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 = [3,3,-1,3,4,-1,-3] 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: \(3\)
Coefficient field: 3.3.1509.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 4 \) 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 \(-2.47735\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.47735 q^{3} +1.00000 q^{4} -1.47735 q^{5} +2.47735 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.13727 q^{9} -1.47735 q^{10} -2.13727 q^{11} +2.47735 q^{12} +1.13727 q^{13} -1.00000 q^{14} -3.65992 q^{15} +1.00000 q^{16} -6.61463 q^{17} +3.13727 q^{18} +0.340078 q^{19} -1.47735 q^{20} -2.47735 q^{21} -2.13727 q^{22} +2.47735 q^{24} -2.81743 q^{25} +1.13727 q^{26} +0.340078 q^{27} -1.00000 q^{28} -5.34008 q^{29} -3.65992 q^{30} -0.340078 q^{31} +1.00000 q^{32} -5.29478 q^{33} -6.61463 q^{34} +1.47735 q^{35} +3.13727 q^{36} -1.38537 q^{37} +0.340078 q^{38} +2.81743 q^{39} -1.47735 q^{40} -8.09198 q^{41} -2.47735 q^{42} -4.34008 q^{43} -2.13727 q^{44} -4.63486 q^{45} +7.43206 q^{47} +2.47735 q^{48} +1.00000 q^{49} -2.81743 q^{50} -16.3868 q^{51} +1.13727 q^{52} +6.56933 q^{53} +0.340078 q^{54} +3.15751 q^{55} -1.00000 q^{56} +0.842492 q^{57} -5.34008 q^{58} -4.27455 q^{59} -3.65992 q^{60} -7.34008 q^{61} -0.340078 q^{62} -3.13727 q^{63} +1.00000 q^{64} -1.68016 q^{65} -5.29478 q^{66} +9.90941 q^{67} -6.61463 q^{68} +1.47735 q^{70} +0.908021 q^{71} +3.13727 q^{72} -5.27455 q^{73} -1.38537 q^{74} -6.97977 q^{75} +0.340078 q^{76} +2.13727 q^{77} +2.81743 q^{78} -12.8892 q^{79} -1.47735 q^{80} -8.56933 q^{81} -8.09198 q^{82} +11.0920 q^{83} -2.47735 q^{84} +9.77213 q^{85} -4.34008 q^{86} -13.2293 q^{87} -2.13727 q^{88} -15.0000 q^{89} -4.63486 q^{90} -1.13727 q^{91} -0.842492 q^{93} +7.43206 q^{94} -0.502415 q^{95} +2.47735 q^{96} -18.1387 q^{97} +1.00000 q^{98} -6.70522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 6 q^{9} + 4 q^{10} - 3 q^{11} - q^{12} - 3 q^{14} - 16 q^{15} + 3 q^{16} - 8 q^{17} + 6 q^{18} - 4 q^{19} + 4 q^{20} + q^{21}+ \cdots - 42 q^{99}+O(q^{100}) \) 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
\(3\) 2.47735 1.43030 0.715150 − 0.698971i \(-0.246358\pi\)
0.715150 + 0.698971i \(0.246358\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.47735 −0.660692 −0.330346 − 0.943860i \(-0.607165\pi\)
−0.330346 + 0.943860i \(0.607165\pi\)
\(6\) 2.47735 1.01137
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 3.13727 1.04576
\(10\) −1.47735 −0.467180
\(11\) −2.13727 −0.644413 −0.322206 − 0.946669i \(-0.604424\pi\)
−0.322206 + 0.946669i \(0.604424\pi\)
\(12\) 2.47735 0.715150
\(13\) 1.13727 0.315423 0.157712 − 0.987485i \(-0.449588\pi\)
0.157712 + 0.987485i \(0.449588\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.65992 −0.944988
\(16\) 1.00000 0.250000
\(17\) −6.61463 −1.60428 −0.802141 − 0.597134i \(-0.796306\pi\)
−0.802141 + 0.597134i \(0.796306\pi\)
\(18\) 3.13727 0.739463
\(19\) 0.340078 0.0780192 0.0390096 − 0.999239i \(-0.487580\pi\)
0.0390096 + 0.999239i \(0.487580\pi\)
\(20\) −1.47735 −0.330346
\(21\) −2.47735 −0.540603
\(22\) −2.13727 −0.455668
\(23\) 0 0
\(24\) 2.47735 0.505687
\(25\) −2.81743 −0.563486
\(26\) 1.13727 0.223038
\(27\) 0.340078 0.0654480
