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Properties

Label 5625.2.a.z.1.3
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
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] = [5625,2,Mod(1,5625)] 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("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5625, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,14,0,0,-10,0,0,0,0,0,-10,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.9158511370\)
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: \( x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.23762\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.23762 q^{2} -0.468306 q^{4} +2.08634 q^{7} +3.05482 q^{8} -2.65287 q^{11} -0.149728 q^{13} -2.58209 q^{14} -2.84408 q^{16} +5.12810 q^{17} -3.08634 q^{19} +3.28323 q^{22} -2.76265 q^{23} +0.185305 q^{26} -0.977047 q^{28} +6.94530 q^{29} +1.52550 q^{31} -2.58976 q^{32} -6.34662 q^{34} -8.52550 q^{37} +3.81970 q^{38} -9.82722 q^{41} -3.43296 q^{43} +1.24236 q^{44} +3.41911 q^{46} +4.00501 q^{47} -2.64718 q^{49} +0.0701184 q^{52} +0.945455 q^{53} +6.37339 q^{56} -8.59562 q^{58} +8.18766 q^{59} -4.47450 q^{61} -1.88798 q^{62} +8.89328 q^{64} +3.34042 q^{67} -2.40152 q^{68} -12.4801 q^{71} -11.6018 q^{73} +10.5513 q^{74} +1.44535 q^{76} -5.53479 q^{77} +11.3696 q^{79} +12.1623 q^{82} -3.45676 q^{83} +4.24869 q^{86} -8.10403 q^{88} +14.8875 q^{89} -0.312383 q^{91} +1.29377 q^{92} -4.95666 q^{94} +0.0571908 q^{97} +3.27620 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73}+ \cdots - 40 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23762 −0.875127 −0.437563 − 0.899188i \(-0.644158\pi\)
−0.437563 + 0.899188i \(0.644158\pi\)
\(3\) 0 0
\(4\) −0.468306 −0.234153
\(5\) 0 0
\(6\) 0 0
\(7\) 2.08634 0.788563 0.394281 − 0.918990i \(-0.370994\pi\)
0.394281 + 0.918990i \(0.370994\pi\)
\(8\) 3.05482 1.08004
\(9\) 0 0
\(10\) 0 0
\(11\) −2.65287 −0.799870 −0.399935 − 0.916543i \(-0.630967\pi\)
−0.399935 + 0.916543i \(0.630967\pi\)
\(12\) 0 0
\(13\) −0.149728 −0.0415269 −0.0207635 − 0.999784i \(-0.506610\pi\)
−0.0207635 + 0.999784i \(0.506610\pi\)
\(14\) −2.58209 −0.690092
\(15\) 0 0
\(16\) −2.84408 −0.711019
\(17\) 5.12810 1.24375 0.621874 − 0.783118i \(-0.286372\pi\)
0.621874 + 0.783118i \(0.286372\pi\)
\(18\) 0 0
\(19\) −3.08634 −0.708055 −0.354028 − 0.935235i \(-0.615188\pi\)
−0.354028 + 0.935235i \(0.615188\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.28323 0.699988
\(23\) −2.76265 −0.576053 −0.288027 − 0.957622i \(-0.592999\pi\)
−0.288027 + 0.957622i \(0.592999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.185305 0.0363413
\(27\) 0 0
\(28\) −0.977047 −0.184644
\(29\) 6.94530 1.28971 0.644855 − 0.764305i \(-0.276918\pi\)
0.644855 + 0.764305i \(0.276918\pi\)
\(30\) 0 0
