LMFDB - Embedded newform 6422.2.a.bl.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 $ x^9 - x^8 - 12x^7 + 14x^6 + 35x^5 - 35x^4 - 28x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.31454$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.77275 q^{3} +1.00000 q^{4} -2.80860 q^{5} +2.77275 q^{6} -1.68633 q^{7} +1.00000 q^{8} +4.68813 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.77275 q^{3} +1.00000 q^{4} -2.80860 q^{5} +2.77275 q^{6} -1.68633 q^{7} +1.00000 q^{8} +4.68813 q^{9} -2.80860 q^{10} -3.21027 q^{11} +2.77275 q^{12} -1.68633 q^{14} -7.78753 q^{15} +1.00000 q^{16} +1.37380 q^{17} +4.68813 q^{18} -1.00000 q^{19} -2.80860 q^{20} -4.67578 q^{21} -3.21027 q^{22} -3.21156 q^{23} +2.77275 q^{24} +2.88822 q^{25} +4.68075 q^{27} -1.68633 q^{28} +0.713407 q^{29} -7.78753 q^{30} +2.38425 q^{31} +1.00000 q^{32} -8.90127 q^{33} +1.37380 q^{34} +4.73623 q^{35} +4.68813 q^{36} -1.39317 q^{37} -1.00000 q^{38} -2.80860 q^{40} -7.66050 q^{41} -4.67578 q^{42} -12.7942 q^{43} -3.21027 q^{44} -13.1671 q^{45} -3.21156 q^{46} -1.01949 q^{47} +2.77275 q^{48} -4.15628 q^{49} +2.88822 q^{50} +3.80921 q^{51} +1.08090 q^{53} +4.68075 q^{54} +9.01636 q^{55} -1.68633 q^{56} -2.77275 q^{57} +0.713407 q^{58} -11.0891 q^{59} -7.78753 q^{60} +12.9868 q^{61} +2.38425 q^{62} -7.90575 q^{63} +1.00000 q^{64} -8.90127 q^{66} +7.66589 q^{67} +1.37380 q^{68} -8.90485 q^{69} +4.73623 q^{70} -4.86090 q^{71} +4.68813 q^{72} -9.26376 q^{73} -1.39317 q^{74} +8.00830 q^{75} -1.00000 q^{76} +5.41359 q^{77} -10.1589 q^{79} -2.80860 q^{80} -1.08586 q^{81} -7.66050 q^{82} +9.75396 q^{83} -4.67578 q^{84} -3.85846 q^{85} -12.7942 q^{86} +1.97810 q^{87} -3.21027 q^{88} +6.45411 q^{89} -13.1671 q^{90} -3.21156 q^{92} +6.61093 q^{93} -1.01949 q^{94} +2.80860 q^{95} +2.77275 q^{96} -9.34495 q^{97} -4.15628 q^{98} -15.0502 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9	$ 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9 + q^10 - 3 * q^11 + q^12 - 13 * q^14 - 3 * q^15 + 9 * q^16 - 8 * q^17 - 2 * q^18 - 9 * q^19 + q^20 - 24 * q^21 - 3 * q^22 - 10 * q^23 + q^24 + 8 * q^25 + 10 * q^27 - 13 * q^28 - 20 * q^29 - 3 * q^30 - q^31 + 9 * q^32 + 2 * q^33 - 8 * q^34 + 4 * q^35 - 2 * q^36 - 15 * q^37 - 9 * q^38 + q^40 - 19 * q^41 - 24 * q^42 - 16 * q^43 - 3 * q^44 - 15 * q^45 - 10 * q^46 - 18 * q^47 + q^48 + 18 * q^49 + 8 * q^50 - 11 * q^51 + 17 * q^53 + 10 * q^54 - 26 * q^55 - 13 * q^56 - q^57 - 20 * q^58 - 24 * q^59 - 3 * q^60 + 6 * q^61 - q^62 - q^63 + 9 * q^64 + 2 * q^66 - 29 * q^67 - 8 * q^68 - 12 * q^69 + 4 * q^70 - 23 * q^71 - 2 * q^72 - 38 * q^73 - 15 * q^74 + 11 * q^75 - 9 * q^76 - 40 * q^77 - 20 * q^79 + q^80 - 31 * q^81 - 19 * q^82 - 20 * q^83 - 24 * q^84 - 39 * q^85 - 16 * q^86 - 10 * q^87 - 3 * q^88 + 7 * q^89 - 15 * q^90 - 10 * q^92 - 11 * q^93 - 18 * q^94 - q^95 + q^96 - 28 * q^97 + 18 * q^98 - 25 * 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
$3$	2.77275	1.60085	0.800423	−	0.599435i	$-0.204608\pi$
$3$	2.77275	1.60085	0.800423	+	0.599435i	$0.204608\pi$
$4$	1.00000	0.500000
$5$	−2.80860	−1.25604	−0.628021	−	0.778196i	$-0.716135\pi$
$5$	−2.80860	−1.25604	−0.628021	+	0.778196i	$0.716135\pi$
$6$	2.77275	1.13197
$7$	−1.68633	−0.637375	−0.318687	−	0.947860i	$-0.603242\pi$
$7$	−1.68633	−0.637375	−0.318687	+	0.947860i	$0.603242\pi$
$8$	1.00000	0.353553
$9$	4.68813	1.56271
$10$	−2.80860	−0.888156
$11$	−3.21027	−0.967933	−0.483967	−	0.875086i	$-0.660804\pi$
$11$	−3.21027	−0.967933	−0.483967	+	0.875086i	$0.660804\pi$
$12$	2.77275	0.800423
$13$	0	0
$13$	0	0
$14$	−1.68633	−0.450692
$15$	−7.78753	−2.01073
$16$	1.00000	0.250000
$17$	1.37380	0.333196	0.166598	−	0.986025i	$-0.446722\pi$
$17$	1.37380	0.333196	0.166598	+	0.986025i	$0.446722\pi$
$18$	4.68813	1.10500
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.80860	−0.628021
$21$	−4.67578	−1.02034
$22$	−3.21027	−0.684432
$23$	−3.21156	−0.669657	−0.334829	−	0.942279i	$-0.608678\pi$
$23$	−3.21156	−0.669657	−0.334829	+	0.942279i	$0.608678\pi$
$24$	2.77275	0.565985
$25$	2.88822	0.577643
$26$	0	0
$27$	4.68075	0.900810
$28$	−1.68633	−0.318687
$29$	0.713407	0.132476	0.0662382	−	0.997804i	$-0.478900\pi$
$29$	0.713407	0.132476	0.0662382	+	0.997804i	$0.478900\pi$
$30$	−7.78753	−1.42180
$31$	2.38425	0.428224	0.214112	−	0.976809i	$-0.431314\pi$
$31$	2.38425	0.428224	0.214112	+	0.976809i	$0.431314\pi$
$32$	1.00000	0.176777
$33$	−8.90127	−1.54951
$34$	1.37380	0.235605
$35$	4.73623	0.800570
$36$	4.68813	0.781354
$37$	−1.39317	−0.229036	−0.114518	−	0.993421i	$-0.536532\pi$
$37$	−1.39317	−0.229036	−0.114518	+	0.993421i	$0.536532\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.80860	−0.444078
$41$	−7.66050	−1.19637	−0.598184	−	0.801358i	$-0.704111\pi$
$41$	−7.66050	−1.19637	−0.598184	+	0.801358i	$0.704111\pi$
$42$	−4.67578	−0.721488
$43$	−12.7942	−1.95109	−0.975547	−	0.219791i	$-0.929463\pi$
$43$	−12.7942	−1.95109	−0.975547	+	0.219791i	$0.929463\pi$
$44$	−3.21027	−0.483967
$45$	−13.1671	−1.96283
$46$	−3.21156	−0.473519
$47$	−1.01949	−0.148708	−0.0743542	−	0.997232i	$-0.523690\pi$
$47$	−1.01949	−0.148708	−0.0743542	+	0.997232i	$0.523690\pi$
$48$	2.77275	0.400212
$49$	−4.15628	−0.593754
$50$	2.88822	0.408456
$51$	3.80921	0.533396
$52$	0	0
$53$	1.08090	0.148473	0.0742363	−	0.997241i	$-0.476348\pi$
$53$	1.08090	0.148473	0.0742363	+	0.997241i	$0.476348\pi$
$54$	4.68075	0.636969
$55$	9.01636	1.21577
$56$	−1.68633	−0.225346
$57$	−2.77275	−0.367259
$58$	0.713407	0.0936749
$59$	−11.0891	−1.44368	−0.721842	−	0.692058i	$-0.756704\pi$
$59$	−11.0891	−1.44368	−0.721842	+	0.692058i	$0.756704\pi$
$60$	−7.78753	−1.00537
