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Properties

Label 975.2.bu.e
Level $975$
Weight $2$
Character orbit 975.bu
Analytic conductor $7.785$
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] = [975,2,Mod(7,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bu (of order \(12\), degree \(4\), 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: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 52 x^{14} + 1743 x^{12} - 34996 x^{10} + 513409 x^{8} - 5039424 x^{6} + 36142848 x^{4} + \cdots + 429981696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{14} - \beta_{10} + \beta_{6}) q^{2} - \beta_{14} q^{3} + (\beta_{5} - 2 \beta_{4}) q^{4} + ( - \beta_{5} + \beta_{4} - 1) q^{6} + (\beta_{15} - \beta_{14} + \cdots - \beta_{6}) q^{7}+ \cdots + \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{14} - \beta_{10} + \beta_{6}) q^{2} - \beta_{14} q^{3} + (\beta_{5} - 2 \beta_{4}) q^{4} + ( - \beta_{5} + \beta_{4} - 1) q^{6} + (\beta_{15} - \beta_{14} + \cdots - \beta_{6}) q^{7}+ \cdots + ( - \beta_{12} - 2 \beta_{5} + \cdots - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{6} - 12 q^{11} + 8 q^{16} + 32 q^{19} - 4 q^{21} + 16 q^{24} + 8 q^{26} + 24 q^{29} - 48 q^{31} - 8 q^{34} + 24 q^{36} - 8 q^{39} - 36 q^{44} - 12 q^{46} - 48 q^{49} - 16 q^{54} + 12 q^{56} + 24 q^{59} - 24 q^{61} + 64 q^{64} + 24 q^{66} - 12 q^{69} + 16 q^{71} + 72 q^{74} + 24 q^{76} + 8 q^{81} + 12 q^{84} - 68 q^{86} + 24 q^{89} + 200 q^{91} - 12 q^{94} - 72 q^{96} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 52 x^{14} + 1743 x^{12} - 34996 x^{10} + 513409 x^{8} - 5039424 x^{6} + 36142848 x^{4} + \cdots + 429981696 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 80413 \nu^{14} - 46629940 \nu^{12} + 1838431923 \nu^{10} - 50676934420 \nu^{8} + \cdots - 179682282934272 ) / 8512019923968 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5414461 \nu^{14} + 253287040 \nu^{12} - 8226683907 \nu^{10} + 151061140360 \nu^{8} + \cdots + 283912077321216 ) / 331968777034752 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 160465 \nu^{14} - 9422452 \nu^{12} + 318819327 \nu^{10} - 6769384180 \nu^{8} + \cdots - 18371762233344 ) / 2491322904576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45434687 \nu^{14} + 1698296444 \nu^{12} - 48347848497 \nu^{10} + 603811550108 \nu^{8} + \cdots - 22\!\cdots\!08 ) / 663937554069504 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39162473 \nu^{15} - 1938838876 \nu^{13} + 95049841497 \nu^{11} + \cdots - 16\!\cdots\!28 \nu ) / 79\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10467013 \nu^{14} + 1009763092 \nu^{12} - 36707393931 \nu^{10} + 871133650036 \nu^{8} + \cdots + 16\!\cdots\!96 ) / 110656259011584 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 98489365 \nu^{14} + 3008415220 \nu^{12} - 62956091835 \nu^{10} + 117956059156 \nu^{8} + \cdots - 72\!\cdots\!72 ) / 442625036046336 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5414461 \nu^{15} - 253287040 \nu^{13} + 8226683907 \nu^{11} - 151061140360 \nu^{9} + \cdots - 615880854355968 \nu ) / 331968777034752 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 98489365 \nu^{15} + 3008415220 \nu^{13} - 62956091835 \nu^{11} + \cdots - 72\!\cdots\!72 \nu ) / 53\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7743625 \nu^{15} + 526561972 \nu^{13} - 18131157063 \nu^{11} + \cdots + 650114397499392 \nu ) / 408576956350464 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13157803 \nu^{14} + 542913208 \nu^{12} - 15759486069 \nu^{10} + 238872573184 \nu^{8} + \cdots + 75811014153216 ) / 25536059771904 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 243622015 \nu^{15} - 17080252444 \nu^{13} + 616766713329 \nu^{11} + \cdots - 46\!\cdots\!80 \nu ) / 53\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 860602975 \nu^{15} - 36650246620 \nu^{13} + 1086119673873 \nu^{11} + \cdots - 92\!