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Properties

Label 675.5.d.a.674.1
Level $675$
Weight $5$
Character 675.674
Analytic conductor $69.775$
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] = [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,-6,0,-14,0,0,0,138,0,0,0,0,0,0,0,-190,828,0,608] 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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 674.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 675.674
Dual form 675.5.d.a.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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} -7.00000 q^{4} -19.0000i q^{7} +69.0000 q^{8} -123.000i q^{11} -302.000i q^{13} +57.0000i q^{14} -95.0000 q^{16} +414.000 q^{17} +304.000 q^{19} +369.000i q^{22} -300.000 q^{23} +906.000i q^{26} +133.000i q^{28} -678.000i q^{29} +239.000 q^{31} -819.000 q^{32} -1242.00 q^{34} +740.000i q^{37} -912.000 q^{38} -228.000i q^{41} +982.000i q^{43} +861.000i q^{44} +900.000 q^{46} +2166.00 q^{47} +2040.00 q^{49} +2114.00i q^{52} +1593.00 q^{53} -1311.00i q^{56} +2034.00i q^{58} -2922.00i q^{59} -316.000 q^{61} -717.000 q^{62} +3977.00 q^{64} +4622.00i q^{67} -2898.00 q^{68} -1818.00i q^{71} +3031.00i q^{73} -2220.00i q^{74} -2128.00 q^{76} -2337.00 q^{77} +10450.0 q^{79} +684.000i q^{82} -12633.0 q^{83} -2946.00i q^{86} -8487.00i q^{88} -7002.00i q^{89} -5738.00 q^{91} +2100.00 q^{92} -6498.00 q^{94} -6517.00i q^{97} -6120.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} - 14 q^{4} + 138 q^{8} - 190 q^{16} + 828 q^{17} + 608 q^{19} - 600 q^{23} + 478 q^{31} - 1638 q^{32} - 2484 q^{34} - 1824 q^{38} + 1800 q^{46} + 4332 q^{47} + 4080 q^{49} + 3186 q^{53} - 632 q^{61}+ \cdots - 12240 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −0.750000 −0.375000 − 0.927025i \(-0.622357\pi\)
−0.375000 + 0.927025i \(0.622357\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.437500
\(5\) 0 0
\(6\) 0 0
\(7\) − 19.0000i − 0.387755i −0.981026 − 0.193878i \(-0.937894\pi\)
0.981026 − 0.193878i \(-0.0621064\pi\)
\(8\) 69.0000 1.07812
\(9\) 0 0
\(10\) 0 0
\(11\) − 123.000i − 1.01653i −0.861201 − 0.508264i \(-0.830287\pi\)
0.861201 − 0.508264i \(-0.169713\pi\)
\(12\) 0 0
\(13\) − 302.000i − 1.78698i −0.449081 − 0.893491i \(-0.648248\pi\)
0.449081 − 0.893491i \(-0.351752\pi\)
\(14\) 57.0000i 0.290816i
\(15\) 0 0
\(16\) −95.0000 −0.371094
\(17\) 414.000 1.43253 0.716263 − 0.697830i \(-0.245851\pi\)
0.716263 + 0.697830i \(0.245851\pi\)
\(18\) 0 0
\(19\) 304.000 0.842105 0.421053 − 0.907036i \(-0.361661\pi\)
0.421053 + 0.907036i \(0.361661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 369.000i 0.762397i
\(23\) −300.000 −0.567108 −0.283554 − 0.958956i \(-0.591513\pi\)
−0.283554 + 0.958956i \(0.591513\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 906.000i 1.34024i
\(27\) 0 0
\(28\) 133.000i 0.169643i
\(29\) − 678.000i − 0.806183i −0.915160 − 0.403092i \(-0.867936\pi\)
0.915160 − 0.403092i \(-0.132064\pi\)
\(30\) 0 0
\(31\) 239.000 0.248699 0.124350 − 0.992238i \(-0.460316\pi\)
0.124350 + 0.992238i \(0.460316\pi\)
\(32\) −819.000 −0.799805
\(33\) 0 0
\(34\) −1242.00 −1.07439
\(35\) 0 0
\(36\) 0 0
\(37\) 740.000i 0.540541i 0.962784 + 0.270270i \(0.0871131\pi\)
