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Properties

Label 675.2.r.a.46.8
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 46.8
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.955162 - 1.06081i) q^{2} +(-0.00393676 + 0.0374558i) q^{4} +(-0.397449 + 2.20046i) q^{5} +(1.20474 - 2.08666i) q^{7} +(-2.26620 + 1.64649i) q^{8} +(2.71391 - 1.68018i) q^{10} +(0.0211445 + 0.0234833i) q^{11} +(-4.22250 + 4.68956i) q^{13} +(-3.36428 + 0.715100i) q^{14} +(3.98488 + 0.847013i) q^{16} +(-2.72065 + 1.97667i) q^{17} +(1.81267 - 1.31698i) q^{19} +(-0.0808554 - 0.0235495i) q^{20} +(0.00471506 - 0.0448608i) q^{22} +(1.88117 - 0.399856i) q^{23} +(-4.68407 - 1.74914i) q^{25} +9.00793 q^{26} +(0.0734149 + 0.0533391i) q^{28} +(-8.54224 + 3.80325i) q^{29} +(-3.43471 - 1.52923i) q^{31} +(-0.106511 - 0.184482i) q^{32} +(4.69554 + 0.998067i) q^{34} +(4.11281 + 3.48032i) q^{35} +(3.22975 + 9.94014i) q^{37} +(-3.12847 - 0.664976i) q^{38} +(-2.72234 - 5.64107i) q^{40} +(-1.08631 + 1.20647i) q^{41} +(-5.58816 + 9.67898i) q^{43} +(-0.000962828 + 0.000699535i) q^{44} +(-2.22100 - 1.61365i) q^{46} +(2.88640 - 1.28511i) q^{47} +(0.597221 + 1.03442i) q^{49} +(2.61853 + 6.63964i) q^{50} +(-0.159028 - 0.176619i) q^{52} +(-1.23376 - 0.896379i) q^{53} +(-0.0600780 + 0.0371942i) q^{55} +(0.705499 + 6.71237i) q^{56} +(12.1938 + 5.42901i) q^{58} +(8.58158 - 9.53081i) q^{59} +(7.10677 + 7.89287i) q^{61} +(1.65848 + 5.10426i) q^{62} +(2.42385 - 7.45983i) q^{64} +(-8.64097 - 11.1553i) q^{65} +(-2.07993 - 0.926044i) q^{67} +(-0.0633271 - 0.109686i) q^{68} +(-0.236421 - 7.68719i) q^{70} +(3.63556 + 2.64139i) q^{71} +(0.380414 - 1.17080i) q^{73} +(7.45971 - 12.9206i) q^{74} +(0.0421926 + 0.0730797i) q^{76} +(0.0744753 - 0.0158302i) q^{77} +(0.840545 - 0.374235i) q^{79} +(-3.44761 + 8.43194i) q^{80} +2.31745 q^{82} +(0.581163 + 5.52940i) q^{83} +(-3.26826 - 6.77231i) q^{85} +(15.6052 - 3.31699i) q^{86} +(-0.0865826 - 0.0184037i) q^{88} +(0.932477 - 2.86987i) q^{89} +(4.69854 + 14.4606i) q^{91} +(0.00757118 + 0.0720350i) q^{92} +(-4.12024 - 1.83445i) q^{94} +(2.17753 + 4.51214i) q^{95} +(-9.49506 + 4.22747i) q^{97} +(0.526882 - 1.62158i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.955162 − 1.06081i −0.675401 − 0.750109i 0.303859 − 0.952717i \(-0.401725\pi\)
−0.979260 + 0.202608i \(0.935058\pi\)
\(3\) 0 0
\(4\) −0.00393676 + 0.0374558i −0.00196838 + 0.0187279i
\(5\) −0.397449 + 2.20046i −0.177745 + 0.984077i
\(6\) 0 0
\(7\) 1.20474 − 2.08666i 0.455348 − 0.788685i −0.543361 − 0.839499i \(-0.682848\pi\)
0.998708 + 0.0508144i \(0.0161817\pi\)
\(8\) −2.26620 + 1.64649i −0.801221 + 0.582121i
\(9\) 0 0
\(10\) 2.71391 − 1.68018i 0.858214 − 0.531319i
\(11\) 0.0211445 + 0.0234833i 0.00637530 + 0.00708049i 0.746324 − 0.665583i \(-0.231817\pi\)
