LMFDB - Embedded newform 6422.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.273891$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.27389 q^{3} +1.00000 q^{4} +2.37720 q^{5} -1.27389 q^{6} +0.273891 q^{7} +1.00000 q^{8} -1.37720 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.27389 q^{3} +1.00000 q^{4} +2.37720 q^{5} -1.27389 q^{6} +0.273891 q^{7} +1.00000 q^{8} -1.37720 q^{9} +2.37720 q^{10} -4.37720 q^{11} -1.27389 q^{12} +0.273891 q^{14} -3.02830 q^{15} +1.00000 q^{16} +0.547781 q^{17} -1.37720 q^{18} +1.00000 q^{19} +2.37720 q^{20} -0.348907 q^{21} -4.37720 q^{22} -4.57608 q^{23} -1.27389 q^{24} +0.651093 q^{25} +5.57608 q^{27} +0.273891 q^{28} +1.67939 q^{29} -3.02830 q^{30} +2.85772 q^{31} +1.00000 q^{32} +5.57608 q^{33} +0.547781 q^{34} +0.651093 q^{35} -1.37720 q^{36} -8.05659 q^{37} +1.00000 q^{38} +2.37720 q^{40} +9.02830 q^{41} -0.348907 q^{42} -0.452219 q^{43} -4.37720 q^{44} -3.27389 q^{45} -4.57608 q^{46} +0.472765 q^{47} -1.27389 q^{48} -6.92498 q^{49} +0.651093 q^{50} -0.697813 q^{51} -2.85772 q^{53} +5.57608 q^{54} -10.4055 q^{55} +0.273891 q^{56} -1.27389 q^{57} +1.67939 q^{58} -13.5294 q^{59} -3.02830 q^{60} +4.19887 q^{61} +2.85772 q^{62} -0.377203 q^{63} +1.00000 q^{64} +5.57608 q^{66} +10.0566 q^{67} +0.547781 q^{68} +5.82942 q^{69} +0.651093 q^{70} -3.00000 q^{71} -1.37720 q^{72} -11.1132 q^{73} -8.05659 q^{74} -0.829422 q^{75} +1.00000 q^{76} -1.19887 q^{77} +2.36945 q^{79} +2.37720 q^{80} -2.97170 q^{81} +9.02830 q^{82} -16.7643 q^{83} -0.348907 q^{84} +1.30219 q^{85} -0.452219 q^{86} -2.13936 q^{87} -4.37720 q^{88} -2.04672 q^{89} -3.27389 q^{90} -4.57608 q^{92} -3.64042 q^{93} +0.472765 q^{94} +2.37720 q^{95} -1.27389 q^{96} -5.45222 q^{97} -6.92498 q^{98} +6.02830 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} - q^{7} + 3 q^{8} + q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 - 2 * q^6 - q^7 + 3 * q^8 + q^9	$ 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} - q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - 8 q^{11} - 2 q^{12} - q^{14} + 3 q^{15} + 3 q^{16} - 2 q^{17} + q^{18} + 3 q^{19} + 2 q^{20} - 8 q^{21} - 8 q^{22} + 2 q^{23} - 2 q^{24} - 5 q^{25} + q^{27} - q^{28} - 14 q^{29} + 3 q^{30} - 5 q^{31} + 3 q^{32} + q^{33} - 2 q^{34} - 5 q^{35} + q^{36} + 3 q^{38} + 2 q^{40} + 15 q^{41} - 8 q^{42} - 5 q^{43} - 8 q^{44} - 8 q^{45} + 2 q^{46} - 11 q^{47} - 2 q^{48} - 12 q^{49} - 5 q^{50} - 16 q^{51} + 5 q^{53} + q^{54} - 14 q^{55} - q^{56} - 2 q^{57} - 14 q^{58} - 4 q^{59} + 3 q^{60} + 2 q^{61} - 5 q^{62} + 4 q^{63} + 3 q^{64} + q^{66} + 6 q^{67} - 2 q^{68} + 16 q^{69} - 5 q^{70} - 9 q^{71} + q^{72} + 15 q^{73} - q^{75} + 3 q^{76} + 7 q^{77} - 2 q^{79} + 2 q^{80} - 21 q^{81} + 15 q^{82} + 5 q^{83} - 8 q^{84} - 10 q^{85} - 5 q^{86} + 5 q^{87} - 8 q^{88} - 27 q^{89} - 8 q^{90} + 2 q^{92} + 25 q^{93} - 11 q^{94} + 2 q^{95} - 2 q^{96} - 20 q^{97} - 12 q^{98} + 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 - 2 * q^6 - q^7 + 3 * q^8 + q^9 + 2 * q^10 - 8 * q^11 - 2 * q^12 - q^14 + 3 * q^15 + 3 * q^16 - 2 * q^17 + q^18 + 3 * q^19 + 2 * q^20 - 8 * q^21 - 8 * q^22 + 2 * q^23 - 2 * q^24 - 5 * q^25 + q^27 - q^28 - 14 * q^29 + 3 * q^30 - 5 * q^31 + 3 * q^32 + q^33 - 2 * q^34 - 5 * q^35 + q^36 + 3 * q^38 + 2 * q^40 + 15 * q^41 - 8 * q^42 - 5 * q^43 - 8 * q^44 - 8 * q^45 + 2 * q^46 - 11 * q^47 - 2 * q^48 - 12 * q^49 - 5 * q^50 - 16 * q^51 + 5 * q^53 + q^54 - 14 * q^55 - q^56 - 2 * q^57 - 14 * q^58 - 4 * q^59 + 3 * q^60 + 2 * q^61 - 5 * q^62 + 4 * q^63 + 3 * q^64 + q^66 + 6 * q^67 - 2 * q^68 + 16 * q^69 - 5 * q^70 - 9 * q^71 + q^72 + 15 * q^73 - q^75 + 3 * q^76 + 7 * q^77 - 2 * q^79 + 2 * q^80 - 21 * q^81 + 15 * q^82 + 5 * q^83 - 8 * q^84 - 10 * q^85 - 5 * q^86 + 5 * q^87 - 8 * q^88 - 27 * q^89 - 8 * q^90 + 2 * q^92 + 25 * q^93 - 11 * q^94 + 2 * q^95 - 2 * q^96 - 20 * q^97 - 12 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.27389	−0.735481	−0.367741	−	0.929928i	$-0.619869\pi$
$3$	−1.27389	−0.735481	−0.367741	+	0.929928i	$0.619869\pi$
$4$	1.00000	0.500000
$5$	2.37720	1.06312	0.531559	−	0.847021i	$-0.321606\pi$
$5$	2.37720	1.06312	0.531559	+	0.847021i	$0.321606\pi$
$6$	−1.27389	−0.520064
$7$	0.273891	0.103521	0.0517604	−	0.998660i	$-0.483517\pi$
$7$	0.273891	0.103521	0.0517604	+	0.998660i	$0.483517\pi$
$8$	1.00000	0.353553
$9$	−1.37720	−0.459068
$10$	2.37720	0.751738
$11$	−4.37720	−1.31978	−0.659888	−	0.751364i	$-0.729396\pi$
$11$	−4.37720	−1.31978	−0.659888	+	0.751364i	$0.729396\pi$
$12$	−1.27389	−0.367741
$13$	0	0
$13$	0	0
$14$	0.273891	0.0732003
$15$	−3.02830	−0.781903
$16$	1.00000	0.250000
$17$	0.547781	0.132856	0.0664282	−	0.997791i	$-0.478840\pi$
$17$	0.547781	0.132856	0.0664282	+	0.997791i	$0.478840\pi$
$18$	−1.37720	−0.324610
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.37720	0.531559
$21$	−0.348907	−0.0761377
$22$	−4.37720	−0.933223
$23$	−4.57608	−0.954178	−0.477089	−	0.878855i	$-0.658308\pi$
$23$	−4.57608	−0.954178	−0.477089	+	0.878855i	$0.658308\pi$
$24$	−1.27389	−0.260032
$25$	0.651093	0.130219
$26$	0	0
$27$	5.57608	1.07312
$28$	0.273891	0.0517604
$29$	1.67939	0.311855	0.155927	−	0.987769i	$-0.450163\pi$
$29$	1.67939	0.311855	0.155927	+	0.987769i	$0.450163\pi$
$30$	−3.02830	−0.552889
$31$	2.85772	0.513261	0.256631	−	0.966510i	$-0.417388\pi$
$31$	2.85772	0.513261	0.256631	+	0.966510i	$0.417388\pi$
$32$	1.00000	0.176777
$33$	5.57608	0.970670
$34$	0.547781	0.0939437
$35$	0.651093	0.110055
$36$	−1.37720	−0.229534
$37$	−8.05659	−1.32450	−0.662248	−	0.749285i	$-0.730397\pi$
$37$	−8.05659	−1.32450	−0.662248	+	0.749285i	$0.730397\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	2.37720	0.375869
$41$	9.02830	1.40998	0.704991	−	0.709216i	$-0.250951\pi$
$41$	9.02830	1.40998	0.704991	+	0.709216i	$0.250951\pi$
$42$	−0.348907	−0.0538375
$43$	−0.452219	−0.0689627	−0.0344814	−	0.999405i	$-0.510978\pi$
$43$	−0.452219	−0.0689627	−0.0344814	+	0.999405i	$0.510978\pi$
