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Properties

Label 5472.2.f.c
Level $5472$
Weight $2$
Character orbit 5472.f
Analytic conductor $43.694$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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] = [5472,2,Mod(1025,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5472.1025"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,28,0,0,0,-16,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(59)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(43.6941399860\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 38 x^{18} + 528 x^{16} + 3442 x^{14} + 11480 x^{12} + 20550 x^{10} + 20369 x^{8} + 11136 x^{6} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + \beta_{3} q^{7} + \beta_{14} q^{11} - \beta_{4} q^{13} - \beta_{13} q^{17} - \beta_{10} q^{19} - \beta_1 q^{23} + \beta_{17} q^{25} + ( - \beta_{17} + \beta_{6}) q^{29} + (\beta_{12} + \beta_1) q^{31}+ \cdots + ( - \beta_{13} + \beta_{8} + \cdots + \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{25} - 16 q^{41} + 28 q^{49} - 16 q^{53} + 16 q^{65} + 16 q^{73} - 32 q^{85} + 48 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 38 x^{18} + 528 x^{16} + 3442 x^{14} + 11480 x^{12} + 20550 x^{10} + 20369 x^{8} + 11136 x^{6} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3375463 \nu^{19} + 109994564 \nu^{17} + 1092479040 \nu^{15} + 2150159350 \nu^{13} + \cdots - 7927734152 \nu ) / 145581952 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1662489 \nu^{18} - 58136340 \nu^{16} - 692107628 \nu^{14} - 3275216302 \nu^{12} + \cdots + 17509352 ) / 36395488 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3906181 \nu^{18} + 139161012 \nu^{16} + 1721910880 \nu^{14} + 8984534898 \nu^{12} + \cdots - 502294392 ) / 72790976 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7168371 \nu^{19} + 274879792 \nu^{17} + 3872946720 \nu^{15} + 25754466246 \nu^{13} + \cdots + 2031272456 \nu ) / 72790976 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13275921 \nu^{19} + 496732304 \nu^{17} + 6718512348 \nu^{15} + 41732925126 \nu^{13} + \cdots + 1288480072 \nu ) / 72790976 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2355976 \nu^{18} + 88732259 \nu^{16} + 1213945952 \nu^{14} + 7696989450 \nu^{12} + 24413784572 \nu^{10} + \cdots + 62079052 ) / 9098872 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21669045 \nu^{18} - 825917620 \nu^{16} - 11535889808 \nu^{14} - 75890015202 \nu^{12} + \cdots - 3692682184 ) / 72790976 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 44248481 \nu^{19} - 1652166192 \nu^{17} - 22277172540 \nu^{15} - 137823430086 \nu^{13} + \cdots - 3242421576 \nu ) / 72790976 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 77070210 \nu^{19} + 20511945 \nu^{18} + 2920062096 \nu^{17} + 776456516 \nu^{16} + \cdots + 453365192 ) / 145581952 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 77070210 \nu^{19} - 20511945 \nu^{18} + 2920062096 \nu^{17} - 776456516 \nu^{16} + \cdots - 453365192 ) / 145581952 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 22040181 \nu^{18} - 830526256 \nu^{16} - 11371525148 \nu^{14} - 72181436510 \nu^{12} + \cdots - 872099144 ) / 36395488 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 122515039 \nu^{19} - 4620791100 \nu^{17} - 63368702128 \nu^{15} - 403435929574 \nu^{13} + \cdots - 7528244184 \nu ) / 145581952 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15371807 \nu^{19} + 578330812 \nu^{17} + 7897953657 \nu^{15} + 49922156229 \nu^{13} + \cdots + 109629868 \nu ) / 18197744 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 123894347 \nu^{19} + 4674498956 \nu^{17} + 64150284256 \nu^{15} + 409016534510 \nu^{13} + \cdots + 10422744952 \nu ) / 145581952 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 51865393 \nu^{18} - 1955036468 \nu^{16} - 26786864688 \nu^{14} - 170311149114 \nu^{12} + \cdots - 3173055080 ) / 72790976 