LMFDB - Embedded newform 7203.2.a.d.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7203,2,Mod(1,7203)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7203, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7203.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7203 = 3 \cdot 7^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7203.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,2,12,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.5162445759$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 16 x^{10} + 30 x^{9} + 92 x^{8} - 154 x^{7} - 244 x^{6} + 343 x^{5} + 295 x^{4} + \cdots + 1 $ x^12 - 2x^11 - 16x^10 + 30x^9 + 92x^8 - 154x^7 - 244x^6 + 343x^5 + 295x^4 - 331x^3 - 121x^2 + 114*x + 1
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			1.4
Root			$-1.39887$ of defining polynomial
Character	$\chi$	$=$	7203.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.39887 q^{2} +1.00000 q^{3} -0.0431625 q^{4} +2.18055 q^{5} -1.39887 q^{6} +2.85812 q^{8} +1.00000 q^{9} -3.05030 q^{10} +6.48029 q^{11} -0.0431625 q^{12} +0.782484 q^{13} +2.18055 q^{15} -3.91181 q^{16} -5.53556 q^{17} -1.39887 q^{18} -0.140576 q^{19} -0.0941180 q^{20} -9.06509 q^{22} +6.13670 q^{23} +2.85812 q^{24} -0.245206 q^{25} -1.09459 q^{26} +1.00000 q^{27} -2.43168 q^{29} -3.05030 q^{30} +2.14201 q^{31} -0.244121 q^{32} +6.48029 q^{33} +7.74353 q^{34} -0.0431625 q^{36} -7.46623 q^{37} +0.196647 q^{38} +0.782484 q^{39} +6.23227 q^{40} +12.1118 q^{41} -8.75698 q^{43} -0.279706 q^{44} +2.18055 q^{45} -8.58445 q^{46} +9.39324 q^{47} -3.91181 q^{48} +0.343012 q^{50} -5.53556 q^{51} -0.0337740 q^{52} +3.55148 q^{53} -1.39887 q^{54} +14.1306 q^{55} -0.140576 q^{57} +3.40160 q^{58} -1.08597 q^{59} -0.0941180 q^{60} -4.30119 q^{61} -2.99639 q^{62} +8.16512 q^{64} +1.70624 q^{65} -9.06509 q^{66} +6.16236 q^{67} +0.238929 q^{68} +6.13670 q^{69} +5.52885 q^{71} +2.85812 q^{72} -5.14349 q^{73} +10.4443 q^{74} -0.245206 q^{75} +0.00606761 q^{76} -1.09459 q^{78} +13.4125 q^{79} -8.52990 q^{80} +1.00000 q^{81} -16.9429 q^{82} +17.0840 q^{83} -12.0706 q^{85} +12.2499 q^{86} -2.43168 q^{87} +18.5214 q^{88} -4.54396 q^{89} -3.05030 q^{90} -0.264876 q^{92} +2.14201 q^{93} -13.1399 q^{94} -0.306533 q^{95} -0.244121 q^{96} -6.51297 q^{97} +6.48029 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{2} + 12 q^{3} + 12 q^{4} + 3 q^{5} + 2 q^{6} + 6 q^{8} + 12 q^{9} - 3 q^{11} + 12 q^{12} + 13 q^{13} + 3 q^{15} + 8 q^{16} + 3 q^{17} + 2 q^{18} - 3 q^{20} - 11 q^{22} + 18 q^{23} + 6 q^{24}+ \cdots - 3 q^{99}+O(q^{100}) $ 12 * q + 2 * q^2 + 12 * q^3 + 12 * q^4 + 3 * q^5 + 2 * q^6 + 6 * q^8 + 12 * q^9 - 3 * q^11 + 12 * q^12 + 13 * q^13 + 3 * q^15 + 8 * q^16 + 3 * q^17 + 2 * q^18 - 3 * q^20 - 11 * q^22 + 18 * q^23 + 6 * q^24 + 5 * q^25 + 26 * q^26 + 12 * q^27 - 13 * q^29 - 6 * q^31 - 6 * q^32 - 3 * q^33 + 2 * q^34 + 12 * q^36 + 15 * q^37 + 43 * q^38 + 13 * q^39 - 6 * q^40 + 38 * q^41 + 21 * q^43 - 38 * q^44 + 3 * q^45 - 16 * q^46 + 28 * q^47 + 8 * q^48 + 7 * q^50 + 3 * q^51 + 40 * q^52 + 11 * q^53 + 2 * q^54 + 50 * q^55 + 42 * q^58 - 6 * q^59 - 3 * q^60 + 26 * q^61 + 47 * q^62 + 28 * q^64 - 5 * q^65 - 11 * q^66 + 14 * q^67 - 14 * q^68 + 18 * q^69 - 31 * q^71 + 6 * q^72 + 48 * q^73 + 20 * q^74 + 5 * q^75 - 9 * q^76 + 26 * q^78 + 5 * q^79 - 47 * q^80 + 12 * q^81 - 29 * q^82 + 57 * q^83 + 4 * q^85 + 82 * q^86 - 13 * q^87 + 30 * q^88 - 15 * q^89 + 35 * q^92 - 6 * q^93 + 50 * q^94 - 55 * q^95 - 6 * q^96 + 6 * q^97 - 3 * 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 $

$2$	−1.39887	−0.989151	−0.494575	−	0.869135i	$-0.664676\pi$
$2$	−1.39887	−0.989151	−0.494575	+	0.869135i	$0.664676\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.0431625	−0.0215813
$5$	2.18055	0.975171	0.487586	−	0.873075i	$-0.337878\pi$
$5$	2.18055	0.975171	0.487586	+	0.873075i	$0.337878\pi$
$6$	−1.39887	−0.571086
$7$	0	0
$7$	0	0
$8$	2.85812	1.01050
$9$	1.00000	0.333333
$10$	−3.05030	−0.964591
$11$	6.48029	1.95388	0.976941	−	0.213510i	$-0.0684897\pi$
$11$	6.48029	1.95388	0.976941	+	0.213510i	$0.0684897\pi$
$12$	−0.0431625	−0.0124600
$13$	0.782484	0.217022	0.108511	−	0.994095i	$-0.465392\pi$
$13$	0.782484	0.217022	0.108511	+	0.994095i	$0.465392\pi$
$14$	0	0
$15$	2.18055	0.563015
$16$	−3.91181	−0.977953
$17$	−5.53556	−1.34257	−0.671286	−	0.741199i	$-0.734258\pi$
$17$	−5.53556	−1.34257	−0.671286	+	0.741199i	$0.734258\pi$
$18$	−1.39887	−0.329717
$19$	−0.140576	−0.0322503	−0.0161252	−	0.999870i	$-0.505133\pi$
$19$	−0.140576	−0.0322503	−0.0161252	+	0.999870i	$0.505133\pi$
$20$	−0.0941180	−0.0210454
$21$	0	0
$22$	−9.06509	−1.93268
$23$	6.13670	1.27959	0.639796	−	0.768545i	$-0.279019\pi$
$23$	6.13670	1.27959	0.639796	+	0.768545i	$0.279019\pi$
$24$	2.85812	0.583411
$25$	−0.245206	−0.0490413
$26$	−1.09459	−0.214667
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.43168	−0.451551	−0.225776	−	0.974179i	$-0.572492\pi$
$29$	−2.43168	−0.451551	−0.225776	+	0.974179i	$0.572492\pi$
$30$	−3.05030	−0.556907
$31$	2.14201	0.384716	0.192358	−	0.981325i	$-0.438387\pi$
$31$	2.14201	0.384716	0.192358	+	0.981325i	$0.438387\pi$
$32$	−0.244121	−0.0431549
$33$	6.48029	1.12807
$34$	7.74353	1.32801
$35$	0	0
$36$	−0.0431625	−0.00719376
$37$	−7.46623	−1.22744	−0.613720	−	0.789524i	$-0.710328\pi$
$37$	−7.46623	−1.22744	−0.613720	+	0.789524i	$0.710328\pi$
$38$	0.196647	0.0319004
$39$	0.782484	0.125298
$40$	6.23227	0.985408
$41$	12.1118	1.89155	0.945776	−	0.324819i	$-0.105303\pi$
$41$	12.1118	1.89155	0.945776	+	0.324819i	$0.105303\pi$
$42$	0	0
$43$	−8.75698	−1.33543	−0.667713	−	0.744419i	$-0.732727\pi$
$43$	−8.75698	−1.33543	−0.667713	+	0.744419i	$0.732727\pi$
