LMFDB - Embedded newform 6002.2.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6002,2,Mod(1,6002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6002 = 2 \cdot 3001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6002.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:	$47.9262112932$
Analytic rank:	$1$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.14369 q^{3} +1.00000 q^{4} -1.86034 q^{5} +2.14369 q^{6} +1.91377 q^{7} -1.00000 q^{8} +1.59541 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.14369 q^{3} +1.00000 q^{4} -1.86034 q^{5} +2.14369 q^{6} +1.91377 q^{7} -1.00000 q^{8} +1.59541 q^{9} +1.86034 q^{10} +4.74287 q^{11} -2.14369 q^{12} +1.69796 q^{13} -1.91377 q^{14} +3.98799 q^{15} +1.00000 q^{16} -0.0172067 q^{17} -1.59541 q^{18} -4.50724 q^{19} -1.86034 q^{20} -4.10253 q^{21} -4.74287 q^{22} -8.01985 q^{23} +2.14369 q^{24} -1.53914 q^{25} -1.69796 q^{26} +3.01100 q^{27} +1.91377 q^{28} +7.99280 q^{29} -3.98799 q^{30} +10.2484 q^{31} -1.00000 q^{32} -10.1672 q^{33} +0.0172067 q^{34} -3.56026 q^{35} +1.59541 q^{36} -3.48625 q^{37} +4.50724 q^{38} -3.63990 q^{39} +1.86034 q^{40} -3.03695 q^{41} +4.10253 q^{42} -11.2808 q^{43} +4.74287 q^{44} -2.96801 q^{45} +8.01985 q^{46} +5.63767 q^{47} -2.14369 q^{48} -3.33749 q^{49} +1.53914 q^{50} +0.0368859 q^{51} +1.69796 q^{52} -0.0585337 q^{53} -3.01100 q^{54} -8.82335 q^{55} -1.91377 q^{56} +9.66213 q^{57} -7.99280 q^{58} -9.29402 q^{59} +3.98799 q^{60} -3.96328 q^{61} -10.2484 q^{62} +3.05325 q^{63} +1.00000 q^{64} -3.15878 q^{65} +10.1672 q^{66} -4.63853 q^{67} -0.0172067 q^{68} +17.1921 q^{69} +3.56026 q^{70} +4.21614 q^{71} -1.59541 q^{72} -1.73576 q^{73} +3.48625 q^{74} +3.29943 q^{75} -4.50724 q^{76} +9.07675 q^{77} +3.63990 q^{78} +6.16438 q^{79} -1.86034 q^{80} -11.2409 q^{81} +3.03695 q^{82} -1.23768 q^{83} -4.10253 q^{84} +0.0320103 q^{85} +11.2808 q^{86} -17.1341 q^{87} -4.74287 q^{88} +0.798011 q^{89} +2.96801 q^{90} +3.24950 q^{91} -8.01985 q^{92} -21.9693 q^{93} -5.63767 q^{94} +8.38500 q^{95} +2.14369 q^{96} -7.14399 q^{97} +3.33749 q^{98} +7.56683 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 56 q^{2} - 11 q^{3} + 56 q^{4} + 11 q^{6} - 21 q^{7} - 56 q^{8} + 53 q^{9}+O(q^{10}) $ 56 * q - 56 * q^2 - 11 * q^3 + 56 * q^4 + 11 * q^6 - 21 * q^7 - 56 * q^8 + 53 * q^9	\( 56 q - 56 q^{2} - 11 q^{3} + 56 q^{4} + 11 q^{6} - 21 q^{7} - 56 q^{8} + 53 q^{9} + 12 q^{11} - 11 q^{12} - 31 q^{13} + 21 q^{14} - 22 q^{15} + 56 q^{16} - 4 q^{17} - 53 q^{18} - 9 q^{19} + 13 q^{21} - 12 q^{22} - 39 q^{23} + 11 q^{24} + 8 q^{25} + 31 q^{26} - 44 q^{27} - 21 q^{28} + 13 q^{29} + 22 q^{30} - 35 q^{31} - 56 q^{32} - 26 q^{33} + 4 q^{34} - 7 q^{35} + 53 q^{36} - 65 q^{37} + 9 q^{38} - 27 q^{39} + 38 q^{41} - 13 q^{42} - 76 q^{43} + 12 q^{44} - 21 q^{45} + 39 q^{46} - 43 q^{47} - 11 q^{48} + 9 q^{49} - 8 q^{50} - 19 q^{51} - 31 q^{52} - 26 q^{53} + 44 q^{54} - 67 q^{55} + 21 q^{56} - 26 q^{57} - 13 q^{58} + 11 q^{59} - 22 q^{60} - 17 q^{61} + 35 q^{62} - 67 q^{63} + 56 q^{64} + 31 q^{65} + 26 q^{66} - 93 q^{67} - 4 q^{68} - 13 q^{69} + 7 q^{70} - 33 q^{71} - 53 q^{72} - 41 q^{73} + 65 q^{74} - 21 q^{75} - 9 q^{76} + 5 q^{77} + 27 q^{78} - 69 q^{79} + 36 q^{81} - 38 q^{82} + 4 q^{83} + 13 q^{84} - 40 q^{85} + 76 q^{86} - 69 q^{87} - 12 q^{88} + 40 q^{89} + 21 q^{90} - 64 q^{91} - 39 q^{92} - 57 q^{93} + 43 q^{94} - 22 q^{95} + 11 q^{96} - 71 q^{97} - 9 q^{98} + 17 q^{99}+O(q^{100}) \) 56 * q - 56 * q^2 - 11 * q^3 + 56 * q^4 + 11 * q^6 - 21 * q^7 - 56 * q^8 + 53 * q^9 + 12 * q^11 - 11 * q^12 - 31 * q^13 + 21 * q^14 - 22 * q^15 + 56 * q^16 - 4 * q^17 - 53 * q^18 - 9 * q^19 + 13 * q^21 - 12 * q^22 - 39 * q^23 + 11 * q^24 + 8 * q^25 + 31 * q^26 - 44 * q^27 - 21 * q^28 + 13 * q^29 + 22 * q^30 - 35 * q^31 - 56 * q^32 - 26 * q^33 + 4 * q^34 - 7 * q^35 + 53 * q^36 - 65 * q^37 + 9 * q^38 - 27 * q^39 + 38 * q^41 - 13 * q^42 - 76 * q^43 + 12 * q^44 - 21 * q^45 + 39 * q^46 - 43 * q^47 - 11 * q^48 + 9 * q^49 - 8 * q^50 - 19 * q^51 - 31 * q^52 - 26 * q^53 + 44 * q^54 - 67 * q^55 + 21 * q^56 - 26 * q^57 - 13 * q^58 + 11 * q^59 - 22 * q^60 - 17 * q^61 + 35 * q^62 - 67 * q^63 + 56 * q^64 + 31 * q^65 + 26 * q^66 - 93 * q^67 - 4 * q^68 - 13 * q^69 + 7 * q^70 - 33 * q^71 - 53 * q^72 - 41 * q^73 + 65 * q^74 - 21 * q^75 - 9 * q^76 + 5 * q^77 + 27 * q^78 - 69 * q^79 + 36 * q^81 - 38 * q^82 + 4 * q^83 + 13 * q^84 - 40 * q^85 + 76 * q^86 - 69 * q^87 - 12 * q^88 + 40 * q^89 + 21 * q^90 - 64 * q^91 - 39 * q^92 - 57 * q^93 + 43 * q^94 - 22 * q^95 + 11 * q^96 - 71 * q^97 - 9 * q^98 + 17 * 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
$3$	−2.14369	−1.23766	−0.618830	−	0.785525i	$-0.712393\pi$
$3$	−2.14369	−1.23766	−0.618830	+	0.785525i	$0.712393\pi$
$4$	1.00000	0.500000
$5$	−1.86034	−0.831969	−0.415985	−	0.909372i	$-0.636563\pi$
$5$	−1.86034	−0.831969	−0.415985	+	0.909372i	$0.636563\pi$
$6$	2.14369	0.875158
$7$	1.91377	0.723336	0.361668	−	0.932307i	$-0.382207\pi$
$7$	1.91377	0.723336	0.361668	+	0.932307i	$0.382207\pi$
$8$	−1.00000	−0.353553
$9$	1.59541	0.531804
$10$	1.86034	0.588291
$11$	4.74287	1.43003	0.715014	−	0.699110i	$-0.246420\pi$
$11$	4.74287	1.43003	0.715014	+	0.699110i	$0.246420\pi$
$12$	−2.14369	−0.618830
$13$	1.69796	0.470929	0.235464	−	0.971883i	$-0.424339\pi$
$13$	1.69796	0.470929	0.235464	+	0.971883i	$0.424339\pi$
$14$	−1.91377	−0.511476
$15$	3.98799	1.02970
$16$	1.00000	0.250000
$17$	−0.0172067	−0.00417324	−0.00208662	−	0.999998i	$-0.500664\pi$
$17$	−0.0172067	−0.00417324	−0.00208662	+	0.999998i	$0.500664\pi$
$18$	−1.59541	−0.376042
$19$	−4.50724	−1.03403	−0.517016	−	0.855976i	$-0.672957\pi$
$19$	−4.50724	−1.03403	−0.517016	+	0.855976i	$0.672957\pi$
$20$	−1.86034	−0.415985
$21$	−4.10253	−0.895245
$22$	−4.74287	−1.01118
$23$	−8.01985	−1.67225	−0.836127	−	0.548536i	$-0.815185\pi$
$23$	−8.01985	−1.67225	−0.836127	+	0.548536i	$0.815185\pi$
$24$	2.14369	0.437579
$25$	−1.53914	−0.307827
$26$	−1.69796	−0.332997
$27$	3.01100	0.579467
$28$	1.91377	0.361668
$29$	7.99280	1.48423	0.742113	−	0.670275i	$-0.233824\pi$
$29$	7.99280	1.48423	0.742113	+	0.670275i	$0.233824\pi$
$30$	−3.98799	−0.728105
$31$	10.2484	1.84066	0.920329	−	0.391144i	$-0.127921\pi$
