LMFDB - Embedded newform 4368.2.a.bq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.77846 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.77846 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.49828 q^{11} -1.00000 q^{13} +1.77846 q^{15} -2.94137 q^{17} +4.83709 q^{19} +1.00000 q^{21} +5.77846 q^{23} -1.83709 q^{25} +1.00000 q^{27} -2.83709 q^{29} -6.27674 q^{31} +6.49828 q^{33} +1.77846 q^{35} +9.55691 q^{37} -1.00000 q^{39} -3.05863 q^{41} -2.71982 q^{43} +1.77846 q^{45} +8.71982 q^{47} +1.00000 q^{49} -2.94137 q^{51} +6.39400 q^{53} +11.5569 q^{55} +4.83709 q^{57} -1.55691 q^{59} +3.88273 q^{61} +1.00000 q^{63} -1.77846 q^{65} -5.67418 q^{67} +5.77846 q^{69} -10.0552 q^{71} -15.8337 q^{73} -1.83709 q^{75} +6.49828 q^{77} +1.28018 q^{79} +1.00000 q^{81} +2.83709 q^{83} -5.23109 q^{85} -2.83709 q^{87} -7.66119 q^{89} -1.00000 q^{91} -6.27674 q^{93} +8.60256 q^{95} -17.7164 q^{97} +6.49828 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 3 q^{15} - 8 q^{17} + 7 q^{19} + 3 q^{21} + 9 q^{23} + 2 q^{25} + 3 q^{27} - q^{29} + 7 q^{31} + 2 q^{33} - 3 q^{35} + 12 q^{37} - 3 q^{39} - 10 q^{41} + q^{43} - 3 q^{45} + 17 q^{47} + 3 q^{49} - 8 q^{51} - 5 q^{53} + 18 q^{55} + 7 q^{57} + 12 q^{59} + 10 q^{61} + 3 q^{63} + 3 q^{65} - 2 q^{67} + 9 q^{69} + 4 q^{71} - 5 q^{73} + 2 q^{75} + 2 q^{77} + 13 q^{79} + 3 q^{81} + q^{83} + 16 q^{85} - q^{87} - 13 q^{89} - 3 q^{91} + 7 q^{93} + 15 q^{95} - 9 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 3 * q^15 - 8 * q^17 + 7 * q^19 + 3 * q^21 + 9 * q^23 + 2 * q^25 + 3 * q^27 - q^29 + 7 * q^31 + 2 * q^33 - 3 * q^35 + 12 * q^37 - 3 * q^39 - 10 * q^41 + q^43 - 3 * q^45 + 17 * q^47 + 3 * q^49 - 8 * q^51 - 5 * q^53 + 18 * q^55 + 7 * q^57 + 12 * q^59 + 10 * q^61 + 3 * q^63 + 3 * q^65 - 2 * q^67 + 9 * q^69 + 4 * q^71 - 5 * q^73 + 2 * q^75 + 2 * q^77 + 13 * q^79 + 3 * q^81 + q^83 + 16 * q^85 - q^87 - 13 * q^89 - 3 * q^91 + 7 * q^93 + 15 * q^95 - 9 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.77846	0.795350	0.397675	−	0.917526i	$-0.369817\pi$
$5$	1.77846	0.795350	0.397675	+	0.917526i	$0.369817\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.49828	1.95931	0.979653	−	0.200700i	$-0.0643217\pi$
$11$	6.49828	1.95931	0.979653	+	0.200700i	$0.0643217\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.77846	0.459196
$16$	0	0
$17$	−2.94137	−0.713386	−0.356693	−	0.934222i	$-0.616096\pi$
$17$	−2.94137	−0.713386	−0.356693	+	0.934222i	$0.616096\pi$
$18$	0	0
$19$	4.83709	1.10970	0.554852	−	0.831949i	$-0.312775\pi$
$19$	4.83709	1.10970	0.554852	+	0.831949i	$0.312775\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	5.77846	1.20489	0.602446	−	0.798160i	$-0.294193\pi$
$23$	5.77846	1.20489	0.602446	+	0.798160i	$0.294193\pi$
$24$	0	0
$25$	−1.83709	−0.367418
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.83709	−0.526834	−0.263417	−	0.964682i	$-0.584850\pi$
$29$	−2.83709	−0.526834	−0.263417	+	0.964682i	$0.584850\pi$
$30$	0	0
$31$	−6.27674	−1.12734	−0.563668	−	0.826002i	$-0.690610\pi$
$31$	−6.27674	−1.12734	−0.563668	+	0.826002i	$0.690610\pi$
$32$	0	0
$33$	6.49828	1.13121
$34$	0	0
$35$	1.77846	0.300614
$36$	0	0
$37$	9.55691	1.57115	0.785574	−	0.618768i	$-0.212368\pi$
$37$	9.55691	1.57115	0.785574	+	0.618768i	$0.212368\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−3.05863	−0.477678	−0.238839	−	0.971059i	$-0.576767\pi$
$41$	−3.05863	−0.477678	−0.238839	+	0.971059i	$0.576767\pi$
$42$	0	0
$43$	−2.71982	−0.414769	−0.207385	−	0.978259i	$-0.566495\pi$
$43$	−2.71982	−0.414769	−0.207385	+	0.978259i	$0.566495\pi$
$44$	0	0
$45$	1.77846	0.265117
$46$	0	0
$47$	8.71982	1.27192	0.635959	−	0.771723i	$-0.280605\pi$
$47$	8.71982	1.27192	0.635959	+	0.771723i	$0.280605\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.94137	−0.411874
$52$	0	0
$53$	6.39400	0.878284	0.439142	−	0.898418i	$-0.355282\pi$
$53$	6.39400	0.878284	0.439142	+	0.898418i	$0.355282\pi$
$54$	0	0
$55$	11.5569	1.55833
$56$	0	0
$57$	4.83709	0.640688
$58$	0	0
$59$	−1.55691	−0.202693	−0.101346	−	0.994851i	$-0.532315\pi$
$59$	−1.55691	−0.202693	−0.101346	+	0.994851i	$0.532315\pi$
$60$	0	0
$61$	3.88273	0.497133	0.248567	−	0.968615i	$-0.420041\pi$
$61$	3.88273	0.497133	0.248567	+	0.968615i	$0.420041\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.77846	−0.220590
$66$	0	0
$67$	−5.67418	−0.693211	−0.346606	−	0.938011i	$-0.612666\pi$
$67$	−5.67418	−0.693211	−0.346606	+	0.938011i	$0.612666\pi$
$68$	0	0
$69$	5.77846	0.695644
$70$	0	0
$71$	−10.0552	−1.19333	−0.596666	−	0.802490i	$-0.703508\pi$
$71$	−10.0552	−1.19333	−0.596666	+	0.802490i	$0.703508\pi$
$72$	0	0
$73$	−15.8337	−1.85319	−0.926594	−	0.376062i	$-0.877278\pi$
