Properties

Label 225.12.a.h
Level 225
Weight 12
Character orbit 225.a
Self dual yes
Analytic conductor 172.877
Analytic rank 1
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.877215626\)
Analytic rank: \(1\)
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.

\(f(q)\) \(=\) \( q + ( -10 + \beta ) q^{2} + ( 3488 - 20 \beta ) q^{4} + ( -28950 - 176 \beta ) q^{7} + ( -123120 + 1640 \beta ) q^{8} +O(q^{10})\) \( q + ( -10 + \beta ) q^{2} + ( 3488 - 20 \beta ) q^{4} + ( -28950 - 176 \beta ) q^{7} + ( -123120 + 1640 \beta ) q^{8} + ( 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} + ( 44745920 + 221088 \beta ) q^{22} + ( 29471970 + 11152 \beta ) q^{23} + ( 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} + ( -10501610 - 6344576 \beta ) q^{37} + ( 470876120 + 1747460 \beta ) q^{38} + ( 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} + ( -6420660800 + 49099144 \beta ) q^{52} + ( 569160290 - 62251328 \beta ) q^{53} + ( 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} + ( -7867145070 + 30563824 \beta ) q^{67} + ( -2286887360 + 133930488 \beta ) q^{68} + ( -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} + ( 37315257820 + 297367658 \beta ) q^{82} + ( 6649551210 - 366939408 \beta ) q^{83} + ( 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} + ( 1540351870 - 1515290176 \beta ) q^{97} + ( 65103401470 - 1072742707 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 20q^{2} + 6976q^{4} - 57900q^{7} - 246240q^{8} + O(q^{10}) \) \( 2q - 20q^{2} + 6976q^{4} - 57900q^{7} - 246240q^{8} + 618176q^{11} - 3414260q^{13} - 1334472q^{14} + 6005632q^{16} + 1317940q^{17} + 5325320q^{19} + 89491840q^{22} + 58943940q^{23} + 80761736q^{26} - 163685760q^{28} - 94140380q^{29} + 244543464q^{31} - 627301120q^{32} + 445358072q^{34} - 21003220q^{37} + 941752240q^{38} + 745743316q^{41} - 629950100q^{43} + 242725888q^{44} - 468194856q^{46} - 1402061540q^{47} - 1941677414q^{49} - 12841321600q^{52} + 1138320580q^{53} + 3990553920q^{56} - 7387417960q^{58} - 7317515560q^{59} - 1516425676q^{61} - 28564327440q^{62} + 819531776q^{64} - 15734290140q^{67} - 4573774720q^{68} - 32938471544q^{71} + 29982848860q^{73} - 68768198072q^{74} - 1325392640q^{76} - 34734748800q^{77} - 3302823120q^{79} + 74630515640q^{82} + 13299102420q^{83} + 56706093896q^{86} + 80794874880q^{88} + 12674770860q^{89} + 90637859064q^{91} + 203171571840q^{92} - 60289765528q^{94} + 3080703740q^{97} + 130206802940q^{98} + O(q^{100}) \)

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
−83.7292 0 4962.58 0 0 −15973.7 −244036. 0 0
1.2 63.7292 0 2013.42 0 0 −41926.3 −2204.06 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 225.12.a.h 2
3.b odd 2 1 25.12.a.c 2
5.b even 2 1 45.12.a.d 2
5.c odd 4 2 225.12.b.f 4
15.d odd 2 1 5.12.a.b 2
15.e even 4 2 25.12.b.c 4
60.h even 2 1 80.12.a.j 2
105.g 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 15.d odd 2 1
25.12.a.c 2 3.b odd 2 1
25.12.b.c 4 15.e even 4 2
45.12.a.d 2 5.b even 2 1
80.12.a.j 2 60.h even 2 1
225.12.a.h 2 1.a even 1 1 trivial
225.12.b.f 4 5.c odd 4 2
245.12.a.b 2 105.g even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} + 20 T_{2} - 5336 \)
\( T_{7}^{2} + 57900 T_{7} + 669716964 \)

Hecke Characteristic Polynomials

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