LMFDB - Embedded newform 2667.2.a.o.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2667,2,Mod(1,2667)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2667, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("2667.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2667 = 3 \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2667.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,5,-16,19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$21.2961022191$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.823041$ of defining polynomial
Character	$\chi$	$=$	2667.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.823041 q^{2} -1.00000 q^{3} -1.32260 q^{4} -3.74037 q^{5} +0.823041 q^{6} -1.00000 q^{7} +2.73464 q^{8} +1.00000 q^{9} +3.07848 q^{10} +5.51001 q^{11} +1.32260 q^{12} +3.05214 q^{13} +0.823041 q^{14} +3.74037 q^{15} +0.394484 q^{16} -2.99620 q^{17} -0.823041 q^{18} -8.18146 q^{19} +4.94702 q^{20} +1.00000 q^{21} -4.53496 q^{22} -8.49401 q^{23} -2.73464 q^{24} +8.99033 q^{25} -2.51203 q^{26} -1.00000 q^{27} +1.32260 q^{28} +2.18992 q^{29} -3.07848 q^{30} -10.0713 q^{31} -5.79396 q^{32} -5.51001 q^{33} +2.46600 q^{34} +3.74037 q^{35} -1.32260 q^{36} +8.15061 q^{37} +6.73368 q^{38} -3.05214 q^{39} -10.2286 q^{40} +5.46666 q^{41} -0.823041 q^{42} -4.04983 q^{43} -7.28755 q^{44} -3.74037 q^{45} +6.99092 q^{46} +3.81563 q^{47} -0.394484 q^{48} +1.00000 q^{49} -7.39942 q^{50} +2.99620 q^{51} -4.03676 q^{52} -5.00929 q^{53} +0.823041 q^{54} -20.6094 q^{55} -2.73464 q^{56} +8.18146 q^{57} -1.80240 q^{58} -13.3668 q^{59} -4.94702 q^{60} -8.19365 q^{61} +8.28911 q^{62} -1.00000 q^{63} +3.97970 q^{64} -11.4161 q^{65} +4.53496 q^{66} -5.69594 q^{67} +3.96278 q^{68} +8.49401 q^{69} -3.07848 q^{70} +7.02554 q^{71} +2.73464 q^{72} -2.25827 q^{73} -6.70829 q^{74} -8.99033 q^{75} +10.8208 q^{76} -5.51001 q^{77} +2.51203 q^{78} -1.56489 q^{79} -1.47551 q^{80} +1.00000 q^{81} -4.49929 q^{82} +5.39693 q^{83} -1.32260 q^{84} +11.2069 q^{85} +3.33318 q^{86} -2.18992 q^{87} +15.0679 q^{88} +2.98116 q^{89} +3.07848 q^{90} -3.05214 q^{91} +11.2342 q^{92} +10.0713 q^{93} -3.14042 q^{94} +30.6017 q^{95} +5.79396 q^{96} -7.99742 q^{97} -0.823041 q^{98} +5.51001 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 5 q^{2} - 16 q^{3} + 19 q^{4} - q^{5} - 5 q^{6} - 16 q^{7} + 6 q^{8} + 16 q^{9} - 12 q^{10} + 11 q^{11} - 19 q^{12} + 18 q^{13} - 5 q^{14} + q^{15} + 25 q^{16} - 5 q^{17} + 5 q^{18} - 11 q^{19}+ \cdots + 11 q^{99}+O(q^{100}) $ 16 * q + 5 * q^2 - 16 * q^3 + 19 * q^4 - q^5 - 5 * q^6 - 16 * q^7 + 6 * q^8 + 16 * q^9 - 12 * q^10 + 11 * q^11 - 19 * q^12 + 18 * q^13 - 5 * q^14 + q^15 + 25 * q^16 - 5 * q^17 + 5 * q^18 - 11 * q^19 - q^20 + 16 * q^21 + q^22 + 13 * q^23 - 6 * q^24 + 33 * q^25 + 8 * q^26 - 16 * q^27 - 19 * q^28 + 24 * q^29 + 12 * q^30 - 42 * q^31 + 42 * q^32 - 11 * q^33 + 9 * q^34 + q^35 + 19 * q^36 + 40 * q^37 + 38 * q^38 - 18 * q^39 - 61 * q^40 + 9 * q^41 + 5 * q^42 + 7 * q^43 + 3 * q^44 - q^45 + 24 * q^46 + 31 * q^47 - 25 * q^48 + 16 * q^49 + 6 * q^50 + 5 * q^51 + 52 * q^52 + 66 * q^53 - 5 * q^54 - 36 * q^55 - 6 * q^56 + 11 * q^57 + 19 * q^58 - 7 * q^59 + q^60 + 6 * q^61 + 52 * q^62 - 16 * q^63 + 10 * q^64 + 51 * q^65 - q^66 + 16 * q^67 + 14 * q^68 - 13 * q^69 + 12 * q^70 + 46 * q^71 + 6 * q^72 + 39 * q^73 + 72 * q^74 - 33 * q^75 + 24 * q^76 - 11 * q^77 - 8 * q^78 + 4 * q^79 - 2 * q^80 + 16 * q^81 - 18 * q^82 + 15 * q^83 + 19 * q^84 - 4 * q^85 + 14 * q^86 - 24 * q^87 + 58 * q^88 - q^89 - 12 * q^90 - 18 * q^91 + 26 * q^92 + 42 * q^93 + 5 * q^94 + 44 * q^95 - 42 * q^96 + 41 * q^97 + 5 * 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 $

$2$	−0.823041	−0.581978	−0.290989	−	0.956726i	$-0.593984\pi$
$2$	−0.823041	−0.581978	−0.290989	+	0.956726i	$0.593984\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.32260	−0.661301
$5$	−3.74037	−1.67274	−0.836371	−	0.548164i	$-0.815327\pi$
$5$	−3.74037	−1.67274	−0.836371	+	0.548164i	$0.815327\pi$
$6$	0.823041	0.336005
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.73464	0.966841
$9$	1.00000	0.333333
$10$	3.07848	0.973500
$11$	5.51001	1.66133	0.830665	−	0.556773i	$-0.187961\pi$
$11$	5.51001	1.66133	0.830665	+	0.556773i	$0.187961\pi$
$12$	1.32260	0.381803
$13$	3.05214	0.846510	0.423255	−	0.906011i	$-0.360887\pi$
$13$	3.05214	0.846510	0.423255	+	0.906011i	$0.360887\pi$
$14$	0.823041	0.219967
$15$	3.74037	0.965758
$16$	0.394484	0.0986209
$17$	−2.99620	−0.726685	−0.363342	−	0.931656i	$-0.618364\pi$
$17$	−2.99620	−0.726685	−0.363342	+	0.931656i	$0.618364\pi$
$18$	−0.823041	−0.193993
$19$	−8.18146	−1.87696	−0.938478	−	0.345338i	$-0.887764\pi$
$19$	−8.18146	−1.87696	−0.938478	+	0.345338i	$0.887764\pi$
$20$	4.94702	1.10619
$21$	1.00000	0.218218
$22$	−4.53496	−0.966858
$23$	−8.49401	−1.77112	−0.885562	−	0.464521i	$-0.846226\pi$
$23$	−8.49401	−1.77112	−0.885562	+	0.464521i	$0.846226\pi$
$24$	−2.73464	−0.558206
$25$	8.99033	1.79807
$26$	−2.51203	−0.492650
$27$	−1.00000	−0.192450
$28$	1.32260	0.249948
$29$	2.18992	0.406658	0.203329	−	0.979110i	$-0.434824\pi$
$29$	2.18992	0.406658	0.203329	+	0.979110i	$0.434824\pi$
$30$	−3.07848	−0.562050
$31$	−10.0713	−1.80886	−0.904431	−	0.426621i	$-0.859704\pi$
$31$	−10.0713	−1.80886	−0.904431	+	0.426621i	$0.859704\pi$
$32$	−5.79396	−1.02424
$33$	−5.51001	−0.959169
$34$	2.46600	0.422915
$35$	3.74037	0.632237
$36$	−1.32260	−0.220434
$37$	8.15061	1.33995	0.669976	−	0.742383i	$-0.266304\pi$
$37$	8.15061	1.33995	0.669976	+	0.742383i	$0.266304\pi$
$38$	6.73368	1.09235
$39$	−3.05214	−0.488733
$40$	−10.2286	−1.61728
$41$	5.46666	0.853749	0.426874	−	0.904311i	$-0.359615\pi$
$41$	5.46666	0.853749	0.426874	+	0.904311i	$0.359615\pi$
$42$	−0.823041	−0.126998
$43$	−4.04983	−0.617594	−0.308797	−	0.951128i	$-0.599926\pi$
$43$	−4.04983	−0.617594	−0.308797	+	0.951128i	$0.599926\pi$
$44$	−7.28755	−1.09864
$45$	−3.74037	−0.557581
$46$	6.99092	1.03076
