Properties

Label 10.22.a.c.1.2
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [10,22,Mod(1,10)] 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("10.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2048,100308] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(27.9477344287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{474529}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 118632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-343.930\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +194815. q^{3} +1.04858e6 q^{4} +9.76562e6 q^{5} -1.99490e8 q^{6} +9.63461e8 q^{7} -1.07374e9 q^{8} +2.74924e10 q^{9} -1.00000e10 q^{10} +6.10882e10 q^{11} +2.04278e11 q^{12} +5.25383e10 q^{13} -9.86584e11 q^{14} +1.90249e12 q^{15} +1.09951e12 q^{16} -1.13179e13 q^{17} -2.81522e13 q^{18} -4.53240e13 q^{19} +1.02400e13 q^{20} +1.87696e14 q^{21} -6.25543e13 q^{22} +2.47091e14 q^{23} -2.09181e14 q^{24} +9.53674e13 q^{25} -5.37992e13 q^{26} +3.31810e15 q^{27} +1.01026e15 q^{28} -2.58718e15 q^{29} -1.94815e15 q^{30} -1.35601e15 q^{31} -1.12590e15 q^{32} +1.19009e16 q^{33} +1.15895e16 q^{34} +9.40880e15 q^{35} +2.88279e16 q^{36} -1.35920e16 q^{37} +4.64117e16 q^{38} +1.02352e16 q^{39} -1.04858e16 q^{40} +6.72480e16 q^{41} -1.92201e17 q^{42} -3.70641e16 q^{43} +6.40556e16 q^{44} +2.68481e17 q^{45} -2.53021e17 q^{46} +3.17182e17 q^{47} +2.14201e17 q^{48} +3.69712e17 q^{49} -9.76562e16 q^{50} -2.20489e18 q^{51} +5.50904e16 q^{52} +1.44909e18 q^{53} -3.39773e18 q^{54} +5.96564e17 q^{55} -1.03451e18 q^{56} -8.82978e18 q^{57} +2.64928e18 q^{58} -1.08957e18 q^{59} +1.99490e18 q^{60} -1.59498e17 q^{61} +1.38855e18 q^{62} +2.64879e19 q^{63} +1.15292e18 q^{64} +5.13069e17 q^{65} -1.21865e19 q^{66} +8.71262e18 q^{67} -1.18677e19 q^{68} +4.81370e19 q^{69} -9.63461e18 q^{70} -3.06845e19 q^{71} -2.95198e19 q^{72} +2.36491e19 q^{73} +1.39182e19 q^{74} +1.85790e19 q^{75} -4.75256e19 q^{76} +5.88561e19 q^{77} -1.04809e19 q^{78} -9.35354e19 q^{79} +1.07374e19 q^{80} +3.58834e20 q^{81} -6.88619e19 q^{82} -2.00855e20 q^{83} +1.96814e20 q^{84} -1.10526e20 q^{85} +3.79536e19 q^{86} -5.04022e20 q^{87} -6.55929e19 q^{88} +3.85510e19 q^{89} -2.74924e20 q^{90} +5.06186e19 q^{91} +2.59094e20 q^{92} -2.64170e20 q^{93} -3.24795e20 q^{94} -4.42617e20 q^{95} -2.19342e20 q^{96} -5.46564e20 q^{97} -3.78585e20 q^{98} +1.67946e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{2} + 100308 q^{3} + 2097152 q^{4} + 19531250 q^{5} - 102715392 q^{6} + 1328895316 q^{7} - 2147483648 q^{8} + 25963598826 q^{9} - 20000000000 q^{10} + 21869194224 q^{11} + 105180561408 q^{12}+ \cdots + 17\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −0.707107
\(3\) 194815. 1.90480 0.952398 − 0.304856i \(-0.0986084\pi\)
0.952398 + 0.304856i \(0.0986084\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 9.76562e6 0.447214
\(6\) −1.99490e8 −1.34689
\(7\) 9.63461e8 1.28915 0.644577 − 0.764539i \(-0.277033\pi\)
0.644577 + 0.764539i \(0.277033\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) 2.74924e10 2.62825
\(10\) −1.00000e10 −0.316228
\(11\) 6.10882e10 0.710124 0.355062 − 0.934843i \(-0.384460\pi\)
0.355062 + 0.934843i \(0.384460\pi\)
\(12\) 2.04278e11 0.952398
\(13\) 5.25383e10 0.105699 0.0528495 − 0.998602i \(-0.483170\pi\)
0.0528495 + 0.998602i \(0.483170\pi\)
\(14\) −9.86584e11 −0.911570
\(15\) 1.90249e12 0.851851
\(16\) 1.09951e12 0.250000
\(17\) −1.13179e13 −1.36161 −0.680803 − 0.732466i \(-0.738369\pi\)
−0.680803 + 0.732466i \(0.738369\pi\)
\(18\) −2.81522e13 −1.85845
\(19\) −4.53240e13 −1.69596 −0.847979 − 0.530030i \(-0.822181\pi\)
−0.847979 + 0.530030i \(0.822181\pi\)
\(20\) 1.02400e13 0.223607
\(21\) 1.87696e14 2.45558
\(22\) −6.25543e13 −0.502133
\(23\) 2.47091e14 1.24370 0.621849 − 0.783137i \(-0.286382\pi\)
