LMFDB - Embedded newform 755.2.a.j.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [755,2,Mod(1,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("755.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 755 = 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	755.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.02870535261$
Analytic rank:	$0$
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} - 2 x^{14} - 22 x^{13} + 48 x^{12} + 171 x^{11} - 423 x^{10} - 527 x^{9} + 1641 x^{8} + 400 x^{7} + \cdots + 8 $ x^15 - 2x^14 - 22x^13 + 48x^12 + 171x^11 - 423x^10 - 527x^9 + 1641x^8 + 400x^7 - 2651x^6 + 421x^5 + 1591x^4 - 396x^3 - 288x^2 + 60x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.37394$ of defining polynomial
Character	$\chi$	$=$	755.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.37394 q^{2} +2.59317 q^{3} -0.112300 q^{4} +1.00000 q^{5} +3.56285 q^{6} +0.630227 q^{7} -2.90216 q^{8} +3.72455 q^{9} +1.37394 q^{10} +5.18068 q^{11} -0.291213 q^{12} -0.855835 q^{13} +0.865892 q^{14} +2.59317 q^{15} -3.76279 q^{16} +3.18557 q^{17} +5.11729 q^{18} -8.22633 q^{19} -0.112300 q^{20} +1.63429 q^{21} +7.11793 q^{22} -4.70418 q^{23} -7.52582 q^{24} +1.00000 q^{25} -1.17586 q^{26} +1.87888 q^{27} -0.0707744 q^{28} +7.90859 q^{29} +3.56285 q^{30} -7.84010 q^{31} +0.634499 q^{32} +13.4344 q^{33} +4.37677 q^{34} +0.630227 q^{35} -0.418266 q^{36} +8.34026 q^{37} -11.3025 q^{38} -2.21933 q^{39} -2.90216 q^{40} +2.59716 q^{41} +2.24541 q^{42} -3.75161 q^{43} -0.581790 q^{44} +3.72455 q^{45} -6.46324 q^{46} +0.869890 q^{47} -9.75756 q^{48} -6.60281 q^{49} +1.37394 q^{50} +8.26073 q^{51} +0.0961102 q^{52} -0.171583 q^{53} +2.58146 q^{54} +5.18068 q^{55} -1.82902 q^{56} -21.3323 q^{57} +10.8659 q^{58} -3.54781 q^{59} -0.291213 q^{60} -10.8752 q^{61} -10.7718 q^{62} +2.34731 q^{63} +8.39734 q^{64} -0.855835 q^{65} +18.4580 q^{66} +2.07416 q^{67} -0.357739 q^{68} -12.1988 q^{69} +0.865892 q^{70} +12.0172 q^{71} -10.8093 q^{72} -10.8558 q^{73} +11.4590 q^{74} +2.59317 q^{75} +0.923816 q^{76} +3.26501 q^{77} -3.04922 q^{78} +5.39876 q^{79} -3.76279 q^{80} -6.30139 q^{81} +3.56834 q^{82} -16.9079 q^{83} -0.183530 q^{84} +3.18557 q^{85} -5.15447 q^{86} +20.5083 q^{87} -15.0352 q^{88} -1.13731 q^{89} +5.11729 q^{90} -0.539371 q^{91} +0.528279 q^{92} -20.3307 q^{93} +1.19517 q^{94} -8.22633 q^{95} +1.64536 q^{96} -6.66313 q^{97} -9.07184 q^{98} +19.2957 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 2 q^{2} + 5 q^{3} + 18 q^{4} + 15 q^{5} - 4 q^{6} + 11 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 11 q^{13} - 9 q^{14} + 5 q^{15} + 40 q^{16} + 25 q^{17} - 15 q^{18} - 3 q^{19}+ \cdots - 11 q^{99}+O(q^{100}) $ 15 * q - 2 * q^2 + 5 * q^3 + 18 * q^4 + 15 * q^5 - 4 * q^6 + 11 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 11 * q^13 - 9 * q^14 + 5 * q^15 + 40 * q^16 + 25 * q^17 - 15 * q^18 - 3 * q^19 + 18 * q^20 - 11 * q^21 + 13 * q^22 + 8 * q^23 - 21 * q^24 + 15 * q^25 - 45 * q^26 + 20 * q^27 + 29 * q^28 + 24 * q^29 - 4 * q^30 - 11 * q^31 + 27 * q^32 + 26 * q^33 - 11 * q^34 + 11 * q^35 - 22 * q^36 + 35 * q^37 + 2 * q^38 + 11 * q^39 + 12 * q^40 - 20 * q^41 - 17 * q^42 - 13 * q^43 - 32 * q^44 + 10 * q^45 - 5 * q^46 + 6 * q^47 - 11 * q^48 + 40 * q^49 - 2 * q^50 + 23 * q^51 + 27 * q^52 + 38 * q^53 - 30 * q^54 + 2 * q^55 - 36 * q^56 + 4 * q^57 + 6 * q^58 - 15 * q^59 + 4 * q^60 + 33 * q^61 - 13 * q^62 + 2 * q^63 + 74 * q^64 + 11 * q^65 + 8 * q^66 + 24 * q^67 + 8 * q^68 + 13 * q^69 - 9 * q^70 + 19 * q^71 + 56 * q^72 + 4 * q^73 - 28 * q^74 + 5 * q^75 + 19 * q^76 + 14 * q^77 - 15 * q^78 - 4 * q^79 + 40 * q^80 - 13 * q^81 + 33 * q^82 + 19 * q^83 - 71 * q^84 + 25 * q^85 - 59 * q^86 + 49 * q^87 - 48 * q^88 - 35 * q^89 - 15 * q^90 - q^91 + 18 * q^92 + 2 * q^93 - 70 * q^94 - 3 * q^95 - 35 * q^96 + 55 * q^97 - 64 * q^98 - 11 * 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 $

$2$	1.37394	0.971519	0.485760	−	0.874092i	$-0.338543\pi$
$2$	1.37394	0.971519	0.485760	+	0.874092i	$0.338543\pi$
$3$	2.59317	1.49717	0.748585	−	0.663039i	$-0.230734\pi$
$3$	2.59317	1.49717	0.748585	+	0.663039i	$0.230734\pi$
$4$	−0.112300	−0.0561499
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	3.56285	1.45453
$7$	0.630227	0.238203	0.119102	−	0.992882i	$-0.461999\pi$
$7$	0.630227	0.238203	0.119102	+	0.992882i	$0.461999\pi$
$8$	−2.90216	−1.02607
$9$	3.72455	1.24152
$10$	1.37394	0.434477
$11$	5.18068	1.56203	0.781017	−	0.624509i	$-0.214701\pi$
$11$	5.18068	1.56203	0.781017	+	0.624509i	$0.214701\pi$
$12$	−0.291213	−0.0840660
$13$	−0.855835	−0.237366	−0.118683	−	0.992932i	$-0.537867\pi$
$13$	−0.855835	−0.237366	−0.118683	+	0.992932i	$0.537867\pi$
$14$	0.865892	0.231419
$15$	2.59317	0.669554
$16$	−3.76279	−0.940697
$17$	3.18557	0.772614	0.386307	−	0.922370i	$-0.373751\pi$
$17$	3.18557	0.772614	0.386307	+	0.922370i	$0.373751\pi$
$18$	5.11729	1.20616
$19$	−8.22633	−1.88725	−0.943625	−	0.331017i	$-0.892608\pi$
$19$	−8.22633	−1.88725	−0.943625	+	0.331017i	$0.892608\pi$
$20$	−0.112300	−0.0251110
$21$	1.63429	0.356631
$22$	7.11793	1.51755
$23$	−4.70418	−0.980890	−0.490445	−	0.871472i	$-0.663166\pi$
$23$	−4.70418	−0.980890	−0.490445	+	0.871472i	$0.663166\pi$
$24$	−7.52582	−1.53620
$25$	1.00000	0.200000
$26$	−1.17586	−0.230606
$27$	1.87888	0.361590
$28$	−0.0707744	−0.0133751
$29$	7.90859	1.46859	0.734294	−	0.678832i	$-0.237513\pi$
$29$	7.90859	1.46859	0.734294	+	0.678832i	$0.237513\pi$
$30$	3.56285	0.650485
$31$	−7.84010	−1.40812	−0.704062	−	0.710139i	$-0.748632\pi$
$31$	−7.84010	−1.40812	−0.704062	+	0.710139i	$0.748632\pi$
$32$	0.634499	0.112165
$33$	13.4344	2.33863
$34$	4.37677	0.750610
$35$	0.630227	0.106528
$36$	−0.418266	−0.0697111
$37$	8.34026	1.37113	0.685565	−	0.728011i	$-0.259555\pi$
$37$	8.34026	1.37113	0.685565	+	0.728011i	$0.259555\pi$
$38$	−11.3025	−1.83350
$39$	−2.21933	−0.355377
$40$	−2.90216	−0.458873
$41$	2.59716	0.405609	0.202804	−	0.979219i	$-0.434994\pi$
$41$	2.59716	0.405609	0.202804	+	0.979219i	$0.434994\pi$
