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Properties

Label 810.4.c.c
Level $810$
Weight $4$
Character orbit 810.c
Analytic conductor $47.792$
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] = [810,4,Mod(649,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.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: \(47.7915471046\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 12278x^{12} + 41020729x^{8} + 21383909028x^{4} + 1134276120576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 4 q^{4} + ( - \beta_{5} + \beta_{3}) q^{5} + (\beta_{14} - \beta_{12}) q^{7} + 4 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - 4 q^{4} + ( - \beta_{5} + \beta_{3}) q^{5} + (\beta_{14} - \beta_{12}) q^{7} + 4 \beta_{3} q^{8} + ( - \beta_{8} + \beta_{2} + 3) q^{10} + (\beta_{13} + 3 \beta_{10} + \cdots + \beta_{5}) q^{11}+ \cdots + ( - 10 \beta_{7} + 28 \beta_{6} + \cdots + 23 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 64 q^{4} + 44 q^{10} + 256 q^{16} + 84 q^{19} - 426 q^{25} - 176 q^{31} + 920 q^{34} - 176 q^{40} - 344 q^{46} - 12 q^{49} - 1120 q^{55} - 2596 q^{61} - 1024 q^{64} - 1072 q^{70} - 336 q^{76} + 2208 q^{79} + 530 q^{85} - 8892 q^{91} + 4408 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 12278x^{12} + 41020729x^{8} + 21383909028x^{4} + 1134276120576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1813703\nu^{12} - 6908461381\nu^{8} + 29572350462244\nu^{4} + 26678557933471572 ) / 932938161091950 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4083787\nu^{12} + 44720816204\nu^{8} + 125279794820329\nu^{4} + 28657292206728492 ) / 1119525793310340 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -887273\nu^{14} - 10991920102\nu^{10} - 37711773834401\nu^{6} - 25475577293148804\nu^{2} ) / 89918690685863616 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1655821847 \nu^{14} + 21952476729594 \nu^{10} + \cdots + 17\!\cdots\!72 \nu^{2} ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 362861962241 \nu^{15} + 11775249778644 \nu^{14} - 2424210113856 \nu^{13} + \cdots - 54\!\cdots\!76 \nu ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 362861962241 \nu^{15} - 11775249778644 \nu^{14} - 2424210113856 \nu^{13} + \cdots - 54\!\cdots\!76 \nu ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1814309811205 \nu^{15} + 134098797109968 \nu^{14} + 12121050569280 \nu^{13} + \cdots + 27\!\cdots\!80 \nu ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 362861962241 \nu^{15} - 2424210113856 \nu^{13} + \cdots - 54\!\cdots\!76 \nu ) / 51\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9684749401465 \nu^{15} - 143729982875040 \nu^{13} + 917368618939776 \nu^{12} + \cdots + 23\!\cdots\!76 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19723509665 \nu^{15} + 248847531872 \nu^{13} - 235853062453558 \nu^{11} + \cdots + 24\!\cdots\!56 \nu ) / 15\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19723509665 \nu^{15} + 248847531872 \nu^{13} + 235853062453558 \nu^{11} + \cdots + 24\!\cdots\!56 \nu ) / 75\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26643184246649 \nu^{15} - 184772022160224 \nu^{13} + \cdots - 37\!\cdots\!24 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 26643184246649 \nu^{15} + 184772022160224 \nu^{13} + \cdots + 37\!\cdots\!24 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 185572737899 \nu^{15} + 606052528464 \nu^{13} + \cdots + 32\!\cdots\!24 \nu ) / 64\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 708767801597 \nu^{15} + 399457309152 \nu^{13} + \cdots + 13\!\cdots\!52 \nu ) / 15\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + 2\beta_{14} + \beta_{10} + 4\beta_{8} + 4\beta_{6} + 4\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - 4\beta_{5} - 8\beta_{4} - 135\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -29\beta_{15} + 58\beta_{14} + 16\beta_{11} - 61\beta_{10} + 296\beta_{8} - 296\beta_{6} - 296\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 189\beta_{11} + 378\beta_{9} + 189\beta_{8} - 57\beta_{2} + 67\beta _1 - 9209 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1321 \beta_{15} - 2642 \beta_{14} - 1320 \beta_{13} + 1320 \beta_{12} - 1780 \beta_{11} + \cdots - 22648 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8010\beta_{7} - 11078\beta_{6} + 19088\beta_{5} + 62200\beta_{4} + 657927\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 78893 \beta_{15} - 157786 \beta_{14} + 180960 \beta_{13} + 180960 \beta_{12} - 117744 \beta_{11} + \cdots + 1741904 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1243339\beta_{11} - 2486678\beta_{9} - 1243339\beta_{8} + 490131\beta_{2} - 224687\beta _1 + 51454911 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5307289 \beta_{15} + 10614578 \beta_{14} + 19055160 \beta_{13} - 19055160 \beta_{12} + \cdots + 134247448 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -22359298\beta_{7} + 100227038\beta_{6} - 122586336\beta_{5} - 394737792\beta_{4} - 3785720391\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 372721133 \beta_{15} + 745442266 \beta_{14} - 1831803600 \beta_{13} - 1831803600 \beta_{12} + \cdots - 10363229312 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 7817560925 \beta_{11} + 15635121850 \beta_{9} + 7817560925 \beta_{8} - 2796309573 \beta_{2} + \cdots - 302017677157 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 26530683337 \beta_{15} - 53061366674 \beta_{14} - 168175514280 \beta_{13} + \cdots - 801215052904 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 6028062022 \beta_{7} - 828231804422 \beta_{6} + 822203742400 \beta_{5} + \cdots + 22203349626663 \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1894208773709 \beta_{15} - 3788417547418 \beta_{14} + 15001827194880 \beta_{13} + \cdots + 62036059093712 \beta_{5} ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
6.30156 − 6.30156i
6.04699 − 6.04699i
−1.96615 + 1.96615i
−3.44363 + 3.44363i
3.44363 − 3.44363i
1.96615 − 1.96615i
−6.04699 + 6.04699i
−6.30156 + 6.30156i
6.30156 + 6.30156i
6.04699 + 6.04699i
−1.96615 − 1.96615i
−3.44363 − 3.44363i
3.44363 + 3.44363i
1.96615 + 1.96615i
−6.04699 − 6.04699i
−6.30156 − 6.30156i
− 2.00000i 0 −4.00000 −11.0525 − 1.68568i 0 − 2.48700i 8.00000i 0 −3.37137 + 22.1051i
649.2 − 2.00000i 0 −4.00000 −8.43367 + 7.33984i 0 11.9571i 8.00000i 0 14.6797 + 16.8673i
649.3 − 2.00000i 0 −4.00000 −1.85578 + 11.0252i 0 31.4869i 8.00000i 0 22.0505 + 3.71156i
649.4 − 2.00000i 0 −4.00000 −0.144522 − 11.1794i 0 15.3107i 8.00000i 0 −22.3588 + 0.289045i
649.5 − 2.00000i 0 −4.00000 0.144522 − 11.1794i 0 − 15.3107i 8.00000i 0 −22.3588 − 0.289045i
649.6 − 2.00000i 0 −4.00000 1.85578 + 11.0252i 0 − 31.4869i 8.00000i 0 22.0505 − 3.71156i
649.7 − 2.00000i 0 −4.00000 8.43367 + 7.33984i 0 − 11.9571i 8.00000i 0 14.6797 − 16.8673i
649.8 − 2.00000i 0 −4.00000 11.0525 − 1.68568i 0 2.48700i 8.00000i 0 −3.37137 − 22.1051i
649.9 2.00000i 0 −4.00000 −11.0525 + 1.68568i 0 2.48700i − 8.00000i 0 −3.37137 − 22.1051i
649.10 2.00000i 0 −4.00000 −8.43367 − 7.33984i 0 − 11.9571i − 8.00000i 0 14.6797 − 16.8673i
649.11 2.00000i 0 −4.00000 −1.85578 − 11.0252i 0 − 31.4869i − 8.00000i 0 22.0505 − 3.71156i
649.12 2.00000i 0 −4.00000 −0.144522 + 11.1794i 0 − 15.3107i − 8.00000i 0 −22.3588 − 0.289045i
649.13 2.00000i 0 −4.00000 0.144522 + 11.1794i 0 15.3107i − 8.00000i 0 −22.3588 + 0.289045i
649.14 2.00000i 0 −4.00000 1.85578 − 11.0252i 0 31.4869i − 8.00000i 0 22.0505 + 3.71156i
649.15 2.00000i 0 −4.00000 8.43367 − 7.33984i 0 11.9571i − 8.00000i 0 14.6797 + 16.8673i
649.16 2.00000i 0 −4.00000 11.0525 + 1.68568i 0 − 2.48700i − 8.00000i 0 −3.37137 + 22.1051i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.4.c.c ✓ 16
3.b odd 2 1 inner 810.4.c.c ✓ 16
5.b even 2 1 inner 810.4.c.c ✓ 16
15.d odd 2 1 inner 810.4.c.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.4.c.c ✓ 16 1.a even 1 1 trivial
810.4.c.c ✓ 16 3.b odd 2 1 inner
810.4.c.c ✓ 16 5.b even 2 1 inner
810.4.c.c ✓ 16 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{8} + 1375T_{7}^{6} + 416136T_{7}^{4} + 35749420T_{7}^{2} + 205520896 \) Copy content Toggle raw display
\( T_{11}^{8} - 1618T_{11}^{6} + 945921T_{11}^{4} - 236044060T_{11}^{2} + 21197030464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 1375 T^{6} + \cdots + 205520896)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 1618 T^{6} + \cdots + 21197030464)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 40264507266624)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 430749104214016)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 21 T^{3} + \cdots + 3027024)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 22\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 44 T^{3} + \cdots - 44225672)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 90\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 71\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 39\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 22\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 23\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 649 T^{3} + \cdots + 29686564768)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 58\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 44\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 70\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 552 T^{3} + \cdots - 24683356800)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 48\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 75\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
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