\pi$
$701$	−24.8853	−0.939905	−0.469952	+	0.882692i	$0.655729\pi$
$702$	0	0
$703$	41.9758	1.58315
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−78.2646	−2.94344
$708$	0	0
$709$	40.7284	1.52959	0.764793	−	0.644276i	$-0.222841\pi$
$709$	40.7284	1.52959	0.764793	+	0.644276i	$0.222841\pi$
$710$	0	0
$711$	−20.3284	−0.762375
$712$	0	0
$713$	−1.29590	−0.0485317
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−38.4282	−1.43513
$718$	0	0
$719$	7.01208	0.261507	0.130753	−	0.991415i	$-0.458260\pi$
$719$	7.01208	0.261507	0.130753	+	0.991415i	$0.458260\pi$
$720$	0	0
$721$	−79.0025	−2.94221
$722$	0	0
$723$	−0.371961	−0.0138334
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.3840	0.718914	0.359457	−	0.933162i	$-0.382962\pi$
$727$	19.3840	0.718914	0.359457	+	0.933162i	$0.382962\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	0	0
$731$	−48.1473	−1.78079
$732$	0	0
$733$	−2.71379	−0.100236	−0.0501181	−	0.998743i	$-0.515960\pi$
$733$	−2.71379	−0.100236	−0.0501181	+	0.998743i	$0.515960\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27.7345	1.02161
$738$	0	0
$739$	−11.6920	−0.430098	−0.215049	−	0.976603i	$-0.568991\pi$
$739$	−11.6920	−0.430098	−0.215049	+	0.976603i	$0.568991\pi$
$740$	0	0
$741$	49.3793	1.81399
$742$	0	0
$743$	−4.64609	−0.170448	−0.0852242	−	0.996362i	$-0.527161\pi$
$743$	−4.64609	−0.170448	−0.0852242	+	0.996362i	$0.527161\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.2828	1.18116
$748$	0	0
$749$	57.1353	2.08768
$750$	0	0
$751$	37.7017	1.37575	0.687877	−	0.725827i	$-0.258543\pi$
$751$	37.7017	1.37575	0.687877	+	0.725827i	$0.258543\pi$
$752$	0	0
$753$	35.2271	1.28375
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.4034	1.21407	0.607034	−	0.794676i	$-0.292359\pi$
$757$	33.4034	1.21407	0.607034	+	0.794676i	$0.292359\pi$
$758$	0	0
$759$	−7.60388	−0.276003
$760$	0	0
$761$	15.0110	0.544149	0.272074	−	0.962276i	$-0.412290\pi$
$761$	15.0110	0.544149	0.272074	+	0.962276i	$0.412290\pi$
$762$	0	0
$763$	23.9517	0.867109
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.4088	1.02578
$768$	0	0
$769$	−3.35988	−0.121160	−0.0605802	−	0.998163i	$-0.519295\pi$
$769$	−3.35988	−0.121160	−0.0605802	+	0.998163i	$0.519295\pi$
$770$	0	0
$771$	−40.4282	−1.45599
$772$	0	0
$773$	22.1608	0.797067	0.398533	−	0.917154i	$-0.369519\pi$
$773$	22.1608	0.797067	0.398533	+	0.917154i	$0.369519\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	58.8068	2.10968
$778$	0	0
$779$	26.1715	0.937692
$780$	0	0
$781$	11.8055	0.422434
$782$	0	0
$783$	−11.7875	−0.421250
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−0.371961	−0.0132590	−0.00662950	−	0.999978i	$-0.502110\pi$
$787$	−0.371961	−0.0132590	−0.00662950	+	0.999978i	$0.502110\pi$
$788$	0	0
$789$	21.7995	0.776084
$790$	0	0
$791$	27.9517	0.993847
$792$	0	0
$793$	40.5676	1.44060
