Properties

Label 75.22.a.c.1.1
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $0$
Dimension $1$
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] = [75,22,Mod(1,75)] 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("75.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2844] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2844.00 q^{2} +59049.0 q^{3} +5.99118e6 q^{4} +1.67935e8 q^{6} -3.63304e8 q^{7} +1.10746e10 q^{8} +3.48678e9 q^{9} +1.45818e10 q^{11} +3.53773e11 q^{12} -1.13351e11 q^{13} -1.03324e12 q^{14} +1.89318e13 q^{16} +8.58939e12 q^{17} +9.91641e12 q^{18} -2.92029e13 q^{19} -2.14527e13 q^{21} +4.14707e13 q^{22} +1.55899e14 q^{23} +6.53946e14 q^{24} -3.22370e14 q^{26} +2.05891e14 q^{27} -2.17662e15 q^{28} +2.40079e15 q^{29} +2.23982e15 q^{31} +3.06169e16 q^{32} +8.61043e14 q^{33} +2.44282e16 q^{34} +2.08900e16 q^{36} +3.07851e16 q^{37} -8.30532e16 q^{38} -6.69325e15 q^{39} -1.03208e17 q^{41} -6.10116e16 q^{42} +1.65557e17 q^{43} +8.73624e16 q^{44} +4.43377e17 q^{46} +6.65872e16 q^{47} +1.11790e18 q^{48} -4.26556e17 q^{49} +5.07195e17 q^{51} -6.79105e17 q^{52} -4.35423e17 q^{53} +5.85554e17 q^{54} -4.02346e18 q^{56} -1.72440e18 q^{57} +6.82784e18 q^{58} +5.53437e18 q^{59} -7.17621e18 q^{61} +6.37005e18 q^{62} -1.26676e18 q^{63} +4.73716e19 q^{64} +2.44881e18 q^{66} +1.57554e19 q^{67} +5.14606e19 q^{68} +9.20569e18 q^{69} +2.64579e19 q^{71} +3.86148e19 q^{72} -1.34712e19 q^{73} +8.75527e19 q^{74} -1.74960e20 q^{76} -5.29764e18 q^{77} -1.90356e19 q^{78} -1.68861e19 q^{79} +1.21577e19 q^{81} -2.93522e20 q^{82} +1.70688e20 q^{83} -1.28527e20 q^{84} +4.70845e20 q^{86} +1.41764e20 q^{87} +1.61488e20 q^{88} -3.12592e20 q^{89} +4.11808e19 q^{91} +9.34021e20 q^{92} +1.32259e20 q^{93} +1.89374e20 q^{94} +1.80790e21 q^{96} -9.49015e20 q^{97} -1.21313e21 q^{98} +5.08437e19 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2844.00 1.96388 0.981939 − 0.189196i \(-0.0605882\pi\)
0.981939 + 0.189196i \(0.0605882\pi\)
\(3\) 59049.0 0.577350
\(4\) 5.99118e6 2.85682
\(5\) 0 0
\(6\) 1.67935e8 1.13385
\(7\) −3.63304e8 −0.486117 −0.243058 − 0.970012i \(-0.578151\pi\)
−0.243058 + 0.970012i \(0.578151\pi\)
\(8\) 1.10746e10 3.64657
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 1.45818e10 0.169508 0.0847538 − 0.996402i \(-0.472990\pi\)
0.0847538 + 0.996402i \(0.472990\pi\)
\(12\) 3.53773e11 1.64939
\(13\) −1.13351e11 −0.228044 −0.114022 − 0.993478i \(-0.536373\pi\)
−0.114022 + 0.993478i \(0.536373\pi\)
\(14\) −1.03324e12 −0.954674
\(15\) 0 0
\(16\) 1.89318e13 4.30460
\(17\) 8.58939e12 1.03335 0.516676 − 0.856181i \(-0.327169\pi\)
0.516676 + 0.856181i \(0.327169\pi\)
\(18\) 9.91641e12 0.654626
\(19\) −2.92029e13 −1.09273 −0.546366 − 0.837546i \(-0.683989\pi\)
−0.546366 + 0.837546i \(0.683989\pi\)
\(20\) 0 0
\(21\) −2.14527e13 −0.280660
\(22\) 4.14707e13 0.332892
\(23\) 1.55899e14 0.784696 0.392348 − 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(24\) 6.53946e14 2.10535
\(25\) 0 0
\(26\) −3.22370e14 −0.447851
\(27\) 2.05891e14 0.192450
\(28\) −2.17662e15 −1.38875
\(29\) 2.40079e15 1.05968 0.529840 − 0.848097i \(-0.322252\pi\)
0.529840 + 0.848097i \(0.322252\pi\)
\(30\) 0 0