$73$	−15.8337	−1.85319	−0.926594	+	0.376062i	$0.877278\pi$
$74$	0	0
$75$	−1.83709	−0.212129
$76$	0	0
$77$	6.49828	0.740548
$78$	0	0
$79$	1.28018	0.144031	0.0720155	−	0.997404i	$-0.477057\pi$
$79$	1.28018	0.144031	0.0720155	+	0.997404i	$0.477057\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.83709	0.311411	0.155706	−	0.987804i	$-0.450235\pi$
$83$	2.83709	0.311411	0.155706	+	0.987804i	$0.450235\pi$
$84$	0	0
$85$	−5.23109	−0.567392
$86$	0	0
$87$	−2.83709	−0.304168
$88$	0	0
$89$	−7.66119	−0.812085	−0.406042	−	0.913854i	$-0.633091\pi$
$89$	−7.66119	−0.812085	−0.406042	+	0.913854i	$0.633091\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−6.27674	−0.650867
$94$	0	0
$95$	8.60256	0.882604
$96$	0	0
$97$	−17.7164	−1.79883	−0.899413	−	0.437099i	$-0.856006\pi$
$97$	−17.7164	−1.79883	−0.899413	+	0.437099i	$0.856006\pi$
$98$	0	0
$99$	6.49828	0.653102
$100$	0	0
$101$	6.73281	0.669940	0.334970	−	0.942229i	$-0.391274\pi$
$101$	6.73281	0.669940	0.334970	+	0.942229i	$0.391274\pi$
$102$	0	0
$103$	−10.8793	−1.07197	−0.535984	−	0.844228i	$-0.680059\pi$
$103$	−10.8793	−1.07197	−0.535984	+	0.844228i	$0.680059\pi$
$104$	0	0
$105$	1.77846	0.173560
$106$	0	0
$107$	8.49828	0.821560	0.410780	−	0.911735i	$-0.365256\pi$
$107$	8.49828	0.821560	0.410780	+	0.911735i	$0.365256\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	9.55691	0.907102
$112$	0	0
$113$	14.6026	1.37369	0.686847	−	0.726802i	$-0.258994\pi$
$113$	14.6026	1.37369	0.686847	+	0.726802i	$0.258994\pi$
$114$	0	0
$115$	10.2767	0.958311
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−2.94137	−0.269635
$120$	0	0
$121$	31.2277	2.83888
$122$	0	0
$123$	−3.05863	−0.275788
$124$	0	0
$125$	−12.1595	−1.08758
$126$	0	0
$127$	9.88273	0.876951	0.438475	−	0.898743i	$-0.355519\pi$
$127$	9.88273	0.876951	0.438475	+	0.898743i	$0.355519\pi$
$128$	0	0
$129$	−2.71982	−0.239467
$130$	0	0
$131$	−3.76547	−0.328990	−0.164495	−	0.986378i	$-0.552600\pi$
$131$	−3.76547	−0.328990	−0.164495	+	0.986378i	$0.552600\pi$
$132$	0	0
$133$	4.83709	0.419429
$134$	0	0
$135$	1.77846	0.153065
$136$	0	0
$137$	−16.9966	−1.45211	−0.726057	−	0.687634i	$-0.758649\pi$
$137$	−16.9966	−1.45211	−0.726057	+	0.687634i	$0.758649\pi$
$138$	0	0
$139$	8.55348	0.725496	0.362748	−	0.931887i	$-0.381839\pi$
$139$	8.55348	0.725496	0.362748	+	0.931887i	$0.381839\pi$
$140$	0	0
$141$	8.71982	0.734342
$142$	0	0
$143$	−6.49828	−0.543414
$144$	0	0
$145$	−5.04564	−0.419018
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−15.5569	−1.27447	−0.637236	−	0.770669i	$-0.719922\pi$
$149$	−15.5569	−1.27447	−0.637236	+	0.770669i	$0.719922\pi$
$150$	0	0
$151$	−4.99656	−0.406614	−0.203307	−	0.979115i	$-0.565169\pi$
$151$	−4.99656	−0.406614	−0.203307	+	0.979115i	$0.565169\pi$
$152$	0	0
$153$	−2.94137	−0.237795
$154$	0	0
$155$	−11.1629	−0.896626
$156$	0	0
$157$	−18.7880	−1.49945	−0.749723	−	0.661752i	$-0.769813\pi$
$157$	−18.7880	−1.49945	−0.749723	+	0.661752i	$0.769813\pi$
$158$	0	0
$159$	6.39400	0.507078
$160$	0	0
$161$	5.77846	0.455406
$162$	0	0
$163$	9.88273	0.774075	0.387038	−	0.922064i	$-0.373498\pi$
$163$	9.88273	0.774075	0.387038	+	0.922064i	$0.373498\pi$
$164$	0	0
$165$	11.5569	0.899705
$166$	0	0
$167$	7.04564	0.545208	0.272604	−	0.962126i	$-0.412115\pi$
$167$	7.04564	0.545208	0.272604	+	0.962126i	$0.412115\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.83709	0.369902
$172$	0	0
$173$	−14.2897	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$173$	−14.2897	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$174$	0	0
$175$	−1.83709	−0.138871
$176$	0	0
$177$	−1.55691	−0.117025
$178$	0	0
$179$	8.65775	0.647111	0.323555	−	0.946209i	$-0.395122\pi$
$179$	8.65775	0.647111	0.323555	+	0.946209i	$0.395122\pi$
$180$	0	0
$181$	26.3449	1.95820	0.979101	−	0.203373i	$-0.0651904\pi$
$181$	26.3449	1.95820	0.979101	+	0.203373i	$0.0651904\pi$
$182$	0	0
$183$	3.88273	0.287020
$184$	0	0
$185$	16.9966	1.24961
$186$	0	0
$187$	−19.1138	−1.39774
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−6.61555	−0.478684	−0.239342	−	0.970935i	$-0.576932\pi$
$191$	−6.61555	−0.478684	−0.239342	+	0.970935i	$0.576932\pi$
$192$	0	0
$193$	11.8827	0.855338	0.427669	−	0.903935i	$-0.359335\pi$
$193$	11.8827	0.855338	0.427669	+	0.903935i	$0.359335\pi$
$194$	0	0
$195$	−1.77846	−0.127358
$196$	0	0
$197$	11.5569	0.823396	0.411698	−	0.911320i	$-0.364936\pi$
$197$	11.5569	0.823396	0.411698	+	0.911320i	$0.364936\pi$
$198$	0	0
$199$	−0.996562	−0.0706444	−0.0353222	−	0.999376i	$-0.511246\pi$
$199$	−0.996562	−0.0706444	−0.0353222	+	0.999376i	$0.511246\pi$
$200$	0	0
$201$	−5.67418	−0.400226
$202$	0	0
$203$	−2.83709	−0.199125
$204$	0	0
$205$	−5.43965	−0.379921
