↑

Properties

Label 495.4.c.a.199.3
Level $495$
Weight $4$
Character 495.199
Analytic conductor $29.206$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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] = [495,4,Mod(199,495)] 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("495.199"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,4,4,0,0,0,0,26,66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 22x^{4} + 101x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(-0.405276i\) of defining polynomial
Character \(\chi\) \(=\) 495.199
Dual form 495.4.c.a.199.4

$q$-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)
 
\(f(q)\) \(=\) \(q-0.405276i q^{2} +7.83575 q^{4} +(0.472938 + 11.1703i) q^{5} +27.4984i q^{7} -6.41784i q^{8} +(4.52706 - 0.191670i) q^{10} +11.0000 q^{11} -27.6235i q^{13} +11.1444 q^{14} +60.0850 q^{16} +9.63773i q^{17} +112.560 q^{19} +(3.70582 + 87.5280i) q^{20} -4.45803i q^{22} +154.630i q^{23} +(-124.553 + 10.5657i) q^{25} -11.1951 q^{26} +215.471i q^{28} -236.372 q^{29} -45.9456 q^{31} -75.6937i q^{32} +3.90594 q^{34} +(-307.167 + 13.0050i) q^{35} -235.321i q^{37} -45.6177i q^{38} +(71.6894 - 3.03524i) q^{40} +55.5664 q^{41} +306.070i q^{43} +86.1933 q^{44} +62.6679 q^{46} +133.999i q^{47} -413.164 q^{49} +(4.28204 + 50.4781i) q^{50} -216.451i q^{52} +315.357i q^{53} +(5.20231 + 122.874i) q^{55} +176.481 q^{56} +95.7959i q^{58} +509.192 q^{59} -656.843 q^{61} +18.6206i q^{62} +450.003 q^{64} +(308.564 - 13.0642i) q^{65} +173.172i q^{67} +75.5189i q^{68} +(5.27062 + 124.487i) q^{70} +183.924 q^{71} +83.2810i q^{73} -95.3697 q^{74} +881.991 q^{76} +302.483i q^{77} -798.912 q^{79} +(28.4165 + 671.170i) q^{80} -22.5197i q^{82} -442.779i q^{83} +(-107.657 + 4.55805i) q^{85} +124.043 q^{86} -70.5963i q^{88} -448.553 q^{89} +759.604 q^{91} +1211.65i q^{92} +54.3067 q^{94} +(53.2338 + 1257.33i) q^{95} -307.267i q^{97} +167.445i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{4} + 4 q^{5} + 26 q^{10} + 66 q^{11} - 18 q^{14} - 108 q^{16} + 258 q^{19} - 194 q^{20} + 106 q^{25} + 356 q^{26} - 494 q^{29} - 514 q^{31} + 6 q^{34} - 570 q^{35} + 422 q^{40} + 824 q^{41} + 44 q^{44}+ \cdots + 2884 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.405276i − 0.143287i −0.997430 − 0.0716433i \(-0.977176\pi\)
0.997430 − 0.0716433i \(-0.0228243\pi\)
\(3\) 0 0
\(4\) 7.83575 0.979469
\(5\) 0.472938 + 11.1703i 0.0423008 + 0.999105i
\(6\) 0 0
\(7\) 27.4984i 1.48478i 0.669970 + 0.742388i \(0.266307\pi\)
−0.669970 + 0.742388i \(0.733693\pi\)
\(8\) − 6.41784i − 0.283631i
\(9\) 0 0
\(10\) 4.52706 − 0.191670i 0.143158 − 0.00606114i
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) − 27.6235i − 0.589338i −0.955599 − 0.294669i \(-0.904791\pi\)
0.955599 − 0.294669i \(-0.0952093\pi\)
\(14\) 11.1444 0.212748
\(15\) 0 0
\(16\) 60.0850 0.938828
\(17\) 9.63773i 0.137500i 0.997634 + 0.0687498i \(0.0219010\pi\)
−0.997634 + 0.0687498i \(0.978099\pi\)
\(18\) 0 0
\(19\) 112.560 1.35911 0.679553 − 0.733627i \(-0.262174\pi\)
0.679553 + 0.733627i \(0.262174\pi\)
\(20\) 3.70582 + 87.5280i 0.0414323 + 0.978592i
\(21\) 0 0
\(22\) − 4.45803i − 0.0432025i
\(23\) 154.630i 1.40186i 0.713232 + 0.700928i \(0.247230\pi\)
−0.713232 + 0.700928i \(0.752770\pi\)
\(24\) 0 0
\(25\) −124.553 + 10.5657i −0.996421 + 0.0845259i
\(26\) −11.1951 −0.0844442
\(27\) 0 0
\(28\) 215.471i 1.45429i
\(29\) −236.372 −1.51356 −0.756780 − 0.653670i \(-0.773229\pi\)
−0.756780 + 0.653670i \(0.773229\pi\)
\(30\) 0 0
