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Properties

Label 504.4.c.f
Level $504$
Weight $4$
Character orbit 504.c
Analytic conductor $29.737$
Analytic rank $0$
Dimension $20$
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] = [504,4,Mod(253,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.253");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.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: \(29.7369626429\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} - 2 x^{16} - 244 x^{14} - 2704 x^{12} + 19328 x^{10} - 173056 x^{8} - 999424 x^{6} + \cdots + 1073741824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{27}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{9} q^{5} + 7 q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{9} q^{5} + 7 q^{7} + \beta_{3} q^{8} + (\beta_{7} - 2) q^{10} + ( - \beta_{16} - \beta_{3} - 2 \beta_1) q^{11} + (\beta_{13} + \beta_{2}) q^{13} + 7 \beta_1 q^{14} + \beta_{4} q^{16} + ( - \beta_{10} - \beta_{8} - 4 \beta_1) q^{17} + ( - \beta_{13} - \beta_{12} + \cdots + 4 \beta_{2}) q^{19}+ \cdots + 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4} + 140 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{4} + 140 q^{7} - 48 q^{10} + 10 q^{16} + 356 q^{22} - 756 q^{25} + 14 q^{28} + 264 q^{31} - 680 q^{34} + 724 q^{40} + 368 q^{46} + 980 q^{49} - 1644 q^{52} - 360 q^{55} + 204 q^{58} + 1478 q^{64} - 336 q^{70} - 1304 q^{73} - 4872 q^{76} - 288 q^{79} + 1232 q^{82} + 6064 q^{88} + 2592 q^{94} + 2584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{18} - 2 x^{16} - 244 x^{14} - 2704 x^{12} + 19328 x^{10} - 173056 x^{8} - 999424 x^{6} + \cdots + 1073741824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 257 \nu^{18} + 1535 \nu^{16} + 45822 \nu^{14} + 33036 \nu^{12} - 65168 \nu^{10} + \cdots - 69860327424 ) / 696254464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{18} + \nu^{16} + 2 \nu^{14} + 244 \nu^{12} + 2704 \nu^{10} - 19328 \nu^{8} + \cdots + 14680064 ) / 2097152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 87 \nu^{18} - 905 \nu^{16} - 8050 \nu^{14} - 8532 \nu^{12} + 209008 \nu^{10} + \cdots + 16957571072 ) / 174063616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{19} + \nu^{17} + 2 \nu^{15} + 244 \nu^{13} + 2704 \nu^{11} - 19328 \nu^{9} + \cdots + 16777216 \nu ) / 16777216 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 495 \nu^{19} + 2289 \nu^{17} + 27970 \nu^{15} + 136820 \nu^{13} - 1065456 \nu^{11} + \cdots - 73870082048 \nu ) / 5570035712 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 515 \nu^{19} - 35 \nu^{17} + 24090 \nu^{15} - 119452 \nu^{13} + 765648 \nu^{11} + \cdots + 16424894464 \nu ) / 5570035712 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 915 \nu^{18} - 1517 \nu^{16} - 47962 \nu^{14} - 325860 \nu^{12} + 1156784 \nu^{10} + \cdots + 137355067392 ) / 696254464 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 141 \nu^{18} - 1195 \nu^{16} - 7598 \nu^{14} - 15148 \nu^{12} + 688 \nu^{10} + \cdots + 16318988288 ) / 87031808 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1423 \nu^{18} + 9713 \nu^{16} + 68962 \nu^{14} + 284852 \nu^{12} - 1700976 \nu^{10} + \cdots - 184465489920 ) / 696254464 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1845 \nu^{18} - 13579 \nu^{16} - 92502 \nu^{14} - 308476 \nu^{12} + 2274512 \nu^{10} + \cdots + 213188083712 ) / 696254464 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1597 \nu^{19} - 17027 \nu^{17} - 49862 \nu^{15} - 255068 \nu^{13} + 3992400 \nu^{11} + \cdots + 186612973568 \nu ) / 5570035712 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 55 \nu^{19} - 282 \nu^{17} - 2785 \nu^{15} - 15442 \nu^{13} + 48996 \nu^{11} + \cdots + 7791968256 \nu ) / 174063616 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3489 \nu^{19} - 23263 \nu^{17} - 111678 \nu^{15} - 358796 \nu^{13} + 1932944 \nu^{11} + \cdots + 316233744384 \nu ) / 5570035712 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 529 \nu^{19} - 4499 \nu^{17} - 42938 \nu^{15} - 73988 \nu^{13} + 690912 \nu^{11} + \cdots + 87711285248 \nu ) / 696254464 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 7695 \nu^{19} + 62609 \nu^{17} + 402754 \nu^{15} + 1589364 \nu^{13} - 9742064 \nu^{11} + \cdots - 1039767961600 \nu ) / 5570035712 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{17} + 2\beta_{16} + \beta_{15} - \beta_{10} + 5\beta_{9} + \beta_{8} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - 4\beta_{13} - 3\beta_{12} - 3\beta_{11} - 7\beta_{7} - 4\beta_{5} + 2\beta_{4} + 3\beta_{2} + 69 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{19} - 6 \beta_{18} - 11 \beta_{17} - 8 \beta_{16} - 9 \beta_{15} - 9 \beta_{10} + \cdots + 88 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33\beta_{14} + 40\beta_{13} - 3\beta_{12} + 5\beta_{11} - 11\beta_{7} - 4\beta_{6} + 55\beta_{2} + 1169 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12 \beta_{19} - 14 \beta_{18} + 27 \beta_{17} - 108 \beta_{16} + 33 \beta_{15} - 107 \beta_{10} + \cdots + 1092 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 83 \beta_{14} - 96 \beta_{13} - 345 \beta_{12} - 49 \beta_{11} + 407 \beta_{7} + 76 \beta_{6} + \cdots - 7845 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 140 \beta_{19} - 26 \beta_{18} - 623 \beta_{17} - 724 \beta_{16} + 451 \beta_{15} + 399 \beta_{10} + \cdots - 8868 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 119 \beta_{14} + 1824 \beta_{13} + 1477 \beta_{12} - 515 \beta_{11} + 1749 \beta_{7} + 2756 \beta_{6} + \cdots + 116929 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1052 \beta_{19} + 3602 \beta_{18} + 1211 \beta_{17} - 4652 \beta_{16} + \beta_{15} - 619 \beta_{10} + \cdots + 120548 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1485 \beta_{14} - 15808 \beta_{13} - 3961 \beta_{12} - 7505 \beta_{11} - 19049 \beta_{7} + \cdots + 1281659 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 12364 \beta_{19} - 32474 \beta_{18} + 12521 \beta_{17} - 10820 \beta_{16} + 14443 \beta_{15} + \cdots + 1289452 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 22897 \beta_{14} + 149888 \beta_{13} + 30221 \beta_{12} + 116549 \beta_{11} - 7299 \beta_{7} + \cdots + 1821897 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 219964 \beta_{19} + 46498 \beta_{18} - 24509 \beta_{17} + 536212 \beta_{16} + 36441 \beta_{15} + \cdots + 1489060 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 249445 \beta_{14} - 1161600 \beta_{13} - 1011377 \beta_{12} - 772425 \beta_{11} + 483103 \beta_{7} + \cdots + 15729507 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 2235244 \beta_{19} + 22390 \beta_{18} - 4813471 \beta_{17} + 123612 \beta_{16} + \cdots + 22401420 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
253.1
−2.82836 − 0.0196938i
−2.82836 + 0.0196938i
−2.53906 − 1.24627i
−2.53906 + 1.24627i
−1.82605 − 2.15998i
−1.82605 + 2.15998i
−1.54032 − 2.37222i
−1.54032 + 2.37222i
−0.310655 − 2.81132i
−0.310655 + 2.81132i
0.310655 − 2.81132i
0.310655 + 2.81132i
1.54032 − 2.37222i
1.54032 + 2.37222i
1.82605 − 2.15998i
1.82605 + 2.15998i
2.53906 − 1.24627i
2.53906 + 1.24627i
2.82836 − 0.0196938i
2.82836 + 0.0196938i
−2.82836 − 0.0196938i 0 7.99922 + 0.111402i 8.53716i 0 7.00000 −22.6225 − 0.472621i 0 0.168129 − 24.1462i
253.2 −2.82836 + 0.0196938i 0 7.99922 − 0.111402i − 8.53716i 0 7.00000 −22.6225 + 0.472621i 0 0.168129 + 24.1462i
253.3 −2.53906 − 1.24627i 0 4.89365 + 6.32868i 15.5203i 0 7.00000 −4.53804 − 22.1677i 0 19.3424 − 39.4069i
253.4 −2.53906 + 1.24627i 0 4.89365 − 6.32868i − 15.5203i 0 7.00000 −4.53804 + 22.1677i 0 19.3424 + 39.4069i
253.5 −1.82605 − 2.15998i 0 −1.33106 + 7.88849i − 20.7652i 0 7.00000 19.4696 − 11.5297i 0 −44.8524 + 37.9183i