$44$	−4.37720	−0.659888
$45$	−3.27389	−0.488043
$46$	−4.57608	−0.674706
$47$	0.472765	0.0689599	0.0344799	−	0.999405i	$-0.489023\pi$
$47$	0.472765	0.0689599	0.0344799	+	0.999405i	$0.489023\pi$
$48$	−1.27389	−0.183870
$49$	−6.92498	−0.989283
$50$	0.651093	0.0920785
$51$	−0.697813	−0.0977134
$52$	0	0
$53$	−2.85772	−0.392538	−0.196269	−	0.980550i	$-0.562883\pi$
$53$	−2.85772	−0.392538	−0.196269	+	0.980550i	$0.562883\pi$
$54$	5.57608	0.758808
$55$	−10.4055	−1.40308
$56$	0.273891	0.0366002
$57$	−1.27389	−0.168731
$58$	1.67939	0.220515
$59$	−13.5294	−1.76137	−0.880686	−	0.473700i	$-0.842918\pi$
$59$	−13.5294	−1.76137	−0.880686	+	0.473700i	$0.842918\pi$
$60$	−3.02830	−0.390951
$61$	4.19887	0.537611	0.268805	−	0.963195i	$-0.413371\pi$
$61$	4.19887	0.537611	0.268805	+	0.963195i	$0.413371\pi$
$62$	2.85772	0.362931
$63$	−0.377203	−0.0475231
$64$	1.00000	0.125000
$65$	0	0
$66$	5.57608	0.686368
$67$	10.0566	1.22861	0.614304	−	0.789069i	$-0.289437\pi$
$67$	10.0566	1.22861	0.614304	+	0.789069i	$0.289437\pi$
$68$	0.547781	0.0664282
$69$	5.82942	0.701780
$70$	0.651093	0.0778205
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	−1.37720	−0.162305
$73$	−11.1132	−1.30070	−0.650350	−	0.759635i	$-0.725378\pi$
$73$	−11.1132	−1.30070	−0.650350	+	0.759635i	$0.725378\pi$
$74$	−8.05659	−0.936560
$75$	−0.829422	−0.0957734
$76$	1.00000	0.114708
$77$	−1.19887	−0.136624
$78$	0	0
$79$	2.36945	0.266584	0.133292	−	0.991077i	$-0.457445\pi$
$79$	2.36945	0.266584	0.133292	+	0.991077i	$0.457445\pi$
$80$	2.37720	0.265779
$81$	−2.97170	−0.330189
$82$	9.02830	0.997009
$83$	−16.7643	−1.84012	−0.920059	−	0.391779i	$-0.871860\pi$
$83$	−16.7643	−1.84012	−0.920059	+	0.391779i	$0.871860\pi$
$84$	−0.348907	−0.0380688
$85$	1.30219	0.141242
$86$	−0.452219	−0.0487640
$87$	−2.13936	−0.229363
$88$	−4.37720	−0.466611
$89$	−2.04672	−0.216952	−0.108476	−	0.994099i	$-0.534597\pi$
$89$	−2.04672	−0.216952	−0.108476	+	0.994099i	$0.534597\pi$
$90$	−3.27389	−0.345098
$91$	0	0
$92$	−4.57608	−0.477089
$93$	−3.64042	−0.377494
$94$	0.472765	0.0487620
$95$	2.37720	0.243896
$96$	−1.27389	−0.130016
$97$	−5.45222	−0.553589	−0.276794	−	0.960929i	$-0.589272\pi$
$97$	−5.45222	−0.553589	−0.276794	+	0.960929i	$0.589272\pi$
$98$	−6.92498	−0.699529
$99$	6.02830	0.605867
$100$	0.651093	0.0651093
$101$	8.15215	0.811170	0.405585	−	0.914057i	$-0.367068\pi$
$101$	8.15215	0.811170	0.405585	+	0.914057i	$0.367068\pi$
$102$	−0.697813	−0.0690938
$103$	−16.3510	−1.61111	−0.805557	−	0.592518i	$-0.798134\pi$
$103$	−16.3510	−1.61111	−0.805557	+	0.592518i	$0.798134\pi$
$104$	0	0
$105$	−0.829422	−0.0809433
$106$	−2.85772	−0.277566
$107$	−7.02830	−0.679451	−0.339726	−	0.940525i	$-0.610334\pi$
$107$	−7.02830	−0.679451	−0.339726	+	0.940525i	$0.610334\pi$
$108$	5.57608	0.536558
$109$	−4.64334	−0.444752	−0.222376	−	0.974961i	$-0.571381\pi$
$109$	−4.64334	−0.444752	−0.222376	+	0.974961i	$0.571381\pi$
$110$	−10.4055	−0.992125
$111$	10.2632	0.974141
$112$	0.273891	0.0258802
$113$	10.6588	1.00270	0.501350	−	0.865245i	$-0.332837\pi$
$113$	10.6588	1.00270	0.501350	+	0.865245i	$0.332837\pi$
$114$	−1.27389	−0.119311
$115$	−10.8783	−1.01440
$116$	1.67939	0.155927
$117$	0	0
$118$	−13.5294	−1.24548
$119$	0.150032	0.0137534
$120$	−3.02830	−0.276444
$121$	8.15990	0.741810
$122$	4.19887	0.380148
$123$	−11.5011	−1.03702
$124$	2.85772	0.256631
$125$	−10.3382	−0.924680
$126$	−0.377203	−0.0336039
$127$	−16.1599	−1.43396	−0.716980	−	0.697094i	$-0.754476\pi$
$127$	−16.1599	−1.43396	−0.716980	+	0.697094i	$0.754476\pi$
$128$	1.00000	0.0883883
$129$	0.576077	0.0507208
$130$	0	0
$131$	−15.9250	−1.39137	−0.695686	−	0.718346i	$-0.744900\pi$
$131$	−15.9250	−1.39137	−0.695686	+	0.718346i	$0.744900\pi$
$132$	5.57608	0.485335
$133$	0.273891	0.0237493
$134$	10.0566	0.868757
$135$	13.2555	1.14085
$136$	0.547781	0.0469718
$137$	−13.1805	−1.12608	−0.563041	−	0.826429i	$-0.690369\pi$
$137$	−13.1805	−1.12608	−0.563041	+	0.826429i	$0.690369\pi$
$138$	5.82942	0.496233
$139$	2.50106	0.212137	0.106069	−	0.994359i	$-0.466174\pi$
$139$	2.50106	0.212137	0.106069	+	0.994359i	$0.466174\pi$
$140$	0.651093	0.0550274
$141$	−0.602251	−0.0507187
$142$	−3.00000	−0.251754
$143$	0	0
$144$	−1.37720	−0.114767
$145$	3.99225	0.331538
$146$	−11.1132	−0.919734
$147$	8.82167	0.727599
$148$	−8.05659	−0.662248
$149$	−9.42392	−0.772038	−0.386019	−	0.922491i	$-0.626150\pi$
$149$	−9.42392	−0.772038	−0.386019	+	0.922491i	$0.626150\pi$
$150$	−0.829422	−0.0677220
$151$	−11.8217	−0.962034	−0.481017	−	0.876711i	$-0.659732\pi$
$151$	−11.8217	−0.962034	−0.481017	+	0.876711i	$0.659732\pi$
$152$	1.00000	0.0811107
$153$	−0.754406	−0.0609901
$154$	−1.19887	−0.0966081
$155$	6.79338	0.545657
$156$	0	0
$157$	1.13161	0.0903122	0.0451561	−	0.998980i	$-0.485621\pi$
$157$	1.13161	0.0903122	0.0451561	+	0.998980i	$0.485621\pi$
$158$	2.36945	0.188504
$159$	3.64042	0.288704
$160$	2.37720	0.187934
$161$	−1.25334	−0.0987774
$162$	−2.97170	−0.233479
$163$	1.37720	0.107871	0.0539354	−	0.998544i	$-0.482823\pi$
$163$	1.37720	0.107871	0.0539354	+	0.998544i	$0.482823\pi$
$164$	9.02830	0.704991
$165$	13.2555	1.03194
$166$	−16.7643	−1.30116
$167$	−0.416173	−0.0322044	−0.0161022	−	0.999870i	$-0.505126\pi$
$167$	−0.416173	−0.0322044	−0.0161022	+	0.999870i	$0.505126\pi$
$168$	−0.348907	−0.0269187
$169$	0	0
$170$	1.30219	0.0998732
$171$	−1.37720	−0.105317
$172$	−0.452219	−0.0344814
$173$	−4.21942	−0.320797	−0.160398	−	0.987052i	$-0.551278\pi$
$173$	−4.21942	−0.320797	−0.160398	+	0.987052i	$0.551278\pi$
$174$	−2.13936	−0.162184
$175$	0.178328	0.0134804
$176$	−4.37720	−0.329944
$177$	17.2349	1.29546
$178$	−2.04672	−0.153408
$179$	20.9143	1.56321	0.781604	−	0.623774i	$-0.214402\pi$
$179$	20.9143	1.56321	0.781604	+	0.623774i	$0.214402\pi$
$180$	−3.27389	−0.244021
$181$	6.87051	0.510681	0.255341	−	0.966851i	$-0.417812\pi$
$181$	6.87051	0.510681	0.255341	+	0.966851i	$0.417812\pi$
$182$	0	0
$183$	−5.34891	−0.395403
$184$	−4.57608	−0.337353
