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Properties

Label 1305.4.a.d.1.1
Level $1305$
Weight $4$
Character 1305.1
Self dual yes
Analytic conductor $76.997$
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] = [1305,4,Mod(1,1305)] 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("1305.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1305.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(76.9974925575\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1305.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -4.00000 q^{4} -5.00000 q^{5} +29.0000 q^{7} -24.0000 q^{8} -10.0000 q^{10} +15.0000 q^{11} +3.00000 q^{13} +58.0000 q^{14} -16.0000 q^{16} -121.000 q^{17} -40.0000 q^{19} +20.0000 q^{20} +30.0000 q^{22} +116.000 q^{23} +25.0000 q^{25} +6.00000 q^{26} -116.000 q^{28} -29.0000 q^{29} -116.000 q^{31} +160.000 q^{32} -242.000 q^{34} -145.000 q^{35} +36.0000 q^{37} -80.0000 q^{38} +120.000 q^{40} +170.000 q^{41} +230.000 q^{43} -60.0000 q^{44} +232.000 q^{46} -231.000 q^{47} +498.000 q^{49} +50.0000 q^{50} -12.0000 q^{52} -456.000 q^{53} -75.0000 q^{55} -696.000 q^{56} -58.0000 q^{58} -576.000 q^{59} +342.000 q^{61} -232.000 q^{62} +448.000 q^{64} -15.0000 q^{65} -269.000 q^{67} +484.000 q^{68} -290.000 q^{70} -302.000 q^{71} -372.000 q^{73} +72.0000 q^{74} +160.000 q^{76} +435.000 q^{77} -348.000 q^{79} +80.0000 q^{80} +340.000 q^{82} +512.000 q^{83} +605.000 q^{85} +460.000 q^{86} -360.000 q^{88} -1525.00 q^{89} +87.0000 q^{91} -464.000 q^{92} -462.000 q^{94} +200.000 q^{95} -560.000 q^{97} +996.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 29.0000 1.56585 0.782926 − 0.622114i \(-0.213726\pi\)
0.782926 + 0.622114i \(0.213726\pi\)
\(8\) −24.0000 −1.06066
\(9\) 0 0
\(10\) −10.0000 −0.316228
\(11\) 15.0000 0.411152 0.205576 − 0.978641i \(-0.434093\pi\)
0.205576 + 0.978641i \(0.434093\pi\)
\(12\) 0 0
\(13\) 3.00000 0.0640039 0.0320019 − 0.999488i \(-0.489812\pi\)
0.0320019 + 0.999488i \(0.489812\pi\)
\(14\) 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 \(0.831506\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 − 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) 30.0000 0.290728
\(23\) 116.000 1.05164 0.525819 − 0.850597i \(-0.323759\pi\)
0.525819 + 0.850597i \(0.323759\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 6.00000 0.0452576
\(27\) 0 0
\(28\) −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
\(33\) 0 0
\(34\) −242.000 −1.22067
\(35\) −145.000 −0.700271
\(36\) 0 0
\(37\) 36.0000 0.159956 0.0799779 − 0.996797i \(-0.474515\pi\)
0.0799779 + 0.996797i \(0.474515\pi\)
\(38\) −80.0000 −0.341519
\(39\) 0 0
\(40\) 120.000 0.474342
\(41\) 170.000 0.647550 0.323775 − 0.946134i \(-0.395048\pi\)
0.323775 + 0.946134i \(0.395048\pi\)
\(42\) 0 0
\(43\) 230.000 0.815690 0.407845 − 0.913051i \(-0.366280\pi\)
0.407845 + 0.913051i \(0.366280\pi\)
\(44\) −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 \(0.616697\pi\)
\(48\) 0 0
\(49\) 498.000 1.45190
\(50\) 50.0000 0.141421
\(51\) 0 0
\(52\) −12.0000 −0.0320019
\(53\) −456.000 −1.18182 −0.590910 − 0.806738i \(-0.701231\pi\)
−0.590910 + 0.806738i \(0.701231\pi\)
\(54\) 0 0
\(55\) −75.0000 −0.183873
\(56\) −696.000 −1.66084
\(57\) 0 0
\(58\) −58.0000 −0.131306
\(59\) −576.000 −1.27100 −0.635498 − 0.772102i \(-0.719205\pi\)
−0.635498 + 0.772102i \(0.719205\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 448.000 0.875000
\(65\) −15.0000 −0.0286234
\(66\) 0 0
\(67\) −269.000 −0.490501 −0.245251 − 0.969460i \(-0.578870\pi\)
−0.245251 + 0.969460i \(0.578870\pi\)
\(68\) 484.000 0.863141
\(69\) 0 0
\(70\) −290.000 −0.495166
\(71\) −302.000 −0.504800 −0.252400 − 0.967623i \(-0.581220\pi\)
−0.252400 + 0.967623i \(0.581220\pi\)
\(72\) 0 0
\(73\) −372.000 −0.596429 −0.298214 − 0.954499i \(-0.596391\pi\)
−0.298214 + 0.954499i \(0.596391\pi\)
\(74\) 72.0000 0.113106
\(75\) 0 0
\(76\) 160.000 0.241490
\(77\) 435.000 0.643803
\(78\) 0 0
\(79\) −348.000 −0.495608 −0.247804 − 0.968810i \(-0.579709\pi\)
−0.247804 + 0.968810i \(0.579709\pi\)
\(80\) 80.0000 0.111803
\(81\) 0 0
\(82\) 340.000 0.457887
\(83\) 512.000 0.677100 0.338550 − 0.940948i \(-0.390064\pi\)
0.338550 + 0.940948i \(0.390064\pi\)
\(84\) 0 0
\(85\) 605.000 0.772017
\(86\) 460.000 0.576780
\(87\) 0 0
\(88\) −360.000 −0.436092
\(89\) −1525.00 −1.81629 −0.908144 − 0.418657i \(-0.862501\pi\)
−0.908144 + 0.418657i \(0.862501\pi\)
\(90\) 0 0
\(91\) 87.0000 0.100221
\(92\) −464.000 −0.525819
\(93\) 0 0
\(94\) −462.000 −0.506933
\(95\) 200.000 0.215995
\(96\) 0 0
\(97\) −560.000 −0.586179 −0.293090 − 0.956085i \(-0.594683\pi\)
−0.293090 + 0.956085i \(0.594683\pi\)
\(98\) 996.000 1.02664
\(99\) 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 1305.4.a.d.1.1 1
3.2 odd 2 435.4.a.a.1.1 ✓ 1
15.14 odd 2 2175.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.a.1.1 ✓ 1 3.2 odd 2
1305.4.a.d.1.1 1 1.1 even 1 trivial
2175.4.a.c.1.1 1 15.14 odd 2