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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 13.15
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.15

$q$-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+(-23.6711 - 21.5332i) q^{2} +(155.126 - 155.126i) q^{3} +(96.6435 + 1019.43i) q^{4} +(1092.16 + 2927.94i) q^{5} +(-7012.36 + 331.648i) q^{6} +(15010.7 - 15010.7i) q^{7} +(19663.9 - 26212.1i) q^{8} +10920.9i q^{9} +(37195.1 - 92825.2i) q^{10} +125087. i q^{11} +(173132. + 143148. i) q^{12} +(-403409. + 403409. i) q^{13} +(-678548. + 32091.8i) q^{14} +(623621. + 284776. i) q^{15} +(-1.02990e6 + 197042. i) q^{16} +(-558653. + 558653. i) q^{17} +(235163. - 258511. i) q^{18} -1.60049e6 q^{19} +(-2.87927e6 + 1.39635e6i) q^{20} -4.65710e6i q^{21} +(2.69351e6 - 2.96094e6i) q^{22} +(3.84512e6 + 3.84512e6i) q^{23} +(-1.01579e6 - 7.11655e6i) q^{24} +(-7.37999e6 + 6.39556e6i) q^{25} +(1.82358e7 - 862458. i) q^{26} +(1.08541e7 + 1.08541e7i) q^{27} +(1.67530e7 + 1.38517e7i) q^{28} -3.05852e7 q^{29} +(-8.62967e6 - 2.01695e7i) q^{30} -3.41873e6 q^{31} +(2.86217e7 + 1.75127e7i) q^{32} +(1.94042e7 + 1.94042e7i) q^{33} +(2.52535e7 - 1.19436e6i) q^{34} +(6.03445e7 + 2.75562e7i) q^{35} +(-1.11331e7 + 1.05544e6i) q^{36} +(2.25988e7 + 2.25988e7i) q^{37} +(3.78853e7 + 3.44636e7i) q^{38} +1.25158e8i q^{39} +(9.82234e7 + 2.89468e7i) q^{40} +1.87606e8 q^{41} +(-1.00282e8 + 1.10239e8i) q^{42} +(3.44463e7 - 3.44463e7i) q^{43} +(-1.27517e8 + 1.20888e7i) q^{44} +(-3.19758e7 + 1.19274e7i) q^{45} +(-8.22059e6 - 1.73816e8i) q^{46} +(2.54572e8 - 2.54572e8i) q^{47} +(-1.29197e8 + 1.90330e8i) q^{48} -1.68167e8i q^{49} +(3.12409e8 + 7.52460e6i) q^{50} +1.73323e8i q^{51} +(-4.50233e8 - 3.72260e8i) q^{52} +(-4.94632e8 + 4.94632e8i) q^{53} +(-2.32054e7 - 4.90654e8i) q^{54} +(-3.66245e8 + 1.36615e8i) q^{55} +(-9.82926e7 - 6.88631e8i) q^{56} +(-2.48277e8 + 2.48277e8i) q^{57} +(7.23985e8 + 6.58597e8i) q^{58} +3.91800e8 q^{59} +(-2.30040e8 + 6.63259e8i) q^{60} +9.12121e8i q^{61} +(8.09252e7 + 7.36162e7i) q^{62} +(1.63931e8 + 1.63931e8i) q^{63} +(-3.00403e8 - 1.03086e9i) q^{64} +(-1.62174e9 - 7.40567e8i) q^{65} +(-4.14846e7 - 8.77151e8i) q^{66} +(4.79330e8 + 4.79330e8i) q^{67} +(-6.23498e8 - 5.15517e8i) q^{68} +1.19296e9 q^{69} +(-8.35048e8 - 1.95170e9i) q^{70} -1.67029e9 q^{71} +(2.86260e8 + 2.14748e8i) q^{72} +(5.60179e8 + 5.60179e8i) q^{73} +(-4.83145e7 - 1.02156e9i) q^{74} +(-1.52710e8 + 2.13694e9i) q^{75} +(-1.54677e8 - 1.63158e9i) q^{76} +(1.87764e9 + 