LMFDB - Embedded newform 8624.2.a.cj.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.85577$ 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)

$f(q)$	$=$	$q-2.85577 q^{3} +4.15544 q^{5} +5.15544 q^{9} +O(q^{10})$	$q-2.85577 q^{3} +4.15544 q^{5} +5.15544 q^{9} -1.00000 q^{11} -5.29966 q^{13} -11.8670 q^{15} -4.41188 q^{17} -5.71155 q^{19} +3.55611 q^{23} +12.2677 q^{25} -6.15544 q^{27} -0.599328 q^{29} +4.56732 q^{31} +2.85577 q^{33} +6.15544 q^{37} +15.1346 q^{39} +4.41188 q^{41} -3.71155 q^{43} +21.4231 q^{45} +5.01121 q^{47} +12.5993 q^{51} +8.31087 q^{53} -4.15544 q^{55} +16.3109 q^{57} -5.14423 q^{59} -13.6105 q^{61} -22.0224 q^{65} +0.443892 q^{67} -10.1554 q^{69} -4.75476 q^{71} +8.41188 q^{73} -35.0336 q^{75} +8.02242 q^{79} +2.11222 q^{81} +8.82376 q^{83} -18.3333 q^{85} +1.71155 q^{87} -7.57853 q^{89} -13.0432 q^{93} -23.7340 q^{95} -13.8670 q^{97} -5.15544 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + q^{5} + 4 q^{9}+O(q^{10}) $ 3 * q - q^3 + q^5 + 4 * q^9	$ 3 q - q^{3} + q^{5} + 4 q^{9} - 3 q^{11} - 12 q^{13} - 9 q^{15} - 2 q^{17} - 2 q^{19} + 7 q^{23} + 18 q^{25} - 7 q^{27} + 6 q^{29} - 9 q^{31} + q^{33} + 7 q^{37} + 2 q^{41} + 4 q^{43} + 34 q^{45} - 4 q^{47} + 30 q^{51} + 2 q^{53} - q^{55} + 26 q^{57} - 23 q^{59} - 14 q^{61} - 28 q^{65} + 5 q^{67} - 19 q^{69} + 5 q^{71} + 14 q^{73} - 48 q^{75} - 14 q^{79} - q^{81} + 4 q^{83} + 6 q^{85} - 10 q^{87} + 19 q^{89} - 35 q^{93} - 18 q^{95} - 15 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - q^3 + q^5 + 4 * q^9 - 3 * q^11 - 12 * q^13 - 9 * q^15 - 2 * q^17 - 2 * q^19 + 7 * q^23 + 18 * q^25 - 7 * q^27 + 6 * q^29 - 9 * q^31 + q^33 + 7 * q^37 + 2 * q^41 + 4 * q^43 + 34 * q^45 - 4 * q^47 + 30 * q^51 + 2 * q^53 - q^55 + 26 * q^57 - 23 * q^59 - 14 * q^61 - 28 * q^65 + 5 * q^67 - 19 * q^69 + 5 * q^71 + 14 * q^73 - 48 * q^75 - 14 * q^79 - q^81 + 4 * q^83 + 6 * q^85 - 10 * q^87 + 19 * q^89 - 35 * q^93 - 18 * q^95 - 15 * q^97 - 4 * 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
$3$	−2.85577	−1.64878	−0.824391	−	0.566021i	$-0.808482\pi$
$3$	−2.85577	−1.64878	−0.824391	+	0.566021i	$0.808482\pi$
$4$	0	0
$5$	4.15544	1.85837	0.929184	−	0.369618i	$-0.120511\pi$
$5$	4.15544	1.85837	0.929184	+	0.369618i	$0.120511\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.15544	1.71848
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.29966	−1.46986	−0.734931	−	0.678142i	$-0.762786\pi$
$13$	−5.29966	−1.46986	−0.734931	+	0.678142i	$0.762786\pi$
$14$	0	0
$15$	−11.8670	−3.06404
$16$	0	0
$17$	−4.41188	−1.07004	−0.535019	−	0.844840i	$-0.679696\pi$
$17$	−4.41188	−1.07004	−0.535019	+	0.844840i	$0.679696\pi$
$18$	0	0
$19$	−5.71155	−1.31032	−0.655159	−	0.755491i	$-0.727398\pi$
$19$	−5.71155	−1.31032	−0.655159	+	0.755491i	$0.727398\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.55611	0.741500	0.370750	−	0.928733i	$-0.379101\pi$
$23$	3.55611	0.741500	0.370750	+	0.928733i	$0.379101\pi$
$24$	0	0
$25$	12.2677	2.45353
$26$	0	0
$27$	−6.15544	−1.18461
$28$	0	0
$29$	−0.599328	−0.111292	−0.0556462	−	0.998451i	$-0.517722\pi$
$29$	−0.599328	−0.111292	−0.0556462	+	0.998451i	$0.517722\pi$
$30$	0	0
$31$	4.56732	0.820314	0.410157	−	0.912015i	$-0.365474\pi$
$31$	4.56732	0.820314	0.410157	+	0.912015i	$0.365474\pi$
$32$	0	0
$33$	2.85577	0.497126
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.15544	1.01195	0.505974	−	0.862549i	$-0.331133\pi$
$37$	6.15544	1.01195	0.505974	+	0.862549i	$0.331133\pi$
$38$	0	0
$39$	15.1346	2.42348
$40$	0	0
$41$	4.41188	0.689020	0.344510	−	0.938783i	$-0.388045\pi$
$41$	4.41188	0.689020	0.344510	+	0.938783i	$0.388045\pi$
$42$	0	0
$43$	−3.71155	−0.566005	−0.283003	−	0.959119i	$-0.591331\pi$
$43$	−3.71155	−0.566005	−0.283003	+	0.959119i	$0.591331\pi$
$44$	0	0
$45$	21.4231	3.19357
$46$	0	0
$47$	5.01121	0.730960	0.365480	−	0.930819i	$-0.380905\pi$
$47$	5.01121	0.730960	0.365480	+	0.930819i	$0.380905\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.5993	1.76426
$52$	0	0
$53$	8.31087	1.14159	0.570793	−	0.821094i	$-0.306636\pi$
$53$	8.31087	1.14159	0.570793	+	0.821094i	$0.306636\pi$
$54$	0	0
$55$	−4.15544	−0.560319
$56$	0	0
$57$	16.3109	2.16043
$58$	0	0
$59$	−5.14423	−0.669721	−0.334861	−	0.942268i	$-0.608689\pi$
$59$	−5.14423	−0.669721	−0.334861	+	0.942268i	$0.608689\pi$
$60$	0	0
$61$	−13.6105	−1.74265	−0.871325	−	0.490706i	$-0.836739\pi$
$61$	−13.6105	−1.74265	−0.871325	+	0.490706i	$0.836739\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−22.0224	−2.73154
$66$	0	0
$67$	0.443892	0.0542300	0.0271150	−	0.999632i	$-0.491368\pi$
$67$	0.443892	0.0542300	0.0271150	+	0.999632i	$0.491368\pi$
$68$	0	0
$69$	−10.1554	−1.22257
$70$	0	0
$71$	−4.75476	−0.564287	−0.282143	−	0.959372i	$-0.591045\pi$
