↑

Properties

Label 1045.2.b.b
Level $1045$
Weight $2$
Character orbit 1045.b
Analytic conductor $8.344$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1045,2,Mod(419,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.419");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.b (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: \(8.34436701122\)
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: \( x^{16} + 19x^{14} + 144x^{12} + 552x^{10} + 1119x^{8} + 1146x^{6} + 524x^{4} + 83x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{11} q^{3} + \beta_{2} q^{4} + (\beta_{13} + \beta_{3}) q^{5} + (\beta_{9} - \beta_{8} - \beta_{6} - 1) q^{6} - \beta_{5} q^{7} + \beta_{3} q^{8} + ( - \beta_{8} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{11} q^{3} + \beta_{2} q^{4} + (\beta_{13} + \beta_{3}) q^{5} + (\beta_{9} - \beta_{8} - \beta_{6} - 1) q^{6} - \beta_{5} q^{7} + \beta_{3} q^{8} + ( - \beta_{8} - 1) q^{9} + ( - \beta_{10} + \beta_{4} - \beta_{2}) q^{10} - q^{11} + (\beta_{15} + \beta_{13} + \cdots + \beta_{5}) q^{12}+ \cdots + (\beta_{8} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 19x^{14} + 144x^{12} + 552x^{10} + 1119x^{8} + 1146x^{6} + 524x^{4} + 83x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 6\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 7\nu^{4} + 12\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{12} - 14\nu^{10} - 72\nu^{8} - 165\nu^{6} - 163\nu^{4} - 60\nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{12} + 15\nu^{10} + 84\nu^{8} + 215\nu^{6} + 246\nu^{4} + 104\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( \nu^{12} + 15\nu^{10} + 85\nu^{8} + 225\nu^{6} + 277\nu^{4} + 134\nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{15} - 17\nu^{13} - 114\nu^{11} - 382\nu^{9} - 667\nu^{7} - 572\nu^{5} - 198\nu^{3} - 15\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} - 19\nu^{13} - 144\nu^{11} - 552\nu^{9} - 1117\nu^{7} - 1128\nu^{5} - 476\nu^{3} - 47\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} + 2 \nu^{14} + 19 \nu^{13} + 34 \nu^{12} + 144 \nu^{11} + 228 \nu^{10} + 552 \nu^{9} + \cdots + 30 ) / 4 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 19 \nu^{13} + 34 \nu^{12} - 144 \nu^{11} + 228 \nu^{10} - 552 \nu^{9} + \cdots + 30 ) / 4 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( -\nu^{15} - 19\nu^{13} - 143\nu^{11} - 538\nu^{9} - 1047\nu^{7} - 981\nu^{5} - 362\nu^{3} - 28\nu \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -\nu^{15} - 19\nu^{13} - 144\nu^{11} - 551\nu^{9} - 1107\nu^{7} - 1096\nu^{5} - 440\nu^{3} - 34\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 5\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 6\beta_{3} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 7\beta_{4} + 23\beta_{2} - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{13} + \beta_{12} + \beta_{11} - 9\beta_{5} + 30\beta_{3} - 75\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{9} - \beta_{8} - 10\beta_{6} + 39\beta_{4} - 105\beta_{2} + 117 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{15} + 10\beta_{13} - 10\beta_{12} - 12\beta_{11} + 58\beta_{5} - 144\beta_{3} + 337\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -12\beta_{9} + 13\beta_{8} + \beta_{7} + 70\beta_{6} - 201\beta_{4} + 481\beta_{2} - 498 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -14\beta_{15} + \beta_{14} - 70\beta_{13} + 70\beta_{12} + 96\beta_{11} - 329\beta_{5} + 684\beta_{3} - 1530\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 96\beta_{9} - 110\beta_{8} - 15\beta_{7} - 425\beta_{6} + 998\beta_{4} - 2214\beta_{2} + 2141 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 125 \beta_{15} - 15 \beta_{14} + 425 \beta_{13} - 425 \beta_{12} - 646 \beta_{11} + \beta_{10} + \cdots + 6994 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( \beta_{13} + \beta_{12} - 646 \beta_{9} + 770 \beta_{8} + 141 \beta_{7} + 2398 \beta_{6} - 4853 \beta_{4} + \cdots - 9266 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 911 \beta_{15} + 141 \beta_{14} - 2398 \beta_{13} + 2398 \beta_{12} + 3955 \beta_{11} + \cdots - 32134 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
− 2.18657i
− 2.10564i
− 2.05535i
− 1.78099i
− 1.19140i
