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Properties

Label 765.4.g.a.271.2
Level $765$
Weight $4$
Character 765.271
Analytic conductor $45.136$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [765,4,Mod(271,765)] 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("765.271"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 765.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(45.1364611544\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 271.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 765.271
Dual form 765.4.g.a.271.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -7.00000 q^{4} +5.00000i q^{5} +14.0000i q^{7} +15.0000 q^{8} -5.00000i q^{10} +20.0000i q^{11} -58.0000 q^{13} -14.0000i q^{14} +41.0000 q^{16} +(17.0000 + 68.0000i) q^{17} +80.0000 q^{19} -35.0000i q^{20} -20.0000i q^{22} +118.000i q^{23} -25.0000 q^{25} +58.0000 q^{26} -98.0000i q^{28} +126.000i q^{29} -70.0000i q^{31} -161.000 q^{32} +(-17.0000 - 68.0000i) q^{34} -70.0000 q^{35} +134.000i q^{37} -80.0000 q^{38} +75.0000i q^{40} -100.000i q^{41} +272.000 q^{43} -140.000i q^{44} -118.000i q^{46} +464.000 q^{47} +147.000 q^{49} +25.0000 q^{50} +406.000 q^{52} -642.000 q^{53} -100.000 q^{55} +210.000i q^{56} -126.000i q^{58} +180.000 q^{59} -110.000i q^{61} +70.0000i q^{62} -167.000 q^{64} -290.000i q^{65} -924.000 q^{67} +(-119.000 - 476.000i) q^{68} +70.0000 q^{70} +90.0000i q^{71} -828.000i q^{73} -134.000i q^{74} -560.000 q^{76} -280.000 q^{77} +1334.00i q^{79} +205.000i q^{80} +100.000i q^{82} -552.000 q^{83} +(-340.000 + 85.0000i) q^{85} -272.000 q^{86} +300.000i q^{88} -1490.00 q^{89} -812.000i q^{91} -826.000i q^{92} -464.000 q^{94} +400.000i q^{95} -1376.00i q^{97} -147.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 14 q^{4} + 30 q^{8} - 116 q^{13} + 82 q^{16} + 34 q^{17} + 160 q^{19} - 50 q^{25} + 116 q^{26} - 322 q^{32} - 34 q^{34} - 140 q^{35} - 160 q^{38} + 544 q^{43} + 928 q^{47} + 294 q^{49} + 50 q^{50}+ \cdots - 294 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/765\mathbb{Z}\right)^\times\).

\(n\) \(307\) \(496\) \(596\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 − 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.875000
\(5\) 5.00000i 0.447214i
\(6\) 0 0
\(7\) 14.0000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 15.0000 0.662913
\(9\) 0 0
\(10\) − 5.00000i − 0.158114i
\(11\) 20.0000i 0.548202i 0.961701 + 0.274101i \(0.0883803\pi\)
−0.961701 + 0.274101i \(0.911620\pi\)
\(12\) 0 0
\(13\) −58.0000 −1.23741 −0.618704 − 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) − 14.0000i − 0.267261i
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) 17.0000 + 68.0000i 0.242536 + 0.970143i
\(18\) 0 0
\(19\) 80.0000 0.965961 0.482980 − 0.875631i \(-0.339554\pi\)
0.482980 + 0.875631i \(0.339554\pi\)
\(20\) − 35.0000i − 0.391312i
\(21\) 0 0
\(22\) − 20.0000i − 0.193819i
\(23\) 118.000i 1.06977i 0.844925 + 0.534885i \(0.179645\pi\)
−0.844925 + 0.534885i \(0.820355\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 58.0000 0.437490
\(27\) 0 0
\(28\) − 98.0000i − 0.661438i
\(29\) 126.000i 0.806814i 0.915021 + 0.403407i \(0.132174\pi\)
−0.915021 + 0.403407i \(0.867826\pi\)
\(30\) 0 0
\(31\) − 70.0000i − 0.405560i −0.979224 − 0.202780i \(-0.935002\pi\)
0.979224 − 0.202780i \(-0.0649977\pi\)
\(32\) −161.000 −0.889408
\(33\) 0 0
