LMFDB - Embedded newform 5070.2.b.s.1351.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.6
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.s.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.55496i q^{11} +1.00000 q^{12} -1.69202 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.35690 q^{17} +1.00000i q^{18} -6.51573i q^{19} -1.00000i q^{20} -1.69202i q^{21} -4.55496 q^{22} -8.94869 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.69202i q^{28} +9.07606 q^{29} +1.00000 q^{30} -10.6528i q^{31} +1.00000i q^{32} -4.55496i q^{33} -2.35690i q^{34} -1.69202 q^{35} -1.00000 q^{36} -6.18598i q^{37} +6.51573 q^{38} +1.00000 q^{40} +3.00969i q^{41} +1.69202 q^{42} +4.93900 q^{43} -4.55496i q^{44} +1.00000i q^{45} -8.94869i q^{46} +4.28621i q^{47} -1.00000 q^{48} +4.13706 q^{49} -1.00000i q^{50} +2.35690 q^{51} -3.40581 q^{53} -1.00000i q^{54} -4.55496 q^{55} +1.69202 q^{56} +6.51573i q^{57} +9.07606i q^{58} -4.32304i q^{59} +1.00000i q^{60} -14.3666 q^{61} +10.6528 q^{62} +1.69202i q^{63} -1.00000 q^{64} +4.55496 q^{66} -3.24698i q^{67} +2.35690 q^{68} +8.94869 q^{69} -1.69202i q^{70} -14.9487i q^{71} -1.00000i q^{72} +6.72886i q^{73} +6.18598 q^{74} +1.00000 q^{75} +6.51573i q^{76} -7.70709 q^{77} +5.67994 q^{79} +1.00000i q^{80} +1.00000 q^{81} -3.00969 q^{82} -7.71917i q^{83} +1.69202i q^{84} -2.35690i q^{85} +4.93900i q^{86} -9.07606 q^{87} +4.55496 q^{88} +9.12498i q^{89} -1.00000 q^{90} +8.94869 q^{92} +10.6528i q^{93} -4.28621 q^{94} +6.51573 q^{95} -1.00000i q^{96} -4.40581i q^{97} +4.13706i q^{98} +4.55496i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} + 6 q^{12} + 6 q^{16} - 6 q^{17} - 28 q^{22} + 10 q^{23} - 6 q^{25} - 6 q^{27} + 24 q^{29} + 6 q^{30} - 6 q^{36} + 14 q^{38} + 6 q^{40} + 10 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 6 q^{53} - 28 q^{55} - 34 q^{61} + 28 q^{62} - 6 q^{64} + 28 q^{66} + 6 q^{68} - 10 q^{69} + 8 q^{74} + 6 q^{75} + 14 q^{77} - 14 q^{79} + 6 q^{81} + 26 q^{82} - 24 q^{87} + 28 q^{88} - 6 q^{90} - 10 q^{92} - 42 q^{94} + 14 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 28 * q^22 + 10 * q^23 - 6 * q^25 - 6 * q^27 + 24 * q^29 + 6 * q^30 - 6 * q^36 + 14 * q^38 + 6 * q^40 + 10 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 6 * q^53 - 28 * q^55 - 34 * q^61 + 28 * q^62 - 6 * q^64 + 28 * q^66 + 6 * q^68 - 10 * q^69 + 8 * q^74 + 6 * q^75 + 14 * q^77 - 14 * q^79 + 6 * q^81 + 26 * q^82 - 24 * q^87 + 28 * q^88 - 6 * q^90 - 10 * q^92 - 42 * q^94 + 14 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 1.00000i	− 0.408248i
$7$	1.69202i	0.639524i	0.947498	+	0.319762i	$0.103603\pi$
$7$	1.69202i	0.639524i	−0.947498	+	0.319762i	$0.896397\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.55496i	1.37337i	0.726954	+	0.686686i	$0.240935\pi$
$11$	4.55496i	1.37337i	−0.726954	+	0.686686i	$0.759065\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−1.69202	−0.452212
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	−2.35690	−0.571631	−0.285816	−	0.958285i	$-0.592264\pi$
$17$	−2.35690	−0.571631	−0.285816	+	0.958285i	$0.592264\pi$
$18$	1.00000i	0.235702i
$19$	− 6.51573i	− 1.49481i	−0.664368	−	0.747405i	$-0.731299\pi$
$19$	− 6.51573i	− 1.49481i	0.664368	−	0.747405i	$-0.268701\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 1.69202i	− 0.369229i
$22$	−4.55496	−0.971120
$23$	−8.94869	−1.86593	−0.932965	−	0.359966i	$-0.882788\pi$
$23$	−8.94869	−1.86593	−0.932965	+	0.359966i	$0.882788\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 1.69202i	− 0.319762i
$29$	9.07606	1.68538	0.842691	−	0.538397i	$-0.180970\pi$
$29$	9.07606	1.68538	0.842691	+	0.538397i	$0.180970\pi$
$30$	1.00000	0.182574
$31$	− 10.6528i	− 1.91330i	−0.291243	−	0.956649i	$-0.594069\pi$
$31$	− 10.6528i	− 1.91330i	0.291243	−	0.956649i	$-0.405931\pi$
$32$	1.00000i	0.176777i
$33$	− 4.55496i	− 0.792916i
$34$	− 2.35690i	− 0.404204i
$35$	−1.69202	−0.286004
$36$	−1.00000	−0.166667
$37$	− 6.18598i	− 1.01697i	−0.861071	−	0.508484i	$-0.830206\pi$
$37$	− 6.18598i	− 1.01697i	0.861071	−	0.508484i	$-0.169794\pi$
$38$	6.51573	1.05699
$39$	0	0
$40$	1.00000	0.158114
$41$	3.00969i	0.470034i	0.971991	+	0.235017i	$0.0755146\pi$
$41$	3.00969i	0.470034i	−0.971991	+	0.235017i	$0.924485\pi$
$42$	1.69202	0.261085
$43$	4.93900	0.753191	0.376595	−	0.926378i	$-0.377095\pi$
$43$	4.93900	0.753191	0.376595	+	0.926378i	$0.377095\pi$
$44$	− 4.55496i	− 0.686686i
$45$	1.00000i	0.149071i
$46$	− 8.94869i	− 1.31941i
$47$	4.28621i	0.625208i	0.949884	+	0.312604i	$0.101201\pi$
$47$	4.28621i	0.625208i	−0.949884	+	0.312604i	$0.898799\pi$
$48$	−1.00000	−0.144338
$49$	4.13706	0.591009
$50$	− 1.00000i	− 0.141421i
$51$	2.35690	0.330031
$52$	0	0
$53$	−3.40581	−0.467824	−0.233912	−	0.972258i	$-0.575153\pi$
$53$	−3.40581	−0.467824	−0.233912	+	0.972258i	$0.575153\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−4.55496	−0.614190
$56$	1.69202	0.226106
$57$	6.51573i	0.863029i
$58$	9.07606i	1.19175i
$59$	− 4.32304i	− 0.562812i	−0.959589	−	0.281406i	$-0.909199\pi$
$59$	− 4.32304i	− 0.562812i	0.959589	−	0.281406i	$-0.0908008\pi$
$60$	1.00000i	0.129099i
$61$	−14.3666	−1.83945	−0.919726	−	0.392560i	$-0.871589\pi$
$61$	−14.3666	−1.83945	−0.919726	+	0.392560i	$0.871589\pi$
$62$	10.6528	1.35291
$63$	1.69202i	0.213175i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.55496	0.560677
$67$	− 3.24698i	− 0.396682i	−0.980133	−	0.198341i	$-0.936445\pi$
$67$	− 3.24698i	− 0.396682i	0.980133	−	0.198341i	$-0.0635553\pi$
$68$	2.35690	0.285816
$69$	8.94869	1.07730
$70$	− 1.69202i	− 0.202235i
$71$	− 14.9487i	− 1.77408i	−0.461690	−	0.887042i	$-0.652757\pi$
$71$	− 14.9487i	− 1.77408i	0.461690	−	0.887042i	$-0.347243\pi$
$72$	− 1.00000i	− 0.117851i
$73$	6.72886i	0.787553i	0.919206	+	0.393777i	$0.128832\pi$
