LMFDB - Embedded newform 6422.2.a.r.1.1

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:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.24698$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.69202 q^{3} +1.00000 q^{4} +2.04892 q^{5} -2.69202 q^{6} -3.60388 q^{7} +1.00000 q^{8} +4.24698 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.69202 q^{3} +1.00000 q^{4} +2.04892 q^{5} -2.69202 q^{6} -3.60388 q^{7} +1.00000 q^{8} +4.24698 q^{9} +2.04892 q^{10} +0.890084 q^{11} -2.69202 q^{12} -3.60388 q^{14} -5.51573 q^{15} +1.00000 q^{16} +5.96077 q^{17} +4.24698 q^{18} -1.00000 q^{19} +2.04892 q^{20} +9.70171 q^{21} +0.890084 q^{22} -8.98792 q^{23} -2.69202 q^{24} -0.801938 q^{25} -3.35690 q^{27} -3.60388 q^{28} +5.20775 q^{29} -5.51573 q^{30} -2.02177 q^{31} +1.00000 q^{32} -2.39612 q^{33} +5.96077 q^{34} -7.38404 q^{35} +4.24698 q^{36} -9.38404 q^{37} -1.00000 q^{38} +2.04892 q^{40} -1.82371 q^{41} +9.70171 q^{42} +4.71379 q^{43} +0.890084 q^{44} +8.70171 q^{45} -8.98792 q^{46} -2.69202 q^{48} +5.98792 q^{49} -0.801938 q^{50} -16.0465 q^{51} +3.87800 q^{53} -3.35690 q^{54} +1.82371 q^{55} -3.60388 q^{56} +2.69202 q^{57} +5.20775 q^{58} -12.3666 q^{59} -5.51573 q^{60} +7.82908 q^{61} -2.02177 q^{62} -15.3056 q^{63} +1.00000 q^{64} -2.39612 q^{66} -3.13169 q^{67} +5.96077 q^{68} +24.1957 q^{69} -7.38404 q^{70} +15.8388 q^{71} +4.24698 q^{72} -4.80731 q^{73} -9.38404 q^{74} +2.15883 q^{75} -1.00000 q^{76} -3.20775 q^{77} +13.3448 q^{79} +2.04892 q^{80} -3.70410 q^{81} -1.82371 q^{82} +4.17629 q^{83} +9.70171 q^{84} +12.2131 q^{85} +4.71379 q^{86} -14.0194 q^{87} +0.890084 q^{88} +5.42758 q^{89} +8.70171 q^{90} -8.98792 q^{92} +5.44265 q^{93} -2.04892 q^{95} -2.69202 q^{96} -3.50604 q^{97} +5.98792 q^{98} +3.78017 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 8 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 2 q^{14} - 4 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} - 3 q^{19} - 3 q^{20} + 2 q^{21} + 2 q^{22} - 8 q^{23} - 3 q^{24} + 2 q^{25} - 6 q^{27} - 2 q^{28} - 2 q^{29} - 4 q^{30} - 3 q^{31} + 3 q^{32} - 16 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 18 q^{37} - 3 q^{38} - 3 q^{40} + 2 q^{41} + 2 q^{42} + 6 q^{43} + 2 q^{44} - q^{45} - 8 q^{46} - 3 q^{48} - q^{49} + 2 q^{50} + 2 q^{51} - 8 q^{53} - 6 q^{54} - 2 q^{55} - 2 q^{56} + 3 q^{57} - 2 q^{58} - 11 q^{59} - 4 q^{60} + 13 q^{61} - 3 q^{62} - 10 q^{63} + 3 q^{64} - 16 q^{66} - 7 q^{67} + 5 q^{68} + 36 q^{69} - 12 q^{70} + 15 q^{71} + 8 q^{72} - 7 q^{73} - 18 q^{74} - 2 q^{75} - 3 q^{76} + 8 q^{77} + 17 q^{79} - 3 q^{80} - 25 q^{81} + 2 q^{82} + 20 q^{83} + 2 q^{84} + 16 q^{85} + 6 q^{86} + 2 q^{87} + 2 q^{88} - q^{90} - 8 q^{92} - 25 q^{93} + 3 q^{95} - 3 q^{96} - 20 q^{97} - q^{98} + 10 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 8 * q^9 - 3 * q^10 + 2 * q^11 - 3 * q^12 - 2 * q^14 - 4 * q^15 + 3 * q^16 + 5 * q^17 + 8 * q^18 - 3 * q^19 - 3 * q^20 + 2 * q^21 + 2 * q^22 - 8 * q^23 - 3 * q^24 + 2 * q^25 - 6 * q^27 - 2 * q^28 - 2 * q^29 - 4 * q^30 - 3 * q^31 + 3 * q^32 - 16 * q^33 + 5 * q^34 - 12 * q^35 + 8 * q^36 - 18 * q^37 - 3 * q^38 - 3 * q^40 + 2 * q^41 + 2 * q^42 + 6 * q^43 + 2 * q^44 - q^45 - 8 * q^46 - 3 * q^48 - q^49 + 2 * q^50 + 2 * q^51 - 8 * q^53 - 6 * q^54 - 2 * q^55 - 2 * q^56 + 3 * q^57 - 2 * q^58 - 11 * q^59 - 4 * q^60 + 13 * q^61 - 3 * q^62 - 10 * q^63 + 3 * q^64 - 16 * q^66 - 7 * q^67 + 5 * q^68 + 36 * q^69 - 12 * q^70 + 15 * q^71 + 8 * q^72 - 7 * q^73 - 18 * q^74 - 2 * q^75 - 3 * q^76 + 8 * q^77 + 17 * q^79 - 3 * q^80 - 25 * q^81 + 2 * q^82 + 20 * q^83 + 2 * q^84 + 16 * q^85 + 6 * q^86 + 2 * q^87 + 2 * q^88 - q^90 - 8 * q^92 - 25 * q^93 + 3 * q^95 - 3 * q^96 - 20 * q^97 - q^98 + 10 * 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
$3$	−2.69202	−1.55424	−0.777120	−	0.629353i	$-0.783320\pi$
$3$	−2.69202	−1.55424	−0.777120	+	0.629353i	$0.783320\pi$
$4$	1.00000	0.500000
$5$	2.04892	0.916304	0.458152	−	0.888874i	$-0.348512\pi$
$5$	2.04892	0.916304	0.458152	+	0.888874i	$0.348512\pi$
$6$	−2.69202	−1.09901
$7$	−3.60388	−1.36214	−0.681068	−	0.732220i	$-0.738484\pi$
$7$	−3.60388	−1.36214	−0.681068	+	0.732220i	$0.738484\pi$
$8$	1.00000	0.353553
$9$	4.24698	1.41566
$10$	2.04892	0.647925
$11$	0.890084	0.268370	0.134185	−	0.990956i	$-0.457158\pi$
$11$	0.890084	0.268370	0.134185	+	0.990956i	$0.457158\pi$
$12$	−2.69202	−0.777120
$13$	0	0
$13$	0	0
$14$	−3.60388	−0.963176
$15$	−5.51573	−1.42416
$16$	1.00000	0.250000
$17$	5.96077	1.44570	0.722850	−	0.691005i	$-0.242832\pi$
$17$	5.96077	1.44570	0.722850	+	0.691005i	$0.242832\pi$
$18$	4.24698	1.00102
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.04892	0.458152
$21$	9.70171	2.11709
$22$	0.890084	0.189766
$23$	−8.98792	−1.87411	−0.937055	−	0.349181i	$-0.886460\pi$
$23$	−8.98792	−1.87411	−0.937055	+	0.349181i	$0.886460\pi$
$24$	−2.69202	−0.549507
$25$	−0.801938	−0.160388
$26$	0	0
$27$	−3.35690	−0.646035
$28$	−3.60388	−0.681068
$29$	5.20775	0.967055	0.483528	−	0.875329i	$-0.339355\pi$
$29$	5.20775	0.967055	0.483528	+	0.875329i	$0.339355\pi$
$30$	−5.51573	−1.00703
$31$	−2.02177	−0.363121	−0.181560	−	0.983380i	$-0.558115\pi$
$31$	−2.02177	−0.363121	−0.181560	+	0.983380i	$0.558115\pi$
$32$	1.00000	0.176777
$33$	−2.39612	−0.417112
$34$	5.96077	1.02226
$35$	−7.38404	−1.24813
$36$	4.24698	0.707830
$37$	−9.38404	−1.54273	−0.771364	−	0.636395i	$-0.780425\pi$
$37$	−9.38404	−1.54273	−0.771364	+	0.636395i	$0.780425\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	2.04892	0.323962
$41$	−1.82371	−0.284815	−0.142408	−	0.989808i	$-0.545484\pi$
$41$	−1.82371	−0.284815	−0.142408	+	0.989808i	$0.545484\pi$
$42$	9.70171	1.49701
$43$	4.71379	0.718847	0.359423	−	0.933175i	$-0.382973\pi$
$43$	4.71379	0.718847	0.359423	+	0.933175i	$0.382973\pi$
