Properties

Label 555.2.c.b
Level $555$
Weight $2$
Character orbit 555.c
Analytic conductor $4.432$
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] = [555,2,Mod(334,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.334");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.c (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.309760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{7} + 44\nu^{6} - 130\nu^{5} + 211\nu^{4} - 186\nu^{3} + 50\nu^{2} + 26\nu + 403 ) / 245 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{7} + 13\nu^{6} - 152\nu^{5} + 451\nu^{4} - 222\nu^{3} - 32\nu^{2} + \nu + 530 ) / 245 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{7} - 47\nu^{6} + 101\nu^{5} - 55\nu^{4} + 143\nu^{3} - 69\nu^{2} - 30\nu + 319 ) / 245 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -61\nu^{7} + 262\nu^{6} - 538\nu^{5} + 194\nu^{4} + 162\nu^{3} - 23\nu^{2} - 941\nu - 155 ) / 245 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 74\nu^{7} - 309\nu^{6} + 639\nu^{5} - 249\nu^{4} - 19\nu^{3} - 291\nu^{2} + 1401\nu + 229 ) / 245 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 87\nu^{7} - 356\nu^{6} + 740\nu^{5} - 304\nu^{4} + 124\nu^{3} - 360\nu^{2} + 1861\nu + 303 ) / 245 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 178\nu^{7} - 734\nu^{6} + 1545\nu^{5} - 591\nu^{4} - 149\nu^{3} - 10\nu^{2} + 3219\nu + 527 ) / 245 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 4\beta_{6} - 5\beta_{5} - 3\beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{3} - 2\beta_{2} + 7\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} - 21\beta_{6} + 24\beta_{5} + 25\beta_{4} + 21\beta_{3} - 9\beta_{2} + 25\beta _1 - 49 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{7} - 35\beta_{6} + 39\beta_{5} + 47\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 64\beta_{7} - 121\beta_{6} + 127\beta_{5} + 168\beta_{4} - 121\beta_{3} + 64\beta_{2} - 168\beta _1 + 295 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(112\) \(371\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
334.1
1.16407 + 1.16407i
−0.164066 0.164066i
−0.748606 + 0.748606i
1.74861 1.74861i
−0.748606 0.748606i
1.74861 + 1.74861i
1.16407 1.16407i
−0.164066 + 0.164066i
1.61803i 1.00000i −0.618034 −2.14896 + 0.618034i 1.61803 0.671869i 2.23607i −1.00000 1.00000 + 3.47709i
334.2 1.61803i 1.00000i −0.618034 2.14896 + 0.618034i 1.61803 3.32813i 2.23607i −1.00000 1.00000 3.47709i
334.3 0.618034i 1.00000i 1.61803 −1.54336 + 1.61803i −0.618034 4.49721i 2.23607i −1.00000 1.00000 + 0.953850i
334.4 0.618034i 1.00000i 1.61803 1.54336 + 1.61803i −0.618034 0.497212i 2.23607i −1.00000 1.00000 0.953850i
334.5 0.618034i 1.00000i 1.61803 −1.54336 1.61803i −0.618034 4.49721i 2.23607i −1.00000 1.00000 0.953850i
334.6 0.618034i 1.00000i 1.61803 1.54336 1.61803i −0.618034 0.497212i 2.23607i −1.00000 1.00000 + 0.953850i
334.7 1.61803i 1.00000i −0.618034 −2.14896 0.618034i 1.61803 0.671869i 2.23607i −1.00000 1.00000 3.47709i
334.8 1.61803i 1.00000i −0.618034 2.14896 0.618034i 1.61803 3.32813i 2.23607i −1.00000 1.00000 + 3.47709i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 334.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.c.b 8
3.b odd 2 1 1665.2.c.d 8
5.b even 2 1 inner 555.2.c.b 8
5.c odd 4 1 2775.2.a.w 4
5.c odd 4 1 2775.2.a.y 4
15.d odd 2 1 1665.2.c.d 8
15.e even 4 1 8325.2.a.bt 4
15.e even 4 1 8325.2.a.bw 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.c.b 8 1.a even 1 1 trivial
555.2.c.b 8 5.b even 2 1 inner
1665.2.c.d 8 3.b odd 2 1
1665.2.c.d 8 15.d odd 2 1
2775.2.a.w 4 5.c odd 4 1
2775.2.a.y 4 5.c odd 4 1
8325.2.a.bt 4 15.e even 4 1
8325.2.a.bw 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{6} + 46 T^{4} - 200 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{6} + 246 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} - T^{2} - 10 T + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 32 T^{6} + 166 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{8} + 62 T^{6} + 1171 T^{4} + \cdots + 21025 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} - 11 T^{2} + 30 T - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 84 T^{6} + 1846 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$29$ \( (T^{4} - 12 T^{3} + 9 T^{2} + 112 T - 49)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} - 28 T^{2} - 14 T + 59)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 26 T^{3} + 238 T^{2} + 922 T + 1289)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 78 T^{6} + 691 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$47$ \( T^{8} + 180 T^{6} + 5878 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( (T^{4} + 68 T^{2} + 176)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 12 T^{3} - 41 T^{2} + 812 T - 1949)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} - 16 T^{2} - 90 T - 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 142 T^{6} + 6351 T^{4} + \cdots + 429025 \) Copy content Toggle raw display
$71$ \( (T^{4} + 34 T^{3} + 391 T^{2} + 1784 T + 2801)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 290 T^{6} + \cdots + 13140625 \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} - 221 T^{2} - 2440 T - 6245)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 126 T^{6} + 3391 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{4} - 24 T^{3} + 104 T^{2} + 120 T - 725)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 200 T^{6} + 12238 T^{4} + \cdots + 398161 \) Copy content Toggle raw display
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