LMFDB - Embedded newform 6422.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.16609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} - x + 9 $ x^4 - 7*x^2 - x + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.47192$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.83345 q^{3} +1.00000 q^{4} -0.638476 q^{5} -1.83345 q^{6} +1.53214 q^{7} -1.00000 q^{8} +0.361524 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.83345 q^{3} +1.00000 q^{4} -0.638476 q^{5} -1.83345 q^{6} +1.53214 q^{7} -1.00000 q^{8} +0.361524 q^{9} +0.638476 q^{10} -2.06021 q^{11} +1.83345 q^{12} -1.53214 q^{14} -1.17061 q^{15} +1.00000 q^{16} +3.23082 q^{17} -0.361524 q^{18} -1.00000 q^{19} -0.638476 q^{20} +2.80909 q^{21} +2.06021 q^{22} +3.33717 q^{23} -1.83345 q^{24} -4.59235 q^{25} -4.83750 q^{27} +1.53214 q^{28} -6.28101 q^{29} +1.17061 q^{30} +6.78135 q^{31} -1.00000 q^{32} -3.77729 q^{33} -3.23082 q^{34} -0.978231 q^{35} +0.361524 q^{36} -10.1429 q^{37} +1.00000 q^{38} +0.638476 q^{40} +11.5864 q^{41} -2.80909 q^{42} -10.8659 q^{43} -2.06021 q^{44} -0.230824 q^{45} -3.33717 q^{46} -9.41982 q^{47} +1.83345 q^{48} -4.65256 q^{49} +4.59235 q^{50} +5.92354 q^{51} -7.61671 q^{53} +4.83750 q^{54} +1.31540 q^{55} -1.53214 q^{56} -1.83345 q^{57} +6.28101 q^{58} +1.27695 q^{59} -1.17061 q^{60} +6.12043 q^{61} -6.78135 q^{62} +0.553904 q^{63} +1.00000 q^{64} +3.77729 q^{66} -1.88769 q^{67} +3.23082 q^{68} +6.11851 q^{69} +0.978231 q^{70} -6.91610 q^{71} -0.361524 q^{72} +0.943844 q^{73} +10.1429 q^{74} -8.41982 q^{75} -1.00000 q^{76} -3.15653 q^{77} +10.1285 q^{79} -0.638476 q^{80} -9.95387 q^{81} -11.5864 q^{82} +3.63509 q^{83} +2.80909 q^{84} -2.06280 q^{85} +10.8659 q^{86} -11.5159 q^{87} +2.06021 q^{88} +4.61412 q^{89} +0.230824 q^{90} +3.33717 q^{92} +12.4332 q^{93} +9.41982 q^{94} +0.638476 q^{95} -1.83345 q^{96} -14.3460 q^{97} +4.65256 q^{98} -0.744817 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 7 * q^7 - 4 * q^8 + 2 * q^9	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 7 q^{14} + 9 q^{15} + 4 q^{16} - 8 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} - 3 q^{21} + q^{22} + 5 q^{23} - 2 q^{24} + 2 q^{25} + 5 q^{27} - 7 q^{28} - 5 q^{29} - 9 q^{30} - 9 q^{31} - 4 q^{32} + 2 q^{33} + 8 q^{34} + 7 q^{35} + 2 q^{36} - 5 q^{37} + 4 q^{38} + 2 q^{40} + 15 q^{41} + 3 q^{42} - 9 q^{43} - q^{44} + 20 q^{45} - 5 q^{46} - q^{47} + 2 q^{48} + 9 q^{49} - 2 q^{50} - 16 q^{51} - 19 q^{53} - 5 q^{54} - 14 q^{55} + 7 q^{56} - 2 q^{57} + 5 q^{58} + 4 q^{59} + 9 q^{60} + 10 q^{61} + 9 q^{62} + 4 q^{64} - 2 q^{66} + 16 q^{67} - 8 q^{68} - 20 q^{69} - 7 q^{70} + 6 q^{71} - 2 q^{72} - 8 q^{73} + 5 q^{74} + 3 q^{75} - 4 q^{76} - 26 q^{77} - 12 q^{79} - 2 q^{80} - 20 q^{81} - 15 q^{82} + q^{83} - 3 q^{84} + q^{85} + 9 q^{86} + 14 q^{87} + q^{88} + 9 q^{89} - 20 q^{90} + 5 q^{92} + 18 q^{93} + q^{94} + 2 q^{95} - 2 q^{96} + 21 q^{97} - 9 q^{98} - 15 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 7 * q^7 - 4 * q^8 + 2 * q^9 + 2 * q^10 - q^11 + 2 * q^12 + 7 * q^14 + 9 * q^15 + 4 * q^16 - 8 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^20 - 3 * q^21 + q^22 + 5 * q^23 - 2 * q^24 + 2 * q^25 + 5 * q^27 - 7 * q^28 - 5 * q^29 - 9 * q^30 - 9 * q^31 - 4 * q^32 + 2 * q^33 + 8 * q^34 + 7 * q^35 + 2 * q^36 - 5 * q^37 + 4 * q^38 + 2 * q^40 + 15 * q^41 + 3 * q^42 - 9 * q^43 - q^44 + 20 * q^45 - 5 * q^46 - q^47 + 2 * q^48 + 9 * q^49 - 2 * q^50 - 16 * q^51 - 19 * q^53 - 5 * q^54 - 14 * q^55 + 7 * q^56 - 2 * q^57 + 5 * q^58 + 4 * q^59 + 9 * q^60 + 10 * q^61 + 9 * q^62 + 4 * q^64 - 2 * q^66 + 16 * q^67 - 8 * q^68 - 20 * q^69 - 7 * q^70 + 6 * q^71 - 2 * q^72 - 8 * q^73 + 5 * q^74 + 3 * q^75 - 4 * q^76 - 26 * q^77 - 12 * q^79 - 2 * q^80 - 20 * q^81 - 15 * q^82 + q^83 - 3 * q^84 + q^85 + 9 * q^86 + 14 * q^87 + q^88 + 9 * q^89 - 20 * q^90 + 5 * q^92 + 18 * q^93 + q^94 + 2 * q^95 - 2 * q^96 + 21 * q^97 - 9 * q^98 - 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.83345	1.05854	0.529270	−	0.848453i	$-0.322466\pi$
$3$	1.83345	1.05854	0.529270	+	0.848453i	$0.322466\pi$
$4$	1.00000	0.500000
$5$	−0.638476	−0.285535	−0.142768	−	0.989756i	$-0.545600\pi$
$5$	−0.638476	−0.285535	−0.142768	+	0.989756i	$0.545600\pi$
$6$	−1.83345	−0.748501
$7$	1.53214	0.579093	0.289546	−	0.957164i	$-0.406496\pi$
$7$	1.53214	0.579093	0.289546	+	0.957164i	$0.406496\pi$
$8$	−1.00000	−0.353553
$9$	0.361524	0.120508
$10$	0.638476	0.201904
$11$	−2.06021	−0.621178	−0.310589	−	0.950544i	$-0.600526\pi$
$11$	−2.06021	−0.621178	−0.310589	+	0.950544i	$0.600526\pi$
$12$	1.83345	0.529270
$13$	0	0
$13$	0	0
$14$	−1.53214	−0.409480
$15$	−1.17061	−0.302250
$16$	1.00000	0.250000
$17$	3.23082	0.783590	0.391795	−	0.920053i	$-0.371854\pi$
$17$	3.23082	0.783590	0.391795	+	0.920053i	$0.371854\pi$
$18$	−0.361524	−0.0852120
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.638476	−0.142768
$21$	2.80909	0.612993
$22$	2.06021	0.439239
$23$	3.33717	0.695847	0.347924	−	0.937523i	$-0.386887\pi$
$23$	3.33717	0.695847	0.347924	+	0.937523i	$0.386887\pi$
$24$	−1.83345	−0.374251
$25$	−4.59235	−0.918470
$26$	0	0
$27$	−4.83750	−0.930978
$28$	1.53214	0.289546
$29$	−6.28101	−1.16635	−0.583177	−	0.812345i	$-0.698191\pi$
$29$	−6.28101	−1.16635	−0.583177	+	0.812345i	$0.698191\pi$
$30$	1.17061	0.213723
$31$	6.78135	1.21797	0.608983	−	0.793183i	$-0.291578\pi$
$31$	6.78135	1.21797	0.608983	+	0.793183i	$0.291578\pi$
$32$	−1.00000	−0.176777
$33$	−3.77729	−0.657542
$34$	−3.23082	−0.554082
$35$	−0.978231	−0.165351
$36$	0.361524	0.0602540
$37$	−10.1429	−1.66748	−0.833739	−	0.552159i	$-0.813804\pi$
$37$	−10.1429	−1.66748	−0.833739	+	0.552159i	$0.813804\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0.638476	0.100952
$41$	11.5864	1.80949	0.904744	−	0.425956i	$-0.140062\pi$
$41$	11.5864	1.80949	0.904744	+	0.425956i	$0.140062\pi$
$42$	−2.80909	−0.433452
$43$	−10.8659	−1.65704	−0.828519	−	0.559962i	$-0.810816\pi$
