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Properties

Label 1155.2.i.c
Level $1155$
Weight $2$
Character orbit 1155.i
Analytic conductor $9.223$
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] = [1155,2,Mod(76,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.i (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: \(9.22272143346\)
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: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + \beta_{9} q^{3} + ( - \beta_{3} - 2) q^{4} - \beta_{9} q^{5} - \beta_{14} q^{6} + (\beta_{14} - \beta_{7}) q^{7} + (\beta_{8} - \beta_{4}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + \beta_{9} q^{3} + ( - \beta_{3} - 2) q^{4} - \beta_{9} q^{5} - \beta_{14} q^{6} + (\beta_{14} - \beta_{7}) q^{7} + (\beta_{8} - \beta_{4}) q^{8} - q^{9} + \beta_{14} q^{10} + ( - \beta_{10} + \beta_1 + 1) q^{11} + ( - 2 \beta_{9} + \beta_{5}) q^{12} + (2 \beta_{15} - \beta_{11}) q^{13} + (4 \beta_{9} - \beta_{5} - \beta_1 + 1) q^{14} + q^{15} + (\beta_{3} + 2 \beta_1 + 2) q^{16} + (\beta_{15} + \beta_{14} + 2 \beta_{13}) q^{17} + \beta_{8} q^{18} + (\beta_{15} - \beta_{14} - 2 \beta_{13}) q^{19} + (2 \beta_{9} - \beta_{5}) q^{20} + ( - \beta_{13} - \beta_{8}) q^{21} + ( - \beta_{10} - \beta_{8} + \cdots - \beta_1) q^{22}+ \cdots + (\beta_{10} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{4} - 16 q^{9} + 16 q^{11} + 16 q^{14} + 16 q^{15} + 24 q^{16} + 40 q^{23} - 16 q^{25} + 24 q^{36} - 40 q^{37} - 56 q^{42} - 24 q^{44} + 8 q^{53} + 8 q^{56} + 72 q^{58} - 24 q^{60} + 8 q^{64} + 80 q^{67} + 56 q^{70} - 24 q^{71} - 16 q^{78} + 16 q^{81} - 8 q^{86} - 40 q^{88} + 16 q^{91} - 160 q^{92} + 40 q^{93} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4976 \nu^{14} + 6064 \nu^{12} - 53806 \nu^{10} + 1664 \nu^{8} + 584304 \nu^{6} + \cdots - 33766125 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 52907 \nu^{15} + 3769822 \nu^{13} + 7571087 \nu^{11} - 23218303 \nu^{9} - 220485308 \nu^{7} + \cdots + 5720425875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 75631 \nu^{14} + 26336 \nu^{12} - 423219 \nu^{10} - 3371514 \nu^{8} + 2776621 \nu^{6} + \cdots + 32055875 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1098448 \nu^{14} - 1820672 \nu^{12} - 3507762 \nu^{10} - 38951322 \nu^{8} + 145662708 \nu^{6} + \cdots - 101116250 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 739864 \nu^{15} - 254886 \nu^{13} - 6858931 \nu^{11} - 26926811 \nu^{9} + 104639904 \nu^{7} + \cdots - 1014860875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64321 \nu^{14} + 44136 \nu^{12} - 365249 \nu^{10} - 2994144 \nu^{8} + 2128291 \nu^{6} + \cdots + 30499350 ) / 3123275 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3708119 \nu^{14} + 2186986 \nu^{12} + 13453081 \nu^{10} + 157291286 \nu^{8} + \cdots - 1320364000 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1073735 \nu^{14} - 529596 \nu^{12} - 3783171 \nu^{10} - 46979616 \nu^{8} + 81866729 \nu^{6} + \cdots + 434987125 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + \cdots + 955700 \nu ) / 633875 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1446284 \nu^{14} + 355734 \nu^{12} + 4966964 \nu^{10} + 65045254 \nu^{8} - 95756546 \nu^{6} + \cdots - 730658725 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 515378 \nu^{15} + 520818 \nu^{13} - 2369422 \nu^{11} - 27002382 \nu^{9} + 8880998 \nu^{7} + \cdots + 379881750 