↑

Properties

Label 975.2.bc.m
Level $975$
Weight $2$
Character orbit 975.bc
Analytic conductor $7.785$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(751,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("975.751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bc (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: \(7.78541419707\)
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: \( x^{16} + 26x^{14} + 269x^{12} + 1420x^{10} + 4080x^{8} + 6272x^{6} + 4672x^{4} + 1308x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_1) q^{2} + ( - \beta_{3} - 1) q^{3} + ( - \beta_{13} - \beta_{3}) q^{4} + \beta_{5} q^{6} + \beta_{14} q^{7} + ( - \beta_{9} + \beta_{8} - \beta_1) q^{8} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_1) q^{2} + ( - \beta_{3} - 1) q^{3} + ( - \beta_{13} - \beta_{3}) q^{4} + \beta_{5} q^{6} + \beta_{14} q^{7} + ( - \beta_{9} + \beta_{8} - \beta_1) q^{8} + \beta_{3} q^{9} + ( - \beta_{15} - \beta_{14} - \beta_{11} + \cdots - 1) q^{11}+ \cdots + (\beta_{15} + \beta_{11} + \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} + 10 q^{4} - 6 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} + 10 q^{4} - 6 q^{7} - 8 q^{9} - 6 q^{11} - 20 q^{12} - 4 q^{14} - 14 q^{16} + 10 q^{17} - 2 q^{22} + 12 q^{23} + 26 q^{26} + 16 q^{27} - 24 q^{28} + 12 q^{29} - 30 q^{32} + 6 q^{33} + 10 q^{36} - 6 q^{37} + 16 q^{38} - 18 q^{41} + 2 q^{42} + 16 q^{43} - 24 q^{46} - 14 q^{48} + 14 q^{49} - 20 q^{51} - 24 q^{52} + 40 q^{53} - 28 q^{56} - 72 q^{58} - 42 q^{59} - 16 q^{61} + 34 q^{62} + 6 q^{63} - 68 q^{64} + 4 q^{66} + 18 q^{67} - 28 q^{68} + 12 q^{69} + 6 q^{71} - 20 q^{74} + 24 q^{76} - 48 q^{77} - 16 q^{78} + 52 q^{79} - 8 q^{81} + 28 q^{82} + 24 q^{84} + 12 q^{87} + 68 q^{88} + 24 q^{89} + 6 q^{91} + 108 q^{92} - 6 q^{93} + 4 q^{94} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 26x^{14} + 269x^{12} + 1420x^{10} + 4080x^{8} + 6272x^{6} + 4672x^{4} + 1308x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{13} - 23\nu^{11} - 200\nu^{9} - 820\nu^{7} - 1610\nu^{5} - 1312\nu^{3} - 236\nu - 20 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17 \nu^{15} - 458 \nu^{13} - 4741 \nu^{11} - 23260 \nu^{9} - 52830 \nu^{7} - 41334 \nu^{5} + \cdots - 2580 ) / 1720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{14} - 23\nu^{12} - 200\nu^{10} - 820\nu^{8} - 1610\nu^{6} - 1312\nu^{4} - 236\nu^{2} - 20\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15 \nu^{15} - 18 \nu^{14} - 351 \nu^{13} - 404 \nu^{12} - 3088 \nu^{11} - 3310 \nu^{10} + \cdots + 7180 ) / 1720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 34 \nu^{15} - \nu^{14} + 830 \nu^{13} + 97 \nu^{12} + 7934 \nu^{11} + 2420 \nu^{10} + 37920 \nu^{9} + \cdots + 9620 ) / 1720 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{14} + 117 \nu^{12} + 1058 \nu^{10} + 4719 \nu^{8} + 11089 \nu^{6} + 13442 \nu^{4} + 86 \nu^{3} + \cdots + 1006 ) / 172 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5 \nu^{14} + 117 \nu^{12} + 1058 \nu^{10} + 4719 \nu^{8} + 11089 \nu^{6} + 13442 \nu^{4} - 86 \nu^{3} + \cdots + 1006 ) / 172 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19 \nu^{15} - 17 \nu^{14} + 479 \nu^{13} - 501 \nu^{12} + 4846 \nu^{11} - 5730 \nu^{10} + 25440 \nu^{9} + \cdots - 2440 ) / 1720 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15 \nu^{15} - 68 \nu^{14} + 351 \nu^{13} - 1574 \nu^{12} + 3088 \nu^{11} - 13890 \nu^{10} + \cdots - 2880 ) / 1720 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30 \nu^{15} - 21 \nu^{14} + 788 \nu^{13} - 543 \nu^{12} + 8154 \nu^{11} - 5510 \nu^{10} + 42160 \nu^{9} + \cdots + 780 ) / 1720 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} + 26 \nu^{13} + 269 \nu^{11} + 1420 \nu^{9} + 4070 \nu^{7} + 6142 \nu^{5} + 4172 \nu^{3} + \cdots + 60 ) / 40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 61 \nu^{15} + 4 \nu^{14} + 1522 \nu^{13} + 42 \nu^{12} + 14877 \nu^{11} - 220 \nu^{10} + 72510 \nu^{9} + \cdots - 9240 ) / 1720 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 61 \nu^{15} - 1522 \nu^{13} - 14877 \nu^{11} - 72510 \nu^{9} - 186000 \nu^{7} - 242562 \nu^{5} + \cdots - 24124 \nu ) / 860 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{11} + \beta_{10} + \beta_{8} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 2\beta_{11} + 8\beta_{9} - 10\beta_{8} + 2\beta_{6} - \beta_{4} + 4\beta_{3} - \beta_{2} + 30\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{15} + 20 \beta_{14} - 12 \beta_{13} + 24 \beta_{12} - 12 \beta_{11} - 11 \beta_{10} + \cdots - 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{15} - 15 \beta_{13} + 23 \beta_{11} - \beta_{10} - 58 \beta_{9} + 81 \beta_{8} - \beta_{7} + \cdots - 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 80 \beta_{15} - 160 \beta_{14} + 106 \beta_{13} - 212 \beta_{12} + 110 \beta_{11} + 93 \beta_{10} + \cdots + 497 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 32 \beta_{15} + 163 \beta_{13} - 197 \beta_{11} + 11 \beta_{10} + 414 \beta_{9} - 611 \beta_{8} + \cdots + 235 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 600 \beta_{15} + 1200 \beta_{14} - 840 \beta_{13} + 1680 \beta_{12} - 914 \beta_{11} - 719 \beta_{10} + \cdots - 3333 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 358 \beta_{15} - 1555 \beta_{13} + 1517 \beta_{11} - 77 \beta_{10} - 2954 \beta_{9} + 4471 \beta_{8} + \cdots - 1999 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4394 \beta_{15} - 8788 \beta_{14} + 6346 \beta_{13} - 12692 \beta_{12} + 7254 \beta_{11} + \cdots + 23193 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3474 \beta_{15} + 13855 \beta_{13} - 11131 \beta_{11} + 391 \beta_{10} + 21134 \beta_{9} - 32265 \beta_{8} + \cdots + 16237 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 31874 \beta_{15} + 63748 \beta_{14} - 46870 \beta_{13} + 93740 \beta_{12} - 56234 \beta_{11} + \cdots - 164849 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 31322 \beta_{15} - 118447 \beta_{13} + 79747 \beta_{11} - 1003 \beta_{10} - 151642 \beta_{9} + \cdots - 128665 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
− 2.61023i
− 2.01123i
− 1.46673i
− 0.347385i
0.592086i
1.22320i
1.89212i
2.72815i
2.61023i
2.01123i
1.46673i
0.347385i
− 0.592086i
− 1.22320i
− 1.89212i
− 2.72815i
−2.26052 + 1.30511i −0.500000 − 0.866025i 2.40664 − 4.16842i 0 2.26052 + 1.30511i −0.645251 − 0.372536i 7.34329i −0.500000 + 0.866025i 0
751.2 −1.74177 + 1.00561i −0.500000 − 0.866025i 1.02251 − 1.77105i 0 1.74177 + 1.00561i −1.92561 − 1.11175i 0.0905636i −0.500000 + 0.866025i 0
751.3 −1.27022 + 0.733363i −0.500000 − 0.866025i 0.0756426 − 0.131017i 0 1.27022 + 0.733363i 3.75091 + 2.16559i − 2.71156i −0.500000 + 0.866025i 0
751.4 −0.300844 + 0.173693i −0.500000 − 0.866025i −0.939662 + 1.62754i 0 0.300844 + 0.173693i −3.62329 − 2.09191i − 1.34762i −0.500000 + 0.866025i 0
751.5 0.512762 − 0.296043i −0.500000 − 0.866025i −0.824717 + 1.42845i 0 −0.512762 − 0.296043i −1.43536 − 0.828705i 2.16078i −0.500000 + 0.866025i 0
751.6 1.05932 − 0.611600i −0.500000 − 0.866025i −0.251890 + 0.436286i 0 −1.05932 − 0.611600i 1.72820 + 0.997778i 3.06263i −0.500000 + 0.866025i 0
751.7 1.63863 − 0.946062i −0.500000 − 0.866025i 0.790065 − 1.36843i 0 −1.63863 − 0.946062i 2.38324 + 1.37597i 0.794445i −0.500000 + 0.866025i 0
751.8 2.36265 − 1.36408i −0.500000 − 0.866025i 2.72141 − 4.71362i 0 −2.36265 − 1.36408i −3.23285 − 1.86649i − 9.39252i −0.500000 + 0.866025i 0
