LMFDB - Embedded newform 8281.2.a.co.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1 $ x^12 - 16x^10 + 88x^8 - 197x^6 + 172x^4 - 36*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.37905$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.37905 q^{2} -2.88120 q^{3} -0.0982074 q^{4} -0.805948 q^{5} -3.97334 q^{6} -2.89354 q^{8} +5.30133 q^{9} +O(q^{10})$	\(q+1.37905 q^{2} -2.88120 q^{3} -0.0982074 q^{4} -0.805948 q^{5} -3.97334 q^{6} -2.89354 q^{8} +5.30133 q^{9} -1.11145 q^{10} +5.27158 q^{11} +0.282955 q^{12} +2.32210 q^{15} -3.79394 q^{16} +0.560102 q^{17} +7.31083 q^{18} -5.84469 q^{19} +0.0791501 q^{20} +7.26980 q^{22} -1.60488 q^{23} +8.33689 q^{24} -4.35045 q^{25} -6.63060 q^{27} +2.28015 q^{29} +3.20230 q^{30} -3.47590 q^{31} +0.555034 q^{32} -15.1885 q^{33} +0.772411 q^{34} -0.520630 q^{36} +1.24196 q^{37} -8.06014 q^{38} +2.33205 q^{40} -0.927702 q^{41} +4.44711 q^{43} -0.517708 q^{44} -4.27260 q^{45} -2.21321 q^{46} +3.84418 q^{47} +10.9311 q^{48} -5.99951 q^{50} -1.61377 q^{51} +5.45454 q^{53} -9.14396 q^{54} -4.24862 q^{55} +16.8397 q^{57} +3.14446 q^{58} +10.9940 q^{59} -0.228047 q^{60} -7.30215 q^{61} -4.79346 q^{62} +8.35330 q^{64} -20.9458 q^{66} -7.34556 q^{67} -0.0550061 q^{68} +4.62397 q^{69} +9.31460 q^{71} -15.3396 q^{72} +5.00146 q^{73} +1.71273 q^{74} +12.5345 q^{75} +0.573991 q^{76} +11.3687 q^{79} +3.05772 q^{80} +3.20012 q^{81} -1.27935 q^{82} +5.81962 q^{83} -0.451413 q^{85} +6.13281 q^{86} -6.56959 q^{87} -15.2535 q^{88} -5.00946 q^{89} -5.89215 q^{90} +0.157611 q^{92} +10.0148 q^{93} +5.30133 q^{94} +4.71051 q^{95} -1.59917 q^{96} +10.6483 q^{97} +27.9464 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 + 8 * q^4 + 2 * q^9	$ 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9} - 24 q^{10} + 2 q^{12} + 16 q^{16} - 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} - 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} - 38 q^{38} - 2 q^{40} + 22 q^{43} + 38 q^{48} + 8 q^{51} + 16 q^{53} - 30 q^{55} + 10 q^{61} - 82 q^{62} - 2 q^{64} - 68 q^{66} - 22 q^{68} - 14 q^{69} + 66 q^{74} - 2 q^{75} + 70 q^{79} - 28 q^{81} - 10 q^{82} + 20 q^{87} - 28 q^{88} - 66 q^{92} + 2 q^{94} + 4 q^{95}+O(q^{100}) $ 12 * q - 6 * q^3 + 8 * q^4 + 2 * q^9 - 24 * q^10 + 2 * q^12 + 16 * q^16 - 34 * q^17 + 30 * q^22 + 6 * q^23 - 10 * q^25 - 12 * q^27 + 2 * q^29 + 22 * q^30 - 26 * q^36 - 38 * q^38 - 2 * q^40 + 22 * q^43 + 38 * q^48 + 8 * q^51 + 16 * q^53 - 30 * q^55 + 10 * q^61 - 82 * q^62 - 2 * q^64 - 68 * q^66 - 22 * q^68 - 14 * q^69 + 66 * q^74 - 2 * q^75 + 70 * q^79 - 28 * q^81 - 10 * q^82 + 20 * q^87 - 28 * q^88 - 66 * q^92 + 2 * q^94 + 4 * q^95

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 $

$2$	1.37905	0.975139	0.487570	−	0.873084i	$-0.337884\pi$
$2$	1.37905	0.975139	0.487570	+	0.873084i	$0.337884\pi$
$3$	−2.88120	−1.66346	−0.831732	−	0.555178i	$-0.812650\pi$
$3$	−2.88120	−1.66346	−0.831732	+	0.555178i	$0.812650\pi$
$4$	−0.0982074	−0.0491037
$5$	−0.805948	−0.360431	−0.180216	−	0.983627i	$-0.557680\pi$
$5$	−0.805948	−0.360431	−0.180216	+	0.983627i	$0.557680\pi$
$6$	−3.97334	−1.62211
$7$	0	0
$7$	0	0
$8$	−2.89354	−1.02302
$9$	5.30133	1.76711
$10$	−1.11145	−0.351470
$11$	5.27158	1.58944	0.794720	−	0.606976i	$-0.207618\pi$
$11$	5.27158	1.58944	0.794720	+	0.606976i	$0.207618\pi$
$12$	0.282955	0.0816822
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.32210	0.599564
$16$	−3.79394	−0.948485
$17$	0.560102	0.135845	0.0679223	−	0.997691i	$-0.478363\pi$
$17$	0.560102	0.135845	0.0679223	+	0.997691i	$0.478363\pi$
$18$	7.31083	1.72318
$19$	−5.84469	−1.34086	−0.670431	−	0.741972i	$-0.733891\pi$
$19$	−5.84469	−1.34086	−0.670431	+	0.741972i	$0.733891\pi$
$20$	0.0791501	0.0176985
$21$	0	0
$22$	7.26980	1.54993
$23$	−1.60488	−0.334640	−0.167320	−	0.985903i	$-0.553511\pi$
$23$	−1.60488	−0.334640	−0.167320	+	0.985903i	$0.553511\pi$
$24$	8.33689	1.70176
$25$	−4.35045	−0.870089
$26$	0	0
$27$	−6.63060	−1.27606
$28$	0	0
$29$	2.28015	0.423414	0.211707	−	0.977333i	$-0.432098\pi$
$29$	2.28015	0.423414	0.211707	+	0.977333i	$0.432098\pi$
$30$	3.20230	0.584658
$31$	−3.47590	−0.624290	−0.312145	−	0.950034i	$-0.601048\pi$
$31$	−3.47590	−0.624290	−0.312145	+	0.950034i	$0.601048\pi$
$32$	0.555034	0.0981171
$33$	−15.1885	−2.64398
$34$	0.772411	0.132467
$35$	0	0
$36$	−0.520630	−0.0867716
$37$	1.24196	0.204177	0.102088	−	0.994775i	$-0.467448\pi$
$37$	1.24196	0.204177	0.102088	+	0.994775i	$0.467448\pi$
$38$	−8.06014	−1.30753
$39$	0	0
$40$	2.33205	0.368729
$41$	−0.927702	−0.144883	−0.0724413	−	0.997373i	$-0.523079\pi$
$41$	−0.927702	−0.144883	−0.0724413	+	0.997373i	$0.523079\pi$
$42$	0	0
$43$	4.44711	0.678179	0.339089	−	0.940754i	$-0.389881\pi$
$43$	4.44711	0.678179	0.339089	+	0.940754i	$0.389881\pi$
$44$	−0.517708	−0.0780474
$45$	−4.27260	−0.636921
$46$	−2.21321	−0.326320
$47$	3.84418	0.560731	0.280365	−	0.959893i	$-0.409544\pi$
$47$	3.84418	0.560731	0.280365	+	0.959893i	$0.409544\pi$
$48$	10.9311	1.57777
$49$	0	0
$50$	−5.99951	−0.848458
$51$	−1.61377	−0.225973
$52$	0	0
$53$	5.45454	0.749239	0.374620	−	0.927179i	$-0.377773\pi$
$53$	5.45454	0.749239	0.374620	+	0.927179i	$0.377773\pi$
$54$	−9.14396	−1.24434
$55$	−4.24862	−0.572884
$56$	0	0
$57$	16.8397	2.23048
$58$	3.14446	0.412887
$59$	10.9940	1.43129	0.715646	−	0.698463i	$-0.246132\pi$
$59$	10.9940	1.43129	0.715646	+	0.698463i	$0.246132\pi$
$60$	−0.228047	−0.0294408
$61$	−7.30215	−0.934944	−0.467472	−	0.884008i	$-0.654835\pi$
$61$	−7.30215	−0.934944	−0.467472	+	0.884008i	$0.654835\pi$
$62$	−4.79346	−0.608770
