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Properties

Label 8030.2.a.bh
Level $8030$
Weight $2$
Character orbit 8030.a
Self dual yes
Analytic conductor $64.120$
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] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.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: \(64.1198728231\)
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: \( x^{17} - x^{16} - 37 x^{15} + 36 x^{14} + 545 x^{13} - 496 x^{12} - 4124 x^{11} + 3309 x^{10} + \cdots - 280 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + \beta_{11} q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + \beta_{11} q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + q^{10} - q^{11} - \beta_1 q^{12} - \beta_{13} q^{13} - \beta_{11} q^{14} + \beta_1 q^{15} + q^{16} + \beta_{10} q^{17} + ( - \beta_{2} - 1) q^{18} - \beta_{14} q^{19} - q^{20} + (\beta_{13} + \beta_{12} + \cdots + \beta_{3}) q^{21}+ \cdots + ( - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 17 q^{2} - q^{3} + 17 q^{4} - 17 q^{5} + q^{6} + q^{7} - 17 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 17 q^{2} - q^{3} + 17 q^{4} - 17 q^{5} + q^{6} + q^{7} - 17 q^{8} + 24 q^{9} + 17 q^{10} - 17 q^{11} - q^{12} + 4 q^{13} - q^{14} + q^{15} + 17 q^{16} + 4 q^{17} - 24 q^{18} + 2 q^{19} - 17 q^{20} - 4 q^{21} + 17 q^{22} - 17 q^{23} + q^{24} + 17 q^{25} - 4 q^{26} + 2 q^{27} + q^{28} + 11 q^{29} - q^{30} + 6 q^{31} - 17 q^{32} + q^{33} - 4 q^{34} - q^{35} + 24 q^{36} - 3 q^{37} - 2 q^{38} + 11 q^{39} + 17 q^{40} + 3 q^{41} + 4 q^{42} - 7 q^{43} - 17 q^{44} - 24 q^{45} + 17 q^{46} - 30 q^{47} - q^{48} + 44 q^{49} - 17 q^{50} + 4 q^{51} + 4 q^{52} - 16 q^{53} - 2 q^{54} + 17 q^{55} - q^{56} + 26 q^{57} - 11 q^{58} + 4 q^{59} + q^{60} + 32 q^{61} - 6 q^{62} - 7 q^{63} + 17 q^{64} - 4 q^{65} - q^{66} - 6 q^{67} + 4 q^{68} + 9 q^{69} + q^{70} - 26 q^{71} - 24 q^{72} + 17 q^{73} + 3 q^{74} - q^{75} + 2 q^{76} - q^{77} - 11 q^{78} + 50 q^{79} - 17 q^{80} + 41 q^{81} - 3 q^{82} - 50 q^{83} - 4 q^{84} - 4 q^{85} + 7 q^{86} + 7 q^{87} + 17 q^{88} + 10 q^{89} + 24 q^{90} + 13 q^{91} - 17 q^{92} - 11 q^{93} + 30 q^{94} - 2 q^{95} + q^{96} + 25 q^{97} - 44 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - x^{16} - 37 x^{15} + 36 x^{14} + 545 x^{13} - 496 x^{12} - 4124 x^{11} + 3309 x^{10} + \cdots - 280 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5480399918128 \nu^{16} + 22513103089541 \nu^{15} + 181267142667560 \nu^{14} + \cdots - 854688964817456 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10535072621233 \nu^{16} + 26272773347426 \nu^{15} + 359846686243862 \nu^{14} + \cdots + 90\!\cdots\!08 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23092425852475 \nu^{16} - 55485450710479 \nu^{15} + 866479507006742 \nu^{14} + \cdots - 25\!\cdots\!76 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23103968391359 \nu^{16} - 35569173308422 \nu^{15} - 820936874070856 \nu^{14} + \cdots - 30\!\cdots\!30 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27548752610518 \nu^{16} + 27354732769355 \nu^{15} + \cdots + 94\!\cdots\!92 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33023844999707 \nu^{16} + 24892579860466 \nu^{15} + \cdots - 12\!\cdots\!14 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33083525256926 \nu^{16} + 76170803482489 \nu^{15} + \cdots + 11\!\cdots\!96 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11903156261155 \nu^{16} + 13114845648070 \nu^{15} + 415878145202378 \nu^{14} + \cdots + 92\!