Properties

Label 3.11.b.a
Level $3$
Weight $11$
Character orbit 3.b
Analytic conductor $1.906$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.90607175802\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (9 \beta - 27) q^{3} + 304 q^{4} - 106 \beta q^{5} + ( - 27 \beta - 6480) q^{6} + 17234 q^{7} + 1328 \beta q^{8} + ( - 486 \beta - 57591) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (9 \beta - 27) q^{3} + 304 q^{4} - 106 \beta q^{5} + ( - 27 \beta - 6480) q^{6} + 17234 q^{7} + 1328 \beta q^{8} + ( - 486 \beta - 57591) q^{9} + 76320 q^{10} - 6962 \beta q^{11} + (2736 \beta - 8208) q^{12} - 169654 q^{13} + 17234 \beta q^{14} + (2862 \beta + 686880) q^{15} - 644864 q^{16} - 12792 \beta q^{17} + ( - 57591 \beta + 349920) q^{18} - 949462 q^{19} - 32224 \beta q^{20} + (155106 \beta - 465318) q^{21} + 5012640 q^{22} + 99044 \beta q^{23} + ( - 35856 \beta - 8605440) q^{24} + 1675705 q^{25} - 169654 \beta q^{26} + ( - 505197 \beta + 4704237) q^{27} + 5239136 q^{28} + 118594 \beta q^{29} + (686880 \beta - 2060640) q^{30} - 29793118 q^{31} + 715008 \beta q^{32} + (187974 \beta + 45113760) q^{33} + 9210240 q^{34} - 1826804 \beta q^{35} + ( - 147744 \beta - 17507664) q^{36} - 60811846 q^{37} - 949462 \beta q^{38} + ( - 1526886 \beta + 4580658) q^{39} + 101352960 q^{40} + 6770372 \beta q^{41} + ( - 465318 \beta - 111676320) q^{42} + 107419706 q^{43} - 2116448 \beta q^{44} + (6104646 \beta - 37091520) q^{45} - 71311680 q^{46} - 9987608 \beta q^{47} + ( - 5803776 \beta + 17411328) q^{48} + 14535507 q^{49} + 1675705 \beta q^{50} + (345384 \beta + 82892160) q^{51} - 51574816 q^{52} + 7158798 \beta q^{53} + (4704237 \beta + 363741840) q^{54} - 531339840 q^{55} + 22886752 \beta q^{56} + ( - 8545158 \beta + 25635474) q^{57} - 85387680 q^{58} - 24192682 \beta q^{59} + (870048 \beta + 208811520) q^{60} + 1030793642 q^{61} - 29793118 \beta q^{62} + ( - 8375724 \beta - 992523294) q^{63} - 1175146496 q^{64} + 17983324 \beta q^{65} + (45113760 \beta - 135341280) q^{66} + 1876742474 q^{67} - 3888768 \beta q^{68} + ( - 2674188 \beta - 641805120) q^{69} + 1315298880 q^{70} + 100003596 \beta q^{71} + ( - 76480848 \beta + 464693760) q^{72} - 2846528494 q^{73} - 60811846 \beta q^{74} + (15081345 \beta - 45244035) q^{75} - 288636448 q^{76} - 119983108 \beta q^{77} + (4580658 \beta + 1099357920) q^{78} + 1488647618 q^{79} + 68355584 \beta q^{80} + (55978452 \beta + 3146662161) q^{81} - 4874667840 q^{82} + 47175562 \beta q^{83} + (47152224 \beta - 141456672) q^{84} - 976285440 q^{85} + 107419706 \beta q^{86} + ( - 3202038 \beta - 768489120) q^{87} + 6656785920 q^{88} - 224371428 \beta q^{89} + ( - 37091520 \beta - 4395345120) q^{90} - 2923817036 q^{91} + 30109376 \beta q^{92} + ( - 268138062 \beta + 804414186) q^{93} + 7191077760 q^{94} + 100642972 \beta q^{95} + ( - 19305216 \beta - 4633251840) q^{96} - 1592948926 q^{97} + 14535507 \beta q^{98} + (400948542 \beta - 2436143040) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{3} + 608 q^{4} - 12960 q^{6} + 34468 q^{7} - 115182 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 54 q^{3} + 608 q^{4} - 12960 q^{6} + 34468 q^{7} - 115182 q^{9} + 152640 q^{10} - 16416 q^{12} - 339308 q^{13} + 1373760 q^{15} - 1289728 q^{16} + 699840 q^{18} - 1898924 q^{19} - 930636 q^{21} + 10025280 q^{22} - 17210880 q^{24} + 3351410 q^{25} + 9408474 q^{27} + 10478272 q^{28} - 4121280 q^{30} - 59586236 q^{31} + 90227520 q^{33} + 18420480 q^{34} - 35015328 q^{36} - 