LMFDB - Embedded newform 9808.2.a.f.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9808,2,Mod(1,9808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9808, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9808 = 2^{4} \cdot 613 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9808.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:	$78.3172743025$
Analytic rank:	$1$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 4 x^{16} - 28 x^{15} + 120 x^{14} + 291 x^{13} - 1382 x^{12} - 1398 x^{11} + 7700 x^{10} + \cdots - 320 $ x^17 - 4x^16 - 28x^15 + 120x^14 + 291x^13 - 1382x^12 - 1398x^11 + 7700x^10 + 3456x^9 - 21974x^8 - 6336x^7 + 33062x^6 + 9753x^5 - 24446x^4 - 8368x^3 + 7000x^2 + 2384x - 320
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1226)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-0.926840$ of defining polynomial
Character	$\chi$	$=$	9808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.926840 q^{3} +0.613322 q^{5} +4.30069 q^{7} -2.14097 q^{9} +O(q^{10})$	$q+0.926840 q^{3} +0.613322 q^{5} +4.30069 q^{7} -2.14097 q^{9} +0.812094 q^{11} -5.56948 q^{13} +0.568452 q^{15} -0.247362 q^{17} -5.43006 q^{19} +3.98605 q^{21} +0.410501 q^{23} -4.62384 q^{25} -4.76486 q^{27} +6.26185 q^{29} -8.45616 q^{31} +0.752681 q^{33} +2.63771 q^{35} +3.11260 q^{37} -5.16202 q^{39} +5.83087 q^{41} +4.28708 q^{43} -1.31310 q^{45} +0.735067 q^{47} +11.4959 q^{49} -0.229265 q^{51} +0.537458 q^{53} +0.498075 q^{55} -5.03279 q^{57} -3.05642 q^{59} -12.3973 q^{61} -9.20763 q^{63} -3.41588 q^{65} +6.79764 q^{67} +0.380469 q^{69} -0.183966 q^{71} +8.53553 q^{73} -4.28556 q^{75} +3.49256 q^{77} -14.5278 q^{79} +2.00664 q^{81} +0.337683 q^{83} -0.151713 q^{85} +5.80373 q^{87} -18.0121 q^{89} -23.9526 q^{91} -7.83751 q^{93} -3.33037 q^{95} -0.706123 q^{97} -1.73867 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 4 q^{3} - 5 q^{5} - 7 q^{7} + 21 q^{9}+O(q^{10}) $ 17 * q - 4 * q^3 - 5 * q^5 - 7 * q^7 + 21 * q^9	$ 17 q - 4 q^{3} - 5 q^{5} - 7 q^{7} + 21 q^{9} - 8 q^{11} + 9 q^{13} + 4 q^{15} - q^{17} - 32 q^{19} + 6 q^{21} + 5 q^{23} + 30 q^{25} - 16 q^{27} + 3 q^{29} - 27 q^{31} - 14 q^{33} - 25 q^{35} + 7 q^{37} - 27 q^{39} - 2 q^{41} - 36 q^{43} - q^{45} + 3 q^{47} + 52 q^{49} - 40 q^{51} - 20 q^{53} - 48 q^{55} + 12 q^{57} - 34 q^{59} + 49 q^{61} - 27 q^{63} - 6 q^{65} - 36 q^{67} + 18 q^{69} + q^{71} + 24 q^{73} - 35 q^{75} - 6 q^{77} - 43 q^{79} + 37 q^{81} - 10 q^{83} + 16 q^{85} - 28 q^{87} - 12 q^{89} - 42 q^{91} + 3 q^{93} + 10 q^{95} + 26 q^{97} - 34 q^{99}+O(q^{100}) $ 17 * q - 4 * q^3 - 5 * q^5 - 7 * q^7 + 21 * q^9 - 8 * q^11 + 9 * q^13 + 4 * q^15 - q^17 - 32 * q^19 + 6 * q^21 + 5 * q^23 + 30 * q^25 - 16 * q^27 + 3 * q^29 - 27 * q^31 - 14 * q^33 - 25 * q^35 + 7 * q^37 - 27 * q^39 - 2 * q^41 - 36 * q^43 - q^45 + 3 * q^47 + 52 * q^49 - 40 * q^51 - 20 * q^53 - 48 * q^55 + 12 * q^57 - 34 * q^59 + 49 * q^61 - 27 * q^63 - 6 * q^65 - 36 * q^67 + 18 * q^69 + q^71 + 24 * q^73 - 35 * q^75 - 6 * q^77 - 43 * q^79 + 37 * q^81 - 10 * q^83 + 16 * q^85 - 28 * q^87 - 12 * q^89 - 42 * q^91 + 3 * q^93 + 10 * q^95 + 26 * q^97 - 34 * 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
$3$	0.926840	0.535111	0.267556	−	0.963542i	$-0.413784\pi$
$3$	0.926840	0.535111	0.267556	+	0.963542i	$0.413784\pi$
$4$	0	0
$5$	0.613322	0.274286	0.137143	−	0.990551i	$-0.456208\pi$
$5$	0.613322	0.274286	0.137143	+	0.990551i	$0.456208\pi$
$6$	0	0
$7$	4.30069	1.62551	0.812753	−	0.582608i	$-0.197968\pi$
$7$	4.30069	1.62551	0.812753	+	0.582608i	$0.197968\pi$
$8$	0	0
$9$	−2.14097	−0.713656
$10$	0	0
$11$	0.812094	0.244855	0.122428	−	0.992477i	$-0.460932\pi$
$11$	0.812094	0.244855	0.122428	+	0.992477i	$0.460932\pi$
$12$	0	0
$13$	−5.56948	−1.54470	−0.772348	−	0.635200i	$-0.780918\pi$
$13$	−5.56948	−1.54470	−0.772348	+	0.635200i	$0.780918\pi$
$14$	0	0
$15$	0.568452	0.146774
$16$	0	0
$17$	−0.247362	−0.0599941	−0.0299970	−	0.999550i	$-0.509550\pi$
$17$	−0.247362	−0.0599941	−0.0299970	+	0.999550i	$0.509550\pi$
$18$	0	0
$19$	−5.43006	−1.24574	−0.622870	−	0.782325i	$-0.714034\pi$
$19$	−5.43006	−1.24574	−0.622870	+	0.782325i	$0.714034\pi$
$20$	0	0
$21$	3.98605	0.869827
$22$	0	0
$23$	0.410501	0.0855953	0.0427977	−	0.999084i	$-0.486373\pi$
$23$	0.410501	0.0855953	0.0427977	+	0.999084i	$0.486373\pi$
$24$	0	0
$25$	−4.62384	−0.924767
$26$	0	0
$27$	−4.76486	−0.916997
$28$	0	0
$29$	6.26185	1.16280	0.581398	−	0.813619i	$-0.302506\pi$
$29$	6.26185	1.16280	0.581398	+	0.813619i	$0.302506\pi$
$30$	0	0
$31$	−8.45616	−1.51877	−0.759386	−	0.650641i	$-0.774500\pi$
$31$	−8.45616	−1.51877	−0.759386	+	0.650641i	$0.774500\pi$
$32$	0	0
$33$	0.752681	0.131025
$34$	0	0
$35$	2.63771	0.445854
$36$	0	0
$37$	3.11260	0.511708	0.255854	−	0.966715i	$-0.417643\pi$
$37$	3.11260	0.511708	0.255854	+	0.966715i	$0.417643\pi$
$38$	0	0
$39$	−5.16202	−0.826585
$40$	0	0
$41$	5.83087	0.910629	0.455315	−	0.890331i	$-0.349527\pi$
