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Properties

Label 150.2.e.b.107.3
Level $150$
Weight $2$
Character 150.107
Analytic conductor $1.198$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [150,2,Mod(107,150)] 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("150.107"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 107.3
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 150.107
Dual form 150.2.e.b.143.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{2} +(0.448288 + 1.67303i) q^{3} -1.00000i q^{4} +(1.50000 + 0.866025i) q^{6} +(2.44949 + 2.44949i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-2.59808 + 1.50000i) q^{9} -5.19615i q^{11} +(1.67303 - 0.448288i) q^{12} +3.46410 q^{14} -1.00000 q^{16} +(-2.12132 + 2.12132i) q^{17} +(-0.776457 + 2.89778i) q^{18} -1.00000i q^{19} +(-3.00000 + 5.19615i) q^{21} +(-3.67423 - 3.67423i) q^{22} +(-4.24264 - 4.24264i) q^{23} +(0.866025 - 1.50000i) q^{24} +(-3.67423 - 3.67423i) q^{27} +(2.44949 - 2.44949i) q^{28} -2.00000 q^{31} +(-0.707107 + 0.707107i) q^{32} +(8.69333 - 2.32937i) q^{33} +3.00000i q^{34} +(1.50000 + 2.59808i) q^{36} +(-2.44949 - 2.44949i) q^{37} +(-0.707107 - 0.707107i) q^{38} +5.19615i q^{41} +(1.55291 + 5.79555i) q^{42} +(-2.44949 + 2.44949i) q^{43} -5.19615 q^{44} -6.00000 q^{46} +(-0.448288 - 1.67303i) q^{48} +5.00000i q^{49} +(-4.50000 - 2.59808i) q^{51} +(4.24264 + 4.24264i) q^{53} -5.19615 q^{54} -3.46410i q^{56} +(1.67303 - 0.448288i) q^{57} +10.3923 q^{59} +14.0000 q^{61} +(-1.41421 + 1.41421i) q^{62} +(-10.0382 - 2.68973i) q^{63} +1.00000i q^{64} +(4.50000 - 7.79423i) q^{66} +(3.67423 + 3.67423i) q^{67} +(2.12132 + 2.12132i) q^{68} +(5.19615 - 9.00000i) q^{69} +(2.89778 + 0.776457i) q^{72} +(-6.12372 + 6.12372i) q^{73} -3.46410 q^{74} -1.00000 q^{76} +(12.7279 - 12.7279i) q^{77} -14.0000i q^{79} +(4.50000 - 7.79423i) q^{81} +(3.67423 + 3.67423i) q^{82} +(2.12132 + 2.12132i) q^{83} +(5.19615 + 3.00000i) q^{84} +3.46410i q^{86} +(-3.67423 + 3.67423i) q^{88} -15.5885 q^{89} +(-4.24264 + 4.24264i) q^{92} +(-0.896575 - 3.34607i) q^{93} +(-1.50000 - 0.866025i) q^{96} +(4.89898 + 4.89898i) q^{97} +(3.53553 + 3.53553i) q^{98} +(7.79423 + 13.5000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{6} - 8 q^{16} - 24 q^{21} - 16 q^{31} + 12 q^{36} - 48 q^{46} - 36 q^{51} + 112 q^{61} + 36 q^{66} - 8 q^{76} + 36 q^{81} - 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 − 0.707107i 0.500000 − 0.500000i
\(3\) 0.448288 + 1.67303i 0.258819 + 0.965926i
\(4\) − 1.00000i − 0.500000i
\(5\) 0 0
\(6\) 1.50000 + 0.866025i 0.612372 + 0.353553i
\(7\) 2.44949 + 2.44949i 0.925820 + 0.925820i 0.997433 − 0.0716124i \(-0.0228145\pi\)
