LMFDB - Embedded newform 8624.2.a.db.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:	$1$
Dimension:	$5$
Coefficient field:	5.5.559701.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 10x - 3 $ x^5 - x^4 - 7x^3 + 4x^2 + 10*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.49300$ 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)

$f(q)$	$=$	$q-1.49300 q^{3} +1.26396 q^{5} -0.770961 q^{9} +O(q^{10})$	$q-1.49300 q^{3} +1.26396 q^{5} -0.770961 q^{9} +1.00000 q^{11} -6.62065 q^{13} -1.88708 q^{15} +1.02554 q^{17} -2.10212 q^{19} +1.84754 q^{23} -3.40241 q^{25} +5.63003 q^{27} +4.86370 q^{29} +1.60912 q^{31} -1.49300 q^{33} -2.16042 q^{37} +9.88461 q^{39} +6.49373 q^{41} +12.3528 q^{43} -0.974463 q^{45} +5.02554 q^{47} -1.53112 q^{51} +2.52174 q^{53} +1.26396 q^{55} +3.13845 q^{57} -3.54051 q^{59} +2.94966 q^{61} -8.36822 q^{65} +9.84506 q^{67} -2.75837 q^{69} -8.86228 q^{71} -15.2251 q^{73} +5.07979 q^{75} +8.31503 q^{79} -6.09273 q^{81} -4.54870 q^{83} +1.29624 q^{85} -7.26148 q^{87} -6.62313 q^{89} -2.40241 q^{93} -2.65699 q^{95} -1.10674 q^{97} -0.770961 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 4 q^{5}+O(q^{10}) $ 5 * q - q^3 - 4 * q^5	$ 5 q - q^{3} - 4 q^{5} + 5 q^{11} - q^{13} - 7 q^{15} - 9 q^{17} + q^{19} + 8 q^{23} + 5 q^{25} - 4 q^{27} + 9 q^{29} + 3 q^{31} - q^{33} + 2 q^{37} + 7 q^{39} - 15 q^{41} - 14 q^{43} - 19 q^{45} + 11 q^{47} + 14 q^{51} - 9 q^{53} - 4 q^{55} + 4 q^{57} + 4 q^{59} + 2 q^{61} + 7 q^{65} + 8 q^{67} - 11 q^{69} - 15 q^{71} - 26 q^{73} + 27 q^{75} + 3 q^{79} - 19 q^{81} - q^{83} + 13 q^{85} + 14 q^{87} - 41 q^{89} + 10 q^{93} + 19 q^{95} - 7 q^{97}+O(q^{100}) $ 5 * q - q^3 - 4 * q^5 + 5 * q^11 - q^13 - 7 * q^15 - 9 * q^17 + q^19 + 8 * q^23 + 5 * q^25 - 4 * q^27 + 9 * q^29 + 3 * q^31 - q^33 + 2 * q^37 + 7 * q^39 - 15 * q^41 - 14 * q^43 - 19 * q^45 + 11 * q^47 + 14 * q^51 - 9 * q^53 - 4 * q^55 + 4 * q^57 + 4 * q^59 + 2 * q^61 + 7 * q^65 + 8 * q^67 - 11 * q^69 - 15 * q^71 - 26 * q^73 + 27 * q^75 + 3 * q^79 - 19 * q^81 - q^83 + 13 * q^85 + 14 * q^87 - 41 * q^89 + 10 * q^93 + 19 * q^95 - 7 * q^97

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
$3$	−1.49300	−0.861982	−0.430991	−	0.902356i	$-0.641836\pi$
$3$	−1.49300	−0.861982	−0.430991	+	0.902356i	$0.641836\pi$
$4$	0	0
$5$	1.26396	0.565259	0.282630	−	0.959229i	$-0.408793\pi$
$5$	1.26396	0.565259	0.282630	+	0.959229i	$0.408793\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.770961	−0.256987
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.62065	−1.83624	−0.918119	−	0.396305i	$-0.870292\pi$
$13$	−6.62065	−1.83624	−0.918119	+	0.396305i	$0.870292\pi$
$14$	0	0
$15$	−1.88708	−0.487243
$16$	0	0
$17$	1.02554	0.248729	0.124365	−	0.992237i	$-0.460311\pi$
$17$	1.02554	0.248729	0.124365	+	0.992237i	$0.460311\pi$
$18$	0	0
$19$	−2.10212	−0.482259	−0.241129	−	0.970493i	$-0.577518\pi$
$19$	−2.10212	−0.482259	−0.241129	+	0.970493i	$0.577518\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.84754	0.385239	0.192619	−	0.981274i	$-0.438302\pi$
$23$	1.84754	0.385239	0.192619	+	0.981274i	$0.438302\pi$
$24$	0	0
$25$	−3.40241	−0.680482
$26$	0	0
$27$	5.63003	1.08350
$28$	0	0
$29$	4.86370	0.903166	0.451583	−	0.892229i	$-0.350860\pi$
$29$	4.86370	0.903166	0.451583	+	0.892229i	$0.350860\pi$
$30$	0	0
$31$	1.60912	0.289006	0.144503	−	0.989504i	$-0.453842\pi$
$31$	1.60912	0.289006	0.144503	+	0.989504i	$0.453842\pi$
$32$	0	0
$33$	−1.49300	−0.259897
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.16042	−0.355172	−0.177586	−	0.984105i	$-0.556829\pi$
$37$	−2.16042	−0.355172	−0.177586	+	0.984105i	$0.556829\pi$
$38$	0	0
$39$	9.88461	1.58280
$40$	0	0
$41$	6.49373	1.01415	0.507075	−	0.861902i	$-0.330727\pi$
$41$	6.49373	1.01415	0.507075	+	0.861902i	$0.330727\pi$
$42$	0	0
$43$	12.3528	1.88378	0.941892	−	0.335916i	$-0.109046\pi$
$43$	12.3528	1.88378	0.941892	+	0.335916i	$0.109046\pi$
$44$	0	0
$45$	−0.974463	−0.145264
$46$	0	0
$47$	5.02554	0.733050	0.366525	−	0.930408i	$-0.380547\pi$
$47$	5.02554	0.733050	0.366525	+	0.930408i	$0.380547\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.53112	−0.214400
$52$	0	0
$53$	2.52174	0.346388	0.173194	−	0.984888i	$-0.444591\pi$
$53$	2.52174	0.346388	0.173194	+	0.984888i	$0.444591\pi$
$54$	0	0
$55$	1.26396	0.170432
$56$	0	0
$57$	3.13845	0.415698
$58$	0	0
$59$	−3.54051	−0.460935	−0.230467	−	0.973080i	$-0.574025\pi$
$59$	−3.54051	−0.460935	−0.230467	+	0.973080i	$0.574025\pi$
$60$	0	0
$61$	2.94966	0.377665	0.188832	−	0.982009i	$-0.439530\pi$
$61$	2.94966	0.377665	0.188832	+	0.982009i	$0.439530\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.36822	−1.03795
$66$	0	0
$67$	9.84506	1.20277	0.601383	−	0.798961i	$-0.294617\pi$
