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Properties

Label 490.2.l.c
Level $490$
Weight $2$
Character orbit 490.l
Analytic conductor $3.913$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,2,Mod(117,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.l (of order \(12\), degree \(4\), not 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: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{15} - \beta_{7}) q^{2} + ( - \beta_{13} + \beta_{4}) q^{3} + (\beta_{14} - \beta_{11}) q^{4} + (\beta_{10} - \beta_{8} + \cdots + \beta_{5}) q^{5}+ \cdots + (\beta_{15} - 2 \beta_{14} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} - \beta_{7}) q^{2} + ( - \beta_{13} + \beta_{4}) q^{3} + (\beta_{14} - \beta_{11}) q^{4} + (\beta_{10} - \beta_{8} + \cdots + \beta_{5}) q^{5}+ \cdots + ( - 3 \beta_{12} - 2 \beta_{11} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{5} + 12 q^{10} - 12 q^{11} + 16 q^{15} + 8 q^{16} + 36 q^{17} - 8 q^{18} - 8 q^{22} - 4 q^{23} + 12 q^{25} - 12 q^{26} + 20 q^{30} - 24 q^{31} - 48 q^{33} - 8 q^{36} + 4 q^{37} - 24 q^{38} - 8 q^{43} + 12 q^{45} - 8 q^{46} - 12 q^{47} - 32 q^{50} - 16 q^{51} - 28 q^{53} + 8 q^{57} - 32 q^{58} + 8 q^{60} + 12 q^{61} - 8 q^{65} + 32 q^{67} + 36 q^{68} + 16 q^{71} - 8 q^{72} + 12 q^{73} + 48 q^{75} + 16 q^{78} + 12 q^{80} + 48 q^{82} + 24 q^{85} + 12 q^{86} + 24 q^{87} - 4 q^{88} + 8 q^{92} + 28 q^{93} + 20 q^{95} - 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 171 \nu^{14} + 2802 \nu^{12} + 20266 \nu^{10} + 96110 \nu^{8} + 343988 \nu^{6} + 866714 \nu^{4} + \cdots + 1302910 ) / 1020740 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6279 \nu^{15} - 6023 \nu^{14} - 94829 \nu^{13} - 319853 \nu^{12} - 291088 \nu^{11} + \cdots - 204554077 ) / 224562800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6279 \nu^{15} - 6023 \nu^{14} + 94829 \nu^{13} - 319853 \nu^{12} + 291088 \nu^{11} + \cdots - 204554077 ) / 224562800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 657161 \nu^{14} + 6218761 \nu^{12} + 35853172 \nu^{10} + 144339958 \nu^{8} + 433352890 \nu^{6} + \cdots + 1019822349 ) / 785969800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 623003 \nu^{14} - 5366258 \nu^{12} - 30938216 \nu^{10} - 125675214 \nu^{8} + \cdots - 1273004222 ) / 392984900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4174573 \nu^{15} + 307671 \nu^{14} + 41450603 \nu^{13} + 4646621 \nu^{12} + 238976156 \nu^{11} + \cdots + 8000085009 ) / 11003577200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4174573 \nu^{15} + 307671 \nu^{14} - 41450603 \nu^{13} + 4646621 \nu^{12} + \cdots + 8000085009 ) / 11003577200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5413027 \nu^{15} - 6673919 \nu^{14} - 53207257 \nu^{13} - 67697189 \nu^{12} + \cdots - 11101746201 ) / 11003577200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5413027 \nu^{15} + 6673919 \nu^{14} - 53207257 \nu^{13} + 67697189 \nu^{12} + \cdots + 11101746201 ) / 11003577200 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26590 \nu^{15} + 257521 \nu^{13} + 1484692 \nu^{11} + 6079906 \nu^{9} + 17945290 \nu^{7} + \cdots + 42231189 \nu ) / 50016260 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 154017 \nu^{15} - 1553547 \nu^{13} - 9597064 \nu^{11} - 41588666 \nu^{9} + \cdots - 477500443 \nu ) / 239208200 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7738601 \nu^{15} - 12843957 \nu^{14} + 58163261 \nu^{13} - 106723897 \nu^{12} + \cdots - 17501784573 ) / 11003577200 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5179651 \nu^{15} - 39396031 \nu^{13} - 212402552 \nu^{11} - 802913678 \nu^{9} + \cdots - 6839548919 \nu ) / 5501788600 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 850941 \nu^{15} - 939402 \nu^{14} - 7115276 \nu^{13} - 7955037 \nu^{12} - 39448752 \nu^{11} + \cdots - 1821562113 ) / 785969800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{13} + \beta_{7} - 2\beta_{6} - \beta_{5} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{15} - 