LMFDB - Embedded newform 7728.2.a.ca.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.256549.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 12x^{2} + 11x + 36 $ x^4 - 2x^3 - 12x^2 + 11*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.73320$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.73320 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.73320 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.20357 q^{11} +0.470368 q^{13} -3.73320 q^{15} -4.20357 q^{17} -4.20357 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.93676 q^{25} -1.00000 q^{27} -0.651155 q^{29} -6.20357 q^{31} -4.20357 q^{33} -3.73320 q^{35} -5.75598 q^{37} -0.470368 q^{39} -3.75598 q^{41} -11.9368 q^{43} +3.73320 q^{45} -6.61168 q^{47} +1.00000 q^{49} +4.20357 q^{51} -3.28561 q^{53} +15.6927 q^{55} +4.20357 q^{57} -8.58792 q^{59} +9.08204 q^{61} -1.00000 q^{63} +1.75598 q^{65} -5.81921 q^{67} +1.00000 q^{69} -9.85075 q^{71} +3.84199 q^{73} -8.93676 q^{75} -4.20357 q^{77} -15.5514 q^{79} +1.00000 q^{81} -10.3677 q^{83} -15.6927 q^{85} +0.651155 q^{87} +12.7915 q^{89} -0.470368 q^{91} +6.20357 q^{93} -15.6927 q^{95} -4.11755 q^{97} +4.20357 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} - 2 q^{15} - 2 q^{17} - 2 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} - 4 q^{27} - q^{29} - 10 q^{31} - 2 q^{33} - 2 q^{35} + 5 q^{37} + 13 q^{41} - 20 q^{43} + 2 q^{45} - 17 q^{47} + 4 q^{49} + 2 q^{51} + 13 q^{53} + 7 q^{55} + 2 q^{57} - 5 q^{59} + 25 q^{61} - 4 q^{63} - 21 q^{65} - 23 q^{67} + 4 q^{69} + q^{71} - 8 q^{75} - 2 q^{77} - 14 q^{79} + 4 q^{81} - 4 q^{83} - 7 q^{85} + q^{87} + 7 q^{89} + 10 q^{93} - 7 q^{95} + 11 q^{97} + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 - 2 * q^15 - 2 * q^17 - 2 * q^19 + 4 * q^21 - 4 * q^23 + 8 * q^25 - 4 * q^27 - q^29 - 10 * q^31 - 2 * q^33 - 2 * q^35 + 5 * q^37 + 13 * q^41 - 20 * q^43 + 2 * q^45 - 17 * q^47 + 4 * q^49 + 2 * q^51 + 13 * q^53 + 7 * q^55 + 2 * q^57 - 5 * q^59 + 25 * q^61 - 4 * q^63 - 21 * q^65 - 23 * q^67 + 4 * q^69 + q^71 - 8 * q^75 - 2 * q^77 - 14 * q^79 + 4 * q^81 - 4 * q^83 - 7 * q^85 + q^87 + 7 * q^89 + 10 * q^93 - 7 * q^95 + 11 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.73320	1.66954	0.834768	−	0.550601i	$-0.185602\pi$
$5$	3.73320	1.66954	0.834768	+	0.550601i	$0.185602\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.20357	1.26742	0.633711	−	0.773570i	$-0.281531\pi$
$11$	4.20357	1.26742	0.633711	+	0.773570i	$0.281531\pi$
$12$	0	0
$13$	0.470368	0.130457	0.0652283	−	0.997870i	$-0.479222\pi$
$13$	0.470368	0.130457	0.0652283	+	0.997870i	$0.479222\pi$
$14$	0	0
$15$	−3.73320	−0.963907
$16$	0	0
$17$	−4.20357	−1.01951	−0.509757	−	0.860318i	$-0.670265\pi$
$17$	−4.20357	−1.01951	−0.509757	+	0.860318i	$0.670265\pi$
$18$	0	0
$19$	−4.20357	−0.964364	−0.482182	−	0.876071i	$-0.660156\pi$
$19$	−4.20357	−0.964364	−0.482182	+	0.876071i	$0.660156\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	8.93676	1.78735
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.651155	−0.120916	−0.0604582	−	0.998171i	$-0.519256\pi$
$29$	−0.651155	−0.120916	−0.0604582	+	0.998171i	$0.519256\pi$
$30$	0	0
$31$	−6.20357	−1.11419	−0.557097	−	0.830448i	$-0.688085\pi$
$31$	−6.20357	−1.11419	−0.557097	+	0.830448i	$0.688085\pi$
$32$	0	0
$33$	−4.20357	−0.731747
$34$	0	0
$35$	−3.73320	−0.631026
$36$	0	0
$37$	−5.75598	−0.946277	−0.473138	−	0.880988i	$-0.656879\pi$
$37$	−5.75598	−0.946277	−0.473138	+	0.880988i	$0.656879\pi$
$38$	0	0
$39$	−0.470368	−0.0753191
$40$	0	0
$41$	−3.75598	−0.586585	−0.293292	−	0.956023i	$-0.594751\pi$
$41$	−3.75598	−0.586585	−0.293292	+	0.956023i	$0.594751\pi$
$42$	0	0
$43$	−11.9368	−1.82034	−0.910170	−	0.414236i	$-0.864049\pi$
$43$	−11.9368	−1.82034	−0.910170	+	0.414236i	$0.864049\pi$
$44$	0	0
$45$	3.73320	0.556512
$46$	0	0
$47$	−6.61168	−0.964412	−0.482206	−	0.876058i	$-0.660164\pi$
$47$	−6.61168	−0.964412	−0.482206	+	0.876058i	$0.660164\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.20357	0.588617
$52$	0	0
$53$	−3.28561	−0.451313	−0.225656	−	0.974207i	$-0.572453\pi$
$53$	−3.28561	−0.451313	−0.225656	+	0.974207i	$0.572453\pi$
$54$	0	0
$55$	15.6927	2.11601
$56$	0	0
$57$	4.20357	0.556776
$58$	0	0
$59$	−8.58792	−1.11805	−0.559026	−	0.829150i	$-0.688825\pi$
$59$	−8.58792	−1.11805	−0.559026	+	0.829150i	$0.688825\pi$
$60$	0	0
$61$	9.08204	1.16284	0.581418	−	0.813605i	$-0.302498\pi$
$61$	9.08204	1.16284	0.581418	+	0.813605i	$0.302498\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.75598	0.217802
$66$	0	0
$67$	−5.81921	−0.710930	−0.355465	−	0.934690i	$-0.615677\pi$
