LMFDB - Embedded newform 7865.2.a.bg.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7865,2,Mod(1,7865)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7865, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("7865.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7865 = 5 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7865.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,-2,-4,16,18,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.8023411897$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} - 24 x^{16} + 48 x^{15} + 230 x^{14} - 466 x^{13} - 1116 x^{12} + 2346 x^{11} + \cdots + 13 $ x^18 - 2x^17 - 24x^16 + 48x^15 + 230x^14 - 466x^13 - 1116x^12 + 2346x^11 + 2845x^10 - 6506x^9 - 3493x^8 + 9722x^7 + 1412x^6 - 7174x^5 + 302x^4 + 2336x^3 - 232x^2 - 244*x + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-0.379916$ of defining polynomial
Character	$\chi$	$=$	7865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.379916 q^{2} +2.63503 q^{3} -1.85566 q^{4} +1.00000 q^{5} +1.00109 q^{6} -2.79558 q^{7} -1.46483 q^{8} +3.94337 q^{9} +0.379916 q^{10} -4.88972 q^{12} +1.00000 q^{13} -1.06209 q^{14} +2.63503 q^{15} +3.15482 q^{16} -0.676249 q^{17} +1.49815 q^{18} -6.56466 q^{19} -1.85566 q^{20} -7.36644 q^{21} +7.06520 q^{23} -3.85986 q^{24} +1.00000 q^{25} +0.379916 q^{26} +2.48580 q^{27} +5.18766 q^{28} -3.29561 q^{29} +1.00109 q^{30} -7.21879 q^{31} +4.12822 q^{32} -0.256918 q^{34} -2.79558 q^{35} -7.31756 q^{36} +11.4594 q^{37} -2.49402 q^{38} +2.63503 q^{39} -1.46483 q^{40} +0.830856 q^{41} -2.79863 q^{42} +1.65091 q^{43} +3.94337 q^{45} +2.68418 q^{46} -12.3173 q^{47} +8.31302 q^{48} +0.815286 q^{49} +0.379916 q^{50} -1.78193 q^{51} -1.85566 q^{52} -9.52603 q^{53} +0.944395 q^{54} +4.09505 q^{56} -17.2980 q^{57} -1.25206 q^{58} +9.39476 q^{59} -4.88972 q^{60} -11.4509 q^{61} -2.74254 q^{62} -11.0240 q^{63} -4.74125 q^{64} +1.00000 q^{65} +1.29322 q^{67} +1.25489 q^{68} +18.6170 q^{69} -1.06209 q^{70} +6.13995 q^{71} -5.77636 q^{72} -12.5125 q^{73} +4.35362 q^{74} +2.63503 q^{75} +12.1818 q^{76} +1.00109 q^{78} +2.74002 q^{79} +3.15482 q^{80} -5.27996 q^{81} +0.315656 q^{82} +15.2622 q^{83} +13.6696 q^{84} -0.676249 q^{85} +0.627207 q^{86} -8.68403 q^{87} -4.93482 q^{89} +1.49815 q^{90} -2.79558 q^{91} -13.1106 q^{92} -19.0217 q^{93} -4.67954 q^{94} -6.56466 q^{95} +10.8780 q^{96} -8.71138 q^{97} +0.309740 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{2} - 4 q^{3} + 16 q^{4} + 18 q^{5} - 2 q^{6} - 20 q^{7} + 10 q^{9} - 2 q^{10} - 20 q^{12} + 18 q^{13} + 10 q^{14} - 4 q^{15} + 8 q^{16} - 6 q^{17} - 38 q^{19} + 16 q^{20} - 22 q^{21} - 2 q^{23}+ \cdots - 36 q^{98}+O(q^{100}) $ 18 * q - 2 * q^2 - 4 * q^3 + 16 * q^4 + 18 * q^5 - 2 * q^6 - 20 * q^7 + 10 * q^9 - 2 * q^10 - 20 * q^12 + 18 * q^13 + 10 * q^14 - 4 * q^15 + 8 * q^16 - 6 * q^17 - 38 * q^19 + 16 * q^20 - 22 * q^21 - 2 * q^23 - 16 * q^24 + 18 * q^25 - 2 * q^26 - 10 * q^27 - 10 * q^28 - 20 * q^29 - 2 * q^30 - 8 * q^31 - 22 * q^32 - 10 * q^34 - 20 * q^35 + 12 * q^36 + 8 * q^37 - 4 * q^39 - 12 * q^41 + 2 * q^42 - 20 * q^43 + 10 * q^45 - 58 * q^46 - 28 * q^47 - 12 * q^48 + 16 * q^49 - 2 * q^50 - 28 * q^51 + 16 * q^52 - 20 * q^53 - 44 * q^54 + 18 * q^57 + 22 * q^58 + 2 * q^59 - 20 * q^60 - 32 * q^61 - 22 * q^62 + 2 * q^63 - 16 * q^64 + 18 * q^65 + 12 * q^67 - 36 * q^68 + 24 * q^69 + 10 * q^70 + 16 * q^71 + 30 * q^72 - 28 * q^73 - 4 * q^74 - 4 * q^75 - 114 * q^76 - 2 * q^78 - 56 * q^79 + 8 * q^80 - 38 * q^81 + 2 * q^82 + 4 * q^83 - 24 * q^84 - 6 * q^85 - 24 * q^86 - 50 * q^87 + 26 * q^89 - 20 * q^91 - 28 * q^92 + 40 * q^93 - 40 * q^94 - 38 * q^95 + 56 * q^96 - 8 * q^97 - 36 * 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 $

$2$	0.379916	0.268641	0.134321	−	0.990938i	$-0.457115\pi$
$2$	0.379916	0.268641	0.134321	+	0.990938i	$0.457115\pi$
$3$	2.63503	1.52133	0.760667	−	0.649143i	$-0.224872\pi$
$3$	2.63503	1.52133	0.760667	+	0.649143i	$0.224872\pi$
$4$	−1.85566	−0.927832
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00109	0.408693
$7$	−2.79558	−1.05663	−0.528316	−	0.849048i	$-0.677176\pi$
$7$	−2.79558	−1.05663	−0.528316	+	0.849048i	$0.677176\pi$
$8$	−1.46483	−0.517895
$9$	3.94337	1.31446
$10$	0.379916	0.120140
$11$	0	0
$11$	0	0
$12$	−4.88972	−1.41154
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.06209	−0.283855
$15$	2.63503	0.680361
$16$	3.15482	0.788704
$17$	−0.676249	−0.164015	−0.0820073	−	0.996632i	$-0.526133\pi$
$17$	−0.676249	−0.164015	−0.0820073	+	0.996632i	$0.526133\pi$
$18$	1.49815	0.353117
$19$	−6.56466	−1.50604	−0.753018	−	0.658000i	$-0.771403\pi$
$19$	−6.56466	−1.50604	−0.753018	+	0.658000i	$0.771403\pi$
$20$	−1.85566	−0.414939
$21$	−7.36644	−1.60749
$22$	0	0
$23$	7.06520	1.47320	0.736598	−	0.676330i	$-0.236431\pi$
$23$	7.06520	1.47320	0.736598	+	0.676330i	$0.236431\pi$
$24$	−3.85986	−0.787891
$25$	1.00000	0.200000
$26$	0.379916	0.0745077
$27$	2.48580	0.478392
$28$	5.18766	0.980376
$29$	−3.29561	−0.611980	−0.305990	−	0.952035i	$-0.598987\pi$
$29$	−3.29561	−0.611980	−0.305990	+	0.952035i	$0.598987\pi$
$30$	1.00109	0.182773
$31$	−7.21879	−1.29653	−0.648267	−	0.761413i	$-0.724506\pi$
$31$	−7.21879	−1.29653	−0.648267	+	0.761413i	$0.724506\pi$
$32$	4.12822	0.729773
$33$	0	0
$34$	−0.256918	−0.0440611
$35$	−2.79558	−0.472540
$36$	−7.31756	−1.21959
$37$	11.4594	1.88392	0.941959	−	0.335729i	$-0.108983\pi$
$37$	11.4594	1.88392	0.941959	+	0.335729i	$0.108983\pi$
$38$	−2.49402	−0.404583
$39$	2.63503	0.421942
$40$	−1.46483	−0.231610
$41$	0.830856	0.129758	0.0648790	−	0.997893i	$-0.479334\pi$
$41$	0.830856	0.129758	0.0648790	+	0.997893i	$0.479334\pi$
$42$	−2.79863	−0.431838
$43$	1.65091	0.251761	0.125881	−	0.992045i	$-0.459824\pi$
