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Properties

Label 675.2.r.a.46.13
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 46.13
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.13

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.349998 - 0.388712i) q^{2} +(0.180458 - 1.71695i) q^{4} +(0.130882 - 2.23223i) q^{5} +(-1.68864 + 2.92481i) q^{7} +(-1.57689 + 1.14568i) q^{8} +(-0.913505 + 0.730402i) q^{10} +(-2.76572 - 3.07165i) q^{11} +(-3.39970 + 3.77575i) q^{13} +(1.72793 - 0.367283i) q^{14} +(-2.38011 - 0.505907i) q^{16} +(-0.0810977 + 0.0589209i) q^{17} +(2.55513 - 1.85641i) q^{19} +(-3.80901 - 0.627543i) q^{20} +(-0.225989 + 2.15014i) q^{22} +(-1.92791 + 0.409791i) q^{23} +(-4.96574 - 0.584319i) q^{25} +2.65757 q^{26} +(4.71702 + 3.42711i) q^{28} +(-6.19799 + 2.75952i) q^{29} +(2.22457 + 0.990444i) q^{31} +(2.58553 + 4.47827i) q^{32} +(0.0512873 + 0.0109015i) q^{34} +(6.30785 + 4.15225i) q^{35} +(0.930291 + 2.86314i) q^{37} +(-1.61590 - 0.343471i) q^{38} +(2.35104 + 3.66994i) q^{40} +(3.88818 - 4.31827i) q^{41} +(5.61908 - 9.73253i) q^{43} +(-5.77296 + 4.19430i) q^{44} +(0.834057 + 0.605978i) q^{46} +(-8.50718 + 3.78764i) q^{47} +(-2.20301 - 3.81573i) q^{49} +(1.51087 + 2.13475i) q^{50} +(5.86925 + 6.51846i) q^{52} +(-1.84804 - 1.34268i) q^{53} +(-7.21862 + 5.77172i) q^{55} +(-0.688092 - 6.54676i) q^{56} +(3.24195 + 1.44341i) q^{58} +(-5.66566 + 6.29235i) q^{59} +(2.32441 + 2.58152i) q^{61} +(-0.393599 - 1.21137i) q^{62} +(-0.668023 + 2.05596i) q^{64} +(7.98339 + 8.08310i) q^{65} +(-12.3948 - 5.51853i) q^{67} +(0.0865293 + 0.149873i) q^{68} +(-0.593706 - 3.90522i) q^{70} +(-6.75154 - 4.90528i) q^{71} +(3.28345 - 10.1054i) q^{73} +(0.787338 - 1.36371i) q^{74} +(-2.72627 - 4.72203i) q^{76} +(13.6543 - 2.90231i) q^{77} +(-13.1297 + 5.84572i) q^{79} +(-1.44082 + 5.24674i) q^{80} -3.03942 q^{82} +(-0.648356 - 6.16870i) q^{83} +(0.120911 + 0.188741i) q^{85} +(-5.74982 + 1.22216i) q^{86} +(7.88038 + 1.67503i) q^{88} +(2.14728 - 6.60864i) q^{89} +(-5.30248 - 16.3193i) q^{91} +(0.355681 + 3.38408i) q^{92} +(4.44980 + 1.98118i) q^{94} +(-3.80953 - 5.94663i) q^{95} +(-1.59492 + 0.710105i) q^{97} +(-0.712171 + 2.19184i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.349998 − 0.388712i −0.247486 − 0.274861i 0.606584 − 0.795020i \(-0.292539\pi\)
−0.854070 + 0.520159i \(0.825873\pi\)
\(3\) 0 0
\(4\) 0.180458 − 1.71695i 0.0902292 − 0.858473i
\(5\) 0.130882 − 2.23223i 0.0585323 − 0.998286i
\(6\) 0 0
\(7\) −1.68864 + 2.92481i −0.638246 + 1.10547i 0.347571 + 0.937654i \(0.387006\pi\)
−0.985817 + 0.167821i \(0.946327\pi\)
\(8\) −1.57689 + 1.14568i −0.557516 + 0.405059i
\(9\) 0 0
\(10\) −0.913505 + 0.730402i −0.288876 + 0.230973i
