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Properties

Label 150.2.h.b
Level $150$
Weight $2$
Character orbit 150.h
Analytic conductor $1.198$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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(19,150)] 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("150.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + \cdots + 11105 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + \beta_{6} q^{3} - \beta_{10} q^{4} + (\beta_{9} - \beta_{5} + \beta_{3} + 1) q^{5} - \beta_{5} q^{6} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{3}) q^{7} + \beta_{13} q^{8}+ \cdots + (\beta_{14} + \beta_{13} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{9} + 2 q^{10} + 2 q^{11} + 20 q^{13} + 2 q^{14} - 2 q^{15} - 4 q^{16} - 30 q^{17} - 4 q^{20} - 2 q^{21} - 20 q^{22} - 10 q^{23} - 16 q^{24} + 24 q^{25} + 4 q^{26}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + \cdots + 11105 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 21\!\cdots\!17 \nu^{15} + \cdots - 36\!\cdots\!85 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26\!\cdots\!57 \nu^{15} + \cdots - 91\!\cdots\!10 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70\!\cdots\!83 \nu^{15} + \cdots + 49\!\cdots\!40 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!78 \nu^{15} + \cdots - 10\!\cdots\!35 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!46 \nu^{15} + \cdots + 21\!\cdots\!00 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31\!\cdots\!03 \nu^{15} + \cdots - 54\!\cdots\!40 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\!\cdots\!94 \nu^{15} + \cdots + 68\!\cdots\!70 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39\!\cdots\!47 \nu^{15} + \cdots - 35\!\cdots\!10 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!10 \nu^{15} + \cdots + 26\!\cdots\!20 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!10 \nu^{15} + \cdots + 10\!\cdots\!70 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 83\!\cdots\!46 \nu^{15} + \cdots + 17\!\cdots\!20 ) / 60\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 90\!\cdots\!78 \nu^{15} + \cdots + 43\!\cdots\!75 ) / 60\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 65\!\cdots\!33 \nu^{15} + \cdots - 61\!\cdots\!15 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 93\!\cdots\!97 \nu^{15} + \cdots - 86\!\cdots\!85 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!00 \nu^{15} + \cdots - 25\!\cdots\!15 ) / 60\!\cdots\!75 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2 \beta_{15} - \beta_{14} - \beta_{13} + 4 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} + \cdots - 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} + \cdots + 16 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17 \beta_{15} - 6 \beta_{14} - 26 \beta_{13} + 29 \beta_{12} - 23 \beta_{11} - 53 \beta_{10} - 2 \beta_{9} + \cdots + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12 \beta_{15} + 31 \beta_{14} - 89 \beta_{13} + 21 \beta_{12} - 77 \beta_{11} - 107 \beta_{10} + \cdots + 66 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 102 \beta_{15} - 31 \beta_{14} - 356 \beta_{13} + 179 \beta_{12} - 158 \beta_{11} - 593 \beta_{10} + \cdots + 69 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 67 \beta_{15} + 396 \beta_{14} - 1679 \beta_{13} + 411 \beta_{12} - 757 \beta_{11} - 1647 \beta_{10} + \cdots - 104 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 322 \beta_{15} + 209 \beta_{14} - 4661 \beta_{13} + 1249 \beta_{12} - 1198 \beta_{11} - 6368 \beta_{10} + \cdots + 659 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 722 \beta_{15} + 4821 \beta_{14} - 