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Properties

Label 15.18.e.a.8.3
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 8.3
Character \(\chi\) \(=\) 15.8
Dual form 15.18.e.a.2.3

$q$-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+(-431.672 - 431.672i) q^{2} +(-6125.41 + 9571.81i) q^{3} +241609. i q^{4} +(298404. + 820911. i) q^{5} +(6.77605e6 - 1.48771e6i) q^{6} +(9.48710e6 - 9.48710e6i) q^{7} +(4.77157e7 - 4.77157e7i) q^{8} +(-5.40989e7 - 1.17262e8i) q^{9} +(2.25552e8 - 4.83176e8i) q^{10} +3.89280e8i q^{11} +(-2.31263e9 - 1.47995e9i) q^{12} +(3.08766e9 + 3.08766e9i) q^{13} -8.19063e9 q^{14} +(-9.68545e9 - 2.17215e9i) q^{15} -9.52685e9 q^{16} +(-1.31519e10 - 1.31519e10i) q^{17} +(-2.72660e10 + 7.39719e10i) q^{18} -1.06970e11i q^{19} +(-1.98339e11 + 7.20970e10i) q^{20} +(3.26963e10 + 1.48921e11i) q^{21} +(1.68041e11 - 1.68041e11i) q^{22} +(3.27975e10 - 3.27975e10i) q^{23} +(1.64447e11 + 7.49004e11i) q^{24} +(-5.84850e11 + 4.89926e11i) q^{25} -2.66571e12i q^{26} +(1.45379e12 + 2.00457e11i) q^{27} +(2.29217e12 + 2.29217e12i) q^{28} +2.84559e12 q^{29} +(3.24328e12 + 5.11859e12i) q^{30} +7.19499e12 q^{31} +(-2.14172e12 - 2.14172e12i) q^{32} +(-3.72611e12 - 2.38450e12i) q^{33} +1.13546e13i q^{34} +(1.06190e13 + 4.95708e12i) q^{35} +(2.83317e13 - 1.30708e13i) q^{36} +(-2.24710e13 + 2.24710e13i) q^{37} +(-4.61757e13 + 4.61757e13i) q^{38} +(-4.84676e13 + 1.06413e13i) q^{39} +(5.34089e13 + 2.49318e13i) q^{40} -3.01524e13i q^{41} +(5.01709e13 - 7.83991e13i) q^{42} +(1.73987e13 + 1.73987e13i) q^{43} -9.40535e13 q^{44} +(8.01188e13 - 7.94019e13i) q^{45} -2.83155e13 q^{46} +(1.43307e14 + 1.43307e14i) q^{47} +(5.83559e13 - 9.11892e13i) q^{48} +5.26204e13i q^{49} +(4.63950e14 + 4.09761e13i) q^{50} +(2.06448e14 - 4.53267e13i) q^{51} +(-7.46005e14 + 7.46005e14i) q^{52} +(-6.60882e12 + 6.60882e12i) q^{53} +(-5.41029e14 - 7.14092e14i) q^{54} +(-3.19564e14 + 1.16163e14i) q^{55} -9.05367e14i q^{56} +(1.02389e15 + 6.55232e14i) q^{57} +(-1.22836e15 - 1.22836e15i) q^{58} +3.73189e14 q^{59} +(5.24812e14 - 2.34009e15i) q^{60} +2.96117e14 q^{61} +(-3.10587e15 - 3.10587e15i) q^{62} +(-1.62572e15 - 5.99240e14i) q^{63} +3.09774e15i q^{64} +(-1.61332e15 + 3.45606e15i) q^{65} +(5.79137e14 + 2.63778e15i) q^{66} +(-3.31730e15 + 3.31730e15i) q^{67} +(3.17762e15 - 3.17762e15i) q^{68} +(1.13033e14 + 5.14829e14i) q^{69} +(-2.44411e15 - 6.72377e15i) q^{70} +8.74855e15i q^{71} +(-8.17663e15 - 3.01390e15i) q^{72} +(4.49203e15 + 4.49203e15i) q^{73} +1.94002e16 q^{74} +(-1.10703e15 - 8.59907e15i) q^{75} +2.58448e16 q^{76} +(3.69314e15 + 3.69314e15i) q^{77} +(2.55156e16 + 1.63285e16i) q^{78} -1.83639e16i q^{79} +(-2.84285e15 - 7.82070e15i) q^{80} +(-1.08238e16 + 1.26875e16i) q^{81} +(-1.30159e16 + 1.30159e16i) q^{82} +(-8.95523e15 + 8.95523e15i) q^{83} +(-3.59807e16 + 7.89973e15i) q^{84} +(6.87196e15 - 1.47211e16i) q^{85} -1.50210e16i q^{86} +(-1.74304e16 + 2.72375e16i) q^{87} +(1.85748e16 + 1.85748e16i) q^{88} -2.51697e16 q^{89} +(-6.88606e16 - 3.09442e14i) q^{90} +5.85858e16 q^{91} +(7.92417e15 + 7.92417e15i) q^{92} +(-4.40723e16 + 6.88691e16i) q^{93} -1.23723e17i q^{94} +(8.78124e16 - 3.19201e16i) q^{95} +(3.36190e16 - 7.38121e15i) q^{96} +(-8.44607e16 + 8.44607e16i) q^{97} +(2.27147e16 - 2.27147e16i) q^{98} +(4.56479e16 - 2.10596e16i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −431.672 − 431.672i −1.19234 − 1.19234i −0.976410 − 0.215926i \(-0.930723\pi\)
−0.215926 − 0.976410i \(-0.569277\pi\)
\(3\) −6125.41 + 9571.81i −0.539020 + 0.842293i
\(4\) 241609.i 1.84333i
\(5\) 298404. + 820911.i 0.341632 + 0.939834i
\(6\) 6.77605e6 − 1.48771e6i 1.64699 − 0.361604i
\(7\) 9.48710e6 − 9.48710e6i 0.622014 − 0.622014i −0.324032 − 0.946046i \(-0.605039\pi\)
0.946046 + 0.324032i \(0.105039\pi\)
\(8\) 4.77157e7 − 4.77157e7i 1.00553 − 1.00553i
\(9\) −5.40989e7 − 1.17262e8i −0.418916 − 0.908025i
\(10\) 2.25552e8 − 4.83176e8i 0.713257 − 1.52794i
\(11\) 3.89280e8i 0.547551i 0.961794 + 0.273775i \(0.0882725\pi\)
−0.961794 + 0.273775i \(0.911728\pi\)
\(12\) −2.31263e9 − 1.47995e9i −1.55262 − 0.993591i
\(13\) 3.08766e9 + 3.08766e9i 1.04981 + 1.04981i 0.998693 + 0.0511164i \(0.0162780\pi\)
0.0511164 + 0.998693i \(0.483722\pi\)
\(14\) −8.19063e9 −1.48330
\(15\) −9.68545e9 − 2.17215e9i −0.975762 − 0.218834i
\(16\) −9.52685e9 −0.554536
\(17\) −1.31519e10 − 1.31519e10i −0.457270 − 0.457270i 0.440488 − 0.897758i \(-0.354805\pi\)
−0.897758 + 0.440488i \(0.854805\pi\)
\(18\) −2.72660e10 + 7.39719e10i −0.583182 + 1.58216i
\(19\) − 1.06970e11i − 1.44496i −0.691394 − 0.722478i \(-0.743003\pi\)
0.691394 − 0.722478i \(-0.256997\pi\)
\(20\) −1.98339e11 + 7.20970e10i −1.73242 + 0.629741i
\(21\) 3.26963e10 + 1.48921e11i 0.188641 + 0.859196i
\(22\) 1.68041e11 − 1.68041e11i 0.652864 − 0.652864i
\(23\) 3.27975e10 − 3.27975e10i 0.0873281 − 0.0873281i −0.662093 − 0.749421i \(-0.730332\pi\)
0.749421 + 0.662093i \(0.230332\pi\)
\(24\) 1.64447e11 + 7.49004e11i 0.304952 + 1.38896i
\(25\) −5.84850e11 + 4.89926e11i −0.766575 + 0.642155i
\(26\) − 2.66571e12i − 2.50345i