$67$	−5.81921	−0.710930	−0.355465	+	0.934690i	$0.615677\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−9.85075	−1.16907	−0.584534	−	0.811369i	$-0.698723\pi$
$71$	−9.85075	−1.16907	−0.584534	+	0.811369i	$0.698723\pi$
$72$	0	0
$73$	3.84199	0.449671	0.224836	−	0.974397i	$-0.427816\pi$
$73$	3.84199	0.449671	0.224836	+	0.974397i	$0.427816\pi$
$74$	0	0
$75$	−8.93676	−1.03193
$76$	0	0
$77$	−4.20357	−0.479041
$78$	0	0
$79$	−15.5514	−1.74967	−0.874836	−	0.484419i	$-0.839031\pi$
$79$	−15.5514	−1.74967	−0.874836	+	0.484419i	$0.839031\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.3677	−1.13800	−0.568999	−	0.822338i	$-0.692669\pi$
$83$	−10.3677	−1.13800	−0.568999	+	0.822338i	$0.692669\pi$
$84$	0	0
$85$	−15.6927	−1.70212
$86$	0	0
$87$	0.651155	0.0698111
$88$	0	0
$89$	12.7915	1.35589	0.677947	−	0.735111i	$-0.262870\pi$
$89$	12.7915	1.35589	0.677947	+	0.735111i	$0.262870\pi$
$90$	0	0
$91$	−0.470368	−0.0493079
$92$	0	0
$93$	6.20357	0.643280
$94$	0	0
$95$	−15.6927	−1.61004
$96$	0	0
$97$	−4.11755	−0.418074	−0.209037	−	0.977908i	$-0.567033\pi$
$97$	−4.11755	−0.418074	−0.209037	+	0.977908i	$0.567033\pi$
$98$	0	0
$99$	4.20357	0.422474
$100$	0	0
$101$	−13.6927	−1.36248	−0.681239	−	0.732061i	$-0.738559\pi$
$101$	−13.6927	−1.36248	−0.681239	+	0.732061i	$0.738559\pi$
$102$	0	0
$103$	16.3667	1.61266	0.806328	−	0.591469i	$-0.201452\pi$
$103$	16.3667	1.61266	0.806328	+	0.591469i	$0.201452\pi$
$104$	0	0
$105$	3.73320	0.364323
$106$	0	0
$107$	−1.27953	−0.123697	−0.0618484	−	0.998086i	$-0.519700\pi$
$107$	−1.27953	−0.123697	−0.0618484	+	0.998086i	$0.519700\pi$
$108$	0	0
$109$	0.266803	0.0255551	0.0127775	−	0.999918i	$-0.495933\pi$
$109$	0.266803	0.0255551	0.0127775	+	0.999918i	$0.495933\pi$
$110$	0	0
$111$	5.75598	0.546333
$112$	0	0
$113$	−17.1996	−1.61800	−0.809001	−	0.587808i	$-0.799991\pi$
$113$	−17.1996	−1.61800	−0.809001	+	0.587808i	$0.799991\pi$
$114$	0	0
$115$	−3.73320	−0.348122
$116$	0	0
$117$	0.470368	0.0434855
$118$	0	0
$119$	4.20357	0.385340
$120$	0	0
$121$	6.66996	0.606360
$122$	0	0
$123$	3.75598	0.338665
$124$	0	0
$125$	14.6967	1.31451
$126$	0	0
$127$	10.3439	0.917872	0.458936	−	0.888469i	$-0.348231\pi$
$127$	10.3439	0.917872	0.458936	+	0.888469i	$0.348231\pi$
$128$	0	0
$129$	11.9368	1.05097
$130$	0	0
$131$	6.20357	0.542008	0.271004	−	0.962578i	$-0.412644\pi$
$131$	6.20357	0.542008	0.271004	+	0.962578i	$0.412644\pi$
$132$	0	0
$133$	4.20357	0.364495
$134$	0	0
$135$	−3.73320	−0.321302
$136$	0	0
$137$	1.23032	0.105113	0.0525565	−	0.998618i	$-0.483263\pi$
$137$	1.23032	0.105113	0.0525565	+	0.998618i	$0.483263\pi$
$138$	0	0
$139$	16.5475	1.40354	0.701769	−	0.712405i	$-0.252394\pi$
$139$	16.5475	1.40354	0.701769	+	0.712405i	$0.252394\pi$
$140$	0	0
$141$	6.61168	0.556803
$142$	0	0
$143$	1.97722	0.165344
$144$	0	0
$145$	−2.43089	−0.201874
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	4.04556	0.331425	0.165712	−	0.986174i	$-0.447008\pi$
$149$	4.04556	0.331425	0.165712	+	0.986174i	$0.447008\pi$
$150$	0	0
$151$	8.32111	0.677163	0.338581	−	0.940937i	$-0.390053\pi$
$151$	8.32111	0.677163	0.338581	+	0.940937i	$0.390053\pi$
$152$	0	0
$153$	−4.20357	−0.339838
$154$	0	0
$155$	−23.1591	−1.86019
$156$	0	0
$157$	−14.5712	−1.16291	−0.581455	−	0.813579i	$-0.697516\pi$
$157$	−14.5712	−1.16291	−0.581455	+	0.813579i	$0.697516\pi$
$158$	0	0
$159$	3.28561	0.260566
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−15.3312	−1.20083	−0.600415	−	0.799689i	$-0.704998\pi$
$163$	−15.3312	−1.20083	−0.600415	+	0.799689i	$0.704998\pi$
$164$	0	0
$165$	−15.6927	−1.22168
$166$	0	0
$167$	17.7155	1.37087	0.685434	−	0.728135i	$-0.259613\pi$
$167$	17.7155	1.37087	0.685434	+	0.728135i	$0.259613\pi$
$168$	0	0
$169$	−12.7788	−0.982981
$170$	0	0
$171$	−4.20357	−0.321455
$172$	0	0
$173$	−2.90126	−0.220578	−0.110289	−	0.993900i	$-0.535178\pi$
$173$	−2.90126	−0.220578	−0.110289	+	0.993900i	$0.535178\pi$
$174$	0	0
$175$	−8.93676	−0.675556
$176$	0	0
$177$	8.58792	0.645507
$178$	0	0
$179$	10.2589	0.766783	0.383391	−	0.923586i	$-0.374756\pi$
$179$	10.2589	0.766783	0.383391	+	0.923586i	$0.374756\pi$
$180$	0	0
$181$	−13.0908	−0.973031	−0.486516	−	0.873672i	$-0.661732\pi$
$181$	−13.0908	−0.973031	−0.486516	+	0.873672i	$0.661732\pi$
$182$	0	0
$183$	−9.08204	−0.671364
$184$	0	0
$185$	−21.4882	−1.57984
$186$	0	0
$187$	−17.6700	−1.29216
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	8.76870	0.634481	0.317241	−	0.948345i	$-0.397244\pi$
$191$	8.76870	0.634481	0.317241	+	0.948345i	$0.397244\pi$
$192$	0	0
$193$	10.3211	0.742930	0.371465	−	0.928447i	$-0.378856\pi$
$193$	10.3211	0.742930	0.371465	+	0.928447i	$0.378856\pi$
$194$	0	0
$195$	−1.75598	−0.125748
$196$	0	0
$197$	−11.4032	−0.812441	−0.406221	−	0.913775i	$-0.633154\pi$
