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Properties

Label 7104.2.a.bc.1.2
Level $7104$
Weight $2$
Character 7104.1
Self dual yes
Analytic conductor $56.726$
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] = [7104,2,Mod(1,7104)] 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("7104.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7104 = 2^{6} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7104.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-2,0,-8,0,2,0,4] 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: yes
Analytic conductor: \(56.7257255959\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 888)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 7104.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.23607 q^{5} -4.00000 q^{7} +1.00000 q^{9} -2.47214 q^{11} -4.47214 q^{13} -1.23607 q^{15} -5.23607 q^{17} +4.00000 q^{21} -3.23607 q^{23} -3.47214 q^{25} -1.00000 q^{27} -3.70820 q^{29} -10.4721 q^{31} +2.47214 q^{33} -4.94427 q^{35} -1.00000 q^{37} +4.47214 q^{39} +6.94427 q^{41} +6.47214 q^{43} +1.23607 q^{45} -8.00000 q^{47} +9.00000 q^{49} +5.23607 q^{51} +0.472136 q^{53} -3.05573 q^{55} +12.1803 q^{59} +14.9443 q^{61} -4.00000 q^{63} -5.52786 q^{65} -4.94427 q^{67} +3.23607 q^{69} -12.9443 q^{71} -0.472136 q^{73} +3.47214 q^{75} +9.88854 q^{77} -8.94427 q^{79} +1.00000 q^{81} -5.52786 q^{83} -6.47214 q^{85} +3.70820 q^{87} +4.29180 q^{89} +17.8885 q^{91} +10.4721 q^{93} +13.4164 q^{97} -2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 8 q^{7} + 2 q^{9} + 4 q^{11} + 2 q^{15} - 6 q^{17} + 8 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 12 q^{31} - 4 q^{33} + 8 q^{35} - 2 q^{37} - 4 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.23607 0.552786 0.276393 − 0.961045i \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.47214 −0.745377 −0.372689 − 0.927957i \(-0.621564\pi\)
−0.372689 + 0.927957i \(0.621564\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 − 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) −1.23607 −0.319151
\(16\) 0 0
\(17\) −5.23607 −1.26993 −0.634967 − 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −3.23607 −0.674767 −0.337383 − 0.941367i \(-0.609542\pi\)
−0.337383 + 0.941367i \(0.609542\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.70820 −0.688596 −0.344298 − 0.938860i \(-0.611883\pi\)
−0.344298 + 0.938860i \(0.611883\pi\)
\(30\) 0 0
\(31\) −10.4721 −1.88085 −0.940426 − 0.340000i \(-0.889573\pi\)
−0.940426 + 0.340000i \(0.889573\pi\)
\(32\) 0 0
\(33\) 2.47214 0.430344
\(34\) 0 0
\(35\) −4.94427 −0.835734
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) 6.94427 1.08451 0.542257 − 0.840213i \(-0.317570\pi\)
0.542257 + 0.840213i \(0.317570\pi\)
\(42\) 0 0
\(43\) 6.47214 0.986991 0.493496 − 0.869748i \(-0.335719\pi\)
0.493496 + 0.869748i \(0.335719\pi\)
\(44\) 0 0
\(45\) 1.23607 0.184262
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 5.23607 0.733196
\(52\) 0 0
\(53\) 0.472136 0.0648529 0.0324264 − 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) 0 0
\(55\) −3.05573 −0.412034
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.1803 1.58575 0.792873 − 0.609387i \(-0.208585\pi\)
0.792873 + 0.609387i \(0.208585\pi\)
\(60\) 0 0
\(61\) 14.9443 1.91342 0.956709 − 0.291046i \(-0.0940034\pi\)
0.956709 + 0.291046i \(0.0940034\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) −5.52786 −0.685647
\(66\) 0 0
\(67\) −4.94427 −0.604039 −0.302019 − 0.953302i \(-0.597661\pi\)
−0.302019 + 0.953302i \(0.597661\pi\)
\(68\) 0 0
\(69\) 3.23607 0.389577
\(70\) 0 0
\(71\) −12.9443 −1.53620 −0.768101 − 0.640328i \(-0.778798\pi\)
−0.768101 + 0.640328i \(0.778798\pi\)
\(72\) 0 0
\(73\) −0.472136 −0.0552593 −0.0276297 − 0.999618i \(-0.508796\pi\)
−0.0276297 + 0.999618i \(0.508796\pi\)
\(74\) 0 0
\(75\) 3.47214 0.400928
\(76\) 0 0
\(77\) 9.88854 1.12690
\(78\) 0 0
\(79\) −8.94427 −1.00631 −0.503155 − 0.864196i \(-0.667827\pi\)
−0.503155 + 0.864196i \(0.667827\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.52786 −0.606762 −0.303381 − 0.952869i \(-0.598115\pi\)
−0.303381 + 0.952869i \(0.598115\pi\)
\(84\) 0 0
\(85\) −6.47214 −0.702002
\(86\) 0 0
\(87\) 3.70820 0.397561
\(88\) 0 0
\(89\) 4.29180 0.454929 0.227465 − 0.973786i \(-0.426956\pi\)
0.227465 + 0.973786i \(0.426956\pi\)
\(90\) 0 0
\(91\) 17.8885 1.87523
\(92\) 0 0
\(93\) 10.4721 1.08591
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.4164 1.36223 0.681115 − 0.732177i \(-0.261495\pi\)
0.681115 + 0.732177i \(0.261495\pi\)
\(98\) 0 0
\(99\) −2.47214 −0.248459
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 7104.2.a.bc.1.2 2
4.3 odd 2 7104.2.a.bj.1.2 2
8.3 odd 2 888.2.a.f.1.1 ✓ 2
8.5 even 2 1776.2.a.p.1.1 2
24.5 odd 2 5328.2.a.ba.1.2 2
24.11 even 2 2664.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.f.1.1 ✓ 2 8.3 odd 2
1776.2.a.p.1.1 2 8.5 even 2
2664.2.a.j.1.2 2 24.11 even 2
5328.2.a.ba.1.2 2 24.5 odd 2
7104.2.a.bc.1.2 2 1.1 even 1 trivial
7104.2.a.bj.1.2 2 4.3 odd 2