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Properties

Label 95.3.c.a
Level $95$
Weight $3$
Character orbit 95.c
Analytic conductor $2.589$
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] = [95,3,Mod(56,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.56");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 95.c (of order \(2\), degree \(1\), 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: \(2.58856251142\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} - 1) q^{4} + \beta_{5} q^{5} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2) q^{6} + (\beta_{7} - \beta_{5} + 2) q^{7} + ( - \beta_{6} + \beta_{4}) q^{8} + ( - \beta_{10} - \beta_{5} - \beta_{2} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} - 1) q^{4} + \beta_{5} q^{5} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2) q^{6} + (\beta_{7} - \beta_{5} + 2) q^{7} + ( - \beta_{6} + \beta_{4}) q^{8} + ( - \beta_{10} - \beta_{5} - \beta_{2} - 4) q^{9} + \beta_{6} q^{10} + (\beta_{10} - \beta_{7} - 2 \beta_{3} + \cdots + 3) q^{11}+ \cdots + ( - 5 \beta_{10} + 5 \beta_{7} + \cdots - 29) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 28 q^{6} + 20 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 28 q^{6} + 20 q^{7} - 48 q^{9} + 32 q^{11} - 44 q^{16} - 44 q^{17} + 8 q^{19} + 36 q^{23} + 100 q^{24} + 60 q^{25} + 108 q^{26} - 36 q^{28} - 40 q^{30} - 40 q^{35} - 80 q^{36} - 44 q^{38} + 76 q^{39} + 100 q^{42} + 320 q^{43} - 256 q^{44} - 40 q^{45} - 56 q^{47} + 72 q^{49} - 76 q^{54} + 60 q^{57} + 68 q^{58} - 296 q^{61} - 376 q^{62} - 96 q^{63} + 188 q^{64} + 152 q^{66} - 340 q^{68} - 244 q^{73} + 136 q^{74} + 248 q^{76} - 200 q^{77} + 200 q^{80} - 372 q^{81} + 424 q^{82} - 160 q^{83} + 160 q^{85} + 444 q^{87} + 716 q^{92} + 296 q^{93} - 80 q^{95} - 44 q^{96} - 312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{10} - 435\nu^{8} - 5828\nu^{6} - 30456\nu^{4} - 49397\nu^{2} - 1209 ) / 2672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{11} + 685\nu^{9} + 6444\nu^{7} + 24280\nu^{5} + 34535\nu^{3} + 28071\nu ) / 2672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\nu^{10} + 685\nu^{8} + 6444\nu^{6} + 24280\nu^{4} + 31863\nu^{2} + 6695 ) / 2672 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\nu^{11} + 685\nu^{9} + 6444\nu^{7} + 24280\nu^{5} + 31863\nu^{3} + 6695\nu ) / 2672 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -49\nu^{10} - 1209\nu^{8} - 10172\nu^{6} - 35832\nu^{4} - 52191\nu^{2} - 17531 ) / 2672 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\nu^{11} + 697\nu^{9} + 7692\nu^{7} + 37568\nu^{5} + 76929\nu^{3} + 46707\nu ) / 1336 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\nu^{11} + 405\nu^{9} + 3376\nu^{7} + 9928\nu^{5} + 2864\nu^{3} - 15321\nu ) / 668 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -89\nu^{10} - 2305\nu^{8} - 19948\nu^{6} - 63992\nu^{4} - 48663\nu^{2} + 14509 ) / 2672 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -127\nu^{11} - 3747\nu^{9} - 40324\nu^{7} - 194952\nu^{5} - 412177\nu^{3} - 281705\nu ) / 2672 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{4} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + 4\beta_{5} + \beta_{3} - 11\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - \beta_{9} - 3\beta_{8} + 16\beta_{6} - 13\beta_{4} + 73\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{10} - 5\beta_{7} - 72\beta_{5} - 20\beta_{3} + 118\beta_{2} - 370 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15\beta_{11} + 20\beta_{9} + 50\beta_{8} - 215\beta_{6} + 148\beta_{4} - 724\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 188\beta_{10} + 112\beta_{7} + 1012\beta_{5} + 280\beta_{3} - 1280\beta_{2} + 3705 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -168\beta_{11} - 300\beta_{9} - 616\beta_{8} + 2704\beta_{6} - 1656\beta_{4} + 7565\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2256\beta_{10} - 1780\beta_{7} - 12948\beta_{5} - 3488\beta_{3} + 14065\beta_{2} - 38889 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1708\beta_{11} + 4036\beta_{9} + 6904\beta_{8} - 32829\beta_{6} + 18577\beta_{4} - 81632\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1
