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Properties

Label 400.4.a.d.1.1
Level $400$
Weight $4$
Character 400.1
Self dual yes
Analytic conductor $23.601$
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] = [400,4,Mod(1,400)] 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("400.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-7,0,0,0,34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 400.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.00000 q^{3} +34.0000 q^{7} +22.0000 q^{9} -27.0000 q^{11} -28.0000 q^{13} +21.0000 q^{17} -35.0000 q^{19} -238.000 q^{21} +78.0000 q^{23} +35.0000 q^{27} -120.000 q^{29} -182.000 q^{31} +189.000 q^{33} +146.000 q^{37} +196.000 q^{39} +357.000 q^{41} +148.000 q^{43} +84.0000 q^{47} +813.000 q^{49} -147.000 q^{51} +702.000 q^{53} +245.000 q^{57} +840.000 q^{59} -238.000 q^{61} +748.000 q^{63} -461.000 q^{67} -546.000 q^{69} +708.000 q^{71} -133.000 q^{73} -918.000 q^{77} -650.000 q^{79} -839.000 q^{81} +903.000 q^{83} +840.000 q^{87} +735.000 q^{89} -952.000 q^{91} +1274.00 q^{93} +1106.00 q^{97} -594.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −7.00000 −1.34715 −0.673575 − 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 34.0000 1.83583 0.917914 − 0.396780i \(-0.129872\pi\)
0.917914 + 0.396780i \(0.129872\pi\)
\(8\) 0 0
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) −27.0000 −0.740073 −0.370037 − 0.929017i \(-0.620655\pi\)
−0.370037 + 0.929017i \(0.620655\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 − 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 21.0000 0.299603 0.149801 − 0.988716i \(-0.452137\pi\)
0.149801 + 0.988716i \(0.452137\pi\)
\(18\) 0 0
\(19\) −35.0000 −0.422608 −0.211304 − 0.977420i \(-0.567771\pi\)
−0.211304 + 0.977420i \(0.567771\pi\)
\(20\) 0 0
\(21\) −238.000 −2.47314
\(22\) 0 0
\(23\) 78.0000 0.707136 0.353568 − 0.935409i \(-0.384968\pi\)
0.353568 + 0.935409i \(0.384968\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 35.0000 0.249472
\(28\) 0 0
\(29\) −120.000 −0.768395 −0.384197 − 0.923251i \(-0.625522\pi\)
−0.384197 + 0.923251i \(0.625522\pi\)
\(30\) 0 0
\(31\) −182.000 −1.05446 −0.527228 − 0.849724i \(-0.676769\pi\)
−0.527228 + 0.849724i \(0.676769\pi\)
\(32\) 0 0
\(33\) 189.000 0.996990
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 146.000 0.648710 0.324355 − 0.945936i \(-0.394853\pi\)
0.324355 + 0.945936i \(0.394853\pi\)
\(38\) 0 0
\(39\) 196.000 0.804747
\(40\) 0 0
\(41\) 357.000 1.35985 0.679927 − 0.733280i \(-0.262011\pi\)
0.679927 + 0.733280i \(0.262011\pi\)
\(42\) 0 0
\(43\) 148.000 0.524879 0.262439 − 0.964948i \(-0.415473\pi\)
0.262439 + 0.964948i \(0.415473\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 84.0000 0.260695 0.130347 − 0.991468i \(-0.458391\pi\)
0.130347 + 0.991468i \(0.458391\pi\)
