Properties

Label 8036.2.a.p
Level $8036$
Weight $2$
Character orbit 8036.a
Self dual yes
Analytic conductor $64.168$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 18 x^{8} + 32 x^{7} + 110 x^{6} - 154 x^{5} - 282 x^{4} + 256 x^{3} + 253 x^{2} - 126 x - 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{9} + \beta_{6} q^{11} + ( 1 + \beta_{7} ) q^{13} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{15} + ( 1 - \beta_{4} + \beta_{9} ) q^{17} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{19} + ( -\beta_{4} + \beta_{5} ) q^{23} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{27} + ( 1 + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{31} + ( \beta_{3} + \beta_{6} - \beta_{9} ) q^{33} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{39} + q^{41} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{43} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{45} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{7} + 2 \beta_{9} ) q^{51} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{53} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{55} + ( -3 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{57} + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{59} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{8} ) q^{61} + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{65} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{67} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{69} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{71} + ( 4 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} + ( -9 + 6 \beta_{1} - 6 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{75} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{79} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{81} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{83} + ( \beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{85} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{87} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{89} + ( 2 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{6} + 3 \beta_{7} - 3 \beta_{9} ) q^{93} + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{95} + ( 6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} ) q^{97} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 2q^{3} + 4q^{5} + 10q^{9} + O(q^{10}) \) \( 10q + 2q^{3} + 4q^{5} + 10q^{9} + 6q^{13} - 4q^{15} + 12q^{17} + 8q^{19} + 4q^{23} + 16q^{25} + 8q^{27} + 2q^{29} + 8q^{31} - 6q^{33} - 2q^{37} - 2q^{39} + 10q^{41} - 2q^{43} + 44q^{45} - 14q^{47} + 14q^{51} + 8q^{53} + 8q^{55} - 10q^{57} + 24q^{59} + 14q^{61} + 2q^{65} - 8q^{67} + 16q^{69} + 10q^{71} + 44q^{73} - 50q^{75} + 10q^{79} - 14q^{81} + 20q^{83} + 8q^{85} + 20q^{87} + 6q^{89} + 8q^{93} + 4q^{95} + 46q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 18 x^{8} + 32 x^{7} + 110 x^{6} - 154 x^{5} - 282 x^{4} + 256 x^{3} + 253 x^{2} - 126 x - 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{9} - 3 \nu^{8} + 36 \nu^{7} + 41 \nu^{6} - 213 \nu^{5} - 154 \nu^{4} + 445 \nu^{3} + 132 \nu^{2} - 275 \nu - 28 \)\()/21\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{9} + 3 \nu^{8} - 36 \nu^{7} - 41 \nu^{6} + 213 \nu^{5} + 154 \nu^{4} - 445 \nu^{3} - 111 \nu^{2} + 275 \nu - 56 \)\()/21\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{9} - \nu^{8} + 47 \nu^{7} + 16 \nu^{6} - 218 \nu^{5} - 84 \nu^{4} + 293 \nu^{3} + 72 \nu^{2} - 73 \nu - 7 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( 13 \nu^{9} + 9 \nu^{8} - 213 \nu^{7} - 151 \nu^{6} + 1080 \nu^{5} + 812 \nu^{4} - 1811 \nu^{3} - 1173 \nu^{2} + 895 \nu + 371 \)\()/21\)