$31$	10.2484	1.84066	0.920329	+	0.391144i	$0.127921\pi$
$32$	−1.00000	−0.176777
$33$	−10.1672	−1.76989
$34$	0.0172067	0.00295093
$35$	−3.56026	−0.601793
$36$	1.59541	0.265902
$37$	−3.48625	−0.573135	−0.286568	−	0.958060i	$-0.592514\pi$
$37$	−3.48625	−0.573135	−0.286568	+	0.958060i	$0.592514\pi$
$38$	4.50724	0.731171
$39$	−3.63990	−0.582850
$40$	1.86034	0.294146
$41$	−3.03695	−0.474292	−0.237146	−	0.971474i	$-0.576212\pi$
$41$	−3.03695	−0.474292	−0.237146	+	0.971474i	$0.576212\pi$
$42$	4.10253	0.633034
$43$	−11.2808	−1.72030	−0.860152	−	0.510037i	$-0.829632\pi$
$43$	−11.2808	−1.72030	−0.860152	+	0.510037i	$0.829632\pi$
$44$	4.74287	0.715014
$45$	−2.96801	−0.442445
$46$	8.01985	1.18246
$47$	5.63767	0.822338	0.411169	−	0.911559i	$-0.365121\pi$
$47$	5.63767	0.822338	0.411169	+	0.911559i	$0.365121\pi$
$48$	−2.14369	−0.309415
$49$	−3.33749	−0.476785
$50$	1.53914	0.217667
$51$	0.0368859	0.00516505
$52$	1.69796	0.235464
$53$	−0.0585337	−0.00804022	−0.00402011	−	0.999992i	$-0.501280\pi$
$53$	−0.0585337	−0.00804022	−0.00402011	+	0.999992i	$0.501280\pi$
$54$	−3.01100	−0.409745
$55$	−8.82335	−1.18974
$56$	−1.91377	−0.255738
$57$	9.66213	1.27978
$58$	−7.99280	−1.04951
$59$	−9.29402	−1.20998	−0.604989	−	0.796234i	$-0.706823\pi$
$59$	−9.29402	−1.20998	−0.604989	+	0.796234i	$0.706823\pi$
$60$	3.98799	0.514848
$61$	−3.96328	−0.507445	−0.253723	−	0.967277i	$-0.581655\pi$
$61$	−3.96328	−0.507445	−0.253723	+	0.967277i	$0.581655\pi$
$62$	−10.2484	−1.30154
$63$	3.05325	0.384673
$64$	1.00000	0.125000
$65$	−3.15878	−0.391798
$66$	10.1672	1.25150
$67$	−4.63853	−0.566687	−0.283344	−	0.959018i	$-0.591444\pi$
$67$	−4.63853	−0.566687	−0.283344	+	0.959018i	$0.591444\pi$
$68$	−0.0172067	−0.00208662
$69$	17.1921	2.06968
$70$	3.56026	0.425532
$71$	4.21614	0.500363	0.250182	−	0.968199i	$-0.419510\pi$
$71$	4.21614	0.500363	0.250182	+	0.968199i	$0.419510\pi$
$72$	−1.59541	−0.188021
$73$	−1.73576	−0.203155	−0.101577	−	0.994828i	$-0.532389\pi$
$73$	−1.73576	−0.203155	−0.101577	+	0.994828i	$0.532389\pi$
$74$	3.48625	0.405268
$75$	3.29943	0.380986
$76$	−4.50724	−0.517016
$77$	9.07675	1.03439
$78$	3.63990	0.412137
$79$	6.16438	0.693547	0.346774	−	0.937949i	$-0.387277\pi$
$79$	6.16438	0.693547	0.346774	+	0.937949i	$0.387277\pi$
$80$	−1.86034	−0.207992
$81$	−11.2409	−1.24899
$82$	3.03695	0.335375
$83$	−1.23768	−0.135853	−0.0679266	−	0.997690i	$-0.521638\pi$
$83$	−1.23768	−0.135853	−0.0679266	+	0.997690i	$0.521638\pi$
$84$	−4.10253	−0.447622
$85$	0.0320103	0.00347201
$86$	11.2808	1.21644
$87$	−17.1341	−1.83697
$88$	−4.74287	−0.505591
$89$	0.798011	0.0845890	0.0422945	−	0.999105i	$-0.486533\pi$
$89$	0.798011	0.0845890	0.0422945	+	0.999105i	$0.486533\pi$
$90$	2.96801	0.312856
$91$	3.24950	0.340640
$92$	−8.01985	−0.836127
$93$	−21.9693	−2.27811
$94$	−5.63767	−0.581481
$95$	8.38500	0.860282
$96$	2.14369	0.218790
$97$	−7.14399	−0.725362	−0.362681	−	0.931913i	$-0.618139\pi$
$97$	−7.14399	−0.725362	−0.362681	+	0.931913i	$0.618139\pi$
$98$	3.33749	0.337138
$99$	7.56683	0.760496
$100$	−1.53914	−0.153914
$101$	−1.52326	−0.151570	−0.0757848	−	0.997124i	$-0.524146\pi$
$101$	−1.52326	−0.151570	−0.0757848	+	0.997124i	$0.524146\pi$
$102$	−0.0368859	−0.00365224
$103$	−2.17475	−0.214285	−0.107142	−	0.994244i	$-0.534170\pi$
$103$	−2.17475	−0.214285	−0.107142	+	0.994244i	$0.534170\pi$
$104$	−1.69796	−0.166498
$105$	7.63209	0.744816
$106$	0.0585337	0.00568529
$107$	17.7309	1.71411	0.857056	−	0.515223i	$-0.172291\pi$
$107$	17.7309	1.71411	0.857056	+	0.515223i	$0.172291\pi$
$108$	3.01100	0.289734
$109$	11.7475	1.12521	0.562603	−	0.826727i	$-0.309800\pi$
$109$	11.7475	1.12521	0.562603	+	0.826727i	$0.309800\pi$
$110$	8.82335	0.841273
$111$	7.47344	0.709347
$112$	1.91377	0.180834
$113$	−7.12556	−0.670316	−0.335158	−	0.942162i	$-0.608790\pi$
$113$	−7.12556	−0.670316	−0.335158	+	0.942162i	$0.608790\pi$
$114$	−9.66213	−0.904942
$115$	14.9196	1.39126
$116$	7.99280	0.742113
$117$	2.70894	0.250442
$118$	9.29402	0.855584
$119$	−0.0329296	−0.00301865
$120$	−3.98799	−0.364052
$121$	11.4948	1.04498
$122$	3.96328	0.358818
$123$	6.51028	0.587012
$124$	10.2484	0.920329
$125$	12.1650	1.08807
$126$	−3.05325	−0.272005
$127$	7.07560	0.627858	0.313929	−	0.949446i	$-0.398355\pi$
$127$	7.07560	0.627858	0.313929	+	0.949446i	$0.398355\pi$
$128$	−1.00000	−0.0883883
$129$	24.1825	2.12915
$130$	3.15878	0.277043
$131$	8.44998	0.738278	0.369139	−	0.929374i	$-0.379653\pi$
$131$	8.44998	0.738278	0.369139	+	0.929374i	$0.379653\pi$
$132$	−10.1672	−0.884945
$133$	−8.62581	−0.747953
$134$	4.63853	0.400708
$135$	−5.60148	−0.482099
$136$	0.0172067	0.00147546
$137$	2.53289	0.216399	0.108200	−	0.994129i	$-0.465491\pi$
$137$	2.53289	0.216399	0.108200	+	0.994129i	$0.465491\pi$
$138$	−17.1921	−1.46349
$139$	−18.2630	−1.54904	−0.774522	−	0.632547i	$-0.782010\pi$
$139$	−18.2630	−1.54904	−0.774522	+	0.632547i	$0.782010\pi$
$140$	−3.56026	−0.300897
$141$	−12.0854	−1.01778
$142$	−4.21614	−0.353810
$143$	8.05319	0.673441
$144$	1.59541	0.132951
$145$	−14.8693	−1.23483
$146$	1.73576	0.143652
$147$	7.15456	0.590098
$148$	−3.48625	−0.286568
$149$	18.5461	1.51936	0.759680	−	0.650297i	$-0.225356\pi$
$149$	18.5461	1.51936	0.759680	+	0.650297i	$0.225356\pi$
$150$	−3.29943	−0.269398
$151$	1.60434	0.130559	0.0652796	−	0.997867i	$-0.479206\pi$
$151$	1.60434	0.130559	0.0652796	+	0.997867i	$0.479206\pi$
$152$	4.50724	0.365585
$153$	−0.0274518	−0.00221935
$154$	−9.07675	−0.731425
$155$	−19.0654	−1.53137
$156$	−3.63990	−0.291425
$157$	11.3900	0.909024	0.454512	−	0.890741i	$-0.349814\pi$
$157$	11.3900	0.909024	0.454512	+	0.890741i	$0.349814\pi$
$158$	−6.16438	−0.490412
$159$	0.125478	0.00995107
$160$	1.86034	0.147073
$161$	−15.3481	−1.20960
$162$	11.2409	0.883168
$163$	−13.0728	−1.02394	−0.511969	−	0.859004i	$-0.671084\pi$
$163$	−13.0728	−1.02394	−0.511969	+	0.859004i	$0.671084\pi$
$164$	−3.03695	−0.237146
$165$	18.9145	1.47249
$166$	1.23768	0.0960627
$167$	5.73327	0.443654	0.221827	−	0.975086i	$-0.428798\pi$
$167$	5.73327	0.443654	0.221827	+	0.975086i	$0.428798\pi$
$168$	4.10253	0.316517
$169$	−10.1169	−0.778226
$170$	−0.0320103	−0.00245508
$171$	−7.19091	−0.549903
$172$	−11.2808	−0.860152
$173$	12.2228	0.929286	0.464643	−	0.885498i	$-0.346183\pi$