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 599 \nu^{15} + 90 \nu^{13} - 3000 \nu^{11} - 26110 \nu^{9} + 31110 \nu^{7} + 237486 \nu^{5} + \cdots + 249500 \nu ) / 74525 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1468861 \nu^{15} + 201274 \nu^{13} + 5426879 \nu^{11} + 66369199 \nu^{9} + \cdots - 722601875 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2100604 \nu^{15} - 547346 \nu^{13} - 7374841 \nu^{11} - 94005021 \nu^{9} + 141150094 \nu^{7} + \cdots + 977629125 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2699561 \nu^{15} + 1648844 \nu^{13} + 8983874 \nu^{11} + 116269094 \nu^{9} + \cdots - 1087173250 \nu ) / 78081875 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{9} + \beta_{5} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - 3\beta_{8} - 3\beta_{7} - \beta_{6} - \beta_{4} + 3\beta_{3} + 2\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - 7\beta_{14} - 9\beta_{13} + \beta_{12} - 7\beta_{9} + 2\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{10} + \beta_{8} + 11\beta_{7} - \beta_{6} + 9\beta_{4} + 9\beta_{3} + 2\beta _1 + 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{15} - \beta_{14} + 7\beta_{13} + 10\beta_{12} + 6\beta_{11} - 18\beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -15\beta_{10} - 31\beta_{8} - 13\beta_{7} + 2\beta_{6} + 16\beta_{4} - 16\beta_{3} + 29\beta _1 + 59 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 12 \beta_{15} + 4 \beta_{14} + 34 \beta_{13} + 11 \beta_{12} + 23 \beta_{11} - 58 \beta_{9} + \cdots + 23 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -53\beta_{10} - 191\beta_{8} - 167\beta_{7} - 15\beta_{6} + 15\beta_{4} + 69\beta_{3} - 38\beta _1 - 61 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 39 \beta_{15} - 153 \beta_{14} - 343 \beta_{13} + 191 \beta_{12} - 152 \beta_{11} - 457 \beta_{9} + \cdots + 38 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -124\beta_{10} - 28\beta_{8} + 276\beta_{7} + 248\beta_{4} - 85\beta _1 - 199 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 150 \beta_{15} + 838 \beta_{14} + 954 \beta_{13} + 857 \beta_{12} + 97 \beta_{11} - 1942 \beta_{9} + \cdots + 19 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -571\beta_{10} - 1453\beta_{8} - 843\beta_{7} + 494\beta_{6} + 494\beta_{4} - 2092\beta_{3} + 77\beta _1 - 299 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1104 \beta_{15} + 2808 \beta_{14} + 1666 \beta_{13} - 169 \beta_{12} - 935 \beta_{11} + \cdots + 935 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1018 \beta_{10} - 5734 \beta_{8} - 6634 \beta_{7} + 169 \beta_{6} - 731 \beta_{4} - 731 \beta_{3} + \cdots - 13386 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1546\beta_{15} + 367\beta_{14} - 3459\beta_{13} + 1040\beta_{12} - 3092\beta_{11} - 2334\beta_{9} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−1.86824 + 0.357358i
1.86824 − 0.357358i
−0.0566033 − 1.17421i
0.0566033 + 1.17421i
−0.644389 + 0.983224i
0.644389 − 0.983224i
0.917186 − 1.66637i
−0.917186 + 1.66637i
−0.917186 − 1.66637i
0.917186 + 1.66637i
0.644389 + 0.983224i
−0.644389 − 0.983224i
0.0566033 − 1.17421i
−0.0566033 + 1.17421i
1.86824 + 0.357358i
−1.86824 − 0.357358i
− 2.60278i − 1.00000i −4.77447 1.00000i −2.60278 2.60278 − 0.474903i 7.22133i −1.00000 2.60278
76.2 − 2.60278i 1.00000i −4.77447 − 1.00000i 2.60278 −2.60278 − 0.474903i 7.22133i −1.00000 −2.60278
76.3 − 2.19849i − 1.00000i −2.83337 1.00000i −2.19849 2.19849 + 1.47195i 1.83215i −1.00000 2.19849