$44$	−0.279706	−0.0421673
$45$	2.18055	0.325057
$46$	−8.58445	−1.26571
$47$	9.39324	1.37015	0.685073	−	0.728475i	$-0.259770\pi$
$47$	9.39324	1.37015	0.685073	+	0.728475i	$0.259770\pi$
$48$	−3.91181	−0.564621
$49$	0	0
$50$	0.343012	0.0485092
$51$	−5.53556	−0.775134
$52$	−0.0337740	−0.00468361
$53$	3.55148	0.487834	0.243917	−	0.969796i	$-0.421568\pi$
$53$	3.55148	0.487834	0.243917	+	0.969796i	$0.421568\pi$
$54$	−1.39887	−0.190362
$55$	14.1306	1.90537
$56$	0	0
$57$	−0.140576	−0.0186197
$58$	3.40160	0.446652
$59$	−1.08597	−0.141381	−0.0706905	−	0.997498i	$-0.522520\pi$
$59$	−1.08597	−0.141381	−0.0706905	+	0.997498i	$0.522520\pi$
$60$	−0.0941180	−0.0121506
$61$	−4.30119	−0.550711	−0.275355	−	0.961342i	$-0.588796\pi$
$61$	−4.30119	−0.550711	−0.275355	+	0.961342i	$0.588796\pi$
$62$	−2.99639	−0.380542
$63$	0	0
$64$	8.16512	1.02064
$65$	1.70624	0.211634
$66$	−9.06509	−1.11584
$67$	6.16236	0.752852	0.376426	−	0.926447i	$-0.377153\pi$
$67$	6.16236	0.752852	0.376426	+	0.926447i	$0.377153\pi$
$68$	0.238929	0.0289744
$69$	6.13670	0.738772
$70$	0	0
$71$	5.52885	0.656154	0.328077	−	0.944651i	$-0.393600\pi$
$71$	5.52885	0.656154	0.328077	+	0.944651i	$0.393600\pi$
$72$	2.85812	0.336833
$73$	−5.14349	−0.602000	−0.301000	−	0.953624i	$-0.597320\pi$
$73$	−5.14349	−0.602000	−0.301000	+	0.953624i	$0.597320\pi$
$74$	10.4443	1.21412
$75$	−0.245206	−0.0283140
$76$	0.00606761	0.000696003 0
$77$	0	0
$78$	−1.09459	−0.123938
$79$	13.4125	1.50902	0.754510	−	0.656288i	$-0.227874\pi$
$79$	13.4125	1.50902	0.754510	+	0.656288i	$0.227874\pi$
$80$	−8.52990	−0.953672
$81$	1.00000	0.111111
$82$	−16.9429	−1.87103
$83$	17.0840	1.87521	0.937605	−	0.347702i	$-0.113038\pi$
$83$	17.0840	1.87521	0.937605	+	0.347702i	$0.113038\pi$
$84$	0	0
$85$	−12.0706	−1.30924
$86$	12.2499	1.32094
$87$	−2.43168	−0.260703
$88$	18.5214	1.97439
$89$	−4.54396	−0.481659	−0.240829	−	0.970567i	$-0.577419\pi$
$89$	−4.54396	−0.481659	−0.240829	+	0.970567i	$0.577419\pi$
$90$	−3.05030	−0.321530
$91$	0	0
$92$	−0.264876	−0.0276152
$93$	2.14201	0.222116
$94$	−13.1399	−1.35528
$95$	−0.306533	−0.0314496
$96$	−0.244121	−0.0249155
$97$	−6.51297	−0.661292	−0.330646	−	0.943755i	$-0.607267\pi$
$97$	−6.51297	−0.661292	−0.330646	+	0.943755i	$0.607267\pi$
$98$	0	0
$99$	6.48029	0.651294
$100$	0.0105837	0.00105837
$101$	−2.93471	−0.292014	−0.146007	−	0.989284i	$-0.546642\pi$
$101$	−2.93471	−0.292014	−0.146007	+	0.989284i	$0.546642\pi$
$102$	7.74353	0.766724
$103$	10.8509	1.06918	0.534588	−	0.845113i	$-0.320467\pi$
$103$	10.8509	1.06918	0.534588	+	0.845113i	$0.320467\pi$
$104$	2.23643	0.219300
$105$	0	0
$106$	−4.96806	−0.482541
$107$	3.45208	0.333725	0.166863	−	0.985980i	$-0.446636\pi$
$107$	3.45208	0.333725	0.166863	+	0.985980i	$0.446636\pi$
$108$	−0.0431625	−0.00415332
$109$	−16.9387	−1.62243	−0.811217	−	0.584745i	$-0.801195\pi$
$109$	−16.9387	−1.62243	−0.811217	+	0.584745i	$0.801195\pi$
$110$	−19.7669	−1.88470
$111$	−7.46623	−0.708663
$112$	0	0
$113$	5.89023	0.554106	0.277053	−	0.960855i	$-0.410642\pi$
$113$	5.89023	0.554106	0.277053	+	0.960855i	$0.410642\pi$
$114$	0.196647	0.0184177
$115$	13.3814	1.24782
$116$	0.104957	0.00974505
$117$	0.782484	0.0723407
$118$	1.51913	0.139847
$119$	0	0
$120$	6.23227	0.568926
$121$	30.9942	2.81765
$122$	6.01680	0.544736
$123$	12.1118	1.09209
$124$	−0.0924545	−0.00830266
$125$	−11.4374	−1.02299
$126$	0	0
$127$	2.93903	0.260797	0.130399	−	0.991462i	$-0.458374\pi$
$127$	2.93903	0.260797	0.130399	+	0.991462i	$0.458374\pi$
$128$	−10.9337	−0.966411
$129$	−8.75698	−0.771009
$130$	−2.38681	−0.209337
$131$	12.3407	1.07821	0.539106	−	0.842238i	$-0.318762\pi$
$131$	12.3407	1.07821	0.539106	+	0.842238i	$0.318762\pi$
$132$	−0.279706	−0.0243453
$133$	0	0
$134$	−8.62035	−0.744684
$135$	2.18055	0.187672
$136$	−15.8213	−1.35667
$137$	−7.51568	−0.642108	−0.321054	−	0.947061i	$-0.604037\pi$
$137$	−7.51568	−0.642108	−0.321054	+	0.947061i	$0.604037\pi$
$138$	−8.58445	−0.730757
$139$	1.73208	0.146914	0.0734568	−	0.997298i	$-0.476597\pi$
$139$	1.73208	0.146914	0.0734568	+	0.997298i	$0.476597\pi$
$140$	0	0
$141$	9.39324	0.791054
$142$	−7.73414	−0.649035
$143$	5.07073	0.424035
$144$	−3.91181	−0.325984
$145$	−5.30239	−0.440340
$146$	7.19508	0.595469
$147$	0	0
$148$	0.322261	0.0264897
$149$	−8.09135	−0.662869	−0.331435	−	0.943478i	$-0.607533\pi$
$149$	−8.09135	−0.662869	−0.331435	+	0.943478i	$0.607533\pi$
$150$	0.343012	0.0280068
$151$	−5.19672	−0.422903	−0.211452	−	0.977388i	$-0.567819\pi$
$151$	−5.19672	−0.422903	−0.211452	+	0.977388i	$0.567819\pi$
$152$	−0.401783	−0.0325889
$153$	−5.53556	−0.447524
$154$	0	0
$155$	4.67075	0.375164
$156$	−0.0337740	−0.00270408
$157$	1.55862	0.124391	0.0621957	−	0.998064i	$-0.480190\pi$
$157$	1.55862	0.124391	0.0621957	+	0.998064i	$0.480190\pi$
$158$	−18.7623	−1.49265
$159$	3.55148	0.281651
$160$	−0.532318	−0.0420835
$161$	0	0
$162$	−1.39887	−0.109906
$163$	3.77617	0.295773	0.147886	−	0.989004i	$-0.452753\pi$
$163$	3.77617	0.295773	0.147886	+	0.989004i	$0.452753\pi$
$164$	−0.522778	−0.0408221
$165$	14.1306	1.10007
$166$	−23.8983	−1.85487
$167$	21.3137	1.64931	0.824653	−	0.565639i	$-0.191370\pi$
$167$	21.3137	1.64931	0.824653	+	0.565639i	$0.191370\pi$
$168$	0	0
$169$	−12.3877	−0.952901
$170$	16.8852	1.29503
$171$	−0.140576	−0.0107501
$172$	0.377973	0.0288202
$173$	22.3191	1.69689	0.848446	−	0.529282i	$-0.177539\pi$
$173$	22.3191	1.69689	0.848446	+	0.529282i	$0.177539\pi$
$174$	3.40160	0.257875
$175$	0	0
$176$	−25.3497	−1.91080
$177$	−1.08597	−0.0816264
$178$	6.35641	0.476433
$179$	−13.3676	−0.999140	−0.499570	−	0.866273i	$-0.666509\pi$
$179$	−13.3676	−0.999140	−0.499570	+	0.866273i	$0.666509\pi$
$180$	−0.0941180	−0.00701514
