LMFDB - Embedded newform 9200.2.a.cu.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.13955077.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 14x^{3} - x^{2} + 32x + 16 $ x^5 - 14x^3 - x^2 + 32x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93283$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93283 q^{3} +2.38236 q^{7} +0.735829 q^{9} +O(q^{10})$	$q+1.93283 q^{3} +2.38236 q^{7} +0.735829 q^{9} -5.33368 q^{11} +4.53752 q^{13} -1.81464 q^{17} -7.00233 q^{19} +4.60469 q^{21} -1.00000 q^{23} -4.37626 q^{27} -0.118188 q^{29} +0.884147 q^{31} -10.3091 q^{33} -7.51903 q^{37} +8.77026 q^{39} -1.45186 q^{41} +10.4389 q^{47} -1.32437 q^{49} -3.50739 q^{51} +9.42167 q^{53} -13.5343 q^{57} -7.79239 q^{59} -2.80533 q^{61} +1.75301 q^{63} -3.11134 q^{67} -1.93283 q^{69} +13.5909 q^{71} -12.4389 q^{73} -12.7067 q^{77} -6.80169 q^{79} -10.6660 q^{81} -13.5190 q^{83} -0.228437 q^{87} +2.89906 q^{89} +10.8100 q^{91} +1.70890 q^{93} +1.97774 q^{97} -3.92468 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{7} + 13 q^{9}+O(q^{10}) $ 5 * q - 2 * q^7 + 13 * q^9	$ 5 q - 2 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 4 q^{17} - 7 q^{19} + 6 q^{21} - 5 q^{23} + 3 q^{27} + 4 q^{29} - 19 q^{31} - 17 q^{33} - 15 q^{37} - 19 q^{39} + 25 q^{41} - 11 q^{47} + 25 q^{49} - 19 q^{51} - 3 q^{53} - 48 q^{57} + q^{59} - 5 q^{61} - 41 q^{63} + 9 q^{67} - q^{71} + q^{73} - 18 q^{77} + 2 q^{79} + 57 q^{81} - 45 q^{83} - 9 q^{87} + 6 q^{89} - 11 q^{91} + 39 q^{93} - 25 q^{97} + 65 q^{99}+O(q^{100}) $ 5 * q - 2 * q^7 + 13 * q^9 + q^11 - 4 * q^13 - 4 * q^17 - 7 * q^19 + 6 * q^21 - 5 * q^23 + 3 * q^27 + 4 * q^29 - 19 * q^31 - 17 * q^33 - 15 * q^37 - 19 * q^39 + 25 * q^41 - 11 * q^47 + 25 * q^49 - 19 * q^51 - 3 * q^53 - 48 * q^57 + q^59 - 5 * q^61 - 41 * q^63 + 9 * q^67 - q^71 + q^73 - 18 * q^77 + 2 * q^79 + 57 * q^81 - 45 * q^83 - 9 * q^87 + 6 * q^89 - 11 * q^91 + 39 * q^93 - 25 * q^97 + 65 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.93283	1.11592	0.557960	−	0.829868i	$-0.311584\pi$
$3$	1.93283	1.11592	0.557960	+	0.829868i	$0.311584\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.38236	0.900447	0.450223	−	0.892916i	$-0.351344\pi$
$7$	2.38236	0.900447	0.450223	+	0.892916i	$0.351344\pi$
$8$	0	0
$9$	0.735829	0.245276
$10$	0	0
$11$	−5.33368	−1.60816	−0.804082	−	0.594519i	$-0.797343\pi$
$11$	−5.33368	−1.60816	−0.804082	+	0.594519i	$0.797343\pi$
$12$	0	0
$13$	4.53752	1.25848	0.629241	−	0.777210i	$-0.283366\pi$
$13$	4.53752	1.25848	0.629241	+	0.777210i	$0.283366\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.81464	−0.440115	−0.220058	−	0.975487i	$-0.570625\pi$
$17$	−1.81464	−0.440115	−0.220058	+	0.975487i	$0.570625\pi$
$18$	0	0
$19$	−7.00233	−1.60645	−0.803223	−	0.595679i	$-0.796883\pi$
$19$	−7.00233	−1.60645	−0.803223	+	0.595679i	$0.796883\pi$
$20$	0	0
$21$	4.60469	1.00483
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.37626	−0.842211
$28$	0	0
$29$	−0.118188	−0.0219469	−0.0109735	−	0.999940i	$-0.503493\pi$
$29$	−0.118188	−0.0219469	−0.0109735	+	0.999940i	$0.503493\pi$
$30$	0	0
$31$	0.884147	0.158797	0.0793987	−	0.996843i	$-0.474700\pi$
$31$	0.884147	0.158797	0.0793987	+	0.996843i	$0.474700\pi$
$32$	0	0
$33$	−10.3091	−1.79458
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.51903	−1.23612	−0.618061	−	0.786130i	$-0.712081\pi$
$37$	−7.51903	−1.23612	−0.618061	+	0.786130i	$0.712081\pi$
$38$	0	0
$39$	8.77026	1.40436
$40$	0	0
$41$	−1.45186	−0.226743	−0.113371	−	0.993553i	$-0.536165\pi$
$41$	−1.45186	−0.226743	−0.113371	+	0.993553i	$0.536165\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.4389	1.52267	0.761336	−	0.648357i	$-0.224544\pi$
$47$	10.4389	1.52267	0.761336	+	0.648357i	$0.224544\pi$
$48$	0	0
$49$	−1.32437	−0.189195
$50$	0	0
$51$	−3.50739	−0.491133
$52$	0	0
$53$	9.42167	1.29417	0.647083	−	0.762420i	$-0.275989\pi$
$53$	9.42167	1.29417	0.647083	+	0.762420i	$0.275989\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.5343	−1.79266
$58$	0	0
$59$	−7.79239	−1.01448	−0.507241	−	0.861804i	$-0.669335\pi$
$59$	−7.79239	−1.01448	−0.507241	+	0.861804i	$0.669335\pi$
$60$	0	0
$61$	−2.80533	−0.359186	−0.179593	−	0.983741i	$-0.557478\pi$
$61$	−2.80533	−0.359186	−0.179593	+	0.983741i	$0.557478\pi$
$62$	0	0
$63$	1.75301	0.220858
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.11134	−0.380111	−0.190055	−	0.981773i	$-0.560867\pi$
$67$	−3.11134	−0.380111	−0.190055	+	0.981773i	$0.560867\pi$
$68$	0	0
$69$	−1.93283	−0.232685
$70$	0	0
$71$	13.5909	1.61294	0.806470	−	0.591275i	$-0.201375\pi$
$71$	13.5909	1.61294	0.806470	+	0.591275i	$0.201375\pi$
$72$	0	0
