LMFDB - Embedded newform 8624.2.a.dd.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [5,0,3,0,-4,0,0,0,4,0,-5,0,-7,0,-7,0,-5,0,13,0,0,0,-8,0,1,0,0, 0,7,0,11,0,-3,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.352076.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 3x^{2} + 8x - 4 $ x^5 - x^4 - 8x^3 + 3x^2 + 8*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			1.4
Root			$0.844040$ 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)

$f(q)$	$=$	$q+1.52551 q^{3} +2.88640 q^{5} -0.672805 q^{9} -1.00000 q^{11} -1.40780 q^{13} +4.40325 q^{15} +2.97568 q^{17} +5.25731 q^{19} +2.65715 q^{23} +3.33131 q^{25} -5.60292 q^{27} +3.44356 q^{29} +2.46582 q^{31} -1.52551 q^{33} -1.88229 q^{37} -2.14762 q^{39} -5.68353 q^{41} +10.6647 q^{43} -1.94199 q^{45} +8.02671 q^{47} +4.53944 q^{51} +9.94950 q^{53} -2.88640 q^{55} +8.02010 q^{57} -1.17670 q^{59} -8.46993 q^{61} -4.06348 q^{65} +1.02394 q^{67} +4.05352 q^{69} -11.9791 q^{71} +5.90168 q^{73} +5.08196 q^{75} +8.60817 q^{79} -6.52892 q^{81} +6.32399 q^{83} +8.58899 q^{85} +5.25319 q^{87} -13.3043 q^{89} +3.76164 q^{93} +15.1747 q^{95} -0.175517 q^{97} +0.672805 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 4 q^{5} + 4 q^{9} - 5 q^{11} - 7 q^{13} - 7 q^{15} - 5 q^{17} + 13 q^{19} - 8 q^{23} + q^{25} + 7 q^{29} + 11 q^{31} - 3 q^{33} - 14 q^{37} + 7 q^{39} + 7 q^{41} - 12 q^{43} - 11 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 4 * q^5 + 4 * q^9 - 5 * q^11 - 7 * q^13 - 7 * q^15 - 5 * q^17 + 13 * q^19 - 8 * q^23 + q^25 + 7 * q^29 + 11 * q^31 - 3 * q^33 - 14 * q^37 + 7 * q^39 + 7 * q^41 - 12 * q^43 - 11 * q^45 + 11 * q^47 + 6 * q^51 + 7 * q^53 + 4 * q^55 + 16 * q^57 - 18 * q^61 - 9 * q^65 + 12 * q^67 + 29 * q^69 - 19 * q^71 + 12 * q^73 + 31 * q^75 - 3 * q^79 - 7 * q^81 + 23 * q^83 + 27 * q^85 + 36 * q^87 - 5 * q^89 - 42 * q^93 - 3 * q^95 + 13 * q^97 - 4 * q^99

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
$3$	1.52551	0.880756	0.440378	−	0.897812i	$-0.354844\pi$
$3$	1.52551	0.880756	0.440378	+	0.897812i	$0.354844\pi$
$4$	0	0
$5$	2.88640	1.29084	0.645419	−	0.763829i	$-0.276683\pi$
$5$	2.88640	1.29084	0.645419	+	0.763829i	$0.276683\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.672805	−0.224268
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.40780	−0.390454	−0.195227	−	0.980758i	$-0.562544\pi$
$13$	−1.40780	−0.390454	−0.195227	+	0.980758i	$0.562544\pi$
$14$	0	0
$15$	4.40325	1.13691
$16$	0	0
$17$	2.97568	0.721708	0.360854	−	0.932622i	$-0.382485\pi$
$17$	2.97568	0.721708	0.360854	+	0.932622i	$0.382485\pi$
$18$	0	0
$19$	5.25731	1.20611	0.603055	−	0.797700i	$-0.293950\pi$
$19$	5.25731	1.20611	0.603055	+	0.797700i	$0.293950\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.65715	0.554054	0.277027	−	0.960862i	$-0.410651\pi$
$23$	2.65715	0.554054	0.277027	+	0.960862i	$0.410651\pi$
$24$	0	0
$25$	3.33131	0.666262
$26$	0	0
$27$	−5.60292	−1.07828
$28$	0	0
$29$	3.44356	0.639452	0.319726	−	0.947510i	$-0.396409\pi$
$29$	3.44356	0.639452	0.319726	+	0.947510i	$0.396409\pi$
$30$	0	0
$31$	2.46582	0.442874	0.221437	−	0.975175i	$-0.428925\pi$
$31$	2.46582	0.442874	0.221437	+	0.975175i	$0.428925\pi$
$32$	0	0
$33$	−1.52551	−0.265558
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.88229	−0.309446	−0.154723	−	0.987958i	$-0.549449\pi$
$37$	−1.88229	−0.309446	−0.154723	+	0.987958i	$0.549449\pi$
$38$	0	0
$39$	−2.14762	−0.343895
$40$	0	0
$41$	−5.68353	−0.887617	−0.443809	−	0.896122i	$-0.646373\pi$
$41$	−5.68353	−0.887617	−0.443809	+	0.896122i	$0.646373\pi$
$42$	0	0
$43$	10.6647	1.62635	0.813173	−	0.582022i	$-0.197738\pi$
$43$	10.6647	1.62635	0.813173	+	0.582022i	$0.197738\pi$
$44$	0	0
$45$	−1.94199	−0.289494
$46$	0	0
$47$	8.02671	1.17082	0.585408	−	0.810739i	$-0.300934\pi$
$47$	8.02671	1.17082	0.585408	+	0.810739i	$0.300934\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.53944	0.635648
$52$	0	0
$53$	9.94950	1.36667	0.683335	−	0.730105i	$-0.260529\pi$
$53$	9.94950	1.36667	0.683335	+	0.730105i	$0.260529\pi$
$54$	0	0
$55$	−2.88640	−0.389202
$56$	0	0
$57$	8.02010	1.06229
$58$	0	0
$59$	−1.17670	−0.153193	−0.0765967	−	0.997062i	$-0.524405\pi$
$59$	−1.17670	−0.153193	−0.0765967	+	0.997062i	$0.524405\pi$
$60$	0	0
$61$	−8.46993	−1.08446	−0.542232	−	0.840229i	$-0.682420\pi$
$61$	−8.46993	−1.08446	−0.542232	+	0.840229i	$0.682420\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.06348	−0.504013
$66$	0	0
$67$	1.02394	0.125095	0.0625474	−	0.998042i	$-0.480078\pi$
$67$	1.02394	0.125095	0.0625474	+	0.998042i	$0.480078\pi$
$68$	0	0
$69$	4.05352	0.487987
$70$	0	0
$71$	−11.9791	−1.42166	−0.710828	−	0.703366i	$-0.751679\pi$
