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Properties

Label 2205.2.b.d
Level $2205$
Weight $2$
Character orbit 2205.b
Analytic conductor $17.607$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,2,Mod(881,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.b (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: \(17.6070136457\)
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: \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 790 x^{12} - 2192 x^{11} + 5036 x^{10} - 9472 x^{9} + \cdots + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{2}) q^{2} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1) q^{4}+ \cdots + (\beta_{14} - \beta_{13} + \cdots - \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_{2}) q^{2} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1) q^{4}+ \cdots + ( - 4 \beta_{15} + \beta_{13} + \cdots - 3 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{5} - 16 q^{20} - 16 q^{22} + 16 q^{25} + 32 q^{26} - 16 q^{38} - 32 q^{41} + 32 q^{43} - 16 q^{46} + 32 q^{47} + 48 q^{58} - 32 q^{59} - 32 q^{62} + 16 q^{64} - 16 q^{68} + 32 q^{79} + 16 q^{88} + 64 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 790 x^{12} - 2192 x^{11} + 5036 x^{10} - 9472 x^{9} + \cdots + 17 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2 \nu^{14} + 14 \nu^{13} - 77 \nu^{12} + 280 \nu^{11} - 722 \nu^{10} + 1377 \nu^{9} - 1026 \nu^{8} + \cdots + 2284 ) / 31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3300 \nu^{15} + 24750 \nu^{14} - 158936 \nu^{13} + 657709 \nu^{12} - 2264384 \nu^{11} + \cdots + 53448 ) / 3689 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 352 \nu^{14} + 2464 \nu^{13} - 15753 \nu^{12} + 62486 \nu^{11} - 211764 \nu^{10} + 544757 \nu^{9} + \cdots - 27738 ) / 217 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 439 \nu^{14} - 3073 \nu^{13} + 19583 \nu^{12} - 77549 \nu^{11} + 261709 \nu^{10} - 670919 \nu^{9} + \cdots + 22438 ) / 217 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16922 \nu^{15} - 126915 \nu^{14} + 802453 \nu^{13} - 3291067 \nu^{12} + 11095997 \nu^{11} + \cdots + 923066 ) / 3689 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1725 \nu^{14} + 12075 \nu^{13} - 77092 \nu^{12} + 305577 \nu^{11} - 1033382 \nu^{10} + 2653575 \nu^{9} + \cdots - 96200 ) / 217 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1893 \nu^{14} - 13251 \nu^{13} + 84645 \nu^{12} - 335607 \nu^{11} + 1135694 \nu^{10} - 2917888 \nu^{9} + \cdots + 112664 ) / 217 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 276 \nu^{14} + 1932 \nu^{13} - 12331 \nu^{12} + 48870 \nu^{11} - 165201 \nu^{10} + 424076 \nu^{9} + \cdots - 14772 ) / 31 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2869 \nu^{14} - 20083 \nu^{13} + 128235 \nu^{12} - 508331 \nu^{11} + 1719228 \nu^{10} - 4415084 \nu^{9} + \cdots + 158898 ) / 217 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7992 \nu^{15} + 59940 \nu^{14} - 384458 \nu^{13} + 1589887 \nu^{12} - 5467116 \nu^{11} + \cdots + 144755 ) / 527 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10622 \nu^{15} - 79665 \nu^{14} + 512061 \nu^{13} - 2120144 \nu^{12} + 7311197 \nu^{11} + \cdots - 314483 ) / 527 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2620 \nu^{15} + 19650 \nu^{14} - 126500 \nu^{13} + 524225 \nu^{12} - 1811504 \nu^{11} + \cdots + 98226 ) / 119 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 136352 \nu^{15} - 1022640 \nu^{14} + 6578168 \nu^{13} - 27248052 \nu^{12} + 94057368 \nu^{11} + \cdots - 4501328 ) / 3689 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 7220 \nu^{15} + 54150 \nu^{14} - 348076 \nu^{13} + 1441219 \nu^{12} - 4970262 \nu^{11} + \cdots + 213146 ) / 119 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 383272 \nu^{15} - 2874540 \nu^{14} + 18466320 \nu^{13} - 76433890 \nu^{12} + 263381182 \nu^{11} + \cdots - 10132816 ) / 3689 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{15} - 2\beta_{14} + \beta_{13} + 2\beta_{11} + 2\beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} - 2 \beta_{14} + \beta_{13} + 2 \beta_{11} + 6 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} + \cdots - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} - 17 \beta_{11} + 9 \beta_{10} + 9 \beta_{8} + \cdots - 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24 \beta_{15} + 18 \beta_{14} - 5 \beta_{13} + 6 \beta_{12} - 36 \beta_{11} + 18 \beta_{10} + 12 \beta_{9} + \cdots + 50 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 62 \beta_{15} - 44 \beta_{14} - 18 \beta_{13} - 22 \beta_{12} + 120 \beta_{11} - 94 \beta_{10} + \cdots + 152 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 247 \beta_{15} - 178 \beta_{14} - 41 \beta_{13} - 81 \beta_{12} + 451 \beta_{11} - 327 \beta_{10} + \cdots - 238 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 298 \beta_{15} + 226 \beta_{14} + 203 \beta_{13} + 126 \beta_{12} - 630 \beta_{11} + 604 \beta_{10} + \cdots - 1384 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2402 \beta_{15} + 