LMFDB - Embedded newform 1849.2.a.r.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} + \cdots - 43 $ x^20 - 5x^19 - 19x^18 + 127x^17 + 95x^16 - 1293x^15 + 329x^14 + 6765x^13 - 5068x^12 - 19360x^11 + 20357x^10 + 29618x^9 - 37889x^8 - 21700x^7 + 32885x^6 + 6093x^5 - 11330x^4 - 1004x^3 + 1386x^2 + 87*x - 43
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			1.16
Root			$2.31191$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.31191 q^{2} +0.910509 q^{3} +3.34494 q^{4} +2.04187 q^{5} +2.10502 q^{6} -2.45270 q^{7} +3.10939 q^{8} -2.17097 q^{9} +O(q^{10})$	\(q+2.31191 q^{2} +0.910509 q^{3} +3.34494 q^{4} +2.04187 q^{5} +2.10502 q^{6} -2.45270 q^{7} +3.10939 q^{8} -2.17097 q^{9} +4.72063 q^{10} +6.34679 q^{11} +3.04560 q^{12} +1.40584 q^{13} -5.67043 q^{14} +1.85914 q^{15} +0.498758 q^{16} +6.68943 q^{17} -5.01910 q^{18} -0.707650 q^{19} +6.82994 q^{20} -2.23321 q^{21} +14.6732 q^{22} +1.78915 q^{23} +2.83113 q^{24} -0.830761 q^{25} +3.25019 q^{26} -4.70822 q^{27} -8.20414 q^{28} -6.38701 q^{29} +4.29818 q^{30} -1.00414 q^{31} -5.06570 q^{32} +5.77881 q^{33} +15.4654 q^{34} -5.00810 q^{35} -7.26178 q^{36} +5.42485 q^{37} -1.63603 q^{38} +1.28003 q^{39} +6.34898 q^{40} -0.748756 q^{41} -5.16298 q^{42} +21.2296 q^{44} -4.43285 q^{45} +4.13636 q^{46} -4.05984 q^{47} +0.454124 q^{48} -0.984262 q^{49} -1.92065 q^{50} +6.09079 q^{51} +4.70247 q^{52} +6.12798 q^{53} -10.8850 q^{54} +12.9593 q^{55} -7.62641 q^{56} -0.644322 q^{57} -14.7662 q^{58} +9.78062 q^{59} +6.21872 q^{60} -3.52420 q^{61} -2.32148 q^{62} +5.32475 q^{63} -12.7090 q^{64} +2.87055 q^{65} +13.3601 q^{66} -10.2096 q^{67} +22.3758 q^{68} +1.62904 q^{69} -11.5783 q^{70} +7.49541 q^{71} -6.75041 q^{72} +1.37573 q^{73} +12.5418 q^{74} -0.756416 q^{75} -2.36705 q^{76} -15.5668 q^{77} +2.95933 q^{78} +7.46275 q^{79} +1.01840 q^{80} +2.22605 q^{81} -1.73106 q^{82} -12.8225 q^{83} -7.46994 q^{84} +13.6590 q^{85} -5.81543 q^{87} +19.7346 q^{88} -15.4830 q^{89} -10.2484 q^{90} -3.44811 q^{91} +5.98460 q^{92} -0.914276 q^{93} -9.38600 q^{94} -1.44493 q^{95} -4.61236 q^{96} -0.288669 q^{97} -2.27553 q^{98} -13.7787 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9}+O(q^{10}) $ 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9} + 21 q^{11} - 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} + 9 q^{18} - 7 q^{19} - 17 q^{20} + 8 q^{21} + 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} + 58 q^{26} - 22 q^{27} - 28 q^{28} - 12 q^{29} - 24 q^{30} + 26 q^{31} + 3 q^{32} - 17 q^{33} + 54 q^{34} + 26 q^{35} + 14 q^{36} - 19 q^{37} + 5 q^{38} - 7 q^{39} - 16 q^{40} + 27 q^{41} - 52 q^{42} + 32 q^{44} + 75 q^{45} + 59 q^{46} + 45 q^{47} - 66 q^{48} + 20 q^{49} + 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} - 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} + 30 q^{61} + 72 q^{62} - 21 q^{63} - 7 q^{64} - 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} - 35 q^{69} - 80 q^{70} - 31 q^{71} - 15 q^{72} - 26 q^{73} + 87 q^{74} + 73 q^{75} - 68 q^{76} - 21 q^{77} - 50 q^{78} + 39 q^{79} - 60 q^{80} + 16 q^{81} + 45 q^{82} + 13 q^{83} + 31 q^{84} - 22 q^{85} + 61 q^{87} + 50 q^{88} - 4 q^{89} - 13 q^{90} - 25 q^{91} + 5 q^{92} + 67 q^{93} + 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} - 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9 + 21 * q^11 - 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 + 9 * q^18 - 7 * q^19 - 17 * q^20 + 8 * q^21 + 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 + 58 * q^26 - 22 * q^27 - 28 * q^28 - 12 * q^29 - 24 * q^30 + 26 * q^31 + 3 * q^32 - 17 * q^33 + 54 * q^34 + 26 * q^35 + 14 * q^36 - 19 * q^37 + 5 * q^38 - 7 * q^39 - 16 * q^40 + 27 * q^41 - 52 * q^42 + 32 * q^44 + 75 * q^45 + 59 * q^46 + 45 * q^47 - 66 * q^48 + 20 * q^49 + 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 - 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 + 30 * q^61 + 72 * q^62 - 21 * q^63 - 7 * q^64 - 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 - 35 * q^69 - 80 * q^70 - 31 * q^71 - 15 * q^72 - 26 * q^73 + 87 * q^74 + 73 * q^75 - 68 * q^76 - 21 * q^77 - 50 * q^78 + 39 * q^79 - 60 * q^80 + 16 * q^81 + 45 * q^82 + 13 * q^83 + 31 * q^84 - 22 * q^85 + 61 * q^87 + 50 * q^88 - 4 * q^89 - 13 * q^90 - 25 * q^91 + 5 * q^92 + 67 * q^93 + 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 - 35 * q^98 + 13 * 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 $

$2$	2.31191	1.63477	0.817385	−	0.576092i	$-0.195423\pi$
$2$	2.31191	1.63477	0.817385	+	0.576092i	$0.195423\pi$
$3$	0.910509	0.525683	0.262841	−	0.964839i	$-0.415340\pi$
$3$	0.910509	0.525683	0.262841	+	0.964839i	$0.415340\pi$
$4$	3.34494	1.67247
$5$	2.04187	0.913153	0.456576	−	0.889684i	$-0.349076\pi$
$5$	2.04187	0.913153	0.456576	+	0.889684i	$0.349076\pi$
$6$	2.10502	0.859370
$7$	−2.45270	−0.927034	−0.463517	−	0.886088i	$-0.653413\pi$
$7$	−2.45270	−0.927034	−0.463517	+	0.886088i	$0.653413\pi$
$8$	3.10939	1.09934
$9$	−2.17097	−0.723658
$10$	4.72063	1.49279
$11$	6.34679	1.91363	0.956814	−	0.290700i	$-0.0938882\pi$
$11$	6.34679	1.91363	0.956814	+	0.290700i	$0.0938882\pi$
$12$	3.04560	0.879189
$13$	1.40584	0.389911	0.194956	−	0.980812i	$-0.437544\pi$
$13$	1.40584	0.389911	0.194956	+	0.980812i	$0.437544\pi$
$14$	−5.67043	−1.51549
$15$	1.85914	0.480028
$16$	0.498758	0.124690
$17$	6.68943	1.62242	0.811212	−	0.584751i	$-0.198808\pi$
$17$	6.68943	1.62242	0.811212	+	0.584751i	$0.198808\pi$
$18$	−5.01910	−1.18301
$19$	−0.707650	−0.162346	−0.0811731	−	0.996700i	$-0.525867\pi$
$19$	−0.707650	−0.162346	−0.0811731	+	0.996700i	$0.525867\pi$
$20$	6.82994	1.52722
$21$	−2.23321	−0.487325
$22$	14.6732	3.12834
$23$	1.78915	0.373063	0.186532	−	0.982449i	$-0.440275\pi$
$23$	1.78915	0.373063	0.186532	+	0.982449i	$0.440275\pi$
$24$	2.83113	0.577902
$25$	−0.830761	−0.166152
$26$	3.25019	0.637415
$27$	−4.70822	−0.906097
$28$	−8.20414	−1.55044
$29$	−6.38701	−1.18604	−0.593019	−	0.805188i	$-0.702064\pi$
$29$	−6.38701	−1.18604	−0.593019	+	0.805188i	$0.702064\pi$
$30$	4.29818	0.784736
$31$	−1.00414	−0.180348	−0.0901742	−	0.995926i	$-0.528742\pi$
