Properties

Label 20.7.f.a
Level 20
Weight 7
Character orbit 20.f
Analytic conductor 4.601
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 20.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(4.6010816724\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{7}\cdot 5 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( 5 + 5 \beta_{1} + \beta_{4} ) q^{3} \) \( + ( -25 + 20 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{5} \) \( + ( -45 + 45 \beta_{1} - 6 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{7} \) \( + ( 293 \beta_{1} + 21 \beta_{2} + 2 \beta_{3} + 23 \beta_{4} - 4 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 5 + 5 \beta_{1} + \beta_{4} ) q^{3} \) \( + ( -25 + 20 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{5} \) \( + ( -45 + 45 \beta_{1} - 6 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{7} \) \( + ( 293 \beta_{1} + 21 \beta_{2} + 2 \beta_{3} + 23 \beta_{4} - 4 \beta_{5} ) q^{9} \) \( + ( 350 - 27 \beta_{2} - 4 \beta_{3} + 29 \beta_{4} - 2 \beta_{5} ) q^{11} \) \( + ( 155 + 155 \beta_{1} - \beta_{2} + 3 \beta_{3} - 39 \beta_{4} - \beta_{5} ) q^{13} \) \( + ( -1245 - 1305 \beta_{1} + 65 \beta_{2} - \beta_{3} - 92 \beta_{4} + 9 \beta_{5} ) q^{15} \) \( + ( -595 + 595 \beta_{1} - 116 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} + 30 \beta_{5} ) q^{17} \) \( + ( -2772 \beta_{1} + 44 \beta_{2} - 22 \beta_{3} + 22 \beta_{4} + 44 \beta_{5} ) q^{19} \) \( + ( 5546 + 49 \beta_{2} + 48 \beta_{3} - 73 \beta_{4} + 24 \beta_{5} ) q^{21} \) \( + ( 3355 + 3355 \beta_{1} + 13 \beta_{2} - 39 \beta_{3} - 34 \beta_{4} + 13 \beta_{5} ) q^{23} \) \( + ( -4195 - 9860 \beta_{1} - 289 \beta_{2} + 14 \beta_{3} + 217 \beta_{4} - 22 \beta_{5} ) q^{25} \) \( + ( -19040 + 19040 \beta_{1} + 548 \beta_{2} - 32 \beta_{3} + 32 \beta_{4} - 96 \beta_{5} ) q^{27} \) \( + ( -10892 \beta_{1} - 422 \beta_{2} + 86 \beta_{3} - 336 \beta_{4} - 172 \beta_{5} ) q^{29} \) \( + ( 17646 + 115 \beta_{2} - 220 \beta_{3} - 5 \beta_{4} - 110 \beta_{5} ) q^{31} \) \( + ( 29130 + 29130 \beta_{1} - 68 \beta_{2} + 204 \beta_{3} + 1066 \beta_{4} - 68 \beta_{5} ) q^{33} \) \( + ( -19515 - 31265 \beta_{1} + 371 \beta_{2} - 82 \beta_{3} + 990 \beta_{4} - 28 \beta_{5} ) q^{35} \) \( + ( -40305 + 40305 \beta_{1} - 129 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} ) q^{37} \) \( + ( -36522 \beta_{1} - 345 \beta_{2} - 90 \beta_{3} - 435 \beta_{4} + 180 \beta_{5} ) q^{39} \) \( + ( 58806 + 473 \beta_{2} + 396 \beta_{3} - 671 \beta_{4} + 198 \beta_{5} ) q^{41} \) \( + ( 11065 + 11065 \beta_{1} + 166 \beta_{2} - 498 \beta_{3} - 2337 \beta_{4} + 166 \beta_{5} ) q^{43} \) \( + ( -47560 - 81225 \beta_{1} - 1298 \beta_{2} + 247 \beta_{3} - 2958 \beta_{4} + 170 \beta_{5} ) q^{45} \) \( + ( -59625 + 59625 \beta_{1} - 720 \beta_{2} + 177 \beta_{3} - 177 \beta_{4} + 531 \beta_{5} ) q^{47} \) \( + ( -41829 \beta_{1} + 3047 \beta_{2} - 286 \beta_{3} + 2761 \beta_{4} + 572 \beta_{5} ) q^{49} \) \( + ( 105162 - 2613 \beta_{2} + 224 \beta_{3} + 2501 \beta_{4} + 112 \beta_{5} ) q^{51} \) \( + ( 57635 + 57635 \beta_{1} - 97 \beta_{2} + 291 \beta_{3} + 67 \beta_{4} - 97 \beta_{5} ) q^{53} \) \( + ( -49850 - 47700 \beta_{1} + 5035 \beta_{2} - 330 \beta_{3} - 195 \beta_{4} + 10 \beta_{5} ) q^{55} \) \( + ( -20020 + 20020 \beta_{1} - 4246 \beta_{2} - 198 \beta_{3} + 198 \beta_{4} - 