$71$	−11.9791	−1.42166	−0.710828	+	0.703366i	$0.751679\pi$
$72$	0	0
$73$	5.90168	0.690739	0.345369	−	0.938467i	$-0.387754\pi$
$73$	5.90168	0.690739	0.345369	+	0.938467i	$0.387754\pi$
$74$	0	0
$75$	5.08196	0.586814
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.60817	0.968495	0.484248	−	0.874931i	$-0.339093\pi$
$79$	8.60817	0.968495	0.484248	+	0.874931i	$0.339093\pi$
$80$	0	0
$81$	−6.52892	−0.725435
$82$	0	0
$83$	6.32399	0.694148	0.347074	−	0.937838i	$-0.387175\pi$
$83$	6.32399	0.694148	0.347074	+	0.937838i	$0.387175\pi$
$84$	0	0
$85$	8.58899	0.931607
$86$	0	0
$87$	5.25319	0.563202
$88$	0	0
$89$	−13.3043	−1.41025	−0.705125	−	0.709083i	$-0.749109\pi$
$89$	−13.3043	−1.41025	−0.705125	+	0.709083i	$0.749109\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.76164	0.390064
$94$	0	0
$95$	15.1747	1.55689
$96$	0	0
$97$	−0.175517	−0.0178210	−0.00891050	−	0.999960i	$-0.502836\pi$
$97$	−0.175517	−0.0178210	−0.00891050	+	0.999960i	$0.502836\pi$
$98$	0	0
$99$	0.672805	0.0676195
$100$	0	0
$101$	−6.52160	−0.648923	−0.324462	−	0.945899i	$-0.605183\pi$
$101$	−6.52160	−0.648923	−0.324462	+	0.945899i	$0.605183\pi$
$102$	0	0
$103$	16.3994	1.61588	0.807940	−	0.589264i	$-0.200582\pi$
$103$	16.3994	1.61588	0.807940	+	0.589264i	$0.200582\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.22887	−0.505494	−0.252747	−	0.967532i	$-0.581334\pi$
$107$	−5.22887	−0.505494	−0.252747	+	0.967532i	$0.581334\pi$
$108$	0	0
$109$	−16.1531	−1.54718	−0.773592	−	0.633684i	$-0.781542\pi$
$109$	−16.1531	−1.54718	−0.773592	+	0.633684i	$0.781542\pi$
$110$	0	0
$111$	−2.87146	−0.272547
$112$	0	0
$113$	14.1000	1.32642	0.663208	−	0.748435i	$-0.269194\pi$
$113$	14.1000	1.32642	0.663208	+	0.748435i	$0.269194\pi$
$114$	0	0
$115$	7.66960	0.715194
$116$	0	0
$117$	0.947176	0.0875665
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.67030	−0.781775
$124$	0	0
$125$	−4.81651	−0.430802
$126$	0	0
$127$	−8.77095	−0.778296	−0.389148	−	0.921175i	$-0.627231\pi$
$127$	−8.77095	−0.778296	−0.389148	+	0.921175i	$0.627231\pi$
$128$	0	0
$129$	16.2691	1.43241
$130$	0	0
$131$	−17.2954	−1.51111	−0.755554	−	0.655086i	$-0.772632\pi$
$131$	−17.2954	−1.51111	−0.755554	+	0.655086i	$0.772632\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−16.1723	−1.39189
$136$	0	0
$137$	12.6046	1.07688	0.538441	−	0.842663i	$-0.319013\pi$
$137$	12.6046	1.07688	0.538441	+	0.842663i	$0.319013\pi$
$138$	0	0
$139$	−1.02464	−0.0869091	−0.0434546	−	0.999055i	$-0.513836\pi$
$139$	−1.02464	−0.0869091	−0.0434546	+	0.999055i	$0.513836\pi$
$140$	0	0
$141$	12.2449	1.03120
$142$	0	0
$143$	1.40780	0.117726
$144$	0	0
$145$	9.93948	0.825429
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.9332	1.63299	0.816495	−	0.577352i	$-0.195914\pi$
$149$	19.9332	1.63299	0.816495	+	0.577352i	$0.195914\pi$
$150$	0	0
$151$	4.45715	0.362718	0.181359	−	0.983417i	$-0.441950\pi$
$151$	4.45715	0.362718	0.181359	+	0.983417i	$0.441950\pi$
$152$	0	0
$153$	−2.00205	−0.161856
$154$	0	0
$155$	7.11733	0.571678
$156$	0	0
$157$	−9.49778	−0.758005	−0.379003	−	0.925396i	$-0.623733\pi$
$157$	−9.49778	−0.758005	−0.379003	+	0.925396i	$0.623733\pi$
$158$	0	0
$159$	15.1781	1.20370
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.3350	1.35779	0.678893	−	0.734238i	$-0.262460\pi$
$163$	17.3350	1.35779	0.678893	+	0.734238i	$0.262460\pi$
$164$	0	0
$165$	−4.40325	−0.342792
$166$	0	0
$167$	24.6484	1.90735	0.953674	−	0.300842i	$-0.0972676\pi$
$167$	24.6484	1.90735	0.953674	+	0.300842i	$0.0972676\pi$
$168$	0	0
$169$	−11.0181	−0.847546
$170$	0	0
$171$	−3.53714	−0.270492
$172$	0	0
$173$	−2.10813	−0.160278	−0.0801392	−	0.996784i	$-0.525536\pi$
$173$	−2.10813	−0.160278	−0.0801392	+	0.996784i	$0.525536\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.79507	−0.134926
$178$	0	0
$179$	9.61928	0.718979	0.359489	−	0.933149i	$-0.382951\pi$
$179$	9.61928	0.718979	0.359489	+	0.933149i	$0.382951\pi$
$180$	0	0
$181$	18.9565	1.40903	0.704514	−	0.709690i	$-0.251165\pi$
$181$	18.9565	1.40903	0.704514	+	0.709690i	$0.251165\pi$
$182$	0	0
$183$	−12.9210	−0.955148
$184$	0	0
$185$	−5.43303	−0.399445
$186$	0	0
$187$	−2.97568	−0.217603
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.1758	−1.02572	−0.512862	−	0.858471i	$-0.671415\pi$
$191$	−14.1758	−1.02572	−0.512862	+	0.858471i	$0.671415\pi$
$192$	0	0
$193$	2.93348	0.211157	0.105578	−	0.994411i	$-0.466331\pi$
$193$	2.93348	0.211157	0.105578	+	0.994411i	$0.466331\pi$
$194$	0	0
$195$	−6.19890	−0.443912
$196$	0	0
$197$	27.5186	1.96062	0.980309	−	0.197471i	$-0.0632730\pi$
$197$	27.5186	1.96062	0.980309	+	0.197471i	$0.0632730\pi$
$198$	0	0
