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Properties

Label 1859.4.a.f
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $1$
Dimension $17$
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: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 4 x^{16} - 99 x^{15} + 375 x^{14} + 3949 x^{13} - 13998 x^{12} - 81750 x^{11} + 267574 x^{10} + \cdots + 2596992 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} + 5) q^{4} + ( - \beta_{6} - 1) q^{5} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{15} + \beta_{8} + 2 \beta_1 + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} + 5) q^{4} + ( - \beta_{6} - 1) q^{5} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots - 1) q^{6}+ \cdots + ( - 11 \beta_{15} + 11 \beta_{8} + \cdots + 77) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 4 q^{2} - 6 q^{3} + 78 q^{4} - 16 q^{5} - 14 q^{6} + 6 q^{7} - 63 q^{8} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 4 q^{2} - 6 q^{3} + 78 q^{4} - 16 q^{5} - 14 q^{6} + 6 q^{7} - 63 q^{8} + 135 q^{9} + 2 q^{10} + 187 q^{11} - 95 q^{12} - 60 q^{14} + 28 q^{15} + 350 q^{16} + 118 q^{17} - 478 q^{18} - 403 q^{19} - 98 q^{20} - 220 q^{21} - 44 q^{22} - 215 q^{23} - 26 q^{24} + 319 q^{25} - 384 q^{27} + 396 q^{28} - 7 q^{29} - 1269 q^{30} - 682 q^{31} - 813 q^{32} - 66 q^{33} - 738 q^{34} + 10 q^{35} + 560 q^{36} - 1084 q^{37} + 410 q^{38} + 95 q^{40} - 240 q^{41} + 393 q^{42} - 435 q^{43} + 858 q^{44} - 1242 q^{45} - 1671 q^{46} - 549 q^{47} + 894 q^{48} + 403 q^{49} + 651 q^{50} + 1552 q^{51} - 566 q^{53} - 311 q^{54} - 176 q^{55} - 1925 q^{56} + 534 q^{57} - 618 q^{58} - 2010 q^{59} + 411 q^{60} + 460 q^{61} - 823 q^{62} - 820 q^{63} + 3171 q^{64} - 154 q^{66} + 232 q^{67} + 1795 q^{68} - 1608 q^{69} - 207 q^{70} - 489 q^{71} - 2556 q^{72} - 290 q^{73} + 2653 q^{74} - 2852 q^{75} - 2421 q^{76} + 66 q^{77} - 732 q^{79} - 4915 q^{80} + 2393 q^{81} - 1772 q^{82} + 117 q^{83} - 4161 q^{84} - 4858 q^{85} - 1034 q^{86} + 3032 q^{87} - 693 q^{88} - 4113 q^{89} + 15145 q^{90} - 3554 q^{92} - 802 q^{93} + 2325 q^{94} - 3924 q^{95} - 2601 q^{96} - 2793 q^{97} - 533 q^{98} + 1485 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 4 x^{16} - 99 x^{15} + 375 x^{14} + 3949 x^{13} - 13998 x^{12} - 81750 x^{11} + 267574 x^{10} + \cdots + 2596992 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 20\nu + 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32\!\cdots\!77 \nu^{16} + \cdots - 40\!\cdots\!80 ) / 19\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38\!\cdots\!51 \nu^{16} + \cdots + 16\!\cdots\!60 ) / 19\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 83\!\cdots\!63 \nu^{16} + \cdots + 94\!\cdots\!32 ) / 19\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!43 \nu^{16} + \cdots + 18\!\cdots\!00 ) / 33\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 79\!\cdots\!35 \nu^{16} + \cdots - 31\!\cdots\!24 ) / 99\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!27 \nu^{16} + \cdots + 44\!\cdots\!40 ) / 16\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 40\!\cdots\!89 \nu^{16} + \cdots - 46\!\cdots\!32 ) / 16\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 48\!\cdots\!93 \nu^{16} + \cdots - 71\!\cdots\!56 ) / 19\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!39 \nu^{16} + \cdots - 51\!\cdots\!52 ) / 49\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 35\!\cdots\!11 \nu^{16} + \cdots + 65\!\cdots\!92 ) / 99\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!67 \nu^{16} + \cdots - 57\!\cdots\!84 ) / 30\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 57\!\cdots\!55 \nu^{16} + \cdots + 42\!\cdots\!68 ) / 99\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 69\!\cdots\!81 \nu^{16} + \cdots - 21\!\cdots\!96 ) / 99\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{8} + \beta_{6} - \beta_{5} + \cdots + 270 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} + \beta_{11} - 4 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \cdots + 141 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{16} - \beta_{15} - 38 \beta_{14} + 45 \beta_{13} - 42 \beta_{12} - 38 \beta_{11} + 51 \beta_{10} + \cdots + 6530 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{16} + 24 \beta_{15} - 13 \beta_{14} + 5 \beta_{13} + 63 \beta_{12} + 75 \beta_{11} + \cdots + 5057 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 76 \beta_{16} - 28 \beta_{15} - 1170 \beta_{14} + 1574 \beta_{13} - 1413 \beta_{12} - 1173 \beta_{11} + \cdots + 169491 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 345 \beta_{16} + 