$71$	−4.75476	−0.564287	−0.282143	+	0.959372i	$0.591045\pi$
$72$	0	0
$73$	8.41188	0.984536	0.492268	−	0.870444i	$-0.336168\pi$
$73$	8.41188	0.984536	0.492268	+	0.870444i	$0.336168\pi$
$74$	0	0
$75$	−35.0336	−4.04533
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.02242	0.902593	0.451296	−	0.892374i	$-0.350962\pi$
$79$	8.02242	0.902593	0.451296	+	0.892374i	$0.350962\pi$
$80$	0	0
$81$	2.11222	0.234691
$82$	0	0
$83$	8.82376	0.968534	0.484267	−	0.874920i	$-0.339086\pi$
$83$	8.82376	0.968534	0.484267	+	0.874920i	$0.339086\pi$
$84$	0	0
$85$	−18.3333	−1.98852
$86$	0	0
$87$	1.71155	0.183497
$88$	0	0
$89$	−7.57853	−0.803322	−0.401661	−	0.915788i	$-0.631567\pi$
$89$	−7.57853	−0.803322	−0.401661	+	0.915788i	$0.631567\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−13.0432	−1.35252
$94$	0	0
$95$	−23.7340	−2.43505
$96$	0	0
$97$	−13.8670	−1.40798	−0.703989	−	0.710211i	$-0.748600\pi$
$97$	−13.8670	−1.40798	−0.703989	+	0.710211i	$0.748600\pi$
$98$	0	0
$99$	−5.15544	−0.518141
$100$	0	0
$101$	2.70034	0.268693	0.134347	−	0.990934i	$-0.457106\pi$
$101$	2.70034	0.268693	0.134347	+	0.990934i	$0.457106\pi$
$102$	0	0
$103$	6.41188	0.631781	0.315891	−	0.948796i	$-0.397697\pi$
$103$	6.41188	0.631781	0.315891	+	0.948796i	$0.397697\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.31087	−0.803442	−0.401721	−	0.915762i	$-0.631588\pi$
$107$	−8.31087	−0.803442	−0.401721	+	0.915762i	$0.631588\pi$
$108$	0	0
$109$	−15.1346	−1.44964	−0.724818	−	0.688941i	$-0.758076\pi$
$109$	−15.1346	−1.44964	−0.724818	+	0.688941i	$0.758076\pi$
$110$	0	0
$111$	−17.5785	−1.66848
$112$	0	0
$113$	−15.8894	−1.49475	−0.747375	−	0.664403i	$-0.768686\pi$
$113$	−15.8894	−1.49475	−0.747375	+	0.664403i	$0.768686\pi$
$114$	0	0
$115$	14.7772	1.37798
$116$	0	0
$117$	−27.3221	−2.52593
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.5993	−1.13604
$124$	0	0
$125$	30.2003	2.70119
$126$	0	0
$127$	−15.1122	−1.34099	−0.670496	−	0.741913i	$-0.733919\pi$
$127$	−15.1122	−1.34099	−0.670496	+	0.741913i	$0.733919\pi$
$128$	0	0
$129$	10.5993	0.933219
$130$	0	0
$131$	−3.68913	−0.322321	−0.161160	−	0.986928i	$-0.551524\pi$
$131$	−3.68913	−0.322321	−0.161160	+	0.986928i	$0.551524\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−25.5785	−2.20145
$136$	0	0
$137$	3.04322	0.260000	0.130000	−	0.991514i	$-0.458502\pi$
$137$	3.04322	0.260000	0.130000	+	0.991514i	$0.458502\pi$
$138$	0	0
$139$	10.9102	0.925391	0.462696	−	0.886517i	$-0.346882\pi$
$139$	10.9102	0.925391	0.462696	+	0.886517i	$0.346882\pi$
$140$	0	0
$141$	−14.3109	−1.20519
$142$	0	0
$143$	5.29966	0.443180
$144$	0	0
$145$	−2.49047	−0.206822
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.19866	0.589737	0.294868	−	0.955538i	$-0.404724\pi$
$149$	7.19866	0.589737	0.294868	+	0.955538i	$0.404724\pi$
$150$	0	0
$151$	8.02242	0.652855	0.326428	−	0.945222i	$-0.394155\pi$
$151$	8.02242	0.652855	0.326428	+	0.945222i	$0.394155\pi$
$152$	0	0
$153$	−22.7452	−1.83884
$154$	0	0
$155$	18.9792	1.52445
$156$	0	0
$157$	−3.84456	−0.306830	−0.153415	−	0.988162i	$-0.549027\pi$
$157$	−3.84456	−0.306830	−0.153415	+	0.988162i	$0.549027\pi$
$158$	0	0
$159$	−23.7340	−1.88223
$160$	0	0
$161$	0	0
$162$	0	0
$163$	14.0224	1.09832	0.549160	−	0.835717i	$-0.314948\pi$
$163$	14.0224	1.09832	0.549160	+	0.835717i	$0.314948\pi$
$164$	0	0
$165$	11.8670	0.923843
$166$	0	0
$167$	19.1122	1.47895	0.739474	−	0.673185i	$-0.235074\pi$
$167$	19.1122	1.47895	0.739474	+	0.673185i	$0.235074\pi$
$168$	0	0
$169$	15.0864	1.16050
$170$	0	0
$171$	−29.4455	−2.25175
$172$	0	0
$173$	−19.3221	−1.46903	−0.734515	−	0.678592i	$-0.762590\pi$
$173$	−19.3221	−1.46903	−0.734515	+	0.678592i	$0.762590\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.6907	1.10422
$178$	0	0
$179$	−6.73235	−0.503199	−0.251600	−	0.967831i	$-0.580957\pi$
$179$	−6.73235	−0.503199	−0.251600	+	0.967831i	$0.580957\pi$
$180$	0	0
$181$	7.64255	0.568066	0.284033	−	0.958814i	$-0.408327\pi$
$181$	7.64255	0.568066	0.284033	+	0.958814i	$0.408327\pi$
$182$	0	0
$183$	38.8686	2.87325
$184$	0	0
$185$	25.5785	1.88057
$186$	0	0
$187$	4.41188	0.322629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.84456	0.133468	0.0667340	−	0.997771i	$-0.478742\pi$
$191$	1.84456	0.133468	0.0667340	+	0.997771i	$0.478742\pi$
$192$	0	0
$193$	−14.6217	−1.05250	−0.526248	−	0.850331i	$-0.676402\pi$
$193$	−14.6217	−1.05250	−0.526248	+	0.850331i	$0.676402\pi$
$194$	0	0
$195$	62.8910	4.50372
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−10.2099	−0.723758	−0.361879	−	0.932225i	$-0.617865\pi$
$199$	−10.2099	−0.723758	−0.361879	+	0.932225i	$0.617865\pi$
$200$	0	0
$201$	−1.26765	−0.0894134
$202$	0	0
$203$	0	0
$204$	0	0
$205$	18.3333	1.28045
$206$	0	0
$207$	18.3333	1.27425
