LMFDB - Embedded newform 409.2.a.b.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [409,2,Mod(1,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 409 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.26588144267$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 126 x^{17} + 100 x^{16} - 1283 x^{15} + 247 x^{14} + 6767 x^{13} + \cdots - 4 $ x^20 - 5x^19 - 19x^18 + 126x^17 + 100x^16 - 1283x^15 + 247x^14 + 6767x^13 - 4554x^12 - 19689x^11 + 18771x^10 + 31011x^9 - 35515x^8 - 23548x^7 + 31466x^6 + 5354x^5 - 10552x^4 + 1129x^3 + 523x^2 - 54*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.84704$ of defining polynomial
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.84704 q^{2} +2.27493 q^{3} +1.41156 q^{4} +0.866439 q^{5} -4.20189 q^{6} +4.45724 q^{7} +1.08688 q^{8} +2.17533 q^{9} -1.60035 q^{10} +0.938829 q^{11} +3.21120 q^{12} -4.86713 q^{13} -8.23270 q^{14} +1.97109 q^{15} -4.83062 q^{16} +5.33791 q^{17} -4.01792 q^{18} -2.46577 q^{19} +1.22303 q^{20} +10.1399 q^{21} -1.73406 q^{22} +3.66124 q^{23} +2.47258 q^{24} -4.24928 q^{25} +8.98978 q^{26} -1.87608 q^{27} +6.29165 q^{28} -1.09032 q^{29} -3.64068 q^{30} -7.99217 q^{31} +6.74859 q^{32} +2.13578 q^{33} -9.85934 q^{34} +3.86192 q^{35} +3.07060 q^{36} +9.64459 q^{37} +4.55438 q^{38} -11.0724 q^{39} +0.941713 q^{40} -1.26932 q^{41} -18.7288 q^{42} +3.56408 q^{43} +1.32521 q^{44} +1.88479 q^{45} -6.76245 q^{46} -2.87779 q^{47} -10.9893 q^{48} +12.8670 q^{49} +7.84860 q^{50} +12.1434 q^{51} -6.87023 q^{52} +9.88144 q^{53} +3.46519 q^{54} +0.813438 q^{55} +4.84448 q^{56} -5.60946 q^{57} +2.01387 q^{58} +7.19174 q^{59} +2.78231 q^{60} -7.34223 q^{61} +14.7619 q^{62} +9.69595 q^{63} -2.80368 q^{64} -4.21707 q^{65} -3.94486 q^{66} -12.3049 q^{67} +7.53477 q^{68} +8.32907 q^{69} -7.13313 q^{70} -1.89104 q^{71} +2.36432 q^{72} +5.02515 q^{73} -17.8139 q^{74} -9.66684 q^{75} -3.48057 q^{76} +4.18459 q^{77} +20.4512 q^{78} +6.32284 q^{79} -4.18544 q^{80} -10.7939 q^{81} +2.34449 q^{82} -15.2910 q^{83} +14.3131 q^{84} +4.62498 q^{85} -6.58300 q^{86} -2.48041 q^{87} +1.02039 q^{88} -5.43146 q^{89} -3.48128 q^{90} -21.6940 q^{91} +5.16804 q^{92} -18.1817 q^{93} +5.31540 q^{94} -2.13644 q^{95} +15.3526 q^{96} -2.27190 q^{97} -23.7658 q^{98} +2.04226 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) $ 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

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 $

$2$	−1.84704	−1.30605	−0.653027	−	0.757334i	$-0.726501\pi$
$2$	−1.84704	−1.30605	−0.653027	+	0.757334i	$0.726501\pi$
$3$	2.27493	1.31343	0.656717	−	0.754137i	$-0.271945\pi$
$3$	2.27493	1.31343	0.656717	+	0.754137i	$0.271945\pi$
$4$	1.41156	0.705778
$5$	0.866439	0.387483	0.193742	−	0.981053i	$-0.437938\pi$
$5$	0.866439	0.387483	0.193742	+	0.981053i	$0.437938\pi$
$6$	−4.20189	−1.71542
$7$	4.45724	1.68468	0.842339	−	0.538948i	$-0.181178\pi$
$7$	4.45724	1.68468	0.842339	+	0.538948i	$0.181178\pi$
$8$	1.08688	0.384270
$9$	2.17533	0.725109
$10$	−1.60035	−0.506074
$11$	0.938829	0.283068	0.141534	−	0.989933i	$-0.454797\pi$
$11$	0.938829	0.283068	0.141534	+	0.989933i	$0.454797\pi$
$12$	3.21120	0.926993
$13$	−4.86713	−1.34990	−0.674949	−	0.737864i	$-0.735835\pi$
$13$	−4.86713	−1.34990	−0.674949	+	0.737864i	$0.735835\pi$
$14$	−8.23270	−2.20028
$15$	1.97109	0.508934
$16$	−4.83062	−1.20766
$17$	5.33791	1.29463	0.647317	−	0.762221i	$-0.275891\pi$
$17$	5.33791	1.29463	0.647317	+	0.762221i	$0.275891\pi$
$18$	−4.01792	−0.947032
$19$	−2.46577	−0.565686	−0.282843	−	0.959166i	$-0.591278\pi$
$19$	−2.46577	−0.565686	−0.282843	+	0.959166i	$0.591278\pi$
$20$	1.22303	0.273477
$21$	10.1399	2.21271
$22$	−1.73406	−0.369702
$23$	3.66124	0.763421	0.381710	−	0.924282i	$-0.375335\pi$
$23$	3.66124	0.763421	0.381710	+	0.924282i	$0.375335\pi$
$24$	2.47258	0.504713
$25$	−4.24928	−0.849857
$26$	8.98978	1.76304
$27$	−1.87608	−0.361051
$28$	6.29165	1.18901
$29$	−1.09032	−0.202468	−0.101234	−	0.994863i	$-0.532279\pi$
$29$	−1.09032	−0.202468	−0.101234	+	0.994863i	$0.532279\pi$
$30$	−3.64068	−0.664695
$31$	−7.99217	−1.43544	−0.717718	−	0.696334i	$-0.754813\pi$
$31$	−7.99217	−1.43544	−0.717718	+	0.696334i	$0.754813\pi$
$32$	6.74859	1.19299
$33$	2.13578	0.371791
$34$	−9.85934	−1.69086
$35$	3.86192	0.652784
$36$	3.07060	0.511766
$37$	9.64459	1.58556	0.792780	−	0.609508i	$-0.208633\pi$
$37$	9.64459	1.58556	0.792780	+	0.609508i	$0.208633\pi$
$38$	4.55438	0.738817
$39$	−11.0724	−1.77300
$40$	0.941713	0.148898
$41$	−1.26932	−0.198235	−0.0991175	−	0.995076i	$-0.531602\pi$
$41$	−1.26932	−0.198235	−0.0991175	+	0.995076i	$0.531602\pi$
$42$	−18.7288	−2.88992
$43$	3.56408	0.543518	0.271759	−	0.962365i	$-0.412395\pi$
$43$	3.56408	0.543518	0.271759	+	0.962365i	$0.412395\pi$
$44$	1.32521	0.199783
$45$	1.88479	0.280967
$46$	−6.76245	−0.997069
$47$	−2.87779	−0.419769	−0.209885	−	0.977726i	$-0.567309\pi$
$47$	−2.87779	−0.419769	−0.209885	+	0.977726i	$0.567309\pi$
$48$	−10.9893	−1.58618
$49$	12.8670	1.83814
$50$	7.84860	1.10996
$51$	12.1434	1.70042
$52$	−6.87023	−0.952729
$53$	9.88144	1.35732	0.678660	−	0.734452i	$-0.262561\pi$
$53$	9.88144	1.35732	0.678660	+	0.734452i	$0.262561\pi$
$54$	3.46519	0.471553
$55$	0.813438	0.109684
$56$	4.84448	0.647370
$57$	−5.60946	−0.742992
$58$	2.01387	0.264434
$59$	7.19174	0.936285	0.468142	−	0.883653i	$-0.344923\pi$
$59$	7.19174	0.936285	0.468142	+	0.883653i	$0.344923\pi$
$60$	2.78231	0.359194
$61$	−7.34223	−0.940076	−0.470038	−	0.882646i	$-0.655760\pi$
$61$	−7.34223	−0.940076	−0.470038	+	0.882646i	$0.655760\pi$
$62$	14.7619	1.87476
$63$	9.69595	1.22157
