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Properties

Label 555.2.i.d
Level $555$
Weight $2$
Character orbit 555.i
Analytic conductor $4.432$
Analytic rank $0$
Dimension $10$
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] = [555,2,Mod(121,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.i (of order \(3\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 2x^{7} + 35x^{6} - 15x^{5} + 45x^{4} + 8x^{3} + 29x^{2} - 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} - \beta_{6} q^{3} + ( - \beta_{9} - \beta_{6} + \beta_{3}) q^{4} - \beta_{6} q^{5} - \beta_{2} q^{6} + ( - \beta_{9} + \beta_{7} + \cdots + \beta_{3}) q^{7}+ \cdots + (\beta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} - \beta_{6} q^{3} + ( - \beta_{9} - \beta_{6} + \beta_{3}) q^{4} - \beta_{6} q^{5} - \beta_{2} q^{6} + ( - \beta_{9} + \beta_{7} + \cdots + \beta_{3}) q^{7}+ \cdots + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 5 q^{3} - 3 q^{4} - 5 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - 5 q^{3} - 3 q^{4} - 5 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} - 5 q^{9} - 2 q^{10} + 4 q^{11} - 3 q^{12} - 3 q^{13} - 18 q^{14} - 5 q^{15} - 3 q^{16} + 6 q^{17} + q^{18} + 16 q^{19} - 3 q^{20} - q^{21} - 4 q^{22} - 8 q^{23} + 3 q^{24} - 5 q^{25} - 10 q^{26} + 10 q^{27} - 17 q^{28} - 8 q^{29} + q^{30} + 6 q^{31} + 7 q^{32} - 2 q^{33} - 10 q^{34} - q^{35} + 6 q^{36} - 10 q^{37} - 20 q^{38} - 3 q^{39} + 3 q^{40} + 16 q^{41} + 9 q^{42} - 26 q^{43} - 12 q^{44} + 10 q^{45} + 8 q^{46} + 24 q^{47} + 6 q^{48} + 2 q^{49} + q^{50} - 12 q^{51} + 9 q^{52} + 8 q^{53} + q^{54} - 2 q^{55} + 19 q^{56} + 16 q^{57} + 16 q^{58} + 14 q^{59} + 6 q^{60} + 14 q^{61} + 13 q^{62} + 2 q^{63} - 26 q^{64} - 3 q^{65} + 8 q^{66} + 5 q^{67} + 36 q^{68} + 4 q^{69} + 9 q^{70} + 14 q^{71} + 3 q^{72} - 10 q^{73} - 29 q^{74} + 10 q^{75} + 4 q^{76} - 10 q^{77} + 5 q^{78} + 3 q^{79} + 6 q^{80} - 5 q^{81} - 36 q^{82} - 6 q^{83} + 34 q^{84} - 12 q^{85} + q^{86} + 4 q^{87} - 72 q^{88} + 30 q^{89} + q^{90} + 9 q^{91} + 24 q^{92} - 3 q^{93} + 28 q^{94} + 16 q^{95} + 7 q^{96} - 2 q^{97} + 20 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 2x^{7} + 35x^{6} - 15x^{5} + 45x^{4} + 8x^{3} + 29x^{2} - 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2479 \nu^{9} + 7794 \nu^{8} - 27712 \nu^{7} + 41790 \nu^{6} - 123838 \nu^{5} + 201778 \nu^{4} + \cdots + 48076 ) / 249923 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5315 \nu^{9} + 10359 \nu^{8} - 36832 \nu^{7} + 37073 \nu^{6} - 164593 \nu^{5} + 268183 \nu^{4} + \cdots + 747290 ) / 249923 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12395 \nu^{9} - 38970 \nu^{8} + 138560 \nu^{7} - 208950 \nu^{6} + 619190 \nu^{5} - 1008890 \nu^{4} + \cdots - 240380 ) / 249923 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28595 \nu^{9} + 66312 \nu^{8} - 235776 \nu^{7} + 322653 \nu^{6} - 1053624 \nu^{5} + \cdots + 626635 ) / 249923 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48076 \nu^{9} - 45597 \nu^{8} + 328738 \nu^{7} - 68440 \nu^{6} + 1640870 \nu^{5} - 597302 \nu^{4} + \cdots + 36389 ) / 249923 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 74621 \nu^{9} - 39026 \nu^{8} + 499759 \nu^{7} + 69917 \nu^{6} + 2647233 \nu^{5} - 38417 \nu^{4} + \cdots + 65698 ) / 249923 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 81192 \nu^{9} + 60391 \nu^{8} - 520185 \nu^{7} - 8848 \nu^{6} - 2597930 \nu^{5} + 452687 \nu^{4} + \cdots + 240079 ) / 249923 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 144228 \nu^{9} + 136791 \nu^{8} - 986214 \nu^{7} + 205320 \nu^{6} - 4922610 \nu^{5} + \cdots + 640602 ) / 249923 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{6} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{9} + \beta_{8} - \beta_{7} - 14\beta_{6} - \beta_{4} + 5\beta_{3} - \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{9} + 7\beta_{8} - \beta_{7} - 9\beta_{6} + \beta_{5} + 20\beta_{2} - 20\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{5} + 9\beta_{4} - 26\beta_{3} + 20\beta_{2} + 65 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55\beta_{9} - 42\beta_{8} + 9\beta_{7} + 70\beta_{6} + 42\beta_{4} - 13\beta_{3} + 42\beta_{2} + 107\beta _1 - 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 204\beta_{9} - 64\beta_{8} + 42\beta_{7} + 409\beta_{6} - 42\beta_{5} - 92\beta_{2} + 92\beta _1 - 409 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -64\beta_{5} - 246\beta_{4} + 114\beta_{3} - 837\beta_{2} - 256 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\) \(371\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−1.06555 − 1.84559i
