Properties

Label 1400.2.bh.h
Level $1400$
Weight $2$
Character orbit 1400.bh
Analytic conductor $11.179$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.bh (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \(x^{12} - x^{6} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{36}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{36} + \zeta_{36}^{3} - \zeta_{36}^{7} + \zeta_{36}^{11} ) q^{3} + ( -\zeta_{36} + 3 \zeta_{36}^{7} ) q^{7} + ( -2 \zeta_{36}^{2} + \zeta_{36}^{4} + \zeta_{36}^{8} - 2 \zeta_{36}^{10} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{36} + \zeta_{36}^{3} - \zeta_{36}^{7} + \zeta_{36}^{11} ) q^{3} + ( -\zeta_{36} + 3 \zeta_{36}^{7} ) q^{7} + ( -2 \zeta_{36}^{2} + \zeta_{36}^{4} + \zeta_{36}^{8} - 2 \zeta_{36}^{10} ) q^{9} + ( 2 - \zeta_{36}^{2} - \zeta_{36}^{4} - 2 \zeta_{36}^{6} + 2 \zeta_{36}^{8} + 2 \zeta_{36}^{10} ) q^{11} + ( -3 \zeta_{36} + 3 \zeta_{36}^{5} + 4 \zeta_{36}^{7} + \zeta_{36}^{11} ) q^{13} + ( 2 \zeta_{36}^{3} - 3 \zeta_{36}^{5} - 3 \zeta_{36}^{7} + 3 \zeta_{36}^{11} ) q^{17} + ( -3 \zeta_{36}^{2} + 2 \zeta_{36}^{4} + 2 \zeta_{36}^{6} + 2 \zeta_{36}^{8} - 3 \zeta_{36}^{10} ) q^{19} + ( -2 + 2 \zeta_{36}^{2} - \zeta_{36}^{4} - \zeta_{36}^{6} + \zeta_{36}^{8} + 3 \zeta_{36}^{10} ) q^{21} + ( \zeta_{36} - \zeta_{36}^{3} + 3 \zeta_{36}^{5} - 3 \zeta_{36}^{7} + \zeta_{36}^{9} - \zeta_{36}^{11} ) q^{23} + ( -3 \zeta_{36}^{7} - 3 \zeta_{36}^{11} ) q^{27} + ( -4 - \zeta_{36}^{2} - \zeta_{36}^{4} - \zeta_{36}^{8} + 2 \zeta_{36}^{10} ) q^{29} + ( 1 - 4 \zeta_{36}^{2} + \zeta_{36}^{4} - \zeta_{36}^{6} + 3 \zeta_{36}^{8} + 3 \zeta_{36}^{10} ) q^{31} + ( -3 \zeta_{36} + 2 \zeta_{36}^{3} + 2 \zeta_{36}^{5} - 2 \zeta_{36}^{7} - 2 \zeta_{36}^{9} + 3 \zeta_{36}^{11} ) q^{33} + ( -4 \zeta_{36} - 2 \zeta_{36}^{3} - \zeta_{36}^{5} + \zeta_{36}^{7} + 2 \zeta_{36}^{9} + 4 \zeta_{36}^{11} ) q^{37} + ( 2 - 7 \zeta_{36}^{4} - 2 \zeta_{36}^{6} + 7 \zeta_{36}^{8} + 7 \zeta_{36}^{10} ) q^{39} + ( 1 + \zeta_{36}^{2} + \zeta_{36}^{4} + 2 \zeta_{36}^{8} - 3 \zeta_{36}^{10} ) q^{41} + ( -4 \zeta_{36} + 4 \zeta_{36}^{5} + 3 \zeta_{36}^{7} + 3 \zeta_{36}^{9} - \zeta_{36}^{11} ) q^{43} + ( 3 \zeta_{36} + 6 \zeta_{36}^{3} + \zeta_{36}^{5} - \zeta_{36}^{7} - 6 \zeta_{36}^{9} - 3 \zeta_{36}^{11} ) q^{47} + ( -8 \zeta_{36}^{2} + 3 \zeta_{36}^{8} ) q^{49} + ( -8 \zeta_{36}^{2} + 2 \zeta_{36}^{4} + 5 \zeta_{36}^{6} + 2 \zeta_{36}^{8} - 8 \zeta_{36}^{10} ) q^{51} + ( 9 \zeta_{36} + 2 \zeta_{36}^{3} + 5 \zeta_{36}^{5} - 4 \zeta_{36}^{7} + 4 \zeta_{36}^{11} ) q^{53} + ( 6 \zeta_{36} - 6 \zeta_{36}^{5} - 4 \zeta_{36}^{7} + 7 \zeta_{36}^{9} + 2 \zeta_{36}^{11} ) q^{57} + ( -1 - 3 \zeta_{36}^{2} + 3 \zeta_{36}^{4} + \zeta_{36}^{6} ) q^{59} + ( -5 \zeta_{36}^{2} + 6 \zeta_{36}^{4} - 3 \zeta_{36}^{6} + 6 \zeta_{36}^{8} - 5 \zeta_{36}^{10} ) q^{61} + ( -\zeta_{36}^{3} + 5 \zeta_{36}^{5} - 4 \zeta_{36}^{9} - \zeta_{36}^{11} ) q^{63} + ( -6 \zeta_{36} - 