\(27\) 1.45379e12 + 2.00457e11i 0.990627 + 0.136593i
\(28\) 2.29217e12 + 2.29217e12i 1.14658 + 1.14658i
\(29\) 2.84559e12 1.05631 0.528153 − 0.849149i \(-0.322885\pi\)
0.528153 + 0.849149i \(0.322885\pi\)
\(30\) 3.24328e12 + 5.11859e12i 0.902513 + 1.42436i
\(31\) 7.19499e12 1.51515 0.757575 − 0.652748i \(-0.226384\pi\)
0.757575 + 0.652748i \(0.226384\pi\)
\(32\) −2.14172e12 − 2.14172e12i −0.344340 − 0.344340i
\(33\) −3.72611e12 − 2.38450e12i −0.461198 − 0.295140i
\(34\) 1.13546e13i 1.09044i
\(35\) 1.06190e13 + 4.95708e12i 0.797090 + 0.372090i
\(36\) 2.83317e13 − 1.30708e13i 1.67379 − 0.772200i
\(37\) −2.24710e13 + 2.24710e13i −1.05174 + 1.05174i −0.0531541 + 0.998586i \(0.516927\pi\)
−0.998586 + 0.0531541i \(0.983073\pi\)
\(38\) −4.61757e13 + 4.61757e13i −1.72287 + 1.72287i
\(39\) −4.84676e13 + 1.06413e13i −1.45011 + 0.318380i
\(40\) 5.34089e13 + 2.49318e13i 1.28856 + 0.601511i
\(41\) − 3.01524e13i − 0.589738i −0.955538 − 0.294869i \(-0.904724\pi\)
0.955538 − 0.294869i \(-0.0952760\pi\)
\(42\) 5.01709e13 − 7.83991e13i 0.799527 − 1.24937i
\(43\) 1.73987e13 + 1.73987e13i 0.227004 + 0.227004i 0.811440 − 0.584436i \(-0.198684\pi\)
−0.584436 + 0.811440i \(0.698684\pi\)
\(44\) −9.40535e13 −1.00932
\(45\) 8.01188e13 − 7.94019e13i 0.710277 − 0.703922i
\(46\) −2.83155e13 −0.208249
\(47\) 1.43307e14 + 1.43307e14i 0.877882 + 0.877882i 0.993315 − 0.115434i \(-0.0368257\pi\)
−0.115434 + 0.993315i \(0.536826\pi\)
\(48\) 5.83559e13 − 9.11892e13i 0.298906 − 0.467082i
\(49\) 5.26204e13i 0.226197i
\(50\) 4.63950e14 + 4.09761e13i 1.67968 + 0.148349i
\(51\) 2.06448e14 − 4.53267e13i 0.631633 − 0.138678i
\(52\) −7.46005e14 + 7.46005e14i −1.93514 + 1.93514i
\(53\) −6.60882e12 + 6.60882e12i −0.0145807 + 0.0145807i −0.714360 − 0.699779i \(-0.753282\pi\)
0.699779 + 0.714360i \(0.253282\pi\)
\(54\) −5.41029e14 − 7.14092e14i −1.01830 − 1.34403i
\(55\) −3.19564e14 + 1.16163e14i −0.514607 + 0.187061i
\(56\) − 9.05367e14i − 1.25091i
\(57\) 1.02389e15 + 6.55232e14i 1.21708 + 0.778859i
\(58\) −1.22836e15 − 1.22836e15i −1.25947 − 1.25947i
\(59\) 3.73189e14 0.330892 0.165446 − 0.986219i \(-0.447094\pi\)
0.165446 + 0.986219i \(0.447094\pi\)
\(60\) 5.24812e14 − 2.34009e15i 0.403383 − 1.79865i
\(61\) 2.96117e14 0.197770 0.0988849 − 0.995099i \(-0.468472\pi\)
0.0988849 + 0.995099i \(0.468472\pi\)
\(62\) −3.10587e15 − 3.10587e15i −1.80657 − 1.80657i
\(63\) −1.62572e15 − 5.99240e14i −0.825376 − 0.304233i
\(64\) 3.09774e15i 1.37567i
\(65\) −1.61332e15 + 3.45606e15i −0.627997 + 1.34529i
\(66\) 5.79137e14 + 2.63778e15i 0.197997 + 0.901810i