$64$	−2.80368	−0.350460
$65$	−4.21707	−0.523063
$66$	−3.94486	−0.485579
$67$	−12.3049	−1.50328	−0.751641	−	0.659573i	$-0.770737\pi$
$67$	−12.3049	−1.50328	−0.751641	+	0.659573i	$0.770737\pi$
$68$	7.53477	0.913725
$69$	8.32907	1.00270
$70$	−7.13313	−0.852572
$71$	−1.89104	−0.224425	−0.112213	−	0.993684i	$-0.535794\pi$
$71$	−1.89104	−0.224425	−0.112213	+	0.993684i	$0.535794\pi$
$72$	2.36432	0.278637
$73$	5.02515	0.588149	0.294074	−	0.955783i	$-0.404989\pi$
$73$	5.02515	0.588149	0.294074	+	0.955783i	$0.404989\pi$
$74$	−17.8139	−2.07083
$75$	−9.66684	−1.11623
$76$	−3.48057	−0.399249
$77$	4.18459	0.476878
$78$	20.4512	2.31564
$79$	6.32284	0.711375	0.355688	−	0.934605i	$-0.384247\pi$
$79$	6.32284	0.711375	0.355688	+	0.934605i	$0.384247\pi$
$80$	−4.18544	−0.467946
$81$	−10.7939	−1.19933
$82$	2.34449	0.258906
$83$	−15.2910	−1.67841	−0.839203	−	0.543819i	$-0.816978\pi$
$83$	−15.2910	−1.67841	−0.839203	+	0.543819i	$0.816978\pi$
$84$	14.3131	1.56169
$85$	4.62498	0.501649
$86$	−6.58300	−0.709864
$87$	−2.48041	−0.265928
$88$	1.02039	0.108774
$89$	−5.43146	−0.575734	−0.287867	−	0.957670i	$-0.592946\pi$
$89$	−5.43146	−0.575734	−0.287867	+	0.957670i	$0.592946\pi$
$90$	−3.48128	−0.366959
$91$	−21.6940	−2.27415
$92$	5.16804	0.538806
$93$	−18.1817	−1.88535
$94$	5.31540	0.548241
$95$	−2.13644	−0.219194
$96$	15.3526	1.56692
$97$	−2.27190	−0.230676	−0.115338	−	0.993326i	$-0.536795\pi$
$97$	−2.27190	−0.230676	−0.115338	+	0.993326i	$0.536795\pi$
$98$	−23.7658	−2.40071
$99$	2.04226	0.205255
$100$	−5.99811	−0.599811
$101$	−16.2815	−1.62007	−0.810036	−	0.586380i	$-0.800552\pi$
$101$	−16.2815	−1.62007	−0.810036	+	0.586380i	$0.800552\pi$
$102$	−22.4294	−2.22084
$103$	11.3321	1.11658	0.558292	−	0.829645i	$-0.311457\pi$
$103$	11.3321	1.11658	0.558292	+	0.829645i	$0.311457\pi$
$104$	−5.28998	−0.518725
$105$	8.78562	0.857389
$106$	−18.2514	−1.77273
$107$	0.767827	0.0742287	0.0371143	−	0.999311i	$-0.488183\pi$
$107$	0.767827	0.0742287	0.0371143	+	0.999311i	$0.488183\pi$
$108$	−2.64819	−0.254822
$109$	−6.85306	−0.656404	−0.328202	−	0.944608i	$-0.606443\pi$
$109$	−6.85306	−0.656404	−0.328202	+	0.944608i	$0.606443\pi$
$110$	−1.50245	−0.143253
$111$	21.9408	2.08253
$112$	−21.5312	−2.03451
$113$	11.3449	1.06724	0.533619	−	0.845725i	$-0.320832\pi$
$113$	11.3449	1.06724	0.533619	+	0.845725i	$0.320832\pi$
$114$	10.3609	0.970388
$115$	3.17224	0.295813
$116$	−1.53905	−0.142897
$117$	−10.5876	−0.978824
$118$	−13.2834	−1.22284
$119$	23.7924	2.18104
$120$	2.14234	0.195568
$121$	−10.1186	−0.919873
$122$	13.5614	1.22779
$123$	−2.88763	−0.260369
$124$	−11.2814	−1.01310
$125$	−8.01394	−0.716788
$126$	−17.9088	−1.59544
$127$	−15.5441	−1.37932	−0.689660	−	0.724133i	$-0.742240\pi$
$127$	−15.5441	−1.37932	−0.689660	+	0.724133i	$0.742240\pi$
$128$	−8.31868	−0.735274
$129$	8.10806	0.713875
$130$	7.78910	0.683149
$131$	−18.0098	−1.57352	−0.786761	−	0.617257i	$-0.788244\pi$
$131$	−18.0098	−1.57352	−0.786761	+	0.617257i	$0.788244\pi$
$132$	3.01477	0.262402
$133$	−10.9905	−0.952999
$134$	22.7276	1.96337
$135$	−1.62551	−0.139901
$136$	5.80166	0.497489
$137$	9.39001	0.802243	0.401121	−	0.916025i	$-0.368621\pi$
$137$	9.39001	0.802243	0.401121	+	0.916025i	$0.368621\pi$
$138$	−15.3841	−1.30958
$139$	4.54500	0.385502	0.192751	−	0.981248i	$-0.438259\pi$
$139$	4.54500	0.385502	0.192751	+	0.981248i	$0.438259\pi$
$140$	5.45132	0.460721
$141$	−6.54679	−0.551339
$142$	3.49283	0.293112
$143$	−4.56941	−0.382113
$144$	−10.5082	−0.875682
$145$	−0.944698	−0.0784529
$146$	−9.28165	−0.768155
$147$	29.2715	2.41428
$148$	13.6139	1.11905
$149$	18.0061	1.47512	0.737560	−	0.675282i	$-0.235978\pi$
$149$	18.0061	1.47512	0.737560	+	0.675282i	$0.235978\pi$
$150$	17.8550	1.45786
$151$	7.22938	0.588319	0.294159	−	0.955756i	$-0.404960\pi$
$151$	7.22938	0.588319	0.294159	+	0.955756i	$0.404960\pi$
$152$	−2.67999	−0.217376
$153$	11.6117	0.938751
$154$	−7.72910	−0.622829
$155$	−6.92473	−0.556208
$156$	−15.6293	−1.25135
$157$	−21.0164	−1.67729	−0.838647	−	0.544676i	$-0.816653\pi$
$157$	−21.0164	−1.67729	−0.838647	+	0.544676i	$0.816653\pi$
$158$	−11.6785	−0.929095
$159$	22.4796	1.78275
$160$	5.84724	0.462265
$161$	16.3190	1.28612
$162$	19.9368	1.56639
$163$	12.0619	0.944762	0.472381	−	0.881394i	$-0.343395\pi$
$163$	12.0619	0.944762	0.472381	+	0.881394i	$0.343395\pi$
$164$	−1.79172	−0.139910
$165$	1.85052	0.144063
$166$	28.2431	2.19209
$167$	−5.06681	−0.392082	−0.196041	−	0.980596i	$-0.562809\pi$
$167$	−5.06681	−0.392082	−0.196041	+	0.980596i	$0.562809\pi$
$168$	11.0209	0.850278
$169$	10.6890	0.822227
$170$	−8.54251	−0.655181
$171$	−5.36385	−0.410184
$172$	5.03091	0.383603
$173$	−14.2589	−1.08409	−0.542043	−	0.840351i	$-0.682349\pi$
$173$	−14.2589	−1.08409	−0.542043	+	0.840351i	$0.682349\pi$
$174$	4.58142	0.347317
$175$	−18.9401	−1.43174
$176$	−4.53513	−0.341848
$177$	16.3607	1.22975
$178$	10.0321	0.751940
$179$	1.09536	0.0818709	0.0409354	−	0.999162i	$-0.486966\pi$
$179$	1.09536	0.0818709	0.0409354	+	0.999162i	$0.486966\pi$
$180$	2.66048	0.198301
$181$	−10.2107	−0.758959	−0.379480	−	0.925200i	$-0.623897\pi$
$181$	−10.2107	−0.758959	−0.379480	+	0.925200i	$0.623897\pi$
$182$	40.0696	2.97016
$183$	−16.7031	−1.23473
$184$	3.97932	0.293359
$185$	8.35644	0.614378
$186$	33.5823	2.46237
$187$	5.01139	0.366469
$188$	−4.06217	−0.296264
$189$	−8.36213	−0.608255
$190$	3.94609	0.286279
$191$	12.5330	0.906858	0.453429	−	0.891292i	$-0.350201\pi$
$191$	12.5330	0.906858	0.453429	+	0.891292i	$0.350201\pi$
$192$	−6.37819	−0.460306
$193$	−14.1538	−1.01881	−0.509406	−	0.860527i	$-0.670135\pi$
