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Properties

Label 975.2.n.o
Level $975$
Weight $2$
Character orbit 975.n
Analytic conductor $7.785$
Analytic rank $0$
Dimension $8$
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] = [975,2,Mod(749,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.n (of order \(4\), degree \(2\), not 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.619810816.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{4} - \beta_{3} + 1) q^{2} + (\beta_{5} - \beta_{2} + \beta_1) q^{3} + (\beta_{7} + \beta_{6} + \cdots - 2 \beta_{3}) q^{4}+ \cdots + (\beta_{7} - \beta_{5} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{4} - \beta_{3} + 1) q^{2} + (\beta_{5} - \beta_{2} + \beta_1) q^{3} + (\beta_{7} + \beta_{6} + \cdots - 2 \beta_{3}) q^{4}+ \cdots + ( - 2 \beta_{7} + \beta_{6} + 3 \beta_{4} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 2 q^{3} - 4 q^{6} + 14 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} + 2 q^{3} - 4 q^{6} + 14 q^{7} - 12 q^{8} + 4 q^{9} - 4 q^{11} + 2 q^{12} + 14 q^{13} - 8 q^{16} + 30 q^{18} + 2 q^{19} - 8 q^{21} + 34 q^{24} + 32 q^{26} - 10 q^{27} - 24 q^{28} - 12 q^{31} - 18 q^{32} - 18 q^{33} - 10 q^{34} + 28 q^{36} + 2 q^{37} + 20 q^{39} - 38 q^{41} - 34 q^{42} - 4 q^{43} - 10 q^{44} + 34 q^{46} - 22 q^{47} + 54 q^{48} + 26 q^{51} - 18 q^{52} + 28 q^{53} - 12 q^{54} - 76 q^{56} + 6 q^{57} + 18 q^{58} - 14 q^{59} - 8 q^{61} - 10 q^{63} + 18 q^{66} - 40 q^{68} + 10 q^{71} + 4 q^{72} - 10 q^{73} + 26 q^{76} + 2 q^{78} + 56 q^{79} + 8 q^{81} - 16 q^{82} - 38 q^{83} + 58 q^{84} + 34 q^{86} + 34 q^{87} + 8 q^{88} + 28 q^{89} + 8 q^{91} + 72 q^{92} + 22 q^{93} - 20 q^{94} + 72 q^{96} + 4 q^{97} - 26 q^{98} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\nu^{7} + 4\nu^{6} + \nu^{5} + 48\nu^{4} + 236\nu^{3} - 69\nu^{2} - 65\nu + 574 ) / 319 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 59\nu + 63 ) / 319 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1582\nu + 268 ) / 319 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 126\nu^{7} - 128\nu^{6} - 32\nu^{5} - 260\nu^{4} + 2018\nu^{3} - 2577\nu^{2} + 804\nu + 134 ) / 319 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -255\nu^{7} + 16\nu^{6} + 4\nu^{5} + 511\nu^{4} - 3522\nu^{3} + 2276\nu^{2} - 579\nu - 575 ) / 319 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{4} + 4\beta_{2} + \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - 4\beta_{5} + 6\beta_{3} - \beta_{2} - 15\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} - 15\beta_{6} + \beta_{5} + 7\beta_{4} - 28\beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{7} + 8\beta_{6} - 43\beta_{4} + 30\beta_{3} + 8\beta_{2} - 30 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
−0.252709 + 0.252709i
−1.49094 + 1.49094i
1.18254 − 1.18254i
0.561103 − 0.561103i
−0.252709 − 0.252709i
−1.49094 − 1.49094i
1.18254 + 1.18254i
0.561103 + 0.561103i
−0.978559 − 0.978559i −0.146426 − 1.72585i − 0.0848427i 0 −1.54556 + 1.83213i 2.39913 + 2.39913i −2.04014 + 2.04014i −2.95712 + 0.505418i 0
749.2 0.664640 + 0.664640i 1.29021 + 1.15558i − 1.11651i 0 0.0894818 + 1.62557i 2.20073 + 2.20073i 2.07136 − 2.07136i 0.329281 + 2.98187i 0
749.3 1.42282 + 1.42282i 1.55654 − 0.759725i 2.04882i 0 3.29562 + 1.13372i −0.739083 − 0.739083i −0.0694623 + 0.0694623i 1.84564 − 2.36509i 0
749.4 1.89110 + 1.89110i −1.70032 + 0.329998i 5.15253i 0 −3.83954 − 2.59143i 3.13922 + 3.13922i −5.96175 + 5.96175i 2.78220 − 1.12221i 0
824.1 −0.978559 + 0.978559i −0.146426 + 1.72585i 0.0848427i 0 −1.54556 − 1.83213i 2.39913 − 2.39913i −2.04014 − 2.04014i −2.95712 − 0.505418i 0
824.2 0.664640 − 0.664640i 1.29021 − 1.15558i 1.11651i 0 0.0894818 − 1.62557i 2.20073 − 2.20073i 2.07136 + 2.07136i 0.329281 − 2.98187i 0
824.3 1.42282 − 1.42282i 1.55654 + 0.759725i − 2.04882i 0 3.29562 − 1.13372i −0.739083 + 0.739083i −0.0694623 − 0.0694623i 1.84564 + 2.36509i 0
824.4 1.89110 − 1.89110i −1.70032 − 0.329998i − 5.15253i 0 −3.83954 + 2.59143i 3.13922 − 3.13922i −5.96175 − 5.96175i 2.78220 + 1.12221i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
195.n even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.n.o 8
3.b odd 2 1 975.2.n.n 8
5.b even 2 1 975.2.n.m 8
5.c odd 4 1 975.2.o.l ✓ 8
5.c odd 4 1 975.2.o.o yes 8
13.d odd 4 1 975.2.n.p 8
15.d odd 2 1 975.2.n.p 8
15.e even 4 1 975.2.o.m yes 8
15.e even 4 1 975.2.o.n yes 8
39.f even 4 1 975.2.n.m 8
65.f even 4 1 975.2.o.m yes 8
65.g odd 4 1 975.2.n.n 8
65.k even 4 1 975.2.o.n yes 8
195.j odd 4 1 975.2.o.l ✓ 8
195.n even 4 1 inner 975.2.n.o 8
195.u odd 4 1 975.2.o.o yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.n.m 8 5.b even 2 1
975.2.n.m 8 39.f even 4 1
975.2.n.n 8 3.b odd 2 1
975.2.n.n 8 65.g odd 4 1
975.2.n.o 8 1.a even 1 1 trivial
975.2.n.o 8 195.n even 4 1 inner
975.2.n.p 8 13.d odd 4 1
975.2.n.p 8 15.d odd 2 1
975.2.o.l ✓ 8 5.c odd 4 1
975.2.o.l ✓ 8 195.j odd 4 1
975.2.o.m yes 8 15.e even 4 1
975.2.o.m yes 8 65.f even 4 1
975.2.o.n yes 8 15.e even 4 1
975.2.o.n yes 8 65.k even 4 1
975.2.o.o yes 8 5.c odd 4 1
975.2.o.o yes 8 195.u odd 4 1

