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Properties

Label 1400.2.q.o
Level $1400$
Weight $2$
Character orbit 1400.q
Analytic conductor $11.179$
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] = [1400,2,Mod(401,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.q (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: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 13 x^{10} - 4 x^{9} + 117 x^{8} - 41 x^{7} + 453 x^{6} - 72 x^{5} + 1249 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 280)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{3} + \beta_{4} q^{7} + ( - \beta_{10} + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{3} + \beta_{4} q^{7} + ( - \beta_{10} + \beta_{6}) q^{9} + (\beta_{11} - \beta_{8} + \beta_{2}) q^{11} + (\beta_{11} + \beta_{9} + \beta_{8} + \cdots - 2) q^{13}+ \cdots + (2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{3} + 3 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{3} + 3 q^{7} - 7 q^{9} - q^{11} - 14 q^{13} + 4 q^{17} + 5 q^{19} - 5 q^{21} - 6 q^{23} - 14 q^{27} - 6 q^{29} + 2 q^{31} + 16 q^{33} - 9 q^{37} - 10 q^{39} + 12 q^{41} - 6 q^{43} + 27 q^{47} + 15 q^{49} + 5 q^{53} - 52 q^{57} + 24 q^{59} - 9 q^{61} + 29 q^{63} - 17 q^{67} + 30 q^{69} + 8 q^{71} + 18 q^{73} + 53 q^{77} + 22 q^{79} + 6 q^{81} - 18 q^{83} + 39 q^{87} - 15 q^{89} + 22 q^{91} - 10 q^{93} - 24 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 13 x^{10} - 4 x^{9} + 117 x^{8} - 41 x^{7} + 453 x^{6} - 72 x^{5} + 1249 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 206469013 \nu^{11} - 977194173 \nu^{10} + 2606565394 \nu^{9} - 10264551516 \nu^{8} + \cdots - 20333876464 ) / 127774806084 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 62053540 \nu^{11} - 26650830 \nu^{10} + 754667545 \nu^{9} + 195891912 \nu^{8} + \cdots + 8760972344 ) / 31943701521 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 339726152 \nu^{11} - 471751391 \nu^{10} - 3666914179 \nu^{9} - 8983597792 \nu^{8} + \cdots - 250533343068 ) / 127774806084 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 806427899 \nu^{11} + 273751289 \nu^{10} - 8418119451 \nu^{9} - 5271318656 \nu^{8} + \cdots - 446266392752 ) / 255549612168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1095121543 \nu^{11} - 1591549863 \nu^{10} + 14449786699 \nu^{9} - 10417826532 \nu^{8} + \cdots - 149604693184 ) / 255549612168 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 802781337 \nu^{11} + 1147491601 \nu^{10} - 10807837730 \nu^{9} + 6729649118 \nu^{8} + \cdots - 118706336696 ) / 127774806084 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1775911397 \nu^{11} - 2365219133 \nu^{10} + 24024942927 \nu^{9} - 15757490752 \nu^{8} + \cdots + 208071017648 ) / 255549612168 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1027949708 \nu^{11} - 797480469 \nu^{10} + 12934671980 \nu^{9} - 157771638 \nu^{8} + \cdots + 551901731128 ) / 127774806084 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1024316123 \nu^{11} - 1487492913 \nu^{10} + 13561574555 \nu^{9} - 9937379460 \nu^{8} + \cdots - 141661840064 ) / 63887403042 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2177508516 \nu^{11} + 1908851780 \nu^{10} - 27379778881 \nu^{9} + 4439963542 \nu^{8} + \cdots - 256082112148 ) / 127774806084 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 4\beta_{6} - \beta_{5} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{4} + 6\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - 8\beta_{10} + 2\beta_{8} + 2\beta_{7} + 27\beta_{6} + \beta_{5} + \beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 24 \beta_{11} - 14 \beta_{10} - 14 \beta_{9} - 4 \beta_{8} + 18 \beta_{6} + 14 \beta_{5} - 43 \beta_{3} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 63 \beta_{11} - 67 \beta_{9} - 77 \beta_{8} - 28 \beta_{7} + 77 \beta_{5} - 42 \beta_{4} - 46 \beta_{3} + \cdots + 269 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 143\beta_{11} + 151\beta_{10} - 8\beta_{8} - 60\beta_{7} - 227\beta_{6} - 91\beta_{5} - 143\beta_{4} + 276\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 745 \beta_{11} + 586 \beta_{10} + 586 \beta_{9} + 427 \beta_{8} - 1609 \beta_{6} - 812 \beta_{5} + \cdots - 2195 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 980 \beta_{11} + 1500 \beta_{9} + 848 \beta_{8} + 680 \beta_{7} - 848 \beta_{5} + 1492 \beta_{4} + \cdots - 4000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2500 \beta_{11} - 5253 \beta_{10} + 1984 \beta_{8} + 2824 \beta_{7} + 13408 \beta_{6} + \cdots - 2676 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 21634 \beta_{11} - 14397 \beta_{10} - 14397 \beta_{9} - 7160 \beta_{8} + 25765 \beta_{6} + \cdots + 40162 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{6}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−1.27215 + 2.20343i
