Properties

Label 4020.2.a.d
Level 4020
Weight 2
Character orbit 4020.a
Self dual yes
Analytic conductor 32.100
Analytic rank 1
Dimension 4
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0998616126\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{4} - x^{3} - 4 x^{2} + 4 x + 1\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( 2 - 3 \beta - \beta^{2} + \beta^{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} + ( 2 - 3 \beta - \beta^{2} + \beta^{3} ) q^{7} + q^{9} + ( -3 + 10 \beta - 3 \beta^{3} ) q^{11} + ( -3 - 8 \beta + \beta^{2} + 2 \beta^{3} ) q^{13} + q^{15} + ( -2 + 12 \beta - \beta^{2} - 3 \beta^{3} ) q^{17} + ( -1 - 3 \beta ) q^{19} + ( 2 - 3 \beta - \beta^{2} + \beta^{3} ) q^{21} + ( -7 - 4 \beta + 2 \beta^{2} + 2 \beta^{3} ) q^{23} + q^{25} + q^{27} + ( -7 - 11 \beta + 3 \beta^{2} + 3 \beta^{3} ) q^{29} + ( -10 - 7 \beta + 4 \beta^{2} + 2 \beta^{3} ) q^{31} + ( -3 + 10 \beta - 3 \beta^{3} ) q^{33} + ( 2 - 3 \beta - \beta^{2} + \beta^{3} ) q^{35} + ( -9 - 4 \beta + 2 \beta^{2} + 2 \beta^{3} ) q^{37} + ( -3 - 8 \beta + \beta^{2} + 2 \beta^{3} ) q^{39} + ( -4 + 7 \beta - 2 \beta^{3} ) q^{41} + ( 8 - 6 \beta - 3 \beta^{2} + 2 \beta^{3} ) q^{43} + q^{45} + ( -5 + 8 \beta + \beta^{2} - 2 \beta^{3} ) q^{47} + ( -1 - 3 \beta ) q^{49} + ( -2 + 12 \beta - \beta^{2} - 3 \beta^{3} ) q^{51} + ( 3 + 13 \beta - 4 \beta^{2} - 4 \beta^{3} ) q^{53} + ( -3 + 10 \beta - 3 \beta^{3} ) q^{55} + ( -1 - 3 \beta ) q^{57} + ( -7 + 12 \beta + 3 \beta^{2} - 4 \beta^{3} ) q^{59} + ( 7 + 3 \beta - 4 \beta^{2} - 2 \beta^{3} ) q^{61} + ( 2 - 3 \beta - \beta^{2} + \beta^{3} ) q^{63} + ( -3 - 8 \beta + \beta^{2} + 2 \beta^{3} ) q^{65} + q^{67} + ( -7 - 4 \beta + 2 \beta^{2} + 2 \beta^{3} ) q^{69} + ( 15 + 5 \beta - 7 \beta^{2} - 2 \beta^{3} ) q^{71} + ( 2 - 10 \beta - 3 \beta^{2} + 3 \beta^{3} ) q^{73} + q^{75} + ( -13 + \beta + 4 \beta^{2} ) q^{77} + ( 2 - 12 \beta - \beta^{2} + 4 \beta^{3} ) q^{79} + q^{81} + ( 3 + 11 \beta - 3 \beta^{2} - 4 \beta^{3} ) q^{83} + ( -2 + 12 \beta - \beta^{2} - 3 \beta^{3} ) q^{85} + ( -7 - 11 \beta + 3 \beta^{2} + 3 \beta^{3} ) q^{87} + ( -1 + 33 \beta - \beta^{2} - 10 \beta^{3} ) q^{89} + ( 16 \beta - \beta^{2} - 4 \beta^{3} ) q^{91} + ( -10 - 7 \beta + 4 \beta^{2} + 2 \beta^{3} ) q^{93} + ( -1 - 3 \beta ) q^{95} + ( -2 + 22 \beta - 2 \beta^{2} - 7 \beta^{3} ) q^{97} + ( -3 + 10 \beta - 3 \beta^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{5} - 3q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{5} - 3q^{7} + 4q^{9} - 5q^{11} - 9q^{13} + 4q^{15} - 8q^{17} - 7q^{19} - 3q^{21} - 12q^{23} + 4q^{25} + 4q^{27} - 9q^{29} - 9q^{31} - 5q^{33} - 3q^{35} - 20q^{37} - 9q^{39} - 11q^{41} + q^{43} + 4q^{45} - 5q^{47} - 7q^{49} - 8q^{51} - 15q^{53} - 5q^{55} - 7q^{57} + 7q^{59} - 7q^{61} - 3q^{63} - 9q^{65} + 4q^{67} - 12q^{69} - 26q^{73} + 4q^{75} - 15q^{77} - 9q^{79} + 4q^{81} - 8q^{83} - 8q^{85} - 9q^{87} + 10q^{89} + 3q^{91} - 9q^{93} - 7q^{95} - 11q^{97} - 5q^{99} + O(q^{100}) \)

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
−1.95630
1.33826
1.82709
−0.209057
0 1.00000 0 1.00000 0 −3.44512 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −1.40898 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 −0.720227 0 1.00000 0
