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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [760,2,Mod(267,760)] 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("760.267"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [108,0] 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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 267.6
Character \(\chi\) \(=\) 760.267
Dual form 760.2.w.d.723.6

$q$-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+(-1.38164 + 0.301766i) q^{2} +(2.05043 - 2.05043i) q^{3} +(1.81787 - 0.833866i) q^{4} +(1.90605 - 1.16918i) q^{5} +(-2.21421 + 3.45171i) q^{6} +(1.97140 - 1.97140i) q^{7} +(-2.26002 + 1.70068i) q^{8} -5.40854i q^{9} +(-2.28066 + 2.19057i) q^{10} -2.36931 q^{11} +(2.01764 - 5.43721i) q^{12} +(5.02569 + 5.02569i) q^{13} +(-2.12887 + 3.31868i) q^{14} +(1.51091 - 6.30554i) q^{15} +(2.60933 - 3.03173i) q^{16} +(-0.294068 - 0.294068i) q^{17} +(1.63211 + 7.47267i) q^{18} +1.00000i q^{19} +(2.49002 - 3.71481i) q^{20} -8.08446i q^{21} +(3.27355 - 0.714979i) q^{22} +(-2.13287 - 2.13287i) q^{23} +(-1.14689 + 8.12114i) q^{24} +(2.26605 - 4.45702i) q^{25} +(-8.46030 - 5.42713i) q^{26} +(-4.93854 - 4.93854i) q^{27} +(1.93988 - 5.22765i) q^{28} +4.11854 q^{29} +(-0.184739 + 9.16795i) q^{30} +7.97583i q^{31} +(-2.69029 + 4.97618i) q^{32} +(-4.85812 + 4.85812i) q^{33} +(0.495037 + 0.317557i) q^{34} +(1.45268 - 6.06251i) q^{35} +(-4.51000 - 9.83204i) q^{36} +(-2.00664 + 2.00664i) q^{37} +(-0.301766 - 1.38164i) q^{38} +20.6097 q^{39} +(-2.31932 + 5.88394i) q^{40} -2.40341 q^{41} +(2.43962 + 11.1698i) q^{42} +(-8.86767 + 8.86767i) q^{43} +(-4.30712 + 1.97569i) q^{44} +(-6.32353 - 10.3089i) q^{45} +(3.59049 + 2.30323i) q^{46} +(-8.27530 + 8.27530i) q^{47} +(-0.866092 - 11.5666i) q^{48} -0.772871i q^{49} +(-1.78590 + 6.84182i) q^{50} -1.20593 q^{51} +(13.3268 + 4.94532i) q^{52} +(-3.32898 - 3.32898i) q^{53} +(8.31358 + 5.33301i) q^{54} +(-4.51603 + 2.77015i) q^{55} +(-1.10269 + 7.80814i) q^{56} +(2.05043 + 2.05043i) q^{57} +(-5.69035 + 1.24284i) q^{58} -10.0011i q^{59} +(-2.51133 - 12.7226i) q^{60} +0.342326i q^{61} +(-2.40684 - 11.0197i) q^{62} +(-10.6624 - 10.6624i) q^{63} +(2.21538 - 7.68714i) q^{64} +(15.4551 + 3.70330i) q^{65} +(5.24617 - 8.17820i) q^{66} +(-10.0633 - 10.0633i) q^{67} +(-0.779792 - 0.289365i) q^{68} -8.74659 q^{69} +(-0.177619 + 8.81460i) q^{70} +11.2920i q^{71} +(9.19818 + 12.2234i) q^{72} +(1.09839 - 1.09839i) q^{73} +(2.16693 - 