$206$	0	0
$207$	5.77846	0.401631
$208$	0	0
$209$	31.4328	2.17425
$210$	0	0
$211$	−26.8302	−1.84707	−0.923534	−	0.383516i	$-0.874713\pi$
$211$	−26.8302	−1.84707	−0.923534	+	0.383516i	$0.874713\pi$
$212$	0	0
$213$	−10.0552	−0.688971
$214$	0	0
$215$	−4.83709	−0.329887
$216$	0	0
$217$	−6.27674	−0.426093
$218$	0	0
$219$	−15.8337	−1.06994
$220$	0	0
$221$	2.94137	0.197858
$222$	0	0
$223$	2.92838	0.196099	0.0980493	−	0.995182i	$-0.468740\pi$
$223$	2.92838	0.196099	0.0980493	+	0.995182i	$0.468740\pi$
$224$	0	0
$225$	−1.83709	−0.122473
$226$	0	0
$227$	9.79145	0.649881	0.324941	−	0.945734i	$-0.394656\pi$
$227$	9.79145	0.649881	0.324941	+	0.945734i	$0.394656\pi$
$228$	0	0
$229$	3.88273	0.256578	0.128289	−	0.991737i	$-0.459051\pi$
$229$	3.88273	0.256578	0.128289	+	0.991737i	$0.459051\pi$
$230$	0	0
$231$	6.49828	0.427556
$232$	0	0
$233$	−15.8337	−1.03730	−0.518649	−	0.854988i	$-0.673565\pi$
$233$	−15.8337	−1.03730	−0.518649	+	0.854988i	$0.673565\pi$
$234$	0	0
$235$	15.5078	1.01162
$236$	0	0
$237$	1.28018	0.0831564
$238$	0	0
$239$	6.94137	0.449000	0.224500	−	0.974474i	$-0.427925\pi$
$239$	6.94137	0.449000	0.224500	+	0.974474i	$0.427925\pi$
$240$	0	0
$241$	7.28018	0.468957	0.234479	−	0.972121i	$-0.424662\pi$
$241$	7.28018	0.468957	0.234479	+	0.972121i	$0.424662\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.77846	0.113621
$246$	0	0
$247$	−4.83709	−0.307777
$248$	0	0
$249$	2.83709	0.179793
$250$	0	0
$251$	−23.3224	−1.47210	−0.736048	−	0.676930i	$-0.763310\pi$
$251$	−23.3224	−1.47210	−0.736048	+	0.676930i	$0.763310\pi$
$252$	0	0
$253$	37.5500	2.36075
$254$	0	0
$255$	−5.23109	−0.327584
$256$	0	0
$257$	15.4948	0.966542	0.483271	−	0.875471i	$-0.339449\pi$
$257$	15.4948	0.966542	0.483271	+	0.875471i	$0.339449\pi$
$258$	0	0
$259$	9.55691	0.593838
$260$	0	0
$261$	−2.83709	−0.175611
$262$	0	0
$263$	−3.42666	−0.211297	−0.105648	−	0.994404i	$-0.533692\pi$
$263$	−3.42666	−0.211297	−0.105648	+	0.994404i	$0.533692\pi$
$264$	0	0
$265$	11.3715	0.698543
$266$	0	0
$267$	−7.66119	−0.468857
$268$	0	0
$269$	−0.172462	−0.0105152	−0.00525759	−	0.999986i	$-0.501674\pi$
$269$	−0.172462	−0.0105152	−0.00525759	+	0.999986i	$0.501674\pi$
$270$	0	0
$271$	25.9931	1.57897	0.789485	−	0.613770i	$-0.210348\pi$
$271$	25.9931	1.57897	0.789485	+	0.613770i	$0.210348\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	−11.9379	−0.719884
$276$	0	0
$277$	3.72326	0.223709	0.111855	−	0.993725i	$-0.464321\pi$
$277$	3.72326	0.223709	0.111855	+	0.993725i	$0.464321\pi$
$278$	0	0
$279$	−6.27674	−0.375778
$280$	0	0
$281$	−21.5500	−1.28557	−0.642784	−	0.766048i	$-0.722221\pi$
$281$	−21.5500	−1.28557	−0.642784	+	0.766048i	$0.722221\pi$
$282$	0	0
$283$	31.8759	1.89482	0.947412	−	0.320018i	$-0.103689\pi$
$283$	31.8759	1.89482	0.947412	+	0.320018i	$0.103689\pi$
$284$	0	0
$285$	8.60256	0.509572
$286$	0	0
$287$	−3.05863	−0.180545
$288$	0	0
$289$	−8.34836	−0.491080
$290$	0	0
$291$	−17.7164	−1.03855
$292$	0	0
$293$	3.42666	0.200188	0.100094	−	0.994978i	$-0.468086\pi$
$293$	3.42666	0.200188	0.100094	+	0.994978i	$0.468086\pi$
$294$	0	0
$295$	−2.76891	−0.161212
$296$	0	0
$297$	6.49828	0.377069
$298$	0	0
$299$	−5.77846	−0.334177
$300$	0	0
$301$	−2.71982	−0.156768
$302$	0	0
$303$	6.73281	0.386790
$304$	0	0
$305$	6.90528	0.395395
$306$	0	0
$307$	3.39744	0.193902	0.0969511	−	0.995289i	$-0.469091\pi$
$307$	3.39744	0.193902	0.0969511	+	0.995289i	$0.469091\pi$
$308$	0	0
$309$	−10.8793	−0.618902
$310$	0	0
$311$	0.443086	0.0251251	0.0125625	−	0.999921i	$-0.496001\pi$
$311$	0.443086	0.0251251	0.0125625	+	0.999921i	$0.496001\pi$
$312$	0	0
$313$	−19.8827	−1.12384	−0.561919	−	0.827192i	$-0.689937\pi$
$313$	−19.8827	−1.12384	−0.561919	+	0.827192i	$0.689937\pi$
$314$	0	0
$315$	1.77846	0.100205
$316$	0	0
$317$	9.46563	0.531643	0.265821	−	0.964022i	$-0.414357\pi$
$317$	9.46563	0.531643	0.265821	+	0.964022i	$0.414357\pi$
$318$	0	0
$319$	−18.4362	−1.03223
$320$	0	0
$321$	8.49828	0.474328
$322$	0	0
$323$	−14.2277	−0.791648
$324$	0	0
$325$	1.83709	0.101903
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	8.71982	0.480739
$330$	0	0
$331$	27.4328	1.50784	0.753921	−	0.656965i	$-0.228160\pi$
$331$	27.4328	1.50784	0.753921	+	0.656965i	$0.228160\pi$
$332$	0	0
$333$	9.55691	0.523716
$334$	0	0
$335$	−10.0913	−0.551346
$336$	0	0
$337$	−20.2767	−1.10454	−0.552272	−	0.833664i	$-0.686239\pi$
$337$	−20.2767	−1.10454	−0.552272	+	0.833664i	$0.686239\pi$
$338$	0	0
$339$	14.6026	0.793102
$340$	0	0
$341$	−40.7880	−2.20879
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	10.2767	0.553281
$346$	0	0
$347$	−16.7328	−0.898265	−0.449132	−	0.893465i	$-0.648267\pi$
$347$	−16.7328	−0.898265	−0.449132	+	0.893465i	$0.648267\pi$
