Properties

 Label 225.4.f.c Level $225$ Weight $4$ Character orbit 225.f Analytic conductor $13.275$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.f (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16 x^{10} - 14 x^{8} - 512 x^{6} + 3889 x^{4} + 126224 x^{2} + 506944$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{11}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{2} + ( -6 \beta_{1} + \beta_{5} ) q^{4} + ( -2 - 2 \beta_{1} + \beta_{4} ) q^{7} + ( -3 \beta_{7} - 7 \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} +O(q^{10})$$ $$q -\beta_{6} q^{2} + ( -6 \beta_{1} + \beta_{5} ) q^{4} + ( -2 - 2 \beta_{1} + \beta_{4} ) q^{7} + ( -3 \beta_{7} - 7 \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} + ( -7 \beta_{6} - 7 \beta_{8} + 3 \beta_{9} + \beta_{11} ) q^{11} + ( 9 - 9 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{13} + ( -3 \beta_{6} - 7 \beta_{7} + 3 \beta_{8} - \beta_{10} ) q^{14} + ( -54 + 9 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} ) q^{16} + ( -8 \beta_{6} - 12 \beta_{7} + 12 \beta_{9} + \beta_{10} + \beta_{11} ) q^{17} + ( -20 \beta_{1} + 5 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} ) q^{19} + ( -88 - 88 \beta_{1} + 6 \beta_{2} - \beta_{4} + 6 \beta_{5} ) q^{22} + ( -2 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} ) q^{23} + ( -6 \beta_{6} - 6 \beta_{8} + 33 \beta_{9} - 7 \beta_{11} ) q^{26} + ( 48 - 48 \beta_{1} + 6 \beta_{2} - \beta_{3} - 6 \beta_{5} ) q^{28} + ( -25 \beta_{6} - 4 \beta_{7} + 25 \beta_{8} + 5 \beta_{10} ) q^{29} + ( -104 - 4 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} ) q^{31} + ( 29 \beta_{6} - 38 \beta_{7} + 38 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{32} + ( -56 \beta_{1} - 10 \beta_{3} - 10 \beta_{4} - 12 \beta_{5} ) q^{34} + ( -69 - 69 \beta_{1} - 8 \beta_{2} + 7 \beta_{4} - 8 \beta_{5} ) q^{37} + ( -47 \beta_{7} - 6 \beta_{8} - 47 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} ) q^{38} + ( 38 \beta_{6} + 38 \beta_{8} + 29 \beta_{9} + 16 \beta_{11} ) q^{41} + ( 8 - 8 \beta_{1} - 16 \beta_{2} + 14 \beta_{3} + 16 \beta_{5} ) q^{43} + ( 91 \beta_{6} - 53 \beta_{7} - 91 \beta_{8} - 3 \beta_{10} ) q^{44} + ( 56 - 28 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} ) q^{46} + ( -58 \beta_{6} - 23 \beta_{7} + 23 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} ) q^{47} + ( -111 \beta_{1} + 16 \beta_{5} ) q^{49} + ( 26 + 26 \beta_{1} - 25 \beta_{2} - 23 \beta_{4} - 25 \beta_{5} ) q^{52} + ( 