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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 13.11
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.11

$q$-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+(-26.9779 + 17.2103i) q^{2} +(31.2791 - 31.2791i) q^{3} +(431.610 - 928.595i) q^{4} +(1815.55 - 2543.50i) q^{5} +(-305.520 + 1382.17i) q^{6} +(-13255.7 + 13255.7i) q^{7} +(4337.49 + 32479.7i) q^{8} +57092.2i q^{9} +(-5205.37 + 99864.4i) q^{10} -150910. i q^{11} +(-15545.2 - 42546.0i) q^{12} +(33544.9 - 33544.9i) q^{13} +(129475. - 585744. i) q^{14} +(-22769.5 - 136347. i) q^{15} +(-676001. - 801582. i) q^{16} +(-488260. + 488260. i) q^{17} +(-982575. - 1.54023e6i) q^{18} +2.78822e6 q^{19} +(-1.57827e6 - 2.78371e6i) q^{20} +829252. i q^{21} +(2.59721e6 + 4.07123e6i) q^{22} +(1.61145e6 + 1.61145e6i) q^{23} +(1.15161e6 + 880262. i) q^{24} +(-3.17315e6 - 9.23572e6i) q^{25} +(-327651. + 1.48229e6i) q^{26} +(3.63280e6 + 3.63280e6i) q^{27} +(6.58787e6 + 1.80304e7i) q^{28} -1.47099e7 q^{29} +(2.96085e6 + 3.28649e6i) q^{30} +4.06908e7 q^{31} +(3.20326e7 + 9.99077e6i) q^{32} +(-4.72033e6 - 4.72033e6i) q^{33} +(4.76910e6 - 2.15753e7i) q^{34} +(9.64939e6 + 5.77822e7i) q^{35} +(5.30156e7 + 2.46416e7i) q^{36} +(7.22496e7 + 7.22496e7i) q^{37} +(-7.52201e7 + 4.79861e7i) q^{38} -2.09851e6i q^{39} +(9.04869e7 + 4.79362e7i) q^{40} +1.28666e8 q^{41} +(-1.42717e7 - 2.23714e7i) q^{42} +(2.06029e7 - 2.06029e7i) q^{43} +(-1.40134e8 - 6.51343e7i) q^{44} +(1.45214e8 + 1.03654e8i) q^{45} +(-7.12070e7 - 1.57399e7i) q^{46} +(5.20404e7 - 5.20404e7i) q^{47} +(-4.62175e7 - 3.92805e6i) q^{48} -6.89506e7i q^{49} +(2.44554e8 + 1.94549e8i) q^{50} +3.05447e7i q^{51} +(-1.66713e7 - 4.56279e7i) q^{52} +(-2.51475e7 + 2.51475e7i) q^{53} +(-1.60527e8 - 3.54835e7i) q^{54} +(-3.83839e8 - 2.73985e8i) q^{55} +(-4.88036e8 - 3.73043e8i) q^{56} +(8.72130e7 - 8.72130e7i) q^{57} +(3.96842e8 - 2.53162e8i) q^{58} +9.05959e8 q^{59} +(-1.36439e8 - 3.77053e7i) q^{60} -6.35746e8i q^{61} +(-1.09775e9 + 7.00302e8i) q^{62} +(-7.56796e8 - 7.56796e8i) q^{63} +(-1.03611e9 + 2.81761e8i) q^{64} +(-2.44188e7 - 1.46224e8i) q^{65} +(2.08583e8 + 4.61061e7i) q^{66} +(9.08725e8 + 9.08725e8i) q^{67} +(2.42658e8 + 6.64133e8i) q^{68} +1.00809e8 q^{69} +(-1.25477e9 - 1.39277e9i) q^{70} +1.73091e9 q^{71} +(-1.85434e9 + 2.47637e8i) q^{72} +(1.36206e9 + 1.36206e9i) q^{73} +(-3.19258e9 - 7.05701e8i) q^{74} +(-3.88139e8 - 1.89632e8i) q^{75} +(1.20342e9 - 2.58912e9i) q^{76} +(2.00041e9 + 