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Properties

Label 400.4.c.d.49.2
Level $400$
Weight $4$
Character 400.49
Analytic conductor $23.601$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(49,400)] 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("400.49"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-44,0,-54] 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: no
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.4.c.d.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+7.00000i q^{3} +34.0000i q^{7} -22.0000 q^{9} -27.0000 q^{11} +28.0000i q^{13} +21.0000i q^{17} +35.0000 q^{19} -238.000 q^{21} -78.0000i q^{23} +35.0000i q^{27} +120.000 q^{29} -182.000 q^{31} -189.000i q^{33} +146.000i q^{37} -196.000 q^{39} +357.000 q^{41} -148.000i q^{43} +84.0000i q^{47} -813.000 q^{49} -147.000 q^{51} -702.000i q^{53} +245.000i q^{57} -840.000 q^{59} -238.000 q^{61} -748.000i q^{63} -461.000i q^{67} +546.000 q^{69} +708.000 q^{71} +133.000i q^{73} -918.000i q^{77} +650.000 q^{79} -839.000 q^{81} -903.000i q^{83} +840.000i q^{87} -735.000 q^{89} -952.000 q^{91} -1274.00i q^{93} +1106.00i q^{97} +594.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 44 q^{9} - 54 q^{11} + 70 q^{19} - 476 q^{21} + 240 q^{29} - 364 q^{31} - 392 q^{39} + 714 q^{41} - 1626 q^{49} - 294 q^{51} - 1680 q^{59} - 476 q^{61} + 1092 q^{69} + 1416 q^{71} + 1300 q^{79} - 1678 q^{81}+ \cdots + 1188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 7.00000i 1.34715i 0.739119 + 0.673575i \(0.235242\pi\)
−0.739119 + 0.673575i \(0.764758\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 34.0000i 1.83583i 0.396780 + 0.917914i \(0.370128\pi\)
−0.396780 + 0.917914i \(0.629872\pi\)
\(8\) 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
\(13\) 28.0000i 0.597369i 0.954352 + 0.298685i \(0.0965479\pi\)
−0.954352 + 0.298685i \(0.903452\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 21.0000i 0.299603i 0.988716 + 0.149801i \(0.0478634\pi\)
−0.988716 + 0.149801i \(0.952137\pi\)
\(18\) 0 0
\(19\) 35.0000 0.422608 0.211304 − 0.977420i \(-0.432229\pi\)
0.211304 + 0.977420i \(0.432229\pi\)
\(20\) 0 0
\(21\) −238.000 −2.47314
\(22\) 0 0
\(23\) − 78.0000i − 0.707136i −0.935409 − 0.353568i \(-0.884968\pi\)
0.935409 − 0.353568i \(-0.115032\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 35.0000i 0.249472i
\(28\) 0 0
\(29\) 120.000 0.768395 0.384197 − 0.923251i \(-0.374478\pi\)
0.384197 + 0.923251i \(0.374478\pi\)
\(30\) 0 0
\(31\) −182.000 −1.05446 −0.527228 − 0.849724i \(-0.676769\pi\)
−0.527228 + 0.849724i \(0.676769\pi\)
\(32\) 0 0
\(33\) − 189.000i − 0.996990i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 146.000i 0.648710i 0.945936 + 0.324355i \(0.105147\pi\)
−0.945936 + 0.324355i \(0.894853\pi\)
\(38\) 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
\(43\) − 148.000i − 0.524879i −0.964948 − 0.262439i \(-0.915473\pi\)
