LMFDB - Embedded newform 8624.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +3.56155 q^{5} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} +3.56155 q^{5} -0.561553 q^{9} -1.00000 q^{11} +5.12311 q^{13} +5.56155 q^{15} +2.00000 q^{17} +3.12311 q^{19} +5.56155 q^{23} +7.68466 q^{25} -5.56155 q^{27} -2.00000 q^{29} -6.43845 q^{31} -1.56155 q^{33} +0.438447 q^{37} +8.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -2.00000 q^{45} +10.2462 q^{47} +3.12311 q^{51} +12.2462 q^{53} -3.56155 q^{55} +4.87689 q^{57} -9.56155 q^{59} -12.2462 q^{61} +18.2462 q^{65} -1.56155 q^{67} +8.68466 q^{69} +8.68466 q^{71} -12.2462 q^{73} +12.0000 q^{75} +3.12311 q^{79} -7.00000 q^{81} -8.00000 q^{83} +7.12311 q^{85} -3.12311 q^{87} +8.43845 q^{89} -10.0540 q^{93} +11.1231 q^{95} -4.43845 q^{97} +0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 3 * q^5 + 3 * q^9	$ 2 q - q^{3} + 3 q^{5} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 7 q^{15} + 4 q^{17} - 2 q^{19} + 7 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 17 q^{31} + q^{33} + 5 q^{37} + 16 q^{39} + 20 q^{41} - 8 q^{43} - 4 q^{45} + 4 q^{47} - 2 q^{51} + 8 q^{53} - 3 q^{55} + 18 q^{57} - 15 q^{59} - 8 q^{61} + 20 q^{65} + q^{67} + 5 q^{69} + 5 q^{71} - 8 q^{73} + 24 q^{75} - 2 q^{79} - 14 q^{81} - 16 q^{83} + 6 q^{85} + 2 q^{87} + 21 q^{89} + 17 q^{93} + 14 q^{95} - 13 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q - q^3 + 3 * q^5 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 7 * q^15 + 4 * q^17 - 2 * q^19 + 7 * q^23 + 3 * q^25 - 7 * q^27 - 4 * q^29 - 17 * q^31 + q^33 + 5 * q^37 + 16 * q^39 + 20 * q^41 - 8 * q^43 - 4 * q^45 + 4 * q^47 - 2 * q^51 + 8 * q^53 - 3 * q^55 + 18 * q^57 - 15 * q^59 - 8 * q^61 + 20 * q^65 + q^67 + 5 * q^69 + 5 * q^71 - 8 * q^73 + 24 * q^75 - 2 * q^79 - 14 * q^81 - 16 * q^83 + 6 * q^85 + 2 * q^87 + 21 * q^89 + 17 * q^93 + 14 * q^95 - 13 * q^97 - 3 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.12311	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$14$	0	0
$15$	5.56155	1.43599
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	3.12311	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.56155	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$23$	5.56155	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−6.43845	−1.15638	−0.578190	−	0.815902i	$-0.696241\pi$
$31$	−6.43845	−1.15638	−0.578190	+	0.815902i	$0.696241\pi$
$32$	0	0
$33$	−1.56155	−0.271831
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.438447	0.0720803	0.0360401	−	0.999350i	$-0.488526\pi$
$37$	0.438447	0.0720803	0.0360401	+	0.999350i	$0.488526\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	10.2462	1.49456	0.747282	−	0.664507i	$-0.231359\pi$
$47$	10.2462	1.49456	0.747282	+	0.664507i	$0.231359\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.12311	0.437322
$52$	0	0
$53$	12.2462	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$54$	0	0
$55$	−3.56155	−0.480240
$56$	0	0
$57$	4.87689	0.645960
$58$	0	0
$59$	−9.56155	−1.24481	−0.622404	−	0.782696i	$-0.713844\pi$
$59$	−9.56155	−1.24481	−0.622404	+	0.782696i	$0.713844\pi$
$60$	0	0
$61$	−12.2462	−1.56797	−0.783983	−	0.620782i	$-0.786815\pi$
$61$	−12.2462	−1.56797	−0.783983	+	0.620782i	$0.786815\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.2462	2.26316
$66$	0	0
$67$	−1.56155	−0.190774	−0.0953870	−	0.995440i	$-0.530409\pi$
$67$	−1.56155	−0.190774	−0.0953870	+	0.995440i	$0.530409\pi$
$68$	0	0
$69$	8.68466	1.04551
$70$	0	0
$71$	8.68466	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$71$	8.68466	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$72$	0	0
$73$	−12.2462	−1.43331	−0.716655	−	0.697428i	$-0.754328\pi$
$73$	−12.2462	−1.43331	−0.716655	+	0.697428i	$0.754328\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.12311	0.351377	0.175688	−	0.984446i	$-0.443785\pi$
$79$	3.12311	0.351377	0.175688	+	0.984446i	$0.443785\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	7.12311	0.772609
$86$	0	0
$87$	−3.12311	−0.334832
$88$	0	0
$89$	8.43845	0.894474	0.447237	−	0.894416i	$-0.352408\pi$
$89$	8.43845	0.894474	0.447237	+	0.894416i	$0.352408\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.0540	−1.04255
$94$	0	0
$95$	11.1231	1.14121
$96$	0	0
