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Properties

Label 880.2.cd.c
Level $880$
Weight $2$
Character orbit 880.cd
Analytic conductor $7.027$
Analytic rank $0$
Dimension $16$
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] = [880,2,Mod(49,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.cd (of order \(10\), degree \(4\), not 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{13} - \beta_{5}) q^{3} + (\beta_{10} + \beta_{6} - \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{12} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{11} + \beta_{9} - \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} - \beta_{13} - \beta_{5}) q^{3} + (\beta_{10} + \beta_{6} - \beta_{3}) q^{5} + ( - \beta_{15} - \beta_{12} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{11} + 5 \beta_{9} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{5} + 2 q^{9} + 6 q^{11} + 16 q^{15} - 6 q^{19} + 8 q^{21} - 16 q^{25} + 2 q^{29} - 8 q^{31} - 22 q^{35} - 30 q^{39} - 52 q^{41} + 12 q^{45} - 10 q^{49} + 42 q^{51} + 8 q^{55} - 2 q^{59} - 40 q^{61} - 40 q^{65} + 26 q^{69} - 36 q^{71} + 20 q^{75} - 38 q^{79} + 68 q^{81} + 58 q^{85} + 24 q^{89} + 20 q^{91} - 48 q^{95} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1829973 \nu^{15} - 24476532 \nu^{13} + 152757677 \nu^{11} - 557238290 \nu^{9} + \cdots - 7498263355 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 608673 \nu^{14} + 4779625 \nu^{12} - 9344761 \nu^{10} - 2560494 \nu^{8} - 9430964 \nu^{6} + \cdots + 31775447 ) / 609644662 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2744937 \nu^{14} - 22433610 \nu^{12} + 99837091 \nu^{10} - 276654262 \nu^{8} + 814713604 \nu^{6} + \cdots - 771926881 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2744937 \nu^{15} - 22433610 \nu^{13} + 99837091 \nu^{11} - 276654262 \nu^{9} + \cdots - 771926881 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2924455 \nu^{14} + 43736988 \nu^{12} - 193036323 \nu^{10} + 488836470 \nu^{8} + \cdots - 236512683 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1351905 \nu^{14} + 5558819 \nu^{12} - 7678521 \nu^{10} - 13223046 \nu^{8} - 42401020 \nu^{6} + \cdots - 629543387 ) / 609644662 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6960689 \nu^{14} + 30587176 \nu^{12} - 61040185 \nu^{10} + 38446538 \nu^{8} + \cdots - 1738523017 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3962283 \nu^{14} + 31992860 \nu^{12} - 118526613 \nu^{10} + 271533274 \nu^{8} + \cdots - 383811549 ) / 1219289324 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8234811 \nu^{15} - 67300830 \nu^{13} + 299511273 \nu^{11} - 829962786 \nu^{9} + \cdots - 2315780643 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 770947 \nu^{14} - 4704418 \nu^{12} + 13758165 \nu^{10} - 28071690 \nu^{8} + 126275508 \nu^{6} + \cdots - 243303327 ) / 221688968 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10669503 \nu^{15} - 86419330 \nu^{13} + 336890317 \nu^{11} - 819720810 \nu^{9} + \cdots - 4303783 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8815096 \nu^{15} + 58503253 \nu^{13} - 198965251 \nu^{11} + 429083674 \nu^{9} + \cdots + 352179707 \nu ) / 1219289324 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19639683 \nu^{15} + 162786416 \nu^{13} - 643887411 \nu^{11} + 1627587846 \nu^{9} + \cdots + 4755604557 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1496498 \nu^{15} + 9707543 \nu^{13} - 31032777 \nu^{11} + 60570826 \nu^{9} + \cdots - 172220447 \nu ) / 110844484 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{9} - 