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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 13.2
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.2

$q$-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+(-31.7327 + 4.12737i) q^{2} +(-33.5937 + 33.5937i) q^{3} +(989.930 - 261.945i) q^{4} +(-939.110 + 2980.55i) q^{5} +(927.364 - 1204.67i) q^{6} +(5089.54 - 5089.54i) q^{7} +(-30332.0 + 12398.0i) q^{8} +56791.9i q^{9} +(17498.7 - 98457.1i) q^{10} +120768. i q^{11} +(-24455.7 + 42055.0i) q^{12} +(378933. - 378933. i) q^{13} +(-140498. + 182511. i) q^{14} +(-68579.5 - 131676. i) q^{15} +(911345. - 518615. i) q^{16} +(-421025. + 421025. i) q^{17} +(-234401. - 1.80216e6i) q^{18} +3.72211e6 q^{19} +(-148911. + 3.19653e6i) q^{20} +341952. i q^{21} +(-498453. - 3.83229e6i) q^{22} +(6.50259e6 + 6.50259e6i) q^{23} +(602467. - 1.43546e6i) q^{24} +(-8.00177e6 - 5.59813e6i) q^{25} +(-1.04606e7 + 1.35886e7i) q^{26} +(-3.89152e6 - 3.89152e6i) q^{27} +(3.70510e6 - 6.37147e6i) q^{28} -2.73192e7 q^{29} +(2.71969e6 + 3.89538e6i) q^{30} -5.47309e7 q^{31} +(-2.67789e7 + 2.02185e7i) q^{32} +(-4.05703e6 - 4.05703e6i) q^{33} +(1.16225e7 - 1.50980e7i) q^{34} +(1.03900e7 + 1.99493e7i) q^{35} +(1.48764e7 + 5.62200e7i) q^{36} +(7.83954e7 + 7.83954e7i) q^{37} +(-1.18113e8 + 1.53625e7i) q^{38} +2.54595e7i q^{39} +(-8.46793e6 - 1.02049e8i) q^{40} -1.38841e8 q^{41} +(-1.41136e6 - 1.08511e7i) q^{42} +(-4.60496e7 + 4.60496e7i) q^{43} +(3.16345e7 + 1.19552e8i) q^{44} +(-1.69271e8 - 5.33339e7i) q^{45} +(-2.33183e8 - 1.79506e8i) q^{46} +(-1.14448e8 + 1.14448e8i) q^{47} +(-1.31933e7 + 4.80376e7i) q^{48} +2.30668e8i q^{49} +(2.77023e8 + 1.44618e8i) q^{50} -2.82875e7i q^{51} +(2.75857e8 - 4.74377e8i) q^{52} +(3.43507e8 - 3.43507e8i) q^{53} +(1.39550e8 + 1.07427e8i) q^{54} +(-3.59955e8 - 1.13414e8i) q^{55} +(-9.12756e7 + 2.17476e8i) q^{56} +(-1.25039e8 + 1.25039e8i) q^{57} +(8.66912e8 - 1.12756e8i) q^{58} -4.06566e8 q^{59} +(-1.02381e8 - 1.12386e8i) q^{60} -1.18617e8i q^{61} +(1.73676e9 - 2.25895e8i) q^{62} +(2.89045e8 + 2.89045e8i) q^{63} +(7.66319e8 - 7.52115e8i) q^{64} +(7.73570e8 + 1.48529e9i) q^{65} +(1.45485e8 + 1.11996e8i) q^{66} +(-6.69240e8 - 6.69240e8i) q^{67} +(-3.06500e8 + 5.27071e8i) q^{68} -4.36891e8 q^{69} +(-4.12041e8 - 5.90161e8i) q^{70} -7.06085e8 q^{71} +(-7.04109e8 - 1.72261e9i) q^{72} +(-8.44893e8 - 8.44893e8i) q^{73} +(-2.81127e9 - 2.16413e9i) q^{74} +(4.56870e8 - 8.07469e7i) q^{75} +(3.68462e9 - 9.74989e8i) q^{76} +(6.14652e8 + 