↑

Properties

Label 108.5.d.b
Level $108$
Weight $5$
Character orbit 108.d
Analytic conductor $11.164$
Analytic rank $0$
Dimension $16$
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] = [108,5,Mod(55,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.55");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.d (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: \(11.1639560131\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 38x^{14} + 1016x^{12} + 13512x^{10} + 130640x^{8} + 569472x^{6} + 1783808x^{4} + 352256x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{24} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{9} - \beta_{7}) q^{5} + \beta_{5} q^{7} + ( - \beta_{10} - 2 \beta_{7}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{9} - \beta_{7}) q^{5} + \beta_{5} q^{7} + ( - \beta_{10} - 2 \beta_{7}) q^{8} + (\beta_{6} - \beta_{5} + 11) q^{10} + ( - \beta_{11} + \beta_{7}) q^{11} + ( - \beta_{2} - \beta_1 + 11) q^{13} + (\beta_{12} - \beta_{11} + \cdots - \beta_{7}) q^{14}+ \cdots + (6 \beta_{14} - 116 \beta_{13} + \cdots + 540 \beta_{7}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 28 q^{4} + 176 q^{10} + 176 q^{13} + 88 q^{16} + 384 q^{22} + 2736 q^{25} + 1812 q^{28} + 1520 q^{34} + 80 q^{37} - 688 q^{40} - 1824 q^{46} - 7904 q^{49} - 5236 q^{52} - 11584 q^{58} - 1648 q^{61} + 5056 q^{64} + 26688 q^{70} + 80 q^{73} - 8388 q^{76} - 38464 q^{82} - 16832 q^{85} - 29520 q^{88} - 4512 q^{94} + 14864 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 38x^{14} + 1016x^{12} + 13512x^{10} + 130640x^{8} + 569472x^{6} + 1783808x^{4} + 352256x^{2} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 31526031 \nu^{14} - 1478004994 \nu^{12} - 34969469560 \nu^{10} - 518036651256 \nu^{8} + \cdots - 560383182118912 ) / 9072826085376 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22125963 \nu^{14} + 810251098 \nu^{12} + 21483034168 \nu^{10} + 275776021848 \nu^{8} + \cdots + 35527188072448 ) / 2268206521344 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55772295 \nu^{14} + 2016497426 \nu^{12} + 53270230904 \nu^{10} + 669847068600 \nu^{8} + \cdots - 54341410430464 ) / 567051630336 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 336655503 \nu^{14} + 12700746162 \nu^{12} + 338874029208 \nu^{10} + 4466455855992 \nu^{8} + \cdots + 61283674128384 ) / 2016183574528 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5389537731 \nu^{14} + 205346757098 \nu^{12} + 5490588520184 \nu^{10} + \cdots + 975274043457536 ) / 18145652170752 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5752396299 \nu^{14} + 214867475674 \nu^{12} + 5697961181752 \nu^{10} + \cdots - 975085761093632 ) / 18145652170752 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 736076377 \nu^{15} + 27811759278 \nu^{13} + 742057419752 \nu^{11} + \cdots - 12225716432896 \nu ) / 27218478256128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3544178403 \nu^{14} + 135483347818 \nu^{12} + 3637189030264 \nu^{10} + 48837038113560 \nu^{8} + \cdots + 21\!\cdots\!52 ) / 9072826085376 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21174584519 \nu^{15} + 801681331074 \nu^{13} + 21390001763416 \nu^{11} + \cdots + 323296419487744 \nu ) / 108873913024512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4319670871 \nu^{15} + 163390571778 \nu^{13} + 4358708917496 \nu^{11} + \cdots - 530311192121344 \nu ) / 13609239128064 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 39104187809 \nu^{15} - 1500703996398 \nu^{13} - 40295155484776 \nu^{11} + \cdots - 31\!