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Properties

Label 15.18.e.a.2.1
Level $15$
Weight $18$
Character 15.2
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 2.1
Character \(\chi\) \(=\) 15.2
Dual form 15.18.e.a.8.1

$q$-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 \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

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

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 −0.490710 + 0.871323i \(0.663263\pi\)
−0.871323 + 0.490710i \(0.836737\pi\)
\(3\) −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 0.911596 − 0.411088i \(-0.134851\pi\)
−0.411088 + 0.911596i \(0.634851\pi\)
\(8\) 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 \(0.0356610\pi\)
−0.993731 + 0.111798i \(0.964339\pi\)
\(12\) 2.24192e9 + 3.35719e9i 1.50515 + 2.25390i
\(13\) −2.74089e9 + 2.74089e9i −0.931910 + 0.931910i −0.997825 − 0.0659155i \(-0.979003\pi\)
0.0659155 + 0.997825i \(0.479003\pi\)
\(14\) −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 −0.696892 − 0.717176i \(-0.745434\pi\)
0.717176 + 0.696892i \(0.245434\pi\)
\(18\) 3.44197e10 + 8.32202e10i 0.736192 + 1.77997i
\(19\) 8.96182e10i 1.21057i 0.796008 + 0.605286i \(0.206941\pi\)
−0.796008 + 0.605286i \(0.793059\pi\)
\(20\) 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 0.999999 + 0.00119894i \(0.000381635\pi\)
0.00119894 + 0.999999i \(0.499618\pi\)
\(24\) −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 0.729368 − 0.684121i \(-0.239814\pi\)
−0.684121 + 0.729368i \(0.739814\pi\)
\(38\) −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 \(0.154104\pi\)
−0.885080 + 0.465439i \(0.845896\pi\)
\(42\) 7.11493e13 − 4.75134e13i 1.13384 − 0.757177i
\(43\) 4.22297e13 − 4.22297e13i 0.550980 − 0.550980i −0.375743 − 0.926724i \(-0.622613\pi\)
0.926724 + 0.375743i \(0.122613\pi\)
\(44\) 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 −0.0781927 + 0.996938i \(0.524915\pi\)
−0.996938 + 0.0781927i \(0.975085\pi\)
\(48\) 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 0.716139 − 0.697958i \(-0.245908\pi\)
−0.697958 + 0.716139i \(0.745908\pi\)
\(54\) −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 −0.921644 − 0.388037i \(-0.873153\pi\)
−0.388037 − 0.921644i \(-0.626847\pi\)
\(68\) −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 \(0.0602950\pi\)
−0.982113 + 0.188292i \(0.939705\pi\)
\(72\) 1.86553e16 − 7.71581e15i 3.04422 − 1.25909i
\(73\) 5.28323e14 − 5.28323e14i 0.0766752 − 0.0766752i −0.667729 − 0.744404i \(-0.732734\pi\)
0.744404 + 0.667729i \(0.232734\pi\)
\(74\) −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 \(-0.921296\pi\)
0.969588 − 0.244745i \(-0.0787042\pi\)
\(80\) −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 0.622772 − 0.782403i \(-0.286006\pi\)
−0.782403 + 0.622772i \(0.786006\pi\)
\(84\) −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 0.928984 − 0.370120i \(-0.120684\pi\)
−0.370120 + 0.928984i \(0.620684\pi\)
\(98\) 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.2.1 ✓ 64
3.2 odd 2 inner 15.18.e.a.2.32 yes 64
5.3 odd 4 inner 15.18.e.a.8.32 yes 64
15.8 even 4 inner 15.18.e.a.8.1 yes 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.e.a.2.1 ✓ 64 1.1 even 1 trivial
15.18.e.a.2.32 yes 64 3.2 odd 2 inner
15.18.e.a.8.1 yes 64 15.8 even 4 inner
15.18.e.a.8.32 yes 64 5.3 odd 4 inner