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Properties

Label 435.2.c.e
Level $435$
Weight $2$
Character orbit 435.c
Analytic conductor $3.473$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(349,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.3899266318336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} + 6x^{7} + 19x^{6} - 12x^{5} + 4x^{4} + 2x^{3} + 9x^{2} - 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4}) q^{2} - \beta_{6} q^{3} + (\beta_{5} - \beta_{2} - 1) q^{4} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{5} + (\beta_{2} + 1) q^{6} + (\beta_{7} + 2 \beta_{6}) q^{7}+ \cdots + ( - \beta_{9} + \beta_{8} - \beta_{5} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9} + 4 q^{10} + 24 q^{11} - 12 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{19} + 8 q^{20} + 16 q^{21} - 18 q^{24} + 2 q^{25} - 10 q^{29} - 18 q^{30} + 4 q^{31} - 8 q^{34} + 2 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} + 6x^{7} + 19x^{6} - 12x^{5} + 4x^{4} + 2x^{3} + 9x^{2} - 6x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4381 \nu^{9} + 19149 \nu^{8} - 24628 \nu^{7} - 15014 \nu^{6} - 7864 \nu^{5} + 263961 \nu^{4} + \cdots - 8160 ) / 57533 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4394 \nu^{9} - 9304 \nu^{8} + 11897 \nu^{7} + 22741 \nu^{6} + 82151 \nu^{5} - 49465 \nu^{4} + \cdots - 25960 ) / 57533 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4908 \nu^{9} + 9502 \nu^{8} - 13053 \nu^{7} - 14036 \nu^{6} - 116141 \nu^{5} + 39251 \nu^{4} + \cdots + 20774 ) / 57533 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9291 \nu^{9} + 31562 \nu^{8} - 40148 \nu^{7} - 39090 \nu^{6} - 86752 \nu^{5} + 380853 \nu^{4} + \cdots + 124078 ) / 57533 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12980 \nu^{9} - 21566 \nu^{8} + 16656 \nu^{7} + 89777 \nu^{6} + 269361 \nu^{5} - 73609 \nu^{4} + \cdots - 38951 ) / 57533 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35262 \nu^{9} - 61938 \nu^{8} + 58262 \nu^{7} + 216331 \nu^{6} + 737014 \nu^{5} - 235934 \nu^{4} + \cdots - 124636 ) / 57533 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49059 \nu^{9} + 93221 \nu^{8} - 75860 \nu^{7} - 322605 \nu^{6} - 948470 \nu^{5} + 584107 \nu^{4} + \cdots + 303094 ) / 57533 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49338 \nu^{9} + 67808 \nu^{8} - 46042 \nu^{7} - 342393 \nu^{6} - 1131012 \nu^{5} - 62807 \nu^{4} + \cdots - 26786 ) / 57533 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} - 2\beta_{7} + 2\beta_{5} - \beta_{4} + 7\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} + 8\beta_{5} + 10\beta_{3} - 7\beta_{2} - 10\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{9} + 16\beta_{7} - 5\beta_{6} + 16\beta_{5} + 10\beta_{4} - 10\beta_{2} - 48\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{9} + 10\beta_{8} + 61\beta_{7} - 44\beta_{6} + 48\beta_{4} - 82\beta_{3} - 82\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 48\beta_{8} + 120\beta_{7} - 55\beta_{6} - 120\beta_{5} + 82\beta_{4} - 345\beta_{3} + 82\beta_{2} + 55 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -82\beta_{9} + 82\beta_{8} - 461\beta_{5} - 636\beta_{3} + 345\beta_{2} + 636\beta _1 + 286 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -345\beta_{9} - 899\beta_{7} + 466\beta_{6} - 899\beta_{5} - 636\beta_{4} + 636\beta_{2} + 2545\beta _1 + 466 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/435\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(146\) \(262\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
−1.20964 + 1.20964i
1.93833 − 1.93833i
−0.604479 − 0.604479i
0.313948 − 0.313948i
0.561843 + 0.561843i
0.561843 − 0.561843i
0.313948 + 0.313948i
−0.604479 + 0.604479i
1.93833 + 1.93833i
−1.20964 − 1.20964i
