↑

Properties

Label 490.3.f.p
Level $490$
Weight $3$
Character orbit 490.f
Analytic conductor $13.352$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,3,Mod(197,490)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 490.f (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: \(13.3515329537\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 730x^{4} - 1570x^{3} + 1682x^{2} + 4930x + 7225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + \beta_{4} q^{3} - 2 \beta_{2} q^{4} + (\beta_{7} - \beta_{2}) q^{5} + (\beta_{4} + \beta_1) q^{6} + ( - 2 \beta_{2} - 2) q^{8} + ( - \beta_{7} - \beta_{6} + \cdots + 8 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + \beta_{4} q^{3} - 2 \beta_{2} q^{4} + (\beta_{7} - \beta_{2}) q^{5} + (\beta_{4} + \beta_1) q^{6} + ( - 2 \beta_{2} - 2) q^{8} + ( - \beta_{7} - \beta_{6} + \cdots + 8 \beta_{2}) q^{9}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots - 16 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 2 q^{3} - 2 q^{5} + 4 q^{6} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 2 q^{3} - 2 q^{5} + 4 q^{6} - 16 q^{8} - 6 q^{10} - 40 q^{11} + 4 q^{12} - 8 q^{13} - 10 q^{15} - 32 q^{16} + 46 q^{17} + 52 q^{18} - 8 q^{20} - 40 q^{22} - 54 q^{23} + 26 q^{25} - 16 q^{26} + 26 q^{27} + 22 q^{30} - 208 q^{31} - 32 q^{32} - 22 q^{33} + 104 q^{36} + 38 q^{37} + 36 q^{38} - 4 q^{40} + 36 q^{41} + 72 q^{43} + 254 q^{45} - 108 q^{46} + 46 q^{47} - 8 q^{48} - 30 q^{50} + 136 q^{51} - 16 q^{52} - 30 q^{53} + 96 q^{55} - 246 q^{57} - 132 q^{58} + 64 q^{60} - 120 q^{61} - 208 q^{62} - 230 q^{65} - 44 q^{66} + 74 q^{67} - 92 q^{68} + 8 q^{71} + 104 q^{72} - 54 q^{73} + 300 q^{75} + 72 q^{76} + 84 q^{78} + 8 q^{80} - 244 q^{81} + 36 q^{82} - 32 q^{83} + 272 q^{85} + 144 q^{86} - 236 q^{87} + 80 q^{88} + 524 q^{90} - 108 q^{92} - 142 q^{93} + 396 q^{95} - 16 q^{96} - 68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 730x^{4} - 1570x^{3} + 1682x^{2} + 4930x + 7225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97051 \nu^{7} - 446535 \nu^{6} + 463297 \nu^{5} - 89849 \nu^{4} + 71050227 \nu^{3} + \cdots + 240484975 ) / 902966900 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12774 \nu^{7} + 127265 \nu^{6} - 148192 \nu^{5} + 542399 \nu^{4} + 6727998 \nu^{3} + \cdots + 248587175 ) / 82087900 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14849 \nu^{7} + 15835 \nu^{6} - 16703 \nu^{5} + 11941 \nu^{4} - 12601053 \nu^{3} + \cdots - 41246675 ) / 53115700 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41711 \nu^{7} - 175970 \nu^{6} - 91253 \nu^{5} + 518316 \nu^{4} + 30505487 \nu^{3} + \cdots + 326962700 ) / 82087900 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 483948 \nu^{7} - 1970335 \nu^{6} + 4587876 \nu^{5} + 8505713 \nu^{4} + 353799316 \nu^{3} + \cdots + 4307870125 ) / 902966900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 847612 \nu^{7} - 2285175 \nu^{6} + 2661844 \nu^{5} - 9768233 \nu^{4} + 592502164 \nu^{3} + \cdots - 1081756325 ) / 902966900 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - 17\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{7} - \beta_{6} - \beta_{5} - 21\beta_{4} - 5\beta_{3} + 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -34\beta_{7} + 34\beta_{6} + 24\beta_{5} + 8\beta_{4} + 24\beta_{3} + 8\beta _1 - 365 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30\beta_{7} + 160\beta_{6} - 160\beta_{5} - 30\beta_{3} - 260\beta_{2} - 529\beta _1 + 70 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -619\beta_{7} - 619\beta_{6} - 949\beta_{5} - 320\beta_{4} + 949\beta_{3} + 10773\beta_{2} + 320\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4415\beta_{7} + 929\beta_{6} + 929\beta_{5} + 13909\beta_{4} + 4415\beta_{3} - 8494\beta_{2} - 3150 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−3.69730 − 3.69730i