$348$	0	0
$349$	22.3940	1.19872	0.599362	−	0.800478i	$-0.295421\pi$
$349$	22.3940	1.19872	0.599362	+	0.800478i	$0.295421\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−27.1690	−1.44606	−0.723031	−	0.690816i	$-0.757251\pi$
$353$	−27.1690	−1.44606	−0.723031	+	0.690816i	$0.757251\pi$
$354$	0	0
$355$	−17.8827	−0.949117
$356$	0	0
$357$	−2.94137	−0.155674
$358$	0	0
$359$	29.0586	1.53366	0.766828	−	0.641853i	$-0.221834\pi$
$359$	29.0586	1.53366	0.766828	+	0.641853i	$0.221834\pi$
$360$	0	0
$361$	4.39744	0.231444
$362$	0	0
$363$	31.2277	1.63903
$364$	0	0
$365$	−28.1595	−1.47393
$366$	0	0
$367$	4.44309	0.231927	0.115964	−	0.993253i	$-0.463004\pi$
$367$	4.44309	0.231927	0.115964	+	0.993253i	$0.463004\pi$
$368$	0	0
$369$	−3.05863	−0.159226
$370$	0	0
$371$	6.39400	0.331960
$372$	0	0
$373$	31.3415	1.62280	0.811400	−	0.584491i	$-0.198706\pi$
$373$	31.3415	1.62280	0.811400	+	0.584491i	$0.198706\pi$
$374$	0	0
$375$	−12.1595	−0.627912
$376$	0	0
$377$	2.83709	0.146118
$378$	0	0
$379$	11.5569	0.593639	0.296819	−	0.954934i	$-0.404074\pi$
$379$	11.5569	0.593639	0.296819	+	0.954934i	$0.404074\pi$
$380$	0	0
$381$	9.88273	0.506308
$382$	0	0
$383$	25.6673	1.31154	0.655769	−	0.754962i	$-0.272345\pi$
$383$	25.6673	1.31154	0.655769	+	0.754962i	$0.272345\pi$
$384$	0	0
$385$	11.5569	0.588995
$386$	0	0
$387$	−2.71982	−0.138256
$388$	0	0
$389$	−24.2277	−1.22839	−0.614195	−	0.789154i	$-0.710519\pi$
$389$	−24.2277	−1.22839	−0.614195	+	0.789154i	$0.710519\pi$
$390$	0	0
$391$	−16.9966	−0.859553
$392$	0	0
$393$	−3.76547	−0.189943
$394$	0	0
$395$	2.27674	0.114555
$396$	0	0
$397$	−14.3680	−0.721111	−0.360555	−	0.932738i	$-0.617413\pi$
$397$	−14.3680	−0.721111	−0.360555	+	0.932738i	$0.617413\pi$
$398$	0	0
$399$	4.83709	0.242157
$400$	0	0
$401$	−21.8827	−1.09277	−0.546386	−	0.837534i	$-0.683997\pi$
$401$	−21.8827	−1.09277	−0.546386	+	0.837534i	$0.683997\pi$
$402$	0	0
$403$	6.27674	0.312667
$404$	0	0
$405$	1.77846	0.0883722
$406$	0	0
$407$	62.1035	3.07836
$408$	0	0
$409$	7.39057	0.365440	0.182720	−	0.983165i	$-0.441510\pi$
$409$	7.39057	0.365440	0.182720	+	0.983165i	$0.441510\pi$
$410$	0	0
$411$	−16.9966	−0.838379
$412$	0	0
$413$	−1.55691	−0.0766107
$414$	0	0
$415$	5.04564	0.247681
$416$	0	0
$417$	8.55348	0.418866
$418$	0	0
$419$	−33.7846	−1.65048	−0.825242	−	0.564779i	$-0.808961\pi$
$419$	−33.7846	−1.65048	−0.825242	+	0.564779i	$0.808961\pi$
$420$	0	0
$421$	−35.5500	−1.73260	−0.866301	−	0.499522i	$-0.833509\pi$
$421$	−35.5500	−1.73260	−0.866301	+	0.499522i	$0.833509\pi$
$422$	0	0
$423$	8.71982	0.423972
$424$	0	0
$425$	5.40356	0.262111
$426$	0	0
$427$	3.88273	0.187899
$428$	0	0
$429$	−6.49828	−0.313740
$430$	0	0
$431$	−23.7294	−1.14300	−0.571502	−	0.820601i	$-0.693639\pi$
$431$	−23.7294	−1.14300	−0.571502	+	0.820601i	$0.693639\pi$
$432$	0	0
$433$	−5.55691	−0.267048	−0.133524	−	0.991046i	$-0.542629\pi$
$433$	−5.55691	−0.267048	−0.133524	+	0.991046i	$0.542629\pi$
$434$	0	0
$435$	−5.04564	−0.241920
$436$	0	0
$437$	27.9509	1.33707
$438$	0	0
$439$	−5.32926	−0.254352	−0.127176	−	0.991880i	$-0.540591\pi$
$439$	−5.32926	−0.254352	−0.127176	+	0.991880i	$0.540591\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−5.33537	−0.253491	−0.126746	−	0.991935i	$-0.540453\pi$
$443$	−5.33537	−0.253491	−0.126746	+	0.991935i	$0.540453\pi$
$444$	0	0
$445$	−13.6251	−0.645892
$446$	0	0
$447$	−15.5569	−0.735817
$448$	0	0
$449$	26.2277	1.23776	0.618880	−	0.785486i	$-0.287587\pi$
$449$	26.2277	1.23776	0.618880	+	0.785486i	$0.287587\pi$
$450$	0	0
$451$	−19.8759	−0.935918
$452$	0	0
$453$	−4.99656	−0.234759
$454$	0	0
$455$	−1.77846	−0.0833754
$456$	0	0
$457$	11.4396	0.535124	0.267562	−	0.963541i	$-0.413782\pi$
$457$	11.4396	0.535124	0.267562	+	0.963541i	$0.413782\pi$
$458$	0	0
$459$	−2.94137	−0.137291
$460$	0	0
$461$	−28.0552	−1.30666	−0.653330	−	0.757073i	$-0.726629\pi$
$461$	−28.0552	−1.30666	−0.653330	+	0.757073i	$0.726629\pi$
$462$	0	0
$463$	10.5604	0.490781	0.245391	−	0.969424i	$-0.421084\pi$
$463$	10.5604	0.490781	0.245391	+	0.969424i	$0.421084\pi$
$464$	0	0
$465$	−11.1629	−0.517668
$466$	0	0
$467$	−16.6776	−0.771748	−0.385874	−	0.922551i	$-0.626100\pi$
$467$	−16.6776	−0.771748	−0.385874	+	0.922551i	$0.626100\pi$
$468$	0	0
$469$	−5.67418	−0.262009
$470$	0	0
$471$	−18.7880	−0.865706
$472$	0	0
$473$	−17.6742	−0.812660
$474$	0	0
$475$	−8.88617	−0.407726
$476$	0	0
$477$	6.39400	0.292761
$478$	0	0
$479$	−4.06819	−0.185880	−0.0929401	−	0.995672i	$-0.529626\pi$
$479$	−4.06819	−0.185880	−0.0929401	+	0.995672i	$0.529626\pi$
$480$	0	0
$481$	−9.55691	−0.435758
$482$	0	0
$483$	5.77846	0.262929
$484$	0	0
$485$	−31.5078	−1.43070
$486$	0	0
$487$	0.443086	0.0200781	0.0100391	−	0.999950i	$-0.496804\pi$