$193$	−14.1538	−1.01881	−0.509406	+	0.860527i	$0.670135\pi$
$194$	4.19628	0.301276
$195$	−9.59356	−0.687009
$196$	18.1625	1.29732
$197$	25.8579	1.84230	0.921150	−	0.389208i	$-0.127251\pi$
$197$	25.8579	1.84230	0.921150	+	0.389208i	$0.127251\pi$
$198$	−3.77214	−0.268074
$199$	−14.1260	−1.00137	−0.500683	−	0.865631i	$-0.666918\pi$
$199$	−14.1260	−1.00137	−0.500683	+	0.865631i	$0.666918\pi$
$200$	−4.61845	−0.326574
$201$	−27.9928	−1.97446
$202$	30.0726	2.11590
$203$	−4.85983	−0.341093
$204$	17.1411	1.20012
$205$	−1.09979	−0.0768127
$206$	−20.9308	−1.45832
$207$	7.96439	0.553563
$208$	23.5113	1.63021
$209$	−2.31494	−0.160128
$210$	−16.2274	−1.11980
$211$	11.5656	0.796209	0.398105	−	0.917340i	$-0.369668\pi$
$211$	11.5656	0.796209	0.398105	+	0.917340i	$0.369668\pi$
$212$	13.9482	0.957968
$213$	−4.30200	−0.294768
$214$	−1.41821	−0.0969467
$215$	3.08806	0.210604
$216$	−2.03907	−0.138741
$217$	−35.6230	−2.41825
$218$	12.6579	0.857300
$219$	11.4319	0.772495
$220$	1.14821	0.0774126
$221$	−25.9803	−1.74763
$222$	−40.5255	−2.71990
$223$	14.9415	1.00056	0.500278	−	0.865865i	$-0.333231\pi$
$223$	14.9415	1.00056	0.500278	+	0.865865i	$0.333231\pi$
$224$	30.0801	2.00981
$225$	−9.24358	−0.616239
$226$	−20.9545	−1.39387
$227$	−24.2298	−1.60819	−0.804094	−	0.594502i	$-0.797349\pi$
$227$	−24.2298	−1.60819	−0.804094	+	0.594502i	$0.797349\pi$
$228$	−7.91808	−0.524387
$229$	29.3018	1.93632	0.968158	−	0.250340i	$-0.0805424\pi$
$229$	29.3018	1.93632	0.968158	+	0.250340i	$0.0805424\pi$
$230$	−5.85925	−0.386347
$231$	9.51966	0.626348
$232$	−1.18505	−0.0778022
$233$	1.98456	0.130013	0.0650065	−	0.997885i	$-0.479293\pi$
$233$	1.98456	0.130013	0.0650065	+	0.997885i	$0.479293\pi$
$234$	19.5557	1.27840
$235$	−2.49343	−0.162653
$236$	10.1515	0.660809
$237$	14.3841	0.934345
$238$	−43.9454	−2.84856
$239$	15.4918	1.00208	0.501041	−	0.865424i	$-0.332951\pi$
$239$	15.4918	1.00208	0.501041	+	0.865424i	$0.332951\pi$
$240$	−9.52159	−0.614616
$241$	−19.3640	−1.24734	−0.623672	−	0.781687i	$-0.714360\pi$
$241$	−19.3640	−1.24734	−0.623672	+	0.781687i	$0.714360\pi$
$242$	18.6895	1.20140
$243$	−18.9273	−1.21418
$244$	−10.3640	−0.663485
$245$	11.1484	0.712248
$246$	5.33356	0.340056
$247$	12.0012	0.763619
$248$	−8.68652	−0.551595
$249$	−34.7860	−2.20447
$250$	14.8021	0.936165
$251$	18.5739	1.17238	0.586188	−	0.810175i	$-0.300628\pi$
$251$	18.5739	1.17238	0.586188	+	0.810175i	$0.300628\pi$
$252$	13.6864	0.862161
$253$	3.43728	0.216100
$254$	28.7107	1.80147
$255$	10.5215	0.658883
$256$	20.9723	1.31077
$257$	9.19948	0.573848	0.286924	−	0.957953i	$-0.407367\pi$
$257$	9.19948	0.573848	0.286924	+	0.957953i	$0.407367\pi$
$258$	−14.9759	−0.932359
$259$	42.9882	2.67116
$260$	−5.95263	−0.369167
$261$	−2.37181	−0.146811
$262$	33.2648	2.05511
$263$	8.98787	0.554216	0.277108	−	0.960839i	$-0.410624\pi$
$263$	8.98787	0.554216	0.277108	+	0.960839i	$0.410624\pi$
$264$	2.32133	0.142868
$265$	8.56167	0.525939
$266$	20.2999	1.24467
$267$	−12.3562	−0.756189
$268$	−17.3690	−1.06098
$269$	11.9084	0.726067	0.363033	−	0.931776i	$-0.381741\pi$
$269$	11.9084	0.726067	0.363033	+	0.931776i	$0.381741\pi$
$270$	3.00238	0.182719
$271$	12.2528	0.744304	0.372152	−	0.928172i	$-0.378620\pi$
$271$	12.2528	0.744304	0.372152	+	0.928172i	$0.378620\pi$
$272$	−25.7854	−1.56347
$273$	−49.3523	−2.98694
$274$	−17.3437	−1.04777
$275$	−3.98935	−0.240567
$276$	11.7570	0.707686
$277$	2.76906	0.166377	0.0831884	−	0.996534i	$-0.473490\pi$
$277$	2.76906	0.166377	0.0831884	+	0.996534i	$0.473490\pi$
$278$	−8.39481	−0.503487
$279$	−17.3856	−1.04085
$280$	4.19744	0.250845
$281$	16.3524	0.975505	0.487752	−	0.872982i	$-0.337817\pi$
$281$	16.3524	0.975505	0.487752	+	0.872982i	$0.337817\pi$
$282$	12.0922	0.720079
$283$	−14.2886	−0.849370	−0.424685	−	0.905341i	$-0.639615\pi$
$283$	−14.2886	−0.849370	−0.424685	+	0.905341i	$0.639615\pi$
$284$	−2.66931	−0.158395
$285$	−4.86026	−0.287897
$286$	8.43987	0.499060
$287$	−5.65768	−0.333962
$288$	14.6804	0.865051
$289$	11.4933	0.676078
$290$	1.74489	0.102464
$291$	−5.16841	−0.302978
$292$	7.09328	0.415103
$293$	−17.4789	−1.02113	−0.510564	−	0.859840i	$-0.670563\pi$
$293$	−17.4789	−1.02113	−0.510564	+	0.859840i	$0.670563\pi$
$294$	−54.0657	−3.15318
$295$	6.23120	0.362795
$296$	10.4825	0.609283
$297$	−1.76132	−0.102202
$298$	−33.2580	−1.92659
$299$	−17.8197	−1.03054
$300$	−13.6453	−0.787812
$301$	15.8860	0.915652
$302$	−13.3530	−0.768376
$303$	−37.0394	−2.12786
$304$	11.9112	0.683154
$305$	−6.36159	−0.364264
$306$	−21.4473	−1.22606
$307$	−23.4803	−1.34009	−0.670047	−	0.742319i	$-0.733726\pi$
$307$	−23.4803	−1.34009	−0.670047	+	0.742319i	$0.733726\pi$
$308$	5.90678	0.336570
$309$	25.7798	1.46656
$310$	12.7902	0.726437
$311$	2.47003	0.140063	0.0700313	−	0.997545i	$-0.477690\pi$
$311$	2.47003	0.140063	0.0700313	+	0.997545i	$0.477690\pi$
$312$	−12.0344	−0.681311
$313$	−21.8998	−1.23785	−0.618924	−	0.785451i	$-0.712431\pi$
$313$	−21.8998	−1.23785	−0.618924	+	0.785451i	$0.712431\pi$
$314$	38.8182	2.19064
$315$	8.40095	0.473340
$316$	8.92505	0.502073
$317$	−17.0485	−0.957542	−0.478771	−	0.877940i	$-0.658918\pi$
$317$	−17.0485	−0.957542	−0.478771	+	0.877940i	$0.658918\pi$
$318$	−41.5208	−2.32837
$319$	−1.02363	−0.0573121
$320$	−2.42922	−0.135797
$321$	1.74676	0.0974944
$322$	−30.1419	−1.67974
$323$	−13.1621	−0.732357
$324$	−15.2362	−0.846458
$325$	20.6818	1.14722
$326$	−22.2788	−1.23391
$327$	−15.5903	−0.862144
$328$	−1.37960	−0.0761757
$329$	−12.8270	−0.707176
$330$	−3.41798	−0.188154
$331$	3.12807	0.171934	0.0859671	−	0.996298i	$-0.472602\pi$
