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Properties

Label 2304.2.d.d.1153.1
Level $2304$
Weight $2$
Character 2304.1153
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
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] = [2304,2,Mod(1153,2304)] 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("2304.1153"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1153.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1153
Dual form 2304.2.d.d.1153.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{7} -4.00000i q^{11} +6.00000i q^{13} -6.00000 q^{17} +4.00000 q^{23} +5.00000 q^{25} -4.00000i q^{29} +10.0000 q^{31} -2.00000i q^{37} -2.00000 q^{41} -8.00000i q^{43} +12.0000 q^{47} -3.00000 q^{49} -12.0000i q^{53} -4.00000i q^{59} +2.00000i q^{61} +4.00000i q^{67} -4.00000 q^{71} +10.0000 q^{73} +8.00000i q^{77} -6.00000 q^{79} -12.0000i q^{83} +2.00000 q^{89} -12.0000i q^{91} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} - 12 q^{17} + 8 q^{23} + 10 q^{25} + 20 q^{31} - 4 q^{41} + 24 q^{47} - 6 q^{49} - 8 q^{71} + 20 q^{73} - 12 q^{79} + 4 q^{89} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.00000i − 0.742781i −0.928477 − 0.371391i \(-0.878881\pi\)
0.928477 − 0.371391i \(-0.121119\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 − 0.609994i \(-0.791172\pi\)
0.792406 − 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 12.0000i − 1.64833i −0.566352 − 0.824163i \(-0.691646\pi\)
0.566352 − 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 − 0.260378i \(-0.916153\pi\)
0.965507 − 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000i 0.911685i
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) − 12.0000i − 1.25794i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 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 2304.2.d.d.1153.1 2
3.2 odd 2 768.2.d.b.385.1 2
4.3 odd 2 2304.2.d.m.1153.2 2
8.3 odd 2 2304.2.d.m.1153.1 2
8.5 even 2 inner 2304.2.d.d.1153.2 2
12.11 even 2 768.2.d.g.385.2 2
16.3 odd 4 1152.2.a.j.1.1 1
16.5 even 4 1152.2.a.l.1.1 1
16.11 odd 4 1152.2.a.i.1.1 1
16.13 even 4 1152.2.a.k.1.1 1
24.5 odd 2 768.2.d.b.385.2 2
24.11 even 2 768.2.d.g.385.1 2
48.5 odd 4 384.2.a.c.1.1 yes 1
48.11 even 4 384.2.a.f.1.1 yes 1
48.29 odd 4 384.2.a.g.1.1 yes 1
48.35 even 4 384.2.a.b.1.1 ✓ 1
240.29 odd 4 9600.2.a.h.1.1 1
240.59 even 4 9600.2.a.w.1.1 1
240.149 odd 4 9600.2.a.bh.1.1 1
240.179 even 4 9600.2.a.bw.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.a.b.1.1 ✓ 1 48.35 even 4
384.2.a.c.1.1 yes 1 48.5 odd 4
384.2.a.f.1.1 yes 1 48.11 even 4
384.2.a.g.1.1 yes 1 48.29 odd 4
768.2.d.b.385.1 2 3.2 odd 2
768.2.d.b.385.2 2 24.5 odd 2
768.2.d.g.385.1 2 24.11 even 2
768.2.d.g.385.2 2 12.11 even 2
1152.2.a.i.1.1 1 16.11 odd 4
1152.2.a.j.1.1 1 16.3 odd 4
1152.2.a.k.1.1 1 16.13 even 4
1152.2.a.l.1.1 1 16.5 even 4
2304.2.d.d.1153.1 2 1.1 even 1 trivial
2304.2.d.d.1153.2 2 8.5 even 2 inner
2304.2.d.m.1153.1 2 8.3 odd 2
2304.2.d.m.1153.2 2 4.3 odd 2
9600.2.a.h.1.1 1 240.29 odd 4
9600.2.a.w.1.1 1 240.59 even 4
9600.2.a.bh.1.1 1 240.149 odd 4
9600.2.a.bw.1.1 1 240.179 even 4