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Properties

Label 675.2.r.a.46.19
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 46.19
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.727367 + 0.807823i) q^{2} +(0.0855417 - 0.813875i) q^{4} +(-0.934297 + 2.03152i) q^{5} +(1.73969 - 3.01324i) q^{7} +(2.47854 - 1.80077i) q^{8} +(-2.32069 + 0.722917i) q^{10} +(-2.35402 - 2.61440i) q^{11} +(1.45133 - 1.61187i) q^{13} +(3.69956 - 0.786365i) q^{14} +(1.65656 + 0.352114i) q^{16} +(1.54674 - 1.12377i) q^{17} +(-0.970198 + 0.704890i) q^{19} +(1.57349 + 0.934181i) q^{20} +(0.399739 - 3.80326i) q^{22} +(1.96627 - 0.417944i) q^{23} +(-3.25418 - 3.79609i) q^{25} +2.35776 q^{26} +(-2.30358 - 1.67365i) q^{28} +(3.26898 - 1.45544i) q^{29} +(5.79773 + 2.58132i) q^{31} +(-2.14316 - 3.71207i) q^{32} +(2.03286 + 0.432098i) q^{34} +(4.49607 + 6.34948i) q^{35} +(2.46324 + 7.58107i) q^{37} +(-1.27512 - 0.271034i) q^{38} +(1.34261 + 6.71767i) q^{40} +(-1.73464 + 1.92652i) q^{41} +(5.34941 - 9.26545i) q^{43} +(-2.32917 + 1.69224i) q^{44} +(1.76783 + 1.28440i) q^{46} +(-11.9942 + 5.34016i) q^{47} +(-2.55306 - 4.42203i) q^{49} +(0.699589 - 5.38996i) q^{50} +(-1.18771 - 1.31909i) q^{52} +(11.2674 + 8.18624i) q^{53} +(7.51058 - 2.33962i) q^{55} +(-1.11423 - 10.6012i) q^{56} +(3.55349 + 1.58212i) q^{58} +(3.64544 - 4.04867i) q^{59} +(0.353232 + 0.392304i) q^{61} +(2.13183 + 6.56111i) q^{62} +(2.48651 - 7.65270i) q^{64} +(1.91857 + 4.45438i) q^{65} +(-9.83507 - 4.37885i) q^{67} +(-0.782301 - 1.35499i) q^{68} +(-1.85897 + 8.25044i) q^{70} +(-6.38098 - 4.63605i) q^{71} +(-2.75054 + 8.46529i) q^{73} +(-4.33248 + 7.50408i) q^{74} +(0.490700 + 0.849917i) q^{76} +(-11.9731 + 2.54496i) q^{77} +(6.17366 - 2.74869i) q^{79} +(-2.26305 + 3.03637i) q^{80} -2.81801 q^{82} +(-0.372037 - 3.53969i) q^{83} +(0.837858 + 4.19218i) q^{85} +(11.3758 - 2.41801i) q^{86} +(-10.5425 - 2.24087i) q^{88} +(-2.71707 + 8.36227i) q^{89} +(-2.33207 - 7.17736i) q^{91} +(-0.171956 - 1.63605i) q^{92} +(-13.0381 - 5.80493i) q^{94} +(-0.525548 - 2.62956i) q^{95} +(-15.6914 + 6.98627i) q^{97} +(1.71520 - 5.27886i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.727367 + 0.807823i 0.514326 + 0.571217i 0.943233 − 0.332131i \(-0.107768\pi\)
−0.428907 + 0.903349i \(0.641101\pi\)
\(3\) 0 0
\(4\) 0.0855417 − 0.813875i 0.0427709 − 0.406938i
\(5\) −0.934297 + 2.03152i −0.417830 + 0.908525i
\(6\) 0 0
\(7\) 1.73969 − 3.01324i 0.657542 − 1.13890i −0.323708 − 0.946157i \(-0.604930\pi\)
0.981250 − 0.192739i \(-0.0617370\pi\)
\(8\) 2.47854 − 1.80077i 0.876298 − 0.636668i
\(9\) 0 0
\(10\) −2.32069 + 0.722917i −0.733866 + 0.228607i
\(11\) −2.35402 − 2.61440i −0.709764 − 0.788272i 0.275135 − 0.961406i \(-0.411278\pi\)