22869 \beta_{13} + 6081 \beta_{12} - 6637 \beta_{11} - 22292 \beta_{10} + \cdots - 6464 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4648 \beta_{15} + 10089 \beta_{14} - 61441 \beta_{13} + 10719 \beta_{12} - 10283 \beta_{11} + \cdots + 1354 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 17042 \beta_{15} + 61161 \beta_{14} - 277069 \beta_{13} + 72921 \beta_{12} - 50587 \beta_{11} + \cdots - 101659 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 134263 \beta_{15} + 202349 \beta_{14} - 797386 \beta_{13} + 111324 \beta_{12} - 86558 \beta_{11} + \cdots - 106706 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 378617 \beta_{15} + 805976 \beta_{14} - 3193009 \beta_{13} + 749601 \beta_{12} - 291027 \beta_{11} + \cdots - 1309799 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2303643 \beta_{15} + 3165794 \beta_{14} - 9996001 \beta_{13} + 1254869 \beta_{12} - 527138 \beta_{11} + \cdots - 2885261 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 6955377 \beta_{15} + 10771406 \beta_{14} - 35942739 \beta_{13} + 6786036 \beta_{12} - 208367 \beta_{11} + \cdots - 16212534 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 33676153 \beta_{15} + 43741979 \beta_{14} - 119837181 \beta_{13} + 13712654 \beta_{12} + \cdots - 51277636 ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{10}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.80334 − 0.309017i
3.42137 − 0.309017i
2.32349 + 0.309017i
−0.705457 + 0.309017i
−1.80334 + 0.309017i
3.42137 + 0.309017i
2.32349 − 0.309017i
−0.705457 − 0.309017i
2.17199 − 0.809017i
−2.79002 − 0.809017i
0.543374 + 0.809017i
−1.16141 + 0.809017i
2.17199 + 0.809017i
−2.79002 + 0.809017i
0.543374 − 0.809017i
−1.16141 − 0.809017i
−0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 − 0.587785i −1.73558 + 1.40988i −0.809017 − 0.587785i 2.61995i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.21496 − 1.87720i
19.2 −0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 − 0.587785i 2.23558 − 0.0466062i −0.809017 − 0.587785i − 3.52206i −0.587785 + 0.809017i −0.309017 + 0.951057i −2.11176 + 0.735158i
19.3 0.951057 − 0.309017i −0.587785 − 0.809017i 0.809017 − 0.587785i −1.47959 − 1.67655i −0.809017 − 0.587785i − 3.23143i 0.587785 − 0.809017i −0.309017 + 0.951057i −1.92526 − 1.13727i
19.4 0.951057 − 0.309017i −0.587785 − 0.809017i 0.809017 − 0.587785i 1.97959 + 1.03982i −0.809017 − 0.587785i 0.329315i 0.587785 − 0.809017i −0.309017 + 0.951057i 2.20402 + 0.377200i
79.1 −0.951057 − 0.309017i 0.587785 − 0.809017i 0.809017 + 0.587785i −1.73558 − 1.40988i −0.809017 + 0.587785i − 2.61995i −0.587785 − 0.809017i −0.309017 − 0.951057i 1.21496 + 1.87720i
79.2 −0.951057 − 0.309017i 0.587785 − 0.809017i 0.809017 + 0.587785i 2.23558 + 0.0466062i −0.809017 + 0.587785i 3.52206i −0.587785 − 0.809017i −0.309017 − 0.951057i −2.11176 − 0.735158i
79.3 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i −1.47959 + 1.67655i −0.809017 + 0.587785i 3.23143i 0.587785 + 0.809017i −0.309017 − 0.951057i −1.92526 + 1.13727i
79.4 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 1.97959 − 1.03982i −0.809017 + 0.587785i − 0.329315i 0.587785 + 0.809017i −0.309017 − 0.951057i 2.20402 − 0.377200i
109.1 −0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 − 0.951057i −1.53938 − 1.62182i 0.309017 − 0.951057i − 4.63137i 0.951057 + 0.309017i 0.809017 − 0.587785i 2.21691 − 0.292102i
109.2 −0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 − 0.951057i 2.03938 − 0.917020i 0.309017 − 0.951057i 4.80694i 0.951057 + 0.309017i 0.809017 − 0.587785i −0.456833 + 2.18890i