$185$	−19.1522	−1.40809
$186$	−3.64042	−0.266929
$187$	−2.39775	−0.175341
$188$	0.472765	0.0344799
$189$	1.52723	0.111090
$190$	2.37720	0.172460
$191$	12.3305	0.892202	0.446101	−	0.894983i	$-0.352812\pi$
$191$	12.3305	0.892202	0.446101	+	0.894983i	$0.352812\pi$
$192$	−1.27389	−0.0919351
$193$	20.3121	1.46209	0.731047	−	0.682327i	$-0.239032\pi$
$193$	20.3121	1.46209	0.731047	+	0.682327i	$0.239032\pi$
$194$	−5.45222	−0.391447
$195$	0	0
$196$	−6.92498	−0.494642
$197$	25.0849	1.78722	0.893612	−	0.448840i	$-0.148163\pi$
$197$	25.0849	1.78722	0.893612	+	0.448840i	$0.148163\pi$
$198$	6.02830	0.428412
$199$	−11.3404	−0.803897	−0.401948	−	0.915662i	$-0.631667\pi$
$199$	−11.3404	−0.803897	−0.401948	+	0.915662i	$0.631667\pi$
$200$	0.651093	0.0460393
$201$	−12.8110	−0.903618
$202$	8.15215	0.573584
$203$	0.459969	0.0322835
$204$	−0.697813	−0.0488567
$205$	21.4621	1.49898
$206$	−16.3510	−1.13923
$207$	6.30219	0.438032
$208$	0	0
$209$	−4.37720	−0.302777
$210$	−0.829422	−0.0572355
$211$	−7.51173	−0.517129	−0.258565	−	0.965994i	$-0.583249\pi$
$211$	−7.51173	−0.517129	−0.258565	+	0.965994i	$0.583249\pi$
$212$	−2.85772	−0.196269
$213$	3.82167	0.261857
$214$	−7.02830	−0.480444
$215$	−1.07502	−0.0733155
$216$	5.57608	0.379404
$217$	0.782702	0.0531333
$218$	−4.64334	−0.314487
$219$	14.1570	0.956640
$220$	−10.4055	−0.701539
$221$	0	0
$222$	10.2632	0.688822
$223$	−1.15003	−0.0770118	−0.0385059	−	0.999258i	$-0.512260\pi$
$223$	−1.15003	−0.0770118	−0.0385059	+	0.999258i	$0.512260\pi$
$224$	0.273891	0.0183001
$225$	−0.896688	−0.0597792
$226$	10.6588	0.709016
$227$	−17.9816	−1.19348	−0.596740	−	0.802435i	$-0.703537\pi$
$227$	−17.9816	−1.19348	−0.596740	+	0.802435i	$0.703537\pi$
$228$	−1.27389	−0.0843655
$229$	16.7437	1.10646	0.553228	−	0.833030i	$-0.313396\pi$
$229$	16.7437	1.10646	0.553228	+	0.833030i	$0.313396\pi$
$230$	−10.8783	−0.717291
$231$	1.52723	0.100485
$232$	1.67939	0.110257
$233$	−0.253344	−0.0165971	−0.00829857	−	0.999966i	$-0.502642\pi$
$233$	−0.253344	−0.0165971	−0.00829857	+	0.999966i	$0.502642\pi$
$234$	0	0
$235$	1.12386	0.0733124
$236$	−13.5294	−0.880686
$237$	−3.01842	−0.196068
$238$	0.150032	0.00972513
$239$	24.1025	1.55906	0.779531	−	0.626364i	$-0.215458\pi$
$239$	24.1025	1.55906	0.779531	+	0.626364i	$0.215458\pi$
$240$	−3.02830	−0.195476
$241$	2.07502	0.133664	0.0668318	−	0.997764i	$-0.478711\pi$
$241$	2.07502	0.133664	0.0668318	+	0.997764i	$0.478711\pi$
$242$	8.15990	0.524539
$243$	−12.9426	−0.830269
$244$	4.19887	0.268805
$245$	−16.4621	−1.05172
$246$	−11.5011	−0.733281
$247$	0	0
$248$	2.85772	0.181465
$249$	21.3559	1.35337
$250$	−10.3382	−0.653847
$251$	−7.57608	−0.478198	−0.239099	−	0.970995i	$-0.576852\pi$
$251$	−7.57608	−0.478198	−0.239099	+	0.970995i	$0.576852\pi$
$252$	−0.377203	−0.0237615
$253$	20.0304	1.25930
$254$	−16.1599	−1.01396
$255$	−1.65884	−0.103881
$256$	1.00000	0.0625000
$257$	8.06434	0.503040	0.251520	−	0.967852i	$-0.419070\pi$
$257$	8.06434	0.503040	0.251520	+	0.967852i	$0.419070\pi$
$258$	0.576077	0.0358650
$259$	−2.20662	−0.137113
$260$	0	0
$261$	−2.31286	−0.143162
$262$	−15.9250	−0.983849
$263$	−16.0078	−0.987080	−0.493540	−	0.869723i	$-0.664297\pi$
$263$	−16.0078	−0.987080	−0.493540	+	0.869723i	$0.664297\pi$
$264$	5.57608	0.343184
$265$	−6.79338	−0.417314
$266$	0.273891	0.0167933
$267$	2.60730	0.159564
$268$	10.0566	0.614304
$269$	−15.6893	−0.956591	−0.478296	−	0.878199i	$-0.658745\pi$
$269$	−15.6893	−0.956591	−0.478296	+	0.878199i	$0.658745\pi$
$270$	13.2555	0.806702
$271$	−21.3793	−1.29870	−0.649351	−	0.760489i	$-0.724959\pi$
$271$	−21.3793	−1.29870	−0.649351	+	0.760489i	$0.724959\pi$
$272$	0.547781	0.0332141
$273$	0	0
$274$	−13.1805	−0.796260
$275$	−2.84997	−0.171860
$276$	5.82942	0.350890
$277$	−6.76508	−0.406474	−0.203237	−	0.979130i	$-0.565146\pi$
$277$	−6.76508	−0.406474	−0.203237	+	0.979130i	$0.565146\pi$
$278$	2.50106	0.150004
$279$	−3.93566	−0.235622
$280$	0.651093	0.0389103
$281$	25.3227	1.51063	0.755314	−	0.655363i	$-0.227484\pi$
$281$	25.3227	1.51063	0.755314	+	0.655363i	$0.227484\pi$
$282$	−0.602251	−0.0358635
$283$	−15.8938	−0.944786	−0.472393	−	0.881388i	$-0.656610\pi$
$283$	−15.8938	−0.944786	−0.472393	+	0.881388i	$0.656610\pi$
$284$	−3.00000	−0.178017
$285$	−3.02830	−0.179381
$286$	0	0
$287$	2.47277	0.145963
$288$	−1.37720	−0.0811525
$289$	−16.6999	−0.982349
$290$	3.99225	0.234433
$291$	6.94553	0.407154
$292$	−11.1132	−0.650350
$293$	33.3014	1.94549	0.972744	−	0.231882i	$-0.0744884\pi$
$293$	33.3014	1.94549	0.972744	+	0.231882i	$0.0744884\pi$
$294$	8.82167	0.514490
$295$	−32.1620	−1.87255
$296$	−8.05659	−0.468280
$297$	−24.4076	−1.41627
$298$	−9.42392	−0.545913
$299$	0	0
$300$	−0.829422	−0.0478867
$301$	−0.123858	−0.00713908
$302$	−11.8217	−0.680261
$303$	−10.3850	−0.596600
$304$	1.00000	0.0573539
$305$	9.98158	0.571543
$306$	−0.754406	−0.0431265
$307$	8.65884	0.494186	0.247093	−	0.968992i	$-0.420525\pi$
$307$	8.65884	0.494186	0.247093	+	0.968992i	$0.420525\pi$
$308$	−1.19887	−0.0683122
$309$	20.8294	1.18494
$310$	6.79338	0.385838
$311$	8.93273	0.506529	0.253264	−	0.967397i	$-0.418496\pi$
$311$	8.93273	0.506529	0.253264	+	0.967397i	$0.418496\pi$
$312$	0	0
$313$	−11.6150	−0.656521	−0.328261	−	0.944587i	$-0.606462\pi$
$313$	−11.6150	−0.656521	−0.328261	+	0.944587i	$0.606462\pi$
$314$	1.13161	0.0638604
$315$	−0.896688	−0.0505226
$316$	2.36945	0.133292
$317$	−32.5958	−1.83076	−0.915382	−	0.402587i	$-0.868111\pi$
$317$	−32.5958	−1.83076	−0.915382	+	0.402587i	$0.868111\pi$
$318$	3.64042	0.204145
$319$	−7.35103	−0.411579
$320$	2.37720	0.132890
$321$	8.95328	0.499723
$322$	−1.25334	−0.0698462
$323$	0.547781	0.0304794
$324$	−2.97170	−0.165095
$325$	0	0
$326$	1.37720	0.0762762
$327$	5.91511	0.327106
$328$	9.02830	0.498504
$329$	0.129486	0.00713879
$330$	13.2555	0.729689
$331$	7.33823	0.403346	0.201673	−	0.979453i	$-0.435362\pi$
$331$	7.33823	0.403346	0.201673	+	0.979453i	$0.435362\pi$
$332$	−16.7643	−0.920059