$67$	9.84506	1.20277	0.601383	+	0.798961i	$0.294617\pi$
$68$	0	0
$69$	−2.75837	−0.332069
$70$	0	0
$71$	−8.86228	−1.05176	−0.525880	−	0.850559i	$-0.676264\pi$
$71$	−8.86228	−1.05176	−0.525880	+	0.850559i	$0.676264\pi$
$72$	0	0
$73$	−15.2251	−1.78197	−0.890984	−	0.454034i	$-0.849984\pi$
$73$	−15.2251	−1.78197	−0.890984	+	0.454034i	$0.849984\pi$
$74$	0	0
$75$	5.07979	0.586563
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.31503	0.935514	0.467757	−	0.883857i	$-0.345062\pi$
$79$	8.31503	0.935514	0.467757	+	0.883857i	$0.345062\pi$
$80$	0	0
$81$	−6.09273	−0.676971
$82$	0	0
$83$	−4.54870	−0.499284	−0.249642	−	0.968338i	$-0.580313\pi$
$83$	−4.54870	−0.499284	−0.249642	+	0.968338i	$0.580313\pi$
$84$	0	0
$85$	1.29624	0.140597
$86$	0	0
$87$	−7.26148	−0.778513
$88$	0	0
$89$	−6.62313	−0.702050	−0.351025	−	0.936366i	$-0.614167\pi$
$89$	−6.62313	−0.702050	−0.351025	+	0.936366i	$0.614167\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.40241	−0.249118
$94$	0	0
$95$	−2.65699	−0.272601
$96$	0	0
$97$	−1.10674	−0.112373	−0.0561863	−	0.998420i	$-0.517894\pi$
$97$	−1.10674	−0.112373	−0.0561863	+	0.998420i	$0.517894\pi$
$98$	0	0
$99$	−0.770961	−0.0774845
$100$	0	0
$101$	−19.5905	−1.94933	−0.974665	−	0.223670i	$-0.928196\pi$
$101$	−19.5905	−1.94933	−0.974665	+	0.223670i	$0.928196\pi$
$102$	0	0
$103$	6.68038	0.658237	0.329118	−	0.944289i	$-0.393248\pi$
$103$	6.68038	0.658237	0.329118	+	0.944289i	$0.393248\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.84295	−0.564859	−0.282430	−	0.959288i	$-0.591140\pi$
$107$	−5.84295	−0.564859	−0.282430	+	0.959288i	$0.591140\pi$
$108$	0	0
$109$	−0.270699	−0.0259282	−0.0129641	−	0.999916i	$-0.504127\pi$
$109$	−0.270699	−0.0259282	−0.0129641	+	0.999916i	$0.504127\pi$
$110$	0	0
$111$	3.22551	0.306151
$112$	0	0
$113$	−11.2445	−1.05779	−0.528897	−	0.848686i	$-0.677394\pi$
$113$	−11.2445	−1.05779	−0.528897	+	0.848686i	$0.677394\pi$
$114$	0	0
$115$	2.33521	0.217760
$116$	0	0
$117$	5.10427	0.471889
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.69511	−0.874179
$124$	0	0
$125$	−10.6203	−0.949908
$126$	0	0
$127$	−3.26643	−0.289849	−0.144925	−	0.989443i	$-0.546294\pi$
$127$	−3.26643	−0.289849	−0.144925	+	0.989443i	$0.546294\pi$
$128$	0	0
$129$	−18.4427	−1.62379
$130$	0	0
$131$	15.2835	1.33532	0.667661	−	0.744466i	$-0.267296\pi$
$131$	15.2835	1.33532	0.667661	+	0.744466i	$0.267296\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	7.11612	0.612458
$136$	0	0
$137$	−15.1961	−1.29829	−0.649144	−	0.760665i	$-0.724873\pi$
$137$	−15.1961	−1.29829	−0.649144	+	0.760665i	$0.724873\pi$
$138$	0	0
$139$	−17.3356	−1.47039	−0.735193	−	0.677858i	$-0.762908\pi$
$139$	−17.3356	−1.47039	−0.735193	+	0.677858i	$0.762908\pi$
$140$	0	0
$141$	−7.50311	−0.631876
$142$	0	0
$143$	−6.62065	−0.553647
$144$	0	0
$145$	6.14751	0.510523
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.9587	1.88085	0.940425	−	0.340000i	$-0.110427\pi$
$149$	22.9587	1.88085	0.940425	+	0.340000i	$0.110427\pi$
$150$	0	0
$151$	−9.39231	−0.764335	−0.382168	−	0.924093i	$-0.624822\pi$
$151$	−9.39231	−0.764335	−0.382168	+	0.924093i	$0.624822\pi$
$152$	0	0
$153$	−0.790650	−0.0639202
$154$	0	0
$155$	2.03386	0.163364
$156$	0	0
$157$	1.01474	0.0809850	0.0404925	−	0.999180i	$-0.487107\pi$
$157$	1.01474	0.0809850	0.0404925	+	0.999180i	$0.487107\pi$
$158$	0	0
$159$	−3.76495	−0.298580
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.9137	1.32478	0.662392	−	0.749157i	$-0.269541\pi$
$163$	16.9137	1.32478	0.662392	+	0.749157i	$0.269541\pi$
$164$	0	0
$165$	−1.88708	−0.146909
$166$	0	0
$167$	−3.48220	−0.269461	−0.134730	−	0.990882i	$-0.543017\pi$
$167$	−3.48220	−0.269461	−0.134730	+	0.990882i	$0.543017\pi$
$168$	0	0
$169$	30.8330	2.37177
$170$	0	0
$171$	1.62065	0.123934
$172$	0	0
$173$	−24.2108	−1.84071	−0.920357	−	0.391079i	$-0.872102\pi$
$173$	−24.2108	−1.84071	−0.920357	+	0.391079i	$0.872102\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.28596	0.397317
$178$	0	0
$179$	−15.7706	−1.17875	−0.589376	−	0.807859i	$-0.700626\pi$
$179$	−15.7706	−1.17875	−0.589376	+	0.807859i	$0.700626\pi$
$180$	0	0
$181$	13.5710	1.00873	0.504364	−	0.863491i	$-0.331727\pi$
$181$	13.5710	1.00873	0.504364	+	0.863491i	$0.331727\pi$
$182$	0	0
$183$	−4.40383	−0.325540
$184$	0	0
$185$	−2.73069	−0.200764
$186$	0	0
$187$	1.02554	0.0749947
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.0609	−1.37920	−0.689598	−	0.724193i	$-0.742213\pi$
$191$	−19.0609	−1.37920	−0.689598	+	0.724193i	$0.742213\pi$
$192$	0	0
$193$	−7.40135	−0.532761	−0.266381	−	0.963868i	$-0.585828\pi$
$193$	−7.40135	−0.532761	−0.266381	+	0.963868i	$0.585828\pi$
$194$	0	0
$195$	12.4937	0.894694
$196$	0	0
$197$	2.01629	0.143655	0.0718273	−	0.997417i	$-0.477117\pi$
