↑

Properties

Label 608.2.i.f
Level $608$
Weight $2$
Character orbit 608.i
Analytic conductor $4.855$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,2,Mod(353,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.353");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.i (of order \(3\), degree \(2\), 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: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 28 x^{9} - 121 x^{8} - 238 x^{7} + 392 x^{6} + 2534 x^{5} + 5589 x^{4} + 6426 x^{3} + \cdots + 676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{5}) q^{3} - \beta_{10} q^{5} + ( - \beta_{5} + \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{5}) q^{3} - \beta_{10} q^{5} + ( - \beta_{5} + \beta_{2}) q^{7} + ( - \beta_{9} + \beta_{7} + \beta_{2}) q^{11} - \beta_{11} q^{13} + ( - \beta_{9} - 2 \beta_{7}) q^{15} + ( - \beta_{11} - 2 \beta_{6} + \beta_1) q^{17} + ( - \beta_{7} + \beta_{5} + \cdots - \beta_{2}) q^{19}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{5} - 2 q^{13} - 10 q^{17} + 12 q^{21} - 20 q^{25} + 2 q^{29} - 12 q^{33} + 8 q^{37} - 14 q^{41} + 44 q^{49} + 26 q^{53} - 18 q^{57} - 26 q^{61} + 12 q^{65} - 60 q^{69} - 10 q^{73} + 8 q^{77} + 54 q^{81} + 2 q^{85} - 58 q^{89} - 24 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 28 x^{9} - 121 x^{8} - 238 x^{7} + 392 x^{6} + 2534 x^{5} + 5589 x^{4} + 6426 x^{3} + \cdots + 676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1227034840 \nu^{11} + 101709860 \nu^{10} + 3024377264 \nu^{9} + 27020327559 \nu^{8} + \cdots - 7946464064633 ) / 1660564373271 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43190891843 \nu^{11} - 42580911256 \nu^{10} + 48924888740 \nu^{9} - 1165706224452 \nu^{8} + \cdots + 42889573836382 ) / 43174673705046 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5212444375 \nu^{11} + 654304868 \nu^{10} + 13056453209 \nu^{9} + 114424956057 \nu^{8} + \cdots - 11558706401042 ) / 3321128746542 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79599768961 \nu^{11} + 19218229433 \nu^{10} + 6041005724 \nu^{9} + 2283878513406 \nu^{8} + \cdots - 96447984015110 ) / 43174673705046 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 315296 \nu^{11} - 45500 \nu^{10} + 140569 \nu^{9} + 8516067 \nu^{8} + 40529036 \nu^{7} + \cdots - 493442950 ) / 127865478 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3583477 \nu^{11} - 2844485 \nu^{10} + 2271166 \nu^{9} - 102646614 \nu^{8} - 351381781 \nu^{7} + \cdots + 2978552900 ) / 484200138 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 358703589464 \nu^{11} + 532399265008 \nu^{10} - 587898375077 \nu^{9} + \cdots - 202860182574868 ) / 43174673705046 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 421851970136 \nu^{11} - 349562722288 \nu^{10} + 298893822344 \nu^{9} + \cdots + 390913901632618 ) / 43174673705046 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 548393989394 \nu^{11} + 233674112776 \nu^{10} - 106115831198 \nu^{9} + \cdots - 638363686296154 ) / 43174673705046 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 70390426103 \nu^{11} + 50668925104 \nu^{10} - 17094253049 \nu^{9} + 1950217136898 \nu^{8} + \cdots - 56869280436520 ) / 3321128746542 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 36046267051 \nu^{11} - 28424148571 \nu^{10} + 20976972330 \nu^{9} - 1028608939136 \nu^{8} + \cdots + 25104497288110 ) / 1107042915514 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{8} - 4\beta_{6} + \beta_{5} - 3\beta_{4} - 2\beta_{3} + 3\beta_{2} + \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{8} + 2\beta_{5} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 16 \beta_{11} - 10 \beta_{10} + 6 \beta_{9} + 20 \beta_{8} + 12 \beta_{7} + 46 \beta_{6} + 5 \beta_{5} + \cdots + 16 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{11} + 13\beta_{6} - 8\beta_{3} + 15\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 109 \beta_{11} + 37 \beta_{10} - 138 \beta_{9} - 263 \beta_{8} - 66 \beta_{7} - 382 \beta_{6} + \cdots + 752 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -65\beta_{9} - 64\beta_{8} + 21\beta_{7} + 211\beta_{5} - 100\beta_{4} + 310\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2383 \beta_{11} - 1039 \beta_{10} - 210 \beta_{9} + 1277 \beta_{8} + 1134 \beta_{7} + 7738 \beta_{6} + \cdots - 1034 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -873\beta_{11} - 436\beta_{10} + 2687\beta_{6} - 540\beta_{3} + 2096\beta _1 + 3953 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9304 \beta_{11} + 4582 \beta_{10} - 15816 \beta_{9} - 36380 \beta_{8} - 11094 \beta_{7} - 28810 \beta_{6} + \cdots + 94130 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -11498\beta_{9} - 15565\beta_{8} - 35\beta_{7} + 30909\beta_{5} - 21454\beta_{4} + 42394\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 347605 \beta_{11} - 162019 \beta_{10} - 59736 \beta_{9} + 89339 \beta_{8} + 125850 \beta_{7} + \cdots - 354662 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1
−1.56354 + 1.15100i
3.40664 − 0.180756i
−0.111052 + 0.761807i
−1.15100 − 1.56354i
0.180756 + 3.40664i
−0.761807 − 0.111052i
−1.56354 − 1.15100i
3.40664 + 0.180756i
−0.111052 − 0.761807i
−1.15100 + 1.56354i
0.180756 − 3.40664i
−0.761807 + 0.111052i
0 −0.866025 + 1.50000i 0 −2.09926 + 3.63603i 0 −4.28918 0 0 0
353.2 0 −0.866025 + 1.50000i 0 0.268461 − 0.464989i 0 2.98767 0 0 0
353.3 0 −0.866025 + 1.50000i 0 1.33080 − 2.30501i 0 −2.16259 0 0 0
353.4 0 0.866025 − 1.50000i 0 −2.09926 + 3.63603i 0 4.28918 0 0 0
353.5 0 0.866025 − 1.50000i 0 0.268461 − 0.464989i 0 −2.98767 0 0 0
353.6 0 0.866025 − 1.50000i 0 1.33080 − 2.30501i 0 2.16259 0 0 0
577.1 0 −0.866025 − 1.50000i 0 −2.09926 − 3.63603i 0 −4.28918 0 0 0
577.2 0 −0.866025 − 1.50000i 0 0.268461 + 0.464989i 0 2.98767 0 0 0
577.3 0 −0.866025 − 1.50000i 0 1.33080 + 2.30501i 0 −2.16259 0 0 0
577.4 0 0.866025 + 1.50000i 0 −2.09926 − 3.63603i 0 4.28918 0 0 0
577.5 0 0.866025 + 1.50000i 0 0.268461 + 0.464989i 0 −2.98767 0 0 0
577.6 0 0.866025 + 1.50000i 0 1.33080 + 2.30501i 0 2.16259 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.i.f ✓ 12
4.b odd 2 1 inner 608.2.i.f ✓ 12
8.b even 2 1 1216.2.i.r 12
8.d odd 2 1 1216.2.i.r 12
19.c even 3 1 inner 608.2.i.f ✓ 12
76.g odd 6 1 inner 608.2.i.f ✓ 12
152.k odd 6 1 1216.2.i.r 12
152.p even 6 1 1216.2.i.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.i.f ✓ 12 1.a even 1 1 trivial
608.2.i.f ✓ 12 4.b odd 2 1 inner
608.2.i.f ✓ 12 19.c even 3 1 inner
608.2.i.f ✓ 12 76.g odd 6 1 inner
1216.2.i.r 12 8.b even 2 1
1216.2.i.r 12 8.d odd 2 1
1216.2.i.r 12 152.k odd 6 1
1216.2.i.r 12 152.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{3} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{5} + 13 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 32 T^{4} + \cdots - 768)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 62 T^{4} + \cdots - 8112)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + T^{5} + 33 T^{4} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 5 T^{5} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 40 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 333135504 \) Copy content Toggle raw display
$29$ \( (T^{6} - T^{5} + 13 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 44 T^{4} + \cdots - 48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 2 T^{2} + \cdots + 128)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 7 T^{5} + \cdots + 33489)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 59 T^{10} + \cdots + 2985984 \) Copy content Toggle raw display
$47$ \( T^{12} + 107 T^{10} + \cdots + 12027024 \) Copy content Toggle raw display
$53$ \( (T^{6} - 13 T^{5} + \cdots + 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 694200576969 \) Copy content Toggle raw display
$61$ \( (T^{6} + 13 T^{5} + \cdots + 1444)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 137 T^{10} + \cdots + 1172889 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 15587023104 \) Copy content Toggle raw display
$73$ \( (T^{6} + 5 T^{5} + 66 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 49262690304 \) Copy content Toggle raw display
$83$ \( (T^{6} - 258 T^{4} + \cdots - 350892)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 29 T^{5} + \cdots + 295936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 3 T^{5} + \cdots + 505521)^{2} \) Copy content Toggle raw display
show more
show less