LMFDB - Embedded newform 6223.2.a.j.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6223,2,Mod(1,6223)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6223, 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("6223.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.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:	$49.6909051778$
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} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-2.51564$ of defining polynomial
Character	$\chi$	$=$	6223.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51564 q^{2} -2.69647 q^{3} +4.32847 q^{4} +1.19101 q^{5} -6.78336 q^{6} +5.85760 q^{8} +4.27094 q^{9} +O(q^{10})$	\(q+2.51564 q^{2} -2.69647 q^{3} +4.32847 q^{4} +1.19101 q^{5} -6.78336 q^{6} +5.85760 q^{8} +4.27094 q^{9} +2.99615 q^{10} -4.18912 q^{11} -11.6716 q^{12} -1.43411 q^{13} -3.21151 q^{15} +6.07870 q^{16} -1.76349 q^{17} +10.7442 q^{18} +4.92899 q^{19} +5.15524 q^{20} -10.5383 q^{22} -0.661413 q^{23} -15.7948 q^{24} -3.58150 q^{25} -3.60772 q^{26} -3.42705 q^{27} -0.521062 q^{29} -8.07903 q^{30} -3.67252 q^{31} +3.57666 q^{32} +11.2958 q^{33} -4.43630 q^{34} +18.4866 q^{36} +1.30196 q^{37} +12.3996 q^{38} +3.86704 q^{39} +6.97644 q^{40} -8.43934 q^{41} +4.56744 q^{43} -18.1325 q^{44} +5.08672 q^{45} -1.66388 q^{46} -0.797594 q^{47} -16.3910 q^{48} -9.00978 q^{50} +4.75518 q^{51} -6.20752 q^{52} -9.82487 q^{53} -8.62124 q^{54} -4.98928 q^{55} -13.2909 q^{57} -1.31081 q^{58} -12.7579 q^{59} -13.9009 q^{60} -0.0608610 q^{61} -9.23874 q^{62} -3.15981 q^{64} -1.70804 q^{65} +28.4163 q^{66} -7.96005 q^{67} -7.63319 q^{68} +1.78348 q^{69} +6.48720 q^{71} +25.0175 q^{72} +9.61580 q^{73} +3.27527 q^{74} +9.65740 q^{75} +21.3350 q^{76} +9.72810 q^{78} +15.3521 q^{79} +7.23978 q^{80} -3.57189 q^{81} -21.2304 q^{82} -5.82288 q^{83} -2.10033 q^{85} +11.4900 q^{86} +1.40503 q^{87} -24.5382 q^{88} -12.1345 q^{89} +12.7964 q^{90} -2.86290 q^{92} +9.90282 q^{93} -2.00646 q^{94} +5.87047 q^{95} -9.64434 q^{96} -0.0612387 q^{97} -17.8915 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 4 q^{3} + 14 q^{4} - 7 q^{5} - 8 q^{6} + 9 q^{9}+O(q^{10}) $ 15 * q - 4 * q^3 + 14 * q^4 - 7 * q^5 - 8 * q^6 + 9 * q^9	$ 15 q - 4 q^{3} + 14 q^{4} - 7 q^{5} - 8 q^{6} + 9 q^{9} - 10 q^{10} + 14 q^{11} - 10 q^{12} - 6 q^{13} + 6 q^{15} + 20 q^{16} - 10 q^{17} + q^{18} - 13 q^{19} - 8 q^{20} - 11 q^{22} + 15 q^{23} - 34 q^{24} - 22 q^{26} - 22 q^{27} + 16 q^{29} + 7 q^{30} - 22 q^{31} - 14 q^{33} + 15 q^{34} + 20 q^{36} - 14 q^{37} + 6 q^{38} + 29 q^{39} - 22 q^{40} - 19 q^{41} - q^{43} + 25 q^{44} + 8 q^{45} - 28 q^{46} - 49 q^{47} + 14 q^{48} + 24 q^{50} - 8 q^{51} + 17 q^{52} - 28 q^{53} - 13 q^{54} - 39 q^{55} - 12 q^{57} - 10 q^{58} - 43 q^{59} - 60 q^{60} - 27 q^{61} - 14 q^{62} + 18 q^{64} - 8 q^{65} + 36 q^{66} + 3 q^{67} - 13 q^{68} + 17 q^{69} + 55 q^{71} - 21 q^{72} + 3 q^{73} - 12 q^{74} - 8 q^{75} + 20 q^{76} - 6 q^{78} + 18 q^{79} - 29 q^{80} - 17 q^{81} - 14 q^{82} - 17 q^{83} + 7 q^{85} + 4 q^{86} - 35 q^{87} - 114 q^{88} - 36 q^{89} + 39 q^{90} + 45 q^{92} + 15 q^{93} + 15 q^{94} + 59 q^{95} - 85 q^{96} + 2 q^{97} + 35 q^{99}+O(q^{100}) $ 15 * q - 4 * q^3 + 14 * q^4 - 7 * q^5 - 8 * q^6 + 9 * q^9 - 10 * q^10 + 14 * q^11 - 10 * q^12 - 6 * q^13 + 6 * q^15 + 20 * q^16 - 10 * q^17 + q^18 - 13 * q^19 - 8 * q^20 - 11 * q^22 + 15 * q^23 - 34 * q^24 - 22 * q^26 - 22 * q^27 + 16 * q^29 + 7 * q^30 - 22 * q^31 - 14 * q^33 + 15 * q^34 + 20 * q^36 - 14 * q^37 + 6 * q^38 + 29 * q^39 - 22 * q^40 - 19 * q^41 - q^43 + 25 * q^44 + 8 * q^45 - 28 * q^46 - 49 * q^47 + 14 * q^48 + 24 * q^50 - 8 * q^51 + 17 * q^52 - 28 * q^53 - 13 * q^54 - 39 * q^55 - 12 * q^57 - 10 * q^58 - 43 * q^59 - 60 * q^60 - 27 * q^61 - 14 * q^62 + 18 * q^64 - 8 * q^65 + 36 * q^66 + 3 * q^67 - 13 * q^68 + 17 * q^69 + 55 * q^71 - 21 * q^72 + 3 * q^73 - 12 * q^74 - 8 * q^75 + 20 * q^76 - 6 * q^78 + 18 * q^79 - 29 * q^80 - 17 * q^81 - 14 * q^82 - 17 * q^83 + 7 * q^85 + 4 * q^86 - 35 * q^87 - 114 * q^88 - 36 * q^89 + 39 * q^90 + 45 * q^92 + 15 * q^93 + 15 * q^94 + 59 * q^95 - 85 * q^96 + 2 * q^97 + 35 * 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 $

$2$	2.51564	1.77883	0.889415	−	0.457101i	$-0.151112\pi$
$2$	2.51564	1.77883	0.889415	+	0.457101i	$0.151112\pi$
$3$	−2.69647	−1.55681	−0.778403	−	0.627765i	$-0.783970\pi$
$3$	−2.69647	−1.55681	−0.778403	+	0.627765i	$0.783970\pi$
$4$	4.32847	2.16423
$5$	1.19101	0.532635	0.266317	−	0.963885i	$-0.414193\pi$
$5$	1.19101	0.532635	0.266317	+	0.963885i	$0.414193\pi$
$6$	−6.78336	−2.76929
$7$	0	0
$7$	0	0
$8$	5.85760	2.07097
$9$	4.27094	1.42365
$10$	2.99615	0.947466
$11$	−4.18912	−1.26307	−0.631534	−	0.775348i	$-0.717574\pi$
$11$	−4.18912	−1.26307	−0.631534	+	0.775348i	$0.717574\pi$
$12$	−11.6716	−3.36929
$13$	−1.43411	−0.397752	−0.198876	−	0.980025i	$-0.563729\pi$
$13$	−1.43411	−0.397752	−0.198876	+	0.980025i	$0.563729\pi$
$14$	0	0
$15$	−3.21151	−0.829209
$16$	6.07870	1.51968
$17$	−1.76349	−0.427708	−0.213854	−	0.976866i	$-0.568602\pi$
$17$	−1.76349	−0.427708	−0.213854	+	0.976866i	$0.568602\pi$
$18$	10.7442	2.53242
$19$	4.92899	1.13079	0.565394	−	0.824821i	$-0.308724\pi$
$19$	4.92899	1.13079	0.565394	+	0.824821i	$0.308724\pi$
$20$	5.15524	1.15275
$21$	0	0
$22$	−10.5383	−2.24678
$23$	−0.661413	−0.137914	−0.0689570	−	0.997620i	$-0.521967\pi$
$23$	−0.661413	−0.137914	−0.0689570	+	0.997620i	$0.521967\pi$
$24$	−15.7948	−3.22411
$25$	−3.58150	−0.716300
$26$	−3.60772	−0.707532
$27$	−3.42705	−0.659536
$28$	0	0
$29$	−0.521062	−0.0967588	−0.0483794	−	0.998829i	$-0.515406\pi$
$29$	−0.521062	−0.0967588	−0.0483794	+	0.998829i	$0.515406\pi$
$30$	−8.07903	−1.47502
$31$	−3.67252	−0.659603	−0.329802	−	0.944050i	$-0.606982\pi$
$31$	−3.67252	−0.659603	−0.329802	+	0.944050i	$0.606982\pi$
