LMFDB - Embedded newform 6422.2.a.bb.1.4

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:	4.4.10273.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + x + 2 $ x^4 - 2x^3 - 5x^2 + x + 2
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			1.4
Root			$-1.38266$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.38266 q^{3} +1.00000 q^{4} +1.23060 q^{5} +1.38266 q^{6} -0.936179 q^{7} +1.00000 q^{8} -1.08824 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.38266 q^{3} +1.00000 q^{4} +1.23060 q^{5} +1.38266 q^{6} -0.936179 q^{7} +1.00000 q^{8} -1.08824 q^{9} +1.23060 q^{10} -2.67708 q^{11} +1.38266 q^{12} -0.936179 q^{14} +1.70151 q^{15} +1.00000 q^{16} -6.67301 q^{17} -1.08824 q^{18} +1.00000 q^{19} +1.23060 q^{20} -1.29442 q^{21} -2.67708 q^{22} -0.617337 q^{23} +1.38266 q^{24} -3.48562 q^{25} -5.65266 q^{27} -0.936179 q^{28} +5.86829 q^{29} +1.70151 q^{30} -5.61326 q^{31} +1.00000 q^{32} -3.70151 q^{33} -6.67301 q^{34} -1.15206 q^{35} -1.08824 q^{36} -4.67301 q^{37} +1.00000 q^{38} +1.23060 q^{40} -4.84386 q^{41} -1.29442 q^{42} +1.21181 q^{43} -2.67708 q^{44} -1.33919 q^{45} -0.617337 q^{46} -10.5642 q^{47} +1.38266 q^{48} -6.12357 q^{49} -3.48562 q^{50} -9.22653 q^{51} -2.29442 q^{53} -5.65266 q^{54} -3.29442 q^{55} -0.936179 q^{56} +1.38266 q^{57} +5.86829 q^{58} -11.5701 q^{59} +1.70151 q^{60} +0.485623 q^{61} -5.61326 q^{62} +1.01879 q^{63} +1.00000 q^{64} -3.70151 q^{66} +15.2184 q^{67} -6.67301 q^{68} -0.853569 q^{69} -1.15206 q^{70} +12.3107 q^{71} -1.08824 q^{72} +12.3460 q^{73} -4.67301 q^{74} -4.81944 q^{75} +1.00000 q^{76} +2.50623 q^{77} -2.33262 q^{79} +1.23060 q^{80} -4.55100 q^{81} -4.84386 q^{82} +5.74090 q^{83} -1.29442 q^{84} -8.21181 q^{85} +1.21181 q^{86} +8.11386 q^{87} -2.67708 q^{88} +18.0663 q^{89} -1.33919 q^{90} -0.617337 q^{92} -7.76125 q^{93} -10.5642 q^{94} +1.23060 q^{95} +1.38266 q^{96} +3.61708 q^{97} -6.12357 q^{98} +2.91332 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9	$ 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9} + q^{10} - 2 q^{11} - 2 q^{12} - q^{14} - 11 q^{15} + 4 q^{16} + q^{17} + 2 q^{18} + 4 q^{19} + q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} - 2 q^{24} + 3 q^{25} - 23 q^{27} - q^{28} - q^{29} - 11 q^{30} - 11 q^{31} + 4 q^{32} + 3 q^{33} + q^{34} - q^{35} + 2 q^{36} + 9 q^{37} + 4 q^{38} + q^{40} - 4 q^{41} - 4 q^{42} - 15 q^{43} - 2 q^{44} + 33 q^{45} - 10 q^{46} - 25 q^{47} - 2 q^{48} - 11 q^{49} + 3 q^{50} - 14 q^{51} - 8 q^{53} - 23 q^{54} - 12 q^{55} - q^{56} - 2 q^{57} - q^{58} - 28 q^{59} - 11 q^{60} - 15 q^{61} - 11 q^{62} + 20 q^{63} + 4 q^{64} + 3 q^{66} + q^{68} + 18 q^{69} - q^{70} + q^{71} + 2 q^{72} - 6 q^{73} + 9 q^{74} - 13 q^{75} + 4 q^{76} - 11 q^{77} - 12 q^{79} + q^{80} + 40 q^{81} - 4 q^{82} + 17 q^{83} - 4 q^{84} - 13 q^{85} - 15 q^{86} + 25 q^{87} - 2 q^{88} + 15 q^{89} + 33 q^{90} - 10 q^{92} + 3 q^{93} - 25 q^{94} + q^{95} - 2 q^{96} - 2 q^{97} - 11 q^{98} - 26 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9 + q^10 - 2 * q^11 - 2 * q^12 - q^14 - 11 * q^15 + 4 * q^16 + q^17 + 2 * q^18 + 4 * q^19 + q^20 - 4 * q^21 - 2 * q^22 - 10 * q^23 - 2 * q^24 + 3 * q^25 - 23 * q^27 - q^28 - q^29 - 11 * q^30 - 11 * q^31 + 4 * q^32 + 3 * q^33 + q^34 - q^35 + 2 * q^36 + 9 * q^37 + 4 * q^38 + q^40 - 4 * q^41 - 4 * q^42 - 15 * q^43 - 2 * q^44 + 33 * q^45 - 10 * q^46 - 25 * q^47 - 2 * q^48 - 11 * q^49 + 3 * q^50 - 14 * q^51 - 8 * q^53 - 23 * q^54 - 12 * q^55 - q^56 - 2 * q^57 - q^58 - 28 * q^59 - 11 * q^60 - 15 * q^61 - 11 * q^62 + 20 * q^63 + 4 * q^64 + 3 * q^66 + q^68 + 18 * q^69 - q^70 + q^71 + 2 * q^72 - 6 * q^73 + 9 * q^74 - 13 * q^75 + 4 * q^76 - 11 * q^77 - 12 * q^79 + q^80 + 40 * q^81 - 4 * q^82 + 17 * q^83 - 4 * q^84 - 13 * q^85 - 15 * q^86 + 25 * q^87 - 2 * q^88 + 15 * q^89 + 33 * q^90 - 10 * q^92 + 3 * q^93 - 25 * q^94 + q^95 - 2 * q^96 - 2 * q^97 - 11 * q^98 - 26 * 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
$3$	1.38266	0.798281	0.399141	−	0.916890i	$-0.369309\pi$
$3$	1.38266	0.798281	0.399141	+	0.916890i	$0.369309\pi$
$4$	1.00000	0.500000
$5$	1.23060	0.550341	0.275171	−	0.961395i	$-0.411266\pi$
$5$	1.23060	0.550341	0.275171	+	0.961395i	$0.411266\pi$
$6$	1.38266	0.564470
$7$	−0.936179	−0.353843	−0.176921	−	0.984225i	$-0.556614\pi$
$7$	−0.936179	−0.353843	−0.176921	+	0.984225i	$0.556614\pi$
$8$	1.00000	0.353553
$9$	−1.08824	−0.362747
$10$	1.23060	0.389150
$11$	−2.67708	−0.807171	−0.403586	−	0.914942i	$-0.632236\pi$
$11$	−2.67708	−0.807171	−0.403586	+	0.914942i	$0.632236\pi$
$12$	1.38266	0.399141
$13$	0	0
$13$	0	0
$14$	−0.936179	−0.250204
$15$	1.70151	0.439327
$16$	1.00000	0.250000
$17$	−6.67301	−1.61844	−0.809221	−	0.587504i	$-0.800111\pi$
$17$	−6.67301	−1.61844	−0.809221	+	0.587504i	$0.800111\pi$
$18$	−1.08824	−0.256501
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.23060	0.275171
$21$	−1.29442	−0.282466
$22$	−2.67708	−0.570756
$23$	−0.617337	−0.128724	−0.0643618	−	0.997927i	$-0.520501\pi$
$23$	−0.617337	−0.128724	−0.0643618	+	0.997927i	$0.520501\pi$
$24$	1.38266	0.282235
$25$	−3.48562	−0.697125
$26$	0	0
$27$	−5.65266	−1.08786
$28$	−0.936179	−0.176921
$29$	5.86829	1.08971	0.544857	−	0.838529i	$-0.316584\pi$
$29$	5.86829	1.08971	0.544857	+	0.838529i	$0.316584\pi$
$30$	1.70151	0.310651
$31$	−5.61326	−1.00817	−0.504086	−	0.863653i	$-0.668171\pi$
$31$	−5.61326	−1.00817	−0.504086	+	0.863653i	$0.668171\pi$
$32$	1.00000	0.176777
$33$	−3.70151	−0.644349
$34$	−6.67301	−1.14441
$35$	−1.15206	−0.194734
$36$	−1.08824	−0.181374
$37$	−4.67301	−0.768238	−0.384119	−	0.923284i	$-0.625495\pi$
$37$	−4.67301	−0.768238	−0.384119	+	0.923284i	$0.625495\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	1.23060	0.194575
$41$	−4.84386	−0.756484	−0.378242	−	0.925707i	$-0.623471\pi$
$41$	−4.84386	−0.756484	−0.378242	+	0.925707i	$0.623471\pi$
$42$	−1.29442	−0.199733
$43$	1.21181	0.184799	0.0923997	−	0.995722i	$-0.470546\pi$
