Properties

Label 20.12.c.a
Level 20
Weight 12
Character orbit 20.c
Analytic conductor 15.367
Analytic rank 0
Dimension 6
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 20.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(15.3668636112\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{19}\cdot 5^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + ( -688 + 2 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( 28 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} \) \( + ( -83586 - \beta_{1} - 10 \beta_{2} + \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + ( -688 + 2 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( 28 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} \) \( + ( -83586 - \beta_{1} - 10 \beta_{2} + \beta_{5} ) q^{9} \) \( + ( 216368 + 2 \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{11} \) \( + ( 21 - 333 \beta_{1} + 78 \beta_{2} - 12 \beta_{3} - 3 \beta_{4} ) q^{13} \) \( + ( -551796 - 2566 \beta_{1} - 5 \beta_{2} - 45 \beta_{3} - 9 \beta_{4} + 5 \beta_{5} ) q^{15} \) \( + ( 154 - 358 \beta_{1} + 364 \beta_{2} + 120 \beta_{3} - 22 \beta_{4} ) q^{17} \) \( + ( -41220 - 132 \beta_{1} - 1218 \beta_{2} - 51 \beta_{4} - 21 \beta_{5} ) q^{19} \) \( + ( -7420659 - 299 \beta_{1} - 2828 \beta_{2} - 81 \beta_{4} + 56 \beta_{5} ) q^{21} \) \( + ( 980 + 37366 \beta_{1} + 3089 \beta_{2} - 9 \beta_{3} - 140 \beta_{4} ) q^{23} \) \( + ( 5785650 - 62725 \beta_{1} - 152 \beta_{2} + 320 \beta_{3} - 186 \beta_{4} - 105 \beta_{5} ) q^{25} \) \( + ( 1512 - 128212 \beta_{1} + 6030 \beta_{2} - 1278 \beta_{3} - 216 \beta_{4} ) q^{27} \) \( + ( -23158506 - 504 \beta_{1} - 4836 \beta_{2} - 102 \beta_{4} + 198 \beta_{5} ) q^{29} \) \( + ( 35742772 + 276 \beta_{1} + 2856 \beta_{2} - 48 \beta_{4} - 420 \beta_{5} ) q^{31} \) \( + ( -2772 + 788624 \beta_{1} - 10296 \beta_{2} + 1584 \beta_{3} + 396 \beta_{4} ) q^{33} \) \( + ( -51475884 - 543589 \beta_{1} - 5700 \beta_{2} + 870 \beta_{3} + 799 \beta_{4} + 945 \beta_{5} ) q^{35} \) \( + ( -10731 - 251069 \beta_{1} - 36478 \beta_{2} + 2752 \beta_{3} + 1533 \beta_{4} ) q^{37} \) \( + ( 90473904 + 5808 \beta_{1} + 54948 \beta_{2} + 1566 \beta_{4} - 1110 \beta_{5} ) q^{39} \) \( + ( 17790137 + 7383 \beta_{1} + 67970 \beta_{2} + 2930 \beta_{4} + 1407 \beta_{5} ) q^{41} \) \( + ( -14112 + 2023091 \beta_{1} - 32174 \beta_{2} - 12178 \beta_{3} + 2016 \beta_{4} ) q^{43} \) \( + ( 551892957 - 1656453 \beta_{1} + 59369 \beta_{2} - 15300 \beta_{3} + 2565 \beta_{4} - 4550 \beta_{5} ) q^{45} \) \( + ( -2380 - 367460 \beta_{1} - 23353 \beta_{2} + 15873 \beta_{3} + 340 \beta_{4} ) q^{47} \) \( + ( -751149018 + 1947 \beta_{1} + 15414 \beta_{2} + 2028 \beta_{4} + 4137 \beta_{5} ) q^{49} \) \( + ( 95119644 - 15860 \beta_{1} - 147872 \beta_{2} - 5364 \beta_{4} - 232 \beta_{5} ) q^{51} \) \( + ( 16611 - 648627 \beta_{1} + 10890 \beta_{2} + 41316 \beta_{3} - 2373 \beta_{4} ) q^{53} \) \( + ( -290336560 + 3401240 \beta_{1} - 181910 \beta_{2} + 53900 \beta_{3} - 12345 \beta_{4} + 