6.14652e8i) q^{77} +(-1.05081e8 - 8.07898e8i) q^{78} +5.80430e9i q^{79} +(6.89906e8 + 3.20335e9i) q^{80} -3.09205e9 q^{81} +(4.40580e9 - 5.73048e8i) q^{82} +(-9.30327e8 + 9.30327e8i) q^{83} +(8.95728e7 + 3.38509e8i) q^{84} +(-8.59499e8 - 1.65028e9i) q^{85} +(1.27122e9 - 1.65134e9i) q^{86} +(9.17751e8 - 9.17751e8i) q^{87} +(-1.49728e9 - 3.66313e9i) q^{88} -2.22155e9i q^{89} +(5.59157e9 + 9.93782e8i) q^{90} -3.85719e9i q^{91} +(8.14042e9 + 4.73378e9i) q^{92} +(1.83861e9 - 1.83861e9i) q^{93} +(3.15938e9 - 4.10412e9i) q^{94} +(-3.49547e9 + 1.10939e10i) q^{95} +(2.20389e8 - 1.57882e9i) q^{96} +(-3.03256e9 + 3.03256e9i) q^{97} +(-9.52054e8 - 7.31973e9i) q^{98} -6.85863e9 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.7327 + 4.12737i −0.991647 + 0.128980i
\(3\) −33.5937 + 33.5937i −0.138245 + 0.138245i −0.772843 − 0.634597i \(-0.781166\pi\)
0.634597 + 0.772843i \(0.281166\pi\)
\(4\) 989.930 − 261.945i 0.966728 − 0.255806i
\(5\) −939.110 + 2980.55i −0.300515 + 0.953777i
\(6\) 927.364 − 1204.67i 0.119260 − 0.154922i
\(7\) 5089.54 − 5089.54i 0.302823 − 0.302823i −0.539295 − 0.842117i \(-0.681309\pi\)
0.842117 + 0.539295i \(0.181309\pi\)
\(8\) −30332.0 + 12398.0i −0.925659 + 0.378358i
\(9\) 56791.9i 0.961776i
\(10\) 17498.7 − 98457.1i 0.174987 − 0.984571i
\(11\) 120768.i 0.749873i 0.927051 + 0.374936i \(0.122335\pi\)
−0.927051 + 0.374936i \(0.877665\pi\)
\(12\) −24455.7 + 42055.0i −0.0982818 + 0.169010i
\(13\) 378933. − 378933.i 1.02058 − 1.02058i 0.0207929 − 0.999784i \(-0.493381\pi\)
0.999784 − 0.0207929i \(-0.00661907\pi\)
\(14\) −140498. + 182511.i −0.261235 + 0.339351i
\(15\) −68579.5 − 131676.i −0.0903105 − 0.173400i
\(16\) 911345. − 518615.i 0.869127 − 0.494590i
\(17\) −421025. + 421025.i −0.296527 + 0.296527i −0.839652 − 0.543125i \(-0.817241\pi\)
0.543125 + 0.839652i \(0.317241\pi\)
\(18\) −234401. − 1.80216e6i −0.124050 − 0.953743i
\(19\) 3.72211e6 1.50321 0.751607 − 0.659611i \(-0.229279\pi\)
0.751607 + 0.659611i \(0.229279\pi\)
\(20\) −148911. + 3.19653e6i −0.0465346 + 0.998917i
\(21\) 341952.i 0.0837277i
\(22\) −498453. − 3.83229e6i −0.0967188 − 0.743609i
\(23\) 6.50259e6 + 6.50259e6i 1.01029 + 1.01029i 0.999946 + 0.0103455i \(0.00329313\pi\)
0.0103455 + 0.999946i \(0.496707\pi\)
\(24\) 602467. − 1.43546e6i 0.0756619 − 0.180275i
\(25\) −8.00177e6 − 5.59813e6i −0.819381 − 0.573249i
\(26\) −1.04606e7 + 1.35886e7i −0.880418 + 1.14369i
\(27\) −3.89152e6 − 3.89152e6i −0.271207 − 0.271207i
\(28\) 3.70510e6 − 6.37147e6i 0.215283 − 0.370211i
