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Properties

Label 8464.2.a.ch
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8464,2,Mod(1,8464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8464, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8464.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

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: yes
Analytic conductor: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{10} - \beta_{5} - \beta_1) q^{5} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{13} - \beta_{9} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{10} - \beta_{5} - \beta_1) q^{5} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{7}+ \cdots + (3 \beta_{14} - 3 \beta_{13} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 615851531698 \nu^{14} - 271143233261 \nu^{13} + 18085178921598 \nu^{12} + \cdots - 203857349469810 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1079221482331 \nu^{14} - 519404024256 \nu^{13} + 31662129440180 \nu^{12} + \cdots - 308470260318530 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1182522700597 \nu^{14} + 364745185307 \nu^{13} - 34492031163809 \nu^{12} + \cdots + 261422455058242 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1870920828717 \nu^{14} + 928061342350 \nu^{13} - 55080793513950 \nu^{12} + \cdots + 566932887627147 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1968913316011 \nu^{14} + 789002277665 \nu^{13} - 57748971108638 \nu^{12} + \cdots + 543559168048149 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2724000474053 \nu^{14} + 1225644969138 \nu^{13} - 79912157870857 \nu^{12} + \cdots + 816383999692376 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2757915593676 \nu^{14} - 1318428371692 \nu^{13} + 81218002403666 \nu^{12} + \cdots - 824974679408609 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2870423960860 \nu^{14} - 1288452326477 \nu^{13} + 84394542781667 \nu^{12} + \cdots - 803219842343713 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2873234287466 \nu^{14} - 1338545904423 \nu^{13} + 84449706694623 \nu^{12} + \cdots - 858385706392526 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3448560973112 \nu^{14} + 1529169221715 \nu^{13} - 101136267853122 \nu^{12} + \cdots + 948171466654919 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3493721110566 \nu^{14} + 1763647857787 \nu^{13} - 102668560860953 \nu^{12} + \cdots + 11\!\cdots\!29 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3949645443191 \nu^{14} + 1807856350733 \nu^{13} - 116056672221847 \nu^{12} + \cdots + 11\!\cdots\!07 ) / 7416523991491 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5092346617153 \nu^{14} + 2478162887537 \nu^{13} - 149600458824963 \nu^{12} + \cdots + 15\!\cdots\!18 ) / 7416523991491 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} - \beta_{9} - \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{13} - \beta_{11} - 9\beta_{9} - 3\beta_{7} - 2\beta_{6} + 2\beta_{4} - 13\beta_{3} - 8\beta_{2} + \beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 26 \beta_{14} - 13 \beta_{13} - 17 \beta_{12} + \beta_{11} + 8 \beta_{10} + \beta_{9} - 15 \beta_{8} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 62 \beta_{13} - 4 \beta_{12} - 15 \beta_{11} - 7 \beta_{10} - 79 \beta_{9} - 47 \beta_{7} + \cdots + 262 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 283 \beta_{14} - 133 \beta_{13} - 216 \beta_{12} + 15 \beta_{11} + 57 \beta_{10} + 23 \beta_{9} + \cdots + 129 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9 \beta_{14} - 506 \beta_{13} - 76 \beta_{12} - 188 \beta_{11} - 123 \beta_{10} - 724 \beta_{9} + \cdots + 2426 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2931 \beta_{14} - 1298 \beta_{13} - 2453 \beta_{12} + 170 \beta_{11} + 411 \beta_{10} + 319 \beta_{9} + \cdots + 1321 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 190 \beta_{14} - 4359 \beta_{13} - 1016 \beta_{12} - 2152 \beta_{11} - 1543 \beta_{10} - 6850 \beta_{9} + \cdots + 23118 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 29753 \beta_{14} - 12595 \beta_{13} - 26346 \beta_{12} + 1767 \beta_{11} + 3088 \beta_{10} + \cdots + 13324 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2796 \beta_{14} - 39242 \beta_{13} - 11907 \beta_{12} - 23322 \beta_{11} - 17131 \beta_{10} + \cdots + 223772 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 299025 \beta_{14} - 122721 \beta_{13} - 274312 \beta_{12} + 17775 \beta_{11} + 24330 \beta_{10} + \cdots + 134252 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 35841 \beta_{14} - 365027 \beta_{13} - 131268 \beta_{12} - 244369 \beta_{11} - 180000 \beta_{10} + \cdots + 2185593 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.15594
2.54853
2.49141
1.98867
1.44882
1.44536
1.07060
−0.159078
−0.593309
−0.851979
−1.67059
−2.01779
−2.12777
−2.59403
−3.13479
0 −3.15594 0 0.0614634 0 2.55213 0 6.95994 0
1.2 0 −2.54853 0 −0.818556 0 2.13290 0 3.49500 0
1.3 0 −2.49141 0 −2.70623 0 −3.48198 0 3.20713 0
1.4 0 −1.98867 0 −3.85556 0 0.534429 0 0.954809 0
1.5 0 −1.44882 0 2.99678 0 −1.69224 0 −0.900926 0
1.6 0 −1.44536 0 2.02221 0 4.89726 0 −0.910921 0
1.7 0 −1.07060 0 3.54589 0 3.82947 0 −1.85382 0
1.8 0 0.159078 0 0.0374616 0 −3.17851 0 −2.97469 0
1.9 0 0.593309 0 −2.48229 0 −3.58382 0 −2.64799 0
1.10 0 0.851979 0 −3.06283 0 1.27761 0 −2.27413 0
1.11 0 1.67059 0 3.30085 0 1.48378 0 −0.209129 0
1.12 0 2.01779 0 0.684021 0 −2.16510 0 1.07147 0
1.13 0 2.12777 0 0.858783 0 3.76707 0 1.52739 0
1.14 0 2.59403 0 −3.58336 0 4.34127 0 3.72897 0
1.15 0 3.13479 0 3.00137 0 −0.714288 0 6.82690 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8464.2.a.ch 15
4.b odd 2 1 4232.2.a.ba 15
23.b odd 2 1 8464.2.a.cg 15
23.d odd 22 2 368.2.m.e 30
92.b even 2 1 4232.2.a.bb 15
92.h even 22 2 184.2.i.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.i.b ✓ 30 92.h even 22 2
368.2.m.e 30 23.d odd 22 2
4232.2.a.ba 15 4.b odd 2 1
4232.2.a.bb 15 92.b even 2 1
8464.2.a.cg 15 23.b odd 2 1
8464.2.a.ch 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):