$197$	2.01629	0.143655	0.0718273	+	0.997417i	$0.477117\pi$
$198$	0	0
$199$	2.29746	0.162863	0.0814313	−	0.996679i	$-0.474051\pi$
$199$	2.29746	0.162863	0.0814313	+	0.996679i	$0.474051\pi$
$200$	0	0
$201$	−14.6986	−1.03676
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.20780	0.573258
$206$	0	0
$207$	−1.42438	−0.0990014
$208$	0	0
$209$	−2.10212	−0.145406
$210$	0	0
$211$	−20.3438	−1.40052	−0.700262	−	0.713886i	$-0.746933\pi$
$211$	−20.3438	−1.40052	−0.700262	+	0.713886i	$0.746933\pi$
$212$	0	0
$213$	13.2314	0.906597
$214$	0	0
$215$	15.6134	1.06483
$216$	0	0
$217$	0	0
$218$	0	0
$219$	22.7311	1.53602
$220$	0	0
$221$	−6.78972	−0.456726
$222$	0	0
$223$	18.0165	1.20648	0.603238	−	0.797561i	$-0.293877\pi$
$223$	18.0165	1.20648	0.603238	+	0.797561i	$0.293877\pi$
$224$	0	0
$225$	2.62313	0.174875
$226$	0	0
$227$	7.81692	0.518827	0.259414	−	0.965766i	$-0.416471\pi$
$227$	7.81692	0.518827	0.259414	+	0.965766i	$0.416471\pi$
$228$	0	0
$229$	3.26858	0.215994	0.107997	−	0.994151i	$-0.465556\pi$
$229$	3.26858	0.215994	0.107997	+	0.994151i	$0.465556\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.1099	1.77603	0.888014	−	0.459817i	$-0.152085\pi$
$233$	27.1099	1.77603	0.888014	+	0.459817i	$0.152085\pi$
$234$	0	0
$235$	6.35207	0.414363
$236$	0	0
$237$	−12.4143	−0.806396
$238$	0	0
$239$	−23.3429	−1.50992	−0.754962	−	0.655769i	$-0.772345\pi$
$239$	−23.3429	−1.50992	−0.754962	+	0.655769i	$0.772345\pi$
$240$	0	0
$241$	−19.8257	−1.27709	−0.638544	−	0.769585i	$-0.720463\pi$
$241$	−19.8257	−1.27709	−0.638544	+	0.769585i	$0.720463\pi$
$242$	0	0
$243$	−7.79366	−0.499964
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.9174	0.885542
$248$	0	0
$249$	6.79119	0.430374
$250$	0	0
$251$	−23.7297	−1.49780	−0.748902	−	0.662681i	$-0.769419\pi$
$251$	−23.7297	−1.49780	−0.748902	+	0.662681i	$0.769419\pi$
$252$	0	0
$253$	1.84754	0.116154
$254$	0	0
$255$	−1.93528	−0.121192
$256$	0	0
$257$	−19.3664	−1.20804	−0.604022	−	0.796967i	$-0.706436\pi$
$257$	−19.3664	−1.20804	−0.604022	+	0.796967i	$0.706436\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.74972	−0.232102
$262$	0	0
$263$	−1.36547	−0.0841987	−0.0420994	−	0.999113i	$-0.513405\pi$
$263$	−1.36547	−0.0841987	−0.0420994	+	0.999113i	$0.513405\pi$
$264$	0	0
$265$	3.18738	0.195799
$266$	0	0
$267$	9.88831	0.605154
$268$	0	0
$269$	−11.5113	−0.701854	−0.350927	−	0.936403i	$-0.614134\pi$
$269$	−11.5113	−0.701854	−0.350927	+	0.936403i	$0.614134\pi$
$270$	0	0
$271$	−8.76267	−0.532294	−0.266147	−	0.963932i	$-0.585751\pi$
$271$	−8.76267	−0.532294	−0.266147	+	0.963932i	$0.585751\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.40241	−0.205173
$276$	0	0
$277$	10.3381	0.621154	0.310577	−	0.950548i	$-0.399478\pi$
$277$	10.3381	0.621154	0.310577	+	0.950548i	$0.399478\pi$
$278$	0	0
$279$	−1.24057	−0.0742709
$280$	0	0
$281$	19.2331	1.14735	0.573676	−	0.819083i	$-0.305517\pi$
$281$	19.2331	1.14735	0.573676	+	0.819083i	$0.305517\pi$
$282$	0	0
$283$	−11.6008	−0.689598	−0.344799	−	0.938677i	$-0.612053\pi$
$283$	−11.6008	−0.689598	−0.344799	+	0.938677i	$0.612053\pi$
$284$	0	0
$285$	3.96687	0.234977
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.9483	−0.938134
$290$	0	0
$291$	1.65236	0.0968631
$292$	0	0
$293$	0.737833	0.0431047	0.0215523	−	0.999768i	$-0.493139\pi$
$293$	0.737833	0.0431047	0.0215523	+	0.999768i	$0.493139\pi$
$294$	0	0
$295$	−4.47505	−0.260547
$296$	0	0
$297$	5.63003	0.326688
$298$	0	0
$299$	−12.2319	−0.707390
$300$	0	0
$301$	0	0
$302$	0	0
$303$	29.2486	1.68029
$304$	0	0
$305$	3.72824	0.213479
$306$	0	0
$307$	−15.0009	−0.856144	−0.428072	−	0.903745i	$-0.640807\pi$
$307$	−15.0009	−0.856144	−0.428072	+	0.903745i	$0.640807\pi$
$308$	0	0
$309$	−9.97378	−0.567388
$310$	0	0
$311$	−15.9493	−0.904403	−0.452202	−	0.891916i	$-0.649361\pi$
$311$	−15.9493	−0.904403	−0.452202	+	0.891916i	$0.649361\pi$
$312$	0	0
$313$	−22.7613	−1.28654	−0.643271	−	0.765639i	$-0.722423\pi$
$313$	−22.7613	−1.28654	−0.643271	+	0.765639i	$0.722423\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.35756	−0.413242	−0.206621	−	0.978421i	$-0.566247\pi$
$317$	−7.35756	−0.413242	−0.206621	+	0.978421i	$0.566247\pi$
$318$	0	0
$319$	4.86370	0.272315
$320$	0	0
$321$	8.72350	0.486898
$322$	0	0
$323$	−2.15580	−0.119952
$324$	0	0
$325$	22.5262	1.24953
$326$	0	0
$327$	0.404152	0.0223497
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.9204	1.47968	0.739839	−	0.672784i	$-0.234902\pi$
$331$	26.9204	1.47968	0.739839	+	0.672784i	$0.234902\pi$
$332$	0	0
$333$	1.66560	0.0912745
$334$	0	0
$335$	12.4437	0.679874
$336$	0	0
$337$	24.4273	1.33064	0.665319	−	0.746559i	$-0.268296\pi$
$337$	24.4273	1.33064	0.665319	+	0.746559i	$0.268296\pi$
$338$	0	0
$339$	16.7880	0.911800
