LMFDB - Embedded newform 7803.2.a.by.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7803,2,Mod(1,7803)] 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("7803.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7803, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-6,0,12,3,0,3,-18,0,0,-6,0,-3,-3,0,6,0,0,-3,-6,0,-12,3,0, 6,24,0,9,6,0,0,-42,0,0,-33,0,0,-36,0,15,0,0,-3,18,0,12,-24,0,18,-42,0, -12,-48,0,-3,15,0,-12,-18,0,0,-63,0,30,-24,0,12,0,0,51,21,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(73)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.3072686972$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 6 x^{14} - 3 x^{13} + 76 x^{12} - 69 x^{11} - 354 x^{10} + 523 x^{9} + 720 x^{8} - 1437 x^{7} + \cdots - 51 $ x^15 - 6x^14 - 3x^13 + 76x^12 - 69x^11 - 354x^10 + 523x^9 + 720x^8 - 1437x^7 - 529x^6 + 1752x^5 - 111x^4 - 902x^3 + 258x^2 + 135x - 51
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.641769$ of defining polynomial
Character	$\chi$	$=$	7803.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.641769 q^{2} -1.58813 q^{4} +2.68329 q^{5} +4.44365 q^{7} +2.30275 q^{8} -1.72205 q^{10} -6.33442 q^{11} -5.38327 q^{13} -2.85180 q^{14} +1.69843 q^{16} -3.40555 q^{19} -4.26142 q^{20} +4.06524 q^{22} +2.35045 q^{23} +2.20003 q^{25} +3.45482 q^{26} -7.05711 q^{28} -0.405468 q^{29} +3.62200 q^{31} -5.69550 q^{32} +11.9236 q^{35} +4.96694 q^{37} +2.18558 q^{38} +6.17895 q^{40} -5.68257 q^{41} +5.89136 q^{43} +10.0599 q^{44} -1.50844 q^{46} +1.84042 q^{47} +12.7460 q^{49} -1.41191 q^{50} +8.54935 q^{52} -2.31475 q^{53} -16.9971 q^{55} +10.2326 q^{56} +0.260217 q^{58} +12.6166 q^{59} -9.13811 q^{61} -2.32449 q^{62} +0.258335 q^{64} -14.4449 q^{65} +0.233694 q^{67} -7.65219 q^{70} -9.29208 q^{71} +3.05704 q^{73} -3.18763 q^{74} +5.40847 q^{76} -28.1480 q^{77} -5.59075 q^{79} +4.55738 q^{80} +3.64690 q^{82} -9.49892 q^{83} -3.78089 q^{86} -14.5866 q^{88} +14.3614 q^{89} -23.9214 q^{91} -3.73282 q^{92} -1.18112 q^{94} -9.13807 q^{95} -17.6601 q^{97} -8.18002 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 6 q^{2} + 12 q^{4} + 3 q^{5} + 3 q^{7} - 18 q^{8} - 6 q^{11} - 3 q^{13} - 3 q^{14} + 6 q^{16} - 3 q^{19} - 6 q^{20} - 12 q^{22} + 3 q^{23} + 6 q^{25} + 24 q^{26} + 9 q^{28} + 6 q^{29} - 42 q^{32}+ \cdots - 15 q^{98}+O(q^{100}) $ 15 * q - 6 * q^2 + 12 * q^4 + 3 * q^5 + 3 * q^7 - 18 * q^8 - 6 * q^11 - 3 * q^13 - 3 * q^14 + 6 * q^16 - 3 * q^19 - 6 * q^20 - 12 * q^22 + 3 * q^23 + 6 * q^25 + 24 * q^26 + 9 * q^28 + 6 * q^29 - 42 * q^32 - 33 * q^35 - 36 * q^38 + 15 * q^40 - 3 * q^43 + 18 * q^44 + 12 * q^46 - 24 * q^47 + 18 * q^49 - 42 * q^50 - 12 * q^52 - 48 * q^53 - 3 * q^55 + 15 * q^56 - 12 * q^58 - 18 * q^59 - 63 * q^62 + 30 * q^64 - 24 * q^65 + 12 * q^67 + 51 * q^70 + 21 * q^71 - 18 * q^73 + 48 * q^74 + 60 * q^76 - 54 * q^77 - 39 * q^79 - 6 * q^80 + 39 * q^82 - 51 * q^83 - 48 * q^86 - 48 * q^88 - 36 * q^89 + 9 * q^91 + 3 * q^92 + 36 * q^94 - 39 * q^95 + 3 * q^97 - 15 * q^98

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 $

$2$	−0.641769	−0.453799	−0.226900	−	0.973918i	$-0.572859\pi$
$2$	−0.641769	−0.453799	−0.226900	+	0.973918i	$0.572859\pi$
$3$	0	0
$3$	0	0
$4$	−1.58813	−0.794066
$5$	2.68329	1.20000	0.600001	−	0.799999i	$-0.295167\pi$
$5$	2.68329	1.20000	0.600001	+	0.799999i	$0.295167\pi$
$6$	0	0
$7$	4.44365	1.67954	0.839771	−	0.542940i	$-0.182689\pi$
$7$	4.44365	1.67954	0.839771	+	0.542940i	$0.182689\pi$
$8$	2.30275	0.814146
$9$	0	0
$10$	−1.72205	−0.544560
$11$	−6.33442	−1.90990	−0.954950	−	0.296766i	$-0.904092\pi$
$11$	−6.33442	−1.90990	−0.954950	+	0.296766i	$0.904092\pi$
$12$	0	0
$13$	−5.38327	−1.49305	−0.746526	−	0.665356i	$-0.768280\pi$
$13$	−5.38327	−1.49305	−0.746526	+	0.665356i	$0.768280\pi$
$14$	−2.85180	−0.762175
$15$	0	0
$16$	1.69843	0.424608
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−3.40555	−0.781287	−0.390643	−	0.920542i	$-0.627747\pi$
$19$	−3.40555	−0.781287	−0.390643	+	0.920542i	$0.627747\pi$
$20$	−4.26142	−0.952882
$21$	0	0
$22$	4.06524	0.866711
$23$	2.35045	0.490102	0.245051	−	0.969510i	$-0.421195\pi$
$23$	2.35045	0.490102	0.245051	+	0.969510i	$0.421195\pi$
$24$	0	0
$25$	2.20003	0.440007
$26$	3.45482	0.677546
$27$	0	0
$28$	−7.05711	−1.33367
$29$	−0.405468	−0.0752935	−0.0376467	−	0.999291i	$-0.511986\pi$
$29$	−0.405468	−0.0752935	−0.0376467	+	0.999291i	$0.511986\pi$
$30$	0	0
$31$	3.62200	0.650530	0.325265	−	0.945623i	$-0.394546\pi$
$31$	3.62200	0.650530	0.325265	+	0.945623i	$0.394546\pi$
$32$	−5.69550	−1.00683
$33$	0	0
$34$	0	0
$35$	11.9236	2.01546
$36$	0	0
$37$	4.96694	0.816559	0.408280	−	0.912857i	$-0.366129\pi$
$37$	4.96694	0.816559	0.408280	+	0.912857i	$0.366129\pi$
$38$	2.18558	0.354547
$39$	0	0
$40$	6.17895	0.976977
$41$	−5.68257	−0.887469	−0.443735	−	0.896158i	$-0.646347\pi$
$41$	−5.68257	−0.887469	−0.443735	+	0.896158i	$0.646347\pi$
$42$	0	0
$43$	5.89136	0.898424	0.449212	−	0.893425i	$-0.351705\pi$
$43$	5.89136	0.898424	0.449212	+	0.893425i	$0.351705\pi$
$44$	10.0599	1.51659
$45$	0	0
$46$	−1.50844	−0.222408