$32$	3.57666	0.632270
$33$	11.2958	1.96635
$34$	−4.43630	−0.760820
$35$	0	0
$36$	18.4866	3.08111
$37$	1.30196	0.214041	0.107021	−	0.994257i	$-0.465869\pi$
$37$	1.30196	0.214041	0.107021	+	0.994257i	$0.465869\pi$
$38$	12.3996	2.01148
$39$	3.86704	0.619222
$40$	6.97644	1.10307
$41$	−8.43934	−1.31800	−0.659002	−	0.752142i	$-0.729021\pi$
$41$	−8.43934	−1.31800	−0.659002	+	0.752142i	$0.729021\pi$
$42$	0	0
$43$	4.56744	0.696527	0.348264	−	0.937397i	$-0.386771\pi$
$43$	4.56744	0.696527	0.348264	+	0.937397i	$0.386771\pi$
$44$	−18.1325	−2.73358
$45$	5.08672	0.758284
$46$	−1.66388	−0.245326
$47$	−0.797594	−0.116341	−0.0581705	−	0.998307i	$-0.518527\pi$
$47$	−0.797594	−0.116341	−0.0581705	+	0.998307i	$0.518527\pi$
$48$	−16.3910	−2.36584
$49$	0	0
$50$	−9.00978	−1.27418
$51$	4.75518	0.665859
$52$	−6.20752	−0.860828
$53$	−9.82487	−1.34955	−0.674775	−	0.738023i	$-0.735759\pi$
$53$	−9.82487	−1.34955	−0.674775	+	0.738023i	$0.735759\pi$
$54$	−8.62124	−1.17320
$55$	−4.98928	−0.672754
$56$	0	0
$57$	−13.2909	−1.76042
$58$	−1.31081	−0.172117
$59$	−12.7579	−1.66093	−0.830466	−	0.557070i	$-0.811926\pi$
$59$	−12.7579	−1.66093	−0.830466	+	0.557070i	$0.811926\pi$
$60$	−13.9009	−1.79460
$61$	−0.0608610	−0.00779246	−0.00389623	−	0.999992i	$-0.501240\pi$
$61$	−0.0608610	−0.00779246	−0.00389623	+	0.999992i	$0.501240\pi$
$62$	−9.23874	−1.17332
$63$	0	0
$64$	−3.15981	−0.394976
$65$	−1.70804	−0.211856
$66$	28.4163	3.49781
$67$	−7.96005	−0.972474	−0.486237	−	0.873827i	$-0.661631\pi$
$67$	−7.96005	−0.972474	−0.486237	+	0.873827i	$0.661631\pi$
$68$	−7.63319	−0.925661
$69$	1.78348	0.214705
$70$	0	0
$71$	6.48720	0.769889	0.384945	−	0.922940i	$-0.374221\pi$
$71$	6.48720	0.769889	0.384945	+	0.922940i	$0.374221\pi$
$72$	25.0175	2.94834
$73$	9.61580	1.12544	0.562722	−	0.826646i	$-0.309754\pi$
$73$	9.61580	1.12544	0.562722	+	0.826646i	$0.309754\pi$
$74$	3.27527	0.380743
$75$	9.65740	1.11514
$76$	21.3350	2.44729
$77$	0	0
$78$	9.72810	1.10149
$79$	15.3521	1.72725	0.863625	−	0.504135i	$-0.168189\pi$
$79$	15.3521	1.72725	0.863625	+	0.504135i	$0.168189\pi$
$80$	7.23978	0.809432
$81$	−3.57189	−0.396877
$82$	−21.2304	−2.34450
$83$	−5.82288	−0.639144	−0.319572	−	0.947562i	$-0.603539\pi$
$83$	−5.82288	−0.639144	−0.319572	+	0.947562i	$0.603539\pi$
$84$	0	0
$85$	−2.10033	−0.227812
$86$	11.4900	1.23900
$87$	1.40503	0.150635
$88$	−24.5382	−2.61578
$89$	−12.1345	−1.28625	−0.643125	−	0.765761i	$-0.722362\pi$
$89$	−12.1345	−1.28625	−0.643125	+	0.765761i	$0.722362\pi$
$90$	12.7964	1.34886
$91$	0	0
$92$	−2.86290	−0.298478
$93$	9.90282	1.02687
$94$	−2.00646	−0.206951
$95$	5.87047	0.602297
$96$	−9.64434	−0.984322
$97$	−0.0612387	−0.00621785	−0.00310893	−	0.999995i	$-0.500990\pi$
$97$	−0.0612387	−0.00621785	−0.00310893	+	0.999995i	$0.500990\pi$
$98$	0	0
$99$	−17.8915	−1.79816
$100$	−15.5024	−1.55024
$101$	9.53969	0.949235	0.474617	−	0.880192i	$-0.342586\pi$
$101$	9.53969	0.949235	0.474617	+	0.880192i	$0.342586\pi$
$102$	11.9624	1.18445
$103$	2.73565	0.269552	0.134776	−	0.990876i	$-0.456969\pi$
$103$	2.73565	0.269552	0.134776	+	0.990876i	$0.456969\pi$
$104$	−8.40046	−0.823733
$105$	0	0
$106$	−24.7159	−2.40062
$107$	−6.98917	−0.675669	−0.337834	−	0.941206i	$-0.609694\pi$
$107$	−6.98917	−0.675669	−0.337834	+	0.941206i	$0.609694\pi$
$108$	−14.8339	−1.42739
$109$	−1.71721	−0.164479	−0.0822395	−	0.996613i	$-0.526207\pi$
$109$	−1.71721	−0.164479	−0.0822395	+	0.996613i	$0.526207\pi$
$110$	−12.5512	−1.19671
$111$	−3.51070	−0.333221
$112$	0	0
$113$	−19.9992	−1.88136	−0.940682	−	0.339290i	$-0.889813\pi$
$113$	−19.9992	−1.88136	−0.940682	+	0.339290i	$0.889813\pi$
$114$	−33.4351	−3.13148
$115$	−0.787747	−0.0734578
$116$	−2.25540	−0.209409
$117$	−6.12501	−0.566258
$118$	−32.0942	−2.95451
$119$	0	0
$120$	−18.8118	−1.71727
$121$	6.54875	0.595341
$122$	−0.153105	−0.0138615
$123$	22.7564	2.05188
$124$	−15.8964	−1.42754
$125$	−10.2206	−0.914161
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−15.1023	−1.33486
$129$	−12.3159	−1.08436
$130$	−4.29682	−0.376856
$131$	−10.3792	−0.906832	−0.453416	−	0.891299i	$-0.649795\pi$
$131$	−10.3792	−0.906832	−0.453416	+	0.891299i	$0.649795\pi$
$132$	48.8937	4.25565
$133$	0	0
$134$	−20.0246	−1.72987
$135$	−4.08164	−0.351292
$136$	−10.3298	−0.885773
$137$	13.4955	1.15300	0.576500	−	0.817097i	$-0.304418\pi$
$137$	13.4955	1.15300	0.576500	+	0.817097i	$0.304418\pi$
$138$	4.48660	0.381924
$139$	13.9654	1.18453	0.592263	−	0.805745i	$-0.298235\pi$
$139$	13.9654	1.18453	0.592263	+	0.805745i	$0.298235\pi$
$140$	0	0
$141$	2.15069	0.181121
$142$	16.3195	1.36950
$143$	6.00768	0.502387
$144$	25.9618	2.16348
$145$	−0.620589	−0.0515371
$146$	24.1899	2.00197
$147$	0	0
$148$	5.63550	0.463235
$149$	−14.4117	−1.18065	−0.590325	−	0.807166i	$-0.701000\pi$
$149$	−14.4117	−1.18065	−0.590325	+	0.807166i	$0.701000\pi$
$150$	24.2946	1.98365
$151$	−9.13744	−0.743595	−0.371797	−	0.928314i	$-0.621258\pi$
$151$	−9.13744	−0.743595	−0.371797	+	0.928314i	$0.621258\pi$
$152$	28.8721	2.34183
$153$	−7.53174	−0.608905
$154$	0	0
$155$	−4.37399	−0.351328
$156$	16.7384	1.34014
$157$	−9.89040	−0.789340	−0.394670	−	0.918823i	$-0.629141\pi$
$157$	−9.89040	−0.789340	−0.394670	+	0.918823i	$0.629141\pi$
$158$	38.6205	3.07248
$159$	26.4925	2.10099
$160$	4.25983	0.336769
$161$	0	0
$162$	−8.98561	−0.705976
$163$	12.7487	0.998553	0.499276	−	0.866443i	$-0.333599\pi$
$163$	12.7487	0.998553	0.499276	+	0.866443i	$0.333599\pi$
$164$	−36.5294	−2.85247
$165$	13.4534	1.04735
$166$	−14.6483	−1.13693
$167$	−20.4473	−1.58226	−0.791129	−	0.611649i	$-0.790506\pi$
$167$	−20.4473	−1.58226	−0.791129	+	0.611649i	$0.790506\pi$
$168$	0	0
$169$	−10.9433	−0.841794
$170$	−5.28367	−0.405239
$171$	21.0514	1.60984
$172$	19.7700	1.50745
$173$	−21.7709	−1.65521	−0.827606	−	0.561309i	$-0.810298\pi$