$340$	0	0
$341$	1.60912	0.0871387
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.48647	−0.187705
$346$	0	0
$347$	34.8484	1.87076	0.935381	−	0.353641i	$-0.115057\pi$
$347$	34.8484	1.87076	0.935381	+	0.353641i	$0.115057\pi$
$348$	0	0
$349$	−9.58943	−0.513311	−0.256655	−	0.966503i	$-0.582620\pi$
$349$	−9.58943	−0.513311	−0.256655	+	0.966503i	$0.582620\pi$
$350$	0	0
$351$	−37.2745	−1.98956
$352$	0	0
$353$	−14.2205	−0.756881	−0.378441	−	0.925626i	$-0.623540\pi$
$353$	−14.2205	−0.756881	−0.378441	+	0.925626i	$0.623540\pi$
$354$	0	0
$355$	−11.2015	−0.594516
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.4672	0.657992	0.328996	−	0.944331i	$-0.393290\pi$
$359$	12.4672	0.657992	0.328996	+	0.944331i	$0.393290\pi$
$360$	0	0
$361$	−14.5811	−0.767427
$362$	0	0
$363$	−1.49300	−0.0783620
$364$	0	0
$365$	−19.2439	−1.00727
$366$	0	0
$367$	−6.95408	−0.363000	−0.181500	−	0.983391i	$-0.558095\pi$
$367$	−6.95408	−0.363000	−0.181500	+	0.983391i	$0.558095\pi$
$368$	0	0
$369$	−5.00641	−0.260623
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13.5762	0.702947	0.351473	−	0.936198i	$-0.385681\pi$
$373$	13.5762	0.702947	0.351473	+	0.936198i	$0.385681\pi$
$374$	0	0
$375$	15.8561	0.818803
$376$	0	0
$377$	−32.2008	−1.65843
$378$	0	0
$379$	18.6650	0.958757	0.479378	−	0.877608i	$-0.340862\pi$
$379$	18.6650	0.958757	0.479378	+	0.877608i	$0.340862\pi$
$380$	0	0
$381$	4.87678	0.249845
$382$	0	0
$383$	4.28024	0.218710	0.109355	−	0.994003i	$-0.465121\pi$
$383$	4.28024	0.218710	0.109355	+	0.994003i	$0.465121\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.52353	−0.484108
$388$	0	0
$389$	3.88827	0.197143	0.0985716	−	0.995130i	$-0.468573\pi$
$389$	3.88827	0.197143	0.0985716	+	0.995130i	$0.468573\pi$
$390$	0	0
$391$	1.89472	0.0958202
$392$	0	0
$393$	−22.8181	−1.15102
$394$	0	0
$395$	10.5099	0.528808
$396$	0	0
$397$	−36.4734	−1.83055	−0.915273	−	0.402835i	$-0.868025\pi$
$397$	−36.4734	−1.83055	−0.915273	+	0.402835i	$0.868025\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.745063	−0.0372067	−0.0186033	−	0.999827i	$-0.505922\pi$
$401$	−0.745063	−0.0372067	−0.0186033	+	0.999827i	$0.505922\pi$
$402$	0	0
$403$	−10.6534	−0.530685
$404$	0	0
$405$	−7.70096	−0.382664
$406$	0	0
$407$	−2.16042	−0.107088
$408$	0	0
$409$	−8.55772	−0.423152	−0.211576	−	0.977362i	$-0.567860\pi$
$409$	−8.55772	−0.423152	−0.211576	+	0.977362i	$0.567860\pi$
$410$	0	0
$411$	22.6877	1.11910
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.74936	−0.282225
$416$	0	0
$417$	25.8820	1.26745
$418$	0	0
$419$	12.1216	0.592181	0.296091	−	0.955160i	$-0.404317\pi$
$419$	12.1216	0.592181	0.296091	+	0.955160i	$0.404317\pi$
$420$	0	0
$421$	3.61874	0.176367	0.0881833	−	0.996104i	$-0.471894\pi$
$421$	3.61874	0.176367	0.0881833	+	0.996104i	$0.471894\pi$
$422$	0	0
$423$	−3.87449	−0.188384
$424$	0	0
$425$	−3.48930	−0.169256
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9.88461	0.477233
$430$	0	0
$431$	−31.2526	−1.50538	−0.752692	−	0.658372i	$-0.771245\pi$
$431$	−31.2526	−1.50538	−0.752692	+	0.658372i	$0.771245\pi$
$432$	0	0
$433$	−2.21202	−0.106303	−0.0531514	−	0.998586i	$-0.516927\pi$
$433$	−2.21202	−0.106303	−0.0531514	+	0.998586i	$0.516927\pi$
$434$	0	0
$435$	−9.17821	−0.440061
$436$	0	0
$437$	−3.88375	−0.185785
$438$	0	0
$439$	16.4508	0.785153	0.392577	−	0.919719i	$-0.371584\pi$
$439$	16.4508	0.785153	0.392577	+	0.919719i	$0.371584\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.6790	−1.31507	−0.657534	−	0.753425i	$-0.728400\pi$
$443$	−27.6790	−1.31507	−0.657534	+	0.753425i	$0.728400\pi$
$444$	0	0
$445$	−8.37135	−0.396840
$446$	0	0
$447$	−34.2773	−1.62126
$448$	0	0
$449$	7.75897	0.366169	0.183084	−	0.983097i	$-0.441392\pi$
$449$	7.75897	0.366169	0.183084	+	0.983097i	$0.441392\pi$
$450$	0	0
$451$	6.49373	0.305778
$452$	0	0
$453$	14.0227	0.658843
$454$	0	0
$455$	0	0
$456$	0	0
$457$	36.5480	1.70965	0.854823	−	0.518920i	$-0.173666\pi$
$457$	36.5480	1.70965	0.854823	+	0.518920i	$0.173666\pi$
$458$	0	0
$459$	5.77381	0.269498
$460$	0	0
$461$	15.1821	0.707099	0.353550	−	0.935416i	$-0.384975\pi$
$461$	15.1821	0.707099	0.353550	+	0.935416i	$0.384975\pi$
$462$	0	0
$463$	−20.5267	−0.953958	−0.476979	−	0.878915i	$-0.658268\pi$
$463$	−20.5267	−0.953958	−0.476979	+	0.878915i	$0.658268\pi$
$464$	0	0
$465$	−3.03655	−0.140816
$466$	0	0
$467$	14.7387	0.682026	0.341013	−	0.940058i	$-0.389230\pi$
$467$	14.7387	0.682026	0.341013	+	0.940058i	$0.389230\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.51500	−0.0698076
$472$	0	0
$473$	12.3528	0.567982
$474$	0	0
$475$	7.15226	0.328168
$476$	0	0
$477$	−1.94417	−0.0890172
$478$	0	0
$479$	−24.7905	−1.13271	−0.566353	−	0.824162i	$-0.691646\pi$
