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 13.19
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.19

$q$-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+(-17.2103 + 26.9779i) 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} +(32479.7 + 4337.49i) q^{8} +57092.2i q^{9} +(-37371.9 - 92754.2i) q^{10} +150910. i q^{11} +(42546.0 + 15545.2i) 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} +(-1.54023e6 - 982575. i) q^{18} -2.78822e6 q^{19} +(3.14549e6 + 588115. i) q^{20} -829252. i q^{21} +(-4.07123e6 - 2.59721e6i) 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} +(1.80304e7 + 6.58787e6i) q^{28} +1.47099e7 q^{29} +(4.07023e6 + 1.73231e6i) q^{30} +4.06908e7 q^{31} +(-9.99077e6 - 3.20326e7i) 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} +(4.79861e7 - 7.52201e7i) q^{38} -2.09851e6i q^{39} +(-7.00010e7 + 7.47370e7i) q^{40} +1.28666e8 q^{41} +(2.23714e7 + 1.42717e7i) 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} +(-3.92805e6 - 4.62175e7i) q^{48} -6.89506e7i q^{49} +(3.03771e8 + 7.33450e7i) q^{50} -3.05447e7i q^{51} +(4.56279e7 + 1.66713e7i) 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} +(-2.53162e8 + 3.96842e8i) q^{58} -9.05959e8 q^{59} +(-1.16784e8 + 7.99925e7i) q^{60} +6.35746e8i q^{61} +(-7.00302e8 + 1.09775e9i) 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} +(6.64133e8 + 2.42658e8i) q^{68} -1.00809e8 q^{69} +(1.72491e9 + 7.34130e8i) q^{70} +1.73091e9 q^{71} +(-2.47637e8 + 1.85434e9i) 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} +(5.66133e7 + 3.61160e7i) q^{78} +1.93358e9i q^{79} +(-8.11505e8 - 3.17472e9i) q^{80} -3.14398e9 q^{81} +(-2.21439e9 + 3.47115e9i) 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} +(-6.54571e8 + 4.90150e9i) q^{88} -1.03269e10i q^{89} +(5.29554e9 - 2.13364e9i) q^{90} -8.89320e8i q^{91} +(8.00866e8 - 2.19190e9i) 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} +(1.86014e9 + 1.18666e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −17.2103 + 26.9779i −0.537822 + 0.843058i
\(3\) −31.2791 + 31.2791i −0.128721 + 0.128721i −0.768532 − 0.639811i \(-0.779012\pi\)
0.639811 + 0.768532i \(0.279012\pi\)
\(4\) −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 \(0.438314\pi\)
\(8\) 32479.7 + 4337.49i 0.991200 + 0.132370i
\(9\) 57092.2i 0.966862i
\(10\) −37371.9 − 92754.2i −0.373719 − 0.927542i
\(11\) 150910.i 0.937032i 0.883455 + 0.468516i \(0.155211\pi\)
−0.883455 + 0.468516i \(0.844789\pi\)
\(12\) 42546.0 + 15545.2i 0.170983 + 0.0624729i
\(13\) −33544.9 + 33544.9i −0.0903462 + 0.0903462i −0.750835 − 0.660489i \(-0.770349\pi\)
0.660489 + 0.750835i \(0.270349\pi\)
\(14\) −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 \(0.328184\pi\)
\(18\) −1.54023e6 − 982575.i −0.815121 − 0.520000i
\(19\) −2.78822e6 −1.12605 −0.563026 − 0.826439i \(-0.690363\pi\)
−0.563026 + 0.826439i \(0.690363\pi\)
\(20\) 3.14549e6 + 588115.i 0.982966 + 0.183786i
\(21\) − 829252.i − 0.203044i
\(22\) −4.07123e6 − 2.59721e6i −0.789973 − 0.503957i
\(23\) 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
\(28\) 1.80304e7 + 6.58787e6i 1.04765 + 0.382785i
\(29\) 1.47099e7 0.717167 0.358583 − 0.933498i \(-0.383260\pi\)
0.358583 + 0.933498i \(0.383260\pi\)
\(30\) 4.07023e6 + 1.73231e6i 0.167499 + 0.0712885i
\(31\) 4.06908e7 1.42131 0.710654 − 0.703542i \(-0.248399\pi\)