$487$	0.443086	0.0200781	0.0100391	+	0.999950i	$0.496804\pi$
$488$	0	0
$489$	9.88273	0.446913
$490$	0	0
$491$	−20.7328	−0.935659	−0.467829	−	0.883819i	$-0.654964\pi$
$491$	−20.7328	−0.935659	−0.467829	+	0.883819i	$0.654964\pi$
$492$	0	0
$493$	8.34492	0.375836
$494$	0	0
$495$	11.5569	0.519445
$496$	0	0
$497$	−10.0552	−0.451037
$498$	0	0
$499$	13.9931	0.626418	0.313209	−	0.949684i	$-0.398596\pi$
$499$	13.9931	0.626418	0.313209	+	0.949684i	$0.398596\pi$
$500$	0	0
$501$	7.04564	0.314776
$502$	0	0
$503$	−5.67418	−0.252999	−0.126500	−	0.991967i	$-0.540374\pi$
$503$	−5.67418	−0.252999	−0.126500	+	0.991967i	$0.540374\pi$
$504$	0	0
$505$	11.9740	0.532837
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−6.22154	−0.275765	−0.137883	−	0.990449i	$-0.544030\pi$
$509$	−6.22154	−0.275765	−0.137883	+	0.990449i	$0.544030\pi$
$510$	0	0
$511$	−15.8337	−0.700440
$512$	0	0
$513$	4.83709	0.213563
$514$	0	0
$515$	−19.3484	−0.852591
$516$	0	0
$517$	56.6639	2.49207
$518$	0	0
$519$	−14.2897	−0.627249
$520$	0	0
$521$	−10.2897	−0.450801	−0.225401	−	0.974266i	$-0.572369\pi$
$521$	−10.2897	−0.450801	−0.225401	+	0.974266i	$0.572369\pi$
$522$	0	0
$523$	2.32582	0.101701	0.0508505	−	0.998706i	$-0.483807\pi$
$523$	2.32582	0.101701	0.0508505	+	0.998706i	$0.483807\pi$
$524$	0	0
$525$	−1.83709	−0.0801772
$526$	0	0
$527$	18.4622	0.804226
$528$	0	0
$529$	10.3906	0.451764
$530$	0	0
$531$	−1.55691	−0.0675643
$532$	0	0
$533$	3.05863	0.132484
$534$	0	0
$535$	15.1138	0.653428
$536$	0	0
$537$	8.65775	0.373610
$538$	0	0
$539$	6.49828	0.279901
$540$	0	0
$541$	5.55691	0.238910	0.119455	−	0.992840i	$-0.461885\pi$
$541$	5.55691	0.238910	0.119455	+	0.992840i	$0.461885\pi$
$542$	0	0
$543$	26.3449	1.13057
$544$	0	0
$545$	17.7846	0.761807
$546$	0	0
$547$	5.48873	0.234681	0.117341	−	0.993092i	$-0.462563\pi$
$547$	5.48873	0.234681	0.117341	+	0.993092i	$0.462563\pi$
$548$	0	0
$549$	3.88273	0.165711
$550$	0	0
$551$	−13.7233	−0.584631
$552$	0	0
$553$	1.28018	0.0544386
$554$	0	0
$555$	16.9966	0.721464
$556$	0	0
$557$	−22.9897	−0.974104	−0.487052	−	0.873373i	$-0.661928\pi$
$557$	−22.9897	−0.974104	−0.487052	+	0.873373i	$0.661928\pi$
$558$	0	0
$559$	2.71982	0.115036
$560$	0	0
$561$	−19.1138	−0.806986
$562$	0	0
$563$	32.7620	1.38075	0.690377	−	0.723449i	$-0.257444\pi$
$563$	32.7620	1.38075	0.690377	+	0.723449i	$0.257444\pi$
$564$	0	0
$565$	25.9700	1.09257
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	36.1526	1.51560	0.757798	−	0.652489i	$-0.226275\pi$
$569$	36.1526	1.51560	0.757798	+	0.652489i	$0.226275\pi$
$570$	0	0
$571$	−6.48529	−0.271401	−0.135700	−	0.990750i	$-0.543328\pi$
$571$	−6.48529	−0.271401	−0.135700	+	0.990750i	$0.543328\pi$
$572$	0	0
$573$	−6.61555	−0.276368
$574$	0	0
$575$	−10.6155	−0.442699
$576$	0	0
$577$	3.99312	0.166236	0.0831180	−	0.996540i	$-0.473512\pi$
$577$	3.99312	0.166236	0.0831180	+	0.996540i	$0.473512\pi$
$578$	0	0
$579$	11.8827	0.493830
$580$	0	0
$581$	2.83709	0.117702
$582$	0	0
$583$	41.5500	1.72083
$584$	0	0
$585$	−1.77846	−0.0735302
$586$	0	0
$587$	−4.16635	−0.171964	−0.0859818	−	0.996297i	$-0.527403\pi$
$587$	−4.16635	−0.171964	−0.0859818	+	0.996297i	$0.527403\pi$
$588$	0	0
$589$	−30.3611	−1.25101
$590$	0	0
$591$	11.5569	0.475388
$592$	0	0
$593$	16.1303	0.662390	0.331195	−	0.943562i	$-0.392548\pi$
$593$	16.1303	0.662390	0.331195	+	0.943562i	$0.392548\pi$
$594$	0	0
$595$	−5.23109	−0.214454
$596$	0	0
$597$	−0.996562	−0.0407866
$598$	0	0
$599$	33.2372	1.35804	0.679018	−	0.734122i	$-0.262406\pi$
$599$	33.2372	1.35804	0.679018	+	0.734122i	$0.262406\pi$
$600$	0	0
$601$	−28.9897	−1.18251	−0.591257	−	0.806483i	$-0.701368\pi$
$601$	−28.9897	−1.18251	−0.591257	+	0.806483i	$0.701368\pi$
$602$	0	0
$603$	−5.67418	−0.231070
$604$	0	0
$605$	55.5370	2.25790
$606$	0	0
$607$	25.7846	1.04656	0.523282	−	0.852160i	$-0.324708\pi$
$607$	25.7846	1.04656	0.523282	+	0.852160i	$0.324708\pi$
$608$	0	0
$609$	−2.83709	−0.114965
$610$	0	0
$611$	−8.71982	−0.352766
$612$	0	0
$613$	−7.43965	−0.300485	−0.150242	−	0.988649i	$-0.548005\pi$
$613$	−7.43965	−0.300485	−0.150242	+	0.988649i	$0.548005\pi$
$614$	0	0
$615$	−5.43965	−0.219348
$616$	0	0
$617$	−2.67074	−0.107520	−0.0537600	−	0.998554i	$-0.517121\pi$
$617$	−2.67074	−0.107520	−0.0537600	+	0.998554i	$0.517121\pi$
$618$	0	0
$619$	4.46907	0.179627	0.0898134	−	0.995959i	$-0.471373\pi$
$619$	4.46907	0.179627	0.0898134	+	0.995959i	$0.471373\pi$
$620$	0	0
$621$	5.77846	0.231881
$622$	0	0
$623$	−7.66119	−0.306939
$624$	0	0
$625$	−12.4396	−0.497586
$626$	0	0
$627$	31.4328	1.25530
$628$	0	0
$629$	−28.1104	−1.12083
$630$	0	0
$631$	−23.0878	−0.919113	−0.459556	−	0.888149i	$-0.651992\pi$
$631$	−23.0878	−0.919113	−0.459556	+	0.888149i	$0.651992\pi$
