# Properties

 Label 45.12.a.d Level 45 Weight 12 Character orbit 45.a Self dual yes Analytic conductor 34.575 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 45.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.5754431252$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{151})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{151}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 10 + \beta ) q^{2} + ( 3488 + 20 \beta ) q^{4} + 3125 q^{5} + ( 28950 - 176 \beta ) q^{7} + ( 123120 + 1640 \beta ) q^{8} +O(q^{10})$$ $$q + ( 10 + \beta ) q^{2} + ( 3488 + 20 \beta ) q^{4} + 3125 q^{5} + ( 28950 - 176 \beta ) q^{7} + ( 123120 + 1640 \beta ) q^{8} + ( 31250 + 3125 \beta ) q^{10} + ( 309088 - 8800 \beta ) q^{11} + ( 1707130 + 4288 \beta ) q^{13} + ( -667236 + 27190 \beta ) q^{14} + ( 3002816 + 98560 \beta ) q^{16} + ( -658970 + 42176 \beta ) q^{17} + ( 2662660 - 91520 \beta ) q^{19} + ( 10900000 + 62500 \beta ) q^{20} + ( -44745920 + 221088 \beta ) q^{22} + ( -29471970 + 11152 \beta ) q^{23} + 9765625 q^{25} + ( 40380868 + 1750010 \beta ) q^{26} + ( 81842880 - 34888 \beta ) q^{28} + ( -47070190 + 766080 \beta ) q^{29} + ( 122271732 + 2402400 \beta ) q^{31} + ( 313650560 + 629696 \beta ) q^{32} + ( 222679036 - 237210 \beta ) q^{34} + ( 90468750 - 550000 \beta ) q^{35} + ( 10501610 - 6344576 \beta ) q^{37} + ( -470876120 + 1747460 \beta ) q^{38} + ( 384750000 + 5125000 \beta ) q^{40} + ( 372871658 - 7550400 \beta ) q^{41} + ( 314975050 + 4636368 \beta ) q^{43} + ( 121362944 - 24512640 \beta ) q^{44} + ( -234097428 - 29360450 \beta ) q^{46} + ( 701030770 - 6835024 \beta ) q^{47} + ( -970838707 - 10190400 \beta ) q^{49} + ( 97656250 + 9765625 \beta ) q^{50} + ( 6420660800 + 49099144 \beta ) q^{52} + ( -569160290 - 62251328 \beta ) q^{53} + ( 965900000 - 27500000 \beta ) q^{55} + ( 1995276960 + 25808880 \beta ) q^{56} + ( 3693708980 - 39409390 \beta ) q^{58} + ( -3658757780 + 58605760 \beta ) q^{59} + ( -758212838 + 17856000 \beta ) q^{61} + ( 14282163720 + 146295732 \beta ) q^{62} + ( 409765888 + 118096640 \beta ) q^{64} + ( 5334781250 + 13400000 \beta ) q^{65} + ( 7867145070 + 30563824 \beta ) q^{67} + ( 2286887360 + 133930488 \beta ) q^{68} + ( -2085112500 + 84968750 \beta ) q^{70} + ( -16469235772 - 18268000 \beta ) q^{71} + ( -14991424430 - 113205952 \beta ) q^{73} + ( -34384099036 - 52944150 \beta ) q^{74} + ( -662696320 - 265968560 \beta ) q^{76} + ( 17367374400 - 309159488 \beta ) q^{77} + ( -1651411560 + 191878720 \beta ) q^{79} + ( 9383800000 + 308000000 \beta ) q^{80} + ( -37315257820 + 297367658 \beta ) q^{82} + ( -6649551210 - 366939408 \beta ) q^{83} + ( -2059281250 + 131800000 \beta ) q^{85} + ( 28353046948 + 361338730 \beta ) q^{86} + ( -40397437440 - 576551680 \beta ) q^{88} + ( 6337385430 - 485093760 \beta ) q^{89} + ( 45318929532 - 176317280 \beta ) q^{91} + ( -101585785920 - 550541224 \beta ) q^{92} + ( -30144882764 + 632680530 \beta ) q^{94} + ( 8320812500 - 286000000 \beta ) q^{95} + ( -1540351870 - 1515290176 \beta ) q^{97} + ( -65103401470 - 1072742707 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 20q^{2} + 6976q^{4} + 6250q^{5} + 57900q^{7} + 246240q^{8} + O(q^{10})$$ $$2q + 20q^{2} + 6976q^{4} + 6250q^{5} + 57900q^{7} + 246240q^{8} + 62500q^{10} + 618176q^{11} + 3414260q^{13} - 1334472q^{14} + 6005632q^{16} - 1317940q^{17} + 5325320q^{19} + 21800000q^{20} - 89491840q^{22} - 58943940q^{23} + 19531250q^{25} + 80761736q^{26} + 163685760q^{28} - 94140380q^{29} + 244543464q^{31} + 627301120q^{32} + 445358072q^{34} + 180937500q^{35} + 21003220q^{37} - 941752240q^{38} + 769500000q^{40} + 745743316q^{41} + 629950100q^{43} + 242725888q^{44} - 468194856q^{46} + 1402061540q^{47} - 1941677414q^{49} + 195312500q^{50} + 12841321600q^{52} - 1138320580q^{53} + 1931800000q^{55} + 3990553920q^{56} + 7387417960q^{58} - 7317515560q^{59} - 1516425676q^{61} + 28564327440q^{62} + 819531776q^{64} + 10669562500q^{65} + 15734290140q^{67} + 4573774720q^{68} - 4170225000q^{70} - 32938471544q^{71} - 29982848860q^{73} - 68768198072q^{74} - 1325392640q^{76} + 34734748800q^{77} - 3302823120q^{79} + 18767600000q^{80} - 74630515640q^{82} - 13299102420q^{83} - 4118562500q^{85} + 56706093896q^{86} - 80794874880q^{88} + 12674770860q^{89} + 90637859064q^{91} - 203171571840q^{92} - 60289765528q^{94} + 16641625000q^{95} - 3080703740q^{97} - 130206802940q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −12.2882 12.2882
−63.7292 0 2013.42 3125.00 0 41926.3 2204.06 0 −199154.
1.2 83.7292 0 4962.58 3125.00 0 15973.7 244036. 0 261654.
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.12.a.d 2
3.b odd 2 1 5.12.a.b 2
5.b even 2 1 225.12.a.h 2
5.c odd 4 2 225.12.b.f 4
12.b even 2 1 80.12.a.j 2
15.d odd 2 1 25.12.a.c 2
15.e even 4 2 25.12.b.c 4
21.c even 2 1 245.12.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.a.b 2 3.b odd 2 1
25.12.a.c 2 15.d odd 2 1
25.12.b.c 4 15.e even 4 2
45.12.a.d 2 1.a even 1 1 trivial
80.12.a.j 2 12.b even 2 1
225.12.a.h 2 5.b even 2 1
225.12.b.f 4 5.c odd 4 2
245.12.a.b 2 21.c even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 20 T_{2} - 5336$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(45))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 20 T - 1240 T^{2} - 40960 T^{3} + 4194304 T^{4}$$