$61$	12.9868	1.66278	0.831392	−	0.555686i	$-0.187544\pi$
$61$	12.9868	1.66278	0.831392	+	0.555686i	$0.187544\pi$
$62$	2.38425	0.302800
$63$	−7.90575	−0.996031
$64$	1.00000	0.125000
$65$	0	0
$66$	−8.90127	−1.09567
$67$	7.66589	0.936538	0.468269	−	0.883586i	$-0.344878\pi$
$67$	7.66589	0.936538	0.468269	+	0.883586i	$0.344878\pi$
$68$	1.37380	0.166598
$69$	−8.90485	−1.07202
$70$	4.73623	0.566088
$71$	−4.86090	−0.576882	−0.288441	−	0.957498i	$-0.593137\pi$
$71$	−4.86090	−0.576882	−0.288441	+	0.957498i	$0.593137\pi$
$72$	4.68813	0.552501
$73$	−9.26376	−1.08424	−0.542121	−	0.840301i	$-0.682378\pi$
$73$	−9.26376	−1.08424	−0.542121	+	0.840301i	$0.682378\pi$
$74$	−1.39317	−0.161953
$75$	8.00830	0.924718
$76$	−1.00000	−0.114708
$77$	5.41359	0.616936
$78$	0	0
$79$	−10.1589	−1.14296	−0.571480	−	0.820616i	$-0.693631\pi$
$79$	−10.1589	−1.14296	−0.571480	+	0.820616i	$0.693631\pi$
$80$	−2.80860	−0.314011
$81$	−1.08586	−0.120651
$82$	−7.66050	−0.845961
$83$	9.75396	1.07064	0.535318	−	0.844651i	$-0.320192\pi$
$83$	9.75396	1.07064	0.535318	+	0.844651i	$0.320192\pi$
$84$	−4.67578	−0.510169
$85$	−3.85846	−0.418508
$86$	−12.7942	−1.37963
$87$	1.97810	0.212074
$88$	−3.21027	−0.342216
$89$	6.45411	0.684134	0.342067	−	0.939676i	$-0.388873\pi$
$89$	6.45411	0.684134	0.342067	+	0.939676i	$0.388873\pi$
$90$	−13.1671	−1.38793
$91$	0	0
$92$	−3.21156	−0.334829
$93$	6.61093	0.685521
$94$	−1.01949	−0.105153
$95$	2.80860	0.288156
$96$	2.77275	0.282992
$97$	−9.34495	−0.948836	−0.474418	−	0.880300i	$-0.657341\pi$
$97$	−9.34495	−0.948836	−0.474418	+	0.880300i	$0.657341\pi$
$98$	−4.15628	−0.419847
$99$	−15.0502	−1.51260
$100$	2.88822	0.288822
$101$	−8.39872	−0.835703	−0.417852	−	0.908515i	$-0.637217\pi$
$101$	−8.39872	−0.835703	−0.417852	+	0.908515i	$0.637217\pi$
$102$	3.80921	0.377168
$103$	−17.1843	−1.69322	−0.846611	−	0.532213i	$-0.821361\pi$
$103$	−17.1843	−1.69322	−0.846611	+	0.532213i	$0.821361\pi$
$104$	0	0
$105$	13.1324	1.28159
$106$	1.08090	0.104986
$107$	13.4013	1.29555	0.647777	−	0.761830i	$-0.275699\pi$
$107$	13.4013	1.29555	0.647777	+	0.761830i	$0.275699\pi$
$108$	4.68075	0.450405
$109$	−10.5139	−1.00705	−0.503524	−	0.863981i	$-0.667963\pi$
$109$	−10.5139	−1.00705	−0.503524	+	0.863981i	$0.667963\pi$
$110$	9.01636	0.859676
$111$	−3.86291	−0.366651
$112$	−1.68633	−0.159344
$113$	12.8500	1.20883	0.604415	−	0.796670i	$-0.293407\pi$
$113$	12.8500	1.20883	0.604415	+	0.796670i	$0.293407\pi$
$114$	−2.77275	−0.259692
$115$	9.01998	0.841118
$116$	0.713407	0.0662382
$117$	0	0
$118$	−11.0891	−1.02084
$119$	−2.31669	−0.212371
$120$	−7.78753	−0.710901
$121$	−0.694159	−0.0631054
$122$	12.9868	1.17577
$123$	−21.2406	−1.91520
$124$	2.38425	0.214112
$125$	5.93115	0.530498
$126$	−7.90575	−0.704300
$127$	14.7699	1.31061	0.655306	−	0.755363i	$-0.272540\pi$
$127$	14.7699	1.31061	0.655306	+	0.755363i	$0.272540\pi$
$128$	1.00000	0.0883883
$129$	−35.4750	−3.12340
$130$	0	0
$131$	−13.4579	−1.17582	−0.587910	−	0.808926i	$-0.700049\pi$
$131$	−13.4579	−1.17582	−0.587910	+	0.808926i	$0.700049\pi$
$132$	−8.90127	−0.774756
$133$	1.68633	0.146224
$134$	7.66589	0.662232
$135$	−13.1463	−1.13146
$136$	1.37380	0.117803
$137$	1.96856	0.168185	0.0840926	−	0.996458i	$-0.473201\pi$
$137$	1.96856	0.168185	0.0840926	+	0.996458i	$0.473201\pi$
$138$	−8.90485	−0.758031
$139$	−10.7343	−0.910471	−0.455235	−	0.890371i	$-0.650445\pi$
$139$	−10.7343	−0.910471	−0.455235	+	0.890371i	$0.650445\pi$
$140$	4.73623	0.400285
$141$	−2.82680	−0.238059
$142$	−4.86090	−0.407917
$143$	0	0
$144$	4.68813	0.390677
$145$	−2.00367	−0.166396
$146$	−9.26376	−0.766675
$147$	−11.5243	−0.950508
$148$	−1.39317	−0.114518
$149$	3.35331	0.274714	0.137357	−	0.990522i	$-0.456139\pi$
$149$	3.35331	0.274714	0.137357	+	0.990522i	$0.456139\pi$
$150$	8.00830	0.653875
$151$	0.0490519	0.00399179	0.00199589	−	0.999998i	$-0.499365\pi$
$151$	0.0490519	0.00399179	0.00199589	+	0.999998i	$0.499365\pi$
$152$	−1.00000	−0.0811107
$153$	6.44056	0.520688
$154$	5.41359	0.436240
$155$	−6.69640	−0.537868
$156$	0	0
$157$	5.40262	0.431176	0.215588	−	0.976484i	$-0.430833\pi$
$157$	5.40262	0.431176	0.215588	+	0.976484i	$0.430833\pi$
$158$	−10.1589	−0.808195
$159$	2.99705	0.237682
$160$	−2.80860	−0.222039
$161$	5.41577	0.426822
$162$	−1.08586	−0.0853128
$163$	−24.0973	−1.88745	−0.943723	−	0.330737i	$-0.892703\pi$
$163$	−24.0973	−1.88745	−0.943723	+	0.330737i	$0.892703\pi$
$164$	−7.66050	−0.598184
$165$	25.0001	1.94625
$166$	9.75396	0.757054
$167$	19.8776	1.53818	0.769088	−	0.639143i	$-0.220711\pi$
$167$	19.8776	1.53818	0.769088	+	0.639143i	$0.220711\pi$
$168$	−4.67578	−0.360744
$169$	0	0
$170$	−3.85846	−0.295930
$171$	−4.68813	−0.358510
$172$	−12.7942	−0.975547
$173$	12.2546	0.931699	0.465850	−	0.884864i	$-0.345749\pi$
$173$	12.2546	0.931699	0.465850	+	0.884864i	$0.345749\pi$
$174$	1.97810	0.149959
$175$	−4.87050	−0.368175
$176$	−3.21027	−0.241983
$177$	−30.7474	−2.31112
$178$	6.45411	0.483756
$179$	−20.6703	−1.54497	−0.772484	−	0.635034i	$-0.780986\pi$
$179$	−20.6703	−1.54497	−0.772484	+	0.635034i	$0.780986\pi$
$180$	−13.1671	−0.981414
$181$	−10.9365	−0.812904	−0.406452	−	0.913672i	$-0.633234\pi$
$181$	−10.9365	−0.812904	−0.406452	+	0.913672i	$0.633234\pi$
$182$	0	0
$183$	36.0090	2.66186
$184$	−3.21156	−0.236760
$185$	3.91285	0.287679
$186$	6.61093	0.484737
$187$	−4.41028	−0.322511
$188$	−1.01949	−0.0743542
$189$	−7.89330	−0.574153
$190$	2.80860	0.203757
$191$	−21.2959	−1.54092	−0.770459	−	0.637489i	$-0.779973\pi$
$191$	−21.2959	−1.54092	−0.770459	+	0.637489i	$0.779973\pi$
$192$	2.77275	0.200106
$193$	15.1123	1.08781	0.543903	−	0.839148i	$-0.316946\pi$
