Properties

 Label 225.12.b.f Level 225 Weight 12 Character orbit 225.b Analytic conductor 172.877 Analytic rank 0 Dimension 4 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 225.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$172.877215626$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{151})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{3} ) q^{2} + ( -3488 + 2 \beta_{2} ) q^{4} + ( 2895 \beta_{1} + 176 \beta_{3} ) q^{7} + ( -12312 \beta_{1} + 1640 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{3} ) q^{2} + ( -3488 + 2 \beta_{2} ) q^{4} + ( 2895 \beta_{1} + 176 \beta_{3} ) q^{7} + ( -12312 \beta_{1} + 1640 \beta_{3} ) q^{8} + ( 309088 + 880 \beta_{2} ) q^{11} + ( -170713 \beta_{1} + 4288 \beta_{3} ) q^{13} + ( 667236 + 2719 \beta_{2} ) q^{14} + ( 3002816 - 9856 \beta_{2} ) q^{16} + ( -65897 \beta_{1} - 42176 \beta_{3} ) q^{17} + ( -2662660 - 9152 \beta_{2} ) q^{19} + ( -4474592 \beta_{1} - 221088 \beta_{3} ) q^{22} + ( 2947197 \beta_{1} + 11152 \beta_{3} ) q^{23} + ( 40380868 - 175001 \beta_{2} ) q^{26} + ( -8184288 \beta_{1} - 34888 \beta_{3} ) q^{28} + ( 47070190 + 76608 \beta_{2} ) q^{29} + ( 122271732 - 240240 \beta_{2} ) q^{31} + ( 31365056 \beta_{1} - 629696 \beta_{3} ) q^{32} + ( -222679036 - 23721 \beta_{2} ) q^{34} + ( 1050161 \beta_{1} + 6344576 \beta_{3} ) q^{37} + ( 47087612 \beta_{1} + 1747460 \beta_{3} ) q^{38} + ( 372871658 + 755040 \beta_{2} ) q^{41} + ( -31497505 \beta_{1} + 4636368 \beta_{3} ) q^{43} + ( -121362944 - 2451264 \beta_{2} ) q^{44} + ( -234097428 + 2936045 \beta_{2} ) q^{46} + ( 70103077 \beta_{1} + 6835024 \beta_{3} ) q^{47} + ( 970838707 - 1019040 \beta_{2} ) q^{49} + ( 642066080 \beta_{1} - 49099144 \beta_{3} ) q^{52} + ( 56916029 \beta_{1} - 62251328 \beta_{3} ) q^{53} + ( 1995276960 - 2580888 \beta_{2} ) q^{56} + ( -369370898 \beta_{1} - 39409390 \beta_{3} ) q^{58} + ( 3658757780 + 5860576 \beta_{2} ) q^{59} + ( -758212838 - 1785600 \beta_{2} ) q^{61} + ( 1428216372 \beta_{1} - 146295732 \beta_{3} ) q^{62} + ( -409765888 + 11809664 \beta_{2} ) q^{64} + ( 786714507 \beta_{1} - 30563824 \beta_{3} ) q^{67} + ( -228688736 \beta_{1} + 133930488 \beta_{3} ) q^{68} + ( -16469235772 + 1826800 \beta_{2} ) q^{71} + ( 1499142443 \beta_{1} - 113205952 \beta_{3} ) q^{73} + ( 34384099036 - 5294415 \beta_{2} ) q^{74} + ( -662696320 + 26596856 \beta_{2} ) q^{76} + ( 1736737440 \beta_{1} + 309159488 \beta_{3} ) q^{77} + ( 1651411560 + 19187872 \beta_{2} ) q^{79} + ( -3731525782 \beta_{1} - 297367658 \beta_{3} ) q^{82} + ( 664955121 \beta_{1} - 366939408 \beta_{3} ) q^{83} + ( 28353046948 - 36133873 \beta_{2} ) q^{86} + ( 4039743744 \beta_{1} - 