$43$	1.65091	0.251761	0.125881	+	0.992045i	$0.459824\pi$
$44$	0	0
$45$	3.94337	0.587842
$46$	2.68418	0.395761
$47$	−12.3173	−1.79666	−0.898331	−	0.439320i	$-0.855220\pi$
$47$	−12.3173	−1.79666	−0.898331	+	0.439320i	$0.855220\pi$
$48$	8.31302	1.19988
$49$	0.815286	0.116469
$50$	0.379916	0.0537282
$51$	−1.78193	−0.249521
$52$	−1.85566	−0.257334
$53$	−9.52603	−1.30850	−0.654250	−	0.756278i	$-0.727016\pi$
$53$	−9.52603	−1.30850	−0.654250	+	0.756278i	$0.727016\pi$
$54$	0.944395	0.128516
$55$	0	0
$56$	4.09505	0.547224
$57$	−17.2980	−2.29118
$58$	−1.25206	−0.164403
$59$	9.39476	1.22309	0.611547	−	0.791208i	$-0.290548\pi$
$59$	9.39476	1.22309	0.611547	+	0.791208i	$0.290548\pi$
$60$	−4.88972	−0.631261
$61$	−11.4509	−1.46614	−0.733071	−	0.680152i	$-0.761914\pi$
$61$	−11.4509	−1.46614	−0.733071	+	0.680152i	$0.761914\pi$
$62$	−2.74254	−0.348302
$63$	−11.0240	−1.38889
$64$	−4.74125	−0.592657
$65$	1.00000	0.124035
$66$	0	0
$67$	1.29322	0.157991	0.0789957	−	0.996875i	$-0.474829\pi$
$67$	1.29322	0.157991	0.0789957	+	0.996875i	$0.474829\pi$
$68$	1.25489	0.152178
$69$	18.6170	2.24122
$70$	−1.06209	−0.126944
$71$	6.13995	0.728678	0.364339	−	0.931266i	$-0.381295\pi$
$71$	6.13995	0.728678	0.364339	+	0.931266i	$0.381295\pi$
$72$	−5.77636	−0.680750
$73$	−12.5125	−1.46448	−0.732239	−	0.681047i	$-0.761525\pi$
$73$	−12.5125	−1.46448	−0.732239	+	0.681047i	$0.761525\pi$
$74$	4.35362	0.506098
$75$	2.63503	0.304267
$76$	12.1818	1.39735
$77$	0	0
$78$	1.00109	0.113351
$79$	2.74002	0.308276	0.154138	−	0.988049i	$-0.450740\pi$
$79$	2.74002	0.308276	0.154138	+	0.988049i	$0.450740\pi$
$80$	3.15482	0.352719
$81$	−5.27996	−0.586662
$82$	0.315656	0.0348583
$83$	15.2622	1.67524	0.837620	−	0.546254i	$-0.183947\pi$
$83$	15.2622	1.67524	0.837620	+	0.546254i	$0.183947\pi$
$84$	13.6696	1.49148
$85$	−0.676249	−0.0733495
$86$	0.627207	0.0676334
$87$	−8.68403	−0.931025
$88$	0	0
$89$	−4.93482	−0.523090	−0.261545	−	0.965191i	$-0.584232\pi$
$89$	−4.93482	−0.523090	−0.261545	+	0.965191i	$0.584232\pi$
$90$	1.49815	0.157919
$91$	−2.79558	−0.293057
$92$	−13.1106	−1.36688
$93$	−19.0217	−1.97246
$94$	−4.67954	−0.482657
$95$	−6.56466	−0.673520
$96$	10.8780	1.11023
$97$	−8.71138	−0.884507	−0.442253	−	0.896890i	$-0.645821\pi$
$97$	−8.71138	−0.884507	−0.442253	+	0.896890i	$0.645821\pi$
$98$	0.309740	0.0312885
$99$	0	0
$100$	−1.85566	−0.185566
$101$	−9.23797	−0.919212	−0.459606	−	0.888123i	$-0.652009\pi$
$101$	−9.23797	−0.919212	−0.459606	+	0.888123i	$0.652009\pi$
$102$	−0.676986	−0.0670316
$103$	−12.6179	−1.24328	−0.621639	−	0.783304i	$-0.713533\pi$
$103$	−12.6179	−1.24328	−0.621639	+	0.783304i	$0.713533\pi$
$104$	−1.46483	−0.143638
$105$	−7.36644	−0.718891
$106$	−3.61909	−0.351517
$107$	−17.4313	−1.68515	−0.842574	−	0.538580i	$-0.818961\pi$
$107$	−17.4313	−1.68515	−0.842574	+	0.538580i	$0.818961\pi$
$108$	−4.61281	−0.443867
$109$	4.99577	0.478508	0.239254	−	0.970957i	$-0.423097\pi$
$109$	4.99577	0.478508	0.239254	+	0.970957i	$0.423097\pi$
$110$	0	0
$111$	30.1959	2.86607
$112$	−8.81955	−0.833369
$113$	−17.4325	−1.63991	−0.819955	−	0.572428i	$-0.806002\pi$
$113$	−17.4325	−1.63991	−0.819955	+	0.572428i	$0.806002\pi$
$114$	−6.57181	−0.615506
$115$	7.06520	0.658834
$116$	6.11555	0.567814
$117$	3.94337	0.364564
$118$	3.56922	0.328573
$119$	1.89051	0.173303
$120$	−3.85986	−0.352356
$121$	0	0
$122$	−4.35039	−0.393866
$123$	2.18933	0.197405
$124$	13.3957	1.20296
$125$	1.00000	0.0894427
$126$	−4.18820	−0.373114
$127$	4.21549	0.374064	0.187032	−	0.982354i	$-0.440113\pi$
$127$	4.21549	0.374064	0.187032	+	0.982354i	$0.440113\pi$
$128$	−10.0577	−0.888986
$129$	4.35019	0.383013
$130$	0.379916	0.0333208
$131$	−1.15638	−0.101033	−0.0505167	−	0.998723i	$-0.516087\pi$
$131$	−1.15638	−0.101033	−0.0505167	+	0.998723i	$0.516087\pi$
$132$	0	0
$133$	18.3520	1.59132
$134$	0.491314	0.0424430
$135$	2.48580	0.213943
$136$	0.990589	0.0849423
$137$	−6.41338	−0.547932	−0.273966	−	0.961739i	$-0.588336\pi$
$137$	−6.41338	−0.547932	−0.273966	+	0.961739i	$0.588336\pi$
$138$	7.07290	0.602085
$139$	−0.774269	−0.0656726	−0.0328363	−	0.999461i	$-0.510454\pi$
$139$	−0.774269	−0.0656726	−0.0328363	+	0.999461i	$0.510454\pi$
$140$	5.18766	0.438438
$141$	−32.4564	−2.73332
$142$	2.33267	0.195753
$143$	0	0
$144$	12.4406	1.03672
$145$	−3.29561	−0.273686
$146$	−4.75370	−0.393419
$147$	2.14830	0.177189
$148$	−21.2648	−1.74796
$149$	7.70984	0.631615	0.315807	−	0.948823i	$-0.397725\pi$
$149$	7.70984	0.631615	0.315807	+	0.948823i	$0.397725\pi$
$150$	1.00109	0.0817386
$151$	−16.3851	−1.33340	−0.666699	−	0.745327i	$-0.732293\pi$
$151$	−16.3851	−1.33340	−0.666699	+	0.745327i	$0.732293\pi$
$152$	9.61610	0.779968
$153$	−2.66670	−0.215590
$154$	0	0
$155$	−7.21879	−0.579827
$156$	−4.88972	−0.391491
$157$	−17.0645	−1.36190	−0.680948	−	0.732332i	$-0.738432\pi$
$157$	−17.0645	−1.36190	−0.680948	+	0.732332i	$0.738432\pi$
$158$	1.04098	0.0828157
$159$	−25.1013	−1.99067
$160$	4.12822	0.326365
$161$	−19.7514	−1.55663
$162$	−2.00594	−0.157602
$163$	15.0436	1.17830	0.589152	−	0.808023i	$-0.299462\pi$
$163$	15.0436	1.17830	0.589152	+	0.808023i	$0.299462\pi$
$164$	−1.54179	−0.120394
$165$	0	0
$166$	5.79834	0.450038
$167$	−5.07446	−0.392673	−0.196337	−	0.980537i	$-0.562905\pi$
$167$	−5.07446	−0.392673	−0.196337	+	0.980537i	$0.562905\pi$
$168$	10.7906	0.832510
$169$	1.00000	0.0769231
$170$	−0.256918	−0.0197047
$171$	−25.8868	−1.97962
$172$	−3.06353	−0.233592
$173$	0.719542	0.0547058	0.0273529	−	0.999626i	$-0.491292\pi$
$173$	0.719542	0.0547058	0.0273529	+	0.999626i	$0.491292\pi$
$174$	−3.29920	−0.250112
$175$	−2.79558	−0.211326
$176$	0	0
$177$	24.7555	1.86073
$178$	−1.87482	−0.140523
$179$	9.73814	0.727863	0.363931	−	0.931426i	$-0.381434\pi$
$179$	9.73814	0.727863	0.363931	+	0.931426i	$0.381434\pi$