$193$	15.1123	1.08781	0.543903	+	0.839148i	$0.316946\pi$
$194$	−9.34495	−0.670928
$195$	0	0
$196$	−4.15628	−0.296877
$197$	3.29700	0.234902	0.117451	−	0.993079i	$-0.462528\pi$
$197$	3.29700	0.234902	0.117451	+	0.993079i	$0.462528\pi$
$198$	−15.0502	−1.06957
$199$	21.2085	1.50343	0.751715	−	0.659488i	$-0.229227\pi$
$199$	21.2085	1.50343	0.751715	+	0.659488i	$0.229227\pi$
$200$	2.88822	0.204228
$201$	21.2556	1.49925
$202$	−8.39872	−0.590932
$203$	−1.20304	−0.0844370
$204$	3.80921	0.266698
$205$	21.5153	1.50269
$206$	−17.1843	−1.19729
$207$	−15.0562	−1.04648
$208$	0	0
$209$	3.21027	0.222059
$210$	13.1324	0.906220
$211$	7.51794	0.517556	0.258778	−	0.965937i	$-0.416680\pi$
$211$	7.51794	0.517556	0.258778	+	0.965937i	$0.416680\pi$
$212$	1.08090	0.0742363
$213$	−13.4780	−0.923500
$214$	13.4013	0.916095
$215$	35.9337	2.45066
$216$	4.68075	0.318484
$217$	−4.02065	−0.272939
$218$	−10.5139	−0.712091
$219$	−25.6861	−1.73570
$220$	9.01636	0.607883
$221$	0	0
$222$	−3.86291	−0.259261
$223$	−10.9813	−0.735364	−0.367682	−	0.929952i	$-0.619849\pi$
$223$	−10.9813	−0.735364	−0.367682	+	0.929952i	$0.619849\pi$
$224$	−1.68633	−0.112673
$225$	13.5403	0.902688
$226$	12.8500	0.854771
$227$	0.822859	0.0546151	0.0273075	−	0.999627i	$-0.491307\pi$
$227$	0.822859	0.0546151	0.0273075	+	0.999627i	$0.491307\pi$
$228$	−2.77275	−0.183630
$229$	20.6513	1.36468	0.682339	−	0.731036i	$-0.260963\pi$
$229$	20.6513	1.36468	0.682339	+	0.731036i	$0.260963\pi$
$230$	9.01998	0.594760
$231$	15.0105	0.987620
$232$	0.713407	0.0468374
$233$	14.0657	0.921472	0.460736	−	0.887537i	$-0.347585\pi$
$233$	14.0657	0.921472	0.460736	+	0.887537i	$0.347585\pi$
$234$	0	0
$235$	2.86335	0.186784
$236$	−11.0891	−0.721842
$237$	−28.1679	−1.82970
$238$	−2.31669	−0.150169
$239$	6.36506	0.411722	0.205861	−	0.978581i	$-0.434001\pi$
$239$	6.36506	0.411722	0.205861	+	0.978581i	$0.434001\pi$
$240$	−7.78753	−0.502683
$241$	15.5870	1.00404	0.502022	−	0.864855i	$-0.332590\pi$
$241$	15.5870	1.00404	0.502022	+	0.864855i	$0.332590\pi$
$242$	−0.694159	−0.0446222
$243$	−17.0530	−1.09395
$244$	12.9868	0.831392
$245$	11.6733	0.745780
$246$	−21.2406	−1.35425
$247$	0	0
$248$	2.38425	0.151400
$249$	27.0453	1.71392
$250$	5.93115	0.375119
$251$	14.3393	0.905089	0.452544	−	0.891742i	$-0.350516\pi$
$251$	14.3393	0.905089	0.452544	+	0.891742i	$0.350516\pi$
$252$	−7.90575	−0.498015
$253$	10.3100	0.648183
$254$	14.7699	0.926743
$255$	−10.6985	−0.669968
$256$	1.00000	0.0625000
$257$	−0.357357	−0.0222913	−0.0111457	−	0.999938i	$-0.503548\pi$
$257$	−0.357357	−0.0222913	−0.0111457	+	0.999938i	$0.503548\pi$
$258$	−35.4750	−2.20858
$259$	2.34935	0.145982
$260$	0	0
$261$	3.34454	0.207022
$262$	−13.4579	−0.831431
$263$	21.7941	1.34388	0.671941	−	0.740605i	$-0.265461\pi$
$263$	21.7941	1.34388	0.671941	+	0.740605i	$0.265461\pi$
$264$	−8.90127	−0.547835
$265$	−3.03580	−0.186488
$266$	1.68633	0.103396
$267$	17.8956	1.09519
$268$	7.66589	0.468269
$269$	−0.152445	−0.00929472	−0.00464736	−	0.999989i	$-0.501479\pi$
$269$	−0.152445	−0.00929472	−0.00464736	+	0.999989i	$0.501479\pi$
$270$	−13.1463	−0.800060
$271$	−28.0000	−1.70088	−0.850441	−	0.526071i	$-0.823665\pi$
$271$	−28.0000	−1.70088	−0.850441	+	0.526071i	$0.823665\pi$
$272$	1.37380	0.0832990
$273$	0	0
$274$	1.96856	0.118925
$275$	−9.27196	−0.559120
$276$	−8.90485	−0.536009
$277$	8.90957	0.535324	0.267662	−	0.963513i	$-0.413749\pi$
$277$	8.90957	0.535324	0.267662	+	0.963513i	$0.413749\pi$
$278$	−10.7343	−0.643800
$279$	11.1777	0.669190
$280$	4.73623	0.283044
$281$	15.1811	0.905628	0.452814	−	0.891605i	$-0.350420\pi$
$281$	15.1811	0.905628	0.452814	+	0.891605i	$0.350420\pi$
$282$	−2.82680	−0.168333
$283$	−31.5990	−1.87837	−0.939183	−	0.343416i	$-0.888416\pi$
$283$	−31.5990	−1.87837	−0.939183	+	0.343416i	$0.888416\pi$
$284$	−4.86090	−0.288441
$285$	7.78753	0.461293
$286$	0	0
$287$	12.9182	0.762535
$288$	4.68813	0.276250
$289$	−15.1127	−0.888980
$290$	−2.00367	−0.117660
$291$	−25.9112	−1.51894
$292$	−9.26376	−0.542121
$293$	−31.3905	−1.83385	−0.916927	−	0.399055i	$-0.869338\pi$
$293$	−31.3905	−1.83385	−0.916927	+	0.399055i	$0.869338\pi$
$294$	−11.5243	−0.672111
$295$	31.1449	1.81333
$296$	−1.39317	−0.0809763
$297$	−15.0265	−0.871924
$298$	3.35331	0.194252
$299$	0	0
$300$	8.00830	0.462359
$301$	21.5753	1.24358
$302$	0.0490519	0.00282262
$303$	−23.2875	−1.33783
$304$	−1.00000	−0.0573539
$305$	−36.4746	−2.08853
$306$	6.44056	0.368182
$307$	−15.5456	−0.887236	−0.443618	−	0.896216i	$-0.646305\pi$
$307$	−15.5456	−0.887236	−0.443618	+	0.896216i	$0.646305\pi$
$308$	5.41359	0.308468
$309$	−47.6478	−2.71059
$310$	−6.69640	−0.380330
$311$	5.78974	0.328306	0.164153	−	0.986435i	$-0.447511\pi$
$311$	5.78974	0.328306	0.164153	+	0.986435i	$0.447511\pi$
$312$	0	0
$313$	−12.2365	−0.691646	−0.345823	−	0.938300i	$-0.612400\pi$
$313$	−12.2365	−0.691646	−0.345823	+	0.938300i	$0.612400\pi$
$314$	5.40262	0.304888
$315$	22.2041	1.25106
$316$	−10.1589	−0.571480
$317$	12.8275	0.720465	0.360232	−	0.932863i	$-0.382697\pi$
$317$	12.8275	0.720465	0.360232	+	0.932863i	$0.382697\pi$
$318$	2.99705	0.168066
$319$	−2.29023	−0.128228
$320$	−2.80860	−0.157005
$321$	37.1584	2.07398
$322$	5.41577	0.301809
$323$	−1.37380	−0.0764404
$324$	−1.08586	−0.0603253
$325$	0	0
$326$	−24.0973	−1.33463
$327$	−29.1524	−1.61213
$328$	−7.66050	−0.422980
$329$	1.71921	0.0947830
$330$	25.0001	1.37621
$331$	−1.53705	−0.0844839	−0.0422420	−	0.999107i	$-0.513450\pi$
$331$	−1.53705	−0.0844839	−0.0422420	+	0.999107i	$0.513450\pi$
$332$	9.75396	0.535318
$333$	−6.53135	−0.357916
$334$	19.8776	1.08765
$335$	−21.5304	−1.17633
$336$	−4.67578	−0.255085