\(\beta_{6}\)\(=\)\((\)\( -22 \nu^{9} - 12 \nu^{8} + 354 \nu^{7} + 199 \nu^{6} - 1734 \nu^{5} - 1043 \nu^{4} + 2711 \nu^{3} + 1221 \nu^{2} - 1240 \nu - 266 \)\()/21\)
\(\beta_{7}\)\(=\)\((\)\( -29 \nu^{9} - 12 \nu^{8} + 459 \nu^{7} + 206 \nu^{6} - 2196 \nu^{5} - 1162 \nu^{4} + 3334 \nu^{3} + 1452 \nu^{2} - 1415 \nu - 343 \)\()/21\)
\(\beta_{8}\)\(=\)\((\)\( 4 \nu^{9} + 3 \nu^{8} - 63 \nu^{7} - 49 \nu^{6} + 297 \nu^{5} + 251 \nu^{4} - 425 \nu^{3} - 309 \nu^{2} + 175 \nu + 74 \)\()/3\)
\(\beta_{9}\)\(=\)\((\)\( 12 \nu^{9} + 11 \nu^{8} - 195 \nu^{7} - 176 \nu^{6} + 970 \nu^{5} + 868 \nu^{4} - 1529 \nu^{3} - 1072 \nu^{2} + 691 \nu + 252 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{5} - \beta_{4} + 9 \beta_{3} + 8 \beta_{2} - \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(10 \beta_{9} - 10 \beta_{8} - 11 \beta_{7} + 10 \beta_{6} - 11 \beta_{5} + 2 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 53 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-14 \beta_{9} + 14 \beta_{8} + 14 \beta_{7} - 2 \beta_{6} + 25 \beta_{5} - 13 \beta_{4} + 77 \beta_{3} + 61 \beta_{2} - 20 \beta_{1} + 194\)
\(\nu^{7}\)\(=\)\(87 \beta_{9} - 86 \beta_{8} - 101 \beta_{7} + 88 \beta_{6} - 102 \beta_{5} + 27 \beta_{4} - 108 \beta_{3} - 34 \beta_{2} + 417 \beta_{1} - 50\)
\(\nu^{8}\)\(=\)\(-149 \beta_{9} + 151 \beta_{8} + 150 \beta_{7} - 33 \beta_{6} + 249 \beta_{5} - 129 \beta_{4} + 655 \beta_{3} + 474 \beta_{2} - 269 \beta_{1} + 1522\)
\(\nu^{9}\)\(=\)\(737 \beta_{9} - 722 \beta_{8} - 884 \beta_{7} + 750 \beta_{6} - 902 \beta_{5} + 277 \beta_{4} - 1026 \beta_{3} - 420 \beta_{2} + 3352 \beta_{1} - 745\)

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
−2.96176
−1.91097
−1.73395
−1.05099
−0.284914
0.729469
1.20067
2.58799
2.70664
2.71781
0 −2.96176 0 4.34462 0 0 0 5.77201 0
1.2 0 −1.91097 0 1.95825 0 0 0 0.651806 0
1.3 0 −1.73395 0 −3.41343 0 0 0 0.00658743 0
1.4 0 −1.05099 0 −1.02904 0 0 0 −1.89542 0
1.5 0 −0.284914 0 2.38934 0 0 0 −2.91882 0
1.6 0 0.729469 0 −3.02160 0 0 0 −2.46788 0
1.7 0 1.20067 0 −0.960855 0 0 0 −1.55839 0
1.8 0 2.58799 0 3.78481 0 0 0 3.69770 0
1.9 0 2.70664 0 0.469317 0 0 0 4.32591 0
1.10 0 2.71781 0 −0.521406 0 0 0 4.38650 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
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

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 8036.2.a.p yes 10
7.b odd 2 1 8036.2.a.o 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8036.2.a.o 10 7.b odd 2 1
8036.2.a.p yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)
\(T_{11}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( -49 - 126 T + 253 T^{2} + 256 T^{3} - 282 T^{4} - 154 T^{5} + 110 T^{6} + 32 T^{7} - 18 T^{8} - 2 T^{9} + T^{10} \)
