Properties

Label 2312.4.a.n
Level $2312$
Weight $4$
Character orbit 2312.a
Self dual yes
Analytic conductor $136.412$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2312,4,Mod(1,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2312.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2312.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: \(136.412415933\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 294 x^{16} - 14 x^{15} + 34371 x^{14} + 2670 x^{13} - 2054705 x^{12} - 160284 x^{11} + 67981059 x^{10} + 2824200 x^{9} - 1279285428 x^{8} + \cdots - 176969301147 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15}\cdot 17^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - \beta_{5} q^{5} + (\beta_{10} - \beta_{4} - 3) q^{7} + (\beta_{2} - 2 \beta_1 + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - \beta_{5} q^{5} + (\beta_{10} - \beta_{4} - 3) q^{7} + (\beta_{2} - 2 \beta_1 + 7) q^{9} + ( - \beta_{7} - \beta_{5} - \beta_{3} + \beta_1 - 7) q^{11} + ( - \beta_{16} - \beta_{15} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{4} - \beta_1 + 2) q^{13} + ( - \beta_{17} - \beta_{16} + \beta_{13} + \beta_{9} + 2 \beta_{7} + 2 \beta_{5} - \beta_1 + 5) q^{15} + (\beta_{17} + \beta_{15} + \beta_{12} + \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{4} + 3) q^{19} + (\beta_{17} + \beta_{16} - \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \cdots + 10) q^{21}+ \cdots + (5 \beta_{17} + 11 \beta_{16} - 7 \beta_{15} + \beta_{14} - 17 \beta_{13} - 35 \beta_{12} + \cdots - 86) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9} - 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} - 153 q^{23} + 306 q^{25} - 768 q^{27} - 51 q^{29} - 303 q^{31} + 525 q^{33} - 255 q^{35} - 717 q^{37} + 216 q^{39} + 393 q^{41} - 390 q^{43} - 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} - 810 q^{57} - 204 q^{59} - 534 q^{61} - 2556 q^{63} + 2127 q^{65} - 405 q^{67} + 2547 q^{69} + 426 q^{71} - 1149 q^{73} - 2226 q^{75} - 357 q^{77} - 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} - 6090 q^{91} + 606 q^{93} - 2109 q^{95} - 2349 q^{97} - 1428 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 294 x^{16} - 14 x^{15} + 34371 x^{14} + 2670 x^{13} - 2054705 x^{12} - 160284 x^{11} + 67981059 x^{10} + 2824200 x^{9} - 1279285428 x^{8} + \cdots - 176969301147 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\!\cdots\!61 \nu^{17} + \cdots - 12\!\cdots\!47 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!48 \nu^{17} + \cdots - 31\!\cdots\!33 ) / 34\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!64 \nu^{17} + \cdots - 10\!\cdots\!46 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!51 \nu^{17} + \cdots - 14\!\cdots\!35 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 80\!\cdots\!13 \nu^{17} + \cdots + 31\!\cdots\!74 ) / 34\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!39 \nu^{17} + \cdots + 25\!\cdots\!78 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 77\!\cdots\!29 \nu^{17} + \cdots + 36\!\cdots\!85 ) / 18\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 70\!\cdots\!39 \nu^{17} + \cdots - 70\!\cdots\!02 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 79\!\cdots\!65 \nu^{17} + \cdots + 32\!\cdots\!55 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!55 \nu^{17} + \cdots + 26\!\cdots\!63 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 20\!\cdots\!47 \nu^{17} + \cdots - 73\!\cdots\!02 ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!43 \nu^{17} + \cdots + 17\!\cdots\!12 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28\!\cdots\!65 \nu^{17} + \cdots - 11\!\cdots\!63 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 97\!\cdots\!54 \nu^{17} + \cdots + 32\!\cdots\!45 ) / 37\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 34\!\cdots\!27 \nu^{17} + \cdots - 13\!