−0.405892 − 0.703025i
0.0905354 + 0.156812i
0.646561 + 1.11988i
1.23435 + 2.13795i
−1.06555 + 1.84559i
−0.405892 + 0.703025i
0.0905354 − 0.156812i
0.646561 − 1.11988i
1.23435 − 2.13795i
−1.06555 + 1.84559i −0.500000 − 0.866025i −1.27080 − 2.20109i −0.500000 − 0.866025i 2.13110 0.330932 + 0.573191i 1.15422 −0.500000 + 0.866025i 2.13110
121.2 −0.405892 + 0.703025i −0.500000 − 0.866025i 0.670504 + 1.16135i −0.500000 − 0.866025i 0.811783 −0.710036 − 1.22982i −2.71217 −0.500000 + 0.866025i 0.811783
121.3 0.0905354 − 0.156812i −0.500000 − 0.866025i 0.983607 + 1.70366i −0.500000 − 0.866025i −0.181071 2.17082 + 3.75996i 0.718346 −0.500000 + 0.866025i −0.181071
121.4 0.646561 − 1.11988i −0.500000 − 0.866025i 0.163918 + 0.283914i −0.500000 − 0.866025i −1.29312 −0.759900 − 1.31619i 3.01018 −0.500000 + 0.866025i −1.29312
121.5 1.23435 − 2.13795i −0.500000 − 0.866025i −2.04723 − 3.54590i −0.500000 − 0.866025i −2.46869 −1.53181 − 2.65317i −5.17056 −0.500000 + 0.866025i −2.46869
211.1 −1.06555 − 1.84559i −0.500000 + 0.866025i −1.27080 + 2.20109i −0.500000 + 0.866025i 2.13110 0.330932 − 0.573191i 1.15422 −0.500000 − 0.866025i 2.13110
211.2 −0.405892 − 0.703025i −0.500000 + 0.866025i 0.670504 − 1.16135i −0.500000 + 0.866025i 0.811783 −0.710036 + 1.22982i −2.71217 −0.500000 − 0.866025i 0.811783
211.3 0.0905354 + 0.156812i −0.500000 + 0.866025i 0.983607 − 1.70366i −0.500000 + 0.866025i −0.181071 2.17082 − 3.75996i 0.718346 −0.500000 − 0.866025i −0.181071
211.4 0.646561 + 1.11988i −0.500000 + 0.866025i 0.163918 − 0.283914i −0.500000 + 0.866025i −1.29312 −0.759900 + 1.31619i 3.01018 −0.500000 − 0.866025i −1.29312
211.5 1.23435 + 2.13795i −0.500000 + 0.866025i −2.04723 + 3.54590i −0.500000 + 0.866025i −2.46869 −1.53181 + 2.65317i −5.17056 −0.500000 − 0.866025i −2.46869
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.i.d ✓ 10
37.c even 3 1 inner 555.2.i.d ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.i.d ✓ 10 1.a even 1 1 trivial
555.2.i.d ✓ 10 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - T_{2}^{9} + 7T_{2}^{8} - 2T_{2}^{7} + 35T_{2}^{6} - 15T_{2}^{5} + 45T_{2}^{4} + 8T_{2}^{3} + 29T_{2}^{2} - 5T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( (T^{5} - 2 T^{4} - 18 T^{3} + \cdots - 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 3 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} - 6 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{10} - 16 T^{9} + \cdots + 1290496 \) Copy content Toggle raw display
$23$ \( (T^{5} + 4 T^{4} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 4 T^{4} + \cdots - 448)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 3 T^{4} - 22 T^{3} + \cdots - 71)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 10 T^{9} + \cdots + 69343957 \) Copy content Toggle raw display
$41$ \( T^{10} - 16 T^{9} + \cdots + 2085136 \) Copy content Toggle raw display
$43$ \( (T^{5} + 13 T^{4} + \cdots - 1161)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} - 12 T^{4} + \cdots + 2476)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 114233344 \) Copy content Toggle raw display
$59$ \( T^{10} - 14 T^{9} + \cdots + 698896 \) Copy content Toggle raw display
$61$ \( T^{10} - 14 T^{9} + \cdots + 58247424 \) Copy content Toggle raw display
$67$ \( T^{10} - 5 T^{9} + \cdots + 43335889 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 748569600 \) Copy content Toggle raw display
$73$ \( (T^{5} + 5 T^{4} + \cdots - 69089)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 3 T^{9} + \cdots + 40030929 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 280093696 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 978688656 \) Copy content Toggle raw display
$97$ \( (T^{5} + T^{4} - 228 T^{3} + \cdots - 5597)^{2} \) Copy content Toggle raw display
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