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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,11,Mod(13,40)] 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("40.13"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 40.i (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: \(25.4142901069\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(58\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.10
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.10

$q$-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+(-27.5829 - 16.2229i) q^{2} +(-281.304 + 281.304i) q^{3} +(497.633 + 894.951i) q^{4} +(-1103.56 + 2923.66i) q^{5} +(12322.8 - 3195.61i) q^{6} +(5272.38 - 5272.38i) q^{7} +(792.570 - 32758.4i) q^{8} -99215.2i q^{9} +(77869.7 - 62740.1i) q^{10} -20164.5i q^{11} +(-391740. - 111767. i) q^{12} +(-31587.3 + 31587.3i) q^{13} +(-230961. + 59894.0i) q^{14} +(-512002. - 1.13287e6i) q^{15} +(-553299. + 890714. i) q^{16} +(1.29754e6 - 1.29754e6i) q^{17} +(-1.60956e6 + 2.73664e6i) q^{18} +1.63462e6 q^{19} +(-3.16570e6 + 467279. i) q^{20} +2.96628e6i q^{21} +(-327127. + 556195. i) q^{22} +(-4.63569e6 - 4.63569e6i) q^{23} +(8.99213e6 + 9.43803e6i) q^{24} +(-7.32994e6 - 6.45286e6i) q^{25} +(1.38371e6 - 358831. i) q^{26} +(1.12989e7 + 1.12989e7i) q^{27} +(7.34223e6 + 2.09481e6i) q^{28} +1.40288e7 q^{29} +(-4.25602e6 + 3.95541e7i) q^{30} +5.01128e7 q^{31} +(2.97116e7 - 1.55924e7i) q^{32} +(5.67236e6 + 5.67236e6i) q^{33} +(-5.68399e7 + 1.47400e7i) q^{34} +(9.59626e6 + 2.12330e7i) q^{35} +(8.87927e7 - 4.93727e7i) q^{36} +(-3.78656e7 - 3.78656e7i) q^{37} +(-4.50875e7 - 2.65183e7i) q^{38} -1.77713e7i q^{39} +(9.48998e7 + 3.84680e7i) q^{40} +1.94964e8 q^{41} +(4.81218e7 - 8.18187e7i) q^{42} +(-1.45718e8 + 1.45718e8i) q^{43} +(1.80462e7 - 1.00345e7i) q^{44} +(2.90071e8 + 1.09490e8i) q^{45} +(5.26613e7 + 2.03070e8i) q^{46} +(-2.47227e8 + 2.47227e8i) q^{47} +(-9.49164e7 - 4.06207e8i) q^{48} +2.26879e8i q^{49} +(9.74969e7 + 2.96902e8i) q^{50} +7.30008e8i q^{51} +(-4.39880e7 - 1.25502e7i) q^{52} +(4.16498e8 - 4.16498e8i) q^{53} +(-1.28355e8 - 4.94958e8i) q^{54} +(5.89541e7 + 2.22527e7i) q^{55} +(-1.68536e8 - 1.76893e8i) q^{56} +(-4.59825e8 + 4.59825e8i) q^{57} +(-3.86956e8 - 2.27589e8i) q^{58} -9.99896e8 q^{59} +(7.59077e8 - 1.02197e9i) q^{60} -4.88290e8i q^{61} +(-1.38226e9 - 8.12977e8i) q^{62} +(-5.23100e8 - 5.23100e8i) q^{63} +(-1.07249e9 - 5.19267e7i) q^{64} +(-5.74921e7 - 1.27209e8i) q^{65} +(-6.44378e7 - 2.48482e8i) q^{66} +(7.31096e8 + 7.31096e8i) q^{67} +(1.80694e9 + 5.15537e8i) q^{68} +2.60808e9 q^{69} +(7.97689e7 - 7.41347e8i) q^{70} -1.48850e8 q^{71} +(-3.25013e9 - 