$31$	−1.00414	−0.180348	−0.0901742	+	0.995926i	$0.528742\pi$
$32$	−5.06570	−0.895497
$33$	5.77881	1.00596
$34$	15.4654	2.65229
$35$	−5.00810	−0.846523
$36$	−7.26178	−1.21030
$37$	5.42485	0.891840	0.445920	−	0.895073i	$-0.352877\pi$
$37$	5.42485	0.891840	0.445920	+	0.895073i	$0.352877\pi$
$38$	−1.63603	−0.265399
$39$	1.28003	0.204969
$40$	6.34898	1.00386
$41$	−0.748756	−0.116936	−0.0584680	−	0.998289i	$-0.518622\pi$
$41$	−0.748756	−0.116936	−0.0584680	+	0.998289i	$0.518622\pi$
$42$	−5.16298	−0.796665
$43$	0	0
$43$	0	0
$44$	21.2296	3.20049
$45$	−4.43285	−0.660810
$46$	4.13636	0.609873
$47$	−4.05984	−0.592188	−0.296094	−	0.955159i	$-0.595684\pi$
$47$	−4.05984	−0.592188	−0.296094	+	0.955159i	$0.595684\pi$
$48$	0.454124	0.0655471
$49$	−0.984262	−0.140609
$50$	−1.92065	−0.271621
$51$	6.09079	0.852881
$52$	4.70247	0.652115
$53$	6.12798	0.841742	0.420871	−	0.907120i	$-0.361724\pi$
$53$	6.12798	0.841742	0.420871	+	0.907120i	$0.361724\pi$
$54$	−10.8850	−1.48126
$55$	12.9593	1.74743
$56$	−7.62641	−1.01912
$57$	−0.644322	−0.0853425
$58$	−14.7662	−1.93890
$59$	9.78062	1.27333	0.636664	−	0.771141i	$-0.280314\pi$
$59$	9.78062	1.27333	0.636664	+	0.771141i	$0.280314\pi$
$60$	6.21872	0.802834
$61$	−3.52420	−0.451227	−0.225614	−	0.974217i	$-0.572439\pi$
$61$	−3.52420	−0.451227	−0.225614	+	0.974217i	$0.572439\pi$
$62$	−2.32148	−0.294828
$63$	5.32475	0.670855
$64$	−12.7090	−1.58862
$65$	2.87055	0.356048
$66$	13.3601	1.64451
$67$	−10.2096	−1.24730	−0.623652	−	0.781702i	$-0.714352\pi$
$67$	−10.2096	−1.24730	−0.623652	+	0.781702i	$0.714352\pi$
$68$	22.3758	2.71346
$69$	1.62904	0.196113
$70$	−11.5783	−1.38387
$71$	7.49541	0.889541	0.444771	−	0.895645i	$-0.353285\pi$
$71$	7.49541	0.889541	0.444771	+	0.895645i	$0.353285\pi$
$72$	−6.75041	−0.795543
$73$	1.37573	0.161017	0.0805085	−	0.996754i	$-0.474346\pi$
$73$	1.37573	0.161017	0.0805085	+	0.996754i	$0.474346\pi$
$74$	12.5418	1.45795
$75$	−0.756416	−0.0873433
$76$	−2.36705	−0.271519
$77$	−15.5668	−1.77400
$78$	2.95933	0.335078
$79$	7.46275	0.839625	0.419812	−	0.907611i	$-0.362096\pi$
$79$	7.46275	0.839625	0.419812	+	0.907611i	$0.362096\pi$
$80$	1.01840	0.113861
$81$	2.22605	0.247339
$82$	−1.73106	−0.191163
$83$	−12.8225	−1.40745	−0.703726	−	0.710472i	$-0.748482\pi$
$83$	−12.8225	−1.40745	−0.703726	+	0.710472i	$0.748482\pi$
$84$	−7.46994	−0.815038
$85$	13.6590	1.48152
$86$	0	0
$87$	−5.81543	−0.623480
$88$	19.7346	2.10372
$89$	−15.4830	−1.64120	−0.820598	−	0.571506i	$-0.806359\pi$
$89$	−15.4830	−1.64120	−0.820598	+	0.571506i	$0.806359\pi$
$90$	−10.2484	−1.08027
$91$	−3.44811	−0.361461
$92$	5.98460	0.623938
$93$	−0.914276	−0.0948060
$94$	−9.38600	−0.968092
$95$	−1.44493	−0.148247
$96$	−4.61236	−0.470747
$97$	−0.288669	−0.0293099	−0.0146550	−	0.999893i	$-0.504665\pi$
$97$	−0.288669	−0.0293099	−0.0146550	+	0.999893i	$0.504665\pi$
$98$	−2.27553	−0.229863
$99$	−13.7787	−1.38481
$100$	−2.77885	−0.277885
$101$	−12.2140	−1.21534	−0.607669	−	0.794190i	$-0.707895\pi$
$101$	−12.2140	−1.21534	−0.607669	+	0.794190i	$0.707895\pi$
$102$	14.0814	1.39426
$103$	−2.16466	−0.213290	−0.106645	−	0.994297i	$-0.534011\pi$
$103$	−2.16466	−0.213290	−0.106645	+	0.994297i	$0.534011\pi$
$104$	4.37132	0.428643
$105$	−4.55992	−0.445002
$106$	14.1673	1.37605
$107$	−6.57548	−0.635676	−0.317838	−	0.948145i	$-0.602957\pi$
$107$	−6.57548	−0.635676	−0.317838	+	0.948145i	$0.602957\pi$
$108$	−15.7487	−1.51542
$109$	−0.909544	−0.0871185	−0.0435593	−	0.999051i	$-0.513870\pi$
$109$	−0.909544	−0.0871185	−0.0435593	+	0.999051i	$0.513870\pi$
$110$	29.9608	2.85665
$111$	4.93937	0.468825
$112$	−1.22330	−0.115591
$113$	1.74932	0.164563	0.0822813	−	0.996609i	$-0.473779\pi$
$113$	1.74932	0.164563	0.0822813	+	0.996609i	$0.473779\pi$
$114$	−1.48962	−0.139515
$115$	3.65321	0.340664
$116$	−21.3642	−1.98362
$117$	−3.05205	−0.282162
$118$	22.6119	2.08160
$119$	−16.4072	−1.50404
$120$	5.78080	0.527713
$121$	29.2817	2.66197
$122$	−8.14764	−0.737653
$123$	−0.681749	−0.0614712
$124$	−3.35878	−0.301628
$125$	−11.9057	−1.06488
$126$	12.3104	1.09669
$127$	3.16793	0.281108	0.140554	−	0.990073i	$-0.455112\pi$
$127$	3.16793	0.281108	0.140554	+	0.990073i	$0.455112\pi$
$128$	−19.2506	−1.70153
$129$	0	0
$130$	6.63647	0.582057
$131$	9.84378	0.860055	0.430028	−	0.902816i	$-0.358504\pi$
$131$	9.84378	0.860055	0.430028	+	0.902816i	$0.358504\pi$
$132$	19.3298	1.68244
$133$	1.73565	0.150500
$134$	−23.6038	−2.03906
$135$	−9.61357	−0.827405
$136$	20.8001	1.78359
$137$	−4.94440	−0.422428	−0.211214	−	0.977440i	$-0.567742\pi$
$137$	−4.94440	−0.422428	−0.211214	+	0.977440i	$0.567742\pi$
$138$	3.76619	0.320599
$139$	−3.10229	−0.263133	−0.131566	−	0.991307i	$-0.542001\pi$
$139$	−3.10229	−0.263133	−0.131566	+	0.991307i	$0.542001\pi$
$140$	−16.7518	−1.41579
$141$	−3.69652	−0.311303
$142$	17.3287	1.45419
$143$	8.92260	0.746145
$144$	−1.08279	−0.0902326
$145$	−13.0415	−1.08303
$146$	3.18057	0.263226
$147$	−0.896179	−0.0739156
$148$	18.1458	1.49158
$149$	−9.62541	−0.788544	−0.394272	−	0.918994i	$-0.629003\pi$
$149$	−9.62541	−0.788544	−0.394272	+	0.918994i	$0.629003\pi$
$150$	−1.74877	−0.142786
$151$	1.65716	0.134857	0.0674287	−	0.997724i	$-0.478520\pi$
$151$	1.65716	0.134857	0.0674287	+	0.997724i	$0.478520\pi$
$152$	−2.20036	−0.178473
$153$	−14.5226	−1.17408
$154$	−35.9890	−2.90008
$155$	−2.05032	−0.164686
$156$	4.28164	0.342806
$157$	−13.8628	−1.10637	−0.553185	−	0.833059i	$-0.686588\pi$
$157$	−13.8628	−1.10637	−0.553185	+	0.833059i	$0.686588\pi$
$158$	17.2532	1.37259
$159$	5.57958	0.442489
$160$	−10.3435	−0.817726
$161$	−4.38825	−0.345842
$162$	5.14643	0.404341
$163$	−9.47410	−0.742068	−0.371034	−	0.928619i	$-0.620997\pi$
$163$	−9.47410	−0.742068	−0.371034	+	0.928619i	$0.620997\pi$
$164$	−2.50455	−0.195572
$165$	11.7996	0.918596
$166$	−29.6445	−2.30086
$167$	10.7402	0.831100	0.415550	−	0.909570i	$-0.363589\pi$
$167$	10.7402	0.831100	0.415550	+	0.909570i	$0.363589\pi$
$168$	−6.94391	−0.535734
$169$	−11.0236	−0.847969
$170$	31.5783	2.42195
$171$	1.53629	0.117483
$172$	0	0
$173$	−24.8095	−1.88623	−0.943115	−	0.332465i	$-0.892120\pi$