$181$	−0.159415	−0.0118492	−0.00592460	−	0.999982i	$-0.501886\pi$
$181$	−0.159415	−0.0118492	−0.00592460	+	0.999982i	$0.501886\pi$
$182$	0	0
$183$	−4.30119	−0.317953
$184$	17.5394	1.29302
$185$	−16.2805	−1.19696
$186$	−2.99639	−0.219706
$187$	−35.8721	−2.62323
$188$	−0.405436	−0.0295695
$189$	0	0
$190$	0.428799	0.0311084
$191$	−23.6260	−1.70952	−0.854758	−	0.519028i	$-0.826294\pi$
$191$	−23.6260	−1.70952	−0.854758	+	0.519028i	$0.826294\pi$
$192$	8.16512	0.589267
$193$	−0.126858	−0.00913145	−0.00456573	−	0.999990i	$-0.501453\pi$
$193$	−0.126858	−0.00913145	−0.00456573	+	0.999990i	$0.501453\pi$
$194$	9.11080	0.654117
$195$	1.70624	0.122187
$196$	0	0
$197$	20.8057	1.48234	0.741172	−	0.671316i	$-0.234270\pi$
$197$	20.8057	1.48234	0.741172	+	0.671316i	$0.234270\pi$
$198$	−9.06509	−0.644228
$199$	−10.8225	−0.767190	−0.383595	−	0.923501i	$-0.625314\pi$
$199$	−10.8225	−0.767190	−0.383595	+	0.923501i	$0.625314\pi$
$200$	−0.700829	−0.0495561
$201$	6.16236	0.434660
$202$	4.10527	0.288846
$203$	0	0
$204$	0.238929	0.0167284
$205$	26.4105	1.84459
$206$	−15.1791	−1.05758
$207$	6.13670	0.426530
$208$	−3.06093	−0.212237
$209$	−0.910973	−0.0630133
$210$	0	0
$211$	−4.04533	−0.278492	−0.139246	−	0.990258i	$-0.544468\pi$
$211$	−4.04533	−0.278492	−0.139246	+	0.990258i	$0.544468\pi$
$212$	−0.153291	−0.0105281
$213$	5.52885	0.378830
$214$	−4.82901	−0.330105
$215$	−19.0950	−1.30227
$216$	2.85812	0.194470
$217$	0	0
$218$	23.6951	1.60483
$219$	−5.14349	−0.347565
$220$	−0.609912	−0.0411203
$221$	−4.33149	−0.291368
$222$	10.4443	0.700974
$223$	−15.3809	−1.02998	−0.514991	−	0.857196i	$-0.672205\pi$
$223$	−15.3809	−1.02998	−0.514991	+	0.857196i	$0.672205\pi$
$224$	0	0
$225$	−0.245206	−0.0163471
$226$	−8.23967	−0.548095
$227$	−14.0751	−0.934199	−0.467099	−	0.884205i	$-0.654701\pi$
$227$	−14.0751	−0.934199	−0.467099	+	0.884205i	$0.654701\pi$
$228$	0.00606761	0.000401838 0
$229$	−4.31492	−0.285138	−0.142569	−	0.989785i	$-0.545536\pi$
$229$	−4.31492	−0.285138	−0.142569	+	0.989785i	$0.545536\pi$
$230$	−18.7188	−1.23428
$231$	0	0
$232$	−6.95003	−0.456292
$233$	−16.3554	−1.07148	−0.535739	−	0.844384i	$-0.679967\pi$
$233$	−16.3554	−1.07148	−0.535739	+	0.844384i	$0.679967\pi$
$234$	−1.09459	−0.0715558
$235$	20.4824	1.33613
$236$	0.0468732	0.00305118
$237$	13.4125	0.871233
$238$	0	0
$239$	−11.0575	−0.715247	−0.357624	−	0.933866i	$-0.616413\pi$
$239$	−11.0575	−0.715247	−0.357624	+	0.933866i	$0.616413\pi$
$240$	−8.52990	−0.550602
$241$	−20.6777	−1.33197	−0.665983	−	0.745967i	$-0.731988\pi$
$241$	−20.6777	−1.33197	−0.665983	+	0.745967i	$0.731988\pi$
$242$	−43.3568	−2.78708
$243$	1.00000	0.0641500
$244$	0.185650	0.0118850
$245$	0	0
$246$	−16.9429	−1.08024
$247$	−0.109998	−0.00699903
$248$	6.12211	0.388754
$249$	17.0840	1.08265
$250$	15.9995	1.01190
$251$	24.6124	1.55352	0.776760	−	0.629796i	$-0.216862\pi$
$251$	24.6124	1.55352	0.776760	+	0.629796i	$0.216862\pi$
$252$	0	0
$253$	39.7676	2.50017
$254$	−4.11133	−0.257968
$255$	−12.0706	−0.755888
$256$	−1.03542	−0.0647134
$257$	−13.4398	−0.838353	−0.419176	−	0.907905i	$-0.637681\pi$
$257$	−13.4398	−0.838353	−0.419176	+	0.907905i	$0.637681\pi$
$258$	12.2499	0.762644
$259$	0	0
$260$	−0.0736459	−0.00456732
$261$	−2.43168	−0.150517
$262$	−17.2631	−1.06651
$263$	11.2887	0.696089	0.348045	−	0.937478i	$-0.386846\pi$
$263$	11.2887	0.696089	0.348045	+	0.937478i	$0.386846\pi$
$264$	18.5214	1.13992
$265$	7.74418	0.475721
$266$	0	0
$267$	−4.54396	−0.278086
$268$	−0.265983	−0.0162475
$269$	−14.0668	−0.857666	−0.428833	−	0.903384i	$-0.641075\pi$
$269$	−14.0668	−0.857666	−0.428833	+	0.903384i	$0.641075\pi$
$270$	−3.05030	−0.185636
$271$	23.1738	1.40771	0.703854	−	0.710345i	$-0.251461\pi$
$271$	23.1738	1.40771	0.703854	+	0.710345i	$0.251461\pi$
$272$	21.6541	1.31297
$273$	0	0
$274$	10.5135	0.635142
$275$	−1.58901	−0.0958209
$276$	−0.264876	−0.0159436
$277$	−4.33550	−0.260495	−0.130247	−	0.991482i	$-0.541577\pi$
$277$	−4.33550	−0.260495	−0.130247	+	0.991482i	$0.541577\pi$
$278$	−2.42296	−0.145320
$279$	2.14201	0.128239
$280$	0	0
$281$	24.4371	1.45780	0.728899	−	0.684621i	$-0.240032\pi$
$281$	24.4371	1.45780	0.728899	+	0.684621i	$0.240032\pi$
$282$	−13.1399	−0.782471
$283$	20.7126	1.23124	0.615618	−	0.788045i	$-0.288906\pi$
$283$	20.7126	1.23124	0.615618	+	0.788045i	$0.288906\pi$
$284$	−0.238639	−0.0141606
$285$	−0.306533	−0.0181574
$286$	−7.09329	−0.419435
$287$	0	0
$288$	−0.244121	−0.0143850
$289$	13.6425	0.802498
$290$	7.41736	0.435562
$291$	−6.51297	−0.381797
$292$	0.222006	0.0129919
$293$	12.8577	0.751156	0.375578	−	0.926791i	$-0.377444\pi$
$293$	12.8577	0.751156	0.375578	+	0.926791i	$0.377444\pi$
$294$	0	0
$295$	−2.36801	−0.137871
$296$	−21.3394	−1.24033
$297$	6.48029	0.376025
$298$	11.3188	0.655678
$299$	4.80187	0.277699
$300$	0.0105837	0.000611052 0
$301$	0	0
$302$	7.26954	0.418315
$303$	−2.93471	−0.168594
$304$	0.549906	0.0315393
$305$	−9.37895	−0.537037
$306$	7.74353	0.442668
$307$	17.1879	0.980964	0.490482	−	0.871451i	$-0.336821\pi$
$307$	17.1879	0.980964	0.490482	+	0.871451i	$0.336821\pi$
$308$	0	0
$309$	10.8509	0.617289
$310$	−6.53377	−0.371093
$311$	−14.7890	−0.838606	−0.419303	−	0.907846i	$-0.637726\pi$
$311$	−14.7890	−0.838606	−0.419303	+	0.907846i	$0.637726\pi$
$312$	2.23643	0.126613
$313$	31.7838	1.79653	0.898264	−	0.439457i	$-0.144829\pi$
$313$	31.7838	1.79653	0.898264	+	0.439457i	$0.144829\pi$
$314$	−2.18031	−0.123042
$315$	0	0
$316$	−0.578916	−0.0325666
$317$	−6.52577	−0.366524	−0.183262	−	0.983064i	$-0.558666\pi$
$317$	−6.52577	−0.366524	−0.183262	+	0.983064i	$0.558666\pi$
$318$	−4.96806	−0.278595
$319$	−15.7580	−0.882278
$320$	17.8044	0.995298
$321$	3.45208	0.192676