$197$	−11.4032	−0.812441	−0.406221	+	0.913775i	$0.633154\pi$
$198$	0	0
$199$	14.6660	1.03964	0.519822	−	0.854275i	$-0.325998\pi$
$199$	14.6660	1.03964	0.519822	+	0.854275i	$0.325998\pi$
$200$	0	0
$201$	5.81921	0.410456
$202$	0	0
$203$	0.651155	0.0457021
$204$	0	0
$205$	−14.0218	−0.979325
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−17.6700	−1.22226
$210$	0	0
$211$	13.7155	0.944215	0.472108	−	0.881541i	$-0.343493\pi$
$211$	13.7155	0.944215	0.472108	+	0.881541i	$0.343493\pi$
$212$	0	0
$213$	9.85075	0.674962
$214$	0	0
$215$	−44.5623	−3.03912
$216$	0	0
$217$	6.20357	0.421125
$218$	0	0
$219$	−3.84199	−0.259618
$220$	0	0
$221$	−1.97722	−0.133002
$222$	0	0
$223$	26.0464	1.74419	0.872097	−	0.489333i	$-0.162760\pi$
$223$	26.0464	1.74419	0.872097	+	0.489333i	$0.162760\pi$
$224$	0	0
$225$	8.93676	0.595784
$226$	0	0
$227$	16.3044	1.08216	0.541081	−	0.840971i	$-0.318015\pi$
$227$	16.3044	1.08216	0.541081	+	0.840971i	$0.318015\pi$
$228$	0	0
$229$	14.9240	0.986208	0.493104	−	0.869970i	$-0.335862\pi$
$229$	14.9240	0.986208	0.493104	+	0.869970i	$0.335862\pi$
$230$	0	0
$231$	4.20357	0.276574
$232$	0	0
$233$	0.172841	0.0113232	0.00566158	−	0.999984i	$-0.498198\pi$
$233$	0.172841	0.0113232	0.00566158	+	0.999984i	$0.498198\pi$
$234$	0	0
$235$	−24.6827	−1.61012
$236$	0	0
$237$	15.5514	1.01017
$238$	0	0
$239$	−28.0999	−1.81763	−0.908815	−	0.417200i	$-0.863011\pi$
$239$	−28.0999	−1.81763	−0.908815	+	0.417200i	$0.863011\pi$
$240$	0	0
$241$	16.1175	1.03822	0.519111	−	0.854707i	$-0.326263\pi$
$241$	16.1175	1.03822	0.519111	+	0.854707i	$0.326263\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.73320	0.238505
$246$	0	0
$247$	−1.97722	−0.125808
$248$	0	0
$249$	10.3677	0.657023
$250$	0	0
$251$	21.4259	1.35239	0.676197	−	0.736721i	$-0.263627\pi$
$251$	21.4259	1.35239	0.676197	+	0.736721i	$0.263627\pi$
$252$	0	0
$253$	−4.20357	−0.264276
$254$	0	0
$255$	15.6927	0.982717
$256$	0	0
$257$	−10.3616	−0.646337	−0.323169	−	0.946341i	$-0.604748\pi$
$257$	−10.3616	−0.646337	−0.323169	+	0.946341i	$0.604748\pi$
$258$	0	0
$259$	5.75598	0.357659
$260$	0	0
$261$	−0.651155	−0.0403055
$262$	0	0
$263$	−0.487069	−0.0300340	−0.0150170	−	0.999887i	$-0.504780\pi$
$263$	−0.487069	−0.0300340	−0.0150170	+	0.999887i	$0.504780\pi$
$264$	0	0
$265$	−12.2658	−0.753484
$266$	0	0
$267$	−12.7915	−0.782826
$268$	0	0
$269$	10.1869	0.621104	0.310552	−	0.950556i	$-0.399486\pi$
$269$	10.1869	0.621104	0.310552	+	0.950556i	$0.399486\pi$
$270$	0	0
$271$	−9.26283	−0.562677	−0.281338	−	0.959609i	$-0.590778\pi$
$271$	−9.26283	−0.562677	−0.281338	+	0.959609i	$0.590778\pi$
$272$	0	0
$273$	0.470368	0.0284680
$274$	0	0
$275$	37.5663	2.26533
$276$	0	0
$277$	26.4220	1.58754	0.793771	−	0.608217i	$-0.208115\pi$
$277$	26.4220	1.58754	0.793771	+	0.608217i	$0.208115\pi$
$278$	0	0
$279$	−6.20357	−0.371398
$280$	0	0
$281$	14.3211	0.854326	0.427163	−	0.904175i	$-0.359513\pi$
$281$	14.3211	0.854326	0.427163	+	0.904175i	$0.359513\pi$
$282$	0	0
$283$	−6.71439	−0.399129	−0.199565	−	0.979885i	$-0.563953\pi$
$283$	−6.71439	−0.399129	−0.199565	+	0.979885i	$0.563953\pi$
$284$	0	0
$285$	15.6927	0.929558
$286$	0	0
$287$	3.75598	0.221708
$288$	0	0
$289$	0.669960	0.0394094
$290$	0	0
$291$	4.11755	0.241375
$292$	0	0
$293$	−1.22335	−0.0714689	−0.0357344	−	0.999361i	$-0.511377\pi$
$293$	−1.22335	−0.0714689	−0.0357344	+	0.999361i	$0.511377\pi$
$294$	0	0
$295$	−32.0604	−1.86663
$296$	0	0
$297$	−4.20357	−0.243916
$298$	0	0
$299$	−0.470368	−0.0272021
$300$	0	0
$301$	11.9368	0.688024
$302$	0	0
$303$	13.6927	0.786627
$304$	0	0
$305$	33.9051	1.94140
$306$	0	0
$307$	−26.9774	−1.53968	−0.769840	−	0.638237i	$-0.779664\pi$
$307$	−26.9774	−1.53968	−0.769840	+	0.638237i	$0.779664\pi$
$308$	0	0
$309$	−16.3667	−0.931067
$310$	0	0
$311$	12.9100	0.732060	0.366030	−	0.930603i	$-0.380717\pi$
$311$	12.9100	0.732060	0.366030	+	0.930603i	$0.380717\pi$
$312$	0	0
$313$	−7.47434	−0.422475	−0.211237	−	0.977435i	$-0.567749\pi$
$313$	−7.47434	−0.422475	−0.211237	+	0.977435i	$0.567749\pi$
$314$	0	0
$315$	−3.73320	−0.210342
$316$	0	0
$317$	−25.7788	−1.44788	−0.723940	−	0.689863i	$-0.757671\pi$
$317$	−25.7788	−1.44788	−0.723940	+	0.689863i	$0.757671\pi$
$318$	0	0
$319$	−2.73717	−0.153252
$320$	0	0
$321$	1.27953	0.0714164
$322$	0	0
$323$	17.6700	0.983183
$324$	0	0
$325$	4.20357	0.233172
$326$	0	0
$327$	−0.266803	−0.0147542
$328$	0	0
$329$	6.61168	0.364513
$330$	0	0
$331$	35.1029	1.92943	0.964714	−	0.263300i	$-0.0848110\pi$
$331$	35.1029	1.92943	0.964714	+	0.263300i	$0.0848110\pi$
$332$	0	0
$333$	−5.75598	−0.315426
$334$	0	0
$335$	−21.7243	−1.18692
$336$	0	0
$337$	34.8746	1.89974	0.949872	−	0.312640i	$-0.101213\pi$
$337$	34.8746	1.89974	0.949872	+	0.312640i	$0.101213\pi$
