Show commands: Magma / Pari/GP / SageMath

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 13.4
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.4

$q$-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+(-31.2076 - 7.07732i) q^{2} +(28.6547 - 28.6547i) q^{3} +(923.823 + 441.732i) q^{4} +(2754.83 + 1475.31i) q^{5} +(-1097.04 + 691.445i) q^{6} +(-18007.5 + 18007.5i) q^{7} +(-25704.0 - 20323.6i) q^{8} +57406.8i q^{9} +(-75530.3 - 65537.5i) q^{10} +2594.74i q^{11} +(39129.6 - 13814.2i) q^{12} +(-40536.9 + 40536.9i) q^{13} +(689416. - 434526. i) q^{14} +(121213. - 36664.4i) q^{15} +(658322. + 816164. i) q^{16} +(1.21994e6 - 1.21994e6i) q^{17} +(406286. - 1.79153e6i) q^{18} -1.91642e6 q^{19} +(1.89329e6 + 2.57982e6i) q^{20} +1.03200e6i q^{21} +(18363.8 - 80975.4i) q^{22} +(-7.15073e6 - 7.15073e6i) q^{23} +(-1.31891e6 + 154175. i) q^{24} +(5.41256e6 + 8.12844e6i) q^{25} +(1.55195e6 - 978166. i) q^{26} +(3.33701e6 + 3.33701e6i) q^{27} +(-2.45903e7 + 8.68128e6i) q^{28} -1.16465e6 q^{29} +(-4.04226e6 + 286340. i) q^{30} -5.06034e6 q^{31} +(-1.47684e7 - 3.01296e7i) q^{32} +(74351.5 + 74351.5i) q^{33} +(-4.67053e7 + 2.94374e7i) q^{34} +(-7.61744e7 + 2.30411e7i) q^{35} +(-2.53584e7 + 5.30337e7i) q^{36} +(-5.86796e7 - 5.86796e7i) q^{37} +(5.98067e7 + 1.35631e7i) q^{38} +2.32315e6i q^{39} +(-4.08267e7 - 9.39092e7i) q^{40} -1.14523e8 q^{41} +(7.30381e6 - 3.22063e7i) q^{42} +(-1.10604e8 + 1.10604e8i) q^{43} +(-1.14618e6 + 2.39708e6i) q^{44} +(-8.46927e7 + 1.58146e8i) q^{45} +(1.72549e8 + 2.73765e8i) q^{46} +(-4.14561e7 + 4.14561e7i) q^{47} +(4.22510e7 + 4.52290e6i) q^{48} -3.66068e8i q^{49} +(-1.11385e8 - 2.91975e8i) q^{50} -6.99141e7i q^{51} +(-5.53554e7 + 1.95425e7i) q^{52} +(-1.05712e7 + 1.05712e7i) q^{53} +(-8.05228e7 - 1.27757e8i) q^{54} +(-3.82803e6 + 7.14806e6i) q^{55} +(8.28843e8 - 9.68883e7i) q^{56} +(-5.49144e7 + 5.49144e7i) q^{57} +(3.63459e7 + 8.24260e6i) q^{58} +1.22232e9 q^{59} +(1.28176e8 + 1.96724e7i) q^{60} +8.17653e8i q^{61} +(1.57921e8 + 3.58136e7i) q^{62} +(-1.03376e9 - 1.03376e9i) q^{63} +(2.47648e8 + 1.04479e9i) q^{64} +(-1.71477e8 + 5.18680e7i) q^{65} +(-1.79412e6 - 2.84654e6i) q^{66} +(-7.09912e8 - 7.09912e8i) q^{67} +(1.66590e9 - 5.88123e8i) q^{68} -4.09804e8 q^{69} +(2.54029e9 - 1.79945e8i) q^{70} -2.21569e9 q^{71} +(1.16671e9 - 1.47558e9i) q^{72} +(-7.22252e8 - 7.22252e8i) q^{73} +(1.41595e9 + 2.24654e9i) q^{74} +(3.88014e8 + 7.78228e7i) q^{75} +(-1.77043e9 - 8.46543e8i) q^{76} +(-4.67248e7 - 4.67248e7i) q^{77} +(1.64417e7 - 