$180$	−7.31756	−0.545419
$181$	−0.447941	−0.0332952	−0.0166476	−	0.999861i	$-0.505299\pi$
$181$	−0.447941	−0.0332952	−0.0166476	+	0.999861i	$0.505299\pi$
$182$	−1.06209	−0.0787271
$183$	−30.1735	−2.23049
$184$	−10.3493	−0.762961
$185$	11.4594	0.842513
$186$	−7.22665	−0.529884
$187$	0	0
$188$	22.8568	1.66700
$189$	−6.94926	−0.505484
$190$	−2.49402	−0.180935
$191$	2.74882	0.198897	0.0994487	−	0.995043i	$-0.468292\pi$
$191$	2.74882	0.198897	0.0994487	+	0.995043i	$0.468292\pi$
$192$	−12.4933	−0.901628
$193$	18.9326	1.36280	0.681398	−	0.731913i	$-0.261372\pi$
$193$	18.9326	1.36280	0.681398	+	0.731913i	$0.261372\pi$
$194$	−3.30959	−0.237615
$195$	2.63503	0.188698
$196$	−1.51290	−0.108064
$197$	−19.3449	−1.37827	−0.689134	−	0.724634i	$-0.742009\pi$
$197$	−19.3449	−1.37827	−0.689134	+	0.724634i	$0.742009\pi$
$198$	0	0
$199$	9.23227	0.654458	0.327229	−	0.944945i	$-0.393885\pi$
$199$	9.23227	0.654458	0.327229	+	0.944945i	$0.393885\pi$
$200$	−1.46483	−0.103579
$201$	3.40766	0.240358
$202$	−3.50965	−0.246938
$203$	9.21316	0.646637
$204$	3.30667	0.231513
$205$	0.830856	0.0580295
$206$	−4.79374	−0.333996
$207$	27.8607	1.93645
$208$	3.15482	0.218747
$209$	0	0
$210$	−2.79863	−0.193124
$211$	−25.4086	−1.74920	−0.874601	−	0.484844i	$-0.838876\pi$
$211$	−25.4086	−1.74920	−0.874601	+	0.484844i	$0.838876\pi$
$212$	17.6771	1.21407
$213$	16.1789	1.10856
$214$	−6.62243	−0.452700
$215$	1.65091	0.112591
$216$	−3.64127	−0.247757
$217$	20.1807	1.36996
$218$	1.89797	0.128547
$219$	−32.9708	−2.22796
$220$	0	0
$221$	−0.676249	−0.0454894
$222$	11.4719	0.769944
$223$	6.66720	0.446469	0.223234	−	0.974765i	$-0.428339\pi$
$223$	6.66720	0.446469	0.223234	+	0.974765i	$0.428339\pi$
$224$	−11.5408	−0.771101
$225$	3.94337	0.262891
$226$	−6.62288	−0.440547
$227$	−4.30099	−0.285467	−0.142733	−	0.989761i	$-0.545589\pi$
$227$	−4.30099	−0.285467	−0.142733	+	0.989761i	$0.545589\pi$
$228$	32.0994	2.12583
$229$	17.7403	1.17231	0.586155	−	0.810199i	$-0.300641\pi$
$229$	17.7403	1.17231	0.586155	+	0.810199i	$0.300641\pi$
$230$	2.68418	0.176990
$231$	0	0
$232$	4.82751	0.316941
$233$	−1.38481	−0.0907221	−0.0453610	−	0.998971i	$-0.514444\pi$
$233$	−1.38481	−0.0907221	−0.0453610	+	0.998971i	$0.514444\pi$
$234$	1.49815	0.0979370
$235$	−12.3173	−0.803492
$236$	−17.4335	−1.13483
$237$	7.22003	0.468991
$238$	0.718235	0.0465563
$239$	9.92601	0.642060	0.321030	−	0.947069i	$-0.395971\pi$
$239$	9.92601	0.642060	0.321030	+	0.947069i	$0.395971\pi$
$240$	8.31302	0.536603
$241$	−7.23892	−0.466300	−0.233150	−	0.972441i	$-0.574903\pi$
$241$	−7.23892	−0.466300	−0.233150	+	0.972441i	$0.574903\pi$
$242$	0	0
$243$	−21.3702	−1.37090
$244$	21.2491	1.36033
$245$	0.815286	0.0520867
$246$	0.831761	0.0530311
$247$	−6.56466	−0.417699
$248$	10.5743	0.671468
$249$	40.2162	2.54860
$250$	0.379916	0.0240280
$251$	−17.2498	−1.08880	−0.544400	−	0.838826i	$-0.683243\pi$
$251$	−17.2498	−1.08880	−0.544400	+	0.838826i	$0.683243\pi$
$252$	20.4569	1.28866
$253$	0	0
$254$	1.60153	0.100489
$255$	−1.78193	−0.111589
$256$	5.66142	0.353839
$257$	10.8029	0.673863	0.336932	−	0.941529i	$-0.390611\pi$
$257$	10.8029	0.673863	0.336932	+	0.941529i	$0.390611\pi$
$258$	1.65271	0.102893
$259$	−32.0358	−1.99061
$260$	−1.85566	−0.115083
$261$	−12.9958	−0.804420
$262$	−0.439328	−0.0271417
$263$	−13.9304	−0.858986	−0.429493	−	0.903070i	$-0.641308\pi$
$263$	−13.9304	−0.858986	−0.429493	+	0.903070i	$0.641308\pi$
$264$	0	0
$265$	−9.52603	−0.585179
$266$	6.97224	0.427495
$267$	−13.0034	−0.795794
$268$	−2.39977	−0.146590
$269$	−4.96474	−0.302705	−0.151353	−	0.988480i	$-0.548363\pi$
$269$	−4.96474	−0.302705	−0.151353	+	0.988480i	$0.548363\pi$
$270$	0.944395	0.0574740
$271$	26.2536	1.59479	0.797396	−	0.603457i	$-0.206210\pi$
$271$	26.2536	1.59479	0.797396	+	0.603457i	$0.206210\pi$
$272$	−2.13344	−0.129359
$273$	−7.36644	−0.445837
$274$	−2.43654	−0.147197
$275$	0	0
$276$	−34.5469	−2.07948
$277$	−16.8047	−1.00970	−0.504848	−	0.863208i	$-0.668452\pi$
$277$	−16.8047	−1.00970	−0.504848	+	0.863208i	$0.668452\pi$
$278$	−0.294157	−0.0176424
$279$	−28.4663	−1.70424
$280$	4.09505	0.244726
$281$	10.4812	0.625253	0.312627	−	0.949876i	$-0.398791\pi$
$281$	10.4812	0.625253	0.312627	+	0.949876i	$0.398791\pi$
$282$	−12.3307	−0.734283
$283$	−9.72330	−0.577990	−0.288995	−	0.957331i	$-0.593321\pi$
$283$	−9.72330	−0.577990	−0.288995	+	0.957331i	$0.593321\pi$
$284$	−11.3937	−0.676091
$285$	−17.2980	−1.02465
$286$	0	0
$287$	−2.32273	−0.137106
$288$	16.2791	0.959255
$289$	−16.5427	−0.973099
$290$	−1.25206	−0.0735233
$291$	−22.9547	−1.34563
$292$	23.2190	1.35879
$293$	21.0853	1.23182	0.615908	−	0.787818i	$-0.288789\pi$
$293$	21.0853	1.23182	0.615908	+	0.787818i	$0.288789\pi$
$294$	0.816174	0.0476002
$295$	9.39476	0.546984
$296$	−16.7861	−0.975672
$297$	0	0
$298$	2.92909	0.169678
$299$	7.06520	0.408591
$300$	−4.88972	−0.282308
$301$	−4.61525	−0.266019
$302$	−6.22495	−0.358205
$303$	−24.3423	−1.39843
$304$	−20.7103	−1.18782
$305$	−11.4509	−0.655679
$306$	−1.01312	−0.0579163
$307$	14.6255	0.834719	0.417359	−	0.908742i	$-0.362956\pi$
$307$	14.6255	0.834719	0.417359	+	0.908742i	$0.362956\pi$
$308$	0	0
$309$	−33.2485	−1.89144
$310$	−2.74254	−0.155766
$311$	8.01582	0.454535	0.227268	−	0.973832i	$-0.427021\pi$
$311$	8.01582	0.454535	0.227268	+	0.973832i	$0.427021\pi$
$312$	−3.85986	−0.218522
$313$	5.29875	0.299503	0.149751	−	0.988724i	$-0.452153\pi$
$313$	5.29875	0.299503	0.149751	+	0.988724i	$0.452153\pi$
$314$	−6.48308	−0.365862
$315$	−11.0240	−0.621133
$316$	−5.08456	−0.286029
$317$	9.94587	0.558616	0.279308	−	0.960202i	$-0.409895\pi$
$317$	9.94587	0.558616	0.279308	+	0.960202i	$0.409895\pi$
$318$	−9.53640	−0.534775
$319$	0	0
$320$	−4.74125	−0.265044
$321$	−45.9320	−2.56367
$322$	−7.50386	−0.418174
