Properties

Label 1344.4.b.h
Level $1344$
Weight $4$
Character orbit 1344.b
Analytic conductor $79.299$
Analytic rank $0$
Dimension $12$
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] = [1344,4,Mod(895,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.895");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), 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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: no (minimal twist has level 84)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} - \beta_{7} q^{5} + (\beta_{2} + 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - \beta_{7} q^{5} + (\beta_{2} + 1) q^{7} + 9 q^{9} - \beta_{11} q^{11} - \beta_{9} q^{13} - 3 \beta_{7} q^{15} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{3} + \beta_{2}) q^{17} + (\beta_{3} - \beta_{2} - \beta_1 + 7) q^{19} + (3 \beta_{2} + 3) q^{21} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2}) q^{23} + (\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 19) q^{25} + 27 q^{27} + (\beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} - 17) q^{29} + (\beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - 32) q^{31} - 3 \beta_{11} q^{33} + ( - 2 \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} - 4 \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 7) q^{35}+ \cdots - 9 \beta_{11} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9} + 84 q^{19} + 30 q^{21} - 216 q^{25} + 324 q^{27} - 200 q^{29} - 384 q^{31} - 84 q^{35} + 244 q^{37} + 280 q^{47} - 424 q^{49} + 16 q^{53} + 212 q^{55} + 252 q^{57} - 1168 q^{59} + 90 q^{63} + 280 q^{65} - 648 q^{75} - 968 q^{77} + 972 q^{81} + 968 q^{83} + 852 q^{85} - 600 q^{87} - 1648 q^{91} - 1152 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - \nu^{10} - 2 \nu^{9} + 122 \nu^{8} - 128 \nu^{7} - 200 \nu^{6} - 1216 \nu^{5} - 7744 \nu^{4} + 15360 \nu^{3} - 44032 \nu^{2} + 65536 \nu - 24576 ) / 8192 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13 \nu^{11} - 47 \nu^{10} - 74 \nu^{9} - 26 \nu^{8} + 40 \nu^{7} - 1048 \nu^{6} + 5792 \nu^{5} + 3904 \nu^{4} + 8960 \nu^{3} + 74752 \nu^{2} + 372736 \nu + 1671168 ) / 49152 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7 \nu^{11} - 11 \nu^{10} - 32 \nu^{9} - 50 \nu^{8} + 172 \nu^{7} - 904 \nu^{6} + 1232 \nu^{5} - 2240 \nu^{4} - 1408 \nu^{3} + 31744 \nu^{2} + 120832 \nu + 786432 ) / 24576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{11} - 8 \nu^{10} - 49 \nu^{9} - 88 \nu^{8} - 130 \nu^{7} - 184 \nu^{6} - 1880 \nu^{5} + 640 \nu^{4} + 9536 \nu^{3} + 44032 \nu^{2} - 142336 \nu + 241664 ) / 12288 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{11} - 85 \nu^{10} + 190 \nu^{9} + 226 \nu^{8} + 1744 \nu^{7} - 488 \nu^{6} - 9472 \nu^{5} + 38336 \nu^{4} - 95744 \nu^{3} + 195584 \nu^{2} - 401408 \nu + 360448 ) / 49152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7 \nu^{11} - \nu^{10} - 74 \nu^{9} - 102 \nu^{8} + 1008 \nu^{7} + 184 \nu^{6} - 896 \nu^{5} + 12480 \nu^{4} + 66048 \nu^{3} + 97280 \nu^{2} - 262144 \nu + 425984 ) / 16384 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6 \nu^{11} + 11 \nu^{10} + 19 \nu^{9} + 10 \nu^{8} + 58 \nu^{7} + 688 \nu^{6} - 1480 \nu^{5} - 1984 \nu^{4} - 4160 \nu^{3} - 22528 \nu^{2} - 115712 \nu - 598016 ) / 12288 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9 \nu^{11} + 4 \nu^{10} - 25 \nu^{9} - 16 \nu^{8} - 130 \nu^{7} - 856 \nu^{6} + 3496 \nu^{5} - 4736 \nu^{4} + 9536 \nu^{3} + 80896 \nu^{2} + 349184 \nu + 561152 ) / 12288 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{11} - 