\cdots\!55 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 33 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{17} + 2 \beta_{16} + \beta_{14} + \beta_{13} + 3 \beta_{12} - \beta_{11} + 2 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{5} - 5 \beta_{4} - \beta_{3} + \beta_{2} + 61 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{17} + 5 \beta_{16} + 7 \beta_{15} + 4 \beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} - 17 \beta_{9} + \beta_{8} - 2 \beta_{7} - 8 \beta_{6} - 5 \beta_{5} - \beta_{4} + 4 \beta_{3} + 78 \beta_{2} + 14 \beta _1 + 1993 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 192 \beta_{17} + 168 \beta_{16} - 5 \beta_{15} + 96 \beta_{14} + 105 \beta_{13} + 272 \beta_{12} - 127 \beta_{11} + 162 \beta_{10} + 359 \beta_{9} - 130 \beta_{8} - 81 \beta_{7} + 28 \beta_{6} - 141 \beta_{5} - 709 \beta_{4} - 91 \beta_{3} + \cdots + 182 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 233 \beta_{17} + 498 \beta_{16} + 739 \beta_{15} + 401 \beta_{14} - 147 \beta_{13} + 15 \beta_{12} + 101 \beta_{11} - 326 \beta_{10} - 2157 \beta_{9} + 50 \beta_{8} - 547 \beta_{7} - 1294 \beta_{6} - 749 \beta_{5} - 552 \beta_{4} + \cdots + 142256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14975 \beta_{17} + 12437 \beta_{16} - 48 \beta_{15} + 7756 \beta_{14} + 9932 \beta_{13} + 21830 \beta_{12} - 12367 \beta_{11} + 11975 \beta_{10} + 28248 \beta_{9} - 7221 \beta_{8} + 906 \beta_{7} + 2521 \beta_{6} + \cdots + 21832 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 23274 \beta_{17} + 42900 \beta_{16} + 64589 \beta_{15} + 32547 \beta_{14} - 14961 \beta_{13} - 6263 \beta_{12} - 1957 \beta_{11} - 48099 \beta_{10} - 216279 \beta_{9} + 2346 \beta_{8} - 73784 \beta_{7} + \cdots + 10702118 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1117144 \beta_{17} + 901157 \beta_{16} + 43452 \beta_{15} + 609813 \beta_{14} + 912376 \beta_{13} + 1728511 \beta_{12} - 1112657 \beta_{11} + 870885 \beta_{10} + 2140680 \beta_{9} + \cdots + 1882304 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2175868 \beta_{17} + 3625093 \beta_{16} + 5424290 \beta_{15} + 2526761 \beta_{14} - 1326725 \beta_{13} - 1184239 \beta_{12} - 1095182 \beta_{11} - 5489384 \beta_{10} - 20018254 \beta_{9} + \cdots + 823162079 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 83250855 \beta_{17} + 65638449 \beta_{16} + 7600850 \beta_{15} + 47875251 \beta_{14} + 82195461 \beta_{13} + 137780716 \beta_{12} - 96581378 \beta_{11} + 62630406 \beta_{10} + \cdots + 145362022 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 193057620 \beta_{17} + 304713971 \beta_{16} + 451441853 \beta_{15} + 195608420 \beta_{14} - 109072487 \beta_{13} - 154904841 \beta_{12} - 166555830 \beta_{11} + \cdots + 64125253972 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6281255886 \beta_{17} + 4847116728 \beta_{16} + 934883629 \beta_{15} + 3770548980 \beta_{14} + 7296407469 \beta_{13} + 11090875364 \beta_{12} - 8217747447 \beta_{11} + \cdots + 11042666994 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 16491161343 \beta_{17} + 25533927354 \beta_{16} + 37519910045 \beta_{15} + 15287240859 \beta_{14} - 8509949307 \beta_{13} - 17528278769 \beta_{12} + \cdots + 5040883216059 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 481366765297 \beta_{17} + 363702220321 \beta_{16} + 100090475110 \beta_{15} + 298122962603 \beta_{14} + 640493018483 \beta_{13} + 900679638785 \beta_{12} + \cdots + 876973681704 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1374193229770 \beta_{17} + 2136893339083 \beta_{16} + 3119704695556 \beta_{15} + 1209922214297 \beta_{14} - 635938234544 \beta_{13} + \cdots + 399120882111311 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 37446869764524 \beta_{17} + 27723023855907 \beta_{16} + 9967011782631 \beta_{15} + 23666049873957 \beta_{14} + 55745904925983 \beta_{13} + \cdots + 75272308039426 \) Copy content Toggle raw display

