Properties

Label 25.10.b.b
Level $25$
Weight $10$
Character orbit 25.b
Analytic conductor $12.876$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.8758959041\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{1009})\)
Defining polynomial: \(x^{4} + 505 x^{2} + 63504\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 5)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( -2 \beta_{1} + 14 \beta_{2} ) q^{3} + ( -522 + \beta_{3} ) q^{4} + ( -2668 + 14 \beta_{3} ) q^{6} + ( 214 \beta_{1} - 192 \beta_{2} ) q^{7} + ( 60 \beta_{1} - 1044 \beta_{2} ) q^{8} + ( -1253 + 52 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( -2 \beta_{1} + 14 \beta_{2} ) q^{3} + ( -522 + \beta_{3} ) q^{4} + ( -2668 + 14 \beta_{3} ) q^{6} + ( 214 \beta_{1} - 192 \beta_{2} ) q^{7} + ( 60 \beta_{1} - 1044 \beta_{2} ) q^{8} + ( -1253 + 52 \beta_{3} ) q^{9} + ( 11992 + 190 \beta_{3} ) q^{11} + ( 2344 \beta_{1} - 9976 \beta_{2} ) q^{12} + ( -1352 \beta_{1} + 6427 \beta_{2} ) q^{13} + ( 220176 - 192 \beta_{3} ) q^{14} + ( -156024 - 532 \beta_{3} ) q^{16} + ( -12856 \beta_{1} - 14213 \beta_{2} ) q^{17} + ( 3853 \beta_{1} - 55021 \beta_{2} ) q^{18} + ( 148260 + 284 \beta_{3} ) q^{19} + ( 542352 - 2952 \beta_{3} ) q^{21} + ( -2492 \beta_{1} - 184468 \beta_{2} ) q^{22} + ( 19398 \beta_{1} - 62160 \beta_{2} ) q^{23} + ( 1439280 - 2808 \beta_{3} ) q^{24} + ( -1651718 + 6427 \beta_{3} ) q^{26} + ( 30740 \beta_{1} + 119284 \beta_{2} ) q^{27} + ( -120208 \beta_{1} + 320400 \beta_{2} ) q^{28} + ( 1833490 + 10696 \beta_{3} ) q^{29} + ( 806572 + 15470 \beta_{3} ) q^{31} + ( 160144 \beta_{1} - 140464 \beta_{2} ) q^{32} + ( 223016 \beta_{1} - 339032 \beta_{2} ) q^{33} + ( -11939654 - 14213 \beta_{3} ) q^{34} + ( 5900866 - 28397 \beta_{3} ) q^{36} + ( -205296 \beta_{1} + 1158745 \beta_{2} ) q^{37} + ( -134060 \beta_{1} - 145396 \beta_{2} ) q^{38} + ( -10204636 + 29078 \beta_{3} ) q^{39} + ( -13478638 - 15580 \beta_{3} ) q^{41} + ( -689952 \beta_{1} + 3594720 \beta_{2} ) q^{42} + ( 25798 \beta_{1} + 2631586 \beta_{2} ) q^{43} + ( 12911176 - 87188 \beta_{3} ) q^{44} + ( 22195632 - 62160 \beta_{3} ) q^{46} + ( 523334 \beta_{1} - 3182276 \beta_{2} ) q^{47} + ( -379552 \beta_{1} - 764960 \beta_{2} ) q^{48} + ( -6577057 + 36380 \beta_{3} ) q^{49} + ( 889892 + 125846 \beta_{3} ) q^{51} + ( 1280844 \beta_{1} - 5006612 \beta_{2} ) q^{52} + ( 1137448 \beta_{1} - 2520481 \beta_{2} ) q^{53} + ( 24283960 + 119284 \beta_{3} ) q^{54} + ( -21574560 + 222096 \beta_{3} ) q^{56} + ( 72680 \beta_{1} + 1317928 \beta_{2} ) q^{57} + ( -1298690 \beta_{1} - 9226174 \beta_{2} ) q^{58} + ( 27497780 + 154472 \beta_{3} ) q^{59} + ( -137289858 - 69200 \beta_{3} ) q^{61} + ( -33072 \beta_{1} - 15189408 \beta_{2} ) q^{62} + ( -710142 \beta_{1} + 11689728 \beta_{2} ) q^{63} + ( 84720608 - 412848 \beta_{3} ) q^{64} + ( 236399344 - 339032 \beta_{3} ) q^{66} + ( -2416706 \beta_{1} + 1224282 \beta_{2} ) q^{67} + ( 4646732 \beta_{1} - 4520468 \beta_{2} ) q^{68} + ( 107344464 - 357096 \beta_{3} ) q^{69} + ( -3565468 - 627850 \beta_{3} ) q^{71} + ( -5347980 \beta_{1} + 7092612 \beta_{2} ) q^{72} + ( -8830952 \beta_{1} + 