$173$	−24.8095	−1.88623	−0.943115	+	0.332465i	$0.892120\pi$
$174$	−13.4448	−1.01925
$175$	2.03761	0.154029
$176$	3.16551	0.238609
$177$	8.90534	0.669366
$178$	−35.7954	−2.68298
$179$	−11.6400	−0.870012	−0.435006	−	0.900428i	$-0.643254\pi$
$179$	−11.6400	−0.870012	−0.435006	+	0.900428i	$0.643254\pi$
$180$	−14.8276	−1.10519
$181$	20.5083	1.52437	0.762184	−	0.647361i	$-0.224127\pi$
$181$	20.5083	1.52437	0.762184	+	0.647361i	$0.224127\pi$
$182$	−7.97174	−0.590905
$183$	−3.20881	−0.237202
$184$	5.56317	0.410122
$185$	11.0768	0.814386
$186$	−2.11373	−0.154986
$187$	42.4564	3.10472
$188$	−13.5799	−0.990418
$189$	11.5478	0.839982
$190$	−3.34056	−0.242349
$191$	−19.2993	−1.39645	−0.698224	−	0.715880i	$-0.746026\pi$
$191$	−19.2993	−1.39645	−0.698224	+	0.715880i	$0.746026\pi$
$192$	−11.5716	−0.835111
$193$	11.5762	0.833271	0.416635	−	0.909074i	$-0.363209\pi$
$193$	11.5762	0.833271	0.416635	+	0.909074i	$0.363209\pi$
$194$	−0.667378	−0.0479150
$195$	2.61366	0.187168
$196$	−3.29230	−0.235164
$197$	16.1457	1.15033	0.575166	−	0.818036i	$-0.304937\pi$
$197$	16.1457	1.15033	0.575166	+	0.818036i	$0.304937\pi$
$198$	−31.8552	−2.26385
$199$	−15.9222	−1.12870	−0.564349	−	0.825536i	$-0.690873\pi$
$199$	−15.9222	−1.12870	−0.564349	+	0.825536i	$0.690873\pi$
$200$	−2.58316	−0.182657
$201$	−9.29596	−0.655686
$202$	−28.2377	−1.98680
$203$	15.6654	1.09950
$204$	20.3733	1.42642
$205$	−1.52886	−0.106780
$206$	−5.00450	−0.348680
$207$	−3.88420	−0.269970
$208$	0.701176	0.0486178
$209$	−4.49131	−0.310670
$210$	−10.5421	−0.727476
$211$	−3.82377	−0.263239	−0.131619	−	0.991300i	$-0.542018\pi$
$211$	−3.82377	−0.263239	−0.131619	+	0.991300i	$0.542018\pi$
$212$	20.4977	1.40779
$213$	6.82463	0.467616
$214$	−15.2019	−1.03918
$215$	0	0
$216$	−14.6397	−0.996105
$217$	2.46285	0.167189
$218$	−2.10279	−0.142419
$219$	1.25261	0.0846438
$220$	43.3482	2.92254
$221$	9.40430	0.632601
$222$	11.4194	0.766420
$223$	24.0546	1.61082	0.805409	−	0.592720i	$-0.201946\pi$
$223$	24.0546	1.61082	0.805409	+	0.592720i	$0.201946\pi$
$224$	12.4246	0.830156
$225$	1.80356	0.120237
$226$	4.04429	0.269022
$227$	−7.24312	−0.480743	−0.240371	−	0.970681i	$-0.577269\pi$
$227$	−7.24312	−0.480743	−0.240371	+	0.970681i	$0.577269\pi$
$228$	−2.15522	−0.142733
$229$	6.96410	0.460201	0.230100	−	0.973167i	$-0.426095\pi$
$229$	6.96410	0.460201	0.230100	+	0.973167i	$0.426095\pi$
$230$	8.44591	0.556907
$231$	−14.1737	−0.932560
$232$	−19.8597	−1.30385
$233$	5.50612	0.360718	0.180359	−	0.983601i	$-0.442274\pi$
$233$	5.50612	0.360718	0.180359	+	0.983601i	$0.442274\pi$
$234$	−7.05608	−0.461270
$235$	−8.28967	−0.540758
$236$	32.7156	2.12960
$237$	6.79490	0.441376
$238$	−37.9319	−2.45876
$239$	3.48155	0.225203	0.112601	−	0.993640i	$-0.464082\pi$
$239$	3.48155	0.225203	0.112601	+	0.993640i	$0.464082\pi$
$240$	0.927262	0.0598545
$241$	−1.62945	−0.104962	−0.0524812	−	0.998622i	$-0.516713\pi$
$241$	−1.62945	−0.104962	−0.0524812	+	0.998622i	$0.516713\pi$
$242$	67.6968	4.35171
$243$	16.1515	1.03612
$244$	−11.7882	−0.754665
$245$	−2.00974	−0.128397
$246$	−1.57614	−0.100491
$247$	−0.994846	−0.0633005
$248$	−3.12226	−0.198264
$249$	−11.6750	−0.739873
$250$	−27.5249	−1.74083
$251$	14.8497	0.937306	0.468653	−	0.883382i	$-0.344739\pi$
$251$	14.8497	0.937306	0.468653	+	0.883382i	$0.344739\pi$
$252$	17.8110	1.12199
$253$	11.3554	0.713905
$254$	7.32397	0.459547
$255$	12.4366	0.778810
$256$	−19.0879	−1.19299
$257$	3.05501	0.190566	0.0952832	−	0.995450i	$-0.469624\pi$
$257$	3.05501	0.190566	0.0952832	+	0.995450i	$0.469624\pi$
$258$	0	0
$259$	−13.3055	−0.826765
$260$	9.60184	0.595481
$261$	13.8660	0.858286
$262$	22.7580	1.40599
$263$	7.41792	0.457408	0.228704	−	0.973496i	$-0.426551\pi$
$263$	7.41792	0.457408	0.228704	+	0.973496i	$0.426551\pi$
$264$	17.9686	1.10589
$265$	12.5125	0.768639
$266$	4.01268	0.246033
$267$	−14.0974	−0.862748
$268$	−34.1506	−2.08608
$269$	16.2509	0.990834	0.495417	−	0.868655i	$-0.335015\pi$
$269$	16.2509	0.990834	0.495417	+	0.868655i	$0.335015\pi$
$270$	−22.2258	−1.35262
$271$	26.6634	1.61969	0.809845	−	0.586645i	$-0.199551\pi$
$271$	26.6634	1.61969	0.809845	+	0.586645i	$0.199551\pi$
$272$	3.33641	0.202299
$273$	−3.13954	−0.190014
$274$	−11.4310	−0.690573
$275$	−5.27267	−0.317954
$276$	5.44903	0.327993
$277$	18.3038	1.09977	0.549886	−	0.835240i	$-0.314671\pi$
$277$	18.3038	1.09977	0.549886	+	0.835240i	$0.314671\pi$
$278$	−7.17222	−0.430161
$279$	2.17996	0.130511
$280$	−15.5721	−0.930613
$281$	−11.8249	−0.705417	−0.352708	−	0.935733i	$-0.614739\pi$
$281$	−11.8249	−0.705417	−0.352708	+	0.935733i	$0.614739\pi$
$282$	−8.54603	−0.508909
$283$	1.73915	0.103382	0.0516910	−	0.998663i	$-0.483539\pi$
$283$	1.73915	0.103382	0.0516910	+	0.998663i	$0.483539\pi$
$284$	25.0717	1.48773
$285$	−1.31562	−0.0779308
$286$	20.6283	1.21978
$287$	1.83647	0.108404
$288$	10.9975	0.648034
$289$	27.7485	1.63226
$290$	−30.1507	−1.77051
$291$	−0.262836	−0.0154077
$292$	4.60174	0.269296
$293$	10.6521	0.622303	0.311152	−	0.950360i	$-0.399285\pi$
$293$	10.6521	0.622303	0.311152	+	0.950360i	$0.399285\pi$
$294$	−2.07189	−0.120835
$295$	19.9708	1.16274
$296$	16.8680	0.980431
$297$	−29.8821	−1.73393
$298$	−22.2531	−1.28909
$299$	2.51527	0.145462
$300$	−2.53017	−0.146079
$301$	0	0
$302$	3.83120	0.220461
$303$	−11.1210	−0.638882
$304$	−0.352946	−0.0202429
$305$	−7.19596	−0.412040
$306$	−33.5749	−1.91935
$307$	5.49017	0.313341	0.156670	−	0.987651i	$-0.449924\pi$
$307$	5.49017	0.313341	0.156670	+	0.987651i	$0.449924\pi$
$308$	−52.0700	−2.96696
$309$	−1.97094	−0.112123
$310$	−4.74016	−0.269223
$311$	−6.50028	−0.368597	−0.184299	−	0.982870i	$-0.559001\pi$
$311$	−6.50028	−0.368597	−0.184299	+	0.982870i	$0.559001\pi$
$312$	3.98013	0.225330
$313$	19.7542	1.11657	0.558285	−	0.829649i	$-0.311459\pi$
$313$	19.7542	1.11657	0.558285	+	0.829649i	$0.311459\pi$
$314$	−32.0495	−1.80866
$315$	10.8724	0.612593
$316$	24.9625	1.40425
$317$	13.8919	0.780250	0.390125	−	0.920762i	$-0.372432\pi$
$317$	13.8919	0.780250	0.390125	+	0.920762i	$0.372432\pi$
$318$	12.8995	0.723368
$319$	−40.5370	−2.26964
