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Properties

Label 1058.4.a.w
Level $1058$
Weight $4$
Character orbit 1058.a
Self dual yes
Analytic conductor $62.424$
Analytic rank $0$
Dimension $15$
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] = [1058,4,Mod(1,1058)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1058, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1058.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1058 = 2 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1058.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: \(62.4240207861\)
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: \( x^{15} - 264 x^{13} - 21 x^{12} + 27588 x^{11} + 3762 x^{10} - 1439134 x^{9} - 201553 x^{8} + \cdots - 774554923 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 23^{4} \)
Twist minimal: no (minimal twist has level 46)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + ( - \beta_{10} + 3) q^{5} + 2 \beta_1 q^{6} + ( - \beta_{14} + \beta_{10} - \beta_{7} + \cdots + 6) q^{7}+ \cdots + (\beta_{14} + \beta_{11} - \beta_{10} + \cdots + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + ( - \beta_{10} + 3) q^{5} + 2 \beta_1 q^{6} + ( - \beta_{14} + \beta_{10} - \beta_{7} + \cdots + 6) q^{7}+ \cdots + (51 \beta_{14} + 31 \beta_{13} + \cdots + 183) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 30 q^{2} + 3 q^{3} + 60 q^{4} + 50 q^{5} + 6 q^{6} + 84 q^{7} + 120 q^{8} + 128 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 30 q^{2} + 3 q^{3} + 60 q^{4} + 50 q^{5} + 6 q^{6} + 84 q^{7} + 120 q^{8} + 128 q^{9} + 100 q^{10} + 89 q^{11} + 12 q^{12} + 4 q^{13} + 168 q^{14} + 226 q^{15} + 240 q^{16} + 316 q^{17} + 256 q^{18} + 373 q^{19} + 200 q^{20} + 389 q^{21} + 178 q^{22} + 24 q^{24} + 115 q^{25} + 8 q^{26} + 117 q^{27} + 336 q^{28} - 276 q^{29} + 452 q^{30} - 86 q^{31} + 480 q^{32} + 1314 q^{33} + 632 q^{34} - 992 q^{35} + 512 q^{36} + 1322 q^{37} + 746 q^{38} - 431 q^{39} + 400 q^{40} + 181 q^{41} + 778 q^{42} + 1500 q^{43} + 356 q^{44} + 1298 q^{45} - 874 q^{47} + 48 q^{48} + 1345 q^{49} + 230 q^{50} + 1407 q^{51} + 16 q^{52} + 1817 q^{53} + 234 q^{54} + 104 q^{55} + 672 q^{56} + 1837 q^{57} - 552 q^{58} + 1456 q^{59} + 904 q^{60} + 2284 q^{61} - 172 q^{62} + 1875 q^{63} + 960 q^{64} + 1186 q^{65} + 2628 q^{66} + 2908 q^{67} + 1264 q^{68} - 1984 q^{70} - 178 q^{71} + 1024 q^{72} + 441 q^{73} + 2644 q^{74} + 301 q^{75} + 1492 q^{76} - 3894 q^{77} - 862 q^{78} + 2012 q^{79} + 800 q^{80} - 4621 q^{81} + 362 q^{82} + 5004 q^{83} + 1556 q^{84} - 1986 q^{85} + 3000 q^{86} - 2593 q^{87} + 712 q^{88} + 5282 q^{89} + 2596 q^{90} + 3091 q^{91} - 4367 q^{93} - 1748 q^{94} + 4422 q^{95} + 96 q^{96} + 5776 q^{97} + 2690 q^{98} + 2991 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 264 x^{13} - 21 x^{12} + 27588 x^{11} + 3762 x^{10} - 1439134 x^{9} - 201553 x^{8} + \cdots - 774554923 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 76\!\cdots\!91 \nu^{14} + \cdots + 29\!\cdots\!17 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!38 \nu^{14} + \cdots + 50\!\cdots\!80 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 38\!\cdots\!86 \nu^{14} + \cdots + 64\!\cdots\!30 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 49\!\cdots\!30 \nu^{14} + \cdots + 11\!\cdots\!05 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\!\cdots\!99 \nu^{14} + \cdots - 31\!\cdots\!85 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57\!\cdots\!48 \nu^{14} + \cdots + 38\!\cdots\!29 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 67\!\cdots\!60 \nu^{14} + \cdots + 28\!\cdots\!54 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!75 \nu^{14} + \cdots + 49\!\cdots\!79 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!30 \nu^{14} + \cdots - 12\!\cdots\!90 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!35 \nu^{14} + \cdots - 65\!\cdots\!64 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!89 \nu^{14} + \cdots - 32\!\cdots\!82 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!04 \nu^{14} + \cdots - 46\!\cdots\!37 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!67 \nu^{14} + \cdots + 19\!\cdots\!40 ) / 21\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!20 \nu^{14} + \cdots - 17\!\cdots\!43 ) / 21\!\cdots\!78 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{13} + \beta_{12} + \beta_{9} + \beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{3} - 23\beta _1 + 4 ) / 23 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 22 \beta_{14} - \beta_{12} + 23 \beta_{11} - 23 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 2 \beta_{6} + \cdots + 808 ) / 23 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 37 \beta_{14} + 90 \beta_{13} + 49 \beta_{12} + 92 \beta_{11} + 23 \beta_{10} + 118 \beta_{9} + \cdots + 279 ) / 23 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1464 \beta_{14} - 33 \beta_{13} - 16 \beta_{12} + 1288 \beta_{11} - 1058 \beta_{10} - 294 \beta_{9} + \cdots + 44619 ) / 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3337 \beta_{14} + 6672 \beta_{13} + 4836 \beta_{12} + 8372 \beta_{11} + 1219 \beta_{10} + \cdots + 22850 ) / 23 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 