$63$	0	0
$64$	8.35330	1.04416
$65$	0	0
$66$	−20.9458	−2.57824
$67$	−7.34556	−0.897403	−0.448701	−	0.893682i	$-0.648113\pi$
$67$	−7.34556	−0.897403	−0.448701	+	0.893682i	$0.648113\pi$
$68$	−0.0550061	−0.00667047
$69$	4.62397	0.556661
$70$	0	0
$71$	9.31460	1.10544	0.552720	−	0.833367i	$-0.313590\pi$
$71$	9.31460	1.10544	0.552720	+	0.833367i	$0.313590\pi$
$72$	−15.3396	−1.80779
$73$	5.00146	0.585376	0.292688	−	0.956208i	$-0.405450\pi$
$73$	5.00146	0.585376	0.292688	+	0.956208i	$0.405450\pi$
$74$	1.71273	0.199101
$75$	12.5345	1.44736
$76$	0.573991	0.0658413
$77$	0	0
$78$	0	0
$79$	11.3687	1.27908	0.639542	−	0.768756i	$-0.279124\pi$
$79$	11.3687	1.27908	0.639542	+	0.768756i	$0.279124\pi$
$80$	3.05772	0.341863
$81$	3.20012	0.355568
$82$	−1.27935	−0.141281
$83$	5.81962	0.638786	0.319393	−	0.947622i	$-0.396521\pi$
$83$	5.81962	0.638786	0.319393	+	0.947622i	$0.396521\pi$
$84$	0	0
$85$	−0.451413	−0.0489626
$86$	6.13281	0.661318
$87$	−6.56959	−0.704334
$88$	−15.2535	−1.62603
$89$	−5.00946	−0.531001	−0.265501	−	0.964111i	$-0.585537\pi$
$89$	−5.00946	−0.531001	−0.265501	+	0.964111i	$0.585537\pi$
$90$	−5.89215	−0.621087
$91$	0	0
$92$	0.157611	0.0164320
$93$	10.0148	1.03848
$94$	5.30133	0.546791
$95$	4.71051	0.483289
$96$	−1.59917	−0.163214
$97$	10.6483	1.08117	0.540586	−	0.841289i	$-0.318202\pi$
$97$	10.6483	1.08117	0.540586	+	0.841289i	$0.318202\pi$
$98$	0	0
$99$	27.9464	2.80872
$100$	0.427246	0.0427246
$101$	−3.91554	−0.389611	−0.194805	−	0.980842i	$-0.562408\pi$
$101$	−3.91554	−0.389611	−0.194805	+	0.980842i	$0.562408\pi$
$102$	−2.22547	−0.220355
$103$	−8.45379	−0.832977	−0.416488	−	0.909141i	$-0.636739\pi$
$103$	−8.45379	−0.832977	−0.416488	+	0.909141i	$0.636739\pi$
$104$	0	0
$105$	0	0
$106$	7.52212	0.730613
$107$	−9.67522	−0.935339	−0.467670	−	0.883903i	$-0.654906\pi$
$107$	−9.67522	−0.935339	−0.467670	+	0.883903i	$0.654906\pi$
$108$	0.651174	0.0626592
$109$	−14.5638	−1.39496	−0.697478	−	0.716606i	$-0.745695\pi$
$109$	−14.5638	−1.39496	−0.697478	+	0.716606i	$0.745695\pi$
$110$	−5.85908	−0.558641
$111$	−3.57833	−0.339640
$112$	0	0
$113$	19.5114	1.83548	0.917741	−	0.397180i	$-0.130011\pi$
$113$	19.5114	1.83548	0.917741	+	0.397180i	$0.130011\pi$
$114$	23.2229	2.17502
$115$	1.29345	0.120615
$116$	−0.223928	−0.0207912
$117$	0	0
$118$	15.1613	1.39571
$119$	0	0
$120$	−6.71910	−0.613367
$121$	16.7895	1.52632
$122$	−10.0701	−0.911701
$123$	2.67290	0.241007
$124$	0.341359	0.0306550
$125$	7.53598	0.674038
$126$	0	0
$127$	−1.91731	−0.170134	−0.0850670	−	0.996375i	$-0.527110\pi$
$127$	−1.91731	−0.170134	−0.0850670	+	0.996375i	$0.527110\pi$
$128$	10.4096	0.920087
$129$	−12.8130	−1.12813
$130$	0	0
$131$	−15.5816	−1.36137	−0.680684	−	0.732577i	$-0.738317\pi$
$131$	−15.5816	−1.36137	−0.680684	+	0.732577i	$0.738317\pi$
$132$	1.49162	0.129829
$133$	0	0
$134$	−10.1299	−0.875093
$135$	5.34392	0.459932
$136$	−1.62068	−0.138972
$137$	−7.85105	−0.670761	−0.335380	−	0.942083i	$-0.608865\pi$
$137$	−7.85105	−0.670761	−0.335380	+	0.942083i	$0.608865\pi$
$138$	6.37671	0.542822
$139$	−9.92481	−0.841812	−0.420906	−	0.907104i	$-0.638288\pi$
$139$	−9.92481	−0.841812	−0.420906	+	0.907104i	$0.638288\pi$
$140$	0	0
$141$	−11.0759	−0.932755
$142$	12.8453	1.07796
$143$	0	0
$144$	−20.1129	−1.67608
$145$	−1.83769	−0.152612
$146$	6.89728	0.570823
$147$	0	0
$148$	−0.121969	−0.0100258
$149$	7.91925	0.648770	0.324385	−	0.945925i	$-0.394843\pi$
$149$	7.91925	0.648770	0.324385	+	0.945925i	$0.394843\pi$
$150$	17.2858	1.41138
$151$	−1.50116	−0.122163	−0.0610815	−	0.998133i	$-0.519455\pi$
$151$	−1.50116	−0.122163	−0.0610815	+	0.998133i	$0.519455\pi$
$152$	16.9118	1.37173
$153$	2.96928	0.240052
$154$	0	0
$155$	2.80140	0.225014
$156$	0	0
$157$	−3.85692	−0.307816	−0.153908	−	0.988085i	$-0.549186\pi$
$157$	−3.85692	−0.307816	−0.153908	+	0.988085i	$0.549186\pi$
$158$	15.6781	1.24728
$159$	−15.7156	−1.24633
$160$	−0.447329	−0.0353644
$161$	0	0
$162$	4.41314	0.346729
$163$	−14.3608	−1.12483	−0.562414	−	0.826856i	$-0.690127\pi$
$163$	−14.3608	−1.12483	−0.562414	+	0.826856i	$0.690127\pi$
$164$	0.0911072	0.00711427
$165$	12.2411	0.952971
$166$	8.02557	0.622905
$167$	−4.52138	−0.349875	−0.174937	−	0.984580i	$-0.555972\pi$
$167$	−4.52138	−0.349875	−0.174937	+	0.984580i	$0.555972\pi$
$168$	0	0
$169$	0	0
$170$	−0.622523	−0.0477454
$171$	−30.9846	−2.36945
$172$	−0.436739	−0.0333011
$173$	19.5179	1.48392	0.741960	−	0.670444i	$-0.233896\pi$
$173$	19.5179	1.48392	0.741960	+	0.670444i	$0.233896\pi$
$174$	−9.05982	−0.686823
$175$	0	0
$176$	−20.0001	−1.50756
$177$	−31.6759	−2.38090
$178$	−6.90832	−0.517800
$179$	20.8196	1.55613	0.778065	−	0.628183i	$-0.216201\pi$
$179$	20.8196	1.55613	0.778065	+	0.628183i	$0.216201\pi$
$180$	0.419601	0.0312752
$181$	16.5522	1.23031	0.615157	−	0.788405i	$-0.289093\pi$
$181$	16.5522	1.23031	0.615157	+	0.788405i	$0.289093\pi$
$182$	0	0
$183$	21.0390	1.55525
$184$	4.64378	0.342344
$185$	−1.00095	−0.0735916
$186$	13.8109	1.01267
$187$	2.95262	0.215917
$188$	−0.377527	−0.0275340
$189$	0	0
$190$	6.49606	0.471274
$191$	−4.25008	−0.307525	−0.153762	−	0.988108i	$-0.549139\pi$
$191$	−4.25008	−0.307525	−0.153762	+	0.988108i	$0.549139\pi$
$192$	−24.0676	−1.73693
$193$	−11.5972	−0.834787	−0.417393	−	0.908726i	$-0.637056\pi$
$193$	−11.5972	−0.834787	−0.417393	+	0.908726i	$0.637056\pi$
$194$	14.6846	1.05429
$195$	0	0
$196$	0	0
$197$	−14.4213	−1.02748	−0.513738	−	0.857947i	$-0.671740\pi$
$197$	−14.4213	−1.02748	−0.513738	+	0.857947i	$0.671740\pi$
$198$	38.5396	2.73889
$199$	−7.05924	−0.500416	−0.250208	−	0.968192i	$-0.580499\pi$