$479$	−24.7905	−1.13271	−0.566353	+	0.824162i	$0.691646\pi$
$480$	0	0
$481$	14.3034	0.652180
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.39887	−0.0635196
$486$	0	0
$487$	−27.0906	−1.22759	−0.613796	−	0.789464i	$-0.710358\pi$
$487$	−27.0906	−1.22759	−0.613796	+	0.789464i	$0.710358\pi$
$488$	0	0
$489$	−25.2521	−1.14194
$490$	0	0
$491$	23.3679	1.05458	0.527289	−	0.849686i	$-0.323208\pi$
$491$	23.3679	1.05458	0.527289	+	0.849686i	$0.323208\pi$
$492$	0	0
$493$	4.98790	0.224644
$494$	0	0
$495$	−0.974463	−0.0437988
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.73287	0.0775738	0.0387869	−	0.999248i	$-0.487651\pi$
$499$	1.73287	0.0775738	0.0387869	+	0.999248i	$0.487651\pi$
$500$	0	0
$501$	5.19891	0.232270
$502$	0	0
$503$	17.9511	0.800399	0.400199	−	0.916428i	$-0.368941\pi$
$503$	17.9511	0.800399	0.400199	+	0.916428i	$0.368941\pi$
$504$	0	0
$505$	−24.7616	−1.10188
$506$	0	0
$507$	−46.0336	−2.04442
$508$	0	0
$509$	26.8142	1.18852	0.594260	−	0.804273i	$-0.297445\pi$
$509$	26.8142	1.18852	0.594260	+	0.804273i	$0.297445\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−11.8350	−0.522527
$514$	0	0
$515$	8.44371	0.372074
$516$	0	0
$517$	5.02554	0.221023
$518$	0	0
$519$	36.1467	1.58666
$520$	0	0
$521$	−10.2430	−0.448756	−0.224378	−	0.974502i	$-0.572035\pi$
$521$	−10.2430	−0.448756	−0.224378	+	0.974502i	$0.572035\pi$
$522$	0	0
$523$	−40.3432	−1.76409	−0.882043	−	0.471169i	$-0.843832\pi$
$523$	−40.3432	−1.76409	−0.882043	+	0.471169i	$0.843832\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.65021	0.0718844
$528$	0	0
$529$	−19.5866	−0.851591
$530$	0	0
$531$	2.72959	0.118454
$532$	0	0
$533$	−42.9927	−1.86222
$534$	0	0
$535$	−7.38524	−0.319292
$536$	0	0
$537$	23.5455	1.01606
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16.5351	−0.710898	−0.355449	−	0.934696i	$-0.615672\pi$
$541$	−16.5351	−0.710898	−0.355449	+	0.934696i	$0.615672\pi$
$542$	0	0
$543$	−20.2615	−0.869505
$544$	0	0
$545$	−0.342152	−0.0146562
$546$	0	0
$547$	−28.5561	−1.22097	−0.610486	−	0.792027i	$-0.709026\pi$
$547$	−28.5561	−1.22097	−0.610486	+	0.792027i	$0.709026\pi$
$548$	0	0
$549$	−2.27407	−0.0970550
$550$	0	0
$551$	−10.2241	−0.435559
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.07690	0.173055
$556$	0	0
$557$	−33.7581	−1.43037	−0.715187	−	0.698933i	$-0.753659\pi$
$557$	−33.7581	−1.43037	−0.715187	+	0.698933i	$0.753659\pi$
$558$	0	0
$559$	−81.7836	−3.45908
$560$	0	0
$561$	−1.53112	−0.0646441
$562$	0	0
$563$	5.24519	0.221059	0.110529	−	0.993873i	$-0.464745\pi$
$563$	5.24519	0.221059	0.110529	+	0.993873i	$0.464745\pi$
$564$	0	0
$565$	−14.2126	−0.597928
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.96331	0.166150	0.0830752	−	0.996543i	$-0.473526\pi$
$569$	3.96331	0.166150	0.0830752	+	0.996543i	$0.473526\pi$
$570$	0	0
$571$	−24.6918	−1.03332	−0.516660	−	0.856190i	$-0.672825\pi$
$571$	−24.6918	−1.03332	−0.516660	+	0.856190i	$0.672825\pi$
$572$	0	0
$573$	28.4578	1.18884
$574$	0	0
$575$	−6.28609	−0.262148
$576$	0	0
$577$	−9.43803	−0.392910	−0.196455	−	0.980513i	$-0.562943\pi$
$577$	−9.43803	−0.392910	−0.196455	+	0.980513i	$0.562943\pi$
$578$	0	0
$579$	11.0502	0.459230
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.52174	0.104440
$584$	0	0
$585$	6.45158	0.266740
$586$	0	0
$587$	18.7860	0.775383	0.387691	−	0.921789i	$-0.373273\pi$
$587$	18.7860	0.775383	0.387691	+	0.921789i	$0.373273\pi$
$588$	0	0
$589$	−3.38256	−0.139376
$590$	0	0
$591$	−3.01031	−0.123828
$592$	0	0
$593$	3.55368	0.145932	0.0729661	−	0.997334i	$-0.476754\pi$
$593$	3.55368	0.145932	0.0729661	+	0.997334i	$0.476754\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.43010	−0.140385
$598$	0	0
$599$	−16.2369	−0.663422	−0.331711	−	0.943381i	$-0.607626\pi$
$599$	−16.2369	−0.663422	−0.331711	+	0.943381i	$0.607626\pi$
$600$	0	0
$601$	−45.1827	−1.84304	−0.921521	−	0.388329i	$-0.873052\pi$
$601$	−45.1827	−1.84304	−0.921521	+	0.388329i	$0.873052\pi$
$602$	0	0
$603$	−7.59016	−0.309095
$604$	0	0
$605$	1.26396	0.0513872
$606$	0	0
$607$	6.65845	0.270258	0.135129	−	0.990828i	$-0.456855\pi$
$607$	6.65845	0.270258	0.135129	+	0.990828i	$0.456855\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−33.2723	−1.34605
$612$	0	0
$613$	27.5866	1.11421	0.557106	−	0.830442i	$-0.311912\pi$
$613$	27.5866	1.11421	0.557106	+	0.830442i	$0.311912\pi$
$614$	0	0
$615$	−12.2542	−0.494138
$616$	0	0
$617$	2.16360	0.0871032	0.0435516	−	0.999051i	$-0.486133\pi$
$617$	2.16360	0.0871032	0.0435516	+	0.999051i	$0.486133\pi$
$618$	0	0
$619$	21.8246	0.877204	0.438602	−	0.898681i	$-0.355474\pi$
$619$	21.8246	0.877204	0.438602	+	0.898681i	$0.355474\pi$
$620$	0	0
$621$	10.4017	0.417406
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.58845	0.143538
$626$	0	0