$331$	3.12807	0.171934	0.0859671	+	0.996298i	$0.472602\pi$
$332$	−21.5841	−1.18458
$333$	20.9801	1.14970
$334$	9.35860	0.512080
$335$	−10.6614	−0.582496
$336$	−48.9821	−2.67219
$337$	−21.3553	−1.16330	−0.581650	−	0.813439i	$-0.697593\pi$
$337$	−21.3553	−1.16330	−0.581650	+	0.813439i	$0.697593\pi$
$338$	−19.7429	−1.07387
$339$	25.8089	1.40175
$340$	6.52841	0.354053
$341$	−7.50329	−0.406326
$342$	9.90725	0.535723
$343$	26.1505	1.41200
$344$	3.87373	0.208857
$345$	7.21663	0.388530
$346$	26.3368	1.41587
$347$	−13.1040	−0.703460	−0.351730	−	0.936102i	$-0.614406\pi$
$347$	−13.1040	−0.703460	−0.351730	+	0.936102i	$0.614406\pi$
$348$	−3.50124	−0.187686
$349$	−2.76530	−0.148023	−0.0740117	−	0.997257i	$-0.523580\pi$
$349$	−2.76530	−0.148023	−0.0740117	+	0.997257i	$0.523580\pi$
$350$	34.9831	1.86992
$351$	9.13112	0.487383
$352$	6.33578	0.337698
$353$	23.4166	1.24634	0.623171	−	0.782086i	$-0.285844\pi$
$353$	23.4166	1.24634	0.623171	+	0.782086i	$0.285844\pi$
$354$	−30.2189	−1.60612
$355$	−1.63847	−0.0869611
$356$	−7.66682	−0.406341
$357$	54.1261	2.86465
$358$	−2.02317	−0.106928
$359$	−9.02335	−0.476234	−0.238117	−	0.971236i	$-0.576530\pi$
$359$	−9.02335	−0.476234	−0.238117	+	0.971236i	$0.576530\pi$
$360$	2.04853	0.107967
$361$	−12.9200	−0.679999
$362$	18.8597	0.991242
$363$	−23.0191	−1.20819
$364$	−30.6223	−1.60504
$365$	4.35398	0.227898
$366$	30.8513	1.61262
$367$	25.4984	1.33101	0.665504	−	0.746394i	$-0.268217\pi$
$367$	25.4984	1.33101	0.665504	+	0.746394i	$0.268217\pi$
$368$	−17.6860	−0.921949
$369$	−2.76119	−0.143742
$370$	−15.4347	−0.802411
$371$	44.0440	2.28665
$372$	−25.6645	−1.33064
$373$	−10.7580	−0.557027	−0.278513	−	0.960432i	$-0.589842\pi$
$373$	−10.7580	−0.557027	−0.278513	+	0.960432i	$0.589842\pi$
$374$	−9.25624	−0.478629
$375$	−18.2312	−0.941454
$376$	−3.12781	−0.161305
$377$	5.30674	0.273311
$378$	15.4452	0.794415
$379$	4.85291	0.249277	0.124639	−	0.992202i	$-0.460223\pi$
$379$	4.85291	0.249277	0.124639	+	0.992202i	$0.460223\pi$
$380$	−3.01570	−0.154702
$381$	−35.3619	−1.81165
$382$	−23.1490	−1.18441
$383$	−3.85060	−0.196757	−0.0983783	−	0.995149i	$-0.531366\pi$
$383$	−3.85060	−0.196757	−0.0983783	+	0.995149i	$0.531366\pi$
$384$	−18.9244	−0.965734
$385$	3.62569	0.184782
$386$	26.1426	1.33062
$387$	7.75305	0.394109
$388$	−3.20691	−0.162806
$389$	−27.1136	−1.37471	−0.687357	−	0.726320i	$-0.741229\pi$
$389$	−27.1136	−1.37471	−0.687357	+	0.726320i	$0.741229\pi$
$390$	17.7197	0.897271
$391$	19.5434	0.988351
$392$	13.9848	0.706341
$393$	−40.9711	−2.06672
$394$	−47.7606	−2.40614
$395$	5.47836	0.275646
$396$	2.88277	0.144864
$397$	6.22911	0.312630	0.156315	−	0.987707i	$-0.450038\pi$
$397$	6.22911	0.312630	0.156315	+	0.987707i	$0.450038\pi$
$398$	26.0913	1.30784
$399$	−25.0027	−1.25170
$400$	20.5267	1.02633
$401$	−17.4167	−0.869749	−0.434874	−	0.900491i	$-0.643207\pi$
$401$	−17.4167	−0.869749	−0.434874	+	0.900491i	$0.643207\pi$
$402$	51.7038	2.57875
$403$	38.8989	1.93769
$404$	−22.9823	−1.14341
$405$	−9.35228	−0.464719
$406$	8.97630	0.445486
$407$	9.05462	0.448821
$408$	13.1984	0.653418
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	2.03136	0.100322
$411$	21.3617	1.05369
$412$	15.9959	0.788061
$413$	32.0553	1.57734
$414$	−14.7105	−0.722984
$415$	−13.2487	−0.650354
$416$	−32.8463	−1.61042
$417$	10.3396	0.506332
$418$	4.27578	0.209135
$419$	17.9232	0.875604	0.437802	−	0.899071i	$-0.355757\pi$
$419$	17.9232	0.875604	0.437802	+	0.899071i	$0.355757\pi$
$420$	12.4014	0.605127
$421$	25.4335	1.23955	0.619777	−	0.784778i	$-0.287223\pi$
$421$	25.4335	1.23955	0.619777	+	0.784778i	$0.287223\pi$
$422$	−21.3621	−1.03989
$423$	−6.26014	−0.304378
$424$	10.7399	0.521577
$425$	−22.6823	−1.10025
$426$	7.94596	0.384983
$427$	−32.7261	−1.58373
$428$	1.08383	0.0523890
$429$	−10.3951	−0.501880
$430$	−5.70377	−0.275060
$431$	17.2801	0.832355	0.416177	−	0.909283i	$-0.363370\pi$
$431$	17.2801	0.832355	0.416177	+	0.909283i	$0.363370\pi$
$432$	9.06262	0.436026
$433$	33.1324	1.59224	0.796120	−	0.605139i	$-0.206883\pi$
$433$	33.1324	1.59224	0.796120	+	0.605139i	$0.206883\pi$
$434$	65.7972	3.15836
$435$	−2.14912	−0.103043
$436$	−9.67349	−0.463276
$437$	−9.02777	−0.431857
$438$	−21.1151	−1.00892
$439$	20.8425	0.994757	0.497379	−	0.867534i	$-0.334296\pi$
$439$	20.8425	0.994757	0.497379	+	0.867534i	$0.334296\pi$
$440$	0.884108	0.0421482
$441$	27.9899	1.33285
$442$	47.9867	2.28249
$443$	−41.5223	−1.97279	−0.986393	−	0.164406i	$-0.947429\pi$
$443$	−41.5223	−1.97279	−0.986393	+	0.164406i	$0.947429\pi$
$444$	30.9707	1.46980
$445$	−4.70603	−0.223087
$446$	−27.5975	−1.30678
$447$	40.9628	1.93747
$448$	−12.4967	−0.590412
$449$	−29.6564	−1.39957	−0.699785	−	0.714353i	$-0.746721\pi$
$449$	−29.6564	−1.39957	−0.699785	+	0.714353i	$0.746721\pi$
$450$	17.0733	0.804841
$451$	−1.19168	−0.0561140
$452$	16.0140	0.753234
$453$	16.4464	0.772718
$454$	44.7534	2.10038
$455$	−18.7965	−0.881193
$456$	−6.09681	−0.285509
$457$	−2.44867	−0.114544	−0.0572719	−	0.998359i	$-0.518240\pi$
$457$	−2.44867	−0.114544	−0.0572719	+	0.998359i	$0.518240\pi$
$458$	−54.1216	−2.52893
$459$	−10.0143	−0.467430
$460$	4.47779	0.208778
$461$	6.58791	0.306830	0.153415	−	0.988162i	$-0.450973\pi$
$461$	6.58791	0.306830	0.153415	+	0.988162i	$0.450973\pi$
$462$	−17.5832	−0.818044
$463$	20.8134	0.967280	0.483640	−	0.875267i	$-0.339314\pi$
$463$	20.8134	0.967280	0.483640	+	0.875267i	$0.339314\pi$
$464$	5.26693	0.244511
$465$	−15.7533	−0.730542
$466$	−3.66557	−0.169804
$467$	17.6134	0.815050	0.407525	−	0.913194i	$-0.366392\pi$
$467$	17.6134	0.815050	0.407525	+	0.913194i	$0.366392\pi$
$468$	−14.9450	−0.690833
