Properties

Label 225.10.a.m
Level $225$
Weight $10$
Character orbit 225.a
Self dual yes
Analytic conductor $115.883$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 652 x + 4000\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -11 - \beta_{1} ) q^{2} + ( 114 + 9 \beta_{1} - \beta_{2} ) q^{4} + ( 1744 + 132 \beta_{1} + 26 \beta_{2} ) q^{7} + ( 24 + 333 \beta_{1} + 33 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -11 - \beta_{1} ) q^{2} + ( 114 + 9 \beta_{1} - \beta_{2} ) q^{4} + ( 1744 + 132 \beta_{1} + 26 \beta_{2} ) q^{7} + ( 24 + 333 \beta_{1} + 33 \beta_{2} ) q^{8} + ( 18298 - 370 \beta_{1} - 195 \beta_{2} ) q^{11} + ( 71808 - 2592 \beta_{1} + 460 \beta_{2} ) q^{13} + ( -90810 + 678 \beta_{1} - 492 \beta_{2} ) q^{14} + ( -233100 - 1227 \beta_{1} + 53 \beta_{2} ) q^{16} + ( -111605 + 7016 \beta_{1} - 168 \beta_{2} ) q^{17} + ( 273798 - 20934 \beta_{1} - 2549 \beta_{2} ) q^{19} + ( 22817 - 35223 \beta_{1} + 4310 \beta_{2} ) q^{22} + ( -1174476 + 17556 \beta_{1} - 3426 \beta_{2} ) q^{23} + ( 431212 - 38812 \beta_{1} - 13632 \beta_{2} ) q^{26} + ( -142436 - 16254 \beta_{1} - 826 \beta_{2} ) q^{28} + ( -727252 - 119984 \beta_{1} + 6276 \beta_{2} ) q^{29} + ( 1425052 - 56940 \beta_{1} - 1090 \beta_{2} ) q^{31} + ( 3161324 + 64549 \beta_{1} - 19395 \beta_{2} ) q^{32} + ( -2283337 + 111693 \beta_{1} + 11048 \beta_{2} ) q^{34} + ( -3447538 - 105792 \beta_{1} + 37572 \beta_{2} ) q^{37} + ( 8046751 - 527233 \beta_{1} + 40242 \beta_{2} ) q^{38} + ( -1962517 + 961840 \beta_{1} - 38760 \beta_{2} ) q^{41} + ( -8142408 + 200976 \beta_{1} - 2732 \beta_{2} ) q^{43} + ( 7344842 + 453907 \beta_{1} - 38823 \beta_{2} ) q^{44} + ( 4707822 + 925230 \beta_{1} + 99780 \beta_{2} ) q^{46} + ( -22283108 - 651880 \beta_{1} - 9384 \beta_{2} ) q^{47} + ( -2224403 + 264480 \beta_{1} + 219280 \beta_{2} ) q^{49} + ( -19305256 - 313188 \beta_{1} + 52836 \beta_{2} ) q^{52} + ( -44311790 + 1064528 \beta_{1} + 314856 \beta_{2} ) q^{53} + ( 56427552 - 305766 \beta_{1} + 255474 \beta_{2} ) q^{56} + ( 67392976 + 1008192 \beta_{1} - 270608 \beta_{2} ) q^{58} + ( -1775144 + 399752 \beta_{1} - 345528 \beta_{2} ) q^{59} + ( 41835342 + 4664400 \beta_{1} - 199100 \beta_{2} ) q^{61} + ( 13287318 - 1629402 \beta_{1} - 30780 \beta_{2} ) q^{62} + ( 55679836 - 4013787 \beta_{1} + 502893 \beta_{2} ) q^{64} + ( 29717410 - 7831506 \beta_{1} - 322747 \beta_{2} ) q^{67} + ( 23743334 - 168485 \beta_{1} - 67443 \beta_{2} ) q^{68} + ( -98809132 + 4103800 \beta_{1} - 1123200 \beta_{2} ) q^{71} + ( 60459531 + 11107704 \beta_{1} - 56864 \beta_{2} ) q^{73} + ( 84171626 + 6354430 \beta_{1} - 1007520 \beta_{2} ) q^{74} + ( 29867606 + 4957077 \beta_{1} - 187953 \beta_{2} ) q^{76} + ( -186837678 + 837696 \beta_{1} - 701052 \beta_{2} ) q^{77} + ( -103098848 + 15961884 \beta_{1} - 728626 \beta_{2} ) q^{79} + ( -456738353 + 669117 \beta_{1} + 1892080 \beta_{2} ) q^{82} + ( 243751566 + 8354490 \beta_{1} - 165897 \beta_{2} ) q^{83} + ( -11404580 + 8317604 \beta_{1} + 266544 \beta_{2} ) q^{86} + ( -314283408 + 8374839 \beta_{1} - 821061 \beta_{2} ) q^{88} + ( 367574409 - 9709272 \beta_{1} + 1136808 \beta_{2} ) q^{89} + ( 393086120 + 20580336 \beta_{1} + 3929296 \beta_{2} ) q^{91} + ( 63246540 - 3564294 \beta_{1} + 284622 \beta_{2} ) q^{92} + ( 576105932 + 20200476 \beta_{1} - 426664 \beta_{2} ) q^{94} + ( -110724742 - 33078720 \beta_{1} - 62616 \beta_{2} ) q^{97} + ( -150976447 + 20953603 \beta_{1} - 4998240 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 33q^{2} + 341q^{4} + 5258q^{7} + 105q^{8} + O(q^{10}) \) \( 3q - 33q^{2} + 341q^{4} + 5258q^{7} + 105q^{8} + 54699q^{11} + 215884q^{13} - 272922q^{14} - 699247q^{16} - 334983q^{17} + 818845q^{19} + 72761q^{22} - 3526854q^{23} + 1280004q^{26} - 428134q^{28} - 2175480q^{29} + 4274066q^{31} + 9464577q^{32} - 6838963q^{34} - 10305042q^{37} + 24180495q^{38} - 5926311q^{41} - 24429956q^{43} + 21995703q^{44} + 14223246q^{46} - 66858708q^{47} - 6453929q^{49} - 57862932q^{52} - 132620514q^{53} + 169538130q^{56} + 201908320q^{58} - 5670960q^{59} + 125306926q^{61} + 39831174q^{62} + 167542401q^{64} + 88829483q^{67} + 71162559q^{68} - 297550596q^{71} + 181321729q^{73} + 251507358q^{74} + 89414865q^{76} - 561214086q^{77} - 310025170q^{79} - 1368322979q^{82} + 731088801q^{83} - 33947196q^{86} - 943671285q^{88} + 1103860035q^{89} + 1183187656q^{91} + 190024242q^{92} + 1727891132q^{94} - 332236842q^{97} - 457927581q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 652 x + 4000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 3 \nu - 436 \)\()/12\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + 33 \nu - 445 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 4 \beta_{1} + 3\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{2} + 132 \beta_{1} + 4351\)\()/10\)