$333$	11.0956	0.608033
$334$	−0.416173	−0.0227719
$335$	23.9066	1.30615
$336$	−0.348907	−0.0190344
$337$	3.48052	0.189596	0.0947979	−	0.995497i	$-0.469780\pi$
$337$	3.48052	0.189596	0.0947979	+	0.995497i	$0.469780\pi$
$338$	0	0
$339$	−13.5782	−0.737467
$340$	1.30219	0.0706210
$341$	−12.5088	−0.677390
$342$	−1.37720	−0.0744706
$343$	−3.81392	−0.205932
$344$	−0.452219	−0.0243820
$345$	13.8577	0.746074
$346$	−4.21942	−0.226837
$347$	12.9709	0.696315	0.348157	−	0.937436i	$-0.386808\pi$
$347$	12.9709	0.696315	0.348157	+	0.937436i	$0.386808\pi$
$348$	−2.13936	−0.114682
$349$	−16.5059	−0.883540	−0.441770	−	0.897128i	$-0.645649\pi$
$349$	−16.5059	−0.883540	−0.441770	+	0.897128i	$0.645649\pi$
$350$	0.178328	0.00953205
$351$	0	0
$352$	−4.37720	−0.233306
$353$	23.4415	1.24767	0.623834	−	0.781557i	$-0.285574\pi$
$353$	23.4415	1.24767	0.623834	+	0.781557i	$0.285574\pi$
$354$	17.2349	0.916026
$355$	−7.13161	−0.378506
$356$	−2.04672	−0.108476
$357$	−0.191124	−0.0101154
$358$	20.9143	1.10536
$359$	−18.6794	−0.985860	−0.492930	−	0.870069i	$-0.664074\pi$
$359$	−18.6794	−0.985860	−0.492930	+	0.870069i	$0.664074\pi$
$360$	−3.27389	−0.172549
$361$	1.00000	0.0526316
$362$	6.87051	0.361106
$363$	−10.3948	−0.545587
$364$	0	0
$365$	−26.4183	−1.38280
$366$	−5.34891	−0.279592
$367$	−22.3977	−1.16915	−0.584576	−	0.811339i	$-0.698739\pi$
$367$	−22.3977	−1.16915	−0.584576	+	0.811339i	$0.698739\pi$
$368$	−4.57608	−0.238545
$369$	−12.4338	−0.647278
$370$	−19.1522	−0.995673
$371$	−0.782702	−0.0406359
$372$	−3.64042	−0.188747
$373$	33.3559	1.72710	0.863550	−	0.504263i	$-0.168236\pi$
$373$	33.3559	1.72710	0.863550	+	0.504263i	$0.168236\pi$
$374$	−2.39775	−0.123985
$375$	13.1698	0.680084
$376$	0.472765	0.0243810
$377$	0	0
$378$	1.52723	0.0785525
$379$	−34.8337	−1.78929	−0.894643	−	0.446782i	$-0.852570\pi$
$379$	−34.8337	−1.78929	−0.894643	+	0.446782i	$0.852570\pi$
$380$	2.37720	0.121948
$381$	20.5860	1.05465
$382$	12.3305	0.630882
$383$	18.1834	0.929127	0.464564	−	0.885540i	$-0.346211\pi$
$383$	18.1834	0.929127	0.464564	+	0.885540i	$0.346211\pi$
$384$	−1.27389	−0.0650080
$385$	−2.84997	−0.145248
$386$	20.3121	1.03386
$387$	0.622797	0.0316586
$388$	−5.45222	−0.276794
$389$	0.894565	0.0453562	0.0226781	−	0.999743i	$-0.492781\pi$
$389$	0.894565	0.0453562	0.0226781	+	0.999743i	$0.492781\pi$
$390$	0	0
$391$	−2.50669	−0.126769
$392$	−6.92498	−0.349765
$393$	20.2867	1.02333
$394$	25.0849	1.26376
$395$	5.63267	0.283410
$396$	6.02830	0.302933
$397$	37.3249	1.87328	0.936640	−	0.350292i	$-0.113918\pi$
$397$	37.3249	1.87328	0.936640	+	0.350292i	$0.113918\pi$
$398$	−11.3404	−0.568441
$399$	−0.348907	−0.0174672
$400$	0.651093	0.0325547
$401$	14.8500	0.741572	0.370786	−	0.928718i	$-0.379088\pi$
$401$	14.8500	0.741572	0.370786	+	0.928718i	$0.379088\pi$
$402$	−12.8110	−0.638955
$403$	0	0
$404$	8.15215	0.405585
$405$	−7.06434	−0.351030
$406$	0.459969	0.0228279
$407$	35.2653	1.74804
$408$	−0.697813	−0.0345469
$409$	−29.4047	−1.45397	−0.726984	−	0.686654i	$-0.759079\pi$
$409$	−29.4047	−1.45397	−0.726984	+	0.686654i	$0.759079\pi$
$410$	21.4621	1.05994
$411$	16.7905	0.828212
$412$	−16.3510	−0.805557
$413$	−3.70556	−0.182339
$414$	6.30219	0.309736
$415$	−39.8521	−1.95626
$416$	0	0
$417$	−3.18608	−0.156023
$418$	−4.37720	−0.214096
$419$	−24.9221	−1.21752	−0.608761	−	0.793354i	$-0.708333\pi$
$419$	−24.9221	−1.21752	−0.608761	+	0.793354i	$0.708333\pi$
$420$	−0.829422	−0.0404716
$421$	5.12174	0.249618	0.124809	−	0.992181i	$-0.460168\pi$
$421$	5.12174	0.249618	0.124809	+	0.992181i	$0.460168\pi$
$422$	−7.51173	−0.365666
$423$	−0.651093	−0.0316572
$424$	−2.85772	−0.138783
$425$	0.356657	0.0173004
$426$	3.82167	0.185161
$427$	1.15003	0.0556540
$428$	−7.02830	−0.339726
$429$	0	0
$430$	−1.07502	−0.0518419
$431$	20.3121	0.978397	0.489199	−	0.872172i	$-0.337289\pi$
$431$	20.3121	0.978397	0.489199	+	0.872172i	$0.337289\pi$
$432$	5.57608	0.268279
$433$	−27.8705	−1.33937	−0.669686	−	0.742645i	$-0.733571\pi$
$433$	−27.8705	−1.33937	−0.669686	+	0.742645i	$0.733571\pi$
$434$	0.782702	0.0375709
$435$	−5.08569	−0.243840
$436$	−4.64334	−0.222376
$437$	−4.57608	−0.218903
$438$	14.1570	0.676447
$439$	−5.88389	−0.280823	−0.140411	−	0.990093i	$-0.544843\pi$
$439$	−5.88389	−0.280823	−0.140411	+	0.990093i	$0.544843\pi$
$440$	−10.4055	−0.496063
$441$	9.53711	0.454148
$442$	0	0
$443$	−24.0878	−1.14445	−0.572223	−	0.820098i	$-0.693919\pi$
$443$	−24.0878	−1.14445	−0.572223	+	0.820098i	$0.693919\pi$
$444$	10.2632	0.487071
$445$	−4.86547	−0.230645
$446$	−1.15003	−0.0544556
$447$	12.0050	0.567819
$448$	0.273891	0.0129401
$449$	−18.4026	−0.868471	−0.434236	−	0.900799i	$-0.642981\pi$
$449$	−18.4026	−0.868471	−0.434236	+	0.900799i	$0.642981\pi$
$450$	−0.896688	−0.0422703
$451$	−39.5187	−1.86086
$452$	10.6588	0.501350
$453$	15.0595	0.707558
$454$	−17.9816	−0.843917
$455$	0	0
$456$	−1.27389	−0.0596554
$457$	−1.82460	−0.0853510	−0.0426755	−	0.999089i	$-0.513588\pi$
$457$	−1.82460	−0.0853510	−0.0426755	+	0.999089i	$0.513588\pi$
$458$	16.7437	0.782383
$459$	3.05447	0.142570
$460$	−10.8783	−0.507202
$461$	−21.9730	−1.02339	−0.511693	−	0.859168i	$-0.670981\pi$
$461$	−21.9730	−1.02339	−0.511693	+	0.859168i	$0.670981\pi$
$462$	1.52723	0.0710534
$463$	6.74373	0.313408	0.156704	−	0.987646i	$-0.449913\pi$
$463$	6.74373	0.313408	0.156704	+	0.987646i	$0.449913\pi$
$464$	1.67939	0.0779637
$465$	−8.65402	−0.401320
$466$	−0.253344	−0.0117360
$467$	24.0227	1.11164	0.555818	−	0.831304i	$-0.312405\pi$
$467$	24.0227	1.11164	0.555818	+	0.831304i	$0.312405\pi$
$468$	0	0
$469$	2.75441	0.127187
$470$	1.12386	0.0518397
$471$	−1.44155	−0.0664229
$472$	−13.5294	−0.622739
$473$	1.97945	0.0910154
$474$	−3.01842	−0.138641
$475$	0.651093	0.0298742
$476$	0.150032	0.00687671
$477$	3.93566	0.180201
$478$	24.1025	1.10242
$479$	−14.2066	−0.649117	−0.324559	−	0.945866i	$-0.605216\pi$
$479$	−14.2066	−0.649117	−0.324559	+	0.945866i	$0.605216\pi$
$480$	−3.02830	−0.138222
$481$	0	0
