Properties

Label 1911.4.a.h
Level $1911$
Weight $4$
Character orbit 1911.a
Self dual yes
Analytic conductor $112.753$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(112.752650021\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + 3 q^{3} + (2 \beta + 7) q^{4} + (2 \beta - 12) q^{5} + (3 \beta + 3) q^{6} + (\beta + 27) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + 3 q^{3} + (2 \beta + 7) q^{4} + (2 \beta - 12) q^{5} + (3 \beta + 3) q^{6} + (\beta + 27) q^{8} + 9 q^{9} + ( - 10 \beta + 16) q^{10} + ( - 12 \beta - 22) q^{11} + (6 \beta + 21) q^{12} + 13 q^{13} + (6 \beta - 36) q^{15} + (12 \beta - 15) q^{16} + ( - 4 \beta - 82) q^{17} + (9 \beta + 9) q^{18} + ( - 2 \beta - 24) q^{19} + ( - 10 \beta - 28) q^{20} + ( - 34 \beta - 190) q^{22} + (48 \beta + 4) q^{23} + (3 \beta + 81) q^{24} + ( - 48 \beta + 75) q^{25} + (13 \beta + 13) q^{26} + 27 q^{27} + ( - 24 \beta + 202) q^{29} + ( - 30 \beta + 48) q^{30} + (26 \beta - 20) q^{31} + ( - 11 \beta - 63) q^{32} + ( - 36 \beta - 66) q^{33} + ( - 86 \beta - 138) q^{34} + (18 \beta + 63) q^{36} + (28 \beta - 50) q^{37} + ( - 26 \beta - 52) q^{38} + 39 q^{39} + (42 \beta - 296) q^{40} + ( - 94 \beta - 100) q^{41} + (52 \beta - 308) q^{43} + ( - 128 \beta - 490) q^{44} + (18 \beta - 108) q^{45} + (52 \beta + 676) q^{46} + ( - 32 \beta + 162) q^{47} + (36 \beta - 45) q^{48} + (27 \beta - 597) q^{50} + ( - 12 \beta - 246) q^{51} + (26 \beta + 91) q^{52} + ( - 120 \beta - 82) q^{53} + (27 \beta + 27) q^{54} + (100 \beta - 72) q^{55} + ( - 6 \beta - 72) q^{57} + (178 \beta - 134) q^{58} + ( - 40 \beta - 70) q^{59} + ( - 30 \beta - 84) q^{60} + ( - 136 \beta - 314) q^{61} + (6 \beta + 344) q^{62} + ( - 170 \beta - 97) q^{64} + (26 \beta - 156) q^{65} + ( - 102 \beta - 570) q^{66} + ( - 170 \beta - 236) q^{67} + ( - 192 \beta - 686) q^{68} + (144 \beta + 12) q^{69} + ( - 84 \beta + 214) q^{71} + (9 \beta + 243) q^{72} + ( - 76 \beta + 450) q^{73} + ( - 22 \beta + 342) q^{74} + ( - 144 \beta + 225) q^{75} + ( - 62 \beta - 224) q^{76} + (39 \beta + 39) q^{78} + ( - 88 \beta - 216) q^{79} + ( - 174 \beta + 516) q^{80} + 81 q^{81} + ( - 194 \beta - 1416) q^{82} + ( - 64 \beta + 694) q^{83} + ( - 116 \beta + 872) q^{85} + ( - 256 \beta + 420) q^{86} + ( - 72 \beta + 606) q^{87} + ( - 346 \beta - 762) q^{88} + (190 \beta - 480) q^{89} + ( - 90 \beta + 144) q^{90} + (344 \beta + 1372) q^{92} + (78 \beta - 60) q^{93} + (130 \beta - 286) q^{94} + ( - 24 \beta + 232) q^{95} + ( - 33 \beta - 189) q^{96} + (220 \beta + 266) q^{97} + ( - 108 \beta - 198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} + 54 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} + 54 q^{8} + 18 q^{9} + 32 q^{10} - 44 q^{11} + 42 q^{12} + 26 q^{13} - 72 q^{15} - 30 q^{16} - 164 q^{17} + 18 q^{18} - 48 q^{19} - 56 q^{20} - 380 q^{22} + 8 q^{23} + 162 q^{24} + 150 q^{25} + 26 q^{26} + 54 q^{27} + 404 q^{29} + 96 q^{30} - 40 q^{31} - 126 q^{32} - 132 q^{33} - 276 q^{34} + 126 q^{36} - 100 q^{37} - 104 q^{38} + 78 q^{39} - 592 q^{40} - 200 q^{41} - 616 q^{43} - 980 q^{44} - 216 q^{45} + 1352 