5 \beta_{7} + 34 \beta_{8} + 5 \beta_{9} - 10 \beta_{10} + 10 \beta_{11} ) q^{53} + ( 25 \beta_{6} + 25 \beta_{8} - 13 \beta_{9} - 5 \beta_{11} ) q^{56} + ( 322 - 322 \beta_{1} - 31 \beta_{2} + 6 \beta_{3} + 31 \beta_{5} ) q^{58} + ( -41 \beta_{6} + 31 \beta_{7} + 41 \beta_{8} - 17 \beta_{10} ) q^{59} + ( 8 - 48 \beta_{2} + 15 \beta_{3} - 15 \beta_{4} ) q^{61} + ( 118 \beta_{6} + 47 \beta_{7} - 47 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} ) q^{62} + ( 78 \beta_{1} - 10 \beta_{3} - 10 \beta_{4} - 57 \beta_{5} ) q^{64} + ( -136 - 136 \beta_{1} + 44 \beta_{2} - 2 \beta_{4} + 44 \beta_{5} ) q^{67} + ( 10 \beta_{7} + 16 \beta_{8} + 10 \beta_{9} + 30 \beta_{10} - 30 \beta_{11} ) q^{68} + ( -16 \beta_{6} - 16 \beta_{8} + 4 \beta_{9} - 32 \beta_{11} ) q^{71} + ( -331 + 331 \beta_{1} + 18 \beta_{2} + 12 \beta_{3} - 18 \beta_{5} ) q^{73} + ( -38 \beta_{6} - \beta_{7} + 38 \beta_{8} + 9 \beta_{10} ) q^{74} + ( -40 + 104 \beta_{2} - 25 \beta_{3} + 25 \beta_{4} ) q^{76} + ( -32 \beta_{6} - 12 \beta_{7} + 12 \beta_{9} - 24 \beta_{10} - 24 \beta_{11} ) q^{77} + ( 672 \beta_{1} + 25 \beta_{3} + 25 \beta_{4} + 28 \beta_{5} ) q^{79} + ( 654 + 654 \beta_{1} - 35 \beta_{2} + 3 \beta_{4} - 35 \beta_{5} ) q^{82} + ( 153 \beta_{7} + 78 \beta_{8} + 153 \beta_{9} + \beta_{10} - \beta_{11} ) q^{83} + ( -82 \beta_{6} - 82 \beta_{8} - 194 \beta_{9} + 46 \beta_{11} ) q^{86} + ( -664 + 664 \beta_{1} + 102 \beta_{2} - 67 \beta_{3} - 102 \beta_{5} ) q^{88} + ( 36 \beta_{6} + 163 \beta_{7} - 36 \beta_{8} + 42 \beta_{10} ) q^{89} + ( 396 + 24 \beta_{2} - 25 \beta_{3} + 25 \beta_{4} ) q^{91} + ( -144 \beta_{6} + 170 \beta_{7} - 170 \beta_{9} - 10 \beta_{10} - 10 \beta_{11} ) q^{92} + ( -648 \beta_{1} - 5 \beta_{3} - 5 \beta_{4} + 48 \beta_{5} ) q^{94} + ( -231 - 231 \beta_{1} + 50 \beta_{2} + 78 \beta_{4} + 50 \beta_{5} ) q^{97} + ( -48 \beta_{7} - 255 \beta_{8} - 48 \beta_{9} - 16 \beta_{10} + 16 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 24 q^{7} + O(q^{10})$$ $$12 q - 24 q^{7} + 108 q^{13} - 648 q^{16} - 1056 q^{22} + 576 q^{28} - 1248 q^{31} - 828 q^{37} + 96 q^{43} + 672 q^{46} + 312 q^{52} + 3864 q^{58} + 96 q^{61} - 1632 q^{67} - 3972 q^{73} - 480 q^{76} + 7848 q^{82} - 7968 q^{88} + 4752 q^{91} - 2772 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16 x^{10} - 14 x^{8} - 512 x^{6} + 3889 x^{4} + 126224 x^{2} + 506944$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{10} - 56 \nu^{8} + 634 \nu^{6} + 3248 \nu^{4} + 83197 \nu^{2} + 418024$$$$)/182400$$ $$\beta_{2}$$ $$=$$ $$($$$$-71 \nu^{10} + 2538 \nu^{8} - 4502 \nu^{6} - 356014 \nu^{4} - 1896111 \nu^{2} - 1578712$$$$)/3620640$$ $$\beta_{3}$$ $$=$$ $$($$$$1029 \nu^{10} - 22972 \nu^{8} + 158738 \nu^{6} - 2560724 \nu^{4} + 22729229 \nu^{2} + 78379688$$$$)/9051600$$ $$\beta_{4}$$ $$=$$ $$($$$$5723 \nu^{10} - 110024 \nu^{8} + 167406 \nu^{6} - 2434608 \nu^{4} + 44277523 \nu^{2} + 559571096$$$$)/18103200$$ $$\beta_{5}$$ $$=$$ $$($$$$1265 \nu^{10} - 2936 \nu^{8} - 23478 \nu^{6} - 1362480 \nu^{4} - 16244903 \nu^{2} - 51726904$$$$)/2896512$$ $$\beta_{6}$$ $$=$$ $$($$$$-113621 \nu^{11} - 12179272 \nu^{9} + 76074438 \nu^{7} + 253832976 \nu^{5} + 12758183579 \nu^{3} + 93476563288 \nu$$$$)/ 25778956800$$ $$\beta_{7}$$ $$=$$ $$($$$$6773 \nu^{11} - 146104 \nu^{9} + 577306 \nu^{7} - 5223568 \nu^{5} + 79774373 \nu^{3} + 554840616 \nu$$$$)/ 226131200$$ $$\beta_{8}$$ $$=$$ $$($$$$-1093919 \nu^{11} + 7804552 \nu^{9} - 15466318 \nu^{7} + 1380835984 \nu^{5} + 6018524881 \nu^{3} + 28260515592 \nu$$$$)/ 25778956800$$ $$\beta_{9}$$ $$=$$ $$($$$$-1095 \nu^{11} + 7908 \nu^{9} + 18534 \nu^{7} + 745404 \nu^{5} + 8928141 \nu^{3} - 11971272 \nu$$$$)/21482464$$ $$\beta_{10}$$ $$=$$ $$($$$$72101 \nu^{11} - 1867040 \nu^{9} + 15340242 \nu^{7} - 118083000 \nu^{5} + 659420461 \nu^{3} + 7004722280 \nu$$$$)/ 1288947840$$ $$\beta_{11}$$ $$=$$ $$($$$$12569 \nu^{11} - 59505 \nu^{9} - 670717 \nu^{7} - 13483505 \nu^{5} - 70039446 \nu^{3} + 713598880 \nu$$$$)/ 161118480$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{11} + 3 \beta_{10} - \beta_{9} + 12 \beta_{8} + \beta_{7} - 6 \beta_{6}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{5} - 3 \beta_{4} + 11 \beta_{3} - 20 \beta_{2} - 64 \beta_{1} + 64$$$$)/24$$ $$\nu^{3}$$ $$=$$ $$($$$$9 \beta_{11} + 3 \beta_{10} + 53 \beta_{9} + 66 \beta_{8} + 137 \beta_{7} - 120 \beta_{6}$$$$)/24$$ $$\nu^{4}$$ $$=$$ $$($$$$12 \beta_{5} - 35 \beta_{4} + 5 \beta_{3} - 268 \beta_{2} + 1136$$$$)/24$$ $$\nu^{5}$$ $$=$$ $$($$$$-171 \beta_{11} + 177 \beta_{10} - 615 \beta_{9} + 1704 \beta_{8} + 1259 \beta_{7} - 1518 \beta_{6}$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$676 \beta_{5} - 1157 \beta_{4} + 909 \beta_{3} - 3212 \beta_{2} + 5824 \beta_{1} + 25216$$$$)/24$$ $$\nu^{7}$$ $$=$$ $$($$$$-489 \beta_{11} + 6429 \beta_{10} - 2133 \beta_{9} + 21438 \beta_{8} + 12439 \beta_{7} - 24264 \beta_{6}$$$$)/24$$ $$\nu^{8}$$ $$=$$ $$($$$$252 \beta_{5} - 4255 \beta_{4} + 5385 \beta_{3} - 14588 \beta_{2} - 20480 \beta_{1} + 129968$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$24939 \beta_{11} + 63663 \beta_{10} + 24551 \beta_{9} + 292920 \beta_{8} + 220021 \beta_{7} - 359538 \beta_{6}$$$$)/24$$ $$\nu^{10}$$ $$=$$ $$($$$$30812 \beta_{5} - 127323 \beta_{4} + 201011 \beta_{3} - 706676 \beta_{2} - 856384 \beta_{1} + 4399744$$$$)/24$$ $$\nu^{11}$$ $$=$$ $$($$$$96009 \beta_{11} + 680739 \beta_{10} - 305227 \beta_{9} + 4045218 \beta_{8} + 3762665 \beta_{7} - 4953432 \beta_{6}$$$$)/24$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$\beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −3.78139 + 0.0336790i −0.347140 − 2.27426i 2.02004 − 2.30794i −2.02004 + 2.30794i 0.347140 + 2.27426i 3.78139 − 0.0336790i −3.78139 − 0.0336790i −0.347140 + 2.27426i 2.02004 + 2.30794i −2.02004 − 2.30794i 0.347140 − 2.27426i 3.78139 + 0.0336790i
−3.74771 + 3.74771i 0 20.0907i 0 0 −1.80948 1.80948i 45.3126 + 45.3126i 0 0
107.2 −2.62140 + 2.62140i 0 5.74350i 0 0 10.8652 + 10.8652i −5.91519 5.91519i 0 0
107.3 −0.287902 + 0.287902i 0 7.83422i 0 0 −15.0557 15.0557i −4.55871 4.55871i 0 0
107.4 0.287902 0.287902i 0 7.83422i 0 0 −15.0557 15.0557i 4.55871 + 4.55871i 0 0
107.5 2.62140 2.62140i 0 5.74350i 0 0 10.8652 + 10.8652i 5.91519 + 5.91519i 0 0
107.6 3.74771 3.74771i 0 20.0907i 0 0 −1.80948 1.80948i −45.3126 45.3126i 0 0
143.1 −3.74771 3.74771i 0 20.0907i 0 0 −1.80948 + 1.80948i 45.3126 45.3126i 0 0
143.2 −2.62140 2.62140i 0 5.74350i 0 0 10.8652 10.8652i −5.91519 + 5.91519i 0 0
143.3 −0.287902 0.287902i 0 7.83422i 0 0 −15.0557 + 15.0557i −4.55871 + 4.55871i 0 0
143.4 0.287902 + 0.287902i 0 7.83422i 0 0 −15.0557 + 15.0557i 4.55871 4.55871i 0 0
143.5 2.62140 + 2.62140i 0 5.74350i 0 0 10.8652 10.8652i 5.91519 5.91519i 0 0
143.6 3.74771 + 3.74771i 0 20.0907i 0 0 −1.80948 + 1.80948i −45.3126 + 45.3126i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 143.6 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.c odd 4 1 inner
15.e even 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.f.c 12
3.b odd 2 1 inner 225.4.f.c 12
5.b even 2 1 45.4.f.a 12
5.c odd 4 1 45.4.f.a 12
5.c odd 4 1 inner 225.4.f.c 12
15.d odd 2 1 45.4.f.a 12
15.e even 4 1 45.4.f.a 12
15.e even 4 1 inner 225.4.f.c 12
20.d odd 2 1 720.4.w.d 12
20.e even 4 1 720.4.w.d 12
60.h even 2 1 720.4.w.d 12
60.l odd 4 1 720.4.w.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.f.a 12 5.b even 2 1
45.4.f.a 12 5.c odd 4 1
45.4.f.a 12 15.d odd 2 1
45.4.f.a 12 15.e even 4 1
225.4.f.c 12 1.a even 1 1 trivial
