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Properties

Label 80.11.i.a.13.12
Level $80$
Weight $11$
Character 80.13
Analytic conductor $50.829$
Analytic rank $0$
Dimension $236$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(13,80)] 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("80.13"); 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, 3, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.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: \(50.8285802139\)
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 13.12
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.12

$q$-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+(-31.0538 + 7.72412i) q^{2} -201.522i q^{3} +(904.676 - 479.726i) q^{4} +(1379.87 - 2803.85i) q^{5} +(1556.58 + 6258.01i) q^{6} +(4671.61 + 4671.61i) q^{7} +(-24388.2 + 21885.1i) q^{8} +18438.0 q^{9} +(-21192.8 + 97728.5i) q^{10} +(-123128. + 123128. i) q^{11} +(-96675.2 - 182312. i) q^{12} -543474. i q^{13} +(-181155. - 108987. i) q^{14} +(-565037. - 278073. i) q^{15} +(588302. - 867994. i) q^{16} +(1.73799e6 - 1.73799e6i) q^{17} +(-572570. + 142417. i) q^{18} +(-2.22474e6 + 2.22474e6i) q^{19} +(-96750.2 - 3.19854e6i) q^{20} +(941430. - 941430. i) q^{21} +(2.87254e6 - 4.77466e6i) q^{22} +(-2.69000e6 + 2.69000e6i) q^{23} +(4.41033e6 + 4.91475e6i) q^{24} +(-5.95757e6 - 7.73788e6i) q^{25} +(4.19785e6 + 1.68769e7i) q^{26} -1.56153e7i q^{27} +(6.46738e6 + 1.98520e6i) q^{28} +(1.86559e7 - 1.86559e7i) q^{29} +(1.96944e7 + 4.27081e6i) q^{30} +8.40566e6 q^{31} +(-1.15645e7 + 3.14986e7i) q^{32} +(2.48130e7 + 2.48130e7i) q^{33} +(-4.05467e7 + 6.73955e7i) q^{34} +(1.95447e7 - 6.65231e6i) q^{35} +(1.66804e7 - 8.84519e6i) q^{36} +4.53311e7i q^{37} +(5.19026e7 - 8.62710e7i) q^{38} -1.09522e8 q^{39} +(2.77103e7 + 9.85794e7i) q^{40} -6.20396e7i q^{41} +(-2.19633e7 + 3.65067e7i) q^{42} +2.26948e8 q^{43} +(-5.23234e7 + 1.70459e8i) q^{44} +(2.54420e7 - 5.16974e7i) q^{45} +(6.27568e7 - 1.04312e8i) q^{46} +(-1.06147e8 + 1.06147e8i) q^{47} +(-1.74920e8 - 1.18556e8i) q^{48} -2.38827e8i q^{49} +(2.44773e8 + 1.94274e8i) q^{50} +(-3.50242e8 - 3.50242e8i) q^{51} +(-2.60718e8 - 4.91668e8i) q^{52} -3.10769e8 q^{53} +(1.20614e8 + 4.84915e8i) q^{54} +(1.75333e8 + 5.15134e8i) q^{55} +(-2.16171e8 - 1.16931e7i) q^{56} +(4.48334e8 + 4.48334e8i) q^{57} +(-4.35235e8 + 7.23435e8i) q^{58} +(-5.37371e8 - 5.37371e8i) q^{59} +(-6.44575e8 + 1.94973e7i) q^{60} +(-3.03567e8 - 3.03567e8i) q^{61} +(-2.61028e8 + 6.49263e7i) q^{62} +(8.61350e7 + 8.61350e7i) q^{63} +(1.15823e8 - 1.06748e9i) q^{64} +(-1.52382e9 - 7.49921e8i) q^{65} +(-9.62197e8 - 5.78880e8i) q^{66} -1.14845e9 q^{67} +(7.38558e8 - 