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Properties

Label 2175.4.a.c.1.1
Level $2175$
Weight $4$
Character 2175.1
Self dual yes
Analytic conductor $128.329$
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] = [2175,4,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,3] 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: \(128.329154262\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2175.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +3.00000 q^{3} -4.00000 q^{4} +6.00000 q^{6} -29.0000 q^{7} -24.0000 q^{8} +9.00000 q^{9} -15.0000 q^{11} -12.0000 q^{12} -3.00000 q^{13} -58.0000 q^{14} -16.0000 q^{16} -121.000 q^{17} +18.0000 q^{18} -40.0000 q^{19} -87.0000 q^{21} -30.0000 q^{22} +116.000 q^{23} -72.0000 q^{24} -6.00000 q^{26} +27.0000 q^{27} +116.000 q^{28} +29.0000 q^{29} -116.000 q^{31} +160.000 q^{32} -45.0000 q^{33} -242.000 q^{34} -36.0000 q^{36} -36.0000 q^{37} -80.0000 q^{38} -9.00000 q^{39} -170.000 q^{41} -174.000 q^{42} -230.000 q^{43} +60.0000 q^{44} +232.000 q^{46} -231.000 q^{47} -48.0000 q^{48} +498.000 q^{49} -363.000 q^{51} +12.0000 q^{52} -456.000 q^{53} +54.0000 q^{54} +696.000 q^{56} -120.000 q^{57} +58.0000 q^{58} +576.000 q^{59} +342.000 q^{61} -232.000 q^{62} -261.000 q^{63} +448.000 q^{64} -90.0000 q^{66} +269.000 q^{67} +484.000 q^{68} +348.000 q^{69} +302.000 q^{71} -216.000 q^{72} +372.000 q^{73} -72.0000 q^{74} +160.000 q^{76} +435.000 q^{77} -18.0000 q^{78} -348.000 q^{79} +81.0000 q^{81} -340.000 q^{82} +512.000 q^{83} +348.000 q^{84} -460.000 q^{86} +87.0000 q^{87} +360.000 q^{88} +1525.00 q^{89} +87.0000 q^{91} -464.000 q^{92} -348.000 q^{93} -462.000 q^{94} +480.000 q^{96} +560.000 q^{97} +996.000 q^{98} -135.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 3.00000 0.577350
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 6.00000 0.408248
\(7\) −29.0000 −1.56585 −0.782926 − 0.622114i \(-0.786274\pi\)
−0.782926 + 0.622114i \(0.786274\pi\)
\(8\) −24.0000 −1.06066
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −15.0000 −0.411152 −0.205576 − 0.978641i \(-0.565907\pi\)
−0.205576 + 0.978641i \(0.565907\pi\)
\(12\) −12.0000 −0.288675
\(13\) −3.00000 −0.0640039 −0.0320019 − 0.999488i \(-0.510188\pi\)
−0.0320019 + 0.999488i \(0.510188\pi\)
\(14\) −58.0000 −1.10723
\(15\) 0 0
\(16\) −16.0000 −0.250000
\(17\) −121.000 −1.72628 −0.863141 − 0.504962i \(-0.831506\pi\)
−0.863141 + 0.504962i \(0.831506\pi\)
\(18\) 18.0000 0.235702
\(19\) −40.0000 −0.482980 −0.241490 − 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 0 0
\(21\) −87.0000 −0.904046
\(22\) −30.0000 −0.290728
\(23\) 116.000 1.05164 0.525819 − 0.850597i \(-0.323759\pi\)
0.525819 + 0.850597i \(0.323759\pi\)
\(24\) −72.0000 −0.612372
\(25\) 0 0
\(26\) −6.00000 −0.0452576
\(27\) 27.0000 0.192450
\(28\) 116.000 0.782926
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −116.000 −0.672071 −0.336036 − 0.941849i \(-0.609086\pi\)
