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Properties

Label 450.2.h.f
Level $450$
Weight $2$
Character orbit 450.h
Analytic conductor $3.593$
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] = [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: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 6 x^{10} - 26 x^{9} + 61 x^{8} - 120 x^{7} + 465 x^{6} - 600 x^{5} + 1525 x^{4} + \cdots + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (\beta_{7} - \beta_{6} + \beta_{4} - 1) q^{4} + \beta_1 q^{5} + (\beta_{10} - \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots - \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + (\beta_{7} - \beta_{6} + \beta_{4} - 1) q^{4} + \beta_1 q^{5} + (\beta_{10} - \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - 3 \beta_{11} + 2 \beta_{10} + \cdots - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 3 q^{4} + q^{5} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 3 q^{4} + q^{5} - 2 q^{7} - 3 q^{8} + q^{10} + q^{11} + 4 q^{13} + 8 q^{14} - 3 q^{16} - 8 q^{17} - 8 q^{19} + q^{20} - 4 q^{22} - 11 q^{25} - 16 q^{26} - 7 q^{28} - 6 q^{29} - 3 q^{31} + 12 q^{32} + 2 q^{34} - 18 q^{35} - 8 q^{37} + 2 q^{38} + q^{40} + 20 q^{41} + 32 q^{43} - 4 q^{44} - 10 q^{46} + 34 q^{49} + 9 q^{50} + 4 q^{52} + 2 q^{53} + 44 q^{55} - 7 q^{56} - 6 q^{58} - 19 q^{59} - 26 q^{61} + 2 q^{62} - 3 q^{64} + 16 q^{65} - 16 q^{67} + 12 q^{68} - 23 q^{70} + 48 q^{71} - 30 q^{73} - 8 q^{74} + 12 q^{76} - 39 q^{77} - 18 q^{79} - 4 q^{80} - 40 q^{82} - 29 q^{83} - 4 q^{85} + 12 q^{86} + q^{88} + 62 q^{89} - 26 q^{91} - 10 q^{92} + 6 q^{95} + 23 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 6 x^{10} - 26 x^{9} + 61 x^{8} - 120 x^{7} + 465 x^{6} - 600 x^{5} + 1525 x^{4} + \cdots + 15625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 27 \nu^{10} - 17 \nu^{9} - 8 \nu^{8} - 467 \nu^{7} - 162 \nu^{6} - 195 \nu^{5} + \cdots - 99375 ) / 4750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6 \nu^{11} + 124 \nu^{10} + 31 \nu^{9} + 199 \nu^{8} - 1339 \nu^{7} + 35 \nu^{6} - 885 \nu^{5} + \cdots - 228125 ) / 23750 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{11} + 134 \nu^{10} - 204 \nu^{9} + 2659 \nu^{8} - 2374 \nu^{7} + 21445 \nu^{6} + \cdots + 1503125 ) / 118750 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - \nu^{10} + 6 \nu^{9} - 26 \nu^{8} + 61 \nu^{7} - 120 \nu^{6} + 465 \nu^{5} - 600 \nu^{4} + \cdots - 3125 ) / 3125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 73 \nu^{11} - 43 \nu^{10} + 1058 \nu^{9} - 1743 \nu^{8} + 5448 \nu^{7} - 15455 \nu^{6} + \cdots - 568750 ) / 118750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 117 \nu^{11} - 43 \nu^{10} - 367 \nu^{9} + 157 \nu^{8} - 1677 \nu^{7} - 6145 \nu^{6} + \cdots - 925000 ) / 118750 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 32 \nu^{11} + 67 \nu^{10} - 577 \nu^{9} + 1092 \nu^{8} - 4037 \nu^{7} + 11150 \nu^{6} + \cdots + 365625 ) / 23750 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 182 \nu^{11} + 183 \nu^{10} + 877 \nu^{9} + 558 \nu^{8} + 2387 \nu^{7} + 5400 \nu^{6} + \cdots + 759375 ) / 118750 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 296 \nu^{11} + 881 \nu^{10} - 1561 \nu^{9} + 9531 \nu^{8} - 18841 \nu^{7} + 43905 \nu^{6} + \cdots + 131250 ) / 118750 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 481 \nu^{11} + 636 \nu^{10} - 2216 \nu^{9} + 11486 \nu^{8} - 16046 \nu^{7} + 45850 \nu^{6} + \cdots + 643750 ) / 118750 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{6} - \beta_{5} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - 2\beta_{10} + \beta_{8} + 4\beta_{7} + 4\beta_{6} + 2\beta_{5} + 4\beta_{4} + \beta_{2} - 2\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{11} - \beta_{9} - \beta_{8} + 13 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{10} + 17\beta_{9} + 7\beta_{8} - 4\beta_{6} - \beta_{5} - 32\beta_{4} + 8\beta_{3} - 12\beta_{2} - 7\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{11} + \beta_{10} - 3 \beta_{9} + 13 \beta_{8} - 18 \beta_{7} + 48 \beta_{6} + 45 \beta_{5} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{11} - 14 \beta_{10} - 71 \beta_{9} - 36 \beta_{8} - 4 \beta_{7} - 52 \beta_{6} + 36 \beta_{5} + \cdots - 240 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 86 \beta_{11} + 82 \beta_{10} + 85 \beta_{9} - 91 \beta_{8} + 156 \beta_{7} - 339 \beta_{6} + \cdots - 184 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 222 \beta_{11} + 400 \beta_{10} - 124 \beta_{9} - 74 \beta_{8} - 188 \beta_{7} + 80 \beta_{6} + \cdots + 688 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 250 \beta_{11} + 48 \beta_{10} - 692 \beta_{9} - 252 \beta_{8} - 2250 \beta_{7} - 346 \beta_{6} + \cdots - 317 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 292 \beta_{11} - 376 \beta_{10} + 978 \beta_{9} - 548 \beta_{8} - 532 \beta_{7} - 1448 \beta_{6} + \cdots + 36 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4} - \beta_{6} + \beta_{7}\)

