↑

Properties

Label 765.4.a.a.1.1
Level $765$
Weight $4$
Character 765.1
Self dual yes
Analytic conductor $45.136$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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] = [765,4,Mod(1,765)] 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("765.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 765.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-3,0,1,-5,0,-22,21,0,15,30] 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: yes
Analytic conductor: \(45.1364611544\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 765.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} -22.0000 q^{7} +21.0000 q^{8} +15.0000 q^{10} +30.0000 q^{11} -46.0000 q^{13} +66.0000 q^{14} -71.0000 q^{16} -17.0000 q^{17} +104.000 q^{19} -5.00000 q^{20} -90.0000 q^{22} -42.0000 q^{23} +25.0000 q^{25} +138.000 q^{26} -22.0000 q^{28} +66.0000 q^{29} +194.000 q^{31} +45.0000 q^{32} +51.0000 q^{34} +110.000 q^{35} +206.000 q^{37} -312.000 q^{38} -105.000 q^{40} +126.000 q^{41} -388.000 q^{43} +30.0000 q^{44} +126.000 q^{46} +540.000 q^{47} +141.000 q^{49} -75.0000 q^{50} -46.0000 q^{52} -78.0000 q^{53} -150.000 q^{55} -462.000 q^{56} -198.000 q^{58} -432.000 q^{59} -610.000 q^{61} -582.000 q^{62} +433.000 q^{64} +230.000 q^{65} +848.000 q^{67} -17.0000 q^{68} -330.000 q^{70} +174.000 q^{71} +362.000 q^{73} -618.000 q^{74} +104.000 q^{76} -660.000 q^{77} +398.000 q^{79} +355.000 q^{80} -378.000 q^{82} -828.000 q^{83} +85.0000 q^{85} +1164.00 q^{86} +630.000 q^{88} -630.000 q^{89} +1012.00 q^{91} -42.0000 q^{92} -1620.00 q^{94} -520.000 q^{95} -1486.00 q^{97} -423.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 − 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −22.0000 −1.18789 −0.593944 − 0.804506i \(-0.702430\pi\)
−0.593944 + 0.804506i \(0.702430\pi\)
\(8\) 21.0000 0.928078
\(9\) 0 0
\(10\) 15.0000 0.474342
\(11\) 30.0000 0.822304 0.411152 − 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) 0 0
\(13\) −46.0000 −0.981393 −0.490696 − 0.871331i \(-0.663258\pi\)
−0.490696 + 0.871331i \(0.663258\pi\)
\(14\) 66.0000 1.25995
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) 104.000 1.25575 0.627875 − 0.778314i \(-0.283925\pi\)
0.627875 + 0.778314i \(0.283925\pi\)
\(20\) −5.00000 −0.0559017
\(21\) 0 0
\(22\) −90.0000 −0.872185
\(23\) −42.0000 −0.380765 −0.190383 − 0.981710i \(-0.560973\pi\)
−0.190383 + 0.981710i \(0.560973\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 138.000 1.04092
\(27\) 0 0
\(28\) −22.0000 −0.148486
\(29\) 66.0000 0.422617 0.211308 − 0.977419i \(-0.432228\pi\)
0.211308 + 0.977419i \(0.432228\pi\)
\(30\) 0 0
\(31\) 194.000 1.12398 0.561991 − 0.827143i \(-0.310036\pi\)
0.561991 + 0.827143i \(0.310036\pi\)
\(32\) 45.0000 0.248592
\(33\) 0 0
\(34\) 51.0000 0.257248
\(35\) 110.000 0.531240