$47$	3.81563	0.556567	0.278283	−	0.960499i	$-0.410235\pi$
$47$	3.81563	0.556567	0.278283	+	0.960499i	$0.410235\pi$
$48$	−0.394484	−0.0569388
$49$	1.00000	0.142857
$50$	−7.39942	−1.04644
$51$	2.99620	0.419552
$52$	−4.03676	−0.559798
$53$	−5.00929	−0.688079	−0.344039	−	0.938955i	$-0.611795\pi$
$53$	−5.00929	−0.688079	−0.344039	+	0.938955i	$0.611795\pi$
$54$	0.823041	0.112002
$55$	−20.6094	−2.77898
$56$	−2.73464	−0.365432
$57$	8.18146	1.08366
$58$	−1.80240	−0.236666
$59$	−13.3668	−1.74021	−0.870103	−	0.492871i	$-0.835948\pi$
$59$	−13.3668	−1.74021	−0.870103	+	0.492871i	$0.835948\pi$
$60$	−4.94702	−0.638657
$61$	−8.19365	−1.04909	−0.524545	−	0.851383i	$-0.675764\pi$
$61$	−8.19365	−1.04909	−0.524545	+	0.851383i	$0.675764\pi$
$62$	8.28911	1.05272
$63$	−1.00000	−0.125988
$64$	3.97970	0.497462
$65$	−11.4161	−1.41599
$66$	4.53496	0.558216
$67$	−5.69594	−0.695869	−0.347935	−	0.937519i	$-0.613117\pi$
$67$	−5.69594	−0.695869	−0.347935	+	0.937519i	$0.613117\pi$
$68$	3.96278	0.480558
$69$	8.49401	1.02256
$70$	−3.07848	−0.367948
$71$	7.02554	0.833779	0.416889	−	0.908957i	$-0.363120\pi$
$71$	7.02554	0.833779	0.416889	+	0.908957i	$0.363120\pi$
$72$	2.73464	0.322280
$73$	−2.25827	−0.264311	−0.132155	−	0.991229i	$-0.542190\pi$
$73$	−2.25827	−0.264311	−0.132155	+	0.991229i	$0.542190\pi$
$74$	−6.70829	−0.779823
$75$	−8.99033	−1.03811
$76$	10.8208	1.24123
$77$	−5.51001	−0.627924
$78$	2.51203	0.284432
$79$	−1.56489	−0.176064	−0.0880318	−	0.996118i	$-0.528058\pi$
$79$	−1.56489	−0.176064	−0.0880318	+	0.996118i	$0.528058\pi$
$80$	−1.47551	−0.164967
$81$	1.00000	0.111111
$82$	−4.49929	−0.496863
$83$	5.39693	0.592390	0.296195	−	0.955127i	$-0.404282\pi$
$83$	5.39693	0.592390	0.296195	+	0.955127i	$0.404282\pi$
$84$	−1.32260	−0.144308
$85$	11.2069	1.21556
$86$	3.33318	0.359426
$87$	−2.18992	−0.234784
$88$	15.0679	1.60624
$89$	2.98116	0.316002	0.158001	−	0.987439i	$-0.449495\pi$
$89$	2.98116	0.316002	0.158001	+	0.987439i	$0.449495\pi$
$90$	3.07848	0.324500
$91$	−3.05214	−0.319951
$92$	11.2342	1.17125
$93$	10.0713	1.04435
$94$	−3.14042	−0.323910
$95$	30.6017	3.13966
$96$	5.79396	0.591343
$97$	−7.99742	−0.812015	−0.406007	−	0.913870i	$-0.633079\pi$
$97$	−7.99742	−0.812015	−0.406007	+	0.913870i	$0.633079\pi$
$98$	−0.823041	−0.0831397
$99$	5.51001	0.553777
$100$	−11.8906	−1.18906
$101$	14.8974	1.48235	0.741173	−	0.671314i	$-0.234270\pi$
$101$	14.8974	1.48235	0.741173	+	0.671314i	$0.234270\pi$
$102$	−2.46600	−0.244170
$103$	5.23013	0.515340	0.257670	−	0.966233i	$-0.417045\pi$
$103$	5.23013	0.515340	0.257670	+	0.966233i	$0.417045\pi$
$104$	8.34649	0.818441
$105$	−3.74037	−0.365022
$106$	4.12285	0.400447
$107$	−2.88042	−0.278460	−0.139230	−	0.990260i	$-0.544463\pi$
$107$	−2.88042	−0.278460	−0.139230	+	0.990260i	$0.544463\pi$
$108$	1.32260	0.127268
$109$	−10.4821	−1.00400	−0.502000	−	0.864867i	$-0.667402\pi$
$109$	−10.4821	−1.00400	−0.502000	+	0.864867i	$0.667402\pi$
$110$	16.9624	1.61730
$111$	−8.15061	−0.773622
$112$	−0.394484	−0.0372752
$113$	7.63990	0.718701	0.359351	−	0.933203i	$-0.382998\pi$
$113$	7.63990	0.718701	0.359351	+	0.933203i	$0.382998\pi$
$114$	−6.73368	−0.630667
$115$	31.7707	2.96263
$116$	−2.89640	−0.268924
$117$	3.05214	0.282170
$118$	11.0014	1.01276
$119$	2.99620	0.274661
$120$	10.2286	0.933735
$121$	19.3602	1.76002
$122$	6.74371	0.610547
$123$	−5.46666	−0.492912
$124$	13.3203	1.19620
$125$	−14.9253	−1.33496
$126$	0.823041	0.0733224
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	8.31245	0.734724
$129$	4.04983	0.356568
$130$	9.39592	0.824077
$131$	−9.80510	−0.856675	−0.428338	−	0.903619i	$-0.640901\pi$
$131$	−9.80510	−0.856675	−0.428338	+	0.903619i	$0.640901\pi$
$132$	7.28755	0.634300
$133$	8.18146	0.709423
$134$	4.68799	0.404981
$135$	3.74037	0.321919
$136$	−8.19352	−0.702589
$137$	17.6377	1.50689	0.753445	−	0.657511i	$-0.228391\pi$
$137$	17.6377	1.50689	0.753445	+	0.657511i	$0.228391\pi$
$138$	−6.99092	−0.595107
$139$	−15.2050	−1.28967	−0.644835	−	0.764322i	$-0.723074\pi$
$139$	−15.2050	−1.28967	−0.644835	+	0.764322i	$0.723074\pi$
$140$	−4.94702	−0.418099
$141$	−3.81563	−0.321334
$142$	−5.78231	−0.485241
$143$	16.8173	1.40633
$144$	0.394484	0.0328736
$145$	−8.19111	−0.680234
$146$	1.85865	0.153823
$147$	−1.00000	−0.0824786
$148$	−10.7800	−0.886112
$149$	−4.13281	−0.338573	−0.169286	−	0.985567i	$-0.554146\pi$
$149$	−4.13281	−0.338573	−0.169286	+	0.985567i	$0.554146\pi$
$150$	7.39942	0.604160
$151$	−7.72102	−0.628328	−0.314164	−	0.949369i	$-0.601724\pi$
$151$	−7.72102	−0.628328	−0.314164	+	0.949369i	$0.601724\pi$
$152$	−22.3734	−1.81472
$153$	−2.99620	−0.242228
$154$	4.53496	0.365438
$155$	37.6704	3.02576
$156$	4.03676	0.323200
$157$	8.14560	0.650090	0.325045	−	0.945699i	$-0.394621\pi$
$157$	8.14560	0.650090	0.325045	+	0.945699i	$0.394621\pi$
$158$	1.28797	0.102465
$159$	5.00929	0.397263
$160$	21.6715	1.71328
$161$	8.49401	0.669422
$162$	−0.823041	−0.0646642
$163$	−6.83265	−0.535175	−0.267587	−	0.963534i	$-0.586226\pi$
$163$	−6.83265	−0.535175	−0.267587	+	0.963534i	$0.586226\pi$
$164$	−7.23022	−0.564585
$165$	20.6094	1.60444
$166$	−4.44190	−0.344758
$167$	18.1213	1.40227	0.701136	−	0.713028i	$-0.252677\pi$
$167$	18.1213	1.40227	0.701136	+	0.713028i	$0.252677\pi$
$168$	2.73464	0.210982
$169$	−3.68447	−0.283421
$170$	−9.22372	−0.707427
$171$	−8.18146	−0.625652
$172$	5.35632	0.408416
$173$	23.7119	1.80279	0.901393	−	0.433002i	$-0.142546\pi$
$173$	23.7119	1.80279	0.901393	+	0.433002i	$0.142546\pi$
$174$	1.80240	0.136639
$175$	−8.99033	−0.679605
$176$	2.17361	0.163842
$177$	13.3668	1.00471
$178$	−2.45362	−0.183906
$179$	9.28257	0.693812	0.346906	−	0.937900i	$-0.387232\pi$
$179$	9.28257	0.693812	0.346906	+	0.937900i	$0.387232\pi$
$180$	4.94702	0.368729
$181$	−11.9070	−0.885037	−0.442518	−	0.896759i	$-0.645915\pi$
$181$	−11.9070	−0.885037	−0.442518	+	0.896759i	$0.645915\pi$
$182$	2.51203	0.186204
$183$	8.19365	0.605692
