# Properties

 Label 1200.3.c.l Level $1200$ Weight $3$ Character orbit 1200.c Analytic conductor $32.698$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 18 x^{10} + 75 x^{8} + 1270 x^{6} + 14397 x^{4} - 7740 x^{2} + 39204$$ Coefficient ring: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$2^{11}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 600) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{6} - \beta_{10} ) q^{3} + ( -2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{7} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{6} - \beta_{10} ) q^{3} + ( -2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{7} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{9} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{11} + ( -4 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{13} + ( -\beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{17} + ( 7 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{19} + ( -4 - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{21} + ( 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{23} + ( -3 \beta_{6} - 4 \beta_{7} + \beta_{8} - 3 \beta_{9} + 7 \beta_{10} - 3 \beta_{11} ) q^{27} + ( 2 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{29} + ( 20 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{31} + ( -12 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{33} + ( 12 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 10 \beta_{10} + 4 \beta_{11} ) q^{37} + ( -2 + 6 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{39} + ( 1 - 6 \beta_{1} - 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{41} + ( 3 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} - 11 \beta_{10} + 5 \beta_{11} ) q^{43} + ( 2 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} ) q^{47} + ( -17 - \beta_{1} - 9 \beta_{2} - 9 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} ) q^{49} + ( -3 - 6 \beta_{1} - 11 \beta_{2} - \beta_{3} + 5 \beta_{4} + 3 \beta_{5} ) q^{51} + ( 6 \beta_{6} - 6 \beta_{7} - 3 \beta_{9} + 15 \beta_{10} ) q^{53} + ( 26 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 11 \beta_{10} + 3 \beta_{11} ) q^{57} + ( -12 \beta_{1} - 12 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( 14 - 13 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{61} + ( -24 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} + 3 \beta_{11} ) q^{63} + ( -71 \beta_{6} - \beta_{8} - 4 \beta_{10} + \beta_{11} ) q^{67} + ( -24 - 15 \beta_{1} - 13 \beta_{2} - 14 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} ) q^{69} + ( -2 + 17 \beta_{1} + 13 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{71} + ( -17 \beta_{6} - 7 \beta_{7} + 3 \beta_{8} - 7 \beta_{9} - 9 \beta_{10} - 3 \beta_{11} ) q^{73} + ( 4 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} + 4 \beta_{11} ) q^{77} + ( -16 + 26 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{79} + ( -36 + 2 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{81} + ( -3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - 9 \beta_{10} - 2 \beta_{11} ) q^{83} + ( 24 \beta_{6} - 2 \beta_{7} - 17 \beta_{9} + 7 \beta_{10} - 6 \beta_{11} ) q^{87} + ( 7 - 11 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} - \beta_{4} + \beta_{5} ) q^{89} + ( 54 - 5 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{91} + ( -26 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 14 \beta_{10} - 9 \beta_{11} ) q^{93} + ( -84 \beta_{6} - 5 \beta_{7} - \beta_{8} - 5 \beta_{9} - 19 \beta_{10} + \beta_{11} ) q^{97} + ( -42 + 7 \beta_{1} - 8 