1.87764e9i) q^{77} +(2.69506e9 - 2.96263e9i) q^{78} -5.62732e7i q^{79} +(-1.70174e9 - 2.80027e9i) q^{80} +2.72265e9 q^{81} +(-4.44084e9 - 4.03975e9i) q^{82} +(8.33030e7 - 8.33030e7i) q^{83} +(4.74758e9 - 4.50078e8i) q^{84} +(-2.24584e9 - 1.02556e9i) q^{85} +(-1.55712e9 + 7.36436e7i) q^{86} +(-4.74455e9 + 4.74455e9i) q^{87} +(3.27878e9 + 2.45969e9i) q^{88} -6.37191e9i q^{89} +(1.01374e9 + 4.06205e8i) q^{90} +1.21109e10i q^{91} +(-3.54823e9 + 4.29144e9i) q^{92} +(-5.30334e8 + 5.30334e8i) q^{93} +(-1.15077e10 + 5.44256e8i) q^{94} +(-1.74799e9 - 4.68612e9i) q^{95} +(7.15665e9 - 1.72329e9i) q^{96} +(-3.76919e8 + 3.76919e8i) q^{97} +(-3.62118e9 + 3.98071e9i) q^{98} -1.36606e9 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −23.6711 − 21.5332i −0.739722 − 0.672912i
\(3\) 155.126 − 155.126i 0.638378 − 0.638378i −0.311777 − 0.950155i \(-0.600924\pi\)
0.950155 + 0.311777i \(0.100924\pi\)
\(4\) 96.6435 + 1019.43i 0.0943784 + 0.995536i
\(5\) 1092.16 + 2927.94i 0.349492 + 0.936939i
\(6\) −7012.36 + 331.648i −0.901795 + 0.0426502i
\(7\) 15010.7 − 15010.7i 0.893122 − 0.893122i −0.101694 − 0.994816i \(-0.532426\pi\)
0.994816 + 0.101694i \(0.0324261\pi\)
\(8\) 19663.9 − 26212.1i 0.600095 − 0.799929i
\(9\) 10920.9i 0.184947i
\(10\) 37195.1 − 92825.2i 0.371951 − 0.928252i
\(11\) 125087.i 0.776689i 0.921514 + 0.388344i \(0.126953\pi\)
−0.921514 + 0.388344i \(0.873047\pi\)
\(12\) 173132. + 143148.i 0.695778 + 0.575279i
\(13\) −403409. + 403409.i −1.08650 + 1.08650i −0.0906103 + 0.995886i \(0.528882\pi\)
−0.995886 + 0.0906103i \(0.971118\pi\)
\(14\) −678548. + 32091.8i −1.26166 + 0.0596697i
\(15\) 623621. + 284776.i 0.821229 + 0.375014i
\(16\) −1.02990e6 + 197042.i −0.982185 + 0.187914i
\(17\) −558653. + 558653.i −0.393458 + 0.393458i −0.875918 − 0.482460i \(-0.839743\pi\)
0.482460 + 0.875918i \(0.339743\pi\)
\(18\) 235163. − 258511.i 0.124453 − 0.136809i
\(19\) −1.60049e6 −0.646374 −0.323187 − 0.946335i \(-0.604754\pi\)
−0.323187 + 0.946335i \(0.604754\pi\)
\(20\) −2.87927e6 + 1.39635e6i −0.899773 + 0.436359i
\(21\) − 4.65710e6i − 1.14030i
\(22\) 2.69351e6 − 2.96094e6i 0.522643 − 0.574534i
\(23\) 3.84512e6 + 3.84512e6i 0.597408 + 0.597408i 0.939622 − 0.342214i \(-0.111177\pi\)
−0.342214 + 0.939622i \(0.611177\pi\)
\(24\) −1.01579e6 − 7.11655e6i −0.127570 − 0.893744i
\(25\) −7.37999e6 + 6.39556e6i −0.755711 + 0.654906i
\(26\) 1.82358e7 − 862458.i 1.53482 − 0.0725891i
\(27\) 1.08541e7 + 1.08541e7i 0.756444 + 0.756444i
\(28\) 1.67530e7 + 1.38517e7i 0.973427 + 0.804844i
