Properties

Label 1148.2.i.d
Level $1148$
Weight $2$
Character orbit 1148.i
Analytic conductor $9.167$
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] = [1148,2,Mod(165,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.i (of order \(3\), 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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 14 x^{14} - 8 x^{13} + 136 x^{12} - 87 x^{11} + 706 x^{10} - 568 x^{9} + 2685 x^{8} - 2100 x^{7} + 5529 x^{6} - 4919 x^{5} + 8145 x^{4} - 5182 x^{3} + 2775 x^{2} + \cdots + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{14} q^{5} + (\beta_{8} - \beta_{5}) q^{7} + (\beta_{10} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{14} q^{5} + (\beta_{8} - \beta_{5}) q^{7} + (\beta_{10} + \beta_{2}) q^{9} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_1) q^{11} + (\beta_{13} + \beta_{4} - 1) q^{13} + (\beta_{8} - \beta_{4}) q^{15} + ( - \beta_{9} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_1) q^{17} + ( - \beta_{15} + \beta_{10} + \beta_{2}) q^{19} + (\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{21}+ \cdots + ( - \beta_{13} - \beta_{9} - 2 \beta_{3} + 2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{9} + 8 q^{11} - 14 q^{13} - 2 q^{15} + q^{17} - 4 q^{19} + 13 q^{21} + 3 q^{23} + 4 q^{25} - 24 q^{27} - 8 q^{29} - 4 q^{31} - 23 q^{33} + 12 q^{35} + 31 q^{37} - 5 q^{39} + 16 q^{41} - 16 q^{43} - q^{45} - 24 q^{47} + 16 q^{49} + 23 q^{51} + q^{53} + 4 q^{55} - 30 q^{57} - 4 q^{59} + 4 q^{61} + 23 q^{63} + 24 q^{65} - 42 q^{69} + 16 q^{71} - 11 q^{73} + 15 q^{75} + 25 q^{77} - 14 q^{79} + 28 q^{81} - 84 q^{83} - 40 q^{85} - 25 q^{87} + 11 q^{89} + 7 q^{91} + 27 q^{93} + 15 q^{95} - 32 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 14 x^{14} - 8 x^{13} + 136 x^{12} - 87 x^{11} + 706 x^{10} - 568 x^{9} + 2685 x^{8} - 2100 x^{7} + 5529 x^{6} - 4919 x^{5} + 8145 x^{4} - 5182 x^{3} + 2775 x^{2} + \cdots + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 780476607889860 \nu^{15} - 90657693823740 \nu^{14} + \cdots + 80\!\cdots\!13 ) / 27\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 913292836654949 \nu^{15} + 780476607889860 \nu^{14} + \cdots - 62\!\cdots\!80 ) / 27\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45\!\cdots\!47 \nu^{15} + \cdots - 40\!\cdots\!34 ) / 40\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!63 \nu^{15} + \cdots - 54\!\cdots\!60 ) / 89\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 80\!\cdots\!97 \nu^{15} + \cdots - 49\!\cdots\!60 ) / 44\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 57\!\cdots\!80 \nu^{15} + \cdots - 96\!\cdots\!46 ) / 29\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!15 \nu^{15} + \cdots - 11\!\cdots\!43 ) / 40\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28\!\cdots\!87 \nu^{15} + \cdots + 13\!\cdots\!73 ) / 81\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!00 \nu^{15} + \cdots + 30\!\cdots\!57 ) / 29\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 62\!\cdots\!10 \nu^{15} + \cdots + 39\!\cdots\!20 ) / 89\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 86\!\cdots\!44 \nu^{15} + \cdots - 47\!\cdots\!83 ) / 81\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 97\!\cdots\!42 \nu^{15} + \cdots + 45\!\cdots\!45 ) / 81\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 70\!\cdots\!19 \nu^{15} + \cdots + 42\!\cdots\!40 ) / 44\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 64\!\cdots\!38 \nu^{15} + \cdots + 38\!\cdots\!00 ) / 40\!