−0.962784 + 0.270270i \(0.912887\pi\)
\(38\) −912.000 −0.631579
\(39\) 0 0
\(40\) 0 0
\(41\) − 228.000i − 0.135634i −0.997698 − 0.0678168i \(-0.978397\pi\)
0.997698 − 0.0678168i \(-0.0216033\pi\)
\(42\) 0 0
\(43\) 982.000i 0.531098i 0.964097 + 0.265549i \(0.0855532\pi\)
−0.964097 + 0.265549i \(0.914447\pi\)
\(44\) 861.000i 0.444731i
\(45\) 0 0
\(46\) 900.000 0.425331
\(47\) 2166.00 0.980534 0.490267 − 0.871572i \(-0.336899\pi\)
0.490267 + 0.871572i \(0.336899\pi\)
\(48\) 0 0
\(49\) 2040.00 0.849646
\(50\) 0 0
\(51\) 0 0
\(52\) 2114.00i 0.781805i
\(53\) 1593.00 0.567106 0.283553 − 0.958957i \(-0.408487\pi\)
0.283553 + 0.958957i \(0.408487\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 1311.00i − 0.418048i
\(57\) 0 0
\(58\) 2034.00i 0.604637i
\(59\) − 2922.00i − 0.839414i −0.907660 − 0.419707i \(-0.862133\pi\)
0.907660 − 0.419707i \(-0.137867\pi\)
\(60\) 0 0
\(61\) −316.000 −0.0849234 −0.0424617 − 0.999098i \(-0.513520\pi\)
−0.0424617 + 0.999098i \(0.513520\pi\)
\(62\) −717.000 −0.186524
\(63\) 0 0
\(64\) 3977.00 0.970947
\(65\) 0 0
\(66\) 0 0
\(67\) 4622.00i 1.02963i 0.857302 + 0.514814i \(0.172139\pi\)
−0.857302 + 0.514814i \(0.827861\pi\)
\(68\) −2898.00 −0.626730
\(69\) 0 0
\(70\) 0 0
\(71\) − 1818.00i − 0.360643i −0.983608 − 0.180321i \(-0.942286\pi\)
0.983608 − 0.180321i \(-0.0577138\pi\)
\(72\) 0 0
\(73\) 3031.00i 0.568775i 0.958709 + 0.284387i \(0.0917902\pi\)
−0.958709 + 0.284387i \(0.908210\pi\)
\(74\) − 2220.00i − 0.405405i
\(75\) 0 0
\(76\) −2128.00 −0.368421
\(77\) −2337.00 −0.394164
\(78\) 0 0
\(79\) 10450.0 1.67441 0.837206 − 0.546888i \(-0.184188\pi\)
0.837206 + 0.546888i \(0.184188\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 684.000i 0.101725i
\(83\) −12633.0 −1.83379 −0.916897 − 0.399125i \(-0.869314\pi\)
−0.916897 + 0.399125i \(0.869314\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 2946.00i − 0.398323i
\(87\) 0 0
\(88\) − 8487.00i − 1.09595i
\(89\) − 7002.00i − 0.883979i −0.897020 − 0.441990i \(-0.854273\pi\)
0.897020 − 0.441990i \(-0.145727\pi\)
\(90\) 0 0
\(91\) −5738.00 −0.692911
\(92\) 2100.00 0.248110
\(93\) 0 0
\(94\) −6498.00 −0.735401
\(95\) 0 0
\(96\) 0 0
\(97\) − 6517.00i − 0.692635i −0.938117 − 0.346317i \(-0.887432\pi\)
0.938117 − 0.346317i \(-0.112568\pi\)
\(98\) −6120.00 −0.637234
\(99\) 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.a.674.1 2
3.2 odd 2 675.5.d.d.674.1 2
5.2 odd 4 675.5.c.h.26.1 2
5.3 odd 4 27.5.b.c.26.2 yes 2
5.4 even 2 675.5.d.d.674.2 2
15.2 even 4 675.5.c.h.26.2 2
15.8 even 4 27.5.b.c.26.1 ✓ 2
15.14 odd 2 inner 675.5.d.a.674.2 2
20.3 even 4 432.5.e.e.161.2 2
45.13 odd 12 81.5.d.b.26.1 4
45.23 even 12 81.5.d.b.26.2 4
45.38 even 12 81.5.d.b.53.1 4
45.43 odd 12 81.5.d.b.53.2 4
60.23 odd 4 432.5.e.e.161.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.c.26.1 ✓ 2 15.8 even 4
27.5.b.c.26.2 yes 2 5.3 odd 4
81.5.d.b.26.1 4 45.13 odd 12
81.5.d.b.26.2 4 45.23 even 12
81.5.d.b.53.1 4 45.38 even 12
81.5.d.b.53.2 4 45.43 odd 12
432.5.e.e.161.1 2 60.23 odd 4
432.5.e.e.161.2 2 20.3 even 4
675.5.c.h.26.1 2 5.2 odd 4
675.5.c.h.26.2 2 15.2 even 4
675.5.d.a.674.1 2 1.1 even 1 trivial
675.5.d.a.674.2 2 15.14 odd 2 inner
675.5.d.d.674.1 2 3.2 odd 2
675.5.d.d.674.2 2 5.4 even 2