\(29\) −3.05852e7 −1.49115 −0.745575 − 0.666422i \(-0.767825\pi\)
−0.745575 + 0.666422i \(0.767825\pi\)
\(30\) −8.62967e6 − 2.01695e7i −0.355131 − 0.830021i
\(31\) −3.41873e6 −0.119414 −0.0597072 − 0.998216i \(-0.519017\pi\)
−0.0597072 + 0.998216i \(0.519017\pi\)
\(32\) 2.86217e7 + 1.75127e7i 0.852994 + 0.521920i
\(33\) 1.94042e7 + 1.94042e7i 0.495821 + 0.495821i
\(34\) 2.52535e7 − 1.19436e6i 0.555812 − 0.0262870i
\(35\) 6.03445e7 + 2.75562e7i 1.14894 + 0.524662i
\(36\) −1.11331e7 + 1.05544e6i −0.184121 + 0.0174550i
\(37\) 2.25988e7 + 2.25988e7i 0.325894 + 0.325894i 0.851023 − 0.525129i \(-0.175983\pi\)
−0.525129 + 0.851023i \(0.675983\pi\)
\(38\) 3.78853e7 + 3.44636e7i 0.478137 + 0.434953i
\(39\) 1.25158e8i 1.38719i
\(40\) 9.82234e7 + 2.89468e7i 0.959213 + 0.282684i
\(41\) 1.87606e8 1.61930 0.809650 − 0.586913i \(-0.199657\pi\)
0.809650 + 0.586913i \(0.199657\pi\)
\(42\) −1.00282e8 + 1.10239e8i −0.767321 + 0.843505i
\(43\) 3.44463e7 − 3.44463e7i 0.234315 − 0.234315i −0.580176 − 0.814491i \(-0.697016\pi\)
0.814491 + 0.580176i \(0.197016\pi\)
\(44\) −1.27517e8 + 1.20888e7i −0.773222 + 0.0733026i
\(45\) −3.19758e7 + 1.19274e7i −0.173284 + 0.0646375i
\(46\) −8.22059e6 − 1.73816e8i −0.0399130 − 0.843920i
\(47\) 2.54572e8 − 2.54572e8i 1.10999 − 1.10999i 0.116844 − 0.993150i \(-0.462722\pi\)
0.993150 − 0.116844i \(-0.0372779\pi\)
\(48\) −1.29197e8 + 1.90330e8i −0.507045 + 0.746966i
\(49\) − 1.68167e8i − 0.595334i
\(50\) 3.12409e8 + 7.52460e6i 0.999710 + 0.0240787i
\(51\) 1.73323e8i 0.502349i
\(52\) −4.50233e8 − 3.72260e8i −1.18419 − 0.979105i
\(53\) −4.94632e8 + 4.94632e8i −1.18278 + 1.18278i −0.203755 + 0.979022i \(0.565314\pi\)
−0.979022 + 0.203755i \(0.934686\pi\)
\(54\) −2.32054e7 − 4.90654e8i −0.0505382 − 1.06858i
\(55\) −3.66245e8 + 1.36615e8i −0.727710 + 0.271447i
\(56\) −9.82926e7 − 6.88631e8i −0.178476 − 1.25039i
\(57\) −2.48277e8 + 2.48277e8i −0.412631 + 0.412631i
\(58\) 7.23985e8 + 6.58597e8i 1.10304 + 1.00341i
\(59\) 3.91800e8 0.548030 0.274015 − 0.961725i \(-0.411648\pi\)
0.274015 + 0.961725i \(0.411648\pi\)
\(60\) −2.30040e8 + 6.63259e8i −0.295833 + 0.852957i
\(61\) 9.12121e8i 1.07995i 0.841681 + 0.539974i \(0.181566\pi\)
−0.841681 + 0.539974i \(0.818434\pi\)
\(62\) 8.09252e7 + 7.36162e7i 0.0883334 + 0.0803554i
\(63\) 1.63931e8 + 1.63931e8i 0.165180 + 0.165180i
\(64\) −3.00403e8 − 1.03086e9i −0.279773 − 0.960066i
\(65\) −1.62174e9 − 7.40567e8i −1.39770 − 0.638260i
\(66\) −4.14846e7 − 8.77151e8i −0.0331259 − 0.700414i