$199$	1.80488	0.127945	0.0639724	−	0.997952i	$-0.479623\pi$
$199$	1.80488	0.127945	0.0639724	+	0.997952i	$0.479623\pi$
$200$	0	0
$201$	1.56204	0.110178
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−16.4049	−1.14577
$206$	0	0
$207$	−1.78774	−0.124257
$208$	0	0
$209$	−5.25731	−0.363656
$210$	0	0
$211$	2.07447	0.142813	0.0714063	−	0.997447i	$-0.477251\pi$
$211$	2.07447	0.142813	0.0714063	+	0.997447i	$0.477251\pi$
$212$	0	0
$213$	−18.2743	−1.25213
$214$	0	0
$215$	30.7825	2.09935
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.00309	0.608373
$220$	0	0
$221$	−4.18916	−0.281794
$222$	0	0
$223$	2.64778	0.177309	0.0886543	−	0.996062i	$-0.471743\pi$
$223$	2.64778	0.177309	0.0886543	+	0.996062i	$0.471743\pi$
$224$	0	0
$225$	−2.24132	−0.149421
$226$	0	0
$227$	22.3739	1.48501	0.742503	−	0.669843i	$-0.233639\pi$
$227$	22.3739	1.48501	0.742503	+	0.669843i	$0.233639\pi$
$228$	0	0
$229$	−13.4240	−0.887082	−0.443541	−	0.896254i	$-0.646278\pi$
$229$	−13.4240	−0.887082	−0.443541	+	0.896254i	$0.646278\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.33488	−0.611548	−0.305774	−	0.952104i	$-0.598915\pi$
$233$	−9.33488	−0.611548	−0.305774	+	0.952104i	$0.598915\pi$
$234$	0	0
$235$	23.1683	1.51133
$236$	0	0
$237$	13.1319	0.853008
$238$	0	0
$239$	4.62821	0.299374	0.149687	−	0.988733i	$-0.452173\pi$
$239$	4.62821	0.299374	0.149687	+	0.988733i	$0.452173\pi$
$240$	0	0
$241$	−14.1558	−0.911858	−0.455929	−	0.890016i	$-0.650693\pi$
$241$	−14.1558	−0.911858	−0.455929	+	0.890016i	$0.650693\pi$
$242$	0	0
$243$	6.84880	0.439350
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.40125	−0.470930
$248$	0	0
$249$	9.64734	0.611375
$250$	0	0
$251$	24.6241	1.55426	0.777130	−	0.629341i	$-0.216675\pi$
$251$	24.6241	1.55426	0.777130	+	0.629341i	$0.216675\pi$
$252$	0	0
$253$	−2.65715	−0.167054
$254$	0	0
$255$	13.1026	0.820519
$256$	0	0
$257$	11.7915	0.735537	0.367768	−	0.929917i	$-0.380122\pi$
$257$	11.7915	0.735537	0.367768	+	0.929917i	$0.380122\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.31684	−0.143409
$262$	0	0
$263$	23.4924	1.44860	0.724302	−	0.689483i	$-0.242162\pi$
$263$	23.4924	1.44860	0.724302	+	0.689483i	$0.242162\pi$
$264$	0	0
$265$	28.7182	1.76415
$266$	0	0
$267$	−20.2959	−1.24209
$268$	0	0
$269$	11.2832	0.687951	0.343976	−	0.938979i	$-0.388226\pi$
$269$	11.2832	0.687951	0.343976	+	0.938979i	$0.388226\pi$
$270$	0	0
$271$	−4.02897	−0.244742	−0.122371	−	0.992484i	$-0.539050\pi$
$271$	−4.02897	−0.244742	−0.122371	+	0.992484i	$0.539050\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.33131	−0.200885
$276$	0	0
$277$	19.2567	1.15702	0.578511	−	0.815674i	$-0.303634\pi$
$277$	19.2567	1.15702	0.578511	+	0.815674i	$0.303634\pi$
$278$	0	0
$279$	−1.65901	−0.0993226
$280$	0	0
$281$	−13.1218	−0.782783	−0.391391	−	0.920224i	$-0.628006\pi$
$281$	−13.1218	−0.782783	−0.391391	+	0.920224i	$0.628006\pi$
$282$	0	0
$283$	30.2728	1.79953	0.899767	−	0.436371i	$-0.143737\pi$
$283$	30.2728	1.79953	0.899767	+	0.436371i	$0.143737\pi$
$284$	0	0
$285$	23.1492	1.37124
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.14535	−0.479138
$290$	0	0
$291$	−0.267753	−0.0156960
$292$	0	0
$293$	1.35020	0.0788794	0.0394397	−	0.999222i	$-0.487443\pi$
$293$	1.35020	0.0788794	0.0394397	+	0.999222i	$0.487443\pi$
$294$	0	0
$295$	−3.39643	−0.197748
$296$	0	0
$297$	5.60292	0.325114
$298$	0	0
$299$	−3.74074	−0.216333
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−9.94880	−0.571543
$304$	0	0
$305$	−24.4476	−1.39987
$306$	0	0
$307$	23.1157	1.31928	0.659640	−	0.751582i	$-0.270709\pi$
$307$	23.1157	1.31928	0.659640	+	0.751582i	$0.270709\pi$
$308$	0	0
$309$	25.0175	1.42320
$310$	0	0
$311$	−28.0970	−1.59323	−0.796617	−	0.604484i	$-0.793379\pi$
$311$	−28.0970	−1.59323	−0.796617	+	0.604484i	$0.793379\pi$
$312$	0	0
$313$	−24.4235	−1.38050	−0.690249	−	0.723572i	$-0.742499\pi$
$313$	−24.4235	−1.38050	−0.690249	+	0.723572i	$0.742499\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.6001	−1.71867	−0.859337	−	0.511411i	$-0.829123\pi$
$317$	−30.6001	−1.71867	−0.859337	+	0.511411i	$0.829123\pi$
$318$	0	0
$319$	−3.44356	−0.192802
$320$	0	0
$321$	−7.97672	−0.445217
$322$	0	0
$323$	15.6440	0.870458
$324$	0	0
$325$	−4.68982	−0.260144
$326$	0	0
$327$	−24.6418	−1.36269
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.7266	0.699517	0.349758	−	0.936840i	$-0.386264\pi$
$331$	12.7266	0.699517	0.349758	+	0.936840i	$0.386264\pi$
$332$	0	0
$333$	1.26641	0.0693990
$334$	0	0
$335$	2.95551	0.161477
$336$	0	0
$337$	−23.7142	−1.29179	−0.645896	−	0.763425i	$-0.723516\pi$
$337$	−23.7142	−1.29179	−0.645896	+	0.763425i	$0.723516\pi$
$338$	0	0
$339$	21.5097	1.16825
