Properties

Label 225.10.b.m
Level $225$
Weight $10$
Character orbit 225.b
Analytic conductor $115.883$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \(x^{6} + 1305 x^{4} + 433104 x^{2} + 16000000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( -114 + \beta_{1} ) q^{4} + ( -140 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} ) q^{7} + ( 279 \beta_{2} - 27 \beta_{3} - 33 \beta_{4} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( -114 + \beta_{1} ) q^{4} + ( -140 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} ) q^{7} + ( 279 \beta_{2} - 27 \beta_{3} - 33 \beta_{4} ) q^{8} + ( 18264 - 76 \beta_{1} - 17 \beta_{5} ) q^{11} + ( -928 \beta_{2} + 832 \beta_{3} - 460 \beta_{4} ) q^{13} + ( 90870 + 282 \beta_{1} + 30 \beta_{5} ) q^{14} + ( -233112 + 95 \beta_{1} - 6 \beta_{5} ) q^{16} + ( -3974 \beta_{2} + 1521 \beta_{3} - 168 \beta_{4} ) q^{17} + ( -273096 + 92 \beta_{1} + 351 \beta_{5} ) q^{19} + ( 28107 \beta_{2} - 3558 \beta_{3} + 4310 \beta_{4} ) q^{22} + ( -4764 \beta_{2} - 11160 \beta_{3} + 3426 \beta_{4} ) q^{23} + ( 428628 - 4588 \beta_{1} - 1292 \beta_{5} ) q^{26} + ( -15778 \beta_{2} + 238 \beta_{3} + 826 \beta_{4} ) q^{28} + ( 728268 - 9832 \beta_{1} + 508 \beta_{5} ) q^{29} + ( 1423984 + 2648 \beta_{1} - 534 \beta_{5} ) q^{31} + ( -101287 \beta_{2} - 18369 \beta_{3} - 19395 \beta_{4} ) q^{32} + ( 2279959 + 775 \beta_{1} - 1689 \beta_{5} ) q^{34} + ( 137524 \beta_{2} + 15866 \beta_{3} + 37572 \beta_{4} ) q^{37} + ( -300541 \beta_{2} + 113346 \beta_{3} - 40242 \beta_{4} ) q^{38} + ( -1952709 - 73088 \beta_{1} + 4904 \beta_{5} ) q^{41} + ( 34976 \beta_{2} - 83000 \beta_{3} + 2732 \beta_{4} ) q^{43} + ( -7346514 + 44675 \beta_{1} - 836 \beta_{5} ) q^{44} + ( 4736994 - 2322 \beta_{1} + 14586 \beta_{5} ) q^{46} + ( 894880 \beta_{2} + 121500 \beta_{3} - 9384 \beta_{4} ) q^{47} + ( 2188595 - 93952 \beta_{1} - 17904 \beta_{5} ) q^{49} + ( 560188 \beta_{2} + 123500 \beta_{3} + 52836 \beta_{4} ) q^{52} + ( 208484 \beta_{2} - 428022 \beta_{3} - 314856 \beta_{4} ) q^{53} + ( 56459448 + 143838 \beta_{1} + 15948 \beta_{5} ) q^{56} + ( 1874400 \beta_{2} + 433104 \beta_{3} + 270608 \beta_{4} ) q^{58} + ( 1818504 + 193768 \beta_{1} + 21680 \beta_{5} ) q^{59} + ( 41881302 - 359960 \beta_{1} + 22980 \beta_{5} ) q^{61} + ( 1133970 \beta_{2} - 247716 \beta_{3} - 30780 \beta_{4} ) q^{62} + ( -55688032 - 474207 \beta_{1} - 4098 \beta_{5} ) q^{64} + ( 6018994 \beta_{2} - 906256 \beta_{3} - 322747 \beta_{4} ) q^{67} + ( 232429 \beta_{2} + 200457 \beta_{3} + 67443 \beta_{4} ) q^{68} + ( -98905212 - 786920 \beta_{1} - 48040 \beta_{5} ) q^{71} + ( 10112822 \beta_{2} - 497441 \beta_{3} + 56864 \beta_{4} ) q^{73} + ( -84128214 + 855578 \beta_{1} + 21706 \beta_{5} ) q^{74} + ( 29919854 - 370821 \beta_{1} + 26124 \beta_{5} ) q^{76} + ( 2388876 \beta_{2} + 1613286 \beta_{3} - 701052 \beta_{4} ) q^{77} + ( 102948380 + 1255264 \beta_{1} - 