\(31\) 1.52550 0.273987 0.136994 − 0.990572i \(-0.456256\pi\)
0.136994 + 0.990572i \(0.456256\pi\)
\(32\) −2.58976 −0.457809
\(33\) 0 0
\(34\) −6.34662 −1.08844
\(35\) 0 0
\(36\) 0 0
\(37\) −8.52550 −1.40158 −0.700792 − 0.713366i \(-0.747170\pi\)
−0.700792 + 0.713366i \(0.747170\pi\)
\(38\) 3.81970 0.619638
\(39\) 0 0
\(40\) 0 0
\(41\) −9.82722 −1.53475 −0.767377 − 0.641196i \(-0.778438\pi\)
−0.767377 + 0.641196i \(0.778438\pi\)
\(42\) 0 0
\(43\) −3.43296 −0.523522 −0.261761 − 0.965133i \(-0.584303\pi\)
−0.261761 + 0.965133i \(0.584303\pi\)
\(44\) 1.24236 0.187292
\(45\) 0 0
\(46\) 3.41911 0.504120
\(47\) 4.00501 0.584191 0.292095 − 0.956389i \(-0.405648\pi\)
0.292095 + 0.956389i \(0.405648\pi\)
\(48\) 0 0
\(49\) −2.64718 −0.378169
\(50\) 0 0
\(51\) 0 0
\(52\) 0.0701184 0.00972367
\(53\) 0.945455 0.129868 0.0649341 − 0.997890i \(-0.479316\pi\)
0.0649341 + 0.997890i \(0.479316\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.37339 0.851679
\(57\) 0 0
\(58\) −8.59562 −1.12866
\(59\) 8.18766 1.06594 0.532971 − 0.846133i \(-0.321075\pi\)
0.532971 + 0.846133i \(0.321075\pi\)
\(60\) 0 0
\(61\) −4.47450 −0.572901 −0.286451 − 0.958095i \(-0.592475\pi\)
−0.286451 + 0.958095i \(0.592475\pi\)
\(62\) −1.88798 −0.239774
\(63\) 0 0
\(64\) 8.89328 1.11166
\(65\) 0 0
\(66\) 0 0
\(67\) 3.34042 0.408098 0.204049 − 0.978961i \(-0.434590\pi\)
0.204049 + 0.978961i \(0.434590\pi\)
\(68\) −2.40152 −0.291227
\(69\) 0 0
\(70\) 0 0
\(71\) −12.4801 −1.48111 −0.740557 − 0.671994i \(-0.765438\pi\)
−0.740557 + 0.671994i \(0.765438\pi\)
\(72\) 0 0
\(73\) −11.6018 −1.35789 −0.678945 − 0.734189i \(-0.737562\pi\)
−0.678945 + 0.734189i \(0.737562\pi\)
\(74\) 10.5513 1.22656
\(75\) 0 0
\(76\) 1.44535 0.165793
\(77\) −5.53479 −0.630748
\(78\) 0 0
\(79\) 11.3696 1.27918 0.639588 − 0.768717i \(-0.279105\pi\)
0.639588 + 0.768717i \(0.279105\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 12.1623 1.34310
\(83\) −3.45676 −0.379429 −0.189715 − 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.24869 0.458148
\(87\) 0 0
\(88\) −8.10403 −0.863892
\(89\) 14.8875 1.57807 0.789034 − 0.614349i \(-0.210581\pi\)
0.789034 + 0.614349i \(0.210581\pi\)
\(90\) 0 0
\(91\) −0.312383 −0.0327466
\(92\) 1.29377 0.134885
\(93\) 0 0
\(94\) −4.95666 −0.511241
\(95\) 0 0
\(96\) 0 0
\(97\) 0.0571908 0.00580685 0.00290342 − 0.999996i \(-0.499076\pi\)
0.00290342 + 0.999996i \(0.499076\pi\)
\(98\) 3.27620 0.330946
\(99\) 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 5625.2.a.z.1.3 ✓ 8
3.2 odd 2 inner 5625.2.a.z.1.6 yes 8
5.4 even 2 5625.2.a.bb.1.6 yes 8
15.14 odd 2 5625.2.a.bb.1.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.3 ✓ 8 1.1 even 1 trivial
5625.2.a.z.1.6 yes 8 3.2 odd 2 inner
5625.2.a.bb.1.3 yes 8 15.14 odd 2
5625.2.a.bb.1.6 yes 8 5.4 even 2