LMFDB - Embedded newform 6422.2.a.bo.1.12

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + \cdots + 392 $ x^15 - 31x^13 - 4x^12 + 373x^11 + 85x^10 - 2208x^9 - 636x^8 + 6791x^7 + 2005x^6 - 10779x^5 - 2515x^4 + 8205x^3 + 686x^2 - 2352*x + 392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.75888$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.75888 q^{3} +1.00000 q^{4} -3.17158 q^{5} -1.75888 q^{6} -5.12706 q^{7} -1.00000 q^{8} +0.0936587 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.75888 q^{3} +1.00000 q^{4} -3.17158 q^{5} -1.75888 q^{6} -5.12706 q^{7} -1.00000 q^{8} +0.0936587 q^{9} +3.17158 q^{10} +1.69164 q^{11} +1.75888 q^{12} +5.12706 q^{14} -5.57844 q^{15} +1.00000 q^{16} +5.66473 q^{17} -0.0936587 q^{18} +1.00000 q^{19} -3.17158 q^{20} -9.01788 q^{21} -1.69164 q^{22} -6.00680 q^{23} -1.75888 q^{24} +5.05894 q^{25} -5.11191 q^{27} -5.12706 q^{28} +8.22342 q^{29} +5.57844 q^{30} +6.60158 q^{31} -1.00000 q^{32} +2.97540 q^{33} -5.66473 q^{34} +16.2609 q^{35} +0.0936587 q^{36} +6.76618 q^{37} -1.00000 q^{38} +3.17158 q^{40} +9.51097 q^{41} +9.01788 q^{42} +4.69323 q^{43} +1.69164 q^{44} -0.297046 q^{45} +6.00680 q^{46} -4.27974 q^{47} +1.75888 q^{48} +19.2867 q^{49} -5.05894 q^{50} +9.96357 q^{51} -12.7549 q^{53} +5.11191 q^{54} -5.36519 q^{55} +5.12706 q^{56} +1.75888 q^{57} -8.22342 q^{58} -1.43278 q^{59} -5.57844 q^{60} -9.84276 q^{61} -6.60158 q^{62} -0.480193 q^{63} +1.00000 q^{64} -2.97540 q^{66} -9.48923 q^{67} +5.66473 q^{68} -10.5652 q^{69} -16.2609 q^{70} +4.44702 q^{71} -0.0936587 q^{72} -7.63745 q^{73} -6.76618 q^{74} +8.89807 q^{75} +1.00000 q^{76} -8.67316 q^{77} +3.47518 q^{79} -3.17158 q^{80} -9.27220 q^{81} -9.51097 q^{82} +11.0932 q^{83} -9.01788 q^{84} -17.9662 q^{85} -4.69323 q^{86} +14.4640 q^{87} -1.69164 q^{88} -15.8716 q^{89} +0.297046 q^{90} -6.00680 q^{92} +11.6114 q^{93} +4.27974 q^{94} -3.17158 q^{95} -1.75888 q^{96} +9.57367 q^{97} -19.2867 q^{98} +0.158437 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9	$ 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9} + q^{10} - 4 q^{11} + 18 q^{14} - 23 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} + 15 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} - 18 q^{28} - 20 q^{29} + 23 q^{30} - 30 q^{31} - 15 q^{32} - 36 q^{33} - 2 q^{34} + 32 q^{35} + 17 q^{36} - 35 q^{37} - 15 q^{38} + q^{40} - 15 q^{41} + 2 q^{42} + q^{43} - 4 q^{44} + 11 q^{45} - 17 q^{46} + 29 q^{49} - 8 q^{50} - q^{51} - q^{53} - 12 q^{54} - 6 q^{55} + 18 q^{56} + 20 q^{58} + 7 q^{59} - 23 q^{60} - 2 q^{61} + 30 q^{62} - 42 q^{63} + 15 q^{64} + 36 q^{66} - 34 q^{67} + 2 q^{68} - 12 q^{69} - 32 q^{70} - 4 q^{71} - 17 q^{72} - 12 q^{73} + 35 q^{74} + 31 q^{75} + 15 q^{76} - 20 q^{77} + 23 q^{79} - q^{80} + 7 q^{81} + 15 q^{82} + 3 q^{83} - 2 q^{84} - 46 q^{85} - q^{86} + 22 q^{87} + 4 q^{88} - 17 q^{89} - 11 q^{90} + 17 q^{92} - 60 q^{93} - q^{95} - 18 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) $ 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9 + q^10 - 4 * q^11 + 18 * q^14 - 23 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 + 15 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 - 18 * q^28 - 20 * q^29 + 23 * q^30 - 30 * q^31 - 15 * q^32 - 36 * q^33 - 2 * q^34 + 32 * q^35 + 17 * q^36 - 35 * q^37 - 15 * q^38 + q^40 - 15 * q^41 + 2 * q^42 + q^43 - 4 * q^44 + 11 * q^45 - 17 * q^46 + 29 * q^49 - 8 * q^50 - q^51 - q^53 - 12 * q^54 - 6 * q^55 + 18 * q^56 + 20 * q^58 + 7 * q^59 - 23 * q^60 - 2 * q^61 + 30 * q^62 - 42 * q^63 + 15 * q^64 + 36 * q^66 - 34 * q^67 + 2 * q^68 - 12 * q^69 - 32 * q^70 - 4 * q^71 - 17 * q^72 - 12 * q^73 + 35 * q^74 + 31 * q^75 + 15 * q^76 - 20 * q^77 + 23 * q^79 - q^80 + 7 * q^81 + 15 * q^82 + 3 * q^83 - 2 * q^84 - 46 * q^85 - q^86 + 22 * q^87 + 4 * q^88 - 17 * q^89 - 11 * q^90 + 17 * q^92 - 60 * q^93 - q^95 - 18 * q^97 - 29 * q^98 + 6 * 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
$3$	1.75888	1.01549	0.507745	−	0.861507i	$-0.330479\pi$
$3$	1.75888	1.01549	0.507745	+	0.861507i	$0.330479\pi$
$4$	1.00000	0.500000
$5$	−3.17158	−1.41838	−0.709188	−	0.705020i	$-0.750938\pi$
$5$	−3.17158	−1.41838	−0.709188	+	0.705020i	$0.750938\pi$
$6$	−1.75888	−0.718060
$7$	−5.12706	−1.93785	−0.968923	−	0.247363i	$-0.920436\pi$
$7$	−5.12706	−1.93785	−0.968923	+	0.247363i	$0.920436\pi$
$8$	−1.00000	−0.353553
$9$	0.0936587	0.0312196
$10$	3.17158	1.00294
$11$	1.69164	0.510050	0.255025	−	0.966934i	$-0.417916\pi$
$11$	1.69164	0.510050	0.255025	+	0.966934i	$0.417916\pi$
$12$	1.75888	0.507745
$13$	0	0
$13$	0	0
$14$	5.12706	1.37026
$15$	−5.57844	−1.44035
$16$	1.00000	0.250000
$17$	5.66473	1.37390	0.686949	−	0.726706i	$-0.258950\pi$
$17$	5.66473	1.37390	0.686949	+	0.726706i	$0.258950\pi$
$18$	−0.0936587	−0.0220756
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.17158	−0.709188
$21$	−9.01788	−1.96786
$22$	−1.69164	−0.360660
$23$	−6.00680	−1.25250	−0.626252	−	0.779621i	$-0.715412\pi$
$23$	−6.00680	−1.25250	−0.626252	+	0.779621i	$0.715412\pi$
$24$	−1.75888	−0.359030
$25$	5.05894	1.01179
$26$	0	0
$27$	−5.11191	−0.983787
$28$	−5.12706	−0.968923
$29$	8.22342	1.52705	0.763525	−	0.645778i	$-0.223467\pi$
$29$	8.22342	1.52705	0.763525	+	0.645778i	$0.223467\pi$
$30$	5.57844	1.01848
$31$	6.60158	1.18568	0.592839	−	0.805321i	$-0.298007\pi$
$31$	6.60158	1.18568	0.592839	+	0.805321i	$0.298007\pi$
$32$	−1.00000	−0.176777
$33$	2.97540	0.517950
$34$	−5.66473	−0.971492
$35$	16.2609	2.74859
$36$	0.0936587	0.0156098
$37$	6.76618	1.11235	0.556177	−	0.831064i	$-0.312268\pi$
$37$	6.76618	1.11235	0.556177	+	0.831064i	$0.312268\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	3.17158	0.501471
$41$	9.51097	1.48536	0.742682	−	0.669645i	$-0.233554\pi$
$41$	9.51097	1.48536	0.742682	+	0.669645i	$0.233554\pi$
$42$	9.01788	1.39149
$43$	4.69323	0.715711	0.357856	−	0.933777i	$-0.383508\pi$
