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Properties

Label 1550.3.c.b
Level $1550$
Weight $3$
Character orbit 1550.c
Analytic conductor $42.234$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1550,3,Mod(1301,1550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1550.1301");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1550.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: \(42.2344409758\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 114 x^{18} + 5385 x^{16} + 136558 x^{14} + 2010600 x^{12} + 17320280 x^{10} + 83177711 x^{8} + \cdots + 8649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_1 q^{3} + 2 q^{4} + \beta_{4} q^{6} + (\beta_{8} - 1) q^{7} + 2 \beta_{3} q^{8} + (\beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_1 q^{3} + 2 q^{4} + \beta_{4} q^{6} + (\beta_{8} - 1) q^{7} + 2 \beta_{3} q^{8} + (\beta_{2} - 2) q^{9} + ( - \beta_{16} + \beta_1) q^{11} + 2 \beta_1 q^{12} - \beta_{12} q^{13} + ( - \beta_{11} - \beta_{3}) q^{14} + 4 q^{16} + (\beta_{17} + \beta_{16} + \cdots + \beta_{4}) q^{17}+ \cdots + ( - 5 \beta_{19} + \beta_{18} + \cdots - 18 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 40 q^{4} - 24 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 40 q^{4} - 24 q^{7} - 48 q^{9} + 8 q^{14} + 80 q^{16} + 16 q^{18} + 24 q^{19} - 48 q^{28} + 8 q^{31} - 124 q^{33} - 96 q^{36} + 104 q^{38} - 68 q^{39} - 80 q^{41} - 96 q^{47} + 12 q^{49} + 116 q^{51} + 16 q^{56} - 200 q^{59} - 40 q^{62} + 248 q^{63} + 160 q^{64} + 128 q^{66} + 396 q^{67} + 232 q^{69} - 240 q^{71} + 32 q^{72} + 48 q^{76} - 48 q^{78} - 84 q^{81} + 112 q^{82} - 212 q^{87} - 188 q^{93} - 48 q^{94} - 196 q^{97} - 80 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 114 x^{18} + 5385 x^{16} + 136558 x^{14} + 2010600 x^{12} + 17320280 x^{10} + 83177711 x^{8} + \cdots + 8649 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 188016873065 \nu^{18} + 21600091315746 \nu^{16} + \cdots + 16\!\cdots\!63 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 188016873065 \nu^{19} + 21600091315746 \nu^{17} + \cdots + 16\!\cdots\!63 \nu ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!35 \nu^{18} + \cdots - 24\!\cdots\!09 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2234353390051 \nu^{18} + 253222487626992 \nu^{16} + \cdots + 17\!\cdots\!38 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73\!\cdots\!82 \nu^{18} + \cdots - 29\!\cdots\!51 ) / 35\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!33 \nu^{18} + \cdots - 87\!\cdots\!71 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!21 \nu^{18} + \cdots + 40\!\cdots\!96 ) / 35\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!33 \nu^{18} + \cdots - 96\!\cdots\!99 ) / 35\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 47\!\cdots\!05 \nu^{18} + \cdots + 12\!\cdots\!32 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 55\!\cdots\!29 \nu^{19} + \cdots - 10\!\cdots\!86 \nu ) / 70\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 58\!\cdots\!25 \nu^{19} + \cdots - 93\!\cdots\!07 \nu ) / 35\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 94\!\cdots\!54 \nu^{19} + \cdots + 22\!\cdots\!00 \nu ) / 35\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 37\!\cdots\!00 \nu^{19} + \cdots - 92\!\cdots\!11 \nu ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 21\!\cdots\!31 \nu^{19} + \cdots + 49\!\cdots\!40 \nu ) / 58\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 18\!\cdots\!15 \nu^{19} + \cdots - 42\!\cdots\!84 \nu ) / 35\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 61\!\cdots\!11 \nu^{19} + \cdots - 14\!\cdots\!42 \nu ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 88\!\cdots\!54 \nu^{19} + \cdots + 21\!\cdots\!56 \nu ) / 11\!\cdots\!70 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} + \beta_{17} + 3\beta_{16} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{4} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + \beta_{10} + 2\beta_{9} - 2\beta_{8} + \beta_{7} + 3\beta_{6} + \beta_{5} - 5\beta_{3} - 25\beta_{2} + 213 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 34 \beta_{19} - 45 \beta_{17} - 112 \beta_{16} + \beta_{15} - 39 \beta_{14} - 31 \beta_{13} + \cdots + 478 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 63 \beta_{11} - 48 \beta_{10} - 83 \beta_{9} + 68 \beta_{8} - 43 \beta_{7} - 129 \beta_{6} + \cdots - 4717 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1006 \beta_{19} - 83 \beta_{18} + 1566 \beta_{17} + 3470 \beta_{16} - 25 \beta_{15} + \cdots - 11557 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1647 \beta_{11} + 1642 \beta_{10} + 2756 \beta_{9} - 1888 \beta_{8} + 1472 \beta_{7} + 4517 \beta_{6} + \cdots + 112575 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 28974 \beta_{19} + 5187 \beta_{18} - 49889 \beta_{17} - 101929 \beta_{16} + 650 \beta_{15} + \cdots + 291870 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 41221 \beta_{11} - 49935 \beta_{10} - 84746 \beta_{9} + 49604 \beta_{8} - 46813 \beta_{7} + \cdots - 2823052 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 829011 \beta_{19} - 219855 \beta_{18} + 1525253 \beta_{17} + 2938042 \beta_{16} + \cdots - 7617170 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1030208 \beta_{11} + 1446918 \beta_{10} + 2515901 \beta_{9} - 1293484 \beta_{8} + 1438899 \beta_{7} + \cdots + 