$632$	0	0
$633$	−26.8302	−1.06641
$634$	0	0
$635$	17.5760	0.697483
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−10.0552	−0.397777
$640$	0	0
$641$	41.8268	1.65206	0.826029	−	0.563627i	$-0.190595\pi$
$641$	41.8268	1.65206	0.826029	+	0.563627i	$0.190595\pi$
$642$	0	0
$643$	−3.11383	−0.122797	−0.0613987	−	0.998113i	$-0.519556\pi$
$643$	−3.11383	−0.122797	−0.0613987	+	0.998113i	$0.519556\pi$
$644$	0	0
$645$	−4.83709	−0.190460
$646$	0	0
$647$	−30.4362	−1.19657	−0.598285	−	0.801283i	$-0.704151\pi$
$647$	−30.4362	−1.19657	−0.598285	+	0.801283i	$0.704151\pi$
$648$	0	0
$649$	−10.1173	−0.397137
$650$	0	0
$651$	−6.27674	−0.246005
$652$	0	0
$653$	3.99312	0.156263	0.0781315	−	0.996943i	$-0.475105\pi$
$653$	3.99312	0.156263	0.0781315	+	0.996943i	$0.475105\pi$
$654$	0	0
$655$	−6.69672	−0.261663
$656$	0	0
$657$	−15.8337	−0.617730
$658$	0	0
$659$	−32.6578	−1.27217	−0.636083	−	0.771621i	$-0.719446\pi$
$659$	−32.6578	−1.27217	−0.636083	+	0.771621i	$0.719446\pi$
$660$	0	0
$661$	43.7355	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$661$	43.7355	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$662$	0	0
$663$	2.94137	0.114233
$664$	0	0
$665$	8.60256	0.333593
$666$	0	0
$667$	−16.3940	−0.634778
$668$	0	0
$669$	2.92838	0.113218
$670$	0	0
$671$	25.2311	0.974036
$672$	0	0
$673$	−9.63198	−0.371285	−0.185643	−	0.982617i	$-0.559437\pi$
$673$	−9.63198	−0.371285	−0.185643	+	0.982617i	$0.559437\pi$
$674$	0	0
$675$	−1.83709	−0.0707096
$676$	0	0
$677$	16.4914	0.633816	0.316908	−	0.948456i	$-0.397355\pi$
$677$	16.4914	0.633816	0.316908	+	0.948456i	$0.397355\pi$
$678$	0	0
$679$	−17.7164	−0.679892
$680$	0	0
$681$	9.79145	0.375209
$682$	0	0
$683$	−41.9311	−1.60445	−0.802224	−	0.597024i	$-0.796350\pi$
$683$	−41.9311	−1.60445	−0.802224	+	0.597024i	$0.796350\pi$
$684$	0	0
$685$	−30.2277	−1.15494
$686$	0	0
$687$	3.88273	0.148136
$688$	0	0
$689$	−6.39400	−0.243592
$690$	0	0
$691$	−38.1786	−1.45238	−0.726191	−	0.687493i	$-0.758711\pi$
$691$	−38.1786	−1.45238	−0.726191	+	0.687493i	$0.758711\pi$
$692$	0	0
$693$	6.49828	0.246849
$694$	0	0
$695$	15.2120	0.577024
$696$	0	0
$697$	8.99656	0.340769
$698$	0	0
$699$	−15.8337	−0.598884
$700$	0	0
$701$	16.5957	0.626810	0.313405	−	0.949620i	$-0.398530\pi$
$701$	16.5957	0.626810	0.313405	+	0.949620i	$0.398530\pi$
$702$	0	0
$703$	46.2277	1.74351
$704$	0	0
$705$	15.5078	0.584059
$706$	0	0
$707$	6.73281	0.253214
$708$	0	0
$709$	21.7655	0.817419	0.408710	−	0.912664i	$-0.365979\pi$
$709$	21.7655	0.817419	0.408710	+	0.912664i	$0.365979\pi$
$710$	0	0
$711$	1.28018	0.0480104
$712$	0	0
$713$	−36.2699	−1.35832
$714$	0	0
$715$	−11.5569	−0.432204
$716$	0	0
$717$	6.94137	0.259230
$718$	0	0
$719$	−36.7880	−1.37196	−0.685981	−	0.727620i	$-0.740627\pi$
$719$	−36.7880	−1.37196	−0.685981	+	0.727620i	$0.740627\pi$
$720$	0	0
$721$	−10.8793	−0.405166
$722$	0	0
$723$	7.28018	0.270753
$724$	0	0
$725$	5.21199	0.193568
$726$	0	0
$727$	32.3189	1.19864	0.599322	−	0.800508i	$-0.295437\pi$
$727$	32.3189	1.19864	0.599322	+	0.800508i	$0.295437\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−12.7198	−0.469817	−0.234909	−	0.972017i	$-0.575479\pi$
$733$	−12.7198	−0.469817	−0.234909	+	0.972017i	$0.575479\pi$
$734$	0	0
$735$	1.77846	0.0655994
$736$	0	0
$737$	−36.8724	−1.35821
$738$	0	0
$739$	−23.1982	−0.853361	−0.426681	−	0.904402i	$-0.640317\pi$
$739$	−23.1982	−0.853361	−0.426681	+	0.904402i	$0.640317\pi$
$740$	0	0
$741$	−4.83709	−0.177695
$742$	0	0
$743$	−31.8138	−1.16713	−0.583567	−	0.812065i	$-0.698344\pi$
$743$	−31.8138	−1.16713	−0.583567	+	0.812065i	$0.698344\pi$
$744$	0	0
$745$	−27.6673	−1.01365
$746$	0	0
$747$	2.83709	0.103804
$748$	0	0
$749$	8.49828	0.310520
$750$	0	0
$751$	0.863070	0.0314939	0.0157469	−	0.999876i	$-0.494987\pi$
$751$	0.863070	0.0314939	0.0157469	+	0.999876i	$0.494987\pi$
$752$	0	0
$753$	−23.3224	−0.849915
$754$	0	0
$755$	−8.88617	−0.323401
$756$	0	0
$757$	−13.3974	−0.486938	−0.243469	−	0.969909i	$-0.578285\pi$
$757$	−13.3974	−0.486938	−0.243469	+	0.969909i	$0.578285\pi$
$758$	0	0
$759$	37.5500	1.36298
$760$	0	0
$761$	−35.5630	−1.28916	−0.644579	−	0.764537i	$-0.722967\pi$
$761$	−35.5630	−1.28916	−0.644579	+	0.764537i	$0.722967\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	−5.23109	−0.189131
$766$	0	0
$767$	1.55691	0.0562169
$768$	0	0
$769$	−48.3871	−1.74488	−0.872442	−	0.488717i	$-0.837465\pi$
$769$	−48.3871	−1.74488	−0.872442	+	0.488717i	$0.837465\pi$
$770$	0	0
$771$	15.4948	0.558033
$772$	0	0
$773$	17.9379	0.645182	0.322591	−	0.946538i	$-0.395446\pi$
$773$	17.9379	0.645182	0.322591	+	0.946538i	$0.395446\pi$
$774$	0	0
$775$	11.5309	0.414203
$776$	0	0
$777$	9.55691	0.342852
