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Properties

Label 4012.2.a.h
Level $4012$
Weight $2$
Character orbit 4012.a
Self dual yes
Analytic conductor $32.036$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(1\)
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: \( x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} - \beta_{8} q^{5} + ( - \beta_{9} - 1) q^{7} + (\beta_{14} - \beta_{12} + \beta_{10} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} - \beta_{8} q^{5} + ( - \beta_{9} - 1) q^{7} + (\beta_{14} - \beta_{12} + \beta_{10} + 1) q^{9} + ( - \beta_{14} + \beta_{12} + \beta_{7} + \cdots - 1) q^{11}+ \cdots + (\beta_{14} - \beta_{13} - 2 \beta_{12} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1642110 \nu^{14} + 5493041 \nu^{13} + 44856541 \nu^{12} - 160648719 \nu^{11} + \cdots + 189641702 ) / 95351182 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2499271 \nu^{14} + 5394424 \nu^{13} - 96824570 \nu^{12} - 88043849 \nu^{11} + 1318014655 \nu^{10} + \cdots + 131157754 ) / 95351182 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6589235 \nu^{14} + 21534662 \nu^{13} + 149175107 \nu^{12} - 577550108 \nu^{11} + \cdots + 11524587 ) / 47675591 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29448859 \nu^{14} - 118865162 \nu^{13} - 583179274 \nu^{12} + 3043456985 \nu^{11} + \cdots + 129215570 ) / 95351182 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 36873004 \nu^{14} + 144992745 \nu^{13} + 729566385 \nu^{12} - 3708988537 \nu^{11} + \cdots - 133290618 ) / 95351182 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 40746102 \nu^{14} - 141104307 \nu^{13} - 875338947 \nu^{12} + 3685347851 \nu^{11} + \cdots - 338210566 ) / 95351182 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 95633725 \nu^{14} - 359609000 \nu^{13} - 1951460100 \nu^{12} + 9276232291 \nu^{11} + \cdots + 443495554 ) / 190702364 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 97204111 \nu^{14} + 408284470 \nu^{13} + 1824797366 \nu^{12} - 10322410111 \nu^{11} + \cdots - 167261742 ) / 190702364 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25842475 \nu^{14} + 101169385 \nu^{13} + 521083651 \nu^{12} - 2604048863 \nu^{11} + \cdots + 106478707 ) / 47675591 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 58215653 \nu^{14} + 202536060 \nu^{13} + 1239397782 \nu^{12} - 5282011523 \nu^{11} + \cdots + 74376318 ) / 95351182 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 121421599 \nu^{14} + 480687854 \nu^{13} + 2412644590 \nu^{12} - 12328562947 \nu^{11} + \cdots + 210174674 ) / 190702364 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 166794261 \nu^{14} + 629928074 \nu^{13} + 3417073022 \nu^{12} - 16278312837 \nu^{11} + \cdots - 204973042 ) / 190702364 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 169653351 \nu^{14} + 605403962 \nu^{13} + 3601868958 \nu^{12} - 15798177543 \nu^{11} + \cdots - 898759938 ) / 190702364 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{12} + \beta_{9} + \beta_{4} + \beta_{3} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{13} - \beta_{11} - 3\beta_{10} - \beta_{9} + 2\beta_{8} - 3\beta_{7} + 4\beta_{3} - 3\beta_{2} + 9\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11 \beta_{14} - 17 \beta_{13} - 9 \beta_{12} + 2 \beta_{11} - \beta_{10} + 15 \beta_{9} - 6 \beta_{8} + \cdots + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{14} + 64 \beta_{13} + 5 \beta_{12} - 21 \beta_{11} - 52 \beta_{10} - 28 \beta_{9} + \cdots - 120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 159 \beta_{14} - 293 \beta_{13} - 127 \beta_{12} + 44 \beta_{11} + 7 \beta_{10} + 244 \beta_{9} + \cdots + 726 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 251 \beta_{14} + 1195 \beta_{13} + 149 \beta_{12} - 368 \beta_{11} - 819 \beta_{10} - 593 \beta_{9} + \cdots - 2367 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2546 \beta_{14} - 5203 \beta_{13} - 2027 \beta_{12} + 893 \beta_{11} + 592 \beta_{10} + 4113 \beta_{9} + \cdots + 12440 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5510 \beta_{14} + 21747 \beta_{13} + 3494 \beta_{12} - 6292 \beta_{11} - 12997 \beta_{10} + \cdots - 44481 