901.1 −2.26052 − 1.30511i −0.500000 + 0.866025i 2.40664 + 4.16842i 0 2.26052 − 1.30511i −0.645251 + 0.372536i − 7.34329i −0.500000 − 0.866025i 0
901.2 −1.74177 − 1.00561i −0.500000 + 0.866025i 1.02251 + 1.77105i 0 1.74177 − 1.00561i −1.92561 + 1.11175i − 0.0905636i −0.500000 − 0.866025i 0
901.3 −1.27022 − 0.733363i −0.500000 + 0.866025i 0.0756426 + 0.131017i 0 1.27022 − 0.733363i 3.75091 − 2.16559i 2.71156i −0.500000 − 0.866025i 0
901.4 −0.300844 − 0.173693i −0.500000 + 0.866025i −0.939662 − 1.62754i 0 0.300844 − 0.173693i −3.62329 + 2.09191i 1.34762i −0.500000 − 0.866025i 0
901.5 0.512762 + 0.296043i −0.500000 + 0.866025i −0.824717 − 1.42845i 0 −0.512762 + 0.296043i −1.43536 + 0.828705i − 2.16078i −0.500000 − 0.866025i 0
901.6 1.05932 + 0.611600i −0.500000 + 0.866025i −0.251890 − 0.436286i 0 −1.05932 + 0.611600i 1.72820 − 0.997778i − 3.06263i −0.500000 − 0.866025i 0
901.7 1.63863 + 0.946062i −0.500000 + 0.866025i 0.790065 + 1.36843i 0 −1.63863 + 0.946062i 2.38324 − 1.37597i − 0.794445i −0.500000 − 0.866025i 0
901.8 2.36265 + 1.36408i −0.500000 + 0.866025i 2.72141 + 4.71362i 0 −2.36265 + 1.36408i −3.23285 + 1.86649i 9.39252i −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bc.m 16
5.b even 2 1 975.2.bc.n 16
5.c odd 4 2 195.2.v.a ✓ 32
13.e even 6 1 inner 975.2.bc.m 16
15.e even 4 2 585.2.bf.c 32
65.l even 6 1 975.2.bc.n 16
65.r odd 12 2 195.2.v.a ✓ 32
195.bf even 12 2 585.2.bf.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.v.a ✓ 32 5.c odd 4 2
195.2.v.a ✓ 32 65.r odd 12 2
585.2.bf.c 32 15.e even 4 2
585.2.bf.c 32 195.bf even 12 2
975.2.bc.m 16 1.a even 1 1 trivial
975.2.bc.m 16 13.e even 6 1 inner
975.2.bc.n 16 5.b even 2 1
975.2.bc.n 16 65.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{16} - 13 T_{2}^{14} + 119 T_{2}^{12} + 6 T_{2}^{11} - 530 T_{2}^{10} - 18 T_{2}^{9} + 1710 T_{2}^{8} + \cdots + 100 \) Copy content Toggle raw display
\( T_{7}^{16} + 6 T_{7}^{15} - 17 T_{7}^{14} - 174 T_{7}^{13} + 276 T_{7}^{12} + 3792 T_{7}^{11} + \cdots + 1040400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 13 T^{14} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 6 T^{15} + \cdots + 1040400 \) Copy content Toggle raw display
$11$ \( T^{16} + 6 T^{15} + \cdots + 25600 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 10 T^{15} + \cdots + 3132900 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 396806400 \) Copy content Toggle raw display
$23$ \( T^{16} - 12 T^{15} + \cdots + 2663424 \) Copy content Toggle raw display
$29$ \( T^{16} - 12 T^{15} + \cdots + 11492100 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 221414400 \) Copy content Toggle raw display
$37$ \( T^{16} + 6 T^{15} + \cdots + 518400 \) Copy content Toggle raw display
$41$ \( T^{16} + 18 T^{15} + \cdots + 65577604 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 235904490000 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 107661376 \) Copy content Toggle raw display
$53$ \( (T^{8} - 20 T^{7} + \cdots - 1920)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 5717779456 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 17954660025 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 25929889779600 \) Copy content Toggle raw display
$71$ \( T^{16} - 6 T^{15} + \cdots + 25000000 \) Copy content Toggle raw display
$73$ \( T^{16} + 620 T^{14} + \cdots + 31640625 \) Copy content Toggle raw display
$79$ \( (T^{8} - 26 T^{7} + \cdots - 1909440)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 9285427840000 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 154952704 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 153648320400 \) Copy content Toggle raw display
show more
show less