5\beta_{13} + 5\beta_{8} + 8\beta_{6} + 2\beta_{4} + 2\beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{15} + 22\beta_{14} + 14\beta_{13} + 3\beta_{12} - 3\beta_{10} - 3\beta_{9} - 14\beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{15} + 14 \beta_{13} - 14 \beta_{10} + 14 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} - 22 \beta_{6} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 64\beta_{15} - 94\beta_{14} - 64\beta_{13} + 94\beta_{11} + 64\beta_{8} + 2\beta_{4} - 2\beta_{3} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -75\beta_{15} - 75\beta_{13} + 62\beta_{10} - 62\beta_{9} - 75\beta_{7} + 112\beta_{6} - 87\beta_{5} + 112 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 75 \beta_{15} + 112 \beta_{14} + 75 \beta_{13} + 112 \beta_{12} - 385 \beta_{11} - 75 \beta_{10} + \cdots + 112 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 337\beta_{15} + 337\beta_{13} - 337\beta_{8} - 486\beta_{6} + 198\beta_{4} + 198\beta_{3} + 273\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -332\beta_{15} + 456\beta_{14} + 332\beta_{13} - 759\beta_{12} + 535\beta_{10} + 535\beta_{9} - 332\beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 332 \beta_{15} + 332 \beta_{13} - 332 \beta_{10} + 332 \beta_{9} + 2364 \beta_{8} + 2696 \beta_{7} + \cdots - 3803 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1452 \beta_{15} - 2032 \beta_{14} - 1452 \beta_{13} + 2032 \beta_{11} + 1452 \beta_{8} + \cdots - 3803 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 10647 \beta_{15} - 10647 \beta_{13} - 1244 \beta_{10} + 1244 \beta_{9} - 10647 \beta_{7} + \cdots + 15030 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 10647 \beta_{15} + 15030 \beta_{14} + 10647 \beta_{13} + 15030 \beta_{12} + 7801 \beta_{11} + \cdots + 15030 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1 + \beta_{6}\) \(\beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
0.144868 + 1.25092i
−1.01089 − 0.750919i
−1.45333 − 1.51725i
0.587308 + 2.01725i
1.45333 − 1.51725i
−0.587308 + 2.01725i
−0.144868 + 1.25092i
1.01089 − 0.750919i
1.45333 + 1.51725i
−0.587308 − 2.01725i
−0.144868 − 1.25092i
1.01089 + 0.750919i
0.144868 − 1.25092i
−1.01089 + 0.750919i
−1.45333 + 1.51725i
0.587308 − 2.01725i
−0.965926 + 0.258819i −0.523277 + 1.95290i 0.866025 − 0.500000i 2.03078 + 0.935904i − 2.02179i 0 −0.707107 + 0.707107i −0.941911 − 0.543813i −2.20382 − 0.378409i
117.2 −0.965926 + 0.258819i 0.0749894 − 0.279864i 0.866025 − 0.500000i −2.20382 + 0.378409i 0.289737i 0 −0.707107 + 0.707107i 2.52538 + 1.45803i 2.03078 − 0.935904i
117.3 0.965926 − 0.258819i −0.304013 + 1.13459i 0.866025 − 0.500000i 1.79038 − 1.33961i 1.17462i 0 0.707107 − 0.707107i 1.40320 + 0.810140i 1.38266 − 1.75735i
117.4 0.965926 − 0.258819i 0.752300 − 2.80762i 0.866025 − 0.500000i 1.38266 + 1.75735i − 2.90667i 0 0.707107 − 0.707107i −4.71872 − 2.72435i 1.79038 + 1.33961i
227.1 −0.258819 + 0.965926i −1.13459 + 0.304013i −0.866025 − 0.500000i −0.264946 − 2.22032i − 1.17462i 0 0.707107 − 0.707107i −1.40320 + 0.810140i 2.21323 + 0.318742i
227.2 −0.258819 + 0.965926i 2.80762 − 0.752300i −0.866025 − 0.500000i 2.21323 − 0.318742i 2.90667i 0 0.707107 − 0.707107i 4.71872 − 2.72435i −0.264946 + 2.22032i
227.3 0.258819 − 0.965926i −1.95290 + 0.523277i −0.866025 − 0.500000i 1.82591 − 1.29076i 2.02179i 0 −0.707107 + 0.707107i 0.941911 − 0.543813i −0.774197 − 2.09777i
227.4 0.258819 − 0.965926i 0.279864 − 0.0749894i −0.866025 − 0.500000i −0.774197 + 2.09777i − 0.289737i 0 −0.707107 + 0.707107i −2.52538 + 1.45803i 1.82591 + 1.29076i
313.1 −0.258819 − 0.965926i −1.13459 − 0.304013i −0.866025 + 0.500000i −0.264946 + 2.22032i 1.17462i 0 0.707107 + 0.707107i −1.40320 − 0.810140i 2.21323 − 0.318742i
313.2 −0.258819 − 0.965926i 2.80762 + 0.752300i −0.866025 + 0.500000i 2.21323 + 0.318742i − 2.90667i 0 0.707107 + 0.707107i 4.71872 + 2.72435i −0.264946 − 2.22032i
313.3 0.258819 + 0.965926i −1.95290 − 0.523277i −0.866025 + 0.500000i 1.82591 + 1.29076i − 2.02179i 0 −0.707107 − 0.707107i 0.941911 + 0.543813i −0.774197 + 2.09777i
