Properties

 Label 1380.2.f.b Level $1380$ Weight $2$ Character orbit 1380.f Analytic conductor $11.019$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.f (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$11.0193554789$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 27 x^{12} + 283 x^{10} + 1441 x^{8} + 3596 x^{6} + 3740 x^{4} + 772 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{3} + \beta_{10} q^{5} + \beta_{13} q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{3} + \beta_{10} q^{5} + \beta_{13} q^{7} - q^{9} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{11} + ( -\beta_{1} + \beta_{7} + \beta_{10} + \beta_{12} ) q^{13} + \beta_{8} q^{15} + ( \beta_{1} + \beta_{8} + \beta_{9} - \beta_{12} - \beta_{13} ) q^{17} + ( \beta_{3} - \beta_{4} ) q^{19} -\beta_{2} q^{21} + \beta_{6} q^{23} + ( -1 + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{13} ) q^{25} + \beta_{6} q^{27} + ( -2 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{29} + ( 1 - \beta_{3} + \beta_{5} + 2 \beta_{7} - 2 \beta_{10} ) q^{31} + ( -\beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{33} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{12} ) q^{35} + ( \beta_{1} + 3 \beta_{6} + \beta_{11} ) q^{37} + ( -\beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{39} + ( 3 - \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{41} + ( \beta_{1} + \beta_{6} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{43} -\beta_{10} q^{45} + ( \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{47} + ( -1 - \beta_{2} - 2 \beta_{3} + \beta_{7} - \beta_{10} ) q^{49} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{10} ) q^{51} + ( -\beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{12} ) q^{55} + ( -\beta_{1} - \beta_{12} ) q^{57} + ( -2 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{59} + ( 4 + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{61} -\beta_{13} q^{63} + ( -3 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{65} + ( -\beta_{1} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{67} + q^{69} + ( 3 + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{71} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - 2 \beta_{13} ) q^{73} + ( -1 - \beta_{2} + \beta_{6} + \beta_{11} + \beta_{13} ) q^{75} + ( -2 \beta_{1} + 3 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{77} + ( -2 - 2 \beta_{2} + 2 \beta_{4} + \beta_{8} - \beta_{9} ) q^{79} + q^{81} + ( \beta_{1} - 4 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} ) q^{83} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{85} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{87} + ( -2 + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{89} + ( -\beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 3 \beta_{9} ) q^{91} + ( \beta_{1} - \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{93} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{95} + ( 