225.4.f.c 12 3.b odd 2 1 inner
225.4.f.c 12 5.c odd 4 1 inner
225.4.f.c 12 15.e even 4 1 inner
720.4.w.d 12 20.d odd 2 1
720.4.w.d 12 20.e even 4 1
720.4.w.d 12 60.h even 2 1
720.4.w.d 12 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{12} + 978 T_{2}^{8} + 149073 T_{2}^{4} + 4096$$ $$T_{7}^{6} + 12 T_{7}^{5} + 72 T_{7}^{4} - 2560 T_{7}^{3} + 97344 T_{7}^{2} + 369408 T_{7} + 700928$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4096 + 149073 T^{4} + 978 T^{8} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$( 700928 + 369408 T + 97344 T^{2} - 2560 T^{3} + 72 T^{4} + 12 T^{5} + T^{6} )^{2}$$
$11$ $$( 2390999552 + 5840832 T^{2} + 4344 T^{4} + T^{6} )^{2}$$
$13$ $$( 40033352 + 17341224 T + 3755844 T^{2} + 113600 T^{3} + 1458 T^{4} - 54 T^{5} + T^{6} )^{2}$$
$17$ $$81\!\cdots\!96$$$$+ 4537848559116288 T^{4} + 138301968 T^{8} + T^{12}$$
$19$ $$( 604661760000 + 226022400 T^{2} + 26640 T^{4} + T^{6} )^{2}$$
$23$ $$44\!\cdots\!96$$$$+ 54082669119602688 T^{4} + 946426368 T^{8} + T^{12}$$
$29$ $$( -3276052522632 + 815112972 T^{2} - 58854 T^{4} + T^{6} )^{2}$$
$31$ $$( 333184 + 19728 T + 312 T^{2} + T^{3} )^{4}$$
$37$ $$( 431026413512 - 30656156424 T + 1090188324 T^{2} - 14597920 T^{3} + 85698 T^{4} + 414 T^{5} + T^{6} )^{2}$$
$41$ $$( 1313254395137288 + 39690796812 T^{2} + 367686 T^{4} + T^{6} )^{2}$$
$43$ $$( 12363096915968 + 421035245568 T + 7169347584 T^{2} + 9036800 T^{3} + 1152 T^{4} - 48 T^{5} + T^{6} )^{2}$$
$47$ $$11\!\cdots\!96$$$$+$$$$10\!\cdots\!88$$$$T^{4} + 22535002368 T^{8} + T^{12}$$
$53$ $$95\!\cdots\!76$$$$+ 48541234921077547008 T^{4} + 14798523408 T^{8} + T^{12}$$
$59$ $$( -1998258773488128 + 52376651712 T^{2} - 421176 T^{4} + T^{6} )^{2}$$
$61$ $$( -70019072 - 341088 T - 24 T^{2} + T^{3} )^{4}$$
$67$ $$( 25774917628854272 + 153730046214144 T + 458448159744 T^{2} - 325457920 T^{3} + 332928 T^{4} + 816 T^{5} + T^{6} )^{2}$$
$71$ $$( 826210599600128 + 88146496512 T^{2} + 602976 T^{4} + T^{6} )^{2}$$
$73$ $$( 3552805641982472 - 34127756459256 T + 163913239044 T^{2} + 888350720 T^{3} + 1972098 T^{4} + 1986 T^{5} + T^{6} )^{2}$$
$79$ $$( 23809566408966144 + 446739648768 T^{2} + 1996512 T^{4} + T^{6} )^{2}$$
$83$ $$40\!\cdots\!76$$$$+$$$$30\!\cdots\!48$$$$T^{4} + 4125230408448 T^{8} + T^{12}$$
$89$ $$( -17269984128833928 + 928594032012 T^{2} - 2356326 T^{4} + T^{6} )^{2}$$
$97$ $$( 1300409140717475912 + 2088546858965544 T + 1677175223364 T^{2} - 182245120 T^{3} + 960498 T^{4} + 1386 T^{5} + T^{6} )^{2}$$