LMFDB - Embedded newform 418.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [418,2,Mod(1,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, 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("418.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 418 = 2 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	418.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:	$3.33774680449$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
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			1.3
Root			$2.39138$ of defining polynomial
Character	$\chi$	$=$	418.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.39138 q^{3} +1.00000 q^{4} -0.391382 q^{5} +2.39138 q^{6} -1.71871 q^{7} +1.00000 q^{8} +2.71871 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.39138 q^{3} +1.00000 q^{4} -0.391382 q^{5} +2.39138 q^{6} -1.71871 q^{7} +1.00000 q^{8} +2.71871 q^{9} -0.391382 q^{10} -1.00000 q^{11} +2.39138 q^{12} +3.71871 q^{13} -1.71871 q^{14} -0.935945 q^{15} +1.00000 q^{16} +5.43742 q^{17} +2.71871 q^{18} -1.00000 q^{19} -0.391382 q^{20} -4.11009 q^{21} -1.00000 q^{22} -3.43742 q^{23} +2.39138 q^{24} -4.84682 q^{25} +3.71871 q^{26} -0.672673 q^{27} -1.71871 q^{28} -3.82880 q^{29} -0.935945 q^{30} -3.06406 q^{31} +1.00000 q^{32} -2.39138 q^{33} +5.43742 q^{34} +0.672673 q^{35} +2.71871 q^{36} -4.65465 q^{37} -1.00000 q^{38} +8.89286 q^{39} -0.391382 q^{40} +1.06406 q^{41} -4.11009 q^{42} -3.73673 q^{43} -1.00000 q^{44} -1.06406 q^{45} -3.43742 q^{46} -8.22018 q^{47} +2.39138 q^{48} -4.04604 q^{49} -4.84682 q^{50} +13.0029 q^{51} +3.71871 q^{52} -1.21724 q^{53} -0.672673 q^{54} +0.391382 q^{55} -1.71871 q^{56} -2.39138 q^{57} -3.82880 q^{58} +12.4404 q^{59} -0.935945 q^{60} +2.00000 q^{61} -3.06406 q^{62} -4.67267 q^{63} +1.00000 q^{64} -1.45544 q^{65} -2.39138 q^{66} -1.71871 q^{67} +5.43742 q^{68} -8.22018 q^{69} +0.672673 q^{70} +1.04604 q^{71} +2.71871 q^{72} +13.6576 q^{73} -4.65465 q^{74} -11.5906 q^{75} -1.00000 q^{76} +1.71871 q^{77} +8.89286 q^{78} +13.0029 q^{79} -0.391382 q^{80} -9.76475 q^{81} +1.06406 q^{82} +8.51949 q^{83} -4.11009 q^{84} -2.12811 q^{85} -3.73673 q^{86} -9.15613 q^{87} -1.00000 q^{88} +17.6936 q^{89} -1.06406 q^{90} -6.39138 q^{91} -3.43742 q^{92} -7.32733 q^{93} -8.22018 q^{94} +0.391382 q^{95} +2.39138 q^{96} -4.65465 q^{97} -4.04604 q^{98} -2.71871 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 + 3 * q^8 + 2 * q^9	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9} + 5 q^{10} - 3 q^{11} + q^{12} + 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} + 4 q^{17} + 2 q^{18} - 3 q^{19} + 5 q^{20} - 3 q^{22} + 2 q^{23} + q^{24} + 4 q^{25} + 5 q^{26} - 2 q^{27} + q^{28} + 7 q^{29} - 9 q^{30} - 3 q^{31} + 3 q^{32} - q^{33} + 4 q^{34} + 2 q^{35} + 2 q^{36} - 14 q^{37} - 3 q^{38} + 2 q^{39} + 5 q^{40} - 3 q^{41} - 5 q^{43} - 3 q^{44} + 3 q^{45} + 2 q^{46} + q^{48} - 6 q^{49} + 4 q^{50} + 2 q^{51} + 5 q^{52} - 16 q^{53} - 2 q^{54} - 5 q^{55} + q^{56} - q^{57} + 7 q^{58} - 12 q^{59} - 9 q^{60} + 6 q^{61} - 3 q^{62} - 14 q^{63} + 3 q^{64} + 8 q^{65} - q^{66} + q^{67} + 4 q^{68} + 2 q^{70} - 3 q^{71} + 2 q^{72} + 4 q^{73} - 14 q^{74} - 41 q^{75} - 3 q^{76} - q^{77} + 2 q^{78} + 2 q^{79} + 5 q^{80} - 17 q^{81} - 3 q^{82} + 7 q^{83} + 6 q^{85} - 5 q^{86} - 9 q^{87} - 3 q^{88} + 16 q^{89} + 3 q^{90} - 13 q^{91} + 2 q^{92} - 22 q^{93} - 5 q^{95} + q^{96} - 14 q^{97} - 6 q^{98} - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 + 3 * q^8 + 2 * q^9 + 5 * q^10 - 3 * q^11 + q^12 + 5 * q^13 + q^14 - 9 * q^15 + 3 * q^16 + 4 * q^17 + 2 * q^18 - 3 * q^19 + 5 * q^20 - 3 * q^22 + 2 * q^23 + q^24 + 4 * q^25 + 5 * q^26 - 2 * q^27 + q^28 + 7 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 - q^33 + 4 * q^34 + 2 * q^35 + 2 * q^36 - 14 * q^37 - 3 * q^38 + 2 * q^39 + 5 * q^40 - 3 * q^41 - 5 * q^43 - 3 * q^44 + 3 * q^45 + 2 * q^46 + q^48 - 6 * q^49 + 4 * q^50 + 2 * q^51 + 5 * q^52 - 16 * q^53 - 2 * q^54 - 5 * q^55 + q^56 - q^57 + 7 * q^58 - 12 * q^59 - 9 * q^60 + 6 * q^61 - 3 * q^62 - 14 * q^63 + 3 * q^64 + 8 * q^65 - q^66 + q^67 + 4 * q^68 + 2 * q^70 - 3 * q^71 + 2 * q^72 + 4 * q^73 - 14 * q^74 - 41 * q^75 - 3 * q^76 - q^77 + 2 * q^78 + 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 + 7 * q^83 + 6 * q^85 - 5 * q^86 - 9 * q^87 - 3 * q^88 + 16 * q^89 + 3 * q^90 - 13 * q^91 + 2 * q^92 - 22 * q^93 - 5 * q^95 + q^96 - 14 * q^97 - 6 * q^98 - 2 * 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
$3$	2.39138	1.38067	0.690333	−	0.723492i	$-0.257464\pi$
$3$	2.39138	1.38067	0.690333	+	0.723492i	$0.257464\pi$
$4$	1.00000	0.500000
$5$	−0.391382	−0.175032	−0.0875158	−	0.996163i	$-0.527893\pi$
$5$	−0.391382	−0.175032	−0.0875158	+	0.996163i	$0.527893\pi$
$6$	2.39138	0.976278
$7$	−1.71871	−0.649611	−0.324806	−	0.945781i	$-0.605299\pi$
$7$	−1.71871	−0.649611	−0.324806	+	0.945781i	$0.605299\pi$
$8$	1.00000	0.353553
$9$	2.71871	0.906237
$10$	−0.391382	−0.123766
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.39138	0.690333
$13$	3.71871	1.03138	0.515692	−	0.856774i	$-0.327535\pi$
$13$	3.71871	1.03138	0.515692	+	0.856774i	$0.327535\pi$
$14$	−1.71871	−0.459344
$15$	−0.935945	−0.241660
$16$	1.00000	0.250000
$17$	5.43742	1.31877	0.659384	−	0.751806i	$-0.270817\pi$
$17$	5.43742	1.31877	0.659384	+	0.751806i	$0.270817\pi$
$18$	2.71871	0.640806
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.391382	−0.0875158
$21$	−4.11009	−0.896896
$22$	−1.00000	−0.213201
$23$	−3.43742	−0.716751	−0.358376	−	0.933577i	$-0.616669\pi$
$23$	−3.43742	−0.716751	−0.358376	+	0.933577i	$0.616669\pi$
$24$	2.39138	0.488139
$25$	−4.84682	−0.969364
$26$	3.71871	0.729299
$27$	−0.672673	−0.129456
$28$	−1.71871	−0.324806
$29$	−3.82880	−0.710991	−0.355495	−	0.934678i	$-0.615688\pi$
$29$	−3.82880	−0.710991	−0.355495	+	0.934678i	$0.615688\pi$
$30$	−0.935945	−0.170879
$31$	−3.06406	−0.550321	−0.275160	−	0.961398i	$-0.588731\pi$
$31$	−3.06406	−0.550321	−0.275160	+	0.961398i	$0.588731\pi$
$32$	1.00000	0.176777
$33$	−2.39138	−0.416286
$34$	5.43742	0.932510
$35$	0.672673	0.113702
$36$	2.71871	0.453118
$37$	−4.65465	−0.765221	−0.382610	−	0.923910i	$-0.624975\pi$
