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Properties

Label 15.18.e.a.8.1
Level $15$
Weight $18$
Character 15.8
Analytic conductor $27.483$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [15,18,Mod(2,15)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15.2"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.4833131017\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 8.1
Character \(\chi\) \(=\) 15.8
Dual form 15.18.e.a.2.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-493.109 - 493.109i) q^{2} +(-9450.47 - 6311.01i) q^{3} +355240. i q^{4} +(425179. - 762996. i) q^{5} +(1.54809e6 + 7.77212e6i) q^{6} +(7.63387e6 - 7.63387e6i) q^{7} +(1.10539e8 - 1.10539e8i) q^{8} +(4.94824e7 + 1.19284e8i) q^{9} +(-5.85899e8 + 1.66581e8i) q^{10} -1.58965e8i q^{11} +(2.24192e9 - 3.35719e9i) q^{12} +(-2.74089e9 - 2.74089e9i) q^{13} -7.52865e9 q^{14} +(-8.83341e9 + 4.52736e9i) q^{15} -6.24537e10 q^{16} +(5.83404e8 + 5.83404e8i) q^{17} +(3.44197e10 - 8.32202e10i) q^{18} -8.96182e10i q^{19} +(2.71047e11 + 1.51041e11i) q^{20} +(-1.20321e11 + 2.39662e10i) q^{21} +(-7.83872e10 + 7.83872e10i) q^{22} +(3.76016e11 - 3.76016e11i) q^{23} +(-1.74226e12 + 3.47033e11i) q^{24} +(-4.01386e11 - 6.48819e11i) q^{25} +2.70312e12i q^{26} +(2.85170e11 - 1.43957e12i) q^{27} +(2.71186e12 + 2.71186e12i) q^{28} +9.43058e11 q^{29} +(6.58831e12 + 2.12335e12i) q^{30} -2.34217e12 q^{31} +(1.63078e13 + 1.63078e13i) q^{32} +(-1.00323e12 + 1.50230e12i) q^{33} -5.75363e11i q^{34} +(-2.57885e12 - 9.07037e12i) q^{35} +(-4.23745e13 + 1.75782e13i) q^{36} +(9.66735e11 - 9.66735e11i) q^{37} +(-4.41915e13 + 4.41915e13i) q^{38} +(8.60491e12 + 4.32005e13i) q^{39} +(-3.73421e13 - 1.31340e14i) q^{40} -4.75944e13i q^{41} +(7.11493e13 + 4.75134e13i) q^{42} +(4.22297e13 + 4.22297e13i) q^{43} +5.64709e13 q^{44} +(1.12052e14 + 1.29621e13i) q^{45} -3.70834e14 q^{46} +(-1.75506e14 - 1.75506e14i) q^{47} +(5.90216e14 + 3.94146e14i) q^{48} +1.16079e14i q^{49} +(-1.22011e14 + 5.17865e14i) q^{50} +(-1.83157e12 - 9.19531e12i) q^{51} +(9.73676e14 - 9.73676e14i) q^{52} +(8.24074e12 - 8.24074e12i) q^{53} +(-8.50486e14 + 5.69246e14i) q^{54} +(-1.21290e14 - 6.75887e13i) q^{55} -1.68768e15i q^{56} +(-5.65581e14 + 8.46933e14i) q^{57} +(-4.65030e14 - 4.65030e14i) q^{58} +1.62877e15 q^{59} +(-1.60830e15 - 3.13798e15i) q^{60} -6.24051e14 q^{61} +(1.15494e15 + 1.15494e15i) q^{62} +(1.28834e15 + 5.32856e14i) q^{63} -7.89715e15i q^{64} +(-3.25666e15 + 9.25921e14i) q^{65} +(1.23550e15 - 2.46093e14i) q^{66} +(-4.35313e15 + 4.35313e15i) q^{67} +(-2.07249e14 + 2.07249e14i) q^{68} +(-5.92657e15 + 1.18049e15i) q^{69} +(-3.20102e15 + 5.74433e15i) q^{70} -2.04907e15i