$47$	1.84042	0.268452	0.134226	−	0.990951i	$-0.457145\pi$
$47$	1.84042	0.268452	0.134226	+	0.990951i	$0.457145\pi$
$48$	0	0
$49$	12.7460	1.82086
$50$	−1.41191	−0.199675
$51$	0	0
$52$	8.54935	1.18558
$53$	−2.31475	−0.317956	−0.158978	−	0.987282i	$-0.550820\pi$
$53$	−2.31475	−0.317956	−0.158978	+	0.987282i	$0.550820\pi$
$54$	0	0
$55$	−16.9971	−2.29189
$56$	10.2326	1.36739
$57$	0	0
$58$	0.260217	0.0341681
$59$	12.6166	1.64254	0.821270	−	0.570539i	$-0.193266\pi$
$59$	12.6166	1.64254	0.821270	+	0.570539i	$0.193266\pi$
$60$	0	0
$61$	−9.13811	−1.17002	−0.585008	−	0.811028i	$-0.698909\pi$
$61$	−9.13811	−1.17002	−0.585008	+	0.811028i	$0.698909\pi$
$62$	−2.32449	−0.295210
$63$	0	0
$64$	0.258335	0.0322919
$65$	−14.4449	−1.79167
$66$	0	0
$67$	0.233694	0.0285503	0.0142752	−	0.999898i	$-0.495456\pi$
$67$	0.233694	0.0285503	0.0142752	+	0.999898i	$0.495456\pi$
$68$	0	0
$69$	0	0
$70$	−7.65219	−0.914612
$71$	−9.29208	−1.10277	−0.551383	−	0.834252i	$-0.685900\pi$
$71$	−9.29208	−1.10277	−0.551383	+	0.834252i	$0.685900\pi$
$72$	0	0
$73$	3.05704	0.357800	0.178900	−	0.983867i	$-0.442746\pi$
$73$	3.05704	0.357800	0.178900	+	0.983867i	$0.442746\pi$
$74$	−3.18763	−0.370554
$75$	0	0
$76$	5.40847	0.620394
$77$	−28.1480	−3.20776
$78$	0	0
$79$	−5.59075	−0.629008	−0.314504	−	0.949256i	$-0.601838\pi$
$79$	−5.59075	−0.629008	−0.314504	+	0.949256i	$0.601838\pi$
$80$	4.55738	0.509530
$81$	0	0
$82$	3.64690	0.402733
$83$	−9.49892	−1.04264	−0.521321	−	0.853361i	$-0.674561\pi$
$83$	−9.49892	−1.04264	−0.521321	+	0.853361i	$0.674561\pi$
$84$	0	0
$85$	0	0
$86$	−3.78089	−0.407704
$87$	0	0
$88$	−14.5866	−1.55494
$89$	14.3614	1.52231	0.761154	−	0.648571i	$-0.224633\pi$
$89$	14.3614	1.52231	0.761154	+	0.648571i	$0.224633\pi$
$90$	0	0
$91$	−23.9214	−2.50764
$92$	−3.73282	−0.389174
$93$	0	0
$94$	−1.18112	−0.121823
$95$	−9.13807	−0.937546
$96$	0	0
$97$	−17.6601	−1.79311	−0.896556	−	0.442930i	$-0.853939\pi$
$97$	−17.6601	−1.79311	−0.896556	+	0.442930i	$0.853939\pi$
$98$	−8.18002	−0.826306
$99$	0	0
$100$	−3.49394	−0.349394
$101$	−8.30139	−0.826019	−0.413009	−	0.910727i	$-0.635522\pi$
$101$	−8.30139	−0.826019	−0.413009	+	0.910727i	$0.635522\pi$
$102$	0	0
$103$	−16.4829	−1.62410	−0.812052	−	0.583585i	$-0.801650\pi$
$103$	−16.4829	−1.62410	−0.812052	+	0.583585i	$0.801650\pi$
$104$	−12.3963	−1.21556
$105$	0	0
$106$	1.48554	0.144288
$107$	−5.39238	−0.521301	−0.260651	−	0.965433i	$-0.583937\pi$
$107$	−5.39238	−0.521301	−0.260651	+	0.965433i	$0.583937\pi$
$108$	0	0
$109$	3.67449	0.351952	0.175976	−	0.984394i	$-0.443692\pi$
$109$	3.67449	0.351952	0.175976	+	0.984394i	$0.443692\pi$
$110$	10.9082	1.04006
$111$	0	0
$112$	7.54724	0.713147
$113$	13.0724	1.22975	0.614873	−	0.788626i	$-0.289207\pi$
$113$	13.0724	1.22975	0.614873	+	0.788626i	$0.289207\pi$
$114$	0	0
$115$	6.30692	0.588124
$116$	0.643936	0.0597880
$117$	0	0
$118$	−8.09694	−0.745383
$119$	0	0
$120$	0	0
$121$	29.1249	2.64772
$122$	5.86456	0.530952
$123$	0	0
$124$	−5.75222	−0.516564
$125$	−7.51312	−0.671994
$126$	0	0
$127$	−4.99469	−0.443207	−0.221604	−	0.975137i	$-0.571129\pi$
$127$	−4.99469	−0.443207	−0.221604	+	0.975137i	$0.571129\pi$
$128$	11.2252	0.992178
$129$	0	0
$130$	9.27027	0.813057
$131$	−12.7397	−1.11307	−0.556535	−	0.830824i	$-0.687870\pi$
$131$	−12.7397	−1.11307	−0.556535	+	0.830824i	$0.687870\pi$
$132$	0	0
$133$	−15.1331	−1.31220
$134$	−0.149978	−0.0129561
$135$	0	0
$136$	0	0
$137$	−6.30237	−0.538448	−0.269224	−	0.963078i	$-0.586767\pi$
$137$	−6.30237	−0.538448	−0.269224	+	0.963078i	$0.586767\pi$
$138$	0	0
$139$	−11.7494	−0.996573	−0.498287	−	0.867012i	$-0.666037\pi$
$139$	−11.7494	−0.996573	−0.498287	+	0.867012i	$0.666037\pi$
$140$	−18.9363	−1.60041
$141$	0	0
$142$	5.96337	0.500435
$143$	34.0999	2.85158
$144$	0	0
$145$	−1.08799	−0.0903523
$146$	−1.96192	−0.162369
$147$	0	0
$148$	−7.88816	−0.648402
$149$	−17.0428	−1.39620	−0.698102	−	0.715999i	$-0.745972\pi$
$149$	−17.0428	−1.39620	−0.698102	+	0.715999i	$0.745972\pi$
$150$	0	0
$151$	5.68000	0.462232	0.231116	−	0.972926i	$-0.425762\pi$
$151$	5.68000	0.462232	0.231116	+	0.972926i	$0.425762\pi$
$152$	−7.84214	−0.636081
$153$	0	0
$154$	18.0645	1.45568
$155$	9.71887	0.780638
$156$	0	0
$157$	12.6032	1.00585	0.502923	−	0.864331i	$-0.332258\pi$
$157$	12.6032	1.00585	0.502923	+	0.864331i	$0.332258\pi$
$158$	3.58797	0.285443
$159$	0	0
$160$	−15.2827	−1.20820
$161$	10.4446	0.823147
$162$	0	0
$163$	20.6595	1.61817	0.809087	−	0.587689i	$-0.199962\pi$
$163$	20.6595	1.61817	0.809087	+	0.587689i	$0.199962\pi$
$164$	9.02468	0.704709
$165$	0	0
$166$	6.09611	0.473150
$167$	−15.2557	−1.18052	−0.590260	−	0.807213i	$-0.700975\pi$
$167$	−15.2557	−1.18052	−0.590260	+	0.807213i	$0.700975\pi$
$168$	0	0
$169$	15.9796	1.22920
$170$	0	0
$171$	0	0
$172$	−9.35626	−0.713408
$173$	−6.13201	−0.466208	−0.233104	−	0.972452i	$-0.574888\pi$
$173$	−6.13201	−0.466208	−0.233104	+	0.972452i	$0.574888\pi$
$174$	0	0
$175$	9.77618	0.739010
$176$	−10.7586	−0.810959
$177$	0	0
$178$	−9.21672	−0.690822
$179$	−22.1716	−1.65718	−0.828592	−	0.559853i	$-0.810858\pi$
$179$	−22.1716	−1.65718	−0.828592	+	0.559853i	$0.810858\pi$