\cdots\!16 \nu ) / 108873913024512 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 40828295945 \nu^{15} + 1588698201534 \nu^{13} + 42879935422312 \nu^{11} + \cdots + 68\!\cdots\!60 \nu ) / 108873913024512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5825845645 \nu^{15} - 220083807270 \nu^{13} - 5872149996680 \nu^{11} + \cdots - 363609072300032 \nu ) / 13609239128064 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 826772793 \nu^{15} + 31470960102 \nu^{13} + 842058646200 \nu^{11} + \cdots + 505026864599040 \nu ) / 1008091787264 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30611053565 \nu^{15} - 1158020226150 \nu^{13} - 30900770575624 \nu^{11} + \cdots - 32\!\cdots\!00 \nu ) / 27218478256128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{15} - 2\beta_{14} - 13\beta_{13} + 2\beta_{12} + 2\beta_{11} - 2\beta_{10} - 206\beta_{7} ) / 432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{8} - 6\beta_{6} - 6\beta_{5} + 40\beta_{4} - 9\beta_{3} + 24\beta_{2} - 1020 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 38 \beta_{15} + 26 \beta_{14} - 5 \beta_{13} - 26 \beta_{12} - 26 \beta_{11} + 2 \beta_{10} + \cdots + 14 \beta_{7} ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 66\beta_{8} + 78\beta_{6} - 6\beta_{5} - 308\beta_{4} - 9\beta_{3} + 372\beta_{2} + 12\beta _1 - 7848 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 290 \beta_{15} - 2 \beta_{14} + 1061 \beta_{13} - 46 \beta_{12} - 70 \beta_{11} + \cdots + 12286 \beta_{7} ) / 108 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22\beta_{8} - 66\beta_{6} + 22\beta_{5} + 285\beta_{3} - 2328\beta_{2} - 136\beta _1 + 45388 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 778 \beta_{15} - 1594 \beta_{14} - 8285 \beta_{13} + 2410 \beta_{12} + 2962 \beta_{11} + \cdots - 119614 \beta_{7} ) / 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7578 \beta_{8} - 6354 \beta_{6} + 3870 \beta_{5} + 23576 \beta_{4} - 5877 \beta_{3} + 48744 \beta_{2} + \cdots - 621288 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 36574 \beta_{15} + 12274 \beta_{14} - 23467 \beta_{13} - 12274 \beta_{12} - 12274 \beta_{11} + \cdots + 154522 \beta_{7} ) / 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 140844 \beta_{8} + 163236 \beta_{6} - 103236 \beta_{5} - 439880 \beta_{4} - 53370 \beta_{3} + \cdots - 11942760 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 569636 \beta_{15} + 38524 \beta_{14} + 1958798 \beta_{13} - 274396 \beta_{12} + \cdots + 18980692 \beta_{7} ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 69512 \beta_{8} - 208536 \beta_{6} + 69512 \beta_{5} + 1029588 \beta_{3} - 8805984 \beta_{2} + \cdots + 152703488 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3273608 \beta_{15} - 4697768 \beta_{14} - 27463084 \beta_{13} + 9781736 \beta_{12} + \cdots - 450395432 \beta_{7} ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 60988272 \beta_{8} - 53055984 \beta_{6} + 42608208 \beta_{5} + 163331456 \beta_{4} + \cdots - 4437300384 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 292631920 \beta_{15} + 72188176 \beta_{14} - 237941896 \beta_{13} - 72188176 \beta_{12} + \cdots + 1808260144 \beta_{7} ) / 27 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.222504 − 0.385387i
−0.222504 + 0.385387i
−1.12787 − 1.95353i
−1.12787 + 1.95353i
1.78073 − 3.08431i
1.78073 + 3.08431i
−2.23772 − 3.87585i
−2.23772 + 3.87585i
2.23772 − 3.87585i
2.23772 + 3.87585i
−1.78073 − 3.08431i
−1.78073 + 3.08431i
1.12787 − 1.95353i
1.12787 + 1.95353i
0.222504 − 0.385387i
0.222504 + 0.385387i
