Properties

Label 7406.2.a.t.1.2
Level $7406$
Weight $2$
Character 7406.1
Self dual yes
Analytic conductor $59.137$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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,-2,3,-2,-2,-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: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.806063 q^{3} +1.00000 q^{4} -4.15633 q^{5} -0.806063 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.35026 q^{9} -4.15633 q^{10} +1.61213 q^{11} -0.806063 q^{12} -1.00000 q^{14} +3.35026 q^{15} +1.00000 q^{16} -1.19394 q^{17} -2.35026 q^{18} +1.35026 q^{19} -4.15633 q^{20} +0.806063 q^{21} +1.61213 q^{22} -0.806063 q^{24} +12.2750 q^{25} +4.31265 q^{27} -1.00000 q^{28} -2.96239 q^{29} +3.35026 q^{30} +9.50659 q^{31} +1.00000 q^{32} -1.29948 q^{33} -1.19394 q^{34} +4.15633 q^{35} -2.35026 q^{36} +0.962389 q^{37} +1.35026 q^{38} -4.15633 q^{40} +4.70052 q^{41} +0.806063 q^{42} +5.27504 q^{43} +1.61213 q^{44} +9.76845 q^{45} -6.80606 q^{47} -0.806063 q^{48} +1.00000 q^{49} +12.2750 q^{50} +0.962389 q^{51} +8.96239 q^{53} +4.31265 q^{54} -6.70052 q^{55} -1.00000 q^{56} -1.08840 q^{57} -2.96239 q^{58} +13.4314 q^{59} +3.35026 q^{60} -11.6932 q^{61} +9.50659 q^{62} +2.35026 q^{63} +1.00000 q^{64} -1.29948 q^{66} -9.08840 q^{67} -1.19394 q^{68} +4.15633 q^{70} +5.61213 q^{71} -2.35026 q^{72} -10.6253 q^{73} +0.962389 q^{74} -9.89446 q^{75} +1.35026 q^{76} -1.61213 q^{77} -14.5745 q^{79} -4.15633 q^{80} +3.57452 q^{81} +4.70052 q^{82} -9.35026 q^{83} +0.806063 q^{84} +4.96239 q^{85} +5.27504 q^{86} +2.38787 q^{87} +1.61213 q^{88} -14.5950 q^{89} +9.76845 q^{90} -7.66291 q^{93} -6.80606 q^{94} -5.61213 q^{95} -0.806063 q^{96} -8.73084 q^{97} +1.00000 q^{98} -3.78892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} - 3 q^{14} + 3 q^{16} - 4 q^{17} + 3 q^{18} - 6 q^{19} - 2 q^{20} + 2 q^{21} + 4 q^{22}+ \cdots + 28 q^{99}+O(q^{100}) \) 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
\(3\) −0.806063 −0.465381 −0.232690 − 0.972551i \(-0.574753\pi\)
−0.232690 + 0.972551i \(0.574753\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.15633 −1.85877 −0.929383 − 0.369118i \(-0.879660\pi\)
−0.929383 + 0.369118i \(0.879660\pi\)
\(6\) −0.806063 −0.329074
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.35026 −0.783421
\(10\) −4.15633 −1.31435
\(11\) 1.61213 0.486075 0.243037 − 0.970017i \(-0.421856\pi\)
0.243037 + 0.970017i \(0.421856\pi\)
\(12\) −0.806063 −0.232690
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.35026 0.865034
\(16\) 1.00000 0.250000
\(17\) −1.19394 −0.289572 −0.144786 − 0.989463i \(-0.546249\pi\)
−0.144786 + 0.989463i \(0.546249\pi\)
\(18\) −2.35026 −0.553962
\(19\) 1.35026 0.309771 0.154886 − 0.987932i \(-0.450499\pi\)
0.154886 + 0.987932i \(0.450499\pi\)
\(20\) −4.15633 −0.929383
\(21\) 0.806063 0.175897
\(22\) 1.61213 0.343707
\(23\) 0 0
\(24\) −0.806063 −0.164537
\(25\) 12.2750 2.45501
\(26\) 0 0
\(27\) 4.31265 0.829970
\(28\) −1.00000 −0.188982
\(29\) −2.96239 −0.550102 −0.275051 − 0.961430i \(-0.588695\pi\)