$41$	5.83087	0.910629	0.455315	+	0.890331i	$0.349527\pi$
$42$	0	0
$43$	4.28708	0.653774	0.326887	−	0.945063i	$-0.394000\pi$
$43$	4.28708	0.653774	0.326887	+	0.945063i	$0.394000\pi$
$44$	0	0
$45$	−1.31310	−0.195746
$46$	0	0
$47$	0.735067	0.107221	0.0536103	−	0.998562i	$-0.482927\pi$
$47$	0.735067	0.107221	0.0536103	+	0.998562i	$0.482927\pi$
$48$	0	0
$49$	11.4959	1.64227
$50$	0	0
$51$	−0.229265	−0.0321035
$52$	0	0
$53$	0.537458	0.0738256	0.0369128	−	0.999318i	$-0.488248\pi$
$53$	0.537458	0.0738256	0.0369128	+	0.999318i	$0.488248\pi$
$54$	0	0
$55$	0.498075	0.0671604
$56$	0	0
$57$	−5.03279	−0.666610
$58$	0	0
$59$	−3.05642	−0.397912	−0.198956	−	0.980008i	$-0.563755\pi$
$59$	−3.05642	−0.397912	−0.198956	+	0.980008i	$0.563755\pi$
$60$	0	0
$61$	−12.3973	−1.58732	−0.793658	−	0.608364i	$-0.791826\pi$
$61$	−12.3973	−1.58732	−0.793658	+	0.608364i	$0.791826\pi$
$62$	0	0
$63$	−9.20763	−1.16005
$64$	0	0
$65$	−3.41588	−0.423688
$66$	0	0
$67$	6.79764	0.830464	0.415232	−	0.909716i	$-0.363700\pi$
$67$	6.79764	0.830464	0.415232	+	0.909716i	$0.363700\pi$
$68$	0	0
$69$	0.380469	0.0458031
$70$	0	0
$71$	−0.183966	−0.0218327	−0.0109164	−	0.999940i	$-0.503475\pi$
$71$	−0.183966	−0.0218327	−0.0109164	+	0.999940i	$0.503475\pi$
$72$	0	0
$73$	8.53553	0.999008	0.499504	−	0.866312i	$-0.333516\pi$
$73$	8.53553	0.999008	0.499504	+	0.866312i	$0.333516\pi$
$74$	0	0
$75$	−4.28556	−0.494854
$76$	0	0
$77$	3.49256	0.398014
$78$	0	0
$79$	−14.5278	−1.63450	−0.817251	−	0.576282i	$-0.804503\pi$
$79$	−14.5278	−1.63450	−0.817251	+	0.576282i	$0.804503\pi$
$80$	0	0
$81$	2.00664	0.222960
$82$	0	0
$83$	0.337683	0.0370655	0.0185328	−	0.999828i	$-0.494101\pi$
$83$	0.337683	0.0370655	0.0185328	+	0.999828i	$0.494101\pi$
$84$	0	0
$85$	−0.151713	−0.0164555
$86$	0	0
$87$	5.80373	0.622226
$88$	0	0
$89$	−18.0121	−1.90928	−0.954638	−	0.297769i	$-0.903758\pi$
$89$	−18.0121	−1.90928	−0.954638	+	0.297769i	$0.903758\pi$
$90$	0	0
$91$	−23.9526	−2.51091
$92$	0	0
$93$	−7.83751	−0.812712
$94$	0	0
$95$	−3.33037	−0.341689
$96$	0	0
$97$	−0.706123	−0.0716960	−0.0358480	−	0.999357i	$-0.511413\pi$
$97$	−0.706123	−0.0716960	−0.0358480	+	0.999357i	$0.511413\pi$
$98$	0	0
$99$	−1.73867	−0.174742
$100$	0	0
$101$	−9.69344	−0.964534	−0.482267	−	0.876024i	$-0.660186\pi$
$101$	−9.69344	−0.964534	−0.482267	+	0.876024i	$0.660186\pi$
$102$	0	0
$103$	−4.50697	−0.444085	−0.222043	−	0.975037i	$-0.571272\pi$
$103$	−4.50697	−0.444085	−0.222043	+	0.975037i	$0.571272\pi$
$104$	0	0
$105$	2.44473	0.238581
$106$	0	0
$107$	−16.8135	−1.62543	−0.812713	−	0.582664i	$-0.802010\pi$
$107$	−16.8135	−1.62543	−0.812713	+	0.582664i	$0.802010\pi$
$108$	0	0
$109$	12.0827	1.15731	0.578656	−	0.815572i	$-0.303577\pi$
$109$	12.0827	1.15731	0.578656	+	0.815572i	$0.303577\pi$
$110$	0	0
$111$	2.88488	0.273821
$112$	0	0
$113$	13.5418	1.27390	0.636952	−	0.770904i	$-0.280195\pi$
$113$	13.5418	1.27390	0.636952	+	0.770904i	$0.280195\pi$
$114$	0	0
$115$	0.251769	0.0234776
$116$	0	0
$117$	11.9241	1.10238
$118$	0	0
$119$	−1.06383	−0.0975208
$120$	0	0
$121$	−10.3405	−0.940046
$122$	0	0
$123$	5.40429	0.487288
$124$	0	0
$125$	−5.90251	−0.527937
$126$	0	0
$127$	−19.6307	−1.74194	−0.870971	−	0.491334i	$-0.836509\pi$
$127$	−19.6307	−1.74194	−0.870971	+	0.491334i	$0.836509\pi$
$128$	0	0
$129$	3.97344	0.349842
$130$	0	0
$131$	−20.6647	−1.80548	−0.902740	−	0.430186i	$-0.858448\pi$
$131$	−20.6647	−1.80548	−0.902740	+	0.430186i	$0.858448\pi$
$132$	0	0
$133$	−23.3530	−2.02496
$134$	0	0
$135$	−2.92239	−0.251519
$136$	0	0
$137$	−9.46216	−0.808407	−0.404203	−	0.914669i	$-0.632451\pi$
$137$	−9.46216	−0.808407	−0.404203	+	0.914669i	$0.632451\pi$
$138$	0	0
$139$	6.86593	0.582361	0.291180	−	0.956668i	$-0.405952\pi$
$139$	6.86593	0.582361	0.291180	+	0.956668i	$0.405952\pi$
$140$	0	0
$141$	0.681290	0.0573749
$142$	0	0
$143$	−4.52294	−0.378227
$144$	0	0
$145$	3.84053	0.318939
$146$	0	0
$147$	10.6549	0.878799
$148$	0	0
$149$	−6.43770	−0.527397	−0.263699	−	0.964605i	$-0.584942\pi$
$149$	−6.43770	−0.527397	−0.263699	+	0.964605i	$0.584942\pi$
$150$	0	0
$151$	−8.34403	−0.679028	−0.339514	−	0.940601i	$-0.610263\pi$
$151$	−8.34403	−0.679028	−0.339514	+	0.940601i	$0.610263\pi$
$152$	0	0
$153$	0.529594	0.0428151
$154$	0	0
$155$	−5.18635	−0.416578
$156$	0	0
$157$	−2.38130	−0.190048	−0.0950241	−	0.995475i	$-0.530293\pi$
$157$	−2.38130	−0.190048	−0.0950241	+	0.995475i	$0.530293\pi$
$158$	0	0
$159$	0.498138	0.0395049
$160$	0	0
$161$	1.76544	0.139136
$162$	0	0
$163$	−16.9060	−1.32418	−0.662090	−	0.749424i	$-0.730330\pi$
$163$	−16.9060	−1.32418	−0.662090	+	0.749424i	$0.730330\pi$
$164$	0	0
$165$	0.461636	0.0359383
$166$	0	0
$167$	17.8651	1.38244	0.691220	−	0.722645i	$-0.257074\pi$
$167$	17.8651	1.38244	0.691220	+	0.722645i	$0.257074\pi$
$168$	0	0
$169$	18.0191	1.38609
$170$	0	0
$171$	11.6256	0.889030
$172$	0	0
$173$	0.837202	0.0636513	0.0318256	−	0.999493i	$-0.489868\pi$
