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Properties

Label 3520.2.f.l
Level $3520$
Weight $2$
Character orbit 3520.f
Analytic conductor $28.107$
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] = [3520,2,Mod(2111,3520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3520.2111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3520.f (of order \(2\), degree \(1\), not 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: \(28.1073415115\)
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: \( x^{16} - 14x^{14} - 7x^{12} + 560x^{10} + 2104x^{8} + 2240x^{6} - 112x^{4} - 896x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 1760)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - q^{5} - \beta_{9} q^{7} + (\beta_{2} - \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - q^{5} - \beta_{9} q^{7} + (\beta_{2} - \beta_1 - 2) q^{9} - \beta_{12} q^{11} + (\beta_{8} + \beta_{7}) q^{13} - \beta_{4} q^{15} - \beta_{3} q^{17} + ( - \beta_{14} - \beta_{6}) q^{19} + ( - \beta_{8} + \beta_{3}) q^{21} + (\beta_{15} - \beta_{4}) q^{23} + q^{25} + ( - \beta_{13} + \beta_{12} + \cdots - 2 \beta_{4}) q^{27}+ \cdots + ( - \beta_{15} - 3 \beta_{14} + \cdots - 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{5} - 24 q^{9} + 16 q^{25} - 16 q^{33} + 72 q^{37} + 24 q^{45} + 40 q^{49} - 40 q^{53} + 80 q^{69} + 56 q^{77} + 112 q^{81} + 56 q^{89} + 8 q^{93} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 14x^{14} - 7x^{12} + 560x^{10} + 2104x^{8} + 2240x^{6} - 112x^{4} - 896x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - 148\nu^{12} + 2783\nu^{10} - 7698\nu^{8} - 78108\nu^{6} - 81528\nu^{4} + 78464\nu^{2} - 63360 ) / 32768 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 209 \nu^{14} + 2828 \nu^{12} + 3407 \nu^{10} - 125522 \nu^{8} - 480604 \nu^{6} - 374392 \nu^{4} + \cdots + 141440 ) / 65536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 209 \nu^{14} - 2924 \nu^{12} - 1423 \nu^{10} + 116082 \nu^{8} + 439516 \nu^{6} + 516856 \nu^{4} + \cdots - 98432 ) / 65536 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 67 \nu^{15} - 960 \nu^{13} - 133 \nu^{11} + 37226 \nu^{9} + 129412 \nu^{7} + 117304 \nu^{5} + \cdots - 2176 \nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 297 \nu^{14} + 4172 \nu^{12} + 2231 \nu^{10} - 171746 \nu^{8} - 611708 \nu^{6} - 457016 \nu^{4} + \cdots + 597120 ) / 65536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 71 \nu^{15} - 1240 \nu^{13} + 3343 \nu^{11} + 35178 \nu^{9} + 21524 \nu^{7} - 166152 \nu^{5} + \cdots + 142208 \nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -41\nu^{14} + 588\nu^{12} + 55\nu^{10} - 22498\nu^{8} - 79100\nu^{6} - 80696\nu^{4} - 13184\nu^{2} + 13440 ) / 4096 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 175 \nu^{14} - 2584 \nu^{12} + 823 \nu^{10} + 96474 \nu^{8} + 292340 \nu^{6} + 214072 \nu^{4} + \cdots - 20608 ) / 16384 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 93 \nu^{15} + 1354 \nu^{13} - 137 \nu^{11} - 51540 \nu^{9} - 167044 \nu^{7} - 133056 \nu^{5} + \cdots + 72192 \nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 471 \nu^{14} - 5484 \nu^{12} - 19737 \nu^{10} + 270022 \nu^{8} + 1596388 \nu^{6} + 2916712 \nu^{4} + \cdots - 717184 ) / 32768 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 119 \nu^{15} - 1692 \nu^{13} - 409 \nu^{11} + 66070 \nu^{9} + 233764 \nu^{7} + 251560 \nu^{5} + \cdots + 25216 \nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 133 \nu^{15} - 1732 \nu^{13} - 2891 \nu^{11} + 75730 \nu^{9} + 350636 \nu^{7} + 493624 \nu^{5} + \cdots - 148608 \nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 503 \nu^{15} - 6956 \nu^{13} - 4217 \nu^{11} + 272838 \nu^{9} + 1120868 \nu^{7} + 1563496 \nu^{5} + \cdots - 323968 \nu ) / 65536 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 119 \nu^{15} - 1748 \nu^{13} + 407 \nu^{11} + 65854 \nu^{9} + 204676 \nu^{7} + 148808 \nu^{5} + \cdots - 76672 \nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1033 \nu^{15} - 14036 \nu^{13} - 12679 \nu^{11} + 567354 \nu^{9} + 2423196 \nu^{7} + \cdots - 740992 \nu ) / 65536 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + 2\beta_{9} + 2\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 2\beta_{5} - 2\beta_{3} + 4\beta_{2} - 2\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{15} + 12\beta_{14} - 5\beta_{13} + \beta_{12} - 3\beta_{11} + 16\beta_{9} + 2\beta_{6} + 2\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{10} + 11\beta_{8} - 18\beta_{5} - 52\beta_{3} + 24\beta_{2} - 6\beta _1 + 102 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 51 \beta_{15} + 102 \beta_{14} - 77 \beta_{13} - 13 \beta_{12} - 41 \beta_{11} + 136 \beta_{9} + \cdots + 102 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 35\beta_{10} + 115\beta_{8} - 70\beta_{7} - 130\beta_{5} - 760\beta_{3} + 188\beta_{2} - 66\beta _1 + 650 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 701 \beta_{15} + 550 \beta_{14} - 1083 \beta_{13} - 263 \beta_{12} - 443 \beta_{11} + \cdots + 1314 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 477\beta_{10} + 1653\beta_{8} - 678\beta_{7} - 394\beta_{5} - 