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Properties

Label 7600.2.a.ch.1.1
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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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] = [7600,2,Mod(1,7600)] 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("7600.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-2,0,0,0,-2,0,6,0,-3,0,-1] 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: \(60.6863055362\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3800)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.26143\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.26143 q^{3} -4.07225 q^{7} +7.63693 q^{9} +0.786366 q^{11} +1.07974 q^{13} -1.90793 q^{17} +1.00000 q^{19} +13.2813 q^{21} +1.41383 q^{23} -15.1230 q^{27} -7.26439 q^{29} -2.22003 q^{31} -2.56468 q^{33} +9.14283 q^{37} -3.52151 q^{39} -6.11703 q^{41} -8.40634 q^{43} -3.56468 q^{47} +9.58319 q^{49} +6.22257 q^{51} -8.57472 q^{53} -3.26143 q^{57} +13.4043 q^{59} +12.7768 q^{61} -31.0994 q^{63} +5.10008 q^{67} -4.61112 q^{69} -1.65535 q^{71} +10.3302 q^{73} -3.20228 q^{77} +16.2256 q^{79} +26.4119 q^{81} -9.35104 q^{83} +23.6923 q^{87} +3.10419 q^{89} -4.39698 q^{91} +7.24046 q^{93} +4.55874 q^{97} +6.00542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 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 0
\(3\) −3.26143 −1.88299 −0.941494 − 0.337031i \(-0.890577\pi\)
−0.941494 + 0.337031i \(0.890577\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.07225 −1.53916 −0.769582 − 0.638548i \(-0.779536\pi\)
−0.769582 + 0.638548i \(0.779536\pi\)
\(8\) 0 0
\(9\) 7.63693 2.54564
\(10\) 0 0
\(11\) 0.786366 0.237098 0.118549 − 0.992948i \(-0.462176\pi\)
0.118549 + 0.992948i \(0.462176\pi\)
\(12\) 0 0
\(13\) 1.07974 0.299467 0.149733 − 0.988726i \(-0.452158\pi\)
0.149733 + 0.988726i \(0.452158\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.90793 −0.462741 −0.231370 − 0.972866i \(-0.574321\pi\)
−0.231370 + 0.972866i \(0.574321\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 13.2813 2.89823
\(22\) 0 0
\(23\) 1.41383 0.294805 0.147402 − 0.989077i \(-0.452909\pi\)
0.147402 + 0.989077i \(0.452909\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −15.1230 −2.91042
\(28\) 0 0
\(29\) −7.26439 −1.34896 −0.674482 − 0.738291i \(-0.735633\pi\)
−0.674482 + 0.738291i \(0.735633\pi\)
\(30\) 0 0
\(31\) −2.22003 −0.398728 −0.199364 − 0.979925i \(-0.563888\pi\)
−0.199364 + 0.979925i \(0.563888\pi\)
\(32\) 0 0
\(33\) −2.56468 −0.446453
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.14283 1.50307 0.751536 − 0.659692i \(-0.229313\pi\)
0.751536 + 0.659692i \(0.229313\pi\)
\(38\) 0 0
\(39\) −3.52151 −0.563893
\(40\) 0 0
\(41\) −6.11703 −0.955319 −0.477660 − 0.878545i \(-0.658515\pi\)
−0.477660 + 0.878545i \(0.658515\pi\)
\(42\) 0 0
\(43\) −8.40634 −1.28195 −0.640977 − 0.767560i \(-0.721471\pi\)
−0.640977 + 0.767560i \(0.721471\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.56468 −0.519962 −0.259981 − 0.965614i \(-0.583716\pi\)
−0.259981 + 0.965614i \(0.583716\pi\)
\(48\) 0 0
\(49\) 9.58319 1.36903
\(50\) 0 0
\(51\) 6.22257 0.871335
\(52\) 0 0
\(53\) −8.57472 −1.17783 −0.588914 − 0.808195i \(-0.700445\pi\)
−0.588914 + 0.808195i \(0.700445\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.26143 −0.431987
\(58\) 0 0
\(59\) 13.4043 1.74509 0.872543 − 0.488537i \(-0.162469\pi\)
0.872543 + 0.488537i \(0.162469\pi\)
\(60\) 0 0
\(61\) 12.7768 1.63590 0.817950 − 0.575289i \(-0.195110\pi\)
0.817950 + 0.575289i \(0.195110\pi\)
\(62\) 0 0
\(63\) −31.0994 −3.91816
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.10008 0.623073 0.311537 − 0.950234i \(-0.399156\pi\)
0.311537 + 0.950234i \(0.399156\pi\)
\(68\) 0 0
\(69\) −4.61112 −0.555114
\(70\) 0 0
\(71\) −1.65535 −0.196454 −0.0982268 − 0.995164i \(-0.531317\pi\)
−0.0982268 + 0.995164i \(0.531317\pi\)
\(72\) 0 0
\(73\) 10.3302 1.20906 0.604532 − 0.796581i \(-0.293360\pi\)
0.604532 + 0.796581i \(0.293360\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.20228 −0.364933
\(78\) 0 0
\(79\) 16.2256 1.82553 0.912764 − 0.408488i \(-0.133944\pi\)
0.912764 + 0.408488i \(0.133944\pi\)
\(80\) 0 0
\(81\) 26.4119 2.93465
\(82\) 0 0
\(83\) −9.35104 −1.02641 −0.513205 − 0.858266i \(-0.671542\pi\)
−0.513205 + 0.858266i \(0.671542\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 23.6923 2.54008
\(88\) 0 0
\(89\) 3.10419 0.329044 0.164522 − 0.986373i \(-0.447392\pi\)
0.164522 + 0.986373i \(0.447392\pi\)
\(90\) 0 0
\(91\) −4.39698 −0.460929
\(92\) 0 0
\(93\) 7.24046 0.750801
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.55874 0.462870 0.231435 − 0.972850i \(-0.425658\pi\)
0.231435 + 0.972850i \(0.425658\pi\)
\(98\) 0 0
\(99\) 6.00542 0.603567
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 7600.2.a.ch.1.1 6
4.3 odd 2 3800.2.a.bc.1.6 yes 6
5.4 even 2 7600.2.a.cl.1.6 6
20.3 even 4 3800.2.d.q.3649.12 12
20.7 even 4 3800.2.d.q.3649.1 12
20.19 odd 2 3800.2.a.ba.1.1 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.1 ✓ 6 20.19 odd 2
3800.2.a.bc.1.6 yes 6 4.3 odd 2
3800.2.d.q.3649.1 12 20.7 even 4
3800.2.d.q.3649.12 12 20.3 even 4
7600.2.a.ch.1.1 6 1.1 even 1 trivial
7600.2.a.cl.1.6 6 5.4 even 2