0.710654 + 0.703542i \(0.248399\pi\)
\(32\) −9.99077e6 − 3.20326e7i −0.297748 − 0.954644i
\(33\) −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 \(-0.986366\pi\)
−0.0428187 − 0.999083i \(-0.513634\pi\)
\(38\) 4.79861e7 − 7.52201e7i 0.605616 − 0.949327i
\(39\) − 2.09851e6i − 0.0232588i
\(40\) −7.00010e7 + 7.47370e7i −0.683604 + 0.729854i
\(41\) 1.28666e8 1.11057 0.555285 − 0.831660i \(-0.312609\pi\)
0.555285 + 0.831660i \(0.312609\pi\)
\(42\) 2.23714e7 + 1.42717e7i 0.171178 + 0.109202i
\(43\) −2.06029e7 + 2.06029e7i −0.140148 + 0.140148i −0.773700 − 0.633552i \(-0.781596\pi\)
0.633552 + 0.773700i \(0.281596\pi\)
\(44\) 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 \(0.198706\pi\)
\(48\) −3.92805e6 − 4.62175e7i −0.0154160 − 0.181385i
\(49\) − 6.89506e7i − 0.244094i
\(50\) 3.03771e8 + 7.33450e7i 0.972067 + 0.234704i
\(51\) − 3.05447e7i − 0.0885288i
\(52\) 4.56279e7 + 1.66713e7i 0.120009 + 0.0438483i
\(53\) 2.51475e7 − 2.51475e7i 0.0601334 − 0.0601334i −0.676401 − 0.736534i \(-0.736461\pi\)
0.736534 + 0.676401i \(0.236461\pi\)
\(54\) 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
\(58\) −2.53162e8 + 3.96842e8i −0.385708 + 0.604613i
\(59\) −9.05959e8 −1.26721 −0.633605 − 0.773657i \(-0.718425\pi\)
−0.633605 + 0.773657i \(0.718425\pi\)
\(60\) −1.16784e8 + 7.99925e7i −0.150185 + 0.102871i
\(61\) 6.35746e8i 0.752721i 0.926473 + 0.376361i \(0.122825\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(62\) −7.00302e8 + 1.09775e9i −0.764411 + 1.19825i
\(63\) −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 0.285355 − 0.958422i \(-0.407889\pi\)
−0.958422 + 0.285355i \(0.907889\pi\)
\(68\) 6.64133e8 + 2.42658e8i 0.456784 + 0.166897i
\(69\) −1.00809e8 −0.0644549
\(70\) 1.72491e9 + 7.34130e8i 1.02630 + 0.436800i
\(71\) 1.73091e9 0.959361 0.479681 − 0.877443i \(-0.340752\pi\)
0.479681 + 0.877443i \(0.340752\pi\)
\(72\) −2.47637e8 + 1.85434e9i −0.127983 + 0.958354i
\(73\) 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
\(78\) 5.66133e7 + 3.61160e7i 0.0196085 + 0.0125091i
\(79\) 1.93358e9i 0.628387i 0.949359 + 0.314194i \(0.101734\pi\)
−0.949359 + 0.314194i \(0.898266\pi\)
\(80\) −8.11505e8 − 3.17472e9i −0.247652 − 0.968849i
\(81\) −3.14398e9 −0.901684
\(82\) −2.21439e9 + 3.47115e9i −0.597290 + 0.936275i
\(83\) −4.07435e9 + 4.07435e9i −1.03435 + 1.03435i −0.0349634 + 0.999389i \(0.511131\pi\)
−0.999389 + 0.0349634i \(0.988869\pi\)
\(84\) −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
\(88\) −6.54571e8 + 4.90150e9i −0.124035 + 0.928787i
\(89\) − 1.03269e10i − 1.84936i −0.380750 − 0.924678i \(-0.624334\pi\)
0.380750 − 0.924678i \(-0.375666\pi\)
\(90\) 5.29554e9 − 2.13364e9i 0.896805 − 0.361335i
\(91\) − 8.89320e8i − 0.142512i
\(92\) 8.00866e8 − 2.19190e9i 0.121512 − 0.332569i
\(93\) −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 \(0.315471\pi\)
\(98\) 1.86014e9 + 1.18666e9i 0.205786 + 0.131279i
\(99\) −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.13.19 yes 116
5.2 odd 4 inner 40.11.i.a.37.11 yes 116
8.5 even 2 inner 40.11.i.a.13.11 ✓ 116
40.37 odd 4 inner 40.11.i.a.37.19 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.11 ✓ 116 8.5 even 2 inner
40.11.i.a.13.19 yes 116 1.1 even 1 trivial
40.11.i.a.37.11 yes 116 5.2 odd 4 inner
40.11.i.a.37.19 yes 116 40.37 odd 4 inner