$340$	0	0
$341$	−2.46582	−0.133531
$342$	0	0
$343$	0	0
$344$	0	0
$345$	11.7001	0.629912
$346$	0	0
$347$	−7.11278	−0.381834	−0.190917	−	0.981606i	$-0.561146\pi$
$347$	−7.11278	−0.381834	−0.190917	+	0.981606i	$0.561146\pi$
$348$	0	0
$349$	8.04330	0.430548	0.215274	−	0.976554i	$-0.430936\pi$
$349$	8.04330	0.430548	0.215274	+	0.976554i	$0.430936\pi$
$350$	0	0
$351$	7.88780	0.421020
$352$	0	0
$353$	−4.07633	−0.216961	−0.108480	−	0.994099i	$-0.534598\pi$
$353$	−4.07633	−0.216961	−0.108480	+	0.994099i	$0.534598\pi$
$354$	0	0
$355$	−34.5764	−1.83513
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.3821	−0.706282	−0.353141	−	0.935570i	$-0.614886\pi$
$359$	−13.3821	−0.706282	−0.353141	+	0.935570i	$0.614886\pi$
$360$	0	0
$361$	8.63928	0.454699
$362$	0	0
$363$	1.52551	0.0800688
$364$	0	0
$365$	17.0346	0.891632
$366$	0	0
$367$	−10.1638	−0.530546	−0.265273	−	0.964173i	$-0.585462\pi$
$367$	−10.1638	−0.530546	−0.265273	+	0.964173i	$0.585462\pi$
$368$	0	0
$369$	3.82391	0.199065
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−32.5858	−1.68723	−0.843614	−	0.536949i	$-0.819577\pi$
$373$	−32.5858	−1.68723	−0.843614	+	0.536949i	$0.819577\pi$
$374$	0	0
$375$	−7.34766	−0.379432
$376$	0	0
$377$	−4.84784	−0.249677
$378$	0	0
$379$	−9.57653	−0.491913	−0.245957	−	0.969281i	$-0.579102\pi$
$379$	−9.57653	−0.491913	−0.245957	+	0.969281i	$0.579102\pi$
$380$	0	0
$381$	−13.3802	−0.685489
$382$	0	0
$383$	−12.7692	−0.652476	−0.326238	−	0.945288i	$-0.605781\pi$
$383$	−12.7692	−0.652476	−0.326238	+	0.945288i	$0.605781\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.17524	−0.364738
$388$	0	0
$389$	−18.3424	−0.929995	−0.464997	−	0.885312i	$-0.653945\pi$
$389$	−18.3424	−0.929995	−0.464997	+	0.885312i	$0.653945\pi$
$390$	0	0
$391$	7.90682	0.399865
$392$	0	0
$393$	−26.3844	−1.33092
$394$	0	0
$395$	24.8466	1.25017
$396$	0	0
$397$	−38.4937	−1.93194	−0.965971	−	0.258649i	$-0.916723\pi$
$397$	−38.4937	−1.93194	−0.965971	+	0.258649i	$0.916723\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.9618	−1.39635	−0.698173	−	0.715929i	$-0.746003\pi$
$401$	−27.9618	−1.39635	−0.698173	+	0.715929i	$0.746003\pi$
$402$	0	0
$403$	−3.47138	−0.172922
$404$	0	0
$405$	−18.8451	−0.936419
$406$	0	0
$407$	1.88229	0.0933015
$408$	0	0
$409$	33.8914	1.67582	0.837912	−	0.545806i	$-0.183776\pi$
$409$	33.8914	1.67582	0.837912	+	0.545806i	$0.183776\pi$
$410$	0	0
$411$	19.2285	0.948471
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.2536	0.896033
$416$	0	0
$417$	−1.56311	−0.0765457
$418$	0	0
$419$	−9.52680	−0.465415	−0.232707	−	0.972547i	$-0.574758\pi$
$419$	−9.52680	−0.465415	−0.232707	+	0.972547i	$0.574758\pi$
$420$	0	0
$421$	−8.12998	−0.396231	−0.198116	−	0.980179i	$-0.563482\pi$
$421$	−8.12998	−0.396231	−0.198116	+	0.980179i	$0.563482\pi$
$422$	0	0
$423$	−5.40041	−0.262577
$424$	0	0
$425$	9.91290	0.480846
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.14762	0.103688
$430$	0	0
$431$	−9.32183	−0.449017	−0.224508	−	0.974472i	$-0.572078\pi$
$431$	−9.32183	−0.449017	−0.224508	+	0.974472i	$0.572078\pi$
$432$	0	0
$433$	39.2565	1.88655	0.943275	−	0.332014i	$-0.107728\pi$
$433$	39.2565	1.88655	0.943275	+	0.332014i	$0.107728\pi$
$434$	0	0
$435$	15.1628	0.727002
$436$	0	0
$437$	13.9695	0.668250
$438$	0	0
$439$	7.55968	0.360804	0.180402	−	0.983593i	$-0.442260\pi$
$439$	7.55968	0.360804	0.180402	+	0.983593i	$0.442260\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.2623	−1.15274	−0.576368	−	0.817190i	$-0.695531\pi$
$443$	−24.2623	−1.15274	−0.576368	+	0.817190i	$0.695531\pi$
$444$	0	0
$445$	−38.4015	−1.82040
$446$	0	0
$447$	30.4084	1.43827
$448$	0	0
$449$	−27.8025	−1.31208	−0.656041	−	0.754725i	$-0.727770\pi$
$449$	−27.8025	−1.31208	−0.656041	+	0.754725i	$0.727770\pi$
$450$	0	0
$451$	5.68353	0.267627
$452$	0	0
$453$	6.79944	0.319466
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.1896	−0.710541	−0.355271	−	0.934763i	$-0.615611\pi$
$457$	−15.1896	−0.710541	−0.355271	+	0.934763i	$0.615611\pi$
$458$	0	0
$459$	−16.6725	−0.778204
$460$	0	0
$461$	5.43434	0.253102	0.126551	−	0.991960i	$-0.459609\pi$
$461$	5.43434	0.253102	0.126551	+	0.991960i	$0.459609\pi$
$462$	0	0
$463$	22.0510	1.02479	0.512397	−	0.858748i	$-0.328758\pi$
$463$	22.0510	1.02479	0.512397	+	0.858748i	$0.328758\pi$
$464$	0	0
$465$	10.8576	0.503509
$466$	0	0
$467$	−8.00715	−0.370527	−0.185263	−	0.982689i	$-0.559314\pi$
$467$	−8.00715	−0.370527	−0.185263	+	0.982689i	$0.559314\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−14.4890	−0.667618
$472$	0	0
$473$	−10.6647	−0.490362
$474$	0	0
$475$	17.5137	0.803584
$476$	0	0
$477$	−6.69408	−0.306501
$478$	0	0
