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Properties

Label 150.11.f.h
Level $150$
Weight $11$
Character orbit 150.f
Analytic conductor $95.304$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,11,Mod(7,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.7");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 150.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(95.3035879011\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 1321585038 x^{10} - 4746832718600 x^{9} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{24}\cdot 5^{14} \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (16 \beta_1 - 16) q^{2} - \beta_{2} q^{3} - 512 \beta_1 q^{4} + ( - 16 \beta_{3} + 16 \beta_{2}) q^{6} + (\beta_{8} - 25 \beta_{3} + 946 \beta_1 - 946) q^{7} + (8192 \beta_1 + 8192) q^{8} + 19683 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (16 \beta_1 - 16) q^{2} - \beta_{2} q^{3} - 512 \beta_1 q^{4} + ( - 16 \beta_{3} + 16 \beta_{2}) q^{6} + (\beta_{8} - 25 \beta_{3} + 946 \beta_1 - 946) q^{7} + (8192 \beta_1 + 8192) q^{8} + 19683 \beta_1 q^{9} + (2 \beta_{11} - 2 \beta_{10} + \cdots + 33836) q^{11}+ \cdots + (39366 \beta_{11} + \cdots + 665993988 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 192 q^{2} - 11352 q^{7} + 98304 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 192 q^{2} - 11352 q^{7} + 98304 q^{8} + 406032 q^{11} + 112404 q^{13} - 3145728 q^{16} + 4207428 q^{17} - 3779136 q^{18} - 5917536 q^{21} - 6496512 q^{22} - 22928832 q^{23} - 3596928 q^{26} + 5812224 q^{28} + 110039760 q^{31} + 50331648 q^{32} + 55056024 q^{33} + 120932352 q^{36} + 139253964 q^{37} + 97581312 q^{38} + 379041456 q^{41} + 94680576 q^{42} + 706743456 q^{43} + 733722624 q^{46} + 196217472 q^{47} + 993337344 q^{51} + 57550848 q^{52} + 452349900 q^{53} - 185991168 q^{56} + 331109856 q^{57} + 1372339968 q^{58} + 4086356592 q^{61} - 1760636160 q^{62} - 223441416 q^{63} - 1761792768 q^{66} + 1954389408 q^{67} - 2154203136 q^{68} - 6569583648 q^{71} - 1934917632 q^{72} - 3913379676 q^{73} - 3122601984 q^{76} + 3631480944 q^{77} - 2739702528 q^{78} - 4649045868 q^{81} - 6064663296 q^{82} - 1508760096 q^{83} - 22615790592 q^{86} + 943152984 q^{87} + 3326214144 q^{88} + 27259739904 q^{91} - 11739561984 q^{92} - 21947561712 q^{93} + 20528028324 q^{97} + 30689842752 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 1321585038 x^{10} - 4746832718600 x^{9} + \cdots + 30\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\!\cdots\!33 \nu^{11} + \cdots + 62\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 37\!\cdots\!03 \nu^{11} + \cdots + 19\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39\!\cdots\!97 \nu^{11} + \cdots + 20\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!57 \nu^{11} + \cdots - 35\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 50\!\cdots\!31 \nu^{11} + \cdots - 17\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!52 \nu^{11} + \cdots + 26\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!07 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!53 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53\!\cdots\!44 \nu^{11} + \cdots - 19\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\!\cdots\!51 \nu^{11} + \cdots + 98\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!69 \nu^{11} + \cdots + 99\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} - \beta_{7} - 25\beta_{3} + 25\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2475 \beta_{11} + 2475 \beta_{10} + 7338 \beta_{8} - 7342 \beta_{7} + 2038 \beta_{5} + \cdots + 440528346 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 21489727 \beta_{11} - 21504577 \beta_{10} - 6114 \beta_{9} + 351172542 \beta_{8} - 351216582 \beta_{7} + \cdots + 2373416359300 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1152478183985 \beta_{11} + 1152650161201 \beta_{10} - 29005776 \beta_{9} + 3324397112542 \beta_{8} + \cdots + 15\!