−0.827941 − 0.827941i
1.93183 + 1.93183i
3.59340 + 3.59340i
−3.69730 + 3.69730i
−0.827941 + 0.827941i
1.93183 − 1.93183i
3.59340 − 3.59340i
1.00000 + 1.00000i −3.69730 + 3.69730i 2.00000i 1.35792 + 4.81207i −7.39459 0 −2.00000 + 2.00000i − 18.3400i −3.45416 + 6.16999i
197.2 1.00000 + 1.00000i −0.827941 + 0.827941i 2.00000i −2.46354 − 4.35097i −1.65588 0 −2.00000 + 2.00000i 7.62903i 1.88743 − 6.81451i
197.3 1.00000 + 1.00000i 1.93183 − 1.93183i 2.00000i −4.87575 + 1.10773i 3.86367 0 −2.00000 + 2.00000i 1.53604i −5.98348 − 3.76802i
197.4 1.00000 + 1.00000i 3.59340 − 3.59340i 2.00000i 4.98137 + 0.431169i 7.18681 0 −2.00000 + 2.00000i − 16.8251i 4.55021 + 5.41254i
393.1 1.00000 − 1.00000i −3.69730 − 3.69730i − 2.00000i 1.35792 − 4.81207i −7.39459 0 −2.00000 − 2.00000i 18.3400i −3.45416 − 6.16999i
393.2 1.00000 − 1.00000i −0.827941 − 0.827941i − 2.00000i −2.46354 + 4.35097i −1.65588 0 −2.00000 − 2.00000i − 7.62903i 1.88743 + 6.81451i
393.3 1.00000 − 1.00000i 1.93183 + 1.93183i − 2.00000i −4.87575 − 1.10773i 3.86367 0 −2.00000 − 2.00000i − 1.53604i −5.98348 + 3.76802i
393.4 1.00000 − 1.00000i 3.59340 + 3.59340i − 2.00000i 4.98137 − 0.431169i 7.18681 0 −2.00000 − 2.00000i 16.8251i 4.55021 − 5.41254i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.3.f.p 8
5.c odd 4 1 inner 490.3.f.p 8
7.b odd 2 1 490.3.f.o 8
7.d odd 6 2 70.3.l.c ✓ 16
35.f even 4 1 490.3.f.o 8
35.i odd 6 2 350.3.p.e 16
35.k even 12 2 70.3.l.c ✓ 16
35.k even 12 2 350.3.p.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.l.c ✓ 16 7.d odd 6 2
70.3.l.c ✓ 16 35.k even 12 2
350.3.p.e 16 35.i odd 6 2
350.3.p.e 16 35.k even 12 2
490.3.f.o 8 7.b odd 2 1
490.3.f.o 8 35.f even 4 1
490.3.f.p 8 1.a even 1 1 trivial
490.3.f.p 8 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} + 2T_{3}^{5} + 730T_{3}^{4} - 1570T_{3}^{3} + 1682T_{3}^{2} + 4930T_{3} + 7225 \) Copy content Toggle raw display
\( T_{11}^{4} + 20T_{11}^{3} - 137T_{11}^{2} - 2856T_{11} - 2570 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 7225 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 20 T^{3} + \cdots - 2570)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 409600 \) Copy content Toggle raw display
$17$ \( T^{8} - 46 T^{7} + \cdots + 347449600 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 8377740900 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 42985314241 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 10488217744 \) Copy content Toggle raw display
$31$ \( (T^{4} + 104 T^{3} + \cdots + 188960)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 77765130496 \) Copy content Toggle raw display
$41$ \( (T^{4} - 18 T^{3} + \cdots - 367408)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 321477660100 \) Copy content Toggle raw display
$47$ \( T^{8} - 46 T^{7} + \cdots + 347449600 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 116618152036 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 485948958062500 \) Copy content Toggle raw display
$61$ \( (T^{4} + 60 T^{3} + \cdots - 623755)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 5370412282225 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots + 2549080)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 24586240000 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 531805562500 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 468971508062500 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 1591975210225 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
show more
show less