$338$	0	0
$339$	17.1996	0.934154
$340$	0	0
$341$	−26.0771	−1.41215
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.73320	0.200989
$346$	0	0
$347$	21.4575	1.15190	0.575949	−	0.817486i	$-0.304633\pi$
$347$	21.4575	1.15190	0.575949	+	0.817486i	$0.304633\pi$
$348$	0	0
$349$	−15.0881	−0.807649	−0.403824	−	0.914837i	$-0.632319\pi$
$349$	−15.0881	−0.807649	−0.403824	+	0.914837i	$0.632319\pi$
$350$	0	0
$351$	−0.470368	−0.0251064
$352$	0	0
$353$	−0.651155	−0.0346575	−0.0173287	−	0.999850i	$-0.505516\pi$
$353$	−0.651155	−0.0346575	−0.0173287	+	0.999850i	$0.505516\pi$
$354$	0	0
$355$	−36.7748	−1.95180
$356$	0	0
$357$	−4.20357	−0.222476
$358$	0	0
$359$	−25.5988	−1.35105	−0.675526	−	0.737336i	$-0.736083\pi$
$359$	−25.5988	−1.35105	−0.675526	+	0.737336i	$0.736083\pi$
$360$	0	0
$361$	−1.33004	−0.0700021
$362$	0	0
$363$	−6.66996	−0.350082
$364$	0	0
$365$	14.3429	0.750742
$366$	0	0
$367$	19.5217	1.01902	0.509512	−	0.860464i	$-0.329826\pi$
$367$	19.5217	1.01902	0.509512	+	0.860464i	$0.329826\pi$
$368$	0	0
$369$	−3.75598	−0.195528
$370$	0	0
$371$	3.28561	0.170580
$372$	0	0
$373$	12.7292	0.659094	0.329547	−	0.944139i	$-0.393104\pi$
$373$	12.7292	0.659094	0.329547	+	0.944139i	$0.393104\pi$
$374$	0	0
$375$	−14.6967	−0.758935
$376$	0	0
$377$	−0.306282	−0.0157743
$378$	0	0
$379$	−0.250102	−0.0128469	−0.00642343	−	0.999979i	$-0.502045\pi$
$379$	−0.250102	−0.0128469	−0.00642343	+	0.999979i	$0.502045\pi$
$380$	0	0
$381$	−10.3439	−0.529934
$382$	0	0
$383$	7.55143	0.385860	0.192930	−	0.981213i	$-0.438201\pi$
$383$	7.55143	0.385860	0.192930	+	0.981213i	$0.438201\pi$
$384$	0	0
$385$	−15.6927	−0.799776
$386$	0	0
$387$	−11.9368	−0.606780
$388$	0	0
$389$	−36.1937	−1.83509	−0.917546	−	0.397630i	$-0.869833\pi$
$389$	−36.1937	−1.83509	−0.917546	+	0.397630i	$0.869833\pi$
$390$	0	0
$391$	4.20357	0.212583
$392$	0	0
$393$	−6.20357	−0.312928
$394$	0	0
$395$	−58.0566	−2.92114
$396$	0	0
$397$	−37.3855	−1.87632	−0.938162	−	0.346198i	$-0.887473\pi$
$397$	−37.3855	−1.87632	−0.938162	+	0.346198i	$0.887473\pi$
$398$	0	0
$399$	−4.20357	−0.210441
$400$	0	0
$401$	2.19562	0.109644	0.0548220	−	0.998496i	$-0.482541\pi$
$401$	2.19562	0.109644	0.0548220	+	0.998496i	$0.482541\pi$
$402$	0	0
$403$	−2.91796	−0.145354
$404$	0	0
$405$	3.73320	0.185504
$406$	0	0
$407$	−24.1956	−1.19933
$408$	0	0
$409$	7.13635	0.352870	0.176435	−	0.984312i	$-0.443543\pi$
$409$	7.13635	0.352870	0.176435	+	0.984312i	$0.443543\pi$
$410$	0	0
$411$	−1.23032	−0.0606870
$412$	0	0
$413$	8.58792	0.422584
$414$	0	0
$415$	−38.7045	−1.89993
$416$	0	0
$417$	−16.5475	−0.810332
$418$	0	0
$419$	−5.81921	−0.284287	−0.142144	−	0.989846i	$-0.545399\pi$
$419$	−5.81921	−0.284287	−0.142144	+	0.989846i	$0.545399\pi$
$420$	0	0
$421$	28.5535	1.39161	0.695807	−	0.718229i	$-0.255047\pi$
$421$	28.5535	1.39161	0.695807	+	0.718229i	$0.255047\pi$
$422$	0	0
$423$	−6.61168	−0.321471
$424$	0	0
$425$	−37.5663	−1.82223
$426$	0	0
$427$	−9.08204	−0.439511
$428$	0	0
$429$	−1.97722	−0.0954612
$430$	0	0
$431$	−15.6937	−0.755940	−0.377970	−	0.925818i	$-0.623378\pi$
$431$	−15.6937	−0.755940	−0.377970	+	0.925818i	$0.623378\pi$
$432$	0	0
$433$	15.3074	0.735627	0.367814	−	0.929900i	$-0.380106\pi$
$433$	15.3074	0.735627	0.367814	+	0.929900i	$0.380106\pi$
$434$	0	0
$435$	2.43089	0.116552
$436$	0	0
$437$	4.20357	0.201084
$438$	0	0
$439$	−29.8595	−1.42512	−0.712558	−	0.701613i	$-0.752464\pi$
$439$	−29.8595	−1.42512	−0.712558	+	0.701613i	$0.752464\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−11.4715	−0.545027	−0.272514	−	0.962152i	$-0.587855\pi$
$443$	−11.4715	−0.545027	−0.272514	+	0.962152i	$0.587855\pi$
$444$	0	0
$445$	47.7531	2.26372
$446$	0	0
$447$	−4.04556	−0.191348
$448$	0	0
$449$	−19.9358	−0.940828	−0.470414	−	0.882446i	$-0.655895\pi$
$449$	−19.9358	−0.940828	−0.470414	+	0.882446i	$0.655895\pi$
$450$	0	0
$451$	−15.7885	−0.743451
$452$	0	0
$453$	−8.32111	−0.390960
$454$	0	0
$455$	−1.75598	−0.0823214
$456$	0	0
$457$	−34.2263	−1.60104	−0.800520	−	0.599305i	$-0.795443\pi$
$457$	−34.2263	−1.60104	−0.800520	+	0.599305i	$0.795443\pi$
$458$	0	0
$459$	4.20357	0.196206
$460$	0	0
$461$	7.12152	0.331682	0.165841	−	0.986152i	$-0.446966\pi$
$461$	7.12152	0.331682	0.165841	+	0.986152i	$0.446966\pi$
$462$	0	0
$463$	−24.2341	−1.12626	−0.563128	−	0.826370i	$-0.690402\pi$
$463$	−24.2341	−1.12626	−0.563128	+	0.826370i	$0.690402\pi$
$464$	0	0
$465$	23.1591	1.07398
$466$	0	0
$467$	23.8666	1.10441	0.552206	−	0.833707i	$-0.313786\pi$
$467$	23.8666	1.10441	0.552206	+	0.833707i	$0.313786\pi$
$468$	0	0
$469$	5.81921	0.268706
$470$	0	0
$471$	14.5712	0.671406
$472$	0	0
$473$	−50.1770	−2.30714
$474$	0	0
$475$	−37.5663	−1.72366
$476$	0	0
$477$	−3.28561	−0.150438
