Properties

Label 8325.2.a.co.1.2
Level $8325$
Weight $2$
Character 8325.1
Self dual yes
Analytic conductor $66.475$
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] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,12,0,0,-6,0,0,0,0,0,2] 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: \(66.4754596827\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.77658083584.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 41x^{4} - 39x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.16541\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.18361 q^{2} +2.76814 q^{4} -4.31293 q^{7} -1.67731 q^{8} -4.00813 q^{11} +4.18662 q^{13} +9.41775 q^{14} -1.87368 q^{16} -3.22601 q^{17} -7.97553 q^{19} +8.75218 q^{22} +7.08693 q^{23} -9.14193 q^{26} -11.9388 q^{28} +0.506293 q^{29} -3.00000 q^{31} +7.44602 q^{32} +7.04434 q^{34} -1.00000 q^{37} +17.4154 q^{38} +1.46544 q^{41} +10.8993 q^{43} -11.0951 q^{44} -15.4751 q^{46} -3.28997 q^{47} +11.6014 q^{49} +11.5891 q^{52} +13.3426 q^{53} +7.23414 q^{56} -1.10554 q^{58} +3.32481 q^{59} +2.88702 q^{61} +6.55082 q^{62} -12.5118 q^{64} +0.164120 q^{67} -8.93005 q^{68} -0.883970 q^{71} -6.13965 q^{73} +2.18361 q^{74} -22.0774 q^{76} +17.2868 q^{77} -11.5374 q^{79} -3.19995 q^{82} -1.12190 q^{83} -23.7997 q^{86} +6.72290 q^{88} +10.7701 q^{89} -18.0566 q^{91} +19.6176 q^{92} +7.18399 q^{94} -3.46002 q^{97} -25.3329 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} - 6 q^{7} + 2 q^{13} - 12 q^{16} - 18 q^{19} + 16 q^{22} - 10 q^{28} - 24 q^{31} - 22 q^{34} - 8 q^{37} + 32 q^{43} - 18 q^{46} - 10 q^{49} - 4 q^{52} - 16 q^{58} - 26 q^{61} - 34 q^{64}+ \cdots + 2 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18361 −1.54404 −0.772022 − 0.635596i \(-0.780754\pi\)
−0.772022 + 0.635596i \(0.780754\pi\)
\(3\) 0 0
\(4\) 2.76814 1.38407
\(5\) 0 0
\(6\) 0 0
\(7\) −4.31293 −1.63014 −0.815068 − 0.579366i \(-0.803300\pi\)
−0.815068 + 0.579366i \(0.803300\pi\)
\(8\) −1.67731 −0.593020
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00813 −1.20850 −0.604249 − 0.796796i \(-0.706527\pi\)
−0.604249 + 0.796796i \(0.706527\pi\)
\(12\) 0 0
\(13\) 4.18662 1.16116 0.580579 − 0.814204i \(-0.302826\pi\)
0.580579 + 0.814204i \(0.302826\pi\)
\(14\) 9.41775 2.51700
\(15\) 0 0
\(16\) −1.87368 −0.468421
\(17\) −3.22601 −0.782423 −0.391212 − 0.920301i \(-0.627944\pi\)
−0.391212 + 0.920301i \(0.627944\pi\)
\(18\) 0 0
\(19\) −7.97553 −1.82971 −0.914856 − 0.403781i \(-0.867696\pi\)
−0.914856 + 0.403781i \(0.867696\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8.75218 1.86597
\(23\) 7.08693 1.47773 0.738864 − 0.673855i \(-0.235363\pi\)
0.738864 + 0.673855i \(0.235363\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −9.14193 −1.79288
\(27\) 0 0
\(28\) −11.9388 −2.25622
\(29\) 0.506293 0.0940162 0.0470081 − 0.998895i \(-0.485031\pi\)