\(11\) −2.76572 − 3.07165i −0.833897 − 0.926137i 0.164284 − 0.986413i \(-0.447469\pi\)
−0.998182 + 0.0602761i \(0.980802\pi\)
\(12\) 0 0
\(13\) −3.39970 + 3.77575i −0.942906 + 1.04720i 0.0559039 + 0.998436i \(0.482196\pi\)
−0.998810 + 0.0487673i \(0.984471\pi\)
\(14\) 1.72793 − 0.367283i 0.461809 − 0.0981605i
\(15\) 0 0
\(16\) −2.38011 − 0.505907i −0.595027 − 0.126477i
\(17\) −0.0810977 + 0.0589209i −0.0196691 + 0.0142904i −0.597576 − 0.801812i \(-0.703870\pi\)
0.577907 + 0.816102i \(0.303870\pi\)
\(18\) 0 0
\(19\) 2.55513 − 1.85641i 0.586188 − 0.425890i −0.254762 − 0.967004i \(-0.581997\pi\)
0.840950 + 0.541113i \(0.181997\pi\)
\(20\) −3.80901 − 0.627543i −0.851720 − 0.140323i
\(21\) 0 0
\(22\) −0.225989 + 2.15014i −0.0481810 + 0.458412i
\(23\) −1.92791 + 0.409791i −0.401998 + 0.0854473i −0.404474 − 0.914550i \(-0.632545\pi\)
0.00247582 + 0.999997i \(0.499212\pi\)
\(24\) 0 0
\(25\) −4.96574 − 0.584319i −0.993148 − 0.116864i
\(26\) 2.65757 0.521192
\(27\) 0 0
\(28\) 4.71702 + 3.42711i 0.891432 + 0.647663i
\(29\) −6.19799 + 2.75952i −1.15094 + 0.512431i −0.891362 − 0.453292i \(-0.850250\pi\)
−0.259576 + 0.965723i \(0.583583\pi\)
\(30\) 0 0
\(31\) 2.22457 + 0.990444i 0.399545 + 0.177889i 0.596665 − 0.802490i \(-0.296492\pi\)
−0.197120 + 0.980379i \(0.563159\pi\)
\(32\) 2.58553 + 4.47827i 0.457061 + 0.791653i
\(33\) 0 0
\(34\) 0.0512873 + 0.0109015i 0.00879570 + 0.00186958i
\(35\) 6.30785 + 4.15225i 1.06622 + 0.701858i
\(36\) 0 0
\(37\) 0.930291 + 2.86314i 0.152939 + 0.470698i 0.997946 − 0.0640571i \(-0.0204040\pi\)
−0.845007 + 0.534755i \(0.820404\pi\)
\(38\) −1.61590 − 0.343471i −0.262134 − 0.0557183i
\(39\) 0 0
\(40\) 2.35104 + 3.66994i 0.371732 + 0.580269i
\(41\) 3.88818 − 4.31827i 0.607232 − 0.674400i −0.358623 − 0.933483i \(-0.616754\pi\)
0.965855 + 0.259083i \(0.0834203\pi\)
\(42\) 0 0
\(43\) 5.61908 − 9.73253i 0.856902 − 1.48420i −0.0179676 − 0.999839i \(-0.505720\pi\)
0.874869 − 0.484359i \(-0.160947\pi\)
\(44\) −5.77296 + 4.19430i −0.870306 + 0.632314i
\(45\) 0 0
\(46\) 0.834057 + 0.605978i 0.122975 + 0.0893465i
\(47\) −8.50718 + 3.78764i −1.24090 + 0.552484i −0.918988 − 0.394286i \(-0.870992\pi\)
−0.321912 + 0.946770i \(0.604325\pi\)
\(48\) 0 0
\(49\) −2.20301 − 3.81573i −0.314716 − 0.545105i
\(50\) 1.51087 + 2.13475i 0.213669 + 0.301900i
\(51\) 0 0
\(52\) 5.86925 + 6.51846i 0.813919 + 0.903948i
\(53\) −1.84804 − 1.34268i −0.253848 − 0.184432i 0.453582 − 0.891214i \(-0.350146\pi\)
−0.707431 + 0.706783i \(0.750146\pi\)
\(54\) 0 0
\(55\) −7.21862 + 5.77172i −0.973359 + 0.778259i
\(56\) −0.688092 − 6.54676i −0.0919502 − 0.874847i
\(57\) 0 0
\(58\) 3.24195 + 1.44341i 0.425689 + 0.189529i
\(59\) −5.66566 + 6.29235i −0.737606 + 0.819195i −0.988879 − 0.148722i \(-0.952484\pi\)