−0.0716124 + 0.997433i \(0.522814\pi\)
\(8\) −0.707107 − 0.707107i −0.250000 − 0.250000i
\(9\) −2.59808 + 1.50000i −0.866025 + 0.500000i
\(10\) 0 0
\(11\) − 5.19615i − 1.56670i −0.621582 − 0.783349i \(-0.713510\pi\)
0.621582 − 0.783349i \(-0.286490\pi\)
\(12\) 1.67303 − 0.448288i 0.482963 − 0.129410i
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.12132 + 2.12132i −0.514496 + 0.514496i −0.915901 − 0.401405i \(-0.868522\pi\)
0.401405 + 0.915901i \(0.368522\pi\)
\(18\) −0.776457 + 2.89778i −0.183013 + 0.683013i
\(19\) − 1.00000i − 0.229416i −0.993399 − 0.114708i \(-0.963407\pi\)
0.993399 − 0.114708i \(-0.0365932\pi\)
\(20\) 0 0
\(21\) −3.00000 + 5.19615i −0.654654 + 1.13389i
\(22\) −3.67423 − 3.67423i −0.783349 − 0.783349i
\(23\) −4.24264 − 4.24264i −0.884652 − 0.884652i 0.109351 − 0.994003i \(-0.465123\pi\)
−0.994003 + 0.109351i \(0.965123\pi\)
\(24\) 0.866025 − 1.50000i 0.176777 − 0.306186i
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 − 3.67423i −0.707107 − 0.707107i
\(28\) 2.44949 − 2.44949i 0.462910 − 0.462910i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −0.707107 + 0.707107i −0.125000 + 0.125000i
\(33\) 8.69333 − 2.32937i 1.51331 − 0.405492i
\(34\) 3.00000i 0.514496i
\(35\) 0 0
\(36\) 1.50000 + 2.59808i 0.250000 + 0.433013i
\(37\) −2.44949 − 2.44949i −0.402694 − 0.402694i 0.476488 − 0.879181i \(-0.341910\pi\)
−0.879181 + 0.476488i \(0.841910\pi\)
\(38\) −0.707107 − 0.707107i −0.114708 − 0.114708i
\(39\) 0 0
\(40\) 0 0
\(41\) 5.19615i 0.811503i 0.913984 + 0.405751i \(0.132990\pi\)
−0.913984 + 0.405751i \(0.867010\pi\)
\(42\) 1.55291 + 5.79555i 0.239620 + 0.894274i
\(43\) −2.44949 + 2.44949i −0.373544 + 0.373544i −0.868766 − 0.495222i \(-0.835087\pi\)
0.495222 + 0.868766i \(0.335087\pi\)
\(44\) −5.19615 −0.783349
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) −0.448288 − 1.67303i −0.0647048 − 0.241481i
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) −4.50000 − 2.59808i −0.630126 − 0.363803i
\(52\) 0 0
\(53\) 4.24264 + 4.24264i 0.582772 + 0.582772i 0.935664 − 0.352892i \(-0.114802\pi\)
−0.352892 + 0.935664i \(0.614802\pi\)
\(54\) −5.19615 −0.707107
\(55\) 0 0
\(56\) − 3.46410i − 0.462910i
\(57\) 1.67303 − 0.448288i 0.221599 − 0.0593772i
\(58\) 0 0
\(59\) 10.3923 1.35296 0.676481 − 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −1.41421 + 1.41421i −0.179605 + 0.179605i
\(63\) −10.0382 − 2.68973i −1.26469 − 0.338874i
\(64\) 1.00000i 0.125000i
\(65\) 0 0
\(66\) 4.50000 − 7.79423i 0.553912 − 0.959403i
\(67\) 3.67423 + 3.67423i 0.448879 + 0.448879i 0.894982 − 0.446103i \(-0.147188\pi\)
−0.446103 + 0.894982i \(0.647188\pi\)
