Properties

Label 25.12.a.c
Level 25
Weight 12
Character orbit 25.a
Self dual yes
Analytic conductor 19.209
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 25 = 5^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2085795140\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{151}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{151}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 10 + 3 \beta ) q^{2} + ( 110 + 16 \beta ) q^{3} + ( 3488 + 60 \beta ) q^{4} + ( 30092 + 490 \beta ) q^{6} + ( -28950 + 528 \beta ) q^{7} + ( 123120 + 4920 \beta ) q^{8} + ( -10423 + 3520 \beta ) q^{9} +O(q^{10})\) \( q + ( 10 + 3 \beta ) q^{2} + ( 110 + 16 \beta ) q^{3} + ( 3488 + 60 \beta ) q^{4} + ( 30092 + 490 \beta ) q^{6} + ( -28950 + 528 \beta ) q^{7} + ( 123120 + 4920 \beta ) q^{8} + ( -10423 + 3520 \beta ) q^{9} + ( -309088 + 26400 \beta ) q^{11} + ( 963520 + 62408 \beta ) q^{12} + ( -1707130 - 12864 \beta ) q^{13} + ( 667236 - 81570 \beta ) q^{14} + ( 3002816 + 295680 \beta ) q^{16} + ( -658970 + 126528 \beta ) q^{17} + ( 6274010 + 3931 \beta ) q^{18} + ( 2662660 - 274560 \beta ) q^{19} + ( 1918092 - 405120 \beta ) q^{21} + ( 44745920 - 663264 \beta ) q^{22} + ( -29471970 + 33456 \beta ) q^{23} + ( 61090080 + 2511120 \beta ) q^{24} + ( -40380868 - 5250030 \beta ) q^{26} + ( 13384580 - 2613920 \beta ) q^{27} + ( -81842880 + 104664 \beta ) q^{28} + ( 47070190 - 2298240 \beta ) q^{29} + ( 122271732 + 7207200 \beta ) q^{31} + ( 313650560 + 1889088 \beta ) q^{32} + ( 221129920 - 2041408 \beta ) q^{33} + ( 222679036 - 711630 \beta ) q^{34} + ( 91209376 + 11652380 \beta ) q^{36} + ( -10501610 + 19033728 \beta ) q^{37} + ( -470876120 + 5242380 \beta ) q^{38} + ( -312101996 - 28729120 \beta ) q^{39} + ( -372871658 + 22651200 \beta ) q^{41} + ( -714896520 + 1703076 \beta ) q^{42} + ( -314975050 - 13909104 \beta ) q^{43} + ( -121362944 + 73537920 \beta ) q^{44} + ( -234097428 - 88081350 \beta ) q^{46} + ( 701030770 - 20505072 \beta ) q^{47} + ( 3187761280 + 80569856 \beta ) q^{48} + ( -970838707 - 30571200 \beta ) q^{49} + ( 1150279892 + 3374560 \beta ) q^{51} + ( -6420660800 - 147297432 \beta ) q^{52} + ( -569160290 - 186753984 \beta ) q^{53} + ( -4602577240 + 14014540 \beta ) q^{54} + ( -1995276960 - 77426640 \beta ) q^{56} + ( -2360455240 + 12400960 \beta ) q^{57} + ( -3693708980 + 118228170 \beta ) q^{58} + ( 3658757780 - 175817280 \beta ) q^{59} + ( -758212838 + 53568000 \beta ) q^{61} + ( 14282163720 + 438887196 \beta ) q^{62} + ( 1424316090 - 107407344 \beta ) q^{63} + ( 409765888 + 354289920 \beta ) q^{64} + ( -1487732096 + 642975680 \beta ) q^{66} + ( -7867145070 - 91691472 \beta ) q^{67} + ( 2286887360 + 401791464 \beta ) q^{68} + ( -2918597916 - 467871360 \beta ) q^{69} + ( 16469235772 + 54804000 \beta ) q^{71} + ( 9177033840 + 382101240 \beta ) q^{72} + ( 14991424430 + 339617856 \beta ) q^{73} + ( 34384099036 + 158832450 \beta ) q^{74} + ( -662696320 - 797905680 \beta ) q^{76} + ( 17367374400 - 927478464 \beta ) q^{77} + ( -55178185400 - 1223597188 \beta ) q^{78} + ( -1651411560 + 575636160 \beta ) q^{79} + ( -21942215899 - 696935360 \beta ) q^{81} + ( 37315257820 - 892102974 \beta ) q^{82} + ( -6649551210 - 1100818224 \beta ) q^{83} + ( -7991243904 - 1297973040 \beta ) q^{84} + ( -28353046948 - 1084016190 \beta ) q^{86} + ( -17032470460 + 500316640 \beta ) q^{87} + ( 40397437440 + 1729655040 \beta ) q^{88} + ( -6337385430 + 1455281280 \beta ) q^{89} + ( 45318929532 - 528951840 \beta ) q^{91} + ( -101585785920 - 1651623672 \beta ) q^{92} + ( 83100271320 + 2749139712 \beta ) q^{93} + ( -30144882764 + 1898041590 \beta ) q^{94} + ( 52757708032 + 5226208640 \beta ) q^{96} + ( 1540351870 + 4545870528 \beta ) q^{97} + ( -65103401470 - 3218228121 \beta ) q^{98} + ( 59350136224 - 1363156960 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 20q^{2} + 220q^{3} + 6976q^{4} + 60184q^{6} - 57900q^{7} + 246240q^{8} - 20846q^{9} + O(q^{10}) \) \( 2q + 20q^{2} + 220q^{3} + 6976q^{4} + 60184q^{6} - 57900q^{7} + 246240q^{8} - 20846q^{9} - 618176q^{11} + 1927040q^{12} - 3414260q^{13} + 1334472q^{14} + 6005632q^{16} - 1317940q^{17} + 12548020q^{18} + 5325320q^{19} + 3836184q^{21} + 89491840q^{22} - 58943940q^{23} + 122180160q^{24} - 80761736q^{26} + 26769160q^{27} - 163685760q^{28} + 94140380q^{29} + 244543464q^{31} + 627301120q^{32} + 442259840q^{33} + 445358072q^{34} + 182418752q^{36} - 