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Properties

Label 2112.2.d.n
Level $2112$
Weight $2$
Character orbit 2112.d
Analytic conductor $16.864$
Analytic rank $0$
Dimension $20$
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] = [2112,2,Mod(1343,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2112.1343");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.d (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: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 8 x^{18} - 4 x^{17} - 6 x^{16} + 4 x^{15} + 8 x^{14} + 60 x^{13} - 183 x^{12} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{23} \)
Twist minimal: no (minimal twist has level 1056)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} - \beta_{18} q^{5} - \beta_{5} q^{7} + \beta_{11} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} - \beta_{18} q^{5} - \beta_{5} q^{7} + \beta_{11} q^{9} + q^{11} + \beta_{10} q^{13} + (\beta_{13} + 1) q^{15} + \beta_{2} q^{17} + (\beta_{19} - \beta_{11} - \beta_{4}) q^{19} + (\beta_{16} + \beta_{14} + \cdots - \beta_{3}) q^{21}+ \cdots + \beta_{11} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{11} - 8 q^{13} + 12 q^{15} - 8 q^{21} + 8 q^{23} - 20 q^{25} + 12 q^{27} + 16 q^{35} + 24 q^{37} - 16 q^{39} + 20 q^{45} - 72 q^{47} - 28 q^{49} - 24 q^{51} - 16 q^{57} - 24 q^{59} - 8 q^{61} - 4 q^{69} + 16 q^{71} + 8 q^{73} + 12 q^{75} + 40 q^{81} + 32 q^{83} - 16 q^{85} - 40 q^{87} - 12 q^{93} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 8 x^{18} - 4 x^{17} - 6 x^{16} + 4 x^{15} + 8 x^{14} + 60 x^{13} - 183 x^{12} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 90377 \nu^{19} + 232262 \nu^{18} - 177259 \nu^{17} - 402100 \nu^{16} + 251847 \nu^{15} + \cdots + 601433748 ) / 44719776 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 746345 \nu^{19} - 2545268 \nu^{18} + 2775523 \nu^{17} + 2460004 \nu^{16} - 5171415 \nu^{15} + \cdots - 16874393364 ) / 223598880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 260421 \nu^{19} - 561812 \nu^{18} + 120833 \nu^{17} + 1266068 \nu^{16} + 189653 \nu^{15} + \cdots + 424706652 ) / 74532960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 54941 \nu^{19} - 38188 \nu^{18} + 327347 \nu^{17} - 454423 \nu^{16} - 739893 \nu^{15} + \cdots - 3323061207 ) / 12422160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 535060 \nu^{19} + 1314409 \nu^{18} - 694304 \nu^{17} - 2724962 \nu^{16} + 894120 \nu^{15} + \cdots + 313628922 ) / 111799440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1157047 \nu^{19} - 823669 \nu^{18} - 3248839 \nu^{17} + 8034611 \nu^{16} + 7967151 \nu^{15} + \cdots + 38301405579 ) / 223598880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1668283 \nu^{19} - 6189269 \nu^{18} + 22067221 \nu^{17} - 19685549 \nu^{16} + \cdots - 198789777261 ) / 223598880 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 587701 \nu^{19} + 2209543 \nu^{18} - 2564939 \nu^{17} - 2074751 \nu^{16} + 4741347 \nu^{15} + \cdots + 15673710681 ) / 74532960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 886231 \nu^{19} - 861155 \nu^{18} + 6101059 \nu^{17} - 7706477 \nu^{16} - 13839903 \nu^{15} + \cdots - 57286800693 ) / 111799440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8751 \nu^{19} + 66940 \nu^{18} - 119851 \nu^{17} + 8498 \nu^{16} + 242177 \nu^{15} + \cdots + 905168682 ) / 1049760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 939071 \nu^{19} + 4515548 \nu^{18} - 6481399 \nu^{17} - 1884121 \nu^{16} + 12459417 \nu^{15} + \cdots + 46113273351 ) / 111799440 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1972277 \nu^{19} + 15532427 \nu^{18} - 27680029 \nu^{17} + 1918205 \nu^{16} + \cdots + 210101361165 ) / 223598880 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 667777 \nu^{19} + 879400 \nu^{18} + 857743 \nu^{17} - 4054214 \nu^{16} - 2404641 \nu^{15} + \cdots - 13976806446 ) / 74532960 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 767603 \nu^{19} - 3296240 \nu^{18} + 4317163 \nu^{17} + 2021626 \nu^{16} - 8069421 \nu^{15} + \cdots - 30133662606 ) / 74532960 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 9617 \nu^{19} + 71540 \nu^{18} - 126757 \nu^{17} + 7331 \nu^{16} + 256479 \nu^{15} + \cdots + 954094059 ) / 787320 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 150675 \nu^{19} - 433871 \nu^{18} + 365261 \nu^{17} + 633368 \nu^{16} - 597325 \nu^{15} + \cdots - 1660510368 ) / 9316620 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 2159161 \nu^{19} + 12347680 \nu^{18} - 19499891 \nu^{17} - 2748287 \nu^{16} + \cdots + 141254242497 ) / 111799440 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 1586445 \nu^{19} + 4239088 \nu^{18} - 2930623 \nu^{17} - 7359304 \nu^{16} + 4450955 \nu^{15} + \cdots + 9336722904 ) / 74532960 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 6715375 \nu^{19} + 18726526 \nu^{18} - 14351501 \nu^{17} - 29728988 \nu^{16} + \cdots + 57212103708 ) / 223598880 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{19} - \beta_{18} - \beta_{16} - \beta_{15} + 2 \beta_{13} - 2 \beta_{9} + 2 \beta_{8} + \cdots + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{18} + \beta_{16} + 2\beta_{11} + 2\beta_{10} + 2\beta_{7} - 2\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7 \beta_{19} + \beta_{18} - 4 \beta_{17} - \beta_{16} + \beta_{15} - 2 \beta_{14} - 10 \beta_{13} + \cdots - 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} + 2 \beta_{16} + \beta_{15} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{7} - 2 \beta_{5} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{19} + 3 \beta_{18} + 4 \beta_{17} - 7 \beta_{16} - 17 \beta_{15} + 14 \beta_{14} + 10 \beta_{13} + \cdots + 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10 \beta_{19} - 6 \beta_{18} + 12 \beta_{17} - \beta_{16} + 2 \beta_{15} + 8 \beta_{14} - 4 \beta_{13} + \cdots + 36 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 29 \beta_{19} - 27 \beta_{18} + 4 \beta_{17} - 27 \beta_{16} - 13 \beta_{15} + 6 \beta_{14} + \cdots + 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5 \beta_{19} - 20 \beta_{18} + 8 \beta_{17} - 6 \beta_{16} - 5 \beta_{15} - 6 \beta_{14} + 2 \beta_{13} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 105 \beta_{19} + 63 \beta_{18} - 4 \beta_{17} - 61 \beta_{16} - 27 \beta_{15} - 118 \beta_{14} + \cdots - 588 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 140 \beta_{19} - 46 \beta_{18} - 72 \beta_{17} + 115 \beta_{16} + 132 \beta_{15} + 16 \beta_{14} + \cdots - 104 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 133 \beta_{19} - 201 \beta_{18} + 16 \beta_{17} - 373 \beta_{16} - 29 \beta_{15} - 40 \beta_{14} + \cdots - 60 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 126 \beta_{19} - 32 \beta_{18} - 24 \beta_{17} + 50 \beta_{16} + 16 \beta_{15} + 150 \beta_{14} + \cdots + 223 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 431 \beta_{19} + 713 \beta_{18} - 1008 \beta_{17} - 1071 \beta_{16} + 1385 \beta_{15} + 872 \beta_{14} + \cdots - 340 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 548 \beta_{19} - 570 \beta_{18} - 248 \beta_{17} + 147 \beta_{16} + 28 \beta_{15} - 576 \beta_{14} + \cdots - 1752 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1769 \beta_{19} + 6183 \beta_{18} - 652 \beta_{17} - 799 \beta_{16} - 2169 \beta_{15} + 714 \beta_{14} + \cdots - 6628 ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 145 \beta_{19} + 468 \beta_{18} - 152 \beta_{17} + 504 \beta_{16} + 547 \beta_{15} - 234 \beta_{14} + \cdots + 114 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 5621 \beta_{19} - 5227 \beta_{18} - 212 \beta_{17} - 1449 \beta_{16} - 2871 \beta_{15} - 3862 \beta_{14} + \cdots + 37764 ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 1074 \beta_{19} - 7486 \beta_{18} + 4604 \beta_{17} + 5405 \beta_{16} - 2134 \beta_{15} + 8000 \beta_{14} + \cdots - 2060 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 31763 \beta_{19} + 17011 \beta_{18} - 4148 \beta_{17} + 10459 \beta_{16} + 21861 \beta_{15} + \cdots - 54940 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(1409\) \(1729\) \(2047\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
−1.46205 − 0.928665i
−1.46205 + 0.928665i
0.839507 − 1.51500i
0.839507 + 1.51500i
1.67342 − 0.446851i
1.67342 + 0.446851i
−0.601061 − 1.62442i
−0.601061 + 1.62442i
0.781644 + 1.54565i
0.781644 − 1.54565i
−1.01682 + 1.40217i
−1.01682 − 1.40217i
0.812509 − 1.52965i
0.812509 + 1.52965i
−1.72437 + 0.162916i
−1.72437 − 0.162916i
1.54864 − 0.775710i
1.54864 + 0.775710i
1.14859 + 1.29644i
1.14859 − 1.29644i
0 −1.69049 − 0.377158i 0 − 1.28720i 0 − 2.23802i 0 2.71550 + 1.27516i 0
1343.2 0 −1.69049 + 0.377158i 0 1.28720i 0 2.23802i 0 2.71550 − 1.27516i 0
1343.3 0 −1.66489 − 0.477646i 0 − 3.40584i 0 3.74766i 0 2.54371 + 1.59046i 0
1343.4 0 −1.66489 + 0.477646i 0 3.40584i 0 − 3.74766i 0 2.54371 − 1.59046i 0
1343.5 0 −0.867313 − 1.49926i 0 − 2.05003i 0 0.172010i 0 −1.49554 + 2.60065i 0
1343.6 0 −0.867313 + 1.49926i 0 2.05003i 0 − 0.172010i 0 −1.49554 − 2.60065i 0
1343.7 0 −0.723621 − 1.57365i 0 0.558351i 0 − 1.52065i 0 −1.95275 + 2.27745i 0
1343.8 0 −0.723621 + 1.57365i 0 − 0.558351i 0 1.52065i 0 −1.95275 − 2.27745i 0
1343.9 0 −0.540233 − 1.64565i 0 4.23832i 0 − 1.55637i 0 −2.41630 + 1.77806i 0
1343.10 0 −0.540233 + 1.64565i 0 − 4.23832i 0 1.55637i 0 −2.41630 − 1.77806i 0
1343.11 0 0.272482 − 1.71048i 0 − 1.35060i 0 − 3.25589i 0 −2.85151 − 0.932151i 0
1343.12 0 0.272482 + 1.71048i 0 1.35060i 0 3.25589i 0 −2.85151 + 0.932151i 0
1343.13 0 0.507095 − 1.65616i 0 2.77142i 0 4.37454i 0 −2.48571 − 1.67966i 0
1343.14 0 0.507095 + 1.65616i 0 − 2.77142i 0 − 4.37454i 0 −2.48571 + 1.67966i 0
1343.15 0 1.33451 − 1.10412i 0 0.740451i 0 0.905333i 0 0.561854 − 2.94692i 0
1343.16 0 1.33451 + 1.10412i 0 − 0.740451i 0 − 0.905333i 0 0.561854 + 2.94692i 0
1343.17 0 1.64356 − 0.546541i 0 − 0.288845i 0 − 4.89769i 0 2.40259 − 1.79655i 0
1343.18 0 1.64356 + 0.546541i 0 0.288845i 0 4.89769i 0 2.40259 + 1.79655i 0