$37$	−4.65465	−0.765221	−0.382610	+	0.923910i	$0.624975\pi$
$38$	−1.00000	−0.162221
$39$	8.89286	1.42400
$40$	−0.391382	−0.0618830
$41$	1.06406	0.166177	0.0830887	−	0.996542i	$-0.473522\pi$
$41$	1.06406	0.166177	0.0830887	+	0.996542i	$0.473522\pi$
$42$	−4.11009	−0.634201
$43$	−3.73673	−0.569846	−0.284923	−	0.958550i	$-0.591968\pi$
$43$	−3.73673	−0.569846	−0.284923	+	0.958550i	$0.591968\pi$
$44$	−1.00000	−0.150756
$45$	−1.06406	−0.158620
$46$	−3.43742	−0.506820
$47$	−8.22018	−1.19904	−0.599519	−	0.800361i	$-0.704641\pi$
$47$	−8.22018	−1.19904	−0.599519	+	0.800361i	$0.704641\pi$
$48$	2.39138	0.345166
$49$	−4.04604	−0.578005
$50$	−4.84682	−0.685444
$51$	13.0029	1.82078
$52$	3.71871	0.515692
$53$	−1.21724	−0.167200	−0.0836001	−	0.996499i	$-0.526642\pi$
$53$	−1.21724	−0.167200	−0.0836001	+	0.996499i	$0.526642\pi$
$54$	−0.672673	−0.0915392
$55$	0.391382	0.0527740
$56$	−1.71871	−0.229672
$57$	−2.39138	−0.316746
$58$	−3.82880	−0.502746
$59$	12.4404	1.61960	0.809799	−	0.586707i	$-0.199576\pi$
$59$	12.4404	1.61960	0.809799	+	0.586707i	$0.199576\pi$
$60$	−0.935945	−0.120830
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−3.06406	−0.389135
$63$	−4.67267	−0.588701
$64$	1.00000	0.125000
$65$	−1.45544	−0.180525
$66$	−2.39138	−0.294359
$67$	−1.71871	−0.209974	−0.104987	−	0.994474i	$-0.533480\pi$
$67$	−1.71871	−0.209974	−0.104987	+	0.994474i	$0.533480\pi$
$68$	5.43742	0.659384
$69$	−8.22018	−0.989594
$70$	0.672673	0.0803998
$71$	1.04604	0.124142	0.0620709	−	0.998072i	$-0.480230\pi$
$71$	1.04604	0.124142	0.0620709	+	0.998072i	$0.480230\pi$
$72$	2.71871	0.320403
$73$	13.6576	1.59850	0.799251	−	0.600998i	$-0.205230\pi$
$73$	13.6576	1.59850	0.799251	+	0.600998i	$0.205230\pi$
$74$	−4.65465	−0.541093
$75$	−11.5906	−1.33837
$76$	−1.00000	−0.114708
$77$	1.71871	0.195865
$78$	8.89286	1.00692
$79$	13.0029	1.46295	0.731473	−	0.681870i	$-0.238833\pi$
$79$	13.0029	1.46295	0.731473	+	0.681870i	$0.238833\pi$
$80$	−0.391382	−0.0437579
$81$	−9.76475	−1.08497
$82$	1.06406	0.117505
$83$	8.51949	0.935136	0.467568	−	0.883957i	$-0.345130\pi$
$83$	8.51949	0.935136	0.467568	+	0.883957i	$0.345130\pi$
$84$	−4.11009	−0.448448
$85$	−2.12811	−0.230826
$86$	−3.73673	−0.402942
$87$	−9.15613	−0.981640
$88$	−1.00000	−0.106600
$89$	17.6936	1.87552	0.937761	−	0.347281i	$-0.112895\pi$
$89$	17.6936	1.87552	0.937761	+	0.347281i	$0.112895\pi$
$90$	−1.06406	−0.112161
$91$	−6.39138	−0.669999
$92$	−3.43742	−0.358376
$93$	−7.32733	−0.759808
$94$	−8.22018	−0.847847
$95$	0.391382	0.0401550
$96$	2.39138	0.244069
$97$	−4.65465	−0.472609	−0.236304	−	0.971679i	$-0.575936\pi$
$97$	−4.65465	−0.472609	−0.236304	+	0.971679i	$0.575936\pi$
$98$	−4.04604	−0.408711
$99$	−2.71871	−0.273241
$100$	−4.84682	−0.484682
$101$	−9.65760	−0.960967	−0.480484	−	0.877004i	$-0.659539\pi$
$101$	−9.65760	−0.960967	−0.480484	+	0.877004i	$0.659539\pi$
$102$	13.0029	1.28748
$103$	−6.95396	−0.685194	−0.342597	−	0.939482i	$-0.611307\pi$
$103$	−6.95396	−0.685194	−0.342597	+	0.939482i	$0.611307\pi$
$104$	3.71871	0.364649
$105$	1.60862	0.156985
$106$	−1.21724	−0.118228
$107$	2.65465	0.256635	0.128318	−	0.991733i	$-0.459042\pi$
$107$	2.65465	0.256635	0.128318	+	0.991733i	$0.459042\pi$
$108$	−0.672673	−0.0647280
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0.391382	0.0373168
$111$	−11.1311	−1.05651
$112$	−1.71871	−0.162403
$113$	7.52949	0.708315	0.354158	−	0.935186i	$-0.384768\pi$
$113$	7.52949	0.708315	0.354158	+	0.935186i	$0.384768\pi$
$114$	−2.39138	−0.223973
$115$	1.34535	0.125454
$116$	−3.82880	−0.355495
$117$	10.1101	0.934678
$118$	12.4404	1.14523
$119$	−9.34535	−0.856686
$120$	−0.935945	−0.0854397
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	2.54456	0.229435
$124$	−3.06406	−0.275160
$125$	3.85387	0.344701
$126$	−4.67267	−0.416275
$127$	−5.00295	−0.443940	−0.221970	−	0.975054i	$-0.571249\pi$
$127$	−5.00295	−0.443940	−0.221970	+	0.975054i	$0.571249\pi$
$128$	1.00000	0.0883883
$129$	−8.93594	−0.786766
$130$	−1.45544	−0.127650
$131$	−6.50147	−0.568036	−0.284018	−	0.958819i	$-0.591668\pi$
$131$	−6.50147	−0.568036	−0.284018	+	0.958819i	$0.591668\pi$
$132$	−2.39138	−0.208143
$133$	1.71871	0.149031
$134$	−1.71871	−0.148474
$135$	0.263272	0.0226589
$136$	5.43742	0.466255
$137$	15.7187	1.34294	0.671470	−	0.741032i	$-0.265663\pi$
$137$	15.7187	1.34294	0.671470	+	0.741032i	$0.265663\pi$
$138$	−8.22018	−0.699749
$139$	−11.1381	−0.944722	−0.472361	−	0.881405i	$-0.656598\pi$
$139$	−11.1381	−0.944722	−0.472361	+	0.881405i	$0.656598\pi$
$140$	0.672673	0.0568512
$141$	−19.6576	−1.65547
$142$	1.04604	0.0877815
$143$	−3.71871	−0.310974
$144$	2.71871	0.226559
$145$	1.49853	0.124446
$146$	13.6576	1.13031
$147$	−9.67562	−0.798032
$148$	−4.65465	−0.382610
$149$	11.3453	0.929447	0.464723	−	0.885456i	$-0.346154\pi$
$149$	11.3453	0.929447	0.464723	+	0.885456i	$0.346154\pi$
$150$	−11.5906	−0.946368
$151$	−14.6547	−1.19258	−0.596289	−	0.802770i	$-0.703359\pi$
$151$	−14.6547	−1.19258	−0.596289	+	0.802770i	$0.703359\pi$
$152$	−1.00000	−0.0811107
$153$	14.7828	1.19512
$154$	1.71871	0.138498
$155$	1.19922	0.0963234
$156$	8.89286	0.711998
$157$	−10.4835	−0.836671	−0.418335	−	0.908293i	$-0.637386\pi$
$157$	−10.4835	−0.836671	−0.418335	+	0.908293i	$0.637386\pi$
$158$	13.0029	1.03446
$159$	−2.91087	−0.230847
$160$	−0.391382	−0.0309415
$161$	5.90793	0.465610
$162$	−9.76475	−0.767191
$163$	−3.47346	−0.272062	−0.136031	−	0.990705i	$-0.543435\pi$
$163$	−3.47346	−0.272062	−0.136031	+	0.990705i	$0.543435\pi$
$164$	1.06406	0.0830887
$165$	0.935945	0.0728632
$166$	8.51949	0.661241
$167$	23.0950	1.78715	0.893573	−	0.448917i	$-0.148190\pi$
$167$	23.0950	1.78715	0.893573	+	0.448917i	$0.148190\pi$
$168$	−4.11009	−0.317100
$169$	0.828802	0.0637540
$170$	−2.12811	−0.163219
$171$	−2.71871	−0.207905
$172$	−3.73673	−0.284923
$173$	−18.1771	−1.38198	−0.690990	−	0.722865i	$-0.742825\pi$
$173$	−18.1771	−1.38198	−0.690990	+	0.722865i	$0.742825\pi$
$174$	−9.15613	−0.694124
$175$	8.33028	0.629710
$176$	−1.00000	−0.0753778
$177$	29.7497	2.23612
$178$	17.6936	1.32619
$179$	11.2842	0.843424	0.421712	−	0.906730i	$-0.361429\pi$