q^{71} +(1.86553e16 + 7.71581e15i) q^{72} +(5.28323e14 + 5.28323e14i) q^{73} -9.53411e14 q^{74} +(-3.01422e14 + 8.66479e15i) q^{75} +3.18360e16 q^{76} +(-1.21352e15 - 1.21352e15i) q^{77} +(1.70594e16 - 2.55457e16i) q^{78} +6.60046e15i q^{79} +(-2.65540e16 + 4.76519e16i) q^{80} +(-1.17802e16 + 1.18049e16i) q^{81} +(-2.34692e16 + 2.34692e16i) q^{82} +(-3.27553e15 + 3.27553e15i) q^{83} +(-8.51375e15 - 4.27429e16i) q^{84} +(6.93186e14 - 1.97084e14i) q^{85} -4.16477e16i q^{86} +(-8.91233e15 - 5.95165e15i) q^{87} +(-1.75719e16 - 1.75719e16i) q^{88} +1.15171e16 q^{89} +(-4.88621e16 - 6.16456e16i) q^{90} -4.18472e16 q^{91} +(1.33576e17 + 1.33576e17i) q^{92} +(2.21346e16 + 1.47814e16i) q^{93} +1.73088e17i q^{94} +(-6.83783e16 - 3.81037e16i) q^{95} +(-5.11978e16 - 2.57036e17i) q^{96} +(4.31385e16 - 4.31385e16i) q^{97} +(5.72394e16 - 5.72394e16i) q^{98} +(1.89620e16 - 7.86599e15i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q + 3600 q^{3} + 4768764 q^{6} + 23750080 q^{7} - 1042530380 q^{10} - 685700220 q^{12} - 4247345480 q^{13} - 14329253640 q^{15} - 182697348548 q^{16} - 18665173200 q^{18} - 458167880616 q^{21} - 307416701980 q^{22}+ \cdots - 16\!\cdots\!00 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −493.109 − 493.109i −1.36203 − 1.36203i −0.871323 − 0.490710i \(-0.836737\pi\)
−0.490710 − 0.871323i \(-0.663263\pi\)
\(3\) −9450.47 − 6311.01i −0.831615 − 0.555352i
\(4\) 355240.i 2.71027i
\(5\) 425179. − 762996.i 0.486773 − 0.873529i
\(6\) 1.54809e6 + 7.77212e6i 0.376280 + 1.88910i
\(7\) 7.63387e6 − 7.63387e6i 0.500508 − 0.500508i −0.411088 − 0.911596i \(-0.634851\pi\)
0.911596 + 0.411088i \(0.134851\pi\)
\(8\) 1.10539e8 − 1.10539e8i 2.32944 − 2.32944i
\(9\) 4.94824e7 + 1.19284e8i 0.383168 + 0.923678i
\(10\) −5.85899e8 + 1.66581e8i −1.85278 + 0.526774i
\(11\) − 1.58965e8i − 0.223596i −0.993731 − 0.111798i \(-0.964339\pi\)
0.993731 − 0.111798i \(-0.0356610\pi\)
\(12\) 2.24192e9 − 3.35719e9i 1.50515 − 2.25390i
\(13\) −2.74089e9 − 2.74089e9i −0.931910 − 0.931910i 0.0659155 − 0.997825i \(-0.479003\pi\)
−0.997825 + 0.0659155i \(0.979003\pi\)
\(14\) −7.52865e9 −1.36342
\(15\) −8.83341e9 + 4.52736e9i −0.889924 + 0.456110i
\(16\) −6.24537e10 −3.63528
\(17\) 5.83404e8 + 5.83404e8i 0.0202840 + 0.0202840i 0.717176 − 0.696892i \(-0.245434\pi\)
−0.696892 + 0.717176i \(0.745434\pi\)
\(18\) 3.44197e10 − 8.32202e10i 0.736192 − 1.77997i
\(19\) − 8.96182e10i − 1.21057i −0.796008 − 0.605286i \(-0.793059\pi\)
0.796008 − 0.605286i \(-0.206941\pi\)
\(20\) 2.71047e11 + 1.51041e11i 2.36750 + 1.31928i
\(21\) −1.20321e11 + 2.39662e10i −0.694189 + 0.138272i
\(22\) −7.83872e10 + 7.83872e10i −0.304546 + 0.304546i
