Properties

Label 7406.2.a.ba.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: 4.4.449797.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 10x^{2} - x + 4 \) 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 \(0.594177\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.594177 q^{3} +1.00000 q^{4} -1.59418 q^{5} +0.594177 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.64695 q^{9} -1.59418 q^{10} -5.13782 q^{11} +0.594177 q^{12} +2.64695 q^{13} -1.00000 q^{14} -0.947224 q^{15} +1.00000 q^{16} +6.54364 q^{17} -2.64695 q^{18} +5.73200 q^{19} -1.59418 q^{20} -0.594177 q^{21} -5.13782 q^{22} +0.594177 q^{24} -2.45860 q^{25} +2.64695 q^{26} -3.35529 q^{27} -1.00000 q^{28} +7.54364 q^{29} -0.947224 q^{30} +4.24113 q^{31} +1.00000 q^{32} -3.05278 q^{33} +6.54364 q^{34} +1.59418 q^{35} -2.64695 q^{36} -6.24113 q^{37} +5.73200 q^{38} +1.57276 q^{39} -1.59418 q^{40} -5.83531 q^{41} -0.594177 q^{42} -10.9204 q^{43} -5.13782 q^{44} +4.21971 q^{45} -2.08505 q^{47} +0.594177 q^{48} +1.00000 q^{49} -2.45860 q^{50} +3.88808 q^{51} +2.64695 q^{52} -9.54364 q^{53} -3.35529 q^{54} +8.19060 q^{55} -1.00000 q^{56} +3.40582 q^{57} +7.54364 q^{58} +8.78478 q^{59} -0.947224 q^{60} -13.5436 q^{61} +4.24113 q^{62} +2.64695 q^{63} +1.00000 q^{64} -4.21971 q^{65} -3.05278 q^{66} -1.49087 q^{67} +6.54364 q^{68} +1.59418 q^{70} -1.67382 q^{71} -2.64695 q^{72} +10.4640 q^{73} -6.24113 q^{74} -1.46084 q^{75} +5.73200 q^{76} +5.13782 q^{77} +1.57276 q^{78} -9.42948 q^{79} -1.59418 q^{80} +5.94722 q^{81} -5.83531 q^{82} -7.64695 q^{83} -0.594177 q^{84} -10.4317 q^{85} -10.9204 q^{86} +4.48226 q^{87} -5.13782 q^{88} +1.67922 q^{89} +4.21971 q^{90} -2.64695 q^{91} +2.51998 q^{93} -2.08505 q^{94} -9.13782 q^{95} +0.594177 q^{96} -6.80304 q^{97} +1.00000 q^{98} +13.5996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{7} + 4 q^{8} + 8 q^{9} - 4 q^{10} + 3 q^{11} - 8 q^{13} - 4 q^{14} - 20 q^{15} + 4 q^{16} + 5 q^{17} + 8 q^{18} - 3 q^{19} - 4 q^{20} + 3 q^{22} + 4 q^{25} - 8 q^{26}+ \cdots + 14 q^{99}+O(q^{100}) \) 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
\(3\) 0.594177 0.343048 0.171524 − 0.985180i \(-0.445131\pi\)
0.171524 + 0.985180i \(0.445131\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.59418 −0.712938 −0.356469 − 0.934307i \(-0.616019\pi\)
−0.356469 + 0.934307i \(0.616019\pi\)
\(6\) 0.594177 0.242572
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.64695 −0.882318
\(10\) −1.59418 −0.504123
\(11\) −5.13782 −1.54911 −0.774556 − 0.632506i \(-0.782027\pi\)
−0.774556 + 0.632506i \(0.782027\pi\)
\(12\) 0.594177 0.171524
\(13\) 2.64695 0.734133 0.367066 − 0.930195i \(-0.380362\pi\)
0.367066 + 0.930195i \(0.380362\pi\)
\(14\) −1.00000 −0.267261
\(15\) −0.947224 −0.244572
\(16\) 1.00000 0.250000
\(17\) 6.54364 1.58707 0.793534 − 0.608527i \(-0.208239\pi\)
0.793534 + 0.608527i \(0.208239\pi\)
\(18\) −2.64695 −0.623893
\(19\) 5.73200 1.31501 0.657505 − 0.753450i \(-0.271612\pi\)
0.657505 + 0.753450i \(0.271612\pi\)
\(20\) −1.59418 −0.356469
\(21\) −0.594177 −0.129660
\(22\) −5.13782 −1.09539
\(23\) 0 0
\(24\) 0.594177 0.121286
\(25\) −2.45860 −0.491720
\(26\) 2.64695 0.519110
\(27\) −3.35529 −0.645726