$42$	2.24541	0.346474
$43$	−3.75161	−0.572115	−0.286058	−	0.958212i	$-0.592345\pi$
$43$	−3.75161	−0.572115	−0.286058	+	0.958212i	$0.592345\pi$
$44$	−0.581790	−0.0877082
$45$	3.72455	0.555223
$46$	−6.46324	−0.952953
$47$	0.869890	0.126887	0.0634433	−	0.997985i	$-0.479792\pi$
$47$	0.869890	0.126887	0.0634433	+	0.997985i	$0.479792\pi$
$48$	−9.75756	−1.40838
$49$	−6.60281	−0.943259
$50$	1.37394	0.194304
$51$	8.26073	1.15673
$52$	0.0961102	0.0133281
$53$	−0.171583	−0.0235688	−0.0117844	−	0.999931i	$-0.503751\pi$
$53$	−0.171583	−0.0235688	−0.0117844	+	0.999931i	$0.503751\pi$
$54$	2.58146	0.351292
$55$	5.18068	0.698563
$56$	−1.82902	−0.244413
$57$	−21.3323	−2.82553
$58$	10.8659	1.42676
$59$	−3.54781	−0.461885	−0.230943	−	0.972967i	$-0.574181\pi$
$59$	−3.54781	−0.461885	−0.230943	+	0.972967i	$0.574181\pi$
$60$	−0.291213	−0.0375954
$61$	−10.8752	−1.39243	−0.696213	−	0.717835i	$-0.745133\pi$
$61$	−10.8752	−1.39243	−0.696213	+	0.717835i	$0.745133\pi$
$62$	−10.7718	−1.36802
$63$	2.34731	0.295733
$64$	8.39734	1.04967
$65$	−0.855835	−0.106153
$66$	18.4580	2.27202
$67$	2.07416	0.253399	0.126700	−	0.991941i	$-0.459562\pi$
$67$	2.07416	0.253399	0.126700	+	0.991941i	$0.459562\pi$
$68$	−0.357739	−0.0433822
$69$	−12.1988	−1.46856
$70$	0.865892	0.103494
$71$	12.0172	1.42618	0.713089	−	0.701074i	$-0.247296\pi$
$71$	12.0172	1.42618	0.713089	+	0.701074i	$0.247296\pi$
$72$	−10.8093	−1.27388
$73$	−10.8558	−1.27058	−0.635290	−	0.772274i	$-0.719119\pi$
$73$	−10.8558	−1.27058	−0.635290	+	0.772274i	$0.719119\pi$
$74$	11.4590	1.33208
$75$	2.59317	0.299434
$76$	0.923816	0.105969
$77$	3.26501	0.372082
$78$	−3.04922	−0.345256
$79$	5.39876	0.607408	0.303704	−	0.952766i	$-0.401777\pi$
$79$	5.39876	0.607408	0.303704	+	0.952766i	$0.401777\pi$
$80$	−3.76279	−0.420693
$81$	−6.30139	−0.700154
$82$	3.56834	0.394057
$83$	−16.9079	−1.85588	−0.927941	−	0.372726i	$-0.878423\pi$
$83$	−16.9079	−1.85588	−0.927941	+	0.372726i	$0.878423\pi$
$84$	−0.183530	−0.0200248
$85$	3.18557	0.345524
$86$	−5.15447	−0.555821
$87$	20.5083	2.19872
$88$	−15.0352	−1.60276
$89$	−1.13731	−0.120554	−0.0602771	−	0.998182i	$-0.519198\pi$
$89$	−1.13731	−0.120554	−0.0602771	+	0.998182i	$0.519198\pi$
$90$	5.11729	0.539410
$91$	−0.539371	−0.0565414
$92$	0.528279	0.0550769
$93$	−20.3307	−2.10820
$94$	1.19517	0.123273
$95$	−8.22633	−0.844004
$96$	1.64536	0.167929
$97$	−6.66313	−0.676538	−0.338269	−	0.941049i	$-0.609841\pi$
$97$	−6.66313	−0.676538	−0.338269	+	0.941049i	$0.609841\pi$
$98$	−9.07184	−0.916395
$99$	19.2957	1.93929
$100$	−0.112300	−0.0112300
$101$	−10.5050	−1.04528	−0.522641	−	0.852553i	$-0.675053\pi$
$101$	−10.5050	−1.04528	−0.522641	+	0.852553i	$0.675053\pi$
$102$	11.3497	1.12379
$103$	6.75436	0.665527	0.332763	−	0.943010i	$-0.392019\pi$
$103$	6.75436	0.665527	0.332763	+	0.943010i	$0.392019\pi$
$104$	2.48378	0.243554
$105$	1.63429	0.159490
$106$	−0.235745	−0.0228976
$107$	−9.85539	−0.952756	−0.476378	−	0.879241i	$-0.658051\pi$
$107$	−9.85539	−0.952756	−0.476378	+	0.879241i	$0.658051\pi$
$108$	−0.210998	−0.0203033
$109$	15.2846	1.46400	0.732001	−	0.681303i	$-0.238587\pi$
$109$	15.2846	1.46400	0.732001	+	0.681303i	$0.238587\pi$
$110$	7.11793	0.678668
$111$	21.6277	2.05281
$112$	−2.37141	−0.224077
$113$	10.4181	0.980056	0.490028	−	0.871707i	$-0.336987\pi$
$113$	10.4181	0.980056	0.490028	+	0.871707i	$0.336987\pi$
$114$	−29.3092	−2.74506
$115$	−4.70418	−0.438667
$116$	−0.888134	−0.0824611
$117$	−3.18760	−0.294694
$118$	−4.87446	−0.448730
$119$	2.00763	0.184039
$120$	−7.52582	−0.687010
$121$	15.8395	1.43995
$122$	−14.9418	−1.35277
$123$	6.73490	0.607265
$124$	0.880443	0.0790661
$125$	1.00000	0.0894427
$126$	3.22505	0.287311
$127$	0.896603	0.0795607	0.0397803	−	0.999208i	$-0.487334\pi$
$127$	0.896603	0.0795607	0.0397803	+	0.999208i	$0.487334\pi$
$128$	10.2684	0.907608
$129$	−9.72858	−0.856553
$130$	−1.17586	−0.103130
$131$	18.2492	1.59444	0.797220	−	0.603689i	$-0.206303\pi$
$131$	18.2492	1.59444	0.797220	+	0.603689i	$0.206303\pi$
$132$	−1.50868	−0.131314
$133$	−5.18446	−0.449549
$134$	2.84977	0.246182
$135$	1.87888	0.161708
$136$	−9.24505	−0.792756
$137$	−7.97837	−0.681638	−0.340819	−	0.940129i	$-0.610704\pi$
$137$	−7.97837	−0.681638	−0.340819	+	0.940129i	$0.610704\pi$
$138$	−16.7603	−1.42673
$139$	−21.2734	−1.80439	−0.902193	−	0.431333i	$-0.858044\pi$
$139$	−21.2734	−1.80439	−0.902193	+	0.431333i	$0.858044\pi$
$140$	−0.0707744	−0.00598153
$141$	2.25578	0.189971
$142$	16.5108	1.38556
$143$	−4.43381	−0.370774
$144$	−14.0147	−1.16789
$145$	7.90859	0.656773
$146$	−14.9152	−1.23439
$147$	−17.1222	−1.41222
$148$	−0.936611	−0.0769889
$149$	−3.39424	−0.278067	−0.139033	−	0.990288i	$-0.544400\pi$
$149$	−3.39424	−0.278067	−0.139033	+	0.990288i	$0.544400\pi$
$150$	3.56285	0.290906
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	23.8742	1.93645
$153$	11.8648	0.959213
$154$	4.48591	0.361485
$155$	−7.84010	−0.629732
$156$	0.249230	0.0199544
$157$	3.54156	0.282648	0.141324	−	0.989963i	$-0.454864\pi$
$157$	3.54156	0.282648	0.141324	+	0.989963i	$0.454864\pi$
$158$	7.41755	0.590109
$159$	−0.444946	−0.0352865
$160$	0.634499	0.0501615
$161$	−2.96470	−0.233651
$162$	−8.65770	−0.680213
$163$	15.9154	1.24659	0.623296	−	0.781986i	$-0.285793\pi$
$163$	15.9154	1.24659	0.623296	+	0.781986i	$0.285793\pi$
$164$	−0.291661	−0.0227749
$165$	13.4344	1.04587
$166$	−23.2304	−1.80303
$167$	5.08284	0.393322	0.196661	−	0.980472i	$-0.436990\pi$
$167$	5.08284	0.393322	0.196661	+	0.980472i	$0.436990\pi$
$168$	−4.74297	−0.365928
$169$	−12.2675	−0.943657
$170$	4.37677	0.335683
$171$	−30.6394	−2.34305
$172$	0.421305	0.0321242
$173$	19.5934	1.48966	0.744829	−	0.667256i	$-0.232531\pi$
$173$	19.5934	1.48966	0.744829	+	0.667256i	$0.232531\pi$
$174$	28.1771	2.13610
$175$	0.630227	0.0476407
$176$	−19.4938	−1.46940
$177$	−9.20008	−0.691520
$178$	−1.56259	−0.117121
$179$	0.314200	0.0234844	0.0117422	−	0.999931i	$-0.496262\pi$
$179$	0.314200	0.0234844	0.0117422	+	0.999931i	$0.496262\pi$
$180$	−0.418266	−0.0311757
