Properties

Label 3724.2.g.f
Level $3724$
Weight $2$
Character orbit 3724.g
Analytic conductor $29.736$
Analytic rank $0$
Dimension $16$
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] = [3724,2,Mod(1861,3724)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3724, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3724.1861");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3724.g (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: \(29.7362897127\)
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: \( x^{16} + 25x^{14} + 413x^{12} + 3916x^{10} + 26956x^{8} + 112304x^{6} + 333008x^{4} + 476096x^{2} + 473344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 532)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{9} q^{5} + (\beta_{4} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{9} q^{5} + (\beta_{4} + 3) q^{9} + (\beta_{12} + \beta_{4}) q^{11} + \beta_{14} q^{13} + (\beta_{10} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3}) q^{15} + ( - \beta_{7} + \beta_1) q^{17} + (\beta_{13} + \beta_{9} - \beta_{7}) q^{19} + (\beta_{12} + \beta_{4} - 2) q^{23} + (\beta_{12} - \beta_{4} - \beta_{2} - 1) q^{25} + ( - 2 \beta_{13} - \beta_{9} - 2 \beta_{6} + \beta_1) q^{27} + (\beta_{10} + \beta_{8} - \beta_{5}) q^{29} - \beta_{6} q^{31} + ( - \beta_{14} - 2 \beta_{13} - \beta_{9} - 2 \beta_{6} + \beta_1) q^{33} + (\beta_{10} - \beta_{8} - \beta_{5} - \beta_{4}) q^{37} + ( - 4 \beta_{12} + 2 \beta_{2} + 2) q^{39} + 2 \beta_{6} q^{41} + (\beta_{4} - \beta_{2} + 1) q^{43} + (2 \beta_{11} + 4 \beta_{9} + 2 \beta_{7} + \beta_1) q^{45} + (\beta_{11} + 2 \beta_{9} - 2 \beta_{7} - 2 \beta_1) q^{47} + ( - \beta_{10} - \beta_{8} + \beta_{3}) q^{51} + ( - \beta_{5} - \beta_{3}) q^{53} + (\beta_{11} - \beta_{9} + 3 \beta_{7} + 2 \beta_1) q^{55} + ( - \beta_{12} - \beta_{8} - 3 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{57} + (\beta_{15} + \beta_{6}) q^{59} + (2 \beta_{11} + \beta_{7}) q^{61} + (3 \beta_{10} - \beta_{8} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{65} + ( - 2 \beta_{10} + \beta_{4}) q^{67} + ( - \beta_{14} - 2 \beta_{13} - \beta_{9} + \beta_1) q^{69} + ( - 2 \beta_{10} - \beta_{5} + \beta_{4}) q^{71} + ( - 2 \beta_{9} - \beta_{7}) q^{73} + (\beta_{15} - 2 \beta_{14} + 2 \beta_{13} + \beta_{9} + 3 \beta_{6} - \beta_1) q^{75} + ( - 2 \beta_{10} + 2 \beta_{8} - \beta_{5} + 2 \beta_{4}) q^{79} + (2 \beta_{12} + 4 \beta_{4} + 2 \beta_{2} + 1) q^{81} + (\beta_{11} - \beta_{9} - 4 \beta_{7} + \beta_1) q^{83} + ( - \beta_{4} - 2) q^{85} + (2 \beta_{11} - 2 \beta_{7} - 6 \beta_1) q^{87} + (\beta_{15} + \beta_{14} + \beta_{6}) q^{89} + (\beta_{4} + 6) q^{93} + (\beta_{12} + \beta_{10} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 4) q^{95} + ( - \beta_{15} - 2 \beta_{13} - \beta_{9} + \beta_{6} + \beta_1) q^{97} + (3 \beta_{12} + 4 \beta_{4} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 52 q^{9} + 8 q^{11} - 24 q^{23} - 20 q^{25} + 24 q^{39} + 16 q^{43} - 2 q^{57} + 48 q^{81} - 36 q^{85} + 100 q^{93} - 62 q^{95} + 156 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 25x^{14} + 413x^{12} + 3916x^{10} + 26956x^{8} + 112304x^{6} + 333008x^{4} + 476096x^{2} + 473344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6524233 \nu^{14} - 585002714 \nu^{12} - 12463481842 \nu^{10} - 174767581263 \nu^{8} - 1179032842436 \nu^{6} + \cdots - 11749889152128 ) / 7487095708464 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 291447 \nu^{14} - 3114862 \nu^{12} - 97530264 \nu^{10} - 772054101 \nu^{8} - 7962457574 \nu^{6} - 10790555724 \nu^{4} + \cdots + 328135179744 ) / 116079003232 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58056497 \nu^{15} - 444070661 \nu^{13} - 24306840849 \nu^{11} - 518793627858 \nu^{9} - 5004274732212 \nu^{7} + \cdots - 287104419417600 \nu ) / 29948382833856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 313336 \nu^{14} + 6647101 \nu^{12} + 104855232 \nu^{10} + 830038888 \nu^{8} + 5303734399 \nu^{6} + 11600975712 \nu^{4} + \cdots - 175953881128 ) / 43529626212 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 123871913 \nu^{15} + 2881222657 \nu^{13} + 46585894581 \nu^{11} + 412942011692 \nu^{9} + 2768024531884 \nu^{7} + \cdots + 77301509525120 \nu ) / 29948382833856 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 123871913 \nu^{15} + 2881222657 \nu^{13} + 46585894581 \nu^{11} + 