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.7439	1.51407	0.757034	−	0.653376i	$-0.226648\pi$
$797$	42.7439	1.51407	0.757034	+	0.653376i	$0.226648\pi$
$798$	0	0
$799$	9.26205	0.327668
$800$	0	0
$801$	−0.249964	−0.00883203
$802$	0	0
$803$	29.5206	1.04176
$804$	0	0
$805$	0	0
$806$	0	0
$807$	48.2301	1.69778
$808$	0	0
$809$	28.5603	1.00413	0.502064	−	0.864830i	$-0.332574\pi$
$809$	28.5603	1.00413	0.502064	+	0.864830i	$0.332574\pi$
$810$	0	0
$811$	21.8558	0.767459	0.383730	−	0.923445i	$-0.374639\pi$
$811$	21.8558	0.767459	0.383730	+	0.923445i	$0.374639\pi$
$812$	0	0
$813$	−4.24698	−0.148948
$814$	0	0
$815$	0	0
$816$	0	0
$817$	77.2223	2.70167
$818$	0	0
$819$	28.0737	0.980973
$820$	0	0
$821$	−27.0242	−0.943150	−0.471575	−	0.881826i	$-0.656314\pi$
$821$	−27.0242	−0.943150	−0.471575	+	0.881826i	$0.656314\pi$
$822$	0	0
$823$	−6.33406	−0.220791	−0.110396	−	0.993888i	$-0.535212\pi$
$823$	−6.33406	−0.220791	−0.110396	+	0.993888i	$0.535212\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	50.4107	1.75295	0.876476	−	0.481445i	$-0.159888\pi$
$827$	50.4107	1.75295	0.876476	+	0.481445i	$0.159888\pi$
$828$	0	0
$829$	−33.8552	−1.17584	−0.587919	−	0.808920i	$-0.700053\pi$
$829$	−33.8552	−1.17584	−0.587919	+	0.808920i	$0.700053\pi$
$830$	0	0
$831$	28.2325	0.979375
$832$	0	0
$833$	−59.3008	−2.05465
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.76941	−0.0957249
$838$	0	0
$839$	−34.9590	−1.20692	−0.603459	−	0.797394i	$-0.706211\pi$
$839$	−34.9590	−1.20692	−0.603459	+	0.797394i	$0.706211\pi$
$840$	0	0
$841$	1.42327	0.0490783
$842$	0	0
$843$	−49.7754	−1.71436
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.03013	−0.0697561
$848$	0	0
$849$	58.6848	2.01406
$850$	0	0
$851$	5.82371	0.199634
$852$	0	0
$853$	54.4999	1.86604	0.933021	−	0.359822i	$-0.117163\pi$
$853$	54.4999	1.86604	0.933021	+	0.359822i	$0.117163\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.93123	−0.270926	−0.135463	−	0.990782i	$-0.543252\pi$
$857$	−7.93123	−0.270926	−0.135463	+	0.990782i	$0.543252\pi$
$858$	0	0
$859$	12.7885	0.436339	0.218169	−	0.975911i	$-0.429991\pi$
$859$	12.7885	0.436339	0.218169	+	0.975911i	$0.429991\pi$
$860$	0	0
$861$	36.6655	1.24956
$862$	0	0
$863$	−47.3279	−1.61106	−0.805531	−	0.592554i	$-0.798120\pi$
$863$	−47.3279	−1.61106	−0.805531	+	0.592554i	$0.798120\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.18060	−0.243866
$868$	0	0
$869$	−33.5749	−1.13895
$870$	0	0
$871$	−24.9879	−0.846683
$872$	0	0
$873$	−31.9711	−1.08206
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.13275	0.0382503	0.0191251	−	0.999817i	$-0.493912\pi$
$877$	1.13275	0.0382503	0.0191251	+	0.999817i	$0.493912\pi$
$878$	0	0
$879$	72.4650	2.44418
$880$	0	0
$881$	24.9530	0.840688	0.420344	−	0.907365i	$-0.361909\pi$
$881$	24.9530	0.840688	0.420344	+	0.907365i	$0.361909\pi$