1579 \beta_{15} - 723 \beta_{14} + 250 \beta_{13} + 2655 \beta_{12} + 3491 \beta_{11} + \cdots + 163801 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3718 \beta_{16} - 190 \beta_{15} - 34032 \beta_{14} + 50710 \beta_{13} - 44323 \beta_{12} + \cdots + 4568062 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 18009 \beta_{16} + 70427 \beta_{15} - 27372 \beta_{14} + 9395 \beta_{13} + 94656 \beta_{12} + \cdots + 5080318 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 153116 \beta_{16} + 18360 \beta_{15} - 970618 \beta_{14} + 1577410 \beta_{13} - 1347192 \beta_{12} + \cdots + 125942716 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 764378 \beta_{16} + 2660158 \beta_{15} - 879847 \beta_{14} + 342061 \beta_{13} + 3096891 \beta_{12} + \cdots + 154334378 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5757958 \beta_{16} + 1212598 \beta_{15} - 27452667 \beta_{14} + 48227045 \beta_{13} + \cdots + 3524491166 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 29143851 \beta_{16} + 91765041 \beta_{15} - 25797947 \beta_{14} + 12824784 \beta_{13} + \cdots + 4638375275 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 204737235 \beta_{16} + 52327109 \beta_{15} - 772903021 \beta_{14} + 1460641142 \beta_{13} + \cdots + 99659560772 \) 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
5.45173
5.30056
4.70666
3.98991
2.79502
2.74484
2.02345
1.75097
−0.114461
−0.549217
−1.07101
−2.47248
−3.17483
−3.21160
−4.18927
−4.61453
−5.36576
−5.45173 6.71552 21.7213 −3.97680 −36.6112 −10.9778 −74.8051 18.0982 21.6804
1.2 −5.30056 −2.59422 20.0959 4.49343 13.7508 33.2644 −64.1151 −20.2700 −23.8177
1.3 −4.70666 −10.1964 14.1527 −19.7628 47.9909 2.86520 −28.9586 76.9660 93.0167
1.4 −3.98991 1.67407 7.91939 20.8270 −6.67941 −9.96761 0.321630 −24.1975 −83.0979
1.5 −2.79502 10.1456 −0.187860 −5.45712 −28.3572 −7.73594 22.8852 75.9332 15.2528
1.6 −2.74484 −4.19208 −0.465832 −5.57379 11.5066 6.88148 23.2374 −9.42645 15.2992
1.7 −2.02345 −7.47211 −3.90564 6.22433 15.1195 −28.5679 24.0905 28.8324 −12.5946
1.8 −1.75097 3.47555 −4.93410 8.23074 −6.08559 17.9650 22.6472 −14.9205 −14.4118
1.9 0.114461 5.80602 −7.98690 −11.2238 0.664561 15.9656 −1.82987 6.70989 −1.28469
1.10 0.549217 −0.561969 −7.69836 −16.1600 −0.308643 −31.1335 −8.62181 −26.6842 −8.87537
1.11 1.07101 −5.99193 −6.85294 −0.973634 −6.41740 28.3191 −15.9076 8.90321 −1.04277
1.12 2.47248 6.49330 −1.88686 10.7576 16.0545 −24.9760 −24.4450 15.1629 26.5979
1.13 3.17483 2.79441 2.07954 8.68483 8.87177 6.73544 −18.7965 −19.1913 27.5728
1.14 3.21160 −6.73941 2.31437 −9.24473 −21.6443 −8.56428 −18.2600 18.4196 −29.6904
1.15 4.18927 −8.20774 9.54998 18.8743 −34.3844 14.8122 6.49330 40.3670 79.0697
1.16 4.61453 3.68758 13.2939 −19.7910 17.0165 18.7869 24.4287 −13.4017 −91.3259
1.17 5.36576 −0.836231 20.7914 −1.92861 −4.48702 −17.6723 68.6355 −26.3007 −10.3485
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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.f 17
13.b even 2 1 1859.4.a.i 17
13.e even 6 2 143.4.e.a ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.e.a ✓ 34 13.e even 6 2
1859.4.a.f 17 1.a even 1 1 trivial
1859.4.a.i 17 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} + 4 T_{2}^{16} - 99 T_{2}^{15} - 375 T_{2}^{14} + 3949 T_{2}^{13} + 13998 T_{2}^{12} + \cdots - 2596992 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + 4 T^{16} + \cdots - 2596992 \) Copy content Toggle raw display
$3$ \( T^{17} + \cdots + 19875034496 \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T - 11)^{17} \) Copy content Toggle raw display
$13$ \( T^{17} \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 40\!\cdots\!57 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 59\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 64\!\cdots\!74 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 87\!\cdots\!50 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 89\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 74\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 44\!\cdots\!54 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 68\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 13\!\cdots\!50 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 28\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 30\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 30\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 81\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 12\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
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