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 42402 \beta_{14} - 93352 \beta_{13} - 33426 \beta_{12} + 17563 \beta_{11} + 17148 \beta_{10} + \cdots + 218298 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 111435 \beta_{14} + 393291 \beta_{13} + 74135 \beta_{12} - 108079 \beta_{11} - 210704 \beta_{10} + \cdots - 821757 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 721157 \beta_{14} - 1680103 \beta_{13} - 561369 \beta_{12} + 336971 \beta_{11} + 394848 \beta_{10} + \cdots + 3867318 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2164334 \beta_{14} + 7100193 \beta_{13} + 1487194 \beta_{12} - 1874470 \beta_{11} - 3488558 \beta_{10} + \cdots - 15058170 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12443114 \beta_{14} - 30267026 \beta_{13} - 9571250 \beta_{12} + 6348517 \beta_{11} + 8244380 \beta_{10} + \cdots + 68870090 \) 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
1.28051
2.10632
−4.24628
−1.49960
−1.20072
2.23095
1.18786
−0.0375146
0.346366
−1.64466
3.62663
−0.829690
1.66914
0.186961
1.82373
0 −3.40908 0 1.59062 0 2.94577 0 8.62180 0
1.2 0 −2.74294 0 −0.560055 0 1.03063 0 4.52374 0
1.3 0 −2.34833 0 3.93023 0 −4.81217 0 2.51465 0
1.4 0 −2.30933 0 2.18242 0 −0.264202 0 2.33302 0
1.5 0 −2.21367 0 −3.70336 0 −3.08925 0 1.90033 0
1.6 0 −0.598056 0 −0.722695 0 −0.897746 0 −2.64233 0
1.7 0 −0.460784 0 −0.730164 0 3.58746 0 −2.78768 0
1.8 0 0.249278 0 −1.04042 0 −3.44893 0 −2.93786 0
1.9 0 0.915888 0 1.84475 0 −3.15913 0 −2.16115 0
1.10 0 1.48779 0 1.28557 0 0.0101628 0 −0.786472 0
1.11 0 1.57548 0 3.44709 0 −4.23195 0 −0.517858 0
1.12 0 1.84628 0 −2.98774 0 −0.920155 0 0.408737 0
1.13 0 2.17000 0 −3.20913 0 1.70252 0 1.70890 0
1.14 0 2.17219 0 −1.55586 0 3.67456 0 1.71840 0
1.15 0 2.66529 0 1.22875 0 −3.12758 0 4.10376 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
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

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 4012.2.a.h ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.h ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\( T_{3}^{15} + T_{3}^{14} - 30 T_{3}^{13} - 16 T_{3}^{12} + 366 T_{3}^{11} + 39 T_{3}^{10} - 2286 T_{3}^{9} + \cdots + 384 \) Copy content Toggle raw display
\( T_{5}^{15} - T_{5}^{14} - 39 T_{5}^{13} + 36 T_{5}^{12} + 562 T_{5}^{11} - 487 T_{5}^{10} - 3751 T_{5}^{9} + \cdots - 2328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + T^{14} + \cdots + 384 \) Copy content Toggle raw display
$5$ \( T^{15} - T^{14} + \cdots - 2328 \) Copy content Toggle raw display
$7$ \( T^{15} + 11 T^{14} + \cdots + 324 \) Copy content Toggle raw display
$11$ \( T^{15} + 12 T^{14} + \cdots + 7639278 \) Copy content Toggle raw display
$13$ \( T^{15} + 10 T^{14} + \cdots - 1585184 \) Copy content Toggle raw display
$17$ \( (T - 1)^{15} \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 347452256 \) Copy content Toggle raw display
$23$ \( T^{15} + 21 T^{14} + \cdots + 19250678 \) Copy content Toggle raw display
$29$ \( T^{15} - 23 T^{14} + \cdots + 62464408 \) Copy content Toggle raw display
$31$ \( T^{15} + 31 T^{14} + \cdots - 21026608 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 9635282176 \) Copy content Toggle raw display
$41$ \( T^{15} + 15 T^{14} + \cdots - 94183424 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 43780477024 \) Copy content Toggle raw display
$47$ \( T^{15} + 47 T^{14} + \cdots - 71415712 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 8473161344476 \) Copy content Toggle raw display
$59$ \( (T + 1)^{15} \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 156112752 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 3701877072 \) Copy content Toggle raw display
$71$ \( T^{15} + 13 T^{14} + \cdots - 4970808 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 433700088328 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 133136280092 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 19\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 1821158154000 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 334266043544 \) Copy content Toggle raw display
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