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Properties

Label 675.2.r.a.46.6
Level $675$
Weight $2$
Character 675.46
Analytic conductor $5.390$
Analytic rank $0$
Dimension $224$
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] = [675,2,Mod(46,675)] 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("675.46"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.r (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 46.6
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.15183 - 1.27924i) q^{2} +(-0.100677 + 0.957875i) q^{4} +(0.0644453 - 2.23514i) q^{5} +(-2.11497 + 3.66323i) q^{7} +(-1.44394 + 1.04909i) q^{8} +(-2.93350 + 2.49206i) q^{10} +(3.66270 + 4.06784i) q^{11} +(1.55472 - 1.72669i) q^{13} +(7.12222 - 1.51387i) q^{14} +(4.88941 + 1.03928i) q^{16} +(1.13466 - 0.824379i) q^{17} +(4.74872 - 3.45015i) q^{19} +(2.13449 + 0.286757i) q^{20} +(0.984923 - 9.37092i) q^{22} +(3.41812 - 0.726543i) q^{23} +(-4.99169 - 0.288088i) q^{25} -3.99961 q^{26} +(-3.29599 - 2.39468i) q^{28} +(3.94221 - 1.75518i) q^{29} +(-2.89216 - 1.28767i) q^{31} +(-2.51747 - 4.36039i) q^{32} +(-2.36151 - 0.501954i) q^{34} +(8.05154 + 4.96333i) q^{35} +(0.930443 + 2.86361i) q^{37} +(-9.88326 - 2.10075i) q^{38} +(2.25180 + 3.29502i) q^{40} +(-3.09385 + 3.43607i) q^{41} +(-2.17619 + 3.76927i) q^{43} +(-4.26523 + 3.09887i) q^{44} +(-4.86650 - 3.53572i) q^{46} +(2.62209 - 1.16743i) q^{47} +(-5.44619 - 9.43308i) q^{49} +(5.38104 + 6.71738i) q^{50} +(1.49743 + 1.66306i) q^{52} +(-4.61926 - 3.35609i) q^{53} +(9.32824 - 7.92450i) q^{55} +(-0.789153 - 7.50829i) q^{56} +(-6.78604 - 3.02134i) q^{58} +(7.75922 - 8.61749i) q^{59} +(7.25373 + 8.05609i) q^{61} +(1.68404 + 5.18293i) q^{62} +(0.411065 - 1.26513i) q^{64} +(-3.75920 - 3.58629i) q^{65} +(10.3055 + 4.58831i) q^{67} +(0.675418 + 1.16986i) q^{68} +(-2.92473 - 16.0167i) q^{70} +(5.34318 + 3.88205i) q^{71} +(1.20170 - 3.69845i) q^{73} +(2.59152 - 4.48864i) q^{74} +(2.82672 + 4.89603i) q^{76} +(-22.6480 + 4.81398i) q^{77} +(-2.62186 + 1.16733i) q^{79} +(2.63803 - 10.8615i) q^{80} +7.95913 q^{82} +(0.713599 + 6.78944i) q^{83} +(-1.76948 - 2.58925i) q^{85} +(7.32838 - 1.55770i) q^{86} +(-9.55625 - 2.03124i) q^{88} +(1.41471 - 4.35403i) q^{89} +(3.03709 + 9.34720i) q^{91} +(0.351813 + 3.34727i) q^{92} +(-4.51362 - 2.00959i) q^{94} +(-7.40552 - 10.8364i) q^{95} +(12.6316 - 5.62394i) q^{97} +(-5.79405 + 17.8322i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.15183 − 1.27924i −0.814466 − 0.904556i 0.182436 − 0.983218i \(-0.441602\pi\)
−0.996901 + 0.0786620i \(0.974935\pi\)
\(3\) 0 0
\(4\) −0.100677 + 0.957875i −0.0503383 + 0.478937i
\(5\) 0.0644453 − 2.23514i 0.0288208 − 0.999585i
\(6\) 0 0
\(7\) −2.11497 + 3.66323i −0.799383 + 1.38457i 0.120635 + 0.992697i \(0.461507\pi\)
