Properties

Label 1323.4.a.bl.1.3
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1323,4,Mod(1,1323)] 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("1323.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,44,0,0,0,0,0,-132,0,0,-336] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.49657\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.49657 q^{2} -1.76715 q^{4} -14.5857 q^{5} +24.3844 q^{8} +36.4142 q^{10} -0.0794330 q^{11} -86.9394 q^{13} -46.7400 q^{16} -93.0605 q^{17} +147.953 q^{19} +25.7751 q^{20} +0.198310 q^{22} +154.273 q^{23} +87.7423 q^{25} +217.050 q^{26} +205.403 q^{29} -271.281 q^{31} -78.3853 q^{32} +232.332 q^{34} +48.3171 q^{37} -369.375 q^{38} -355.663 q^{40} -52.3857 q^{41} +48.0508 q^{43} +0.140370 q^{44} -385.152 q^{46} +266.115 q^{47} -219.055 q^{50} +153.635 q^{52} +11.8224 q^{53} +1.15859 q^{55} -512.802 q^{58} +542.677 q^{59} -84.4131 q^{61} +677.270 q^{62} +569.614 q^{64} +1268.07 q^{65} -51.0586 q^{67} +164.452 q^{68} +495.085 q^{71} -71.1990 q^{73} -120.627 q^{74} -261.455 q^{76} +378.650 q^{79} +681.735 q^{80} +130.785 q^{82} +486.792 q^{83} +1357.35 q^{85} -119.962 q^{86} -1.93692 q^{88} +1168.36 q^{89} -272.623 q^{92} -664.374 q^{94} -2158.00 q^{95} -1310.92 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 44 q^{4} - 132 q^{10} - 336 q^{13} + 204 q^{16} + 288 q^{19} + 484 q^{22} + 152 q^{25} - 120 q^{31} - 1008 q^{34} + 592 q^{37} - 1620 q^{40} - 1872 q^{43} - 1644 q^{46} - 2400 q^{52} - 1344 q^{55}+ \cdots - 5712 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.49657 −0.882670 −0.441335 − 0.897342i \(-0.645495\pi\)
−0.441335 + 0.897342i \(0.645495\pi\)
\(3\) 0 0
\(4\) −1.76715 −0.220894
\(5\) −14.5857 −1.30458 −0.652292 − 0.757968i \(-0.726192\pi\)
−0.652292 + 0.757968i \(0.726192\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 24.3844 1.07765
\(9\) 0 0
\(10\) 36.4142 1.15152
\(11\) −0.0794330 −0.00217727 −0.00108863 − 0.999999i \(-0.500347\pi\)
−0.00108863 + 0.999999i \(0.500347\pi\)
\(12\) 0 0
\(13\) −86.9394 −1.85482 −0.927409 − 0.374048i \(-0.877970\pi\)
−0.927409 + 0.374048i \(0.877970\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −46.7400 −0.730312
\(17\) −93.0605 −1.32768 −0.663838 − 0.747877i \(-0.731074\pi\)
−0.663838 + 0.747877i \(0.731074\pi\)
\(18\) 0 0
\(19\) 147.953 1.78646 0.893230 − 0.449599i \(-0.148433\pi\)
0.893230 + 0.449599i \(0.148433\pi\)
\(20\) 25.7751 0.288175
\(21\) 0 0
\(22\) 0.198310 0.00192181
\(23\) 154.273 1.39861 0.699306 − 0.714823i \(-0.253493\pi\)
0.699306 + 0.714823i \(0.253493\pi\)
\(24\) 0 0
\(25\) 87.7423 0.701938
\(26\) 217.050 1.63719
\(27\) 0 0
\(28\) 0 0
\(29\) 205.403 1.31525 0.657626 − 0.753344i \(-0.271561\pi\)
0.657626 + 0.753344i \(0.271561\pi\)
\(30\) 0 0
\(31\) −271.281 −1.57172 −0.785862 − 0.618402i \(-0.787780\pi\)
