↑

Properties

Label 675.2.r.a.46.1
Level $675$
Weight $2$
Character 675.46
Analytic conductor $5.390$
Analytic rank $0$
Dimension $224$
Inner twists $4$

Related objects

Downloads

Learn more

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 46.1
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.77417 - 1.97042i) q^{2} +(-0.525803 + 5.00268i) q^{4} +(1.61775 + 1.54366i) q^{5} +(1.02644 - 1.77785i) q^{7} +(6.50010 - 4.72260i) q^{8} +(0.171493 - 5.92636i) q^{10} +(3.84053 + 4.26534i) q^{11} +(-1.39743 + 1.55200i) q^{13} +(-5.32421 + 1.13169i) q^{14} +(-10.9972 - 2.33752i) q^{16} +(0.0247468 - 0.0179796i) q^{17} +(-5.30003 + 3.85070i) q^{19} +(-8.57307 + 7.28142i) q^{20} +(1.59074 - 15.1349i) q^{22} +(-3.94065 + 0.837611i) q^{23} +(0.234221 + 4.99451i) q^{25} +5.53738 q^{26} +(8.35433 + 6.06978i) q^{28} +(-2.99836 + 1.33495i) q^{29} +(2.93669 + 1.30750i) q^{31} +(6.87040 + 11.8999i) q^{32} +(-0.0793325 - 0.0168626i) q^{34} +(4.40493 - 1.29164i) q^{35} +(1.37908 + 4.24438i) q^{37} +(16.9906 + 3.61147i) q^{38} +(17.8056 + 2.39397i) q^{40} +(-1.29130 + 1.43413i) q^{41} +(0.309512 - 0.536091i) q^{43} +(-23.3575 + 16.9702i) q^{44} +(8.64184 + 6.27867i) q^{46} +(2.37044 - 1.05539i) q^{47} +(1.39282 + 2.41244i) q^{49} +(9.42573 - 9.32264i) q^{50} +(-7.02940 - 7.80695i) q^{52} +(2.83942 + 2.06296i) q^{53} +(-0.371228 + 12.8287i) q^{55} +(-1.72410 - 16.4037i) q^{56} +(7.95002 + 3.53958i) q^{58} +(5.43616 - 6.03746i) q^{59} +(-5.51851 - 6.12893i) q^{61} +(-2.63388 - 8.10625i) q^{62} +(4.31002 - 13.2649i) q^{64} +(-4.65646 + 0.353593i) q^{65} +(-0.642929 - 0.286250i) q^{67} +(0.0769343 + 0.133254i) q^{68} +(-10.3602 - 6.38797i) q^{70} +(-3.81216 - 2.76970i) q^{71} +(2.55263 - 7.85620i) q^{73} +(5.91647 - 10.2476i) q^{74} +(-16.4770 - 28.5391i) q^{76} +(11.5252 - 2.44977i) q^{77} +(10.9723 - 4.88520i) q^{79} +(-14.1823 - 20.7574i) q^{80} +5.11682 q^{82} +(1.37616 + 13.0933i) q^{83} +(0.0677885 + 0.00911420i) q^{85} +(-1.60545 + 0.341249i) q^{86} +(45.1073 + 9.58785i) q^{88} +(-1.99904 + 6.15242i) q^{89} +(1.32485 + 4.07747i) q^{91} +(-2.11830 - 20.1543i) q^{92} +(-6.28513 - 2.79832i) q^{94} +(-14.5183 - 1.95199i) q^{95} +(-9.33612 + 4.15671i) q^{97} +(2.28241 - 7.02454i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.77417 − 1.97042i −1.25453 − 1.39330i −0.886003 − 0.463679i \(-0.846529\pi\)
−0.368527 − 0.929617i \(-0.620138\pi\)
\(3\) 0 0
\(4\) −0.525803 + 5.00268i −0.262902 + 2.50134i
\(5\) 1.61775 + 1.54366i 0.723479 + 0.690346i
\(6\) 0 0
\(7\) 1.02644 − 1.77785i 0.387959 − 0.671966i −0.604215 − 0.796821i \(-0.706513\pi\)
0.992175 + 0.124855i \(0.0398467\pi\)
\(8\) 6.50010 − 4.72260i 2.29813 − 1.66969i
\(9\) 0 0
\(10\) 0.171493 − 5.92636i 0.0542308 − 1.87408i