$43$	1.21181	0.184799	0.0923997	+	0.995722i	$0.470546\pi$
$44$	−2.67708	−0.403586
$45$	−1.33919	−0.199635
$46$	−0.617337	−0.0910213
$47$	−10.5642	−1.54094	−0.770471	−	0.637476i	$-0.779979\pi$
$47$	−10.5642	−1.54094	−0.770471	+	0.637476i	$0.779979\pi$
$48$	1.38266	0.199570
$49$	−6.12357	−0.874795
$50$	−3.48562	−0.492941
$51$	−9.22653	−1.29197
$52$	0	0
$53$	−2.29442	−0.315163	−0.157581	−	0.987506i	$-0.550370\pi$
$53$	−2.29442	−0.315163	−0.157581	+	0.987506i	$0.550370\pi$
$54$	−5.65266	−0.769230
$55$	−3.29442	−0.444220
$56$	−0.936179	−0.125102
$57$	1.38266	0.183138
$58$	5.86829	0.770544
$59$	−11.5701	−1.50629	−0.753146	−	0.657853i	$-0.771465\pi$
$59$	−11.5701	−1.50629	−0.753146	+	0.657853i	$0.771465\pi$
$60$	1.70151	0.219663
$61$	0.485623	0.0621776	0.0310888	−	0.999517i	$-0.490103\pi$
$61$	0.485623	0.0621776	0.0310888	+	0.999517i	$0.490103\pi$
$62$	−5.61326	−0.712885
$63$	1.01879	0.128355
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.70151	−0.455624
$67$	15.2184	1.85922	0.929611	−	0.368543i	$-0.120143\pi$
$67$	15.2184	1.85922	0.929611	+	0.368543i	$0.120143\pi$
$68$	−6.67301	−0.809221
$69$	−0.853569	−0.102758
$70$	−1.15206	−0.137698
$71$	12.3107	1.46101	0.730505	−	0.682907i	$-0.239285\pi$
$71$	12.3107	1.46101	0.730505	+	0.682907i	$0.239285\pi$
$72$	−1.08824	−0.128251
$73$	12.3460	1.44499	0.722496	−	0.691375i	$-0.242995\pi$
$73$	12.3460	1.44499	0.722496	+	0.691375i	$0.242995\pi$
$74$	−4.67301	−0.543227
$75$	−4.81944	−0.556501
$76$	1.00000	0.114708
$77$	2.50623	0.285612
$78$	0	0
$79$	−2.33262	−0.262440	−0.131220	−	0.991353i	$-0.541889\pi$
$79$	−2.33262	−0.262440	−0.131220	+	0.991353i	$0.541889\pi$
$80$	1.23060	0.137585
$81$	−4.55100	−0.505667
$82$	−4.84386	−0.534915
$83$	5.74090	0.630146	0.315073	−	0.949067i	$-0.397971\pi$
$83$	5.74090	0.630146	0.315073	+	0.949067i	$0.397971\pi$
$84$	−1.29442	−0.141233
$85$	−8.21181	−0.890696
$86$	1.21181	0.130673
$87$	8.11386	0.869897
$88$	−2.67708	−0.285378
$89$	18.0663	1.91503	0.957513	−	0.288390i	$-0.0931200\pi$
$89$	18.0663	1.91503	0.957513	+	0.288390i	$0.0931200\pi$
$90$	−1.33919	−0.141163
$91$	0	0
$92$	−0.617337	−0.0643618
$93$	−7.76125	−0.804804
$94$	−10.5642	−1.08961
$95$	1.23060	0.126257
$96$	1.38266	0.141117
$97$	3.61708	0.367259	0.183629	−	0.982996i	$-0.441215\pi$
$97$	3.61708	0.367259	0.183629	+	0.982996i	$0.441215\pi$
$98$	−6.12357	−0.618574
$99$	2.91332	0.292799
$100$	−3.48562	−0.348562
$101$	10.9349	1.08806	0.544030	−	0.839066i	$-0.316898\pi$
$101$	10.9349	1.08806	0.544030	+	0.839066i	$0.316898\pi$
$102$	−9.22653	−0.913562
$103$	−13.9880	−1.37828	−0.689141	−	0.724627i	$-0.742012\pi$
$103$	−13.9880	−1.37828	−0.689141	+	0.724627i	$0.742012\pi$
$104$	0	0
$105$	−1.59291	−0.155453
$106$	−2.29442	−0.222854
$107$	−1.11266	−0.107565	−0.0537826	−	0.998553i	$-0.517128\pi$
$107$	−1.11266	−0.107565	−0.0537826	+	0.998553i	$0.517128\pi$
$108$	−5.65266	−0.543928
$109$	−10.0066	−0.958456	−0.479228	−	0.877691i	$-0.659083\pi$
$109$	−10.0066	−0.958456	−0.479228	+	0.877691i	$0.659083\pi$
$110$	−3.29442	−0.314111
$111$	−6.46120	−0.613270
$112$	−0.936179	−0.0884606
$113$	−19.7572	−1.85860	−0.929300	−	0.369327i	$-0.879588\pi$
$113$	−19.7572	−1.85860	−0.929300	+	0.369327i	$0.879588\pi$
$114$	1.38266	0.129498
$115$	−0.759695	−0.0708419
$116$	5.86829	0.544857
$117$	0	0
$118$	−11.5701	−1.06511
$119$	6.24714	0.572674
$120$	1.70151	0.155326
$121$	−3.83322	−0.348475
$122$	0.485623	0.0439662
$123$	−6.69743	−0.603887
$124$	−5.61326	−0.504086
$125$	−10.4424	−0.933998
$126$	1.01879	0.0907610
$127$	−10.2021	−0.905290	−0.452645	−	0.891691i	$-0.649520\pi$
$127$	−10.2021	−0.905290	−0.452645	+	0.891691i	$0.649520\pi$
$128$	1.00000	0.0883883
$129$	1.67553	0.147522
$130$	0	0
$131$	0.613524	0.0536038	0.0268019	−	0.999641i	$-0.491468\pi$
$131$	0.613524	0.0536038	0.0268019	+	0.999641i	$0.491468\pi$
$132$	−3.70151	−0.322175
$133$	−0.936179	−0.0811771
$134$	15.2184	1.31467
$135$	−6.95617	−0.598692
$136$	−6.67301	−0.572206
$137$	−10.9074	−0.931884	−0.465942	−	0.884815i	$-0.654284\pi$
$137$	−10.9074	−0.931884	−0.465942	+	0.884815i	$0.654284\pi$
$138$	−0.853569	−0.0726606
$139$	−0.904927	−0.0767549	−0.0383774	−	0.999263i	$-0.512219\pi$
$139$	−0.904927	−0.0767549	−0.0383774	+	0.999263i	$0.512219\pi$
$140$	−1.15206	−0.0973671
$141$	−14.6067	−1.23010
$142$	12.3107	1.03309
$143$	0	0
$144$	−1.08824	−0.0906869
$145$	7.22152	0.599714
$146$	12.3460	1.02176
$147$	−8.46683	−0.698333
$148$	−4.67301	−0.384119
$149$	13.5456	1.10970	0.554851	−	0.831950i	$-0.312775\pi$
$149$	13.5456	1.10970	0.554851	+	0.831950i	$0.312775\pi$
$150$	−4.81944	−0.393506
$151$	5.70808	0.464517	0.232258	−	0.972654i	$-0.425389\pi$
$151$	5.70808	0.464517	0.232258	+	0.972654i	$0.425389\pi$
$152$	1.00000	0.0811107
$153$	7.26185	0.587086
$154$	2.50623	0.201958
$155$	−6.90768	−0.554839
$156$	0	0
$157$	−14.0448	−1.12089	−0.560447	−	0.828190i	$-0.689371\pi$
$157$	−14.0448	−1.12089	−0.560447	+	0.828190i	$0.689371\pi$
$158$	−2.33262	−0.185573
$159$	−3.17241	−0.251589
$160$	1.23060	0.0972875
$161$	0.577938	0.0455479
$162$	−4.55100	−0.357560
$163$	−8.19302	−0.641727	−0.320863	−	0.947125i	$-0.603973\pi$
$163$	−8.19302	−0.641727	−0.320863	+	0.947125i	$0.603973\pi$
$164$	−4.84386	−0.378242
$165$	−4.55507	−0.354612
$166$	5.74090	0.445581
$167$	8.19302	0.633995	0.316997	−	0.948426i	$-0.397325\pi$
$167$	8.19302	0.633995	0.316997	+	0.948426i	$0.397325\pi$
$168$	−1.29442	−0.0998667
$169$	0	0
$170$	−8.21181	−0.629817
$171$	−1.08824	−0.0832200
$172$	1.21181	0.0923997
$173$	7.97151	0.606062	0.303031	−	0.952981i	$-0.402001\pi$
$173$	7.97151	0.606062	0.303031	+	0.952981i	$0.402001\pi$
$174$	8.11386	0.615110
$175$	3.26317	0.246672
$176$	−2.67708	−0.201793
$177$	−15.9975	−1.20244
$178$	18.0663	1.35413
$179$	21.9799	1.64285	0.821427	−	0.570313i	$-0.193178\pi$
$179$	21.9799	1.64285	0.821427	+	0.570313i	$0.193178\pi$
$180$	−1.33919	−0.0998174
$181$	12.0025	0.892139	0.446069	−	0.894998i	$-0.352823\pi$