$469$	−54.8458	−2.53255
$470$	4.60547	0.212434
$471$	−47.8110	−2.20301
$472$	7.81655	0.359786
$473$	3.34607	0.153852
$474$	−26.5679	−1.22030
$475$	10.4778	0.480752
$476$	33.5843	1.53933
$477$	21.4954	0.984205
$478$	−28.6140	−1.30877
$479$	31.6259	1.44503	0.722513	−	0.691357i	$-0.242987\pi$
$479$	31.6259	1.44503	0.722513	+	0.691357i	$0.242987\pi$
$480$	13.3021	0.607155
$481$	−46.9415	−2.14035
$482$	35.7660	1.62910
$483$	37.1247	1.68923
$484$	−14.2830	−0.649226
$485$	−1.96846	−0.0893831
$486$	34.9594	1.58579
$487$	23.3900	1.05990	0.529951	−	0.848028i	$-0.322210\pi$
$487$	23.3900	1.05990	0.529951	+	0.848028i	$0.322210\pi$
$488$	−7.98011	−0.361243
$489$	27.4401	1.24088
$490$	−20.5916	−0.930235
$491$	−11.4607	−0.517216	−0.258608	−	0.965982i	$-0.583264\pi$
$491$	−11.4607	−0.517216	−0.258608	+	0.965982i	$0.583264\pi$
$492$	−4.07605	−0.183763
$493$	−5.82005	−0.262122
$494$	−22.1667	−0.997329
$495$	1.76949	0.0795328
$496$	38.6072	1.73351
$497$	−8.42883	−0.378085
$498$	64.2512	2.87916
$499$	11.2619	0.504150	0.252075	−	0.967708i	$-0.418887\pi$
$499$	11.2619	0.504150	0.252075	+	0.967708i	$0.418887\pi$
$500$	−11.3121	−0.505894
$501$	−11.5267	−0.514973
$502$	−34.3068	−1.53119
$503$	2.88287	0.128541	0.0642703	−	0.997933i	$-0.479528\pi$
$503$	2.88287	0.128541	0.0642703	+	0.997933i	$0.479528\pi$
$504$	10.5383	0.469414
$505$	−14.1069	−0.627751
$506$	−6.34879	−0.282238
$507$	24.3167	1.07994
$508$	−21.9414	−0.973494
$509$	−1.17239	−0.0519654	−0.0259827	−	0.999662i	$-0.508271\pi$
$509$	−1.17239	−0.0519654	−0.0259827	+	0.999662i	$0.508271\pi$
$510$	−19.4337	−0.860537
$511$	22.3983	0.990842
$512$	−22.0993	−0.976660
$513$	4.62598	0.204242
$514$	−16.9918	−0.749477
$515$	9.81856	0.432657
$516$	11.4450	0.503837
$517$	−2.70176	−0.118823
$518$	−79.4010	−3.48868
$519$	−32.4381	−1.42387
$520$	−4.58344	−0.200997
$521$	−33.0757	−1.44907	−0.724537	−	0.689236i	$-0.757946\pi$
$521$	−33.0757	−1.44907	−0.724537	+	0.689236i	$0.757946\pi$
$522$	4.38082	0.191743
$523$	27.0778	1.18403	0.592014	−	0.805928i	$-0.298333\pi$
$523$	27.0778	1.18403	0.592014	+	0.805928i	$0.298333\pi$
$524$	−25.4218	−1.11056
$525$	−43.0874	−1.88049
$526$	−16.6010	−0.723836
$527$	−42.6615	−1.85837
$528$	−10.3171	−0.448995
$529$	−9.59534	−0.417189
$530$	−15.8137	−0.686905
$531$	15.6444	0.678908
$532$	−15.5137	−0.672606
$533$	6.17796	0.267597
$534$	22.8224	0.987623
$535$	0.665275	0.0287624
$536$	−13.3739	−0.577665
$537$	2.49187	0.107532
$538$	−21.9952	−0.948283
$539$	12.0799	0.520318
$540$	−2.29449	−0.0987393
$541$	26.8275	1.15341	0.576703	−	0.816954i	$-0.304339\pi$
$541$	26.8275	1.15341	0.576703	+	0.816954i	$0.304339\pi$
$542$	−22.6314	−0.972102
$543$	−23.2288	−0.996843
$544$	36.0234	1.54449
$545$	−5.93776	−0.254346
$546$	91.1557	3.90111
$547$	−9.25166	−0.395572	−0.197786	−	0.980245i	$-0.563375\pi$
$547$	−9.25166	−0.395572	−0.197786	+	0.980245i	$0.563375\pi$
$548$	13.2545	0.566206
$549$	−15.9717	−0.681658
$550$	7.36849	0.314194
$551$	2.68848	0.114533
$552$	9.05269	0.385308
$553$	28.1824	1.19844
$554$	−5.11457	−0.217297
$555$	19.0104	0.806945
$556$	6.41553	0.272079
$557$	−3.09487	−0.131134	−0.0655669	−	0.997848i	$-0.520886\pi$
$557$	−3.09487	−0.131134	−0.0655669	+	0.997848i	$0.520886\pi$
$558$	32.1119	1.35940
$559$	−17.3469	−0.733694
$560$	−18.6555	−0.788338
$561$	11.4006	0.481333
$562$	−30.2036	−1.27406
$563$	−32.4478	−1.36751	−0.683755	−	0.729712i	$-0.739654\pi$
$563$	−32.4478	−1.36751	−0.683755	+	0.729712i	$0.739654\pi$
$564$	−9.24116	−0.389123
$565$	9.82966	0.413537
$566$	26.3916	1.10932
$567$	−48.1111	−2.02048
$568$	−2.05533	−0.0862398
$569$	−7.93403	−0.332612	−0.166306	−	0.986074i	$-0.553184\pi$
$569$	−7.93403	−0.332612	−0.166306	+	0.986074i	$0.553184\pi$
$570$	8.97709	0.376009
$571$	−7.49748	−0.313760	−0.156880	−	0.987618i	$-0.550144\pi$
$571$	−7.49748	−0.313760	−0.156880	+	0.987618i	$0.550144\pi$
$572$	−6.44997	−0.269687
$573$	28.5118	1.19110
$574$	10.4500	0.436173
$575$	−15.5576	−0.648798
$576$	−6.09892	−0.254122
$577$	33.2185	1.38291	0.691453	−	0.722421i	$-0.256971\pi$
$577$	33.2185	1.38291	0.691453	+	0.722421i	$0.256971\pi$
$578$	−21.2286	−0.882995
$579$	−32.1989	−1.33814
$580$	−1.33349	−0.0553703
$581$	−68.1557	−2.82757
$582$	9.54627	0.395706
$583$	9.27699	0.384214
$584$	5.46172	0.226008
$585$	−9.17350	−0.379278
$586$	32.2842	1.33365
$587$	41.5823	1.71629	0.858143	−	0.513411i	$-0.171618\pi$
$587$	41.5823	1.71629	0.858143	+	0.513411i	$0.171618\pi$
$588$	41.3184	1.70394
$589$	19.7069	0.812007
$590$	−11.5093	−0.473829
$591$	58.8250	2.41974
$592$	−46.5894	−1.91481
$593$	19.7111	0.809438	0.404719	−	0.914441i	$-0.367369\pi$
$593$	19.7111	0.809438	0.404719	+	0.914441i	$0.367369\pi$
$594$	3.25322	0.133481
$595$	20.6146	0.845117
$596$	25.4167	1.04111
$597$	−32.1357	−1.31523
$598$	32.9137	1.34594
$599$	16.9989	0.694558	0.347279	−	0.937762i	$-0.387106\pi$
$599$	16.9989	0.694558	0.347279	+	0.937762i	$0.387106\pi$
$600$	−10.5067	−0.428933
$601$	−33.7387	−1.37623	−0.688115	−	0.725602i	$-0.741562\pi$
$601$	−33.7387	−1.37623	−0.688115	+	0.725602i	$0.741562\pi$
$602$	−29.3420	−1.19589
$603$	−26.7672	−1.09004
$604$	10.2047	0.415223
$605$	−8.76715	−0.356435
$606$	68.4132	2.77910
$607$	−12.1806	−0.494396	−0.247198	−	0.968965i	$-0.579510\pi$
$607$	−12.1806	−0.494396	−0.247198	+	0.968965i	$0.579510\pi$
$608$	−16.6405	−0.674861
$609$	−11.0558	−0.448003
$610$	11.7501	0.475748
$611$	14.0066	0.566646
$612$	16.3906	0.662550
$613$	−14.4438	−0.583379	−0.291690	−	0.956513i	$-0.594217\pi$
$613$	−14.4438	−0.583379	−0.291690	+	0.956513i	$0.594217\pi$
$614$	43.3691	1.75024
