Properties

Label 1840.4.a.ba
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + 2423016 x^{2} + 864576 x + 57408 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + 5 q^{5} + (\beta_{6} - \beta_1 - 1) q^{7} + (\beta_{2} + \beta_1 + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + 5 q^{5} + (\beta_{6} - \beta_1 - 1) q^{7} + (\beta_{2} + \beta_1 + 13) q^{9} + (\beta_{5} - \beta_1 + 6) q^{11} + (\beta_{7} + 5) q^{13} - 5 \beta_1 q^{15} + ( - \beta_{4} - \beta_1 + 24) q^{17} + ( - \beta_{9} + \beta_{7} + 2 \beta_{6} + \beta_{5} - 4 \beta_1 - 7) q^{19} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{2} + 3 \beta_1 + 33) q^{21} + 23 q^{23} + 25 q^{25} + (\beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 18 \beta_1 - 36) q^{27} + ( - 3 \beta_{8} - \beta_{7} - \beta_{6} + 7 \beta_1 + 29) q^{29} + ( - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 12) q^{31} + ( - \beta_{9} + 3 \beta_{8} - \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_1 + 51) q^{33} + (5 \beta_{6} - 5 \beta_1 - 5) q^{35} + (4 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{2} + 8 \beta_1 + 54) q^{37} + (4 \beta_{9} - 5 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + 2 \beta_{4} + \cdots - 7) q^{39}+ \cdots + ( - 12 \beta_{9} + 14 \beta_{8} + 16 \beta_{7} + 3 \beta_{6} + 16 \beta_{5} + \cdots - 80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9} + 56 q^{11} + 49 q^{13} - 25 q^{15} + 240 q^{17} - 88 q^{19} + 346 q^{21} + 230 q^{23} + 250 q^{25} - 449 q^{27} + 319 q^{29} + 109 q^{31} + 504 q^{33} - 70 q^{35} + 580 q^{37} - 107 q^{39} + 259 q^{41} - 330 q^{43} + 695 q^{45} - 227 q^{47} + 630 q^{49} + 192 q^{51} - 186 q^{53} + 280 q^{55} + 1708 q^{57} - 262 q^{59} + 1000 q^{61} - 722 q^{63} + 245 q^{65} - 354 q^{67} - 115 q^{69} - 599 q^{71} + 355 q^{73} - 125 q^{75} + 1776 q^{77} + 1068 q^{79} + 3490 q^{81} - 754 q^{83} + 1200 q^{85} - 2675 q^{87} + 1740 q^{89} - 690 q^{91} + 1669 q^{93} - 440 q^{95} + 2592 q^{97} - 916 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + 2423016 x^{2} + 864576 x + 57408 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13492675 \nu^{9} - 90064369 \nu^{8} - 2424760468 \nu^{7} + 14223759494 \nu^{6} + 137945915486 \nu^{5} - 675980973244 \nu^{4} + \cdots - 13787240781840 ) / 189638109072 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 133928635 \nu^{9} - 932238831 \nu^{8} - 23988456900 \nu^{7} + 148152752502 \nu^{6} + 1380501328554 \nu^{5} + \cdots + 46471969427424 ) / 1706742981648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 74568380 \nu^{9} + 381349725 \nu^{8} + 14353347492 \nu^{7} - 60353430060 \nu^{6} - 914845733946 \nu^{5} + \cdots + 7611310191408 ) / 853371490824 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 73632521 \nu^{9} - 380531223 \nu^{8} - 13939762884 \nu^{7} + 57963763818 \nu^{6} + 869421171810 \nu^{5} - 2581999841436 \nu^{4} + \cdots + 30379739386224 ) / 568914327216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 157136465 \nu^{9} - 873082230 \nu^{8} - 30030328044 \nu^{7} + 136872598350 \nu^{6} + 1903576710780 \nu^{5} + \cdots + 44170193069544 ) / 853371490824 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 88362863 \nu^{9} + 431383173 \nu^{8} + 17149709286 \nu^{7} - 66106410600 \nu^{6} - 1106918855226 \nu^{5} + \cdots - 61611158517060 ) / 426685745412 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 230763412 \nu^{9} + 1153582707 \nu^{8} + 44674875612 \nu^{7} - 177340372980 \nu^{6} - 2890571942574 \nu^{5} + \cdots - 98477372031504 ) / 853371490824 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} - 2\beta_{7} - 2\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 72\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 10\beta_{8} - 8\beta_{7} - 7\beta_{6} + 11\beta_{4} - 4\beta_{3} + 99\beta_{2} + 145\beta _1 + 2829 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 157 \beta_{9} - 91 \beta_{8} - 266 \beta_{7} - 245 \beta_{6} - 9 \beta_{5} + 160 \beta_{4} + 85 \beta_{3} + 190 \beta_{2} + 6003 \beta _1 + 5097 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34 \beta_{9} - 