$173$	12.2228	0.929286	0.464643	+	0.885498i	$0.346183\pi$
$174$	17.1341	1.29893
$175$	−2.94555	−0.222663
$176$	4.74287	0.357507
$177$	19.9235	1.49754
$178$	−0.798011	−0.0598134
$179$	16.7468	1.25171	0.625856	−	0.779939i	$-0.284750\pi$
$179$	16.7468	1.25171	0.625856	+	0.779939i	$0.284750\pi$
$180$	−2.96801	−0.221222
$181$	−5.76973	−0.428861	−0.214430	−	0.976739i	$-0.568790\pi$
$181$	−5.76973	−0.428861	−0.214430	+	0.976739i	$0.568790\pi$
$182$	−3.24950	−0.240869
$183$	8.49604	0.628045
$184$	8.01985	0.591231
$185$	6.48560	0.476831
$186$	21.9693	1.61087
$187$	−0.0816091	−0.00596785
$188$	5.63767	0.411169
$189$	5.76236	0.419150
$190$	−8.38500	−0.608312
$191$	24.8359	1.79706	0.898532	−	0.438907i	$-0.144634\pi$
$191$	24.8359	1.79706	0.898532	+	0.438907i	$0.144634\pi$
$192$	−2.14369	−0.154708
$193$	−5.21355	−0.375279	−0.187640	−	0.982238i	$-0.560084\pi$
$193$	−5.21355	−0.375279	−0.187640	+	0.982238i	$0.560084\pi$
$194$	7.14399	0.512909
$195$	6.77144	0.484913
$196$	−3.33749	−0.238392
$197$	−23.7296	−1.69066	−0.845332	−	0.534242i	$-0.820597\pi$
$197$	−23.7296	−1.69066	−0.845332	+	0.534242i	$0.820597\pi$
$198$	−7.56683	−0.537752
$199$	26.0864	1.84921	0.924607	−	0.380922i	$-0.124393\pi$
$199$	26.0864	1.84921	0.924607	+	0.380922i	$0.124393\pi$
$200$	1.53914	0.108833
$201$	9.94359	0.701367
$202$	1.52326	0.107176
$203$	15.2964	1.07359
$204$	0.0368859	0.00258253
$205$	5.64976	0.394596
$206$	2.17475	0.151522
$207$	−12.7950	−0.889312
$208$	1.69796	0.117732
$209$	−21.3772	−1.47869
$210$	−7.63209	−0.526665
$211$	−3.02501	−0.208251	−0.104125	−	0.994564i	$-0.533204\pi$
$211$	−3.02501	−0.208251	−0.104125	+	0.994564i	$0.533204\pi$
$212$	−0.0585337	−0.00402011
$213$	−9.03810	−0.619280
$214$	−17.7309	−1.21206
$215$	20.9861	1.43124
$216$	−3.01100	−0.204873
$217$	19.6130	1.33142
$218$	−11.7475	−0.795641
$219$	3.72093	0.251437
$220$	−8.82335	−0.594870
$221$	−0.0292162	−0.00196530
$222$	−7.47344	−0.501584
$223$	−4.18889	−0.280509	−0.140254	−	0.990115i	$-0.544792\pi$
$223$	−4.18889	−0.280509	−0.140254	+	0.990115i	$0.544792\pi$
$224$	−1.91377	−0.127869
$225$	−2.45556	−0.163704
$226$	7.12556	0.473985
$227$	−8.29637	−0.550650	−0.275325	−	0.961351i	$-0.588785\pi$
$227$	−8.29637	−0.550650	−0.275325	+	0.961351i	$0.588785\pi$
$228$	9.66213	0.639890
$229$	4.48606	0.296447	0.148224	−	0.988954i	$-0.452644\pi$
$229$	4.48606	0.296447	0.148224	+	0.988954i	$0.452644\pi$
$230$	−14.9196	−0.983772
$231$	−19.4577	−1.28023
$232$	−7.99280	−0.524753
$233$	−22.3189	−1.46216	−0.731080	−	0.682292i	$-0.760983\pi$
$233$	−22.3189	−1.46216	−0.731080	+	0.682292i	$0.760983\pi$
$234$	−2.70894	−0.177089
$235$	−10.4880	−0.684160
$236$	−9.29402	−0.604989
$237$	−13.2145	−0.858376
$238$	0.0329296	0.00213451
$239$	−21.9317	−1.41864	−0.709322	−	0.704885i	$-0.750999\pi$
$239$	−21.9317	−1.41864	−0.709322	+	0.704885i	$0.750999\pi$
$240$	3.98799	0.257424
$241$	−6.00107	−0.386563	−0.193282	−	0.981143i	$-0.561913\pi$
$241$	−6.00107	−0.386563	−0.193282	+	0.981143i	$0.561913\pi$
$242$	−11.4948	−0.738914
$243$	15.0640	0.966357
$244$	−3.96328	−0.253723
$245$	6.20887	0.396670
$246$	−6.51028	−0.415080
$247$	−7.65310	−0.486955
$248$	−10.2484	−0.650771
$249$	2.65321	0.168140
$250$	−12.1650	−0.769383
$251$	−22.0724	−1.39320	−0.696599	−	0.717460i	$-0.745304\pi$
$251$	−22.0724	−1.39320	−0.696599	+	0.717460i	$0.745304\pi$
$252$	3.05325	0.192337
$253$	−38.0371	−2.39137
$254$	−7.07560	−0.443963
$255$	−0.0686202	−0.00429717
$256$	1.00000	0.0625000
$257$	16.7365	1.04399	0.521996	−	0.852948i	$-0.325187\pi$
$257$	16.7365	1.04399	0.521996	+	0.852948i	$0.325187\pi$
$258$	−24.1825	−1.50554
$259$	−6.67187	−0.414570
$260$	−3.15878	−0.195899
$261$	12.7518	0.789317
$262$	−8.44998	−0.522042
$263$	−29.4381	−1.81523	−0.907615	−	0.419804i	$-0.862099\pi$
$263$	−29.4381	−1.81523	−0.907615	+	0.419804i	$0.862099\pi$
$264$	10.1672	0.625751
$265$	0.108893	0.00668921
$266$	8.62581	0.528882
$267$	−1.71069	−0.104692
$268$	−4.63853	−0.283344
$269$	−30.2519	−1.84449	−0.922247	−	0.386601i	$-0.873649\pi$
$269$	−30.2519	−1.84449	−0.922247	+	0.386601i	$0.873649\pi$
$270$	5.60148	0.340895
$271$	−14.0062	−0.850818	−0.425409	−	0.905001i	$-0.639870\pi$
$271$	−14.0062	−0.850818	−0.425409	+	0.905001i	$0.639870\pi$
$272$	−0.0172067	−0.00104331
$273$	−6.96592	−0.421596
$274$	−2.53289	−0.153018
$275$	−7.29992	−0.440202
$276$	17.1921	1.03484
$277$	−3.53381	−0.212326	−0.106163	−	0.994349i	$-0.533857\pi$
$277$	−3.53381	−0.212326	−0.106163	+	0.994349i	$0.533857\pi$
$278$	18.2630	1.09534
$279$	16.3504	0.978871
$280$	3.56026	0.212766
$281$	−12.8002	−0.763599	−0.381799	−	0.924245i	$-0.624695\pi$
$281$	−12.8002	−0.763599	−0.381799	+	0.924245i	$0.624695\pi$
$282$	12.0854	0.719676
$283$	−6.86680	−0.408189	−0.204095	−	0.978951i	$-0.565425\pi$
$283$	−6.86680	−0.408189	−0.204095	+	0.978951i	$0.565425\pi$
$284$	4.21614	0.250182
$285$	−17.9748	−1.06474
$286$	−8.05319	−0.476195
$287$	−5.81201	−0.343072
$288$	−1.59541	−0.0940106
$289$	−16.9997	−0.999983
$290$	14.8693	0.873156
$291$	15.3145	0.897752
$292$	−1.73576	−0.101577
$293$	26.7353	1.56189	0.780947	−	0.624597i	$-0.214737\pi$
$293$	26.7353	1.56189	0.780947	+	0.624597i	$0.214737\pi$
$294$	−7.15456	−0.417262
$295$	17.2900	1.00666
$296$	3.48625	0.202634
$297$	14.2808	0.828655
$298$	−18.5461	−1.07435
$299$	−13.6174	−0.787512
$300$	3.29943	0.190493
$301$	−21.5888	−1.24436
$302$	−1.60434	−0.0923193
$303$	3.26539	0.187592
$304$	−4.50724	−0.258508
$305$	7.37304	0.422179
$306$	0.0274518	0.00156931
$307$	−23.2089	−1.32460	−0.662300	−	0.749239i	$-0.730420\pi$
$307$	−23.2089	−1.32460	−0.662300	+	0.749239i	$0.730420\pi$
$308$	9.07675	0.517196
$309$	4.66200	0.265212
$310$	19.0654	1.08284
$311$	−22.7881	−1.29219	−0.646096	−	0.763256i	$-0.723600\pi$
$311$	−22.7881	−1.29219	−0.646096	+	0.763256i	$0.723600\pi$
$312$	3.63990	0.206069
$313$	−10.1645	−0.574530	−0.287265	−	0.957851i	$-0.592746\pi$
$313$	−10.1645	−0.574530	−0.287265	+	0.957851i	$0.592746\pi$
$314$	−11.3900	−0.642777
$315$	−5.68008	−0.320036
$316$	6.16438	0.346774
$317$	28.0423	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$317$	28.0423	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$318$	−0.125478	−0.00703647
$319$	37.9088	2.12248
$320$	−1.86034	−0.103996