\cdots\!71 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 3\beta_{7} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{9} - \beta_{8} + 5\beta_{3} - \beta_{2} - 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + 3 \beta_{11} - 8 \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - 15 \beta_{7} + 2 \beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{15} + 10 \beta_{14} - 12 \beta_{11} + 13 \beta_{10} + 9 \beta_{7} - \beta_{6} - 10 \beta_{5} - 28 \beta_{3} + 13 \beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{13} - 15\beta_{12} + 23\beta_{9} - 36\beta_{8} + 2\beta_{4} + 15\beta_{3} - 62\beta_{2} - 15\beta _1 + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 83 \beta_{15} - 85 \beta_{14} + 85 \beta_{13} + 83 \beta_{12} + 116 \beta_{11} - 126 \beta_{10} - 86 \beta_{9} + 116 \beta_{8} - 84 \beta_{7} + 14 \beta_{6} + 86 \beta_{5} - 14 \beta_{4} + 173 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 159 \beta_{15} + 173 \beta_{14} - 341 \beta_{11} + 484 \beta_{10} + 559 \beta_{7} - 32 \beta_{6} - 211 \beta_{5} - 167 \beta_{3} + 484 \beta_{2} - 559 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 698 \beta_{13} - 666 \beta_{12} + 709 \beta_{9} - 1030 \beta_{8} + 144 \beta_{4} + 1157 \beta_{3} - 1119 \beta_{2} - 1157 \beta _1 + 774 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1483 \beta_{15} - 1627 \beta_{14} + 1627 \beta_{13} + 1483 \beta_{12} + 2991 \beta_{11} - 3828 \beta_{10} - 1818 \beta_{9} + 2991 \beta_{8} - 3881 \beta_{7} + 353 \beta_{6} + 1818 \beta_{5} - 353 \beta_{4} + \cdots + 1637 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5336 \beta_{15} + 5689 \beta_{14} - 8799 \beta_{11} + 9586 \beta_{10} + 6921 \beta_{7} - 1317 \beta_{6} - 5778 \beta_{5} - 8236 \beta_{3} + 9586 \beta_{2} - 6921 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14366 \beta_{13} - 13049 \beta_{12} + 15309 \beta_{9} - 25391 \beta_{8} + 3374 \beta_{4} + 14979 \beta_{3} - 30640 \beta_{2} - 14979 \beta _1 + 28516 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 42982 \beta_{15} - 46356 \beta_{14} + 46356 \beta_{13} + 42982 \beta_{12} + 73771 \beta_{11} - 80685 \beta_{10} - 47023 \beta_{9} + 73771 \beta_{8} - 60268 \beta_{7} + 11399 \beta_{6} + 47023 \beta_{5} + \cdots + 61438 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 111474 \beta_{15} + 122873 \beta_{14} - 212211 \beta_{11} + 247477 \beta_{10} + 217976 \beta_{7} - 30122 \beta_{6} - 127617 \beta_{5} - 131633 \beta_{3} + 247477 \beta_{2} + \cdots - 217976 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 378336 \beta_{13} - 348214 \beta_{12} + 383301 \beta_{9} - 612683 \beta_{8} + 95993 \beta_{4} + 473839 \beta_{3} - 672673 \beta_{2} - 473839 \beta _1 + 514759 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(-1 + \beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
1.14440 1.98216i
1.12452 1.94772i
0.666692 1.15474i
0.251812 0.436150i
0.154058 0.266837i
−0.856454 + 1.48342i
−1.05150 + 1.82125i
−1.43353 + 2.48295i
1.14440 + 1.98216i
1.12452 + 1.94772i
0.666692 + 1.15474i
0.251812 + 0.436150i
0.154058 + 0.266837i
−0.856454 1.48342i
−1.05150 1.82125i
−1.43353 2.48295i
0 −1.14440 + 1.98216i 0 −0.175017 0.303139i 0 −0.303923 + 2.62824i 0 −1.11932 1.93871i 0
165.2 0 −1.12452 + 1.94772i 0 0.735486 + 1.27390i 0 0.221110 2.63650i 0 −1.02909 1.78243i 0
165.3 0 −0.666692 + 1.15474i 0 −0.883435 1.53015i 0 2.62366 0.341202i 0 0.611043 + 1.05836i 0
165.4 0 −0.251812 + 0.436150i 0 −1.11864 1.93755i 0 −1.70106 2.02642i 0 1.37318 + 2.37842i 0
165.5 0 −0.154058 + 0.266837i 0 1.65062 + 2.85896i 0 −2.57062 0.626020i 0 1.45253 + 2.51586i 0
165.6 0 0.856454 1.48342i 0 −1.07560 1.86299i 0 1.55192 + 2.14279i 0 0.0329726 + 0.0571103i 0
165.7 0 1.05150 1.82125i 0 1.46762 + 2.54199i 0 2.64377 0.102404i 0 −0.711302 1.23201i 0
165.8 0 1.43353 2.48295i 0 −0.601031 1.04102i 0 −2.46485 + 0.961519i 0 −2.61002 4.52069i 0
821.1 0 −1.14440 1.98216i 0 −0.175017 + 0.303139i 0 −0.303923 2.62824i 0 −1.11932 + 1.93871i 0
