Properties

Label 684.5.y.c
Level $684$
Weight $5$
Character orbit 684.y
Analytic conductor $70.705$
Analytic rank $0$
Dimension $12$
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] = [684,5,Mod(145,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.145");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), 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: \(70.7050547493\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + \cdots + 90728724573 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{5} - \beta_{3} - 1) q^{5} + (\beta_1 - 4) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} + \beta_{5} - \beta_{3} - 1) q^{5} + (\beta_1 - 4) q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{3} + \beta_{2} - 1) q^{11} + (\beta_{11} - \beta_{10} + \beta_{7} + 5 \beta_{5} - \beta_{4} - \beta_{2} - 10) q^{13} + (2 \beta_{11} + 3 \beta_{10} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - 78 \beta_{5} + \cdots + 78) q^{17}+ \cdots + (8 \beta_{11} + 9 \beta_{10} - 59 \beta_{9} + 39 \beta_{8} + 17 \beta_{7} + \cdots - 1930) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{5} - 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 9 q^{5} - 52 q^{7} - 6 q^{11} - 93 q^{13} + 483 q^{17} - 533 q^{19} - 531 q^{23} - 217 q^{25} - 2025 q^{29} + 1128 q^{35} + 1692 q^{41} - 63 q^{43} + 3471 q^{47} + 420 q^{49} + 3771 q^{53} - 2014 q^{55} + 9594 q^{59} + 1229 q^{61} + 7590 q^{67} - 963 q^{71} - 2838 q^{73} + 15408 q^{77} + 11073 q^{79} + 14202 q^{83} + 9455 q^{85} - 6525 q^{89} - 7686 q^{91} - 1521 q^{95} - 34110 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + \cdots + 90728724573 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16530610 \nu^{10} + 82653050 \nu^{9} - 6854019055 \nu^{8} + 26920157920 \nu^{7} - 468707180930 \nu^{6} + \cdots + 19\!\cdots\!29 ) / 689119197082182 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5704 \nu^{10} + 28520 \nu^{9} - 3465505 \nu^{8} + 13690900 \nu^{7} - 741062408 \nu^{6} + 2175388858 \nu^{5} - 66731757496 \nu^{4} + \cdots - 27588762045072 ) / 123100964109 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5176266 \nu^{10} + 25881330 \nu^{9} - 3116634049 \nu^{8} + 12311248216 \nu^{7} - 646970256701 \nu^{6} + \cdots - 18\!\cdots\!52 ) / 25522933225266 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 162445852 \nu^{11} - 5113227722 \nu^{10} - 69019867025 \nu^{9} - 3210361481925 \nu^{8} - 6648576376768 \nu^{7} + \cdots - 28\!\cdots\!57 ) / 25\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 324891704 \nu^{11} - 1786904372 \nu^{10} + 198106533130 \nu^{9} - 878077616295 \nu^{8} + 42131954279636 \nu^{7} + \cdots - 58\!\cdots\!35 ) / 25\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1328468937428 \nu^{11} - 47554498066674 \nu^{10} - 659027444905734 \nu^{9} + \cdots - 60\!\cdots\!40 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1328468937428 \nu^{11} - 62167656378382 \nu^{10} + \cdots - 98\!\cdots\!33 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4824263493526 \nu^{11} - 42886275115864 \nu^{10} + \cdots - 64\!\cdots\!65 ) / 16\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22328146096228 \nu^{11} - 128607403047369 \nu^{10} + \cdots + 27\!\cdots\!15 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42974864378431 \nu^{11} + 52922250531799 \nu^{10} + \cdots + 11\!\cdots\!23 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 42974864378431 \nu^{11} - 525645758694540 \nu^{10} + \cdots - 12\!\cdots\!54 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} + \beta_{5} + 2\beta_{4} - 3\beta_{3} - \beta_{2} - 3\beta _1 - 305 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{11} - 9 \beta_{10} + 51 \beta_{9} - 33 \beta_{8} + 3 \beta_{7} + 6 \beta_{6} + 37 \beta_{5} - 310 \beta_{4} + 12 \beta_{3} + 155 \beta_{2} - 30 \beta _1 - 476 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 78 \beta_{11} + 42 \beta_{10} + 102 \beta_{9} - 66 \beta_{8} - 750 \beta_{7} - 744 \beta_{6} + 73 \beta_{5} - 622 \beta_{4} + 1020 \beta_{3} + 806 \beta_{2} + 702 \beta _1 + 52372 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2892 \beta_{11} + 3192 \beta_{10} - 14280 \beta_{9} + 8898 \beta_{8} - 636 \beta_{7} - 3114 \beta_{6} - 93110 \beta_{5} + 60161 \beta_{4} - 1974 \beta_{3} - 28843 \beta_{2} + 9030 \beta _1 + 178339 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 28569 \beta_{11} - 10227 \beta_{10} - 43095 \beta_{9} + 26859 \beta_{8} + 170931 \beta_{7} + 163482 \beta_{6} - 279512 \beta_{5} + 182039 \beta_{4} - 251046 \beta_{3} - 281599 \beta_{2} + \cdots - 10495799 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 