1778 \beta_{14} + 991 \beta_{13} + 896 \beta_{12} - 4710 \beta_{11} + 3984 \beta_{10} + \cdots + 746 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 406 \beta_{15} - 408 \beta_{14} - 857 \beta_{13} - 278 \beta_{12} + 1068 \beta_{11} - 1530 \beta_{10} + \cdots + 12332 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 21897 \beta_{15} - 16714 \beta_{14} - 11978 \beta_{13} - 8707 \beta_{12} + 44005 \beta_{11} + \cdots + 4570 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 17183 \beta_{15} - 12460 \beta_{14} - 6119 \beta_{13} - 6231 \beta_{12} + 33523 \beta_{11} + \cdots - 104894 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 91621 \beta_{15} + 71327 \beta_{14} + 56105 \beta_{13} + 37528 \beta_{12} - 185579 \beta_{11} + \cdots - 69317 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 335686 \beta_{15} + 257000 \beta_{14} + 186253 \beta_{13} + 133974 \beta_{12} - 674908 \beta_{11} + \cdots + 819834 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1358332 \beta_{15} - 1068632 \beta_{14} - 872305 \beta_{13} - 563682 \beta_{12} + 2757860 \beta_{11} + \cdots + 2051158 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4434185 \beta_{15} - 3452832 \beta_{14} - 2701154 \beta_{13} - 1813567 \beta_{12} + 8970011 \beta_{11} + \cdots - 5538976 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
0.500000 + 0.229342i
0.500000 − 3.00459i
0.500000 + 0.0544047i
0.500000 + 2.14927i
0.500000 − 2.12518i
0.500000 + 1.71572i
0.500000 − 0.878189i
0.500000 − 0.257385i
0.500000 + 0.257385i
0.500000 + 0.878189i
0.500000 − 1.71572i
0.500000 + 2.12518i
0.500000 − 2.14927i
0.500000 − 0.0544047i
0.500000 + 3.00459i
0.500000 − 0.229342i
− 2.65421i 0 −5.04484 1.00000 0 0 8.08165i 0 − 2.65421i
881.2 − 2.14110i 0 −2.58431 1.00000 0 0 1.25106i 0 − 2.14110i
881.3 − 2.12992i 0 −2.53654 1.00000 0 0 1.14279i 0 − 2.12992i
881.4 − 1.80349i 0 −1.25258 1.00000 0 0 − 1.34796i 0 − 1.80349i
881.5 − 1.69637i 0 −0.877672 1.00000 0 0 − 1.90388i 0 − 1.69637i
881.6 − 1.15464i 0 0.666795 1.00000 0 0 − 3.07920i 0 − 1.15464i
881.7 − 0.607761i 0 1.63063 1.00000 0 0 − 2.20655i 0 − 0.607761i
881.8 − 0.0384813i 0 1.99852 1.00000 0 0 − 0.153868i 0 − 0.0384813i
881.9 0.0384813i 0 1.99852 1.00000 0 0 0.153868i 0 0.0384813i
881.10 0.607761i 0 1.63063 1.00000 0 0 2.20655i 0 0.607761i
881.11 1.15464i 0 0.666795 1.00000 0 0 3.07920i 0 1.15464i
881.12 1.69637i 0 −0.877672 1.00000 0 0 1.90388i 0 1.69637i
881.13 1.80349i 0 −1.25258 1.00000 0 0 1.34796i 0 1.80349i
881.14 2.12992i 0 −2.53654 1.00000 0 0 − 1.14279i 0 2.12992i
881.15 2.14110i 0 −2.58431 1.00000 0 0 − 1.25106i 0 2.14110i
881.16 2.65421i 0 −5.04484 1.00000 0 0 − 8.08165i 0 2.65421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.2.b.d yes 16
3.b odd 2 1 2205.2.b.c ✓ 16
7.b odd 2 1 2205.2.b.c ✓ 16
21.c even 2 1 inner 2205.2.b.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2205.2.b.c ✓ 16 3.b odd 2 1
2205.2.b.c ✓ 16 7.b odd 2 1
2205.2.b.d yes 16 1.a even 1 1 trivial
2205.2.b.d yes 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2205, [\chi])\):

\( T_{2}^{16} + 24T_{2}^{14} + 232T_{2}^{12} + 1160T_{2}^{10} + 3186T_{2}^{8} + 4664T_{2}^{6} + 3176T_{2}^{4} + 680T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{8} - 72T_{17}^{6} + 32T_{17}^{5} + 1160T_{17}^{4} + 384T_{17}^{3} - 4064T_{17}^{2} - 1152T_{17} + 4112 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 24 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 96 T^{14} + \cdots + 565504 \) Copy content Toggle raw display
$13$ \( T^{16} + 80 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( (T^{8} - 72 T^{6} + \cdots + 4112)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 176 T^{14} + \cdots + 59043856 \) Copy content Toggle raw display
$23$ \( T^{16} + 176 T^{14} + \cdots + 14776336 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 913490176 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 490533904 \) Copy content Toggle raw display
$37$ \( (T^{8} - 96 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 16 T^{7} + \cdots + 531968)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 16 T^{7} + \cdots + 332032)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 16 T^{7} + \cdots - 1268464)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 2196833837584 \) Copy content Toggle raw display
$59$ \( (T^{8} + 16 T^{7} + \cdots + 1170496)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 750356507028496 \) Copy content Toggle raw display
$67$ \( (T^{8} - 208 T^{6} + \cdots - 1984)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 3204787396864 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 238643974144 \) Copy content Toggle raw display
$79$ \( (T^{8} - 16 T^{7} + \cdots - 40040432)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 360 T^{6} + \cdots + 1595408)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 32 T^{7} + \cdots + 442432)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
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