109.3 0.587785 − 0.809017i 0.951057 − 0.309017i −0.309017 − 0.951057i −1.36682 − 1.76969i 0.309017 − 0.951057i − 0.533559i −0.951057 − 0.309017i 0.809017 − 0.587785i −2.23511 + 0.0655797i
109.4 0.587785 − 0.809017i 0.951057 − 0.309017i −0.309017 − 0.951057i 1.86682 + 1.23085i 0.309017 − 0.951057i 2.70913i −0.951057 − 0.309017i 0.809017 − 0.587785i 2.09307 − 0.786811i
139.1 −0.587785 − 0.809017i −0.951057 − 0.309017i −0.309017 + 0.951057i −1.53938 + 1.62182i 0.309017 + 0.951057i 4.63137i 0.951057 − 0.309017i 0.809017 + 0.587785i 2.21691 + 0.292102i
139.2 −0.587785 − 0.809017i −0.951057 − 0.309017i −0.309017 + 0.951057i 2.03938 + 0.917020i 0.309017 + 0.951057i − 4.80694i 0.951057 − 0.309017i 0.809017 + 0.587785i −0.456833 − 2.18890i
139.3 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i −1.36682 + 1.76969i 0.309017 + 0.951057i 0.533559i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.23511 − 0.0655797i
139.4 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 1.86682 − 1.23085i 0.309017 + 0.951057i − 2.70913i −0.951057 + 0.309017i 0.809017 + 0.587785i 2.09307 + 0.786811i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.2.h.b ✓ 16
3.b odd 2 1 450.2.l.c 16
5.b even 2 1 750.2.h.d 16
5.c odd 4 1 750.2.g.f 16
5.c odd 4 1 750.2.g.g 16
25.d even 5 1 750.2.h.d 16
25.d even 5 1 3750.2.c.k 16
25.e even 10 1 inner 150.2.h.b ✓ 16
25.e even 10 1 3750.2.c.k 16
25.f odd 20 1 750.2.g.f 16
25.f odd 20 1 750.2.g.g 16
25.f odd 20 1 3750.2.a.u 8
25.f odd 20 1 3750.2.a.v 8
75.h odd 10 1 450.2.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b ✓ 16 1.a even 1 1 trivial
150.2.h.b ✓ 16 25.e even 10 1 inner
450.2.l.c 16 3.b odd 2 1
450.2.l.c 16 75.h odd 10 1
750.2.g.f 16 5.c odd 4 1
750.2.g.f 16 25.f odd 20 1
750.2.g.g 16 5.c odd 4 1
750.2.g.g 16 25.f odd 20 1
750.2.h.d 16 5.b even 2 1
750.2.h.d 16 25.d even 5 1
3750.2.a.u 8 25.f odd 20 1
3750.2.a.v 8 25.f odd 20 1
3750.2.c.k 16 25.d even 5 1
3750.2.c.k 16 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 82 T_{7}^{14} + 2683 T_{7}^{12} + 44874 T_{7}^{10} + 407105 T_{7}^{8} + 1927704 T_{7}^{6} + \cdots + 99856 \) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 82 T^{14} + \cdots + 99856 \) Copy content Toggle raw display
$11$ \( T^{16} - 2 T^{15} + \cdots + 524176 \) Copy content Toggle raw display
$13$ \( T^{16} - 20 T^{15} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{16} + 30 T^{15} + \cdots + 3748096 \) Copy content Toggle raw display
$19$ \( T^{16} + 100 T^{14} + \cdots + 2560000 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 20533743616 \) Copy content Toggle raw display
$29$ \( T^{16} + 10 T^{15} + \cdots + 40960000 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 205176976 \) Copy content Toggle raw display
$37$ \( T^{16} - 20 T^{15} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 78050949376 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 15083769856 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 172199901270016 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 111534721 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1600000000 \) Copy content Toggle raw display
$61$ \( T^{16} - 12 T^{15} + \cdots + 10137856 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 201983672713216 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 3398330023936 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 4778526048256 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 110872476160000 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 25573127056 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 36100000000 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 91700905971481 \) Copy content Toggle raw display
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