Properties

Label 1849.4.a.d
Level $1849$
Weight $4$
Character orbit 1849.a
Self dual yes
Analytic conductor $109.095$
Analytic rank $0$
Dimension $10$
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] = [1849,4,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, 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("1849.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.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: \(109.094531601\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 59x^{8} + 42x^{7} + 1187x^{6} - 541x^{5} - 9389x^{4} + 2180x^{3} + 22676x^{2} - 320x - 768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 43)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{7} + 2) q^{5} + (\beta_{5} + \beta_{3} - \beta_1 - 1) q^{6} + (\beta_{6} + \beta_{2} + 5) q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{6} - 3 \beta_1 - 4) q^{8} + (\beta_{8} - \beta_{7} - \beta_{4} + \beta_1 + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{7} + 2) q^{5} + (\beta_{5} + \beta_{3} - \beta_1 - 1) q^{6} + (\beta_{6} + \beta_{2} + 5) q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{6} - 3 \beta_1 - 4) q^{8} + (\beta_{8} - \beta_{7} - \beta_{4} + \beta_1 + 12) q^{9} + ( - \beta_{8} + \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 3) q^{10} + ( - \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 + 3) q^{11} + ( - \beta_{9} + \beta_{8} - 4 \beta_{5} + \beta_{4} + 4 \beta_{2} + \beta_1 + 6) q^{12} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{9} - 2 \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} - 10 \beta_1 - 10) q^{14} + ( - \beta_{9} + 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \cdots - 6) q^{15}+ \cdots + ( - 5 \beta_{9} + 30 \beta_{8} - 42 \beta_{7} - 45 \beta_{6} - 79 \beta_{5} + \cdots + 318) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 5 q^{3} + 39 q^{4} + 19 q^{5} - 15 q^{6} + 51 q^{7} - 36 q^{8} + 117 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 5 q^{3} + 39 q^{4} + 19 q^{5} - 15 q^{6} + 51 q^{7} - 36 q^{8} + 117 q^{9} - 27 q^{10} + 27 q^{11} + 72 q^{12} + 15 q^{13} - 96 q^{14} - 65 q^{15} + 67 q^{16} + 82 q^{17} - 247 q^{18} - 78 q^{19} + 495 q^{20} - 9 q^{21} + 190 q^{22} + 61 q^{23} - 202 q^{24} + 151 q^{25} + 21 q^{26} - 97 q^{27} + 794 q^{28} + 53 q^{29} - 627 q^{30} - 253 q^{31} - 399 q^{32} + 424 q^{33} + 231 q^{34} + 355 q^{35} + 1092 q^{36} + 129 q^{37} + 854 q^{38} + 691 q^{39} - 1345 q^{40} + 391 q^{41} + 31 q^{42} + 377 q^{44} + 944 q^{45} + 40 q^{46} - 334 q^{47} + 2401 q^{48} + 115 q^{49} - 424 q^{50} + 795 q^{51} + 564 q^{52} - 773 q^{53} + 182 q^{54} + 1242 q^{55} + 923 q^{56} + 765 q^{57} - 1328 q^{58} - 1483 q^{59} + 1075 q^{60} - 437 q^{61} - 1509 q^{62} + 2222 q^{63} - 738 q^{64} - 1063 q^{65} - 1483 q^{66} + 642 q^{67} + 1052 q^{68} + 3503 q^{69} - 85 q^{70} + 1545 q^{71} - 3834 q^{72} - 1292 q^{73} + 2232 q^{74} + 82 q^{75} + 252 q^{76} - 1448 q^{77} + 2822 q^{78} + 1405 q^{79} + 3157 q^{80} - 974 q^{81} + 3304 q^{82} - 543 q^{83} + 3652 q^{84} + 973 q^{85} + 1409 q^{87} - 2686 q^{88} + 2196 q^{89} - 742 q^{90} + 3513 q^{91} - 2629 q^{92} + 983 q^{93} + 4939 q^{94} + 149 q^{95} - 3540 q^{96} - 425 q^{97} + 213 q^{98} + 3181 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 