121623692 q^{37} + 9161316 q^{39} + 202705920 q^{40} - 223352640 q^{42} + 214839412 q^{43} - 74183040 q^{45} - 142623360 q^{46} + 34822656 q^{48} + 29071014 q^{49} + 165784320 q^{51} - 103149632 q^{52} + 727483680 q^{54} - 1062679680 q^{55} + 51270948 q^{57} - 170775360 q^{58} + 417623040 q^{60} + 2061587284 q^{61} - 1985046588 q^{63} - 2350292992 q^{64} - 270682560 q^{66} + 3753484948 q^{67} - 1283610240 q^{69} + 2630597760 q^{70} + 929387520 q^{72} - 5693056988 q^{73} - 90488070 q^{75} - 577272896 q^{76} + 2198715840 q^{78} + 2977295236 q^{79} + 6293324322 q^{81} - 9749335680 q^{82} - 282913344 q^{84} - 1952570880 q^{85} - 1536978240 q^{87} + 13313571840 q^{88} - 8790690240 q^{90} - 5847634072 q^{91} + 1608828372 q^{93} + 14382155520 q^{94} - 9266503680 q^{96} - 3185897852 q^{97} - 4872286080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
2.23607i
2.23607i
26.8328i −27.0000 241.495i 304.000 2844.28i −6480.00 + 724.486i 17234.0 35634.0i −57591.0 + 13040.7i 76320.0
2.2 26.8328i −27.0000 + 241.495i 304.000 2844.28i −6480.00 724.486i 17234.0 35634.0i −57591.0 13040.7i 76320.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.11.b.a 2
3.b odd 2 1 inner 3.11.b.a 2
4.b odd 2 1 48.11.e.c 2
5.b even 2 1 75.11.c.d 2
5.c odd 4 2 75.11.d.b 4
8.b even 2 1 192.11.e.e 2
8.d odd 2 1 192.11.e.d 2
9.c even 3 2 81.11.d.d 4
9.d odd 6 2 81.11.d.d 4
12.b even 2 1 48.11.e.c 2
15.d odd 2 1 75.11.c.d 2
15.e even 4 2 75.11.d.b 4
24.f even 2 1 192.11.e.d 2
24.h odd 2 1 192.11.e.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.11.b.a 2 1.a even 1 1 trivial
3.11.b.a 2 3.b odd 2 1 inner
48.11.e.c 2 4.b odd 2 1
48.11.e.c 2 12.b even 2 1
75.11.c.d 2 5.b even 2 1
75.11.c.d 2 15.d odd 2 1
75.11.d.b 4 5.c odd 4 2
75.11.d.b 4 15.e even 4 2
81.11.d.d 4 9.c even 3 2
81.11.d.d 4 9.d odd 6 2
192.11.e.d 2 8.d odd 2 1
192.11.e.d 2 24.f even 2 1
192.11.e.e 2 8.b even 2 1
192.11.e.e 2 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 720 \) Copy content Toggle raw display
$3$ \( T^{2} + 54T + 59049 \) Copy content Toggle raw display
$5$ \( T^{2} + 8089920 \) Copy content Toggle raw display
$7$ \( (T - 17234)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 34897999680 \) Copy content Toggle raw display
$13$ \( (T + 169654)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 117817390080 \) Copy content Toggle raw display
$19$ \( (T + 949462)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 7062994033920 \) Copy content Toggle raw display
$29$ \( T^{2} + 10126466521920 \) Copy content Toggle raw display
$31$ \( (T + 29793118)^{2} \) Copy content Toggle raw display
$37$ \( (T + 60811846)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 33\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( (T - 107419706)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 71\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{2} + 36\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} + 42\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( (T - 1030793642)^{2} \) Copy content Toggle raw display
$67$ \( (T - 1876742474)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 72\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( (T + 2846528494)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1488647618)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{2} + 36\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( (T + 1592948926)^{2} \) Copy content Toggle raw display
show more
show less