$173$	−21.7709	−1.65521	−0.827606	+	0.561309i	$0.810298\pi$
$174$	3.53455	0.267953
$175$	0	0
$176$	−25.4644	−1.91945
$177$	34.4012	2.58575
$178$	−30.5260	−2.28802
$179$	3.27383	0.244698	0.122349	−	0.992487i	$-0.460957\pi$
$179$	3.27383	0.244698	0.122349	+	0.992487i	$0.460957\pi$
$180$	22.0177	1.64110
$181$	16.5329	1.22888	0.614440	−	0.788964i	$-0.289382\pi$
$181$	16.5329	1.22888	0.614440	+	0.788964i	$0.289382\pi$
$182$	0	0
$183$	0.164110	0.0121313
$184$	−3.87429	−0.285616
$185$	1.55065	0.114006
$186$	24.9120	1.82663
$187$	7.38746	0.540225
$188$	−3.45236	−0.251789
$189$	0	0
$190$	14.7680	1.07138
$191$	23.8442	1.72530	0.862651	−	0.505799i	$-0.168802\pi$
$191$	23.8442	1.72530	0.862651	+	0.505799i	$0.168802\pi$
$192$	8.52032	0.614901
$193$	2.23275	0.160717	0.0803586	−	0.996766i	$-0.474393\pi$
$193$	2.23275	0.160717	0.0803586	+	0.996766i	$0.474393\pi$
$194$	−0.154055	−0.0110605
$195$	4.60568	0.329819
$196$	0	0
$197$	20.4153	1.45453	0.727266	−	0.686355i	$-0.240791\pi$
$197$	20.4153	1.45453	0.727266	+	0.686355i	$0.240791\pi$
$198$	−45.0086	−3.19862
$199$	−9.51842	−0.674743	−0.337372	−	0.941372i	$-0.609538\pi$
$199$	−9.51842	−0.674743	−0.337372	+	0.941372i	$0.609538\pi$
$200$	−20.9790	−1.48344
$201$	21.4640	1.51395
$202$	23.9985	1.68853
$203$	0	0
$204$	20.5827	1.44107
$205$	−10.0513	−0.702014
$206$	6.88193	0.479487
$207$	−2.82485	−0.196341
$208$	−8.71755	−0.604453
$209$	−20.6482	−1.42826
$210$	0	0
$211$	−12.1258	−0.834773	−0.417386	−	0.908729i	$-0.637054\pi$
$211$	−12.1258	−0.834773	−0.417386	+	0.908729i	$0.637054\pi$
$212$	−42.5267	−2.92074
$213$	−17.4925	−1.19857
$214$	−17.5823	−1.20190
$215$	5.43985	0.370995
$216$	−20.0743	−1.36588
$217$	0	0
$218$	−4.31989	−0.292580
$219$	−25.9287	−1.75210
$220$	−21.5959	−1.45600
$221$	2.52904	0.170122
$222$	−8.83166	−0.592743
$223$	−6.01349	−0.402693	−0.201347	−	0.979520i	$-0.564532\pi$
$223$	−6.01349	−0.402693	−0.201347	+	0.979520i	$0.564532\pi$
$224$	0	0
$225$	−15.2964	−1.01976
$226$	−50.3108	−3.34662
$227$	20.1839	1.33965	0.669825	−	0.742519i	$-0.266369\pi$
$227$	20.1839	1.33965	0.669825	+	0.742519i	$0.266369\pi$
$228$	−57.5291	−3.80996
$229$	−10.8054	−0.714043	−0.357021	−	0.934096i	$-0.616208\pi$
$229$	−10.8054	−0.714043	−0.357021	+	0.934096i	$0.616208\pi$
$230$	−1.98169	−0.130669
$231$	0	0
$232$	−3.05217	−0.200385
$233$	−17.7238	−1.16112	−0.580561	−	0.814217i	$-0.697167\pi$
$233$	−17.7238	−1.16112	−0.580561	+	0.814217i	$0.697167\pi$
$234$	−15.4084	−1.00728
$235$	−0.949941	−0.0619673
$236$	−55.2220	−3.59465
$237$	−41.3965	−2.68899
$238$	0	0
$239$	16.3003	1.05438	0.527189	−	0.849748i	$-0.323246\pi$
$239$	16.3003	1.05438	0.527189	+	0.849748i	$0.323246\pi$
$240$	−19.5218	−1.26013
$241$	16.4252	1.05804	0.529020	−	0.848609i	$-0.322560\pi$
$241$	16.4252	1.05804	0.529020	+	0.848609i	$0.322560\pi$
$242$	16.4743	1.05901
$243$	19.9126	1.27740
$244$	−0.263435	−0.0168647
$245$	0	0
$246$	57.2470	3.64994
$247$	−7.06874	−0.449773
$248$	−21.5121	−1.36602
$249$	15.7012	0.995024
$250$	−25.7115	−1.62614
$251$	10.0670	0.635421	0.317711	−	0.948188i	$-0.397086\pi$
$251$	10.0670	0.635421	0.317711	+	0.948188i	$0.397086\pi$
$252$	0	0
$253$	2.77074	0.174195
$254$	2.51564	0.157846
$255$	5.66346	0.354660
$256$	−31.6723	−1.97952
$257$	−12.9478	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$257$	−12.9478	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$258$	−30.9825	−1.92889
$259$	0	0
$260$	−7.39320	−0.458507
$261$	−2.22542	−0.137750
$262$	−26.1103	−1.61310
$263$	−26.8400	−1.65503	−0.827513	−	0.561446i	$-0.810245\pi$
$263$	−26.8400	−1.65503	−0.827513	+	0.561446i	$0.810245\pi$
$264$	66.1665	4.07227
$265$	−11.7015	−0.718817
$266$	0	0
$267$	32.7202	2.00244
$268$	−34.4548	−2.10466
$269$	17.1313	1.04451	0.522256	−	0.852789i	$-0.325090\pi$
$269$	17.1313	1.04451	0.522256	+	0.852789i	$0.325090\pi$
$270$	−10.2680	−0.624888
$271$	−25.5813	−1.55395	−0.776975	−	0.629531i	$-0.783247\pi$
$271$	−25.5813	−1.55395	−0.776975	+	0.629531i	$0.783247\pi$
$272$	−10.7197	−0.649978
$273$	0	0
$274$	33.9499	2.05099
$275$	15.0033	0.904736
$276$	7.71973	0.464673
$277$	−14.5241	−0.872669	−0.436334	−	0.899785i	$-0.643724\pi$
$277$	−14.5241	−0.872669	−0.436334	+	0.899785i	$0.643724\pi$
$278$	35.1319	2.10707
$279$	−15.6851	−0.939042
$280$	0	0
$281$	10.3069	0.614857	0.307429	−	0.951571i	$-0.400531\pi$
$281$	10.3069	0.614857	0.307429	+	0.951571i	$0.400531\pi$
$282$	5.41037	0.322183
$283$	19.8952	1.18264	0.591322	−	0.806435i	$-0.298606\pi$
$283$	19.8952	1.18264	0.591322	+	0.806435i	$0.298606\pi$
$284$	28.0796	1.66622
$285$	−15.8295	−0.937660
$286$	15.1132	0.893661
$287$	0	0
$288$	15.2757	0.900129
$289$	−13.8901	−0.817066
$290$	−1.56118	−0.0916757
$291$	0.165128	0.00967999
$292$	41.6217	2.43572
$293$	−8.29939	−0.484855	−0.242428	−	0.970169i	$-0.577944\pi$
$293$	−8.29939	−0.484855	−0.242428	+	0.970169i	$0.577944\pi$
$294$	0	0
$295$	−15.1947	−0.884670
$296$	7.62637	0.443274
$297$	14.3563	0.833039
$298$	−36.2547	−2.10018
$299$	0.948541	0.0548555
$300$	41.8018	2.41343
$301$	0	0
$302$	−22.9866	−1.32273
$303$	−25.7235	−1.47778
$304$	29.9619	1.71843
$305$	−0.0724859	−0.00415053
$306$	−18.9472	−1.08314
$307$	16.4163	0.936927	0.468464	−	0.883483i	$-0.344808\pi$
$307$	16.4163	0.936927	0.468464	+	0.883483i	$0.344808\pi$
$308$	0	0
$309$	−7.37660	−0.419640
$310$	−11.0034	−0.624952
$311$	9.12922	0.517671	0.258835	−	0.965921i	$-0.416661\pi$
$311$	9.12922	0.517671	0.258835	+	0.965921i	$0.416661\pi$
$312$	22.6516	1.28239
$313$	3.23344	0.182765	0.0913824	−	0.995816i	$-0.470871\pi$
$313$	3.23344	0.182765	0.0913824	+	0.995816i	$0.470871\pi$
$314$	−24.8807	−1.40410
$315$	0	0
$316$	66.4512	3.73817
$317$	−12.6938	−0.712953	−0.356477	−	0.934304i	$-0.616022\pi$
$317$	−12.6938	−0.712953	−0.356477	+	0.934304i	$0.616022\pi$
$318$	66.6456	3.73730