$179$	11.2842	0.843424	0.421712	+	0.906730i	$0.361429\pi$
$180$	−1.06406	−0.0793100
$181$	21.8778	1.62616	0.813082	−	0.582150i	$-0.197788\pi$
$181$	21.8778	1.62616	0.813082	+	0.582150i	$0.197788\pi$
$182$	−6.39138	−0.473761
$183$	4.78276	0.353552
$184$	−3.43742	−0.253410
$185$	1.82175	0.133938
$186$	−7.32733	−0.537266
$187$	−5.43742	−0.397623
$188$	−8.22018	−0.599519
$189$	1.15613	0.0840960
$190$	0.391382	0.0283939
$191$	−7.09502	−0.513378	−0.256689	−	0.966494i	$-0.582632\pi$
$191$	−7.09502	−0.513378	−0.256689	+	0.966494i	$0.582632\pi$
$192$	2.39138	0.172583
$193$	12.6476	0.910394	0.455197	−	0.890391i	$-0.349569\pi$
$193$	12.6476	0.910394	0.455197	+	0.890391i	$0.349569\pi$
$194$	−4.65465	−0.334185
$195$	−3.48051	−0.249244
$196$	−4.04604	−0.289003
$197$	8.43447	0.600931	0.300466	−	0.953793i	$-0.402858\pi$
$197$	8.43447	0.600931	0.300466	+	0.953793i	$0.402858\pi$
$198$	−2.71871	−0.193210
$199$	−2.09207	−0.148303	−0.0741516	−	0.997247i	$-0.523625\pi$
$199$	−2.09207	−0.148303	−0.0741516	+	0.997247i	$0.523625\pi$
$200$	−4.84682	−0.342722
$201$	−4.11009	−0.289904
$202$	−9.65760	−0.679507
$203$	6.58060	0.461867
$204$	13.0029	0.910389
$205$	−0.416452	−0.0290863
$206$	−6.95396	−0.484506
$207$	−9.34535	−0.649546
$208$	3.71871	0.257846
$209$	1.00000	0.0691714
$210$	1.60862	0.111005
$211$	−6.09207	−0.419396	−0.209698	−	0.977766i	$-0.567248\pi$
$211$	−6.09207	−0.419396	−0.209698	+	0.977766i	$0.567248\pi$
$212$	−1.21724	−0.0836001
$213$	2.50147	0.171398
$214$	2.65465	0.181468
$215$	1.46249	0.0997409
$216$	−0.672673	−0.0457696
$217$	5.26622	0.357494
$218$	6.00000	0.406371
$219$	32.6606	2.20700
$220$	0.391382	0.0263870
$221$	20.2202	1.36016
$222$	−11.1311	−0.747068
$223$	−18.8748	−1.26395	−0.631976	−	0.774988i	$-0.717756\pi$
$223$	−18.8748	−1.26395	−0.631976	+	0.774988i	$0.717756\pi$
$224$	−1.71871	−0.114836
$225$	−13.1771	−0.878473
$226$	7.52949	0.500854
$227$	20.9669	1.39162	0.695811	−	0.718225i	$-0.255045\pi$
$227$	20.9669	1.39162	0.695811	+	0.718225i	$0.255045\pi$
$228$	−2.39138	−0.158373
$229$	13.8108	0.912642	0.456321	−	0.889815i	$-0.349167\pi$
$229$	13.8108	0.912642	0.456321	+	0.889815i	$0.349167\pi$
$230$	1.34535	0.0887094
$231$	4.11009	0.270424
$232$	−3.82880	−0.251373
$233$	4.91087	0.321722	0.160861	−	0.986977i	$-0.448573\pi$
$233$	4.91087	0.321722	0.160861	+	0.986977i	$0.448573\pi$
$234$	10.1101	0.660917
$235$	3.21724	0.209869
$236$	12.4404	0.809799
$237$	31.0950	2.01984
$238$	−9.34535	−0.605769
$239$	−14.0490	−0.908753	−0.454377	−	0.890810i	$-0.650138\pi$
$239$	−14.0490	−0.908753	−0.454377	+	0.890810i	$0.650138\pi$
$240$	−0.935945	−0.0604150
$241$	22.7397	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$241$	22.7397	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$242$	1.00000	0.0642824
$243$	−21.3332	−1.36853
$244$	2.00000	0.128037
$245$	1.58355	0.101169
$246$	2.54456	0.162235
$247$	−3.71871	−0.236616
$248$	−3.06406	−0.194568
$249$	20.3734	1.29111
$250$	3.85387	0.243740
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−4.67267	−0.294351
$253$	3.43742	0.216109
$254$	−5.00295	−0.313913
$255$	−5.08913	−0.318693
$256$	1.00000	0.0625000
$257$	−19.4433	−1.21284	−0.606420	−	0.795144i	$-0.707395\pi$
$257$	−19.4433	−1.21284	−0.606420	+	0.795144i	$0.707395\pi$
$258$	−8.93594	−0.556328
$259$	8.00000	0.497096
$260$	−1.45544	−0.0902624
$261$	−10.4094	−0.644326
$262$	−6.50147	−0.401662
$263$	14.6476	0.903210	0.451605	−	0.892218i	$-0.350852\pi$
$263$	14.6476	0.903210	0.451605	+	0.892218i	$0.350852\pi$
$264$	−2.39138	−0.147179
$265$	0.476404	0.0292653
$266$	1.71871	0.105381
$267$	42.3123	2.58947
$268$	−1.71871	−0.104987
$269$	8.34829	0.509004	0.254502	−	0.967072i	$-0.418088\pi$
$269$	8.34829	0.509004	0.254502	+	0.967072i	$0.418088\pi$
$270$	0.263272	0.0160222
$271$	−10.5375	−0.640108	−0.320054	−	0.947399i	$-0.603701\pi$
$271$	−10.5375	−0.640108	−0.320054	+	0.947399i	$0.603701\pi$
$272$	5.43742	0.329692
$273$	−15.2842	−0.925044
$274$	15.7187	0.949602
$275$	4.84682	0.292274
$276$	−8.22018	−0.494797
$277$	−22.8807	−1.37477	−0.687385	−	0.726293i	$-0.741242\pi$
$277$	−22.8807	−1.37477	−0.687385	+	0.726293i	$0.741242\pi$
$278$	−11.1381	−0.668020
$279$	−8.33028	−0.498721
$280$	0.672673	0.0401999
$281$	−4.39138	−0.261968	−0.130984	−	0.991384i	$-0.541814\pi$
$281$	−4.39138	−0.261968	−0.130984	+	0.991384i	$0.541814\pi$
$282$	−19.6576	−1.17059
$283$	−24.0670	−1.43063	−0.715317	−	0.698800i	$-0.753718\pi$
$283$	−24.0670	−1.43063	−0.715317	+	0.698800i	$0.753718\pi$
$284$	1.04604	0.0620709
$285$	0.935945	0.0554406
$286$	−3.71871	−0.219892
$287$	−1.82880	−0.107951
$288$	2.71871	0.160202
$289$	12.5655	0.739149
$290$	1.49853	0.0879965
$291$	−11.1311	−0.652514
$292$	13.6576	0.799251
$293$	−27.2901	−1.59431	−0.797153	−	0.603777i	$-0.793662\pi$
$293$	−27.2901	−1.59431	−0.797153	+	0.603777i	$0.793662\pi$
$294$	−9.67562	−0.564294
$295$	−4.86894	−0.283481
$296$	−4.65465	−0.270546
$297$	0.672673	0.0390324
$298$	11.3453	0.657218
$299$	−12.7828	−0.739246
$300$	−11.5906	−0.669184
$301$	6.42235	0.370178
$302$	−14.6547	−0.843281
$303$	−23.0950	−1.32677
$304$	−1.00000	−0.0573539
$305$	−0.782765	−0.0448210
$306$	14.7828	0.845074
$307$	−9.22313	−0.526392	−0.263196	−	0.964742i	$-0.584777\pi$
$307$	−9.22313	−0.526392	−0.263196	+	0.964742i	$0.584777\pi$
$308$	1.71871	0.0979326
$309$	−16.6296	−0.946024
$310$	1.19922	0.0681110
$311$	6.34829	0.359979	0.179989	−	0.983669i	$-0.442394\pi$
$311$	6.34829	0.359979	0.179989	+	0.983669i	$0.442394\pi$
$312$	8.89286	0.503459
$313$	13.3943	0.757092	0.378546	−	0.925582i	$-0.376424\pi$
$313$	13.3943	0.757092	0.378546	+	0.925582i	$0.376424\pi$
$314$	−10.4835	−0.591616
$315$	1.82880	0.103041
$316$	13.0029	0.731473
$317$	31.0029	1.74130	0.870650	−	0.491904i	$-0.163699\pi$
$317$	31.0029	1.74130	0.870650	+	0.491904i	$0.163699\pi$
$318$	−2.91087	−0.163234
$319$	3.82880	0.214372
$320$	−0.391382	−0.0218789
$321$	6.34829	0.354327
$322$	5.90793	0.329236
$323$	−5.43742	−0.302546
$324$	−9.76475	−0.542486
$325$	−18.0239	−0.999787
$326$	−3.47346	−0.192377
$327$	14.3483	0.793462
$328$	1.06406	0.0587526
$329$	14.1281	0.778908
$330$	0.935945	0.0515221
