Properties

Label 4.19.b.a
Level $4$
Weight $19$
Character orbit 4.b
Analytic conductor $8.215$
Analytic rank $0$
Dimension $8$
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] = [4,19,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 4.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: \(8.21544550839\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3 x^{7} - 10969 x^{6} - 835887 x^{5} + 20786973 x^{4} - 17082875145 x^{3} - 2740871824195 x^{2} + 109263554839035 x + 71\!\cdots\!30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 11) q^{2} - \beta_{2} q^{3} + (\beta_{4} - \beta_{2} - 9 \beta_1 + 44007) q^{4} + (\beta_{5} - \beta_{4} + 412 \beta_1 + 107404) q^{5} + (\beta_{7} + 2 \beta_{5} + 43 \beta_{2} + 9 \beta_1 + 38865) q^{6} + ( - 4 \beta_{7} - \beta_{6} + \beta_{5} + 37 \beta_{4} - 212 \beta_{2} - 1625 \beta_1 + 830) q^{7} + (4 \beta_{7} - 4 \beta_{6} - 52 \beta_{5} + 25 \beta_{4} + \beta_{3} + \cdots - 19247535) q^{8}+ \cdots + ( - 16 \beta_{6} + 2 \beta_{5} - 1062 \beta_{4} - 4 \beta_{3} + \cdots - 91144375) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 11) q^{2} - \beta_{2} q^{3} + (\beta_{4} - \beta_{2} - 9 \beta_1 + 44007) q^{4} + (\beta_{5} - \beta_{4} + 412 \beta_1 + 107404) q^{5} + (\beta_{7} + 2 \beta_{5} + 43 \beta_{2} + 9 \beta_1 + 38865) q^{6} + ( - 4 \beta_{7} - \beta_{6} + \beta_{5} + 37 \beta_{4} - 212 \beta_{2} - 1625 \beta_1 + 830) q^{7} + (4 \beta_{7} - 4 \beta_{6} - 52 \beta_{5} + 25 \beta_{4} + \beta_{3} + \cdots - 19247535) q^{8}+ \cdots + ( - 80498540832 \beta_{7} + \cdots - 299966892248208) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 84 q^{2} + 352016 q^{4} + 860880 q^{5} + 310944 q^{6} - 154160064 q^{8} - 729541560 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 84 q^{2} + 352016 q^{4} + 860880 q^{5} + 310944 q^{6} - 154160064 q^{8} - 729541560 q^{9} - 853591160 q^{10} - 2983758720 q^{12} - 6658778288 q^{13} - 3330240576 q^{14} + 35282993408 q^{16} + 213854181648 q^{17} + 197420552436 q^{18} - 421824543840 q^{20} - 771822339072 q^{21} - 586750684320 q^{22} - 3940602195456 q^{24} + 528925732440 q^{25} + 12314194404552 q^{26} - 20860043224320 q^{28} + 11238056568912 q^{29} + 63814514963520 q^{30} - 133632603995136 q^{32} - 21541938424320 q^{33} + 295093712425768 q^{34} - 595880470532208 q^{36} - 158886968816432 q^{37} + 830666716492320 q^{38} - 16\!\cdots\!80 q^{40}+ \cdots + 24\!\cdots\!76 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3 x^{7} - 10969 x^{6} - 835887 x^{5} + 20786973 x^{4} - 17082875145 x^{3} - 2740871824195 x^{2} + 109263554839035 x + 71\!\cdots\!30 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{7} + 10969 \nu^{5} + 868794 \nu^{4} - 18180591 \nu^{3} + 17028333372 \nu^{2} + 2791956824311 \nu - 88793056460566 ) / 1099511627776 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 349 \nu^{7} + 53248 \nu^{6} + 525867 \nu^{5} + 212100462 \nu^{4} - 9178232109 \nu^{3} - 5640029463756 \nu^{2} + \cdots - 18\!\cdots\!30 ) / 5772436045824 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11435 \nu^{7} + 1163264 \nu^{6} - 138717939 \nu^{5} - 27079663422 \nu^{4} - 970946160987 \nu^{3} - 174632082951444 \nu^{2} + \cdots - 11\!\cdots\!98 ) / 23089744183296 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2971 \nu^{7} - 1163264 \nu^{6} - 19301475 \nu^{5} + 14563817058 \nu^{4} + 1232855754933 \nu^{3} - 70678087605588 \nu^{2} + \cdots + 23\!\cdots\!98 ) / 23089744183296 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27353 \nu^{7} + 3555328 \nu^{6} - 477828111 \nu^{5} - 15638480214 \nu^{4} + 18432717602409 \nu^{3} + 162963069743580 \nu^{2} + \cdots - 31\!\cdots\!14 ) / 23089744183296 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3817 \nu^{7} - 3604480 \nu^{6} - 1646666367 \nu^{5} - 190583341494 \nu^{4} + 18428744008569 \nu^{3} + \cdots - 29\!\cdots\!26 ) / 23089744183296 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 683035 \nu^{7} + 75218944 \nu^{6} + 7547727645 \nu^{5} - 422692546206 \nu^{4} + 33370477920309 \nu^{3} + \cdots - 38\!