$181$	12.0025	0.892139	0.446069	+	0.894998i	$0.352823\pi$
$182$	0	0
$183$	0.671453	0.0496352
$184$	−0.617337	−0.0455107
$185$	−5.75061	−0.422793
$186$	−7.76125	−0.569083
$187$	17.8642	1.30636
$188$	−10.5642	−0.770471
$189$	5.29191	0.384930
$190$	1.23060	0.0892771
$191$	−12.7928	−0.925652	−0.462826	−	0.886449i	$-0.653165\pi$
$191$	−12.7928	−0.925652	−0.462826	+	0.886449i	$0.653165\pi$
$192$	1.38266	0.0997851
$193$	19.1858	1.38103	0.690513	−	0.723320i	$-0.257385\pi$
$193$	19.1858	1.38103	0.690513	+	0.723320i	$0.257385\pi$
$194$	3.61708	0.259691
$195$	0	0
$196$	−6.12357	−0.437398
$197$	−9.73094	−0.693301	−0.346650	−	0.937994i	$-0.612681\pi$
$197$	−9.73094	−0.693301	−0.346650	+	0.937994i	$0.612681\pi$
$198$	2.91332	0.207040
$199$	−0.317024	−0.0224732	−0.0112366	−	0.999937i	$-0.503577\pi$
$199$	−0.317024	−0.0224732	−0.0112366	+	0.999937i	$0.503577\pi$
$200$	−3.48562	−0.246471
$201$	21.0419	1.48418
$202$	10.9349	0.769374
$203$	−5.49377	−0.385587
$204$	−9.22653	−0.645986
$205$	−5.96086	−0.416325
$206$	−13.9880	−0.974593
$207$	0.671812	0.0466942
$208$	0	0
$209$	−2.67708	−0.185178
$210$	−1.59291	−0.109922
$211$	−24.7737	−1.70549	−0.852746	−	0.522325i	$-0.825065\pi$
$211$	−24.7737	−1.70549	−0.852746	+	0.522325i	$0.825065\pi$
$212$	−2.29442	−0.157581
$213$	17.0215	1.16630
$214$	−1.11266	−0.0760602
$215$	1.49125	0.101703
$216$	−5.65266	−0.384615
$217$	5.25502	0.356734
$218$	−10.0066	−0.677731
$219$	17.0704	1.15351
$220$	−3.29442	−0.222110
$221$	0	0
$222$	−6.46120	−0.433647
$223$	23.8789	1.59905	0.799526	−	0.600632i	$-0.205084\pi$
$223$	23.8789	1.59905	0.799526	+	0.600632i	$0.205084\pi$
$224$	−0.936179	−0.0625511
$225$	3.79320	0.252880
$226$	−19.7572	−1.31423
$227$	8.44830	0.560734	0.280367	−	0.959893i	$-0.409544\pi$
$227$	8.44830	0.560734	0.280367	+	0.959893i	$0.409544\pi$
$228$	1.38266	0.0915691
$229$	−21.4793	−1.41939	−0.709696	−	0.704508i	$-0.751168\pi$
$229$	−21.4793	−1.41939	−0.709696	+	0.704508i	$0.751168\pi$
$230$	−0.759695	−0.0500928
$231$	3.46527	0.227998
$232$	5.86829	0.385272
$233$	21.2064	1.38928	0.694639	−	0.719358i	$-0.255564\pi$
$233$	21.2064	1.38928	0.694639	+	0.719358i	$0.255564\pi$
$234$	0	0
$235$	−13.0003	−0.848043
$236$	−11.5701	−0.753146
$237$	−3.22523	−0.209501
$238$	6.24714	0.404942
$239$	−11.9486	−0.772893	−0.386447	−	0.922312i	$-0.626298\pi$
$239$	−11.9486	−0.772893	−0.386447	+	0.922312i	$0.626298\pi$
$240$	1.70151	0.109832
$241$	9.69837	0.624727	0.312364	−	0.949963i	$-0.398879\pi$
$241$	9.69837	0.624727	0.312364	+	0.949963i	$0.398879\pi$
$242$	−3.83322	−0.246409
$243$	10.6655	0.684191
$244$	0.485623	0.0310888
$245$	−7.53567	−0.481436
$246$	−6.69743	−0.427013
$247$	0	0
$248$	−5.61326	−0.356443
$249$	7.93774	0.503034
$250$	−10.4424	−0.660436
$251$	−9.99749	−0.631036	−0.315518	−	0.948920i	$-0.602178\pi$
$251$	−9.99749	−0.631036	−0.315518	+	0.948920i	$0.602178\pi$
$252$	1.01879	0.0641777
$253$	1.65266	0.103902
$254$	−10.2021	−0.640137
$255$	−11.3542	−0.711026
$256$	1.00000	0.0625000
$257$	−27.0994	−1.69041	−0.845207	−	0.534439i	$-0.820523\pi$
$257$	−27.0994	−1.69041	−0.845207	+	0.534439i	$0.820523\pi$
$258$	1.67553	0.104314
$259$	4.37478	0.271835
$260$	0	0
$261$	−6.38612	−0.395291
$262$	0.613524	0.0379036
$263$	17.5334	1.08116	0.540578	−	0.841294i	$-0.318206\pi$
$263$	17.5334	1.08116	0.540578	+	0.841294i	$0.318206\pi$
$264$	−3.70151	−0.227812
$265$	−2.82352	−0.173447
$266$	−0.936179	−0.0574008
$267$	24.9796	1.52873
$268$	15.2184	0.929611
$269$	3.05567	0.186308	0.0931539	−	0.995652i	$-0.470305\pi$
$269$	3.05567	0.186308	0.0931539	+	0.995652i	$0.470305\pi$
$270$	−6.95617	−0.423339
$271$	5.36413	0.325848	0.162924	−	0.986639i	$-0.447907\pi$
$271$	5.36413	0.325848	0.162924	+	0.986639i	$0.447907\pi$
$272$	−6.67301	−0.404611
$273$	0	0
$274$	−10.9074	−0.658942
$275$	9.33131	0.562699
$276$	−0.853569	−0.0513788
$277$	−20.4370	−1.22794	−0.613971	−	0.789329i	$-0.710429\pi$
$277$	−20.4370	−1.22794	−0.613971	+	0.789329i	$0.710429\pi$
$278$	−0.904927	−0.0542739
$279$	6.10859	0.365712
$280$	−1.15206	−0.0688489
$281$	−14.3707	−0.857284	−0.428642	−	0.903474i	$-0.641008\pi$
$281$	−14.3707	−0.857284	−0.428642	+	0.903474i	$0.641008\pi$
$282$	−14.6067	−0.869815
$283$	−2.04978	−0.121847	−0.0609235	−	0.998142i	$-0.519405\pi$
$283$	−2.04978	−0.121847	−0.0609235	+	0.998142i	$0.519405\pi$
$284$	12.3107	0.730505
$285$	1.70151	0.100789
$286$	0	0
$287$	4.53473	0.267676
$288$	−1.08824	−0.0641253
$289$	27.5291	1.61936
$290$	7.22152	0.424062
$291$	5.00120	0.293175
$292$	12.3460	0.722496
$293$	−13.6424	−0.797000	−0.398500	−	0.917168i	$-0.630469\pi$
$293$	−13.6424	−0.797000	−0.398500	+	0.917168i	$0.630469\pi$
$294$	−8.46683	−0.493796
$295$	−14.2381	−0.828975
$296$	−4.67301	−0.271613
$297$	15.1327	0.878086
$298$	13.5456	0.784677
$299$	0	0
$300$	−4.81944	−0.278251
$301$	−1.13447	−0.0653899
$302$	5.70808	0.328463
$303$	15.1192	0.868577
$304$	1.00000	0.0573539
$305$	0.597607	0.0342189
$306$	7.26185	0.415133
$307$	8.12607	0.463779	0.231890	−	0.972742i	$-0.425509\pi$
$307$	8.12607	0.463779	0.231890	+	0.972742i	$0.425509\pi$
$308$	2.50623	0.142806
$309$	−19.3408	−1.10026
$310$	−6.90768	−0.392330
$311$	−15.6786	−0.889054	−0.444527	−	0.895765i	$-0.646628\pi$
$311$	−15.6786	−0.889054	−0.444527	+	0.895765i	$0.646628\pi$
$312$	0	0
$313$	19.4665	1.10031	0.550155	−	0.835063i	$-0.314569\pi$
$313$	19.4665	1.10031	0.550155	+	0.835063i	$0.314569\pi$
$314$	−14.0448	−0.792592
$315$	1.25372	0.0706393
$316$	−2.33262	−0.131220
$317$	9.36100	0.525766	0.262883	−	0.964828i	$-0.415327\pi$
$317$	9.36100	0.525766	0.262883	+	0.964828i	$0.415327\pi$
$318$	−3.17241	−0.177900
$319$	−15.7099	−0.879585
$320$	1.23060	0.0687927
$321$	−1.53844	−0.0858673
$322$	0.577938	0.0322072
$323$	−6.67301	−0.371296
$324$	−4.55100	−0.252833
$325$	0	0
$326$	−8.19302	−0.453769
$327$	−13.8357	−0.765117
$328$	−4.84386	−0.267458
$329$	9.88995	0.545251
$330$	−4.55507	−0.250749
$331$	17.2147	0.946205	0.473102	−	0.881007i	$-0.343134\pi$