$331$	−16.2993	−0.895891	−0.447946	−	0.894061i	$-0.647844\pi$
$331$	−16.2993	−0.895891	−0.447946	+	0.894061i	$0.647844\pi$
$332$	8.51949	0.467568
$333$	−12.6547	−0.693471
$334$	23.0950	1.26370
$335$	0.672673	0.0367520
$336$	−4.11009	−0.224224
$337$	−3.19217	−0.173888	−0.0869442	−	0.996213i	$-0.527710\pi$
$337$	−3.19217	−0.173888	−0.0869442	+	0.996213i	$0.527710\pi$
$338$	0.828802	0.0450809
$339$	18.0059	0.977946
$340$	−2.12811	−0.115413
$341$	3.06406	0.165928
$342$	−2.71871	−0.147011
$343$	18.9849	1.02509
$344$	−3.73673	−0.201471
$345$	3.21724	0.173210
$346$	−18.1771	−0.977207
$347$	24.6966	1.32578	0.662891	−	0.748716i	$-0.269329\pi$
$347$	24.6966	1.32578	0.662891	+	0.748716i	$0.269329\pi$
$348$	−9.15613	−0.490820
$349$	−4.43447	−0.237372	−0.118686	−	0.992932i	$-0.537868\pi$
$349$	−4.43447	−0.237372	−0.118686	+	0.992932i	$0.537868\pi$
$350$	8.33028	0.445272
$351$	−2.50147	−0.133519
$352$	−1.00000	−0.0533002
$353$	25.3152	1.34739	0.673696	−	0.739008i	$-0.264706\pi$
$353$	25.3152	1.34739	0.673696	+	0.739008i	$0.264706\pi$
$354$	29.7497	1.58118
$355$	−0.409400	−0.0217287
$356$	17.6936	0.937761
$357$	−22.3483	−1.18280
$358$	11.2842	0.596391
$359$	−13.9749	−0.737569	−0.368784	−	0.929515i	$-0.620226\pi$
$359$	−13.9749	−0.737569	−0.368784	+	0.929515i	$0.620226\pi$
$360$	−1.06406	−0.0560806
$361$	1.00000	0.0526316
$362$	21.8778	1.14987
$363$	2.39138	0.125515
$364$	−6.39138	−0.334999
$365$	−5.34535	−0.279788
$366$	4.78276	0.249999
$367$	−25.3453	−1.32302	−0.661508	−	0.749938i	$-0.730083\pi$
$367$	−25.3453	−1.32302	−0.661508	+	0.749938i	$0.730083\pi$
$368$	−3.43742	−0.179188
$369$	2.89286	0.150596
$370$	1.82175	0.0947083
$371$	2.09207	0.108615
$372$	−7.32733	−0.379904
$373$	8.95396	0.463619	0.231809	−	0.972761i	$-0.425535\pi$
$373$	8.95396	0.463619	0.231809	+	0.972761i	$0.425535\pi$
$374$	−5.43742	−0.281162
$375$	9.21608	0.475916
$376$	−8.22018	−0.423924
$377$	−14.2382	−0.733305
$378$	1.15613	0.0594649
$379$	−25.8527	−1.32796	−0.663982	−	0.747748i	$-0.731135\pi$
$379$	−25.8527	−1.32796	−0.663982	+	0.747748i	$0.731135\pi$
$380$	0.391382	0.0200775
$381$	−11.9640	−0.612932
$382$	−7.09502	−0.363013
$383$	−20.6296	−1.05412	−0.527061	−	0.849827i	$-0.676706\pi$
$383$	−20.6296	−1.05412	−0.527061	+	0.849827i	$0.676706\pi$
$384$	2.39138	0.122035
$385$	−0.672673	−0.0342826
$386$	12.6476	0.643746
$387$	−10.1591	−0.516415
$388$	−4.65465	−0.236304
$389$	15.6086	0.791388	0.395694	−	0.918382i	$-0.370504\pi$
$389$	15.6086	0.791388	0.395694	+	0.918382i	$0.370504\pi$
$390$	−3.48051	−0.176242
$391$	−18.6907	−0.945229
$392$	−4.04604	−0.204356
$393$	−15.5475	−0.784268
$394$	8.43447	0.424922
$395$	−5.08913	−0.256062
$396$	−2.71871	−0.136620
$397$	−34.2512	−1.71902	−0.859508	−	0.511122i	$-0.829230\pi$
$397$	−34.2512	−1.71902	−0.859508	+	0.511122i	$0.829230\pi$
$398$	−2.09207	−0.104866
$399$	4.11009	0.205762
$400$	−4.84682	−0.242341
$401$	4.09207	0.204348	0.102174	−	0.994767i	$-0.467420\pi$
$401$	4.09207	0.204348	0.102174	+	0.994767i	$0.467420\pi$
$402$	−4.11009	−0.204993
$403$	−11.3943	−0.567592
$404$	−9.65760	−0.480484
$405$	3.82175	0.189904
$406$	6.58060	0.326590
$407$	4.65465	0.230723
$408$	13.0029	0.643742
$409$	2.48346	0.122799	0.0613995	−	0.998113i	$-0.480444\pi$
$409$	2.48346	0.122799	0.0613995	+	0.998113i	$0.480444\pi$
$410$	−0.416452	−0.0205671
$411$	37.5894	1.85415
$412$	−6.95396	−0.342597
$413$	−21.3814	−1.05211
$414$	−9.34535	−0.459299
$415$	−3.33438	−0.163678
$416$	3.71871	0.182325
$417$	−26.6355	−1.30435
$418$	1.00000	0.0489116
$419$	35.0950	1.71450	0.857252	−	0.514897i	$-0.172170\pi$
$419$	35.0950	1.71450	0.857252	+	0.514897i	$0.172170\pi$
$420$	1.60862	0.0784925
$421$	−12.3483	−0.601819	−0.300910	−	0.953653i	$-0.597290\pi$
$421$	−12.3483	−0.601819	−0.300910	+	0.953653i	$0.597290\pi$
$422$	−6.09207	−0.296558
$423$	−22.3483	−1.08661
$424$	−1.21724	−0.0591142
$425$	−26.3542	−1.27837
$426$	2.50147	0.121197
$427$	−3.43742	−0.166348
$428$	2.65465	0.128318
$429$	−8.89286	−0.429351
$430$	1.46249	0.0705275
$431$	−6.94691	−0.334621	−0.167310	−	0.985904i	$-0.553508\pi$
$431$	−6.94691	−0.334621	−0.167310	+	0.985904i	$0.553508\pi$
$432$	−0.672673	−0.0323640
$433$	−13.2533	−0.636912	−0.318456	−	0.947938i	$-0.603164\pi$
$433$	−13.2533	−0.636912	−0.318456	+	0.947938i	$0.603164\pi$
$434$	5.26622	0.252787
$435$	3.58355	0.171818
$436$	6.00000	0.287348
$437$	3.43742	0.164434
$438$	32.6606	1.56058
$439$	15.3152	0.730955	0.365477	−	0.930820i	$-0.380906\pi$
$439$	15.3152	0.730955	0.365477	+	0.930820i	$0.380906\pi$
$440$	0.391382	0.0186584
$441$	−11.0000	−0.523810
$442$	20.2202	0.961776
$443$	−20.8807	−0.992074	−0.496037	−	0.868301i	$-0.665212\pi$
$443$	−20.8807	−0.992074	−0.496037	+	0.868301i	$0.665212\pi$
$444$	−11.1311	−0.528257
$445$	−6.92498	−0.328275
$446$	−18.8748	−0.893750
$447$	27.1311	1.28326
$448$	−1.71871	−0.0812014
$449$	−28.7887	−1.35862	−0.679310	−	0.733851i	$-0.737721\pi$
$449$	−28.7887	−1.35862	−0.679310	+	0.733851i	$0.737721\pi$
$450$	−13.1771	−0.621174
$451$	−1.06406	−0.0501044
$452$	7.52949	0.354158
$453$	−35.0449	−1.64655
$454$	20.9669	0.984026
$455$	2.50147	0.117271
$456$	−2.39138	−0.111987
$457$	−32.7526	−1.53210	−0.766052	−	0.642779i	$-0.777781\pi$
$457$	−32.7526	−1.53210	−0.766052	+	0.642779i	$0.777781\pi$
$458$	13.8108	0.645336
$459$	−3.65760	−0.170722
$460$	1.34535	0.0627271
$461$	20.2261	0.942023	0.471011	−	0.882127i	$-0.343889\pi$
$461$	20.2261	0.942023	0.471011	+	0.882127i	$0.343889\pi$
$462$	4.11009	0.191219
$463$	−18.6907	−0.868630	−0.434315	−	0.900761i	$-0.643010\pi$
$463$	−18.6907	−0.868630	−0.434315	+	0.900761i	$0.643010\pi$
$464$	−3.82880	−0.177748
$465$	2.86779	0.132990
$466$	4.91087	0.227492
$467$	34.5685	1.59964	0.799819	−	0.600241i	$-0.204929\pi$
$467$	34.5685	1.59964	0.799819	+	0.600241i	$0.204929\pi$
$468$	10.1101	0.467339
$469$	2.95396	0.136401
$470$	3.21724	0.148400
$471$	−25.0700	−1.15516
$472$	12.4404	0.572614
$473$	3.73673	0.171815
$474$	31.0950	1.42824
$475$	4.84682	0.222387
$476$	−9.34535	−0.428343
$477$	−3.30931	−0.151523
$478$	−14.0490	−0.642586
$479$	−17.7187	−0.809589	−0.404794	−	0.914408i	$-0.632657\pi$