$43$	−10.8659	−1.65704	−0.828519	+	0.559962i	$0.810816\pi$
$44$	−2.06021	−0.310589
$45$	−0.230824	−0.0344093
$46$	−3.33717	−0.492038
$47$	−9.41982	−1.37402	−0.687011	−	0.726647i	$-0.741078\pi$
$47$	−9.41982	−1.37402	−0.687011	+	0.726647i	$0.741078\pi$
$48$	1.83345	0.264635
$49$	−4.65256	−0.664652
$50$	4.59235	0.649456
$51$	5.92354	0.829462
$52$	0	0
$53$	−7.61671	−1.04624	−0.523118	−	0.852261i	$-0.675231\pi$
$53$	−7.61671	−1.04624	−0.523118	+	0.852261i	$0.675231\pi$
$54$	4.83750	0.658301
$55$	1.31540	0.177368
$56$	−1.53214	−0.204740
$57$	−1.83345	−0.242846
$58$	6.28101	0.824737
$59$	1.27695	0.166245	0.0831225	−	0.996539i	$-0.473511\pi$
$59$	1.27695	0.166245	0.0831225	+	0.996539i	$0.473511\pi$
$60$	−1.17061	−0.151125
$61$	6.12043	0.783640	0.391820	−	0.920042i	$-0.371846\pi$
$61$	6.12043	0.783640	0.391820	+	0.920042i	$0.371846\pi$
$62$	−6.78135	−0.861232
$63$	0.553904	0.0697853
$64$	1.00000	0.125000
$65$	0	0
$66$	3.77729	0.464952
$67$	−1.88769	−0.230618	−0.115309	−	0.993330i	$-0.536786\pi$
$67$	−1.88769	−0.230618	−0.115309	+	0.993330i	$0.536786\pi$
$68$	3.23082	0.391795
$69$	6.11851	0.736582
$70$	0.978231	0.116921
$71$	−6.91610	−0.820790	−0.410395	−	0.911908i	$-0.634609\pi$
$71$	−6.91610	−0.820790	−0.410395	+	0.911908i	$0.634609\pi$
$72$	−0.361524	−0.0426060
$73$	0.943844	0.110469	0.0552343	−	0.998473i	$-0.482409\pi$
$73$	0.943844	0.110469	0.0552343	+	0.998473i	$0.482409\pi$
$74$	10.1429	1.17908
$75$	−8.41982	−0.972237
$76$	−1.00000	−0.114708
$77$	−3.15653	−0.359719
$78$	0	0
$79$	10.1285	1.13955	0.569775	−	0.821801i	$-0.307030\pi$
$79$	10.1285	1.13955	0.569775	+	0.821801i	$0.307030\pi$
$80$	−0.638476	−0.0713838
$81$	−9.95387	−1.10599
$82$	−11.5864	−1.27950
$83$	3.63509	0.399003	0.199502	−	0.979897i	$-0.436068\pi$
$83$	3.63509	0.399003	0.199502	+	0.979897i	$0.436068\pi$
$84$	2.80909	0.306497
$85$	−2.06280	−0.223742
$86$	10.8659	1.17170
$87$	−11.5159	−1.23463
$88$	2.06021	0.219619
$89$	4.61412	0.489095	0.244548	−	0.969637i	$-0.421361\pi$
$89$	4.61412	0.489095	0.244548	+	0.969637i	$0.421361\pi$
$90$	0.230824	0.0243310
$91$	0	0
$92$	3.33717	0.347924
$93$	12.4332	1.28927
$94$	9.41982	0.971581
$95$	0.638476	0.0655062
$96$	−1.83345	−0.187125
$97$	−14.3460	−1.45661	−0.728306	−	0.685253i	$-0.759692\pi$
$97$	−14.3460	−1.45661	−0.728306	+	0.685253i	$0.759692\pi$
$98$	4.65256	0.469980
$99$	−0.744817	−0.0748569
$100$	−4.59235	−0.459235
$101$	13.5546	1.34873	0.674366	−	0.738398i	$-0.264417\pi$
$101$	13.5546	1.34873	0.674366	+	0.738398i	$0.264417\pi$
$102$	−5.92354	−0.586518
$103$	−9.07238	−0.893929	−0.446964	−	0.894552i	$-0.647495\pi$
$103$	−9.07238	−0.893929	−0.446964	+	0.894552i	$0.647495\pi$
$104$	0	0
$105$	−1.79353	−0.175031
$106$	7.61671	0.739800
$107$	−6.62415	−0.640380	−0.320190	−	0.947353i	$-0.603747\pi$
$107$	−6.62415	−0.640380	−0.320190	+	0.947353i	$0.603747\pi$
$108$	−4.83750	−0.465489
$109$	1.63250	0.156366	0.0781828	−	0.996939i	$-0.475088\pi$
$109$	1.63250	0.156366	0.0781828	+	0.996939i	$0.475088\pi$
$110$	−1.31540	−0.125418
$111$	−18.5964	−1.76509
$112$	1.53214	0.144773
$113$	4.61074	0.433742	0.216871	−	0.976200i	$-0.430415\pi$
$113$	4.61074	0.433742	0.216871	+	0.976200i	$0.430415\pi$
$114$	1.83345	0.171718
$115$	−2.13070	−0.198689
$116$	−6.28101	−0.583177
$117$	0	0
$118$	−1.27695	−0.117553
$119$	4.95006	0.453771
$120$	1.17061	0.106862
$121$	−6.75552	−0.614138
$122$	−6.12043	−0.554117
$123$	21.2430	1.91542
$124$	6.78135	0.608983
$125$	6.12448	0.547790
$126$	−0.553904	−0.0493457
$127$	9.45421	0.838926	0.419463	−	0.907773i	$-0.362218\pi$
$127$	9.45421	0.838926	0.419463	+	0.907773i	$0.362218\pi$
$128$	−1.00000	−0.0883883
$129$	−19.9221	−1.75404
$130$	0	0
$131$	−10.4841	−0.915999	−0.458000	−	0.888952i	$-0.651434\pi$
$131$	−10.4841	−0.915999	−0.458000	+	0.888952i	$0.651434\pi$
$132$	−3.77729	−0.328771
$133$	−1.53214	−0.132853
$134$	1.88769	0.163071
$135$	3.08863	0.265827
$136$	−3.23082	−0.277041
$137$	−12.5620	−1.07325	−0.536623	−	0.843822i	$-0.680300\pi$
$137$	−12.5620	−1.07325	−0.536623	+	0.843822i	$0.680300\pi$
$138$	−6.11851	−0.520842
$139$	19.7610	1.67611	0.838055	−	0.545586i	$-0.183693\pi$
$139$	19.7610	1.67611	0.838055	+	0.545586i	$0.183693\pi$
$140$	−0.978231	−0.0826756
$141$	−17.2707	−1.45446
$142$	6.91610	0.580387
$143$	0	0
$144$	0.361524	0.0301270
$145$	4.01027	0.333035
$146$	−0.943844	−0.0781131
$147$	−8.53022	−0.703561
$148$	−10.1429	−0.833739
$149$	12.6506	1.03638	0.518191	−	0.855265i	$-0.326606\pi$
$149$	12.6506	1.03638	0.518191	+	0.855265i	$0.326606\pi$
$150$	8.41982	0.687476
$151$	−17.8131	−1.44961	−0.724807	−	0.688952i	$-0.758071\pi$
$151$	−17.8131	−1.44961	−0.724807	+	0.688952i	$0.758071\pi$
$152$	1.00000	0.0811107
$153$	1.16802	0.0944289
$154$	3.15653	0.254360
$155$	−4.32973	−0.347772
$156$	0	0
$157$	−14.3981	−1.14909	−0.574545	−	0.818473i	$-0.694821\pi$
$157$	−14.3981	−1.14909	−0.574545	+	0.818473i	$0.694821\pi$
$158$	−10.1285	−0.805783
$159$	−13.9648	−1.10748
$160$	0.638476	0.0504760
$161$	5.11299	0.402960
$162$	9.95387	0.782050
$163$	−20.6506	−1.61748	−0.808742	−	0.588164i	$-0.799851\pi$
$163$	−20.6506	−1.61748	−0.808742	+	0.588164i	$0.799851\pi$
$164$	11.5864	0.904744
$165$	2.41171	0.187751
$166$	−3.63509	−0.282138
$167$	−2.57082	−0.198936	−0.0994682	−	0.995041i	$-0.531714\pi$
$167$	−2.57082	−0.198936	−0.0994682	+	0.995041i	$0.531714\pi$
$168$	−2.80909	−0.216726
$169$	0	0
$170$	2.06280	0.158210
$171$	−0.361524	−0.0276464
$172$	−10.8659	−0.828519
$173$	−17.3737	−1.32090	−0.660449	−	0.750871i	$-0.729634\pi$
$173$	−17.3737	−1.32090	−0.660449	+	0.750871i	$0.729634\pi$
$174$	11.5159	0.873017
$175$	−7.03610	−0.531879
$176$	−2.06021	−0.155294
$177$	2.34122	0.175977
$178$	−4.61412	−0.345843
$179$	4.87121	0.364092	0.182046	−	0.983290i	$-0.441728\pi$
$179$	4.87121	0.364092	0.182046	+	0.983290i	$0.441728\pi$
$180$	−0.230824	−0.0172046
$181$	−4.86333	−0.361488	−0.180744	−	0.983530i	$-0.557851\pi$
$181$	−4.86333	−0.361488	−0.180744	+	0.983530i	$0.557851\pi$