$181$	18.7645	1.39476	0.697378	−	0.716704i	$-0.254350\pi$
$181$	18.7645	1.39476	0.697378	+	0.716704i	$0.254350\pi$
$182$	−0.741061	−0.0549311
$183$	−28.2013	−2.08470
$184$	13.6523	1.00646
$185$	8.34026	0.613188
$186$	−27.9331	−2.04816
$187$	16.5034	1.20685
$188$	−0.0976886	−0.00712467
$189$	1.18412	0.0861320
$190$	−11.3025	−0.819966
$191$	20.1843	1.46049	0.730244	−	0.683187i	$-0.239407\pi$
$191$	20.1843	1.46049	0.730244	+	0.683187i	$0.239407\pi$
$192$	21.7758	1.57153
$193$	17.7680	1.27897	0.639486	−	0.768803i	$-0.279147\pi$
$193$	17.7680	1.27897	0.639486	+	0.768803i	$0.279147\pi$
$194$	−9.15471	−0.657270
$195$	−2.21933	−0.158929
$196$	0.741495	0.0529640
$197$	18.6907	1.33166	0.665829	−	0.746105i	$-0.268078\pi$
$197$	18.6907	1.33166	0.665829	+	0.746105i	$0.268078\pi$
$198$	26.5111	1.88406
$199$	−2.33943	−0.165838	−0.0829189	−	0.996556i	$-0.526424\pi$
$199$	−2.33943	−0.165838	−0.0829189	+	0.996556i	$0.526424\pi$
$200$	−2.90216	−0.205214
$201$	5.37866	0.379382
$202$	−14.4331	−1.01551
$203$	4.98421	0.349823
$204$	−0.927680	−0.0649506
$205$	2.59716	0.181394
$206$	9.28006	0.646572
$207$	−17.5209	−1.21779
$208$	3.22033	0.223290
$209$	−42.6180	−2.94795
$210$	2.24541	0.154948
$211$	−2.54512	−0.175213	−0.0876065	−	0.996155i	$-0.527922\pi$
$211$	−2.54512	−0.175213	−0.0876065	+	0.996155i	$0.527922\pi$
$212$	0.0192688	0.00132339
$213$	31.1626	2.13523
$214$	−13.5407	−0.925621
$215$	−3.75161	−0.255858
$216$	−5.45281	−0.371017
$217$	−4.94104	−0.335420
$218$	21.0001	1.42231
$219$	−28.1511	−1.90227
$220$	−0.581790	−0.0392243
$221$	−2.72632	−0.183392
$222$	29.7151	1.99435
$223$	−24.5770	−1.64580	−0.822898	−	0.568188i	$-0.807644\pi$
$223$	−24.5770	−1.64580	−0.822898	+	0.568188i	$0.807644\pi$
$224$	0.399878	0.0267180
$225$	3.72455	0.248303
$226$	14.3139	0.952144
$227$	7.66549	0.508776	0.254388	−	0.967102i	$-0.418126\pi$
$227$	7.66549	0.508776	0.254388	+	0.967102i	$0.418126\pi$
$228$	2.39562	0.158653
$229$	10.7688	0.711622	0.355811	−	0.934558i	$-0.384205\pi$
$229$	10.7688	0.711622	0.355811	+	0.934558i	$0.384205\pi$
$230$	−6.46324	−0.426174
$231$	8.46673	0.557070
$232$	−22.9520	−1.50687
$233$	−4.78465	−0.313453	−0.156726	−	0.987642i	$-0.550094\pi$
$233$	−4.78465	−0.313453	−0.156726	+	0.987642i	$0.550094\pi$
$234$	−4.37956	−0.286301
$235$	0.869890	0.0567454
$236$	0.398418	0.0259348
$237$	13.9999	0.909393
$238$	2.75836	0.178798
$239$	−15.6444	−1.01195	−0.505976	−	0.862547i	$-0.668868\pi$
$239$	−15.6444	−1.01195	−0.505976	+	0.862547i	$0.668868\pi$
$240$	−9.75756	−0.629848
$241$	−4.44919	−0.286597	−0.143299	−	0.989679i	$-0.545771\pi$
$241$	−4.44919	−0.286597	−0.143299	+	0.989679i	$0.545771\pi$
$242$	21.7624	1.39894
$243$	−21.9772	−1.40984
$244$	1.22128	0.0781846
$245$	−6.60281	−0.421838
$246$	9.25332	0.589970
$247$	7.04038	0.447969
$248$	22.7533	1.44483
$249$	−43.8451	−2.77857
$250$	1.37394	0.0868953
$251$	9.24794	0.583725	0.291862	−	0.956460i	$-0.405725\pi$
$251$	9.24794	0.583725	0.291862	+	0.956460i	$0.405725\pi$
$252$	−0.263603	−0.0166054
$253$	−24.3709	−1.53218
$254$	1.23188	0.0772947
$255$	8.26073	0.517307
$256$	−2.68654	−0.167909
$257$	9.02608	0.563031	0.281516	−	0.959557i	$-0.409163\pi$
$257$	9.02608	0.563031	0.281516	+	0.959557i	$0.409163\pi$
$258$	−13.3664	−0.832158
$259$	5.25626	0.326608
$260$	0.0961102	0.00596050
$261$	29.4559	1.82328
$262$	25.0732	1.54903
$263$	2.84760	0.175590	0.0877952	−	0.996139i	$-0.472018\pi$
$263$	2.84760	0.175590	0.0877952	+	0.996139i	$0.472018\pi$
$264$	−38.9889	−2.39960
$265$	−0.171583	−0.0105403
$266$	−7.12311	−0.436746
$267$	−2.94923	−0.180490
$268$	−0.232928	−0.0142284
$269$	−2.69331	−0.164214	−0.0821069	−	0.996624i	$-0.526165\pi$
$269$	−2.69331	−0.164214	−0.0821069	+	0.996624i	$0.526165\pi$
$270$	2.58146	0.157103
$271$	−5.79191	−0.351833	−0.175917	−	0.984405i	$-0.556289\pi$
$271$	−5.79191	−0.351833	−0.175917	+	0.984405i	$0.556289\pi$
$272$	−11.9866	−0.726796
$273$	−1.39868	−0.0846520
$274$	−10.9618	−0.662224
$275$	5.18068	0.312407
$276$	1.36992	0.0824595
$277$	−26.7346	−1.60632	−0.803162	−	0.595760i	$-0.796851\pi$
$277$	−26.7346	−1.60632	−0.803162	+	0.595760i	$0.796851\pi$
$278$	−29.2283	−1.75300
$279$	−29.2008	−1.74821
$280$	−1.82902	−0.109305
$281$	12.9751	0.774031	0.387016	−	0.922073i	$-0.373506\pi$
$281$	12.9751	0.774031	0.387016	+	0.922073i	$0.373506\pi$
$282$	3.09929	0.184560
$283$	25.7387	1.53000	0.765002	−	0.644028i	$-0.222738\pi$
$283$	25.7387	1.53000	0.765002	+	0.644028i	$0.222738\pi$
$284$	−1.34953	−0.0800798
$285$	−21.3323	−1.26362
$286$	−6.09177	−0.360214
$287$	1.63680	0.0966174
$288$	2.36322	0.139254
$289$	−6.85215	−0.403067
$290$	10.8659	0.638067
$291$	−17.2786	−1.01289
$292$	1.21911	0.0713430
$293$	17.3208	1.01189	0.505946	−	0.862565i	$-0.331144\pi$
$293$	17.3208	1.01189	0.505946	+	0.862565i	$0.331144\pi$
$294$	−23.5249	−1.37200
$295$	−3.54781	−0.206561
$296$	−24.2048	−1.40688
$297$	9.73387	0.564816
$298$	−4.66347	−0.270147
$299$	4.02600	0.232830
$300$	−0.291213	−0.0168132
$301$	−2.36437	−0.136280
$302$	−1.37394	−0.0790611
$303$	−27.2412	−1.56496
$304$	30.9539	1.77533
$305$	−10.8752	−0.622712
$306$	16.3015	0.931894
$307$	2.77705	0.158495	0.0792474	−	0.996855i	$-0.474748\pi$
$307$	2.77705	0.158495	0.0792474	+	0.996855i	$0.474748\pi$
$308$	−0.366660	−0.0208924
$309$	17.5152	0.996406
$310$	−10.7718	−0.611797
$311$	22.2633	1.26243	0.631217	−	0.775606i	$-0.282556\pi$
$311$	22.2633	1.26243	0.631217	+	0.775606i	$0.282556\pi$
$312$	6.44086	0.364642
$313$	11.6916	0.660846	0.330423	−	0.943833i	$-0.392809\pi$
$313$	11.6916	0.660846	0.330423	+	0.943833i	$0.392809\pi$
$314$	4.86588	0.274598
$315$	2.34731	0.132256
$316$	−0.606280	−0.0341059
$317$	−27.4049	−1.53921	−0.769606	−	0.638519i	$-0.779547\pi$
$317$	−27.4049	−1.53921	−0.769606	+	0.638519i	$0.779547\pi$
$318$	−0.611327	−0.0342815
$319$	40.9719	2.29399
$320$	8.39734	0.469425
$321$	−25.5567	−1.42644
$322$	−4.07331	−0.226997
$323$	−26.2055	−1.45812
$324$	0.707645	0.0393136
$325$	−0.855835	−0.0474732
$326$	21.8668	1.21109