$184$	−23.2281	−1.71240
$185$	−30.4863	−2.24139
$186$	−8.28911	−0.607787
$187$	−16.5091	−1.20726
$188$	−5.04656	−0.368058
$189$	1.00000	0.0727393
$190$	−25.1864	−1.82722
$191$	14.5880	1.05555	0.527774	−	0.849385i	$-0.323027\pi$
$191$	14.5880	1.05555	0.527774	+	0.849385i	$0.323027\pi$
$192$	−3.97970	−0.287210
$193$	8.15977	0.587353	0.293676	−	0.955905i	$-0.405121\pi$
$193$	8.15977	0.587353	0.293676	+	0.955905i	$0.405121\pi$
$194$	6.58221	0.472575
$195$	11.4161	0.817524
$196$	−1.32260	−0.0944716
$197$	0.326739	0.0232792	0.0116396	−	0.999932i	$-0.496295\pi$
$197$	0.326739	0.0232792	0.0116396	+	0.999932i	$0.496295\pi$
$198$	−4.53496	−0.322286
$199$	13.5317	0.959238	0.479619	−	0.877477i	$-0.340775\pi$
$199$	13.5317	0.959238	0.479619	+	0.877477i	$0.340775\pi$
$200$	24.5853	1.73844
$201$	5.69594	0.401760
$202$	−12.2612	−0.862694
$203$	−2.18992	−0.153702
$204$	−3.96278	−0.277450
$205$	−20.4473	−1.42810
$206$	−4.30461	−0.299917
$207$	−8.49401	−0.590375
$208$	1.20402	0.0834836
$209$	−45.0799	−3.11824
$210$	3.07848	0.212435
$211$	12.5212	0.861995	0.430998	−	0.902353i	$-0.358162\pi$
$211$	12.5212	0.861995	0.430998	+	0.902353i	$0.358162\pi$
$212$	6.62530	0.455028
$213$	−7.02554	−0.481382
$214$	2.37070	0.162058
$215$	15.1479	1.03308
$216$	−2.73464	−0.186069
$217$	10.0713	0.683685
$218$	8.62718	0.584306
$219$	2.25827	0.152600
$220$	27.2581	1.83774
$221$	−9.14480	−0.615146
$222$	6.70829	0.450231
$223$	−10.2034	−0.683271	−0.341636	−	0.939832i	$-0.610981\pi$
$223$	−10.2034	−0.683271	−0.341636	+	0.939832i	$0.610981\pi$
$224$	5.79396	0.387125
$225$	8.99033	0.599356
$226$	−6.28795	−0.418269
$227$	15.0175	0.996745	0.498373	−	0.866963i	$-0.333931\pi$
$227$	15.0175	0.996745	0.498373	+	0.866963i	$0.333931\pi$
$228$	−10.8208	−0.716627
$229$	12.2701	0.810834	0.405417	−	0.914132i	$-0.367126\pi$
$229$	12.2701	0.810834	0.405417	+	0.914132i	$0.367126\pi$
$230$	−26.1486	−1.72419
$231$	5.51001	0.362532
$232$	5.98865	0.393174
$233$	−22.0349	−1.44355	−0.721776	−	0.692127i	$-0.756674\pi$
$233$	−22.0349	−1.44355	−0.721776	+	0.692127i	$0.756674\pi$
$234$	−2.51203	−0.164217
$235$	−14.2719	−0.930993
$236$	17.6789	1.15080
$237$	1.56489	0.101650
$238$	−2.46600	−0.159847
$239$	−11.3726	−0.735632	−0.367816	−	0.929899i	$-0.619894\pi$
$239$	−11.3726	−0.735632	−0.367816	+	0.929899i	$0.619894\pi$
$240$	1.47551	0.0952439
$241$	1.83314	0.118083	0.0590415	−	0.998256i	$-0.481196\pi$
$241$	1.83314	0.118083	0.0590415	+	0.998256i	$0.481196\pi$
$242$	−15.9342	−1.02429
$243$	−1.00000	−0.0641500
$244$	10.8369	0.693764
$245$	−3.74037	−0.238963
$246$	4.49929	0.286864
$247$	−24.9709	−1.58886
$248$	−27.5414	−1.74888
$249$	−5.39693	−0.342017
$250$	12.2841	0.776918
$251$	−2.74118	−0.173022	−0.0865110	−	0.996251i	$-0.527572\pi$
$251$	−2.74118	−0.173022	−0.0865110	+	0.996251i	$0.527572\pi$
$252$	1.32260	0.0833161
$253$	−46.8021	−2.94242
$254$	−0.823041	−0.0516422
$255$	−11.2069	−0.701802
$256$	−14.8009	−0.925056
$257$	−30.4318	−1.89828	−0.949142	−	0.314847i	$-0.898047\pi$
$257$	−30.4318	−1.89828	−0.949142	+	0.314847i	$0.898047\pi$
$258$	−3.33318	−0.207515
$259$	−8.15061	−0.506454
$260$	15.0990	0.936398
$261$	2.18992	0.135553
$262$	8.07000	0.498566
$263$	−15.1790	−0.935975	−0.467987	−	0.883735i	$-0.655021\pi$
$263$	−15.1790	−0.935975	−0.467987	+	0.883735i	$0.655021\pi$
$264$	−15.0679	−0.927364
$265$	18.7366	1.15098
$266$	−6.73368	−0.412869
$267$	−2.98116	−0.182444
$268$	7.53346	0.460179
$269$	22.2608	1.35726	0.678631	−	0.734479i	$-0.262573\pi$
$269$	22.2608	1.35726	0.678631	+	0.734479i	$0.262573\pi$
$270$	−3.07848	−0.187350
$271$	22.6860	1.37808	0.689038	−	0.724725i	$-0.258033\pi$
$271$	22.6860	1.37808	0.689038	+	0.724725i	$0.258033\pi$
$272$	−1.18195	−0.0716663
$273$	3.05214	0.184724
$274$	−14.5166	−0.876977
$275$	49.5368	2.98718
$276$	−11.2342	−0.676220
$277$	25.6782	1.54286	0.771428	−	0.636317i	$-0.219543\pi$
$277$	25.6782	1.54286	0.771428	+	0.636317i	$0.219543\pi$
$278$	12.5143	0.750560
$279$	−10.0713	−0.602954
$280$	10.2286	0.611273
$281$	8.18323	0.488170	0.244085	−	0.969754i	$-0.421512\pi$
$281$	8.18323	0.488170	0.244085	+	0.969754i	$0.421512\pi$
$282$	3.14042	0.187009
$283$	−13.1927	−0.784226	−0.392113	−	0.919917i	$-0.628256\pi$
$283$	−13.1927	−0.784226	−0.392113	+	0.919917i	$0.628256\pi$
$284$	−9.29200	−0.551379
$285$	−30.6017	−1.81269
$286$	−13.8413	−0.818455
$287$	−5.46666	−0.322687
$288$	−5.79396	−0.341412
$289$	−8.02280	−0.471929
$290$	6.74162	0.395882
$291$	7.99742	0.468817
$292$	2.98680	0.174789
$293$	−2.08917	−0.122051	−0.0610254	−	0.998136i	$-0.519437\pi$
$293$	−2.08917	−0.122051	−0.0610254	+	0.998136i	$0.519437\pi$
$294$	0.823041	0.0480008
$295$	49.9966	2.91091
$296$	22.2890	1.29552
$297$	−5.51001	−0.319723
$298$	3.40147	0.197042
$299$	−25.9249	−1.49927
$300$	11.8906	0.686506
$301$	4.04983	0.233429
$302$	6.35472	0.365673
$303$	−14.8974	−0.855833
$304$	−3.22745	−0.185107
$305$	30.6472	1.75486
$306$	2.46600	0.140972
$307$	27.3452	1.56068	0.780338	−	0.625359i	$-0.215047\pi$
$307$	27.3452	1.56068	0.780338	+	0.625359i	$0.215047\pi$
$308$	7.28755	0.415247
$309$	−5.23013	−0.297532
$310$	−31.0043	−1.76093
$311$	18.2259	1.03349	0.516747	−	0.856138i	$-0.327143\pi$
$311$	18.2259	1.03349	0.516747	+	0.856138i	$0.327143\pi$
$312$	−8.34649	−0.472527
$313$	−26.2831	−1.48561	−0.742803	−	0.669510i	$-0.766504\pi$
$313$	−26.2831	−1.48561	−0.742803	+	0.669510i	$0.766504\pi$
$314$	−6.70417	−0.378338
$315$	3.74037	0.210746
$316$	2.06972	0.116431
$317$	20.4264	1.14726	0.573631	−	0.819114i	$-0.305534\pi$
$317$	20.4264	1.14726	0.573631	+	0.819114i	$0.305534\pi$
$318$	−4.12285	−0.231198
$319$	12.0665	0.675593
$320$	−14.8855	−0.832126
$321$	2.88042	0.160769
$322$	−6.99092	−0.389589
$323$	24.5133	1.36396
$324$	−1.32260	−0.0734779
$325$	27.4397	1.52208
$326$	5.62356	0.311460
$327$	10.4821	0.579660
$328$	14.9493	0.825439
$329$	−3.81563	−0.210363
$330$	−16.9624	−0.933751