$182$	0	0
$183$	11.2215	0.829515
$184$	−3.33717	−0.246019
$185$	6.47598	0.476123
$186$	−12.4332	−0.911649
$187$	−6.65619	−0.486749
$188$	−9.41982	−0.687011
$189$	−7.41171	−0.539122
$190$	−0.638476	−0.0463199
$191$	−2.39141	−0.173036	−0.0865181	−	0.996250i	$-0.527574\pi$
$191$	−2.39141	−0.173036	−0.0865181	+	0.996250i	$0.527574\pi$
$192$	1.83345	0.132318
$193$	12.0331	0.866165	0.433083	−	0.901354i	$-0.357426\pi$
$193$	12.0331	0.866165	0.433083	+	0.901354i	$0.357426\pi$
$194$	14.3460	1.02998
$195$	0	0
$196$	−4.65256	−0.332326
$197$	19.8246	1.41245	0.706224	−	0.707989i	$-0.250397\pi$
$197$	19.8246	1.41245	0.706224	+	0.707989i	$0.250397\pi$
$198$	0.744817	0.0529318
$199$	18.0047	1.27632	0.638161	−	0.769903i	$-0.279695\pi$
$199$	18.0047	1.27632	0.638161	+	0.769903i	$0.279695\pi$
$200$	4.59235	0.324728
$201$	−3.46097	−0.244118
$202$	−13.5546	−0.953697
$203$	−9.62335	−0.675427
$204$	5.92354	0.414731
$205$	−7.39762	−0.516672
$206$	9.07238	0.632103
$207$	1.20647	0.0838552
$208$	0	0
$209$	2.06021	0.142508
$210$	1.79353	0.123766
$211$	−8.41577	−0.579365	−0.289683	−	0.957123i	$-0.593550\pi$
$211$	−8.41577	−0.579365	−0.289683	+	0.957123i	$0.593550\pi$
$212$	−7.61671	−0.523118
$213$	−12.6803	−0.868840
$214$	6.62415	0.452817
$215$	6.93763	0.473142
$216$	4.83750	0.329150
$217$	10.3899	0.705315
$218$	−1.63250	−0.110567
$219$	1.73049	0.116935
$220$	1.31540	0.0886840
$221$	0	0
$222$	18.5964	1.24811
$223$	0.953197	0.0638308	0.0319154	−	0.999491i	$-0.489839\pi$
$223$	0.953197	0.0638308	0.0319154	+	0.999491i	$0.489839\pi$
$224$	−1.53214	−0.102370
$225$	−1.66024	−0.110683
$226$	−4.61074	−0.306702
$227$	−14.0568	−0.932985	−0.466492	−	0.884525i	$-0.654482\pi$
$227$	−14.0568	−0.932985	−0.466492	+	0.884525i	$0.654482\pi$
$228$	−1.83345	−0.121423
$229$	−27.8213	−1.83848	−0.919240	−	0.393697i	$-0.871196\pi$
$229$	−27.8213	−1.83848	−0.919240	+	0.393697i	$0.871196\pi$
$230$	2.13070	0.140494
$231$	−5.78732	−0.380778
$232$	6.28101	0.412368
$233$	−3.23298	−0.211800	−0.105900	−	0.994377i	$-0.533772\pi$
$233$	−3.23298	−0.211800	−0.105900	+	0.994377i	$0.533772\pi$
$234$	0	0
$235$	6.01433	0.392332
$236$	1.27695	0.0831225
$237$	18.5701	1.20626
$238$	−4.95006	−0.320865
$239$	−21.2721	−1.37598	−0.687988	−	0.725722i	$-0.741506\pi$
$239$	−21.2721	−1.37598	−0.687988	+	0.725722i	$0.741506\pi$
$240$	−1.17061	−0.0755626
$241$	−14.2274	−0.916470	−0.458235	−	0.888831i	$-0.651518\pi$
$241$	−14.2274	−0.916470	−0.458235	+	0.888831i	$0.651518\pi$
$242$	6.75552	0.434261
$243$	−3.73738	−0.239753
$244$	6.12043	0.391820
$245$	2.97055	0.189781
$246$	−21.2430	−1.35440
$247$	0	0
$248$	−6.78135	−0.430616
$249$	6.66475	0.422361
$250$	−6.12448	−0.387346
$251$	0.324994	0.0205134	0.0102567	−	0.999947i	$-0.496735\pi$
$251$	0.324994	0.0205134	0.0102567	+	0.999947i	$0.496735\pi$
$252$	0.553904	0.0348927
$253$	−6.87527	−0.432245
$254$	−9.45421	−0.593210
$255$	−3.78204	−0.236840
$256$	1.00000	0.0625000
$257$	10.8740	0.678304	0.339152	−	0.940732i	$-0.389860\pi$
$257$	10.8740	0.678304	0.339152	+	0.940732i	$0.389860\pi$
$258$	19.9221	1.24029
$259$	−15.5402	−0.965624
$260$	0	0
$261$	−2.27074	−0.140555
$262$	10.4841	0.647709
$263$	15.1042	0.931364	0.465682	−	0.884952i	$-0.345809\pi$
$263$	15.1042	0.931364	0.465682	+	0.884952i	$0.345809\pi$
$264$	3.77729	0.232476
$265$	4.86308	0.298737
$266$	1.53214	0.0939412
$267$	8.45973	0.517727
$268$	−1.88769	−0.115309
$269$	2.44159	0.148866	0.0744332	−	0.997226i	$-0.476285\pi$
$269$	2.44159	0.148866	0.0744332	+	0.997226i	$0.476285\pi$
$270$	−3.08863	−0.187968
$271$	−4.63994	−0.281856	−0.140928	−	0.990020i	$-0.545009\pi$
$271$	−4.63994	−0.281856	−0.140928	+	0.990020i	$0.545009\pi$
$272$	3.23082	0.195898
$273$	0	0
$274$	12.5620	0.758899
$275$	9.46122	0.570533
$276$	6.11851	0.368291
$277$	−5.73860	−0.344799	−0.172400	−	0.985027i	$-0.555152\pi$
$277$	−5.73860	−0.344799	−0.172400	+	0.985027i	$0.555152\pi$
$278$	−19.7610	−1.18519
$279$	2.45162	0.146775
$280$	0.978231	0.0584605
$281$	11.6784	0.696674	0.348337	−	0.937369i	$-0.386746\pi$
$281$	11.6784	0.696674	0.348337	+	0.937369i	$0.386746\pi$
$282$	17.2707	1.02846
$283$	14.1653	0.842041	0.421020	−	0.907051i	$-0.361672\pi$
$283$	14.1653	0.842041	0.421020	+	0.907051i	$0.361672\pi$
$284$	−6.91610	−0.410395
$285$	1.17061	0.0693410
$286$	0	0
$287$	17.7519	1.04786
$288$	−0.361524	−0.0213030
$289$	−6.56177	−0.385987
$290$	−4.01027	−0.235491
$291$	−26.3025	−1.54188
$292$	0.943844	0.0552343
$293$	−0.800973	−0.0467933	−0.0233967	−	0.999726i	$-0.507448\pi$
$293$	−0.800973	−0.0467933	−0.0233967	+	0.999726i	$0.507448\pi$
$294$	8.53022	0.497493
$295$	−0.815303	−0.0474688
$296$	10.1429	0.589542
$297$	9.96629	0.578303
$298$	−12.6506	−0.732832
$299$	0	0
$300$	−8.41982	−0.486119
$301$	−16.6481	−0.959578
$302$	17.8131	1.02503
$303$	24.8516	1.42769
$304$	−1.00000	−0.0573539
$305$	−3.90775	−0.223757
$306$	−1.16802	−0.0667713
$307$	−15.1766	−0.866173	−0.433087	−	0.901352i	$-0.642576\pi$
$307$	−15.1766	−0.866173	−0.433087	+	0.901352i	$0.642576\pi$
$308$	−3.15653	−0.179860
$309$	−16.6337	−0.946260
$310$	4.32973	0.245912
$311$	1.57826	0.0894951	0.0447475	−	0.998998i	$-0.485752\pi$
$311$	1.57826	0.0894951	0.0447475	+	0.998998i	$0.485752\pi$
$312$	0	0
$313$	−34.7684	−1.96522	−0.982612	−	0.185670i	$-0.940554\pi$
$313$	−34.7684	−1.96522	−0.982612	+	0.185670i	$0.940554\pi$
$314$	14.3981	0.812529
$315$	−0.353654	−0.0199262
$316$	10.1285	0.569775
$317$	−3.64683	−0.204827	−0.102413	−	0.994742i	$-0.532656\pi$
$317$	−3.64683	−0.204827	−0.102413	+	0.994742i	$0.532656\pi$
$318$	13.9648	0.783108
$319$	12.9402	0.724513
$320$	−0.638476	−0.0356919
$321$	−12.1450	−0.677869
$322$	−5.11299	−0.284936
$323$	−3.23082	−0.179768
$324$	−9.95387	−0.552993
$325$	0	0
$326$	20.6506	1.14373
$327$	2.99311	0.165519
$328$	−11.5864	−0.639751
$329$	−14.4324	−0.795686
$330$	−2.41171	−0.132760
$331$	−30.5701	−1.68029	−0.840143	−	0.542365i	$-0.817529\pi$