$479$	−14.4371	−0.659647	−0.329824	−	0.944043i	$-0.606989\pi$
$479$	−14.4371	−0.659647	−0.329824	+	0.944043i	$0.606989\pi$
$480$	0	0
$481$	2.64989	0.120824
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.506611	−0.0230040
$486$	0	0
$487$	−42.3171	−1.91757	−0.958785	−	0.284132i	$-0.908295\pi$
$487$	−42.3171	−1.91757	−0.958785	+	0.284132i	$0.908295\pi$
$488$	0	0
$489$	26.4449	1.19588
$490$	0	0
$491$	12.9662	0.585158	0.292579	−	0.956241i	$-0.405486\pi$
$491$	12.9662	0.585158	0.292579	+	0.956241i	$0.405486\pi$
$492$	0	0
$493$	10.2469	0.461498
$494$	0	0
$495$	1.94199	0.0872858
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−36.2920	−1.62465	−0.812327	−	0.583202i	$-0.801800\pi$
$499$	−36.2920	−1.62465	−0.812327	+	0.583202i	$0.801800\pi$
$500$	0	0
$501$	37.6014	1.67991
$502$	0	0
$503$	−11.5706	−0.515906	−0.257953	−	0.966157i	$-0.583048\pi$
$503$	−11.5706	−0.515906	−0.257953	+	0.966157i	$0.583048\pi$
$504$	0	0
$505$	−18.8240	−0.837655
$506$	0	0
$507$	−16.8083	−0.746481
$508$	0	0
$509$	−25.5300	−1.13160	−0.565799	−	0.824543i	$-0.691432\pi$
$509$	−25.5300	−1.13160	−0.565799	+	0.824543i	$0.691432\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−29.4563	−1.30053
$514$	0	0
$515$	47.3352	2.08584
$516$	0	0
$517$	−8.02671	−0.353014
$518$	0	0
$519$	−3.21599	−0.141166
$520$	0	0
$521$	21.4443	0.939491	0.469746	−	0.882802i	$-0.344346\pi$
$521$	21.4443	0.939491	0.469746	+	0.882802i	$0.344346\pi$
$522$	0	0
$523$	41.9213	1.83309	0.916546	−	0.399930i	$-0.130966\pi$
$523$	41.9213	1.83309	0.916546	+	0.399930i	$0.130966\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.33747	0.319625
$528$	0	0
$529$	−15.9395	−0.693024
$530$	0	0
$531$	0.791690	0.0343564
$532$	0	0
$533$	8.00128	0.346574
$534$	0	0
$535$	−15.0926	−0.652511
$536$	0	0
$537$	14.6744	0.633245
$538$	0	0
$539$	0	0
$540$	0	0
$541$	36.7453	1.57980	0.789902	−	0.613234i	$-0.210132\pi$
$541$	36.7453	1.57980	0.789902	+	0.613234i	$0.210132\pi$
$542$	0	0
$543$	28.9185	1.24101
$544$	0	0
$545$	−46.6242	−1.99716
$546$	0	0
$547$	−35.4244	−1.51464	−0.757318	−	0.653046i	$-0.773491\pi$
$547$	−35.4244	−1.51464	−0.757318	+	0.653046i	$0.773491\pi$
$548$	0	0
$549$	5.69861	0.243211
$550$	0	0
$551$	18.1038	0.771249
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8.28817	−0.351813
$556$	0	0
$557$	36.0780	1.52867	0.764337	−	0.644818i	$-0.223067\pi$
$557$	36.0780	1.52867	0.764337	+	0.644818i	$0.223067\pi$
$558$	0	0
$559$	−15.0137	−0.635013
$560$	0	0
$561$	−4.53944	−0.191655
$562$	0	0
$563$	−17.9753	−0.757570	−0.378785	−	0.925485i	$-0.623658\pi$
$563$	−17.9753	−0.757570	−0.378785	+	0.925485i	$0.623658\pi$
$564$	0	0
$565$	40.6982	1.71219
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.9606	0.459491	0.229746	−	0.973251i	$-0.426211\pi$
$569$	10.9606	0.459491	0.229746	+	0.973251i	$0.426211\pi$
$570$	0	0
$571$	23.6018	0.987706	0.493853	−	0.869545i	$-0.335588\pi$
$571$	23.6018	0.987706	0.493853	+	0.869545i	$0.335588\pi$
$572$	0	0
$573$	−21.6254	−0.903413
$574$	0	0
$575$	8.85179	0.369145
$576$	0	0
$577$	38.2864	1.59388	0.796941	−	0.604057i	$-0.206450\pi$
$577$	38.2864	1.59388	0.796941	+	0.604057i	$0.206450\pi$
$578$	0	0
$579$	4.47507	0.185978
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.94950	−0.412066
$584$	0	0
$585$	2.73393	0.113034
$586$	0	0
$587$	−11.1318	−0.459457	−0.229728	−	0.973255i	$-0.573784\pi$
$587$	−11.1318	−0.459457	−0.229728	+	0.973255i	$0.573784\pi$
$588$	0	0
$589$	12.9636	0.534154
$590$	0	0
$591$	41.9800	1.72683
$592$	0	0
$593$	−10.9940	−0.451471	−0.225736	−	0.974189i	$-0.572479\pi$
$593$	−10.9940	−0.451471	−0.225736	+	0.974189i	$0.572479\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.75338	0.112688
$598$	0	0
$599$	7.18363	0.293515	0.146758	−	0.989172i	$-0.453116\pi$
$599$	7.18363	0.293515	0.146758	+	0.989172i	$0.453116\pi$
$600$	0	0
$601$	−30.0535	−1.22591	−0.612954	−	0.790119i	$-0.710019\pi$
$601$	−30.0535	−1.22591	−0.612954	+	0.790119i	$0.710019\pi$
$602$	0	0
$603$	−0.688915	−0.0280548
$604$	0	0
$605$	2.88640	0.117349
$606$	0	0
$607$	−8.85867	−0.359562	−0.179781	−	0.983707i	$-0.557539\pi$
$607$	−8.85867	−0.359562	−0.179781	+	0.983707i	$0.557539\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.3000	−0.457150
$612$	0	0
$613$	−41.8922	−1.69201	−0.846004	−	0.533176i	$-0.820998\pi$
$613$	−41.8922	−1.69201	−0.846004	+	0.533176i	$0.820998\pi$
$614$	0	0
$615$	−25.0260	−1.00914
$616$	0	0
$617$	22.3850	0.901185	0.450592	−	0.892730i	$-0.351213\pi$
$617$	22.3850	0.901185	0.450592	+	0.892730i	$0.351213\pi$
$618$	0	0
$619$	2.18536	0.0878369	0.0439185	−	0.999035i	$-0.486016\pi$
$619$	2.18536	0.0878369	0.0439185	+	0.999035i	$0.486016\pi$