3 \beta_{1} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{97} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q - 14q^{9} + O(q^{10})$$ $$14q - 14q^{9} + 2q^{15} + 4q^{19} - 2q^{21} - 6q^{25} - 30q^{29} + 6q^{31} - 14q^{35} + 4q^{39} + 46q^{41} - 20q^{49} + 2q^{51} - 16q^{55} - 10q^{59} + 64q^{61} - 36q^{65} + 14q^{69} + 42q^{71} - 16q^{75} - 32q^{79} + 14q^{81} - 42q^{85} - 52q^{89} + 28q^{91} - 44q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 27 x^{12} + 283 x^{10} + 1441 x^{8} + 3596 x^{6} + 3740 x^{4} + 772 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 6 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{12} + 6 \nu^{10} - 153 \nu^{8} - 1732 \nu^{6} - 5726 \nu^{4} - 5036 \nu^{2} + 136$$$$)/372$$ $$\beta_{4}$$ $$=$$ $$($$$$15 \nu^{12} + 338 \nu^{10} + 2851 \nu^{8} + 11282 \nu^{6} + 21122 \nu^{4} + 15228 \nu^{2} + 800$$$$)/372$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{12} - 86 \nu^{10} - 659 \nu^{8} - 2124 \nu^{6} - 2423 \nu^{4} - 6 \nu^{2} + 200$$$$)/62$$ $$\beta_{6}$$ $$=$$ $$($$$$6 \nu^{13} + 160 \nu^{11} + 1655 \nu^{9} + 8332 \nu^{7} + 20731 \nu^{5} + 22050 \nu^{3} + 5218 \nu$$$$)/372$$ $$\beta_{7}$$ $$=$$ $$($$$$-21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 12 \nu^{10} - 6459 \nu^{9} - 306 \nu^{8} - 34215 \nu^{7} - 3464 \nu^{6} - 88260 \nu^{5} - 11824 \nu^{4} - 93078 \nu^{3} - 12676 \nu^{2} - 16992 \nu - 1216$$$$)/744$$ $$\beta_{8}$$ $$=$$ $$($$$$-21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 74 \nu^{10} - 6459 \nu^{9} - 934 \nu^{8} - 34215 \nu^{7} - 5154 \nu^{6} - 88260 \nu^{5} - 12604 \nu^{4} - 93078 \nu^{3} - 11504 \nu^{2} - 16248 \nu - 1264$$$$)/744$$ $$\beta_{9}$$ $$=$$ $$($$$$-21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 74 \nu^{10} - 6459 \nu^{9} + 934 \nu^{8} - 34215 \nu^{7} + 5154 \nu^{6} - 88260 \nu^{5} + 12604 \nu^{4} - 93078 \nu^{3} + 11504 \nu^{2} - 16248 \nu + 1264$$$$)/744$$ $$\beta_{10}$$ $$=$$ $$($$$$-21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 12 \nu^{10} - 6459 \nu^{9} + 306 \nu^{8} - 34215 \nu^{7} + 3464 \nu^{6} - 88260 \nu^{5} + 11824 \nu^{4} - 93078 \nu^{3} + 12676 \nu^{2} - 16992 \nu + 1216$$$$)/744$$ $$\beta_{11}$$ $$=$$ $$($$$$16 \nu^{13} + 468 \nu^{11} + 5271 \nu^{9} + 28460 \nu^{7} + 74017 \nu^{5} + 78826 \nu^{3} + 16870 \nu$$$$)/372$$ $$\beta_{12}$$ $$=$$ $$($$$$-27 \nu^{13} - 689 \nu^{11} - 6781 \nu^{9} - 32255 \nu^{7} - 74798 \nu^{5} - 71046 \nu^{3} - 10244 \nu$$$$)/372$$ $$\beta_{13}$$ $$=$$ $$($$$$-22 \nu^{13} - 597 \nu^{11} - 6306 \nu^{9} - 32483 \nu^{7} - 82534 \nu^{5} - 88786 \nu^{3} - 20848 \nu$$$$)/372$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{10} - 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{10} - \beta_{7} + 2 \beta_{3} - 7 \beta_{2} + 24$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{13} + \beta_{11} - 19 \beta_{10} + 21 \beta_{9} + 21 \beta_{8} - 19 \beta_{7} - 3 \beta_{6} - 11 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-17 \beta_{10} + 4 \beta_{9} - 4 \beta_{8} + 17 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 24 \beta_{3} + 49 \beta_{2} - 156$$ $$\nu^{7}$$ $$=$$ $$30 \beta_{13} + 2 \beta_{12} - 15 \beta_{11} + 128 \beta_{10} - 156 \beta_{9} - 156 \beta_{8} + 128 \beta_{7} + 61 \beta_{6} + 99 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$197 \beta_{10} - 68 \beta_{9} + 68 \beta_{8} - 197 \beta_{7} - 30 \beta_{5} - 28 \beta_{4} + 230 \beta_{3} - 353 \beta_{2} + 1072$$ $$\nu^{9}$$ $$=$$ $$-338 \beta_{13} - 32 \beta_{12} + 157 \beta_{11} - 904 \beta_{10} + 1194 \beta_{9} + 1194 \beta_{8} - 904 \beta_{7} - 787 \beta_{6} - 841 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-1965 \beta_{10} + 810 \beta_{9} - 810 \beta_{8} + 1965 \beta_{7} + 322 \beta_{5} + 282 \beta_{4} - 2052 \beta_{3} + 2617 \beta_{2} - 7692$$ $$\nu^{11}$$ $$=$$ $$3390 \beta_{13} + 362 \beta_{12} - 1437 \beta_{11} + 6624 \beta_{10} - 9320 \beta_{9} - 9320 \beta_{8} + 6624 \beta_{7} + 8455 \beta_{6} + 7003 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$18213 \beta_{10} - 8336 \beta_{9} + 8336 \beta_{8} - 18213 \beta_{7} - 3058 \beta_{5} - 2512 \beta_{4} + 17758 \beta_{3} - 19889 \beta_{2} + 57120$$ $$\nu^{13}$$ $$=$$ $$-31918 \beta_{13} - 3604 \beta_{12} + 12389 \beta_{11} - 49978 \beta_{10} + 73852 \beta_{9} + 73852 \beta_{8} - 49978 \beta_{7} - 82667 \beta_{6} - 57917 \beta_{1}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$461$$ $$691$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1
 2.39625i 0.511427i 2.47928i − 2.04690i − 2.88373i 0.0729221i − 1.52925i − 2.39625i − 0.511427i − 2.47928i 2.04690i 2.88373i − 0.0729221i 1.52925i
0 1.00000i 0 −2.22987 0.166381i 0 1.74202i 0 −1.00000 0
829.2 0 1.00000i 0 −1.81604 + 1.30461i 0 3.73844i 0 −1.00000 0
829.3 0 1.00000i 0 −0.258163 2.22111i 0 2.14682i 0 −1.00000 0
829.4 0 1.00000i 0 −0.181772 + 2.22867i 0 0.189781i 0 −1.00000 0
829.5 0 1.00000i 0 0.792997 + 2.09073i 0 4.31589i 0 −1.00000 0
829.6 0 1.00000i 0 1.54426 1.61718i 0 3.99468i 0 −1.00000 0
829.7 0 1.00000i 0 2.14859 0.619333i 0 1.66138i 0 −1.00000 0
829.8 0 1.00000i 0 −2.22987 + 0.166381i 0 1.74202i 0 −1.00000 0
829.9 0 1.00000i 0 −1.81604 1.30461i 0 3.73844i 0 −1.00000 0
829.10 0 1.00000i 0 −0.258163 + 2.22111i 0 2.14682i 0 −1.00000 0
829.11 0 1.00000i 0 −0.181772 2.22867i 0 0.189781i 0 −1.00000 0
829.12 0 1.00000i 0 0.792997 2.09073i 0 4.31589i 0 −1.00000 0
829.13 0 1.00000i 0 1.54426 + 1.61718i 0 3.99468i 0 −1.00000 0
829.14 0 1.00000i 0 2.14859 + 0.619333i 0 1.66138i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 829.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.f.b 14
3.b odd 2 1 4140.2.f.c 14
5.b even 2 1 inner 1380.2.f.b 14
5.c odd 4 1 6900.2.a.bc 7
5.c odd 4 1 6900.2.a.bd 7
15.d odd 2 1 4140.2.f.c 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.b 14 1.a even 1 1 trivial
1380.2.f.b 14 5.b even 2 1 inner
4140.2.f.c 14 3.b odd 2 1