$331$	−10.6077	−0.583055	−0.291527	−	0.956563i	$-0.594163\pi$
$331$	−10.6077	−0.583055	−0.291527	+	0.956563i	$0.594163\pi$
$332$	−7.13800	−0.391749
$333$	8.15061	0.446651
$334$	−14.9146	−0.816091
$335$	21.3049	1.16401
$336$	0.394484	0.0215208
$337$	−6.43774	−0.350686	−0.175343	−	0.984507i	$-0.556103\pi$
$337$	−6.43774	−0.350686	−0.175343	+	0.984507i	$0.556103\pi$
$338$	3.03247	0.164945
$339$	−7.63990	−0.414942
$340$	−14.8222	−0.803849
$341$	−55.4930	−3.00511
$342$	6.73368	0.364116
$343$	−1.00000	−0.0539949
$344$	−11.0748	−0.597115
$345$	−31.7707	−1.71048
$346$	−19.5159	−1.04918
$347$	13.9202	0.747278	0.373639	−	0.927574i	$-0.378110\pi$
$347$	13.9202	0.747278	0.373639	+	0.927574i	$0.378110\pi$
$348$	2.89640	0.155263
$349$	−4.72177	−0.252750	−0.126375	−	0.991983i	$-0.540334\pi$
$349$	−4.72177	−0.252750	−0.126375	+	0.991983i	$0.540334\pi$
$350$	7.39942	0.395515
$351$	−3.05214	−0.162911
$352$	−31.9247	−1.70159
$353$	1.15223	0.0613271	0.0306635	−	0.999530i	$-0.490238\pi$
$353$	1.15223	0.0613271	0.0306635	+	0.999530i	$0.490238\pi$
$354$	−11.0014	−0.584718
$355$	−26.2781	−1.39470
$356$	−3.94289	−0.208973
$357$	−2.99620	−0.158576
$358$	−7.63994	−0.403783
$359$	12.1816	0.642921	0.321460	−	0.946923i	$-0.395826\pi$
$359$	12.1816	0.642921	0.321460	+	0.946923i	$0.395826\pi$
$360$	−10.2286	−0.539092
$361$	47.9363	2.52297
$362$	9.79992	0.515072
$363$	−19.3602	−1.01615
$364$	4.03676	0.211584
$365$	8.44676	0.442124
$366$	−6.74371	−0.352500
$367$	20.3114	1.06025	0.530124	−	0.847920i	$-0.322145\pi$
$367$	20.3114	1.06025	0.530124	+	0.847920i	$0.322145\pi$
$368$	−3.35075	−0.174670
$369$	5.46666	0.284583
$370$	25.0915	1.30444
$371$	5.00929	0.260069
$372$	−13.3203	−0.690628
$373$	37.7582	1.95504	0.977522	−	0.210833i	$-0.0676176\pi$
$373$	37.7582	1.95504	0.977522	+	0.210833i	$0.0676176\pi$
$374$	13.5877	0.702601
$375$	14.9253	0.770739
$376$	10.4344	0.538112
$377$	6.68394	0.344240
$378$	−0.823041	−0.0423327
$379$	−21.5022	−1.10449	−0.552246	−	0.833681i	$-0.686229\pi$
$379$	−21.5022	−1.10449	−0.552246	+	0.833681i	$0.686229\pi$
$380$	−40.4738	−2.07626
$381$	−1.00000	−0.0512316
$382$	−12.0065	−0.614306
$383$	25.0241	1.27867	0.639335	−	0.768928i	$-0.279210\pi$
$383$	25.0241	1.27867	0.639335	+	0.768928i	$0.279210\pi$
$384$	−8.31245	−0.424193
$385$	20.6094	1.05035
$386$	−6.71583	−0.341827
$387$	−4.04983	−0.205865
$388$	10.5774	0.536987
$389$	−11.2191	−0.568833	−0.284416	−	0.958701i	$-0.591800\pi$
$389$	−11.2191	−0.568833	−0.284416	+	0.958701i	$0.591800\pi$
$390$	−9.39592	−0.475781
$391$	25.4497	1.28705
$392$	2.73464	0.138120
$393$	9.80510	0.494602
$394$	−0.268919	−0.0135480
$395$	5.85325	0.294509
$396$	−7.28755	−0.366213
$397$	−20.5557	−1.03166	−0.515831	−	0.856690i	$-0.672517\pi$
$397$	−20.5557	−1.03166	−0.515831	+	0.856690i	$0.672517\pi$
$398$	−11.1372	−0.558256
$399$	−8.18146	−0.409585
$400$	3.54654	0.177327
$401$	−10.9879	−0.548711	−0.274355	−	0.961628i	$-0.588464\pi$
$401$	−10.9879	−0.548711	−0.274355	+	0.961628i	$0.588464\pi$
$402$	−4.68799	−0.233816
$403$	−30.7390	−1.53122
$404$	−19.7033	−0.980278
$405$	−3.74037	−0.185860
$406$	1.80240	0.0894514
$407$	44.9099	2.22610
$408$	8.19352	0.405640
$409$	−11.9000	−0.588417	−0.294208	−	0.955741i	$-0.595056\pi$
$409$	−11.9000	−0.588417	−0.294208	+	0.955741i	$0.595056\pi$
$410$	16.8290	0.831124
$411$	−17.6377	−0.870003
$412$	−6.91738	−0.340795
$413$	13.3668	0.657736
$414$	6.99092	0.343585
$415$	−20.1865	−0.990917
$416$	−17.6839	−0.867026
$417$	15.2050	0.744591
$418$	37.1026	1.81475
$419$	−36.0723	−1.76225	−0.881123	−	0.472887i	$-0.843212\pi$
$419$	−36.0723	−1.76225	−0.881123	+	0.472887i	$0.843212\pi$
$420$	4.94702	0.241390
$421$	2.95291	0.143916	0.0719580	−	0.997408i	$-0.477075\pi$
$421$	2.95291	0.143916	0.0719580	+	0.997408i	$0.477075\pi$
$422$	−10.3055	−0.501663
$423$	3.81563	0.185522
$424$	−13.6986	−0.665263
$425$	−26.9368	−1.30663
$426$	5.78231	0.280154
$427$	8.19365	0.396519
$428$	3.80965	0.184146
$429$	−16.8173	−0.811946
$430$	−12.4673	−0.601227
$431$	−35.1328	−1.69229	−0.846143	−	0.532956i	$-0.821081\pi$
$431$	−35.1328	−1.69229	−0.846143	+	0.532956i	$0.821081\pi$
$432$	−0.394484	−0.0189796
$433$	38.5473	1.85246	0.926232	−	0.376955i	$-0.123029\pi$
$433$	38.5473	1.85246	0.926232	+	0.376955i	$0.123029\pi$
$434$	−8.28911	−0.397890
$435$	8.19111	0.392733
$436$	13.8636	0.663947
$437$	69.4934	3.32432
$438$	−1.85865	−0.0888098
$439$	−18.5170	−0.883766	−0.441883	−	0.897073i	$-0.645689\pi$
$439$	−18.5170	−0.883766	−0.441883	+	0.897073i	$0.645689\pi$
$440$	−56.3594	−2.68683
$441$	1.00000	0.0476190
$442$	7.52655	0.358002
$443$	11.0423	0.524637	0.262318	−	0.964981i	$-0.415513\pi$
$443$	11.0423	0.524637	0.262318	+	0.964981i	$0.415513\pi$
$444$	10.7800	0.511597
$445$	−11.1506	−0.528590
$446$	8.39783	0.397649
$447$	4.13281	0.195475
$448$	−3.97970	−0.188023
$449$	−17.0853	−0.806304	−0.403152	−	0.915133i	$-0.632085\pi$
$449$	−17.0853	−0.806304	−0.403152	+	0.915133i	$0.632085\pi$
$450$	−7.39942	−0.348812
$451$	30.1213	1.41836
$452$	−10.1046	−0.475278
$453$	7.72102	0.362765
$454$	−12.3600	−0.580084
$455$	11.4161	0.535195
$456$	22.3734	1.04773
$457$	−34.4820	−1.61300	−0.806500	−	0.591234i	$-0.798641\pi$
$457$	−34.4820	−1.61300	−0.806500	+	0.591234i	$0.798641\pi$
$458$	−10.0988	−0.471888
$459$	2.99620	0.139851
$460$	−42.0200	−1.95919
$461$	9.52206	0.443487	0.221743	−	0.975105i	$-0.428825\pi$
$461$	9.52206	0.443487	0.221743	+	0.975105i	$0.428825\pi$
$462$	−4.53496	−0.210986
$463$	−20.5419	−0.954662	−0.477331	−	0.878723i	$-0.658396\pi$
$463$	−20.5419	−0.954662	−0.477331	+	0.878723i	$0.658396\pi$
$464$	0.863888	0.0401050
$465$	−37.6704	−1.74692
$466$	18.1356	0.840116
$467$	33.3542	1.54345	0.771725	−	0.635956i	$-0.219394\pi$
$467$	33.3542	1.54345	0.771725	+	0.635956i	$0.219394\pi$
$468$	−4.03676	−0.186599
$469$	5.69594	0.263014
$470$	11.7463	0.541818
$471$	−8.14560	−0.375330
$472$	−36.5533	−1.68250
$473$	−22.3146	−1.02603
$474$	−1.28797	−0.0591583
