Properties

 Label 225.10.a.j Level 225 Weight 10 Character orbit 225.a Self dual yes Analytic conductor 115.883 Analytic rank 0 Dimension 2 CM no Inner twists 1

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$115.883063137$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{4729})$$ Defining polynomial: $$x^{2} - x - 1182$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{4729})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 10 - \beta ) q^{2} + ( 770 - 19 \beta ) q^{4} + ( 5964 - 56 \beta ) q^{7} + ( 25038 - 429 \beta ) q^{8} +O(q^{10})$$ $$q + ( 10 - \beta ) q^{2} + ( 770 - 19 \beta ) q^{4} + ( 5964 - 56 \beta ) q^{7} + ( 25038 - 429 \beta ) q^{8} + ( -16768 - 1952 \beta ) q^{11} + ( -71146 - 1384 \beta ) q^{13} + ( 125832 - 6468 \beta ) q^{14} + ( 363218 - 19171 \beta ) q^{16} + ( 193678 - 2200 \beta ) q^{17} + ( -201164 - 968 \beta ) q^{19} + ( 2139584 - 800 \beta ) q^{22} + ( 79368 + 64968 \beta ) q^{23} + ( 924428 + 58690 \beta ) q^{26} + ( 5849928 - 155372 \beta ) q^{28} + ( 31078 + 12416 \beta ) q^{29} + ( -2475696 - 75736 \beta ) q^{31} + ( 13472846 - 316109 \beta ) q^{32} + ( 4537180 - 213478 \beta ) q^{34} + ( -2599466 - 174696 \beta ) q^{37} + ( -867464 + 192452 \beta ) q^{38} + ( -7340714 + 470096 \beta ) q^{41} + ( -13950652 + 152384 \beta ) q^{43} + ( 30926656 - 1147360 \beta ) q^{44} + ( -75998496 + 505344 \beta ) q^{46} + ( 48198904 - 431368 \beta ) q^{47} + ( -1077559 - 664832 \beta ) q^{49} + ( -23700548 + 312390 \beta ) q^{52} + ( -31687862 - 929872 \beta ) q^{53} + ( 177723000 - 3936660 \beta ) q^{56} + ( -14364932 + 80666 \beta ) q^{58} + ( -93124864 - 1613408 \beta ) q^{59} + ( 75911686 + 2256688 \beta ) q^{61} + ( 64762992 + 1794072 \beta ) q^{62} + ( 322401682 - 6502275 \beta ) q^{64} + ( -13074108 - 7444160 \beta ) q^{67} + ( 198539660 - 5332082 \beta ) q^{68} + ( 110604928 + 7061120 \beta ) q^{71} + ( 19801262 - 6480208 \beta ) q^{73} + ( 180496012 + 1027202 \beta ) q^{74} + ( -133156936 + 3095148 \beta ) q^{76} + ( 29202432 - 10593408 \beta ) q^{77} + ( -467102400 + 1798040 \beta ) q^{79} + ( -629060612 + 11571578 \beta ) q^{82} + ( 105100620 - 3161088 \beta ) q^{83} + ( -319624408 + 15322108 \beta ) q^{86} + ( 569979072 - 40843296 \beta ) q^{88} + ( -107605986 - 9306192 \beta ) q^{89} + ( -332705016 - 4192496 \beta ) q^{91} + ( -1397937984 + 47282976 \beta ) q^{92} + ( 991866016 - 52081216 \beta ) q^{94} + ( -215586818 + 44039040 \beta ) q^{97} + ( 775055834 - 4905929 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 19q^{2} + 1521q^{4} + 11872q^{7} + 49647q^{8} + O(q^{10})$$ $$2q + 19q^{2} + 1521q^{4} + 11872q^{7} + 49647q^{8} - 35488q^{11} - 143676q^{13} + 245196q^{14} + 707265q^{16} + 385156q^{17} - 403296q^{19} + 4278368q^{22} + 223704q^{23} + 1907546q^{26} + 11544484q^{28} + 74572q^{29} - 5027128q^{31} + 26629583q^{32} + 8860882q^{34} - 5373628q^{37} - 1542476q^{38} - 14211332q^{41} - 27748920q^{43} + 60705952q^{44} - 151491648q^{46} + 95966440q^{47} - 2819950q^{49} - 47088706q^{52} - 64305596q^{53} + 351509340q^{56} - 28649198q^{58} - 187863136q^{59} + 154080060q^{61} + 131320056q^{62} + 638301089q^{64} - 33592376q^{67} + 391747238q^{68} + 228270976q^{71} + 33122316q^{73} + 362019226q^{74} - 263218724q^{76} + 47811456q^{77} - 932406760q^{79} - 1246549646q^{82} + 207040152q^{83} - 623926708q^{86} + 1099114848q^{88} - 224518164q^{89} - 669602528q^{91} - 2748592992q^{92} + 1931650816q^{94} - 387134596q^{97} + 1545205739q^{98} + O(q^{100})$$