\cdots\!68 \nu ) / 15\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 232913 \nu^{15} + 7349828 \nu^{13} - 193353663 \nu^{11} + 1553108516 \nu^{9} + \cdots - 17555843598336 \nu ) / 3175468572672 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{8} + \beta_{4} - 14\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - 12\beta_{14} + 2\beta_{13} - 14\beta_{10} + 14\beta_{9} + \beta_{6} + 14\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 27\beta_{12} - 28\beta_{8} - 27\beta_{7} + 12\beta_{5} + 40\beta_{4} - 223\beta_{3} - 28\beta_{2} - 195 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 40\beta_{15} - 324\beta_{14} + 40\beta_{13} - 324\beta_{11} - 376\beta_{10} + 223\beta_{9} - 376\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -52\beta_{12} - 52\beta_{8} - 639\beta_{7} + 960\beta_{5} - 428\beta_{4} - 587\beta_{2} - 3315 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2238\beta_{15} + 624\beta_{14} - 1119\beta_{13} - 7668\beta_{11} + 495\beta_{10} - 9282\beta_{6} - 3902\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13808 \beta_{12} + 12065 \beta_{8} - 1743 \beta_{7} + 13428 \beta_{5} - 38921 \beta_{4} + 72697 \beta_{3} + \cdots - 1743 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 27236 \beta_{15} + 165696 \beta_{14} - 54472 \beta_{13} - 20916 \beta_{11} + 199252 \beta_{10} + \cdots - 72697 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 244713 \beta_{12} + 292865 \beta_{8} + 244713 \beta_{7} - 326832 \beta_{5} - 619697 \beta_{4} + \cdots + 1117077 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 619697 \beta_{15} + 2936556 \beta_{14} - 619697 \beta_{13} + 2936556 \beta_{11} + \cdots + 4134077 \beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1197521 \beta_{12} + 1197521 \beta_{8} + 6163716 \beta_{7} - 14872728 \beta_{5} + 6238843 \beta_{4} + \cdots + 23083020 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 27200160 \beta_{15} - 14370252 \beta_{14} + 13600080 \beta_{13} + 73964592 \beta_{11} + \cdots + 28049215 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 129213967 \beta_{12} - 101243635 \beta_{8} + 27970332 \beta_{7} - 163200960 \beta_{5} + 427645555 \beta_{4} + \cdots + 27970332 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 292414927 \beta_{15} - 1550567604 \beta_{14} + 584829854 \beta_{13} + 335643984 \beta_{11} + \cdots + 567048182 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.27793 + 1.31516i
−3.95096 − 2.28109i
3.95096 + 2.28109i
−2.27793 − 1.31516i
−3.23251 + 1.86629i
2.78422 − 1.60747i
−2.78422 + 1.60747i
3.23251 − 1.86629i
2.27793 − 1.31516i
−3.95096 + 2.28109i
3.95096 − 2.28109i
−2.27793 + 1.31516i
−3.23251 − 1.86629i
2.78422 + 1.60747i
−2.78422 − 1.60747i
3.23251 + 1.86629i
−0.448288 − 0.258819i −0.258819 − 0.965926i −0.866025 − 1.50000i 0 −0.133975 + 0.500000i −2.28109 − 3.95096i 1.93185i −0.866025 + 0.500000i 0
7.2 −0.448288 − 0.258819i −0.258819 − 0.965926i −0.866025 − 1.50000i 0 −0.133975 + 0.500000i 1.31516 + 2.27793i 1.93185i −0.866025 + 0.500000i 0
7.3 0.448288 + 0.258819i 0.258819 + 0.965926i −0.866025 − 1.50000i 0 −0.133975 + 0.500000i −1.31516 − 2.27793i − 1.93185i −0.866025 + 0.500000i 0
7.4 0.448288 + 0.258819i 0.258819 + 0.965926i −0.866025 − 1.50000i 0 −0.133975 + 0.500000i 2.28109 + 3.95096i − 1.93185i −0.866025 + 0.500000i 0
232.1 −1.67303 + 0.965926i 0.965926 + 0.258819i 0.866025 − 1.50000i 0 −1.86603 + 0.500000i −1.60747 + 2.78422i − 0.517638i 0.866025 + 0.500000i 0
232.2 −1.67303 + 0.965926i 0.965926 + 0.258819i 0.866025 − 1.50000i 0 −1.86603 + 0.500000i 1.86629 − 3.23251i − 0.517638i 0.866025 + 0.500000i 0
232.3 1.67303 − 0.965926i −0.965926 − 0.258819i 0.866025 − 1.50000i 0 −1.86603 + 0.500000i −1.86629 + 3.23251i 0.517638i 0.866025 + 0.500000i 0