\(36\) 0 0
\(37\) 206.000 0.915302 0.457651 − 0.889132i \(-0.348691\pi\)
0.457651 + 0.889132i \(0.348691\pi\)
\(38\) −312.000 −1.33192
\(39\) 0 0
\(40\) −105.000 −0.415049
\(41\) 126.000 0.479949 0.239974 − 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(42\) 0 0
\(43\) −388.000 −1.37603 −0.688017 − 0.725695i \(-0.741518\pi\)
−0.688017 + 0.725695i \(0.741518\pi\)
\(44\) 30.0000 0.102788
\(45\) 0 0
\(46\) 126.000 0.403863
\(47\) 540.000 1.67590 0.837948 − 0.545750i \(-0.183755\pi\)
0.837948 + 0.545750i \(0.183755\pi\)
\(48\) 0 0
\(49\) 141.000 0.411079
\(50\) −75.0000 −0.212132
\(51\) 0 0
\(52\) −46.0000 −0.122674
\(53\) −78.0000 −0.202153 −0.101077 − 0.994879i \(-0.532229\pi\)
−0.101077 + 0.994879i \(0.532229\pi\)
\(54\) 0 0
\(55\) −150.000 −0.367745
\(56\) −462.000 −1.10245
\(57\) 0 0
\(58\) −198.000 −0.448253
\(59\) −432.000 −0.953248 −0.476624 − 0.879107i \(-0.658140\pi\)
−0.476624 + 0.879107i \(0.658140\pi\)
\(60\) 0 0
\(61\) −610.000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −582.000 −1.19216
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 230.000 0.438892
\(66\) 0 0
\(67\) 848.000 1.54626 0.773132 − 0.634245i \(-0.218689\pi\)
0.773132 + 0.634245i \(0.218689\pi\)
\(68\) −17.0000 −0.0303170
\(69\) 0 0
\(70\) −330.000 −0.563465
\(71\) 174.000 0.290845 0.145423 − 0.989370i \(-0.453546\pi\)
0.145423 + 0.989370i \(0.453546\pi\)
\(72\) 0 0
\(73\) 362.000 0.580396 0.290198 − 0.956967i \(-0.406279\pi\)
0.290198 + 0.956967i \(0.406279\pi\)
\(74\) −618.000 −0.970825
\(75\) 0 0
\(76\) 104.000 0.156969
\(77\) −660.000 −0.976805
\(78\) 0 0
\(79\) 398.000 0.566816 0.283408 − 0.958999i \(-0.408535\pi\)
0.283408 + 0.958999i \(0.408535\pi\)
\(80\) 355.000 0.496128
\(81\) 0 0
\(82\) −378.000 −0.509062
\(83\) −828.000 −1.09500 −0.547499 − 0.836806i \(-0.684420\pi\)
−0.547499 + 0.836806i \(0.684420\pi\)
\(84\) 0 0
\(85\) 85.0000 0.108465
\(86\) 1164.00 1.45950
\(87\) 0 0
\(88\) 630.000 0.763162
\(89\) −630.000 −0.750336 −0.375168 − 0.926957i \(-0.622415\pi\)
−0.375168 + 0.926957i \(0.622415\pi\)
\(90\) 0 0
\(91\) 1012.00 1.16578
\(92\) −42.0000 −0.0475957
\(93\) 0 0
\(94\) −1620.00 −1.77756
\(95\) −520.000 −0.561588
\(96\) 0 0
\(97\) −1486.00 −1.55547 −0.777734 − 0.628593i \(-0.783631\pi\)
−0.777734 + 0.628593i \(0.783631\pi\)
\(98\) −423.000 −0.436015
\(99\) 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 765.4.a.a.1.1 1
3.2 odd 2 85.4.a.c.1.1 ✓ 1
12.11 even 2 1360.4.a.a.1.1 1
15.2 even 4 425.4.b.d.324.2 2
15.8 even 4 425.4.b.d.324.1 2
15.14 odd 2 425.4.a.a.1.1 1
51.50 odd 2 1445.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.c.1.1 ✓ 1 3.2 odd 2
425.4.a.a.1.1 1 15.14 odd 2
425.4.b.d.324.1 2 15.8 even 4
425.4.b.d.324.2 2 15.2 even 4
765.4.a.a.1.1 1 1.1 even 1 trivial
1360.4.a.a.1.1 1 12.11 even 2
1445.4.a.f.1.1 1 51.50 odd 2