73381991 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 23680048 \beta_{19} + 7933668 \beta_{18} - 45612043 \beta_{17} - 84120399 \beta_{16} + \cdots + 203860016 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 26131460 \beta_{11} - 41099185 \beta_{10} - 73375716 \beta_{9} + 34108912 \beta_{8} + \cdots - 1959733382 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 676374820 \beta_{19} - 263314820 \beta_{18} + 1346691925 \beta_{17} + 2404266384 \beta_{16} + \cdots - 5562193727 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 676595031 \beta_{11} + 1159326890 \beta_{10} + 2120606259 \beta_{9} - 915199314 \beta_{8} + \cdots + 53417648780 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 19333240503 \beta_{19} + 8326301777 \beta_{18} - 39455513827 \beta_{17} - 68740981695 \beta_{16} + \cdots + 154000719754 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 17892995777 \beta_{11} - 32674909163 \beta_{10} - 61016832956 \beta_{9} + 24991073342 \beta_{8} + \cdots - 1478573944886 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 553223293032 \beta_{19} - 255550013573 \beta_{18} + 1150443662310 \beta_{17} + \cdots - 4311086898318 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1550\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(1427\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
− 5.37481i
− 4.17035i
− 2.59623i
− 2.57015i
− 0.449306i
0.449306i
2.57015i
2.59623i
4.17035i
5.37481i
− 4.66268i
− 4.44783i
− 3.40711i
− 1.01932i
− 0.0192141i
0.0192141i
1.01932i
3.40711i
4.44783i
4.66268i
−1.41421 − 5.37481i 2.00000 0 7.60114i −2.30213 −2.82843 −19.8886 0
1301.2 −1.41421 − 4.17035i 2.00000 0 5.89776i −1.05083 −2.82843 −8.39181 0
1301.3 −1.41421 − 2.59623i 2.00000 0 3.67163i 8.26552 −2.82843 2.25958 0
1301.4 −1.41421 − 2.57015i 2.00000 0 3.63475i −13.6281 −2.82843 2.39431 0
1301.5 −1.41421 − 0.449306i 2.00000 0 0.635415i 1.30135 −2.82843 8.79812 0
1301.6 −1.41421 0.449306i 2.00000 0 − 0.635415i 1.30135 −2.82843 8.79812 0
1301.7 −1.41421 2.57015i 2.00000 0 − 3.63475i −13.6281 −2.82843 2.39431 0
1301.8 −1.41421 2.59623i 2.00000 0 − 3.67163i 8.26552 −2.82843 2.25958 0
1301.9 −1.41421 4.17035i 2.00000 0 − 5.89776i −1.05083 −2.82843 −8.39181 0
1301.10 −1.41421 5.37481i 2.00000 0 − 7.60114i −2.30213 −2.82843 −19.8886 0
1301.11 1.41421 − 4.66268i 2.00000 0 − 6.59403i 0.654661 2.82843 −12.7406 0
1301.12 1.41421 − 4.44783i 2.00000 0 − 6.29018i −8.62322 2.82843 −10.7832 0
1301.13 1.41421 − 3.40711i 2.00000 0 − 4.81838i 2.92414 2.82843 −2.60838 0
1301.14 1.41421 − 1.01932i 2.00000 0 − 1.44154i 8.90150 2.82843 7.96098 0
1301.15 1.41421 − 0.0192141i 2.00000 0 − 0.0271728i −8.44287 2.82843 8.99963 0
1301.16 1.41421 0.0192141i 2.00000 0 0.0271728i −8.44287 2.82843 8.99963 0
1301.17 1.41421 1.01932i 2.00000 0 1.44154i 8.90150 2.82843 7.96098 0
1301.18 1.41421 3.40711i 2.00000 0 4.81838i 2.92414 2.82843 −2.60838 0
1301.19 1.41421 4.44783i 2.00000 0 6.29018i −8.62322 2.82843 −10.7832 0
1301.20 1.41421 4.66268i 2.00000 0 6.59403i 0.654661 2.82843 −12.7406 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1301.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1550.3.c.b ✓ 20
5.b even 2 1 1550.3.c.c yes 20
5.c odd 4 2 1550.3.d.b 40
31.b odd 2 1 inner 1550.3.c.b ✓ 20
155.c odd 2 1 1550.3.c.c yes 20
155.f even 4 2 1550.3.d.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1550.3.c.b ✓ 20 1.a even 1 1 trivial
1550.3.c.b ✓ 20 31.b odd 2 1 inner
1550.3.c.c yes 20 5.b even 2 1
1550.3.c.c yes 20 155.c odd 2 1
1550.3.d.b 40 5.c odd 4 2
1550.3.d.b 40 155.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1550, [\chi])\):

\( T_{3}^{20} + 114 T_{3}^{18} + 5385 T_{3}^{16} + 136558 T_{3}^{14} + 2010600 T_{3}^{12} + 17320280 T_{3}^{10} + \cdots + 8649 \) Copy content Toggle raw display
\( T_{7}^{10} + 12 T_{7}^{9} - 176 T_{7}^{8} - 1858 T_{7}^{7} + 9681 T_{7}^{6} + 78336 T_{7}^{5} + \cdots - 439943 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + 114 T^{18} + \cdots + 8649 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + 12 T^{9} + \cdots - 439943)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 73\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 72\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 63\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 480348657407)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 16\!\cdots\!29 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 22\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 39\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 799695379111041)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 88\!\cdots\!09 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 102555165223161)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 39\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 51\!\cdots\!75)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 21\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 13\!\cdots\!29)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 42\!\cdots\!47)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 28\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 53\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 55\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 80\!\cdots\!83)^{2} \) Copy content Toggle raw display
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