$180$	0	0
$181$	−3.73212	−0.277407	−0.138703	−	0.990334i	$-0.544293\pi$
$181$	−3.73212	−0.277407	−0.138703	+	0.990334i	$0.544293\pi$
$182$	15.3520	1.13797
$183$	0	0
$184$	5.41250	0.399014
$185$	13.3277	0.979874
$186$	0	0
$187$	0	0
$188$	−2.92283	−0.213169
$189$	0	0
$190$	5.86453	0.425458
$191$	−6.20059	−0.448659	−0.224329	−	0.974513i	$-0.572019\pi$
$191$	−6.20059	−0.448659	−0.224329	+	0.974513i	$0.572019\pi$
$192$	0	0
$193$	−6.58343	−0.473885	−0.236943	−	0.971524i	$-0.576145\pi$
$193$	−6.58343	−0.473885	−0.236943	+	0.971524i	$0.576145\pi$
$194$	11.3337	0.813713
$195$	0	0
$196$	−20.2424	−1.44589
$197$	−6.73707	−0.479997	−0.239998	−	0.970773i	$-0.577147\pi$
$197$	−6.73707	−0.479997	−0.239998	+	0.970773i	$0.577147\pi$
$198$	0	0
$199$	−6.42392	−0.455380	−0.227690	−	0.973734i	$-0.573117\pi$
$199$	−6.42392	−0.455380	−0.227690	+	0.973734i	$0.573117\pi$
$200$	5.06613	0.358229
$201$	0	0
$202$	5.32757	0.374847
$203$	−1.80176	−0.126459
$204$	0	0
$205$	−15.2480	−1.06497
$206$	10.5782	0.737017
$207$	0	0
$208$	−9.14312	−0.633961
$209$	21.5722	1.49218
$210$	0	0
$211$	−3.61442	−0.248827	−0.124414	−	0.992230i	$-0.539705\pi$
$211$	−3.61442	−0.248827	−0.124414	+	0.992230i	$0.539705\pi$
$212$	3.67613	0.252478
$213$	0	0
$214$	3.46066	0.236566
$215$	15.8082	1.07811
$216$	0	0
$217$	16.0949	1.09259
$218$	−2.35817	−0.159716
$219$	0	0
$220$	26.9936	1.81991
$221$	0	0
$222$	0	0
$223$	6.74600	0.451745	0.225873	−	0.974157i	$-0.427477\pi$
$223$	6.74600	0.451745	0.225873	+	0.974157i	$0.427477\pi$
$224$	−25.3088	−1.69102
$225$	0	0
$226$	−8.38944	−0.558058
$227$	−16.6281	−1.10365	−0.551823	−	0.833961i	$-0.686068\pi$
$227$	−16.6281	−1.10365	−0.551823	+	0.833961i	$0.686068\pi$
$228$	0	0
$229$	−3.97644	−0.262771	−0.131385	−	0.991331i	$-0.541943\pi$
$229$	−3.97644	−0.262771	−0.131385	+	0.991331i	$0.541943\pi$
$230$	−4.04759	−0.266890
$231$	0	0
$232$	−0.933691	−0.0612998
$233$	3.45045	0.226047	0.113023	−	0.993592i	$-0.463947\pi$
$233$	3.45045	0.226047	0.113023	+	0.993592i	$0.463947\pi$
$234$	0	0
$235$	4.93837	0.322144
$236$	−20.0368	−1.30429
$237$	0	0
$238$	0	0
$239$	−30.6355	−1.98165	−0.990824	−	0.135160i	$-0.956845\pi$
$239$	−30.6355	−1.98165	−0.990824	+	0.135160i	$0.956845\pi$
$240$	0	0
$241$	12.7141	0.818985	0.409493	−	0.912313i	$-0.365706\pi$
$241$	12.7141	0.818985	0.409493	+	0.912313i	$0.365706\pi$
$242$	−18.6915	−1.20153
$243$	0	0
$244$	14.5125	0.929070
$245$	34.2013	2.18504
$246$	0	0
$247$	18.3330	1.16650
$248$	8.34057	0.529626
$249$	0	0
$250$	4.82169	0.304950
$251$	11.8594	0.748556	0.374278	−	0.927317i	$-0.377891\pi$
$251$	11.8594	0.748556	0.374278	+	0.927317i	$0.377891\pi$
$252$	0	0
$253$	−14.8887	−0.936046
$254$	3.20544	0.201127
$255$	0	0
$256$	−7.72066	−0.482542
$257$	20.1673	1.25800	0.629000	−	0.777405i	$-0.283465\pi$
$257$	20.1673	1.25800	0.629000	+	0.777405i	$0.283465\pi$
$258$	0	0
$259$	22.0713	1.37145
$260$	22.9404	1.42270
$261$	0	0
$262$	8.17593	0.505110
$263$	−25.4907	−1.57182	−0.785912	−	0.618338i	$-0.787806\pi$
$263$	−25.4907	−1.57182	−0.785912	+	0.618338i	$0.787806\pi$
$264$	0	0
$265$	−6.21115	−0.381548
$266$	9.71194	0.595477
$267$	0	0
$268$	−0.371138	−0.0226708
$269$	−5.05965	−0.308492	−0.154246	−	0.988032i	$-0.549295\pi$
$269$	−5.05965	−0.308492	−0.154246	+	0.988032i	$0.549295\pi$
$270$	0	0
$271$	−3.07396	−0.186730	−0.0933650	−	0.995632i	$-0.529762\pi$
$271$	−3.07396	−0.186730	−0.0933650	+	0.995632i	$0.529762\pi$
$272$	0	0
$273$	0	0
$274$	4.04467	0.244347
$275$	−13.9359	−0.840369
$276$	0	0
$277$	−10.7230	−0.644282	−0.322141	−	0.946692i	$-0.604403\pi$
$277$	−10.7230	−0.644282	−0.322141	+	0.946692i	$0.604403\pi$
$278$	7.54042	0.452244
$279$	0	0
$280$	27.4571	1.64087
$281$	−10.0742	−0.600976	−0.300488	−	0.953786i	$-0.597149\pi$
$281$	−10.0742	−0.600976	−0.300488	+	0.953786i	$0.597149\pi$
$282$	0	0
$283$	31.8828	1.89524	0.947618	−	0.319405i	$-0.103483\pi$
$283$	31.8828	1.89524	0.947618	+	0.319405i	$0.103483\pi$
$284$	14.7571	0.875670
$285$	0	0
$286$	−21.8843	−1.29404
$287$	−25.2514	−1.49054
$288$	0	0
$289$	0	0
$290$	0.698236	0.0410018
$291$	0	0
$292$	−4.85499	−0.284117
$293$	−0.223998	−0.0130861	−0.00654304	−	0.999979i	$-0.502083\pi$
$293$	−0.223998	−0.0130861	−0.00654304	+	0.999979i	$0.502083\pi$
$294$	0	0
$295$	33.8539	1.97105
$296$	11.4376	0.664798
$297$	0	0
$298$	10.9376	0.633596
$299$	−12.6531	−0.731748
$300$	0	0
$301$	26.1792	1.50894
$302$	−3.64525	−0.209761
$303$	0	0
$304$	−5.78409	−0.331740
$305$	−24.5202	−1.40402
$306$	0	0
$307$	−14.6597	−0.836676	−0.418338	−	0.908291i	$-0.637387\pi$
$307$	−14.6597	−0.836676	−0.418338	+	0.908291i	$0.637387\pi$
$308$	44.7027	2.54717
$309$	0	0
$310$	−6.23727	−0.354253
$311$	−33.1970	−1.88243	−0.941215	−	0.337807i	$-0.890315\pi$
$311$	−33.1970	−1.88243	−0.941215	+	0.337807i	$0.890315\pi$
$312$	0	0
$313$	23.9346	1.35287	0.676433	−	0.736504i	$-0.263525\pi$
$313$	23.9346	1.35287	0.676433	+	0.736504i	$0.263525\pi$
$314$	−8.08835	−0.456452
$315$	0	0
$316$	8.87885	0.499474
$317$	−26.9926	−1.51606	−0.758029	−	0.652221i	$-0.773837\pi$
