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Properties

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

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} -7.00000 q^{4} +19.0000i q^{7} -69.0000 q^{8} -123.000i q^{11} +302.000i q^{13} +57.0000i q^{14} -95.0000 q^{16} -414.000 q^{17} +304.000 q^{19} -369.000i q^{22} +300.000 q^{23} +906.000i q^{26} -133.000i q^{28} -678.000i q^{29} +239.000 q^{31} +819.000 q^{32} -1242.00 q^{34} -740.000i q^{37} +912.000 q^{38} -228.000i q^{41} -982.000i q^{43} +861.000i q^{44} +900.000 q^{46} -2166.00 q^{47} +2040.00 q^{49} -2114.00i q^{52} -1593.00 q^{53} -1311.00i q^{56} -2034.00i q^{58} -2922.00i q^{59} -316.000 q^{61} +717.000 q^{62} +3977.00 q^{64} -4622.00i q^{67} +2898.00 q^{68} -1818.00i q^{71} -3031.00i q^{73} -2220.00i q^{74} -2128.00 q^{76} +2337.00 q^{77} +10450.0 q^{79} -684.000i q^{82} +12633.0 q^{83} -2946.00i q^{86} +8487.00i q^{88} -7002.00i q^{89} -5738.00 q^{91} -2100.00 q^{92} -6498.00 q^{94} +6517.00i q^{97} +6120.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} - 14 q^{4} - 138 q^{8} - 190 q^{16} - 828 q^{17} + 608 q^{19} + 600 q^{23} + 478 q^{31} + 1638 q^{32} - 2484 q^{34} + 1824 q^{38} + 1800 q^{46} - 4332 q^{47} + 4080 q^{49} - 3186 q^{53} - 632 q^{61}+ \cdots + 12240 q^{98}+O(q^{100}) \) 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 \(-0.377643\pi\)
0.375000 + 0.927025i \(0.377643\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.437500
\(5\) 0 0
\(6\) 0 0
\(7\) 19.0000i 0.387755i 0.981026 + 0.193878i \(0.0621064\pi\)
−0.981026 + 0.193878i \(0.937894\pi\)
\(8\) −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 \(0.351752\pi\)
−0.449081 + 0.893491i \(0.648248\pi\)
\(14\) 57.0000i 0.290816i
\(15\) 0 0
\(16\) −95.0000 −0.371094
\(17\) −414.000 −1.43253 −0.716263 − 0.697830i \(-0.754149\pi\)
−0.716263 + 0.697830i \(0.754149\pi\)
\(18\) 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 \(-0.408487\pi\)
0.283554 + 0.958956i \(0.408487\pi\)
\(24\) 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 \(-0.912887\pi\)
0.962784 − 0.270270i \(-0.0871131\pi\)
\(38\) 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 \(-0.914447\pi\)
0.964097 − 0.265549i \(-0.0855532\pi\)
\(44\) 861.000i 0.444731i
\(45\) 0 0
\(46\) 900.000 0.425331
\(47\) −2166.00 −0.980534 −0.490267 − 0.871572i \(-0.663101\pi\)
−0.490267 + 0.871572i \(0.663101\pi\)
\(48\) 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 \(-0.591513\pi\)
−0.283553 + 0.958957i \(0.591513\pi\)
\(54\) 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 \(-0.827861\pi\)
0.857302 − 0.514814i \(-0.172139\pi\)
\(68\) 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 \(-0.908210\pi\)
0.958709 − 0.284387i \(-0.0917902\pi\)
\(74\) − 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 \(-0.130686\pi\)
0.916897 + 0.399125i \(0.130686\pi\)
\(84\) 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 \(0.112568\pi\)
−0.938117 + 0.346317i \(0.887432\pi\)
\(98\) 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.d.674.2 2
3.2 odd 2 675.5.d.a.674.2 2
5.2 odd 4 27.5.b.c.26.2 yes 2
5.3 odd 4 675.5.c.h.26.1 2
5.4 even 2 675.5.d.a.674.1 2
15.2 even 4 27.5.b.c.26.1 ✓ 2
15.8 even 4 675.5.c.h.26.2 2
15.14 odd 2 inner 675.5.d.d.674.1 2
20.7 even 4 432.5.e.e.161.2 2
45.2 even 12 81.5.d.b.53.1 4
45.7 odd 12 81.5.d.b.53.2 4
45.22 odd 12 81.5.d.b.26.1 4
45.32 even 12 81.5.d.b.26.2 4
60.47 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 15.2 even 4
27.5.b.c.26.2 yes 2 5.2 odd 4
81.5.d.b.26.1 4 45.22 odd 12
81.5.d.b.26.2 4 45.32 even 12
81.5.d.b.53.1 4 45.2 even 12
81.5.d.b.53.2 4 45.7 odd 12
432.5.e.e.161.1 2 60.47 odd 4
432.5.e.e.161.2 2 20.7 even 4
675.5.c.h.26.1 2 5.3 odd 4
675.5.c.h.26.2 2 15.8 even 4
675.5.d.a.674.1 2 5.4 even 2
675.5.d.a.674.2 2 3.2 odd 2
675.5.d.d.674.1 2 15.14 odd 2 inner
675.5.d.d.674.2 2 1.1 even 1 trivial