$73$	−12.4389	−1.45586	−0.727932	−	0.685649i	$-0.759519\pi$
$73$	−12.4389	−1.45586	−0.727932	+	0.685649i	$0.759519\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.7067	−1.44807
$78$	0	0
$79$	−6.80169	−0.765250	−0.382625	−	0.923904i	$-0.624980\pi$
$79$	−6.80169	−0.765250	−0.382625	+	0.923904i	$0.624980\pi$
$80$	0	0
$81$	−10.6660	−1.18512
$82$	0	0
$83$	−13.5190	−1.48391	−0.741953	−	0.670451i	$-0.766100\pi$
$83$	−13.5190	−1.48391	−0.741953	+	0.670451i	$0.766100\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.228437	−0.0244910
$88$	0	0
$89$	2.89906	0.307300	0.153650	−	0.988125i	$-0.450897\pi$
$89$	2.89906	0.307300	0.153650	+	0.988125i	$0.450897\pi$
$90$	0	0
$91$	10.8100	1.13320
$92$	0	0
$93$	1.70890	0.177205
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.97774	0.200809	0.100405	−	0.994947i	$-0.467986\pi$
$97$	1.97774	0.200809	0.100405	+	0.994947i	$0.467986\pi$
$98$	0	0
$99$	−3.92468	−0.394445
$100$	0	0
$101$	−12.2838	−1.22228	−0.611139	−	0.791523i	$-0.709289\pi$
$101$	−12.2838	−1.22228	−0.611139	+	0.791523i	$0.709289\pi$
$102$	0	0
$103$	−12.5061	−1.23226	−0.616131	−	0.787644i	$-0.711301\pi$
$103$	−12.5061	−1.23226	−0.616131	+	0.787644i	$0.711301\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.3847	−1.68064	−0.840321	−	0.542089i	$-0.817633\pi$
$107$	−17.3847	−1.68064	−0.840321	+	0.542089i	$0.817633\pi$
$108$	0	0
$109$	4.30908	0.412735	0.206368	−	0.978475i	$-0.433836\pi$
$109$	4.30908	0.412735	0.206368	+	0.978475i	$0.433836\pi$
$110$	0	0
$111$	−14.5330	−1.37941
$112$	0	0
$113$	12.1494	1.14292	0.571460	−	0.820630i	$-0.306377\pi$
$113$	12.1494	1.14292	0.571460	+	0.820630i	$0.306377\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.33884	0.308676
$118$	0	0
$119$	−4.32313	−0.396300
$120$	0	0
$121$	17.4481	1.58619
$122$	0	0
$123$	−2.80620	−0.253027
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.22953	−0.109103	−0.0545515	−	0.998511i	$-0.517373\pi$
$127$	−1.22953	−0.109103	−0.0545515	+	0.998511i	$0.517373\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.97345	−0.172421	−0.0862104	−	0.996277i	$-0.527476\pi$
$131$	−1.97345	−0.172421	−0.0862104	+	0.996277i	$0.527476\pi$
$132$	0	0
$133$	−16.6821	−1.44652
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.7671	−1.17620	−0.588099	−	0.808789i	$-0.700124\pi$
$137$	−13.7671	−1.17620	−0.588099	+	0.808789i	$0.700124\pi$
$138$	0	0
$139$	−5.30280	−0.449778	−0.224889	−	0.974384i	$-0.572202\pi$
$139$	−5.30280	−0.449778	−0.224889	+	0.974384i	$0.572202\pi$
$140$	0	0
$141$	20.1766	1.69918
$142$	0	0
$143$	−24.2017	−2.02385
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.55978	−0.211127
$148$	0	0
$149$	7.73256	0.633476	0.316738	−	0.948513i	$-0.397412\pi$
$149$	7.73256	0.633476	0.316738	+	0.948513i	$0.397412\pi$
$150$	0	0
$151$	−21.4156	−1.74277	−0.871387	−	0.490596i	$-0.836779\pi$
$151$	−21.4156	−1.74277	−0.871387	+	0.490596i	$0.836779\pi$
$152$	0	0
$153$	−1.33527	−0.107950
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.04738	0.323016	0.161508	−	0.986871i	$-0.448364\pi$
$157$	4.04738	0.323016	0.161508	+	0.986871i	$0.448364\pi$
$158$	0	0
$159$	18.2105	1.44418
$160$	0	0
$161$	−2.38236	−0.187756
$162$	0	0
$163$	9.04725	0.708635	0.354318	−	0.935125i	$-0.384713\pi$
$163$	9.04725	0.708635	0.354318	+	0.935125i	$0.384713\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.5467	−0.970893	−0.485446	−	0.874266i	$-0.661343\pi$
$167$	−12.5467	−0.970893	−0.485446	+	0.874266i	$0.661343\pi$
$168$	0	0
$169$	7.58910	0.583777
$170$	0	0
$171$	−5.15252	−0.394023
$172$	0	0
$173$	14.1354	1.07469	0.537346	−	0.843362i	$-0.319427\pi$
$173$	14.1354	1.07469	0.537346	+	0.843362i	$0.319427\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−15.0614	−1.13208
$178$	0	0
$179$	26.2442	1.96158	0.980791	−	0.195061i	$-0.0624904\pi$
$179$	26.2442	1.96158	0.980791	+	0.195061i	$0.0624904\pi$
$180$	0	0
$181$	−18.3461	−1.36365	−0.681826	−	0.731514i	$-0.738814\pi$
$181$	−18.3461	−1.36365	−0.681826	+	0.731514i	$0.738814\pi$
$182$	0	0
$183$	−5.42223	−0.400823
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.67871	0.707777
$188$	0	0
$189$	−10.4258	−0.758366
$190$	0	0
$191$	3.86566	0.279709	0.139855	−	0.990172i	$-0.455336\pi$
$191$	3.86566	0.279709	0.139855	+	0.990172i	$0.455336\pi$
$192$	0	0
$193$	23.8599	1.71747	0.858737	−	0.512417i	$-0.171250\pi$
$193$	23.8599	1.71747	0.858737	+	0.512417i	$0.171250\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.7356	−0.836128	−0.418064	−	0.908418i	$-0.637291\pi$
$197$	−11.7356	−0.836128	−0.418064	+	0.908418i	$0.637291\pi$
$198$	0	0
$199$	8.89906	0.630838	0.315419	−	0.948953i	$-0.397855\pi$
$199$	8.89906	0.630838	0.315419	+	0.948953i	$0.397855\pi$