$321$	−38.0096	−2.12149
$322$	15.3481	0.855317
$323$	0.0775547	0.00431526
$324$	−11.2409	−0.624494
$325$	−2.61339	−0.144965
$326$	13.0728	0.724034
$327$	−25.1830	−1.39262
$328$	3.03695	0.167687
$329$	10.7892	0.594827
$330$	−18.9145	−1.04121
$331$	−6.12204	−0.336498	−0.168249	−	0.985745i	$-0.553811\pi$
$331$	−6.12204	−0.336498	−0.168249	+	0.985745i	$0.553811\pi$
$332$	−1.23768	−0.0679266
$333$	−5.56200	−0.304796
$334$	−5.73327	−0.313710
$335$	8.62925	0.471466
$336$	−4.10253	−0.223811
$337$	−33.6649	−1.83385	−0.916923	−	0.399065i	$-0.869335\pi$
$337$	−33.6649	−1.83385	−0.916923	+	0.399065i	$0.869335\pi$
$338$	10.1169	0.550289
$339$	15.2750	0.829624
$340$	0.0320103	0.00173600
$341$	48.6066	2.63220
$342$	7.19091	0.388840
$343$	−19.7836	−1.06821
$344$	11.2808	0.608219
$345$	−31.9831	−1.72191
$346$	−12.2228	−0.657104
$347$	12.2162	0.655797	0.327899	−	0.944713i	$-0.393660\pi$
$347$	12.2162	0.655797	0.327899	+	0.944713i	$0.393660\pi$
$348$	−17.1341	−0.918484
$349$	26.8446	1.43696	0.718479	−	0.695549i	$-0.244839\pi$
$349$	26.8446	1.43696	0.718479	+	0.695549i	$0.244839\pi$
$350$	2.94555	0.157446
$351$	5.11255	0.272888
$352$	−4.74287	−0.252796
$353$	12.8500	0.683939	0.341969	−	0.939711i	$-0.388906\pi$
$353$	12.8500	0.683939	0.341969	+	0.939711i	$0.388906\pi$
$354$	−19.9235	−1.05892
$355$	−7.84345	−0.416287
$356$	0.798011	0.0422945
$357$	0.0705910	0.00373607
$358$	−16.7468	−0.885094
$359$	33.2433	1.75451	0.877256	−	0.480022i	$-0.159371\pi$
$359$	33.2433	1.75451	0.877256	+	0.480022i	$0.159371\pi$
$360$	2.96801	0.156428
$361$	1.31521	0.0692215
$362$	5.76973	0.303250
$363$	−24.6413	−1.29333
$364$	3.24950	0.170320
$365$	3.22910	0.169019
$366$	−8.49604	−0.444095
$367$	−16.7117	−0.872343	−0.436172	−	0.899864i	$-0.643666\pi$
$367$	−16.7117	−0.872343	−0.436172	+	0.899864i	$0.643666\pi$
$368$	−8.01985	−0.418063
$369$	−4.84519	−0.252230
$370$	−6.48560	−0.337170
$371$	−0.112020	−0.00581578
$372$	−21.9693	−1.13906
$373$	7.82884	0.405362	0.202681	−	0.979245i	$-0.435035\pi$
$373$	7.82884	0.405362	0.202681	+	0.979245i	$0.435035\pi$
$374$	0.0816091	0.00421991
$375$	−26.0780	−1.34666
$376$	−5.63767	−0.290740
$377$	13.5714	0.698964
$378$	−5.76236	−0.296384
$379$	−9.96902	−0.512074	−0.256037	−	0.966667i	$-0.582417\pi$
$379$	−9.96902	−0.512074	−0.256037	+	0.966667i	$0.582417\pi$
$380$	8.38500	0.430141
$381$	−15.1679	−0.777075
$382$	−24.8359	−1.27072
$383$	6.37519	0.325757	0.162878	−	0.986646i	$-0.447922\pi$
$383$	6.37519	0.325757	0.162878	+	0.986646i	$0.447922\pi$
$384$	2.14369	0.109395
$385$	−16.8858	−0.860582
$386$	5.21355	0.265363
$387$	−17.9975	−0.914865
$388$	−7.14399	−0.362681
$389$	−36.2814	−1.83954	−0.919771	−	0.392456i	$-0.871626\pi$
$389$	−36.2814	−1.83954	−0.919771	+	0.392456i	$0.871626\pi$
$390$	−6.77144	−0.342885
$391$	0.137995	0.00697871
$392$	3.33749	0.168569
$393$	−18.1142	−0.913738
$394$	23.7296	1.19548
$395$	−11.4678	−0.577010
$396$	7.56683	0.380248
$397$	12.4048	0.622579	0.311289	−	0.950315i	$-0.399239\pi$
$397$	12.4048	0.622579	0.311289	+	0.950315i	$0.399239\pi$
$398$	−26.0864	−1.30759
$399$	18.4911	0.925712
$400$	−1.53914	−0.0769568
$401$	−13.0181	−0.650094	−0.325047	−	0.945698i	$-0.605380\pi$
$401$	−13.0181	−0.650094	−0.325047	+	0.945698i	$0.605380\pi$
$402$	−9.94359	−0.495941
$403$	17.4013	0.866819
$404$	−1.52326	−0.0757848
$405$	20.9119	1.03912
$406$	−15.2964	−0.759145
$407$	−16.5348	−0.819600
$408$	−0.0368859	−0.00182612
$409$	17.1129	0.846176	0.423088	−	0.906089i	$-0.360946\pi$
$409$	17.1129	0.846176	0.423088	+	0.906089i	$0.360946\pi$
$410$	−5.64976	−0.279022
$411$	−5.42974	−0.267829
$412$	−2.17475	−0.107142
$413$	−17.7866	−0.875221
$414$	12.7950	0.628838
$415$	2.30251	0.113026
$416$	−1.69796	−0.0832492
$417$	39.1502	1.91719
$418$	21.3772	1.04560
$419$	−15.8161	−0.772668	−0.386334	−	0.922359i	$-0.626259\pi$
$419$	−15.8161	−0.772668	−0.386334	+	0.922359i	$0.626259\pi$
$420$	7.63209	0.372408
$421$	27.0663	1.31913	0.659566	−	0.751647i	$-0.270740\pi$
$421$	27.0663	1.31913	0.659566	+	0.751647i	$0.270740\pi$
$422$	3.02501	0.147255
$423$	8.99441	0.437323
$424$	0.0585337	0.00284265
$425$	0.0264835	0.00128464
$426$	9.03810	0.437897
$427$	−7.58479	−0.367054
$428$	17.7309	0.857056
$429$	−17.2636	−0.833492
$430$	−20.9861	−1.01204
$431$	−30.9543	−1.49101	−0.745507	−	0.666498i	$-0.767793\pi$
$431$	−30.9543	−1.49101	−0.745507	+	0.666498i	$0.767793\pi$
$432$	3.01100	0.144867
$433$	0.322810	0.0155133	0.00775664	−	0.999970i	$-0.497531\pi$
$433$	0.322810	0.0155133	0.00775664	+	0.999970i	$0.497531\pi$
$434$	−19.6130	−0.941453
$435$	31.8752	1.52830
$436$	11.7475	0.562603
$437$	36.1474	1.72916
$438$	−3.72093	−0.177793
$439$	2.35586	0.112439	0.0562195	−	0.998418i	$-0.482095\pi$
$439$	2.35586	0.112439	0.0562195	+	0.998418i	$0.482095\pi$
$440$	8.82335	0.420636
$441$	−5.32468	−0.253556
$442$	0.0292162	0.00138968
$443$	35.7998	1.70090	0.850449	−	0.526057i	$-0.176330\pi$
$443$	35.7998	1.70090	0.850449	+	0.526057i	$0.176330\pi$
$444$	7.47344	0.354674
$445$	−1.48457	−0.0703754
$446$	4.18889	0.198350
$447$	−39.7572	−1.88045
$448$	1.91377	0.0904170
$449$	−23.3236	−1.10071	−0.550354	−	0.834931i	$-0.685507\pi$
$449$	−23.3236	−1.10071	−0.550354	+	0.834931i	$0.685507\pi$
$450$	2.45556	0.115756
$451$	−14.4038	−0.678251
$452$	−7.12556	−0.335158
$453$	−3.43920	−0.161588
$454$	8.29637	0.389368
$455$	−6.04517	−0.283402
$456$	−9.66213	−0.452471
$457$	−1.74298	−0.0815331	−0.0407666	−	0.999169i	$-0.512980\pi$
$457$	−1.74298	−0.0815331	−0.0407666	+	0.999169i	$0.512980\pi$
$458$	−4.48606	−0.209620
$459$	−0.0518094	−0.00241826
$460$	14.9196	0.695632
$461$	11.9285	0.555567	0.277784	−	0.960644i	$-0.410400\pi$
$461$	11.9285	0.555567	0.277784	+	0.960644i	$0.410400\pi$
$462$	19.4577	0.905256
$463$	−30.3980	−1.41271	−0.706357	−	0.707856i	$-0.749663\pi$
$463$	−30.3980	−1.41271	−0.706357	+	0.707856i	$0.749663\pi$
$464$	7.99280	0.371056
$465$	40.8704	1.89532
$466$	22.3189	1.03390
$467$	7.86245	0.363831	0.181915	−	0.983314i	$-0.441770\pi$
$467$	7.86245	0.363831	0.181915	+	0.983314i	$0.441770\pi$
$468$	2.70894	0.125221
$469$	−8.87708	−0.409905
$470$	10.4880	0.483774
$471$	−24.4167	−1.12506
$472$	9.29402	0.427792
$473$	−53.5033	−2.46008
$474$	13.2145	0.606964
$475$	6.93726	0.318303