$327$	39.6357	2.19186
$328$	−7.53740	−0.416183
$329$	0.548228	0.0302248
$330$	18.4580	1.01608
$331$	−16.9726	−0.932900	−0.466450	−	0.884548i	$-0.654467\pi$
$331$	−16.9726	−0.932900	−0.466450	+	0.884548i	$0.654467\pi$
$332$	1.89875	0.104208
$333$	31.0637	1.70228
$334$	6.98350	0.382120
$335$	2.07416	0.113324
$336$	−6.14948	−0.335482
$337$	33.6340	1.83216	0.916080	−	0.400995i	$-0.131336\pi$
$337$	33.6340	1.83216	0.916080	+	0.400995i	$0.131336\pi$
$338$	−16.8548	−0.916782
$339$	27.0160	1.46731
$340$	−0.357739	−0.0194011
$341$	−40.6171	−2.19954
$342$	−42.0965	−2.27632
$343$	−8.57286	−0.462891
$344$	10.8878	0.587030
$345$	−12.1988	−0.656759
$346$	26.9201	1.44723
$347$	−14.8627	−0.797874	−0.398937	−	0.916978i	$-0.630621\pi$
$347$	−14.8627	−0.797874	−0.398937	+	0.916978i	$0.630621\pi$
$348$	−2.30308	−0.123458
$349$	−8.82000	−0.472124	−0.236062	−	0.971738i	$-0.575857\pi$
$349$	−8.82000	−0.472124	−0.236062	+	0.971738i	$0.575857\pi$
$350$	0.865892	0.0462839
$351$	−1.60801	−0.0858292
$352$	3.28714	0.175205
$353$	−5.03612	−0.268046	−0.134023	−	0.990978i	$-0.542790\pi$
$353$	−5.03612	−0.268046	−0.134023	+	0.990978i	$0.542790\pi$
$354$	−12.6403	−0.671825
$355$	12.0172	0.637806
$356$	0.127719	0.00676912
$357$	5.20614	0.275538
$358$	0.431690	0.0228155
$359$	24.7446	1.30597	0.652986	−	0.757370i	$-0.273516\pi$
$359$	24.7446	1.30597	0.652986	+	0.757370i	$0.273516\pi$
$360$	−10.8093	−0.569698
$361$	48.6725	2.56171
$362$	25.7812	1.35503
$363$	41.0745	2.15585
$364$	0.0605713	0.00317480
$365$	−10.8558	−0.568220
$366$	−38.7467	−2.02532
$367$	0.467081	0.0243814	0.0121907	−	0.999926i	$-0.496119\pi$
$367$	0.467081	0.0243814	0.0121907	+	0.999926i	$0.496119\pi$
$368$	17.7008	0.922720
$369$	9.67326	0.503570
$370$	11.4590	0.595724
$371$	−0.108137	−0.00561417
$372$	2.28314	0.118375
$373$	25.5913	1.32507	0.662534	−	0.749032i	$-0.269481\pi$
$373$	25.5913	1.32507	0.662534	+	0.749032i	$0.269481\pi$
$374$	22.6747	1.17248
$375$	2.59317	0.133911
$376$	−2.52456	−0.130194
$377$	−6.76845	−0.348593
$378$	1.62690	0.0836789
$379$	14.9668	0.768792	0.384396	−	0.923168i	$-0.374410\pi$
$379$	14.9668	0.768792	0.384396	+	0.923168i	$0.374410\pi$
$380$	0.923816	0.0473908
$381$	2.32505	0.119116
$382$	27.7320	1.41889
$383$	11.6224	0.593876	0.296938	−	0.954897i	$-0.404035\pi$
$383$	11.6224	0.593876	0.296938	+	0.954897i	$0.404035\pi$
$384$	26.6278	1.35884
$385$	3.26501	0.166400
$386$	24.4122	1.24255
$387$	−13.9731	−0.710290
$388$	0.748268	0.0379876
$389$	−27.8265	−1.41086	−0.705430	−	0.708779i	$-0.749246\pi$
$389$	−27.8265	−1.41086	−0.705430	+	0.708779i	$0.749246\pi$
$390$	−3.04922	−0.154403
$391$	−14.9855	−0.757849
$392$	19.1625	0.967850
$393$	47.3233	2.38715
$394$	25.6798	1.29373
$395$	5.39876	0.271641
$396$	−2.16690	−0.108891
$397$	29.5489	1.48302	0.741509	−	0.670943i	$-0.234110\pi$
$397$	29.5489	1.48302	0.741509	+	0.670943i	$0.234110\pi$
$398$	−3.21423	−0.161115
$399$	−13.4442	−0.673051
$400$	−3.76279	−0.188139
$401$	−21.9047	−1.09387	−0.546934	−	0.837175i	$-0.684205\pi$
$401$	−21.9047	−1.09387	−0.546934	+	0.837175i	$0.684205\pi$
$402$	7.38994	0.368577
$403$	6.70984	0.334241
$404$	1.17971	0.0586926
$405$	−6.30139	−0.313118
$406$	6.84798	0.339860
$407$	43.2082	2.14175
$408$	−23.9740	−1.18689
$409$	28.3917	1.40388	0.701939	−	0.712237i	$-0.252318\pi$
$409$	28.3917	1.40388	0.701939	+	0.712237i	$0.252318\pi$
$410$	3.56834	0.176228
$411$	−20.6893	−1.02053
$412$	−0.758514	−0.0373693
$413$	−2.23592	−0.110023
$414$	−24.0727	−1.18311
$415$	−16.9079	−0.829976
$416$	−0.543026	−0.0266241
$417$	−55.1656	−2.70147
$418$	−58.5544	−2.86399
$419$	−29.5636	−1.44428	−0.722139	−	0.691748i	$-0.756841\pi$
$419$	−29.5636	−1.44428	−0.722139	+	0.691748i	$0.756841\pi$
$420$	−0.183530	−0.00895537
$421$	9.26381	0.451491	0.225745	−	0.974186i	$-0.427518\pi$
$421$	9.26381	0.451491	0.225745	+	0.974186i	$0.427518\pi$
$422$	−3.49683	−0.170223
$423$	3.23995	0.157532
$424$	0.497964	0.0241832
$425$	3.18557	0.154523
$426$	42.8155	2.07442
$427$	−6.85384	−0.331681
$428$	1.10676	0.0534972
$429$	−11.4976	−0.555111
$430$	−5.15447	−0.248571
$431$	−26.9482	−1.29805	−0.649024	−	0.760768i	$-0.724823\pi$
$431$	−26.9482	−1.29805	−0.649024	+	0.760768i	$0.724823\pi$
$432$	−7.06982	−0.340147
$433$	27.2498	1.30954	0.654770	−	0.755828i	$-0.272765\pi$
$433$	27.2498	1.30954	0.654770	+	0.755828i	$0.272765\pi$
$434$	−6.78868	−0.325867
$435$	20.5083	0.983300
$436$	−1.71646	−0.0822036
$437$	38.6982	1.85118
$438$	−38.6778	−1.84809
$439$	−20.0464	−0.956763	−0.478381	−	0.878152i	$-0.658776\pi$
$439$	−20.0464	−0.956763	−0.478381	+	0.878152i	$0.658776\pi$
$440$	−15.0352	−0.716775
$441$	−24.5925	−1.17107
$442$	−3.74579	−0.178169
$443$	15.4823	0.735588	0.367794	−	0.929907i	$-0.380113\pi$
$443$	15.4823	0.735588	0.367794	+	0.929907i	$0.380113\pi$
$444$	−2.42879	−0.115265
$445$	−1.13731	−0.0539135
$446$	−33.7672	−1.59892
$447$	−8.80185	−0.416313
$448$	5.29223	0.250034
$449$	6.44934	0.304363	0.152182	−	0.988353i	$-0.451370\pi$
$449$	6.44934	0.304363	0.152182	+	0.988353i	$0.451370\pi$
$450$	5.11729	0.241231
$451$	13.4551	0.633575
$452$	−1.16996	−0.0550301
$453$	−2.59317	−0.121838
$454$	10.5319	0.494286
$455$	−0.539371	−0.0252861
$456$	61.9099	2.89919
$457$	28.6199	1.33878	0.669391	−	0.742911i	$-0.266555\pi$
$457$	28.6199	1.33878	0.669391	+	0.742911i	$0.266555\pi$
$458$	14.7956	0.691355
$459$	5.98530	0.279370
$460$	0.528279	0.0246311
$461$	12.3582	0.575581	0.287791	−	0.957693i	$-0.407079\pi$
$461$	12.3582	0.575581	0.287791	+	0.957693i	$0.407079\pi$
$462$	11.6327	0.541204
$463$	13.8354	0.642983	0.321492	−	0.946912i	$-0.395816\pi$
$463$	13.8354	0.642983	0.321492	+	0.946912i	$0.395816\pi$
$464$	−29.7583	−1.38150
$465$	−20.3307	−0.942816
$466$	−6.57380	−0.304525
$467$	−28.0863	−1.29968	−0.649840	−	0.760071i	$-0.725164\pi$
$467$	−28.0863	−1.29968	−0.649840	+	0.760071i	$0.725164\pi$
$468$	0.357967	0.0165470
$469$	1.30719	0.0603606
$470$	1.19517	0.0551292
$471$	9.18389	0.423171
$472$	10.2963	0.473927
$473$	−19.4359	−0.893664