$173$	0.837202	0.0636513	0.0318256	+	0.999493i	$0.489868\pi$
$174$	0	0
$175$	−19.8857	−1.50322
$176$	0	0
$177$	−2.83281	−0.212927
$178$	0	0
$179$	10.9794	0.820637	0.410318	−	0.911942i	$-0.365418\pi$
$179$	10.9794	0.820637	0.410318	+	0.911942i	$0.365418\pi$
$180$	0	0
$181$	−5.16051	−0.383578	−0.191789	−	0.981436i	$-0.561429\pi$
$181$	−5.16051	−0.383578	−0.191789	+	0.981436i	$0.561429\pi$
$182$	0	0
$183$	−11.4904	−0.849391
$184$	0	0
$185$	1.90903	0.140354
$186$	0	0
$187$	−0.200881	−0.0146899
$188$	0	0
$189$	−20.4922	−1.49058
$190$	0	0
$191$	22.8080	1.65033	0.825166	−	0.564890i	$-0.191081\pi$
$191$	22.8080	1.65033	0.825166	+	0.564890i	$0.191081\pi$
$192$	0	0
$193$	24.9859	1.79853	0.899263	−	0.437409i	$-0.144104\pi$
$193$	24.9859	1.79853	0.899263	+	0.437409i	$0.144104\pi$
$194$	0	0
$195$	−3.16598	−0.226721
$196$	0	0
$197$	−5.85052	−0.416832	−0.208416	−	0.978040i	$-0.566831\pi$
$197$	−5.85052	−0.416832	−0.208416	+	0.978040i	$0.566831\pi$
$198$	0	0
$199$	−15.0230	−1.06495	−0.532476	−	0.846445i	$-0.678738\pi$
$199$	−15.0230	−1.06495	−0.532476	+	0.846445i	$0.678738\pi$
$200$	0	0
$201$	6.30033	0.444391
$202$	0	0
$203$	26.9303	1.89013
$204$	0	0
$205$	3.57620	0.249773
$206$	0	0
$207$	−0.878869	−0.0610856
$208$	0	0
$209$	−4.40971	−0.305026
$210$	0	0
$211$	−15.5580	−1.07106	−0.535528	−	0.844517i	$-0.679887\pi$
$211$	−15.5580	−1.07106	−0.535528	+	0.844517i	$0.679887\pi$
$212$	0	0
$213$	−0.170507	−0.0116829
$214$	0	0
$215$	2.62936	0.179321
$216$	0	0
$217$	−36.3673	−2.46877
$218$	0	0
$219$	7.91107	0.534580
$220$	0	0
$221$	1.37768	0.0926726
$222$	0	0
$223$	−7.31293	−0.489710	−0.244855	−	0.969560i	$-0.578740\pi$
$223$	−7.31293	−0.489710	−0.244855	+	0.969560i	$0.578740\pi$
$224$	0	0
$225$	9.89948	0.659965
$226$	0	0
$227$	−7.49736	−0.497617	−0.248809	−	0.968553i	$-0.580039\pi$
$227$	−7.49736	−0.497617	−0.248809	+	0.968553i	$0.580039\pi$
$228$	0	0
$229$	−22.2152	−1.46802	−0.734011	−	0.679138i	$-0.762354\pi$
$229$	−22.2152	−1.46802	−0.734011	+	0.679138i	$0.762354\pi$
$230$	0	0
$231$	3.23705	0.212982
$232$	0	0
$233$	10.7762	0.705974	0.352987	−	0.935628i	$-0.385166\pi$
$233$	10.7762	0.705974	0.352987	+	0.935628i	$0.385166\pi$
$234$	0	0
$235$	0.450833	0.0294091
$236$	0	0
$237$	−13.4649	−0.874641
$238$	0	0
$239$	16.2655	1.05213	0.526065	−	0.850444i	$-0.323667\pi$
$239$	16.2655	1.05213	0.526065	+	0.850444i	$0.323667\pi$
$240$	0	0
$241$	22.9590	1.47892	0.739458	−	0.673203i	$-0.235082\pi$
$241$	22.9590	1.47892	0.739458	+	0.673203i	$0.235082\pi$
$242$	0	0
$243$	16.1544	1.03631
$244$	0	0
$245$	7.05069	0.450452
$246$	0	0
$247$	30.2426	1.92429
$248$	0	0
$249$	0.312978	0.0198342
$250$	0	0
$251$	−4.77513	−0.301404	−0.150702	−	0.988579i	$-0.548153\pi$
$251$	−4.77513	−0.301404	−0.150702	+	0.988579i	$0.548153\pi$
$252$	0	0
$253$	0.333365	0.0209585
$254$	0	0
$255$	−0.140613	−0.00880555
$256$	0	0
$257$	−8.89232	−0.554687	−0.277344	−	0.960771i	$-0.589454\pi$
$257$	−8.89232	−0.554687	−0.277344	+	0.960771i	$0.589454\pi$
$258$	0	0
$259$	13.3863	0.831786
$260$	0	0
$261$	−13.4064	−0.829836
$262$	0	0
$263$	22.7327	1.40176	0.700879	−	0.713281i	$-0.252791\pi$
$263$	22.7327	1.40176	0.700879	+	0.713281i	$0.252791\pi$
$264$	0	0
$265$	0.329635	0.0202493
$266$	0	0
$267$	−16.6943	−1.02168
$268$	0	0
$269$	−15.9281	−0.971152	−0.485576	−	0.874194i	$-0.661390\pi$
$269$	−15.9281	−0.971152	−0.485576	+	0.874194i	$0.661390\pi$
$270$	0	0
$271$	−18.2010	−1.10563	−0.552816	−	0.833303i	$-0.686447\pi$
$271$	−18.2010	−1.10563	−0.552816	+	0.833303i	$0.686447\pi$
$272$	0	0
$273$	−22.2002	−1.34362
$274$	0	0
$275$	−3.75499	−0.226434
$276$	0	0
$277$	−18.0087	−1.08204	−0.541019	−	0.841010i	$-0.681961\pi$
$277$	−18.0087	−1.08204	−0.541019	+	0.841010i	$0.681961\pi$
$278$	0	0
$279$	18.1044	1.08388
$280$	0	0
$281$	−9.93003	−0.592376	−0.296188	−	0.955130i	$-0.595716\pi$
$281$	−9.93003	−0.592376	−0.296188	+	0.955130i	$0.595716\pi$
$282$	0	0
$283$	−14.9457	−0.888432	−0.444216	−	0.895920i	$-0.646518\pi$
$283$	−14.9457	−0.888432	−0.444216	+	0.895920i	$0.646518\pi$
$284$	0	0
$285$	−3.08672	−0.182842
$286$	0	0
$287$	25.0768	1.48023
$288$	0	0
$289$	−16.9388	−0.996401
$290$	0	0
$291$	−0.654464	−0.0383653
$292$	0	0
$293$	0.705749	0.0412303	0.0206152	−	0.999787i	$-0.493438\pi$
$293$	0.705749	0.0412303	0.0206152	+	0.999787i	$0.493438\pi$
$294$	0	0
$295$	−1.87457	−0.109142
$296$	0	0
$297$	−3.86951	−0.224532
$298$	0	0
$299$	−2.28628	−0.132219
$300$	0	0
$301$	18.4374	1.06271
$302$	0	0
$303$	−8.98427	−0.516133
$304$	0	0
$305$	−7.60356	−0.435379
$306$	0	0
$307$	3.38803	0.193365	0.0966827	−	0.995315i	$-0.469177\pi$
$307$	3.38803	0.193365	0.0966827	+	0.995315i	$0.469177\pi$
$308$	0	0
$309$	−4.17724	−0.237635
$310$	0	0
$311$	−8.16099	−0.462767	−0.231384	−	0.972863i	$-0.574325\pi$
$311$	−8.16099	−0.462767	−0.231384	+	0.972863i	$0.574325\pi$
$312$	0	0
$313$	−3.17712	−0.179582	−0.0897908	−	0.995961i	$-0.528620\pi$
$313$	−3.17712	−0.179582	−0.0897908	+	0.995961i	$0.528620\pi$