\(67\) 4.79330e8 + 4.79330e8i 0.355027 + 0.355027i 0.861976 − 0.506949i \(-0.169227\pi\)
−0.506949 + 0.861976i \(0.669227\pi\)
\(68\) −6.23498e8 − 5.15517e8i −0.428835 − 0.354567i
\(69\) 1.19296e9 0.762745
\(70\) −8.35048e8 − 1.95170e9i −0.496845 − 1.16124i
\(71\) −1.67029e9 −0.925766 −0.462883 − 0.886419i \(-0.653185\pi\)
−0.462883 + 0.886419i \(0.653185\pi\)
\(72\) 2.86260e8 + 2.14748e8i 0.147944 + 0.110986i
\(73\) 5.60179e8 + 5.60179e8i 0.270217 + 0.270217i 0.829187 − 0.558971i \(-0.188804\pi\)
−0.558971 + 0.829187i \(0.688804\pi\)
\(74\) −4.83145e7 − 1.02156e9i −0.0217730 − 0.460369i
\(75\) −1.52710e8 + 2.13694e9i −0.0643518 + 0.900506i
\(76\) −1.54677e8 − 1.63158e9i −0.0610037 − 0.643489i
\(77\) 1.87764e9 + 1.87764e9i 0.693678 + 0.693678i
\(78\) 2.69506e9 − 2.96263e9i 0.933458 − 1.02614i
\(79\) − 5.62732e7i − 0.0182880i −0.999958 − 0.00914400i \(-0.997089\pi\)
0.999958 − 0.00914400i \(-0.00291066\pi\)
\(80\) −1.70174e9 − 2.80027e9i −0.519330 − 0.854574i
\(81\) 2.72265e9 0.780848
\(82\) −4.44084e9 − 4.03975e9i −1.19783 − 1.08965i
\(83\) 8.33030e7 − 8.33030e7i 0.0211481 − 0.0211481i −0.696454 − 0.717602i \(-0.745240\pi\)
0.717602 + 0.696454i \(0.245240\pi\)
\(84\) 4.74758e9 − 4.50078e8i 1.13521 − 0.107620i
\(85\) −2.24584e9 − 1.02556e9i −0.506156 − 0.231136i
\(86\) −1.55712e9 + 7.36436e7i −0.331001 + 0.0156546i
\(87\) −4.74455e9 + 4.74455e9i −0.951917 + 0.951917i
\(88\) 3.27878e9 + 2.45969e9i 0.621296 + 0.466087i
\(89\) − 6.37191e9i − 1.14109i −0.821267 − 0.570545i \(-0.806732\pi\)
0.821267 − 0.570545i \(-0.193268\pi\)
\(90\) 1.01374e9 + 4.06205e8i 0.171677 + 0.0687912i
\(91\) 1.21109e10i 1.94075i
\(92\) −3.54823e9 + 4.29144e9i −0.538359 + 0.651124i
\(93\) −5.30334e8 + 5.30334e8i −0.0762315 + 0.0762315i
\(94\) −1.15077e10 + 5.44256e8i −1.56802 + 0.0741589i
\(95\) −1.74799e9 − 4.68612e9i −0.225903 − 0.605613i
\(96\) 7.15665e9 − 1.72329e9i 0.877715 − 0.211350i
\(97\) −3.76919e8 + 3.76919e8i −0.0438924 + 0.0438924i −0.728712 − 0.684820i \(-0.759881\pi\)
0.684820 + 0.728712i \(0.259881\pi\)
\(98\) −3.62118e9 + 3.98071e9i −0.400608 + 0.440382i
\(99\) −1.36606e9 −0.143646
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.13.15 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.44 yes 116
8.5 even 2 inner 40.11.i.a.13.44 yes 116
40.37 odd 4 inner 40.11.i.a.37.15 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.15 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.44 yes 116 8.5 even 2 inner
40.11.i.a.37.15 yes 116 40.37 odd 4 inner
40.11.i.a.37.44 yes 116 5.2 odd 4 inner