$620$	0	0
$621$	−14.8878	−0.597427
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.5589	−1.22236
$626$	0	0
$627$	−8.02010	−0.320292
$628$	0	0
$629$	−5.60108	−0.223330
$630$	0	0
$631$	−24.5466	−0.977185	−0.488593	−	0.872512i	$-0.662490\pi$
$631$	−24.5466	−0.977185	−0.488593	+	0.872512i	$0.662490\pi$
$632$	0	0
$633$	3.16464	0.125783
$634$	0	0
$635$	−25.3165	−1.00465
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.05959	0.318832
$640$	0	0
$641$	21.9140	0.865553	0.432776	−	0.901501i	$-0.357534\pi$
$641$	21.9140	0.865553	0.432776	+	0.901501i	$0.357534\pi$
$642$	0	0
$643$	−14.1152	−0.556648	−0.278324	−	0.960487i	$-0.589779\pi$
$643$	−14.1152	−0.556648	−0.278324	+	0.960487i	$0.589779\pi$
$644$	0	0
$645$	46.9592	1.84901
$646$	0	0
$647$	36.1452	1.42101	0.710507	−	0.703690i	$-0.248466\pi$
$647$	36.1452	1.42101	0.710507	+	0.703690i	$0.248466\pi$
$648$	0	0
$649$	1.17670	0.0461895
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.03967	−0.118951	−0.0594757	−	0.998230i	$-0.518943\pi$
$653$	−3.03967	−0.118951	−0.0594757	+	0.998230i	$0.518943\pi$
$654$	0	0
$655$	−49.9215	−1.95060
$656$	0	0
$657$	−3.97068	−0.154911
$658$	0	0
$659$	33.9030	1.32067	0.660337	−	0.750969i	$-0.270413\pi$
$659$	33.9030	1.32067	0.660337	+	0.750969i	$0.270413\pi$
$660$	0	0
$661$	15.1763	0.590288	0.295144	−	0.955453i	$-0.404632\pi$
$661$	15.1763	0.590288	0.295144	+	0.955453i	$0.404632\pi$
$662$	0	0
$663$	−6.39063	−0.248191
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.15005	0.354291
$668$	0	0
$669$	4.03923	0.156166
$670$	0	0
$671$	8.46993	0.326978
$672$	0	0
$673$	36.8310	1.41973	0.709866	−	0.704337i	$-0.248756\pi$
$673$	36.8310	1.41973	0.709866	+	0.704337i	$0.248756\pi$
$674$	0	0
$675$	−18.6650	−0.718418
$676$	0	0
$677$	−15.0318	−0.577719	−0.288859	−	0.957372i	$-0.593276\pi$
$677$	−15.0318	−0.577719	−0.288859	+	0.957372i	$0.593276\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	34.1317	1.30793
$682$	0	0
$683$	−29.2316	−1.11852	−0.559258	−	0.828994i	$-0.688914\pi$
$683$	−29.2316	−1.11852	−0.559258	+	0.828994i	$0.688914\pi$
$684$	0	0
$685$	36.3819	1.39008
$686$	0	0
$687$	−20.4785	−0.781303
$688$	0	0
$689$	−14.0069	−0.533621
$690$	0	0
$691$	0.514316	0.0195655	0.00978275	−	0.999952i	$-0.496886\pi$
$691$	0.514316	0.0195655	0.00978275	+	0.999952i	$0.496886\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.95753	−0.112186
$696$	0	0
$697$	−16.9123	−0.640600
$698$	0	0
$699$	−14.2405	−0.538625
$700$	0	0
$701$	−24.3123	−0.918264	−0.459132	−	0.888368i	$-0.651840\pi$
$701$	−24.3123	−0.918264	−0.459132	+	0.888368i	$0.651840\pi$
$702$	0	0
$703$	−9.89576	−0.373226
$704$	0	0
$705$	35.3436	1.33112
$706$	0	0
$707$	0	0
$708$	0	0
$709$	19.2767	0.723952	0.361976	−	0.932188i	$-0.382102\pi$
$709$	19.2767	0.723952	0.361976	+	0.932188i	$0.382102\pi$
$710$	0	0
$711$	−5.79162	−0.217203
$712$	0	0
$713$	6.55205	0.245376
$714$	0	0
$715$	4.06348	0.151966
$716$	0	0
$717$	7.06041	0.263676
$718$	0	0
$719$	−24.5173	−0.914342	−0.457171	−	0.889379i	$-0.651137\pi$
$719$	−24.5173	−0.914342	−0.457171	+	0.889379i	$0.651137\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−21.5949	−0.803124
$724$	0	0
$725$	11.4715	0.426042
$726$	0	0
$727$	−41.9241	−1.55488	−0.777439	−	0.628959i	$-0.783481\pi$
$727$	−41.9241	−1.55488	−0.777439	+	0.628959i	$0.783481\pi$
$728$	0	0
$729$	30.0347	1.11240
$730$	0	0
$731$	31.7346	1.17375
$732$	0	0
$733$	−27.9311	−1.03166	−0.515829	−	0.856691i	$-0.672516\pi$
$733$	−27.9311	−1.03166	−0.515829	+	0.856691i	$0.672516\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.02394	−0.0377175
$738$	0	0
$739$	3.27117	0.120332	0.0601659	−	0.998188i	$-0.480837\pi$
$739$	3.27117	0.120332	0.0601659	+	0.998188i	$0.480837\pi$
$740$	0	0
$741$	−11.2907	−0.414775
$742$	0	0
$743$	17.1416	0.628866	0.314433	−	0.949280i	$-0.398186\pi$
$743$	17.1416	0.628866	0.314433	+	0.949280i	$0.398186\pi$
$744$	0	0
$745$	57.5352	2.10793
$746$	0	0
$747$	−4.25481	−0.155676
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28.8280	−1.05195	−0.525975	−	0.850500i	$-0.676300\pi$
$751$	−28.8280	−1.05195	−0.525975	+	0.850500i	$0.676300\pi$
$752$	0	0
$753$	37.5644	1.36892
$754$	0	0
$755$	12.8651	0.468209
$756$	0	0
$757$	−20.5337	−0.746309	−0.373154	−	0.927769i	$-0.621724\pi$
$757$	−20.5337	−0.746309	−0.373154	+	0.927769i	$0.621724\pi$
$758$	0	0
$759$	−4.05352	−0.147134
$760$	0	0
$761$	−1.22762	−0.0445011	−0.0222506	−	0.999752i	$-0.507083\pi$
$761$	−1.22762	−0.0445011	−0.0222506	+	0.999752i	$0.507083\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5.77872	−0.208930
$766$	0	0
$767$	1.65656	0.0598150
$768$	0	0
$769$	36.6863	1.32294	0.661471	−	0.749971i	$-0.269933\pi$