821.2 0 −1.12452 1.94772i 0 0.735486 1.27390i 0 0.221110 + 2.63650i 0 −1.02909 + 1.78243i 0
821.3 0 −0.666692 1.15474i 0 −0.883435 + 1.53015i 0 2.62366 + 0.341202i 0 0.611043 1.05836i 0
821.4 0 −0.251812 0.436150i 0 −1.11864 + 1.93755i 0 −1.70106 + 2.02642i 0 1.37318 2.37842i 0
821.5 0 −0.154058 0.266837i 0 1.65062 2.85896i 0 −2.57062 + 0.626020i 0 1.45253 2.51586i 0
821.6 0 0.856454 + 1.48342i 0 −1.07560 + 1.86299i 0 1.55192 2.14279i 0 0.0329726 0.0571103i 0
821.7 0 1.05150 + 1.82125i 0 1.46762 2.54199i 0 2.64377 + 0.102404i 0 −0.711302 + 1.23201i 0
821.8 0 1.43353 + 2.48295i 0 −0.601031 + 1.04102i 0 −2.46485 0.961519i 0 −2.61002 + 4.52069i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.i.d 16
7.c even 3 1 inner 1148.2.i.d 16
7.c even 3 1 8036.2.a.m 8
7.d odd 6 1 8036.2.a.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.d 16 1.a even 1 1 trivial
1148.2.i.d 16 7.c even 3 1 inner
8036.2.a.m 8 7.c even 3 1
8036.2.a.n 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\):

\( T_{3}^{16} + 14 T_{3}^{14} + 8 T_{3}^{13} + 136 T_{3}^{12} + 87 T_{3}^{11} + 706 T_{3}^{10} + 568 T_{3}^{9} + 2685 T_{3}^{8} + 2100 T_{3}^{7} + 5529 T_{3}^{6} + 4919 T_{3}^{5} + 8145 T_{3}^{4} + 5182 T_{3}^{3} + 2775 T_{3}^{2} + \cdots + 121 \) Copy content Toggle raw display
\( T_{11}^{16} - 8 T_{11}^{15} + 63 T_{11}^{14} - 196 T_{11}^{13} + 884 T_{11}^{12} - 2156 T_{11}^{11} + 8265 T_{11}^{10} - 14203 T_{11}^{9} + 32607 T_{11}^{8} - 29699 T_{11}^{7} + 65663 T_{11}^{6} - 61266 T_{11}^{5} + 57974 T_{11}^{4} + \cdots + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 14 T^{14} + 8 T^{13} + 136 T^{12} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{16} + 18 T^{14} + 24 T^{13} + \cdots + 2601 \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{14} - 9 T^{13} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{15} + 63 T^{14} - 196 T^{13} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{8} + 7 T^{7} - 17 T^{6} - 223 T^{5} + \cdots - 634)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - T^{15} + 50 T^{14} + \cdots + 10673289 \) Copy content Toggle raw display
$19$ \( T^{16} + 4 T^{15} + 50 T^{14} + \cdots + 314721 \) Copy content Toggle raw display
$23$ \( T^{16} - 3 T^{15} + 56 T^{14} - 459 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( (T^{8} + 4 T^{7} - 94 T^{6} - 296 T^{5} + \cdots + 486)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 4 T^{15} + \cdots + 3567553441 \) Copy content Toggle raw display
$37$ \( T^{16} - 31 T^{15} + 614 T^{14} + \cdots + 1547536 \) Copy content Toggle raw display
$41$ \( (T - 1)^{16} \) Copy content Toggle raw display
$43$ \( (T^{8} + 8 T^{7} - 56 T^{6} - 372 T^{5} + \cdots - 20588)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 24 T^{15} + \cdots + 539770865481 \) Copy content Toggle raw display
$53$ \( T^{16} - T^{15} + 145 T^{14} + \cdots + 5363072289 \) Copy content Toggle raw display
$59$ \( T^{16} + 4 T^{15} + 147 T^{14} + \cdots + 126765081 \) Copy content Toggle raw display
$61$ \( T^{16} - 4 T^{15} + \cdots + 5224976656 \) Copy content Toggle raw display
$67$ \( T^{16} + 196 T^{14} + \cdots + 4942280319376 \) Copy content Toggle raw display
$71$ \( (T^{8} - 8 T^{7} - 227 T^{6} + \cdots + 896976)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 11 T^{15} + 231 T^{14} + \cdots + 7458361 \) Copy content Toggle raw display
$79$ \( T^{16} + 14 T^{15} + \cdots + 215074265121 \) Copy content Toggle raw display
$83$ \( (T^{8} + 42 T^{7} + 536 T^{6} + \cdots - 3922452)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 11 T^{15} + \cdots + 5289507441 \) Copy content Toggle raw display
$97$ \( (T^{8} + 16 T^{7} - 83 T^{6} - 2761 T^{5} + \cdots - 69682)^{2} \) Copy content Toggle raw display
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