106484 \beta_{14} - 882 \beta_{13} - 7013 \beta_{12} + 77165 \beta_{11} - 48668 \beta_{10} + \cdots + 2674545 ) / 23 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 350862 \beta_{14} + 439694 \beta_{13} + 438623 \beta_{12} + 649773 \beta_{11} + 91195 \beta_{10} + \cdots + 1488156 ) / 23 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 8072635 \beta_{14} + 270040 \beta_{13} - 990379 \beta_{12} + 4873861 \beta_{11} - 2230701 \beta_{10} + \cdots + 166815834 ) / 23 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 34171638 \beta_{14} + 27225632 \beta_{13} + 37107923 \beta_{12} + 47648019 \beta_{11} + \cdots + 77501558 ) / 23 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 620756956 \beta_{14} + 49987881 \beta_{13} - 104413066 \beta_{12} + 320696337 \beta_{11} + \cdots + 10707335140 ) / 23 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3088960342 \beta_{14} + 1615226289 \beta_{13} + 3006156399 \beta_{12} + 3399636210 \beta_{11} + \cdots + 2862110120 ) / 23 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 47803502160 \beta_{14} + 5859619949 \beta_{13} - 9718020897 \beta_{12} + 21737401821 \beta_{11} + \cdots + 703496298758 ) / 23 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 264762406330 \beta_{14} + 92247308683 \beta_{13} + 236810900088 \beta_{12} + 239218066477 \beta_{11} + \cdots - 517850304 ) / 23 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 3668265151671 \beta_{14} + 575315604423 \beta_{13} - 845847730945 \beta_{12} + 1502968240106 \beta_{11} + \cdots + 47134119591660 ) / 23 \) 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
7.43953
7.57713
8.42732
7.56978
3.59012
3.07989
1.40474
−0.610597
−2.54832
−0.524256
−5.85108
−7.64102
−8.63715
−6.83983
−6.43626
2.00000 −8.27036 4.00000 0.333259 −16.5407 9.14559 8.00000 41.3989 0.666518
1.2 2.00000 −7.29250 4.00000 −3.26224 −14.5850 −12.2745 8.00000 26.1806 −6.52448
1.3 2.00000 −7.11760 4.00000 −7.99776 −14.2352 6.41042 8.00000 23.6603 −15.9955
1.4 2.00000 −5.65079 4.00000 16.2190 −11.3016 23.8810 8.00000 4.93144 32.4380
1.5 2.00000 −5.27263 4.00000 20.5153 −10.5453 −28.5051 8.00000 0.800647 41.0306
1.6 2.00000 −4.76240 4.00000 −11.2255 −9.52480 17.1614 8.00000 −4.31955 −22.4510
1.7 2.00000 −0.0950139 4.00000 11.6344 −0.190028 −9.72620 8.00000 −26.9910 23.2688
1.8 2.00000 0.895226 4.00000 −11.2981 1.79045 25.2874 8.00000 −26.1986 −22.5962
1.9 2.00000 1.71749 4.00000 1.62671 3.43499 −32.6133 8.00000 −24.0502 3.25343
1.10 2.00000 2.44324 4.00000 −12.9438 4.88648 21.0607 8.00000 −21.0306 −25.8876
1.11 2.00000 6.13571 4.00000 13.5611 12.2714 −0.0778932 8.00000 10.6469 27.1222
1.12 2.00000 6.81019 4.00000 10.4771 13.6204 30.8168 8.00000 19.3787 20.9542
1.13 2.00000 6.95465 4.00000 5.17204 13.9093 9.74027 8.00000 21.3671 10.3441
1.14 2.00000 8.14955 4.00000 −1.50521 16.2991 33.8782 8.00000 39.4152 −3.01041
1.15 2.00000 8.35524 4.00000 18.6937 16.7105 −10.1848 8.00000 42.8101 37.3875
\(n\): 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 1058.4.a.w 15
23.b odd 2 1 1058.4.a.v 15
23.d odd 22 2 46.4.c.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.c.a ✓ 30 23.d odd 22 2
1058.4.a.v 15 23.b odd 2 1
1058.4.a.w 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):

\( T_{3}^{15} - 3 T_{3}^{14} - 262 T_{3}^{13} + 702 T_{3}^{12} + 27690 T_{3}^{11} - 65560 T_{3}^{10} + \cdots + 430200319 \) Copy content Toggle raw display
\( T_{5}^{15} - 50 T_{5}^{14} + 255 T_{5}^{13} + 22510 T_{5}^{12} - 292443 T_{5}^{11} + \cdots - 1858594139159 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 430200319 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 1858594139159 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 11\!\cdots\!51 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 32\!\cdots\!63 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 11\!\cdots\!33 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 41\!\cdots\!73 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 18\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{15} \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 15\!\cdots\!19 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 90\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 16\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 12\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 27\!\cdots\!97 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 13\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 14\!\cdots\!33 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 16\!\cdots\!43 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 10\!\cdots\!67 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 13\!\cdots\!67 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 21\!\cdots\!03 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 54\!\cdots\!59 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 14\!\cdots\!83 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 11\!\cdots\!43 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 38\!\cdots\!87 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 12\!\cdots\!93 \) Copy content Toggle raw display
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