$478$	0	0
$479$	32.8203	1.49960	0.749800	−	0.661665i	$-0.230150\pi$
$479$	32.8203	1.49960	0.749800	+	0.661665i	$0.230150\pi$
$480$	0	0
$481$	−2.70743	−0.123448
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−15.3716	−0.697989
$486$	0	0
$487$	−25.2224	−1.14293	−0.571467	−	0.820625i	$-0.693626\pi$
$487$	−25.2224	−1.14293	−0.571467	+	0.820625i	$0.693626\pi$
$488$	0	0
$489$	15.3312	0.693299
$490$	0	0
$491$	17.6612	0.797039	0.398520	−	0.917160i	$-0.369524\pi$
$491$	17.6612	0.797039	0.398520	+	0.917160i	$0.369524\pi$
$492$	0	0
$493$	2.73717	0.123276
$494$	0	0
$495$	15.6927	0.705336
$496$	0	0
$497$	9.85075	0.441866
$498$	0	0
$499$	8.50896	0.380913	0.190457	−	0.981696i	$-0.439003\pi$
$499$	8.50896	0.380913	0.190457	+	0.981696i	$0.439003\pi$
$500$	0	0
$501$	−17.7155	−0.791471
$502$	0	0
$503$	20.9495	0.934092	0.467046	−	0.884233i	$-0.345318\pi$
$503$	20.9495	0.934092	0.467046	+	0.884233i	$0.345318\pi$
$504$	0	0
$505$	−51.1177	−2.27471
$506$	0	0
$507$	12.7788	0.567524
$508$	0	0
$509$	32.8853	1.45761	0.728807	−	0.684719i	$-0.240075\pi$
$509$	32.8853	1.45761	0.728807	+	0.684719i	$0.240075\pi$
$510$	0	0
$511$	−3.84199	−0.169960
$512$	0	0
$513$	4.20357	0.185592
$514$	0	0
$515$	61.1000	2.69239
$516$	0	0
$517$	−27.7926	−1.22232
$518$	0	0
$519$	2.90126	0.127351
$520$	0	0
$521$	−5.27685	−0.231183	−0.115592	−	0.993297i	$-0.536876\pi$
$521$	−5.27685	−0.231183	−0.115592	+	0.993297i	$0.536876\pi$
$522$	0	0
$523$	−28.9328	−1.26514	−0.632571	−	0.774502i	$-0.718000\pi$
$523$	−28.9328	−1.26514	−0.632571	+	0.774502i	$0.718000\pi$
$524$	0	0
$525$	8.93676	0.390032
$526$	0	0
$527$	26.0771	1.13594
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.58792	−0.372684
$532$	0	0
$533$	−1.76669	−0.0765238
$534$	0	0
$535$	−4.77674	−0.206516
$536$	0	0
$537$	−10.2589	−0.442702
$538$	0	0
$539$	4.20357	0.181060
$540$	0	0
$541$	1.46639	0.0630452	0.0315226	−	0.999503i	$-0.489964\pi$
$541$	1.46639	0.0630452	0.0315226	+	0.999503i	$0.489964\pi$
$542$	0	0
$543$	13.0908	0.561780
$544$	0	0
$545$	0.996027	0.0426651
$546$	0	0
$547$	30.8606	1.31951	0.659753	−	0.751483i	$-0.270661\pi$
$547$	30.8606	1.31951	0.659753	+	0.751483i	$0.270661\pi$
$548$	0	0
$549$	9.08204	0.387612
$550$	0	0
$551$	2.73717	0.116607
$552$	0	0
$553$	15.5514	0.661314
$554$	0	0
$555$	21.4882	0.912123
$556$	0	0
$557$	36.1997	1.53383	0.766916	−	0.641747i	$-0.221790\pi$
$557$	36.1997	1.53383	0.766916	+	0.641747i	$0.221790\pi$
$558$	0	0
$559$	−5.61467	−0.237475
$560$	0	0
$561$	17.6700	0.746026
$562$	0	0
$563$	−1.41110	−0.0594709	−0.0297355	−	0.999558i	$-0.509466\pi$
$563$	−1.41110	−0.0594709	−0.0297355	+	0.999558i	$0.509466\pi$
$564$	0	0
$565$	−64.2095	−2.70131
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−20.9732	−0.879244	−0.439622	−	0.898183i	$-0.644888\pi$
$569$	−20.9732	−0.879244	−0.439622	+	0.898183i	$0.644888\pi$
$570$	0	0
$571$	−21.6165	−0.904621	−0.452310	−	0.891861i	$-0.649400\pi$
$571$	−21.6165	−0.904621	−0.452310	+	0.891861i	$0.649400\pi$
$572$	0	0
$573$	−8.76870	−0.366318
$574$	0	0
$575$	−8.93676	−0.372689
$576$	0	0
$577$	−43.6285	−1.81628	−0.908140	−	0.418668i	$-0.862497\pi$
$577$	−43.6285	−1.81628	−0.908140	+	0.418668i	$0.862497\pi$
$578$	0	0
$579$	−10.3211	−0.428931
$580$	0	0
$581$	10.3677	0.430123
$582$	0	0
$583$	−13.8113	−0.572004
$584$	0	0
$585$	1.75598	0.0726007
$586$	0	0
$587$	17.5701	0.725195	0.362598	−	0.931946i	$-0.381890\pi$
$587$	17.5701	0.725195	0.362598	+	0.931946i	$0.381890\pi$
$588$	0	0
$589$	26.0771	1.07449
$590$	0	0
$591$	11.4032	0.469063
$592$	0	0
$593$	2.99335	0.122922	0.0614611	−	0.998109i	$-0.480424\pi$
$593$	2.99335	0.122922	0.0614611	+	0.998109i	$0.480424\pi$
$594$	0	0
$595$	15.6927	0.643340
$596$	0	0
$597$	−14.6660	−0.600239
$598$	0	0
$599$	18.1314	0.740829	0.370414	−	0.928867i	$-0.379216\pi$
$599$	18.1314	0.740829	0.370414	+	0.928867i	$0.379216\pi$
$600$	0	0
$601$	−22.6335	−0.923239	−0.461619	−	0.887078i	$-0.652731\pi$
$601$	−22.6335	−0.923239	−0.461619	+	0.887078i	$0.652731\pi$
$602$	0	0
$603$	−5.81921	−0.236977
$604$	0	0
$605$	24.9003	1.01234
$606$	0	0
$607$	−40.3430	−1.63747	−0.818736	−	0.574170i	$-0.805325\pi$
$607$	−40.3430	−1.63747	−0.818736	+	0.574170i	$0.805325\pi$
$608$	0	0
$609$	−0.651155	−0.0263861
$610$	0	0
$611$	−3.10992	−0.125814
$612$	0	0
$613$	−16.4992	−0.666397	−0.333199	−	0.942857i	$-0.608128\pi$
$613$	−16.4992	−0.666397	−0.333199	+	0.942857i	$0.608128\pi$
$614$	0	0
$615$	14.0218	0.565413
$616$	0	0
$617$	−24.7125	−0.994889	−0.497444	−	0.867496i	$-0.665728\pi$
$617$	−24.7125	−0.994889	−0.497444	+	0.867496i	$0.665728\pi$
$618$	0	0
$619$	−4.38435	−0.176222	−0.0881110	−	0.996111i	$-0.528083\pi$