$199$	−7.05924	−0.500416	−0.250208	+	0.968192i	$0.580499\pi$
$200$	12.5882	0.890121
$201$	21.1640	1.49280
$202$	−5.39974	−0.379925
$203$	0	0
$204$	0.158484	0.0110961
$205$	0.747680	0.0522202
$206$	−11.6582	−0.812268
$207$	−8.50798	−0.591345
$208$	0	0
$209$	−30.8107	−2.13122
$210$	0	0
$211$	−26.4226	−1.81901	−0.909505	−	0.415693i	$-0.863539\pi$
$211$	−26.4226	−1.81901	−0.909505	+	0.415693i	$0.863539\pi$
$212$	−0.535677	−0.0367904
$213$	−26.8372	−1.83886
$214$	−13.3427	−0.912086
$215$	−3.58414	−0.244437
$216$	19.1859	1.30544
$217$	0	0
$218$	−20.0842	−1.36028
$219$	−14.4102	−0.973752
$220$	0.417246	0.0281307
$221$	0	0
$222$	−4.93472	−0.331197
$223$	−23.0005	−1.54023	−0.770115	−	0.637905i	$-0.779801\pi$
$223$	−23.0005	−1.54023	−0.770115	+	0.637905i	$0.779801\pi$
$224$	0	0
$225$	−23.0632	−1.53754
$226$	26.9073	1.78985
$227$	0.453367	0.0300911	0.0150455	−	0.999887i	$-0.495211\pi$
$227$	0.453367	0.0300911	0.0150455	+	0.999887i	$0.495211\pi$
$228$	−1.65379	−0.109525
$229$	−17.3335	−1.14543	−0.572714	−	0.819755i	$-0.694110\pi$
$229$	−17.3335	−1.14543	−0.572714	+	0.819755i	$0.694110\pi$
$230$	1.78373	0.117616
$231$	0	0
$232$	−6.59772	−0.433162
$233$	7.81511	0.511985	0.255992	−	0.966679i	$-0.417598\pi$
$233$	7.81511	0.511985	0.255992	+	0.966679i	$0.417598\pi$
$234$	0	0
$235$	−3.09821	−0.202105
$236$	−1.07969	−0.0702818
$237$	−32.7557	−2.12771
$238$	0	0
$239$	−13.5314	−0.875276	−0.437638	−	0.899151i	$-0.644185\pi$
$239$	−13.5314	−0.875276	−0.437638	+	0.899151i	$0.644185\pi$
$240$	−8.80991	−0.568677
$241$	−22.5592	−1.45317	−0.726583	−	0.687078i	$-0.758893\pi$
$241$	−22.5592	−1.45317	−0.726583	+	0.687078i	$0.758893\pi$
$242$	23.1537	1.48838
$243$	10.6716	0.684585
$244$	0.717125	0.0459092
$245$	0	0
$246$	3.68607	0.235015
$247$	0	0
$248$	10.0577	0.638663
$249$	−16.7675	−1.06260
$250$	10.3925	0.657281
$251$	6.73236	0.424943	0.212471	−	0.977167i	$-0.431849\pi$
$251$	6.73236	0.424943	0.212471	+	0.977167i	$0.431849\pi$
$252$	0	0
$253$	−8.46023	−0.531890
$254$	−2.64408	−0.165904
$255$	1.30061	0.0814475
$256$	−2.35120	−0.146950
$257$	−16.5381	−1.03162	−0.515811	−	0.856703i	$-0.672509\pi$
$257$	−16.5381	−1.03162	−0.515811	+	0.856703i	$0.672509\pi$
$258$	−17.6699	−1.10008
$259$	0	0
$260$	0	0
$261$	12.0878	0.748219
$262$	−21.4878	−1.32752
$263$	−10.0227	−0.618028	−0.309014	−	0.951057i	$-0.599999\pi$
$263$	−10.0227	−0.618028	−0.309014	+	0.951057i	$0.599999\pi$
$264$	43.9485	2.70485
$265$	−4.39608	−0.270049
$266$	0	0
$267$	14.4333	0.883302
$268$	0.721388	0.0440658
$269$	15.7230	0.958647	0.479323	−	0.877638i	$-0.340882\pi$
$269$	15.7230	0.958647	0.479323	+	0.877638i	$0.340882\pi$
$270$	7.36956	0.448497
$271$	5.21618	0.316860	0.158430	−	0.987370i	$-0.449357\pi$
$271$	5.21618	0.316860	0.158430	+	0.987370i	$0.449357\pi$
$272$	−2.12499	−0.128847
$273$	0	0
$274$	−10.8270	−0.654085
$275$	−22.9337	−1.38296
$276$	−0.454108	−0.0273341
$277$	−19.2724	−1.15797	−0.578983	−	0.815340i	$-0.696550\pi$
$277$	−19.2724	−1.15797	−0.578983	+	0.815340i	$0.696550\pi$
$278$	−13.6869	−0.820884
$279$	−18.4269	−1.10319
$280$	0	0
$281$	2.14283	0.127831	0.0639153	−	0.997955i	$-0.479641\pi$
$281$	2.14283	0.127831	0.0639153	+	0.997955i	$0.479641\pi$
$282$	−15.2742	−0.909566
$283$	15.7502	0.936255	0.468127	−	0.883661i	$-0.344929\pi$
$283$	15.7502	0.936255	0.468127	+	0.883661i	$0.344929\pi$
$284$	−0.914762	−0.0542812
$285$	−13.5719	−0.803933
$286$	0	0
$287$	0	0
$288$	2.94242	0.173384
$289$	−16.6863	−0.981546
$290$	−2.53427	−0.148817
$291$	−30.6800	−1.79849
$292$	−0.491180	−0.0287441
$293$	23.1487	1.35236	0.676182	−	0.736735i	$-0.263633\pi$
$293$	23.1487	1.35236	0.676182	+	0.736735i	$0.263633\pi$
$294$	0	0
$295$	−8.86057	−0.515882
$296$	−3.59366	−0.208877
$297$	−34.9537	−2.02822
$298$	10.9211	0.632641
$299$	0	0
$300$	−1.23098	−0.0710708
$301$	0	0
$302$	−2.07019	−0.119126
$303$	11.2815	0.648103
$304$	22.1744	1.27179
$305$	5.88515	0.336983
$306$	4.09481	0.234085
$307$	4.23590	0.241756	0.120878	−	0.992667i	$-0.461429\pi$
$307$	4.23590	0.241756	0.120878	+	0.992667i	$0.461429\pi$
$308$	0	0
$309$	24.3571	1.38563
$310$	3.86328	0.219420
$311$	−27.2501	−1.54521	−0.772606	−	0.634885i	$-0.781047\pi$
$311$	−27.2501	−1.54521	−0.772606	+	0.634885i	$0.781047\pi$
$312$	0	0
$313$	−2.69697	−0.152442	−0.0762209	−	0.997091i	$-0.524285\pi$
$313$	−2.69697	−0.152442	−0.0762209	+	0.997091i	$0.524285\pi$
$314$	−5.31891	−0.300163
$315$	0	0
$316$	−1.11649	−0.0628077
$317$	−24.0705	−1.35193	−0.675966	−	0.736933i	$-0.736273\pi$
$317$	−24.0705	−1.35193	−0.675966	+	0.736933i	$0.736273\pi$
$318$	−21.6727	−1.21535
$319$	12.0200	0.672991
$320$	−6.73233	−0.376349
$321$	27.8763	1.55590
$322$	0	0
$323$	−3.27362	−0.182149
$324$	−0.314275	−0.0174597
$325$	0	0
$326$	−19.8044	−1.09686
$327$	41.9612	2.32046
$328$	2.68434	0.148218
$329$	0	0
$330$	16.8812	0.929279
$331$	0.619723	0.0340631	0.0170315	−	0.999855i	$-0.494578\pi$
$331$	0.619723	0.0340631	0.0170315	+	0.999855i	$0.494578\pi$
$332$	−0.571530	−0.0313668
$333$	6.58403	0.360803
$334$	−6.23523	−0.341177
$335$	5.92014	0.323452
$336$	0	0
$337$	5.72118	0.311652	0.155826	−	0.987784i	$-0.450196\pi$
$337$	5.72118	0.311652	0.155826	+	0.987784i	$0.450196\pi$
$338$	0	0
$339$	−56.2164	−3.05326
$340$	0.0443321	0.00240425
$341$	−18.3235	−0.992272
$342$	−42.7295	−2.31055
$343$	0	0
$344$	−12.8679	−0.693792
$345$	−3.72668	−0.200638
$346$	26.9163	1.44703
$347$	−1.86486	−0.100111	−0.0500554	−	0.998746i	$-0.515940\pi$
$347$	−1.86486	−0.100111	−0.0500554	+	0.998746i	$0.515940\pi$
$348$	0.645182	0.0345854
$349$	−22.3172	−1.19461	−0.597307	−	0.802012i	$-0.703763\pi$
