LMFDB - Embedded newform 6422.2.a.bq.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + \cdots + 392 $ x^15 - 31x^13 - 4x^12 + 373x^11 + 85x^10 - 2208x^9 - 636x^8 + 6791x^7 + 2005x^6 - 10779x^5 - 2515x^4 + 8205x^3 + 686x^2 - 2352*x + 392
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			1.9
Root			$0.503505$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.503505 q^{3} +1.00000 q^{4} -1.68542 q^{5} +0.503505 q^{6} -4.13863 q^{7} +1.00000 q^{8} -2.74648 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.503505 q^{3} +1.00000 q^{4} -1.68542 q^{5} +0.503505 q^{6} -4.13863 q^{7} +1.00000 q^{8} -2.74648 q^{9} -1.68542 q^{10} -0.105212 q^{11} +0.503505 q^{12} -4.13863 q^{14} -0.848619 q^{15} +1.00000 q^{16} -7.67425 q^{17} -2.74648 q^{18} -1.00000 q^{19} -1.68542 q^{20} -2.08382 q^{21} -0.105212 q^{22} +2.21700 q^{23} +0.503505 q^{24} -2.15936 q^{25} -2.89338 q^{27} -4.13863 q^{28} +2.83626 q^{29} -0.848619 q^{30} +7.51861 q^{31} +1.00000 q^{32} -0.0529748 q^{33} -7.67425 q^{34} +6.97534 q^{35} -2.74648 q^{36} +8.42298 q^{37} -1.00000 q^{38} -1.68542 q^{40} -1.72166 q^{41} -2.08382 q^{42} -6.26415 q^{43} -0.105212 q^{44} +4.62898 q^{45} +2.21700 q^{46} +11.7044 q^{47} +0.503505 q^{48} +10.1283 q^{49} -2.15936 q^{50} -3.86403 q^{51} -6.72006 q^{53} -2.89338 q^{54} +0.177327 q^{55} -4.13863 q^{56} -0.503505 q^{57} +2.83626 q^{58} -8.40051 q^{59} -0.848619 q^{60} +10.3621 q^{61} +7.51861 q^{62} +11.3667 q^{63} +1.00000 q^{64} -0.0529748 q^{66} +8.21110 q^{67} -7.67425 q^{68} +1.11627 q^{69} +6.97534 q^{70} -3.68131 q^{71} -2.74648 q^{72} +9.69867 q^{73} +8.42298 q^{74} -1.08725 q^{75} -1.00000 q^{76} +0.435434 q^{77} +2.34585 q^{79} -1.68542 q^{80} +6.78261 q^{81} -1.72166 q^{82} +7.43747 q^{83} -2.08382 q^{84} +12.9344 q^{85} -6.26415 q^{86} +1.42807 q^{87} -0.105212 q^{88} -13.6150 q^{89} +4.62898 q^{90} +2.21700 q^{92} +3.78566 q^{93} +11.7044 q^{94} +1.68542 q^{95} +0.503505 q^{96} +15.0152 q^{97} +10.1283 q^{98} +0.288963 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 + 15 * q^4 + q^5 + 18 * q^7 + 15 * q^8 + 17 * q^9	$ 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9} + q^{10} + 4 q^{11} + 18 q^{14} + 23 q^{15} + 15 q^{16} + 2 q^{17} + 17 q^{18} - 15 q^{19} + q^{20} + 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} + 18 q^{28} - 20 q^{29} + 23 q^{30} + 30 q^{31} + 15 q^{32} + 36 q^{33} + 2 q^{34} + 32 q^{35} + 17 q^{36} + 35 q^{37} - 15 q^{38} + q^{40} + 15 q^{41} + 2 q^{42} + q^{43} + 4 q^{44} - 11 q^{45} + 17 q^{46} + 29 q^{49} + 8 q^{50} - q^{51} - q^{53} + 12 q^{54} - 6 q^{55} + 18 q^{56} - 20 q^{58} - 7 q^{59} + 23 q^{60} - 2 q^{61} + 30 q^{62} + 42 q^{63} + 15 q^{64} + 36 q^{66} + 34 q^{67} + 2 q^{68} - 12 q^{69} + 32 q^{70} + 4 q^{71} + 17 q^{72} + 12 q^{73} + 35 q^{74} + 31 q^{75} - 15 q^{76} - 20 q^{77} + 23 q^{79} + q^{80} + 7 q^{81} + 15 q^{82} - 3 q^{83} + 2 q^{84} + 46 q^{85} + q^{86} + 22 q^{87} + 4 q^{88} + 17 q^{89} - 11 q^{90} + 17 q^{92} + 60 q^{93} - q^{95} + 18 q^{97} + 29 q^{98} - 6 q^{99}+O(q^{100}) $ 15 * q + 15 * q^2 + 15 * q^4 + q^5 + 18 * q^7 + 15 * q^8 + 17 * q^9 + q^10 + 4 * q^11 + 18 * q^14 + 23 * q^15 + 15 * q^16 + 2 * q^17 + 17 * q^18 - 15 * q^19 + q^20 + 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 + 18 * q^28 - 20 * q^29 + 23 * q^30 + 30 * q^31 + 15 * q^32 + 36 * q^33 + 2 * q^34 + 32 * q^35 + 17 * q^36 + 35 * q^37 - 15 * q^38 + q^40 + 15 * q^41 + 2 * q^42 + q^43 + 4 * q^44 - 11 * q^45 + 17 * q^46 + 29 * q^49 + 8 * q^50 - q^51 - q^53 + 12 * q^54 - 6 * q^55 + 18 * q^56 - 20 * q^58 - 7 * q^59 + 23 * q^60 - 2 * q^61 + 30 * q^62 + 42 * q^63 + 15 * q^64 + 36 * q^66 + 34 * q^67 + 2 * q^68 - 12 * q^69 + 32 * q^70 + 4 * q^71 + 17 * q^72 + 12 * q^73 + 35 * q^74 + 31 * q^75 - 15 * q^76 - 20 * q^77 + 23 * q^79 + q^80 + 7 * q^81 + 15 * q^82 - 3 * q^83 + 2 * q^84 + 46 * q^85 + q^86 + 22 * q^87 + 4 * q^88 + 17 * q^89 - 11 * q^90 + 17 * q^92 + 60 * q^93 - q^95 + 18 * q^97 + 29 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.503505	0.290699	0.145349	−	0.989380i	$-0.453569\pi$
$3$	0.503505	0.290699	0.145349	+	0.989380i	$0.453569\pi$
$4$	1.00000	0.500000
$5$	−1.68542	−0.753743	−0.376872	−	0.926266i	$-0.623000\pi$
$5$	−1.68542	−0.753743	−0.376872	+	0.926266i	$0.623000\pi$
$6$	0.503505	0.205555
$7$	−4.13863	−1.56426	−0.782128	−	0.623118i	$-0.785866\pi$
$7$	−4.13863	−1.56426	−0.782128	+	0.623118i	$0.785866\pi$
$8$	1.00000	0.353553
$9$	−2.74648	−0.915494
$10$	−1.68542	−0.532977
$11$	−0.105212	−0.0317226	−0.0158613	−	0.999874i	$-0.505049\pi$
$11$	−0.105212	−0.0317226	−0.0158613	+	0.999874i	$0.505049\pi$
$12$	0.503505	0.145349
$13$	0	0
$13$	0	0
$14$	−4.13863	−1.10610
$15$	−0.848619	−0.219112
$16$	1.00000	0.250000
$17$	−7.67425	−1.86128	−0.930640	−	0.365936i	$-0.880749\pi$
$17$	−7.67425	−1.86128	−0.930640	+	0.365936i	$0.880749\pi$
$18$	−2.74648	−0.647352
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.68542	−0.376872
$21$	−2.08382	−0.454727
$22$	−0.105212	−0.0224313
$23$	2.21700	0.462276	0.231138	−	0.972921i	$-0.425755\pi$
$23$	2.21700	0.462276	0.231138	+	0.972921i	$0.425755\pi$
$24$	0.503505	0.102778
$25$	−2.15936	−0.431871
$26$	0	0
$27$	−2.89338	−0.556832
$28$	−4.13863	−0.782128
$29$	2.83626	0.526681	0.263340	−	0.964703i	$-0.415176\pi$
$29$	2.83626	0.526681	0.263340	+	0.964703i	$0.415176\pi$
$30$	−0.848619	−0.154936
$31$	7.51861	1.35038	0.675191	−	0.737643i	$-0.264061\pi$
$31$	7.51861	1.35038	0.675191	+	0.737643i	$0.264061\pi$
$32$	1.00000	0.176777
$33$	−0.0529748	−0.00922174
$34$	−7.67425	−1.31612
$35$	6.97534	1.17905
$36$	−2.74648	−0.457747
$37$	8.42298	1.38473	0.692365	−	0.721548i	$-0.256569\pi$
$37$	8.42298	1.38473	0.692365	+	0.721548i	$0.256569\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−1.68542	−0.266488
$41$	−1.72166	−0.268878	−0.134439	−	0.990922i	$-0.542923\pi$
$41$	−1.72166	−0.268878	−0.134439	+	0.990922i	$0.542923\pi$
$42$	−2.08382	−0.321541