$314$	0	0
$315$	−5.64724	−0.318186
$316$	0	0
$317$	23.8445	1.33924	0.669621	−	0.742703i	$-0.266456\pi$
$317$	23.8445	1.33924	0.669621	+	0.742703i	$0.266456\pi$
$318$	0	0
$319$	5.08521	0.284717
$320$	0	0
$321$	−15.5835	−0.869784
$322$	0	0
$323$	1.34319	0.0747371
$324$	0	0
$325$	25.7524	1.42848
$326$	0	0
$327$	11.1987	0.619291
$328$	0	0
$329$	3.16129	0.174288
$330$	0	0
$331$	−18.6849	−1.02702	−0.513508	−	0.858085i	$-0.671654\pi$
$331$	−18.6849	−1.02702	−0.513508	+	0.858085i	$0.671654\pi$
$332$	0	0
$333$	−6.66398	−0.365184
$334$	0	0
$335$	4.16914	0.227785
$336$	0	0
$337$	−6.75341	−0.367882	−0.183941	−	0.982937i	$-0.558885\pi$
$337$	−6.75341	−0.367882	−0.183941	+	0.982937i	$0.558885\pi$
$338$	0	0
$339$	12.5511	0.681680
$340$	0	0
$341$	−6.86719	−0.371879
$342$	0	0
$343$	19.3355	1.04402
$344$	0	0
$345$	0.233350	0.0125631
$346$	0	0
$347$	−11.4668	−0.615570	−0.307785	−	0.951456i	$-0.599588\pi$
$347$	−11.4668	−0.615570	−0.307785	+	0.951456i	$0.599588\pi$
$348$	0	0
$349$	28.5903	1.53040	0.765202	−	0.643790i	$-0.222639\pi$
$349$	28.5903	1.53040	0.765202	+	0.643790i	$0.222639\pi$
$350$	0	0
$351$	26.5378	1.41648
$352$	0	0
$353$	27.9111	1.48556	0.742779	−	0.669537i	$-0.233507\pi$
$353$	27.9111	1.48556	0.742779	+	0.669537i	$0.233507\pi$
$354$	0	0
$355$	−0.112830	−0.00598840
$356$	0	0
$357$	−0.985997	−0.0521845
$358$	0	0
$359$	−34.8379	−1.83867	−0.919336	−	0.393474i	$-0.871273\pi$
$359$	−34.8379	−1.83867	−0.919336	+	0.393474i	$0.871273\pi$
$360$	0	0
$361$	10.4855	0.551869
$362$	0	0
$363$	−9.58400	−0.503029
$364$	0	0
$365$	5.23503	0.274014
$366$	0	0
$367$	21.7737	1.13658	0.568288	−	0.822829i	$-0.307606\pi$
$367$	21.7737	1.13658	0.568288	+	0.822829i	$0.307606\pi$
$368$	0	0
$369$	−12.4837	−0.649876
$370$	0	0
$371$	2.31144	0.120004
$372$	0	0
$373$	15.3161	0.793035	0.396518	−	0.918027i	$-0.370219\pi$
$373$	15.3161	0.793035	0.396518	+	0.918027i	$0.370219\pi$
$374$	0	0
$375$	−5.47068	−0.282505
$376$	0	0
$377$	−34.8752	−1.79617
$378$	0	0
$379$	−4.14465	−0.212897	−0.106448	−	0.994318i	$-0.533948\pi$
$379$	−4.14465	−0.212897	−0.106448	+	0.994318i	$0.533948\pi$
$380$	0	0
$381$	−18.1945	−0.932133
$382$	0	0
$383$	−9.14276	−0.467173	−0.233587	−	0.972336i	$-0.575046\pi$
$383$	−9.14276	−0.467173	−0.233587	+	0.972336i	$0.575046\pi$
$384$	0	0
$385$	2.14206	0.109170
$386$	0	0
$387$	−9.17850	−0.466570
$388$	0	0
$389$	−23.7310	−1.20321	−0.601605	−	0.798793i	$-0.705472\pi$
$389$	−23.7310	−1.20321	−0.601605	+	0.798793i	$0.705472\pi$
$390$	0	0
$391$	−0.101542	−0.00513522
$392$	0	0
$393$	−19.1528	−0.966133
$394$	0	0
$395$	−8.91020	−0.448321
$396$	0	0
$397$	8.35856	0.419504	0.209752	−	0.977755i	$-0.432734\pi$
$397$	8.35856	0.419504	0.209752	+	0.977755i	$0.432734\pi$
$398$	0	0
$399$	−21.6445	−1.08358
$400$	0	0
$401$	25.3323	1.26503	0.632517	−	0.774546i	$-0.282022\pi$
$401$	25.3323	1.26503	0.632517	+	0.774546i	$0.282022\pi$
$402$	0	0
$403$	47.0964	2.34604
$404$	0	0
$405$	1.23072	0.0611548
$406$	0	0
$407$	2.52772	0.125295
$408$	0	0
$409$	−3.67900	−0.181915	−0.0909575	−	0.995855i	$-0.528993\pi$
$409$	−3.67900	−0.181915	−0.0909575	+	0.995855i	$0.528993\pi$
$410$	0	0
$411$	−8.76991	−0.432588
$412$	0	0
$413$	−13.1447	−0.646808
$414$	0	0
$415$	0.207108	0.0101666
$416$	0	0
$417$	6.36362	0.311628
$418$	0	0
$419$	−34.8876	−1.70437	−0.852185	−	0.523240i	$-0.824723\pi$
$419$	−34.8876	−1.70437	−0.852185	+	0.523240i	$0.824723\pi$
$420$	0	0
$421$	19.8674	0.968279	0.484139	−	0.874991i	$-0.339133\pi$
$421$	19.8674	0.968279	0.484139	+	0.874991i	$0.339133\pi$
$422$	0	0
$423$	−1.57375	−0.0765186
$424$	0	0
$425$	1.14376	0.0554806
$426$	0	0
$427$	−53.3171	−2.58019
$428$	0	0
$429$	−4.19204	−0.202394
$430$	0	0
$431$	38.4689	1.85298	0.926491	−	0.376316i	$-0.122809\pi$
$431$	38.4689	1.85298	0.926491	+	0.376316i	$0.122809\pi$
$432$	0	0
$433$	24.0036	1.15354	0.576769	−	0.816907i	$-0.304313\pi$
$433$	24.0036	1.15354	0.576769	+	0.816907i	$0.304313\pi$
$434$	0	0
$435$	3.55956	0.170668
$436$	0	0
$437$	−2.22904	−0.106630
$438$	0	0
$439$	−29.3726	−1.40188	−0.700938	−	0.713222i	$-0.747235\pi$
$439$	−29.3726	−1.40188	−0.700938	+	0.713222i	$0.747235\pi$
$440$	0	0
$441$	−24.6124	−1.17202
$442$	0	0
$443$	2.48716	0.118168	0.0590842	−	0.998253i	$-0.481182\pi$
$443$	2.48716	0.118168	0.0590842	+	0.998253i	$0.481182\pi$
$444$	0	0
$445$	−11.0472	−0.523688
$446$	0	0
$447$	−5.96672	−0.282216
$448$	0	0
$449$	−23.8800	−1.12697	−0.563483	−	0.826128i	$-0.690539\pi$
$449$	−23.8800	−1.12697	−0.563483	+	0.826128i	$0.690539\pi$
$450$	0	0
$451$	4.73522	0.222973
$452$	0	0
$453$	−7.73358	−0.363355
$454$	0	0
$455$	−14.6907	−0.688708
$456$	0	0
$457$	3.36625	0.157466	0.0787332	−	0.996896i	$-0.474912\pi$
$457$	3.36625	0.157466	0.0787332	+	0.996896i	$0.474912\pi$
$458$	0	0
$459$	1.17864	0.0550144
$460$	0	0
$461$	27.7553	1.29270	0.646348	−	0.763043i	$-0.276296\pi$
$461$	27.7553	1.29270	0.646348	+	0.763043i	$0.276296\pi$