1.4 0 1.00000 0 1.00000 0 2.57433 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 4020.2.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.d 4 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 3 T_{7}^{3} - 6 T_{7}^{2} - 18 T_{7} - 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{4} \)
$5$ \( ( 1 - T )^{4} \)
$7$ \( 1 + 3 T + 22 T^{2} + 45 T^{3} + 201 T^{4} + 315 T^{5} + 1078 T^{6} + 1029 T^{7} + 2401 T^{8} \)
$11$ \( 1 + 5 T + 34 T^{2} + 115 T^{3} + 501 T^{4} + 1265 T^{5} + 4114 T^{6} + 6655 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 9 T + 63 T^{2} + 330 T^{3} + 1271 T^{4} + 4290 T^{5} + 10647 T^{6} + 19773 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 8 T + 52 T^{2} + 205 T^{3} + 921 T^{4} + 3485 T^{5} + 15028 T^{6} + 39304 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 7 T + 55 T^{2} + 232 T^{3} + 1309 T^{4} + 4408 T^{5} + 19855 T^{6} + 48013 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 12 T + 106 T^{2} + 576 T^{3} + 3099 T^{4} + 13248 T^{5} + 56074 T^{6} + 146004 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 9 T + 97 T^{2} + 642 T^{3} + 4125 T^{4} + 18618 T^{5} + 81577 T^{6} + 219501 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 9 T + 90 T^{2} + 591 T^{3} + 4349 T^{4} + 18321 T^{5} + 86490 T^{6} + 268119 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 20 T + 258 T^{2} + 2200 T^{3} + 15299 T^{4} + 81400 T^{5} + 353202 T^{6} + 1013060 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 11 T + 200 T^{2} + 1379 T^{3} + 13009 T^{4} + 56539 T^{5} + 336200 T^{6} + 758131 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - T + 108 T^{2} - 215 T^{3} + 5561 T^{4} - 9245 T^{5} + 199692 T^{6} - 79507 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 5 T + 173 T^{2} + 680 T^{3} + 11869 T^{4} + 31960 T^{5} + 382157 T^{6} + 519115 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 15 T + 227 T^{2} + 2070 T^{3} + 18489 T^{4} + 109710 T^{5} + 637643 T^{6} + 2233155 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 7 T + 145 T^{2} - 692 T^{3} + 9879 T^{4} - 40828 T^{5} + 504745 T^{6} - 1437653 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 7 T + 163 T^{2} + 1264 T^{3} + 12505 T^{4} + 77104 T^{5} + 606523 T^{6} + 1588867 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 - T )^{4} \)
$71$ \( 1 + 94 T^{2} - 285 T^{3} + 7761 T^{4} - 20235 T^{5} + 473854 T^{6} + 25411681 T^{8} \)
$73$ \( 1 + 26 T + 468 T^{2} + 5485 T^{3} + 53951 T^{4} + 400405 T^{5} + 2493972 T^{6} + 10114442 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 9 T + 292 T^{2} + 1917 T^{3} + 33915 T^{4} + 151443 T^{5} + 1822372 T^{6} + 4437351 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 8 T + 296 T^{2} + 2009 T^{3} + 35359 T^{4} + 166747 T^{5} + 2039144 T^{6} + 4574296 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 10 T + 196 T^{2} - 2345 T^{3} + 22821 T^{4} - 208705 T^{5} + 1552516 T^{6} - 7049690 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 11 T + 339 T^{2} + 2662 T^{3} + 46799 T^{4} + 258214 T^{5} + 3189651 T^{6} + 10039403 T^{7} + 88529281 T^{8} \)
show more
show less