$627$	3.13845	0.125338
$628$	0	0
$629$	−2.21560	−0.0883416
$630$	0	0
$631$	35.8552	1.42737	0.713687	−	0.700465i	$-0.247024\pi$
$631$	35.8552	1.42737	0.713687	+	0.700465i	$0.247024\pi$
$632$	0	0
$633$	30.3732	1.20723
$634$	0	0
$635$	−4.12864	−0.163840
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.83247	0.270289
$640$	0	0
$641$	−20.2748	−0.800806	−0.400403	−	0.916339i	$-0.631130\pi$
$641$	−20.2748	−0.800806	−0.400403	+	0.916339i	$0.631130\pi$
$642$	0	0
$643$	−10.2348	−0.403620	−0.201810	−	0.979425i	$-0.564682\pi$
$643$	−10.2348	−0.403620	−0.201810	+	0.979425i	$0.564682\pi$
$644$	0	0
$645$	−23.3108	−0.917861
$646$	0	0
$647$	20.9968	0.825471	0.412735	−	0.910851i	$-0.364573\pi$
$647$	20.9968	0.825471	0.412735	+	0.910851i	$0.364573\pi$
$648$	0	0
$649$	−3.54051	−0.138977
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.37617	0.288652	0.144326	−	0.989530i	$-0.453899\pi$
$653$	7.37617	0.288652	0.144326	+	0.989530i	$0.453899\pi$
$654$	0	0
$655$	19.3176	0.754803
$656$	0	0
$657$	11.7380	0.457943
$658$	0	0
$659$	13.0961	0.510152	0.255076	−	0.966921i	$-0.417900\pi$
$659$	13.0961	0.510152	0.255076	+	0.966921i	$0.417900\pi$
$660$	0	0
$661$	−47.4950	−1.84734	−0.923671	−	0.383186i	$-0.874827\pi$
$661$	−47.4950	−1.84734	−0.923671	+	0.383186i	$0.874827\pi$
$662$	0	0
$663$	10.1370	0.393690
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.98587	0.347934
$668$	0	0
$669$	−26.8986	−1.03996
$670$	0	0
$671$	2.94966	0.113870
$672$	0	0
$673$	−13.9244	−0.536745	−0.268372	−	0.963315i	$-0.586486\pi$
$673$	−13.9244	−0.536745	−0.268372	+	0.963315i	$0.586486\pi$
$674$	0	0
$675$	−19.1557	−0.737303
$676$	0	0
$677$	51.3953	1.97528	0.987642	−	0.156728i	$-0.0500948\pi$
$677$	51.3953	1.97528	0.987642	+	0.156728i	$0.0500948\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11.6706	−0.447220
$682$	0	0
$683$	27.1874	1.04030	0.520149	−	0.854076i	$-0.325876\pi$
$683$	27.1874	1.04030	0.520149	+	0.854076i	$0.325876\pi$
$684$	0	0
$685$	−19.2072	−0.733869
$686$	0	0
$687$	−4.87998	−0.186183
$688$	0	0
$689$	−16.6956	−0.636051
$690$	0	0
$691$	−24.2262	−0.921610	−0.460805	−	0.887502i	$-0.652439\pi$
$691$	−24.2262	−0.921610	−0.460805	+	0.887502i	$0.652439\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.9115	−0.831149
$696$	0	0
$697$	6.65956	0.252249
$698$	0	0
$699$	−40.4750	−1.53090
$700$	0	0
$701$	−50.4982	−1.90729	−0.953645	−	0.300935i	$-0.902701\pi$
$701$	−50.4982	−1.90729	−0.953645	+	0.300935i	$0.902701\pi$
$702$	0	0
$703$	4.54146	0.171285
$704$	0	0
$705$	−9.48361	−0.357174
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−31.3941	−1.17903	−0.589516	−	0.807757i	$-0.700681\pi$
$709$	−31.3941	−1.17903	−0.589516	+	0.807757i	$0.700681\pi$
$710$	0	0
$711$	−6.41057	−0.240415
$712$	0	0
$713$	2.97291	0.111337
$714$	0	0
$715$	−8.36822	−0.312954
$716$	0	0
$717$	34.8508	1.30153
$718$	0	0
$719$	−17.0611	−0.636270	−0.318135	−	0.948045i	$-0.603057\pi$
$719$	−17.0611	−0.636270	−0.318135	+	0.948045i	$0.603057\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	29.5998	1.10083
$724$	0	0
$725$	−16.5483	−0.614588
$726$	0	0
$727$	−24.4693	−0.907517	−0.453758	−	0.891125i	$-0.649917\pi$
$727$	−24.4693	−0.907517	−0.453758	+	0.891125i	$0.649917\pi$
$728$	0	0
$729$	29.9141	1.10793
$730$	0	0
$731$	12.6683	0.468552
$732$	0	0
$733$	24.9260	0.920664	0.460332	−	0.887747i	$-0.347730\pi$
$733$	24.9260	0.920664	0.460332	+	0.887747i	$0.347730\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.84506	0.362648
$738$	0	0
$739$	−1.83293	−0.0674255	−0.0337128	−	0.999432i	$-0.510733\pi$
$739$	−1.83293	−0.0674255	−0.0337128	+	0.999432i	$0.510733\pi$
$740$	0	0
$741$	−20.7786	−0.763321
$742$	0	0
$743$	18.5966	0.682242	0.341121	−	0.940019i	$-0.389193\pi$
$743$	18.5966	0.682242	0.341121	+	0.940019i	$0.389193\pi$
$744$	0	0
$745$	29.0188	1.06317
$746$	0	0
$747$	3.50687	0.128310
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−47.1642	−1.72105	−0.860523	−	0.509411i	$-0.829863\pi$
$751$	−47.1642	−1.72105	−0.860523	+	0.509411i	$0.829863\pi$
$752$	0	0
$753$	35.4283	1.29108
$754$	0	0
$755$	−11.8715	−0.432048
$756$	0	0
$757$	−14.8217	−0.538703	−0.269351	−	0.963042i	$-0.586809\pi$
$757$	−14.8217	−0.538703	−0.269351	+	0.963042i	$0.586809\pi$
$758$	0	0
$759$	−2.75837	−0.100123
$760$	0	0
$761$	7.76647	0.281534	0.140767	−	0.990043i	$-0.455043\pi$
$761$	7.76647	0.281534	0.140767	+	0.990043i	$0.455043\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.999348	−0.0361315
$766$	0	0
$767$	23.4404	0.846386
$768$	0	0
$769$	22.0677	0.795783	0.397891	−	0.917432i	$-0.369742\pi$
$769$	22.0677	0.795783	0.397891	+	0.917432i	$0.369742\pi$
$770$	0	0
$771$	28.9140	1.04131
$772$	0	0
$773$	−29.6946	−1.06804	−0.534020	−	0.845472i	$-0.679319\pi$
