# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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