$200$	0	0
$201$	−6.01369	−0.424173
$202$	0	0
$203$	−0.281566	−0.0197620
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.735829	−0.0511437
$208$	0	0
$209$	37.3482	2.58343
$210$	0	0
$211$	−23.1224	−1.59181	−0.795906	−	0.605420i	$-0.793005\pi$
$211$	−23.1224	−1.59181	−0.795906	+	0.605420i	$0.793005\pi$
$212$	0	0
$213$	26.2688	1.79991
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.10635	0.142989
$218$	0	0
$219$	−24.0423	−1.62463
$220$	0	0
$221$	−8.23398	−0.553877
$222$	0	0
$223$	1.50863	0.101026	0.0505128	−	0.998723i	$-0.483914\pi$
$223$	1.50863	0.101026	0.0505128	+	0.998723i	$0.483914\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.0047	−1.46050	−0.730251	−	0.683179i	$-0.760597\pi$
$227$	−22.0047	−1.46050	−0.730251	+	0.683179i	$0.760597\pi$
$228$	0	0
$229$	−11.2931	−0.746266	−0.373133	−	0.927778i	$-0.621717\pi$
$229$	−11.2931	−0.746266	−0.373133	+	0.927778i	$0.621717\pi$
$230$	0	0
$231$	−24.5599	−1.61593
$232$	0	0
$233$	17.6789	1.15818	0.579091	−	0.815263i	$-0.303408\pi$
$233$	17.6789	1.15818	0.579091	+	0.815263i	$0.303408\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.1465	−0.853958
$238$	0	0
$239$	25.2583	1.63382	0.816911	−	0.576763i	$-0.195684\pi$
$239$	25.2583	1.63382	0.816911	+	0.576763i	$0.195684\pi$
$240$	0	0
$241$	0.790615	0.0509280	0.0254640	−	0.999676i	$-0.491894\pi$
$241$	0.790615	0.0509280	0.0254640	+	0.999676i	$0.491894\pi$
$242$	0	0
$243$	−7.48688	−0.480283
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−31.7732	−2.02168
$248$	0	0
$249$	−26.1300	−1.65592
$250$	0	0
$251$	−12.3461	−0.779276	−0.389638	−	0.920968i	$-0.627400\pi$
$251$	−12.3461	−0.779276	−0.389638	+	0.920968i	$0.627400\pi$
$252$	0	0
$253$	5.33368	0.335325
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.40442	−0.399497	−0.199748	−	0.979847i	$-0.564012\pi$
$257$	−6.40442	−0.399497	−0.199748	+	0.979847i	$0.564012\pi$
$258$	0	0
$259$	−17.9130	−1.11306
$260$	0	0
$261$	−0.0869661	−0.00538307
$262$	0	0
$263$	19.6914	1.21422	0.607111	−	0.794617i	$-0.292328\pi$
$263$	19.6914	1.21422	0.607111	+	0.794617i	$0.292328\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.60338	0.342922
$268$	0	0
$269$	−3.42494	−0.208822	−0.104411	−	0.994534i	$-0.533296\pi$
$269$	−3.42494	−0.208822	−0.104411	+	0.994534i	$0.533296\pi$
$270$	0	0
$271$	−13.7197	−0.833411	−0.416705	−	0.909042i	$-0.636815\pi$
$271$	−13.7197	−0.833411	−0.416705	+	0.909042i	$0.636815\pi$
$272$	0	0
$273$	20.8939	1.26456
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.636129	−0.0382213	−0.0191107	−	0.999817i	$-0.506083\pi$
$277$	−0.636129	−0.0382213	−0.0191107	+	0.999817i	$0.506083\pi$
$278$	0	0
$279$	0.650581	0.0389493
$280$	0	0
$281$	18.4124	1.09839	0.549195	−	0.835694i	$-0.314935\pi$
$281$	18.4124	1.09839	0.549195	+	0.835694i	$0.314935\pi$
$282$	0	0
$283$	−7.95729	−0.473012	−0.236506	−	0.971630i	$-0.576002\pi$
$283$	−7.95729	−0.473012	−0.236506	+	0.971630i	$0.576002\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.45886	−0.204170
$288$	0	0
$289$	−13.7071	−0.806299
$290$	0	0
$291$	3.82264	0.224087
$292$	0	0
$293$	20.9166	1.22196	0.610981	−	0.791645i	$-0.290775\pi$
$293$	20.9166	1.22196	0.610981	+	0.791645i	$0.290775\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	23.3415	1.35441
$298$	0	0
$299$	−4.53752	−0.262412
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−23.7424	−1.36396
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.78807	−0.330342	−0.165171	−	0.986265i	$-0.552818\pi$
$307$	−5.78807	−0.330342	−0.165171	+	0.986265i	$0.552818\pi$
$308$	0	0
$309$	−24.1721	−1.37510
$310$	0	0
$311$	−18.3674	−1.04152	−0.520761	−	0.853702i	$-0.674352\pi$
$311$	−18.3674	−1.04152	−0.520761	+	0.853702i	$0.674352\pi$
$312$	0	0
$313$	−9.85869	−0.557246	−0.278623	−	0.960401i	$-0.589878\pi$
$313$	−9.85869	−0.557246	−0.278623	+	0.960401i	$0.589878\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.9488	1.85059	0.925294	−	0.379250i	$-0.123818\pi$
$317$	32.9488	1.85059	0.925294	+	0.379250i	$0.123818\pi$
$318$	0	0
$319$	0.630376	0.0352943
$320$	0	0
$321$	−33.6016	−1.87546
$322$	0	0
$323$	12.7067	0.707021
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.32873	0.460580
$328$	0	0
$329$	24.8692	1.37109
$330$	0	0
$331$	6.85308	0.376679	0.188340	−	0.982104i	$-0.439689\pi$
$331$	6.85308	0.376679	0.188340	+	0.982104i	$0.439689\pi$
$332$	0	0
$333$	−5.53273	−0.303192
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−35.8432	−1.95250	−0.976251	−	0.216641i	$-0.930490\pi$
$337$	−35.8432	−1.95250	−0.976251	+	0.216641i	$0.930490\pi$
$338$	0	0
$339$	23.4827	1.27541
$340$	0	0
$341$	−4.71575	−0.255372
$342$	0	0
