↑

Properties

Label 1050.3.f.d
Level $1050$
Weight $3$
Character orbit 1050.f
Analytic conductor $28.610$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,3,Mod(601,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1050.601");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.f (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: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 103 x^{10} + 14 x^{9} + 7600 x^{8} + 1474 x^{7} + 229961 x^{6} + 442438 x^{5} + \cdots + 69739201 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{2} q^{3} + 2 q^{4} - \beta_{4} q^{6} + ( - \beta_{3} + 1) q^{7} - 2 \beta_{5} q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{2} q^{3} + 2 q^{4} - \beta_{4} q^{6} + ( - \beta_{3} + 1) q^{7} - 2 \beta_{5} q^{8} - 3 q^{9} + (\beta_{8} + \beta_{6} - 2 \beta_{5} - 1) q^{11} + 2 \beta_{2} q^{12} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{4}) q^{13}+ \cdots + ( - 3 \beta_{8} - 3 \beta_{6} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{4} + 10 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{4} + 10 q^{7} - 36 q^{9} - 16 q^{11} + 8 q^{14} + 48 q^{16} - 6 q^{21} + 48 q^{22} + 20 q^{28} + 48 q^{29} - 72 q^{36} - 64 q^{37} - 12 q^{39} - 24 q^{42} + 108 q^{43} - 32 q^{44} + 80 q^{46} + 118 q^{49} + 72 q^{51} - 176 q^{53} + 16 q^{56} + 132 q^{57} - 128 q^{58} - 30 q^{63} + 96 q^{64} + 4 q^{67} + 248 q^{71} - 64 q^{74} + 396 q^{77} - 48 q^{78} - 208 q^{79} + 108 q^{81} - 12 q^{84} + 128 q^{86} + 96 q^{88} - 158 q^{91} - 252 q^{93} + 240 q^{98} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 103 x^{10} + 14 x^{9} + 7600 x^{8} + 1474 x^{7} + 229961 x^{6} + 442438 x^{5} + \cdots + 69739201 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!17 \nu^{11} + \cdots - 43\!\cdots\!34 ) / 25\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!88 \nu^{11} + \cdots - 18\!\cdots\!95 ) / 15\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 87\!\cdots\!62 \nu^{11} + \cdots + 15\!\cdots\!05 ) / 30\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 958718783 \nu^{11} + 2012568579 \nu^{10} - 103184081077 \nu^{9} - 19016500460 \nu^{8} + \cdots + 22\!\cdots\!16 ) / 27\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 397109320631 \nu^{11} + 851720091837 \nu^{10} - 39749913778739 \nu^{9} + \cdots + 56\!\cdots\!78 ) / 90\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!32 \nu^{11} + \cdots - 82\!\cdots\!47 ) / 30\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!44 \nu^{11} + \cdots + 51\!\cdots\!90 ) / 60\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 40\!\cdots\!30 \nu^{11} + \cdots - 30\!\cdots\!29 ) / 30\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49\!\cdots\!59 \nu^{11} + \cdots - 12\!\cdots\!18 ) / 30\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!74 \nu^{11} + \cdots - 37\!\cdots\!69 ) / 30\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13\!\cdots\!53 \nu^{11} + \cdots + 13\!\cdots\!07 ) / 30\!\cdots\!30 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{5} - \beta_{3} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{11} + \beta_{10} - \beta_{9} - 4 \beta_{7} + \beta_{6} - 10 \beta_{5} - 9 \beta_{4} + 5 \beta_{3} + \cdots - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{11} - 6\beta_{10} - 8\beta_{8} + 51\beta_{6} - 7\beta_{5} + 59\beta_{3} + 77\beta _1 - 133 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 78 \beta_{11} - 234 \beta_{10} + 78 \beta_{9} - 280 \beta_{8} + 240 \beta_{7} + 78 \beta_{6} + \cdots - 1845 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3935 \beta_{11} - 2943 \beta_{10} - 1951 \beta_{9} - 2484 \beta_{8} - 972 \beta_{7} - 2943 \beta_{6} + \cdots + 10251 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5557 \beta_{11} + 5557 \beta_{10} + 8488 \beta_{8} - 5413 \beta_{6} + 68046 \beta_{5} - 13901 \beta_{3} + \cdots + 113528 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 183289 \beta_{11} + 