$43$	4.69323	0.715711	0.357856	+	0.933777i	$0.383508\pi$
$44$	1.69164	0.255025
$45$	−0.297046	−0.0442811
$46$	6.00680	0.885654
$47$	−4.27974	−0.624264	−0.312132	−	0.950039i	$-0.601043\pi$
$47$	−4.27974	−0.624264	−0.312132	+	0.950039i	$0.601043\pi$
$48$	1.75888	0.253872
$49$	19.2867	2.75525
$50$	−5.05894	−0.715443
$51$	9.96357	1.39518
$52$	0	0
$53$	−12.7549	−1.75202	−0.876011	−	0.482291i	$-0.839805\pi$
$53$	−12.7549	−1.75202	−0.876011	+	0.482291i	$0.839805\pi$
$54$	5.11191	0.695642
$55$	−5.36519	−0.723442
$56$	5.12706	0.685132
$57$	1.75888	0.232969
$58$	−8.22342	−1.07979
$59$	−1.43278	−0.186532	−0.0932659	−	0.995641i	$-0.529731\pi$
$59$	−1.43278	−0.186532	−0.0932659	+	0.995641i	$0.529731\pi$
$60$	−5.57844	−0.720173
$61$	−9.84276	−1.26024	−0.630118	−	0.776499i	$-0.716994\pi$
$61$	−9.84276	−1.26024	−0.630118	+	0.776499i	$0.716994\pi$
$62$	−6.60158	−0.838401
$63$	−0.480193	−0.0604987
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.97540	−0.366246
$67$	−9.48923	−1.15929	−0.579647	−	0.814868i	$-0.696809\pi$
$67$	−9.48923	−1.15929	−0.579647	+	0.814868i	$0.696809\pi$
$68$	5.66473	0.686949
$69$	−10.5652	−1.27191
$70$	−16.2609	−1.94355
$71$	4.44702	0.527764	0.263882	−	0.964555i	$-0.414997\pi$
$71$	4.44702	0.527764	0.263882	+	0.964555i	$0.414997\pi$
$72$	−0.0936587	−0.0110378
$73$	−7.63745	−0.893896	−0.446948	−	0.894560i	$-0.647489\pi$
$73$	−7.63745	−0.893896	−0.446948	+	0.894560i	$0.647489\pi$
$74$	−6.76618	−0.786553
$75$	8.89807	1.02746
$76$	1.00000	0.114708
$77$	−8.67316	−0.988398
$78$	0	0
$79$	3.47518	0.390989	0.195494	−	0.980705i	$-0.437369\pi$
$79$	3.47518	0.390989	0.195494	+	0.980705i	$0.437369\pi$
$80$	−3.17158	−0.354594
$81$	−9.27220	−1.03024
$82$	−9.51097	−1.05031
$83$	11.0932	1.21763	0.608816	−	0.793311i	$-0.291645\pi$
$83$	11.0932	1.21763	0.608816	+	0.793311i	$0.291645\pi$
$84$	−9.01788	−0.983931
$85$	−17.9662	−1.94870
$86$	−4.69323	−0.506084
$87$	14.4640	1.55070
$88$	−1.69164	−0.180330
$89$	−15.8716	−1.68239	−0.841195	−	0.540732i	$-0.818147\pi$
$89$	−15.8716	−1.68239	−0.841195	+	0.540732i	$0.818147\pi$
$90$	0.297046	0.0313114
$91$	0	0
$92$	−6.00680	−0.626252
$93$	11.6114	1.20404
$94$	4.27974	0.441421
$95$	−3.17158	−0.325398
$96$	−1.75888	−0.179515
$97$	9.57367	0.972059	0.486030	−	0.873942i	$-0.338445\pi$
$97$	9.57367	0.972059	0.486030	+	0.873942i	$0.338445\pi$
$98$	−19.2867	−1.94825
$99$	0.158437	0.0159235
$100$	5.05894	0.505894
$101$	−13.5457	−1.34785	−0.673926	−	0.738799i	$-0.735393\pi$
$101$	−13.5457	−1.34785	−0.673926	+	0.738799i	$0.735393\pi$
$102$	−9.96357	−0.986541
$103$	−8.59368	−0.846760	−0.423380	−	0.905952i	$-0.639156\pi$
$103$	−8.59368	−0.846760	−0.423380	+	0.905952i	$0.639156\pi$
$104$	0	0
$105$	28.6010	2.79117
$106$	12.7549	1.23887
$107$	5.22430	0.505052	0.252526	−	0.967590i	$-0.418739\pi$
$107$	5.22430	0.505052	0.252526	+	0.967590i	$0.418739\pi$
$108$	−5.11191	−0.491893
$109$	−2.63385	−0.252277	−0.126139	−	0.992013i	$-0.540258\pi$
$109$	−2.63385	−0.252277	−0.126139	+	0.992013i	$0.540258\pi$
$110$	5.36519	0.511551
$111$	11.9009	1.12958
$112$	−5.12706	−0.484461
$113$	0.168875	0.0158864	0.00794321	−	0.999968i	$-0.497472\pi$
$113$	0.168875	0.0158864	0.00794321	+	0.999968i	$0.497472\pi$
$114$	−1.75888	−0.164734
$115$	19.0511	1.77652
$116$	8.22342	0.763525
$117$	0	0
$118$	1.43278	0.131898
$119$	−29.0434	−2.66240
$120$	5.57844	0.509239
$121$	−8.13834	−0.739849
$122$	9.84276	0.891122
$123$	16.7286	1.50837
$124$	6.60158	0.592839
$125$	−0.186944	−0.0167207
$126$	0.480193	0.0427790
$127$	2.44334	0.216812	0.108406	−	0.994107i	$-0.465425\pi$
$127$	2.44334	0.216812	0.108406	+	0.994107i	$0.465425\pi$
$128$	−1.00000	−0.0883883
$129$	8.25483	0.726797
$130$	0	0
$131$	−0.295598	−0.0258265	−0.0129132	−	0.999917i	$-0.504111\pi$
$131$	−0.295598	−0.0258265	−0.0129132	+	0.999917i	$0.504111\pi$
$132$	2.97540	0.258975
$133$	−5.12706	−0.444572
$134$	9.48923	0.819744
$135$	16.2128	1.39538
$136$	−5.66473	−0.485746
$137$	−13.6298	−1.16447	−0.582235	−	0.813021i	$-0.697822\pi$
$137$	−13.6298	−1.16447	−0.582235	+	0.813021i	$0.697822\pi$
$138$	10.5652	0.899373
$139$	2.41868	0.205150	0.102575	−	0.994725i	$-0.467292\pi$
$139$	2.41868	0.205150	0.102575	+	0.994725i	$0.467292\pi$
$140$	16.2609	1.37430
$141$	−7.52754	−0.633933
$142$	−4.44702	−0.373186
$143$	0	0
$144$	0.0936587	0.00780489
$145$	−26.0813	−2.16593
$146$	7.63745	0.632080
$147$	33.9230	2.79792
$148$	6.76618	0.556177
$149$	18.9769	1.55465	0.777325	−	0.629100i	$-0.216576\pi$
$149$	18.9769	1.55465	0.777325	+	0.629100i	$0.216576\pi$
$150$	−8.89807	−0.726525
$151$	9.28558	0.755649	0.377825	−	0.925877i	$-0.376672\pi$
$151$	9.28558	0.755649	0.377825	+	0.925877i	$0.376672\pi$
$152$	−1.00000	−0.0811107
$153$	0.530551	0.0428925
$154$	8.67316	0.698903
$155$	−20.9375	−1.68174
$156$	0	0
$157$	−12.2130	−0.974704	−0.487352	−	0.873206i	$-0.662037\pi$
$157$	−12.2130	−0.974704	−0.487352	+	0.873206i	$0.662037\pi$
$158$	−3.47518	−0.276471
$159$	−22.4344	−1.77916
$160$	3.17158	0.250736
$161$	30.7972	2.42716
$162$	9.27220	0.728493
$163$	−20.2182	−1.58362	−0.791808	−	0.610771i	$-0.790860\pi$
$163$	−20.2182	−1.58362	−0.791808	+	0.610771i	$0.790860\pi$
$164$	9.51097	0.742682
$165$	−9.43673	−0.734648
$166$	−11.0932	−0.860996
$167$	−9.96580	−0.771177	−0.385589	−	0.922671i	$-0.626002\pi$
$167$	−9.96580	−0.771177	−0.385589	+	0.922671i	$0.626002\pi$
$168$	9.01788	0.695744
$169$	0	0
$170$	17.9662	1.37794
$171$	0.0936587	0.00716226
$172$	4.69323	0.357856
$173$	3.98084	0.302658	0.151329	−	0.988483i	$-0.451645\pi$
$173$	3.98084	0.302658	0.151329	+	0.988483i	$0.451645\pi$
$174$	−14.4640	−1.09651
$175$	−25.9375	−1.96069
$176$	1.69164	0.127512
$177$	−2.52009	−0.189421
$178$	15.8716	1.18963
$179$	10.6606	0.796814	0.398407	−	0.917209i	$-0.369563\pi$
$179$	10.6606	0.796814	0.398407	+	0.917209i	$0.369563\pi$
$180$	−0.297046	−0.0221405
