# Properties

 Label 1800.2.k.u Level 1800 Weight 2 Character orbit 1800.k Analytic conductor 14.373 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3730723638$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.180227832610816.1 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 120) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{4} q^{4} + ( -\beta_{1} - \beta_{8} ) q^{7} + ( -\beta_{3} + \beta_{8} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{4} q^{4} + ( -\beta_{1} - \beta_{8} ) q^{7} + ( -\beta_{3} + \beta_{8} ) q^{8} + ( -\beta_{4} + \beta_{6} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{7} - \beta_{9} ) q^{13} + ( -2 - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} ) q^{14} + ( -\beta_{5} + \beta_{6} - \beta_{10} ) q^{16} + ( -\beta_{1} - \beta_{2} - \beta_{7} ) q^{17} + ( -2 \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} ) q^{19} + ( -\beta_{2} + 2 \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{22} + ( -2 \beta_{2} + \beta_{3} - \beta_{7} ) q^{23} + ( 2 - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{26} + ( -2 \beta_{1} - 2 \beta_{7} ) q^{28} + ( -2 \beta_{4} + \beta_{5} - \beta_{10} + \beta_{11} ) q^{29} + ( -2 + \beta_{6} + \beta_{10} + \beta_{11} ) q^{31} + ( \beta_{3} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{32} + ( -2 - \beta_{4} - \beta_{5} + \beta_{11} ) q^{34} + ( -\beta_{1} - \beta_{2} + \beta_{7} + \beta_{9} ) q^{37} + ( 2 \beta_{7} - 2 \beta_{8} ) q^{38} + ( 2 + 2 \beta_{6} + 2 \beta_{10} + 2 \beta_{11} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( 4 + 2 \beta_{10} ) q^{44} + ( -2 \beta_{5} + 2 \beta_{11} ) q^{46} + ( \beta_{3} + \beta_{7} - 2 \beta_{8} ) q^{47} + ( 1 + 2 \beta_{4} - 2 \beta_{10} ) q^{49} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{8} - 2 \beta_{9} ) q^{52} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{9} ) q^{53} + ( -4 - 2 \beta_{4} + 2 \beta_{11} ) q^{56} + ( \beta_{2} + 4 \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{58} + ( -\beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{59} + ( 2 \beta_{6} + 2 \beta_{10} - 2 \beta_{11} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{9} ) q^{62} + ( 4 + \beta_{5} + \beta_{6} + 3 \beta_{10} - 2 \beta_{11} ) q^{64} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{8} + 2 \beta_{9} ) q^{68} + ( -4 - 2 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{71} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{7} - 2 \beta_{8} ) q^{73} + ( 2 - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{74} + ( 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{10} - 4 \beta_{11} ) q^{76} + 2 \beta_{9} q^{77} + ( -2 + \beta_{6} + \beta_{10} + \beta_{11} ) q^{79} + ( 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{9} ) q^{82} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{7} + 2 \beta_{9} ) q^{83} + ( -4 + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{11} ) q^{86} + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} + 2 \beta_{9} ) q^{88} + ( -2 + 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{10} + 2 \beta_{11} ) q^{89} + ( -2 \beta_{4} - 2 \beta_{6} - 4 \beta_{10} + 4 \beta_{11} ) q^{91} + ( -2 \beta_{3} - 2 \beta_{8} + 4 \beta_{9} ) q^{92} + ( 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{10} - 2 \beta_{11} ) q^{94} + ( 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{7} - 4 \beta_{8} ) q^{97} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 2q^{4} + O(q^{10})$$ $$12q - 2q^{4} - 20q^{14} + 2q^{16} + 28q^{26} - 32q^{31} - 24q^{34} + 8q^{41} + 44q^{44} - 4q^{46} + 12q^{49} - 52q^{56} + 46q^{64} - 32q^{71} + 36q^{74} + 12q^{76} - 32q^{79} - 40q^{86} - 40q^{89} + 4q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + x^{10} - 8 x^{6} + 16 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{11} - \nu^{9} + 8 \nu^{5} - 16 \nu$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{9} - \nu^{7} + 2 \nu^{5} + 4 \nu^{3} + 8 \nu$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{9} + 6 \nu^{7} + 4 \nu^{5} - 8 \nu^{3}$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{10} - \nu^{8} + 8 \nu^{4} - 16$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{8} - \nu^{6} + 2 \nu^{4} + 4 \nu^{2} + 8$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - \nu^{4} + 4 \nu^{2} + 4$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{11} - \nu^{9} - 6 \nu^{7} - 4 \nu^{5} + 8 \nu^{3} + 64 \nu$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{9} + \nu^{7} - 2 \nu^{5} + 12 \nu^{3} + 8 \nu$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{11} - \nu^{9} + 2 \nu^{7} + 4 \nu^{5} + 8 \nu^{3}$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{10} + \nu^{8} + 2 \nu^{6} - 24 \nu^{2} - 16$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{10} + \nu^{8} + 6 \nu^{6} + 4 \nu^{4} - 8 \nu^{2} - 32$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{6}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{8} - \beta_{7} - \beta_{3} + 2 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{10} - 3 \beta_{6} + 2 \beta_{5} + 2 \beta_{4}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{3} + 2 \beta_{2} + 4 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{11} - 3 \beta_{10} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 8$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$4 \beta_{9} + 2 \beta_{8} - \beta_{7} + 7 \beta_{3} - 2 \beta_{2} - 4 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$3 \beta_{11} + 3 \beta_{10} + \beta_{6} + 10 \beta_{5} - 6 \beta_{4} - 8$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-4 \beta_{9} - 2 \beta_{8} - 7 \beta_{7} + \beta_{3} + 18 \beta_{2} - 12 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-11 \beta_{11} - 11 \beta_{10} - 25 \beta_{6} + 6 \beta_{5} - 10 \beta_{4} - 24$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$36 \beta_{9} - 14 \beta_{8} - \beta_{7} - 9 \beta_{3} - 2 \beta_{2} - 20 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 −1.37729 + 0.321037i −1.37729 − 0.321037i −0.806504 + 1.16170i −0.806504 − 1.16170i −0.450129 + 1.34067i −0.450129 − 1.34067i 0.450129 + 1.34067i 0.450129 − 1.34067i 0.806504 + 1.16170i 0.806504 − 1.16170i 1.37729 + 0.321037i 1.37729 − 0.321037i
−1.37729 0.321037i 0 1.79387 + 0.884323i 0 0 4.05705 −2.18678 1.79387i 0 0
901.2 −1.37729 + 0.321037i 0 1.79387 0.884323i 0 0 4.05705 −2.18678 + 1.79387i 0 0
901.3 −0.806504 1.16170i 0 −0.699104 + 1.87383i 0 0 0.746175 2.74067 0.699104i 0 0
901.4 −0.806504 + 1.16170i 0 −0.699104 1.87383i 0 0 0.746175 2.74067 + 0.699104i 0 0
901.5 −0.450129 1.34067i 0 −1.59477 + 1.20695i 0 0 −2.64265 2.33596 + 1.59477i 0 0
901.6 −0.450129 + 1.34067i 0 −1.59477 1.20695i 0 0 −2.64265 2.33596 1.59477i 0 0
901.7 0.450129 1.34067i 0 −1.59477 1.20695i 0 0 2.64265 −2.33596 + 1.59477i 0 0
901.8 0.450129 + 1.34067i 0 −1.59477 + 1.20695i 0 0 2.64265 −2.33596 1.59477i 0 0
901.9 0.806504 1.16170i 0 −0.699104 1.87383i 0 0 −0.746175 −2.74067 0.699104i 0 0
901.10 0.806504 + 1.16170i 0 −0.699104 + 1.87383i 0 0 −0.746175 −2.74067 + 0.699104i 0 0
901.11 1.37729 0.321037i 0 1.79387 0.884323i 0 0 −4.05705 2.18678 1.79387i 0 0
901.12 1.37729 + 0.321037i 0 1.79387 + 0.884323i 0 0 −4.05705 2.18678 + 1.79387i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.k.u 12
3.b odd 2 1 600.2.k.f 12
4.b odd 2 1 7200.2.k.u 12
5.b even 2 1 inner 1800.2.k.u 12
5.c odd 4 1 360.2.d.e 6
5.c odd 4 1 360.2.d.f 6
8.b even 2 1 inner 1800.2.k.u 12
8.d odd 2 1 7200.2.k.u 12
12.b even 2 1 2400.2.k.f 12
15.d odd 2 1 600.2.k.f 12
15.e even 4 1 120.2.d.a 6
15.e even 4 1 120.2.d.b yes 6
20.d odd 2 1 7200.2.k.u 12
20.e even 4 1 1440.2.d.e 6
20.e even 4 1 1440.2.d.f 6
24.f even 2 1 2400.2.k.f 12
24.h odd 2 1 600.2.k.f 12
40.e odd 2 1 7200.2.k.u 12
40.f even 2 1 inner 1800.2.k.u 12
40.i odd 4 1 360.2.d.e 6
40.i odd 4 1 360.2.d.f 6
40.k even 4 1 1440.2.d.e 6
40.k even 4 1 1440.2.d.f 6
60.h even 2 1 2400.2.k.f 12
60.l odd 4 1 480.2.d.a 6
