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Properties

Label 4600.2.e.w.4049.6
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
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] = [4600,2,Mod(4049,4600)] 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("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.278379347567616.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.6
Root \(0.794805i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.5

$q$-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)
 
\(f(q)\) \(=\) \(q+0.794805i q^{3} +2.47193i q^{7} +2.36829 q^{9} -2.29993 q^{11} +3.84022i q^{13} -7.74682i q^{17} +2.29993 q^{19} -1.96471 q^{21} +1.00000i q^{23} +4.26674i q^{27} -5.28380 q^{29} -6.40148 q^{31} -1.82800i q^{33} +8.56457i q^{37} -3.05223 q^{39} +4.27699 q^{41} +1.88954i q^{43} +12.3432i q^{47} +0.889540 q^{49} +6.15721 q^{51} +7.57482i q^{53} +1.82800i q^{57} -6.07180 q^{59} -0.635155 q^{61} +5.85425i q^{63} +11.1333i q^{67} -0.794805 q^{69} +8.58163 q^{71} -16.5849i q^{73} -5.68528i q^{77} -0.335225 q^{79} +3.71363 q^{81} -15.1937i q^{83} -4.19959i q^{87} -5.55735 q^{89} -9.49277 q^{91} -5.08792i q^{93} +6.42786i q^{97} -5.44689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0.794805i 0.458881i 0.973323 + 0.229440i \(0.0736896\pi\)
−0.973323 + 0.229440i \(0.926310\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.47193i 0.934303i 0.884177 + 0.467152i \(0.154720\pi\)
−0.884177 + 0.467152i \(0.845280\pi\)
\(8\) 0 0
\(9\) 2.36829 0.789428
\(10\) 0 0
\(11\) −2.29993 −0.693455 −0.346728 − 0.937966i \(-0.612707\pi\)
−0.346728 + 0.937966i \(0.612707\pi\)
\(12\) 0 0
\(13\) 3.84022i 1.06509i 0.846403 + 0.532543i \(0.178763\pi\)
−0.846403 + 0.532543i \(0.821237\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.74682i − 1.87888i −0.342713 − 0.939440i \(-0.611346\pi\)
0.342713 − 0.939440i \(-0.388654\pi\)
\(18\) 0 0
\(19\) 2.29993 0.527640 0.263820 − 0.964572i \(-0.415017\pi\)
0.263820 + 0.964572i \(0.415017\pi\)
\(20\) 0 0
\(21\) −1.96471 −0.428734
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.26674i 0.821134i
\(28\) 0 0
\(29\) −5.28380 −0.981177 −0.490589 − 0.871391i \(-0.663218\pi\)
−0.490589 + 0.871391i \(0.663218\pi\)
\(30\) 0 0
\(31\) −6.40148 −1.14974 −0.574870 − 0.818245i \(-0.694947\pi\)
−0.574870 + 0.818245i \(0.694947\pi\)
\(32\) 0 0
\(33\) − 1.82800i − 0.318213i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.56457i 1.40801i 0.710197 + 0.704003i \(0.248606\pi\)
−0.710197 + 0.704003i \(0.751394\pi\)
\(38\) 0 0
\(39\) −3.05223 −0.488747
\(40\) 0 0
\(41\) 4.27699 0.667954 0.333977 − 0.942581i \(-0.391609\pi\)
0.333977 + 0.942581i \(0.391609\pi\)
\(42\) 0 0
\(43\) 1.88954i 0.288152i 0.989567 + 0.144076i \(0.0460210\pi\)
−0.989567 + 0.144076i \(0.953979\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.3432i 1.80045i 0.435428 + 0.900223i \(0.356597\pi\)
−0.435428 + 0.900223i \(0.643403\pi\)
\(48\) 0 0
\(49\) 0.889540 0.127077
\(50\) 0 0
\(51\) 6.15721 0.862182
\(52\) 0 0
\(53\) 7.57482i 1.04048i 0.854020 + 0.520241i \(0.174158\pi\)
−0.854020 + 0.520241i \(0.825842\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.82800i 0.242124i
\(58\) 0 0
\(59\) −6.07180 −0.790480 −0.395240 − 0.918578i \(-0.629339\pi\)
−0.395240 + 0.918578i \(0.629339\pi\)
\(60\) 0 0
\(61\) −0.635155 −0.0813233 −0.0406616 − 0.999173i \(-0.512947\pi\)
−0.0406616 + 0.999173i \(0.512947\pi\)
\(62\) 0 0
\(63\) 5.85425i 0.737566i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.1333i 1.36015i 0.733141 + 0.680077i \(0.238054\pi\)
−0.733141 + 0.680077i \(0.761946\pi\)
\(68\) 0 0
\(69\) −0.794805 −0.0956833
\(70\) 0 0
\(71\) 8.58163 1.01845 0.509226 − 0.860633i \(-0.329932\pi\)
0.509226 + 0.860633i \(0.329932\pi\)
\(72\) 0 0
\(73\) − 16.5849i − 1.94112i −0.240859 − 0.970560i \(-0.577429\pi\)
0.240859 − 0.970560i \(-0.422571\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.68528i − 0.647897i
\(78\) 0 0
\(79\) −0.335225 −0.0377157 −0.0188579 − 0.999822i \(-0.506003\pi\)
−0.0188579 + 0.999822i \(0.506003\pi\)
\(80\) 0 0
\(81\) 3.71363 0.412626
\(82\) 0 0
\(83\) − 15.1937i − 1.66773i −0.551971 − 0.833863i \(-0.686124\pi\)
0.551971 − 0.833863i \(-0.313876\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.19959i − 0.450243i
\(88\) 0 0
\(89\) −5.55735 −0.589078 −0.294539 − 0.955639i \(-0.595166\pi\)
−0.294539 + 0.955639i \(0.595166\pi\)
\(90\) 0 0
\(91\) −9.49277 −0.995113
\(92\) 0 0
\(93\) − 5.08792i − 0.527593i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.42786i 0.652650i 0.945258 + 0.326325i \(0.105810\pi\)
−0.945258 + 0.326325i \(0.894190\pi\)
\(98\) 0 0
\(99\) −5.44689 −0.547433
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 4600.2.e.w.4049.6 10
5.2 odd 4 4600.2.a.bd.1.3 ✓ 5
5.3 odd 4 4600.2.a.bf.1.3 yes 5
5.4 even 2 inner 4600.2.e.w.4049.5 10
20.3 even 4 9200.2.a.ct.1.3 5
20.7 even 4 9200.2.a.cv.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.3 ✓ 5 5.2 odd 4
4600.2.a.bf.1.3 yes 5 5.3 odd 4
4600.2.e.w.4049.5 10 5.4 even 2 inner
4600.2.e.w.4049.6 10 1.1 even 1 trivial
9200.2.a.ct.1.3 5 20.3 even 4
9200.2.a.cv.1.3 5 20.7 even 4