Properties

Label 1859.4.a.d
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $1$
Dimension $9$
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] = [1859,4,Mod(1,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, 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("1859.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1859.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: \(109.684550701\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 59x^{7} - 12x^{6} + 1144x^{5} + 345x^{4} - 7888x^{3} - 2245x^{2} + 9710x - 2988 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + 5) q^{4} + ( - \beta_{8} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{5} + ( - \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_1 - 4) q^{6} + (\beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 2) q^{7} + ( - \beta_{8} - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{2} - 3 \beta_1 - 5) q^{8} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + 5) q^{4} + ( - \beta_{8} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{5} + ( - \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_1 - 4) q^{6} + (\beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 2) q^{7} + ( - \beta_{8} - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{2} - 3 \beta_1 - 5) q^{8} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 10) q^{9} + ( - \beta_{8} + 2 \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_1 - 4) q^{10} + 11 q^{11} + ( - 3 \beta_{8} - \beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 20) q^{12} + (4 \beta_{8} - 2 \beta_{7} + \beta_{6} + 4 \beta_{3} - \beta_{2} + 6 \beta_1 + 2) q^{14} + (6 \beta_{8} - \beta_{7} + 2 \beta_{6} + 3 \beta_{4} - 6 \beta_{2} + \beta_1 - 37) q^{15} + ( - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + \cdots + 11) q^{16}+ \cdots + ( - 22 \beta_{8} - 11 \beta_{7} - 11 \beta_{6} + 11 \beta_{5} - 11 \beta_{4} + \cdots + 110) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 8 q^{3} + 46 q^{4} - 30 q^{5} - 34 q^{6} - 25 q^{7} - 36 q^{8} + 91 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 8 q^{3} + 46 q^{4} - 30 q^{5} - 34 q^{6} - 25 q^{7} - 36 q^{8} + 91 q^{9} - 22 q^{10} + 99 q^{11} + 181 q^{12} - 351 q^{15} + 130 q^{16} + 53 q^{17} - 33 q^{18} - 69 q^{19} - 282 q^{20} - 463 q^{21} + 216 q^{23} + 121 q^{24} + 617 q^{25} + 275 q^{27} - 279 q^{28} - 91 q^{29} + 29 q^{30} - 636 q^{31} - 663 q^{32} + 88 q^{33} - 423 q^{34} - 358 q^{35} - 252 q^{36} - 967 q^{37} - 101 q^{38} + 652 q^{40} + 226 q^{41} - 1186 q^{42} + 42 q^{43} + 506 q^{44} - 5 q^{45} + 1127 q^{46} + 269 q^{47} - 1820 q^{48} + 228 q^{49} + 1374 q^{50} - 589 q^{51} + 1227 q^{53} + 2438 q^{54} - 330 q^{55} - 659 q^{56} + 71 q^{57} - 471 q^{58} + 613 q^{59} + 859 q^{60} + 427 q^{61} - 1714 q^{62} - 305 q^{63} - 1194 q^{64} - 374 q^{66} + 271 q^{67} - 2835 q^{68} - 846 q^{69} + 102 q^{70} - 2279 q^{71} + 2400 q^{72} - 3602 q^{73} - 4955 q^{74} - 883 q^{75} - 1126 q^{76} - 275 q^{77} - 1182 q^{79} + 2360 q^{80} + 2697 q^{81} + 1007 q^{82} + 1877 q^{83} - 1618 q^{84} + 441 q^{85} - 830 q^{86} + 1942 q^{87} - 396 q^{88} - 1258 q^{89} - 5669 q^{90} + 1046 q^{92} - 1556 q^{93} + 1439 q^{94} + 2032 q^{95} + 3417 q^{96} - 4002 q^{97} + 1855 q^{98} + 1001 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 59x^{7} - 12x^{6} + 1144x^{5} + 345x^{4} - 7888x^{3} - 2245x^{2} + 9710x - 2988 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{8} + 357 \nu^{7} - 350 \nu^{6} - 14878 \nu^{5} + 9822 \nu^{4} + 154641 \nu^{3} - 82789 \nu^{2} - 171682 \nu + 322364 ) / 37760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 87 \nu^{8} - 621 \nu^{7} + 6990 \nu^{6} + 29774 \nu^{5} - 168206 \nu^{4} - 418873 \nu^{3} + 1290477 \nu^{2} + 1596466 \nu - 945052 ) / 75520 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 89 \nu^{8} + 93 \nu^{7} + 6290 \nu^{6} + 18 \nu^{5} - 148562 \nu^{4} - 109591 \nu^{3} + 1200419 \nu^{2} + 1253102 \nu - 1282084 ) / 75520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53 \nu^{8} - 119 \nu^{7} + 4810 \nu^{6} + 6346 \nu^{5} - 133194 \nu^{4} - 131707 \nu^{3} + 1212663 \nu^{2} + 1011254 \nu - 1429268 ) / 37760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 283 \nu^{8} + 551 \nu^{7} + 14550 \nu^{6} - 21674 \nu^{5} - 237974 \nu^{4} + 240843 \nu^{3} + 1287513 \nu^{2} - 549206 \nu - 612268 ) / 75520 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 311 \nu^{8} + 653 \nu^{7} - 16270 \nu^{6} - 35982 \nu^{5} + 263118 \nu^{4} + 525529 \nu^{3} - 1433101 \nu^{2} - 