Properties

Label 7406.2.a.bb.1.3
Level $7406$
Weight $2$
Character 7406.1
Self dual yes
Analytic conductor $59.137$
Analytic rank $1$
Dimension $4$
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] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,4,-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: \(59.1372077370\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{10 +4 \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.74912\) of defining polynomial
Character \(\chi\) \(=\) 7406.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.334904 q^{3} +1.00000 q^{4} +2.22274 q^{5} +0.334904 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.88784 q^{9} +2.22274 q^{10} -0.585786 q^{11} +0.334904 q^{12} -5.97186 q^{13} +1.00000 q^{14} +0.744406 q^{15} +1.00000 q^{16} -4.22274 q^{17} -2.88784 q^{18} -2.66981 q^{19} +2.22274 q^{20} +0.334904 q^{21} -0.585786 q^{22} +0.334904 q^{24} -0.0594122 q^{25} -5.97186 q^{26} -1.97186 q^{27} +1.00000 q^{28} -4.43882 q^{29} +0.744406 q^{30} +6.11058 q^{31} +1.00000 q^{32} -0.196182 q^{33} -4.22274 q^{34} +2.22274 q^{35} -2.88784 q^{36} +3.53304 q^{37} -2.66981 q^{38} -2.00000 q^{39} +2.22274 q^{40} +2.27744 q^{41} +0.334904 q^{42} +5.78510 q^{43} -0.585786 q^{44} -6.41893 q^{45} +3.04451 q^{47} +0.334904 q^{48} +1.00000 q^{49} -0.0594122 q^{50} -1.41421 q^{51} -5.97186 q^{52} -9.19932 q^{53} -1.97186 q^{54} -1.30205 q^{55} +1.00000 q^{56} -0.894129 q^{57} -4.43882 q^{58} -4.11058 q^{59} +0.744406 q^{60} -1.23570 q^{61} +6.11058 q^{62} -2.88784 q^{63} +1.00000 q^{64} -13.2739 q^{65} -0.196182 q^{66} -6.46696 q^{67} -4.22274 q^{68} +2.22274 q^{70} +1.21137 q^{71} -2.88784 q^{72} -12.6948 q^{73} +3.53304 q^{74} -0.0198974 q^{75} -2.66981 q^{76} -0.585786 q^{77} -2.00000 q^{78} -12.7672 q^{79} +2.22274 q^{80} +8.00313 q^{81} +2.27744 q^{82} +9.97186 q^{83} +0.334904 q^{84} -9.38607 q^{85} +5.78510 q^{86} -1.48658 q^{87} -0.585786 q^{88} -14.6588 q^{89} -6.41893 q^{90} -5.97186 q^{91} +2.04646 q^{93} +3.04451 q^{94} -5.93430 q^{95} +0.334904 q^{96} -2.85499 q^{97} +1.00000 q^{98} +1.69166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8} - 4 q^{10} - 8 q^{11} - 4 q^{13} + 4 q^{14} + 4 q^{16} - 4 q^{17} - 8 q^{19} - 4 q^{20} - 8 q^{22} - 4 q^{26} + 12 q^{27} + 4 q^{28} - 4 q^{29} + 4 q^{32}+ \cdots - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.334904 0.193357 0.0966785 − 0.995316i \(-0.469178\pi\)
0.0966785 + 0.995316i \(0.469178\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.22274 0.994041 0.497021 − 0.867739i \(-0.334427\pi\)
0.497021 + 0.867739i \(0.334427\pi\)
\(6\) 0.334904 0.136724
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.88784 −0.962613
\(10\) 2.22274 0.702893
\(11\) −0.585786 −0.176621 −0.0883106 − 0.996093i \(-0.528147\pi\)
−0.0883106 + 0.996093i \(0.528147\pi\)
\(12\) 0.334904 0.0966785
\(13\) −5.97186 −1.65630 −0.828148 − 0.560509i \(-0.810605\pi\)
−0.828148 + 0.560509i \(0.810605\pi\)
\(14\) 1.00000 0.267261
\(15\) 0.744406 0.192205
\(16\) 1.00000 0.250000
\(17\) −4.22274 −1.02417 −0.512083 − 0.858936i \(-0.671126\pi\)
−0.512083 + 0.858936i \(0.671126\pi\)
\(18\) −2.88784 −0.680670
\(19\) −2.66981 −0.612496 −0.306248 − 0.951952i \(-0.599074\pi\)
−0.306248 + 0.951952i \(0.599074\pi\)
\(20\) 2.22274 0.497021
\(21\) 0.334904 0.0730820
\(22\) −0.585786 −0.124890
\(23\) 0 0
\(24\) 0.334904 0.0683620
