↑

Properties

Label 3420.3.h.e
Level $3420$
Weight $3$
Character orbit 3420.h
Analytic conductor $93.188$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3420,3,Mod(2089,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.2089");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3420.h (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: \(93.1882504112\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19}\cdot 3 \)
Twist minimal: no (minimal twist has level 380)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - 1) q^{5} - \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - 1) q^{5} - \beta_{9} q^{7} + (\beta_{4} - \beta_{3} + 2) q^{11} - \beta_{8} q^{13} + ( - \beta_{9} - \beta_{4} - \beta_{3}) q^{17} + (\beta_{2} - \beta_1 + 6) q^{19} + 5 \beta_{9} q^{23} + (2 \beta_{10} - \beta_{9} - 2 \beta_{4} + \cdots - 6) q^{25}+ \cdots + ( - \beta_{11} + 4 \beta_{8} - 6 \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} + 24 q^{11} + 68 q^{19} - 76 q^{25} + 32 q^{35} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 408 q^{85} - 124 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 28618\nu^{10} + 524952\nu^{8} + 4391039\nu^{6} + 54167333\nu^{4} + 1825258945\nu^{2} - 558303007 ) / 908842237 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60512 \nu^{10} + 2443821 \nu^{8} + 32340835 \nu^{6} + 100053868 \nu^{4} + 517724371 \nu^{2} + 21947164823 ) / 1817684474 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 194997239 \nu^{11} + 17200092 \nu^{10} - 5048505213 \nu^{9} + 7318528889 \nu^{8} + \cdots + 187896630501135 ) / 57260696299948 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 194997239 \nu^{11} - 17200092 \nu^{10} - 5048505213 \nu^{9} - 7318528889 \nu^{8} + \cdots - 187896630501135 ) / 57260696299948 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29187003 \nu^{11} - 586455864 \nu^{9} - 2439785325 \nu^{7} - 11649718743 \nu^{5} + \cdots + 6195603639189 \nu ) / 1506860428946 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1114636881 \nu^{11} + 1291775986 \nu^{10} + 17000858399 \nu^{9} + 18377506607 \nu^{8} + \cdots - 104748473351889 ) / 57260696299948 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1114636881 \nu^{11} + 13046909770 \nu^{10} - 17000858399 \nu^{9} + \cdots - 679853386340457 ) / 57260696299948 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29478096 \nu^{11} + 308326157 \nu^{9} - 7284318313 \nu^{7} - 56455260952 \nu^{5} + \cdots - 8909774942657 \nu ) / 1506860428946 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 651724712 \nu^{11} + 15674641213 \nu^{9} + 65363306369 \nu^{7} - 565819505556 \nu^{5} + \cdots + 25942480991957 \nu ) / 28630348149974 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1021426750 \nu^{11} + 23103082157 \nu^{9} + 96267253819 \nu^{7} - 418256401478 \nu^{5} + \cdots + 61986227495459 \nu ) / 28630348149974 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49502747 \nu^{11} + 398592455 \nu^{9} - 12831012835 \nu^{7} - 122204283727 \nu^{5} + \cdots - 15978219794293 \nu ) / 753430214473 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{10} - 3\beta_{9} + 2\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{4} + 3\beta_{3} - \beta_{2} + \beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{11} - 21\beta_{10} + 15\beta_{9} - 18\beta_{8} - 2\beta_{5} - 48\beta_{4} - 48\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{8} + \beta_{7} - 21\beta_{6} - 11\beta_{5} + 37\beta_{4} - 37\beta_{3} + 9\beta_{2} - \beta _1 + 107 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -156\beta_{11} + 351\beta_{10} - 579\beta_{9} + 522\beta_{8} - 76\beta_{5} + 114\beta_{4} + 114\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 205 \beta_{8} - 67 \beta_{7} + 343 \beta_{6} + 205 \beta_{5} - 152 \beta_{4} + 152 \beta_{3} + \cdots - 1504 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2532\beta_{11} + 183\beta_{10} + 9\beta_{9} - 10716\beta_{8} - 2186\beta_{5} + 852\beta_{4} + 852\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3663 \beta_{8} + 1169 \beta_{7} - 6157 \beta_{6} - 3663 \beta_{5} - 4487 \beta_{4} + 4487 \beta_{3} + \cdots - 25057 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 47250 \beta_{11} - 126297 \beta_{10} + 172287 \beta_{9} + 172206 \beta_{8} + 9578 \beta_{5} + \cdots - 74172 \beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 56558 \beta_{8} - 13056 \beta_{7} + 100060 \beta_{6} + 56558 \beta_{5} + 226937 \beta_{4} + \cdots + 973351 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 581364 \beta_{11} + 3784371 \beta_{10} - 4414851 \beta_{9} - 2116794 \beta_{8} + \cdots + 3122778 \beta_{3} ) / 12 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2089.1
−1.63363 + 1.34185i
1.63363 + 1.34185i
−1.63363 − 1.34185i
1.63363 − 1.34185i
2.80718 + 1.46321i
−2.80718 + 1.46321i
2.80718 − 1.46321i
−2.80718 − 1.46321i
0.975196 − 4.19028i
−0.975196 − 4.19028i
0.975196 + 4.19028i
−0.975196 + 4.19028i
0 0 0 −4.26535 − 2.60898i 0 − 6.30368i 0 0 0
2089.2 0 0 0 −4.26535 − 2.60898i 0 − 6.30368i 0 0 0
2089.3 0 0 0 −4.26535 + 2.60898i 0 6.30368i 0 0 0
2089.4 0 0 0 −4.26535 + 2.60898i 0 6.30368i 0 0 0
2089.5 0 0 0 −1.48938 − 4.77302i 0 7.03410i 0 0 0
2089.6 0 0 0 −1.48938 − 4.77302i 0 7.03410i 0 0 0
2089.7 0 0 0 −1.48938 + 4.77302i 0 − 7.03410i 0 0 0
2089.8 0 0 0 −1.48938 + 4.77302i 0 − 7.03410i 0 0 0
2089.9 0 0 0 2.75473 − 4.17271i 0 − 2.18749i 0 0 0
2089.10 0 0 0 2.75473 − 4.17271i 0 − 2.18749i 0 0 0
2089.11 0 0 0 2.75473 + 4.17271i 0 2.18749i 0 0 0
2089.12 0 0 0 2.75473 + 4.17271i 0 2.18749i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2089.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.b odd 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.3.h.e 12
3.b odd 2 1 380.3.g.c ✓ 12
5.b even 2 1 inner 3420.3.h.e 12
15.d odd 2 1 380.3.g.c ✓ 12
15.e even 4 2 1900.3.e.e 12
19.b odd 2 1 inner 3420.3.h.e 12
57.d even 2 1 380.3.g.c ✓ 12
95.d odd 2 1 inner 3420.3.h.e 12
285.b even 2 1 380.3.g.c ✓ 12
285.j odd 4 2 1900.3.e.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.g.c ✓ 12 3.b odd 2 1
380.3.g.c ✓ 12 15.d odd 2 1
380.3.g.c ✓ 12 57.d even 2 1
380.3.g.c ✓ 12 285.b even 2 1
1900.3.e.e 12 15.e even 4 2
1900.3.e.e 12 285.j odd 4 2
3420.3.h.e 12 1.a even 1 1 trivial
3420.3.h.e 12 5.b even 2 1 inner
3420.3.h.e 12 19.b odd 2 1 inner
3420.3.h.e 12 95.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3420, [\chi])\):