594 \beta_{5} ) q^{57} \) \( + ( 46916 \beta_{1} - 1632 \beta_{2} + 766 \beta_{3} - 866 \beta_{4} - 1532 \beta_{5} ) q^{59} \) \( + ( -80098 + 2089 \beta_{2} - 1972 \beta_{3} - 1103 \beta_{4} - 986 \beta_{5} ) q^{61} \) \( + ( -725 - 725 \beta_{1} - 463 \beta_{2} + 1389 \beta_{3} + 1930 \beta_{4} - 463 \beta_{5} ) q^{63} \) \( + ( -2555 + 61205 \beta_{1} - 3135 \beta_{2} - 214 \beta_{3} + 2857 \beta_{4} - 654 \beta_{5} ) q^{65} \) \( + ( -35455 + 35455 \beta_{1} + 7401 \beta_{2} - 462 \beta_{3} + 462 \beta_{4} - 1386 \beta_{5} ) q^{67} \) \( + ( 2634 \beta_{1} + 1199 \beta_{2} + 88 \beta_{3} + 1287 \beta_{4} - 176 \beta_{5} ) q^{69} \) \( + ( 26238 - 3101 \beta_{2} + 2148 \beta_{3} + 2027 \beta_{4} + 1074 \beta_{5} ) q^{71} \) \( + ( -55395 - 55395 \beta_{1} + 1154 \beta_{2} - 3462 \beta_{3} + 3426 \beta_{4} + 1154 \beta_{5} ) q^{73} \) \( + ( 303865 + 134545 \beta_{1} - 10182 \beta_{2} + 1602 \beta_{3} + 4211 \beta_{4} - 66 \beta_{5} ) q^{75} \) \( + ( 271590 - 271590 \beta_{1} + 7172 \beta_{2} + 594 \beta_{3} - 594 \beta_{4} + 1782 \beta_{5} ) q^{77} \) \( + ( 52144 \beta_{1} - 10552 \beta_{2} - 1624 \beta_{3} - 12176 \beta_{4} + 3248 \beta_{5} ) q^{79} \) \( + ( -504211 + 15721 \beta_{2} + 1492 \beta_{3} - 16467 \beta_{4} + 746 \beta_{5} ) q^{81} \) \( + ( -364475 - 364475 \beta_{1} - 824 \beta_{2} + 2472 \beta_{3} - 4515 \beta_{4} - 824 \beta_{5} ) q^{83} \) \( + ( -24905 + 445145 \beta_{1} + 8632 \beta_{2} - 2689 \beta_{3} + 1540 \beta_{4} + 2029 \beta_{5} ) q^{85} \) \( + ( 429900 - 429900 \beta_{1} - 13630 \beta_{2} + 1274 \beta_{3} - 1274 \beta_{4} + 3822 \beta_{5} ) q^{87} \) \( + ( 26392 \beta_{1} - 7956 \beta_{2} - 172 \beta_{3} - 8128 \beta_{4} + 344 \beta_{5} ) q^{89} \) \( + ( -367246 - 5753 \beta_{2} - 3256 \beta_{3} + 7381 \beta_{4} - 1628 \beta_{5} ) q^{91} \) \( + ( 38930 + 38930 \beta_{1} - 540 \beta_{2} + 1620 \beta_{3} + 2626 \beta_{4} - 540 \beta_{5} ) q^{93} \) \( + ( 585640 + 249700 \beta_{1} + 9152 \beta_{2} + 1782 \beta_{3} - 8338 \beta_{4} + 880 \beta_{5} ) q^{95} \) \( + ( 560785 - 560785 \beta_{1} + 10008 \beta_{2} - 1048 \beta_{3} + 1048 \beta_{4} - 3144 \beta_{5} ) q^{97} \) \( + ( 1061150 \beta_{1} + 33477 \beta_{2} + 2774 \beta_{3} + 36251 \beta_{4} - 5548 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 32q^{3} \) \(\mathstrut -\mathstrut 156q^{5} \) \(\mathstrut -\mathstrut 264q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 32q^{3} \) \(\mathstrut -\mathstrut 156q^{5} \) \(\mathstrut -\mathstrut 264q^{7} \) \(\mathstrut +\mathstrut 2200q^{11} \) \(\mathstrut +\mathstrut 858q^{13} \) \(\mathstrut -\mathstrut 7768q^{15} \) \(\mathstrut -\mathstrut 3278q^{17} \) \(\mathstrut +\mathstrut 33176q^{21} \) \(\mathstrut +\mathstrut 19984q^{23} \) \(\mathstrut -\mathstrut 24174q^{25} \) \(\mathstrut -\mathstrut 115528q^{27} \) \(\mathstrut +\mathstrut 104976q^{31} \) \(\mathstrut +\mathstrut 177320q^{33} \) \(\mathstrut -\mathstrut 116072q^{35} \) \(\mathstrut -\mathstrut 241554q^{37} \) \(\mathstrut +\mathstrut 351736q^{41} \) \(\mathstrut +\mathstrut 60720q^{43} \) \(\mathstrut -\mathstrut 287846q^{45} \) \(\mathstrut -\mathstrut 355248q^{47} \) \(\mathstrut +\mathstrut 641872q^{51} \) \(\mathstrut +\mathstrut 346526q^{53} \) \(\mathstrut -\mathstrut 310200q^{55} \) \(\mathstrut -\mathstrut 112816q^{57} \) \(\mathstrut -\mathstrut 