$331$	−30.5701	−1.68029	−0.840143	+	0.542365i	$0.817529\pi$
$332$	3.63509	0.199502
$333$	−3.66689	−0.200944
$334$	2.57082	0.140669
$335$	1.20524	0.0658495
$336$	2.80909	0.153248
$337$	15.6052	0.850069	0.425035	−	0.905177i	$-0.360262\pi$
$337$	15.6052	0.850069	0.425035	+	0.905177i	$0.360262\pi$
$338$	0	0
$339$	8.45353	0.459133
$340$	−2.06280	−0.111871
$341$	−13.9710	−0.756573
$342$	0.361524	0.0195490
$343$	−17.8533	−0.963988
$344$	10.8659	0.585851
$345$	−3.90652	−0.210320
$346$	17.3737	0.934016
$347$	−25.7175	−1.38059	−0.690294	−	0.723529i	$-0.742519\pi$
$347$	−25.7175	−1.38059	−0.690294	+	0.723529i	$0.742519\pi$
$348$	−11.5159	−0.617317
$349$	−7.22271	−0.386623	−0.193311	−	0.981137i	$-0.561923\pi$
$349$	−7.22271	−0.386623	−0.193311	+	0.981137i	$0.561923\pi$
$350$	7.03610	0.376095
$351$	0	0
$352$	2.06021	0.109810
$353$	13.2052	0.702844	0.351422	−	0.936217i	$-0.385698\pi$
$353$	13.2052	0.702844	0.351422	+	0.936217i	$0.385698\pi$
$354$	−2.34122	−0.124435
$355$	4.41577	0.234365
$356$	4.61412	0.244548
$357$	9.07567	0.480335
$358$	−4.87121	−0.257452
$359$	3.93641	0.207756	0.103878	−	0.994590i	$-0.466875\pi$
$359$	3.93641	0.207756	0.103878	+	0.994590i	$0.466875\pi$
$360$	0.230824	0.0121655
$361$	1.00000	0.0526316
$362$	4.86333	0.255611
$363$	−12.3859	−0.650090
$364$	0	0
$365$	−0.602622	−0.0315427
$366$	−11.2215	−0.586556
$367$	18.2356	0.951888	0.475944	−	0.879475i	$-0.342106\pi$
$367$	18.2356	0.951888	0.475944	+	0.879475i	$0.342106\pi$
$368$	3.33717	0.173962
$369$	4.18875	0.218058
$370$	−6.47598	−0.336670
$371$	−11.6698	−0.605867
$372$	12.4332	0.644633
$373$	8.69125	0.450016	0.225008	−	0.974357i	$-0.427759\pi$
$373$	8.69125	0.450016	0.225008	+	0.974357i	$0.427759\pi$
$374$	6.65619	0.344183
$375$	11.2289	0.579858
$376$	9.41982	0.485790
$377$	0	0
$378$	7.41171	0.381217
$379$	20.9887	1.07812	0.539059	−	0.842268i	$-0.318780\pi$
$379$	20.9887	1.07812	0.539059	+	0.842268i	$0.318780\pi$
$380$	0.638476	0.0327531
$381$	17.3338	0.888037
$382$	2.39141	0.122355
$383$	28.3335	1.44778	0.723888	−	0.689917i	$-0.242353\pi$
$383$	28.3335	1.44778	0.723888	+	0.689917i	$0.242353\pi$
$384$	−1.83345	−0.0935626
$385$	2.01537	0.102713
$386$	−12.0331	−0.612471
$387$	−3.92829	−0.199686
$388$	−14.3460	−0.728306
$389$	−32.5185	−1.64875	−0.824376	−	0.566042i	$-0.808474\pi$
$389$	−32.5185	−1.64875	−0.824376	+	0.566042i	$0.808474\pi$
$390$	0	0
$391$	10.7818	0.545259
$392$	4.65256	0.234990
$393$	−19.2220	−0.969623
$394$	−19.8246	−0.998751
$395$	−6.46683	−0.325382
$396$	−0.744817	−0.0374284
$397$	−26.8731	−1.34872	−0.674361	−	0.738401i	$-0.735581\pi$
$397$	−26.8731	−1.34872	−0.674361	+	0.738401i	$0.735581\pi$
$398$	−18.0047	−0.902496
$399$	−2.80909	−0.140630
$400$	−4.59235	−0.229617
$401$	23.1345	1.15528	0.577641	−	0.816291i	$-0.303973\pi$
$401$	23.1345	1.15528	0.577641	+	0.816291i	$0.303973\pi$
$402$	3.46097	0.172618
$403$	0	0
$404$	13.5546	0.674366
$405$	6.35531	0.315798
$406$	9.62335	0.477599
$407$	20.8965	1.03580
$408$	−5.92354	−0.293259
$409$	13.0805	0.646789	0.323395	−	0.946264i	$-0.395176\pi$
$409$	13.0805	0.646789	0.323395	+	0.946264i	$0.395176\pi$
$410$	7.39762	0.365343
$411$	−23.0318	−1.13607
$412$	−9.07238	−0.446964
$413$	1.95646	0.0962712
$414$	−1.20647	−0.0592945
$415$	−2.32092	−0.113930
$416$	0	0
$417$	36.2308	1.77423
$418$	−2.06021	−0.100768
$419$	10.6251	0.519068	0.259534	−	0.965734i	$-0.416431\pi$
$419$	10.6251	0.519068	0.259534	+	0.965734i	$0.416431\pi$
$420$	−1.79353	−0.0875155
$421$	−26.5409	−1.29353	−0.646763	−	0.762691i	$-0.723878\pi$
$421$	−26.5409	−1.29353	−0.646763	+	0.762691i	$0.723878\pi$
$422$	8.41577	0.409673
$423$	−3.40549	−0.165581
$424$	7.61671	0.369900
$425$	−14.8371	−0.719704
$426$	12.6803	0.614363
$427$	9.37732	0.453800
$428$	−6.62415	−0.320190
$429$	0	0
$430$	−6.93763	−0.334562
$431$	17.4000	0.838127	0.419064	−	0.907957i	$-0.362358\pi$
$431$	17.4000	0.838127	0.419064	+	0.907957i	$0.362358\pi$
$432$	−4.83750	−0.232744
$433$	8.93087	0.429190	0.214595	−	0.976703i	$-0.431157\pi$
$433$	8.93087	0.429190	0.214595	+	0.976703i	$0.431157\pi$
$434$	−10.3899	−0.498733
$435$	7.35262	0.352531
$436$	1.63250	0.0781828
$437$	−3.33717	−0.159638
$438$	−1.73049	−0.0826858
$439$	−35.1412	−1.67720	−0.838598	−	0.544751i	$-0.816624\pi$
$439$	−35.1412	−1.67720	−0.838598	+	0.544751i	$0.816624\pi$
$440$	−1.31540	−0.0627091
$441$	−1.68201	−0.0800959
$442$	0	0
$443$	29.1517	1.38504	0.692519	−	0.721400i	$-0.256501\pi$
$443$	29.1517	1.38504	0.692519	+	0.721400i	$0.256501\pi$
$444$	−18.5964	−0.882546
$445$	−2.94600	−0.139654
$446$	−0.953197	−0.0451352
$447$	23.1943	1.09705
$448$	1.53214	0.0723866
$449$	40.8480	1.92773	0.963867	−	0.266383i	$-0.0858286\pi$
$449$	40.8480	1.92773	0.963867	+	0.266383i	$0.0858286\pi$
$450$	1.66024	0.0782647
$451$	−23.8704	−1.12401
$452$	4.61074	0.216871
$453$	−32.6594	−1.53447
$454$	14.0568	0.659720
$455$	0	0
$456$	1.83345	0.0858590
$457$	30.4711	1.42538	0.712689	−	0.701480i	$-0.247477\pi$
$457$	30.4711	1.42538	0.712689	+	0.701480i	$0.247477\pi$
$458$	27.8213	1.30000
$459$	−15.6291	−0.729505
$460$	−2.13070	−0.0993444
$461$	−9.64107	−0.449029	−0.224515	−	0.974471i	$-0.572080\pi$
$461$	−9.64107	−0.449029	−0.224515	+	0.974471i	$0.572080\pi$
$462$	5.78732	0.269250
$463$	8.58208	0.398843	0.199421	−	0.979914i	$-0.436094\pi$
$463$	8.58208	0.398843	0.199421	+	0.979914i	$0.436094\pi$
$464$	−6.28101	−0.291589
$465$	−7.93832	−0.368131
$466$	3.23298	0.149765
$467$	−20.2002	−0.934756	−0.467378	−	0.884058i	$-0.654801\pi$
$467$	−20.2002	−0.934756	−0.467378	+	0.884058i	$0.654801\pi$
$468$	0	0
$469$	−2.89219	−0.133549
$470$	−6.01433	−0.277420
$471$	−26.3981	−1.21636
$472$	−1.27695	−0.0587765
$473$	22.3861	1.02931
$474$	−18.5701	−0.852954
$475$	4.59235	0.210711
$476$	4.95006	0.226886
$477$	−2.75362	−0.126080
$478$	21.2721	0.972963
$479$	−33.6401	−1.53706	−0.768528	−	0.639816i	$-0.779011\pi$
$479$	−33.6401	−1.53706	−0.768528	+	0.639816i	$0.779011\pi$
$480$	1.17061	0.0534308