7.86349e7i) q^{72} +(-1.70951e9 - 1.70951e9i) q^{73} +(4.30152e8 + 1.65873e9i) q^{74} +(3.87716e9 - 2.46728e8i) q^{75} +(8.13440e8 + 1.46290e9i) q^{76} +(-1.06315e8 - 1.06315e8i) q^{77} +(-2.88303e8 + 4.90184e8i) q^{78} +2.85001e9i q^{79} +(-1.99355e9 - 2.60061e9i) q^{80} -4.98308e8 q^{81} +(-5.37767e9 - 3.16289e9i) q^{82} +(3.69805e9 - 3.69805e9i) q^{83} +(-2.65468e9 + 1.47612e9i) q^{84} +(2.36166e9 + 5.22548e9i) q^{85} +(6.38329e9 - 1.65535e9i) q^{86} +(-3.94637e9 + 3.94637e9i) q^{87} +(-6.60557e8 - 1.59818e7i) q^{88} +4.29039e8i q^{89} +(-6.22477e9 - 7.72585e9i) q^{90} +3.33081e8i q^{91} +(1.84184e9 - 6.45558e9i) q^{92} +(-1.40969e10 + 1.40969e10i) q^{93} +(1.08300e10 - 2.80849e9i) q^{94} +(-1.80390e9 + 4.77907e9i) q^{95} +(-3.97180e9 + 1.27442e10i) q^{96} +(2.47448e9 - 2.47448e9i) q^{97} +(3.68065e9 - 6.25799e9i) q^{98} -2.00062e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 116 q - 2 q^{2} - 4864 q^{6} - 4 q^{7} + 69124 q^{8} + 256166 q^{10} + 502036 q^{12} - 4 q^{15} + 3502536 q^{16} - 905772 q^{17} - 5688470 q^{18} + 4385256 q^{20} + 9808012 q^{22} - 4 q^{23} - 1476988 q^{25}+ \cdots - 65612488734 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −27.5829 − 16.2229i −0.861966 − 0.506967i
\(3\) −281.304 + 281.304i −1.15763 + 1.15763i −0.172647 + 0.984984i \(0.555232\pi\)
−0.984984 + 0.172647i \(0.944768\pi\)
\(4\) 497.633 + 894.951i 0.485970 + 0.873976i
\(5\) −1103.56 + 2923.66i −0.353139 + 0.935571i
\(6\) 12322.8 − 3195.61i 1.58472 − 0.410958i
\(7\) 5272.38 − 5272.38i 0.313701 − 0.313701i −0.532640 − 0.846342i \(-0.678800\pi\)
0.846342 + 0.532640i \(0.178800\pi\)
\(8\) 792.570 − 32758.4i 0.0241873 − 0.999707i
\(9\) − 99215.2i − 1.68022i
\(10\) 77869.7 − 62740.1i 0.778697 − 0.627401i
\(11\) − 20164.5i − 0.125206i −0.998039 − 0.0626028i \(-0.980060\pi\)
0.998039 − 0.0626028i \(-0.0199401\pi\)
\(12\) −391740. − 111767.i −1.57431 − 0.449168i
\(13\) −31587.3 + 31587.3i −0.0850739 + 0.0850739i −0.748363 − 0.663289i \(-0.769160\pi\)
0.663289 + 0.748363i \(0.269160\pi\)
\(14\) −230961. + 59894.0i −0.429436 + 0.111364i
\(15\) −512002. − 1.13287e6i −0.674242 − 1.49185i
\(16\) −553299. + 890714.i −0.527667 + 0.849451i
\(17\) 1.29754e6 − 1.29754e6i 0.913854 − 0.913854i −0.0827194 − 0.996573i \(-0.526361\pi\)
0.996573 + 0.0827194i \(0.0263605\pi\)
\(18\) −1.60956e6 + 2.73664e6i −0.851814 + 1.44829i
\(19\) 1.63462e6 0.660159 0.330079 − 0.943953i \(-0.392925\pi\)
0.330079 + 0.943953i \(0.392925\pi\)
\(20\) −3.16570e6 + 467279.i −0.989281 + 0.146025i
\(21\) 2.96628e6i 0.726300i
\(22\) −327127. + 556195.i −0.0634751 + 0.107923i
\(23\) −4.63569e6 − 4.63569e6i −0.720236 − 0.720236i 0.248417 − 0.968653i \(-0.420090\pi\)