4 \zeta_{36}^{3} + 6 \zeta_{36}^{7} - 6 \zeta_{36}^{11} ) q^{67} + ( 4 - \zeta_{36}^{2} - \zeta_{36}^{4} + \zeta_{36}^{10} ) q^{69} + ( 3 - 6 \zeta_{36}^{2} - 6 \zeta_{36}^{4} + 6 \zeta_{36}^{10} ) q^{71} + ( 4 \zeta_{36} - 4 \zeta_{36}^{3} + 4 \zeta_{36}^{5} ) q^{73} + ( 4 \zeta_{36} - 5 \zeta_{36}^{3} - 5 \zeta_{36}^{5} + 2 \zeta_{36}^{7} + \zeta_{36}^{9} + \zeta_{36}^{11} ) q^{77} + ( \zeta_{36}^{4} - \zeta_{36}^{6} + \zeta_{36}^{8} ) q^{79} + ( 3 + 3 \zeta_{36}^{2} - 3 \zeta_{36}^{4} - 3 \zeta_{36}^{6} ) q^{81} + ( -5 \zeta_{36} + 5 \zeta_{36}^{5} + 3 \zeta_{36}^{7} - 2 \zeta_{36}^{11} ) q^{83} + ( -4 \zeta_{36} - 7 \zeta_{36}^{3} + 4 \zeta_{36}^{7} - 4 \zeta_{36}^{11} ) q^{87} + ( -6 \zeta_{36}^{2} + 7 \zeta_{36}^{4} + 2 \zeta_{36}^{6} + 7 \zeta_{36}^{8} - 6 \zeta_{36}^{10} ) q^{89} + ( -11 - 9 \zeta_{36}^{2} + 5 \zeta_{36}^{6} - \zeta_{36}^{8} ) q^{91} + ( -2 \zeta_{36} - 4 \zeta_{36}^{3} + \zeta_{36}^{5} - \zeta_{36}^{7} + 4 \zeta_{36}^{9} + 2 \zeta_{36}^{11} ) q^{93} + ( -3 \zeta_{36} + 3 \zeta_{36}^{5} + 6 \zeta_{36}^{7} - 7 \zeta_{36}^{9} + 3 \zeta_{36}^{11} ) q^{97} + ( 3 - 2 \zeta_{36}^{2} - 2 \zeta_{36}^{4} + 7 \zeta_{36}^{8} - 5 \zeta_{36}^{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q + 12q^{11} + 12q^{19} - 30q^{21} - 48q^{29} + 6q^{31} + 12q^{39} + 12q^{41} + 30q^{51} - 6q^{59} - 18q^{61} + 48q^{69} + 36q^{71} - 6q^{79} + 18q^{81} + 12q^{89} - 102q^{91} + 36q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{36}^{6}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
0.642788 0.766044i
−0.984808 0.173648i
−0.342020 0.939693i
0.342020 + 0.939693i
0.984808 + 0.173648i
−0.642788 + 0.766044i
0.642788 + 0.766044i
−0.984808 + 0.173648i
−0.342020 + 0.939693i
0.342020 0.939693i
0.984808 0.173648i
−0.642788 0.766044i
0 −2.19285 1.26604i 0 0 0 2.31164 + 1.28699i 0 1.70574 + 2.95442i 0
249.2 0 −1.16679 0.673648i 0 0 0 −0.0412527 2.64543i 0 −0.592396 1.02606i 0
249.3 0 −0.761570 0.439693i 0 0 0 2.27038 1.35844i 0 −1.11334 1.92836i 0
249.4 0 0.761570 + 0.439693i 0 0 0 −2.27038 + 1.35844i 0 −1.11334 1.92836i 0
249.5 0 1.16679 + 0.673648i 0 0 0 0.0412527 + 2.64543i 0 −0.592396 1.02606i 0
249.6 0 2.19285 + 1.26604i 0 0 0 −2.31164 1.28699i 0 1.70574 + 2.95442i 0
849.1 0 −2.19285 + 1.26604i 0 0 0 2.31164 1.28699i 0 1.70574 2.95442i 0
849.2 0 −1.16679 + 0.673648i 0 0 0 −0.0412527 + 2.64543i 0 −0.592396 + 1.02606i 0
849.3 0 −0.761570 + 0.439693i 0 0 0 2.27038 + 1.35844i 0 −1.11334 + 1.92836i 0
849.4 0 0.761570 0.439693i 0 0 0 −2.27038 1.35844i 0 −1.11334 + 1.92836i 0
849.5 0 1.16679 0.673648i 0 0 0 0.0412527 2.64543i 0 −0.592396 + 1.02606i 0
849.6 0 2.19285 1.26604i 0 0 0 −2.31164 + 1.28699i 0 1.70574 2.95442i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 849.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.bh.h 12
5.b even 2 1 inner 1400.2.bh.h 12
5.c odd 4 1 1400.2.q.i 6
5.c odd 4 1 1400.2.q.k yes 6
7.c even 3 1 inner 1400.2.bh.h 12
