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Properties

Label 8030.2.a.bf
Level $8030$
Weight $2$
Character orbit 8030.a
Self dual yes
Analytic conductor $64.120$
Analytic rank $1$
Dimension $15$
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: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 4 x^{14} - 23 x^{13} + 99 x^{12} + 191 x^{11} - 922 x^{10} - 702 x^{9} + 4108 x^{8} + 957 x^{7} + \cdots + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 23 x^{13} + 99 x^{12} + 191 x^{11} - 922 x^{10} - 702 x^{9} + 4108 x^{8} + 957 x^{7} + \cdots + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 366035 \nu^{14} - 1741279 \nu^{13} - 10239084 \nu^{12} + 56953737 \nu^{11} + 90005878 \nu^{10} + \cdots + 261843832 ) / 85835268 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 221319 \nu^{14} - 1050335 \nu^{13} - 1195170 \nu^{12} + 13572861 \nu^{11} - 36046464 \nu^{10} + \cdots - 70626556 ) / 28611756 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 960341 \nu^{14} - 2079073 \nu^{13} - 24085422 \nu^{12} + 48433869 \nu^{11} + 218289340 \nu^{10} + \cdots + 498906736 ) / 85835268 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1401323 \nu^{14} - 3706999 \nu^{13} - 32508126 \nu^{12} + 80618787 \nu^{11} + 269290000 \nu^{10} + \cdots + 246176608 ) / 85835268 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3305617 \nu^{14} + 10474127 \nu^{13} + 80134722 \nu^{12} - 252534357 \nu^{11} + \cdots + 60835336 ) / 85835268 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1280809 \nu^{14} + 4426957 \nu^{13} + 33773790 \nu^{12} - 119493219 \nu^{11} - 333339020 \nu^{10} + \cdots - 18158464 ) / 28611756 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2319073 \nu^{14} - 9327080 \nu^{13} - 53426214 \nu^{12} + 228472740 \nu^{11} + 453199208 \nu^{10} + \cdots - 66381772 ) / 42917634 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5483185 \nu^{14} + 25183331 \nu^{13} + 106053138 \nu^{12} - 580526073 \nu^{11} + \cdots - 214919840 ) / 85835268 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1064697 \nu^{14} - 3945448 \nu^{13} - 25941384 \nu^{12} + 100332648 \nu^{11} + 233803794 \nu^{10} + \cdots + 23240302 ) / 14305878 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7527395 \nu^{14} - 29036557 \nu^{13} - 178480998 \nu^{12} + 733439763 \nu^{11} + 1531273792 \nu^{10} + \cdots + 771923284 ) / 85835268 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 438707 \nu^{14} + 1765566 \nu^{13} + 10104105 \nu^{12} - 43697817 \nu^{11} - 84343819 \nu^{10} + \cdots - 15724668 ) / 4768626 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3990373 \nu^{14} + 16003502 \nu^{13} + 91665171 \nu^{12} - 395775153 \nu^{11} - 759666443 \nu^{10} + \cdots - 1774460 ) / 42917634 \) 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
\(\nu^{3}\)\(=\) \( - \beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{11} - \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{3} + 10\beta_{2} + 2\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{14} - 3 \beta_{13} + 13 \beta_{12} - 4 \beta_{11} + 13 \beta_{10} - 16 \beta_{9} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7 \beta_{14} - 5 \beta_{13} + 13 \beta_{12} - 23 \beta_{11} - 9 \beta_{9} - 20 \beta_{8} + 31 \beta_{7} + \cdots + 240 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 163 \beta_{14} - 60 \beta_{13} + 141 \beta_{12} - 79 \beta_{11} + 137 \beta_{10} - 208 \beta_{9} + \cdots + 359 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 147 \beta_{14} - 110 \beta_{13} + 151 \beta_{12} - 348 \beta_{11} - 2 \beta_{10} - 213 \beta_{9} + \cdots + 2305 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1806 \beta_{14} - 849 \beta_{13} + 1479 \beta_{12} - 1140 \beta_{11} + 1356 \beta_{10} - 2511 \beta_{9} + \cdots + 4104 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2195 \beta_{14} - 1700 \beta_{13} + 1761 \beta_{12} - 4531 \beta_{11} - 57 \beta_{10} - 3498 \beta_{9} + \cdots + 23478 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 19682 \beta_{14} - 10588 \beta_{13} + 15478 \beta_{12} - 14625 \beta_{11} + 13103 \beta_{10} + \cdots + 46607 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 28912 \beta_{14} - 22917 \beta_{13} + 20741 \beta_{12} - 55160 \beta_{11} - 1004 \beta_{10} + \cdots + 246608 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 213402 \beta_{14} - 124866 \beta_{13} + 162850 \beta_{12} - 177555 \beta_{11} + 125423 \beta_{10} + \cdots + 530114 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 359398 \beta_{14} - 289003 \beta_{13} + 244941 \beta_{12} - 649452 \beta_{11} - 14003 \beta_{10} + \cdots + 2635917 \) 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
3.35557
2.96469
2.44568
2.39402
1.82644
1.49088
0.440451
0.142263
−0.0958822
−0.205398
−1.67023
−1.82374
−1.99698
−2.16715
−3.10061
1.00000 −3.35557 1.00000 −1.00000 −3.35557 3.99638 1.00000 8.25982 −1.00000
1.2 1.00000 −2.96469 1.00000 −1.00000 −2.96469 −3.63894 1.00000 5.78937 −1.00000
1.3 1.00000 −2.44568 1.00000 −1.00000 −2.44568 −0.745785 1.00000 2.98137 −1.00000
1.4 1.00000 −2.39402 1.00000 −1.00000 −2.39402 −0.390362 1.00000 2.73133 −1.00000
1.5 1.00000 −1.82644 1.00000 −1.00000 −1.82644 3.00453 1.00000 0.335869 −1.00000
1.6 1.00000 −1.49088 1.00000 −1.00000 −1.49088 −3.98101 1.00000 −0.777287 −1.00000
1.7 1.00000 −0.440451 1.00000 −1.00000 −0.440451 3.12770 1.00000 −2.80600 −1.00000
1.8 1.00000 −0.142263 1.00000 −1.00000 −0.142263 −2.21205 1.00000 −2.97976 −1.00000
1.9 1.00000 0.0958822 1.00000 −1.00000 0.0958822 −1.44050 1.00000 −2.99081 −1.00000
1.10 1.00000 0.205398 1.00000 −1.00000 0.205398 3.29352 1.00000 −2.95781 −1.00000
1.11 1.00000 1.67023 1.00000 −1.00000 1.67023 1.44867 1.00000 −0.210343 −1.00000
1.12 1.00000 1.82374 1.00000 −1.00000 1.82374 −3.76421 1.00000 0.326027 −1.00000
1.13 1.00000 1.99698 1.00000 −1.00000 1.99698 0.970755 1.00000 0.987937 −1.00000
1.14 1.00000 2.16715 1.00000 −1.00000 2.16715 −3.23897 1.00000 1.69652 −1.00000
1.15 1.00000 3.10061 1.00000 −1.00000 3.10061 −2.42975 1.00000 6.61377 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.bf ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.bf ✓ 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{15} + 4 T_{3}^{14} - 23 T_{3}^{13} - 99 T_{3}^{12} + 191 T_{3}^{11} + 922 T_{3}^{10} - 702 T_{3}^{9} + \cdots - 8 \) Copy content Toggle raw display
\( T_{7}^{15} + 6 T_{7}^{14} - 40 T_{7}^{13} - 290 T_{7}^{12} + 485 T_{7}^{11} + 5342 T_{7}^{10} + \cdots + 69248 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + 4 T^{14} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 6 T^{14} + \cdots + 69248 \) Copy content Toggle raw display
$11$ \( (T + 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} + 6 T^{14} + \cdots + 409824 \) Copy content Toggle raw display
$17$ \( T^{15} - 2 T^{14} + \cdots - 2419624 \) Copy content Toggle raw display
$19$ \( T^{15} + 8 T^{14} + \cdots + 640016 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 1258312704 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 16355612232 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 13085051904 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 5024668504 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 339120384 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 3547846656 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 5851271168 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 13187262528 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 18676833024 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 161445847616 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 1868436469376 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 1771619712 \) Copy content Toggle raw display
$73$ \( (T - 1)^{15} \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 904257536 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 5743243221504 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 1609884514194 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 73536688375296 \) Copy content Toggle raw display
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