$73$	6.72886i	0.787553i	−0.919206	+	0.393777i	$0.871168\pi$
$74$	6.18598	0.719106
$75$	1.00000	0.115470
$76$	6.51573i	0.747405i
$77$	−7.70709	−0.878304
$78$	0	0
$79$	5.67994	0.639043	0.319522	−	0.947579i	$-0.396478\pi$
$79$	5.67994	0.639043	0.319522	+	0.947579i	$0.396478\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−3.00969	−0.332365
$83$	− 7.71917i	− 0.847289i	−0.905829	−	0.423644i	$-0.860751\pi$
$83$	− 7.71917i	− 0.847289i	0.905829	−	0.423644i	$-0.139249\pi$
$84$	1.69202i	0.184615i
$85$	− 2.35690i	− 0.255641i
$86$	4.93900i	0.532586i
$87$	−9.07606	−0.973056
$88$	4.55496	0.485560
$89$	9.12498i	0.967246i	0.875276	+	0.483623i	$0.160679\pi$
$89$	9.12498i	0.967246i	−0.875276	+	0.483623i	$0.839321\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	8.94869	0.932965
$93$	10.6528i	1.10464i
$94$	−4.28621	−0.442089
$95$	6.51573	0.668500
$96$	− 1.00000i	− 0.102062i
$97$	− 4.40581i	− 0.447343i	−0.974665	−	0.223671i	$-0.928196\pi$
$97$	− 4.40581i	− 0.447343i	0.974665	−	0.223671i	$-0.0718042\pi$
$98$	4.13706i	0.417907i
$99$	4.55496i	0.457791i
$100$	1.00000	0.100000
$101$	−2.31767	−0.230617	−0.115308	−	0.993330i	$-0.536786\pi$
$101$	−2.31767	−0.230617	−0.115308	+	0.993330i	$0.536786\pi$
$102$	2.35690i	0.233367i
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	0	0
$105$	1.69202	0.165124
$106$	− 3.40581i	− 0.330802i
$107$	−3.06100	−0.295918	−0.147959	−	0.988994i	$-0.547270\pi$
$107$	−3.06100	−0.295918	−0.147959	+	0.988994i	$0.547270\pi$
$108$	1.00000	0.0962250
$109$	7.45473i	0.714034i	0.934098	+	0.357017i	$0.116206\pi$
$109$	7.45473i	0.714034i	−0.934098	+	0.357017i	$0.883794\pi$
$110$	− 4.55496i	− 0.434298i
$111$	6.18598i	0.587147i
$112$	1.69202i	0.159881i
$113$	−19.9812	−1.87967	−0.939837	−	0.341623i	$-0.889024\pi$
$113$	−19.9812	−1.87967	−0.939837	+	0.341623i	$0.889024\pi$
$114$	−6.51573	−0.610254
$115$	− 8.94869i	− 0.834470i
$116$	−9.07606	−0.842691
$117$	0	0
$118$	4.32304	0.397968
$119$	− 3.98792i	− 0.365572i
$120$	−1.00000	−0.0912871
$121$	−9.74764	−0.886149
$122$	− 14.3666i	− 1.30069i
$123$	− 3.00969i	− 0.271374i
$124$	10.6528i	0.956649i
$125$	− 1.00000i	− 0.0894427i
$126$	−1.69202	−0.150737
$127$	3.14244	0.278846	0.139423	−	0.990233i	$-0.455475\pi$
$127$	3.14244	0.278846	0.139423	+	0.990233i	$0.455475\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−4.93900	−0.434855
$130$	0	0
$131$	−3.77479	−0.329805	−0.164902	−	0.986310i	$-0.552731\pi$
$131$	−3.77479	−0.329805	−0.164902	+	0.986310i	$0.552731\pi$
$132$	4.55496i	0.396458i
$133$	11.0248	0.955967
$134$	3.24698	0.280496
$135$	− 1.00000i	− 0.0860663i
$136$	2.35690i	0.202102i
$137$	− 14.5894i	− 1.24646i	−0.782040	−	0.623228i	$-0.785821\pi$
$137$	− 14.5894i	− 1.24646i	0.782040	−	0.623228i	$-0.214179\pi$
$138$	8.94869i	0.761763i
$139$	−14.3599	−1.21799	−0.608995	−	0.793174i	$-0.708427\pi$
$139$	−14.3599	−1.21799	−0.608995	+	0.793174i	$0.708427\pi$
$140$	1.69202	0.143002
$141$	− 4.28621i	− 0.360964i
$142$	14.9487	1.25447
$143$	0	0
$144$	1.00000	0.0833333
$145$	9.07606i	0.753726i
$146$	−6.72886	−0.556884
$147$	−4.13706	−0.341219
$148$	6.18598i	0.508484i
$149$	0.252356i	0.0206738i	0.999947	+	0.0103369i	$0.00329040\pi$
$149$	0.252356i	0.0206738i	−0.999947	+	0.0103369i	$0.996710\pi$
$150$	1.00000i	0.0816497i
$151$	− 11.0804i	− 0.901708i	−0.892598	−	0.450854i	$-0.851119\pi$
$151$	− 11.0804i	− 0.901708i	0.892598	−	0.450854i	$-0.148881\pi$
$152$	−6.51573	−0.528495
$153$	−2.35690	−0.190544
$154$	− 7.70709i	− 0.621055i
$155$	10.6528	0.855653
$156$	0	0
$157$	5.03923	0.402174	0.201087	−	0.979573i	$-0.435553\pi$
$157$	5.03923	0.402174	0.201087	+	0.979573i	$0.435553\pi$
$158$	5.67994i	0.451872i
$159$	3.40581	0.270099
$160$	−1.00000	−0.0790569
$161$	− 15.1414i	− 1.19331i
$162$	1.00000i	0.0785674i
$163$	− 20.9825i	− 1.64348i	−0.569863	−	0.821740i	$-0.693004\pi$
$163$	− 20.9825i	− 1.64348i	0.569863	−	0.821740i	$-0.306996\pi$
$164$	− 3.00969i	− 0.235017i
$165$	4.55496	0.354603
$166$	7.71917	0.599124
$167$	22.2325i	1.72040i	0.509954	+	0.860201i	$0.329662\pi$
$167$	22.2325i	1.72040i	−0.509954	+	0.860201i	$0.670338\pi$
$168$	−1.69202	−0.130542
$169$	0	0
$170$	2.35690	0.180766
$171$	− 6.51573i	− 0.498270i
$172$	−4.93900	−0.376595
$173$	11.6286	0.884108	0.442054	−	0.896988i	$-0.354250\pi$
$173$	11.6286	0.884108	0.442054	+	0.896988i	$0.354250\pi$
$174$	− 9.07606i	− 0.688055i
$175$	− 1.69202i	− 0.127905i
$176$	4.55496i	0.343343i
$177$	4.32304i	0.324940i
$178$	−9.12498	−0.683946
$179$	15.7875	1.18001	0.590005	−	0.807399i	$-0.299126\pi$
$179$	15.7875	1.18001	0.590005	+	0.807399i	$0.299126\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	1.91185	0.142107	0.0710535	−	0.997473i	$-0.477364\pi$
$181$	1.91185	0.142107	0.0710535	+	0.997473i	$0.477364\pi$
$182$	0	0
$183$	14.3666	1.06201
$184$	8.94869i	0.659706i
$185$	6.18598	0.454802
$186$	−10.6528	−0.781101
$187$	− 10.7356i	− 0.785062i
$188$	− 4.28621i	− 0.312604i
$189$	− 1.69202i	− 0.123076i
$190$	6.51573i	0.472701i
$191$	24.3260	1.76017	0.880085	−	0.474817i	$-0.157486\pi$
$191$	24.3260	1.76017	0.880085	+	0.474817i	$0.157486\pi$
$192$	1.00000	0.0721688
$193$	0.521106i	0.0375101i	0.999824	+	0.0187550i	$0.00597026\pi$
$193$	0.521106i	0.0375101i	−0.999824	+	0.0187550i	$0.994030\pi$
$194$	4.40581	0.316319
$195$	0	0
$196$	−4.13706	−0.295505
$197$	− 4.87502i	− 0.347331i	−0.984805	−	0.173665i	$-0.944439\pi$
$197$	− 4.87502i	− 0.347331i	0.984805	−	0.173665i	$-0.0555611\pi$
$198$	−4.55496	−0.323707
$199$	18.4983	1.31131	0.655654	−	0.755062i	$-0.272393\pi$
$199$	18.4983	1.31131	0.655654	+	0.755062i	$0.272393\pi$
$200$	1.00000i	0.0707107i
$201$	3.24698i	0.229024i
$202$	− 2.31767i	− 0.163070i
$203$	15.3569i	1.07784i
$204$	−2.35690	−0.165016
$205$	−3.00969	−0.210206
$206$	9.00000i	0.627060i
$207$	−8.94869	−0.621977
$208$	0	0
