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Properties

Label 3420.2.bj.c
Level $3420$
Weight $2$
Character orbit 3420.bj
Analytic conductor $27.309$
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] = [3420,2,Mod(1189,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bj (of order \(6\), degree \(2\), 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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} + (\beta_{18} - \beta_{15}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{5} + (\beta_{18} - \beta_{15}) q^{7} + \beta_{6} q^{11} + (\beta_{19} - \beta_{17} + \beta_{9}) q^{13} + (\beta_{15} + \beta_{9} + \cdots - \beta_{7}) q^{17}+ \cdots + (\beta_{16} - \beta_{15} + \cdots - 2 \beta_{7}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 257269301703 \nu^{18} - 4826223210105 \nu^{16} + 61286511972345 \nu^{14} + \cdots + 63\!\cdots\!36 ) / 16\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58379368812683 \nu^{18} + \cdots - 23\!\cdots\!96 ) / 10\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 63\!\cdots\!85 \nu^{18} + \cdots - 25\!\cdots\!84 ) / 88\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 108802723986440 \nu^{18} + \cdots - 49\!\cdots\!91 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 123113231093090 \nu^{18} + \cdots - 29\!\cdots\!75 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112685518955184 \nu^{19} + \cdots + 22\!\cdots\!52 ) / 88\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 112685518955184 \nu^{19} + \cdots + 22\!\cdots\!52 ) / 88\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 149437719142871 \nu^{19} + \cdots + 22\!\cdots\!45 \nu ) / 55\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 54263059985435 \nu^{18} + \cdots - 12\!\cdots\!72 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 58379368812683 \nu^{19} + \cdots - 23\!\cdots\!96 \nu ) / 10\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21\!\cdots\!13 \nu^{18} + \cdots - 91\!\cdots\!84 ) / 88\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 41\!\cdots\!91 \nu^{19} + \cdots + 70\!\cdots\!92 ) / 35\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 40\!\cdots\!55 \nu^{19} + \cdots + 18\!\cdots\!16 ) / 35\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 352957477246353 \nu^{19} + \cdots - 15\!\cdots\!77 \nu ) / 27\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 395664181329051 \nu^{19} + \cdots + 31\!\cdots\!37 \nu ) / 27\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 86\!\cdots\!67 \nu^{19} + \cdots - 35\!\cdots\!92 \nu ) / 35\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 44\!\cdots\!75 \nu^{19} + \cdots - 12\!\cdots\!08 \nu ) / 17\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 99\!\cdots\!71 \nu^{19} + \cdots + 25\!\cdots\!64 \nu ) / 17\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + 4\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} - \beta_{16} - \beta_{15} + 6\beta_{11} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{12} - 9\beta_{10} - \beta_{8} - \beta_{7} + 2\beta_{5} - \beta_{4} + 25\beta_{3} - 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{19} + 11\beta_{18} + 3\beta_{14} + 3\beta_{13} + 40\beta_{11} - 3\beta_{7} - 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{14} - 8\beta_{13} - 8\beta_{8} - 16\beta_{6} + 27\beta_{5} + 72\beta_{2} - 177 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 88\beta_{16} + 99\beta_{15} + 6\beta_{9} + 46\beta_{8} - 46\beta_{7} - 283\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 47 \beta_{14} - 47 \beta_{13} - 279 \beta_{12} + 569 \beta_{10} + 47 \beta_{7} - 174 \beta_{6} + \cdots + 569 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 743 \beta_{19} - 848 \beta_{18} + 116 \beta_{17} + 743 \beta_{16} + 848 \beta_{15} + \cdots + 511 \beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2613 \beta_{12} + 4522 \beta_{10} + 221 \beta_{8} + 221 \beta_{7} - 2613 \beta_{5} + 1649 \beta_{4} + \cdots + 10318 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6171 \beta_{19} - 7135 \beta_{18} + 1486 \beta_{17} - 5005 \beta_{14} - 5005 \beta_{13} + \cdots + 15785 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -644\beta_{14} + 644\beta_{13} + 644\beta_{8} + 14695\beta_{6} - 23316\beta_{5} - 36226\beta_{2} + 81771 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -50921\beta_{16} - 59542\beta_{15} - 15954\beta_{9} - 45988\beta_{8} + 45988\beta_{7} + 122286\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2400 \beta_{14} - 2400 \beta_{13} + 202439 \beta_{12} - 292291 \beta_{10} + 2400 \beta_{7} + \cdots - 292291 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 419234 \beta_{19} + 494730 \beta_{18} - 155792 \beta_{17} - 419234 \beta_{16} - 494730 \beta_{15} + \cdots - 407278 \beta_{8} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1728520 \beta_{12} - 2371957 \beta_{10} + 68340 \beta_{8} + 68340 \beta_{7} + 1728520 \beta_{5} + \cdots - 5318130 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3449955 \beta_{19} + 4100477 \beta_{18} - 1437724 \beta_{17} + 3525380 \beta_{14} + 3525380 \beta_{13} + \cdots - 7683002 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 862627 \beta_{14} + 862627 \beta_{13} + 862627 \beta_{8} - 9062991 \beta_{6} + 14601192 \beta_{5} + \cdots - 43320969 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 28396902 \beta_{16} + 33935103 \beta_{15} + 12801656 \beta_{9} + 30065011 \beta_{8} + \cdots - 61849398 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1