$615$	−2.50195	−0.100888
$616$	4.54814	0.183250
$617$	8.77290	0.353183	0.176592	−	0.984284i	$-0.443493\pi$
$617$	8.77290	0.353183	0.176592	+	0.984284i	$0.443493\pi$
$618$	−47.6162	−1.91541
$619$	19.9600	0.802260	0.401130	−	0.916021i	$-0.368618\pi$
$619$	19.9600	0.802260	0.401130	+	0.916021i	$0.368618\pi$
$620$	−9.77465	−0.392559
$621$	−6.86877	−0.275634
$622$	−4.56225	−0.182929
$623$	−24.2093	−0.969926
$624$	53.4866	2.14118
$625$	14.3028	0.572113
$626$	40.4497	1.61670
$627$	−5.26633	−0.210317
$628$	−29.6659	−1.18380
$629$	51.4820	2.05272
$630$	−15.5169	−0.618207
$631$	12.2024	0.485770	0.242885	−	0.970055i	$-0.421906\pi$
$631$	12.2024	0.485770	0.242885	+	0.970055i	$0.421906\pi$
$632$	6.87216	0.273360
$633$	26.3110	1.04577
$634$	31.4893	1.25060
$635$	−13.4681	−0.534463
$636$	31.7313	1.25823
$637$	−62.6253	−2.48130
$638$	1.89068	0.0748527
$639$	−4.11363	−0.162733
$640$	−7.20762	−0.284906
$641$	7.53030	0.297429	0.148715	−	0.988880i	$-0.452486\pi$
$641$	7.53030	0.297429	0.148715	+	0.988880i	$0.452486\pi$
$642$	−3.22633	−0.127333
$643$	32.3440	1.27552	0.637761	−	0.770234i	$-0.279861\pi$
$643$	32.3440	1.27552	0.637761	+	0.770234i	$0.279861\pi$
$644$	23.0352	0.907714
$645$	7.02513	0.276614
$646$	24.3109	0.956498
$647$	46.0306	1.80965	0.904825	−	0.425784i	$-0.140002\pi$
$647$	46.0306	1.80965	0.904825	+	0.425784i	$0.140002\pi$
$648$	−11.7317	−0.460864
$649$	6.75182	0.265032
$650$	−38.2001	−1.49833
$651$	−81.0400	−3.17621
$652$	17.0261	0.666793
$653$	16.0341	0.627463	0.313731	−	0.949512i	$-0.398421\pi$
$653$	16.0341	0.627463	0.313731	+	0.949512i	$0.398421\pi$
$654$	28.7958	1.12601
$655$	−15.6044	−0.609714
$656$	6.13162	0.239400
$657$	10.9313	0.426472
$658$	23.6920	0.923610
$659$	17.3962	0.677659	0.338829	−	0.940848i	$-0.389969\pi$
$659$	17.3962	0.677659	0.338829	+	0.940848i	$0.389969\pi$
$660$	2.61211	0.101676
$661$	−26.6419	−1.03625	−0.518124	−	0.855305i	$-0.673370\pi$
$661$	−26.6419	−1.03625	−0.518124	+	0.855305i	$0.673370\pi$
$662$	−5.77767	−0.224555
$663$	−59.1035	−2.29539
$664$	−16.6195	−0.644960
$665$	−9.52262	−0.369271
$666$	−38.7511	−1.50158
$667$	−3.99193	−0.154568
$668$	−7.15209	−0.276723
$669$	33.9909	1.31416
$670$	19.6921	0.760772
$671$	−6.89310	−0.266105
$672$	68.4302	2.63975
$673$	−4.96395	−0.191346	−0.0956732	−	0.995413i	$-0.530500\pi$
$673$	−4.96395	−0.191346	−0.0956732	+	0.995413i	$0.530500\pi$
$674$	39.4442	1.51933
$675$	7.97199	0.306842
$676$	15.0881	0.580310
$677$	−4.27993	−0.164491	−0.0822456	−	0.996612i	$-0.526209\pi$
$677$	−4.27993	−0.164491	−0.0822456	+	0.996612i	$0.526209\pi$
$678$	−47.6701	−1.83076
$679$	−10.1264	−0.388615
$680$	5.02679	0.192768
$681$	−55.1212	−2.11225
$682$	13.8589	0.530684
$683$	42.2433	1.61640	0.808198	−	0.588911i	$-0.200443\pi$
$683$	42.2433	1.61640	0.808198	+	0.588911i	$0.200443\pi$
$684$	−7.57138	−0.289499
$685$	8.13587	0.310856
$686$	−48.3011	−1.84414
$687$	66.6596	2.54322
$688$	−17.2167	−0.656382
$689$	−48.0943	−1.83225
$690$	−13.3294	−0.507442
$691$	−48.1303	−1.83096	−0.915482	−	0.402359i	$-0.868190\pi$
$691$	−48.1303	−1.83096	−0.915482	+	0.402359i	$0.868190\pi$
$692$	−20.1273	−0.765124
$693$	9.10284	0.345788
$694$	24.2036	0.918757
$695$	3.93797	0.149376
$696$	−2.69591	−0.102188
$697$	−6.77554	−0.256642
$698$	5.10763	0.193327
$699$	4.51475	0.170764
$700$	−26.7350	−1.01049
$701$	51.2784	1.93676	0.968378	−	0.249486i	$-0.0802618\pi$
$701$	51.2784	1.93676	0.968378	+	0.249486i	$0.0802618\pi$
$702$	−16.8655	−0.636549
$703$	−23.7813	−0.896930
$704$	−2.63218	−0.0992039
$705$	−5.67239	−0.213635
$706$	−43.2514	−1.62779
$707$	−72.5706	−2.72930
$708$	23.0941	0.867930
$709$	−17.0289	−0.639532	−0.319766	−	0.947497i	$-0.603604\pi$
$709$	−17.0289	−0.639532	−0.319766	+	0.947497i	$0.603604\pi$
$710$	3.02632	0.113576
$711$	13.7542	0.515825
$712$	−5.90334	−0.221237
$713$	−29.2612	−1.09584
$714$	−99.9730	−3.74140
$715$	−3.95911	−0.148062
$716$	1.54616	0.0577827
$717$	35.2428	1.31617
$718$	16.6665	0.621988
$719$	22.4148	0.835930	0.417965	−	0.908463i	$-0.362744\pi$
$719$	22.4148	0.835930	0.417965	+	0.908463i	$0.362744\pi$
$720$	−9.10469	−0.339312
$721$	50.5098	1.88108
$722$	23.8637	0.888116
$723$	−44.0518	−1.63830
$724$	−14.4131	−0.535657
$725$	4.63309	0.172069
$726$	42.5173	1.57796
$727$	43.7327	1.62196	0.810978	−	0.585076i	$-0.198935\pi$
$727$	43.7327	1.62196	0.810978	+	0.585076i	$0.198935\pi$
$728$	−23.5787	−0.873885
$729$	−10.6765	−0.395425
$730$	−8.04198	−0.297647
$731$	19.0248	0.703657
$732$	−23.5774	−0.871444
$733$	0.746662	0.0275786	0.0137893	−	0.999905i	$-0.495611\pi$
$733$	0.746662	0.0275786	0.0137893	+	0.999905i	$0.495611\pi$
$734$	−47.0966	−1.73837
$735$	25.3620	0.935491
$736$	24.7082	0.910756
$737$	−11.5522	−0.425530
$738$	5.10004	0.187735
$739$	−13.1420	−0.483438	−0.241719	−	0.970346i	$-0.577711\pi$
$739$	−13.1420	−0.483438	−0.241719	+	0.970346i	$0.577711\pi$
$740$	11.7956	0.433615
$741$	27.3020	1.00296
$742$	−81.3510	−2.98649
$743$	34.0373	1.24871	0.624354	−	0.781141i	$-0.285362\pi$
$743$	34.0373	1.24871	0.624354	+	0.781141i	$0.285362\pi$
$744$	−19.7613	−0.724483
$745$	15.6012	0.571584
$746$	19.8704	0.727507
$747$	−33.2629	−1.21703
$748$	7.07386	0.258646
$749$	3.42239	0.125051
$750$	33.6737	1.22959
$751$	−4.82573	−0.176093	−0.0880466	−	0.996116i	$-0.528062\pi$
$751$	−4.82573	−0.176093	−0.0880466	+	0.996116i	$0.528062\pi$
$752$	13.9015	0.506936
$753$	42.2545	1.53984
$754$	−9.80176	−0.356959
$755$	6.26382	0.227964
$756$	−11.8036	−0.429293
$757$	53.9676	1.96148	0.980742	−	0.195307i	$-0.0625705\pi$
$757$	53.9676	1.96148	0.980742	+	0.195307i	$0.0625705\pi$