$773$	−29.6946	−1.06804	−0.534020	+	0.845472i	$0.679319\pi$
$774$	0	0
$775$	−5.47489	−0.196664
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.6506	−0.489083
$780$	0	0
$781$	−8.86228	−0.317117
$782$	0	0
$783$	27.3828	0.978580
$784$	0	0
$785$	1.28259	0.0457775
$786$	0	0
$787$	12.5462	0.447224	0.223612	−	0.974678i	$-0.428215\pi$
$787$	12.5462	0.447224	0.223612	+	0.974678i	$0.428215\pi$
$788$	0	0
$789$	2.03865	0.0725778
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−19.5286	−0.693483
$794$	0	0
$795$	−4.75874	−0.168775
$796$	0	0
$797$	−2.10555	−0.0745825	−0.0372913	−	0.999304i	$-0.511873\pi$
$797$	−2.10555	−0.0745825	−0.0372913	+	0.999304i	$0.511873\pi$
$798$	0	0
$799$	5.15388	0.182331
$800$	0	0
$801$	5.10617	0.180418
$802$	0	0
$803$	−15.2251	−0.537284
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.1863	0.604986
$808$	0	0
$809$	−20.1294	−0.707713	−0.353857	−	0.935300i	$-0.615130\pi$
$809$	−20.1294	−0.707713	−0.353857	+	0.935300i	$0.615130\pi$
$810$	0	0
$811$	46.7505	1.64163	0.820817	−	0.571192i	$-0.193519\pi$
$811$	46.7505	1.64163	0.820817	+	0.571192i	$0.193519\pi$
$812$	0	0
$813$	13.0826	0.458828
$814$	0	0
$815$	21.3782	0.748846
$816$	0	0
$817$	−25.9670	−0.908471
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.1335	−0.807366	−0.403683	−	0.914899i	$-0.632270\pi$
$821$	−23.1335	−0.807366	−0.403683	+	0.914899i	$0.632270\pi$
$822$	0	0
$823$	−10.3392	−0.360400	−0.180200	−	0.983630i	$-0.557675\pi$
$823$	−10.3392	−0.360400	−0.180200	+	0.983630i	$0.557675\pi$
$824$	0	0
$825$	5.07979	0.176855
$826$	0	0
$827$	−48.1690	−1.67500	−0.837500	−	0.546437i	$-0.815984\pi$
$827$	−48.1690	−1.67500	−0.837500	+	0.546437i	$0.815984\pi$
$828$	0	0
$829$	45.7897	1.59034	0.795170	−	0.606386i	$-0.207381\pi$
$829$	45.7897	1.59034	0.795170	+	0.606386i	$0.207381\pi$
$830$	0	0
$831$	−15.4347	−0.535423
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.40135	−0.152315
$836$	0	0
$837$	9.05940	0.313139
$838$	0	0
$839$	15.4999	0.535116	0.267558	−	0.963542i	$-0.413783\pi$
$839$	15.4999	0.535116	0.267558	+	0.963542i	$0.413783\pi$
$840$	0	0
$841$	−5.34446	−0.184292
$842$	0	0
$843$	−28.7150	−0.988996
$844$	0	0
$845$	38.9716	1.34066
$846$	0	0
$847$	0	0
$848$	0	0
$849$	17.3200	0.594421
$850$	0	0
$851$	−3.99147	−0.136826
$852$	0	0
$853$	21.4586	0.734729	0.367364	−	0.930077i	$-0.380260\pi$
$853$	21.4586	0.734729	0.367364	+	0.930077i	$0.380260\pi$
$854$	0	0
$855$	2.04843	0.0700550
$856$	0	0
$857$	−13.3095	−0.454645	−0.227322	−	0.973820i	$-0.572997\pi$
$857$	−13.3095	−0.454645	−0.227322	+	0.973820i	$0.572997\pi$
$858$	0	0
$859$	−36.9452	−1.26055	−0.630277	−	0.776371i	$-0.717059\pi$
$859$	−36.9452	−1.26055	−0.630277	+	0.776371i	$0.717059\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−7.34232	−0.249936	−0.124968	−	0.992161i	$-0.539883\pi$
$863$	−7.34232	−0.249936	−0.124968	+	0.992161i	$0.539883\pi$
$864$	0	0
$865$	−30.6014	−1.04048
$866$	0	0
$867$	23.8107	0.808654
$868$	0	0
$869$	8.31503	0.282068
$870$	0	0
$871$	−65.1807	−2.20856
$872$	0	0
$873$	0.853255	0.0288783
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.3452	0.349332	0.174666	−	0.984628i	$-0.444115\pi$
$877$	10.3452	0.349332	0.174666	+	0.984628i	$0.444115\pi$
$878$	0	0
$879$	−1.10158	−0.0371555
$880$	0	0
$881$	17.0322	0.573830	0.286915	−	0.957956i	$-0.407370\pi$
$881$	17.0322	0.573830	0.286915	+	0.957956i	$0.407370\pi$
$882$	0	0
$883$	2.22157	0.0747618	0.0373809	−	0.999301i	$-0.488099\pi$
$883$	2.22157	0.0747618	0.0373809	+	0.999301i	$0.488099\pi$
$884$	0	0
$885$	6.68123	0.224587
$886$	0	0
$887$	−13.7718	−0.462412	−0.231206	−	0.972905i	$-0.574267\pi$
$887$	−13.7718	−0.462412	−0.231206	+	0.972905i	$0.574267\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.09273	−0.204114
$892$	0	0
$893$	−10.5643	−0.353520
$894$	0	0
$895$	−19.9334	−0.666301
$896$	0	0
$897$	18.2622	0.609758
$898$	0	0
$899$	7.82627	0.261021
$900$	0	0
$901$	2.58614	0.0861568
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.1532	0.570192
$906$	0	0
$907$	40.9581	1.35999	0.679995	−	0.733217i	$-0.261982\pi$
$907$	40.9581	1.35999	0.679995	+	0.733217i	$0.261982\pi$
$908$	0	0
$909$	15.1035	0.500953
$910$	0	0
$911$	−15.9473	−0.528359	−0.264180	−	0.964474i	$-0.585101\pi$
$911$	−15.9473	−0.528359	−0.264180	+	0.964474i	$0.585101\pi$
$912$	0	0
$913$	−4.54870	−0.150540
$914$	0	0
$915$	−5.56625	−0.184015
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37.7132	−1.24404	−0.622022	−	0.783000i	$-0.713689\pi$
$919$	−37.7132	−1.24404	−0.622022	+	0.783000i	$0.713689\pi$
$920$	0	0
$921$	22.3962	0.737981
$922$	0	0
$923$	58.6741	1.93128
$924$	0	0
$925$	7.35065	0.241688
$926$	0	0
$927$	−5.15031	−0.169158
$928$	0	0
$929$	28.9034	0.948290	0.474145	−	0.880447i	$-0.342757\pi$