$343$	−19.8316	−1.07081
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.09198	−0.219669	−0.109835	−	0.993950i	$-0.535032\pi$
$347$	−4.09198	−0.219669	−0.109835	+	0.993950i	$0.535032\pi$
$348$	0	0
$349$	−11.3229	−0.606101	−0.303051	−	0.952974i	$-0.598005\pi$
$349$	−11.3229	−0.606101	−0.303051	+	0.952974i	$0.598005\pi$
$350$	0	0
$351$	−19.8574	−1.05991
$352$	0	0
$353$	−26.1443	−1.39152	−0.695761	−	0.718273i	$-0.744933\pi$
$353$	−26.1443	−1.39152	−0.695761	+	0.718273i	$0.744933\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−8.35587	−0.442239
$358$	0	0
$359$	12.2058	0.644198	0.322099	−	0.946706i	$-0.395612\pi$
$359$	12.2058	0.644198	0.322099	+	0.946706i	$0.395612\pi$
$360$	0	0
$361$	30.0327	1.58067
$362$	0	0
$363$	33.7242	1.77006
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.15299	−0.164585	−0.0822923	−	0.996608i	$-0.526224\pi$
$367$	−3.15299	−0.164585	−0.0822923	+	0.996608i	$0.526224\pi$
$368$	0	0
$369$	−1.06832	−0.0556147
$370$	0	0
$371$	22.4458	1.16533
$372$	0	0
$373$	−9.59872	−0.497003	−0.248501	−	0.968632i	$-0.579938\pi$
$373$	−9.59872	−0.497003	−0.248501	+	0.968632i	$0.579938\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.536280	−0.0276198
$378$	0	0
$379$	−32.7841	−1.68401	−0.842003	−	0.539473i	$-0.818624\pi$
$379$	−32.7841	−1.68401	−0.842003	+	0.539473i	$0.818624\pi$
$380$	0	0
$381$	−2.37647	−0.121750
$382$	0	0
$383$	34.7378	1.77502	0.887508	−	0.460792i	$-0.152435\pi$
$383$	34.7378	1.77502	0.887508	+	0.460792i	$0.152435\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.8965	0.704581	0.352290	−	0.935891i	$-0.385403\pi$
$389$	13.8965	0.704581	0.352290	+	0.935891i	$0.385403\pi$
$390$	0	0
$391$	1.81464	0.0917704
$392$	0	0
$393$	−3.81434	−0.192408
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.1135	1.31060	0.655299	−	0.755370i	$-0.272543\pi$
$397$	26.1135	1.31060	0.655299	+	0.755370i	$0.272543\pi$
$398$	0	0
$399$	−32.2436	−1.61420
$400$	0	0
$401$	18.0715	0.902446	0.451223	−	0.892411i	$-0.350988\pi$
$401$	18.0715	0.902446	0.451223	+	0.892411i	$0.350988\pi$
$402$	0	0
$403$	4.01183	0.199844
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.1041	1.98789
$408$	0	0
$409$	−0.865532	−0.0427978	−0.0213989	−	0.999771i	$-0.506812\pi$
$409$	−0.865532	−0.0427978	−0.0213989	+	0.999771i	$0.506812\pi$
$410$	0	0
$411$	−26.6094	−1.31254
$412$	0	0
$413$	−18.5643	−0.913487
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.2494	−0.501916
$418$	0	0
$419$	20.5675	1.00479	0.502394	−	0.864639i	$-0.332453\pi$
$419$	20.5675	1.00479	0.502394	+	0.864639i	$0.332453\pi$
$420$	0	0
$421$	26.2423	1.27897	0.639485	−	0.768803i	$-0.279147\pi$
$421$	26.2423	1.27897	0.639485	+	0.768803i	$0.279147\pi$
$422$	0	0
$423$	7.68126	0.373476
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.68331	−0.323428
$428$	0	0
$429$	−46.7777	−2.25845
$430$	0	0
$431$	−23.2654	−1.12066	−0.560328	−	0.828271i	$-0.689325\pi$
$431$	−23.2654	−1.12066	−0.560328	+	0.828271i	$0.689325\pi$
$432$	0	0
$433$	34.7365	1.66933	0.834665	−	0.550758i	$-0.185661\pi$
$433$	34.7365	1.66933	0.834665	+	0.550758i	$0.185661\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.00233	0.334967
$438$	0	0
$439$	−28.5238	−1.36137	−0.680684	−	0.732577i	$-0.738317\pi$
$439$	−28.5238	−1.36137	−0.680684	+	0.732577i	$0.738317\pi$
$440$	0	0
$441$	−0.974509	−0.0464052
$442$	0	0
$443$	26.3428	1.25158	0.625792	−	0.779990i	$-0.284776\pi$
$443$	26.3428	1.25158	0.625792	+	0.779990i	$0.284776\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.9457	0.706908
$448$	0	0
$449$	29.5411	1.39413	0.697065	−	0.717008i	$-0.254489\pi$
$449$	29.5411	1.39413	0.697065	+	0.717008i	$0.254489\pi$
$450$	0	0
$451$	7.74377	0.364640
$452$	0	0
$453$	−41.3926	−1.94480
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.16668	−0.148131	−0.0740655	−	0.997253i	$-0.523597\pi$
$457$	−3.16668	−0.148131	−0.0740655	+	0.997253i	$0.523597\pi$
$458$	0	0
$459$	7.94133	0.370670
$460$	0	0
$461$	8.43891	0.393039	0.196520	−	0.980500i	$-0.437036\pi$
$461$	8.43891	0.393039	0.196520	+	0.980500i	$0.437036\pi$
$462$	0	0
$463$	9.62461	0.447294	0.223647	−	0.974670i	$-0.428204\pi$
$463$	9.62461	0.447294	0.223647	+	0.974670i	$0.428204\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.1041	−1.06913	−0.534564	−	0.845128i	$-0.679524\pi$
$467$	−23.1041	−1.06913	−0.534564	+	0.845128i	$0.679524\pi$
$468$	0	0
$469$	−7.41233	−0.342270
$470$	0	0
$471$	7.82289	0.360460
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.93274	0.317428
$478$	0	0
$479$	−20.5981	−0.941150	−0.470575	−	0.882360i	$-0.655953\pi$
$479$	−20.5981	−0.941150	−0.470575	+	0.882360i	$0.655953\pi$
$480$	0	0