3.37800i) q^{74} +(-4.49242 - 13.7852i) q^{75} +(0.833866 + 1.81787i) q^{76} +(-4.67088 + 4.67088i) q^{77} +(-28.4752 + 6.21930i) q^{78} -2.68484 q^{79} +(1.42890 - 8.82940i) q^{80} -4.02665 q^{81} +(3.32066 - 0.725270i) q^{82} +(0.229034 - 0.229034i) q^{83} +(-6.74136 - 14.6965i) q^{84} +(-0.904326 - 0.216691i) q^{85} +(9.57599 - 14.9279i) q^{86} +(8.44478 - 8.44478i) q^{87} +(5.35470 - 4.02944i) q^{88} -5.49535i q^{89} +(11.8478 + 12.3350i) q^{90} +19.8153 q^{91} +(-5.65581 - 2.09876i) q^{92} +(16.3539 + 16.3539i) q^{93} +(8.93631 - 13.9307i) q^{94} +(1.16918 + 1.90605i) q^{95} +(4.68704 + 15.7196i) q^{96} +(9.15978 + 9.15978i) q^{97} +(0.233227 + 1.06783i) q^{98} +12.8145i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 8 q^{6} + 8 q^{10} + 18 q^{12} + 16 q^{16} + 4 q^{17} - 36 q^{22} - 4 q^{25} - 40 q^{26} + 16 q^{28} - 12 q^{30} - 10 q^{32} + 24 q^{35} - 80 q^{36} + 28 q^{40} + 108 q^{42} - 32 q^{43} + 64 q^{46}+ \cdots + 76 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38164 + 0.301766i −0.976969 + 0.213381i
\(3\) 2.05043 − 2.05043i 1.18382 − 1.18382i 0.205070 − 0.978747i \(-0.434258\pi\)
0.978747 − 0.205070i \(-0.0657421\pi\)
\(4\) 1.81787 − 0.833866i 0.908937 − 0.416933i
\(5\) 1.90605 − 1.16918i 0.852412 − 0.522871i
\(6\) −2.21421 + 3.45171i −0.903949 + 1.40916i
\(7\) 1.97140 − 1.97140i 0.745121 − 0.745121i −0.228438 − 0.973559i \(-0.573362\pi\)
0.973559 + 0.228438i \(0.0733617\pi\)
\(8\) −2.26002 + 1.70068i −0.799038 + 0.601281i
\(9\) − 5.40854i − 1.80285i
\(10\) −2.28066 + 2.19057i −0.721209 + 0.692718i
\(11\) −2.36931 −0.714375 −0.357188 − 0.934033i \(-0.616264\pi\)
−0.357188 + 0.934033i \(0.616264\pi\)
\(12\) 2.01764 − 5.43721i 0.582443 − 1.56959i
\(13\) 5.02569 + 5.02569i 1.39388 + 1.39388i 0.816425 + 0.577451i \(0.195953\pi\)
0.577451 + 0.816425i \(0.304047\pi\)
\(14\) −2.12887 + 3.31868i −0.568965 + 0.886955i
\(15\) 1.51091 − 6.30554i 0.390115 − 1.62808i
\(16\) 2.60933 − 3.03173i 0.652333 − 0.757932i
\(17\) −0.294068 − 0.294068i −0.0713220 − 0.0713220i 0.670546 − 0.741868i \(-0.266060\pi\)
−0.741868 + 0.670546i \(0.766060\pi\)
\(18\) 1.63211 + 7.47267i 0.384693 + 1.76132i
\(19\) 1.00000i 0.229416i
\(20\) 2.49002 − 3.71481i 0.556786 − 0.830656i
\(21\) − 8.08446i − 1.76417i
\(22\) 3.27355 − 0.714979i 0.697922 − 0.152434i
\(23\) −2.13287 − 2.13287i −0.444733 − 0.444733i 0.448866 − 0.893599i \(-0.351828\pi\)
−0.893599 + 0.448866i \(0.851828\pi\)
\(24\) −1.14689 + 8.12114i −0.234108 + 1.65772i
