LMFDB - Embedded newform 725.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.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:	$5.78915414654$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.517638 q^{2} +1.41421 q^{3} -1.73205 q^{4} +0.732051 q^{6} -2.44949 q^{7} -1.93185 q^{8} -1.00000 q^{9} -1.26795 q^{11} -2.44949 q^{12} -1.79315 q^{13} -1.26795 q^{14} +2.46410 q^{16} +1.41421 q^{17} -0.517638 q^{18} -3.26795 q^{19} -3.46410 q^{21} -0.656339 q^{22} -6.31319 q^{23} -2.73205 q^{24} -0.928203 q^{26} -5.65685 q^{27} +4.24264 q^{28} +1.00000 q^{29} -8.73205 q^{31} +5.13922 q^{32} -1.79315 q^{33} +0.732051 q^{34} +1.73205 q^{36} +9.14162 q^{37} -1.69161 q^{38} -2.53590 q^{39} -6.92820 q^{41} -1.79315 q^{42} +9.14162 q^{43} +2.19615 q^{44} -3.26795 q^{46} +1.41421 q^{47} +3.48477 q^{48} -1.00000 q^{49} +2.00000 q^{51} +3.10583 q^{52} +5.93426 q^{53} -2.92820 q^{54} +4.73205 q^{56} -4.62158 q^{57} +0.517638 q^{58} -10.3923 q^{59} +2.92820 q^{61} -4.52004 q^{62} +2.44949 q^{63} -2.26795 q^{64} -0.928203 q^{66} +4.24264 q^{67} -2.44949 q^{68} -8.92820 q^{69} +3.46410 q^{71} +1.93185 q^{72} -7.34847 q^{73} +4.73205 q^{74} +5.66025 q^{76} +3.10583 q^{77} -1.31268 q^{78} -4.19615 q^{79} -5.00000 q^{81} -3.58630 q^{82} +10.1769 q^{83} +6.00000 q^{84} +4.73205 q^{86} +1.41421 q^{87} +2.44949 q^{88} +10.3923 q^{89} +4.39230 q^{91} +10.9348 q^{92} -12.3490 q^{93} +0.732051 q^{94} +7.26795 q^{96} -10.9348 q^{97} -0.517638 q^{98} +1.26795 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{6} - 4 q^{9} - 12 q^{11} - 12 q^{14} - 4 q^{16} - 20 q^{19} - 4 q^{24} + 24 q^{26} + 4 q^{29} - 28 q^{31} - 4 q^{34} - 24 q^{39} - 12 q^{44} - 20 q^{46} - 4 q^{49} + 8 q^{51} + 16 q^{54} + 12 q^{56}+ \cdots + 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^6 - 4 * q^9 - 12 * q^11 - 12 * q^14 - 4 * q^16 - 20 * q^19 - 4 * q^24 + 24 * q^26 + 4 * q^29 - 28 * q^31 - 4 * q^34 - 24 * q^39 - 12 * q^44 - 20 * q^46 - 4 * q^49 + 8 * q^51 + 16 * q^54 + 12 * q^56 - 16 * q^61 - 16 * q^64 + 24 * q^66 - 8 * q^69 + 12 * q^74 - 12 * q^76 + 4 * q^79 - 20 * q^81 + 24 * q^84 + 12 * q^86 - 24 * q^91 - 4 * q^94 + 36 * q^96 + 12 * 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 $

$2$	0.517638	0.366025	0.183013	−	0.983111i	$-0.441415\pi$
$2$	0.517638	0.366025	0.183013	+	0.983111i	$0.441415\pi$
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	−1.73205	−0.866025
$5$	0	0
$5$	0	0
$6$	0.732051	0.298858
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	−1.93185	−0.683013
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	−2.44949	−0.707107
$13$	−1.79315	−0.497331	−0.248665	−	0.968589i	$-0.579992\pi$
$13$	−1.79315	−0.497331	−0.248665	+	0.968589i	$0.579992\pi$
$14$	−1.26795	−0.338874
$15$	0	0
$16$	2.46410	0.616025
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	−0.517638	−0.122008
$19$	−3.26795	−0.749719	−0.374859	−	0.927082i	$-0.622309\pi$
$19$	−3.26795	−0.749719	−0.374859	+	0.927082i	$0.622309\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	−0.656339	−0.139932
$23$	−6.31319	−1.31639	−0.658196	−	0.752847i	$-0.728680\pi$
$23$	−6.31319	−1.31639	−0.658196	+	0.752847i	$0.728680\pi$
$24$	−2.73205	−0.557678
$25$	0	0
$26$	−0.928203	−0.182036
$27$	−5.65685	−1.08866
$28$	4.24264	0.801784
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.73205	−1.56832	−0.784161	−	0.620557i	$-0.786907\pi$
$31$	−8.73205	−1.56832	−0.784161	+	0.620557i	$0.786907\pi$
$32$	5.13922	0.908494
$33$	−1.79315	−0.312148
$34$	0.732051	0.125546
$35$	0	0
$36$	1.73205	0.288675
$37$	9.14162	1.50287	0.751437	−	0.659805i	$-0.229361\pi$
$37$	9.14162	1.50287	0.751437	+	0.659805i	$0.229361\pi$
$38$	−1.69161	−0.274416
$39$	−2.53590	−0.406069
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	−1.79315	−0.276689
$43$	9.14162	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$43$	9.14162	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$44$	2.19615	0.331082
$45$	0	0
$46$	−3.26795	−0.481833
$47$	1.41421	0.206284	0.103142	−	0.994667i	$-0.467110\pi$
$47$	1.41421	0.206284	0.103142	+	0.994667i	$0.467110\pi$
$48$	3.48477	0.502983
$49$	−1.00000	−0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	3.10583	0.430701
$53$	5.93426	0.815133	0.407566	−	0.913176i	$-0.366378\pi$
$53$	5.93426	0.815133	0.407566	+	0.913176i	$0.366378\pi$
$54$	−2.92820	−0.398478
$55$	0	0
$56$	4.73205	0.632347
$57$	−4.62158	−0.612143
$58$	0.517638	0.0679692
$59$	−10.3923	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	2.92820	0.374918	0.187459	−	0.982272i	$-0.439975\pi$
$61$	2.92820	0.374918	0.187459	+	0.982272i	$0.439975\pi$
$62$	−4.52004	−0.574046
$63$	2.44949	0.308607
$64$	−2.26795	−0.283494
$65$	0	0
$66$	−0.928203	−0.114254
$67$	4.24264	0.518321	0.259161	−	0.965834i	$-0.416554\pi$
$67$	4.24264	0.518321	0.259161	+	0.965834i	$0.416554\pi$
$68$	−2.44949	−0.297044
$69$	−8.92820	−1.07483
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	1.93185	0.227671
$73$	−7.34847	−0.860073	−0.430037	−	0.902811i	$-0.641499\pi$
$73$	−7.34847	−0.860073	−0.430037	+	0.902811i	$0.641499\pi$
$74$	4.73205	0.550090
$75$	0	0
$76$	5.66025	0.649276
$77$	3.10583	0.353942
$78$	−1.31268	−0.148631