\cdots\!58 ) / 722434804854214 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 58808126343037 \nu^{16} + 72517311449276 \nu^{15} + \cdots + 80\!\cdots\!24 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66449062099424 \nu^{16} + 156248250179140 \nu^{15} + \cdots + 38\!\cdots\!74 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 84004558254643 \nu^{16} - 154483454730872 \nu^{15} + \cdots - 41\!\cdots\!96 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 88370776945300 \nu^{16} + 153608392858745 \nu^{15} + \cdots + 42\!\cdots\!62 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 123032165609735 \nu^{16} - 228359058610078 \nu^{15} + \cdots - 39\!\cdots\!42 ) / 21\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 68438411725867 \nu^{16} - 141749949625154 \nu^{15} + \cdots - 27\!\cdots\!08 ) / 722434804854214 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 7 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{14} - 3 \beta_{13} - \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{5} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} + 12 \beta_{14} + 14 \beta_{13} + 15 \beta_{12} - 9 \beta_{11} - 14 \beta_{10} + \beta_{9} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{16} + 14 \beta_{15} + 13 \beta_{14} - 50 \beta_{13} - 21 \beta_{12} - 55 \beta_{11} + \cdots + 236 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 25 \beta_{16} - 7 \beta_{15} + 129 \beta_{14} + 176 \beta_{13} + 187 \beta_{12} - 53 \beta_{11} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 92 \beta_{16} + 173 \beta_{15} + 128 \beta_{14} - 670 \beta_{13} - 309 \beta_{12} - 768 \beta_{11} + \cdots + 2159 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 426 \beta_{16} - 176 \beta_{15} + 1379 \beta_{14} + 2142 \beta_{13} + 2203 \beta_{12} - 66 \beta_{11} + \cdots - 578 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1512 \beta_{16} + 2104 \beta_{15} + 1112 \beta_{14} - 8340 \beta_{13} - 4033 \beta_{12} + \cdots + 20630 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6237 \beta_{16} - 3041 \beta_{15} + 14867 \beta_{14} + 25669 \beta_{13} + 25367 \beta_{12} + 4601 \beta_{11} + \cdots - 7056 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 21755 \beta_{16} + 25462 \beta_{15} + 8613 \beta_{14} - 100418 \beta_{13} - 50075 \beta_{12} + \cdots + 202418 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 84404 \beta_{16} - 45048 \beta_{15} + 161590 \beta_{14} + 304845 \beta_{13} + 289248 \beta_{12} + \cdots - 86225 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 291809 \beta_{16} + 306176 \beta_{15} + 54992 \beta_{14} - 1188386 \beta_{13} - 607146 \beta_{12} + \cdots + 2022421 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1089746 \beta_{16} - 615512 \beta_{15} + 1766843 \beta_{14} + 3598627 \beta_{13} + 3285162 \beta_{12} + \cdots - 1056984 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3749897 \beta_{16} + 3653849 \beta_{15} + 182006 \beta_{14} - 13926126 \beta_{13} - 7267863 \beta_{12} + \cdots + 20488783 \) 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
3.21234
3.07211
2.45288
2.20538
1.98561
1.48163
1.42314
0.302920
0.0740730
−0.776156
−0.804548
−1.15772
−1.38127
−2.27573
−2.38236
−3.03941
−3.39289
−1.00000 −3.21234 1.00000 −1.00000 3.21234 0.386703 −1.00000 7.31913 1.00000
1.2 −1.00000 −3.07211 1.00000 −1.00000 3.07211 −4.13515 −1.00000 6.43787 1.00000
