Properties

Label 684.3.e.a
Level $684$
Weight $3$
Character orbit 684.e
Analytic conductor $18.638$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 156 x^{10} + 8721 x^{8} + 208784 x^{6} + 2024760 x^{4} + 7117056 x^{2} + 6533136\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( 1 + \beta_{4} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( 1 + \beta_{4} ) q^{7} -\beta_{10} q^{11} + ( -1 - \beta_{2} + \beta_{3} ) q^{13} + ( -\beta_{1} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{17} -\beta_{2} q^{19} + ( -\beta_{1} - \beta_{8} ) q^{23} + ( -1 - 3 \beta_{2} - \beta_{4} + \beta_{6} ) q^{25} + ( \beta_{1} + \beta_{7} + \beta_{9} - 2 \beta_{11} ) q^{29} + ( -3 - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{31} + ( \beta_{1} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{35} + ( -1 - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{8} - 2 \beta_{11} ) q^{41} + ( 6 - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{43} + ( -2 \beta_{1} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{47} + ( -1 - 7 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} ) q^{53} + ( -7 - 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{55} + ( \beta_{1} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{59} + ( -5 - 6 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} ) q^{61} + ( -3 \beta_{1} + 3 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} ) q^{65} + ( -6 - 6 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{67} + ( 9 \beta_{1} - 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} ) q^{71} + ( 12 - 5 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{73} + ( 3 \beta_{1} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 6 \beta_{10} - \beta_{11} ) q^{77} + ( -5 - 3 \beta_{2} + \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{79} + ( -5 \beta_{1} + \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} - 5 \beta_{11} ) q^{83} + ( 19 - 8 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{85} + ( 9 \beta_{1} + 3 \beta_{7} - 4 \beta_{8} - 3 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{89} + ( 2 + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{91} + ( -2 \beta_{1} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{95} + ( 4 - 6 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 16q^{7} + O(q^{10}) \) \( 12q + 16q^{7} - 16q^{13} - 12q^{25} - 40q^{31} - 32q^{37} + 92q^{43} - 84q^{55} - 48q^{61} - 88q^{67} + 148q^{73} - 56q^{79} + 228q^{85} - 8q^{91} + 72q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 156 x^{10} + 8721 x^{8} + 208784 x^{6} + 2024760 x^{4} + 7117056 x^{2} + 6533136\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5283 \nu^{10} + 747468 \nu^{8} + 35746495 \nu^{6} + 640600728 \nu^{4} + 2908500948 \nu^{2} - 2437936256 \)\()/ 1226974736 \)
\(\beta_{3}\)\(=\)\((\)\( -10699 \nu^{10} - 1453458 \nu^{8} - 71124471 \nu^{6} - 1579317638 \nu^{4} - 16445727660 \nu^{2} - 50749430904 \)\()/ 1698888096 \)
\(\beta_{4}\)\(=\)\((\)\( 247303 \nu^{10} + 39300927 \nu^{8} + 2207100159 \nu^{6} + 51133155599 \nu^{4} + 423439865172 \nu^{2} + 789616257444 \)\()/ 27606931560 \)
