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Properties

Label 765.2.be.b.586.4
Level $765$
Weight $2$
Character 765.586
Analytic conductor $6.109$
Analytic rank $0$
Dimension $24$
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] = [765,2,Mod(406,765)] 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("765.406"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.be (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.10855575463\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 586.4
Character \(\chi\) \(=\) 765.586
Dual form 765.2.be.b.406.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.254738 - 0.254738i) q^{2} +1.87022i q^{4} +(-0.923880 - 0.382683i) q^{5} +(0.275980 - 0.114315i) q^{7} +(0.985893 + 0.985893i) q^{8} +(-0.332832 + 0.137863i) q^{10} +(1.05900 + 2.55665i) q^{11} -1.97956i q^{13} +(0.0411823 - 0.0994229i) q^{14} -3.23814 q^{16} +(1.21202 + 3.94094i) q^{17} +(-1.99331 + 1.99331i) q^{19} +(0.715701 - 1.72785i) q^{20} +(0.921046 + 0.381510i) q^{22} +(2.57919 + 6.22672i) q^{23} +(0.707107 + 0.707107i) q^{25} +(-0.504269 - 0.504269i) q^{26} +(0.213793 + 0.516142i) q^{28} +(-4.36632 - 1.80859i) q^{29} +(1.15808 - 2.79584i) q^{31} +(-2.79667 + 2.79667i) q^{32} +(1.31266 + 0.695159i) q^{34} -0.298718 q^{35} +(-3.60537 + 8.70414i) q^{37} +1.01554i q^{38} +(-0.533561 - 1.28813i) q^{40} +(-2.87301 + 1.19004i) q^{41} +(5.78771 + 5.78771i) q^{43} +(-4.78150 + 1.98056i) q^{44} +(2.24320 + 0.929166i) q^{46} -1.08341i q^{47} +(-4.88665 + 4.88665i) q^{49} +0.360254 q^{50} +3.70220 q^{52} +(1.89858 - 1.89858i) q^{53} -2.76730i q^{55} +(0.384788 + 0.159384i) q^{56} +(-1.57299 + 0.651553i) q^{58} +(6.47310 + 6.47310i) q^{59} +(10.3418 - 4.28372i) q^{61} +(-0.417202 - 1.00722i) q^{62} -5.05145i q^{64} +(-0.757544 + 1.82887i) q^{65} -12.5585 q^{67} +(-7.37041 + 2.26675i) q^{68} +(-0.0760950 + 0.0760950i) q^{70} +(2.25315 - 5.43960i) q^{71} +(-0.200173 - 0.0829144i) q^{73} +(1.29885 + 3.13570i) q^{74} +(-3.72792 - 3.72792i) q^{76} +(0.584525 + 0.584525i) q^{77} +(-4.07771 - 9.84447i) q^{79} +(2.99165 + 1.23918i) q^{80} +(-0.428717 + 1.03501i) q^{82} +(11.0129 - 11.0129i) q^{83} +(0.388367 - 4.10477i) q^{85} +2.94870 q^{86} +(-1.47653 + 3.56465i) q^{88} +1.55264i q^{89} +(-0.226292 - 0.546318i) q^{91} +(-11.6453 + 4.82365i) q^{92} +(-0.275986 - 0.275986i) q^{94} +(2.60438 - 1.07877i) q^{95} +(8.28752 + 3.43280i) q^{97} +2.48963i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 8 q^{11} - 24 q^{16} + 8 q^{17} - 8 q^{19} - 32 q^{22} + 16 q^{23} - 16 q^{26} + 48 q^{28} + 8 q^{29} + 16 q^{34} + 32 q^{35} + 24 q^{37} + 16 q^{40} - 16 q^{41} + 8 q^{43} - 16 q^{44} + 8 q^{46}+ \cdots - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(307\) \(496\) \(596\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{8}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.254738 − 0.254738i 0.180127 − 0.180127i −0.611284 − 0.791411i \(-0.709347\pi\)
0.791411 + 0.611284i \(0.209347\pi\)
\(3\) 0 0
