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Properties

Label 210.4.b.a
Level $210$
Weight $4$
Character orbit 210.b
Analytic conductor $12.390$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [210,4,Mod(41,210)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("210.41"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.3904011012\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 13 x^{14} + 14 x^{13} - 377 x^{12} - 3822 x^{11} + 11594 x^{10} + \cdots + 172160384320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{2} q^{3} - 4 q^{4} + 5 q^{5} + ( - \beta_{12} - 2) q^{6} + (\beta_{10} + \beta_{5} - 1) q^{7} - 4 \beta_{5} q^{8} + ( - \beta_{9} - 2 \beta_{5} + 2) q^{9} + 5 \beta_{5} q^{10}+ \cdots + (14 \beta_{15} - 12 \beta_{14} + \cdots - 253) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 64 q^{4} + 80 q^{5} - 28 q^{6} - 10 q^{7} + 30 q^{9} + 16 q^{12} - 36 q^{14} - 20 q^{15} + 256 q^{16} - 144 q^{17} + 112 q^{18} - 320 q^{20} + 234 q^{21} + 112 q^{24} + 400 q^{25} + 24 q^{26}+ \cdots - 3858 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 13 x^{14} + 14 x^{13} - 377 x^{12} - 3822 x^{11} + 11594 x^{10} + \cdots + 172160384320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\!\cdots\!71 \nu^{15} + \cdots - 60\!\cdots\!20 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!11 \nu^{15} + \cdots + 40\!\cdots\!20 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!29 \nu^{15} + \cdots + 76\!\cdots\!80 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40\!\cdots\!89 \nu^{15} + \cdots + 12\!\cdots\!32 ) / 17\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!71 \nu^{15} + \cdots - 59\!\cdots\!20 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26\!\cdots\!99 \nu^{15} + \cdots - 89\!\cdots\!80 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 97\!\cdots\!71 \nu^{15} + \cdots + 35\!\cdots\!20 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50\!\cdots\!41 \nu^{15} + \cdots - 19\!\cdots\!20 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!62 \nu^{15} + \cdots - 62\!\cdots\!40 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39\!\cdots\!79 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!13 \nu^{15} + \cdots + 49\!\cdots\!60 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 29\!\cdots\!03 \nu^{15} + \cdots - 10\!\cdots\!60 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!41 \nu^{15} + \cdots - 47\!\cdots\!20 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\!\cdots\!89 \nu^{15} + \cdots - 64\!\cdots\!80 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18\!\cdots\!89 \nu^{15} + \cdots - 62\!\cdots\!80 ) / 43\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + 2\beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 3\beta_{13} + 3\beta_{9} - 3\beta_{8} + \beta_{7} + 6\beta_{5} + 3\beta_{4} + 3\beta_{2} - \beta _1 + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} - 15 \beta_{14} + 3 \beta_{13} + 36 \beta_{12} + 12 \beta_{11} - 33 \beta_{10} + \cdots + 60 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{15} - 2 \beta_{14} + 16 \beta_{13} - 6 \beta_{12} + \beta_{11} + 25 \beta_{10} - 14 \beta_{9} + \cdots + 150 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 39 \beta_{15} + 429 \beta_{14} - 261 \beta_{13} - 1467 \beta_{12} + 24 \beta_{11} - 507 \beta_{10} + \cdots + 11948 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1339 \beta_{15} + 24 \beta_{14} - 1509 \beta_{13} - 765 \beta_{12} - 2010 \beta_{11} + 564 \beta_{10} + \cdots + 25307 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8501 \beta_{15} + 987 \beta_{14} - 13848 \beta_{13} + 56889 \beta_{12} + 2922 \beta_{11} + \cdots - 318096 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3952 \beta_{15} - 13366 \beta_{14} + 1295 \beta_{13} + 7056 \beta_{12} + 14231 \beta_{11} + \cdots + 172775 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 281706 \beta_{15} - 213756 \beta_{14} + 918189 \beta_{13} - 286542 \beta_{12} + 399450 \beta_{11} + \cdots + 10078712 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 513895 \beta_{15} + 1932402 \beta_{14} - 1028259 \beta_{13} - 5907861 \beta_{12} - 1448961 \beta_{11} + \cdots + 15710624 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7597531 \beta_{15} + 1600857 \beta_{14} - 33376353 \beta_{13} + 558045 \beta_{12} - 14142960 \beta_{11} + \cdots + 145610544 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4453848 \beta_{15} - 14825985 \beta_{14} - 4413981 \beta_{13} + 86440818 \beta_{12} + \cdots - 325938563 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 557470557 \beta_{15} - 146817081 \beta_{14} + 738418572 \beta_{13} + 410141169 \beta_{12} + \cdots - 3654706156 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1744859368 \beta_{15} - 298277136 \beta_{14} + 1645054926 \beta_{13} - 7583669100 \beta_{12} + \cdots + 65623445837 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 13910959294 \beta_{15} + 23361458484 \beta_{14} - 7994610519 \beta_{13} - 38226575034 \beta_{12} + \cdots + 166179473808 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
5.13413 + 0.199526i
5.08074 − 0.0890798i
2.51851 − 3.54501i
1.48800 + 5.97854i
0.409319 − 4.18001i
−3.71477 − 2.63325i
−3.73137 + 4.61620i
−5.18456 + 0.653078i
5.13413 − 0.199526i
5.08074 + 0.0890798i
2.51851 + 3.54501i
1.48800 − 5.97854i
0.409319 + 4.18001i
−3.71477 + 2.63325i
−3.73137 − 4.61620i
−5.18456 − 0.653078i
− 2.00000i −5.13413 − 0.800474i −4.00000 5.00000 −1.60095 + 10.2683i 5.52391 − 17.6773i 8.00000i 25.7185 + 8.21946i − 10.0000i
41.2 − 2.00000i −5.08074 − 1.08908i −4.00000 5.00000 −2.17816 + 10.1615i −13.0046 + 13.1863i 8.00000i 24.6278 + 11.0667i − 10.0000i
41.3 − 2.00000i −2.51851 − 4.54501i −4.00000 5.00000 −9.09001 + 5.03703i 15.8708 + 9.54556i 8.00000i −14.3142 + 22.8933i − 10.0000i
41.4 − 2.00000i −1.48800 + 4.97854i −4.00000 5.00000 9.95708 + 2.97599i −4.51401 − 17.9617i 8.00000i −22.5717 − 14.8161i − 10.0000i
41.5 − 2.00000i −0.409319 − 5.18001i −4.00000 5.00000 −10.3600 + 0.818638i −9.13843 − 16.1087i 8.00000i −26.6649 + 4.24055i − 10.0000i
41.6 − 2.00000i 3.71477 − 3.63325i −4.00000 5.00000 −7.26650 − 7.42953i −0.448691 + 18.5148i 8.00000i 0.598988 − 26.9934i − 10.0000i
41.7 − 2.00000i 3.73137 + 3.61620i −4.00000 5.00000 7.23239 − 7.46274i −16.3649 + 8.67128i 8.00000i 0.846237 + 26.9867i − 10.0000i
41.8 − 2.00000i 5.18456 − 0.346922i −4.00000 5.00000 −0.693844 − 10.3691i 17.0759 − 7.17033i 8.00000i 26.7593 − 3.59728i − 10.0000i
41.9 2.00000i −5.13413 + 0.800474i −4.00000 5.00000 −1.60095 − 10.2683i 5.52391 + 17.6773i − 8.00000i 25.7185 − 8.21946i 10.0000i
41.10 2.00000i −5.08074 + 1.08908i −4.00000 5.00000 −2.17816 − 10.1615i −13.0046 − 13.1863i − 8.00000i 24.6278 − 11.0667i 10.0000i
41.11 2.00000i −2.51851 + 4.54501i −4.00000 5.00000 −9.09001 − 5.03703i 15.8708 − 9.54556i − 8.00000i −14.3142 − 22.8933i 10.0000i
41.12 2.00000i −1.48800 − 4.97854i −4.00000 5.00000 9.95708 − 2.97599i −4.51401 + 17.9617i − 8.00000i −22.5717 + 14.8161i 10.0000i
41.13 2.00000i −0.409319 + 5.18001i −4.00000 5.00000 −10.3600 − 0.818638i −9.13843 + 16.1087i − 8.00000i −26.6649 − 4.24055i 10.0000i
41.14 2.00000i 3.71477 + 3.63325i −4.00000 5.00000 −7.26650 + 7.42953i −0.448691 − 18.5148i − 8.00000i 0.598988 + 26.9934i 10.0000i
41.15 2.00000i 3.73137 − 3.61620i −4.00000 5.00000 7.23239 + 7.46274i −16.3649 − 8.67128i − 8.00000i 0.846237 − 26.9867i 10.0000i
41.16 2.00000i 5.18456 + 0.346922i −4.00000 5.00000 −0.693844 + 10.3691i 17.0759 + 7.17033i − 8.00000i 26.7593 + 3.59728i 10.0000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.4.b.a ✓ 16
3.b odd 2 1 210.4.b.b yes 16
7.b odd 2 1 210.4.b.b yes 16
21.c even 2 1 inner 210.4.b.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.b.a ✓ 16 1.a even 1 1 trivial
210.4.b.a ✓ 16 21.c even 2 1 inner
210.4.b.b yes 16 3.b odd 2 1
210.4.b.b yes 16 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{8} + 72 T_{17}^{7} - 19671 T_{17}^{6} - 1290894 T_{17}^{5} + 86295756 T_{17}^{4} + \cdots - 23849968865472 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T - 5)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 23849968865472)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 69\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 18\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 27\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 77\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 17\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 55\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
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