$97$	−4.43845	−0.450656	−0.225328	−	0.974283i	$-0.572345\pi$
$97$	−4.43845	−0.450656	−0.225328	+	0.974283i	$0.572345\pi$
$98$	0	0
$99$	0.561553	0.0564382
$100$	0	0
$101$	11.3693	1.13129	0.565645	−	0.824649i	$-0.308627\pi$
$101$	11.3693	1.13129	0.565645	+	0.824649i	$0.308627\pi$
$102$	0	0
$103$	−16.4924	−1.62505	−0.812523	−	0.582929i	$-0.801907\pi$
$103$	−16.4924	−1.62505	−0.812523	+	0.582929i	$0.801907\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.87689	−0.858162	−0.429081	−	0.903266i	$-0.641162\pi$
$107$	−8.87689	−0.858162	−0.429081	+	0.903266i	$0.641162\pi$
$108$	0	0
$109$	15.3693	1.47211	0.736057	−	0.676920i	$-0.236686\pi$
$109$	15.3693	1.47211	0.736057	+	0.676920i	$0.236686\pi$
$110$	0	0
$111$	0.684658	0.0649849
$112$	0	0
$113$	−6.68466	−0.628840	−0.314420	−	0.949284i	$-0.601810\pi$
$113$	−6.68466	−0.628840	−0.314420	+	0.949284i	$0.601810\pi$
$114$	0	0
$115$	19.8078	1.84708
$116$	0	0
$117$	−2.87689	−0.265969
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.6155	1.40800
$124$	0	0
$125$	9.56155	0.855211
$126$	0	0
$127$	−3.12311	−0.277131	−0.138565	−	0.990353i	$-0.544249\pi$
$127$	−3.12311	−0.277131	−0.138565	+	0.990353i	$0.544249\pi$
$128$	0	0
$129$	−6.24621	−0.549948
$130$	0	0
$131$	4.87689	0.426096	0.213048	−	0.977042i	$-0.431661\pi$
$131$	4.87689	0.426096	0.213048	+	0.977042i	$0.431661\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−19.8078	−1.70478
$136$	0	0
$137$	−14.6847	−1.25460	−0.627298	−	0.778780i	$-0.715839\pi$
$137$	−14.6847	−1.25460	−0.627298	+	0.778780i	$0.715839\pi$
$138$	0	0
$139$	3.12311	0.264898	0.132449	−	0.991190i	$-0.457716\pi$
$139$	3.12311	0.264898	0.132449	+	0.991190i	$0.457716\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	0	0
$143$	−5.12311	−0.428416
$144$	0	0
$145$	−7.12311	−0.591542
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	3.12311	0.254155	0.127077	−	0.991893i	$-0.459440\pi$
$151$	3.12311	0.254155	0.127077	+	0.991893i	$0.459440\pi$
$152$	0	0
$153$	−1.12311	−0.0907977
$154$	0	0
$155$	−22.9309	−1.84185
$156$	0	0
$157$	−12.4384	−0.992696	−0.496348	−	0.868124i	$-0.665326\pi$
$157$	−12.4384	−0.992696	−0.496348	+	0.868124i	$0.665326\pi$
$158$	0	0
$159$	19.1231	1.51656
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.4924	1.29179	0.645893	−	0.763428i	$-0.276485\pi$
$163$	16.4924	1.29179	0.645893	+	0.763428i	$0.276485\pi$
$164$	0	0
$165$	−5.56155	−0.432966
$166$	0	0
$167$	17.3693	1.34408	0.672039	−	0.740516i	$-0.265419\pi$
$167$	17.3693	1.34408	0.672039	+	0.740516i	$0.265419\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	−1.75379	−0.134116
$172$	0	0
$173$	14.8769	1.13107	0.565535	−	0.824725i	$-0.308670\pi$
$173$	14.8769	1.13107	0.565535	+	0.824725i	$0.308670\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.9309	−1.12227
$178$	0	0
$179$	0.192236	0.0143684	0.00718419	−	0.999974i	$-0.497713\pi$
$179$	0.192236	0.0143684	0.00718419	+	0.999974i	$0.497713\pi$
$180$	0	0
$181$	20.9309	1.55578	0.777890	−	0.628401i	$-0.216290\pi$
$181$	20.9309	1.55578	0.777890	+	0.628401i	$0.216290\pi$
$182$	0	0
$183$	−19.1231	−1.41362
$184$	0	0
$185$	1.56155	0.114808
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.0540	−0.727480	−0.363740	−	0.931500i	$-0.618500\pi$
$191$	−10.0540	−0.727480	−0.363740	+	0.931500i	$0.618500\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	28.4924	2.04038
$196$	0	0
$197$	−3.75379	−0.267446	−0.133723	−	0.991019i	$-0.542693\pi$
$197$	−3.75379	−0.267446	−0.133723	+	0.991019i	$0.542693\pi$
$198$	0	0
$199$	2.24621	0.159230	0.0796148	−	0.996826i	$-0.474631\pi$
$199$	2.24621	0.159230	0.0796148	+	0.996826i	$0.474631\pi$
$200$	0	0
$201$	−2.43845	−0.171995
$202$	0	0
$203$	0	0
$204$	0	0
$205$	35.6155	2.48750
$206$	0	0
$207$	−3.12311	−0.217071
$208$	0	0
$209$	−3.12311	−0.216030
$210$	0	0
$211$	−16.8769	−1.16185	−0.580927	−	0.813956i	$-0.697310\pi$
$211$	−16.8769	−1.16185	−0.580927	+	0.813956i	$0.697310\pi$
$212$	0	0
$213$	13.5616	0.929222
$214$	0	0
$215$	−14.2462	−0.971584
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−19.1231	−1.29222
$220$	0	0
$221$	10.2462	0.689235
$222$	0	0
$223$	−22.4384	−1.50259	−0.751295	−	0.659967i	$-0.770570\pi$
$223$	−22.4384	−1.50259	−0.751295	+	0.659967i	$0.770570\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	0	0