−0.968653 + 0.248417i \(0.920090\pi\)
\(24\) 8.99213e6 + 9.43803e6i 1.12929 + 1.18529i
\(25\) −7.32994e6 − 6.45286e6i −0.750586 − 0.660772i
\(26\) 1.38371e6 − 358831.i 0.116460 − 0.0302011i
\(27\) 1.12989e7 + 1.12989e7i 0.787441 + 0.787441i
\(28\) 7.34223e6 + 2.09481e6i 0.426616 + 0.121718i
\(29\) 1.40288e7 0.683961 0.341980 − 0.939707i \(-0.388902\pi\)
0.341980 + 0.939707i \(0.388902\pi\)
\(30\) −4.25602e6 + 3.95541e7i −0.175145 + 1.62774i
\(31\) 5.01128e7 1.75041 0.875206 − 0.483751i \(-0.160726\pi\)
0.875206 + 0.483751i \(0.160726\pi\)
\(32\) 2.97116e7 − 1.55924e7i 0.885474 − 0.464688i
\(33\) 5.67236e6 + 5.67236e6i 0.144942 + 0.144942i
\(34\) −5.68399e7 + 1.47400e7i −1.25100 + 0.324417i
\(35\) 9.59626e6 + 2.12330e7i 0.182710 + 0.404270i
\(36\) 8.87927e7 − 4.93727e7i 1.46847 − 0.816535i
\(37\) −3.78656e7 − 3.78656e7i −0.546055 − 0.546055i 0.379243 − 0.925297i \(-0.376185\pi\)
−0.925297 + 0.379243i \(0.876185\pi\)
\(38\) −4.50875e7 − 2.65183e7i −0.569034 − 0.334678i
\(39\) − 1.77713e7i − 0.196968i
\(40\) 9.48998e7 + 3.84680e7i 0.926756 + 0.375664i
\(41\) 1.94964e8 1.68281 0.841405 − 0.540405i \(-0.181729\pi\)
0.841405 + 0.540405i \(0.181729\pi\)
\(42\) 4.81218e7 − 8.18187e7i 0.368210 − 0.626046i
\(43\) −1.45718e8 + 1.45718e8i −0.991220 + 0.991220i −0.999962 − 0.00874182i \(-0.997217\pi\)
0.00874182 + 0.999962i \(0.497217\pi\)
\(44\) 1.80462e7 − 1.00345e7i 0.109427 − 0.0608461i
\(45\) 2.90071e8 + 1.09490e8i 1.57196 + 0.593350i
\(46\) 5.26613e7 + 2.03070e8i 0.255683 + 0.985955i
\(47\) −2.47227e8 + 2.47227e8i −1.07797 + 1.07797i −0.0812778 + 0.996691i \(0.525900\pi\)
−0.996691 + 0.0812778i \(0.974100\pi\)
\(48\) −9.49164e7 − 4.06207e8i −0.372507 − 1.59419i
\(49\) 2.26879e8i 0.803183i
\(50\) 9.74969e7 + 2.96902e8i 0.311990 + 0.950085i
\(51\) 7.30008e8i 2.11581i
\(52\) −4.39880e7 − 1.25502e7i −0.115696 − 0.0330092i
\(53\) 4.16498e8 − 4.16498e8i 0.995942 − 0.995942i −0.00405022 − 0.999992i \(-0.501289\pi\)
0.999992 + 0.00405022i \(0.00128923\pi\)
\(54\) −1.28355e8 − 4.94958e8i −0.279541 − 1.07795i
\(55\) 5.89541e7 + 2.22527e7i 0.117139 + 0.0442149i
\(56\) −1.68536e8 − 1.76893e8i −0.306022 − 0.321197i
\(57\) −4.59825e8 + 4.59825e8i −0.764220 + 0.764220i
\(58\) −3.86956e8 − 2.27589e8i −0.589551 − 0.346745i
\(59\) −9.99896e8 −1.39860 −0.699302 − 0.714827i \(-0.746506\pi\)
−0.699302 + 0.714827i \(0.746506\pi\)
\(60\) 7.59077e8 − 1.02197e9i 0.976179 − 1.31426i
\(61\) − 4.88290e8i − 0.578135i −0.957309 − 0.289067i \(-0.906655\pi\)
0.957309 − 0.289067i \(-0.0933452\pi\)
\(62\) −1.38226e9 − 8.12977e8i −1.50879 − 0.887400i
\(63\) −5.23100e8 − 5.23100e8i −0.527086 − 0.527086i
\(64\) −1.07249e9 − 5.19267e7i −0.998830 − 0.0483605i