2.40607e9i) q^{68} +(5.42093e8 + 5.42093e8i) q^{69} +(-5.55554e8 + 3.57545e8i) q^{70} -2.18924e9i q^{71} +(-4.49669e8 + 4.03518e8i) q^{72} +(1.27300e9 - 1.27300e9i) q^{73} +(-3.50143e8 - 1.40770e9i) q^{74} +(-1.55935e9 + 1.20058e9i) q^{75} +(-9.45405e8 + 3.07994e9i) q^{76} -1.15041e9 q^{77} +(3.40107e9 - 8.45959e8i) q^{78} +3.32501e9i q^{79} +(-1.62195e9 - 2.84723e9i) q^{80} -2.05808e9 q^{81} +(4.79201e8 + 1.92657e9i) q^{82} -1.71712e9i q^{83} +(4.00061e8 - 1.30332e9i) q^{84} +(-2.47487e9 - 7.27125e9i) q^{85} +(-7.04760e9 + 1.75297e9i) q^{86} +(-3.75956e9 - 3.75956e9i) q^{87} +(3.08193e8 - 5.69755e9i) q^{88} -7.02195e9 q^{89} +(-3.90752e8 + 1.80192e9i) q^{90} +(2.53889e9 - 2.53889e9i) q^{91} +(-1.14311e9 + 3.72404e9i) q^{92} -1.69392e9i q^{93} +(2.47638e9 - 4.11617e9i) q^{94} +(3.16801e9 + 9.30771e9i) q^{95} +(6.34765e9 + 2.33050e9i) q^{96} +(-7.54228e9 + 7.54228e9i) q^{97} +(1.84473e9 + 7.41650e9i) q^{98} +(-2.27024e9 + 2.27024e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} + 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} - 4487724 q^{9} + 2046 q^{10} - 4 q^{11} + 738232 q^{12} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} + 5924662 q^{18} - 5107040 q^{19} + 2913160 q^{20}+ \cdots - 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.0538 + 7.72412i −0.970431 + 0.241379i
\(3\) − 201.522i − 0.829308i −0.909979 − 0.414654i \(-0.863903\pi\)
0.909979 − 0.414654i \(-0.136097\pi\)
\(4\) 904.676 − 479.726i 0.883473 − 0.468483i
\(5\) 1379.87 − 2803.85i 0.441557 − 0.897233i
\(6\) 1556.58 + 6258.01i 0.200177 + 0.804786i
\(7\) 4671.61 + 4671.61i 0.277956 + 0.277956i 0.832293 − 0.554337i \(-0.187028\pi\)
−0.554337 + 0.832293i \(0.687028\pi\)
\(8\) −24388.2 + 21885.1i −0.744268 + 0.667881i
\(9\) 18438.0 0.312249
\(10\) −21192.8 + 97728.5i −0.211928 + 0.977285i
\(11\) −123128. + 123128.i −0.764530 + 0.764530i −0.977138 − 0.212608i \(-0.931804\pi\)
0.212608 + 0.977138i \(0.431804\pi\)
\(12\) −96675.2 − 182312.i −0.388516 − 0.732671i
\(13\) − 543474.i − 1.46373i −0.681448 − 0.731866i \(-0.738650\pi\)
0.681448 − 0.731866i \(-0.261350\pi\)
\(14\) −181155. − 108987.i −0.336830 − 0.202644i
\(15\) −565037. − 278073.i −0.744082 − 0.366187i
\(16\) 588302. − 867994.i 0.561048 − 0.827783i
\(17\) 1.73799e6 − 1.73799e6i 1.22406 − 1.22406i 0.257882 − 0.966177i \(-0.416976\pi\)
0.966177 − 0.257882i \(-0.0830244\pi\)
\(18\) −572570. + 142417.i −0.303016 + 0.0753702i
\(19\) −2.22474e6 + 2.22474e6i −0.898488 + 0.898488i −0.995302 − 0.0968145i \(-0.969135\pi\)
0.0968145 + 0.995302i \(0.469135\pi\)
\(20\) −96750.2 − 3.19854e6i −0.0302344 − 0.999543i
\(21\) 941430. − 941430.i 0.230511 − 0.230511i
\(22\) 2.87254e6 − 4.77466e6i 0.557382 − 0.926465i