$474$	19.2350	0.883493
$475$	−8.22633	−0.377450
$476$	−0.225457	−0.0103338
$477$	−0.639071	−0.0292610
$478$	−21.4944	−0.983131
$479$	−15.9302	−0.727869	−0.363935	−	0.931424i	$-0.618567\pi$
$479$	−15.9302	−0.727869	−0.363935	+	0.931424i	$0.618567\pi$
$480$	1.64536	0.0751003
$481$	−7.13789	−0.325460
$482$	−6.11290	−0.278435
$483$	−7.68799	−0.349816
$484$	−1.77877	−0.0808532
$485$	−6.66313	−0.302557
$486$	−30.1953	−1.36969
$487$	13.3874	0.606640	0.303320	−	0.952889i	$-0.401905\pi$
$487$	13.3874	0.606640	0.303320	+	0.952889i	$0.401905\pi$
$488$	31.5616	1.42873
$489$	41.2715	1.86636
$490$	−9.07184	−0.409824
$491$	−22.0823	−0.996560	−0.498280	−	0.867016i	$-0.666035\pi$
$491$	−22.0823	−0.996560	−0.498280	+	0.867016i	$0.666035\pi$
$492$	−0.756328	−0.0340979
$493$	25.1934	1.13465
$494$	9.67304	0.435210
$495$	19.2957	0.867277
$496$	29.5006	1.32462
$497$	7.57355	0.339720
$498$	−60.2404	−2.69944
$499$	17.0256	0.762170	0.381085	−	0.924540i	$-0.375550\pi$
$499$	17.0256	0.762170	0.381085	+	0.924540i	$0.375550\pi$
$500$	−0.112300	−0.00502220
$501$	13.1807	0.588869
$502$	12.7061	0.567100
$503$	19.3446	0.862531	0.431266	−	0.902225i	$-0.358067\pi$
$503$	19.3446	0.862531	0.431266	+	0.902225i	$0.358067\pi$
$504$	−6.81228	−0.303443
$505$	−10.5050	−0.467465
$506$	−33.4840	−1.48855
$507$	−31.8119	−1.41281
$508$	−0.100688	−0.00446733
$509$	−4.72064	−0.209239	−0.104619	−	0.994512i	$-0.533362\pi$
$509$	−4.72064	−0.209239	−0.104619	+	0.994512i	$0.533362\pi$
$510$	11.3497	0.502574
$511$	−6.84164	−0.302656
$512$	−24.2280	−1.07073
$513$	−15.4563	−0.682411
$514$	12.4013	0.546996
$515$	6.75436	0.297633
$516$	1.09252	0.0480954
$517$	4.50662	0.198201
$518$	7.22176	0.317306
$519$	50.8090	2.23027
$520$	2.48378	0.108921
$521$	−5.67236	−0.248511	−0.124255	−	0.992250i	$-0.539654\pi$
$521$	−5.67236	−0.248511	−0.124255	+	0.992250i	$0.539654\pi$
$522$	40.4705	1.77135
$523$	−14.4840	−0.633340	−0.316670	−	0.948536i	$-0.602565\pi$
$523$	−14.4840	−0.633340	−0.316670	+	0.948536i	$0.602565\pi$
$524$	−2.04938	−0.0895277
$525$	1.63429	0.0713262
$526$	3.91242	0.170589
$527$	−24.9752	−1.08794
$528$	−50.5508	−2.19994
$529$	−0.870673	−0.0378553
$530$	−0.235745	−0.0102401
$531$	−13.2140	−0.573438
$532$	0.582214	0.0252422
$533$	−2.22274	−0.0962777
$534$	−4.05206	−0.175350
$535$	−9.85539	−0.426086
$536$	−6.01956	−0.260006
$537$	0.814775	0.0351601
$538$	−3.70043	−0.159537
$539$	−34.2071	−1.47340
$540$	−0.210998	−0.00907990
$541$	−36.6339	−1.57501	−0.787507	−	0.616305i	$-0.788629\pi$
$541$	−36.6339	−1.57501	−0.787507	+	0.616305i	$0.788629\pi$
$542$	−7.95771	−0.341813
$543$	48.6596	2.08818
$544$	2.02124	0.0866599
$545$	15.2846	0.654722
$546$	−1.92170	−0.0822411
$547$	−28.4935	−1.21829	−0.609147	−	0.793057i	$-0.708488\pi$
$547$	−28.4935	−1.21829	−0.609147	+	0.793057i	$0.708488\pi$
$548$	0.895970	0.0382739
$549$	−40.5052	−1.72872
$550$	7.11793	0.303509
$551$	−65.0587	−2.77159
$552$	35.4028	1.50684
$553$	3.40245	0.144687
$554$	−36.7316	−1.56058
$555$	21.6277	0.918047
$556$	2.38900	0.101316
$557$	−17.8421	−0.755994	−0.377997	−	0.925807i	$-0.623387\pi$
$557$	−17.8421	−0.755994	−0.377997	+	0.925807i	$0.623387\pi$
$558$	−40.1201	−1.69842
$559$	3.21076	0.135801
$560$	−2.37141	−0.100210
$561$	42.7962	1.80686
$562$	17.8270	0.751987
$563$	−32.8159	−1.38303	−0.691513	−	0.722364i	$-0.743056\pi$
$563$	−32.8159	−1.38303	−0.691513	+	0.722364i	$0.743056\pi$
$564$	−0.253323	−0.0106668
$565$	10.4181	0.438294
$566$	35.3633	1.48643
$567$	−3.97130	−0.166779
$568$	−34.8758	−1.46336
$569$	23.3893	0.980531	0.490265	−	0.871573i	$-0.336900\pi$
$569$	23.3893	0.980531	0.490265	+	0.871573i	$0.336900\pi$
$570$	−29.3092	−1.22763
$571$	−7.61103	−0.318512	−0.159256	−	0.987237i	$-0.550909\pi$
$571$	−7.61103	−0.318512	−0.159256	+	0.987237i	$0.550909\pi$
$572$	0.497916	0.0208189
$573$	52.3415	2.18660
$574$	2.24886	0.0938657
$575$	−4.70418	−0.196178
$576$	31.2763	1.30318
$577$	−11.5248	−0.479783	−0.239891	−	0.970800i	$-0.577112\pi$
$577$	−11.5248	−0.479783	−0.239891	+	0.970800i	$0.577112\pi$
$578$	−9.41441	−0.391588
$579$	46.0756	1.91484
$580$	−0.888134	−0.0368777
$581$	−10.6558	−0.442078
$582$	−23.7397	−0.984044
$583$	−0.888920	−0.0368153
$584$	31.5054	1.30370
$585$	−3.18760	−0.131791
$586$	23.7977	0.983072
$587$	4.83954	0.199749	0.0998745	−	0.995000i	$-0.468156\pi$
$587$	4.83954	0.199749	0.0998745	+	0.995000i	$0.468156\pi$
$588$	1.92283	0.0792960
$589$	64.4953	2.65748
$590$	−4.87446	−0.200678
$591$	48.4682	1.99372
$592$	−31.3826	−1.28982
$593$	1.70954	0.0702026	0.0351013	−	0.999384i	$-0.488825\pi$
$593$	1.70954	0.0702026	0.0351013	+	0.999384i	$0.488825\pi$
$594$	13.3737	0.548730
$595$	2.00763	0.0823049
$596$	0.381173	0.0156134
$597$	−6.06654	−0.248287
$598$	5.53147	0.226199
$599$	8.09099	0.330589	0.165294	−	0.986244i	$-0.447143\pi$
$599$	8.09099	0.330589	0.165294	+	0.986244i	$0.447143\pi$
$600$	−7.52582	−0.307240
$601$	−28.3274	−1.15550	−0.577750	−	0.816214i	$-0.696069\pi$
$601$	−28.3274	−1.15550	−0.577750	+	0.816214i	$0.696069\pi$
$602$	−3.24849	−0.132398
$603$	7.72532	0.314599
$604$	0.112300	0.00456942
$605$	15.8395	0.643966
$606$	−37.4276	−1.52039
$607$	37.5155	1.52271	0.761354	−	0.648336i	$-0.224535\pi$
$607$	37.5155	1.52271	0.761354	+	0.648336i	$0.224535\pi$
$608$	−5.21959	−0.211683
$609$	12.9249	0.523744
$610$	−14.9418	−0.604977
$611$	−0.744483	−0.0301185
$612$	−1.33242	−0.0538597
$613$	−31.3968	−1.26810	−0.634052	−	0.773290i	$-0.718610\pi$
$613$	−31.3968	−1.26810	−0.634052	+	0.773290i	$0.718610\pi$
$614$	3.81549	0.153981
$615$	6.73490	0.271577
$616$	−9.47559	−0.381782
$617$	−6.57291	−0.264616	−0.132308	−	0.991209i	$-0.542239\pi$
$617$	−6.57291	−0.264616	−0.132308	+	0.991209i	$0.542239\pi$
$618$	24.0648	0.968028
$619$	42.9193	1.72507	0.862537	−	0.505994i	$-0.168874\pi$
$619$	42.9193	1.72507	0.862537	+	0.505994i	$0.168874\pi$
$620$	0.880443	0.0353594
$621$	−8.83858	−0.354680
$622$	30.5883	1.22648
$623$	−0.716762	−0.0287164
$624$	8.35087	0.334302
$625$	1.00000	0.0400000
$626$	16.0635	0.642025