\(25\) 2.26605 − 4.45702i 0.453211 − 0.891403i
\(26\) −8.46030 − 5.42713i −1.65920 − 1.06435i
\(27\) −4.93854 − 4.93854i −0.950422 − 0.950422i
\(28\) 1.93988 − 5.22765i 0.366602 − 0.987934i
\(29\) 4.11854 0.764793 0.382397 − 0.923998i \(-0.375099\pi\)
0.382397 + 0.923998i \(0.375099\pi\)
\(30\) −0.184739 + 9.16795i −0.0337285 + 1.67383i
\(31\) 7.97583i 1.43250i 0.697843 + 0.716250i \(0.254143\pi\)
−0.697843 + 0.716250i \(0.745857\pi\)
\(32\) −2.69029 + 4.97618i −0.475581 + 0.879672i
\(33\) −4.85812 + 4.85812i −0.845690 + 0.845690i
\(34\) 0.495037 + 0.317557i 0.0848981 + 0.0544606i
\(35\) 1.45268 − 6.06251i 0.245547 − 1.02475i
\(36\) −4.51000 − 9.83204i −0.751666 − 1.63867i
\(37\) −2.00664 + 2.00664i −0.329890 + 0.329890i −0.852545 − 0.522654i \(-0.824942\pi\)
0.522654 + 0.852545i \(0.324942\pi\)
\(38\) −0.301766 − 1.38164i −0.0489530 − 0.224132i
\(39\) 20.6097 3.30019
\(40\) −2.31932 + 5.88394i −0.366717 + 0.930333i
\(41\) −2.40341 −0.375350 −0.187675 − 0.982231i \(-0.560095\pi\)
−0.187675 + 0.982231i \(0.560095\pi\)
\(42\) 2.43962 + 11.1698i 0.376441 + 1.72354i
\(43\) −8.86767 + 8.86767i −1.35231 + 1.35231i −0.469233 + 0.883074i \(0.655470\pi\)
−0.883074 + 0.469233i \(0.844530\pi\)
\(44\) −4.30712 + 1.97569i −0.649322 + 0.297847i
\(45\) −6.32353 − 10.3089i −0.942656 − 1.53677i
\(46\) 3.59049 + 2.30323i 0.529388 + 0.339593i
\(47\) −8.27530 + 8.27530i −1.20708 + 1.20708i −0.235108 + 0.971969i \(0.575544\pi\)
−0.971969 + 0.235108i \(0.924456\pi\)
\(48\) −0.866092 − 11.5666i −0.125010 − 1.66950i
\(49\) − 0.772871i − 0.110410i
\(50\) −1.78590 + 6.84182i −0.252565 + 0.967580i
\(51\) −1.20593 −0.168864
\(52\) 13.3268 + 4.94532i 1.84810 + 0.685793i
\(53\) −3.32898 − 3.32898i −0.457270 − 0.457270i 0.440488 − 0.897758i \(-0.354805\pi\)
−0.897758 + 0.440488i \(0.854805\pi\)
\(54\) 8.31358 + 5.33301i 1.13133 + 0.725731i
\(55\) −4.51603 + 2.77015i −0.608942 + 0.373526i
\(56\) −1.10269 + 7.80814i −0.147353 + 1.04341i
\(57\) 2.05043 + 2.05043i 0.271586 + 0.271586i
\(58\) −5.69035 + 1.24284i −0.747179 + 0.163192i
\(59\) − 10.0011i − 1.30203i −0.759063 − 0.651017i \(-0.774343\pi\)
0.759063 − 0.651017i \(-0.225657\pi\)
\(60\) −2.51133 − 12.7226i −0.324212 − 1.64248i
\(61\) 0.342326i 0.0438303i 0.999760 + 0.0219152i \(0.00697637\pi\)
−0.999760 + 0.0219152i \(0.993024\pi\)
\(62\) −2.40684 − 11.0197i −0.305668 − 1.39951i
\(63\) −10.6624 − 10.6624i −1.34334 − 1.34334i
\(64\) 2.21538 − 7.68714i 0.276923 − 0.960892i