\(11\) 3.84053 + 4.26534i 1.15796 + 1.28605i 0.951519 + 0.307590i \(0.0995226\pi\)
0.206445 + 0.978458i \(0.433811\pi\)
\(12\) 0 0
\(13\) −1.39743 + 1.55200i −0.387577 + 0.430448i −0.905085 − 0.425230i \(-0.860193\pi\)
0.517508 + 0.855678i \(0.326860\pi\)
\(14\) −5.32421 + 1.13169i −1.42295 + 0.302458i
\(15\) 0 0
\(16\) −10.9972 − 2.33752i −2.74929 − 0.584379i
\(17\) 0.0247468 − 0.0179796i 0.00600198 − 0.00436070i −0.584780 − 0.811192i \(-0.698819\pi\)
0.590782 + 0.806831i \(0.298819\pi\)
\(18\) 0 0
\(19\) −5.30003 + 3.85070i −1.21591 + 0.883410i −0.995754 − 0.0920546i \(-0.970657\pi\)
−0.220156 + 0.975465i \(0.570657\pi\)
\(20\) −8.57307 + 7.28142i −1.91700 + 1.62818i
\(21\) 0 0
\(22\) 1.59074 − 15.1349i 0.339147 − 3.22677i
\(23\) −3.94065 + 0.837611i −0.821683 + 0.174654i −0.599524 − 0.800357i \(-0.704644\pi\)
−0.222158 + 0.975011i \(0.571310\pi\)
\(24\) 0 0
\(25\) 0.234221 + 4.99451i 0.0468442 + 0.998902i
\(26\) 5.53738 1.08597
\(27\) 0 0
\(28\) 8.35433 + 6.06978i 1.57882 + 1.14708i
\(29\) −2.99836 + 1.33495i −0.556781 + 0.247895i −0.665781 − 0.746147i \(-0.731902\pi\)
0.109000 + 0.994042i \(0.465235\pi\)
\(30\) 0 0
\(31\) 2.93669 + 1.30750i 0.527446 + 0.234834i 0.653143 − 0.757235i \(-0.273450\pi\)
−0.125697 + 0.992069i \(0.540117\pi\)
\(32\) 6.87040 + 11.8999i 1.21453 + 2.10362i
\(33\) 0 0
\(34\) −0.0793325 − 0.0168626i −0.0136054 − 0.00289192i
\(35\) 4.40493 − 1.29164i 0.744569 − 0.218327i
\(36\) 0 0
\(37\) 1.37908 + 4.24438i 0.226720 + 0.697771i 0.998112 + 0.0614120i \(0.0195604\pi\)
−0.771393 + 0.636359i \(0.780440\pi\)
\(38\) 16.9906 + 3.61147i 2.75625 + 0.585858i
\(39\) 0 0
\(40\) 17.8056 + 2.39397i 2.81531 + 0.378520i
\(41\) −1.29130 + 1.43413i −0.201667 + 0.223974i −0.835492 − 0.549503i \(-0.814817\pi\)
0.633825 + 0.773476i \(0.281484\pi\)
\(42\) 0 0
\(43\) 0.309512 − 0.536091i 0.0472002 − 0.0817531i −0.841460 − 0.540319i \(-0.818303\pi\)
0.888660 + 0.458566i \(0.151637\pi\)
\(44\) −23.3575 + 16.9702i −3.52128 + 2.55836i
\(45\) 0 0
\(46\) 8.64184 + 6.27867i 1.27417 + 0.925739i
\(47\) 2.37044 − 1.05539i 0.345764 − 0.153944i −0.226505 − 0.974010i \(-0.572730\pi\)
0.572269 + 0.820066i \(0.306063\pi\)
\(48\) 0 0
\(49\) 1.39282 + 2.41244i 0.198975 + 0.344635i
\(50\) 9.42573 − 9.32264i 1.33300 − 1.31842i
\(51\) 0 0
\(52\) −7.02940 − 7.80695i −0.974803 − 1.08263i
\(53\) 2.83942 + 2.06296i 0.390024 + 0.283369i 0.765465 − 0.643477i \(-0.222509\pi\)
−0.375441 + 0.926846i \(0.622509\pi\)
\(54\) 0 0
\(55\) −0.371228 + 12.8287i −0.0500564 + 1.72983i
\(56\) −1.72410 − 16.4037i −0.230392 − 2.19204i
\(57\) 0 0
\(58\) 7.95002 + 3.53958i 1.04389 + 0.464769i
\(59\) 5.43616 − 6.03746i 0.707727 − 0.786011i −0.276859 − 0.960911i \(-0.589293\pi\)
