# Properties

 Label 1344.4.c.h Level $1344$ Weight $4$ Character orbit 1344.c Analytic conductor $79.299$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.2985670477$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 386 x^{10} + 54793 x^{8} + 3447408 x^{6} + 90154296 x^{4} + 707138208 x^{2} + 525876624$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{22}$$ 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.

 $$f(q)$$ $$=$$ $$q -3 \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + 7 q^{7} -9 q^{9} +O(q^{10})$$ $$q -3 \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + 7 q^{7} -9 q^{9} + ( -11 \beta_{1} - \beta_{3} - \beta_{8} - \beta_{9} ) q^{11} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{11} ) q^{13} + ( 3 + 3 \beta_{4} ) q^{15} + ( -\beta_{4} + 5 \beta_{5} ) q^{17} + ( -10 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{19} -21 \beta_{1} q^{21} + ( -6 - 2 \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{10} ) q^{23} + ( -47 - 6 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{25} + 27 \beta_{1} q^{27} + ( -20 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + \beta_{8} + 3 \beta_{9} - \beta_{11} ) q^{29} + ( 50 + 2 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{10} ) q^{31} + ( -33 + 3 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{33} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{35} + ( 62 \beta_{1} + 3 \beta_{3} - 5 \beta_{8} + 9 \beta_{9} - 3 \beta_{11} ) q^{37} + ( -9 + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{10} ) q^{39} + ( 114 - 11 \beta_{4} + \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{41} + ( -39 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 7 \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{43} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{45} + ( 122 - 5 \beta_{4} + 8 \beta_{5} - 10 \beta_{7} - 5 \beta_{10} ) q^{47} + 49 q^{49} + ( -15 \beta_{2} - 3 \beta_{3} ) q^{51} + ( -246 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} - \beta_{8} - 13 \beta_{9} ) q^{53} + ( -112 - 12 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{55} + ( -30 + 3 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} - 3 \beta_{10} ) q^{57} + ( 8 \beta_{1} - 10 \beta_{2} - 13 \beta_{3} - 6 \beta_{8} + 9 \beta_{11} ) q^{59} + ( -263 \beta_{1} + 5 \beta_{2} + 11 \beta_{3} + 22 \beta_{8} + 8 \beta_{9} - 7 \beta_{11} ) q^{61} -63 q^{63} + ( -228 + 11 \beta_{4} - 22 \beta_{5} + 44 \beta_{6} + 10 \beta_{7} + 3 \beta_{10} ) q^{65} + ( 73 \beta_{1} + 3 \beta_{2} + 18 \beta_{3} - 13 \beta_{8} + 11 \beta_{9} - 6 \beta_{11} ) q^{67} + ( 18 \beta_{1} + 15 \beta_{2} - 6 \beta_{3} - 6 \beta_{8} + 6 \beta_{9} + 3 \beta_{11} ) q^{69} + ( -112 + 10 \beta_{4} - \beta_{5} - 10 \beta_{6} - 6 \beta_{7} + \beta_{10} ) q^{71} + ( 204 - 18 \beta_{4} - 12 \beta_{5} + 30 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 141 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} - 9 \beta_{8} + 12 \beta_{9} ) q^{75} + ( -77 \beta_{1} - 7 \beta_{3} - 7 \beta_{8} - 7 \beta_{9} ) q^{77} + ( 54 - 10 \beta_{4} - 20 \beta_{5} - 8 \beta_{6} + 18 \beta_{7} + 16 \beta_{10} ) q^{79} + 81 q^{81} + ( -70 \beta_{1} - 2 \beta_{2} + 39 \beta_{3} + 12 \beta_{8} + 6 \beta_{9} - 9 \beta_{11} ) q^{83} + ( -216 \beta_{1} - 26 \beta_{2} - 35 \beta_{3} - 7 \beta_{8} - 14 \beta_{9} - 5 \beta_{11} ) q^{85} + ( -60 + 15 \beta_{4} + 12 \beta_{5} - 9 \beta_{6} + 3 \beta_{7} - 3 \beta_{10} ) q^{87} + ( 70 + 11 \beta_{4} + 31 \beta_{5} + 32 \beta_{6} + 14 \beta_{7} + 8 \beta_{10} ) q^{89} + ( -21 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 7 \beta_{11} ) q^{91} + ( -150 \beta_{1} - 30 \beta_{2} + 6 \beta_{3} - 3 \beta_{8} - 6 \beta_{9} + 6 \beta_{11} ) q^{93} + ( -194 + 12 \beta_{4} - 4 \beta_{5} + 34 \beta_{6} + 20 \beta_{7} - 6 \beta_{10} ) q^{95} + ( -244 - 42 \beta_{4} + 12 \beta_{5} + 36 \beta_{6} - 8 \beta_{7} ) q^{97} + ( 99 \beta_{1} + 9 \beta_{3} + 9 \beta_{8} + 9 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 84q^{7} - 108q^{9} + O(q^{10})$$ $$12q + 84q^{7} - 108q^{9} + 24q^{15} + 24q^{17} - 80q^{23} - 564q^{25} + 640q^{31} - 408q^{33} - 120q^{39} + 1416q^{41} + 1536q^{47} + 588q^{49} - 1392q^{55} - 336q^{57} - 756q^{63} - 2880q^{65} - 1392q^{71} + 2472q^{73} + 544q^{79} + 972q^{81} - 720q^{87} + 888q^{89} - 2368q^{95} - 2712q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 386 x^{10} + 54793 x^{8} + 3447408 x^{6} + 90154296 x^{4} + 707138208 x^{2} + 525876624$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$415171 \nu^{11} + 174672590 \nu^{9} + 26906273167 \nu^{7} + 1820198588256 \nu^{5} + 51364871210556 \nu^{3} + 517741765875936 \nu$$$$)/ 433631923478208$$ $$\beta_{2}$$ $$=$$ $$($$$$415171 \nu^{11} + 174672590 \nu^{9} + 26906273167 \nu^{7} + 1820198588256 \nu^{5} + 51364871210556 \nu^{3} + 1385005612832352 \nu$$$$)/ 433631923478208$$ $$\beta_{3}$$ $$=$$ $$($$$$150398 \nu^{11} - 907043 \nu^{9} - 11496542836 \nu^{7} - 1860503278707 \nu^{5} - 103808484356868 \nu^{3} - 1609312635149940 \nu$$$$)/ 108407980869552$$ $$\beta_{4}$$ $$=$$ $$($$$$-897179 \nu^{10} - 395288260 \nu^{8} - 58374438995 \nu^{6} - 3268567476990 \nu^{4} - 46608391182636 \nu^{2} + 227912499930024$$$$)/ 20649139213248$$ $$\beta_{5}$$ $$=$$ $$($$$$-1886 \nu^{10} - 543931 \nu^{8} - 50881052 \nu^{6} - 1823053635 \nu^{4} - 29324763072 \nu^{2} + 56926306332$$$$)/ 28364202216$$ $$\beta_{6}$$ $$=$$ $$($$$$-149 \nu^{10} - 34288 \nu^{8} - 1766423 \nu^{6} + 59669136 \nu^{4} + 4564567404 \nu^{2} + 23304782592$$$$)/ 1412003502$$ $$\beta_{7}$$ $$=$$ $$($$$$725843 \nu^{10} + 180719902 \nu^{8} + 15930365615 \nu^{6} + 713417388048 \nu^{4} + 17320042967436 \nu^{2} + 48026973574944$$$$)/ 5162284803312$$ $$\beta_{8}$$ $$=$$ $$($$$$-597544 \nu^{11} - 88086239 \nu^{9} + 6707059802 \nu^{7} + 1485391973733 \nu^{5} + 56665055702952 \nu^{3} + 308637875104020 \nu$$$$)/ 36135993623184$$ $$\beta_{9}$$ $$=$$ $$($$$$-26 \nu^{11} - 9595 \nu^{9} - 1316132 \nu^{7} - 80656935 \nu^{5} - 2006939520 \nu^{3} - 11659276188 \nu$$$$)/ 660235212$$ $$\beta_{10}$$ $$=$$ $$($$$$12881299 \nu^{10} + 4283247980 \nu^{8} + 503759280859 \nu^{6} + 24505239122262 \nu^{4} + 429014116275084 \nu^{2} + 1529865809906616$$$$)/ 20649139213248$$ $$\beta_{11}$$ $$=$$ $$($$$$19037951 \nu^{11} + 7224960640 \nu^{9} + 991200314147 \nu^{7} + 58547526756570 \nu^{5} + 1355342997312936 \nu^{3} + 7952916982828608 \nu$$$$)/ 108407980869552$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{10} - \beta_{7} + 4 \beta_{5} + 5 \beta_{4} - 128$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} + 9 \beta_{9} - 5 \beta_{8} + 5 \beta_{3} - 100 \beta_{2} + 560 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-115 \beta_{10} + 109 \beta_{7} + 72 \beta_{6} - 634 \beta_{5} - 503 \beta_{4} + 13142$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$271 \beta_{11} - 945 \beta_{9} + 635 \beta_{8} - 1715 \beta_{3} + 11122 \beta_{2} - 86246 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$13207 \beta_{10} - 9385 \beta_{7} - 10566 \beta_{6} + 86194 \beta_{5} + 53003 \beta_{4} - 1474790$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-51523 \beta_{11} + 63999 \beta_{9} - 73367 \beta_{8} + 289943 \beta_{3} - 1285582 \beta_{2} + 11681138 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-1521847 \beta_{10} + 670873 \beta_{7} + 1377972 \beta_{6} - 11179762 \beta_{5} - 5916587 \beta_{4} + 171508166$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$8320819 \beta_{11} - 1246689 \beta_{9} + 8889023 \beta_{8} - 41548295 \beta_{3} + 151579342 \beta_{2} - 1515355250 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$178218511 \beta_{10} - 30105001 \beta_{7} - 181957770 \beta_{6} + 1419494314 \beta_{5} + 684909251 \beta_{4} - 20329210046$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-1226087659 \beta_{11} - 593510517 \beta_{9} - 1150460591 \beta_{8} + 5590175999 \beta_{3} - 18093843574 \beta_{2} + 192693279482 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$449$$ $$577$$ $$1093$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
673.1
 6.47819i 9.31944i − 11.2296i − 10.8906i 0.910182i 3.41237i − 3.41237i − 0.910182i 10.8906i 11.2296i − 9.31944i − 6.47819i
0 3.00000i 0 16.7959i 0 7.00000 0 −9.00000 0
673.2 0 3.00000i 0 8.40898i 0 7.00000 0 −9.00000 0
673.3 0 3.00000i 0 6.85537i 0 7.00000 0 −9.00000 0
673.4 0 3.00000i 0 1.28149i 0 7.00000 0 −9.00000 0
673.5 0 3.00000i 0 13.8002i 0 7.00000 0 −9.00000 0
673.6 0 3.00000i 0 20.9786i 0 7.00000 0 −9.00000 0
673.7 0 3.00000i 0 20.9786i 0 7.00000 0 −9.00000 0
673.8 0 3.00000i 0 13.8002i 0 7.00000 0 −9.00000 0
673.9 0 3.00000i 0 1.28149i 0 7.00000 0 −9.00000 0
673.10 0 3.00000i 0 6.85537i 0 7.00000 0 −9.00000 0
673.11 0 3.00000i 0 8.40898i 0 7.00000 0 −9.00000 0
673.12 0 3.00000i 0 16.7959i 0 7.00000 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.c.h yes 12
4.b odd 2 1 1344.4.c.e 12
8.b even 2 1 inner 1344.4.c.h yes 12
8.d odd 2 1 1344.4.c.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.c.e 12 4.b odd 2 1
1344.4.c.e 12 8.d odd 2 1
1344.4.c.h yes 12 1.a even 1 1 trivial
1344.4.c.h yes 12 8.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{12} + 1032 T_{5}^{10} + 374136 T_{5}^{8} + 58092608 T_{5}^{6} + 3747196176 T_{5}^{4} + 84573340800 T_{5}^{2} + 129036134656$$ $$T_{23}^{6} + 40 T_{23}^{5} - 36968 T_{23}^{4} - 638448 T_{23}^{3} + 361912548 T_{23}^{2} + 2746383552 T_{23} - 854202265344$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( 9 + T^{2} )^{6}$$