$481$	0	0
$482$	14.2274	0.648042
$483$	9.37439	0.426549
$484$	−6.75552	−0.307069
$485$	9.15955	0.415914
$486$	3.73738	0.169531
$487$	−13.0160	−0.589811	−0.294906	−	0.955526i	$-0.595288\pi$
$487$	−13.0160	−0.589811	−0.294906	+	0.955526i	$0.595288\pi$
$488$	−6.12043	−0.277059
$489$	−37.8618	−1.71217
$490$	−2.97055	−0.134196
$491$	−17.8447	−0.805320	−0.402660	−	0.915350i	$-0.631914\pi$
$491$	−17.8447	−0.805320	−0.402660	+	0.915350i	$0.631914\pi$
$492$	21.2430	0.957708
$493$	−20.2928	−0.913943
$494$	0	0
$495$	0.475548	0.0213743
$496$	6.78135	0.304491
$497$	−10.5964	−0.475314
$498$	−6.66475	−0.298655
$499$	9.23704	0.413507	0.206753	−	0.978393i	$-0.433710\pi$
$499$	9.23704	0.413507	0.206753	+	0.978393i	$0.433710\pi$
$500$	6.12448	0.273895
$501$	−4.71347	−0.210582
$502$	−0.324994	−0.0145052
$503$	−6.53768	−0.291501	−0.145750	−	0.989321i	$-0.546560\pi$
$503$	−6.53768	−0.291501	−0.145750	+	0.989321i	$0.546560\pi$
$504$	−0.553904	−0.0246728
$505$	−8.65427	−0.385110
$506$	6.87527	0.305643
$507$	0	0
$508$	9.45421	0.419463
$509$	37.0456	1.64202	0.821008	−	0.570917i	$-0.193412\pi$
$509$	37.0456	1.64202	0.821008	+	0.570917i	$0.193412\pi$
$510$	3.78204	0.167472
$511$	1.44610	0.0639715
$512$	−1.00000	−0.0441942
$513$	4.83750	0.213581
$514$	−10.8740	−0.479633
$515$	5.79250	0.255248
$516$	−19.9221	−0.877020
$517$	19.4068	0.853512
$518$	15.5402	0.682799
$519$	−31.8537	−1.39822
$520$	0	0
$521$	3.07049	0.134520	0.0672602	−	0.997735i	$-0.478574\pi$
$521$	3.07049	0.134520	0.0672602	+	0.997735i	$0.478574\pi$
$522$	2.27074	0.0993874
$523$	−27.2062	−1.18964	−0.594822	−	0.803857i	$-0.702778\pi$
$523$	−27.2062	−1.18964	−0.594822	+	0.803857i	$0.702778\pi$
$524$	−10.4841	−0.458000
$525$	−12.9003	−0.563016
$526$	−15.1042	−0.658574
$527$	21.9093	0.954386
$528$	−3.77729	−0.164385
$529$	−11.8633	−0.515797
$530$	−4.86308	−0.211239
$531$	0.461649	0.0200339
$532$	−1.53214	−0.0664265
$533$	0	0
$534$	−8.45973	−0.366089
$535$	4.22936	0.182851
$536$	1.88769	0.0815357
$537$	8.93111	0.385406
$538$	−2.44159	−0.105264
$539$	9.58527	0.412867
$540$	3.08863	0.132913
$541$	−3.24786	−0.139636	−0.0698182	−	0.997560i	$-0.522242\pi$
$541$	−3.24786	−0.139636	−0.0698182	+	0.997560i	$0.522242\pi$
$542$	4.63994	0.199303
$543$	−8.91665	−0.382650
$544$	−3.23082	−0.138520
$545$	−1.04231	−0.0446479
$546$	0	0
$547$	−30.2361	−1.29280	−0.646401	−	0.762998i	$-0.723727\pi$
$547$	−30.2361	−1.29280	−0.646401	+	0.762998i	$0.723727\pi$
$548$	−12.5620	−0.536623
$549$	2.21268	0.0944349
$550$	−9.46122	−0.403428
$551$	6.28101	0.267580
$552$	−6.11851	−0.260421
$553$	15.5183	0.659905
$554$	5.73860	0.243810
$555$	11.8734	0.503996
$556$	19.7610	0.838055
$557$	−1.37832	−0.0584011	−0.0292006	−	0.999574i	$-0.509296\pi$
$557$	−1.37832	−0.0584011	−0.0292006	+	0.999574i	$0.509296\pi$
$558$	−2.45162	−0.103785
$559$	0	0
$560$	−0.978231	−0.0413378
$561$	−12.2038	−0.515243
$562$	−11.6784	−0.492623
$563$	−22.6580	−0.954920	−0.477460	−	0.878654i	$-0.658442\pi$
$563$	−22.6580	−0.954920	−0.477460	+	0.878654i	$0.658442\pi$
$564$	−17.2707	−0.727229
$565$	−2.94384	−0.123848
$566$	−14.1653	−0.595413
$567$	−15.2507	−0.640468
$568$	6.91610	0.290193
$569$	3.53214	0.148075	0.0740374	−	0.997255i	$-0.476412\pi$
$569$	3.53214	0.148075	0.0740374	+	0.997255i	$0.476412\pi$
$570$	−1.17061	−0.0490315
$571$	44.4717	1.86108	0.930541	−	0.366189i	$-0.119338\pi$
$571$	44.4717	1.86108	0.930541	+	0.366189i	$0.119338\pi$
$572$	0	0
$573$	−4.38452	−0.183166
$574$	−17.7519	−0.740950
$575$	−15.3254	−0.639114
$576$	0.361524	0.0150635
$577$	−27.6195	−1.14982	−0.574908	−	0.818218i	$-0.694962\pi$
$577$	−27.6195	−1.14982	−0.574908	+	0.818218i	$0.694962\pi$
$578$	6.56177	0.272934
$579$	22.0621	0.916871
$580$	4.01027	0.166518
$581$	5.56946	0.231060
$582$	26.3025	1.09028
$583$	15.6920	0.649898
$584$	−0.943844	−0.0390565
$585$	0	0
$586$	0.800973	0.0330879
$587$	−25.7605	−1.06325	−0.531625	−	0.846980i	$-0.678418\pi$
$587$	−25.7605	−1.06325	−0.531625	+	0.846980i	$0.678418\pi$
$588$	−8.53022	−0.351780
$589$	−6.78135	−0.279421
$590$	0.815303	0.0335655
$591$	36.3474	1.49513
$592$	−10.1429	−0.416869
$593$	30.5266	1.25358	0.626789	−	0.779189i	$-0.284369\pi$
$593$	30.5266	1.25358	0.626789	+	0.779189i	$0.284369\pi$
$594$	−9.96629	−0.408922
$595$	−3.16049	−0.129568
$596$	12.6506	0.518191
$597$	33.0107	1.35104
$598$	0	0
$599$	4.62268	0.188878	0.0944388	−	0.995531i	$-0.469894\pi$
$599$	4.62268	0.188878	0.0944388	+	0.995531i	$0.469894\pi$
$600$	8.41982	0.343738
$601$	26.1104	1.06506	0.532532	−	0.846410i	$-0.321240\pi$
$601$	26.1104	1.06506	0.532532	+	0.846410i	$0.321240\pi$
$602$	16.6481	0.678524
$603$	−0.682444	−0.0277913
$604$	−17.8131	−0.724807
$605$	4.31324	0.175358
$606$	−24.8516	−1.00953
$607$	−11.5478	−0.468711	−0.234356	−	0.972151i	$-0.575298\pi$
$607$	−11.5478	−0.468711	−0.234356	+	0.972151i	$0.575298\pi$
$608$	1.00000	0.0405554
$609$	−17.6439	−0.714967
$610$	3.90775	0.158220
$611$	0	0
$612$	1.16802	0.0472144
$613$	23.8537	0.963443	0.481722	−	0.876324i	$-0.340012\pi$
$613$	23.8537	0.963443	0.481722	+	0.876324i	$0.340012\pi$
$614$	15.1766	0.612477
$615$	−13.5631	−0.546919
$616$	3.15653	0.127180
$617$	15.3286	0.617106	0.308553	−	0.951207i	$-0.400155\pi$
$617$	15.3286	0.617106	0.308553	+	0.951207i	$0.400155\pi$
$618$	16.6337	0.669107
$619$	−3.41079	−0.137091	−0.0685456	−	0.997648i	$-0.521836\pi$
$619$	−3.41079	−0.137091	−0.0685456	+	0.997648i	$0.521836\pi$
$620$	−4.32973	−0.173886
$621$	−16.1435	−0.647818
$622$	−1.57826	−0.0632826
$623$	7.06945	0.283232
$624$	0	0
$625$	19.0514	0.762056
$626$	34.7684	1.38962
$627$	3.77729	0.150850
$628$	−14.3981	−0.574545
$629$	−32.7698	−1.30662
$630$	0.353654	0.0140899
$631$	16.3517	0.650952	0.325476	−	0.945550i	$-0.394475\pi$
$631$	16.3517	0.650952	0.325476	+	0.945550i	$0.394475\pi$
$632$	−10.1285	−0.402892
$633$	−15.4299	−0.613282
$634$	3.64683	0.144834
$635$	−6.03629	−0.239543
$636$	−13.9648	−0.553741
$637$	0	0