$929$	28.9034	0.948290	0.474145	+	0.880447i	$0.342757\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	23.8123	0.779579
$934$	0	0
$935$	1.29624	0.0423915
$936$	0	0
$937$	−14.8484	−0.485075	−0.242538	−	0.970142i	$-0.577980\pi$
$937$	−14.8484	−0.485075	−0.242538	+	0.970142i	$0.577980\pi$
$938$	0	0
$939$	33.9825	1.10898
$940$	0	0
$941$	−43.2503	−1.40992	−0.704960	−	0.709247i	$-0.749035\pi$
$941$	−43.2503	−1.40992	−0.704960	+	0.709247i	$0.749035\pi$
$942$	0	0
$943$	11.9974	0.390690
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.92145	0.257413	0.128706	−	0.991683i	$-0.458918\pi$
$947$	7.92145	0.257413	0.128706	+	0.991683i	$0.458918\pi$
$948$	0	0
$949$	100.800	3.27212
$950$	0	0
$951$	10.9848	0.356207
$952$	0	0
$953$	−13.5952	−0.440390	−0.220195	−	0.975456i	$-0.570669\pi$
$953$	−13.5952	−0.440390	−0.220195	+	0.975456i	$0.570669\pi$
$954$	0	0
$955$	−24.0921	−0.779603
$956$	0	0
$957$	−7.26148	−0.234730
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.4107	−0.916475
$962$	0	0
$963$	4.50469	0.145162
$964$	0	0
$965$	−9.35500	−0.301148
$966$	0	0
$967$	−19.1495	−0.615807	−0.307903	−	0.951418i	$-0.599627\pi$
$967$	−19.1495	−0.615807	−0.307903	+	0.951418i	$0.599627\pi$
$968$	0	0
$969$	3.21860	0.103396
$970$	0	0
$971$	−32.6427	−1.04755	−0.523777	−	0.851856i	$-0.675477\pi$
$971$	−32.6427	−1.04755	−0.523777	+	0.851856i	$0.675477\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−33.6315	−1.07707
$976$	0	0
$977$	−36.3994	−1.16452	−0.582259	−	0.813003i	$-0.697831\pi$
$977$	−36.3994	−1.16452	−0.582259	+	0.813003i	$0.697831\pi$
$978$	0	0
$979$	−6.62313	−0.211676
$980$	0	0
$981$	0.208698	0.00666322
$982$	0	0
$983$	24.4637	0.780271	0.390136	−	0.920757i	$-0.372428\pi$
$983$	24.4637	0.780271	0.390136	+	0.920757i	$0.372428\pi$
$984$	0	0
$985$	2.54850	0.0812020
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.8223	0.725707
$990$	0	0
$991$	−29.0250	−0.922010	−0.461005	−	0.887398i	$-0.652511\pi$
$991$	−29.0250	−0.922010	−0.461005	+	0.887398i	$0.652511\pi$
$992$	0	0
$993$	−40.1921	−1.27546
$994$	0	0
$995$	2.90389	0.0920596
$996$	0	0
$997$	55.4481	1.75606	0.878029	−	0.478607i	$-0.158858\pi$
$997$	55.4481	1.75606	0.878029	+	0.478607i	$0.158858\pi$
$998$	0	0
$999$	−12.1633	−0.384828

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.db.1.2		5
4.3	odd	2		4312.2.a.bg.1.4		5
7.3	odd	6		1232.2.q.o.177.2		10
7.5	odd	6		1232.2.q.o.529.2		10
7.6	odd	2		8624.2.a.dc.1.4		5
28.3	even	6		616.2.q.f.177.4	✓	10
28.19	even	6		616.2.q.f.529.4	yes	10
28.27	even	2		4312.2.a.bf.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.q.f.177.4	✓	10	28.3	even	6
616.2.q.f.529.4	yes	10	28.19	even	6
1232.2.q.o.177.2		10	7.3	odd	6
1232.2.q.o.529.2		10	7.5	odd	6
4312.2.a.bf.1.2		5	28.27	even	2
4312.2.a.bg.1.4		5	4.3	odd	2
8624.2.a.db.1.2		5	1.1	even	1	trivial
8624.2.a.dc.1.4		5	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-1.49300 q^{3} +1.26396 q^{5} -0.770961 q^{9} +O(q^{10})\)	\(q-1.49300 q^{3} +1.26396 q^{5} -0.770961 q^{9} +1.00000 q^{11} -6.62065 q^{13} -1.88708 q^{15} +1.02554 q^{17} -2.10212 q^{19} +1.84754 q^{23} -3.40241 q^{25} +5.63003 q^{27} +4.86370 q^{29} +1.60912 q^{31} -1.49300 q^{33} -2.16042 q^{37} +9.88461 q^{39} +6.49373 q^{41} +12.3528 q^{43} -0.974463 q^{45} +5.02554 q^{47} -1.53112 q^{51} +2.52174 q^{53} +1.26396 q^{55} +3.13845 q^{57} -3.54051 q^{59} +2.94966 q^{61} -8.36822 q^{65} +9.84506 q^{67} -2.75837 q^{69} -8.86228 q^{71} -15.2251 q^{73} +5.07979 q^{75} +8.31503 q^{79} -6.09273 q^{81} -4.54870 q^{83} +1.29624 q^{85} -7.26148 q^{87} -6.62313 q^{89} -2.40241 q^{93} -2.65699 q^{95} -1.10674 q^{97} -0.770961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 4 q^{5}+O(q^{10}) \) 5 * q - q^3 - 4 * q^5	\( 5 q - q^{3} - 4 q^{5} + 5 q^{11} - q^{13} - 7 q^{15} - 9 q^{17} + q^{19} + 8 q^{23} + 5 q^{25} - 4 q^{27} + 9 q^{29} + 3 q^{31} - q^{33} + 2 q^{37} + 7 q^{39} - 15 q^{41} - 14 q^{43} - 19 q^{45} + 11 q^{47} + 14 q^{51} - 9 q^{53} - 4 q^{55} + 4 q^{57} + 4 q^{59} + 2 q^{61} + 7 q^{65} + 8 q^{67} - 11 q^{69} - 15 q^{71} - 26 q^{73} + 27 q^{75} + 3 q^{79} - 19 q^{81} - q^{83} + 13 q^{85} + 14 q^{87} - 41 q^{89} + 10 q^{93} + 19 q^{95} - 7 q^{97}+O(q^{100}) \) 5 * q - q^3 - 4 * q^5 + 5 * q^11 - q^13 - 7 * q^15 - 9 * q^17 + q^19 + 8 * q^23 + 5 * q^25 - 4 * q^27 + 9 * q^29 + 3 * q^31 - q^33 + 2 * q^37 + 7 * q^39 - 15 * q^41 - 14 * q^43 - 19 * q^45 + 11 * q^47 + 14 * q^51 - 9 * q^53 - 4 * q^55 + 4 * q^57 + 4 * q^59 + 2 * q^61 + 7 * q^65 + 8 * q^67 - 11 * q^69 - 15 * q^71 - 26 * q^73 + 27 * q^75 + 3 * q^79 - 19 * q^81 - q^83 + 13 * q^85 + 14 * q^87 - 41 * q^89 + 10 * q^93 + 19 * q^95 - 7 * q^97

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