$208$	0	0
$209$	5.71155	0.395076
$210$	0	0
$211$	−16.3109	−1.12289	−0.561443	−	0.827515i	$-0.689754\pi$
$211$	−16.3109	−1.12289	−0.561443	+	0.827515i	$0.689754\pi$
$212$	0	0
$213$	13.5785	0.930385
$214$	0	0
$215$	−15.4231	−1.05185
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−24.0224	−1.62328
$220$	0	0
$221$	23.3815	1.57281
$222$	0	0
$223$	−13.0802	−0.875915	−0.437958	−	0.898996i	$-0.644298\pi$
$223$	−13.0802	−0.875915	−0.437958	+	0.898996i	$0.644298\pi$
$224$	0	0
$225$	63.2451	4.21634
$226$	0	0
$227$	−16.6217	−1.10322	−0.551612	−	0.834101i	$-0.685987\pi$
$227$	−16.6217	−1.10322	−0.551612	+	0.834101i	$0.685987\pi$
$228$	0	0
$229$	−22.1779	−1.46555	−0.732777	−	0.680469i	$-0.761776\pi$
$229$	−22.1779	−1.46555	−0.732777	+	0.680469i	$0.761776\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.5993	−0.825409	−0.412705	−	0.910865i	$-0.635416\pi$
$233$	−12.5993	−0.825409	−0.412705	+	0.910865i	$0.635416\pi$
$234$	0	0
$235$	20.8238	1.35839
$236$	0	0
$237$	−22.9102	−1.48818
$238$	0	0
$239$	−14.2244	−0.920102	−0.460051	−	0.887892i	$-0.652169\pi$
$239$	−14.2244	−0.920102	−0.460051	+	0.887892i	$0.652169\pi$
$240$	0	0
$241$	0.101008	0.00650648	0.00325324	−	0.999995i	$-0.498964\pi$
$241$	0.101008	0.00650648	0.00325324	+	0.999995i	$0.498964\pi$
$242$	0	0
$243$	12.4343	0.797661
$244$	0	0
$245$	0	0
$246$	0	0
$247$	30.2693	1.92599
$248$	0	0
$249$	−25.1987	−1.59690
$250$	0	0
$251$	−25.9680	−1.63908	−0.819542	−	0.573018i	$-0.805772\pi$
$251$	−25.9680	−1.63908	−0.819542	+	0.573018i	$0.805772\pi$
$252$	0	0
$253$	−3.55611	−0.223571
$254$	0	0
$255$	52.3557	3.27864
$256$	0	0
$257$	−2.57691	−0.160743	−0.0803716	−	0.996765i	$-0.525611\pi$
$257$	−2.57691	−0.160743	−0.0803716	+	0.996765i	$0.525611\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.08980	−0.191254
$262$	0	0
$263$	−20.3109	−1.25242	−0.626211	−	0.779654i	$-0.715395\pi$
$263$	−20.3109	−1.25242	−0.626211	+	0.779654i	$0.715395\pi$
$264$	0	0
$265$	34.5353	2.12149
$266$	0	0
$267$	21.6425	1.32450
$268$	0	0
$269$	−2.82376	−0.172168	−0.0860839	−	0.996288i	$-0.527435\pi$
$269$	−2.82376	−0.172168	−0.0860839	+	0.996288i	$0.527435\pi$
$270$	0	0
$271$	4.82376	0.293023	0.146511	−	0.989209i	$-0.453196\pi$
$271$	4.82376	0.293023	0.146511	+	0.989209i	$0.453196\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−12.2677	−0.739767
$276$	0	0
$277$	13.7340	0.825194	0.412597	−	0.910914i	$-0.364622\pi$
$277$	13.7340	0.825194	0.412597	+	0.910914i	$0.364622\pi$
$278$	0	0
$279$	23.5465	1.40969
$280$	0	0
$281$	−20.9102	−1.24740	−0.623699	−	0.781665i	$-0.714371\pi$
$281$	−20.9102	−1.24740	−0.623699	+	0.781665i	$0.714371\pi$
$282$	0	0
$283$	−5.19866	−0.309028	−0.154514	−	0.987991i	$-0.549381\pi$
$283$	−5.19866	−0.309028	−0.154514	+	0.987991i	$0.549381\pi$
$284$	0	0
$285$	67.7788	4.01487
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.46469	0.144982
$290$	0	0
$291$	39.6009	2.32145
$292$	0	0
$293$	17.8574	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$293$	17.8574	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$294$	0	0
$295$	−21.3765	−1.24459
$296$	0	0
$297$	6.15544	0.357175
$298$	0	0
$299$	−18.8462	−1.08990
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.71155	−0.443017
$304$	0	0
$305$	−56.5577	−3.23849
$306$	0	0
$307$	−32.6217	−1.86182	−0.930911	−	0.365247i	$-0.880985\pi$
$307$	−32.6217	−1.86182	−0.930911	+	0.365247i	$0.880985\pi$
$308$	0	0
$309$	−18.3109	−1.04167
$310$	0	0
$311$	−25.0560	−1.42080	−0.710399	−	0.703799i	$-0.751485\pi$
$311$	−25.0560	−1.42080	−0.710399	+	0.703799i	$0.751485\pi$
$312$	0	0
$313$	19.8894	1.12422	0.562108	−	0.827064i	$-0.309991\pi$
$313$	19.8894	1.12422	0.562108	+	0.827064i	$0.309991\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.04322	−0.507918	−0.253959	−	0.967215i	$-0.581733\pi$
$317$	−9.04322	−0.507918	−0.253959	+	0.967215i	$0.581733\pi$
$318$	0	0
$319$	0.599328	0.0335559
$320$	0	0
$321$	23.7340	1.32470
$322$	0	0
$323$	25.1987	1.40209
$324$	0	0
$325$	−65.0144	−3.60635
$326$	0	0
$327$	43.2211	2.39013
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−29.8894	−1.64287	−0.821435	−	0.570302i	$-0.806826\pi$
$331$	−29.8894	−1.64287	−0.821435	+	0.570302i	$0.806826\pi$
$332$	0	0
$333$	31.7340	1.73901
$334$	0	0
$335$	1.84456	0.100779
$336$	0	0
$337$	3.97758	0.216673	0.108336	−	0.994114i	$-0.465448\pi$
$337$	3.97758	0.216673	0.108336	+	0.994114i	$0.465448\pi$
$338$	0	0
$339$	45.3765	2.46451
$340$	0	0
$341$	−4.56732	−0.247334
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−42.2003	−2.27199
$346$	0	0
$347$	2.82376	0.151587	0.0757937	−	0.997124i	$-0.475851\pi$
$347$	2.82376	0.151587	0.0757937	+	0.997124i	$0.475851\pi$
$348$	0	0
$349$	12.7868	0.684460	0.342230	−	0.939616i	$-0.388818\pi$