$337$	−28.6821	−1.56241	−0.781206	−	0.624273i	$-0.785395\pi$
$337$	−28.6821	−1.56241	−0.781206	+	0.624273i	$0.785395\pi$
$338$	0	0
$339$	35.6299	1.93515
$340$	−3.85846	−0.209254
$341$	−7.65410	−0.414493
$342$	−4.68813	−0.253505
$343$	18.8132	1.01582
$344$	−12.7942	−0.689816
$345$	25.0101	1.34650
$346$	12.2546	0.658811
$347$	−23.2771	−1.24958	−0.624789	−	0.780793i	$-0.714815\pi$
$347$	−23.2771	−1.24958	−0.624789	+	0.780793i	$0.714815\pi$
$348$	1.97810	0.106037
$349$	26.0360	1.39367	0.696837	−	0.717230i	$-0.254590\pi$
$349$	26.0360	1.39367	0.696837	+	0.717230i	$0.254590\pi$
$350$	−4.87050	−0.260339
$351$	0	0
$352$	−3.21027	−0.171108
$353$	12.2732	0.653236	0.326618	−	0.945156i	$-0.394091\pi$
$353$	12.2732	0.653236	0.326618	+	0.945156i	$0.394091\pi$
$354$	−30.7474	−1.63421
$355$	13.6523	0.724589
$356$	6.45411	0.342067
$357$	−6.42360	−0.339973
$358$	−20.6703	−1.09246
$359$	−25.7095	−1.35690	−0.678449	−	0.734648i	$-0.737347\pi$
$359$	−25.7095	−1.35690	−0.678449	+	0.734648i	$0.737347\pi$
$360$	−13.1671	−0.693965
$361$	1.00000	0.0526316
$362$	−10.9365	−0.574810
$363$	−1.92473	−0.101022
$364$	0	0
$365$	26.0182	1.36185
$366$	36.0090	1.88222
$367$	5.23093	0.273052	0.136526	−	0.990636i	$-0.456406\pi$
$367$	5.23093	0.273052	0.136526	+	0.990636i	$0.456406\pi$
$368$	−3.21156	−0.167414
$369$	−35.9134	−1.86958
$370$	3.91285	0.203419
$371$	−1.82275	−0.0946327
$372$	6.61093	0.342761
$373$	30.2581	1.56670	0.783352	−	0.621579i	$-0.213508\pi$
$373$	30.2581	1.56670	0.783352	+	0.621579i	$0.213508\pi$
$374$	−4.41028	−0.228050
$375$	16.4456	0.849245
$376$	−1.01949	−0.0525764
$377$	0	0
$378$	−7.89330	−0.405988
$379$	21.8147	1.12055	0.560273	−	0.828308i	$-0.310696\pi$
$379$	21.8147	1.12055	0.560273	+	0.828308i	$0.310696\pi$
$380$	2.80860	0.144078
$381$	40.9531	2.09809
$382$	−21.2959	−1.08959
$383$	30.3085	1.54869	0.774345	−	0.632764i	$-0.218079\pi$
$383$	30.3085	1.54869	0.774345	+	0.632764i	$0.218079\pi$
$384$	2.77275	0.141496
$385$	−15.2046	−0.774898
$386$	15.1123	0.769195
$387$	−59.9807	−3.04899
$388$	−9.34495	−0.474418
$389$	8.43202	0.427520	0.213760	−	0.976886i	$-0.431429\pi$
$389$	8.43202	0.427520	0.213760	+	0.976886i	$0.431429\pi$
$390$	0	0
$391$	−4.41205	−0.223127
$392$	−4.15628	−0.209924
$393$	−37.3153	−1.88231
$394$	3.29700	0.166101
$395$	28.5321	1.43561
$396$	−15.0502	−0.756299
$397$	6.11388	0.306847	0.153423	−	0.988161i	$-0.450970\pi$
$397$	6.11388	0.306847	0.153423	+	0.988161i	$0.450970\pi$
$398$	21.2085	1.06309
$399$	4.67578	0.234082
$400$	2.88822	0.144411
$401$	4.62043	0.230733	0.115367	−	0.993323i	$-0.463196\pi$
$401$	4.62043	0.230733	0.115367	+	0.993323i	$0.463196\pi$
$402$	21.2556	1.06013
$403$	0	0
$404$	−8.39872	−0.417852
$405$	3.04973	0.151542
$406$	−1.20304	−0.0597060
$407$	4.47245	0.221691
$408$	3.80921	0.188584
$409$	−16.9426	−0.837757	−0.418878	−	0.908042i	$-0.637577\pi$
$409$	−16.9426	−0.837757	−0.418878	+	0.908042i	$0.637577\pi$
$410$	21.5153	1.06256
$411$	5.45831	0.269239
$412$	−17.1843	−0.846611
$413$	18.7000	0.920167
$414$	−15.0562	−0.739972
$415$	−27.3949	−1.34476
$416$	0	0
$417$	−29.7635	−1.45752
$418$	3.21027	0.157019
$419$	−6.90928	−0.337541	−0.168770	−	0.985655i	$-0.553980\pi$
$419$	−6.90928	−0.337541	−0.168770	+	0.985655i	$0.553980\pi$
$420$	13.1324	0.640795
$421$	8.65379	0.421760	0.210880	−	0.977512i	$-0.432367\pi$
$421$	8.65379	0.421760	0.210880	+	0.977512i	$0.432367\pi$
$422$	7.51794	0.365967
$423$	−4.77952	−0.232388
$424$	1.08090	0.0524930
$425$	3.96784	0.192468
$426$	−13.4780	−0.653013
$427$	−21.9000	−1.05982
$428$	13.4013	0.647777
$429$	0	0
$430$	35.9337	1.73288
$431$	9.89779	0.476760	0.238380	−	0.971172i	$-0.423384\pi$
$431$	9.89779	0.476760	0.238380	+	0.971172i	$0.423384\pi$
$432$	4.68075	0.225202
$433$	13.5225	0.649850	0.324925	−	0.945740i	$-0.394661\pi$
$433$	13.5225	0.649850	0.324925	+	0.945740i	$0.394661\pi$
$434$	−4.02065	−0.192997
$435$	−5.55568	−0.266374
$436$	−10.5139	−0.503524
$437$	3.21156	0.153630
$438$	−25.6861	−1.22733
$439$	−23.5131	−1.12222	−0.561109	−	0.827742i	$-0.689625\pi$
$439$	−23.5131	−1.12222	−0.561109	+	0.827742i	$0.689625\pi$
$440$	9.01636	0.429838
$441$	−19.4851	−0.927864
$442$	0	0
$443$	−14.3234	−0.680525	−0.340263	−	0.940330i	$-0.610516\pi$
$443$	−14.3234	−0.680525	−0.340263	+	0.940330i	$0.610516\pi$
$444$	−3.86291	−0.183325
$445$	−18.1270	−0.859302
$446$	−10.9813	−0.519981
$447$	9.29788	0.439775
$448$	−1.68633	−0.0796718
$449$	3.37875	0.159453	0.0797265	−	0.996817i	$-0.474595\pi$
$449$	3.37875	0.159453	0.0797265	+	0.996817i	$0.474595\pi$
$450$	13.5403	0.638297
$451$	24.5923	1.15801
$452$	12.8500	0.604415
$453$	0.136009	0.00639024
$454$	0.822859	0.0386187
$455$	0	0
$456$	−2.77275	−0.129846
$457$	12.8600	0.601564	0.300782	−	0.953693i	$-0.402752\pi$
$457$	12.8600	0.601564	0.300782	+	0.953693i	$0.402752\pi$
$458$	20.6513	0.964973
$459$	6.43042	0.300146
$460$	9.01998	0.420559
$461$	−9.58002	−0.446186	−0.223093	−	0.974797i	$-0.571615\pi$
$461$	−9.58002	−0.446186	−0.223093	+	0.974797i	$0.571615\pi$
$462$	15.0105	0.698353
$463$	13.7638	0.639656	0.319828	−	0.947476i	$-0.396375\pi$
$463$	13.7638	0.639656	0.319828	+	0.947476i	$0.396375\pi$
$464$	0.713407	0.0331191
$465$	−18.5674	−0.861044
$466$	14.0657	0.651579
$467$	27.0616	1.25226	0.626130	−	0.779718i	$-0.284638\pi$
$467$	27.0616	1.25226	0.626130	+	0.779718i	$0.284638\pi$
$468$	0	0
$469$	−12.9273	−0.596926
$470$	2.86335	0.132076
$471$	14.9801	0.690247
$472$	−11.0891	−0.510419
$473$	41.0728	1.88853
$474$	−28.1679	−1.29380
$475$	−2.88822	−0.132521
$476$	−2.31669	−0.106185
$477$	5.06738	0.232019
$478$	6.36506	0.291131
$479$	42.4758	1.94077	0.970385	−	0.241562i	$-0.0776596\pi$