0.0470081 + 0.998895i \(0.485031\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 − 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 7.44602 1.31628
\(33\) 0 0
\(34\) 7.04434 1.20810
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 17.4154 2.82515
\(39\) 0 0
\(40\) 0 0
\(41\) 1.46544 0.228864 0.114432 − 0.993431i \(-0.463495\pi\)
0.114432 + 0.993431i \(0.463495\pi\)
\(42\) 0 0
\(43\) 10.8993 1.66212 0.831062 − 0.556180i \(-0.187734\pi\)
0.831062 + 0.556180i \(0.187734\pi\)
\(44\) −11.0951 −1.67264
\(45\) 0 0
\(46\) −15.4751 −2.28168
\(47\) −3.28997 −0.479891 −0.239945 − 0.970786i \(-0.577130\pi\)
−0.239945 + 0.970786i \(0.577130\pi\)
\(48\) 0 0
\(49\) 11.6014 1.65734
\(50\) 0 0
\(51\) 0 0
\(52\) 11.5891 1.60712
\(53\) 13.3426 1.83275 0.916375 − 0.400320i \(-0.131101\pi\)
0.916375 + 0.400320i \(0.131101\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.23414 0.966703
\(57\) 0 0
\(58\) −1.10554 −0.145165
\(59\) 3.32481 0.432853 0.216427 − 0.976299i \(-0.430560\pi\)
0.216427 + 0.976299i \(0.430560\pi\)
\(60\) 0 0
\(61\) 2.88702 0.369644 0.184822 − 0.982772i \(-0.440829\pi\)
0.184822 + 0.982772i \(0.440829\pi\)
\(62\) 6.55082 0.831955
\(63\) 0 0
\(64\) −12.5118 −1.56398
\(65\) 0 0
\(66\) 0 0
\(67\) 0.164120 0.0200504 0.0100252 − 0.999950i \(-0.496809\pi\)
0.0100252 + 0.999950i \(0.496809\pi\)
\(68\) −8.93005 −1.08293
\(69\) 0 0
\(70\) 0 0
\(71\) −0.883970 −0.104908 −0.0524540 − 0.998623i \(-0.516704\pi\)
−0.0524540 + 0.998623i \(0.516704\pi\)
\(72\) 0 0
\(73\) −6.13965 −0.718591 −0.359296 − 0.933224i \(-0.616983\pi\)
−0.359296 + 0.933224i \(0.616983\pi\)
\(74\) 2.18361 0.253839
\(75\) 0 0
\(76\) −22.0774 −2.53245
\(77\) 17.2868 1.97001
\(78\) 0 0
\(79\) −11.5374 −1.29805 −0.649027 − 0.760765i \(-0.724824\pi\)
−0.649027 + 0.760765i \(0.724824\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.19995 −0.353375
\(83\) −1.12190 −0.123144 −0.0615721 − 0.998103i \(-0.519611\pi\)
−0.0615721 + 0.998103i \(0.519611\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −23.7997 −2.56639
\(87\) 0 0
\(88\) 6.72290 0.716663
\(89\) 10.7701 1.14163 0.570815 − 0.821079i \(-0.306627\pi\)
0.570815 + 0.821079i \(0.306627\pi\)
\(90\) 0 0
\(91\) −18.0566 −1.89285
\(92\) 19.6176 2.04528
\(93\) 0 0
\(94\) 7.18399 0.740972
\(95\) 0 0
\(96\) 0 0
\(97\) −3.46002 −0.351312 −0.175656 − 0.984452i \(-0.556205\pi\)
−0.175656 + 0.984452i \(0.556205\pi\)
\(98\) −25.3329 −2.55901
\(99\) 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 8325.2.a.co.1.2 ✓ 8
3.2 odd 2 inner 8325.2.a.co.1.7 yes 8
5.4 even 2 8325.2.a.cp.1.7 yes 8
15.14 odd 2 8325.2.a.cp.1.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8325.2.a.co.1.2 ✓ 8 1.1 even 1 trivial
8325.2.a.co.1.7 yes 8 3.2 odd 2 inner
8325.2.a.cp.1.2 yes 8 15.14 odd 2
8325.2.a.cp.1.7 yes 8 5.4 even 2