264461 \beta_{10} + 102117 \beta_{9} + 316554 \beta_{8} + 70042 \beta_{7} + \cdots + 749771 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1153764 \beta_{11} + 379548 \beta_{10} - 394668 \beta_{9} + 341320 \beta_{8} - 797976 \beta_{7} + \cdots - 7331813 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3248552 \beta_{11} - 3248552 \beta_{10} - 4810800 \beta_{8} + 11912105 \beta_{6} - 27500585 \beta_{5} + \cdots - 55482497 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 27264133 \beta_{11} - 80953743 \beta_{10} + 26425477 \beta_{9} - 107767100 \beta_{8} + \cdots - 486145062 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1301713519 \beta_{11} - 794670131 \beta_{10} - 287626743 \beta_{9} - 1011637682 \beta_{8} + \cdots + 4149103773 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
0.854122 + 1.47938i
4.23503 + 7.33529i
−3.88205 − 6.72390i
0.854122 − 1.47938i
4.23503 − 7.33529i
−3.88205 + 6.72390i
−1.37625 − 2.38373i
−2.07881 − 3.60061i
3.24795 + 5.62562i
−1.37625 + 2.38373i
−2.07881 + 3.60061i
3.24795 − 5.62562i
−1.41421 − 1.73205i 2.00000 0 2.44949i −6.06258 + 3.49930i −2.82843 −3.00000 0
601.2 −1.41421 − 1.73205i 2.00000 0 2.44949i 2.03595 − 6.69738i −2.82843 −3.00000 0
601.3 −1.41421 − 1.73205i 2.00000 0 2.44949i 5.11242 + 4.78154i −2.82843 −3.00000 0
601.4 −1.41421 1.73205i 2.00000 0 − 2.44949i −6.06258 − 3.49930i −2.82843 −3.00000 0
601.5 −1.41421 1.73205i 2.00000 0 − 2.44949i 2.03595 + 6.69738i −2.82843 −3.00000 0
601.6 −1.41421 1.73205i 2.00000 0 − 2.44949i 5.11242 − 4.78154i −2.82843 −3.00000 0
601.7 1.41421 − 1.73205i 2.00000 0 − 2.44949i −6.84490 + 1.46536i 2.82843 −3.00000 0
601.8 1.41421 − 1.73205i 2.00000 0 − 2.44949i 3.84151 − 5.85173i 2.82843 −3.00000 0
601.9 1.41421 − 1.73205i 2.00000 0 − 2.44949i 6.91761 + 1.07086i 2.82843 −3.00000 0
601.10 1.41421 1.73205i 2.00000 0 2.44949i −6.84490 − 1.46536i 2.82843 −3.00000 0
601.11 1.41421 1.73205i 2.00000 0 2.44949i 3.84151 + 5.85173i 2.82843 −3.00000 0
601.12 1.41421 1.73205i 2.00000 0 2.44949i 6.91761 − 1.07086i 2.82843 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.f.d yes 12
5.b even 2 1 1050.3.f.c ✓ 12
5.c odd 4 2 1050.3.h.c 24
7.b odd 2 1 inner 1050.3.f.d yes 12
35.c odd 2 1 1050.3.f.c ✓ 12
35.f even 4 2 1050.3.h.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.f.c ✓ 12 5.b even 2 1
1050.3.f.c ✓ 12 35.c odd 2 1
1050.3.f.d yes 12 1.a even 1 1 trivial
1050.3.f.d yes 12 7.b odd 2 1 inner
1050.3.h.c 24 5.c odd 4 2
1050.3.h.c 24 35.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 \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{6} + 8T_{11}^{5} - 462T_{11}^{4} - 4780T_{11}^{3} + 27217T_{11}^{2} + 422940T_{11} + 1141308 \) Copy content Toggle raw display
\( T_{23}^{6} - 1294T_{23}^{4} - 7740T_{23}^{3} + 320689T_{23}^{2} + 1942260T_{23} - 14928228 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} + 8 T^{5} + \cdots + 1141308)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 6029833104 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 33708960000 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 7157315699856 \) Copy content Toggle raw display
$23$ \( (T^{6} - 1294 T^{4} + \cdots - 14928228)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 24 T^{5} + \cdots - 186322500)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 50797435981824 \) Copy content Toggle raw display
$37$ \( (T^{6} + 32 T^{5} + \cdots + 348454564)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{6} - 54 T^{5} + \cdots + 822485377)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} + 88 T^{5} + \cdots + 2495949696)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T^{6} - 2 T^{5} + \cdots - 372707559)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 124 T^{5} + \cdots - 37762911432)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{6} + 104 T^{5} + \cdots - 128832956)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
show more
show less