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Properties

Label 1573.4.a.h
Level $1573$
Weight $4$
Character orbit 1573.a
Self dual yes
Analytic conductor $92.810$
Analytic rank $1$
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] = [1573,4,Mod(1,1573)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1573, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1573.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1573 = 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1573.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: \(92.8100044390\)
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: \( x^{15} - 6 x^{14} - 69 x^{13} + 418 x^{12} + 1806 x^{11} - 10742 x^{10} - 24098 x^{9} + 129758 x^{8} + \cdots + 47232 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + \beta_1 + 3) q^{4} + \beta_{4} q^{5} + (\beta_{10} - \beta_{5} + 2) q^{6} + ( - \beta_{6} - \beta_{5} - 2) q^{7} + ( - \beta_{3} - 4 \beta_1 - 6) q^{8} + (\beta_{8} + 2 \beta_1 + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + \beta_1 + 3) q^{4} + \beta_{4} q^{5} + (\beta_{10} - \beta_{5} + 2) q^{6} + ( - \beta_{6} - \beta_{5} - 2) q^{7} + ( - \beta_{3} - 4 \beta_1 - 6) q^{8} + (\beta_{8} + 2 \beta_1 + 8) q^{9} + ( - \beta_{13} - \beta_{5} - \beta_{4} + 1) q^{10} + (\beta_{13} - \beta_{12} + \cdots - 2 \beta_1) q^{12}+ \cdots + ( - \beta_{14} - 8 \beta_{13} + \cdots + 16) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 6 q^{2} + 2 q^{3} + 54 q^{4} - 2 q^{5} + 25 q^{6} - 28 q^{7} - 108 q^{8} + 133 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 6 q^{2} + 2 q^{3} + 54 q^{4} - 2 q^{5} + 25 q^{6} - 28 q^{7} - 108 q^{8} + 133 q^{9} + 16 q^{10} - 3 q^{12} - 195 q^{13} + 60 q^{14} + 202 q^{15} + 282 q^{16} - 136 q^{17} - 351 q^{18} - 136 q^{19} - 62 q^{20} - 625 q^{21} - 176 q^{23} + 227 q^{24} + 479 q^{25} + 78 q^{26} - 13 q^{27} + 86 q^{28} - 344 q^{29} - 542 q^{30} + 275 q^{31} + 170 q^{32} - 1505 q^{34} - 655 q^{35} + 623 q^{36} + 346 q^{37} - 848 q^{38} - 26 q^{39} - 256 q^{40} - 38 q^{41} + 575 q^{42} - 534 q^{43} - 633 q^{45} + 375 q^{46} + 276 q^{47} - 723 q^{48} + 773 q^{49} - 1684 q^{50} - 1111 q^{51} - 702 q^{52} + 706 q^{53} + 178 q^{54} - 2056 q^{56} - 1320 q^{57} - 983 q^{58} - 174 q^{59} + 2004 q^{60} - 1078 q^{61} - 1365 q^{62} + 2055 q^{63} + 1338 q^{64} + 26 q^{65} - 260 q^{67} - 559 q^{68} + 154 q^{69} - 1161 q^{70} + 2910 q^{71} - 6171 q^{72} - 1000 q^{73} - 688 q^{74} - 553 q^{75} - 178 q^{76} - 325 q^{78} + 386 q^{79} + 1702 q^{80} - 2805 q^{81} + 3380 q^{82} - 4318 q^{83} - 6681 q^{84} + 1391 q^{85} + 4139 q^{86} - 3144 q^{87} - 810 q^{89} + 3789 q^{90} + 364 q^{91} - 4857 q^{92} + 394 q^{93} + 116 q^{94} - 5204 q^{95} + 893 q^{96} - 1860 q^{97} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 6 x^{14} - 69 x^{13} + 418 x^{12} + 1806 x^{11} - 10742 x^{10} - 24098 x^{9} + 129758 x^{8} + \cdots + 47232 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 20\nu - 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2169611941 \nu^{14} - 73637278605 \nu^{13} + 126737365518 \nu^{12} + 4660917615648 \nu^{11} + \cdots + 14\!\cdots\!20 ) / 193702542066176 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6525284091 \nu^{14} + 81211424019 \nu^{13} + 149033628446 \nu^{12} + \cdots - 236754999077504 ) / 193702542066176 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7004702884 \nu^{14} + 48567493227 \nu^{13} + 495886455282 \nu^{12} + \cdots + 21\!\cdots\!92 ) / 72638453274816 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 91872753529 \nu^{14} + 726548029185 \nu^{13} + 7047200789562 \nu^{12} + \cdots + 79\!\cdots\!08 ) / 581107626198528 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8631219019 \nu^{14} - 31349803705 \nu^{13} - 575222836390 \nu^{12} + 1957775452708 \nu^{11} + \cdots - 16\!\cdots\!28 ) / 48425635516544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 134468972641 \nu^{14} + 1014508706673 \nu^{13} + 5632684296138 \nu^{12} + \cdots - 27\!\cdots\!16 ) / 581107626198528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12146250891 \nu^{14} + 95605599463 \nu^{13} + 543204289610 \nu^{12} + \cdots - 233090575348992 ) / 48425635516544 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9299204477 \nu^{14} + 34277516031 \nu^{13} + 662868386340 \nu^{12} + \cdots + 20\!\cdots\!04 ) / 36319226637408 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 83713336549 \nu^{14} + 168236004057 \nu^{13} + 6332159308122 \nu^{12} + \cdots + 69\!\cdots\!24 ) / 290553813099264 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 56263934809 \nu^{14} + 268866444033 \nu^{13} + 3478248830346 \nu^{12} + \cdots - 11\!\cdots\!52 ) / 193702542066176 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 87701416867 \nu^{14} + 511921177185 \nu^{13} + 5743481895618 \nu^{12} + \cdots + 76\!