59x^{8} + 42x^{7} + 1187x^{6} - 541x^{5} - 9389x^{4} + 2180x^{3} + 22676x^{2} - 320x - 768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 225 \nu^{9} + 1607 \nu^{8} - 30915 \nu^{7} - 48526 \nu^{6} + 948499 \nu^{5} + 145275 \nu^{4} - 8738005 \nu^{3} + 2790076 \nu^{2} + 16689012 \nu - 1568544 ) / 704512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 345 \nu^{9} + 8335 \nu^{8} - 47403 \nu^{7} - 385566 \nu^{6} + 1554171 \nu^{5} + 5330467 \nu^{4} - 16950189 \nu^{3} - 23679268 \nu^{2} + 48345556 \nu + 21830112 ) / 704512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 585 \nu^{9} + 225 \nu^{8} + 36347 \nu^{7} - 19138 \nu^{6} - 762059 \nu^{5} + 502925 \nu^{4} + 6140765 \nu^{3} - 3872540 \nu^{2} - 14347924 \nu + 1823776 ) / 704512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 827 \nu^{9} - 3069 \nu^{8} + 51985 \nu^{7} + 139514 \nu^{6} - 1039745 \nu^{5} - 1620089 \nu^{4} + 6893047 \nu^{3} + 2149420 \nu^{2} - 5101372 \nu + 10152032 ) / 704512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1749 \nu^{9} + 3213 \nu^{8} + 95007 \nu^{7} - 129306 \nu^{6} - 1729711 \nu^{5} + 1424585 \nu^{4} + 12258905 \nu^{3} - 3579884 \nu^{2} - 26340356 \nu + 18848 ) / 352256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4325 \nu^{9} + 3357 \nu^{8} + 241999 \nu^{7} - 119098 \nu^{6} - 4499167 \nu^{5} + 1229081 \nu^{4} + 32115369 \nu^{3} - 5010348 \nu^{2} - 71167812 \nu + 7371680 ) / 704512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 141 \nu^{9} - 213 \nu^{8} - 9191 \nu^{7} + 10586 \nu^{6} + 207911 \nu^{5} - 170401 \nu^{4} - 1838177 \nu^{3} + 949964 \nu^{2} + 4671652 \nu - 704672 ) / 22016 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{6} + 19\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 2\beta_{8} - 3\beta_{7} + \beta_{6} - 5\beta_{5} + \beta_{4} + \beta_{3} + 26\beta_{2} + 3\beta _1 + 229 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + 33 \beta_{8} - 31 \beta_{7} - 30 \beta_{6} - 10 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 411 \beta _1 + 165 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 39 \beta_{9} + 85 \beta_{8} - 114 \beta_{7} + 32 \beta_{6} - 261 \beta_{5} + 35 \beta_{4} + 29 \beta_{3} + 658 \beta_{2} + 165 \beta _1 + 4959 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26 \beta_{9} + 941 \beta_{8} - 858 \beta_{7} - 797 \beta_{6} - 573 \beta_{5} + 89 \beta_{4} - 328 \beta_{3} + 423 \beta_{2} + 9604 \beta _1 + 5588 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1214 \beta_{9} + 2809 \beta_{8} - 3510 \beta_{7} + 731 \beta_{6} - 9313 \beta_{5} + 1071 \beta_{4} + 602 \beta_{3} + 16880 \beta_{2} + 6767 \beta _1 + 116228 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 262 \beta_{9} + 25994 \beta_{8} - 23623 \beta_{7} - 20842 \beta_{6} - 23121 \beta_{5} + 3051 \beta_{4} - 11118 \beta_{3} + 16559 \beta_{2} + 236018 \beta _1 + 177264 \) 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
5.31336
4.23473
3.55840
2.14781
0.190911
−0.179767
−1.92278
−3.59365
−3.82341
−4.92559
−5.31336 8.05642 20.2317 15.9193 −42.8066 16.6323 −64.9916 37.9059 −84.5852
1.2 −4.23473 −8.85858 9.93292 5.96488 37.5137 22.5055 −8.18537 51.4744 −25.2597
1.3 −3.55840 −0.389088 4.66219 −1.72780 1.38453 −21.1861 11.8773 −26.8486 6.14819
