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Properties

Label 4.17.b.b
Level $4$
Weight $17$
Character orbit 4.b
Analytic conductor $6.493$
Analytic rank $0$
Dimension $6$
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] = [4,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 17 \)
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: \(6.49298175427\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5152x^{4} + 242526x^{3} + 17329473x^{2} + 402444531x + 64957563630 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 27) q^{2} + (\beta_{2} - 3 \beta_1 + 1) q^{3} + ( - \beta_{4} - \beta_{2} + \cdots - 26741) q^{4}+ \cdots + ( - 258 \beta_{5} - 704 \beta_{4} + \cdots - 22951713) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 27) q^{2} + (\beta_{2} - 3 \beta_1 + 1) q^{3} + ( - \beta_{4} - \beta_{2} + \cdots - 26741) q^{4}+ \cdots + ( - 796777344 \beta_{5} + \cdots - 4219215747417) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 164 q^{2} - 160368 q^{4} - 506740 q^{5} - 1187136 q^{6} - 33829184 q^{8} - 137574522 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 164 q^{2} - 160368 q^{4} - 506740 q^{5} - 1187136 q^{6} - 33829184 q^{8} - 137574522 q^{9} - 142269000 q^{10} + 254142720 q^{12} + 2544478092 q^{13} + 1073417856 q^{14} + 145019136 q^{16} + 1579205132 q^{17} - 22662764964 q^{18} + 4421361440 q^{20} - 27228321792 q^{21} - 138624795840 q^{22} + 403778497536 q^{24} + 271424476050 q^{25} - 190529582152 q^{26} + 1617685224960 q^{28} - 1158411768436 q^{29} - 4663806986880 q^{30} + 4663321578496 q^{32} - 767957621760 q^{33} - 5461346085192 q^{34} + 14104690258320 q^{36} + 8581446019212 q^{37} - 20462346561600 q^{38} + 18971054755200 q^{40} + 1840369253132 q^{41} - 20386577111040 q^{42} + 3644055863040 q^{44} - 34166370110580 q^{45} + 5034653652864 q^{46} - 56248898088960 q^{48} - 5527245758202 q^{49} + 129240587956500 q^{50} - 103374802254048 q^{52} + 130668269409932 q^{53} + 366965883430272 q^{54} - 342617610295296 q^{56} - 122486852367360 q^{57} + 69923529928632 q^{58} - 641633781542400 q^{60} - 429486008315508 q^{61} + 619512054551040 q^{62} - 516434037731328 q^{64} + 12\!\cdots\!00 q^{65}+ \cdots + 13\!\cdots\!56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5152x^{4} + 242526x^{3} + 17329473x^{2} + 402444531x + 64957563630 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 6\nu^{4} + 5194\nu^{3} + 278884\nu^{2} + 19281661\nu + 424169950 ) / 4194304 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -361\nu^{5} + 1930\nu^{4} + 275366\nu^{3} - 20309508\nu^{2} - 346364613\nu + 262341750258 ) / 37748736 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1655\nu^{5} - 161482\nu^{4} - 12663398\nu^{3} - 113266044\nu^{2} - 47656971675\nu - 1037716791858 ) / 37748736 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -191\nu^{5} + 34694\nu^{4} - 1050422\nu^{3} + 121467492\nu^{2} + 6308187453\nu + 313280271198 ) / 9437184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{5} - 9038\nu^{4} - 52930\nu^{3} - 47309076\nu^{2} - 332488665\nu - 104406897222 ) / 2359296 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 108\beta _1 + 135 ) / 1024 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -31\beta_{5} + \beta_{4} + 32\beta_{3} + 64\beta_{2} + 7820\beta _1 - 1760985 ) / 1024 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1669\beta_{5} - 3077\beta_{4} - 896\beta_{3} + 16128\beta_{2} + 182372\beta _1 - 126872611 ) / 1024 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -130347\beta_{5} - 19019\beta_{4} - 157472\beta_{3} - 285760\beta_{2} - 45046468\beta _1 - 2924369197 ) / 1024 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1185389 \beta_{5} - 3464493 \beta_{4} - 3325632 \beta_{3} - 99902848 \beta_{2} + \cdots + 730680044421 ) / 1024 \) Copy content Toggle raw display

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

Character values

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

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

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
−46.2446 + 35.5107i
−46.2446 − 35.5107i
11.8147 + 63.8186i
11.8147 − 63.8186i
34.9299 + 57.5840i
34.9299 − 57.5840i
−212.978 − 142.043i 2908.07i 25183.6 + 60504.2i 20884.7 413070. − 619356.i − 8.01505e6i 3.23062e6 − 1.64632e7i 3.45899e7 −4.44799e6 − 2.96652e6i
3.2 −212.978 + 142.043i − 2908.07i 25183.6 − 60504.2i 20884.7 413070. + 619356.i 8.01505e6i 3.23062e6 + 1.64632e7i 3.45899e7 −4.44799e6 + 2.96652e6i
3.3 19.2587 − 255.275i − 11183.9i −64794.2 − 9832.53i 389860. −2.85496e6 − 215387.i 1.06777e6i −3.75785e6 + 1.63509e7i −8.20322e7 7.50821e6 − 9.95214e7i
3.4 19.2587 + 255.275i 11183.9i −64794.2 + 9832.53i 389860. −2.85496e6 + 215387.i − 1.06777e6i −3.75785e6 − 1.63509e7i −8.20322e7 7.50821e6 + 9.95214e7i
3.5 111.720 − 230.336i 8024.44i −40573.4 − 51466.2i −664115. 1.84832e6 + 896488.i 6.08943e6i −1.63874e7 + 3.59574e6i −2.13450e7 −7.41947e7 + 1.52970e8i
3.6 111.720 + 230.336i − 8024.44i −40573.4 + 51466.2i −664115. 1.84832e6 − 896488.i − 6.08943e6i −1.63874e7 − 3.59574e6i −2.13450e7 −7.41947e7 − 1.52970e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.6
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 4.17.b.b ✓ 6
3.b odd 2 1 36.17.d.b 6
4.b odd 2 1 inner 4.17.b.b ✓ 6
8.b even 2 1 64.17.c.d 6
8.d odd 2 1 64.17.c.d 6
12.b even 2 1 36.17.d.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.17.b.b ✓ 6 1.a even 1 1 trivial
4.17.b.b ✓ 6 4.b odd 2 1 inner
36.17.d.b 6 3.b odd 2 1
36.17.d.b 6 12.b even 2 1
64.17.c.d 6 8.b even 2 1
64.17.c.d 6 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 197927424T_{3}^{4} + 9656364500582400T_{3}^{2} + 68111840207423731138560 \) acting on \(S_{17}^{\mathrm{new}}(4, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + \cdots + 281474976710656 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 68\!\cdots\!60 \) Copy content Toggle raw display
$5$ \( (T^{3} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots + 12\!\cdots\!20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots - 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots - 40\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots + 48\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 13\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots + 14\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 43\!\cdots\!88)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 75\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 36\!\cdots\!20)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 47\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 37\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 76\!\cdots\!40)^{2} \) Copy content Toggle raw display
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