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 32q^{9} + O(q^{10})$$ $$12q - 32q^{9} + 100q^{19} - 36q^{21} + 228q^{31} - 12q^{39} - 152q^{49} + 12q^{51} + 124q^{61} - 312q^{69} - 152q^{79} - 448q^{81} + 620q^{91} - 500q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 18 x^{10} + 75 x^{8} + 1270 x^{6} + 14397 x^{4} - 7740 x^{2} + 39204$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-39 \nu^{10} - 1068 \nu^{8} - 5305 \nu^{6} - 45816 \nu^{4} - 1125197 \nu^{2} - 291510$$$$)/745170$$ $$\beta_{2}$$ $$=$$ $$($$$$-39 \nu^{10} - 1068 \nu^{8} - 5305 \nu^{6} - 45816 \nu^{4} - 752612 \nu^{2} + 453660$$$$)/372585$$ $$\beta_{3}$$ $$=$$ $$($$$$-339 \nu^{10} - 5462 \nu^{8} - 19363 \nu^{6} - 589316 \nu^{4} - 5294253 \nu^{2} + 6304968$$$$)/2235510$$ $$\beta_{4}$$ $$=$$ $$($$$$111 \nu^{10} + 1129 \nu^{8} + 1724 \nu^{6} + 101739 \nu^{4} + 462551 \nu^{2} - 1354239$$$$)/372585$$ $$\beta_{5}$$ $$=$$ $$($$$$-1766 \nu^{10} - 31165 \nu^{8} - 161246 \nu^{6} - 2437674 \nu^{4} - 24014925 \nu^{2} - 9938619$$$$)/3353265$$ $$\beta_{6}$$ $$=$$ $$($$$$-617 \nu^{11} - 12492 \nu^{9} - 67131 \nu^{7} - 852428 \nu^{5} - 11366133 \nu^{3} - 12637134 \nu$$$$)/30008880$$ $$\beta_{7}$$ $$=$$ $$($$$$13087 \nu^{11} + 174956 \nu^{9} + 330589 \nu^{7} + 19080948 \nu^{5} + 121414299 \nu^{3} - 2193841134 \nu$$$$)/ 590174640$$ $$\beta_{8}$$ $$=$$ $$($$$$136583 \nu^{11} + 1982436 \nu^{9} + 16375125 \nu^{7} + 221022452 \nu^{5} + 1301198379 \nu^{3} + 5815099890 \nu$$$$)/ 1770523920$$ $$\beta_{9}$$ $$=$$ $$($$$$35393 \nu^{11} + 495372 \nu^{9} + 917817 \nu^{7} + 29751422 \nu^{5} + 293274123 \nu^{3} - 1096750422 \nu$$$$)/ 442630980$$ $$\beta_{10}$$ $$=$$ $$($$$$-37019 \nu^{11} - 585756 \nu^{9} - 1956771 \nu^{7} - 45762506 \nu^{5} - 442706289 \nu^{3} + 540370386 \nu$$$$)/ 442630980$$ $$\beta_{11}$$ $$=$$ $$($$$$-616375 \nu^{11} - 11391684 \nu^{9} - 44271093 \nu^{7} - 730709380 \nu^{5} - 8809153131 \nu^{3} + 11124243918 \nu$$$$)/ 1770523920$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{10} - \beta_{9} - 4 \beta_{7} + 4 \beta_{6}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2 \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{11} - \beta_{10} + 5 \beta_{9} - 3 \beta_{8} + 6 \beta_{7} - 49 \beta_{6}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{4} - 9 \beta_{3} - 4 \beta_{2} + 17 \beta_{1} + 26$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{11} - 110 \beta_{10} - 96 \beta_{9} + 6 \beta_{8} - 22 \beta_{7} + 293 \beta_{6}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-81 \beta_{5} - 36 \beta_{4} + 135 \beta_{3} + 115 \beta_{2} - 11 \beta_{1} - 896$$ $$\nu^{7}$$ $$=$$ $$($$$$91 \beta_{11} + 1328 \beta_{10} + 654 \beta_{9} + 261 \beta_{8} + 688 \beta_{7} - 2678 \beta_{6}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$567 \beta_{5} + 102 \beta_{4} - 810 \beta_{3} - 2179 \beta_{2} + 1580 \beta_{1} + 7607$$ $$\nu^{9}$$ $$=$$ $$($$$$-6568 \beta_{11} - 2348 \beta_{10} - 11100 \beta_{9} + 885 \beta_{8} - 11458 \beta_{7} + 68555 \beta_{6}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$-4509 \beta_{5} + 5628 \beta_{4} + 14391 \beta_{3} + 19876 \beta_{2} - 23147 \beta_{1} - 66752$$ $$\nu^{11}$$ $$=$$ $$($$$$67351 \beta_{11} + 83684 \beta_{10} + 197514 \beta_{9} + 660 \beta_{8} + 90646 \beta_{7} - 709685 \beta_{6}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.994180 − 0.788754i 0.994180 + 0.788754i 0.139571 − 3.56627i 0.139571 + 3.56627i 2.54797 − 1.77752i 2.54797 + 1.77752i −2.54797 − 1.77752i −2.54797 + 1.77752i −0.139571 − 3.56627i −0.139571 + 3.56627i −0.994180 − 0.788754i −0.994180 + 0.788754i