$79$	−4.19615	−0.472104	−0.236052	−	0.971740i	$-0.575854\pi$
$79$	−4.19615	−0.472104	−0.236052	+	0.971740i	$0.575854\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	−3.58630	−0.396041
$83$	10.1769	1.11706	0.558530	−	0.829484i	$-0.311366\pi$
$83$	10.1769	1.11706	0.558530	+	0.829484i	$0.311366\pi$
$84$	6.00000	0.654654
$85$	0	0
$86$	4.73205	0.510270
$87$	1.41421	0.151620
$88$	2.44949	0.261116
$89$	10.3923	1.10158	0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923	1.10158	0.550791	+	0.834643i	$0.314326\pi$
$90$	0	0
$91$	4.39230	0.460439
$92$	10.9348	1.14003
$93$	−12.3490	−1.28053
$94$	0.732051	0.0755053
$95$	0	0
$96$	7.26795	0.741782
$97$	−10.9348	−1.11026	−0.555129	−	0.831764i	$-0.687331\pi$
$97$	−10.9348	−1.11026	−0.555129	+	0.831764i	$0.687331\pi$
$98$	−0.517638	−0.0522893
$99$	1.26795	0.127434
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	1.03528	0.102508
$103$	9.14162	0.900751	0.450375	−	0.892839i	$-0.351290\pi$
$103$	9.14162	0.900751	0.450375	+	0.892839i	$0.351290\pi$
$104$	3.46410	0.339683
$105$	0	0
$106$	3.07180	0.298359
$107$	−18.6622	−1.80414	−0.902070	−	0.431589i	$-0.857953\pi$
$107$	−18.6622	−1.80414	−0.902070	+	0.431589i	$0.857953\pi$
$108$	9.79796	0.942809
$109$	17.8564	1.71033	0.855167	−	0.518353i	$-0.173455\pi$
$109$	17.8564	1.71033	0.855167	+	0.518353i	$0.173455\pi$
$110$	0	0
$111$	12.9282	1.22709
$112$	−6.03579	−0.570329
$113$	−13.0053	−1.22344	−0.611719	−	0.791075i	$-0.709522\pi$
$113$	−13.0053	−1.22344	−0.611719	+	0.791075i	$0.709522\pi$
$114$	−2.39230	−0.224060
$115$	0	0
$116$	−1.73205	−0.160817
$117$	1.79315	0.165777
$118$	−5.37945	−0.495219
$119$	−3.46410	−0.317554
$120$	0	0
$121$	−9.39230	−0.853846
$122$	1.51575	0.137230
$123$	−9.79796	−0.883452
$124$	15.1244	1.35821
$125$	0	0
$126$	1.26795	0.112958
$127$	4.24264	0.376473	0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264	0.376473	0.188237	+	0.982124i	$0.439723\pi$
$128$	−11.4524	−1.01226
$129$	12.9282	1.13826
$130$	0	0
$131$	−4.73205	−0.413441	−0.206721	−	0.978400i	$-0.566279\pi$
$131$	−4.73205	−0.413441	−0.206721	+	0.978400i	$0.566279\pi$
$132$	3.10583	0.270328
$133$	8.00481	0.694105
$134$	2.19615	0.189719
$135$	0	0
$136$	−2.73205	−0.234271
$137$	14.7985	1.26432	0.632159	−	0.774838i	$-0.282169\pi$
$137$	14.7985	1.26432	0.632159	+	0.774838i	$0.282169\pi$
$138$	−4.62158	−0.393415
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	1.79315	0.150478
$143$	2.27362	0.190130
$144$	−2.46410	−0.205342
$145$	0	0
$146$	−3.80385	−0.314809
$147$	−1.41421	−0.116642
$148$	−15.8338	−1.30153
$149$	−0.928203	−0.0760414	−0.0380207	−	0.999277i	$-0.512105\pi$
$149$	−0.928203	−0.0760414	−0.0380207	+	0.999277i	$0.512105\pi$
$150$	0	0
$151$	−18.7846	−1.52867	−0.764335	−	0.644819i	$-0.776933\pi$
$151$	−18.7846	−1.52867	−0.764335	+	0.644819i	$0.776933\pi$
$152$	6.31319	0.512068
$153$	−1.41421	−0.114332
$154$	1.60770	0.129552
$155$	0	0
$156$	4.39230	0.351666
$157$	10.9348	0.872690	0.436345	−	0.899780i	$-0.356273\pi$
$157$	10.9348	0.872690	0.436345	+	0.899780i	$0.356273\pi$
$158$	−2.17209	−0.172802
$159$	8.39230	0.665553
$160$	0	0
$161$	15.4641	1.21874
$162$	−2.58819	−0.203347
$163$	14.0406	1.09974	0.549872	−	0.835249i	$-0.314676\pi$
$163$	14.0406	1.09974	0.549872	+	0.835249i	$0.314676\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	5.26795	0.408872
$167$	11.2122	0.867624	0.433812	−	0.901003i	$-0.357168\pi$
$167$	11.2122	0.867624	0.433812	+	0.901003i	$0.357168\pi$
$168$	6.69213	0.516309
$169$	−9.78461	−0.752662
$170$	0	0
$171$	3.26795	0.249906
$172$	−15.8338	−1.20731
$173$	4.62158	0.351372	0.175686	−	0.984446i	$-0.443786\pi$
$173$	4.62158	0.351372	0.175686	+	0.984446i	$0.443786\pi$
$174$	0.732051	0.0554966
$175$	0	0
$176$	−3.12436	−0.235507
$177$	−14.6969	−1.10469
$178$	5.37945	0.403207
$179$	2.53590	0.189542	0.0947710	−	0.995499i	$-0.469788\pi$
$179$	2.53590	0.189542	0.0947710	+	0.995499i	$0.469788\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	2.27362	0.168532
$183$	4.14110	0.306119
$184$	12.1962	0.899112
$185$	0	0
$186$	−6.39230	−0.468707
$187$	−1.79315	−0.131128
$188$	−2.44949	−0.178647
$189$	13.8564	1.00791
$190$	0	0
$191$	15.1244	1.09436	0.547180	−	0.837015i	$-0.315701\pi$
$191$	15.1244	1.09436	0.547180	+	0.837015i	$0.315701\pi$
$192$	−3.20736	−0.231472
$193$	−0.656339	−0.0472443	−0.0236222	−	0.999721i	$-0.507520\pi$
$193$	−0.656339	−0.0472443	−0.0236222	+	0.999721i	$0.507520\pi$
$194$	−5.66025	−0.406383
$195$	0	0
$196$	1.73205	0.123718
$197$	−22.4243	−1.59767	−0.798834	−	0.601551i	$-0.794550\pi$
$197$	−22.4243	−1.59767	−0.798834	+	0.601551i	$0.794550\pi$
$198$	0.656339	0.0466440
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	−3.10583	−0.218525
$203$	−2.44949	−0.171920
$204$	−3.46410	−0.242536
$205$	0	0
$206$	4.73205	0.329698
$207$	6.31319	0.438797
$208$	−4.41851	−0.306368
$209$	4.14359	0.286618
$210$	0	0
$211$	−14.7321	−1.01420	−0.507098	−	0.861888i	$-0.669282\pi$
$211$	−14.7321	−1.01420	−0.507098	+	0.861888i	$0.669282\pi$
$212$	−10.2784	−0.705926
$213$	4.89898	0.335673
$214$	−9.66025	−0.660361
$215$	0	0
$216$	10.9282	0.743570
$217$	21.3891	1.45198
$218$	9.24316	0.626026
$219$	−10.3923	−0.702247
$220$	0	0
$221$	−2.53590	−0.170583