Hecke kernels

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

\( T_{2}^{8} - 6T_{2}^{7} + 18T_{2}^{6} - 24T_{2}^{5} + 18T_{2}^{4} - 18T_{2}^{3} + 72T_{2}^{2} - 84T_{2} + 49 \) Copy content Toggle raw display
\( T_{7}^{8} - 14T_{7}^{7} + 98T_{7}^{6} - 384T_{7}^{5} + 882T_{7}^{4} - 910T_{7}^{3} + 32T_{7}^{2} + 392T_{7} + 2401 \) Copy content Toggle raw display
\( T_{11}^{8} + 4T_{11}^{7} + 8T_{11}^{6} - 54T_{11}^{5} + 134T_{11}^{4} - 92T_{11}^{3} + 18T_{11}^{2} + 30T_{11} + 25 \) Copy content Toggle raw display
\( T_{37}^{8} - 2T_{37}^{7} + 2T_{37}^{6} + 12T_{37}^{5} + 68T_{37}^{4} - 68T_{37}^{3} + 72T_{37}^{2} + 456T_{37} + 1444 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 14 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 68 T^{6} + \cdots + 2209 \) Copy content Toggle raw display
$19$ \( T^{8} - 2 T^{7} + \cdots + 1444 \) Copy content Toggle raw display
$23$ \( T^{8} + 92 T^{6} + \cdots + 98596 \) Copy content Toggle raw display
$29$ \( T^{8} + 116 T^{6} + \cdots + 133225 \) Copy content Toggle raw display
$31$ \( T^{8} + 12 T^{7} + \cdots + 10609 \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{7} + \cdots + 1444 \) Copy content Toggle raw display
$41$ \( T^{8} + 38 T^{7} + \cdots + 206116 \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} + \cdots + 218)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 22 T^{7} + \cdots + 22801 \) Copy content Toggle raw display
$53$ \( (T^{4} - 14 T^{3} + \cdots - 8795)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 14 T^{7} + \cdots + 5340721 \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots + 1801)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 66 T^{5} + \cdots + 64529089 \) Copy content Toggle raw display
$71$ \( T^{8} - 10 T^{7} + \cdots + 925444 \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{7} + \cdots + 198916 \) Copy content Toggle raw display
$79$ \( (T^{4} - 28 T^{3} + \cdots - 2654)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 38 T^{7} + \cdots + 375769 \) Copy content Toggle raw display
$89$ \( T^{8} - 28 T^{7} + \cdots + 234256 \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 17707264 \) Copy content Toggle raw display
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