492888q^{61} \) \(\mathstrut +\mathstrut 2288q^{63} \) \(\mathstrut -\mathstrut 5082q^{65} \) \(\mathstrut -\mathstrut 230304q^{67} \) \(\mathstrut +\mathstrut 174128q^{71} \) \(\mathstrut -\mathstrut 332442q^{73} \) \(\mathstrut +\mathstrut 1855048q^{75} \) \(\mathstrut +\mathstrut 1618760q^{77} \) \(\mathstrut -\mathstrut 3085166q^{81} \) \(\mathstrut -\mathstrut 2190936q^{83} \) \(\mathstrut -\mathstrut 164934q^{85} \) \(\mathstrut +\mathstrut 2614304q^{87} \) \(\mathstrut -\mathstrut 2186976q^{91} \) \(\mathstrut +\mathstrut 242072q^{93} \) \(\mathstrut +\mathstrut 3484184q^{95} \) \(\mathstrut +\mathstrut 3338406q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut +\mathstrut \) \(2\) \(x^{4}\mathstrut -\mathstrut \) \(450\) \(x^{3}\mathstrut +\mathstrut \) \(23409\) \(x^{2}\mathstrut -\mathstrut \) \(115668\) \(x\mathstrut +\mathstrut \) \(285768\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 38 \nu^{5} + 17 \nu^{4} + 25 \nu^{3} - 2685 \nu^{2} + 868617 \nu - 2248722 \)\()/2216970\)
\(\beta_{2}\)\(=\)\((\)\( -325 \nu^{5} + 1326 \nu^{4} + 18050 \nu^{3} + 251030 \nu^{2} - 7760025 \nu + 37081044 \)\()/738990\)
\(\beta_{3}\)\(=\)\((\)\( 1327 \nu^{5} + 14824 \nu^{4} + 12140 \nu^{3} - 844020 \nu^{2} + 38955033 \nu + 130369176 \)\()/2216970\)
\(\beta_{4}\)\(=\)\((\)\( -1375 \nu^{5} - 5338 \nu^{4} - 7850 \nu^{3} + 843090 \nu^{2} - 22977675 \nu + 9970128 \)\()/2216970\)
\(\beta_{5}\)\(=\)\((\)\( -4289 \nu^{5} + 3298 \nu^{4} - 72430 \nu^{3} + 2589630 \nu^{2} - 100691181 \nu + 349337772 \)\()/2216970\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/40\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(410\) \(\beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(429\) \(\beta_{5}\mathstrut +\mathstrut \) \(143\) \(\beta_{4}\mathstrut -\mathstrut \) \(143\) \(\beta_{3}\mathstrut +\mathstrut \) \(1121\) \(\beta_{2}\mathstrut -\mathstrut \) \(9490\) \(\beta_{1}\mathstrut +\mathstrut \) \(9490\)\()/40\)
\(\nu^{4}\)\(=\)\((\)\(100\) \(\beta_{5}\mathstrut -\mathstrut \) \(165\) \(\beta_{4}\mathstrut +\mathstrut \) \(200\) \(\beta_{3}\mathstrut +\mathstrut \) \(65\) \(\beta_{2}\mathstrut -\mathstrut \) \(30038\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(23659\) \(\beta_{5}\mathstrut -\mathstrut \) \(157213\) \(\beta_{4}\mathstrut -\mathstrut \) \(70977\) \(\beta_{3}\mathstrut +\mathstrut \) \(23659\) \(\beta_{2}\mathstrut +\mathstrut \) \(2401010\) \(\beta_{1}\mathstrut +\mathstrut \) \(2401010\)\()/40\)

Character Values

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

\(n\) \(11\) \(17\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
2.61494 + 2.61494i
−9.34732 9.34732i
7.73238 + 7.73238i
2.61494 2.61494i
−9.34732 + 9.34732i
7.73238 7.73238i
0 −16.7249 16.7249i 0 −22.1946 + 123.014i 0 −422.022 + 422.022i 0 169.553i 0
13.2 0 −2.90948 2.90948i 0 59.6880 109.829i 0 236.070 236.070i 0 712.070i 0
13.3 0 35.6344 + 35.6344i 0 −115.493 + 47.8149i 0 53.9521 53.9521i 0 1810.62i 0
17.1 0 −16.7249 + 16.7249i 0 −22.1946 123.014i 0 −422.022 422.022i 0 169.553i 0
17.2 0 −2.90948 + 2.90948i 0 59.6880 + 109.829i 0 236.070 + 236.070i 0 712.070i 0
17.3 0 35.6344 35.6344i 0 −115.493 47.8149i 0 53.9521 + 53.9521i 0 1810.62i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.3
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{7}^{\mathrm{new}}(20, [\chi])\).