$476$	−0.0329296	−0.00150933
$477$	−0.0933854	−0.00427582
$478$	21.9317	1.00313
$479$	14.1341	0.645803	0.322901	−	0.946433i	$-0.395342\pi$
$479$	14.1341	0.645803	0.322901	+	0.946433i	$0.395342\pi$
$480$	−3.98799	−0.182026
$481$	−5.91950	−0.269906
$482$	6.00107	0.273341
$483$	32.9016	1.49708
$484$	11.4948	0.522491
$485$	13.2902	0.603479
$486$	−15.0640	−0.683317
$487$	−33.9859	−1.54005	−0.770024	−	0.638014i	$-0.779756\pi$
$487$	−33.9859	−1.54005	−0.770024	+	0.638014i	$0.779756\pi$
$488$	3.96328	0.179409
$489$	28.0240	1.26729
$490$	−6.20887	−0.280488
$491$	−32.2462	−1.45525	−0.727625	−	0.685975i	$-0.759376\pi$
$491$	−32.2462	−1.45525	−0.727625	+	0.685975i	$0.759376\pi$
$492$	6.51028	0.293506
$493$	−0.137530	−0.00619403
$494$	7.65310	0.344329
$495$	−14.0769	−0.632709
$496$	10.2484	0.460165
$497$	8.06871	0.361931
$498$	−2.65321	−0.118893
$499$	−25.4960	−1.14136	−0.570680	−	0.821173i	$-0.693320\pi$
$499$	−25.4960	−1.14136	−0.570680	+	0.821173i	$0.693320\pi$
$500$	12.1650	0.544036
$501$	−12.2904	−0.549093
$502$	22.0724	0.985140
$503$	−19.3039	−0.860717	−0.430358	−	0.902658i	$-0.641613\pi$
$503$	−19.3039	−0.860717	−0.430358	+	0.902658i	$0.641613\pi$
$504$	−3.05325	−0.136003
$505$	2.83377	0.126101
$506$	38.0371	1.69095
$507$	21.6876	0.963180
$508$	7.07560	0.313929
$509$	30.7634	1.36356	0.681781	−	0.731556i	$-0.261206\pi$
$509$	30.7634	1.36356	0.681781	+	0.731556i	$0.261206\pi$
$510$	0.0686202	0.00303855
$511$	−3.32184	−0.146949
$512$	−1.00000	−0.0441942
$513$	−13.5713	−0.599188
$514$	−16.7365	−0.738214
$515$	4.04578	0.178278
$516$	24.1825	1.06458
$517$	26.7387	1.17597
$518$	6.67187	0.293145
$519$	−26.2020	−1.15014
$520$	3.15878	0.138522
$521$	−0.825773	−0.0361778	−0.0180889	−	0.999836i	$-0.505758\pi$
$521$	−0.825773	−0.0361778	−0.0180889	+	0.999836i	$0.505758\pi$
$522$	−12.7518	−0.558132
$523$	−30.0291	−1.31308	−0.656540	−	0.754291i	$-0.727981\pi$
$523$	−30.0291	−1.31308	−0.656540	+	0.754291i	$0.727981\pi$
$524$	8.44998	0.369139
$525$	6.31435	0.275581
$526$	29.4381	1.28356
$527$	−0.176340	−0.00768151
$528$	−10.1672	−0.442473
$529$	41.3179	1.79643
$530$	−0.108893	−0.00472999
$531$	−14.8278	−0.643472
$532$	−8.62581	−0.373976
$533$	−5.15661	−0.223358
$534$	1.71069	0.0740287
$535$	−32.9855	−1.42609
$536$	4.63853	0.200354
$537$	−35.8999	−1.54920
$538$	30.2519	1.30425
$539$	−15.8293	−0.681816
$540$	−5.60148	−0.241049
$541$	3.72042	0.159953	0.0799766	−	0.996797i	$-0.474515\pi$
$541$	3.72042	0.159953	0.0799766	+	0.996797i	$0.474515\pi$
$542$	14.0062	0.601619
$543$	12.3685	0.530784
$544$	0.0172067	0.000737731 0
$545$	−21.8543	−0.936137
$546$	6.96592	0.298114
$547$	−17.0448	−0.728785	−0.364392	−	0.931246i	$-0.618723\pi$
$547$	−17.0448	−0.728785	−0.364392	+	0.931246i	$0.618723\pi$
$548$	2.53289	0.108200
$549$	−6.32306	−0.269862
$550$	7.29992	0.311270
$551$	−36.0255	−1.53474
$552$	−17.1921	−0.731743
$553$	11.7972	0.501668
$554$	3.53381	0.150137
$555$	−13.9031	−0.590155
$556$	−18.2630	−0.774522
$557$	−4.21574	−0.178627	−0.0893134	−	0.996004i	$-0.528467\pi$
$557$	−4.21574	−0.178627	−0.0893134	+	0.996004i	$0.528467\pi$
$558$	−16.3504	−0.692166
$559$	−19.1543	−0.810141
$560$	−3.56026	−0.150448
$561$	0.174945	0.00738617
$562$	12.8002	0.539946
$563$	7.84130	0.330471	0.165236	−	0.986254i	$-0.447162\pi$
$563$	7.84130	0.330471	0.165236	+	0.986254i	$0.447162\pi$
$564$	−12.0854	−0.508888
$565$	13.2560	0.557682
$566$	6.86680	0.288633
$567$	−21.5125	−0.903439
$568$	−4.21614	−0.176905
$569$	−29.4742	−1.23562	−0.617811	−	0.786327i	$-0.711980\pi$
$569$	−29.4742	−1.23562	−0.617811	+	0.786327i	$0.711980\pi$
$570$	17.9748	0.752883
$571$	22.0363	0.922189	0.461094	−	0.887351i	$-0.347457\pi$
$571$	22.0363	0.922189	0.461094	+	0.887351i	$0.347457\pi$
$572$	8.05319	0.336721
$573$	−53.2406	−2.22416
$574$	5.81201	0.242589
$575$	12.3436	0.514765
$576$	1.59541	0.0664755
$577$	−35.5368	−1.47942	−0.739708	−	0.672928i	$-0.765036\pi$
$577$	−35.5368	−1.47942	−0.739708	+	0.672928i	$0.765036\pi$
$578$	16.9997	0.707094
$579$	11.1762	0.464469
$580$	−14.8693	−0.617415
$581$	−2.36863	−0.0982675
$582$	−15.3145	−0.634807
$583$	−0.277618	−0.0114977
$584$	1.73576	0.0718261
$585$	−5.03955	−0.208360
$586$	−26.7353	−1.10443
$587$	−47.5084	−1.96088	−0.980440	−	0.196819i	$-0.936939\pi$
$587$	−47.5084	−1.96088	−0.980440	+	0.196819i	$0.936939\pi$
$588$	7.15456	0.295049
$589$	−46.1918	−1.90330
$590$	−17.2900	−0.711819
$591$	50.8689	2.09247
$592$	−3.48625	−0.143284
$593$	18.2271	0.748496	0.374248	−	0.927329i	$-0.377901\pi$
$593$	18.2271	0.748496	0.374248	+	0.927329i	$0.377901\pi$
$594$	−14.2808	−0.585948
$595$	0.0612603	0.00251143
$596$	18.5461	0.759680
$597$	−55.9211	−2.28870
$598$	13.6174	0.556855
$599$	−17.5743	−0.718067	−0.359034	−	0.933325i	$-0.616894\pi$
$599$	−17.5743	−0.718067	−0.359034	+	0.933325i	$0.616894\pi$
$600$	−3.29943	−0.134699
$601$	16.0911	0.656370	0.328185	−	0.944613i	$-0.393563\pi$
$601$	16.0911	0.656370	0.328185	+	0.944613i	$0.393563\pi$
$602$	21.5888	0.879894
$603$	−7.40038	−0.301367
$604$	1.60434	0.0652796
$605$	−21.3842	−0.869393
$606$	−3.26539	−0.132647
$607$	14.0206	0.569080	0.284540	−	0.958664i	$-0.408159\pi$
$607$	14.0206	0.569080	0.284540	+	0.958664i	$0.408159\pi$
$608$	4.50724	0.182793
$609$	−32.7907	−1.32875
$610$	−7.37304	−0.298526
$611$	9.57252	0.387263
$612$	−0.0274518	−0.00110967
$613$	−30.4459	−1.22970	−0.614849	−	0.788645i	$-0.710783\pi$
$613$	−30.4459	−1.22970	−0.614849	+	0.788645i	$0.710783\pi$
$614$	23.2089	0.936634
$615$	−12.1113	−0.488376
$616$	−9.07675	−0.365713
$617$	27.5799	1.11032	0.555162	−	0.831742i	$-0.312656\pi$
$617$	27.5799	1.11032	0.555162	+	0.831742i	$0.312656\pi$
$618$	−4.66200	−0.187533
$619$	−10.3937	−0.417758	−0.208879	−	0.977942i	$-0.566981\pi$
$619$	−10.3937	−0.417758	−0.208879	+	0.977942i	$0.566981\pi$
$620$	−19.0654	−0.765686
$621$	−24.1478	−0.969016
$622$	22.7881	0.913718
$623$	1.52721	0.0611863
$624$	−3.63990	−0.145712
$625$	−14.9354	−0.597415
$626$	10.1645	0.406254
$627$	45.8262	1.83012
$628$	11.3900	0.454512
$629$	0.0599868	0.00239183
$630$	5.68008	0.226300
$631$	−16.7641	−0.667369	−0.333685	−	0.942685i	$-0.608292\pi$
$631$	−16.7641	−0.667369	−0.333685	+	0.942685i	$0.608292\pi$
$632$	−6.16438	−0.245206
$633$	6.48470	0.257744
