Properties

Label 450.2.h.d
Level $450$
Weight $2$
Character orbit 450.h
Analytic conductor $3.593$
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] = [450,2,Mod(91,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.h (of order \(5\), degree \(4\), 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: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1064390625.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 3x^{6} - 5x^{5} + 36x^{4} - 35x^{3} + 23x^{2} - 171x + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{4} + (\beta_{4} + \beta_{2} + 1) q^{5} + ( - \beta_{7} - \beta_{6} - 1) q^{7} + \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{4} + (\beta_{4} + \beta_{2} + 1) q^{5} + ( - \beta_{7} - \beta_{6} - 1) q^{7} + \beta_{5} q^{8} + ( - \beta_{3} - \beta_1 + 1) q^{10} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{11} + ( - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 1) q^{13} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{14} + \beta_{2} q^{16} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{2} - \beta_1) q^{17} + (2 \beta_{5} - 2 \beta_{3} + 2) q^{19} + ( - \beta_{7} - \beta_{5} - \beta_{2} - 1) q^{20} + ( - 2 \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{22} + ( - 4 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{23} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 2) q^{25} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} - 2) q^{26} + (\beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{28} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - \beta_{4} - 3 \beta_{3} + 5 \beta_{2} + 5) q^{29} + ( - \beta_{7} - \beta_{4} - 2 \beta_{3} + 1) q^{31} + q^{32} + (\beta_{7} - 2 \beta_{6} + \beta_{2} + 2 \beta_1 - 1) q^{34} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5} + 3 \beta_{3} - \beta_{2} - \beta_1) q^{35} + (\beta_{7} - 2 \beta_{6} + 6 \beta_{5} + 7 \beta_{2} + 2 \beta_1 + 5) q^{37} + ( - 2 \beta_{5} - 2) q^{38} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_1 - 1) q^{40} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - 6 \beta_{2} + 2 \beta_1 - 3) q^{41} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 - 4) q^{43} + ( - 2 \beta_{6} - 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{44} + (2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{46} + ( - 4 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{47} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{49} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{50} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1) q^{52} + (2 \beta_{7} + \beta_{6} + 8 \beta_{5} - \beta_{4} - 7 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 4) q^{53} + (2 \beta_{6} - 5 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} - 7) q^{55} + (\beta_{7} + \beta_{4} + 1) q^{56} + ( - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{58} + (\beta_{7} + 3 \beta_{5} - 2 \beta_{4} - 5 \beta_{2} + 2 \beta_1 + 2) q^{59} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - \beta_1) q^{61} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{62} - \beta_{3} q^{64} + ( - \beta_{6} - 5 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - \beta_1 - 7) q^{65} + ( - 2 \beta_{7} - 8 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4) q^{67} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 + 1) q^{68} + ( - \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} + 8 \beta_{2} + \beta_1 + 2) q^{70} + (4 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{71} + (\beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 2) q^{73} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 