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Properties

Label 1859.4.a.i
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $0$
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: \(0\)
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
−5.36576
−4.61453
−4.18927
−3.21160
−3.17483
−2.47248
−1.07101
−0.549217
−0.114461
1.75097
2.02345
2.74484
2.79502
3.98991
4.70666
5.30056
5.45173
−5.36576 −0.836231 20.7914 1.92861 4.48702 17.6723 −68.6355 −26.3007 −10.3485
1.2 −4.61453 3.68758 13.2939 19.7910 −17.0165 −18.7869 −24.4287 −13.4017 −91.3259
1.3 −4.18927 −8.20774 9.54998 −18.8743 34.3844 −14.8122 −6.49330 40.3670 79.0697
1.4 −3.21160 −6.73941 2.31437 9.24473 21.6443 8.56428 18.2600 18.4196 −29.6904
1.5 −3.17483 2.79441 2.07954 −8.68483 −8.87177 −6.73544 18.7965 −19.1913 27.5728
1.6 −2.47248 6.49330 −1.88686 −10.7576 −16.0545 24.9760 24.4450 15.1629 26.5979
1.7 −1.07101 −5.99193 −6.85294 0.973634 6.41740 −28.3191 15.9076 8.90321 −1.04277
1.8 −0.549217 −0.561969 −7.69836 16.1600 0.308643 31.1335 8.62181 −26.6842 −8.87537
1.9 −0.114461 5.80602 −7.98690 11.2238 −0.664561 −15.9656 1.82987 6.70989 −1.28469
1.10 1.75097 3.47555 −4.93410 −8.23074 6.08559 −17.9650 −22.6472 −14.9205 −14.4118
1.11 2.02345 −7.47211 −3.90564 −6.22433 −15.1195 28.5679 −24.0905 28.8324 −12.5946
1.12 2.74484 −4.19208 −0.465832 5.57379 −11.5066 −6.88148 −23.2374 −9.42645 15.2992
1.13 2.79502 10.1456 −0.187860 5.45712 28.3572 7.73594 −22.8852 75.9332 15.2528
1.14 3.98991 1.67407 7.91939 −20.8270 6.67941 9.96761 −0.321630 −24.1975 −83.0979
1.15 4.70666 −10.1964 14.1527 19.7628 −47.9909 −2.86520 28.9586 76.9660 93.0167
1.16 5.30056 −2.59422 20.0959 −4.49343 −13.7508 −33.2644 64.1151 −20.2700 −23.8177
1.17 5.45173 6.71552 21.7213 3.97680 36.6112 10.9778 74.8051 18.0982 21.6804
\(n\): 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.i 17
13.b even 2 1 1859.4.a.f 17
13.c even 3 2 143.4.e.a ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.e.a ✓ 34 13.c even 3 2
1859.4.a.f 17 13.b even 2 1
1859.4.a.i 17 1.a even 1 1 trivial

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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