$481$	−34.1178	−1.55564
$482$	0	0
$483$	−4.60469	−0.209521
$484$	0	0
$485$	0	0
$486$	0	0
$487$	40.1145	1.81776	0.908881	−	0.417056i	$-0.136938\pi$
$487$	40.1145	1.81776	0.908881	+	0.417056i	$0.136938\pi$
$488$	0	0
$489$	17.4868	0.790780
$490$	0	0
$491$	18.4799	0.833985	0.416993	−	0.908910i	$-0.363084\pi$
$491$	18.4799	0.833985	0.416993	+	0.908910i	$0.363084\pi$
$492$	0	0
$493$	0.214469	0.00965918
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.3783	1.45237
$498$	0	0
$499$	−0.0506452	−0.00226719	−0.00113360	−	0.999999i	$-0.500361\pi$
$499$	−0.0506452	−0.00226719	−0.00113360	+	0.999999i	$0.500361\pi$
$500$	0	0
$501$	−24.2506	−1.08344
$502$	0	0
$503$	−31.1887	−1.39064	−0.695318	−	0.718702i	$-0.744737\pi$
$503$	−31.1887	−1.39064	−0.695318	+	0.718702i	$0.744737\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.6684	0.651448
$508$	0	0
$509$	3.42426	0.151778	0.0758888	−	0.997116i	$-0.475821\pi$
$509$	3.42426	0.151778	0.0758888	+	0.997116i	$0.475821\pi$
$510$	0	0
$511$	−29.6340	−1.31093
$512$	0	0
$513$	30.6440	1.35297
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−55.6778	−2.44871
$518$	0	0
$519$	27.3213	1.19927
$520$	0	0
$521$	−9.78442	−0.428663	−0.214332	−	0.976761i	$-0.568757\pi$
$521$	−9.78442	−0.428663	−0.214332	+	0.976761i	$0.568757\pi$
$522$	0	0
$523$	6.22884	0.272368	0.136184	−	0.990684i	$-0.456516\pi$
$523$	6.22884	0.272368	0.136184	+	0.990684i	$0.456516\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.60441	−0.0698892
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.73387	−0.248829
$532$	0	0
$533$	−6.58786	−0.285352
$534$	0	0
$535$	0	0
$536$	0	0
$537$	50.7255	2.18897
$538$	0	0
$539$	7.06375	0.304257
$540$	0	0
$541$	19.8477	0.853319	0.426660	−	0.904412i	$-0.359690\pi$
$541$	19.8477	0.853319	0.426660	+	0.904412i	$0.359690\pi$
$542$	0	0
$543$	−35.4598	−1.52173
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.68024	−0.328383	−0.164192	−	0.986428i	$-0.552502\pi$
$547$	−7.68024	−0.328383	−0.164192	+	0.986428i	$0.552502\pi$
$548$	0	0
$549$	−2.06425	−0.0880999
$550$	0	0
$551$	0.827591	0.0352566
$552$	0	0
$553$	−16.2041	−0.689067
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.1260	−0.768023	−0.384012	−	0.923328i	$-0.625458\pi$
$557$	−18.1260	−0.768023	−0.384012	+	0.923328i	$0.625458\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	18.7073	0.789823
$562$	0	0
$563$	−16.8772	−0.711287	−0.355644	−	0.934622i	$-0.615738\pi$
$563$	−16.8772	−0.711287	−0.355644	+	0.934622i	$0.615738\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−25.4103	−1.06713
$568$	0	0
$569$	36.0022	1.50929	0.754645	−	0.656133i	$-0.227809\pi$
$569$	36.0022	1.50929	0.754645	+	0.656133i	$0.227809\pi$
$570$	0	0
$571$	−26.6267	−1.11429	−0.557147	−	0.830414i	$-0.688104\pi$
$571$	−26.6267	−1.11429	−0.557147	+	0.830414i	$0.688104\pi$
$572$	0	0
$573$	7.47166	0.312133
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−30.0190	−1.24971	−0.624854	−	0.780742i	$-0.714842\pi$
$577$	−30.0190	−1.24971	−0.624854	+	0.780742i	$0.714842\pi$
$578$	0	0
$579$	46.1171	1.91656
$580$	0	0
$581$	−32.2072	−1.33618
$582$	0	0
$583$	−50.2521	−2.08123
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.1814	−1.28699	−0.643497	−	0.765449i	$-0.722517\pi$
$587$	−31.1814	−1.28699	−0.643497	+	0.765449i	$0.722517\pi$
$588$	0	0
$589$	−6.19109	−0.255099
$590$	0	0
$591$	−22.6829	−0.933051
$592$	0	0
$593$	−15.0334	−0.617348	−0.308674	−	0.951168i	$-0.599885\pi$
$593$	−15.0334	−0.617348	−0.308674	+	0.951168i	$0.599885\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.2004	0.703964
$598$	0	0
$599$	−20.3534	−0.831617	−0.415808	−	0.909452i	$-0.636501\pi$
$599$	−20.3534	−0.831617	−0.415808	+	0.909452i	$0.636501\pi$
$600$	0	0
$601$	46.5338	1.89816	0.949078	−	0.315043i	$-0.102019\pi$
$601$	46.5338	1.89816	0.949078	+	0.315043i	$0.102019\pi$
$602$	0	0
$603$	−2.28942	−0.0932323
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.52725	−0.305522	−0.152761	−	0.988263i	$-0.548816\pi$
$607$	−7.52725	−0.305522	−0.152761	+	0.988263i	$0.548816\pi$
$608$	0	0
$609$	−0.544219	−0.0220529
$610$	0	0
$611$	47.3668	1.91626
$612$	0	0
$613$	−32.2643	−1.30314	−0.651572	−	0.758587i	$-0.725890\pi$
$613$	−32.2643	−1.30314	−0.651572	+	0.758587i	$0.725890\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.52158	−0.182032	−0.0910161	−	0.995849i	$-0.529011\pi$
$617$	−4.52158	−0.182032	−0.0910161	+	0.995849i	$0.529011\pi$
$618$	0	0
$619$	−17.7807	−0.714668	−0.357334	−	0.933977i	$-0.616314\pi$
$619$	−17.7807	−0.714668	−0.357334	+	0.933977i	$0.616314\pi$
$620$	0	0
$621$	4.37626	0.175613
$622$	0	0
$623$	6.90660	0.276707
$624$	0	0
$625$	0	0
$626$	0	0