$227$	6.24621	0.414576	0.207288	−	0.978280i	$-0.433536\pi$
$227$	6.24621	0.414576	0.207288	+	0.978280i	$0.433536\pi$
$228$	0	0
$229$	14.6847	0.970390	0.485195	−	0.874406i	$-0.338749\pi$
$229$	14.6847	0.970390	0.485195	+	0.874406i	$0.338749\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	30.4924	1.99763	0.998813	−	0.0487193i	$-0.0155140\pi$
$233$	30.4924	1.99763	0.998813	+	0.0487193i	$0.0155140\pi$
$234$	0	0
$235$	36.4924	2.38050
$236$	0	0
$237$	4.87689	0.316788
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	19.3693	1.24769	0.623844	−	0.781549i	$-0.285570\pi$
$241$	19.3693	1.24769	0.623844	+	0.781549i	$0.285570\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	−12.4924	−0.791675
$250$	0	0
$251$	−22.0540	−1.39203	−0.696017	−	0.718025i	$-0.745046\pi$
$251$	−22.0540	−1.39203	−0.696017	+	0.718025i	$0.745046\pi$
$252$	0	0
$253$	−5.56155	−0.349652
$254$	0	0
$255$	11.1231	0.696556
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.12311	0.0695185
$262$	0	0
$263$	31.6155	1.94950	0.974748	−	0.223306i	$-0.0716848\pi$
$263$	31.6155	1.94950	0.974748	+	0.223306i	$0.0716848\pi$
$264$	0	0
$265$	43.6155	2.67928
$266$	0	0
$267$	13.1771	0.806424
$268$	0	0
$269$	−24.2462	−1.47832	−0.739159	−	0.673531i	$-0.764777\pi$
$269$	−24.2462	−1.47832	−0.739159	+	0.673531i	$0.764777\pi$
$270$	0	0
$271$	1.75379	0.106535	0.0532675	−	0.998580i	$-0.483036\pi$
$271$	1.75379	0.106535	0.0532675	+	0.998580i	$0.483036\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.68466	−0.463402
$276$	0	0
$277$	17.1231	1.02883	0.514414	−	0.857542i	$-0.328010\pi$
$277$	17.1231	1.02883	0.514414	+	0.857542i	$0.328010\pi$
$278$	0	0
$279$	3.61553	0.216456
$280$	0	0
$281$	−27.8617	−1.66209	−0.831046	−	0.556204i	$-0.812257\pi$
$281$	−27.8617	−1.66209	−0.831046	+	0.556204i	$0.812257\pi$
$282$	0	0
$283$	−4.49242	−0.267047	−0.133523	−	0.991046i	$-0.542629\pi$
$283$	−4.49242	−0.267047	−0.133523	+	0.991046i	$0.542629\pi$
$284$	0	0
$285$	17.3693	1.02887
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−6.93087	−0.406295
$292$	0	0
$293$	−4.24621	−0.248066	−0.124033	−	0.992278i	$-0.539583\pi$
$293$	−4.24621	−0.248066	−0.124033	+	0.992278i	$0.539583\pi$
$294$	0	0
$295$	−34.0540	−1.98270
$296$	0	0
$297$	5.56155	0.322714
$298$	0	0
$299$	28.4924	1.64776
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.7538	1.01993
$304$	0	0
$305$	−43.6155	−2.49742
$306$	0	0
$307$	−12.4924	−0.712980	−0.356490	−	0.934299i	$-0.616027\pi$
$307$	−12.4924	−0.712980	−0.356490	+	0.934299i	$0.616027\pi$
$308$	0	0
$309$	−25.7538	−1.46508
$310$	0	0
$311$	−30.7386	−1.74303	−0.871514	−	0.490371i	$-0.836861\pi$
$311$	−30.7386	−1.74303	−0.871514	+	0.490371i	$0.836861\pi$
$312$	0	0
$313$	12.9309	0.730896	0.365448	−	0.930832i	$-0.380916\pi$
$313$	12.9309	0.730896	0.365448	+	0.930832i	$0.380916\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.0540	1.35101	0.675503	−	0.737357i	$-0.263927\pi$
$317$	24.0540	1.35101	0.675503	+	0.737357i	$0.263927\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	−13.8617	−0.773687
$322$	0	0
$323$	6.24621	0.347548
$324$	0	0
$325$	39.3693	2.18382
$326$	0	0
$327$	24.0000	1.32720
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7.80776	0.429154	0.214577	−	0.976707i	$-0.431163\pi$
$331$	7.80776	0.429154	0.214577	+	0.976707i	$0.431163\pi$
$332$	0	0
$333$	−0.246211	−0.0134923
$334$	0	0
$335$	−5.56155	−0.303860
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	−10.4384	−0.566939
$340$	0	0
$341$	6.43845	0.348661
$342$	0	0
$343$	0	0
$344$	0	0
$345$	30.9309	1.66526
$346$	0	0
$347$	−15.1231	−0.811851	−0.405925	−	0.913906i	$-0.633051\pi$
$347$	−15.1231	−0.811851	−0.405925	+	0.913906i	$0.633051\pi$
$348$	0	0
$349$	−4.24621	−0.227294	−0.113647	−	0.993521i	$-0.536253\pi$
$349$	−4.24621	−0.227294	−0.113647	+	0.993521i	$0.536253\pi$
$350$	0	0
$351$	−28.4924	−1.52081
$352$	0	0
$353$	16.4384	0.874930	0.437465	−	0.899235i	$-0.355876\pi$
$353$	16.4384	0.874930	0.437465	+	0.899235i	$0.355876\pi$
$354$	0	0
$355$	30.9309	1.64164
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.2462	0.751886	0.375943	−	0.926643i	$-0.377319\pi$
$359$	14.2462	0.751886	0.375943	+	0.926643i	$0.377319\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	1.56155	0.0819603
$364$	0	0
$365$	−43.6155	−2.28294
$366$	0	0