774531 \beta_{11} - 911367 \beta_{10} + 3407337 \beta_{9} - 1932201 \beta_{8} + 195768 \beta_{7} + 987813 \beta_{6} + 35577745 \beta_{5} - 12599905 \beta_{4} + 125775 \beta_{3} + \cdots - 55475093 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8178912 \beta_{11} + 1349640 \beta_{10} + 13830696 \beta_{9} - 7854300 \beta_{8} - 38395032 \beta_{7} - 35192076 \beta_{6} + 143615539 \beta_{5} - 51250588 \beta_{4} + \cdots + 2187500545 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 196776792 \beta_{11} + 240477552 \beta_{10} - 782134572 \beta_{9} + 384009816 \beta_{8} - 64316148 \beta_{7} - 273943092 \beta_{6} - 10460379527 \beta_{5} + \cdots + 15623748910 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2192902131 \beta_{11} + 44717703 \beta_{10} - 4014705255 \beta_{9} + 1979144673 \beta_{8} + 8590586568 \beta_{7} + 7518377505 \beta_{6} - 53380971125 \beta_{5} + \cdots - 460182452948 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 48603415992 \beta_{11} - 60820529142 \beta_{10} + 176993509674 \beta_{9} - 72407261352 \beta_{8} + 19895172393 \beta_{7} + 71817900399 \beta_{6} + \cdots - 4167581620361 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 14.3199i
0.500000 + 4.96177i
0.500000 4.44379i
0.500000 + 15.2283i
0.500000 9.58497i
0.500000 + 6.42649i
0.500000 + 14.3199i
0.500000 4.96177i
0.500000 + 4.44379i
0.500000 15.2283i
0.500000 + 9.58497i
0.500000 6.42649i
0 0 0 −19.9265 34.5138i 0 −53.3663 0 0 0
145.2 0 0 0 −15.5891 27.0012i 0 −2.67956 0 0 0
145.3 0 0 0 1.89169 + 3.27650i 0 36.8385 0 0 0
145.4 0 0 0 3.11411 + 5.39380i 0 49.9415 0 0 0
145.5 0 0 0 12.7505 + 22.0845i 0 27.7589 0 0 0
145.6 0 0 0 13.2594 + 22.9660i 0 −84.4930 0 0 0
217.1 0 0 0 −19.9265 + 34.5138i 0 −53.3663 0 0 0
217.2 0 0 0 −15.5891 + 27.0012i 0 −2.67956 0 0 0
217.3 0 0 0 1.89169 3.27650i 0 36.8385 0 0 0
217.4 0 0 0 3.11411 5.39380i 0 49.9415 0 0 0
217.5 0 0 0 12.7505 22.0845i 0 27.7589 0 0 0
217.6 0 0 0 13.2594 22.9660i 0 −84.4930 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.5.y.c 12
3.b odd 2 1 76.5.h.a 12
12.b even 2 1 304.5.r.b 12
19.d odd 6 1 inner 684.5.y.c 12
57.f even 6 1 76.5.h.a 12
228.n odd 6 1 304.5.r.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.h.a 12 3.b odd 2 1
76.5.h.a 12 57.f even 6 1
304.5.r.b 12 12.b even 2 1
304.5.r.b 12 228.n odd 6 1
684.5.y.c 12 1.a even 1 1 trivial
684.5.y.c 12 19.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{12} + 9 T_{5}^{11} + 2024 T_{5}^{10} - 20745 T_{5}^{9} + 2795952 T_{5}^{8} - 29332341 T_{5}^{7} + 1995775355 T_{5}^{6} - 32461763778 T_{5}^{5} + 983335559548 T_{5}^{4} + \cdots + 392044989615876 \) Copy content Toggle raw display
\( T_{7}^{6} + 26T_{7}^{5} - 6970T_{7}^{4} - 658T_{7}^{3} + 12165932T_{7}^{2} - 197809864T_{7} - 617046128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 392044989615876 \) Copy content Toggle raw display
$7$ \( (T^{6} + 26 T^{5} - 6970 T^{4} + \cdots - 617046128)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 3 T^{5} - 37505 T^{4} + \cdots - 781272216)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 93 T^{11} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} - 483 T^{11} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{12} + 533 T^{11} + \cdots + 48\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{12} + 531 T^{11} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{12} + 2025 T^{11} + \cdots + 43\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{12} + 5102232 T^{10} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{12} + 13373304 T^{10} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{12} - 1692 T^{11} + \cdots + 48\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{12} + 63 T^{11} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} - 3471 T^{11} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} - 3771 T^{11} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{12} - 9594 T^{11} + \cdots + 48\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{12} - 1229 T^{11} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{12} - 7590 T^{11} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{12} + 963 T^{11} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{12} + 2838 T^{11} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{12} - 11073 T^{11} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{6} - 7101 T^{5} + \cdots + 48\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 6525 T^{11} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{12} + 34110 T^{11} + \cdots + 49\!\cdots\!69 \) Copy content Toggle raw display
show more
show less