−0.920018 + 0.391876i \(0.871826\pi\)
\(8\) −1.44394 + 1.04909i −0.510511 + 0.370908i
\(9\) 0 0
\(10\) −2.93350 + 2.49206i −0.927654 + 0.788057i
\(11\) 3.66270 + 4.06784i 1.10435 + 1.22650i 0.971921 + 0.235308i \(0.0756100\pi\)
0.132426 + 0.991193i \(0.457723\pi\)
\(12\) 0 0
\(13\) 1.55472 − 1.72669i 0.431201 − 0.478898i −0.487911 − 0.872893i \(-0.662241\pi\)
0.919112 + 0.393996i \(0.128908\pi\)
\(14\) 7.12222 − 1.51387i 1.90349 − 0.404600i
\(15\) 0 0
\(16\) 4.88941 + 1.03928i 1.22235 + 0.259819i
\(17\) 1.13466 − 0.824379i 0.275195 − 0.199941i −0.441624 − 0.897200i \(-0.645597\pi\)
0.716819 + 0.697259i \(0.245597\pi\)
\(18\) 0 0
\(19\) 4.74872 − 3.45015i 1.08943 − 0.791518i 0.110128 − 0.993917i \(-0.464874\pi\)
0.979303 + 0.202400i \(0.0648740\pi\)
\(20\) 2.13449 + 0.286757i 0.477288 + 0.0641208i
\(21\) 0 0
\(22\) 0.984923 − 9.37092i 0.209986 − 1.99789i
\(23\) 3.41812 − 0.726543i 0.712727 − 0.151495i 0.162741 − 0.986669i \(-0.447966\pi\)
0.549986 + 0.835174i \(0.314633\pi\)
\(24\) 0 0
\(25\) −4.99169 − 0.288088i −0.998339 − 0.0576177i
\(26\) −3.99961 −0.784388
\(27\) 0 0
\(28\) −3.29599 − 2.39468i −0.622884 − 0.452552i
\(29\) 3.94221 − 1.75518i 0.732050 − 0.325929i −0.00663753 − 0.999978i \(-0.502113\pi\)
0.738687 + 0.674049i \(0.235446\pi\)
\(30\) 0 0
\(31\) −2.89216 − 1.28767i −0.519448 − 0.231273i 0.130229 − 0.991484i \(-0.458429\pi\)
−0.649676 + 0.760211i \(0.725096\pi\)
\(32\) −2.51747 − 4.36039i −0.445030 − 0.770815i
\(33\) 0 0
\(34\) −2.36151 − 0.501954i −0.404995 − 0.0860844i
\(35\) 8.05154 + 4.96333i 1.36096 + 0.838956i
\(36\) 0 0
\(37\) 0.930443 + 2.86361i 0.152964 + 0.470775i 0.997949 − 0.0640154i \(-0.0203907\pi\)
−0.844985 + 0.534790i \(0.820391\pi\)
\(38\) −9.88326 − 2.10075i −1.60328 − 0.340787i
\(39\) 0 0
\(40\) 2.25180 + 3.29502i 0.356041 + 0.520989i
\(41\) −3.09385 + 3.43607i −0.483178 + 0.536624i −0.934606 − 0.355684i \(-0.884248\pi\)
0.451428 + 0.892308i \(0.350915\pi\)
\(42\) 0 0
\(43\) −2.17619 + 3.76927i −0.331866 + 0.574809i −0.982878 − 0.184259i \(-0.941011\pi\)
0.651012 + 0.759068i \(0.274345\pi\)
\(44\) −4.26523 + 3.09887i −0.643008 + 0.467173i
\(45\) 0 0
\(46\) −4.86650 − 3.53572i −0.717527 − 0.521314i
\(47\) 2.62209 − 1.16743i 0.382471 − 0.170287i −0.206489 − 0.978449i \(-0.566204\pi\)
0.588961 + 0.808162i \(0.299537\pi\)
\(48\) 0 0
\(49\) −5.44619 − 9.43308i −0.778027 − 1.34758i
\(50\) 5.38104 + 6.71738i 0.760994 + 0.949981i
\(51\) 0 0
\(52\) 1.49743 + 1.66306i 0.207656 + 0.230625i
\(53\) −4.61926 − 3.35609i −0.634504 − 0.460994i 0.223454 − 0.974715i \(-0.428267\pi\)
−0.857958 + 0.513720i \(0.828267\pi\)
\(54\) 0 0
\(55\) 9.32824 − 7.92450i 1.25782 − 1.06854i
\(56\) −0.789153 − 7.50829i −0.105455 − 1.00334i
\(57\) 0 0
\(58\) −6.78604 − 3.02134i −0.891051 − 0.396721i