$181$	−17.8056	−1.32348	−0.661740	−	0.749734i	$-0.730182\pi$
$181$	−17.8056	−1.32348	−0.661740	+	0.749734i	$0.730182\pi$
$182$	0	0
$183$	−17.3122	−1.27976
$184$	6.00680	0.442827
$185$	−21.4595	−1.57773
$186$	−11.6114	−0.851388
$187$	9.58270	0.700756
$188$	−4.27974	−0.312132
$189$	26.2090	1.90643
$190$	3.17158	0.230091
$191$	12.5096	0.905164	0.452582	−	0.891723i	$-0.350503\pi$
$191$	12.5096	0.905164	0.452582	+	0.891723i	$0.350503\pi$
$192$	1.75888	0.126936
$193$	−18.1300	−1.30503	−0.652514	−	0.757777i	$-0.726286\pi$
$193$	−18.1300	−1.30503	−0.652514	+	0.757777i	$0.726286\pi$
$194$	−9.57367	−0.687350
$195$	0	0
$196$	19.2867	1.37762
$197$	−21.5634	−1.53633	−0.768163	−	0.640254i	$-0.778829\pi$
$197$	−21.5634	−1.53633	−0.768163	+	0.640254i	$0.778829\pi$
$198$	−0.158437	−0.0112596
$199$	6.11442	0.433440	0.216720	−	0.976234i	$-0.430464\pi$
$199$	6.11442	0.433440	0.216720	+	0.976234i	$0.430464\pi$
$200$	−5.05894	−0.357721
$201$	−16.6904	−1.17725
$202$	13.5457	0.953075
$203$	−42.1619	−2.95919
$204$	9.96357	0.697590
$205$	−30.1648	−2.10680
$206$	8.59368	0.598750
$207$	−0.562589	−0.0391026
$208$	0	0
$209$	1.69164	0.117013
$210$	−28.6010	−1.97365
$211$	−0.703588	−0.0484370	−0.0242185	−	0.999707i	$-0.507710\pi$
$211$	−0.703588	−0.0484370	−0.0242185	+	0.999707i	$0.507710\pi$
$212$	−12.7549	−0.876011
$213$	7.82178	0.535939
$214$	−5.22430	−0.357126
$215$	−14.8850	−1.01515
$216$	5.11191	0.347821
$217$	−33.8467	−2.29766
$218$	2.63385	0.178387
$219$	−13.4334	−0.907742
$220$	−5.36519	−0.361721
$221$	0	0
$222$	−11.9009	−0.798736
$223$	12.8201	0.858497	0.429248	−	0.903186i	$-0.358778\pi$
$223$	12.8201	0.858497	0.429248	+	0.903186i	$0.358778\pi$
$224$	5.12706	0.342566
$225$	0.473814	0.0315876
$226$	−0.168875	−0.0112334
$227$	1.87315	0.124325	0.0621626	−	0.998066i	$-0.480200\pi$
$227$	1.87315	0.124325	0.0621626	+	0.998066i	$0.480200\pi$
$228$	1.75888	0.116485
$229$	−2.22829	−0.147249	−0.0736247	−	0.997286i	$-0.523457\pi$
$229$	−2.22829	−0.147249	−0.0736247	+	0.997286i	$0.523457\pi$
$230$	−19.0511	−1.25619
$231$	−15.2550	−1.00371
$232$	−8.22342	−0.539894
$233$	−10.2432	−0.671055	−0.335527	−	0.942030i	$-0.608914\pi$
$233$	−10.2432	−0.671055	−0.335527	+	0.942030i	$0.608914\pi$
$234$	0	0
$235$	13.5735	0.885440
$236$	−1.43278	−0.0932659
$237$	6.11243	0.397045
$238$	29.0434	1.88260
$239$	−7.78009	−0.503252	−0.251626	−	0.967825i	$-0.580965\pi$
$239$	−7.78009	−0.503252	−0.251626	+	0.967825i	$0.580965\pi$
$240$	−5.57844	−0.360086
$241$	7.45240	0.480052	0.240026	−	0.970767i	$-0.422844\pi$
$241$	7.45240	0.480052	0.240026	+	0.970767i	$0.422844\pi$
$242$	8.13834	0.523152
$243$	−0.972978	−0.0624165
$244$	−9.84276	−0.630118
$245$	−61.1694	−3.90797
$246$	−16.7286	−1.06658
$247$	0	0
$248$	−6.60158	−0.419201
$249$	19.5115	1.23649
$250$	0.186944	0.0118233
$251$	10.5077	0.663239	0.331619	−	0.943413i	$-0.392405\pi$
$251$	10.5077	0.663239	0.331619	+	0.943413i	$0.392405\pi$
$252$	−0.480193	−0.0302493
$253$	−10.1614	−0.638840
$254$	−2.44334	−0.153309
$255$	−31.6003	−1.97889
$256$	1.00000	0.0625000
$257$	2.84632	0.177549	0.0887744	−	0.996052i	$-0.471705\pi$
$257$	2.84632	0.177549	0.0887744	+	0.996052i	$0.471705\pi$
$258$	−8.25483	−0.513923
$259$	−34.6906	−2.15557
$260$	0	0
$261$	0.770194	0.0476738
$262$	0.295598	0.0182621
$263$	−10.2565	−0.632441	−0.316221	−	0.948686i	$-0.602414\pi$
$263$	−10.2565	−0.632441	−0.316221	+	0.948686i	$0.602414\pi$
$264$	−2.97540	−0.183123
$265$	40.4533	2.48503
$266$	5.12706	0.314360
$267$	−27.9163	−1.70845
$268$	−9.48923	−0.579647
$269$	−23.0933	−1.40802	−0.704011	−	0.710189i	$-0.748609\pi$
$269$	−23.0933	−1.40802	−0.704011	+	0.710189i	$0.748609\pi$
$270$	−16.2128	−0.986682
$271$	−10.6214	−0.645204	−0.322602	−	0.946535i	$-0.604558\pi$
$271$	−10.6214	−0.645204	−0.322602	+	0.946535i	$0.604558\pi$
$272$	5.66473	0.343474
$273$	0	0
$274$	13.6298	0.823404
$275$	8.55793	0.516063
$276$	−10.5652	−0.635953
$277$	1.04905	0.0630314	0.0315157	−	0.999503i	$-0.489967\pi$
$277$	1.04905	0.0630314	0.0315157	+	0.999503i	$0.489967\pi$
$278$	−2.41868	−0.145063
$279$	0.618295	0.0370164
$280$	−16.2609	−0.971774
$281$	17.3656	1.03594	0.517972	−	0.855397i	$-0.326687\pi$
$281$	17.3656	1.03594	0.517972	+	0.855397i	$0.326687\pi$
$282$	7.52754	0.448258
$283$	−10.9067	−0.648338	−0.324169	−	0.945999i	$-0.605085\pi$
$283$	−10.9067	−0.648338	−0.324169	+	0.945999i	$0.605085\pi$
$284$	4.44702	0.263882
$285$	−5.57844	−0.330438
$286$	0	0
$287$	−48.7633	−2.87841
$288$	−0.0936587	−0.00551889
$289$	15.0891	0.887595
$290$	26.0813	1.53154
$291$	16.8389	0.987116
$292$	−7.63745	−0.446948
$293$	20.0576	1.17178	0.585889	−	0.810391i	$-0.300745\pi$
$293$	20.0576	1.17178	0.585889	+	0.810391i	$0.300745\pi$
$294$	−33.9230	−1.97843
$295$	4.54418	0.264572
$296$	−6.76618	−0.393276
$297$	−8.64752	−0.501780
$298$	−18.9769	−1.09930
$299$	0	0
$300$	8.89807	0.513731
$301$	−24.0625	−1.38694
$302$	−9.28558	−0.534325
$303$	−23.8253	−1.36873
$304$	1.00000	0.0573539
$305$	31.2171	1.78749
$306$	−0.530551	−0.0303296
$307$	22.7077	1.29600	0.648000	−	0.761641i	$-0.275606\pi$
$307$	22.7077	1.29600	0.648000	+	0.761641i	$0.275606\pi$
$308$	−8.67316	−0.494199
$309$	−15.1152	−0.859876
$310$	20.9375	1.18917
$311$	11.3189	0.641835	0.320917	−	0.947107i	$-0.396009\pi$
$311$	11.3189	0.641835	0.320917	+	0.947107i	$0.396009\pi$
$312$	0	0
$313$	5.69862	0.322105	0.161053	−	0.986946i	$-0.448511\pi$
$313$	5.69862	0.322105	0.161053	+	0.986946i	$0.448511\pi$
$314$	12.2130	0.689220
$315$	1.52297	0.0858099
$316$	3.47518	0.195494
$317$	−22.9385	−1.28836	−0.644178	−	0.764876i	$-0.722800\pi$
$317$	−22.9385	−1.28836	−0.644178	+	0.764876i	$0.722800\pi$
$318$	22.4344	1.25806
$319$	13.9111	0.778872
$320$	−3.17158	−0.177297
$321$	9.18891	0.512875
$322$	−30.7972	−1.71626
$323$	5.66473	0.315194
$324$	−9.27220	−0.515122
$325$	0	0
$326$	20.2182	1.11979
$327$	−4.63263	−0.256185
$328$	−9.51097	−0.525155
$329$	21.9424	1.20973