\(29\) −2.73192e7 −1.33192 −0.665960 − 0.745988i \(-0.731978\pi\)
−0.665960 + 0.745988i \(0.731978\pi\)
\(30\) 2.71969e6 + 3.89538e6i 0.111921 + 0.160304i
\(31\) −5.47309e7 −1.91172 −0.955859 − 0.293826i \(-0.905071\pi\)
−0.955859 + 0.293826i \(0.905071\pi\)
\(32\) −2.67789e7 + 2.02185e7i −0.798075 + 0.602559i
\(33\) −4.05703e6 − 4.05703e6i −0.103666 − 0.103666i
\(34\) 1.16225e7 − 1.50980e7i 0.255804 − 0.332296i
\(35\) 1.03900e7 + 1.99493e7i 0.197822 + 0.379828i
\(36\) 1.48764e7 + 5.62200e7i 0.246028 + 0.929776i
\(37\) 7.83954e7 + 7.83954e7i 1.13053 + 1.13053i 0.990089 + 0.140440i \(0.0448519\pi\)
0.140440 + 0.990089i \(0.455148\pi\)
\(38\) −1.18113e8 + 1.53625e7i −1.49066 + 0.193885i
\(39\) 2.54595e7i 0.282180i
\(40\) −8.46793e6 − 1.02049e8i −0.0826947 − 0.996575i
\(41\) −1.38841e8 −1.19839 −0.599195 − 0.800603i \(-0.704513\pi\)
−0.599195 + 0.800603i \(0.704513\pi\)
\(42\) −1.41136e6 − 1.08511e7i −0.0107992 − 0.0830283i
\(43\) −4.60496e7 + 4.60496e7i −0.313245 + 0.313245i −0.846165 − 0.532921i \(-0.821094\pi\)
0.532921 + 0.846165i \(0.321094\pi\)
\(44\) 3.16345e7 + 1.19552e8i 0.191822 + 0.724923i
\(45\) −1.69271e8 − 5.33339e7i −0.917320 − 0.289028i
\(46\) −2.33183e8 − 1.79506e8i −1.13216 − 0.871545i
\(47\) −1.14448e8 + 1.14448e8i −0.499022 + 0.499022i −0.911133 − 0.412112i \(-0.864791\pi\)
0.412112 + 0.911133i \(0.364791\pi\)
\(48\) −1.31933e7 + 4.80376e7i −0.0517780 + 0.188528i
\(49\) 2.30668e8i 0.816597i
\(50\) 2.77023e8 + 1.44618e8i 0.886475 + 0.462777i
\(51\) − 2.82875e7i − 0.0819869i
\(52\) 2.75857e8 − 4.74377e8i 0.725551 − 1.24769i
\(53\) 3.43507e8 − 3.43507e8i 0.821403 − 0.821403i −0.164906 − 0.986309i \(-0.552732\pi\)
0.986309 + 0.164906i \(0.0527322\pi\)
\(54\) 1.39550e8 + 1.07427e8i 0.303922 + 0.233961i
\(55\) −3.59955e8 − 1.13414e8i −0.715211 − 0.225348i
\(56\) −9.12756e7 + 2.17476e8i −0.165735 + 0.394886i
\(57\) −1.25039e8 + 1.25039e8i −0.207813 + 0.207813i
\(58\) 8.66912e8 − 1.12756e8i 1.32079 − 0.171791i
\(59\) −4.06566e8 −0.568684 −0.284342 − 0.958723i \(-0.591775\pi\)
−0.284342 + 0.958723i \(0.591775\pi\)
\(60\) −1.02381e8 − 1.12386e8i −0.131663 − 0.144529i
\(61\) − 1.18617e8i − 0.140442i −0.997531 − 0.0702210i \(-0.977630\pi\)
0.997531 − 0.0702210i \(-0.0223704\pi\)
\(62\) 1.73676e9 − 2.25895e8i 1.89575 − 0.246574i
\(63\) 2.89045e8 + 2.89045e8i 0.291248 + 0.291248i
\(64\) 7.66319e8 − 7.52115e8i 0.713690 − 0.700461i
\(65\) 7.73570e8 + 1.48529e9i 0.666704 + 1.28010i
\(66\) 1.45485e8 + 1.11996e8i 0.116172 + 0.0894296i
