Properties

Label 441.4.a.s
Level 441
Weight 4
Character orbit 441.a
Self dual yes
Analytic conductor 26.020
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 8 + \beta_{1} + \beta_{2} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} ) q^{5} + ( -10 - 9 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 8 + \beta_{1} + \beta_{2} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} ) q^{5} + ( -10 - 9 \beta_{1} - \beta_{2} ) q^{8} + ( -22 + 11 \beta_{1} - \beta_{2} ) q^{10} + ( -12 + \beta_{1} + 3 \beta_{2} ) q^{11} + ( 19 + 5 \beta_{1} - \beta_{2} ) q^{13} + ( 74 + 19 \beta_{1} + \beta_{2} ) q^{16} + ( -16 - 4 \beta_{2} ) q^{17} + ( -65 - 7 \beta_{1} - \beta_{2} ) q^{19} + ( -150 + 11 \beta_{1} - 3 \beta_{2} ) q^{20} + ( 2 - 13 \beta_{1} - \beta_{2} ) q^{22} + ( -80 + 24 \beta_{1} + 4 \beta_{2} ) q^{23} + ( 53 - 29 \beta_{1} + \beta_{2} ) q^{25} + ( -86 - 16 \beta_{1} - 5 \beta_{2} ) q^{26} + ( -26 + 25 \beta_{1} - 5 \beta_{2} ) q^{29} + ( -39 + 22 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -218 - 29 \beta_{1} - 11 \beta_{2} ) q^{32} + ( -24 + 48 \beta_{1} ) q^{34} + ( 81 + 19 \beta_{1} + \beta_{2} ) q^{37} + ( 106 + 80 \beta_{1} + 7 \beta_{2} ) q^{38} + ( -18 + 75 \beta_{1} - 3 \beta_{2} ) q^{40} + ( -82 + 2 \beta_{1} + 14 \beta_{2} ) q^{41} + ( 143 - 69 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 298 + 11 \beta_{1} - 11 \beta_{2} ) q^{44} + ( -360 + 24 \beta_{1} - 24 \beta_{2} ) q^{46} + ( 46 + 72 \beta_{1} + 28 \beta_{2} ) q^{47} + ( 470 - 32 \beta_{1} + 29 \beta_{2} ) q^{50} + ( 74 + 102 \beta_{1} + 24 \beta_{2} ) q^{52} + ( -154 + 69 \beta_{1} + 11 \beta_{2} ) q^{53} + ( -350 + 19 \beta_{1} + 25 \beta_{2} ) q^{55} + ( -430 + 41 \beta_{1} - 25 \beta_{2} ) q^{58} + ( -358 - 69 \beta_{1} + 29 \beta_{2} ) q^{59} + ( 10 - 100 \beta_{1} + 20 \beta_{2} ) q^{61} + ( -364 + 33 \beta_{1} - 22 \beta_{2} ) q^{62} + ( -194 + 183 \beta_{1} + 21 \beta_{2} ) q^{64} + ( 174 - 50 \beta_{1} - 18 \beta_{2} ) q^{65} + ( -215 + 17 \beta_{1} + 47 \beta_{2} ) q^{67} + ( -640 - 24 \beta_{1} - 16 \beta_{2} ) q^{68} + ( -66 - 120 \beta_{1} - 12 \beta_{2} ) q^{71} + ( 363 - 101 \beta_{1} - 23 \beta_{2} ) q^{73} + ( -298 - 108 \beta_{1} - 19 \beta_{2} ) q^{74} + ( -718 - 186 \beta_{1} - 72 \beta_{2} ) q^{76} + ( 299 - 36 \beta_{1} + 48 \beta_{2} ) q^{79} + ( -18 - 121 \beta_{1} - 51 \beta_{2} ) q^{80} + ( 52 - 32 \beta_{1} - 2 \beta_{2} ) q^{82} + ( -156 - 51 \beta_{1} + 27 \beta_{2} ) q^{83} + ( 624 - 72 \beta_{1} ) q^{85} + ( 1086 - 50 \beta_{1} + 69 \beta_{2} ) q^{86} + ( -258 - 117 \beta_{1} - 3 \beta_{2} ) q^{88} + ( -532 - 170 \beta_{1} - 22 \beta_{2} ) q^{89} + ( 112 + 336 \beta_{1} - 56 \beta_{2} ) q^{92} + ( -984 - 342 \beta_{1} - 72 \beta_{2} ) q^{94} + ( 246 - 2 \beta_{1} + 54 \beta_{2} ) q^{95} + ( 24 - 53 \beta_{1} + 49 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} + O(q^{10}) \) \( 3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} - 55q^{10} - 35q^{11} + 62q^{13} + 241q^{16} - 48q^{17} - 202q^{19} - 439q^{20} - 7q^{22} - 216q^{23} + 130q^{25} - 274q^{26} - 53q^{29} - 95q^{31} - 683q^{32} - 24q^{34} + 262q^{37} + 398q^{38} + 21q^{40} - 244q^{41} + 360q^{43} + 905q^{44} - 1056q^{46} + 210q^{47} + 1378q^{50} + 324q^{52} - 393q^{53} - 1031q^{55} - 1249q^{58} - 1143q^{59} - 70q^{61} - 1059q^{62} - 399q^{64} + 472q^{65} - 628q^{67} - 1944q^{68} - 318q^{71} + 988q^{73} - 1002q^{74} - 2340q^{76} + 861q^{79} - 175q^{80} + 124q^{82} - 519q^{83} + 1800q^{85} + 3208q^{86} - 891q^{88} - 1766q^{89} + 672q^{92} - 3294q^{94} + 736q^{95} + 19q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 24 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 16\)

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
5.30829
0.248072
−4.55637