\(31\) −45.9456 −0.266196 −0.133098 − 0.991103i \(-0.542492\pi\)
−0.133098 + 0.991103i \(0.542492\pi\)
\(32\) − 75.6937i − 0.418153i
\(33\) 0 0
\(34\) 3.90594 0.0197018
\(35\) −307.167 + 13.0050i −1.48345 + 0.0628072i
\(36\) 0 0
\(37\) − 235.321i − 1.04558i −0.852461 − 0.522790i \(-0.824891\pi\)
0.852461 − 0.522790i \(-0.175109\pi\)
\(38\) − 45.6177i − 0.194741i
\(39\) 0 0
\(40\) 71.6894 − 3.03524i 0.283377 − 0.0119978i
\(41\) 55.5664 0.211659 0.105829 − 0.994384i \(-0.466250\pi\)
0.105829 + 0.994384i \(0.466250\pi\)
\(42\) 0 0
\(43\) 306.070i 1.08547i 0.839904 + 0.542736i \(0.182611\pi\)
−0.839904 + 0.542736i \(0.817389\pi\)
\(44\) 86.1933 0.295321
\(45\) 0 0
\(46\) 62.6679 0.200867
\(47\) 133.999i 0.415868i 0.978143 + 0.207934i \(0.0666740\pi\)
−0.978143 + 0.207934i \(0.933326\pi\)
\(48\) 0 0
\(49\) −413.164 −1.20456
\(50\) 4.28204 + 50.4781i 0.0121114 + 0.142774i
\(51\) 0 0
\(52\) − 216.451i − 0.577238i
\(53\) 315.357i 0.817315i 0.912688 + 0.408657i \(0.134003\pi\)
−0.912688 + 0.408657i \(0.865997\pi\)
\(54\) 0 0
\(55\) 5.20231 + 122.874i 0.0127542 + 0.301241i
\(56\) 176.481 0.421129
\(57\) 0 0
\(58\) 95.7959i 0.216873i
\(59\) 509.192 1.12358 0.561790 − 0.827280i \(-0.310113\pi\)
0.561790 + 0.827280i \(0.310113\pi\)
\(60\) 0 0
\(61\) −656.843 −1.37869 −0.689345 − 0.724433i \(-0.742101\pi\)
−0.689345 + 0.724433i \(0.742101\pi\)
\(62\) 18.6206i 0.0381423i
\(63\) 0 0
\(64\) 450.003 0.878913
\(65\) 308.564 − 13.0642i 0.588810 − 0.0249295i
\(66\) 0 0
\(67\) 173.172i 0.315766i 0.987458 + 0.157883i \(0.0504669\pi\)
−0.987458 + 0.157883i \(0.949533\pi\)
\(68\) 75.5189i 0.134677i
\(69\) 0 0
\(70\) 5.27062 + 124.487i 0.00899943 + 0.212558i
\(71\) 183.924 0.307433 0.153716 − 0.988115i \(-0.450876\pi\)
0.153716 + 0.988115i \(0.450876\pi\)
\(72\) 0 0
\(73\) 83.2810i 0.133525i 0.997769 + 0.0667624i \(0.0212669\pi\)
−0.997769 + 0.0667624i \(0.978733\pi\)
\(74\) −95.3697 −0.149818
\(75\) 0 0
\(76\) 881.991 1.33120
\(77\) 302.483i 0.447677i
\(78\) 0 0
\(79\) −798.912 −1.13778 −0.568890 − 0.822414i \(-0.692627\pi\)
−0.568890 + 0.822414i \(0.692627\pi\)
\(80\) 28.4165 + 671.170i 0.0397132 + 0.937988i
\(81\) 0 0
\(82\) − 22.5197i − 0.0303278i
\(83\) − 442.779i − 0.585558i −0.956180 − 0.292779i \(-0.905420\pi\)
0.956180 − 0.292779i \(-0.0945799\pi\)
\(84\) 0 0
\(85\) −107.657 + 4.55805i −0.137377 + 0.00581635i
\(86\) 124.043 0.155533
\(87\) 0 0
\(88\) − 70.5963i − 0.0855180i
\(89\) −448.553 −0.534230 −0.267115 − 0.963665i \(-0.586070\pi\)
−0.267115 + 0.963665i \(0.586070\pi\)
\(90\) 0 0
\(91\) 759.604 0.875034
\(92\) 1211.65i 1.37307i
\(93\) 0 0
\(94\) 54.3067 0.0595883
\(95\) 53.2338 + 1257.33i 0.0574913 + 1.35789i
\(96\) 0 0
\(97\) − 307.267i − 0.321632i −0.986984 − 0.160816i \(-0.948587\pi\)
0.986984 − 0.160816i \(-0.0514125\pi\)
\(98\) 167.445i 0.172597i
\(99\) 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 495.4.c.a.199.3 6
3.2 odd 2 55.4.b.a.34.4 yes 6
5.2 odd 4 2475.4.a.bn.1.4 6
5.3 odd 4 2475.4.a.bn.1.3 6
5.4 even 2 inner 495.4.c.a.199.4 6
12.11 even 2 880.4.b.f.529.1 6
15.2 even 4 275.4.a.j.1.3 6
15.8 even 4 275.4.a.j.1.4 6
15.14 odd 2 55.4.b.a.34.3 ✓ 6
60.59 even 2 880.4.b.f.529.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.4.b.a.34.3 ✓ 6 15.14 odd 2
55.4.b.a.34.4 yes 6 3.2 odd 2
275.4.a.j.1.3 6 15.2 even 4
275.4.a.j.1.4 6 15.8 even 4
495.4.c.a.199.3 6 1.1 even 1 trivial
495.4.c.a.199.4 6 5.4 even 2 inner
880.4.b.f.529.1 6 12.11 even 2
880.4.b.f.529.6 6 60.59 even 2
2475.4.a.bn.1.3 6 5.3 odd 4
2475.4.a.bn.1.4 6 5.2 odd 4