$317$	−26.9926	−1.51606	−0.758029	+	0.652221i	$0.773837\pi$
$318$	0	0
$319$	2.56840	0.143803
$320$	0.693187	0.0387504
$321$	0	0
$322$	−6.70300	−0.373544
$323$	0	0
$324$	0	0
$325$	−11.8434	−0.656952
$326$	−13.2586	−0.734326
$327$	0	0
$328$	−13.0856	−0.722529
$329$	8.17817	0.450877
$330$	0	0
$331$	25.5352	1.40354	0.701770	−	0.712404i	$-0.252394\pi$
$331$	25.5352	1.40354	0.701770	+	0.712404i	$0.252394\pi$
$332$	15.0855	0.827927
$333$	0	0
$334$	9.79062	0.535719
$335$	0.627069	0.0342604
$336$	0	0
$337$	0.987585	0.0537972	0.0268986	−	0.999638i	$-0.491437\pi$
$337$	0.987585	0.0537972	0.0268986	+	0.999638i	$0.491437\pi$
$338$	−10.2552	−0.557811
$339$	0	0
$340$	0	0
$341$	−22.9433	−1.24245
$342$	0	0
$343$	25.5334	1.37868
$344$	13.5663	0.731448
$345$	0	0
$346$	3.93533	0.211565
$347$	20.5814	1.10487	0.552434	−	0.833557i	$-0.313699\pi$
$347$	20.5814	1.10487	0.552434	+	0.833557i	$0.313699\pi$
$348$	0	0
$349$	−19.8215	−1.06102	−0.530509	−	0.847679i	$-0.677999\pi$
$349$	−19.8215	−1.06102	−0.530509	+	0.847679i	$0.677999\pi$
$350$	−6.27405	−0.335362
$351$	0	0
$352$	36.0777	1.92295
$353$	−14.0227	−0.746355	−0.373177	−	0.927760i	$-0.621732\pi$
$353$	−14.0227	−0.746355	−0.373177	+	0.927760i	$0.621732\pi$
$354$	0	0
$355$	−24.9333	−1.32332
$356$	−22.8079	−1.20881
$357$	0	0
$358$	14.2290	0.752029
$359$	1.22037	0.0644086	0.0322043	−	0.999481i	$-0.489747\pi$
$359$	1.22037	0.0644086	0.0322043	+	0.999481i	$0.489747\pi$
$360$	0	0
$361$	−7.40222	−0.389591
$362$	2.39516	0.125887
$363$	0	0
$364$	37.9904	1.99124
$365$	8.20293	0.429361
$366$	0	0
$367$	16.7935	0.876613	0.438307	−	0.898825i	$-0.355578\pi$
$367$	16.7935	0.876613	0.438307	+	0.898825i	$0.355578\pi$
$368$	3.99207	0.208101
$369$	0	0
$370$	−8.55332	−0.444666
$371$	−10.2860	−0.534020
$372$	0	0
$373$	−13.2048	−0.683720	−0.341860	−	0.939751i	$-0.611057\pi$
$373$	−13.2048	−0.683720	−0.341860	+	0.939751i	$0.611057\pi$
$374$	0	0
$375$	0	0
$376$	4.23802	0.218559
$377$	2.18274	0.112417
$378$	0	0
$379$	18.8775	0.969672	0.484836	−	0.874605i	$-0.338879\pi$
$379$	18.8775	0.969672	0.484836	+	0.874605i	$0.338879\pi$
$380$	14.5125	0.744474
$381$	0	0
$382$	3.97935	0.203601
$383$	−2.00369	−0.102384	−0.0511919	−	0.998689i	$-0.516302\pi$
$383$	−2.00369	−0.102384	−0.0511919	+	0.998689i	$0.516302\pi$
$384$	0	0
$385$	−75.5291	−3.84932
$386$	4.22504	0.215049
$387$	0	0
$388$	28.0466	1.42385
$389$	−13.0743	−0.662893	−0.331447	−	0.943474i	$-0.607537\pi$
$389$	−13.0743	−0.662893	−0.331447	+	0.943474i	$0.607537\pi$
$390$	0	0
$391$	0	0
$392$	29.3510	1.48245
$393$	0	0
$394$	4.32364	0.217822
$395$	−15.0016	−0.754812
$396$	0	0
$397$	3.60996	0.181179	0.0905895	−	0.995888i	$-0.471125\pi$
$397$	3.60996	0.181179	0.0905895	+	0.995888i	$0.471125\pi$
$398$	4.12267	0.206651
$399$	0	0
$400$	3.73660	0.186830
$401$	27.4185	1.36922	0.684608	−	0.728912i	$-0.259974\pi$
$401$	27.4185	1.36922	0.684608	+	0.728912i	$0.259974\pi$
$402$	0	0
$403$	−19.4982	−0.971275
$404$	13.1837	0.655914
$405$	0	0
$406$	1.15631	0.0573868
$407$	−31.4627	−1.55955
$408$	0	0
$409$	−17.4351	−0.862112	−0.431056	−	0.902325i	$-0.641859\pi$
$409$	−17.4351	−0.862112	−0.431056	+	0.902325i	$0.641859\pi$
$410$	9.78568	0.483280
$411$	0	0
$412$	26.1770	1.28965
$413$	56.0638	2.75872
$414$	0	0
$415$	−25.4883	−1.25117
$416$	30.6605	1.50325
$417$	0	0
$418$	−13.8444	−0.677150
$419$	19.4358	0.949501	0.474751	−	0.880120i	$-0.342538\pi$
$419$	19.4358	0.949501	0.474751	+	0.880120i	$0.342538\pi$
$420$	0	0
$421$	−25.0871	−1.22267	−0.611335	−	0.791372i	$-0.709367\pi$
$421$	−25.0871	−1.22267	−0.611335	+	0.791372i	$0.709367\pi$
$422$	2.31962	0.112918
$423$	0	0
$424$	−5.33030	−0.258862
$425$	0	0
$426$	0	0
$427$	−40.6066	−1.96509
$428$	8.56382	0.413948
$429$	0	0
$430$	−10.1452	−0.489246
$431$	21.4510	1.03326	0.516630	−	0.856209i	$-0.327186\pi$
$431$	21.4510	1.03326	0.516630	+	0.856209i	$0.327186\pi$
$432$	0	0
$433$	16.4261	0.789386	0.394693	−	0.918813i	$-0.370851\pi$
$433$	16.4261	0.789386	0.394693	+	0.918813i	$0.370851\pi$
$434$	−10.3292	−0.495818
$435$	0	0
$436$	−5.83557	−0.279473
$437$	−8.00457	−0.382910
$438$	0	0
$439$	1.02381	0.0488637	0.0244319	−	0.999701i	$-0.492222\pi$
$439$	1.02381	0.0488637	0.0244319	+	0.999701i	$0.492222\pi$
$440$	−39.1401	−1.86593
$441$	0	0
$442$	0	0
$443$	11.8564	0.563316	0.281658	−	0.959515i	$-0.409116\pi$
$443$	11.8564	0.563316	0.281658	+	0.959515i	$0.409116\pi$
$444$	0	0
$445$	38.5359	1.82677
$446$	−4.32937	−0.205002
$447$	0	0
$448$	1.14795	0.0542356
$449$	−15.7799	−0.744701	−0.372350	−	0.928092i	$-0.621448\pi$
$449$	−15.7799	−0.744701	−0.372350	+	0.928092i	$0.621448\pi$
$450$	0	0
$451$	35.9958	1.69498
$452$	−20.7607	−0.976500
$453$	0	0
$454$	10.6714	0.500833
$455$	−64.1880	−3.00918
$456$	0	0
$457$	9.57499	0.447899	0.223950	−	0.974601i	$-0.428105\pi$
$457$	9.57499	0.447899	0.223950	+	0.974601i	$0.428105\pi$
$458$	2.55196	0.119245
$459$	0	0
$460$	−10.0162	−0.467009
$461$	−20.3019	−0.945555	−0.472777	−	0.881182i	$-0.656749\pi$
$461$	−20.3019	−0.945555	−0.472777	+	0.881182i	$0.656749\pi$
$462$	0	0
$463$	−2.14003	−0.0994556	−0.0497278	−	0.998763i	$-0.515835\pi$