9816\beta_{3} + 560\beta_{2} - 182\beta _1 + 2030 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8367 \beta_{15} - 1142 \beta_{14} - 12897 \beta_{13} - 4429 \beta_{12} - 4041 \beta_{11} + \cdots + 15766 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5247 \beta_{10} + 17919 \beta_{8} - 8162 \beta_{7} + 8406 \beta_{5} - 109296 \beta_{3} - 11996 \beta_{2} + \cdots - 42998 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 87293 \beta_{15} - 97154 \beta_{14} - 134507 \beta_{13} - 58959 \beta_{12} - 29475 \beta_{11} + \cdots + 164594 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 51005 \beta_{10} + 174885 \beta_{8} - 77630 \beta_{7} + 230238 \beta_{5} - 1059320 \beta_{3} + \cdots - 1179810 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 774375 \beta_{15} - 1917654 \beta_{14} - 1193657 \beta_{13} - 684117 \beta_{12} + \cdots + 1459142 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 403351 \beta_{10} + 1381575 \beta_{8} - 617266 \beta_{7} + 3816302 \beta_{5} - 8383232 \beta_{3} + \cdots - 19547206 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5177973 \beta_{15} - 28271026 \beta_{14} - 7979875 \beta_{13} - 6927335 \beta_{12} + \cdots + 9760322 \beta_{4} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(1541\) \(2751\) \(2817\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2111.1
−0.303877 − 1.66348i
0.303877 − 1.66348i
0.212539 − 1.16348i
−0.212539 − 1.16348i
3.32004 − 0.606491i
−3.32004 − 0.606491i
0.582949 − 0.106491i
−0.582949 − 0.106491i
0.582949 + 0.106491i
−0.582949 + 0.106491i
3.32004 + 0.606491i
−3.32004 + 0.606491i
0.212539 + 1.16348i
−0.212539 + 1.16348i
−0.303877 + 1.66348i
0.303877 + 1.66348i
0 − 3.32695i 0 −1.00000 0 −0.806460 0 −8.06861 0
2111.2 0 − 3.32695i 0 −1.00000 0 0.806460 0 −8.06861 0
2111.3 0 − 2.32695i 0 −1.00000 0 −1.83929 0 −2.41471 0
2111.4 0 − 2.32695i 0 −1.00000 0 1.83929 0 −2.41471 0
2111.5 0 − 1.21298i 0 −1.00000 0 −5.22586 0 1.52868 0
2111.6 0 − 1.21298i 0 −1.00000 0 5.22586 0 1.52868 0
2111.7 0 − 0.212981i 0 −1.00000 0 −2.58011 0 2.95464 0
2111.8 0 − 0.212981i 0 −1.00000 0 2.58011 0 2.95464 0
2111.9 0 0.212981i 0 −1.00000 0 −2.58011 0 2.95464 0
2111.10 0 0.212981i 0 −1.00000 0 2.58011 0 2.95464 0
2111.11 0 1.21298i 0 −1.00000 0 −5.22586 0 1.52868 0
2111.12 0 1.21298i 0 −1.00000 0 5.22586 0 1.52868 0
2111.13 0 2.32695i 0 −1.00000 0 −1.83929 0 −2.41471 0
2111.14 0 2.32695i 0 −1.00000 0 1.83929 0 −2.41471 0
2111.15 0 3.32695i 0 −1.00000 0 −0.806460 0 −8.06861 0
2111.16 0 3.32695i 0 −1.00000 0 0.806460 0 −8.06861 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2111.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.2.f.l 16
4.b odd 2 1 inner 3520.2.f.l 16
8.b even 2 1 1760.2.f.c ✓ 16
8.d odd 2 1 1760.2.f.c ✓ 16
11.b odd 2 1 inner 3520.2.f.l 16
44.c even 2 1 inner 3520.2.f.l 16
88.b odd 2 1 1760.2.f.c ✓ 16
88.g even 2 1 1760.2.f.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1760.2.f.c ✓ 16 8.b even 2 1
1760.2.f.c ✓ 16 8.d odd 2 1
1760.2.f.c ✓ 16 88.b odd 2 1
1760.2.f.c ✓ 16 88.g even 2 1
3520.2.f.l 16 1.a even 1 1 trivial
3520.2.f.l 16 4.b odd 2 1 inner
3520.2.f.l 16 11.b odd 2 1 inner
3520.2.f.l 16 44.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3520, [\chi])\):

\( T_{3}^{8} + 18T_{3}^{6} + 85T_{3}^{4} + 92T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} - 38T_{7}^{6} + 321T_{7}^{4} - 808T_{7}^{2} + 400 \) Copy content Toggle raw display
\( T_{19}^{8} - 134T_{19}^{6} + 5617T_{19}^{4} - 70872T_{19}^{2} + 48400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 18 T^{6} + 85 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 38 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 6 T^{2} + 121)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 52 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 46 T^{6} + 85 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 134 T^{6} + \cdots + 48400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 100 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 98 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 78 T^{6} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 18 T^{3} + 85 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 144 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 200 T^{6} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 308 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 10 T^{3} + \cdots + 3140)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 368 T^{6} + \cdots + 49561600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 514 T^{6} + \cdots + 203689984)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 204 T^{6} + \cdots + 1000000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 190 T^{6} + \cdots + 246016)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 52 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 376 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 72 T^{2} + 400)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 14 T^{3} + \cdots + 2480)^{4} \) Copy content Toggle raw display
$97$ \( (T - 6)^{16} \) Copy content Toggle raw display
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