$627$	72.1877	2.88290
$628$	0	0
$629$	13.6444	0.544036
$630$	0	0
$631$	−27.4554	−1.09298	−0.546490	−	0.837466i	$-0.684036\pi$
$631$	−27.4554	−1.09298	−0.546490	+	0.837466i	$0.684036\pi$
$632$	0	0
$633$	−44.6917	−1.77634
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.00935	−0.238099
$638$	0	0
$639$	10.0006	0.395616
$640$	0	0
$641$	−48.9270	−1.93250	−0.966250	−	0.257605i	$-0.917067\pi$
$641$	−48.9270	−1.93250	−0.966250	+	0.257605i	$0.917067\pi$
$642$	0	0
$643$	2.89333	0.114102	0.0570508	−	0.998371i	$-0.481830\pi$
$643$	2.89333	0.114102	0.0570508	+	0.998371i	$0.481830\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.0877	0.593157	0.296578	−	0.955009i	$-0.404154\pi$
$647$	15.0877	0.593157	0.296578	+	0.955009i	$0.404154\pi$
$648$	0	0
$649$	41.5621	1.63145
$650$	0	0
$651$	4.07122	0.159564
$652$	0	0
$653$	13.3255	0.521468	0.260734	−	0.965411i	$-0.416035\pi$
$653$	13.3255	0.521468	0.260734	+	0.965411i	$0.416035\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.15292	−0.357089
$658$	0	0
$659$	39.9000	1.55428	0.777142	−	0.629325i	$-0.216669\pi$
$659$	39.9000	1.55428	0.777142	+	0.629325i	$0.216669\pi$
$660$	0	0
$661$	−37.3060	−1.45104	−0.725518	−	0.688203i	$-0.758400\pi$
$661$	−37.3060	−1.45104	−0.725518	+	0.688203i	$0.758400\pi$
$662$	0	0
$663$	−15.9149	−0.618082
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.118188	0.00457625
$668$	0	0
$669$	2.91593	0.112736
$670$	0	0
$671$	14.9627	0.577630
$672$	0	0
$673$	18.6641	0.719447	0.359724	−	0.933059i	$-0.382871\pi$
$673$	18.6641	0.719447	0.359724	+	0.933059i	$0.382871\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.4978	−0.749361	−0.374681	−	0.927154i	$-0.622248\pi$
$677$	−19.4978	−0.749361	−0.374681	+	0.927154i	$0.622248\pi$
$678$	0	0
$679$	4.71169	0.180818
$680$	0	0
$681$	−42.5313	−1.62980
$682$	0	0
$683$	−41.6331	−1.59305	−0.796523	−	0.604608i	$-0.793330\pi$
$683$	−41.6331	−1.59305	−0.796523	+	0.604608i	$0.793330\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−21.8276	−0.832773
$688$	0	0
$689$	42.7510	1.62868
$690$	0	0
$691$	9.53192	0.362611	0.181306	−	0.983427i	$-0.441968\pi$
$691$	9.53192	0.362611	0.181306	+	0.983427i	$0.441968\pi$
$692$	0	0
$693$	−9.34998	−0.355177
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.63461	0.0997930
$698$	0	0
$699$	34.1702	1.29244
$700$	0	0
$701$	27.9897	1.05716	0.528579	−	0.848884i	$-0.322725\pi$
$701$	27.9897	1.05716	0.528579	+	0.848884i	$0.322725\pi$
$702$	0	0
$703$	52.6508	1.98576
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−29.2643	−1.10060
$708$	0	0
$709$	27.6832	1.03966	0.519831	−	0.854269i	$-0.325995\pi$
$709$	27.6832	1.03966	0.519831	+	0.854269i	$0.325995\pi$
$710$	0	0
$711$	−5.00489	−0.187698
$712$	0	0
$713$	−0.884147	−0.0331115
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.8200	1.82322
$718$	0	0
$719$	−50.8364	−1.89588	−0.947938	−	0.318454i	$-0.896836\pi$
$719$	−50.8364	−1.89588	−0.947938	+	0.318454i	$0.896836\pi$
$720$	0	0
$721$	−29.7940	−1.10959
$722$	0	0
$723$	1.52812	0.0568316
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.5061	1.39102	0.695512	−	0.718514i	$-0.255177\pi$
$727$	37.5061	1.39102	0.695512	+	0.718514i	$0.255177\pi$
$728$	0	0
$729$	17.5273	0.649158
$730$	0	0
$731$	0	0
$732$	0	0
$733$	36.9374	1.36431	0.682157	−	0.731206i	$-0.261042\pi$
$733$	36.9374	1.36431	0.682157	+	0.731206i	$0.261042\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.5949	0.611281
$738$	0	0
$739$	12.3287	0.453517	0.226759	−	0.973951i	$-0.427187\pi$
$739$	12.3287	0.453517	0.226759	+	0.973951i	$0.427187\pi$
$740$	0	0
$741$	−61.4123	−2.25604
$742$	0	0
$743$	−29.7534	−1.09154	−0.545772	−	0.837933i	$-0.683764\pi$
$743$	−29.7534	−1.09154	−0.545772	+	0.837933i	$0.683764\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.94770	−0.363967
$748$	0	0
$749$	−41.4166	−1.51333
$750$	0	0
$751$	41.9834	1.53200	0.765999	−	0.642842i	$-0.222245\pi$
$751$	41.9834	1.53200	0.765999	+	0.642842i	$0.222245\pi$
$752$	0	0
$753$	−23.8628	−0.869610
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.4708	−0.416915	−0.208458	−	0.978031i	$-0.566844\pi$
$757$	−11.4708	−0.416915	−0.208458	+	0.978031i	$0.566844\pi$
$758$	0	0
$759$	10.3091	0.374196
$760$	0	0
$761$	−13.4122	−0.486193	−0.243097	−	0.970002i	$-0.578163\pi$
$761$	−13.4122	−0.486193	−0.243097	+	0.970002i	$0.578163\pi$
$762$	0	0
$763$	10.2658	0.371646
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35.3581	−1.27671
$768$	0	0
$769$	2.18327	0.0787307	0.0393654	−	0.999225i	$-0.487466\pi$
$769$	2.18327	0.0787307	0.0393654	+	0.999225i	$0.487466\pi$
$770$	0	0
$771$	−12.3787	−0.445806
$772$	0	0
$773$	−5.73306	−0.206204	−0.103102	−	0.994671i	$-0.532877\pi$