\( T_{3}^{15} + T_{3}^{14} - 30 T_{3}^{13} - 28 T_{3}^{12} + 354 T_{3}^{11} + 302 T_{3}^{10} - 2111 T_{3}^{9} + \cdots + 419 \) Copy content Toggle raw display
\( T_{5}^{15} - 49 T_{5}^{13} + 8 T_{5}^{12} + 945 T_{5}^{11} - 296 T_{5}^{10} - 9000 T_{5}^{9} + 4085 T_{5}^{8} + \cdots - 67 \) Copy content Toggle raw display
\( T_{7}^{15} - 10 T_{7}^{14} - 15 T_{7}^{13} + 415 T_{7}^{12} - 503 T_{7}^{11} - 6368 T_{7}^{10} + \cdots - 175571 \) Copy content Toggle raw display
\( T_{13}^{15} - 104 T_{13}^{13} + 28 T_{13}^{12} + 3915 T_{13}^{11} - 673 T_{13}^{10} - 66033 T_{13}^{9} + \cdots + 871309 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + T^{14} + \cdots + 419 \) Copy content Toggle raw display
$5$ \( T^{15} - 49 T^{13} + \cdots - 67 \) Copy content Toggle raw display
$7$ \( T^{15} - 10 T^{14} + \cdots - 175571 \) Copy content Toggle raw display
$11$ \( T^{15} - 23 T^{14} + \cdots - 3257 \) Copy content Toggle raw display
$13$ \( T^{15} - 104 T^{13} + \cdots + 871309 \) Copy content Toggle raw display
$17$ \( T^{15} - 111 T^{13} + \cdots - 4570369 \) Copy content Toggle raw display
$19$ \( T^{15} - 29 T^{14} + \cdots + 11296 \) Copy content Toggle raw display
$23$ \( T^{15} \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 362817731 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 68503394189 \) Copy content Toggle raw display
$37$ \( T^{15} + 24 T^{14} + \cdots + 4028177 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 14398188331 \) Copy content Toggle raw display
$43$ \( T^{15} - 48 T^{14} + \cdots - 71260463 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 4016448512 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 137376723859 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 17476247953 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 13648427511181 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 32183473849 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 430038223 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 153653405267 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 4831892800571 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 387804979937 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 15846191483 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 107540873749 \) Copy content Toggle raw display
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