$462$	0	0
$463$	28.8211	1.33943	0.669714	−	0.742619i	$-0.266417\pi$
$463$	28.8211	1.33943	0.669714	+	0.742619i	$0.266417\pi$
$464$	0	0
$465$	−4.80692	−0.222915
$466$	0	0
$467$	18.1652	0.840583	0.420292	−	0.907389i	$-0.361928\pi$
$467$	18.1652	0.840583	0.420292	+	0.907389i	$0.361928\pi$
$468$	0	0
$469$	29.2345	1.34992
$470$	0	0
$471$	−2.20708	−0.101697
$472$	0	0
$473$	3.48151	0.160080
$474$	0	0
$475$	25.1077	1.15202
$476$	0	0
$477$	−1.15068	−0.0526860
$478$	0	0
$479$	36.4654	1.66615	0.833074	−	0.553162i	$-0.186579\pi$
$479$	36.4654	1.66615	0.833074	+	0.553162i	$0.186579\pi$
$480$	0	0
$481$	−17.3356	−0.790434
$482$	0	0
$483$	1.63628	0.0744532
$484$	0	0
$485$	−0.433081	−0.0196652
$486$	0	0
$487$	32.8608	1.48906	0.744532	−	0.667587i	$-0.232673\pi$
$487$	32.8608	1.48906	0.744532	+	0.667587i	$0.232673\pi$
$488$	0	0
$489$	−15.6692	−0.708584
$490$	0	0
$491$	11.8666	0.535532	0.267766	−	0.963484i	$-0.413715\pi$
$491$	11.8666	0.535532	0.267766	+	0.963484i	$0.413715\pi$
$492$	0	0
$493$	−1.54894	−0.0697609
$494$	0	0
$495$	−1.06636	−0.0479294
$496$	0	0
$497$	−0.791179	−0.0354892
$498$	0	0
$499$	3.50427	0.156873	0.0784363	−	0.996919i	$-0.475007\pi$
$499$	3.50427	0.156873	0.0784363	+	0.996919i	$0.475007\pi$
$500$	0	0
$501$	16.5581	0.739759
$502$	0	0
$503$	11.4687	0.511364	0.255682	−	0.966761i	$-0.417700\pi$
$503$	11.4687	0.511364	0.255682	+	0.966761i	$0.417700\pi$
$504$	0	0
$505$	−5.94520	−0.264558
$506$	0	0
$507$	16.7008	0.741710
$508$	0	0
$509$	4.30197	0.190682	0.0953408	−	0.995445i	$-0.469606\pi$
$509$	4.30197	0.190682	0.0953408	+	0.995445i	$0.469606\pi$
$510$	0	0
$511$	36.7086	1.62389
$512$	0	0
$513$	25.8734	1.14234
$514$	0	0
$515$	−2.76423	−0.121806
$516$	0	0
$517$	0.596943	0.0262535
$518$	0	0
$519$	0.775953	0.0340605
$520$	0	0
$521$	30.7835	1.34865	0.674326	−	0.738434i	$-0.264434\pi$
$521$	30.7835	1.34865	0.674326	+	0.738434i	$0.264434\pi$
$522$	0	0
$523$	−13.9326	−0.609229	−0.304614	−	0.952476i	$-0.598528\pi$
$523$	−13.9326	−0.609229	−0.304614	+	0.952476i	$0.598528\pi$
$524$	0	0
$525$	−18.4308	−0.804388
$526$	0	0
$527$	2.09173	0.0911173
$528$	0	0
$529$	−22.8315	−0.992673
$530$	0	0
$531$	6.54369	0.283972
$532$	0	0
$533$	−32.4749	−1.40665
$534$	0	0
$535$	−10.3121	−0.445832
$536$	0	0
$537$	10.1761	0.439132
$538$	0	0
$539$	9.33575	0.402119
$540$	0	0
$541$	1.54382	0.0663739	0.0331869	−	0.999449i	$-0.489434\pi$
$541$	1.54382	0.0663739	0.0331869	+	0.999449i	$0.489434\pi$
$542$	0	0
$543$	−4.78297	−0.205257
$544$	0	0
$545$	7.41058	0.317434
$546$	0	0
$547$	8.68580	0.371378	0.185689	−	0.982609i	$-0.440548\pi$
$547$	8.68580	0.371378	0.185689	+	0.982609i	$0.440548\pi$
$548$	0	0
$549$	26.5423	1.13280
$550$	0	0
$551$	−34.0022	−1.44854
$552$	0	0
$553$	−62.4794	−2.65689
$554$	0	0
$555$	1.76936	0.0751053
$556$	0	0
$557$	−34.4895	−1.46137	−0.730684	−	0.682716i	$-0.760799\pi$
$557$	−34.4895	−1.46137	−0.730684	+	0.682716i	$0.760799\pi$
$558$	0	0
$559$	−23.8768	−1.00988
$560$	0	0
$561$	−0.186185	−0.00786072
$562$	0	0
$563$	−29.9981	−1.26427	−0.632134	−	0.774859i	$-0.717821\pi$
$563$	−29.9981	−1.26427	−0.632134	+	0.774859i	$0.717821\pi$
$564$	0	0
$565$	8.30547	0.349414
$566$	0	0
$567$	8.62994	0.362423
$568$	0	0
$569$	21.7035	0.909859	0.454929	−	0.890527i	$-0.349664\pi$
$569$	21.7035	0.909859	0.454929	+	0.890527i	$0.349664\pi$
$570$	0	0
$571$	−0.0233525	−0.000977271 0	−0.000488635	−	1.00000i	$-0.500156\pi$
$571$	−0.0233525	−0.000977271 0	−0.000488635		1.00000i	$0.500156\pi$
$572$	0	0
$573$	21.1394	0.883112
$574$	0	0
$575$	−1.89809	−0.0791558
$576$	0	0
$577$	5.73179	0.238618	0.119309	−	0.992857i	$-0.461932\pi$
$577$	5.73179	0.238618	0.119309	+	0.992857i	$0.461932\pi$
$578$	0	0
$579$	23.1580	0.962412
$580$	0	0
$581$	1.45227	0.0602503
$582$	0	0
$583$	0.436467	0.0180766
$584$	0	0
$585$	7.31330	0.302368
$586$	0	0
$587$	−15.4386	−0.637219	−0.318609	−	0.947886i	$-0.603216\pi$
$587$	−15.4386	−0.637219	−0.318609	+	0.947886i	$0.603216\pi$
$588$	0	0
$589$	45.9174	1.89199
$590$	0	0
$591$	−5.42250	−0.223052
$592$	0	0
$593$	−5.38781	−0.221251	−0.110625	−	0.993862i	$-0.535285\pi$
$593$	−5.38781	−0.221251	−0.110625	+	0.993862i	$0.535285\pi$
$594$	0	0
$595$	−0.652468	−0.0267486
$596$	0	0
$597$	−13.9239	−0.569868
$598$	0	0
$599$	2.26577	0.0925768	0.0462884	−	0.998928i	$-0.485261\pi$
$599$	2.26577	0.0925768	0.0462884	+	0.998928i	$0.485261\pi$
$600$	0	0
$601$	−11.0329	−0.450043	−0.225021	−	0.974354i	$-0.572245\pi$
$601$	−11.0329	−0.450043	−0.225021	+	0.974354i	$0.572245\pi$
$602$	0	0
$603$	−14.5535	−0.592665
$604$	0	0
$605$	−6.34206	−0.257841
$606$	0	0
$607$	−23.2240	−0.942634	−0.471317	−	0.881964i	$-0.656221\pi$
$607$	−23.2240	−0.942634	−0.471317	+	0.881964i	$0.656221\pi$
$608$	0	0
$609$	24.9600	1.01143
$610$	0	0
$611$	−4.09394	−0.165623
$612$	0	0
$613$	1.00000	0.0403896
$613$	1.00000	0.0403896
$614$	0	0
$615$	3.31457	0.133656
$616$	0	0
$617$	−16.3274	−0.657318	−0.328659	−	0.944449i	$-0.606597\pi$