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
 34.8839 −33.8839
−24.8839 0 107.207 0 0 4010.50 10072.8 0 0
1.2 43.8839 0 1413.79 0 0 7861.50 39574.2 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

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

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 225.10.a.j 2
3.b odd 2 1 75.10.a.g 2
5.b even 2 1 45.10.a.e 2
5.c odd 4 2 225.10.b.g 4
15.d odd 2 1 15.10.a.c 2
15.e even 4 2 75.10.b.e 4
60.h even 2 1 240.10.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.c 2 15.d odd 2 1
45.10.a.e 2 5.b even 2 1
75.10.a.g 2 3.b odd 2 1
75.10.b.e 4 15.e even 4 2
225.10.a.j 2 1.a even 1 1 trivial
225.10.b.g 4 5.c odd 4 2
240.10.a.m 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{2} - 19 T_{2} - 1092$$ $$T_{7}^{2} - 11872 T_{7} + 31528560$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 19 T - 68 T^{2} - 9728 T^{3} + 262144 T^{4}$$
$3$ 1
$5$ 1
$7$ $$1 - 11872 T + 112235774 T^{2} - 479078022304 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 + 35488 T + 526013014 T^{2} + 83678847658208 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 + 143676 T + 24105149134 T^{2} + 1523612051915148 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 - 385156 T + 268539949078 T^{2} - 45674832160078532 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 + 403296 T + 684929514838 T^{2} + 130138657763479584 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 - 223704 T - 1375273107794 T^{2} - 402925054979918952 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 - 74572 T + 28833430018078 T^{2} - 1081826889712503068 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 + 5027128 T + 52415931233342 T^{2} +$$$$13\!\cdots\!88$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 + 5373628 T + 231061724951934 T^{2} +$$$$69\!\cdots\!56$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 + 14211332 T + 443988635955862 T^{2} +$$$$46\!\cdots\!52$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 + 27748920 T + 1170232974699430 T^{2} +$$$$13\!\cdots\!60$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 95966440 T + 4320659216802910 T^{2} -$$$$10\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 64305596 T + 6611083028543086 T^{2} +$$$$21\!\cdots\!68$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 + 187863136 T + 23071633420288438 T^{2} +$$$$16\!\cdots\!04$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 154080060 T + 23302683905802238 T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 + 33592376 T - 10819815556424362 T^{2} +$$$$91\!\cdots\!72$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 - 228270976 T + 45777616900481806 T^{2} -$$$$10\!\cdots\!56$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 - 33122316 T + 68371107952007926 T^{2} -$$$$19\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 + 932406760 T + 453226630902929438 T^{2} +$$$$11\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 - 207040152 T + 372783310330485238 T^{2} -$$$$38\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 + 224518164 T + 610925899926766678 T^{2} +$$$$78\!\cdots\!76$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 + 387134596 T - 734969029248610362 T^{2} +$$$$29\!\cdots\!32$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$