21003220q^{37} - 941752240q^{38} - 624203992q^{39} - 745743316q^{41} - 1429793040q^{42} - 629950100q^{43} - 242725888q^{44} - 468194856q^{46} + 1402061540q^{47} + 6375522560q^{48} - 1941677414q^{49} + 2300559784q^{51} - 12841321600q^{52} - 1138320580q^{53} - 9205154480q^{54} - 3990553920q^{56} - 4720910480q^{57} - 7387417960q^{58} + 7317515560q^{59} - 1516425676q^{61} + 28564327440q^{62} + 2848632180q^{63} + 819531776q^{64} - 2975464192q^{66} - 15734290140q^{67} + 4573774720q^{68} - 5837195832q^{69} + 32938471544q^{71} + 18354067680q^{72} + 29982848860q^{73} + 68768198072q^{74} - 1325392640q^{76} + 34734748800q^{77} - 110356370800q^{78} - 3302823120q^{79} - 43884431798q^{81} + 74630515640q^{82} - 13299102420q^{83} - 15982487808q^{84} - 56706093896q^{86} - 34064940920q^{87} + 80794874880q^{88} - 12674770860q^{89} + 90637859064q^{91} - 203171571840q^{92} + 166200542640q^{93} - 60289765528q^{94} + 105515416064q^{96} + 3080703740q^{97} - 130206802940q^{98} + 118700272448q^{99} + O(q^{100}) \)

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
−63.7292 −283.223 2013.42 0 18049.6 −41926.3 2204.06 −96932.0 0
1.2 83.7292 503.223 4962.58 0 42134.4 −15973.7 244036. 76086.0 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 25.12.a.c 2
3.b odd 2 1 225.12.a.h 2
5.b even 2 1 5.12.a.b 2
5.c odd 4 2 25.12.b.c 4
15.d odd 2 1 45.12.a.d 2
15.e even 4 2 225.12.b.f 4
20.d odd 2 1 80.12.a.j 2
35.c odd 2 1 245.12.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.a.b 2 5.b even 2 1
25.12.a.c 2 1.a even 1 1 trivial
25.12.b.c 4 5.c odd 4 2
45.12.a.d 2 15.d odd 2 1
80.12.a.j 2 20.d odd 2 1
225.12.a.h 2 3.b odd 2 1
225.12.b.f 4 15.e even 4 2
245.12.a.b 2 35.c odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 20 T_{2} - 5336 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 20 T - 1240 T^{2} - 40960 T^{3} + 4194304 T^{4} \)
$3$ \( 1 - 220 T + 211770 T^{2} - 38972340 T^{3} + 31381059609 T^{4} \)
$5$ 1
$7$ \( 1 + 57900 T + 4624370450 T^{2} + 114487218419700 T^{3} + 3909821048582988049 T^{4} \)
$11$ \( 1 + 618176 T + 245194892966 T^{2} + 176372827291625536 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \)
$13$ \( 1 + 3414260 T + 6398662197390 T^{2} + 6118901546944767620 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \)
$17$ \( 1 + 1317940 T + 59308395866630 T^{2} + 45168303019681836020 T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \)
$19$ \( 1 - 5325320 T + 194538827137638 T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \)
$23$ \( 1 + 58943940 T + 2773540471931410 T^{2} + \)\(56\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 - 94140380 T + 23426350431097358 T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \)
$31$ \( 1 - 244543464 T + 34393316207729486 T^{2} - \)\(62\!\cdots\!84\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \)
$37$ \( 1 + 21003220 T + 137126715218410590 T^{2} + \)\(37\!\cdots\!60\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \)
$41$ \( 1 + 745743316 T + 929792912462405846 T^{2} + \)\(41\!\cdots\!56\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \)
$43$ \( 1 + 629950100 T + 1840945003918927050 T^{2} + \)\(58\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 1402061540 T + 5181805952108806370 T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 + 1138320580 T - 2203723231625575330 T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 - 7317515560 T + 55027608950440780118 T^{2} - \)\(22\!\cdots\!40\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \)
$61$ \( 1 + 1516425676 T + 85869525433683691566 T^{2} + \)\(65\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \)
$67$ \( 1 + 15734290140 T + \)\(30\!\cdots\!30\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \)
$71$ \( 1 - 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} - \)\(76\!\cdots\!24\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \)
$73$ \( 1 - 29982848860 T + \)\(78\!\cdots\!10\)\( T^{2} - \)\(94\!\cdots\!20\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \)
$79$ \( 1 + 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 + 13299102420 T + \)\(18\!\cdots\!30\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 + 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \)
$97$ \( 1 - 3080703740 T + \)\(18\!\cdots\!70\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \)
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