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Properties

Label 624.4.a.r.1.2
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $2$
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] = [624,4,Mod(1,624)] 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("624.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,0,24,0,0] 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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.74166\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +19.4833 q^{5} -7.48331 q^{7} +9.00000 q^{9} -22.8999 q^{11} -13.0000 q^{13} +58.4499 q^{15} +67.0334 q^{17} -16.5167 q^{19} -22.4499 q^{21} +175.600 q^{23} +254.600 q^{25} +27.0000 q^{27} +291.800 q^{29} -117.283 q^{31} -68.6997 q^{33} -145.800 q^{35} -154.766 q^{37} -39.0000 q^{39} -251.716 q^{41} +502.566 q^{43} +175.350 q^{45} +281.733 q^{47} -287.000 q^{49} +201.100 q^{51} +366.999 q^{53} -446.166 q^{55} -49.5501 q^{57} +79.6663 q^{59} -194.865 q^{61} -67.3498 q^{63} -253.283 q^{65} -400.082 q^{67} +526.799 q^{69} -528.299 q^{71} -734.366 q^{73} +763.799 q^{75} +171.367 q^{77} -113.266 q^{79} +81.0000 q^{81} +933.466 q^{83} +1306.03 q^{85} +875.399 q^{87} +1190.91 q^{89} +97.2831 q^{91} -351.849 q^{93} -321.800 q^{95} +557.165 q^{97} -206.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 24 q^{5} + 18 q^{9} + 44 q^{11} - 26 q^{13} + 72 q^{15} + 164 q^{17} - 48 q^{19} - 8 q^{23} + 150 q^{25} + 54 q^{27} + 404 q^{29} - 40 q^{31} + 132 q^{33} - 112 q^{35} - 100 q^{37} - 78 q^{39}+ \cdots + 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 19.4833 1.74264 0.871320 − 0.490715i \(-0.163264\pi\)
0.871320 + 0.490715i \(0.163264\pi\)
\(6\) 0 0
\(7\) −7.48331 −0.404061 −0.202031 − 0.979379i \(-0.564754\pi\)
−0.202031 + 0.979379i \(0.564754\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −22.8999 −0.627689 −0.313844 − 0.949474i \(-0.601617\pi\)
−0.313844 + 0.949474i \(0.601617\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 58.4499 1.00611
\(16\) 0 0
\(17\) 67.0334 0.956352 0.478176 − 0.878264i \(-0.341298\pi\)
0.478176 + 0.878264i \(0.341298\pi\)
\(18\) 0 0
\(19\) −16.5167 −0.199431 −0.0997155 − 0.995016i \(-0.531793\pi\)
−0.0997155 + 0.995016i \(0.531793\pi\)
\(20\) 0 0
\(21\) −22.4499 −0.233285
\(22\) 0 0
\(23\) 175.600 1.59196 0.795979 − 0.605324i \(-0.206956\pi\)
0.795979 + 0.605324i \(0.206956\pi\)
\(24\) 0 0
\(25\) 254.600 2.03680
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 291.800 1.86848 0.934239 − 0.356648i \(-0.116080\pi\)
0.934239 + 0.356648i \(0.116080\pi\)
\(30\) 0 0
\(31\) −117.283 −0.679505 −0.339753 − 0.940515i \(-0.610343\pi\)
−0.339753 + 0.940515i \(0.610343\pi\)
\(32\) 0 0
\(33\) −68.6997 −0.362396
\(34\) 0 0
\(35\) −145.800 −0.704133
\(36\) 0 0
\(37\) −154.766 −0.687661 −0.343830 − 0.939032i \(-0.611724\pi\)