$44$	0.890084	0.134185
$45$	8.70171	1.29717
$46$	−8.98792	−1.32520
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.69202	−0.388560
$49$	5.98792	0.855417
$50$	−0.801938	−0.113411
$51$	−16.0465	−2.24696
$52$	0	0
$53$	3.87800	0.532685	0.266342	−	0.963879i	$-0.414185\pi$
$53$	3.87800	0.532685	0.266342	+	0.963879i	$0.414185\pi$
$54$	−3.35690	−0.456816
$55$	1.82371	0.245909
$56$	−3.60388	−0.481588
$57$	2.69202	0.356567
$58$	5.20775	0.683811
$59$	−12.3666	−1.60999	−0.804996	−	0.593280i	$-0.797833\pi$
$59$	−12.3666	−1.60999	−0.804996	+	0.593280i	$0.797833\pi$
$60$	−5.51573	−0.712078
$61$	7.82908	1.00241	0.501206	−	0.865328i	$-0.332890\pi$
$61$	7.82908	1.00241	0.501206	+	0.865328i	$0.332890\pi$
$62$	−2.02177	−0.256765
$63$	−15.3056	−1.92832
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.39612	−0.294943
$67$	−3.13169	−0.382596	−0.191298	−	0.981532i	$-0.561270\pi$
$67$	−3.13169	−0.382596	−0.191298	+	0.981532i	$0.561270\pi$
$68$	5.96077	0.722850
$69$	24.1957	2.91282
$70$	−7.38404	−0.882562
$71$	15.8388	1.87972	0.939858	−	0.341565i	$-0.110957\pi$
$71$	15.8388	1.87972	0.939858	+	0.341565i	$0.110957\pi$
$72$	4.24698	0.500511
$73$	−4.80731	−0.562654	−0.281327	−	0.959612i	$-0.590774\pi$
$73$	−4.80731	−0.562654	−0.281327	+	0.959612i	$0.590774\pi$
$74$	−9.38404	−1.09087
$75$	2.15883	0.249281
$76$	−1.00000	−0.114708
$77$	−3.20775	−0.365557
$78$	0	0
$79$	13.3448	1.50141	0.750704	−	0.660638i	$-0.229714\pi$
$79$	13.3448	1.50141	0.750704	+	0.660638i	$0.229714\pi$
$80$	2.04892	0.229076
$81$	−3.70410	−0.411567
$82$	−1.82371	−0.201395
$83$	4.17629	0.458408	0.229204	−	0.973378i	$-0.426388\pi$
$83$	4.17629	0.458408	0.229204	+	0.973378i	$0.426388\pi$
$84$	9.70171	1.05854
$85$	12.2131	1.32470
$86$	4.71379	0.508301
$87$	−14.0194	−1.50304
$88$	0.890084	0.0948832
$89$	5.42758	0.575323	0.287661	−	0.957732i	$-0.407122\pi$
$89$	5.42758	0.575323	0.287661	+	0.957732i	$0.407122\pi$
$90$	8.70171	0.917241
$91$	0	0
$92$	−8.98792	−0.937055
$93$	5.44265	0.564376
$94$	0	0
$95$	−2.04892	−0.210214
$96$	−2.69202	−0.274753
$97$	−3.50604	−0.355985	−0.177992	−	0.984032i	$-0.556960\pi$
$97$	−3.50604	−0.355985	−0.177992	+	0.984032i	$0.556960\pi$
$98$	5.98792	0.604871
$99$	3.78017	0.379921
$100$	−0.801938	−0.0801938
$101$	−18.2446	−1.81540	−0.907702	−	0.419615i	$-0.862165\pi$
$101$	−18.2446	−1.81540	−0.907702	+	0.419615i	$0.862165\pi$
$102$	−16.0465	−1.58884
$103$	−14.8605	−1.46425	−0.732126	−	0.681169i	$-0.761472\pi$
$103$	−14.8605	−1.46425	−0.732126	+	0.681169i	$0.761472\pi$
$104$	0	0
$105$	19.8780	1.93989
$106$	3.87800	0.376665
$107$	−4.57673	−0.442449	−0.221225	−	0.975223i	$-0.571005\pi$
$107$	−4.57673	−0.442449	−0.221225	+	0.975223i	$0.571005\pi$
$108$	−3.35690	−0.323017
$109$	3.82371	0.366245	0.183122	−	0.983090i	$-0.441380\pi$
$109$	3.82371	0.366245	0.183122	+	0.983090i	$0.441380\pi$
$110$	1.82371	0.173884
$111$	25.2620	2.39777
$112$	−3.60388	−0.340534
$113$	2.71379	0.255292	0.127646	−	0.991820i	$-0.459258\pi$
$113$	2.71379	0.255292	0.127646	+	0.991820i	$0.459258\pi$
$114$	2.69202	0.252131
$115$	−18.4155	−1.71725
$116$	5.20775	0.483528
$117$	0	0
$118$	−12.3666	−1.13844
$119$	−21.4819	−1.96924
$120$	−5.51573	−0.503515
$121$	−10.2078	−0.927977
$122$	7.82908	0.708812
$123$	4.90946	0.442671
$124$	−2.02177	−0.181560
$125$	−11.8877	−1.06327
$126$	−15.3056	−1.36353
$127$	−7.75840	−0.688446	−0.344223	−	0.938888i	$-0.611858\pi$
$127$	−7.75840	−0.688446	−0.344223	+	0.938888i	$0.611858\pi$
$128$	1.00000	0.0883883
$129$	−12.6896	−1.11726
$130$	0	0
$131$	16.5918	1.44963	0.724816	−	0.688943i	$-0.241925\pi$
$131$	16.5918	1.44963	0.724816	+	0.688943i	$0.241925\pi$
$132$	−2.39612	−0.208556
$133$	3.60388	0.312496
$134$	−3.13169	−0.270537
$135$	−6.87800	−0.591964
$136$	5.96077	0.511132
$137$	13.6286	1.16437	0.582186	−	0.813055i	$-0.302197\pi$
$137$	13.6286	1.16437	0.582186	+	0.813055i	$0.302197\pi$
$138$	24.1957	2.05967
$139$	−15.0315	−1.27495	−0.637476	−	0.770470i	$-0.720021\pi$
$139$	−15.0315	−1.27495	−0.637476	+	0.770470i	$0.720021\pi$
$140$	−7.38404	−0.624066
$141$	0	0
$142$	15.8388	1.32916
$143$	0	0
$144$	4.24698	0.353915
$145$	10.6703	0.886116
$146$	−4.80731	−0.397856
$147$	−16.1196	−1.32952
$148$	−9.38404	−0.771364
$149$	−12.6746	−1.03834	−0.519170	−	0.854671i	$-0.673759\pi$
$149$	−12.6746	−1.03834	−0.519170	+	0.854671i	$0.673759\pi$
$150$	2.15883	0.176268
$151$	1.78448	0.145219	0.0726094	−	0.997360i	$-0.476867\pi$
$151$	1.78448	0.145219	0.0726094	+	0.997360i	$0.476867\pi$
$152$	−1.00000	−0.0811107
$153$	25.3153	2.04662
$154$	−3.20775	−0.258488
$155$	−4.14244	−0.332729
$156$	0	0
$157$	3.16852	0.252876	0.126438	−	0.991975i	$-0.459646\pi$
$157$	3.16852	0.252876	0.126438	+	0.991975i	$0.459646\pi$
$158$	13.3448	1.06166
$159$	−10.4397	−0.827919
$160$	2.04892	0.161981
$161$	32.3913	2.55280
$162$	−3.70410	−0.291022
$163$	−22.0737	−1.72894	−0.864472	−	0.502681i	$-0.832347\pi$
$163$	−22.0737	−1.72894	−0.864472	+	0.502681i	$0.832347\pi$
$164$	−1.82371	−0.142408
$165$	−4.90946	−0.382201
$166$	4.17629	0.324143
$167$	−18.0218	−1.39457	−0.697283	−	0.716796i	$-0.745608\pi$
$167$	−18.0218	−1.39457	−0.697283	+	0.716796i	$0.745608\pi$
$168$	9.70171	0.748503
$169$	0	0
$170$	12.2131	0.936704
$171$	−4.24698	−0.324775
$172$	4.71379	0.359423
$173$	−2.19567	−0.166934	−0.0834668	−	0.996511i	$-0.526599\pi$
$173$	−2.19567	−0.166934	−0.0834668	+	0.996511i	$0.526599\pi$
$174$	−14.0194	−1.06281
$175$	2.89008	0.218470
$176$	0.890084	0.0670926
$177$	33.2911	2.50231
$178$	5.42758	0.406815
$179$	−13.1250	−0.981007	−0.490504	−	0.871439i	$-0.663187\pi$
$179$	−13.1250	−0.981007	−0.490504	+	0.871439i	$0.663187\pi$
$180$	8.70171	0.648587
$181$	9.74525	0.724359	0.362179	−	0.932108i	$-0.382033\pi$
$181$	9.74525	0.724359	0.362179	+	0.932108i	$0.382033\pi$
$182$	0	0
$183$	−21.0761	−1.55799
