↑

Properties

Label 1125.4.a.q.1.2
Level $1125$
Weight $4$
Character 1125.1
Self dual yes
Analytic conductor $66.377$
Analytic rank $1$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1125,4,Mod(1,1125)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1125.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1125 = 3^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1125.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,54,0,0,0,0,0,0,-170] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.3771487565\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 75x^{10} + 2040x^{8} - 24475x^{6} + 126325x^{4} - 223000x^{2} + 122000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.87111\) of defining polynomial
Character \(\chi\) \(=\) 1125.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.87111 q^{2} +15.7277 q^{4} +10.1198 q^{7} -37.6425 q^{8} -23.1354 q^{11} +83.8762 q^{13} -49.2948 q^{14} +57.5390 q^{16} -35.8405 q^{17} -49.6884 q^{19} +112.695 q^{22} +6.79916 q^{23} -408.570 q^{26} +159.162 q^{28} +133.378 q^{29} -200.312 q^{31} +20.8610 q^{32} +174.583 q^{34} +199.563 q^{37} +242.038 q^{38} -432.178 q^{41} -171.181 q^{43} -363.867 q^{44} -33.1195 q^{46} -385.466 q^{47} -240.589 q^{49} +1319.18 q^{52} +162.487 q^{53} -380.935 q^{56} -649.697 q^{58} -484.210 q^{59} +693.456 q^{61} +975.742 q^{62} -561.928 q^{64} +387.845 q^{67} -563.688 q^{68} -545.697 q^{71} -657.942 q^{73} -972.094 q^{74} -781.484 q^{76} -234.127 q^{77} +559.373 q^{79} +2105.19 q^{82} -1360.57 q^{83} +833.843 q^{86} +870.875 q^{88} +751.021 q^{89} +848.812 q^{91} +106.935 q^{92} +1877.64 q^{94} +1138.82 q^{97} +1171.94 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 54 q^{4} - 170 q^{11} - 290 q^{14} + 258 q^{16} + 152 q^{19} - 360 q^{26} - 630 q^{29} - 454 q^{31} - 780 q^{34} - 1260 q^{41} - 3340 q^{44} - 640 q^{46} - 16 q^{49} - 4410 q^{56} - 2970 q^{59} - 256 q^{61}+ \cdots + 2200 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.87111 −1.72220 −0.861099 − 0.508438i \(-0.830223\pi\)
−0.861099 + 0.508438i \(0.830223\pi\)
\(3\) 0 0
\(4\) 15.7277 1.96596
\(5\) 0 0
\(6\) 0 0
\(7\) 10.1198 0.546419 0.273210 − 0.961955i \(-0.411915\pi\)
0.273210 + 0.961955i \(0.411915\pi\)
\(8\) −37.6425 −1.66358
\(9\) 0 0
\(10\) 0 0
\(11\) −23.1354 −0.634145 −0.317072 − 0.948401i \(-0.602700\pi\)
−0.317072 + 0.948401i \(0.602700\pi\)
\(12\) 0 0
\(13\) 83.8762 1.78947 0.894733 − 0.446601i \(-0.147366\pi\)
0.894733 + 0.446601i \(0.147366\pi\)
\(14\) −49.2948 −0.941042
\(15\) 0 0
\(16\) 57.5390 0.899047
\(17\) −35.8405 −0.511329 −0.255664 − 0.966766i \(-0.582294\pi\)
−0.255664 + 0.966766i \(0.582294\pi\)
\(18\) 0 0
\(19\) −49.6884 −0.599963 −0.299981 − 0.953945i \(-0.596981\pi\)
−0.299981 + 0.953945i \(0.596981\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 112.695 1.09212
\(23\) 6.79916 0.0616402 0.0308201 − 0.999525i \(-0.490188\pi\)
0.0308201 + 0.999525i \(0.490188\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −408.570 −3.08181