\cdots\!48 ) / 145276906549632 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 20\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \cdots + 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{14} - 2 \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \cdots + 173 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 40 \beta_{14} + 81 \beta_{13} - 36 \beta_{12} + 4 \beta_{11} + 36 \beta_{10} - 40 \beta_{9} + \cdots + 5021 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 95 \beta_{14} - 104 \beta_{13} - 34 \beta_{12} - 54 \beta_{11} - 61 \beta_{10} + 45 \beta_{9} + \cdots + 4134 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1270 \beta_{14} + 2579 \beta_{13} - 1060 \beta_{12} + 221 \beta_{11} + 1029 \beta_{10} + \cdots + 126287 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3356 \beta_{14} - 3936 \beta_{13} - 868 \beta_{12} - 1958 \beta_{11} - 2494 \beta_{10} + 1632 \beta_{9} + \cdots + 89012 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 37576 \beta_{14} + 76174 \beta_{13} - 29614 \beta_{12} + 8466 \beta_{11} + 27662 \beta_{10} + \cdots + 3295761 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 106898 \beta_{14} - 132150 \beta_{13} - 19272 \beta_{12} - 61632 \beta_{11} - 87938 \beta_{10} + \cdots + 1688462 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1081437 \beta_{14} + 2183508 \beta_{13} - 813859 \beta_{12} + 281690 \beta_{11} + 731971 \beta_{10} + \cdots + 87578478 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3255360 \beta_{14} - 4193244 \beta_{13} - 373021 \beta_{12} - 1825945 \beta_{11} - 2884383 \beta_{10} + \cdots + 24762737 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 30749908 \beta_{14} + 61770579 \beta_{13} - 22257758 \beta_{12} + 8766538 \beta_{11} + \cdots + 2350069099 \) 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
5.11405
5.08953
5.00561
3.43729
3.13862
2.65412
0.460593
0.122071
−0.500005
−1.11582
−2.09746
−3.04830
−3.07124
−3.84909
−5.33997
−5.11405 7.70661 18.1535 4.71951 −39.4119 −22.4944 −51.9253 32.3918 −24.1358
1.2 −5.08953 −2.18836 17.9033 14.1461 11.1377 12.0816 −50.4033 −22.2111 −71.9971
1.3 −5.00561 −9.04285 17.0562 −15.8707 45.2650 23.4660 −45.3316 54.7732 79.4425
1.4 −3.43729 −6.00304 3.81493 −4.47666 20.6342 −28.7105 14.3853 9.03648 15.3875
1.5 −3.13862 4.96398 1.85094 −20.3862 −15.5801 −8.13132 19.2996 −2.35886 63.9846
1.6 −2.65412 7.25625 −0.955648 11.9209 −19.2590 3.65834 23.7694 25.6532 −31.6396
1.7 −0.460593 −6.09765 −7.78785 15.2965 2.80854 15.1681 7.27177 10.1814 −7.04544
1.8 −0.122071 −2.14956 −7.98510 8.59769 0.262400 −30.6142 1.95132 −22.3794 −1.04953
1.9 0.500005 3.71996 −7.74999 −5.62008 1.86000 8.40124 −7.87508 −13.1619 −2.81007
1.10 1.11582 −5.11180 −6.75495 −21.8247 −5.70385 −5.96280 −16.4639 −0.869465 −24.3524
1.11 2.09746 7.98226 −3.60065 17.0941 16.7425 −31.2892 −24.3319 36.7164 35.8541
1.12 3.04830 7.84090 1.29211 −10.4483 23.9014 17.4647 −20.4476 34.4796 −31.8495
1.13 3.07124 −8.32314 1.43251 −1.48177 −25.5623 32.7757 −20.1703 42.2747 −4.55086
1.14 3.84909 −0.121722 6.81551 8.75953 −0.468519 −0.407809 −4.55922 −26.9852 33.7162
1.15 5.33997 1.56817 20.5152 −2.42597 8.37400 −13.4053 66.8310 −24.5408 −12.9546
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(13\) \(1\)

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 1573.4.a.h ✓ 15
11.b odd 2 1 1573.4.a.k yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1573.4.a.h ✓ 15 1.a even 1 1 trivial
1573.4.a.k yes 15 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 6 T_{2}^{14} - 69 T_{2}^{13} - 418 T_{2}^{12} + 1806 T_{2}^{11} + 10742 T_{2}^{10} + \cdots - 47232 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1573))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 6 T^{14} + \cdots - 47232 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 817272823 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 104575737010304 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{15} \) Copy content Toggle raw display
$13$ \( (T + 13)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 48\!\cdots\!63 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 42\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 84\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 73\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 11\!\cdots\!29 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 45\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 57\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 36\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 34\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 43\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 52\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 38\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 48\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 68\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
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