\cdots\!66 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 94\!\cdots\!25 \beta_{11} + \cdots + 11\!\cdots\!00 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 45\!\cdots\!31 \beta_{11} + \cdots + 60\!\cdots\!86 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34\!\cdots\!07 \beta_{11} + \cdots + 50\!\cdots\!00 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 17\!\cdots\!41 \beta_{11} + \cdots + 23\!\cdots\!06 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 12\!\cdots\!13 \beta_{11} + \cdots + 21\!\cdots\!00 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 69\!\cdots\!55 \beta_{11} + \cdots + 95\!\cdots\!26 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 41\!\cdots\!83 \beta_{11} + \cdots + 93\!\cdots\!00 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−8248.68 − 1.00000i
−3112.91 − 1.00000i
18817.8 − 1.00000i
−19397.0 − 1.00000i
−8899.16 − 1.00000i
20839.9 − 1.00000i
−8248.68 + 1.00000i
−3112.91 + 1.00000i
18817.8 + 1.00000i
−19397.0 + 1.00000i
−8899.16 + 1.00000i
20839.9 + 1.00000i
−16.0000 − 16.0000i −99.2043 + 99.2043i 512.000i 0 3174.54 −9194.68 − 9194.68i 8192.00 − 8192.00i − 19683.0i 0
7.2 −16.0000 − 16.0000i −99.2043 + 99.2043i 512.000i 0 3174.54 −4058.91 − 4058.91i 8192.00 − 8192.00i − 19683.0i 0
7.3 −16.0000 − 16.0000i −99.2043 + 99.2043i 512.000i 0 3174.54 17871.8 + 17871.8i 8192.00 − 8192.00i − 19683.0i 0
7.4 −16.0000 − 16.0000i 99.2043 − 99.2043i 512.000i 0 −3174.54 −20343.0 − 20343.0i 8192.00 − 8192.00i − 19683.0i 0
7.5 −16.0000 − 16.0000i 99.2043 − 99.2043i 512.000i 0 −3174.54 −9845.16 − 9845.16i 8192.00 − 8192.00i − 19683.0i 0
7.6 −16.0000 − 16.0000i 99.2043 − 99.2043i 512.000i 0 −3174.54 19893.9 + 19893.9i 8192.00 − 8192.00i − 19683.0i 0
43.1 −16.0000 + 16.0000i −99.2043 − 99.2043i − 512.000i 0 3174.54 −9194.68 + 9194.68i 8192.00 + 8192.00i 19683.0i 0
43.2 −16.0000 + 16.0000i −99.2043 − 99.2043i − 512.000i 0 3174.54 −4058.91 + 4058.91i 8192.00 + 8192.00i 19683.0i 0
43.3 −16.0000 + 16.0000i −99.2043 − 99.2043i − 512.000i 0 3174.54 17871.8 − 17871.8i 8192.00 + 8192.00i 19683.0i 0
43.4 −16.0000 + 16.0000i 99.2043 + 99.2043i − 512.000i 0 −3174.54 −20343.0 + 20343.0i 8192.00 + 8192.00i 19683.0i 0
43.5 −16.0000 + 16.0000i 99.2043 + 99.2043i − 512.000i 0 −3174.54 −9845.16 + 9845.16i 8192.00 + 8192.00i 19683.0i 0
43.6 −16.0000 + 16.0000i 99.2043 + 99.2043i − 512.000i 0 −3174.54 19893.9 − 19893.9i 8192.00 + 8192.00i 19683.0i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.11.f.h 12
5.b even 2 1 30.11.f.b ✓ 12
5.c odd 4 1 30.11.f.b ✓ 12
5.c odd 4 1 inner 150.11.f.h 12
15.d odd 2 1 90.11.g.g 12
15.e even 4 1 90.11.g.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.11.f.b ✓ 12 5.b even 2 1
30.11.f.b ✓ 12 5.c odd 4 1
90.11.g.g 12 15.d odd 2 1
90.11.g.g 12 15.e even 4 1
150.11.f.h 12 1.a even 1 1 trivial
150.11.f.h 12 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 11352 T_{7}^{11} + 64433952 T_{7}^{10} + 4729832980784 T_{7}^{9} + \cdots + 45\!\cdots\!16 \) acting on \(S_{11}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32 T + 512)^{6} \) Copy content Toggle raw display
$3$ \( (T^{4} + 387420489)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 83\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
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