7.24998e7i) q^{78} +1.36664e9i q^{79} +(6.09474e8 + 3.21962e9i) q^{80} -3.19857e9 q^{81} +(3.57397e9 + 8.10513e8i) q^{82} +(-4.26026e9 + 4.26026e9i) q^{83} +(-4.55868e8 + 9.53387e8i) q^{84} +(5.16052e9 - 1.56094e9i) q^{85} +(4.23446e9 - 2.66890e9i) q^{86} +(-3.33727e7 + 3.33727e7i) q^{87} +(5.27343e7 - 6.66951e7i) q^{88} -1.76875e9i q^{89} +(3.76230e9 - 4.33596e9i) q^{90} -1.45994e9i q^{91} +(-3.44731e9 - 9.76471e9i) q^{92} +(-1.45003e8 + 1.45003e8i) q^{93} +(1.58714e9 - 1.00035e9i) q^{94} +(-5.27941e9 - 2.82731e9i) q^{95} +(-1.28654e9 - 4.40172e8i) q^{96} +(8.31552e9 - 8.31552e9i) q^{97} +(-2.59078e9 + 1.14241e10i) q^{98} -1.48956e8 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.2076 − 7.07732i −0.975236 − 0.221166i
\(3\) 28.6547 − 28.6547i 0.117921 − 0.117921i −0.645684 − 0.763605i \(-0.723428\pi\)
0.763605 + 0.645684i \(0.223428\pi\)
\(4\) 923.823 + 441.732i 0.902171 + 0.431379i
\(5\) 2754.83 + 1475.31i 0.881546 + 0.472098i
\(6\) −1097.04 + 691.445i −0.141081 + 0.0889204i
\(7\) −18007.5 + 18007.5i −1.07143 + 1.07143i −0.0741867 + 0.997244i \(0.523636\pi\)
−0.997244 + 0.0741867i \(0.976364\pi\)
\(8\) −25704.0 − 20323.6i −0.784423 − 0.620226i
\(9\) 57406.8i 0.972189i
\(10\) −75530.3 − 65537.5i −0.755303 − 0.655375i
\(11\) 2594.74i 0.0161113i 0.999968 + 0.00805564i \(0.00256422\pi\)
−0.999968 + 0.00805564i \(0.997436\pi\)
\(12\) 39129.6 − 13814.2i 0.157253 − 0.0555161i
\(13\) −40536.9 + 40536.9i −0.109178 + 0.109178i −0.759585 − 0.650408i \(-0.774598\pi\)
0.650408 + 0.759585i \(0.274598\pi\)
\(14\) 689416. − 434526.i 1.28186 − 0.807934i
\(15\) 121213. − 36664.4i 0.159623 − 0.0482823i
\(16\) 658322. + 816164.i 0.627825 + 0.778354i
\(17\) 1.21994e6 − 1.21994e6i 0.859199 − 0.859199i −0.132044 − 0.991244i \(-0.542154\pi\)
0.991244 + 0.132044i \(0.0421542\pi\)
\(18\) 406286. − 1.79153e6i 0.215015 − 0.948114i
\(19\) −1.91642e6 −0.773967 −0.386983 − 0.922087i \(-0.626483\pi\)
−0.386983 + 0.922087i \(0.626483\pi\)
\(20\) 1.89329e6 + 2.57982e6i 0.591652 + 0.806193i
\(21\) 1.03200e6i 0.252688i
\(22\) 18363.8 − 80975.4i 0.00356327 − 0.0157123i
\(23\) −7.15073e6 − 7.15073e6i −1.11099 − 1.11099i −0.993016 − 0.117976i \(-0.962359\pi\)
−0.117976 − 0.993016i \(-0.537641\pi\)
\(24\) −1.31891e6 + 154175.i −0.165637 + 0.0193623i
\(25\) 5.41256e6 + 8.12844e6i 0.554246 + 0.832353i
\(26\) 1.55195e6 − 978166.i 0.130620 − 0.0823276i
\(27\) 3.33701e6 + 3.33701e6i 0.232562 + 0.232562i
\(28\) −2.45903e7 + 8.68128e6i −1.42881 + 0.504422i
\(29\) −1.16465e6 −0.0567813 −0.0283907 − 0.999597i \(-0.509038\pi\)