−0.785862 + 0.618402i \(0.787780\pi\)
\(32\) −78.3853 −0.433022
\(33\) 0 0
\(34\) 232.332 1.17190
\(35\) 0 0
\(36\) 0 0
\(37\) 48.3171 0.214683 0.107342 − 0.994222i \(-0.465766\pi\)
0.107342 + 0.994222i \(0.465766\pi\)
\(38\) −369.375 −1.57685
\(39\) 0 0
\(40\) −355.663 −1.40588
\(41\) −52.3857 −0.199543 −0.0997717 − 0.995010i \(-0.531811\pi\)
−0.0997717 + 0.995010i \(0.531811\pi\)
\(42\) 0 0
\(43\) 48.0508 0.170411 0.0852056 − 0.996363i \(-0.472845\pi\)
0.0852056 + 0.996363i \(0.472845\pi\)
\(44\) 0.140370 0.000480945 0
\(45\) 0 0
\(46\) −385.152 −1.23451
\(47\) 266.115 0.825891 0.412945 − 0.910756i \(-0.364500\pi\)
0.412945 + 0.910756i \(0.364500\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −219.055 −0.619580
\(51\) 0 0
\(52\) 153.635 0.409718
\(53\) 11.8224 0.0306403 0.0153202 − 0.999883i \(-0.495123\pi\)
0.0153202 + 0.999883i \(0.495123\pi\)
\(54\) 0 0
\(55\) 1.15859 0.00284043
\(56\) 0 0
\(57\) 0 0
\(58\) −512.802 −1.16093
\(59\) 542.677 1.19747 0.598733 − 0.800948i \(-0.295671\pi\)
0.598733 + 0.800948i \(0.295671\pi\)
\(60\) 0 0
\(61\) −84.4131 −0.177180 −0.0885901 − 0.996068i \(-0.528236\pi\)
−0.0885901 + 0.996068i \(0.528236\pi\)
\(62\) 677.270 1.38731
\(63\) 0 0
\(64\) 569.614 1.11253
\(65\) 1268.07 2.41977
\(66\) 0 0
\(67\) −51.0586 −0.0931015 −0.0465507 − 0.998916i \(-0.514823\pi\)
−0.0465507 + 0.998916i \(0.514823\pi\)
\(68\) 164.452 0.293275
\(69\) 0 0
\(70\) 0 0
\(71\) 495.085 0.827547 0.413773 − 0.910380i \(-0.364211\pi\)
0.413773 + 0.910380i \(0.364211\pi\)
\(72\) 0 0
\(73\) −71.1990 −0.114154 −0.0570768 − 0.998370i \(-0.518178\pi\)
−0.0570768 + 0.998370i \(0.518178\pi\)
\(74\) −120.627 −0.189494
\(75\) 0 0
\(76\) −261.455 −0.394618
\(77\) 0 0
\(78\) 0 0
\(79\) 378.650 0.539259 0.269629 − 0.962964i \(-0.413099\pi\)
0.269629 + 0.962964i \(0.413099\pi\)
\(80\) 681.735 0.952753
\(81\) 0 0
\(82\) 130.785 0.176131
\(83\) 486.792 0.643763 0.321882 − 0.946780i \(-0.395685\pi\)
0.321882 + 0.946780i \(0.395685\pi\)
\(84\) 0 0
\(85\) 1357.35 1.73206
\(86\) −119.962 −0.150417
\(87\) 0 0
\(88\) −1.93692 −0.00234633
\(89\) 1168.36 1.39152 0.695762 − 0.718273i \(-0.255067\pi\)
0.695762 + 0.718273i \(0.255067\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −272.623 −0.308945
\(93\) 0 0
\(94\) −664.374 −0.728989
\(95\) −2158.00 −2.33059
\(96\) 0 0
\(97\) −1310.92 −1.37221 −0.686104 − 0.727504i \(-0.740680\pi\)
−0.686104 + 0.727504i \(0.740680\pi\)
\(98\) 0 0
\(99\) 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 1323.4.a.bl.1.3 ✓ 8
3.2 odd 2 inner 1323.4.a.bl.1.6 yes 8
7.6 odd 2 1323.4.a.bm.1.3 yes 8
21.20 even 2 1323.4.a.bm.1.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bl.1.3 ✓ 8 1.1 even 1 trivial
1323.4.a.bl.1.6 yes 8 3.2 odd 2 inner
1323.4.a.bm.1.3 yes 8 7.6 odd 2
1323.4.a.bm.1.6 yes 8 21.20 even 2