$43$	−6.26415	−0.955274	−0.477637	−	0.878557i	$-0.658507\pi$
$43$	−6.26415	−0.955274	−0.477637	+	0.878557i	$0.658507\pi$
$44$	−0.105212	−0.0158613
$45$	4.62898	0.690048
$46$	2.21700	0.326878
$47$	11.7044	1.70727	0.853635	−	0.520872i	$-0.174393\pi$
$47$	11.7044	1.70727	0.853635	+	0.520872i	$0.174393\pi$
$48$	0.503505	0.0726747
$49$	10.1283	1.44689
$50$	−2.15936	−0.305379
$51$	−3.86403	−0.541072
$52$	0	0
$53$	−6.72006	−0.923072	−0.461536	−	0.887122i	$-0.652701\pi$
$53$	−6.72006	−0.923072	−0.461536	+	0.887122i	$0.652701\pi$
$54$	−2.89338	−0.393740
$55$	0.177327	0.0239107
$56$	−4.13863	−0.553048
$57$	−0.503505	−0.0666909
$58$	2.83626	0.372420
$59$	−8.40051	−1.09365	−0.546827	−	0.837246i	$-0.684164\pi$
$59$	−8.40051	−1.09365	−0.546827	+	0.837246i	$0.684164\pi$
$60$	−0.848619	−0.109556
$61$	10.3621	1.32673	0.663363	−	0.748298i	$-0.269129\pi$
$61$	10.3621	1.32673	0.663363	+	0.748298i	$0.269129\pi$
$62$	7.51861	0.954864
$63$	11.3667	1.43207
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.0529748	−0.00652075
$67$	8.21110	1.00315	0.501573	−	0.865115i	$-0.332755\pi$
$67$	8.21110	1.00315	0.501573	+	0.865115i	$0.332755\pi$
$68$	−7.67425	−0.930640
$69$	1.11627	0.134383
$70$	6.97534	0.833712
$71$	−3.68131	−0.436891	−0.218446	−	0.975849i	$-0.570099\pi$
$71$	−3.68131	−0.436891	−0.218446	+	0.975849i	$0.570099\pi$
$72$	−2.74648	−0.323676
$73$	9.69867	1.13514	0.567571	−	0.823324i	$-0.307883\pi$
$73$	9.69867	1.13514	0.567571	+	0.823324i	$0.307883\pi$
$74$	8.42298	0.979152
$75$	−1.08725	−0.125544
$76$	−1.00000	−0.114708
$77$	0.435434	0.0496223
$78$	0	0
$79$	2.34585	0.263928	0.131964	−	0.991254i	$-0.457872\pi$
$79$	2.34585	0.263928	0.131964	+	0.991254i	$0.457872\pi$
$80$	−1.68542	−0.188436
$81$	6.78261	0.753624
$82$	−1.72166	−0.190126
$83$	7.43747	0.816368	0.408184	−	0.912900i	$-0.366162\pi$
$83$	7.43747	0.816368	0.408184	+	0.912900i	$0.366162\pi$
$84$	−2.08382	−0.227364
$85$	12.9344	1.40293
$86$	−6.26415	−0.675481
$87$	1.42807	0.153106
$88$	−0.105212	−0.0112156
$89$	−13.6150	−1.44318	−0.721591	−	0.692319i	$-0.756589\pi$
$89$	−13.6150	−1.44318	−0.721591	+	0.692319i	$0.756589\pi$
$90$	4.62898	0.487937
$91$	0	0
$92$	2.21700	0.231138
$93$	3.78566	0.392555
$94$	11.7044	1.20722
$95$	1.68542	0.172921
$96$	0.503505	0.0513888
$97$	15.0152	1.52456	0.762280	−	0.647247i	$-0.224080\pi$
$97$	15.0152	1.52456	0.762280	+	0.647247i	$0.224080\pi$
$98$	10.1283	1.02311
$99$	0.288963	0.0290419
$100$	−2.15936	−0.215936
$101$	7.02420	0.698934	0.349467	−	0.936949i	$-0.386363\pi$
$101$	7.02420	0.698934	0.349467	+	0.936949i	$0.386363\pi$
$102$	−3.86403	−0.382596
$103$	−5.17818	−0.510221	−0.255111	−	0.966912i	$-0.582112\pi$
$103$	−5.17818	−0.510221	−0.255111	+	0.966912i	$0.582112\pi$
$104$	0	0
$105$	3.51212	0.342748
$106$	−6.72006	−0.652710
$107$	−15.0101	−1.45108	−0.725542	−	0.688178i	$-0.758411\pi$
$107$	−15.0101	−1.45108	−0.725542	+	0.688178i	$0.758411\pi$
$108$	−2.89338	−0.278416
$109$	19.5883	1.87622	0.938109	−	0.346340i	$-0.112576\pi$
$109$	19.5883	1.87622	0.938109	+	0.346340i	$0.112576\pi$
$110$	0.177327	0.0169074
$111$	4.24102	0.402539
$112$	−4.13863	−0.391064
$113$	3.55168	0.334114	0.167057	−	0.985947i	$-0.446574\pi$
$113$	3.55168	0.334114	0.167057	+	0.985947i	$0.446574\pi$
$114$	−0.503505	−0.0471576
$115$	−3.73657	−0.348437
$116$	2.83626	0.263340
$117$	0	0
$118$	−8.40051	−0.773330
$119$	31.7609	2.91152
$120$	−0.848619	−0.0774679
$121$	−10.9889	−0.998994
$122$	10.3621	0.938137
$123$	−0.866865	−0.0781626
$124$	7.51861	0.675191
$125$	12.0665	1.07926
$126$	11.3667	1.01262
$127$	−4.46649	−0.396337	−0.198169	−	0.980168i	$-0.563499\pi$
$127$	−4.46649	−0.396337	−0.198169	+	0.980168i	$0.563499\pi$
$128$	1.00000	0.0883883
$129$	−3.15403	−0.277697
$130$	0	0
$131$	2.14974	0.187824	0.0939118	−	0.995581i	$-0.470063\pi$
$131$	2.14974	0.187824	0.0939118	+	0.995581i	$0.470063\pi$
$132$	−0.0529748	−0.00461087
$133$	4.13863	0.358865
$134$	8.21110	0.709331
$135$	4.87657	0.419708
$136$	−7.67425	−0.658062
$137$	−1.76415	−0.150722	−0.0753608	−	0.997156i	$-0.524011\pi$
$137$	−1.76415	−0.150722	−0.0753608	+	0.997156i	$0.524011\pi$
$138$	1.11627	0.0950232
$139$	−17.4235	−1.47784	−0.738921	−	0.673793i	$-0.764664\pi$
$139$	−17.4235	−1.47784	−0.738921	+	0.673793i	$0.764664\pi$
$140$	6.97534	0.589523
$141$	5.89325	0.496301
$142$	−3.68131	−0.308929
$143$	0	0
$144$	−2.74648	−0.228874
$145$	−4.78030	−0.396982
$146$	9.69867	0.802667
$147$	5.09963	0.420611
$148$	8.42298	0.692365
$149$	−2.13533	−0.174933	−0.0874664	−	0.996167i	$-0.527877\pi$
$149$	−2.13533	−0.174933	−0.0874664	+	0.996167i	$0.527877\pi$
$150$	−1.08725	−0.0887733
$151$	−7.28016	−0.592451	−0.296225	−	0.955118i	$-0.595728\pi$
$151$	−7.28016	−0.592451	−0.296225	+	0.955118i	$0.595728\pi$
$152$	−1.00000	−0.0811107
$153$	21.0772	1.70399
$154$	0.435434	0.0350883
$155$	−12.6720	−1.01784
$156$	0	0
$157$	11.5683	0.923254	0.461627	−	0.887074i	$-0.347266\pi$
$157$	11.5683	0.923254	0.461627	+	0.887074i	$0.347266\pi$
$158$	2.34585	0.186626
$159$	−3.38359	−0.268336
$160$	−1.68542	−0.133244
$161$	−9.17533	−0.723117
$162$	6.78261	0.532892
$163$	−9.51467	−0.745247	−0.372623	−	0.927983i	$-0.621542\pi$
$163$	−9.51467	−0.745247	−0.372623	+	0.927983i	$0.621542\pi$
$164$	−1.72166	−0.134439
$165$	0.0892849	0.00695082
$166$	7.43747	0.577259
$167$	−7.54977	−0.584219	−0.292109	−	0.956385i	$-0.594357\pi$
$167$	−7.54977	−0.584219	−0.292109	+	0.956385i	$0.594357\pi$
$168$	−2.08382	−0.160770
$169$	0	0
$170$	12.9344	0.992019
$171$	2.74648	0.210029
$172$	−6.26415	−0.477637
$173$	4.26730	0.324437	0.162219	−	0.986755i	$-0.448135\pi$
$173$	4.26730	0.324437	0.162219	+	0.986755i	$0.448135\pi$
$174$	1.42807	0.108262
$175$	8.93678	0.675557
$176$	−0.105212	−0.00793066
$177$	−4.22970	−0.317924
$178$	−13.6150	−1.02048
$179$	7.20267	0.538353	0.269176	−	0.963091i	$-0.413249\pi$
$179$	7.20267	0.538353	0.269176	+	0.963091i	$0.413249\pi$
$180$	4.62898	0.345024
