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 = [16,0,0,20,0,0,20,0,0,0,0,0,20] 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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 26x^{14} + 271x^{12} - 1454x^{10} + 4306x^{8} - 7062x^{6} + 6123x^{4} - 2486x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1665)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(0.747627\) 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)
 
\(f(q)\) \(=\) \(q+0.747627 q^{2} -1.44105 q^{4} -3.07627 q^{7} -2.57262 q^{8} -2.70718 q^{11} +4.25952 q^{13} -2.29990 q^{14} +0.958745 q^{16} -3.18208 q^{17} -4.77146 q^{19} -2.02396 q^{22} -2.20232 q^{23} +3.18453 q^{26} +4.43307 q^{28} -9.00471 q^{29} +8.95460 q^{31} +5.86203 q^{32} -2.37901 q^{34} -1.00000 q^{37} -3.56727 q^{38} +5.38222 q^{41} -6.38988 q^{43} +3.90119 q^{44} -1.64651 q^{46} -8.04729 q^{47} +2.46342 q^{49} -6.13820 q^{52} -11.0048 q^{53} +7.91408 q^{56} -6.73216 q^{58} -3.94184 q^{59} -15.0209 q^{61} +6.69470 q^{62} +2.46512 q^{64} +8.51155 q^{67} +4.58555 q^{68} -5.25460 q^{71} +8.04281 q^{73} -0.747627 q^{74} +6.87593 q^{76} +8.32800 q^{77} -2.93453 q^{79} +4.02389 q^{82} +7.70175 q^{83} -4.77725 q^{86} +6.96455 q^{88} +9.91984 q^{89} -13.1034 q^{91} +3.17366 q^{92} -6.01637 q^{94} -7.58278 q^{97} +1.84172 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 20 q^{7} + 20 q^{13} + 20 q^{16} - 8 q^{19} + 12 q^{22} + 72 q^{28} + 8 q^{31} + 12 q^{34} - 16 q^{37} + 24 q^{43} - 4 q^{46} + 40 q^{49} + 52 q^{52} + 64 q^{58} - 8 q^{61} + 40 q^{64}+ \cdots + 28 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.747627 0.528652 0.264326 − 0.964433i \(-0.414851\pi\)
0.264326 + 0.964433i \(0.414851\pi\)
\(3\) 0 0
\(4\) −1.44105 −0.720527
\(5\) 0 0
\(6\) 0 0
\(7\) −3.07627 −1.16272 −0.581360 − 0.813647i \(-0.697479\pi\)
−0.581360 + 0.813647i \(0.697479\pi\)
\(8\) −2.57262 −0.909560
\(9\) 0 0
\(10\) 0 0
\(11\) −2.70718 −0.816245 −0.408122 − 0.912927i \(-0.633816\pi\)
−0.408122 + 0.912927i \(0.633816\pi\)
\(12\) 0 0
\(13\) 4.25952 1.18138 0.590690 − 0.806899i \(-0.298856\pi\)
0.590690 + 0.806899i \(0.298856\pi\)
\(14\) −2.29990 −0.614674
\(15\) 0 0
\(16\) 0.958745 0.239686
\(17\) −3.18208 −0.771768 −0.385884 − 0.922547i \(-0.626103\pi\)
−0.385884 + 0.922547i \(0.626103\pi\)
\(18\) 0 0
\(19\) −4.77146 −1.09465 −0.547324 − 0.836921i \(-0.684353\pi\)
−0.547324 + 0.836921i \(0.684353\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.02396 −0.431509
\(23\) −2.20232 −0.459214 −0.229607 − 0.973283i \(-0.573744\pi\)
−0.229607 + 0.973283i \(0.573744\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.18453 0.624538
\(27\) 0 0
\(28\) 4.43307 0.837771
\(29\) −9.00471 −1.67213 −0.836066 − 0.548629i \(-0.815150\pi\)
−0.836066 + 0.548629i \(0.815150\pi\)
\(30\) 0 0
\(31\) 8.95460 1.60829 0.804147 − 0.594430i \(-0.202622\pi\)
0.804147 + 0.594430i \(0.202622\pi\)
\(32\) 5.86203 1.03627