$619$	−4.38435	−0.176222	−0.0881110	+	0.996111i	$0.528083\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−12.7915	−0.512480
$624$	0	0
$625$	10.1819	0.407276
$626$	0	0
$627$	17.6700	0.705670
$628$	0	0
$629$	24.1956	0.964743
$630$	0	0
$631$	5.71552	0.227531	0.113766	−	0.993508i	$-0.463709\pi$
$631$	5.71552	0.227531	0.113766	+	0.993508i	$0.463709\pi$
$632$	0	0
$633$	−13.7155	−0.545143
$634$	0	0
$635$	38.6158	1.53242
$636$	0	0
$637$	0.470368	0.0186367
$638$	0	0
$639$	−9.85075	−0.389690
$640$	0	0
$641$	−7.35760	−0.290608	−0.145304	−	0.989387i	$-0.546416\pi$
$641$	−7.35760	−0.290608	−0.145304	+	0.989387i	$0.546416\pi$
$642$	0	0
$643$	−13.1075	−0.516909	−0.258455	−	0.966023i	$-0.583213\pi$
$643$	−13.1075	−0.516909	−0.258455	+	0.966023i	$0.583213\pi$
$644$	0	0
$645$	44.5623	1.75464
$646$	0	0
$647$	−40.3369	−1.58581	−0.792904	−	0.609346i	$-0.791432\pi$
$647$	−40.3369	−1.58581	−0.792904	+	0.609346i	$0.791432\pi$
$648$	0	0
$649$	−36.0999	−1.41704
$650$	0	0
$651$	−6.20357	−0.243137
$652$	0	0
$653$	−21.1540	−0.827821	−0.413911	−	0.910317i	$-0.635837\pi$
$653$	−21.1540	−0.827821	−0.413911	+	0.910317i	$0.635837\pi$
$654$	0	0
$655$	23.1591	0.904902
$656$	0	0
$657$	3.84199	0.149890
$658$	0	0
$659$	25.7410	1.00273	0.501363	−	0.865237i	$-0.332832\pi$
$659$	25.7410	1.00273	0.501363	+	0.865237i	$0.332832\pi$
$660$	0	0
$661$	20.5377	0.798825	0.399412	−	0.916771i	$-0.369214\pi$
$661$	20.5377	0.798825	0.399412	+	0.916771i	$0.369214\pi$
$662$	0	0
$663$	1.97722	0.0767889
$664$	0	0
$665$	15.6927	0.608538
$666$	0	0
$667$	0.651155	0.0252128
$668$	0	0
$669$	−26.0464	−1.00701
$670$	0	0
$671$	38.1770	1.47381
$672$	0	0
$673$	40.6808	1.56813	0.784065	−	0.620679i	$-0.213143\pi$
$673$	40.6808	1.56813	0.784065	+	0.620679i	$0.213143\pi$
$674$	0	0
$675$	−8.93676	−0.343976
$676$	0	0
$677$	−4.50092	−0.172984	−0.0864922	−	0.996253i	$-0.527566\pi$
$677$	−4.50092	−0.172984	−0.0864922	+	0.996253i	$0.527566\pi$
$678$	0	0
$679$	4.11755	0.158017
$680$	0	0
$681$	−16.3044	−0.624786
$682$	0	0
$683$	−14.0771	−0.538645	−0.269322	−	0.963050i	$-0.586800\pi$
$683$	−14.0771	−0.538645	−0.269322	+	0.963050i	$0.586800\pi$
$684$	0	0
$685$	4.59301	0.175490
$686$	0	0
$687$	−14.9240	−0.569387
$688$	0	0
$689$	−1.54544	−0.0588767
$690$	0	0
$691$	−15.5376	−0.591077	−0.295539	−	0.955331i	$-0.595499\pi$
$691$	−15.5376	−0.591077	−0.295539	+	0.955331i	$0.595499\pi$
$692$	0	0
$693$	−4.20357	−0.159680
$694$	0	0
$695$	61.7749	2.34326
$696$	0	0
$697$	15.7885	0.598032
$698$	0	0
$699$	−0.172841	−0.00653743
$700$	0	0
$701$	−6.92403	−0.261517	−0.130759	−	0.991414i	$-0.541741\pi$
$701$	−6.92403	−0.261517	−0.130759	+	0.991414i	$0.541741\pi$
$702$	0	0
$703$	24.1956	0.912555
$704$	0	0
$705$	24.6827	0.929604
$706$	0	0
$707$	13.6927	0.514968
$708$	0	0
$709$	11.0721	0.415823	0.207911	−	0.978148i	$-0.433333\pi$
$709$	11.0721	0.415823	0.207911	+	0.978148i	$0.433333\pi$
$710$	0	0
$711$	−15.5514	−0.583224
$712$	0	0
$713$	6.20357	0.232325
$714$	0	0
$715$	7.38136	0.276047
$716$	0	0
$717$	28.0999	1.04941
$718$	0	0
$719$	−17.2204	−0.642213	−0.321106	−	0.947043i	$-0.604055\pi$
$719$	−17.2204	−0.642213	−0.321106	+	0.947043i	$0.604055\pi$
$720$	0	0
$721$	−16.3667	−0.609527
$722$	0	0
$723$	−16.1175	−0.599418
$724$	0	0
$725$	−5.81921	−0.216120
$726$	0	0
$727$	−16.7273	−0.620380	−0.310190	−	0.950675i	$-0.600393\pi$
$727$	−16.7273	−0.620380	−0.310190	+	0.950675i	$0.600393\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	50.1770	1.85586
$732$	0	0
$733$	−31.3855	−1.15925	−0.579625	−	0.814884i	$-0.696801\pi$
$733$	−31.3855	−1.15925	−0.579625	+	0.814884i	$0.696801\pi$
$734$	0	0
$735$	−3.73320	−0.137701
$736$	0	0
$737$	−24.4614	−0.901049
$738$	0	0
$739$	−25.4645	−0.936728	−0.468364	−	0.883536i	$-0.655156\pi$
$739$	−25.4645	−0.936728	−0.468364	+	0.883536i	$0.655156\pi$
$740$	0	0
$741$	1.97722	0.0726351
$742$	0	0
$743$	33.8568	1.24209	0.621043	−	0.783776i	$-0.286709\pi$
$743$	33.8568	1.24209	0.621043	+	0.783776i	$0.286709\pi$
$744$	0	0
$745$	15.1029	0.553326
$746$	0	0
$747$	−10.3677	−0.379333
$748$	0	0
$749$	1.27953	0.0467530
$750$	0	0
$751$	8.37033	0.305438	0.152719	−	0.988270i	$-0.451197\pi$
$751$	8.37033	0.305438	0.152719	+	0.988270i	$0.451197\pi$
$752$	0	0
$753$	−21.4259	−0.780804
$754$	0	0
$755$	31.0644	1.13055
$756$	0	0
$757$	−6.12461	−0.222603	−0.111301	−	0.993787i	$-0.535502\pi$
$757$	−6.12461	−0.222603	−0.111301	+	0.993787i	$0.535502\pi$
$758$	0	0
$759$	4.20357	0.152580
$760$	0	0
$761$	29.4310	1.06687	0.533437	−	0.845840i	$-0.320900\pi$
$761$	29.4310	1.06687	0.533437	+	0.845840i	$0.320900\pi$
$762$	0	0
$763$	−0.266803	−0.00965890
$764$	0	0
$765$	−15.6927	−0.567372
$766$	0	0
$767$	−4.03948	−0.145857
$768$	0	0
$769$	−48.4498	−1.74715	−0.873573	−	0.486693i	$-0.838203\pi$