1390 \beta_{8} - 1430 \beta_{7} - 1243 \beta_{6} - 555 \beta_{5} + 1757 \beta_{4} - 574 \beta_{3} + 9330 \beta_{2} + 18382 \beta _1 + 232185 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 17788 \beta_{9} - 7819 \beta_{8} - 30512 \beta_{7} - 24080 \beta_{6} - 3465 \beta_{5} + 20305 \beta_{4} + 5755 \beta_{3} + 26830 \beta_{2} + 536847 \beta _1 + 653064 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7706 \beta_{9} - 156058 \beta_{8} - 199196 \beta_{7} - 151990 \beta_{6} - 110946 \beta_{5} + 218168 \beta_{4} - 68386 \beta_{3} + 886527 \beta_{2} + 2198497 \beta _1 + 20541678 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1822591 \beta_{9} - 746035 \beta_{8} - 3374474 \beta_{7} - 2265212 \beta_{6} - 686178 \beta_{5} + 2362249 \beta_{4} + 321355 \beta_{3} + 3408271 \beta_{2} + 50118384 \beta _1 + 79069830 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
10.3184
8.50847
6.17442
4.57895
−0.0875767
−0.292216
−2.78233
−4.67252
−7.60722
−9.13838
0 −10.3184 0 5.00000 0 −13.9200 0 79.4692 0
1.2 0 −8.50847 0 5.00000 0 −14.9751 0 45.3940 0
1.3 0 −6.17442 0 5.00000 0 21.9366 0 11.1235 0
1.4 0 −4.57895 0 5.00000 0 −22.7390 0 −6.03324 0
1.5 0 0.0875767 0 5.00000 0 28.3995 0 −26.9923 0
1.6 0 0.292216 0 5.00000 0 −23.7015 0 −26.9146 0
1.7 0 2.78233 0 5.00000 0 11.8146 0 −19.2586 0
1.8 0 4.67252 0 5.00000 0 −26.1910 0 −5.16760 0
1.9 0 7.60722 0 5.00000 0 20.6846 0 30.8698 0
1.10 0 9.13838 0 5.00000 0 4.69126 0 56.5099 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 1840.4.a.ba 10
4.b odd 2 1 920.4.a.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.h 10 4.b odd 2 1
1840.4.a.ba 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{10} + 5 T_{3}^{9} - 192 T_{3}^{8} - 762 T_{3}^{7} + 12246 T_{3}^{6} + 33828 T_{3}^{5} - 298243 T_{3}^{4} - 383603 T_{3}^{3} + 2423016 T_{3}^{2} - 864576 T_{3} + 57408 \) Copy content Toggle raw display
\( T_{7}^{10} + 14 T_{7}^{9} - 1932 T_{7}^{8} - 26286 T_{7}^{7} + 1329984 T_{7}^{6} + 16807960 T_{7}^{5} - 395834563 T_{7}^{4} - 4276110998 T_{7}^{3} + 48504293896 T_{7}^{2} + \cdots - 2101566628800 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 5 T^{9} - 192 T^{8} + \cdots + 57408 \) Copy content Toggle raw display
$5$ \( (T - 5)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 14 T^{9} + \cdots - 2101566628800 \) Copy content Toggle raw display
$11$ \( T^{10} - 56 T^{9} + \cdots + 80040845736576 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 923318227263000 \) Copy content Toggle raw display
$17$ \( T^{10} - 240 T^{9} + \cdots - 83\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{10} + 88 T^{9} + \cdots + 65\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( (T - 23)^{10} \) Copy content Toggle raw display
$29$ \( T^{10} - 319 T^{9} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{10} - 109 T^{9} + \cdots - 90\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{10} - 580 T^{9} + \cdots - 71\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{10} - 259 T^{9} + \cdots - 66\!\cdots\!14 \) Copy content Toggle raw display
$43$ \( T^{10} + 330 T^{9} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + 227 T^{9} + \cdots - 34\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{10} + 186 T^{9} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{10} + 262 T^{9} + \cdots - 43\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{10} - 1000 T^{9} + \cdots - 21\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{10} + 354 T^{9} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{10} + 599 T^{9} + \cdots + 79\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{10} - 355 T^{9} + \cdots + 41\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{10} - 1068 T^{9} + \cdots - 30\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{10} + 754 T^{9} + \cdots - 83\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{10} - 1740 T^{9} + \cdots - 56\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( T^{10} - 2592 T^{9} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
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