0.251273 + 0.967916i \(0.419151\pi\)
\(60\) 0 0
\(61\) 2.32441 + 2.58152i 0.297610 + 0.330530i 0.873341 − 0.487110i \(-0.161949\pi\)
−0.575730 + 0.817640i \(0.695282\pi\)
\(62\) −0.393599 − 1.21137i −0.0499871 − 0.153844i
\(63\) 0 0
\(64\) −0.668023 + 2.05596i −0.0835028 + 0.256995i
\(65\) 7.98339 + 8.08310i 0.990218 + 1.00258i
\(66\) 0 0
\(67\) −12.3948 − 5.51853i −1.51427 − 0.674195i −0.529538 − 0.848286i \(-0.677635\pi\)
−0.984730 + 0.174091i \(0.944301\pi\)
\(68\) 0.0865293 + 0.149873i 0.0104932 + 0.0181748i
\(69\) 0 0
\(70\) −0.593706 − 3.90522i −0.0709615 − 0.466763i
\(71\) −6.75154 − 4.90528i −0.801260 − 0.582149i 0.110024 − 0.993929i \(-0.464907\pi\)
−0.911283 + 0.411780i \(0.864907\pi\)
\(72\) 0 0
\(73\) 3.28345 − 10.1054i 0.384299 − 1.18275i −0.552689 − 0.833387i \(-0.686398\pi\)
0.936988 − 0.349362i \(-0.113602\pi\)
\(74\) 0.787338 − 1.36371i 0.0915262 − 0.158528i
\(75\) 0 0
\(76\) −2.72627 − 4.72203i −0.312724 − 0.541654i
\(77\) 13.6543 − 2.90231i 1.55605 − 0.330749i
\(78\) 0 0
\(79\) −13.1297 + 5.84572i −1.47721 + 0.657695i −0.977965 − 0.208767i \(-0.933055\pi\)
−0.499243 + 0.866462i \(0.666388\pi\)
\(80\) −1.44082 + 5.24674i −0.161088 + 0.586604i
\(81\) 0 0
\(82\) −3.03942 −0.335648
\(83\) −0.648356 − 6.16870i −0.0711664 − 0.677103i −0.970707 − 0.240266i \(-0.922765\pi\)
0.899541 − 0.436837i \(-0.143901\pi\)
\(84\) 0 0
\(85\) 0.120911 + 0.188741i 0.0131146 + 0.0204718i
\(86\) −5.74982 + 1.22216i −0.620019 + 0.131789i
\(87\) 0 0
\(88\) 7.88038 + 1.67503i 0.840051 + 0.178558i
\(89\) 2.14728 − 6.60864i 0.227611 − 0.700514i −0.770405 − 0.637555i \(-0.779946\pi\)
0.998016 − 0.0629597i \(-0.0200540\pi\)
\(90\) 0 0
\(91\) −5.30248 − 16.3193i −0.555851 − 1.71073i
\(92\) 0.355681 + 3.38408i 0.0370823 + 0.352814i
\(93\) 0 0
\(94\) 4.44980 + 1.98118i 0.458962 + 0.204343i
\(95\) −3.80953 − 5.94663i −0.390849 − 0.610111i
\(96\) 0 0
\(97\) −1.59492 + 0.710105i −0.161940 + 0.0721002i −0.486106 − 0.873900i \(-0.661583\pi\)
0.324166 + 0.946000i \(0.394916\pi\)
\(98\) −0.712171 + 2.19184i −0.0719402 + 0.221409i
\(99\) 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.46.13 224
3.2 odd 2 225.2.q.a.196.16 yes 224
9.4 even 3 inner 675.2.r.a.496.16 224
9.5 odd 6 225.2.q.a.121.13 yes 224
25.6 even 5 inner 675.2.r.a.181.16 224
75.56 odd 10 225.2.q.a.106.13 yes 224
225.31 even 15 inner 675.2.r.a.631.13 224
225.131 odd 30 225.2.q.a.31.16 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.16 ✓ 224 225.131 odd 30
225.2.q.a.106.13 yes 224 75.56 odd 10
225.2.q.a.121.13 yes 224 9.5 odd 6
225.2.q.a.196.16 yes 224 3.2 odd 2
675.2.r.a.46.13 224 1.1 even 1 trivial
675.2.r.a.181.16 224 25.6 even 5 inner
675.2.r.a.496.16 224 9.4 even 3 inner
675.2.r.a.631.13 224 225.31 even 15 inner