$320$	−25.9501	−1.45065
$321$	−5.98703	−0.334164
$322$	−10.1452	−0.565373
$323$	−4.73378	−0.263394
$324$	7.44600	0.413667
$325$	−1.16792	−0.0647846
$326$	−21.9033	−1.21311
$327$	−0.828148	−0.0457967
$328$	−2.32818	−0.128552
$329$	9.95757	0.548979
$330$	27.2796	1.50169
$331$	−14.6176	−0.803459	−0.401729	−	0.915758i	$-0.631591\pi$
$331$	−14.6176	−0.803459	−0.401729	+	0.915758i	$0.631591\pi$
$332$	−42.8905	−2.35392
$333$	−11.7772	−0.645387
$334$	24.8304	1.35866
$335$	−20.8467	−1.13898
$336$	−1.11383	−0.0607644
$337$	−24.4065	−1.32951	−0.664753	−	0.747063i	$-0.731463\pi$
$337$	−24.4065	−1.32951	−0.664753	+	0.747063i	$0.731463\pi$
$338$	−25.4856	−1.38623
$339$	1.59278	0.0865077
$340$	45.6884	2.47780
$341$	−6.37305	−0.345120
$342$	3.55177	0.192058
$343$	19.5830	1.05738
$344$	0	0
$345$	3.32628	0.179081
$346$	−57.3574	−3.08355
$347$	−7.52857	−0.404155	−0.202077	−	0.979370i	$-0.564769\pi$
$347$	−7.52857	−0.404155	−0.202077	+	0.979370i	$0.564769\pi$
$348$	−19.4523	−1.04275
$349$	−13.5757	−0.726692	−0.363346	−	0.931654i	$-0.618366\pi$
$349$	−13.5757	−0.726692	−0.363346	+	0.931654i	$0.618366\pi$
$350$	4.71077	0.251801
$351$	−6.61902	−0.353297
$352$	−32.1509	−1.71365
$353$	−30.2021	−1.60750	−0.803748	−	0.594970i	$-0.797164\pi$
$353$	−30.2021	−1.60750	−0.803748	+	0.594970i	$0.797164\pi$
$354$	20.5884	1.09426
$355$	15.3047	0.812287
$356$	−51.7898	−2.74485
$357$	−14.9389	−0.790649
$358$	−26.9106	−1.42227
$359$	−12.0540	−0.636187	−0.318093	−	0.948059i	$-0.603043\pi$
$359$	−12.0540	−0.636187	−0.318093	+	0.948059i	$0.603043\pi$
$360$	−13.7835	−0.726452
$361$	−18.4992	−0.973644
$362$	47.4133	2.49199
$363$	26.6613	1.39935
$364$	−11.5337	−0.604533
$365$	2.80906	0.147033
$366$	−7.41850	−0.387771
$367$	31.5354	1.64614	0.823068	−	0.567943i	$-0.192261\pi$
$367$	31.5354	1.64614	0.823068	+	0.567943i	$0.192261\pi$
$368$	0.892353	0.0465171
$369$	1.62553	0.0846217
$370$	25.6087	1.33133
$371$	−15.0301	−0.780323
$372$	−3.05820	−0.158560
$373$	19.3174	1.00022	0.500109	−	0.865963i	$-0.333293\pi$
$373$	19.3174	1.00022	0.500109	+	0.865963i	$0.333293\pi$
$374$	98.1555	5.07550
$375$	−10.8402	−0.559786
$376$	−12.6236	−0.651014
$377$	−8.97914	−0.462449
$378$	26.6976	1.37318
$379$	−19.4777	−1.00050	−0.500250	−	0.865881i	$-0.666759\pi$
$379$	−19.4777	−1.00050	−0.500250	+	0.865881i	$0.666759\pi$
$380$	−4.83321	−0.247939
$381$	2.88442	0.147774
$382$	−44.6183	−2.28287
$383$	23.8663	1.21951	0.609755	−	0.792590i	$-0.291268\pi$
$383$	23.8663	1.21951	0.609755	+	0.792590i	$0.291268\pi$
$384$	−17.5279	−0.894466
$385$	−31.7853	−1.61993
$386$	26.7631	1.36221
$387$	0	0
$388$	−0.965582	−0.0490200
$389$	4.95237	0.251095	0.125547	−	0.992088i	$-0.459931\pi$
$389$	4.95237	0.251095	0.125547	+	0.992088i	$0.459931\pi$
$390$	6.04257	0.305977
$391$	11.9684	0.605267
$392$	−3.06046	−0.154576
$393$	8.96285	0.452116
$394$	37.3274	1.88053
$395$	15.2380	0.766706
$396$	−46.0890	−2.31606
$397$	8.51276	0.427243	0.213622	−	0.976916i	$-0.431474\pi$
$397$	8.51276	0.427243	0.213622	+	0.976916i	$0.431474\pi$
$398$	−36.8108	−1.84516
$399$	1.58033	0.0791154
$400$	−0.414349	−0.0207174
$401$	5.71984	0.285635	0.142818	−	0.989749i	$-0.454384\pi$
$401$	5.71984	0.285635	0.142818	+	0.989749i	$0.454384\pi$
$402$	−21.4914	−1.07190
$403$	−1.41166	−0.0703199
$404$	−40.8551	−2.03262
$405$	4.54530	0.225858
$406$	36.2171	1.79742
$407$	34.4304	1.70665
$408$	18.9386	0.937602
$409$	−17.5608	−0.868325	−0.434163	−	0.900835i	$-0.642956\pi$
$409$	−17.5608	−0.868325	−0.434163	+	0.900835i	$0.642956\pi$
$410$	−3.53460	−0.174561
$411$	−4.50192	−0.222063
$412$	−7.24066	−0.356722
$413$	−23.9889	−1.18042
$414$	−8.97992	−0.441339
$415$	−26.1819	−1.28522
$416$	−7.12158	−0.349164
$417$	−2.82466	−0.138324
$418$	−10.3835	−0.507874
$419$	29.9639	1.46383	0.731917	−	0.681393i	$-0.238626\pi$
$419$	29.9639	1.46383	0.731917	+	0.681393i	$0.238626\pi$
$420$	−15.2527	−0.744254
$421$	37.1590	1.81102	0.905510	−	0.424325i	$-0.139489\pi$
$421$	37.1590	1.81102	0.905510	+	0.424325i	$0.139489\pi$
$422$	−8.84021	−0.430335
$423$	8.81380	0.428542
$424$	19.0543	0.925357
$425$	−5.55732	−0.269570
$426$	15.7780	0.764445
$427$	8.64380	0.418303
$428$	−21.9946	−1.06315
$429$	8.12410	0.392235
$430$	0	0
$431$	25.0938	1.20872	0.604362	−	0.796710i	$-0.293428\pi$
$431$	25.0938	1.20872	0.604362	+	0.796710i	$0.293428\pi$
$432$	−2.34826	−0.112981
$433$	−29.2470	−1.40552	−0.702760	−	0.711427i	$-0.748049\pi$
$433$	−29.2470	−1.40552	−0.702760	+	0.711427i	$0.748049\pi$
$434$	5.69389	0.273316
$435$	−11.8744	−0.569332
$436$	−3.04237	−0.145703
$437$	−1.26609	−0.0605654
$438$	2.89594	0.138373
$439$	10.5083	0.501534	0.250767	−	0.968047i	$-0.419317\pi$
$439$	10.5083	0.501534	0.250767	+	0.968047i	$0.419317\pi$
$440$	40.2956	1.92102
$441$	2.13681	0.101753
$442$	21.7419	1.03416
$443$	−8.51248	−0.404440	−0.202220	−	0.979340i	$-0.564816\pi$
$443$	−8.51248	−0.404440	−0.202220	+	0.979340i	$0.564816\pi$
$444$	16.5219	0.784096
$445$	−31.6143	−1.49866
$446$	55.6122	2.63332
$447$	−8.76402	−0.414524
$448$	31.1713	1.47271
$449$	−34.8622	−1.64525	−0.822626	−	0.568583i	$-0.807492\pi$
$449$	−34.8622	−1.64525	−0.822626	+	0.568583i	$0.807492\pi$
$450$	4.16968	0.196560
$451$	−4.75219	−0.223772
$452$	5.85139	0.275226
$453$	1.50886	0.0708922
$454$	−16.7455	−0.785904
$455$	−7.04061	−0.330069
$456$	−2.00345	−0.0938201
$457$	−32.7436	−1.53168	−0.765841	−	0.643030i	$-0.777677\pi$
$457$	−32.7436	−1.53168	−0.765841	+	0.643030i	$0.777677\pi$
$458$	16.1004	0.752322
$459$	−31.4953	−1.47007
$460$	12.2198	0.569751
$461$	−39.7806	−1.85277	−0.926383	−	0.376582i	$-0.877099\pi$
$461$	−39.7806	−1.85277	−0.926383	+	0.376582i	$0.877099\pi$
$462$	−32.7683	−1.52452
$463$	12.6595	0.588336	0.294168	−	0.955754i	$-0.404958\pi$
$463$	12.6595	0.588336	0.294168	+	0.955754i	$0.404958\pi$
$464$	−3.18557	−0.147887
$465$	−1.86683	−0.0865724
$466$	12.7297	0.589690
$467$	34.9711	1.61827	0.809136	−	0.587622i	$-0.199936\pi$
$467$	34.9711	1.61827	0.809136	+	0.587622i	$0.199936\pi$
$468$	−10.2089	−0.471908
$469$	25.0412	1.15629
$470$	−19.1650	−0.884015
$471$	−12.6222	−0.581599