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
−1.38239 + 1.75756i
0.220100 − 2.22521i
1.97131 + 1.05544i
−1.24741 − 1.85579i
−1.01542 + 1.99221i
1.95382 − 1.08748i
−1.24741 + 1.85579i
−1.01542 − 1.99221i
1.95382 + 1.08748i
−1.38239 − 1.75756i
0.220100 + 2.22521i
1.97131 − 1.05544i
−0.809017 − 0.587785i 0 0.309017 + 0.951057i −1.38239 + 1.75756i 0 −0.447412 0.309017 − 0.951057i 0 2.15144 − 0.609344i
91.2 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.220100 − 2.22521i 0 1.64173 0.309017 − 0.951057i 0 −1.48601 + 1.67086i
91.3 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 1.97131 + 1.05544i 0 −5.04842 0.309017 − 0.951057i 0 −0.974449 − 2.01257i
181.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −1.24741 − 1.85579i 0 −2.68284 −0.809017 − 0.587785i 0 1.37949 − 1.75983i
181.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −1.01542 + 1.99221i 0 4.77988 −0.809017 − 0.587785i 0 −2.20849 − 0.350097i
181.3 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 1.95382 − 1.08748i 0 0.757055 −0.809017 − 0.587785i 0 1.63801 + 1.52214i
271.1 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −1.24741 + 1.85579i 0 −2.68284 −0.809017 + 0.587785i 0 1.37949 + 1.75983i
271.2 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −1.01542 − 1.99221i 0 4.77988 −0.809017 + 0.587785i 0 −2.20849 + 0.350097i
271.3 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 1.95382 + 1.08748i 0 0.757055 −0.809017 + 0.587785i 0 1.63801 − 1.52214i
361.1 −0.809017 + 0.587785i 0 0.309017 − 0.951057i −1.38239 − 1.75756i 0 −0.447412 0.309017 + 0.951057i 0 2.15144 + 0.609344i
361.2 −0.809017 + 0.587785i 0 0.309017 − 0.951057i 0.220100 + 2.22521i 0 1.64173 0.309017 + 0.951057i 0 −1.48601 − 1.67086i
361.3 −0.809017 + 0.587785i 0 0.309017 − 0.951057i 1.97131 − 1.05544i 0 −5.04842 0.309017 + 0.951057i 0 −0.974449 + 2.01257i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.3
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.f ✓ 12
3.b odd 2 1 450.2.h.g yes 12
25.d even 5 1 inner 450.2.h.f ✓ 12
75.j odd 10 1 450.2.h.g yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.h.f ✓ 12 1.a even 1 1 trivial
450.2.h.f ✓ 12 25.d even 5 1 inner
450.2.h.g yes 12 3.b odd 2 1
450.2.h.g yes 12 75.j odd 10 1

Hecke kernels

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

\( T_{7}^{6} + T_{7}^{5} - 29T_{7}^{4} - 18T_{7}^{3} + 124T_{7}^{2} - 24T_{7} - 36 \) Copy content Toggle raw display
\( T_{11}^{12} - T_{11}^{11} - 8 T_{11}^{10} + 28 T_{11}^{9} + 693 T_{11}^{8} - 1222 T_{11}^{7} + 8798 T_{11}^{6} + \cdots + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{5} - 29 T^{4} + \cdots - 36)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - T^{11} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} + \cdots + 126736 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots + 467856 \) Copy content Toggle raw display
$19$ \( T^{12} + 8 T^{11} + \cdots + 92416 \) Copy content Toggle raw display
$23$ \( T^{12} + 70 T^{10} + \cdots + 2560000 \) Copy content Toggle raw display
$29$ \( T^{12} + 6 T^{11} + \cdots + 1936 \) Copy content Toggle raw display
$31$ \( T^{12} + 3 T^{11} + \cdots + 26896 \) Copy content Toggle raw display
$37$ \( T^{12} + 8 T^{11} + \cdots + 68492176 \) Copy content Toggle raw display
$41$ \( T^{12} - 20 T^{11} + \cdots + 24010000 \) Copy content Toggle raw display
$43$ \( (T^{6} - 16 T^{5} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 6724000000 \) Copy content Toggle raw display
$53$ \( T^{12} - 2 T^{11} + \cdots + 31798321 \) Copy content Toggle raw display
$59$ \( T^{12} + 19 T^{11} + \cdots + 70291456 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 32290652416 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 3550253056 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 4180398336 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 3906250000 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 15083769856 \) Copy content Toggle raw display
$83$ \( T^{12} + 29 T^{11} + \cdots + 27081616 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1387115536 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 786185521 \) Copy content Toggle raw display
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