$349$	−22.3172	−1.19461	−0.597307	+	0.802012i	$0.703763\pi$
$350$	0	0
$351$	0	0
$352$	2.92591	0.155951
$353$	2.33199	0.124119	0.0620597	−	0.998072i	$-0.480233\pi$
$353$	2.33199	0.124119	0.0620597	+	0.998072i	$0.480233\pi$
$354$	−43.6827	−2.32171
$355$	−7.50708	−0.398435
$356$	0.491966	0.0260741
$357$	0	0
$358$	28.7114	1.51744
$359$	−3.27105	−0.172639	−0.0863197	−	0.996267i	$-0.527511\pi$
$359$	−3.27105	−0.172639	−0.0863197	+	0.996267i	$0.527511\pi$
$360$	12.3629	0.651585
$361$	15.1603	0.797913
$362$	22.8264	1.19973
$363$	−48.3741	−2.53898
$364$	0	0
$365$	−4.03092	−0.210988
$366$	29.0139	1.51658
$367$	−4.15290	−0.216780	−0.108390	−	0.994108i	$-0.534569\pi$
$367$	−4.15290	−0.216780	−0.108390	+	0.994108i	$0.534569\pi$
$368$	6.08880	0.317401
$369$	−4.91805	−0.256024
$370$	−1.38037	−0.0717620
$371$	0	0
$372$	−0.983525	−0.0509934
$373$	−11.1089	−0.575198	−0.287599	−	0.957751i	$-0.592857\pi$
$373$	−11.1089	−0.575198	−0.287599	+	0.957751i	$0.592857\pi$
$374$	4.07183	0.210549
$375$	−21.7127	−1.12124
$376$	−11.1233	−0.573640
$377$	0	0
$378$	0	0
$379$	−4.64030	−0.238356	−0.119178	−	0.992873i	$-0.538026\pi$
$379$	−4.64030	−0.238356	−0.119178	+	0.992873i	$0.538026\pi$
$380$	−0.462607	−0.0237312
$381$	5.52416	0.283012
$382$	−5.86109	−0.299880
$383$	3.66933	0.187494	0.0937469	−	0.995596i	$-0.470116\pi$
$383$	3.66933	0.187494	0.0937469	+	0.995596i	$0.470116\pi$
$384$	−29.9922	−1.53053
$385$	0	0
$386$	−15.9932	−0.814033
$387$	23.5756	1.19842
$388$	−1.04574	−0.0530896
$389$	16.8831	0.856008	0.428004	−	0.903777i	$-0.359217\pi$
$389$	16.8831	0.856008	0.428004	+	0.903777i	$0.359217\pi$
$390$	0	0
$391$	−0.898894	−0.0454590
$392$	0	0
$393$	44.8937	2.26459
$394$	−19.8878	−1.00193
$395$	−9.16262	−0.461021
$396$	−2.74454	−0.137918
$397$	−16.7086	−0.838578	−0.419289	−	0.907853i	$-0.637721\pi$
$397$	−16.7086	−0.838578	−0.419289	+	0.907853i	$0.637721\pi$
$398$	−9.73508	−0.487976
$399$	0	0
$400$	16.5053	0.825267
$401$	25.3134	1.26409	0.632046	−	0.774931i	$-0.282215\pi$
$401$	25.3134	1.26409	0.632046	+	0.774931i	$0.282215\pi$
$402$	29.1864	1.45568
$403$	0	0
$404$	0.384535	0.0191313
$405$	−2.57913	−0.128158
$406$	0	0
$407$	6.54708	0.324527
$408$	4.66950	0.231175
$409$	−5.73343	−0.283500	−0.141750	−	0.989903i	$-0.545273\pi$
$409$	−5.73343	−0.283500	−0.141750	+	0.989903i	$0.545273\pi$
$410$	1.03109	0.0509220
$411$	22.6205	1.11579
$412$	0.830225	0.0409022
$413$	0	0
$414$	−11.7330	−0.576644
$415$	−4.69031	−0.230238
$416$	0	0
$417$	28.5954	1.40032
$418$	−42.4897	−2.07824
$419$	34.3458	1.67790	0.838950	−	0.544208i	$-0.183170\pi$
$419$	34.3458	1.67790	0.838950	+	0.544208i	$0.183170\pi$
$420$	0	0
$421$	−2.94167	−0.143368	−0.0716842	−	0.997427i	$-0.522837\pi$
$421$	−2.94167	−0.143368	−0.0716842	+	0.997427i	$0.522837\pi$
$422$	−36.4383	−1.77379
$423$	20.3793	0.990873
$424$	−15.7830	−0.766488
$425$	−2.43669	−0.118197
$426$	−37.0100	−1.79314
$427$	0	0
$428$	0.950178	0.0459286
$429$	0	0
$430$	−4.94273	−0.238360
$431$	−39.6955	−1.91207	−0.956033	−	0.293258i	$-0.905261\pi$
$431$	−39.6955	−1.91207	−0.956033	+	0.293258i	$0.905261\pi$
$432$	25.1561	1.21032
$433$	9.83653	0.472714	0.236357	−	0.971666i	$-0.424047\pi$
$433$	9.83653	0.472714	0.236357	+	0.971666i	$0.424047\pi$
$434$	0	0
$435$	5.29475	0.253864
$436$	1.43027	0.0684975
$437$	9.37999	0.448706
$438$	−19.8725	−0.949544
$439$	−28.5465	−1.36245	−0.681226	−	0.732073i	$-0.738553\pi$
$439$	−28.5465	−1.36245	−0.681226	+	0.732073i	$0.738553\pi$
$440$	12.2936	0.586073
$441$	0	0
$442$	0	0
$443$	3.33901	0.158641	0.0793207	−	0.996849i	$-0.474725\pi$
$443$	3.33901	0.158641	0.0793207	+	0.996849i	$0.474725\pi$
$444$	0.351419	0.0166776
$445$	4.03736	0.191389
$446$	−31.7190	−1.50194
$447$	−22.8170	−1.07921
$448$	0	0
$449$	18.1851	0.858206	0.429103	−	0.903256i	$-0.358830\pi$
$449$	18.1851	0.858206	0.429103	+	0.903256i	$0.358830\pi$
$450$	−31.8054	−1.49932
$451$	−4.89045	−0.230282
$452$	−1.91617	−0.0901289
$453$	4.32516	0.203214
$454$	0.625219	0.0293430
$455$	0	0
$456$	−48.7265	−2.28183
$457$	−8.72932	−0.408341	−0.204170	−	0.978935i	$-0.565450\pi$
$457$	−8.72932	−0.408341	−0.204170	+	0.978935i	$0.565450\pi$
$458$	−23.9038	−1.11695
$459$	−3.71381	−0.173346
$460$	−0.127026	−0.00592262
$461$	−2.27124	−0.105782	−0.0528910	−	0.998600i	$-0.516844\pi$
$461$	−2.27124	−0.105782	−0.0528910	+	0.998600i	$0.516844\pi$
$462$	0	0
$463$	5.48326	0.254829	0.127414	−	0.991850i	$-0.459332\pi$
$463$	5.48326	0.254829	0.127414	+	0.991850i	$0.459332\pi$
$464$	−8.65077	−0.401602
$465$	−8.07139	−0.374302
$466$	10.7775	0.499257
$467$	−18.8819	−0.873750	−0.436875	−	0.899522i	$-0.643915\pi$
$467$	−18.8819	−0.873750	−0.436875	+	0.899522i	$0.643915\pi$
$468$	0	0
$469$	0	0
$470$	−4.27260	−0.197080
$471$	11.1126	0.512040
$472$	−31.8115	−1.46424
$473$	23.4433	1.07792
$474$	−45.1718	−2.07481
$475$	25.4270	1.16667
$476$	0	0
$477$	28.9163	1.32399
$478$	−18.6606	−0.853516
$479$	33.1354	1.51399	0.756997	−	0.653418i	$-0.226666\pi$
$479$	33.1354	1.51399	0.756997	+	0.653418i	$0.226666\pi$
$480$	1.28884	0.0588275
$481$	0	0
$482$	−31.1104	−1.41704
$483$	0	0
$484$	−1.64886	−0.0749480
$485$	−8.58199	−0.389688
$486$	14.7168	0.667565
$487$	−15.9563	−0.723048	−0.361524	−	0.932363i	$-0.617743\pi$
$487$	−15.9563	−0.723048	−0.361524	+	0.932363i	$0.617743\pi$
$488$	21.1291	0.956469
$489$	41.3765	1.87111
$490$	0	0
$491$	−31.6928	−1.43028	−0.715138	−	0.698983i	$-0.753636\pi$
$491$	−31.6928	−1.43028	−0.715138	+	0.698983i	$0.753636\pi$
$492$	−0.262498	−0.0118343
$493$	1.27712	0.0575185
$494$	0	0
$495$	−22.5233	−1.01235