$482$	2.07502	0.0945144
$483$	1.59662	0.0726489
$484$	8.15990	0.370905
$485$	−12.9610	−0.588530
$486$	−12.9426	−0.587089
$487$	12.0021	0.543868	0.271934	−	0.962316i	$-0.412337\pi$
$487$	12.0021	0.543868	0.271934	+	0.962316i	$0.412337\pi$
$488$	4.19887	0.190074
$489$	−1.75441	−0.0793370
$490$	−16.4621	−0.743681
$491$	−28.4047	−1.28189	−0.640943	−	0.767588i	$-0.721457\pi$
$491$	−28.4047	−1.28189	−0.640943	+	0.767588i	$0.721457\pi$
$492$	−11.5011	−0.518508
$493$	0.919938	0.0414319
$494$	0	0
$495$	14.3305	0.644107
$496$	2.85772	0.128315
$497$	−0.821672	−0.0368570
$498$	21.3559	0.956979
$499$	0.452219	0.0202441	0.0101220	−	0.999949i	$-0.496778\pi$
$499$	0.452219	0.0202441	0.0101220	+	0.999949i	$0.496778\pi$
$500$	−10.3382	−0.462340
$501$	0.530158	0.0236857
$502$	−7.57608	−0.338137
$503$	−1.57125	−0.0700586	−0.0350293	−	0.999386i	$-0.511152\pi$
$503$	−1.57125	−0.0700586	−0.0350293	+	0.999386i	$0.511152\pi$
$504$	−0.377203	−0.0168020
$505$	19.3793	0.862369
$506$	20.0304	0.890461
$507$	0	0
$508$	−16.1599	−0.716980
$509$	−30.5422	−1.35376	−0.676879	−	0.736095i	$-0.736668\pi$
$509$	−30.5422	−1.35376	−0.676879	+	0.736095i	$0.736668\pi$
$510$	−1.65884	−0.0734548
$511$	−3.04380	−0.134650
$512$	1.00000	0.0441942
$513$	5.57608	0.246190
$514$	8.06434	0.355703
$515$	−38.8697	−1.71280
$516$	0.576077	0.0253604
$517$	−2.06939	−0.0910116
$518$	−2.20662	−0.0969535
$519$	5.37508	0.235940
$520$	0	0
$521$	0.405499	0.0177652	0.00888262	−	0.999961i	$-0.497173\pi$
$521$	0.405499	0.0177652	0.00888262	+	0.999961i	$0.497173\pi$
$522$	−2.31286	−0.101231
$523$	−2.74958	−0.120231	−0.0601153	−	0.998191i	$-0.519147\pi$
$523$	−2.74958	−0.120231	−0.0601153	+	0.998191i	$0.519147\pi$
$524$	−15.9250	−0.695686
$525$	−0.227171	−0.00991455
$526$	−16.0078	−0.697971
$527$	1.56540	0.0681901
$528$	5.57608	0.242668
$529$	−2.05952	−0.0895442
$530$	−6.79338	−0.295085
$531$	18.6327	0.808589
$532$	0.273891	0.0118747
$533$	0	0
$534$	2.60730	0.112829
$535$	−16.7077	−0.722336
$536$	10.0566	0.434379
$537$	−26.6425	−1.14971
$538$	−15.6893	−0.676412
$539$	30.3121	1.30563
$540$	13.2555	0.570424
$541$	24.0128	1.03239	0.516195	−	0.856471i	$-0.327348\pi$
$541$	24.0128	1.03239	0.516195	+	0.856471i	$0.327348\pi$
$542$	−21.3793	−0.918321
$543$	−8.75228	−0.375596
$544$	0.547781	0.0234859
$545$	−11.0382	−0.472823
$546$	0	0
$547$	−43.2525	−1.84935	−0.924673	−	0.380763i	$-0.875661\pi$
$547$	−43.2525	−1.84935	−0.924673	+	0.380763i	$0.875661\pi$
$548$	−13.1805	−0.563041
$549$	−5.78270	−0.246800
$550$	−2.84997	−0.121523
$551$	1.67939	0.0715444
$552$	5.82942	0.248117
$553$	0.648971	0.0275970
$554$	−6.76508	−0.287421
$555$	24.3977	1.03563
$556$	2.50106	0.106069
$557$	28.0849	1.18999	0.594997	−	0.803728i	$-0.297153\pi$
$557$	28.0849	1.18999	0.594997	+	0.803728i	$0.297153\pi$
$558$	−3.93566	−0.166610
$559$	0	0
$560$	0.651093	0.0275137
$561$	3.05447	0.128960
$562$	25.3227	1.06818
$563$	35.1669	1.48211	0.741053	−	0.671446i	$-0.234327\pi$
$563$	35.1669	1.48211	0.741053	+	0.671446i	$0.234327\pi$
$564$	−0.602251	−0.0253593
$565$	25.3382	1.06599
$566$	−15.8938	−0.668065
$567$	−0.813922	−0.0341815
$568$	−3.00000	−0.125877
$569$	−26.6922	−1.11899	−0.559497	−	0.828832i	$-0.689006\pi$
$569$	−26.6922	−1.11899	−0.559497	+	0.828832i	$0.689006\pi$
$570$	−3.02830	−0.126841
$571$	18.1882	0.761153	0.380576	−	0.924750i	$-0.375726\pi$
$571$	18.1882	0.761153	0.380576	+	0.924750i	$0.375726\pi$
$572$	0	0
$573$	−15.7077	−0.656198
$574$	2.47277	0.103211
$575$	−2.97945	−0.124252
$576$	−1.37720	−0.0573835
$577$	27.1076	1.12850	0.564251	−	0.825603i	$-0.309165\pi$
$577$	27.1076	1.12850	0.564251	+	0.825603i	$0.309165\pi$
$578$	−16.6999	−0.694626
$579$	−25.8753	−1.07534
$580$	3.99225	0.165769
$581$	−4.59158	−0.190491
$582$	6.94553	0.287901
$583$	12.5088	0.518062
$584$	−11.1132	−0.459867
$585$	0	0
$586$	33.3014	1.37567
$587$	16.3540	0.675000	0.337500	−	0.941326i	$-0.390419\pi$
$587$	16.3540	0.675000	0.337500	+	0.941326i	$0.390419\pi$
$588$	8.82167	0.363800
$589$	2.85772	0.117750
$590$	−32.1620	−1.32409
$591$	−31.9554	−1.31447
$592$	−8.05659	−0.331124
$593$	37.0021	1.51950	0.759748	−	0.650218i	$-0.225323\pi$
$593$	37.0021	1.51950	0.759748	+	0.650218i	$0.225323\pi$
$594$	−24.4076	−1.00146
$595$	0.356657	0.0146215
$596$	−9.42392	−0.386019
$597$	14.4464	0.591251
$598$	0	0
$599$	17.5830	0.718423	0.359211	−	0.933256i	$-0.383046\pi$
$599$	17.5830	0.718423	0.359211	+	0.933256i	$0.383046\pi$
$600$	−0.829422	−0.0338610
$601$	36.8930	1.50490	0.752448	−	0.658652i	$-0.228873\pi$
$601$	36.8930	1.50490	0.752448	+	0.658652i	$0.228873\pi$
$602$	−0.123858	−0.00504809
$603$	−13.8500	−0.564014
$604$	−11.8217	−0.481017
$605$	19.3977	0.788631
$606$	−10.3850	−0.421860
$607$	−22.8967	−0.929348	−0.464674	−	0.885482i	$-0.653828\pi$
$607$	−22.8967	−0.929348	−0.464674	+	0.885482i	$0.653828\pi$
$608$	1.00000	0.0405554
$609$	−0.585950	−0.0237439
$610$	9.98158	0.404142
$611$	0	0
$612$	−0.754406	−0.0304950
$613$	−24.9114	−1.00616	−0.503081	−	0.864239i	$-0.667800\pi$
$613$	−24.9114	−1.00616	−0.503081	+	0.864239i	$0.667800\pi$
$614$	8.65884	0.349442
$615$	−27.3404	−1.10247
$616$	−1.19887	−0.0483040
$617$	−7.65402	−0.308139	−0.154070	−	0.988060i	$-0.549238\pi$
$617$	−7.65402	−0.308139	−0.154070	+	0.988060i	$0.549238\pi$
$618$	20.8294	0.837882
$619$	−10.4140	−0.418576	−0.209288	−	0.977854i	$-0.567115\pi$
$619$	−10.4140	−0.418576	−0.209288	+	0.977854i	$0.567115\pi$
$620$	6.79338	0.272829
$621$	−25.5166	−1.02394
$622$	8.93273	0.358170
$623$	−0.560577	−0.0224591
$624$	0	0
$625$	−27.8315	−1.11326
$626$	−11.6150	−0.464231
$627$	5.57608	0.222687
$628$	1.13161	0.0451561
$629$	−4.41325	−0.175968
$630$	−0.896688	−0.0357249
$631$	28.8500	1.14850	0.574250	−	0.818680i	$-0.305294\pi$
$631$	28.8500	1.14850	0.574250	+	0.818680i	$0.305294\pi$
$632$	2.36945	0.0942518
$633$	9.56913	0.380339
$634$	−32.5958	−1.29455
$635$	−38.4154	−1.52447
$636$	3.64042	0.144352
$637$	0	0
$638$	−7.35103	−0.291030