$330$	9.43673	0.519475
$331$	−8.44810	−0.464349	−0.232175	−	0.972674i	$-0.574584\pi$
$331$	−8.44810	−0.464349	−0.232175	+	0.972674i	$0.574584\pi$
$332$	11.0932	0.608816
$333$	0.633712	0.0347272
$334$	9.96580	0.545305
$335$	30.0959	1.64431
$336$	−9.01788	−0.491966
$337$	8.19053	0.446166	0.223083	−	0.974799i	$-0.428388\pi$
$337$	8.19053	0.446166	0.223083	+	0.974799i	$0.428388\pi$
$338$	0	0
$339$	0.297031	0.0161325
$340$	−17.9662	−0.974351
$341$	11.1675	0.604755
$342$	−0.0936587	−0.00506448
$343$	−62.9947	−3.40139
$344$	−4.69323	−0.253042
$345$	33.5085	1.80404
$346$	−3.98084	−0.214011
$347$	−3.31206	−0.177801	−0.0889003	−	0.996041i	$-0.528335\pi$
$347$	−3.31206	−0.177801	−0.0889003	+	0.996041i	$0.528335\pi$
$348$	14.4640	0.775352
$349$	−28.5146	−1.52635	−0.763175	−	0.646192i	$-0.776361\pi$
$349$	−28.5146	−1.52635	−0.763175	+	0.646192i	$0.776361\pi$
$350$	25.9375	1.38642
$351$	0	0
$352$	−1.69164	−0.0901649
$353$	11.6407	0.619574	0.309787	−	0.950806i	$-0.399742\pi$
$353$	11.6407	0.619574	0.309787	+	0.950806i	$0.399742\pi$
$354$	2.52009	0.133941
$355$	−14.1041	−0.748568
$356$	−15.8716	−0.841195
$357$	−51.0838	−2.70364
$358$	−10.6606	−0.563433
$359$	31.9255	1.68497	0.842483	−	0.538724i	$-0.181093\pi$
$359$	31.9255	1.68497	0.842483	+	0.538724i	$0.181093\pi$
$360$	0.297046	0.0156557
$361$	1.00000	0.0526316
$362$	17.8056	0.935841
$363$	−14.3144	−0.751309
$364$	0	0
$365$	24.2228	1.26788
$366$	17.3122	0.904925
$367$	−12.9511	−0.676043	−0.338022	−	0.941138i	$-0.609758\pi$
$367$	−12.9511	−0.676043	−0.338022	+	0.941138i	$0.609758\pi$
$368$	−6.00680	−0.313126
$369$	0.890785	0.0463724
$370$	21.4595	1.11563
$371$	65.3952	3.39515
$372$	11.6114	0.602022
$373$	11.3513	0.587750	0.293875	−	0.955844i	$-0.405055\pi$
$373$	11.3513	0.587750	0.293875	+	0.955844i	$0.405055\pi$
$374$	−9.58270	−0.495510
$375$	−0.328811	−0.0169797
$376$	4.27974	0.220710
$377$	0	0
$378$	−26.2090	−1.34805
$379$	6.88201	0.353505	0.176752	−	0.984255i	$-0.443441\pi$
$379$	6.88201	0.353505	0.176752	+	0.984255i	$0.443441\pi$
$380$	−3.17158	−0.162699
$381$	4.29755	0.220170
$382$	−12.5096	−0.640047
$383$	16.5966	0.848049	0.424024	−	0.905651i	$-0.360617\pi$
$383$	16.5966	0.848049	0.424024	+	0.905651i	$0.360617\pi$
$384$	−1.75888	−0.0897575
$385$	27.5076	1.40192
$386$	18.1300	0.922794
$387$	0.439562	0.0223442
$388$	9.57367	0.486030
$389$	−23.9314	−1.21337	−0.606686	−	0.794941i	$-0.707502\pi$
$389$	−23.9314	−1.21337	−0.606686	+	0.794941i	$0.707502\pi$
$390$	0	0
$391$	−34.0269	−1.72081
$392$	−19.2867	−0.974126
$393$	−0.519921	−0.0262265
$394$	21.5634	1.08635
$395$	−11.0218	−0.554569
$396$	0.158437	0.00796177
$397$	−18.4339	−0.925171	−0.462586	−	0.886575i	$-0.653078\pi$
$397$	−18.4339	−0.925171	−0.462586	+	0.886575i	$0.653078\pi$
$398$	−6.11442	−0.306488
$399$	−9.01788	−0.451459
$400$	5.05894	0.252947
$401$	7.52372	0.375717	0.187858	−	0.982196i	$-0.439845\pi$
$401$	7.52372	0.375717	0.187858	+	0.982196i	$0.439845\pi$
$402$	16.6904	0.832442
$403$	0	0
$404$	−13.5457	−0.673926
$405$	29.4076	1.46127
$406$	42.1619	2.09246
$407$	11.4460	0.567356
$408$	−9.96357	−0.493270
$409$	−16.3972	−0.810791	−0.405395	−	0.914141i	$-0.632866\pi$
$409$	−16.3972	−0.810791	−0.405395	+	0.914141i	$0.632866\pi$
$410$	30.1648	1.48973
$411$	−23.9731	−1.18251
$412$	−8.59368	−0.423380
$413$	7.34594	0.361470
$414$	0.562589	0.0276497
$415$	−35.1829	−1.72706
$416$	0	0
$417$	4.25417	0.208328
$418$	−1.69164	−0.0827410
$419$	−14.4806	−0.707424	−0.353712	−	0.935354i	$-0.615081\pi$
$419$	−14.4806	−0.707424	−0.353712	+	0.935354i	$0.615081\pi$
$420$	28.6010	1.39558
$421$	−2.50015	−0.121850	−0.0609248	−	0.998142i	$-0.519405\pi$
$421$	−2.50015	−0.121850	−0.0609248	+	0.998142i	$0.519405\pi$
$422$	0.703588	0.0342501
$423$	−0.400834	−0.0194892
$424$	12.7549	0.619433
$425$	28.6575	1.39009
$426$	−7.82178	−0.378966
$427$	50.4644	2.44214
$428$	5.22430	0.252526
$429$	0	0
$430$	14.8850	0.717817
$431$	12.6330	0.608511	0.304255	−	0.952590i	$-0.401592\pi$
$431$	12.6330	0.608511	0.304255	+	0.952590i	$0.401592\pi$
$432$	−5.11191	−0.245947
$433$	16.5242	0.794100	0.397050	−	0.917797i	$-0.370034\pi$
$433$	16.5242	0.794100	0.397050	+	0.917797i	$0.370034\pi$
$434$	33.8467	1.62469
$435$	−45.8738	−2.19948
$436$	−2.63385	−0.126139
$437$	−6.00680	−0.287344
$438$	13.4334	0.641871
$439$	−18.4009	−0.878226	−0.439113	−	0.898432i	$-0.644707\pi$
$439$	−18.4009	−0.878226	−0.439113	+	0.898432i	$0.644707\pi$
$440$	5.36519	0.255775
$441$	1.80637	0.0860176
$442$	0	0
$443$	0.483841	0.0229880	0.0114940	−	0.999934i	$-0.496341\pi$
$443$	0.483841	0.0229880	0.0114940	+	0.999934i	$0.496341\pi$
$444$	11.9009	0.564792
$445$	50.3382	2.38626
$446$	−12.8201	−0.607049
$447$	33.3781	1.57873
$448$	−5.12706	−0.242231
$449$	−33.7865	−1.59448	−0.797241	−	0.603661i	$-0.793708\pi$
$449$	−33.7865	−1.59448	−0.797241	+	0.603661i	$0.793708\pi$
$450$	−0.473814	−0.0223358
$451$	16.0892	0.757609
$452$	0.168875	0.00794321
$453$	16.3322	0.767354
$454$	−1.87315	−0.0879112
$455$	0	0
$456$	−1.75888	−0.0823671
$457$	0.891560	0.0417054	0.0208527	−	0.999783i	$-0.493362\pi$
$457$	0.891560	0.0417054	0.0208527	+	0.999783i	$0.493362\pi$
$458$	2.22829	0.104121
$459$	−28.9575	−1.35162
$460$	19.0511	0.888261
$461$	3.48758	0.162433	0.0812165	−	0.996696i	$-0.474119\pi$
$461$	3.48758	0.162433	0.0812165	+	0.996696i	$0.474119\pi$
$462$	15.2550	0.709729
$463$	−6.81285	−0.316620	−0.158310	−	0.987389i	$-0.550605\pi$
$463$	−6.81285	−0.316620	−0.158310	+	0.987389i	$0.550605\pi$
$464$	8.22342	0.381763
$465$	−36.8265	−1.70779
$466$	10.2432	0.474507
$467$	−8.96725	−0.414955	−0.207477	−	0.978240i	$-0.566525\pi$
$467$	−8.96725	−0.414955	−0.207477	+	0.978240i	$0.566525\pi$
$468$	0	0
$469$	48.6518	2.24653
$470$	−13.5735	−0.626101
$471$	−21.4812	−0.989802
$472$	1.43278	0.0659490
$473$	7.93928	0.365048
$474$	−6.11243	−0.280753
$475$	5.05894	0.232120
$476$	−29.0434	−1.33120