76.4 − 2.19849i 1.00000i −2.83337 − 1.00000i 2.19849 −2.19849 + 1.47195i 1.83215i −1.00000 −2.19849
76.5 − 1.47195i − 1.00000i −0.166634 1.00000i −1.47195 1.47195 + 2.19849i − 2.69862i −1.00000 1.47195
76.6 − 1.47195i 1.00000i −0.166634 − 1.00000i 1.47195 −1.47195 + 2.19849i − 2.69862i −1.00000 −1.47195
76.7 − 0.474903i − 1.00000i 1.77447 1.00000i −0.474903 0.474903 − 2.60278i − 1.79251i −1.00000 0.474903
76.8 − 0.474903i 1.00000i 1.77447 − 1.00000i 0.474903 −0.474903 − 2.60278i − 1.79251i −1.00000 −0.474903
76.9 0.474903i − 1.00000i 1.77447 1.00000i 0.474903 −0.474903 + 2.60278i 1.79251i −1.00000 −0.474903
76.10 0.474903i 1.00000i 1.77447 − 1.00000i −0.474903 0.474903 + 2.60278i 1.79251i −1.00000 0.474903
76.11 1.47195i − 1.00000i −0.166634 1.00000i 1.47195 −1.47195 − 2.19849i 2.69862i −1.00000 −1.47195
76.12 1.47195i 1.00000i −0.166634 − 1.00000i −1.47195 1.47195 − 2.19849i 2.69862i −1.00000 1.47195
76.13 2.19849i − 1.00000i −2.83337 1.00000i 2.19849 −2.19849 − 1.47195i − 1.83215i −1.00000 −2.19849
76.14 2.19849i 1.00000i −2.83337 − 1.00000i −2.19849 2.19849 − 1.47195i − 1.83215i −1.00000 2.19849
76.15 2.60278i − 1.00000i −4.77447 1.00000i 2.60278 −2.60278 + 0.474903i − 7.22133i −1.00000 −2.60278
76.16 2.60278i 1.00000i −4.77447 − 1.00000i −2.60278 2.60278 + 0.474903i − 7.22133i −1.00000 2.60278
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.i.c ✓ 16
7.b odd 2 1 inner 1155.2.i.c ✓ 16
11.b odd 2 1 inner 1155.2.i.c ✓ 16
77.b even 2 1 inner 1155.2.i.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.i.c ✓ 16 1.a even 1 1 trivial
1155.2.i.c ✓ 16 7.b odd 2 1 inner
1155.2.i.c ✓ 16 11.b odd 2 1 inner
1155.2.i.c ✓ 16 77.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 14T_{2}^{6} + 61T_{2}^{4} + 84T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 14 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 84 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 74 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 74 T^{6} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 10 T^{3} + \cdots - 20)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 106 T^{6} + \cdots + 215296)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 100 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 10 T^{3} + \cdots + 80)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 236 T^{6} + \cdots + 3041536)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 346 T^{6} + \cdots + 29767936)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 300 T^{6} + \cdots + 1638400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + \cdots + 5776)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 106 T^{6} + \cdots + 215296)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 226 T^{6} + \cdots + 4946176)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 20 T^{3} + \cdots - 320)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} - 84 T^{2} + \cdots - 64)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 176 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 300 T^{6} + \cdots + 5382400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 466 T^{6} + \cdots + 88435216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 146 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 274 T^{6} + \cdots + 1547536)^{2} \) Copy content Toggle raw display
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