$181$	−4.87492	−0.362350	−0.181175	−	0.983451i	$-0.557990\pi$
$181$	−4.87492	−0.362350	−0.181175	+	0.983451i	$0.557990\pi$
$182$	0	0
$183$	5.21735	0.385678
$184$	2.21700	0.163439
$185$	−14.1963	−1.04373
$186$	3.78566	0.277578
$187$	0.807424	0.0590447
$188$	11.7044	0.853635
$189$	11.9746	0.871028
$190$	1.68542	0.122273
$191$	11.2354	0.812965	0.406482	−	0.913659i	$-0.366755\pi$
$191$	11.2354	0.812965	0.406482	+	0.913659i	$0.366755\pi$
$192$	0.503505	0.0363374
$193$	−3.90822	−0.281320	−0.140660	−	0.990058i	$-0.544922\pi$
$193$	−3.90822	−0.281320	−0.140660	+	0.990058i	$0.544922\pi$
$194$	15.0152	1.07803
$195$	0	0
$196$	10.1283	0.723447
$197$	−19.4807	−1.38794	−0.693972	−	0.720002i	$-0.744141\pi$
$197$	−19.4807	−1.38794	−0.693972	+	0.720002i	$0.744141\pi$
$198$	0.288963	0.0205357
$199$	−23.6053	−1.67334	−0.836668	−	0.547711i	$-0.815499\pi$
$199$	−23.6053	−1.67334	−0.836668	+	0.547711i	$0.815499\pi$
$200$	−2.15936	−0.152689
$201$	4.13433	0.291613
$202$	7.02420	0.494221
$203$	−11.7382	−0.823863
$204$	−3.86403	−0.270536
$205$	2.90172	0.202665
$206$	−5.17818	−0.360781
$207$	−6.08894	−0.423211
$208$	0	0
$209$	0.105212	0.00727767
$210$	3.51212	0.242359
$211$	15.7872	1.08683	0.543417	−	0.839463i	$-0.317130\pi$
$211$	15.7872	1.08683	0.543417	+	0.839463i	$0.317130\pi$
$212$	−6.72006	−0.461536
$213$	−1.85356	−0.127004
$214$	−15.0101	−1.02607
$215$	10.5577	0.720031
$216$	−2.89338	−0.196870
$217$	−31.1167	−2.11234
$218$	19.5883	1.32669
$219$	4.88333	0.329985
$220$	0.177327	0.0119554
$221$	0	0
$222$	4.24102	0.284638
$223$	15.1196	1.01248	0.506240	−	0.862393i	$-0.331035\pi$
$223$	15.1196	1.01248	0.506240	+	0.862393i	$0.331035\pi$
$224$	−4.13863	−0.276524
$225$	5.93063	0.395375
$226$	3.55168	0.236254
$227$	20.1459	1.33713	0.668565	−	0.743654i	$-0.266909\pi$
$227$	20.1459	1.33713	0.668565	+	0.743654i	$0.266909\pi$
$228$	−0.503505	−0.0333455
$229$	−27.2632	−1.80161	−0.900803	−	0.434228i	$-0.857021\pi$
$229$	−27.2632	−1.80161	−0.900803	+	0.434228i	$0.857021\pi$
$230$	−3.73657	−0.246382
$231$	0.219243	0.0144252
$232$	2.83626	0.186210
$233$	22.0907	1.44721	0.723605	−	0.690215i	$-0.242484\pi$
$233$	22.0907	1.44721	0.723605	+	0.690215i	$0.242484\pi$
$234$	0	0
$235$	−19.7269	−1.28684
$236$	−8.40051	−0.546827
$237$	1.18115	0.0767237
$238$	31.7609	2.05875
$239$	10.5421	0.681915	0.340957	−	0.940079i	$-0.389249\pi$
$239$	10.5421	0.681915	0.340957	+	0.940079i	$0.389249\pi$
$240$	−0.848619	−0.0547781
$241$	5.76879	0.371600	0.185800	−	0.982588i	$-0.440512\pi$
$241$	5.76879	0.371600	0.185800	+	0.982588i	$0.440512\pi$
$242$	−10.9889	−0.706395
$243$	12.0952	0.775910
$244$	10.3621	0.663363
$245$	−17.0704	−1.09059
$246$	−0.866865	−0.0552693
$247$	0	0
$248$	7.51861	0.477432
$249$	3.74480	0.237317
$250$	12.0665	0.763154
$251$	26.7972	1.69142	0.845711	−	0.533641i	$-0.179177\pi$
$251$	26.7972	1.69142	0.845711	+	0.533641i	$0.179177\pi$
$252$	11.3667	0.716033
$253$	−0.233255	−0.0146646
$254$	−4.46649	−0.280253
$255$	6.51251	0.407829
$256$	1.00000	0.0625000
$257$	−7.26588	−0.453233	−0.226617	−	0.973984i	$-0.572766\pi$
$257$	−7.26588	−0.453233	−0.226617	+	0.973984i	$0.572766\pi$
$258$	−3.15403	−0.196362
$259$	−34.8596	−2.16607
$260$	0	0
$261$	−7.78975	−0.482173
$262$	2.14974	0.132811
$263$	7.65776	0.472197	0.236099	−	0.971729i	$-0.424131\pi$
$263$	7.65776	0.472197	0.236099	+	0.971729i	$0.424131\pi$
$264$	−0.0529748	−0.00326038
$265$	11.3261	0.695759
$266$	4.13863	0.253756
$267$	−6.85521	−0.419532
$268$	8.21110	0.501573
$269$	26.2048	1.59773	0.798866	−	0.601508i	$-0.205433\pi$
$269$	26.2048	1.59773	0.798866	+	0.601508i	$0.205433\pi$
$270$	4.87657	0.296779
$271$	−25.8382	−1.56956	−0.784780	−	0.619775i	$-0.787224\pi$
$271$	−25.8382	−1.56956	−0.784780	+	0.619775i	$0.787224\pi$
$272$	−7.67425	−0.465320
$273$	0	0
$274$	−1.76415	−0.106576
$275$	0.227190	0.0137001
$276$	1.11627	0.0671915
$277$	1.69135	0.101624	0.0508118	−	0.998708i	$-0.483819\pi$
$277$	1.69135	0.101624	0.0508118	+	0.998708i	$0.483819\pi$
$278$	−17.4235	−1.04499
$279$	−20.6497	−1.23627
$280$	6.97534	0.416856
$281$	4.02634	0.240191	0.120096	−	0.992762i	$-0.461680\pi$
$281$	4.02634	0.240191	0.120096	+	0.992762i	$0.461680\pi$
$282$	5.89325	0.350938
$283$	−1.42614	−0.0847755	−0.0423877	−	0.999101i	$-0.513496\pi$
$283$	−1.42614	−0.0847755	−0.0423877	+	0.999101i	$0.513496\pi$
$284$	−3.68131	−0.218446
$285$	0.848619	0.0502678
$286$	0	0
$287$	7.12532	0.420594
$288$	−2.74648	−0.161838
$289$	41.8942	2.46436
$290$	−4.78030	−0.280709
$291$	7.56022	0.443188
$292$	9.69867	0.567571
$293$	1.95366	0.114134	0.0570671	−	0.998370i	$-0.481825\pi$
$293$	1.95366	0.114134	0.0570671	+	0.998370i	$0.481825\pi$
$294$	5.09963	0.297417
$295$	14.1584	0.824334
$296$	8.42298	0.489576
$297$	0.304419	0.0176642
$298$	−2.13533	−0.123696
$299$	0	0
$300$	−1.08725	−0.0627722
$301$	25.9250	1.49429
$302$	−7.28016	−0.418926
$303$	3.53672	0.203179
$304$	−1.00000	−0.0573539
$305$	−17.4644	−1.00001
$306$	21.0772	1.20490
$307$	10.9314	0.623888	0.311944	−	0.950100i	$-0.399020\pi$
$307$	10.9314	0.623888	0.311944	+	0.950100i	$0.399020\pi$
$308$	0.435434	0.0248112
$309$	−2.60724	−0.148321
$310$	−12.6720	−0.719722
$311$	−1.96306	−0.111315	−0.0556576	−	0.998450i	$-0.517726\pi$
$311$	−1.96306	−0.111315	−0.0556576	+	0.998450i	$0.517726\pi$
$312$	0	0
$313$	−13.1657	−0.744170	−0.372085	−	0.928199i	$-0.621357\pi$
$313$	−13.1657	−0.744170	−0.372085	+	0.928199i	$0.621357\pi$
$314$	11.5683	0.652839
$315$	−19.1576	−1.07941
$316$	2.34585	0.131964
$317$	−3.60302	−0.202366	−0.101183	−	0.994868i	$-0.532263\pi$
$317$	−3.60302	−0.202366	−0.101183	+	0.994868i	$0.532263\pi$
$318$	−3.38359	−0.189742
$319$	−0.298409	−0.0167077
$320$	−1.68542	−0.0942179
$321$	−7.55768	−0.421828
$322$	−9.17533	−0.511321
$323$	7.67425	0.427007
$324$	6.78261	0.376812
$325$	0	0
$326$	−9.51467	−0.526969
$327$	9.86281	0.545415
$328$	−1.72166	−0.0950628
$329$	−48.4404	−2.67060
$330$	0.0892849	0.00491497