$639$	4.13161	0.163444
$640$	2.37720	0.0939672
$641$	−6.12174	−0.241794	−0.120897	−	0.992665i	$-0.538577\pi$
$641$	−6.12174	−0.241794	−0.120897	+	0.992665i	$0.538577\pi$
$642$	8.95328	0.353358
$643$	31.0078	1.22283	0.611413	−	0.791312i	$-0.290601\pi$
$643$	31.0078	1.22283	0.611413	+	0.791312i	$0.290601\pi$
$644$	−1.25334	−0.0493887
$645$	1.36945	0.0539222
$646$	0.547781	0.0215522
$647$	−33.2341	−1.30657	−0.653284	−	0.757113i	$-0.726609\pi$
$647$	−33.2341	−1.30657	−0.653284	+	0.757113i	$0.726609\pi$
$648$	−2.97170	−0.116740
$649$	59.2207	2.32462
$650$	0	0
$651$	−0.997077	−0.0390785
$652$	1.37720	0.0539354
$653$	−23.4047	−0.915897	−0.457948	−	0.888979i	$-0.651416\pi$
$653$	−23.4047	−0.915897	−0.457948	+	0.888979i	$0.651416\pi$
$654$	5.91511	0.231299
$655$	−37.8569	−1.47919
$656$	9.02830	0.352496
$657$	15.3051	0.597109
$658$	0.129486	0.00504789
$659$	12.7232	0.495625	0.247812	−	0.968808i	$-0.420288\pi$
$659$	12.7232	0.495625	0.247812	+	0.968808i	$0.420288\pi$
$660$	13.2555	0.515968
$661$	40.3842	1.57076	0.785381	−	0.619013i	$-0.212467\pi$
$661$	40.3842	1.57076	0.785381	+	0.619013i	$0.212467\pi$
$662$	7.33823	0.285209
$663$	0	0
$664$	−16.7643	−0.650580
$665$	0.651093	0.0252483
$666$	11.0956	0.429944
$667$	−7.68502	−0.297565
$668$	−0.416173	−0.0161022
$669$	1.46501	0.0566408
$670$	23.9066	0.923591
$671$	−18.3793	−0.709526
$672$	−0.348907	−0.0134594
$673$	−7.16978	−0.276375	−0.138187	−	0.990406i	$-0.544128\pi$
$673$	−7.16978	−0.276375	−0.138187	+	0.990406i	$0.544128\pi$
$674$	3.48052	0.134064
$675$	3.63055	0.139740
$676$	0	0
$677$	−24.6220	−0.946300	−0.473150	−	0.880982i	$-0.656883\pi$
$677$	−24.6220	−0.946300	−0.473150	+	0.880982i	$0.656883\pi$
$678$	−13.5782	−0.521468
$679$	−1.49331	−0.0573080
$680$	1.30219	0.0499366
$681$	22.9066	0.877781
$682$	−12.5088	−0.478987
$683$	−26.0673	−0.997436	−0.498718	−	0.866764i	$-0.666196\pi$
$683$	−26.0673	−0.997436	−0.498718	+	0.866764i	$0.666196\pi$
$684$	−1.37720	−0.0526587
$685$	−31.3326	−1.19716
$686$	−3.81392	−0.145616
$687$	−21.3297	−0.813778
$688$	−0.452219	−0.0172407
$689$	0	0
$690$	13.8577	0.527554
$691$	19.0849	0.726023	0.363012	−	0.931785i	$-0.381748\pi$
$691$	19.0849	0.726023	0.363012	+	0.931785i	$0.381748\pi$
$692$	−4.21942	−0.160398
$693$	1.65109	0.0627199
$694$	12.9709	0.492369
$695$	5.94553	0.225527
$696$	−2.13936	−0.0810922
$697$	4.94553	0.187325
$698$	−16.5059	−0.624757
$699$	0.322733	0.0122069
$700$	0.178328	0.00674018
$701$	23.1834	0.875624	0.437812	−	0.899067i	$-0.355754\pi$
$701$	23.1834	0.875624	0.437812	+	0.899067i	$0.355754\pi$
$702$	0	0
$703$	−8.05659	−0.303860
$704$	−4.37720	−0.164972
$705$	−1.43167	−0.0539199
$706$	23.4415	0.882234
$707$	2.23280	0.0839730
$708$	17.2349	0.647728
$709$	19.3404	0.726342	0.363171	−	0.931722i	$-0.381694\pi$
$709$	19.3404	0.726342	0.363171	+	0.931722i	$0.381694\pi$
$710$	−7.13161	−0.267645
$711$	−3.26322	−0.122380
$712$	−2.04672	−0.0767041
$713$	−13.0771	−0.489743
$714$	−0.191124	−0.00715265
$715$	0	0
$716$	20.9143	0.781604
$717$	−30.7040	−1.14666
$718$	−18.6794	−0.697109
$719$	49.9001	1.86096	0.930480	−	0.366342i	$-0.119390\pi$
$719$	49.9001	1.86096	0.930480	+	0.366342i	$0.119390\pi$
$720$	−3.27389	−0.122011
$721$	−4.47839	−0.166784
$722$	1.00000	0.0372161
$723$	−2.64334	−0.0983070
$724$	6.87051	0.255341
$725$	1.09344	0.0406093
$726$	−10.3948	−0.385788
$727$	3.20100	0.118718	0.0593592	−	0.998237i	$-0.481094\pi$
$727$	3.20100	0.118718	0.0593592	+	0.998237i	$0.481094\pi$
$728$	0	0
$729$	25.4026	0.940836
$730$	−26.4183	−0.977785
$731$	−0.247717	−0.00916214
$732$	−5.34891	−0.197701
$733$	25.6823	0.948598	0.474299	−	0.880364i	$-0.342702\pi$
$733$	25.6823	0.948598	0.474299	+	0.880364i	$0.342702\pi$
$734$	−22.3977	−0.826716
$735$	20.9709	0.773523
$736$	−4.57608	−0.168676
$737$	−44.0197	−1.62149
$738$	−12.4338	−0.457694
$739$	36.5598	1.34487	0.672437	−	0.740155i	$-0.265248\pi$
$739$	36.5598	1.34487	0.672437	+	0.740155i	$0.265248\pi$
$740$	−19.1522	−0.704047
$741$	0	0
$742$	−0.782702	−0.0287339
$743$	25.6999	0.942839	0.471420	−	0.881909i	$-0.343742\pi$
$743$	25.6999	0.942839	0.471420	+	0.881909i	$0.343742\pi$
$744$	−3.64042	−0.133464
$745$	−22.4026	−0.820767
$746$	33.3559	1.22124
$747$	23.0878	0.844739
$748$	−2.39775	−0.0876704
$749$	−1.92498	−0.0703374
$750$	13.1698	0.480892
$751$	−46.3326	−1.69070	−0.845350	−	0.534212i	$-0.820608\pi$
$751$	−46.3326	−1.69070	−0.845350	+	0.534212i	$0.820608\pi$
$752$	0.472765	0.0172400
$753$	9.65109	0.351705
$754$	0	0
$755$	−28.1025	−1.02276
$756$	1.52723	0.0555450
$757$	−28.6511	−1.04134	−0.520671	−	0.853757i	$-0.674318\pi$
$757$	−28.6511	−1.04134	−0.520671	+	0.853757i	$0.674318\pi$
$758$	−34.8337	−1.26522
$759$	−25.5166	−0.926193
$760$	2.37720	0.0862302
$761$	−31.1004	−1.12739	−0.563694	−	0.825984i	$-0.690620\pi$
$761$	−31.1004	−1.12739	−0.563694	+	0.825984i	$0.690620\pi$
$762$	20.5860	0.745750
$763$	−1.27177	−0.0460411
$764$	12.3305	0.446101
$765$	−1.79338	−0.0648396
$766$	18.1834	0.656992
$767$	0	0
$768$	−1.27389	−0.0459676
$769$	49.4301	1.78249	0.891247	−	0.453518i	$-0.149831\pi$
$769$	49.4301	1.78249	0.891247	+	0.453518i	$0.149831\pi$
$770$	−2.84997	−0.102706
$771$	−10.2731	−0.369976
$772$	20.3121	0.731047
$773$	8.98450	0.323150	0.161575	−	0.986860i	$-0.448343\pi$
$773$	8.98450	0.323150	0.161575	+	0.986860i	$0.448343\pi$
$774$	0.622797	0.0223860
$775$	1.86064	0.0668362
$776$	−5.45222	−0.195723
$777$	2.81100	0.100844
$778$	0.894565	0.0320717
$779$	9.02830	0.323472
$780$	0	0
$781$	13.1316	0.469886
$782$	−2.50669	−0.0896390
$783$	9.36441	0.334657
$784$	−6.92498	−0.247321
$785$	2.69006	0.0960125
$786$	20.2867	0.723602
$787$	−25.0403	−0.892590	−0.446295	−	0.894886i	$-0.647257\pi$
$787$	−25.0403	−0.892590	−0.446295	+	0.894886i	$0.647257\pi$
$788$	25.0849	0.893612
$789$	20.3921	0.725979
$790$	5.63267	0.200401
$791$	2.91936	0.103800
$792$	6.02830	0.214206
$793$	0	0
$794$	37.3249	1.32461
$795$	8.65402	0.306926
$796$	−11.3404	−0.401948