232.4 1.67303 − 0.965926i −0.965926 − 0.258819i 0.866025 − 1.50000i 0 −1.86603 + 0.500000i 1.60747 − 2.78422i 0.517638i 0.866025 + 0.500000i 0
418.1 −0.448288 + 0.258819i −0.258819 + 0.965926i −0.866025 + 1.50000i 0 −0.133975 − 0.500000i −2.28109 + 3.95096i − 1.93185i −0.866025 − 0.500000i 0
418.2 −0.448288 + 0.258819i −0.258819 + 0.965926i −0.866025 + 1.50000i 0 −0.133975 − 0.500000i 1.31516 − 2.27793i − 1.93185i −0.866025 − 0.500000i 0
418.3 0.448288 − 0.258819i 0.258819 − 0.965926i −0.866025 + 1.50000i 0 −0.133975 − 0.500000i −1.31516 + 2.27793i 1.93185i −0.866025 − 0.500000i 0
418.4 0.448288 − 0.258819i 0.258819 − 0.965926i −0.866025 + 1.50000i 0 −0.133975 − 0.500000i 2.28109 − 3.95096i 1.93185i −0.866025 − 0.500000i 0
643.1 −1.67303 − 0.965926i 0.965926 − 0.258819i 0.866025 + 1.50000i 0 −1.86603 − 0.500000i −1.60747 − 2.78422i 0.517638i 0.866025 − 0.500000i 0
643.2 −1.67303 − 0.965926i 0.965926 − 0.258819i 0.866025 + 1.50000i 0 −1.86603 − 0.500000i 1.86629 + 3.23251i 0.517638i 0.866025 − 0.500000i 0
643.3 1.67303 + 0.965926i −0.965926 + 0.258819i 0.866025 + 1.50000i 0 −1.86603 − 0.500000i −1.86629 − 3.23251i − 0.517638i 0.866025 − 0.500000i 0
643.4 1.67303 + 0.965926i −0.965926 + 0.258819i 0.866025 + 1.50000i 0 −1.86603 − 0.500000i 1.60747 + 2.78422i − 0.517638i 0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.o even 12 1 inner
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bu.e yes 16
5.b even 2 1 inner 975.2.bu.e yes 16
5.c odd 4 2 975.2.bl.g ✓ 16
13.f odd 12 1 975.2.bl.g ✓ 16
65.o even 12 1 inner 975.2.bu.e yes 16
65.s odd 12 1 975.2.bl.g ✓ 16
65.t even 12 1 inner 975.2.bu.e yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bl.g ✓ 16 5.c odd 4 2
975.2.bl.g ✓ 16 13.f odd 12 1
975.2.bl.g ✓ 16 65.s odd 12 1
975.2.bu.e yes 16 1.a even 1 1 trivial
975.2.bu.e yes 16 5.b even 2 1 inner
975.2.bu.e yes 16 65.o even 12 1 inner
975.2.bu.e yes 16 65.t even 12 1 inner

Hecke kernels

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

\( T_{2}^{8} - 4T_{2}^{6} + 15T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{16} + 52 T_{7}^{14} + 1743 T_{7}^{12} + 34996 T_{7}^{10} + 513409 T_{7}^{8} + 5039424 T_{7}^{6} + \cdots + 429981696 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 429981696 \) Copy content Toggle raw display
$11$ \( (T^{8} + 6 T^{7} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 110075314176 \) Copy content Toggle raw display
$19$ \( (T^{8} - 16 T^{7} + \cdots + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 9721171216 \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} + 14 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + \cdots + 36)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 454371856 \) Copy content Toggle raw display
$41$ \( (T^{8} + 12 T^{6} + \cdots + 3564544)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 196741925136 \) Copy content Toggle raw display
$47$ \( (T^{8} - 212 T^{6} + \cdots + 222784)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 116319195136 \) Copy content Toggle raw display
$59$ \( (T^{4} - 6 T^{3} + 90 T^{2} + \cdots + 36)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + 12 T^{7} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 64 T^{14} + \cdots + 37015056 \) Copy content Toggle raw display
$71$ \( (T^{8} - 8 T^{7} + \cdots + 183184)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 156 T^{2} + 4761)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 466 T^{6} + \cdots + 106543684)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 128 T^{6} + \cdots + 98596)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 12 T^{7} + \cdots + 524176)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 32193413557776 \) Copy content Toggle raw display
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