$634$	−28.0423	−1.11370
$635$	−13.1630	−0.522359
$636$	0.125478	0.00497553
$637$	−5.66692	−0.224532
$638$	−37.9088	−1.50082
$639$	6.72648	0.266095
$640$	1.86034	0.0735364
$641$	32.3397	1.27734	0.638671	−	0.769480i	$-0.279484\pi$
$641$	32.3397	1.27734	0.638671	+	0.769480i	$0.279484\pi$
$642$	38.0096	1.50012
$643$	−4.92716	−0.194308	−0.0971541	−	0.995269i	$-0.530974\pi$
$643$	−4.92716	−0.194308	−0.0971541	+	0.995269i	$0.530974\pi$
$644$	−15.3481	−0.604801
$645$	−44.9877	−1.77139
$646$	−0.0775547	−0.00305135
$647$	−22.0675	−0.867564	−0.433782	−	0.901018i	$-0.642821\pi$
$647$	−22.0675	−0.867564	−0.433782	+	0.901018i	$0.642821\pi$
$648$	11.2409	0.441584
$649$	−44.0803	−1.73030
$650$	2.61339	0.102506
$651$	−42.0442	−1.64784
$652$	−13.0728	−0.511969
$653$	−21.6660	−0.847857	−0.423929	−	0.905696i	$-0.639349\pi$
$653$	−21.6660	−0.847857	−0.423929	+	0.905696i	$0.639349\pi$
$654$	25.1830	0.984733
$655$	−15.7198	−0.614225
$656$	−3.03695	−0.118573
$657$	−2.76925	−0.108039
$658$	−10.7892	−0.420606
$659$	46.3979	1.80741	0.903703	−	0.428160i	$-0.140838\pi$
$659$	46.3979	1.80741	0.903703	+	0.428160i	$0.140838\pi$
$660$	18.9145	0.736247
$661$	32.0263	1.24568	0.622839	−	0.782350i	$-0.285979\pi$
$661$	32.0263	1.24568	0.622839	+	0.782350i	$0.285979\pi$
$662$	6.12204	0.237940
$663$	0.0626306	0.00243237
$664$	1.23768	0.0480313
$665$	16.0469	0.622273
$666$	5.56200	0.215523
$667$	−64.1010	−2.48200
$668$	5.73327	0.221827
$669$	8.97969	0.347175
$670$	−8.62925	−0.333377
$671$	−18.7973	−0.725661
$672$	4.10253	0.158258
$673$	15.8760	0.611974	0.305987	−	0.952036i	$-0.401014\pi$
$673$	15.8760	0.611974	0.305987	+	0.952036i	$0.401014\pi$
$674$	33.6649	1.29672
$675$	−4.63434	−0.178376
$676$	−10.1169	−0.389113
$677$	−24.2351	−0.931432	−0.465716	−	0.884934i	$-0.654203\pi$
$677$	−24.2351	−0.931432	−0.465716	+	0.884934i	$0.654203\pi$
$678$	−15.2750	−0.586633
$679$	−13.6719	−0.524681
$680$	−0.0320103	−0.00122754
$681$	17.7849	0.681518
$682$	−48.6066	−1.86124
$683$	−0.891233	−0.0341021	−0.0170510	−	0.999855i	$-0.505428\pi$
$683$	−0.891233	−0.0341021	−0.0170510	+	0.999855i	$0.505428\pi$
$684$	−7.19091	−0.274951
$685$	−4.71204	−0.180038
$686$	19.7836	0.755340
$687$	−9.61673	−0.366901
$688$	−11.2808	−0.430076
$689$	−0.0993877	−0.00378637
$690$	31.9831	1.21758
$691$	−25.6476	−0.975682	−0.487841	−	0.872932i	$-0.662215\pi$
$691$	−25.6476	−0.975682	−0.487841	+	0.872932i	$0.662215\pi$
$692$	12.2228	0.464643
$693$	14.4812	0.550094
$694$	−12.2162	−0.463719
$695$	33.9753	1.28876
$696$	17.1341	0.649466
$697$	0.0522559	0.00197933
$698$	−26.8446	−1.01608
$699$	47.8448	1.80966
$700$	−2.94555	−0.111331
$701$	15.1600	0.572584	0.286292	−	0.958142i	$-0.407577\pi$
$701$	15.1600	0.572584	0.286292	+	0.958142i	$0.407577\pi$
$702$	−5.11255	−0.192961
$703$	15.7133	0.592640
$704$	4.74287	0.178754
$705$	22.4830	0.846758
$706$	−12.8500	−0.483618
$707$	−2.91516	−0.109636
$708$	19.9235	0.748772
$709$	33.5588	1.26033	0.630164	−	0.776462i	$-0.282987\pi$
$709$	33.5588	1.26033	0.630164	+	0.776462i	$0.282987\pi$
$710$	7.84345	0.294359
$711$	9.83474	0.368831
$712$	−0.798011	−0.0299067
$713$	−82.1902	−3.07805
$714$	−0.0705910	−0.00264180
$715$	−14.9817	−0.560282
$716$	16.7468	0.625856
$717$	47.0148	1.75580
$718$	−33.2433	−1.24063
$719$	34.4827	1.28599	0.642994	−	0.765871i	$-0.277692\pi$
$719$	34.4827	1.28599	0.642994	+	0.765871i	$0.277692\pi$
$720$	−2.96801	−0.110611
$721$	−4.16197	−0.155000
$722$	−1.31521	−0.0489470
$723$	12.8645	0.478434
$724$	−5.76973	−0.214430
$725$	−12.3020	−0.456885
$726$	24.6413	0.914525
$727$	−7.47085	−0.277079	−0.138539	−	0.990357i	$-0.544241\pi$
$727$	−7.47085	−0.277079	−0.138539	+	0.990357i	$0.544241\pi$
$728$	−3.24950	−0.120434
$729$	1.43010	0.0529666
$730$	−3.22910	−0.119514
$731$	0.194105	0.00717924
$732$	8.49604	0.314023
$733$	−29.5784	−1.09250	−0.546251	−	0.837622i	$-0.683945\pi$
$733$	−29.5784	−1.09250	−0.546251	+	0.837622i	$0.683945\pi$
$734$	16.7117	0.616840
$735$	−13.3099	−0.490943
$736$	8.01985	0.295615
$737$	−22.0000	−0.810379
$738$	4.84519	0.178354
$739$	41.0230	1.50905	0.754527	−	0.656269i	$-0.227866\pi$
$739$	41.0230	1.50905	0.754527	+	0.656269i	$0.227866\pi$
$740$	6.48560	0.238415
$741$	16.4059	0.602685
$742$	0.112020	0.00411238
$743$	11.7842	0.432320	0.216160	−	0.976358i	$-0.430647\pi$
$743$	11.7842	0.432320	0.216160	+	0.976358i	$0.430647\pi$
$744$	21.9693	0.805434
$745$	−34.5021	−1.26406
$746$	−7.82884	−0.286634
$747$	−1.97461	−0.0722473
$748$	−0.0816091	−0.00298393
$749$	33.9329	1.23988
$750$	26.0780	0.952235
$751$	−26.7037	−0.974432	−0.487216	−	0.873281i	$-0.661988\pi$
$751$	−26.7037	−0.974432	−0.487216	+	0.873281i	$0.661988\pi$
$752$	5.63767	0.205585
$753$	47.3164	1.72431
$754$	−13.5714	−0.494242
$755$	−2.98461	−0.108621
$756$	5.76236	0.209575
$757$	−34.0031	−1.23586	−0.617931	−	0.786232i	$-0.712029\pi$
$757$	−34.0031	−1.23586	−0.617931	+	0.786232i	$0.712029\pi$
$758$	9.96902	0.362091
$759$	81.5397	2.95971
$760$	−8.38500	−0.304156
$761$	−21.7083	−0.786926	−0.393463	−	0.919340i	$-0.628723\pi$
$761$	−21.7083	−0.786926	−0.393463	+	0.919340i	$0.628723\pi$
$762$	15.1679	0.549475
$763$	22.4820	0.813902
$764$	24.8359	0.898532
$765$	0.0510697	0.00184643
$766$	−6.37519	−0.230345
$767$	−15.7809	−0.569814
$768$	−2.14369	−0.0773538
$769$	−30.7638	−1.10937	−0.554685	−	0.832060i	$-0.687161\pi$
$769$	−30.7638	−1.10937	−0.554685	+	0.832060i	$0.687161\pi$
$770$	16.8858	0.608523
$771$	−35.8778	−1.29211
$772$	−5.21355	−0.187640
$773$	−10.6643	−0.383567	−0.191784	−	0.981437i	$-0.561427\pi$
$773$	−10.6643	−0.383567	−0.191784	+	0.981437i	$0.561427\pi$
$774$	17.9975	0.646908
$775$	−15.7736	−0.566605
$776$	7.14399	0.256454
$777$	14.3024	0.513096
$778$	36.2814	1.30075
$779$	13.6883	0.490433
$780$	6.77144	0.242457
$781$	19.9966	0.715534
$782$	−0.137995	−0.00493469
$783$	24.0663	0.860060
$784$	−3.33749	−0.119196
$785$	−21.1893	−0.756280
$786$	18.1142	0.646110
$787$	47.1825	1.68188	0.840938	−	0.541132i	$-0.182004\pi$
$787$	47.1825	1.68188	0.840938	+	0.541132i	$0.182004\pi$
$788$	−23.7296	−0.845332
$789$	63.1061	2.24664
$790$	11.4678	0.408008
$791$	−13.6367	−0.484864
$792$	−7.56683	−0.268876
$793$	−6.72947	−0.238971
$794$	−12.4048	−0.440230
$795$	−0.233432	−0.00827898