6 \beta_{3} + 5 \beta_{2} - \beta_1 + 7) q^{74} + (2 \beta_{3} - 2 \beta_{2}) q^{76} + ( - \beta_{7} + 9 \beta_{5} + 3 \beta_{3} + 10 \beta_{2} + \beta_1 - 1) q^{77} + (2 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - \beta_{4} + 6 \beta_{3} - 5 \beta_{2} - 5) q^{79} + (\beta_{6} + \beta_{3}) q^{80} + (\beta_{7} + \beta_{6} + 3 \beta_{3} - 3 \beta_{2} - 6) q^{82} + ( - \beta_{7} + 4 \beta_{6} - 8 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{83}+ \cdots + (\beta_{6} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{4} + 4 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{4} + 4 q^{5} - 2 q^{7} - 2 q^{8} + 4 q^{10} + 5 q^{11} + 6 q^{13} - 2 q^{14} - 2 q^{16} + 2 q^{17} + 8 q^{19} - q^{20} + 20 q^{23} + 14 q^{25} - 14 q^{26} + 3 q^{28} + 18 q^{29} + 9 q^{31} + 8 q^{32} - 3 q^{34} + 4 q^{35} + 21 q^{37} - 12 q^{38} - 6 q^{40} - 2 q^{41} - 32 q^{43} - 20 q^{46} + 10 q^{47} + 22 q^{49} - 26 q^{50} + 6 q^{52} - 7 q^{53} - 40 q^{55} + 3 q^{56} + 18 q^{58} + 25 q^{59} + 10 q^{61} - 6 q^{62} - 2 q^{64} - 37 q^{65} - 2 q^{67} + 2 q^{68} - 11 q^{70} - 24 q^{73} + 26 q^{74} + 8 q^{76} - 35 q^{77} - 6 q^{79} - q^{80} - 42 q^{82} - 11 q^{83} + q^{85} - 2 q^{86} + 5 q^{88} - 9 q^{89} - 4 q^{91} - 20 q^{92} + 10 q^{94} + 4 q^{95} + q^{97} + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 3x^{6} - 5x^{5} + 36x^{4} - 35x^{3} + 23x^{2} - 171x + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27571 \nu^{7} + 156 \nu^{6} - 50800 \nu^{5} + 197116 \nu^{4} - 261151 \nu^{3} + 1281772 \nu^{2} - 1105295 \nu + 4276691 ) / 4238558 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45697 \nu^{7} - 18958 \nu^{6} + 87368 \nu^{5} + 389890 \nu^{4} - 351515 \nu^{3} + 1275844 \nu^{2} - 687257 \nu + 5880747 ) / 4238558 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2894\nu^{7} + 7027\nu^{6} - 3119\nu^{5} - 38495\nu^{4} - 16673\nu^{3} + 24798\nu^{2} + 23050\nu - 523849 ) / 223082 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 60697 \nu^{7} + 29865 \nu^{6} - 107003 \nu^{5} - 736723 \nu^{4} + 989612 \nu^{3} + 38090 \nu^{2} + 939005 \nu - 13281190 ) / 4238558 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 72436 \nu^{7} - 26109 \nu^{6} + 188067 \nu^{5} + 265983 \nu^{4} - 1082509 \nu^{3} + 501044 \nu^{2} + 3422970 \nu + 7026371 ) / 4238558 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5808 \nu^{7} + 2617 \nu^{6} - 8495 \nu^{5} - 68083 \nu^{4} + 17029 \nu^{3} - 19146 \nu^{2} + 101760 \nu - 868243 ) / 223082 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + 4\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + 5\beta_{5} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} - 7\beta_{5} + \beta_{4} + 7\beta_{3} - 4\beta_{2} + 8\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + 4\beta_{4} + 39\beta_{3} - 39\beta_{2} - 4\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -39\beta_{7} + 16\beta_{5} + 39\beta_{4} - 26\beta_{3} + 6\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 71\beta_{7} - 100\beta_{6} - 101\beta_{5} + 164\beta_{3} - 101\beta_{2} + 71\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2} + \beta_{3} - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−1.86886 1.45788i
2.36886 0.0809628i
−0.815575 + 1.64827i
1.31557 1.28500i
−0.815575 1.64827i
1.31557 + 1.28500i
−1.86886 + 1.45788i
2.36886 + 0.0809628i
−0.809017 0.587785i 0 0.309017 + 0.951057i −2.17787 + 0.506822i 0 −2.31003 0.309017 0.951057i 0 2.05984 + 0.870094i
91.2 −0.809017 0.587785i 0 0.309017 + 0.951057i 2.05984 0.870094i 0 2.92807 0.309017 0.951057i 0 −2.17787 0.506822i