$769$	36.6863	1.32294	0.661471	+	0.749971i	$0.269933\pi$
$770$	0	0
$771$	17.9882	0.647828
$772$	0	0
$773$	4.67427	0.168122	0.0840608	−	0.996461i	$-0.473211\pi$
$773$	4.67427	0.168122	0.0840608	+	0.996461i	$0.473211\pi$
$774$	0	0
$775$	8.21439	0.295070
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−29.8800	−1.07056
$780$	0	0
$781$	11.9791	0.428645
$782$	0	0
$783$	−19.2940	−0.689510
$784$	0	0
$785$	−27.4144	−0.978461
$786$	0	0
$787$	10.6198	0.378555	0.189278	−	0.981924i	$-0.439385\pi$
$787$	10.6198	0.378555	0.189278	+	0.981924i	$0.439385\pi$
$788$	0	0
$789$	35.8380	1.27587
$790$	0	0
$791$	0	0
$792$	0	0
$793$	11.9240	0.423433
$794$	0	0
$795$	43.8101	1.55378
$796$	0	0
$797$	0.0387939	0.00137415	0.000687075	−	1.00000i	$-0.499781\pi$
$797$	0.0387939	0.00137415	0.000687075		1.00000i	$0.499781\pi$
$798$	0	0
$799$	23.8849	0.844986
$800$	0	0
$801$	8.95119	0.316275
$802$	0	0
$803$	−5.90168	−0.208266
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.2127	0.605917
$808$	0	0
$809$	42.9113	1.50868	0.754341	−	0.656483i	$-0.227957\pi$
$809$	42.9113	1.50868	0.754341	+	0.656483i	$0.227957\pi$
$810$	0	0
$811$	6.80324	0.238894	0.119447	−	0.992841i	$-0.461888\pi$
$811$	6.80324	0.238894	0.119447	+	0.992841i	$0.461888\pi$
$812$	0	0
$813$	−6.14625	−0.215558
$814$	0	0
$815$	50.0359	1.75268
$816$	0	0
$817$	56.0674	1.96155
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.0163904	0.000572029 0	0.000286015	−	1.00000i	$-0.499909\pi$
$821$	0.0163904	0.000572029 0	0.000286015		1.00000i	$0.499909\pi$
$822$	0	0
$823$	−44.2656	−1.54300	−0.771501	−	0.636229i	$-0.780493\pi$
$823$	−44.2656	−1.54300	−0.771501	+	0.636229i	$0.780493\pi$
$824$	0	0
$825$	−5.08196	−0.176931
$826$	0	0
$827$	20.6075	0.716592	0.358296	−	0.933608i	$-0.383358\pi$
$827$	20.6075	0.716592	0.358296	+	0.933608i	$0.383358\pi$
$828$	0	0
$829$	−34.8163	−1.20922	−0.604610	−	0.796522i	$-0.706671\pi$
$829$	−34.8163	−1.20922	−0.604610	+	0.796522i	$0.706671\pi$
$830$	0	0
$831$	29.3764	1.01906
$832$	0	0
$833$	0	0
$834$	0	0
$835$	71.1451	2.46208
$836$	0	0
$837$	−13.8158	−0.477543
$838$	0	0
$839$	−27.8559	−0.961692	−0.480846	−	0.876805i	$-0.659670\pi$
$839$	−27.8559	−0.961692	−0.480846	+	0.876805i	$0.659670\pi$
$840$	0	0
$841$	−17.1419	−0.591101
$842$	0	0
$843$	−20.0175	−0.689441
$844$	0	0
$845$	−31.8026	−1.09404
$846$	0	0
$847$	0	0
$848$	0	0
$849$	46.1817	1.58495
$850$	0	0
$851$	−5.00152	−0.171450
$852$	0	0
$853$	−3.28041	−0.112319	−0.0561595	−	0.998422i	$-0.517886\pi$
$853$	−3.28041	−0.112319	−0.0561595	+	0.998422i	$0.517886\pi$
$854$	0	0
$855$	−10.2096	−0.349161
$856$	0	0
$857$	1.98505	0.0678079	0.0339039	−	0.999425i	$-0.489206\pi$
$857$	1.98505	0.0678079	0.0339039	+	0.999425i	$0.489206\pi$
$858$	0	0
$859$	−31.4170	−1.07194	−0.535968	−	0.844238i	$-0.680053\pi$
$859$	−31.4170	−1.07194	−0.535968	+	0.844238i	$0.680053\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.86734	0.131646	0.0658229	−	0.997831i	$-0.479033\pi$
$863$	3.86734	0.131646	0.0658229	+	0.997831i	$0.479033\pi$
$864$	0	0
$865$	−6.08492	−0.206893
$866$	0	0
$867$	−12.4258	−0.422004
$868$	0	0
$869$	−8.60817	−0.292012
$870$	0	0
$871$	−1.44151	−0.0488437
$872$	0	0
$873$	0.118088	0.00399669
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.2536	−0.346238	−0.173119	−	0.984901i	$-0.555385\pi$
$877$	−10.2536	−0.346238	−0.173119	+	0.984901i	$0.555385\pi$
$878$	0	0
$879$	2.05975	0.0694735
$880$	0	0
$881$	24.9545	0.840737	0.420369	−	0.907353i	$-0.361901\pi$
$881$	24.9545	0.840737	0.420369	+	0.907353i	$0.361901\pi$
$882$	0	0
$883$	12.2842	0.413398	0.206699	−	0.978405i	$-0.433728\pi$
$883$	12.2842	0.413398	0.206699	+	0.978405i	$0.433728\pi$
$884$	0	0
$885$	−5.18130	−0.174168
$886$	0	0
$887$	−26.0831	−0.875783	−0.437892	−	0.899028i	$-0.644275\pi$
$887$	−26.0831	−0.875783	−0.437892	+	0.899028i	$0.644275\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.52892	0.218727
$892$	0	0
$893$	42.1989	1.41213
$894$	0	0
$895$	27.7651	0.928085
$896$	0	0
$897$	−5.70656	−0.190536
$898$	0	0
$899$	8.49118	0.283197
$900$	0	0
$901$	29.6065	0.986335
$902$	0	0
$903$	0	0
$904$	0	0
$905$	54.7162	1.81883
$906$	0	0
$907$	−27.2168	−0.903719	−0.451860	−	0.892089i	$-0.649239\pi$
$907$	−27.2168	−0.903719	−0.451860	+	0.892089i	$0.649239\pi$
$908$	0	0
$909$	4.38777	0.145533
$910$	0	0
$911$	−29.0677	−0.963055	−0.481528	−	0.876431i	$-0.659918\pi$
$911$	−29.0677	−0.963055	−0.481528	+	0.876431i	$0.659918\pi$
$912$	0	0
$913$	−6.32399	−0.209294
$914$	0	0
$915$	−37.2952	−1.23294
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.4226	0.541732	0.270866	−	0.962617i	$-0.412690\pi$