$184$	−8.98792	−0.662598
$185$	−19.2271	−1.41361
$186$	5.44265	0.399074
$187$	5.30559	0.387983
$188$	0	0
$189$	12.0978	0.879988
$190$	−2.04892	−0.148644
$191$	−20.4698	−1.48114	−0.740571	−	0.671978i	$-0.765445\pi$
$191$	−20.4698	−1.48114	−0.740571	+	0.671978i	$0.765445\pi$
$192$	−2.69202	−0.194280
$193$	−2.96854	−0.213680	−0.106840	−	0.994276i	$-0.534073\pi$
$193$	−2.96854	−0.213680	−0.106840	+	0.994276i	$0.534073\pi$
$194$	−3.50604	−0.251719
$195$	0	0
$196$	5.98792	0.427708
$197$	−13.9366	−0.992942	−0.496471	−	0.868053i	$-0.665371\pi$
$197$	−13.9366	−0.992942	−0.496471	+	0.868053i	$0.665371\pi$
$198$	3.78017	0.268645
$199$	−7.10992	−0.504009	−0.252004	−	0.967726i	$-0.581090\pi$
$199$	−7.10992	−0.504009	−0.252004	+	0.967726i	$0.581090\pi$
$200$	−0.801938	−0.0567056
$201$	8.43057	0.594646
$202$	−18.2446	−1.28368
$203$	−18.7681	−1.31726
$204$	−16.0465	−1.12348
$205$	−3.73663	−0.260977
$206$	−14.8605	−1.03538
$207$	−38.1715	−2.65310
$208$	0	0
$209$	−0.890084	−0.0615684
$210$	19.8780	1.37171
$211$	9.50365	0.654258	0.327129	−	0.944980i	$-0.393919\pi$
$211$	9.50365	0.654258	0.327129	+	0.944980i	$0.393919\pi$
$212$	3.87800	0.266342
$213$	−42.6383	−2.92153
$214$	−4.57673	−0.312859
$215$	9.65817	0.658682
$216$	−3.35690	−0.228408
$217$	7.28621	0.494620
$218$	3.82371	0.258974
$219$	12.9414	0.874498
$220$	1.82371	0.122954
$221$	0	0
$222$	25.2620	1.69548
$223$	19.0465	1.27545	0.637725	−	0.770264i	$-0.279876\pi$
$223$	19.0465	1.27545	0.637725	+	0.770264i	$0.279876\pi$
$224$	−3.60388	−0.240794
$225$	−3.40581	−0.227054
$226$	2.71379	0.180519
$227$	10.2959	0.683363	0.341681	−	0.939816i	$-0.389004\pi$
$227$	10.2959	0.683363	0.341681	+	0.939816i	$0.389004\pi$
$228$	2.69202	0.178283
$229$	8.71917	0.576179	0.288089	−	0.957604i	$-0.406980\pi$
$229$	8.71917	0.576179	0.288089	+	0.957604i	$0.406980\pi$
$230$	−18.4155	−1.21428
$231$	8.63533	0.568163
$232$	5.20775	0.341906
$233$	−16.4450	−1.07735	−0.538675	−	0.842513i	$-0.681075\pi$
$233$	−16.4450	−1.07735	−0.538675	+	0.842513i	$0.681075\pi$
$234$	0	0
$235$	0	0
$236$	−12.3666	−0.804996
$237$	−35.9245	−2.33355
$238$	−21.4819	−1.39246
$239$	−25.7453	−1.66532	−0.832661	−	0.553783i	$-0.813184\pi$
$239$	−25.7453	−1.66532	−0.832661	+	0.553783i	$0.813184\pi$
$240$	−5.51573	−0.356039
$241$	2.67025	0.172006	0.0860030	−	0.996295i	$-0.472591\pi$
$241$	2.67025	0.172006	0.0860030	+	0.996295i	$0.472591\pi$
$242$	−10.2078	−0.656179
$243$	20.0422	1.28571
$244$	7.82908	0.501206
$245$	12.2687	0.783822
$246$	4.90946	0.313016
$247$	0	0
$248$	−2.02177	−0.128383
$249$	−11.2427	−0.712475
$250$	−11.8877	−0.751844
$251$	−11.6039	−0.732430	−0.366215	−	0.930530i	$-0.619347\pi$
$251$	−11.6039	−0.732430	−0.366215	+	0.930530i	$0.619347\pi$
$252$	−15.3056	−0.964161
$253$	−8.00000	−0.502956
$254$	−7.75840	−0.486805
$255$	−32.8780	−2.05890
$256$	1.00000	0.0625000
$257$	2.79225	0.174176	0.0870879	−	0.996201i	$-0.472244\pi$
$257$	2.79225	0.174176	0.0870879	+	0.996201i	$0.472244\pi$
$258$	−12.6896	−0.790022
$259$	33.8189	2.10141
$260$	0	0
$261$	22.1172	1.36902
$262$	16.5918	1.02504
$263$	12.7681	0.787314	0.393657	−	0.919257i	$-0.371210\pi$
$263$	12.7681	0.787314	0.393657	+	0.919257i	$0.371210\pi$
$264$	−2.39612	−0.147471
$265$	7.94571	0.488101
$266$	3.60388	0.220968
$267$	−14.6112	−0.894189
$268$	−3.13169	−0.191298
$269$	−11.5603	−0.704846	−0.352423	−	0.935841i	$-0.614642\pi$
$269$	−11.5603	−0.704846	−0.352423	+	0.935841i	$0.614642\pi$
$270$	−6.87800	−0.418582
$271$	7.10992	0.431897	0.215948	−	0.976405i	$-0.430716\pi$
$271$	7.10992	0.431897	0.215948	+	0.976405i	$0.430716\pi$
$272$	5.96077	0.361425
$273$	0	0
$274$	13.6286	0.823336
$275$	−0.713792	−0.0430433
$276$	24.1957	1.45641
$277$	9.59850	0.576718	0.288359	−	0.957522i	$-0.406890\pi$
$277$	9.59850	0.576718	0.288359	+	0.957522i	$0.406890\pi$
$278$	−15.0315	−0.901527
$279$	−8.58642	−0.514055
$280$	−7.38404	−0.441281
$281$	12.8552	0.766875	0.383437	−	0.923567i	$-0.374740\pi$
$281$	12.8552	0.766875	0.383437	+	0.923567i	$0.374740\pi$
$282$	0	0
$283$	−3.83446	−0.227935	−0.113968	−	0.993484i	$-0.536356\pi$
$283$	−3.83446	−0.227935	−0.113968	+	0.993484i	$0.536356\pi$
$284$	15.8388	0.939858
$285$	5.51573	0.326724
$286$	0	0
$287$	6.57242	0.387957
$288$	4.24698	0.250256
$289$	18.5308	1.09005
$290$	10.6703	0.626579
$291$	9.43834	0.553285
$292$	−4.80731	−0.281327
$293$	−30.2935	−1.76977	−0.884883	−	0.465814i	$-0.845762\pi$
$293$	−30.2935	−1.76977	−0.884883	+	0.465814i	$0.845762\pi$
$294$	−16.1196	−0.940114
$295$	−25.3381	−1.47524
$296$	−9.38404	−0.545436
$297$	−2.98792	−0.173377
$298$	−12.6746	−0.734218
$299$	0	0
$300$	2.15883	0.124640
$301$	−16.9879	−0.979167
$302$	1.78448	0.102685
$303$	49.1148	2.82157
$304$	−1.00000	−0.0573539
$305$	16.0411	0.918513
$306$	25.3153	1.44718
$307$	−30.0084	−1.71267	−0.856334	−	0.516423i	$-0.827263\pi$
$307$	−30.0084	−1.71267	−0.856334	+	0.516423i	$0.827263\pi$
$308$	−3.20775	−0.182779
$309$	40.0049	2.27580
$310$	−4.14244	−0.235275
$311$	22.3478	1.26723	0.633614	−	0.773650i	$-0.281571\pi$
$311$	22.3478	1.26723	0.633614	+	0.773650i	$0.281571\pi$
$312$	0	0
$313$	−31.2131	−1.76427	−0.882135	−	0.470996i	$-0.843894\pi$
$313$	−31.2131	−1.76427	−0.882135	+	0.470996i	$0.843894\pi$
$314$	3.16852	0.178810
$315$	−31.3599	−1.76693
$316$	13.3448	0.750704
$317$	−8.54825	−0.480118	−0.240059	−	0.970758i	$-0.577167\pi$
$317$	−8.54825	−0.480118	−0.240059	+	0.970758i	$0.577167\pi$
$318$	−10.4397	−0.585427
$319$	4.63533	0.259529
$320$	2.04892	0.114538
$321$	12.3207	0.687672
$322$	32.3913	1.80510
$323$	−5.96077	−0.331666
$324$	−3.70410	−0.205784
$325$	0	0
$326$	−22.0737	−1.22255
$327$	−10.2935	−0.569232
$328$	−1.82371	−0.100697
$329$	0	0
$330$	−4.90946	−0.270257
$331$	−1.79118	−0.0984524	−0.0492262	−	0.998788i	$-0.515676\pi$