$331$	17.2147	0.946205	0.473102	+	0.881007i	$0.343134\pi$
$332$	5.74090	0.315073
$333$	5.08537	0.278676
$334$	8.19302	0.448302
$335$	18.7277	1.02321
$336$	−1.29442	−0.0706165
$337$	−17.1709	−0.935356	−0.467678	−	0.883899i	$-0.654909\pi$
$337$	−17.1709	−0.935356	−0.467678	+	0.883899i	$0.654909\pi$
$338$	0	0
$339$	−27.3175	−1.48368
$340$	−8.21181	−0.445348
$341$	15.0272	0.813767
$342$	−1.08824	−0.0588454
$343$	12.2860	0.663382
$344$	1.21181	0.0653364
$345$	−1.05040	−0.0565518
$346$	7.97151	0.428551
$347$	−16.0093	−0.859426	−0.429713	−	0.902966i	$-0.641385\pi$
$347$	−16.0093	−0.859426	−0.429713	+	0.902966i	$0.641385\pi$
$348$	8.11386	0.434949
$349$	7.57768	0.405624	0.202812	−	0.979218i	$-0.434992\pi$
$349$	7.57768	0.405624	0.202812	+	0.979218i	$0.434992\pi$
$350$	3.26317	0.174424
$351$	0	0
$352$	−2.67708	−0.142689
$353$	−6.04096	−0.321528	−0.160764	−	0.986993i	$-0.551396\pi$
$353$	−6.04096	−0.321528	−0.160764	+	0.986993i	$0.551396\pi$
$354$	−15.9975	−0.850257
$355$	15.1495	0.804055
$356$	18.0663	0.957513
$357$	8.63769	0.457155
$358$	21.9799	1.16167
$359$	14.9226	0.787587	0.393794	−	0.919199i	$-0.371162\pi$
$359$	14.9226	0.787587	0.393794	+	0.919199i	$0.371162\pi$
$360$	−1.33919	−0.0705816
$361$	1.00000	0.0526316
$362$	12.0025	0.630837
$363$	−5.30005	−0.278181
$364$	0	0
$365$	15.1930	0.795239
$366$	0.671453	0.0350974
$367$	−33.6633	−1.75721	−0.878605	−	0.477550i	$-0.841525\pi$
$367$	−33.6633	−1.75721	−0.878605	+	0.477550i	$0.841525\pi$
$368$	−0.617337	−0.0321809
$369$	5.27130	0.274413
$370$	−5.75061	−0.298960
$371$	2.14799	0.111518
$372$	−7.76125	−0.402402
$373$	−0.203421	−0.0105327	−0.00526637	−	0.999986i	$-0.501676\pi$
$373$	−0.203421	−0.0105327	−0.00526637	+	0.999986i	$0.501676\pi$
$374$	17.8642	0.923736
$375$	−14.4383	−0.745593
$376$	−10.5642	−0.544805
$377$	0	0
$378$	5.29191	0.272186
$379$	−13.7043	−0.703941	−0.351970	−	0.936011i	$-0.614488\pi$
$379$	−13.7043	−0.703941	−0.351970	+	0.936011i	$0.614488\pi$
$380$	1.23060	0.0631285
$381$	−14.1061	−0.722676
$382$	−12.7928	−0.654535
$383$	22.9522	1.17280	0.586401	−	0.810021i	$-0.300544\pi$
$383$	22.9522	1.17280	0.586401	+	0.810021i	$0.300544\pi$
$384$	1.38266	0.0705587
$385$	3.08417	0.157184
$386$	19.1858	0.976532
$387$	−1.31874	−0.0670355
$388$	3.61708	0.183629
$389$	−8.38448	−0.425110	−0.212555	−	0.977149i	$-0.568178\pi$
$389$	−8.38448	−0.425110	−0.212555	+	0.977149i	$0.568178\pi$
$390$	0	0
$391$	4.11949	0.208332
$392$	−6.12357	−0.309287
$393$	0.848297	0.0427909
$394$	−9.73094	−0.490238
$395$	−2.87052	−0.144432
$396$	2.91332	0.146400
$397$	−11.9478	−0.599641	−0.299821	−	0.953996i	$-0.596927\pi$
$397$	−11.9478	−0.599641	−0.299821	+	0.953996i	$0.596927\pi$
$398$	−0.317024	−0.0158910
$399$	−1.29442	−0.0648021
$400$	−3.48562	−0.174281
$401$	25.9255	1.29466	0.647329	−	0.762210i	$-0.275886\pi$
$401$	25.9255	1.29466	0.647329	+	0.762210i	$0.275886\pi$
$402$	21.0419	1.04947
$403$	0	0
$404$	10.9349	0.544030
$405$	−5.60046	−0.278289
$406$	−5.49377	−0.272651
$407$	12.5100	0.620100
$408$	−9.22653	−0.456781
$409$	−6.08217	−0.300744	−0.150372	−	0.988629i	$-0.548047\pi$
$409$	−6.08217	−0.300744	−0.150372	+	0.988629i	$0.548047\pi$
$410$	−5.96086	−0.294386
$411$	−15.0813	−0.743905
$412$	−13.9880	−0.689141
$413$	10.8316	0.532990
$414$	0.671812	0.0330178
$415$	7.06476	0.346795
$416$	0	0
$417$	−1.25121	−0.0612720
$418$	−2.67708	−0.130940
$419$	22.2456	1.08677	0.543383	−	0.839485i	$-0.317143\pi$
$419$	22.2456	1.08677	0.543383	+	0.839485i	$0.317143\pi$
$420$	−1.59291	−0.0777263
$421$	−13.8757	−0.676262	−0.338131	−	0.941099i	$-0.609795\pi$
$421$	−13.8757	−0.676262	−0.338131	+	0.941099i	$0.609795\pi$
$422$	−24.7737	−1.20597
$423$	11.4964	0.558972
$424$	−2.29442	−0.111427
$425$	23.2596	1.12826
$426$	17.0215	0.824697
$427$	−0.454630	−0.0220011
$428$	−1.11266	−0.0537826
$429$	0	0
$430$	1.49125	0.0719147
$431$	−15.4333	−0.743397	−0.371699	−	0.928353i	$-0.621225\pi$
$431$	−15.4333	−0.743397	−0.371699	+	0.928353i	$0.621225\pi$
$432$	−5.65266	−0.271964
$433$	32.2761	1.55109	0.775546	−	0.631291i	$-0.217475\pi$
$433$	32.2761	1.55109	0.775546	+	0.631291i	$0.217475\pi$
$434$	5.25502	0.252249
$435$	9.98492	0.478740
$436$	−10.0066	−0.479228
$437$	−0.617337	−0.0295312
$438$	17.0704	0.815655
$439$	−19.8156	−0.945748	−0.472874	−	0.881130i	$-0.656783\pi$
$439$	−19.8156	−0.945748	−0.472874	+	0.881130i	$0.656783\pi$
$440$	−3.29442	−0.157055
$441$	6.66393	0.317330
$442$	0	0
$443$	−16.7528	−0.795952	−0.397976	−	0.917396i	$-0.630287\pi$
$443$	−16.7528	−0.795952	−0.397976	+	0.917396i	$0.630287\pi$
$444$	−6.46120	−0.306635
$445$	22.2324	1.05392
$446$	23.8789	1.13070
$447$	18.7290	0.885853
$448$	−0.936179	−0.0442303
$449$	−22.5080	−1.06222	−0.531110	−	0.847303i	$-0.678225\pi$
$449$	−22.5080	−1.06222	−0.531110	+	0.847303i	$0.678225\pi$
$450$	3.79320	0.178813
$451$	12.9674	0.610613
$452$	−19.7572	−0.929300
$453$	7.89235	0.370815
$454$	8.44830	0.396498
$455$	0	0
$456$	1.38266	0.0647491
$457$	1.14429	0.0535278	0.0267639	−	0.999642i	$-0.491480\pi$
$457$	1.14429	0.0535278	0.0267639	+	0.999642i	$0.491480\pi$
$458$	−21.4793	−1.00366
$459$	37.7203	1.76063
$460$	−0.759695	−0.0354210
$461$	21.2738	0.990820	0.495410	−	0.868659i	$-0.335018\pi$
$461$	21.2738	0.990820	0.495410	+	0.868659i	$0.335018\pi$
$462$	3.46527	0.161219
$463$	−31.4594	−1.46204	−0.731020	−	0.682356i	$-0.760956\pi$
$463$	−31.4594	−1.46204	−0.731020	+	0.682356i	$0.760956\pi$
$464$	5.86829	0.272428
$465$	−9.55100	−0.442917
$466$	21.2064	0.982369
$467$	−33.1449	−1.53376	−0.766881	−	0.641790i	$-0.778192\pi$
$467$	−33.1449	−1.53376	−0.766881	+	0.641790i	$0.778192\pi$
$468$	0	0
$469$	−14.2471	−0.657872
$470$	−13.0003	−0.599657
$471$	−19.4192	−0.894789
$472$	−11.5701	−0.532555
$473$	−3.24412	−0.149165
$474$	−3.22523	−0.148140
$475$	−3.48562	−0.159931
$476$	6.24714	0.286337
$477$	2.49689	0.114325
$478$	−11.9486	−0.546518
$479$	12.1358	0.554498	0.277249	−	0.960798i	$-0.410577\pi$