$222$	6.69213	0.449146
$223$	−15.3533	−1.02813	−0.514066	−	0.857751i	$-0.671861\pi$
$223$	−15.3533	−1.02813	−0.514066	+	0.857751i	$0.671861\pi$
$224$	−12.5885	−0.841102
$225$	0	0
$226$	−6.73205	−0.447809
$227$	−20.4553	−1.35767	−0.678834	−	0.734292i	$-0.737514\pi$
$227$	−20.4553	−1.35767	−0.678834	+	0.734292i	$0.737514\pi$
$228$	8.00481	0.530131
$229$	18.7846	1.24132	0.620661	−	0.784079i	$-0.286864\pi$
$229$	18.7846	1.24132	0.620661	+	0.784079i	$0.286864\pi$
$230$	0	0
$231$	4.39230	0.288992
$232$	−1.93185	−0.126832
$233$	−3.38323	−0.221643	−0.110821	−	0.993840i	$-0.535348\pi$
$233$	−3.38323	−0.221643	−0.110821	+	0.993840i	$0.535348\pi$
$234$	0.928203	0.0606785
$235$	0	0
$236$	18.0000	1.17170
$237$	−5.93426	−0.385471
$238$	−1.79315	−0.116233
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	0	0
$241$	11.4641	0.738468	0.369234	−	0.929337i	$-0.379620\pi$
$241$	11.4641	0.738468	0.369234	+	0.929337i	$0.379620\pi$
$242$	−4.86181	−0.312529
$243$	9.89949	0.635053
$244$	−5.07180	−0.324689
$245$	0	0
$246$	−5.07180	−0.323366
$247$	5.85993	0.372858
$248$	16.8690	1.07118
$249$	14.3923	0.912075
$250$	0	0
$251$	15.1244	0.954641	0.477320	−	0.878729i	$-0.341608\pi$
$251$	15.1244	0.954641	0.477320	+	0.878729i	$0.341608\pi$
$252$	−4.24264	−0.267261
$253$	8.00481	0.503258
$254$	2.19615	0.137799
$255$	0	0
$256$	−1.39230	−0.0870191
$257$	−9.52056	−0.593876	−0.296938	−	0.954897i	$-0.595965\pi$
$257$	−9.52056	−0.593876	−0.296938	+	0.954897i	$0.595965\pi$
$258$	6.69213	0.416634
$259$	−22.3923	−1.39139
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−2.44949	−0.151330
$263$	4.79744	0.295823	0.147912	−	0.989001i	$-0.452745\pi$
$263$	4.79744	0.295823	0.147912	+	0.989001i	$0.452745\pi$
$264$	3.46410	0.213201
$265$	0	0
$266$	4.14359	0.254060
$267$	14.6969	0.899438
$268$	−7.34847	−0.448879
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−26.7321	−1.62386	−0.811928	−	0.583757i	$-0.801582\pi$
$271$	−26.7321	−1.62386	−0.811928	+	0.583757i	$0.801582\pi$
$272$	3.48477	0.211295
$273$	6.21166	0.375947
$274$	7.66025	0.462773
$275$	0	0
$276$	15.4641	0.930830
$277$	18.7637	1.12740	0.563701	−	0.825979i	$-0.309377\pi$
$277$	18.7637	1.12740	0.563701	+	0.825979i	$0.309377\pi$
$278$	−4.14110	−0.248367
$279$	8.73205	0.522774
$280$	0	0
$281$	32.7846	1.95577	0.977883	−	0.209153i	$-0.0670707\pi$
$281$	32.7846	1.95577	0.977883	+	0.209153i	$0.0670707\pi$
$282$	1.03528	0.0616498
$283$	−27.4249	−1.63024	−0.815119	−	0.579293i	$-0.803329\pi$
$283$	−27.4249	−1.63024	−0.815119	+	0.579293i	$0.803329\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	1.17691	0.0695924
$287$	16.9706	1.00174
$288$	−5.13922	−0.302831
$289$	−15.0000	−0.882353
$290$	0	0
$291$	−15.4641	−0.906522
$292$	12.7279	0.744845
$293$	30.7338	1.79549	0.897743	−	0.440520i	$-0.145206\pi$
$293$	30.7338	1.79549	0.897743	+	0.440520i	$0.145206\pi$
$294$	−0.732051	−0.0426941
$295$	0	0
$296$	−17.6603	−1.02648
$297$	7.17260	0.416197
$298$	−0.480473	−0.0278331
$299$	11.3205	0.654682
$300$	0	0
$301$	−22.3923	−1.29067
$302$	−9.72363	−0.559532
$303$	−8.48528	−0.487467
$304$	−8.05256	−0.461846
$305$	0	0
$306$	−0.732051	−0.0418486
$307$	10.4543	0.596658	0.298329	−	0.954463i	$-0.403571\pi$
$307$	10.4543	0.596658	0.298329	+	0.954463i	$0.403571\pi$
$308$	−5.37945	−0.306523
$309$	12.9282	0.735460
$310$	0	0
$311$	−6.58846	−0.373597	−0.186799	−	0.982398i	$-0.559811\pi$
$311$	−6.58846	−0.373597	−0.186799	+	0.982398i	$0.559811\pi$
$312$	4.89898	0.277350
$313$	−23.6627	−1.33749	−0.668747	−	0.743490i	$-0.733169\pi$
$313$	−23.6627	−1.33749	−0.668747	+	0.743490i	$0.733169\pi$
$314$	5.66025	0.319427
$315$	0	0
$316$	7.26795	0.408854
$317$	−5.75839	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$317$	−5.75839	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$318$	4.34418	0.243609
$319$	−1.26795	−0.0709915
$320$	0	0
$321$	−26.3923	−1.47307
$322$	8.00481	0.446091
$323$	−4.62158	−0.257151
$324$	8.66025	0.481125
$325$	0	0
$326$	7.26795	0.402534
$327$	25.2528	1.39648
$328$	13.3843	0.739022
$329$	−3.46410	−0.190982
$330$	0	0
$331$	−18.1962	−1.00015	−0.500075	−	0.865982i	$-0.666694\pi$
$331$	−18.1962	−1.00015	−0.500075	+	0.865982i	$0.666694\pi$
$332$	−17.6269	−0.967402
$333$	−9.14162	−0.500958
$334$	5.80385	0.317572
$335$	0	0
$336$	−8.53590	−0.465671
$337$	−27.9053	−1.52010	−0.760050	−	0.649864i	$-0.774826\pi$
$337$	−27.9053	−1.52010	−0.760050	+	0.649864i	$0.774826\pi$
$338$	−5.06489	−0.275494
$339$	−18.3923	−0.998933
$340$	0	0
$341$	11.0718	0.599571
$342$	1.69161	0.0914721
$343$	19.5959	1.05808
$344$	−17.6603	−0.952177
$345$	0	0
$346$	2.39230	0.128611
$347$	21.4906	1.15368	0.576838	−	0.816859i	$-0.304286\pi$
$347$	21.4906	1.15368	0.576838	+	0.816859i	$0.304286\pi$
$348$	−2.44949	−0.131306
$349$	−22.7846	−1.21963	−0.609816	−	0.792543i	$-0.708757\pi$
$349$	−22.7846	−1.21963	−0.609816	+	0.792543i	$0.708757\pi$
$350$	0	0
$351$	10.1436	0.541425
$352$	−6.51626	−0.347318
$353$	−31.9449	−1.70026	−0.850128	−	0.526576i	$-0.823475\pi$
$353$	−31.9449	−1.70026	−0.850128	+	0.526576i	$0.823475\pi$
$354$	−7.60770	−0.404344
$355$	0	0
$356$	−18.0000	−0.953998
$357$	−4.89898	−0.259281
$358$	1.31268	0.0693772
$359$	−7.26795	−0.383588	−0.191794	−	0.981435i	$-0.561430\pi$