75234 \beta_{5} ) q^{79} + ( 6514275 \beta_{2} + 3591696 \beta_{3} + 1892080 \beta_{4} ) q^{82} + ( 10762890 \beta_{2} + 1204200 \beta_{3} + 165897 \beta_{4} ) q^{83} + ( -11233116 - 333580 \beta_{1} + 85732 \beta_{5} ) q^{86} + ( 1767237 \beta_{2} - 3303801 \beta_{3} + 821061 \beta_{4} ) q^{88} + ( -367582761 - 1107576 \beta_{1} - 4176 \beta_{5} ) q^{89} + ( 393981224 + 796432 \beta_{1} + 447552 \beta_{5} ) q^{91} + ( 1888602 \beta_{2} - 837846 \beta_{3} + 284622 \beta_{4} ) q^{92} + ( -576367700 + 1342852 \beta_{1} - 130884 \beta_{5} ) q^{94} + ( 29036476 \beta_{2} - 2021122 \beta_{3} - 62616 \beta_{4} ) q^{97} + ( 14210371 \beta_{2} - 3371616 \beta_{3} + 4998240 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 682q^{4} + O(q^{10}) \) \( 6q - 682q^{4} + 109398q^{11} + 545844q^{14} - 1398494q^{16} - 1637690q^{19} + 2560008q^{26} + 4350960q^{29} + 8548132q^{31} + 13677926q^{34} - 11852622q^{41} - 43991406q^{44} + 28446492q^{46} + 12907858q^{49} + 339076260q^{56} + 11341920q^{59} + 250613852q^{61} - 335084802q^{64} - 595101192q^{71} - 503014716q^{74} + 178829730q^{76} + 620050340q^{79} - 67894392q^{86} - 2207720070q^{89} + 2366375312q^{91} - 3455782264q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 1305 x^{4} + 433104 x^{2} + 16000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -35 \nu^{4} - 17275 \nu^{2} + 2252704 \)\()/13392\)
\(\beta_{2}\)\(=\)\((\)\( 77 \nu^{5} + 71485 \nu^{3} + 13296008 \nu \)\()/3348000\)
\(\beta_{3}\)\(=\)\((\)\( -491 \nu^{5} - 908755 \nu^{3} - 378730064 \nu \)\()/13392000\)
\(\beta_{4}\)\(=\)\((\)\( -1177 \nu^{5} - 1291985 \nu^{3} - 198655408 \nu \)\()/13392000\)
\(\beta_{5}\)\(=\)\((\)\( -185 \nu^{4} - 171025 \nu^{2} - 22789928 \)\()/6696\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(25 \beta_{4} - 11 \beta_{3} + 78 \beta_{2}\)\()/250\)
\(\nu^{2}\)\(=\)\((\)\(-21 \beta_{5} + 222 \beta_{1} - 108817\)\()/250\)
\(\nu^{3}\)\(=\)\((\)\(-16225 \beta_{4} - 253 \beta_{3} - 62406 \beta_{2}\)\()/250\)
\(\nu^{4}\)\(=\)\((\)\(2073 \beta_{5} - 41046 \beta_{1} + 13959941\)\()/50\)
\(\nu^{5}\)\(=\)\((\)\(10746025 \beta_{4} + 2134309 \beta_{3} + 55337718 \beta_{2}\)\()/250\)

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
27.7229i
22.2334i
6.48955i
6.48955i
22.2334i
27.7229i
31.7828i 0 −498.143 0 0 637.237i 440.406i 0 0
199.2 21.4187i 0 53.2406 0 0 9905.49i 12106.7i 0 0
199.3 20.2014i 0 103.903 0 0 4010.25i 12442.1i 0 0
199.4 20.2014i 0 103.903 0 0 4010.25i 12442.1i 0 0
199.5 21.4187i 0 53.2406 0 0 9905.49i 12106.7i 0 0
199.6 31.7828i 0 −498.143 0 0 637.237i 440.406i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
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.m 6
3.b odd 2 1 25.10.b.c 6
5.b even 2 1 inner 225.10.b.m 6
5.c odd 4 1 225.10.a.m 3
5.c odd 4 1 225.10.a.p 3
12.b even 2 1 400.10.c.q 6
15.d odd 2 1 25.10.b.c 6
15.e even 4 1 25.10.a.c 3
15.e even 4 1 25.10.a.d yes 3
