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Properties

Label 608.3.e.b
Level $608$
Weight $3$
Character orbit 608.e
Analytic conductor $16.567$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,3,Mod(417,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.417");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 608.e (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: \(16.5668000731\)
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: \( x^{20} - 8 x^{19} - 196 x^{18} + 1676 x^{17} + 16346 x^{16} - 161824 x^{15} - 667200 x^{14} + \cdots + 1135285065792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{27} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + \beta_{6} q^{5} + \beta_1 q^{7} + (\beta_{4} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + \beta_{6} q^{5} + \beta_1 q^{7} + (\beta_{4} - 3) q^{9} - \beta_{5} q^{11} - \beta_{12} q^{13} + ( - \beta_{14} + \beta_{11}) q^{15} + ( - \beta_{9} - \beta_{6} + 3) q^{17} - \beta_{10} q^{19} + ( - \beta_{16} + \beta_{12}) q^{21} + ( - \beta_{5} - \beta_{2}) q^{23} + ( - \beta_{9} - \beta_{6} + \beta_{4} + \cdots + 2) q^{25}+ \cdots + ( - 2 \beta_{15} - 2 \beta_{10} + \cdots + 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} - 196 x^{18} + 1676 x^{17} + 16346 x^{16} - 161824 x^{15} - 667200 x^{14} + \cdots + 1135285065792 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 45\!\cdots\!22 \nu^{19} + \cdots - 87\!\cdots\!20 ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36\!\cdots\!36 \nu^{19} + \cdots - 19\!\cdots\!40 ) / 65\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70\!\cdots\!78 \nu^{19} + \cdots - 12\!\cdots\!20 ) / 11\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27\!\cdots\!84 \nu^{19} + \cdots + 35\!\cdots\!68 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 54\!\cdots\!34 \nu^{19} + \cdots + 20\!\cdots\!80 ) / 65\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!96 \nu^{19} + \cdots + 91\!\cdots\!20 ) / 17\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 84\!\cdots\!49 \nu^{19} + \cdots - 66\!\cdots\!76 ) / 65\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!70 \nu^{19} + \cdots - 41\!\cdots\!24 ) / 50\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 66\!\cdots\!96 \nu^{19} + \cdots - 74\!\cdots\!36 ) / 88\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25\!\cdots\!87 \nu^{19} + \cdots - 87\!\cdots\!40 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!01 \nu^{19} + \cdots + 21\!\cdots\!76 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!68 \nu^{19} + \cdots - 34\!\cdots\!24 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!74 \nu^{19} + \cdots + 66\!\cdots\!80 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!91 \nu^{19} + \cdots + 16\!\cdots\!24 ) / 65\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 90\!\cdots\!83 \nu^{19} + \cdots + 10\!\cdots\!40 ) / 45\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 87\!\cdots\!67 \nu^{19} + \cdots - 16\!\cdots\!96 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 31\!\cdots\!69 \nu^{19} + \cdots + 29\!\cdots\!72 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 55\!\cdots\!51 \nu^{19} + \cdots - 10\!\cdots\!88 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 62\!\cdots\!57 \nu^{19} + \cdots - 93\!\cdots\!16 ) / 14\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + \beta_{3} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{9} + \cdots + 92 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{18} - 9 \beta_{17} - 12 \beta_{16} + 11 \beta_{15} - 9 \beta_{14} - 9 \beta_{13} + 6 \beta_{12} + \cdots + 70 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{19} + 8 \beta_{18} - 11 \beta_{17} + 134 \beta_{16} - 64 \beta_{15} + 90 \beta_{14} + \cdots + 2394 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 90 \beta_{19} + 580 \beta_{18} - 515 \beta_{17} - 1160 \beta_{16} + 1147 \beta_{15} - 221 \beta_{14} + \cdots + 18294 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 369 \beta_{19} + 807 \beta_{18} - 347 \beta_{17} + 6554 \beta_{16} - 2982 \beta_{15} + 6480 \beta_{14} + \cdots + 51878 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6293 \beta_{19} + 18620 \beta_{18} - 8673 \beta_{17} - 26320 \beta_{16} + 44301 \beta_{15} + \cdots + 1066043 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 65684 \beta_{19} + 95126 \beta_{18} + 707 \beta_{17} + 238318 \beta_{16} - 42358 \beta_{15} + \cdots + 732858 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1054482 \beta_{19} + 2150484 \beta_{18} - 423993 \beta_{17} - 581448 \beta_{16} + 5464597 \beta_{15} + \cdots + 164629678 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6714915 \beta_{19} + 9918931 \beta_{18} + 324869 \beta_{17} + 4332940 \beta_{16} + 8137643 \beta_{15} + \cdots + 49322922 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 74424262 \beta_{19} + 128990092 \beta_{18} - 10104281 \beta_{17} + 131398916 \beta_{16} + \cdots + 10050905834 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 272446377 \beta_{19} + 428173619 \beta_{18} - 5848221 \beta_{17} - 88319898 \beta_{16} + \cdots + 4736782377 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2631167123 \beta_{19} + 4399712018 \beta_{18} - 304698030 \beta_{17} + 7473905504 \beta_{16} + \cdots + 258992571489 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 38560424569 \beta_{19} + 63865802775 \beta_{18} - 3699963342 \beta_{17} - 19433215762 \beta_{16} + \cdots + 1117828047194 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 402873857542 \beta_{19} + 678345620244 \beta_{18} - 59983825151 \beta_{17} + 1049413194212 \beta_{16} + \cdots + 23543093198274 ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 2506153848480 \beta_{19} + 4318240876494 \beta_{18} - 393409897143 \beta_{17} - 608758026798 \beta_{16} + \cdots + 96964206019162 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 32329817221434 \beta_{19} + 54527387352628 \beta_{18} - 5104591630351 \beta_{17} + 57354589714184 \beta_{16} + \cdots + 10\!\cdots\!50 ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 156123507579291 \beta_{19} + 276824764082093 \beta_{18} - 31610369671175 \beta_{17} + \cdots + 68\!\cdots\!50 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 12\!\cdots\!65 \beta_{19} + \cdots + 23\!\cdots\!83 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
−7.02110 + 2.30731i
3.53771 + 2.30731i
2.72330 + 2.14790i
5.68070 + 2.14790i
−6.67959 + 1.97398i
4.75152 + 1.97398i
−4.46201 + 0.741526i
−2.07038 + 0.741526i
−0.454273 + 0.341165i
7.99412 + 0.341165i
−0.454273 − 0.341165i
7.99412 − 0.341165i
−4.46201 − 0.741526i
−2.07038 − 0.741526i
−6.67959 − 1.97398i
4.75152 − 1.97398i
2.72330 − 2.14790i
5.68070 − 2.14790i
−7.02110 − 2.30731i
3.53771 − 2.30731i
0 − 4.61461i 0 3.04266 0 −10.5588 0 −12.2947 0
417.2 0 − 4.61461i 0 3.04266 0 10.5588 0 −12.2947 0
417.3 0 − 4.29580i 0 2.32882 0 −2.95740 0 −9.45390 0
417.4 0 − 4.29580i 0 2.32882 0 2.95740 0 −9.45390 0
417.5 0 − 3.94796i 0 −7.51133 0 −11.4311 0 −6.58642 0
417.6 0 − 3.94796i 0 −7.51133 0 11.4311 0 −6.58642 0
417.7 0 − 1.48305i 0 −4.43504 0 −2.39163 0 6.80056 0
417.8 0 − 1.48305i 0 −4.43504 0 2.39163 0 6.80056 0
417.9 0 − 0.682329i 0 6.57489 0 −8.44839 0 8.53443 0
417.10 0 − 0.682329i 0 6.57489 0 8.44839 0 8.53443 0
417.11 0 0.682329i 0 6.57489 0 −8.44839 0 8.53443 0
417.12 0 0.682329i 0 6.57489 0 8.44839 0 8.53443 0
417.13 0 1.48305i 0 −4.43504 0 −2.39163 0 6.80056 0
417.14 0 1.48305i 0 −4.43504 0 2.39163 0 6.80056 0
417.15 0 3.94796i 0 −7.51133 0 −11.4311 0 −6.58642 0
417.16 0 3.94796i 0 −7.51133 0 11.4311 0 −6.58642 0
417.17 0 4.29580i 0 2.32882 0 −2.95740 0 −9.45390 0
417.18 0 4.29580i 0 2.32882 0 2.95740 0 −9.45390 0
417.19 0 4.61461i 0 3.04266 0 −10.5588 0 −12.2947 0
417.20 0 4.61461i 0 3.04266 0 10.5588 0 −12.2947 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.3.e.b ✓ 20
4.b odd 2 1 inner 608.3.e.b ✓ 20
8.b even 2 1 1216.3.e.p 20
8.d odd 2 1 1216.3.e.p 20
19.b odd 2 1 inner 608.3.e.b ✓ 20
76.d even 2 1 inner 608.3.e.b ✓ 20
152.b even 2 1 1216.3.e.p 20
152.g odd 2 1 1216.3.e.p 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.3.e.b ✓ 20 1.a even 1 1 trivial
608.3.e.b ✓ 20 4.b odd 2 1 inner
608.3.e.b ✓ 20 19.b odd 2 1 inner
608.3.e.b ✓ 20 76.d even 2 1 inner
1216.3.e.p 20 8.b even 2 1
1216.3.e.p 20 8.d odd 2 1
1216.3.e.p 20 152.b even 2 1
1216.3.e.p 20 152.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 58T_{3}^{8} + 1161T_{3}^{6} + 8880T_{3}^{4} + 17360T_{3}^{2} + 6272 \) acting on \(S_{3}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + 58 T^{8} + \cdots + 6272)^{2} \) Copy content Toggle raw display
$5$ \( (T^{5} - 67 T^{3} + \cdots - 1552)^{4} \) Copy content Toggle raw display
$7$ \( (T^{10} - 328 T^{8} + \cdots - 52019100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 742 T^{8} + \cdots - 33292224)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 1258 T^{8} + \cdots + 42732419072)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 14 T^{4} + \cdots - 353826)^{4} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 161581959616)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 29563502310272)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 4164913594368)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 12\!\cdots\!52)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 15\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 2427003129600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 752855556992)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 8 T^{4} + \cdots + 1052928)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 31\!\cdots\!48)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 13\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 38 T^{4} + \cdots - 35915530)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 711012765331456)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 50\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 52\!\cdots\!12)^{2} \) Copy content Toggle raw display
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