$359$	−7.26795	−0.383588	−0.191794	+	0.981435i	$0.561430\pi$
$360$	0	0
$361$	−8.32051	−0.437921
$362$	−8.28221	−0.435303
$363$	−13.2827	−0.697162
$364$	−7.60770	−0.398752
$365$	0	0
$366$	2.14359	0.112047
$367$	−33.6365	−1.75581	−0.877906	−	0.478833i	$-0.841060\pi$
$367$	−33.6365	−1.75581	−0.877906	+	0.478833i	$0.841060\pi$
$368$	−15.5563	−0.810931
$369$	6.92820	0.360668
$370$	0	0
$371$	−14.5359	−0.754666
$372$	21.3891	1.10897
$373$	−19.5959	−1.01464	−0.507319	−	0.861758i	$-0.669363\pi$
$373$	−19.5959	−1.01464	−0.507319	+	0.861758i	$0.669363\pi$
$374$	−0.928203	−0.0479962
$375$	0	0
$376$	−2.73205	−0.140895
$377$	−1.79315	−0.0923520
$378$	7.17260	0.368919
$379$	−16.1962	−0.831940	−0.415970	−	0.909378i	$-0.636558\pi$
$379$	−16.1962	−0.831940	−0.415970	+	0.909378i	$0.636558\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	7.82894	0.400564
$383$	−0.933740	−0.0477119	−0.0238559	−	0.999715i	$-0.507594\pi$
$383$	−0.933740	−0.0477119	−0.0238559	+	0.999715i	$0.507594\pi$
$384$	−16.1962	−0.826506
$385$	0	0
$386$	−0.339746	−0.0172926
$387$	−9.14162	−0.464695
$388$	18.9396	0.961511
$389$	−20.7846	−1.05382	−0.526911	−	0.849921i	$-0.676650\pi$
$389$	−20.7846	−1.05382	−0.526911	+	0.849921i	$0.676650\pi$
$390$	0	0
$391$	−8.92820	−0.451519
$392$	1.93185	0.0975732
$393$	−6.69213	−0.337573
$394$	−11.6077	−0.584787
$395$	0	0
$396$	−2.19615	−0.110361
$397$	4.89898	0.245873	0.122936	−	0.992415i	$-0.460769\pi$
$397$	4.89898	0.245873	0.122936	+	0.992415i	$0.460769\pi$
$398$	−7.24693	−0.363256
$399$	11.3205	0.566734
$400$	0	0
$401$	37.8564	1.89046	0.945229	−	0.326407i	$-0.105838\pi$
$401$	37.8564	1.89046	0.945229	+	0.326407i	$0.105838\pi$
$402$	3.10583	0.154905
$403$	15.6579	0.779975
$404$	10.3923	0.517036
$405$	0	0
$406$	−1.26795	−0.0629273
$407$	−11.5911	−0.574550
$408$	−3.86370	−0.191282
$409$	9.07180	0.448571	0.224286	−	0.974523i	$-0.427995\pi$
$409$	9.07180	0.448571	0.224286	+	0.974523i	$0.427995\pi$
$410$	0	0
$411$	20.9282	1.03231
$412$	−15.8338	−0.780073
$413$	25.4558	1.25260
$414$	3.26795	0.160611
$415$	0	0
$416$	−9.21539	−0.451822
$417$	−11.3137	−0.554035
$418$	2.14488	0.104910
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	5.46410	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$422$	−7.62587	−0.371222
$423$	−1.41421	−0.0687614
$424$	−11.4641	−0.556746
$425$	0	0
$426$	2.53590	0.122865
$427$	−7.17260	−0.347107
$428$	32.3238	1.56243
$429$	3.21539	0.155241
$430$	0	0
$431$	−37.1769	−1.79075	−0.895374	−	0.445314i	$-0.853092\pi$
$431$	−37.1769	−1.79075	−0.895374	+	0.445314i	$0.853092\pi$
$432$	−13.9391	−0.670644
$433$	−2.92996	−0.140805	−0.0704025	−	0.997519i	$-0.522428\pi$
$433$	−2.92996	−0.140805	−0.0704025	+	0.997519i	$0.522428\pi$
$434$	11.0718	0.531463
$435$	0	0
$436$	−30.9282	−1.48119
$437$	20.6312	0.986924
$438$	−5.37945	−0.257040
$439$	−7.07180	−0.337518	−0.168759	−	0.985657i	$-0.553976\pi$
$439$	−7.07180	−0.337518	−0.168759	+	0.985657i	$0.553976\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−1.31268	−0.0624377
$443$	18.1817	0.863839	0.431919	−	0.901912i	$-0.357836\pi$
$443$	18.1817	0.863839	0.431919	+	0.901912i	$0.357836\pi$
$444$	−22.3923	−1.06269
$445$	0	0
$446$	−7.94744	−0.376322
$447$	−1.31268	−0.0620875
$448$	5.55532	0.262464
$449$	−25.8564	−1.22024	−0.610120	−	0.792309i	$-0.708879\pi$
$449$	−25.8564	−1.22024	−0.610120	+	0.792309i	$0.708879\pi$
$450$	0	0
$451$	8.78461	0.413651
$452$	22.5259	1.05953
$453$	−26.5654	−1.24815
$454$	−10.5885	−0.496941
$455$	0	0
$456$	8.92820	0.418101
$457$	−23.6627	−1.10689	−0.553447	−	0.832884i	$-0.686688\pi$
$457$	−23.6627	−1.10689	−0.553447	+	0.832884i	$0.686688\pi$
$458$	9.72363	0.454355
$459$	−8.00000	−0.373408
$460$	0	0
$461$	24.9282	1.16102	0.580511	−	0.814252i	$-0.302853\pi$
$461$	24.9282	1.16102	0.580511	+	0.814252i	$0.302853\pi$
$462$	2.27362	0.105779
$463$	6.86800	0.319183	0.159591	−	0.987183i	$-0.448982\pi$
$463$	6.86800	0.319183	0.159591	+	0.987183i	$0.448982\pi$
$464$	2.46410	0.114393
$465$	0	0
$466$	−1.75129	−0.0811269
$467$	21.0101	0.972233	0.486116	−	0.873894i	$-0.338413\pi$
$467$	21.0101	0.972233	0.486116	+	0.873894i	$0.338413\pi$
$468$	−3.10583	−0.143567
$469$	−10.3923	−0.479872
$470$	0	0
$471$	15.4641	0.712548
$472$	20.0764	0.924091
$473$	−11.5911	−0.532960
$474$	−3.07180	−0.141092
$475$	0	0
$476$	6.00000	0.275010
$477$	−5.93426	−0.271711
$478$	5.37945	0.246050
$479$	5.66025	0.258624	0.129312	−	0.991604i	$-0.458723\pi$
$479$	5.66025	0.258624	0.129312	+	0.991604i	$0.458723\pi$
$480$	0	0
$481$	−16.3923	−0.747425
$482$	5.93426	0.270298
$483$	21.8695	0.995099
$484$	16.2679	0.739452
$485$	0	0
$486$	5.12436	0.232445
$487$	−10.9348	−0.495502	−0.247751	−	0.968824i	$-0.579691\pi$
$487$	−10.9348	−0.495502	−0.247751	+	0.968824i	$0.579691\pi$
$488$	−5.65685	−0.256074
$489$	19.8564	0.897938
$490$	0	0
$491$	7.26795	0.327998	0.163999	−	0.986461i	$-0.447561\pi$
$491$	7.26795	0.327998	0.163999	+	0.986461i	$0.447561\pi$
$492$	16.9706	0.765092
$493$	1.41421	0.0636930
$494$	3.03332	0.136476
$495$	0	0
$496$	−21.5167	−0.966127
$497$	−8.48528	−0.380617
$498$	7.45001	0.333843
$499$	−1.07180	−0.0479802	−0.0239901	−	0.999712i	$-0.507637\pi$