−0.739949 + 0.672663i \(0.765150\pi\)
\(12\) 0 0
\(13\) −4.22250 + 4.68956i −1.17111 + 1.30065i −0.225914 + 0.974147i \(0.572537\pi\)
−0.945196 + 0.326503i \(0.894130\pi\)
\(14\) −3.36428 + 0.715100i −0.899142 + 0.191119i
\(15\) 0 0
\(16\) 3.98488 + 0.847013i 0.996220 + 0.211753i
\(17\) −2.72065 + 1.97667i −0.659854 + 0.479412i −0.866614 − 0.498980i \(-0.833708\pi\)
0.206760 + 0.978392i \(0.433708\pi\)
\(18\) 0 0
\(19\) 1.81267 − 1.31698i 0.415855 − 0.302136i −0.360113 − 0.932909i \(-0.617262\pi\)
0.775968 + 0.630772i \(0.217262\pi\)
\(20\) −0.0808554 − 0.0235495i −0.0180798 − 0.00526582i
\(21\) 0 0
\(22\) 0.00471506 − 0.0448608i 0.00100525 − 0.00956434i
\(23\) 1.88117 − 0.399856i 0.392252 − 0.0833756i −0.00756369 − 0.999971i \(-0.502408\pi\)
0.399815 + 0.916596i \(0.369074\pi\)
\(24\) 0 0
\(25\) −4.68407 − 1.74914i −0.936814 − 0.349829i
\(26\) 9.00793 1.76660
\(27\) 0 0
\(28\) 0.0734149 + 0.0533391i 0.0138741 + 0.0100801i
\(29\) −8.54224 + 3.80325i −1.58625 + 0.706246i −0.994966 − 0.100215i \(-0.968047\pi\)
−0.591288 + 0.806460i \(0.701380\pi\)
\(30\) 0 0
\(31\) −3.43471 − 1.52923i −0.616893 − 0.274658i 0.0744052 − 0.997228i \(-0.476294\pi\)
−0.691298 + 0.722570i \(0.742961\pi\)
\(32\) −0.106511 − 0.184482i −0.0188286 − 0.0326121i
\(33\) 0 0
\(34\) 4.69554 + 0.998067i 0.805278 + 0.171167i
\(35\) 4.11281 + 3.48032i 0.695191 + 0.588281i
\(36\) 0 0
\(37\) 3.22975 + 9.94014i 0.530967 + 1.63415i 0.752205 + 0.658929i \(0.228990\pi\)
−0.221238 + 0.975220i \(0.571010\pi\)
\(38\) −3.12847 − 0.664976i −0.507504 − 0.107873i
\(39\) 0 0
\(40\) −2.72234 − 5.64107i −0.430439 − 0.891932i
\(41\) −1.08631 + 1.20647i −0.169654 + 0.188420i −0.821976 − 0.569522i \(-0.807128\pi\)
0.652322 + 0.757942i \(0.273795\pi\)
\(42\) 0 0
\(43\) −5.58816 + 9.67898i −0.852186 + 1.47603i 0.0270444 + 0.999634i \(0.491390\pi\)
−0.879231 + 0.476396i \(0.841943\pi\)
\(44\) −0.000962828 0 0.000699535i −0.000145152 0 0.000105459i
\(45\) 0 0
\(46\) −2.22100 − 1.61365i −0.327468 − 0.237919i
\(47\) 2.88640 − 1.28511i 0.421025 − 0.187452i −0.185278 − 0.982686i \(-0.559319\pi\)
0.606303 + 0.795234i \(0.292652\pi\)
\(48\) 0 0
\(49\) 0.597221 + 1.03442i 0.0853173 + 0.147774i
\(50\) 2.61853 + 6.63964i 0.370316 + 0.938987i
\(51\) 0 0
\(52\) −0.159028 − 0.176619i −0.0220533 − 0.0244926i
\(53\) −1.23376 − 0.896379i −0.169470 − 0.123127i 0.499817 − 0.866131i \(-0.333400\pi\)
−0.669287 + 0.743004i \(0.733400\pi\)
\(54\) 0 0
\(55\) −0.0600780 + 0.0371942i −0.00810092 + 0.00501527i
\(56\) 0.705499 + 6.71237i 0.0942763 + 0.896979i
\(57\) 0 0
\(58\) 12.1938 + 5.42901i 1.60112 + 0.712864i
\(59\) 8.58158 − 9.53081i 1.11723 − 1.24081i 0.149511 − 0.988760i \(-0.452230\pi\)