60.l odd 4 1 480.2.d.b 6
120.i odd 2 1 600.2.k.f 12
120.m even 2 1 2400.2.k.f 12
120.q odd 4 1 480.2.d.a 6
120.q odd 4 1 480.2.d.b 6
120.w even 4 1 120.2.d.a 6
120.w even 4 1 120.2.d.b yes 6
240.z odd 4 2 3840.2.f.m 12
240.bb even 4 2 3840.2.f.l 12
240.bd odd 4 2 3840.2.f.m 12
240.bf even 4 2 3840.2.f.l 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 15.e even 4 1
120.2.d.a 6 120.w even 4 1
120.2.d.b yes 6 15.e even 4 1
120.2.d.b yes 6 120.w even 4 1
360.2.d.e 6 5.c odd 4 1
360.2.d.e 6 40.i odd 4 1
360.2.d.f 6 5.c odd 4 1
360.2.d.f 6 40.i odd 4 1
480.2.d.a 6 60.l odd 4 1
480.2.d.a 6 120.q odd 4 1
480.2.d.b 6 60.l odd 4 1
480.2.d.b 6 120.q odd 4 1
600.2.k.f 12 3.b odd 2 1
600.2.k.f 12 15.d odd 2 1
600.2.k.f 12 24.h odd 2 1
600.2.k.f 12 120.i odd 2 1
1440.2.d.e 6 20.e even 4 1
1440.2.d.e 6 40.k even 4 1
1440.2.d.f 6 20.e even 4 1
1440.2.d.f 6 40.k even 4 1
1800.2.k.u 12 1.a even 1 1 trivial
1800.2.k.u 12 5.b even 2 1 inner
1800.2.k.u 12 8.b even 2 1 inner
1800.2.k.u 12 40.f even 2 1 inner
2400.2.k.f 12 12.b even 2 1
2400.2.k.f 12 24.f even 2 1
2400.2.k.f 12 60.h even 2 1
2400.2.k.f 12 120.m even 2 1
3840.2.f.l 12 240.bb even 4 2
3840.2.f.l 12 240.bf even 4 2
3840.2.f.m 12 240.z odd 4 2
3840.2.f.m 12 240.bd odd 4 2
7200.2.k.u 12 4.b odd 2 1
7200.2.k.u 12 8.d odd 2 1
7200.2.k.u 12 20.d odd 2 1
7200.2.k.u 12 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{6} - 24 T_{7}^{4} + 128 T_{7}^{2} - 64$$ $$T_{11}^{6} + 32 T_{11}^{4} + 96 T_{11}^{2} + 64$$ $$T_{17}^{6} - 36 T_{17}^{4} + 368 T_{17}^{2} - 1024$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} - 8 T^{6} + 16 T^{10} + 64 T^{12}$$
$3$ 1
$5$ 1
$7$ $$( 1 + 18 T^{2} + 191 T^{4} + 1532 T^{6} + 9359 T^{8} + 43218 T^{10} + 117649 T^{12} )^{2}$$
$11$ $$( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 60863 T^{8} - 497794 T^{10} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 - 30 T^{2} + 743 T^{4} - 10436 T^{6} + 125567 T^{8} - 856830 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 + 66 T^{2} + 2255 T^{4} + 47324 T^{6} + 651695 T^{8} + 5512386 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 493487 T^{8} - 7037334 T^{10} + 47045881 T^{12} )^{2}$$
$23$ $$( 1 + 46 T^{2} + 1775 T^{4} + 40932 T^{6} + 938975 T^{8} + 12872686 T^{10} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 2697087 T^{8} - 46680546 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 + 8 T + 89 T^{2} + 432 T^{3} + 2759 T^{4} + 7688 T^{5} + 29791 T^{6} )^{4}$$
$37$ $$( 1 - 158 T^{2} + 11191 T^{4} - 496772 T^{6} + 15320479 T^{8} - 296117438 T^{10} + 2565726409 T^{12} )^{2}$$
$41$ $$( 1 - 2 T + 23 T^{2} - 220 T^{3} + 943 T^{4} - 3362 T^{5} + 68921 T^{6} )^{4}$$
$43$ $$( 1 - 130 T^{2} + 9815 T^{4} - 518268 T^{6} + 18147935 T^{8} - 444444130 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 + 222 T^{2} + 22367 T^{4} + 1328324 T^{6} + 49408703 T^{8} + 1083289182 T^{10} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 - 238 T^{2} + 26391 T^{4} - 1758052 T^{6} + 74132319 T^{8} - 1877934478 T^{10} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 71593727 T^{8} - 2156890258 T^{10} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 74565119 T^{8} - 2630709790 T^{10} + 51520374361 T^{12} )^{2}$$
$67$ $$( 1 - 274 T^{2} + 37127 T^{4} - 3112476 T^{6} + 166663103 T^{8} - 5521407154 T^{10} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 9443 T^{4} + 40328 T^{5} + 357911 T^{6} )^{4}$$
$73$ $$( 1 + 54 T^{2} + 2367 T^{4} + 531700 T^{6} + 12613743 T^{8} + 1533505014 T^{10} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 18407 T^{4} + 49928 T^{5} + 493039 T^{6} )^{4}$$
$83$ $$( 1 - 306 T^{2} + 50855 T^{4} - 5168732 T^{6} + 350340095 T^{8} - 14522246226 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 + 10 T + 103 T^{2} + 396 T^{3} + 9167 T^{4} + 79210 T^{5} + 704969 T^{6} )^{4}$$
$97$ $$( 1 + 246 T^{2} + 39183 T^{4} + 4535476 T^{6} + 368672847 T^{8} + 21778203126 T^{10} + 832972004929 T^{12} )^{2}$$