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 = [6,6,2,6,4,2,6] 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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.84770496.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 24x^{3} + 6x^{2} - 28x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.13793\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.13793 q^{3} +1.00000 q^{4} +3.90332 q^{5} -3.13793 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.84662 q^{9} +3.90332 q^{10} +6.13793 q^{11} -3.13793 q^{12} -2.06916 q^{13} +1.00000 q^{14} -12.2483 q^{15} +1.00000 q^{16} +0.890760 q^{17} +6.84662 q^{18} +0.719979 q^{19} +3.90332 q^{20} -3.13793 q^{21} +6.13793 q^{22} -3.13793 q^{24} +10.2359 q^{25} -2.06916 q^{26} -12.0704 q^{27} +1.00000 q^{28} +5.88660 q^{29} -12.2483 q^{30} +1.19502 q^{31} +1.00000 q^{32} -19.2604 q^{33} +0.890760 q^{34} +3.90332 q^{35} +6.84662 q^{36} -6.18999 q^{37} +0.719979 q^{38} +6.49289 q^{39} +3.90332 q^{40} -1.42818 q^{41} -3.13793 q^{42} +10.0079 q^{43} +6.13793 q^{44} +26.7245 q^{45} +11.6127 q^{47} -3.13793 q^{48} +1.00000 q^{49} +10.2359 q^{50} -2.79514 q^{51} -2.06916 q^{52} -11.9117 q^{53} -12.0704 q^{54} +23.9583 q^{55} +1.00000 q^{56} -2.25924 q^{57} +5.88660 q^{58} -1.70975 q^{59} -12.2483 q^{60} +6.59425 q^{61} +1.19502 q^{62} +6.84662 q^{63} +1.00000 q^{64} -8.07659 q^{65} -19.2604 q^{66} +2.57134 q^{67} +0.890760 q^{68} +3.90332 q^{70} +11.0070 q^{71} +6.84662 q^{72} -14.0704 q^{73} -6.18999 q^{74} -32.1195 q^{75} +0.719979 q^{76} +6.13793 q^{77} +6.49289 q^{78} +4.94437 q^{79} +3.90332 q^{80} +17.3363 q^{81} -1.42818 q^{82} +4.50886 q^{83} -3.13793 q^{84} +3.47692 q^{85} +10.0079 q^{86} -18.4718 q^{87} +6.13793 q^{88} -8.11443 q^{89} +26.7245 q^{90} -2.06916 q^{91} -3.74990 q^{93} +11.6127 q^{94} +2.81031 q^{95} -3.13793 q^{96} -14.3573 q^{97} +1.00000 q^{98} +42.0241 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9} + 4 q^{10} + 16 q^{11} + 2 q^{12} + 6 q^{14} + 6 q^{16} + 2 q^{17} + 6 q^{18} + 18 q^{19} + 4 q^{20} + 2 q^{21} + 16 q^{22}+ \cdots + 28 q^{99}+O(q^{100}) \) 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
\(3\) −3.13793 −1.81169 −0.905843 − 0.423614i \(-0.860761\pi\)
−0.905843 + 0.423614i \(0.860761\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.90332 1.74562 0.872808 − 0.488064i \(-0.162297\pi\)
0.872808 + 0.488064i \(0.162297\pi\)
\(6\) −3.13793 −1.28106
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 6.84662 2.28221
\(10\) 3.90332 1.23434
\(11\) 6.13793 1.85066 0.925328 − 0.379167i \(-0.123790\pi\)
0.925328 + 0.379167i \(0.123790\pi\)
\(12\) −3.13793 −0.905843
\(13\) −2.06916 −0.573882 −0.286941 − 0.957948i \(-0.592638\pi\)
−0.286941 + 0.957948i \(0.592638\pi\)
\(14\) 1.00000 0.267261
\(15\) −12.2483 −3.16251
\(16\) 1.00000 0.250000
\(17\) 0.890760 0.216041 0.108020 − 0.994149i \(-0.465549\pi\)
0.108020 + 0.994149i \(0.465549\pi\)
\(18\) 6.84662 1.61376
\(19\) 0.719979 0.165174 0.0825872 − 0.996584i \(-0.473682\pi\)
0.0825872 + 0.996584i \(0.473682\pi\)
\(20\) 3.90332 0.872808
\(21\) −3.13793 −0.684753
\(22\) 6.13793 1.30861
\(23\) 0 0
\(24\) −3.13793 −0.640528
\(25\) 10.2359 2.04718
\(26\) −2.06916 −0.405796