$349$	12.7868	0.684460	0.342230	+	0.939616i	$0.388818\pi$
$350$	0	0
$351$	32.6217	1.74122
$352$	0	0
$353$	−33.9758	−1.80835	−0.904176	−	0.427161i	$-0.859514\pi$
$353$	−33.9758	−1.80835	−0.904176	+	0.427161i	$0.859514\pi$
$354$	0	0
$355$	−19.7581	−1.04865
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.1346	0.587664	0.293832	−	0.955857i	$-0.405069\pi$
$359$	11.1346	0.587664	0.293832	+	0.955857i	$0.405069\pi$
$360$	0	0
$361$	13.6217	0.716934
$362$	0	0
$363$	−2.85577	−0.149889
$364$	0	0
$365$	34.9550	1.82963
$366$	0	0
$367$	24.3653	1.27186	0.635929	−	0.771747i	$-0.280617\pi$
$367$	24.3653	1.27186	0.635929	+	0.771747i	$0.280617\pi$
$368$	0	0
$369$	22.7452	1.18407
$370$	0	0
$371$	0	0
$372$	0	0
$373$	21.1122	1.09315	0.546575	−	0.837410i	$-0.315931\pi$
$373$	21.1122	1.09315	0.546575	+	0.837410i	$0.315931\pi$
$374$	0	0
$375$	−86.2451	−4.45368
$376$	0	0
$377$	3.17624	0.163585
$378$	0	0
$379$	−10.0914	−0.518361	−0.259181	−	0.965829i	$-0.583452\pi$
$379$	−10.0914	−0.518361	−0.259181	+	0.965829i	$0.583452\pi$
$380$	0	0
$381$	43.1571	2.21100
$382$	0	0
$383$	37.4359	1.91289	0.956443	−	0.291919i	$-0.0942939\pi$
$383$	37.4359	1.91289	0.956443	+	0.291919i	$0.0942939\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−19.1346	−0.972668
$388$	0	0
$389$	17.3541	0.879887	0.439944	−	0.898025i	$-0.354998\pi$
$389$	17.3541	0.879887	0.439944	+	0.898025i	$0.354998\pi$
$390$	0	0
$391$	−15.6891	−0.793433
$392$	0	0
$393$	10.5353	0.531436
$394$	0	0
$395$	33.3367	1.67735
$396$	0	0
$397$	18.0448	0.905644	0.452822	−	0.891601i	$-0.350417\pi$
$397$	18.0448	0.905644	0.452822	+	0.891601i	$0.350417\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.57691	0.128685	0.0643424	−	0.997928i	$-0.479505\pi$
$401$	2.57691	0.128685	0.0643424	+	0.997928i	$0.479505\pi$
$402$	0	0
$403$	−24.2052	−1.20575
$404$	0	0
$405$	8.77718	0.436142
$406$	0	0
$407$	−6.15544	−0.305114
$408$	0	0
$409$	25.0336	1.23783	0.618917	−	0.785457i	$-0.287572\pi$
$409$	25.0336	1.23783	0.618917	+	0.785457i	$0.287572\pi$
$410$	0	0
$411$	−8.69074	−0.428683
$412$	0	0
$413$	0	0
$414$	0	0
$415$	36.6666	1.79989
$416$	0	0
$417$	−31.1571	−1.52577
$418$	0	0
$419$	3.56570	0.174196	0.0870979	−	0.996200i	$-0.472241\pi$
$419$	3.56570	0.174196	0.0870979	+	0.996200i	$0.472241\pi$
$420$	0	0
$421$	4.31087	0.210099	0.105050	−	0.994467i	$-0.466500\pi$
$421$	4.31087	0.210099	0.105050	+	0.994467i	$0.466500\pi$
$422$	0	0
$423$	25.8350	1.25614
$424$	0	0
$425$	−54.1234	−2.62537
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−15.1346	−0.730707
$430$	0	0
$431$	−14.5577	−0.701221	−0.350610	−	0.936521i	$-0.614026\pi$
$431$	−14.5577	−0.701221	−0.350610	+	0.936521i	$0.614026\pi$
$432$	0	0
$433$	0.357452	0.0171780	0.00858902	−	0.999963i	$-0.497266\pi$
$433$	0.357452	0.0171780	0.00858902	+	0.999963i	$0.497266\pi$
$434$	0	0
$435$	7.11222	0.341005
$436$	0	0
$437$	−20.3109	−0.971601
$438$	0	0
$439$	−14.2885	−0.681951	−0.340975	−	0.940072i	$-0.610757\pi$
$439$	−14.2885	−0.681951	−0.340975	+	0.940072i	$0.610757\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.95678	−0.425549	−0.212775	−	0.977101i	$-0.568250\pi$
$443$	−8.95678	−0.425549	−0.212775	+	0.977101i	$0.568250\pi$
$444$	0	0
$445$	−31.4921	−1.49287
$446$	0	0
$447$	−20.5577	−0.972347
$448$	0	0
$449$	0.379870	0.0179272	0.00896359	−	0.999960i	$-0.497147\pi$
$449$	0.379870	0.0179272	0.00896359	+	0.999960i	$0.497147\pi$
$450$	0	0
$451$	−4.41188	−0.207747
$452$	0	0
$453$	−22.9102	−1.07642
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.3557	−1.23287	−0.616434	−	0.787407i	$-0.711423\pi$
$457$	−26.3557	−1.23287	−0.616434	+	0.787407i	$0.711423\pi$
$458$	0	0
$459$	27.1571	1.26758
$460$	0	0
$461$	−38.4343	−1.79006	−0.895032	−	0.446002i	$-0.852847\pi$
$461$	−38.4343	−1.79006	−0.895032	+	0.446002i	$0.852847\pi$
$462$	0	0
$463$	1.64255	0.0763357	0.0381678	−	0.999271i	$-0.487848\pi$
$463$	1.64255	0.0763357	0.0381678	+	0.999271i	$0.487848\pi$
$464$	0	0
$465$	−54.2003	−2.51348
$466$	0	0
$467$	5.65712	0.261780	0.130890	−	0.991397i	$-0.458217\pi$
$467$	5.65712	0.261780	0.130890	+	0.991397i	$0.458217\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10.9792	0.505895
$472$	0	0
$473$	3.71155	0.170657
$474$	0	0
$475$	−70.0673	−3.21491
$476$	0	0
$477$	42.8462	1.96179
$478$	0	0
$479$	32.8238	1.49976	0.749878	−	0.661576i	$-0.230112\pi$
$479$	32.8238	1.49976	0.749878	+	0.661576i	$0.230112\pi$
$480$	0	0
$481$	−32.6217	−1.48742
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−57.6234	−2.61654
$486$	0	0
$487$	6.53033	0.295918	0.147959	−	0.988994i	$-0.452730\pi$
$487$	6.53033	0.295918	0.147959	+	0.988994i	$0.452730\pi$
$488$	0	0
$489$	−40.0448	−1.81089
$490$	0	0
$491$	−31.4455	−1.41912	−0.709558	−	0.704647i	$-0.751105\pi$