$627$	−110.516	−4.41358
$628$	−0.397717	−0.0158706
$629$	26.5685	1.05935
$630$	3.22505	0.128489
$631$	4.93114	0.196305	0.0981527	−	0.995171i	$-0.468707\pi$
$631$	4.93114	0.196305	0.0981527	+	0.995171i	$0.468707\pi$
$632$	−15.6681	−0.623244
$633$	−6.59993	−0.262324
$634$	−37.6526	−1.49537
$635$	0.896603	0.0355806
$636$	0.0499674	0.00198133
$637$	5.65092	0.223898
$638$	56.2927	2.22865
$639$	44.7586	1.77062
$640$	10.2684	0.405894
$641$	−15.3889	−0.607823	−0.303912	−	0.952700i	$-0.598293\pi$
$641$	−15.3889	−0.607823	−0.303912	+	0.952700i	$0.598293\pi$
$642$	−35.1133	−1.38581
$643$	−19.8546	−0.782991	−0.391495	−	0.920180i	$-0.628042\pi$
$643$	−19.8546	−0.782991	−0.391495	+	0.920180i	$0.628042\pi$
$644$	0.332936	0.0131195
$645$	−9.72858	−0.383062
$646$	−36.0047	−1.41659
$647$	−2.11286	−0.0830652	−0.0415326	−	0.999137i	$-0.513224\pi$
$647$	−2.11286	−0.0830652	−0.0415326	+	0.999137i	$0.513224\pi$
$648$	18.2877	0.718407
$649$	−18.3801	−0.721480
$650$	−1.17586	−0.0461211
$651$	−12.8130	−0.502180
$652$	−1.78730	−0.0699961
$653$	31.8431	1.24612	0.623059	−	0.782175i	$-0.285890\pi$
$653$	31.8431	1.24612	0.623059	+	0.782175i	$0.285890\pi$
$654$	54.4569	2.12943
$655$	18.2492	0.713055
$656$	−9.77258	−0.381555
$657$	−40.4331	−1.57744
$658$	0.753231	0.0293640
$659$	15.8190	0.616220	0.308110	−	0.951351i	$-0.400304\pi$
$659$	15.8190	0.616220	0.308110	+	0.951351i	$0.400304\pi$
$660$	−1.50868	−0.0587254
$661$	−13.5311	−0.526300	−0.263150	−	0.964755i	$-0.584761\pi$
$661$	−13.5311	−0.526300	−0.263150	+	0.964755i	$0.584761\pi$
$662$	−23.3193	−0.906331
$663$	−7.06983	−0.274569
$664$	49.0695	1.90427
$665$	−5.18446	−0.201045
$666$	42.6795	1.65380
$667$	−37.2034	−1.44052
$668$	−0.570802	−0.0220850
$669$	−63.7324	−2.46404
$670$	2.84977	0.110096
$671$	−56.3409	−2.17502
$672$	1.03695	0.0400013
$673$	−40.7493	−1.57077	−0.785384	−	0.619009i	$-0.787535\pi$
$673$	−40.7493	−1.57077	−0.785384	+	0.619009i	$0.787535\pi$
$674$	46.2109	1.77998
$675$	1.87888	0.0723180
$676$	1.37764	0.0529863
$677$	−7.96692	−0.306193	−0.153097	−	0.988211i	$-0.548925\pi$
$677$	−7.96692	−0.306193	−0.153097	+	0.988211i	$0.548925\pi$
$678$	37.1183	1.42552
$679$	−4.19928	−0.161154
$680$	−9.24505	−0.354531
$681$	19.8779	0.761724
$682$	−55.8053	−2.13689
$683$	30.4257	1.16421	0.582103	−	0.813115i	$-0.302230\pi$
$683$	30.4257	1.16421	0.582103	+	0.813115i	$0.302230\pi$
$684$	3.44080	0.131562
$685$	−7.97837	−0.304838
$686$	−11.7786	−0.449708
$687$	27.9254	1.06542
$688$	14.1165	0.538187
$689$	0.146847	0.00559443
$690$	−16.7603	−0.638054
$691$	−22.1449	−0.842434	−0.421217	−	0.906960i	$-0.638397\pi$
$691$	−22.1449	−0.842434	−0.421217	+	0.906960i	$0.638397\pi$
$692$	−2.20034	−0.0836442
$693$	12.1607	0.461946
$694$	−20.4205	−0.775150
$695$	−21.2734	−0.806946
$696$	−59.5186	−2.25605
$697$	8.27345	0.313379
$698$	−12.1181	−0.458677
$699$	−12.4074	−0.469292
$700$	−0.0707744	−0.00267502
$701$	2.69074	0.101628	0.0508139	−	0.998708i	$-0.483818\pi$
$701$	2.69074	0.101628	0.0508139	+	0.998708i	$0.483818\pi$
$702$	−2.20930	−0.0833848
$703$	−68.6098	−2.58767
$704$	43.5039	1.63962
$705$	2.25578	0.0849574
$706$	−6.91931	−0.260411
$707$	−6.62051	−0.248990
$708$	1.03317	0.0388288
$709$	−33.3983	−1.25430	−0.627149	−	0.778899i	$-0.715778\pi$
$709$	−33.3983	−1.25430	−0.627149	+	0.778899i	$0.715778\pi$
$710$	16.5108	0.619641
$711$	20.1079	0.754107
$712$	3.30065	0.123697
$713$	36.8813	1.38121
$714$	7.15290	0.267691
$715$	−4.43381	−0.165815
$716$	−0.0352846	−0.00131865
$717$	−40.5686	−1.51506
$718$	33.9976	1.26878
$719$	4.60117	0.171595	0.0857974	−	0.996313i	$-0.472656\pi$
$719$	4.60117	0.171595	0.0857974	+	0.996313i	$0.472656\pi$
$720$	−14.0147	−0.522297
$721$	4.25678	0.158531
$722$	66.8729	2.48875
$723$	−11.5375	−0.429085
$724$	−2.10725	−0.0783154
$725$	7.90859	0.293718
$726$	56.4337	2.09445
$727$	−21.5538	−0.799385	−0.399692	−	0.916649i	$-0.630883\pi$
$727$	−21.5538	−0.799385	−0.399692	+	0.916649i	$0.630883\pi$
$728$	1.56534	0.0580154
$729$	−38.0866	−1.41061
$730$	−14.9152	−0.552037
$731$	−11.9510	−0.442024
$732$	3.16700	0.117056
$733$	−14.8487	−0.548447	−0.274224	−	0.961666i	$-0.588421\pi$
$733$	−14.8487	−0.548447	−0.274224	+	0.961666i	$0.588421\pi$
$734$	0.641739	0.0236870
$735$	−17.1222	−0.631563
$736$	−2.98480	−0.110021
$737$	10.7456	0.395819
$738$	13.2904	0.489228
$739$	−41.7268	−1.53495	−0.767473	−	0.641081i	$-0.778486\pi$
$739$	−41.7268	−1.53495	−0.767473	+	0.641081i	$0.778486\pi$
$740$	−0.936611	−0.0344305
$741$	18.2569	0.670685
$742$	−0.148573	−0.00545428
$743$	−27.5240	−1.00976	−0.504880	−	0.863190i	$-0.668463\pi$
$743$	−27.5240	−1.00976	−0.504880	+	0.863190i	$0.668463\pi$
$744$	59.0032	2.16316
$745$	−3.39424	−0.124355
$746$	35.1608	1.28733
$747$	−62.9743	−2.30411
$748$	−1.85333	−0.0677646
$749$	−6.21113	−0.226950
$750$	3.56285	0.130097
$751$	23.2700	0.849135	0.424568	−	0.905396i	$-0.360426\pi$
$751$	23.2700	0.849135	0.424568	+	0.905396i	$0.360426\pi$
$752$	−3.27321	−0.119362
$753$	23.9815	0.873935
$754$	−9.29942	−0.338665
$755$	−1.00000	−0.0363937
$756$	−0.132977	−0.00483631
$757$	44.3136	1.61061	0.805303	−	0.592863i	$-0.202003\pi$
$757$	44.3136	1.61061	0.805303	+	0.592863i	$0.202003\pi$
$758$	20.5634	0.746896
$759$	−63.1979	−2.29394
$760$	23.8742	0.866007
$761$	−10.8530	−0.393422	−0.196711	−	0.980462i	$-0.563026\pi$
$761$	−10.8530	−0.393422	−0.196711	+	0.980462i	$0.563026\pi$
$762$	3.19447	0.115723
$763$	9.63279	0.348730
$764$	−2.26670	−0.0820063
$765$	11.8648	0.428973
$766$	15.9684	0.576962
$767$	3.03634	0.109636
$768$	−6.96667	−0.251388
$769$	−48.9883	−1.76656	−0.883282	−	0.468843i	$-0.844671\pi$
$769$	−48.9883	−1.76656	−0.883282	+	0.468843i	$0.844671\pi$
$770$	4.48591	0.161661
$771$	23.4062	0.842953
$772$	−1.99535	−0.0718142
$773$	−14.1530	−0.509046	−0.254523	−	0.967067i	$-0.581919\pi$
$773$	−14.1530	−0.509046	−0.254523	+	0.967067i	$0.581919\pi$
$774$	−19.1981	−0.690061
$775$	−7.84010	−0.281625
$776$	19.3375	0.694176
$777$	13.6304	0.488988
$778$	−38.2319	−1.37068