−0.343830 + 0.939032i \(0.611724\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) −251.716 −0.958815 −0.479407 − 0.877592i \(-0.659148\pi\)
−0.479407 + 0.877592i \(0.659148\pi\)
\(42\) 0 0
\(43\) 502.566 1.78234 0.891170 − 0.453669i \(-0.149885\pi\)
0.891170 + 0.453669i \(0.149885\pi\)
\(44\) 0 0
\(45\) 175.350 0.580880
\(46\) 0 0
\(47\) 281.733 0.874361 0.437181 − 0.899374i \(-0.355977\pi\)
0.437181 + 0.899374i \(0.355977\pi\)
\(48\) 0 0
\(49\) −287.000 −0.836735
\(50\) 0 0
\(51\) 201.100 0.552150
\(52\) 0 0
\(53\) 366.999 0.951154 0.475577 − 0.879674i \(-0.342239\pi\)
0.475577 + 0.879674i \(0.342239\pi\)
\(54\) 0 0
\(55\) −446.166 −1.09384
\(56\) 0 0
\(57\) −49.5501 −0.115141
\(58\) 0 0
\(59\) 79.6663 0.175791 0.0878955 − 0.996130i \(-0.471986\pi\)
0.0878955 + 0.996130i \(0.471986\pi\)
\(60\) 0 0
\(61\) −194.865 −0.409016 −0.204508 − 0.978865i \(-0.565559\pi\)
−0.204508 + 0.978865i \(0.565559\pi\)
\(62\) 0 0
\(63\) −67.3498 −0.134687
\(64\) 0 0
\(65\) −253.283 −0.483322
\(66\) 0 0
\(67\) −400.082 −0.729519 −0.364759 − 0.931102i \(-0.618849\pi\)
−0.364759 + 0.931102i \(0.618849\pi\)
\(68\) 0 0
\(69\) 526.799 0.919117
\(70\) 0 0
\(71\) −528.299 −0.883065 −0.441532 − 0.897245i \(-0.645565\pi\)
−0.441532 + 0.897245i \(0.645565\pi\)
\(72\) 0 0
\(73\) −734.366 −1.17741 −0.588706 − 0.808347i \(-0.700362\pi\)
−0.588706 + 0.808347i \(0.700362\pi\)
\(74\) 0 0
\(75\) 763.799 1.17594
\(76\) 0 0
\(77\) 171.367 0.253625
\(78\) 0 0
\(79\) −113.266 −0.161309 −0.0806545 − 0.996742i \(-0.525701\pi\)
−0.0806545 + 0.996742i \(0.525701\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 933.466 1.23447 0.617236 − 0.786778i \(-0.288252\pi\)
0.617236 + 0.786778i \(0.288252\pi\)
\(84\) 0 0
\(85\) 1306.03 1.66658
\(86\) 0 0
\(87\) 875.399 1.07877
\(88\) 0 0
\(89\) 1190.91 1.41839 0.709195 − 0.705012i \(-0.249059\pi\)
0.709195 + 0.705012i \(0.249059\pi\)
\(90\) 0 0
\(91\) 97.2831 0.112066
\(92\) 0 0
\(93\) −351.849 −0.392313
\(94\) 0 0
\(95\) −321.800 −0.347536
\(96\) 0 0
\(97\) 557.165 0.583211 0.291606 − 0.956539i \(-0.405811\pi\)
0.291606 + 0.956539i \(0.405811\pi\)
\(98\) 0 0
\(99\) −206.099 −0.209230
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 624.4.a.r.1.2 2
3.2 odd 2 1872.4.a.t.1.1 2
4.3 odd 2 39.4.a.b.1.1 ✓ 2
8.3 odd 2 2496.4.a.bc.1.1 2
8.5 even 2 2496.4.a.s.1.1 2
12.11 even 2 117.4.a.c.1.2 2
20.19 odd 2 975.4.a.j.1.2 2
28.27 even 2 1911.4.a.h.1.1 2
52.31 even 4 507.4.b.f.337.3 4
52.47 even 4 507.4.b.f.337.2 4
52.51 odd 2 507.4.a.f.1.2 2
156.155 even 2 1521.4.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 ✓ 2 4.3 odd 2
117.4.a.c.1.2 2 12.11 even 2
507.4.a.f.1.2 2 52.51 odd 2
507.4.b.f.337.2 4 52.47 even 4
507.4.b.f.337.3 4 52.31 even 4
624.4.a.r.1.2 2 1.1 even 1 trivial
975.4.a.j.1.2 2 20.19 odd 2
1521.4.a.s.1.1 2 156.155 even 2
1872.4.a.t.1.1 2 3.2 odd 2
1911.4.a.h.1.1 2 28.27 even 2
2496.4.a.s.1.1 2 8.5 even 2
2496.4.a.bc.1.1 2 8.3 odd 2