$919$	16.4226	0.541732	0.270866	+	0.962617i	$0.412690\pi$
$920$	0	0
$921$	35.2633	1.16196
$922$	0	0
$923$	16.8642	0.555091
$924$	0	0
$925$	−6.27048	−0.206172
$926$	0	0
$927$	−11.0336	−0.362391
$928$	0	0
$929$	56.8238	1.86433	0.932163	−	0.362038i	$-0.117919\pi$
$929$	56.8238	1.86433	0.932163	+	0.362038i	$0.117919\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−42.8624	−1.40325
$934$	0	0
$935$	−8.58899	−0.280890
$936$	0	0
$937$	−14.6797	−0.479565	−0.239783	−	0.970827i	$-0.577076\pi$
$937$	−14.6797	−0.479565	−0.239783	+	0.970827i	$0.577076\pi$
$938$	0	0
$939$	−37.2584	−1.21588
$940$	0	0
$941$	2.93243	0.0955944	0.0477972	−	0.998857i	$-0.484780\pi$
$941$	2.93243	0.0955944	0.0477972	+	0.998857i	$0.484780\pi$
$942$	0	0
$943$	−15.1020	−0.491788
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.8324	−0.806943	−0.403472	−	0.914992i	$-0.632197\pi$
$947$	−24.8324	−0.806943	−0.403472	+	0.914992i	$0.632197\pi$
$948$	0	0
$949$	−8.30839	−0.269702
$950$	0	0
$951$	−46.6809	−1.51373
$952$	0	0
$953$	−10.9698	−0.355348	−0.177674	−	0.984089i	$-0.556857\pi$
$953$	−10.9698	−0.355348	−0.177674	+	0.984089i	$0.556857\pi$
$954$	0	0
$955$	−40.9170	−1.32404
$956$	0	0
$957$	−5.25319	−0.169812
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.9197	−0.803863
$962$	0	0
$963$	3.51801	0.113366
$964$	0	0
$965$	8.46721	0.272569
$966$	0	0
$967$	3.89737	0.125331	0.0626655	−	0.998035i	$-0.480040\pi$
$967$	3.89737	0.125331	0.0626655	+	0.998035i	$0.480040\pi$
$968$	0	0
$969$	23.8652	0.766661
$970$	0	0
$971$	−0.227641	−0.00730536	−0.00365268	−	0.999993i	$-0.501163\pi$
$971$	−0.227641	−0.00730536	−0.00365268	+	0.999993i	$0.501163\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.15439	−0.229124
$976$	0	0
$977$	18.0990	0.579039	0.289520	−	0.957172i	$-0.406504\pi$
$977$	18.0990	0.579039	0.289520	+	0.957172i	$0.406504\pi$
$978$	0	0
$979$	13.3043	0.425207
$980$	0	0
$981$	10.8679	0.346985
$982$	0	0
$983$	4.79175	0.152833	0.0764166	−	0.997076i	$-0.475652\pi$
$983$	4.79175	0.152833	0.0764166	+	0.997076i	$0.475652\pi$
$984$	0	0
$985$	79.4296	2.53084
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.3376	0.901084
$990$	0	0
$991$	−48.8918	−1.55310	−0.776549	−	0.630057i	$-0.783032\pi$
$991$	−48.8918	−1.55310	−0.776549	+	0.630057i	$0.783032\pi$
$992$	0	0
$993$	19.4146	0.616104
$994$	0	0
$995$	5.20962	0.165156
$996$	0	0
$997$	−9.96604	−0.315628	−0.157814	−	0.987469i	$-0.550445\pi$
$997$	−9.96604	−0.315628	−0.157814	+	0.987469i	$0.550445\pi$
$998$	0	0
$999$	10.5463	0.333670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.dd.1.4		5
4.3	odd	2		4312.2.a.be.1.2		5
7.3	odd	6		1232.2.q.p.177.4		10
7.5	odd	6		1232.2.q.p.529.4		10
7.6	odd	2		8624.2.a.da.1.2		5
28.3	even	6		616.2.q.e.177.2	✓	10
28.19	even	6		616.2.q.e.529.2	yes	10
28.27	even	2		4312.2.a.bh.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.q.e.177.2	✓	10	28.3	even	6
616.2.q.e.529.2	yes	10	28.19	even	6
1232.2.q.p.177.4		10	7.3	odd	6
1232.2.q.p.529.4		10	7.5	odd	6
4312.2.a.be.1.2		5	4.3	odd	2
4312.2.a.bh.1.4		5	28.27	even	2
8624.2.a.da.1.2		5	7.6	odd	2
8624.2.a.dd.1.4		5	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.52551 q^{3} +2.88640 q^{5} -0.672805 q^{9} -1.00000 q^{11} -1.40780 q^{13} +4.40325 q^{15} +2.97568 q^{17} +5.25731 q^{19} +2.65715 q^{23} +3.33131 q^{25} -5.60292 q^{27} +3.44356 q^{29} +2.46582 q^{31} -1.52551 q^{33} -1.88229 q^{37} -2.14762 q^{39} -5.68353 q^{41} +10.6647 q^{43} -1.94199 q^{45} +8.02671 q^{47} +4.53944 q^{51} +9.94950 q^{53} -2.88640 q^{55} +8.02010 q^{57} -1.17670 q^{59} -8.46993 q^{61} -4.06348 q^{65} +1.02394 q^{67} +4.05352 q^{69} -11.9791 q^{71} +5.90168 q^{73} +5.08196 q^{75} +8.60817 q^{79} -6.52892 q^{81} +6.32399 q^{83} +8.58899 q^{85} +5.25319 q^{87} -13.3043 q^{89} +3.76164 q^{93} +15.1747 q^{95} -0.175517 q^{97} +0.672805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 4 q^{5} + 4 q^{9} - 5 q^{11} - 7 q^{13} - 7 q^{15} - 5 q^{17} + 13 q^{19} - 8 q^{23} + q^{25} + 7 q^{29} + 11 q^{31} - 3 q^{33} - 14 q^{37} + 7 q^{39} + 7 q^{41} - 12 q^{43} - 11 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 4 * q^5 + 4 * q^9 - 5 * q^11 - 7 * q^13 - 7 * q^15 - 5 * q^17 + 13 * q^19 - 8 * q^23 + q^25 + 7 * q^29 + 11 * q^31 - 3 * q^33 - 14 * q^37 + 7 * q^39 + 7 * q^41 - 12 * q^43 - 11 * q^45 + 11 * q^47 + 6 * q^51 + 7 * q^53 + 4 * q^55 + 16 * q^57 - 18 * q^61 - 9 * q^65 + 12 * q^67 + 29 * q^69 - 19 * q^71 + 12 * q^73 + 31 * q^75 - 3 * q^79 - 7 * q^81 + 23 * q^83 + 27 * q^85 + 36 * q^87 - 5 * q^89 - 42 * q^93 - 3 * q^95 + 13 * q^97 - 4 * q^99

Introduction

L-functions

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