\( T_{7}^{6} + 94T_{7}^{4} + 2393T_{7}^{2} + 9408 \) Copy content Toggle raw display
\( T_{11}^{3} - 6T_{11}^{2} - 38T_{11} + 44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 6 T^{5} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 94 T^{4} + \cdots + 9408)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 6 T^{2} - 38 T + 44)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} - 682 T^{4} + \cdots - 7200000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 314 T^{4} + \cdots + 12288)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 34 T^{5} + \cdots + 47045881)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 2350 T^{4} + \cdots + 147000000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 2306 T^{4} + \cdots + 51671040)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 4652 T^{4} + \cdots + 1345781760)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 5312 T^{4} + \cdots - 5202247680)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 7140 T^{4} + \cdots + 318504960)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 7540 T^{4} + \cdots + 95158272)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 14056 T^{4} + \cdots + 75606592512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 3130 T^{4} + \cdots - 136869120)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 18086 T^{4} + \cdots + 7228354560)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 150 T^{2} + \cdots + 18964)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 22430 T^{4} + \cdots - 301871934720)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 11980 T^{4} + \cdots + 23313776640)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 12714 T^{4} + \cdots + 4401589248)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 26440 T^{4} + \cdots + 4282122240)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 19616 T^{4} + \cdots + 216760320000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 18508 T^{4} + \cdots + 187730165760)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 22008 T^{4} + \cdots - 318730752000)^{2} \) Copy content Toggle raw display
show more
show less