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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(53,80)] 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("80.53"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.t (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: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(118\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 53.12
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.12

$q$-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+(-30.4979 - 9.68897i) q^{2} +476.454 q^{3} +(836.248 + 590.987i) q^{4} +(-1108.33 - 2921.85i) q^{5} +(-14530.9 - 4616.35i) q^{6} +(-17569.8 - 17569.8i) q^{7} +(-19777.8 - 26126.3i) q^{8} +167960. q^{9} +(5492.17 + 99849.1i) q^{10} +(107181. + 107181. i) q^{11} +(398434. + 281579. i) q^{12} +233597. q^{13} +(365609. + 706075. i) q^{14} +(-528071. - 1.39213e6i) q^{15} +(350044. + 988423. i) q^{16} +(775339. - 775339. i) q^{17} +(-5.12243e6 - 1.62736e6i) q^{18} +(1.39381e6 + 1.39381e6i) q^{19} +(799935. - 3.09840e6i) q^{20} +(-8.37119e6 - 8.37119e6i) q^{21} +(-2.23033e6 - 4.30729e6i) q^{22} +(3.65447e6 - 3.65447e6i) q^{23} +(-9.42320e6 - 1.24480e7i) q^{24} +(-7.30881e6 + 6.47678e6i) q^{25} +(-7.12423e6 - 2.26331e6i) q^{26} +5.18911e7 q^{27} +(-4.30917e6 - 2.50762e7i) q^{28} +(-1.17796e7 - 1.17796e7i) q^{29} +(2.61677e6 + 4.75735e7i) q^{30} -1.86548e7 q^{31} +(-1.09881e6 - 3.35364e7i) q^{32} +(5.10671e7 + 5.10671e7i) q^{33} +(-3.11585e7 + 1.61340e7i) q^{34} +(-3.18631e7 + 7.08094e7i) q^{35} +(1.40456e8 + 9.92621e7i) q^{36} -2.66727e7 q^{37} +(-2.90038e7 - 5.60131e7i) q^{38} +1.11298e8 q^{39} +(-5.44167e7 + 8.67443e7i) q^{40} -1.69788e8i q^{41} +(1.74196e8 + 3.36412e8i) q^{42} -2.38078e8i q^{43} +(2.62873e7 + 1.52973e8i) q^{44} +(-1.86156e8 - 4.90754e8i) q^{45} +(-1.46862e8 + 7.60458e7i) q^{46} +(-8.92513e7 + 8.92513e7i) q^{47} +(1.66780e8 + 4.70939e8i) q^{48} +3.34918e8i q^{49} +(2.85657e8 - 1.26713e8i) q^{50} +(3.69414e8 - 3.69414e8i) q^{51} +(1.95345e8 + 1.38053e8i) q^{52} -3.51357e8i q^{53} +(-1.58257e9 - 5.02771e8i) q^{54} +(1.94375e8 - 4.31961e8i) q^{55} +(-1.11542e8 + 8.06523e8i) q^{56} +(6.64089e8 + 6.64089e8i) q^{57} +(2.45121e8 + 4.73386e8i) q^{58} +(-7.61707e8 + 7.61707e8i) q^{59} +(3.81133e8 - 1.47625e9i) q^{60} +(6.17929e7 - 6.17929e7i) q^{61} +(5.68933e8 + 1.80746e8i) q^{62} +(-2.95102e9 - 2.95102e9i) q^{63} +(-2.91422e8 + 1.03344e9i) q^{64} +(-2.58904e8 - 6.82536e8i) q^{65} +(-1.06265e9 - 2.05223e9i) q^{66} -8.82809e8i q^{67} +(1.10659e9 - 1.90160e8i) q^{68} +(1.74119e9 - 1.74119e9i) q^{69} +(1.65783e9 - 1.85082e9i) q^{70} -1.88675e8i q^{71} +(-3.32187e9 - 4.38816e9i) q^{72} +(8.04938e8 - 8.04938e8i) q^{73} +(8.13462e8 + 2.58431e8i) q^{74} +(-3.48232e9 + 3.08589e9i) q^{75} +(3.41847e8 + 1.98930e9i) q^{76} -3.76630e9i q^{77} +(-3.39437e9 - 1.07837e9i) q^{78} +9.53750e7i q^{79} +(2.50006e9 - 2.11828e9i) q^{80} +1.48059e10 q^{81} +(-1.64507e9 + 5.17819e9i) q^{82} -4.41176e9 q^{83} +(-2.05312e9 - 1.19477e10i) q^{84} +(-3.12476e9 - 1.40609e9i) q^{85} +(-2.30673e9 + 7.26088e9i) q^{86} +(-5.61244e9 - 5.61244e9i) q^{87} +(6.80442e8 - 4.92006e9i) q^{88} -6.63828e8 q^{89} +(9.22464e8 + 1.67706e10i) q^{90} +(-4.10424e9 - 4.10424e9i) q^{91} +(5.21579e9 - 8.96297e8i) q^{92} -8.88817e9 q^{93} +(3.58673e9 - 1.85723e9i) q^{94} +(2.52771e9 - 5.61733e9i) q^{95} +(-5.23535e8 - 1.59786e10i) q^{96} +(-4.18928e9 + 4.18928e9i) q^{97} +(3.24501e9 - 1.02143e10i) q^{98} +(1.80022e10 + 1.80022e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} - 4 q^{3} - 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} + 4487724 q^{9} - 2050 q^{10} - 4 q^{11} + 502036 q^{12} - 4 q^{13} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} - 5928762 q^{18} + 5107040 q^{19}+ \cdots + 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30.4979 − 9.68897i −0.953060 − 0.302780i
\(3\) 476.454 1.96072 0.980359 − 0.197221i \(-0.0631917\pi\)
0.980359 + 0.197221i \(0.0631917\pi\)
\(4\) 836.248 + 590.987i 0.816648 + 0.577136i
\(5\) −1108.33 − 2921.85i −0.354667 − 0.934993i
\(6\) −14530.9 − 4616.35i −1.86868 − 0.593667i
\(7\) −17569.8 − 17569.8i −1.04538 − 1.04538i −0.998920 − 0.0464639i \(-0.985205\pi\)
−0.0464639 − 0.998920i \(-0.514795\pi\)
\(8\) −19777.8 − 26126.3i −0.603569 − 0.797310i
\(9\) 167960. 2.84442
\(10\) 5492.17 + 99849.1i 0.0549217 + 0.998491i
\(11\) 107181. + 107181.i 0.665512 + 0.665512i 0.956674 − 0.291162i \(-0.0940418\pi\)
−0.291162 + 0.956674i \(0.594042\pi\)
\(12\) 398434. + 281579.i 1.60122 + 1.13160i
\(13\) 233597. 0.629145 0.314572 − 0.949234i \(-0.398139\pi\)
0.314572 + 0.949234i \(0.398139\pi\)
\(14\) 365609. + 706075.i 0.679792 + 1.31284i
\(15\) −528071. − 1.39213e6i −0.695402 − 1.83326i
\(16\) 350044. + 988423.i 0.333828 + 0.942634i
\(17\) 775339. − 775339.i 0.546068 − 0.546068i −0.379233 − 0.925301i \(-0.623812\pi\)
0.925301 + 0.379233i \(0.123812\pi\)
\(18\) −5.12243e6 − 1.62736e6i −2.71090 − 0.861233i
\(19\) 1.39381e6 + 1.39381e6i 0.562907 + 0.562907i 0.930132 − 0.367225i \(-0.119692\pi\)
−0.367225 + 0.930132i \(0.619692\pi\)
\(20\) 799935. − 3.09840e6i 0.249980 − 0.968251i
\(21\) −8.37119e6 − 8.37119e6i −2.04970 − 2.04970i
\(22\) −2.23033e6 − 4.30729e6i −0.432769 − 0.835777i
\(23\) 3.65447e6 − 3.65447e6i 0.567787 − 0.567787i −0.363721 − 0.931508i \(-0.618494\pi\)