$367$	−12.6847	−0.662134	−0.331067	−	0.943607i	$-0.607409\pi$
$367$	−12.6847	−0.662134	−0.331067	+	0.943607i	$0.607409\pi$
$368$	0	0
$369$	−5.61553	−0.292333
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.6155	1.53343	0.766717	−	0.641985i	$-0.221889\pi$
$373$	29.6155	1.53343	0.766717	+	0.641985i	$0.221889\pi$
$374$	0	0
$375$	14.9309	0.771027
$376$	0	0
$377$	−10.2462	−0.527707
$378$	0	0
$379$	−2.93087	−0.150549	−0.0752743	−	0.997163i	$-0.523983\pi$
$379$	−2.93087	−0.150549	−0.0752743	+	0.997163i	$0.523983\pi$
$380$	0	0
$381$	−4.87689	−0.249851
$382$	0	0
$383$	−18.9309	−0.967322	−0.483661	−	0.875255i	$-0.660693\pi$
$383$	−18.9309	−0.967322	−0.483661	+	0.875255i	$0.660693\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.24621	0.114181
$388$	0	0
$389$	8.05398	0.408353	0.204176	−	0.978934i	$-0.434548\pi$
$389$	8.05398	0.408353	0.204176	+	0.978934i	$0.434548\pi$
$390$	0	0
$391$	11.1231	0.562520
$392$	0	0
$393$	7.61553	0.384153
$394$	0	0
$395$	11.1231	0.559664
$396$	0	0
$397$	−22.4924	−1.12886	−0.564431	−	0.825480i	$-0.690904\pi$
$397$	−22.4924	−1.12886	−0.564431	+	0.825480i	$0.690904\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−32.9848	−1.64309
$404$	0	0
$405$	−24.9309	−1.23882
$406$	0	0
$407$	−0.438447	−0.0217330
$408$	0	0
$409$	−1.50758	−0.0745449	−0.0372725	−	0.999305i	$-0.511867\pi$
$409$	−1.50758	−0.0745449	−0.0372725	+	0.999305i	$0.511867\pi$
$410$	0	0
$411$	−22.9309	−1.13110
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−28.4924	−1.39864
$416$	0	0
$417$	4.87689	0.238823
$418$	0	0
$419$	−30.7386	−1.50168	−0.750840	−	0.660484i	$-0.770351\pi$
$419$	−30.7386	−1.50168	−0.750840	+	0.660484i	$0.770351\pi$
$420$	0	0
$421$	−34.9848	−1.70506	−0.852529	−	0.522681i	$-0.824932\pi$
$421$	−34.9848	−1.70506	−0.852529	+	0.522681i	$0.824932\pi$
$422$	0	0
$423$	−5.75379	−0.279759
$424$	0	0
$425$	15.3693	0.745521
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−34.7386	−1.67330	−0.836651	−	0.547737i	$-0.815489\pi$
$431$	−34.7386	−1.67330	−0.836651	+	0.547737i	$0.815489\pi$
$432$	0	0
$433$	−10.6847	−0.513472	−0.256736	−	0.966482i	$-0.582647\pi$
$433$	−10.6847	−0.513472	−0.256736	+	0.966482i	$0.582647\pi$
$434$	0	0
$435$	−11.1231	−0.533312
$436$	0	0
$437$	17.3693	0.830887
$438$	0	0
$439$	−23.6155	−1.12711	−0.563554	−	0.826079i	$-0.690566\pi$
$439$	−23.6155	−1.12711	−0.563554	+	0.826079i	$0.690566\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.1771	1.57629	0.788145	−	0.615489i	$-0.211042\pi$
$443$	33.1771	1.57629	0.788145	+	0.615489i	$0.211042\pi$
$444$	0	0
$445$	30.0540	1.42470
$446$	0	0
$447$	−15.6155	−0.738589
$448$	0	0
$449$	−5.31534	−0.250846	−0.125423	−	0.992103i	$-0.540029\pi$
$449$	−5.31534	−0.250846	−0.125423	+	0.992103i	$0.540029\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	0	0
$453$	4.87689	0.229136
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.8617	−0.929093	−0.464546	−	0.885549i	$-0.653783\pi$
$457$	−19.8617	−0.929093	−0.464546	+	0.885549i	$0.653783\pi$
$458$	0	0
$459$	−11.1231	−0.519182
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−30.5464	−1.41961	−0.709806	−	0.704397i	$-0.751217\pi$
$463$	−30.5464	−1.41961	−0.709806	+	0.704397i	$0.751217\pi$
$464$	0	0
$465$	−35.8078	−1.66055
$466$	0	0
$467$	31.4233	1.45410	0.727048	−	0.686586i	$-0.240892\pi$
$467$	31.4233	1.45410	0.727048	+	0.686586i	$0.240892\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−19.4233	−0.894978
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	−6.87689	−0.314871
$478$	0	0
$479$	−12.4924	−0.570793	−0.285397	−	0.958409i	$-0.592125\pi$
$479$	−12.4924	−0.570793	−0.285397	+	0.958409i	$0.592125\pi$
$480$	0	0
$481$	2.24621	0.102418
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.8078	−0.717794
$486$	0	0
$487$	21.5616	0.977047	0.488524	−	0.872551i	$-0.337536\pi$
$487$	21.5616	0.977047	0.488524	+	0.872551i	$0.337536\pi$
$488$	0	0
$489$	25.7538	1.16463
$490$	0	0
$491$	8.87689	0.400609	0.200304	−	0.979734i	$-0.435807\pi$
$491$	8.87689	0.400609	0.200304	+	0.979734i	$0.435807\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.492423	−0.0220439	−0.0110219	−	0.999939i	$-0.503508\pi$
$499$	−0.492423	−0.0220439	−0.0110219	+	0.999939i	$0.503508\pi$
$500$	0	0
$501$	27.1231	1.21177
$502$	0	0