$617$	−16.3274	−0.657318	−0.328659	+	0.944449i	$0.606597\pi$
$618$	0	0
$619$	25.9989	1.04499	0.522493	−	0.852644i	$-0.325002\pi$
$619$	25.9989	1.04499	0.522493	+	0.852644i	$0.325002\pi$
$620$	0	0
$621$	−1.95598	−0.0784907
$622$	0	0
$623$	−77.4643	−3.10354
$624$	0	0
$625$	19.4990	0.779962
$626$	0	0
$627$	−4.08710	−0.163223
$628$	0	0
$629$	−0.769939	−0.0306995
$630$	0	0
$631$	−26.3784	−1.05011	−0.525054	−	0.851069i	$-0.675955\pi$
$631$	−26.3784	−1.05011	−0.525054	+	0.851069i	$0.675955\pi$
$632$	0	0
$633$	−14.4198	−0.573134
$634$	0	0
$635$	−12.0399	−0.477790
$636$	0	0
$637$	−64.0262	−2.53681
$638$	0	0
$639$	0.393864	0.0155810
$640$	0	0
$641$	−21.7817	−0.860324	−0.430162	−	0.902752i	$-0.641544\pi$
$641$	−21.7817	−0.860324	−0.430162	+	0.902752i	$0.641544\pi$
$642$	0	0
$643$	−0.624181	−0.0246153	−0.0123077	−	0.999924i	$-0.503918\pi$
$643$	−0.624181	−0.0246153	−0.0123077	+	0.999924i	$0.503918\pi$
$644$	0	0
$645$	2.43700	0.0959567
$646$	0	0
$647$	−47.2241	−1.85657	−0.928285	−	0.371871i	$-0.878717\pi$
$647$	−47.2241	−1.85657	−0.928285	+	0.371871i	$0.878717\pi$
$648$	0	0
$649$	−2.48210	−0.0974309
$650$	0	0
$651$	−33.7067	−1.32107
$652$	0	0
$653$	21.3764	0.836523	0.418261	−	0.908327i	$-0.362640\pi$
$653$	21.3764	0.836523	0.418261	+	0.908327i	$0.362640\pi$
$654$	0	0
$655$	−12.6741	−0.495218
$656$	0	0
$657$	−18.2743	−0.712948
$658$	0	0
$659$	−41.4420	−1.61435	−0.807176	−	0.590311i	$-0.799005\pi$
$659$	−41.4420	−1.61435	−0.807176	+	0.590311i	$0.799005\pi$
$660$	0	0
$661$	−1.74064	−0.0677029	−0.0338514	−	0.999427i	$-0.510777\pi$
$661$	−1.74064	−0.0677029	−0.0338514	+	0.999427i	$0.510777\pi$
$662$	0	0
$663$	1.27689	0.0495902
$664$	0	0
$665$	−14.3229	−0.555418
$666$	0	0
$667$	2.57049	0.0995299
$668$	0	0
$669$	−6.77792	−0.262049
$670$	0	0
$671$	−10.0678	−0.388663
$672$	0	0
$673$	25.3091	0.975595	0.487798	−	0.872957i	$-0.337800\pi$
$673$	25.3091	0.975595	0.487798	+	0.872957i	$0.337800\pi$
$674$	0	0
$675$	22.0319	0.848009
$676$	0	0
$677$	10.9018	0.418989	0.209494	−	0.977810i	$-0.432818\pi$
$677$	10.9018	0.418989	0.209494	+	0.977810i	$0.432818\pi$
$678$	0	0
$679$	−3.03682	−0.116542
$680$	0	0
$681$	−6.94885	−0.266281
$682$	0	0
$683$	−20.2808	−0.776024	−0.388012	−	0.921654i	$-0.626838\pi$
$683$	−20.2808	−0.776024	−0.388012	+	0.921654i	$0.626838\pi$
$684$	0	0
$685$	−5.80335	−0.221735
$686$	0	0
$687$	−20.5899	−0.785555
$688$	0	0
$689$	−2.99336	−0.114038
$690$	0	0
$691$	−13.0102	−0.494933	−0.247466	−	0.968896i	$-0.579598\pi$
$691$	−13.0102	−0.494933	−0.247466	+	0.968896i	$0.579598\pi$
$692$	0	0
$693$	−7.47746	−0.284045
$694$	0	0
$695$	4.21103	0.159733
$696$	0	0
$697$	−1.44234	−0.0546324
$698$	0	0
$699$	9.98783	0.377775
$700$	0	0
$701$	34.4344	1.30057	0.650284	−	0.759691i	$-0.274650\pi$
$701$	34.4344	1.30057	0.650284	+	0.759691i	$0.274650\pi$
$702$	0	0
$703$	−16.9016	−0.637456
$704$	0	0
$705$	0.417850	0.0157371
$706$	0	0
$707$	−41.6885	−1.56786
$708$	0	0
$709$	−10.9346	−0.410657	−0.205328	−	0.978693i	$-0.565826\pi$
$709$	−10.9346	−0.410657	−0.205328	+	0.978693i	$0.565826\pi$
$710$	0	0
$711$	31.1035	1.16647
$712$	0	0
$713$	−3.47126	−0.130000
$714$	0	0
$715$	−2.77402	−0.103742
$716$	0	0
$717$	15.0756	0.563007
$718$	0	0
$719$	−33.5358	−1.25067	−0.625337	−	0.780355i	$-0.715038\pi$
$719$	−33.5358	−1.25067	−0.625337	+	0.780355i	$0.715038\pi$
$720$	0	0
$721$	−19.3831	−0.721864
$722$	0	0
$723$	21.2793	0.791385
$724$	0	0
$725$	−28.9538	−1.07532
$726$	0	0
$727$	−20.4936	−0.760064	−0.380032	−	0.924973i	$-0.624087\pi$
$727$	−20.4936	−0.760064	−0.380032	+	0.924973i	$0.624087\pi$
$728$	0	0
$729$	8.95263	0.331579
$730$	0	0
$731$	−1.06046	−0.0392226
$732$	0	0
$733$	23.5872	0.871213	0.435606	−	0.900137i	$-0.356534\pi$
$733$	23.5872	0.871213	0.435606	+	0.900137i	$0.356534\pi$
$734$	0	0
$735$	6.53487	0.241042
$736$	0	0
$737$	5.52032	0.203344
$738$	0	0
$739$	2.84419	0.104625	0.0523126	−	0.998631i	$-0.483341\pi$
$739$	2.84419	0.104625	0.0523126	+	0.998631i	$0.483341\pi$
$740$	0	0
$741$	28.0301	1.02971
$742$	0	0
$743$	36.0648	1.32309	0.661544	−	0.749906i	$-0.269901\pi$
$743$	36.0648	1.32309	0.661544	+	0.749906i	$0.269901\pi$
$744$	0	0
$745$	−3.94838	−0.144658
$746$	0	0
$747$	−0.722968	−0.0264520
$748$	0	0
$749$	−72.3098	−2.64214
$750$	0	0
$751$	32.8806	1.19983	0.599914	−	0.800064i	$-0.295201\pi$
$751$	32.8806	1.19983	0.599914	+	0.800064i	$0.295201\pi$
$752$	0	0
$753$	−4.42579	−0.161285
$754$	0	0
$755$	−5.11758	−0.186248
$756$	0	0
$757$	6.78162	0.246482	0.123241	−	0.992377i	$-0.460671\pi$
$757$	6.78162	0.246482	0.123241	+	0.992377i	$0.460671\pi$
$758$	0	0
$759$	0.308976	0.0112151
$760$	0	0
$761$	−20.5159	−0.743700	−0.371850	−	0.928293i	$-0.621276\pi$
$761$	−20.5159	−0.743700	−0.371850	+	0.928293i	$0.621276\pi$
$762$	0	0
$763$	51.9639	1.88122
$764$	0	0
$765$	0.324812	0.0117436
$766$	0	0
$767$	17.0227	0.614653
$768$	0	0
$769$	−28.5132	−1.02821	−0.514106	−	0.857727i	$-0.671876\pi$