−1.01918 + 1.76528i
−0.159993 + 0.277116i
0.375714 − 0.650755i
1.05411 − 1.82577i
1.52151 − 2.63533i
−1.27215 − 2.20343i
−1.01918 − 1.76528i
−0.159993 − 0.277116i
0.375714 + 0.650755i
1.05411 + 1.82577i
1.52151 + 2.63533i
0 −1.27215 − 2.20343i 0 0 0 0.0988222 − 2.64391i 0 −1.73675 + 3.00813i 0
401.2 0 −1.01918 − 1.76528i 0 0 0 1.34267 + 2.27974i 0 −0.577473 + 1.00021i 0
401.3 0 −0.159993 − 0.277116i 0 0 0 −2.17465 + 1.50695i 0 1.44880 − 2.50940i 0
401.4 0 0.375714 + 0.650755i 0 0 0 2.58943 + 0.543003i 0 1.21768 − 2.10908i 0
401.5 0 1.05411 + 1.82577i 0 0 0 2.21349 − 1.44930i 0 −0.722285 + 1.25103i 0
401.6 0 1.52151 + 2.63533i 0 0 0 −2.56976 + 0.629537i 0 −3.12998 + 5.42128i 0
1201.1 0 −1.27215 + 2.20343i 0 0 0 0.0988222 + 2.64391i 0 −1.73675 − 3.00813i 0
1201.2 0 −1.01918 + 1.76528i 0 0 0 1.34267 − 2.27974i 0 −0.577473 − 1.00021i 0
1201.3 0 −0.159993 + 0.277116i 0 0 0 −2.17465 − 1.50695i 0 1.44880 + 2.50940i 0
1201.4 0 0.375714 − 0.650755i 0 0 0 2.58943 − 0.543003i 0 1.21768 + 2.10908i 0
1201.5 0 1.05411 − 1.82577i 0 0 0 2.21349 + 1.44930i 0 −0.722285 − 1.25103i 0
1201.6 0 1.52151 − 2.63533i 0 0 0 −2.56976 − 0.629537i 0 −3.12998 − 5.42128i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.6
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 1400.2.q.o 12
5.b even 2 1 1400.2.q.n 12
5.c odd 4 2 280.2.bg.a ✓ 24
7.c even 3 1 inner 1400.2.q.o 12
7.c even 3 1 9800.2.a.cv 6
7.d odd 6 1 9800.2.a.cy 6
20.e even 4 2 560.2.bw.f 24
35.i odd 6 1 9800.2.a.cw 6
35.j even 6 1 1400.2.q.n 12
35.j even 6 1 9800.2.a.cx 6
35.k even 12 2 1960.2.g.e 12
35.l odd 12 2 280.2.bg.a ✓ 24
35.l odd 12 2 1960.2.g.f 12
140.w even 12 2 560.2.bw.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bg.a ✓ 24 5.c odd 4 2
280.2.bg.a ✓ 24 35.l odd 12 2
560.2.bw.f 24 20.e even 4 2
560.2.bw.f 24 140.w even 12 2
1400.2.q.n 12 5.b even 2 1
1400.2.q.n 12 35.j even 6 1
1400.2.q.o 12 1.a even 1 1 trivial
1400.2.q.o 12 7.c even 3 1 inner
1960.2.g.e 12 35.k even 12 2
1960.2.g.f 12 35.l odd 12 2
9800.2.a.cv 6 7.c even 3 1
9800.2.a.cw 6 35.i odd 6 1
9800.2.a.cx 6 35.j even 6 1
9800.2.a.cy 6 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 \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{12} - T_{3}^{11} + 13 T_{3}^{10} - 4 T_{3}^{9} + 117 T_{3}^{8} - 41 T_{3}^{7} + 453 T_{3}^{6} + \cdots + 64 \) Copy content Toggle raw display
\( T_{11}^{12} + T_{11}^{11} + 38 T_{11}^{10} + 101 T_{11}^{9} + 1142 T_{11}^{8} + 2793 T_{11}^{7} + \cdots + 262144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + \cdots + 262144 \) Copy content Toggle raw display
$13$ \( (T^{6} + 7 T^{5} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} + \cdots + 91240704 \) Copy content Toggle raw display
$19$ \( T^{12} - 5 T^{11} + \cdots + 322624 \) Copy content Toggle raw display
$23$ \( T^{12} + 6 T^{11} + \cdots + 113569 \) Copy content Toggle raw display
$29$ \( (T^{6} + 3 T^{5} + \cdots - 7344)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 2 T^{11} + \cdots + 3873024 \) Copy content Toggle raw display
$37$ \( T^{12} + 9 T^{11} + \cdots + 11943936 \) Copy content Toggle raw display
$41$ \( (T^{6} - 6 T^{5} + \cdots + 2764)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 3 T^{5} + \cdots - 95736)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 27 T^{11} + \cdots + 1937664 \) Copy content Toggle raw display
$53$ \( T^{12} - 5 T^{11} + \cdots + 118336 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 15551087616 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11194909636 \) Copy content Toggle raw display
$67$ \( T^{12} + 17 T^{11} + \cdots + 14760964 \) Copy content Toggle raw display
$71$ \( (T^{6} - 4 T^{5} + \cdots - 183296)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 32066781184 \) Copy content Toggle raw display
$79$ \( T^{12} - 22 T^{11} + \cdots + 73984 \) Copy content Toggle raw display
$83$ \( (T^{6} + 9 T^{5} + \cdots - 136)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13352726916 \) Copy content Toggle raw display
$97$ \( (T^{6} + 12 T^{5} + \cdots - 8192)^{2} \) Copy content Toggle raw display
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