\(25\) −0.0594122 −0.0118824
\(26\) −5.97186 −1.17118
\(27\) −1.97186 −0.379485
\(28\) 1.00000 0.188982
\(29\) −4.43882 −0.824269 −0.412134 − 0.911123i \(-0.635217\pi\)
−0.412134 + 0.911123i \(0.635217\pi\)
\(30\) 0.744406 0.135909
\(31\) 6.11058 1.09749 0.548747 − 0.835989i \(-0.315105\pi\)
0.548747 + 0.835989i \(0.315105\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.196182 −0.0341509
\(34\) −4.22274 −0.724195
\(35\) 2.22274 0.375712
\(36\) −2.88784 −0.481307
\(37\) 3.53304 0.580828 0.290414 − 0.956901i \(-0.406207\pi\)
0.290414 + 0.956901i \(0.406207\pi\)
\(38\) −2.66981 −0.433100
\(39\) −2.00000 −0.320256
\(40\) 2.22274 0.351447
\(41\) 2.27744 0.355677 0.177838 − 0.984060i \(-0.443090\pi\)
0.177838 + 0.984060i \(0.443090\pi\)
\(42\) 0.334904 0.0516768
\(43\) 5.78510 0.882220 0.441110 − 0.897453i \(-0.354585\pi\)
0.441110 + 0.897453i \(0.354585\pi\)
\(44\) −0.585786 −0.0883106
\(45\) −6.41893 −0.956877
\(46\) 0 0
\(47\) 3.04451 0.444087 0.222043 − 0.975037i \(-0.428727\pi\)
0.222043 + 0.975037i \(0.428727\pi\)
\(48\) 0.334904 0.0483392
\(49\) 1.00000 0.142857
\(50\) −0.0594122 −0.00840215
\(51\) −1.41421 −0.198030
\(52\) −5.97186 −0.828148
\(53\) −9.19932 −1.26362 −0.631812 − 0.775122i \(-0.717688\pi\)
−0.631812 + 0.775122i \(0.717688\pi\)
\(54\) −1.97186 −0.268336
\(55\) −1.30205 −0.175569
\(56\) 1.00000 0.133631
\(57\) −0.894129 −0.118430
\(58\) −4.43882 −0.582846
\(59\) −4.11058 −0.535152 −0.267576 − 0.963537i \(-0.586223\pi\)
−0.267576 + 0.963537i \(0.586223\pi\)
\(60\) 0.744406 0.0961023
\(61\) −1.23570 −0.158215 −0.0791074 − 0.996866i \(-0.525207\pi\)
−0.0791074 + 0.996866i \(0.525207\pi\)
\(62\) 6.11058 0.776045
\(63\) −2.88784 −0.363834
\(64\) 1.00000 0.125000
\(65\) −13.2739 −1.64643
\(66\) −0.196182 −0.0241484
\(67\) −6.46696 −0.790065 −0.395033 − 0.918667i \(-0.629267\pi\)
−0.395033 + 0.918667i \(0.629267\pi\)
\(68\) −4.22274 −0.512083
\(69\) 0 0
\(70\) 2.22274 0.265669
\(71\) 1.21137 0.143763 0.0718814 − 0.997413i \(-0.477100\pi\)
0.0718814 + 0.997413i \(0.477100\pi\)
\(72\) −2.88784 −0.340335
\(73\) −12.6948 −1.48581 −0.742906 − 0.669395i \(-0.766553\pi\)
−0.742906 + 0.669395i \(0.766553\pi\)
\(74\) 3.53304 0.410707
\(75\) −0.0198974 −0.00229755
\(76\) −2.66981 −0.306248
\(77\) −0.585786 −0.0667566
\(78\) −2.00000 −0.226455
\(79\) −12.7672 −1.43642 −0.718209 − 0.695828i \(-0.755038\pi\)
−0.718209 + 0.695828i \(0.755038\pi\)
\(80\) 2.22274 0.248510
\(81\) 8.00313 0.889237
\(82\) 2.27744 0.251502
\(83\) 9.97186 1.09455 0.547277 − 0.836952i \(-0.315664\pi\)
0.547277 + 0.836952i \(0.315664\pi\)
\(84\) 0.334904 0.0365410
\(85\) −9.38607 −1.01806
\(86\) 5.78510 0.623824
\(87\) −1.48658 −0.159378
\(88\) −0.585786 −0.0624450
\(89\) −14.6588 −1.55383 −0.776915 − 0.629605i \(-0.783217\pi\)
−0.776915 + 0.629605i \(0.783217\pi\)
\(90\) −6.41893 −0.676614
\(91\) −5.97186 −0.626021
\(92\) 0 0
\(93\) 2.04646 0.212208
\(94\) 3.04451 0.314017
\(95\) −5.93430 −0.608846
\(96\) 0.334904 0.0341810
\(97\) −2.85499 −0.289880 −0.144940 − 0.989440i \(-0.546299\pi\)
−0.144940 + 0.989440i \(0.546299\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.69166 0.170018
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 7406.2.a.bb.1.3 ✓ 4
23.22 odd 2 7406.2.a.bc.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.bb.1.3 ✓ 4 1.1 even 1 trivial
7406.2.a.bc.1.3 yes 4 23.22 odd 2