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Properties

Label 1152.5.b.m
Level $1152$
Weight $5$
Character orbit 1152.b
Analytic conductor $119.082$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1152,5,Mod(703,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.703"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1152.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-480] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(119.082197473\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + \cdots + 362673936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{5} + \beta_{3} q^{7} + (\beta_{9} + 2 \beta_{5}) q^{11} + (\beta_{14} + 3 \beta_{11} + \beta_{6}) q^{13} + (\beta_{7} - 30) q^{17} + ( - \beta_{13} + \beta_{9} + 5 \beta_{5}) q^{19} + (5 \beta_{3} - 3 \beta_{2} - \beta_1) q^{23}+ \cdots + (24 \beta_{10} - 2 \beta_{8} + \cdots + 318) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 480 q^{17} + 144 q^{25} + 3552 q^{41} - 20080 q^{49} - 23040 q^{65} + 34400 q^{73} + 15648 q^{89} + 5088 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + \cdots + 362673936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 21203481 \nu^{14} - 939198146 \nu^{12} - 26999699747 \nu^{10} - 451004763042 \nu^{8} + \cdots - 125937410468616 ) / 4614201420840 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4108 \nu^{14} - 76360 \nu^{12} - 1460868 \nu^{10} - 9399208 \nu^{8} - 100952428 \nu^{6} + \cdots - 19234287648 ) / 752642685 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 57146483 \nu^{14} - 2115064406 \nu^{12} - 52703126049 \nu^{10} - 806141495414 \nu^{8} + \cdots - 673855067434968 ) / 4614201420840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3543459 \nu^{14} + 146316662 \nu^{12} + 4063494065 \nu^{10} + 68539689750 \nu^{8} + \cdots + 157301735600088 ) / 242852706360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 556 \nu^{15} + 18699 \nu^{13} + 428444 \nu^{11} + 5765809 \nu^{9} + 58126704 \nu^{7} + \cdots - 4265951220 \nu ) / 794991780 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1601473 \nu^{15} - 98155168 \nu^{13} - 2885978475 \nu^{11} - 54364746640 \nu^{9} + \cdots - 118184013684432 \nu ) / 2285023191660 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 72757 \nu^{14} - 4260382 \nu^{12} - 121527975 \nu^{10} - 2311227190 \nu^{8} + \cdots - 7044868877688 ) / 2955295530 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31371 \nu^{14} + 1579778 \nu^{12} + 43486169 \nu^{10} + 829237674 \nu^{8} + 10759730579 \nu^{6} + \cdots + 2463773203656 ) / 985098510 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 151456 \nu^{15} - 9818815 \nu^{13} - 237370488 \nu^{11} - 3778979053 \nu^{9} + \cdots + 1358375631012 \nu ) / 135943594380 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 44399 \nu^{14} - 1891290 \nu^{12} - 45803557 \nu^{10} - 744244322 \nu^{8} + \cdots - 1592996663592 ) / 985098510 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6590197 \nu^{15} + 282332912 \nu^{13} + 7259403175 \nu^{11} + 122888600320 \nu^{9} + \cdots + 276231209943888 \nu ) / 4823937849060 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 116582967 \nu^{15} - 6636652714 \nu^{13} - 198004463629 \nu^{11} + \cdots - 30\!\cdots\!60 \nu ) / 53063316339660 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1638308 \nu^{15} - 42500159 \nu^{13} - 863735652 \nu^{11} - 6047196077 \nu^{9} + \cdots + 76376622312612 \nu ) / 407830783140 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2256580781 \nu^{15} - 111612544475 \nu^{13} - 3164107444263 \nu^{11} + \cdots - 15\!\cdots\!88 \nu ) / 477569847056940 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 893983 \nu^{15} - 28765249 \nu^{13} - 726540813 \nu^{11} - 10463518723 \nu^{9} + \cdots - 7232245440804 \nu ) / 101957695785 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6 \beta_{15} - 16 \beta_{14} - 6 \beta_{13} + 16 \beta_{12} - 31 \beta_{11} + 6 \beta_{9} + \cdots + 36 \beta_{5} ) / 1536 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{10} + \beta_{8} + 12\beta_{7} + 47\beta_{4} + 36\beta_{3} + 24\beta_{2} + 35\beta _1 - 8832 ) / 1536 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{15} - 2\beta_{14} + 12\beta_{13} - \beta_{12} - 28\beta_{11} - 18\beta_{9} + 37\beta_{6} - 10\beta_{5} ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 76 \beta_{10} - 128 \beta_{8} - 276 \beta_{7} - 475 \beta_{4} + 732 \beta_{3} - 1092 \beta_{2} + \cdots - 97152 ) / 1536 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 93 \beta_{15} + 5608 \beta_{14} - 2667 \beta_{13} - 2716 \beta_{12} + 29710 \beta_{11} + \cdots - 42632 \beta_{5} ) / 3072 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 555\beta_{10} + 111\beta_{8} - 225\beta_{4} - 