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Properties

Label 1575.2.bk.g
Level $1575$
Weight $2$
Character orbit 1575.bk
Analytic conductor $12.576$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(26,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bk (of order \(6\), 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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10x^{14} + 70x^{12} + 244x^{10} + 619x^{8} + 820x^{6} + 754x^{4} + 28x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{9} - \beta_{8}) q^{2} + ( - \beta_{5} - 2 \beta_{3} + 2) q^{4} + ( - \beta_{6} + \beta_{5} - 1) q^{7} + ( - \beta_{12} + \beta_{11} + \cdots - \beta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} - \beta_{9} - \beta_{8}) q^{2} + ( - \beta_{5} - 2 \beta_{3} + 2) q^{4} + ( - \beta_{6} + \beta_{5} - 1) q^{7} + ( - \beta_{12} + \beta_{11} + \cdots - \beta_{8}) q^{8}+ \cdots + (2 \beta_{15} - 2 \beta_{14} + \cdots - 4 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} - 8 q^{7} + 8 q^{16} - 36 q^{19} - 8 q^{22} + 12 q^{28} - 24 q^{31} + 32 q^{37} + 56 q^{43} + 4 q^{46} + 16 q^{49} - 48 q^{52} + 4 q^{58} + 12 q^{61} + 64 q^{64} + 20 q^{67} - 60 q^{73} - 40 q^{79} + 72 q^{82} - 52 q^{88} - 36 q^{91} - 84 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10x^{14} + 70x^{12} + 244x^{10} + 619x^{8} + 820x^{6} + 754x^{4} + 28x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6070 \nu^{14} - 54200 \nu^{12} - 368449 \nu^{10} - 1086530 \nu^{8} - 2593830 \nu^{6} + \cdots + 5675948 ) / 2892729 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23216 \nu^{14} - 274018 \nu^{12} - 1987757 \nu^{10} - 8012636 \nu^{8} - 20899043 \nu^{6} + \cdots - 8946120 ) / 2892729 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 109510 \nu^{14} - 1089030 \nu^{12} - 7611500 \nu^{10} - 26351991 \nu^{8} - 66700160 \nu^{6} + \cdots - 117100 ) / 2892729 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 169677 \nu^{14} + 1697755 \nu^{12} + 11938607 \nu^{10} + 41943099 \nu^{8} + 108099107 \nu^{6} + \cdots + 6206481 ) / 2892729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 212950 \nu^{14} - 2123860 \nu^{12} - 14854551 \nu^{10} - 51617452 \nu^{8} - 130806490 \nu^{6} + \cdots - 5910148 ) / 2892729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 222813 \nu^{14} - 2240522 \nu^{12} - 15742508 \nu^{10} - 55311415 \nu^{8} - 141103658 \nu^{6} + \cdots - 1339946 ) / 2892729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 231943 \nu^{14} + 2333165 \nu^{12} + 16296699 \nu^{10} + 56945685 \nu^{8} + 143138553 \nu^{6} + \cdots + 5175775 ) / 2892729 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 743929 \nu^{15} + 7610081 \nu^{13} + 53738253 \nu^{11} + 192946357 \nu^{9} + 498777584 \nu^{7} + \cdots + 125495945 \nu ) / 8678187 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 296333 \nu^{15} - 3006603 \nu^{13} - 21169415 \nu^{11} - 75221196 \nu^{9} - 193104480 \nu^{7} + \cdots - 29623836 \nu ) / 2892729 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 377443 \nu^{15} + 3745144 \nu^{13} + 26092792 \nu^{11} + 89739886 \nu^{9} + 224538051 \nu^{7} + \cdots - 19981186 \nu ) / 2892729 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1370 \nu^{15} - 13717 \nu^{13} - 96042 \nu^{11} - 335411 \nu^{9} - 852064 \nu^{7} + \cdots - 77068 \nu ) / 8919 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1459190 \nu^{15} + 14478088 \nu^{13} + 101107992 \nu^{11} + 348941123 \nu^{9} + \cdots - 35219237 \nu ) / 8678187 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1661540 \nu^{15} + 16613731 \nu^{13} + 116283366 \nu^{11} + 405410876 \nu^{9} + \cdots + 92694838 \nu ) / 8678187 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1787720 \nu^{15} - 17745178 \nu^{13} - 123942492 \nu^{11} - 427997096 \nu^{9} + \cdots + 43546124 \nu ) / 8678187 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 809413 \nu^{15} + 8047064 \nu^{13} + 56170343 \nu^{11} + 194061320 \nu^{9} + 488744861 \nu^{7} + \cdots - 8051380 \nu ) / 2892729 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{14} + 4\beta_{13} - 11\beta_{12} + 4\beta_{11} - 3\beta_{9} - 3\beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 6\beta_{5} - \beta_{4} + 8\beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 24\beta_{15} + 19\beta_{14} - 38\beta_{13} + 16\beta_{12} - 32\beta_{11} - 24\beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} + 8\beta_{6} + 7\beta_{4} - 7\beta_{2} - 33\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 147 \beta_{15} + 97 \beta_{14} + 97 \beta_{13} + 217 \beta_{12} + 70 \beta_{11} + \cdots + 147 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 40\beta_{7} - 40\beta_{6} + 178\beta_{5} + 10\beta_{4} - 185\beta_{3} + 50\beta_{2} + 178\beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 30 \beta_{15} - 1022 \beta_{14} + 511 \beta_{13} - 1526 \beta_{12} + 331 \beta_{11} + \cdots - 864 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -288\beta_{7} - 70\beta_{6} - 957\beta_{5} - 288\beta_{4} + 962\beta_{3} - 70\beta_{2} - 1032 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4809 \beta_{15} + 2728 \beta_{14} - 5456 \beta_{13} + 1864 \beta_{12} - 3308 \beta_{11} + \cdots + 210 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 428\beta_{7} + 1603\beta_{6} + 1175\beta_{4} - 1175\beta_{2} - 5148\beta _1 + 3923 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 25062 \beta_{15} + 14647 \beta_{14} + 14647 \beta_{13} + 33616 \beta_{12} + 8554 \beta_{11} + \cdots + 25062 \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6323 \beta_{7} - 6323 \beta_{6} + 27721 \beta_{5} + 2459 \beta_{4} - 27263 \beta_{3} + 8782 \beta_{2} + \cdots + 8782 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 7377 \beta_{15} - 157682 \beta_{14} + 78841 \beta_{13} - 233468 \beta_{12} + 45118 \beta_{11} + \cdots - 143232 \beta_{8} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.0964065 + 0.166981i
−0.671503 − 1.16308i
−0.831322 − 1.43989i
1.16133 + 2.01149i
−1.16133 − 2.01149i
0.831322 + 1.43989i
0.671503 + 1.16308i
−0.0964065 − 0.166981i
0.0964065 − 0.166981i
−0.671503 + 1.16308i
−0.831322 + 1.43989i
1.16133 − 2.01149i
−1.16133 + 2.01149i
0.831322 − 1.43989i
0.671503 − 1.16308i
−0.0964065 + 0.166981i
−2.11474 − 1.22094i 0 1.98141 + 3.43191i 0 0 −2.64390 + 0.0990746i − 4.79300i 0 0
26.2 −1.77405 − 1.02425i 0 1.09817 + 1.90208i 0 0 −1.26228 − 2.32522i − 0.402191i 0 0
26.3 −1.55779 − 0.899391i 0 0.617809 + 1.07008i 0 0 2.62636 − 0.319700i 1.37496i 0 0
26.4 −0.673735 − 0.388981i 0 −0.697387 − 1.20791i 0 0 −0.720188 + 2.54585i 2.64101i 0 0
26.5 0.673735 + 0.388981i 0 −0.697387 − 1.20791i 0 0 −0.720188 + 2.54585i − 2.64101i 0 0
26.6 1.55779 + 0.899391i 0 0.617809 + 1.07008i 0 0 2.62636 − 0.319700i − 1.37496i 0 0
26.7 1.77405 + 1.02425i 0 1.09817 + 1.90208i 0 0 −1.26228 − 2.32522i 0.402191i 0 0
26.8 2.11474 + 1.22094i 0 1.98141 + 3.43191i 0 0 −2.64390 + 0.0990746i 4.79300i 0 0
1151.1 −2.11474 + 1.22094i 0 1.98141 − 3.43191i 0 0 −2.64390 − 0.0990746i 4.79300i 0 0