10458385 \beta_{2} ) q^{73} + ( -259948514 + 1158745 \beta_{3} ) q^{74} + ( -48736120 + 12 \beta_{3} ) q^{76} + ( 951288 \beta_{1} + 39530976 \beta_{2} ) q^{77} + ( 11658536 \beta_{1} - 40271288 \beta_{2} ) q^{78} + ( -3438760 - 1877564 \beta_{3} ) q^{79} + ( -137679679 + 893204 \beta_{3} ) q^{81} + ( 12699638 \beta_{1} + 2631082 \beta_{2} ) q^{82} + ( -2748402 \beta_{1} + 71491638 \beta_{2} ) q^{83} + ( -580964544 + 2083296 \beta_{3} ) q^{84} + ( -106194068 + 2631586 \beta_{3} ) q^{86} + ( 10237820 \beta_{1} - 2868068 \beta_{2} ) q^{87} + ( -18546480 \beta_{1} + 8615952 \beta_{2} ) q^{88} + ( -415044330 + 1338168 \beta_{3} ) q^{89} + ( 340815452 - 1345634 \beta_{3} ) q^{91} + ( -15371856 \beta_{1} + 54643152 \beta_{2} ) q^{92} + ( 18497856 \beta_{1} - 29981952 \beta_{2} ) q^{93} + ( 674074456 - 3182276 \beta_{3} ) q^{94} + ( 401680192 - 2202656 \beta_{3} ) q^{96} + ( -2622216 \beta_{1} - 30608621 \beta_{2} ) q^{97} + ( 8396057 \beta_{1} - 44193977 \beta_{2} ) q^{98} + ( 981866024 + 385514 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2088q^{4} - 10672q^{6} - 5012q^{9} + O(q^{10}) \) \( 4q - 2088q^{4} - 10672q^{6} - 5012q^{9} + 47968q^{11} + 880704q^{14} - 624096q^{16} + 593040q^{19} + 2169408q^{21} + 5757120q^{24} - 6606872q^{26} + 7333960q^{29} + 3226288q^{31} - 47758616q^{34} + 23603464q^{36} - 40818544q^{39} - 53914552q^{41} + 51644704q^{44} + 88782528q^{46} - 26308228q^{49} + 3559568q^{51} + 97135840q^{54} - 86298240q^{56} + 109991120q^{59} - 549159432q^{61} + 338882432q^{64} + 945597376q^{66} + 429377856q^{69} - 14261872q^{71} - 1039794056q^{74} - 194944480q^{76} - 13755040q^{79} - 550718716q^{81} - 2323858176q^{84} - 424776272q^{86} - 1660177320q^{89} + 1363261808q^{91} + 2696297824q^{94} + 1606720768q^{96} + 3927464096q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 505 x^{2} + 63504\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 337 \nu \)\()/42\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{3} - 1265 \nu \)\()/126\)
\(\beta_{3}\)\(=\)\( 20 \nu^{2} + 5050 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{2} - 5 \beta_{1}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 5050\)\()/20\)
\(\nu^{3}\)\(=\)\((\)\(-1011 \beta_{2} + 1265 \beta_{1}\)\()/10\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
16.3824i
15.3824i
15.3824i
16.3824i
36.7648i 193.530i −839.648 0 −7115.07 7647.66i 12045.9i −17770.7 0
24.2 26.7648i 66.4705i −204.352 0 1779.07 5947.66i 8234.11i 15264.7 0
24.3 26.7648i 66.4705i −204.352 0 1779.07 5947.66i 8234.11i 15264.7 0
24.4 36.7648i 193.530i −839.648 0 −7115.07 7647.66i 12045.9i −17770.7 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 25.10.b.b 4
3.b odd 2 1 225.10.b.h 4
4.b odd 2 1 400.10.c.p 4
5.b even 2 1 inner 25.10.b.b 4
5.c odd 4 1 5.10.a.b 2
5.c odd 4 1 25.10.a.b 2
15.d odd 2 1 225.10.b.h 4
15.e even 4 1 45.10.a.f 2
15.e even 4 1 225.10.a.h 2
20.d odd 2 1 400.10.c.p 4
20.e even 4 1 80.10.a.f 2
20.e even 4 1 400.10.a.t 2
35.f even 4 1 245.10.a.d 2
40.i odd 4 1 320.10.a.k 2
40.k even 4 1 320.10.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.b 2 5.c odd 4 1
25.10.a.b 2 5.c odd 4 1
