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Properties

Label 312.4.a.a.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.46410 q^{5} +12.3923 q^{7} +9.00000 q^{9} -6.78461 q^{11} -13.0000 q^{13} +16.3923 q^{15} +72.0666 q^{17} -99.2436 q^{19} -37.1769 q^{21} +120.708 q^{23} -95.1436 q^{25} -27.0000 q^{27} -185.138 q^{29} -85.4641 q^{31} +20.3538 q^{33} -67.7128 q^{35} -340.928 q^{37} +39.0000 q^{39} -427.587 q^{41} +64.9179 q^{43} -49.1769 q^{45} -39.2820 q^{47} -189.431 q^{49} -216.200 q^{51} -21.4462 q^{53} +37.0718 q^{55} +297.731 q^{57} +62.4205 q^{59} -423.149 q^{61} +111.531 q^{63} +71.0333 q^{65} +451.643 q^{67} -362.123 q^{69} +335.779 q^{71} -1016.60 q^{73} +285.431 q^{75} -84.0770 q^{77} -398.390 q^{79} +81.0000 q^{81} -865.672 q^{83} -393.779 q^{85} +555.415 q^{87} +641.577 q^{89} -161.100 q^{91} +256.392 q^{93} +542.277 q^{95} +1381.71 q^{97} -61.0615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 4 q^{5} + 4 q^{7} + 18 q^{9} + 28 q^{11} - 26 q^{13} + 12 q^{15} - 36 q^{17} + 44 q^{19} - 12 q^{21} - 8 q^{23} - 218 q^{25} - 54 q^{27} - 204 q^{29} - 164 q^{31} - 84 q^{33} - 80 q^{35}+ \cdots + 252 q^{99}+O(q^{100}) \) 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
\(5\) −5.46410 −0.488724 −0.244362 − 0.969684i \(-0.578579\pi\)
−0.244362 + 0.969684i \(0.578579\pi\)
\(6\) 0 0
\(7\) 12.3923 0.669122 0.334561 − 0.942374i \(-0.391412\pi\)
0.334561 + 0.942374i \(0.391412\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −6.78461 −0.185967 −0.0929835 − 0.995668i \(-0.529640\pi\)
−0.0929835 + 0.995668i \(0.529640\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 16.3923 0.282165
\(16\) 0 0
\(17\) 72.0666 1.02816 0.514080 − 0.857742i \(-0.328133\pi\)
0.514080 + 0.857742i \(0.328133\pi\)
\(18\) 0 0
\(19\) −99.2436 −1.19832 −0.599159 − 0.800630i \(-0.704498\pi\)
−0.599159 + 0.800630i \(0.704498\pi\)
\(20\) 0 0
\(21\) −37.1769 −0.386318
\(22\) 0 0
\(23\) 120.708 1.09432 0.547158 − 0.837029i \(-0.315710\pi\)
0.547158 + 0.837029i \(0.315710\pi\)
\(24\) 0 0
\(25\) −95.1436 −0.761149
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −185.138 −1.18549 −0.592747 − 0.805388i \(-0.701957\pi\)
−0.592747 + 0.805388i \(0.701957\pi\)
\(30\) 0 0
\(31\) −85.4641 −0.495155 −0.247578 − 0.968868i \(-0.579634\pi\)
−0.247578 + 0.968868i \(0.579634\pi\)
\(32\) 0 0
\(33\) 20.3538 0.107368
\(34\) 0 0
\(35\) −67.7128 −0.327016
\(36\) 0 0
\(37\) −340.928 −1.51482 −0.757409 − 0.652941i \(-0.773535\pi\)