$797$	−14.0614	−0.498081	−0.249041	−	0.968493i	$-0.580115\pi$
$797$	−14.0614	−0.498081	−0.249041	+	0.968493i	$0.580115\pi$
$798$	−0.348907	−0.0123512
$799$	0.258972	0.00916176
$800$	0.651093	0.0230196
$801$	2.81875	0.0995956
$802$	14.8500	0.524371
$803$	48.6447	1.71663
$804$	−12.8110	−0.451809
$805$	−2.97945	−0.105012
$806$	0	0
$807$	19.9864	0.703555
$808$	8.15215	0.286792
$809$	3.86839	0.136005	0.0680027	−	0.997685i	$-0.478337\pi$
$809$	3.86839	0.136005	0.0680027	+	0.997685i	$0.478337\pi$
$810$	−7.06434	−0.248216
$811$	21.7467	0.763628	0.381814	−	0.924239i	$-0.375299\pi$
$811$	21.7467	0.763628	0.381814	+	0.924239i	$0.375299\pi$
$812$	0.459969	0.0161417
$813$	27.2349	0.955170
$814$	35.2653	1.23605
$815$	3.27389	0.114679
$816$	−0.697813	−0.0244283
$817$	−0.452219	−0.0158211
$818$	−29.4047	−1.02811
$819$	0	0
$820$	21.4621	0.749489
$821$	36.4925	1.27360	0.636799	−	0.771030i	$-0.280258\pi$
$821$	36.4925	1.27360	0.636799	+	0.771030i	$0.280258\pi$
$822$	16.7905	0.585634
$823$	−22.0390	−0.768230	−0.384115	−	0.923285i	$-0.625493\pi$
$823$	−22.0390	−0.768230	−0.384115	+	0.923285i	$0.625493\pi$
$824$	−16.3510	−0.569615
$825$	3.63055	0.126399
$826$	−3.70556	−0.128933
$827$	−3.61292	−0.125634	−0.0628168	−	0.998025i	$-0.520008\pi$
$827$	−3.61292	−0.125634	−0.0628168	+	0.998025i	$0.520008\pi$
$828$	6.30219	0.219016
$829$	−2.26614	−0.0787063	−0.0393532	−	0.999225i	$-0.512530\pi$
$829$	−2.26614	−0.0787063	−0.0393532	+	0.999225i	$0.512530\pi$
$830$	−39.8521	−1.38329
$831$	8.61797	0.298954
$832$	0	0
$833$	−3.79338	−0.131433
$834$	−3.18608	−0.110325
$835$	−0.989327	−0.0342371
$836$	−4.37720	−0.151389
$837$	15.9349	0.550789
$838$	−24.9221	−0.860918
$839$	−23.9370	−0.826396	−0.413198	−	0.910641i	$-0.635588\pi$
$839$	−23.9370	−0.826396	−0.413198	+	0.910641i	$0.635588\pi$
$840$	−0.829422	−0.0286178
$841$	−26.1797	−0.902747
$842$	5.12174	0.176507
$843$	−32.2584	−1.11104
$844$	−7.51173	−0.258565
$845$	0	0
$846$	−0.651093	−0.0223851
$847$	2.23492	0.0767928
$848$	−2.85772	−0.0981344
$849$	20.2469	0.694872
$850$	0.356657	0.0122332
$851$	36.8676	1.26380
$852$	3.82167	0.130928
$853$	−44.1484	−1.51161	−0.755807	−	0.654795i	$-0.772755\pi$
$853$	−44.1484	−1.51161	−0.755807	+	0.654795i	$0.772755\pi$
$854$	1.15003	0.0393533
$855$	−3.27389	−0.111965
$856$	−7.02830	−0.240222
$857$	29.5344	1.00888	0.504438	−	0.863448i	$-0.331700\pi$
$857$	29.5344	1.00888	0.504438	+	0.863448i	$0.331700\pi$
$858$	0	0
$859$	22.2808	0.760212	0.380106	−	0.924943i	$-0.375887\pi$
$859$	22.2808	0.760212	0.380106	+	0.924943i	$0.375887\pi$
$860$	−1.07502	−0.0366577
$861$	−3.15003	−0.107353
$862$	20.3121	0.691831
$863$	26.2427	0.893311	0.446655	−	0.894706i	$-0.352615\pi$
$863$	26.2427	0.893311	0.446655	+	0.894706i	$0.352615\pi$
$864$	5.57608	0.189702
$865$	−10.0304	−0.341044
$866$	−27.8705	−0.947079
$867$	21.2739	0.722499
$868$	0.782702	0.0265666
$869$	−10.3716	−0.351832
$870$	−5.08569	−0.172421
$871$	0	0
$872$	−4.64334	−0.157243
$873$	7.50881	0.254135
$874$	−4.57608	−0.154788
$875$	−2.83154	−0.0957237
$876$	14.1570	0.478320
$877$	42.2683	1.42730	0.713649	−	0.700504i	$-0.247041\pi$
$877$	42.2683	1.42730	0.713649	+	0.700504i	$0.247041\pi$
$878$	−5.88389	−0.198572
$879$	−42.4223	−1.43087
$880$	−10.4055	−0.350769
$881$	31.8726	1.07382	0.536908	−	0.843641i	$-0.319592\pi$
$881$	31.8726	1.07382	0.536908	+	0.843641i	$0.319592\pi$
$882$	9.53711	0.321131
$883$	20.3121	0.683555	0.341778	−	0.939781i	$-0.388971\pi$
$883$	20.3121	0.683555	0.341778	+	0.939781i	$0.388971\pi$
$884$	0	0
$885$	40.9709	1.37722
$886$	−24.0878	−0.809246
$887$	42.5577	1.42895	0.714473	−	0.699663i	$-0.246666\pi$
$887$	42.5577	1.42895	0.714473	+	0.699663i	$0.246666\pi$
$888$	10.2632	0.344411
$889$	−4.42605	−0.148445
$890$	−4.86547	−0.163091
$891$	13.0078	0.435776
$892$	−1.15003	−0.0385059
$893$	0.472765	0.0158205
$894$	12.0050	0.401509
$895$	49.7176	1.66187
$896$	0.273891	0.00915004
$897$	0	0
$898$	−18.4026	−0.614102
$899$	4.79922	0.160063
$900$	−0.896688	−0.0298896
$901$	−1.56540	−0.0521512
$902$	−39.5187	−1.31583
$903$	0.157782	0.00525066
$904$	10.6588	0.354508
$905$	16.3326	0.542914
$906$	15.0595	0.500319
$907$	4.77203	0.158453	0.0792263	−	0.996857i	$-0.474755\pi$
$907$	4.77203	0.158453	0.0792263	+	0.996857i	$0.474755\pi$
$908$	−17.9816	−0.596740
$909$	−11.2272	−0.372382
$910$	0	0
$911$	−1.66367	−0.0551199	−0.0275599	−	0.999620i	$-0.508774\pi$
$911$	−1.66367	−0.0551199	−0.0275599	+	0.999620i	$0.508774\pi$
$912$	−1.27389	−0.0421827
$913$	73.3806	2.42854
$914$	−1.82460	−0.0603522
$915$	−12.7154	−0.420359
$916$	16.7437	0.553228
$917$	−4.36170	−0.144036
$918$	3.05447	0.100813
$919$	13.9653	0.460672	0.230336	−	0.973111i	$-0.426017\pi$
$919$	13.9653	0.460672	0.230336	+	0.973111i	$0.426017\pi$
$920$	−10.8783	−0.358646
$921$	−11.0304	−0.363465
$922$	−21.9730	−0.723643
$923$	0	0
$924$	1.52723	0.0502423
$925$	−5.24559	−0.172474
$926$	6.74373	0.221613
$927$	22.5187	0.739611
$928$	1.67939	0.0551287
$929$	−48.3142	−1.58514	−0.792568	−	0.609783i	$-0.791257\pi$
$929$	−48.3142	−1.58514	−0.792568	+	0.609783i	$0.791257\pi$
$930$	−8.65402	−0.283776
$931$	−6.92498	−0.226957
$932$	−0.253344	−0.00829857
$933$	−11.3793	−0.372542
$934$	24.0227	0.786046
$935$	−5.69994	−0.186408
$936$	0	0
$937$	40.6524	1.32806	0.664028	−	0.747707i	$-0.268845\pi$
$937$	40.6524	1.32806	0.664028	+	0.747707i	$0.268845\pi$
$938$	2.75441	0.0899345
$939$	14.7963	0.482859
$940$	1.12386	0.0366562
$941$	47.7643	1.55707	0.778535	−	0.627601i	$-0.215963\pi$
$941$	47.7643	1.55707	0.778535	+	0.627601i	$0.215963\pi$
$942$	−1.44155	−0.0469681
$943$	−41.3142	−1.34537
$944$	−13.5294	−0.440343
$945$	3.63055	0.118102
$946$	1.97945	0.0643576
$947$	−4.82438	−0.156771	−0.0783856	−	0.996923i	$-0.524977\pi$
$947$	−4.82438	−0.156771	−0.0783856	+	0.996923i	$0.524977\pi$
$948$	−3.01842	−0.0980338
$949$	0	0
$950$	0.651093	0.0211243
$951$	41.5235	1.34649
$952$	0.150032	0.00486257
$953$	−50.5782	−1.63839	−0.819194	−	0.573516i	$-0.805579\pi$