25.10.b.b 4 1.a even 1 1 trivial
25.10.b.b 4 5.b even 2 1 inner
45.10.a.f 2 15.e even 4 1
80.10.a.f 2 20.e even 4 1
225.10.a.h 2 15.e even 4 1
225.10.b.h 4 3.b odd 2 1
225.10.b.h 4 15.d odd 2 1
245.10.a.d 2 35.f even 4 1
320.10.a.k 2 40.i odd 4 1
320.10.a.s 2 40.k even 4 1
400.10.a.t 2 20.e even 4 1
400.10.c.p 4 4.b odd 2 1
400.10.c.p 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2068 T_{2}^{2} + 968256 \) acting on \(S_{10}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 20 T^{2} + 423488 T^{4} + 5242880 T^{6} + 68719476736 T^{8} \)
$3$ \( 1 - 36860 T^{2} + 841672278 T^{4} - 14280319224540 T^{6} + 150094635296999121 T^{8} \)
$5$ 1
$7$ \( 1 - 67553100 T^{2} + 4264140931763398 T^{4} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 - 23984 T + 1217213446 T^{2} - 56553017420944 T^{3} + 5559917313492231481 T^{4} )^{2} \)
$13$ \( 1 - 32114487020 T^{2} + \)\(45\!\cdots\!58\)\( T^{4} - \)\(36\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 - 55612876540 T^{2} + \)\(47\!\cdots\!18\)\( T^{4} - \)\(78\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \)
$19$ \( ( 1 - 296520 T + 659218232758 T^{2} - 95683356145429080 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \)
$23$ \( 1 - 5894841450380 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 - 3666980 T + 20832571957438 T^{2} - 53197414150592105620 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$31$ \( ( 1 - 1613144 T + 29382323902526 T^{2} - 42650917850753459624 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \)
$37$ \( 1 - 211727253209420 T^{2} + \)\(26\!\cdots\!58\)\( T^{4} - \)\(35\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 + 26957276 T + 811945448362966 T^{2} + \)\(88\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \)
$43$ \( 1 - 610367211431900 T^{2} + \)\(59\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 2217843742562860 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 - 9826319201242060 T^{2} + \)\(43\!\cdots\!78\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 - 54995560 T + 15674484224932678 T^{2} - \)\(47\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$61$ \( ( 1 + 274579716 T + 41753623519328446 T^{2} + \)\(32\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \)
$67$ \( 1 - 97040022635671740 T^{2} + \)\(38\!\cdots\!18\)\( T^{4} - \)\(71\!\cdots\!60\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 + 7130936 T + 51935375688707086 T^{2} + \)\(32\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$73$ \( 1 - 70807827705933980 T^{2} + \)\(70\!\cdots\!38\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 + 6877520 T - 115982362290712162 T^{2} + \)\(82\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \)
$83$ \( 1 + 250773367170805060 T^{2} + \)\(70\!\cdots\!18\)\( T^{4} + \)\(87\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 + 830088660 T + 692293619421117718 T^{2} + \)\(29\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \)
$97$ \( 1 - 2823274612928748860 T^{2} + \)\(31\!\cdots\!78\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \)
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