13 \nu^{10} - 30 \nu^{9} + 18 \nu^{8} - 136 \nu^{7} - 264 \nu^{6} + 736 \nu^{5} + 3264 \nu^{4} + 2304 \nu^{3} + 13312 \nu^{2} + 200704 \nu + 589824 ) / 8192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27 \nu^{11} - 29 \nu^{10} - 114 \nu^{9} - 14 \nu^{8} - 240 \nu^{7} - 1704 \nu^{6} + 6912 \nu^{5} - 2624 \nu^{4} + 13824 \nu^{3} + 150528 \nu^{2} + 393216 \nu + 2260992 ) / 16384 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 31 \nu^{11} + 61 \nu^{10} + 70 \nu^{9} + 174 \nu^{8} + 280 \nu^{7} + 1928 \nu^{6} - 6176 \nu^{5} - 4544 \nu^{4} - 14080 \nu^{3} - 158720 \nu^{2} - 708608 \nu - 2686976 ) / 16384 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{3} + \beta_{2} + 3 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{9} + 3\beta_{8} + 3\beta_{7} + 2\beta_{5} + 2\beta_{4} - 5\beta_{3} + \beta_{2} + 15 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{11} - 3 \beta_{10} - 5 \beta_{9} + \beta_{8} - 9 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 17 \beta_{3} + \beta_{2} + 4 \beta _1 + 67 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{11} - 11 \beta_{10} + 19 \beta_{9} + 5 \beta_{8} - 57 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 33 \beta_{3} - 3 \beta_{2} - 12 \beta _1 + 119 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20 \beta_{11} + 21 \beta_{10} - 61 \beta_{9} - 7 \beta_{8} - 57 \beta_{7} - 4 \beta_{6} - 22 \beta_{5} - 36 \beta_{4} - 97 \beta_{3} + 197 \beta_{2} - 20 \beta _1 - 361 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 108 \beta_{11} + 101 \beta_{10} + 83 \beta_{9} - 79 \beta_{8} + 807 \beta_{7} + 28 \beta_{6} - 30 \beta_{5} - 68 \beta_{4} - 257 \beta_{3} + 205 \beta_{2} - 20 \beta _1 + 7279 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 132 \beta_{11} - 171 \beta_{10} + 67 \beta_{9} + 89 \beta_{8} + 631 \beta_{7} + 172 \beta_{6} + 82 \beta_{5} - 252 \beta_{4} + 1359 \beta_{3} + 717 \beta_{2} - 4 \beta _1 - 929 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1284 \beta_{11} + 965 \beta_{10} + 531 \beta_{9} + 473 \beta_{8} - 537 \beta_{7} - 20 \beta_{6} + 898 \beta_{5} - 220 \beta_{4} - 2241 \beta_{3} + 1485 \beta_{2} + 636 \beta _1 + 12831 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3612 \beta_{11} - 971 \beta_{10} - 4765 \beta_{9} + 25 \beta_{8} - 3785 \beta_{7} + 12 \beta_{6} + 1026 \beta_{5} - 2028 \beta_{4} - 7665 \beta_{3} - 4899 \beta_{2} + 924 \beta _1 + 146031 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5796 \beta_{11} + 293 \beta_{10} + 5747 \beta_{9} + 6761 \beta_{8} - 8121 \beta_{7} + 1228 \beta_{6} - 1854 \beta_{5} - 1964 \beta_{4} + 5535 \beta_{3} - 23763 \beta_{2} - 3876 \beta _1 + 431839 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15908 \beta_{11} - 25067 \beta_{10} - 3197 \beta_{9} + 26105 \beta_{8} - 13929 \beta_{7} - 4020 \beta_{6} + 2658 \beta_{5} + 19252 \beta_{4} - 11921 \beta_{3} + 97501 \beta_{2} - 676 \beta _1 + 1199151 ) / 32 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
0.965027 + 2.65871i
2.82801 + 0.0488466i
−0.951271 2.66366i
−1.72458 + 2.24184i
−2.78362 0.501431i
2.16644 1.81839i
2.16644 + 1.81839i
−2.78362 + 0.501431i
−1.72458 2.24184i
−0.951271 + 2.66366i
2.82801 0.0488466i
0.965027 2.65871i
0 3.00000 0 19.4608i 0 1.95109 + 18.4172i 0 9.00000 0
895.2 0 3.00000 0 16.6517i 0 15.0420 10.8045i 0 9.00000 0
895.3 0 3.00000 0 10.8462i 0 −15.1344 10.6748i 0 9.00000 0
895.4 0 3.00000 0 6.58775i 0 15.1925 10.5918i 0 9.00000 0
895.5 0 3.00000 0 4.57514i 0 2.93118 18.2868i 0 9.00000 0
