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Properties

Label 490.2.g.b
Level $490$
Weight $2$
Character orbit 490.g
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(97,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.g (of order \(4\), degree \(2\), 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: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 32x^{12} + 128x^{10} + 223x^{8} + 128x^{6} + 32x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + (\beta_{13} - \beta_{12} - \beta_{11}) q^{3} + \beta_{8} q^{4} + (\beta_{11} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{14} - \beta_{11} + \cdots - \beta_{4}) q^{6}+ \cdots + (\beta_{10} - 2 \beta_{9} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + (\beta_{13} - \beta_{12} - \beta_{11}) q^{3} + \beta_{8} q^{4} + (\beta_{11} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{14} - \beta_{11} + \cdots - \beta_{4}) q^{6}+ \cdots + ( - 5 \beta_{10} - 4 \beta_{9} + \cdots + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{11} - 24 q^{15} - 16 q^{16} - 16 q^{18} + 16 q^{22} + 16 q^{23} + 32 q^{25} + 40 q^{30} - 48 q^{36} - 48 q^{37} + 16 q^{43} - 16 q^{46} - 32 q^{50} - 80 q^{51} + 32 q^{53} + 48 q^{57} + 16 q^{58} + 40 q^{60} + 32 q^{65} + 16 q^{67} + 16 q^{71} - 16 q^{72} - 112 q^{81} + 24 q^{85} + 48 q^{86} - 16 q^{88} + 16 q^{92} + 64 q^{93} + 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 32x^{12} + 128x^{10} + 223x^{8} + 128x^{6} + 32x^{4} - 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{14} - 24\nu^{12} - 33\nu^{10} + 1737\nu^{8} + 547\nu^{6} - 1909\nu^{4} - 3916\nu^{2} + 523 ) / 1085 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 349 \nu^{14} - 1132 \nu^{12} - 3695 \nu^{10} + 110989 \nu^{8} + 236219 \nu^{6} + 218975 \nu^{4} + \cdots - 33216 ) / 24955 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 541 \nu^{14} + 6570 \nu^{12} - 35628 \nu^{10} + 4719 \nu^{8} + 162905 \nu^{6} + 335939 \nu^{4} + \cdots + 36398 ) / 24955 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1455 \nu^{15} + 9035 \nu^{13} - 25726 \nu^{11} - 269420 \nu^{9} - 659642 \nu^{7} + \cdots - 84174 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1440 \nu^{14} + 11322 \nu^{12} - 44809 \nu^{10} - 188028 \nu^{8} - 357624 \nu^{6} - 262908 \nu^{4} + \cdots - 5598 ) / 24955 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1857 \nu^{15} + 15460 \nu^{13} - 64664 \nu^{11} - 214178 \nu^{9} - 357832 \nu^{7} + \cdots + 94708 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3092 \nu^{15} - 25784 \nu^{13} + 108166 \nu^{11} + 355937 \nu^{9} + 577886 \nu^{7} + \cdots + 35502 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 584 \nu^{14} + 4479 \nu^{12} - 17084 \nu^{10} - 81440 \nu^{8} - 152700 \nu^{6} - 111808 \nu^{4} + \cdots + 2608 ) / 4991 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4016 \nu^{14} + 32574 \nu^{12} - 132232 \nu^{10} - 498436 \nu^{8} - 844649 \nu^{6} - 427772 \nu^{4} + \cdots + 25314 ) / 24955 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5093 \nu^{14} - 41378 \nu^{12} + 167828 \nu^{10} + 634835 \nu^{8} + 1034367 \nu^{6} + 540319 \nu^{4} + \cdots + 7235 ) / 24955 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6317 \nu^{15} + 52016 \nu^{13} - 213217 \nu^{11} - 767965 \nu^{9} - 1189519 \nu^{7} + \cdots + 93832 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 8116 \nu^{15} + 62194 \nu^{13} - 236940 \nu^{11} - 1134177 \nu^{9} - 2125636 \nu^{7} + \cdots - 33169 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8937 \nu^{15} + 70563 \nu^{13} - 277314 \nu^{11} - 1184534 \nu^{9} - 2064630 \nu^{7} + \cdots + 41666 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 9973 \nu^{15} + 77654 \nu^{13} - 301604 \nu^{11} - 1348355 \nu^{9} - 2483468 \nu^{7} + \cdots + 111449 \nu ) / 24955 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15601 \nu^{15} + 124939 \nu^{13} - 501391 \nu^{11} - 1983556 \nu^{9} - 3500127 \nu^{7} + \cdots + 57668 \nu ) / 24955 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{12} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - 6\beta_{5} - 2\beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{15} + 3\beta_{13} - 4\beta_{12} + 4\beta_{11} - 6\beta_{7} + 6\beta_{6} + 9\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{10} - 4\beta_{9} - 8\beta_{8} - 27\beta_{5} - 7\beta_{3} - 7\beta_{2} + 16\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -36\beta_{15} - 27\beta_{14} + 68\beta_{13} - 27\beta_{12} - 54\beta_{7} + 95\beta_{6} + 68\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -26\beta_{10} - 27\beta_{9} - 18\beta_{8} + 26\beta_{3} - 62\beta_{2} + 62\beta _1 - 106 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -203\beta_{15} - 284\beta_{14} + 521\beta_{13} - 203\beta_{11} - 307\beta_{7} + 735\beta_{6} + 214\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 397\beta_{10} - 284\beta_{9} + 1637\beta_{5} + 956\beta_{3} - 956\beta_{2} + 397\beta _1 - 2318 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -1556\beta_{14} + 1657\beta_{13} + 1556\beta_{12} - 2196\beta_{11} + 2350\beta_{6} - 1657\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 7362\beta_{10} + 2196\beta_{8} + 17836\beta_{5} + 7362\beta_{3} - 3051\beta_{2} - 3051\beta _1 - 12609 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11969 \beta_{15} - 12771 \beta_{13} + 16920 \beta_{12} - 11969 \beta_{11} + 18071 \beta_{7} + \cdots - 30842 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 28340\beta_{10} + 8460\beta_{9} + 11969\beta_{8} + 48540\beta_{5} + 11740\beta_{3} + 11740\beta_{2} - 28340\beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 130280 \beta_{15} + 92129 \beta_{14} - 237429 \beta_{13} + 92129 \beta_{12} + 196680 \beta_{7} + \cdots - 237429 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 180749 \beta_{10} + 184258 \beta_{9} + 130280 \beta_{8} - 180749 \beta_{3} + 436358 \beta_{2} + \cdots + 747387 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 709236 \beta_{15} + 1002996 \beta_{14} - 1827841 \beta_{13} + 709236 \beta_{11} + \cdots - 757107 \beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
2.56342 − 1.06180i
0.332975 − 0.137923i
−0.332975 + 0.137923i
−2.56342 + 1.06180i
−0.470061 − 1.13483i
−0.311548 − 0.752144i
0.311548 + 0.752144i
0.470061 + 1.13483i
2.56342 + 1.06180i
0.332975 + 0.137923i
−0.332975 − 0.137923i
−2.56342 − 1.06180i
−0.470061 + 1.13483i
−0.311548 + 0.752144i
0.311548 − 0.752144i
0.470061 − 1.13483i
−0.707107 − 0.707107i −2.12360 − 2.12360i 1.00000i 1.93597 − 1.11893i 3.00323i 0 0.707107 − 0.707107i 6.01939i −2.16014 − 0.577736i
97.2 −0.707107 − 0.707107i −0.275845 − 0.275845i 1.00000i 2.16014 − 0.577736i 0.390104i 0 0.707107 − 0.707107i − 2.84782i −1.93597 − 1.11893i
97.3 −0.707107 − 0.707107i 0.275845 + 0.275845i 1.00000i −2.16014 + 0.577736i − 0.390104i 0 0.707107 − 0.707107i − 2.84782i 1.93597 + 1.11893i
97.4 −0.707107 − 0.707107i 2.12360 + 2.12360i 1.00000i −1.93597 + 1.11893i − 3.00323i 0 0.707107 − 0.707107i 6.01939i 2.16014 + 0.577736i
97.5 0.707107 + 0.707107i −2.26965 − 2.26965i 1.00000i 2.12984 + 0.681009i − 3.20978i 0 −0.707107 + 0.707107i 7.30266i 1.02448 + 1.98757i
97.6 0.707107 + 0.707107i −1.50429 − 1.50429i 1.00000i −1.02448 + 1.98757i − 2.12738i 0 −0.707107 + 0.707107i 1.52576i −2.12984 + 0.681009i
97.7 0.707107 + 0.707107i 1.50429 + 1.50429i 1.00000i 1.02448 − 1.98757i 2.12738i 0 −0.707107 + 0.707107i 1.52576i 2.12984 − 0.681009i