$331$	29.4952	1.62120	0.810601	−	0.585599i	$-0.199141\pi$
$331$	29.4952	1.62120	0.810601	+	0.585599i	$0.199141\pi$
$332$	7.43747	0.408184
$333$	−23.1336	−1.26771
$334$	−7.54977	−0.413105
$335$	−13.8392	−0.756114
$336$	−2.08382	−0.113682
$337$	10.2346	0.557515	0.278758	−	0.960361i	$-0.410077\pi$
$337$	10.2346	0.557515	0.278758	+	0.960361i	$0.410077\pi$
$338$	0	0
$339$	1.78829	0.0971264
$340$	12.9344	0.701464
$341$	−0.791048	−0.0428377
$342$	2.74648	0.148513
$343$	−12.9467	−0.699058
$344$	−6.26415	−0.337740
$345$	−1.88138	−0.101290
$346$	4.26730	0.229412
$347$	27.1776	1.45897	0.729486	−	0.683996i	$-0.239759\pi$
$347$	27.1776	1.45897	0.729486	+	0.683996i	$0.239759\pi$
$348$	1.42807	0.0765528
$349$	8.44407	0.452001	0.226000	−	0.974127i	$-0.427435\pi$
$349$	8.44407	0.452001	0.226000	+	0.974127i	$0.427435\pi$
$350$	8.93678	0.477691
$351$	0	0
$352$	−0.105212	−0.00560782
$353$	−21.1378	−1.12505	−0.562527	−	0.826779i	$-0.690171\pi$
$353$	−21.1378	−1.12505	−0.562527	+	0.826779i	$0.690171\pi$
$354$	−4.22970	−0.224806
$355$	6.20456	0.329304
$356$	−13.6150	−0.721591
$357$	15.9918	0.846375
$358$	7.20267	0.380673
$359$	−18.5109	−0.976966	−0.488483	−	0.872574i	$-0.662449\pi$
$359$	−18.5109	−0.976966	−0.488483	+	0.872574i	$0.662449\pi$
$360$	4.62898	0.243969
$361$	1.00000	0.0526316
$362$	−4.87492	−0.256220
$363$	−5.53298	−0.290406
$364$	0	0
$365$	−16.3463	−0.855606
$366$	5.21735	0.272715
$367$	−4.27536	−0.223172	−0.111586	−	0.993755i	$-0.535593\pi$
$367$	−4.27536	−0.223172	−0.111586	+	0.993755i	$0.535593\pi$
$368$	2.21700	0.115569
$369$	4.72851	0.246156
$370$	−14.1963	−0.738029
$371$	27.8119	1.44392
$372$	3.78566	0.196277
$373$	32.4058	1.67791	0.838955	−	0.544201i	$-0.183167\pi$
$373$	32.4058	1.67791	0.838955	+	0.544201i	$0.183167\pi$
$374$	0.807424	0.0417509
$375$	6.07556	0.313741
$376$	11.7044	0.603611
$377$	0	0
$378$	11.9746	0.615910
$379$	−3.42614	−0.175989	−0.0879945	−	0.996121i	$-0.528046\pi$
$379$	−3.42614	−0.175989	−0.0879945	+	0.996121i	$0.528046\pi$
$380$	1.68542	0.0864603
$381$	−2.24890	−0.115215
$382$	11.2354	0.574853
$383$	−0.702625	−0.0359024	−0.0179512	−	0.999839i	$-0.505714\pi$
$383$	−0.702625	−0.0359024	−0.0179512	+	0.999839i	$0.505714\pi$
$384$	0.503505	0.0256944
$385$	−0.733890	−0.0374025
$386$	−3.90822	−0.198923
$387$	17.2044	0.874548
$388$	15.0152	0.762280
$389$	−28.5651	−1.44831	−0.724153	−	0.689639i	$-0.757769\pi$
$389$	−28.5651	−1.44831	−0.724153	+	0.689639i	$0.757769\pi$
$390$	0	0
$391$	−17.0138	−0.860424
$392$	10.1283	0.511555
$393$	1.08241	0.0546001
$394$	−19.4807	−0.981425
$395$	−3.95374	−0.198934
$396$	0.288963	0.0145209
$397$	−12.4904	−0.626876	−0.313438	−	0.949609i	$-0.601481\pi$
$397$	−12.4904	−0.626876	−0.313438	+	0.949609i	$0.601481\pi$
$398$	−23.6053	−1.18323
$399$	2.08382	0.104322
$400$	−2.15936	−0.107968
$401$	37.8153	1.88841	0.944203	−	0.329364i	$-0.106834\pi$
$401$	37.8153	1.88841	0.944203	+	0.329364i	$0.106834\pi$
$402$	4.13433	0.206202
$403$	0	0
$404$	7.02420	0.349467
$405$	−11.4316	−0.568039
$406$	−11.7382	−0.582559
$407$	−0.886199	−0.0439273
$408$	−3.86403	−0.191298
$409$	19.1438	0.946601	0.473300	−	0.880901i	$-0.343063\pi$
$409$	19.1438	0.946601	0.473300	+	0.880901i	$0.343063\pi$
$410$	2.90172	0.143306
$411$	−0.888259	−0.0438146
$412$	−5.17818	−0.255111
$413$	34.7666	1.71075
$414$	−6.08894	−0.299255
$415$	−12.5353	−0.615332
$416$	0	0
$417$	−8.77282	−0.429607
$418$	0.105212	0.00514609
$419$	−27.7446	−1.35541	−0.677707	−	0.735332i	$-0.737026\pi$
$419$	−27.7446	−1.35541	−0.677707	+	0.735332i	$0.737026\pi$
$420$	3.51212	0.171374
$421$	−20.1628	−0.982675	−0.491337	−	0.870969i	$-0.663492\pi$
$421$	−20.1628	−0.982675	−0.491337	+	0.870969i	$0.663492\pi$
$422$	15.7872	0.768508
$423$	−32.1461	−1.56299
$424$	−6.72006	−0.326355
$425$	16.5714	0.803833
$426$	−1.85356	−0.0898052
$427$	−42.8847	−2.07534
$428$	−15.0101	−0.725542
$429$	0	0
$430$	10.5577	0.509139
$431$	11.8978	0.573098	0.286549	−	0.958066i	$-0.407492\pi$
$431$	11.8978	0.573098	0.286549	+	0.958066i	$0.407492\pi$
$432$	−2.89338	−0.139208
$433$	−38.9700	−1.87278	−0.936389	−	0.350964i	$-0.885854\pi$
$433$	−38.9700	−1.87278	−0.936389	+	0.350964i	$0.885854\pi$
$434$	−31.1167	−1.49365
$435$	−2.40691	−0.115402
$436$	19.5883	0.938109
$437$	−2.21700	−0.106053
$438$	4.88333	0.233335
$439$	34.2618	1.63523	0.817613	−	0.575769i	$-0.195297\pi$
$439$	34.2618	1.63523	0.817613	+	0.575769i	$0.195297\pi$
$440$	0.177327	0.00845372
$441$	−27.8171	−1.32462
$442$	0	0
$443$	−29.7007	−1.41112	−0.705561	−	0.708649i	$-0.749305\pi$
$443$	−29.7007	−1.41112	−0.705561	+	0.708649i	$0.749305\pi$
$444$	4.24102	0.201270
$445$	22.9469	1.08779
$446$	15.1196	0.715932
$447$	−1.07515	−0.0508528
$448$	−4.13863	−0.195532
$449$	−16.7941	−0.792561	−0.396280	−	0.918130i	$-0.629699\pi$
$449$	−16.7941	−0.792561	−0.396280	+	0.918130i	$0.629699\pi$
$450$	5.93063	0.279573
$451$	0.181139	0.00852953
$452$	3.55168	0.167057
$453$	−3.66560	−0.172225
$454$	20.1459	0.945493
$455$	0	0
$456$	−0.503505	−0.0235788
$457$	24.6585	1.15348	0.576738	−	0.816929i	$-0.304326\pi$
$457$	24.6585	1.15348	0.576738	+	0.816929i	$0.304326\pi$
$458$	−27.2632	−1.27393
$459$	22.2046	1.03642
$460$	−3.73657	−0.174219
$461$	−12.9611	−0.603660	−0.301830	−	0.953362i	$-0.597598\pi$
$461$	−12.9611	−0.603660	−0.301830	+	0.953362i	$0.597598\pi$
$462$	0.219243	0.0102001
$463$	24.3921	1.13360	0.566799	−	0.823856i	$-0.308182\pi$
$463$	24.3921	1.13360	0.566799	+	0.823856i	$0.308182\pi$
$464$	2.83626	0.131670
$465$	−6.38043	−0.295885
$466$	22.0907	1.02333
$467$	−4.51918	−0.209123	−0.104561	−	0.994518i	$-0.533344\pi$
$467$	−4.51918	−0.209123	−0.104561	+	0.994518i	$0.533344\pi$
$468$	0	0
$469$	−33.9827	−1.56918
$470$	−19.7269	−0.909935
$471$	5.82472	0.268389
$472$	−8.40051	−0.386665
$473$	0.659064	0.0303038
$474$	1.18115	0.0542519
$475$	2.15936	0.0990780
$476$	31.7609	1.45576
$477$	18.4565	0.845067
$478$	10.5421	0.482187