−0.336036 + 0.941849i \(0.609086\pi\)
\(32\) 160.000 0.883883
\(33\) −45.0000 −0.237379
\(34\) −242.000 −1.22067
\(35\) 0 0
\(36\) −36.0000 −0.166667
\(37\) −36.0000 −0.159956 −0.0799779 − 0.996797i \(-0.525485\pi\)
−0.0799779 + 0.996797i \(0.525485\pi\)
\(38\) −80.0000 −0.341519
\(39\) −9.00000 −0.0369527
\(40\) 0 0
\(41\) −170.000 −0.647550 −0.323775 − 0.946134i \(-0.604952\pi\)
−0.323775 + 0.946134i \(0.604952\pi\)
\(42\) −174.000 −0.639257
\(43\) −230.000 −0.815690 −0.407845 − 0.913051i \(-0.633720\pi\)
−0.407845 + 0.913051i \(0.633720\pi\)
\(44\) 60.0000 0.205576
\(45\) 0 0
\(46\) 232.000 0.743620
\(47\) −231.000 −0.716911 −0.358455 − 0.933547i \(-0.616697\pi\)
−0.358455 + 0.933547i \(0.616697\pi\)
\(48\) −48.0000 −0.144338
\(49\) 498.000 1.45190
\(50\) 0 0
\(51\) −363.000 −0.996670
\(52\) 12.0000 0.0320019
\(53\) −456.000 −1.18182 −0.590910 − 0.806738i \(-0.701231\pi\)
−0.590910 + 0.806738i \(0.701231\pi\)
\(54\) 54.0000 0.136083
\(55\) 0 0
\(56\) 696.000 1.66084
\(57\) −120.000 −0.278849
\(58\) 58.0000 0.131306
\(59\) 576.000 1.27100 0.635498 − 0.772102i \(-0.280795\pi\)
0.635498 + 0.772102i \(0.280795\pi\)
\(60\) 0 0
\(61\) 342.000 0.717846 0.358923 − 0.933367i \(-0.383144\pi\)
0.358923 + 0.933367i \(0.383144\pi\)
\(62\) −232.000 −0.475226
\(63\) −261.000 −0.521951
\(64\) 448.000 0.875000
\(65\) 0 0
\(66\) −90.0000 −0.167852
\(67\) 269.000 0.490501 0.245251 − 0.969460i \(-0.421130\pi\)
0.245251 + 0.969460i \(0.421130\pi\)
\(68\) 484.000 0.863141
\(69\) 348.000 0.607163
\(70\) 0 0
\(71\) 302.000 0.504800 0.252400 − 0.967623i \(-0.418780\pi\)
0.252400 + 0.967623i \(0.418780\pi\)
\(72\) −216.000 −0.353553
\(73\) 372.000 0.596429 0.298214 − 0.954499i \(-0.403609\pi\)
0.298214 + 0.954499i \(0.403609\pi\)
\(74\) −72.0000 −0.113106
\(75\) 0 0
\(76\) 160.000 0.241490
\(77\) 435.000 0.643803
\(78\) −18.0000 −0.0261295
\(79\) −348.000 −0.495608 −0.247804 − 0.968810i \(-0.579709\pi\)
−0.247804 + 0.968810i \(0.579709\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −340.000 −0.457887
\(83\) 512.000 0.677100 0.338550 − 0.940948i \(-0.390064\pi\)
0.338550 + 0.940948i \(0.390064\pi\)
\(84\) 348.000 0.452023
\(85\) 0 0
\(86\) −460.000 −0.576780
\(87\) 87.0000 0.107211
\(88\) 360.000 0.436092
\(89\) 1525.00 1.81629 0.908144 − 0.418657i \(-0.137499\pi\)
0.908144 + 0.418657i \(0.137499\pi\)
\(90\) 0 0
\(91\) 87.0000 0.100221
\(92\) −464.000 −0.525819
\(93\) −348.000 −0.388021
\(94\) −462.000 −0.506933
\(95\) 0 0
\(96\) 480.000 0.510310
\(97\) 560.000 0.586179 0.293090 − 0.956085i \(-0.405317\pi\)
0.293090 + 0.956085i \(0.405317\pi\)
\(98\) 996.000 1.02664
\(99\) −135.000 −0.137051
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 2175.4.a.c.1.1 1
5.4 even 2 435.4.a.a.1.1 ✓ 1
15.14 odd 2 1305.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.a.1.1 ✓ 1 5.4 even 2
1305.4.a.d.1.1 1 15.14 odd 2
2175.4.a.c.1.1 1 1.1 even 1 trivial