$773$	−5.73306	−0.206204	−0.103102	+	0.994671i	$0.532877\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−34.6228	−1.24209
$778$	0	0
$779$	10.1664	0.364250
$780$	0	0
$781$	−72.4893	−2.59387
$782$	0	0
$783$	0.517220	0.0184839
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−8.13502	−0.289982	−0.144991	−	0.989433i	$-0.546315\pi$
$787$	−8.13502	−0.289982	−0.144991	+	0.989433i	$0.546315\pi$
$788$	0	0
$789$	38.0601	1.35497
$790$	0	0
$791$	28.9442	1.02914
$792$	0	0
$793$	−12.7293	−0.452029
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.5139	1.47050	0.735250	−	0.677797i	$-0.237065\pi$
$797$	41.5139	1.47050	0.735250	+	0.677797i	$0.237065\pi$
$798$	0	0
$799$	−18.9429	−0.670151
$800$	0	0
$801$	2.13321	0.0753733
$802$	0	0
$803$	66.3451	2.34127
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.61982	−0.233029
$808$	0	0
$809$	−22.8566	−0.803596	−0.401798	−	0.915728i	$-0.631615\pi$
$809$	−22.8566	−0.803596	−0.401798	+	0.915728i	$0.631615\pi$
$810$	0	0
$811$	42.1917	1.48155	0.740776	−	0.671753i	$-0.234458\pi$
$811$	42.1917	1.48155	0.740776	+	0.671753i	$0.234458\pi$
$812$	0	0
$813$	−26.5178	−0.930020
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	7.95432	0.277946
$820$	0	0
$821$	27.0195	0.942985	0.471493	−	0.881870i	$-0.343715\pi$
$821$	27.0195	0.942985	0.471493	+	0.881870i	$0.343715\pi$
$822$	0	0
$823$	27.2435	0.949650	0.474825	−	0.880080i	$-0.342511\pi$
$823$	27.2435	0.949650	0.474825	+	0.880080i	$0.342511\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.5732	0.367667	0.183834	−	0.982957i	$-0.441149\pi$
$827$	10.5732	0.367667	0.183834	+	0.982957i	$0.441149\pi$
$828$	0	0
$829$	−1.16445	−0.0404429	−0.0202215	−	0.999796i	$-0.506437\pi$
$829$	−1.16445	−0.0404429	−0.0202215	+	0.999796i	$0.506437\pi$
$830$	0	0
$831$	−1.22953	−0.0426519
$832$	0	0
$833$	2.40325	0.0832678
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.86925	−0.133741
$838$	0	0
$839$	20.2920	0.700556	0.350278	−	0.936646i	$-0.386087\pi$
$839$	20.2920	0.700556	0.350278	+	0.936646i	$0.386087\pi$
$840$	0	0
$841$	−28.9860	−0.999518
$842$	0	0
$843$	35.5880	1.22571
$844$	0	0
$845$	0	0
$846$	0	0
$847$	41.5676	1.42828
$848$	0	0
$849$	−15.3801	−0.527843
$850$	0	0
$851$	7.51903	0.257749
$852$	0	0
$853$	−26.6154	−0.911295	−0.455648	−	0.890160i	$-0.650592\pi$
$853$	−26.6154	−0.911295	−0.455648	+	0.890160i	$0.650592\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.4342	0.356427	0.178214	−	0.983992i	$-0.442968\pi$
$857$	10.4342	0.356427	0.178214	+	0.983992i	$0.442968\pi$
$858$	0	0
$859$	24.6623	0.841466	0.420733	−	0.907185i	$-0.361773\pi$
$859$	24.6623	0.841466	0.420733	+	0.907185i	$0.361773\pi$
$860$	0	0
$861$	−6.68538	−0.227837
$862$	0	0
$863$	8.48317	0.288771	0.144385	−	0.989522i	$-0.453880\pi$
$863$	8.48317	0.288771	0.144385	+	0.989522i	$0.453880\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−26.4934	−0.899764
$868$	0	0
$869$	36.2780	1.23065
$870$	0	0
$871$	−14.1178	−0.478363
$872$	0	0
$873$	1.45528	0.0492538
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.08796	−0.104273	−0.0521365	−	0.998640i	$-0.516603\pi$
$877$	−3.08796	−0.104273	−0.0521365	+	0.998640i	$0.516603\pi$
$878$	0	0
$879$	40.4282	1.36361
$880$	0	0
$881$	42.5879	1.43482	0.717412	−	0.696649i	$-0.245327\pi$
$881$	42.5879	1.43482	0.717412	+	0.696649i	$0.245327\pi$
$882$	0	0
$883$	−48.3342	−1.62657	−0.813287	−	0.581863i	$-0.802324\pi$
$883$	−48.3342	−1.62657	−0.813287	+	0.581863i	$0.802324\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.12173	−0.0712408	−0.0356204	−	0.999365i	$-0.511341\pi$
$887$	−2.12173	−0.0712408	−0.0356204	+	0.999365i	$0.511341\pi$
$888$	0	0
$889$	−2.92918	−0.0982415
$890$	0	0
$891$	56.8892	1.90586
$892$	0	0
$893$	−73.0968	−2.44609
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.77026	−0.292830
$898$	0	0
$899$	−0.104495	−0.00348512
$900$	0	0
$901$	−17.0970	−0.569582
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.6914	0.952684	0.476342	−	0.879260i	$-0.341962\pi$
$907$	28.6914	0.952684	0.476342	+	0.879260i	$0.341962\pi$
$908$	0	0
$909$	−9.03875	−0.299796
$910$	0	0
$911$	1.14899	0.0380679	0.0190339	−	0.999819i	$-0.493941\pi$
$911$	1.14899	0.0380679	0.0190339	+	0.999819i	$0.493941\pi$
$912$	0	0
$913$	72.1061	2.38636
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.70146	−0.155256
$918$	0	0
$919$	55.4798	1.83011	0.915055	−	0.403329i	$-0.132147\pi$
$919$	55.4798	1.83011	0.915055	+	0.403329i	$0.132147\pi$
$920$	0	0
$921$	−11.1873	−0.368636
$922$	0	0
$923$	61.6689	2.02986
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−9.20235	−0.302245
$928$	0	0