$463$	−2.14003	−0.0994556	−0.0497278	+	0.998763i	$0.515835\pi$
$464$	−0.688659	−0.0319702
$465$	0	0
$466$	−2.21439	−0.102580
$467$	8.65689	0.400593	0.200296	−	0.979735i	$-0.435809\pi$
$467$	8.65689	0.400593	0.200296	+	0.979735i	$0.435809\pi$
$468$	0	0
$469$	1.03846	0.0479515
$470$	−3.16929	−0.146188
$471$	0	0
$472$	29.0529	1.33727
$473$	−37.3184	−1.71590
$474$	0	0
$475$	−7.49232	−0.343771
$476$	0	0
$477$	0	0
$478$	19.6609	0.899270
$479$	−22.2385	−1.01610	−0.508052	−	0.861326i	$-0.669634\pi$
$479$	−22.2385	−1.01610	−0.508052	+	0.861326i	$0.669634\pi$
$480$	0	0
$481$	−26.7384	−1.21917
$482$	−8.15949	−0.371655
$483$	0	0
$484$	−46.2542	−2.10247
$485$	−47.3872	−2.15174
$486$	0	0
$487$	42.3969	1.92119	0.960593	−	0.277959i	$-0.0896578\pi$
$487$	42.3969	1.92119	0.960593	+	0.277959i	$0.0896578\pi$
$488$	−21.0428	−0.952563
$489$	0	0
$490$	−21.9493	−0.991570
$491$	−29.8116	−1.34538	−0.672689	−	0.739926i	$-0.734861\pi$
$491$	−29.8116	−1.34538	−0.672689	+	0.739926i	$0.734861\pi$
$492$	0	0
$493$	0	0
$494$	−11.7656	−0.529357
$495$	0	0
$496$	6.15172	0.276220
$497$	−41.2908	−1.85214
$498$	0	0
$499$	−35.2350	−1.57733	−0.788667	−	0.614821i	$-0.789228\pi$
$499$	−35.2350	−1.57733	−0.788667	+	0.614821i	$0.789228\pi$
$500$	11.9318	0.533608
$501$	0	0
$502$	−7.61096	−0.339694
$503$	−12.8548	−0.573169	−0.286585	−	0.958055i	$-0.592520\pi$
$503$	−12.8548	−0.573169	−0.286585	+	0.958055i	$0.592520\pi$
$504$	0	0
$505$	−22.2750	−0.991225
$506$	9.55512	0.424777
$507$	0	0
$508$	7.93224	0.351936
$509$	−18.7529	−0.831208	−0.415604	−	0.909546i	$-0.636430\pi$
$509$	−18.7529	−0.831208	−0.415604	+	0.909546i	$0.636430\pi$
$510$	0	0
$511$	13.5844	0.600940
$512$	−17.4956	−0.773201
$513$	0	0
$514$	−12.9427	−0.570879
$515$	−44.2283	−1.94893
$516$	0	0
$517$	−11.6580	−0.512717
$518$	−14.1647	−0.622361
$519$	0	0
$520$	−33.2630	−1.45868
$521$	−12.5246	−0.548714	−0.274357	−	0.961628i	$-0.588465\pi$
$521$	−12.5246	−0.548714	−0.274357	+	0.961628i	$0.588465\pi$
$522$	0	0
$523$	12.0664	0.527628	0.263814	−	0.964574i	$-0.415020\pi$
$523$	12.0664	0.527628	0.263814	+	0.964574i	$0.415020\pi$
$524$	20.2323	0.883852
$525$	0	0
$526$	16.3591	0.713293
$527$	0	0
$528$	0	0
$529$	−17.4754	−0.759800
$530$	3.98612	0.173146
$531$	0	0
$532$	24.0333	1.04198
$533$	30.5909	1.32504
$534$	0	0
$535$	−14.4693	−0.625563
$536$	0.538140	0.0232441
$537$	0	0
$538$	3.24713	0.139994
$539$	−80.7389	−3.47767
$540$	0	0
$541$	−28.0602	−1.20640	−0.603202	−	0.797589i	$-0.706109\pi$
$541$	−28.0602	−1.20640	−0.603202	+	0.797589i	$0.706109\pi$
$542$	1.97277	0.0847379
$543$	0	0
$544$	0	0
$545$	9.85971	0.422343
$546$	0	0
$547$	−1.62765	−0.0695931	−0.0347965	−	0.999394i	$-0.511078\pi$
$547$	−1.62765	−0.0695931	−0.0347965	+	0.999394i	$0.511078\pi$
$548$	10.0090	0.427563
$549$	0	0
$550$	8.94365	0.381359
$551$	1.38084	0.0588258
$552$	0	0
$553$	−24.8433	−1.05645
$554$	6.88168	0.292375
$555$	0	0
$556$	18.6596	0.791345
$557$	−7.21691	−0.305790	−0.152895	−	0.988242i	$-0.548860\pi$
$557$	−7.21691	−0.305790	−0.152895	+	0.988242i	$0.548860\pi$
$558$	0	0
$559$	−31.7148	−1.34139
$560$	20.2514	0.855778
$561$	0	0
$562$	6.46530	0.272722
$563$	−4.90951	−0.206911	−0.103456	−	0.994634i	$-0.532990\pi$
$563$	−4.90951	−0.206911	−0.103456	+	0.994634i	$0.532990\pi$
$564$	0	0
$565$	35.0769	1.47570
$566$	−20.4614	−0.860057
$567$	0	0
$568$	−21.3974	−0.897813
$569$	25.2147	1.05706	0.528528	−	0.848916i	$-0.322744\pi$
$569$	25.2147	1.05706	0.528528	+	0.848916i	$0.322744\pi$
$570$	0	0
$571$	12.4087	0.519288	0.259644	−	0.965704i	$-0.416395\pi$
$571$	12.4087	0.519288	0.259644	+	0.965704i	$0.416395\pi$
$572$	−54.1552	−2.26434
$573$	0	0
$574$	16.2056	0.676407
$575$	5.17106	0.215648
$576$	0	0
$577$	5.07039	0.211083	0.105542	−	0.994415i	$-0.466342\pi$
$577$	5.07039	0.211083	0.105542	+	0.994415i	$0.466342\pi$
$578$	0	0
$579$	0	0
$580$	1.72787	0.0717458
$581$	−42.2099	−1.75116
$582$	0	0
$583$	14.6626	0.607264
$584$	7.03961	0.291301
$585$	0	0
$586$	0.143755	0.00593845
$587$	−18.2018	−0.751270	−0.375635	−	0.926768i	$-0.622575\pi$
$587$	−18.2018	−0.751270	−0.375635	+	0.926768i	$0.622575\pi$
$588$	0	0
$589$	−12.3349	−0.508251
$590$	−21.7264	−0.894462
$591$	0	0
$592$	8.43600	0.346717
$593$	17.5786	0.721865	0.360932	−	0.932592i	$-0.382459\pi$
$593$	17.5786	0.721865	0.360932	+	0.932592i	$0.382459\pi$
$594$	0	0
$595$	0	0
$596$	27.0663	1.10868
$597$	0	0
$598$	8.12037	0.332066
$599$	3.12165	0.127547	0.0637736	−	0.997964i	$-0.479686\pi$
$599$	3.12165	0.127547	0.0637736	+	0.997964i	$0.479686\pi$
$600$	0	0
$601$	−20.5455	−0.838070	−0.419035	−	0.907970i	$-0.637632\pi$
$601$	−20.5455	−0.838070	−0.419035	+	0.907970i	$0.637632\pi$
$602$	−16.8010	−0.684756
$603$	0	0
$604$	−9.02060	−0.367043
$605$	78.1505	3.17727
$606$	0	0
$607$	−17.3208	−0.703029	−0.351515	−	0.936182i	$-0.614333\pi$
$607$	−17.3208	−0.703029	−0.351515	+	0.936182i	$0.614333\pi$
$608$	19.3963	0.786625
$609$	0	0
$610$	15.7363	0.637144
$611$	−9.90747	−0.400813
$612$	0	0
$613$	23.1375	0.934513	0.467256	−	0.884122i	$-0.345242\pi$
$613$	23.1375	0.934513	0.467256	+	0.884122i	$0.345242\pi$