2.00041e9i) q^{77} +(3.61160e7 + 5.66133e7i) q^{78} +1.93358e9i q^{79} +(-3.26614e9 + 2.64093e8i) q^{80} -3.14398e9 q^{81} +(-3.47115e9 + 2.21439e9i) q^{82} +(4.07435e9 - 4.07435e9i) q^{83} +(7.70039e8 + 3.57913e8i) q^{84} +(3.55426e8 + 2.12835e9i) q^{85} +(-2.01240e8 + 9.10406e8i) q^{86} +(-4.60113e8 + 4.60113e8i) q^{87} +(4.90150e9 - 6.54571e8i) q^{88} -1.03269e10i q^{89} +(-5.70148e9 - 2.97186e8i) q^{90} +8.89320e8i q^{91} +(2.19190e9 - 8.00866e8i) q^{92} +(1.27277e9 - 1.27277e9i) q^{93} +(-5.08308e8 + 2.29957e9i) q^{94} +(5.06216e9 - 7.09183e9i) q^{95} +(1.31445e9 - 6.89447e8i) q^{96} +(-2.48030e9 + 2.48030e9i) q^{97} +(1.18666e9 + 1.86014e9i) q^{98} +8.61579e9 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −26.9779 + 17.2103i −0.843058 + 0.537822i
\(3\) 31.2791 − 31.2791i 0.128721 − 0.128721i −0.639811 − 0.768532i \(-0.720988\pi\)
0.768532 + 0.639811i \(0.220988\pi\)
\(4\) 431.610 − 928.595i 0.421494 − 0.906831i
\(5\) 1815.55 − 2543.50i 0.580977 − 0.813920i
\(6\) −305.520 + 1382.17i −0.0392902 + 0.177748i
\(7\) −13255.7 + 13255.7i −0.788700 + 0.788700i −0.981281 − 0.192581i \(-0.938314\pi\)
0.192581 + 0.981281i \(0.438314\pi\)
\(8\) 4337.49 + 32479.7i 0.132370 + 0.991200i
\(9\) 57092.2i 0.966862i
\(10\) −5205.37 + 99864.4i −0.0520537 + 0.998644i
\(11\) − 150910.i − 0.937032i −0.883455 − 0.468516i \(-0.844789\pi\)
0.883455 − 0.468516i \(-0.155211\pi\)
\(12\) −15545.2 − 42546.0i −0.0624729 − 0.170983i
\(13\) 33544.9 − 33544.9i 0.0903462 − 0.0903462i −0.660489 − 0.750835i \(-0.729651\pi\)
0.750835 + 0.660489i \(0.229651\pi\)
\(14\) 129475. − 585744.i 0.240739 − 1.08910i
\(15\) −22769.5 − 136347.i −0.0299845 − 0.179552i
\(16\) −676001. − 801582.i −0.644685 − 0.764448i
\(17\) −488260. + 488260.i −0.343879 + 0.343879i −0.857824 − 0.513944i \(-0.828184\pi\)
0.513944 + 0.857824i \(0.328184\pi\)
\(18\) −982575. − 1.54023e6i −0.520000 − 0.815121i
\(19\) 2.78822e6 1.12605 0.563026 − 0.826439i \(-0.309637\pi\)
0.563026 + 0.826439i \(0.309637\pi\)
\(20\) −1.57827e6 − 2.78371e6i −0.493209 − 0.869911i
\(21\) 829252.i 0.203044i
\(22\) 2.59721e6 + 4.07123e6i 0.503957 + 0.789973i
\(23\) 1.61145e6 + 1.61145e6i 0.250367 + 0.250367i 0.821121 − 0.570754i \(-0.193349\pi\)
−0.570754 + 0.821121i \(0.693349\pi\)
\(24\) 1.15161e6 + 880262.i 0.144627 + 0.110549i
\(25\) −3.17315e6 − 9.23572e6i −0.324930 − 0.945738i
\(26\) −327651. + 1.48229e6i −0.0275769 + 0.124757i
\(27\) 3.63280e6 + 3.63280e6i 0.253176 + 0.253176i
\(28\) 6.58787e6 + 1.80304e7i 0.382785 + 1.04765i