$479$	−23.7974	−1.08733	−0.543666	−	0.839302i	$-0.682964\pi$
$479$	−23.7974	−1.08733	−0.543666	+	0.839302i	$0.682964\pi$
$480$	−0.848619	−0.0387340
$481$	0	0
$482$	5.76879	0.262761
$483$	−4.61983	−0.210209
$484$	−10.9889	−0.499497
$485$	−25.3069	−1.14913
$486$	12.0952	0.548651
$487$	37.2022	1.68579	0.842895	−	0.538078i	$-0.180849\pi$
$487$	37.2022	1.68579	0.842895	+	0.538078i	$0.180849\pi$
$488$	10.3621	0.469068
$489$	−4.79069	−0.216642
$490$	−17.0704	−0.771162
$491$	−3.79803	−0.171402	−0.0857012	−	0.996321i	$-0.527313\pi$
$491$	−3.79803	−0.171402	−0.0857012	+	0.996321i	$0.527313\pi$
$492$	−0.866865	−0.0390813
$493$	−21.7662	−0.980301
$494$	0	0
$495$	−0.487025	−0.0218901
$496$	7.51861	0.337595
$497$	15.2356	0.683409
$498$	3.74480	0.167809
$499$	−4.06253	−0.181864	−0.0909320	−	0.995857i	$-0.528985\pi$
$499$	−4.06253	−0.181864	−0.0909320	+	0.995857i	$0.528985\pi$
$500$	12.0665	0.539632
$501$	−3.80135	−0.169832
$502$	26.7972	1.19602
$503$	−13.2381	−0.590257	−0.295129	−	0.955457i	$-0.595363\pi$
$503$	−13.2381	−0.590257	−0.295129	+	0.955457i	$0.595363\pi$
$504$	11.3667	0.506312
$505$	−11.8387	−0.526817
$506$	−0.233255	−0.0103694
$507$	0	0
$508$	−4.46649	−0.198169
$509$	−4.59071	−0.203480	−0.101740	−	0.994811i	$-0.532441\pi$
$509$	−4.59071	−0.203480	−0.101740	+	0.994811i	$0.532441\pi$
$510$	6.51251	0.288379
$511$	−40.1392	−1.77565
$512$	1.00000	0.0441942
$513$	2.89338	0.127746
$514$	−7.26588	−0.320484
$515$	8.72741	0.384576
$516$	−3.15403	−0.138849
$517$	−1.23145	−0.0541591
$518$	−34.8596	−1.53164
$519$	2.14861	0.0943135
$520$	0	0
$521$	−31.7856	−1.39255	−0.696277	−	0.717773i	$-0.745161\pi$
$521$	−31.7856	−1.39255	−0.696277	+	0.717773i	$0.745161\pi$
$522$	−7.78975	−0.340948
$523$	26.3819	1.15360	0.576800	−	0.816885i	$-0.304301\pi$
$523$	26.3819	1.15360	0.576800	+	0.816885i	$0.304301\pi$
$524$	2.14974	0.0939118
$525$	4.49971	0.196384
$526$	7.65776	0.333894
$527$	−57.6997	−2.51344
$528$	−0.0529748	−0.00230543
$529$	−18.0849	−0.786301
$530$	11.3261	0.491976
$531$	23.0719	1.00123
$532$	4.13863	0.179432
$533$	0	0
$534$	−6.85521	−0.296654
$535$	25.2984	1.09374
$536$	8.21110	0.354666
$537$	3.62658	0.156499
$538$	26.2048	1.12977
$539$	−1.06562	−0.0458993
$540$	4.87657	0.209854
$541$	−9.07616	−0.390215	−0.195107	−	0.980782i	$-0.562505\pi$
$541$	−9.07616	−0.390215	−0.195107	+	0.980782i	$0.562505\pi$
$542$	−25.8382	−1.10985
$543$	−2.45455	−0.105335
$544$	−7.67425	−0.329031
$545$	−33.0145	−1.41419
$546$	0	0
$547$	26.0681	1.11459	0.557295	−	0.830315i	$-0.311839\pi$
$547$	26.0681	1.11459	0.557295	+	0.830315i	$0.311839\pi$
$548$	−1.76415	−0.0753608
$549$	−28.4592	−1.21461
$550$	0.227190	0.00968743
$551$	−2.83626	−0.120829
$552$	1.11627	0.0475116
$553$	−9.70859	−0.412851
$554$	1.69135	0.0718588
$555$	−7.14790	−0.303411
$556$	−17.4235	−0.738921
$557$	−13.9336	−0.590387	−0.295193	−	0.955437i	$-0.595384\pi$
$557$	−13.9336	−0.590387	−0.295193	+	0.955437i	$0.595384\pi$
$558$	−20.6497	−0.874172
$559$	0	0
$560$	6.97534	0.294762
$561$	0.406542	0.0171642
$562$	4.02634	0.169841
$563$	16.6253	0.700672	0.350336	−	0.936624i	$-0.386067\pi$
$563$	16.6253	0.700672	0.350336	+	0.936624i	$0.386067\pi$
$564$	5.89325	0.248151
$565$	−5.98607	−0.251836
$566$	−1.42614	−0.0599453
$567$	−28.0707	−1.17886
$568$	−3.68131	−0.154464
$569$	−26.0979	−1.09408	−0.547040	−	0.837107i	$-0.684245\pi$
$569$	−26.0979	−1.09408	−0.547040	+	0.837107i	$0.684245\pi$
$570$	0.848619	0.0355447
$571$	18.7099	0.782985	0.391493	−	0.920181i	$-0.371959\pi$
$571$	18.7099	0.782985	0.391493	+	0.920181i	$0.371959\pi$
$572$	0	0
$573$	5.65708	0.236328
$574$	7.12532	0.297405
$575$	−4.78728	−0.199644
$576$	−2.74648	−0.114437
$577$	−12.0977	−0.503636	−0.251818	−	0.967775i	$-0.581028\pi$
$577$	−12.0977	−0.503636	−0.251818	+	0.967775i	$0.581028\pi$
$578$	41.8942	1.74257
$579$	−1.96781	−0.0817794
$580$	−4.78030	−0.198491
$581$	−30.7809	−1.27701
$582$	7.56022	0.313381
$583$	0.707032	0.0292823
$584$	9.69867	0.401334
$585$	0	0
$586$	1.95366	0.0807051
$587$	23.4033	0.965957	0.482979	−	0.875632i	$-0.339555\pi$
$587$	23.4033	0.965957	0.482979	+	0.875632i	$0.339555\pi$
$588$	5.09963	0.210305
$589$	−7.51861	−0.309799
$590$	14.1584	0.582892
$591$	−9.80865	−0.403474
$592$	8.42298	0.346182
$593$	40.0823	1.64598	0.822992	−	0.568052i	$-0.192303\pi$
$593$	40.0823	1.64598	0.822992	+	0.568052i	$0.192303\pi$
$594$	0.304419	0.0124905
$595$	−53.5305	−2.19454
$596$	−2.13533	−0.0874664
$597$	−11.8854	−0.486437
$598$	0	0
$599$	−20.2143	−0.825933	−0.412967	−	0.910746i	$-0.635507\pi$
$599$	−20.2143	−0.825933	−0.412967	+	0.910746i	$0.635507\pi$
$600$	−1.08725	−0.0443867
$601$	−43.3937	−1.77007	−0.885033	−	0.465529i	$-0.845864\pi$
$601$	−43.3937	−1.77007	−0.885033	+	0.465529i	$0.845864\pi$
$602$	25.9250	1.05662
$603$	−22.5516	−0.918374
$604$	−7.28016	−0.296225
$605$	18.5210	0.752985
$606$	3.53672	0.143670
$607$	4.30877	0.174887	0.0874437	−	0.996169i	$-0.472130\pi$
$607$	4.30877	0.174887	0.0874437	+	0.996169i	$0.472130\pi$
$608$	−1.00000	−0.0405554
$609$	−5.91027	−0.239496
$610$	−17.4644	−0.707114
$611$	0	0
$612$	21.0772	0.851995
$613$	−2.33577	−0.0943410	−0.0471705	−	0.998887i	$-0.515020\pi$
$613$	−2.33577	−0.0943410	−0.0471705	+	0.998887i	$0.515020\pi$
$614$	10.9314	0.441155
$615$	1.46103	0.0589145
$616$	0.435434	0.0175441
$617$	32.2319	1.29761	0.648804	−	0.760956i	$-0.275270\pi$
$617$	32.2319	1.29761	0.648804	+	0.760956i	$0.275270\pi$
$618$	−2.60724	−0.104879
$619$	−0.299204	−0.0120260	−0.00601301	−	0.999982i	$-0.501914\pi$
$619$	−0.299204	−0.0120260	−0.00601301	+	0.999982i	$0.501914\pi$
$620$	−12.6720	−0.508921
$621$	−6.41462	−0.257410
$622$	−1.96306	−0.0787117
$623$	56.3473	2.25751
$624$	0	0
$625$	−9.54041	−0.381616
$626$	−13.1657	−0.526208
$627$	0.0529748	0.00211561
$628$	11.5683	0.461627
$629$	−64.6401	−2.57737
$630$	−19.1576	−0.763259
$631$	34.2791	1.36463	0.682314	−	0.731059i	$-0.260974\pi$
$631$	34.2791	1.36463	0.682314	+	0.731059i	$0.260974\pi$