−5.30829 0 20.1780 −5.56140 0 0 −64.6443 0 29.5215
1.2 −0.248072 0 −7.93846 12.4346 0 0 3.95388 0 −3.08468
1.3 4.55637 0 12.7605 −17.8732 0 0 21.6905 0 −81.4369
\(n\): 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 441.4.a.s 3
3.b odd 2 1 147.4.a.l 3
7.b odd 2 1 441.4.a.t 3
7.c even 3 2 63.4.e.c 6
7.d odd 6 2 441.4.e.w 6
12.b even 2 1 2352.4.a.ci 3
21.c even 2 1 147.4.a.m 3
21.g even 6 2 147.4.e.n 6
21.h odd 6 2 21.4.e.b 6
84.h odd 2 1 2352.4.a.cg 3
84.n even 6 2 336.4.q.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 21.h odd 6 2
63.4.e.c 6 7.c even 3 2
147.4.a.l 3 3.b odd 2 1
147.4.a.m 3 21.c even 2 1
147.4.e.n 6 21.g even 6 2
336.4.q.k 6 84.n even 6 2
441.4.a.s 3 1.a even 1 1 trivial
441.4.a.t 3 7.b odd 2 1
441.4.e.w 6 7.d odd 6 2
2352.4.a.cg 3 84.h odd 2 1
2352.4.a.ci 3 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{3} + T_{2}^{2} - 24 T_{2} - 6 \)
\( T_{5}^{3} + 11 T_{5}^{2} - 192 T_{5} - 1236 \)
\( T_{13}^{3} - 62 T_{13}^{2} + 425 T_{13} + 18452 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 10 T^{3} + 64 T^{5} + 512 T^{6} \)
$3$ \( \)
$5$ \( 1 + 11 T + 183 T^{2} + 1514 T^{3} + 22875 T^{4} + 171875 T^{5} + 1953125 T^{6} \)
$7$ \( \)
$11$ \( 1 + 35 T + 2625 T^{2} + 102734 T^{3} + 3493875 T^{4} + 62004635 T^{5} + 2357947691 T^{6} \)
$13$ \( 1 - 62 T + 7016 T^{2} - 253976 T^{3} + 15414152 T^{4} - 299262158 T^{5} + 10604499373 T^{6} \)
$17$ \( 1 + 48 T + 12339 T^{2} + 358752 T^{3} + 60621507 T^{4} + 1158603312 T^{5} + 118587876497 T^{6} \)
$19$ \( 1 + 202 T + 32858 T^{2} + 3004840 T^{3} + 225373022 T^{4} + 9503267962 T^{5} + 322687697779 T^{6} \)
$23$ \( 1 + 216 T + 35829 T^{2} + 3675600 T^{3} + 435931443 T^{4} + 31975752024 T^{5} + 1801152661463 T^{6} \)
$29$ \( 1 + 53 T + 52695 T^{2} + 3410210 T^{3} + 1285178355 T^{4} + 31525636013 T^{5} + 14507145975869 T^{6} \)
$31$ \( 1 + 95 T + 79372 T^{2} + 5648467 T^{3} + 2364571252 T^{4} + 84312849695 T^{5} + 26439622160671 T^{6} \)
$37$ \( 1 - 262 T + 166048 T^{2} - 26591324 T^{3} + 8410829344 T^{4} - 672220319158 T^{5} + 129961739795077 T^{6} \)
$41$ \( 1 + 244 T + 187983 T^{2} + 33933832 T^{3} + 12955976343 T^{4} + 1159025434804 T^{5} + 327381934393961 T^{6} \)
$43$ \( 1 - 360 T + 166158 T^{2} - 38975294 T^{3} + 13210724106 T^{4} - 2275690697640 T^{5} + 502592611936843 T^{6} \)
$47$ \( 1 - 210 T + 64953 T^{2} - 48724788 T^{3} + 6743615319 T^{4} - 2263635219090 T^{5} + 1119130473102767 T^{6} \)
$53$ \( 1 + 393 T + 365895 T^{2} + 83847930 T^{3} + 54473349915 T^{4} + 8710593923697 T^{5} + 3299763591802133 T^{6} \)
$59$ \( 1 + 1143 T + 749241 T^{2} + 369027450 T^{3} + 153878367339 T^{4} + 48212349951663 T^{5} + 8662995818654939 T^{6} \)
$61$ \( 1 + 70 T + 340043 T^{2} - 52853660 T^{3} + 77183300183 T^{4} + 3606426205270 T^{5} + 11694146092834141 T^{6} \)
$67$ \( 1 + 628 T + 597326 T^{2} + 405751330 T^{3} + 179653559738 T^{4} + 56807864002132 T^{5} + 27206534396294947 T^{6} \)
$71$ \( 1 + 318 T + 742929 T^{2} + 256167372 T^{3} + 265902461319 T^{4} + 40735890286878 T^{5} + 45848500718449031 T^{6} \)
$73$ \( 1 - 988 T + 1162696 T^{2} - 625490474 T^{3} + 452308509832 T^{4} - 149518215573532 T^{5} + 58871586708267913 T^{6} \)
$79$ \( 1 - 861 T + 1221216 T^{2} - 655056821 T^{3} + 602107115424 T^{4} - 209298299203581 T^{5} + 119851595982618319 T^{6} \)
$83$ \( 1 + 519 T + 1583745 T^{2} + 545598870 T^{3} + 905564802315 T^{4} + 169682053778511 T^{5} + 186940255267540403 T^{6} \)
$89$ \( 1 + 1766 T + 2392827 T^{2} + 2476945964 T^{3} + 1686868857363 T^{4} + 877668959837126 T^{5} + 350356403707485209 T^{6} \)
$97$ \( 1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 1979057473987 T^{4} - 15826468093651 T^{5} + 760231058654565217 T^{6} \)
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