$882$	0	0
$883$	−22.7453	−0.765439	−0.382719	−	0.923865i	$-0.625012\pi$
$883$	−22.7453	−0.765439	−0.382719	+	0.923865i	$0.625012\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.0452	1.57962	0.789812	−	0.613350i	$-0.210178\pi$
$887$	47.0452	1.57962	0.789812	+	0.613350i	$0.210178\pi$
$888$	0	0
$889$	9.38404	0.314731
$890$	0	0
$891$	−37.0508	−1.24125
$892$	0	0
$893$	−14.8552	−0.497109
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.85086	0.228743
$898$	0	0
$899$	7.14782	0.238393
$900$	0	0
$901$	−13.4276	−0.447338
$902$	0	0
$903$	108.186	3.60021
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−54.3081	−1.80327	−0.901635	−	0.432497i	$-0.857633\pi$
$907$	−54.3081	−1.80327	−0.901635	+	0.432497i	$0.857633\pi$
$908$	0	0
$909$	35.6829	1.18353
$910$	0	0
$911$	−54.2693	−1.79802	−0.899012	−	0.437925i	$-0.855714\pi$
$911$	−54.2693	−1.79802	−0.899012	+	0.437925i	$0.855714\pi$
$912$	0	0
$913$	53.3190	1.76460
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−48.5133	−1.60205
$918$	0	0
$919$	15.5362	0.512491	0.256246	−	0.966612i	$-0.417514\pi$
$919$	15.5362	0.512491	0.256246	+	0.966612i	$0.417514\pi$
$920$	0	0
$921$	−25.1836	−0.829827
$922$	0	0
$923$	−10.6364	−0.350101
$924$	0	0
$925$	0	0
$926$	0	0
$927$	36.0194	1.18303
$928$	0	0
$929$	−20.4989	−0.672546	−0.336273	−	0.941765i	$-0.609166\pi$
$929$	−20.4989	−0.672546	−0.336273	+	0.941765i	$0.609166\pi$
$930$	0	0
$931$	95.1111	3.11714
$932$	0	0
$933$	53.5230	1.75227
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.933624	−0.0305002	−0.0152501	−	0.999884i	$-0.504854\pi$
$937$	−0.933624	−0.0305002	−0.0152501	+	0.999884i	$0.504854\pi$
$938$	0	0
$939$	−3.23191	−0.105470
$940$	0	0
$941$	−9.01208	−0.293785	−0.146893	−	0.989152i	$-0.546927\pi$
$941$	−9.01208	−0.293785	−0.146893	+	0.989152i	$0.546927\pi$
$942$	0	0
$943$	3.63102	0.118242
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.8562	0.547754	0.273877	−	0.961765i	$-0.411694\pi$
$947$	16.8562	0.547754	0.273877	+	0.961765i	$0.411694\pi$
$948$	0	0
$949$	−26.5972	−0.863381
$950$	0	0
$951$	−29.9215	−0.970272
$952$	0	0
$953$	11.9711	0.387780	0.193890	−	0.981023i	$-0.437889\pi$
$953$	11.9711	0.387780	0.193890	+	0.981023i	$0.437889\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	41.9409	1.35576
$958$	0	0
$959$	−10.7681	−0.347720
$960$	0	0
$961$	−29.3207	−0.945827
$962$	0	0
$963$	−26.0495	−0.839434
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−32.1976	−1.03540	−0.517702	−	0.855561i	$-0.673213\pi$
$967$	−32.1976	−1.03540	−0.517702	+	0.855561i	$0.673213\pi$
$968$	0	0
$969$	72.7827	2.33812
$970$	0	0
$971$	30.6547	0.983757	0.491878	−	0.870664i	$-0.336311\pi$
$971$	30.6547	0.983757	0.491878	+	0.870664i	$0.336311\pi$
$972$	0	0
$973$	45.2137	1.44949
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.10646	−0.259349	−0.129674	−	0.991557i	$-0.541393\pi$