$632$	2.34585	0.0933128
$633$	7.94893	0.315942
$634$	−3.60302	−0.143094
$635$	7.52792	0.298736
$636$	−3.38359	−0.134168
$637$	0	0
$638$	−0.298409	−0.0118141
$639$	10.1107	0.399971
$640$	−1.68542	−0.0666221
$641$	−25.6616	−1.01357	−0.506785	−	0.862072i	$-0.669166\pi$
$641$	−25.6616	−1.01357	−0.506785	+	0.862072i	$0.669166\pi$
$642$	−7.55768	−0.298278
$643$	−29.8045	−1.17537	−0.587687	−	0.809088i	$-0.699961\pi$
$643$	−29.8045	−1.17537	−0.587687	+	0.809088i	$0.699961\pi$
$644$	−9.17533	−0.361559
$645$	5.31587	0.209312
$646$	7.67425	0.301939
$647$	1.50554	0.0591889	0.0295945	−	0.999562i	$-0.490578\pi$
$647$	1.50554	0.0591889	0.0295945	+	0.999562i	$0.490578\pi$
$648$	6.78261	0.266446
$649$	0.883835	0.0346936
$650$	0	0
$651$	−15.6674	−0.614056
$652$	−9.51467	−0.372623
$653$	−6.18241	−0.241936	−0.120968	−	0.992656i	$-0.538600\pi$
$653$	−6.18241	−0.241936	−0.120968	+	0.992656i	$0.538600\pi$
$654$	9.86281	0.385666
$655$	−3.62322	−0.141571
$656$	−1.72166	−0.0672195
$657$	−26.6372	−1.03922
$658$	−48.4404	−1.88840
$659$	15.9141	0.619926	0.309963	−	0.950749i	$-0.399683\pi$
$659$	15.9141	0.619926	0.309963	+	0.950749i	$0.399683\pi$
$660$	0.0892849	0.00347541
$661$	39.4955	1.53620	0.768098	−	0.640332i	$-0.221203\pi$
$661$	39.4955	1.53620	0.768098	+	0.640332i	$0.221203\pi$
$662$	29.4952	1.14636
$663$	0	0
$664$	7.43747	0.288630
$665$	−6.97534	−0.270492
$666$	−23.1336	−0.896408
$667$	6.28799	0.243472
$668$	−7.54977	−0.292109
$669$	7.61278	0.294327
$670$	−13.8392	−0.534654
$671$	−1.09021	−0.0420872
$672$	−2.08382	−0.0803852
$673$	22.8201	0.879651	0.439825	−	0.898083i	$-0.355040\pi$
$673$	22.8201	0.879651	0.439825	+	0.898083i	$0.355040\pi$
$674$	10.2346	0.394223
$675$	6.24785	0.240480
$676$	0	0
$677$	45.6646	1.75503	0.877516	−	0.479547i	$-0.159199\pi$
$677$	45.6646	1.75503	0.877516	+	0.479547i	$0.159199\pi$
$678$	1.78829	0.0686788
$679$	−62.1423	−2.38480
$680$	12.9344	0.496010
$681$	10.1436	0.388702
$682$	−0.791048	−0.0302908
$683$	43.3849	1.66008	0.830039	−	0.557706i	$-0.188318\pi$
$683$	43.3849	1.66008	0.830039	+	0.557706i	$0.188318\pi$
$684$	2.74648	0.105014
$685$	2.97334	0.113605
$686$	−12.9467	−0.494308
$687$	−13.7272	−0.523725
$688$	−6.26415	−0.238819
$689$	0	0
$690$	−1.88138	−0.0716231
$691$	−35.7113	−1.35852	−0.679262	−	0.733896i	$-0.737700\pi$
$691$	−35.7113	−1.35852	−0.679262	+	0.733896i	$0.737700\pi$
$692$	4.26730	0.162219
$693$	−1.19591	−0.0454289
$694$	27.1776	1.03165
$695$	29.3659	1.11391
$696$	1.42807	0.0541310
$697$	13.2125	0.500458
$698$	8.44407	0.319613
$699$	11.1228	0.420702
$700$	8.93678	0.337778
$701$	26.8620	1.01456	0.507282	−	0.861780i	$-0.330650\pi$
$701$	26.8620	1.01456	0.507282	+	0.861780i	$0.330650\pi$
$702$	0	0
$703$	−8.42298	−0.317679
$704$	−0.105212	−0.00396533
$705$	−9.93261	−0.374084
$706$	−21.1378	−0.795533
$707$	−29.0706	−1.09331
$708$	−4.22970	−0.158962
$709$	1.22437	0.0459820	0.0229910	−	0.999736i	$-0.492681\pi$
$709$	1.22437	0.0459820	0.0229910	+	0.999736i	$0.492681\pi$
$710$	6.20456	0.232853
$711$	−6.44283	−0.241625
$712$	−13.6150	−0.510242
$713$	16.6687	0.624249
$714$	15.9918	0.598477
$715$	0	0
$716$	7.20267	0.269176
$717$	5.30803	0.198232
$718$	−18.5109	−0.690819
$719$	53.2571	1.98615	0.993077	−	0.117463i	$-0.0374762\pi$
$719$	53.2571	1.98615	0.993077	+	0.117463i	$0.0374762\pi$
$720$	4.62898	0.172512
$721$	21.4306	0.798116
$722$	1.00000	0.0372161
$723$	2.90461	0.108024
$724$	−4.87492	−0.181175
$725$	−6.12450	−0.227458
$726$	−5.53298	−0.205348
$727$	13.2509	0.491447	0.245723	−	0.969340i	$-0.420974\pi$
$727$	13.2509	0.491447	0.245723	+	0.969340i	$0.420974\pi$
$728$	0	0
$729$	−14.2578	−0.528068
$730$	−16.3463	−0.605005
$731$	48.0727	1.77803
$732$	5.21735	0.192839
$733$	18.7782	0.693587	0.346794	−	0.937941i	$-0.387270\pi$
$733$	18.7782	0.693587	0.346794	+	0.937941i	$0.387270\pi$
$734$	−4.27536	−0.157806
$735$	−8.59503	−0.317033
$736$	2.21700	0.0817196
$737$	−0.863907	−0.0318224
$738$	4.72851	0.174059
$739$	25.6218	0.942514	0.471257	−	0.881996i	$-0.343800\pi$
$739$	25.6218	0.942514	0.471257	+	0.881996i	$0.343800\pi$
$740$	−14.1963	−0.521865
$741$	0	0
$742$	27.8119	1.02101
$743$	−6.00483	−0.220296	−0.110148	−	0.993915i	$-0.535132\pi$
$743$	−6.00483	−0.220296	−0.110148	+	0.993915i	$0.535132\pi$
$744$	3.78566	0.138789
$745$	3.59892	0.131854
$746$	32.4058	1.18646
$747$	−20.4269	−0.747380
$748$	0.807424	0.0295224
$749$	62.1213	2.26986
$750$	6.07556	0.221848
$751$	−28.7640	−1.04961	−0.524806	−	0.851222i	$-0.675862\pi$
$751$	−28.7640	−1.04961	−0.524806	+	0.851222i	$0.675862\pi$
$752$	11.7044	0.426817
$753$	13.4925	0.491695
$754$	0	0
$755$	12.2701	0.446556
$756$	11.9746	0.435514
$757$	17.3593	0.630933	0.315466	−	0.948937i	$-0.397839\pi$
$757$	17.3593	0.630933	0.315466	+	0.948937i	$0.397839\pi$
$758$	−3.42614	−0.124443
$759$	−0.117445	−0.00426298
$760$	1.68542	0.0611367
$761$	8.39422	0.304290	0.152145	−	0.988358i	$-0.451382\pi$
$761$	8.39422	0.304290	0.152145	+	0.988358i	$0.451382\pi$
$762$	−2.24890	−0.0814691
$763$	−81.0687	−2.93488
$764$	11.2354	0.406482
$765$	−35.5240	−1.28437
$766$	−0.702625	−0.0253869
$767$	0	0
$768$	0.503505	0.0181687
$769$	−44.1506	−1.59211	−0.796056	−	0.605223i	$-0.793084\pi$
$769$	−44.1506	−1.59211	−0.796056	+	0.605223i	$0.793084\pi$
$770$	−0.733890	−0.0264475
$771$	−3.65841	−0.131754
$772$	−3.90822	−0.140660
$773$	2.83554	0.101987	0.0509936	−	0.998699i	$-0.483761\pi$
$773$	2.83554	0.101987	0.0509936	+	0.998699i	$0.483761\pi$
$774$	17.2044	0.618399
$775$	−16.2353	−0.583191
$776$	15.0152	0.539013
$777$	−17.5520	−0.629674
$778$	−28.5651	−1.02411
$779$	1.72166	0.0616849
$780$	0	0
$781$	0.387318	0.0138593
$782$	−17.0138	−0.608412
$783$	−8.20640	−0.293273
$784$	10.1283	0.361724
$785$	−19.4975	−0.695896
$786$	1.08241	0.0386081
$787$	30.1709	1.07548	0.537738	−	0.843112i	$-0.319279\pi$
$787$	30.1709	1.07548	0.537738	+	0.843112i	$0.319279\pi$
$788$	−19.4807	−0.693972