$209$	29.6789	2.05293
$210$	1.69202i	0.116761i
$211$	18.2500	1.25638	0.628190	−	0.778060i	$-0.283796\pi$
$211$	18.2500	1.25638	0.628190	+	0.778060i	$0.283796\pi$
$212$	3.40581	0.233912
$213$	14.9487i	1.02427i
$214$	− 3.06100i	− 0.209246i
$215$	4.93900i	0.336837i
$216$	1.00000i	0.0680414i
$217$	18.0248	1.22360
$218$	−7.45473	−0.504898
$219$	− 6.72886i	− 0.454694i
$220$	4.55496	0.307095
$221$	0	0
$222$	−6.18598	−0.415176
$223$	8.54048i	0.571913i	0.958243	+	0.285957i	$0.0923113\pi$
$223$	8.54048i	0.571913i	−0.958243	+	0.285957i	$0.907689\pi$
$224$	−1.69202	−0.113053
$225$	−1.00000	−0.0666667
$226$	− 19.9812i	− 1.32913i
$227$	− 9.61596i	− 0.638233i	−0.947715	−	0.319117i	$-0.896614\pi$
$227$	− 9.61596i	− 0.638233i	0.947715	−	0.319117i	$-0.103386\pi$
$228$	− 6.51573i	− 0.431515i
$229$	13.4058i	0.885881i	0.896551	+	0.442941i	$0.146065\pi$
$229$	13.4058i	0.885881i	−0.896551	+	0.442941i	$0.853935\pi$
$230$	8.94869	0.590059
$231$	7.70709	0.507089
$232$	− 9.07606i	− 0.595873i
$233$	0.153457	0.0100533	0.00502664	−	0.999987i	$-0.498400\pi$
$233$	0.153457	0.0100533	0.00502664	+	0.999987i	$0.498400\pi$
$234$	0	0
$235$	−4.28621	−0.279601
$236$	4.32304i	0.281406i
$237$	−5.67994	−0.368952
$238$	3.98792	0.258498
$239$	14.5851i	0.943431i	0.881751	+	0.471715i	$0.156365\pi$
$239$	14.5851i	0.943431i	−0.881751	+	0.471715i	$0.843635\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 3.40044i	− 0.219041i	−0.993985	−	0.109521i	$-0.965068\pi$
$241$	− 3.40044i	− 0.219041i	0.993985	−	0.109521i	$-0.0349316\pi$
$242$	− 9.74764i	− 0.626602i
$243$	−1.00000	−0.0641500
$244$	14.3666	0.919726
$245$	4.13706i	0.264307i
$246$	3.00969	0.191891
$247$	0	0
$248$	−10.6528	−0.676453
$249$	7.71917i	0.489182i
$250$	1.00000	0.0632456
$251$	−1.80864	−0.114161	−0.0570803	−	0.998370i	$-0.518179\pi$
$251$	−1.80864	−0.114161	−0.0570803	+	0.998370i	$0.518179\pi$
$252$	− 1.69202i	− 0.106587i
$253$	− 40.7609i	− 2.56262i
$254$	3.14244i	0.197174i
$255$	2.35690i	0.147595i
$256$	1.00000	0.0625000
$257$	22.2664	1.38894	0.694469	−	0.719523i	$-0.255640\pi$
$257$	22.2664	1.38894	0.694469	+	0.719523i	$0.255640\pi$
$258$	− 4.93900i	− 0.307489i
$259$	10.4668	0.650376
$260$	0	0
$261$	9.07606	0.561794
$262$	− 3.77479i	− 0.233207i
$263$	8.08815	0.498736	0.249368	−	0.968409i	$-0.419777\pi$
$263$	8.08815	0.498736	0.249368	+	0.968409i	$0.419777\pi$
$264$	−4.55496	−0.280338
$265$	− 3.40581i	− 0.209217i
$266$	11.0248i	0.675971i
$267$	− 9.12498i	− 0.558440i
$268$	3.24698i	0.198341i
$269$	16.6843	1.01726	0.508628	−	0.860986i	$-0.330153\pi$
$269$	16.6843	1.01726	0.508628	+	0.860986i	$0.330153\pi$
$270$	1.00000	0.0608581
$271$	− 27.4077i	− 1.66490i	−0.554099	−	0.832451i	$-0.686937\pi$
$271$	− 27.4077i	− 1.66490i	0.554099	−	0.832451i	$-0.313063\pi$
$272$	−2.35690	−0.142908
$273$	0	0
$274$	14.5894	0.881378
$275$	− 4.55496i	− 0.274674i
$276$	−8.94869	−0.538648
$277$	16.0411	0.963819	0.481910	−	0.876221i	$-0.339943\pi$
$277$	16.0411	0.963819	0.481910	+	0.876221i	$0.339943\pi$
$278$	− 14.3599i	− 0.861248i
$279$	− 10.6528i	− 0.637766i
$280$	1.69202i	0.101118i
$281$	10.0935i	0.602129i	0.953604	+	0.301065i	$0.0973420\pi$
$281$	10.0935i	0.602129i	−0.953604	+	0.301065i	$0.902658\pi$
$282$	4.28621	0.255240
$283$	−12.1021	−0.719398	−0.359699	−	0.933068i	$-0.617121\pi$
$283$	−12.1021	−0.719398	−0.359699	+	0.933068i	$0.617121\pi$
$284$	14.9487i	0.887042i
$285$	−6.51573	−0.385959
$286$	0	0
$287$	−5.09246	−0.300598
$288$	1.00000i	0.0589256i
$289$	−11.4450	−0.673238
$290$	−9.07606	−0.532965
$291$	4.40581i	0.258273i
$292$	− 6.72886i	− 0.393777i
$293$	− 4.01075i	− 0.234311i	−0.993114	−	0.117155i	$-0.962622\pi$
$293$	− 4.01075i	− 0.234311i	0.993114	−	0.117155i	$-0.0373775\pi$
$294$	− 4.13706i	− 0.241278i
$295$	4.32304	0.251697
$296$	−6.18598	−0.359553
$297$	− 4.55496i	− 0.264305i
$298$	−0.252356	−0.0146186
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	8.35690i	0.481683i
$302$	11.0804	0.637604
$303$	2.31767	0.133147
$304$	− 6.51573i	− 0.373703i
$305$	− 14.3666i	− 0.822628i
$306$	− 2.35690i	− 0.134735i
$307$	15.2784i	0.871987i	0.899950	+	0.435993i	$0.143603\pi$
$307$	15.2784i	0.871987i	−0.899950	+	0.435993i	$0.856397\pi$
$308$	7.70709	0.439152
$309$	−9.00000	−0.511992
$310$	10.6528i	0.605038i
$311$	−25.4330	−1.44217	−0.721085	−	0.692846i	$-0.756356\pi$
$311$	−25.4330	−1.44217	−0.721085	+	0.692846i	$0.756356\pi$
$312$	0	0
$313$	−19.1226	−1.08087	−0.540436	−	0.841385i	$-0.681741\pi$
$313$	−19.1226	−1.08087	−0.540436	+	0.841385i	$0.681741\pi$
$314$	5.03923i	0.284380i
$315$	−1.69202	−0.0953346
$316$	−5.67994	−0.319522
$317$	− 20.3230i	− 1.14146i	−0.821139	−	0.570728i	$-0.806661\pi$
$317$	− 20.3230i	− 1.14146i	0.821139	−	0.570728i	$-0.193339\pi$
$318$	3.40581i	0.190989i
$319$	41.3411i	2.31466i
$320$	− 1.00000i	− 0.0559017i
$321$	3.06100	0.170848
$322$	15.1414	0.843796
$323$	15.3569i	0.854481i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.9825	1.16212
$327$	− 7.45473i	− 0.412248i
$328$	3.00969	0.166182
$329$	−7.25236	−0.399835
$330$	4.55496i	0.250742i
$331$	− 22.3588i	− 1.22895i	−0.788936	−	0.614476i	$-0.789368\pi$
$331$	− 22.3588i	− 1.22895i	0.788936	−	0.614476i	$-0.210632\pi$
$332$	7.71917i	0.423644i
$333$	− 6.18598i	− 0.338990i
$334$	−22.2325	−1.21651
$335$	3.24698	0.177401
$336$	− 1.69202i	− 0.0923073i
$337$	3.67025	0.199931	0.0999657	−	0.994991i	$-0.468127\pi$
$337$	3.67025	0.199931	0.0999657	+	0.994991i	$0.468127\pi$
$338$	0	0
$339$	19.9812	1.08523
$340$	2.35690i	0.127821i
$341$	48.5230	2.62767
$342$	6.51573	0.352330
$343$	18.8442i	1.01749i
$344$	− 4.93900i	− 0.266293i
$345$	8.94869i	0.481781i
$346$	11.6286i	0.625159i
$347$	−21.7832	−1.16938	−0.584690	−	0.811257i	$-0.698784\pi$
$347$	−21.7832	−1.16938	−0.584690	+	0.811257i	$0.698784\pi$
$348$	9.07606	0.486528