1343.19 0 1.72889 − 0.104545i 0 3.75886i 0 − 2.37436i 0 2.97814 − 0.361494i 0
1343.20 0 1.72889 + 0.104545i 0 − 3.75886i 0 2.37436i 0 2.97814 + 0.361494i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1343.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2112.2.d.n 20
3.b odd 2 1 2112.2.d.m 20
4.b odd 2 1 2112.2.d.m 20
8.b even 2 1 1056.2.d.a ✓ 20
8.d odd 2 1 1056.2.d.b yes 20
12.b even 2 1 inner 2112.2.d.n 20
24.f even 2 1 1056.2.d.a ✓ 20
24.h odd 2 1 1056.2.d.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1056.2.d.a ✓ 20 8.b even 2 1
1056.2.d.a ✓ 20 24.f even 2 1
1056.2.d.b yes 20 8.d odd 2 1
1056.2.d.b yes 20 24.h odd 2 1
2112.2.d.m 20 3.b odd 2 1
2112.2.d.m 20 4.b odd 2 1
2112.2.d.n 20 1.a even 1 1 trivial
2112.2.d.n 20 12.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 \(S_{2}^{\mathrm{new}}(2112, [\chi])\):

\( T_{5}^{20} + 60 T_{5}^{18} + 1430 T_{5}^{16} + 17372 T_{5}^{14} + 115297 T_{5}^{12} + 420896 T_{5}^{10} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{7}^{20} + 84 T_{7}^{18} + 2868 T_{7}^{16} + 51712 T_{7}^{14} + 537088 T_{7}^{12} + 3300608 T_{7}^{10} + \cdots + 262144 \) Copy content Toggle raw display
\( T_{23}^{10} - 4 T_{23}^{9} - 82 T_{23}^{8} + 108 T_{23}^{7} + 2121 T_{23}^{6} + 464 T_{23}^{5} + \cdots - 26368 \) Copy content Toggle raw display
\( T_{47}^{10} + 36 T_{47}^{9} + 396 T_{47}^{8} - 224 T_{47}^{7} - 33624 T_{47}^{6} - 209120 T_{47}^{5} + \cdots - 1236992 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 4 T^{17} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} + 60 T^{18} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( T^{20} + 84 T^{18} + \cdots + 262144 \) Copy content Toggle raw display
$11$ \( (T - 1)^{20} \) Copy content Toggle raw display
$13$ \( (T^{10} + 4 T^{9} + \cdots - 512)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 4848615424 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 4920180736 \) Copy content Toggle raw display
$23$ \( (T^{10} - 4 T^{9} + \cdots - 26368)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 40411347288064 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 19452474966016 \) Copy content Toggle raw display
$37$ \( (T^{10} - 12 T^{9} + \cdots + 204800)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + 260 T^{18} + \cdots + 16777216 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 347288633344 \) Copy content Toggle raw display
$47$ \( (T^{10} + 36 T^{9} + \cdots - 1236992)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 4294967296 \) Copy content Toggle raw display
$59$ \( (T^{10} + 12 T^{9} + \cdots + 913408)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 4 T^{9} + \cdots - 41025536)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 82780300263424 \) Copy content Toggle raw display
$71$ \( (T^{10} - 8 T^{9} + \cdots - 739879648)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 4 T^{9} + \cdots - 19856384)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 1015805968384 \) Copy content Toggle raw display
$83$ \( (T^{10} - 16 T^{9} + \cdots + 160415744)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 631427301376 \) Copy content Toggle raw display
$97$ \( (T^{10} - 4 T^{9} + \cdots - 538264576)^{2} \) Copy content Toggle raw display
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