0.621849 + 0.783137i \(0.286382\pi\)
\(24\) −2.09181e14 −0.673447
\(25\) 9.53674e13 0.200000
\(26\) −5.37992e13 −0.0747404
\(27\) 3.31810e15 3.10149
\(28\) 1.01026e15 0.644577
\(29\) −2.58718e15 −1.14195 −0.570977 − 0.820966i \(-0.693435\pi\)
−0.570977 + 0.820966i \(0.693435\pi\)
\(30\) −1.94815e15 −0.602350
\(31\) −1.35601e15 −0.297142 −0.148571 − 0.988902i \(-0.547467\pi\)
−0.148571 + 0.988902i \(0.547467\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 1.19009e16 1.35264
\(34\) 1.15895e16 0.962801
\(35\) 9.40880e15 0.576527
\(36\) 2.88279e16 1.31413
\(37\) −1.35920e16 −0.464692 −0.232346 − 0.972633i \(-0.574640\pi\)
−0.232346 + 0.972633i \(0.574640\pi\)
\(38\) 4.64117e16 1.19922
\(39\) 1.02352e16 0.201335
\(40\) −1.04858e16 −0.158114
\(41\) 6.72480e16 0.782436 0.391218 − 0.920298i \(-0.372054\pi\)
0.391218 + 0.920298i \(0.372054\pi\)
\(42\) −1.92201e17 −1.73636
\(43\) −3.70641e16 −0.261538 −0.130769 − 0.991413i \(-0.541745\pi\)
−0.130769 + 0.991413i \(0.541745\pi\)
\(44\) 6.40556e16 0.355062
\(45\) 2.68481e17 1.17539
\(46\) −2.53021e17 −0.879427
\(47\) 3.17182e17 0.879592 0.439796 − 0.898098i \(-0.355051\pi\)
0.439796 + 0.898098i \(0.355051\pi\)
\(48\) 2.14201e17 0.476199
\(49\) 3.69712e17 0.661919
\(50\) −9.76562e16 −0.141421
\(51\) −2.20489e18 −2.59358
\(52\) 5.50904e16 0.0528495
\(53\) 1.44909e18 1.13815 0.569074 − 0.822286i \(-0.307302\pi\)
0.569074 + 0.822286i \(0.307302\pi\)
\(54\) −3.39773e18 −2.19308
\(55\) 5.96564e17 0.317577
\(56\) −1.03451e18 −0.455785
\(57\) −8.82978e18 −3.23046
\(58\) 2.64928e18 0.807483
\(59\) −1.08957e18 −0.277529 −0.138764 − 0.990325i \(-0.544313\pi\)
−0.138764 + 0.990325i \(0.544313\pi\)
\(60\) 1.99490e18 0.425925
\(61\) −1.59498e17 −0.0286281 −0.0143140 − 0.999898i \(-0.504556\pi\)
−0.0143140 + 0.999898i \(0.504556\pi\)
\(62\) 1.38855e18 0.210111
\(63\) 2.64879e19 3.38822
\(64\) 1.15292e18 0.125000
\(65\) 5.13069e17 0.0472700
\(66\) −1.21865e19 −0.956462
\(67\) 8.71262e18 0.583934 0.291967 − 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(68\) −1.18677e19 −0.680803
\(69\) 4.81370e19 2.36899
\(70\) −9.63461e18 −0.407666
\(71\) −3.06845e19 −1.11868 −0.559340 − 0.828938i \(-0.688946\pi\)
−0.559340 + 0.828938i \(0.688946\pi\)
\(72\) −2.95198e19 −0.929227
\(73\) 2.36491e19 0.644059 0.322029 − 0.946730i \(-0.395635\pi\)
0.322029 + 0.946730i \(0.395635\pi\)
\(74\) 1.39182e19 0.328587
\(75\) 1.85790e19 0.380959
\(76\) −4.75256e19 −0.847979
\(77\) 5.88561e19 0.915459
\(78\) −1.04809e19 −0.142365
\(79\) −9.35354e19 −1.11145 −0.555727 − 0.831365i \(-0.687560\pi\)
−0.555727 + 0.831365i \(0.687560\pi\)
\(80\) 1.07374e19 0.111803
\(81\) 3.58834e20 3.27945
\(82\) −6.88619e19 −0.553266
\(83\) −2.00855e20 −1.42090 −0.710448 − 0.703750i \(-0.751508\pi\)
−0.710448 + 0.703750i \(0.751508\pi\)
\(84\) 1.96814e20 1.22779
\(85\) −1.10526e20 −0.608929
\(86\) 3.79536e19 0.184935
\(87\) −5.04022e20 −2.17519
\(88\) −6.55929e19 −0.251067
\(89\) 3.85510e19 0.131051 0.0655255 − 0.997851i \(-0.479128\pi\)
0.0655255 + 0.997851i \(0.479128\pi\)
\(90\) −2.74924e20 −0.831126
\(91\) 5.06186e19 0.136262
\(92\) 2.59094e20 0.621849
\(93\) −2.64170e20 −0.565996
\(94\) −3.24795e20 −0.621965
\(95\) −4.42617e20 −0.758456
\(96\) −2.19342e20 −0.336724
\(97\) −5.46564e20 −0.752554 −0.376277 − 0.926507i \(-0.622796\pi\)
−0.376277 + 0.926507i \(0.622796\pi\)
\(98\) −3.78585e20 −0.468047
\(99\) 1.67946e21 1.86638
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 10.22.a.c.1.2 ✓ 2
4.3 odd 2 80.22.a.b.1.1 2
5.2 odd 4 50.22.b.d.49.1 4
5.3 odd 4 50.22.b.d.49.4 4
5.4 even 2 50.22.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.c.1.2 ✓ 2 1.1 even 1 trivial
50.22.a.e.1.1 2 5.4 even 2
50.22.b.d.49.1 4 5.2 odd 4
50.22.b.d.49.4 4 5.3 odd 4
80.22.a.b.1.1 2 4.3 odd 2