Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4275,2,Mod(1,4275)] 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("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,8,0,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.1360468641\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.16717036.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 26x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.53035\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.53035 q^{2} +4.40268 q^{4} -2.62981 q^{7} +6.07962 q^{8} -4.12400 q^{11} -3.54573 q^{13} -6.65435 q^{14} +6.57822 q^{16} -3.91124 q^{17} -1.00000 q^{19} -10.4352 q^{22} -0.936703 q^{23} -8.97195 q^{26} -11.5782 q^{28} +1.01892 q^{29} -3.17554 q^{31} +4.48597 q^{32} -9.89682 q^{34} -7.54573 q^{37} -2.53035 q^{38} +1.95562 q^{41} +6.35109 q^{43} -18.1566 q^{44} -2.37019 q^{46} +8.97195 q^{47} -0.0840822 q^{49} -15.6107 q^{52} -11.1403 q^{53} -15.9883 q^{56} +2.57822 q^{58} -7.01632 q^{59} +12.6107 q^{61} -8.03524 q^{62} -1.80536 q^{64} +3.98090 q^{67} -17.2199 q^{68} +7.01632 q^{71} -7.16816 q^{73} -19.0934 q^{74} -4.40268 q^{76} +10.8454 q^{77} -11.5266 q^{79} +4.94841 q^{82} -14.2454 q^{83} +16.0705 q^{86} -25.0724 q^{88} +13.1782 q^{89} +9.32461 q^{91} -4.12400 q^{92} +22.7022 q^{94} -3.54573 q^{97} -0.212757 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} - 16 q^{13} - 6 q^{19} - 10 q^{22} - 30 q^{28} + 2 q^{31} - 12 q^{34} - 40 q^{37} - 4 q^{43} - 30 q^{46} + 10 q^{49} - 20 q^{52} - 24 q^{58} + 2 q^{61} + 26 q^{64} - 34 q^{67} - 22 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.53035 1.78923 0.894614 − 0.446839i \(-0.147450\pi\)
0.894614 + 0.446839i \(0.147450\pi\)
\(3\) 0 0
\(4\) 4.40268 2.20134
\(5\) 0 0
\(6\) 0 0
\(7\) −2.62981 −0.993976 −0.496988 − 0.867757i \(-0.665561\pi\)
−0.496988 + 0.867757i \(0.665561\pi\)
\(8\) 6.07962 2.14947
\(9\) 0 0
\(10\) 0 0
\(11\) −4.12400 −1.24343 −0.621716 − 0.783242i \(-0.713564\pi\)
−0.621716 + 0.783242i \(0.713564\pi\)
\(12\) 0 0
\(13\) −3.54573 −0.983409 −0.491704 − 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(14\) −6.65435 −1.77845
\(15\) 0 0
\(16\) 6.57822 1.64456
\(17\) −3.91124 −0.948616 −0.474308 − 0.880359i \(-0.657302\pi\)
−0.474308 + 0.880359i \(0.657302\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −10.4352 −2.22479
\(23\) −0.936703 −0.195316 −0.0976580 − 0.995220i \(-0.531135\pi\)
−0.0976580 + 0.995220i \(0.531135\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.97195 −1.75954
\(27\) 0 0
\(28\) −11.5782 −2.18808
\(29\) 1.01892 0.189209 0.0946043 − 0.995515i \(-0.469841\pi\)
0.0946043 + 0.995515i \(0.469841\pi\)
\(30\) 0 0
\(31\) −3.17554 −0.570345 −0.285172 − 0.958476i \(-0.592051\pi\)
−0.285172 + 0.958476i \(0.592051\pi\)