$5$ $$129036134656 + 84573340800 T^{2} + 3747196176 T^{4} + 58092608 T^{6} + 374136 T^{8} + 1032 T^{10} + T^{12}$$
$7$ $$( -7 + T )^{12}$$
$11$ $$14244025305968896 + 384068248308864 T^{2} + 3402486747024 T^{4} + 11327993600 T^{6} + 14027544 T^{8} + 6648 T^{10} + T^{12}$$
$13$ $$1137605934311276544 + 23337990659506176 T^{2} + 131817561591808 T^{4} + 156265730560 T^{6} + 73687824 T^{8} + 14824 T^{10} + T^{12}$$
$17$ $$( 2432891472 - 1998849936 T + 99246964 T^{2} + 305088 T^{3} - 20108 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$19$ $$69243751964353757184 + 524838336943915008 T^{2} + 1158734298122496 T^{4} + 961524703488 T^{6} + 300209632 T^{8} + 31184 T^{10} + T^{12}$$
$23$ $$( -854202265344 + 2746383552 T + 361912548 T^{2} - 638448 T^{3} - 36968 T^{4} + 40 T^{5} + T^{6} )^{2}$$
$29$ $$29\!\cdots\!44$$$$+ 67553614148077584384 T^{2} + 53582944195197184 T^{4} + 17226248248576 T^{6} + 2020706016 T^{8} + 79888 T^{10} + T^{12}$$
$31$ $$( -2770620689408 - 54901088000 T + 1442292176 T^{2} + 14561920 T^{3} - 57496 T^{4} - 320 T^{5} + T^{6} )^{2}$$
$37$ $$72\!\cdots\!36$$$$+$$$$21\!\cdots\!56$$$$T^{2} + 40487707216763261184 T^{4} + 1888267555539200 T^{6} + 36556297824 T^{8} + 315120 T^{10} + T^{12}$$
$41$ $$( 6866119763664 - 338248101744 T - 1600619372 T^{2} + 43114560 T^{3} + 30052 T^{4} - 708 T^{5} + T^{6} )^{2}$$
$43$ $$10\!\cdots\!04$$$$+$$$$28\!\cdots\!76$$$$T^{2} + 29506731386584076032 T^{4} + 1454306619393280 T^{6} + 34564857072 T^{8} + 355864 T^{10} + T^{12}$$
$47$ $$( 1526303560077312 - 10553680232448 T - 16503838976 T^{2} + 196291584 T^{3} - 133136 T^{4} - 768 T^{5} + T^{6} )^{2}$$
$53$ $$12\!\cdots\!16$$$$+$$$$15\!\cdots\!44$$$$T^{2} +$$$$10\!\cdots\!56$$$$T^{4} + 23973349702622976 T^{6} + 220789272672 T^{8} + 835152 T^{10} + T^{12}$$
$59$ $$51\!\cdots\!36$$$$+$$$$11\!\cdots\!88$$$$T^{2} +$$$$33\!\cdots\!56$$$$T^{4} + 26436275678904320 T^{6} + 412503680256 T^{8} + 1278240 T^{10} + T^{12}$$
$61$ $$77\!\cdots\!04$$$$+$$$$13\!\cdots\!96$$$$T^{2} +$$$$88\!\cdots\!56$$$$T^{4} + 2703531180779678720 T^{6} + 4219515453840 T^{8} + 3270984 T^{10} + T^{12}$$
$67$ $$50\!\cdots\!24$$$$+$$$$12\!\cdots\!56$$$$T^{2} +$$$$28\!\cdots\!12$$$$T^{4} + 247266396767604992 T^{6} + 932858309616 T^{8} + 1587576 T^{10} + T^{12}$$
$71$ $$( 14258685155328 - 35853629568 T - 5598358844 T^{2} - 45613104 T^{3} - 12248 T^{4} + 696 T^{5} + T^{6} )^{2}$$
$73$ $$( -28788786341568 - 4771165902912 T - 92811620304 T^{2} + 487954080 T^{3} - 206340 T^{4} - 1236 T^{5} + T^{6} )^{2}$$
$79$ $$( -137542941144956928 - 67375337852928 T + 1190863313728 T^{2} + 314252800 T^{3} - 2207424 T^{4} - 272 T^{5} + T^{6} )^{2}$$
$83$ $$36\!\cdots\!16$$$$+$$$$99\!\cdots\!76$$$$T^{2} +$$$$72\!\cdots\!88$$$$T^{4} + 729106373996986368 T^{6} + 2328839784064 T^{8} + 2676608 T^{10} + T^{12}$$
$89$ $$( 30429865537350864 + 453774698468208 T + 1275225530644 T^{2} + 128101440 T^{3} - 2181404 T^{4} - 444 T^{5} + T^{6} )^{2}$$
$97$ $$( -73207203980856512 + 81548580109248 T + 852602854704 T^{2} - 1224442336 T^{3} - 1313892 T^{4} + 1356 T^{5} + T^{6} )^{2}$$