\(29\) −1.47099e7 −0.717167 −0.358583 − 0.933498i \(-0.616740\pi\)
−0.358583 + 0.933498i \(0.616740\pi\)
\(30\) 2.96085e6 + 3.28649e6i 0.121846 + 0.135247i
\(31\) 4.06908e7 1.42131 0.710654 − 0.703542i \(-0.248399\pi\)
0.710654 + 0.703542i \(0.248399\pi\)
\(32\) 3.20326e7 + 9.99077e6i 0.954644 + 0.297748i
\(33\) −4.72033e6 − 4.72033e6i −0.120615 − 0.120615i
\(34\) 4.76910e6 − 2.15753e7i 0.104964 − 0.474856i
\(35\) 9.64939e6 + 5.77822e7i 0.183721 + 1.10015i
\(36\) 5.30156e7 + 2.46416e7i 0.876780 + 0.407527i
\(37\) 7.22496e7 + 7.22496e7i 1.04190 + 1.04190i 0.999083 + 0.0428187i \(0.0136338\pi\)
0.0428187 + 0.999083i \(0.486366\pi\)
\(38\) −7.52201e7 + 4.79861e7i −0.949327 + 0.605616i
\(39\) − 2.09851e6i − 0.0232588i
\(40\) 9.04869e7 + 4.79362e7i 0.883661 + 0.468127i
\(41\) 1.28666e8 1.11057 0.555285 − 0.831660i \(-0.312609\pi\)
0.555285 + 0.831660i \(0.312609\pi\)
\(42\) −1.42717e7 − 2.23714e7i −0.109202 − 0.171178i
\(43\) 2.06029e7 − 2.06029e7i 0.140148 − 0.140148i −0.633552 − 0.773700i \(-0.718404\pi\)
0.773700 + 0.633552i \(0.218404\pi\)
\(44\) −1.40134e8 − 6.51343e7i −0.849730 − 0.394954i
\(45\) 1.45214e8 + 1.03654e8i 0.786948 + 0.561725i
\(46\) −7.12070e7 − 1.57399e7i −0.345727 − 0.0764211i
\(47\) 5.20404e7 − 5.20404e7i 0.226909 − 0.226909i −0.584491 − 0.811400i \(-0.698706\pi\)
0.811400 + 0.584491i \(0.198706\pi\)
\(48\) −4.62175e7 − 3.92805e6i −0.181385 − 0.0154160i
\(49\) − 6.89506e7i − 0.244094i
\(50\) 2.44554e8 + 1.94549e8i 0.782574 + 0.622557i
\(51\) 3.05447e7i 0.0885288i
\(52\) −1.66713e7 − 4.56279e7i −0.0438483 − 0.120009i
\(53\) −2.51475e7 + 2.51475e7i −0.0601334 + 0.0601334i −0.736534 − 0.676401i \(-0.763539\pi\)
0.676401 + 0.736534i \(0.263539\pi\)
\(54\) −1.60527e8 − 3.54835e7i −0.349605 − 0.0772783i
\(55\) −3.83839e8 − 2.73985e8i −0.762669 − 0.544395i
\(56\) −4.88036e8 − 3.73043e8i −0.886159 − 0.677359i
\(57\) 8.72130e7 − 8.72130e7i 0.144946 − 0.144946i
\(58\) 3.96842e8 − 2.53162e8i 0.604613 − 0.385708i
\(59\) 9.05959e8 1.26721 0.633605 − 0.773657i \(-0.281575\pi\)
0.633605 + 0.773657i \(0.281575\pi\)
\(60\) −1.36439e8 − 3.77053e7i −0.175462 − 0.0484893i
\(61\) − 6.35746e8i − 0.752721i −0.926473 − 0.376361i \(-0.877175\pi\)
0.926473 − 0.376361i \(-0.122825\pi\)
\(62\) −1.09775e9 + 7.00302e8i −1.19825 + 0.764411i
\(63\) −7.56796e8 − 7.56796e8i −0.762564 − 0.762564i
\(64\) −1.03611e9 + 2.81761e8i −0.964956 + 0.262410i
\(65\) −2.44188e7 − 1.46224e8i −0.0210454 − 0.126024i
\(66\) 2.08583e8 + 4.61061e7i 0.166555 + 0.0368161i