\(27\) −12.0704 −2.32295
\(28\) 1.00000 0.188982
\(29\) 5.88660 1.09311 0.546557 − 0.837422i \(-0.315938\pi\)
0.546557 + 0.837422i \(0.315938\pi\)
\(30\) −12.2483 −2.23623
\(31\) 1.19502 0.214632 0.107316 − 0.994225i \(-0.465774\pi\)
0.107316 + 0.994225i \(0.465774\pi\)
\(32\) 1.00000 0.176777
\(33\) −19.2604 −3.35281
\(34\) 0.890760 0.152764
\(35\) 3.90332 0.659781
\(36\) 6.84662 1.14110
\(37\) −6.18999 −1.01763 −0.508814 − 0.860877i \(-0.669916\pi\)
−0.508814 + 0.860877i \(0.669916\pi\)
\(38\) 0.719979 0.116796
\(39\) 6.49289 1.03969
\(40\) 3.90332 0.617169
\(41\) −1.42818 −0.223044 −0.111522 − 0.993762i \(-0.535573\pi\)
−0.111522 + 0.993762i \(0.535573\pi\)
\(42\) −3.13793 −0.484193
\(43\) 10.0079 1.52619 0.763096 − 0.646285i \(-0.223678\pi\)
0.763096 + 0.646285i \(0.223678\pi\)
\(44\) 6.13793 0.925328
\(45\) 26.7245 3.98386
\(46\) 0 0
\(47\) 11.6127 1.69389 0.846944 − 0.531681i \(-0.178440\pi\)
0.846944 + 0.531681i \(0.178440\pi\)
\(48\) −3.13793 −0.452921
\(49\) 1.00000 0.142857
\(50\) 10.2359 1.44757
\(51\) −2.79514 −0.391398
\(52\) −2.06916 −0.286941
\(53\) −11.9117 −1.63620 −0.818100 − 0.575076i \(-0.804972\pi\)
−0.818100 + 0.575076i \(0.804972\pi\)
\(54\) −12.0704 −1.64258
\(55\) 23.9583 3.23054
\(56\) 1.00000 0.133631
\(57\) −2.25924 −0.299244
\(58\) 5.88660 0.772949
\(59\) −1.70975 −0.222591 −0.111295 − 0.993787i \(-0.535500\pi\)
−0.111295 + 0.993787i \(0.535500\pi\)
\(60\) −12.2483 −1.58125
\(61\) 6.59425 0.844307 0.422154 − 0.906524i \(-0.361274\pi\)
0.422154 + 0.906524i \(0.361274\pi\)
\(62\) 1.19502 0.151768
\(63\) 6.84662 0.862593
\(64\) 1.00000 0.125000
\(65\) −8.07659 −1.00178
\(66\) −19.2604 −2.37079
\(67\) 2.57134 0.314139 0.157069 − 0.987588i \(-0.449795\pi\)
0.157069 + 0.987588i \(0.449795\pi\)
\(68\) 0.890760 0.108020
\(69\) 0 0
\(70\) 3.90332 0.466536
\(71\) 11.0070 1.30629 0.653144 − 0.757234i \(-0.273450\pi\)
0.653144 + 0.757234i \(0.273450\pi\)
\(72\) 6.84662 0.806882
\(73\) −14.0704 −1.64682 −0.823409 − 0.567448i \(-0.807931\pi\)
−0.823409 + 0.567448i \(0.807931\pi\)
\(74\) −6.18999 −0.719572
\(75\) −32.1195 −3.70884
\(76\) 0.719979 0.0825872
\(77\) 6.13793 0.699482
\(78\) 6.49289 0.735175
\(79\) 4.94437 0.556285 0.278142 − 0.960540i \(-0.410281\pi\)
0.278142 + 0.960540i \(0.410281\pi\)
\(80\) 3.90332 0.436404
\(81\) 17.3363 1.92626
\(82\) −1.42818 −0.157716
\(83\) 4.50886 0.494912 0.247456 − 0.968899i \(-0.420405\pi\)
0.247456 + 0.968899i \(0.420405\pi\)
\(84\) −3.13793 −0.342376
\(85\) 3.47692 0.377125
\(86\) 10.0079 1.07918
\(87\) −18.4718 −1.98038
\(88\) 6.13793 0.654306
\(89\) −8.11443 −0.860128 −0.430064 − 0.902798i \(-0.641509\pi\)
−0.430064 + 0.902798i \(0.641509\pi\)
\(90\) 26.7245 2.81701
\(91\) −2.06916 −0.216907
\(92\) 0 0
\(93\) −3.74990 −0.388846
\(94\) 11.6127 1.19776
\(95\) 2.81031 0.288331
\(96\) −3.13793 −0.320264
\(97\) −14.3573 −1.45777 −0.728883 − 0.684639i \(-0.759960\pi\)
−0.728883 + 0.684639i \(0.759960\pi\)
\(98\) 1.00000 0.101015
\(99\) 42.0241 4.22358
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.bl.1.1 yes 6
23.22 odd 2 7406.2.a.bk.1.1 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.bk.1.1 ✓ 6 23.22 odd 2
7406.2.a.bl.1.1 yes 6 1.1 even 1 trivial