\(23\) 3.76016e11 − 3.76016e11i 1.00120 − 1.00120i 0.00119894 − 0.999999i \(-0.499618\pi\)
0.999999 − 0.00119894i \(-0.000381635\pi\)
\(24\) −1.74226e12 + 3.47033e11i −3.23086 + 0.643540i
\(25\) −4.01386e11 − 6.48819e11i −0.526104 − 0.850420i
\(26\) 2.70312e12i 2.53858i
\(27\) 2.85170e11 − 1.43957e12i 0.194318 − 0.980939i
\(28\) 2.71186e12 + 2.71186e12i 1.35651 + 1.35651i
\(29\) 9.43058e11 0.350070 0.175035 − 0.984562i \(-0.443996\pi\)
0.175035 + 0.984562i \(0.443996\pi\)
\(30\) 6.58831e12 + 2.12335e12i 1.83334 + 0.590869i
\(31\) −2.34217e12 −0.493223 −0.246611 − 0.969114i \(-0.579317\pi\)
−0.246611 + 0.969114i \(0.579317\pi\)
\(32\) 1.63078e13 + 1.63078e13i 2.62193 + 2.62193i
\(33\) −1.00323e12 + 1.50230e12i −0.124175 + 0.185946i
\(34\) − 5.75363e11i − 0.0552550i
\(35\) −2.57885e12 − 9.07037e12i −0.193574 − 0.680842i
\(36\) −4.23745e13 + 1.75782e13i −2.50342 + 1.03849i
\(37\) 9.66735e11 − 9.66735e11i 0.0452473 − 0.0452473i −0.684121 − 0.729368i \(-0.739814\pi\)
0.729368 + 0.684121i \(0.239814\pi\)
\(38\) −4.41915e13 + 4.41915e13i −1.64884 + 1.64884i
\(39\) 8.60491e12 + 4.32005e13i 0.257453 + 1.29253i
\(40\) −3.73421e13 − 1.31340e14i −0.900925 − 3.16874i
\(41\) − 4.75944e13i − 0.930879i −0.885080 − 0.465439i \(-0.845896\pi\)
0.885080 − 0.465439i \(-0.154104\pi\)
\(42\) 7.11493e13 + 4.75134e13i 1.13384 + 0.757177i
\(43\) 4.22297e13 + 4.22297e13i 0.550980 + 0.550980i 0.926724 − 0.375743i \(-0.122613\pi\)
−0.375743 + 0.926724i \(0.622613\pi\)
\(44\) 5.64709e13 0.606006
\(45\) 1.12052e14 + 1.29621e13i 0.993376 + 0.114913i
\(46\) −3.70834e14 −2.72733
\(47\) −1.75506e14 − 1.75506e14i −1.07513 − 1.07513i −0.996938 − 0.0781927i \(-0.975085\pi\)
−0.0781927 − 0.996938i \(-0.524915\pi\)
\(48\) 5.90216e14 + 3.94146e14i 3.02316 + 2.01886i
\(49\) 1.16079e14i 0.498983i
\(50\) −1.22011e14 + 5.17865e14i −0.441729 + 1.87487i
\(51\) −1.83157e12 − 9.19531e12i −0.00560373 − 0.0281332i
\(52\) 9.73676e14 − 9.73676e14i 2.52572 − 2.52572i
\(53\) 8.24074e12 − 8.24074e12i 0.0181811 − 0.0181811i −0.697958 − 0.716139i \(-0.745908\pi\)
0.716139 + 0.697958i \(0.245908\pi\)
\(54\) −8.50486e14 + 5.69246e14i −1.60074 + 1.07140i
\(55\) −1.21290e14 − 6.75887e13i −0.195318 − 0.108841i
\(56\) − 1.68768e15i − 2.33181i
\(57\) −5.65581e14 + 8.46933e14i −0.672293 + 1.00673i
\(58\) −4.65030e14 − 4.65030e14i −0.476807 − 0.476807i
\(59\) 1.62877e15 1.44417 0.722085 − 0.691805i \(-0.243184\pi\)
0.722085 + 0.691805i \(0.243184\pi\)
\(60\) −1.60830e15 − 3.13798e15i −1.23618 − 2.41193i
\(61\) −6.24051e14 −0.416790 −0.208395 − 0.978045i \(-0.566824\pi\)
−0.208395 + 0.978045i \(0.566824\pi\)
\(62\) 1.15494e15 + 1.15494e15i 0.671786 + 0.671786i
\(63\) 1.28834e15 + 5.32856e14i 0.654088 + 0.270530i
