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Properties

Label 1575.4.a.l.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
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] = [1575,4,Mod(1,1575)] 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("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,5,0,17,0,0,-7,45,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1575.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{2} +17.0000 q^{4} -7.00000 q^{7} +45.0000 q^{8} -12.0000 q^{11} -30.0000 q^{13} -35.0000 q^{14} +89.0000 q^{16} -134.000 q^{17} -92.0000 q^{19} -60.0000 q^{22} +112.000 q^{23} -150.000 q^{26} -119.000 q^{28} +58.0000 q^{29} -224.000 q^{31} +85.0000 q^{32} -670.000 q^{34} +146.000 q^{37} -460.000 q^{38} -18.0000 q^{41} -340.000 q^{43} -204.000 q^{44} +560.000 q^{46} +208.000 q^{47} +49.0000 q^{49} -510.000 q^{52} -754.000 q^{53} -315.000 q^{56} +290.000 q^{58} -380.000 q^{59} +718.000 q^{61} -1120.00 q^{62} -287.000 q^{64} -412.000 q^{67} -2278.00 q^{68} +960.000 q^{71} -1066.00 q^{73} +730.000 q^{74} -1564.00 q^{76} +84.0000 q^{77} +896.000 q^{79} -90.0000 q^{82} +436.000 q^{83} -1700.00 q^{86} -540.000 q^{88} +1038.00 q^{89} +210.000 q^{91} +1904.00 q^{92} +1040.00 q^{94} +702.000 q^{97} +245.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.00000 1.76777 0.883883 − 0.467707i \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) 0 0
\(4\) 17.0000 2.12500
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 45.0000 1.98874
\(9\) 0 0
\(10\) 0 0
\(11\) −12.0000 −0.328921 −0.164461 − 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) 0 0
\(13\) −30.0000 −0.640039 −0.320019 − 0.947411i \(-0.603689\pi\)
−0.320019 + 0.947411i \(0.603689\pi\)
\(14\) −35.0000 −0.668153
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) −134.000 −1.91175 −0.955876 − 0.293771i \(-0.905090\pi\)
−0.955876 + 0.293771i \(0.905090\pi\)
\(18\) 0 0
\(19\) −92.0000 −1.11086 −0.555428 − 0.831565i \(-0.687445\pi\)
−0.555428 + 0.831565i \(0.687445\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −60.0000 −0.581456
\(23\) 112.000 1.01537 0.507687 − 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −150.000 −1.13144
\(27\) 0 0
\(28\) −119.000 −0.803175
\(29\) 58.0000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −224.000 −1.29779 −0.648897 − 0.760877i \(-0.724769\pi\)
−0.648897 + 0.760877i \(0.724769\pi\)
\(32\) 85.0000 0.469563
\(33\) 0 0
\(34\) −670.000 −3.37953
\(35\) 0 0
\(36\) 0 0
\(37\) 146.000 0.648710 0.324355 − 0.945936i \(-0.394853\pi\)
0.324355 + 0.945936i \(0.394853\pi\)
\(38\) −460.000 −1.96373
\(39\) 0 0
\(40\) 0 0
\(41\) −18.0000 −0.0685641 −0.0342820 − 0.999412i \(-0.510914\pi\)
−0.0342820 + 0.999412i \(0.510914\pi\)
\(42\) 0 0
\(43\) −340.000 −1.20580 −0.602901 − 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(44\) −204.000 −0.698958
\(45\) 0 0
\(46\) 560.000 1.79495
\(47\) 208.000 0.645530 0.322765 − 0.946479i \(-0.395388\pi\)
0.322765 + 0.946479i \(0.395388\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −510.000 −1.36008
\(53\) −754.000 −1.95415 −0.977074 − 0.212899i \(-0.931709\pi\)
−0.977074 + 0.212899i \(0.931709\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −315.000 −0.751672
\(57\) 0 0
\(58\) 290.000 0.656532
\(59\) −380.000 −0.838505 −0.419252 − 0.907870i \(-0.637708\pi\)
−0.419252 + 0.907870i \(0.637708\pi\)
\(60\) 0 0
\(61\) 718.000 1.50706 0.753529 − 0.657415i \(-0.228350\pi\)
0.753529 + 0.657415i \(0.228350\pi\)
\(62\) −1120.00 −2.29420
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) 0 0
\(67\) −412.000 −0.751251 −0.375625 − 0.926772i \(-0.622572\pi\)
−0.375625 + 0.926772i \(0.622572\pi\)
\(68\) −2278.00 −4.06247
\(69\) 0 0
\(70\) 0 0
\(71\) 960.000 1.60466 0.802331 − 0.596879i \(-0.203593\pi\)
0.802331 + 0.596879i \(0.203593\pi\)
\(72\) 0 0
\(73\) −1066.00 −1.70912 −0.854561 − 0.519352i \(-0.826174\pi\)
−0.854561 + 0.519352i \(0.826174\pi\)
\(74\) 730.000 1.14677
\(75\) 0 0
\(76\) −1564.00 −2.36057
\(77\) 84.0000 0.124321
\(78\) 0 0
\(79\) 896.000 1.27605 0.638025 − 0.770016i \(-0.279752\pi\)
0.638025 + 0.770016i \(0.279752\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −90.0000 −0.121205
\(83\) 436.000 0.576593 0.288296 − 0.957541i \(-0.406911\pi\)
0.288296 + 0.957541i \(0.406911\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1700.00 −2.13158
\(87\) 0 0
\(88\) −540.000 −0.654139
\(89\) 1038.00 1.23627 0.618134 − 0.786073i \(-0.287889\pi\)
0.618134 + 0.786073i \(0.287889\pi\)
\(90\) 0 0
\(91\) 210.000 0.241912
\(92\) 1904.00 2.15767
\(93\) 0 0
\(94\) 1040.00 1.14115
\(95\) 0 0
\(96\) 0 0
\(97\) 702.000 0.734818 0.367409 − 0.930060i \(-0.380245\pi\)
0.367409 + 0.930060i \(0.380245\pi\)
\(98\) 245.000 0.252538
\(99\) 0 0
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 1575.4.a.l.1.1 1
3.2 odd 2 525.4.a.a.1.1 1
5.4 even 2 315.4.a.a.1.1 1
15.2 even 4 525.4.d.a.274.1 2
15.8 even 4 525.4.d.a.274.2 2
15.14 odd 2 105.4.a.b.1.1 ✓ 1
35.34 odd 2 2205.4.a.b.1.1 1
60.59 even 2 1680.4.a.u.1.1 1
105.104 even 2 735.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.b.1.1 ✓ 1 15.14 odd 2
315.4.a.a.1.1 1 5.4 even 2
525.4.a.a.1.1 1 3.2 odd 2
525.4.d.a.274.1 2 15.2 even 4
525.4.d.a.274.2 2 15.8 even 4
735.4.a.j.1.1 1 105.104 even 2
1575.4.a.l.1.1 1 1.1 even 1 trivial
1680.4.a.u.1.1 1 60.59 even 2
2205.4.a.b.1.1 1 35.34 odd 2