1848178 \nu + 1310876 ) / 37760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 145 \nu^{8} - 181 \nu^{7} - 7810 \nu^{6} + 5246 \nu^{5} + 134978 \nu^{4} - 6305 \nu^{3} - 783307 \nu^{2} - 483326 \nu + 495876 ) / 15104 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{6} - \beta_{5} + 3\beta_{4} - \beta_{2} + 19\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} + 2\beta_{7} - \beta_{6} - 5\beta_{5} + 28\beta_{4} - 21\beta_{3} - 29\beta_{2} + 8\beta _1 + 259 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 33\beta_{8} - \beta_{7} + 60\beta_{6} - 43\beta_{5} + 127\beta_{4} - 4\beta_{3} - 37\beta_{2} + 404\beta _1 + 232 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 66 \beta_{8} + 90 \beta_{7} - 2 \beta_{6} - 220 \beta_{5} + 804 \beta_{4} - 436 \beta_{3} - 804 \beta_{2} + 391 \beta _1 + 5756 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 928\beta_{8} + 1662\beta_{6} - 1458\beta_{5} + 4309\beta_{4} - 275\beta_{3} - 1287\beta_{2} + 9292\beta _1 + 8519 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1581 \beta_{8} + 3022 \beta_{7} + 814 \beta_{6} - 7503 \beta_{5} + 23557 \beta_{4} - 9536 \beta_{3} - 22023 \beta_{2} + 14755 \beta _1 + 138197 \) 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
5.41479
4.08298
3.71870
0.765277
0.388321
−1.62159
−3.76323
−4.11495
−4.87031
−5.41479 −1.18631 21.3199 −5.73789 6.42360 −3.58372 −72.1247 −25.5927 31.0694
1.2 −4.08298 7.19985 8.67073 7.90460 −29.3968 −23.1330 −2.73856 24.8378 −32.2743
1.3 −3.71870 7.61710 5.82875 −15.9808 −28.3257 21.9580 8.07423 31.0202 59.4279
1.4 −0.765277 −9.54214 −7.41435 17.1562 7.30238 4.60754 11.7962 64.0525 −13.1292
1.5 −0.388321 3.09988 −7.84921 −16.8933 −1.20375 −26.1569 6.15457 −17.3907 6.56001
1.6 1.62159 −2.83913 −5.37046 8.40999 −4.60389 9.04976 −21.6813 −18.9393 13.6375
1.7 3.76323 −4.70664 6.16188 −20.6048 −17.7122 28.6315 −6.91727 −4.84752 −77.5406
1.8 4.11495 9.51427 8.93279 −14.5196 39.1507 −21.0410 3.83839 63.5214 −59.7475
1.9 4.87031 −1.15688 15.7199 10.2656 −5.63435 −15.3321 37.5984 −25.6616 49.9968
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 1859.4.a.d 9
13.b even 2 1 143.4.a.c 9
39.d odd 2 1 1287.4.a.k 9
52.b odd 2 1 2288.4.a.r 9
143.d odd 2 1 1573.4.a.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.a.c 9 13.b even 2 1
1287.4.a.k 9 39.d odd 2 1
1573.4.a.e 9 143.d odd 2 1
1859.4.a.d 9 1.a even 1 1 trivial
2288.4.a.r 9 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 59T_{2}^{7} + 12T_{2}^{6} + 1144T_{2}^{5} - 345T_{2}^{4} - 7888T_{2}^{3} + 2245T_{2}^{2} + 9710T_{2} + 2988 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 59 T^{7} + 12 T^{6} + \cdots + 2988 \) Copy content Toggle raw display
$3$ \( T^{9} - 8 T^{8} - 135 T^{7} + \cdots + 283048 \) Copy content Toggle raw display
$5$ \( T^{9} + 30 T^{8} + \cdots + 5425892224 \) Copy content Toggle raw display
$7$ \( T^{9} + 25 T^{8} + \cdots + 18338418984 \) Copy content Toggle raw display
$11$ \( (T - 11)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 53 T^{8} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{9} + 69 T^{8} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{9} - 216 T^{8} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{9} + 91 T^{8} + \cdots + 27\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{9} + 636 T^{8} + \cdots - 56\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{9} + 967 T^{8} + \cdots + 72\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{9} - 226 T^{8} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{9} - 42 T^{8} + \cdots + 51\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{9} - 269 T^{8} + \cdots + 51\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{9} - 1227 T^{8} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{9} - 613 T^{8} + \cdots - 15\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{9} - 427 T^{8} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{9} - 271 T^{8} + \cdots - 89\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{9} + 2279 T^{8} + \cdots - 15\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{9} + 3602 T^{8} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{9} + 1182 T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{9} - 1877 T^{8} + \cdots - 41\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{9} + 1258 T^{8} + \cdots + 94\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{9} + 4002 T^{8} + \cdots - 56\!\cdots\!64 \) Copy content Toggle raw display
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