$491$	−31.4455	−1.41912	−0.709558	+	0.704647i	$0.751105\pi$
$492$	0	0
$493$	2.64416	0.119087
$494$	0	0
$495$	−21.4231	−0.962896
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−23.6699	−1.05961	−0.529806	−	0.848119i	$-0.677735\pi$
$499$	−23.6699	−1.05961	−0.529806	+	0.848119i	$0.677735\pi$
$500$	0	0
$501$	−54.5801	−2.43846
$502$	0	0
$503$	5.57355	0.248512	0.124256	−	0.992250i	$-0.460346\pi$
$503$	5.57355	0.248512	0.124256	+	0.992250i	$0.460346\pi$
$504$	0	0
$505$	11.2211	0.499331
$506$	0	0
$507$	−43.0834	−1.91340
$508$	0	0
$509$	9.82215	0.435359	0.217679	−	0.976020i	$-0.430151\pi$
$509$	9.82215	0.435359	0.217679	+	0.976020i	$0.430151\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	35.1571	1.55222
$514$	0	0
$515$	26.6442	1.17408
$516$	0	0
$517$	−5.01121	−0.220393
$518$	0	0
$519$	55.1795	2.42211
$520$	0	0
$521$	−40.1363	−1.75840	−0.879201	−	0.476452i	$-0.841923\pi$
$521$	−40.1363	−1.75840	−0.879201	+	0.476452i	$0.841923\pi$
$522$	0	0
$523$	−18.2244	−0.796899	−0.398449	−	0.917190i	$-0.630452\pi$
$523$	−18.2244	−0.796899	−0.398449	+	0.917190i	$0.630452\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.1505	−0.877768
$528$	0	0
$529$	−10.3541	−0.450178
$530$	0	0
$531$	−26.5207	−1.15090
$532$	0	0
$533$	−23.3815	−1.01276
$534$	0	0
$535$	−34.5353	−1.49309
$536$	0	0
$537$	19.2261	0.829665
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.68913	−0.0726212	−0.0363106	−	0.999341i	$-0.511561\pi$
$541$	−1.68913	−0.0726212	−0.0363106	+	0.999341i	$0.511561\pi$
$542$	0	0
$543$	−21.8254	−0.936617
$544$	0	0
$545$	−62.8910	−2.69396
$546$	0	0
$547$	25.0706	1.07194	0.535971	−	0.844236i	$-0.319946\pi$
$547$	25.0706	1.07194	0.535971	+	0.844236i	$0.319946\pi$
$548$	0	0
$549$	−70.1683	−2.99471
$550$	0	0
$551$	3.42309	0.145829
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−73.0465	−3.10065
$556$	0	0
$557$	26.0448	1.10356	0.551778	−	0.833991i	$-0.313950\pi$
$557$	26.0448	1.10356	0.551778	+	0.833991i	$0.313950\pi$
$558$	0	0
$559$	19.6699	0.831950
$560$	0	0
$561$	−12.5993	−0.531944
$562$	0	0
$563$	−37.4455	−1.57814	−0.789070	−	0.614303i	$-0.789437\pi$
$563$	−37.4455	−1.57814	−0.789070	+	0.614303i	$0.789437\pi$
$564$	0	0
$565$	−66.0274	−2.77779
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.97758	−0.334438	−0.167219	−	0.985920i	$-0.553479\pi$
$569$	−7.97758	−0.334438	−0.167219	+	0.985920i	$0.553479\pi$
$570$	0	0
$571$	−9.19866	−0.384952	−0.192476	−	0.981302i	$-0.561652\pi$
$571$	−9.19866	−0.384952	−0.192476	+	0.981302i	$0.561652\pi$
$572$	0	0
$573$	−5.26765	−0.220059
$574$	0	0
$575$	43.6251	1.81929
$576$	0	0
$577$	2.75476	0.114682	0.0573412	−	0.998355i	$-0.481738\pi$
$577$	2.75476	0.114682	0.0573412	+	0.998355i	$0.481738\pi$
$578$	0	0
$579$	41.7564	1.73534
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−8.31087	−0.344201
$584$	0	0
$585$	−113.535	−4.69410
$586$	0	0
$587$	6.20987	0.256309	0.128154	−	0.991754i	$-0.459095\pi$
$587$	6.20987	0.256309	0.128154	+	0.991754i	$0.459095\pi$
$588$	0	0
$589$	−26.0864	−1.07487
$590$	0	0
$591$	17.1346	0.704825
$592$	0	0
$593$	25.2356	1.03630	0.518152	−	0.855289i	$-0.326620\pi$
$593$	25.2356	1.03630	0.518152	+	0.855289i	$0.326620\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.1571	1.19332
$598$	0	0
$599$	14.6442	0.598344	0.299172	−	0.954199i	$-0.403290\pi$
$599$	14.6442	0.598344	0.299172	+	0.954199i	$0.403290\pi$
$600$	0	0
$601$	9.87657	0.402874	0.201437	−	0.979501i	$-0.435439\pi$
$601$	9.87657	0.402874	0.201437	+	0.979501i	$0.435439\pi$
$602$	0	0
$603$	2.28845	0.0931931
$604$	0	0
$605$	4.15544	0.168943
$606$	0	0
$607$	47.1571	1.91405	0.957023	−	0.290012i	$-0.0936595\pi$
$607$	47.1571	1.91405	0.957023	+	0.290012i	$0.0936595\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26.5577	−1.07441
$612$	0	0
$613$	6.10886	0.246734	0.123367	−	0.992361i	$-0.460631\pi$
$613$	6.10886	0.246734	0.123367	+	0.992361i	$0.460631\pi$
$614$	0	0
$615$	−52.3557	−2.11119
$616$	0	0
$617$	−29.3815	−1.18285	−0.591427	−	0.806359i	$-0.701435\pi$
$617$	−29.3815	−1.18285	−0.591427	+	0.806359i	$0.701435\pi$
$618$	0	0
$619$	−17.4551	−0.701580	−0.350790	−	0.936454i	$-0.614087\pi$
$619$	−17.4551	−0.701580	−0.350790	+	0.936454i	$0.614087\pi$
$620$	0	0
$621$	−21.8894	−0.878391
$622$	0	0
$623$	0	0
$624$	0	0
$625$	64.1571	2.56628
$626$	0	0
$627$	−16.3109	−0.651394
$628$	0	0
$629$	−27.1571	−1.08282
$630$	0	0
$631$	49.6234	1.97547	0.987737	−	0.156124i	$-0.0499000\pi$
$631$	49.6234	1.97547	0.987737	+	0.156124i	$0.0499000\pi$
$632$	0	0
$633$	46.5801	1.85139
$634$	0	0
$635$	−62.7979	−2.49206
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−24.5129	−0.969715
$640$	0	0
$641$	22.1554	0.875087	0.437544	−	0.899197i	$-0.355849\pi$