$322$	0	0
$323$	0.778167	0.0432984
$324$	−0.0431625	−0.00239792
$325$	−0.191870	−0.0106430
$326$	−5.28237	−0.292564
$327$	−16.9387	−0.936713
$328$	34.6171	1.91141
$329$	0	0
$330$	−19.7669	−1.08813
$331$	24.4510	1.34395	0.671974	−	0.740575i	$-0.265447\pi$
$331$	24.4510	1.34395	0.671974	+	0.740575i	$0.265447\pi$
$332$	−0.737388	−0.0404694
$333$	−7.46623	−0.409147
$334$	−29.8151	−1.63141
$335$	13.4373	0.734160
$336$	0	0
$337$	4.31055	0.234811	0.117405	−	0.993084i	$-0.462542\pi$
$337$	4.31055	0.234811	0.117405	+	0.993084i	$0.462542\pi$
$338$	17.3288	0.942563
$339$	5.89023	0.319913
$340$	0.520996	0.0282550
$341$	13.8808	0.751689
$342$	0.196647	0.0106335
$343$	0	0
$344$	−25.0285	−1.34945
$345$	13.3814	0.720429
$346$	−31.2216	−1.67848
$347$	25.7393	1.38176	0.690879	−	0.722971i	$-0.257224\pi$
$347$	25.7393	1.38176	0.690879	+	0.722971i	$0.257224\pi$
$348$	0.104957	0.00562631
$349$	34.3791	1.84027	0.920136	−	0.391598i	$-0.128078\pi$
$349$	34.3791	1.84027	0.920136	+	0.391598i	$0.128078\pi$
$350$	0	0
$351$	0.782484	0.0417659
$352$	−1.58198	−0.0843197
$353$	−9.63624	−0.512885	−0.256443	−	0.966559i	$-0.582551\pi$
$353$	−9.63624	−0.512885	−0.256443	+	0.966559i	$0.582551\pi$
$354$	1.51913	0.0807408
$355$	12.0559	0.639862
$356$	0.196129	0.0103948
$357$	0	0
$358$	18.6995	0.988300
$359$	−29.8557	−1.57572	−0.787861	−	0.615853i	$-0.788811\pi$
$359$	−29.8557	−1.57572	−0.787861	+	0.615853i	$0.788811\pi$
$360$	6.23227	0.328469
$361$	−18.9802	−0.998960
$362$	0.223000	0.0117206
$363$	30.9942	1.62677
$364$	0	0
$365$	−11.2156	−0.587053
$366$	6.01680	0.314503
$367$	−20.6975	−1.08040	−0.540202	−	0.841536i	$-0.681652\pi$
$367$	−20.6975	−1.08040	−0.540202	+	0.841536i	$0.681652\pi$
$368$	−24.0056	−1.25138
$369$	12.1118	0.630517
$370$	22.7743	1.18398
$371$	0	0
$372$	−0.0924545	−0.00479354
$373$	−31.5100	−1.63153	−0.815764	−	0.578385i	$-0.803683\pi$
$373$	−31.5100	−1.63153	−0.815764	+	0.578385i	$0.803683\pi$
$374$	50.1804	2.59477
$375$	−11.4374	−0.590626
$376$	26.8470	1.38453
$377$	−1.90275	−0.0979966
$378$	0	0
$379$	27.4692	1.41100	0.705500	−	0.708710i	$-0.250722\pi$
$379$	27.4692	1.41100	0.705500	+	0.708710i	$0.250722\pi$
$380$	0.0132307	0.000678722 0
$381$	2.93903	0.150571
$382$	33.0497	1.69097
$383$	23.5967	1.20574	0.602869	−	0.797840i	$-0.294024\pi$
$383$	23.5967	1.20574	0.602869	+	0.797840i	$0.294024\pi$
$384$	−10.9337	−0.557958
$385$	0	0
$386$	0.177458	0.00903238
$387$	−8.75698	−0.445142
$388$	0.281116	0.0142715
$389$	−21.5794	−1.09412	−0.547059	−	0.837094i	$-0.684253\pi$
$389$	−21.5794	−1.09412	−0.547059	+	0.837094i	$0.684253\pi$
$390$	−2.38681	−0.120861
$391$	−33.9701	−1.71794
$392$	0	0
$393$	12.3407	0.622507
$394$	−29.1044	−1.46626
$395$	29.2465	1.47155
$396$	−0.279706	−0.0140558
$397$	−25.2434	−1.26693	−0.633464	−	0.773772i	$-0.718368\pi$
$397$	−25.2434	−1.26693	−0.633464	+	0.773772i	$0.718368\pi$
$398$	15.1393	0.758866
$399$	0	0
$400$	0.959201	0.0479601
$401$	24.9827	1.24758	0.623789	−	0.781593i	$-0.285592\pi$
$401$	24.9827	1.24758	0.623789	+	0.781593i	$0.285592\pi$
$402$	−8.62035	−0.429944
$403$	1.67609	0.0834918
$404$	0.126669	0.00630204
$405$	2.18055	0.108352
$406$	0	0
$407$	−48.3833	−2.39827
$408$	−15.8213	−0.783271
$409$	2.34152	0.115781	0.0578904	−	0.998323i	$-0.481563\pi$
$409$	2.34152	0.115781	0.0578904	+	0.998323i	$0.481563\pi$
$410$	−36.9448	−1.82457
$411$	−7.51568	−0.370721
$412$	−0.468354	−0.0230742
$413$	0	0
$414$	−8.58445	−0.421903
$415$	37.2524	1.82865
$416$	−0.191021	−0.00936557
$417$	1.73208	0.0848206
$418$	1.27433	0.0623297
$419$	−33.3174	−1.62766	−0.813830	−	0.581104i	$-0.802621\pi$
$419$	−33.3174	−1.62766	−0.813830	+	0.581104i	$0.802621\pi$
$420$	0	0
$421$	2.28334	0.111283	0.0556417	−	0.998451i	$-0.482280\pi$
$421$	2.28334	0.111283	0.0556417	+	0.998451i	$0.482280\pi$
$422$	5.65890	0.275471
$423$	9.39324	0.456715
$424$	10.1506	0.492955
$425$	1.35736	0.0658414
$426$	−7.73414	−0.374720
$427$	0	0
$428$	−0.149001	−0.00720222
$429$	5.07073	0.244817
$430$	26.7114	1.28814
$431$	15.5680	0.749884	0.374942	−	0.927048i	$-0.377663\pi$
$431$	15.5680	0.749884	0.374942	+	0.927048i	$0.377663\pi$
$432$	−3.91181	−0.188207
$433$	−23.6011	−1.13420	−0.567098	−	0.823650i	$-0.691934\pi$
$433$	−23.6011	−1.13420	−0.567098	+	0.823650i	$0.691934\pi$
$434$	0	0
$435$	−5.30239	−0.254230
$436$	0.731118	0.0350142
$437$	−0.862673	−0.0412672
$438$	7.19508	0.343794
$439$	20.0248	0.955730	0.477865	−	0.878433i	$-0.341411\pi$
$439$	20.0248	0.955730	0.477865	+	0.878433i	$0.341411\pi$
$440$	40.3869	1.92537
$441$	0	0
$442$	6.05919	0.288206
$443$	−5.84019	−0.277476	−0.138738	−	0.990329i	$-0.544305\pi$
$443$	−5.84019	−0.277476	−0.138738	+	0.990329i	$0.544305\pi$
$444$	0.322261	0.0152939
$445$	−9.90832	−0.469700
$446$	21.5159	1.01881
$447$	−8.09135	−0.382708
$448$	0	0
$449$	3.94040	0.185959	0.0929795	−	0.995668i	$-0.470361\pi$
$449$	3.94040	0.185959	0.0929795	+	0.995668i	$0.470361\pi$
$450$	0.343012	0.0161697
$451$	78.4883	3.69587
$452$	−0.254237	−0.0119583
$453$	−5.19672	−0.244163
$454$	19.6893	0.924063
$455$	0	0
$456$	−0.401783	−0.0188152
$457$	7.32815	0.342796	0.171398	−	0.985202i	$-0.445172\pi$
$457$	7.32815	0.342796	0.171398	+	0.985202i	$0.445172\pi$
$458$	6.03602	0.282045
$459$	−5.53556	−0.258378
$460$	−0.577574	−0.0269295
$461$	−31.6277	−1.47305	−0.736525	−	0.676410i	$-0.763535\pi$
$461$	−31.6277	−1.47305	−0.736525	+	0.676410i	$0.763535\pi$
$462$	0	0
$463$	16.2417	0.754816	0.377408	−	0.926047i	$-0.376815\pi$
$463$	16.2417	0.754816	0.377408	+	0.926047i	$0.376815\pi$
$464$	9.51227	0.441596
$465$	4.67075	0.216601
$466$	22.8791	1.05985
$467$	−22.7406	−1.05231	−0.526155	−	0.850389i	$-0.676367\pi$