$479$	42.4758	1.94077	0.970385	+	0.241562i	$0.0776596\pi$
$480$	−7.78753	−0.355450
$481$	0	0
$482$	15.5870	0.709967
$483$	15.0166	0.683277
$484$	−0.694159	−0.0315527
$485$	26.2462	1.19178
$486$	−17.0530	−0.773541
$487$	−32.0478	−1.45222	−0.726112	−	0.687577i	$-0.758674\pi$
$487$	−32.0478	−1.45222	−0.726112	+	0.687577i	$0.758674\pi$
$488$	12.9868	0.587883
$489$	−66.8157	−3.02151
$490$	11.6733	0.527346
$491$	−1.05374	−0.0475545	−0.0237773	−	0.999717i	$-0.507569\pi$
$491$	−1.05374	−0.0475545	−0.0237773	+	0.999717i	$0.507569\pi$
$492$	−21.2406	−0.957601
$493$	0.980080	0.0441406
$494$	0	0
$495$	42.2698	1.89989
$496$	2.38425	0.107056
$497$	8.19710	0.367690
$498$	27.0453	1.21193
$499$	22.0474	0.986976	0.493488	−	0.869753i	$-0.335722\pi$
$499$	22.0474	0.986976	0.493488	+	0.869753i	$0.335722\pi$
$500$	5.93115	0.265249
$501$	55.1156	2.46238
$502$	14.3393	0.639994
$503$	−30.1549	−1.34454	−0.672271	−	0.740305i	$-0.734681\pi$
$503$	−30.1549	−1.34454	−0.672271	+	0.740305i	$0.734681\pi$
$504$	−7.90575	−0.352150
$505$	23.5886	1.04968
$506$	10.3100	0.458335
$507$	0	0
$508$	14.7699	0.655306
$509$	−18.5708	−0.823135	−0.411567	−	0.911379i	$-0.635019\pi$
$509$	−18.5708	−0.823135	−0.411567	+	0.911379i	$0.635019\pi$
$510$	−10.6985	−0.473739
$511$	15.6218	0.691068
$512$	1.00000	0.0441942
$513$	−4.68075	−0.206660
$514$	−0.357357	−0.0157623
$515$	48.2638	2.12676
$516$	−35.4750	−1.56170
$517$	3.27285	0.143940
$518$	2.34935	0.103225
$519$	33.9789	1.49151
$520$	0	0
$521$	−24.8430	−1.08839	−0.544195	−	0.838958i	$-0.683165\pi$
$521$	−24.8430	−1.08839	−0.544195	+	0.838958i	$0.683165\pi$
$522$	3.34454	0.146387
$523$	23.3001	1.01884	0.509422	−	0.860517i	$-0.329859\pi$
$523$	23.3001	1.01884	0.509422	+	0.860517i	$0.329859\pi$
$524$	−13.4579	−0.587910
$525$	−13.5047	−0.589392
$526$	21.7941	0.950267
$527$	3.27549	0.142683
$528$	−8.90127	−0.387378
$529$	−12.6859	−0.551559
$530$	−3.03580	−0.131867
$531$	−51.9873	−2.25606
$532$	1.68633	0.0731119
$533$	0	0
$534$	17.8956	0.774419
$535$	−37.6389	−1.62727
$536$	7.66589	0.331116
$537$	−57.3134	−2.47326
$538$	−0.152445	−0.00657236
$539$	13.3428	0.574714
$540$	−13.1463	−0.565728
$541$	−19.7964	−0.851114	−0.425557	−	0.904932i	$-0.639922\pi$
$541$	−19.7964	−0.851114	−0.425557	+	0.904932i	$0.639922\pi$
$542$	−28.0000	−1.20270
$543$	−30.3242	−1.30133
$544$	1.37380	0.0589013
$545$	29.5293	1.26490
$546$	0	0
$547$	5.36539	0.229408	0.114704	−	0.993400i	$-0.463408\pi$
$547$	5.36539	0.229408	0.114704	+	0.993400i	$0.463408\pi$
$548$	1.96856	0.0840926
$549$	60.8836	2.59845
$550$	−9.27196	−0.395358
$551$	−0.713407	−0.0303921
$552$	−8.90485	−0.379016
$553$	17.1312	0.728494
$554$	8.90957	0.378532
$555$	10.8493	0.460529
$556$	−10.7343	−0.455235
$557$	34.0560	1.44300	0.721500	−	0.692414i	$-0.243453\pi$
$557$	34.0560	1.44300	0.721500	+	0.692414i	$0.243453\pi$
$558$	11.1777	0.473189
$559$	0	0
$560$	4.73623	0.200142
$561$	−12.2286	−0.516291
$562$	15.1811	0.640376
$563$	41.5003	1.74903	0.874515	−	0.484998i	$-0.161180\pi$
$563$	41.5003	1.74903	0.874515	+	0.484998i	$0.161180\pi$
$564$	−2.82680	−0.119030
$565$	−36.0906	−1.51834
$566$	−31.5990	−1.32821
$567$	1.83112	0.0768996
$568$	−4.86090	−0.203959
$569$	17.7514	0.744177	0.372088	−	0.928197i	$-0.378642\pi$
$569$	17.7514	0.744177	0.372088	+	0.928197i	$0.378642\pi$
$570$	7.78753	0.326184
$571$	−30.1369	−1.26119	−0.630595	−	0.776112i	$-0.717189\pi$
$571$	−30.1369	−1.26119	−0.630595	+	0.776112i	$0.717189\pi$
$572$	0	0
$573$	−59.0482	−2.46677
$574$	12.9182	0.539194
$575$	−9.27569	−0.386823
$576$	4.68813	0.195339
$577$	−32.4954	−1.35280	−0.676401	−	0.736534i	$-0.736461\pi$
$577$	−32.4954	−1.35280	−0.676401	+	0.736534i	$0.736461\pi$
$578$	−15.1127	−0.628604
$579$	41.9025	1.74141
$580$	−2.00367	−0.0831980
$581$	−16.4484	−0.682396
$582$	−25.9112	−1.07405
$583$	−3.46997	−0.143712
$584$	−9.26376	−0.383337
$585$	0	0
$586$	−31.3905	−1.29673
$587$	−36.0366	−1.48739	−0.743695	−	0.668519i	$-0.766929\pi$
$587$	−36.0366	−1.48739	−0.743695	+	0.668519i	$0.766929\pi$
$588$	−11.5243	−0.475254
$589$	−2.38425	−0.0982414
$590$	31.1449	1.28222
$591$	9.14176	0.376042
$592$	−1.39317	−0.0572589
$593$	−14.2371	−0.584646	−0.292323	−	0.956320i	$-0.594428\pi$
$593$	−14.2371	−0.584646	−0.292323	+	0.956320i	$0.594428\pi$
$594$	−15.0265	−0.616543
$595$	6.50665	0.266747
$596$	3.35331	0.137357
$597$	58.8058	2.40676
$598$	0	0
$599$	−1.25939	−0.0514572	−0.0257286	−	0.999669i	$-0.508191\pi$
$599$	−1.25939	−0.0514572	−0.0257286	+	0.999669i	$0.508191\pi$
$600$	8.00830	0.326937
$601$	23.1555	0.944534	0.472267	−	0.881456i	$-0.343436\pi$
$601$	23.1555	0.944534	0.472267	+	0.881456i	$0.343436\pi$
$602$	21.5753	0.879342
$603$	35.9387	1.46354
$604$	0.0490519	0.00199589
$605$	1.94961	0.0792630
$606$	−23.2875	−0.945991
$607$	15.9039	0.645521	0.322761	−	0.946481i	$-0.395389\pi$
$607$	15.9039	0.645521	0.322761	+	0.946481i	$0.395389\pi$
$608$	−1.00000	−0.0405554
$609$	−3.33573	−0.135171
$610$	−36.4746	−1.47681
$611$	0	0
$612$	6.44056	0.260344
$613$	−18.7164	−0.755949	−0.377974	−	0.925816i	$-0.623379\pi$
$613$	−18.7164	−0.755949	−0.377974	+	0.925816i	$0.623379\pi$
$614$	−15.5456	−0.627371
$615$	59.6564	2.40558
$616$	5.41359	0.218120
$617$	−14.0571	−0.565919	−0.282960	−	0.959132i	$-0.591316\pi$
$617$	−14.0571	−0.565919	−0.282960	+	0.959132i	$0.591316\pi$
$618$	−47.6478	−1.91667
$619$	−11.0937	−0.445892	−0.222946	−	0.974831i	$-0.571567\pi$
$619$	−11.0937	−0.445892	−0.222946	+	0.974831i	$0.571567\pi$
$620$	−6.69640	−0.268934
$621$	−15.0325	−0.603234
$622$	5.78974	0.232147
$623$	−10.8838	−0.436050
$624$	0	0
$625$	−31.0993	−1.24397
$626$	−12.2365	−0.489068