0.984586 + 0.174900i \(0.0559601\pi\)
\(60\) 0 0
\(61\) −5.51851 − 6.12893i −0.706573 − 0.784729i 0.277835 − 0.960629i \(-0.410383\pi\)
−0.984408 + 0.175900i \(0.943716\pi\)
\(62\) −2.63388 − 8.10625i −0.334503 − 1.02949i
\(63\) 0 0
\(64\) 4.31002 − 13.2649i 0.538753 − 1.65811i
\(65\) −4.65646 + 0.353593i −0.577562 + 0.0438578i
\(66\) 0 0
\(67\) −0.642929 − 0.286250i −0.0785463 − 0.0349710i 0.367087 − 0.930186i \(-0.380355\pi\)
−0.445634 + 0.895215i \(0.647022\pi\)
\(68\) 0.0769343 + 0.133254i 0.00932966 + 0.0161594i
\(69\) 0 0
\(70\) −10.3602 − 6.38797i −1.23828 − 0.763509i
\(71\) −3.81216 − 2.76970i −0.452421 − 0.328703i 0.338130 − 0.941099i \(-0.390206\pi\)
−0.790551 + 0.612397i \(0.790206\pi\)
\(72\) 0 0
\(73\) 2.55263 − 7.85620i 0.298763 − 0.919499i −0.683168 − 0.730261i \(-0.739398\pi\)
0.981931 − 0.189237i \(-0.0606016\pi\)
\(74\) 5.91647 − 10.2476i 0.687776 − 1.19126i
\(75\) 0 0
\(76\) −16.4770 − 28.5391i −1.89005 − 3.27366i
\(77\) 11.5252 − 2.44977i 1.31342 − 0.279177i
\(78\) 0 0
\(79\) 10.9723 − 4.88520i 1.23448 − 0.549628i 0.317389 − 0.948295i \(-0.397194\pi\)
0.917095 + 0.398668i \(0.130527\pi\)
\(80\) −14.1823 − 20.7574i −1.58563 − 2.32075i
\(81\) 0 0
\(82\) 5.11682 0.565058
\(83\) 1.37616 + 13.0933i 0.151053 + 1.43717i 0.763064 + 0.646323i \(0.223694\pi\)
−0.612011 + 0.790849i \(0.709639\pi\)
\(84\) 0 0
\(85\) 0.0677885 + 0.00911420i 0.00735270 + 0.000988573i
\(86\) −1.60545 + 0.341249i −0.173120 + 0.0367979i
\(87\) 0 0
\(88\) 45.1073 + 9.58785i 4.80845 + 1.02207i
\(89\) −1.99904 + 6.15242i −0.211898 + 0.652155i 0.787461 + 0.616364i \(0.211395\pi\)
−0.999359 + 0.0357910i \(0.988605\pi\)
\(90\) 0 0
\(91\) 1.32485 + 4.07747i 0.138882 + 0.427435i
\(92\) −2.11830 − 20.1543i −0.220848 − 2.10123i
\(93\) 0 0
\(94\) −6.28513 − 2.79832i −0.648262 − 0.288625i
\(95\) −14.5183 − 1.95199i −1.48954 − 0.200270i
\(96\) 0 0
\(97\) −9.33612 + 4.15671i −0.947940 + 0.422050i −0.821681 − 0.569947i \(-0.806964\pi\)
−0.126258 + 0.991997i \(0.540297\pi\)
\(98\) 2.28241 − 7.02454i 0.230558 − 0.709585i
\(99\) 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.46.1 224
3.2 odd 2 225.2.q.a.196.28 yes 224
9.4 even 3 inner 675.2.r.a.496.28 224
9.5 odd 6 225.2.q.a.121.1 yes 224
25.6 even 5 inner 675.2.r.a.181.28 224
75.56 odd 10 225.2.q.a.106.1 yes 224
225.31 even 15 inner 675.2.r.a.631.1 224
225.131 odd 30 225.2.q.a.31.28 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.28 ✓ 224 225.131 odd 30
225.2.q.a.106.1 yes 224 75.56 odd 10
225.2.q.a.121.1 yes 224 9.5 odd 6
225.2.q.a.196.28 yes 224 3.2 odd 2
675.2.r.a.46.1 224 1.1 even 1 trivial
675.2.r.a.181.28 224 25.6 even 5 inner
675.2.r.a.496.28 224 9.4 even 3 inner
675.2.r.a.631.1 224 225.31 even 15 inner