$323$	4.43934	0.247012
$324$	9.79782	0.544324
$325$	1.00000	0.0554700
$326$	5.71529	0.316541
$327$	13.1640	0.727971
$328$	−1.21706	−0.0672010
$329$	34.4340	1.89841
$330$	0	0
$331$	−8.98625	−0.493929	−0.246965	−	0.969024i	$-0.579433\pi$
$331$	−8.98625	−0.493929	−0.246965	+	0.969024i	$0.579433\pi$
$332$	−28.3214	−1.55434
$333$	45.1887	2.47633
$334$	−1.92787	−0.105488
$335$	1.29322	0.0706559
$336$	−23.2398	−1.26783
$337$	−35.7894	−1.94958	−0.974788	−	0.223134i	$-0.928371\pi$
$337$	−35.7894	−1.94958	−0.974788	+	0.223134i	$0.928371\pi$
$338$	0.379916	0.0206647
$339$	−45.9351	−2.49485
$340$	1.25489	0.0680560
$341$	0	0
$342$	−9.83483	−0.531807
$343$	17.2899	0.933566
$344$	−2.41830	−0.130386
$345$	18.6170	1.00231
$346$	0.273366	0.0146962
$347$	−35.4471	−1.90290	−0.951449	−	0.307806i	$-0.900405\pi$
$347$	−35.4471	−1.90290	−0.951449	+	0.307806i	$0.900405\pi$
$348$	16.1146	0.863835
$349$	1.34082	0.0717725	0.0358863	−	0.999356i	$-0.488575\pi$
$349$	1.34082	0.0717725	0.0358863	+	0.999356i	$0.488575\pi$
$350$	−1.06209	−0.0567709
$351$	2.48580	0.132682
$352$	0	0
$353$	−2.94069	−0.156517	−0.0782586	−	0.996933i	$-0.524936\pi$
$353$	−2.94069	−0.156517	−0.0782586	+	0.996933i	$0.524936\pi$
$354$	9.40499	0.499870
$355$	6.13995	0.325875
$356$	9.15736	0.485339
$357$	4.98155	0.263651
$358$	3.69967	0.195534
$359$	21.5054	1.13501	0.567506	−	0.823369i	$-0.307908\pi$
$359$	21.5054	1.13501	0.567506	+	0.823369i	$0.307908\pi$
$360$	−5.77636	−0.304441
$361$	24.0947	1.26814
$362$	−0.170180	−0.00894447
$363$	0	0
$364$	5.18766	0.271907
$365$	−12.5125	−0.654935
$366$	−11.4634	−0.599202
$367$	−20.9797	−1.09513	−0.547566	−	0.836763i	$-0.684445\pi$
$367$	−20.9797	−1.09513	−0.547566	+	0.836763i	$0.684445\pi$
$368$	22.2894	1.16192
$369$	3.27637	0.170561
$370$	4.35362	0.226334
$371$	26.6308	1.38260
$372$	35.2979	1.83011
$373$	15.3589	0.795251	0.397626	−	0.917548i	$-0.369834\pi$
$373$	15.3589	0.795251	0.397626	+	0.917548i	$0.369834\pi$
$374$	0	0
$375$	2.63503	0.136072
$376$	18.0427	0.930482
$377$	−3.29561	−0.169733
$378$	−2.64013	−0.135794
$379$	6.11986	0.314356	0.157178	−	0.987570i	$-0.449760\pi$
$379$	6.11986	0.314356	0.157178	+	0.987570i	$0.449760\pi$
$380$	12.1818	0.624913
$381$	11.1079	0.569076
$382$	1.04432	0.0534321
$383$	−37.5335	−1.91787	−0.958935	−	0.283626i	$-0.908462\pi$
$383$	−37.5335	−1.91787	−0.958935	+	0.283626i	$0.908462\pi$
$384$	−26.5024	−1.35244
$385$	0	0
$386$	7.19279	0.366103
$387$	6.51014	0.330929
$388$	16.1654	0.820674
$389$	−11.9505	−0.605915	−0.302957	−	0.953004i	$-0.597974\pi$
$389$	−11.9505	−0.605915	−0.302957	+	0.953004i	$0.597974\pi$
$390$	1.00109	0.0506921
$391$	−4.77784	−0.241626
$392$	−1.19425	−0.0603190
$393$	−3.04709	−0.153706
$394$	−7.34944	−0.370260
$395$	2.74002	0.137865
$396$	0	0
$397$	18.2610	0.916492	0.458246	−	0.888825i	$-0.348478\pi$
$397$	18.2610	0.916492	0.458246	+	0.888825i	$0.348478\pi$
$398$	3.50749	0.175814
$399$	48.3581	2.42093
$400$	3.15482	0.157741
$401$	11.4536	0.571966	0.285983	−	0.958235i	$-0.407680\pi$
$401$	11.4536	0.571966	0.285983	+	0.958235i	$0.407680\pi$
$402$	1.29462	0.0645700
$403$	−7.21879	−0.359594
$404$	17.1426	0.852874
$405$	−5.27996	−0.262363
$406$	3.50023	0.173713
$407$	0	0
$408$	2.61023	0.129226
$409$	27.6610	1.36775	0.683873	−	0.729601i	$-0.260294\pi$
$409$	27.6610	1.36775	0.683873	+	0.729601i	$0.260294\pi$
$410$	0.315656	0.0155891
$411$	−16.8994	−0.833587
$412$	23.4146	1.15355
$413$	−26.2638	−1.29236
$414$	10.5847	0.520211
$415$	15.2622	0.749190
$416$	4.12822	0.202403
$417$	−2.04022	−0.0999099
$418$	0	0
$419$	−24.0090	−1.17292	−0.586459	−	0.809979i	$-0.699478\pi$
$419$	−24.0090	−1.17292	−0.586459	+	0.809979i	$0.699478\pi$
$420$	13.6696	0.667010
$421$	−15.4557	−0.753267	−0.376634	−	0.926362i	$-0.622918\pi$
$421$	−15.4557	−0.753267	−0.376634	+	0.926362i	$0.622918\pi$
$422$	−9.65314	−0.469908
$423$	−48.5716	−2.36163
$424$	13.9540	0.677666
$425$	−0.676249	−0.0328029
$426$	6.14664	0.297806
$427$	32.0120	1.54917
$428$	32.3467	1.56353
$429$	0	0
$430$	0.627207	0.0302466
$431$	−1.53751	−0.0740592	−0.0370296	−	0.999314i	$-0.511790\pi$
$431$	−1.53751	−0.0740592	−0.0370296	+	0.999314i	$0.511790\pi$
$432$	7.84223	0.377310
$433$	18.2571	0.877379	0.438690	−	0.898639i	$-0.355443\pi$
$433$	18.2571	0.877379	0.438690	+	0.898639i	$0.355443\pi$
$434$	7.66699	0.368027
$435$	−8.68403	−0.416367
$436$	−9.27048	−0.443975
$437$	−46.3806	−2.21869
$438$	−12.5261	−0.598522
$439$	−31.3633	−1.49689	−0.748445	−	0.663197i	$-0.769199\pi$
$439$	−31.3633	−1.49689	−0.748445	+	0.663197i	$0.769199\pi$
$440$	0	0
$441$	3.21497	0.153094
$442$	−0.256918	−0.0122203
$443$	2.20624	0.104822	0.0524108	−	0.998626i	$-0.483309\pi$
$443$	2.20624	0.104822	0.0524108	+	0.998626i	$0.483309\pi$
$444$	−56.0334	−2.65923
$445$	−4.93482	−0.233933
$446$	2.53298	0.119940
$447$	20.3156	0.960897
$448$	13.2546	0.626220
$449$	−4.14789	−0.195751	−0.0978755	−	0.995199i	$-0.531205\pi$
$449$	−4.14789	−0.195751	−0.0978755	+	0.995199i	$0.531205\pi$
$450$	1.49815	0.0706234
$451$	0	0
$452$	32.3488	1.52156
$453$	−43.1751	−2.02854
$454$	−1.63401	−0.0766881
$455$	−2.79558	−0.131059
$456$	25.3387	1.18659
$457$	36.8736	1.72487	0.862436	−	0.506166i	$-0.168938\pi$
$457$	36.8736	1.72487	0.862436	+	0.506166i	$0.168938\pi$
$458$	6.73982	0.314931
$459$	−1.68102	−0.0784632
$460$	−13.1106	−0.611287
$461$	−20.3735	−0.948889	−0.474445	−	0.880285i	$-0.657351\pi$
$461$	−20.3735	−0.948889	−0.474445	+	0.880285i	$0.657351\pi$
$462$	0	0
$463$	29.0166	1.34851	0.674257	−	0.738497i	$-0.264464\pi$
$463$	29.0166	1.34851	0.674257	+	0.738497i	$0.264464\pi$
$464$	−10.3970	−0.482671
$465$	−19.0217	−0.882111
$466$	−0.526113	−0.0243717
$467$	0.557245	0.0257862	0.0128931	−	0.999917i	$-0.495896\pi$
