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Properties

Label 1575.4.a.ba.1.3
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $3$
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 = [3,-3,0,13,0,0,-21,-15,0,0,74] 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: \(3\)
Coefficient field: 3.3.14360.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 17x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.48565\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+3.48565 q^{2} +4.14976 q^{4} -7.00000 q^{7} -13.4206 q^{8} +6.90764 q^{11} +22.1364 q^{13} -24.3996 q^{14} -79.9776 q^{16} +88.3030 q^{17} +36.9560 q^{19} +24.0776 q^{22} -95.5283 q^{23} +77.1598 q^{26} -29.0483 q^{28} -269.029 q^{29} +197.114 q^{31} -171.409 q^{32} +307.793 q^{34} -2.14546 q^{37} +128.816 q^{38} -174.127 q^{41} +17.0345 q^{43} +28.6650 q^{44} -332.978 q^{46} -528.029 q^{47} +49.0000 q^{49} +91.8608 q^{52} -641.114 q^{53} +93.9441 q^{56} -937.742 q^{58} +642.975 q^{59} +142.967 q^{61} +687.070 q^{62} +42.3480 q^{64} -478.797 q^{67} +366.436 q^{68} -105.550 q^{71} -986.512 q^{73} -7.47834 q^{74} +153.358 q^{76} -48.3534 q^{77} -1099.86 q^{79} -606.947 q^{82} -1236.62 q^{83} +59.3763 q^{86} -92.7045 q^{88} +711.698 q^{89} -154.955 q^{91} -396.420 q^{92} -1840.52 q^{94} +636.553 q^{97} +170.797 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8} + 74 q^{11} - 44 q^{13} + 21 q^{14} - 79 q^{16} - 52 q^{17} + 168 q^{19} - 184 q^{22} - 124 q^{23} + 446 q^{26} - 91 q^{28} - 332 q^{29} + 320 q^{31} - 183 q^{32}+ \cdots - 147 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.48565 1.23236 0.616182 − 0.787604i \(-0.288679\pi\)
0.616182 + 0.787604i \(0.288679\pi\)
\(3\) 0 0
\(4\) 4.14976 0.518720
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −13.4206 −0.593112
\(9\) 0 0
\(10\) 0 0
\(11\) 6.90764 0.189339 0.0946696 − 0.995509i \(-0.469821\pi\)
0.0946696 + 0.995509i \(0.469821\pi\)
\(12\) 0 0
\(13\) 22.1364 0.472272 0.236136 − 0.971720i \(-0.424119\pi\)
0.236136 + 0.971720i \(0.424119\pi\)
\(14\) −24.3996 −0.465790
\(15\) 0 0
\(16\) −79.9776 −1.24965
\(17\) 88.3030 1.25980 0.629901 − 0.776676i \(-0.283096\pi\)
0.629901 + 0.776676i \(0.283096\pi\)
\(18\) 0 0
\(19\) 36.9560 0.446225 0.223113 − 0.974793i \(-0.428378\pi\)
0.223113 + 0.974793i \(0.428378\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 24.0776 0.233335
\(23\) −95.5283 −0.866045 −0.433022 − 0.901383i \(-0.642553\pi\)
−0.433022 + 0.901383i \(0.642553\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 77.1598 0.582010
\(27\) 0 0
\(28\) −29.0483 −0.196058
\(29\) −269.029 −1.72267 −0.861336 − 0.508035i \(-0.830372\pi\)
−0.861336 + 0.508035i \(0.830372\pi\)
\(30\) 0 0
\(31\) 197.114 1.14202 0.571012 − 0.820942i \(-0.306551\pi\)
0.571012 + 0.820942i \(0.306551\pi\)
\(32\) −171.409 −0.946911
\(33\) 0 0
\(34\) 307.793 1.55253
\(35\) 0 0
\(36\) 0 0
\(37\) −2.14546 −0.00953276 −0.00476638 − 0.999989i \(-0.501517\pi\)
−0.00476638 + 0.999989i \(0.501517\pi\)
\(38\) 128.816 0.549912
\(39\) 0 0
\(40\) 0 0
\(41\) −174.127 −0.663271 −0.331636 − 0.943408i \(-0.607600\pi\)