$479$	12.1358	0.554498	0.277249	+	0.960798i	$0.410577\pi$
$480$	1.70151	0.0776628
$481$	0	0
$482$	9.69837	0.441749
$483$	0.799094	0.0363600
$484$	−3.83322	−0.174237
$485$	4.45118	0.202118
$486$	10.6655	0.483796
$487$	−25.7350	−1.16616	−0.583082	−	0.812413i	$-0.698153\pi$
$487$	−25.7350	−1.16616	−0.583082	+	0.812413i	$0.698153\pi$
$488$	0.485623	0.0219831
$489$	−11.3282	−0.512278
$490$	−7.53567	−0.340427
$491$	−24.8233	−1.12026	−0.560129	−	0.828405i	$-0.689248\pi$
$491$	−24.8233	−1.12026	−0.560129	+	0.828405i	$0.689248\pi$
$492$	−6.69743	−0.301944
$493$	−39.1591	−1.76364
$494$	0	0
$495$	3.58513	0.161140
$496$	−5.61326	−0.252043
$497$	−11.5250	−0.516968
$498$	7.93774	0.355699
$499$	−3.63994	−0.162946	−0.0814730	−	0.996676i	$-0.525962\pi$
$499$	−3.63994	−0.162946	−0.0814730	+	0.996676i	$0.525962\pi$
$500$	−10.4424	−0.466999
$501$	11.3282	0.506106
$502$	−9.99749	−0.446210
$503$	25.3810	1.13168	0.565841	−	0.824515i	$-0.308552\pi$
$503$	25.3810	1.13168	0.565841	+	0.824515i	$0.308552\pi$
$504$	1.01879	0.0453805
$505$	13.4564	0.598804
$506$	1.65266	0.0734698
$507$	0	0
$508$	−10.2021	−0.452645
$509$	36.5954	1.62206	0.811031	−	0.585003i	$-0.198907\pi$
$509$	36.5954	1.62206	0.811031	+	0.585003i	$0.198907\pi$
$510$	−11.3542	−0.502771
$511$	−11.5581	−0.511300
$512$	1.00000	0.0441942
$513$	−5.65266	−0.249571
$514$	−27.0994	−1.19530
$515$	−17.2137	−0.758526
$516$	1.67553	0.0737609
$517$	28.2811	1.24380
$518$	4.37478	0.192217
$519$	11.0219	0.483808
$520$	0	0
$521$	21.2998	0.933161	0.466580	−	0.884479i	$-0.345486\pi$
$521$	21.2998	0.933161	0.466580	+	0.884479i	$0.345486\pi$
$522$	−6.38612	−0.279513
$523$	−34.4311	−1.50557	−0.752784	−	0.658267i	$-0.771290\pi$
$523$	−34.4311	−1.50557	−0.752784	+	0.658267i	$0.771290\pi$
$524$	0.613524	0.0268019
$525$	4.51186	0.196914
$526$	17.5334	0.764493
$527$	37.4574	1.63167
$528$	−3.70151	−0.161087
$529$	−22.6189	−0.983430
$530$	−2.82352	−0.122646
$531$	12.5910	0.546404
$532$	−0.936179	−0.0405885
$533$	0	0
$534$	24.9796	1.08097
$535$	−1.36924	−0.0591976
$536$	15.2184	0.657334
$537$	30.3908	1.31146
$538$	3.05567	0.131739
$539$	16.3933	0.706110
$540$	−6.95617	−0.299346
$541$	−8.30281	−0.356966	−0.178483	−	0.983943i	$-0.557119\pi$
$541$	−8.30281	−0.356966	−0.178483	+	0.983943i	$0.557119\pi$
$542$	5.36413	0.230409
$543$	16.5954	0.712177
$544$	−6.67301	−0.286103
$545$	−12.3141	−0.527478
$546$	0	0
$547$	18.7694	0.802521	0.401260	−	0.915964i	$-0.368572\pi$
$547$	18.7694	0.802521	0.401260	+	0.915964i	$0.368572\pi$
$548$	−10.9074	−0.465942
$549$	−0.528475	−0.0225548
$550$	9.33131	0.397888
$551$	5.86829	0.249997
$552$	−0.853569	−0.0363303
$553$	2.18375	0.0928626
$554$	−20.4370	−0.868286
$555$	−7.95116	−0.337508
$556$	−0.904927	−0.0383774
$557$	−41.3854	−1.75356	−0.876778	−	0.480895i	$-0.840312\pi$
$557$	−41.3854	−1.75356	−0.876778	+	0.480895i	$0.840312\pi$
$558$	6.10859	0.258597
$559$	0	0
$560$	−1.15206	−0.0486835
$561$	24.7002	1.04284
$562$	−14.3707	−0.606192
$563$	27.9367	1.17739	0.588695	−	0.808355i	$-0.299642\pi$
$563$	27.9367	1.17739	0.588695	+	0.808355i	$0.299642\pi$
$564$	−14.6067	−0.615052
$565$	−24.3132	−1.02286
$566$	−2.04978	−0.0861588
$567$	4.26055	0.178926
$568$	12.3107	0.516545
$569$	23.0620	0.966809	0.483405	−	0.875397i	$-0.339400\pi$
$569$	23.0620	0.966809	0.483405	+	0.875397i	$0.339400\pi$
$570$	1.70151	0.0712682
$571$	−24.9466	−1.04398	−0.521991	−	0.852951i	$-0.674810\pi$
$571$	−24.9466	−1.04398	−0.521991	+	0.852951i	$0.674810\pi$
$572$	0	0
$573$	−17.6881	−0.738930
$574$	4.53473	0.189276
$575$	2.15180	0.0897364
$576$	−1.08824	−0.0453434
$577$	32.5263	1.35409	0.677044	−	0.735942i	$-0.263261\pi$
$577$	32.5263	1.35409	0.677044	+	0.735942i	$0.263261\pi$
$578$	27.5291	1.14506
$579$	26.5275	1.10245
$580$	7.22152	0.299857
$581$	−5.37452	−0.222973
$582$	5.00120	0.207306
$583$	6.14236	0.254390
$584$	12.3460	0.510882
$585$	0	0
$586$	−13.6424	−0.563564
$587$	−3.39445	−0.140104	−0.0700519	−	0.997543i	$-0.522316\pi$
$587$	−3.39445	−0.140104	−0.0700519	+	0.997543i	$0.522316\pi$
$588$	−8.46683	−0.349166
$589$	−5.61326	−0.231291
$590$	−14.2381	−0.586174
$591$	−13.4546	−0.553449
$592$	−4.67301	−0.192060
$593$	12.7494	0.523556	0.261778	−	0.965128i	$-0.415691\pi$
$593$	12.7494	0.523556	0.261778	+	0.965128i	$0.415691\pi$
$594$	15.1327	0.620900
$595$	7.68773	0.315166
$596$	13.5456	0.554851
$597$	−0.438338	−0.0179400
$598$	0	0
$599$	15.4865	0.632761	0.316381	−	0.948632i	$-0.397532\pi$
$599$	15.4865	0.632761	0.316381	+	0.948632i	$0.397532\pi$
$600$	−4.81944	−0.196753
$601$	32.3488	1.31953	0.659767	−	0.751470i	$-0.270655\pi$
$601$	32.3488	1.31953	0.659767	+	0.751470i	$0.270655\pi$
$602$	−1.13447	−0.0462376
$603$	−16.5613	−0.674428
$604$	5.70808	0.232258
$605$	−4.71716	−0.191780
$606$	15.1192	0.614177
$607$	−27.7669	−1.12702	−0.563512	−	0.826108i	$-0.690550\pi$
$607$	−27.7669	−1.12702	−0.563512	+	0.826108i	$0.690550\pi$
$608$	1.00000	0.0405554
$609$	−7.59603	−0.307807
$610$	0.597607	0.0241964
$611$	0	0
$612$	7.26185	0.293543
$613$	5.40621	0.218355	0.109177	−	0.994022i	$-0.465178\pi$
$613$	5.40621	0.218355	0.109177	+	0.994022i	$0.465178\pi$
$614$	8.12607	0.327941
$615$	−8.24186	−0.332344
$616$	2.50623	0.100979
$617$	−7.79746	−0.313914	−0.156957	−	0.987605i	$-0.550168\pi$
$617$	−7.79746	−0.313914	−0.156957	+	0.987605i	$0.550168\pi$
$618$	−19.3408	−0.777999
$619$	−6.17016	−0.248000	−0.124000	−	0.992282i	$-0.539572\pi$
$619$	−6.17016	−0.248000	−0.124000	+	0.992282i	$0.539572\pi$
$620$	−6.90768	−0.277419
$621$	3.48960	0.140033
$622$	−15.6786	−0.628656
$623$	−16.9133	−0.677618
$624$	0	0
$625$	4.57768	0.183107
$626$	19.4665	0.778036
$627$	−3.70151	−0.147824
$628$	−14.0448	−0.560447
$629$	31.1831	1.24335
$630$	1.25372	0.0499495
$631$	28.9539	1.15264	0.576318	−	0.817225i	$-0.304489\pi$
$631$	28.9539	1.15264	0.576318	+	0.817225i	$0.304489\pi$
$632$	−2.33262	−0.0927867
$633$	−34.2537	−1.36146
$634$	9.36100	0.371773
$635$	−12.5547	−0.498219