$349$	1.74333i	0.0933184i	0.998911	+	0.0466592i	$0.0148575\pi$
$349$	1.74333i	0.0933184i	−0.998911	+	0.0466592i	$0.985143\pi$
$350$	1.69202	0.0904424
$351$	0	0
$352$	−4.55496	−0.242780
$353$	6.09544i	0.324428i	0.986756	+	0.162214i	$0.0518634\pi$
$353$	6.09544i	0.324428i	−0.986756	+	0.162214i	$0.948137\pi$
$354$	−4.32304	−0.229767
$355$	14.9487	0.793394
$356$	− 9.12498i	− 0.483623i
$357$	3.98792i	0.211063i
$358$	15.7875i	0.834393i
$359$	− 25.9627i	− 1.37026i	−0.728422	−	0.685129i	$-0.759746\pi$
$359$	− 25.9627i	− 1.37026i	0.728422	−	0.685129i	$-0.240254\pi$
$360$	1.00000	0.0527046
$361$	−23.4547	−1.23446
$362$	1.91185i	0.100485i
$363$	9.74764	0.511619
$364$	0	0
$365$	−6.72886	−0.352204
$366$	14.3666i	0.750953i
$367$	27.8799	1.45532	0.727660	−	0.685938i	$-0.240608\pi$
$367$	27.8799	1.45532	0.727660	+	0.685938i	$0.240608\pi$
$368$	−8.94869	−0.466483
$369$	3.00969i	0.156678i
$370$	6.18598i	0.321594i
$371$	− 5.76271i	− 0.299185i
$372$	− 10.6528i	− 0.552322i
$373$	−25.1400	−1.30170	−0.650851	−	0.759205i	$-0.725588\pi$
$373$	−25.1400	−1.30170	−0.650851	+	0.759205i	$0.725588\pi$
$374$	10.7356	0.555123
$375$	1.00000i	0.0516398i
$376$	4.28621	0.221044
$377$	0	0
$378$	1.69202	0.0870282
$379$	− 15.7463i	− 0.808834i	−0.914575	−	0.404417i	$-0.867474\pi$
$379$	− 15.7463i	− 0.808834i	0.914575	−	0.404417i	$-0.132526\pi$
$380$	−6.51573	−0.334250
$381$	−3.14244	−0.160992
$382$	24.3260i	1.24463i
$383$	11.8183i	0.603889i	0.953326	+	0.301944i	$0.0976357\pi$
$383$	11.8183i	0.603889i	−0.953326	+	0.301944i	$0.902364\pi$
$384$	1.00000i	0.0510310i
$385$	− 7.70709i	− 0.392790i
$386$	−0.521106	−0.0265236
$387$	4.93900	0.251064
$388$	4.40581i	0.223671i
$389$	6.14675	0.311653	0.155826	−	0.987784i	$-0.450196\pi$
$389$	6.14675	0.311653	0.155826	+	0.987784i	$0.450196\pi$
$390$	0	0
$391$	21.0911	1.06662
$392$	− 4.13706i	− 0.208953i
$393$	3.77479	0.190413
$394$	4.87502	0.245600
$395$	5.67994i	0.285789i
$396$	− 4.55496i	− 0.228895i
$397$	5.92825i	0.297530i	0.988873	+	0.148765i	$0.0475298\pi$
$397$	5.92825i	0.297530i	−0.988873	+	0.148765i	$0.952470\pi$
$398$	18.4983i	0.927235i
$399$	−11.0248	−0.551928
$400$	−1.00000	−0.0500000
$401$	− 9.49934i	− 0.474374i	−0.971464	−	0.237187i	$-0.923775\pi$
$401$	− 9.49934i	− 0.474374i	0.971464	−	0.237187i	$-0.0762254\pi$
$402$	−3.24698	−0.161945
$403$	0	0
$404$	2.31767	0.115308
$405$	1.00000i	0.0496904i
$406$	−15.3569	−0.762150
$407$	28.1769	1.39668
$408$	− 2.35690i	− 0.116684i
$409$	− 9.00000i	− 0.445021i	−0.974930	−	0.222511i	$-0.928575\pi$
$409$	− 9.00000i	− 0.445021i	0.974930	−	0.222511i	$-0.0714252\pi$
$410$	− 3.00969i	− 0.148638i
$411$	14.5894i	0.719642i
$412$	−9.00000	−0.443398
$413$	7.31468	0.359932
$414$	− 8.94869i	− 0.439804i
$415$	7.71917	0.378919
$416$	0	0
$417$	14.3599	0.703206
$418$	29.6789i	1.45164i
$419$	30.6450	1.49711	0.748554	−	0.663074i	$-0.230749\pi$
$419$	30.6450	1.49711	0.748554	+	0.663074i	$0.230749\pi$
$420$	−1.69202	−0.0825622
$421$	4.03252i	0.196533i	0.995160	+	0.0982666i	$0.0313298\pi$
$421$	4.03252i	0.196533i	−0.995160	+	0.0982666i	$0.968670\pi$
$422$	18.2500i	0.888394i
$423$	4.28621i	0.208403i
$424$	3.40581i	0.165401i
$425$	2.35690	0.114326
$426$	−14.9487	−0.724266
$427$	− 24.3086i	− 1.17637i
$428$	3.06100	0.147959
$429$	0	0
$430$	−4.93900	−0.238180
$431$	− 27.6179i	− 1.33031i	−0.746707	−	0.665153i	$-0.768366\pi$
$431$	− 27.6179i	− 1.33031i	0.746707	−	0.665153i	$-0.231634\pi$
$432$	−1.00000	−0.0481125
$433$	−10.6756	−0.513038	−0.256519	−	0.966539i	$-0.582576\pi$
$433$	−10.6756	−0.513038	−0.256519	+	0.966539i	$0.582576\pi$
$434$	18.0248i	0.865216i
$435$	− 9.07606i	− 0.435164i
$436$	− 7.45473i	− 0.357017i
$437$	58.3072i	2.78921i
$438$	6.72886	0.321517
$439$	−25.9299	−1.23757	−0.618783	−	0.785562i	$-0.712374\pi$
$439$	−25.9299	−1.23757	−0.618783	+	0.785562i	$0.712374\pi$
$440$	4.55496i	0.217149i
$441$	4.13706	0.197003
$442$	0	0
$443$	−35.3749	−1.68071	−0.840357	−	0.542033i	$-0.817655\pi$
$443$	−35.3749	−1.68071	−0.840357	+	0.542033i	$0.817655\pi$
$444$	− 6.18598i	− 0.293574i
$445$	−9.12498	−0.432566
$446$	−8.54048	−0.404404
$447$	− 0.252356i	− 0.0119360i
$448$	− 1.69202i	− 0.0799405i
$449$	6.24698i	0.294813i	0.989076	+	0.147407i	$0.0470926\pi$
$449$	6.24698i	0.294813i	−0.989076	+	0.147407i	$0.952907\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	−13.7090	−0.645532
$452$	19.9812	0.939837
$453$	11.0804i	0.520601i
$454$	9.61596	0.451299
$455$	0	0
$456$	6.51573	0.305127
$457$	− 29.6383i	− 1.38642i	−0.720735	−	0.693211i	$-0.756195\pi$
$457$	− 29.6383i	− 1.38642i	0.720735	−	0.693211i	$-0.243805\pi$
$458$	−13.4058	−0.626413
$459$	2.35690	0.110010
$460$	8.94869i	0.417235i
$461$	34.2911i	1.59710i	0.601931	+	0.798548i	$0.294398\pi$
$461$	34.2911i	1.59710i	−0.601931	+	0.798548i	$0.705602\pi$
$462$	7.70709i	0.358566i
$463$	− 24.1769i	− 1.12360i	−0.827275	−	0.561798i	$-0.810110\pi$
$463$	− 24.1769i	− 1.12360i	0.827275	−	0.561798i	$-0.189890\pi$
$464$	9.07606	0.421346
$465$	−10.6528	−0.494011
$466$	0.153457i	0.00710875i
$467$	0.521106	0.0241139	0.0120570	−	0.999927i	$-0.496162\pi$
$467$	0.521106	0.0241139	0.0120570	+	0.999927i	$0.496162\pi$
$468$	0	0
$469$	5.49396	0.253687
$470$	− 4.28621i	− 0.197708i
$471$	−5.03923	−0.232195
$472$	−4.32304	−0.198984
$473$	22.4969i	1.03441i
$474$	− 5.67994i	− 0.260888i
$475$	6.51573i	0.298962i
$476$	3.98792i	0.182786i
$477$	−3.40581	−0.155941
$478$	−14.5851	−0.667106
$479$	− 10.4168i	− 0.475957i	−0.971270	−	0.237979i	$-0.923515\pi$
$479$	− 10.4168i	− 0.475957i	0.971270	−	0.237979i	$-0.0764848\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	3.40044	0.154886
$483$	15.1414i	0.688956i
$484$	9.74764	0.443075
$485$	4.40581	0.200058
$486$	− 1.00000i	− 0.0453609i
$487$	30.2349i	1.37007i	0.728508	+	0.685037i	$0.240214\pi$