\(67\) −6.69240e8 − 6.69240e8i −0.495688 − 0.495688i 0.414405 − 0.910093i \(-0.363990\pi\)
−0.910093 + 0.414405i \(0.863990\pi\)
\(68\) −3.06500e8 + 5.27071e8i −0.210807 + 0.362514i
\(69\) −4.36891e8 −0.279337
\(70\) −4.12041e8 − 5.90161e8i −0.245160 − 0.351140i
\(71\) −7.06085e8 −0.391350 −0.195675 − 0.980669i \(-0.562690\pi\)
−0.195675 + 0.980669i \(0.562690\pi\)
\(72\) −7.04109e8 − 1.72261e9i −0.363896 − 0.890277i
\(73\) −8.44893e8 − 8.44893e8i −0.407556 − 0.407556i 0.473329 − 0.880886i \(-0.343052\pi\)
−0.880886 + 0.473329i \(0.843052\pi\)
\(74\) −2.81127e9 − 2.16413e9i −1.26690 − 0.975270i
\(75\) 4.56870e8 − 8.07469e7i 0.192525 − 0.0340267i
\(76\) 3.68462e9 − 9.74989e8i 1.45320 − 0.384531i
\(77\) 6.14652e8 + 6.14652e8i 0.227078 + 0.227078i
\(78\) −1.05081e8 − 8.07898e8i −0.0363957 − 0.279823i
\(79\) 5.80430e9i 1.88632i 0.332346 + 0.943158i \(0.392160\pi\)
−0.332346 + 0.943158i \(0.607840\pi\)
\(80\) 6.89906e8 + 3.20335e9i 0.210543 + 0.977585i
\(81\) −3.09205e9 −0.886790
\(82\) 4.40580e9 − 5.73048e8i 1.18838 − 0.154569i
\(83\) −9.30327e8 + 9.30327e8i −0.236181 + 0.236181i −0.815267 − 0.579086i \(-0.803410\pi\)
0.579086 + 0.815267i \(0.303410\pi\)
\(84\) 8.95728e7 + 3.38509e8i 0.0214180 + 0.0809419i
\(85\) −8.59499e8 − 1.65028e9i −0.193709 − 0.371931i
\(86\) 1.27122e9 − 1.65134e9i 0.270226 − 0.351031i
\(87\) 9.17751e8 − 9.17751e8i 0.184132 − 0.184132i
\(88\) −1.49728e9 − 3.66313e9i −0.283720 − 0.694127i
\(89\) − 2.22155e9i − 0.397838i −0.980016 − 0.198919i \(-0.936257\pi\)
0.980016 − 0.198919i \(-0.0637431\pi\)
\(90\) 5.59157e9 + 9.93782e8i 0.946937 + 0.168298i
\(91\) − 3.85719e9i − 0.618107i
\(92\) 8.14042e9 + 4.73378e9i 1.23512 + 0.718239i
\(93\) 1.83861e9 − 1.83861e9i 0.264286 − 0.264286i
\(94\) 3.15938e9 − 4.10412e9i 0.430489 − 0.559217i
\(95\) −3.49547e9 + 1.10939e10i −0.451739 + 1.43373i
\(96\) 2.20389e8 − 1.57882e9i 0.0270292 − 0.193631i
\(97\) −3.03256e9 + 3.03256e9i −0.353143 + 0.353143i −0.861278 − 0.508135i \(-0.830335\pi\)
0.508135 + 0.861278i \(0.330335\pi\)
\(98\) −9.52054e8 − 7.31973e9i −0.105325 − 0.809776i
\(99\) −6.85863e9 −0.721210
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.13.2 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.27 yes 116
8.5 even 2 inner 40.11.i.a.13.27 yes 116
40.37 odd 4 inner 40.11.i.a.37.2 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.2 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.27 yes 116 8.5 even 2 inner
40.11.i.a.37.2 yes 116 40.37 odd 4 inner
40.11.i.a.37.27 yes 116 5.2 odd 4 inner