$331$	−1.79118	−0.0984524	−0.0492262	+	0.998788i	$0.515676\pi$
$332$	4.17629	0.229204
$333$	−39.8538	−2.18398
$334$	−18.0218	−0.986107
$335$	−6.41657	−0.350574
$336$	9.70171	0.529272
$337$	10.5181	0.572959	0.286479	−	0.958086i	$-0.407515\pi$
$337$	10.5181	0.572959	0.286479	+	0.958086i	$0.407515\pi$
$338$	0	0
$339$	−7.30559	−0.396785
$340$	12.2131	0.662350
$341$	−1.79954	−0.0974508
$342$	−4.24698	−0.229650
$343$	3.64742	0.196942
$344$	4.71379	0.254151
$345$	49.5749	2.66902
$346$	−2.19567	−0.118040
$347$	−25.4034	−1.36373	−0.681864	−	0.731479i	$-0.738830\pi$
$347$	−25.4034	−1.36373	−0.681864	+	0.731479i	$0.738830\pi$
$348$	−14.0194	−0.751518
$349$	22.7549	1.21804	0.609022	−	0.793153i	$-0.291562\pi$
$349$	22.7549	1.21804	0.609022	+	0.793153i	$0.291562\pi$
$350$	2.89008	0.154481
$351$	0	0
$352$	0.890084	0.0474416
$353$	−21.5144	−1.14510	−0.572548	−	0.819871i	$-0.694045\pi$
$353$	−21.5144	−1.14510	−0.572548	+	0.819871i	$0.694045\pi$
$354$	33.2911	1.76940
$355$	32.4523	1.72239
$356$	5.42758	0.287661
$357$	57.8297	3.06067
$358$	−13.1250	−0.693677
$359$	28.2198	1.48939	0.744693	−	0.667407i	$-0.232596\pi$
$359$	28.2198	1.48939	0.744693	+	0.667407i	$0.232596\pi$
$360$	8.70171	0.458620
$361$	1.00000	0.0526316
$362$	9.74525	0.512199
$363$	27.4795	1.44230
$364$	0	0
$365$	−9.84979	−0.515562
$366$	−21.0761	−1.10166
$367$	−31.0858	−1.62266	−0.811332	−	0.584586i	$-0.801257\pi$
$367$	−31.0858	−1.62266	−0.811332	+	0.584586i	$0.801257\pi$
$368$	−8.98792	−0.468528
$369$	−7.74525	−0.403202
$370$	−19.2271	−0.999571
$371$	−13.9758	−0.725589
$372$	5.44265	0.282188
$373$	13.2862	0.687934	0.343967	−	0.938982i	$-0.388229\pi$
$373$	13.2862	0.687934	0.343967	+	0.938982i	$0.388229\pi$
$374$	5.30559	0.274345
$375$	32.0019	1.65257
$376$	0	0
$377$	0	0
$378$	12.0978	0.622245
$379$	−21.7802	−1.11877	−0.559386	−	0.828907i	$-0.688963\pi$
$379$	−21.7802	−1.11877	−0.559386	+	0.828907i	$0.688963\pi$
$380$	−2.04892	−0.105107
$381$	20.8858	1.07001
$382$	−20.4698	−1.04733
$383$	−1.88338	−0.0962362	−0.0481181	−	0.998842i	$-0.515322\pi$
$383$	−1.88338	−0.0962362	−0.0481181	+	0.998842i	$0.515322\pi$
$384$	−2.69202	−0.137377
$385$	−6.57242	−0.334961
$386$	−2.96854	−0.151095
$387$	20.0194	1.01764
$388$	−3.50604	−0.177992
$389$	−39.2707	−1.99110	−0.995551	−	0.0942245i	$-0.969963\pi$
$389$	−39.2707	−1.99110	−0.995551	+	0.0942245i	$0.969963\pi$
$390$	0	0
$391$	−53.5749	−2.70940
$392$	5.98792	0.302436
$393$	−44.6655	−2.25307
$394$	−13.9366	−0.702116
$395$	27.3424	1.37575
$396$	3.78017	0.189961
$397$	−5.65758	−0.283946	−0.141973	−	0.989871i	$-0.545345\pi$
$397$	−5.65758	−0.283946	−0.141973	+	0.989871i	$0.545345\pi$
$398$	−7.10992	−0.356388
$399$	−9.70171	−0.485693
$400$	−0.801938	−0.0400969
$401$	−29.5362	−1.47497	−0.737483	−	0.675366i	$-0.763986\pi$
$401$	−29.5362	−1.47497	−0.737483	+	0.675366i	$0.763986\pi$
$402$	8.43057	0.420478
$403$	0	0
$404$	−18.2446	−0.907702
$405$	−7.58940	−0.377120
$406$	−18.7681	−0.931444
$407$	−8.35258	−0.414022
$408$	−16.0465	−0.794421
$409$	−17.3927	−0.860012	−0.430006	−	0.902826i	$-0.641489\pi$
$409$	−17.3927	−0.860012	−0.430006	+	0.902826i	$0.641489\pi$
$410$	−3.73663	−0.184539
$411$	−36.6886	−1.80971
$412$	−14.8605	−0.732126
$413$	44.5676	2.19303
$414$	−38.1715	−1.87603
$415$	8.55688	0.420041
$416$	0	0
$417$	40.4650	1.98158
$418$	−0.890084	−0.0435354
$419$	−2.34183	−0.114406	−0.0572030	−	0.998363i	$-0.518218\pi$
$419$	−2.34183	−0.114406	−0.0572030	+	0.998363i	$0.518218\pi$
$420$	19.8780	0.969947
$421$	−33.3357	−1.62468	−0.812342	−	0.583182i	$-0.801808\pi$
$421$	−33.3357	−1.62468	−0.812342	+	0.583182i	$0.801808\pi$
$422$	9.50365	0.462630
$423$	0	0
$424$	3.87800	0.188332
$425$	−4.78017	−0.231872
$426$	−42.6383	−2.06583
$427$	−28.2150	−1.36542
$428$	−4.57673	−0.221225
$429$	0	0
$430$	9.65817	0.465758
$431$	−2.61356	−0.125891	−0.0629455	−	0.998017i	$-0.520049\pi$
$431$	−2.61356	−0.125891	−0.0629455	+	0.998017i	$0.520049\pi$
$432$	−3.35690	−0.161509
$433$	30.3236	1.45726	0.728630	−	0.684907i	$-0.240157\pi$
$433$	30.3236	1.45726	0.728630	+	0.684907i	$0.240157\pi$
$434$	7.28621	0.349749
$435$	−28.7245	−1.37724
$436$	3.82371	0.183122
$437$	8.98792	0.429950
$438$	12.9414	0.618364
$439$	26.3744	1.25878	0.629390	−	0.777090i	$-0.283305\pi$
$439$	26.3744	1.25878	0.629390	+	0.777090i	$0.283305\pi$
$440$	1.82371	0.0869419
$441$	25.4306	1.21098
$442$	0	0
$443$	−27.0965	−1.28739	−0.643697	−	0.765281i	$-0.722600\pi$
$443$	−27.0965	−1.28739	−0.643697	+	0.765281i	$0.722600\pi$
$444$	25.2620	1.19888
$445$	11.1207	0.527170
$446$	19.0465	0.901879
$447$	34.1202	1.61383
$448$	−3.60388	−0.170267
$449$	−20.4940	−0.967170	−0.483585	−	0.875297i	$-0.660666\pi$
$449$	−20.4940	−0.967170	−0.483585	+	0.875297i	$0.660666\pi$
$450$	−3.40581	−0.160552
$451$	−1.62325	−0.0764360
$452$	2.71379	0.127646
$453$	−4.80386	−0.225705
$454$	10.2959	0.483210
$455$	0	0
$456$	2.69202	0.126065
$457$	−5.23729	−0.244990	−0.122495	−	0.992469i	$-0.539090\pi$
$457$	−5.23729	−0.244990	−0.122495	+	0.992469i	$0.539090\pi$
$458$	8.71917	0.407420
$459$	−20.0097	−0.933972
$460$	−18.4155	−0.858627
$461$	−17.0659	−0.794838	−0.397419	−	0.917637i	$-0.630094\pi$
$461$	−17.0659	−0.794838	−0.397419	+	0.917637i	$0.630094\pi$
$462$	8.63533	0.401752
$463$	−8.74392	−0.406365	−0.203182	−	0.979141i	$-0.565128\pi$
$463$	−8.74392	−0.406365	−0.203182	+	0.979141i	$0.565128\pi$
$464$	5.20775	0.241764
$465$	11.1515	0.517140
$466$	−16.4450	−0.761802
$467$	−37.8974	−1.75368	−0.876841	−	0.480781i	$-0.840353\pi$
$467$	−37.8974	−1.75368	−0.876841	+	0.480781i	$0.840353\pi$
$468$	0	0
$469$	11.2862	0.521149
$470$	0	0
$471$	−8.52973	−0.393029
$472$	−12.3666	−0.569218
$473$	4.19567	0.192917
$474$	−35.9245	−1.65007
$475$	0.801938	0.0367954
$476$	−21.4819	−0.984620
$477$	16.4698	0.754100
$478$	−25.7453	−1.17756