\cdots\!18 ) / 23089744183296 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 686\beta _1 + 5801 ) / 16384 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16\beta_{6} - 16\beta_{5} + 53\beta_{4} + 5\beta_{3} + 800\beta_{2} + 81334\beta _1 + 44906797 ) / 16384 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1024 \beta_{7} + 320 \beta_{6} + 14016 \beta_{5} + 58307 \beta_{4} + 3 \beta_{3} - 271744 \beta_{2} + 5177930 \beta _1 + 5335373179 ) / 16384 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 140288 \beta_{7} - 8464 \beta_{6} - 120560 \beta_{5} + 10208305 \beta_{4} + 1889 \beta_{3} + 107155168 \beta_{2} + 4164164766 \beta _1 + 341814123305 ) / 16384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 28823552 \beta_{7} + 1727008 \beta_{6} - 78969376 \beta_{5} + 366137859 \beta_{4} - 3783773 \beta_{3} - 6811399616 \beta_{2} + 636988879210 \beta _1 + 271636965004379 ) / 16384 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 700712960 \beta_{7} - 57019888 \beta_{6} + 11801042416 \beta_{5} - 95357690295 \beta_{4} + 16939288601 \beta_{3} + 1212327792416 \beta_{2} + \cdots + 41\!\cdots\!33 ) / 16384 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 175667244032 \beta_{7} + 278225623168 \beta_{6} - 1498229387392 \beta_{5} + 15519483083131 \beta_{4} + 2837180895227 \beta_{3} + \cdots + 25\!\cdots\!83 ) / 16384 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
130.763 7.78570i
130.763 + 7.78570i
54.8957 117.008i
54.8957 + 117.008i
−67.8767 106.586i
−67.8767 + 106.586i
−116.282 46.4306i
−116.282 + 46.4306i
−511.052 31.1428i 30905.2i 260204. + 31831.2i 991387. 962474. 1.57942e7i 6.49702e7i −1.31987e8 2.43709e7i −5.67711e8 −5.06650e8 3.08746e7i
3.2 −511.052 + 31.1428i 30905.2i 260204. 31831.2i 991387. 962474. + 1.57942e7i 6.49702e7i −1.31987e8 + 2.43709e7i −5.67711e8 −5.06650e8 + 3.08746e7i
3.3 −207.583 468.031i 4128.11i −175963. + 194311.i −1.32216e6 −1.93208e6 + 856924.i 1.88890e7i 1.27470e8 + 4.20206e7i 3.70379e8 2.74457e8 + 6.18811e8i
3.4 −207.583 + 468.031i 4128.11i −175963. 194311.i −1.32216e6 −1.93208e6 856924.i 1.88890e7i 1.27470e8 4.20206e7i 3.70379e8 2.74457e8 6.18811e8i
3.5 283.507 426.343i 14439.2i −101392. 241742.i 2.88086e6 6.15603e6 + 4.09360e6i 4.40005e7i −1.31810e8 2.53077e7i 1.78931e8 8.16744e8 1.22823e9i
3.6 283.507 + 426.343i 14439.2i −101392. + 241742.i 2.88086e6 6.15603e6 4.09360e6i 4.40005e7i −1.31810e8 + 2.53077e7i 1.78931e8 8.16744e8 + 1.22823e9i
3.7 477.128 185.722i 27088.6i 193159. 177227.i −2.11965e6 −5.03095e6 1.29247e7i 3.35455e7i 5.92464e7 1.20434e8i −3.46370e8 −1.01135e9 + 3.93667e8i
3.8 477.128 + 185.722i 27088.6i 193159. + 177227.i −2.11965e6 −5.03095e6 + 1.29247e7i 3.35455e7i 5.92464e7 + 1.20434e8i −3.46370e8 −1.01135e9 3.93667e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.19.b.a 8
3.b odd 2 1 36.19.d.c 8
4.b odd 2 1 inner 4.19.b.a 8
8.b even 2 1 64.19.c.e 8
8.d odd 2 1 64.19.c.e 8
12.b even 2 1 36.19.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.19.b.a 8 1.a even 1 1 trivial
4.19.b.a 8 4.b odd 2 1 inner
36.19.d.c 8 3.b odd 2 1
36.19.d.c 8 12.b even 2 1
64.19.c.e 8 8.b even 2 1
64.19.c.e 8 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{19}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 84 T^{7} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( T^{8} + 1914452736 T^{6} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{4} - 430440 T^{3} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + 3329389144 T^{3} + \cdots - 57\!\cdots\!60)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 106927090824 T^{3} + \cdots - 57\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} - 5619028284456 T^{3} + \cdots - 48\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + 79443484408216 T^{3} + \cdots - 30\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 225754992362568 T^{3} + \cdots - 21\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} - 608192380043496 T^{3} + \cdots + 18\!\cdots\!20)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} - 92296457552168 T^{3} + \cdots + 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 16\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 45\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
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