\(65\) −5.74921e7 − 1.27209e8i −0.0495498 − 0.109636i
\(66\) −6.44378e7 − 2.48482e8i −0.0514542 − 0.198416i
\(67\) 7.31096e8 + 7.31096e8i 0.541502 + 0.541502i 0.923969 − 0.382467i \(-0.124925\pi\)
−0.382467 + 0.923969i \(0.624925\pi\)
\(68\) 1.80694e9 + 5.15537e8i 1.24279 + 0.354581i
\(69\) 2.60808e9 1.66753
\(70\) 7.97689e7 − 7.41347e8i 0.0474617 − 0.441094i
\(71\) −1.48850e8 −0.0825005 −0.0412502 − 0.999149i \(-0.513134\pi\)
−0.0412502 + 0.999149i \(0.513134\pi\)
\(72\) −3.25013e9 − 7.86349e7i −1.67973 − 0.0406399i
\(73\) −1.70951e9 − 1.70951e9i −0.824625 − 0.824625i 0.162143 − 0.986767i \(-0.448160\pi\)
−0.986767 + 0.162143i \(0.948160\pi\)
\(74\) 4.30152e8 + 1.65873e9i 0.193849 + 0.747512i
\(75\) 3.87716e9 − 2.46728e8i 1.63383 − 0.103971i
\(76\) 8.13440e8 + 1.46290e9i 0.320817 + 0.576963i
\(77\) −1.06315e8 − 1.06315e8i −0.0392771 − 0.0392771i
\(78\) −2.88303e8 + 4.90184e8i −0.0998563 + 0.169780i
\(79\) 2.85001e9i 0.926214i 0.886302 + 0.463107i \(0.153265\pi\)
−0.886302 + 0.463107i \(0.846735\pi\)
\(80\) −1.99355e9 − 2.60061e9i −0.608382 − 0.793644i
\(81\) −4.98308e8 −0.142913
\(82\) −5.37767e9 − 3.16289e9i −1.45052 − 0.853129i
\(83\) 3.69805e9 − 3.69805e9i 0.938819 − 0.938819i −0.0594142 − 0.998233i \(-0.518923\pi\)
0.998233 + 0.0594142i \(0.0189233\pi\)
\(84\) −2.65468e9 + 1.47612e9i −0.634769 + 0.352960i
\(85\) 2.36166e9 + 5.22548e9i 0.532258 + 1.17769i
\(86\) 6.38329e9 − 1.65535e9i 1.35691 − 0.351882i
\(87\) −3.94637e9 + 3.94637e9i −0.791774 + 0.791774i
\(88\) −6.60557e8 − 1.59818e7i −0.125169 − 0.00302839i
\(89\) 4.29039e8i 0.0768327i 0.999262 + 0.0384164i \(0.0122313\pi\)
−0.999262 + 0.0384164i \(0.987769\pi\)
\(90\) −6.22477e9 − 7.72585e9i −1.05417 − 1.30838i
\(91\) 3.33081e8i 0.0533755i
\(92\) 1.84184e9 − 6.45558e9i 0.279456 − 0.979482i
\(93\) −1.40969e10 + 1.40969e10i −2.02633 + 2.02633i
\(94\) 1.08300e10 − 2.80849e9i 1.47567 − 0.382678i
\(95\) −1.80390e9 + 4.77907e9i −0.233127 + 0.617625i
\(96\) −3.97180e9 + 1.27442e10i −0.487115 + 1.56299i
\(97\) 2.47448e9 − 2.47448e9i 0.288154 − 0.288154i −0.548196 − 0.836350i \(-0.684685\pi\)
0.836350 + 0.548196i \(0.184685\pi\)
\(98\) 3.68065e9 − 6.25799e9i 0.407187 − 0.692316i
\(99\) −2.00062e9 −0.210373
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 40.11.i.a.13.10 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.39 yes 116
8.5 even 2 inner 40.11.i.a.13.39 yes 116
40.37 odd 4 inner 40.11.i.a.37.10 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.10 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.39 yes 116 8.5 even 2 inner
40.11.i.a.37.10 yes 116 40.37 odd 4 inner
40.11.i.a.37.39 yes 116 5.2 odd 4 inner