$467$	−22.7406	−1.05231	−0.526155	+	0.850389i	$0.676367\pi$
$468$	−0.0337740	−0.00156120
$469$	0	0
$470$	−28.6523	−1.32163
$471$	1.55862	0.0718174
$472$	−3.10383	−0.142865
$473$	−56.7478	−2.60926
$474$	−18.7623	−0.861781
$475$	0.0344701	0.00158160
$476$	0	0
$477$	3.55148	0.162611
$478$	15.4679	0.707487
$479$	23.7726	1.08620	0.543098	−	0.839669i	$-0.317251\pi$
$479$	23.7726	1.08620	0.543098	+	0.839669i	$0.317251\pi$
$480$	−0.532318	−0.0242969
$481$	−5.84221	−0.266382
$482$	28.9254	1.31751
$483$	0	0
$484$	−1.33779	−0.0608085
$485$	−14.2018	−0.644873
$486$	−1.39887	−0.0634540
$487$	10.5562	0.478349	0.239175	−	0.970977i	$-0.423123\pi$
$487$	10.5562	0.478349	0.239175	+	0.970977i	$0.423123\pi$
$488$	−12.2933	−0.556492
$489$	3.77617	0.170764
$490$	0	0
$491$	−12.0238	−0.542628	−0.271314	−	0.962491i	$-0.587458\pi$
$491$	−12.0238	−0.542628	−0.271314	+	0.962491i	$0.587458\pi$
$492$	−0.522778	−0.0235687
$493$	13.4607	0.606240
$494$	0.153873	0.00692309
$495$	14.1306	0.635123
$496$	−8.37913	−0.376234
$497$	0	0
$498$	−23.8983	−1.07091
$499$	9.96682	0.446176	0.223088	−	0.974798i	$-0.428386\pi$
$499$	9.96682	0.446176	0.223088	+	0.974798i	$0.428386\pi$
$500$	0.493669	0.0220775
$501$	21.3137	0.952227
$502$	−34.4295	−1.53667
$503$	−23.4438	−1.04531	−0.522654	−	0.852545i	$-0.675058\pi$
$503$	−23.4438	−1.04531	−0.522654	+	0.852545i	$0.675058\pi$
$504$	0	0
$505$	−6.39927	−0.284764
$506$	−55.6297	−2.47304
$507$	−12.3877	−0.550158
$508$	−0.126856	−0.00562833
$509$	−18.5688	−0.823048	−0.411524	−	0.911399i	$-0.635003\pi$
$509$	−18.5688	−0.823048	−0.411524	+	0.911399i	$0.635003\pi$
$510$	16.8852	0.747687
$511$	0	0
$512$	23.3158	1.03042
$513$	−0.140576	−0.00620658
$514$	18.8006	0.829257
$515$	23.6610	1.04263
$516$	0.377973	0.0166393
$517$	60.8710	2.67710
$518$	0	0
$519$	22.3191	0.979701
$520$	4.87665	0.213855
$521$	−44.4176	−1.94597	−0.972985	−	0.230868i	$-0.925844\pi$
$521$	−44.4176	−1.94597	−0.972985	+	0.230868i	$0.925844\pi$
$522$	3.40160	0.148884
$523$	0.264831	0.0115803	0.00579013	−	0.999983i	$-0.498157\pi$
$523$	0.264831	0.0115803	0.00579013	+	0.999983i	$0.498157\pi$
$524$	−0.532656	−0.0232692
$525$	0	0
$526$	−15.7914	−0.688537
$527$	−11.8572	−0.516508
$528$	−25.3497	−1.10320
$529$	14.6591	0.637353
$530$	−10.8331	−0.470560
$531$	−1.08597	−0.0471270
$532$	0	0
$533$	9.47732	0.410508
$534$	6.35641	0.275069
$535$	7.52743	0.325439
$536$	17.6128	0.760756
$537$	−13.3676	−0.576854
$538$	19.6776	0.848361
$539$	0	0
$540$	−0.0941180	−0.00405020
$541$	4.96320	0.213385	0.106692	−	0.994292i	$-0.465974\pi$
$541$	4.96320	0.213385	0.106692	+	0.994292i	$0.465974\pi$
$542$	−32.4171	−1.39243
$543$	−0.159415	−0.00684114
$544$	1.35135	0.0579386
$545$	−36.9357	−1.58215
$546$	0	0
$547$	−32.0123	−1.36875	−0.684374	−	0.729131i	$-0.739924\pi$
$547$	−32.0123	−1.36875	−0.684374	+	0.729131i	$0.739924\pi$
$548$	0.324396	0.0138575
$549$	−4.30119	−0.183570
$550$	2.22282	0.0947813
$551$	0.341835	0.0145627
$552$	17.5394	0.746528
$553$	0	0
$554$	6.06480	0.257669
$555$	−16.2805	−0.691068
$556$	−0.0747612	−0.00317058
$557$	6.46297	0.273845	0.136922	−	0.990582i	$-0.456279\pi$
$557$	6.46297	0.273845	0.136922	+	0.990582i	$0.456279\pi$
$558$	−2.99639	−0.126847
$559$	−6.85219	−0.289817
$560$	0	0
$561$	−35.8721	−1.51452
$562$	−34.1844	−1.44198
$563$	34.4790	1.45312	0.726558	−	0.687105i	$-0.241119\pi$
$563$	34.4790	1.45312	0.726558	+	0.687105i	$0.241119\pi$
$564$	−0.405436	−0.0170719
$565$	12.8439	0.540349
$566$	−28.9742	−1.21788
$567$	0	0
$568$	15.8021	0.663042
$569$	−39.9096	−1.67310	−0.836548	−	0.547893i	$-0.815430\pi$
$569$	−39.9096	−1.67310	−0.836548	+	0.547893i	$0.815430\pi$
$570$	0.428799	0.0179604
$571$	−2.84961	−0.119252	−0.0596262	−	0.998221i	$-0.518991\pi$
$571$	−2.84961	−0.119252	−0.0596262	+	0.998221i	$0.518991\pi$
$572$	−0.218865	−0.00915122
$573$	−23.6260	−0.986989
$574$	0	0
$575$	−1.50476	−0.0627528
$576$	8.16512	0.340213
$577$	19.6366	0.817483	0.408742	−	0.912650i	$-0.365968\pi$
$577$	19.6366	0.817483	0.408742	+	0.912650i	$0.365968\pi$
$578$	−19.0840	−0.793792
$579$	−0.126858	−0.00527205
$580$	0.228865	0.00950310
$581$	0	0
$582$	9.11080	0.377655
$583$	23.0146	0.953169
$584$	−14.7007	−0.608320
$585$	1.70624	0.0705445
$586$	−17.9863	−0.743007
$587$	41.7448	1.72299	0.861496	−	0.507765i	$-0.169528\pi$
$587$	41.7448	1.72299	0.861496	+	0.507765i	$0.169528\pi$
$588$	0	0
$589$	−0.301115	−0.0124072
$590$	3.31253	0.136375
$591$	20.8057	0.855831
$592$	29.2065	1.20038
$593$	7.20053	0.295690	0.147845	−	0.989011i	$-0.452766\pi$
$593$	7.20053	0.295690	0.147845	+	0.989011i	$0.452766\pi$
$594$	−9.06509	−0.371945
$595$	0	0
$596$	0.349243	0.0143056
$597$	−10.8225	−0.442937
$598$	−6.71720	−0.274687
$599$	−31.4482	−1.28494	−0.642469	−	0.766312i	$-0.722090\pi$
$599$	−31.4482	−1.28494	−0.642469	+	0.766312i	$0.722090\pi$
$600$	−0.700829	−0.0286112
$601$	2.88150	0.117539	0.0587694	−	0.998272i	$-0.481282\pi$
$601$	2.88150	0.117539	0.0587694	+	0.998272i	$0.481282\pi$
$602$	0	0
$603$	6.16236	0.250951
$604$	0.224304	0.00912679
$605$	67.5843	2.74769
$606$	4.10527	0.166765
$607$	28.4147	1.15332	0.576658	−	0.816985i	$-0.304356\pi$
$607$	28.4147	1.15332	0.576658	+	0.816985i	$0.304356\pi$
$608$	0.0343176	0.00139176
$609$	0	0
$610$	13.1199	0.531211
$611$	7.35006	0.297352
$612$	0.238929	0.00965813
$613$	28.1992	1.13896	0.569478	−	0.822007i	$-0.307145\pi$
$613$	28.1992	1.13896	0.569478	+	0.822007i	$0.307145\pi$
$614$	−24.0436	−0.970321
$615$	26.4105	1.06497
$616$	0	0
$617$	1.53526	0.0618074	0.0309037	−	0.999522i	$-0.490161\pi$
$617$	1.53526	0.0618074	0.0309037	+	0.999522i	$0.490161\pi$
$618$	−15.1791	−0.610591
$619$	−0.440861	−0.0177197	−0.00885986	−	0.999961i	$-0.502820\pi$