$479$	−36.7004	−1.67688	−0.838442	−	0.544991i	$-0.816533\pi$
$479$	−36.7004	−1.67688	−0.838442	+	0.544991i	$0.816533\pi$
$480$	−5.51573	−0.251757
$481$	0	0
$482$	2.67025	0.121627
$483$	−87.1982	−3.96765
$484$	−10.2078	−0.463989
$485$	−7.18359	−0.326190
$486$	20.0422	0.909133
$487$	−28.1497	−1.27559	−0.637793	−	0.770208i	$-0.720153\pi$
$487$	−28.1497	−1.27559	−0.637793	+	0.770208i	$0.720153\pi$
$488$	7.82908	0.354406
$489$	59.4228	2.68719
$490$	12.2687	0.554246
$491$	27.0315	1.21991	0.609956	−	0.792435i	$-0.291187\pi$
$491$	27.0315	1.21991	0.609956	+	0.792435i	$0.291187\pi$
$492$	4.90946	0.221336
$493$	31.0422	1.39807
$494$	0	0
$495$	7.74525	0.348123
$496$	−2.02177	−0.0907802
$497$	−57.0810	−2.56043
$498$	−11.2427	−0.503796
$499$	7.53617	0.337365	0.168683	−	0.985670i	$-0.446049\pi$
$499$	7.53617	0.337365	0.168683	+	0.985670i	$0.446049\pi$
$500$	−11.8877	−0.531634
$501$	48.5150	2.16749
$502$	−11.6039	−0.517906
$503$	27.2078	1.21313	0.606567	−	0.795033i	$-0.292546\pi$
$503$	27.2078	1.21313	0.606567	+	0.795033i	$0.292546\pi$
$504$	−15.3056	−0.681765
$505$	−37.3817	−1.66346
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−7.75840	−0.344223
$509$	16.2392	0.719790	0.359895	−	0.932993i	$-0.382812\pi$
$509$	16.2392	0.719790	0.359895	+	0.932993i	$0.382812\pi$
$510$	−32.8780	−1.45586
$511$	17.3250	0.766411
$512$	1.00000	0.0441942
$513$	3.35690	0.148211
$514$	2.79225	0.123161
$515$	−30.4480	−1.34170
$516$	−12.6896	−0.558630
$517$	0	0
$518$	33.8189	1.48592
$519$	5.91079	0.259455
$520$	0	0
$521$	−10.2500	−0.449059	−0.224530	−	0.974467i	$-0.572085\pi$
$521$	−10.2500	−0.449059	−0.224530	+	0.974467i	$0.572085\pi$
$522$	22.1172	0.968044
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	16.5918	0.724816
$525$	−7.78017	−0.339554
$526$	12.7681	0.556715
$527$	−12.0513	−0.524963
$528$	−2.39612	−0.104278
$529$	57.7827	2.51229
$530$	7.94571	0.345139
$531$	−52.5206	−2.27920
$532$	3.60388	0.156248
$533$	0	0
$534$	−14.6112	−0.632287
$535$	−9.37734	−0.405418
$536$	−3.13169	−0.135268
$537$	35.3327	1.52472
$538$	−11.5603	−0.498401
$539$	5.32975	0.229569
$540$	−6.87800	−0.295982
$541$	−6.55927	−0.282005	−0.141003	−	0.990009i	$-0.545033\pi$
$541$	−6.55927	−0.282005	−0.141003	+	0.990009i	$0.545033\pi$
$542$	7.10992	0.305397
$543$	−26.2344	−1.12583
$544$	5.96077	0.255566
$545$	7.83446	0.335591
$546$	0	0
$547$	−6.71810	−0.287245	−0.143623	−	0.989633i	$-0.545875\pi$
$547$	−6.71810	−0.287245	−0.143623	+	0.989633i	$0.545875\pi$
$548$	13.6286	0.582186
$549$	33.2500	1.41907
$550$	−0.713792	−0.0304362
$551$	−5.20775	−0.221858
$552$	24.1957	1.02984
$553$	−48.0930	−2.04512
$554$	9.59850	0.407801
$555$	51.7598	2.19708
$556$	−15.0315	−0.637476
$557$	6.23383	0.264136	0.132068	−	0.991241i	$-0.457838\pi$
$557$	6.23383	0.264136	0.132068	+	0.991241i	$0.457838\pi$
$558$	−8.58642	−0.363492
$559$	0	0
$560$	−7.38404	−0.312033
$561$	−14.2828	−0.603018
$562$	12.8552	0.542262
$563$	7.72827	0.325708	0.162854	−	0.986650i	$-0.447930\pi$
$563$	7.72827	0.325708	0.162854	+	0.986650i	$0.447930\pi$
$564$	0	0
$565$	5.56033	0.233925
$566$	−3.83446	−0.161174
$567$	13.3491	0.560611
$568$	15.8388	0.664580
$569$	−15.6039	−0.654148	−0.327074	−	0.944999i	$-0.606063\pi$
$569$	−15.6039	−0.654148	−0.327074	+	0.944999i	$0.606063\pi$
$570$	5.51573	0.231028
$571$	18.7633	0.785220	0.392610	−	0.919705i	$-0.371572\pi$
$571$	18.7633	0.785220	0.392610	+	0.919705i	$0.371572\pi$
$572$	0	0
$573$	55.1051	2.30205
$574$	6.57242	0.274327
$575$	7.20775	0.300584
$576$	4.24698	0.176957
$577$	−35.1812	−1.46461	−0.732306	−	0.680976i	$-0.761556\pi$
$577$	−35.1812	−1.46461	−0.732306	+	0.680976i	$0.761556\pi$
$578$	18.5308	0.770779
$579$	7.99138	0.332110
$580$	10.6703	0.443058
$581$	−15.0508	−0.624414
$582$	9.43834	0.391232
$583$	3.45175	0.142957
$584$	−4.80731	−0.198928
$585$	0	0
$586$	−30.2935	−1.25141
$587$	32.8504	1.35588	0.677940	−	0.735117i	$-0.262873\pi$
$587$	32.8504	1.35588	0.677940	+	0.735117i	$0.262873\pi$
$588$	−16.1196	−0.664761
$589$	2.02177	0.0833056
$590$	−25.3381	−1.04315
$591$	37.5176	1.54327
$592$	−9.38404	−0.385682
$593$	1.36467	0.0560401	0.0280200	−	0.999607i	$-0.491080\pi$
$593$	1.36467	0.0560401	0.0280200	+	0.999607i	$0.491080\pi$
$594$	−2.98792	−0.122596
$595$	−44.0146	−1.80442
$596$	−12.6746	−0.519170
$597$	19.1400	0.783350
$598$	0	0
$599$	2.46144	0.100572	0.0502858	−	0.998735i	$-0.483987\pi$
$599$	2.46144	0.100572	0.0502858	+	0.998735i	$0.483987\pi$
$600$	2.15883	0.0881340
$601$	−20.6655	−0.842962	−0.421481	−	0.906837i	$-0.638490\pi$
$601$	−20.6655	−0.842962	−0.421481	+	0.906837i	$0.638490\pi$
$602$	−16.9879	−0.692376
$603$	−13.3002	−0.541626
$604$	1.78448	0.0726094
$605$	−20.9148	−0.850309
$606$	49.1148	1.99515
$607$	47.5174	1.92867	0.964336	−	0.264683i	$-0.0852672\pi$
$607$	47.5174	1.92867	0.964336	+	0.264683i	$0.0852672\pi$
$608$	−1.00000	−0.0405554
$609$	50.5241	2.04734
$610$	16.0411	0.649487
$611$	0	0
$612$	25.3153	1.02331
$613$	−3.11231	−0.125705	−0.0628525	−	0.998023i	$-0.520020\pi$
$613$	−3.11231	−0.125705	−0.0628525	+	0.998023i	$0.520020\pi$
$614$	−30.0084	−1.21104
$615$	10.0591	0.405621
$616$	−3.20775	−0.129244
$617$	−23.9667	−0.964865	−0.482432	−	0.875933i	$-0.660247\pi$
$617$	−23.9667	−0.964865	−0.482432	+	0.875933i	$0.660247\pi$
$618$	40.0049	1.60923
$619$	12.9965	0.522375	0.261188	−	0.965288i	$-0.415886\pi$
$619$	12.9965	0.522375	0.261188	+	0.965288i	$0.415886\pi$
$620$	−4.14244	−0.166364
$621$	30.1715	1.21074
$622$	22.3478	0.896065
$623$	−19.5603	−0.783668
$624$	0	0
$625$	−20.3472	−0.813888
$626$	−31.2131	−1.24753
$627$	2.39612	0.0956920
$628$	3.16852	0.126438
$629$	−55.9361	−2.23032
$630$	−31.3599	−1.24941
$631$	4.20642	0.167455	0.0837275	−	0.996489i	$-0.473317\pi$
$631$	4.20642	0.167455	0.0837275	+	0.996489i	$0.473317\pi$
$632$	13.3448	0.530828
$633$	−25.5840	−1.01687