−0.757409 + 0.652941i \(0.773535\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −427.587 −1.62873 −0.814364 − 0.580354i \(-0.802914\pi\)
−0.814364 + 0.580354i \(0.802914\pi\)
\(42\) 0 0
\(43\) 64.9179 0.230230 0.115115 − 0.993352i \(-0.463276\pi\)
0.115115 + 0.993352i \(0.463276\pi\)
\(44\) 0 0
\(45\) −49.1769 −0.162908
\(46\) 0 0
\(47\) −39.2820 −0.121912 −0.0609561 − 0.998140i \(-0.519415\pi\)
−0.0609561 + 0.998140i \(0.519415\pi\)
\(48\) 0 0
\(49\) −189.431 −0.552276
\(50\) 0 0
\(51\) −216.200 −0.593609
\(52\) 0 0
\(53\) −21.4462 −0.0555824 −0.0277912 − 0.999614i \(-0.508847\pi\)
−0.0277912 + 0.999614i \(0.508847\pi\)
\(54\) 0 0
\(55\) 37.0718 0.0908865
\(56\) 0 0
\(57\) 297.731 0.691849
\(58\) 0 0
\(59\) 62.4205 0.137736 0.0688682 − 0.997626i \(-0.478061\pi\)
0.0688682 + 0.997626i \(0.478061\pi\)
\(60\) 0 0
\(61\) −423.149 −0.888175 −0.444087 − 0.895984i \(-0.646472\pi\)
−0.444087 + 0.895984i \(0.646472\pi\)
\(62\) 0 0
\(63\) 111.531 0.223041
\(64\) 0 0
\(65\) 71.0333 0.135548
\(66\) 0 0
\(67\) 451.643 0.823538 0.411769 − 0.911288i \(-0.364911\pi\)
0.411769 + 0.911288i \(0.364911\pi\)
\(68\) 0 0
\(69\) −362.123 −0.631804
\(70\) 0 0
\(71\) 335.779 0.561263 0.280632 − 0.959816i \(-0.409456\pi\)
0.280632 + 0.959816i \(0.409456\pi\)
\(72\) 0 0
\(73\) −1016.60 −1.62992 −0.814959 − 0.579519i \(-0.803241\pi\)
−0.814959 + 0.579519i \(0.803241\pi\)
\(74\) 0 0
\(75\) 285.431 0.439449
\(76\) 0 0
\(77\) −84.0770 −0.124435
\(78\) 0 0
\(79\) −398.390 −0.567371 −0.283686 − 0.958917i \(-0.591557\pi\)
−0.283686 + 0.958917i \(0.591557\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −865.672 −1.14482 −0.572408 − 0.819969i \(-0.693991\pi\)
−0.572408 + 0.819969i \(0.693991\pi\)
\(84\) 0 0
\(85\) −393.779 −0.502487
\(86\) 0 0
\(87\) 555.415 0.684446
\(88\) 0 0
\(89\) 641.577 0.764124 0.382062 − 0.924137i \(-0.375214\pi\)
0.382062 + 0.924137i \(0.375214\pi\)
\(90\) 0 0
\(91\) −161.100 −0.185581
\(92\) 0 0
\(93\) 256.392 0.285878
\(94\) 0 0
\(95\) 542.277 0.585647
\(96\) 0 0
\(97\) 1381.71 1.44630 0.723150 − 0.690691i \(-0.242693\pi\)
0.723150 + 0.690691i \(0.242693\pi\)
\(98\) 0 0
\(99\) −61.0615 −0.0619890
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.1.1 ✓ 2
3.2 odd 2 936.4.a.g.1.2 2
4.3 odd 2 624.4.a.o.1.1 2
8.3 odd 2 2496.4.a.z.1.2 2
8.5 even 2 2496.4.a.bg.1.2 2
12.11 even 2 1872.4.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.a.1.1 ✓ 2 1.1 even 1 trivial
624.4.a.o.1.1 2 4.3 odd 2
936.4.a.g.1.2 2 3.2 odd 2
1872.4.a.be.1.2 2 12.11 even 2
2496.4.a.z.1.2 2 8.3 odd 2
2496.4.a.bg.1.2 2 8.5 even 2