$487$	30.2349i	1.37007i	−0.728508	+	0.685037i	$0.759786\pi$
$488$	14.3666i	0.650345i
$489$	20.9825i	0.948863i
$490$	−4.13706	−0.186893
$491$	31.5883	1.42556	0.712781	−	0.701387i	$-0.247435\pi$
$491$	31.5883	1.42556	0.712781	+	0.701387i	$0.247435\pi$
$492$	3.00969i	0.135687i
$493$	−21.3913	−0.963417
$494$	0	0
$495$	−4.55496	−0.204730
$496$	− 10.6528i	− 0.478325i
$497$	25.2935	1.13457
$498$	−7.71917	−0.345904
$499$	− 27.2862i	− 1.22150i	−0.791824	−	0.610749i	$-0.790868\pi$
$499$	− 27.2862i	− 1.22150i	0.791824	−	0.610749i	$-0.209132\pi$
$500$	1.00000i	0.0447214i
$501$	− 22.2325i	− 0.993275i
$502$	− 1.80864i	− 0.0807237i
$503$	29.6752	1.32315	0.661575	−	0.749879i	$-0.269888\pi$
$503$	29.6752	1.32315	0.661575	+	0.749879i	$0.269888\pi$
$504$	1.69202	0.0753686
$505$	− 2.31767i	− 0.103135i
$506$	40.7609	1.81204
$507$	0	0
$508$	−3.14244	−0.139423
$509$	− 44.1661i	− 1.95763i	−0.204748	−	0.978815i	$-0.565638\pi$
$509$	− 44.1661i	− 1.95763i	0.204748	−	0.978815i	$-0.434362\pi$
$510$	−2.35690	−0.104365
$511$	−11.3854	−0.503659
$512$	1.00000i	0.0441942i
$513$	6.51573i	0.287676i
$514$	22.2664i	0.982127i
$515$	9.00000i	0.396587i
$516$	4.93900	0.217427
$517$	−19.5235	−0.858643
$518$	10.4668i	0.459885i
$519$	−11.6286	−0.510440
$520$	0	0
$521$	20.1153	0.881267	0.440633	−	0.897687i	$-0.354754\pi$
$521$	20.1153	0.881267	0.440633	+	0.897687i	$0.354754\pi$
$522$	9.07606i	0.397249i
$523$	6.46117	0.282527	0.141264	−	0.989972i	$-0.454883\pi$
$523$	6.46117	0.282527	0.141264	+	0.989972i	$0.454883\pi$
$524$	3.77479	0.164902
$525$	1.69202i	0.0738459i
$526$	8.08815i	0.352660i
$527$	25.1075i	1.09370i
$528$	− 4.55496i	− 0.198229i
$529$	57.0790	2.48170
$530$	3.40581	0.147939
$531$	− 4.32304i	− 0.187604i
$532$	−11.0248	−0.477984
$533$	0	0
$534$	9.12498	0.394877
$535$	− 3.06100i	− 0.132339i
$536$	−3.24698	−0.140248
$537$	−15.7875	−0.681279
$538$	16.6843i	0.719309i
$539$	18.8442i	0.811675i
$540$	1.00000i	0.0430331i
$541$	− 34.0097i	− 1.46219i	−0.682275	−	0.731095i	$-0.739009\pi$
$541$	− 34.0097i	− 1.46219i	0.682275	−	0.731095i	$-0.260991\pi$
$542$	27.4077	1.17726
$543$	−1.91185	−0.0820455
$544$	− 2.35690i	− 0.101051i
$545$	−7.45473	−0.319326
$546$	0	0
$547$	14.6907	0.628129	0.314064	−	0.949402i	$-0.398309\pi$
$547$	14.6907	0.628129	0.314064	+	0.949402i	$0.398309\pi$
$548$	14.5894i	0.623228i
$549$	−14.3666	−0.613151
$550$	4.55496	0.194224
$551$	− 59.1372i	− 2.51933i
$552$	− 8.94869i	− 0.380882i
$553$	9.61058i	0.408683i
$554$	16.0411i	0.681523i
$555$	−6.18598	−0.262580
$556$	14.3599	0.608995
$557$	− 7.00969i	− 0.297010i	−0.988912	−	0.148505i	$-0.952554\pi$
$557$	− 7.00969i	− 0.297010i	0.988912	−	0.148505i	$-0.0474461\pi$
$558$	10.6528	0.450969
$559$	0	0
$560$	−1.69202	−0.0715010
$561$	10.7356i	0.453256i
$562$	−10.0935	−0.425770
$563$	30.2131	1.27333	0.636666	−	0.771140i	$-0.280313\pi$
$563$	30.2131	1.27333	0.636666	+	0.771140i	$0.280313\pi$
$564$	4.28621i	0.180482i
$565$	− 19.9812i	− 0.840616i
$566$	− 12.1021i	− 0.508691i
$567$	1.69202i	0.0710582i
$568$	−14.9487	−0.627233
$569$	−14.8436	−0.622274	−0.311137	−	0.950365i	$-0.600710\pi$
$569$	−14.8436	−0.622274	−0.311137	+	0.950365i	$0.600710\pi$
$570$	− 6.51573i	− 0.272914i
$571$	−0.965557	−0.0404073	−0.0202037	−	0.999796i	$-0.506431\pi$
$571$	−0.965557	−0.0404073	−0.0202037	+	0.999796i	$0.506431\pi$
$572$	0	0
$573$	−24.3260	−1.01623
$574$	− 5.09246i	− 0.212555i
$575$	8.94869	0.373186
$576$	−1.00000	−0.0416667
$577$	27.8431i	1.15912i	0.814929	+	0.579561i	$0.196776\pi$
$577$	27.8431i	1.15912i	−0.814929	+	0.579561i	$0.803224\pi$
$578$	− 11.4450i	− 0.476051i
$579$	− 0.521106i	− 0.0216564i
$580$	− 9.07606i	− 0.376863i
$581$	13.0610	0.541862
$582$	−4.40581	−0.182627
$583$	− 15.5133i	− 0.642497i
$584$	6.72886	0.278442
$585$	0	0
$586$	4.01075	0.165683
$587$	− 1.47889i	− 0.0610405i	−0.999534	−	0.0305202i	$-0.990284\pi$
$587$	− 1.47889i	− 0.0610405i	0.999534	−	0.0305202i	$-0.00971640\pi$
$588$	4.13706	0.170610
$589$	−69.4107	−2.86002
$590$	4.32304i	0.177977i
$591$	4.87502i	0.200531i
$592$	− 6.18598i	− 0.254242i
$593$	16.8552i	0.692159i	0.938205	+	0.346079i	$0.112487\pi$
$593$	16.8552i	0.692159i	−0.938205	+	0.346079i	$0.887513\pi$
$594$	4.55496	0.186892
$595$	3.98792	0.163489
$596$	− 0.252356i	− 0.0103369i
$597$	−18.4983	−0.757084
$598$	0	0
$599$	−24.4862	−1.00048	−0.500239	−	0.865887i	$-0.666755\pi$
$599$	−24.4862	−1.00048	−0.500239	+	0.865887i	$0.666755\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−9.97344	−0.406825	−0.203413	−	0.979093i	$-0.565203\pi$
$601$	−9.97344	−0.406825	−0.203413	+	0.979093i	$0.565203\pi$
$602$	−8.35690	−0.340602
$603$	− 3.24698i	− 0.132227i
$604$	11.0804i	0.450854i
$605$	− 9.74764i	− 0.396298i
$606$	2.31767i	0.0941488i
$607$	−35.4476	−1.43877	−0.719386	−	0.694611i	$-0.755577\pi$
$607$	−35.4476	−1.43877	−0.719386	+	0.694611i	$0.755577\pi$
$608$	6.51573	0.264248
$609$	− 15.3569i	− 0.622293i
$610$	14.3666	0.581686
$611$	0	0
$612$	2.35690	0.0952719
$613$	16.1782i	0.653432i	0.945123	+	0.326716i	$0.105942\pi$
$613$	16.1782i	0.653432i	−0.945123	+	0.326716i	$0.894058\pi$
$614$	−15.2784	−0.616588
$615$	3.00969	0.121362
$616$	7.70709i	0.310527i
$617$	− 17.6920i	− 0.712254i	−0.934438	−	0.356127i	$-0.884097\pi$
$617$	− 17.6920i	− 0.712254i	0.934438	−	0.356127i	$-0.115903\pi$
$618$	− 9.00000i	− 0.362033i
$619$	− 1.23968i	− 0.0498271i	−0.999690	−	0.0249136i	$-0.992069\pi$
$619$	− 1.23968i	− 0.0498271i	0.999690	−	0.0249136i	$-0.00793105\pi$
$620$	−10.6528	−0.427826
$621$	8.94869	0.359099
$622$	− 25.4330i	− 1.01977i
$623$	−15.4397	−0.618577
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 19.1226i	− 0.764292i
$627$	−29.6789	−1.18526
$628$	−5.03923	−0.201087
$629$	14.5797i	0.581331i
$630$	− 1.69202i	− 0.0674117i
$631$	− 33.7845i	− 1.34494i	−0.740125	−	0.672469i	$-0.765234\pi$