$496$	13.1874	0.592130
$497$	0	0
$498$	−23.1233	−1.03618
$499$	−24.2184	−1.08417	−0.542083	−	0.840325i	$-0.682364\pi$
$499$	−24.2184	−1.08417	−0.542083	+	0.840325i	$0.682364\pi$
$500$	−0.740089	−0.0330978
$501$	13.0270	0.582004
$502$	9.28429	0.414378
$503$	−0.854498	−0.0381002	−0.0190501	−	0.999819i	$-0.506064\pi$
$503$	−0.854498	−0.0381002	−0.0190501	+	0.999819i	$0.506064\pi$
$504$	0	0
$505$	3.15572	0.140428
$506$	−11.6671	−0.518667
$507$	0	0
$508$	0.188294	0.00835420
$509$	−1.30000	−0.0576215	−0.0288108	−	0.999585i	$-0.509172\pi$
$509$	−1.30000	−0.0576215	−0.0288108	+	0.999585i	$0.509172\pi$
$510$	1.79362	0.0794227
$511$	0	0
$512$	−24.0616	−1.06338
$513$	38.7538	1.71102
$514$	−22.8070	−1.00597
$515$	6.81332	0.300231
$516$	1.25833	0.0553951
$517$	20.2649	0.891248
$518$	0	0
$519$	−56.2351	−2.46845
$520$	0	0
$521$	25.0455	1.09726	0.548632	−	0.836064i	$-0.315149\pi$
$521$	25.0455	1.09726	0.548632	+	0.836064i	$0.315149\pi$
$522$	16.6698	0.729618
$523$	−12.8239	−0.560752	−0.280376	−	0.959890i	$-0.590459\pi$
$523$	−12.8239	−0.560752	−0.280376	+	0.959890i	$0.590459\pi$
$524$	1.53022	0.0668482
$525$	0	0
$526$	−13.8219	−0.602664
$527$	−1.94686	−0.0848065
$528$	57.6242	2.50777
$529$	−20.4244	−0.888016
$530$	−6.06244	−0.263335
$531$	58.2827	2.52925
$532$	0	0
$533$	0	0
$534$	19.9043	0.861342
$535$	7.79773	0.337125
$536$	21.2547	0.918063
$537$	−59.9855	−2.58857
$538$	21.6828	0.934814
$539$	0	0
$540$	−0.524813	−0.0225843
$541$	−28.7449	−1.23584	−0.617920	−	0.786241i	$-0.712025\pi$
$541$	−28.7449	−1.23584	−0.617920	+	0.786241i	$0.712025\pi$
$542$	7.19340	0.308983
$543$	−47.6902	−2.04658
$544$	0.310876	0.0133287
$545$	11.7376	0.502785
$546$	0	0
$547$	−8.88085	−0.379718	−0.189859	−	0.981811i	$-0.560803\pi$
$547$	−8.88085	−0.379718	−0.189859	+	0.981811i	$0.560803\pi$
$548$	0.771031	0.0329368
$549$	−38.7111	−1.65215
$550$	−31.6269	−1.34857
$551$	−13.3268	−0.567740
$552$	−13.3797	−0.569476
$553$	0	0
$554$	−26.5777	−1.12918
$555$	2.88395	0.122417
$556$	0.974690	0.0413361
$557$	38.7273	1.64093	0.820465	−	0.571696i	$-0.193714\pi$
$557$	38.7273	1.64093	0.820465	+	0.571696i	$0.193714\pi$
$558$	−25.4117	−1.07576
$559$	0	0
$560$	0	0
$561$	−8.50710	−0.359170
$562$	2.95508	0.124653
$563$	−6.90882	−0.291172	−0.145586	−	0.989346i	$-0.546507\pi$
$563$	−6.90882	−0.291172	−0.145586	+	0.989346i	$0.546507\pi$
$564$	1.08773	0.0458017
$565$	−15.7252	−0.661565
$566$	21.7205	0.912979
$567$	0	0
$568$	−26.9522	−1.13089
$569$	−2.83745	−0.118952	−0.0594759	−	0.998230i	$-0.518943\pi$
$569$	−2.83745	−0.118952	−0.0594759	+	0.998230i	$0.518943\pi$
$570$	−18.7165	−0.783946
$571$	−46.6724	−1.95318	−0.976589	−	0.215113i	$-0.930988\pi$
$571$	−46.6724	−1.95318	−0.976589	+	0.215113i	$0.930988\pi$
$572$	0	0
$573$	12.2453	0.511557
$574$	0	0
$575$	6.98193	0.291167
$576$	44.2836	1.84515
$577$	11.4088	0.474955	0.237478	−	0.971393i	$-0.423679\pi$
$577$	11.4088	0.474955	0.237478	+	0.971393i	$0.423679\pi$
$578$	−23.0113	−0.957144
$579$	33.4140	1.38864
$580$	0.180474	0.00749379
$581$	0	0
$582$	−42.3094	−1.75378
$583$	28.7541	1.19087
$584$	−14.4719	−0.598853
$585$	0	0
$586$	31.9234	1.31874
$587$	46.4410	1.91683	0.958413	−	0.285384i	$-0.0921211\pi$
$587$	46.4410	1.91683	0.958413	+	0.285384i	$0.0921211\pi$
$588$	0	0
$589$	20.3156	0.837088
$590$	−12.2192	−0.503057
$591$	41.5508	1.70917
$592$	−4.71191	−0.193658
$593$	−20.2606	−0.832002	−0.416001	−	0.909364i	$-0.636569\pi$
$593$	−20.2606	−0.832002	−0.416001	+	0.909364i	$0.636569\pi$
$594$	−48.2031	−1.97780
$595$	0	0
$596$	−0.777728	−0.0318570
$597$	20.3391	0.832424
$598$	0	0
$599$	−38.9876	−1.59299	−0.796494	−	0.604646i	$-0.793315\pi$
$599$	−38.9876	−1.59299	−0.796494	+	0.604646i	$0.793315\pi$
$600$	−36.2692	−1.48068
$601$	−19.1390	−0.780697	−0.390348	−	0.920667i	$-0.627645\pi$
$601$	−19.1390	−0.780697	−0.390348	+	0.920667i	$0.627645\pi$
$602$	0	0
$603$	−38.9412	−1.58581
$604$	0.147425	0.00599865
$605$	−13.5315	−0.550134
$606$	15.5578	0.631991
$607$	−43.3336	−1.75886	−0.879428	−	0.476033i	$-0.842074\pi$
$607$	−43.3336	−1.75886	−0.879428	+	0.476033i	$0.842074\pi$
$608$	−3.24400	−0.131562
$609$	0	0
$610$	8.11595	0.328605
$611$	0	0
$612$	−0.291606	−0.0117875
$613$	−10.3096	−0.416399	−0.208200	−	0.978086i	$-0.566760\pi$
$613$	−10.3096	−0.416399	−0.208200	+	0.978086i	$0.566760\pi$
$614$	5.84154	0.235745
$615$	−2.15422	−0.0868664
$616$	0	0
$617$	11.0699	0.445659	0.222829	−	0.974857i	$-0.428471\pi$
$617$	11.0699	0.445659	0.222829	+	0.974857i	$0.428471\pi$
$618$	33.5898	1.35118
$619$	33.7616	1.35700	0.678498	−	0.734603i	$-0.262631\pi$
$619$	33.7616	1.35700	0.678498	+	0.734603i	$0.262631\pi$
$620$	−0.275118	−0.0110490
$621$	10.6413	0.427020
$622$	−37.5794	−1.50680
$623$	0	0
$624$	0	0
$625$	15.6786	0.627145
$626$	−3.71927	−0.148652
$627$	88.7719	3.54521
$628$	0.378778	0.0151149
$629$	0.695623	0.0277363
$630$	0	0
$631$	38.5975	1.53654	0.768271	−	0.640125i	$-0.221117\pi$
$631$	38.5975	1.53654	0.768271	+	0.640125i	$0.221117\pi$
$632$	−32.8959	−1.30853
$633$	76.1290	3.02586
$634$	−33.1945	−1.31832
$635$	1.54525	0.0613215
$636$	1.54339	0.0611995
$637$	0	0
$638$	16.5763	0.656260
$639$	49.3798	1.95343
$640$	−8.38960	−0.331628
$641$	19.5228	0.771105	0.385553	−	0.922686i	$-0.374011\pi$
$641$	19.5228	0.771105	0.385553	+	0.922686i	$0.374011\pi$
$642$	38.4429	1.51722
$643$	12.4718	0.491839	0.245920	−	0.969290i	$-0.420910\pi$
$643$	12.4718	0.491839	0.245920	+	0.969290i	$0.420910\pi$
$644$	0	0
$645$	10.3266	0.406611
$646$	−4.51450	−0.177621
$647$	−35.9391	−1.41291	−0.706455	−	0.707758i	$-0.749707\pi$