$636$	−3.17241	−0.125794
$637$	0	0
$638$	−15.7099	−0.621961
$639$	−13.3970	−0.529978
$640$	1.23060	0.0486438
$641$	3.48864	0.137793	0.0688965	−	0.997624i	$-0.478052\pi$
$641$	3.48864	0.137793	0.0688965	+	0.997624i	$0.478052\pi$
$642$	−1.53844	−0.0607174
$643$	2.76914	0.109204	0.0546021	−	0.998508i	$-0.482611\pi$
$643$	2.76914	0.109204	0.0546021	+	0.998508i	$0.482611\pi$
$644$	0.577938	0.0227739
$645$	2.06190	0.0811873
$646$	−6.67301	−0.262546
$647$	−30.4782	−1.19822	−0.599110	−	0.800667i	$-0.704479\pi$
$647$	−30.4782	−1.19822	−0.599110	+	0.800667i	$0.704479\pi$
$648$	−4.55100	−0.178780
$649$	30.9740	1.21584
$650$	0	0
$651$	7.26593	0.284774
$652$	−8.19302	−0.320863
$653$	−27.2082	−1.06474	−0.532370	−	0.846512i	$-0.678699\pi$
$653$	−27.2082	−1.06474	−0.532370	+	0.846512i	$0.678699\pi$
$654$	−13.8357	−0.541019
$655$	0.755002	0.0295004
$656$	−4.84386	−0.189121
$657$	−13.4355	−0.524167
$658$	9.88995	0.385550
$659$	33.0589	1.28779	0.643895	−	0.765114i	$-0.277317\pi$
$659$	33.0589	1.28779	0.643895	+	0.765114i	$0.277317\pi$
$660$	−4.55507	−0.177306
$661$	8.63267	0.335772	0.167886	−	0.985806i	$-0.446306\pi$
$661$	8.63267	0.335772	0.167886	+	0.985806i	$0.446306\pi$
$662$	17.2147	0.669068
$663$	0	0
$664$	5.74090	0.222790
$665$	−1.15206	−0.0446751
$666$	5.08537	0.197054
$667$	−3.62271	−0.140272
$668$	8.19302	0.316997
$669$	33.0165	1.27649
$670$	18.7277	0.723516
$671$	−1.30005	−0.0501880
$672$	−1.29442	−0.0499334
$673$	−10.4742	−0.403750	−0.201875	−	0.979411i	$-0.564703\pi$
$673$	−10.4742	−0.403750	−0.201875	+	0.979411i	$0.564703\pi$
$674$	−17.1709	−0.661397
$675$	19.7030	0.758371
$676$	0	0
$677$	7.63994	0.293627	0.146813	−	0.989164i	$-0.453098\pi$
$677$	7.63994	0.293627	0.146813	+	0.989164i	$0.453098\pi$
$678$	−27.3175	−1.04912
$679$	−3.38623	−0.129952
$680$	−8.21181	−0.314909
$681$	11.6812	0.447623
$682$	15.0272	0.575420
$683$	−9.33987	−0.357380	−0.178690	−	0.983905i	$-0.557186\pi$
$683$	−9.33987	−0.357380	−0.178690	+	0.983905i	$0.557186\pi$
$684$	−1.08824	−0.0416100
$685$	−13.4227	−0.512854
$686$	12.2860	0.469082
$687$	−29.6986	−1.13307
$688$	1.21181	0.0461998
$689$	0	0
$690$	−1.05040	−0.0399881
$691$	−20.7190	−0.788187	−0.394093	−	0.919070i	$-0.628941\pi$
$691$	−20.7190	−0.788187	−0.394093	+	0.919070i	$0.628941\pi$
$692$	7.97151	0.303031
$693$	−2.72739	−0.103605
$694$	−16.0093	−0.607706
$695$	−1.11360	−0.0422414
$696$	8.11386	0.307555
$697$	32.3232	1.22433
$698$	7.57768	0.286819
$699$	29.3213	1.10903
$700$	3.26317	0.123336
$701$	−17.8187	−0.673002	−0.336501	−	0.941683i	$-0.609243\pi$
$701$	−17.8187	−0.673002	−0.336501	+	0.941683i	$0.609243\pi$
$702$	0	0
$703$	−4.67301	−0.176246
$704$	−2.67708	−0.100896
$705$	−17.9750	−0.676977
$706$	−6.04096	−0.227354
$707$	−10.2370	−0.385002
$708$	−15.9975	−0.601222
$709$	6.54876	0.245944	0.122972	−	0.992410i	$-0.460757\pi$
$709$	6.54876	0.245944	0.122972	+	0.992410i	$0.460757\pi$
$710$	15.1495	0.568552
$711$	2.53846	0.0951996
$712$	18.0663	0.677064
$713$	3.46527	0.129776
$714$	8.63769	0.323257
$715$	0	0
$716$	21.9799	0.821427
$717$	−16.5209	−0.616986
$718$	14.9226	0.556908
$719$	−39.8147	−1.48484	−0.742419	−	0.669935i	$-0.766322\pi$
$719$	−39.8147	−1.48484	−0.742419	+	0.669935i	$0.766322\pi$
$720$	−1.33919	−0.0499087
$721$	13.0953	0.487695
$722$	1.00000	0.0372161
$723$	13.4096	0.498708
$724$	12.0025	0.446069
$725$	−20.4546	−0.759666
$726$	−5.30005	−0.196703
$727$	−38.2036	−1.41689	−0.708447	−	0.705764i	$-0.750604\pi$
$727$	−38.2036	−1.41689	−0.708447	+	0.705764i	$0.750604\pi$
$728$	0	0
$729$	28.3998	1.05184
$730$	15.1930	0.562319
$731$	−8.08642	−0.299087
$732$	0.671453	0.0248176
$733$	−38.9461	−1.43851	−0.719254	−	0.694747i	$-0.755516\pi$
$733$	−38.9461	−1.43851	−0.719254	+	0.694747i	$0.755516\pi$
$734$	−33.6633	−1.24253
$735$	−10.4193	−0.384321
$736$	−0.617337	−0.0227553
$737$	−40.7409	−1.50071
$738$	5.27130	0.194039
$739$	−33.0523	−1.21585	−0.607924	−	0.793995i	$-0.707997\pi$
$739$	−33.0523	−1.21585	−0.607924	+	0.793995i	$0.707997\pi$
$740$	−5.75061	−0.211397
$741$	0	0
$742$	2.14799	0.0788552
$743$	−18.6508	−0.684232	−0.342116	−	0.939658i	$-0.611144\pi$
$743$	−18.6508	−0.684232	−0.342116	+	0.939658i	$0.611144\pi$
$744$	−7.76125	−0.284541
$745$	16.6693	0.610714
$746$	−0.203421	−0.00744777
$747$	−6.24750	−0.228584
$748$	17.8642	0.653180
$749$	1.04165	0.0380612
$750$	−14.4383	−0.527214
$751$	30.1575	1.10046	0.550231	−	0.835012i	$-0.314540\pi$
$751$	30.1575	1.10046	0.550231	+	0.835012i	$0.314540\pi$
$752$	−10.5642	−0.385235
$753$	−13.8232	−0.503744
$754$	0	0
$755$	7.02436	0.255643
$756$	5.29191	0.192465
$757$	24.1188	0.876613	0.438306	−	0.898826i	$-0.355579\pi$
$757$	24.1188	0.876613	0.438306	+	0.898826i	$0.355579\pi$
$758$	−13.7043	−0.497761
$759$	2.28508	0.0829430
$760$	1.23060	0.0446386
$761$	34.8923	1.26485	0.632423	−	0.774623i	$-0.282060\pi$
$761$	34.8923	1.26485	0.632423	+	0.774623i	$0.282060\pi$
$762$	−14.1061	−0.511009
$763$	9.36795	0.339142
$764$	−12.7928	−0.462826
$765$	8.93644	0.323098
$766$	22.9522	0.829297
$767$	0	0
$768$	1.38266	0.0498926
$769$	18.2083	0.656607	0.328303	−	0.944572i	$-0.393523\pi$
$769$	18.2083	0.656607	0.328303	+	0.944572i	$0.393523\pi$
$770$	3.08417	0.111146
$771$	−37.4693	−1.34942
$772$	19.1858	0.690513
$773$	−29.2901	−1.05349	−0.526746	−	0.850023i	$-0.676588\pi$
$773$	−29.2901	−1.05349	−0.526746	+	0.850023i	$0.676588\pi$
$774$	−1.31874	−0.0474012
$775$	19.5657	0.702821
$776$	3.61708	0.129845
$777$	6.04884	0.217001
$778$	−8.38448	−0.300598
$779$	−4.84386	−0.173549
$780$	0	0
$781$	−32.9568	−1.17929
$782$	4.11949	0.147313
$783$	−33.1714	−1.18545
$784$	−6.12357	−0.218699
$785$	−17.2835	−0.616875
$786$	0.848297	0.0302577
$787$	9.36256	0.333739	0.166870	−	0.985979i	$-0.446634\pi$
$787$	9.36256	0.333739	0.166870	+	0.985979i	$0.446634\pi$
$788$	−9.73094	−0.346650
$789$	24.2428	0.863067
$790$	−2.87052	−0.102129
$791$	18.4963	0.657652
$792$	2.91332	0.103520
$793$	0	0
$794$	−11.9478	−0.424010