$499$	−1.07180	−0.0479802	−0.0239901	+	0.999712i	$0.507637\pi$
$500$	0	0
$501$	15.8564	0.708412
$502$	7.82894	0.349423
$503$	−12.1731	−0.542773	−0.271386	−	0.962471i	$-0.587482\pi$
$503$	−12.1731	−0.542773	−0.271386	+	0.962471i	$0.587482\pi$
$504$	−4.73205	−0.210782
$505$	0	0
$506$	4.14359	0.184205
$507$	−13.8375	−0.614546
$508$	−7.34847	−0.326036
$509$	33.4641	1.48327	0.741635	−	0.670804i	$-0.234051\pi$
$509$	33.4641	1.48327	0.741635	+	0.670804i	$0.234051\pi$
$510$	0	0
$511$	18.0000	0.796273
$512$	22.1841	0.980408
$513$	18.4863	0.816191
$514$	−4.92820	−0.217374
$515$	0	0
$516$	−22.3923	−0.985766
$517$	−1.79315	−0.0788627
$518$	−11.5911	−0.509284
$519$	6.53590	0.286894
$520$	0	0
$521$	4.39230	0.192430	0.0962152	−	0.995361i	$-0.469326\pi$
$521$	4.39230	0.192430	0.0962152	+	0.995361i	$0.469326\pi$
$522$	−0.517638	−0.0226564
$523$	1.61729	0.0707190	0.0353595	−	0.999375i	$-0.488742\pi$
$523$	1.61729	0.0707190	0.0353595	+	0.999375i	$0.488742\pi$
$524$	8.19615	0.358051
$525$	0	0
$526$	2.48334	0.108279
$527$	−12.3490	−0.537930
$528$	−4.41851	−0.192291
$529$	16.8564	0.732887
$530$	0	0
$531$	10.3923	0.450988
$532$	−13.8647	−0.601112
$533$	12.4233	0.538113
$534$	7.60770	0.329217
$535$	0	0
$536$	−8.19615	−0.354020
$537$	3.58630	0.154760
$538$	−6.21166	−0.267804
$539$	1.26795	0.0546144
$540$	0	0
$541$	42.3923	1.82259	0.911294	−	0.411757i	$-0.135085\pi$
$541$	42.3923	1.82259	0.911294	+	0.411757i	$0.135085\pi$
$542$	−13.8375	−0.594373
$543$	−22.6274	−0.971035
$544$	7.26795	0.311611
$545$	0	0
$546$	3.21539	0.137606
$547$	−2.44949	−0.104733	−0.0523663	−	0.998628i	$-0.516676\pi$
$547$	−2.44949	−0.104733	−0.0523663	+	0.998628i	$0.516676\pi$
$548$	−25.6317	−1.09493
$549$	−2.92820	−0.124973
$550$	0	0
$551$	−3.26795	−0.139219
$552$	17.2480	0.734122
$553$	10.2784	0.437083
$554$	9.71281	0.412658
$555$	0	0
$556$	13.8564	0.587643
$557$	−0.554803	−0.0235078	−0.0117539	−	0.999931i	$-0.503741\pi$
$557$	−0.554803	−0.0235078	−0.0117539	+	0.999931i	$0.503741\pi$
$558$	4.52004	0.191349
$559$	−16.3923	−0.693321
$560$	0	0
$561$	−2.53590	−0.107066
$562$	16.9706	0.715860
$563$	13.2827	0.559800	0.279900	−	0.960029i	$-0.409699\pi$
$563$	13.2827	0.559800	0.279900	+	0.960029i	$0.409699\pi$
$564$	−3.46410	−0.145865
$565$	0	0
$566$	−14.1962	−0.596709
$567$	12.2474	0.514344
$568$	−6.69213	−0.280796
$569$	21.4641	0.899822	0.449911	−	0.893073i	$-0.351456\pi$
$569$	21.4641	0.899822	0.449911	+	0.893073i	$0.351456\pi$
$570$	0	0
$571$	−39.3205	−1.64551	−0.822756	−	0.568395i	$-0.807565\pi$
$571$	−39.3205	−1.64551	−0.822756	+	0.568395i	$0.807565\pi$
$572$	−3.93803	−0.164657
$573$	21.3891	0.893541
$574$	8.78461	0.366663
$575$	0	0
$576$	2.26795	0.0944979
$577$	9.14162	0.380571	0.190285	−	0.981729i	$-0.439059\pi$
$577$	9.14162	0.380571	0.190285	+	0.981729i	$0.439059\pi$
$578$	−7.76457	−0.322964
$579$	−0.928203	−0.0385748
$580$	0	0
$581$	−24.9282	−1.03420
$582$	−8.00481	−0.331810
$583$	−7.52433	−0.311626
$584$	14.1962	0.587441
$585$	0	0
$586$	15.9090	0.657193
$587$	−15.5563	−0.642079	−0.321040	−	0.947066i	$-0.604032\pi$
$587$	−15.5563	−0.642079	−0.321040	+	0.947066i	$0.604032\pi$
$588$	2.44949	0.101015
$589$	28.5359	1.17580
$590$	0	0
$591$	−31.7128	−1.30449
$592$	22.5259	0.925808
$593$	25.5302	1.04840	0.524199	−	0.851596i	$-0.324365\pi$
$593$	25.5302	1.04840	0.524199	+	0.851596i	$0.324365\pi$
$594$	3.71281	0.152339
$595$	0	0
$596$	1.60770	0.0658538
$597$	−19.7990	−0.810319
$598$	5.85993	0.239630
$599$	1.26795	0.0518070	0.0259035	−	0.999664i	$-0.491754\pi$
$599$	1.26795	0.0518070	0.0259035	+	0.999664i	$0.491754\pi$
$600$	0	0
$601$	16.7846	0.684659	0.342329	−	0.939580i	$-0.388784\pi$
$601$	16.7846	0.684659	0.342329	+	0.939580i	$0.388784\pi$
$602$	−11.5911	−0.472418
$603$	−4.24264	−0.172774
$604$	32.5359	1.32387
$605$	0	0
$606$	−4.39230	−0.178425
$607$	16.3142	0.662174	0.331087	−	0.943600i	$-0.392585\pi$
$607$	16.3142	0.662174	0.331087	+	0.943600i	$0.392585\pi$
$608$	−16.7947	−0.681115
$609$	−3.46410	−0.140372
$610$	0	0
$611$	−2.53590	−0.102591
$612$	2.44949	0.0990148
$613$	18.2832	0.738453	0.369227	−	0.929339i	$-0.379623\pi$
$613$	18.2832	0.738453	0.369227	+	0.929339i	$0.379623\pi$
$614$	5.41154	0.218392
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−32.0464	−1.29014	−0.645071	−	0.764123i	$-0.723172\pi$
$617$	−32.0464	−1.29014	−0.645071	+	0.764123i	$0.723172\pi$
$618$	6.69213	0.269197
$619$	−23.8038	−0.956757	−0.478379	−	0.878154i	$-0.658775\pi$
$619$	−23.8038	−0.956757	−0.478379	+	0.878154i	$0.658775\pi$
$620$	0	0
$621$	35.7128	1.43311
$622$	−3.41044	−0.136746
$623$	−25.4558	−1.01987
$624$	−6.24871	−0.250149
$625$	0	0
$626$	−12.2487	−0.489557
$627$	5.85993	0.234023
$628$	−18.9396	−0.755771
$629$	12.9282	0.515481
$630$	0	0
$631$	15.6077	0.621333	0.310666	−	0.950519i	$-0.399448\pi$
$631$	15.6077	0.621333	0.310666	+	0.950519i	$0.399448\pi$
$632$	8.10634	0.322453
$633$	−20.8343	−0.828088
$634$	−2.98076	−0.118381
$635$	0	0
$636$	−14.5359	−0.576386
$637$	1.79315	0.0710472
$638$	−0.656339	−0.0259847
$639$	−3.46410	−0.137038
$640$	0	0
$641$	34.6410	1.36824	0.684119	−	0.729370i	$-0.260187\pi$
$641$	34.6410	1.36824	0.684119	+	0.729370i	$0.260187\pi$
$642$	−13.6617	−0.539183