$647$	−35.9391	−1.41291	−0.706455	+	0.707758i	$0.749707\pi$
$648$	−9.25967	−0.363754
$649$	57.9556	2.27496
$650$	0	0
$651$	0	0
$652$	1.41034	0.0552332
$653$	4.85888	0.190143	0.0950713	−	0.995470i	$-0.469692\pi$
$653$	4.85888	0.190143	0.0950713	+	0.995470i	$0.469692\pi$
$654$	57.8668	2.26277
$655$	12.5579	0.490679
$656$	3.51964	0.137419
$657$	26.5144	1.03442
$658$	0	0
$659$	−23.6206	−0.920127	−0.460063	−	0.887886i	$-0.652173\pi$
$659$	−23.6206	−0.920127	−0.460063	+	0.887886i	$0.652173\pi$
$660$	−1.20217	−0.0467944
$661$	−16.3932	−0.637623	−0.318812	−	0.947818i	$-0.603284\pi$
$661$	−16.3932	−0.637623	−0.318812	+	0.947818i	$0.603284\pi$
$662$	0.854633	0.0332162
$663$	0	0
$664$	−16.8393	−0.653492
$665$	0	0
$666$	9.07974	0.351833
$667$	−3.65936	−0.141691
$668$	0.444033	0.0171801
$669$	66.2692	2.56212
$670$	8.16420	0.315410
$671$	−38.4939	−1.48604
$672$	0	0
$673$	−14.2536	−0.549434	−0.274717	−	0.961525i	$-0.588584\pi$
$673$	−14.2536	−0.549434	−0.274717	+	0.961525i	$0.588584\pi$
$674$	7.88982	0.303904
$675$	28.8461	1.11029
$676$	0	0
$677$	10.2715	0.394765	0.197383	−	0.980327i	$-0.436756\pi$
$677$	10.2715	0.394765	0.197383	+	0.980327i	$0.436756\pi$
$678$	−77.5255	−2.97735
$679$	0	0
$680$	1.30618	0.0500898
$681$	−1.30624	−0.0500554
$682$	−25.2691	−0.967604
$683$	−2.22201	−0.0850230	−0.0425115	−	0.999096i	$-0.513536\pi$
$683$	−2.22201	−0.0850230	−0.0425115	+	0.999096i	$0.513536\pi$
$684$	3.04292	0.116349
$685$	6.32754	0.241763
$686$	0	0
$687$	49.9413	1.90538
$688$	−16.8721	−0.643242
$689$	0	0
$690$	−5.13930	−0.195650
$691$	2.64015	0.100436	0.0502179	−	0.998738i	$-0.484008\pi$
$691$	2.64015	0.100436	0.0502179	+	0.998738i	$0.484008\pi$
$692$	−1.91680	−0.0728659
$693$	0	0
$694$	−2.57174	−0.0976220
$695$	7.99889	0.303415
$696$	19.0094	0.720549
$697$	−0.519607	−0.0196815
$698$	−30.7767	−1.16492
$699$	−22.5169	−0.851668
$700$	0	0
$701$	8.89991	0.336145	0.168072	−	0.985775i	$-0.446246\pi$
$701$	8.89991	0.336145	0.168072	+	0.985775i	$0.446246\pi$
$702$	0	0
$703$	−7.25885	−0.273773
$704$	44.0351	1.65964
$705$	8.92656	0.336194
$706$	3.21594	0.121034
$707$	0	0
$708$	3.11080	0.116911
$709$	40.5944	1.52456	0.762278	−	0.647250i	$-0.224081\pi$
$709$	40.5944	1.52456	0.762278	+	0.647250i	$0.224081\pi$
$710$	−10.3527	−0.388529
$711$	60.2695	2.26028
$712$	14.4951	0.543226
$713$	5.57839	0.208912
$714$	0	0
$715$	0	0
$716$	−2.04464	−0.0764118
$717$	38.9868	1.45599
$718$	−4.51096	−0.168348
$719$	−14.5135	−0.541262	−0.270631	−	0.962683i	$-0.587232\pi$
$719$	−14.5135	−0.541262	−0.270631	+	0.962683i	$0.587232\pi$
$720$	16.2100	0.604110
$721$	0	0
$722$	20.9069	0.778076
$723$	64.9977	2.41729
$724$	−1.62555	−0.0604129
$725$	−9.91969	−0.368408
$726$	−66.7105	−2.47586
$727$	30.6942	1.13839	0.569193	−	0.822204i	$-0.307256\pi$
$727$	30.6942	1.13839	0.569193	+	0.822204i	$0.307256\pi$
$728$	0	0
$729$	−40.3475	−1.49435
$730$	−5.55885	−0.205742
$731$	2.49084	0.0921269
$732$	−2.06618	−0.0763683
$733$	−13.2644	−0.489930	−0.244965	−	0.969532i	$-0.578777\pi$
$733$	−13.2644	−0.489930	−0.244965	+	0.969532i	$0.578777\pi$
$734$	−5.72708	−0.211390
$735$	0	0
$736$	−0.890761	−0.0328339
$737$	−38.7227	−1.42637
$738$	−6.78227	−0.249659
$739$	−7.25474	−0.266870	−0.133435	−	0.991058i	$-0.542601\pi$
$739$	−7.25474	−0.266870	−0.133435	+	0.991058i	$0.542601\pi$
$740$	0.0983010	0.00361362
$741$	0	0
$742$	0	0
$743$	−46.2694	−1.69746	−0.848730	−	0.528827i	$-0.822632\pi$
$743$	−46.2694	−1.69746	−0.848730	+	0.528827i	$0.822632\pi$
$744$	−28.9782	−1.06239
$745$	−6.38250	−0.233837
$746$	−15.3198	−0.560898
$747$	30.8517	1.12881
$748$	−0.289969	−0.0106023
$749$	0	0
$750$	−29.9430	−1.09336
$751$	−36.0260	−1.31461	−0.657305	−	0.753625i	$-0.728303\pi$
$751$	−36.0260	−1.31461	−0.657305	+	0.753625i	$0.728303\pi$
$752$	−14.5846	−0.531845
$753$	−19.3973	−0.706877
$754$	0	0
$755$	1.20986	0.0440313
$756$	0	0
$757$	−10.5626	−0.383906	−0.191953	−	0.981404i	$-0.561482\pi$
$757$	−10.5626	−0.383906	−0.191953	+	0.981404i	$0.561482\pi$
$758$	−6.39923	−0.232430
$759$	24.3756	0.884780
$760$	−13.6301	−0.494415
$761$	7.81202	0.283185	0.141593	−	0.989925i	$-0.454778\pi$
$761$	7.81202	0.283185	0.141593	+	0.989925i	$0.454778\pi$
$762$	7.61813	0.275976
$763$	0	0
$764$	0.417389	0.0151006
$765$	−2.39309	−0.0865223
$766$	5.06020	0.182833
$767$	0	0
$768$	6.77429	0.244446
$769$	−25.2915	−0.912033	−0.456017	−	0.889971i	$-0.650724\pi$
$769$	−25.2915	−0.912033	−0.456017	+	0.889971i	$0.650724\pi$
$770$	0	0
$771$	47.6497	1.71606
$772$	1.13893	0.0409911
$773$	46.6004	1.67610	0.838051	−	0.545592i	$-0.183695\pi$
$773$	46.6004	1.67610	0.838051	+	0.545592i	$0.183695\pi$
$774$	32.5121	1.16862
$775$	15.1217	0.543188
$776$	−30.8114	−1.10606
$777$	0	0
$778$	23.2827	0.834727
$779$	5.42212	0.194268
$780$	0	0
$781$	49.1026	1.75703
$782$	−1.23962	−0.0443289
$783$	−15.1188	−0.540302
$784$	0	0
$785$	3.10848	0.110946
$786$	61.9108	2.20829
$787$	39.7332	1.41633	0.708167	−	0.706045i	$-0.249522\pi$
$787$	39.7332	1.41633	0.708167	+	0.706045i	$0.249522\pi$
$788$	1.41628	0.0504529
$789$	28.8775	1.02807
$790$	−12.6358	−0.449560
$791$	0	0
$792$	−80.8641	−2.87338
$793$	0	0
$794$	−23.0420	−0.817730
$795$	12.6660	0.449217
$796$	0.693270	0.0245723
$797$	2.78598	0.0986844	0.0493422	−	0.998782i	$-0.484288\pi$
$797$	2.78598	0.0986844	0.0493422	+	0.998782i	$0.484288\pi$
$798$	0	0
$799$	2.15313	0.0761723
$800$	−2.41465	−0.0853706
$801$	−26.5568	−0.938338
$802$	34.9086	1.23267
$803$	26.3656	0.930421
$804$	−2.07847	−0.0733018
$805$	0	0
$806$	0	0
$807$	−45.3011	−1.59467
$808$	11.3298	0.398580