\(23\) −2.69000e6 + 2.69000e6i −0.417939 + 0.417939i −0.884493 − 0.466554i \(-0.845495\pi\)
0.466554 + 0.884493i \(0.345495\pi\)
\(24\) 4.41033e6 + 4.91475e6i 0.553879 + 0.617227i
\(25\) −5.95757e6 − 7.73788e6i −0.610055 − 0.792359i
\(26\) 4.19785e6 + 1.68769e7i 0.353314 + 1.42045i
\(27\) − 1.56153e7i − 1.08826i
\(28\) 6.46738e6 + 1.98520e6i 0.375784 + 0.115349i
\(29\) 1.86559e7 − 1.86559e7i 0.909547 − 0.909547i −0.0866884 − 0.996235i \(-0.527628\pi\)
0.996235 + 0.0866884i \(0.0276285\pi\)
\(30\) 1.96944e7 + 4.27081e6i 0.810470 + 0.175753i
\(31\) 8.40566e6 0.293605 0.146802 − 0.989166i \(-0.453102\pi\)
0.146802 + 0.989166i \(0.453102\pi\)
\(32\) −1.15645e7 + 3.14986e7i −0.344649 + 0.938731i
\(33\) 2.48130e7 + 2.48130e7i 0.634030 + 0.634030i
\(34\) −4.05467e7 + 6.73955e7i −0.892403 + 1.48333i
\(35\) 1.95447e7 − 6.65231e6i 0.372125 − 0.126658i
\(36\) 1.66804e7 − 8.84519e6i 0.275863 − 0.146283i
\(37\) 4.53311e7i 0.653714i 0.945074 + 0.326857i \(0.105990\pi\)
−0.945074 + 0.326857i \(0.894010\pi\)
\(38\) 5.19026e7 − 8.62710e7i 0.655045 − 1.08880i
\(39\) −1.09522e8 −1.21388
\(40\) 2.77103e7 + 9.85794e7i 0.270609 + 0.962689i
\(41\) − 6.20396e7i − 0.535488i −0.963490 − 0.267744i \(-0.913722\pi\)
0.963490 − 0.267744i \(-0.0862782\pi\)
\(42\) −2.19633e7 + 3.65067e7i −0.168055 + 0.279335i
\(43\) 2.26948e8 1.54378 0.771888 − 0.635759i \(-0.219313\pi\)
0.771888 + 0.635759i \(0.219313\pi\)
\(44\) −5.23234e7 + 1.70459e8i −0.317272 + 1.03361i
\(45\) 2.54420e7 − 5.16974e7i 0.137876 − 0.280160i
\(46\) 6.27568e7 − 1.04312e8i 0.304699 − 0.506462i
\(47\) −1.06147e8 + 1.06147e8i −0.462828 + 0.462828i −0.899581 − 0.436753i \(-0.856128\pi\)
0.436753 + 0.899581i \(0.356128\pi\)
\(48\) −1.74920e8 − 1.18556e8i −0.686487 − 0.465281i
\(49\) − 2.38827e8i − 0.845481i
\(50\) 2.44773e8 + 1.94274e8i 0.783275 + 0.621676i
\(51\) −3.50242e8 − 3.50242e8i −1.01512 − 1.01512i
\(52\) −2.60718e8 − 4.91668e8i −0.685733 − 1.29317i
\(53\) −3.10769e8 −0.743119 −0.371559 − 0.928409i \(-0.621177\pi\)
−0.371559 + 0.928409i \(0.621177\pi\)
\(54\) 1.20614e8 + 4.84915e8i 0.262682 + 1.05608i
\(55\) 1.75333e8 + 5.15134e8i 0.348378 + 1.02354i
\(56\) −2.16171e8 − 1.16931e7i −0.392515 − 0.0212320i
\(57\) 4.48334e8 + 4.48334e8i 0.745123 + 0.745123i
\(58\) −4.35235e8 + 7.23435e8i −0.663107 + 1.10220i
\(59\) −5.37371e8 − 5.37371e8i −0.751647 − 0.751647i 0.223140 − 0.974787i \(-0.428369\pi\)
−0.974787 + 0.223140i \(0.928369\pi\)
\(60\) −6.44575e8 + 1.94973e7i −0.828928 + 0.0250737i
\(61\) −3.03567e8 − 3.03567e8i −0.359423 − 0.359423i 0.504177 − 0.863600i \(-0.331796\pi\)
−0.863600 + 0.504177i \(0.831796\pi\)
\(62\) −2.61028e8 + 6.49263e7i −0.284923 + 0.0708699i