$5$ \( -192 - 192 T + 960 T^{2} + 1072 T^{3} - 592 T^{4} - 614 T^{5} + 201 T^{6} + 92 T^{7} - 25 T^{8} - 4 T^{9} + T^{10} \)
$7$ \( T^{10} \)
$11$ \( 64 + 1248 T + 336 T^{2} - 3280 T^{3} - 1804 T^{4} + 1184 T^{5} + 705 T^{6} - 48 T^{7} - 50 T^{8} + T^{10} \)
$13$ \( 1559 - 3152 T - 6926 T^{2} + 14560 T^{3} - 1304 T^{4} - 4358 T^{5} + 663 T^{6} + 324 T^{7} - 53 T^{8} - 6 T^{9} + T^{10} \)
$17$ \( -18693 + 113046 T - 142484 T^{2} + 35306 T^{3} + 22504 T^{4} - 8966 T^{5} - 827 T^{6} + 586 T^{7} - 17 T^{8} - 12 T^{9} + T^{10} \)
$19$ \( 5067 - 9582 T - 13085 T^{2} + 21600 T^{3} + 6210 T^{4} - 6116 T^{5} - 170 T^{6} + 514 T^{7} - 48 T^{8} - 8 T^{9} + T^{10} \)
$23$ \( 3737 + 38638 T - 120411 T^{2} + 90872 T^{3} - 8272 T^{4} - 10154 T^{5} + 1910 T^{6} + 356 T^{7} - 80 T^{8} - 4 T^{9} + T^{10} \)
$29$ \( 192 - 1152 T - 9552 T^{2} + 20768 T^{3} - 1324 T^{4} - 12994 T^{5} + 3525 T^{6} + 322 T^{7} - 121 T^{8} - 2 T^{9} + T^{10} \)
$31$ \( 3471296 - 4771264 T + 474448 T^{2} + 815632 T^{3} - 110692 T^{4} - 46702 T^{5} + 6249 T^{6} + 1060 T^{7} - 137 T^{8} - 8 T^{9} + T^{10} \)
$37$ \( -11649008 + 1432864 T + 3285076 T^{2} - 179460 T^{3} - 274399 T^{4} + 9348 T^{5} + 9922 T^{6} - 226 T^{7} - 163 T^{8} + 2 T^{9} + T^{10} \)
$41$ \( ( -1 + T )^{10} \)
$43$ \( -980127 - 1869660 T + 3991236 T^{2} - 154050 T^{3} - 409410 T^{4} + 14038 T^{5} + 14517 T^{6} - 306 T^{7} - 209 T^{8} + 2 T^{9} + T^{10} \)
$47$ \( -286272 + 433248 T + 2110024 T^{2} - 421180 T^{3} - 280565 T^{4} + 59596 T^{5} + 8056 T^{6} - 1636 T^{7} - 123 T^{8} + 14 T^{9} + T^{10} \)
$53$ \( -2454336 - 4683360 T - 1246336 T^{2} + 1501248 T^{3} + 294520 T^{4} - 221660 T^{5} + 18389 T^{6} + 2620 T^{7} - 285 T^{8} - 8 T^{9} + T^{10} \)
$59$ \( -7947072 - 2501280 T + 7155648 T^{2} - 2639016 T^{3} - 19632 T^{4} + 207262 T^{5} - 48135 T^{6} + 3968 T^{7} + 39 T^{8} - 24 T^{9} + T^{10} \)
$61$ \( 523584 - 1064448 T + 156240 T^{2} + 408360 T^{3} - 48072 T^{4} - 39366 T^{5} + 3001 T^{6} + 1324 T^{7} - 82 T^{8} - 14 T^{9} + T^{10} \)
$67$ \( 228928 - 1238016 T - 158048 T^{2} + 1123160 T^{3} + 704176 T^{4} + 147448 T^{5} + 3529 T^{6} - 2504 T^{7} - 221 T^{8} + 8 T^{9} + T^{10} \)
$71$ \( 59328 + 380160 T + 619888 T^{2} + 158952 T^{3} - 160008 T^{4} - 52942 T^{5} + 10973 T^{6} + 2128 T^{7} - 266 T^{8} - 10 T^{9} + T^{10} \)
$73$ \( 22060224 - 121263648 T + 152397328 T^{2} - 51085664 T^{3} + 4790412 T^{4} + 525436 T^{5} - 122423 T^{6} + 4400 T^{7} + 490 T^{8} - 44 T^{9} + T^{10} \)
$79$ \( 6294528 - 6008832 T + 393216 T^{2} + 954112 T^{3} - 163776 T^{4} - 51616 T^{5} + 9456 T^{6} + 1208 T^{7} - 176 T^{8} - 10 T^{9} + T^{10} \)
$83$ \( -20754496 + 102250496 T - 102357728 T^{2} + 20258080 T^{3} + 2324252 T^{4} - 807160 T^{5} + 12441 T^{6} + 7444 T^{7} - 302 T^{8} - 20 T^{9} + T^{10} \)
$89$ \( -3358621053 - 1892743680 T + 274484965 T^{2} + 114583092 T^{3} - 14564160 T^{4} - 1244970 T^{5} + 169524 T^{6} + 4766 T^{7} - 712 T^{8} - 6 T^{9} + T^{10} \)
$97$ \( -109176709 - 31519130 T + 177594189 T^{2} - 89184680 T^{3} + 12809534 T^{4} + 543946 T^{5} - 242438 T^{6} + 13628 T^{7} + 322 T^{8} - 46 T^{9} + T^{10} \)
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