$641$	22.1554	0.875087	0.437544	+	0.899197i	$0.355849\pi$
$642$	0	0
$643$	2.07685	0.0819029	0.0409514	−	0.999161i	$-0.486961\pi$
$643$	2.07685	0.0819029	0.0409514	+	0.999161i	$0.486961\pi$
$644$	0	0
$645$	44.0448	1.73426
$646$	0	0
$647$	3.67953	0.144657	0.0723287	−	0.997381i	$-0.476957\pi$
$647$	3.67953	0.144657	0.0723287	+	0.997381i	$0.476957\pi$
$648$	0	0
$649$	5.14423	0.201929
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.0016	−1.36972	−0.684860	−	0.728675i	$-0.740136\pi$
$653$	−35.0016	−1.36972	−0.684860	+	0.728675i	$0.740136\pi$
$654$	0	0
$655$	−15.3299	−0.598990
$656$	0	0
$657$	43.3669	1.69190
$658$	0	0
$659$	−48.0673	−1.87243	−0.936217	−	0.351422i	$-0.885698\pi$
$659$	−48.0673	−1.87243	−0.936217	+	0.351422i	$0.885698\pi$
$660$	0	0
$661$	41.3989	1.61023	0.805116	−	0.593118i	$-0.202103\pi$
$661$	41.3989	1.61023	0.805116	+	0.593118i	$0.202103\pi$
$662$	0	0
$663$	−66.7722	−2.59322
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.13128	−0.0825233
$668$	0	0
$669$	37.3541	1.44419
$670$	0	0
$671$	13.6105	0.525429
$672$	0	0
$673$	17.9360	0.691381	0.345691	−	0.938349i	$-0.387645\pi$
$673$	17.9360	0.691381	0.345691	+	0.938349i	$0.387645\pi$
$674$	0	0
$675$	−75.5128	−2.90649
$676$	0	0
$677$	−18.6363	−0.716252	−0.358126	−	0.933673i	$-0.616584\pi$
$677$	−18.6363	−0.716252	−0.358126	+	0.933673i	$0.616584\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	47.4679	1.81897
$682$	0	0
$683$	1.19866	0.0458653	0.0229327	−	0.999737i	$-0.492700\pi$
$683$	1.19866	0.0458653	0.0229327	+	0.999737i	$0.492700\pi$
$684$	0	0
$685$	12.6459	0.483175
$686$	0	0
$687$	63.3349	2.41638
$688$	0	0
$689$	−44.0448	−1.67797
$690$	0	0
$691$	−47.7884	−1.81796	−0.908978	−	0.416844i	$-0.863136\pi$
$691$	−47.7884	−1.81796	−0.908978	+	0.416844i	$0.863136\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	45.3367	1.71972
$696$	0	0
$697$	−19.4647	−0.737278
$698$	0	0
$699$	35.9808	1.36092
$700$	0	0
$701$	−6.51289	−0.245988	−0.122994	−	0.992407i	$-0.539250\pi$
$701$	−6.51289	−0.245988	−0.122994	+	0.992407i	$0.539250\pi$
$702$	0	0
$703$	−35.1571	−1.32597
$704$	0	0
$705$	−59.4679	−2.23969
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−20.1554	−0.756953	−0.378477	−	0.925611i	$-0.623552\pi$
$709$	−20.1554	−0.756953	−0.378477	+	0.925611i	$0.623552\pi$
$710$	0	0
$711$	41.3591	1.55109
$712$	0	0
$713$	16.2419	0.608263
$714$	0	0
$715$	22.0224	0.823592
$716$	0	0
$717$	40.6217	1.51705
$718$	0	0
$719$	−27.6795	−1.03227	−0.516136	−	0.856507i	$-0.672630\pi$
$719$	−27.6795	−1.03227	−0.516136	+	0.856507i	$0.672630\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−0.288455	−0.0107278
$724$	0	0
$725$	−7.35235	−0.273059
$726$	0	0
$727$	14.3429	0.531948	0.265974	−	0.963980i	$-0.414306\pi$
$727$	14.3429	0.531948	0.265974	+	0.963980i	$0.414306\pi$
$728$	0	0
$729$	−41.8462	−1.54986
$730$	0	0
$731$	16.3749	0.605647
$732$	0	0
$733$	6.16826	0.227830	0.113915	−	0.993491i	$-0.463661\pi$
$733$	6.16826	0.227830	0.113915	+	0.993491i	$0.463661\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.443892	−0.0163510
$738$	0	0
$739$	25.3367	0.932024	0.466012	−	0.884778i	$-0.345690\pi$
$739$	25.3367	0.932024	0.466012	+	0.884778i	$0.345690\pi$
$740$	0	0
$741$	−86.4421	−3.17553
$742$	0	0
$743$	−33.2435	−1.21959	−0.609793	−	0.792561i	$-0.708747\pi$
$743$	−33.2435	−1.21959	−0.609793	+	0.792561i	$0.708747\pi$
$744$	0	0
$745$	29.9136	1.09595
$746$	0	0
$747$	45.4903	1.66440
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.7548	−0.465428	−0.232714	−	0.972545i	$-0.574761\pi$
$751$	−12.7548	−0.465428	−0.232714	+	0.972545i	$0.574761\pi$
$752$	0	0
$753$	74.1587	2.70249
$754$	0	0
$755$	33.3367	1.21324
$756$	0	0
$757$	−21.4679	−0.780265	−0.390133	−	0.920759i	$-0.627571\pi$
$757$	−21.4679	−0.780265	−0.390133	+	0.920759i	$0.627571\pi$
$758$	0	0
$759$	10.1554	0.368619
$760$	0	0
$761$	40.1458	1.45529	0.727643	−	0.685956i	$-0.240616\pi$
$761$	40.1458	1.45529	0.727643	+	0.685956i	$0.240616\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−94.5161	−3.41724
$766$	0	0
$767$	27.2627	0.984398
$768$	0	0
$769$	−0.145844	−0.00525927	−0.00262964	−	0.999997i	$-0.500837\pi$
$769$	−0.145844	−0.00525927	−0.00262964	+	0.999997i	$0.500837\pi$
$770$	0	0
$771$	7.35907	0.265030
$772$	0	0
$773$	20.4713	0.736301	0.368150	−	0.929766i	$-0.379991\pi$
$773$	20.4713	0.736301	0.368150	+	0.929766i	$0.379991\pi$
$774$	0	0
$775$	56.0303	2.01267
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−25.1987	−0.902836
$780$	0	0
$781$	4.75476	0.170139
$782$	0	0
$783$	3.68913	0.131839
$784$	0	0
$785$	−15.9758	−0.570202
$786$	0	0
$787$	5.75638	0.205193	0.102596	−	0.994723i	$-0.467285\pi$
$787$	5.75638	0.205193	0.102596	+	0.994723i	$0.467285\pi$
$788$	0	0
$789$	58.0032	2.06497
