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Properties

Label 150.11.f.d
Level $150$
Weight $11$
Character orbit 150.f
Analytic conductor $95.304$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 61622x^{6} + 1413899071x^{4} + 14314236068250x^{2} + 53958957691955625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{14}\cdot 5^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 16 \beta_1 - 16) q^{2} - \beta_{2} q^{3} + 512 \beta_1 q^{4} + ( - 16 \beta_{3} + 16 \beta_{2}) q^{6} + ( - \beta_{5} + 32 \beta_{3} + \cdots + 3564) q^{7}+ \cdots - 19683 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 16 \beta_1 - 16) q^{2} - \beta_{2} q^{3} + 512 \beta_1 q^{4} + ( - 16 \beta_{3} + 16 \beta_{2}) q^{6} + ( - \beta_{5} + 32 \beta_{3} + \cdots + 3564) q^{7}+ \cdots + ( - 19683 \beta_{7} + \cdots + 925770222 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} + 28512 q^{7} + 65536 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 128 q^{2} + 28512 q^{7} + 65536 q^{8} - 376272 q^{11} + 677376 q^{13} - 2097152 q^{16} + 2319072 q^{17} - 2519424 q^{18} + 5003856 q^{21} + 6020352 q^{22} - 1387488 q^{23} - 21676032 q^{26} - 14598144 q^{28} - 44418640 q^{31} + 33554432 q^{32} - 50458464 q^{33} + 80621568 q^{36} + 93099456 q^{37} - 114420992 q^{38} + 105510384 q^{41} - 80061696 q^{42} - 818747136 q^{43} + 44399616 q^{46} + 625006368 q^{47} + 311358816 q^{51} + 346816512 q^{52} - 712628160 q^{53} + 467140608 q^{56} + 441319104 q^{57} - 216373248 q^{58} - 2322624752 q^{61} + 710698240 q^{62} + 561201696 q^{63} + 1614670848 q^{66} + 2552219712 q^{67} - 1187364864 q^{68} - 6154235712 q^{71} - 1289945088 q^{72} - 1801142784 q^{73} + 3661471744 q^{76} - 11940574464 q^{77} + 2541259008 q^{78} - 3099363912 q^{81} - 1688166144 q^{82} + 16245491136 q^{83} + 26199908352 q^{86} - 5044866624 q^{87} - 3082420224 q^{88} - 21024467424 q^{91} - 710393856 q^{92} + 16014298752 q^{93} + 10686465216 q^{97} + 26429001088 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 61622x^{6} + 1413899071x^{4} + 14314236068250x^{2} + 53958957691955625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15404\nu^{7} + 716934613\nu^{5} + 11043923163209\nu^{3} + 56282285924794575\nu ) / 2326597848825750 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1001589 \nu^{7} - 46458135 \nu^{6} - 61719917358 \nu^{5} - 2147155625295 \nu^{4} + \cdots - 15\!\cdots\!50 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1001589 \nu^{7} + 46458135 \nu^{6} - 61719917358 \nu^{5} + 2147155625295 \nu^{4} + \cdots + 15\!\cdots\!50 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24381\nu^{6} + 1327134477\nu^{4} + 23407943085621\nu^{2} + 133775240473580250 ) / 1018282150 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17791291 \nu^{7} + 566835705135 \nu^{6} + 812940310502 \nu^{5} + \cdots + 20\!\cdots\!50 ) / 47\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17791291 \nu^{7} - 566835705135 \nu^{6} + 812940310502 \nu^{5} + \cdots - 20\!\cdots\!50 ) / 47\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1801709449 \nu^{7} + 82964226126278 \nu^{5} + \cdots + 60\!\cdots\!25 \nu ) / 47\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 9\beta_{6} + 9\beta_{5} - 2\beta_{3} - 2\beta_{2} - 1080\beta_1 ) / 2160 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{6} + 3\beta_{5} - 8134\beta_{3} + 8134\beta_{2} - 11091960 ) / 720 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -305\beta_{7} - 8008\beta_{6} - 8008\beta_{5} + 3779\beta_{3} + 3779\beta_{2} + 2772960\beta_1 ) / 120 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 32031\beta_{6} - 32031\beta_{5} + 1220\beta_{4} + 83541738\beta_{3} - 83541738\beta_{2} + 58168364520 ) / 240 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18795930 \beta_{7} + 261179521 \beta_{6} + 261179521 \beta_{5} - 264781568 \beta_{3} + \cdots - 145411668120 \beta_1 ) / 240 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 391729243 \beta_{6} + 391729243 \beta_{5} - 28192370 \beta_{4} - 972158328524 \beta_{3} + \cdots - 466688068773360 ) / 120 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 437461676900 \beta_{7} - 4326892453551 \beta_{6} - 4326892453551 \beta_{5} + \cdots + 32\!\cdots\!20 \beta_1 ) / 240 \) 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
114.234i
− 115.234i
132.340i
− 133.340i
− 114.234i
115.234i
− 132.340i
133.340i
−16.0000 − 16.0000i −99.2043 + 99.2043i 512.000i 0 3174.54 −13356.6 − 13356.6i 8192.00 − 8192.00i − 19683.0i 0
7.2 −16.0000 − 16.0000i −99.2043 + 99.2043i 512.000i 0 3174.54 14179.6 + 14179.6i 8192.00 − 8192.00i − 19683.0i 0
7.3 −16.0000 − 16.0000i 99.2043 − 99.2043i 512.000i 0 −3174.54 −9224.34 − 9224.34i 8192.00 − 8192.00i − 19683.0i 0
7.4 −16.0000 − 16.0000i 99.2043 − 99.2043i 512.000i 0 −3174.54 22657.3 + 22657.3i 8192.00 − 8192.00i − 19683.0i 0
43.1 −16.0000 + 16.0000i −99.2043 − 99.2043i − 512.000i 0 3174.54 −13356.6 + 13356.6i 8192.00 + 8192.00i 19683.0i 0
43.2 −16.0000 + 16.0000i −99.2043 − 99.2043i − 512.000i 0 3174.54 14179.6 − 14179.6i 8192.00 + 8192.00i 19683.0i 0
43.3 −16.0000 + 16.0000i 99.2043 + 99.2043i − 512.000i 0 −3174.54 −9224.34 + 9224.34i 8192.00 + 8192.00i 19683.0i 0
43.4 −16.0000 + 16.0000i 99.2043 + 99.2043i − 512.000i 0 −3174.54 22657.3 − 22657.3i 8192.00 + 8192.00i 19683.0i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
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.d ✓ 8
5.b even 2 1 150.11.f.e yes 8
5.c odd 4 1 inner 150.11.f.d ✓ 8
5.c odd 4 1 150.11.f.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.11.f.d ✓ 8 1.a even 1 1 trivial
150.11.f.d ✓ 8 5.c odd 4 1 inner
150.11.f.e yes 8 5.b even 2 1
150.11.f.e yes 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 28512 T_{7}^{7} + 406467072 T_{7}^{6} + 11222969510016 T_{7}^{5} + \cdots + 25\!\cdots\!16 \) acting on \(S_{11}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32 T + 512)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 387420489)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 51\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 70\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 61\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 91\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
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