$631$	− 33.7845i	− 1.34494i	0.740125	−	0.672469i	$-0.234766\pi$
$632$	− 5.67994i	− 0.225936i
$633$	−18.2500	−0.725371
$634$	20.3230	0.807131
$635$	3.14244i	0.124704i
$636$	−3.40581	−0.135049
$637$	0	0
$638$	−41.3411	−1.63671
$639$	− 14.9487i	− 0.591361i
$640$	1.00000	0.0395285
$641$	−12.0785	−0.477070	−0.238535	−	0.971134i	$-0.576667\pi$
$641$	−12.0785	−0.477070	−0.238535	+	0.971134i	$0.576667\pi$
$642$	3.06100i	0.120808i
$643$	− 41.1377i	− 1.62231i	−0.584831	−	0.811155i	$-0.698839\pi$
$643$	− 41.1377i	− 1.62231i	0.584831	−	0.811155i	$-0.301161\pi$
$644$	15.1414i	0.596654i
$645$	− 4.93900i	− 0.194473i
$646$	−15.3569	−0.604209
$647$	−23.2054	−0.912297	−0.456148	−	0.889904i	$-0.650771\pi$
$647$	−23.2054	−0.912297	−0.456148	+	0.889904i	$0.650771\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	19.6913	0.772951
$650$	0	0
$651$	−18.0248	−0.706446
$652$	20.9825i	0.821740i
$653$	7.77538	0.304274	0.152137	−	0.988359i	$-0.451384\pi$
$653$	7.77538	0.304274	0.152137	+	0.988359i	$0.451384\pi$
$654$	7.45473	0.291503
$655$	− 3.77479i	− 0.147493i
$656$	3.00969i	0.117509i
$657$	6.72886i	0.262518i
$658$	− 7.25236i	− 0.282726i
$659$	−0.349600	−0.0136185	−0.00680924	−	0.999977i	$-0.502167\pi$
$659$	−0.349600	−0.0136185	−0.00680924	+	0.999977i	$0.502167\pi$
$660$	−4.55496	−0.177302
$661$	− 19.7855i	− 0.769568i	−0.923007	−	0.384784i	$-0.874276\pi$
$661$	− 19.7855i	− 0.769568i	0.923007	−	0.384784i	$-0.125724\pi$
$662$	22.3588	0.869000
$663$	0	0
$664$	−7.71917	−0.299562
$665$	11.0248i	0.427522i
$666$	6.18598	0.239702
$667$	−81.2189	−3.14481
$668$	− 22.2325i	− 0.860201i
$669$	− 8.54048i	− 0.330194i
$670$	3.24698i	0.125442i
$671$	− 65.4392i	− 2.52625i
$672$	1.69202	0.0652711
$673$	16.6146	0.640447	0.320223	−	0.947342i	$-0.396242\pi$
$673$	16.6146	0.640447	0.320223	+	0.947342i	$0.396242\pi$
$674$	3.67025i	0.141373i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−5.85517	−0.225032	−0.112516	−	0.993650i	$-0.535891\pi$
$677$	−5.85517	−0.225032	−0.112516	+	0.993650i	$0.535891\pi$
$678$	19.9812i	0.767374i
$679$	7.45473	0.286086
$680$	−2.35690	−0.0903828
$681$	9.61596i	0.368484i
$682$	48.5230i	1.85804i
$683$	2.19029i	0.0838092i	0.999122	+	0.0419046i	$0.0133426\pi$
$683$	2.19029i	0.0838092i	−0.999122	+	0.0419046i	$0.986657\pi$
$684$	6.51573i	0.249135i
$685$	14.5894	0.557432
$686$	−18.8442	−0.719473
$687$	− 13.4058i	− 0.511464i
$688$	4.93900	0.188298
$689$	0	0
$690$	−8.94869	−0.340671
$691$	11.8576i	0.451083i	0.974234	+	0.225541i	$0.0724151\pi$
$691$	11.8576i	0.451083i	−0.974234	+	0.225541i	$0.927585\pi$
$692$	−11.6286	−0.442054
$693$	−7.70709	−0.292768
$694$	− 21.7832i	− 0.826877i
$695$	− 14.3599i	− 0.544701i
$696$	9.07606i	0.344027i
$697$	− 7.09352i	− 0.268686i
$698$	−1.74333	−0.0659861
$699$	−0.153457	−0.00580427
$700$	1.69202i	0.0639524i
$701$	−22.7627	−0.859736	−0.429868	−	0.902892i	$-0.641440\pi$
$701$	−22.7627	−0.859736	−0.429868	+	0.902892i	$0.641440\pi$
$702$	0	0
$703$	−40.3062	−1.52018
$704$	− 4.55496i	− 0.171671i
$705$	4.28621	0.161428
$706$	−6.09544	−0.229405
$707$	− 3.92154i	− 0.147485i
$708$	− 4.32304i	− 0.162470i
$709$	− 18.5854i	− 0.697988i	−0.937125	−	0.348994i	$-0.886523\pi$
$709$	− 18.5854i	− 0.697988i	0.937125	−	0.348994i	$-0.113477\pi$
$710$	14.9487i	0.561014i
$711$	5.67994	0.213014
$712$	9.12498	0.341973
$713$	95.3285i	3.57008i
$714$	−3.98792	−0.149244
$715$	0	0
$716$	−15.7875	−0.590005
$717$	− 14.5851i	− 0.544690i
$718$	25.9627	0.968919
$719$	36.1575	1.34845	0.674224	−	0.738527i	$-0.264478\pi$
$719$	36.1575	1.34845	0.674224	+	0.738527i	$0.264478\pi$
$720$	1.00000i	0.0372678i
$721$	15.2282i	0.567128i
$722$	− 23.4547i	− 0.872895i
$723$	3.40044i	0.126464i
$724$	−1.91185	−0.0710535
$725$	−9.07606	−0.337077
$726$	9.74764i	0.361769i
$727$	−19.0164	−0.705279	−0.352639	−	0.935759i	$-0.614716\pi$
$727$	−19.0164	−0.705279	−0.352639	+	0.935759i	$0.614716\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 6.72886i	− 0.249046i
$731$	−11.6407	−0.430547
$732$	−14.3666	−0.531004
$733$	8.89248i	0.328451i	0.986423	+	0.164226i	$0.0525125\pi$
$733$	8.89248i	0.328451i	−0.986423	+	0.164226i	$0.947488\pi$
$734$	27.8799i	1.02907i
$735$	− 4.13706i	− 0.152598i
$736$	− 8.94869i	− 0.329853i
$737$	14.7899	0.544791
$738$	−3.00969	−0.110788
$739$	30.8595i	1.13518i	0.823310	+	0.567592i	$0.192125\pi$
$739$	30.8595i	1.13518i	−0.823310	+	0.567592i	$0.807875\pi$
$740$	−6.18598	−0.227401
$741$	0	0
$742$	5.76271	0.211556
$743$	18.6300i	0.683467i	0.939797	+	0.341733i	$0.111014\pi$
$743$	18.6300i	0.683467i	−0.939797	+	0.341733i	$0.888986\pi$
$744$	10.6528	0.390550
$745$	−0.252356	−0.00924562
$746$	− 25.1400i	− 0.920443i
$747$	− 7.71917i	− 0.282430i
$748$	10.7356i	0.392531i
$749$	− 5.17928i	− 0.189247i
$750$	−1.00000	−0.0365148
$751$	−12.5147	−0.456667	−0.228333	−	0.973583i	$-0.573328\pi$
$751$	−12.5147	−0.456667	−0.228333	+	0.973583i	$0.573328\pi$
$752$	4.28621i	0.156302i
$753$	1.80864	0.0659106
$754$	0	0
$755$	11.0804	0.403256
$756$	1.69202i	0.0615382i
$757$	−4.75494	−0.172821	−0.0864106	−	0.996260i	$-0.527540\pi$
$757$	−4.75494	−0.172821	−0.0864106	+	0.996260i	$0.527540\pi$
$758$	15.7463	0.571932
$759$	40.7609i	1.47953i
$760$	− 6.51573i	− 0.236350i
$761$	21.7748i	0.789336i	0.918824	+	0.394668i	$0.129140\pi$
$761$	21.7748i	0.789336i	−0.918824	+	0.394668i	$0.870860\pi$
$762$	− 3.14244i	− 0.113839i
$763$	−12.6136	−0.456642
$764$	−24.3260	−0.880085
$765$	− 2.35690i	− 0.0852137i
$766$	−11.8183	−0.427014
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	11.4045i	0.411256i	0.978630	+	0.205628i	$0.0659237\pi$
$769$	11.4045i	0.411256i	−0.978630	+	0.205628i	$0.934076\pi$
$770$	7.70709	0.277744
$771$	−22.2664	−0.801903
$772$	− 0.521106i	− 0.0187550i
$773$	49.6118i	1.78441i	0.451630	+	0.892206i	$0.350843\pi$