\(48\) 0 0
\(49\) 813.000 2.37026
\(50\) 0 0
\(51\) −147.000 −0.403610
\(52\) 0 0
\(53\) 702.000 1.81938 0.909690 − 0.415288i \(-0.136319\pi\)
0.909690 + 0.415288i \(0.136319\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 245.000 0.569317
\(58\) 0 0
\(59\) 840.000 1.85354 0.926769 − 0.375633i \(-0.122575\pi\)
0.926769 + 0.375633i \(0.122575\pi\)
\(60\) 0 0
\(61\) −238.000 −0.499554 −0.249777 − 0.968303i \(-0.580357\pi\)
−0.249777 + 0.968303i \(0.580357\pi\)
\(62\) 0 0
\(63\) 748.000 1.49586
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −461.000 −0.840599 −0.420299 − 0.907386i \(-0.638075\pi\)
−0.420299 + 0.907386i \(0.638075\pi\)
\(68\) 0 0
\(69\) −546.000 −0.952618
\(70\) 0 0
\(71\) 708.000 1.18344 0.591719 − 0.806144i \(-0.298449\pi\)
0.591719 + 0.806144i \(0.298449\pi\)
\(72\) 0 0
\(73\) −133.000 −0.213239 −0.106620 − 0.994300i \(-0.534003\pi\)
−0.106620 + 0.994300i \(0.534003\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −918.000 −1.35865
\(78\) 0 0
\(79\) −650.000 −0.925705 −0.462853 − 0.886435i \(-0.653174\pi\)
−0.462853 + 0.886435i \(0.653174\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 0 0
\(83\) 903.000 1.19418 0.597091 − 0.802173i \(-0.296323\pi\)
0.597091 + 0.802173i \(0.296323\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 840.000 1.03514
\(88\) 0 0
\(89\) 735.000 0.875392 0.437696 − 0.899123i \(-0.355795\pi\)
0.437696 + 0.899123i \(0.355795\pi\)
\(90\) 0 0
\(91\) −952.000 −1.09667
\(92\) 0 0
\(93\) 1274.00 1.42051
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1106.00 1.15770 0.578852 − 0.815433i \(-0.303501\pi\)
0.578852 + 0.815433i \(0.303501\pi\)
\(98\) 0 0
\(99\) −594.000 −0.603023
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 400.4.a.d.1.1 1
4.3 odd 2 50.4.a.e.1.1 yes 1
5.2 odd 4 400.4.c.d.49.2 2
5.3 odd 4 400.4.c.d.49.1 2
5.4 even 2 400.4.a.r.1.1 1
8.3 odd 2 1600.4.a.f.1.1 1
8.5 even 2 1600.4.a.bv.1.1 1
12.11 even 2 450.4.a.a.1.1 1
20.3 even 4 50.4.b.b.49.1 2
20.7 even 4 50.4.b.b.49.2 2
20.19 odd 2 50.4.a.a.1.1 ✓ 1
28.27 even 2 2450.4.a.y.1.1 1
40.19 odd 2 1600.4.a.bu.1.1 1
40.29 even 2 1600.4.a.g.1.1 1
60.23 odd 4 450.4.c.c.199.2 2
60.47 odd 4 450.4.c.c.199.1 2
60.59 even 2 450.4.a.t.1.1 1
140.139 even 2 2450.4.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.4.a.a.1.1 ✓ 1 20.19 odd 2
50.4.a.e.1.1 yes 1 4.3 odd 2
50.4.b.b.49.1 2 20.3 even 4
50.4.b.b.49.2 2 20.7 even 4
400.4.a.d.1.1 1 1.1 even 1 trivial
400.4.a.r.1.1 1 5.4 even 2
400.4.c.d.49.1 2 5.3 odd 4
400.4.c.d.49.2 2 5.2 odd 4
450.4.a.a.1.1 1 12.11 even 2
450.4.a.t.1.1 1 60.59 even 2
450.4.c.c.199.1 2 60.47 odd 4
450.4.c.c.199.2 2 60.23 odd 4
1600.4.a.f.1.1 1 8.3 odd 2
1600.4.a.g.1.1 1 40.29 even 2
1600.4.a.bu.1.1 1 40.19 odd 2
1600.4.a.bv.1.1 1 8.5 even 2
2450.4.a.t.1.1 1 140.139 even 2
2450.4.a.y.1.1 1 28.27 even 2