$503$	−34.7386	−1.54892	−0.774460	−	0.632623i	$-0.781978\pi$
$503$	−34.7386	−1.54892	−0.774460	+	0.632623i	$0.781978\pi$
$504$	0	0
$505$	40.4924	1.80189
$506$	0	0
$507$	20.6847	0.918638
$508$	0	0
$509$	−5.80776	−0.257425	−0.128712	−	0.991682i	$-0.541084\pi$
$509$	−5.80776	−0.257425	−0.128712	+	0.991682i	$0.541084\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−17.3693	−0.766874
$514$	0	0
$515$	−58.7386	−2.58833
$516$	0	0
$517$	−10.2462	−0.450628
$518$	0	0
$519$	23.2311	1.01973
$520$	0	0
$521$	−34.6847	−1.51956	−0.759781	−	0.650179i	$-0.774694\pi$
$521$	−34.6847	−1.51956	−0.759781	+	0.650179i	$0.774694\pi$
$522$	0	0
$523$	−14.2462	−0.622943	−0.311472	−	0.950255i	$-0.600822\pi$
$523$	−14.2462	−0.622943	−0.311472	+	0.950255i	$0.600822\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.8769	−0.560926
$528$	0	0
$529$	7.93087	0.344820
$530$	0	0
$531$	5.36932	0.233009
$532$	0	0
$533$	51.2311	2.21906
$534$	0	0
$535$	−31.6155	−1.36686
$536$	0	0
$537$	0.300187	0.0129540
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.87689	−0.295661	−0.147830	−	0.989013i	$-0.547229\pi$
$541$	−6.87689	−0.295661	−0.147830	+	0.989013i	$0.547229\pi$
$542$	0	0
$543$	32.6847	1.40263
$544$	0	0
$545$	54.7386	2.34475
$546$	0	0
$547$	14.7386	0.630178	0.315089	−	0.949062i	$-0.397966\pi$
$547$	14.7386	0.630178	0.315089	+	0.949062i	$0.397966\pi$
$548$	0	0
$549$	6.87689	0.293499
$550$	0	0
$551$	−6.24621	−0.266098
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.43845	0.103506
$556$	0	0
$557$	23.7538	1.00648	0.503240	−	0.864147i	$-0.332141\pi$
$557$	23.7538	1.00648	0.503240	+	0.864147i	$0.332141\pi$
$558$	0	0
$559$	−20.4924	−0.866737
$560$	0	0
$561$	−3.12311	−0.131858
$562$	0	0
$563$	18.7386	0.789739	0.394870	−	0.918737i	$-0.370790\pi$
$563$	18.7386	0.789739	0.394870	+	0.918737i	$0.370790\pi$
$564$	0	0
$565$	−23.8078	−1.00160
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	7.50758	0.314182	0.157091	−	0.987584i	$-0.449788\pi$
$571$	7.50758	0.314182	0.157091	+	0.987584i	$0.449788\pi$
$572$	0	0
$573$	−15.6998	−0.655869
$574$	0	0
$575$	42.7386	1.78232
$576$	0	0
$577$	−18.6847	−0.777853	−0.388926	−	0.921269i	$-0.627154\pi$
$577$	−18.6847	−0.777853	−0.388926	+	0.921269i	$0.627154\pi$
$578$	0	0
$579$	3.12311	0.129792
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.2462	−0.507186
$584$	0	0
$585$	−10.2462	−0.423629
$586$	0	0
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	−20.1080	−0.828534
$590$	0	0
$591$	−5.86174	−0.241120
$592$	0	0
$593$	8.24621	0.338631	0.169316	−	0.985562i	$-0.445844\pi$
$593$	8.24621	0.338631	0.169316	+	0.985562i	$0.445844\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.50758	0.143556
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−34.8769	−1.42266	−0.711329	−	0.702859i	$-0.751906\pi$
$601$	−34.8769	−1.42266	−0.711329	+	0.702859i	$0.751906\pi$
$602$	0	0
$603$	0.876894	0.0357099
$604$	0	0
$605$	3.56155	0.144798
$606$	0	0
$607$	42.3542	1.71910	0.859551	−	0.511050i	$-0.170743\pi$
$607$	42.3542	1.71910	0.859551	+	0.511050i	$0.170743\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	52.4924	2.12362
$612$	0	0
$613$	−17.6155	−0.711484	−0.355742	−	0.934584i	$-0.615772\pi$
$613$	−17.6155	−0.711484	−0.355742	+	0.934584i	$0.615772\pi$
$614$	0	0
$615$	55.6155	2.24263
$616$	0	0
$617$	6.49242	0.261375	0.130688	−	0.991424i	$-0.458282\pi$
$617$	6.49242	0.261375	0.130688	+	0.991424i	$0.458282\pi$
$618$	0	0
$619$	47.8078	1.92156	0.960778	−	0.277318i	$-0.0894456\pi$
$619$	47.8078	1.92156	0.960778	+	0.277318i	$0.0894456\pi$
$620$	0	0
$621$	−30.9309	−1.24121
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	−4.87689	−0.194764
$628$	0	0
$629$	0.876894	0.0349641
$630$	0	0
$631$	9.06913	0.361036	0.180518	−	0.983572i	$-0.442223\pi$
$631$	9.06913	0.361036	0.180518	+	0.983572i	$0.442223\pi$
$632$	0	0
$633$	−26.3542	−1.04748
$634$	0	0
$635$	−11.1231	−0.441407
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.87689	−0.192927
$640$	0	0
$641$	14.1922	0.560560	0.280280	−	0.959918i	$-0.409573\pi$
$641$	14.1922	0.560560	0.280280	+	0.959918i	$0.409573\pi$
$642$	0	0
$643$	−26.5464	−1.04689	−0.523444	−	0.852060i	$-0.675353\pi$
$643$	−26.5464	−1.04689	−0.523444	+	0.852060i	$0.675353\pi$
$644$	0	0
$645$	−22.2462	−0.875944
$646$	0	0