1.4 −2.14781 6.84883 −3.38692 −17.1363 −14.7100 22.9786 24.4569 19.9064 36.8056
1.5 −0.190911 1.43836 −7.96355 16.3462 −0.274600 6.23994 3.04762 −24.9311 −3.12068
1.6 0.179767 −6.02248 −7.96768 −10.9703 −1.08264 8.78367 −2.87047 9.27026 −1.97211
1.7 1.92278 8.37832 −4.30290 0.0702257 16.1097 −23.4425 −23.6558 43.1963 0.135029
1.8 3.59365 −1.67792 4.91431 −9.72181 −6.02986 −15.7603 −11.0889 −24.1846 −34.9368
1.9 3.82341 −7.76517 6.61843 18.1323 −29.6894 5.19796 −5.28231 33.2979 69.3271
1.10 4.92559 4.99131 16.2615 2.12330 24.5852 29.0509 40.6926 −2.08684 10.4585
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(1\)

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 1849.4.a.d 10
43.b odd 2 1 1849.4.a.f 10
43.c even 3 2 43.4.c.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.c.a 20 43.c even 3 2
1849.4.a.d 10 1.a even 1 1 trivial
1849.4.a.f 10 43.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + T_{2}^{9} - 59 T_{2}^{8} - 42 T_{2}^{7} + 1187 T_{2}^{6} + 541 T_{2}^{5} - 9389 T_{2}^{4} - 2180 T_{2}^{3} + 22676 T_{2}^{2} + 320 T_{2} - 768 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1849))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} - 59 T^{8} - 42 T^{7} + \cdots - 768 \) Copy content Toggle raw display
$3$ \( T^{10} - 5 T^{9} - 181 T^{8} + \cdots - 897652 \) Copy content Toggle raw display
$5$ \( T^{10} - 19 T^{9} - 520 T^{8} + \cdots + 13252064 \) Copy content Toggle raw display
$7$ \( T^{10} - 51 T^{9} + \cdots - 557226765054 \) Copy content Toggle raw display
$11$ \( T^{10} - 27 T^{9} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{10} - 15 T^{9} + \cdots - 16347624948144 \) Copy content Toggle raw display
$17$ \( T^{10} - 82 T^{9} + \cdots - 55\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{10} + 78 T^{9} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{10} - 61 T^{9} + \cdots - 50\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{10} - 53 T^{9} + \cdots - 99\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{10} + 253 T^{9} + \cdots - 52\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{10} - 129 T^{9} + \cdots - 13\!\cdots\!22 \) Copy content Toggle raw display
$41$ \( T^{10} - 391 T^{9} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{10} \) Copy content Toggle raw display
$47$ \( T^{10} + 334 T^{9} + \cdots - 43\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{10} + 773 T^{9} + \cdots + 90\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{10} + 1483 T^{9} + \cdots + 24\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{10} + 437 T^{9} + \cdots - 87\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{10} - 642 T^{9} + \cdots + 32\!\cdots\!47 \) Copy content Toggle raw display
$71$ \( T^{10} - 1545 T^{9} + \cdots - 57\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{10} + 1292 T^{9} + \cdots + 12\!\cdots\!97 \) Copy content Toggle raw display
$79$ \( T^{10} - 1405 T^{9} + \cdots - 28\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{10} + 543 T^{9} + \cdots + 29\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{10} - 2196 T^{9} + \cdots + 31\!\cdots\!37 \) Copy content Toggle raw display
$97$ \( T^{10} + 425 T^{9} + \cdots - 51\!\cdots\!44 \) Copy content Toggle raw display
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