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1382811 \nu^{19} - 51876312 \nu^{17} - 704981274 \nu^{15} - 4418206044 \nu^{13} + \cdots - 23096304 \nu ) / 1174048 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 19639281 \nu^{18} + 733743728 \nu^{16} + 9902270396 \nu^{14} + 61328367142 \nu^{12} + \cdots + 521896888 ) / 18197744 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 100292835 \nu^{19} - 3760972596 \nu^{17} - 51082942400 \nu^{15} - 319997466254 \nu^{13} + \cdots - 5506509272 \nu ) / 72790976 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 115882331 \nu^{18} - 4354638052 \nu^{16} - 59348265768 \nu^{14} - 373816931734 \nu^{12} + \cdots - 5618817976 ) / 72790976 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{16} - 6\beta_{14} - 3\beta_{12} + 3\beta_{4} - 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{19} - 3\beta_{17} - 5\beta_{15} - \beta_{10} + \beta_{9} + \beta_{7} - 9\beta_{6} + 8\beta_{3} - 48 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 34 \beta_{16} + 60 \beta_{14} + 15 \beta_{13} + 45 \beta_{12} + 9 \beta_{10} + 9 \beta_{9} + \cdots + 39 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{19} + 5 \beta_{17} + 31 \beta_{15} - 2 \beta_{11} - \beta_{10} + \beta_{9} - 9 \beta_{7} + \cdots + 166 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{18} - 508 \beta_{16} - 786 \beta_{14} - 333 \beta_{13} - 705 \beta_{12} - 135 \beta_{10} + \cdots - 501 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 205 \beta_{19} + 9 \beta_{17} - 1477 \beta_{15} - 24 \beta_{11} + 241 \beta_{10} - 241 \beta_{9} + \cdots - 6414 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 54 \beta_{18} + 7508 \beta_{16} + 11394 \beta_{14} + 5919 \beta_{13} + 11235 \beta_{12} + \cdots + 6891 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1493 \beta_{19} - 667 \beta_{17} + 7571 \beta_{15} + 770 \beta_{11} - 1941 \beta_{10} + 1941 \beta_{9} + \cdots + 29822 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3708 \beta_{18} - 112154 \beta_{16} - 171612 \beta_{14} - 98523 \beta_{13} - 178263 \beta_{12} + \cdots - 98949 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 80873 \beta_{19} + 46347 \beta_{17} - 348203 \beta_{15} - 59148 \beta_{11} + 112541 \beta_{10} + \cdots - 1301676 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 94524 \beta_{18} + 1696528 \beta_{16} + 2627370 \beta_{14} + 1594215 \beta_{13} + 2811771 \beta_{12} + \cdots + 1460919 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 453793 \beta_{19} - 284577 \beta_{17} + 1785421 \beta_{15} + 391292 \beta_{11} - 659861 \beta_{10} + \cdots + 6474534 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1886604 \beta_{18} - 25918948 \beta_{16} - 40544862 \beta_{14} - 25418613 \beta_{13} + \cdots - 21974985 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 22175569 \beta_{19} + 14503803 \beta_{17} - 82719127 \beta_{15} - 20932332 \beta_{11} + \cdots - 294471708 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 33802764 \beta_{18} + 398765102 \beta_{16} + 628332804 \beta_{14} + 401864901 \beta_{13} + \cdots + 334693899 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 118316765 \beta_{19} - 79129131 \beta_{17} + 427167671 \beta_{15} + 117768482 \beta_{11} + \cdots + 1503243262 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 572393706 \beta_{18} - 6164104808 \beta_{16} - 9761047206 \beta_{14} - 6321102405 \beta_{13} + \cdots - 5139954453 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 5624759621 \beta_{19} + 3809971185 \beta_{17} - 19897989077 \beta_{15} - 5781712680 \beta_{11} + \cdots - 69514744206 ) / 12 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 9386689542 \beta_{18} + 95582270080 \beta_{16} + 151856386806 \beta_{14} + 99114245331 \beta_{13} + \cdots + 79364933859 \beta_1 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(2053\) \(3745\) \(4447\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
− 1.10730i
0.784661i
− 0.250650i
− 0.517163i