−0.984898 + 0.173133i \(0.944611\pi\)
\(12\) 0 0
\(13\) 1.45133 − 1.61187i 0.402527 − 0.447052i −0.507468 − 0.861671i \(-0.669418\pi\)
0.909995 + 0.414619i \(0.136085\pi\)
\(14\) 3.69956 − 0.786365i 0.988748 − 0.210165i
\(15\) 0 0
\(16\) 1.65656 + 0.352114i 0.414141 + 0.0880284i
\(17\) 1.54674 − 1.12377i 0.375140 − 0.272555i −0.384199 − 0.923250i \(-0.625522\pi\)
0.759339 + 0.650695i \(0.225522\pi\)
\(18\) 0 0
\(19\) −0.970198 + 0.704890i −0.222579 + 0.161713i −0.693487 − 0.720469i \(-0.743926\pi\)
0.470908 + 0.882182i \(0.343926\pi\)
\(20\) 1.57349 + 0.934181i 0.351842 + 0.208889i
\(21\) 0 0
\(22\) 0.399739 − 3.80326i 0.0852247 − 0.810859i
\(23\) 1.96627 − 0.417944i 0.409996 − 0.0871474i 0.00170377 − 0.999999i \(-0.499458\pi\)
0.408293 + 0.912851i \(0.366124\pi\)
\(24\) 0 0
\(25\) −3.25418 − 3.79609i −0.650836 − 0.759219i
\(26\) 2.35776 0.462394
\(27\) 0 0
\(28\) −2.30358 − 1.67365i −0.435336 − 0.316290i
\(29\) 3.26898 − 1.45544i 0.607035 − 0.270269i −0.0801144 − 0.996786i \(-0.525529\pi\)
0.687149 + 0.726516i \(0.258862\pi\)
\(30\) 0 0
\(31\) 5.79773 + 2.58132i 1.04130 + 0.463618i 0.854866 − 0.518849i \(-0.173639\pi\)
0.186437 + 0.982467i \(0.440306\pi\)
\(32\) −2.14316 − 3.71207i −0.378862 − 0.656207i
\(33\) 0 0
\(34\) 2.03286 + 0.432098i 0.348633 + 0.0741042i
\(35\) 4.49607 + 6.34948i 0.759975 + 1.07326i
\(36\) 0 0
\(37\) 2.46324 + 7.58107i 0.404954 + 1.24632i 0.920934 + 0.389719i \(0.127428\pi\)
−0.515980 + 0.856601i \(0.672572\pi\)
\(38\) −1.27512 − 0.271034i −0.206851 − 0.0439676i
\(39\) 0 0
\(40\) 1.34261 + 6.71767i 0.212285 + 1.06216i
\(41\) −1.73464 + 1.92652i −0.270906 + 0.300871i −0.863212 − 0.504842i \(-0.831551\pi\)
0.592306 + 0.805713i \(0.298218\pi\)
\(42\) 0 0
\(43\) 5.34941 − 9.26545i 0.815778 − 1.41297i −0.0929908 − 0.995667i \(-0.529643\pi\)
0.908768 − 0.417301i \(-0.137024\pi\)
\(44\) −2.32917 + 1.69224i −0.351135 + 0.255114i
\(45\) 0 0
\(46\) 1.76783 + 1.28440i 0.260652 + 0.189375i
\(47\) −11.9942 + 5.34016i −1.74953 + 0.778942i −0.757518 + 0.652814i \(0.773588\pi\)
−0.992014 + 0.126128i \(0.959745\pi\)
\(48\) 0 0
\(49\) −2.55306 − 4.42203i −0.364723 − 0.631718i
\(50\) 0.699589 − 5.38996i 0.0989369 − 0.762255i
\(51\) 0 0
\(52\) −1.18771 − 1.31909i −0.164706 − 0.182924i
\(53\) 11.2674 + 8.18624i 1.54770 + 1.12447i 0.945269 + 0.326293i \(0.105800\pi\)
0.602427 + 0.798174i \(0.294200\pi\)
\(54\) 0 0
\(55\) 7.51058 − 2.33962i 1.01273 − 0.315474i
\(56\) −1.11423 − 10.6012i −0.148896 − 1.41665i
\(57\) 0 0
\(58\) 3.55349 + 1.58212i 0.466597 + 0.207742i
\(59\) 3.64544 − 4.04867i 0.474596 − 0.527092i −0.457545 − 0.889186i \(-0.651271\pi\)
0.932142 + 0.362094i \(0.117938\pi\)
