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Properties

Label 675.5.d.c.674.1
Level $675$
Weight $5$
Character 675.674
Analytic conductor $69.775$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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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] = [675,5,Mod(674,675)] 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("675.674"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 675.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-32,0,0,0,0,0,0,0,0,0,0,0,512,0,0,1202] 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: \(69.7747250816\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 674.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 675.674
Dual form 675.5.d.c.674.2

$q$-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)
 
\(f(q)\) \(=\) \(q-16.0000 q^{4} -71.0000i q^{7} -337.000i q^{13} +256.000 q^{16} +601.000 q^{19} +1136.00i q^{28} +194.000 q^{31} +529.000i q^{37} -3214.00i q^{43} -2640.00 q^{49} +5392.00i q^{52} +7199.00 q^{61} -4096.00 q^{64} -2903.00i q^{67} -1249.00i q^{73} -9616.00 q^{76} -4679.00 q^{79} -23927.0 q^{91} -9071.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 512 q^{16} + 1202 q^{19} + 388 q^{31} - 5280 q^{49} + 14398 q^{61} - 8192 q^{64} - 19232 q^{76} - 9358 q^{79} - 47854 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(-1\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −16.0000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) − 71.0000i − 1.44898i −0.689286 − 0.724490i \(-0.742075\pi\)
0.689286 − 0.724490i \(-0.257925\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 337.000i − 1.99408i −0.0768662 − 0.997041i \(-0.524491\pi\)
0.0768662 − 0.997041i \(-0.475509\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 601.000 1.66482 0.832410 − 0.554160i \(-0.186961\pi\)
0.832410 + 0.554160i \(0.186961\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 1136.00i 1.44898i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 194.000 0.201873 0.100937 − 0.994893i \(-0.467816\pi\)
0.100937 + 0.994893i \(0.467816\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 529.000i 0.386413i 0.981158 + 0.193207i \(0.0618888\pi\)
−0.981158 + 0.193207i \(0.938111\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 3214.00i − 1.73824i −0.494604 − 0.869118i \(-0.664687\pi\)
0.494604 − 0.869118i \(-0.335313\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −2640.00 −1.09954
\(50\) 0 0
\(51\) 0 0
\(52\) 5392.00i 1.99408i
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 7199.00 1.93469 0.967347 − 0.253454i \(-0.0815666\pi\)
0.967347 + 0.253454i \(0.0815666\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −4096.00 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 2903.00i − 0.646692i −0.946281 − 0.323346i \(-0.895192\pi\)
0.946281 − 0.323346i \(-0.104808\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 1249.00i − 0.234378i −0.993110 − 0.117189i \(-0.962612\pi\)
0.993110 − 0.117189i \(-0.0373883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −9616.00 −1.66482
\(77\) 0 0
\(78\) 0 0
\(79\) −4679.00 −0.749720 −0.374860 − 0.927082i \(-0.622309\pi\)
−0.374860 + 0.927082i \(0.622309\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −23927.0 −2.88939
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 9071.00i − 0.964077i −0.876150 − 0.482038i \(-0.839897\pi\)
0.876150 − 0.482038i \(-0.160103\pi\)
\(98\) 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 675.5.d.c.674.1 2
3.2 odd 2 CM 675.5.d.c.674.1 2
5.2 odd 4 27.5.b.a.26.1 ✓ 1
5.3 odd 4 675.5.c.b.26.1 1
5.4 even 2 inner 675.5.d.c.674.2 2
15.2 even 4 27.5.b.a.26.1 ✓ 1
15.8 even 4 675.5.c.b.26.1 1
15.14 odd 2 inner 675.5.d.c.674.2 2
20.7 even 4 432.5.e.a.161.1 1
45.2 even 12 81.5.d.a.53.1 2
45.7 odd 12 81.5.d.a.53.1 2
45.22 odd 12 81.5.d.a.26.1 2
45.32 even 12 81.5.d.a.26.1 2
60.47 odd 4 432.5.e.a.161.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.a.26.1 ✓ 1 5.2 odd 4
27.5.b.a.26.1 ✓ 1 15.2 even 4
81.5.d.a.26.1 2 45.22 odd 12
81.5.d.a.26.1 2 45.32 even 12
81.5.d.a.53.1 2 45.2 even 12
81.5.d.a.53.1 2 45.7 odd 12
432.5.e.a.161.1 1 20.7 even 4
432.5.e.a.161.1 1 60.47 odd 4
675.5.c.b.26.1 1 5.3 odd 4
675.5.c.b.26.1 1 15.8 even 4
675.5.d.c.674.1 2 1.1 even 1 trivial
675.5.d.c.674.1 2 3.2 odd 2 CM
675.5.d.c.674.2 2 5.4 even 2 inner
675.5.d.c.674.2 2 15.14 odd 2 inner