− 2.51908i 1.00000i −4.34577 −1.27413 + 1.83755i 2.51908 − 0.173311i 5.90919i −1.00000 4.62893 + 3.20964i
349.2 − 2.15351i 1.00000i −2.63760 −0.0286357 − 2.23588i 2.15351 − 1.51591i 1.37308i −1.00000 −4.81500 + 0.0616673i
349.3 − 1.71457i − 1.00000i −0.939748 1.51903 + 1.64090i −1.71457 − 0.654317i − 1.81788i −1.00000 2.81344 − 2.60448i
349.4 − 0.754474i 1.00000i 1.43077 −2.23474 + 0.0770824i 0.754474 − 4.18524i − 2.58843i −1.00000 0.0581566 + 1.68605i
349.5 − 0.712495i − 1.00000i 1.49235 2.01848 − 0.962154i −0.712495 2.77986i − 2.48828i −1.00000 −0.685530 − 1.43816i
349.6 0.712495i 1.00000i 1.49235 2.01848 + 0.962154i −0.712495 − 2.77986i 2.48828i −1.00000 −0.685530 + 1.43816i
349.7 0.754474i − 1.00000i 1.43077 −2.23474 − 0.0770824i 0.754474 4.18524i 2.58843i −1.00000 0.0581566 − 1.68605i
349.8 1.71457i 1.00000i −0.939748 1.51903 − 1.64090i −1.71457 0.654317i 1.81788i −1.00000 2.81344 + 2.60448i
349.9 2.15351i − 1.00000i −2.63760 −0.0286357 + 2.23588i 2.15351 1.51591i − 1.37308i −1.00000 −4.81500 − 0.0616673i
349.10 2.51908i − 1.00000i −4.34577 −1.27413 − 1.83755i 2.51908 0.173311i − 5.90919i −1.00000 4.62893 − 3.20964i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.10
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 435.2.c.e ✓ 10
3.b odd 2 1 1305.2.c.j 10
5.b even 2 1 inner 435.2.c.e ✓ 10
5.c odd 4 1 2175.2.a.w 5
5.c odd 4 1 2175.2.a.z 5
15.d odd 2 1 1305.2.c.j 10
15.e even 4 1 6525.2.a.bl 5
15.e even 4 1 6525.2.a.bs 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.e ✓ 10 1.a even 1 1 trivial
435.2.c.e ✓ 10 5.b even 2 1 inner
1305.2.c.j 10 3.b odd 2 1
1305.2.c.j 10 15.d odd 2 1
2175.2.a.w 5 5.c odd 4 1
2175.2.a.z 5 5.c odd 4 1
6525.2.a.bl 5 15.e even 4 1
6525.2.a.bs 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(435, [\chi])\):

\( T_{2}^{10} + 15T_{2}^{8} + 77T_{2}^{6} + 157T_{2}^{4} + 111T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{10} + 28T_{7}^{8} + 206T_{7}^{6} + 400T_{7}^{4} + 145T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 15 T^{8} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} - T^{8} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 28 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{5} - 12 T^{4} + 48 T^{3} + \cdots - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 16 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{10} + 56 T^{8} + \cdots + 88804 \) Copy content Toggle raw display
$19$ \( (T^{5} - 2 T^{4} + \cdots + 304)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 76 T^{8} + \cdots + 1600 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} - 2 T^{4} + \cdots - 6304)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 180 T^{8} + \cdots + 341056 \) Copy content Toggle raw display
$41$ \( (T^{5} + 14 T^{4} + \cdots + 6176)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 292 T^{8} + \cdots + 46895104 \) Copy content Toggle raw display
$47$ \( T^{10} + 220 T^{8} + \cdots + 7246864 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 226683136 \) Copy content Toggle raw display
$59$ \( (T^{5} + 4 T^{4} + \cdots - 2000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 12 T^{4} + \cdots + 6872)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 228 T^{8} + \cdots + 1716100 \) Copy content Toggle raw display
$71$ \( (T^{5} - 30 T^{4} + \cdots + 6592)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 420 T^{8} + \cdots + 11343424 \) Copy content Toggle raw display
$79$ \( (T^{5} - 18 T^{4} + \cdots - 52048)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 216 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$89$ \( (T^{5} - 22 T^{4} + \cdots - 40682)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 112 T^{8} + \cdots + 107584 \) Copy content Toggle raw display
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