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
−27.7229
22.2334
6.48955
−31.7828 0 498.143 0 0 −637.237 440.406 0 0
1.2 −21.4187 0 −53.2406 0 0 9905.49 12106.7 0 0
1.3 20.2014 0 −103.903 0 0 −4010.25 −12442.1 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.m 3
3.b odd 2 1 25.10.a.d yes 3
5.b even 2 1 225.10.a.p 3
5.c odd 4 2 225.10.b.m 6
12.b even 2 1 400.10.a.u 3
15.d odd 2 1 25.10.a.c 3
15.e even 4 2 25.10.b.c 6
60.h even 2 1 400.10.a.y 3
60.l odd 4 2 400.10.c.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 15.d odd 2 1
25.10.a.d yes 3 3.b odd 2 1
25.10.b.c 6 15.e even 4 2
225.10.a.m 3 1.a even 1 1 trivial
225.10.a.p 3 5.b even 2 1
225.10.b.m 6 5.c odd 4 2
400.10.a.u 3 12.b even 2 1
400.10.a.y 3 60.h even 2 1
400.10.c.q 6 60.l odd 4 2

Hecke kernels

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

\( T_{2}^{3} + 33 T_{2}^{2} - 394 T_{2} - 13752 \)
\( T_{7}^{3} - 5258 T_{7}^{2} - 43480164 T_{7} - 25313297688 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -13752 - 394 T + 33 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( -25313297688 - 43480164 T - 5258 T^{2} + T^{3} \)
$11$ \( 75283351667163 - 1257655633 T - 54699 T^{2} + T^{3} \)
$13$ \( 1483699912993984 - 588341776 T - 215884 T^{2} + T^{3} \)
$17$ \( -1773847472341707 - 732967429 T + 334983 T^{2} + T^{3} \)
$19$ \( 226123024842854125 - 499889436625 T - 818845 T^{2} + T^{3} \)
$23$ \( 381747315559395816 + 3297138740604 T + 3526854 T^{2} + T^{3} \)
$29$ \( -3964526545895424000 - 11063731072000 T + 2175480 T^{2} + T^{3} \)
$31$ \( -817098195664566648 + 3532627327452 T - 4274066 T^{2} + T^{3} \)
$37$ \( -\)\(17\!\cdots\!28\)\( - 49043023094484 T + 10305042 T^{2} + T^{3} \)
$41$ \( -\)\(15\!\cdots\!47\)\( - 750102076057093 T + 5926311 T^{2} + T^{3} \)
$43$ \( \)\(33\!\cdots\!84\)\( + 168272432263664 T + 24429956 T^{2} + T^{3} \)
$47$ \( \)\(14\!\cdots\!08\)\( + 1159874710756976 T + 66858708 T^{2} + T^{3} \)
$53$ \( -\)\(40\!\cdots\!84\)\( - 690042287731156 T + 132620514 T^{2} + T^{3} \)
$59$ \( \)\(49\!\cdots\!00\)\( - 6651292273432000 T + 5670960 T^{2} + T^{3} \)
$61$ \( \)\(62\!\cdots\!12\)\( - 12890308075143508 T - 125306926 T^{2} + T^{3} \)
$67$ \( -\)\(27\!\cdots\!93\)\( - 51057218268990129 T - 88829483 T^{2} + T^{3} \)
$71$ \( -\)\(11\!\cdots\!32\)\( - 50509406137014928 T + 297550596 T^{2} + T^{3} \)
$73$ \( \)\(13\!\cdots\!09\)\( - 82249507598185621 T - 181321729 T^{2} + T^{3} \)
$79$ \( -\)\(26\!\cdots\!00\)\( - 183462234827962500 T + 310025170 T^{2} + T^{3} \)
$83$ \( \)\(83\!\cdots\!41\)\( + 124612582244716359 T - 731088801 T^{2} + T^{3} \)
$89$ \( \)\(19\!\cdots\!75\)\( + 269595002285863875 T - 1103860035 T^{2} + T^{3} \)
$97$ \( -\)\(34\!\cdots\!08\)\( - 792948149717036724 T + 332236842 T^{2} + T^{3} \)
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