$769$	−28.5132	−1.02821	−0.514106	+	0.857727i	$0.671876\pi$
$770$	0	0
$771$	−8.24176	−0.296820
$772$	0	0
$773$	12.2489	0.440561	0.220280	−	0.975437i	$-0.429303\pi$
$773$	12.2489	0.440561	0.220280	+	0.975437i	$0.429303\pi$
$774$	0	0
$775$	39.0999	1.40451
$776$	0	0
$777$	12.4070	0.445098
$778$	0	0
$779$	−31.6620	−1.13441
$780$	0	0
$781$	−0.149397	−0.00534586
$782$	0	0
$783$	−29.8368	−1.06628
$784$	0	0
$785$	−1.46050	−0.0521275
$786$	0	0
$787$	18.9134	0.674189	0.337095	−	0.941471i	$-0.390556\pi$
$787$	18.9134	0.674189	0.337095	+	0.941471i	$0.390556\pi$
$788$	0	0
$789$	21.0696	0.750096
$790$	0	0
$791$	58.2390	2.07074
$792$	0	0
$793$	69.0467	2.45192
$794$	0	0
$795$	0.305519	0.0108356
$796$	0	0
$797$	−24.4745	−0.866932	−0.433466	−	0.901170i	$-0.642710\pi$
$797$	−24.4745	−0.866932	−0.433466	+	0.901170i	$0.642710\pi$
$798$	0	0
$799$	−0.181828	−0.00643260
$800$	0	0
$801$	38.5633	1.36257
$802$	0	0
$803$	6.93165	0.244612
$804$	0	0
$805$	1.08278	0.0381630
$806$	0	0
$807$	−14.7628	−0.519675
$808$	0	0
$809$	15.0958	0.530742	0.265371	−	0.964146i	$-0.414506\pi$
$809$	15.0958	0.530742	0.265371	+	0.964146i	$0.414506\pi$
$810$	0	0
$811$	−8.01418	−0.281416	−0.140708	−	0.990051i	$-0.544938\pi$
$811$	−8.01418	−0.281416	−0.140708	+	0.990051i	$0.544938\pi$
$812$	0	0
$813$	−16.8694	−0.591637
$814$	0	0
$815$	−10.3688	−0.363204
$816$	0	0
$817$	−23.2791	−0.814433
$818$	0	0
$819$	51.2817	1.79193
$820$	0	0
$821$	39.5232	1.37937	0.689685	−	0.724109i	$-0.257749\pi$
$821$	39.5232	1.37937	0.689685	+	0.724109i	$0.257749\pi$
$822$	0	0
$823$	13.0704	0.455605	0.227803	−	0.973707i	$-0.426846\pi$
$823$	13.0704	0.455605	0.227803	+	0.973707i	$0.426846\pi$
$824$	0	0
$825$	−3.48027	−0.121168
$826$	0	0
$827$	12.0694	0.419695	0.209848	−	0.977734i	$-0.432703\pi$
$827$	12.0694	0.419695	0.209848	+	0.977734i	$0.432703\pi$
$828$	0	0
$829$	15.2448	0.529475	0.264737	−	0.964321i	$-0.414715\pi$
$829$	15.2448	0.529475	0.264737	+	0.964321i	$0.414715\pi$
$830$	0	0
$831$	−16.6912	−0.579011
$832$	0	0
$833$	−2.84365	−0.0985267
$834$	0	0
$835$	10.9570	0.379184
$836$	0	0
$837$	40.2924	1.39271
$838$	0	0
$839$	2.81736	0.0972662	0.0486331	−	0.998817i	$-0.484513\pi$
$839$	2.81736	0.0972662	0.0486331	+	0.998817i	$0.484513\pi$
$840$	0	0
$841$	10.2107	0.352095
$842$	0	0
$843$	−9.20356	−0.316987
$844$	0	0
$845$	11.0515	0.380184
$846$	0	0
$847$	−44.4713	−1.52805
$848$	0	0
$849$	−13.8523	−0.475410
$850$	0	0
$851$	1.27773	0.0437999
$852$	0	0
$853$	10.0437	0.343891	0.171946	−	0.985106i	$-0.444995\pi$
$853$	10.0437	0.343891	0.171946	+	0.985106i	$0.444995\pi$
$854$	0	0
$855$	7.13022	0.243848
$856$	0	0
$857$	−37.5879	−1.28398	−0.641989	−	0.766714i	$-0.721891\pi$
$857$	−37.5879	−1.28398	−0.641989	+	0.766714i	$0.721891\pi$
$858$	0	0
$859$	−21.0944	−0.719732	−0.359866	−	0.933004i	$-0.617178\pi$
$859$	−21.0944	−0.719732	−0.359866	+	0.933004i	$0.617178\pi$
$860$	0	0
$861$	23.2422	0.792090
$862$	0	0
$863$	−54.9243	−1.86964	−0.934822	−	0.355117i	$-0.884441\pi$
$863$	−54.9243	−1.86964	−0.934822	+	0.355117i	$0.884441\pi$
$864$	0	0
$865$	0.513474	0.0174587
$866$	0	0
$867$	−15.6996	−0.533185
$868$	0	0
$869$	−11.7979	−0.400217
$870$	0	0
$871$	−37.8593	−1.28281
$872$	0	0
$873$	1.51179	0.0511662
$874$	0	0
$875$	−25.3849	−0.858165
$876$	0	0
$877$	24.9954	0.844035	0.422017	−	0.906588i	$-0.361322\pi$
$877$	24.9954	0.844035	0.422017	+	0.906588i	$0.361322\pi$
$878$	0	0
$879$	0.654117	0.0220628
$880$	0	0
$881$	−5.61754	−0.189260	−0.0946298	−	0.995513i	$-0.530167\pi$
$881$	−5.61754	−0.189260	−0.0946298	+	0.995513i	$0.530167\pi$
$882$	0	0
$883$	3.79477	0.127704	0.0638521	−	0.997959i	$-0.479661\pi$
$883$	3.79477	0.127704	0.0638521	+	0.997959i	$0.479661\pi$
$884$	0	0
$885$	−1.73743	−0.0584029
$886$	0	0
$887$	−30.5307	−1.02512	−0.512560	−	0.858651i	$-0.671303\pi$
$887$	−30.5307	−1.02512	−0.512560	+	0.858651i	$0.671303\pi$
$888$	0	0
$889$	−84.4254	−2.83154
$890$	0	0
$891$	1.62958	0.0545930
$892$	0	0
$893$	−3.99146	−0.133569
$894$	0	0
$895$	6.73389	0.225089
$896$	0	0
$897$	−2.11901	−0.0707518
$898$	0	0
$899$	−52.9512	−1.76602
$900$	0	0
$901$	−0.132947	−0.00442910
$902$	0	0
$903$	17.0885	0.568671
$904$	0	0
$905$	−3.16506	−0.105210
$906$	0	0
$907$	21.2662	0.706133	0.353067	−	0.935598i	$-0.385139\pi$
$907$	21.2662	0.706133	0.353067	+	0.935598i	$0.385139\pi$
$908$	0	0
$909$	20.7533	0.688345
$910$	0	0
$911$	13.8890	0.460164	0.230082	−	0.973171i	$-0.426101\pi$
$911$	13.8890	0.460164	0.230082	+	0.973171i	$0.426101\pi$
$912$	0	0
$913$	0.274230	0.00907570
$914$	0	0
$915$	−7.04728	−0.232976
$916$	0	0
$917$	−88.8723	−2.93482
$918$	0	0
$919$	32.7378	1.07992	0.539960	−	0.841691i	$-0.318439\pi$
$919$	32.7378	1.07992	0.539960	+	0.841691i	$0.318439\pi$
$920$	0	0
$921$	3.14017	0.103472
$922$	0	0
$923$	1.02459	0.0337249
$924$	0	0
$925$	−14.3922	−0.473211
$926$	0	0
$927$	9.64928	0.316924
$928$	0	0
$929$	7.79703	0.255812	0.127906	−	0.991786i	$-0.459174\pi$