\(4\) 1.87022i 0.935108i
\(5\) −0.923880 − 0.382683i −0.413171 − 0.171141i
\(6\) 0 0
\(7\) 0.275980 − 0.114315i 0.104310 − 0.0432068i −0.329918 − 0.944010i \(-0.607021\pi\)
0.434228 + 0.900803i \(0.357021\pi\)
\(8\) 0.985893 + 0.985893i 0.348566 + 0.348566i
\(9\) 0 0
\(10\) −0.332832 + 0.137863i −0.105251 + 0.0435962i
\(11\) 1.05900 + 2.55665i 0.319301 + 0.770860i 0.999291 + 0.0376394i \(0.0119838\pi\)
−0.679991 + 0.733221i \(0.738016\pi\)
\(12\) 0 0
\(13\) − 1.97956i − 0.549030i −0.961583 − 0.274515i \(-0.911483\pi\)
0.961583 − 0.274515i \(-0.0885174\pi\)
\(14\) 0.0411823 − 0.0994229i 0.0110064 − 0.0265719i
\(15\) 0 0
\(16\) −3.23814 −0.809536
\(17\) 1.21202 + 3.94094i 0.293959 + 0.955818i
\(18\) 0 0
\(19\) −1.99331 + 1.99331i −0.457296 + 0.457296i −0.897767 − 0.440471i \(-0.854812\pi\)
0.440471 + 0.897767i \(0.354812\pi\)
\(20\) 0.715701 − 1.72785i 0.160036 − 0.386360i
\(21\) 0 0
\(22\) 0.921046 + 0.381510i 0.196368 + 0.0813382i
\(23\) 2.57919 + 6.22672i 0.537799 + 1.29836i 0.926256 + 0.376895i \(0.123008\pi\)
−0.388457 + 0.921467i \(0.626992\pi\)
\(24\) 0 0
\(25\) 0.707107 + 0.707107i 0.141421 + 0.141421i
\(26\) −0.504269 − 0.504269i −0.0988953 − 0.0988953i
\(27\) 0 0
\(28\) 0.213793 + 0.516142i 0.0404031 + 0.0975416i
\(29\) −4.36632 − 1.80859i −0.810806 − 0.335847i −0.0615305 − 0.998105i \(-0.519598\pi\)
−0.749276 + 0.662258i \(0.769598\pi\)
\(30\) 0 0
\(31\) 1.15808 − 2.79584i 0.207997 − 0.502148i −0.785111 − 0.619355i \(-0.787394\pi\)
0.993108 + 0.117207i \(0.0373941\pi\)
\(32\) −2.79667 + 2.79667i −0.494385 + 0.494385i
\(33\) 0 0
\(34\) 1.31266 + 0.695159i 0.225119 + 0.119219i
\(35\) −0.298718 −0.0504926
\(36\) 0 0
\(37\) −3.60537 + 8.70414i −0.592719 + 1.43095i 0.288147 + 0.957586i \(0.406961\pi\)
−0.880866 + 0.473365i \(0.843039\pi\)
\(38\) 1.01554i 0.164743i
\(39\) 0 0
\(40\) −0.533561 − 1.28813i −0.0843635 − 0.203671i
\(41\) −2.87301 + 1.19004i −0.448688 + 0.185853i −0.595574 − 0.803301i \(-0.703075\pi\)
0.146885 + 0.989154i \(0.453075\pi\)
\(42\) 0 0
\(43\) 5.78771 + 5.78771i 0.882617 + 0.882617i 0.993800 − 0.111183i \(-0.0354638\pi\)
−0.111183 + 0.993800i \(0.535464\pi\)
\(44\) −4.78150 + 1.98056i −0.720838 + 0.298581i
\(45\) 0 0
\(46\) 2.24320 + 0.929166i 0.330742 + 0.136998i
\(47\) − 1.08341i − 0.158032i −0.996873 − 0.0790159i \(-0.974822\pi\)
0.996873 − 0.0790159i \(-0.0251778\pi\)
\(48\) 0 0
\(49\) −4.88665 + 4.88665i −0.698093 + 0.698093i
\(50\) 0.360254 0.0509477
\(51\) 0 0
\(52\) 3.70220 0.513403
\(53\) 1.89858 − 1.89858i 0.260790 − 0.260790i −0.564585 − 0.825375i \(-0.690964\pi\)
0.825375 + 0.564585i \(0.190964\pi\)
\(54\) 0 0
\(55\) − 2.76730i − 0.373143i
\(56\) 0.384788 + 0.159384i 0.0514195 + 0.0212986i
\(57\) 0 0
\(58\) −1.57299 + 0.651553i −0.206543 + 0.0855531i
\(59\) 6.47310 + 6.47310i 0.842726 + 0.842726i 0.989213 − 0.146487i \(-0.0467966\pi\)
−0.146487 + 0.989213i \(0.546797\pi\)
\(60\) 0 0
\(61\) 10.3418 − 4.28372i 1.32413 − 0.548474i 0.395158 − 0.918613i \(-0.370690\pi\)