$790$	0	0
$791$	0	0
$792$	0	0
$793$	72.1313	2.56146
$794$	0	0
$795$	−98.6250	−3.49787
$796$	0	0
$797$	−33.0432	−1.17045	−0.585225	−	0.810871i	$-0.698994\pi$
$797$	−33.0432	−1.17045	−0.585225	+	0.810871i	$0.698994\pi$
$798$	0	0
$799$	−22.1089	−0.782155
$800$	0	0
$801$	−39.0706	−1.38049
$802$	0	0
$803$	−8.41188	−0.296849
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.06402	0.283867
$808$	0	0
$809$	−29.4679	−1.03604	−0.518019	−	0.855369i	$-0.673330\pi$
$809$	−29.4679	−1.03604	−0.518019	+	0.855369i	$0.673330\pi$
$810$	0	0
$811$	−13.5095	−0.474384	−0.237192	−	0.971463i	$-0.576227\pi$
$811$	−13.5095	−0.474384	−0.237192	+	0.971463i	$0.576227\pi$
$812$	0	0
$813$	−13.7756	−0.483130
$814$	0	0
$815$	58.2693	2.04108
$816$	0	0
$817$	21.1987	0.741647
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.5960	−0.753705	−0.376852	−	0.926273i	$-0.622994\pi$
$821$	−21.5960	−0.753705	−0.376852	+	0.926273i	$0.622994\pi$
$822$	0	0
$823$	−8.64591	−0.301377	−0.150689	−	0.988581i	$-0.548149\pi$
$823$	−8.64591	−0.301377	−0.150689	+	0.988581i	$0.548149\pi$
$824$	0	0
$825$	35.0336	1.21971
$826$	0	0
$827$	10.1089	0.351519	0.175760	−	0.984433i	$-0.443762\pi$
$827$	10.1089	0.351519	0.175760	+	0.984433i	$0.443762\pi$
$828$	0	0
$829$	−20.4663	−0.710824	−0.355412	−	0.934710i	$-0.615659\pi$
$829$	−20.4663	−0.710824	−0.355412	+	0.934710i	$0.615659\pi$
$830$	0	0
$831$	−39.2211	−1.36056
$832$	0	0
$833$	0	0
$834$	0	0
$835$	79.4196	2.74843
$836$	0	0
$837$	−28.1138	−0.971756
$838$	0	0
$839$	−27.6795	−0.955604	−0.477802	−	0.878468i	$-0.658566\pi$
$839$	−27.6795	−0.955604	−0.477802	+	0.878468i	$0.658566\pi$
$840$	0	0
$841$	−28.6408	−0.987614
$842$	0	0
$843$	59.7148	2.05669
$844$	0	0
$845$	62.6907	2.15663
$846$	0	0
$847$	0	0
$848$	0	0
$849$	14.8462	0.509520
$850$	0	0
$851$	21.8894	0.750359
$852$	0	0
$853$	55.1009	1.88662	0.943309	−	0.331915i	$-0.107695\pi$
$853$	55.1009	1.88662	0.943309	+	0.331915i	$0.107695\pi$
$854$	0	0
$855$	−122.359	−4.18459
$856$	0	0
$857$	43.5689	1.48829	0.744143	−	0.668020i	$-0.232858\pi$
$857$	43.5689	1.48829	0.744143	+	0.668020i	$0.232858\pi$
$858$	0	0
$859$	9.45510	0.322604	0.161302	−	0.986905i	$-0.448431\pi$
$859$	9.45510	0.322604	0.161302	+	0.986905i	$0.448431\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.77557	0.332764	0.166382	−	0.986061i	$-0.446791\pi$
$863$	9.77557	0.332764	0.166382	+	0.986061i	$0.446791\pi$
$864$	0	0
$865$	−80.2917	−2.73000
$866$	0	0
$867$	−7.03860	−0.239043
$868$	0	0
$869$	−8.02242	−0.272142
$870$	0	0
$871$	−2.35248	−0.0797106
$872$	0	0
$873$	−71.4903	−2.41958
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.44551	0.116347	0.0581733	−	0.998307i	$-0.481472\pi$
$877$	3.44551	0.116347	0.0581733	+	0.998307i	$0.481472\pi$
$878$	0	0
$879$	−50.9966	−1.72007
$880$	0	0
$881$	11.7805	0.396897	0.198448	−	0.980111i	$-0.436410\pi$
$881$	11.7805	0.396897	0.198448	+	0.980111i	$0.436410\pi$
$882$	0	0
$883$	−16.0448	−0.539952	−0.269976	−	0.962867i	$-0.587016\pi$
$883$	−16.0448	−0.539952	−0.269976	+	0.962867i	$0.587016\pi$
$884$	0	0
$885$	61.0465	2.05205
$886$	0	0
$887$	46.2885	1.55421	0.777107	−	0.629368i	$-0.216686\pi$
$887$	46.2885	1.55421	0.777107	+	0.629368i	$0.216686\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.11222	−0.0707619
$892$	0	0
$893$	−28.6217	−0.957790
$894$	0	0
$895$	−27.9758	−0.935129
$896$	0	0
$897$	53.8204	1.79701
$898$	0	0
$899$	−2.73732	−0.0912948
$900$	0	0
$901$	−36.6666	−1.22154
$902$	0	0
$903$	0	0
$904$	0	0
$905$	31.7581	1.05568
$906$	0	0
$907$	8.37489	0.278084	0.139042	−	0.990286i	$-0.455598\pi$
$907$	8.37489	0.278084	0.139042	+	0.990286i	$0.455598\pi$
$908$	0	0
$909$	13.9214	0.461744
$910$	0	0
$911$	36.2469	1.20091	0.600456	−	0.799658i	$-0.294986\pi$
$911$	36.2469	1.20091	0.600456	+	0.799658i	$0.294986\pi$
$912$	0	0
$913$	−8.82376	−0.292024
$914$	0	0
$915$	161.516	5.33955
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.60269	−0.184816	−0.0924078	−	0.995721i	$-0.529456\pi$
$919$	−5.60269	−0.184816	−0.0924078	+	0.995721i	$0.529456\pi$
$920$	0	0
$921$	93.1603	3.06974
$922$	0	0
$923$	25.1987	0.829424
$924$	0	0
$925$	75.5128	2.48284
$926$	0	0
$927$	33.0560	1.08570
$928$	0	0
$929$	−25.7980	−0.846404	−0.423202	−	0.906035i	$-0.639094\pi$
$929$	−25.7980	−0.846404	−0.423202	+	0.906035i	$0.639094\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	71.5544	2.34258
$934$	0	0
$935$	18.3333	0.599563
$936$	0	0
$937$	−60.8124	−1.98666	−0.993328	−	0.115326i	$-0.963209\pi$
$937$	−60.8124	−1.98666	−0.993328	+	0.115326i	$0.963209\pi$
$938$	0	0
$939$	−56.7996	−1.85358
$940$	0	0
$941$	−42.2771	−1.37819	−0.689097	−	0.724669i	$-0.741993\pi$
$941$	−42.2771	−1.37819	−0.689097	+	0.724669i	$0.741993\pi$
$942$	0	0
$943$	15.6891	0.510908