$929$	7.79703	0.255812	0.127906	+	0.991786i	$0.459174\pi$
$930$	0	0
$931$	−62.4234	−2.04585
$932$	0	0
$933$	−7.56394	−0.247632
$934$	0	0
$935$	−0.123205	−0.00402923
$936$	0	0
$937$	−7.89602	−0.257952	−0.128976	−	0.991648i	$-0.541169\pi$
$937$	−7.89602	−0.257952	−0.128976	+	0.991648i	$0.541169\pi$
$938$	0	0
$939$	−2.94469	−0.0960962
$940$	0	0
$941$	−40.8241	−1.33083	−0.665414	−	0.746475i	$-0.731745\pi$
$941$	−40.8241	−1.33083	−0.665414	+	0.746475i	$0.731745\pi$
$942$	0	0
$943$	2.39358	0.0779456
$944$	0	0
$945$	−12.5683	−0.408846
$946$	0	0
$947$	45.4382	1.47654	0.738272	−	0.674503i	$-0.235642\pi$
$947$	45.4382	1.47654	0.738272	+	0.674503i	$0.235642\pi$
$948$	0	0
$949$	−47.5384	−1.54316
$950$	0	0
$951$	22.1001	0.716644
$952$	0	0
$953$	−31.4201	−1.01780	−0.508898	−	0.860827i	$-0.669947\pi$
$953$	−31.4201	−1.01780	−0.508898	+	0.860827i	$0.669947\pi$
$954$	0	0
$955$	13.9887	0.452663
$956$	0	0
$957$	4.71318	0.152355
$958$	0	0
$959$	−40.6938	−1.31407
$960$	0	0
$961$	40.5066	1.30667
$962$	0	0
$963$	35.9972	1.15999
$964$	0	0
$965$	15.3244	0.493310
$966$	0	0
$967$	−14.8848	−0.478664	−0.239332	−	0.970938i	$-0.576928\pi$
$967$	−14.8848	−0.478664	−0.239332	+	0.970938i	$0.576928\pi$
$968$	0	0
$969$	1.24492	0.0399927
$970$	0	0
$971$	24.6156	0.789951	0.394976	−	0.918692i	$-0.370753\pi$
$971$	24.6156	0.789951	0.394976	+	0.918692i	$0.370753\pi$
$972$	0	0
$973$	29.5282	0.946631
$974$	0	0
$975$	23.8683	0.764398
$976$	0	0
$977$	17.9162	0.573190	0.286595	−	0.958052i	$-0.407477\pi$
$977$	17.9162	0.573190	0.286595	+	0.958052i	$0.407477\pi$
$978$	0	0
$979$	−14.6275	−0.467497
$980$	0	0
$981$	−25.8686	−0.825922
$982$	0	0
$983$	59.3441	1.89278	0.946392	−	0.323022i	$-0.104699\pi$
$983$	59.3441	1.89278	0.946392	+	0.323022i	$0.104699\pi$
$984$	0	0
$985$	−3.58825	−0.114331
$986$	0	0
$987$	2.93001	0.0932634
$988$	0	0
$989$	1.75985	0.0559600
$990$	0	0
$991$	18.3182	0.581897	0.290948	−	0.956739i	$-0.406029\pi$
$991$	18.3182	0.581897	0.290948	+	0.956739i	$0.406029\pi$
$992$	0	0
$993$	−17.3179	−0.549568
$994$	0	0
$995$	−9.21394	−0.292102
$996$	0	0
$997$	11.4964	0.364095	0.182047	−	0.983290i	$-0.441728\pi$
$997$	11.4964	0.364095	0.182047	+	0.983290i	$0.441728\pi$
$998$	0	0
$999$	−14.8311	−0.469235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9808.2.a.f.1.11		17
4.3	odd	2		1226.2.a.e.1.7	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1226.2.a.e.1.7	✓	17	4.3	odd	2
9808.2.a.f.1.11		17	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:	\( N \)	\(=\)	\( 9808 = 2^{4} \cdot 613 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9808.a (trivial)

\(f(q)\)	\(=\)	\(q+0.926840 q^{3} +0.613322 q^{5} +4.30069 q^{7} -2.14097 q^{9} +O(q^{10})\)	\(q+0.926840 q^{3} +0.613322 q^{5} +4.30069 q^{7} -2.14097 q^{9} +0.812094 q^{11} -5.56948 q^{13} +0.568452 q^{15} -0.247362 q^{17} -5.43006 q^{19} +3.98605 q^{21} +0.410501 q^{23} -4.62384 q^{25} -4.76486 q^{27} +6.26185 q^{29} -8.45616 q^{31} +0.752681 q^{33} +2.63771 q^{35} +3.11260 q^{37} -5.16202 q^{39} +5.83087 q^{41} +4.28708 q^{43} -1.31310 q^{45} +0.735067 q^{47} +11.4959 q^{49} -0.229265 q^{51} +0.537458 q^{53} +0.498075 q^{55} -5.03279 q^{57} -3.05642 q^{59} -12.3973 q^{61} -9.20763 q^{63} -3.41588 q^{65} +6.79764 q^{67} +0.380469 q^{69} -0.183966 q^{71} +8.53553 q^{73} -4.28556 q^{75} +3.49256 q^{77} -14.5278 q^{79} +2.00664 q^{81} +0.337683 q^{83} -0.151713 q^{85} +5.80373 q^{87} -18.0121 q^{89} -23.9526 q^{91} -7.83751 q^{93} -3.33037 q^{95} -0.706123 q^{97} -1.73867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 4 q^{3} - 5 q^{5} - 7 q^{7} + 21 q^{9}+O(q^{10}) \) 17 * q - 4 * q^3 - 5 * q^5 - 7 * q^7 + 21 * q^9	\( 17 q - 4 q^{3} - 5 q^{5} - 7 q^{7} + 21 q^{9} - 8 q^{11} + 9 q^{13} + 4 q^{15} - q^{17} - 32 q^{19} + 6 q^{21} + 5 q^{23} + 30 q^{25} - 16 q^{27} + 3 q^{29} - 27 q^{31} - 14 q^{33} - 25 q^{35} + 7 q^{37} - 27 q^{39} - 2 q^{41} - 36 q^{43} - q^{45} + 3 q^{47} + 52 q^{49} - 40 q^{51} - 20 q^{53} - 48 q^{55} + 12 q^{57} - 34 q^{59} + 49 q^{61} - 27 q^{63} - 6 q^{65} - 36 q^{67} + 18 q^{69} + q^{71} + 24 q^{73} - 35 q^{75} - 6 q^{77} - 43 q^{79} + 37 q^{81} - 10 q^{83} + 16 q^{85} - 28 q^{87} - 12 q^{89} - 42 q^{91} + 3 q^{93} + 10 q^{95} + 26 q^{97} - 34 q^{99}+O(q^{100}) \) 17 * q - 4 * q^3 - 5 * q^5 - 7 * q^7 + 21 * q^9 - 8 * q^11 + 9 * q^13 + 4 * q^15 - q^17 - 32 * q^19 + 6 * q^21 + 5 * q^23 + 30 * q^25 - 16 * q^27 + 3 * q^29 - 27 * q^31 - 14 * q^33 - 25 * q^35 + 7 * q^37 - 27 * q^39 - 2 * q^41 - 36 * q^43 - q^45 + 3 * q^47 + 52 * q^49 - 40 * q^51 - 20 * q^53 - 48 * q^55 + 12 * q^57 - 34 * q^59 + 49 * q^61 - 27 * q^63 - 6 * q^65 - 36 * q^67 + 18 * q^69 + q^71 + 24 * q^73 - 35 * q^75 - 6 * q^77 - 43 * q^79 + 37 * q^81 - 10 * q^83 + 16 * q^85 - 28 * q^87 - 12 * q^89 - 42 * q^91 + 3 * q^93 + 10 * q^95 + 26 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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