1151.2 −1.77405 + 1.02425i 0 1.09817 − 1.90208i 0 0 −1.26228 + 2.32522i 0.402191i 0 0
1151.3 −1.55779 + 0.899391i 0 0.617809 − 1.07008i 0 0 2.62636 + 0.319700i − 1.37496i 0 0
1151.4 −0.673735 + 0.388981i 0 −0.697387 + 1.20791i 0 0 −0.720188 − 2.54585i − 2.64101i 0 0
1151.5 0.673735 − 0.388981i 0 −0.697387 + 1.20791i 0 0 −0.720188 − 2.54585i 2.64101i 0 0
1151.6 1.55779 − 0.899391i 0 0.617809 − 1.07008i 0 0 2.62636 + 0.319700i 1.37496i 0 0
1151.7 1.77405 − 1.02425i 0 1.09817 − 1.90208i 0 0 −1.26228 + 2.32522i − 0.402191i 0 0
1151.8 2.11474 − 1.22094i 0 1.98141 − 3.43191i 0 0 −2.64390 − 0.0990746i − 4.79300i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.bk.g ✓ 16
3.b odd 2 1 inner 1575.2.bk.g ✓ 16
5.b even 2 1 1575.2.bk.h yes 16
5.c odd 4 2 1575.2.bc.e 32
7.d odd 6 1 inner 1575.2.bk.g ✓ 16
15.d odd 2 1 1575.2.bk.h yes 16
15.e even 4 2 1575.2.bc.e 32
21.g even 6 1 inner 1575.2.bk.g ✓ 16
35.i odd 6 1 1575.2.bk.h yes 16
35.k even 12 2 1575.2.bc.e 32
105.p even 6 1 1575.2.bk.h yes 16
105.w odd 12 2 1575.2.bc.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.bc.e 32 5.c odd 4 2
1575.2.bc.e 32 15.e even 4 2
1575.2.bc.e 32 35.k even 12 2
1575.2.bc.e 32 105.w odd 12 2
1575.2.bk.g ✓ 16 1.a even 1 1 trivial
1575.2.bk.g ✓ 16 3.b odd 2 1 inner
1575.2.bk.g ✓ 16 7.d odd 6 1 inner
1575.2.bk.g ✓ 16 21.g even 6 1 inner
1575.2.bk.h yes 16 5.b even 2 1
1575.2.bk.h yes 16 15.d odd 2 1
1575.2.bk.h yes 16 35.i odd 6 1
1575.2.bk.h yes 16 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2}^{16} - 14T_{2}^{14} + 130T_{2}^{12} - 692T_{2}^{10} + 2683T_{2}^{8} - 6284T_{2}^{6} + 10222T_{2}^{4} - 5684T_{2}^{2} + 2401 \) Copy content Toggle raw display
\( T_{11}^{16} - 50 T_{11}^{14} + 1768 T_{11}^{12} - 30632 T_{11}^{10} + 383260 T_{11}^{8} + \cdots + 11316496 \) Copy content Toggle raw display
\( T_{37}^{8} - 16 T_{37}^{7} + 202 T_{37}^{6} - 1240 T_{37}^{5} + 6787 T_{37}^{4} - 17464 T_{37}^{3} + \cdots + 744769 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 14 T^{14} + \cdots + 2401 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 50 T^{14} + \cdots + 11316496 \) Copy content Toggle raw display
$13$ \( (T^{8} + 36 T^{6} + 222 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 54 T^{14} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( (T^{8} + 18 T^{7} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 104 T^{6} + \cdots + 222784)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 12 T^{7} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 16 T^{7} + \cdots + 744769)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 162 T^{6} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 14 T^{3} + \cdots - 488)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 84288924810000 \) Copy content Toggle raw display
$53$ \( T^{16} - 98 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2200843458576 \) Copy content Toggle raw display
$61$ \( (T^{8} - 6 T^{7} + \cdots + 23299929)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 10 T^{7} + \cdots + 790321)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 344 T^{6} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 30 T^{7} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 20 T^{7} + \cdots + 4182025)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 690 T^{6} + \cdots + 416894724)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 337147454736 \) Copy content Toggle raw display
$97$ \( (T^{8} + 396 T^{6} + \cdots + 741321)^{2} \) Copy content Toggle raw display
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