$638$	−12.9402	−0.512308
$639$	−2.50034	−0.0989118
$640$	0.638476	0.0252380
$641$	40.8884	1.61499	0.807497	−	0.589872i	$-0.200822\pi$
$641$	40.8884	1.61499	0.807497	+	0.589872i	$0.200822\pi$
$642$	12.1450	0.479325
$643$	−17.5228	−0.691031	−0.345516	−	0.938413i	$-0.612296\pi$
$643$	−17.5228	−0.691031	−0.345516	+	0.938413i	$0.612296\pi$
$644$	5.11299	0.201480
$645$	12.7198	0.500840
$646$	3.23082	0.127115
$647$	0.630909	0.0248036	0.0124018	−	0.999923i	$-0.496052\pi$
$647$	0.630909	0.0248036	0.0124018	+	0.999923i	$0.496052\pi$
$648$	9.95387	0.391025
$649$	−2.63079	−0.103268
$650$	0	0
$651$	19.0494	0.746605
$652$	−20.6506	−0.808742
$653$	44.0286	1.72297	0.861487	−	0.507780i	$-0.169534\pi$
$653$	44.0286	1.72297	0.861487	+	0.507780i	$0.169534\pi$
$654$	−2.99311	−0.117040
$655$	6.69384	0.261550
$656$	11.5864	0.452372
$657$	0.341222	0.0133123
$658$	14.4324	0.562635
$659$	−14.8485	−0.578414	−0.289207	−	0.957267i	$-0.593392\pi$
$659$	−14.8485	−0.578414	−0.289207	+	0.957267i	$0.593392\pi$
$660$	2.41171	0.0938756
$661$	−28.7312	−1.11751	−0.558756	−	0.829332i	$-0.688721\pi$
$661$	−28.7312	−1.11751	−0.558756	+	0.829332i	$0.688721\pi$
$662$	30.5701	1.18814
$663$	0	0
$664$	−3.63509	−0.141069
$665$	0.978231	0.0379342
$666$	3.66689	0.142089
$667$	−20.9608	−0.811604
$668$	−2.57082	−0.0994682
$669$	1.74763	0.0675675
$670$	−1.20524	−0.0465626
$671$	−12.6094	−0.486780
$672$	−2.80909	−0.108363
$673$	9.42794	0.363420	0.181710	−	0.983352i	$-0.441837\pi$
$673$	9.42794	0.363420	0.181710	+	0.983352i	$0.441837\pi$
$674$	−15.6052	−0.601090
$675$	22.2155	0.855075
$676$	0	0
$677$	−37.0785	−1.42504	−0.712521	−	0.701651i	$-0.752447\pi$
$677$	−37.0785	−1.42504	−0.712521	+	0.701651i	$0.752447\pi$
$678$	−8.45353	−0.324656
$679$	−21.9799	−0.843513
$680$	2.06280	0.0791049
$681$	−25.7724	−0.987602
$682$	13.9710	0.534978
$683$	28.3861	1.08616	0.543082	−	0.839680i	$-0.317257\pi$
$683$	28.3861	1.08616	0.543082	+	0.839680i	$0.317257\pi$
$684$	−0.361524	−0.0138232
$685$	8.02055	0.306449
$686$	17.8533	0.681642
$687$	−51.0088	−1.94611
$688$	−10.8659	−0.414259
$689$	0	0
$690$	3.90652	0.148719
$691$	−33.4102	−1.27098	−0.635492	−	0.772107i	$-0.719203\pi$
$691$	−33.4102	−1.27098	−0.635492	+	0.772107i	$0.719203\pi$
$692$	−17.3737	−0.660449
$693$	−1.14116	−0.0433491
$694$	25.7175	0.976223
$695$	−12.6170	−0.478588
$696$	11.5159	0.436509
$697$	37.4335	1.41790
$698$	7.22271	0.273384
$699$	−5.92750	−0.224199
$700$	−7.03610	−0.265940
$701$	22.1734	0.837479	0.418739	−	0.908106i	$-0.362472\pi$
$701$	22.1734	0.837479	0.418739	+	0.908106i	$0.362472\pi$
$702$	0	0
$703$	10.1429	0.382546
$704$	−2.06021	−0.0776472
$705$	11.0269	0.415299
$706$	−13.2052	−0.496986
$707$	20.7674	0.781040
$708$	2.34122	0.0879885
$709$	−20.4429	−0.767751	−0.383875	−	0.923385i	$-0.625411\pi$
$709$	−20.4429	−0.767751	−0.383875	+	0.923385i	$0.625411\pi$
$710$	−4.41577	−0.165721
$711$	3.66171	0.137325
$712$	−4.61412	−0.172921
$713$	22.6305	0.847518
$714$	−9.07567	−0.339648
$715$	0	0
$716$	4.87121	0.182046
$717$	−39.0012	−1.45653
$718$	−3.93641	−0.146905
$719$	29.0656	1.08396	0.541982	−	0.840390i	$-0.317674\pi$
$719$	29.0656	1.08396	0.541982	+	0.840390i	$0.317674\pi$
$720$	−0.230824	−0.00860232
$721$	−13.9001	−0.517668
$722$	−1.00000	−0.0372161
$723$	−26.0852	−0.970121
$724$	−4.86333	−0.180744
$725$	28.8446	1.07126
$726$	12.3859	0.459683
$727$	21.1477	0.784325	0.392163	−	0.919896i	$-0.371727\pi$
$727$	21.1477	0.784325	0.392163	+	0.919896i	$0.371727\pi$
$728$	0	0
$729$	23.0093	0.852198
$730$	0.602622	0.0223040
$731$	−35.1059	−1.29844
$732$	11.2215	0.414758
$733$	30.6609	1.13249	0.566243	−	0.824238i	$-0.308396\pi$
$733$	30.6609	1.13249	0.566243	+	0.824238i	$0.308396\pi$
$734$	−18.2356	−0.673087
$735$	5.44634	0.200891
$736$	−3.33717	−0.123010
$737$	3.88904	0.143255
$738$	−4.18875	−0.154190
$739$	29.5469	1.08690	0.543450	−	0.839442i	$-0.317118\pi$
$739$	29.5469	1.08690	0.543450	+	0.839442i	$0.317118\pi$
$740$	6.47598	0.238062
$741$	0	0
$742$	11.6698	0.428413
$743$	27.2573	0.999974	0.499987	−	0.866033i	$-0.333338\pi$
$743$	27.2573	0.999974	0.499987	+	0.866033i	$0.333338\pi$
$744$	−12.4332	−0.455824
$745$	−8.07713	−0.295923
$746$	−8.69125	−0.318209
$747$	1.31417	0.0480831
$748$	−6.65619	−0.243374
$749$	−10.1491	−0.370840
$750$	−11.2289	−0.410022
$751$	−38.8478	−1.41757	−0.708787	−	0.705422i	$-0.750757\pi$
$751$	−38.8478	−1.41757	−0.708787	+	0.705422i	$0.750757\pi$
$752$	−9.41982	−0.343506
$753$	0.595859	0.0217143
$754$	0	0
$755$	11.3733	0.413915
$756$	−7.41171	−0.269561
$757$	−6.25757	−0.227435	−0.113718	−	0.993513i	$-0.536276\pi$
$757$	−6.25757	−0.227435	−0.113718	+	0.993513i	$0.536276\pi$
$758$	−20.9887	−0.762345
$759$	−12.6054	−0.457548
$760$	−0.638476	−0.0231600
$761$	−34.8123	−1.26195	−0.630973	−	0.775805i	$-0.717344\pi$
$761$	−34.8123	−1.26195	−0.630973	+	0.775805i	$0.717344\pi$
$762$	−17.3338	−0.627937
$763$	2.50122	0.0905501
$764$	−2.39141	−0.0865181
$765$	−0.745753	−0.0269628
$766$	−28.3335	−1.02373
$767$	0	0
$768$	1.83345	0.0661588
$769$	−26.7460	−0.964487	−0.482243	−	0.876037i	$-0.660178\pi$
$769$	−26.7460	−0.964487	−0.482243	+	0.876037i	$0.660178\pi$
$770$	−2.01537	−0.0726287
$771$	19.9370	0.718012
$772$	12.0331	0.433083
$773$	−47.6487	−1.71381	−0.856903	−	0.515478i	$-0.827614\pi$
$773$	−47.6487	−1.71381	−0.856903	+	0.515478i	$0.827614\pi$
$774$	3.92829	0.141200
$775$	−31.1423	−1.11866
$776$	14.3460	0.514990
$777$	−28.4922	−1.02215
$778$	32.5185	1.16584
$779$	−11.5864	−0.415125
$780$	0	0
$781$	14.2486	0.509857
$782$	−10.7818	−0.385556
$783$	30.3844	1.08585
$784$	−4.65256	−0.166163
$785$	9.19281	0.328105
$786$	19.2220	0.685627
$787$	12.0282	0.428758	0.214379	−	0.976751i	$-0.431227\pi$
$787$	12.0282	0.428758	0.214379	+	0.976751i	$0.431227\pi$
$788$	19.8246	0.706224
$789$	27.6927	0.985886
$790$	6.46683	0.230079
$791$	7.06427	0.251177
$792$	0.744817	0.0264659
$793$	0	0
$794$	26.8731	0.953691
$795$	8.91620	0.316225
$796$	18.0047	0.638161