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.4405	0.956689	0.478344	−	0.878172i	$-0.341237\pi$
$947$	29.4405	0.956689	0.478344	+	0.878172i	$0.341237\pi$
$948$	0	0
$949$	−44.5801	−1.44713
$950$	0	0
$951$	25.8254	0.837445
$952$	0	0
$953$	−24.5801	−0.796229	−0.398114	−	0.917336i	$-0.630335\pi$
$953$	−24.5801	−0.796229	−0.398114	+	0.917336i	$0.630335\pi$
$954$	0	0
$955$	7.66497	0.248032
$956$	0	0
$957$	−1.71155	−0.0553264
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−10.1396	−0.327084
$962$	0	0
$963$	−42.8462	−1.38070
$964$	0	0
$965$	−60.7597	−1.95593
$966$	0	0
$967$	12.6442	0.406609	0.203304	−	0.979116i	$-0.434832\pi$
$967$	12.6442	0.406609	0.203304	+	0.979116i	$0.434832\pi$
$968$	0	0
$969$	−71.9616	−2.31174
$970$	0	0
$971$	−39.6795	−1.27338	−0.636688	−	0.771121i	$-0.719696\pi$
$971$	−39.6795	−1.27338	−0.636688	+	0.771121i	$0.719696\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	185.666	5.94609
$976$	0	0
$977$	47.1745	1.50925	0.754623	−	0.656159i	$-0.227820\pi$
$977$	47.1745	1.50925	0.754623	+	0.656159i	$0.227820\pi$
$978$	0	0
$979$	7.57853	0.242211
$980$	0	0
$981$	−78.0257	−2.49117
$982$	0	0
$983$	−51.6795	−1.64832	−0.824161	−	0.566356i	$-0.808353\pi$
$983$	−51.6795	−1.64832	−0.824161	+	0.566356i	$0.808353\pi$
$984$	0	0
$985$	−24.9326	−0.794419
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.1987	−0.419693
$990$	0	0
$991$	−55.2211	−1.75416	−0.877078	−	0.480349i	$-0.840510\pi$
$991$	−55.2211	−1.75416	−0.877078	+	0.480349i	$0.840510\pi$
$992$	0	0
$993$	85.3573	2.70873
$994$	0	0
$995$	−42.4264	−1.34501
$996$	0	0
$997$	46.5432	1.47404	0.737018	−	0.675873i	$-0.236233\pi$
$997$	46.5432	1.47404	0.737018	+	0.675873i	$0.236233\pi$
$998$	0	0
$999$	−37.8894	−1.19877

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cj.1.1		3
4.3	odd	2		2156.2.a.j.1.3		3
7.6	odd	2		1232.2.a.r.1.3		3
28.3	even	6		2156.2.i.m.177.3		6
28.11	odd	6		2156.2.i.k.177.1		6
28.19	even	6		2156.2.i.m.1145.3		6
28.23	odd	6		2156.2.i.k.1145.1		6
28.27	even	2		308.2.a.c.1.1	✓	3
56.13	odd	2		4928.2.a.bx.1.1		3
56.27	even	2		4928.2.a.ca.1.3		3
84.83	odd	2		2772.2.a.s.1.3		3
140.27	odd	4		7700.2.e.p.1849.6		6
140.83	odd	4		7700.2.e.p.1849.1		6
140.139	even	2		7700.2.a.y.1.3		3
308.307	odd	2		3388.2.a.o.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.a.c.1.1	✓	3	28.27	even	2
1232.2.a.r.1.3		3	7.6	odd	2
2156.2.a.j.1.3		3	4.3	odd	2
2156.2.i.k.177.1		6	28.11	odd	6
2156.2.i.k.1145.1		6	28.23	odd	6
2156.2.i.m.177.3		6	28.3	even	6
2156.2.i.m.1145.3		6	28.19	even	6
2772.2.a.s.1.3		3	84.83	odd	2
3388.2.a.o.1.1		3	308.307	odd	2
4928.2.a.bx.1.1		3	56.13	odd	2
4928.2.a.ca.1.3		3	56.27	even	2
7700.2.a.y.1.3		3	140.139	even	2
7700.2.e.p.1849.1		6	140.83	odd	4
7700.2.e.p.1849.6		6	140.27	odd	4
8624.2.a.cj.1.1		3	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)

\(f(q)\)	\(=\)	\(q-2.85577 q^{3} +4.15544 q^{5} +5.15544 q^{9} +O(q^{10})\)	\(q-2.85577 q^{3} +4.15544 q^{5} +5.15544 q^{9} -1.00000 q^{11} -5.29966 q^{13} -11.8670 q^{15} -4.41188 q^{17} -5.71155 q^{19} +3.55611 q^{23} +12.2677 q^{25} -6.15544 q^{27} -0.599328 q^{29} +4.56732 q^{31} +2.85577 q^{33} +6.15544 q^{37} +15.1346 q^{39} +4.41188 q^{41} -3.71155 q^{43} +21.4231 q^{45} +5.01121 q^{47} +12.5993 q^{51} +8.31087 q^{53} -4.15544 q^{55} +16.3109 q^{57} -5.14423 q^{59} -13.6105 q^{61} -22.0224 q^{65} +0.443892 q^{67} -10.1554 q^{69} -4.75476 q^{71} +8.41188 q^{73} -35.0336 q^{75} +8.02242 q^{79} +2.11222 q^{81} +8.82376 q^{83} -18.3333 q^{85} +1.71155 q^{87} -7.57853 q^{89} -13.0432 q^{93} -23.7340 q^{95} -13.8670 q^{97} -5.15544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + q^{5} + 4 q^{9}+O(q^{10}) \) 3 * q - q^3 + q^5 + 4 * q^9	\( 3 q - q^{3} + q^{5} + 4 q^{9} - 3 q^{11} - 12 q^{13} - 9 q^{15} - 2 q^{17} - 2 q^{19} + 7 q^{23} + 18 q^{25} - 7 q^{27} + 6 q^{29} - 9 q^{31} + q^{33} + 7 q^{37} + 2 q^{41} + 4 q^{43} + 34 q^{45} - 4 q^{47} + 30 q^{51} + 2 q^{53} - q^{55} + 26 q^{57} - 23 q^{59} - 14 q^{61} - 28 q^{65} + 5 q^{67} - 19 q^{69} + 5 q^{71} + 14 q^{73} - 48 q^{75} - 14 q^{79} - q^{81} + 4 q^{83} + 6 q^{85} - 10 q^{87} + 19 q^{89} - 35 q^{93} - 18 q^{95} - 15 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - q^3 + q^5 + 4 * q^9 - 3 * q^11 - 12 * q^13 - 9 * q^15 - 2 * q^17 - 2 * q^19 + 7 * q^23 + 18 * q^25 - 7 * q^27 + 6 * q^29 - 9 * q^31 + q^33 + 7 * q^37 + 2 * q^41 + 4 * q^43 + 34 * q^45 - 4 * q^47 + 30 * q^51 + 2 * q^53 - q^55 + 26 * q^57 - 23 * q^59 - 14 * q^61 - 28 * q^65 + 5 * q^67 - 19 * q^69 + 5 * q^71 + 14 * q^73 - 48 * q^75 - 14 * q^79 - q^81 + 4 * q^83 + 6 * q^85 - 10 * q^87 + 19 * q^89 - 35 * q^93 - 18 * q^95 - 15 * q^97 - 4 * q^99

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