$477$	−1.19461	−0.0546974
$478$	7.78009	0.355853
$479$	−26.6153	−1.21609	−0.608043	−	0.793904i	$-0.708045\pi$
$479$	−26.6153	−1.21609	−0.608043	+	0.793904i	$0.708045\pi$
$480$	5.57844	0.254620
$481$	0	0
$482$	−7.45240	−0.339448
$483$	54.1686	2.46476
$484$	−8.13834	−0.369925
$485$	−30.3637	−1.37874
$486$	0.972978	0.0441352
$487$	−26.6839	−1.20916	−0.604581	−	0.796544i	$-0.706659\pi$
$487$	−26.6839	−1.20916	−0.604581	+	0.796544i	$0.706659\pi$
$488$	9.84276	0.445561
$489$	−35.5615	−1.60815
$490$	61.1694	2.76335
$491$	27.3317	1.23346	0.616730	−	0.787175i	$-0.288457\pi$
$491$	27.3317	1.23346	0.616730	+	0.787175i	$0.288457\pi$
$492$	16.7286	0.754186
$493$	46.5834	2.09801
$494$	0	0
$495$	−0.502497	−0.0225855
$496$	6.60158	0.296420
$497$	−22.8001	−1.02273
$498$	−19.5115	−0.874333
$499$	−37.5106	−1.67921	−0.839603	−	0.543200i	$-0.817212\pi$
$499$	−37.5106	−1.67921	−0.839603	+	0.543200i	$0.817212\pi$
$500$	−0.186944	−0.00836037
$501$	−17.5287	−0.783123
$502$	−10.5077	−0.468981
$503$	11.5430	0.514679	0.257340	−	0.966321i	$-0.417154\pi$
$503$	11.5430	0.514679	0.257340	+	0.966321i	$0.417154\pi$
$504$	0.480193	0.0213895
$505$	42.9614	1.91176
$506$	10.1614	0.451728
$507$	0	0
$508$	2.44334	0.108406
$509$	24.2819	1.07628	0.538138	−	0.842857i	$-0.319128\pi$
$509$	24.2819	1.07628	0.538138	+	0.842857i	$0.319128\pi$
$510$	31.6003	1.39928
$511$	39.1577	1.73223
$512$	−1.00000	−0.0441942
$513$	−5.11191	−0.225696
$514$	−2.84632	−0.125546
$515$	27.2556	1.20102
$516$	8.25483	0.363399
$517$	−7.23979	−0.318406
$518$	34.6906	1.52422
$519$	7.00183	0.307346
$520$	0	0
$521$	11.1860	0.490067	0.245034	−	0.969515i	$-0.421201\pi$
$521$	11.1860	0.490067	0.245034	+	0.969515i	$0.421201\pi$
$522$	−0.770194	−0.0337105
$523$	−10.2024	−0.446119	−0.223060	−	0.974805i	$-0.571605\pi$
$523$	−10.2024	−0.446119	−0.223060	+	0.974805i	$0.571605\pi$
$524$	−0.295598	−0.0129132
$525$	−45.6209	−1.99106
$526$	10.2565	0.447204
$527$	37.3961	1.62900
$528$	2.97540	0.129488
$529$	13.0817	0.568767
$530$	−40.4533	−1.75718
$531$	−0.134192	−0.00582344
$532$	−5.12706	−0.222286
$533$	0	0
$534$	27.9163	1.20806
$535$	−16.5693	−0.716353
$536$	9.48923	0.409872
$537$	18.7508	0.809157
$538$	23.0933	0.995622
$539$	32.6263	1.40531
$540$	16.2128	0.697689
$541$	−32.2708	−1.38743	−0.693715	−	0.720250i	$-0.744027\pi$
$541$	−32.2708	−1.38743	−0.693715	+	0.720250i	$0.744027\pi$
$542$	10.6214	0.456228
$543$	−31.3179	−1.34398
$544$	−5.66473	−0.242873
$545$	8.35348	0.357824
$546$	0	0
$547$	18.8113	0.804313	0.402157	−	0.915571i	$-0.368261\pi$
$547$	18.8113	0.804313	0.402157	+	0.915571i	$0.368261\pi$
$548$	−13.6298	−0.582235
$549$	−0.921860	−0.0393440
$550$	−8.55793	−0.364911
$551$	8.22342	0.350329
$552$	10.5652	0.449687
$553$	−17.8175	−0.757676
$554$	−1.04905	−0.0445699
$555$	−37.7447	−1.60217
$556$	2.41868	0.102575
$557$	−14.4643	−0.612872	−0.306436	−	0.951891i	$-0.599137\pi$
$557$	−14.4643	−0.612872	−0.306436	+	0.951891i	$0.599137\pi$
$558$	−0.618295	−0.0261745
$559$	0	0
$560$	16.2609	0.687148
$561$	16.8548	0.711611
$562$	−17.3656	−0.732523
$563$	27.0345	1.13937	0.569684	−	0.821864i	$-0.307066\pi$
$563$	27.0345	1.13937	0.569684	+	0.821864i	$0.307066\pi$
$564$	−7.52754	−0.316967
$565$	−0.535601	−0.0225329
$566$	10.9067	0.458444
$567$	47.5391	1.99646
$568$	−4.44702	−0.186593
$569$	−35.3876	−1.48353	−0.741763	−	0.670662i	$-0.766010\pi$
$569$	−35.3876	−1.48353	−0.741763	+	0.670662i	$0.766010\pi$
$570$	5.57844	0.233655
$571$	32.9658	1.37957	0.689787	−	0.724012i	$-0.257704\pi$
$571$	32.9658	1.37957	0.689787	+	0.724012i	$0.257704\pi$
$572$	0	0
$573$	22.0029	0.919185
$574$	48.7633	2.03534
$575$	−30.3881	−1.26727
$576$	0.0936587	0.00390245
$577$	6.24664	0.260051	0.130025	−	0.991511i	$-0.458494\pi$
$577$	6.24664	0.260051	0.130025	+	0.991511i	$0.458494\pi$
$578$	−15.0891	−0.627624
$579$	−31.8885	−1.32524
$580$	−26.0813	−1.08297
$581$	−56.8753	−2.35958
$582$	−16.8389	−0.697997
$583$	−21.5768	−0.893619
$584$	7.63745	0.316040
$585$	0	0
$586$	−20.0576	−0.828572
$587$	−21.9020	−0.903991	−0.451996	−	0.892020i	$-0.649288\pi$
$587$	−21.9020	−0.903991	−0.451996	+	0.892020i	$0.649288\pi$
$588$	33.9230	1.39896
$589$	6.60158	0.272013
$590$	−4.54418	−0.187081
$591$	−37.9274	−1.56012
$592$	6.76618	0.278088
$593$	−47.8401	−1.96456	−0.982278	−	0.187430i	$-0.939984\pi$
$593$	−47.8401	−1.96456	−0.982278	+	0.187430i	$0.939984\pi$
$594$	8.64752	0.354812
$595$	92.1135	3.77628
$596$	18.9769	0.777325
$597$	10.7545	0.440154
$598$	0	0
$599$	28.2848	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$599$	28.2848	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$600$	−8.89807	−0.363262
$601$	−31.8180	−1.29788	−0.648941	−	0.760839i	$-0.724788\pi$
$601$	−31.8180	−1.29788	−0.648941	+	0.760839i	$0.724788\pi$
$602$	24.0625	0.980713
$603$	−0.888748	−0.0361926
$604$	9.28558	0.377825
$605$	25.8114	1.04938
$606$	23.8253	0.967838
$607$	5.60234	0.227392	0.113696	−	0.993516i	$-0.463731\pi$
$607$	5.60234	0.227392	0.113696	+	0.993516i	$0.463731\pi$
$608$	−1.00000	−0.0405554
$609$	−74.1578	−3.00502
$610$	−31.2171	−1.26395
$611$	0	0
$612$	0.530551	0.0214462
$613$	−26.9611	−1.08895	−0.544475	−	0.838777i	$-0.683271\pi$
$613$	−26.9611	−1.08895	−0.544475	+	0.838777i	$0.683271\pi$
$614$	−22.7077	−0.916410
$615$	−53.0563	−2.13944
$616$	8.67316	0.349451
$617$	−36.6918	−1.47715	−0.738577	−	0.674169i	$-0.764502\pi$
$617$	−36.6918	−1.47715	−0.738577	+	0.674169i	$0.764502\pi$
$618$	15.1152	0.608024
$619$	−13.5267	−0.543685	−0.271843	−	0.962342i	$-0.587633\pi$
$619$	−13.5267	−0.543685	−0.271843	+	0.962342i	$0.587633\pi$
$620$	−20.9375	−0.840868
$621$	30.7062	1.23220
$622$	−11.3189	−0.453846
$623$	81.3748	3.26021
$624$	0	0
$625$	−24.7018	−0.988072
$626$	−5.69862	−0.227763
$627$	2.97540	0.118826
$628$	−12.2130	−0.487352
$629$	38.3286	1.52826
$630$	−1.52297	−0.0606767
$631$	−38.2577	−1.52301	−0.761507	−	0.648157i	$-0.775540\pi$