$795$	−3.90397	−0.138460
$796$	−0.317024	−0.0112366
$797$	7.12029	0.252214	0.126107	−	0.992017i	$-0.459752\pi$
$797$	7.12029	0.252214	0.126107	+	0.992017i	$0.459752\pi$
$798$	−1.29442	−0.0458220
$799$	70.4948	2.49393
$800$	−3.48562	−0.123235
$801$	−19.6605	−0.694671
$802$	25.9255	0.915462
$803$	−33.0513	−1.16636
$804$	21.0419	0.742090
$805$	0.711211	0.0250669
$806$	0	0
$807$	4.22497	0.148726
$808$	10.9349	0.384687
$809$	−13.3648	−0.469882	−0.234941	−	0.972010i	$-0.575490\pi$
$809$	−13.3648	−0.469882	−0.234941	+	0.972010i	$0.575490\pi$
$810$	−5.60046	−0.196780
$811$	−54.7234	−1.92160	−0.960799	−	0.277244i	$-0.910579\pi$
$811$	−54.7234	−1.92160	−0.960799	+	0.277244i	$0.910579\pi$
$812$	−5.49377	−0.192793
$813$	7.41679	0.260118
$814$	12.5100	0.438477
$815$	−10.0823	−0.353169
$816$	−9.22653	−0.322993
$817$	1.21181	0.0423959
$818$	−6.08217	−0.212658
$819$	0	0
$820$	−5.96086	−0.208162
$821$	−21.0172	−0.733506	−0.366753	−	0.930318i	$-0.619531\pi$
$821$	−21.0172	−0.733506	−0.366753	+	0.930318i	$0.619531\pi$
$822$	−15.0813	−0.526021
$823$	7.99655	0.278742	0.139371	−	0.990240i	$-0.455492\pi$
$823$	7.99655	0.278742	0.139371	+	0.990240i	$0.455492\pi$
$824$	−13.9880	−0.487296
$825$	12.9021	0.449192
$826$	10.8316	0.376881
$827$	0.623908	0.0216954	0.0108477	−	0.999941i	$-0.496547\pi$
$827$	0.623908	0.0216954	0.0108477	+	0.999941i	$0.496547\pi$
$828$	0.671812	0.0233471
$829$	3.88112	0.134797	0.0673985	−	0.997726i	$-0.478530\pi$
$829$	3.88112	0.134797	0.0673985	+	0.997726i	$0.478530\pi$
$830$	7.06476	0.245221
$831$	−28.2575	−0.980243
$832$	0	0
$833$	40.8626	1.41581
$834$	−1.25121	−0.0433258
$835$	10.0823	0.348914
$836$	−2.67708	−0.0925889
$837$	31.7299	1.09675
$838$	22.2456	0.768460
$839$	−0.257277	−0.00888218	−0.00444109	−	0.999990i	$-0.501414\pi$
$839$	−0.257277	−0.00888218	−0.00444109	+	0.999990i	$0.501414\pi$
$840$	−1.59291	−0.0549608
$841$	5.43678	0.187475
$842$	−13.8757	−0.478189
$843$	−19.8698	−0.684354
$844$	−24.7737	−0.852746
$845$	0	0
$846$	11.4964	0.395253
$847$	3.58858	0.123305
$848$	−2.29442	−0.0787907
$849$	−2.83416	−0.0972681
$850$	23.2596	0.797798
$851$	2.88482	0.0988904
$852$	17.0215	0.583149
$853$	37.7643	1.29302	0.646512	−	0.762904i	$-0.276227\pi$
$853$	37.7643	1.29302	0.646512	+	0.762904i	$0.276227\pi$
$854$	−0.454630	−0.0155571
$855$	−1.33919	−0.0457994
$856$	−1.11266	−0.0380301
$857$	32.5472	1.11179	0.555896	−	0.831252i	$-0.312375\pi$
$857$	32.5472	1.11179	0.555896	+	0.831252i	$0.312375\pi$
$858$	0	0
$859$	17.0760	0.582626	0.291313	−	0.956628i	$-0.405908\pi$
$859$	17.0760	0.582626	0.291313	+	0.956628i	$0.405908\pi$
$860$	1.49125	0.0508513
$861$	6.27000	0.213681
$862$	−15.4333	−0.525661
$863$	−22.5037	−0.766035	−0.383018	−	0.923741i	$-0.625115\pi$
$863$	−22.5037	−0.766035	−0.383018	+	0.923741i	$0.625115\pi$
$864$	−5.65266	−0.192307
$865$	9.80974	0.333541
$866$	32.2761	1.09679
$867$	38.0634	1.29270
$868$	5.25502	0.178367
$869$	6.24462	0.211834
$870$	9.98492	0.338521
$871$	0	0
$872$	−10.0066	−0.338865
$873$	−3.93626	−0.133222
$874$	−0.617337	−0.0208817
$875$	9.77597	0.330488
$876$	17.0704	0.576755
$877$	39.1279	1.32125	0.660627	−	0.750714i	$-0.270290\pi$
$877$	39.1279	1.32125	0.660627	+	0.750714i	$0.270290\pi$
$878$	−19.8156	−0.668745
$879$	−18.8629	−0.636230
$880$	−3.29442	−0.111055
$881$	−50.3004	−1.69466	−0.847331	−	0.531065i	$-0.821792\pi$
$881$	−50.3004	−1.69466	−0.847331	+	0.531065i	$0.821792\pi$
$882$	6.66393	0.224386
$883$	−5.04728	−0.169855	−0.0849273	−	0.996387i	$-0.527066\pi$
$883$	−5.04728	−0.169855	−0.0849273	+	0.996387i	$0.527066\pi$
$884$	0	0
$885$	−19.6865	−0.661755
$886$	−16.7528	−0.562823
$887$	8.26724	0.277587	0.138793	−	0.990321i	$-0.455678\pi$
$887$	8.26724	0.277587	0.138793	+	0.990321i	$0.455678\pi$
$888$	−6.46120	−0.216824
$889$	9.55100	0.320330
$890$	22.2324	0.745232
$891$	12.1834	0.408160
$892$	23.8789	0.799526
$893$	−10.5642	−0.353516
$894$	18.7290	0.626393
$895$	27.0485	0.904131
$896$	−0.936179	−0.0312756
$897$	0	0
$898$	−22.5080	−0.751103
$899$	−32.9402	−1.09862
$900$	3.79320	0.126440
$901$	15.3107	0.510073
$902$	12.9674	0.431768
$903$	−1.56859	−0.0521995
$904$	−19.7572	−0.657114
$905$	14.7703	0.490981
$906$	7.89235	0.262206
$907$	38.3768	1.27428	0.637141	−	0.770747i	$-0.280117\pi$
$907$	38.3768	1.27428	0.637141	+	0.770747i	$0.280117\pi$
$908$	8.44830	0.280367
$909$	−11.8998	−0.394691
$910$	0	0
$911$	53.5988	1.77581	0.887904	−	0.460030i	$-0.152161\pi$
$911$	53.5988	1.77581	0.887904	+	0.460030i	$0.152161\pi$
$912$	1.38266	0.0457846
$913$	−15.3689	−0.508636
$914$	1.14429	0.0378498
$915$	0.826290	0.0273163
$916$	−21.4793	−0.709696
$917$	−0.574368	−0.0189673
$918$	37.7203	1.24495
$919$	−4.40942	−0.145453	−0.0727267	−	0.997352i	$-0.523170\pi$
$919$	−4.40942	−0.145453	−0.0727267	+	0.997352i	$0.523170\pi$
$920$	−0.759695	−0.0250464
$921$	11.2356	0.370226
$922$	21.2738	0.700616
$923$	0	0
$924$	3.46527	0.113999
$925$	16.2884	0.535558
$926$	−31.4594	−1.03382
$927$	15.2224	0.499968
$928$	5.86829	0.192636
$929$	−39.3100	−1.28972	−0.644860	−	0.764301i	$-0.723084\pi$
$929$	−39.3100	−1.28972	−0.644860	+	0.764301i	$0.723084\pi$
$930$	−9.55100	−0.313190
$931$	−6.12357	−0.200692
$932$	21.2064	0.694639
$933$	−21.6783	−0.709715
$934$	−33.1449	−1.08453
$935$	21.9837	0.718944
$936$	0	0
$937$	−36.8428	−1.20360	−0.601801	−	0.798646i	$-0.705550\pi$
$937$	−36.8428	−1.20360	−0.601801	+	0.798646i	$0.705550\pi$
$938$	−14.2471	−0.465185
$939$	26.9156	0.878356
$940$	−13.0003	−0.424022
$941$	−44.4101	−1.44773	−0.723864	−	0.689943i	$-0.757636\pi$
$941$	−44.4101	−1.44773	−0.723864	+	0.689943i	$0.757636\pi$
$942$	−19.4192	−0.632711
$943$	2.99030	0.0973774
$944$	−11.5701	−0.376573
$945$	6.51222	0.211843
$946$	−3.24412	−0.105475
$947$	−11.1787	−0.363260	−0.181630	−	0.983367i	$-0.558137\pi$
$947$	−11.1787	−0.363260	−0.181630	+	0.983367i	$0.558137\pi$
$948$	−3.22523	−0.104751
$949$	0	0
$950$	−3.48562	−0.113089
$951$	12.9431	0.419709
$952$	6.24714	0.202471