$769$	−48.4498	−1.74715	−0.873573	+	0.486693i	$0.838203\pi$
$770$	0	0
$771$	10.3616	0.373163
$772$	0	0
$773$	12.6432	0.454745	0.227372	−	0.973808i	$-0.426987\pi$
$773$	12.6432	0.454745	0.227372	+	0.973808i	$0.426987\pi$
$774$	0	0
$775$	−55.4398	−1.99146
$776$	0	0
$777$	−5.75598	−0.206494
$778$	0	0
$779$	15.7885	0.565681
$780$	0	0
$781$	−41.4083	−1.48170
$782$	0	0
$783$	0.651155	0.0232704
$784$	0	0
$785$	−54.3972	−1.94152
$786$	0	0
$787$	21.5207	0.767130	0.383565	−	0.923514i	$-0.374696\pi$
$787$	21.5207	0.767130	0.383565	+	0.923514i	$0.374696\pi$
$788$	0	0
$789$	0.487069	0.0173401
$790$	0	0
$791$	17.1996	0.611547
$792$	0	0
$793$	4.27190	0.151700
$794$	0	0
$795$	12.2658	0.435024
$796$	0	0
$797$	12.0071	0.425312	0.212656	−	0.977127i	$-0.431789\pi$
$797$	12.0071	0.425312	0.212656	+	0.977127i	$0.431789\pi$
$798$	0	0
$799$	27.7926	0.983232
$800$	0	0
$801$	12.7915	0.451965
$802$	0	0
$803$	16.1501	0.569923
$804$	0	0
$805$	3.73320	0.131578
$806$	0	0
$807$	−10.1869	−0.358595
$808$	0	0
$809$	−4.88456	−0.171732	−0.0858659	−	0.996307i	$-0.527366\pi$
$809$	−4.88456	−0.171732	−0.0858659	+	0.996307i	$0.527366\pi$
$810$	0	0
$811$	8.65910	0.304062	0.152031	−	0.988376i	$-0.451419\pi$
$811$	8.65910	0.304062	0.152031	+	0.988376i	$0.451419\pi$
$812$	0	0
$813$	9.26283	0.324862
$814$	0	0
$815$	−57.2343	−2.00483
$816$	0	0
$817$	50.1770	1.75547
$818$	0	0
$819$	−0.470368	−0.0164360
$820$	0	0
$821$	−27.5199	−0.960451	−0.480226	−	0.877145i	$-0.659445\pi$
$821$	−27.5199	−0.960451	−0.480226	+	0.877145i	$0.659445\pi$
$822$	0	0
$823$	−41.3617	−1.44178	−0.720889	−	0.693050i	$-0.756266\pi$
$823$	−41.3617	−1.44178	−0.720889	+	0.693050i	$0.756266\pi$
$824$	0	0
$825$	−37.5663	−1.30789
$826$	0	0
$827$	−29.2697	−1.01781	−0.508904	−	0.860823i	$-0.669949\pi$
$827$	−29.2697	−1.01781	−0.508904	+	0.860823i	$0.669949\pi$
$828$	0	0
$829$	53.8340	1.86973	0.934867	−	0.354999i	$-0.115519\pi$
$829$	53.8340	1.86973	0.934867	+	0.354999i	$0.115519\pi$
$830$	0	0
$831$	−26.4220	−0.916568
$832$	0	0
$833$	−4.20357	−0.145645
$834$	0	0
$835$	66.1355	2.28871
$836$	0	0
$837$	6.20357	0.214427
$838$	0	0
$839$	−24.5920	−0.849011	−0.424506	−	0.905425i	$-0.639552\pi$
$839$	−24.5920	−0.849011	−0.424506	+	0.905425i	$0.639552\pi$
$840$	0	0
$841$	−28.5760	−0.985379
$842$	0	0
$843$	−14.3211	−0.493245
$844$	0	0
$845$	−47.7056	−1.64112
$846$	0	0
$847$	−6.66996	−0.229183
$848$	0	0
$849$	6.71439	0.230437
$850$	0	0
$851$	5.75598	0.197312
$852$	0	0
$853$	5.83503	0.199787	0.0998937	−	0.994998i	$-0.468150\pi$
$853$	5.83503	0.199787	0.0998937	+	0.994998i	$0.468150\pi$
$854$	0	0
$855$	−15.6927	−0.536680
$856$	0	0
$857$	33.3450	1.13904	0.569522	−	0.821976i	$-0.307128\pi$
$857$	33.3450	1.13904	0.569522	+	0.821976i	$0.307128\pi$
$858$	0	0
$859$	−1.26381	−0.0431206	−0.0215603	−	0.999768i	$-0.506863\pi$
$859$	−1.26381	−0.0431206	−0.0215603	+	0.999768i	$0.506863\pi$
$860$	0	0
$861$	−3.75598	−0.128003
$862$	0	0
$863$	41.0179	1.39627	0.698133	−	0.715968i	$-0.254014\pi$
$863$	41.0179	1.39627	0.698133	+	0.715968i	$0.254014\pi$
$864$	0	0
$865$	−10.8310	−0.368264
$866$	0	0
$867$	−0.669960	−0.0227530
$868$	0	0
$869$	−65.3715	−2.21757
$870$	0	0
$871$	−2.73717	−0.0927455
$872$	0	0
$873$	−4.11755	−0.139358
$874$	0	0
$875$	−14.6967	−0.496840
$876$	0	0
$877$	21.6031	0.729484	0.364742	−	0.931109i	$-0.381157\pi$
$877$	21.6031	0.729484	0.364742	+	0.931109i	$0.381157\pi$
$878$	0	0
$879$	1.22335	0.0412626
$880$	0	0
$881$	24.8872	0.838472	0.419236	−	0.907877i	$-0.362298\pi$
$881$	24.8872	0.838472	0.419236	+	0.907877i	$0.362298\pi$
$882$	0	0
$883$	−41.7223	−1.40407	−0.702034	−	0.712144i	$-0.747724\pi$
$883$	−41.7223	−1.40407	−0.702034	+	0.712144i	$0.747724\pi$
$884$	0	0
$885$	32.0604	1.07770
$886$	0	0
$887$	−18.8370	−0.632486	−0.316243	−	0.948678i	$-0.602421\pi$
$887$	−18.8370	−0.632486	−0.316243	+	0.948678i	$0.602421\pi$
$888$	0	0
$889$	−10.3439	−0.346923
$890$	0	0
$891$	4.20357	0.140825
$892$	0	0
$893$	27.7926	0.930044
$894$	0	0
$895$	38.2983	1.28017
$896$	0	0
$897$	0.470368	0.0157051
$898$	0	0
$899$	4.03948	0.134724
$900$	0	0
$901$	13.8113	0.460120
$902$	0	0
$903$	−11.9368	−0.397231
$904$	0	0
$905$	−48.8705	−1.62451
$906$	0	0
$907$	−6.72145	−0.223182	−0.111591	−	0.993754i	$-0.535595\pi$
$907$	−6.72145	−0.223182	−0.111591	+	0.993754i	$0.535595\pi$
$908$	0	0
$909$	−13.6927	−0.454159
$910$	0	0
$911$	−10.3587	−0.343200	−0.171600	−	0.985167i	$-0.554894\pi$
$911$	−10.3587	−0.343200	−0.171600	+	0.985167i	$0.554894\pi$
$912$	0	0
$913$	−43.5811	−1.44232
$914$	0	0
$915$	−33.9051	−1.12087
$916$	0	0
$917$	−6.20357	−0.204860
$918$	0	0
$919$	−24.0041	−0.791823	−0.395911	−	0.918289i	$-0.629571\pi$
$919$	−24.0041	−0.791823	−0.395911	+	0.918289i	$0.629571\pi$