$319$	2.18279	0.122213
$320$	−3.76335	−0.210378
$321$	18.8461	1.05189
$322$	0	0
$323$	−8.69221	−0.483647
$324$	−15.4608	−0.858934
$325$	5.13628	0.284910
$326$	32.0711	1.77625
$327$	4.63040	0.256062
$328$	−49.4343	−2.72955
$329$	0	0
$330$	33.8440	1.86305
$331$	2.70814	0.148853	0.0744265	−	0.997227i	$-0.476287\pi$
$331$	2.70814	0.148853	0.0744265	+	0.997227i	$0.476287\pi$
$332$	−25.2041	−1.38326
$333$	5.56060	0.304719
$334$	−51.4381	−2.81457
$335$	−9.48047	−0.517974
$336$	0	0
$337$	28.3780	1.54585	0.772924	−	0.634499i	$-0.218793\pi$
$337$	28.3780	1.54585	0.772924	+	0.634499i	$0.218793\pi$
$338$	−27.5295	−1.49741
$339$	53.9271	2.92892
$340$	−9.09119	−0.493039
$341$	15.3846	0.833124
$342$	52.9579	2.86364
$343$	0	0
$344$	26.7542	1.44249
$345$	2.12414	0.114360
$346$	−54.7679	−2.94434
$347$	−1.19640	−0.0642261	−0.0321131	−	0.999484i	$-0.510224\pi$
$347$	−1.19640	−0.0642261	−0.0321131	+	0.999484i	$0.510224\pi$
$348$	6.08161	0.326009
$349$	6.91527	0.370166	0.185083	−	0.982723i	$-0.440745\pi$
$349$	6.91527	0.370166	0.185083	+	0.982723i	$0.440745\pi$
$350$	0	0
$351$	4.91478	0.262332
$352$	−14.9831	−0.798600
$353$	−21.1758	−1.12707	−0.563537	−	0.826091i	$-0.690560\pi$
$353$	−21.1758	−1.12707	−0.563537	+	0.826091i	$0.690560\pi$
$354$	86.5411	4.59961
$355$	7.72630	0.410070
$356$	−52.5236	−2.78375
$357$	0	0
$358$	8.23580	0.435276
$359$	−22.4862	−1.18678	−0.593388	−	0.804916i	$-0.702210\pi$
$359$	−22.4862	−1.18678	−0.593388	+	0.804916i	$0.702210\pi$
$360$	29.7960	1.57039
$361$	5.29497	0.278682
$362$	41.5908	2.18597
$363$	−17.6585	−0.926831
$364$	0	0
$365$	11.4525	0.599450
$366$	0.412842	0.0215796
$367$	−17.1938	−0.897512	−0.448756	−	0.893654i	$-0.648133\pi$
$367$	−17.1938	−0.897512	−0.448756	+	0.893654i	$0.648133\pi$
$368$	−4.02053	−0.209585
$369$	−36.0439	−1.87637
$370$	3.90087	0.202797
$371$	0	0
$372$	42.8640	2.22240
$373$	8.57902	0.444205	0.222102	−	0.975023i	$-0.428708\pi$
$373$	8.57902	0.444205	0.222102	+	0.975023i	$0.428708\pi$
$374$	18.5842	0.960967
$375$	27.5596	1.42317
$376$	−4.67199	−0.240939
$377$	0.747262	0.0384860
$378$	0	0
$379$	−4.60673	−0.236632	−0.118316	−	0.992976i	$-0.537750\pi$
$379$	−4.60673	−0.236632	−0.118316	+	0.992976i	$0.537750\pi$
$380$	25.4101	1.30351
$381$	−2.69647	−0.138144
$382$	59.9834	3.06902
$383$	21.3698	1.09195	0.545973	−	0.837803i	$-0.316160\pi$
$383$	21.3698	1.09195	0.545973	+	0.837803i	$0.316160\pi$
$384$	40.7228	2.07813
$385$	0	0
$386$	5.61682	0.285888
$387$	19.5072	0.991609
$388$	−0.265070	−0.0134569
$389$	−24.6810	−1.25138	−0.625689	−	0.780073i	$-0.715182\pi$
$389$	−24.6810	−1.25138	−0.625689	+	0.780073i	$0.715182\pi$
$390$	11.5862	0.586692
$391$	1.16639	0.0589870
$392$	0	0
$393$	27.9871	1.41176
$394$	51.3577	2.58737
$395$	18.2845	0.919993
$396$	−77.4428	−3.89165
$397$	−21.0436	−1.05615	−0.528073	−	0.849199i	$-0.677085\pi$
$397$	−21.0436	−1.05615	−0.528073	+	0.849199i	$0.677085\pi$
$398$	−23.9450	−1.20025
$399$	0	0
$400$	−21.7709	−1.08854
$401$	12.7428	0.636346	0.318173	−	0.948033i	$-0.396931\pi$
$401$	12.7428	0.636346	0.318173	+	0.948033i	$0.396931\pi$
$402$	53.9958	2.69307
$403$	5.26681	0.262358
$404$	41.2923	2.05437
$405$	−4.25415	−0.211390
$406$	0	0
$407$	−5.45407	−0.270348
$408$	27.8540	1.37898
$409$	10.4497	0.516703	0.258351	−	0.966051i	$-0.416821\pi$
$409$	10.4497	0.516703	0.258351	+	0.966051i	$0.416821\pi$
$410$	−25.2855	−1.24876
$411$	−36.3902	−1.79500
$412$	11.8412	0.583374
$413$	0	0
$414$	−7.10633	−0.349257
$415$	−6.93509	−0.340430
$416$	−5.12933	−0.251486
$417$	−37.6571	−1.84408
$418$	−51.9434	−2.54064
$419$	−35.8726	−1.75249	−0.876246	−	0.481864i	$-0.839960\pi$
$419$	−35.8726	−1.75249	−0.876246	+	0.481864i	$0.839960\pi$
$420$	0	0
$421$	2.32495	0.113311	0.0566555	−	0.998394i	$-0.481956\pi$
$421$	2.32495	0.113311	0.0566555	+	0.998394i	$0.481956\pi$
$422$	−30.5041	−1.48492
$423$	−3.40648	−0.165629
$424$	−57.5502	−2.79488
$425$	6.31593	0.306368
$426$	−44.0050	−2.13205
$427$	0	0
$428$	−30.2524	−1.46231
$429$	−16.1995	−0.782120
$430$	13.6847	0.659936
$431$	34.8419	1.67828	0.839138	−	0.543919i	$-0.183060\pi$
$431$	34.8419	1.67828	0.839138	+	0.543919i	$0.183060\pi$
$432$	−20.8320	−1.00228
$433$	2.46087	0.118262	0.0591309	−	0.998250i	$-0.481167\pi$
$433$	2.46087	0.118262	0.0591309	+	0.998250i	$0.481167\pi$
$434$	0	0
$435$	1.67340	0.0802333
$436$	−7.43289	−0.355971
$437$	−3.26010	−0.155952
$438$	−65.2274	−3.11668
$439$	−4.86694	−0.232286	−0.116143	−	0.993232i	$-0.537053\pi$
$439$	−4.86694	−0.232286	−0.116143	+	0.993232i	$0.537053\pi$
$440$	−29.2252	−1.39326
$441$	0	0
$442$	6.36217	0.302617
$443$	5.94755	0.282577	0.141288	−	0.989968i	$-0.454876\pi$
$443$	5.94755	0.282577	0.141288	+	0.989968i	$0.454876\pi$
$444$	−15.1959	−0.721167
$445$	−14.4522	−0.685101
$446$	−15.1278	−0.716323
$447$	38.8606	1.83804
$448$	0	0
$449$	−36.7039	−1.73216	−0.866082	−	0.499902i	$-0.833369\pi$
$449$	−36.7039	−1.73216	−0.866082	+	0.499902i	$0.833369\pi$
$450$	−38.4803	−1.81398
$451$	35.3534	1.66473
$452$	−86.5658	−4.07171
$453$	24.6388	1.15763
$454$	50.7754	2.38301
$455$	0	0
$456$	−77.8526	−3.64578
$457$	−30.5030	−1.42687	−0.713436	−	0.700721i	$-0.752862\pi$
$457$	−30.5030	−1.42687	−0.713436	+	0.700721i	$0.752862\pi$
$458$	−27.1826	−1.27016
$459$	6.04355	0.282089
$460$	−3.40974	−0.158980
$461$	−26.1775	−1.21921	−0.609604	−	0.792706i	$-0.708671\pi$
$461$	−26.1775	−1.21921	−0.609604	+	0.792706i	$0.708671\pi$
$462$	0	0
$463$	29.8321	1.38642	0.693208	−	0.720738i	$-0.256197\pi$
$463$	29.8321	1.38642	0.693208	+	0.720738i	$0.256197\pi$
$464$	−3.16738	−0.147042
$465$	11.7943	0.546949
$466$	−44.5867	−2.06544
$467$	26.4147	1.22233	0.611163	−	0.791505i	$-0.290702\pi$
$467$	26.4147	1.22233	0.611163	+	0.791505i	$0.290702\pi$
$468$	−26.5119	−1.22551
$469$	0	0
$470$	−2.38971	−0.110229
$471$	26.6692	1.22885