\(\beta_{5}\)\(=\)\((\)\( 439759 \nu^{10} + 64571058 \nu^{8} + 3297682539 \nu^{6} + 66047797334 \nu^{4} + 385032689052 \nu^{2} + 322447888440 \)\()/ 22085545248 \)
\(\beta_{6}\)\(=\)\((\)\( 1207811 \nu^{10} + 179510034 \nu^{8} + 9239977143 \nu^{6} + 188747409478 \nu^{4} + 1294741221444 \nu^{2} + 2685671561448 \)\()/ 55213863120 \)
\(\beta_{7}\)\(=\)\((\)\( 1217 \nu^{11} + 189923 \nu^{9} + 10410681 \nu^{7} + 240839611 \nu^{5} + 2413128508 \nu^{3} + 11894428236 \nu \)\()/ 2498524080 \)
\(\beta_{8}\)\(=\)\((\)\( 1402267 \nu^{11} + 254053290 \nu^{9} + 18225308799 \nu^{7} + 639808808750 \nu^{5} + 10013639996244 \nu^{3} + 33589159747272 \nu \)\()/ 2352110568912 \)
\(\beta_{9}\)\(=\)\((\)\( 29570539 \nu^{11} + 4740331224 \nu^{9} + 270890863839 \nu^{7} + 6494446864916 \nu^{5} + 58490152517388 \nu^{3} + 151480250456112 \nu \)\()/ 4704221137824 \)
\(\beta_{10}\)\(=\)\((\)\( 87410753 \nu^{11} + 13229974437 \nu^{9} + 707856730569 \nu^{7} + 15696035968309 \nu^{5} + 128584765723092 \nu^{3} + 318148928199684 \nu \)\()/ 11760552844560 \)
\(\beta_{11}\)\(=\)\((\)\(-237670967 \nu^{11} - 36242682558 \nu^{9} - 1941708477411 \nu^{7} - 42485286182386 \nu^{5} - 325489533985788 \nu^{3} - 551649655461096 \nu\)\()/ 23521105689120 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{4} - 3 \beta_{2} - 26\)
\(\nu^{3}\)\(=\)\(3 \beta_{11} + 3 \beta_{10} + \beta_{9} - 2 \beta_{8} + 6 \beta_{7} - 43 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-51 \beta_{6} - 13 \beta_{5} + 61 \beta_{4} - 5 \beta_{3} + 185 \beta_{2} + 1144\)
\(\nu^{5}\)\(=\)\(-203 \beta_{11} - 208 \beta_{10} - 83 \beta_{9} + 151 \beta_{8} - 151 \beta_{7} + 2101 \beta_{1}\)
\(\nu^{6}\)\(=\)\(2403 \beta_{6} + 1212 \beta_{5} - 3273 \beta_{4} + 192 \beta_{3} - 10723 \beta_{2} - 56536\)
\(\nu^{7}\)\(=\)\(12613 \beta_{11} + 13371 \beta_{10} + 5431 \beta_{9} - 8826 \beta_{8} - 1660 \beta_{7} - 106839 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-114005 \beta_{6} - 86295 \beta_{5} + 179075 \beta_{4} + 753 \beta_{3} + 606807 \beta_{2} + 2911516\)
\(\nu^{9}\)\(=\)\(-758925 \beta_{11} - 841572 \beta_{10} - 312221 \beta_{9} + 485389 \beta_{8} + 533019 \beta_{7} + 5551775 \beta_{1}\)
\(\nu^{10}\)\(=\)\(5504145 \beta_{6} + 5585048 \beta_{5} - 10036475 \beta_{4} - 799388 \beta_{3} - 33847777 \beta_{2} - 153338912\)
\(\nu^{11}\)\(=\)\(44764543 \beta_{11} + 52167557 \beta_{10} + 16708381 \beta_{9} - 26164658 \beta_{8} - 48943552 \beta_{7} - 292750157 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
7.53519i
7.04185i
5.71859i
3.29597i
2.16068i
1.18282i
1.18282i
2.16068i
3.29597i
5.71859i
7.04185i
7.53519i
0 0 0 7.53519i 0 5.18161 0 0 0
305.2 0 0 0 7.04185i 0 −3.07744 0 0 0
305.3 0 0 0 5.71859i 0 5.91816 0 0 0
305.4 0 0 0 3.29597i 0 −2.46307 0 0 0
305.5 0 0 0 2.16068i 0 −9.11429 0 0 0
305.6 0 0 0 1.18282i 0 11.5550 0 0 0
305.7 0 0 0 1.18282i 0 11.5550 0 0 0
305.8 0 0 0 2.16068i 0 −9.11429 0 0 0
305.9 0 0 0 3.29597i 0 −2.46307 0 0 0
305.10 0 0 0 5.71859i 0 5.91816 0 0 0
305.11 0 0 0 7.04185i 0 −3.07744 0 0 0
305.12 0 0 0 7.53519i 0 5.18161 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.e.a 12