0.967715 − 0.252046i \(-0.0811034\pi\)
\(60\) 0 0
\(61\) 7.10677 + 7.89287i 0.909929 + 1.01058i 0.999893 + 0.0146335i \(0.00465815\pi\)
−0.0899640 + 0.995945i \(0.528675\pi\)
\(62\) 1.65848 + 5.10426i 0.210627 + 0.648242i
\(63\) 0 0
\(64\) 2.42385 − 7.45983i 0.302981 − 0.932479i
\(65\) −8.64097 − 11.1553i −1.07178 − 1.38365i
\(66\) 0 0
\(67\) −2.07993 − 0.926044i −0.254104 − 0.113134i 0.275732 − 0.961235i \(-0.411080\pi\)
−0.529836 + 0.848100i \(0.677746\pi\)
\(68\) −0.0633271 − 0.109686i −0.00767954 − 0.0133014i
\(69\) 0 0
\(70\) −0.236421 − 7.68719i −0.0282577 − 0.918795i
\(71\) 3.63556 + 2.64139i 0.431462 + 0.313475i 0.782233 − 0.622986i \(-0.214081\pi\)
−0.350771 + 0.936461i \(0.614081\pi\)
\(72\) 0 0
\(73\) 0.380414 − 1.17080i 0.0445241 − 0.137031i −0.926323 − 0.376730i \(-0.877049\pi\)
0.970847 + 0.239698i \(0.0770486\pi\)
\(74\) 7.45971 − 12.9206i 0.867174 − 1.50199i
\(75\) 0 0
\(76\) 0.0421926 + 0.0730797i 0.00483982 + 0.00838281i
\(77\) 0.0744753 − 0.0158302i 0.00848725 − 0.00180402i
\(78\) 0 0
\(79\) 0.840545 − 0.374235i 0.0945687 − 0.0421047i −0.358907 − 0.933373i \(-0.616851\pi\)
0.453476 + 0.891269i \(0.350184\pi\)
\(80\) −3.44761 + 8.43194i −0.385454 + 0.942719i
\(81\) 0 0
\(82\) 2.31745 0.255920
\(83\) 0.581163 + 5.52940i 0.0637909 + 0.606930i 0.978987 + 0.203922i \(0.0653688\pi\)
−0.915196 + 0.403008i \(0.867965\pi\)
\(84\) 0 0
\(85\) −3.26826 − 6.77231i −0.354493 − 0.734560i
\(86\) 15.6052 − 3.31699i 1.68275 − 0.357680i
\(87\) 0 0
\(88\) −0.0865826 − 0.0184037i −0.00922973 − 0.00196184i
\(89\) 0.932477 − 2.86987i 0.0988424 − 0.304206i −0.889394 − 0.457142i \(-0.848873\pi\)
0.988236 + 0.152937i \(0.0488730\pi\)
\(90\) 0 0
\(91\) 4.69854 + 14.4606i 0.492541 + 1.51589i
\(92\) 0.00757118 + 0.0720350i 0.000789350 + 0.00751017i
\(93\) 0 0
\(94\) −4.12024 − 1.83445i −0.424971 − 0.189209i
\(95\) 2.17753 + 4.51214i 0.223409 + 0.462936i
\(96\) 0 0
\(97\) −9.49506 + 4.22747i −0.964077 + 0.429235i −0.827544 − 0.561401i \(-0.810263\pi\)
−0.136533 + 0.990636i \(0.543596\pi\)
\(98\) 0.526882 − 1.62158i 0.0532232 − 0.163804i
\(99\) 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.46.8 224
3.2 odd 2 225.2.q.a.196.21 yes 224
9.4 even 3 inner 675.2.r.a.496.21 224
9.5 odd 6 225.2.q.a.121.8 yes 224
25.6 even 5 inner 675.2.r.a.181.21 224
75.56 odd 10 225.2.q.a.106.8 yes 224
225.31 even 15 inner 675.2.r.a.631.8 224
225.131 odd 30 225.2.q.a.31.21 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.21 ✓ 224 225.131 odd 30
225.2.q.a.106.8 yes 224 75.56 odd 10
225.2.q.a.121.8 yes 224 9.5 odd 6
225.2.q.a.196.21 yes 224 3.2 odd 2
675.2.r.a.46.8 224 1.1 even 1 trivial
675.2.r.a.181.21 224 25.6 even 5 inner
675.2.r.a.496.21 224 9.4 even 3 inner
675.2.r.a.631.8 224 225.31 even 15 inner