$953$	−50.5782	−1.63839	−0.819194	+	0.573516i	$0.805579\pi$
$954$	3.93566	0.127422
$955$	29.3121	0.948516
$956$	24.1025	0.779531
$957$	9.36441	0.302708
$958$	−14.2066	−0.458995
$959$	−3.61000	−0.116573
$960$	−3.02830	−0.0977378
$961$	−22.8334	−0.736563
$962$	0	0
$963$	9.67939	0.311914
$964$	2.07502	0.0668318
$965$	48.2859	1.55438
$966$	1.59662	0.0513705
$967$	28.2555	0.908635	0.454317	−	0.890840i	$-0.349883\pi$
$967$	28.2555	0.908635	0.454317	+	0.890840i	$0.349883\pi$
$968$	8.15990	0.262269
$969$	−0.697813	−0.0224170
$970$	−12.9610	−0.416154
$971$	21.3695	0.685778	0.342889	−	0.939376i	$-0.388594\pi$
$971$	21.3695	0.685778	0.342889	+	0.939376i	$0.388594\pi$
$972$	−12.9426	−0.415134
$973$	0.685017	0.0219606
$974$	12.0021	0.384573
$975$	0	0
$976$	4.19887	0.134403
$977$	−51.1457	−1.63630	−0.818148	−	0.575007i	$-0.804999\pi$
$977$	−51.1457	−1.63630	−0.818148	+	0.575007i	$0.804999\pi$
$978$	−1.75441	−0.0560997
$979$	8.95891	0.286328
$980$	−16.4621	−0.525862
$981$	6.39483	0.204171
$982$	−28.4047	−0.906430
$983$	−15.1004	−0.481628	−0.240814	−	0.970571i	$-0.577414\pi$
$983$	−15.1004	−0.481628	−0.240814	+	0.970571i	$0.577414\pi$
$984$	−11.5011	−0.366640
$985$	59.6319	1.90003
$986$	0.919938	0.0292968
$987$	−0.164951	−0.00525044
$988$	0	0
$989$	2.06939	0.0658027
$990$	14.3305	0.455453
$991$	21.8492	0.694062	0.347031	−	0.937854i	$-0.387190\pi$
$991$	21.8492	0.694062	0.347031	+	0.937854i	$0.387190\pi$
$992$	2.85772	0.0907326
$993$	−9.34811	−0.296653
$994$	−0.821672	−0.0260618
$995$	−26.9583	−0.854636
$996$	21.3559	0.676686
$997$	−14.0390	−0.444619	−0.222309	−	0.974976i	$-0.571359\pi$
$997$	−14.0390	−0.444619	−0.222309	+	0.974976i	$0.571359\pi$
$998$	0.452219	0.0143147
$999$	−44.9242	−1.42134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.u.1.2		3
13.3	even	3		494.2.g.b.191.2	✓	6
13.9	even	3		494.2.g.b.419.2	yes	6
13.12	even	2		6422.2.a.m.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.b.191.2	✓	6	13.3	even	3
494.2.g.b.419.2	yes	6	13.9	even	3
6422.2.a.m.1.2		3	13.12	even	2
6422.2.a.u.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.27389 q^{3} +1.00000 q^{4} +2.37720 q^{5} -1.27389 q^{6} +0.273891 q^{7} +1.00000 q^{8} -1.37720 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.27389 q^{3} +1.00000 q^{4} +2.37720 q^{5} -1.27389 q^{6} +0.273891 q^{7} +1.00000 q^{8} -1.37720 q^{9} +2.37720 q^{10} -4.37720 q^{11} -1.27389 q^{12} +0.273891 q^{14} -3.02830 q^{15} +1.00000 q^{16} +0.547781 q^{17} -1.37720 q^{18} +1.00000 q^{19} +2.37720 q^{20} -0.348907 q^{21} -4.37720 q^{22} -4.57608 q^{23} -1.27389 q^{24} +0.651093 q^{25} +5.57608 q^{27} +0.273891 q^{28} +1.67939 q^{29} -3.02830 q^{30} +2.85772 q^{31} +1.00000 q^{32} +5.57608 q^{33} +0.547781 q^{34} +0.651093 q^{35} -1.37720 q^{36} -8.05659 q^{37} +1.00000 q^{38} +2.37720 q^{40} +9.02830 q^{41} -0.348907 q^{42} -0.452219 q^{43} -4.37720 q^{44} -3.27389 q^{45} -4.57608 q^{46} +0.472765 q^{47} -1.27389 q^{48} -6.92498 q^{49} +0.651093 q^{50} -0.697813 q^{51} -2.85772 q^{53} +5.57608 q^{54} -10.4055 q^{55} +0.273891 q^{56} -1.27389 q^{57} +1.67939 q^{58} -13.5294 q^{59} -3.02830 q^{60} +4.19887 q^{61} +2.85772 q^{62} -0.377203 q^{63} +1.00000 q^{64} +5.57608 q^{66} +10.0566 q^{67} +0.547781 q^{68} +5.82942 q^{69} +0.651093 q^{70} -3.00000 q^{71} -1.37720 q^{72} -11.1132 q^{73} -8.05659 q^{74} -0.829422 q^{75} +1.00000 q^{76} -1.19887 q^{77} +2.36945 q^{79} +2.37720 q^{80} -2.97170 q^{81} +9.02830 q^{82} -16.7643 q^{83} -0.348907 q^{84} +1.30219 q^{85} -0.452219 q^{86} -2.13936 q^{87} -4.37720 q^{88} -2.04672 q^{89} -3.27389 q^{90} -4.57608 q^{92} -3.64042 q^{93} +0.472765 q^{94} +2.37720 q^{95} -1.27389 q^{96} -5.45222 q^{97} -6.92498 q^{98} +6.02830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} - q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 - 2 * q^6 - q^7 + 3 * q^8 + q^9	\( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 2 q^{6} - q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - 8 q^{11} - 2 q^{12} - q^{14} + 3 q^{15} + 3 q^{16} - 2 q^{17} + q^{18} + 3 q^{19} + 2 q^{20} - 8 q^{21} - 8 q^{22} + 2 q^{23} - 2 q^{24} - 5 q^{25} + q^{27} - q^{28} - 14 q^{29} + 3 q^{30} - 5 q^{31} + 3 q^{32} + q^{33} - 2 q^{34} - 5 q^{35} + q^{36} + 3 q^{38} + 2 q^{40} + 15 q^{41} - 8 q^{42} - 5 q^{43} - 8 q^{44} - 8 q^{45} + 2 q^{46} - 11 q^{47} - 2 q^{48} - 12 q^{49} - 5 q^{50} - 16 q^{51} + 5 q^{53} + q^{54} - 14 q^{55} - q^{56} - 2 q^{57} - 14 q^{58} - 4 q^{59} + 3 q^{60} + 2 q^{61} - 5 q^{62} + 4 q^{63} + 3 q^{64} + q^{66} + 6 q^{67} - 2 q^{68} + 16 q^{69} - 5 q^{70} - 9 q^{71} + q^{72} + 15 q^{73} - q^{75} + 3 q^{76} + 7 q^{77} - 2 q^{79} + 2 q^{80} - 21 q^{81} + 15 q^{82} + 5 q^{83} - 8 q^{84} - 10 q^{85} - 5 q^{86} + 5 q^{87} - 8 q^{88} - 27 q^{89} - 8 q^{90} + 2 q^{92} + 25 q^{93} - 11 q^{94} + 2 q^{95} - 2 q^{96} - 20 q^{97} - 12 q^{98} + 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 - 2 * q^6 - q^7 + 3 * q^8 + q^9 + 2 * q^10 - 8 * q^11 - 2 * q^12 - q^14 + 3 * q^15 + 3 * q^16 - 2 * q^17 + q^18 + 3 * q^19 + 2 * q^20 - 8 * q^21 - 8 * q^22 + 2 * q^23 - 2 * q^24 - 5 * q^25 + q^27 - q^28 - 14 * q^29 + 3 * q^30 - 5 * q^31 + 3 * q^32 + q^33 - 2 * q^34 - 5 * q^35 + q^36 + 3 * q^38 + 2 * q^40 + 15 * q^41 - 8 * q^42 - 5 * q^43 - 8 * q^44 - 8 * q^45 + 2 * q^46 - 11 * q^47 - 2 * q^48 - 12 * q^49 - 5 * q^50 - 16 * q^51 + 5 * q^53 + q^54 - 14 * q^55 - q^56 - 2 * q^57 - 14 * q^58 - 4 * q^59 + 3 * q^60 + 2 * q^61 - 5 * q^62 + 4 * q^63 + 3 * q^64 + q^66 + 6 * q^67 - 2 * q^68 + 16 * q^69 - 5 * q^70 - 9 * q^71 + q^72 + 15 * q^73 - q^75 + 3 * q^76 + 7 * q^77 - 2 * q^79 + 2 * q^80 - 21 * q^81 + 15 * q^82 + 5 * q^83 - 8 * q^84 - 10 * q^85 - 5 * q^86 + 5 * q^87 - 8 * q^88 - 27 * q^89 - 8 * q^90 + 2 * q^92 + 25 * q^93 - 11 * q^94 + 2 * q^95 - 2 * q^96 - 20 * q^97 - 12 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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