4\beta_{8} + \beta_{7} - 4\beta_{6} + 4\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 5\beta_{14} - \beta_{13} - 10\beta_{12} - \beta_{10} + 15\beta_{5} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{11} - 22\beta_{8} + 21\beta_{7} + \beta_{4} - 7\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 21\beta_{15} - 21\beta_{14} - 36\beta_{13} - 36\beta_{12} + 29\beta_{5} - 28\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -85\beta_{11} - 78\beta_{9} - 87\beta_{8} + 85\beta_{7} + 49\beta_{6} - 7\beta_{4} - 49\beta_{3} - 87 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 85\beta_{15} - 36\beta_{14} - 136\beta_{13} - 45\beta_{12} + 36\beta_{10} + 36\beta_{5} - 121\beta_{2} - 85\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -342\beta_{11} - 396\beta_{9} - 230\beta_{8} + 166\beta_{7} + 166\beta_{6} - 166\beta_{4} - 529 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 166\beta_{15} - 220\beta_{13} + 342\beta_{10} - 220\beta_{5} - 342\beta_{2} - 475\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -728\beta_{11} - 1653\beta_{9} + 728\beta_{6} - 1170\beta_{4} + 651\beta_{3} - 1653 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 728\beta_{14} + 1002\beta_{12} + 1379\beta_{10} - 2823\beta_{5} - 1002\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -4644\beta_{9} + 3748\beta_{8} - 3109\beta_{7} + 2472\beta_{6} - 3748\beta_{4} + 3109\beta_{3} - 3109 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3109 \beta_{15} + 5581 \beta_{14} + 4385 \beta_{13} + 8392 \beta_{12} + 3109 \beta_{10} + \cdots - 1276 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.972539 + 1.33858i
0.471815 − 0.649397i
−0.471815 + 0.649397i
0.972539 − 1.33858i
1.92464 − 0.625353i
−1.17360 + 0.381325i
1.17360 − 0.381325i
−1.92464 + 0.625353i
−0.972539 − 1.33858i
0.471815 + 0.649397i
−0.471815 − 0.649397i
0.972539 + 1.33858i
1.92464 + 0.625353i
−1.17360 − 0.381325i
1.17360 + 0.381325i
−1.92464 − 0.625353i
0 −1.16075 + 1.59764i 0 1.09069 − 1.95203i 0 −1.31845 − 1.81468i 0 −0.278050 − 0.855749i 0
49.2 0 −1.03494 + 1.42447i 0 −2.16717 + 0.550792i 0 −0.348029 − 0.479022i 0 −0.0309674 − 0.0953077i 0
49.3 0 1.03494 − 1.42447i 0 1.42953 + 1.71943i 0 0.348029 + 0.479022i 0 −0.0309674 − 0.0953077i 0
49.4 0 1.16075 − 1.59764i 0 0.264988 − 2.22031i 0 1.31845 + 1.81468i 0 −0.278050 − 0.855749i 0
289.1 0 −2.49233 + 0.809808i 0 −1.06421 + 1.96658i 0 0.918552 + 0.298456i 0 3.12889 − 2.27327i 0
289.2 0 −0.346168 + 0.112477i 0 −2.11914 + 0.713621i 0 2.45903 + 0.798988i 0 −2.31987 + 1.68548i 0
289.3 0 0.346168 − 0.112477i 0 0.0238439 − 2.23594i 0 −2.45903 − 0.798988i 0 −2.31987 + 1.68548i 0
289.4 0 2.49233 − 0.809808i 0 1.54147 − 1.61983i 0 −0.918552 − 0.298456i 0 3.12889 − 2.27327i 0
449.1 0 −1.16075 − 1.59764i 0 1.09069 + 1.95203i 0 −1.31845 + 1.81468i 0 −0.278050 + 0.855749i 0
449.2 0 −1.03494 − 1.42447i 0 −2.16717 − 0.550792i 0 −0.348029 + 0.479022i 0 −0.0309674 + 0.0953077i 0
449.3 0 1.03494 + 1.42447i 0 1.42953 − 1.71943i 0 0.348029 − 0.479022i 0 −0.0309674 + 0.0953077i 0
449.4 0 1.16075 + 1.59764i 0 0.264988 + 2.22031i 0 1.31845 − 1.81468i 0 −0.278050 + 0.855749i 0
609.1 0 −2.49233 − 0.809808i 0 −1.06421 − 1.96658i 0 0.918552 − 0.298456i 0 3.12889 + 2.27327i 0
609.2 0 −0.346168 − 0.112477i 0 −2.11914 − 0.713621i 0 2.45903 − 0.798988i 0 −2.31987 − 1.68548i 0
609.3 0 0.346168 + 0.112477i 0 0.0238439 + 2.23594i 0 −2.45903 + 0.798988i 0 −2.31987 − 1.68548i 0
609.4 0 2.49233 + 0.809808i 0 1.54147 + 1.61983i 0 −0.918552 + 0.298456i 0 3.12889 + 2.27327i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.cd.c 16