$631$	−38.2577	−1.52301	−0.761507	+	0.648157i	$0.775540\pi$
$632$	−3.47518	−0.138235
$633$	−1.23753	−0.0491873
$634$	22.9385	0.911005
$635$	−7.74927	−0.307520
$636$	−22.4344	−0.889580
$637$	0	0
$638$	−13.9111	−0.550745
$639$	0.416502	0.0164766
$640$	3.17158	0.125368
$641$	38.6131	1.52513	0.762563	−	0.646914i	$-0.223941\pi$
$641$	38.6131	1.52513	0.762563	+	0.646914i	$0.223941\pi$
$642$	−9.18891	−0.362657
$643$	−20.1159	−0.793295	−0.396647	−	0.917971i	$-0.629826\pi$
$643$	−20.1159	−0.793295	−0.396647	+	0.917971i	$0.629826\pi$
$644$	30.7972	1.21358
$645$	−26.1809	−1.03087
$646$	−5.66473	−0.222876
$647$	−7.51894	−0.295600	−0.147800	−	0.989017i	$-0.547219\pi$
$647$	−7.51894	−0.295600	−0.147800	+	0.989017i	$0.547219\pi$
$648$	9.27220	0.364247
$649$	−2.42375	−0.0951406
$650$	0	0
$651$	−59.5322	−2.33325
$652$	−20.2182	−0.791808
$653$	29.6550	1.16049	0.580245	−	0.814442i	$-0.302957\pi$
$653$	29.6550	1.16049	0.580245	+	0.814442i	$0.302957\pi$
$654$	4.63263	0.181150
$655$	0.937513	0.0366316
$656$	9.51097	0.371341
$657$	−0.715314	−0.0279070
$658$	−21.9424	−0.855406
$659$	22.9991	0.895919	0.447959	−	0.894054i	$-0.352151\pi$
$659$	22.9991	0.895919	0.447959	+	0.894054i	$0.352151\pi$
$660$	−9.43673	−0.367324
$661$	29.4632	1.14599	0.572993	−	0.819560i	$-0.305782\pi$
$661$	29.4632	1.14599	0.572993	+	0.819560i	$0.305782\pi$
$662$	8.44810	0.328345
$663$	0	0
$664$	−11.0932	−0.430498
$665$	16.2609	0.630570
$666$	−0.633712	−0.0245558
$667$	−49.3964	−1.91264
$668$	−9.96580	−0.385589
$669$	22.5490	0.871795
$670$	−30.0959	−1.16270
$671$	−16.6505	−0.642784
$672$	9.01788	0.347872
$673$	−12.7361	−0.490940	−0.245470	−	0.969404i	$-0.578942\pi$
$673$	−12.7361	−0.490940	−0.245470	+	0.969404i	$0.578942\pi$
$674$	−8.19053	−0.315487
$675$	−25.8608	−0.995384
$676$	0	0
$677$	−3.10377	−0.119288	−0.0596438	−	0.998220i	$-0.518996\pi$
$677$	−3.10377	−0.119288	−0.0596438	+	0.998220i	$0.518996\pi$
$678$	−0.297031	−0.0114074
$679$	−49.0848	−1.88370
$680$	17.9662	0.688970
$681$	3.29464	0.126251
$682$	−11.1675	−0.427626
$683$	11.8720	0.454269	0.227135	−	0.973863i	$-0.427064\pi$
$683$	11.8720	0.454269	0.227135	+	0.973863i	$0.427064\pi$
$684$	0.0936587	0.00358113
$685$	43.2280	1.65166
$686$	62.9947	2.40515
$687$	−3.91929	−0.149530
$688$	4.69323	0.178928
$689$	0	0
$690$	−33.5085	−1.27565
$691$	−33.0961	−1.25903	−0.629517	−	0.776987i	$-0.716747\pi$
$691$	−33.0961	−1.25903	−0.629517	+	0.776987i	$0.716747\pi$
$692$	3.98084	0.151329
$693$	−0.812316	−0.0308573
$694$	3.31206	0.125724
$695$	−7.67106	−0.290980
$696$	−14.4640	−0.548257
$697$	53.8770	2.04074
$698$	28.5146	1.07929
$699$	−18.0166	−0.681449
$700$	−25.9375	−0.980345
$701$	−34.9161	−1.31876	−0.659382	−	0.751808i	$-0.729182\pi$
$701$	−34.9161	−1.31876	−0.659382	+	0.751808i	$0.729182\pi$
$702$	0	0
$703$	6.76618	0.255191
$704$	1.69164	0.0637562
$705$	23.8742	0.899155
$706$	−11.6407	−0.438105
$707$	69.4498	2.61193
$708$	−2.52009	−0.0947106
$709$	−1.47639	−0.0554472	−0.0277236	−	0.999616i	$-0.508826\pi$
$709$	−1.47639	−0.0554472	−0.0277236	+	0.999616i	$0.508826\pi$
$710$	14.1041	0.529318
$711$	0.325481	0.0122065
$712$	15.8716	0.594815
$713$	−39.6544	−1.48507
$714$	51.0838	1.91176
$715$	0	0
$716$	10.6606	0.398407
$717$	−13.6842	−0.511048
$718$	−31.9255	−1.19145
$719$	27.7457	1.03474	0.517371	−	0.855761i	$-0.326911\pi$
$719$	27.7457	1.03474	0.517371	+	0.855761i	$0.326911\pi$
$720$	−0.297046	−0.0110703
$721$	44.0603	1.64089
$722$	−1.00000	−0.0372161
$723$	13.1079	0.487487
$724$	−17.8056	−0.661740
$725$	41.6018	1.54505
$726$	14.3144	0.531256
$727$	16.2102	0.601204	0.300602	−	0.953750i	$-0.402812\pi$
$727$	16.2102	0.601204	0.300602	+	0.953750i	$0.402812\pi$
$728$	0	0
$729$	26.1053	0.966862
$730$	−24.2228	−0.896527
$731$	26.5859	0.983314
$732$	−17.3122	−0.639879
$733$	23.8476	0.880833	0.440416	−	0.897794i	$-0.354831\pi$
$733$	23.8476	0.880833	0.440416	+	0.897794i	$0.354831\pi$
$734$	12.9511	0.478035
$735$	−107.590	−3.96851
$736$	6.00680	0.221414
$737$	−16.0524	−0.591297
$738$	−0.890785	−0.0327902
$739$	43.1862	1.58863	0.794315	−	0.607506i	$-0.207830\pi$
$739$	43.1862	1.58863	0.794315	+	0.607506i	$0.207830\pi$
$740$	−21.4595	−0.788867
$741$	0	0
$742$	−65.3952	−2.40073
$743$	5.73784	0.210501	0.105250	−	0.994446i	$-0.466436\pi$
$743$	5.73784	0.210501	0.105250	+	0.994446i	$0.466436\pi$
$744$	−11.6114	−0.425694
$745$	−60.1869	−2.20508
$746$	−11.3513	−0.415602
$747$	1.03897	0.0380140
$748$	9.58270	0.350378
$749$	−26.7853	−0.978713
$750$	0.328811	0.0120065
$751$	−13.9381	−0.508609	−0.254305	−	0.967124i	$-0.581847\pi$
$751$	−13.9381	−0.508609	−0.254305	+	0.967124i	$0.581847\pi$
$752$	−4.27974	−0.156066
$753$	18.4817	0.673512
$754$	0	0
$755$	−29.4500	−1.07179
$756$	26.2090	0.953213
$757$	−33.6401	−1.22267	−0.611335	−	0.791372i	$-0.709367\pi$
$757$	−33.6401	−1.22267	−0.611335	+	0.791372i	$0.709367\pi$
$758$	−6.88201	−0.249966
$759$	−17.8726	−0.648735
$760$	3.17158	0.115045
$761$	16.2861	0.590370	0.295185	−	0.955440i	$-0.404619\pi$
$761$	16.2861	0.590370	0.295185	+	0.955440i	$0.404619\pi$
$762$	−4.29755	−0.155684
$763$	13.5039	0.488874
$764$	12.5096	0.452582
$765$	−1.68269	−0.0608376
$766$	−16.5966	−0.599661
$767$	0	0
$768$	1.75888	0.0634681
$769$	−45.6632	−1.64666	−0.823328	−	0.567566i	$-0.807885\pi$
$769$	−45.6632	−1.64666	−0.823328	+	0.567566i	$0.807885\pi$
$770$	−27.5076	−0.991306
$771$	5.00634	0.180299
$772$	−18.1300	−0.652514
$773$	41.5768	1.49541	0.747706	−	0.664030i	$-0.231155\pi$
$773$	41.5768	1.49541	0.747706	+	0.664030i	$0.231155\pi$
$774$	−0.439562	−0.0157997
$775$	33.3970	1.19966
$776$	−9.57367	−0.343675
$777$	−61.0166	−2.18896
$778$	23.9314	0.857984
$779$	9.51097	0.340766
$780$	0	0
$781$	7.52278	0.269186
$782$	34.0269	1.21680
$783$	−42.0373	−1.50229
$784$	19.2867	0.688811
$785$	38.7346	1.38250
$786$	0.519921	0.0185450
$787$	22.6768	0.808340	0.404170	−	0.914684i	$-0.367560\pi$