$643$	−10.4543	−0.412277	−0.206139	−	0.978523i	$-0.566090\pi$
$643$	−10.4543	−0.412277	−0.206139	+	0.978523i	$0.566090\pi$
$644$	−26.7846	−1.05546
$645$	0	0
$646$	−2.39230	−0.0941240
$647$	14.3180	0.562899	0.281449	−	0.959576i	$-0.409185\pi$
$647$	14.3180	0.562899	0.281449	+	0.959576i	$0.409185\pi$
$648$	9.65926	0.379452
$649$	13.1769	0.517239
$650$	0	0
$651$	30.2487	1.18554
$652$	−24.3190	−0.952407
$653$	−13.0053	−0.508938	−0.254469	−	0.967081i	$-0.581901\pi$
$653$	−13.0053	−0.508938	−0.254469	+	0.967081i	$0.581901\pi$
$654$	13.0718	0.511148
$655$	0	0
$656$	−17.0718	−0.666542
$657$	7.34847	0.286691
$658$	−1.79315	−0.0699043
$659$	32.4449	1.26387	0.631936	−	0.775020i	$-0.282260\pi$
$659$	32.4449	1.26387	0.631936	+	0.775020i	$0.282260\pi$
$660$	0	0
$661$	−0.784610	−0.0305178	−0.0152589	−	0.999884i	$-0.504857\pi$
$661$	−0.784610	−0.0305178	−0.0152589	+	0.999884i	$0.504857\pi$
$662$	−9.41902	−0.366081
$663$	−3.58630	−0.139280
$664$	−19.6603	−0.762966
$665$	0	0
$666$	−4.73205	−0.183363
$667$	−6.31319	−0.244448
$668$	−19.4201	−0.751384
$669$	−21.7128	−0.839466
$670$	0	0
$671$	−3.71281	−0.143332
$672$	−17.8028	−0.686757
$673$	46.3644	1.78722	0.893609	−	0.448846i	$-0.148165\pi$
$673$	46.3644	1.78722	0.893609	+	0.448846i	$0.148165\pi$
$674$	−14.4449	−0.556395
$675$	0	0
$676$	16.9474	0.651825
$677$	32.2495	1.23945	0.619725	−	0.784819i	$-0.287244\pi$
$677$	32.2495	1.23945	0.619725	+	0.784819i	$0.287244\pi$
$678$	−9.52056	−0.365635
$679$	26.7846	1.02790
$680$	0	0
$681$	−28.9282	−1.10853
$682$	5.73118	0.219458
$683$	−21.9711	−0.840700	−0.420350	−	0.907362i	$-0.638093\pi$
$683$	−21.9711	−0.840700	−0.420350	+	0.907362i	$0.638093\pi$
$684$	−5.66025	−0.216425
$685$	0	0
$686$	10.1436	0.387284
$687$	26.5654	1.01354
$688$	22.5259	0.858791
$689$	−10.6410	−0.405390
$690$	0	0
$691$	41.7128	1.58683	0.793415	−	0.608681i	$-0.208301\pi$
$691$	41.7128	1.58683	0.793415	+	0.608681i	$0.208301\pi$
$692$	−8.00481	−0.304297
$693$	−3.10583	−0.117981
$694$	11.1244	0.422275
$695$	0	0
$696$	−2.73205	−0.103558
$697$	−9.79796	−0.371124
$698$	−11.7942	−0.446416
$699$	−4.78461	−0.180971
$700$	0	0
$701$	27.7128	1.04670	0.523349	−	0.852118i	$-0.324682\pi$
$701$	27.7128	1.04670	0.523349	+	0.852118i	$0.324682\pi$
$702$	5.25071	0.198175
$703$	−29.8744	−1.12673
$704$	2.87564	0.108380
$705$	0	0
$706$	−16.5359	−0.622337
$707$	14.6969	0.552735
$708$	25.4558	0.956689
$709$	20.3923	0.765849	0.382925	−	0.923780i	$-0.374917\pi$
$709$	20.3923	0.765849	0.382925	+	0.923780i	$0.374917\pi$
$710$	0	0
$711$	4.19615	0.157368
$712$	−20.0764	−0.752395
$713$	55.1271	2.06453
$714$	−2.53590	−0.0949036
$715$	0	0
$716$	−4.39230	−0.164148
$717$	14.6969	0.548867
$718$	−3.76217	−0.140403
$719$	−19.8564	−0.740519	−0.370260	−	0.928928i	$-0.620731\pi$
$719$	−19.8564	−0.740519	−0.370260	+	0.928928i	$0.620731\pi$
$720$	0	0
$721$	−22.3923	−0.833933
$722$	−4.30701	−0.160290
$723$	16.2127	0.602956
$724$	27.7128	1.02994
$725$	0	0
$726$	−6.87564	−0.255179
$727$	−46.6690	−1.73086	−0.865430	−	0.501031i	$-0.832954\pi$
$727$	−46.6690	−1.73086	−0.865430	+	0.501031i	$0.832954\pi$
$728$	−8.48528	−0.314485
$729$	29.0000	1.07407
$730$	0	0
$731$	12.9282	0.478167
$732$	−7.17260	−0.265107
$733$	12.7279	0.470117	0.235058	−	0.971981i	$-0.424472\pi$
$733$	12.7279	0.470117	0.235058	+	0.971981i	$0.424472\pi$
$734$	−17.4115	−0.642672
$735$	0	0
$736$	−32.4449	−1.19593
$737$	−5.37945	−0.198155
$738$	3.58630	0.132014
$739$	−18.9808	−0.698219	−0.349109	−	0.937082i	$-0.613516\pi$
$739$	−18.9808	−0.698219	−0.349109	+	0.937082i	$0.613516\pi$
$740$	0	0
$741$	8.28719	0.304437
$742$	−7.52433	−0.276227
$743$	−9.89949	−0.363177	−0.181589	−	0.983375i	$-0.558124\pi$
$743$	−9.89949	−0.363177	−0.181589	+	0.983375i	$0.558124\pi$
$744$	23.8564	0.874618
$745$	0	0
$746$	−10.1436	−0.371383
$747$	−10.1769	−0.372353
$748$	3.10583	0.113560
$749$	45.7128	1.67031
$750$	0	0
$751$	49.9090	1.82120	0.910602	−	0.413284i	$-0.135618\pi$
$751$	49.9090	1.82120	0.910602	+	0.413284i	$0.135618\pi$
$752$	3.48477	0.127076
$753$	21.3891	0.779461
$754$	−0.928203	−0.0338032
$755$	0	0
$756$	−24.0000	−0.872872
$757$	10.9348	0.397431	0.198716	−	0.980057i	$-0.436323\pi$
$757$	10.9348	0.397431	0.198716	+	0.980057i	$0.436323\pi$
$758$	−8.38375	−0.304511
$759$	11.3205	0.410908
$760$	0	0
$761$	−35.5692	−1.28938	−0.644692	−	0.764443i	$-0.723014\pi$
$761$	−35.5692	−1.28938	−0.644692	+	0.764443i	$0.723014\pi$
$762$	3.10583	0.112512
$763$	−43.7391	−1.58346
$764$	−26.1962	−0.947744
$765$	0	0
$766$	−0.483340	−0.0174638
$767$	18.6350	0.672870
$768$	−1.96902	−0.0710508
$769$	−19.0718	−0.687747	−0.343873	−	0.939016i	$-0.611739\pi$
$769$	−19.0718	−0.687747	−0.343873	+	0.939016i	$0.611739\pi$
$770$	0	0
$771$	−13.4641	−0.484898
$772$	1.13681	0.0409148
$773$	−11.2122	−0.403274	−0.201637	−	0.979460i	$-0.564626\pi$
$773$	−11.2122	−0.403274	−0.201637	+	0.979460i	$0.564626\pi$
$774$	−4.73205	−0.170090
$775$	0	0
$776$	21.1244	0.758320
$777$	−31.6675	−1.13607
$778$	−10.7589	−0.385725
$779$	22.6410	0.811199
$780$	0	0
$781$	−4.39230	−0.157169
$782$	−4.62158	−0.165267
$783$	−5.65685	−0.202159
$784$	−2.46410	−0.0880036
$785$	0	0
$786$	−3.46410	−0.123560
$787$	35.0779	1.25039	0.625197	−	0.780467i	$-0.285019\pi$