576551680 \beta_{3} ) q^{88} + ( -6337385430 - 48509376 \beta_{2} ) q^{89} + ( 45318929532 + 17631728 \beta_{2} ) q^{91} + ( -10158578592 \beta_{1} + 550541224 \beta_{3} ) q^{92} + ( 30144882764 + 63268053 \beta_{2} ) q^{94} + ( -154035187 \beta_{1} + 1515290176 \beta_{3} ) q^{97} + ( 6510340147 \beta_{1} - 1072742707 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 13952q^{4} + O(q^{10})$$ $$4q - 13952q^{4} + 1236352q^{11} + 2668944q^{14} + 12011264q^{16} - 10650640q^{19} + 161523472q^{26} + 188280760q^{29} + 489086928q^{31} - 890716144q^{34} + 1491486632q^{41} - 485451776q^{44} - 936389712q^{46} + 3883354828q^{49} + 7981107840q^{56} + 14635031120q^{59} - 3032851352q^{61} - 1639063552q^{64} - 65876943088q^{71} + 137536396144q^{74} - 2650785280q^{76} + 6605646240q^{79} + 113412187792q^{86} - 25349541720q^{89} + 181275718128q^{91} + 120579531056q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 75 x^{2} + 1444$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$5 \nu^{3} - 185 \nu$$$$)/19$$ $$\beta_{2}$$ $$=$$ $$($$$$-30 \nu^{3} + 3390 \nu$$$$)/19$$ $$\beta_{3}$$ $$=$$ $$12 \nu^{2} - 450$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 6 \beta_{1}$$$$)/120$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 450$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$37 \beta_{2} + 678 \beta_{1}$$$$)/120$$

Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −6.14410 − 0.500000i 6.14410 + 0.500000i 6.14410 − 0.500000i −6.14410 + 0.500000i
83.7292i 0 −4962.58 0 0 15973.7i 244036.i 0 0
199.2 63.7292i 0 −2013.42 0 0 41926.3i 2204.06i 0 0
199.3 63.7292i 0 −2013.42 0 0 41926.3i 2204.06i 0 0
199.4 83.7292i 0 −4962.58 0 0 15973.7i 244036.i 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 225.12.b.f 4
3.b odd 2 1 25.12.b.c 4
5.b even 2 1 inner 225.12.b.f 4
5.c odd 4 1 45.12.a.d 2
5.c odd 4 1 225.12.a.h 2
15.d odd 2 1 25.12.b.c 4
15.e even 4 1 5.12.a.b 2
15.e even 4 1 25.12.a.c 2
60.l odd 4 1 80.12.a.j 2
105.k odd 4 1 245.12.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.a.b 2 15.e even 4 1
25.12.a.c 2 15.e even 4 1
25.12.b.c 4 3.b odd 2 1
25.12.b.c 4 15.d odd 2 1
45.12.a.d 2 5.c odd 4 1
80.12.a.j 2 60.l odd 4 1
225.12.a.h 2 5.c odd 4 1
225.12.b.f 4 1.a even 1 1 trivial
225.12.b.f 4 5.b even 2 1 inner
245.12.a.b 2 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{12}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{4} + 11072 T_{2}^{2} + 28472896$$ $$T_{11}^{2} - 618176 T_{11} - 325428448256$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2880 T^{2} + 8287808 T^{4} + 12079595520 T^{6} + 17592186044416 T^{8}$$
$3$ 1
$5$ 1