$809$	−41.4586	−1.45761	−0.728803	−	0.684723i	$-0.759923\pi$
$809$	−41.4586	−1.45761	−0.728803	+	0.684723i	$0.759923\pi$
$810$	−3.55676	−0.124972
$811$	−27.8622	−0.978375	−0.489188	−	0.872179i	$-0.662707\pi$
$811$	−27.8622	−0.978375	−0.489188	+	0.872179i	$0.662707\pi$
$812$	0	0
$813$	−15.0289	−0.527085
$814$	9.02878	0.316459
$815$	11.5741	0.405423
$816$	6.12254	0.214332
$817$	−25.9920	−0.909344
$818$	−7.90671	−0.276452
$819$	0	0
$820$	−0.0734276	−0.00256420
$821$	22.4202	0.782469	0.391235	−	0.920291i	$-0.372048\pi$
$821$	22.4202	0.782469	0.391235	+	0.920291i	$0.372048\pi$
$822$	31.1949	1.08805
$823$	−2.36166	−0.0823221	−0.0411611	−	0.999153i	$-0.513106\pi$
$823$	−2.36166	−0.0823221	−0.0411611	+	0.999153i	$0.513106\pi$
$824$	24.4614	0.852154
$825$	66.0767	2.30050
$826$	0	0
$827$	−43.3148	−1.50620	−0.753102	−	0.657904i	$-0.771443\pi$
$827$	−43.3148	−1.50620	−0.753102	+	0.657904i	$0.771443\pi$
$828$	0.835546	0.0290372
$829$	54.9280	1.90773	0.953864	−	0.300239i	$-0.0970665\pi$
$829$	54.9280	1.90773	0.953864	+	0.300239i	$0.0970665\pi$
$830$	−6.46820	−0.224514
$831$	55.5277	1.92623
$832$	0	0
$833$	0	0
$834$	39.4346	1.36551
$835$	3.64400	0.126106
$836$	3.02584	0.104651
$837$	23.0473	0.796632
$838$	47.3647	1.63619
$839$	14.7410	0.508917	0.254459	−	0.967084i	$-0.418103\pi$
$839$	14.7410	0.508917	0.254459	+	0.967084i	$0.418103\pi$
$840$	0	0
$841$	−23.8009	−0.820721
$842$	−4.05673	−0.139804
$843$	−6.17393	−0.212642
$844$	2.59490	0.0893201
$845$	0	0
$846$	28.1041	0.966239
$847$	0	0
$848$	−20.6942	−0.710642
$849$	−45.3796	−1.55743
$850$	−3.36033	−0.115258
$851$	−1.99319	−0.0683256
$852$	2.63562	0.0902947
$853$	24.1038	0.825297	0.412649	−	0.910890i	$-0.364604\pi$
$853$	24.1038	0.825297	0.412649	+	0.910890i	$0.364604\pi$
$854$	0	0
$855$	24.9720	0.854024
$856$	27.9957	0.956873
$857$	18.5850	0.634851	0.317425	−	0.948283i	$-0.397182\pi$
$857$	18.5850	0.634851	0.317425	+	0.948283i	$0.397182\pi$
$858$	0	0
$859$	−29.4975	−1.00644	−0.503221	−	0.864158i	$-0.667852\pi$
$859$	−29.4975	−1.00644	−0.503221	+	0.864158i	$0.667852\pi$
$860$	0.351989	0.0120027
$861$	0	0
$862$	−54.7423	−1.86453
$863$	18.6435	0.634631	0.317315	−	0.948320i	$-0.397219\pi$
$863$	18.6435	0.634631	0.317315	+	0.948320i	$0.397219\pi$
$864$	−3.68021	−0.125203
$865$	−15.7304	−0.534851
$866$	13.5651	0.460961
$867$	48.0766	1.63277
$868$	0	0
$869$	59.9312	2.03303
$870$	7.30175	0.247552
$871$	0	0
$872$	42.1409	1.42707
$873$	56.4502	1.91055
$874$	12.9355	0.437551
$875$	0	0
$876$	1.41519	0.0478148
$877$	−37.7518	−1.27479	−0.637395	−	0.770538i	$-0.719988\pi$
$877$	−37.7518	−1.27479	−0.637395	+	0.770538i	$0.719988\pi$
$878$	−39.3673	−1.32858
$879$	−66.6962	−2.24961
$880$	16.1190	0.543372
$881$	−29.8298	−1.00499	−0.502497	−	0.864579i	$-0.667585\pi$
$881$	−29.8298	−1.00499	−0.502497	+	0.864579i	$0.667585\pi$
$882$	0	0
$883$	32.3979	1.09028	0.545138	−	0.838346i	$-0.316477\pi$
$883$	32.3979	1.09028	0.545138	+	0.838346i	$0.316477\pi$
$884$	0	0
$885$	25.5291	0.858151
$886$	4.60468	0.154697
$887$	25.9198	0.870302	0.435151	−	0.900358i	$-0.356695\pi$
$887$	25.9198	0.870302	0.435151	+	0.900358i	$0.356695\pi$
$888$	10.3541	0.347459
$889$	0	0
$890$	5.56775	0.186631
$891$	16.8697	0.565155
$892$	2.25882	0.0756310
$893$	−22.4680	−0.751863
$894$	−31.4658	−1.05238
$895$	−16.7795	−0.560878
$896$	0	0
$897$	0	0
$898$	25.0782	0.836870
$899$	−7.92559	−0.264333
$900$	2.26497	0.0754991
$901$	3.05510	0.101780
$902$	−6.74420	−0.224557
$903$	0	0
$904$	−56.4572	−1.87774
$905$	−13.3402	−0.443443
$906$	5.96463	0.198162
$907$	15.5423	0.516072	0.258036	−	0.966135i	$-0.416925\pi$
$907$	15.5423	0.516072	0.258036	+	0.966135i	$0.416925\pi$
$908$	−0.0445240	−0.00147758
$909$	−20.7576	−0.688485
$910$	0	0
$911$	23.6358	0.783090	0.391545	−	0.920159i	$-0.371941\pi$
$911$	23.6358	0.783090	0.391545	+	0.920159i	$0.371941\pi$
$912$	−63.8889	−2.11557
$913$	30.6786	1.01531
$914$	−12.0382	−0.398189
$915$	−16.9563	−0.560559
$916$	1.70228	0.0562448
$917$	0	0
$918$	−5.12155	−0.169036
$919$	44.4817	1.46732	0.733659	−	0.679518i	$-0.237811\pi$
$919$	44.4817	1.46732	0.733659	+	0.679518i	$0.237811\pi$
$920$	−3.74264	−0.123391
$921$	−12.2045	−0.402151
$922$	−3.13216	−0.103152
$923$	0	0
$924$	0	0
$925$	−5.40307	−0.177652
$926$	7.56171	0.248493
$927$	−44.8163	−1.47196
$928$	1.26556	0.0415441
$929$	2.94270	0.0965470	0.0482735	−	0.998834i	$-0.484628\pi$
$929$	2.94270	0.0965470	0.0482735	+	0.998834i	$0.484628\pi$
$930$	−11.1309	−0.364996
$931$	0	0
$932$	−0.767502	−0.0251404
$933$	78.5131	2.57040
$934$	−26.0392	−0.852028
$935$	−2.37966	−0.0778232
$936$	0	0
$937$	0.951020	0.0310685	0.0155342	−	0.999879i	$-0.495055\pi$
$937$	0.951020	0.0310685	0.0155342	+	0.999879i	$0.495055\pi$
$938$	0	0
$939$	7.77052	0.253581
$940$	0.304267	0.00992409
$941$	22.0692	0.719436	0.359718	−	0.933061i	$-0.382873\pi$
$941$	22.0692	0.719436	0.359718	+	0.933061i	$0.382873\pi$
$942$	15.3248	0.499311
$943$	1.48885	0.0484835
$944$	−41.7105	−1.35756
$945$	0	0
$946$	32.3296	1.05113
$947$	−51.1717	−1.66286	−0.831429	−	0.555631i	$-0.812477\pi$
$947$	−51.1717	−1.66286	−0.831429	+	0.555631i	$0.812477\pi$
$948$	3.21685	0.104478
$949$	0	0
$950$	35.0652	1.13767
$951$	69.3519	2.24889
$952$	0	0
$953$	45.8470	1.48513	0.742565	−	0.669773i	$-0.233609\pi$
$953$	45.8470	1.48513	0.742565	+	0.669773i	$0.233609\pi$
$954$	39.8772	1.29107
$955$	3.42534	0.110842
$956$	1.32889	0.0429793
$957$	−34.6321	−1.11950
$958$	45.6955	1.47635
$959$	0	0
$960$	19.3972	0.626042
$961$	−18.9181	−0.610262
$962$	0	0
$963$	−51.2916	−1.65285
$964$	2.21548	0.0713559
$965$	9.34676	0.300883
$966$	0	0