−0.275051 + 0.961430i \(0.588695\pi\)
\(30\) 3.35026 0.611671
\(31\) 9.50659 1.70743 0.853717 − 0.520738i \(-0.174343\pi\)
0.853717 + 0.520738i \(0.174343\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.29948 −0.226210
\(34\) −1.19394 −0.204758
\(35\) 4.15633 0.702547
\(36\) −2.35026 −0.391710
\(37\) 0.962389 0.158216 0.0791079 − 0.996866i \(-0.474793\pi\)
0.0791079 + 0.996866i \(0.474793\pi\)
\(38\) 1.35026 0.219041
\(39\) 0 0
\(40\) −4.15633 −0.657173
\(41\) 4.70052 0.734098 0.367049 − 0.930202i \(-0.380368\pi\)
0.367049 + 0.930202i \(0.380368\pi\)
\(42\) 0.806063 0.124378
\(43\) 5.27504 0.804436 0.402218 − 0.915544i \(-0.368239\pi\)
0.402218 + 0.915544i \(0.368239\pi\)
\(44\) 1.61213 0.243037
\(45\) 9.76845 1.45619
\(46\) 0 0
\(47\) −6.80606 −0.992766 −0.496383 − 0.868104i \(-0.665339\pi\)
−0.496383 + 0.868104i \(0.665339\pi\)
\(48\) −0.806063 −0.116345
\(49\) 1.00000 0.142857
\(50\) 12.2750 1.73595
\(51\) 0.962389 0.134761
\(52\) 0 0
\(53\) 8.96239 1.23108 0.615539 − 0.788106i \(-0.288938\pi\)
0.615539 + 0.788106i \(0.288938\pi\)
\(54\) 4.31265 0.586877
\(55\) −6.70052 −0.903498
\(56\) −1.00000 −0.133631
\(57\) −1.08840 −0.144162
\(58\) −2.96239 −0.388981
\(59\) 13.4314 1.74861 0.874307 − 0.485373i \(-0.161316\pi\)
0.874307 + 0.485373i \(0.161316\pi\)
\(60\) 3.35026 0.432517
\(61\) −11.6932 −1.49716 −0.748582 − 0.663042i \(-0.769265\pi\)
−0.748582 + 0.663042i \(0.769265\pi\)
\(62\) 9.50659 1.20734
\(63\) 2.35026 0.296105
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.29948 −0.159955
\(67\) −9.08840 −1.11032 −0.555162 − 0.831742i \(-0.687344\pi\)
−0.555162 + 0.831742i \(0.687344\pi\)
\(68\) −1.19394 −0.144786
\(69\) 0 0
\(70\) 4.15633 0.496776
\(71\) 5.61213 0.666037 0.333018 − 0.942920i \(-0.391933\pi\)
0.333018 + 0.942920i \(0.391933\pi\)
\(72\) −2.35026 −0.276981
\(73\) −10.6253 −1.24360 −0.621799 − 0.783177i \(-0.713598\pi\)
−0.621799 + 0.783177i \(0.713598\pi\)
\(74\) 0.962389 0.111875
\(75\) −9.89446 −1.14251
\(76\) 1.35026 0.154886
\(77\) −1.61213 −0.183719
\(78\) 0 0
\(79\) −14.5745 −1.63976 −0.819881 − 0.572535i \(-0.805960\pi\)
−0.819881 + 0.572535i \(0.805960\pi\)
\(80\) −4.15633 −0.464691
\(81\) 3.57452 0.397168
\(82\) 4.70052 0.519086
\(83\) −9.35026 −1.02632 −0.513162 − 0.858292i \(-0.671526\pi\)
−0.513162 + 0.858292i \(0.671526\pi\)
\(84\) 0.806063 0.0879487
\(85\) 4.96239 0.538247
\(86\) 5.27504 0.568822
\(87\) 2.38787 0.256007
\(88\) 1.61213 0.171853
\(89\) −14.5950 −1.54707 −0.773533 − 0.633757i \(-0.781512\pi\)
−0.773533 + 0.633757i \(0.781512\pi\)
\(90\) 9.76845 1.02969
\(91\) 0 0
\(92\) 0 0
\(93\) −7.66291 −0.794607
\(94\) −6.80606 −0.701992
\(95\) −5.61213 −0.575792
\(96\) −0.806063 −0.0822685
\(97\) −8.73084 −0.886483 −0.443241 − 0.896402i \(-0.646172\pi\)
−0.443241 + 0.896402i \(0.646172\pi\)
\(98\) 1.00000 0.101015
\(99\) −3.78892 −0.380801
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.t.1.2 ✓ 3
23.22 odd 2 7406.2.a.u.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.t.1.2 ✓ 3 1.1 even 1 trivial
7406.2.a.u.1.2 yes 3 23.22 odd 2