$614$	9.40817	0.379683
$615$	0	0
$616$	−64.8178	−2.61158
$617$	−21.9578	−0.883989	−0.441994	−	0.897018i	$-0.645729\pi$
$617$	−21.9578	−0.883989	−0.441994	+	0.897018i	$0.645729\pi$
$618$	0	0
$619$	24.9794	1.00400	0.502002	−	0.864866i	$-0.332597\pi$
$619$	24.9794	1.00400	0.502002	+	0.864866i	$0.332597\pi$
$620$	−15.4348	−0.619878
$621$	0	0
$622$	21.3048	0.854245
$623$	63.8172	2.55678
$624$	0	0
$625$	−31.1600	−1.24640
$626$	−15.3605	−0.613929
$627$	0	0
$628$	−20.0156	−0.798708
$629$	0	0
$630$	0	0
$631$	30.7571	1.22442	0.612210	−	0.790696i	$-0.290281\pi$
$631$	30.7571	1.22442	0.612210	+	0.790696i	$0.290281\pi$
$632$	−12.8741	−0.512104
$633$	0	0
$634$	17.3230	0.687985
$635$	−13.4022	−0.531850
$636$	0	0
$637$	−68.6155	−2.71864
$638$	−1.64832	−0.0652577
$639$	0	0
$640$	30.1205	1.19062
$641$	15.7700	0.622876	0.311438	−	0.950266i	$-0.399189\pi$
$641$	15.7700	0.622876	0.311438	+	0.950266i	$0.399189\pi$
$642$	0	0
$643$	39.1574	1.54422	0.772109	−	0.635491i	$-0.219202\pi$
$643$	39.1574	1.54422	0.772109	+	0.635491i	$0.219202\pi$
$644$	−16.5874	−0.653634
$645$	0	0
$646$	0	0
$647$	−6.20318	−0.243872	−0.121936	−	0.992538i	$-0.538910\pi$
$647$	−6.20318	−0.243872	−0.121936	+	0.992538i	$0.538910\pi$
$648$	0	0
$649$	−79.9188	−3.13709
$650$	7.60071	0.298124
$651$	0	0
$652$	−32.8100	−1.28494
$653$	−17.7568	−0.694879	−0.347439	−	0.937702i	$-0.612949\pi$
$653$	−17.7568	−0.694879	−0.347439	+	0.937702i	$0.612949\pi$
$654$	0	0
$655$	−34.1842	−1.33569
$656$	−9.65146	−0.376826
$657$	0	0
$658$	−5.24850	−0.204608
$659$	−34.8652	−1.35816	−0.679078	−	0.734066i	$-0.737620\pi$
$659$	−34.8652	−1.35816	−0.679078	+	0.734066i	$0.737620\pi$
$660$	0	0
$661$	−5.79174	−0.225273	−0.112636	−	0.993636i	$-0.535930\pi$
$661$	−5.79174	−0.225273	−0.112636	+	0.993636i	$0.535930\pi$
$662$	−16.3877	−0.636925
$663$	0	0
$664$	−21.8737	−0.848863
$665$	−40.6064	−1.57465
$666$	0	0
$667$	−0.953030	−0.0369015
$668$	24.2280	0.937411
$669$	0	0
$670$	−0.402433	−0.0155474
$671$	57.8847	2.23461
$672$	0	0
$673$	−23.6619	−0.912099	−0.456050	−	0.889954i	$-0.650736\pi$
$673$	−23.6619	−0.912099	−0.456050	+	0.889954i	$0.650736\pi$
$674$	−0.633802	−0.0244131
$675$	0	0
$676$	−25.3778	−0.976069
$677$	−30.3751	−1.16741	−0.583706	−	0.811965i	$-0.698398\pi$
$677$	−30.3751	−1.16741	−0.583706	+	0.811965i	$0.698398\pi$
$678$	0	0
$679$	−78.4754	−3.01161
$680$	0	0
$681$	0	0
$682$	14.7243	0.563822
$683$	13.5599	0.518855	0.259428	−	0.965763i	$-0.416466\pi$
$683$	13.5599	0.518855	0.259428	+	0.965763i	$0.416466\pi$
$684$	0	0
$685$	−16.9111	−0.646139
$686$	−16.3866	−0.625642
$687$	0	0
$688$	10.0061	0.381478
$689$	12.4609	0.474724
$690$	0	0
$691$	−25.4881	−0.969615	−0.484807	−	0.874621i	$-0.661110\pi$
$691$	−25.4881	−0.969615	−0.484807	+	0.874621i	$0.661110\pi$
$692$	9.73844	0.370200
$693$	0	0
$694$	−13.2085	−0.501388
$695$	−31.5271	−1.19589
$696$	0	0
$697$	0	0
$698$	12.7208	0.481489
$699$	0	0
$700$	−15.5259	−0.586823
$701$	16.0986	0.608034	0.304017	−	0.952667i	$-0.401672\pi$
$701$	16.0986	0.608034	0.304017	+	0.952667i	$0.401672\pi$
$702$	0	0
$703$	−16.9152	−0.637967
$704$	−1.63640	−0.0616743
$705$	0	0
$706$	8.99935	0.338695
$707$	−36.8885	−1.38733
$708$	0	0
$709$	−20.5492	−0.771741	−0.385870	−	0.922553i	$-0.626099\pi$
$709$	−20.5492	−0.771741	−0.385870	+	0.922553i	$0.626099\pi$
$710$	16.0014	0.600523
$711$	0	0
$712$	33.0708	1.23938
$713$	8.51332	0.318826
$714$	0	0
$715$	91.4999	3.42190
$716$	35.2115	1.31591
$717$	0	0
$718$	−0.783195	−0.0292286
$719$	−6.56197	−0.244720	−0.122360	−	0.992486i	$-0.539046\pi$
$719$	−6.56197	−0.244720	−0.122360	+	0.992486i	$0.539046\pi$
$720$	0	0
$721$	−73.2441	−2.72775
$722$	4.75052	0.176796
$723$	0	0
$724$	5.92711	0.220279
$725$	−0.892042	−0.0331296
$726$	0	0
$727$	27.4523	1.01815	0.509076	−	0.860722i	$-0.329987\pi$
$727$	27.4523	1.01815	0.509076	+	0.860722i	$0.329987\pi$
$728$	−55.0851	−2.04159
$729$	0	0
$730$	−5.26438	−0.194844
$731$	0	0
$732$	0	0
$733$	−2.85412	−0.105419	−0.0527097	−	0.998610i	$-0.516786\pi$
$733$	−2.85412	−0.105419	−0.0527097	+	0.998610i	$0.516786\pi$
$734$	−10.7775	−0.397806
$735$	0	0
$736$	−13.3870	−0.493451
$737$	−1.48032	−0.0545283
$738$	0	0
$739$	36.7505	1.35189	0.675944	−	0.736953i	$-0.263736\pi$
$739$	36.7505	1.35189	0.675944	+	0.736953i	$0.263736\pi$
$740$	−21.1662	−0.778085
$741$	0	0
$742$	6.60121	0.242338
$743$	24.6098	0.902846	0.451423	−	0.892310i	$-0.350916\pi$
$743$	24.6098	0.902846	0.451423	+	0.892310i	$0.350916\pi$
$744$	0	0
$745$	−45.7308	−1.67545
$746$	8.47444	0.310271
$747$	0	0
$748$	0	0
$749$	−23.9619	−0.875548
$750$	0	0
$751$	−24.1092	−0.879759	−0.439879	−	0.898057i	$-0.644979\pi$
$751$	−24.1092	−0.879759	−0.439879	+	0.898057i	$0.644979\pi$
$752$	3.12582	0.113987
$753$	0	0
$754$	−1.40082	−0.0510147
$755$	15.2411	0.554680
$756$	0	0
$757$	30.6687	1.11467	0.557336	−	0.830287i	$-0.311824\pi$
$757$	30.6687	1.11467	0.557336	+	0.830287i	$0.311824\pi$
$758$	−12.1150	−0.440036
$759$	0	0
$760$	−21.0427	−0.763299
$761$	26.0947	0.945932	0.472966	−	0.881081i	$-0.343183\pi$
$761$	26.0947	0.945932	0.472966	+	0.881081i	$0.343183\pi$