\(67\) 9.08725e8 + 9.08725e8i 0.673068 + 0.673068i 0.958422 − 0.285355i \(-0.0921113\pi\)
−0.285355 + 0.958422i \(0.592111\pi\)
\(68\) 2.42658e8 + 6.64133e8i 0.166897 + 0.456784i
\(69\) 1.00809e8 0.0644549
\(70\) −1.25477e9 − 1.39277e9i −0.746576 − 0.828685i
\(71\) 1.73091e9 0.959361 0.479681 − 0.877443i \(-0.340752\pi\)
0.479681 + 0.877443i \(0.340752\pi\)
\(72\) −1.85434e9 + 2.47637e8i −0.958354 + 0.127983i
\(73\) 1.36206e9 + 1.36206e9i 0.657026 + 0.657026i 0.954675 − 0.297650i \(-0.0962027\pi\)
−0.297650 + 0.954675i \(0.596203\pi\)
\(74\) −3.19258e9 − 7.05701e8i −1.43874 − 0.318026i
\(75\) −3.88139e8 − 1.89632e8i −0.163561 − 0.0799108i
\(76\) 1.20342e9 − 2.58912e9i 0.474625 − 1.02114i
\(77\) 2.00041e9 + 2.00041e9i 0.739037 + 0.739037i
\(78\) 3.61160e7 + 5.66133e7i 0.0125091 + 0.0196085i
\(79\) 1.93358e9i 0.628387i 0.949359 + 0.314194i \(0.101734\pi\)
−0.949359 + 0.314194i \(0.898266\pi\)
\(80\) −3.26614e9 + 2.64093e8i −0.996747 + 0.0805947i
\(81\) −3.14398e9 −0.901684
\(82\) −3.47115e9 + 2.21439e9i −0.936275 + 0.597290i
\(83\) 4.07435e9 − 4.07435e9i 1.03435 − 1.03435i 0.0349634 − 0.999389i \(-0.488869\pi\)
0.999389 − 0.0349634i \(-0.0111315\pi\)
\(84\) 7.70039e8 + 3.57913e8i 0.184126 + 0.0855818i
\(85\) 3.55426e8 + 2.12835e9i 0.0801040 + 0.479676i
\(86\) −2.01240e8 + 9.10406e8i −0.0427782 + 0.193528i
\(87\) −4.60113e8 + 4.60113e8i −0.0923141 + 0.0923141i
\(88\) 4.90150e9 − 6.54571e8i 0.928787 − 0.124035i
\(89\) − 1.03269e10i − 1.84936i −0.380750 − 0.924678i \(-0.624334\pi\)
0.380750 − 0.924678i \(-0.375666\pi\)
\(90\) −5.70148e9 − 2.97186e8i −0.965551 − 0.0503287i
\(91\) 8.89320e8i 0.142512i
\(92\) 2.19190e9 − 8.00866e8i 0.332569 − 0.121512i
\(93\) 1.27277e9 − 1.27277e9i 0.182952 − 0.182952i
\(94\) −5.08308e8 + 2.29957e9i −0.0692608 + 0.313334i
\(95\) 5.06216e9 − 7.09183e9i 0.654211 − 0.916516i
\(96\) 1.31445e9 − 6.89447e8i 0.161209 − 0.0845561i
\(97\) −2.48030e9 + 2.48030e9i −0.288832 + 0.288832i −0.836618 − 0.547786i \(-0.815471\pi\)
0.547786 + 0.836618i \(0.315471\pi\)
\(98\) 1.18666e9 + 1.86014e9i 0.131279 + 0.205786i
\(99\) 8.61579e9 0.905981
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.13.11 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.19 yes 116
8.5 even 2 inner 40.11.i.a.13.19 yes 116
40.37 odd 4 inner 40.11.i.a.37.11 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.11 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.19 yes 116 8.5 even 2 inner
40.11.i.a.37.11 yes 116 40.37 odd 4 inner
40.11.i.a.37.19 yes 116 5.2 odd 4 inner