\(60\) 0 0
\(61\) 0.353232 + 0.392304i 0.0452268 + 0.0502294i 0.765334 − 0.643634i \(-0.222574\pi\)
−0.720107 + 0.693863i \(0.755907\pi\)
\(62\) 2.13183 + 6.56111i 0.270743 + 0.833262i
\(63\) 0 0
\(64\) 2.48651 − 7.65270i 0.310814 − 0.956588i
\(65\) 1.91857 + 4.45438i 0.237970 + 0.552498i
\(66\) 0 0
\(67\) −9.83507 − 4.37885i −1.20154 − 0.534962i −0.294360 − 0.955695i \(-0.595106\pi\)
−0.907185 + 0.420732i \(0.861773\pi\)
\(68\) −0.782301 − 1.35499i −0.0948680 − 0.164316i
\(69\) 0 0
\(70\) −1.85897 + 8.25044i −0.222189 + 0.986116i
\(71\) −6.38098 − 4.63605i −0.757283 − 0.550198i 0.140793 − 0.990039i \(-0.455035\pi\)
−0.898076 + 0.439841i \(0.855035\pi\)
\(72\) 0 0
\(73\) −2.75054 + 8.46529i −0.321926 + 0.990787i 0.650883 + 0.759178i \(0.274399\pi\)
−0.972809 + 0.231609i \(0.925601\pi\)
\(74\) −4.33248 + 7.50408i −0.503641 + 0.872332i
\(75\) 0 0
\(76\) 0.490700 + 0.849917i 0.0562871 + 0.0974922i
\(77\) −11.9731 + 2.54496i −1.36446 + 0.290025i
\(78\) 0 0
\(79\) 6.17366 − 2.74869i 0.694591 − 0.309252i −0.0289032 − 0.999582i \(-0.509201\pi\)
0.723494 + 0.690330i \(0.242535\pi\)
\(80\) −2.26305 + 3.03637i −0.253017 + 0.339477i
\(81\) 0 0
\(82\) −2.81801 −0.311197
\(83\) −0.372037 − 3.53969i −0.0408363 − 0.388532i −0.995782 − 0.0917511i \(-0.970754\pi\)
0.954946 − 0.296781i \(-0.0959131\pi\)
\(84\) 0 0
\(85\) 0.837858 + 4.19218i 0.0908784 + 0.454706i
\(86\) 11.3758 − 2.41801i 1.22669 − 0.260741i
\(87\) 0 0
\(88\) −10.5425 − 2.24087i −1.12383 − 0.238878i
\(89\) −2.71707 + 8.36227i −0.288008 + 0.886399i 0.697472 + 0.716612i \(0.254308\pi\)
−0.985481 + 0.169787i \(0.945692\pi\)
\(90\) 0 0
\(91\) −2.33207 − 7.17736i −0.244467 − 0.752392i
\(92\) −0.171956 − 1.63605i −0.0179277 − 0.170570i
\(93\) 0 0
\(94\) −13.0381 − 5.80493i −1.34478 − 0.598733i
\(95\) −0.525548 − 2.62956i −0.0539201 − 0.269787i
\(96\) 0 0
\(97\) −15.6914 + 6.98627i −1.59322 + 0.709348i −0.995712 − 0.0925124i \(-0.970510\pi\)
−0.597511 + 0.801861i \(0.703844\pi\)
\(98\) 1.71520 − 5.27886i 0.173262 − 0.533245i
\(99\) 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.46.19 224
3.2 odd 2 225.2.q.a.196.10 yes 224
9.4 even 3 inner 675.2.r.a.496.10 224
9.5 odd 6 225.2.q.a.121.19 yes 224
25.6 even 5 inner 675.2.r.a.181.10 224
75.56 odd 10 225.2.q.a.106.19 yes 224
225.31 even 15 inner 675.2.r.a.631.19 224
225.131 odd 30 225.2.q.a.31.10 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.10 ✓ 224 225.131 odd 30
225.2.q.a.106.19 yes 224 75.56 odd 10
225.2.q.a.121.19 yes 224 9.5 odd 6
225.2.q.a.196.10 yes 224 3.2 odd 2
675.2.r.a.46.19 224 1.1 even 1 trivial
675.2.r.a.181.10 224 25.6 even 5 inner
675.2.r.a.496.10 224 9.4 even 3 inner
675.2.r.a.631.19 224 225.31 even 15 inner