$787$	35.0779	1.25039	0.625197	+	0.780467i	$0.285019\pi$
$788$	38.8401	1.38362
$789$	6.78461	0.241539
$790$	0	0
$791$	31.8564	1.13268
$792$	−2.44949	−0.0870388
$793$	−5.25071	−0.186458
$794$	2.53590	0.0899957
$795$	0	0
$796$	24.2487	0.859473
$797$	29.0149	1.02776	0.513881	−	0.857862i	$-0.328207\pi$
$797$	29.0149	1.02776	0.513881	+	0.857862i	$0.328207\pi$
$798$	5.85993	0.207439
$799$	2.00000	0.0707549
$800$	0	0
$801$	−10.3923	−0.367194
$802$	19.5959	0.691956
$803$	9.31749	0.328807
$804$	−10.3923	−0.366508
$805$	0	0
$806$	8.10512	0.285491
$807$	−16.9706	−0.597392
$808$	11.5911	0.407774
$809$	13.6077	0.478421	0.239211	−	0.970968i	$-0.423111\pi$
$809$	13.6077	0.478421	0.239211	+	0.970968i	$0.423111\pi$
$810$	0	0
$811$	−36.7846	−1.29168	−0.645841	−	0.763472i	$-0.723493\pi$
$811$	−36.7846	−1.29168	−0.645841	+	0.763472i	$0.723493\pi$
$812$	4.24264	0.148888
$813$	−37.8048	−1.32587
$814$	−6.00000	−0.210300
$815$	0	0
$816$	4.92820	0.172522
$817$	−29.8744	−1.04517
$818$	4.69591	0.164189
$819$	−4.39230	−0.153480
$820$	0	0
$821$	9.21539	0.321619	0.160810	−	0.986985i	$-0.448589\pi$
$821$	9.21539	0.321619	0.160810	+	0.986985i	$0.448589\pi$
$822$	10.8332	0.377852
$823$	−40.8091	−1.42252	−0.711258	−	0.702931i	$-0.751874\pi$
$823$	−40.8091	−1.42252	−0.711258	+	0.702931i	$0.751874\pi$
$824$	−17.6603	−0.615224
$825$	0	0
$826$	13.1769	0.458483
$827$	3.68784	0.128239	0.0641193	−	0.997942i	$-0.479576\pi$
$827$	3.68784	0.128239	0.0641193	+	0.997942i	$0.479576\pi$
$828$	−10.9348	−0.380010
$829$	−16.7846	−0.582954	−0.291477	−	0.956578i	$-0.594147\pi$
$829$	−16.7846	−0.582954	−0.291477	+	0.956578i	$0.594147\pi$
$830$	0	0
$831$	26.5359	0.920520
$832$	4.06678	0.140990
$833$	−1.41421	−0.0489996
$834$	−5.85641	−0.202791
$835$	0	0
$836$	−7.17691	−0.248219
$837$	49.3959	1.70737
$838$	−3.10583	−0.107289
$839$	40.9808	1.41481	0.707407	−	0.706807i	$-0.249865\pi$
$839$	40.9808	1.41481	0.707407	+	0.706807i	$0.249865\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	2.82843	0.0974740
$843$	46.3644	1.59688
$844$	25.5167	0.878320
$845$	0	0
$846$	−0.732051	−0.0251684
$847$	23.0064	0.790508
$848$	14.6226	0.502142
$849$	−38.7846	−1.33108
$850$	0	0
$851$	−57.7128	−1.97837
$852$	−8.48528	−0.290701
$853$	32.3238	1.10675	0.553374	−	0.832933i	$-0.313340\pi$
$853$	32.3238	1.10675	0.553374	+	0.832933i	$0.313340\pi$
$854$	−3.71281	−0.127050
$855$	0	0
$856$	36.0526	1.23225
$857$	33.2576	1.13606	0.568029	−	0.823009i	$-0.307706\pi$
$857$	33.2576	1.13606	0.568029	+	0.823009i	$0.307706\pi$
$858$	1.66441	0.0568220
$859$	−18.7321	−0.639129	−0.319565	−	0.947564i	$-0.603537\pi$
$859$	−18.7321	−0.639129	−0.319565	+	0.947564i	$0.603537\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	−19.2442	−0.655460
$863$	1.69161	0.0575832	0.0287916	−	0.999585i	$-0.490834\pi$
$863$	1.69161	0.0575832	0.0287916	+	0.999585i	$0.490834\pi$
$864$	−29.0718	−0.989043
$865$	0	0
$866$	−1.51666	−0.0515382
$867$	−21.2132	−0.720438
$868$	−37.0470	−1.25746
$869$	5.32051	0.180486
$870$	0	0
$871$	−7.60770	−0.257777
$872$	−34.4959	−1.16818
$873$	10.9348	0.370086
$874$	10.6795	0.361239
$875$	0	0
$876$	18.0000	0.608164
$877$	24.9754	0.843358	0.421679	−	0.906745i	$-0.361441\pi$
$877$	24.9754	0.843358	0.421679	+	0.906745i	$0.361441\pi$
$878$	−3.66063	−0.123540
$879$	43.4641	1.46601
$880$	0	0
$881$	−50.5359	−1.70260	−0.851299	−	0.524681i	$-0.824185\pi$
$881$	−50.5359	−1.70260	−0.851299	+	0.524681i	$0.824185\pi$
$882$	0.517638	0.0174298
$883$	1.48854	0.0500935	0.0250467	−	0.999686i	$-0.492027\pi$
$883$	1.48854	0.0500935	0.0250467	+	0.999686i	$0.492027\pi$
$884$	4.39230	0.147729
$885$	0	0
$886$	9.41154	0.316187
$887$	21.0101	0.705451	0.352726	−	0.935727i	$-0.385255\pi$
$887$	21.0101	0.705451	0.352726	+	0.935727i	$0.385255\pi$
$888$	−24.9754	−0.838119
$889$	−10.3923	−0.348547
$890$	0	0
$891$	6.33975	0.212389
$892$	26.5927	0.890388
$893$	−4.62158	−0.154655
$894$	−0.679492	−0.0227256
$895$	0	0
$896$	28.0526	0.937170
$897$	16.0096	0.534546
$898$	−13.3843	−0.446639
$899$	−8.73205	−0.291230
$900$	0	0
$901$	8.39230	0.279588
$902$	4.54725	0.151407
$903$	−31.6675	−1.05383
$904$	25.1244	0.835624
$905$	0	0
$906$	−13.7513	−0.456856
$907$	−45.7081	−1.51771	−0.758856	−	0.651258i	$-0.774242\pi$
$907$	−45.7081	−1.51771	−0.758856	+	0.651258i	$0.774242\pi$
$908$	35.4297	1.17577
$909$	6.00000	0.199007
$910$	0	0
$911$	−9.80385	−0.324816	−0.162408	−	0.986724i	$-0.551926\pi$
$911$	−9.80385	−0.324816	−0.162408	+	0.986724i	$0.551926\pi$
$912$	−11.3880	−0.377096
$913$	−12.9038	−0.427053
$914$	−12.2487	−0.405151
$915$	0	0
$916$	−32.5359	−1.07502
$917$	11.5911	0.382772
$918$	−4.14110	−0.136677
$919$	43.9615	1.45016	0.725078	−	0.688666i	$-0.241803\pi$
$919$	43.9615	1.45016	0.725078	+	0.688666i	$0.241803\pi$
$920$	0	0
$921$	14.7846	0.487169
$922$	12.9038	0.424964
$923$	−6.21166	−0.204459
$924$	−7.60770	−0.250275
$925$	0	0
$926$	3.55514	0.116829
$927$	−9.14162	−0.300250
$928$	5.13922	0.168703
$929$	55.8564	1.83259	0.916295	−	0.400505i	$-0.131165\pi$
$929$	55.8564	1.83259	0.916295	+	0.400505i	$0.131165\pi$
$930$	0	0
$931$	3.26795	0.107103
$932$	5.85993	0.191948
$933$	−9.31749	−0.305041
$934$	10.8756	0.355862
$935$	0	0
$936$	−3.46410	−0.113228