$920$	0	0
$921$	26.9774	0.888934
$922$	0	0
$923$	−4.63347	−0.152513
$924$	0	0
$925$	−51.4398	−1.69133
$926$	0	0
$927$	16.3667	0.537552
$928$	0	0
$929$	10.0473	0.329640	0.164820	−	0.986324i	$-0.447296\pi$
$929$	10.0473	0.329640	0.164820	+	0.986324i	$0.447296\pi$
$930$	0	0
$931$	−4.20357	−0.137766
$932$	0	0
$933$	−12.9100	−0.422655
$934$	0	0
$935$	−65.9654	−2.15730
$936$	0	0
$937$	−29.5304	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$937$	−29.5304	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$938$	0	0
$939$	7.47434	0.243916
$940$	0	0
$941$	15.6333	0.509632	0.254816	−	0.966990i	$-0.417985\pi$
$941$	15.6333	0.509632	0.254816	+	0.966990i	$0.417985\pi$
$942$	0	0
$943$	3.75598	0.122311
$944$	0	0
$945$	3.73320	0.121441
$946$	0	0
$947$	−9.66898	−0.314200	−0.157100	−	0.987583i	$-0.550214\pi$
$947$	−9.66898	−0.314200	−0.157100	+	0.987583i	$0.550214\pi$
$948$	0	0
$949$	1.80715	0.0586625
$950$	0	0
$951$	25.7788	0.835933
$952$	0	0
$953$	15.3627	0.497647	0.248823	−	0.968549i	$-0.419956\pi$
$953$	15.3627	0.497647	0.248823	+	0.968549i	$0.419956\pi$
$954$	0	0
$955$	32.7353	1.05929
$956$	0	0
$957$	2.73717	0.0884802
$958$	0	0
$959$	−1.23032	−0.0397290
$960$	0	0
$961$	7.48422	0.241426
$962$	0	0
$963$	−1.27953	−0.0412323
$964$	0	0
$965$	38.5308	1.24035
$966$	0	0
$967$	−11.5929	−0.372802	−0.186401	−	0.982474i	$-0.559682\pi$
$967$	−11.5929	−0.372802	−0.186401	+	0.982474i	$0.559682\pi$
$968$	0	0
$969$	−17.6700	−0.567641
$970$	0	0
$971$	−2.79943	−0.0898379	−0.0449190	−	0.998991i	$-0.514303\pi$
$971$	−2.79943	−0.0898379	−0.0449190	+	0.998991i	$0.514303\pi$
$972$	0	0
$973$	−16.5475	−0.530487
$974$	0	0
$975$	−4.20357	−0.134622
$976$	0	0
$977$	−27.5137	−0.880243	−0.440121	−	0.897938i	$-0.645065\pi$
$977$	−27.5137	−0.880243	−0.440121	+	0.897938i	$0.645065\pi$
$978$	0	0
$979$	53.7698	1.71849
$980$	0	0
$981$	0.266803	0.00851835
$982$	0	0
$983$	−42.9564	−1.37010	−0.685048	−	0.728498i	$-0.740219\pi$
$983$	−42.9564	−1.37010	−0.685048	+	0.728498i	$0.740219\pi$
$984$	0	0
$985$	−42.5702	−1.35640
$986$	0	0
$987$	−6.61168	−0.210452
$988$	0	0
$989$	11.9368	0.379567
$990$	0	0
$991$	25.7698	0.818606	0.409303	−	0.912399i	$-0.365772\pi$
$991$	25.7698	0.818606	0.409303	+	0.912399i	$0.365772\pi$
$992$	0	0
$993$	−35.1029	−1.11396
$994$	0	0
$995$	54.7510	1.73572
$996$	0	0
$997$	25.1618	0.796883	0.398441	−	0.917194i	$-0.369551\pi$
$997$	25.1618	0.796883	0.398441	+	0.917194i	$0.369551\pi$
$998$	0	0
$999$	5.75598	0.182111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.ca.1.4		4
4.3	odd	2		3864.2.a.u.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.u.1.4	✓	4	4.3	odd	2
7728.2.a.ca.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.73320 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.73320 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.20357 q^{11} +0.470368 q^{13} -3.73320 q^{15} -4.20357 q^{17} -4.20357 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.93676 q^{25} -1.00000 q^{27} -0.651155 q^{29} -6.20357 q^{31} -4.20357 q^{33} -3.73320 q^{35} -5.75598 q^{37} -0.470368 q^{39} -3.75598 q^{41} -11.9368 q^{43} +3.73320 q^{45} -6.61168 q^{47} +1.00000 q^{49} +4.20357 q^{51} -3.28561 q^{53} +15.6927 q^{55} +4.20357 q^{57} -8.58792 q^{59} +9.08204 q^{61} -1.00000 q^{63} +1.75598 q^{65} -5.81921 q^{67} +1.00000 q^{69} -9.85075 q^{71} +3.84199 q^{73} -8.93676 q^{75} -4.20357 q^{77} -15.5514 q^{79} +1.00000 q^{81} -10.3677 q^{83} -15.6927 q^{85} +0.651155 q^{87} +12.7915 q^{89} -0.470368 q^{91} +6.20357 q^{93} -15.6927 q^{95} -4.11755 q^{97} +4.20357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} - 2 q^{15} - 2 q^{17} - 2 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} - 4 q^{27} - q^{29} - 10 q^{31} - 2 q^{33} - 2 q^{35} + 5 q^{37} + 13 q^{41} - 20 q^{43} + 2 q^{45} - 17 q^{47} + 4 q^{49} + 2 q^{51} + 13 q^{53} + 7 q^{55} + 2 q^{57} - 5 q^{59} + 25 q^{61} - 4 q^{63} - 21 q^{65} - 23 q^{67} + 4 q^{69} + q^{71} - 8 q^{75} - 2 q^{77} - 14 q^{79} + 4 q^{81} - 4 q^{83} - 7 q^{85} + q^{87} + 7 q^{89} + 10 q^{93} - 7 q^{95} + 11 q^{97} + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 - 2 * q^15 - 2 * q^17 - 2 * q^19 + 4 * q^21 - 4 * q^23 + 8 * q^25 - 4 * q^27 - q^29 - 10 * q^31 - 2 * q^33 - 2 * q^35 + 5 * q^37 + 13 * q^41 - 20 * q^43 + 2 * q^45 - 17 * q^47 + 4 * q^49 + 2 * q^51 + 13 * q^53 + 7 * q^55 + 2 * q^57 - 5 * q^59 + 25 * q^61 - 4 * q^63 - 21 * q^65 - 23 * q^67 + 4 * q^69 + q^71 - 8 * q^75 - 2 * q^77 - 14 * q^79 + 4 * q^81 - 4 * q^83 - 7 * q^85 + q^87 + 7 * q^89 + 10 * q^93 - 7 * q^95 + 11 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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