# Properties

 Label 45.10.b.b Level 45 Weight 10 Character orbit 45.b Analytic conductor 23.177 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 45.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.1766126274$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.49740556.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -342 - \beta_{3} ) q^{4} + ( -285 + 19 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{5} + ( -133 \beta_{1} + 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -342 - \beta_{3} ) q^{4} + ( -285 + 19 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{5} + ( -133 \beta_{1} + 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} + ( -17290 - 1139 \beta_{1} + 8 \beta_{2} - 19 \beta_{3} ) q^{10} + ( -27492 + 20 \beta_{3} ) q^{11} + ( 1918 \beta_{1} + 110 \beta_{2} ) q^{13} + ( 106134 + 133 \beta_{3} ) q^{14} + ( 407816 + 172 \beta_{3} ) q^{16} + ( -7448 \beta_{1} - 1632 \beta_{2} ) q^{17} + ( -159220 + 476 \beta_{3} ) q^{19} + ( 825570 - 23788 \beta_{1} + 3736 \beta_{2} + 627 \beta_{3} ) q^{20} + ( -10412 \beta_{1} - 160 \beta_{2} ) q^{22} + ( 17005 \beta_{1} + 5199 \beta_{2} ) q^{23} + ( -334475 - 45410 \beta_{1} + 8270 \beta_{2} + 1140 \beta_{3} ) q^{25} + ( -1654692 - 1918 \beta_{3} ) q^{26} + ( 151620 \beta_{1} + 24024 \beta_{2} ) q^{28} + ( 882930 + 1976 \beta_{3} ) q^{29} + ( -2646928 - 760 \beta_{3} ) q^{31} + ( 204496 \beta_{1} + 2720 \beta_{2} ) q^{32} + ( 6608656 + 7448 \beta_{3} ) q^{34} + ( -3407460 + 144039 \beta_{1} + 26817 \beta_{2} + 9044 \beta_{3} ) q^{35} + ( 335370 \beta_{1} + 51378 \beta_{2} ) q^{37} + ( 247284 \beta_{1} - 3808 \beta_{2} ) q^{38} + ( 10894600 + 777860 \beta_{1} - 920 \beta_{2} + 14060 \beta_{3} ) q^{40} + ( 4197138 - 24890 \beta_{3} ) q^{41} + ( -719131 \beta_{1} + 132643 \beta_{2} ) q^{43} + ( -5159736 + 20652 \beta_{3} ) q^{44} + ( -15312518 - 17005 \beta_{3} ) q^{46} + ( -1012111 \beta_{1} + 214259 \beta_{2} ) q^{47} + ( -11730257 + 27930 \beta_{3} ) q^{49} + ( 37523100 + 639085 \beta_{1} - 9120 \beta_{2} + 45410 \beta_{3} ) q^{50} + ( -2310648 \beta_{1} + 71664 \beta_{2} ) q^{52} + ( -1349666 \beta_{1} - 259794 \beta_{2} ) q^{53} + ( -6726780 - 176548 \beta_{1} - 315044 \beta_{2} + 21792 \beta_{3} ) q^{55} + ( -78794520 - 83524 \beta_{3} ) q^{56} + ( 2570434 \beta_{1} - 15808 \beta_{2} ) q^{58} + ( 115207260 + 52972 \beta_{3} ) q^{59} + ( 90122642 - 43150 \beta_{3} ) q^{61} + ( -3295968 \beta_{1} + 6080 \beta_{2} ) q^{62} + ( 33748768 - 116432 \beta_{3} ) q^{64} + ( -45973920 - 2201322 \beta_{1} + 77934 \beta_{2} - 21812 \beta_{3} ) q^{65} + ( -6669647 \beta_{1} - 1448953 \beta_{2} ) q^{67} + ( 9155872 \beta_{1} - 895168 \beta_{2} ) q^{68} + ( -127085490 + 4316116 \beta_{1} - 72352 \beta_{2} - 144039 \beta_{3} ) q^{70} + ( 11902968 + 91200 \beta_{3} ) q^{71} + ( 1847940 \beta_{1} - 90564 \beta_{2} ) q^{73} + ( -294215436 - 335370 \beta_{3} ) q^{74} + ( -292122360 - 3572 \beta_{3} ) q^{76} + ( 1533756 \beta_{1} - 2162748 \beta_{2} ) q^{77} + ( -182010880 - 267976 \beta_{3} ) q^{79} + ( -241460760 + 10722384 \beta_{1} + 1800352 \beta_{2} - 456836 \beta_{3} ) q^{80} + ( -17058922 \beta_{1} + 199120 \beta_{2} ) q^{82} + ( -12737657 \beta_{1} + 1180353 \beta_{2} ) q^{83} + ( 318854960 + 8731336 \beta_{1} - 988192 \beta_{2} - 75544 \beta_{3} ) q^{85} + ( 593976138 + 719131 \beta_{3} ) q^{86} + ( 7146128 \beta_{1} - 247136 \beta_{2} ) q^{88} + ( 395675190 - 185592 \beta_{3} ) q^{89} + ( 118330632 + 357504 \beta_{3} ) q^{91} + ( -21128228 \beta_{1} + 2797928 \beta_{2} ) q^{92} + ( 831775426 + 1012111 \beta_{3} ) q^{94} + ( -301197900 + 5204860 \beta_{1} - 4032420 \beta_{2} + 23560 \beta_{3} ) q^{95} + ( -14786464 \beta_{1} + 1954216 \beta_{2} ) q^{97} + ( 12121963 \beta_{1} - 223440 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 1368q^{4} - 1140q^{5} + O(q^{10})$$ $$4q - 1368q^{4} - 1140q^{5} - 69160q^{10} - 109968q^{11} + 424536q^{14} + 1631264q^{16} - 636880q^{19} + 3302280q^{20} - 1337900q^{25} - 6618768q^{26} + 3531720q^{29} - 10587712q^{31} + 26434624q^{34} - 13629840q^{35} + 43578400q^{40} + 16788552q^{41} - 20638944q^{44} - 61250072q^{46} - 46921028q^{49} + 150092400q^{50} - 26907120q^{55} - 315178080q^{56} + 460829040q^{59} + 360490568q^{61} + 134995072q^{64} - 183895680q^{65} - 508341960q^{70} + 47611872q^{71} - 1176861744q^{74} - 1168489440q^{76} - 728043520q^{79} - 965843040q^{80} + 1275419840q^{85} + 2375904552q^{86} + 1582700760q^{89} + 473322528q^{91} + 3327101704q^{94} - 1204791600q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 45 x^{2} + 304$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 37 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} - 7 \nu$$ $$\beta_{3}$$ $$=$$ $$60 \nu^{2} + 1350$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/30$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 1350$$$$)/60$$ $$\nu^{3}$$ $$=$$ $$($$$$-37 \beta_{2} - 14 \beta_{1}$$$$)/30$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 2.87724i − 6.05982i 6.05982i 2.87724i
41.3193i 0 −1195.29 −1138.29 810.818i 0 5315.22i 28233.0i 0 −33502.5 + 47033.3i
19.2 0.843944i 0 511.288 568.288 1276.78i 0 8712.99i 863.597i 0 −1077.53 479.603i
19.3 0.843944i 0 511.288 568.288 + 1276.78i 0 8712.99i 863.597i 0 −1077.53 + 479.603i
19.4 41.3193i 0 −1195.29 −1138.29 + 810.818i 0 5315.22i 28233.0i 0 −33502.5 47033.3i
 $$n$$: 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 45.10.b.b 4
3.b odd 2 1 5.10.b.a 4
5.b even 2 1 inner 45.10.b.b 4
5.c odd 4 2 225.10.a.s 4
12.b even 2 1 80.10.c.c 4
15.d odd 2 1 5.10.b.a 4
15.e even 4 2 25.10.a.e 4
60.h even 2 1 80.10.c.c 4
60.l odd 4 2 400.10.a.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.b.a 4 3.b odd 2 1
5.10.b.a 4 15.d odd 2 1
25.10.a.e 4 15.e even 4 2
45.10.b.b 4 1.a even 1 1 trivial
45.10.b.b 4 5.b even 2 1 inner
80.10.c.c 4 12.b even 2 1
80.10.c.c 4 60.h even 2 1
225.10.a.s 4 5.c odd 4 2