$937$	2.75410	0.0899724	0.0449862	−	0.998988i	$-0.485676\pi$
$937$	2.75410	0.0899724	0.0449862	+	0.998988i	$0.485676\pi$
$938$	−5.37945	−0.175645
$939$	−33.4641	−1.09206
$940$	0	0
$941$	−7.85641	−0.256112	−0.128056	−	0.991767i	$-0.540874\pi$
$941$	−7.85641	−0.256112	−0.128056	+	0.991767i	$0.540874\pi$
$942$	8.00481	0.260811
$943$	43.7391	1.42434
$944$	−25.6077	−0.833459
$945$	0	0
$946$	−6.00000	−0.195077
$947$	−27.9797	−0.909217	−0.454608	−	0.890691i	$-0.650221\pi$
$947$	−27.9797	−0.909217	−0.454608	+	0.890691i	$0.650221\pi$
$948$	10.2784	0.333828
$949$	13.1769	0.427741
$950$	0	0
$951$	−8.14359	−0.264074
$952$	6.69213	0.216893
$953$	−5.65685	−0.183243	−0.0916217	−	0.995794i	$-0.529205\pi$
$953$	−5.65685	−0.183243	−0.0916217	+	0.995794i	$0.529205\pi$
$954$	−3.07180	−0.0994531
$955$	0	0
$956$	−18.0000	−0.582162
$957$	−1.79315	−0.0579643
$958$	2.92996	0.0946628
$959$	−36.2487	−1.17053
$960$	0	0
$961$	45.2487	1.45964
$962$	−8.48528	−0.273576
$963$	18.6622	0.601380
$964$	−19.8564	−0.639532
$965$	0	0
$966$	11.3205	0.364231
$967$	−1.96902	−0.0633193	−0.0316596	−	0.999499i	$-0.510079\pi$
$967$	−1.96902	−0.0633193	−0.0316596	+	0.999499i	$0.510079\pi$
$968$	18.1445	0.583188
$969$	−6.53590	−0.209963
$970$	0	0
$971$	−49.5167	−1.58907	−0.794533	−	0.607221i	$-0.792284\pi$
$971$	−49.5167	−1.58907	−0.794533	+	0.607221i	$0.792284\pi$
$972$	−17.1464	−0.549972
$973$	19.5959	0.628216
$974$	−5.66025	−0.181366
$975$	0	0
$976$	7.21539	0.230959
$977$	−22.9048	−0.732790	−0.366395	−	0.930459i	$-0.619408\pi$
$977$	−22.9048	−0.732790	−0.366395	+	0.930459i	$0.619408\pi$
$978$	10.2784	0.328668
$979$	−13.1769	−0.421136
$980$	0	0
$981$	−17.8564	−0.570111
$982$	3.76217	0.120056
$983$	30.2533	0.964930	0.482465	−	0.875915i	$-0.339742\pi$
$983$	30.2533	0.964930	0.482465	+	0.875915i	$0.339742\pi$
$984$	18.9282	0.603409
$985$	0	0
$986$	0.732051	0.0233132
$987$	−4.89898	−0.155936
$988$	−10.1497	−0.322905
$989$	−57.7128	−1.83516
$990$	0	0
$991$	34.7846	1.10497	0.552485	−	0.833523i	$-0.313680\pi$
$991$	34.7846	1.10497	0.552485	+	0.833523i	$0.313680\pi$
$992$	−44.8759	−1.42481
$993$	−25.7332	−0.816620
$994$	−4.39230	−0.139315
$995$	0	0
$996$	−24.9282	−0.789880
$997$	26.5927	0.842198	0.421099	−	0.907015i	$-0.361645\pi$
$997$	26.5927	0.842198	0.421099	+	0.907015i	$0.361645\pi$
$998$	−0.554803	−0.0175620
$999$	−51.7128	−1.63612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.a.f.1.3		4
3.2	odd	2		6525.2.a.bj.1.2		4
5.2	odd	4		145.2.b.b.59.3	yes	4
5.3	odd	4		145.2.b.b.59.2	✓	4
5.4	even	2	inner	725.2.a.f.1.2		4
15.2	even	4		1305.2.c.f.784.2		4
15.8	even	4		1305.2.c.f.784.3		4
15.14	odd	2		6525.2.a.bj.1.3		4
20.3	even	4		2320.2.d.f.929.1		4
20.7	even	4		2320.2.d.f.929.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.b.59.2	✓	4	5.3	odd	4
145.2.b.b.59.3	yes	4	5.2	odd	4
725.2.a.f.1.2		4	5.4	even	2	inner
725.2.a.f.1.3		4	1.1	even	1	trivial
1305.2.c.f.784.2		4	15.2	even	4
1305.2.c.f.784.3		4	15.8	even	4
2320.2.d.f.929.1		4	20.3	even	4
2320.2.d.f.929.3		4	20.7	even	4
6525.2.a.bj.1.2		4	3.2	odd	2
6525.2.a.bj.1.3		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)

\(f(q)\)	\(=\)	\(q+0.517638 q^{2} +1.41421 q^{3} -1.73205 q^{4} +0.732051 q^{6} -2.44949 q^{7} -1.93185 q^{8} -1.00000 q^{9} -1.26795 q^{11} -2.44949 q^{12} -1.79315 q^{13} -1.26795 q^{14} +2.46410 q^{16} +1.41421 q^{17} -0.517638 q^{18} -3.26795 q^{19} -3.46410 q^{21} -0.656339 q^{22} -6.31319 q^{23} -2.73205 q^{24} -0.928203 q^{26} -5.65685 q^{27} +4.24264 q^{28} +1.00000 q^{29} -8.73205 q^{31} +5.13922 q^{32} -1.79315 q^{33} +0.732051 q^{34} +1.73205 q^{36} +9.14162 q^{37} -1.69161 q^{38} -2.53590 q^{39} -6.92820 q^{41} -1.79315 q^{42} +9.14162 q^{43} +2.19615 q^{44} -3.26795 q^{46} +1.41421 q^{47} +3.48477 q^{48} -1.00000 q^{49} +2.00000 q^{51} +3.10583 q^{52} +5.93426 q^{53} -2.92820 q^{54} +4.73205 q^{56} -4.62158 q^{57} +0.517638 q^{58} -10.3923 q^{59} +2.92820 q^{61} -4.52004 q^{62} +2.44949 q^{63} -2.26795 q^{64} -0.928203 q^{66} +4.24264 q^{67} -2.44949 q^{68} -8.92820 q^{69} +3.46410 q^{71} +1.93185 q^{72} -7.34847 q^{73} +4.73205 q^{74} +5.66025 q^{76} +3.10583 q^{77} -1.31268 q^{78} -4.19615 q^{79} -5.00000 q^{81} -3.58630 q^{82} +10.1769 q^{83} +6.00000 q^{84} +4.73205 q^{86} +1.41421 q^{87} +2.44949 q^{88} +10.3923 q^{89} +4.39230 q^{91} +10.9348 q^{92} -12.3490 q^{93} +0.732051 q^{94} +7.26795 q^{96} -10.9348 q^{97} -0.517638 q^{98} +1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{6} - 4 q^{9} - 12 q^{11} - 12 q^{14} - 4 q^{16} - 20 q^{19} - 4 q^{24} + 24 q^{26} + 4 q^{29} - 28 q^{31} - 4 q^{34} - 24 q^{39} - 12 q^{44} - 20 q^{46} - 4 q^{49} + 8 q^{51} + 16 q^{54} + 12 q^{56}+ \cdots + 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^6 - 4 * q^9 - 12 * q^11 - 12 * q^14 - 4 * q^16 - 20 * q^19 - 4 * q^24 + 24 * q^26 + 4 * q^29 - 28 * q^31 - 4 * q^34 - 24 * q^39 - 12 * q^44 - 20 * q^46 - 4 * q^49 + 8 * q^51 + 16 * q^54 + 12 * q^56 - 16 * q^61 - 16 * q^64 + 24 * q^66 - 8 * q^69 + 12 * q^74 - 12 * q^76 + 4 * q^79 - 20 * q^81 + 24 * q^84 + 12 * q^86 - 24 * q^91 - 4 * q^94 + 36 * q^96 + 12 * q^99

Introduction

L-functions

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