− 3.38208i
− 2.84623i
− 2.36559i
− 1.88109i
− 1.14149i
− 0.151673i
0.151673i
1.14149i
1.88109i
2.36559i
2.84623i
3.38208i
− 3.38208i − 4.08851i −7.43849 2.23607 −13.8277 4.56923 11.6293i −7.71590 − 7.56257i
56.2 − 2.84623i − 2.18419i −4.10101 −2.23607 −6.21669 −9.15076 0.287487i 4.22932 6.36436i
56.3 − 2.36559i 3.55563i −1.59600 −2.23607 8.41115 11.0785 − 5.68687i −3.64250 5.28961i
56.4 − 1.88109i 0.840697i 0.461500 2.23607 1.58143 2.31342 − 8.39248i 8.29323 − 4.20625i
56.5 − 1.14149i − 4.13699i 2.69701 −2.23607 −4.72232 7.54444 − 7.64455i −8.11468 2.55244i
56.6 − 0.151673i 5.10387i 3.97700 2.23607 0.774118 −6.35478 − 1.20989i −17.0495 − 0.339151i
56.7 0.151673i − 5.10387i 3.97700 2.23607 0.774118 −6.35478 1.20989i −17.0495 0.339151i
56.8 1.14149i 4.13699i 2.69701 −2.23607 −4.72232 7.54444 7.64455i −8.11468 − 2.55244i
56.9 1.88109i − 0.840697i 0.461500 2.23607 1.58143 2.31342 8.39248i 8.29323 4.20625i
56.10 2.36559i − 3.55563i −1.59600 −2.23607 8.41115 11.0785 5.68687i −3.64250 − 5.28961i
56.11 2.84623i 2.18419i −4.10101 −2.23607 −6.21669 −9.15076 − 0.287487i 4.22932 − 6.36436i
56.12 3.38208i 4.08851i −7.43849 2.23607 −13.8277 4.56923 − 11.6293i −7.71590 7.56257i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.3.c.a ✓ 12
3.b odd 2 1 855.3.e.a 12
4.b odd 2 1 1520.3.h.a 12
5.b even 2 1 475.3.c.g 12
5.c odd 4 2 475.3.d.c 24
19.b odd 2 1 inner 95.3.c.a ✓ 12
57.d even 2 1 855.3.e.a 12
76.d even 2 1 1520.3.h.a 12
95.d odd 2 1 475.3.c.g 12
95.g even 4 2 475.3.d.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.3.c.a ✓ 12 1.a even 1 1 trivial
95.3.c.a ✓ 12 19.b odd 2 1 inner
475.3.c.g 12 5.b even 2 1
475.3.c.g 12 95.d odd 2 1
475.3.d.c 24 5.c odd 4 2
475.3.d.c 24 95.g even 4 2
855.3.e.a 12 3.b odd 2 1
855.3.e.a 12 57.d even 2 1
1520.3.h.a 12 4.b odd 2 1
1520.3.h.a 12 76.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(95, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 30 T^{10} + \cdots + 55 \) Copy content Toggle raw display
$3$ \( T^{12} + 78 T^{10} + \cdots + 317680 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 10 T^{5} + \cdots + 51376)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 16 T^{5} + \cdots + 1609984)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1391784557680 \) Copy content Toggle raw display
$17$ \( (T^{6} + 22 T^{5} + \cdots + 1301904)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{6} - 18 T^{5} + \cdots - 97445584)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 63\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 19\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( (T^{6} - 160 T^{5} + \cdots + 10954816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 28 T^{5} + \cdots - 253813824)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 41\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 89\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( (T^{6} + 148 T^{5} + \cdots - 76754624)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 86\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( (T^{6} + 122 T^{5} + \cdots + 5877225616)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( (T^{6} + 80 T^{5} + \cdots + 5457804544)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 52\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
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