$472$	30.4118	1.39982
$473$	0	0
$474$	15.7092	0.721548
$475$	0.587889	0.0269742
$476$	−54.8810	−2.51547
$477$	−13.3037	−0.609133
$478$	8.04904	0.368155
$479$	21.9996	1.00519	0.502593	−	0.864523i	$-0.332380\pi$
$479$	21.9996	1.00519	0.502593	+	0.864523i	$0.332380\pi$
$480$	−9.41785	−0.429864
$481$	7.62649	0.347738
$482$	−3.76716	−0.171589
$483$	−3.99554	−0.181803
$484$	97.9457	4.45208
$485$	−0.589425	−0.0267644
$486$	37.3408	1.69381
$487$	−6.49327	−0.294238	−0.147119	−	0.989119i	$-0.547000\pi$
$487$	−6.49327	−0.294238	−0.147119	+	0.989119i	$0.547000\pi$
$488$	−10.9581	−0.496051
$489$	−8.62625	−0.390092
$490$	−4.64634	−0.209900
$491$	7.19968	0.324917	0.162459	−	0.986715i	$-0.448058\pi$
$491$	7.19968	0.324917	0.162459	+	0.986715i	$0.448058\pi$
$492$	−2.28041	−0.102809
$493$	−42.7255	−1.92426
$494$	−2.30000	−0.103482
$495$	−28.1343	−1.26454
$496$	−0.500822	−0.0224876
$497$	−18.3840	−0.824634
$498$	−26.9916	−1.20952
$499$	37.0430	1.65827	0.829135	−	0.559048i	$-0.188833\pi$
$499$	37.0430	1.65827	0.829135	+	0.559048i	$0.188833\pi$
$500$	−39.8238	−1.78097
$501$	9.77902	0.436895
$502$	34.3313	1.53228
$503$	−5.21326	−0.232448	−0.116224	−	0.993223i	$-0.537079\pi$
$503$	−5.21326	−0.232448	−0.116224	+	0.993223i	$0.537079\pi$
$504$	16.5567	0.737495
$505$	−24.9394	−1.10979
$506$	26.2526	1.16707
$507$	−10.0371	−0.445763
$508$	10.5965	0.470145
$509$	19.2599	0.853680	0.426840	−	0.904327i	$-0.359627\pi$
$509$	19.2599	0.853680	0.426840	+	0.904327i	$0.359627\pi$
$510$	28.7523	1.27318
$511$	−3.37425	−0.149268
$512$	−5.62823	−0.248735
$513$	3.33177	0.147101
$514$	7.06292	0.311532
$515$	−4.41995	−0.194766
$516$	0	0
$517$	−25.7669	−1.13323
$518$	−30.7612	−1.35157
$519$	−22.5893	−0.991559
$520$	8.92567	0.391417
$521$	20.9839	0.919322	0.459661	−	0.888094i	$-0.347971\pi$
$521$	20.9839	0.919322	0.459661	+	0.888094i	$0.347971\pi$
$522$	32.0571	1.40310
$523$	−8.38255	−0.366543	−0.183272	−	0.983062i	$-0.558669\pi$
$523$	−8.38255	−0.366543	−0.183272	+	0.983062i	$0.558669\pi$
$524$	32.9269	1.43842
$525$	1.85526	0.0809702
$526$	17.1496	0.747757
$527$	−6.71711	−0.292602
$528$	2.88223	0.125433
$529$	−19.7989	−0.860824
$530$	28.9279	1.25655
$531$	−21.2335	−0.921454
$532$	5.80566	0.251707
$533$	−1.05263	−0.0455946
$534$	−32.5920	−1.41039
$535$	−13.4263	−0.580469
$536$	−31.7457	−1.37121
$537$	−10.5983	−0.457350
$538$	37.5706	1.61978
$539$	−6.24690	−0.269073
$540$	−32.1569	−1.38381
$541$	34.2947	1.47444	0.737222	−	0.675651i	$-0.236137\pi$
$541$	34.2947	1.47444	0.737222	+	0.675651i	$0.236137\pi$
$542$	61.6436	2.64782
$543$	18.6730	0.801333
$544$	−33.8866	−1.45288
$545$	−1.85717	−0.0795525
$546$	−7.25834	−0.310628
$547$	−1.93445	−0.0827112	−0.0413556	−	0.999144i	$-0.513168\pi$
$547$	−1.93445	−0.0827112	−0.0413556	+	0.999144i	$0.513168\pi$
$548$	−16.5387	−0.706500
$549$	7.65094	0.326534
$550$	−12.1899	−0.519781
$551$	4.51977	0.192549
$552$	5.06531	0.215594
$553$	−18.3039	−0.778360
$554$	42.3169	1.79787
$555$	10.0856	0.428108
$556$	−10.3770	−0.440082
$557$	−26.5867	−1.12651	−0.563257	−	0.826282i	$-0.690452\pi$
$557$	−26.5867	−1.12651	−0.563257	+	0.826282i	$0.690452\pi$
$558$	5.03987	0.213355
$559$	0	0
$560$	−2.49783	−0.105553
$561$	38.6569	1.63210
$562$	−27.3382	−1.15319
$563$	20.4328	0.861139	0.430570	−	0.902557i	$-0.358313\pi$
$563$	20.4328	0.861139	0.430570	+	0.902557i	$0.358313\pi$
$564$	−12.3646	−0.520646
$565$	3.57190	0.150271
$566$	4.02077	0.169006
$567$	−5.45983	−0.229291
$568$	23.3062	0.977905
$569$	32.1027	1.34581	0.672907	−	0.739727i	$-0.265045\pi$
$569$	32.1027	1.34581	0.672907	+	0.739727i	$0.265045\pi$
$570$	−3.04161	−0.127399
$571$	13.4605	0.563303	0.281651	−	0.959517i	$-0.409118\pi$
$571$	13.4605	0.563303	0.281651	+	0.959517i	$0.409118\pi$
$572$	29.8456	1.24791
$573$	−17.5722	−0.734088
$574$	4.24577	0.177215
$575$	−1.48636	−0.0619853
$576$	27.5908	1.14962
$577$	24.7963	1.03228	0.516141	−	0.856504i	$-0.327368\pi$
$577$	24.7963	1.03228	0.516141	+	0.856504i	$0.327368\pi$
$578$	64.1521	2.66837
$579$	10.5402	0.438036
$580$	−43.6229	−1.81134
$581$	31.4497	1.30475
$582$	−0.607654	−0.0251881
$583$	38.8930	1.61078
$584$	4.27768	0.177012
$585$	−6.23190	−0.257657
$586$	24.6268	1.01732
$587$	−12.2593	−0.505998	−0.252999	−	0.967467i	$-0.581417\pi$
$587$	−12.2593	−0.505998	−0.252999	+	0.967467i	$0.581417\pi$
$588$	−2.99767	−0.123622
$589$	0.710578	0.0292789
$590$	46.1707	1.90082
$591$	14.7008	0.604710
$592$	2.70569	0.111203
$593$	33.0810	1.35847	0.679236	−	0.733920i	$-0.262311\pi$
$593$	33.0810	1.35847	0.679236	+	0.733920i	$0.262311\pi$
$594$	−69.0847	−2.83458
$595$	−33.5013	−1.37342
$596$	−32.1965	−1.31882
$597$	−14.4973	−0.593337
$598$	5.81508	0.237796
$599$	35.0641	1.43268	0.716339	−	0.697752i	$-0.245816\pi$
$599$	35.0641	1.43268	0.716339	+	0.697752i	$0.245816\pi$
$600$	−2.35199	−0.0960197
$601$	−37.9483	−1.54794	−0.773972	−	0.633220i	$-0.781733\pi$
$601$	−37.9483	−1.54794	−0.773972	+	0.633220i	$0.781733\pi$
$602$	0	0
$603$	22.1648	0.902622
$604$	5.54309	0.225545
$605$	59.7895	2.43079
$606$	−25.7107	−1.04442
$607$	35.9199	1.45794	0.728971	−	0.684544i	$-0.239999\pi$
$607$	35.9199	1.45794	0.728971	+	0.684544i	$0.239999\pi$
$608$	3.58474	0.145381
$609$	14.2635	0.577986
$610$	−16.6364	−0.673590
$611$	−5.70750	−0.230901
$612$	−48.5772	−1.96362
$613$	3.16067	0.127658	0.0638292	−	0.997961i	$-0.479669\pi$
$613$	3.16067	0.127658	0.0638292	+	0.997961i	$0.479669\pi$
$614$	12.6928	0.512240
$615$	−1.39204	−0.0561326
$616$	−48.4032	−1.95022
$617$	16.3852	0.659642	0.329821	−	0.944044i	$-0.393012\pi$
$617$	16.3852	0.659642	0.329821	+	0.944044i	$0.393012\pi$
$618$	−4.55664	−0.183295
$619$	−4.92740	−0.198049	−0.0990244	−	0.995085i	$-0.531572\pi$
$619$	−4.92740	−0.198049	−0.0990244	+	0.995085i	$0.531572\pi$
$620$	−6.85820	−0.275432
$621$	−8.42370	−0.338032
$622$	−15.0281	−0.602571
$623$	37.9752	1.52144
$624$	0.638427	0.0255575
$625$	−20.1560	−0.806241
$626$	45.6699	1.82534
$627$	−4.08937	−0.163314
$628$	−46.3702	−1.85037
$629$	36.2891	1.44694
$630$	25.1362	1.00145
$631$	−6.97783	−0.277783	−0.138892	−	0.990308i	$-0.544354\pi$