$779$	−21.3651	−0.765485
$780$	0.249230	0.00892388
$781$	62.2572	2.22774
$782$	−20.5891	−0.736265
$783$	14.8593	0.531027
$784$	24.8450	0.887321
$785$	3.54156	0.126404
$786$	65.0192	2.31916
$787$	12.1843	0.434325	0.217162	−	0.976135i	$-0.430320\pi$
$787$	12.1843	0.434325	0.217162	+	0.976135i	$0.430320\pi$
$788$	−2.09896	−0.0747725
$789$	7.38431	0.262888
$790$	7.41755	0.263905
$791$	6.56579	0.233453
$792$	−55.9993	−1.98985
$793$	9.30738	0.330515
$794$	40.5983	1.44078
$795$	−0.444946	−0.0157806
$796$	0.262718	0.00931178
$797$	−20.4340	−0.723809	−0.361904	−	0.932215i	$-0.617873\pi$
$797$	−20.4340	−0.723809	−0.361904	+	0.932215i	$0.617873\pi$
$798$	−18.4715	−0.653883
$799$	2.77110	0.0980343
$800$	0.634499	0.0224329
$801$	−4.23595	−0.149670
$802$	−30.0957	−1.06271
$803$	−56.2406	−1.98469
$804$	−0.604023	−0.0213023
$805$	−2.96470	−0.104492
$806$	9.21888	0.324721
$807$	−6.98421	−0.245856
$808$	30.4871	1.07253
$809$	−47.0787	−1.65520	−0.827600	−	0.561319i	$-0.810294\pi$
$809$	−47.0787	−1.65520	−0.827600	+	0.561319i	$0.810294\pi$
$810$	−8.65770	−0.304201
$811$	−35.7750	−1.25623	−0.628115	−	0.778121i	$-0.716173\pi$
$811$	−35.7750	−1.25623	−0.628115	+	0.778121i	$0.716173\pi$
$812$	−0.559726	−0.0196425
$813$	−15.0194	−0.526754
$814$	59.3654	2.08076
$815$	15.9154	0.557493
$816$	−31.0834	−1.08814
$817$	30.8620	1.07972
$818$	39.0083	1.36389
$819$	−2.00891	−0.0701970
$820$	−0.291661	−0.0101853
$821$	−44.7860	−1.56304	−0.781521	−	0.623879i	$-0.785556\pi$
$821$	−44.7860	−1.56304	−0.781521	+	0.623879i	$0.785556\pi$
$822$	−28.4258	−0.991462
$823$	7.20433	0.251127	0.125564	−	0.992086i	$-0.459926\pi$
$823$	7.20433	0.251127	0.125564	+	0.992086i	$0.459926\pi$
$824$	−19.6023	−0.682877
$825$	13.4344	0.467726
$826$	−3.07202	−0.106889
$827$	−31.5555	−1.09729	−0.548646	−	0.836055i	$-0.684856\pi$
$827$	−31.5555	−1.09729	−0.548646	+	0.836055i	$0.684856\pi$
$828$	1.96760	0.0683789
$829$	33.1698	1.15203	0.576017	−	0.817437i	$-0.304606\pi$
$829$	33.1698	1.15203	0.576017	+	0.817437i	$0.304606\pi$
$830$	−23.2304	−0.806338
$831$	−69.3274	−2.40494
$832$	−7.18674	−0.249155
$833$	−21.0337	−0.728775
$834$	−75.7940	−2.62453
$835$	5.08284	0.175899
$836$	4.78600	0.165527
$837$	−14.7306	−0.509164
$838$	−40.6185	−1.40314
$839$	−23.1918	−0.800669	−0.400335	−	0.916369i	$-0.631106\pi$
$839$	−23.1918	−0.800669	−0.400335	+	0.916369i	$0.631106\pi$
$840$	−4.74297	−0.163648
$841$	33.5458	1.15675
$842$	12.7279	0.438632
$843$	33.6468	1.15886
$844$	0.285816	0.00983820
$845$	−12.2675	−0.422016
$846$	4.45148	0.153045
$847$	9.98246	0.343002
$848$	0.645632	0.0221711
$849$	66.7448	2.29068
$850$	4.37677	0.150122
$851$	−39.2341	−1.34493
$852$	−3.49956	−0.119893
$853$	20.0593	0.686816	0.343408	−	0.939186i	$-0.388419\pi$
$853$	20.0593	0.686816	0.343408	+	0.939186i	$0.388419\pi$
$854$	−9.41674	−0.322234
$855$	−30.6394	−1.04784
$856$	28.6020	0.977595
$857$	9.87547	0.337340	0.168670	−	0.985673i	$-0.446053\pi$
$857$	9.87547	0.337340	0.168670	+	0.985673i	$0.446053\pi$
$858$	−15.7970	−0.539301
$859$	16.2800	0.555467	0.277734	−	0.960658i	$-0.410417\pi$
$859$	16.2800	0.555467	0.277734	+	0.960658i	$0.410417\pi$
$860$	0.421305	0.0143664
$861$	4.24451	0.144653
$862$	−37.0251	−1.26108
$863$	42.0286	1.43067	0.715336	−	0.698781i	$-0.246274\pi$
$863$	42.0286	1.43067	0.715336	+	0.698781i	$0.246274\pi$
$864$	1.19215	0.0405576
$865$	19.5934	0.666195
$866$	37.4394	1.27224
$867$	−17.7688	−0.603460
$868$	0.554879	0.0188338
$869$	27.9693	0.948793
$870$	28.1771	0.955295
$871$	−1.77514	−0.0601484
$872$	−44.3585	−1.50217
$873$	−24.8171	−0.839933
$874$	53.1688	1.79846
$875$	0.630227	0.0213056
$876$	3.16136	0.106812
$877$	20.1657	0.680946	0.340473	−	0.940254i	$-0.389413\pi$
$877$	20.1657	0.680946	0.340473	+	0.940254i	$0.389413\pi$
$878$	−27.5425	−0.929513
$879$	44.9158	1.51497
$880$	−19.4938	−0.657136
$881$	−14.2022	−0.478483	−0.239242	−	0.970960i	$-0.576899\pi$
$881$	−14.2022	−0.478483	−0.239242	+	0.970960i	$0.576899\pi$
$882$	−33.7885	−1.13772
$883$	46.4890	1.56448	0.782240	−	0.622977i	$-0.214077\pi$
$883$	46.4890	1.56448	0.782240	+	0.622977i	$0.214077\pi$
$884$	0.306166	0.0102975
$885$	−9.20008	−0.309257
$886$	21.2717	0.714638
$887$	−15.2221	−0.511108	−0.255554	−	0.966795i	$-0.582258\pi$
$887$	−15.2221	−0.511108	−0.255554	+	0.966795i	$0.582258\pi$
$888$	−62.7673	−2.10633
$889$	0.565064	0.0189516
$890$	−1.56259	−0.0523780
$891$	−32.6455	−1.09367
$892$	2.75999	0.0924114
$893$	−7.15600	−0.239467
$894$	−12.0932	−0.404456
$895$	0.314200	0.0105025
$896$	6.47143	0.216195
$897$	10.4401	0.348586
$898$	8.86099	0.295695
$899$	−62.0041	−2.06795
$900$	−0.418266	−0.0139422
$901$	−0.546591	−0.0182096
$902$	18.4864	0.615530
$903$	−6.13121	−0.204034
$904$	−30.2352	−1.00561
$905$	18.7645	0.623754
$906$	−3.56285	−0.118368
$907$	−6.28561	−0.208710	−0.104355	−	0.994540i	$-0.533278\pi$
$907$	−6.28561	−0.208710	−0.104355	+	0.994540i	$0.533278\pi$
$908$	−0.860833	−0.0285678
$909$	−39.1262	−1.29773
$910$	−0.741061	−0.0245659
$911$	−16.7995	−0.556592	−0.278296	−	0.960495i	$-0.589770\pi$
$911$	−16.7995	−0.556592	−0.278296	+	0.960495i	$0.589770\pi$
$912$	80.2689	2.65797
$913$	−87.5944	−2.89895
$914$	39.3219	1.30065
$915$	−28.2013	−0.932305
$916$	−1.20934	−0.0399576
$917$	11.5011	0.379801
$918$	8.22341	0.271413
$919$	−3.57994	−0.118091	−0.0590457	−	0.998255i	$-0.518806\pi$
$919$	−3.57994	−0.118091	−0.0590457	+	0.998255i	$0.518806\pi$
$920$	13.6523	0.450103
$921$	7.20138	0.237294
$922$	16.9794	0.559188
$923$	−10.2847	−0.338526
$924$	−0.950813	−0.0312794
$925$	8.34026	0.274226
$926$	19.0089	0.624671
$927$	25.1569	0.826262
$928$	5.01799	0.164724
$929$	−3.46043	−0.113533	−0.0567665	−	0.998387i	$-0.518079\pi$
$929$	−3.46043	−0.113533	−0.0567665	+	0.998387i	$0.518079\pi$
$930$	−27.9331	−0.915964
$931$	54.3169	1.78017
$932$	0.537315	0.0176003
$933$	57.7325	1.89008
$934$	−38.5888	−1.26266
$935$	16.5034	0.539720
$936$	9.25094	0.302376
$937$	47.1178	1.53927	0.769635	−	0.638484i	$-0.220438\pi$
$937$	47.1178	1.53927	0.769635	+	0.638484i	$0.220438\pi$