$789$	3.85572	0.137267
$790$	−3.95374	−0.140668
$791$	−14.6991	−0.522639
$792$	0.288963	0.0102679
$793$	0	0
$794$	−12.4904	−0.443268
$795$	5.70277	0.202256
$796$	−23.6053	−0.836668
$797$	−5.21641	−0.184775	−0.0923874	−	0.995723i	$-0.529450\pi$
$797$	−5.21641	−0.184775	−0.0923874	+	0.995723i	$0.529450\pi$
$798$	2.08382	0.0737665
$799$	−89.8229	−3.17771
$800$	−2.15936	−0.0763447
$801$	37.3933	1.32123
$802$	37.8153	1.33530
$803$	−1.02042	−0.0360097
$804$	4.13433	0.145807
$805$	15.4643	0.545045
$806$	0	0
$807$	13.1942	0.464459
$808$	7.02420	0.247110
$809$	−12.0318	−0.423017	−0.211509	−	0.977376i	$-0.567838\pi$
$809$	−12.0318	−0.423017	−0.211509	+	0.977376i	$0.567838\pi$
$810$	−11.4316	−0.401664
$811$	29.5309	1.03697	0.518485	−	0.855086i	$-0.326496\pi$
$811$	29.5309	1.03697	0.518485	+	0.855086i	$0.326496\pi$
$812$	−11.7382	−0.411932
$813$	−13.0097	−0.456269
$814$	−0.886199	−0.0310613
$815$	16.0362	0.561725
$816$	−3.86403	−0.135268
$817$	6.26415	0.219155
$818$	19.1438	0.669348
$819$	0	0
$820$	2.90172	0.101333
$821$	3.99370	0.139381	0.0696905	−	0.997569i	$-0.477799\pi$
$821$	3.99370	0.139381	0.0696905	+	0.997569i	$0.477799\pi$
$822$	−0.888259	−0.0309816
$823$	10.2029	0.355649	0.177825	−	0.984062i	$-0.443094\pi$
$823$	10.2029	0.355649	0.177825	+	0.984062i	$0.443094\pi$
$824$	−5.17818	−0.180390
$825$	0.114392	0.00398260
$826$	34.7666	1.20968
$827$	45.1316	1.56938	0.784690	−	0.619888i	$-0.212822\pi$
$827$	45.1316	1.56938	0.784690	+	0.619888i	$0.212822\pi$
$828$	−6.08894	−0.211605
$829$	−7.57879	−0.263222	−0.131611	−	0.991301i	$-0.542015\pi$
$829$	−7.57879	−0.263222	−0.131611	+	0.991301i	$0.542015\pi$
$830$	−12.5353	−0.435105
$831$	0.851606	0.0295419
$832$	0	0
$833$	−77.7269	−2.69308
$834$	−8.77282	−0.303778
$835$	12.7245	0.440351
$836$	0.105212	0.00363884
$837$	−21.7542	−0.751936
$838$	−27.7446	−0.958422
$839$	−2.46231	−0.0850084	−0.0425042	−	0.999096i	$-0.513534\pi$
$839$	−2.46231	−0.0850084	−0.0425042	+	0.999096i	$0.513534\pi$
$840$	3.51212	0.121180
$841$	−20.9556	−0.722607
$842$	−20.1628	−0.694856
$843$	2.02728	0.0698233
$844$	15.7872	0.543417
$845$	0	0
$846$	−32.1461	−1.10520
$847$	45.4791	1.56268
$848$	−6.72006	−0.230768
$849$	−0.718071	−0.0246441
$850$	16.5714	0.568396
$851$	18.6737	0.640127
$852$	−1.85356	−0.0635019
$853$	7.50414	0.256937	0.128468	−	0.991714i	$-0.458994\pi$
$853$	7.50414	0.256937	0.128468	+	0.991714i	$0.458994\pi$
$854$	−42.8847	−1.46749
$855$	−4.62898	−0.158308
$856$	−15.0101	−0.513035
$857$	11.6146	0.396747	0.198373	−	0.980127i	$-0.436434\pi$
$857$	11.6146	0.396747	0.198373	+	0.980127i	$0.436434\pi$
$858$	0	0
$859$	8.31854	0.283825	0.141912	−	0.989879i	$-0.454675\pi$
$859$	8.31854	0.283825	0.141912	+	0.989879i	$0.454675\pi$
$860$	10.5577	0.360016
$861$	3.58763	0.122266
$862$	11.8978	0.405242
$863$	26.2852	0.894759	0.447380	−	0.894344i	$-0.352357\pi$
$863$	26.2852	0.894759	0.447380	+	0.894344i	$0.352357\pi$
$864$	−2.89338	−0.0984349
$865$	−7.19221	−0.244542
$866$	−38.9700	−1.32425
$867$	21.0939	0.716388
$868$	−31.1167	−1.05617
$869$	−0.246811	−0.00837251
$870$	−2.40691	−0.0816017
$871$	0	0
$872$	19.5883	0.663343
$873$	−41.2389	−1.39573
$874$	−2.21700	−0.0749910
$875$	−49.9389	−1.68824
$876$	4.88333	0.164992
$877$	25.5547	0.862920	0.431460	−	0.902132i	$-0.357999\pi$
$877$	25.5547	0.862920	0.431460	+	0.902132i	$0.357999\pi$
$878$	34.2618	1.15628
$879$	0.983680	0.0331787
$880$	0.177327	0.00597768
$881$	14.6161	0.492427	0.246214	−	0.969216i	$-0.420814\pi$
$881$	14.6161	0.492427	0.246214	+	0.969216i	$0.420814\pi$
$882$	−27.8171	−0.936650
$883$	25.2624	0.850146	0.425073	−	0.905159i	$-0.360248\pi$
$883$	25.2624	0.850146	0.425073	+	0.905159i	$0.360248\pi$
$884$	0	0
$885$	7.12883	0.239633
$886$	−29.7007	−0.997814
$887$	−54.2692	−1.82218	−0.911091	−	0.412206i	$-0.864759\pi$
$887$	−54.2692	−1.82218	−0.911091	+	0.412206i	$0.864759\pi$
$888$	4.24102	0.142319
$889$	18.4852	0.619972
$890$	22.9469	0.769183
$891$	−0.713613	−0.0239069
$892$	15.1196	0.506240
$893$	−11.7044	−0.391674
$894$	−1.07515	−0.0359583
$895$	−12.1395	−0.405780
$896$	−4.13863	−0.138262
$897$	0	0
$898$	−16.7941	−0.560425
$899$	21.3248	0.711220
$900$	5.93063	0.197688
$901$	51.5715	1.71809
$902$	0.181139	0.00603128
$903$	13.0534	0.434389
$904$	3.55168	0.118127
$905$	8.21629	0.273119
$906$	−3.66560	−0.121781
$907$	−8.73036	−0.289887	−0.144943	−	0.989440i	$-0.546300\pi$
$907$	−8.73036	−0.289887	−0.144943	+	0.989440i	$0.546300\pi$
$908$	20.1459	0.668565
$909$	−19.2918	−0.639870
$910$	0	0
$911$	26.2149	0.868539	0.434269	−	0.900783i	$-0.357007\pi$
$911$	26.2149	0.868539	0.434269	+	0.900783i	$0.357007\pi$
$912$	−0.503505	−0.0166727
$913$	−0.782512	−0.0258974
$914$	24.6585	0.815630
$915$	−8.79344	−0.290702
$916$	−27.2632	−0.900803
$917$	−8.89698	−0.293804
$918$	22.2046	0.732860
$919$	46.4850	1.53340	0.766699	−	0.642007i	$-0.221898\pi$
$919$	46.4850	1.53340	0.766699	+	0.642007i	$0.221898\pi$
$920$	−3.73657	−0.123191
$921$	5.50402	0.181364
$922$	−12.9611	−0.426852
$923$	0	0
$924$	0.219243	0.00721258
$925$	−18.1882	−0.598025
$926$	24.3921	0.801575
$927$	14.2218	0.467104
$928$	2.83626	0.0931049
$929$	−25.9042	−0.849889	−0.424944	−	0.905220i	$-0.639706\pi$
$929$	−25.9042	−0.849889	−0.424944	+	0.905220i	$0.639706\pi$
$930$	−6.38043	−0.209223
$931$	−10.1283	−0.331940
$932$	22.0907	0.723605
$933$	−0.988414	−0.0323592
$934$	−4.51918	−0.147872
$935$	−1.36085	−0.0445046
$936$	0	0
$937$	9.41253	0.307494	0.153747	−	0.988110i	$-0.450866\pi$
$937$	9.41253	0.307494	0.153747	+	0.988110i	$0.450866\pi$
$938$	−33.9827	−1.10958
$939$	−6.62901	−0.216330
$940$	−19.7269	−0.643421
$941$	17.0644	0.556282	0.278141	−	0.960540i	$-0.410282\pi$
$941$	17.0644	0.556282	0.278141	+	0.960540i	$0.410282\pi$
$942$	5.82472	0.189780
$943$	−3.81691	−0.124296
$944$	−8.40051	−0.273413
$945$	−20.1823	−0.656531
$946$	0.659064	0.0214280
$947$	25.3286	0.823069	0.411535	−	0.911394i	$-0.364993\pi$