4.b odd 2 1 55.2.j.a ✓ 16
5.b even 2 1 inner 880.2.cd.c 16
11.c even 5 1 inner 880.2.cd.c 16
12.b even 2 1 495.2.ba.a 16
20.d odd 2 1 55.2.j.a ✓ 16
20.e even 4 2 275.2.h.d 16
44.c even 2 1 605.2.j.d 16
44.g even 10 1 605.2.b.f 8
44.g even 10 1 605.2.j.d 16
44.g even 10 2 605.2.j.g 16
44.h odd 10 1 55.2.j.a ✓ 16
44.h odd 10 1 605.2.b.g 8
44.h odd 10 2 605.2.j.h 16
55.j even 10 1 inner 880.2.cd.c 16
60.h even 2 1 495.2.ba.a 16
132.o even 10 1 495.2.ba.a 16
220.g even 2 1 605.2.j.d 16
220.n odd 10 1 55.2.j.a ✓ 16
220.n odd 10 1 605.2.b.g 8
220.n odd 10 2 605.2.j.h 16
220.o even 10 1 605.2.b.f 8
220.o even 10 1 605.2.j.d 16
220.o even 10 2 605.2.j.g 16
220.v even 20 2 275.2.h.d 16
220.v even 20 2 3025.2.a.bl 8
220.w odd 20 2 3025.2.a.bk 8
660.bd even 10 1 495.2.ba.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a ✓ 16 4.b odd 2 1
55.2.j.a ✓ 16 20.d odd 2 1
55.2.j.a ✓ 16 44.h odd 10 1
55.2.j.a ✓ 16 220.n odd 10 1
275.2.h.d 16 20.e even 4 2
275.2.h.d 16 220.v even 20 2
495.2.ba.a 16 12.b even 2 1
495.2.ba.a 16 60.h even 2 1
495.2.ba.a 16 132.o even 10 1
495.2.ba.a 16 660.bd even 10 1
605.2.b.f 8 44.g even 10 1
605.2.b.f 8 220.o even 10 1
605.2.b.g 8 44.h odd 10 1
605.2.b.g 8 220.n odd 10 1
605.2.j.d 16 44.c even 2 1
605.2.j.d 16 44.g even 10 1
605.2.j.d 16 220.g even 2 1
605.2.j.d 16 220.o even 10 1
605.2.j.g 16 44.g even 10 2
605.2.j.g 16 220.o even 10 2
605.2.j.h 16 44.h odd 10 2
605.2.j.h 16 220.n odd 10 2
880.2.cd.c 16 1.a even 1 1 trivial
880.2.cd.c 16 5.b even 2 1 inner
880.2.cd.c 16 11.c even 5 1 inner
880.2.cd.c 16 55.j even 10 1 inner
3025.2.a.bk 8 220.w odd 20 2
3025.2.a.bl 8 220.v even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 7T_{3}^{14} + 30T_{3}^{12} - 77T_{3}^{10} + 969T_{3}^{8} + 637T_{3}^{6} + 6730T_{3}^{4} - 1463T_{3}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 9 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( (T^{8} - 3 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 10 T^{14} + \cdots + 47265625 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 1675346761 \) Copy content Toggle raw display
$19$ \( (T^{8} + 3 T^{7} + \cdots + 121)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 99 T^{6} + \cdots + 26411)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - T^{7} + 25 T^{6} + \cdots + 121)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + 39 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 28 T^{14} + \cdots + 15768841 \) Copy content Toggle raw display
$41$ \( (T^{8} + 26 T^{7} + \cdots + 3876961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 173 T^{6} + \cdots + 212531)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 59639012521 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 239836452361 \) Copy content Toggle raw display
$59$ \( (T^{8} + T^{7} + 129 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 20 T^{7} + \cdots + 43681)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 149 T^{6} + \cdots + 18491)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 18 T^{7} + \cdots + 6305121)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1636073786281 \) Copy content Toggle raw display
$79$ \( (T^{8} + 19 T^{7} + \cdots + 101761)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45169425961 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} + \cdots + 1871)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 25937424601 \) Copy content Toggle raw display
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