\(33\) 0 0
\(34\) −2.37901 −0.407997
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −3.56727 −0.578688
\(39\) 0 0
\(40\) 0 0
\(41\) 5.38222 0.840562 0.420281 − 0.907394i \(-0.361932\pi\)
0.420281 + 0.907394i \(0.361932\pi\)
\(42\) 0 0
\(43\) −6.38988 −0.974448 −0.487224 − 0.873277i \(-0.661990\pi\)
−0.487224 + 0.873277i \(0.661990\pi\)
\(44\) 3.90119 0.588126
\(45\) 0 0
\(46\) −1.64651 −0.242765
\(47\) −8.04729 −1.17382 −0.586909 − 0.809653i \(-0.699655\pi\)
−0.586909 + 0.809653i \(0.699655\pi\)
\(48\) 0 0
\(49\) 2.46342 0.351917
\(50\) 0 0
\(51\) 0 0
\(52\) −6.13820 −0.851216
\(53\) −11.0048 −1.51163 −0.755814 − 0.654786i \(-0.772759\pi\)
−0.755814 + 0.654786i \(0.772759\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.91408 1.05756
\(57\) 0 0
\(58\) −6.73216 −0.883976
\(59\) −3.94184 −0.513184 −0.256592 − 0.966520i \(-0.582600\pi\)
−0.256592 + 0.966520i \(0.582600\pi\)
\(60\) 0 0
\(61\) −15.0209 −1.92323 −0.961616 − 0.274399i \(-0.911521\pi\)
−0.961616 + 0.274399i \(0.911521\pi\)
\(62\) 6.69470 0.850228
\(63\) 0 0
\(64\) 2.46512 0.308140
\(65\) 0 0
\(66\) 0 0
\(67\) 8.51155 1.03985 0.519925 − 0.854212i \(-0.325960\pi\)
0.519925 + 0.854212i \(0.325960\pi\)
\(68\) 4.58555 0.556080
\(69\) 0 0
\(70\) 0 0
\(71\) −5.25460 −0.623607 −0.311803 − 0.950147i \(-0.600933\pi\)
−0.311803 + 0.950147i \(0.600933\pi\)
\(72\) 0 0
\(73\) 8.04281 0.941340 0.470670 − 0.882309i \(-0.344012\pi\)
0.470670 + 0.882309i \(0.344012\pi\)
\(74\) −0.747627 −0.0869099
\(75\) 0 0
\(76\) 6.87593 0.788723
\(77\) 8.32800 0.949064
\(78\) 0 0
\(79\) −2.93453 −0.330160 −0.165080 − 0.986280i \(-0.552788\pi\)
−0.165080 + 0.986280i \(0.552788\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.02389 0.444365
\(83\) 7.70175 0.845377 0.422689 − 0.906275i \(-0.361086\pi\)
0.422689 + 0.906275i \(0.361086\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.77725 −0.515144
\(87\) 0 0
\(88\) 6.96455 0.742424
\(89\) 9.91984 1.05150 0.525751 − 0.850639i \(-0.323784\pi\)
0.525751 + 0.850639i \(0.323784\pi\)
\(90\) 0 0
\(91\) −13.1034 −1.37361
\(92\) 3.17366 0.330876
\(93\) 0 0
\(94\) −6.01637 −0.620541
\(95\) 0 0
\(96\) 0 0
\(97\) −7.58278 −0.769915 −0.384957 − 0.922934i \(-0.625784\pi\)
−0.384957 + 0.922934i \(0.625784\pi\)
\(98\) 1.84172 0.186042
\(99\) 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.cx.1.10 16
3.2 odd 2 inner 8325.2.a.cx.1.7 16
5.2 odd 4 1665.2.c.g.334.20 yes 32
5.3 odd 4 1665.2.c.g.334.14 yes 32
5.4 even 2 8325.2.a.cw.1.7 16
15.2 even 4 1665.2.c.g.334.13 ✓ 32
15.8 even 4 1665.2.c.g.334.19 yes 32
15.14 odd 2 8325.2.a.cw.1.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1665.2.c.g.334.13 ✓ 32 15.2 even 4
1665.2.c.g.334.14 yes 32 5.3 odd 4
1665.2.c.g.334.19 yes 32 15.8 even 4
1665.2.c.g.334.20 yes 32 5.2 odd 4
8325.2.a.cw.1.7 16 5.4 even 2
8325.2.a.cw.1.10 16 15.14 odd 2
8325.2.a.cx.1.7 16 3.2 odd 2 inner
8325.2.a.cx.1.10 16 1.1 even 1 trivial