0.928976 + 0.370139i \(0.120690\pi\)
\(62\) −0.417202 − 1.00722i −0.0529847 − 0.127916i
\(63\) 0 0
\(64\) − 5.05145i − 0.631432i
\(65\) −0.757544 + 1.82887i −0.0939617 + 0.226844i
\(66\) 0 0
\(67\) −12.5585 −1.53427 −0.767133 − 0.641488i \(-0.778317\pi\)
−0.767133 + 0.641488i \(0.778317\pi\)
\(68\) −7.37041 + 2.26675i −0.893793 + 0.274884i
\(69\) 0 0
\(70\) −0.0760950 + 0.0760950i −0.00909509 + 0.00909509i
\(71\) 2.25315 − 5.43960i 0.267400 − 0.645561i −0.731959 − 0.681348i \(-0.761394\pi\)
0.999359 + 0.0357872i \(0.0113939\pi\)
\(72\) 0 0
\(73\) −0.200173 − 0.0829144i −0.0234285 − 0.00970440i 0.370938 − 0.928657i \(-0.379036\pi\)
−0.394367 + 0.918953i \(0.629036\pi\)
\(74\) 1.29885 + 3.13570i 0.150988 + 0.364518i
\(75\) 0 0
\(76\) −3.72792 − 3.72792i −0.427622 − 0.427622i
\(77\) 0.584525 + 0.584525i 0.0666128 + 0.0666128i
\(78\) 0 0
\(79\) −4.07771 − 9.84447i −0.458779 − 1.10759i −0.968892 − 0.247482i \(-0.920397\pi\)
0.510114 − 0.860107i \(-0.329603\pi\)
\(80\) 2.99165 + 1.23918i 0.334477 + 0.138545i
\(81\) 0 0
\(82\) −0.428717 + 1.03501i −0.0473438 + 0.114298i
\(83\) 11.0129 − 11.0129i 1.20883 − 1.20883i 0.237421 − 0.971407i \(-0.423698\pi\)
0.971407 − 0.237421i \(-0.0763020\pi\)
\(84\) 0 0
\(85\) 0.388367 − 4.10477i 0.0421243 − 0.445225i
\(86\) 2.94870 0.317967
\(87\) 0 0
\(88\) −1.47653 + 3.56465i −0.157398 + 0.379993i
\(89\) 1.55264i 0.164579i 0.996608 + 0.0822897i \(0.0262233\pi\)
−0.996608 + 0.0822897i \(0.973777\pi\)
\(90\) 0 0
\(91\) −0.226292 − 0.546318i −0.0237219 − 0.0572696i
\(92\) −11.6453 + 4.82365i −1.21411 + 0.502900i
\(93\) 0 0
\(94\) −0.275986 − 0.275986i −0.0284658 − 0.0284658i
\(95\) 2.60438 − 1.07877i 0.267204 − 0.110680i
\(96\) 0 0
\(97\) 8.28752 + 3.43280i 0.841471 + 0.348549i 0.761433 − 0.648243i \(-0.224496\pi\)
0.0800373 + 0.996792i \(0.474496\pi\)
\(98\) 2.48963i 0.251491i
\(99\) 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 765.2.be.b.586.4 24
3.2 odd 2 85.2.l.a.76.3 yes 24
15.2 even 4 425.2.n.f.399.3 24
15.8 even 4 425.2.n.c.399.4 24
15.14 odd 2 425.2.m.b.76.4 24
17.15 even 8 inner 765.2.be.b.406.4 24
51.11 even 16 1445.2.d.j.866.10 24
51.23 even 16 1445.2.d.j.866.9 24
51.32 odd 8 85.2.l.a.66.3 ✓ 24
51.41 even 16 1445.2.a.q.1.8 12
51.44 even 16 1445.2.a.p.1.8 12
255.32 even 8 425.2.n.c.49.4 24
255.44 even 16 7225.2.a.bs.1.5 12
255.83 even 8 425.2.n.f.49.3 24
255.134 odd 8 425.2.m.b.151.4 24
255.194 even 16 7225.2.a.bq.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.66.3 ✓ 24 51.32 odd 8
85.2.l.a.76.3 yes 24 3.2 odd 2
425.2.m.b.76.4 24 15.14 odd 2
425.2.m.b.151.4 24 255.134 odd 8
425.2.n.c.49.4 24 255.32 even 8
425.2.n.c.399.4 24 15.8 even 4
425.2.n.f.49.3 24 255.83 even 8
425.2.n.f.399.3 24 15.2 even 4
765.2.be.b.406.4 24 17.15 even 8 inner
765.2.be.b.586.4 24 1.1 even 1 trivial
1445.2.a.p.1.8 12 51.44 even 16
1445.2.a.q.1.8 12 51.41 even 16
1445.2.d.j.866.9 24 51.23 even 16
1445.2.d.j.866.10 24 51.11 even 16
7225.2.a.bq.1.5 12 255.194 even 16
7225.2.a.bs.1.5 12 255.44 even 16