$627$	8.90127	0.355482
$628$	5.40262	0.215588
$629$	−1.91394	−0.0763138
$630$	22.2041	0.884631
$631$	13.7826	0.548675	0.274338	−	0.961633i	$-0.411541\pi$
$631$	13.7826	0.548675	0.274338	+	0.961633i	$0.411541\pi$
$632$	−10.1589	−0.404098
$633$	20.8453	0.828528
$634$	12.8275	0.509445
$635$	−41.4826	−1.64619
$636$	2.99705	0.118841
$637$	0	0
$638$	−2.29023	−0.0906710
$639$	−22.7885	−0.901499
$640$	−2.80860	−0.111020
$641$	21.4489	0.847180	0.423590	−	0.905854i	$-0.360770\pi$
$641$	21.4489	0.847180	0.423590	+	0.905854i	$0.360770\pi$
$642$	37.1584	1.46653
$643$	45.2486	1.78443	0.892214	−	0.451612i	$-0.149151\pi$
$643$	45.2486	1.78443	0.892214	+	0.451612i	$0.149151\pi$
$644$	5.41577	0.213411
$645$	99.6350	3.92313
$646$	−1.37380	−0.0540515
$647$	2.02612	0.0796549	0.0398275	−	0.999207i	$-0.487319\pi$
$647$	2.02612	0.0796549	0.0398275	+	0.999207i	$0.487319\pi$
$648$	−1.08586	−0.0426564
$649$	35.5992	1.39739
$650$	0	0
$651$	−11.1482	−0.436934
$652$	−24.0973	−0.943723
$653$	−18.6941	−0.731555	−0.365778	−	0.930702i	$-0.619197\pi$
$653$	−18.6941	−0.731555	−0.365778	+	0.930702i	$0.619197\pi$
$654$	−29.1524	−1.13995
$655$	37.7978	1.47688
$656$	−7.66050	−0.299092
$657$	−43.4297	−1.69435
$658$	1.71921	0.0670217
$659$	1.36146	0.0530348	0.0265174	−	0.999648i	$-0.491558\pi$
$659$	1.36146	0.0530348	0.0265174	+	0.999648i	$0.491558\pi$
$660$	25.0001	0.973127
$661$	3.23169	0.125698	0.0628491	−	0.998023i	$-0.479981\pi$
$661$	3.23169	0.125698	0.0628491	+	0.998023i	$0.479981\pi$
$662$	−1.53705	−0.0597391
$663$	0	0
$664$	9.75396	0.378527
$665$	−4.73623	−0.183663
$666$	−6.53135	−0.253085
$667$	−2.29115	−0.0887137
$668$	19.8776	0.769088
$669$	−30.4484	−1.17721
$670$	−21.5304	−0.831792
$671$	−41.6910	−1.60946
$672$	−4.67578	−0.180372
$673$	47.7907	1.84220	0.921098	−	0.389331i	$-0.127294\pi$
$673$	47.7907	1.84220	0.921098	+	0.389331i	$0.127294\pi$
$674$	−28.6821	−1.10479
$675$	13.5190	0.520347
$676$	0	0
$677$	−3.24195	−0.124598	−0.0622991	−	0.998058i	$-0.519843\pi$
$677$	−3.24195	−0.124598	−0.0622991	+	0.998058i	$0.519843\pi$
$678$	35.6299	1.36836
$679$	15.7587	0.604764
$680$	−3.85846	−0.147965
$681$	2.28158	0.0874303
$682$	−7.65410	−0.293090
$683$	20.5963	0.788095	0.394048	−	0.919090i	$-0.371074\pi$
$683$	20.5963	0.788095	0.394048	+	0.919090i	$0.371074\pi$
$684$	−4.68813	−0.179255
$685$	−5.52888	−0.211248
$686$	18.8132	0.718292
$687$	57.2609	2.18464
$688$	−12.7942	−0.487774
$689$	0	0
$690$	25.0101	0.952120
$691$	38.1088	1.44973	0.724863	−	0.688893i	$-0.241903\pi$
$691$	38.1088	1.44973	0.724863	+	0.688893i	$0.241903\pi$
$692$	12.2546	0.465850
$693$	25.3796	0.964091
$694$	−23.2771	−0.883585
$695$	30.1483	1.14359
$696$	1.97810	0.0749795
$697$	−10.5240	−0.398625
$698$	26.0360	0.985476
$699$	39.0005	1.47514
$700$	−4.87050	−0.184088
$701$	−14.5755	−0.550508	−0.275254	−	0.961372i	$-0.588762\pi$
$701$	−14.5755	−0.550508	−0.275254	+	0.961372i	$0.588762\pi$
$702$	0	0
$703$	1.39317	0.0525444
$704$	−3.21027	−0.120992
$705$	7.93934	0.299013
$706$	12.2732	0.461908
$707$	14.1630	0.532656
$708$	−30.7474	−1.15556
$709$	22.6897	0.852129	0.426065	−	0.904693i	$-0.359900\pi$
$709$	22.6897	0.852129	0.426065	+	0.904693i	$0.359900\pi$
$710$	13.6523	0.512362
$711$	−47.6260	−1.78611
$712$	6.45411	0.241878
$713$	−7.65717	−0.286763
$714$	−6.42360	−0.240397
$715$	0	0
$716$	−20.6703	−0.772484
$717$	17.6487	0.659103
$718$	−25.7095	−0.959472
$719$	22.3986	0.835329	0.417664	−	0.908601i	$-0.362849\pi$
$719$	22.3986	0.835329	0.417664	+	0.908601i	$0.362849\pi$
$720$	−13.1671	−0.490707
$721$	28.9785	1.07922
$722$	1.00000	0.0372161
$723$	43.2187	1.60732
$724$	−10.9365	−0.406452
$725$	2.06047	0.0765241
$726$	−1.92473	−0.0714333
$727$	30.4743	1.13023	0.565115	−	0.825012i	$-0.308832\pi$
$727$	30.4743	1.13023	0.565115	+	0.825012i	$0.308832\pi$
$728$	0	0
$729$	−44.0262	−1.63060
$730$	26.0182	0.962976
$731$	−17.5767	−0.650097
$732$	36.0090	1.33093
$733$	−17.5436	−0.647987	−0.323993	−	0.946059i	$-0.605026\pi$
$733$	−17.5436	−0.647987	−0.323993	+	0.946059i	$0.605026\pi$
$734$	5.23093	0.193077
$735$	32.3671	1.19388
$736$	−3.21156	−0.118380
$737$	−24.6096	−0.906506
$738$	−35.9134	−1.32199
$739$	−11.6092	−0.427050	−0.213525	−	0.976938i	$-0.568494\pi$
$739$	−11.6092	−0.427050	−0.213525	+	0.976938i	$0.568494\pi$
$740$	3.91285	0.143839
$741$	0	0
$742$	−1.82275	−0.0669154
$743$	−16.3983	−0.601596	−0.300798	−	0.953688i	$-0.597253\pi$
$743$	−16.3983	−0.601596	−0.300798	+	0.953688i	$0.597253\pi$
$744$	6.61093	0.242368
$745$	−9.41810	−0.345053
$746$	30.2581	1.10783
$747$	45.7278	1.67309
$748$	−4.41028	−0.161256
$749$	−22.5991	−0.825753
$750$	16.4456	0.600507
$751$	27.0219	0.986042	0.493021	−	0.870017i	$-0.335892\pi$
$751$	27.0219	0.986042	0.493021	+	0.870017i	$0.335892\pi$
$752$	−1.01949	−0.0371771
$753$	39.7593	1.44891
$754$	0	0
$755$	−0.137767	−0.00501386
$756$	−7.89330	−0.287077
$757$	−29.5126	−1.07265	−0.536327	−	0.844010i	$-0.680189\pi$
$757$	−29.5126	−1.07265	−0.536327	+	0.844010i	$0.680189\pi$
$758$	21.8147	0.792346
$759$	28.5870	1.03764
$760$	2.80860	0.101879
$761$	6.06079	0.219704	0.109852	−	0.993948i	$-0.464962\pi$
$761$	6.06079	0.219704	0.109852	+	0.993948i	$0.464962\pi$
$762$	40.9531	1.48357
$763$	17.7299	0.641867
$764$	−21.2959	−0.770459
$765$	−18.0889	−0.654007
$766$	30.3085	1.09509
$767$	0	0
$768$	2.77275	0.100053
$769$	−33.4706	−1.20698	−0.603491	−	0.797370i	$-0.706224\pi$
$769$	−33.4706	−1.20698	−0.603491	+	0.797370i	$0.706224\pi$
$770$	−15.2046	−0.547936
$771$	−0.990861	−0.0356850
$772$	15.1123	0.543903
$773$	6.59677	0.237269	0.118635	−	0.992938i	$-0.462148\pi$
$773$	6.59677	0.237269	0.118635	+	0.992938i	$0.462148\pi$