4140.2.f.c 14 15.d odd 2 1
6900.2.a.bc 7 5.c odd 4 1
6900.2.a.bd 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{14} + 59 T_{7}^{12} + 1323 T_{7}^{10} + 14065 T_{7}^{8} + 72984 T_{7}^{6} + 178488 T_{7}^{4} + 166704 T_{7}^{2} + 5776$$ acting on $$S_{2}^{\mathrm{new}}(1380, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$( 1 + T^{2} )^{7}$$
$5$ $$78125 + 9375 T^{2} + 5000 T^{3} - 1875 T^{4} + 800 T^{5} + 95 T^{6} - 144 T^{7} + 19 T^{8} + 32 T^{9} - 15 T^{10} + 8 T^{11} + 3 T^{12} + T^{14}$$
$7$ $$5776 + 166704 T^{2} + 178488 T^{4} + 72984 T^{6} + 14065 T^{8} + 1323 T^{10} + 59 T^{12} + T^{14}$$
$11$ $$( 9216 - 4608 T - 2112 T^{2} + 1200 T^{3} + 62 T^{4} - 64 T^{5} + T^{7} )^{2}$$
$13$ $$6071296 + 12337152 T^{2} + 5286784 T^{4} + 958912 T^{6} + 88004 T^{8} + 4268 T^{10} + 104 T^{12} + T^{14}$$
$17$ $$10000 + 6370864 T^{2} + 23376664 T^{4} + 4556280 T^{6} + 323441 T^{8} + 10683 T^{10} + 167 T^{12} + T^{14}$$
$19$ $$( -36680 - 35928 T - 2216 T^{2} + 4108 T^{3} + 158 T^{4} - 118 T^{5} - 2 T^{6} + T^{7} )^{2}$$
$23$ $$( 1 + T^{2} )^{7}$$
$29$ $$( -12272 + 20976 T + 6504 T^{2} - 1792 T^{3} - 595 T^{4} + 17 T^{5} + 15 T^{6} + T^{7} )^{2}$$
$31$ $$( 54720 - 23616 T - 13920 T^{2} + 4248 T^{3} + 499 T^{4} - 141 T^{5} - 3 T^{6} + T^{7} )^{2}$$
$37$ $$16126968064 + 4925021696 T^{2} + 590900192 T^{4} + 36366992 T^{6} + 1247449 T^{8} + 24043 T^{10} + 243 T^{12} + T^{14}$$
$41$ $$( 15248 - 45552 T + 41576 T^{2} - 12896 T^{3} + 1185 T^{4} + 103 T^{5} - 23 T^{6} + T^{7} )^{2}$$
$43$ $$389193984 + 483577344 T^{2} + 194891904 T^{4} + 29281296 T^{6} + 1720720 T^{8} + 37928 T^{10} + 336 T^{12} + T^{14}$$
$47$ $$1973013529600 + 304913816576 T^{2} + 18562826624 T^{4} + 577880384 T^{6} + 10016260 T^{8} + 97288 T^{10} + 492 T^{12} + T^{14}$$
$53$ $$374345104 + 274681456 T^{2} + 76576024 T^{4} + 10169496 T^{6} + 662233 T^{8} + 19775 T^{10} + 251 T^{12} + T^{14}$$
$59$ $$( -1483200 - 308160 T + 81312 T^{2} + 16872 T^{3} - 1217 T^{4} - 255 T^{5} + 5 T^{6} + T^{7} )^{2}$$
$61$ $$( 377248 - 459232 T + 192464 T^{2} - 32672 T^{3} + 1226 T^{4} + 274 T^{5} - 32 T^{6} + T^{7} )^{2}$$
$67$ $$300304 + 803120048 T^{2} + 345348536 T^{4} + 45589400 T^{6} + 1979401 T^{8} + 37075 T^{10} + 315 T^{12} + T^{14}$$
$71$ $$( -784 + 5840 T - 3848 T^{2} - 3184 T^{3} + 875 T^{4} + 55 T^{5} - 21 T^{6} + T^{7} )^{2}$$
$73$ $$47127199744 + 28443672576 T^{2} + 3597107200 T^{4} + 192441856 T^{6} + 5104292 T^{8} + 68876 T^{10} + 440 T^{12} + T^{14}$$
$79$ $$( -4755584 - 697856 T + 264784 T^{2} + 23540 T^{3} - 3792 T^{4} - 264 T^{5} + 16 T^{6} + T^{7} )^{2}$$
$83$ $$1638400 + 13500416 T^{2} + 25436672 T^{4} + 11084416 T^{6} + 1252593 T^{8} + 47443 T^{10} + 415 T^{12} + T^{14}$$
$89$ $$( -526112 + 457456 T + 64264 T^{2} - 23724 T^{3} - 4000 T^{4} + 12 T^{5} + 26 T^{6} + T^{7} )^{2}$$
$97$ $$56070144 + 16510058496 T^{2} + 34971683328 T^{4} + 1825932288 T^{6} + 31417360 T^{8} + 237264 T^{10} + 804 T^{12} + T^{14}$$