35.j even 6 1 inner 1400.2.bh.h 12
35.k even 12 1 9800.2.a.cb 3
35.k even 12 1 9800.2.a.ch 3
35.l odd 12 1 1400.2.q.i 6
35.l odd 12 1 1400.2.q.k yes 6
35.l odd 12 1 9800.2.a.cc 3
35.l odd 12 1 9800.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.i 6 5.c odd 4 1
1400.2.q.i 6 35.l odd 12 1
1400.2.q.k yes 6 5.c odd 4 1
1400.2.q.k yes 6 35.l odd 12 1
1400.2.bh.h 12 1.a even 1 1 trivial
1400.2.bh.h 12 5.b even 2 1 inner
1400.2.bh.h 12 7.c even 3 1 inner
1400.2.bh.h 12 35.j even 6 1 inner
9800.2.a.cb 3 35.k even 12 1
9800.2.a.cc 3 35.l odd 12 1
9800.2.a.ch 3 35.k even 12 1
9800.2.a.ci 3 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{12} - 9 T_{3}^{10} + 63 T_{3}^{8} - 144 T_{3}^{6} + 243 T_{3}^{4} - 162 T_{3}^{2} + 81 \)
\( T_{11}^{6} - 6 T_{11}^{5} + 33 T_{11}^{4} - 56 T_{11}^{3} + 123 T_{11}^{2} + 57 T_{11} + 361 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 81 - 162 T^{2} + 243 T^{4} - 144 T^{6} + 63 T^{8} - 9 T^{10} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( 117649 + 683 T^{6} + T^{12} \)
$11$ \( ( 361 + 57 T + 123 T^{2} - 56 T^{3} + 33 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$13$ \( ( 361 + 1521 T^{2} + 78 T^{4} + T^{6} )^{2} \)
$17$ \( 130321 - 163533 T^{2} + 181383 T^{4} - 29176 T^{6} + 3903 T^{8} - 66 T^{10} + T^{12} \)
$19$ \( ( 289 - 153 T + 183 T^{2} + 20 T^{3} + 45 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$23$ \( 83521 - 64158 T^{2} + 36279 T^{4} - 9412 T^{6} + 1803 T^{8} - 45 T^{10} + T^{12} \)
$29$ \( ( 19 + 39 T + 12 T^{2} + T^{3} )^{4} \)
$31$ \( ( 361 - 684 T + 1353 T^{2} + 70 T^{3} + 45 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$37$ \( 6765201 - 3487941 T^{2} + 1564191 T^{4} - 115488 T^{6} + 6759 T^{8} - 90 T^{10} + T^{12} \)
$41$ \( ( 57 - 18 T - 3 T^{2} + T^{3} )^{4} \)
$43$ \( ( 32041 + 3366 T^{2} + 105 T^{4} + T^{6} )^{2} \)
$47$ \( 131079601 - 42555933 T^{2} + 12098739 T^{4} - 534652 T^{6} + 18783 T^{8} - 150 T^{10} + T^{12} \)
$53$ \( 1345435285041 - 48908406285 T^{2} + 1339434063 T^{4} - 13618512 T^{6} + 100719 T^{8} - 378 T^{10} + T^{12} \)
$59$ \( ( 2809 + 1272 T + 735 T^{2} + 34 T^{3} + 33 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$61$ \( ( 1369 - 2442 T + 4023 T^{2} - 668 T^{3} + 147 T^{4} + 9 T^{5} + T^{6} )^{2} \)
$67$ \( 116319195136 - 6008042496 T^{2} + 220284672 T^{4} - 3968512 T^{6} + 52080 T^{8} - 264 T^{10} + T^{12} \)
$71$ \( ( 513 - 81 T - 9 T^{2} + T^{3} )^{4} \)
$73$ \( 1358954496 - 169869312 T^{2} + 15925248 T^{4} - 589824 T^{6} + 16128 T^{8} - 144 T^{10} + T^{12} \)
$79$ \( ( 9 + 9 T^{2} + 6 T^{3} + 9 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$83$ \( ( 26569 + 3249 T^{2} + 114 T^{4} + T^{6} )^{2} \)
$89$ \( ( 488601 - 81783 T + 17883 T^{2} - 696 T^{3} + 153 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$97$ \( ( 218089 + 23970 T^{2} + 309 T^{4} + T^{6} )^{2} \)
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