$475$	−73.5541	−3.37489
$476$	−3.96278	−0.181634
$477$	−5.00929	−0.229360
$478$	9.36012	0.428122
$479$	42.2188	1.92903	0.964514	−	0.264031i	$-0.0850522\pi$
$479$	42.2188	1.92903	0.964514	+	0.264031i	$0.0850522\pi$
$480$	−21.6715	−0.989165
$481$	24.8768	1.13428
$482$	−1.50875	−0.0687217
$483$	−8.49401	−0.386491
$484$	−25.6058	−1.16390
$485$	29.9133	1.35829
$486$	0.823041	0.0373339
$487$	10.6109	0.480826	0.240413	−	0.970671i	$-0.422717\pi$
$487$	10.6109	0.480826	0.240413	+	0.970671i	$0.422717\pi$
$488$	−22.4067	−1.01430
$489$	6.83265	0.308983
$490$	3.07848	0.139071
$491$	27.3680	1.23510	0.617550	−	0.786532i	$-0.288125\pi$
$491$	27.3680	1.23510	0.617550	+	0.786532i	$0.288125\pi$
$492$	7.23022	0.325963
$493$	−6.56144	−0.295512
$494$	20.5521	0.924683
$495$	−20.6094	−0.926325
$496$	−3.97297	−0.178391
$497$	−7.02554	−0.315139
$498$	4.44190	0.199046
$499$	13.3392	0.597146	0.298573	−	0.954387i	$-0.403489\pi$
$499$	13.3392	0.597146	0.298573	+	0.954387i	$0.403489\pi$
$500$	19.7402	0.882811
$501$	−18.1213	−0.809602
$502$	2.25611	0.100695
$503$	13.0642	0.582504	0.291252	−	0.956646i	$-0.405928\pi$
$503$	13.0642	0.582504	0.291252	+	0.956646i	$0.405928\pi$
$504$	−2.73464	−0.121811
$505$	−55.7217	−2.47958
$506$	38.5200	1.71242
$507$	3.68447	0.163633
$508$	−1.32260	−0.0586810
$509$	0.882990	0.0391378	0.0195689	−	0.999809i	$-0.493771\pi$
$509$	0.882990	0.0391378	0.0195689	+	0.999809i	$0.493771\pi$
$510$	9.22372	0.408433
$511$	2.25827	0.0999001
$512$	−4.44316	−0.196362
$513$	8.18146	0.361220
$514$	25.0467	1.10476
$515$	−19.5626	−0.862031
$516$	−5.35632	−0.235799
$517$	21.0242	0.924641
$518$	6.70829	0.294745
$519$	−23.7119	−1.04084
$520$	−31.2189	−1.36904
$521$	35.5113	1.55578	0.777889	−	0.628402i	$-0.216291\pi$
$521$	35.5113	1.55578	0.777889	+	0.628402i	$0.216291\pi$
$522$	−1.80240	−0.0788887
$523$	26.0419	1.13873	0.569367	−	0.822084i	$-0.307188\pi$
$523$	26.0419	1.13873	0.569367	+	0.822084i	$0.307188\pi$
$524$	12.9682	0.566521
$525$	8.99033	0.392370
$526$	12.4929	0.544717
$527$	30.1756	1.31447
$528$	−2.17361	−0.0945941
$529$	49.1482	2.13688
$530$	−15.4210	−0.669845
$531$	−13.3668	−0.580068
$532$	−10.8208	−0.469142
$533$	16.6850	0.722707
$534$	2.45362	0.106178
$535$	10.7738	0.465792
$536$	−15.5763	−0.672795
$537$	−9.28257	−0.400572
$538$	−18.3215	−0.789897
$539$	5.51001	0.237333
$540$	−4.94702	−0.212886
$541$	28.0997	1.20810	0.604049	−	0.796947i	$-0.293553\pi$
$541$	28.0997	1.20810	0.604049	+	0.796947i	$0.293553\pi$
$542$	−18.6715	−0.802010
$543$	11.9070	0.510976
$544$	17.3598	0.744297
$545$	39.2068	1.67943
$546$	−2.51203	−0.107505
$547$	12.7653	0.545803	0.272902	−	0.962042i	$-0.412017\pi$
$547$	12.7653	0.545803	0.272902	+	0.962042i	$0.412017\pi$
$548$	−23.3277	−0.996508
$549$	−8.19365	−0.349696
$550$	−40.7708	−1.73847
$551$	−17.9168	−0.763280
$552$	23.2281	0.988652
$553$	1.56489	0.0665458
$554$	−21.1342	−0.897908
$555$	30.4863	1.29407
$556$	20.1102	0.852860
$557$	−30.7436	−1.30265	−0.651324	−	0.758800i	$-0.725786\pi$
$557$	−30.7436	−1.30265	−0.651324	+	0.758800i	$0.725786\pi$
$558$	8.28911	0.350906
$559$	−12.3606	−0.522799
$560$	1.47551	0.0623518
$561$	16.5091	0.697014
$562$	−6.73514	−0.284105
$563$	−6.17851	−0.260393	−0.130197	−	0.991488i	$-0.541561\pi$
$563$	−6.17851	−0.260393	−0.130197	+	0.991488i	$0.541561\pi$
$564$	5.04656	0.212499
$565$	−28.5760	−1.20220
$566$	10.8582	0.456403
$567$	−1.00000	−0.0419961
$568$	19.2123	0.806132
$569$	11.0785	0.464436	0.232218	−	0.972664i	$-0.425402\pi$
$569$	11.0785	0.464436	0.232218	+	0.972664i	$0.425402\pi$
$570$	25.1864	1.05494
$571$	1.50955	0.0631727	0.0315863	−	0.999501i	$-0.489944\pi$
$571$	1.50955	0.0631727	0.0315863	+	0.999501i	$0.489944\pi$
$572$	−22.2426	−0.930009
$573$	−14.5880	−0.609421
$574$	4.49929	0.187797
$575$	−76.3640	−3.18460
$576$	3.97970	0.165821
$577$	20.1961	0.840776	0.420388	−	0.907344i	$-0.361894\pi$
$577$	20.1961	0.840776	0.420388	+	0.907344i	$0.361894\pi$
$578$	6.60309	0.274653
$579$	−8.15977	−0.339108
$580$	10.8336	0.449840
$581$	−5.39693	−0.223903
$582$	−6.58221	−0.272841
$583$	−27.6012	−1.14313
$584$	−6.17556	−0.255547
$585$	−11.4161	−0.471998
$586$	1.71948	0.0710309
$587$	−33.4825	−1.38197	−0.690985	−	0.722869i	$-0.742823\pi$
$587$	−33.4825	−1.38197	−0.690985	+	0.722869i	$0.742823\pi$
$588$	1.32260	0.0545432
$589$	82.3981	3.39515
$590$	−41.1493	−1.69409
$591$	−0.326739	−0.0134402
$592$	3.21528	0.132147
$593$	26.7210	1.09730	0.548649	−	0.836053i	$-0.315142\pi$
$593$	26.7210	1.09730	0.548649	+	0.836053i	$0.315142\pi$
$594$	4.53496	0.186072
$595$	−11.2069	−0.459437
$596$	5.46606	0.223899
$597$	−13.5317	−0.553817
$598$	21.3372	0.872545
$599$	−27.7249	−1.13281	−0.566405	−	0.824127i	$-0.691666\pi$
$599$	−27.7249	−1.13281	−0.566405	+	0.824127i	$0.691666\pi$
$600$	−24.5853	−1.00369
$601$	−39.7183	−1.62014	−0.810072	−	0.586331i	$-0.800572\pi$
$601$	−39.7183	−1.62014	−0.810072	+	0.586331i	$0.800572\pi$
$602$	−3.33318	−0.135850
$603$	−5.69594	−0.231956
$604$	10.2118	0.415514
$605$	−72.4141	−2.94405
$606$	12.2612	0.498076
$607$	32.9569	1.33768	0.668839	−	0.743407i	$-0.266791\pi$
$607$	32.9569	1.33768	0.668839	+	0.743407i	$0.266791\pi$
$608$	47.4030	1.92245
$609$	2.18992	0.0887401
$610$	−25.2240	−1.02129
$611$	11.6458	0.471140
$612$	3.96278	0.160186
$613$	−13.5673	−0.547980	−0.273990	−	0.961733i	$-0.588343\pi$
$613$	−13.5673	−0.547980	−0.273990	+	0.961733i	$0.588343\pi$
$614$	−22.5063	−0.908279
$615$	20.4473	0.824515
$616$	−15.0679	−0.607102
$617$	38.0240	1.53079	0.765394	−	0.643563i	$-0.222544\pi$
$617$	38.0240	1.53079	0.765394	+	0.643563i	$0.222544\pi$
$618$	4.30461	0.173157
$619$	−4.14786	−0.166717	−0.0833584	−	0.996520i	$-0.526565\pi$
$619$	−4.14786	−0.166717	−0.0833584	+	0.996520i	$0.526565\pi$
$620$	−49.8230	−2.00094
$621$	8.49401	0.340853
$622$	−15.0006	−0.601470
$623$	−2.98116	−0.119438
$624$	−1.20402	−0.0481993
$625$	10.8744	0.434977
$626$	21.6321	0.864591