$619$	−0.440861	−0.0177197	−0.00885986	+	0.999961i	$0.502820\pi$
$620$	−0.201601	−0.00809651
$621$	6.13670	0.246257
$622$	20.6879	0.829508
$623$	0	0
$624$	−3.06093	−0.122535
$625$	−23.7138	−0.948554
$626$	−44.4614	−1.77704
$627$	−0.910973	−0.0363808
$628$	−0.0672740	−0.00268452
$629$	41.3298	1.64793
$630$	0	0
$631$	26.2628	1.04551	0.522753	−	0.852484i	$-0.324905\pi$
$631$	26.2628	1.04551	0.522753	+	0.852484i	$0.324905\pi$
$632$	38.3344	1.52486
$633$	−4.04533	−0.160788
$634$	9.12870	0.362547
$635$	6.40871	0.254322
$636$	−0.153291	−0.00607838
$637$	0	0
$638$	22.0434	0.872706
$639$	5.52885	0.218718
$640$	−23.8415	−0.942416
$641$	−1.97181	−0.0778817	−0.0389408	−	0.999242i	$-0.512398\pi$
$641$	−1.97181	−0.0778817	−0.0389408	+	0.999242i	$0.512398\pi$
$642$	−4.82901	−0.190586
$643$	−10.5785	−0.417174	−0.208587	−	0.978004i	$-0.566886\pi$
$643$	−10.5785	−0.417174	−0.208587	+	0.978004i	$0.566886\pi$
$644$	0	0
$645$	−19.0950	−0.751865
$646$	−1.08855	−0.0428286
$647$	7.08860	0.278682	0.139341	−	0.990244i	$-0.455502\pi$
$647$	7.08860	0.278682	0.139341	+	0.990244i	$0.455502\pi$
$648$	2.85812	0.112278
$649$	−7.03739	−0.276242
$650$	0.268401	0.0105276
$651$	0	0
$652$	−0.162989	−0.00638315
$653$	30.0519	1.17602	0.588012	−	0.808852i	$-0.299911\pi$
$653$	30.0519	1.17602	0.588012	+	0.808852i	$0.299911\pi$
$654$	23.6951	0.926550
$655$	26.9095	1.05144
$656$	−47.3793	−1.84985
$657$	−5.14349	−0.200667
$658$	0	0
$659$	14.0423	0.547009	0.273504	−	0.961871i	$-0.411817\pi$
$659$	14.0423	0.547009	0.273504	+	0.961871i	$0.411817\pi$
$660$	−0.609912	−0.0237408
$661$	−24.6070	−0.957102	−0.478551	−	0.878060i	$-0.658838\pi$
$661$	−24.6070	−0.957102	−0.478551	+	0.878060i	$0.658838\pi$
$662$	−34.2038	−1.32937
$663$	−4.33149	−0.168221
$664$	48.8280	1.89490
$665$	0	0
$666$	10.4443	0.404708
$667$	−14.9225	−0.577801
$668$	−0.919955	−0.0355941
$669$	−15.3809	−0.594660
$670$	−18.7971	−0.726195
$671$	−27.8730	−1.07602
$672$	0	0
$673$	−2.02401	−0.0780197	−0.0390098	−	0.999239i	$-0.512420\pi$
$673$	−2.02401	−0.0780197	−0.0390098	+	0.999239i	$0.512420\pi$
$674$	−6.02990	−0.232263
$675$	−0.245206	−0.00943800
$676$	0.534685	0.0205648
$677$	27.5915	1.06043	0.530214	−	0.847864i	$-0.322112\pi$
$677$	27.5915	1.06043	0.530214	+	0.847864i	$0.322112\pi$
$678$	−8.23967	−0.316443
$679$	0	0
$680$	−34.4991	−1.32298
$681$	−14.0751	−0.539360
$682$	−19.4175	−0.743534
$683$	−3.41209	−0.130560	−0.0652799	−	0.997867i	$-0.520794\pi$
$683$	−3.41209	−0.130560	−0.0652799	+	0.997867i	$0.520794\pi$
$684$	0.00606761	0.000232001 0
$685$	−16.3883	−0.626165
$686$	0	0
$687$	−4.31492	−0.164625
$688$	34.2556	1.30598
$689$	2.77898	0.105871
$690$	−18.7188	−0.712613
$691$	−25.0397	−0.952556	−0.476278	−	0.879295i	$-0.658014\pi$
$691$	−25.0397	−0.952556	−0.476278	+	0.879295i	$0.658014\pi$
$692$	−0.963351	−0.0366211
$693$	0	0
$694$	−36.0059	−1.36677
$695$	3.77690	0.143266
$696$	−6.95003	−0.263440
$697$	−67.0459	−2.53954
$698$	−48.0919	−1.82031
$699$	−16.3554	−0.618618
$700$	0	0
$701$	−4.99223	−0.188554	−0.0942769	−	0.995546i	$-0.530054\pi$
$701$	−4.99223	−0.188554	−0.0942769	+	0.995546i	$0.530054\pi$
$702$	−1.09459	−0.0413128
$703$	1.04957	0.0395854
$704$	52.9123	1.99421
$705$	20.4824	0.771413
$706$	13.4799	0.507321
$707$	0	0
$708$	0.0468732	0.00176160
$709$	−23.3049	−0.875234	−0.437617	−	0.899162i	$-0.644177\pi$
$709$	−23.3049	−0.875234	−0.437617	+	0.899162i	$0.644177\pi$
$710$	−16.8647	−0.632920
$711$	13.4125	0.503007
$712$	−12.9872	−0.486715
$713$	13.1449	0.492279
$714$	0	0
$715$	11.0570	0.413507
$716$	0.576979	0.0215627
$717$	−11.0575	−0.412948
$718$	41.7642	1.55863
$719$	8.05817	0.300519	0.150260	−	0.988647i	$-0.451989\pi$
$719$	8.05817	0.300519	0.150260	+	0.988647i	$0.451989\pi$
$720$	−8.52990	−0.317891
$721$	0	0
$722$	26.5509	0.988122
$723$	−20.6777	−0.769011
$724$	0.00688074	0.000255721 0
$725$	0.596263	0.0221447
$726$	−43.3568	−1.60912
$727$	1.05207	0.0390191	0.0195095	−	0.999810i	$-0.493790\pi$
$727$	1.05207	0.0390191	0.0195095	+	0.999810i	$0.493790\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	15.6892	0.580684
$731$	48.4748	1.79291
$732$	0.185650	0.00686183
$733$	36.5748	1.35092	0.675460	−	0.737397i	$-0.263945\pi$
$733$	36.5748	1.35092	0.675460	+	0.737397i	$0.263945\pi$
$734$	28.9532	1.06868
$735$	0	0
$736$	−1.49810	−0.0552207
$737$	39.9339	1.47098
$738$	−16.9429	−0.623677
$739$	37.5524	1.38139	0.690694	−	0.723147i	$-0.257305\pi$
$739$	37.5524	1.38139	0.690694	+	0.723147i	$0.257305\pi$
$740$	0.702707	0.0258320
$741$	−0.109998	−0.00404089
$742$	0	0
$743$	−46.4337	−1.70349	−0.851744	−	0.523958i	$-0.824455\pi$
$743$	−46.4337	−1.70349	−0.851744	+	0.523958i	$0.824455\pi$
$744$	6.12211	0.224447
$745$	−17.6436	−0.646411
$746$	44.0784	1.61383
$747$	17.0840	0.625070
$748$	1.54833	0.0566125
$749$	0	0
$750$	15.9995	0.584218
$751$	−7.12646	−0.260048	−0.130024	−	0.991511i	$-0.541505\pi$
$751$	−7.12646	−0.260048	−0.130024	+	0.991511i	$0.541505\pi$
$752$	−36.7446	−1.33994
$753$	24.6124	0.896926
$754$	2.66170	0.0969334
$755$	−11.3317	−0.412403
$756$	0	0
$757$	−40.8034	−1.48302	−0.741512	−	0.670940i	$-0.765891\pi$
$757$	−40.8034	−1.48302	−0.741512	+	0.670940i	$0.765891\pi$
$758$	−38.4259	−1.39569
$759$	39.7676	1.44347
$760$	−0.876107	−0.0317797
$761$	−5.37874	−0.194979	−0.0974896	−	0.995237i	$-0.531081\pi$
$761$	−5.37874	−0.194979	−0.0974896	+	0.995237i	$0.531081\pi$
$762$	−4.11133	−0.148938
$763$	0	0
$764$	1.01976	0.0368935
$765$	−12.0706	−0.436412
$766$	−33.0088	−1.19266
$767$	−0.849753	−0.0306828
$768$	−1.03542	−0.0373623
$769$	16.7793	0.605077	0.302539	−	0.953137i	$-0.402166\pi$
$769$	16.7793	0.605077	0.302539	+	0.953137i	$0.402166\pi$
$770$	0	0
$771$	−13.4398	−0.484023