895.6 0 3.00000 0 4.47531i 0 −14.9825 + 10.8869i 0 9.00000 0
895.7 0 3.00000 0 4.47531i 0 −14.9825 10.8869i 0 9.00000 0
895.8 0 3.00000 0 4.57514i 0 2.93118 + 18.2868i 0 9.00000 0
895.9 0 3.00000 0 6.58775i 0 15.1925 + 10.5918i 0 9.00000 0
895.10 0 3.00000 0 10.8462i 0 −15.1344 + 10.6748i 0 9.00000 0
895.11 0 3.00000 0 16.6517i 0 15.0420 + 10.8045i 0 9.00000 0
895.12 0 3.00000 0 19.4608i 0 1.95109 18.4172i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.b.h 12
4.b odd 2 1 1344.4.b.g 12
7.b odd 2 1 1344.4.b.g 12
8.b even 2 1 84.4.b.a 12
8.d odd 2 1 84.4.b.b yes 12
24.f even 2 1 252.4.b.e 12
24.h odd 2 1 252.4.b.f 12
28.d even 2 1 inner 1344.4.b.h 12
56.e even 2 1 84.4.b.a 12
56.h odd 2 1 84.4.b.b yes 12
168.e odd 2 1 252.4.b.f 12
168.i even 2 1 252.4.b.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.b.a 12 8.b even 2 1
84.4.b.a 12 56.e even 2 1
84.4.b.b yes 12 8.d odd 2 1
84.4.b.b yes 12 56.h odd 2 1
252.4.b.e 12 24.f even 2 1
252.4.b.e 12 168.i even 2 1
252.4.b.f 12 24.h odd 2 1
252.4.b.f 12 168.e odd 2 1
1344.4.b.g 12 4.b odd 2 1
1344.4.b.g 12 7.b odd 2 1
1344.4.b.h 12 1.a even 1 1 trivial
1344.4.b.h 12 28.d 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 \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{12} + 858T_{5}^{10} + 249644T_{5}^{8} + 29440120T_{5}^{6} + 1456436096T_{5}^{4} + 30453516288T_{5}^{2} + 224760987648 \) Copy content Toggle raw display
\( T_{19}^{6} - 42T_{19}^{5} - 27084T_{19}^{4} + 1835880T_{19}^{3} + 99824480T_{19}^{2} - 8618904192T_{19} + 111463534080 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T - 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 858 T^{10} + \cdots + 224760987648 \) Copy content Toggle raw display
$7$ \( T^{12} - 10 T^{11} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + 10414 T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + 9976 T^{10} + \cdots + 62\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{12} + 34826 T^{10} + \cdots + 59\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( (T^{6} - 42 T^{5} + \cdots + 111463534080)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 53182 T^{10} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( (T^{6} + 100 T^{5} + \cdots - 3698977199040)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 192 T^{5} + \cdots + 34547086393344)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 122 T^{5} + \cdots + 11485380096)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 373146 T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + 328092 T^{10} + \cdots + 47\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( (T^{6} - 140 T^{5} + \cdots + 118618190512128)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} + \cdots - 24896048217408)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 584 T^{5} + \cdots - 19\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 1590928 T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + 1407700 T^{10} + \cdots + 64\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{12} + 1375838 T^{10} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + 3228968 T^{10} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{12} + 3584020 T^{10} + \cdots + 96\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{6} - 484 T^{5} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 6134874 T^{10} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{12} + 5772328 T^{10} + \cdots + 29\!\cdots\!88 \) Copy content Toggle raw display
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