$479$	−17.7187	−0.809589	−0.404794	+	0.914408i	$0.632657\pi$
$480$	−0.935945	−0.0427198
$481$	−17.3093	−0.789237
$482$	22.7397	1.03576
$483$	14.1281	0.642851
$484$	1.00000	0.0454545
$485$	1.82175	0.0827214
$486$	−21.3332	−0.967695
$487$	37.7426	1.71028	0.855141	−	0.518396i	$-0.173471\pi$
$487$	37.7426	1.71028	0.855141	+	0.518396i	$0.173471\pi$
$488$	2.00000	0.0905357
$489$	−8.30636	−0.375627
$490$	1.58355	0.0715374
$491$	0.373364	0.0168497	0.00842485	−	0.999965i	$-0.497318\pi$
$491$	0.373364	0.0168497	0.00842485	+	0.999965i	$0.497318\pi$
$492$	2.54456	0.114718
$493$	−20.8188	−0.937632
$494$	−3.71871	−0.167313
$495$	1.06406	0.0478257
$496$	−3.06406	−0.137580
$497$	−1.79783	−0.0806439
$498$	20.3734	0.912952
$499$	−14.8388	−0.664276	−0.332138	−	0.943231i	$-0.607770\pi$
$499$	−14.8388	−0.664276	−0.332138	+	0.943231i	$0.607770\pi$
$500$	3.85387	0.172350
$501$	55.2290	2.46745
$502$	−12.0000	−0.535586
$503$	33.2662	1.48327	0.741634	−	0.670805i	$-0.234051\pi$
$503$	33.2662	1.48327	0.741634	+	0.670805i	$0.234051\pi$
$504$	−4.67267	−0.208137
$505$	3.77982	0.168200
$506$	3.43742	0.152812
$507$	1.98198	0.0880229
$508$	−5.00295	−0.221970
$509$	−8.47051	−0.375449	−0.187724	−	0.982222i	$-0.560111\pi$
$509$	−8.47051	−0.375449	−0.187724	+	0.982222i	$0.560111\pi$
$510$	−5.08913	−0.225350
$511$	−23.4735	−1.03840
$512$	1.00000	0.0441942
$513$	0.672673	0.0296992
$514$	−19.4433	−0.857608
$515$	2.72166	0.119931
$516$	−8.93594	−0.393383
$517$	8.22018	0.361523
$518$	8.00000	0.351500
$519$	−43.4684	−1.90805
$520$	−1.45544	−0.0638252
$521$	−37.3152	−1.63481	−0.817404	−	0.576064i	$-0.804588\pi$
$521$	−37.3152	−1.63481	−0.817404	+	0.576064i	$0.804588\pi$
$522$	−10.4094	−0.455607
$523$	−31.6576	−1.38429	−0.692145	−	0.721758i	$-0.743334\pi$
$523$	−31.6576	−1.38429	−0.692145	+	0.721758i	$0.743334\pi$
$524$	−6.50147	−0.284018
$525$	19.9209	0.869418
$526$	14.6476	0.638666
$527$	−16.6606	−0.725745
$528$	−2.39138	−0.104072
$529$	−11.1841	−0.486267
$530$	0.476404	0.0206937
$531$	33.8217	1.46774
$532$	1.71871	0.0745155
$533$	3.95691	0.171393
$534$	42.3123	1.83103
$535$	−1.03899	−0.0449192
$536$	−1.71871	−0.0742370
$537$	26.9849	1.16449
$538$	8.34829	0.359921
$539$	4.04604	0.174275
$540$	0.263272	0.0113294
$541$	28.0921	1.20777	0.603886	−	0.797070i	$-0.293618\pi$
$541$	28.0921	1.20777	0.603886	+	0.797070i	$0.293618\pi$
$542$	−10.5375	−0.452625
$543$	52.3182	2.24519
$544$	5.43742	0.233127
$545$	−2.34829	−0.100590
$546$	−15.2842	−0.654105
$547$	24.1841	1.03404	0.517020	−	0.855973i	$-0.327041\pi$
$547$	24.1841	1.03404	0.517020	+	0.855973i	$0.327041\pi$
$548$	15.7187	0.671470
$549$	5.43742	0.232063
$550$	4.84682	0.206669
$551$	3.82880	0.163112
$552$	−8.22018	−0.349874
$553$	−22.3483	−0.950346
$554$	−22.8807	−0.972109
$555$	4.35650	0.184923
$556$	−11.1381	−0.472361
$557$	39.9699	1.69358	0.846789	−	0.531929i	$-0.178533\pi$
$557$	39.9699	1.69358	0.846789	+	0.531929i	$0.178533\pi$
$558$	−8.33028	−0.352649
$559$	−13.8958	−0.587730
$560$	0.672673	0.0284256
$561$	−13.0029	−0.548985
$562$	−4.39138	−0.185239
$563$	16.4764	0.694398	0.347199	−	0.937792i	$-0.387133\pi$
$563$	16.4764	0.694398	0.347199	+	0.937792i	$0.387133\pi$
$564$	−19.6576	−0.827734
$565$	−2.94691	−0.123977
$566$	−24.0670	−1.01161
$567$	16.7828	0.704810
$568$	1.04604	0.0438907
$569$	−24.9418	−1.04562	−0.522808	−	0.852450i	$-0.675116\pi$
$569$	−24.9418	−1.04562	−0.522808	+	0.852450i	$0.675116\pi$
$570$	0.935945	0.0392024
$571$	−8.50737	−0.356022	−0.178011	−	0.984028i	$-0.556966\pi$
$571$	−8.50737	−0.356022	−0.178011	+	0.984028i	$0.556966\pi$
$572$	−3.71871	−0.155487
$573$	−16.9669	−0.708803
$574$	−1.82880	−0.0763327
$575$	16.6606	0.694793
$576$	2.71871	0.113280
$577$	31.8527	1.32605	0.663023	−	0.748599i	$-0.269273\pi$
$577$	31.8527	1.32605	0.663023	+	0.748599i	$0.269273\pi$
$578$	12.5655	0.522657
$579$	30.2453	1.25695
$580$	1.49853	0.0622229
$581$	−14.6425	−0.607475
$582$	−11.1311	−0.461397
$583$	1.21724	0.0504127
$584$	13.6576	0.565156
$585$	−3.95691	−0.163598
$586$	−27.2901	−1.12735
$587$	2.43447	0.100481	0.0502407	−	0.998737i	$-0.484001\pi$
$587$	2.43447	0.100481	0.0502407	+	0.998737i	$0.484001\pi$
$588$	−9.67562	−0.399016
$589$	3.06406	0.126252
$590$	−4.86894	−0.200451
$591$	20.1700	0.829685
$592$	−4.65465	−0.191305
$593$	−18.1841	−0.746733	−0.373367	−	0.927684i	$-0.621797\pi$
$593$	−18.1841	−0.746733	−0.373367	+	0.927684i	$0.621797\pi$
$594$	0.672673	0.0276001
$595$	3.65760	0.149947
$596$	11.3453	0.464723
$597$	−5.00295	−0.204757
$598$	−12.7828	−0.522726
$599$	4.33534	0.177137	0.0885687	−	0.996070i	$-0.471771\pi$
$599$	4.33534	0.177137	0.0885687	+	0.996070i	$0.471771\pi$
$600$	−11.5906	−0.473184
$601$	16.2453	0.662658	0.331329	−	0.943515i	$-0.392503\pi$
$601$	16.2453	0.662658	0.331329	+	0.943515i	$0.392503\pi$
$602$	6.42235	0.261755
$603$	−4.67267	−0.190286
$604$	−14.6547	−0.596289
$605$	−0.391382	−0.0159120
$606$	−23.0950	−0.938171
$607$	41.6995	1.69253	0.846266	−	0.532761i	$-0.178845\pi$
$607$	41.6995	1.69253	0.846266	+	0.532761i	$0.178845\pi$
$608$	−1.00000	−0.0405554
$609$	15.7367	0.637684
$610$	−0.782765	−0.0316932
$611$	−30.5685	−1.23667
$612$	14.7828	0.597558
$613$	48.1900	1.94638	0.973189	−	0.230008i	$-0.0738751\pi$
$613$	48.1900	1.94638	0.973189	+	0.230008i	$0.0738751\pi$
$614$	−9.22313	−0.372215
$615$	−0.995897	−0.0401584
$616$	1.71871	0.0692488
$617$	15.7187	0.632811	0.316406	−	0.948624i	$-0.397524\pi$
$617$	15.7187	0.632811	0.316406	+	0.948624i	$0.397524\pi$
$618$	−16.6296	−0.668940
$619$	44.6606	1.79506	0.897530	−	0.440954i	$-0.145360\pi$
$619$	44.6606	1.79506	0.897530	+	0.440954i	$0.145360\pi$
$620$	1.19922	0.0481617
$621$	2.31226	0.0927877
$622$	6.34829	0.254543
$623$	−30.4102	−1.21836
$624$	8.89286	0.355999
$625$	22.7258	0.909030
$626$	13.3943	0.535345
$627$	2.39138	0.0955026
$628$	−10.4835	−0.418335
$629$	−25.3093	−1.00915
$630$	1.82880	0.0728612
$631$	2.38433	0.0949187	0.0474593	−	0.998873i	$-0.484888\pi$
$631$	2.38433	0.0949187	0.0474593	+	0.998873i	$0.484888\pi$
$632$	13.0029	0.517230
$633$	−14.5685	−0.579045
$634$	31.0029	1.23128
$635$	1.95807	0.0777035
$636$	−2.91087	−0.115424
$637$	−15.0460	−0.596146