$467$	0.557245	0.0257862	0.0128931	+	0.999917i	$0.495896\pi$
$468$	−7.31756	−0.338255
$469$	−3.61529	−0.166939
$470$	−4.67954	−0.215851
$471$	−44.9655	−2.07190
$472$	−13.7617	−0.633434
$473$	0	0
$474$	2.74300	0.125990
$475$	−6.56466	−0.301207
$476$	−3.50815	−0.160796
$477$	−37.5646	−1.71997
$478$	3.77105	0.172484
$479$	−17.6209	−0.805120	−0.402560	−	0.915394i	$-0.631879\pi$
$479$	−17.6209	−0.805120	−0.402560	+	0.915394i	$0.631879\pi$
$480$	10.8780	0.496509
$481$	11.4594	0.522505
$482$	−2.75018	−0.125267
$483$	−52.0454	−2.36815
$484$	0	0
$485$	−8.71138	−0.395564
$486$	−8.11889	−0.368280
$487$	−6.81012	−0.308596	−0.154298	−	0.988024i	$-0.549312\pi$
$487$	−6.81012	−0.308596	−0.154298	+	0.988024i	$0.549312\pi$
$488$	16.7737	0.759308
$489$	39.6402	1.79259
$490$	0.309740	0.0139926
$491$	30.9237	1.39557	0.697783	−	0.716310i	$-0.254170\pi$
$491$	30.9237	1.39557	0.697783	+	0.716310i	$0.254170\pi$
$492$	−4.06266	−0.183159
$493$	2.22866	0.100374
$494$	−2.49402	−0.112211
$495$	0	0
$496$	−22.7740	−1.02258
$497$	−17.1648	−0.769944
$498$	15.2788	0.684659
$499$	−2.93250	−0.131277	−0.0656384	−	0.997843i	$-0.520908\pi$
$499$	−2.93250	−0.131277	−0.0656384	+	0.997843i	$0.520908\pi$
$500$	−1.85566	−0.0829878
$501$	−13.3713	−0.597387
$502$	−6.55349	−0.292496
$503$	−3.08164	−0.137404	−0.0687018	−	0.997637i	$-0.521886\pi$
$503$	−3.08164	−0.137404	−0.0687018	+	0.997637i	$0.521886\pi$
$504$	16.1483	0.719302
$505$	−9.23797	−0.411084
$506$	0	0
$507$	2.63503	0.117026
$508$	−7.82253	−0.347068
$509$	2.56237	0.113575	0.0567875	−	0.998386i	$-0.481914\pi$
$509$	2.56237	0.113575	0.0567875	+	0.998386i	$0.481914\pi$
$510$	−0.676986	−0.0299774
$511$	34.9798	1.54741
$512$	22.2663	0.984041
$513$	−16.3184	−0.720475
$514$	4.10418	0.181028
$515$	−12.6179	−0.556011
$516$	−8.07249	−0.355371
$517$	0	0
$518$	−12.1709	−0.534759
$519$	1.89601	0.0832257
$520$	−1.46483	−0.0642370
$521$	28.0833	1.23035	0.615176	−	0.788390i	$-0.289085\pi$
$521$	28.0833	1.23035	0.615176	+	0.788390i	$0.289085\pi$
$522$	−4.93732	−0.216100
$523$	9.13242	0.399333	0.199667	−	0.979864i	$-0.436014\pi$
$523$	9.13242	0.399333	0.199667	+	0.979864i	$0.436014\pi$
$524$	2.14585	0.0937420
$525$	−7.36644	−0.321498
$526$	−5.29238	−0.230759
$527$	4.88170	0.212650
$528$	0	0
$529$	26.9171	1.17031
$530$	−3.61909	−0.157203
$531$	37.0470	1.60770
$532$	−34.0552	−1.47648
$533$	0.830856	0.0359884
$534$	−4.94019	−0.213783
$535$	−17.4313	−0.753621
$536$	−1.89434	−0.0818230
$537$	25.6603	1.10732
$538$	−1.88618	−0.0813192
$539$	0	0
$540$	−4.61281	−0.198504
$541$	−8.18441	−0.351875	−0.175938	−	0.984401i	$-0.556296\pi$
$541$	−8.18441	−0.351875	−0.175938	+	0.984401i	$0.556296\pi$
$542$	9.97416	0.428427
$543$	−1.18034	−0.0506531
$544$	−2.79171	−0.119693
$545$	4.99577	0.213995
$546$	−2.79863	−0.119770
$547$	23.3374	0.997834	0.498917	−	0.866650i	$-0.333731\pi$
$547$	23.3374	0.997834	0.498917	+	0.866650i	$0.333731\pi$
$548$	11.9011	0.508389
$549$	−45.1552	−1.92718
$550$	0	0
$551$	21.6346	0.921663
$552$	−27.2707	−1.16072
$553$	−7.65995	−0.325734
$554$	−6.38437	−0.271246
$555$	30.1959	1.28174
$556$	1.43678	0.0609331
$557$	−34.5346	−1.46328	−0.731640	−	0.681691i	$-0.761245\pi$
$557$	−34.5346	−1.46328	−0.731640	+	0.681691i	$0.761245\pi$
$558$	−10.8148	−0.457828
$559$	1.65091	0.0698260
$560$	−8.81955	−0.372694
$561$	0	0
$562$	3.98196	0.167969
$563$	8.69398	0.366408	0.183204	−	0.983075i	$-0.441353\pi$
$563$	8.69398	0.366408	0.183204	+	0.983075i	$0.441353\pi$
$564$	60.2282	2.53606
$565$	−17.4325	−0.733390
$566$	−3.69404	−0.155272
$567$	14.7606	0.619885
$568$	−8.99398	−0.377379
$569$	44.6581	1.87217	0.936083	−	0.351780i	$-0.114423\pi$
$569$	44.6581	1.87217	0.936083	+	0.351780i	$0.114423\pi$
$570$	−6.57181	−0.275263
$571$	10.3823	0.434487	0.217244	−	0.976117i	$-0.430293\pi$
$571$	10.3823	0.434487	0.217244	+	0.976117i	$0.430293\pi$
$572$	0	0
$573$	7.24321	0.302589
$574$	−0.882441	−0.0368324
$575$	7.06520	0.294639
$576$	−18.6965	−0.779021
$577$	19.2380	0.800887	0.400444	−	0.916321i	$-0.368856\pi$
$577$	19.2380	0.800887	0.400444	+	0.916321i	$0.368856\pi$
$578$	−6.28483	−0.261415
$579$	49.8878	2.07327
$580$	6.11555	0.253934
$581$	−42.6666	−1.77011
$582$	−8.72087	−0.361492
$583$	0	0
$584$	18.3287	0.758446
$585$	3.94337	0.163038
$586$	8.01064	0.330916
$587$	8.90534	0.367563	0.183781	−	0.982967i	$-0.441166\pi$
$587$	8.90534	0.367563	0.183781	+	0.982967i	$0.441166\pi$
$588$	−3.98652	−0.164402
$589$	47.3889	1.95263
$590$	3.56922	0.146943
$591$	−50.9744	−2.09680
$592$	36.1524	1.48585
$593$	−34.7057	−1.42519	−0.712595	−	0.701575i	$-0.752480\pi$
$593$	−34.7057	−1.42519	−0.712595	+	0.701575i	$0.752480\pi$
$594$	0	0
$595$	1.89051	0.0775034
$596$	−14.3069	−0.586032
$597$	24.3273	0.995649
$598$	2.68418	0.109764
$599$	−38.8140	−1.58590	−0.792948	−	0.609289i	$-0.791455\pi$
$599$	−38.8140	−1.58590	−0.792948	+	0.609289i	$0.791455\pi$
$600$	−3.85986	−0.157578
$601$	4.91787	0.200604	0.100302	−	0.994957i	$-0.468019\pi$
$601$	4.91787	0.200604	0.100302	+	0.994957i	$0.468019\pi$
$602$	−1.75341	−0.0714636
$603$	5.09963	0.207673
$604$	30.4052	1.23717
$605$	0	0
$606$	−9.24803	−0.375675
$607$	4.06001	0.164791	0.0823954	−	0.996600i	$-0.473743\pi$
$607$	4.06001	0.164791	0.0823954	+	0.996600i	$0.473743\pi$
$608$	−27.1004	−1.09906
$609$	24.2769	0.983751
$610$	−4.35039	−0.176142
$611$	−12.3173	−0.498304
$612$	4.94850	0.200031
$613$	28.6773	1.15827	0.579133	−	0.815233i	$-0.303391\pi$
$613$	28.6773	1.15827	0.579133	+	0.815233i	$0.303391\pi$
$614$	5.55645	0.224240
$615$	2.18933	0.0882822
$616$	0	0
$617$	−19.3172	−0.777682	−0.388841	−	0.921305i	$-0.627125\pi$
$617$	−19.3172	−0.777682	−0.388841	+	0.921305i	$0.627125\pi$
$618$	−12.6316	−0.508119
$619$	33.3454	1.34026	0.670132	−	0.742242i	$-0.266238\pi$