$631$	−6.97783	−0.277783	−0.138892	+	0.990308i	$0.544354\pi$
$632$	23.2046	0.923030
$633$	−3.48157	−0.138380
$634$	32.1170	1.27553
$635$	6.46850	0.256695
$636$	18.6634	0.740051
$637$	−1.38372	−0.0548249
$638$	−93.7180	−3.71033
$639$	−16.2723	−0.643723
$640$	−39.3073	−1.55376
$641$	−26.5976	−1.05054	−0.525271	−	0.850935i	$-0.676036\pi$
$641$	−26.5976	−1.05054	−0.525271	+	0.850935i	$0.676036\pi$
$642$	−13.8415	−0.546281
$643$	8.22463	0.324348	0.162174	−	0.986762i	$-0.448149\pi$
$643$	8.22463	0.324348	0.162174	+	0.986762i	$0.448149\pi$
$644$	−14.6784	−0.578411
$645$	0	0
$646$	−10.9441	−0.430589
$647$	−37.9614	−1.49242	−0.746209	−	0.665712i	$-0.768128\pi$
$647$	−37.9614	−1.49242	−0.746209	+	0.665712i	$0.768128\pi$
$648$	6.92165	0.271908
$649$	62.0755	2.43668
$650$	−2.70013	−0.105908
$651$	2.24245	0.0878884
$652$	−31.6903	−1.24109
$653$	1.96304	0.0768198	0.0384099	−	0.999262i	$-0.487771\pi$
$653$	1.96304	0.0768198	0.0384099	+	0.999262i	$0.487771\pi$
$654$	−1.91461	−0.0748670
$655$	20.0997	0.785362
$656$	−0.373448	−0.0145807
$657$	−2.98667	−0.116521
$658$	23.0210	0.897453
$659$	26.0591	1.01512	0.507560	−	0.861617i	$-0.330548\pi$
$659$	26.0591	1.01512	0.507560	+	0.861617i	$0.330548\pi$
$660$	39.4689	1.53633
$661$	−14.5776	−0.567001	−0.283501	−	0.958972i	$-0.591496\pi$
$661$	−14.5776	−0.567001	−0.283501	+	0.958972i	$0.591496\pi$
$662$	−33.7947	−1.31347
$663$	8.56270	0.332548
$664$	−39.8701	−1.54726
$665$	3.54398	0.137430
$666$	−27.2279	−1.05506
$667$	−11.4273	−0.442467
$668$	35.9253	1.38999
$669$	21.9020	0.846779
$670$	−48.1959	−1.86197
$671$	−22.3673	−0.863482
$672$	11.3127	0.436399
$673$	26.0002	1.00224	0.501118	−	0.865379i	$-0.332922\pi$
$673$	26.0002	1.00224	0.501118	+	0.865379i	$0.332922\pi$
$674$	−56.4257	−2.17344
$675$	3.91140	0.150550
$676$	−36.8733	−1.41820
$677$	−22.0690	−0.848182	−0.424091	−	0.905620i	$-0.639406\pi$
$677$	−22.0690	−0.848182	−0.424091	+	0.905620i	$0.639406\pi$
$678$	3.68236	0.141420
$679$	0.708019	0.0271713
$680$	42.4710	1.62869
$681$	−6.59493	−0.252718
$682$	−14.7339	−0.564192
$683$	4.70251	0.179937	0.0899684	−	0.995945i	$-0.471323\pi$
$683$	4.70251	0.179937	0.0899684	+	0.995945i	$0.471323\pi$
$684$	5.13880	0.196487
$685$	−10.0958	−0.385742
$686$	45.2742	1.72858
$687$	6.34088	0.241920
$688$	0	0
$689$	8.61498	0.328205
$690$	7.69008	0.292756
$691$	36.0380	1.37095	0.685475	−	0.728096i	$-0.259594\pi$
$691$	36.0380	1.37095	0.685475	+	0.728096i	$0.259594\pi$
$692$	−82.9863	−3.15467
$693$	33.7950	1.28377
$694$	−17.4054	−0.660700
$695$	−6.33447	−0.240280
$696$	−18.0825	−0.685414
$697$	−5.00875	−0.189720
$698$	−31.3859	−1.18797
$699$	5.01337	0.189623
$700$	6.81568	0.257609
$701$	−39.9027	−1.50711	−0.753553	−	0.657387i	$-0.771662\pi$
$701$	−39.9027	−1.50711	−0.753553	+	0.657387i	$0.771662\pi$
$702$	−15.3026	−0.577559
$703$	−3.83890	−0.144787
$704$	−80.6611	−3.04003
$705$	−7.54782	−0.284267
$706$	−69.8246	−2.62789
$707$	29.9573	1.12666
$708$	29.7878	1.11950
$709$	−1.96754	−0.0738927	−0.0369463	−	0.999317i	$-0.511763\pi$
$709$	−1.96754	−0.0738927	−0.0369463	+	0.999317i	$0.511763\pi$
$710$	35.3830	1.32790
$711$	−16.2014	−0.607601
$712$	−48.1428	−1.80423
$713$	−1.79655	−0.0672814
$714$	−34.5374	−1.29253
$715$	18.2188	0.681344
$716$	−38.9350	−1.45507
$717$	3.16998	0.118385
$718$	−27.8678	−1.04002
$719$	29.1938	1.08875	0.544373	−	0.838843i	$-0.316768\pi$
$719$	29.1938	1.08875	0.544373	+	0.838843i	$0.316768\pi$
$720$	−2.21092	−0.0823961
$721$	5.30926	0.197727
$722$	−42.7686	−1.59168
$723$	−1.48363	−0.0551769
$724$	68.5990	2.54946
$725$	5.30608	0.197063
$726$	61.6385	2.28762
$727$	−41.5692	−1.54172	−0.770858	−	0.637007i	$-0.780172\pi$
$727$	−41.5692	−1.54172	−0.770858	+	0.637007i	$0.780172\pi$
$728$	−10.7215	−0.397367
$729$	8.02793	0.297331
$730$	6.49431	0.240365
$731$	0	0
$732$	−10.7333	−0.396714
$733$	30.0670	1.11055	0.555275	−	0.831667i	$-0.312613\pi$
$733$	30.0670	1.11055	0.555275	+	0.831667i	$0.312613\pi$
$734$	72.9072	2.69105
$735$	−1.82988	−0.0674962
$736$	−9.06329	−0.334077
$737$	−64.7983	−2.38688
$738$	3.75808	0.138337
$739$	−3.87146	−0.142414	−0.0712071	−	0.997462i	$-0.522685\pi$
$739$	−3.87146	−0.142414	−0.0712071	+	0.997462i	$0.522685\pi$
$740$	37.0514	1.36204
$741$	−0.905816	−0.0332760
$742$	−34.7483	−1.27565
$743$	−52.0970	−1.91125	−0.955626	−	0.294582i	$-0.904820\pi$
$743$	−52.0970	−1.91125	−0.955626	+	0.294582i	$0.904820\pi$
$744$	−2.84284	−0.104224
$745$	−19.6539	−0.720061
$746$	44.6602	1.63512
$747$	27.8373	1.01851
$748$	142.014	5.19255
$749$	16.1277	0.589293
$750$	−25.0616	−0.915121
$751$	29.7160	1.08435	0.542177	−	0.840264i	$-0.317600\pi$
$751$	29.7160	1.08435	0.542177	+	0.840264i	$0.317600\pi$
$752$	−2.02488	−0.0738397
$753$	13.5208	0.492725
$754$	−20.7590	−0.755998
$755$	3.38370	0.123145
$756$	38.6269	1.40485
$757$	−18.1870	−0.661018	−0.330509	−	0.943803i	$-0.607220\pi$
$757$	−18.1870	−0.661018	−0.330509	+	0.943803i	$0.607220\pi$
$758$	−45.0307	−1.63559
$759$	10.3391	0.375287
$760$	−4.49286	−0.162973
$761$	12.4175	0.450133	0.225066	−	0.974343i	$-0.427740\pi$
$761$	12.4175	0.450133	0.225066	+	0.974343i	$0.427740\pi$
$762$	6.66854	0.241576
$763$	2.23084	0.0807618
$764$	−64.5550	−2.33552
$765$	−29.6532	−1.07211
$766$	55.1768	1.99362
$767$	13.7500	0.496485
$768$	−17.3797	−0.627135
$769$	13.6399	0.491868	0.245934	−	0.969287i	$-0.420905\pi$
$769$	13.6399	0.491868	0.245934	+	0.969287i	$0.420905\pi$
$770$	−73.4849	−2.64821
$771$	2.78161	0.100177
$772$	38.7216	1.39362
$773$	−31.9270	−1.14833	−0.574167	−	0.818738i	$-0.694674\pi$
$773$	−31.9270	−1.14833	−0.574167	+	0.818738i	$0.694674\pi$
$774$	0	0
$775$	0.834199	0.0299653
$776$	−0.897586	−0.0322215
$777$	−12.1148	−0.434616
$778$	11.4494	0.410482
$779$	0.529857	0.0189841
$780$	8.74256	0.313034
$781$	47.5718	1.70225
$782$	27.6699	0.989473
$783$	30.0714	1.07467
$784$	−0.490909	−0.0175325
$785$	−28.3060	−1.01028
$786$	20.7213	0.739106
$787$	30.6799	1.09362	0.546811	−	0.837256i	$-0.315842\pi$
$787$	30.6799	1.09362	0.546811	+	0.837256i	$0.315842\pi$
$788$	54.0064	1.92390
$789$	6.75408	0.240452
$790$	35.2289	1.25339
