Properties

Label 225.10.b.g
Level 225
Weight 10
Character orbit 225.b
Analytic conductor 115.883
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{4729})\)
Defining polynomial: \(x^{4} + 2365 x^{2} + 1397124\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - 5 \beta_{2} ) q^{2} + ( -770 + 19 \beta_{3} ) q^{4} + ( -56 \beta_{1} - 2982 \beta_{2} ) q^{7} + ( 429 \beta_{1} + 12519 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{1} - 5 \beta_{2} ) q^{2} + ( -770 + 19 \beta_{3} ) q^{4} + ( -56 \beta_{1} - 2982 \beta_{2} ) q^{7} + ( 429 \beta_{1} + 12519 \beta_{2} ) q^{8} + ( -16768 - 1952 \beta_{3} ) q^{11} + ( 1384 \beta_{1} - 35573 \beta_{2} ) q^{13} + ( -125832 + 6468 \beta_{3} ) q^{14} + ( 363218 - 19171 \beta_{3} ) q^{16} + ( -2200 \beta_{1} - 96839 \beta_{2} ) q^{17} + ( 201164 + 968 \beta_{3} ) q^{19} + ( -800 \beta_{1} - 1069792 \beta_{2} ) q^{22} + ( -64968 \beta_{1} + 39684 \beta_{2} ) q^{23} + ( 924428 + 58690 \beta_{3} ) q^{26} + ( 155372 \beta_{1} + 2924964 \beta_{2} ) q^{28} + ( -31078 - 12416 \beta_{3} ) q^{29} + ( -2475696 - 75736 \beta_{3} ) q^{31} + ( -316109 \beta_{1} - 6736423 \beta_{2} ) q^{32} + ( -4537180 + 213478 \beta_{3} ) q^{34} + ( -174696 \beta_{1} + 1299733 \beta_{2} ) q^{37} + ( -192452 \beta_{1} - 433732 \beta_{2} ) q^{38} + ( -7340714 + 470096 \beta_{3} ) q^{41} + ( -152384 \beta_{1} - 6975326 \beta_{2} ) q^{43} + ( -30926656 + 1147360 \beta_{3} ) q^{44} + ( -75998496 + 505344 \beta_{3} ) q^{46} + ( -431368 \beta_{1} - 24099452 \beta_{2} ) q^{47} + ( 1077559 + 664832 \beta_{3} ) q^{49} + ( 312390 \beta_{1} + 11850274 \beta_{2} ) q^{52} + ( 929872 \beta_{1} - 15843931 \beta_{2} ) q^{53} + ( 177723000 - 3936660 \beta_{3} ) q^{56} + ( -80666 \beta_{1} - 7182466 \beta_{2} ) q^{58} + ( 93124864 + 1613408 \beta_{3} ) q^{59} + ( 75911686 + 2256688 \beta_{3} ) q^{61} + ( 1794072 \beta_{1} - 32381496 \beta_{2} ) q^{62} + ( -322401682 + 6502275 \beta_{3} ) q^{64} + ( -7444160 \beta_{1} + 6537054 \beta_{2} ) q^{67} + ( 5332082 \beta_{1} + 99269830 \beta_{2} ) q^{68} + ( 110604928 + 7061120 \beta_{3} ) q^{71} + ( 6480208 \beta_{1} + 9900631 \beta_{2} ) q^{73} + ( -180496012 - 1027202 \beta_{3} ) q^{74} + ( -133156936 + 3095148 \beta_{3} ) q^{76} + ( -10593408 \beta_{1} - 14601216 \beta_{2} ) q^{77} + ( 467102400 - 1798040 \beta_{3} ) q^{79} + ( 11571578 \beta_{1} + 314530306 \beta_{2} ) q^{82} + ( 3161088 \beta_{1} + 52550310 \beta_{2} ) q^{83} + ( -319624408 + 15322108 \beta_{3} ) q^{86} + ( 40843296 \beta_{1} + 284989536 \beta_{2} ) q^{88} + ( 107605986 + 9306192 \beta_{3} ) q^{89} + ( -332705016 - 4192496 \beta_{3} ) q^{91} + ( 47282976 \beta_{1} + 698968992 \beta_{2} ) q^{92} + ( -991866016 + 52081216 \beta_{3} ) q^{94} + ( 44039040 \beta_{1} + 107793409 \beta_{2} ) q^{97} + ( 4905929 \beta_{1} + 387527917 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3042q^{4} + O(q^{10}) \) \( 4q - 3042q^{4} - 70976q^{11} - 490392q^{14} + 1414530q^{16} + 806592q^{19} + 3815092q^{26} - 149144q^{29} - 10054256q^{31} - 17721764q^{34} - 28422664q^{41} - 121411904q^{44} - 302983296q^{46} + 5639900q^{49} + 703018680q^{56} + 375726272q^{59} + 308160120q^{61} - 1276602178q^{64} + 456541952q^{71} - 724038452q^{74} - 526437448q^{76} + 1864813520q^{79} - 1247853416q^{86} + 449036328q^{89} - 1339205056q^{91} - 3863301632q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2365 x^{2} + 1397124\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 2365 \nu \)\()/1182\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 1183 \nu \)\()/591\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 1183 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} - 2 \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 1183\)
\(\nu^{3}\)\(=\)\((\)\(2365 \beta_{2} + 2366 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
34.8839i
33.8839i
33.8839i
34.8839i
43.8839i 0 −1413.79 0 0 7861.50i 39574.2i 0 0
199.2 24.8839i 0 −107.207 0 0 4010.50i 10072.8i 0 0
199.3 24.8839i 0 −107.207 0 0 4010.50i 10072.8i 0 0
199.4 43.8839i 0 −1413.79 0 0 7861.50i 39574.2i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Hecke kernels