$977$	−8.10646	−0.259349	−0.129674	+	0.991557i	$0.541393\pi$
$978$	0	0
$979$	−0.412846	−0.0131946
$980$	0	0
$981$	−10.9202	−0.348656
$982$	0	0
$983$	−35.3599	−1.12781	−0.563903	−	0.825841i	$-0.690701\pi$
$983$	−35.3599	−1.12781	−0.563903	+	0.825841i	$0.690701\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−20.8116	−0.662441
$988$	0	0
$989$	10.7138	0.340679
$990$	0	0
$991$	−0.940920	−0.0298893	−0.0149447	−	0.999888i	$-0.504757\pi$
$991$	−0.940920	−0.0298893	−0.0149447	+	0.999888i	$0.504757\pi$
$992$	0	0
$993$	−15.9390	−0.505809
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.40342	0.0761171	0.0380585	−	0.999276i	$-0.487883\pi$
$997$	2.40342	0.0761171	0.0380585	+	0.999276i	$0.487883\pi$
$998$	0	0
$999$	12.4456	0.393762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.w.1.1	✓	3
4.3	odd	2		9200.2.a.ch.1.3		3
5.2	odd	4		4600.2.e.s.4049.6		6
5.3	odd	4		4600.2.e.s.4049.1		6
5.4	even	2		4600.2.a.z.1.3	yes	3
20.19	odd	2		9200.2.a.cb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.w.1.1	✓	3	1.1	even	1	trivial
4600.2.a.z.1.3	yes	3	5.4	even	2
4600.2.e.s.4049.1		6	5.3	odd	4
4600.2.e.s.4049.6		6	5.2	odd	4
9200.2.a.cb.1.1		3	20.19	odd	2
9200.2.a.ch.1.3		3	4.3	odd	2

Overview	Random
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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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24698 q^{3} -4.49396 q^{7} +2.04892 q^{9} +3.38404 q^{11} -3.04892 q^{13} -4.49396 q^{17} +7.20775 q^{19} +10.0978 q^{21} +1.00000 q^{23} +2.13706 q^{27} -5.51573 q^{29} -1.29590 q^{31} -7.60388 q^{33} +5.82371 q^{37} +6.85086 q^{39} +3.63102 q^{41} +10.7138 q^{43} -2.06100 q^{47} +13.1957 q^{49} +10.0978 q^{51} +2.98792 q^{53} -16.1957 q^{57} -9.31767 q^{59} -13.3056 q^{61} -9.20775 q^{63} +8.19567 q^{67} -2.24698 q^{69} +3.48858 q^{71} +8.72348 q^{73} -15.2078 q^{77} -9.92154 q^{79} -10.9487 q^{81} +15.7560 q^{83} +12.3937 q^{87} -0.121998 q^{89} +13.7017 q^{91} +2.91185 q^{93} -15.6039 q^{97} +6.93362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 4 q^{7} - 3 q^{9} - 4 q^{17} + 4 q^{19} + 12 q^{21} + 3 q^{23} + q^{27} - 4 q^{29} + 10 q^{31} - 14 q^{33} + 10 q^{37} + 7 q^{39} - 4 q^{41} + 24 q^{43} - 16 q^{47} + 3 q^{49} + 12 q^{51}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 4 * q^7 - 3 * q^9 - 4 * q^17 + 4 * q^19 + 12 * q^21 + 3 * q^23 + q^27 - 4 * q^29 + 10 * q^31 - 14 * q^33 + 10 * q^37 + 7 * q^39 - 4 * q^41 + 24 * q^43 - 16 * q^47 + 3 * q^49 + 12 * q^51 - 10 * q^53 - 12 * q^57 - 11 * q^59 - 4 * q^61 - 10 * q^63 - 12 * q^67 - 2 * q^69 + 4 * q^71 - 4 * q^73 - 28 * q^77 - 4 * q^79 - q^81 + 8 * q^83 + 5 * q^87 - 20 * q^89 + 14 * q^91 + 5 * q^93 - 38 * q^97 + 14 * q^99

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