$3$ 1
$5$ $$( 1 - 3125 T )^{2}$$
$7$ $$1 - 57900 T + 4624370450 T^{2} - 114487218419700 T^{3} + 3909821048582988049 T^{4}$$
$11$ $$1 - 618176 T + 245194892966 T^{2} - 176372827291625536 T^{3} +$$$$81\!\cdots\!21$$$$T^{4}$$
$13$ $$1 - 3414260 T + 6398662197390 T^{2} - 6118901546944767620 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$17$ $$1 + 1317940 T + 59308395866630 T^{2} + 45168303019681836020 T^{3} +$$$$11\!\cdots\!89$$$$T^{4}$$
$19$ $$1 - 5325320 T + 194538827137638 T^{2} -$$$$62\!\cdots\!80$$$$T^{3} +$$$$13\!\cdots\!61$$$$T^{4}$$
$23$ $$1 + 58943940 T + 2773540471931410 T^{2} +$$$$56\!\cdots\!80$$$$T^{3} +$$$$90\!\cdots\!29$$$$T^{4}$$
$29$ $$1 + 94140380 T + 23426350431097358 T^{2} +$$$$11\!\cdots\!20$$$$T^{3} +$$$$14\!\cdots\!41$$$$T^{4}$$
$31$ $$1 - 244543464 T + 34393316207729486 T^{2} -$$$$62\!\cdots\!84$$$$T^{3} +$$$$64\!\cdots\!61$$$$T^{4}$$
$37$ $$1 - 21003220 T + 137126715218410590 T^{2} -$$$$37\!\cdots\!60$$$$T^{3} +$$$$31\!\cdots\!69$$$$T^{4}$$
$41$ $$1 - 745743316 T + 929792912462405846 T^{2} -$$$$41\!\cdots\!56$$$$T^{3} +$$$$30\!\cdots\!81$$$$T^{4}$$
$43$ $$1 - 629950100 T + 1840945003918927050 T^{2} -$$$$58\!\cdots\!00$$$$T^{3} +$$$$86\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 1402061540 T + 5181805952108806370 T^{2} -$$$$34\!\cdots\!20$$$$T^{3} +$$$$61\!\cdots\!09$$$$T^{4}$$
$53$ $$1 + 1138320580 T - 2203723231625575330 T^{2} +$$$$10\!\cdots\!60$$$$T^{3} +$$$$85\!\cdots\!09$$$$T^{4}$$
$59$ $$1 + 7317515560 T + 55027608950440780118 T^{2} +$$$$22\!\cdots\!40$$$$T^{3} +$$$$90\!\cdots\!81$$$$T^{4}$$
$61$ $$1 + 1516425676 T + 85869525433683691566 T^{2} +$$$$65\!\cdots\!36$$$$T^{3} +$$$$18\!\cdots\!21$$$$T^{4}$$
$67$ $$1 - 15734290140 T +$$$$30\!\cdots\!30$$$$T^{2} -$$$$19\!\cdots\!20$$$$T^{3} +$$$$14\!\cdots\!89$$$$T^{4}$$
$71$ $$1 + 32938471544 T +$$$$73\!\cdots\!26$$$$T^{2} +$$$$76\!\cdots\!24$$$$T^{3} +$$$$53\!\cdots\!41$$$$T^{4}$$
$73$ $$1 + 29982848860 T +$$$$78\!\cdots\!10$$$$T^{2} +$$$$94\!\cdots\!20$$$$T^{3} +$$$$98\!\cdots\!29$$$$T^{4}$$
$79$ $$1 + 3302823120 T +$$$$12\!\cdots\!58$$$$T^{2} +$$$$24\!\cdots\!80$$$$T^{3} +$$$$55\!\cdots\!41$$$$T^{4}$$
$83$ $$1 + 13299102420 T +$$$$18\!\cdots\!30$$$$T^{2} +$$$$17\!\cdots\!40$$$$T^{3} +$$$$16\!\cdots\!89$$$$T^{4}$$
$89$ $$1 - 12674770860 T +$$$$43\!\cdots\!78$$$$T^{2} -$$$$35\!\cdots\!40$$$$T^{3} +$$$$77\!\cdots\!21$$$$T^{4}$$
$97$ $$1 + 3080703740 T +$$$$18\!\cdots\!70$$$$T^{2} +$$$$22\!\cdots\!20$$$$T^{3} +$$$$51\!\cdots\!09$$$$T^{4}$$