\(64\) − 7.89715e15i − 3.50704i
\(65\) −3.25666e15 + 9.25921e14i −1.26768 + 0.360421i
\(66\) 1.23550e15 − 2.46093e14i 0.422395 − 0.0841348i
\(67\) −4.35313e15 + 4.35313e15i −1.30968 + 1.30968i −0.388037 + 0.921644i \(0.626847\pi\)
−0.921644 + 0.388037i \(0.873153\pi\)
\(68\) −2.07249e14 + 2.07249e14i −0.0549751 + 0.0549751i
\(69\) −5.92657e15 + 1.18049e15i −1.38863 + 0.276595i
\(70\) −3.20102e15 + 5.74433e15i −0.663675 + 1.19098i
\(71\) − 2.04907e15i − 0.376583i −0.982113 − 0.188292i \(-0.939705\pi\)
0.982113 − 0.188292i \(-0.0602950\pi\)
\(72\) 1.86553e16 + 7.71581e15i 3.04422 + 1.25909i
\(73\) 5.28323e14 + 5.28323e14i 0.0766752 + 0.0766752i 0.744404 − 0.667729i \(-0.232734\pi\)
−0.667729 + 0.744404i \(0.732734\pi\)
\(74\) −9.53411e14 −0.123257
\(75\) −3.01422e14 + 8.66479e15i −0.0347659 + 0.999395i
\(76\) 3.18360e16 3.28097
\(77\) −1.21352e15 − 1.21352e15i −0.111912 − 0.111912i
\(78\) 1.70594e16 − 2.55457e16i 1.40981 − 2.11113i
\(79\) 6.60046e15i 0.489490i 0.969588 + 0.244745i \(0.0787042\pi\)
−0.969588 + 0.244745i \(0.921296\pi\)
\(80\) −2.65540e16 + 4.76519e16i −1.76956 + 3.17552i
\(81\) −1.17802e16 + 1.18049e16i −0.706364 + 0.707849i
\(82\) −2.34692e16 + 2.34692e16i −1.26789 + 1.26789i
\(83\) −3.27553e15 + 3.27553e15i −0.159631 + 0.159631i −0.782403 − 0.622772i \(-0.786006\pi\)
0.622772 + 0.782403i \(0.286006\pi\)
\(84\) −8.51375e15 − 4.27429e16i −0.374755 − 1.88144i
\(85\) 6.93186e14 − 1.97084e14i 0.0275924 − 0.00784495i
\(86\) − 4.16477e16i − 1.50091i
\(87\) −8.91233e15 − 5.95165e15i −0.291124 − 0.194412i
\(88\) −1.75719e16 − 1.75719e16i −0.520854 − 0.520854i
\(89\) 1.15171e16 0.310119 0.155060 − 0.987905i \(-0.450443\pi\)
0.155060 + 0.987905i \(0.450443\pi\)
\(90\) −4.88621e16 − 6.16456e16i −1.19649 − 1.50953i
\(91\) −4.18472e16 −0.932857
\(92\) 1.33576e17 + 1.33576e17i 2.71352 + 2.71352i
\(93\) 2.21346e16 + 1.47814e16i 0.410172 + 0.273912i
\(94\) 1.73088e17i 2.92873i
\(95\) −6.83783e16 − 3.81037e16i −1.05747 − 0.589273i
\(96\) −5.11978e16 − 2.57036e17i −0.724345 − 3.63654i
\(97\) 4.31385e16 − 4.31385e16i 0.558863 − 0.558863i −0.370120 − 0.928984i \(-0.620684\pi\)
0.928984 + 0.370120i \(0.120684\pi\)
\(98\) 5.72394e16 − 5.72394e16i 0.679631 − 0.679631i
\(99\) 1.89620e16 − 7.86599e15i 0.206531 − 0.0856751i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 15.18.e.a.8.1 yes 64
3.2 odd 2 inner 15.18.e.a.8.32 yes 64
5.2 odd 4 inner 15.18.e.a.2.32 yes 64
15.2 even 4 inner 15.18.e.a.2.1 ✓ 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.e.a.2.1 ✓ 64 15.2 even 4 inner
15.18.e.a.2.32 yes 64 5.2 odd 4 inner
15.18.e.a.8.1 yes 64 1.1 even 1 trivial
15.18.e.a.8.32 yes 64 3.2 odd 2 inner