$797$	44.0128	1.55902	0.779508	−	0.626393i	$-0.215469\pi$
$797$	44.0128	1.55902	0.779508	+	0.626393i	$0.215469\pi$
$798$	2.80909	0.0994406
$799$	−30.4338	−1.07667
$800$	4.59235	0.162364
$801$	1.66811	0.0589399
$802$	−23.1345	−0.816908
$803$	−1.94452	−0.0686206
$804$	−3.46097	−0.122059
$805$	−3.26452	−0.115059
$806$	0	0
$807$	4.47653	0.157581
$808$	−13.5546	−0.476848
$809$	−4.01906	−0.141303	−0.0706514	−	0.997501i	$-0.522508\pi$
$809$	−4.01906	−0.141303	−0.0706514	+	0.997501i	$0.522508\pi$
$810$	−6.35531	−0.223303
$811$	18.9438	0.665208	0.332604	−	0.943067i	$-0.392073\pi$
$811$	18.9438	0.665208	0.332604	+	0.943067i	$0.392073\pi$
$812$	−9.62335	−0.337714
$813$	−8.50708	−0.298356
$814$	−20.8965	−0.732421
$815$	13.1849	0.461848
$816$	5.92354	0.207365
$817$	10.8659	0.380150
$818$	−13.0805	−0.457349
$819$	0	0
$820$	−7.39762	−0.258336
$821$	−33.1598	−1.15728	−0.578642	−	0.815582i	$-0.696417\pi$
$821$	−33.1598	−1.15728	−0.578642	+	0.815582i	$0.696417\pi$
$822$	23.0318	0.803326
$823$	−4.80516	−0.167497	−0.0837487	−	0.996487i	$-0.526689\pi$
$823$	−4.80516	−0.167497	−0.0837487	+	0.996487i	$0.526689\pi$
$824$	9.07238	0.316051
$825$	17.3466	0.603932
$826$	−1.95646	−0.0680741
$827$	9.18605	0.319430	0.159715	−	0.987163i	$-0.448942\pi$
$827$	9.18605	0.319430	0.159715	+	0.987163i	$0.448942\pi$
$828$	1.20647	0.0419276
$829$	18.6677	0.648355	0.324178	−	0.945996i	$-0.394912\pi$
$829$	18.6677	0.648355	0.324178	+	0.945996i	$0.394912\pi$
$830$	2.32092	0.0805603
$831$	−10.5214	−0.364984
$832$	0	0
$833$	−15.0316	−0.520814
$834$	−36.2308	−1.25457
$835$	1.64141	0.0568033
$836$	2.06021	0.0712540
$837$	−32.8048	−1.13390
$838$	−10.6251	−0.367037
$839$	−13.5942	−0.469326	−0.234663	−	0.972077i	$-0.575399\pi$
$839$	−13.5942	−0.469326	−0.234663	+	0.972077i	$0.575399\pi$
$840$	1.79353	0.0618828
$841$	10.4511	0.360382
$842$	26.5409	0.914661
$843$	21.4117	0.737458
$844$	−8.41577	−0.289683
$845$	0	0
$846$	3.40549	0.117083
$847$	−10.3504	−0.355643
$848$	−7.61671	−0.261559
$849$	25.9713	0.891334
$850$	14.8371	0.508907
$851$	−33.8484	−1.16031
$852$	−12.6803	−0.434420
$853$	7.93900	0.271826	0.135913	−	0.990721i	$-0.456603\pi$
$853$	7.93900	0.271826	0.135913	+	0.990721i	$0.456603\pi$
$854$	−9.37732	−0.320885
$855$	0.230824	0.00789403
$856$	6.62415	0.226409
$857$	−42.2968	−1.44483	−0.722415	−	0.691459i	$-0.756968\pi$
$857$	−42.2968	−1.44483	−0.722415	+	0.691459i	$0.756968\pi$
$858$	0	0
$859$	27.5072	0.938534	0.469267	−	0.883056i	$-0.344518\pi$
$859$	27.5072	0.938534	0.469267	+	0.883056i	$0.344518\pi$
$860$	6.93763	0.236571
$861$	32.5471	1.10920
$862$	−17.4000	−0.592645
$863$	24.3670	0.829464	0.414732	−	0.909944i	$-0.363875\pi$
$863$	24.3670	0.829464	0.414732	+	0.909944i	$0.363875\pi$
$864$	4.83750	0.164575
$865$	11.0927	0.377163
$866$	−8.93087	−0.303483
$867$	−12.0307	−0.408583
$868$	10.3899	0.352658
$869$	−20.8670	−0.707863
$870$	−7.35262	−0.249277
$871$	0	0
$872$	−1.63250	−0.0552836
$873$	−5.18641	−0.175533
$874$	3.33717	0.112881
$875$	9.38354	0.317221
$876$	1.73049	0.0584677
$877$	25.8815	0.873955	0.436978	−	0.899472i	$-0.356049\pi$
$877$	25.8815	0.873955	0.436978	+	0.899472i	$0.356049\pi$
$878$	35.1412	1.18596
$879$	−1.46854	−0.0495326
$880$	1.31540	0.0443420
$881$	18.5142	0.623760	0.311880	−	0.950122i	$-0.399041\pi$
$881$	18.5142	0.623760	0.311880	+	0.950122i	$0.399041\pi$
$882$	1.68201	0.0566363
$883$	28.6627	0.964577	0.482288	−	0.876013i	$-0.339806\pi$
$883$	28.6627	0.964577	0.482288	+	0.876013i	$0.339806\pi$
$884$	0	0
$885$	−1.49481	−0.0502476
$886$	−29.1517	−0.979369
$887$	−26.6475	−0.894736	−0.447368	−	0.894350i	$-0.647639\pi$
$887$	−26.6475	−0.894736	−0.447368	+	0.894350i	$0.647639\pi$
$888$	18.5964	0.624055
$889$	14.4851	0.485816
$890$	2.94600	0.0987502
$891$	20.5071	0.687014
$892$	0.953197	0.0319154
$893$	9.41982	0.315222
$894$	−23.1943	−0.775733
$895$	−3.11015	−0.103961
$896$	−1.53214	−0.0511850
$897$	0	0
$898$	−40.8480	−1.36311
$899$	−42.5937	−1.42058
$900$	−1.66024	−0.0553415
$901$	−24.6082	−0.819819
$902$	23.8704	0.794798
$903$	−30.5233	−1.01575
$904$	−4.61074	−0.153351
$905$	3.10512	0.103218
$906$	32.6594	1.08504
$907$	−45.7228	−1.51820	−0.759100	−	0.650974i	$-0.774361\pi$
$907$	−45.7228	−1.51820	−0.759100	+	0.650974i	$0.774361\pi$
$908$	−14.0568	−0.466492
$909$	4.90031	0.162533
$910$	0	0
$911$	−33.7481	−1.11812	−0.559062	−	0.829126i	$-0.688839\pi$
$911$	−33.7481	−1.11812	−0.559062	+	0.829126i	$0.688839\pi$
$912$	−1.83345	−0.0607115
$913$	−7.48907	−0.247852
$914$	−30.4711	−1.00789
$915$	−7.16464	−0.236856
$916$	−27.8213	−0.919240
$917$	−16.0630	−0.530449
$918$	15.6291	0.515838
$919$	−11.6380	−0.383903	−0.191952	−	0.981404i	$-0.561482\pi$
$919$	−11.6380	−0.383903	−0.191952	+	0.981404i	$0.561482\pi$
$920$	2.13070	0.0702471
$921$	−27.8254	−0.916880
$922$	9.64107	0.317512
$923$	0	0
$924$	−5.78732	−0.190389
$925$	46.5796	1.53153
$926$	−8.58208	−0.282024
$927$	−3.27989	−0.107726
$928$	6.28101	0.206184
$929$	−30.4262	−0.998252	−0.499126	−	0.866529i	$-0.666346\pi$
$929$	−30.4262	−0.998252	−0.499126	+	0.866529i	$0.666346\pi$
$930$	7.93832	0.260308
$931$	4.65256	0.152482
$932$	−3.23298	−0.105900
$933$	2.89366	0.0947342
$934$	20.2002	0.660972
$935$	4.24982	0.138984
$936$	0	0
$937$	−55.0303	−1.79776	−0.898881	−	0.438193i	$-0.855619\pi$
$937$	−55.0303	−1.79776	−0.898881	+	0.438193i	$0.855619\pi$
$938$	2.89219	0.0944334
$939$	−63.7459	−2.08027
$940$	6.01433	0.196166
$941$	−1.75969	−0.0573644	−0.0286822	−	0.999589i	$-0.509131\pi$
$941$	−1.75969	−0.0573644	−0.0286822	+	0.999589i	$0.509131\pi$
$942$	26.3981	0.860095
$943$	38.6657	1.25913
$944$	1.27695	0.0415612
$945$	4.73220	0.153938
$946$	−22.3861	−0.727835
$947$	−51.0303	−1.65826	−0.829131	−	0.559054i	$-0.811164\pi$
$947$	−51.0303	−1.65826	−0.829131	+	0.559054i	$0.811164\pi$
$948$	18.5701	0.603130
$949$	0	0
$950$	−4.59235	−0.148995
$951$	−6.68627	−0.216817
$952$	−4.95006	−0.160432
$953$	−53.1095	−1.72038	−0.860192	−	0.509970i	$-0.829657\pi$