253.6 −1.82605 + 2.15998i 0 −1.33106 − 7.88849i 20.7652i 0 7.00000 19.4696 + 11.5297i 0 −44.8524 − 37.9183i
253.7 −1.54032 − 2.37222i 0 −3.25483 + 7.30795i 8.05272i 0 7.00000 22.3495 − 3.53543i 0 19.1028 − 12.4038i
253.8 −1.54032 + 2.37222i 0 −3.25483 − 7.30795i − 8.05272i 0 7.00000 22.3495 + 3.53543i 0 19.1028 + 12.4038i
253.9 −0.310655 − 2.81132i 0 −7.80699 + 1.74670i − 2.04919i 0 7.00000 7.33580 + 21.4053i 0 −5.76093 + 0.636592i
253.10 −0.310655 + 2.81132i 0 −7.80699 − 1.74670i 2.04919i 0 7.00000 7.33580 − 21.4053i 0 −5.76093 − 0.636592i
253.11 0.310655 − 2.81132i 0 −7.80699 − 1.74670i − 2.04919i 0 7.00000 −7.33580 + 21.4053i 0 −5.76093 − 0.636592i
253.12 0.310655 + 2.81132i 0 −7.80699 + 1.74670i 2.04919i 0 7.00000 −7.33580 − 21.4053i 0 −5.76093 + 0.636592i
253.13 1.54032 − 2.37222i 0 −3.25483 − 7.30795i 8.05272i 0 7.00000 −22.3495 − 3.53543i 0 19.1028 + 12.4038i
253.14 1.54032 + 2.37222i 0 −3.25483 + 7.30795i − 8.05272i 0 7.00000 −22.3495 + 3.53543i 0 19.1028 − 12.4038i
253.15 1.82605 − 2.15998i 0 −1.33106 − 7.88849i − 20.7652i 0 7.00000 −19.4696 − 11.5297i 0 −44.8524 − 37.9183i
253.16 1.82605 + 2.15998i 0 −1.33106 + 7.88849i 20.7652i 0 7.00000 −19.4696 + 11.5297i 0 −44.8524 + 37.9183i
253.17 2.53906 − 1.24627i 0 4.89365 − 6.32868i 15.5203i 0 7.00000 4.53804 − 22.1677i 0 19.3424 + 39.4069i
253.18 2.53906 + 1.24627i 0 4.89365 + 6.32868i − 15.5203i 0 7.00000 4.53804 + 22.1677i 0 19.3424 − 39.4069i
253.19 2.82836 − 0.0196938i 0 7.99922 − 0.111402i 8.53716i 0 7.00000 22.6225 − 0.472621i 0 0.168129 + 24.1462i
253.20 2.82836 + 0.0196938i 0 7.99922 + 0.111402i − 8.53716i 0 7.00000 22.6225 + 0.472621i 0 0.168129 − 24.1462i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 253.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.4.c.f ✓ 20
3.b odd 2 1 inner 504.4.c.f ✓ 20
4.b odd 2 1 2016.4.c.e 20
8.b even 2 1 inner 504.4.c.f ✓ 20
8.d odd 2 1 2016.4.c.e 20
12.b even 2 1 2016.4.c.e 20
24.f even 2 1 2016.4.c.e 20
24.h odd 2 1 inner 504.4.c.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.4.c.f ✓ 20 1.a even 1 1 trivial
504.4.c.f ✓ 20 3.b odd 2 1 inner
504.4.c.f ✓ 20 8.b even 2 1 inner
504.4.c.f ✓ 20 24.h odd 2 1 inner
2016.4.c.e 20 4.b odd 2 1
2016.4.c.e 20 8.d odd 2 1
2016.4.c.e 20 12.b even 2 1
2016.4.c.e 20 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 814T_{5}^{8} + 204556T_{5}^{6} + 18326344T_{5}^{4} + 564297216T_{5}^{2} + 2061333632 \) acting on \(S_{4}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 1073741824 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + 814 T^{8} + \cdots + 2061333632)^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 271161000722432)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 39\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 22\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 51\!\cdots\!12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 66 T^{4} + \cdots - 18972771200)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 92\!\cdots\!20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 15\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 12\!\cdots\!80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 23\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 40\!\cdots\!40)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 62\!\cdots\!60)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 16\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 57607517656864)^{4} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 5690359808000)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 44\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 65\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 48351375161248)^{4} \) Copy content Toggle raw display
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