\(32\) 4.48597 0.793015
\(33\) 0 0
\(34\) −9.89682 −1.69729
\(35\) 0 0
\(36\) 0 0
\(37\) −7.54573 −1.24051 −0.620255 − 0.784400i \(-0.712971\pi\)
−0.620255 + 0.784400i \(0.712971\pi\)
\(38\) −2.53035 −0.410477
\(39\) 0 0
\(40\) 0 0
\(41\) 1.95562 0.305417 0.152708 − 0.988271i \(-0.451200\pi\)
0.152708 + 0.988271i \(0.451200\pi\)
\(42\) 0 0
\(43\) 6.35109 0.968532 0.484266 − 0.874921i \(-0.339087\pi\)
0.484266 + 0.874921i \(0.339087\pi\)
\(44\) −18.1566 −2.73722
\(45\) 0 0
\(46\) −2.37019 −0.349465
\(47\) 8.97195 1.30869 0.654346 − 0.756195i \(-0.272944\pi\)
0.654346 + 0.756195i \(0.272944\pi\)
\(48\) 0 0
\(49\) −0.0840822 −0.0120117
\(50\) 0 0
\(51\) 0 0
\(52\) −15.6107 −2.16482
\(53\) −11.1403 −1.53024 −0.765121 − 0.643887i \(-0.777321\pi\)
−0.765121 + 0.643887i \(0.777321\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −15.9883 −2.13652
\(57\) 0 0
\(58\) 2.57822 0.338537
\(59\) −7.01632 −0.913448 −0.456724 − 0.889609i \(-0.650977\pi\)
−0.456724 + 0.889609i \(0.650977\pi\)
\(60\) 0 0
\(61\) 12.6107 1.61464 0.807318 − 0.590116i \(-0.200918\pi\)
0.807318 + 0.590116i \(0.200918\pi\)
\(62\) −8.03524 −1.02048
\(63\) 0 0
\(64\) −1.80536 −0.225670
\(65\) 0 0
\(66\) 0 0
\(67\) 3.98090 0.486345 0.243172 − 0.969983i \(-0.421812\pi\)
0.243172 + 0.969983i \(0.421812\pi\)
\(68\) −17.2199 −2.08823
\(69\) 0 0
\(70\) 0 0
\(71\) 7.01632 0.832685 0.416342 − 0.909208i \(-0.363312\pi\)
0.416342 + 0.909208i \(0.363312\pi\)
\(72\) 0 0
\(73\) −7.16816 −0.838970 −0.419485 − 0.907762i \(-0.637789\pi\)
−0.419485 + 0.907762i \(0.637789\pi\)
\(74\) −19.0934 −2.21956
\(75\) 0 0
\(76\) −4.40268 −0.505022
\(77\) 10.8454 1.23594
\(78\) 0 0
\(79\) −11.5266 −1.29685 −0.648424 − 0.761280i \(-0.724571\pi\)
−0.648424 + 0.761280i \(0.724571\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.94841 0.546460
\(83\) −14.2454 −1.56364 −0.781818 − 0.623506i \(-0.785708\pi\)
−0.781818 + 0.623506i \(0.785708\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 16.0705 1.73293
\(87\) 0 0
\(88\) −25.0724 −2.67272
\(89\) 13.1782 1.39688 0.698441 − 0.715667i \(-0.253877\pi\)
0.698441 + 0.715667i \(0.253877\pi\)
\(90\) 0 0
\(91\) 9.32461 0.977485
\(92\) −4.12400 −0.429957
\(93\) 0 0
\(94\) 22.7022 2.34155
\(95\) 0 0
\(96\) 0 0
\(97\) −3.54573 −0.360014 −0.180007 − 0.983665i \(-0.557612\pi\)
−0.180007 + 0.983665i \(0.557612\pi\)
\(98\) −0.212757 −0.0214917
\(99\) 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 4275.2.a.bs.1.6 yes 6
3.2 odd 2 inner 4275.2.a.bs.1.1 ✓ 6
5.4 even 2 4275.2.a.bt.1.1 yes 6
15.14 odd 2 4275.2.a.bt.1.6 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4275.2.a.bs.1.1 ✓ 6 3.2 odd 2 inner
4275.2.a.bs.1.6 yes 6 1.1 even 1 trivial
4275.2.a.bt.1.1 yes 6 5.4 even 2
4275.2.a.bt.1.6 yes 6 15.14 odd 2