$634$	−8.54825	−0.339495
$635$	−15.8963	−0.630826
$636$	−10.4397	−0.413960
$637$	0	0
$638$	4.63533	0.183515
$639$	67.2669	2.66104
$640$	2.04892	0.0809906
$641$	−12.1655	−0.480510	−0.240255	−	0.970710i	$-0.577231\pi$
$641$	−12.1655	−0.480510	−0.240255	+	0.970710i	$0.577231\pi$
$642$	12.3207	0.486257
$643$	44.1124	1.73962	0.869812	−	0.493383i	$-0.164240\pi$
$643$	44.1124	1.73962	0.869812	+	0.493383i	$0.164240\pi$
$644$	32.3913	1.27640
$645$	−26.0000	−1.02375
$646$	−5.96077	−0.234523
$647$	43.2465	1.70020	0.850098	−	0.526625i	$-0.176543\pi$
$647$	43.2465	1.70020	0.850098	+	0.526625i	$0.176543\pi$
$648$	−3.70410	−0.145511
$649$	−11.0073	−0.432074
$650$	0	0
$651$	−19.6146	−0.768758
$652$	−22.0737	−0.864472
$653$	23.1685	0.906654	0.453327	−	0.891344i	$-0.350237\pi$
$653$	23.1685	0.906654	0.453327	+	0.891344i	$0.350237\pi$
$654$	−10.2935	−0.402508
$655$	33.9952	1.32830
$656$	−1.82371	−0.0712038
$657$	−20.4166	−0.796526
$658$	0	0
$659$	19.1239	0.744962	0.372481	−	0.928040i	$-0.378507\pi$
$659$	19.1239	0.744962	0.372481	+	0.928040i	$0.378507\pi$
$660$	−4.90946	−0.191101
$661$	9.09651	0.353813	0.176907	−	0.984228i	$-0.443391\pi$
$661$	9.09651	0.353813	0.176907	+	0.984228i	$0.443391\pi$
$662$	−1.79118	−0.0696163
$663$	0	0
$664$	4.17629	0.162072
$665$	7.38404	0.286341
$666$	−39.8538	−1.54430
$667$	−46.8068	−1.81237
$668$	−18.0218	−0.697283
$669$	−51.2737	−1.98235
$670$	−6.41657	−0.247894
$671$	6.96854	0.269018
$672$	9.70171	0.374252
$673$	43.4878	1.67633	0.838167	−	0.545414i	$-0.183628\pi$
$673$	43.4878	1.67633	0.838167	+	0.545414i	$0.183628\pi$
$674$	10.5181	0.405143
$675$	2.69202	0.103616
$676$	0	0
$677$	−5.28142	−0.202982	−0.101491	−	0.994836i	$-0.532361\pi$
$677$	−5.28142	−0.202982	−0.101491	+	0.994836i	$0.532361\pi$
$678$	−7.30559	−0.280569
$679$	12.6353	0.484900
$680$	12.2131	0.468352
$681$	−27.7168	−1.06211
$682$	−1.79954	−0.0689081
$683$	−43.8195	−1.67671	−0.838354	−	0.545127i	$-0.816481\pi$
$683$	−43.8195	−1.67671	−0.838354	+	0.545127i	$0.816481\pi$
$684$	−4.24698	−0.162387
$685$	27.9239	1.06692
$686$	3.64742	0.139259
$687$	−23.4722	−0.895520
$688$	4.71379	0.179712
$689$	0	0
$690$	49.5749	1.88729
$691$	27.1535	1.03297	0.516483	−	0.856297i	$-0.327241\pi$
$691$	27.1535	1.03297	0.516483	+	0.856297i	$0.327241\pi$
$692$	−2.19567	−0.0834668
$693$	−13.6233	−0.517505
$694$	−25.4034	−0.964301
$695$	−30.7982	−1.16824
$696$	−14.0194	−0.531403
$697$	−10.8707	−0.411757
$698$	22.7549	0.861287
$699$	44.2704	1.67446
$700$	2.89008	0.109235
$701$	48.0084	1.81325	0.906625	−	0.421937i	$-0.138649\pi$
$701$	48.0084	1.81325	0.906625	+	0.421937i	$0.138649\pi$
$702$	0	0
$703$	9.38404	0.353926
$704$	0.890084	0.0335463
$705$	0	0
$706$	−21.5144	−0.809705
$707$	65.7512	2.47283
$708$	33.2911	1.25116
$709$	−5.71081	−0.214474	−0.107237	−	0.994233i	$-0.534200\pi$
$709$	−5.71081	−0.214474	−0.107237	+	0.994233i	$0.534200\pi$
$710$	32.4523	1.21791
$711$	56.6752	2.12548
$712$	5.42758	0.203407
$713$	18.1715	0.680528
$714$	57.8297	2.16422
$715$	0	0
$716$	−13.1250	−0.490504
$717$	69.3068	2.58831
$718$	28.2198	1.05315
$719$	9.45771	0.352713	0.176357	−	0.984326i	$-0.443569\pi$
$719$	9.45771	0.352713	0.176357	+	0.984326i	$0.443569\pi$
$720$	8.70171	0.324294
$721$	53.5555	1.99451
$722$	1.00000	0.0372161
$723$	−7.18837	−0.267338
$724$	9.74525	0.362179
$725$	−4.17629	−0.155104
$726$	27.4795	1.01986
$727$	−27.5362	−1.02126	−0.510630	−	0.859801i	$-0.670588\pi$
$727$	−27.5362	−1.02126	−0.510630	+	0.859801i	$0.670588\pi$
$728$	0	0
$729$	−42.8418	−1.58673
$730$	−9.84979	−0.364557
$731$	28.0978	1.03924
$732$	−21.0761	−0.778994
$733$	−18.9245	−0.698993	−0.349497	−	0.936938i	$-0.613647\pi$
$733$	−18.9245	−0.698993	−0.349497	+	0.936938i	$0.613647\pi$
$734$	−31.0858	−1.14740
$735$	−33.0277	−1.21825
$736$	−8.98792	−0.331299
$737$	−2.78746	−0.102678
$738$	−7.74525	−0.285107
$739$	11.2513	0.413885	0.206943	−	0.978353i	$-0.433649\pi$
$739$	11.2513	0.413885	0.206943	+	0.978353i	$0.433649\pi$
$740$	−19.2271	−0.706803
$741$	0	0
$742$	−13.9758	−0.513069
$743$	42.0218	1.54163	0.770815	−	0.637060i	$-0.219849\pi$
$743$	42.0218	1.54163	0.770815	+	0.637060i	$0.219849\pi$
$744$	5.44265	0.199537
$745$	−25.9691	−0.951435
$746$	13.2862	0.486443
$747$	17.7366	0.648949
$748$	5.30559	0.193991
$749$	16.4940	0.602676
$750$	32.0019	1.16854
$751$	−44.1473	−1.61096	−0.805480	−	0.592623i	$-0.798092\pi$
$751$	−44.1473	−1.61096	−0.805480	+	0.592623i	$0.798092\pi$
$752$	0	0
$753$	31.2379	1.13837
$754$	0	0
$755$	3.65625	0.133065
$756$	12.0978	0.439994
$757$	41.8920	1.52259	0.761295	−	0.648405i	$-0.224564\pi$
$757$	41.8920	1.52259	0.761295	+	0.648405i	$0.224564\pi$
$758$	−21.7802	−0.791091
$759$	21.5362	0.781714
$760$	−2.04892	−0.0743220
$761$	8.18060	0.296547	0.148273	−	0.988946i	$-0.452628\pi$
$761$	8.18060	0.296547	0.148273	+	0.988946i	$0.452628\pi$
$762$	20.8858	0.756612
$763$	−13.7802	−0.498876
$764$	−20.4698	−0.740571
$765$	51.8689	1.87532
$766$	−1.88338	−0.0680492
$767$	0	0
$768$	−2.69202	−0.0971400
$769$	−40.2881	−1.45283	−0.726414	−	0.687258i	$-0.758814\pi$
$769$	−40.2881	−1.45283	−0.726414	+	0.687258i	$0.758814\pi$
$770$	−6.57242	−0.236853
$771$	−7.51679	−0.270711
$772$	−2.96854	−0.106840
$773$	−19.2728	−0.693194	−0.346597	−	0.938014i	$-0.612663\pi$
$773$	−19.2728	−0.693194	−0.346597	+	0.938014i	$0.612663\pi$
$774$	20.0194	0.719582
$775$	1.62133	0.0582400
$776$	−3.50604	−0.125860
$777$	−91.0413	−3.26609
$778$	−39.2707	−1.40792
$779$	1.82371	0.0653411
$780$	0	0
$781$	14.0978	0.504460
$782$	−53.5749	−1.91584
$783$	−17.4819	−0.624751
$784$	5.98792	0.213854
$785$	6.49204	0.231711
$786$	−44.6655	−1.59316
$787$	21.5114	0.766799	0.383400	−	0.923583i	$-0.374753\pi$
$787$	21.5114	0.766799	0.383400	+	0.923583i	$0.374753\pi$
$788$	−13.9366	−0.496471
$789$	−34.3720	−1.22367