$796$	26.0864	0.924607
$797$	6.73088	0.238420	0.119210	−	0.992869i	$-0.461964\pi$
$797$	6.73088	0.238420	0.119210	+	0.992869i	$0.461964\pi$
$798$	−18.4911	−0.654577
$799$	−0.0970057	−0.00343181
$800$	1.53914	0.0544167
$801$	1.27316	0.0449848
$802$	13.0181	0.459686
$803$	−8.23247	−0.290517
$804$	9.94359	0.350683
$805$	28.5527	1.00635
$806$	−17.4013	−0.612934
$807$	64.8508	2.28286
$808$	1.52326	0.0535880
$809$	−32.2235	−1.13292	−0.566459	−	0.824090i	$-0.691687\pi$
$809$	−32.2235	−1.13292	−0.566459	+	0.824090i	$0.691687\pi$
$810$	−20.9119	−0.734769
$811$	−36.8678	−1.29461	−0.647303	−	0.762233i	$-0.724103\pi$
$811$	−36.8678	−1.29461	−0.647303	+	0.762233i	$0.724103\pi$
$812$	15.2964	0.536797
$813$	30.0250	1.05302
$814$	16.5348	0.579545
$815$	24.3198	0.851885
$816$	0.0368859	0.00129126
$817$	50.8452	1.77885
$818$	−17.1129	−0.598337
$819$	5.18429	0.181154
$820$	5.64976	0.197298
$821$	−35.1152	−1.22553	−0.612764	−	0.790266i	$-0.709942\pi$
$821$	−35.1152	−1.22553	−0.612764	+	0.790266i	$0.709942\pi$
$822$	5.42974	0.189384
$823$	22.2733	0.776398	0.388199	−	0.921576i	$-0.373097\pi$
$823$	22.2733	0.776398	0.388199	+	0.921576i	$0.373097\pi$
$824$	2.17475	0.0757610
$825$	15.6488	0.544821
$826$	17.7866	0.618875
$827$	−26.0039	−0.904246	−0.452123	−	0.891956i	$-0.649333\pi$
$827$	−26.0039	−0.904246	−0.452123	+	0.891956i	$0.649333\pi$
$828$	−12.7950	−0.444656
$829$	−8.39066	−0.291420	−0.145710	−	0.989327i	$-0.546547\pi$
$829$	−8.39066	−0.291420	−0.145710	+	0.989327i	$0.546547\pi$
$830$	−2.30251	−0.0799212
$831$	7.57540	0.262788
$832$	1.69796	0.0588661
$833$	0.0574272	0.00198974
$834$	−39.1502	−1.35566
$835$	−10.6658	−0.369106
$836$	−21.3772	−0.739347
$837$	30.8578	1.06660
$838$	15.8161	0.546358
$839$	−42.4771	−1.46647	−0.733237	−	0.679974i	$-0.761991\pi$
$839$	−42.4771	−1.46647	−0.733237	+	0.679974i	$0.761991\pi$
$840$	−7.63209	−0.263332
$841$	34.8848	1.20292
$842$	−27.0663	−0.932767
$843$	27.4398	0.945076
$844$	−3.02501	−0.104125
$845$	18.8209	0.647460
$846$	−8.99441	−0.309234
$847$	21.9984	0.755873
$848$	−0.0585337	−0.00201005
$849$	14.7203	0.505200
$850$	−0.0264835	−0.000908376 0
$851$	27.9592	0.958428
$852$	−9.03810	−0.309640
$853$	−9.19632	−0.314876	−0.157438	−	0.987529i	$-0.550323\pi$
$853$	−9.19632	−0.314876	−0.157438	+	0.987529i	$0.550323\pi$
$854$	7.58479	0.259546
$855$	13.3775	0.457502
$856$	−17.7309	−0.606030
$857$	13.0304	0.445111	0.222556	−	0.974920i	$-0.428560\pi$
$857$	13.0304	0.445111	0.222556	+	0.974920i	$0.428560\pi$
$858$	17.2636	0.589368
$859$	−23.4832	−0.801236	−0.400618	−	0.916245i	$-0.631205\pi$
$859$	−23.4832	−0.801236	−0.400618	+	0.916245i	$0.631205\pi$
$860$	20.9861	0.715620
$861$	12.4592	0.424607
$862$	30.9543	1.05431
$863$	−3.47081	−0.118148	−0.0590738	−	0.998254i	$-0.518815\pi$
$863$	−3.47081	−0.118148	−0.0590738	+	0.998254i	$0.518815\pi$
$864$	−3.01100	−0.102436
$865$	−22.7386	−0.773137
$866$	−0.322810	−0.0109695
$867$	36.4421	1.23764
$868$	19.6130	0.665708
$869$	29.2369	0.991792
$870$	−31.8752	−1.08067
$871$	−7.87603	−0.266869
$872$	−11.7475	−0.397820
$873$	−11.3976	−0.385751
$874$	−36.1474	−1.22270
$875$	23.2810	0.787042
$876$	3.72093	0.125718
$877$	−30.7755	−1.03922	−0.519608	−	0.854405i	$-0.673922\pi$
$877$	−30.7755	−1.03922	−0.519608	+	0.854405i	$0.673922\pi$
$878$	−2.35586	−0.0795064
$879$	−57.3123	−1.93310
$880$	−8.82335	−0.297435
$881$	19.7883	0.666684	0.333342	−	0.942806i	$-0.391824\pi$
$881$	19.7883	0.666684	0.333342	+	0.942806i	$0.391824\pi$
$882$	5.32468	0.179291
$883$	14.9476	0.503027	0.251514	−	0.967854i	$-0.419072\pi$
$883$	14.9476	0.503027	0.251514	+	0.967854i	$0.419072\pi$
$884$	−0.0292162	−0.000982649 0
$885$	−37.0645	−1.24591
$886$	−35.7998	−1.20272
$887$	−11.1668	−0.374944	−0.187472	−	0.982270i	$-0.560029\pi$
$887$	−11.1668	−0.374944	−0.187472	+	0.982270i	$0.560029\pi$
$888$	−7.47344	−0.250792
$889$	13.5411	0.454153
$890$	1.48457	0.0497629
$891$	−53.3141	−1.78609
$892$	−4.18889	−0.140254
$893$	−25.4103	−0.850324
$894$	39.7572	1.32968
$895$	−31.1547	−1.04139
$896$	−1.91377	−0.0639345
$897$	29.1914	0.974673
$898$	23.3236	0.778318
$899$	81.9130	2.73195
$900$	−2.45556	−0.0818520
$901$	0.00100717	3.35538e−5 0
$902$	14.4038	0.479596
$903$	46.2798	1.54009
$904$	7.12556	0.236993
$905$	10.7337	0.356799
$906$	3.43920	0.114260
$907$	−53.9931	−1.79281	−0.896406	−	0.443233i	$-0.853831\pi$
$907$	−53.9931	−1.79281	−0.896406	+	0.443233i	$0.853831\pi$
$908$	−8.29637	−0.275325
$909$	−2.43022	−0.0806054
$910$	6.04517	0.200395
$911$	−23.6385	−0.783179	−0.391589	−	0.920140i	$-0.628075\pi$
$911$	−23.6385	−0.783179	−0.391589	+	0.920140i	$0.628075\pi$
$912$	9.66213	0.319945
$913$	−5.87016	−0.194274
$914$	1.74298	0.0576526
$915$	−15.8055	−0.522514
$916$	4.48606	0.148224
$917$	16.1713	0.534023
$918$	0.0518094	0.00170996
$919$	−29.1025	−0.960002	−0.480001	−	0.877268i	$-0.659364\pi$
$919$	−29.1025	−0.960002	−0.480001	+	0.877268i	$0.659364\pi$
$920$	−14.9196	−0.491886
$921$	49.7527	1.63941
$922$	−11.9285	−0.392845
$923$	7.15882	0.235635
$924$	−19.4577	−0.640113
$925$	5.36581	0.176427
$926$	30.3980	0.998940
$927$	−3.46963	−0.113957
$928$	−7.99280	−0.262376
$929$	45.0045	1.47655	0.738275	−	0.674499i	$-0.235640\pi$
$929$	45.0045	1.47655	0.738275	+	0.674499i	$0.235640\pi$
$930$	−40.8704	−1.34019
$931$	15.0429	0.493011
$932$	−22.3189	−0.731080
$933$	48.8506	1.59930
$934$	−7.86245	−0.257267
$935$	0.151821	0.00496507
$936$	−2.70894	−0.0885446
$937$	24.6940	0.806717	0.403359	−	0.915042i	$-0.367843\pi$
$937$	24.6940	0.806717	0.403359	+	0.915042i	$0.367843\pi$
$938$	8.87708	0.289847
$939$	21.7895	0.711074
$940$	−10.4880	−0.342080
$941$	31.3410	1.02169	0.510844	−	0.859674i	$-0.329333\pi$
$941$	31.3410	1.02169	0.510844	+	0.859674i	$0.329333\pi$
$942$	24.4167	0.795540
$943$	24.3559	0.793136
$944$	−9.29402	−0.302495
$945$	−10.7199	−0.348720
$946$	53.5033	1.73954
$947$	42.4425	1.37920	0.689598	−	0.724192i	$-0.257787\pi$
$947$	42.4425	1.37920	0.689598	+	0.724192i	$0.257787\pi$
$948$	−13.2145	−0.429188
$949$	−2.94724	−0.0956715
$950$	−6.93726	−0.225074
$951$	−60.1140	−1.94933
$952$	0.0329296	0.00106726
$953$	43.4651	1.40797	0.703987	−	0.710213i	$-0.251401\pi$
$953$	43.4651	1.40797	0.703987	+	0.710213i	$0.251401\pi$
$954$	0.0933854	0.00302346