$947$	25.3286	0.823069	0.411535	+	0.911394i	$0.364993\pi$
$948$	1.18115	0.0383619
$949$	0	0
$950$	2.15936	0.0700587
$951$	−1.81414	−0.0588275
$952$	31.7609	1.02938
$953$	−23.4420	−0.759361	−0.379680	−	0.925118i	$-0.623966\pi$
$953$	−23.4420	−0.759361	−0.379680	+	0.925118i	$0.623966\pi$
$954$	18.4565	0.597552
$955$	−18.9364	−0.612767
$956$	10.5421	0.340957
$957$	−0.150251	−0.00485691
$958$	−23.7974	−0.768859
$959$	7.30117	0.235767
$960$	−0.848619	−0.0273890
$961$	25.5295	0.823531
$962$	0	0
$963$	41.2250	1.32846
$964$	5.76879	0.185800
$965$	6.58700	0.212043
$966$	−4.61983	−0.148640
$967$	26.8837	0.864520	0.432260	−	0.901749i	$-0.357716\pi$
$967$	26.8837	0.864520	0.432260	+	0.901749i	$0.357716\pi$
$968$	−10.9889	−0.353198
$969$	3.86403	0.124130
$970$	−25.3069	−0.812556
$971$	39.6985	1.27399	0.636993	−	0.770869i	$-0.280178\pi$
$971$	39.6985	1.27399	0.636993	+	0.770869i	$0.280178\pi$
$972$	12.0952	0.387955
$973$	72.1094	2.31172
$974$	37.2022	1.19203
$975$	0	0
$976$	10.3621	0.331681
$977$	2.66493	0.0852585	0.0426293	−	0.999091i	$-0.486427\pi$
$977$	2.66493	0.0852585	0.0426293	+	0.999091i	$0.486427\pi$
$978$	−4.79069	−0.153189
$979$	1.43246	0.0457816
$980$	−17.0704	−0.545294
$981$	−53.7989	−1.71767
$982$	−3.79803	−0.121200
$983$	−42.1002	−1.34279	−0.671393	−	0.741101i	$-0.734304\pi$
$983$	−42.1002	−1.34279	−0.671393	+	0.741101i	$0.734304\pi$
$984$	−0.866865	−0.0276347
$985$	32.8332	1.04615
$986$	−21.7662	−0.693177
$987$	−24.3900	−0.776342
$988$	0	0
$989$	−13.8876	−0.441600
$990$	−0.487025	−0.0154787
$991$	32.7540	1.04046	0.520232	−	0.854025i	$-0.325846\pi$
$991$	32.7540	1.04046	0.520232	+	0.854025i	$0.325846\pi$
$992$	7.51861	0.238716
$993$	14.8510	0.471282
$994$	15.2356	0.483243
$995$	39.7849	1.26127
$996$	3.74480	0.118659
$997$	10.7227	0.339592	0.169796	−	0.985479i	$-0.445689\pi$
$997$	10.7227	0.339592	0.169796	+	0.985479i	$0.445689\pi$
$998$	−4.06253	−0.128597
$999$	−24.3709	−0.771062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bq.1.9	yes	15
13.12	even	2		6422.2.a.bo.1.9	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bo.1.9	✓	15	13.12	even	2
6422.2.a.bq.1.9	yes	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.503505 q^{3} +1.00000 q^{4} -1.68542 q^{5} +0.503505 q^{6} -4.13863 q^{7} +1.00000 q^{8} -2.74648 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.503505 q^{3} +1.00000 q^{4} -1.68542 q^{5} +0.503505 q^{6} -4.13863 q^{7} +1.00000 q^{8} -2.74648 q^{9} -1.68542 q^{10} -0.105212 q^{11} +0.503505 q^{12} -4.13863 q^{14} -0.848619 q^{15} +1.00000 q^{16} -7.67425 q^{17} -2.74648 q^{18} -1.00000 q^{19} -1.68542 q^{20} -2.08382 q^{21} -0.105212 q^{22} +2.21700 q^{23} +0.503505 q^{24} -2.15936 q^{25} -2.89338 q^{27} -4.13863 q^{28} +2.83626 q^{29} -0.848619 q^{30} +7.51861 q^{31} +1.00000 q^{32} -0.0529748 q^{33} -7.67425 q^{34} +6.97534 q^{35} -2.74648 q^{36} +8.42298 q^{37} -1.00000 q^{38} -1.68542 q^{40} -1.72166 q^{41} -2.08382 q^{42} -6.26415 q^{43} -0.105212 q^{44} +4.62898 q^{45} +2.21700 q^{46} +11.7044 q^{47} +0.503505 q^{48} +10.1283 q^{49} -2.15936 q^{50} -3.86403 q^{51} -6.72006 q^{53} -2.89338 q^{54} +0.177327 q^{55} -4.13863 q^{56} -0.503505 q^{57} +2.83626 q^{58} -8.40051 q^{59} -0.848619 q^{60} +10.3621 q^{61} +7.51861 q^{62} +11.3667 q^{63} +1.00000 q^{64} -0.0529748 q^{66} +8.21110 q^{67} -7.67425 q^{68} +1.11627 q^{69} +6.97534 q^{70} -3.68131 q^{71} -2.74648 q^{72} +9.69867 q^{73} +8.42298 q^{74} -1.08725 q^{75} -1.00000 q^{76} +0.435434 q^{77} +2.34585 q^{79} -1.68542 q^{80} +6.78261 q^{81} -1.72166 q^{82} +7.43747 q^{83} -2.08382 q^{84} +12.9344 q^{85} -6.26415 q^{86} +1.42807 q^{87} -0.105212 q^{88} -13.6150 q^{89} +4.62898 q^{90} +2.21700 q^{92} +3.78566 q^{93} +11.7044 q^{94} +1.68542 q^{95} +0.503505 q^{96} +15.0152 q^{97} +10.1283 q^{98} +0.288963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 + 15 * q^4 + q^5 + 18 * q^7 + 15 * q^8 + 17 * q^9	\( 15 q + 15 q^{2} + 15 q^{4} + q^{5} + 18 q^{7} + 15 q^{8} + 17 q^{9} + q^{10} + 4 q^{11} + 18 q^{14} + 23 q^{15} + 15 q^{16} + 2 q^{17} + 17 q^{18} - 15 q^{19} + q^{20} + 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} + 18 q^{28} - 20 q^{29} + 23 q^{30} + 30 q^{31} + 15 q^{32} + 36 q^{33} + 2 q^{34} + 32 q^{35} + 17 q^{36} + 35 q^{37} - 15 q^{38} + q^{40} + 15 q^{41} + 2 q^{42} + q^{43} + 4 q^{44} - 11 q^{45} + 17 q^{46} + 29 q^{49} + 8 q^{50} - q^{51} - q^{53} + 12 q^{54} - 6 q^{55} + 18 q^{56} - 20 q^{58} - 7 q^{59} + 23 q^{60} - 2 q^{61} + 30 q^{62} + 42 q^{63} + 15 q^{64} + 36 q^{66} + 34 q^{67} + 2 q^{68} - 12 q^{69} + 32 q^{70} + 4 q^{71} + 17 q^{72} + 12 q^{73} + 35 q^{74} + 31 q^{75} - 15 q^{76} - 20 q^{77} + 23 q^{79} + q^{80} + 7 q^{81} + 15 q^{82} - 3 q^{83} + 2 q^{84} + 46 q^{85} + q^{86} + 22 q^{87} + 4 q^{88} + 17 q^{89} - 11 q^{90} + 17 q^{92} + 60 q^{93} - q^{95} + 18 q^{97} + 29 q^{98} - 6 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 + 15 * q^4 + q^5 + 18 * q^7 + 15 * q^8 + 17 * q^9 + q^10 + 4 * q^11 + 18 * q^14 + 23 * q^15 + 15 * q^16 + 2 * q^17 + 17 * q^18 - 15 * q^19 + q^20 + 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 + 18 * q^28 - 20 * q^29 + 23 * q^30 + 30 * q^31 + 15 * q^32 + 36 * q^33 + 2 * q^34 + 32 * q^35 + 17 * q^36 + 35 * q^37 - 15 * q^38 + q^40 + 15 * q^41 + 2 * q^42 + q^43 + 4 * q^44 - 11 * q^45 + 17 * q^46 + 29 * q^49 + 8 * q^50 - q^51 - q^53 + 12 * q^54 - 6 * q^55 + 18 * q^56 - 20 * q^58 - 7 * q^59 + 23 * q^60 - 2 * q^61 + 30 * q^62 + 42 * q^63 + 15 * q^64 + 36 * q^66 + 34 * q^67 + 2 * q^68 - 12 * q^69 + 32 * q^70 + 4 * q^71 + 17 * q^72 + 12 * q^73 + 35 * q^74 + 31 * q^75 - 15 * q^76 - 20 * q^77 + 23 * q^79 + q^80 + 7 * q^81 + 15 * q^82 - 3 * q^83 + 2 * q^84 + 46 * q^85 + q^86 + 22 * q^87 + 4 * q^88 + 17 * q^89 - 11 * q^90 + 17 * q^92 + 60 * q^93 - q^95 + 18 * q^97 + 29 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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