$774$	−59.9807	−2.15596
$775$	6.88624	0.247361
$776$	−9.34495	−0.335464
$777$	6.51415	0.233694
$778$	8.43202	0.302303
$779$	7.66050	0.274466
$780$	0	0
$781$	15.6048	0.558384
$782$	−4.41205	−0.157775
$783$	3.33928	0.119336
$784$	−4.15628	−0.148438
$785$	−15.1738	−0.541576
$786$	−37.3153	−1.33099
$787$	−49.9091	−1.77907	−0.889533	−	0.456870i	$-0.848970\pi$
$787$	−49.9091	−1.77907	−0.889533	+	0.456870i	$0.848970\pi$
$788$	3.29700	0.117451
$789$	60.4295	2.15135
$790$	28.5321	1.01513
$791$	−21.6695	−0.770477
$792$	−15.0502	−0.534784
$793$	0	0
$794$	6.11388	0.216974
$795$	−8.41752	−0.298539
$796$	21.2085	0.751715
$797$	−46.9571	−1.66330	−0.831652	−	0.555297i	$-0.812605\pi$
$797$	−46.9571	−1.66330	−0.831652	+	0.555297i	$0.812605\pi$
$798$	4.67578	0.165521
$799$	−1.40058	−0.0495491
$800$	2.88822	0.102114
$801$	30.2577	1.06910
$802$	4.62043	0.163153
$803$	29.7392	1.04947
$804$	21.2556	0.749627
$805$	−15.2107	−0.536107
$806$	0	0
$807$	−0.422691	−0.0148794
$808$	−8.39872	−0.295466
$809$	42.1561	1.48213	0.741064	−	0.671434i	$-0.234321\pi$
$809$	42.1561	1.48213	0.741064	+	0.671434i	$0.234321\pi$
$810$	3.04973	0.107157
$811$	−1.32060	−0.0463725	−0.0231862	−	0.999731i	$-0.507381\pi$
$811$	−1.32060	−0.0463725	−0.0231862	+	0.999731i	$0.507381\pi$
$812$	−1.20304	−0.0422185
$813$	−77.6370	−2.72285
$814$	4.47245	0.156759
$815$	67.6796	2.37071
$816$	3.80921	0.133349
$817$	12.7942	0.447612
$818$	−16.9426	−0.592383
$819$	0	0
$820$	21.5153	0.751345
$821$	−4.45564	−0.155503	−0.0777515	−	0.996973i	$-0.524774\pi$
$821$	−4.45564	−0.155503	−0.0777515	+	0.996973i	$0.524774\pi$
$822$	5.45831	0.190380
$823$	20.7566	0.723530	0.361765	−	0.932269i	$-0.382174\pi$
$823$	20.7566	0.723530	0.361765	+	0.932269i	$0.382174\pi$
$824$	−17.1843	−0.598644
$825$	−25.7088	−0.895066
$826$	18.7000	0.650657
$827$	12.8148	0.445615	0.222808	−	0.974862i	$-0.428478\pi$
$827$	12.8148	0.445615	0.222808	+	0.974862i	$0.428478\pi$
$828$	−15.0562	−0.523239
$829$	18.4413	0.640494	0.320247	−	0.947334i	$-0.396234\pi$
$829$	18.4413	0.640494	0.320247	+	0.947334i	$0.396234\pi$
$830$	−27.3949	−0.950892
$831$	24.7040	0.856972
$832$	0	0
$833$	−5.70990	−0.197836
$834$	−29.7635	−1.03062
$835$	−55.8282	−1.93201
$836$	3.21027	0.111030
$837$	11.1601	0.385749
$838$	−6.90928	−0.238677
$839$	41.1910	1.42207	0.711035	−	0.703157i	$-0.248227\pi$
$839$	41.1910	1.42207	0.711035	+	0.703157i	$0.248227\pi$
$840$	13.1324	0.453110
$841$	−28.4911	−0.982450
$842$	8.65379	0.298229
$843$	42.0933	1.44977
$844$	7.51794	0.258778
$845$	0	0
$846$	−4.77952	−0.164323
$847$	1.17058	0.0402218
$848$	1.08090	0.0371182
$849$	−87.6161	−3.00698
$850$	3.96784	0.136096
$851$	4.47425	0.153375
$852$	−13.4780	−0.461750
$853$	−30.0868	−1.03015	−0.515077	−	0.857144i	$-0.672237\pi$
$853$	−30.0868	−1.03015	−0.515077	+	0.857144i	$0.672237\pi$
$854$	−21.9000	−0.749403
$855$	13.1671	0.450304
$856$	13.4013	0.458047
$857$	40.0397	1.36773	0.683864	−	0.729609i	$-0.260298\pi$
$857$	40.0397	1.36773	0.683864	+	0.729609i	$0.260298\pi$
$858$	0	0
$859$	−13.8843	−0.473728	−0.236864	−	0.971543i	$-0.576120\pi$
$859$	−13.8843	−0.473728	−0.236864	+	0.971543i	$0.576120\pi$
$860$	35.9337	1.22533
$861$	35.8188	1.22070
$862$	9.89779	0.337120
$863$	10.4812	0.356785	0.178393	−	0.983959i	$-0.442910\pi$
$863$	10.4812	0.356785	0.178393	+	0.983959i	$0.442910\pi$
$864$	4.68075	0.159242
$865$	−34.4182	−1.17025
$866$	13.5225	0.459513
$867$	−41.9036	−1.42312
$868$	−4.02065	−0.136470
$869$	32.6127	1.10631
$870$	−5.55568	−0.188355
$871$	0	0
$872$	−10.5139	−0.356045
$873$	−43.8103	−1.48275
$874$	3.21156	0.108633
$875$	−10.0019	−0.338126
$876$	−25.6861	−0.867852
$877$	−55.2951	−1.86718	−0.933592	−	0.358337i	$-0.883344\pi$
$877$	−55.2951	−1.86718	−0.933592	+	0.358337i	$0.883344\pi$
$878$	−23.5131	−0.793528
$879$	−87.0380	−2.93572
$880$	9.01636	0.303941
$881$	−28.5299	−0.961197	−0.480598	−	0.876941i	$-0.659580\pi$
$881$	−28.5299	−0.961197	−0.480598	+	0.876941i	$0.659580\pi$
$882$	−19.4851	−0.656099
$883$	0.115641	0.00389163	0.00194582	−	0.999998i	$-0.499381\pi$
$883$	0.115641	0.00389163	0.00194582	+	0.999998i	$0.499381\pi$
$884$	0	0
$885$	86.3570	2.90286
$886$	−14.3234	−0.481204
$887$	−10.0121	−0.336173	−0.168087	−	0.985772i	$-0.553759\pi$
$887$	−10.0121	−0.336173	−0.168087	+	0.985772i	$0.553759\pi$
$888$	−3.86291	−0.129631
$889$	−24.9069	−0.835351
$890$	−18.1270	−0.607618
$891$	3.48589	0.116782
$892$	−10.9813	−0.367682
$893$	1.01949	0.0341161
$894$	9.29788	0.310968
$895$	58.0545	1.94055
$896$	−1.68633	−0.0563365
$897$	0	0
$898$	3.37875	0.112750
$899$	1.70094	0.0567296
$900$	13.5403	0.451344
$901$	1.48494	0.0494705
$902$	24.5923	0.818833
$903$	59.8228	1.99078
$904$	12.8500	0.427386
$905$	30.7162	1.02104
$906$	0.136009	0.00451858
$907$	35.7870	1.18829	0.594144	−	0.804359i	$-0.297491\pi$
$907$	35.7870	1.18829	0.594144	+	0.804359i	$0.297491\pi$
$908$	0.822859	0.0273075
$909$	−39.3742	−1.30596
$910$	0	0
$911$	−27.0191	−0.895183	−0.447592	−	0.894238i	$-0.647718\pi$
$911$	−27.0191	−0.895183	−0.447592	+	0.894238i	$0.647718\pi$
$912$	−2.77275	−0.0918148
$913$	−31.3129	−1.03630
$914$	12.8600	0.425370
$915$	−101.135	−3.34341
$916$	20.6513	0.682339
$917$	22.6945	0.749438
$918$	6.43042	0.212235
$919$	−15.3010	−0.504733	−0.252366	−	0.967632i	$-0.581209\pi$
$919$	−15.3010	−0.504733	−0.252366	+	0.967632i	$0.581209\pi$
$920$	9.01998	0.297380
$921$	−43.1041	−1.42033
$922$	−9.58002	−0.315501
$923$	0	0
$924$	15.0105	0.493810
$925$	−4.02378	−0.132301
$926$	13.7638	0.452305
$927$	−80.5622	−2.64601
$928$	0.713407	0.0234187
$929$	32.8187	1.07675	0.538374	−	0.842706i	$-0.319039\pi$