$627$	45.0799	1.80032
$628$	−10.7734	−0.429905
$629$	−24.4208	−0.973723
$630$	−3.07848	−0.122649
$631$	36.7451	1.46280	0.731400	−	0.681949i	$-0.238867\pi$
$631$	36.7451	1.46280	0.731400	+	0.681949i	$0.238867\pi$
$632$	−4.27940	−0.170226
$633$	−12.5212	−0.497673
$634$	−16.8118	−0.667682
$635$	−3.74037	−0.148432
$636$	−6.62530	−0.262710
$637$	3.05214	0.120930
$638$	−9.93122	−0.393181
$639$	7.02554	0.277926
$640$	−31.0916	−1.22900
$641$	−44.5747	−1.76060	−0.880298	−	0.474422i	$-0.842657\pi$
$641$	−44.5747	−1.76060	−0.880298	+	0.474422i	$0.842657\pi$
$642$	−2.37070	−0.0935642
$643$	29.4222	1.16030	0.580150	−	0.814510i	$-0.302994\pi$
$643$	29.4222	1.16030	0.580150	+	0.814510i	$0.302994\pi$
$644$	−11.2342	−0.442690
$645$	−15.1479	−0.596446
$646$	−20.1755	−0.793792
$647$	−45.0622	−1.77158	−0.885789	−	0.464089i	$-0.846382\pi$
$647$	−45.0622	−1.77158	−0.885789	+	0.464089i	$0.846382\pi$
$648$	2.73464	0.107427
$649$	−73.6510	−2.89105
$650$	−22.5840	−0.885818
$651$	−10.0713	−0.394726
$652$	9.03689	0.353912
$653$	42.4383	1.66074	0.830369	−	0.557214i	$-0.188130\pi$
$653$	42.4383	1.66074	0.830369	+	0.557214i	$0.188130\pi$
$654$	−8.62718	−0.337350
$655$	36.6746	1.43300
$656$	2.15651	0.0841974
$657$	−2.25827	−0.0881036
$658$	3.14042	0.122426
$659$	29.3268	1.14241	0.571206	−	0.820807i	$-0.306476\pi$
$659$	29.3268	1.14241	0.571206	+	0.820807i	$0.306476\pi$
$660$	−27.2581	−1.06102
$661$	12.1959	0.474364	0.237182	−	0.971465i	$-0.423776\pi$
$661$	12.1959	0.474364	0.237182	+	0.971465i	$0.423776\pi$
$662$	8.73062	0.339325
$663$	9.14480	0.355155
$664$	14.7587	0.572747
$665$	−30.6017	−1.18668
$666$	−6.70829	−0.259941
$667$	−18.6012	−0.720242
$668$	−23.9673	−0.927324
$669$	10.2034	0.394487
$670$	−17.5348	−0.677429
$671$	−45.1471	−1.74288
$672$	−5.79396	−0.223507
$673$	−7.89440	−0.304307	−0.152153	−	0.988357i	$-0.548621\pi$
$673$	−7.89440	−0.304307	−0.152153	+	0.988357i	$0.548621\pi$
$674$	5.29853	0.204092
$675$	−8.99033	−0.346038
$676$	4.87309	0.187427
$677$	37.0732	1.42484	0.712419	−	0.701755i	$-0.247600\pi$
$677$	37.0732	1.42484	0.712419	+	0.701755i	$0.247600\pi$
$678$	6.28795	0.241487
$679$	7.99742	0.306913
$680$	30.6468	1.17525
$681$	−15.0175	−0.575471
$682$	45.6730	1.74891
$683$	28.3006	1.08289	0.541446	−	0.840735i	$-0.317877\pi$
$683$	28.3006	1.08289	0.541446	+	0.840735i	$0.317877\pi$
$684$	10.8208	0.413745
$685$	−65.9714	−2.52064
$686$	0.823041	0.0314239
$687$	−12.2701	−0.468135
$688$	−1.59759	−0.0609077
$689$	−15.2890	−0.582466
$690$	26.1486	0.995460
$691$	4.33936	0.165077	0.0825386	−	0.996588i	$-0.473697\pi$
$691$	4.33936	0.165077	0.0825386	+	0.996588i	$0.473697\pi$
$692$	−31.3615	−1.19218
$693$	−5.51001	−0.209308
$694$	−11.4569	−0.434899
$695$	56.8722	2.15729
$696$	−5.98865	−0.226999
$697$	−16.3792	−0.620406
$698$	3.88621	0.147095
$699$	22.0349	0.833435
$700$	11.8906	0.449424
$701$	−13.9841	−0.528171	−0.264085	−	0.964499i	$-0.585070\pi$
$701$	−13.9841	−0.528171	−0.264085	+	0.964499i	$0.585070\pi$
$702$	2.51203	0.0948106
$703$	−66.6839	−2.51503
$704$	21.9282	0.826449
$705$	14.2719	0.537509
$706$	−0.948334	−0.0356910
$707$	−14.8974	−0.560274
$708$	−17.6789	−0.664415
$709$	11.7157	0.439993	0.219997	−	0.975501i	$-0.429395\pi$
$709$	11.7157	0.439993	0.219997	+	0.975501i	$0.429395\pi$
$710$	21.6280	0.811683
$711$	−1.56489	−0.0586879
$712$	8.15240	0.305524
$713$	85.5458	3.20372
$714$	2.46600	0.0922876
$715$	−62.9028	−2.35243
$716$	−12.2772	−0.458819
$717$	11.3726	0.424717
$718$	−10.0260	−0.374166
$719$	−7.06966	−0.263654	−0.131827	−	0.991273i	$-0.542084\pi$
$719$	−7.06966	−0.263654	−0.131827	+	0.991273i	$0.542084\pi$
$720$	−1.47551	−0.0549891
$721$	−5.23013	−0.194780
$722$	−39.4536	−1.46831
$723$	−1.83314	−0.0681753
$724$	15.7482	0.585276
$725$	19.6881	0.731199
$726$	15.9342	0.591375
$727$	−48.7194	−1.80690	−0.903451	−	0.428692i	$-0.858975\pi$
$727$	−48.7194	−1.80690	−0.903451	+	0.428692i	$0.858975\pi$
$728$	−8.34649	−0.309342
$729$	1.00000	0.0370370
$730$	−6.95204	−0.257306
$731$	12.1341	0.448796
$732$	−10.8369	−0.400545
$733$	18.8184	0.695073	0.347536	−	0.937666i	$-0.387018\pi$
$733$	18.8184	0.695073	0.347536	+	0.937666i	$0.387018\pi$
$734$	−16.7171	−0.617041
$735$	3.74037	0.137965
$736$	49.2139	1.81405
$737$	−31.3847	−1.15607
$738$	−4.49929	−0.165621
$739$	−15.0604	−0.554004	−0.277002	−	0.960869i	$-0.589341\pi$
$739$	−15.0604	−0.554004	−0.277002	+	0.960869i	$0.589341\pi$
$740$	40.3212	1.48224
$741$	24.9709	0.917330
$742$	−4.12285	−0.151355
$743$	4.11581	0.150994	0.0754972	−	0.997146i	$-0.475946\pi$
$743$	4.11581	0.150994	0.0754972	+	0.997146i	$0.475946\pi$
$744$	27.5414	1.00972
$745$	15.4582	0.566345
$746$	−31.0765	−1.13779
$747$	5.39693	0.197463
$748$	21.8349	0.798365
$749$	2.88042	0.105248
$750$	−12.2841	−0.448554
$751$	−9.97080	−0.363840	−0.181920	−	0.983313i	$-0.558231\pi$
$751$	−9.97080	−0.363840	−0.181920	+	0.983313i	$0.558231\pi$
$752$	1.50520	0.0548891
$753$	2.74118	0.0998943
$754$	−5.50116	−0.200340
$755$	28.8794	1.05103
$756$	−1.32260	−0.0481026
$757$	20.7508	0.754201	0.377101	−	0.926172i	$-0.376921\pi$
$757$	20.7508	0.754201	0.377101	+	0.926172i	$0.376921\pi$
$758$	17.6972	0.642791
$759$	46.8021	1.69881
$760$	83.6845	3.03556
$761$	22.9097	0.830475	0.415237	−	0.909713i	$-0.363699\pi$
$761$	22.9097	0.830475	0.415237	+	0.909713i	$0.363699\pi$
$762$	0.823041	0.0298156
$763$	10.4821	0.379477
$764$	−19.2941	−0.698036
$765$	11.2069	0.405185
$766$	−20.5958	−0.744158
$767$	−40.7972	−1.47310
$768$	14.8009	0.534081
$769$	19.7800	0.713286	0.356643	−	0.934241i	$-0.383921\pi$
$769$	19.7800	0.713286	0.356643	+	0.934241i	$0.383921\pi$
$770$	−16.9624	−0.611283
$771$	30.4318	1.09598
$772$	−10.7921	−0.388417
$773$	41.1828	1.48124	0.740621	−	0.671923i	$-0.234532\pi$
$773$	41.1828	1.48124	0.740621	+	0.671923i	$0.234532\pi$
$774$	3.33318	0.119809
$775$	−90.5445	−3.25245
$776$	−21.8701	−0.785090
$777$	8.15061	0.292402
$778$	9.23381	0.331048