$953$	−53.1095	−1.72038	−0.860192	+	0.509970i	$0.829657\pi$
$954$	2.75362	0.0891518
$955$	1.52686	0.0494079
$956$	−21.2721	−0.687988
$957$	23.7252	0.766926
$958$	33.6401	1.08686
$959$	−19.2467	−0.621509
$960$	−1.17061	−0.0377813
$961$	14.9867	0.483441
$962$	0	0
$963$	−2.39479	−0.0771710
$964$	−14.2274	−0.458235
$965$	−7.68288	−0.247321
$966$	−9.37439	−0.301616
$967$	46.6481	1.50010	0.750050	−	0.661381i	$-0.230029\pi$
$967$	46.6481	1.50010	0.750050	+	0.661381i	$0.230029\pi$
$968$	6.75552	0.217131
$969$	−5.92354	−0.190292
$970$	−9.15955	−0.294095
$971$	−15.6421	−0.501978	−0.250989	−	0.967990i	$-0.580756\pi$
$971$	−15.6421	−0.501978	−0.250989	+	0.967990i	$0.580756\pi$
$972$	−3.73738	−0.119876
$973$	30.2766	0.970623
$974$	13.0160	0.417060
$975$	0	0
$976$	6.12043	0.195910
$977$	33.4619	1.07054	0.535270	−	0.844681i	$-0.320210\pi$
$977$	33.4619	1.07054	0.535270	+	0.844681i	$0.320210\pi$
$978$	37.8618	1.21069
$979$	−9.50607	−0.303815
$980$	2.97055	0.0948907
$981$	0.590190	0.0188433
$982$	17.8447	0.569447
$983$	−7.22654	−0.230491	−0.115245	−	0.993337i	$-0.536765\pi$
$983$	−7.22654	−0.230491	−0.115245	+	0.993337i	$0.536765\pi$
$984$	−21.2430	−0.677202
$985$	−12.6576	−0.403303
$986$	20.2928	0.646256
$987$	−26.4611	−0.842266
$988$	0	0
$989$	−36.2614	−1.15304
$990$	−0.475548	−0.0151139
$991$	2.56202	0.0813852	0.0406926	−	0.999172i	$-0.487044\pi$
$991$	2.56202	0.0813852	0.0406926	+	0.999172i	$0.487044\pi$
$992$	−6.78135	−0.215308
$993$	−56.0487	−1.77865
$994$	10.5964	0.336098
$995$	−11.4956	−0.364435
$996$	6.66475	0.211181
$997$	15.5598	0.492783	0.246391	−	0.969170i	$-0.420755\pi$
$997$	15.5598	0.492783	0.246391	+	0.969170i	$0.420755\pi$
$998$	−9.23704	−0.292393
$999$	49.0662	1.55238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.ba.1.3		4
13.12	even	2		494.2.a.h.1.3	✓	4
39.38	odd	2		4446.2.a.bl.1.2		4
52.51	odd	2		3952.2.a.u.1.2		4
247.246	odd	2		9386.2.a.bf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.a.h.1.3	✓	4	13.12	even	2
3952.2.a.u.1.2		4	52.51	odd	2
4446.2.a.bl.1.2		4	39.38	odd	2
6422.2.a.ba.1.3		4	1.1	even	1	trivial
9386.2.a.bf.1.2		4	247.246	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.83345 q^{3} +1.00000 q^{4} -0.638476 q^{5} -1.83345 q^{6} +1.53214 q^{7} -1.00000 q^{8} +0.361524 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.83345 q^{3} +1.00000 q^{4} -0.638476 q^{5} -1.83345 q^{6} +1.53214 q^{7} -1.00000 q^{8} +0.361524 q^{9} +0.638476 q^{10} -2.06021 q^{11} +1.83345 q^{12} -1.53214 q^{14} -1.17061 q^{15} +1.00000 q^{16} +3.23082 q^{17} -0.361524 q^{18} -1.00000 q^{19} -0.638476 q^{20} +2.80909 q^{21} +2.06021 q^{22} +3.33717 q^{23} -1.83345 q^{24} -4.59235 q^{25} -4.83750 q^{27} +1.53214 q^{28} -6.28101 q^{29} +1.17061 q^{30} +6.78135 q^{31} -1.00000 q^{32} -3.77729 q^{33} -3.23082 q^{34} -0.978231 q^{35} +0.361524 q^{36} -10.1429 q^{37} +1.00000 q^{38} +0.638476 q^{40} +11.5864 q^{41} -2.80909 q^{42} -10.8659 q^{43} -2.06021 q^{44} -0.230824 q^{45} -3.33717 q^{46} -9.41982 q^{47} +1.83345 q^{48} -4.65256 q^{49} +4.59235 q^{50} +5.92354 q^{51} -7.61671 q^{53} +4.83750 q^{54} +1.31540 q^{55} -1.53214 q^{56} -1.83345 q^{57} +6.28101 q^{58} +1.27695 q^{59} -1.17061 q^{60} +6.12043 q^{61} -6.78135 q^{62} +0.553904 q^{63} +1.00000 q^{64} +3.77729 q^{66} -1.88769 q^{67} +3.23082 q^{68} +6.11851 q^{69} +0.978231 q^{70} -6.91610 q^{71} -0.361524 q^{72} +0.943844 q^{73} +10.1429 q^{74} -8.41982 q^{75} -1.00000 q^{76} -3.15653 q^{77} +10.1285 q^{79} -0.638476 q^{80} -9.95387 q^{81} -11.5864 q^{82} +3.63509 q^{83} +2.80909 q^{84} -2.06280 q^{85} +10.8659 q^{86} -11.5159 q^{87} +2.06021 q^{88} +4.61412 q^{89} +0.230824 q^{90} +3.33717 q^{92} +12.4332 q^{93} +9.41982 q^{94} +0.638476 q^{95} -1.83345 q^{96} -14.3460 q^{97} +4.65256 q^{98} -0.744817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 7 * q^7 - 4 * q^8 + 2 * q^9	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 7 q^{14} + 9 q^{15} + 4 q^{16} - 8 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} - 3 q^{21} + q^{22} + 5 q^{23} - 2 q^{24} + 2 q^{25} + 5 q^{27} - 7 q^{28} - 5 q^{29} - 9 q^{30} - 9 q^{31} - 4 q^{32} + 2 q^{33} + 8 q^{34} + 7 q^{35} + 2 q^{36} - 5 q^{37} + 4 q^{38} + 2 q^{40} + 15 q^{41} + 3 q^{42} - 9 q^{43} - q^{44} + 20 q^{45} - 5 q^{46} - q^{47} + 2 q^{48} + 9 q^{49} - 2 q^{50} - 16 q^{51} - 19 q^{53} - 5 q^{54} - 14 q^{55} + 7 q^{56} - 2 q^{57} + 5 q^{58} + 4 q^{59} + 9 q^{60} + 10 q^{61} + 9 q^{62} + 4 q^{64} - 2 q^{66} + 16 q^{67} - 8 q^{68} - 20 q^{69} - 7 q^{70} + 6 q^{71} - 2 q^{72} - 8 q^{73} + 5 q^{74} + 3 q^{75} - 4 q^{76} - 26 q^{77} - 12 q^{79} - 2 q^{80} - 20 q^{81} - 15 q^{82} + q^{83} - 3 q^{84} + q^{85} + 9 q^{86} + 14 q^{87} + q^{88} + 9 q^{89} - 20 q^{90} + 5 q^{92} + 18 q^{93} + q^{94} + 2 q^{95} - 2 q^{96} + 21 q^{97} - 9 q^{98} - 15 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 7 * q^7 - 4 * q^8 + 2 * q^9 + 2 * q^10 - q^11 + 2 * q^12 + 7 * q^14 + 9 * q^15 + 4 * q^16 - 8 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^20 - 3 * q^21 + q^22 + 5 * q^23 - 2 * q^24 + 2 * q^25 + 5 * q^27 - 7 * q^28 - 5 * q^29 - 9 * q^30 - 9 * q^31 - 4 * q^32 + 2 * q^33 + 8 * q^34 + 7 * q^35 + 2 * q^36 - 5 * q^37 + 4 * q^38 + 2 * q^40 + 15 * q^41 + 3 * q^42 - 9 * q^43 - q^44 + 20 * q^45 - 5 * q^46 - q^47 + 2 * q^48 + 9 * q^49 - 2 * q^50 - 16 * q^51 - 19 * q^53 - 5 * q^54 - 14 * q^55 + 7 * q^56 - 2 * q^57 + 5 * q^58 + 4 * q^59 + 9 * q^60 + 10 * q^61 + 9 * q^62 + 4 * q^64 - 2 * q^66 + 16 * q^67 - 8 * q^68 - 20 * q^69 - 7 * q^70 + 6 * q^71 - 2 * q^72 - 8 * q^73 + 5 * q^74 + 3 * q^75 - 4 * q^76 - 26 * q^77 - 12 * q^79 - 2 * q^80 - 20 * q^81 - 15 * q^82 + q^83 - 3 * q^84 + q^85 + 9 * q^86 + 14 * q^87 + q^88 + 9 * q^89 - 20 * q^90 + 5 * q^92 + 18 * q^93 + q^94 + 2 * q^95 - 2 * q^96 + 21 * q^97 - 9 * q^98 - 15 * q^99

Introduction

L-functions

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