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

\( T_{2}^{4} + 2545 T_{2}^{2} + 1192464 \)
\( T_{11}^{2} + 35488 T_{11} - 4189882368 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 497 T^{2} + 159248 T^{4} + 130285568 T^{6} + 68719476736 T^{8} \)
$3$ 1
$5$ 1
$7$ \( 1 - 83527164 T^{2} + 4478467599613798 T^{4} - \)\(13\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 + 35488 T + 526013014 T^{2} + 83678847658208 T^{3} + 5559917313492231481 T^{4} )^{2} \)
$13$ \( 1 - 27567505292 T^{2} + \)\(36\!\cdots\!18\)\( T^{4} - \)\(31\!\cdots\!68\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 - 388734753820 T^{2} + \)\(65\!\cdots\!18\)\( T^{4} - \)\(54\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \)
$19$ \( ( 1 - 403296 T + 684929514838 T^{2} - 130138657763479584 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \)
$23$ \( 1 + 2800589695204 T^{2} + \)\(81\!\cdots\!58\)\( T^{4} + \)\(90\!\cdots\!76\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 + 74572 T + 28833430018078 T^{2} + 1081826889712503068 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$31$ \( ( 1 + 5027128 T + 52415931233342 T^{2} + \)\(13\!\cdots\!88\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \)
$37$ \( 1 - 433247572021484 T^{2} + \)\(79\!\cdots\!78\)\( T^{4} - \)\(73\!\cdots\!36\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 + 14211332 T + 443988635955862 T^{2} + \)\(46\!\cdots\!52\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \)
$43$ \( 1 - 1570463388232460 T^{2} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(39\!\cdots\!40\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 568239172667780 T^{2} + \)\(55\!\cdots\!78\)\( T^{4} + \)\(71\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 - 9086956380170956 T^{2} + \)\(38\!\cdots\!18\)\( T^{4} - \)\(98\!\cdots\!84\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 - 187863136 T + 23071633420288438 T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$61$ \( ( 1 - 154080060 T + 23302683905802238 T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \)
$67$ \( 1 + 22768078838174100 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} + \)\(16\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 - 228270976 T + 45777616900481806 T^{2} - \)\(10\!\cdots\!56\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$73$ \( 1 - 135645128086811996 T^{2} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(47\!\cdots\!24\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 - 932406760 T + 453226630902929438 T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \)
$83$ \( 1 - 702700996120787372 T^{2} + \)\(19\!\cdots\!38\)\( T^{4} - \)\(24\!\cdots\!48\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 - 224518164 T + 610925899926766678 T^{2} - \)\(78\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \)
$97$ \( 1 + 1619811253917303940 T^{2} + \)\(14\!\cdots\!78\)\( T^{4} + \)\(93\!\cdots\!60\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \)
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