$779$	−44.7253	−1.60245
$780$	−15.0990	−0.540630
$781$	38.7108	1.38518
$782$	−20.9462	−0.749034
$783$	−2.18992	−0.0782614
$784$	0.394484	0.0140887
$785$	−30.4675	−1.08743
$786$	−8.07000	−0.287847
$787$	−14.4977	−0.516788	−0.258394	−	0.966040i	$-0.583193\pi$
$787$	−14.4977	−0.516788	−0.258394	+	0.966040i	$0.583193\pi$
$788$	−0.432145	−0.0153945
$789$	15.1790	0.540385
$790$	−4.81747	−0.171398
$791$	−7.63990	−0.271644
$792$	15.0679	0.535414
$793$	−25.0081	−0.888065
$794$	16.9182	0.600405
$795$	−18.7366	−0.664518
$796$	−17.8971	−0.634346
$797$	19.3076	0.683911	0.341955	−	0.939716i	$-0.388911\pi$
$797$	19.3076	0.683911	0.341955	+	0.939716i	$0.388911\pi$
$798$	6.73368	0.238370
$799$	−11.4324	−0.404449
$800$	−52.0896	−1.84165
$801$	2.98116	0.105334
$802$	9.04352	0.319338
$803$	−12.4431	−0.439107
$804$	−7.53346	−0.265685
$805$	−31.7707	−1.11977
$806$	25.2995	0.891136
$807$	−22.2608	−0.783616
$808$	40.7390	1.43319
$809$	50.2043	1.76509	0.882544	−	0.470229i	$-0.155829\pi$
$809$	50.2043	1.76509	0.882544	+	0.470229i	$0.155829\pi$
$810$	3.07848	0.108167
$811$	6.45146	0.226542	0.113271	−	0.993564i	$-0.463867\pi$
$811$	6.45146	0.226542	0.113271	+	0.993564i	$0.463867\pi$
$812$	2.89640	0.101644
$813$	−22.6860	−0.795633
$814$	−36.9627	−1.29554
$815$	25.5566	0.895209
$816$	1.18195	0.0413766
$817$	33.1336	1.15920
$818$	9.79419	0.342446
$819$	−3.05214	−0.106650
$820$	27.0437	0.944405
$821$	−3.21962	−0.112365	−0.0561827	−	0.998421i	$-0.517893\pi$
$821$	−3.21962	−0.112365	−0.0561827	+	0.998421i	$0.517893\pi$
$822$	14.5166	0.506323
$823$	−44.1000	−1.53723	−0.768615	−	0.639712i	$-0.779054\pi$
$823$	−44.1000	−1.53723	−0.768615	+	0.639712i	$0.779054\pi$
$824$	14.3025	0.498252
$825$	−49.5368	−1.72465
$826$	−11.0014	−0.382788
$827$	−12.9386	−0.449920	−0.224960	−	0.974368i	$-0.572225\pi$
$827$	−12.9386	−0.449920	−0.224960	+	0.974368i	$0.572225\pi$
$828$	11.2342	0.390416
$829$	32.0863	1.11440	0.557201	−	0.830378i	$-0.311875\pi$
$829$	32.0863	1.11440	0.557201	+	0.830378i	$0.311875\pi$
$830$	16.6143	0.576692
$831$	−25.6782	−0.890768
$832$	12.1466	0.421107
$833$	−2.99620	−0.103812
$834$	−12.5143	−0.433336
$835$	−67.7804	−2.34564
$836$	59.6228	2.06210
$837$	10.0713	0.348115
$838$	29.6890	1.02559
$839$	−33.1797	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$839$	−33.1797	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$840$	−10.2286	−0.352919
$841$	−24.2042	−0.834629
$842$	−2.43037	−0.0837560
$843$	−8.18323	−0.281845
$844$	−16.5606	−0.570039
$845$	13.7813	0.474090
$846$	−3.14042	−0.107970
$847$	−19.3602	−0.665224
$848$	−1.97608	−0.0678590
$849$	13.1927	0.452773
$850$	22.1701	0.760429
$851$	−69.2314	−2.37322
$852$	9.29200	0.318339
$853$	−29.0122	−0.993360	−0.496680	−	0.867934i	$-0.665448\pi$
$853$	−29.0122	−0.993360	−0.496680	+	0.867934i	$0.665448\pi$
$854$	−6.74371	−0.230765
$855$	30.6017	1.04655
$856$	−7.87690	−0.269227
$857$	−1.38316	−0.0472477	−0.0236239	−	0.999721i	$-0.507520\pi$
$857$	−1.38316	−0.0472477	−0.0236239	+	0.999721i	$0.507520\pi$
$858$	13.8413	0.472535
$859$	4.82470	0.164617	0.0823084	−	0.996607i	$-0.473771\pi$
$859$	4.82470	0.164617	0.0823084	+	0.996607i	$0.473771\pi$
$860$	−20.0346	−0.683174
$861$	5.46666	0.186303
$862$	28.9157	0.984873
$863$	11.2021	0.381325	0.190662	−	0.981656i	$-0.438936\pi$
$863$	11.2021	0.381325	0.190662	+	0.981656i	$0.438936\pi$
$864$	5.79396	0.197114
$865$	−88.6913	−3.01560
$866$	−31.7260	−1.07809
$867$	8.02280	0.272468
$868$	−13.3203	−0.452122
$869$	−8.62254	−0.292500
$870$	−6.74162	−0.228562
$871$	−17.3848	−0.589060
$872$	−28.6647	−0.970709
$873$	−7.99742	−0.270672
$874$	−57.1960	−1.93468
$875$	14.9253	0.504567
$876$	−2.98680	−0.100915
$877$	31.6371	1.06831	0.534155	−	0.845386i	$-0.320630\pi$
$877$	31.6371	1.06831	0.534155	+	0.845386i	$0.320630\pi$
$878$	15.2402	0.514333
$879$	2.08917	0.0704660
$880$	−8.13009	−0.274065
$881$	12.8041	0.431381	0.215690	−	0.976462i	$-0.430800\pi$
$881$	12.8041	0.431381	0.215690	+	0.976462i	$0.430800\pi$
$882$	−0.823041	−0.0277132
$883$	3.88096	0.130605	0.0653023	−	0.997866i	$-0.479199\pi$
$883$	3.88096	0.130605	0.0653023	+	0.997866i	$0.479199\pi$
$884$	12.0949	0.406797
$885$	−49.9966	−1.68062
$886$	−9.08829	−0.305327
$887$	11.5566	0.388032	0.194016	−	0.980998i	$-0.437849\pi$
$887$	11.5566	0.388032	0.194016	+	0.980998i	$0.437849\pi$
$888$	−22.2890	−0.747969
$889$	−1.00000	−0.0335389
$890$	9.17743	0.307628
$891$	5.51001	0.184592
$892$	13.4951	0.451848
$893$	−31.2174	−1.04465
$894$	−3.40147	−0.113762
$895$	−34.7202	−1.16057
$896$	−8.31245	−0.277700
$897$	25.9249	0.865606
$898$	14.0619	0.469251
$899$	−22.0554	−0.735588
$900$	−11.8906	−0.396355
$901$	15.0088	0.500017
$902$	−24.7911	−0.825453
$903$	−4.04983	−0.134770
$904$	20.8924	0.694870
$905$	44.5364	1.48044
$906$	−6.35472	−0.211121
$907$	−23.4079	−0.777248	−0.388624	−	0.921397i	$-0.627049\pi$
$907$	−23.4079	−0.777248	−0.388624	+	0.921397i	$0.627049\pi$
$908$	−19.8622	−0.659149
$909$	14.8974	0.494116
$910$	−9.39592	−0.311472
$911$	12.1098	0.401216	0.200608	−	0.979672i	$-0.435708\pi$
$911$	12.1098	0.401216	0.200608	+	0.979672i	$0.435708\pi$
$912$	3.22745	0.106872
$913$	29.7371	0.984156
$914$	28.3801	0.938731
$915$	−30.6472	−1.01317
$916$	−16.2285	−0.536206
$917$	9.80510	0.323793
$918$	−2.46600	−0.0813900
$919$	−15.2424	−0.502799	−0.251400	−	0.967883i	$-0.580891\pi$
$919$	−15.2424	−0.502799	−0.251400	+	0.967883i	$0.580891\pi$
$920$	86.8814	2.86440
$921$	−27.3452	−0.901056
$922$	−7.83705	−0.258100
$923$	21.4429	0.705802
$924$	−7.28755	−0.239743
$925$	73.2767	2.40932
$926$	16.9068	0.555593
$927$	5.23013	0.171780
$928$	−12.6883	−0.416514
$929$	−13.6188	−0.446818	−0.223409	−	0.974725i	$-0.571719\pi$
$929$	−13.6188	−0.446818	−0.223409	+	0.974725i	$0.571719\pi$
$930$	31.0043	1.01667
$931$	−8.18146	−0.268137
$932$	29.1434	0.954623
$933$	−18.2259	−0.596688
$934$	−27.4519	−0.898254
$935$	61.7500	2.01944
$936$	8.34649	0.272814