$787$	22.6768	0.808340	0.404170	+	0.914684i	$0.367560\pi$
$788$	−21.5634	−0.768163
$789$	−18.0399	−0.642238
$790$	11.0218	0.392139
$791$	−0.865832	−0.0307854
$792$	−0.158437	−0.00562982
$793$	0	0
$794$	18.4339	0.654195
$795$	71.1525	2.52352
$796$	6.11442	0.216720
$797$	25.9570	0.919444	0.459722	−	0.888063i	$-0.347949\pi$
$797$	25.9570	0.919444	0.459722	+	0.888063i	$0.347949\pi$
$798$	9.01788	0.319229
$799$	−24.2435	−0.857674
$800$	−5.05894	−0.178861
$801$	−1.48652	−0.0525235
$802$	−7.52372	−0.265672
$803$	−12.9198	−0.455932
$804$	−16.6904	−0.588625
$805$	−97.6759	−3.44262
$806$	0	0
$807$	−40.6183	−1.42983
$808$	13.5457	0.476537
$809$	−31.8059	−1.11824	−0.559119	−	0.829088i	$-0.688860\pi$
$809$	−31.8059	−1.11824	−0.559119	+	0.829088i	$0.688860\pi$
$810$	−29.4076	−1.03328
$811$	−11.2948	−0.396613	−0.198307	−	0.980140i	$-0.563544\pi$
$811$	−11.2948	−0.396613	−0.198307	+	0.980140i	$0.563544\pi$
$812$	−42.1619	−1.47959
$813$	−18.6818	−0.655198
$814$	−11.4460	−0.401181
$815$	64.1239	2.24616
$816$	9.96357	0.348795
$817$	4.69323	0.164195
$818$	16.3972	0.573316
$819$	0	0
$820$	−30.1648	−1.05340
$821$	28.0177	0.977824	0.488912	−	0.872333i	$-0.337394\pi$
$821$	28.0177	0.977824	0.488912	+	0.872333i	$0.337394\pi$
$822$	23.9731	0.836159
$823$	13.1885	0.459722	0.229861	−	0.973223i	$-0.426173\pi$
$823$	13.1885	0.459722	0.229861	+	0.973223i	$0.426173\pi$
$824$	8.59368	0.299375
$825$	15.0524	0.524056
$826$	−7.34594	−0.255598
$827$	17.0534	0.593003	0.296502	−	0.955032i	$-0.404180\pi$
$827$	17.0534	0.593003	0.296502	+	0.955032i	$0.404180\pi$
$828$	−0.562589	−0.0195513
$829$	53.9714	1.87450	0.937252	−	0.348652i	$-0.113361\pi$
$829$	53.9714	1.87450	0.937252	+	0.348652i	$0.113361\pi$
$830$	35.1829	1.22122
$831$	1.84515	0.0640077
$832$	0	0
$833$	109.254	3.78543
$834$	−4.25417	−0.147310
$835$	31.6074	1.09382
$836$	1.69164	0.0585067
$837$	−33.7466	−1.16645
$838$	14.4806	0.500224
$839$	−9.64862	−0.333107	−0.166554	−	0.986032i	$-0.553264\pi$
$839$	−9.64862	−0.333107	−0.166554	+	0.986032i	$0.553264\pi$
$840$	−28.6010	−0.986827
$841$	38.6246	1.33188
$842$	2.50015	0.0861607
$843$	30.5440	1.05199
$844$	−0.703588	−0.0242185
$845$	0	0
$846$	0.400834	0.0137810
$847$	41.7257	1.43371
$848$	−12.7549	−0.438006
$849$	−19.1836	−0.658380
$850$	−28.6575	−0.982945
$851$	−40.6431	−1.39323
$852$	7.82178	0.267970
$853$	38.4553	1.31668	0.658342	−	0.752719i	$-0.271258\pi$
$853$	38.4553	1.31668	0.658342	+	0.752719i	$0.271258\pi$
$854$	−50.4644	−1.72686
$855$	−0.297046	−0.0101588
$856$	−5.22430	−0.178563
$857$	−1.70561	−0.0582625	−0.0291313	−	0.999576i	$-0.509274\pi$
$857$	−1.70561	−0.0582625	−0.0291313	+	0.999576i	$0.509274\pi$
$858$	0	0
$859$	−37.1800	−1.26857	−0.634283	−	0.773101i	$-0.718704\pi$
$859$	−37.1800	−1.26857	−0.634283	+	0.773101i	$0.718704\pi$
$860$	−14.8850	−0.507573
$861$	−85.7687	−2.92299
$862$	−12.6330	−0.430282
$863$	1.27230	0.0433096	0.0216548	−	0.999766i	$-0.493107\pi$
$863$	1.27230	0.0433096	0.0216548	+	0.999766i	$0.493107\pi$
$864$	5.11191	0.173911
$865$	−12.6256	−0.429283
$866$	−16.5242	−0.561514
$867$	26.5399	0.901344
$868$	−33.8467	−1.14883
$869$	5.87877	0.199424
$870$	45.8738	1.55527
$871$	0	0
$872$	2.63385	0.0891934
$873$	0.896658	0.0303473
$874$	6.00680	0.203183
$875$	0.958470	0.0324022
$876$	−13.4334	−0.453871
$877$	−13.7888	−0.465616	−0.232808	−	0.972523i	$-0.574791\pi$
$877$	−13.7888	−0.465616	−0.232808	+	0.972523i	$0.574791\pi$
$878$	18.4009	0.620999
$879$	35.2789	1.18993
$880$	−5.36519	−0.180861
$881$	−11.0992	−0.373940	−0.186970	−	0.982366i	$-0.559867\pi$
$881$	−11.0992	−0.373940	−0.186970	+	0.982366i	$0.559867\pi$
$882$	−1.80637	−0.0608236
$883$	−50.6746	−1.70533	−0.852667	−	0.522454i	$-0.825017\pi$
$883$	−50.6746	−1.70533	−0.852667	+	0.522454i	$0.825017\pi$
$884$	0	0
$885$	7.99266	0.268670
$886$	−0.483841	−0.0162549
$887$	11.9792	0.402224	0.201112	−	0.979568i	$-0.435545\pi$
$887$	11.9792	0.402224	0.201112	+	0.979568i	$0.435545\pi$
$888$	−11.9009	−0.399368
$889$	−12.5272	−0.420147
$890$	−50.3382	−1.68734
$891$	−15.6853	−0.525476
$892$	12.8201	0.429248
$893$	−4.27974	−0.143216
$894$	−33.3781	−1.11633
$895$	−33.8111	−1.13018
$896$	5.12706	0.171283
$897$	0	0
$898$	33.7865	1.12747
$899$	54.2875	1.81059
$900$	0.473814	0.0157938
$901$	−72.2531	−2.40710
$902$	−16.0892	−0.535711
$903$	−42.3230	−1.40842
$904$	−0.168875	−0.00561670
$905$	56.4719	1.87719
$906$	−16.3322	−0.542601
$907$	−6.32028	−0.209861	−0.104931	−	0.994480i	$-0.533462\pi$
$907$	−6.32028	−0.209861	−0.104931	+	0.994480i	$0.533462\pi$
$908$	1.87315	0.0621626
$909$	−1.26868	−0.0420793
$910$	0	0
$911$	−37.2989	−1.23577	−0.617885	−	0.786269i	$-0.712010\pi$
$911$	−37.2989	−1.23577	−0.617885	+	0.786269i	$0.712010\pi$
$912$	1.75888	0.0582423
$913$	18.7657	0.621053
$914$	−0.891560	−0.0294902
$915$	54.9072	1.81518
$916$	−2.22829	−0.0736247
$917$	1.51555	0.0500477
$918$	28.9575	0.955741
$919$	−47.0380	−1.55164	−0.775821	−	0.630953i	$-0.782664\pi$
$919$	−47.0380	−1.55164	−0.775821	+	0.630953i	$0.782664\pi$
$920$	−19.0511	−0.628095
$921$	39.9402	1.31607
$922$	−3.48758	−0.114857
$923$	0	0
$924$	−15.2550	−0.501854
$925$	34.2297	1.12547
$926$	6.81285	0.223884
$927$	−0.804872	−0.0264355
$928$	−8.22342	−0.269947
$929$	4.13346	0.135615	0.0678073	−	0.997698i	$-0.478400\pi$
$929$	4.13346	0.135615	0.0678073	+	0.997698i	$0.478400\pi$
$930$	36.8265	1.20759
$931$	19.2867	0.632097
$932$	−10.2432	−0.335527
$933$	19.9085	0.651776
$934$	8.96725	0.293417
$935$	−30.3923	−0.993935
$936$	0	0
$937$	−55.0353	−1.79793	−0.898963	−	0.438025i	$-0.855678\pi$
$937$	−55.0353	−1.79793	−0.898963	+	0.438025i	$0.855678\pi$
$938$	−48.6518	−1.58854
$939$	10.0232	0.327095
$940$	13.5735	0.442720
$941$	−2.31917	−0.0756028	−0.0378014	−	0.999285i	$-0.512035\pi$
$941$	−2.31917	−0.0756028	−0.0378014	+	0.999285i	$0.512035\pi$
$942$	21.4812	0.699896
$943$	−57.1305	−1.86042