$7$ $$1 - 5896330900 T^{2} + 15946824262997918598 T^{4} -$$$$23\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 - 618176 T + 245194892966 T^{2} - 176372827291625536 T^{3} +$$$$81\!\cdots\!21$$$$T^{4} )^{2}$$
$13$ $$1 - 1140153047180 T^{2} +$$$$55\!\cdots\!38$$$$T^{4} -$$$$36\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$17$ $$1 - 116879825889660 T^{2} +$$$$57\!\cdots\!78$$$$T^{4} -$$$$13\!\cdots\!40$$$$T^{6} +$$$$13\!\cdots\!21$$$$T^{8}$$
$19$ $$( 1 + 5325320 T + 194538827137638 T^{2} +$$$$62\!\cdots\!80$$$$T^{3} +$$$$13\!\cdots\!61$$$$T^{4} )^{2}$$
$23$ $$1 - 2072692881139220 T^{2} +$$$$28\!\cdots\!58$$$$T^{4} -$$$$18\!\cdots\!80$$$$T^{6} +$$$$82\!\cdots\!41$$$$T^{8}$$
$29$ $$( 1 - 94140380 T + 23426350431097358 T^{2} -$$$$11\!\cdots\!20$$$$T^{3} +$$$$14\!\cdots\!41$$$$T^{4} )^{2}$$
$31$ $$( 1 - 244543464 T + 34393316207729486 T^{2} -$$$$62\!\cdots\!84$$$$T^{3} +$$$$64\!\cdots\!61$$$$T^{4} )^{2}$$
$37$ $$1 - 273812295186452780 T^{2} +$$$$81\!\cdots\!38$$$$T^{4} -$$$$86\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$41$ $$( 1 - 745743316 T + 929792912462405846 T^{2} -$$$$41\!\cdots\!56$$$$T^{3} +$$$$30\!\cdots\!81$$$$T^{4} )^{2}$$
$43$ $$1 - 3285052879347844100 T^{2} +$$$$43\!\cdots\!98$$$$T^{4} -$$$$28\!\cdots\!00$$$$T^{6} +$$$$74\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 8397835342270441140 T^{2} +$$$$29\!\cdots\!18$$$$T^{4} -$$$$51\!\cdots\!60$$$$T^{6} +$$$$37\!\cdots\!81$$$$T^{8}$$
$53$ $$1 + 5703220206102687060 T^{2} +$$$$15\!\cdots\!18$$$$T^{4} +$$$$48\!\cdots\!40$$$$T^{6} +$$$$73\!\cdots\!81$$$$T^{8}$$
$59$ $$( 1 - 7317515560 T + 55027608950440780118 T^{2} -$$$$22\!\cdots\!40$$$$T^{3} +$$$$90\!\cdots\!81$$$$T^{4} )^{2}$$
$61$ $$( 1 + 1516425676 T + 85869525433683691566 T^{2} +$$$$65\!\cdots\!36$$$$T^{3} +$$$$18\!\cdots\!21$$$$T^{4} )^{2}$$
$67$ $$1 -$$$$35\!\cdots\!60$$$$T^{2} +$$$$60\!\cdots\!78$$$$T^{4} -$$$$52\!\cdots\!40$$$$T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$
$71$ $$( 1 + 32938471544 T +$$$$73\!\cdots\!26$$$$T^{2} +$$$$76\!\cdots\!24$$$$T^{3} +$$$$53\!\cdots\!41$$$$T^{4} )^{2}$$
$73$ $$1 -$$$$66\!\cdots\!20$$$$T^{2} +$$$$24\!\cdots\!58$$$$T^{4} -$$$$65\!\cdots\!80$$$$T^{6} +$$$$96\!\cdots\!41$$$$T^{8}$$
$79$ $$( 1 - 3302823120 T +$$$$12\!\cdots\!58$$$$T^{2} -$$$$24\!\cdots\!80$$$$T^{3} +$$$$55\!\cdots\!41$$$$T^{4} )^{2}$$
$83$ $$1 -$$$$35\!\cdots\!60$$$$T^{2} +$$$$64\!\cdots\!78$$$$T^{4} -$$$$59\!\cdots\!40$$$$T^{6} +$$$$27\!\cdots\!21$$$$T^{8}$$
$89$ $$( 1 + 12674770860 T +$$$$43\!\cdots\!78$$$$T^{2} +$$$$35\!\cdots\!40$$$$T^{3} +$$$$77\!\cdots\!21$$$$T^{4} )^{2}$$
$97$ $$1 -$$$$36\!\cdots\!40$$$$T^{2} +$$$$10\!\cdots\!18$$$$T^{4} -$$$$18\!\cdots\!60$$$$T^{6} +$$$$26\!\cdots\!81$$$$T^{8}$$