$790$	27.3424	0.972800
$791$	−9.78017	−0.347743
$792$	3.78017	0.134322
$793$	0	0
$794$	−5.65758	−0.200780
$795$	−21.3900	−0.758626
$796$	−7.10992	−0.252004
$797$	7.02284	0.248762	0.124381	−	0.992235i	$-0.460306\pi$
$797$	7.02284	0.248762	0.124381	+	0.992235i	$0.460306\pi$
$798$	−9.70171	−0.343437
$799$	0	0
$800$	−0.801938	−0.0283528
$801$	23.0508	0.814461
$802$	−29.5362	−1.04296
$803$	−4.27891	−0.151000
$804$	8.43057	0.297323
$805$	66.3672	2.33914
$806$	0	0
$807$	31.1207	1.09550
$808$	−18.2446	−0.641842
$809$	36.9178	1.29796	0.648981	−	0.760805i	$-0.275196\pi$
$809$	36.9178	1.29796	0.648981	+	0.760805i	$0.275196\pi$
$810$	−7.58940	−0.266664
$811$	50.5381	1.77463	0.887316	−	0.461162i	$-0.152567\pi$
$811$	50.5381	1.77463	0.887316	+	0.461162i	$0.152567\pi$
$812$	−18.7681	−0.658631
$813$	−19.1400	−0.671271
$814$	−8.35258	−0.292758
$815$	−45.2271	−1.58424
$816$	−16.0465	−0.561741
$817$	−4.71379	−0.164915
$818$	−17.3927	−0.608120
$819$	0	0
$820$	−3.73663	−0.130489
$821$	32.7579	1.14326	0.571630	−	0.820512i	$-0.306311\pi$
$821$	32.7579	1.14326	0.571630	+	0.820512i	$0.306311\pi$
$822$	−36.6886	−1.27966
$823$	9.63401	0.335820	0.167910	−	0.985802i	$-0.446298\pi$
$823$	9.63401	0.335820	0.167910	+	0.985802i	$0.446298\pi$
$824$	−14.8605	−0.517692
$825$	1.92154	0.0668995
$826$	44.5676	1.55071
$827$	−19.9801	−0.694778	−0.347389	−	0.937721i	$-0.612932\pi$
$827$	−19.9801	−0.694778	−0.347389	+	0.937721i	$0.612932\pi$
$828$	−38.1715	−1.32655
$829$	51.7840	1.79853	0.899266	−	0.437401i	$-0.144101\pi$
$829$	51.7840	1.79853	0.899266	+	0.437401i	$0.144101\pi$
$830$	8.55688	0.297014
$831$	−25.8394	−0.896358
$832$	0	0
$833$	35.6926	1.23668
$834$	40.4650	1.40119
$835$	−36.9251	−1.27785
$836$	−0.890084	−0.0307842
$837$	6.78687	0.234589
$838$	−2.34183	−0.0808972
$839$	44.7881	1.54626	0.773128	−	0.634250i	$-0.218691\pi$
$839$	44.7881	1.54626	0.773128	+	0.634250i	$0.218691\pi$
$840$	19.8780	0.685856
$841$	−1.87933	−0.0648045
$842$	−33.3357	−1.14882
$843$	−34.6064	−1.19191
$844$	9.50365	0.327129
$845$	0	0
$846$	0	0
$847$	36.7875	1.26403
$848$	3.87800	0.133171
$849$	10.3225	0.354266
$850$	−4.78017	−0.163958
$851$	84.3430	2.89124
$852$	−42.6383	−1.46076
$853$	3.63342	0.124406	0.0622029	−	0.998064i	$-0.480187\pi$
$853$	3.63342	0.124406	0.0622029	+	0.998064i	$0.480187\pi$
$854$	−28.2150	−0.965499
$855$	−8.70171	−0.297592
$856$	−4.57673	−0.156429
$857$	−8.76330	−0.299349	−0.149674	−	0.988735i	$-0.547823\pi$
$857$	−8.76330	−0.299349	−0.149674	+	0.988735i	$0.547823\pi$
$858$	0	0
$859$	−46.6064	−1.59019	−0.795095	−	0.606485i	$-0.792579\pi$
$859$	−46.6064	−1.59019	−0.795095	+	0.606485i	$0.792579\pi$
$860$	9.65817	0.329341
$861$	−17.6931	−0.602979
$862$	−2.61356	−0.0890183
$863$	1.81892	0.0619168	0.0309584	−	0.999521i	$-0.490144\pi$
$863$	1.81892	0.0619168	0.0309584	+	0.999521i	$0.490144\pi$
$864$	−3.35690	−0.114204
$865$	−4.49875	−0.152962
$866$	30.3236	1.03044
$867$	−49.8853	−1.69419
$868$	7.28621	0.247310
$869$	11.8780	0.402934
$870$	−28.7245	−0.973853
$871$	0	0
$872$	3.82371	0.129487
$873$	−14.8901	−0.503953
$874$	8.98792	0.304021
$875$	42.8418	1.44832
$876$	12.9414	0.437249
$877$	−34.2258	−1.15572	−0.577862	−	0.816135i	$-0.696113\pi$
$877$	−34.2258	−1.15572	−0.577862	+	0.816135i	$0.696113\pi$
$878$	26.3744	0.890091
$879$	81.5508	2.75064
$880$	1.82371	0.0614772
$881$	12.6088	0.424801	0.212400	−	0.977183i	$-0.431872\pi$
$881$	12.6088	0.424801	0.212400	+	0.977183i	$0.431872\pi$
$882$	25.4306	0.856292
$883$	−5.88663	−0.198101	−0.0990504	−	0.995082i	$-0.531580\pi$
$883$	−5.88663	−0.198101	−0.0990504	+	0.995082i	$0.531580\pi$
$884$	0	0
$885$	68.2107	2.29288
$886$	−27.0965	−0.910325
$887$	−17.9734	−0.603489	−0.301745	−	0.953389i	$-0.597569\pi$
$887$	−17.9734	−0.603489	−0.301745	+	0.953389i	$0.597569\pi$
$888$	25.2620	0.847739
$889$	27.9603	0.937758
$890$	11.1207	0.372766
$891$	−3.29696	−0.110452
$892$	19.0465	0.637725
$893$	0	0
$894$	34.1202	1.14115
$895$	−26.8920	−0.898901
$896$	−3.60388	−0.120397
$897$	0	0
$898$	−20.4940	−0.683892
$899$	−10.5289	−0.351158
$900$	−3.40581	−0.113527
$901$	23.1159	0.770102
$902$	−1.62325	−0.0540484
$903$	45.7318	1.52186
$904$	2.71379	0.0902594
$905$	19.9672	0.663733
$906$	−4.80386	−0.159597
$907$	−29.2707	−0.971917	−0.485958	−	0.873982i	$-0.661529\pi$
$907$	−29.2707	−0.971917	−0.485958	+	0.873982i	$0.661529\pi$
$908$	10.2959	0.341681
$909$	−77.4844	−2.56999
$910$	0	0
$911$	11.6045	0.384473	0.192237	−	0.981349i	$-0.438426\pi$
$911$	11.6045	0.384473	0.192237	+	0.981349i	$0.438426\pi$
$912$	2.69202	0.0891417
$913$	3.71725	0.123023
$914$	−5.23729	−0.173234
$915$	−43.1831	−1.42759
$916$	8.71917	0.288089
$917$	−59.7948	−1.97460
$918$	−20.0097	−0.660418
$919$	1.70171	0.0561342	0.0280671	−	0.999606i	$-0.491065\pi$
$919$	1.70171	0.0561342	0.0280671	+	0.999606i	$0.491065\pi$
$920$	−18.4155	−0.607141
$921$	80.7832	2.66190
$922$	−17.0659	−0.562036
$923$	0	0
$924$	8.63533	0.284082
$925$	7.52542	0.247434
$926$	−8.74392	−0.287343
$927$	−63.1124	−2.07288
$928$	5.20775	0.170953
$929$	−10.2416	−0.336016	−0.168008	−	0.985786i	$-0.553733\pi$
$929$	−10.2416	−0.336016	−0.168008	+	0.985786i	$0.553733\pi$
$930$	11.1515	0.365673
$931$	−5.98792	−0.196246
$932$	−16.4450	−0.538675
$933$	−60.1608	−1.96957
$934$	−37.8974	−1.24004
$935$	10.8707	0.355510
$936$	0	0
$937$	28.8140	0.941313	0.470656	−	0.882317i	$-0.344017\pi$
$937$	28.8140	0.941313	0.470656	+	0.882317i	$0.344017\pi$
$938$	11.2862	0.368508
$939$	84.0264	2.74210
$940$	0	0
$941$	47.3357	1.54310	0.771550	−	0.636169i	$-0.219482\pi$
$941$	47.3357	1.54310	0.771550	+	0.636169i	$0.219482\pi$
$942$	−8.52973	−0.277914
$943$	16.3913	0.533775
$944$	−12.3666	−0.402498
$945$	24.7875	0.806336
$946$	4.19567	0.136413
$947$	−2.08708	−0.0678210	−0.0339105	−	0.999425i	$-0.510796\pi$
