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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 37.19
Character \(\chi\) \(=\) 40.37
Dual form 40.11.i.a.13.19

$q$-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 \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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 0.639811 − 0.768532i \(-0.279012\pi\)
−0.768532 + 0.639811i \(0.779012\pi\)
\(4\) −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 0.192581 − 0.981281i \(-0.438314\pi\)
−0.981281 + 0.192581i \(0.938314\pi\)
\(8\) 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 \(-0.844789\pi\)
0.883455 − 0.468516i \(-0.155211\pi\)
\(12\) 42546.0 − 15545.2i 0.170983 − 0.0624729i
\(13\) −33544.9 − 33544.9i −0.0903462 − 0.0903462i 0.660489 − 0.750835i \(-0.270349\pi\)
−0.750835 + 0.660489i \(0.770349\pi\)
\(14\) −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 0.513944 − 0.857824i \(-0.328184\pi\)
−0.857824 + 0.513944i \(0.828184\pi\)
\(18\) −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 −0.570754 − 0.821121i \(-0.693349\pi\)
0.821121 + 0.570754i \(0.193349\pi\)
\(24\) −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 −0.0428187 + 0.999083i \(0.513634\pi\)
−0.999083 + 0.0428187i \(0.986366\pi\)
\(38\) 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 0.633552 − 0.773700i \(-0.281596\pi\)
−0.773700 + 0.633552i \(0.781596\pi\)
\(44\) 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 0.811400 − 0.584491i \(-0.198706\pi\)
−0.584491 + 0.811400i \(0.698706\pi\)
\(48\) −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 0.736534 − 0.676401i \(-0.236461\pi\)
−0.676401 + 0.736534i \(0.736461\pi\)
\(54\) 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 \(-0.877175\pi\)
0.926473 − 0.376361i \(-0.122825\pi\)
\(62\) −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 −0.958422 − 0.285355i \(-0.907889\pi\)
0.285355 + 0.958422i \(0.407889\pi\)
\(68\) 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 −0.297650 − 0.954675i \(-0.596203\pi\)
0.954675 + 0.297650i \(0.0962027\pi\)
\(74\) 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 \(-0.898266\pi\)
0.949359 − 0.314194i \(-0.101734\pi\)
\(80\) −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 −0.999389 − 0.0349634i \(-0.988869\pi\)
−0.0349634 − 0.999389i \(-0.511131\pi\)
\(84\) −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 \(0.375666\pi\)
−0.380750 + 0.924678i \(0.624334\pi\)
\(90\) 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 0.547786 − 0.836618i \(-0.315471\pi\)
−0.836618 + 0.547786i \(0.815471\pi\)
\(98\) 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.37.19 yes 116
5.3 odd 4 inner 40.11.i.a.13.11 ✓ 116
8.5 even 2 inner 40.11.i.a.37.11 yes 116
40.13 odd 4 inner 40.11.i.a.13.19 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.11 ✓ 116 5.3 odd 4 inner
40.11.i.a.13.19 yes 116 40.13 odd 4 inner
40.11.i.a.37.11 yes 116 8.5 even 2 inner
40.11.i.a.37.19 yes 116 1.1 even 1 trivial