11025 \beta_{5} ) q^{55} \) \( + ( 58212 - 130200 \beta_{1} + 270864 \beta_{2} - 87912 \beta_{3} - 8316 \beta_{4} ) q^{57} \) \( + ( -2469811468 - 66252 \beta_{1} - 611254 \beta_{2} - 25633 \beta_{4} - 10647 \beta_{5} ) q^{59} \) \( + ( 2210592635 - 9231 \beta_{1} - 76440 \beta_{2} - 7935 \beta_{4} - 14574 \beta_{5} ) q^{61} \) \( + ( 227556 - 22417934 \beta_{1} + 791847 \beta_{2} - 76671 \beta_{3} - 32508 \beta_{4} ) q^{63} \) \( + ( 3673735881 + 10426851 \beta_{1} + 78240 \beta_{2} - 48480 \beta_{3} + 1554 \beta_{4} - 1905 \beta_{5} ) q^{65} \) \( + ( 181104 + 6129851 \beta_{1} + 478942 \beta_{2} + 90242 \beta_{3} - 25872 \beta_{4} ) q^{67} \) \( + ( -9638418963 + 68813 \beta_{1} + 631232 \beta_{2} + 28449 \beta_{4} + 16534 \beta_{5} ) q^{69} \) \( + ( 163601676 - 87636 \beta_{1} - 847980 \beta_{2} - 14190 \beta_{4} + 45066 \beta_{5} ) q^{71} \) \( + ( -525504 - 20553268 \beta_{1} - 1784432 \beta_{2} + 132848 \beta_{3} + 75072 \beta_{4} ) q^{73} \) \( + ( 16329299100 + 19853225 \beta_{1} + 559004 \beta_{2} - 89640 \beta_{3} - 11178 \beta_{4} - 66290 \beta_{5} ) q^{75} \) \( + ( -806932 + 40055092 \beta_{1} - 2809588 \beta_{2} + 273516 \beta_{3} + 115276 \beta_{4} ) q^{77} \) \( + ( -26400848028 + 55332 \beta_{1} + 512640 \beta_{2} + 20340 \beta_{4} + 5688 \beta_{5} ) q^{79} \) \( + ( 18934612824 + 457459 \beta_{1} + 4282018 \beta_{2} + 146286 \beta_{4} - 18601 \beta_{5} ) q^{81} \) \( + ( 81872 + 10566547 \beta_{1} + 613790 \beta_{2} - 356478 \beta_{3} - 11696 \beta_{4} ) q^{83} \) \( + ( 18297462962 - 27006098 \beta_{1} - 593140 \beta_{2} - 6260 \beta_{3} + 211248 \beta_{4} + 163890 \beta_{5} ) q^{85} \) \( + ( 479304 - 76561986 \beta_{1} + 1988928 \beta_{2} - 482544 \beta_{3} - 68472 \beta_{4} ) q^{87} \) \( + ( -48111251846 + 497784 \beta_{1} + 4730068 \beta_{2} + 123886 \beta_{4} - 126126 \beta_{5} ) q^{89} \) \( + ( 38147473332 - 174684 \beta_{1} - 1514436 \beta_{2} - 116202 \beta_{4} - 173922 \beta_{5} ) q^{91} \) \( + ( -550368 + 124420408 \beta_{1} - 2158488 \beta_{2} + 428760 \beta_{3} + 78624 \beta_{4} ) q^{93} \) \( + ( 58072039332 - 45273828 \beta_{1} - 1398066 \beta_{2} + 729300 \beta_{3} - 291015 \beta_{4} - 38325 \beta_{5} ) q^{95} \) \( + ( -505554 + 4951262 \beta_{1} - 804452 \beta_{2} - 784432 \beta_{3} + 72222 \beta_{4} ) q^{97} \) \( + ( -167773134876 - 1511324 \beta_{1} - 14345522 \beta_{2} - 383859 \beta_{4} + 359747 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 4126q^{5} \) \(\mathstrut -\mathstrut 501494q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 4126q^{5} \) \(\mathstrut -\mathstrut 501494q^{9} \) \(\mathstrut +\mathstrut 1298200q^{11} \) \(\mathstrut -\mathstrut 3310648q^{15} \) \(\mathstrut -\mathstrut 244824q^{19} \) \(\mathstrut -\mathstrut 44518024q^{21} \) \(\mathstrut +\mathstrut 34713726q^{25} \) \(\mathstrut -\mathstrut 138940764q^{29} \) \(\mathstrut +\mathstrut 214450176q^{31} \) \(\mathstrut -\mathstrut 308845352q^{35} \) \(\mathstrut +\mathstrut 542728176q^{39} \) \(\mathstrut +\mathstrut 106601836q^{41} \) \(\mathstrut +\mathstrut 3311255374q^{45} \) \(\mathstrut -\mathstrut 4506920718q^{49} \) \(\mathstrut +\mathstrut 571023872q^{51} \) \(\mathstrut -\mathstrut 1741716600q^{55} \) \(\mathstrut -\mathstrut 14817616328q^{59} \) \(\mathstrut +\mathstrut 13263695412q^{61} \) \(\mathstrut +\mathstrut 22042348848q^{65} \) \(\mathstrut -\mathstrut 57831800072q^{69} \) \(\mathstrut +\mathstrut 983424528q^{71} \) \(\mathstrut +\mathstrut 97974745648q^{75} \) \(\mathstrut -\mathstrut 158406142752q^{79} \) \(\mathstrut +\mathstrut 113598783134q^{81} \) \(\mathstrut +\mathstrut 109785881856q^{85} \) \(\mathstrut -\mathstrut 288677471236q^{89} \) \(\mathstrut +\mathstrut 228887753424q^{91} \) \(\mathstrut +\mathstrut 348434078904q^{95} \) \(\mathstrut -\mathstrut 1006608631000q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut +\mathstrut \) \(195547\) \(x^{4}\mathstrut +\mathstrut \) \(7296096303\) \(x^{2}\mathstrut +\mathstrut \) \(4158054148149\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 142 \nu^{5} + 9108 \nu^{4} + 28669366 \nu^{3} + 1252445634 \nu^{2} + 1135055801142 \nu + 9845359382133 \)\()/\)\(4163665275\)
\(\beta_{3}\)\(=\)\((\)\( 122 \nu^{5} + 828 \nu^{4} + 22262006 \nu^{3} + 113858694 \nu^{2} + 689757655572 \nu + 895032671103 \)\()/\)\(378515025\)
\(\beta_{4}\)\(=\)\((\)\( -3976 \nu^{5} + 200376 \nu^{4} - 802742248 \nu^{3} + 27553803948 \nu^{2} - 31806544423626 \nu + 216627052063851 \)\()/\)\(4163665275\)
\(\beta_{5}\)\(=\)\((\)\( 284 \nu^{5} + 18216 \nu^{4} + 57338732 \nu^{3} + 5835823488 \nu^{2} + 2271777068394 \nu + 236811706393581 \)\()/\)\(832733055\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(260733\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(108\) \(\beta_{4}\mathstrut -\mathstrut \) \(639\) \(\beta_{3}\mathstrut +\mathstrut \) \(3015\) \(\beta_{2}\mathstrut -\mathstrut \) \(241253\) \(\beta_{1}\mathstrut +\mathstrut \) \(756\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(275021\) \(\beta_{5}\mathstrut +\mathstrut \) \(73143\) \(\beta_{4}\mathstrut +\mathstrut \) \(4798214\) \(\beta_{2}\mathstrut +\mathstrut \) \(494450\) \(\beta_{1}\mathstrut +\mathstrut \) \(63058879734\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(19459143\) \(\beta_{4}\mathstrut +\mathstrut \) \(129012147\) \(\beta_{3}\mathstrut -\mathstrut \) \(557113293\) \(\beta_{2}\mathstrut +\mathstrut \) \(32714504744\) \(\beta_{1}\mathstrut -\mathstrut \) \(136214001\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
381.641i
222.078i
24.0594i
24.0594i
222.078i
381.641i
0 763.281i 0 −5645.73 4117.51i 0 59643.1i 0 −405451. 0
9.2 0 444.156i 0 6461.93 + 2659.24i 0 56488.4i 0 −20127.7 0
9.3 0 48.1188i 0 −2879.20 + 6366.97i 0 37910.4i 0 174832. 0
9.4 0 48.1188i 0 −2879.20 6366.97i 0 37910.4i 0 174832. 0
9.5 0 444.156i 0 6461.93 2659.24i 0 56488.4i 0 −20127.7 0
9.6 0 763.281i 0 −5645.73 + 4117.51i 0 59643.1i 0 −405451. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{12}^{\mathrm{new}}(20, [\chi])\).