97.8 0.707107 + 0.707107i 2.26965 + 2.26965i 1.00000i −2.12984 − 0.681009i 3.20978i 0 −0.707107 + 0.707107i 7.30266i −1.02448 − 1.98757i
293.1 −0.707107 + 0.707107i −2.12360 + 2.12360i − 1.00000i 1.93597 + 1.11893i − 3.00323i 0 0.707107 + 0.707107i − 6.01939i −2.16014 + 0.577736i
293.2 −0.707107 + 0.707107i −0.275845 + 0.275845i − 1.00000i 2.16014 + 0.577736i − 0.390104i 0 0.707107 + 0.707107i 2.84782i −1.93597 + 1.11893i
293.3 −0.707107 + 0.707107i 0.275845 − 0.275845i − 1.00000i −2.16014 − 0.577736i 0.390104i 0 0.707107 + 0.707107i 2.84782i 1.93597 − 1.11893i
293.4 −0.707107 + 0.707107i 2.12360 − 2.12360i − 1.00000i −1.93597 − 1.11893i 3.00323i 0 0.707107 + 0.707107i − 6.01939i 2.16014 − 0.577736i
293.5 0.707107 − 0.707107i −2.26965 + 2.26965i − 1.00000i 2.12984 − 0.681009i 3.20978i 0 −0.707107 − 0.707107i − 7.30266i 1.02448 − 1.98757i
293.6 0.707107 − 0.707107i −1.50429 + 1.50429i − 1.00000i −1.02448 − 1.98757i 2.12738i 0 −0.707107 − 0.707107i − 1.52576i −2.12984 − 0.681009i
293.7 0.707107 − 0.707107i 1.50429 − 1.50429i − 1.00000i 1.02448 + 1.98757i − 2.12738i 0 −0.707107 − 0.707107i − 1.52576i 2.12984 + 0.681009i
293.8 0.707107 − 0.707107i 2.26965 − 2.26965i − 1.00000i −2.12984 + 0.681009i − 3.20978i 0 −0.707107 − 0.707107i − 7.30266i −1.02448 + 1.98757i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.g.b ✓ 16
5.c odd 4 1 inner 490.2.g.b ✓ 16
7.b odd 2 1 inner 490.2.g.b ✓ 16
7.c even 3 2 490.2.l.d 32
7.d odd 6 2 490.2.l.d 32
35.f even 4 1 inner 490.2.g.b ✓ 16
35.k even 12 2 490.2.l.d 32
35.l odd 12 2 490.2.l.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.g.b ✓ 16 1.a even 1 1 trivial
490.2.g.b ✓ 16 5.c odd 4 1 inner
490.2.g.b ✓ 16 7.b odd 2 1 inner
490.2.g.b ✓ 16 35.f even 4 1 inner
490.2.l.d 32 7.c even 3 2
490.2.l.d 32 7.d odd 6 2
490.2.l.d 32 35.k even 12 2
490.2.l.d 32 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 208T_{3}^{12} + 12480T_{3}^{8} + 177152T_{3}^{4} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 208 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( T^{16} - 16 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} - 12 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 2008 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{16} + 1240 T^{12} + \cdots + 38416 \) Copy content Toggle raw display
$19$ \( (T^{8} - 56 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 8 T^{7} + \cdots + 2715904)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 128 T^{6} + \cdots + 85264)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 80 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 24 T^{7} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 152 T^{6} + \cdots + 198916)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 8 T^{7} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 87578116096 \) Copy content Toggle raw display
$53$ \( (T^{8} - 16 T^{7} + \cdots + 4624)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 376 T^{6} + \cdots + 36096064)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 248 T^{6} + \cdots + 8836)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 8 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots + 8176)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 406586896 \) Copy content Toggle raw display
$79$ \( (T^{8} + 440 T^{6} + \cdots + 3625216)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 176319369216 \) Copy content Toggle raw display
$89$ \( (T^{8} - 280 T^{6} + \cdots + 3378244)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 60245477145616 \) Copy content Toggle raw display
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