$953$	−13.1303	−0.425333	−0.212666	−	0.977125i	$-0.568215\pi$
$953$	−13.1303	−0.425333	−0.212666	+	0.977125i	$0.568215\pi$
$954$	2.49689	0.0808397
$955$	−15.7428	−0.509425
$956$	−11.9486	−0.386447
$957$	−21.7215	−0.702156
$958$	12.1358	0.392089
$959$	10.2113	0.329740
$960$	1.70151	0.0549159
$961$	0.508729	0.0164106
$962$	0	0
$963$	1.21085	0.0390190
$964$	9.69837	0.312364
$965$	23.6101	0.760035
$966$	0.799094	0.0257104
$967$	15.4058	0.495416	0.247708	−	0.968835i	$-0.420323\pi$
$967$	15.4058	0.495416	0.247708	+	0.968835i	$0.420323\pi$
$968$	−3.83322	−0.123204
$969$	−9.22653	−0.296399
$970$	4.45118	0.142919
$971$	6.59499	0.211643	0.105822	−	0.994385i	$-0.466253\pi$
$971$	6.59499	0.211643	0.105822	+	0.994385i	$0.466253\pi$
$972$	10.6655	0.342096
$973$	0.847174	0.0271591
$974$	−25.7350	−0.824602
$975$	0	0
$976$	0.485623	0.0155444
$977$	38.3537	1.22704	0.613522	−	0.789678i	$-0.289752\pi$
$977$	38.3537	1.22704	0.613522	+	0.789678i	$0.289752\pi$
$978$	−11.3282	−0.362236
$979$	−48.3651	−1.54575
$980$	−7.53567	−0.240718
$981$	10.8896	0.347677
$982$	−24.8233	−0.792142
$983$	21.2372	0.677360	0.338680	−	0.940902i	$-0.390020\pi$
$983$	21.2372	0.677360	0.338680	+	0.940902i	$0.390020\pi$
$984$	−6.69743	−0.213506
$985$	−11.9749	−0.381552
$986$	−39.1591	−1.24708
$987$	13.6745	0.435263
$988$	0	0
$989$	−0.748095	−0.0237880
$990$	3.58513	0.113943
$991$	−0.474803	−0.0150826	−0.00754130	−	0.999972i	$-0.502400\pi$
$991$	−0.474803	−0.0150826	−0.00754130	+	0.999972i	$0.502400\pi$
$992$	−5.61326	−0.178221
$993$	23.8021	0.755337
$994$	−11.5250	−0.365551
$995$	−0.390130	−0.0123680
$996$	7.93774	0.251517
$997$	−51.3291	−1.62561	−0.812804	−	0.582538i	$-0.802060\pi$
$997$	−51.3291	−1.62561	−0.812804	+	0.582538i	$0.802060\pi$
$998$	−3.63994	−0.115220
$999$	26.4150	0.835732

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bb.1.4		4
13.4	even	6		494.2.g.e.419.1	yes	8
13.10	even	6		494.2.g.e.191.1	✓	8
13.12	even	2		6422.2.a.z.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.e.191.1	✓	8	13.10	even	6
494.2.g.e.419.1	yes	8	13.4	even	6
6422.2.a.z.1.4		4	13.12	even	2
6422.2.a.bb.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.38266 q^{3} +1.00000 q^{4} +1.23060 q^{5} +1.38266 q^{6} -0.936179 q^{7} +1.00000 q^{8} -1.08824 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.38266 q^{3} +1.00000 q^{4} +1.23060 q^{5} +1.38266 q^{6} -0.936179 q^{7} +1.00000 q^{8} -1.08824 q^{9} +1.23060 q^{10} -2.67708 q^{11} +1.38266 q^{12} -0.936179 q^{14} +1.70151 q^{15} +1.00000 q^{16} -6.67301 q^{17} -1.08824 q^{18} +1.00000 q^{19} +1.23060 q^{20} -1.29442 q^{21} -2.67708 q^{22} -0.617337 q^{23} +1.38266 q^{24} -3.48562 q^{25} -5.65266 q^{27} -0.936179 q^{28} +5.86829 q^{29} +1.70151 q^{30} -5.61326 q^{31} +1.00000 q^{32} -3.70151 q^{33} -6.67301 q^{34} -1.15206 q^{35} -1.08824 q^{36} -4.67301 q^{37} +1.00000 q^{38} +1.23060 q^{40} -4.84386 q^{41} -1.29442 q^{42} +1.21181 q^{43} -2.67708 q^{44} -1.33919 q^{45} -0.617337 q^{46} -10.5642 q^{47} +1.38266 q^{48} -6.12357 q^{49} -3.48562 q^{50} -9.22653 q^{51} -2.29442 q^{53} -5.65266 q^{54} -3.29442 q^{55} -0.936179 q^{56} +1.38266 q^{57} +5.86829 q^{58} -11.5701 q^{59} +1.70151 q^{60} +0.485623 q^{61} -5.61326 q^{62} +1.01879 q^{63} +1.00000 q^{64} -3.70151 q^{66} +15.2184 q^{67} -6.67301 q^{68} -0.853569 q^{69} -1.15206 q^{70} +12.3107 q^{71} -1.08824 q^{72} +12.3460 q^{73} -4.67301 q^{74} -4.81944 q^{75} +1.00000 q^{76} +2.50623 q^{77} -2.33262 q^{79} +1.23060 q^{80} -4.55100 q^{81} -4.84386 q^{82} +5.74090 q^{83} -1.29442 q^{84} -8.21181 q^{85} +1.21181 q^{86} +8.11386 q^{87} -2.67708 q^{88} +18.0663 q^{89} -1.33919 q^{90} -0.617337 q^{92} -7.76125 q^{93} -10.5642 q^{94} +1.23060 q^{95} +1.38266 q^{96} +3.61708 q^{97} -6.12357 q^{98} +2.91332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9	\( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} + q^{5} - 2 q^{6} - q^{7} + 4 q^{8} + 2 q^{9} + q^{10} - 2 q^{11} - 2 q^{12} - q^{14} - 11 q^{15} + 4 q^{16} + q^{17} + 2 q^{18} + 4 q^{19} + q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} - 2 q^{24} + 3 q^{25} - 23 q^{27} - q^{28} - q^{29} - 11 q^{30} - 11 q^{31} + 4 q^{32} + 3 q^{33} + q^{34} - q^{35} + 2 q^{36} + 9 q^{37} + 4 q^{38} + q^{40} - 4 q^{41} - 4 q^{42} - 15 q^{43} - 2 q^{44} + 33 q^{45} - 10 q^{46} - 25 q^{47} - 2 q^{48} - 11 q^{49} + 3 q^{50} - 14 q^{51} - 8 q^{53} - 23 q^{54} - 12 q^{55} - q^{56} - 2 q^{57} - q^{58} - 28 q^{59} - 11 q^{60} - 15 q^{61} - 11 q^{62} + 20 q^{63} + 4 q^{64} + 3 q^{66} + q^{68} + 18 q^{69} - q^{70} + q^{71} + 2 q^{72} - 6 q^{73} + 9 q^{74} - 13 q^{75} + 4 q^{76} - 11 q^{77} - 12 q^{79} + q^{80} + 40 q^{81} - 4 q^{82} + 17 q^{83} - 4 q^{84} - 13 q^{85} - 15 q^{86} + 25 q^{87} - 2 q^{88} + 15 q^{89} + 33 q^{90} - 10 q^{92} + 3 q^{93} - 25 q^{94} + q^{95} - 2 q^{96} - 2 q^{97} - 11 q^{98} - 26 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 + q^5 - 2 * q^6 - q^7 + 4 * q^8 + 2 * q^9 + q^10 - 2 * q^11 - 2 * q^12 - q^14 - 11 * q^15 + 4 * q^16 + q^17 + 2 * q^18 + 4 * q^19 + q^20 - 4 * q^21 - 2 * q^22 - 10 * q^23 - 2 * q^24 + 3 * q^25 - 23 * q^27 - q^28 - q^29 - 11 * q^30 - 11 * q^31 + 4 * q^32 + 3 * q^33 + q^34 - q^35 + 2 * q^36 + 9 * q^37 + 4 * q^38 + q^40 - 4 * q^41 - 4 * q^42 - 15 * q^43 - 2 * q^44 + 33 * q^45 - 10 * q^46 - 25 * q^47 - 2 * q^48 - 11 * q^49 + 3 * q^50 - 14 * q^51 - 8 * q^53 - 23 * q^54 - 12 * q^55 - q^56 - 2 * q^57 - q^58 - 28 * q^59 - 11 * q^60 - 15 * q^61 - 11 * q^62 + 20 * q^63 + 4 * q^64 + 3 * q^66 + q^68 + 18 * q^69 - q^70 + q^71 + 2 * q^72 - 6 * q^73 + 9 * q^74 - 13 * q^75 + 4 * q^76 - 11 * q^77 - 12 * q^79 + q^80 + 40 * q^81 - 4 * q^82 + 17 * q^83 - 4 * q^84 - 13 * q^85 - 15 * q^86 + 25 * q^87 - 2 * q^88 + 15 * q^89 + 33 * q^90 - 10 * q^92 + 3 * q^93 - 25 * q^94 + q^95 - 2 * q^96 - 2 * q^97 - 11 * q^98 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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