−0.0283907 + 0.999597i \(0.509038\pi\)
\(30\) −4.04226e6 + 286340.i −0.166348 + 0.0117835i
\(31\) −5.06034e6 −0.176755 −0.0883774 − 0.996087i \(-0.528168\pi\)
−0.0883774 + 0.996087i \(0.528168\pi\)
\(32\) −1.47684e7 − 3.01296e7i −0.440132 − 0.897933i
\(33\) 74351.5 + 74351.5i 0.00189985 + 0.00189985i
\(34\) −4.67053e7 + 2.94374e7i −1.02795 + 0.647896i
\(35\) −7.61744e7 + 2.30411e7i −1.45034 + 0.438695i
\(36\) −2.53584e7 + 5.30337e7i −0.419382 + 0.877081i
\(37\) −5.86796e7 − 5.86796e7i −0.846211 − 0.846211i 0.143447 − 0.989658i \(-0.454181\pi\)
−0.989658 + 0.143447i \(0.954181\pi\)
\(38\) 5.98067e7 + 1.35631e7i 0.754800 + 0.171175i
\(39\) 2.32315e6i 0.0257486i
\(40\) −4.08267e7 − 9.39092e7i −0.398698 − 0.917082i
\(41\) −1.14523e8 −0.988489 −0.494244 − 0.869323i \(-0.664555\pi\)
−0.494244 + 0.869323i \(0.664555\pi\)
\(42\) 7.30381e6 − 3.22063e7i 0.0558860 − 0.246430i
\(43\) −1.10604e8 + 1.10604e8i −0.752365 + 0.752365i −0.974920 − 0.222555i \(-0.928560\pi\)
0.222555 + 0.974920i \(0.428560\pi\)
\(44\) −1.14618e6 + 2.39708e6i −0.00695006 + 0.0145351i
\(45\) −8.46927e7 + 1.58146e8i −0.458969 + 0.857030i
\(46\) 1.72549e8 + 2.73765e8i 0.837766 + 1.32919i
\(47\) −4.14561e7 + 4.14561e7i −0.180759 + 0.180759i −0.791686 − 0.610928i \(-0.790797\pi\)
0.610928 + 0.791686i \(0.290797\pi\)
\(48\) 4.22510e7 + 4.52290e6i 0.165818 + 0.0177505i
\(49\) − 3.66068e8i − 1.29593i
\(50\) −1.11385e8 − 2.91975e8i −0.356433 − 0.934321i
\(51\) − 6.99141e7i − 0.202635i
\(52\) −5.53554e7 + 1.95425e7i −0.145594 + 0.0514000i
\(53\) −1.05712e7 + 1.05712e7i −0.0252782 + 0.0252782i −0.719633 − 0.694355i \(-0.755690\pi\)
0.694355 + 0.719633i \(0.255690\pi\)
\(54\) −8.05228e7 − 1.27757e8i −0.175368 − 0.278238i
\(55\) −3.82803e6 + 7.14806e6i −0.00760611 + 0.0142028i
\(56\) 8.28843e8 − 9.68883e7i 1.50498 − 0.175927i
\(57\) −5.49144e7 + 5.49144e7i −0.0912667 + 0.0912667i
\(58\) 3.63459e7 + 8.24260e6i 0.0553752 + 0.0125581i
\(59\) 1.22232e9 1.70972 0.854860 − 0.518860i \(-0.173643\pi\)
0.854860 + 0.518860i \(0.173643\pi\)
\(60\) 1.28176e8 + 1.96724e7i 0.164835 + 0.0252988i
\(61\) 8.17653e8i 0.968099i 0.875041 + 0.484050i \(0.160835\pi\)
−0.875041 + 0.484050i \(0.839165\pi\)
\(62\) 1.57921e8 + 3.58136e7i 0.172378 + 0.0390922i
\(63\) −1.03376e9 − 1.03376e9i −1.04163 − 1.04163i
\(64\) 2.47648e8 + 1.04479e9i 0.230640 + 0.973039i
\(65\) −1.71477e8 + 5.18680e7i −0.147788 + 0.0447026i
\(66\) −1.79412e6 − 2.84654e6i −0.00143262 − 0.00227299i