\(59\) 7.75922 − 8.61749i 1.01016 − 1.12190i 0.0176411 − 0.999844i \(-0.494384\pi\)
0.992523 − 0.122057i \(-0.0389490\pi\)
\(60\) 0 0
\(61\) 7.25373 + 8.05609i 0.928745 + 1.03148i 0.999423 + 0.0339768i \(0.0108172\pi\)
−0.0706774 + 0.997499i \(0.522516\pi\)
\(62\) 1.68404 + 5.18293i 0.213873 + 0.658233i
\(63\) 0 0
\(64\) 0.411065 − 1.26513i 0.0513831 − 0.158141i
\(65\) −3.75920 − 3.58629i −0.466271 − 0.444824i
\(66\) 0 0
\(67\) 10.3055 + 4.58831i 1.25902 + 0.560551i 0.924264 − 0.381754i \(-0.124680\pi\)
0.334754 + 0.942305i \(0.391347\pi\)
\(68\) 0.675418 + 1.16986i 0.0819064 + 0.141866i
\(69\) 0 0
\(70\) −2.92473 − 16.0167i −0.349572 − 1.91436i
\(71\) 5.34318 + 3.88205i 0.634119 + 0.460714i 0.857825 − 0.513942i \(-0.171816\pi\)
−0.223706 + 0.974657i \(0.571816\pi\)
\(72\) 0 0
\(73\) 1.20170 − 3.69845i 0.140648 − 0.432871i −0.855777 − 0.517344i \(-0.826921\pi\)
0.996426 + 0.0844731i \(0.0269207\pi\)
\(74\) 2.59152 − 4.48864i 0.301258 − 0.521794i
\(75\) 0 0
\(76\) 2.82672 + 4.89603i 0.324247 + 0.561613i
\(77\) −22.6480 + 4.81398i −2.58098 + 0.548603i
\(78\) 0 0
\(79\) −2.62186 + 1.16733i −0.294983 + 0.131335i −0.548892 − 0.835893i \(-0.684950\pi\)
0.253909 + 0.967228i \(0.418284\pi\)
\(80\) 2.63803 − 10.8615i 0.294940 − 1.21436i
\(81\) 0 0
\(82\) 7.95913 0.878938
\(83\) 0.713599 + 6.78944i 0.0783276 + 0.745237i 0.961242 + 0.275704i \(0.0889111\pi\)
−0.882915 + 0.469533i \(0.844422\pi\)
\(84\) 0 0
\(85\) −1.76948 − 2.58925i −0.191927 − 0.280844i
\(86\) 7.32838 − 1.55770i 0.790240 − 0.167971i
\(87\) 0 0
\(88\) −9.55625 − 2.03124i −1.01870 − 0.216531i
\(89\) 1.41471 − 4.35403i 0.149959 − 0.461526i −0.847656 − 0.530545i \(-0.821987\pi\)
0.997615 + 0.0690197i \(0.0219871\pi\)
\(90\) 0 0
\(91\) 3.03709 + 9.34720i 0.318373 + 0.979852i
\(92\) 0.351813 + 3.34727i 0.0366790 + 0.348977i
\(93\) 0 0
\(94\) −4.51362 − 2.00959i −0.465544 − 0.207274i
\(95\) −7.40552 − 10.8364i −0.759791 − 1.11179i
\(96\) 0 0
\(97\) 12.6316 − 5.62394i 1.28254 − 0.571024i 0.351585 − 0.936156i \(-0.385643\pi\)
0.930956 + 0.365132i \(0.118976\pi\)
\(98\) −5.79405 + 17.8322i −0.585287 + 1.80133i
\(99\) 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 675.2.r.a.46.6 224
3.2 odd 2 225.2.q.a.196.23 yes 224
9.4 even 3 inner 675.2.r.a.496.23 224
9.5 odd 6 225.2.q.a.121.6 yes 224
25.6 even 5 inner 675.2.r.a.181.23 224
75.56 odd 10 225.2.q.a.106.6 yes 224
225.31 even 15 inner 675.2.r.a.631.6 224
225.131 odd 30 225.2.q.a.31.23 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.23 ✓ 224 225.131 odd 30
225.2.q.a.106.6 yes 224 75.56 odd 10
225.2.q.a.121.6 yes 224 9.5 odd 6
225.2.q.a.196.23 yes 224 3.2 odd 2
675.2.r.a.46.6 224 1.1 even 1 trivial
675.2.r.a.181.23 224 25.6 even 5 inner
675.2.r.a.496.23 224 9.4 even 3 inner
675.2.r.a.631.6 224 225.31 even 15 inner