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Properties

Label 3380.2.f.j
Level $3380$
Weight $2$
Character orbit 3380.f
Analytic conductor $26.989$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3380,2,Mod(3041,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3380.3041");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.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: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 39x^{16} + 605x^{14} + 4764x^{12} + 20080x^{10} + 43783x^{8} + 43791x^{6} + 14071x^{4} + 378x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{14} q^{5} + ( - \beta_{16} + \beta_{10} - \beta_{7}) q^{7} + ( - \beta_{8} + \beta_{6} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{14} q^{5} + ( - \beta_{16} + \beta_{10} - \beta_{7}) q^{7} + ( - \beta_{8} + \beta_{6} + \beta_{3} + 2) q^{9} + ( - \beta_{16} + \beta_{12} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{17} - 6 \beta_{15} + \cdots + 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{3} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{3} + 24 q^{9} - 26 q^{17} - 24 q^{23} - 18 q^{25} - 8 q^{27} + 32 q^{29} + 2 q^{35} - 2 q^{43} - 40 q^{49} - 22 q^{51} + 60 q^{53} - 14 q^{55} + 42 q^{61} - 30 q^{69} + 2 q^{75} - 92 q^{77} + 62 q^{79} + 82 q^{81} + 56 q^{87} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 39x^{16} + 605x^{14} + 4764x^{12} + 20080x^{10} + 43783x^{8} + 43791x^{6} + 14071x^{4} + 378x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4465 \nu^{16} + 164340 \nu^{14} + 2362668 \nu^{12} + 16811588 \nu^{10} + 61753817 \nu^{8} + \cdots - 282419 ) / 7098676 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 64216 \nu^{16} + 2866647 \nu^{14} + 50151420 \nu^{12} + 436629877 \nu^{10} + 1985264696 \nu^{8} + \cdots - 79342410 ) / 92282788 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 77915 \nu^{16} + 3588696 \nu^{14} + 63670590 \nu^{12} + 551655274 \nu^{10} + 2439238251 \nu^{8} + \cdots + 1273681 ) / 92282788 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 488576 \nu^{16} - 19122613 \nu^{14} - 297638486 \nu^{12} - 2350474613 \nu^{10} + \cdots - 136674392 ) / 92282788 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 557109 \nu^{16} + 21447549 \nu^{14} + 327848720 \nu^{12} + 2538316715 \nu^{10} + 10479912565 \nu^{8} + \cdots - 23485091 ) / 92282788 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 557109 \nu^{17} - 21447549 \nu^{15} - 327848720 \nu^{13} - 2538316715 \nu^{11} + \cdots + 23485091 \nu ) / 92282788 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 621325 \nu^{16} + 24314196 \nu^{14} + 378000140 \nu^{12} + 2974946592 \nu^{10} + 12465177261 \nu^{8} + \cdots + 358586439 ) / 92282788 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 702873 \nu^{16} - 27426046 \nu^{14} - 425620419 \nu^{12} - 3350803582 \nu^{10} + \cdots + 125946598 ) / 92282788 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1045685 \nu^{17} - 40570162 \nu^{15} - 625487206 \nu^{13} - 4888791328 \nu^{11} + \cdots - 113189301 \nu ) / 92282788 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1273681 \nu^{17} + 49595644 \nu^{15} + 766988309 \nu^{13} + 6004145694 \nu^{11} + \cdots - 714795114 \nu ) / 92282788 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2183469 \nu^{17} - 85345401 \nu^{15} - 1328391613 \nu^{13} - 10517013529 \nu^{11} + \cdots - 2826146910 \nu ) / 92282788 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1639059 \nu^{16} - 63977882 \nu^{14} - 993431409 \nu^{12} - 7830947983 \nu^{10} + \cdots - 238797188 ) / 46141394 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 282419 \nu^{17} + 11018806 \nu^{15} + 171027835 \nu^{13} + 1347806784 \nu^{11} + \cdots + 142210062 \nu ) / 7098676 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7995250 \nu^{17} - 311770404 \nu^{15} - 4835711879 \nu^{13} - 38072175486 \nu^{11} + \cdots - 2439514483 \nu ) / 92282788 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 17063199 \nu^{17} + 665280138 \nu^{15} + 10316637619 \nu^{13} + 81197440434 \nu^{11} + \cdots + 6881928752 \nu ) / 92282788 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 27127423 \nu^{17} + 1057647528 \nu^{15} + 16399385740 \nu^{13} + 129035466304 \nu^{11} + \cdots + 5280657055 \nu ) / 92282788 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} + 4\beta_{15} - 2\beta_{14} - 3\beta_{11} + \beta_{10} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - 8\beta_{8} + 10\beta_{6} - 4\beta_{5} - 3\beta_{4} + 14\beta_{3} + \beta_{2} + 45 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{17} - 13 \beta_{16} - 52 \beta_{15} + 34 \beta_{14} + 3 \beta_{12} + 38 \beta_{11} + \cdots + 74 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 21 \beta_{13} + 8 \beta_{9} + 74 \beta_{8} - 106 \beta_{6} + 88 \beta_{5} + 47 \beta_{4} - 162 \beta_{3} + \cdots - 444 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 162 \beta_{17} + 155 \beta_{16} + 586 \beta_{15} - 454 \beta_{14} - 55 \beta_{12} - 418 \beta_{11} + \cdots - 747 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 319 \beta_{13} - 160 \beta_{9} - 747 \beta_{8} + 1179 \beta_{6} - 1374 \beta_{5} - 599 \beta_{4} + \cdots + 4607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1817 \beta_{17} - 1830 \beta_{16} - 6478 \beta_{15} + 5706 \beta_{14} + 759 \beta_{12} + \cdots + 7931 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4282 \beta_{13} + 2332 \beta_{9} + 7931 \beta_{8} - 13469 \beta_{6} + 18714 \beta_{5} + 7186 \beta_{4} + \cdots - 49442 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 20375 \beta_{17} + 21579 \beta_{16} + 71960 \beta_{15} - 69882 \beta_{14} - 9518 \beta_{12} + \cdots - 86883 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 54093 \beta_{13} - 30284 \beta_{9} - 86883 \beta_{8} + 155767 \beta_{6} - 238342 \beta_{5} + \cdots + 543378 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 229877 \beta_{17} - 254016 \beta_{16} - 806718 \beta_{15} + 842378 \beta_{14} + 114581 \beta_{12} + \cdots + 971160 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 661032 \beta_{13} + 373020 \beta_{9} + 971160 \beta_{8} - 1810342 \beta_{6} + 2926706 \beta_{5} + \cdots - 6073279 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2610652 \beta_{17} + 2982602 \beta_{16} + 9122264 \beta_{15} - 10041960 \beta_{14} + \cdots - 10998201 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 7923540 \beta_{13} - 4473056 \beta_{9} - 10998201 \beta_{8} + 21073451 \beta_{6} - 35185966 \beta_{5} + \cdots + 68686421 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 29814825 \beta_{17} - 34928905 \beta_{16} - 103879098 \beta_{15} + 118755852 \beta_{14} + \cdots + 125611728 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3041.1
− 3.40622i
3.40622i
− 2.52382i
2.52382i
− 1.40883i
1.40883i
− 0.715841i
0.715841i
− 0.161999i
0.161999i
− 0.0545075i
0.0545075i
1.65879i
− 1.65879i
2.79462i
− 2.79462i
2.81781i
− 2.81781i
0 −3.40622 0 − 1.00000i 0 2.33790i 0 8.60234 0
3041.2 0 −3.40622 0 1.00000i 0 − 2.33790i 0 8.60234 0
3041.3 0 −2.52382 0 − 1.00000i 0 − 4.54727i 0 3.36968 0
3041.4 0 −2.52382 0 1.00000i 0 4.54727i 0 3.36968 0
3041.5 0 −1.40883 0 − 1.00000i 0 − 2.17051i 0 −1.01520 0
3041.6 0 −1.40883 0 1.00000i 0 2.17051i 0 −1.01520 0
3041.7 0 −0.715841 0 − 1.00000i 0 3.76323i 0 −2.48757 0
3041.8 0 −0.715841 0 1.00000i 0 − 3.76323i 0 −2.48757 0
3041.9 0 −0.161999 0 − 1.00000i 0 1.89771i 0 −2.97376 0
3041.10 0 −0.161999 0 1.00000i 0 − 1.89771i 0 −2.97376 0
3041.11 0 −0.0545075 0 − 1.00000i 0 3.71818i 0 −2.99703 0
3041.12 0 −0.0545075 0 1.00000i 0 − 3.71818i 0 −2.99703 0
3041.13 0 1.65879 0 − 1.00000i 0 − 4.25414i 0 −0.248414 0
3041.14 0 1.65879 0 1.00000i 0 4.25414i 0 −0.248414 0
3041.15 0 2.79462 0 − 1.00000i 0 1.22908i 0 4.80991 0
3041.16 0 2.79462 0 1.00000i 0 − 1.22908i 0 4.80991 0
3041.17 0 2.81781 0 − 1.00000i 0 − 0.974186i 0 4.94004 0
3041.18 0 2.81781 0 1.00000i 0 0.974186i 0 4.94004 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3041.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.f.j 18
13.b even 2 1 inner 3380.2.f.j 18
13.d odd 4 1 3380.2.a.r ✓ 9
13.d odd 4 1 3380.2.a.s yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.2.a.r ✓ 9 13.d odd 4 1
3380.2.a.s yes 9 13.d odd 4 1
3380.2.f.j 18 1.a even 1 1 trivial
3380.2.f.j 18 13.b 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 \(S_{2}^{\mathrm{new}}(3380, [\chi])\):

\( T_{3}^{9} + T_{3}^{8} - 19T_{3}^{7} - 16T_{3}^{6} + 106T_{3}^{5} + 87T_{3}^{4} - 153T_{3}^{3} - 149T_{3}^{2} - 26T_{3} - 1 \) Copy content Toggle raw display
\( T_{19}^{18} + 190 T_{19}^{16} + 13825 T_{19}^{14} + 474625 T_{19}^{12} + 7491320 T_{19}^{10} + \cdots + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{9} + T^{8} - 19 T^{7} + \cdots - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{18} + 83 T^{16} + \cdots + 9740641 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 8839008256 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( (T^{9} + 13 T^{8} + \cdots - 86528)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + 190 T^{16} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{9} + 12 T^{8} + \cdots - 46171)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} - 16 T^{8} + \cdots - 96559)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 11164880896 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 351637504 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 569451407438761 \) Copy content Toggle raw display
$43$ \( (T^{9} + T^{8} + \cdots + 9484117)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 649934807204761 \) Copy content Toggle raw display
$53$ \( (T^{9} - 30 T^{8} + \cdots + 118208)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + 467 T^{16} + \cdots + 262144 \) Copy content Toggle raw display
$61$ \( (T^{9} - 21 T^{8} + \cdots + 1253057)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 23538032263201 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 49655360782336 \) Copy content Toggle raw display
$79$ \( (T^{9} - 31 T^{8} + \cdots + 342428864)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 42\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 61614210678784 \) Copy content Toggle raw display
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