\(28\) −1.00000 −0.188982
\(29\) −5.34008 −0.991628 −0.495814 − 0.868429i \(-0.665130\pi\)
−0.495814 + 0.868429i \(0.665130\pi\)
\(30\) −3.65992 −0.668207
\(31\) −0.340078 −0.0610798 −0.0305399 − 0.999534i \(-0.509723\pi\)
−0.0305399 + 0.999534i \(0.509723\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.29478 −0.921703
\(34\) −6.61463 −1.13440
\(35\) 1.47735 0.249718
\(36\) 3.13727 0.522879
\(37\) −1.38537 −0.227754 −0.113877 − 0.993495i \(-0.536327\pi\)
−0.113877 + 0.993495i \(0.536327\pi\)
\(38\) 0.340078 0.0551679
\(39\) 2.81743 0.451150
\(40\) −1.47735 −0.233590
\(41\) −8.09198 −1.26375 −0.631877 − 0.775068i \(-0.717715\pi\)
−0.631877 + 0.775068i \(0.717715\pi\)
\(42\) −2.47735 −0.382264
\(43\) −4.34008 −0.661856 −0.330928 − 0.943656i \(-0.607362\pi\)
−0.330928 + 0.943656i \(0.607362\pi\)
\(44\) −2.13727 −0.322206
\(45\) −4.63486 −0.690924
\(46\) 0 0
\(47\) 7.43206 1.08408 0.542038 − 0.840354i \(-0.317653\pi\)
0.542038 + 0.840354i \(0.317653\pi\)
\(48\) 2.47735 0.357575
\(49\) 1.00000 0.142857
\(50\) −2.81743 −0.398445
\(51\) −16.3868 −2.29461
\(52\) 1.13727 0.157712
\(53\) 6.56933 0.902367 0.451184 − 0.892431i \(-0.351002\pi\)
0.451184 + 0.892431i \(0.351002\pi\)
\(54\) 0.340078 0.0462787
\(55\) 3.15751 0.425758
\(56\) −1.00000 −0.133631
\(57\) 0.842492 0.111591
\(58\) −5.34008 −0.701187
\(59\) −4.27455 −0.556499 −0.278249 − 0.960509i \(-0.589754\pi\)
−0.278249 + 0.960509i \(0.589754\pi\)
\(60\) −3.65992 −0.472494
\(61\) −7.34008 −0.939801 −0.469900 − 0.882719i \(-0.655710\pi\)
−0.469900 + 0.882719i \(0.655710\pi\)
\(62\) −0.340078 −0.0431899
\(63\) −3.13727 −0.395259
\(64\) 1.00000 0.125000
\(65\) −1.68016 −0.208398
\(66\) −5.29478 −0.651743
\(67\) 9.90941 1.21063 0.605314 − 0.795987i \(-0.293048\pi\)
0.605314 + 0.795987i \(0.293048\pi\)
\(68\) −6.61463 −0.802141
\(69\) 0 0
\(70\) 1.47735 0.176577
\(71\) 0.908021 0.107762 0.0538811 − 0.998547i \(-0.482841\pi\)
0.0538811 + 0.998547i \(0.482841\pi\)
\(72\) 3.13727 0.369731
\(73\) −5.27455 −0.617339 −0.308670 − 0.951169i \(-0.599884\pi\)
−0.308670 + 0.951169i \(0.599884\pi\)
\(74\) −1.38537 −0.161046
\(75\) −6.97977 −0.805954
\(76\) 0.340078 0.0390096
\(77\) 2.13727 0.243565
\(78\) 2.81743 0.319011
\(79\) −12.8892 −1.45015 −0.725073 − 0.688672i \(-0.758194\pi\)
−0.725073 + 0.688672i \(0.758194\pi\)
\(80\) −1.47735 −0.165173
\(81\) −8.56933 −0.952148
\(82\) −8.09198 −0.893610
\(83\) 11.0920 1.21750 0.608751 − 0.793361i \(-0.291671\pi\)
0.608751 + 0.793361i \(0.291671\pi\)
\(84\) −2.47735 −0.270301
\(85\) 9.77213 1.05994
\(86\) −4.34008 −0.468003
\(87\) −13.2293 −1.41832
\(88\) −2.13727 −0.227834
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) −4.63486 −0.488557
\(91\) −1.13727 −0.119219
\(92\) 0 0
\(93\) −0.842492 −0.0873624
\(94\) 7.43206 0.766558
\(95\) −0.502415 −0.0515466
\(96\) 2.47735 0.252844
\(97\) −18.1387 −1.84170 −0.920851 − 0.389914i \(-0.872505\pi\)
−0.920851 + 0.389914i \(0.872505\pi\)
\(98\) 1.00000 0.101015
\(99\) −6.70522 −0.673900
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.w.1.3 yes 3
23.22 odd 2 7406.2.a.v.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.v.1.3 ✓ 3 23.22 odd 2
7406.2.a.w.1.3 yes 3 1.1 even 1 trivial