3748\beta_{3} + 1884\beta_{2} + 2475\beta _1 + 333952 ) / 256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34281 \beta_{15} - 27752 \beta_{14} + 7809 \beta_{13} + 39068 \beta_{12} - 248798 \beta_{11} + \cdots + 760216 \beta_{5} ) / 3072 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 23246 \beta_{10} + 54302 \beta_{8} + 76164 \beta_{7} + 59845 \beta_{4} + 155628 \beta_{3} + \cdots - 9105792 ) / 1536 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5510 \beta_{15} - 34866 \beta_{14} + 11856 \beta_{13} - 17433 \beta_{12} - 57196 \beta_{11} + \cdots - 14250 \beta_{5} ) / 128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 142961 \beta_{10} - 1052881 \beta_{8} - 1127460 \beta_{7} - 100127 \beta_{4} + 2051052 \beta_{3} + \cdots - 58564992 ) / 1536 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7730301 \beta_{15} + 8830808 \beta_{14} - 8432997 \beta_{13} + 3010012 \beta_{12} + \cdots - 166647016 \beta_{5} ) / 3072 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 57750 \beta_{10} + 11550 \beta_{8} - 1035465 \beta_{4} - 8744900 \beta_{3} + 7075860 \beta_{2} + \cdots - 50199424 ) / 256 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 110384487 \beta_{15} + 141741608 \beta_{14} + 130968543 \beta_{13} + 29702692 \beta_{12} + \cdots + 2306971832 \beta_{5} ) / 3072 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 35312813 \beta_{10} + 236287741 \beta_{8} + 212340612 \beta_{7} - 37442803 \beta_{4} + \cdots + 15566656128 ) / 1536 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 19515260 \beta_{15} - 120946422 \beta_{14} - 38717958 \beta_{13} - 60473211 \beta_{12} + \cdots + 45014670 \beta_{5} ) / 128 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
−1.09182 − 2.89109i
1.09182 − 2.89109i
−2.31712 + 3.01338i
2.31712 + 3.01338i
1.45110 + 3.51338i
−1.45110 + 3.51338i
−1.95785 + 2.39109i
1.95785 + 2.39109i
1.95785 − 2.39109i
−1.95785 − 2.39109i
−1.45110 − 3.51338i
1.45110 − 3.51338i
2.31712 − 3.01338i
−2.31712 − 3.01338i
1.09182 + 2.89109i
−1.09182 + 2.89109i
0 0 0 − 39.0495i 0 − 95.6459i 0 0 0
703.2 0 0 0 − 39.0495i 0 95.6459i 0 0 0
703.3 0 0 0 − 23.5183i 0 − 40.3412i 0 0 0
703.4 0 0 0 − 23.5183i 0 40.3412i 0 0 0
703.5 0 0 0 − 19.1142i 0 − 6.40010i 0 0 0
703.6 0 0 0 − 19.1142i 0 6.40010i 0 0 0
703.7 0 0 0 − 4.54640i 0 − 61.7048i 0 0 0
703.8 0 0 0 − 4.54640i 0 61.7048i 0 0 0
703.9 0 0 0 4.54640i 0 − 61.7048i 0 0 0
703.10 0 0 0 4.54640i 0 61.7048i 0 0 0
703.11 0 0 0 19.1142i 0 − 6.40010i 0 0 0
703.12 0 0 0 19.1142i 0 6.40010i 0 0 0
703.13 0 0 0 23.5183i 0 − 40.3412i 0 0 0
703.14 0 0 0 23.5183i 0 40.3412i 0 0 0
703.15 0 0 0 39.0495i 0 − 95.6459i 0 0 0
703.16 0 0 0 39.0495i 0 95.6459i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.5.b.m 16
3.b odd 2 1 384.5.b.d ✓ 16
4.b odd 2 1 inner 1152.5.b.m 16
8.b even 2 1 inner 1152.5.b.m 16
8.d odd 2 1 inner 1152.5.b.m 16
12.b even 2 1 384.5.b.d ✓ 16
24.f even 2 1 384.5.b.d ✓ 16
24.h odd 2 1 384.5.b.d ✓ 16
48.i odd 4 1 768.5.g.h 8
48.i odd 4 1 768.5.g.j 8
48.k even 4 1 768.5.g.h 8
48.k even 4 1 768.5.g.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.d ✓ 16 3.b odd 2 1
384.5.b.d ✓ 16 12.b even 2 1
384.5.b.d ✓ 16 24.f even 2 1
384.5.b.d ✓ 16 24.h odd 2 1
768.5.g.h 8 48.i odd 4 1
768.5.g.h 8 48.k even 4 1
768.5.g.j 8 48.i odd 4 1
768.5.g.j 8 48.k even 4 1
1152.5.b.m 16 1.a even 1 1 trivial
1152.5.b.m 16 4.b odd 2 1 inner
1152.5.b.m 16 8.b even 2 1 inner
1152.5.b.m 16 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{8} + 2464T_{5}^{6} + 1653120T_{5}^{4} + 341272576T_{5}^{2} + 6369316864 \) Copy content Toggle raw display
\( T_{17}^{4} + 120T_{17}^{3} - 283112T_{17}^{2} - 8551968T_{17} + 16654958608 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 2464 T^{6} + \cdots + 6369316864)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 2321893298176)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 47\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 42\!\cdots\!64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 120 T^{3} + \cdots + 16654958608)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 76\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 47\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 4399896700944)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 21\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 16\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 73\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 12\!\cdots\!76)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 23\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 21\!\cdots\!48)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 319894196086800)^{4} \) Copy content Toggle raw display
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