$929$	−4.05780	−0.133132	−0.0665660	−	0.997782i	$-0.521204\pi$
$929$	−4.05780	−0.133132	−0.0665660	+	0.997782i	$0.521204\pi$
$930$	0	0
$931$	9.27367	0.303932
$932$	0	0
$933$	−35.5011	−1.16226
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.0588	−0.524617	−0.262308	−	0.964984i	$-0.584484\pi$
$937$	−16.0588	−0.524617	−0.262308	+	0.964984i	$0.584484\pi$
$938$	0	0
$939$	−19.0552	−0.621842
$940$	0	0
$941$	4.73779	0.154448	0.0772238	−	0.997014i	$-0.475394\pi$
$941$	4.73779	0.154448	0.0772238	+	0.997014i	$0.475394\pi$
$942$	0	0
$943$	1.45186	0.0472792
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.38634	−0.110041	−0.0550207	−	0.998485i	$-0.517522\pi$
$947$	−3.38634	−0.110041	−0.0550207	+	0.998485i	$0.517522\pi$
$948$	0	0
$949$	−56.4418	−1.83218
$950$	0	0
$951$	63.6844	2.06511
$952$	0	0
$953$	−29.2967	−0.949013	−0.474507	−	0.880252i	$-0.657373\pi$
$953$	−29.2967	−0.949013	−0.474507	+	0.880252i	$0.657373\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.21841	0.0393856
$958$	0	0
$959$	−32.7981	−1.05910
$960$	0	0
$961$	−30.2183	−0.974783
$962$	0	0
$963$	−12.7922	−0.412222
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15.5743	−0.500837	−0.250419	−	0.968138i	$-0.580568\pi$
$967$	−15.5743	−0.500837	−0.250419	+	0.968138i	$0.580568\pi$
$968$	0	0
$969$	24.5599	0.788979
$970$	0	0
$971$	−51.5951	−1.65577	−0.827883	−	0.560900i	$-0.810455\pi$
$971$	−51.5951	−1.65577	−0.827883	+	0.560900i	$0.810455\pi$
$972$	0	0
$973$	−12.6332	−0.405001
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.04984	−0.289530	−0.144765	−	0.989466i	$-0.546243\pi$
$977$	−9.04984	−0.289530	−0.144765	+	0.989466i	$0.546243\pi$
$978$	0	0
$979$	−15.4626	−0.494188
$980$	0	0
$981$	3.17075	0.101234
$982$	0	0
$983$	−39.3855	−1.25620	−0.628101	−	0.778132i	$-0.716167\pi$
$983$	−39.3855	−1.25620	−0.628101	+	0.778132i	$0.716167\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	48.0680	1.53002
$988$	0	0
$989$	0	0
$990$	0	0
$991$	18.9152	0.600860	0.300430	−	0.953804i	$-0.402870\pi$
$991$	18.9152	0.600860	0.300430	+	0.953804i	$0.402870\pi$
$992$	0	0
$993$	13.2458	0.420344
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.2966	0.389438	0.194719	−	0.980859i	$-0.437620\pi$
$997$	12.2966	0.389438	0.194719	+	0.980859i	$0.437620\pi$
$998$	0	0
$999$	32.9052	1.04107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cu.1.4		5
4.3	odd	2		4600.2.a.be.1.2		5
5.4	even	2		1840.2.a.v.1.2		5
20.3	even	4		4600.2.e.u.4049.3		10
20.7	even	4		4600.2.e.u.4049.8		10
20.19	odd	2		920.2.a.j.1.4	✓	5
40.19	odd	2		7360.2.a.co.1.2		5
40.29	even	2		7360.2.a.cp.1.4		5
60.59	even	2		8280.2.a.bs.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.4	✓	5	20.19	odd	2
1840.2.a.v.1.2		5	5.4	even	2
4600.2.a.be.1.2		5	4.3	odd	2
4600.2.e.u.4049.3		10	20.3	even	4
4600.2.e.u.4049.8		10	20.7	even	4
7360.2.a.co.1.2		5	40.19	odd	2
7360.2.a.cp.1.4		5	40.29	even	2
8280.2.a.bs.1.4		5	60.59	even	2
9200.2.a.cu.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93283 q^{3} +2.38236 q^{7} +0.735829 q^{9} +O(q^{10})\)	\(q+1.93283 q^{3} +2.38236 q^{7} +0.735829 q^{9} -5.33368 q^{11} +4.53752 q^{13} -1.81464 q^{17} -7.00233 q^{19} +4.60469 q^{21} -1.00000 q^{23} -4.37626 q^{27} -0.118188 q^{29} +0.884147 q^{31} -10.3091 q^{33} -7.51903 q^{37} +8.77026 q^{39} -1.45186 q^{41} +10.4389 q^{47} -1.32437 q^{49} -3.50739 q^{51} +9.42167 q^{53} -13.5343 q^{57} -7.79239 q^{59} -2.80533 q^{61} +1.75301 q^{63} -3.11134 q^{67} -1.93283 q^{69} +13.5909 q^{71} -12.4389 q^{73} -12.7067 q^{77} -6.80169 q^{79} -10.6660 q^{81} -13.5190 q^{83} -0.228437 q^{87} +2.89906 q^{89} +10.8100 q^{91} +1.70890 q^{93} +1.97774 q^{97} -3.92468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{7} + 13 q^{9}+O(q^{10}) \) 5 * q - 2 * q^7 + 13 * q^9	\( 5 q - 2 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 4 q^{17} - 7 q^{19} + 6 q^{21} - 5 q^{23} + 3 q^{27} + 4 q^{29} - 19 q^{31} - 17 q^{33} - 15 q^{37} - 19 q^{39} + 25 q^{41} - 11 q^{47} + 25 q^{49} - 19 q^{51} - 3 q^{53} - 48 q^{57} + q^{59} - 5 q^{61} - 41 q^{63} + 9 q^{67} - q^{71} + q^{73} - 18 q^{77} + 2 q^{79} + 57 q^{81} - 45 q^{83} - 9 q^{87} + 6 q^{89} - 11 q^{91} + 39 q^{93} - 25 q^{97} + 65 q^{99}+O(q^{100}) \) 5 * q - 2 * q^7 + 13 * q^9 + q^11 - 4 * q^13 - 4 * q^17 - 7 * q^19 + 6 * q^21 - 5 * q^23 + 3 * q^27 + 4 * q^29 - 19 * q^31 - 17 * q^33 - 15 * q^37 - 19 * q^39 + 25 * q^41 - 11 * q^47 + 25 * q^49 - 19 * q^51 - 3 * q^53 - 48 * q^57 + q^59 - 5 * q^61 - 41 * q^63 + 9 * q^67 - q^71 + q^73 - 18 * q^77 + 2 * q^79 + 57 * q^81 - 45 * q^83 - 9 * q^87 + 6 * q^89 - 11 * q^91 + 39 * q^93 - 25 * q^97 + 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more