$647$	9.17708	0.360788	0.180394	−	0.983594i	$-0.442263\pi$
$647$	9.17708	0.360788	0.180394	+	0.983594i	$0.442263\pi$
$648$	0	0
$649$	9.56155	0.375324
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.3002	−0.403077	−0.201539	−	0.979481i	$-0.564594\pi$
$653$	−10.3002	−0.403077	−0.201539	+	0.979481i	$0.564594\pi$
$654$	0	0
$655$	17.3693	0.678675
$656$	0	0
$657$	6.87689	0.268293
$658$	0	0
$659$	−19.6155	−0.764112	−0.382056	−	0.924139i	$-0.624784\pi$
$659$	−19.6155	−0.764112	−0.382056	+	0.924139i	$0.624784\pi$
$660$	0	0
$661$	−43.6695	−1.69855	−0.849273	−	0.527953i	$-0.822960\pi$
$661$	−43.6695	−1.69855	−0.849273	+	0.527953i	$0.822960\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.1231	−0.430688
$668$	0	0
$669$	−35.0388	−1.35468
$670$	0	0
$671$	12.2462	0.472760
$672$	0	0
$673$	8.63068	0.332688	0.166344	−	0.986068i	$-0.446804\pi$
$673$	8.63068	0.332688	0.166344	+	0.986068i	$0.446804\pi$
$674$	0	0
$675$	−42.7386	−1.64501
$676$	0	0
$677$	11.7538	0.451735	0.225867	−	0.974158i	$-0.427478\pi$
$677$	11.7538	0.451735	0.225867	+	0.974158i	$0.427478\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.75379	0.373766
$682$	0	0
$683$	−0.492423	−0.0188420	−0.00942101	−	0.999956i	$-0.502999\pi$
$683$	−0.492423	−0.0188420	−0.00942101	+	0.999956i	$0.502999\pi$
$684$	0	0
$685$	−52.3002	−1.99829
$686$	0	0
$687$	22.9309	0.874867
$688$	0	0
$689$	62.7386	2.39015
$690$	0	0
$691$	17.1771	0.653447	0.326723	−	0.945120i	$-0.394055\pi$
$691$	17.1771	0.653447	0.326723	+	0.945120i	$0.394055\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.1231	0.421923
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	47.6155	1.80098
$700$	0	0
$701$	−38.1080	−1.43932	−0.719659	−	0.694328i	$-0.755702\pi$
$701$	−38.1080	−1.43932	−0.719659	+	0.694328i	$0.755702\pi$
$702$	0	0
$703$	1.36932	0.0516448
$704$	0	0
$705$	56.9848	2.14617
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−50.3002	−1.88906	−0.944532	−	0.328421i	$-0.893484\pi$
$709$	−50.3002	−1.88906	−0.944532	+	0.328421i	$0.893484\pi$
$710$	0	0
$711$	−1.75379	−0.0657722
$712$	0	0
$713$	−35.8078	−1.34101
$714$	0	0
$715$	−18.2462	−0.682370
$716$	0	0
$717$	12.4924	0.466538
$718$	0	0
$719$	12.3002	0.458720	0.229360	−	0.973342i	$-0.426337\pi$
$719$	12.3002	0.458720	0.229360	+	0.973342i	$0.426337\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	30.2462	1.12487
$724$	0	0
$725$	−15.3693	−0.570802
$726$	0	0
$727$	23.8078	0.882981	0.441491	−	0.897266i	$-0.354450\pi$
$727$	23.8078	0.882981	0.441491	+	0.897266i	$0.354450\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	14.8769	0.549491	0.274745	−	0.961517i	$-0.411406\pi$
$733$	14.8769	0.549491	0.274745	+	0.961517i	$0.411406\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.56155	0.0575205
$738$	0	0
$739$	45.3693	1.66894	0.834469	−	0.551055i	$-0.185775\pi$
$739$	45.3693	1.66894	0.834469	+	0.551055i	$0.185775\pi$
$740$	0	0
$741$	24.9848	0.917841
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−35.6155	−1.30485
$746$	0	0
$747$	4.49242	0.164369
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−16.3002	−0.594802	−0.297401	−	0.954753i	$-0.596120\pi$
$751$	−16.3002	−0.594802	−0.297401	+	0.954753i	$0.596120\pi$
$752$	0	0
$753$	−34.4384	−1.25501
$754$	0	0
$755$	11.1231	0.404811
$756$	0	0
$757$	38.9848	1.41693	0.708464	−	0.705747i	$-0.249388\pi$
$757$	38.9848	1.41693	0.708464	+	0.705747i	$0.249388\pi$
$758$	0	0
$759$	−8.68466	−0.315233
$760$	0	0
$761$	−34.1080	−1.23641	−0.618206	−	0.786016i	$-0.712140\pi$
$761$	−34.1080	−1.23641	−0.618206	+	0.786016i	$0.712140\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	−48.9848	−1.76874
$768$	0	0
$769$	43.3693	1.56394	0.781969	−	0.623318i	$-0.214216\pi$
$769$	43.3693	1.56394	0.781969	+	0.623318i	$0.214216\pi$
$770$	0	0
$771$	21.8617	0.787331
$772$	0	0
$773$	−46.4924	−1.67222	−0.836108	−	0.548565i	$-0.815174\pi$
$773$	−46.4924	−1.67222	−0.836108	+	0.548565i	$0.815174\pi$
$774$	0	0
$775$	−49.4773	−1.77728
$776$	0	0
$777$	0	0
$778$	0	0
$779$	31.2311	1.11897
$780$	0	0
$781$	−8.68466	−0.310762
$782$	0	0
$783$	11.1231	0.397507
$784$	0	0
$785$	−44.3002	−1.58114
$786$	0	0
$787$	−19.1231	−0.681665	−0.340833	−	0.940124i	$-0.610709\pi$
$787$	−19.1231	−0.681665	−0.340833	+	0.940124i	$0.610709\pi$
$788$	0	0
$789$	49.3693	1.75759