400.10.a.ba 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 1708 T_{2}^{2} + 1216$$ acting on $$S_{10}^{\mathrm{new}}(45, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 340 T^{2} - 174912 T^{4} - 89128960 T^{6} + 68719476736 T^{8}$$
$3$ 1
$5$ $$1 + 1140 T + 1318750 T^{2} + 2226562500 T^{3} + 3814697265625 T^{4}$$
$7$ $$1 - 57246700 T^{2} + 3508143545353398 T^{4} -$$$$93\!\cdots\!00$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 54984 T + 5180465446 T^{2} + 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2}$$
$13$ $$1 - 35613791860 T^{2} +$$$$53\!\cdots\!58$$$$T^{4} -$$$$40\!\cdots\!40$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$17$ $$1 - 285780369220 T^{2} +$$$$48\!\cdots\!18$$$$T^{4} -$$$$40\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!81$$$$T^{8}$$
$19$ $$( 1 + 318440 T + 505756418358 T^{2} + 102756670480744760 T^{3} +$$$$10\!\cdots\!41$$$$T^{4} )^{2}$$
$23$ $$1 - 5779790962540 T^{2} +$$$$14\!\cdots\!38$$$$T^{4} -$$$$18\!\cdots\!60$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 1765860 T + 26950935551038 T^{2} - 25617588792948032340 T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$( 1 + 5293856 T + 59464921598526 T^{2} +$$$$13\!\cdots\!76$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$
$37$ $$1 - 231603274936660 T^{2} +$$$$44\!\cdots\!58$$$$T^{4} -$$$$39\!\cdots\!40$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 - 8394276 T + 221313076168966 T^{2} -$$$$27\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 - 614109141147100 T^{2} +$$$$57\!\cdots\!98$$$$T^{4} -$$$$15\!\cdots\!00$$$$T^{6} +$$$$63\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 1368976020813580 T^{2} +$$$$29\!\cdots\!78$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 7684297973864980 T^{2} +$$$$36\!\cdots\!78$$$$T^{4} -$$$$83\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 - 230414520 T + 28555631923987078 T^{2} -$$$$19\!\cdots\!80$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$61$ $$( 1 - 180245284 T + 30154717014478446 T^{2} -$$$$21\!\cdots\!44$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 + 41160407446058180 T^{2} +$$$$18\!\cdots\!18$$$$T^{4} +$$$$30\!\cdots\!20$$$$T^{6} +$$$$54\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 23805936 T + 85782754020107086 T^{2} -$$$$10\!\cdots\!16$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$73$ $$1 - 229489314868712740 T^{2} +$$$$20\!\cdots\!38$$$$T^{4} -$$$$79\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 + 364021760 T + 220545463862625438 T^{2} +$$$$43\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} )^{2}$$
$83$ $$1 - 434569632367965820 T^{2} +$$$$10\!\cdots\!18$$$$T^{4} -$$$$15\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 - 791350380 T + 832192702699668118 T^{2} -$$$$27\!\cdots\!20$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$1 - 2561123777205326980 T^{2} +$$$$27\!\cdots\!78$$$$T^{4} -$$$$14\!\cdots\!20$$$$T^{6} +$$$$33\!\cdots\!21$$$$T^{8}$$