$955$	−46.2033	−1.49510
$956$	−21.9317	−0.709322
$957$	−81.2647	−2.62692
$958$	−14.1341	−0.456651
$959$	4.84736	0.156530
$960$	3.98799	0.128712
$961$	74.0288	2.38803
$962$	5.91950	0.190852
$963$	28.2881	0.911572
$964$	−6.00107	−0.193282
$965$	9.69897	0.312221
$966$	−32.9016	−1.05859
$967$	−46.0814	−1.48188	−0.740939	−	0.671572i	$-0.765619\pi$
$967$	−46.0814	−1.48188	−0.740939	+	0.671572i	$0.765619\pi$
$968$	−11.4948	−0.369457
$969$	−0.166253	−0.00534083
$970$	−13.2902	−0.426724
$971$	−11.5248	−0.369849	−0.184924	−	0.982753i	$-0.559204\pi$
$971$	−11.5248	−0.369849	−0.184924	+	0.982753i	$0.559204\pi$
$972$	15.0640	0.483178
$973$	−34.9511	−1.12048
$974$	33.9859	1.08898
$975$	5.60230	0.179417
$976$	−3.96328	−0.126861
$977$	3.57200	0.114278	0.0571391	−	0.998366i	$-0.481802\pi$
$977$	3.57200	0.114278	0.0571391	+	0.998366i	$0.481802\pi$
$978$	−28.0240	−0.896108
$979$	3.78486	0.120965
$980$	6.20887	0.198335
$981$	18.7421	0.598389
$982$	32.2462	1.02902
$983$	−43.6657	−1.39272	−0.696360	−	0.717693i	$-0.745198\pi$
$983$	−43.6657	−1.39272	−0.696360	+	0.717693i	$0.745198\pi$
$984$	−6.51028	−0.207540
$985$	44.1451	1.40658
$986$	0.137530	0.00437984
$987$	−23.1287	−0.736194
$988$	−7.65310	−0.243478
$989$	90.4702	2.87678
$990$	14.0769	0.447393
$991$	13.4308	0.426642	0.213321	−	0.976982i	$-0.431572\pi$
$991$	13.4308	0.426642	0.213321	+	0.976982i	$0.431572\pi$
$992$	−10.2484	−0.325386
$993$	13.1238	0.416470
$994$	−8.06871	−0.255924
$995$	−48.5295	−1.53849
$996$	2.65321	0.0840700
$997$	45.3013	1.43471	0.717353	−	0.696710i	$-0.245353\pi$
$997$	45.3013	1.43471	0.717353	+	0.696710i	$0.245353\pi$
$998$	25.4960	0.807063
$999$	−10.4971	−0.332113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6002.2.a.b.1.12	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6002.2.a.b.1.12	✓	56	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:	\( N \)	\(=\)	\( 6002 = 2 \cdot 3001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.14369 q^{3} +1.00000 q^{4} -1.86034 q^{5} +2.14369 q^{6} +1.91377 q^{7} -1.00000 q^{8} +1.59541 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.14369 q^{3} +1.00000 q^{4} -1.86034 q^{5} +2.14369 q^{6} +1.91377 q^{7} -1.00000 q^{8} +1.59541 q^{9} +1.86034 q^{10} +4.74287 q^{11} -2.14369 q^{12} +1.69796 q^{13} -1.91377 q^{14} +3.98799 q^{15} +1.00000 q^{16} -0.0172067 q^{17} -1.59541 q^{18} -4.50724 q^{19} -1.86034 q^{20} -4.10253 q^{21} -4.74287 q^{22} -8.01985 q^{23} +2.14369 q^{24} -1.53914 q^{25} -1.69796 q^{26} +3.01100 q^{27} +1.91377 q^{28} +7.99280 q^{29} -3.98799 q^{30} +10.2484 q^{31} -1.00000 q^{32} -10.1672 q^{33} +0.0172067 q^{34} -3.56026 q^{35} +1.59541 q^{36} -3.48625 q^{37} +4.50724 q^{38} -3.63990 q^{39} +1.86034 q^{40} -3.03695 q^{41} +4.10253 q^{42} -11.2808 q^{43} +4.74287 q^{44} -2.96801 q^{45} +8.01985 q^{46} +5.63767 q^{47} -2.14369 q^{48} -3.33749 q^{49} +1.53914 q^{50} +0.0368859 q^{51} +1.69796 q^{52} -0.0585337 q^{53} -3.01100 q^{54} -8.82335 q^{55} -1.91377 q^{56} +9.66213 q^{57} -7.99280 q^{58} -9.29402 q^{59} +3.98799 q^{60} -3.96328 q^{61} -10.2484 q^{62} +3.05325 q^{63} +1.00000 q^{64} -3.15878 q^{65} +10.1672 q^{66} -4.63853 q^{67} -0.0172067 q^{68} +17.1921 q^{69} +3.56026 q^{70} +4.21614 q^{71} -1.59541 q^{72} -1.73576 q^{73} +3.48625 q^{74} +3.29943 q^{75} -4.50724 q^{76} +9.07675 q^{77} +3.63990 q^{78} +6.16438 q^{79} -1.86034 q^{80} -11.2409 q^{81} +3.03695 q^{82} -1.23768 q^{83} -4.10253 q^{84} +0.0320103 q^{85} +11.2808 q^{86} -17.1341 q^{87} -4.74287 q^{88} +0.798011 q^{89} +2.96801 q^{90} +3.24950 q^{91} -8.01985 q^{92} -21.9693 q^{93} -5.63767 q^{94} +8.38500 q^{95} +2.14369 q^{96} -7.14399 q^{97} +3.33749 q^{98} +7.56683 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 56 q^{2} - 11 q^{3} + 56 q^{4} + 11 q^{6} - 21 q^{7} - 56 q^{8} + 53 q^{9}+O(q^{10}) \) 56 * q - 56 * q^2 - 11 * q^3 + 56 * q^4 + 11 * q^6 - 21 * q^7 - 56 * q^8 + 53 * q^9	\( 56 q - 56 q^{2} - 11 q^{3} + 56 q^{4} + 11 q^{6} - 21 q^{7} - 56 q^{8} + 53 q^{9} + 12 q^{11} - 11 q^{12} - 31 q^{13} + 21 q^{14} - 22 q^{15} + 56 q^{16} - 4 q^{17} - 53 q^{18} - 9 q^{19} + 13 q^{21} - 12 q^{22} - 39 q^{23} + 11 q^{24} + 8 q^{25} + 31 q^{26} - 44 q^{27} - 21 q^{28} + 13 q^{29} + 22 q^{30} - 35 q^{31} - 56 q^{32} - 26 q^{33} + 4 q^{34} - 7 q^{35} + 53 q^{36} - 65 q^{37} + 9 q^{38} - 27 q^{39} + 38 q^{41} - 13 q^{42} - 76 q^{43} + 12 q^{44} - 21 q^{45} + 39 q^{46} - 43 q^{47} - 11 q^{48} + 9 q^{49} - 8 q^{50} - 19 q^{51} - 31 q^{52} - 26 q^{53} + 44 q^{54} - 67 q^{55} + 21 q^{56} - 26 q^{57} - 13 q^{58} + 11 q^{59} - 22 q^{60} - 17 q^{61} + 35 q^{62} - 67 q^{63} + 56 q^{64} + 31 q^{65} + 26 q^{66} - 93 q^{67} - 4 q^{68} - 13 q^{69} + 7 q^{70} - 33 q^{71} - 53 q^{72} - 41 q^{73} + 65 q^{74} - 21 q^{75} - 9 q^{76} + 5 q^{77} + 27 q^{78} - 69 q^{79} + 36 q^{81} - 38 q^{82} + 4 q^{83} + 13 q^{84} - 40 q^{85} + 76 q^{86} - 69 q^{87} - 12 q^{88} + 40 q^{89} + 21 q^{90} - 64 q^{91} - 39 q^{92} - 57 q^{93} + 43 q^{94} - 22 q^{95} + 11 q^{96} - 71 q^{97} - 9 q^{98} + 17 q^{99}+O(q^{100}) \) 56 * q - 56 * q^2 - 11 * q^3 + 56 * q^4 + 11 * q^6 - 21 * q^7 - 56 * q^8 + 53 * q^9 + 12 * q^11 - 11 * q^12 - 31 * q^13 + 21 * q^14 - 22 * q^15 + 56 * q^16 - 4 * q^17 - 53 * q^18 - 9 * q^19 + 13 * q^21 - 12 * q^22 - 39 * q^23 + 11 * q^24 + 8 * q^25 + 31 * q^26 - 44 * q^27 - 21 * q^28 + 13 * q^29 + 22 * q^30 - 35 * q^31 - 56 * q^32 - 26 * q^33 + 4 * q^34 - 7 * q^35 + 53 * q^36 - 65 * q^37 + 9 * q^38 - 27 * q^39 + 38 * q^41 - 13 * q^42 - 76 * q^43 + 12 * q^44 - 21 * q^45 + 39 * q^46 - 43 * q^47 - 11 * q^48 + 9 * q^49 - 8 * q^50 - 19 * q^51 - 31 * q^52 - 26 * q^53 + 44 * q^54 - 67 * q^55 + 21 * q^56 - 26 * q^57 - 13 * q^58 + 11 * q^59 - 22 * q^60 - 17 * q^61 + 35 * q^62 - 67 * q^63 + 56 * q^64 + 31 * q^65 + 26 * q^66 - 93 * q^67 - 4 * q^68 - 13 * q^69 + 7 * q^70 - 33 * q^71 - 53 * q^72 - 41 * q^73 + 65 * q^74 - 21 * q^75 - 9 * q^76 + 5 * q^77 + 27 * q^78 - 69 * q^79 + 36 * q^81 - 38 * q^82 + 4 * q^83 + 13 * q^84 - 40 * q^85 + 76 * q^86 - 69 * q^87 - 12 * q^88 + 40 * q^89 + 21 * q^90 - 64 * q^91 - 39 * q^92 - 57 * q^93 + 43 * q^94 - 22 * q^95 + 11 * q^96 - 71 * q^97 - 9 * q^98 + 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more