−0.331636 + 0.943408i \(0.607600\pi\)
\(42\) 0 0
\(43\) 17.0345 0.0604125 0.0302062 − 0.999544i \(-0.490384\pi\)
0.0302062 + 0.999544i \(0.490384\pi\)
\(44\) 28.6650 0.0982140
\(45\) 0 0
\(46\) −332.978 −1.06728
\(47\) −528.029 −1.63874 −0.819371 − 0.573264i \(-0.805677\pi\)
−0.819371 + 0.573264i \(0.805677\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 91.8608 0.244977
\(53\) −641.114 −1.66158 −0.830790 − 0.556586i \(-0.812111\pi\)
−0.830790 + 0.556586i \(0.812111\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 93.9441 0.224175
\(57\) 0 0
\(58\) −937.742 −2.12296
\(59\) 642.975 1.41878 0.709391 − 0.704815i \(-0.248970\pi\)
0.709391 + 0.704815i \(0.248970\pi\)
\(60\) 0 0
\(61\) 142.967 0.300083 0.150042 − 0.988680i \(-0.452059\pi\)
0.150042 + 0.988680i \(0.452059\pi\)
\(62\) 687.070 1.40739
\(63\) 0 0
\(64\) 42.3480 0.0827109
\(65\) 0 0
\(66\) 0 0
\(67\) −478.797 −0.873050 −0.436525 − 0.899692i \(-0.643791\pi\)
−0.436525 + 0.899692i \(0.643791\pi\)
\(68\) 366.436 0.653484
\(69\) 0 0
\(70\) 0 0
\(71\) −105.550 −0.176430 −0.0882150 − 0.996101i \(-0.528116\pi\)
−0.0882150 + 0.996101i \(0.528116\pi\)
\(72\) 0 0
\(73\) −986.512 −1.58168 −0.790839 − 0.612024i \(-0.790356\pi\)
−0.790839 + 0.612024i \(0.790356\pi\)
\(74\) −7.47834 −0.0117478
\(75\) 0 0
\(76\) 153.358 0.231466
\(77\) −48.3534 −0.0715635
\(78\) 0 0
\(79\) −1099.86 −1.56638 −0.783190 − 0.621783i \(-0.786409\pi\)
−0.783190 + 0.621783i \(0.786409\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −606.947 −0.817391
\(83\) −1236.62 −1.63538 −0.817691 − 0.575657i \(-0.804746\pi\)
−0.817691 + 0.575657i \(0.804746\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 59.3763 0.0744501
\(87\) 0 0
\(88\) −92.7045 −0.112299
\(89\) 711.698 0.847638 0.423819 − 0.905747i \(-0.360689\pi\)
0.423819 + 0.905747i \(0.360689\pi\)
\(90\) 0 0
\(91\) −154.955 −0.178502
\(92\) −396.420 −0.449235
\(93\) 0 0
\(94\) −1840.52 −2.01953
\(95\) 0 0
\(96\) 0 0
\(97\) 636.553 0.666311 0.333156 − 0.942872i \(-0.391887\pi\)
0.333156 + 0.942872i \(0.391887\pi\)
\(98\) 170.797 0.176052
\(99\) 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.ba.1.3 3
3.2 odd 2 175.4.a.f.1.1 3
5.4 even 2 315.4.a.p.1.1 3
15.2 even 4 175.4.b.e.99.2 6
15.8 even 4 175.4.b.e.99.5 6
15.14 odd 2 35.4.a.c.1.3 ✓ 3
21.20 even 2 1225.4.a.y.1.1 3
35.34 odd 2 2205.4.a.bm.1.1 3
60.59 even 2 560.4.a.u.1.2 3
105.44 odd 6 245.4.e.m.116.1 6
105.59 even 6 245.4.e.n.226.1 6
105.74 odd 6 245.4.e.m.226.1 6
105.89 even 6 245.4.e.n.116.1 6
105.104 even 2 245.4.a.l.1.3 3
120.29 odd 2 2240.4.a.bt.1.2 3
120.59 even 2 2240.4.a.bv.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.c.1.3 ✓ 3 15.14 odd 2
175.4.a.f.1.1 3 3.2 odd 2
175.4.b.e.99.2 6 15.2 even 4
175.4.b.e.99.5 6 15.8 even 4
245.4.a.l.1.3 3 105.104 even 2
245.4.e.m.116.1 6 105.44 odd 6
245.4.e.m.226.1 6 105.74 odd 6
245.4.e.n.116.1 6 105.89 even 6
245.4.e.n.226.1 6 105.59 even 6
315.4.a.p.1.1 3 5.4 even 2
560.4.a.u.1.2 3 60.59 even 2
1225.4.a.y.1.1 3 21.20 even 2
1575.4.a.ba.1.3 3 1.1 even 1 trivial
2205.4.a.bm.1.1 3 35.34 odd 2
2240.4.a.bt.1.2 3 120.29 odd 2
2240.4.a.bv.1.2 3 120.59 even 2