60.h even 2 1 400.10.c.q 6
60.l odd 4 1 400.10.a.u 3
60.l odd 4 1 400.10.a.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 15.e even 4 1
25.10.a.d yes 3 15.e even 4 1
25.10.b.c 6 3.b odd 2 1
25.10.b.c 6 15.d odd 2 1
225.10.a.m 3 5.c odd 4 1
225.10.a.p 3 5.c odd 4 1
225.10.b.m 6 1.a even 1 1 trivial
225.10.b.m 6 5.b even 2 1 inner
400.10.a.u 3 60.l odd 4 1
400.10.a.y 3 60.l odd 4 1
400.10.c.q 6 12.b even 2 1
400.10.c.q 6 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 \(S_{10}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{6} + 1877 T_{2}^{4} + 1062868 T_{2}^{2} + 189117504 \)
\( T_{11}^{3} - 54699 T_{11}^{2} - 1257655633 T_{11} + \)\(75\!\cdots\!63\)\( \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 189117504 + 1062868 T^{2} + 1877 T^{4} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( T^{6} \)
$7$ \( \)\(64\!\cdots\!44\)\( + 1624330022979888 T^{2} + 114606892 T^{4} + T^{6} \)
$11$ \( ( 75283351667163 - 1257655633 T - 54699 T^{2} + T^{3} )^{2} \)
$13$ \( \)\(22\!\cdots\!56\)\( + \)\(64\!\cdots\!88\)\( T^{2} + 47782585008 T^{4} + T^{6} \)
$17$ \( \)\(31\!\cdots\!49\)\( + \)\(11\!\cdots\!03\)\( T^{2} + 113679545147 T^{4} + T^{6} \)
$19$ \( ( -226123024842854125 - 499889436625 T + 818845 T^{2} + T^{3} )^{2} \)
$23$ \( \)\(14\!\cdots\!56\)\( + \)\(81\!\cdots\!88\)\( T^{2} + 5844421656108 T^{4} + T^{6} \)
$29$ \( ( 3964526545895424000 - 11063731072000 T - 2175480 T^{2} + T^{3} )^{2} \)
$31$ \( ( -817098195664566648 + 3532627327452 T - 4274066 T^{2} + T^{3} )^{2} \)
$37$ \( \)\(29\!\cdots\!84\)\( + \)\(59\!\cdots\!08\)\( T^{2} + 204279936810732 T^{4} + T^{6} \)
$41$ \( ( -\)\(15\!\cdots\!47\)\( - 750102076057093 T + 5926311 T^{2} + T^{3} )^{2} \)
$43$ \( \)\(10\!\cdots\!56\)\( + \)\(12\!\cdots\!88\)\( T^{2} + 260277885634608 T^{4} + T^{6} \)
$47$ \( \)\(21\!\cdots\!64\)\( + \)\(11\!\cdots\!48\)\( T^{2} + 2150337413915312 T^{4} + T^{6} \)
$53$ \( \)\(16\!\cdots\!56\)\( + \)\(10\!\cdots\!88\)\( T^{2} + 18968285309086508 T^{4} + T^{6} \)
$59$ \( ( -\)\(49\!\cdots\!00\)\( - 6651292273432000 T - 5670960 T^{2} + T^{3} )^{2} \)
$61$ \( ( \)\(62\!\cdots\!12\)\( - 12890308075143508 T - 125306926 T^{2} + T^{3} )^{2} \)
$67$ \( \)\(77\!\cdots\!49\)\( + \)\(21\!\cdots\!03\)\( T^{2} + 110005113588027547 T^{4} + T^{6} \)
$71$ \( ( -\)\(11\!\cdots\!32\)\( - 50509406137014928 T + 297550596 T^{2} + T^{3} )^{2} \)
$73$ \( \)\(19\!\cdots\!81\)\( + \)\(11\!\cdots\!63\)\( T^{2} + 197376584603920683 T^{4} + T^{6} \)
$79$ \( ( \)\(26\!\cdots\!00\)\( - 183462234827962500 T - 310025170 T^{2} + T^{3} )^{2} \)
$83$ \( \)\(70\!\cdots\!81\)\( + \)\(16\!\cdots\!63\)\( T^{2} + 285265670458184883 T^{4} + T^{6} \)
$89$ \( ( -\)\(19\!\cdots\!75\)\( + 269595002285863875 T + 1103860035 T^{2} + T^{3} )^{2} \)
$97$ \( \)\(11\!\cdots\!64\)\( + \)\(85\!\cdots\!48\)\( T^{2} + 1696277618616206412 T^{4} + T^{6} \)
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