$937$	−7.92871	−0.259020	−0.129510	−	0.991578i	$-0.541340\pi$
$937$	−7.92871	−0.259020	−0.129510	+	0.991578i	$0.541340\pi$
$938$	−4.68799	−0.153068
$939$	26.2831	0.857715
$940$	18.8760	0.615667
$941$	−58.4662	−1.90594	−0.952972	−	0.303059i	$-0.901992\pi$
$941$	−58.4662	−1.90594	−0.952972	+	0.303059i	$0.901992\pi$
$942$	6.70417	0.218434
$943$	−46.4339	−1.51209
$944$	−5.27297	−0.171621
$945$	−3.74037	−0.121674
$946$	18.3659	0.597125
$947$	18.5748	0.603601	0.301801	−	0.953371i	$-0.402412\pi$
$947$	18.5748	0.603601	0.301801	+	0.953371i	$0.402412\pi$
$948$	−2.06972	−0.0672215
$949$	−6.89255	−0.223742
$950$	60.5381	1.96411
$951$	−20.4264	−0.662372
$952$	8.19352	0.265554
$953$	−36.6177	−1.18616	−0.593082	−	0.805142i	$-0.702089\pi$
$953$	−36.6177	−1.18616	−0.593082	+	0.805142i	$0.702089\pi$
$954$	4.12285	0.133482
$955$	−54.5643	−1.76566
$956$	15.0414	0.486474
$957$	−12.0665	−0.390054
$958$	−34.7479	−1.12265
$959$	−17.6377	−0.569551
$960$	14.8855	0.480428
$961$	70.4313	2.27198
$962$	−20.4746	−0.660128
$963$	−2.88042	−0.0928201
$964$	−2.42452	−0.0780885
$965$	−30.5205	−0.982490
$966$	6.99092	0.224929
$967$	−15.6972	−0.504788	−0.252394	−	0.967625i	$-0.581218\pi$
$967$	−15.6972	−0.504788	−0.252394	+	0.967625i	$0.581218\pi$
$968$	52.9431	1.70166
$969$	−24.5133	−0.787480
$970$	−24.6199	−0.790496
$971$	−15.0134	−0.481802	−0.240901	−	0.970550i	$-0.577443\pi$
$971$	−15.0134	−0.481802	−0.240901	+	0.970550i	$0.577443\pi$
$972$	1.32260	0.0424225
$973$	15.2050	0.487449
$974$	−8.73322	−0.279830
$975$	−27.4397	−0.878774
$976$	−3.23226	−0.103462
$977$	−47.3491	−1.51483	−0.757417	−	0.652932i	$-0.773539\pi$
$977$	−47.3491	−1.51483	−0.757417	+	0.652932i	$0.773539\pi$
$978$	−5.62356	−0.179822
$979$	16.4262	0.524984
$980$	4.94702	0.158027
$981$	−10.4821	−0.334667
$982$	−22.5250	−0.718801
$983$	−18.8317	−0.600636	−0.300318	−	0.953839i	$-0.597093\pi$
$983$	−18.8317	−0.600636	−0.300318	+	0.953839i	$0.597093\pi$
$984$	−14.9493	−0.476568
$985$	−1.22212	−0.0389400
$986$	5.40034	0.171982
$987$	3.81563	0.121453
$988$	33.0266	1.05072
$989$	34.3993	1.09384
$990$	16.9624	0.539101
$991$	44.4405	1.41170	0.705849	−	0.708363i	$-0.250566\pi$
$991$	44.4405	1.41170	0.705849	+	0.708363i	$0.250566\pi$
$992$	58.3527	1.85270
$993$	10.6077	0.336627
$994$	5.78231	0.183404
$995$	−50.6136	−1.60456
$996$	7.13800	0.226176
$997$	58.1693	1.84224	0.921120	−	0.389280i	$-0.127276\pi$
$997$	58.1693	1.84224	0.921120	+	0.389280i	$0.127276\pi$
$998$	−10.9787	−0.347526
$999$	−8.15061	−0.257874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2667.2.a.o.1.5	✓	16
3.2	odd	2		8001.2.a.r.1.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.5	✓	16	1.1	even	1	trivial
8001.2.a.r.1.12		16	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:	\( N \)	\(=\)	\( 2667 = 3 \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2667.a (trivial)

\(f(q)\)	\(=\)	\(q-0.823041 q^{2} -1.00000 q^{3} -1.32260 q^{4} -3.74037 q^{5} +0.823041 q^{6} -1.00000 q^{7} +2.73464 q^{8} +1.00000 q^{9} +3.07848 q^{10} +5.51001 q^{11} +1.32260 q^{12} +3.05214 q^{13} +0.823041 q^{14} +3.74037 q^{15} +0.394484 q^{16} -2.99620 q^{17} -0.823041 q^{18} -8.18146 q^{19} +4.94702 q^{20} +1.00000 q^{21} -4.53496 q^{22} -8.49401 q^{23} -2.73464 q^{24} +8.99033 q^{25} -2.51203 q^{26} -1.00000 q^{27} +1.32260 q^{28} +2.18992 q^{29} -3.07848 q^{30} -10.0713 q^{31} -5.79396 q^{32} -5.51001 q^{33} +2.46600 q^{34} +3.74037 q^{35} -1.32260 q^{36} +8.15061 q^{37} +6.73368 q^{38} -3.05214 q^{39} -10.2286 q^{40} +5.46666 q^{41} -0.823041 q^{42} -4.04983 q^{43} -7.28755 q^{44} -3.74037 q^{45} +6.99092 q^{46} +3.81563 q^{47} -0.394484 q^{48} +1.00000 q^{49} -7.39942 q^{50} +2.99620 q^{51} -4.03676 q^{52} -5.00929 q^{53} +0.823041 q^{54} -20.6094 q^{55} -2.73464 q^{56} +8.18146 q^{57} -1.80240 q^{58} -13.3668 q^{59} -4.94702 q^{60} -8.19365 q^{61} +8.28911 q^{62} -1.00000 q^{63} +3.97970 q^{64} -11.4161 q^{65} +4.53496 q^{66} -5.69594 q^{67} +3.96278 q^{68} +8.49401 q^{69} -3.07848 q^{70} +7.02554 q^{71} +2.73464 q^{72} -2.25827 q^{73} -6.70829 q^{74} -8.99033 q^{75} +10.8208 q^{76} -5.51001 q^{77} +2.51203 q^{78} -1.56489 q^{79} -1.47551 q^{80} +1.00000 q^{81} -4.49929 q^{82} +5.39693 q^{83} -1.32260 q^{84} +11.2069 q^{85} +3.33318 q^{86} -2.18992 q^{87} +15.0679 q^{88} +2.98116 q^{89} +3.07848 q^{90} -3.05214 q^{91} +11.2342 q^{92} +10.0713 q^{93} -3.14042 q^{94} +30.6017 q^{95} +5.79396 q^{96} -7.99742 q^{97} -0.823041 q^{98} +5.51001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 5 q^{2} - 16 q^{3} + 19 q^{4} - q^{5} - 5 q^{6} - 16 q^{7} + 6 q^{8} + 16 q^{9} - 12 q^{10} + 11 q^{11} - 19 q^{12} + 18 q^{13} - 5 q^{14} + q^{15} + 25 q^{16} - 5 q^{17} + 5 q^{18} - 11 q^{19}+ \cdots + 11 q^{99}+O(q^{100}) \) 16 * q + 5 * q^2 - 16 * q^3 + 19 * q^4 - q^5 - 5 * q^6 - 16 * q^7 + 6 * q^8 + 16 * q^9 - 12 * q^10 + 11 * q^11 - 19 * q^12 + 18 * q^13 - 5 * q^14 + q^15 + 25 * q^16 - 5 * q^17 + 5 * q^18 - 11 * q^19 - q^20 + 16 * q^21 + q^22 + 13 * q^23 - 6 * q^24 + 33 * q^25 + 8 * q^26 - 16 * q^27 - 19 * q^28 + 24 * q^29 + 12 * q^30 - 42 * q^31 + 42 * q^32 - 11 * q^33 + 9 * q^34 + q^35 + 19 * q^36 + 40 * q^37 + 38 * q^38 - 18 * q^39 - 61 * q^40 + 9 * q^41 + 5 * q^42 + 7 * q^43 + 3 * q^44 - q^45 + 24 * q^46 + 31 * q^47 - 25 * q^48 + 16 * q^49 + 6 * q^50 + 5 * q^51 + 52 * q^52 + 66 * q^53 - 5 * q^54 - 36 * q^55 - 6 * q^56 + 11 * q^57 + 19 * q^58 - 7 * q^59 + q^60 + 6 * q^61 + 52 * q^62 - 16 * q^63 + 10 * q^64 + 51 * q^65 - q^66 + 16 * q^67 + 14 * q^68 - 13 * q^69 + 12 * q^70 + 46 * q^71 + 6 * q^72 + 39 * q^73 + 72 * q^74 - 33 * q^75 + 24 * q^76 - 11 * q^77 - 8 * q^78 + 4 * q^79 - 2 * q^80 + 16 * q^81 - 18 * q^82 + 15 * q^83 + 19 * q^84 - 4 * q^85 + 14 * q^86 - 24 * q^87 + 58 * q^88 - q^89 - 12 * q^90 - 18 * q^91 + 26 * q^92 + 42 * q^93 + 5 * q^94 + 44 * q^95 - 42 * q^96 + 41 * q^97 + 5 * q^98 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more