\(34\) −17.0000 − 68.0000i −0.0857493 − 0.342997i
\(35\) −70.0000 −0.338062
\(36\) 0 0
\(37\) 134.000i 0.595391i 0.954661 + 0.297695i \(0.0962180\pi\)
−0.954661 + 0.297695i \(0.903782\pi\)
\(38\) −80.0000 −0.341519
\(39\) 0 0
\(40\) 75.0000i 0.296464i
\(41\) − 100.000i − 0.380912i −0.981696 − 0.190456i \(-0.939003\pi\)
0.981696 − 0.190456i \(-0.0609966\pi\)
\(42\) 0 0
\(43\) 272.000 0.964642 0.482321 − 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(44\) − 140.000i − 0.479677i
\(45\) 0 0
\(46\) − 118.000i − 0.378221i
\(47\) 464.000 1.44003 0.720014 − 0.693959i \(-0.244135\pi\)
0.720014 + 0.693959i \(0.244135\pi\)
\(48\) 0 0
\(49\) 147.000 0.428571
\(50\) 25.0000 0.0707107
\(51\) 0 0
\(52\) 406.000 1.08273
\(53\) −642.000 −1.66388 −0.831939 − 0.554868i \(-0.812769\pi\)
−0.831939 + 0.554868i \(0.812769\pi\)
\(54\) 0 0
\(55\) −100.000 −0.245164
\(56\) 210.000i 0.501115i
\(57\) 0 0
\(58\) − 126.000i − 0.285252i
\(59\) 180.000 0.397187 0.198593 − 0.980082i \(-0.436363\pi\)
0.198593 + 0.980082i \(0.436363\pi\)
\(60\) 0 0
\(61\) − 110.000i − 0.230886i −0.993314 − 0.115443i \(-0.963171\pi\)
0.993314 − 0.115443i \(-0.0368288\pi\)
\(62\) 70.0000i 0.143387i
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) − 290.000i − 0.553386i
\(66\) 0 0
\(67\) −924.000 −1.68484 −0.842422 − 0.538818i \(-0.818871\pi\)
−0.842422 + 0.538818i \(0.818871\pi\)
\(68\) −119.000 − 476.000i −0.212219 − 0.848875i
\(69\) 0 0
\(70\) 70.0000 0.119523
\(71\) 90.0000i 0.150437i 0.997167 + 0.0752186i \(0.0239654\pi\)
−0.997167 + 0.0752186i \(0.976035\pi\)
\(72\) 0 0
\(73\) − 828.000i − 1.32754i −0.747939 − 0.663768i \(-0.768956\pi\)
0.747939 − 0.663768i \(-0.231044\pi\)
\(74\) − 134.000i − 0.210502i
\(75\) 0 0
\(76\) −560.000 −0.845216
\(77\) −280.000 −0.414402
\(78\) 0 0
\(79\) 1334.00i 1.89983i 0.312505 + 0.949916i \(0.398832\pi\)
−0.312505 + 0.949916i \(0.601168\pi\)
\(80\) 205.000i 0.286496i
\(81\) 0 0
\(82\) 100.000i 0.134673i
\(83\) −552.000 −0.729998 −0.364999 − 0.931008i \(-0.618931\pi\)
−0.364999 + 0.931008i \(0.618931\pi\)
\(84\) 0 0
\(85\) −340.000 + 85.0000i −0.433861 + 0.108465i
\(86\) −272.000 −0.341052
\(87\) 0 0
\(88\) 300.000i 0.363410i
\(89\) −1490.00 −1.77460 −0.887302 − 0.461190i \(-0.847423\pi\)
−0.887302 + 0.461190i \(0.847423\pi\)
\(90\) 0 0
\(91\) − 812.000i − 0.935393i
\(92\) − 826.000i − 0.936048i
\(93\) 0 0
\(94\) −464.000 −0.509127
\(95\) 400.000i 0.431991i
\(96\) 0 0
\(97\) − 1376.00i − 1.44033i −0.693805 − 0.720163i \(-0.744067\pi\)
0.693805 − 0.720163i \(-0.255933\pi\)
\(98\) −147.000 −0.151523
\(99\) 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.g.a.271.2 2
3.2 odd 2 85.4.d.a.16.1 ✓ 2
15.2 even 4 425.4.c.a.424.2 2
15.8 even 4 425.4.c.b.424.1 2
15.14 odd 2 425.4.d.a.101.2 2
17.16 even 2 inner 765.4.g.a.271.1 2
51.38 odd 4 1445.4.a.d.1.1 1
51.47 odd 4 1445.4.a.e.1.1 1
51.50 odd 2 85.4.d.a.16.2 yes 2
255.152 even 4 425.4.c.b.424.2 2
255.203 even 4 425.4.c.a.424.1 2
255.254 odd 2 425.4.d.a.101.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.d.a.16.1 ✓ 2 3.2 odd 2
85.4.d.a.16.2 yes 2 51.50 odd 2
425.4.c.a.424.1 2 255.203 even 4
425.4.c.a.424.2 2 15.2 even 4
425.4.c.b.424.1 2 15.8 even 4
425.4.c.b.424.2 2 255.152 even 4
425.4.d.a.101.1 2 255.254 odd 2
425.4.d.a.101.2 2 15.14 odd 2
765.4.g.a.271.1 2 17.16 even 2 inner
765.4.g.a.271.2 2 1.1 even 1 trivial
1445.4.a.d.1.1 1 51.38 odd 4
1445.4.a.e.1.1 1 51.47 odd 4