$772$	0.00547552	0.000197068 0
$773$	39.5951	1.42414	0.712069	−	0.702110i	$-0.247758\pi$
$773$	39.5951	1.42414	0.712069	+	0.702110i	$0.247758\pi$
$774$	12.2499	0.440313
$775$	−0.525234	−0.0188670
$776$	−18.6148	−0.668234
$777$	0	0
$778$	30.1868	1.08225
$779$	−1.70263	−0.0610032
$780$	−0.0736459	−0.00263694
$781$	35.8285	1.28205
$782$	47.5198	1.69930
$783$	−2.43168	−0.0869011
$784$	0	0
$785$	3.39865	0.121303
$786$	−17.2631	−0.615753
$787$	34.8958	1.24390	0.621950	−	0.783057i	$-0.286341\pi$
$787$	34.8958	1.24390	0.621950	+	0.783057i	$0.286341\pi$
$788$	−0.898026	−0.0319909
$789$	11.2887	0.401887
$790$	−40.9121	−1.45559
$791$	0	0
$792$	18.5214	0.658131
$793$	−3.36561	−0.119516
$794$	35.3122	1.25318
$795$	7.74418	0.274658
$796$	0.467128	0.0165569
$797$	26.6968	0.945649	0.472825	−	0.881157i	$-0.343234\pi$
$797$	26.6968	0.945649	0.472825	+	0.881157i	$0.343234\pi$
$798$	0	0
$799$	−51.9969	−1.83952
$800$	0.0598601	0.00211637
$801$	−4.54396	−0.160553
$802$	−34.9476	−1.23404
$803$	−33.3313	−1.17624
$804$	−0.265983	−0.00938051
$805$	0	0
$806$	−2.34463	−0.0825860
$807$	−14.0668	−0.495174
$808$	−8.38774	−0.295080
$809$	−14.6103	−0.513673	−0.256836	−	0.966455i	$-0.582680\pi$
$809$	−14.6103	−0.513673	−0.256836	+	0.966455i	$0.582680\pi$
$810$	−3.05030	−0.107177
$811$	−24.5472	−0.861968	−0.430984	−	0.902360i	$-0.641833\pi$
$811$	−24.5472	−0.861968	−0.430984	+	0.902360i	$0.641833\pi$
$812$	0	0
$813$	23.1738	0.812740
$814$	67.6820	2.37225
$815$	8.23412	0.288429
$816$	21.6541	0.758045
$817$	1.23102	0.0430679
$818$	−3.27549	−0.114525
$819$	0	0
$820$	−1.13994	−0.0398085
$821$	1.98558	0.0692971	0.0346485	−	0.999400i	$-0.488969\pi$
$821$	1.98558	0.0692971	0.0346485	+	0.999400i	$0.488969\pi$
$822$	10.5135	0.366699
$823$	18.5909	0.648040	0.324020	−	0.946050i	$-0.394966\pi$
$823$	18.5909	0.648040	0.324020	+	0.946050i	$0.394966\pi$
$824$	31.0133	1.08040
$825$	−1.58901	−0.0553222
$826$	0	0
$827$	−14.5988	−0.507649	−0.253824	−	0.967250i	$-0.581689\pi$
$827$	−14.5988	−0.507649	−0.253824	+	0.967250i	$0.581689\pi$
$828$	−0.264876	−0.00920507
$829$	20.7737	0.721502	0.360751	−	0.932662i	$-0.382520\pi$
$829$	20.7737	0.721502	0.360751	+	0.932662i	$0.382520\pi$
$830$	−52.1113	−1.80881
$831$	−4.33550	−0.150397
$832$	6.38907	0.221501
$833$	0	0
$834$	−2.42296	−0.0839003
$835$	46.4756	1.60836
$836$	0.0393199	0.00135991
$837$	2.14201	0.0740386
$838$	46.6066	1.61000
$839$	−10.4581	−0.361053	−0.180526	−	0.983570i	$-0.557780\pi$
$839$	−10.4581	−0.361053	−0.180526	+	0.983570i	$0.557780\pi$
$840$	0	0
$841$	−23.0869	−0.796101
$842$	−3.19410	−0.110076
$843$	24.4371	0.841660
$844$	0.174607	0.00601022
$845$	−27.0120	−0.929242
$846$	−13.1399	−0.451760
$847$	0	0
$848$	−13.8927	−0.477078
$849$	20.7126	0.710855
$850$	−1.89876	−0.0651271
$851$	−45.8180	−1.57062
$852$	−0.238639	−0.00817564
$853$	−18.9134	−0.647582	−0.323791	−	0.946129i	$-0.604957\pi$
$853$	−18.9134	−0.647582	−0.323791	+	0.946129i	$0.604957\pi$
$854$	0	0
$855$	−0.306533	−0.0104832
$856$	9.86646	0.337229
$857$	29.5970	1.01101	0.505506	−	0.862823i	$-0.331306\pi$
$857$	29.5970	1.01101	0.505506	+	0.862823i	$0.331306\pi$
$858$	−7.09329	−0.242161
$859$	33.7465	1.15142	0.575709	−	0.817655i	$-0.304726\pi$
$859$	33.7465	1.15142	0.575709	+	0.817655i	$0.304726\pi$
$860$	0.824189	0.0281046
$861$	0	0
$862$	−21.7776	−0.741748
$863$	1.93942	0.0660187	0.0330093	−	0.999455i	$-0.489491\pi$
$863$	1.93942	0.0660187	0.0330093	+	0.999455i	$0.489491\pi$
$864$	−0.244121	−0.00830517
$865$	48.6680	1.65476
$866$	33.0149	1.12189
$867$	13.6425	0.463323
$868$	0	0
$869$	86.9167	2.94845
$870$	7.41736	0.251472
$871$	4.82195	0.163386
$872$	−48.4129	−1.63947
$873$	−6.51297	−0.220431
$874$	1.20677	0.0408195
$875$	0	0
$876$	0.222006	0.00750090
$877$	11.8809	0.401190	0.200595	−	0.979674i	$-0.435712\pi$
$877$	11.8809	0.401190	0.200595	+	0.979674i	$0.435712\pi$
$878$	−28.0120	−0.945361
$879$	12.8577	0.433680
$880$	−55.2762	−1.86336
$881$	−14.6006	−0.491906	−0.245953	−	0.969282i	$-0.579101\pi$
$881$	−14.6006	−0.491906	−0.245953	+	0.969282i	$0.579101\pi$
$882$	0	0
$883$	−23.0586	−0.775984	−0.387992	−	0.921663i	$-0.626831\pi$
$883$	−23.0586	−0.775984	−0.387992	+	0.921663i	$0.626831\pi$
$884$	0.186958	0.00628808
$885$	−2.36801	−0.0795997
$886$	8.16967	0.274466
$887$	−5.55359	−0.186471	−0.0932357	−	0.995644i	$-0.529721\pi$
$887$	−5.55359	−0.186471	−0.0932357	+	0.995644i	$0.529721\pi$
$888$	−21.3394	−0.716102
$889$	0	0
$890$	13.8605	0.464604
$891$	6.48029	0.217098
$892$	0.663879	0.0222283
$893$	−1.32046	−0.0441876
$894$	11.3188	0.378556
$895$	−29.1487	−0.974333
$896$	0	0
$897$	4.80187	0.160330
$898$	−5.51211	−0.183942
$899$	−5.20867	−0.173719
$900$	0.0105837	0.000352791 0
$901$	−19.6595	−0.654951
$902$	−109.795	−3.65577
$903$	0	0
$904$	16.8350	0.559923
$905$	−0.347612	−0.0115550
$906$	7.26954	0.241514
$907$	−21.1469	−0.702172	−0.351086	−	0.936343i	$-0.614188\pi$
$907$	−21.1469	−0.702172	−0.351086	+	0.936343i	$0.614188\pi$
$908$	0.607518	0.0201612
$909$	−2.93471	−0.0973381
$910$	0	0
$911$	33.7471	1.11809	0.559045	−	0.829137i	$-0.311168\pi$
$911$	33.7471	1.11809	0.559045	+	0.829137i	$0.311168\pi$
$912$	0.549906	0.0182092
$913$	110.709	3.66394
$914$	−10.2511	−0.339077
$915$	−9.37895	−0.310059
$916$	0.186243	0.00615364
$917$	0	0
$918$	7.74353	0.255575
$919$	17.9477	0.592038	0.296019	−	0.955182i	$-0.404341\pi$
$919$	17.9477	0.592038	0.296019	+	0.955182i	$0.404341\pi$
$920$	38.2456	1.26092
$921$	17.1879	0.566360
$922$	44.2431	1.45707
$923$	4.32624	0.142400
$924$	0	0
$925$	1.83077	0.0601953
$926$	−22.7200	−0.746627
$927$	10.8509	0.356392
$928$	0.593625	0.0194867
$929$	16.3051	0.534952	0.267476	−	0.963565i	$-0.413810\pi$