\(68\) 2.12132 + 2.12132i 0.257248 + 0.257248i
\(69\) 5.19615 − 9.00000i 0.625543 − 1.08347i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.89778 + 0.776457i 0.341506 + 0.0915064i
\(73\) −6.12372 + 6.12372i −0.716728 + 0.716728i −0.967934 − 0.251206i \(-0.919173\pi\)
0.251206 + 0.967934i \(0.419173\pi\)
\(74\) −3.46410 −0.402694
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 12.7279 − 12.7279i 1.45048 − 1.45048i
\(78\) 0 0
\(79\) − 14.0000i − 1.57512i −0.616236 − 0.787562i \(-0.711343\pi\)
0.616236 − 0.787562i \(-0.288657\pi\)
\(80\) 0 0
\(81\) 4.50000 − 7.79423i 0.500000 − 0.866025i
\(82\) 3.67423 + 3.67423i 0.405751 + 0.405751i
\(83\) 2.12132 + 2.12132i 0.232845 + 0.232845i 0.813879 − 0.581034i \(-0.197352\pi\)
−0.581034 + 0.813879i \(0.697352\pi\)
\(84\) 5.19615 + 3.00000i 0.566947 + 0.327327i
\(85\) 0 0
\(86\) 3.46410i 0.373544i
\(87\) 0 0
\(88\) −3.67423 + 3.67423i −0.391675 + 0.391675i
\(89\) −15.5885 −1.65237 −0.826187 − 0.563397i \(-0.809494\pi\)
−0.826187 + 0.563397i \(0.809494\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.24264 + 4.24264i −0.442326 + 0.442326i
\(93\) −0.896575 − 3.34607i −0.0929705 − 0.346971i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.50000 − 0.866025i −0.153093 − 0.0883883i
\(97\) 4.89898 + 4.89898i 0.497416 + 0.497416i 0.910633 − 0.413217i \(-0.135595\pi\)
−0.413217 + 0.910633i \(0.635595\pi\)
\(98\) 3.53553 + 3.53553i 0.357143 + 0.357143i
\(99\) 7.79423 + 13.5000i 0.783349 + 1.35680i
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 150.2.e.b.107.3 yes 8
3.2 odd 2 inner 150.2.e.b.107.1 ✓ 8
4.3 odd 2 1200.2.v.l.257.2 8
5.2 odd 4 inner 150.2.e.b.143.4 yes 8
5.3 odd 4 inner 150.2.e.b.143.1 yes 8
5.4 even 2 inner 150.2.e.b.107.2 yes 8
12.11 even 2 1200.2.v.l.257.4 8
15.2 even 4 inner 150.2.e.b.143.2 yes 8
15.8 even 4 inner 150.2.e.b.143.3 yes 8
15.14 odd 2 inner 150.2.e.b.107.4 yes 8
20.3 even 4 1200.2.v.l.593.4 8
20.7 even 4 1200.2.v.l.593.1 8
20.19 odd 2 1200.2.v.l.257.3 8
60.23 odd 4 1200.2.v.l.593.2 8
60.47 odd 4 1200.2.v.l.593.3 8
60.59 even 2 1200.2.v.l.257.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.2.e.b.107.1 ✓ 8 3.2 odd 2 inner
150.2.e.b.107.2 yes 8 5.4 even 2 inner
150.2.e.b.107.3 yes 8 1.1 even 1 trivial
150.2.e.b.107.4 yes 8 15.14 odd 2 inner
150.2.e.b.143.1 yes 8 5.3 odd 4 inner
150.2.e.b.143.2 yes 8 15.2 even 4 inner
150.2.e.b.143.3 yes 8 15.8 even 4 inner
150.2.e.b.143.4 yes 8 5.2 odd 4 inner
1200.2.v.l.257.1 8 60.59 even 2
1200.2.v.l.257.2 8 4.3 odd 2
1200.2.v.l.257.3 8 20.19 odd 2
1200.2.v.l.257.4 8 12.11 even 2
1200.2.v.l.593.1 8 20.7 even 4
1200.2.v.l.593.2 8 60.23 odd 4
1200.2.v.l.593.3 8 60.47 odd 4
1200.2.v.l.593.4 8 20.3 even 4