−3.98139 − 0.385387i 0 15.7030 + 3.06876i 22.8093 0 82.1204i −61.3369 − 18.2696i 0 −90.8127 − 8.79041i
55.2 −3.98139 + 0.385387i 0 15.7030 − 3.06876i 22.8093 0 − 82.1204i −61.3369 + 18.2696i 0 −90.8127 + 8.79041i
55.3 −3.49052 − 1.95353i 0 8.36746 + 13.6377i −47.1174 0 − 67.4003i −2.56528 − 63.9486i 0 164.464 + 92.0451i
55.4 −3.49052 + 1.95353i 0 8.36746 − 13.6377i −47.1174 0 67.4003i −2.56528 + 63.9486i 0 164.464 − 92.0451i
55.5 −2.54696 − 3.08431i 0 −3.02598 + 15.7113i 3.58334 0 16.2605i 56.1655 − 30.6829i 0 −9.12663 − 11.0521i
55.6 −2.54696 + 3.08431i 0 −3.02598 − 15.7113i 3.58334 0 − 16.2605i 56.1655 + 30.6829i 0 −9.12663 + 11.0521i
55.7 −0.988828 − 3.87585i 0 −14.0444 + 7.66510i 20.7568 0 − 5.38785i 43.5963 + 46.8547i 0 −20.5250 − 80.4504i
55.8 −0.988828 + 3.87585i 0 −14.0444 − 7.66510i 20.7568 0 5.38785i 43.5963 − 46.8547i 0 −20.5250 + 80.4504i
55.9 0.988828 − 3.87585i 0 −14.0444 − 7.66510i −20.7568 0 5.38785i −43.5963 + 46.8547i 0 −20.5250 + 80.4504i
55.10 0.988828 + 3.87585i 0 −14.0444 + 7.66510i −20.7568 0 − 5.38785i −43.5963 − 46.8547i 0 −20.5250 − 80.4504i
55.11 2.54696 − 3.08431i 0 −3.02598 − 15.7113i −3.58334 0 − 16.2605i −56.1655 − 30.6829i 0 −9.12663 + 11.0521i
55.12 2.54696 + 3.08431i 0 −3.02598 + 15.7113i −3.58334 0 16.2605i −56.1655 + 30.6829i 0 −9.12663 − 11.0521i
55.13 3.49052 − 1.95353i 0 8.36746 − 13.6377i 47.1174 0 67.4003i 2.56528 − 63.9486i 0 164.464 − 92.0451i
55.14 3.49052 + 1.95353i 0 8.36746 + 13.6377i 47.1174 0 − 67.4003i 2.56528 + 63.9486i 0 164.464 + 92.0451i
55.15 3.98139 − 0.385387i 0 15.7030 − 3.06876i −22.8093 0 − 82.1204i 61.3369 − 18.2696i 0 −90.8127 + 8.79041i
55.16 3.98139 + 0.385387i 0 15.7030 + 3.06876i −22.8093 0 82.1204i 61.3369 + 18.2696i 0 −90.8127 − 8.79041i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.5.d.b ✓ 16
3.b odd 2 1 inner 108.5.d.b ✓ 16
4.b odd 2 1 inner 108.5.d.b ✓ 16
12.b even 2 1 inner 108.5.d.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.d.b ✓ 16 1.a even 1 1 trivial
108.5.d.b ✓ 16 3.b odd 2 1 inner
108.5.d.b ✓ 16 4.b odd 2 1 inner
108.5.d.b ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 3184T_{5}^{6} + 2376384T_{5}^{4} - 527623168T_{5}^{2} + 6389764096 \) acting on \(S_{5}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 3184 T^{6} + \cdots + 6389764096)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 11580 T^{6} + \cdots + 235138677921)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 44 T^{3} + \cdots - 8708231)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 555827776000000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 46\!\cdots\!21)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 75\!\cdots\!64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 72\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 29\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 20 T^{3} + \cdots + 341576670649)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 52\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 68\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 20\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 333745064800105)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 51\!\cdots\!41)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 86\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 522411342741841)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 57\!\cdots\!29)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 58\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 39\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 59439054971905)^{4} \) Copy content Toggle raw display
show more
show less