$472$	−74.7304	−3.43975
$473$	−19.1336	−0.879762
$474$	−104.139	−4.78326
$475$	−17.6532	−0.809984
$476$	0	0
$477$	−41.9614	−1.92128
$478$	41.0057	1.87556
$479$	−29.8315	−1.36304	−0.681519	−	0.731801i	$-0.738680\pi$
$479$	−29.8315	−1.36304	−0.681519	+	0.731801i	$0.738680\pi$
$480$	−11.4865	−0.524284
$481$	−1.86716	−0.0851352
$482$	41.3200	1.88207
$483$	0	0
$484$	28.3461	1.28846
$485$	−0.0729358	−0.00331184
$486$	50.0931	2.27227
$487$	30.4878	1.38153	0.690767	−	0.723077i	$-0.257273\pi$
$487$	30.4878	1.38153	0.690767	+	0.723077i	$0.257273\pi$
$488$	−0.356500	−0.0161380
$489$	−34.3764	−1.55455
$490$	0	0
$491$	12.6414	0.570496	0.285248	−	0.958454i	$-0.407924\pi$
$491$	12.6414	0.570496	0.285248	+	0.958454i	$0.407924\pi$
$492$	98.5004	4.44074
$493$	0.918885	0.0413845
$494$	−17.7824	−0.800069
$495$	−21.3089	−0.957764
$496$	−22.3241	−1.00238
$497$	0	0
$498$	39.4987	1.76998
$499$	−32.9525	−1.47516	−0.737578	−	0.675262i	$-0.764031\pi$
$499$	−32.9525	−1.47516	−0.737578	+	0.675262i	$0.764031\pi$
$500$	−44.2397	−1.97846
$501$	55.1354	2.46327
$502$	25.3249	1.13031
$503$	−6.65795	−0.296863	−0.148432	−	0.988923i	$-0.547423\pi$
$503$	−6.65795	−0.296863	−0.148432	+	0.988923i	$0.547423\pi$
$504$	0	0
$505$	11.3618	0.505595
$506$	6.97019	0.309863
$507$	29.5083	1.31051
$508$	4.32847	0.192045
$509$	30.1687	1.33721	0.668603	−	0.743619i	$-0.266893\pi$
$509$	30.1687	1.33721	0.668603	+	0.743619i	$0.266893\pi$
$510$	14.2473	0.630879
$511$	0	0
$512$	−49.4718	−2.18636
$513$	−16.8919	−0.745796
$514$	−32.5722	−1.43670
$515$	3.25818	0.143573
$516$	−53.3092	−2.34681
$517$	3.34122	0.146947
$518$	0	0
$519$	58.7046	2.57685
$520$	−10.0050	−0.438749
$521$	45.2001	1.98025	0.990126	−	0.140184i	$-0.0447693\pi$
$521$	45.2001	1.98025	0.990126	+	0.140184i	$0.0447693\pi$
$522$	−5.59838	−0.245034
$523$	31.0371	1.35716	0.678578	−	0.734528i	$-0.262596\pi$
$523$	31.0371	1.35716	0.678578	+	0.734528i	$0.262596\pi$
$524$	−44.9259	−1.96260
$525$	0	0
$526$	−67.5199	−2.94401
$527$	6.47643	0.282118
$528$	68.6640	2.98822
$529$	−22.5625	−0.980980
$530$	−29.4368	−1.27865
$531$	−54.4880	−2.36458
$532$	0	0
$533$	12.1030	0.524238
$534$	82.3123	3.56200
$535$	−8.32415	−0.359885
$536$	−46.6268	−2.01397
$537$	−8.82779	−0.380947
$538$	43.0962	1.85801
$539$	0	0
$540$	−17.6673	−0.760278
$541$	28.5580	1.22781	0.613903	−	0.789381i	$-0.289599\pi$
$541$	28.5580	1.22781	0.613903	+	0.789381i	$0.289599\pi$
$542$	−64.3533	−2.76421
$543$	−44.5804	−1.91313
$544$	−6.30739	−0.270427
$545$	−2.04521	−0.0876072
$546$	0	0
$547$	44.5596	1.90523	0.952615	−	0.304179i	$-0.0983822\pi$
$547$	44.5596	1.90523	0.952615	+	0.304179i	$0.0983822\pi$
$548$	58.4149	2.49536
$549$	−0.259934	−0.0110937
$550$	37.7431	1.60937
$551$	−2.56831	−0.109414
$552$	10.4469	0.444650
$553$	0	0
$554$	−36.5375	−1.55233
$555$	−4.18127	−0.177485
$556$	60.4486	2.56359
$557$	−4.97061	−0.210612	−0.105306	−	0.994440i	$-0.533582\pi$
$557$	−4.97061	−0.210612	−0.105306	+	0.994440i	$0.533582\pi$
$558$	−39.4581	−1.67040
$559$	−6.55022	−0.277045
$560$	0	0
$561$	−19.9201	−0.841025
$562$	25.9285	1.09373
$563$	−35.6218	−1.50128	−0.750641	−	0.660710i	$-0.770255\pi$
$563$	−35.6218	−1.50128	−0.750641	+	0.660710i	$0.770255\pi$
$564$	9.30918	0.391987
$565$	−23.8192	−1.00208
$566$	50.0492	2.10372
$567$	0	0
$568$	37.9994	1.59442
$569$	−3.49593	−0.146557	−0.0732785	−	0.997312i	$-0.523346\pi$
$569$	−3.49593	−0.146557	−0.0732785	+	0.997312i	$0.523346\pi$
$570$	−39.8215	−1.66794
$571$	−8.82565	−0.369342	−0.184671	−	0.982800i	$-0.559122\pi$
$571$	−8.82565	−0.369342	−0.184671	+	0.982800i	$0.559122\pi$
$572$	26.0040	1.08728
$573$	−64.2950	−2.68596
$574$	0	0
$575$	2.36885	0.0987879
$576$	−13.4953	−0.562306
$577$	23.9481	0.996972	0.498486	−	0.866898i	$-0.333890\pi$
$577$	23.9481	0.996972	0.498486	+	0.866898i	$0.333890\pi$
$578$	−34.9426	−1.45342
$579$	−6.02055	−0.250206
$580$	−2.68620	−0.111538
$581$	0	0
$582$	0.415404	0.0172191
$583$	41.1576	1.70457
$584$	56.3255	2.33077
$585$	−7.29494	−0.301609
$586$	−20.8783	−0.862475
$587$	23.7662	0.980935	0.490468	−	0.871459i	$-0.336826\pi$
$587$	23.7662	0.980935	0.490468	+	0.871459i	$0.336826\pi$
$588$	0	0
$589$	−18.1018	−0.745872
$590$	−38.2245	−1.57368
$591$	−55.0493	−2.26443
$592$	7.91423	0.325273
$593$	6.53535	0.268375	0.134187	−	0.990956i	$-0.457158\pi$
$593$	6.53535	0.268375	0.134187	+	0.990956i	$0.457158\pi$
$594$	36.1154	1.48183
$595$	0	0
$596$	−62.3805	−2.55520
$597$	25.6661	1.05044
$598$	2.38619	0.0975786
$599$	9.00587	0.367970	0.183985	−	0.982929i	$-0.441100\pi$
$599$	9.00587	0.367970	0.183985	+	0.982929i	$0.441100\pi$
$600$	56.5692	2.30943
$601$	−16.6060	−0.677375	−0.338687	−	0.940899i	$-0.609983\pi$
$601$	−16.6060	−0.677375	−0.338687	+	0.940899i	$0.609983\pi$
$602$	0	0
$603$	−33.9969	−1.38446
$604$	−39.5511	−1.60931
$605$	7.79961	0.317099
$606$	−64.7111	−2.62871
$607$	26.6853	1.08312	0.541561	−	0.840661i	$-0.317833\pi$
$607$	26.6853	1.08312	0.541561	+	0.840661i	$0.317833\pi$
$608$	17.6293	0.714963
$609$	0	0
$610$	−0.182349	−0.00738309
$611$	1.14384	0.0462749
$612$	−32.6009	−1.31781
$613$	−7.72787	−0.312126	−0.156063	−	0.987747i	$-0.549880\pi$
$613$	−7.72787	−0.312126	−0.156063	+	0.987747i	$0.549880\pi$
$614$	41.2976	1.66663
$615$	27.1030	1.09290
$616$	0	0
$617$	−5.07794	−0.204430	−0.102215	−	0.994762i	$-0.532593\pi$
$617$	−5.07794	−0.204430	−0.102215	+	0.994762i	$0.532593\pi$
$618$	−18.5569	−0.746468
$619$	27.0052	1.08543	0.542715	−	0.839917i	$-0.317396\pi$
$619$	27.0052	1.08543	0.542715	+	0.839917i	$0.317396\pi$
$620$	−18.9327	−0.760355
$621$	2.26669	0.0909593
$622$	22.9659	0.920848
$623$	0	0
$624$	23.5066	0.941017
$625$	5.73466	0.229386
$626$	8.13418	0.325107
$627$	55.6771	2.22353
$628$	−42.8103	−1.70832
$629$	−2.29599	−0.0915471
$630$	0	0
$631$	−9.84678	−0.391994	−0.195997	−	0.980604i	$-0.562794\pi$