$619$	33.3454	1.34026	0.670132	+	0.742242i	$0.266238\pi$
$620$	13.3957	0.537982
$621$	17.5627	0.704766
$622$	3.04534	0.122107
$623$	13.7957	0.552713
$624$	8.31302	0.332787
$625$	1.00000	0.0400000
$626$	2.01308	0.0804588
$627$	0	0
$628$	31.6660	1.26361
$629$	−7.74942	−0.308990
$630$	−4.18820	−0.166862
$631$	−8.79795	−0.350241	−0.175121	−	0.984547i	$-0.556032\pi$
$631$	−8.79795	−0.350241	−0.175121	+	0.984547i	$0.556032\pi$
$632$	−4.01366	−0.159655
$633$	−66.9524	−2.66112
$634$	3.77860	0.150067
$635$	4.21549	0.167286
$636$	46.5796	1.84700
$637$	0.815286	0.0323028
$638$	0	0
$639$	24.2121	0.957816
$640$	−10.0577	−0.397566
$641$	31.7129	1.25258	0.626292	−	0.779589i	$-0.284572\pi$
$641$	31.7129	1.25258	0.626292	+	0.779589i	$0.284572\pi$
$642$	−17.4503	−0.688708
$643$	−3.87237	−0.152711	−0.0763556	−	0.997081i	$-0.524328\pi$
$643$	−3.87237	−0.152711	−0.0763556	+	0.997081i	$0.524328\pi$
$644$	36.6519	1.44429
$645$	4.35019	0.171289
$646$	1.68658	0.0663575
$647$	10.5551	0.414962	0.207481	−	0.978239i	$-0.433473\pi$
$647$	10.5551	0.414962	0.207481	+	0.978239i	$0.433473\pi$
$648$	7.73423	0.303829
$649$	0	0
$650$	0.379916	0.0149015
$651$	53.1768	2.08416
$652$	−27.9158	−1.09327
$653$	19.8722	0.777660	0.388830	−	0.921310i	$-0.372879\pi$
$653$	19.8722	0.777660	0.388830	+	0.921310i	$0.372879\pi$
$654$	5.00121	0.195563
$655$	−1.15638	−0.0451835
$656$	2.62120	0.102341
$657$	−49.3414	−1.92499
$658$	13.0820	0.509991
$659$	7.62120	0.296880	0.148440	−	0.988921i	$-0.452575\pi$
$659$	7.62120	0.296880	0.148440	+	0.988921i	$0.452575\pi$
$660$	0	0
$661$	17.9251	0.697204	0.348602	−	0.937271i	$-0.386656\pi$
$661$	17.9251	0.697204	0.348602	+	0.937271i	$0.386656\pi$
$662$	−3.41402	−0.132690
$663$	−1.78193	−0.0692046
$664$	−22.3564	−0.867598
$665$	18.3520	0.711662
$666$	17.1679	0.665243
$667$	−23.2842	−0.901567
$668$	9.41649	0.364335
$669$	17.5682	0.679228
$670$	0.491314	0.0189811
$671$	0	0
$672$	−30.4103	−1.17310
$673$	10.7782	0.415467	0.207734	−	0.978185i	$-0.433391\pi$
$673$	10.7782	0.415467	0.207734	+	0.978185i	$0.433391\pi$
$674$	−13.5970	−0.523736
$675$	2.48580	0.0956784
$676$	−1.85566	−0.0713717
$677$	−29.7453	−1.14320	−0.571602	−	0.820531i	$-0.693678\pi$
$677$	−29.7453	−1.14320	−0.571602	+	0.820531i	$0.693678\pi$
$678$	−17.4515	−0.670220
$679$	24.3534	0.934598
$680$	0.990589	0.0379874
$681$	−11.3332	−0.434290
$682$	0	0
$683$	−27.1818	−1.04008	−0.520042	−	0.854141i	$-0.674084\pi$
$683$	−27.1818	−1.04008	−0.520042	+	0.854141i	$0.674084\pi$
$684$	48.0373	1.83675
$685$	−6.41338	−0.245043
$686$	6.56870	0.250794
$687$	46.7461	1.78348
$688$	5.20831	0.198565
$689$	−9.52603	−0.362913
$690$	7.07290	0.269261
$691$	1.93123	0.0734675	0.0367337	−	0.999325i	$-0.488305\pi$
$691$	1.93123	0.0734675	0.0367337	+	0.999325i	$0.488305\pi$
$692$	−1.33523	−0.0507578
$693$	0	0
$694$	−13.4669	−0.511197
$695$	−0.774269	−0.0293697
$696$	12.7206	0.482174
$697$	−0.561866	−0.0212822
$698$	0.509400	0.0192811
$699$	−3.64902	−0.138019
$700$	5.18766	0.196075
$701$	44.3304	1.67434	0.837169	−	0.546944i	$-0.184209\pi$
$701$	44.3304	1.67434	0.837169	+	0.546944i	$0.184209\pi$
$702$	0.944395	0.0356439
$703$	−75.2272	−2.83725
$704$	0	0
$705$	−32.4564	−1.22238
$706$	−1.11722	−0.0420470
$707$	25.8255	0.971268
$708$	−45.9378	−1.72645
$709$	18.5864	0.698025	0.349013	−	0.937118i	$-0.386517\pi$
$709$	18.5864	0.698025	0.349013	+	0.937118i	$0.386517\pi$
$710$	2.33267	0.0875434
$711$	10.8049	0.405216
$712$	7.22866	0.270906
$713$	−51.0022	−1.91005
$714$	1.89257	0.0708276
$715$	0	0
$716$	−18.0707	−0.675334
$717$	26.1553	0.976788
$718$	8.17025	0.304911
$719$	−35.3240	−1.31736	−0.658681	−	0.752423i	$-0.728885\pi$
$719$	−35.3240	−1.31736	−0.658681	+	0.752423i	$0.728885\pi$
$720$	12.4406	0.463634
$721$	35.2744	1.31369
$722$	9.15397	0.340675
$723$	−19.0748	−0.709398
$724$	0.831228	0.0308924
$725$	−3.29561	−0.122396
$726$	0	0
$727$	−51.0753	−1.89428	−0.947139	−	0.320822i	$-0.896041\pi$
$727$	−51.0753	−1.89428	−0.947139	+	0.320822i	$0.896041\pi$
$728$	4.09505	0.151773
$729$	−40.4712	−1.49893
$730$	−4.75370	−0.175942
$731$	−1.11643	−0.0412925
$732$	55.9919	2.06952
$733$	−18.9469	−0.699820	−0.349910	−	0.936783i	$-0.613788\pi$
$733$	−18.9469	−0.699820	−0.349910	+	0.936783i	$0.613788\pi$
$734$	−7.97052	−0.294197
$735$	2.14830	0.0792413
$736$	29.1667	1.07510
$737$	0	0
$738$	1.24475	0.0458197
$739$	−26.9568	−0.991622	−0.495811	−	0.868430i	$-0.665129\pi$
$739$	−26.9568	−0.991622	−0.495811	+	0.868430i	$0.665129\pi$
$740$	−21.2648	−0.781711
$741$	−17.2980	−0.635460
$742$	10.1175	0.371424
$743$	20.1200	0.738133	0.369066	−	0.929403i	$-0.379677\pi$
$743$	20.1200	0.738133	0.369066	+	0.929403i	$0.379677\pi$
$744$	27.8635	1.02153
$745$	7.70984	0.282467
$746$	5.83508	0.213637
$747$	60.1843	2.20203
$748$	0	0
$749$	48.7307	1.78058
$750$	1.00109	0.0365546
$751$	−13.7673	−0.502376	−0.251188	−	0.967938i	$-0.580821\pi$
$751$	−13.7673	−0.502376	−0.251188	+	0.967938i	$0.580821\pi$
$752$	−38.8588	−1.41703
$753$	−45.4538	−1.65643
$754$	−1.25206	−0.0455972
$755$	−16.3851	−0.596313
$756$	12.8955	0.469004
$757$	46.5102	1.69044	0.845220	−	0.534418i	$-0.179469\pi$
$757$	46.5102	1.69044	0.845220	+	0.534418i	$0.179469\pi$
$758$	2.32503	0.0844490
$759$	0	0
$760$	9.61610	0.348812
$761$	−17.2638	−0.625812	−0.312906	−	0.949784i	$-0.601303\pi$
$761$	−17.2638	−0.625812	−0.312906	+	0.949784i	$0.601303\pi$
$762$	4.22008	0.152877
$763$	−13.9661	−0.505607
$764$	−5.10088	−0.184543
$765$	−2.66670	−0.0964147
$766$	−14.2596	−0.515219
$767$	9.39476	0.339225
$768$	14.9180	0.538306
$769$	33.0347	1.19126	0.595631	−	0.803258i	$-0.296902\pi$
$769$	33.0347	1.19126	0.595631	+	0.803258i	$0.296902\pi$
$770$	0	0
$771$	28.4658	1.02517
$772$	−35.1325	−1.26445
$773$	27.6163	0.993291	0.496645	−	0.867954i	$-0.334565\pi$