$791$	−4.29057	−0.152555
$792$	−42.8434	−1.52237
$793$	−4.95448	−0.175939
$794$	19.6808	0.698444
$795$	11.3928	0.404060
$796$	−53.2590	−1.88771
$797$	18.4303	0.652835	0.326417	−	0.945226i	$-0.394159\pi$
$797$	18.4303	0.652835	0.326417	+	0.945226i	$0.394159\pi$
$798$	3.65358	0.129335
$799$	−27.1580	−0.960781
$800$	4.20839	0.148789
$801$	33.6132	1.18766
$802$	13.2238	0.466948
$803$	8.73147	0.308127
$804$	−31.0944	−1.09662
$805$	−8.96024	−0.315807
$806$	−3.26364	−0.114957
$807$	14.7966	0.520864
$808$	−37.9781	−1.33606
$809$	−14.2481	−0.500938	−0.250469	−	0.968125i	$-0.580585\pi$
$809$	−14.2481	−0.500938	−0.250469	+	0.968125i	$0.580585\pi$
$810$	10.5083	0.369225
$811$	6.10231	0.214281	0.107141	−	0.994244i	$-0.465830\pi$
$811$	6.10231	0.214281	0.107141	+	0.994244i	$0.465830\pi$
$812$	52.3999	1.83888
$813$	24.2773	0.851442
$814$	79.6000	2.78998
$815$	−19.3449	−0.677622
$816$	3.03783	0.106345
$817$	0	0
$818$	−40.5990	−1.41951
$819$	7.48577	0.261574
$820$	−5.11396	−0.178587
$821$	34.2987	1.19703	0.598516	−	0.801111i	$-0.295757\pi$
$821$	34.2987	1.19703	0.598516	+	0.801111i	$0.295757\pi$
$822$	−10.4080	−0.363022
$823$	42.4031	1.47808	0.739040	−	0.673662i	$-0.235280\pi$
$823$	42.4031	1.47808	0.739040	+	0.673662i	$0.235280\pi$
$824$	−6.73077	−0.234477
$825$	−4.80081	−0.167143
$826$	−55.4603	−1.92971
$827$	9.85209	0.342591	0.171295	−	0.985220i	$-0.445205\pi$
$827$	9.85209	0.342591	0.171295	+	0.985220i	$0.445205\pi$
$828$	−12.9924	−0.451518
$829$	3.03968	0.105573	0.0527863	−	0.998606i	$-0.483190\pi$
$829$	3.03968	0.105573	0.0527863	+	0.998606i	$0.483190\pi$
$830$	−60.5302	−2.10103
$831$	16.6658	0.578131
$832$	−17.8668	−0.619421
$833$	−6.58415	−0.228127
$834$	−6.53037	−0.226128
$835$	21.9301	0.758921
$836$	−15.0232	−0.519587
$837$	4.72770	0.163413
$838$	69.2740	2.39303
$839$	−46.5265	−1.60627	−0.803137	−	0.595794i	$-0.796837\pi$
$839$	−46.5265	−1.60627	−0.803137	+	0.595794i	$0.796837\pi$
$840$	−14.1786	−0.489207
$841$	11.7939	0.406686
$842$	85.9084	2.96060
$843$	−10.7667	−0.370825
$844$	−12.7903	−0.440260
$845$	−22.5088	−0.774325
$846$	20.3768	0.700567
$847$	−71.8193	−2.46774
$848$	3.05638	0.104956
$849$	1.58352	0.0543461
$850$	−12.8480	−0.440684
$851$	9.70586	0.332713
$852$	22.8280	0.782075
$853$	16.9716	0.581098	0.290549	−	0.956860i	$-0.406162\pi$
$853$	16.9716	0.581098	0.290549	+	0.956860i	$0.406162\pi$
$854$	19.9837	0.683829
$855$	3.13691	0.107280
$856$	−20.4457	−0.698821
$857$	15.9929	0.546308	0.273154	−	0.961970i	$-0.411933\pi$
$857$	15.9929	0.546308	0.273154	+	0.961970i	$0.411933\pi$
$858$	18.7822	0.641214
$859$	−49.8888	−1.70219	−0.851093	−	0.525015i	$-0.824060\pi$
$859$	−49.8888	−1.70219	−0.851093	+	0.525015i	$0.824060\pi$
$860$	0	0
$861$	1.67213	0.0569859
$862$	58.0146	1.97599
$863$	−12.5338	−0.426657	−0.213328	−	0.976981i	$-0.568430\pi$
$863$	−12.5338	−0.426657	−0.213328	+	0.976981i	$0.568430\pi$
$864$	23.8504	0.811407
$865$	−50.6578	−1.72242
$866$	−67.6164	−2.29770
$867$	25.2652	0.858052
$868$	8.23809	0.279619
$869$	47.3645	1.60673
$870$	−27.4525	−0.930727
$871$	−14.3531	−0.486338
$872$	−2.82813	−0.0957725
$873$	0.626693	0.0212104
$874$	−2.92710	−0.0990105
$875$	29.2010	0.987175
$876$	4.18992	0.141564
$877$	−33.8971	−1.14463	−0.572313	−	0.820035i	$-0.693954\pi$
$877$	−33.8971	−1.14463	−0.572313	+	0.820035i	$0.693954\pi$
$878$	24.2943	0.819893
$879$	9.69884	0.327134
$880$	6.46357	0.217887
$881$	38.1336	1.28475	0.642377	−	0.766389i	$-0.277948\pi$
$881$	38.1336	1.28475	0.642377	+	0.766389i	$0.277948\pi$
$882$	4.94011	0.166342
$883$	−9.47384	−0.318820	−0.159410	−	0.987212i	$-0.550959\pi$
$883$	−9.47384	−0.318820	−0.159410	+	0.987212i	$0.550959\pi$
$884$	31.4568	1.05801
$885$	18.1836	0.611234
$886$	−19.6801	−0.661166
$887$	−23.4875	−0.788633	−0.394317	−	0.918975i	$-0.629019\pi$
$887$	−23.4875	−0.788633	−0.394317	+	0.918975i	$0.629019\pi$
$888$	15.3584	0.515396
$889$	−7.76997	−0.260597
$890$	−73.0896	−2.44997
$891$	14.1282	0.473314
$892$	80.4614	2.69405
$893$	2.87295	0.0961395
$894$	−20.2617	−0.677651
$895$	−23.7673	−0.794454
$896$	47.2161	1.57738
$897$	2.29017	0.0764666
$898$	−80.5985	−2.68961
$899$	6.41344	0.213900
$900$	6.03281	0.201094
$901$	40.9927	1.36566
$902$	−10.9867	−0.365816
$903$	0	0
$904$	5.43933	0.180910
$905$	41.8752	1.39198
$906$	3.48834	0.115892
$907$	−56.7976	−1.88593	−0.942966	−	0.332889i	$-0.891977\pi$
$907$	−56.7976	−1.88593	−0.942966	+	0.332889i	$0.891977\pi$
$908$	−24.2278	−0.804029
$909$	26.5163	0.879489
$910$	−16.2773	−0.539586
$911$	−24.2739	−0.804229	−0.402114	−	0.915589i	$-0.631725\pi$
$911$	−24.2739	−0.804229	−0.402114	+	0.915589i	$0.631725\pi$
$912$	−0.321361	−0.0106413
$913$	−81.3816	−2.69334
$914$	−75.7004	−2.50395
$915$	−6.55199	−0.216602
$916$	23.2945	0.769673
$917$	−24.1438	−0.797300
$918$	−72.8144	−2.40323
$919$	9.60014	0.316680	0.158340	−	0.987385i	$-0.449386\pi$
$919$	9.60014	0.316680	0.158340	+	0.987385i	$0.449386\pi$
$920$	11.3593	0.374504
$921$	4.99885	0.164718
$922$	−91.9692	−3.02885
$923$	10.5374	0.346842
$924$	−47.4102	−1.55968
$925$	−4.50675	−0.148181
$926$	29.2676	0.961794
$927$	4.69941	0.154349
$928$	32.3547	1.06209
$929$	−12.0757	−0.396190	−0.198095	−	0.980183i	$-0.563475\pi$
$929$	−12.0757	−0.396190	−0.198095	+	0.980183i	$0.563475\pi$
$930$	−4.31596	−0.141526
$931$	0.696513	0.0228273
$932$	18.4177	0.603290
$933$	−5.91856	−0.193765
$934$	80.8503	2.64550
$935$	86.6905	2.83508
$936$	−9.49002	−0.310191
$937$	48.4179	1.58174	0.790872	−	0.611982i	$-0.209627\pi$
$937$	48.4179	1.58174	0.790872	+	0.611982i	$0.209627\pi$
$938$	57.8930	1.89027
$939$	17.9863	0.586962
$940$	−27.7285	−0.904403
$941$	−26.1750	−0.853279	−0.426640	−	0.904422i	$-0.640303\pi$
$941$	−26.1750	−0.853279	−0.426640	+	0.904422i	$0.640303\pi$
$942$	−29.1814	−0.950781
$943$	−1.33964	−0.0436246
$944$	4.87816	0.158771
$945$	23.5792	0.767032
$946$	0	0
$947$	50.3957	1.63764	0.818820	−	0.574050i	$-0.194628\pi$
$947$	50.3957	1.63764	0.818820	+	0.574050i	$0.194628\pi$
$948$	22.7286	0.738189
$949$	1.93406	0.0627823
$950$	1.35915	0.0440966
$951$	12.6487	0.410164
$952$	−51.0163	−1.65345