$631$	−9.84678	−0.391994	−0.195997	+	0.980604i	$0.562794\pi$
$632$	89.9266	3.57709
$633$	32.6968	1.29958
$634$	−31.9330	−1.26822
$635$	1.19101	0.0472637
$636$	114.672	4.54703
$637$	0	0
$638$	5.49113	0.217396
$639$	27.7064	1.09605
$640$	−17.9869	−0.710995
$641$	37.7457	1.49087	0.745433	−	0.666580i	$-0.232243\pi$
$641$	37.7457	1.49087	0.745433	+	0.666580i	$0.232243\pi$
$642$	47.4100	1.87112
$643$	−47.5016	−1.87328	−0.936640	−	0.350293i	$-0.886082\pi$
$643$	−47.5016	−1.87328	−0.936640	+	0.350293i	$0.886082\pi$
$644$	0	0
$645$	−14.6684	−0.577567
$646$	−21.8665	−0.860326
$647$	−34.5662	−1.35894	−0.679469	−	0.733704i	$-0.737790\pi$
$647$	−34.5662	−1.35894	−0.679469	+	0.733704i	$0.737790\pi$
$648$	−20.9227	−0.821921
$649$	53.4442	2.09787
$650$	12.9211	0.506806
$651$	0	0
$652$	55.1822	2.16110
$653$	−4.89201	−0.191439	−0.0957196	−	0.995408i	$-0.530515\pi$
$653$	−4.89201	−0.191439	−0.0957196	+	0.995408i	$0.530515\pi$
$654$	11.6485	0.455491
$655$	−12.3617	−0.483010
$656$	−51.3002	−2.00294
$657$	41.0685	1.60223
$658$	0	0
$659$	22.2731	0.867635	0.433817	−	0.901001i	$-0.357166\pi$
$659$	22.2731	0.867635	0.433817	+	0.901001i	$0.357166\pi$
$660$	58.2327	2.26671
$661$	−24.9111	−0.968929	−0.484464	−	0.874811i	$-0.660985\pi$
$661$	−24.9111	−0.968929	−0.484464	+	0.874811i	$0.660985\pi$
$662$	6.81273	0.264784
$663$	−6.81948	−0.264846
$664$	−34.1081	−1.32365
$665$	0	0
$666$	13.9885	0.542043
$667$	0.344637	0.0133444
$668$	−88.5054	−3.42438
$669$	16.2152	0.626915
$670$	−23.8495	−0.921387
$671$	0.254954	0.00984240
$672$	0	0
$673$	−18.1608	−0.700048	−0.350024	−	0.936741i	$-0.613827\pi$
$673$	−18.1608	−0.700048	−0.350024	+	0.936741i	$0.613827\pi$
$674$	71.3890	2.74980
$675$	12.2740	0.472426
$676$	−47.3678	−1.82184
$677$	−13.4238	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$677$	−13.4238	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$678$	135.661	5.21005
$679$	0	0
$680$	−12.3029	−0.471793
$681$	−54.4251	−2.08557
$682$	38.7022	1.48199
$683$	37.8944	1.44999	0.724995	−	0.688754i	$-0.241842\pi$
$683$	37.8944	1.44999	0.724995	+	0.688754i	$0.241842\pi$
$684$	91.1205	3.48408
$685$	16.0733	0.614128
$686$	0	0
$687$	29.1365	1.11163
$688$	27.7641	1.05850
$689$	14.0900	0.536786
$690$	5.34357	0.203426
$691$	16.2233	0.617162	0.308581	−	0.951198i	$-0.400146\pi$
$691$	16.2233	0.617162	0.308581	+	0.951198i	$0.400146\pi$
$692$	−94.2348	−3.58227
$693$	0	0
$694$	−3.00972	−0.114247
$695$	16.6328	0.630920
$696$	8.23008	0.311961
$697$	14.8827	0.563721
$698$	17.3964	0.658462
$699$	47.7915	1.80764
$700$	0	0
$701$	38.4189	1.45106	0.725530	−	0.688190i	$-0.241594\pi$
$701$	38.4189	1.45106	0.725530	+	0.688190i	$0.241594\pi$
$702$	12.3638	0.466643
$703$	6.41736	0.242035
$704$	13.2368	0.498881
$705$	2.56148	0.0964711
$706$	−53.2708	−2.00487
$707$	0	0
$708$	148.904	5.59617
$709$	29.6978	1.11532	0.557662	−	0.830068i	$-0.311698\pi$
$709$	29.6978	1.11532	0.557662	+	0.830068i	$0.311698\pi$
$710$	19.4366	0.729444
$711$	65.5680	2.45899
$712$	−71.0788	−2.66379
$713$	2.42905	0.0909686
$714$	0	0
$715$	7.15519	0.267589
$716$	14.1707	0.529583
$717$	−43.9532	−1.64146
$718$	−56.5673	−2.11107
$719$	45.3102	1.68979	0.844893	−	0.534935i	$-0.179664\pi$
$719$	45.3102	1.68979	0.844893	+	0.534935i	$0.179664\pi$
$720$	30.9207	1.15235
$721$	0	0
$722$	13.3203	0.495728
$723$	−44.2900	−1.64716
$724$	71.5620	2.65958
$725$	1.86618	0.0693083
$726$	−44.4225	−1.64867
$727$	39.3597	1.45977	0.729885	−	0.683570i	$-0.239574\pi$
$727$	39.3597	1.45977	0.729885	+	0.683570i	$0.239574\pi$
$728$	0	0
$729$	−42.9781	−1.59178
$730$	28.8104	1.06632
$731$	−8.05461	−0.297911
$732$	0.710344	0.0262551
$733$	27.2312	1.00581	0.502903	−	0.864343i	$-0.332265\pi$
$733$	27.2312	1.00581	0.502903	+	0.864343i	$0.332265\pi$
$734$	−43.2536	−1.59652
$735$	0	0
$736$	−2.36565	−0.0871989
$737$	33.3456	1.22830
$738$	−90.6737	−3.33774
$739$	−22.2361	−0.817967	−0.408984	−	0.912542i	$-0.634117\pi$
$739$	−22.2361	−0.817967	−0.408984	+	0.912542i	$0.634117\pi$
$740$	6.71192	0.246735
$741$	19.0606	0.700209
$742$	0	0
$743$	28.0249	1.02814	0.514068	−	0.857750i	$-0.328138\pi$
$743$	28.0249	1.02814	0.514068	+	0.857750i	$0.328138\pi$
$744$	58.0068	2.12663
$745$	−17.1644	−0.628855
$746$	21.5818	0.790165
$747$	−24.8692	−0.909915
$748$	31.9764	1.16917
$749$	0	0
$750$	69.3302	2.53158
$751$	9.19486	0.335525	0.167763	−	0.985827i	$-0.446346\pi$
$751$	9.19486	0.335525	0.167763	+	0.985827i	$0.446346\pi$
$752$	−4.84834	−0.176801
$753$	−27.1453	−0.989228
$754$	1.87985	0.0684599
$755$	−10.8828	−0.396064
$756$	0	0
$757$	8.04507	0.292403	0.146202	−	0.989255i	$-0.453295\pi$
$757$	8.04507	0.292403	0.146202	+	0.989255i	$0.453295\pi$
$758$	−11.5889	−0.420928
$759$	−7.47121	−0.271188
$760$	34.3868	1.24734
$761$	−7.85377	−0.284699	−0.142350	−	0.989816i	$-0.545466\pi$
$761$	−7.85377	−0.284699	−0.142350	+	0.989816i	$0.545466\pi$
$762$	−6.78336	−0.245735
$763$	0	0
$764$	103.209	3.73396
$765$	−8.97036	−0.324324
$766$	53.7589	1.94239
$767$	18.2962	0.660638
$768$	85.4034	3.08173
$769$	20.0308	0.722331	0.361165	−	0.932502i	$-0.382379\pi$
$769$	20.0308	0.722331	0.361165	+	0.932502i	$0.382379\pi$
$770$	0	0
$771$	34.9134	1.25738
$772$	9.66441	0.347830
$773$	28.9847	1.04251	0.521253	−	0.853402i	$-0.325465\pi$
$773$	28.9847	1.04251	0.521253	+	0.853402i	$0.325465\pi$
$774$	49.0733	1.76390
$775$	13.1531	0.472474
$776$	−0.358712	−0.0128770
$777$	0	0
$778$	−62.0887	−2.22599
$779$	−41.5974	−1.49038
$780$	19.9355	0.713806
$781$	−27.1757	−0.972422
$782$	2.93423	0.104928
$783$	1.78570	0.0638159
$784$	0	0
$785$	−11.7795	−0.420430
$786$	70.4056	2.51128
$787$	24.5826	0.876276	0.438138	−	0.898908i	$-0.355638\pi$
$787$	24.5826	0.876276	0.438138	+	0.898908i	$0.355638\pi$
$788$	88.3671	3.14795
$789$	72.3732	2.57656
$790$	45.9973	1.63651