\(31\) 2.23982e15 0.490812 0.245406 − 0.969420i \(-0.421079\pi\)
0.245406 + 0.969420i \(0.421079\pi\)
\(32\) 3.06169e16 4.80714
\(33\) 8.61043e14 0.0978652
\(34\) 2.44282e16 2.02938
\(35\) 0 0
\(36\) 2.08900e16 0.952273
\(37\) 3.07851e16 1.05250 0.526250 − 0.850330i \(-0.323598\pi\)
0.526250 + 0.850330i \(0.323598\pi\)
\(38\) −8.30532e16 −2.14599
\(39\) −6.69325e15 −0.131661
\(40\) 0 0
\(41\) −1.03208e17 −1.20083 −0.600414 − 0.799689i \(-0.704998\pi\)
−0.600414 + 0.799689i \(0.704998\pi\)
\(42\) −6.10116e16 −0.551182
\(43\) 1.65557e17 1.16823 0.584117 − 0.811670i \(-0.301441\pi\)
0.584117 + 0.811670i \(0.301441\pi\)
\(44\) 8.73624e16 0.484252
\(45\) 0 0
\(46\) 4.43377e17 1.54105
\(47\) 6.65872e16 0.184656 0.0923280 − 0.995729i \(-0.470569\pi\)
0.0923280 + 0.995729i \(0.470569\pi\)
\(48\) 1.11790e18 2.48526
\(49\) −4.26556e17 −0.763690
\(50\) 0 0
\(51\) 5.07195e17 0.596607
\(52\) −6.79105e17 −0.651482
\(53\) −4.35423e17 −0.341991 −0.170995 − 0.985272i \(-0.554698\pi\)
−0.170995 + 0.985272i \(0.554698\pi\)
\(54\) 5.85554e17 0.377949
\(55\) 0 0
\(56\) −4.02346e18 −1.77266
\(57\) −1.72440e18 −0.630889
\(58\) 6.82784e18 2.08108
\(59\) 5.53437e18 1.40968 0.704842 − 0.709364i \(-0.251018\pi\)
0.704842 + 0.709364i \(0.251018\pi\)
\(60\) 0 0
\(61\) −7.17621e18 −1.28805 −0.644023 − 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) 6.37005e18 0.963896
\(63\) −1.26676e18 −0.162039
\(64\) 4.73716e19 5.13604
\(65\) 0 0
\(66\) 2.44881e18 0.192195
\(67\) 1.57554e19 1.05595 0.527977 − 0.849258i \(-0.322951\pi\)
0.527977 + 0.849258i \(0.322951\pi\)
\(68\) 5.14606e19 2.95210
\(69\) 9.20569e18 0.453045
\(70\) 0 0
\(71\) 2.64579e19 0.964588 0.482294 − 0.876009i \(-0.339804\pi\)
0.482294 + 0.876009i \(0.339804\pi\)
\(72\) 3.86148e19 1.21552
\(73\) −1.34712e19 −0.366875 −0.183437 − 0.983031i \(-0.558722\pi\)
−0.183437 + 0.983031i \(0.558722\pi\)
\(74\) 8.75527e19 2.06698
\(75\) 0 0
\(76\) −1.74960e20 −3.12174
\(77\) −5.29764e18 −0.0824005
\(78\) −1.90356e19 −0.258567
\(79\) −1.68861e19 −0.200653 −0.100326 − 0.994955i \(-0.531989\pi\)
−0.100326 + 0.994955i \(0.531989\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) −2.93522e20 −2.35828
\(83\) 1.70688e20 1.20749 0.603744 − 0.797178i \(-0.293675\pi\)
0.603744 + 0.797178i \(0.293675\pi\)
\(84\) −1.28527e20 −0.801794
\(85\) 0 0
\(86\) 4.70845e20 2.29427
\(87\) 1.41764e20 0.611807
\(88\) 1.61488e20 0.618121
\(89\) −3.12592e20 −1.06263 −0.531317 − 0.847173i \(-0.678303\pi\)
−0.531317 + 0.847173i \(0.678303\pi\)
\(90\) 0 0
\(91\) 4.11808e19 0.110856
\(92\) 9.34021e20 2.24174
\(93\) 1.32259e20 0.283371
\(94\) 1.89374e20 0.362642
\(95\) 0 0
\(96\) 1.80790e21 2.77540
\(97\) −9.49015e20 −1.30668 −0.653341 − 0.757064i \(-0.726633\pi\)
−0.653341 + 0.757064i \(0.726633\pi\)
\(98\) −1.21313e21 −1.49980
\(99\) 5.08437e19 0.0565025
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 75.22.a.c.1.1 1
5.2 odd 4 75.22.b.a.49.2 2
5.3 odd 4 75.22.b.a.49.1 2
5.4 even 2 3.22.a.a.1.1 ✓ 1
15.14 odd 2 9.22.a.d.1.1 1
20.19 odd 2 48.22.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 ✓ 1 5.4 even 2
9.22.a.d.1.1 1 15.14 odd 2
48.22.a.e.1.1 1 20.19 odd 2
75.22.a.c.1.1 1 1.1 even 1 trivial
75.22.b.a.49.1 2 5.3 odd 4
75.22.b.a.49.2 2 5.2 odd 4