412942011692 \nu^{9} + 2768024531884 \nu^{7} + \cdots + 17404743857408 \nu ) / 29948382833856 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 123871913 \nu^{14} + 2881222657 \nu^{12} + 46585894581 \nu^{10} + 412942011692 \nu^{8} + 2768024531884 \nu^{6} + \cdots + 32378935274336 ) / 14974191416928 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 85466383 \nu^{15} - 107787584 \nu^{14} + 278524655 \nu^{13} - 2286602744 \nu^{12} - 3909734629 \nu^{11} - 36070199808 \nu^{10} + \cdots + 60528135108032 ) / 29948382833856 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 73640665 \nu^{14} + 2002145881 \nu^{12} + 33198574355 \nu^{10} + 336411300050 \nu^{8} + 2264743206938 \nu^{6} + \cdots + 22839245999744 ) / 7487095708464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 337968981 \nu^{15} + 107787584 \nu^{14} - 6507614479 \nu^{13} + 2286602744 \nu^{12} - 89896337859 \nu^{11} + \cdots - 60528135108032 ) / 29948382833856 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 384838379 \nu^{14} - 10087515194 \nu^{12} - 167384640036 \nu^{10} - 1607238227653 \nu^{8} - 10835607328398 \nu^{6} + \cdots - 117067434278112 ) / 14974191416928 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10673935 \nu^{14} - 238094522 \nu^{12} - 3571941720 \nu^{10} - 28275656605 \nu^{8} - 157555174298 \nu^{6} - 395192575020 \nu^{4} + \cdots + 749007918752 ) / 348237009696 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 247743826 \nu^{15} - 67116432 \nu^{14} - 5762445314 \nu^{13} - 2587148595 \nu^{12} - 93171789162 \nu^{11} + \cdots - 34589135151872 ) / 14974191416928 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 489025224 \nu^{15} + 12749938415 \nu^{13} + 206151123195 \nu^{11} + 1922724612147 \nu^{9} + 12249015960980 \nu^{7} + \cdots + 77019182037760 \nu ) / 14974191416928 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 196732969 \nu^{15} + 4706686266 \nu^{13} + 76101438978 \nu^{11} + 690754338253 \nu^{9} + 4521768915192 \nu^{7} + \cdots + 28431911944704 \nu ) / 3743547854232 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} + \beta_{9} + 6\beta_{7} - \beta_{4} - \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{13} + \beta_{9} + 8\beta_{6} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{12} - 22\beta_{11} - 17\beta_{9} - 46\beta_{7} - 13\beta_{4} - 2\beta_{2} + 11\beta _1 - 46 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} - 26 \beta_{13} - 28 \beta_{10} - 13 \beta_{9} + 6 \beta_{8} - 72 \beta_{6} - 74 \beta_{5} + 17 \beta_{4} - 28 \beta_{3} + 13 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{12} + 141\beta_{4} + 38\beta_{2} + 398 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 38 \beta_{15} + 8 \beta_{14} - 282 \beta_{13} + 304 \beta_{10} - 141 \beta_{9} - 114 \beta_{8} - 680 \beta_{6} + 726 \beta_{5} - 209 \beta_{4} + 328 \beta_{3} + 141 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 326 \beta_{12} + 2070 \beta_{11} + 2313 \beta_{9} + 3662 \beta_{7} - 1461 \beta_{4} - 526 \beta_{2} - 735 \beta _1 - 3662 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 526\beta_{15} - 200\beta_{14} + 2922\beta_{13} + 1461\beta_{9} + 6584\beta_{6} - 1461\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3174 \beta_{12} - 20230 \beta_{11} - 24593 \beta_{9} - 34878 \beta_{7} - 14941 \beta_{4} - 6478 \beta_{2} + 5159 \beta _1 - 34878 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6478 \beta_{15} + 3304 \beta_{14} - 29882 \beta_{13} - 29752 \beta_{10} - 14941 \beta_{9} + 19434 \beta_{8} - 64760 \beta_{6} - 74542 \beta_{5} + 24593 \beta_{4} - 39664 \beta_{3} + 14941 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 29622\beta_{12} + 152421\beta_{4} + 75358\beta_{2} + 339374 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 75358 \beta_{15} + 45736 \beta_{14} - 304842 \beta_{13} + 288728 \beta_{10} - 152421 \beta_{9} - 226074 \beta_{8} - 644216 \beta_{6} + 765310 \beta_{5} - 257401 \beta_{4} + \cdots + 152421 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 272614 \beta_{12} + 1992998 \beta_{11} + 2678113 \beta_{9} + 3350494 \beta_{7} - 1557037 \beta_{4} - 848462 \beta_{2} - 132727 \beta _1 - 3350494 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 848462\beta_{15} - 575848\beta_{14} + 3114074\beta_{13} + 1557037\beta_{9} + 6464568\beta_{6} - 1557037\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times\).

