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Properties

Label 312.4.a.a.1.2
Level $312$
Weight $4$
Character 312.1
Self dual yes
Analytic conductor $18.409$
Analytic rank $1$
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] = [312,4,Mod(1,312)] 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("312.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 312.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-6,0,-4,0,4] 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: \(18.4085959218\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 312.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +1.46410 q^{5} -8.39230 q^{7} +9.00000 q^{9} +34.7846 q^{11} -13.0000 q^{13} -4.39230 q^{15} -108.067 q^{17} +143.244 q^{19} +25.1769 q^{21} -128.708 q^{23} -122.856 q^{25} -27.0000 q^{27} -18.8616 q^{29} -78.5359 q^{31} -104.354 q^{33} -12.2872 q^{35} -327.072 q^{37} +39.0000 q^{39} +327.587 q^{41} -336.918 q^{43} +13.1769 q^{45} +99.2820 q^{47} -272.569 q^{49} +324.200 q^{51} -686.554 q^{53} +50.9282 q^{55} -429.731 q^{57} -242.420 q^{59} -644.851 q^{61} -75.5307 q^{63} -19.0333 q^{65} -871.643 q^{67} +386.123 q^{69} +100.221 q^{71} +604.600 q^{73} +368.569 q^{75} -291.923 q^{77} +1070.39 q^{79} +81.0000 q^{81} +741.672 q^{83} -158.221 q^{85} +56.5847 q^{87} -501.577 q^{89} +109.100 q^{91} +235.608 q^{93} +209.723 q^{95} -1569.71 q^{97} +313.061 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 4 q^{5} + 4 q^{7} + 18 q^{9} + 28 q^{11} - 26 q^{13} + 12 q^{15} - 36 q^{17} + 44 q^{19} - 12 q^{21} - 8 q^{23} - 218 q^{25} - 54 q^{27} - 204 q^{29} - 164 q^{31} - 84 q^{33} - 80 q^{35}+ \cdots + 252 q^{99}+O(q^{100}) \) 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
\(5\) 1.46410 0.130953 0.0654766 − 0.997854i \(-0.479143\pi\)
0.0654766 + 0.997854i \(0.479143\pi\)
\(6\) 0 0
\(7\) −8.39230 −0.453142 −0.226571 − 0.973995i \(-0.572752\pi\)
−0.226571 + 0.973995i \(0.572752\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 34.7846 0.953450 0.476725 − 0.879052i \(-0.341824\pi\)
0.476725 + 0.879052i \(0.341824\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −4.39230 −0.0756059
\(16\) 0 0
\(17\) −108.067 −1.54177 −0.770883 − 0.636977i \(-0.780185\pi\)
−0.770883 + 0.636977i \(0.780185\pi\)
\(18\) 0 0
\(19\) 143.244 1.72960 0.864798 − 0.502120i \(-0.167446\pi\)
0.864798 + 0.502120i \(0.167446\pi\)
\(20\) 0 0
\(21\) 25.1769 0.261622
\(22\) 0 0
\(23\) −128.708 −1.16684 −0.583422 − 0.812169i \(-0.698287\pi\)
−0.583422 + 0.812169i \(0.698287\pi\)
\(24\) 0 0
\(25\) −122.856 −0.982851
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −18.8616 −0.120776 −0.0603880 − 0.998175i \(-0.519234\pi\)
−0.0603880 + 0.998175i \(0.519234\pi\)
\(30\) 0 0
\(31\) −78.5359 −0.455015 −0.227507 − 0.973776i \(-0.573058\pi\)