$773$	27.6163	0.993291	0.496645	+	0.867954i	$0.334565\pi$
$774$	2.47331	0.0889012
$775$	−7.21879	−0.259307
$776$	12.7607	0.458082
$777$	−84.4151	−3.02838
$778$	−4.54019	−0.162774
$779$	−5.45428	−0.195420
$780$	−4.88972	−0.175080
$781$	0	0
$782$	−1.81518	−0.0649106
$783$	−8.19223	−0.292766
$784$	2.57208	0.0918599
$785$	−17.0645	−0.609059
$786$	−1.15764	−0.0412916
$787$	−23.9788	−0.854752	−0.427376	−	0.904074i	$-0.640562\pi$
$787$	−23.9788	−0.854752	−0.427376	+	0.904074i	$0.640562\pi$
$788$	35.8976	1.27880
$789$	−36.7070	−1.30680
$790$	1.04098	0.0370363
$791$	48.7340	1.73278
$792$	0	0
$793$	−11.4509	−0.406635
$794$	6.93764	0.246208
$795$	−25.1013	−0.890253
$796$	−17.1320	−0.607227
$797$	26.4462	0.936774	0.468387	−	0.883523i	$-0.344835\pi$
$797$	26.4462	0.936774	0.468387	+	0.883523i	$0.344835\pi$
$798$	18.3720	0.650363
$799$	8.32956	0.294679
$800$	4.12822	0.145955
$801$	−19.4598	−0.687578
$802$	4.35141	0.153654
$803$	0	0
$804$	−6.32347	−0.223012
$805$	−19.7514	−0.696144
$806$	−2.74254	−0.0966017
$807$	−13.0822	−0.460516
$808$	13.5320	0.476055
$809$	14.5914	0.513005	0.256502	−	0.966544i	$-0.417430\pi$
$809$	14.5914	0.513005	0.256502	+	0.966544i	$0.417430\pi$
$810$	−2.00594	−0.0704816
$811$	−12.2350	−0.429630	−0.214815	−	0.976655i	$-0.568915\pi$
$811$	−12.2350	−0.429630	−0.214815	+	0.976655i	$0.568915\pi$
$812$	−17.0965	−0.599970
$813$	69.1789	2.42621
$814$	0	0
$815$	15.0436	0.526953
$816$	−5.62168	−0.196798
$817$	−10.8376	−0.379161
$818$	10.5088	0.367433
$819$	−11.0240	−0.385210
$820$	−1.54179	−0.0538416
$821$	−31.0426	−1.08339	−0.541697	−	0.840574i	$-0.682218\pi$
$821$	−31.0426	−1.08339	−0.541697	+	0.840574i	$0.682218\pi$
$822$	−6.42036	−0.223936
$823$	27.3242	0.952463	0.476232	−	0.879320i	$-0.342002\pi$
$823$	27.3242	0.952463	0.476232	+	0.879320i	$0.342002\pi$
$824$	18.4831	0.643888
$825$	0	0
$826$	−9.97806	−0.347181
$827$	−14.0412	−0.488261	−0.244130	−	0.969742i	$-0.578502\pi$
$827$	−14.0412	−0.488261	−0.244130	+	0.969742i	$0.578502\pi$
$828$	−51.7001	−1.79670
$829$	36.8622	1.28028	0.640138	−	0.768260i	$-0.278877\pi$
$829$	36.8622	1.28028	0.640138	+	0.768260i	$0.278877\pi$
$830$	5.79834	0.201263
$831$	−44.2808	−1.53608
$832$	−4.74125	−0.164373
$833$	−0.551337	−0.0191027
$834$	−0.775112	−0.0268399
$835$	−5.07446	−0.175609
$836$	0	0
$837$	−17.9445	−0.620251
$838$	−9.12141	−0.315094
$839$	−26.0628	−0.899789	−0.449894	−	0.893082i	$-0.648538\pi$
$839$	−26.0628	−0.899789	−0.449894	+	0.893082i	$0.648538\pi$
$840$	10.7906	0.372310
$841$	−18.1389	−0.625481
$842$	−5.87189	−0.202359
$843$	27.6181	0.951219
$844$	47.1499	1.62296
$845$	1.00000	0.0344010
$846$	−18.4531	−0.634432
$847$	0	0
$848$	−30.0529	−1.03202
$849$	−25.6212	−0.879316
$850$	−0.256918	−0.00881221
$851$	80.9632	2.77538
$852$	−30.0227	−1.02856
$853$	−18.1474	−0.621355	−0.310678	−	0.950515i	$-0.600556\pi$
$853$	−18.1474	−0.621355	−0.310678	+	0.950515i	$0.600556\pi$
$854$	12.1619	0.416171
$855$	−25.8868	−0.885312
$856$	25.5339	0.872730
$857$	38.3335	1.30945	0.654724	−	0.755868i	$-0.272785\pi$
$857$	38.3335	1.30945	0.654724	+	0.755868i	$0.272785\pi$
$858$	0	0
$859$	−23.2643	−0.793767	−0.396883	−	0.917869i	$-0.629908\pi$
$859$	−23.2643	−0.793767	−0.396883	+	0.917869i	$0.629908\pi$
$860$	−3.06353	−0.104466
$861$	−6.12045	−0.208584
$862$	−0.584125	−0.0198954
$863$	−16.9430	−0.576748	−0.288374	−	0.957518i	$-0.593115\pi$
$863$	−16.9430	−0.576748	−0.288374	+	0.957518i	$0.593115\pi$
$864$	10.2619	0.349118
$865$	0.719542	0.0244652
$866$	6.93616	0.235700
$867$	−43.5904	−1.48041
$868$	−37.4487	−1.27109
$869$	0	0
$870$	−3.29920	−0.111853
$871$	1.29322	0.0438190
$872$	−7.31795	−0.247817
$873$	−34.3522	−1.16265
$874$	−17.6207	−0.596031
$875$	−2.79558	−0.0945080
$876$	61.1827	2.06717
$877$	−42.3752	−1.43091	−0.715455	−	0.698659i	$-0.753780\pi$
$877$	−42.3752	−1.43091	−0.715455	+	0.698659i	$0.753780\pi$
$878$	−11.9154	−0.402126
$879$	55.5603	1.87400
$880$	0	0
$881$	49.7712	1.67683	0.838417	−	0.545029i	$-0.183481\pi$
$881$	49.7712	1.67683	0.838417	+	0.545029i	$0.183481\pi$
$882$	1.22142	0.0411273
$883$	−7.00971	−0.235896	−0.117948	−	0.993020i	$-0.537632\pi$
$883$	−7.00971	−0.235896	−0.117948	+	0.993020i	$0.537632\pi$
$884$	1.25489	0.0422066
$885$	24.7555	0.832145
$886$	0.838186	0.0281594
$887$	49.4984	1.66199	0.830997	−	0.556277i	$-0.187771\pi$
$887$	49.4984	1.66199	0.830997	+	0.556277i	$0.187771\pi$
$888$	−44.2318	−1.48432
$889$	−11.7847	−0.395248
$890$	−1.87482	−0.0628440
$891$	0	0
$892$	−12.3721	−0.414248
$893$	80.8588	2.70584
$894$	7.71824	0.258136
$895$	9.73814	0.325510
$896$	28.1172	0.939330
$897$	18.6170	0.621604
$898$	−1.57585	−0.0525868
$899$	23.7903	0.793452
$900$	−7.31756	−0.243919
$901$	6.44197	0.214613
$902$	0	0
$903$	−12.1613	−0.404703
$904$	25.5356	0.849302
$905$	−0.447941	−0.0148901
$906$	−16.4029	−0.544950
$907$	−44.1074	−1.46456	−0.732282	−	0.681002i	$-0.761545\pi$
$907$	−44.1074	−1.46456	−0.732282	+	0.681002i	$0.761545\pi$
$908$	7.98119	0.264865
$909$	−36.4287	−1.20826
$910$	−1.06209	−0.0352078
$911$	−36.9744	−1.22502	−0.612508	−	0.790465i	$-0.709839\pi$
$911$	−36.9744	−1.22502	−0.612508	+	0.790465i	$0.709839\pi$
$912$	−54.5721	−1.80706
$913$	0	0
$914$	14.0089	0.463372
$915$	−30.1735	−0.997506
$916$	−32.9200	−1.08771
$917$	3.23276	0.106755
$918$	−0.638646	−0.0210785
$919$	46.0940	1.52050	0.760251	−	0.649630i	$-0.225076\pi$
$919$	46.0940	1.52050	0.760251	+	0.649630i	$0.225076\pi$
$920$	−10.3493	−0.341207
$921$	38.5385	1.26989
$922$	−7.74022	−0.254911
$923$	6.13995	0.202099
$924$	0	0
$925$	11.4594	0.376783
$926$	11.0239	0.362267
$927$	−49.7570	−1.63423
$928$	−13.6050	−0.446607
$929$	−14.0716	−0.461675	−0.230838	−	0.972992i	$-0.574147\pi$
$929$	−14.0716	−0.461675	−0.230838	+	0.972992i	$0.574147\pi$