$773$	49.6118i	1.78441i	−0.451630	+	0.892206i	$0.649157\pi$
$774$	4.93900i	0.177529i
$775$	10.6528i	0.382660i
$776$	−4.40581	−0.158159
$777$	−10.4668	−0.375495
$778$	6.14675i	0.220372i
$779$	19.6103	0.702613
$780$	0	0
$781$	68.0907	2.43648
$782$	21.0911i	0.754217i
$783$	−9.07606	−0.324352
$784$	4.13706	0.147752
$785$	5.03923i	0.179858i
$786$	3.77479i	0.134642i
$787$	− 26.5532i	− 0.946518i	−0.880923	−	0.473259i	$-0.843078\pi$
$787$	− 26.5532i	− 0.946518i	0.880923	−	0.473259i	$-0.156922\pi$
$788$	4.87502i	0.173665i
$789$	−8.08815	−0.287946
$790$	−5.67994	−0.202083
$791$	− 33.8086i	− 1.20210i
$792$	4.55496	0.161853
$793$	0	0
$794$	−5.92825	−0.210386
$795$	3.40581i	0.120792i
$796$	−18.4983	−0.655654
$797$	−9.44803	−0.334666	−0.167333	−	0.985900i	$-0.553516\pi$
$797$	−9.44803	−0.334666	−0.167333	+	0.985900i	$0.553516\pi$
$798$	− 11.0248i	− 0.390272i
$799$	− 10.1021i	− 0.357388i
$800$	− 1.00000i	− 0.0353553i
$801$	9.12498i	0.322415i
$802$	9.49934	0.335433
$803$	−30.6497	−1.08160
$804$	− 3.24698i	− 0.114512i
$805$	15.1414	0.533663
$806$	0	0
$807$	−16.6843	−0.587313
$808$	2.31767i	0.0815352i
$809$	−27.3376	−0.961140	−0.480570	−	0.876956i	$-0.659570\pi$
$809$	−27.3376	−0.961140	−0.480570	+	0.876956i	$0.659570\pi$
$810$	−1.00000	−0.0351364
$811$	− 2.86592i	− 0.100636i	−0.998733	−	0.0503180i	$-0.983977\pi$
$811$	− 2.86592i	− 0.100636i	0.998733	−	0.0503180i	$-0.0160235\pi$
$812$	− 15.3569i	− 0.538921i
$813$	27.4077i	0.961231i
$814$	28.1769i	0.987599i
$815$	20.9825	0.734986
$816$	2.35690	0.0825079
$817$	− 32.1812i	− 1.12588i
$818$	9.00000	0.314678
$819$	0	0
$820$	3.00969	0.105103
$821$	16.9739i	0.592394i	0.955127	+	0.296197i	$0.0957185\pi$
$821$	16.9739i	0.592394i	−0.955127	+	0.296197i	$0.904282\pi$
$822$	−14.5894	−0.508864
$823$	22.2083	0.774134	0.387067	−	0.922052i	$-0.373488\pi$
$823$	22.2083	0.774134	0.387067	+	0.922052i	$0.373488\pi$
$824$	− 9.00000i	− 0.313530i
$825$	4.55496i	0.158583i
$826$	7.31468i	0.254510i
$827$	− 36.3618i	− 1.26442i	−0.774796	−	0.632212i	$-0.782147\pi$
$827$	− 36.3618i	− 1.26442i	0.774796	−	0.632212i	$-0.217853\pi$
$828$	8.94869	0.310988
$829$	26.5478	0.922042	0.461021	−	0.887389i	$-0.347483\pi$
$829$	26.5478	0.922042	0.461021	+	0.887389i	$0.347483\pi$
$830$	7.71917i	0.267936i
$831$	−16.0411	−0.556461
$832$	0	0
$833$	−9.75063	−0.337839
$834$	14.3599i	0.497242i
$835$	−22.2325	−0.769388
$836$	−29.6789	−1.02647
$837$	10.6528i	0.368214i
$838$	30.6450i	1.05861i
$839$	− 26.7668i	− 0.924091i	−0.886856	−	0.462046i	$-0.847116\pi$
$839$	− 26.7668i	− 0.924091i	0.886856	−	0.462046i	$-0.152884\pi$
$840$	− 1.69202i	− 0.0583803i
$841$	53.3749	1.84052
$842$	−4.03252	−0.138970
$843$	− 10.0935i	− 0.347639i
$844$	−18.2500	−0.628190
$845$	0	0
$846$	−4.28621	−0.147363
$847$	− 16.4932i	− 0.566714i
$848$	−3.40581	−0.116956
$849$	12.1021	0.415345
$850$	2.35690i	0.0808409i
$851$	55.3564i	1.89759i
$852$	− 14.9487i	− 0.512134i
$853$	50.3967i	1.72555i	0.505587	+	0.862775i	$0.331276\pi$
$853$	50.3967i	1.72555i	−0.505587	+	0.862775i	$0.668724\pi$
$854$	24.3086	0.831822
$855$	6.51573	0.222833
$856$	3.06100i	0.104623i
$857$	−27.5114	−0.939772	−0.469886	−	0.882727i	$-0.655705\pi$
$857$	−27.5114	−0.939772	−0.469886	+	0.882727i	$0.655705\pi$
$858$	0	0
$859$	2.26098	0.0771436	0.0385718	−	0.999256i	$-0.487719\pi$
$859$	2.26098	0.0771436	0.0385718	+	0.999256i	$0.487719\pi$
$860$	− 4.93900i	− 0.168419i
$861$	5.09246	0.173550
$862$	27.6179	0.940669
$863$	2.18406i	0.0743463i	0.999309	+	0.0371732i	$0.0118353\pi$
$863$	2.18406i	0.0743463i	−0.999309	+	0.0371732i	$0.988165\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	11.6286i	0.395385i
$866$	− 10.6756i	− 0.362773i
$867$	11.4450	0.388694
$868$	−18.0248	−0.611800
$869$	25.8719i	0.877644i
$870$	9.07606	0.307707
$871$	0	0
$872$	7.45473	0.252449
$873$	− 4.40581i	− 0.149114i
$874$	−58.3072	−1.97227
$875$	1.69202	0.0572008
$876$	6.72886i	0.227347i
$877$	19.9815i	0.674727i	0.941375	+	0.337363i	$0.109535\pi$
$877$	19.9815i	0.674727i	−0.941375	+	0.337363i	$0.890465\pi$
$878$	− 25.9299i	− 0.875092i
$879$	4.01075i	0.135279i
$880$	−4.55496	−0.153548
$881$	38.4801	1.29643	0.648213	−	0.761459i	$-0.275516\pi$
$881$	38.4801	1.29643	0.648213	+	0.761459i	$0.275516\pi$
$882$	4.13706i	0.139302i
$883$	37.5512	1.26370	0.631850	−	0.775091i	$-0.282296\pi$
$883$	37.5512	1.26370	0.631850	+	0.775091i	$0.282296\pi$
$884$	0	0
$885$	−4.32304	−0.145318
$886$	− 35.3749i	− 1.18844i
$887$	18.5676	0.623440	0.311720	−	0.950174i	$-0.399095\pi$
$887$	18.5676	0.623440	0.311720	+	0.950174i	$0.399095\pi$
$888$	6.18598	0.207588
$889$	5.31708i	0.178329i
$890$	− 9.12498i	− 0.305870i
$891$	4.55496i	0.152597i
$892$	− 8.54048i	− 0.285957i
$893$	27.9278	0.934567
$894$	0.252356	0.00844006
$895$	15.7875i	0.527717i
$896$	1.69202	0.0565265
$897$	0	0
$898$	−6.24698	−0.208464
$899$	− 96.6854i	− 3.22464i
$900$	1.00000	0.0333333
$901$	8.02715	0.267423
$902$	− 13.7090i	− 0.456460i
$903$	− 8.35690i	− 0.278100i
$904$	19.9812i	0.664565i
$905$	1.91185i	0.0635522i
$906$	−11.0804	−0.368121
$907$	−60.1648	−1.99774	−0.998870	−	0.0475319i	$-0.984864\pi$
$907$	−60.1648	−1.99774	−0.998870	+	0.0475319i	$0.984864\pi$
$908$	9.61596i	0.319117i
$909$	−2.31767	−0.0768722
$910$	0	0
$911$	−37.0737	−1.22831	−0.614153	−	0.789187i	$-0.710502\pi$
$911$	−37.0737	−1.22831	−0.614153	+	0.789187i	$0.710502\pi$
$912$	6.51573i	0.215757i
$913$	35.1605	1.16364
$914$	29.6383	0.980348
$915$	14.3666i	0.474945i
$916$	− 13.4058i	− 0.442941i
$917$	− 6.38703i	− 0.210918i
$918$	2.35690i	0.0777892i
$919$	−50.5526	−1.66758	−0.833788	−	0.552085i	$-0.813832\pi$
$919$	−50.5526	−1.66758	−0.833788	+	0.552085i	$0.813832\pi$
$920$	−8.94869	−0.295030
$921$	− 15.2784i	− 0.503442i
$922$	−34.2911	−1.12932
$923$	0	0
$924$	−7.70709	−0.253545
$925$	6.18598i	0.203394i
$926$	24.1769	0.794502