$938$	1.79600	0.0586415
$939$	30.3182	0.989398
$940$	−0.0976886	−0.00318625
$941$	−32.0773	−1.04569	−0.522846	−	0.852427i	$-0.675130\pi$
$941$	−32.0773	−1.04569	−0.522846	+	0.852427i	$0.675130\pi$
$942$	12.6181	0.411119
$943$	−12.2175	−0.397858
$944$	13.3496	0.434494
$945$	1.18412	0.0385194
$946$	−26.7037	−0.868212
$947$	−22.9541	−0.745908	−0.372954	−	0.927850i	$-0.621655\pi$
$947$	−22.9541	−0.745908	−0.372954	+	0.927850i	$0.621655\pi$
$948$	−1.57219	−0.0510624
$949$	9.29081	0.301592
$950$	−11.3025	−0.366700
$951$	−71.0657	−2.30446
$952$	−5.82648	−0.188837
$953$	42.5681	1.37892	0.689458	−	0.724325i	$-0.257849\pi$
$953$	42.5681	1.37892	0.689458	+	0.724325i	$0.257849\pi$
$954$	−0.878042	−0.0284277
$955$	20.1843	0.653150
$956$	1.75686	0.0568211
$957$	106.247	3.43448
$958$	−21.8871	−0.707139
$959$	−5.02818	−0.162368
$960$	21.7758	0.702809
$961$	30.4672	0.982813
$962$	−9.80700	−0.316191
$963$	−36.7069	−1.18286
$964$	0.499643	0.0160924
$965$	17.7680	0.571974
$966$	−10.5628	−0.339853
$967$	−21.9537	−0.705983	−0.352991	−	0.935627i	$-0.614836\pi$
$967$	−21.9537	−0.705983	−0.352991	+	0.935627i	$0.614836\pi$
$968$	−45.9688	−1.47749
$969$	−67.9555	−2.18305
$970$	−9.15471	−0.293940
$971$	17.3202	0.555830	0.277915	−	0.960606i	$-0.410357\pi$
$971$	17.3202	0.555830	0.277915	+	0.960606i	$0.410357\pi$
$972$	2.46804	0.0791624
$973$	−13.4071	−0.429811
$974$	18.3934	0.589363
$975$	−2.21933	−0.0710754
$976$	40.9211	1.30985
$977$	−21.9417	−0.701976	−0.350988	−	0.936380i	$-0.614154\pi$
$977$	−21.9417	−0.701976	−0.350988	+	0.936380i	$0.614154\pi$
$978$	56.7043	1.81320
$979$	−5.89203	−0.188310
$980$	0.741495	0.0236862
$981$	56.9283	1.81758
$982$	−30.3396	−0.968177
$983$	56.1811	1.79190	0.895949	−	0.444156i	$-0.146496\pi$
$983$	56.1811	1.79190	0.895949	+	0.444156i	$0.146496\pi$
$984$	−19.5458	−0.623097
$985$	18.6907	0.595535
$986$	34.6141	1.10234
$987$	1.42165	0.0452517
$988$	−0.790634	−0.0251534
$989$	17.6483	0.561182
$990$	26.5111	0.842577
$991$	−39.6659	−1.26003	−0.630014	−	0.776584i	$-0.716951\pi$
$991$	−39.6659	−1.26003	−0.630014	+	0.776584i	$0.716951\pi$
$992$	−4.97453	−0.157942
$993$	−44.0130	−1.39671
$994$	10.4056	0.330045
$995$	−2.33943	−0.0741649
$996$	4.92380	0.156017
$997$	−14.9585	−0.473740	−0.236870	−	0.971541i	$-0.576122\pi$
$997$	−14.9585	−0.473740	−0.236870	+	0.971541i	$0.576122\pi$
$998$	23.3921	0.740463
$999$	15.6703	0.495787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	755.2.a.j.1.12	✓	15
3.2	odd	2		6795.2.a.bh.1.4		15
5.4	even	2		3775.2.a.q.1.4		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.a.j.1.12	✓	15	1.1	even	1	trivial
3775.2.a.q.1.4		15	5.4	even	2
6795.2.a.bh.1.4		15	3.2	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:	\( N \)	\(=\)	\( 755 = 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	755.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37394 q^{2} +2.59317 q^{3} -0.112300 q^{4} +1.00000 q^{5} +3.56285 q^{6} +0.630227 q^{7} -2.90216 q^{8} +3.72455 q^{9} +1.37394 q^{10} +5.18068 q^{11} -0.291213 q^{12} -0.855835 q^{13} +0.865892 q^{14} +2.59317 q^{15} -3.76279 q^{16} +3.18557 q^{17} +5.11729 q^{18} -8.22633 q^{19} -0.112300 q^{20} +1.63429 q^{21} +7.11793 q^{22} -4.70418 q^{23} -7.52582 q^{24} +1.00000 q^{25} -1.17586 q^{26} +1.87888 q^{27} -0.0707744 q^{28} +7.90859 q^{29} +3.56285 q^{30} -7.84010 q^{31} +0.634499 q^{32} +13.4344 q^{33} +4.37677 q^{34} +0.630227 q^{35} -0.418266 q^{36} +8.34026 q^{37} -11.3025 q^{38} -2.21933 q^{39} -2.90216 q^{40} +2.59716 q^{41} +2.24541 q^{42} -3.75161 q^{43} -0.581790 q^{44} +3.72455 q^{45} -6.46324 q^{46} +0.869890 q^{47} -9.75756 q^{48} -6.60281 q^{49} +1.37394 q^{50} +8.26073 q^{51} +0.0961102 q^{52} -0.171583 q^{53} +2.58146 q^{54} +5.18068 q^{55} -1.82902 q^{56} -21.3323 q^{57} +10.8659 q^{58} -3.54781 q^{59} -0.291213 q^{60} -10.8752 q^{61} -10.7718 q^{62} +2.34731 q^{63} +8.39734 q^{64} -0.855835 q^{65} +18.4580 q^{66} +2.07416 q^{67} -0.357739 q^{68} -12.1988 q^{69} +0.865892 q^{70} +12.0172 q^{71} -10.8093 q^{72} -10.8558 q^{73} +11.4590 q^{74} +2.59317 q^{75} +0.923816 q^{76} +3.26501 q^{77} -3.04922 q^{78} +5.39876 q^{79} -3.76279 q^{80} -6.30139 q^{81} +3.56834 q^{82} -16.9079 q^{83} -0.183530 q^{84} +3.18557 q^{85} -5.15447 q^{86} +20.5083 q^{87} -15.0352 q^{88} -1.13731 q^{89} +5.11729 q^{90} -0.539371 q^{91} +0.528279 q^{92} -20.3307 q^{93} +1.19517 q^{94} -8.22633 q^{95} +1.64536 q^{96} -6.66313 q^{97} -9.07184 q^{98} +19.2957 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 2 q^{2} + 5 q^{3} + 18 q^{4} + 15 q^{5} - 4 q^{6} + 11 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 11 q^{13} - 9 q^{14} + 5 q^{15} + 40 q^{16} + 25 q^{17} - 15 q^{18} - 3 q^{19}+ \cdots - 11 q^{99}+O(q^{100}) \) 15 * q - 2 * q^2 + 5 * q^3 + 18 * q^4 + 15 * q^5 - 4 * q^6 + 11 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 11 * q^13 - 9 * q^14 + 5 * q^15 + 40 * q^16 + 25 * q^17 - 15 * q^18 - 3 * q^19 + 18 * q^20 - 11 * q^21 + 13 * q^22 + 8 * q^23 - 21 * q^24 + 15 * q^25 - 45 * q^26 + 20 * q^27 + 29 * q^28 + 24 * q^29 - 4 * q^30 - 11 * q^31 + 27 * q^32 + 26 * q^33 - 11 * q^34 + 11 * q^35 - 22 * q^36 + 35 * q^37 + 2 * q^38 + 11 * q^39 + 12 * q^40 - 20 * q^41 - 17 * q^42 - 13 * q^43 - 32 * q^44 + 10 * q^45 - 5 * q^46 + 6 * q^47 - 11 * q^48 + 40 * q^49 - 2 * q^50 + 23 * q^51 + 27 * q^52 + 38 * q^53 - 30 * q^54 + 2 * q^55 - 36 * q^56 + 4 * q^57 + 6 * q^58 - 15 * q^59 + 4 * q^60 + 33 * q^61 - 13 * q^62 + 2 * q^63 + 74 * q^64 + 11 * q^65 + 8 * q^66 + 24 * q^67 + 8 * q^68 + 13 * q^69 - 9 * q^70 + 19 * q^71 + 56 * q^72 + 4 * q^73 - 28 * q^74 + 5 * q^75 + 19 * q^76 + 14 * q^77 - 15 * q^78 - 4 * q^79 + 40 * q^80 - 13 * q^81 + 33 * q^82 + 19 * q^83 - 71 * q^84 + 25 * q^85 - 59 * q^86 + 49 * q^87 - 48 * q^88 - 35 * q^89 - 15 * q^90 - q^91 + 18 * q^92 + 2 * q^93 - 70 * q^94 - 3 * q^95 - 35 * q^96 + 55 * q^97 - 64 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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