\(67\) −3.31730e15 + 3.31730e15i −0.998042 + 0.998042i −0.999998 − 0.00195629i \(-0.999377\pi\)
0.00195629 + 0.999998i \(0.499377\pi\)
\(68\) 3.17762e15 − 3.17762e15i 0.842900 − 0.842900i
\(69\) 1.13033e14 + 5.14829e14i 0.0264843 + 0.120627i
\(70\) −2.44411e15 − 6.72377e15i −0.506743 − 1.39405i
\(71\) 8.74855e15i 1.60783i 0.594744 + 0.803915i \(0.297254\pi\)
−0.594744 + 0.803915i \(0.702746\pi\)
\(72\) −8.17663e15 − 3.01390e15i −1.33428 − 0.491815i
\(73\) 4.49203e15 + 4.49203e15i 0.651926 + 0.651926i 0.953456 − 0.301531i \(-0.0974976\pi\)
−0.301531 + 0.953456i \(0.597498\pi\)
\(74\) 1.94002e16 2.50806
\(75\) −1.10703e15 − 8.59907e15i −0.127684 − 0.991815i
\(76\) 2.58448e16 2.66353
\(77\) 3.69314e15 + 3.69314e15i 0.340584 + 0.340584i
\(78\) 2.55156e16 + 1.63285e16i 2.10864 + 1.34941i
\(79\) − 1.83639e16i − 1.36187i −0.732344 − 0.680935i \(-0.761574\pi\)
0.732344 − 0.680935i \(-0.238426\pi\)
\(80\) −2.84285e15 − 7.82070e15i −0.189447 − 0.521171i
\(81\) −1.08238e16 + 1.26875e16i −0.649019 + 0.760772i
\(82\) −1.30159e16 + 1.30159e16i −0.703166 + 0.703166i
\(83\) −8.95523e15 + 8.95523e15i −0.436428 + 0.436428i −0.890808 − 0.454380i \(-0.849861\pi\)
0.454380 + 0.890808i \(0.349861\pi\)
\(84\) −3.59807e16 + 7.89973e15i −1.58378 + 0.347727i
\(85\) 6.87196e15 − 1.47211e16i 0.273539 − 0.585976i
\(86\) − 1.50210e16i − 0.541331i
\(87\) −1.74304e16 + 2.72375e16i −0.569370 + 0.889720i
\(88\) 1.85748e16 + 1.85748e16i 0.550580 + 0.550580i
\(89\) −2.51697e16 −0.677740 −0.338870 − 0.940833i \(-0.610045\pi\)
−0.338870 + 0.940833i \(0.610045\pi\)
\(90\) −6.88606e16 − 3.09442e14i −1.68620 − 0.00757736i
\(91\) 5.85858e16 1.30599
\(92\) 7.92417e15 + 7.92417e15i 0.160974 + 0.160974i
\(93\) −4.40723e16 + 6.88691e16i −0.816696 + 1.27620i
\(94\) − 1.23723e17i − 2.09346i
\(95\) 8.78124e16 − 3.19201e16i 1.35802 − 0.493644i
\(96\) 3.36190e16 − 7.38121e15i 0.475641 − 0.104429i
\(97\) −8.44607e16 + 8.44607e16i −1.09420 + 1.09420i −0.0991196 + 0.995076i \(0.531603\pi\)
−0.995076 + 0.0991196i \(0.968397\pi\)
\(98\) 2.27147e16 − 2.27147e16i 0.269703 − 0.269703i
\(99\) 4.56479e16 − 2.10596e16i 0.497190 − 0.229378i
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.8.3 yes 64
3.2 odd 2 inner 15.18.e.a.8.30 yes 64
5.2 odd 4 inner 15.18.e.a.2.30 yes 64
15.2 even 4 inner 15.18.e.a.2.3 ✓ 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.e.a.2.3 ✓ 64 15.2 even 4 inner
15.18.e.a.2.30 yes 64 5.2 odd 4 inner
15.18.e.a.8.3 yes 64 1.1 even 1 trivial
15.18.e.a.8.30 yes 64 3.2 odd 2 inner