0.931508 + 0.363721i \(0.118494\pi\)
\(24\) −9.42320e6 − 1.24480e7i −1.18343 − 1.56330i
\(25\) −7.30881e6 + 6.47678e6i −0.748422 + 0.663222i
\(26\) −7.12423e6 − 2.26331e6i −0.599613 − 0.190493i
\(27\) 5.18911e7 3.61638
\(28\) −4.30917e6 − 2.50762e7i −0.250382 − 1.45704i
\(29\) −1.17796e7 − 1.17796e7i −0.574302 − 0.574302i 0.359025 − 0.933328i \(-0.383109\pi\)
−0.933328 + 0.359025i \(0.883109\pi\)
\(30\) 2.61677e6 + 4.75735e7i 0.107686 + 1.95776i
\(31\) −1.86548e7 −0.651602 −0.325801 − 0.945438i \(-0.605634\pi\)
−0.325801 + 0.945438i \(0.605634\pi\)
\(32\) −1.09881e6 − 3.35364e7i −0.0327472 − 0.999464i
\(33\) 5.10671e7 + 5.10671e7i 1.30488 + 1.30488i
\(34\) −3.11585e7 + 1.61340e7i −0.685775 + 0.355097i
\(35\) −3.18631e7 + 7.08094e7i −0.606663 + 1.34819i
\(36\) 1.40456e8 + 9.92621e7i 2.32289 + 1.64161i
\(37\) −2.66727e7 −0.384643 −0.192322 − 0.981332i \(-0.561602\pi\)
−0.192322 + 0.981332i \(0.561602\pi\)
\(38\) −2.90038e7 − 5.60131e7i −0.366047 − 0.706922i
\(39\) 1.11298e8 1.23358
\(40\) −5.44167e7 + 8.67443e7i −0.531413 + 0.847113i
\(41\) − 1.69788e8i − 1.46551i −0.680494 − 0.732754i \(-0.738235\pi\)
0.680494 − 0.732754i \(-0.261765\pi\)
\(42\) 1.74196e8 + 3.36412e8i 1.33288 + 2.57410i
\(43\) − 2.38078e8i − 1.61948i −0.586786 − 0.809742i \(-0.699607\pi\)
0.586786 − 0.809742i \(-0.300393\pi\)
\(44\) 2.62873e7 + 1.52973e8i 0.159398 + 0.927580i
\(45\) −1.86156e8 − 4.90754e8i −1.00882 − 2.65951i
\(46\) −1.46862e8 + 7.60458e7i −0.713050 + 0.369220i
\(47\) −8.92513e7 + 8.92513e7i −0.389157 + 0.389157i −0.874387 − 0.485229i \(-0.838736\pi\)
0.485229 + 0.874387i \(0.338736\pi\)
\(48\) 1.66780e8 + 4.70939e8i 0.654543 + 1.84824i
\(49\) 3.34918e8i 1.18565i
\(50\) 2.85657e8 − 1.26713e8i 0.914102 − 0.405483i
\(51\) 3.69414e8 − 3.69414e8i 1.07069 − 1.07069i
\(52\) 1.95345e8 + 1.38053e8i 0.513790 + 0.363102i
\(53\) − 3.51357e8i − 0.840175i −0.907484 − 0.420087i \(-0.861999\pi\)
0.907484 − 0.420087i \(-0.138001\pi\)
\(54\) −1.58257e9 − 5.02771e8i −3.44663 − 1.09497i
\(55\) 1.94375e8 − 4.31961e8i 0.386214 − 0.858284i
\(56\) −1.11542e8 + 8.06523e8i −0.202534 + 1.46446i
\(57\) 6.64089e8 + 6.64089e8i 1.10370 + 1.10370i
\(58\) 2.45121e8 + 4.73386e8i 0.373457 + 0.721232i
\(59\) −7.61707e8 + 7.61707e8i −1.06544 + 1.06544i −0.0677336 + 0.997703i \(0.521577\pi\)
−0.997703 + 0.0677336i \(0.978423\pi\)
\(60\) 3.81133e8 − 1.47625e9i 0.490140 − 1.89847i
\(61\) 6.17929e7 − 6.17929e7i 0.0731627 − 0.0731627i −0.669579 − 0.742741i \(-0.733525\pi\)
0.742741 + 0.669579i \(0.233525\pi\)
\(62\) 5.68933e8 + 1.80746e8i 0.621016 + 0.197292i
\(63\) −2.95102e9 − 2.95102e9i −2.97351 − 2.97351i