0 −2.40839 1.78875i 0 0 0 12.2014i 0 2.60072 + 8.61605i 0
449.2 0 −2.40839 + 1.78875i 0 0 0 12.2014i 0 2.60072 8.61605i 0
449.3 0 −1.55378 2.56627i 0 0 0 1.34301i 0 −4.17150 + 7.97487i 0
449.4 0 −1.55378 + 2.56627i 0 0 0 1.34301i 0 −4.17150 7.97487i 0
449.5 0 −1.13375 2.77752i 0 0 0 5.85843i 0 −6.42922 + 6.29803i 0
449.6 0 −1.13375 + 2.77752i 0 0 0 5.85843i 0 −6.42922 6.29803i 0
449.7 0 1.13375 2.77752i 0 0 0 5.85843i 0 −6.42922 6.29803i 0
449.8 0 1.13375 + 2.77752i 0 0 0 5.85843i 0 −6.42922 + 6.29803i 0
449.9 0 1.55378 2.56627i 0 0 0 1.34301i 0 −4.17150 7.97487i 0
449.10 0 1.55378 + 2.56627i 0 0 0 1.34301i 0 −4.17150 + 7.97487i 0
449.11 0 2.40839 1.78875i 0 0 0 12.2014i 0 2.60072 8.61605i 0
449.12 0 2.40839 + 1.78875i 0 0 0 12.2014i 0 2.60072 + 8.61605i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.12 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
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.c.l 12
3.b odd 2 1 inner 1200.3.c.l 12
4.b odd 2 1 600.3.c.c 12
5.b even 2 1 inner 1200.3.c.l 12
5.c odd 4 1 1200.3.l.v 6
5.c odd 4 1 1200.3.l.w 6
12.b even 2 1 600.3.c.c 12
15.d odd 2 1 inner 1200.3.c.l 12
15.e even 4 1 1200.3.l.v 6
15.e even 4 1 1200.3.l.w 6
20.d odd 2 1 600.3.c.c 12
20.e even 4 1 600.3.l.d 6
20.e even 4 1 600.3.l.e yes 6
60.h even 2 1 600.3.c.c 12
60.l odd 4 1 600.3.l.d 6
60.l odd 4 1 600.3.l.e yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.c.c 12 4.b odd 2 1
600.3.c.c 12 12.b even 2 1
600.3.c.c 12 20.d odd 2 1
600.3.c.c 12 60.h even 2 1
600.3.l.d 6 20.e even 4 1
600.3.l.d 6 60.l odd 4 1
600.3.l.e yes 6 20.e even 4 1
600.3.l.e yes 6 60.l odd 4 1
1200.3.c.l 12 1.a even 1 1 trivial
1200.3.c.l 12 3.b odd 2 1 inner
1200.3.c.l 12 5.b even 2 1 inner
1200.3.c.l 12 15.d odd 2 1 inner
1200.3.l.v 6 5.c odd 4 1
1200.3.l.v 6 15.e even 4 1
1200.3.l.w 6 5.c odd 4 1
1200.3.l.w 6 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{6} + 185 T_{7}^{4} + 5440 T_{7}^{2} + 9216$$ $$T_{11}^{6} + 246 T_{11}^{4} + 15129 T_{11}^{2} + 161312$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$531441 + 104976 T^{2} + 19440 T^{4} + 2034 T^{6} + 240 T^{8} + 16 T^{10} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$( 9216 + 5440 T^{2} + 185 T^{4} + T^{6} )^{2}$$
$11$ $$( 161312 + 15129 T^{2} + 246 T^{4} + T^{6} )^{2}$$
$13$ $$( 291600 + 15064 T^{2} + 233 T^{4} + T^{6} )^{2}$$
$17$ $$( -9435168 + 282073 T^{2} - 1174 T^{4} + T^{6} )^{2}$$
$19$ $$( 877 - 205 T - 25 T^{2} + T^{3} )^{4}$$
$23$ $$( -423055872 + 1985680 T^{2} - 2680 T^{4} + T^{6} )^{2}$$
$29$ $$( 37601792 + 1599888 T^{2} + 2616 T^{4} + T^{6} )^{2}$$
$31$ $$( 1804 + 288 T - 57 T^{2} + T^{3} )^{4}$$
$37$ $$( 4857532416 + 8623872 T^{2} + 5092 T^{4} + T^{6} )^{2}$$
$41$ $$( 22418208 + 743337 T^{2} + 4806 T^{4} + T^{6} )^{2}$$
$43$ $$( 3555498384 + 9312280 T^{2} + 6065 T^{4} + T^{6} )^{2}$$
$47$ $$( -187644280832 + 99107472 T^{2} - 17304 T^{4} + T^{6} )^{2}$$
$53$ $$( -24652657152 + 28689552 T^{2} - 9720 T^{4} + T^{6} )^{2}$$
$59$ $$( 91824979968 + 82182400 T^{2} + 17056 T^{4} + T^{6} )^{2}$$
$61$ $$( 117664 - 4288 T - 31 T^{2} + T^{3} )^{4}$$
$67$ $$( 105555461449 + 73000179 T^{2} + 15531 T^{4} + T^{6} )^{2}$$
$71$ $$( 15890452992 + 36573840 T^{2} + 20600 T^{4} + T^{6} )^{2}$$
$73$ $$( 44511716484 + 44642745 T^{2} + 12150 T^{4} + T^{6} )^{2}$$
$79$ $$( 370080 - 14016 T + 38 T^{2} + T^{3} )^{4}$$
$83$ $$( -1152 + 449113 T^{2} - 5206 T^{4} + T^{6} )^{2}$$
$89$ $$( 5739061248 + 17838441 T^{2} + 13478 T^{4} + T^{6} )^{2}$$
$97$ $$( 376549140496 + 353179224 T^{2} + 39129 T^{4} + T^{6} )^{2}$$