$944$	−1.43278	−0.0466330
$945$	−83.1241	−2.70403
$946$	−7.93928	−0.258128
$947$	11.3874	0.370041	0.185021	−	0.982735i	$-0.440765\pi$
$947$	11.3874	0.370041	0.185021	+	0.982735i	$0.440765\pi$
$948$	6.11243	0.198523
$949$	0	0
$950$	−5.05894	−0.164134
$951$	−40.3461	−1.30831
$952$	29.0434	0.941301
$953$	−18.7524	−0.607451	−0.303726	−	0.952760i	$-0.598231\pi$
$953$	−18.7524	−0.607451	−0.303726	+	0.952760i	$0.598231\pi$
$954$	1.19461	0.0386769
$955$	−39.6753	−1.28386
$956$	−7.78009	−0.251626
$957$	24.4679	0.790936
$958$	26.6153	0.859902
$959$	69.8806	2.25656
$960$	−5.57844	−0.180043
$961$	12.5808	0.405833
$962$	0	0
$963$	0.489301	0.0157675
$964$	7.45240	0.240026
$965$	57.5009	1.85102
$966$	−54.1686	−1.74285
$967$	−54.0878	−1.73935	−0.869674	−	0.493627i	$-0.835671\pi$
$967$	−54.0878	−1.73935	−0.869674	+	0.493627i	$0.835671\pi$
$968$	8.13834	0.261576
$969$	9.96357	0.320076
$970$	30.3637	0.974920
$971$	−54.0303	−1.73391	−0.866957	−	0.498383i	$-0.833928\pi$
$971$	−54.0303	−1.73391	−0.866957	+	0.498383i	$0.833928\pi$
$972$	−0.972978	−0.0312083
$973$	−12.4007	−0.397549
$974$	26.6839	0.855007
$975$	0	0
$976$	−9.84276	−0.315059
$977$	11.7204	0.374968	0.187484	−	0.982268i	$-0.439967\pi$
$977$	11.7204	0.374968	0.187484	+	0.982268i	$0.439967\pi$
$978$	35.5615	1.13713
$979$	−26.8492	−0.858103
$980$	−61.1694	−1.95399
$981$	−0.246683	−0.00787598
$982$	−27.3317	−0.872188
$983$	39.3515	1.25512	0.627560	−	0.778569i	$-0.284054\pi$
$983$	39.3515	1.25512	0.627560	+	0.778569i	$0.284054\pi$
$984$	−16.7286	−0.533290
$985$	68.3900	2.17909
$986$	−46.5834	−1.48352
$987$	38.5941	1.22846
$988$	0	0
$989$	−28.1913	−0.896431
$990$	0.502497	0.0159704
$991$	37.7835	1.20023	0.600115	−	0.799913i	$-0.295121\pi$
$991$	37.7835	1.20023	0.600115	+	0.799913i	$0.295121\pi$
$992$	−6.60158	−0.209600
$993$	−14.8592	−0.471542
$994$	22.8001	0.723177
$995$	−19.3924	−0.614780
$996$	19.5115	0.618247
$997$	−23.2518	−0.736392	−0.368196	−	0.929748i	$-0.620024\pi$
$997$	−23.2518	−0.736392	−0.368196	+	0.929748i	$0.620024\pi$
$998$	37.5106	1.18738
$999$	−34.5881	−1.09432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bo.1.12	✓	15
13.12	even	2		6422.2.a.bq.1.12	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bo.1.12	✓	15	1.1	even	1	trivial
6422.2.a.bq.1.12	yes	15	13.12	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.75888 q^{3} +1.00000 q^{4} -3.17158 q^{5} -1.75888 q^{6} -5.12706 q^{7} -1.00000 q^{8} +0.0936587 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.75888 q^{3} +1.00000 q^{4} -3.17158 q^{5} -1.75888 q^{6} -5.12706 q^{7} -1.00000 q^{8} +0.0936587 q^{9} +3.17158 q^{10} +1.69164 q^{11} +1.75888 q^{12} +5.12706 q^{14} -5.57844 q^{15} +1.00000 q^{16} +5.66473 q^{17} -0.0936587 q^{18} +1.00000 q^{19} -3.17158 q^{20} -9.01788 q^{21} -1.69164 q^{22} -6.00680 q^{23} -1.75888 q^{24} +5.05894 q^{25} -5.11191 q^{27} -5.12706 q^{28} +8.22342 q^{29} +5.57844 q^{30} +6.60158 q^{31} -1.00000 q^{32} +2.97540 q^{33} -5.66473 q^{34} +16.2609 q^{35} +0.0936587 q^{36} +6.76618 q^{37} -1.00000 q^{38} +3.17158 q^{40} +9.51097 q^{41} +9.01788 q^{42} +4.69323 q^{43} +1.69164 q^{44} -0.297046 q^{45} +6.00680 q^{46} -4.27974 q^{47} +1.75888 q^{48} +19.2867 q^{49} -5.05894 q^{50} +9.96357 q^{51} -12.7549 q^{53} +5.11191 q^{54} -5.36519 q^{55} +5.12706 q^{56} +1.75888 q^{57} -8.22342 q^{58} -1.43278 q^{59} -5.57844 q^{60} -9.84276 q^{61} -6.60158 q^{62} -0.480193 q^{63} +1.00000 q^{64} -2.97540 q^{66} -9.48923 q^{67} +5.66473 q^{68} -10.5652 q^{69} -16.2609 q^{70} +4.44702 q^{71} -0.0936587 q^{72} -7.63745 q^{73} -6.76618 q^{74} +8.89807 q^{75} +1.00000 q^{76} -8.67316 q^{77} +3.47518 q^{79} -3.17158 q^{80} -9.27220 q^{81} -9.51097 q^{82} +11.0932 q^{83} -9.01788 q^{84} -17.9662 q^{85} -4.69323 q^{86} +14.4640 q^{87} -1.69164 q^{88} -15.8716 q^{89} +0.297046 q^{90} -6.00680 q^{92} +11.6114 q^{93} +4.27974 q^{94} -3.17158 q^{95} -1.75888 q^{96} +9.57367 q^{97} -19.2867 q^{98} +0.158437 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9} + q^{10} - 4 q^{11} + 18 q^{14} - 23 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} + 15 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} - 18 q^{28} - 20 q^{29} + 23 q^{30} - 30 q^{31} - 15 q^{32} - 36 q^{33} - 2 q^{34} + 32 q^{35} + 17 q^{36} - 35 q^{37} - 15 q^{38} + q^{40} - 15 q^{41} + 2 q^{42} + q^{43} - 4 q^{44} + 11 q^{45} - 17 q^{46} + 29 q^{49} - 8 q^{50} - q^{51} - q^{53} - 12 q^{54} - 6 q^{55} + 18 q^{56} + 20 q^{58} + 7 q^{59} - 23 q^{60} - 2 q^{61} + 30 q^{62} - 42 q^{63} + 15 q^{64} + 36 q^{66} - 34 q^{67} + 2 q^{68} - 12 q^{69} - 32 q^{70} - 4 q^{71} - 17 q^{72} - 12 q^{73} + 35 q^{74} + 31 q^{75} + 15 q^{76} - 20 q^{77} + 23 q^{79} - q^{80} + 7 q^{81} + 15 q^{82} + 3 q^{83} - 2 q^{84} - 46 q^{85} - q^{86} + 22 q^{87} + 4 q^{88} - 17 q^{89} - 11 q^{90} + 17 q^{92} - 60 q^{93} - q^{95} - 18 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9 + q^10 - 4 * q^11 + 18 * q^14 - 23 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 + 15 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 - 18 * q^28 - 20 * q^29 + 23 * q^30 - 30 * q^31 - 15 * q^32 - 36 * q^33 - 2 * q^34 + 32 * q^35 + 17 * q^36 - 35 * q^37 - 15 * q^38 + q^40 - 15 * q^41 + 2 * q^42 + q^43 - 4 * q^44 + 11 * q^45 - 17 * q^46 + 29 * q^49 - 8 * q^50 - q^51 - q^53 - 12 * q^54 - 6 * q^55 + 18 * q^56 + 20 * q^58 + 7 * q^59 - 23 * q^60 - 2 * q^61 + 30 * q^62 - 42 * q^63 + 15 * q^64 + 36 * q^66 - 34 * q^67 + 2 * q^68 - 12 * q^69 - 32 * q^70 - 4 * q^71 - 17 * q^72 - 12 * q^73 + 35 * q^74 + 31 * q^75 + 15 * q^76 - 20 * q^77 + 23 * q^79 - q^80 + 7 * q^81 + 15 * q^82 + 3 * q^83 - 2 * q^84 - 46 * q^85 - q^86 + 22 * q^87 + 4 * q^88 - 17 * q^89 - 11 * q^90 + 17 * q^92 - 60 * q^93 - q^95 - 18 * q^97 - 29 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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