1.3 −1.00000 −2.45288 1.00000 −1.00000 2.45288 2.78508 −1.00000 3.01661 1.00000
1.4 −1.00000 −2.20538 1.00000 −1.00000 2.20538 5.07955 −1.00000 1.86371 1.00000
1.5 −1.00000 −1.98561 1.00000 −1.00000 1.98561 −4.35012 −1.00000 0.942630 1.00000
1.6 −1.00000 −1.48163 1.00000 −1.00000 1.48163 3.95593 −1.00000 −0.804770 1.00000
1.7 −1.00000 −1.42314 1.00000 −1.00000 1.42314 −0.985117 −1.00000 −0.974673 1.00000
1.8 −1.00000 −0.302920 1.00000 −1.00000 0.302920 2.86257 −1.00000 −2.90824 1.00000
1.9 −1.00000 −0.0740730 1.00000 −1.00000 0.0740730 −1.71486 −1.00000 −2.99451 1.00000
1.10 −1.00000 0.776156 1.00000 −1.00000 −0.776156 0.919898 −1.00000 −2.39758 1.00000
1.11 −1.00000 0.804548 1.00000 −1.00000 −0.804548 −2.71727 −1.00000 −2.35270 1.00000
1.12 −1.00000 1.15772 1.00000 −1.00000 −1.15772 −2.67230 −1.00000 −1.65968 1.00000
1.13 −1.00000 1.38127 1.00000 −1.00000 −1.38127 −1.56817 −1.00000 −1.09209 1.00000
1.14 −1.00000 2.27573 1.00000 −1.00000 −2.27573 −1.02794 −1.00000 2.17895 1.00000
1.15 −1.00000 2.38236 1.00000 −1.00000 −2.38236 4.85422 −1.00000 2.67562 1.00000
1.16 −1.00000 3.03941 1.00000 −1.00000 −3.03941 3.04264 −1.00000 6.23802 1.00000
1.17 −1.00000 3.39289 1.00000 −1.00000 −3.39289 −3.71566 −1.00000 8.51170 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(11\) \(1\)
\(73\) \(-1\)

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 8030.2.a.bh ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.bh ✓ 17 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{17} + T_{3}^{16} - 37 T_{3}^{15} - 36 T_{3}^{14} + 545 T_{3}^{13} + 496 T_{3}^{12} - 4124 T_{3}^{11} + \cdots + 280 \) Copy content Toggle raw display
\( T_{7}^{17} - T_{7}^{16} - 81 T_{7}^{15} + 39 T_{7}^{14} + 2662 T_{7}^{13} + 6 T_{7}^{12} + \cdots + 1112448 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} + T^{16} + \cdots + 280 \) Copy content Toggle raw display
$5$ \( (T + 1)^{17} \) Copy content Toggle raw display
$7$ \( T^{17} - T^{16} + \cdots + 1112448 \) Copy content Toggle raw display
$11$ \( (T + 1)^{17} \) Copy content Toggle raw display
$13$ \( T^{17} - 4 T^{16} + \cdots + 43221720 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 3718861344 \) Copy content Toggle raw display
$19$ \( T^{17} - 2 T^{16} + \cdots - 39722200 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 4158984960 \) Copy content Toggle raw display
$29$ \( T^{17} - 11 T^{16} + \cdots + 24318240 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 3281526784 \) Copy content Toggle raw display
$37$ \( T^{17} + 3 T^{16} + \cdots + 3468928 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 11183394816 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 258524044160 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 758315743488 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 944378880 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 356112220032 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 641142848384 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 2265271781952 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 1350838560 \) Copy content Toggle raw display
$73$ \( (T - 1)^{17} \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 326534668288 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 34494469056 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 5061603810 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 8791255049216 \) Copy content Toggle raw display
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