\(67\) −7.09912e8 − 7.09912e8i −0.525812 − 0.525812i 0.393509 − 0.919321i \(-0.371261\pi\)
−0.919321 + 0.393509i \(0.871261\pi\)
\(68\) 1.66590e9 − 5.88123e8i 1.14578 − 0.404505i
\(69\) −4.09804e8 −0.262018
\(70\) 2.54029e9 − 1.79945e8i 1.51145 − 0.107066i
\(71\) −2.21569e9 −1.22805 −0.614026 − 0.789286i \(-0.710451\pi\)
−0.614026 + 0.789286i \(0.710451\pi\)
\(72\) 1.16671e9 − 1.47558e9i 0.602977 − 0.762608i
\(73\) −7.22252e8 − 7.22252e8i −0.348397 − 0.348397i 0.511115 − 0.859512i \(-0.329233\pi\)
−0.859512 + 0.511115i \(0.829233\pi\)
\(74\) 1.41595e9 + 2.24654e9i 0.638102 + 1.01241i
\(75\) 3.88014e8 + 7.78228e7i 0.163509 + 0.0327945i
\(76\) −1.77043e9 − 8.46543e8i −0.698250 − 0.333873i
\(77\) −4.67248e7 − 4.67248e7i −0.0172621 − 0.0172621i
\(78\) 1.64417e7 − 7.24998e7i 0.00569472 − 0.0251110i
\(79\) 1.36664e9i 0.444140i 0.975031 + 0.222070i \(0.0712814\pi\)
−0.975031 + 0.222070i \(0.928719\pi\)
\(80\) 6.09474e8 + 3.21962e9i 0.185997 + 0.982550i
\(81\) −3.19857e9 −0.917342
\(82\) 3.57397e9 + 8.10513e8i 0.964010 + 0.218620i
\(83\) −4.26026e9 + 4.26026e9i −1.08155 + 1.08155i −0.0851833 + 0.996365i \(0.527148\pi\)
−0.996365 + 0.0851833i \(0.972852\pi\)
\(84\) −4.55868e8 + 9.53387e8i −0.109004 + 0.227967i
\(85\) 5.16052e9 − 1.56094e9i 1.16305 − 0.351797i
\(86\) 4.23446e9 − 2.66890e9i 0.900131 − 0.567335i
\(87\) −3.33727e7 + 3.33727e7i −0.00669569 + 0.00669569i
\(88\) 5.27343e7 − 6.66951e7i 0.00999263 − 0.0126381i
\(89\) − 1.76875e9i − 0.316750i −0.987379 − 0.158375i \(-0.949375\pi\)
0.987379 − 0.158375i \(-0.0506254\pi\)
\(90\) 3.76230e9 − 4.33596e9i 0.637149 − 0.734298i
\(91\) − 1.45994e9i − 0.233953i
\(92\) −3.44731e9 − 9.76471e9i −0.523047 − 1.48156i
\(93\) −1.45003e8 + 1.45003e8i −0.0208430 + 0.0208430i
\(94\) 1.58714e9 − 1.00035e9i 0.216260 − 0.136305i
\(95\) −5.27941e9 − 2.82731e9i −0.682287 − 0.365388i
\(96\) −1.28654e9 − 4.40172e8i −0.157785 − 0.0539842i
\(97\) 8.31552e9 − 8.31552e9i 0.968346 − 0.968346i −0.0311683 − 0.999514i \(-0.509923\pi\)
0.999514 + 0.0311683i \(0.00992278\pi\)
\(98\) −2.59078e9 + 1.14241e10i −0.286616 + 1.26384i
\(99\) −1.48956e8 −0.0156632
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.13.4 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.34 yes 116
8.5 even 2 inner 40.11.i.a.13.34 yes 116
40.37 odd 4 inner 40.11.i.a.37.4 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.4 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.34 yes 116 8.5 even 2 inner
40.11.i.a.37.4 yes 116 40.37 odd 4 inner
40.11.i.a.37.34 yes 116 5.2 odd 4 inner