$927$	9.00000	0.295599
$928$	9.07606i	0.297936i
$929$	− 12.7269i	− 0.417557i	−0.977963	−	0.208779i	$-0.933051\pi$
$929$	− 12.7269i	− 0.417557i	0.977963	−	0.208779i	$-0.0669488\pi$
$930$	− 10.6528i	− 0.349319i
$931$	− 26.9560i	− 0.883447i
$932$	−0.153457	−0.00502664
$933$	25.4330	0.832638
$934$	0.521106i	0.0170511i
$935$	10.7356	0.351090
$936$	0	0
$937$	−35.3279	−1.15411	−0.577057	−	0.816704i	$-0.695799\pi$
$937$	−35.3279	−1.15411	−0.577057	+	0.816704i	$0.695799\pi$
$938$	5.49396i	0.179384i
$939$	19.1226	0.624042
$940$	4.28621	0.139801
$941$	42.0267i	1.37003i	0.728529	+	0.685015i	$0.240204\pi$
$941$	42.0267i	1.37003i	−0.728529	+	0.685015i	$0.759796\pi$
$942$	− 5.03923i	− 0.164187i
$943$	− 26.9328i	− 0.877052i
$944$	− 4.32304i	− 0.140703i
$945$	1.69202	0.0550415
$946$	−22.4969	−0.731439
$947$	− 4.63640i	− 0.150663i	−0.997159	−	0.0753314i	$-0.975999\pi$
$947$	− 4.63640i	− 0.150663i	0.997159	−	0.0753314i	$-0.0240015\pi$
$948$	5.67994	0.184476
$949$	0	0
$950$	−6.51573	−0.211398
$951$	20.3230i	0.659020i
$952$	−3.98792	−0.129249
$953$	41.8998	1.35727	0.678633	−	0.734477i	$-0.262573\pi$
$953$	41.8998	1.35727	0.678633	+	0.734477i	$0.262573\pi$
$954$	− 3.40581i	− 0.110267i
$955$	24.3260i	0.787172i
$956$	− 14.5851i	− 0.471715i
$957$	− 41.3411i	− 1.33637i
$958$	10.4168	0.336552
$959$	24.6856	0.797139
$960$	1.00000i	0.0322749i
$961$	−82.4820	−2.66071
$962$	0	0
$963$	−3.06100	−0.0986393
$964$	3.40044i	0.109521i
$965$	−0.521106	−0.0167750
$966$	−15.1414	−0.487166
$967$	− 48.3545i	− 1.55498i	−0.628898	−	0.777488i	$-0.716494\pi$
$967$	− 48.3545i	− 1.55498i	0.628898	−	0.777488i	$-0.283506\pi$
$968$	9.74764i	0.313301i
$969$	− 15.3569i	− 0.493335i
$970$	4.40581i	0.141462i
$971$	−18.9326	−0.607575	−0.303787	−	0.952740i	$-0.598251\pi$
$971$	−18.9326	−0.607575	−0.303787	+	0.952740i	$0.598251\pi$
$972$	1.00000	0.0320750
$973$	− 24.2972i	− 0.778933i
$974$	−30.2349	−0.968789
$975$	0	0
$976$	−14.3666	−0.459863
$977$	− 54.7972i	− 1.75312i	−0.481296	−	0.876558i	$-0.659834\pi$
$977$	− 54.7972i	− 1.75312i	0.481296	−	0.876558i	$-0.340166\pi$
$978$	−20.9825	−0.670948
$979$	−41.5639	−1.32839
$980$	− 4.13706i	− 0.132154i
$981$	7.45473i	0.238011i
$982$	31.5883i	1.00802i
$983$	− 24.6431i	− 0.785993i	−0.919540	−	0.392996i	$-0.871438\pi$
$983$	− 24.6431i	− 0.785993i	0.919540	−	0.392996i	$-0.128562\pi$
$984$	−3.00969	−0.0959454
$985$	4.87502	0.155331
$986$	− 21.3913i	− 0.681239i
$987$	7.25236	0.230845
$988$	0	0
$989$	−44.1976	−1.40540
$990$	− 4.55496i	− 0.144766i
$991$	23.1612	0.735741	0.367870	−	0.929877i	$-0.380087\pi$
$991$	23.1612	0.735741	0.367870	+	0.929877i	$0.380087\pi$
$992$	10.6528	0.338227
$993$	22.3588i	0.709536i
$994$	25.2935i	0.802261i
$995$	18.4983i	0.586435i
$996$	− 7.71917i	− 0.244591i
$997$	−11.5700	−0.366426	−0.183213	−	0.983073i	$-0.558650\pi$
$997$	−11.5700	−0.366426	−0.183213	+	0.983073i	$0.558650\pi$
$998$	27.2862	0.863730
$999$	6.18598i	0.195716i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.s.1351.6		6
13.5	odd	4		5070.2.a.bt.1.1	yes	3
13.8	odd	4		5070.2.a.bk.1.3	✓	3
13.12	even	2	inner	5070.2.b.s.1351.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bk.1.3	✓	3	13.8	odd	4
5070.2.a.bt.1.1	yes	3	13.5	odd	4
5070.2.b.s.1351.1		6	13.12	even	2	inner
5070.2.b.s.1351.6		6	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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.55496i q^{11} +1.00000 q^{12} -1.69202 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.35690 q^{17} +1.00000i q^{18} -6.51573i q^{19} -1.00000i q^{20} -1.69202i q^{21} -4.55496 q^{22} -8.94869 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.69202i q^{28} +9.07606 q^{29} +1.00000 q^{30} -10.6528i q^{31} +1.00000i q^{32} -4.55496i q^{33} -2.35690i q^{34} -1.69202 q^{35} -1.00000 q^{36} -6.18598i q^{37} +6.51573 q^{38} +1.00000 q^{40} +3.00969i q^{41} +1.69202 q^{42} +4.93900 q^{43} -4.55496i q^{44} +1.00000i q^{45} -8.94869i q^{46} +4.28621i q^{47} -1.00000 q^{48} +4.13706 q^{49} -1.00000i q^{50} +2.35690 q^{51} -3.40581 q^{53} -1.00000i q^{54} -4.55496 q^{55} +1.69202 q^{56} +6.51573i q^{57} +9.07606i q^{58} -4.32304i q^{59} +1.00000i q^{60} -14.3666 q^{61} +10.6528 q^{62} +1.69202i q^{63} -1.00000 q^{64} +4.55496 q^{66} -3.24698i q^{67} +2.35690 q^{68} +8.94869 q^{69} -1.69202i q^{70} -14.9487i q^{71} -1.00000i q^{72} +6.72886i q^{73} +6.18598 q^{74} +1.00000 q^{75} +6.51573i q^{76} -7.70709 q^{77} +5.67994 q^{79} +1.00000i q^{80} +1.00000 q^{81} -3.00969 q^{82} -7.71917i q^{83} +1.69202i q^{84} -2.35690i q^{85} +4.93900i q^{86} -9.07606 q^{87} +4.55496 q^{88} +9.12498i q^{89} -1.00000 q^{90} +8.94869 q^{92} +10.6528i q^{93} -4.28621 q^{94} +6.51573 q^{95} -1.00000i q^{96} -4.40581i q^{97} +4.13706i q^{98} +4.55496i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} + 6 q^{12} + 6 q^{16} - 6 q^{17} - 28 q^{22} + 10 q^{23} - 6 q^{25} - 6 q^{27} + 24 q^{29} + 6 q^{30} - 6 q^{36} + 14 q^{38} + 6 q^{40} + 10 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 6 q^{53} - 28 q^{55} - 34 q^{61} + 28 q^{62} - 6 q^{64} + 28 q^{66} + 6 q^{68} - 10 q^{69} + 8 q^{74} + 6 q^{75} + 14 q^{77} - 14 q^{79} + 6 q^{81} + 26 q^{82} - 24 q^{87} + 28 q^{88} - 6 q^{90} - 10 q^{92} - 42 q^{94} + 14 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 28 * q^22 + 10 * q^23 - 6 * q^25 - 6 * q^27 + 24 * q^29 + 6 * q^30 - 6 * q^36 + 14 * q^38 + 6 * q^40 + 10 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 6 * q^53 - 28 * q^55 - 34 * q^61 + 28 * q^62 - 6 * q^64 + 28 * q^66 + 6 * q^68 - 10 * q^69 + 8 * q^74 + 6 * q^75 + 14 * q^77 - 14 * q^79 + 6 * q^81 + 26 * q^82 - 24 * q^87 + 28 * q^88 - 6 * q^90 - 10 * q^92 - 42 * q^94 + 14 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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