$778$	0	0
$779$	−14.7949	−0.530082
$780$	0	0
$781$	−65.3415	−2.33810
$782$	0	0
$783$	−2.83709	−0.101389
$784$	0	0
$785$	−33.4137	−1.19258
$786$	0	0
$787$	3.71639	0.132475	0.0662374	−	0.997804i	$-0.478901\pi$
$787$	3.71639	0.132475	0.0662374	+	0.997804i	$0.478901\pi$
$788$	0	0
$789$	−3.42666	−0.121992
$790$	0	0
$791$	14.6026	0.519207
$792$	0	0
$793$	−3.88273	−0.137880
$794$	0	0
$795$	11.3715	0.403304
$796$	0	0
$797$	14.9673	0.530171	0.265085	−	0.964225i	$-0.414600\pi$
$797$	14.9673	0.530171	0.265085	+	0.964225i	$0.414600\pi$
$798$	0	0
$799$	−25.6482	−0.907368
$800$	0	0
$801$	−7.66119	−0.270695
$802$	0	0
$803$	−102.892	−3.63096
$804$	0	0
$805$	10.2767	0.362207
$806$	0	0
$807$	−0.172462	−0.00607094
$808$	0	0
$809$	11.7233	0.412168	0.206084	−	0.978534i	$-0.433928\pi$
$809$	11.7233	0.412168	0.206084	+	0.978534i	$0.433928\pi$
$810$	0	0
$811$	0.886172	0.0311177	0.0155588	−	0.999879i	$-0.495047\pi$
$811$	0.886172	0.0311177	0.0155588	+	0.999879i	$0.495047\pi$
$812$	0	0
$813$	25.9931	0.911619
$814$	0	0
$815$	17.5760	0.615661
$816$	0	0
$817$	−13.1560	−0.460271
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−53.7846	−1.87709	−0.938547	−	0.345151i	$-0.887828\pi$
$821$	−53.7846	−1.87709	−0.938547	+	0.345151i	$0.887828\pi$
$822$	0	0
$823$	35.7655	1.24671	0.623353	−	0.781941i	$-0.285770\pi$
$823$	35.7655	1.24671	0.623353	+	0.781941i	$0.285770\pi$
$824$	0	0
$825$	−11.9379	−0.415625
$826$	0	0
$827$	−7.41043	−0.257686	−0.128843	−	0.991665i	$-0.541126\pi$
$827$	−7.41043	−0.257686	−0.128843	+	0.991665i	$0.541126\pi$
$828$	0	0
$829$	−31.9931	−1.11117	−0.555584	−	0.831461i	$-0.687505\pi$
$829$	−31.9931	−1.11117	−0.555584	+	0.831461i	$0.687505\pi$
$830$	0	0
$831$	3.72326	0.129159
$832$	0	0
$833$	−2.94137	−0.101912
$834$	0	0
$835$	12.5304	0.433631
$836$	0	0
$837$	−6.27674	−0.216956
$838$	0	0
$839$	1.55691	0.0537506	0.0268753	−	0.999639i	$-0.491444\pi$
$839$	1.55691	0.0537506	0.0268753	+	0.999639i	$0.491444\pi$
$840$	0	0
$841$	−20.9509	−0.722445
$842$	0	0
$843$	−21.5500	−0.742223
$844$	0	0
$845$	1.77846	0.0611808
$846$	0	0
$847$	31.2277	1.07299
$848$	0	0
$849$	31.8759	1.09398
$850$	0	0
$851$	55.2242	1.89306
$852$	0	0
$853$	35.4750	1.21464	0.607320	−	0.794457i	$-0.292245\pi$
$853$	35.4750	1.21464	0.607320	+	0.794457i	$0.292245\pi$
$854$	0	0
$855$	8.60256	0.294201
$856$	0	0
$857$	−2.49828	−0.0853397	−0.0426698	−	0.999089i	$-0.513586\pi$
$857$	−2.49828	−0.0853397	−0.0426698	+	0.999089i	$0.513586\pi$
$858$	0	0
$859$	−52.0122	−1.77463	−0.887317	−	0.461160i	$-0.847434\pi$
$859$	−52.0122	−1.77463	−0.887317	+	0.461160i	$0.847434\pi$
$860$	0	0
$861$	−3.05863	−0.104238
$862$	0	0
$863$	−34.8172	−1.18519	−0.592596	−	0.805500i	$-0.701897\pi$
$863$	−34.8172	−1.18519	−0.592596	+	0.805500i	$0.701897\pi$
$864$	0	0
$865$	−25.4137	−0.864091
$866$	0	0
$867$	−8.34836	−0.283525
$868$	0	0
$869$	8.31894	0.282201
$870$	0	0
$871$	5.67418	0.192262
$872$	0	0
$873$	−17.7164	−0.599609
$874$	0	0
$875$	−12.1595	−0.411065
$876$	0	0
$877$	−11.2571	−0.380124	−0.190062	−	0.981772i	$-0.560869\pi$
$877$	−11.2571	−0.380124	−0.190062	+	0.981772i	$0.560869\pi$
$878$	0	0
$879$	3.42666	0.115578
$880$	0	0
$881$	−9.50172	−0.320121	−0.160061	−	0.987107i	$-0.551169\pi$
$881$	−9.50172	−0.320121	−0.160061	+	0.987107i	$0.551169\pi$
$882$	0	0
$883$	−23.7655	−0.799772	−0.399886	−	0.916565i	$-0.630950\pi$
$883$	−23.7655	−0.799772	−0.399886	+	0.916565i	$0.630950\pi$
$884$	0	0
$885$	−2.76891	−0.0930757
$886$	0	0
$887$	−18.3518	−0.616193	−0.308097	−	0.951355i	$-0.599692\pi$
$887$	−18.3518	−0.616193	−0.308097	+	0.951355i	$0.599692\pi$
$888$	0	0
$889$	9.88273	0.331456
$890$	0	0
$891$	6.49828	0.217701
$892$	0	0
$893$	42.1786	1.41145
$894$	0	0
$895$	15.3974	0.514680
$896$	0	0
$897$	−5.77846	−0.192937
$898$	0	0
$899$	17.8077	0.593919
$900$	0	0
$901$	−18.8071	−0.626556
$902$	0	0
$903$	−2.71982	−0.0905101
$904$	0	0
$905$	46.8533	1.55746
$906$	0	0
$907$	−35.7164	−1.18594	−0.592972	−	0.805223i	$-0.702045\pi$
$907$	−35.7164	−1.18594	−0.592972	+	0.805223i	$0.702045\pi$
$908$	0	0
$909$	6.73281	0.223313
$910$	0	0
$911$	16.2147	0.537216	0.268608	−	0.963250i	$-0.413436\pi$
$911$	16.2147	0.537216	0.268608	+	0.963250i	$0.413436\pi$
$912$	0	0
$913$	18.4362	0.610149
$914$	0	0
$915$	6.90528	0.228281
$916$	0	0
$917$	−3.76547	−0.124347
$918$	0	0
$919$	34.4622	1.13680	0.568401	−	0.822751i	$-0.307562\pi$
$919$	34.4622	1.13680	0.568401	+	0.822751i	$0.307562\pi$
$920$	0	0
$921$	3.39744	0.111950
$922$	0	0
$923$	10.0552	0.330971
$924$	0	0
$925$	−17.5569	−0.577268
$926$	0	0
$927$	−10.8793	−0.357323
$928$	0	0
$929$	−28.4492	−0.933388	−0.466694	−	0.884419i	$-0.654555\pi$
$929$	−28.4492	−0.933388	−0.466694	+	0.884419i	$0.654555\pi$