\(n\) \(1863\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1861.1
−1.61383 + 2.79524i
−1.61383 2.79524i
−1.44088 2.49568i
−1.44088 + 2.49568i
−1.06269 1.84063i
−1.06269 + 1.84063i
−0.663412 + 1.14906i
−0.663412 1.14906i
0.663412 1.14906i
0.663412 + 1.14906i
1.06269 + 1.84063i
1.06269 1.84063i
1.44088 + 2.49568i
1.44088 2.49568i
1.61383 2.79524i
1.61383 + 2.79524i
0 −3.22766 0 4.02258i 0 0 0 7.41779 0
1861.2 0 −3.22766 0 4.02258i 0 0 0 7.41779 0
1861.3 0 −2.88176 0 0.688385i 0 0 0 5.30453 0
1861.4 0 −2.88176 0 0.688385i 0 0 0 5.30453 0
1861.5 0 −2.12538 0 2.68010i 0 0 0 1.51723 0
1861.6 0 −2.12538 0 2.68010i 0 0 0 1.51723 0
1861.7 0 −1.32682 0 1.07796i 0 0 0 −1.23954 0
1861.8 0 −1.32682 0 1.07796i 0 0 0 −1.23954 0
1861.9 0 1.32682 0 1.07796i 0 0 0 −1.23954 0
1861.10 0 1.32682 0 1.07796i 0 0 0 −1.23954 0
1861.11 0 2.12538 0 2.68010i 0 0 0 1.51723 0
1861.12 0 2.12538 0 2.68010i 0 0 0 1.51723 0
1861.13 0 2.88176 0 0.688385i 0 0 0 5.30453 0
1861.14 0 2.88176 0 0.688385i 0 0 0 5.30453 0
1861.15 0 3.22766 0 4.02258i 0 0 0 7.41779 0
1861.16 0 3.22766 0 4.02258i 0 0 0 7.41779 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1861.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.b odd 2 1 inner
133.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3724.2.g.f 16
7.b odd 2 1 inner 3724.2.g.f 16
7.c even 3 1 532.2.v.e 16
7.d odd 6 1 532.2.v.e 16
19.b odd 2 1 inner 3724.2.g.f 16
133.c even 2 1 inner 3724.2.g.f 16
133.o even 6 1 532.2.v.e 16
133.r odd 6 1 532.2.v.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.v.e 16 7.c even 3 1
532.2.v.e 16 7.d odd 6 1
532.2.v.e 16 133.o even 6 1
532.2.v.e 16 133.r odd 6 1
3724.2.g.f 16 1.a even 1 1 trivial
3724.2.g.f 16 7.b odd 2 1 inner
3724.2.g.f 16 19.b odd 2 1 inner
3724.2.g.f 16 133.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 \(S_{2}^{\mathrm{new}}(3724, [\chi])\):

\( T_{3}^{8} - 25T_{3}^{6} + 212T_{3}^{4} - 692T_{3}^{2} + 688 \) Copy content Toggle raw display
\( T_{5}^{8} + 25T_{5}^{6} + 155T_{5}^{4} + 203T_{5}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 25 T^{6} + 212 T^{4} - 692 T^{2} + \cdots + 688)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 25 T^{6} + 155 T^{4} + 203 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 22 T^{2} + 38 T + 49)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 124 T^{6} + 5636 T^{4} + \cdots + 795328)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 46 T^{6} + 281 T^{4} + 440 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} - 33 T^{14} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} - 10 T^{2} - 42 T + 37)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 180 T^{6} + 11444 T^{4} + \cdots + 2818048)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 25 T^{6} + 212 T^{4} - 692 T^{2} + \cdots + 688)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 215 T^{6} + 13680 T^{4} + \cdots + 2314432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 100 T^{6} + 3392 T^{4} + \cdots + 176128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} - 51 T^{2} + 56 T + 16)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 168 T^{6} + 6102 T^{4} + \cdots + 110889)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 163 T^{6} + 4764 T^{4} + \cdots + 688)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 293 T^{6} + 28796 T^{4} + \cdots + 12725248)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 140 T^{6} + 5654 T^{4} + \cdots + 152881)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 211 T^{6} + 11568 T^{4} + \cdots + 176128)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 364 T^{6} + 30756 T^{4} + \cdots + 7730368)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 124 T^{6} + 2918 T^{4} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 491 T^{6} + 69444 T^{4} + \cdots + 941872)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 297 T^{6} + 24291 T^{4} + \cdots + 1127844)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 349 T^{6} + 34832 T^{4} + \cdots + 795328)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 576 T^{6} + 106992 T^{4} + \cdots + 14266368)^{2} \) Copy content Toggle raw display
show more
show less