− 0.853553i
− 0.387488i
− 0.301154i
0.301154i
0.387488i
0.853553i
1.19140i
1.78099i
2.05535i
2.10564i
2.18657i
− 2.18657i 1.58567i −2.78109 −0.558280 + 2.16525i 3.46718 2.56330i 1.70792i 0.485645 4.73448 + 1.22072i
419.2 − 2.10564i − 1.90142i −2.43371 1.75301 + 1.38815i −4.00370 0.116917i 0.913237i −0.615392 2.92295 − 3.69120i
419.3 − 2.05535i − 2.80376i −2.22445 −2.02523 + 0.947852i −5.76269 − 1.02917i 0.461316i −4.86106 1.94816 + 4.16256i
419.4 − 1.78099i 0.213501i −1.17191 2.04104 − 0.913326i 0.380243 − 3.50934i − 1.47481i 2.95442 −1.62662 − 3.63506i
419.5 − 1.19140i 2.62369i 0.580557 0.0644373 − 2.23514i 3.12588 0.593512i − 3.07449i −3.88377 −2.66295 − 0.0767708i
419.6 − 0.853553i − 1.84364i 1.27145 1.54048 − 1.62078i −1.57364 2.69678i − 2.79235i −0.399003 −1.38342 − 1.31488i
419.7 − 0.387488i − 0.494325i 1.84985 −1.95299 + 1.08895i −0.191545 2.37207i − 1.49177i 2.75564 0.421956 + 0.756762i
419.8 − 0.301154i 1.85377i 1.90931 0.637551 + 2.14325i 0.558272 1.94668i − 1.17730i −0.436480 0.645450 − 0.192001i
419.9 0.301154i − 1.85377i 1.90931 0.637551 − 2.14325i 0.558272 − 1.94668i 1.17730i −0.436480 0.645450 + 0.192001i
419.10 0.387488i 0.494325i 1.84985 −1.95299 − 1.08895i −0.191545 − 2.37207i 1.49177i 2.75564 0.421956 − 0.756762i
419.11 0.853553i 1.84364i 1.27145 1.54048 + 1.62078i −1.57364 − 2.69678i 2.79235i −0.399003 −1.38342 + 1.31488i
419.12 1.19140i − 2.62369i 0.580557 0.0644373 + 2.23514i 3.12588 − 0.593512i 3.07449i −3.88377 −2.66295 + 0.0767708i
419.13 1.78099i − 0.213501i −1.17191 2.04104 + 0.913326i 0.380243 3.50934i 1.47481i 2.95442 −1.62662 + 3.63506i
419.14 2.05535i 2.80376i −2.22445 −2.02523 − 0.947852i −5.76269 1.02917i − 0.461316i −4.86106 1.94816 − 4.16256i
419.15 2.10564i 1.90142i −2.43371 1.75301 − 1.38815i −4.00370 − 0.116917i − 0.913237i −0.615392 2.92295 + 3.69120i
419.16 2.18657i − 1.58567i −2.78109 −0.558280 − 2.16525i 3.46718 − 2.56330i − 1.70792i 0.485645 4.73448 − 1.22072i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.b ✓ 16
5.b even 2 1 inner 1045.2.b.b ✓ 16
5.c odd 4 2 5225.2.a.z 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.b ✓ 16 1.a even 1 1 trivial
1045.2.b.b ✓ 16 5.b even 2 1 inner
5225.2.a.z 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 19T_{2}^{14} + 144T_{2}^{12} + 552T_{2}^{10} + 1119T_{2}^{8} + 1146T_{2}^{6} + 524T_{2}^{4} + 83T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1045, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 19 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{16} + 28 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{16} - 3 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 37 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T + 1)^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + 81 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{16} + 113 T^{14} + \cdots + 6390784 \) Copy content Toggle raw display
$19$ \( (T + 1)^{16} \) Copy content Toggle raw display
$23$ \( T^{16} + 181 T^{14} + \cdots + 1936 \) Copy content Toggle raw display
$29$ \( (T^{8} - T^{7} - 59 T^{6} + \cdots - 784)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 16 T^{7} + \cdots - 3104)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 476636224 \) Copy content Toggle raw display
$41$ \( (T^{8} - 3 T^{7} + \cdots - 3152)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 277688896 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 37937690176 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32319410176 \) Copy content Toggle raw display
$59$ \( (T^{8} - 12 T^{7} + \cdots + 9973312)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 21 T^{7} + \cdots + 269488)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 3129530826304 \) Copy content Toggle raw display
$71$ \( (T^{8} + 23 T^{7} + \cdots + 1022368)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 3803560873984 \) Copy content Toggle raw display
$79$ \( (T^{8} - 37 T^{7} + \cdots + 143552)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 28036855560256 \) Copy content Toggle raw display
$89$ \( (T^{8} - 7 T^{7} + \cdots - 10084)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 574639907835904 \) Copy content Toggle raw display
show more
show less