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−62.7386	−2.22791
$794$	0	0
$795$	68.1080	2.41554
$796$	0	0
$797$	−40.9309	−1.44985	−0.724923	−	0.688830i	$-0.758125\pi$
$797$	−40.9309	−1.44985	−0.724923	+	0.688830i	$0.758125\pi$
$798$	0	0
$799$	20.4924	0.724970
$800$	0	0
$801$	−4.73863	−0.167431
$802$	0	0
$803$	12.2462	0.432159
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−37.8617	−1.33280
$808$	0	0
$809$	28.7386	1.01040	0.505198	−	0.863003i	$-0.331419\pi$
$809$	28.7386	1.01040	0.505198	+	0.863003i	$0.331419\pi$
$810$	0	0
$811$	3.12311	0.109667	0.0548335	−	0.998496i	$-0.482537\pi$
$811$	3.12311	0.109667	0.0548335	+	0.998496i	$0.482537\pi$
$812$	0	0
$813$	2.73863	0.0960481
$814$	0	0
$815$	58.7386	2.05752
$816$	0	0
$817$	−12.4924	−0.437055
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.7538	−0.689412	−0.344706	−	0.938711i	$-0.612021\pi$
$821$	−19.7538	−0.689412	−0.344706	+	0.938711i	$0.612021\pi$
$822$	0	0
$823$	−25.6695	−0.894783	−0.447391	−	0.894338i	$-0.647647\pi$
$823$	−25.6695	−0.894783	−0.447391	+	0.894338i	$0.647647\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	35.2311	1.22510	0.612552	−	0.790430i	$-0.290143\pi$
$827$	35.2311	1.22510	0.612552	+	0.790430i	$0.290143\pi$
$828$	0	0
$829$	−39.1771	−1.36068	−0.680338	−	0.732898i	$-0.738167\pi$
$829$	−39.1771	−1.36068	−0.680338	+	0.732898i	$0.738167\pi$
$830$	0	0
$831$	26.7386	0.927553
$832$	0	0
$833$	0	0
$834$	0	0
$835$	61.8617	2.14081
$836$	0	0
$837$	35.8078	1.23770
$838$	0	0
$839$	−4.68466	−0.161732	−0.0808662	−	0.996725i	$-0.525769\pi$
$839$	−4.68466	−0.161732	−0.0808662	+	0.996725i	$0.525769\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−43.5076	−1.49848
$844$	0	0
$845$	47.1771	1.62294
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.01515	−0.240759
$850$	0	0
$851$	2.43845	0.0835889
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−6.24621	−0.213616
$856$	0	0
$857$	−41.1231	−1.40474	−0.702369	−	0.711813i	$-0.747874\pi$
$857$	−41.1231	−1.40474	−0.702369	+	0.711813i	$0.747874\pi$
$858$	0	0
$859$	16.1922	0.552472	0.276236	−	0.961090i	$-0.410913\pi$
$859$	16.1922	0.552472	0.276236	+	0.961090i	$0.410913\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.2311	1.06312	0.531559	−	0.847021i	$-0.321606\pi$
$863$	31.2311	1.06312	0.531559	+	0.847021i	$0.321606\pi$
$864$	0	0
$865$	52.9848	1.80154
$866$	0	0
$867$	−20.3002	−0.689430
$868$	0	0
$869$	−3.12311	−0.105944
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	2.49242	0.0843557
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.7538	−0.937179	−0.468589	−	0.883416i	$-0.655238\pi$
$877$	−27.7538	−0.937179	−0.468589	+	0.883416i	$0.655238\pi$
$878$	0	0
$879$	−6.63068	−0.223647
$880$	0	0
$881$	−8.93087	−0.300889	−0.150444	−	0.988618i	$-0.548070\pi$
$881$	−8.93087	−0.300889	−0.150444	+	0.988618i	$0.548070\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	−53.1771	−1.78753
$886$	0	0
$887$	−6.63068	−0.222637	−0.111318	−	0.993785i	$-0.535507\pi$
$887$	−6.63068	−0.222637	−0.111318	+	0.993785i	$0.535507\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.00000	0.234509
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	0.684658	0.0228856
$896$	0	0
$897$	44.4924	1.48556
$898$	0	0
$899$	12.8769	0.429468
$900$	0	0
$901$	24.4924	0.815961
$902$	0	0
$903$	0	0
$904$	0	0
$905$	74.5464	2.47801
$906$	0	0
$907$	−18.2462	−0.605856	−0.302928	−	0.953014i	$-0.597964\pi$
$907$	−18.2462	−0.605856	−0.302928	+	0.953014i	$0.597964\pi$
$908$	0	0
$909$	−6.38447	−0.211760
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	−68.1080	−2.25158
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−10.7386	−0.354235	−0.177117	−	0.984190i	$-0.556677\pi$
$919$	−10.7386	−0.354235	−0.177117	+	0.984190i	$0.556677\pi$
$920$	0	0
$921$	−19.5076	−0.642797
$922$	0	0
$923$	44.4924	1.46449
$924$	0	0
$925$	3.36932	0.110782
$926$	0	0
$927$	9.26137	0.304183
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−48.0000	−1.57145
$934$	0	0
$935$	−7.12311	−0.232950
$936$	0	0
$937$	21.1231	0.690062	0.345031	−	0.938591i	$-0.387868\pi$
$937$	21.1231	0.690062	0.345031	+	0.938591i	$0.387868\pi$
$938$	0	0
$939$	20.1922	0.658949
$940$	0	0
$941$	44.7386	1.45844	0.729219	−	0.684281i	$-0.239884\pi$
$941$	44.7386	1.45844	0.729219	+	0.684281i	$0.239884\pi$