$758$	−8.96351	−0.325569
$759$	7.81958	0.283833
$760$	−2.32205	−0.0842296
$761$	21.9954	0.797333	0.398667	−	0.917096i	$-0.369473\pi$
$761$	21.9954	0.797333	0.398667	+	0.917096i	$0.369473\pi$
$762$	65.3149	2.36611
$763$	−30.5457	−1.10583
$764$	17.6911	0.640041
$765$	10.0608	0.363750
$766$	7.11221	0.256975
$767$	−35.0031	−1.26389
$768$	47.7106	1.72161
$769$	8.32640	0.300258	0.150129	−	0.988666i	$-0.452031\pi$
$769$	8.32640	0.300258	0.150129	+	0.988666i	$0.452031\pi$
$770$	−6.69679	−0.241336
$771$	20.9282	0.753712
$772$	−19.9789	−0.719055
$773$	−13.2469	−0.476459	−0.238230	−	0.971209i	$-0.576567\pi$
$773$	−13.2469	−0.476459	−0.238230	+	0.971209i	$0.576567\pi$
$774$	−14.3202	−0.514728
$775$	33.9610	1.21992
$776$	−2.46927	−0.0886418
$777$	97.7954	3.50839
$778$	50.0799	1.79545
$779$	3.12986	0.112139
$780$	−13.5418	−0.484876
$781$	−1.77537	−0.0635276
$782$	−36.0974	−1.29084
$783$	2.04553	0.0731013
$784$	−62.1555	−2.21984
$785$	−18.2094	−0.649923
$786$	75.6752	2.69925
$787$	−22.6644	−0.807897	−0.403948	−	0.914782i	$-0.632362\pi$
$787$	−22.6644	−0.807897	−0.403948	+	0.914782i	$0.632362\pi$
$788$	36.4999	1.30026
$789$	20.4468	0.727926
$790$	−10.1187	−0.360009
$791$	50.5669	1.79795
$792$	2.21969	0.0788732
$793$	35.7356	1.26901
$794$	−11.5054	−0.408312
$795$	19.4772	0.690786
$796$	−19.9397	−0.706742
$797$	−42.3239	−1.49919	−0.749595	−	0.661897i	$-0.769751\pi$
$797$	−42.3239	−1.49919	−0.749595	+	0.661897i	$0.769751\pi$
$798$	46.1810	1.63479
$799$	−15.3614	−0.543448
$800$	−28.6767	−1.01387
$801$	−11.8152	−0.417470
$802$	32.1694	1.13594
$803$	4.71776	0.166486
$804$	−39.5134	−1.39353
$805$	14.1394	0.498349
$806$	−71.8479	−2.53073
$807$	27.0908	0.953641
$808$	−17.6960	−0.622544
$809$	−1.26438	−0.0444532	−0.0222266	−	0.999753i	$-0.507076\pi$
$809$	−1.26438	−0.0444532	−0.0222266	+	0.999753i	$0.507076\pi$
$810$	17.2740	0.606948
$811$	−17.5652	−0.616798	−0.308399	−	0.951257i	$-0.599793\pi$
$811$	−17.5652	−0.616798	−0.308399	+	0.951257i	$0.599793\pi$
$812$	−6.85992	−0.240736
$813$	27.8743	0.977594
$814$	−16.7243	−0.586185
$815$	10.4509	0.366080
$816$	−58.6602	−2.05352
$817$	−8.78821	−0.307461
$818$	−1.84704	−0.0645802
$819$	−47.1915	−1.64900
$820$	−1.55242	−0.0542128
$821$	31.5653	1.10164	0.550819	−	0.834625i	$-0.314316\pi$
$821$	31.5653	1.10164	0.550819	+	0.834625i	$0.314316\pi$
$822$	−39.4558	−1.37618
$823$	−48.3629	−1.68583	−0.842913	−	0.538050i	$-0.819161\pi$
$823$	−48.3629	−1.68583	−0.842913	+	0.538050i	$0.819161\pi$
$824$	12.3166	0.429069
$825$	−9.07552	−0.315969
$826$	−59.2074	−2.06009
$827$	4.60437	0.160110	0.0800548	−	0.996790i	$-0.474490\pi$
$827$	4.60437	0.160110	0.0800548	+	0.996790i	$0.474490\pi$
$828$	11.2422	0.390693
$829$	−4.35551	−0.151273	−0.0756365	−	0.997135i	$-0.524099\pi$
$829$	−4.35551	−0.151273	−0.0756365	+	0.997135i	$0.524099\pi$
$830$	24.4709	0.849398
$831$	6.29943	0.218525
$832$	13.6459	0.473086
$833$	68.6828	2.37972
$834$	−19.0976	−0.661297
$835$	−4.39008	−0.151925
$836$	−3.26767	−0.113015
$837$	14.9939	0.518266
$838$	−33.1048	−1.14359
$839$	−9.62150	−0.332171	−0.166086	−	0.986111i	$-0.553113\pi$
$839$	−9.62150	−0.332171	−0.166086	+	0.986111i	$0.553113\pi$
$840$	9.54891	0.329469
$841$	−27.8112	−0.959007
$842$	−46.9767	−1.61892
$843$	37.2007	1.28126
$844$	16.3255	0.561947
$845$	9.26132	0.318599
$846$	11.5627	0.397535
$847$	−45.1010	−1.54969
$848$	−47.7335	−1.63918
$849$	−32.5056	−1.11559
$850$	41.8951	1.43699
$851$	35.3111	1.21045
$852$	−6.07251	−0.208041
$853$	53.6073	1.83548	0.917739	−	0.397184i	$-0.130012\pi$
$853$	53.6073	1.83548	0.917739	+	0.397184i	$0.130012\pi$
$854$	60.4464	2.06843
$855$	−4.64745	−0.158939
$856$	0.834535	0.0285238
$857$	49.9719	1.70701	0.853503	−	0.521087i	$-0.174473\pi$
$857$	49.9719	1.70701	0.853503	+	0.521087i	$0.174473\pi$
$858$	19.2002	0.655483
$859$	−20.6492	−0.704541	−0.352270	−	0.935898i	$-0.614590\pi$
$859$	−20.6492	−0.704541	−0.352270	+	0.935898i	$0.614590\pi$
$860$	4.35897	0.148640
$861$	−12.8708	−0.438637
$862$	−31.9171	−1.08710
$863$	38.0634	1.29569	0.647846	−	0.761771i	$-0.275670\pi$
$863$	38.0634	1.29569	0.647846	+	0.761771i	$0.275670\pi$
$864$	−12.6609	−0.430732
$865$	−12.3545	−0.420065
$866$	−61.1968	−2.07955
$867$	26.1466	0.887984
$868$	−50.2839	−1.70675
$869$	5.93607	0.201367
$870$	3.96952	0.134579
$871$	59.8895	2.02928
$872$	−7.44844	−0.252236
$873$	−4.94212	−0.167265
$874$	16.6746	0.564028
$875$	−35.7200	−1.20756
$876$	16.1367	0.545210
$877$	−25.2001	−0.850948	−0.425474	−	0.904971i	$-0.639893\pi$
$877$	−25.2001	−0.850948	−0.425474	+	0.904971i	$0.639893\pi$
$878$	−38.4969	−1.29921
$879$	−39.7633	−1.34118
$880$	−3.92941	−0.132460
$881$	−32.3168	−1.08878	−0.544389	−	0.838833i	$-0.683239\pi$
$881$	−32.3168	−1.08878	−0.544389	+	0.838833i	$0.683239\pi$
$882$	−51.6984	−1.74078
$883$	20.8144	0.700459	0.350230	−	0.936664i	$-0.386104\pi$
$883$	20.8144	0.700459	0.350230	+	0.936664i	$0.386104\pi$
$884$	−36.6727	−1.23344
$885$	14.1756	0.476507
$886$	76.6934	2.57657
$887$	15.8463	0.532068	0.266034	−	0.963964i	$-0.414287\pi$
$887$	15.8463	0.532068	0.266034	+	0.963964i	$0.414287\pi$
$888$	23.8470	0.800253
$889$	−69.2840	−2.32371
$890$	8.69223	0.291364
$891$	−10.1337	−0.339491
$892$	21.0907	0.706170
$893$	7.09597	0.237458
$894$	−75.6599	−2.53044
$895$	0.949060	0.0317236
$896$	−37.0783	−1.23870
$897$	−40.5387	−1.35355
$898$	54.7765	1.82792
$899$	8.71405	0.290630
$900$	−13.0478	−0.434928
$901$	52.7463	1.75723
$902$	2.20108	0.0732879
$903$	36.1395	1.20265
$904$	12.3305	0.410107
$905$	−8.84699	−0.294084
$906$	−30.3771	−1.00921