\(27\) 0 0
\(28\) 159.162 1.07424
\(29\) 133.378 0.854055 0.427027 − 0.904239i \(-0.359561\pi\)
0.427027 + 0.904239i \(0.359561\pi\)
\(30\) 0 0
\(31\) −200.312 −1.16055 −0.580276 − 0.814420i \(-0.697055\pi\)
−0.580276 + 0.814420i \(0.697055\pi\)
\(32\) 20.8610 0.115242
\(33\) 0 0
\(34\) 174.583 0.880609
\(35\) 0 0
\(36\) 0 0
\(37\) 199.563 0.886702 0.443351 − 0.896348i \(-0.353789\pi\)
0.443351 + 0.896348i \(0.353789\pi\)
\(38\) 242.038 1.03325
\(39\) 0 0
\(40\) 0 0
\(41\) −432.178 −1.64622 −0.823109 − 0.567884i \(-0.807762\pi\)
−0.823109 + 0.567884i \(0.807762\pi\)
\(42\) 0 0
\(43\) −171.181 −0.607091 −0.303545 − 0.952817i \(-0.598170\pi\)
−0.303545 + 0.952817i \(0.598170\pi\)
\(44\) −363.867 −1.24671
\(45\) 0 0
\(46\) −33.1195 −0.106157
\(47\) −385.466 −1.19630 −0.598148 − 0.801386i \(-0.704097\pi\)
−0.598148 + 0.801386i \(0.704097\pi\)
\(48\) 0 0
\(49\) −240.589 −0.701426
\(50\) 0 0
\(51\) 0 0
\(52\) 1319.18 3.51802
\(53\) 162.487 0.421119 0.210559 − 0.977581i \(-0.432471\pi\)
0.210559 + 0.977581i \(0.432471\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −380.935 −0.909011
\(57\) 0 0
\(58\) −649.697 −1.47085
\(59\) −484.210 −1.06845 −0.534227 − 0.845341i \(-0.679397\pi\)
−0.534227 + 0.845341i \(0.679397\pi\)
\(60\) 0 0
\(61\) 693.456 1.45554 0.727770 − 0.685821i \(-0.240557\pi\)
0.727770 + 0.685821i \(0.240557\pi\)
\(62\) 975.742 1.99870
\(63\) 0 0
\(64\) −561.928 −1.09752
\(65\) 0 0
\(66\) 0 0
\(67\) 387.845 0.707207 0.353603 − 0.935395i \(-0.384956\pi\)
0.353603 + 0.935395i \(0.384956\pi\)
\(68\) −563.688 −1.00525
\(69\) 0 0
\(70\) 0 0
\(71\) −545.697 −0.912146 −0.456073 − 0.889942i \(-0.650744\pi\)
−0.456073 + 0.889942i \(0.650744\pi\)
\(72\) 0 0
\(73\) −657.942 −1.05488 −0.527440 − 0.849592i \(-0.676848\pi\)
−0.527440 + 0.849592i \(0.676848\pi\)
\(74\) −972.094 −1.52708
\(75\) 0 0
\(76\) −781.484 −1.17951
\(77\) −234.127 −0.346509
\(78\) 0 0
\(79\) 559.373 0.796638 0.398319 − 0.917247i \(-0.369594\pi\)
0.398319 + 0.917247i \(0.369594\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2105.19 2.83511
\(83\) −1360.57 −1.79930 −0.899650 − 0.436612i \(-0.856178\pi\)
−0.899650 + 0.436612i \(0.856178\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 833.843 1.04553
\(87\) 0 0
\(88\) 870.875 1.05495
\(89\) 751.021 0.894473 0.447236 − 0.894416i \(-0.352408\pi\)
0.447236 + 0.894416i \(0.352408\pi\)
\(90\) 0 0
\(91\) 848.812 0.977799
\(92\) 106.935 0.121182
\(93\) 0 0
\(94\) 1877.64 2.06026
\(95\) 0 0
\(96\) 0 0
\(97\) 1138.82 1.19206 0.596029 − 0.802963i \(-0.296744\pi\)
0.596029 + 0.802963i \(0.296744\pi\)
\(98\) 1171.94 1.20799
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1125.4.a.q.1.2 ✓ 12
3.2 odd 2 1125.4.a.r.1.11 yes 12
5.4 even 2 inner 1125.4.a.q.1.11 yes 12
15.14 odd 2 1125.4.a.r.1.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1125.4.a.q.1.2 ✓ 12 1.1 even 1 trivial
1125.4.a.q.1.11 yes 12 5.4 even 2 inner
1125.4.a.r.1.2 yes 12 15.14 odd 2
1125.4.a.r.1.11 yes 12 3.2 odd 2