\(65\) 15.4551 + 3.70330i 1.91697 + 0.459338i
\(66\) 5.24617 − 8.17820i 0.645758 − 1.00667i
\(67\) −10.0633 − 10.0633i −1.22943 − 1.22943i −0.964179 − 0.265252i \(-0.914545\pi\)
−0.265252 − 0.964179i \(-0.585455\pi\)
\(68\) −0.779792 − 0.289365i −0.0945637 − 0.0350907i
\(69\) −8.74659 −1.05297
\(70\) −0.177619 + 8.81460i −0.0212295 + 1.05355i
\(71\) 11.2920i 1.34011i 0.742312 + 0.670054i \(0.233729\pi\)
−0.742312 + 0.670054i \(0.766271\pi\)
\(72\) 9.19818 + 12.2234i 1.08402 + 1.44054i
\(73\) 1.09839 − 1.09839i 0.128557 − 0.128557i −0.639900 − 0.768458i \(-0.721024\pi\)
0.768458 + 0.639900i \(0.221024\pi\)
\(74\) 2.16693 − 3.37800i 0.251900 − 0.392685i
\(75\) −4.49242 − 13.7852i −0.518739 − 1.59178i
\(76\) 0.833866 + 1.81787i 0.0956510 + 0.208524i
\(77\) −4.67088 + 4.67088i −0.532296 + 0.532296i
\(78\) −28.4752 + 6.21930i −3.22418 + 0.704198i
\(79\) −2.68484 −0.302068 −0.151034 − 0.988529i \(-0.548260\pi\)
−0.151034 + 0.988529i \(0.548260\pi\)
\(80\) 1.42890 − 8.82940i 0.159755 − 0.987157i
\(81\) −4.02665 −0.447406
\(82\) 3.32066 − 0.725270i 0.366706 − 0.0800926i
\(83\) 0.229034 − 0.229034i 0.0251397 − 0.0251397i −0.694425 − 0.719565i \(-0.744341\pi\)
0.719565 + 0.694425i \(0.244341\pi\)
\(84\) −6.74136 − 14.6965i −0.735543 − 1.60352i
\(85\) −0.904326 − 0.216691i −0.0980879 − 0.0235035i
\(86\) 9.57599 − 14.9279i 1.03261 − 1.60972i
\(87\) 8.44478 − 8.44478i 0.905375 − 0.905375i
\(88\) 5.35470 − 4.02944i 0.570813 − 0.429540i
\(89\) − 5.49535i − 0.582505i −0.956646 − 0.291253i \(-0.905928\pi\)
0.956646 − 0.291253i \(-0.0940720\pi\)
\(90\) 11.8478 + 12.3350i 1.24886 + 1.30023i
\(91\) 19.8153 2.07721
\(92\) −5.65581 − 2.09876i −0.589659 − 0.218811i
\(93\) 16.3539 + 16.3539i 1.69582 + 1.69582i
\(94\) 8.93631 − 13.9307i 0.921710 − 1.43684i
\(95\) 1.16918 + 1.90605i 0.119955 + 0.195557i
\(96\) 4.68704 + 15.7196i 0.478369 + 1.60437i
\(97\) 9.15978 + 9.15978i 0.930035 + 0.930035i 0.997708 − 0.0676725i \(-0.0215573\pi\)
−0.0676725 + 0.997708i \(0.521557\pi\)
\(98\) 0.233227 + 1.06783i 0.0235594 + 0.107867i
\(99\) 12.8145i 1.28791i
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 760.2.w.d.267.6 ✓ 108
5.3 odd 4 inner 760.2.w.d.723.32 yes 108
8.3 odd 2 inner 760.2.w.d.267.32 yes 108
40.3 even 4 inner 760.2.w.d.723.6 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.w.d.267.6 ✓ 108 1.1 even 1 trivial
760.2.w.d.267.32 yes 108 8.3 odd 2 inner
760.2.w.d.723.6 yes 108 40.3 even 4 inner
760.2.w.d.723.32 yes 108 5.3 odd 4 inner