$929$	32.8187	1.07675	0.538374	+	0.842706i	$0.319039\pi$
$930$	−18.5674	−0.608850
$931$	4.15628	0.136216
$932$	14.0657	0.460736
$933$	16.0535	0.525567
$934$	27.0616	0.885482
$935$	12.3867	0.405088
$936$	0	0
$937$	−49.3064	−1.61077	−0.805386	−	0.592751i	$-0.798042\pi$
$937$	−49.3064	−1.61077	−0.805386	+	0.592751i	$0.798042\pi$
$938$	−12.9273	−0.422090
$939$	−33.9286	−1.10722
$940$	2.86335	0.0933921
$941$	−29.8642	−0.973546	−0.486773	−	0.873529i	$-0.661826\pi$
$941$	−29.8642	−0.973546	−0.486773	+	0.873529i	$0.661826\pi$
$942$	14.9801	0.488078
$943$	24.6022	0.801157
$944$	−11.0891	−0.360921
$945$	22.1691	0.721161
$946$	41.0728	1.33539
$947$	−20.7179	−0.673242	−0.336621	−	0.941640i	$-0.609284\pi$
$947$	−20.7179	−0.673242	−0.336621	+	0.941640i	$0.609284\pi$
$948$	−28.1679	−0.914852
$949$	0	0
$950$	−2.88822	−0.0937061
$951$	35.5674	1.15335
$952$	−2.31669	−0.0750844
$953$	−35.8746	−1.16209	−0.581047	−	0.813870i	$-0.697357\pi$
$953$	−35.8746	−1.16209	−0.581047	+	0.813870i	$0.697357\pi$
$954$	5.06738	0.164063
$955$	59.8116	1.93546
$956$	6.36506	0.205861
$957$	−6.35023	−0.205274
$958$	42.4758	1.37233
$959$	−3.31965	−0.107197
$960$	−7.78753	−0.251341
$961$	−25.3153	−0.816624
$962$	0	0
$963$	62.8270	2.02457
$964$	15.5870	0.502022
$965$	−42.4443	−1.36633
$966$	15.0166	0.483150
$967$	−25.2615	−0.812356	−0.406178	−	0.913794i	$-0.633139\pi$
$967$	−25.2615	−0.812356	−0.406178	+	0.913794i	$0.633139\pi$
$968$	−0.694159	−0.0223111
$969$	−3.80921	−0.122369
$970$	26.2462	0.842715
$971$	−19.4267	−0.623431	−0.311715	−	0.950175i	$-0.600904\pi$
$971$	−19.4267	−0.623431	−0.311715	+	0.950175i	$0.600904\pi$
$972$	−17.0530	−0.546976
$973$	18.1016	0.580311
$974$	−32.0478	−1.02688
$975$	0	0
$976$	12.9868	0.415696
$977$	−9.21259	−0.294737	−0.147368	−	0.989082i	$-0.547080\pi$
$977$	−9.21259	−0.294737	−0.147368	+	0.989082i	$0.547080\pi$
$978$	−66.8157	−2.13653
$979$	−20.7194	−0.662196
$980$	11.6733	0.372890
$981$	−49.2904	−1.57372
$982$	−1.05374	−0.0336261
$983$	−17.7796	−0.567081	−0.283541	−	0.958960i	$-0.591509\pi$
$983$	−17.7796	−0.567081	−0.283541	+	0.958960i	$0.591509\pi$
$984$	−21.2406	−0.677126
$985$	−9.25996	−0.295047
$986$	0.980080	0.0312121
$987$	4.76693	0.151733
$988$	0	0
$989$	41.0893	1.30656
$990$	42.2698	1.34342
$991$	24.3729	0.774229	0.387115	−	0.922032i	$-0.373472\pi$
$991$	24.3729	0.774229	0.387115	+	0.922032i	$0.373472\pi$
$992$	2.38425	0.0757001
$993$	−4.26185	−0.135246
$994$	8.19710	0.259996
$995$	−59.5661	−1.88837
$996$	27.0453	0.856962
$997$	51.4609	1.62978	0.814891	−	0.579614i	$-0.196797\pi$
$997$	51.4609	1.62978	0.814891	+	0.579614i	$0.196797\pi$
$998$	22.0474	0.697898
$999$	−6.52107	−0.206318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bl.1.9	yes	9
13.12	even	2		6422.2.a.bj.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bj.1.9	✓	9	13.12	even	2
6422.2.a.bl.1.9	yes	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.77275 q^{3} +1.00000 q^{4} -2.80860 q^{5} +2.77275 q^{6} -1.68633 q^{7} +1.00000 q^{8} +4.68813 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.77275 q^{3} +1.00000 q^{4} -2.80860 q^{5} +2.77275 q^{6} -1.68633 q^{7} +1.00000 q^{8} +4.68813 q^{9} -2.80860 q^{10} -3.21027 q^{11} +2.77275 q^{12} -1.68633 q^{14} -7.78753 q^{15} +1.00000 q^{16} +1.37380 q^{17} +4.68813 q^{18} -1.00000 q^{19} -2.80860 q^{20} -4.67578 q^{21} -3.21027 q^{22} -3.21156 q^{23} +2.77275 q^{24} +2.88822 q^{25} +4.68075 q^{27} -1.68633 q^{28} +0.713407 q^{29} -7.78753 q^{30} +2.38425 q^{31} +1.00000 q^{32} -8.90127 q^{33} +1.37380 q^{34} +4.73623 q^{35} +4.68813 q^{36} -1.39317 q^{37} -1.00000 q^{38} -2.80860 q^{40} -7.66050 q^{41} -4.67578 q^{42} -12.7942 q^{43} -3.21027 q^{44} -13.1671 q^{45} -3.21156 q^{46} -1.01949 q^{47} +2.77275 q^{48} -4.15628 q^{49} +2.88822 q^{50} +3.80921 q^{51} +1.08090 q^{53} +4.68075 q^{54} +9.01636 q^{55} -1.68633 q^{56} -2.77275 q^{57} +0.713407 q^{58} -11.0891 q^{59} -7.78753 q^{60} +12.9868 q^{61} +2.38425 q^{62} -7.90575 q^{63} +1.00000 q^{64} -8.90127 q^{66} +7.66589 q^{67} +1.37380 q^{68} -8.90485 q^{69} +4.73623 q^{70} -4.86090 q^{71} +4.68813 q^{72} -9.26376 q^{73} -1.39317 q^{74} +8.00830 q^{75} -1.00000 q^{76} +5.41359 q^{77} -10.1589 q^{79} -2.80860 q^{80} -1.08586 q^{81} -7.66050 q^{82} +9.75396 q^{83} -4.67578 q^{84} -3.85846 q^{85} -12.7942 q^{86} +1.97810 q^{87} -3.21027 q^{88} +6.45411 q^{89} -13.1671 q^{90} -3.21156 q^{92} +6.61093 q^{93} -1.01949 q^{94} +2.80860 q^{95} +2.77275 q^{96} -9.34495 q^{97} -4.15628 q^{98} -15.0502 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9	\( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9 + q^10 - 3 * q^11 + q^12 - 13 * q^14 - 3 * q^15 + 9 * q^16 - 8 * q^17 - 2 * q^18 - 9 * q^19 + q^20 - 24 * q^21 - 3 * q^22 - 10 * q^23 + q^24 + 8 * q^25 + 10 * q^27 - 13 * q^28 - 20 * q^29 - 3 * q^30 - q^31 + 9 * q^32 + 2 * q^33 - 8 * q^34 + 4 * q^35 - 2 * q^36 - 15 * q^37 - 9 * q^38 + q^40 - 19 * q^41 - 24 * q^42 - 16 * q^43 - 3 * q^44 - 15 * q^45 - 10 * q^46 - 18 * q^47 + q^48 + 18 * q^49 + 8 * q^50 - 11 * q^51 + 17 * q^53 + 10 * q^54 - 26 * q^55 - 13 * q^56 - q^57 - 20 * q^58 - 24 * q^59 - 3 * q^60 + 6 * q^61 - q^62 - q^63 + 9 * q^64 + 2 * q^66 - 29 * q^67 - 8 * q^68 - 12 * q^69 + 4 * q^70 - 23 * q^71 - 2 * q^72 - 38 * q^73 - 15 * q^74 + 11 * q^75 - 9 * q^76 - 40 * q^77 - 20 * q^79 + q^80 - 31 * q^81 - 19 * q^82 - 20 * q^83 - 24 * q^84 - 39 * q^85 - 16 * q^86 - 10 * q^87 - 3 * q^88 + 7 * q^89 - 15 * q^90 - 10 * q^92 - 11 * q^93 - 18 * q^94 - q^95 + q^96 - 28 * q^97 + 18 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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