$947$	−2.08708	−0.0678210	−0.0339105	+	0.999425i	$0.510796\pi$
$948$	−35.9245	−1.16677
$949$	0	0
$950$	0.801938	0.0260183
$951$	23.0121	0.746218
$952$	−21.4819	−0.696232
$953$	−33.8926	−1.09789	−0.548944	−	0.835859i	$-0.684970\pi$
$953$	−33.8926	−1.09789	−0.548944	+	0.835859i	$0.684970\pi$
$954$	16.4698	0.533229
$955$	−41.9409	−1.35718
$956$	−25.7453	−0.832661
$957$	−12.4784	−0.403370
$958$	−36.7004	−1.18574
$959$	−49.1159	−1.58603
$960$	−5.51573	−0.178019
$961$	−26.9124	−0.868143
$962$	0	0
$963$	−19.4373	−0.626357
$964$	2.67025	0.0860030
$965$	−6.08230	−0.195796
$966$	−87.1982	−2.80556
$967$	−53.2513	−1.71245	−0.856223	−	0.516606i	$-0.827195\pi$
$967$	−53.2513	−1.71245	−0.856223	+	0.516606i	$0.827195\pi$
$968$	−10.2078	−0.328090
$969$	16.0465	0.515489
$970$	−7.18359	−0.230651
$971$	−24.3201	−0.780468	−0.390234	−	0.920716i	$-0.627606\pi$
$971$	−24.3201	−0.780468	−0.390234	+	0.920716i	$0.627606\pi$
$972$	20.0422	0.642854
$973$	54.1715	1.73666
$974$	−28.1497	−0.901976
$975$	0	0
$976$	7.82908	0.250603
$977$	−49.5690	−1.58585	−0.792926	−	0.609318i	$-0.791443\pi$
$977$	−49.5690	−1.58585	−0.792926	+	0.609318i	$0.791443\pi$
$978$	59.4228	1.90013
$979$	4.83100	0.154400
$980$	12.2687	0.391911
$981$	16.2392	0.518478
$982$	27.0315	0.862609
$983$	9.94139	0.317081	0.158541	−	0.987352i	$-0.449321\pi$
$983$	9.94139	0.317081	0.158541	+	0.987352i	$0.449321\pi$
$984$	4.90946	0.156508
$985$	−28.5550	−0.909837
$986$	31.0422	0.988585
$987$	0	0
$988$	0	0
$989$	−42.3672	−1.34720
$990$	7.74525	0.246160
$991$	46.9971	1.49291	0.746457	−	0.665434i	$-0.231754\pi$
$991$	46.9971	1.49291	0.746457	+	0.665434i	$0.231754\pi$
$992$	−2.02177	−0.0641913
$993$	4.82191	0.153019
$994$	−57.0810	−1.81050
$995$	−14.5676	−0.461825
$996$	−11.2427	−0.356238
$997$	43.1075	1.36523	0.682614	−	0.730779i	$-0.260843\pi$
$997$	43.1075	1.36523	0.682614	+	0.730779i	$0.260843\pi$
$998$	7.53617	0.238553
$999$	31.5013	0.996656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.r.1.1	yes	3
13.12	even	2		6422.2.a.l.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.l.1.1	✓	3	13.12	even	2
6422.2.a.r.1.1	yes	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.69202 q^{3} +1.00000 q^{4} +2.04892 q^{5} -2.69202 q^{6} -3.60388 q^{7} +1.00000 q^{8} +4.24698 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.69202 q^{3} +1.00000 q^{4} +2.04892 q^{5} -2.69202 q^{6} -3.60388 q^{7} +1.00000 q^{8} +4.24698 q^{9} +2.04892 q^{10} +0.890084 q^{11} -2.69202 q^{12} -3.60388 q^{14} -5.51573 q^{15} +1.00000 q^{16} +5.96077 q^{17} +4.24698 q^{18} -1.00000 q^{19} +2.04892 q^{20} +9.70171 q^{21} +0.890084 q^{22} -8.98792 q^{23} -2.69202 q^{24} -0.801938 q^{25} -3.35690 q^{27} -3.60388 q^{28} +5.20775 q^{29} -5.51573 q^{30} -2.02177 q^{31} +1.00000 q^{32} -2.39612 q^{33} +5.96077 q^{34} -7.38404 q^{35} +4.24698 q^{36} -9.38404 q^{37} -1.00000 q^{38} +2.04892 q^{40} -1.82371 q^{41} +9.70171 q^{42} +4.71379 q^{43} +0.890084 q^{44} +8.70171 q^{45} -8.98792 q^{46} -2.69202 q^{48} +5.98792 q^{49} -0.801938 q^{50} -16.0465 q^{51} +3.87800 q^{53} -3.35690 q^{54} +1.82371 q^{55} -3.60388 q^{56} +2.69202 q^{57} +5.20775 q^{58} -12.3666 q^{59} -5.51573 q^{60} +7.82908 q^{61} -2.02177 q^{62} -15.3056 q^{63} +1.00000 q^{64} -2.39612 q^{66} -3.13169 q^{67} +5.96077 q^{68} +24.1957 q^{69} -7.38404 q^{70} +15.8388 q^{71} +4.24698 q^{72} -4.80731 q^{73} -9.38404 q^{74} +2.15883 q^{75} -1.00000 q^{76} -3.20775 q^{77} +13.3448 q^{79} +2.04892 q^{80} -3.70410 q^{81} -1.82371 q^{82} +4.17629 q^{83} +9.70171 q^{84} +12.2131 q^{85} +4.71379 q^{86} -14.0194 q^{87} +0.890084 q^{88} +5.42758 q^{89} +8.70171 q^{90} -8.98792 q^{92} +5.44265 q^{93} -2.04892 q^{95} -2.69202 q^{96} -3.50604 q^{97} +5.98792 q^{98} +3.78017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 8 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 2 q^{14} - 4 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} - 3 q^{19} - 3 q^{20} + 2 q^{21} + 2 q^{22} - 8 q^{23} - 3 q^{24} + 2 q^{25} - 6 q^{27} - 2 q^{28} - 2 q^{29} - 4 q^{30} - 3 q^{31} + 3 q^{32} - 16 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 18 q^{37} - 3 q^{38} - 3 q^{40} + 2 q^{41} + 2 q^{42} + 6 q^{43} + 2 q^{44} - q^{45} - 8 q^{46} - 3 q^{48} - q^{49} + 2 q^{50} + 2 q^{51} - 8 q^{53} - 6 q^{54} - 2 q^{55} - 2 q^{56} + 3 q^{57} - 2 q^{58} - 11 q^{59} - 4 q^{60} + 13 q^{61} - 3 q^{62} - 10 q^{63} + 3 q^{64} - 16 q^{66} - 7 q^{67} + 5 q^{68} + 36 q^{69} - 12 q^{70} + 15 q^{71} + 8 q^{72} - 7 q^{73} - 18 q^{74} - 2 q^{75} - 3 q^{76} + 8 q^{77} + 17 q^{79} - 3 q^{80} - 25 q^{81} + 2 q^{82} + 20 q^{83} + 2 q^{84} + 16 q^{85} + 6 q^{86} + 2 q^{87} + 2 q^{88} - q^{90} - 8 q^{92} - 25 q^{93} + 3 q^{95} - 3 q^{96} - 20 q^{97} - q^{98} + 10 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 8 * q^9 - 3 * q^10 + 2 * q^11 - 3 * q^12 - 2 * q^14 - 4 * q^15 + 3 * q^16 + 5 * q^17 + 8 * q^18 - 3 * q^19 - 3 * q^20 + 2 * q^21 + 2 * q^22 - 8 * q^23 - 3 * q^24 + 2 * q^25 - 6 * q^27 - 2 * q^28 - 2 * q^29 - 4 * q^30 - 3 * q^31 + 3 * q^32 - 16 * q^33 + 5 * q^34 - 12 * q^35 + 8 * q^36 - 18 * q^37 - 3 * q^38 - 3 * q^40 + 2 * q^41 + 2 * q^42 + 6 * q^43 + 2 * q^44 - q^45 - 8 * q^46 - 3 * q^48 - q^49 + 2 * q^50 + 2 * q^51 - 8 * q^53 - 6 * q^54 - 2 * q^55 - 2 * q^56 + 3 * q^57 - 2 * q^58 - 11 * q^59 - 4 * q^60 + 13 * q^61 - 3 * q^62 - 10 * q^63 + 3 * q^64 - 16 * q^66 - 7 * q^67 + 5 * q^68 + 36 * q^69 - 12 * q^70 + 15 * q^71 + 8 * q^72 - 7 * q^73 - 18 * q^74 - 2 * q^75 - 3 * q^76 + 8 * q^77 + 17 * q^79 - 3 * q^80 - 25 * q^81 + 2 * q^82 + 20 * q^83 + 2 * q^84 + 16 * q^85 + 6 * q^86 + 2 * q^87 + 2 * q^88 - q^90 - 8 * q^92 - 25 * q^93 + 3 * q^95 - 3 * q^96 - 20 * q^97 - q^98 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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