$930$	−7.22665	−0.236971
$931$	−5.35207	−0.175407
$932$	2.56975	0.0841748
$933$	21.1219	0.691500
$934$	0.211706	0.00692724
$935$	0	0
$936$	−5.77636	−0.188806
$937$	54.5327	1.78151	0.890753	−	0.454487i	$-0.150178\pi$
$937$	54.5327	1.78151	0.890753	+	0.454487i	$0.150178\pi$
$938$	−1.37351	−0.0448466
$939$	13.9623	0.455644
$940$	22.8568	0.745505
$941$	13.9865	0.455946	0.227973	−	0.973668i	$-0.426790\pi$
$941$	13.9865	0.455946	0.227973	+	0.973668i	$0.426790\pi$
$942$	−17.0831	−0.556597
$943$	5.87017	0.191159
$944$	29.6387	0.964659
$945$	−6.94926	−0.226059
$946$	0	0
$947$	58.4058	1.89793	0.948966	−	0.315378i	$-0.102131\pi$
$947$	58.4058	1.89793	0.948966	+	0.315378i	$0.102131\pi$
$948$	−13.3979	−0.435145
$949$	−12.5125	−0.406173
$950$	−2.49402	−0.0809166
$951$	26.2076	0.849841
$952$	−2.76927	−0.0897527
$953$	36.9371	1.19651	0.598255	−	0.801306i	$-0.295861\pi$
$953$	36.9371	1.19651	0.598255	+	0.801306i	$0.295861\pi$
$954$	−14.2714	−0.462054
$955$	2.74882	0.0889496
$956$	−18.4193	−0.595724
$957$	0	0
$958$	−6.69446	−0.216288
$959$	17.9291	0.578962
$960$	−12.4933	−0.403221
$961$	21.1110	0.680999
$962$	4.35362	0.140366
$963$	−68.7381	−2.21505
$964$	13.4330	0.432648
$965$	18.9326	0.609461
$966$	−19.7729	−0.636182
$967$	−29.4133	−0.945867	−0.472934	−	0.881098i	$-0.656805\pi$
$967$	−29.4133	−0.945867	−0.472934	+	0.881098i	$0.656805\pi$
$968$	0	0
$969$	11.6978	0.375787
$970$	−3.30959	−0.106265
$971$	9.21880	0.295846	0.147923	−	0.988999i	$-0.452741\pi$
$971$	9.21880	0.295846	0.147923	+	0.988999i	$0.452741\pi$
$972$	39.6559	1.27197
$973$	2.16453	0.0693917
$974$	−2.58727	−0.0829016
$975$	2.63503	0.0843884
$976$	−36.1256	−1.15635
$977$	21.3092	0.681741	0.340870	−	0.940110i	$-0.389278\pi$
$977$	21.3092	0.681741	0.340870	+	0.940110i	$0.389278\pi$
$978$	15.0600	0.481564
$979$	0	0
$980$	−1.51290	−0.0483277
$981$	19.7002	0.628978
$982$	11.7484	0.374906
$983$	−36.6496	−1.16894	−0.584471	−	0.811415i	$-0.698698\pi$
$983$	−36.6496	−1.16894	−0.584471	+	0.811415i	$0.698698\pi$
$984$	−3.20699	−0.102235
$985$	−19.3449	−0.616380
$986$	0.846702	0.0269645
$987$	90.7346	2.88811
$988$	12.1818	0.387555
$989$	11.6640	0.370894
$990$	0	0
$991$	−2.14700	−0.0682016	−0.0341008	−	0.999418i	$-0.510857\pi$
$991$	−2.14700	−0.0682016	−0.0341008	+	0.999418i	$0.510857\pi$
$992$	−29.8008	−0.946176
$993$	−23.6790	−0.751431
$994$	−6.52117	−0.206839
$995$	9.23227	0.292683
$996$	−74.6277	−2.36467
$997$	1.81700	0.0575450	0.0287725	−	0.999586i	$-0.490840\pi$
$997$	1.81700	0.0575450	0.0287725	+	0.999586i	$0.490840\pi$
$998$	−1.11410	−0.0352663
$999$	28.4858	0.901251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7865.2.a.bg.1.11	✓	18
11.10	odd	2		7865.2.a.bh.1.8	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7865.2.a.bg.1.11	✓	18	1.1	even	1	trivial
7865.2.a.bh.1.8	yes	18	11.10	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:	\( N \)	\(=\)	\( 7865 = 5 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7865.a (trivial)

\(f(q)\)	\(=\)	\(q+0.379916 q^{2} +2.63503 q^{3} -1.85566 q^{4} +1.00000 q^{5} +1.00109 q^{6} -2.79558 q^{7} -1.46483 q^{8} +3.94337 q^{9} +0.379916 q^{10} -4.88972 q^{12} +1.00000 q^{13} -1.06209 q^{14} +2.63503 q^{15} +3.15482 q^{16} -0.676249 q^{17} +1.49815 q^{18} -6.56466 q^{19} -1.85566 q^{20} -7.36644 q^{21} +7.06520 q^{23} -3.85986 q^{24} +1.00000 q^{25} +0.379916 q^{26} +2.48580 q^{27} +5.18766 q^{28} -3.29561 q^{29} +1.00109 q^{30} -7.21879 q^{31} +4.12822 q^{32} -0.256918 q^{34} -2.79558 q^{35} -7.31756 q^{36} +11.4594 q^{37} -2.49402 q^{38} +2.63503 q^{39} -1.46483 q^{40} +0.830856 q^{41} -2.79863 q^{42} +1.65091 q^{43} +3.94337 q^{45} +2.68418 q^{46} -12.3173 q^{47} +8.31302 q^{48} +0.815286 q^{49} +0.379916 q^{50} -1.78193 q^{51} -1.85566 q^{52} -9.52603 q^{53} +0.944395 q^{54} +4.09505 q^{56} -17.2980 q^{57} -1.25206 q^{58} +9.39476 q^{59} -4.88972 q^{60} -11.4509 q^{61} -2.74254 q^{62} -11.0240 q^{63} -4.74125 q^{64} +1.00000 q^{65} +1.29322 q^{67} +1.25489 q^{68} +18.6170 q^{69} -1.06209 q^{70} +6.13995 q^{71} -5.77636 q^{72} -12.5125 q^{73} +4.35362 q^{74} +2.63503 q^{75} +12.1818 q^{76} +1.00109 q^{78} +2.74002 q^{79} +3.15482 q^{80} -5.27996 q^{81} +0.315656 q^{82} +15.2622 q^{83} +13.6696 q^{84} -0.676249 q^{85} +0.627207 q^{86} -8.68403 q^{87} -4.93482 q^{89} +1.49815 q^{90} -2.79558 q^{91} -13.1106 q^{92} -19.0217 q^{93} -4.67954 q^{94} -6.56466 q^{95} +10.8780 q^{96} -8.71138 q^{97} +0.309740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{2} - 4 q^{3} + 16 q^{4} + 18 q^{5} - 2 q^{6} - 20 q^{7} + 10 q^{9} - 2 q^{10} - 20 q^{12} + 18 q^{13} + 10 q^{14} - 4 q^{15} + 8 q^{16} - 6 q^{17} - 38 q^{19} + 16 q^{20} - 22 q^{21} - 2 q^{23}+ \cdots - 36 q^{98}+O(q^{100}) \) 18 * q - 2 * q^2 - 4 * q^3 + 16 * q^4 + 18 * q^5 - 2 * q^6 - 20 * q^7 + 10 * q^9 - 2 * q^10 - 20 * q^12 + 18 * q^13 + 10 * q^14 - 4 * q^15 + 8 * q^16 - 6 * q^17 - 38 * q^19 + 16 * q^20 - 22 * q^21 - 2 * q^23 - 16 * q^24 + 18 * q^25 - 2 * q^26 - 10 * q^27 - 10 * q^28 - 20 * q^29 - 2 * q^30 - 8 * q^31 - 22 * q^32 - 10 * q^34 - 20 * q^35 + 12 * q^36 + 8 * q^37 - 4 * q^39 - 12 * q^41 + 2 * q^42 - 20 * q^43 + 10 * q^45 - 58 * q^46 - 28 * q^47 - 12 * q^48 + 16 * q^49 - 2 * q^50 - 28 * q^51 + 16 * q^52 - 20 * q^53 - 44 * q^54 + 18 * q^57 + 22 * q^58 + 2 * q^59 - 20 * q^60 - 32 * q^61 - 22 * q^62 + 2 * q^63 - 16 * q^64 + 18 * q^65 + 12 * q^67 - 36 * q^68 + 24 * q^69 + 10 * q^70 + 16 * q^71 + 30 * q^72 - 28 * q^73 - 4 * q^74 - 4 * q^75 - 114 * q^76 - 2 * q^78 - 56 * q^79 + 8 * q^80 - 38 * q^81 + 2 * q^82 + 4 * q^83 - 24 * q^84 - 6 * q^85 - 24 * q^86 - 50 * q^87 + 26 * q^89 - 20 * q^91 - 28 * q^92 + 40 * q^93 - 40 * q^94 - 38 * q^95 + 56 * q^96 - 8 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more