3.94907i
− 1.72939i
− 3.15931i
− 0.656771i
2.39458i
− 1.04664i
− 2.39458i
1.04664i
3.15931i
0.656771i
− 3.94907i
1.72939i
0.250650i
0.517163i
1.10730i
− 0.784661i
0 0 0 − 3.50596i 0 −2.78457 0 0 0
1025.2 0 0 0 − 3.50596i 0 2.78457 0 0 0
1025.3 0 0 0 − 2.77362i 0 −2.56051 0 0 0
1025.4 0 0 0 − 2.77362i 0 2.56051 0 0 0
1025.5 0 0 0 − 2.17575i 0 −1.38237 0 0 0
1025.6 0 0 0 − 2.17575i 0 1.38237 0 0 0
1025.7 0 0 0 − 1.06784i 0 −2.36143 0 0 0
1025.8 0 0 0 − 1.06784i 0 2.36143 0 0 0
1025.9 0 0 0 − 0.375576i 0 −4.49473 0 0 0
1025.10 0 0 0 − 0.375576i 0 4.49473 0 0 0
1025.11 0 0 0 0.375576i 0 −4.49473 0 0 0
1025.12 0 0 0 0.375576i 0 4.49473 0 0 0
1025.13 0 0 0 1.06784i 0 −2.36143 0 0 0
1025.14 0 0 0 1.06784i 0 2.36143 0 0 0
1025.15 0 0 0 2.17575i 0 −1.38237 0 0 0
1025.16 0 0 0 2.17575i 0 1.38237 0 0 0
1025.17 0 0 0 2.77362i 0 −2.56051 0 0 0
1025.18 0 0 0 2.77362i 0 2.56051 0 0 0
1025.19 0 0 0 3.50596i 0 −2.78457 0 0 0
1025.20 0 0 0 3.50596i 0 2.78457 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
57.d even 2 1 inner
228.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5472.2.f.c ✓ 20
3.b odd 2 1 5472.2.f.d yes 20
4.b odd 2 1 inner 5472.2.f.c ✓ 20
12.b even 2 1 5472.2.f.d yes 20
19.b odd 2 1 5472.2.f.d yes 20
57.d even 2 1 inner 5472.2.f.c ✓ 20
76.d even 2 1 5472.2.f.d yes 20
228.b odd 2 1 inner 5472.2.f.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5472.2.f.c ✓ 20 1.a even 1 1 trivial
5472.2.f.c ✓ 20 4.b odd 2 1 inner
5472.2.f.c ✓ 20 57.d even 2 1 inner
5472.2.f.c ✓ 20 228.b odd 2 1 inner
5472.2.f.d yes 20 3.b odd 2 1
5472.2.f.d yes 20 12.b even 2 1
5472.2.f.d yes 20 19.b odd 2 1
5472.2.f.d yes 20 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5472, [\chi])\):

\( T_{5}^{10} + 26T_{5}^{8} + 221T_{5}^{6} + 694T_{5}^{4} + 604T_{5}^{2} + 72 \) Copy content Toggle raw display
\( T_{29}^{5} - 72T_{29}^{3} - 16T_{29}^{2} + 1104T_{29} + 1472 \) Copy content Toggle raw display
\( T_{59}^{10} - 488T_{59}^{8} + 91856T_{59}^{6} - 8339968T_{59}^{4} + 365548544T_{59}^{2} - 6188875776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + 26 T^{8} + \cdots + 72)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} - 42 T^{8} + \cdots - 10944)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 70 T^{8} + \cdots + 54872)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 80 T^{8} + \cdots + 165888)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 82 T^{8} + \cdots + 69192)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( (T^{10} + 110 T^{8} + \cdots + 608)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 72 T^{3} + \cdots + 1472)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + 144 T^{8} + \cdots + 2490368)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 256 T^{8} + \cdots + 26615808)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + 4 T^{4} + \cdots + 2432)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} - 242 T^{8} + \cdots - 11206656)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 190 T^{8} + \cdots + 80408)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 4 T^{4} + \cdots + 1024)^{4} \) Copy content Toggle raw display
$59$ \( (T^{10} - 488 T^{8} + \cdots - 6188875776)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 139 T^{3} + \cdots + 192)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + 384 T^{8} + \cdots + 39845888)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 696 T^{8} + \cdots - 2801664)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 4 T^{4} + \cdots - 456)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + 496 T^{8} + \cdots + 294524928)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 230 T^{8} + \cdots + 175712)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} - 12 T^{4} + \cdots + 512)^{4} \) Copy content Toggle raw display
$97$ \( (T^{10} + 416 T^{8} + \cdots + 2654208)^{2} \) Copy content Toggle raw display
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