q^{46} + 324 q^{47} - 90 q^{48} - 1194 q^{50} - 492 q^{51} + 182 q^{52} - 164 q^{53} + 54 q^{54} - 144 q^{55} - 144 q^{57} - 268 q^{58} - 140 q^{59} - 168 q^{60} - 628 q^{61} + 688 q^{62} - 194 q^{64} - 312 q^{65} - 1140 q^{66} - 472 q^{67} - 1372 q^{68} + 24 q^{69} + 428 q^{71} + 486 q^{72} + 900 q^{73} + 684 q^{74} + 450 q^{75} - 448 q^{76} + 78 q^{78} - 432 q^{79} + 1032 q^{80} + 162 q^{81} - 2832 q^{82} + 1388 q^{83} + 1744 q^{85} + 840 q^{86} + 1212 q^{87} - 1524 q^{88} - 960 q^{89} + 288 q^{90} + 2744 q^{92} - 120 q^{93} - 572 q^{94} + 464 q^{95} - 378 q^{96} + 532 q^{97} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−3.74166
3.74166
−2.74166 3.00000 −0.483315 −19.4833 −8.22497 0 23.2583 9.00000 53.4166
1.2 4.74166 3.00000 14.4833 −4.51669 14.2250 0 30.7417 9.00000 −21.4166
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 1911.4.a.h 2
7.b odd 2 1 39.4.a.b 2
21.c even 2 1 117.4.a.c 2
28.d even 2 1 624.4.a.r 2
35.c odd 2 1 975.4.a.j 2
56.e even 2 1 2496.4.a.s 2
56.h odd 2 1 2496.4.a.bc 2
84.h odd 2 1 1872.4.a.t 2
91.b odd 2 1 507.4.a.f 2
91.i even 4 2 507.4.b.f 4
273.g even 2 1 1521.4.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 7.b odd 2 1
117.4.a.c 2 21.c even 2 1
507.4.a.f 2 91.b odd 2 1
507.4.b.f 4 91.i even 4 2
624.4.a.r 2 28.d even 2 1
975.4.a.j 2 35.c odd 2 1
1521.4.a.s 2 273.g even 2 1
1872.4.a.t 2 84.h odd 2 1
1911.4.a.h 2 1.a even 1 1 trivial
2496.4.a.s 2 56.e even 2 1
2496.4.a.bc 2 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{2} - 2T_{2} - 13 \) Copy content Toggle raw display
\( T_{5}^{2} + 24T_{5} + 88 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 13 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 24T + 88 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 44T - 1532 \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 164T + 6500 \) Copy content Toggle raw display
$19$ \( T^{2} + 48T + 520 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T - 32240 \) Copy content Toggle raw display
$29$ \( T^{2} - 404T + 32740 \) Copy content Toggle raw display
$31$ \( T^{2} + 40T - 9064 \) Copy content Toggle raw display
$37$ \( T^{2} + 100T - 8476 \) Copy content Toggle raw display
$41$ \( T^{2} + 200T - 113704 \) Copy content Toggle raw display
$43$ \( T^{2} + 616T + 57008 \) Copy content Toggle raw display
$47$ \( T^{2} - 324T + 11908 \) Copy content Toggle raw display
$53$ \( T^{2} + 164T - 194876 \) Copy content Toggle raw display
$59$ \( T^{2} + 140T - 17500 \) Copy content Toggle raw display
$61$ \( T^{2} + 628T - 160348 \) Copy content Toggle raw display
$67$ \( T^{2} + 472T - 348904 \) Copy content Toggle raw display
$71$ \( T^{2} - 428T - 52988 \) Copy content Toggle raw display
$73$ \( T^{2} - 900T + 121636 \) Copy content Toggle raw display
$79$ \( T^{2} + 432T - 61760 \) Copy content Toggle raw display
$83$ \( T^{2} - 1388 T + 424292 \) Copy content Toggle raw display
$89$ \( T^{2} + 960T - 275000 \) Copy content Toggle raw display
$97$ \( T^{2} - 532T - 606844 \) Copy content Toggle raw display
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