2.10552 + 1.21562i
−1.08802 − 0.628167i
−0.392182 − 0.226426i
−2.48777 − 1.43632i
0.392182 + 0.226426i
1.74361 + 1.00667i
−2.10552 − 1.21562i
−1.74361 − 1.00667i
1.08802 + 0.628167i
2.48777 + 1.43632i
2.10552 − 1.21562i
−1.08802 + 0.628167i
−0.392182 + 0.226426i
−2.48777 + 1.43632i
0.392182 − 0.226426i
1.74361 − 1.00667i
−2.10552 + 1.21562i
−1.74361 + 1.00667i
1.08802 − 0.628167i
2.48777 − 1.43632i
0 0 0 −2.22225 + 0.248224i 0 0.663818i 0 0 0
1189.2 0 0 0 −1.87507 − 1.21824i 0 4.97100i 0 0 0
1189.3 0 0 0 −1.82467 + 1.29251i 0 − 2.54366i 0 0 0
1189.4 0 0 0 −1.38776 − 1.75332i 0 − 3.54568i 0 0 0
1189.5 0 0 0 −0.207009 + 2.22647i 0 2.54366i 0 0 0
1189.6 0 0 0 −0.0408382 − 2.23570i 0 − 1.34403i 0 0 0
1189.7 0 0 0 0.896156 + 2.04863i 0 − 0.663818i 0 0 0
1189.8 0 0 0 1.95659 − 1.08248i 0 1.34403i 0 0 0
1189.9 0 0 0 1.99256 + 1.01474i 0 − 4.97100i 0 0 0
1189.10 0 0 0 2.21230 + 0.325180i 0 3.54568i 0 0 0
2629.1 0 0 0 −2.22225 − 0.248224i 0 − 0.663818i 0 0 0
2629.2 0 0 0 −1.87507 + 1.21824i 0 − 4.97100i 0 0 0
2629.3 0 0 0 −1.82467 − 1.29251i 0 2.54366i 0 0 0
2629.4 0 0 0 −1.38776 + 1.75332i 0 3.54568i 0 0 0
2629.5 0 0 0 −0.207009 − 2.22647i 0 − 2.54366i 0 0 0
2629.6 0 0 0 −0.0408382 + 2.23570i 0 1.34403i 0 0 0
2629.7 0 0 0 0.896156 − 2.04863i 0 0.663818i 0 0 0
2629.8 0 0 0 1.95659 + 1.08248i 0 − 1.34403i 0 0 0
2629.9 0 0 0 1.99256 − 1.01474i 0 4.97100i 0 0 0
2629.10 0 0 0 2.21230 − 0.325180i 0 − 3.54568i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1189.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.2.bj.c 20
3.b odd 2 1 380.2.r.a ✓ 20
5.b even 2 1 inner 3420.2.bj.c 20
15.d odd 2 1 380.2.r.a ✓ 20
15.e even 4 2 1900.2.i.g 20
19.c even 3 1 inner 3420.2.bj.c 20
57.h odd 6 1 380.2.r.a ✓ 20
95.i even 6 1 inner 3420.2.bj.c 20
285.n odd 6 1 380.2.r.a ✓ 20
285.v even 12 2 1900.2.i.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.r.a ✓ 20 3.b odd 2 1
380.2.r.a ✓ 20 15.d odd 2 1
380.2.r.a ✓ 20 57.h odd 6 1
380.2.r.a ✓ 20 285.n odd 6 1
1900.2.i.g 20 15.e even 4 2
1900.2.i.g 20 285.v even 12 2
3420.2.bj.c 20 1.a even 1 1 trivial
3420.2.bj.c 20 5.b even 2 1 inner
3420.2.bj.c 20 19.c even 3 1 inner
3420.2.bj.c 20 95.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\):

\( T_{7}^{10} + 46T_{7}^{8} + 651T_{7}^{6} + 3285T_{7}^{4} + 4956T_{7}^{2} + 1600 \) Copy content Toggle raw display
\( T_{11}^{5} - 27T_{11}^{3} + 29T_{11}^{2} + 107T_{11} - 148 \) 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
$5$ \( T^{20} + T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{10} + 46 T^{8} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 27 T^{3} + \cdots - 148)^{4} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 13051691536 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 9721171216 \) Copy content Toggle raw display
$19$ \( (T^{10} - 7 T^{9} + \cdots + 2476099)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 39691260010000 \) Copy content Toggle raw display
$29$ \( (T^{10} - 8 T^{9} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 2 T^{4} + \cdots + 388)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 86 T^{8} + \cdots + 518400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 13 T^{9} + \cdots + 1368900)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 122963703210000 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 2998219536 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 591687332741376 \) Copy content Toggle raw display
$59$ \( (T^{10} + 2 T^{9} + 62 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - T^{9} + \cdots + 3073009)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 805854925357056 \) Copy content Toggle raw display
$71$ \( (T^{10} - T^{9} + \cdots + 640140601)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 25628906250000 \) Copy content Toggle raw display
$79$ \( (T^{10} + 8 T^{9} + \cdots + 233967616)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 323 T^{8} + \cdots + 54464400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 20 T^{9} + \cdots + 5314993216)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 385136700010000 \) Copy content Toggle raw display
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