\(63\) 8.61350e7 + 8.61350e7i 0.0867915 + 0.0867915i
\(64\) 1.15823e8 − 1.06748e9i 0.107869 − 0.994165i
\(65\) −1.52382e9 − 7.49921e8i −1.31331 − 0.646321i
\(66\) −9.62197e8 − 5.78880e8i −0.768324 − 0.462241i
\(67\) −1.14845e9 −0.850621 −0.425311 − 0.905047i \(-0.639835\pi\)
−0.425311 + 0.905047i \(0.639835\pi\)
\(68\) 7.38558e8 − 2.40607e9i 0.507972 − 1.65487i
\(69\) 5.42093e8 + 5.42093e8i 0.346600 + 0.346600i
\(70\) −5.55554e8 + 3.57545e8i −0.330549 + 0.212736i
\(71\) − 2.18924e9i − 1.21339i −0.794934 − 0.606696i \(-0.792494\pi\)
0.794934 − 0.606696i \(-0.207506\pi\)
\(72\) −4.49669e8 + 4.03518e8i −0.232397 + 0.208545i
\(73\) 1.27300e9 − 1.27300e9i 0.614065 − 0.614065i −0.329938 − 0.944003i \(-0.607028\pi\)
0.944003 + 0.329938i \(0.107028\pi\)
\(74\) −3.50143e8 − 1.40770e9i −0.157793 − 0.634384i
\(75\) −1.55935e9 + 1.20058e9i −0.657110 + 0.505923i
\(76\) −9.45405e8 + 3.07994e9i −0.372864 + 1.21472i
\(77\) −1.15041e9 −0.425011
\(78\) 3.40107e9 − 8.45959e8i 1.17799 − 0.293006i
\(79\) 3.32501e9i 1.08058i 0.841478 + 0.540291i \(0.181686\pi\)
−0.841478 + 0.540291i \(0.818314\pi\)
\(80\) −1.62195e9 − 2.84723e9i −0.494980 − 0.868905i
\(81\) −2.05808e9 −0.590252
\(82\) 4.79201e8 + 1.92657e9i 0.129255 + 0.519654i
\(83\) − 1.71712e9i − 0.435923i −0.975957 − 0.217961i \(-0.930059\pi\)
0.975957 − 0.217961i \(-0.0699406\pi\)
\(84\) 4.00061e8 − 1.30332e9i 0.0956598 − 0.311641i
\(85\) −2.47487e9 − 7.27125e9i −0.557774 − 1.63876i
\(86\) −7.04760e9 + 1.75297e9i −1.49813 + 0.372634i
\(87\) −3.75956e9 − 3.75956e9i −0.754294 − 0.754294i
\(88\) 3.08193e8 − 5.69755e9i 0.0583996 − 1.07963i
\(89\) −7.02195e9 −1.25750 −0.628750 − 0.777608i \(-0.716433\pi\)
−0.628750 + 0.777608i \(0.716433\pi\)
\(90\) −3.90752e8 + 1.80192e9i −0.0661742 + 0.305156i
\(91\) 2.53889e9 − 2.53889e9i 0.406853 − 0.406853i
\(92\) −1.14311e9 + 3.72404e9i −0.173440 + 0.565034i
\(93\) − 1.69392e9i − 0.243489i
\(94\) 2.47638e9 − 4.11617e9i 0.337426 − 0.560860i
\(95\) 3.16801e9 + 9.30771e9i 0.409419 + 1.20289i
\(96\) 6.34765e9 + 2.33050e9i 0.778497 + 0.285820i
\(97\) −7.54228e9 + 7.54228e9i −0.878302 + 0.878302i −0.993359 − 0.115057i \(-0.963295\pi\)
0.115057 + 0.993359i \(0.463295\pi\)
\(98\) 1.84473e9 + 7.41650e9i 0.204081 + 0.820481i
\(99\) −2.27024e9 + 2.27024e9i −0.238724 + 0.238724i
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.i.a.13.12 ✓ 236
5.2 odd 4 80.11.t.a.77.48 yes 236
16.5 even 4 80.11.t.a.53.48 yes 236
80.37 odd 4 inner 80.11.i.a.37.12 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.12 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.12 yes 236 80.37 odd 4 inner
80.11.t.a.53.48 yes 236 16.5 even 4
80.11.t.a.77.48 yes 236 5.2 odd 4