3.b odd 2 1 inner 684.3.e.a 12
4.b odd 2 1 2736.3.h.c 12
12.b even 2 1 2736.3.h.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.e.a 12 1.a even 1 1 trivial
684.3.e.a 12 3.b odd 2 1 inner
2736.3.h.c 12 4.b odd 2 1
2736.3.h.c 12 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( 6533136 + 7117056 T^{2} + 2024760 T^{4} + 208784 T^{6} + 8721 T^{8} + 156 T^{10} + T^{12} \)
$7$ \( ( -24480 - 9600 T + 2472 T^{2} + 728 T^{3} - 115 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$11$ \( 4706508816 + 4858943712 T^{2} + 833089672 T^{4} + 25110056 T^{6} + 248577 T^{8} + 884 T^{10} + T^{12} \)
$13$ \( ( 189408 - 171968 T + 44648 T^{2} - 1472 T^{3} - 464 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$17$ \( 22829551568784 + 3493048196448 T^{2} + 73478610216 T^{4} + 506132136 T^{6} + 1515297 T^{8} + 2036 T^{10} + T^{12} \)
$19$ \( ( -19 + T^{2} )^{6} \)
$23$ \( 5361077160000 + 3337820404800 T^{2} + 59361970480 T^{4} + 393895712 T^{6} + 1224780 T^{8} + 1796 T^{10} + T^{12} \)
$29$ \( 67807760026189824 + 1507454720630784 T^{2} + 7295855047680 T^{4} + 14169401600 T^{6} + 13168272 T^{8} + 5856 T^{10} + T^{12} \)
$31$ \( ( -14066208 + 2029664 T + 189256 T^{2} - 20320 T^{3} - 1208 T^{4} + 20 T^{5} + T^{6} )^{2} \)
$37$ \( ( 641267584 - 136276352 T + 6956600 T^{2} + 23296 T^{3} - 5380 T^{4} + 16 T^{5} + T^{6} )^{2} \)
$41$ \( 81338090857168896 + 2894400788889600 T^{2} + 13756899856384 T^{4} + 24216461312 T^{6} + 19480704 T^{8} + 7232 T^{10} + T^{12} \)
$43$ \( ( 86899920 - 158238688 T + 2166856 T^{2} + 350048 T^{3} - 7031 T^{4} - 46 T^{5} + T^{6} )^{2} \)
$47$ \( 125152140271658256 + 2837898717756288 T^{2} + 19582679908600 T^{4} + 42719640752 T^{6} + 32174385 T^{8} + 9644 T^{10} + T^{12} \)
$53$ \( \)\(12\!\cdots\!00\)\( + 794544535529226240 T^{2} + 1485301451155456 T^{4} + 950257260032 T^{6} + 240390288 T^{8} + 25904 T^{10} + T^{12} \)
$59$ \( 29887462703760998400 + 146799175854440448 T^{2} + 255410384274432 T^{4} + 206682075648 T^{6} + 82669632 T^{8} + 15392 T^{10} + T^{12} \)
$61$ \( ( -12101473952 + 622152960 T + 18008808 T^{2} - 544480 T^{3} - 12387 T^{4} + 24 T^{5} + T^{6} )^{2} \)
$67$ \( ( 7409664 + 6912000 T - 456960 T^{2} - 321408 T^{3} - 7304 T^{4} + 44 T^{5} + T^{6} )^{2} \)
$71$ \( 27058898353678516224 + 292095257834962944 T^{2} + 954592492782592 T^{4} + 947683771904 T^{6} + 268313664 T^{8} + 28640 T^{10} + T^{12} \)
$73$ \( ( -22640659056 - 856474080 T + 22162200 T^{2} + 502688 T^{3} - 7999 T^{4} - 74 T^{5} + T^{6} )^{2} \)
$79$ \( ( -63064430272 + 2122964960 T + 110784664 T^{2} - 454464 T^{3} - 25924 T^{4} + 28 T^{5} + T^{6} )^{2} \)
$83$ \( 803976416463182400 + 43183613419450560 T^{2} + 227863550516976 T^{4} + 410529278880 T^{6} + 258131100 T^{8} + 30476 T^{10} + T^{12} \)
$89$ \( \)\(13\!\cdots\!00\)\( + 3545914950029352960 T^{2} + 10223337422767104 T^{4} + 6237402041088 T^{6} + 921785616 T^{8} + 51680 T^{10} + T^{12} \)
$97$ \( ( -973638592 + 15119808 T + 6840048 T^{2} - 63456 T^{3} - 9348 T^{4} - 36 T^{5} + T^{6} )^{2} \)
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