181.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.00655751 2.23606i 0 2.63925 −0.809017 0.587785i 0 2.12459 0.697217i
181.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 2.12459 + 0.697217i 0 −4.25729 −0.809017 0.587785i 0 −0.00655751 + 2.23606i
271.1 0.309017 0.951057i 0 −0.809017 0.587785i −0.00655751 + 2.23606i 0 2.63925 −0.809017 + 0.587785i 0 2.12459 + 0.697217i
271.2 0.309017 0.951057i 0 −0.809017 0.587785i 2.12459 0.697217i 0 −4.25729 −0.809017 + 0.587785i 0 −0.00655751 2.23606i
361.1 −0.809017 + 0.587785i 0 0.309017 0.951057i −2.17787 0.506822i 0 −2.31003 0.309017 + 0.951057i 0 2.05984 0.870094i
361.2 −0.809017 + 0.587785i 0 0.309017 0.951057i 2.05984 + 0.870094i 0 2.92807 0.309017 + 0.951057i 0 −2.17787 + 0.506822i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.h.d 8
3.b odd 2 1 150.2.g.c 8
15.d odd 2 1 750.2.g.d 8
15.e even 4 2 750.2.h.e 16
25.d even 5 1 inner 450.2.h.d 8
75.h odd 10 1 750.2.g.d 8
75.h odd 10 1 3750.2.a.q 4
75.j odd 10 1 150.2.g.c 8
75.j odd 10 1 3750.2.a.l 4
75.l even 20 2 750.2.h.e 16
75.l even 20 2 3750.2.c.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.c 8 3.b odd 2 1
150.2.g.c 8 75.j odd 10 1
450.2.h.d 8 1.a even 1 1 trivial
450.2.h.d 8 25.d even 5 1 inner
750.2.g.d 8 15.d odd 2 1
750.2.g.d 8 75.h odd 10 1
750.2.h.e 16 15.e even 4 2
750.2.h.e 16 75.l even 20 2
3750.2.a.l 4 75.j odd 10 1
3750.2.a.q 4 75.h odd 10 1
3750.2.c.h 8 75.l even 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{4} + T_{7}^{3} - 19T_{7}^{2} - 4T_{7} + 76 \) Copy content Toggle raw display
\( T_{11}^{8} - 5T_{11}^{7} + 60T_{11}^{6} - 320T_{11}^{5} + 1785T_{11}^{4} - 6050T_{11}^{3} + 12800T_{11}^{2} - 3600T_{11} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + T^{6} + 16 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} - 19 T^{2} - 4 T + 76)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + 60 T^{6} - 320 T^{5} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{8} - 6 T^{7} + 32 T^{6} - 143 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + 78 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 10 T^{3} + 60 T^{2} - 200 T + 400)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 18 T^{7} + 218 T^{6} + \cdots + 1860496 \) Copy content Toggle raw display
$31$ \( T^{8} - 9 T^{7} + 42 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$37$ \( T^{8} - 21 T^{7} + 332 T^{6} + \cdots + 11182336 \) Copy content Toggle raw display
$41$ \( T^{8} + 2 T^{7} + 118 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( (T^{4} + 16 T^{3} - 4 T^{2} - 784 T - 1424)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + 160 T^{6} + \cdots + 774400 \) Copy content Toggle raw display
$53$ \( T^{8} + 7 T^{7} + 168 T^{6} + \cdots + 85359121 \) Copy content Toggle raw display
$59$ \( T^{8} - 25 T^{7} + 330 T^{6} + \cdots + 2310400 \) Copy content Toggle raw display
$61$ \( T^{8} - 10 T^{7} + 60 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$67$ \( T^{8} + 2 T^{7} + 168 T^{6} + \cdots + 6885376 \) Copy content Toggle raw display
$71$ \( T^{8} - 40 T^{6} + 40 T^{5} + \cdots + 102400 \) Copy content Toggle raw display
$73$ \( T^{8} + 24 T^{7} + 372 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$79$ \( T^{8} + 6 T^{7} + 192 T^{6} + \cdots + 331776 \) Copy content Toggle raw display
$83$ \( T^{8} + 11 T^{7} + 162 T^{6} + \cdots + 1008016 \) Copy content Toggle raw display
$89$ \( T^{8} + 9 T^{7} + 12 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$97$ \( T^{8} - T^{7} + 222 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
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