$907$	−53.0221	−1.76057	−0.880286	−	0.474444i	$-0.842649\pi$
$907$	−53.0221	−1.76057	−0.880286	+	0.474444i	$0.842649\pi$
$908$	−34.2017	−1.13502
$909$	−35.4176	−1.17473
$910$	34.7179	1.15089
$911$	−10.3445	−0.342727	−0.171364	−	0.985208i	$-0.554817\pi$
$911$	−10.3445	−0.342727	−0.171364	+	0.985208i	$0.554817\pi$
$912$	27.0972	0.897278
$913$	−14.3556	−0.475102
$914$	4.52278	0.149600
$915$	−14.4722	−0.478436
$916$	41.3611	1.36661
$917$	−80.2739	−2.65088
$918$	18.4969	0.610488
$919$	−51.4484	−1.69713	−0.848563	−	0.529094i	$-0.822532\pi$
$919$	−51.4484	−1.69713	−0.848563	+	0.529094i	$0.822532\pi$
$920$	3.44784	0.113672
$921$	−53.4162	−1.76013
$922$	−12.1681	−0.400736
$923$	9.20395	0.302952
$924$	13.4375	0.442063
$925$	−40.9826	−1.34750
$926$	−38.4432	−1.26332
$927$	24.6510	0.809645
$928$	−7.35814	−0.241543
$929$	2.02854	0.0665541	0.0332770	−	0.999446i	$-0.489406\pi$
$929$	2.02854	0.0665541	0.0332770	+	0.999446i	$0.489406\pi$
$930$	29.0970	0.954128
$931$	−31.7270	−1.03981
$932$	2.80132	0.0917604
$933$	5.61916	0.183963
$934$	−32.5326	−1.06450
$935$	4.34206	0.142001
$936$	−11.5074	−0.376132
$937$	−23.5868	−0.770549	−0.385274	−	0.922802i	$-0.625893\pi$
$937$	−23.5868	−0.770549	−0.385274	+	0.922802i	$0.625893\pi$
$938$	101.302	3.30764
$939$	−49.8205	−1.62583
$940$	−3.51962	−0.114797
$941$	46.1140	1.50327	0.751637	−	0.659577i	$-0.229264\pi$
$941$	46.1140	1.50327	0.751637	+	0.659577i	$0.229264\pi$
$942$	88.3088	2.87726
$943$	−4.64730	−0.151337
$944$	−34.7406	−1.13071
$945$	−7.24527	−0.235689
$946$	−6.18032	−0.200940
$947$	26.0758	0.847350	0.423675	−	0.905814i	$-0.360740\pi$
$947$	26.0758	0.847350	0.423675	+	0.905814i	$0.360740\pi$
$948$	20.3039	0.659440
$949$	−24.4580	−0.793942
$950$	−19.3528	−0.627889
$951$	−38.7843	−1.25767
$952$	25.8594	0.838108
$953$	−60.5603	−1.96174	−0.980870	−	0.194666i	$-0.937638\pi$
$953$	−60.5603	−1.96174	−0.980870	+	0.194666i	$0.937638\pi$
$954$	−39.7028	−1.28543
$955$	10.8591	0.351392
$956$	21.8676	0.707247
$957$	−2.32868	−0.0752757
$958$	−58.4144	−1.88728
$959$	41.8535	1.35152
$960$	−5.52631	−0.178361
$961$	32.8748	1.06048
$962$	86.7028	2.79541
$963$	1.67027	0.0538239
$964$	−27.3333	−0.880348
$965$	−12.2634	−0.394772
$966$	−68.5708	−2.20623
$967$	−36.5038	−1.17388	−0.586941	−	0.809630i	$-0.699668\pi$
$967$	−36.5038	−1.17388	−0.586941	+	0.809630i	$0.699668\pi$
$968$	−10.9977	−0.353479
$969$	−29.9428	−0.961903
$970$	3.63582	0.116739
$971$	2.62614	0.0842770	0.0421385	−	0.999112i	$-0.486583\pi$
$971$	2.62614	0.0842770	0.0421385	+	0.999112i	$0.486583\pi$
$972$	−26.7169	−0.856945
$973$	20.2582	0.649447
$974$	−43.2023	−1.38429
$975$	47.0498	1.50680
$976$	35.4675	1.13529
$977$	19.3688	0.619664	0.309832	−	0.950791i	$-0.399727\pi$
$977$	19.3688	0.619664	0.309832	+	0.950791i	$0.399727\pi$
$978$	−50.6829	−1.62066
$979$	−5.09922	−0.162972
$980$	15.7367	0.502689
$981$	−14.9076	−0.475965
$982$	21.1684	0.675512
$983$	45.2430	1.44303	0.721514	−	0.692400i	$-0.243446\pi$
$983$	45.2430	1.44303	0.721514	+	0.692400i	$0.243446\pi$
$984$	−3.13850	−0.100052
$985$	22.4043	0.713860
$986$	10.7499	0.342345
$987$	−29.1806	−0.928829
$988$	16.9404	0.538946
$989$	13.0490	0.414933
$990$	−3.26833	−0.103874
$991$	22.5861	0.717470	0.358735	−	0.933439i	$-0.383208\pi$
$991$	22.5861	0.717470	0.358735	+	0.933439i	$0.383208\pi$
$992$	−53.9359	−1.71247
$993$	7.11615	0.225824
$994$	15.5684	0.493799
$995$	−12.2393	−0.388012
$996$	−49.1025	−1.55587
$997$	−46.7957	−1.48203	−0.741017	−	0.671487i	$-0.765656\pi$
$997$	−46.7957	−1.48203	−0.741017	+	0.671487i	$0.765656\pi$
$998$	−20.8011	−0.658447
$999$	−18.0940	−0.572469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.4	✓	20
3.2	odd	2		3681.2.a.i.1.17		20
4.3	odd	2		6544.2.a.i.1.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.4	✓	20	1.1	even	1	trivial
3681.2.a.i.1.17		20	3.2	odd	2
6544.2.a.i.1.4		20	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	409.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84704 q^{2} +2.27493 q^{3} +1.41156 q^{4} +0.866439 q^{5} -4.20189 q^{6} +4.45724 q^{7} +1.08688 q^{8} +2.17533 q^{9} -1.60035 q^{10} +0.938829 q^{11} +3.21120 q^{12} -4.86713 q^{13} -8.23270 q^{14} +1.97109 q^{15} -4.83062 q^{16} +5.33791 q^{17} -4.01792 q^{18} -2.46577 q^{19} +1.22303 q^{20} +10.1399 q^{21} -1.73406 q^{22} +3.66124 q^{23} +2.47258 q^{24} -4.24928 q^{25} +8.98978 q^{26} -1.87608 q^{27} +6.29165 q^{28} -1.09032 q^{29} -3.64068 q^{30} -7.99217 q^{31} +6.74859 q^{32} +2.13578 q^{33} -9.85934 q^{34} +3.86192 q^{35} +3.07060 q^{36} +9.64459 q^{37} +4.55438 q^{38} -11.0724 q^{39} +0.941713 q^{40} -1.26932 q^{41} -18.7288 q^{42} +3.56408 q^{43} +1.32521 q^{44} +1.88479 q^{45} -6.76245 q^{46} -2.87779 q^{47} -10.9893 q^{48} +12.8670 q^{49} +7.84860 q^{50} +12.1434 q^{51} -6.87023 q^{52} +9.88144 q^{53} +3.46519 q^{54} +0.813438 q^{55} +4.84448 q^{56} -5.60946 q^{57} +2.01387 q^{58} +7.19174 q^{59} +2.78231 q^{60} -7.34223 q^{61} +14.7619 q^{62} +9.69595 q^{63} -2.80368 q^{64} -4.21707 q^{65} -3.94486 q^{66} -12.3049 q^{67} +7.53477 q^{68} +8.32907 q^{69} -7.13313 q^{70} -1.89104 q^{71} +2.36432 q^{72} +5.02515 q^{73} -17.8139 q^{74} -9.66684 q^{75} -3.48057 q^{76} +4.18459 q^{77} +20.4512 q^{78} +6.32284 q^{79} -4.18544 q^{80} -10.7939 q^{81} +2.34449 q^{82} -15.2910 q^{83} +14.3131 q^{84} +4.62498 q^{85} -6.58300 q^{86} -2.48041 q^{87} +1.02039 q^{88} -5.43146 q^{89} -3.48128 q^{90} -21.6940 q^{91} +5.16804 q^{92} -18.1817 q^{93} +5.31540 q^{94} -2.13644 q^{95} +15.3526 q^{96} -2.27190 q^{97} -23.7658 q^{98} +2.04226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more