$929$	16.3051	0.534952	0.267476	+	0.963565i	$0.413810\pi$
$930$	−6.53377	−0.214251
$931$	0	0
$932$	0.705940	0.0231238
$933$	−14.7890	−0.484170
$934$	31.8112	1.04089
$935$	−78.2208	−2.55809
$936$	2.23643	0.0731001
$937$	−13.5618	−0.443044	−0.221522	−	0.975155i	$-0.571102\pi$
$937$	−13.5618	−0.443044	−0.221522	+	0.975155i	$0.571102\pi$
$938$	0	0
$939$	31.7838	1.03723
$940$	−0.884074	−0.0288353
$941$	−5.78287	−0.188516	−0.0942580	−	0.995548i	$-0.530048\pi$
$941$	−5.78287	−0.188516	−0.0942580	+	0.995548i	$0.530048\pi$
$942$	−2.18031	−0.0710382
$943$	74.3268	2.42041
$944$	4.24810	0.138264
$945$	0	0
$946$	79.3827	2.58096
$947$	1.05397	0.0342493	0.0171246	−	0.999853i	$-0.494549\pi$
$947$	1.05397	0.0342493	0.0171246	+	0.999853i	$0.494549\pi$
$948$	−0.578916	−0.0188023
$949$	−4.02470	−0.130647
$950$	−0.0482192	−0.00156444
$951$	−6.52577	−0.211612
$952$	0	0
$953$	9.16338	0.296831	0.148416	−	0.988925i	$-0.452583\pi$
$953$	9.16338	0.296831	0.148416	+	0.988925i	$0.452583\pi$
$954$	−4.96806	−0.160847
$955$	−51.5176	−1.66707
$956$	0.477268	0.0154359
$957$	−15.7580	−0.509383
$958$	−33.2547	−1.07441
$959$	0	0
$960$	17.8044	0.574636
$961$	−26.4118	−0.851994
$962$	8.17249	0.263492
$963$	3.45208	0.111242
$964$	0.892501	0.0287455
$965$	−0.276620	−0.00890473
$966$	0	0
$967$	−16.5819	−0.533238	−0.266619	−	0.963802i	$-0.585907\pi$
$967$	−16.5819	−0.533238	−0.266619	+	0.963802i	$0.585907\pi$
$968$	88.5851	2.84723
$969$	0.778167	0.0249983
$970$	19.8665	0.637876
$971$	−9.36097	−0.300408	−0.150204	−	0.988655i	$-0.547993\pi$
$971$	−9.36097	−0.300408	−0.150204	+	0.988655i	$0.547993\pi$
$972$	−0.0431625	−0.00138444
$973$	0	0
$974$	−14.7668	−0.473159
$975$	−0.191870	−0.00614476
$976$	16.8254	0.538569
$977$	17.7953	0.569322	0.284661	−	0.958628i	$-0.408119\pi$
$977$	17.7953	0.569322	0.284661	+	0.958628i	$0.408119\pi$
$978$	−5.28237	−0.168912
$979$	−29.4462	−0.941104
$980$	0	0
$981$	−16.9387	−0.540812
$982$	16.8198	0.536741
$983$	12.3653	0.394392	0.197196	−	0.980364i	$-0.436816\pi$
$983$	12.3653	0.394392	0.197196	+	0.980364i	$0.436816\pi$
$984$	34.6171	1.10355
$985$	45.3678	1.44554
$986$	−18.8298	−0.599663
$987$	0	0
$988$	0.00474781	0.000151048 0
$989$	−53.7390	−1.70880
$990$	−19.7669	−0.628232
$991$	51.0709	1.62232	0.811161	−	0.584823i	$-0.198836\pi$
$991$	51.0709	1.62232	0.811161	+	0.584823i	$0.198836\pi$
$992$	−0.522909	−0.0166024
$993$	24.4510	0.775929
$994$	0	0
$995$	−23.5991	−0.748141
$996$	−0.737388	−0.0233650
$997$	−0.725343	−0.0229718	−0.0114859	−	0.999934i	$-0.503656\pi$
$997$	−0.725343	−0.0229718	−0.0114859	+	0.999934i	$0.503656\pi$
$998$	−13.9423	−0.441335
$999$	−7.46623	−0.236221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7203.2.a.d.1.4	yes	12
7.6	odd	2		7203.2.a.c.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7203.2.a.c.1.4	✓	12	7.6	odd	2
7203.2.a.d.1.4	yes	12	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:	\( N \)	\(=\)	\( 7203 = 3 \cdot 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7203.a (trivial)

\(f(q)\)	\(=\)	\(q-1.39887 q^{2} +1.00000 q^{3} -0.0431625 q^{4} +2.18055 q^{5} -1.39887 q^{6} +2.85812 q^{8} +1.00000 q^{9} -3.05030 q^{10} +6.48029 q^{11} -0.0431625 q^{12} +0.782484 q^{13} +2.18055 q^{15} -3.91181 q^{16} -5.53556 q^{17} -1.39887 q^{18} -0.140576 q^{19} -0.0941180 q^{20} -9.06509 q^{22} +6.13670 q^{23} +2.85812 q^{24} -0.245206 q^{25} -1.09459 q^{26} +1.00000 q^{27} -2.43168 q^{29} -3.05030 q^{30} +2.14201 q^{31} -0.244121 q^{32} +6.48029 q^{33} +7.74353 q^{34} -0.0431625 q^{36} -7.46623 q^{37} +0.196647 q^{38} +0.782484 q^{39} +6.23227 q^{40} +12.1118 q^{41} -8.75698 q^{43} -0.279706 q^{44} +2.18055 q^{45} -8.58445 q^{46} +9.39324 q^{47} -3.91181 q^{48} +0.343012 q^{50} -5.53556 q^{51} -0.0337740 q^{52} +3.55148 q^{53} -1.39887 q^{54} +14.1306 q^{55} -0.140576 q^{57} +3.40160 q^{58} -1.08597 q^{59} -0.0941180 q^{60} -4.30119 q^{61} -2.99639 q^{62} +8.16512 q^{64} +1.70624 q^{65} -9.06509 q^{66} +6.16236 q^{67} +0.238929 q^{68} +6.13670 q^{69} +5.52885 q^{71} +2.85812 q^{72} -5.14349 q^{73} +10.4443 q^{74} -0.245206 q^{75} +0.00606761 q^{76} -1.09459 q^{78} +13.4125 q^{79} -8.52990 q^{80} +1.00000 q^{81} -16.9429 q^{82} +17.0840 q^{83} -12.0706 q^{85} +12.2499 q^{86} -2.43168 q^{87} +18.5214 q^{88} -4.54396 q^{89} -3.05030 q^{90} -0.264876 q^{92} +2.14201 q^{93} -13.1399 q^{94} -0.306533 q^{95} -0.244121 q^{96} -6.51297 q^{97} +6.48029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} + 12 q^{3} + 12 q^{4} + 3 q^{5} + 2 q^{6} + 6 q^{8} + 12 q^{9} - 3 q^{11} + 12 q^{12} + 13 q^{13} + 3 q^{15} + 8 q^{16} + 3 q^{17} + 2 q^{18} - 3 q^{20} - 11 q^{22} + 18 q^{23} + 6 q^{24}+ \cdots - 3 q^{99}+O(q^{100}) \) 12 * q + 2 * q^2 + 12 * q^3 + 12 * q^4 + 3 * q^5 + 2 * q^6 + 6 * q^8 + 12 * q^9 - 3 * q^11 + 12 * q^12 + 13 * q^13 + 3 * q^15 + 8 * q^16 + 3 * q^17 + 2 * q^18 - 3 * q^20 - 11 * q^22 + 18 * q^23 + 6 * q^24 + 5 * q^25 + 26 * q^26 + 12 * q^27 - 13 * q^29 - 6 * q^31 - 6 * q^32 - 3 * q^33 + 2 * q^34 + 12 * q^36 + 15 * q^37 + 43 * q^38 + 13 * q^39 - 6 * q^40 + 38 * q^41 + 21 * q^43 - 38 * q^44 + 3 * q^45 - 16 * q^46 + 28 * q^47 + 8 * q^48 + 7 * q^50 + 3 * q^51 + 40 * q^52 + 11 * q^53 + 2 * q^54 + 50 * q^55 + 42 * q^58 - 6 * q^59 - 3 * q^60 + 26 * q^61 + 47 * q^62 + 28 * q^64 - 5 * q^65 - 11 * q^66 + 14 * q^67 - 14 * q^68 + 18 * q^69 - 31 * q^71 + 6 * q^72 + 48 * q^73 + 20 * q^74 + 5 * q^75 - 9 * q^76 + 26 * q^78 + 5 * q^79 - 47 * q^80 + 12 * q^81 - 29 * q^82 + 57 * q^83 + 4 * q^85 + 82 * q^86 - 13 * q^87 + 30 * q^88 - 15 * q^89 + 35 * q^92 - 6 * q^93 + 50 * q^94 - 55 * q^95 - 6 * q^96 + 6 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more