$638$	3.82880	0.151584
$639$	2.84387	0.112502
$640$	−0.391382	−0.0154707
$641$	−45.6936	−1.80479	−0.902395	−	0.430910i	$-0.858193\pi$
$641$	−45.6936	−1.80479	−0.902395	+	0.430910i	$0.858193\pi$
$642$	6.34829	0.250547
$643$	42.2621	1.66666	0.833328	−	0.552779i	$-0.186433\pi$
$643$	42.2621	1.66666	0.833328	+	0.552779i	$0.186433\pi$
$644$	5.90793	0.232805
$645$	3.49737	0.137709
$646$	−5.43742	−0.213932
$647$	−38.7166	−1.52211	−0.761053	−	0.648690i	$-0.775317\pi$
$647$	−38.7166	−1.52211	−0.761053	+	0.648690i	$0.775317\pi$
$648$	−9.76475	−0.383595
$649$	−12.4404	−0.488327
$650$	−18.0239	−0.706956
$651$	12.5935	0.493580
$652$	−3.47346	−0.136031
$653$	38.6915	1.51412	0.757058	−	0.653348i	$-0.226636\pi$
$653$	38.6915	1.51412	0.757058	+	0.653348i	$0.226636\pi$
$654$	14.3483	0.561063
$655$	2.54456	0.0994243
$656$	1.06406	0.0415444
$657$	37.1311	1.44862
$658$	14.1281	0.550771
$659$	12.7467	0.496542	0.248271	−	0.968691i	$-0.420138\pi$
$659$	12.7467	0.496542	0.248271	+	0.968691i	$0.420138\pi$
$660$	0.935945	0.0364316
$661$	−21.1671	−0.823305	−0.411652	−	0.911341i	$-0.635048\pi$
$661$	−21.1671	−0.823305	−0.411652	+	0.911341i	$0.635048\pi$
$662$	−16.2993	−0.633491
$663$	48.3542	1.87792
$664$	8.51949	0.330620
$665$	−0.672673	−0.0260851
$666$	−12.6547	−0.490358
$667$	13.1612	0.509604
$668$	23.0950	0.893573
$669$	−45.1370	−1.74510
$670$	0.672673	0.0259876
$671$	−2.00000	−0.0772091
$672$	−4.11009	−0.158550
$673$	2.25115	0.0867755	0.0433878	−	0.999058i	$-0.486185\pi$
$673$	2.25115	0.0867755	0.0433878	+	0.999058i	$0.486185\pi$
$674$	−3.19217	−0.122958
$675$	3.26032	0.125490
$676$	0.828802	0.0318770
$677$	−9.57848	−0.368131	−0.184065	−	0.982914i	$-0.558926\pi$
$677$	−9.57848	−0.368131	−0.184065	+	0.982914i	$0.558926\pi$
$678$	18.0059	0.691512
$679$	8.00000	0.307012
$680$	−2.12811	−0.0816093
$681$	50.1399	1.92137
$682$	3.06406	0.117329
$683$	−32.0118	−1.22490	−0.612449	−	0.790510i	$-0.709815\pi$
$683$	−32.0118	−1.22490	−0.612449	+	0.790510i	$0.709815\pi$
$684$	−2.71871	−0.103952
$685$	−6.15203	−0.235057
$686$	18.9849	0.724848
$687$	33.0269	1.26005
$688$	−3.73673	−0.142461
$689$	−4.52654	−0.172448
$690$	3.21724	0.122478
$691$	−25.9699	−0.987940	−0.493970	−	0.869479i	$-0.664455\pi$
$691$	−25.9699	−0.987940	−0.493970	+	0.869479i	$0.664455\pi$
$692$	−18.1771	−0.690990
$693$	4.67267	0.177500
$694$	24.6966	0.937470
$695$	4.35926	0.165356
$696$	−9.15613	−0.347062
$697$	5.78571	0.219150
$698$	−4.43447	−0.167847
$699$	11.7438	0.444191
$700$	8.33028	0.314855
$701$	−21.0950	−0.796748	−0.398374	−	0.917223i	$-0.630425\pi$
$701$	−21.0950	−0.796748	−0.398374	+	0.917223i	$0.630425\pi$
$702$	−2.50147	−0.0944121
$703$	4.65465	0.175554
$704$	−1.00000	−0.0376889
$705$	7.69364	0.289759
$706$	25.3152	0.952750
$707$	16.5986	0.624255
$708$	29.7497	1.11806
$709$	29.1981	1.09656	0.548278	−	0.836296i	$-0.315284\pi$
$709$	29.1981	1.09656	0.548278	+	0.836296i	$0.315284\pi$
$710$	−0.409400	−0.0153645
$711$	35.3512	1.32578
$712$	17.6936	0.663097
$713$	10.5324	0.394443
$714$	−22.3483	−0.836364
$715$	1.45544	0.0544303
$716$	11.2842	0.421712
$717$	−33.5965	−1.25468
$718$	−13.9749	−0.521540
$719$	−13.9079	−0.518678	−0.259339	−	0.965786i	$-0.583505\pi$
$719$	−13.9079	−0.518678	−0.259339	+	0.965786i	$0.583505\pi$
$720$	−1.06406	−0.0396550
$721$	11.9518	0.445110
$722$	1.00000	0.0372161
$723$	54.3793	2.02239
$724$	21.8778	0.813082
$725$	18.5575	0.689209
$726$	2.39138	0.0887525
$727$	−42.0059	−1.55791	−0.778956	−	0.627078i	$-0.784251\pi$
$727$	−42.0059	−1.55791	−0.778956	+	0.627078i	$0.784251\pi$
$728$	−6.39138	−0.236880
$729$	−21.7217	−0.804506
$730$	−5.34535	−0.197840
$731$	−20.3182	−0.751494
$732$	4.78276	0.176776
$733$	36.0780	1.33257	0.666285	−	0.745697i	$-0.267883\pi$
$733$	36.0780	1.33257	0.666285	+	0.745697i	$0.267883\pi$
$734$	−25.3453	−0.935514
$735$	3.78687	0.139681
$736$	−3.43742	−0.126705
$737$	1.71871	0.0633095
$738$	2.89286	0.106488
$739$	49.9988	1.83924	0.919619	−	0.392812i	$-0.128498\pi$
$739$	49.9988	1.83924	0.919619	+	0.392812i	$0.128498\pi$
$740$	1.82175	0.0669689
$741$	−8.89286	−0.326687
$742$	2.09207	0.0768025
$743$	0.220184	0.00807777	0.00403889	−	0.999992i	$-0.498714\pi$
$743$	0.220184	0.00807777	0.00403889	+	0.999992i	$0.498714\pi$
$744$	−7.32733	−0.268633
$745$	−4.44037	−0.162683
$746$	8.95396	0.327828
$747$	23.1620	0.847454
$748$	−5.43742	−0.198812
$749$	−4.56258	−0.166713
$750$	9.21608	0.336524
$751$	−23.1311	−0.844064	−0.422032	−	0.906581i	$-0.638683\pi$
$751$	−23.1311	−0.844064	−0.422032	+	0.906581i	$0.638683\pi$
$752$	−8.22018	−0.299759
$753$	−28.6966	−1.04576
$754$	−14.2382	−0.518525
$755$	5.73557	0.208739
$756$	1.15613	0.0420480
$757$	37.4304	1.36043	0.680215	−	0.733013i	$-0.261886\pi$
$757$	37.4304	1.36043	0.680215	+	0.733013i	$0.261886\pi$
$758$	−25.8527	−0.939013
$759$	8.22018	0.298374
$760$	0.391382	0.0141969
$761$	−33.9499	−1.23068	−0.615341	−	0.788261i	$-0.710982\pi$
$761$	−33.9499	−1.23068	−0.615341	+	0.788261i	$0.710982\pi$
$762$	−11.9640	−0.433409
$763$	−10.3123	−0.373329
$764$	−7.09502	−0.256689
$765$	−5.78571	−0.209183
$766$	−20.6296	−0.745377
$767$	46.2621	1.67043
$768$	2.39138	0.0862916
$769$	−44.7025	−1.61201	−0.806006	−	0.591907i	$-0.798375\pi$
$769$	−44.7025	−1.61201	−0.806006	+	0.591907i	$0.798375\pi$
$770$	−0.672673	−0.0242414
$771$	−46.4964	−1.67453
$772$	12.6476	0.455197
$773$	−44.8748	−1.61404	−0.807018	−	0.590527i	$-0.798920\pi$
$773$	−44.8748	−1.61404	−0.807018	+	0.590527i	$0.798920\pi$
$774$	−10.1591	−0.365161
$775$	14.8509	0.533461
$776$	−4.65465	−0.167092
$777$	19.1311	0.686323
$778$	15.6086	0.559596
$779$	−1.06406	−0.0381237
$780$	−3.48051	−0.124622
$781$	−1.04604	−0.0374301
$782$	−18.6907	−0.668378
$783$	2.57553	0.0920419
$784$	−4.04604	−0.144501
$785$	4.10304	0.146444
$786$	−15.5475	−0.554561
$787$	11.8418	0.422113	0.211056	−	0.977474i	$-0.432310\pi$
$787$	11.8418	0.422113	0.211056	+	0.977474i	$0.432310\pi$
$788$	8.43447	0.300466
$789$	35.0280	1.24703
$790$	−5.08913	−0.181063
$791$	−12.9410	−0.460129
$792$	−2.71871	−0.0966051
$793$	7.43742	0.264111
$794$	−34.2512	−1.21553
$795$	1.13927	0.0404056
$796$	−2.09207	−0.0741516