313.4 0.258819 + 0.965926i 0.279864 + 0.0749894i −0.866025 + 0.500000i −0.774197 − 2.09777i 0.289737i 0 −0.707107 − 0.707107i −2.52538 − 1.45803i 1.82591 − 1.29076i
423.1 −0.965926 − 0.258819i −0.523277 − 1.95290i 0.866025 + 0.500000i 2.03078 − 0.935904i 2.02179i 0 −0.707107 − 0.707107i −0.941911 + 0.543813i −2.20382 + 0.378409i
423.2 −0.965926 − 0.258819i 0.0749894 + 0.279864i 0.866025 + 0.500000i −2.20382 − 0.378409i − 0.289737i 0 −0.707107 − 0.707107i 2.52538 − 1.45803i 2.03078 + 0.935904i
423.3 0.965926 + 0.258819i −0.304013 − 1.13459i 0.866025 + 0.500000i 1.79038 + 1.33961i − 1.17462i 0 0.707107 + 0.707107i 1.40320 − 0.810140i 1.38266 + 1.75735i
423.4 0.965926 + 0.258819i 0.752300 + 2.80762i 0.866025 + 0.500000i 1.38266 − 1.75735i 2.90667i 0 0.707107 + 0.707107i −4.71872 + 2.72435i 1.79038 − 1.33961i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.l.c 16
5.c odd 4 1 inner 490.2.l.c 16
7.b odd 2 1 70.2.k.a ✓ 16
7.c even 3 1 70.2.k.a ✓ 16
7.c even 3 1 490.2.g.c 16
7.d odd 6 1 490.2.g.c 16
7.d odd 6 1 inner 490.2.l.c 16
21.c even 2 1 630.2.bv.c 16
21.h odd 6 1 630.2.bv.c 16
28.d even 2 1 560.2.ci.c 16
28.g odd 6 1 560.2.ci.c 16
35.c odd 2 1 350.2.o.c 16
35.f even 4 1 70.2.k.a ✓ 16
35.f even 4 1 350.2.o.c 16
35.j even 6 1 350.2.o.c 16
35.k even 12 1 490.2.g.c 16
35.k even 12 1 inner 490.2.l.c 16
35.l odd 12 1 70.2.k.a ✓ 16
35.l odd 12 1 350.2.o.c 16
35.l odd 12 1 490.2.g.c 16
105.k odd 4 1 630.2.bv.c 16
105.x even 12 1 630.2.bv.c 16
140.j odd 4 1 560.2.ci.c 16
140.w even 12 1 560.2.ci.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.k.a ✓ 16 7.b odd 2 1
70.2.k.a ✓ 16 7.c even 3 1
70.2.k.a ✓ 16 35.f even 4 1
70.2.k.a ✓ 16 35.l odd 12 1
350.2.o.c 16 35.c odd 2 1
350.2.o.c 16 35.f even 4 1
350.2.o.c 16 35.j even 6 1
350.2.o.c 16 35.l odd 12 1
490.2.g.c 16 7.c even 3 1
490.2.g.c 16 7.d odd 6 1
490.2.g.c 16 35.k even 12 1
490.2.g.c 16 35.l odd 12 1
490.2.l.c 16 1.a even 1 1 trivial
490.2.l.c 16 5.c odd 4 1 inner
490.2.l.c 16 7.d odd 6 1 inner
490.2.l.c 16 35.k even 12 1 inner
560.2.ci.c 16 28.d even 2 1
560.2.ci.c 16 28.g odd 6 1
560.2.ci.c 16 140.j odd 4 1
560.2.ci.c 16 140.w even 12 1
630.2.bv.c 16 21.c even 2 1
630.2.bv.c 16 21.h odd 6 1
630.2.bv.c 16 105.k odd 4 1
630.2.bv.c 16 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 45 T_{3}^{12} - 84 T_{3}^{11} + 564 T_{3}^{9} + 2357 T_{3}^{8} + 3780 T_{3}^{7} + 3528 T_{3}^{6} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 45 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} - 12 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 6 T^{7} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 90 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{16} - 36 T^{15} + \cdots + 9834496 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 100000000 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 260144641 \) Copy content Toggle raw display
$29$ \( (T^{8} + 162 T^{6} + \cdots + 329476)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 12 T^{7} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 4 T^{15} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{8} + 140 T^{6} + \cdots + 18769)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 4 T^{7} + \cdots + 784)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 12 T^{15} + \cdots + 9834496 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 41740124416 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 268435456 \) Copy content Toggle raw display
$61$ \( (T^{8} - 6 T^{7} + \cdots + 148996)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 3429742096 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots - 4424)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 61465600000000 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 9971220736 \) Copy content Toggle raw display
$97$ \( (T^{8} + 8136 T^{4} + 3111696)^{2} \) Copy content Toggle raw display
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