$762$	0	0
$763$	16.3281	0.591119
$764$	9.84736	0.356265
$765$	0	0
$766$	1.28591	0.0464616
$767$	−67.9186	−2.45240
$768$	0	0
$769$	2.13990	0.0771669	0.0385834	−	0.999255i	$-0.487715\pi$
$769$	2.13990	0.0771669	0.0385834	+	0.999255i	$0.487715\pi$
$770$	48.4722	1.74682
$771$	0	0
$772$	10.4554	0.376296
$773$	10.3940	0.373847	0.186923	−	0.982374i	$-0.440148\pi$
$773$	10.3940	0.373847	0.186923	+	0.982374i	$0.440148\pi$
$774$	0	0
$775$	7.96852	0.286238
$776$	−40.6669	−1.45985
$777$	0	0
$778$	8.39068	0.300820
$779$	19.3523	0.693368
$780$	0	0
$781$	58.8600	2.10618
$782$	0	0
$783$	0	0
$784$	21.6483	0.773153
$785$	33.8180	1.20702
$786$	0	0
$787$	37.7273	1.34483	0.672417	−	0.740173i	$-0.265256\pi$
$787$	37.7273	1.34483	0.672417	+	0.740173i	$0.265256\pi$
$788$	10.6994	0.381149
$789$	0	0
$790$	9.62755	0.342533
$791$	58.0891	2.06541
$792$	0	0
$793$	49.1930	1.74689
$794$	−2.31676	−0.0822188
$795$	0	0
$796$	10.2020	0.361602
$797$	−2.08990	−0.0740282	−0.0370141	−	0.999315i	$-0.511785\pi$
$797$	−2.08990	−0.0740282	−0.0370141	+	0.999315i	$0.511785\pi$
$798$	0	0
$799$	0	0
$800$	−12.5303	−0.443013
$801$	0	0
$802$	−17.5963	−0.621349
$803$	−19.3646	−0.683362
$804$	0	0
$805$	28.0258	0.987779
$806$	12.5133	0.440764
$807$	0	0
$808$	−19.1160	−0.672500
$809$	18.2226	0.640671	0.320336	−	0.947304i	$-0.396204\pi$
$809$	18.2226	0.640671	0.320336	+	0.947304i	$0.396204\pi$
$810$	0	0
$811$	35.7235	1.25442	0.627211	−	0.778850i	$-0.284196\pi$
$811$	35.7235	1.25442	0.627211	+	0.778850i	$0.284196\pi$
$812$	2.86143	0.100417
$813$	0	0
$814$	20.1918	0.707721
$815$	55.4353	1.94181
$816$	0	0
$817$	−20.0633	−0.701927
$818$	11.1893	0.391226
$819$	0	0
$820$	24.2158	0.845653
$821$	−9.02635	−0.315022	−0.157511	−	0.987517i	$-0.550347\pi$
$821$	−9.02635	−0.315022	−0.157511	+	0.987517i	$0.550347\pi$
$822$	0	0
$823$	−33.7346	−1.17591	−0.587956	−	0.808893i	$-0.700067\pi$
$823$	−33.7346	−1.17591	−0.587956	+	0.808893i	$0.700067\pi$
$824$	−37.9559	−1.32226
$825$	0	0
$826$	−35.9800	−1.25190
$827$	32.2238	1.12053	0.560265	−	0.828314i	$-0.310699\pi$
$827$	32.2238	1.12053	0.560265	+	0.828314i	$0.310699\pi$
$828$	0	0
$829$	−39.8058	−1.38251	−0.691256	−	0.722610i	$-0.742942\pi$
$829$	−39.8058	−1.38251	−0.691256	+	0.722610i	$0.742942\pi$
$830$	16.3576	0.567781
$831$	0	0
$832$	−1.39069	−0.0482135
$833$	0	0
$834$	0	0
$835$	−40.9354	−1.41663
$836$	−34.2595	−1.18489
$837$	0	0
$838$	−12.4733	−0.430883
$839$	−2.30065	−0.0794273	−0.0397137	−	0.999211i	$-0.512645\pi$
$839$	−2.30065	−0.0794273	−0.0397137	+	0.999211i	$0.512645\pi$
$840$	0	0
$841$	−28.8356	−0.994331
$842$	16.1001	0.554847
$843$	0	0
$844$	5.74018	0.197585
$845$	42.8780	1.47505
$846$	0	0
$847$	129.421	4.44696
$848$	−3.93145	−0.135006
$849$	0	0
$850$	0	0
$851$	11.6745	0.400197
$852$	0	0
$853$	−42.6432	−1.46007	−0.730037	−	0.683407i	$-0.760497\pi$
$853$	−42.6432	−1.46007	−0.730037	+	0.683407i	$0.760497\pi$
$854$	26.0601	0.891757
$855$	0	0
$856$	−12.4173	−0.424415
$857$	−3.79418	−0.129607	−0.0648034	−	0.997898i	$-0.520642\pi$
$857$	−3.79418	−0.129607	−0.0648034	+	0.997898i	$0.520642\pi$
$858$	0	0
$859$	−13.2996	−0.453776	−0.226888	−	0.973921i	$-0.572855\pi$
$859$	−13.2996	−0.453776	−0.226888	+	0.973921i	$0.572855\pi$
$860$	−25.1055	−0.856092
$861$	0	0
$862$	−13.7666	−0.468892
$863$	10.8826	0.370447	0.185223	−	0.982696i	$-0.440699\pi$
$863$	10.8826	0.370447	0.185223	+	0.982696i	$0.440699\pi$
$864$	0	0
$865$	−16.4539	−0.559451
$866$	−10.5417	−0.358223
$867$	0	0
$868$	−25.5608	−0.867592
$869$	35.4142	1.20134
$870$	0	0
$871$	−1.25804	−0.0426271
$872$	8.46143	0.286540
$873$	0	0
$874$	5.13708	0.173764
$875$	−33.3857	−1.12864
$876$	0	0
$877$	−26.4117	−0.891860	−0.445930	−	0.895068i	$-0.647127\pi$
$877$	−26.4117	−0.891860	−0.445930	+	0.895068i	$0.647127\pi$
$878$	−0.657049	−0.0221743
$879$	0	0
$880$	−28.8684	−0.973152
$881$	56.5710	1.90593	0.952963	−	0.303086i	$-0.0980171\pi$
$881$	56.5710	1.90593	0.952963	+	0.303086i	$0.0980171\pi$
$882$	0	0
$883$	53.5321	1.80150	0.900749	−	0.434341i	$-0.143019\pi$
$883$	53.5321	1.80150	0.900749	+	0.434341i	$0.143019\pi$
$884$	0	0
$885$	0	0
$886$	−7.60909	−0.255632
$887$	48.6230	1.63260	0.816300	−	0.577628i	$-0.196022\pi$
$887$	48.6230	1.63260	0.816300	+	0.577628i	$0.196022\pi$
$888$	0	0
$889$	−22.1947	−0.744386
$890$	−24.7311	−0.828989
$891$	0	0
$892$	−10.7135	−0.358716
$893$	−6.26763	−0.209738
$894$	0	0
$895$	−59.4928	−1.98863
$896$	49.8810	1.66641
$897$	0	0
$898$	10.1271	0.337945
$899$	−1.46860	−0.0489807
$900$	0	0
$901$	0	0
$902$	−23.1010	−0.769179
$903$	0	0
$904$	30.1024	1.00119
$905$	−10.0144	−0.332889
$906$	0	0
$907$	21.5013	0.713939	0.356969	−	0.934116i	$-0.383810\pi$
$907$	21.5013	0.713939	0.356969	+	0.934116i	$0.383810\pi$
$908$	26.4076	0.876368
$909$	0	0
$910$	41.1939	1.36556
$911$	−24.3047	−0.805251	−0.402625	−	0.915365i	$-0.631902\pi$
$911$	−24.3047	−0.805251	−0.402625	+	0.915365i	$0.631902\pi$
$912$	0	0
$913$	60.1702	1.99134
$914$	−6.14493	−0.203256
$915$	0	0
$916$	6.31512	0.208657
$917$	−56.6107	−1.86945
$918$	0	0
$919$	−38.3537	−1.26517	−0.632586	−	0.774490i	$-0.718007\pi$