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
−9.04549
−8.80003
−7.45229
−5.63767
−3.86015
−3.76043
−3.57628
−1.72316
−1.36423
1.40926
2.18564
2.65826
4.14696
4.29424
4.64908
8.25076
8.44447
9.18106
0 −10.0455 0 13.8476 0 −18.1085 0 73.9118 0
1.2 0 −9.80003 0 −6.52834 0 −26.6805 0 69.0406 0
1.3 0 −8.45229 0 −17.0781 0 29.7278 0 44.4413 0
1.4 0 −6.63767 0 3.97104 0 −24.2054 0 17.0586 0
1.5 0 −4.86015 0 1.88035 0 7.34878 0 −3.37898 0
1.6 0 −4.76043 0 −6.38717 0 20.9631 0 −4.33833 0
1.7 0 −4.57628 0 −17.7945 0 −35.0165 0 −6.05769 0
1.8 0 −2.72316 0 10.7272 0 20.9446 0 −19.5844 0
1.9 0 −2.36423 0 13.8182 0 9.64631 0 −21.4104 0
1.10 0 0.409262 0 15.5974 0 8.71140 0 −26.8325 0
1.11 0 1.18564 0 −19.4950 0 −3.40994 0 −25.5943 0
1.12 0 1.65826 0 −15.8272 0 −7.01175 0 −24.2502 0
1.13 0 3.14696 0 5.44286 0 25.0621 0 −17.0966 0
1.14 0 3.29424 0 18.7619 0 −35.6360 0 −16.1480 0
1.15 0 3.64908 0 3.63285 0 −21.8594 0 −13.6842 0
1.16 0 7.25076 0 −0.749836 0 4.82634 0 25.5735 0
1.17 0 7.44447 0 −7.85422 0 9.10727 0 28.4201 0
1.18 0 8.18106 0 4.03504 0 −15.4098 0 39.9297 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

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 2312.4.a.n 18
17.b even 2 1 2312.4.a.q yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2312.4.a.n 18 1.a even 1 1 trivial
2312.4.a.q yes 18 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 18 T_{3}^{17} - 141 T_{3}^{16} - 3902 T_{3}^{15} + 1941 T_{3}^{14} + 326322 T_{3}^{13} + 584880 T_{3}^{12} - 13368918 T_{3}^{11} - 36576240 T_{3}^{10} + 289119052 T_{3}^{9} + \cdots - 50608026631 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2312))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + 18 T^{17} + \cdots - 50608026631 \) Copy content Toggle raw display
$5$ \( T^{18} - 1278 T^{16} + \cdots + 82\!\cdots\!29 \) Copy content Toggle raw display
$7$ \( T^{18} + 51 T^{17} + \cdots - 10\!\cdots\!83 \) Copy content Toggle raw display
$11$ \( T^{18} + 132 T^{17} + \cdots - 68\!\cdots\!88 \) Copy content Toggle raw display
$13$ \( T^{18} - 30 T^{17} + \cdots + 44\!\cdots\!81 \) Copy content Toggle raw display
$17$ \( T^{18} \) Copy content Toggle raw display
$19$ \( T^{18} - 66 T^{17} + \cdots - 59\!\cdots\!97 \) Copy content Toggle raw display
$23$ \( T^{18} + 153 T^{17} + \cdots - 13\!\cdots\!97 \) Copy content Toggle raw display
$29$ \( T^{18} + 51 T^{17} + \cdots - 35\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{18} + 303 T^{17} + \cdots - 11\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{18} + 717 T^{17} + \cdots + 42\!\cdots\!27 \) Copy content Toggle raw display
$41$ \( T^{18} - 393 T^{17} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{18} + 390 T^{17} + \cdots - 80\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( T^{18} + 633 T^{17} + \cdots - 26\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{18} - 1275 T^{17} + \cdots + 14\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{18} + 204 T^{17} + \cdots + 15\!\cdots\!39 \) Copy content Toggle raw display
$61$ \( T^{18} + 534 T^{17} + \cdots - 97\!\cdots\!19 \) Copy content Toggle raw display
$67$ \( T^{18} + 405 T^{17} + \cdots - 52\!\cdots\!03 \) Copy content Toggle raw display
$71$ \( T^{18} - 426 T^{17} + \cdots + 16\!\cdots\!91 \) Copy content Toggle raw display
$73$ \( T^{18} + 1149 T^{17} + \cdots + 24\!\cdots\!83 \) Copy content Toggle raw display
$79$ \( T^{18} + 1053 T^{17} + \cdots - 45\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{18} - 66 T^{17} + \cdots - 16\!\cdots\!27 \) Copy content Toggle raw display
$89$ \( T^{18} + 4119 T^{17} + \cdots + 38\!\cdots\!89 \) Copy content Toggle raw display
$97$ \( T^{18} + 2349 T^{17} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
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