−0.227507 + 0.973776i \(0.573058\pi\)
\(32\) 0 0
\(33\) −104.354 −0.550475
\(34\) 0 0
\(35\) −12.2872 −0.0593404
\(36\) 0 0
\(37\) −327.072 −1.45325 −0.726625 − 0.687034i \(-0.758912\pi\)
−0.726625 + 0.687034i \(0.758912\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 327.587 1.24782 0.623909 − 0.781497i \(-0.285544\pi\)
0.623909 + 0.781497i \(0.285544\pi\)
\(42\) 0 0
\(43\) −336.918 −1.19487 −0.597436 − 0.801917i \(-0.703814\pi\)
−0.597436 + 0.801917i \(0.703814\pi\)
\(44\) 0 0
\(45\) 13.1769 0.0436511
\(46\) 0 0
\(47\) 99.2820 0.308123 0.154061 − 0.988061i \(-0.450765\pi\)
0.154061 + 0.988061i \(0.450765\pi\)
\(48\) 0 0
\(49\) −272.569 −0.794662
\(50\) 0 0
\(51\) 324.200 0.890139
\(52\) 0 0
\(53\) −686.554 −1.77935 −0.889674 − 0.456597i \(-0.849068\pi\)
−0.889674 + 0.456597i \(0.849068\pi\)
\(54\) 0 0
\(55\) 50.9282 0.124857
\(56\) 0 0
\(57\) −429.731 −0.998583
\(58\) 0 0
\(59\) −242.420 −0.534923 −0.267462 − 0.963569i \(-0.586185\pi\)
−0.267462 + 0.963569i \(0.586185\pi\)
\(60\) 0 0
\(61\) −644.851 −1.35352 −0.676760 − 0.736204i \(-0.736617\pi\)
−0.676760 + 0.736204i \(0.736617\pi\)
\(62\) 0 0
\(63\) −75.5307 −0.151047
\(64\) 0 0
\(65\) −19.0333 −0.0363199
\(66\) 0 0
\(67\) −871.643 −1.58938 −0.794688 − 0.607018i \(-0.792366\pi\)
−0.794688 + 0.607018i \(0.792366\pi\)
\(68\) 0 0
\(69\) 386.123 0.673677
\(70\) 0 0
\(71\) 100.221 0.167521 0.0837605 − 0.996486i \(-0.473307\pi\)
0.0837605 + 0.996486i \(0.473307\pi\)
\(72\) 0 0
\(73\) 604.600 0.969357 0.484678 − 0.874692i \(-0.338937\pi\)
0.484678 + 0.874692i \(0.338937\pi\)
\(74\) 0 0
\(75\) 368.569 0.567449
\(76\) 0 0
\(77\) −291.923 −0.432048
\(78\) 0 0
\(79\) 1070.39 1.52441 0.762204 − 0.647337i \(-0.224117\pi\)
0.762204 + 0.647337i \(0.224117\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 741.672 0.980832 0.490416 − 0.871489i \(-0.336845\pi\)
0.490416 + 0.871489i \(0.336845\pi\)
\(84\) 0 0
\(85\) −158.221 −0.201899
\(86\) 0 0
\(87\) 56.5847 0.0697301
\(88\) 0 0
\(89\) −501.577 −0.597382 −0.298691 − 0.954350i \(-0.596550\pi\)
−0.298691 + 0.954350i \(0.596550\pi\)
\(90\) 0 0
\(91\) 109.100 0.125679
\(92\) 0 0
\(93\) 235.608 0.262703
\(94\) 0 0
\(95\) 209.723 0.226496
\(96\) 0 0
\(97\) −1569.71 −1.64309 −0.821544 − 0.570144i \(-0.806887\pi\)
−0.821544 + 0.570144i \(0.806887\pi\)
\(98\) 0 0
\(99\) 313.061 0.317817
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 312.4.a.a.1.2 ✓ 2
3.2 odd 2 936.4.a.g.1.1 2
4.3 odd 2 624.4.a.o.1.2 2
8.3 odd 2 2496.4.a.z.1.1 2
8.5 even 2 2496.4.a.bg.1.1 2
12.11 even 2 1872.4.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.a.1.2 ✓ 2 1.1 even 1 trivial
624.4.a.o.1.2 2 4.3 odd 2
936.4.a.g.1.1 2 3.2 odd 2
1872.4.a.be.1.1 2 12.11 even 2
2496.4.a.z.1.1 2 8.3 odd 2
2496.4.a.bg.1.1 2 8.5 even 2