$919$	−38.3537	−1.26517	−0.632586	+	0.774490i	$0.718007\pi$
$920$	14.5233	0.478818
$921$	0	0
$922$	13.0291	0.429092
$923$	50.0218	1.64649
$924$	0	0
$925$	10.9274	0.359291
$926$	1.37340	0.0451329
$927$	0	0
$928$	2.30934	0.0758079
$929$	21.5943	0.708486	0.354243	−	0.935153i	$-0.384738\pi$
$929$	21.5943	0.708486	0.354243	+	0.935153i	$0.384738\pi$
$930$	0	0
$931$	−43.4073	−1.42262
$932$	−5.47977	−0.179496
$933$	0	0
$934$	−5.55572	−0.181789
$935$	0	0
$936$	0	0
$937$	−0.378276	−0.0123577	−0.00617887	−	0.999981i	$-0.501967\pi$
$937$	−0.378276	−0.0123577	−0.00617887	+	0.999981i	$0.501967\pi$
$938$	−0.666449	−0.0217603
$939$	0	0
$940$	−7.84278	−0.255803
$941$	−14.8330	−0.483543	−0.241772	−	0.970333i	$-0.577728\pi$
$941$	−14.8330	−0.483543	−0.241772	+	0.970333i	$0.577728\pi$
$942$	0	0
$943$	−13.3566	−0.434950
$944$	21.4284	0.697435
$945$	0	0
$946$	23.9498	0.778674
$947$	16.3020	0.529743	0.264872	−	0.964284i	$-0.414670\pi$
$947$	16.3020	0.529743	0.264872	+	0.964284i	$0.414670\pi$
$948$	0	0
$949$	−16.4569	−0.534214
$950$	4.80834	0.156003
$951$	0	0
$952$	0	0
$953$	53.8486	1.74433	0.872163	−	0.489216i	$-0.162717\pi$
$953$	53.8486	1.74433	0.872163	+	0.489216i	$0.162717\pi$
$954$	0	0
$955$	−16.6380	−0.538392
$956$	48.6533	1.57356
$957$	0	0
$958$	14.2720	0.461107
$959$	−28.0056	−0.904347
$960$	0	0
$961$	−17.8811	−0.576810
$962$	17.1599	0.553256
$963$	0	0
$964$	−20.1916	−0.650328
$965$	−17.6652	−0.568664
$966$	0	0
$967$	5.19509	0.167063	0.0835315	−	0.996505i	$-0.473380\pi$
$967$	5.19509	0.167063	0.0835315	+	0.996505i	$0.473380\pi$
$968$	67.0675	2.15563
$969$	0	0
$970$	30.4116	0.976458
$971$	28.9362	0.928607	0.464304	−	0.885676i	$-0.346305\pi$
$971$	28.9362	0.928607	0.464304	+	0.885676i	$0.346305\pi$
$972$	0	0
$973$	−52.2104	−1.67379
$974$	−27.2090	−0.871833
$975$	0	0
$976$	−15.5205	−0.496798
$977$	−9.04283	−0.289306	−0.144653	−	0.989482i	$-0.546207\pi$
$977$	−9.04283	−0.289306	−0.144653	+	0.989482i	$0.546207\pi$
$978$	0	0
$979$	−90.9714	−2.90746
$980$	−54.3162	−1.73507
$981$	0	0
$982$	19.1321	0.610531
$983$	−5.07951	−0.162011	−0.0810055	−	0.996714i	$-0.525813\pi$
$983$	−5.07951	−0.162011	−0.0810055	+	0.996714i	$0.525813\pi$
$984$	0	0
$985$	−18.0775	−0.575997
$986$	0	0
$987$	0	0
$988$	−29.1153	−0.926280
$989$	13.8473	0.440319
$990$	0	0
$991$	38.4382	1.22103	0.610515	−	0.792005i	$-0.290963\pi$
$991$	38.4382	1.22103	0.610515	+	0.792005i	$0.290963\pi$
$992$	−20.6291	−0.654975
$993$	0	0
$994$	26.4991	0.840501
$995$	−17.2372	−0.546457
$996$	0	0
$997$	−14.7949	−0.468559	−0.234280	−	0.972169i	$-0.575273\pi$
$997$	−14.7949	−0.468559	−0.234280	+	0.972169i	$0.575273\pi$
$998$	22.6127	0.715793
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.by.1.8	yes	15
3.2	odd	2		7803.2.a.bz.1.8	yes	15
17.16	even	2		7803.2.a.bx.1.8	✓	15
51.50	odd	2		7803.2.a.ca.1.8	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7803.2.a.bx.1.8	✓	15	17.16	even	2
7803.2.a.by.1.8	yes	15	1.1	even	1	trivial
7803.2.a.bz.1.8	yes	15	3.2	odd	2
7803.2.a.ca.1.8	yes	15	51.50	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q-0.641769 q^{2} -1.58813 q^{4} +2.68329 q^{5} +4.44365 q^{7} +2.30275 q^{8} -1.72205 q^{10} -6.33442 q^{11} -5.38327 q^{13} -2.85180 q^{14} +1.69843 q^{16} -3.40555 q^{19} -4.26142 q^{20} +4.06524 q^{22} +2.35045 q^{23} +2.20003 q^{25} +3.45482 q^{26} -7.05711 q^{28} -0.405468 q^{29} +3.62200 q^{31} -5.69550 q^{32} +11.9236 q^{35} +4.96694 q^{37} +2.18558 q^{38} +6.17895 q^{40} -5.68257 q^{41} +5.89136 q^{43} +10.0599 q^{44} -1.50844 q^{46} +1.84042 q^{47} +12.7460 q^{49} -1.41191 q^{50} +8.54935 q^{52} -2.31475 q^{53} -16.9971 q^{55} +10.2326 q^{56} +0.260217 q^{58} +12.6166 q^{59} -9.13811 q^{61} -2.32449 q^{62} +0.258335 q^{64} -14.4449 q^{65} +0.233694 q^{67} -7.65219 q^{70} -9.29208 q^{71} +3.05704 q^{73} -3.18763 q^{74} +5.40847 q^{76} -28.1480 q^{77} -5.59075 q^{79} +4.55738 q^{80} +3.64690 q^{82} -9.49892 q^{83} -3.78089 q^{86} -14.5866 q^{88} +14.3614 q^{89} -23.9214 q^{91} -3.73282 q^{92} -1.18112 q^{94} -9.13807 q^{95} -17.6601 q^{97} -8.18002 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 6 q^{2} + 12 q^{4} + 3 q^{5} + 3 q^{7} - 18 q^{8} - 6 q^{11} - 3 q^{13} - 3 q^{14} + 6 q^{16} - 3 q^{19} - 6 q^{20} - 12 q^{22} + 3 q^{23} + 6 q^{25} + 24 q^{26} + 9 q^{28} + 6 q^{29} - 42 q^{32}+ \cdots - 15 q^{98}+O(q^{100}) \) 15 * q - 6 * q^2 + 12 * q^4 + 3 * q^5 + 3 * q^7 - 18 * q^8 - 6 * q^11 - 3 * q^13 - 3 * q^14 + 6 * q^16 - 3 * q^19 - 6 * q^20 - 12 * q^22 + 3 * q^23 + 6 * q^25 + 24 * q^26 + 9 * q^28 + 6 * q^29 - 42 * q^32 - 33 * q^35 - 36 * q^38 + 15 * q^40 - 3 * q^43 + 18 * q^44 + 12 * q^46 - 24 * q^47 + 18 * q^49 - 42 * q^50 - 12 * q^52 - 48 * q^53 - 3 * q^55 + 15 * q^56 - 12 * q^58 - 18 * q^59 - 63 * q^62 + 30 * q^64 - 24 * q^65 + 12 * q^67 + 51 * q^70 + 21 * q^71 - 18 * q^73 + 48 * q^74 + 60 * q^76 - 54 * q^77 - 39 * q^79 - 6 * q^80 + 39 * q^82 - 51 * q^83 - 48 * q^86 - 48 * q^88 - 36 * q^89 + 9 * q^91 + 3 * q^92 + 36 * q^94 - 39 * q^95 + 3 * q^97 - 15 * q^98

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