$953$	−2.40711	−0.0779740	−0.0389870	−	0.999240i	$-0.512413\pi$
$953$	−2.40711	−0.0779740	−0.0389870	+	0.999240i	$0.512413\pi$
$954$	−30.7569	−0.995793
$955$	−39.4067	−1.27517
$956$	11.6456	0.376645
$957$	−36.9093	−1.19311
$958$	50.8611	1.64325
$959$	12.1271	0.391605
$960$	−23.6278	−0.762583
$961$	−29.9917	−0.967474
$962$	17.6318	0.568472
$963$	14.2752	0.460012
$964$	−5.45043	−0.175547
$965$	23.6370	0.760903
$966$	−9.23734	−0.297206
$967$	−31.2564	−1.00514	−0.502568	−	0.864538i	$-0.667611\pi$
$967$	−31.2564	−1.00514	−0.502568	+	0.864538i	$0.667611\pi$
$968$	91.0483	2.92640
$969$	−4.31015	−0.138462
$970$	−1.36270	−0.0437537
$971$	−0.00187696	−6.02345e−5 0	−3.01173e−5	−	1.00000i	$-0.500010\pi$
$971$	−0.00187696	−6.02345e−5 0	−3.01173e−5		1.00000i	$0.500010\pi$
$972$	54.0258	1.73288
$973$	7.60898	0.243933
$974$	−15.0119	−0.481012
$975$	−1.06340	−0.0340561
$976$	−1.75772	−0.0562633
$977$	−25.3298	−0.810372	−0.405186	−	0.914234i	$-0.632793\pi$
$977$	−25.3298	−0.810372	−0.405186	+	0.914234i	$0.632793\pi$
$978$	−19.9431	−0.637711
$979$	−98.2674	−3.14064
$980$	−6.72245	−0.214741
$981$	1.97460	0.0630440
$982$	16.6450	0.531164
$983$	−37.1682	−1.18548	−0.592741	−	0.805393i	$-0.701954\pi$
$983$	−37.1682	−1.18548	−0.592741	+	0.805393i	$0.701954\pi$
$984$	−2.11982	−0.0675775
$985$	32.9674	1.05043
$986$	−98.7776	−3.14572
$987$	9.06646	0.288588
$988$	−3.32770	−0.105868
$989$	0	0
$990$	−65.0442	−2.06724
$991$	−0.390175	−0.0123943	−0.00619716	−	0.999981i	$-0.501973\pi$
$991$	−0.390175	−0.0123943	−0.00619716	+	0.999981i	$0.501973\pi$
$992$	5.08666	0.161502
$993$	−13.3095	−0.422364
$994$	−42.5022	−1.34809
$995$	−32.5112	−1.03067
$996$	−39.0522	−1.23742
$997$	−14.6123	−0.462777	−0.231388	−	0.972861i	$-0.574327\pi$
$997$	−14.6123	−0.462777	−0.231388	+	0.972861i	$0.574327\pi$
$998$	85.6401	2.71089
$999$	−25.5414	−0.808093

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.r.1.16	yes	20
43.42	odd	2		1849.2.a.p.1.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.5	✓	20	43.42	odd	2
1849.2.a.r.1.16	yes	20	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+2.31191 q^{2} +0.910509 q^{3} +3.34494 q^{4} +2.04187 q^{5} +2.10502 q^{6} -2.45270 q^{7} +3.10939 q^{8} -2.17097 q^{9} +O(q^{10})\)	\(q+2.31191 q^{2} +0.910509 q^{3} +3.34494 q^{4} +2.04187 q^{5} +2.10502 q^{6} -2.45270 q^{7} +3.10939 q^{8} -2.17097 q^{9} +4.72063 q^{10} +6.34679 q^{11} +3.04560 q^{12} +1.40584 q^{13} -5.67043 q^{14} +1.85914 q^{15} +0.498758 q^{16} +6.68943 q^{17} -5.01910 q^{18} -0.707650 q^{19} +6.82994 q^{20} -2.23321 q^{21} +14.6732 q^{22} +1.78915 q^{23} +2.83113 q^{24} -0.830761 q^{25} +3.25019 q^{26} -4.70822 q^{27} -8.20414 q^{28} -6.38701 q^{29} +4.29818 q^{30} -1.00414 q^{31} -5.06570 q^{32} +5.77881 q^{33} +15.4654 q^{34} -5.00810 q^{35} -7.26178 q^{36} +5.42485 q^{37} -1.63603 q^{38} +1.28003 q^{39} +6.34898 q^{40} -0.748756 q^{41} -5.16298 q^{42} +21.2296 q^{44} -4.43285 q^{45} +4.13636 q^{46} -4.05984 q^{47} +0.454124 q^{48} -0.984262 q^{49} -1.92065 q^{50} +6.09079 q^{51} +4.70247 q^{52} +6.12798 q^{53} -10.8850 q^{54} +12.9593 q^{55} -7.62641 q^{56} -0.644322 q^{57} -14.7662 q^{58} +9.78062 q^{59} +6.21872 q^{60} -3.52420 q^{61} -2.32148 q^{62} +5.32475 q^{63} -12.7090 q^{64} +2.87055 q^{65} +13.3601 q^{66} -10.2096 q^{67} +22.3758 q^{68} +1.62904 q^{69} -11.5783 q^{70} +7.49541 q^{71} -6.75041 q^{72} +1.37573 q^{73} +12.5418 q^{74} -0.756416 q^{75} -2.36705 q^{76} -15.5668 q^{77} +2.95933 q^{78} +7.46275 q^{79} +1.01840 q^{80} +2.22605 q^{81} -1.73106 q^{82} -12.8225 q^{83} -7.46994 q^{84} +13.6590 q^{85} -5.81543 q^{87} +19.7346 q^{88} -15.4830 q^{89} -10.2484 q^{90} -3.44811 q^{91} +5.98460 q^{92} -0.914276 q^{93} -9.38600 q^{94} -1.44493 q^{95} -4.61236 q^{96} -0.288669 q^{97} -2.27553 q^{98} -13.7787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9}+O(q^{10}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9} + 21 q^{11} - 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} + 9 q^{18} - 7 q^{19} - 17 q^{20} + 8 q^{21} + 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} + 58 q^{26} - 22 q^{27} - 28 q^{28} - 12 q^{29} - 24 q^{30} + 26 q^{31} + 3 q^{32} - 17 q^{33} + 54 q^{34} + 26 q^{35} + 14 q^{36} - 19 q^{37} + 5 q^{38} - 7 q^{39} - 16 q^{40} + 27 q^{41} - 52 q^{42} + 32 q^{44} + 75 q^{45} + 59 q^{46} + 45 q^{47} - 66 q^{48} + 20 q^{49} + 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} - 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} + 30 q^{61} + 72 q^{62} - 21 q^{63} - 7 q^{64} - 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} - 35 q^{69} - 80 q^{70} - 31 q^{71} - 15 q^{72} - 26 q^{73} + 87 q^{74} + 73 q^{75} - 68 q^{76} - 21 q^{77} - 50 q^{78} + 39 q^{79} - 60 q^{80} + 16 q^{81} + 45 q^{82} + 13 q^{83} + 31 q^{84} - 22 q^{85} + 61 q^{87} + 50 q^{88} - 4 q^{89} - 13 q^{90} - 25 q^{91} + 5 q^{92} + 67 q^{93} + 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} - 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9 + 21 * q^11 - 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 + 9 * q^18 - 7 * q^19 - 17 * q^20 + 8 * q^21 + 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 + 58 * q^26 - 22 * q^27 - 28 * q^28 - 12 * q^29 - 24 * q^30 + 26 * q^31 + 3 * q^32 - 17 * q^33 + 54 * q^34 + 26 * q^35 + 14 * q^36 - 19 * q^37 + 5 * q^38 - 7 * q^39 - 16 * q^40 + 27 * q^41 - 52 * q^42 + 32 * q^44 + 75 * q^45 + 59 * q^46 + 45 * q^47 - 66 * q^48 + 20 * q^49 + 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 - 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 + 30 * q^61 + 72 * q^62 - 21 * q^63 - 7 * q^64 - 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 - 35 * q^69 - 80 * q^70 - 31 * q^71 - 15 * q^72 - 26 * q^73 + 87 * q^74 + 73 * q^75 - 68 * q^76 - 21 * q^77 - 50 * q^78 + 39 * q^79 - 60 * q^80 + 16 * q^81 + 45 * q^82 + 13 * q^83 + 31 * q^84 - 22 * q^85 + 61 * q^87 + 50 * q^88 - 4 * q^89 - 13 * q^90 - 25 * q^91 + 5 * q^92 + 67 * q^93 + 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 - 35 * q^98 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more