$791$	0	0
$792$	−104.801	−3.72395
$793$	0.0872816	0.00309946
$794$	−52.9381	−1.87870
$795$	31.5527	1.11906
$796$	−41.2002	−1.46030
$797$	36.3224	1.28661	0.643303	−	0.765611i	$-0.277563\pi$
$797$	36.3224	1.28661	0.643303	+	0.765611i	$0.277563\pi$
$798$	0	0
$799$	1.40655	0.0497600
$800$	−12.8098	−0.452895
$801$	−51.8255	−1.83117
$802$	32.0564	1.13195
$803$	−40.2818	−1.42151
$804$	92.9063	3.27655
$805$	0	0
$806$	13.2494	0.466691
$807$	−46.1939	−1.62610
$808$	55.8797	1.96584
$809$	36.0132	1.26616	0.633079	−	0.774087i	$-0.281791\pi$
$809$	36.0132	1.26616	0.633079	+	0.774087i	$0.281791\pi$
$810$	−10.7019	−0.376027
$811$	26.6085	0.934350	0.467175	−	0.884165i	$-0.345272\pi$
$811$	26.6085	0.934350	0.467175	+	0.884165i	$0.345272\pi$
$812$	0	0
$813$	68.9790	2.41920
$814$	−13.7205	−0.480904
$815$	15.1838	0.531864
$816$	28.9054	1.01189
$817$	22.5129	0.787625
$818$	26.2876	0.919126
$819$	0	0
$820$	−43.5068	−1.51932
$821$	−26.1926	−0.914130	−0.457065	−	0.889433i	$-0.651099\pi$
$821$	−26.1926	−0.914130	−0.457065	+	0.889433i	$0.651099\pi$
$822$	−91.5449	−3.19299
$823$	11.7091	0.408155	0.204077	−	0.978955i	$-0.434581\pi$
$823$	11.7091	0.408155	0.204077	+	0.978955i	$0.434581\pi$
$824$	16.0244	0.558235
$825$	−40.4561	−1.40850
$826$	0	0
$827$	18.5235	0.644126	0.322063	−	0.946718i	$-0.395624\pi$
$827$	18.5235	0.644126	0.322063	+	0.946718i	$0.395624\pi$
$828$	−12.2273	−0.424928
$829$	42.3461	1.47074	0.735370	−	0.677666i	$-0.237008\pi$
$829$	42.3461	1.47074	0.735370	+	0.677666i	$0.237008\pi$
$830$	−17.4462	−0.605567
$831$	39.1638	1.35858
$832$	4.53152	0.157102
$833$	0	0
$834$	−94.7320	−3.28030
$835$	−24.3529	−0.842766
$836$	−89.3749	−3.09110
$837$	12.5859	0.435032
$838$	−90.2428	−3.11738
$839$	44.7000	1.54322	0.771608	−	0.636098i	$-0.219453\pi$
$839$	44.7000	1.54322	0.771608	+	0.636098i	$0.219453\pi$
$840$	0	0
$841$	−28.7285	−0.990638
$842$	5.84875	0.201561
$843$	−27.7922	−0.957214
$844$	−52.4860	−1.80664
$845$	−13.0336	−0.448369
$846$	−8.56949	−0.294625
$847$	0	0
$848$	−59.7225	−2.05088
$849$	−53.6467	−1.84115
$850$	15.8886	0.544976
$851$	−0.861133	−0.0295193
$852$	−75.7159	−2.59398
$853$	−7.80530	−0.267248	−0.133624	−	0.991032i	$-0.542662\pi$
$853$	−7.80530	−0.267248	−0.133624	+	0.991032i	$0.542662\pi$
$854$	0	0
$855$	25.0724	0.857458
$856$	−40.9398	−1.39929
$857$	1.90271	0.0649954	0.0324977	−	0.999472i	$-0.489654\pi$
$857$	1.90271	0.0649954	0.0324977	+	0.999472i	$0.489654\pi$
$858$	−40.7522	−1.39126
$859$	28.4688	0.971343	0.485671	−	0.874142i	$-0.338575\pi$
$859$	28.4688	0.971343	0.485671	+	0.874142i	$0.338575\pi$
$860$	23.5462	0.802919
$861$	0	0
$862$	87.6499	2.98537
$863$	−13.4857	−0.459058	−0.229529	−	0.973302i	$-0.573719\pi$
$863$	−13.4857	−0.459058	−0.229529	+	0.973302i	$0.573719\pi$
$864$	−12.2574	−0.417005
$865$	−25.9293	−0.881624
$866$	6.19067	0.210368
$867$	37.4543	1.27201
$868$	0	0
$869$	−64.3120	−2.18163
$870$	4.20967	0.142721
$871$	11.4156	0.386803
$872$	−10.0587	−0.340632
$873$	−0.261547	−0.00885202
$874$	−8.20125	−0.277411
$875$	0	0
$876$	−112.232	−3.79195
$877$	−44.3003	−1.49592	−0.747958	−	0.663746i	$-0.768965\pi$
$877$	−44.3003	−1.49592	−0.747958	+	0.663746i	$0.768965\pi$
$878$	−12.2435	−0.413198
$879$	22.3790	0.754826
$880$	−30.3283	−1.02237
$881$	6.63739	0.223619	0.111810	−	0.993730i	$-0.464335\pi$
$881$	6.63739	0.223619	0.111810	+	0.993730i	$0.464335\pi$
$882$	0	0
$883$	−2.05879	−0.0692840	−0.0346420	−	0.999400i	$-0.511029\pi$
$883$	−2.05879	−0.0692840	−0.0346420	+	0.999400i	$0.511029\pi$
$884$	10.9469	0.368183
$885$	40.9720	1.37726
$886$	14.9619	0.502655
$887$	30.8935	1.03730	0.518650	−	0.854986i	$-0.326435\pi$
$887$	30.8935	1.03730	0.518650	+	0.854986i	$0.326435\pi$
$888$	−20.5643	−0.690091
$889$	0	0
$890$	−36.3567	−1.21868
$891$	14.9631	0.501282
$892$	−26.0292	−0.871522
$893$	−3.93134	−0.131557
$894$	97.7595	3.26957
$895$	3.89916	0.130335
$896$	0	0
$897$	−2.55771	−0.0853995
$898$	−92.3340	−3.08122
$899$	1.91361	0.0638224
$900$	−66.2099	−2.20700
$901$	17.3260	0.577214
$902$	88.9367	2.96127
$903$	0	0
$904$	−117.147	−3.89625
$905$	19.6908	0.654544
$906$	61.9825	2.05923
$907$	−41.0485	−1.36299	−0.681496	−	0.731822i	$-0.738670\pi$
$907$	−41.0485	−1.36299	−0.681496	+	0.731822i	$0.738670\pi$
$908$	87.3652	2.89931
$909$	40.7435	1.35138
$910$	0	0
$911$	26.5615	0.880022	0.440011	−	0.897992i	$-0.354975\pi$
$911$	26.5615	0.880022	0.440011	+	0.897992i	$0.354975\pi$
$912$	−80.7912	−2.67527
$913$	24.3928	0.807282
$914$	−76.7348	−2.53816
$915$	0.195456	0.00646158
$916$	−46.7710	−1.54536
$917$	0	0
$918$	15.2034	0.501788
$919$	17.7417	0.585244	0.292622	−	0.956228i	$-0.405472\pi$
$919$	17.7417	0.585244	0.292622	+	0.956228i	$0.405472\pi$
$920$	−4.61431	−0.152129
$921$	−44.2660	−1.45861
$922$	−65.8533	−2.16876
$923$	−9.30338	−0.306225
$924$	0	0
$925$	−4.66297	−0.153318
$926$	75.0470	2.46620
$927$	11.6838	0.383747
$928$	−1.86366	−0.0611776
$929$	−7.81515	−0.256407	−0.128203	−	0.991748i	$-0.540921\pi$
$929$	−7.81515	−0.256407	−0.128203	+	0.991748i	$0.540921\pi$
$930$	29.6704	0.972929
$931$	0	0
$932$	−76.7167	−2.51294
$933$	−24.6167	−0.805913
$934$	66.4500	2.17431
$935$	8.79852	0.287742
$936$	−35.8779	−1.17271
$937$	−8.84523	−0.288961	−0.144481	−	0.989508i	$-0.546151\pi$
$937$	−8.84523	−0.288961	−0.144481	+	0.989508i	$0.546151\pi$
$938$	0	0
$939$	−8.71886	−0.284529
$940$	−4.11179	−0.134112
$941$	−49.2706	−1.60618	−0.803089	−	0.595860i	$-0.796811\pi$
$941$	−49.2706	−1.60618	−0.803089	+	0.595860i	$0.796811\pi$
$942$	67.0901	2.18591
$943$	5.58188	0.181771
$944$	−77.5512	−2.52408
$945$	0	0
$946$	−48.1332	−1.56495
$947$	13.1496	0.427304	0.213652	−	0.976910i	$-0.431464\pi$
$947$	13.1496	0.427304	0.213652	+	0.976910i	$0.431464\pi$
$948$	−179.184	−5.81961
$949$	−13.7901	−0.447647
$950$	−44.4092	−1.44082
$951$	34.2284	1.10993