\(28\) −1.00000 −0.188982
\(29\) 7.54364 1.40082 0.700410 − 0.713741i \(-0.253001\pi\)
0.700410 + 0.713741i \(0.253001\pi\)
\(30\) −0.947224 −0.172939
\(31\) 4.24113 0.761730 0.380865 − 0.924631i \(-0.375626\pi\)
0.380865 + 0.924631i \(0.375626\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.05278 −0.531420
\(34\) 6.54364 1.12223
\(35\) 1.59418 0.269465
\(36\) −2.64695 −0.441159
\(37\) −6.24113 −1.02604 −0.513018 − 0.858378i \(-0.671473\pi\)
−0.513018 + 0.858378i \(0.671473\pi\)
\(38\) 5.73200 0.929853
\(39\) 1.57276 0.251843
\(40\) −1.59418 −0.252062
\(41\) −5.83531 −0.911322 −0.455661 − 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(42\) −0.594177 −0.0916835
\(43\) −10.9204 −1.66534 −0.832669 − 0.553771i \(-0.813188\pi\)
−0.832669 + 0.553771i \(0.813188\pi\)
\(44\) −5.13782 −0.774556
\(45\) 4.21971 0.629038
\(46\) 0 0
\(47\) −2.08505 −0.304135 −0.152068 − 0.988370i \(-0.548593\pi\)
−0.152068 + 0.988370i \(0.548593\pi\)
\(48\) 0.594177 0.0857621
\(49\) 1.00000 0.142857
\(50\) −2.45860 −0.347698
\(51\) 3.88808 0.544441
\(52\) 2.64695 0.367066
\(53\) −9.54364 −1.31092 −0.655460 − 0.755230i \(-0.727525\pi\)
−0.655460 + 0.755230i \(0.727525\pi\)
\(54\) −3.35529 −0.456597
\(55\) 8.19060 1.10442
\(56\) −1.00000 −0.133631
\(57\) 3.40582 0.451112
\(58\) 7.54364 0.990529
\(59\) 8.78478 1.14368 0.571840 − 0.820365i \(-0.306230\pi\)
0.571840 + 0.820365i \(0.306230\pi\)
\(60\) −0.947224 −0.122286
\(61\) −13.5436 −1.73409 −0.867043 − 0.498233i \(-0.833982\pi\)
−0.867043 + 0.498233i \(0.833982\pi\)
\(62\) 4.24113 0.538624
\(63\) 2.64695 0.333485
\(64\) 1.00000 0.125000
\(65\) −4.21971 −0.523391
\(66\) −3.05278 −0.375771
\(67\) −1.49087 −0.182139 −0.0910693 − 0.995845i \(-0.529028\pi\)
−0.0910693 + 0.995845i \(0.529028\pi\)
\(68\) 6.54364 0.793534
\(69\) 0 0
\(70\) 1.59418 0.190541
\(71\) −1.67382 −0.198646 −0.0993232 − 0.995055i \(-0.531668\pi\)
−0.0993232 + 0.995055i \(0.531668\pi\)
\(72\) −2.64695 −0.311946
\(73\) 10.4640 1.22472 0.612359 − 0.790580i \(-0.290221\pi\)
0.612359 + 0.790580i \(0.290221\pi\)
\(74\) −6.24113 −0.725517
\(75\) −1.46084 −0.168684
\(76\) 5.73200 0.657505
\(77\) 5.13782 0.585509
\(78\) 1.57276 0.178080
\(79\) −9.42948 −1.06090 −0.530450 − 0.847716i \(-0.677977\pi\)
−0.530450 + 0.847716i \(0.677977\pi\)
\(80\) −1.59418 −0.178234
\(81\) 5.94722 0.660803
\(82\) −5.83531 −0.644402
\(83\) −7.64695 −0.839362 −0.419681 − 0.907672i \(-0.637858\pi\)
−0.419681 + 0.907672i \(0.637858\pi\)
\(84\) −0.594177 −0.0648300
\(85\) −10.4317 −1.13148
\(86\) −10.9204 −1.17757
\(87\) 4.48226 0.480549
\(88\) −5.13782 −0.547694
\(89\) 1.67922 0.177997 0.0889986 − 0.996032i \(-0.471633\pi\)
0.0889986 + 0.996032i \(0.471633\pi\)
\(90\) 4.21971 0.444797
\(91\) −2.64695 −0.277476
\(92\) 0 0
\(93\) 2.51998 0.261310
\(94\) −2.08505 −0.215056
\(95\) −9.13782 −0.937521
\(96\) 0.594177 0.0606429
\(97\) −6.80304 −0.690744 −0.345372 − 0.938466i \(-0.612247\pi\)
−0.345372 + 0.938466i \(0.612247\pi\)
\(98\) 1.00000 0.101015
\(99\) 13.5996 1.36681
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.ba.1.3 ✓ 4
23.22 odd 2 7406.2.a.bd.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.ba.1.3 ✓ 4 1.1 even 1 trivial
7406.2.a.bd.1.3 yes 4 23.22 odd 2