$942$	0	0
$943$	55.6155	1.81109
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.9309	0.355206	0.177603	−	0.984102i	$-0.443166\pi$
$947$	10.9309	0.355206	0.177603	+	0.984102i	$0.443166\pi$
$948$	0	0
$949$	−62.7386	−2.03658
$950$	0	0
$951$	37.5616	1.21802
$952$	0	0
$953$	36.3542	1.17763	0.588813	−	0.808269i	$-0.299595\pi$
$953$	36.3542	1.17763	0.588813	+	0.808269i	$0.299595\pi$
$954$	0	0
$955$	−35.8078	−1.15871
$956$	0	0
$957$	3.12311	0.100956
$958$	0	0
$959$	0	0
$960$	0	0
$961$	10.4536	0.337213
$962$	0	0
$963$	4.98485	0.160634
$964$	0	0
$965$	7.12311	0.229301
$966$	0	0
$967$	28.8769	0.928618	0.464309	−	0.885673i	$-0.346303\pi$
$967$	28.8769	0.928618	0.464309	+	0.885673i	$0.346303\pi$
$968$	0	0
$969$	9.75379	0.313337
$970$	0	0
$971$	−9.94602	−0.319183	−0.159592	−	0.987183i	$-0.551018\pi$
$971$	−9.94602	−0.319183	−0.159592	+	0.987183i	$0.551018\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	61.4773	1.96885
$976$	0	0
$977$	−57.4233	−1.83713	−0.918567	−	0.395265i	$-0.870653\pi$
$977$	−57.4233	−1.83713	−0.918567	+	0.395265i	$0.870653\pi$
$978$	0	0
$979$	−8.43845	−0.269694
$980$	0	0
$981$	−8.63068	−0.275557
$982$	0	0
$983$	35.3153	1.12638	0.563192	−	0.826326i	$-0.309573\pi$
$983$	35.3153	1.12638	0.563192	+	0.826326i	$0.309573\pi$
$984$	0	0
$985$	−13.3693	−0.425982
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−22.2462	−0.707388
$990$	0	0
$991$	36.4924	1.15922	0.579610	−	0.814894i	$-0.303205\pi$
$991$	36.4924	1.15922	0.579610	+	0.814894i	$0.303205\pi$
$992$	0	0
$993$	12.1922	0.386909
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	11.3693	0.360070	0.180035	−	0.983660i	$-0.442379\pi$
$997$	11.3693	0.360070	0.180035	+	0.983660i	$0.442379\pi$
$998$	0	0
$999$	−2.43845	−0.0771491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bi.1.2		2
4.3	odd	2		4312.2.a.t.1.1		2
7.6	odd	2		1232.2.a.o.1.1		2
28.27	even	2		616.2.a.f.1.2	✓	2
56.13	odd	2		4928.2.a.bo.1.2		2
56.27	even	2		4928.2.a.bs.1.1		2
84.83	odd	2		5544.2.a.bf.1.2		2
308.307	odd	2		6776.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.f.1.2	✓	2	28.27	even	2
1232.2.a.o.1.1		2	7.6	odd	2
4312.2.a.t.1.1		2	4.3	odd	2
4928.2.a.bo.1.2		2	56.13	odd	2
4928.2.a.bs.1.1		2	56.27	even	2
5544.2.a.bf.1.2		2	84.83	odd	2
6776.2.a.l.1.2		2	308.307	odd	2
8624.2.a.bi.1.2		2	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +3.56155 q^{5} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} +3.56155 q^{5} -0.561553 q^{9} -1.00000 q^{11} +5.12311 q^{13} +5.56155 q^{15} +2.00000 q^{17} +3.12311 q^{19} +5.56155 q^{23} +7.68466 q^{25} -5.56155 q^{27} -2.00000 q^{29} -6.43845 q^{31} -1.56155 q^{33} +0.438447 q^{37} +8.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -2.00000 q^{45} +10.2462 q^{47} +3.12311 q^{51} +12.2462 q^{53} -3.56155 q^{55} +4.87689 q^{57} -9.56155 q^{59} -12.2462 q^{61} +18.2462 q^{65} -1.56155 q^{67} +8.68466 q^{69} +8.68466 q^{71} -12.2462 q^{73} +12.0000 q^{75} +3.12311 q^{79} -7.00000 q^{81} -8.00000 q^{83} +7.12311 q^{85} -3.12311 q^{87} +8.43845 q^{89} -10.0540 q^{93} +11.1231 q^{95} -4.43845 q^{97} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 3 * q^5 + 3 * q^9	\( 2 q - q^{3} + 3 q^{5} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 7 q^{15} + 4 q^{17} - 2 q^{19} + 7 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 17 q^{31} + q^{33} + 5 q^{37} + 16 q^{39} + 20 q^{41} - 8 q^{43} - 4 q^{45} + 4 q^{47} - 2 q^{51} + 8 q^{53} - 3 q^{55} + 18 q^{57} - 15 q^{59} - 8 q^{61} + 20 q^{65} + q^{67} + 5 q^{69} + 5 q^{71} - 8 q^{73} + 24 q^{75} - 2 q^{79} - 14 q^{81} - 16 q^{83} + 6 q^{85} + 2 q^{87} + 21 q^{89} + 17 q^{93} + 14 q^{95} - 13 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q - q^3 + 3 * q^5 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 7 * q^15 + 4 * q^17 - 2 * q^19 + 7 * q^23 + 3 * q^25 - 7 * q^27 - 4 * q^29 - 17 * q^31 + q^33 + 5 * q^37 + 16 * q^39 + 20 * q^41 - 8 * q^43 - 4 * q^45 + 4 * q^47 - 2 * q^51 + 8 * q^53 - 3 * q^55 + 18 * q^57 - 15 * q^59 - 8 * q^61 + 20 * q^65 + q^67 + 5 * q^69 + 5 * q^71 - 8 * q^73 + 24 * q^75 - 2 * q^79 - 14 * q^81 - 16 * q^83 + 6 * q^85 + 2 * q^87 + 21 * q^89 + 17 * q^93 + 14 * q^95 - 13 * q^97 - 3 * q^99

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