$952$	0	0
$953$	−28.3848	−0.919474	−0.459737	−	0.888055i	$-0.652056\pi$
$953$	−28.3848	−0.919474	−0.459737	+	0.888055i	$0.652056\pi$
$954$	−105.560	−3.41763
$955$	28.3986	0.918956
$956$	70.5552	2.28192
$957$	−5.88583	−0.190262
$958$	−75.0456	−2.42461
$959$	0	0
$960$	10.1478	0.327518
$961$	−17.5126	−0.564924
$962$	−4.69711	−0.151441
$963$	−29.8503	−0.961913
$964$	71.0959	2.28985
$965$	2.65923	0.0856035
$966$	0	0
$967$	0.655032	0.0210644	0.0105322	−	0.999945i	$-0.496647\pi$
$967$	0.655032	0.0210644	0.0105322	+	0.999945i	$0.496647\pi$
$968$	38.3600	1.23294
$969$	23.4383	0.752946
$970$	−0.183481	−0.00589120
$971$	−13.2088	−0.423892	−0.211946	−	0.977281i	$-0.567980\pi$
$971$	−13.2088	−0.423892	−0.211946	+	0.977281i	$0.567980\pi$
$972$	86.1912	2.76458
$973$	0	0
$974$	76.6965	2.45751
$975$	−13.8498	−0.443549
$976$	−0.369956	−0.0118420
$977$	−41.1030	−1.31500	−0.657501	−	0.753454i	$-0.728386\pi$
$977$	−41.1030	−1.31500	−0.657501	+	0.753454i	$0.728386\pi$
$978$	−86.4787	−2.76528
$979$	50.8327	1.62462
$980$	0	0
$981$	−7.33410	−0.234160
$982$	31.8012	1.01482
$983$	−31.5908	−1.00759	−0.503795	−	0.863823i	$-0.668063\pi$
$983$	−31.5908	−1.00759	−0.503795	+	0.863823i	$0.668063\pi$
$984$	133.298	4.24938
$985$	24.3148	0.774735
$986$	2.31159	0.0736160
$987$	0	0
$988$	−30.5968	−0.973414
$989$	−3.02096	−0.0960609
$990$	−53.6056	−1.70370
$991$	−18.8619	−0.599168	−0.299584	−	0.954070i	$-0.596848\pi$
$991$	−18.8619	−0.599168	−0.299584	+	0.954070i	$0.596848\pi$
$992$	−13.1353	−0.417047
$993$	−7.30242	−0.231735
$994$	0	0
$995$	−11.3365	−0.359392
$996$	67.9622	2.15346
$997$	−10.7025	−0.338952	−0.169476	−	0.985534i	$-0.554207\pi$
$997$	−10.7025	−0.338952	−0.169476	+	0.985534i	$0.554207\pi$
$998$	−82.8967	−2.62405
$999$	−4.46188	−0.141168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.j.1.14		15
7.6	odd	2		889.2.a.b.1.14	✓	15
21.20	even	2		8001.2.a.q.1.2		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.14	✓	15	7.6	odd	2
6223.2.a.j.1.14		15	1.1	even	1	trivial
8001.2.a.q.1.2		15	21.20	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:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51564 q^{2} -2.69647 q^{3} +4.32847 q^{4} +1.19101 q^{5} -6.78336 q^{6} +5.85760 q^{8} +4.27094 q^{9} +O(q^{10})\)	\(q+2.51564 q^{2} -2.69647 q^{3} +4.32847 q^{4} +1.19101 q^{5} -6.78336 q^{6} +5.85760 q^{8} +4.27094 q^{9} +2.99615 q^{10} -4.18912 q^{11} -11.6716 q^{12} -1.43411 q^{13} -3.21151 q^{15} +6.07870 q^{16} -1.76349 q^{17} +10.7442 q^{18} +4.92899 q^{19} +5.15524 q^{20} -10.5383 q^{22} -0.661413 q^{23} -15.7948 q^{24} -3.58150 q^{25} -3.60772 q^{26} -3.42705 q^{27} -0.521062 q^{29} -8.07903 q^{30} -3.67252 q^{31} +3.57666 q^{32} +11.2958 q^{33} -4.43630 q^{34} +18.4866 q^{36} +1.30196 q^{37} +12.3996 q^{38} +3.86704 q^{39} +6.97644 q^{40} -8.43934 q^{41} +4.56744 q^{43} -18.1325 q^{44} +5.08672 q^{45} -1.66388 q^{46} -0.797594 q^{47} -16.3910 q^{48} -9.00978 q^{50} +4.75518 q^{51} -6.20752 q^{52} -9.82487 q^{53} -8.62124 q^{54} -4.98928 q^{55} -13.2909 q^{57} -1.31081 q^{58} -12.7579 q^{59} -13.9009 q^{60} -0.0608610 q^{61} -9.23874 q^{62} -3.15981 q^{64} -1.70804 q^{65} +28.4163 q^{66} -7.96005 q^{67} -7.63319 q^{68} +1.78348 q^{69} +6.48720 q^{71} +25.0175 q^{72} +9.61580 q^{73} +3.27527 q^{74} +9.65740 q^{75} +21.3350 q^{76} +9.72810 q^{78} +15.3521 q^{79} +7.23978 q^{80} -3.57189 q^{81} -21.2304 q^{82} -5.82288 q^{83} -2.10033 q^{85} +11.4900 q^{86} +1.40503 q^{87} -24.5382 q^{88} -12.1345 q^{89} +12.7964 q^{90} -2.86290 q^{92} +9.90282 q^{93} -2.00646 q^{94} +5.87047 q^{95} -9.64434 q^{96} -0.0612387 q^{97} -17.8915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 4 q^{3} + 14 q^{4} - 7 q^{5} - 8 q^{6} + 9 q^{9}+O(q^{10}) \) 15 * q - 4 * q^3 + 14 * q^4 - 7 * q^5 - 8 * q^6 + 9 * q^9	\( 15 q - 4 q^{3} + 14 q^{4} - 7 q^{5} - 8 q^{6} + 9 q^{9} - 10 q^{10} + 14 q^{11} - 10 q^{12} - 6 q^{13} + 6 q^{15} + 20 q^{16} - 10 q^{17} + q^{18} - 13 q^{19} - 8 q^{20} - 11 q^{22} + 15 q^{23} - 34 q^{24} - 22 q^{26} - 22 q^{27} + 16 q^{29} + 7 q^{30} - 22 q^{31} - 14 q^{33} + 15 q^{34} + 20 q^{36} - 14 q^{37} + 6 q^{38} + 29 q^{39} - 22 q^{40} - 19 q^{41} - q^{43} + 25 q^{44} + 8 q^{45} - 28 q^{46} - 49 q^{47} + 14 q^{48} + 24 q^{50} - 8 q^{51} + 17 q^{52} - 28 q^{53} - 13 q^{54} - 39 q^{55} - 12 q^{57} - 10 q^{58} - 43 q^{59} - 60 q^{60} - 27 q^{61} - 14 q^{62} + 18 q^{64} - 8 q^{65} + 36 q^{66} + 3 q^{67} - 13 q^{68} + 17 q^{69} + 55 q^{71} - 21 q^{72} + 3 q^{73} - 12 q^{74} - 8 q^{75} + 20 q^{76} - 6 q^{78} + 18 q^{79} - 29 q^{80} - 17 q^{81} - 14 q^{82} - 17 q^{83} + 7 q^{85} + 4 q^{86} - 35 q^{87} - 114 q^{88} - 36 q^{89} + 39 q^{90} + 45 q^{92} + 15 q^{93} + 15 q^{94} + 59 q^{95} - 85 q^{96} + 2 q^{97} + 35 q^{99}+O(q^{100}) \) 15 * q - 4 * q^3 + 14 * q^4 - 7 * q^5 - 8 * q^6 + 9 * q^9 - 10 * q^10 + 14 * q^11 - 10 * q^12 - 6 * q^13 + 6 * q^15 + 20 * q^16 - 10 * q^17 + q^18 - 13 * q^19 - 8 * q^20 - 11 * q^22 + 15 * q^23 - 34 * q^24 - 22 * q^26 - 22 * q^27 + 16 * q^29 + 7 * q^30 - 22 * q^31 - 14 * q^33 + 15 * q^34 + 20 * q^36 - 14 * q^37 + 6 * q^38 + 29 * q^39 - 22 * q^40 - 19 * q^41 - q^43 + 25 * q^44 + 8 * q^45 - 28 * q^46 - 49 * q^47 + 14 * q^48 + 24 * q^50 - 8 * q^51 + 17 * q^52 - 28 * q^53 - 13 * q^54 - 39 * q^55 - 12 * q^57 - 10 * q^58 - 43 * q^59 - 60 * q^60 - 27 * q^61 - 14 * q^62 + 18 * q^64 - 8 * q^65 + 36 * q^66 + 3 * q^67 - 13 * q^68 + 17 * q^69 + 55 * q^71 - 21 * q^72 + 3 * q^73 - 12 * q^74 - 8 * q^75 + 20 * q^76 - 6 * q^78 + 18 * q^79 - 29 * q^80 - 17 * q^81 - 14 * q^82 - 17 * q^83 + 7 * q^85 + 4 * q^86 - 35 * q^87 - 114 * q^88 - 36 * q^89 + 39 * q^90 + 45 * q^92 + 15 * q^93 + 15 * q^94 + 59 * q^95 - 85 * q^96 + 2 * q^97 + 35 * q^99

Introduction

L-functions

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