$967$	19.2609	0.619387	0.309694	−	0.950836i	$-0.399773\pi$
$967$	19.2609	0.619387	0.309694	+	0.950836i	$0.399773\pi$
$968$	−48.5813	−1.56146
$969$	9.43196	0.302998
$970$	−11.8350	−0.380000
$971$	−47.3326	−1.51897	−0.759487	−	0.650522i	$-0.774550\pi$
$971$	−47.3326	−1.51897	−0.759487	+	0.650522i	$0.774550\pi$
$972$	−1.04803	−0.0336156
$973$	0	0
$974$	−22.0046	−0.705072
$975$	0	0
$976$	27.7039	0.886781
$977$	−47.8571	−1.53108	−0.765541	−	0.643387i	$-0.777529\pi$
$977$	−47.8571	−1.53108	−0.765541	+	0.643387i	$0.777529\pi$
$978$	57.0605	1.82459
$979$	−26.4078	−0.843995
$980$	0	0
$981$	−77.2074	−2.46504
$982$	−43.7061	−1.39472
$983$	15.7686	0.502941	0.251471	−	0.967865i	$-0.419086\pi$
$983$	15.7686	0.502941	0.251471	+	0.967865i	$0.419086\pi$
$984$	−7.73414	−0.246555
$985$	11.6228	0.370335
$986$	1.76122	0.0560885
$987$	0	0
$988$	0	0
$989$	−7.13707	−0.226945
$990$	−31.0609	−0.987181
$991$	−12.1378	−0.385571	−0.192786	−	0.981241i	$-0.561752\pi$
$991$	−12.1378	−0.385571	−0.192786	+	0.981241i	$0.561752\pi$
$992$	−1.92924	−0.0612536
$993$	−1.78555	−0.0566627
$994$	0	0
$995$	5.68938	0.180366
$996$	1.64669	0.0521775
$997$	−33.7876	−1.07006	−0.535032	−	0.844832i	$-0.679701\pi$
$997$	−33.7876	−1.07006	−0.535032	+	0.844832i	$0.679701\pi$
$998$	−33.3986	−1.05721
$999$	−8.23493	−0.260542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.co.1.10		12
7.3	odd	6		1183.2.e.j.170.3		24
7.5	odd	6		1183.2.e.j.508.3		24
7.6	odd	2		8281.2.a.cp.1.10		12
13.2	odd	12		637.2.q.i.589.5		12
13.7	odd	12		637.2.q.i.491.5		12
13.12	even	2	inner	8281.2.a.co.1.3		12
91.2	odd	12		637.2.k.i.459.2		12
91.12	odd	6		1183.2.e.j.508.10		24
91.20	even	12		637.2.q.g.491.5		12
91.33	even	12		91.2.u.b.88.2	yes	12
91.38	odd	6		1183.2.e.j.170.10		24
91.41	even	12		637.2.q.g.589.5		12
91.46	odd	12		637.2.k.i.569.5		12
91.54	even	12		91.2.k.b.4.2	✓	12
91.59	even	12		91.2.k.b.23.5	yes	12
91.67	odd	12		637.2.u.g.30.2		12
91.72	odd	12		637.2.u.g.361.2		12
91.80	even	12		91.2.u.b.30.2	yes	12
91.90	odd	2		8281.2.a.cp.1.3		12
273.59	odd	12		819.2.bm.f.478.2		12
273.80	odd	12		819.2.do.e.667.5		12
273.215	odd	12		819.2.do.e.361.5		12
273.236	odd	12		819.2.bm.f.550.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.k.b.4.2	✓	12	91.54	even	12
91.2.k.b.23.5	yes	12	91.59	even	12
91.2.u.b.30.2	yes	12	91.80	even	12
91.2.u.b.88.2	yes	12	91.33	even	12
637.2.k.i.459.2		12	91.2	odd	12
637.2.k.i.569.5		12	91.46	odd	12
637.2.q.g.491.5		12	91.20	even	12
637.2.q.g.589.5		12	91.41	even	12
637.2.q.i.491.5		12	13.7	odd	12
637.2.q.i.589.5		12	13.2	odd	12
637.2.u.g.30.2		12	91.67	odd	12
637.2.u.g.361.2		12	91.72	odd	12
819.2.bm.f.478.2		12	273.59	odd	12
819.2.bm.f.550.5		12	273.236	odd	12
819.2.do.e.361.5		12	273.215	odd	12
819.2.do.e.667.5		12	273.80	odd	12
1183.2.e.j.170.3		24	7.3	odd	6
1183.2.e.j.170.10		24	91.38	odd	6
1183.2.e.j.508.3		24	7.5	odd	6
1183.2.e.j.508.10		24	91.12	odd	6
8281.2.a.co.1.3		12	13.12	even	2	inner
8281.2.a.co.1.10		12	1.1	even	1	trivial
8281.2.a.cp.1.3		12	91.90	odd	2
8281.2.a.cp.1.10		12	7.6	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37905 q^{2} -2.88120 q^{3} -0.0982074 q^{4} -0.805948 q^{5} -3.97334 q^{6} -2.89354 q^{8} +5.30133 q^{9} +O(q^{10})\)	\(q+1.37905 q^{2} -2.88120 q^{3} -0.0982074 q^{4} -0.805948 q^{5} -3.97334 q^{6} -2.89354 q^{8} +5.30133 q^{9} -1.11145 q^{10} +5.27158 q^{11} +0.282955 q^{12} +2.32210 q^{15} -3.79394 q^{16} +0.560102 q^{17} +7.31083 q^{18} -5.84469 q^{19} +0.0791501 q^{20} +7.26980 q^{22} -1.60488 q^{23} +8.33689 q^{24} -4.35045 q^{25} -6.63060 q^{27} +2.28015 q^{29} +3.20230 q^{30} -3.47590 q^{31} +0.555034 q^{32} -15.1885 q^{33} +0.772411 q^{34} -0.520630 q^{36} +1.24196 q^{37} -8.06014 q^{38} +2.33205 q^{40} -0.927702 q^{41} +4.44711 q^{43} -0.517708 q^{44} -4.27260 q^{45} -2.21321 q^{46} +3.84418 q^{47} +10.9311 q^{48} -5.99951 q^{50} -1.61377 q^{51} +5.45454 q^{53} -9.14396 q^{54} -4.24862 q^{55} +16.8397 q^{57} +3.14446 q^{58} +10.9940 q^{59} -0.228047 q^{60} -7.30215 q^{61} -4.79346 q^{62} +8.35330 q^{64} -20.9458 q^{66} -7.34556 q^{67} -0.0550061 q^{68} +4.62397 q^{69} +9.31460 q^{71} -15.3396 q^{72} +5.00146 q^{73} +1.71273 q^{74} +12.5345 q^{75} +0.573991 q^{76} +11.3687 q^{79} +3.05772 q^{80} +3.20012 q^{81} -1.27935 q^{82} +5.81962 q^{83} -0.451413 q^{85} +6.13281 q^{86} -6.56959 q^{87} -15.2535 q^{88} -5.00946 q^{89} -5.89215 q^{90} +0.157611 q^{92} +10.0148 q^{93} +5.30133 q^{94} +4.71051 q^{95} -1.59917 q^{96} +10.6483 q^{97} +27.9464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 + 8 * q^4 + 2 * q^9	\( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9} - 24 q^{10} + 2 q^{12} + 16 q^{16} - 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} - 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} - 38 q^{38} - 2 q^{40} + 22 q^{43} + 38 q^{48} + 8 q^{51} + 16 q^{53} - 30 q^{55} + 10 q^{61} - 82 q^{62} - 2 q^{64} - 68 q^{66} - 22 q^{68} - 14 q^{69} + 66 q^{74} - 2 q^{75} + 70 q^{79} - 28 q^{81} - 10 q^{82} + 20 q^{87} - 28 q^{88} - 66 q^{92} + 2 q^{94} + 4 q^{95}+O(q^{100}) \) 12 * q - 6 * q^3 + 8 * q^4 + 2 * q^9 - 24 * q^10 + 2 * q^12 + 16 * q^16 - 34 * q^17 + 30 * q^22 + 6 * q^23 - 10 * q^25 - 12 * q^27 + 2 * q^29 + 22 * q^30 - 26 * q^36 - 38 * q^38 - 2 * q^40 + 22 * q^43 + 38 * q^48 + 8 * q^51 + 16 * q^53 - 30 * q^55 + 10 * q^61 - 82 * q^62 - 2 * q^64 - 68 * q^66 - 22 * q^68 - 14 * q^69 + 66 * q^74 - 2 * q^75 + 70 * q^79 - 28 * q^81 - 10 * q^82 + 20 * q^87 - 28 * q^88 - 66 * q^92 + 2 * q^94 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more