$797$	0.128110	0.00453789	0.00226895	−	0.999997i	$-0.499278\pi$
$797$	0.128110	0.00453789	0.00226895	+	0.999997i	$0.499278\pi$
$798$	4.11009	0.145496
$799$	−44.6966	−1.58125
$800$	−4.84682	−0.171361
$801$	48.1039	1.69967
$802$	4.09207	0.144496
$803$	−13.6576	−0.481966
$804$	−4.11009	−0.144952
$805$	−2.31226	−0.0814964
$806$	−11.3943	−0.401348
$807$	19.9640	0.702765
$808$	−9.65760	−0.339753
$809$	17.5354	0.616512	0.308256	−	0.951304i	$-0.400255\pi$
$809$	17.5354	0.616512	0.308256	+	0.951304i	$0.400255\pi$
$810$	3.82175	0.134283
$811$	−39.3512	−1.38181	−0.690905	−	0.722946i	$-0.742788\pi$
$811$	−39.3512	−1.38181	−0.690905	+	0.722946i	$0.742788\pi$
$812$	6.58060	0.230934
$813$	−25.1992	−0.883775
$814$	4.65465	0.163146
$815$	1.35945	0.0476194
$816$	13.0029	0.455194
$817$	3.73673	0.130732
$818$	2.48346	0.0868320
$819$	−17.3763	−0.607178
$820$	−0.416452	−0.0145431
$821$	14.0980	0.492023	0.246011	−	0.969267i	$-0.420880\pi$
$821$	14.0980	0.492023	0.246011	+	0.969267i	$0.420880\pi$
$822$	37.5894	1.31108
$823$	−1.12516	−0.0392207	−0.0196103	−	0.999808i	$-0.506243\pi$
$823$	−1.12516	−0.0392207	−0.0196103	+	0.999808i	$0.506243\pi$
$824$	−6.95396	−0.242253
$825$	11.5906	0.403533
$826$	−21.3814	−0.743953
$827$	−33.8217	−1.17610	−0.588049	−	0.808825i	$-0.700104\pi$
$827$	−33.8217	−1.17610	−0.588049	+	0.808825i	$0.700104\pi$
$828$	−9.34535	−0.324773
$829$	−36.1900	−1.25693	−0.628466	−	0.777837i	$-0.716317\pi$
$829$	−36.1900	−1.25693	−0.628466	+	0.777837i	$0.716317\pi$
$830$	−3.33438	−0.115738
$831$	−54.7166	−1.89810
$832$	3.71871	0.128923
$833$	−22.0000	−0.762255
$834$	−26.6355	−0.922311
$835$	−9.03899	−0.312807
$836$	1.00000	0.0345857
$837$	2.06111	0.0712423
$838$	35.0950	1.21234
$839$	−31.8347	−1.09906	−0.549528	−	0.835475i	$-0.685192\pi$
$839$	−31.8347	−1.09906	−0.549528	+	0.835475i	$0.685192\pi$
$840$	1.60862	0.0555026
$841$	−14.3403	−0.494492
$842$	−12.3483	−0.425550
$843$	−10.5015	−0.361690
$844$	−6.09207	−0.209698
$845$	−0.324378	−0.0111590
$846$	−22.3483	−0.768350
$847$	−1.71871	−0.0590556
$848$	−1.21724	−0.0418000
$849$	−57.5534	−1.97523
$850$	−26.3542	−0.903941
$851$	16.0000	0.548473
$852$	2.50147	0.0856991
$853$	−31.1009	−1.06488	−0.532438	−	0.846469i	$-0.678724\pi$
$853$	−31.1009	−1.06488	−0.532438	+	0.846469i	$0.678724\pi$
$854$	−3.43742	−0.117626
$855$	1.06406	0.0363899
$856$	2.65465	0.0907342
$857$	−35.4985	−1.21261	−0.606303	−	0.795234i	$-0.707348\pi$
$857$	−35.4985	−1.21261	−0.606303	+	0.795234i	$0.707348\pi$
$858$	−8.89286	−0.303597
$859$	−5.63760	−0.192352	−0.0961762	−	0.995364i	$-0.530661\pi$
$859$	−5.63760	−0.192352	−0.0961762	+	0.995364i	$0.530661\pi$
$860$	1.46249	0.0498705
$861$	−4.37336	−0.149044
$862$	−6.94691	−0.236613
$863$	31.9949	1.08912	0.544560	−	0.838722i	$-0.316697\pi$
$863$	31.9949	1.08912	0.544560	+	0.838722i	$0.316697\pi$
$864$	−0.672673	−0.0228848
$865$	7.11420	0.241890
$866$	−13.2533	−0.450364
$867$	30.0490	1.02052
$868$	5.26622	0.178747
$869$	−13.0029	−0.441095
$870$	3.58355	0.121494
$871$	−6.39138	−0.216564
$872$	6.00000	0.203186
$873$	−12.6547	−0.428295
$874$	3.43742	0.116272
$875$	−6.62369	−0.223921
$876$	32.6606	1.10350
$877$	−3.23019	−0.109076	−0.0545378	−	0.998512i	$-0.517369\pi$
$877$	−3.23019	−0.109076	−0.0545378	+	0.998512i	$0.517369\pi$
$878$	15.3152	0.516863
$879$	−65.2612	−2.20120
$880$	0.391382	0.0131935
$881$	−24.7217	−0.832894	−0.416447	−	0.909160i	$-0.636725\pi$
$881$	−24.7217	−0.832894	−0.416447	+	0.909160i	$0.636725\pi$
$882$	−11.0000	−0.370389
$883$	11.9138	0.400932	0.200466	−	0.979701i	$-0.435754\pi$
$883$	11.9138	0.400932	0.200466	+	0.979701i	$0.435754\pi$
$884$	20.2202	0.680078
$885$	−11.6435	−0.391392
$886$	−20.8807	−0.701502
$887$	−12.1841	−0.409104	−0.204552	−	0.978856i	$-0.565574\pi$
$887$	−12.1841	−0.409104	−0.204552	+	0.978856i	$0.565574\pi$
$888$	−11.1311	−0.373534
$889$	8.59862	0.288388
$890$	−6.92498	−0.232126
$891$	9.76475	0.327131
$892$	−18.8748	−0.631976
$893$	8.22018	0.275078
$894$	27.1311	0.907398
$895$	−4.41645	−0.147626
$896$	−1.71871	−0.0574181
$897$	−30.5685	−1.02065
$898$	−28.7887	−0.960690
$899$	11.7317	0.391273
$900$	−13.1771	−0.439237
$901$	−6.61862	−0.220498
$902$	−1.06406	−0.0354292
$903$	15.3583	0.511092
$904$	7.52949	0.250427
$905$	−8.56258	−0.284630
$906$	−35.0449	−1.16429
$907$	45.3211	1.50486	0.752431	−	0.658671i	$-0.228881\pi$
$907$	45.3211	1.50486	0.752431	+	0.658671i	$0.228881\pi$
$908$	20.9669	0.695811
$909$	−26.2562	−0.870864
$910$	2.50147	0.0829231
$911$	−13.6376	−0.451834	−0.225917	−	0.974147i	$-0.572538\pi$
$911$	−13.6376	−0.451834	−0.225917	+	0.974147i	$0.572538\pi$
$912$	−2.39138	−0.0791866
$913$	−8.51949	−0.281954
$914$	−32.7526	−1.08336
$915$	−1.87189	−0.0618828
$916$	13.8108	0.456321
$917$	11.1741	0.369003
$918$	−3.65760	−0.120719
$919$	39.5283	1.30392	0.651960	−	0.758254i	$-0.273947\pi$
$919$	39.5283	1.30392	0.651960	+	0.758254i	$0.273947\pi$
$920$	1.34535	0.0443547
$921$	−22.0560	−0.726771
$922$	20.2261	0.666111
$923$	3.88991	0.128038
$924$	4.11009	0.135212
$925$	22.5603	0.741777
$926$	−18.6907	−0.614214
$927$	−18.9058	−0.620948
$928$	−3.82880	−0.125687
$929$	6.89991	0.226379	0.113189	−	0.993573i	$-0.463893\pi$
$929$	6.89991	0.226379	0.113189	+	0.993573i	$0.463893\pi$
$930$	2.86779	0.0940384
$931$	4.04604	0.132604
$932$	4.91087	0.160861
$933$	15.1812	0.497010
$934$	34.5685	1.13112
$935$	2.12811	0.0695966
$936$	10.1101	0.330459
$937$	−21.3873	−0.698692	−0.349346	−	0.936994i	$-0.613596\pi$
$937$	−21.3873	−0.698692	−0.349346	+	0.936994i	$0.613596\pi$
$938$	2.95396	0.0964503
$939$	32.0310	1.04529
$940$	3.21724	0.104935
$941$	−27.5655	−0.898611	−0.449305	−	0.893378i	$-0.648328\pi$
$941$	−27.5655	−0.898611	−0.449305	+	0.893378i	$0.648328\pi$
$942$	−25.0700	−0.816823
$943$	−3.65760	−0.119108
$944$	12.4404	0.404899
$945$	−0.452489	−0.0147195
$946$	3.73673	0.121491
$947$	−9.22313	−0.299712	−0.149856	−	0.988708i	$-0.547881\pi$
$947$	−9.22313	−0.299712	−0.149856	+	0.988708i	$0.547881\pi$
$948$	31.0950	1.00992
$949$	50.7887	1.64867
$950$	4.84682	0.157252
$951$	74.1399	2.40415
$952$	−9.34535	−0.302884