0.964948 − 0.262439i \(-0.0845270\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 84.0000i 0.260695i 0.991468 + 0.130347i \(0.0416093\pi\)
−0.991468 + 0.130347i \(0.958391\pi\)
\(48\) 0 0
\(49\) −813.000 −2.37026
\(50\) 0 0
\(51\) −147.000 −0.403610
\(52\) 0 0
\(53\) − 702.000i − 1.81938i −0.415288 − 0.909690i \(-0.636319\pi\)
0.415288 − 0.909690i \(-0.363681\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 245.000i 0.569317i
\(58\) 0 0
\(59\) −840.000 −1.85354 −0.926769 − 0.375633i \(-0.877425\pi\)
−0.926769 + 0.375633i \(0.877425\pi\)
\(60\) 0 0
\(61\) −238.000 −0.499554 −0.249777 − 0.968303i \(-0.580357\pi\)
−0.249777 + 0.968303i \(0.580357\pi\)
\(62\) 0 0
\(63\) − 748.000i − 1.49586i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 461.000i − 0.840599i −0.907386 − 0.420299i \(-0.861925\pi\)
0.907386 − 0.420299i \(-0.138075\pi\)
\(68\) 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
\(73\) 133.000i 0.213239i 0.994300 + 0.106620i \(0.0340027\pi\)
−0.994300 + 0.106620i \(0.965997\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 918.000i − 1.35865i
\(78\) 0 0
\(79\) 650.000 0.925705 0.462853 − 0.886435i \(-0.346826\pi\)
0.462853 + 0.886435i \(0.346826\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 0 0
\(83\) − 903.000i − 1.19418i −0.802173 − 0.597091i \(-0.796323\pi\)
0.802173 − 0.597091i \(-0.203677\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 840.000i 1.03514i
\(88\) 0 0
\(89\) −735.000 −0.875392 −0.437696 − 0.899123i \(-0.644205\pi\)
−0.437696 + 0.899123i \(0.644205\pi\)
\(90\) 0 0
\(91\) −952.000 −1.09667
\(92\) 0 0
\(93\) − 1274.00i − 1.42051i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1106.00i 1.15770i 0.815433 + 0.578852i \(0.196499\pi\)
−0.815433 + 0.578852i \(0.803501\pi\)
\(98\) 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.c.d.49.2 2
4.3 odd 2 50.4.b.b.49.2 2
5.2 odd 4 400.4.a.r.1.1 1
5.3 odd 4 400.4.a.d.1.1 1
5.4 even 2 inner 400.4.c.d.49.1 2
12.11 even 2 450.4.c.c.199.1 2
20.3 even 4 50.4.a.e.1.1 yes 1
20.7 even 4 50.4.a.a.1.1 ✓ 1
20.19 odd 2 50.4.b.b.49.1 2
40.3 even 4 1600.4.a.f.1.1 1
40.13 odd 4 1600.4.a.bv.1.1 1
40.27 even 4 1600.4.a.bu.1.1 1
40.37 odd 4 1600.4.a.g.1.1 1
60.23 odd 4 450.4.a.a.1.1 1
60.47 odd 4 450.4.a.t.1.1 1
60.59 even 2 450.4.c.c.199.2 2
140.27 odd 4 2450.4.a.t.1.1 1
140.83 odd 4 2450.4.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.4.a.a.1.1 ✓ 1 20.7 even 4
50.4.a.e.1.1 yes 1 20.3 even 4
50.4.b.b.49.1 2 20.19 odd 2
50.4.b.b.49.2 2 4.3 odd 2
400.4.a.d.1.1 1 5.3 odd 4
400.4.a.r.1.1 1 5.2 odd 4
400.4.c.d.49.1 2 5.4 even 2 inner
400.4.c.d.49.2 2 1.1 even 1 trivial
450.4.a.a.1.1 1 60.23 odd 4
450.4.a.t.1.1 1 60.47 odd 4
450.4.c.c.199.1 2 12.11 even 2
450.4.c.c.199.2 2 60.59 even 2
1600.4.a.f.1.1 1 40.3 even 4
1600.4.a.g.1.1 1 40.37 odd 4
1600.4.a.bu.1.1 1 40.27 even 4
1600.4.a.bv.1.1 1 40.13 odd 4
2450.4.a.t.1.1 1 140.27 odd 4
2450.4.a.y.1.1 1 140.83 odd 4