\(64\) −2.91422e8 + 1.03344e9i −0.271408 + 0.962464i
\(65\) −2.58904e8 − 6.82536e8i −0.223137 − 0.588246i
\(66\) −1.06265e9 − 2.05223e9i −0.848538 − 1.63872i
\(67\) − 8.82809e8i − 0.653872i −0.945046 − 0.326936i \(-0.893984\pi\)
0.945046 − 0.326936i \(-0.106016\pi\)
\(68\) 1.10659e9 − 1.90160e8i 0.761101 − 0.130790i
\(69\) 1.74119e9 − 1.74119e9i 1.11327 − 1.11327i
\(70\) 1.65783e9 − 1.85082e9i 0.986392 − 1.10122i
\(71\) − 1.88675e8i − 0.104574i −0.998632 − 0.0522870i \(-0.983349\pi\)
0.998632 − 0.0522870i \(-0.0166511\pi\)
\(72\) −3.32187e9 − 4.38816e9i −1.71680 − 2.26788i
\(73\) 8.04938e8 − 8.04938e8i 0.388283 − 0.388283i −0.485792 − 0.874075i \(-0.661469\pi\)
0.874075 + 0.485792i \(0.161469\pi\)
\(74\) 8.13462e8 + 2.58431e8i 0.366588 + 0.116462i
\(75\) −3.48232e9 + 3.08589e9i −1.46745 + 1.30039i
\(76\) 3.41847e8 + 1.98930e9i 0.134823 + 0.784571i
\(77\) − 3.76630e9i − 1.39143i
\(78\) −3.39437e9 − 1.07837e9i −1.17567 − 0.373502i
\(79\) 9.53750e7i 0.0309955i 0.999880 + 0.0154978i \(0.00493329\pi\)
−0.999880 + 0.0154978i \(0.995067\pi\)
\(80\) 2.50006e9 − 2.11828e9i 0.762958 − 0.646448i
\(81\) 1.48059e10 4.24628
\(82\) −1.64507e9 + 5.17819e9i −0.443727 + 1.39672i
\(83\) −4.41176e9 −1.12001 −0.560004 − 0.828490i \(-0.689201\pi\)
−0.560004 + 0.828490i \(0.689201\pi\)
\(84\) −2.05312e9 − 1.19477e10i −0.490929 − 2.85684i
\(85\) −3.12476e9 − 1.40609e9i −0.704242 − 0.316897i
\(86\) −2.30673e9 + 7.26088e9i −0.490348 + 1.54347i
\(87\) −5.61244e9 − 5.61244e9i −1.12604 − 1.12604i
\(88\) 6.80442e8 − 4.92006e9i 0.128937 − 0.932303i
\(89\) −6.63828e8 −0.118879 −0.0594396 − 0.998232i \(-0.518931\pi\)
−0.0594396 + 0.998232i \(0.518931\pi\)
\(90\) 9.22464e8 + 1.67706e10i 0.156220 + 2.84012i
\(91\) −4.10424e9 − 4.10424e9i −0.657698 − 0.657698i
\(92\) 5.21579e9 − 8.96297e8i 0.791372 − 0.135992i
\(93\) −8.88817e9 −1.27761
\(94\) 3.58673e9 − 1.85723e9i 0.488720 − 0.253061i
\(95\) 2.52771e9 − 5.61733e9i 0.326669 − 0.725959i
\(96\) −5.23535e8 − 1.59786e10i −0.0642081 − 1.95967i
\(97\) −4.18928e9 + 4.18928e9i −0.487843 + 0.487843i −0.907625 − 0.419782i \(-0.862107\pi\)
0.419782 + 0.907625i \(0.362107\pi\)
\(98\) 3.24501e9 − 1.02143e10i 0.358993 − 1.13000i
\(99\) 1.80022e10 + 1.80022e10i 1.89299 + 1.89299i
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 80.11.t.a.53.12 yes 236
5.2 odd 4 80.11.i.a.37.70 yes 236
16.13 even 4 80.11.i.a.13.70 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.12 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.70 ✓ 236 16.13 even 4
80.11.i.a.37.70 yes 236 5.2 odd 4
80.11.t.a.53.12 yes 236 1.1 even 1 trivial
80.11.t.a.77.12 yes 236 80.77 odd 4 inner