$930$	0	0
$931$	4.83709	0.158529
$932$	0	0
$933$	0.443086	0.0145060
$934$	0	0
$935$	−33.9931	−1.11169
$936$	0	0
$937$	2.91215	0.0951358	0.0475679	−	0.998868i	$-0.484853\pi$
$937$	2.91215	0.0951358	0.0475679	+	0.998868i	$0.484853\pi$
$938$	0	0
$939$	−19.8827	−0.648848
$940$	0	0
$941$	−59.0096	−1.92366	−0.961828	−	0.273654i	$-0.911768\pi$
$941$	−59.0096	−1.92366	−0.961828	+	0.273654i	$0.911768\pi$
$942$	0	0
$943$	−17.6742	−0.575551
$944$	0	0
$945$	1.77846	0.0578532
$946$	0	0
$947$	−34.8432	−1.13225	−0.566126	−	0.824319i	$-0.691558\pi$
$947$	−34.8432	−1.13225	−0.566126	+	0.824319i	$0.691558\pi$
$948$	0	0
$949$	15.8337	0.513982
$950$	0	0
$951$	9.46563	0.306944
$952$	0	0
$953$	23.9578	0.776069	0.388035	−	0.921645i	$-0.373154\pi$
$953$	23.9578	0.776069	0.388035	+	0.921645i	$0.373154\pi$
$954$	0	0
$955$	−11.7655	−0.380722
$956$	0	0
$957$	−18.4362	−0.595958
$958$	0	0
$959$	−16.9966	−0.548848
$960$	0	0
$961$	8.39744	0.270885
$962$	0	0
$963$	8.49828	0.273853
$964$	0	0
$965$	21.1329	0.680293
$966$	0	0
$967$	37.3155	1.19999	0.599993	−	0.800005i	$-0.295170\pi$
$967$	37.3155	1.19999	0.599993	+	0.800005i	$0.295170\pi$
$968$	0	0
$969$	−14.2277	−0.457058
$970$	0	0
$971$	−20.7880	−0.667119	−0.333559	−	0.942729i	$-0.608250\pi$
$971$	−20.7880	−0.667119	−0.333559	+	0.942729i	$0.608250\pi$
$972$	0	0
$973$	8.55348	0.274212
$974$	0	0
$975$	1.83709	0.0588340
$976$	0	0
$977$	49.0810	1.57024	0.785120	−	0.619344i	$-0.212601\pi$
$977$	49.0810	1.57024	0.785120	+	0.619344i	$0.212601\pi$
$978$	0	0
$979$	−49.7846	−1.59112
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−10.1855	−0.324865	−0.162433	−	0.986720i	$-0.551934\pi$
$983$	−10.1855	−0.324865	−0.162433	+	0.986720i	$0.551934\pi$
$984$	0	0
$985$	20.5535	0.654888
$986$	0	0
$987$	8.71982	0.277555
$988$	0	0
$989$	−15.7164	−0.499752
$990$	0	0
$991$	−4.34492	−0.138021	−0.0690105	−	0.997616i	$-0.521984\pi$
$991$	−4.34492	−0.138021	−0.0690105	+	0.997616i	$0.521984\pi$
$992$	0	0
$993$	27.4328	0.870553
$994$	0	0
$995$	−1.77234	−0.0561871
$996$	0	0
$997$	44.9637	1.42401	0.712007	−	0.702172i	$-0.247786\pi$
$997$	44.9637	1.42401	0.712007	+	0.702172i	$0.247786\pi$
$998$	0	0
$999$	9.55691	0.302367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bq.1.3		3
4.3	odd	2		273.2.a.d.1.2	✓	3
12.11	even	2		819.2.a.j.1.2		3
20.19	odd	2		6825.2.a.bd.1.2		3
28.27	even	2		1911.2.a.n.1.2		3
52.51	odd	2		3549.2.a.t.1.2		3
84.83	odd	2		5733.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.d.1.2	✓	3	4.3	odd	2
819.2.a.j.1.2		3	12.11	even	2
1911.2.a.n.1.2		3	28.27	even	2
3549.2.a.t.1.2		3	52.51	odd	2
4368.2.a.bq.1.3		3	1.1	even	1	trivial
5733.2.a.bc.1.2		3	84.83	odd	2
6825.2.a.bd.1.2		3	20.19	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.77846 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.77846 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.49828 q^{11} -1.00000 q^{13} +1.77846 q^{15} -2.94137 q^{17} +4.83709 q^{19} +1.00000 q^{21} +5.77846 q^{23} -1.83709 q^{25} +1.00000 q^{27} -2.83709 q^{29} -6.27674 q^{31} +6.49828 q^{33} +1.77846 q^{35} +9.55691 q^{37} -1.00000 q^{39} -3.05863 q^{41} -2.71982 q^{43} +1.77846 q^{45} +8.71982 q^{47} +1.00000 q^{49} -2.94137 q^{51} +6.39400 q^{53} +11.5569 q^{55} +4.83709 q^{57} -1.55691 q^{59} +3.88273 q^{61} +1.00000 q^{63} -1.77846 q^{65} -5.67418 q^{67} +5.77846 q^{69} -10.0552 q^{71} -15.8337 q^{73} -1.83709 q^{75} +6.49828 q^{77} +1.28018 q^{79} +1.00000 q^{81} +2.83709 q^{83} -5.23109 q^{85} -2.83709 q^{87} -7.66119 q^{89} -1.00000 q^{91} -6.27674 q^{93} +8.60256 q^{95} -17.7164 q^{97} +6.49828 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 3 q^{15} - 8 q^{17} + 7 q^{19} + 3 q^{21} + 9 q^{23} + 2 q^{25} + 3 q^{27} - q^{29} + 7 q^{31} + 2 q^{33} - 3 q^{35} + 12 q^{37} - 3 q^{39} - 10 q^{41} + q^{43} - 3 q^{45} + 17 q^{47} + 3 q^{49} - 8 q^{51} - 5 q^{53} + 18 q^{55} + 7 q^{57} + 12 q^{59} + 10 q^{61} + 3 q^{63} + 3 q^{65} - 2 q^{67} + 9 q^{69} + 4 q^{71} - 5 q^{73} + 2 q^{75} + 2 q^{77} + 13 q^{79} + 3 q^{81} + q^{83} + 16 q^{85} - q^{87} - 13 q^{89} - 3 q^{91} + 7 q^{93} + 15 q^{95} - 9 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 3 * q^15 - 8 * q^17 + 7 * q^19 + 3 * q^21 + 9 * q^23 + 2 * q^25 + 3 * q^27 - q^29 + 7 * q^31 + 2 * q^33 - 3 * q^35 + 12 * q^37 - 3 * q^39 - 10 * q^41 + q^43 - 3 * q^45 + 17 * q^47 + 3 * q^49 - 8 * q^51 - 5 * q^53 + 18 * q^55 + 7 * q^57 + 12 * q^59 + 10 * q^61 + 3 * q^63 + 3 * q^65 - 2 * q^67 + 9 * q^69 + 4 * q^71 - 5 * q^73 + 2 * q^75 + 2 * q^77 + 13 * q^79 + 3 * q^81 + q^83 + 16 * q^85 - q^87 - 13 * q^89 - 3 * q^91 + 7 * q^93 + 15 * q^95 - 9 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more