$953$	22.6966	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$953$	22.6966	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$954$	−3.30931	−0.107143
$955$	2.77687	0.0898573
$956$	−14.0490	−0.454377
$957$	9.15613	0.295976
$958$	−17.7187	−0.572466
$959$	−27.0159	−0.872389
$960$	−0.935945	−0.0302075
$961$	−21.6116	−0.697147
$962$	−17.3093	−0.558075
$963$	7.21724	0.232572
$964$	22.7397	0.732396
$965$	−4.95005	−0.159348
$966$	14.1281	0.454564
$967$	−48.6966	−1.56598	−0.782988	−	0.622036i	$-0.786306\pi$
$967$	−48.6966	−1.56598	−0.782988	+	0.622036i	$0.786306\pi$
$968$	1.00000	0.0321412
$969$	−13.0029	−0.417715
$970$	1.82175	0.0584929
$971$	−19.5165	−0.626316	−0.313158	−	0.949701i	$-0.601387\pi$
$971$	−19.5165	−0.626316	−0.313158	+	0.949701i	$0.601387\pi$
$972$	−21.3332	−0.684264
$973$	19.1432	0.613702
$974$	37.7426	1.20935
$975$	−43.1021	−1.38037
$976$	2.00000	0.0640184
$977$	37.3512	1.19497	0.597486	−	0.801879i	$-0.296166\pi$
$977$	37.3512	1.19497	0.597486	+	0.801879i	$0.296166\pi$
$978$	−8.30636	−0.265608
$979$	−17.6936	−0.565491
$980$	1.58355	0.0505846
$981$	16.3123	0.520810
$982$	0.373364	0.0119145
$983$	−45.8888	−1.46362	−0.731812	−	0.681507i	$-0.761325\pi$
$983$	−45.8888	−1.46362	−0.731812	+	0.681507i	$0.761325\pi$
$984$	2.54456	0.0811177
$985$	−3.30110	−0.105182
$986$	−20.8188	−0.663006
$987$	33.7857	1.07541
$988$	−3.71871	−0.118308
$989$	12.8447	0.408438
$990$	1.06406	0.0338179
$991$	−30.9418	−0.982900	−0.491450	−	0.870906i	$-0.663533\pi$
$991$	−30.9418	−0.982900	−0.491450	+	0.870906i	$0.663533\pi$
$992$	−3.06406	−0.0972838
$993$	−38.9779	−1.23693
$994$	−1.79783	−0.0570238
$995$	0.818801	0.0259577
$996$	20.3734	0.645555
$997$	49.8418	1.57850	0.789252	−	0.614069i	$-0.210469\pi$
$997$	49.8418	1.57850	0.789252	+	0.614069i	$0.210469\pi$
$998$	−14.8388	−0.469714
$999$	3.13106	0.0990623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	418.2.a.h.1.3	✓	3
3.2	odd	2		3762.2.a.bd.1.3		3
4.3	odd	2		3344.2.a.p.1.1		3
11.10	odd	2		4598.2.a.bm.1.3		3
19.18	odd	2		7942.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.h.1.3	✓	3	1.1	even	1	trivial
3344.2.a.p.1.1		3	4.3	odd	2
3762.2.a.bd.1.3		3	3.2	odd	2
4598.2.a.bm.1.3		3	11.10	odd	2
7942.2.a.bc.1.1		3	19.18	odd	2

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:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.39138 q^{3} +1.00000 q^{4} -0.391382 q^{5} +2.39138 q^{6} -1.71871 q^{7} +1.00000 q^{8} +2.71871 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.39138 q^{3} +1.00000 q^{4} -0.391382 q^{5} +2.39138 q^{6} -1.71871 q^{7} +1.00000 q^{8} +2.71871 q^{9} -0.391382 q^{10} -1.00000 q^{11} +2.39138 q^{12} +3.71871 q^{13} -1.71871 q^{14} -0.935945 q^{15} +1.00000 q^{16} +5.43742 q^{17} +2.71871 q^{18} -1.00000 q^{19} -0.391382 q^{20} -4.11009 q^{21} -1.00000 q^{22} -3.43742 q^{23} +2.39138 q^{24} -4.84682 q^{25} +3.71871 q^{26} -0.672673 q^{27} -1.71871 q^{28} -3.82880 q^{29} -0.935945 q^{30} -3.06406 q^{31} +1.00000 q^{32} -2.39138 q^{33} +5.43742 q^{34} +0.672673 q^{35} +2.71871 q^{36} -4.65465 q^{37} -1.00000 q^{38} +8.89286 q^{39} -0.391382 q^{40} +1.06406 q^{41} -4.11009 q^{42} -3.73673 q^{43} -1.00000 q^{44} -1.06406 q^{45} -3.43742 q^{46} -8.22018 q^{47} +2.39138 q^{48} -4.04604 q^{49} -4.84682 q^{50} +13.0029 q^{51} +3.71871 q^{52} -1.21724 q^{53} -0.672673 q^{54} +0.391382 q^{55} -1.71871 q^{56} -2.39138 q^{57} -3.82880 q^{58} +12.4404 q^{59} -0.935945 q^{60} +2.00000 q^{61} -3.06406 q^{62} -4.67267 q^{63} +1.00000 q^{64} -1.45544 q^{65} -2.39138 q^{66} -1.71871 q^{67} +5.43742 q^{68} -8.22018 q^{69} +0.672673 q^{70} +1.04604 q^{71} +2.71871 q^{72} +13.6576 q^{73} -4.65465 q^{74} -11.5906 q^{75} -1.00000 q^{76} +1.71871 q^{77} +8.89286 q^{78} +13.0029 q^{79} -0.391382 q^{80} -9.76475 q^{81} +1.06406 q^{82} +8.51949 q^{83} -4.11009 q^{84} -2.12811 q^{85} -3.73673 q^{86} -9.15613 q^{87} -1.00000 q^{88} +17.6936 q^{89} -1.06406 q^{90} -6.39138 q^{91} -3.43742 q^{92} -7.32733 q^{93} -8.22018 q^{94} +0.391382 q^{95} +2.39138 q^{96} -4.65465 q^{97} -4.04604 q^{98} -2.71871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 + 3 * q^8 + 2 * q^9	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9} + 5 q^{10} - 3 q^{11} + q^{12} + 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} + 4 q^{17} + 2 q^{18} - 3 q^{19} + 5 q^{20} - 3 q^{22} + 2 q^{23} + q^{24} + 4 q^{25} + 5 q^{26} - 2 q^{27} + q^{28} + 7 q^{29} - 9 q^{30} - 3 q^{31} + 3 q^{32} - q^{33} + 4 q^{34} + 2 q^{35} + 2 q^{36} - 14 q^{37} - 3 q^{38} + 2 q^{39} + 5 q^{40} - 3 q^{41} - 5 q^{43} - 3 q^{44} + 3 q^{45} + 2 q^{46} + q^{48} - 6 q^{49} + 4 q^{50} + 2 q^{51} + 5 q^{52} - 16 q^{53} - 2 q^{54} - 5 q^{55} + q^{56} - q^{57} + 7 q^{58} - 12 q^{59} - 9 q^{60} + 6 q^{61} - 3 q^{62} - 14 q^{63} + 3 q^{64} + 8 q^{65} - q^{66} + q^{67} + 4 q^{68} + 2 q^{70} - 3 q^{71} + 2 q^{72} + 4 q^{73} - 14 q^{74} - 41 q^{75} - 3 q^{76} - q^{77} + 2 q^{78} + 2 q^{79} + 5 q^{80} - 17 q^{81} - 3 q^{82} + 7 q^{83} + 6 q^{85} - 5 q^{86} - 9 q^{87} - 3 q^{88} + 16 q^{89} + 3 q^{90} - 13 q^{91} + 2 q^{92} - 22 q^{93} - 5 q^{95} + q^{96} - 14 q^{97} - 6 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 + 3 * q^8 + 2 * q^9 + 5 * q^10 - 3 * q^11 + q^12 + 5 * q^13 + q^14 - 9 * q^15 + 3 * q^16 + 4 * q^17 + 2 * q^18 - 3 * q^19 + 5 * q^20 - 3 * q^22 + 2 * q^23 + q^24 + 4 * q^25 + 5 * q^26 - 2 * q^27 + q^28 + 7 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 - q^33 + 4 * q^34 + 2 * q^35 + 2 * q^36 - 14 * q^37 - 3 * q^38 + 2 * q^39 + 5 * q^40 - 3 * q^41 - 5 * q^43 - 3 * q^44 + 3 * q^45 + 2 * q^46 + q^48 - 6 * q^49 + 4 * q^50 + 2 * q^51 + 5 * q^52 - 16 * q^53 - 2 * q^54 - 5 * q^55 + q^56 - q^57 + 7 * q^58 - 12 * q^59 - 9 * q^60 + 6 * q^61 - 3 * q^62 - 14 * q^63 + 3 * q^64 + 8 * q^65 - q^66 + q^67 + 4 * q^68 + 2 * q^70 - 3 * q^71 + 2 * q^72 + 4 * q^73 - 14 * q^74 - 41 * q^75 - 3 * q^76 - q^77 + 2 * q^78 + 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 + 7 * q^83 + 6 * q^85 - 5 * q^86 - 9 * q^87 - 3 * q^88 + 16 * q^89 + 3 * q^90 - 13 * q^91 + 2 * q^92 - 22 * q^93 - 5 * q^95 + q^96 - 14 * q^97 - 6 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more