Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,4,Mod(599,1150)] 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("1150.599"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-40,0,20,0,0,-168,0,-52] 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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 214x^{8} + 15751x^{6} + 460323x^{4} + 4609305x^{2} + 8503056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.1
Root \(-9.27140i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.r.599.10

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -8.27140i q^{3} -4.00000 q^{4} -16.5428 q^{6} -22.8621i q^{7} +8.00000i q^{8} -41.4161 q^{9} -0.987996 q^{11} +33.0856i q^{12} -4.22276i q^{13} -45.7241 q^{14} +16.0000 q^{16} -73.8741i q^{17} +82.8322i q^{18} -71.1586 q^{19} -189.101 q^{21} +1.97599i q^{22} +23.0000i q^{23} +66.1712 q^{24} -8.44552 q^{26} +119.241i q^{27} +91.4483i q^{28} -27.8984 q^{29} -136.861 q^{31} -32.0000i q^{32} +8.17211i q^{33} -147.748 q^{34} +165.664 q^{36} +201.718i q^{37} +142.317i q^{38} -34.9281 q^{39} -204.140 q^{41} +378.203i q^{42} -54.1024i q^{43} +3.95198 q^{44} +46.0000 q^{46} +41.4490i q^{47} -132.342i q^{48} -179.674 q^{49} -611.042 q^{51} +16.8910i q^{52} +428.612i q^{53} +238.483 q^{54} +182.897 q^{56} +588.581i q^{57} +55.7969i q^{58} +164.858 q^{59} +188.248 q^{61} +273.722i q^{62} +946.857i q^{63} -64.0000 q^{64} +16.3442 q^{66} -932.167i q^{67} +295.496i q^{68} +190.242 q^{69} -263.336 q^{71} -331.329i q^{72} +900.401i q^{73} +403.437 q^{74} +284.634 q^{76} +22.5876i q^{77} +69.8563i q^{78} +956.182 q^{79} -131.942 q^{81} +408.279i q^{82} +194.877i q^{83} +756.405 q^{84} -108.205 q^{86} +230.759i q^{87} -7.90397i q^{88} +213.751 q^{89} -96.5410 q^{91} -92.0000i q^{92} +1132.03i q^{93} +82.8981 q^{94} -264.685 q^{96} -628.762i q^{97} +359.348i q^{98} +40.9189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 40 q^{4} + 20 q^{6} - 168 q^{9} - 52 q^{11} + 12 q^{14} + 160 q^{16} - 148 q^{19} - 176 q^{21} - 80 q^{24} - 244 q^{26} - 74 q^{29} + 440 q^{31} - 924 q^{34} + 672 q^{36} - 390 q^{39} + 224 q^{41}+ \cdots - 4794 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) − 8.27140i − 1.59183i −0.605407 − 0.795916i \(-0.706990\pi\)
0.605407 − 0.795916i \(-0.293010\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −16.5428 −1.12560
\(7\) − 22.8621i − 1.23444i −0.786792 − 0.617218i \(-0.788260\pi\)
0.786792 − 0.617218i \(-0.211740\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −41.4161 −1.53393
\(10\) 0 0
\(11\) −0.987996 −0.0270811 −0.0135405 − 0.999908i \(-0.504310\pi\)
−0.0135405 + 0.999908i \(0.504310\pi\)
\(12\) 33.0856i 0.795916i
\(13\) − 4.22276i − 0.0900910i −0.998985 − 0.0450455i \(-0.985657\pi\)
0.998985 − 0.0450455i \(-0.0143433\pi\)
\(14\) −45.7241 −0.872878
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 73.8741i − 1.05395i −0.849882 − 0.526973i \(-0.823327\pi\)
0.849882 − 0.526973i \(-0.176673\pi\)
\(18\) 82.8322i 1.08465i
\(19\) −71.1586 −0.859205 −0.429602 − 0.903018i \(-0.641346\pi\)
−0.429602 + 0.903018i \(0.641346\pi\)
\(20\) 0 0
\(21\) −189.101 −1.96501
\(22\) 1.97599i 0.0191492i
\(23\) 23.0000i 0.208514i
\(24\) 66.1712 0.562798
\(25\) 0 0
\(26\) −8.44552 −0.0637039
\(27\) 119.241i 0.849926i
\(28\) 91.4483i 0.617218i
\(29\) −27.8984 −0.178642 −0.0893209 − 0.996003i \(-0.528470\pi\)
−0.0893209 + 0.996003i \(0.528470\pi\)
\(30\) 0 0
\(31\) −136.861 −0.792935 −0.396468 − 0.918049i \(-0.629764\pi\)
−0.396468 + 0.918049i \(0.629764\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 8.17211i 0.0431085i
\(34\) −147.748 −0.745253
\(35\) 0 0
\(36\) 165.664 0.766965
\(37\) 201.718i 0.896278i 0.893964 + 0.448139i \(0.147913\pi\)
−0.893964 + 0.448139i \(0.852087\pi\)
\(38\) 142.317i 0.607550i
\(39\) −34.9281 −0.143410
\(40\) 0 0
\(41\) −204.140 −0.777592 −0.388796 − 0.921324i \(-0.627109\pi\)
−0.388796 + 0.921324i \(0.627109\pi\)
\(42\) 378.203i 1.38947i
\(43\) − 54.1024i − 0.191873i −0.995387 − 0.0959365i \(-0.969415\pi\)
0.995387 − 0.0959365i \(-0.0305846\pi\)
\(44\) 3.95198 0.0135405
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 41.4490i 0.128638i 0.997929 + 0.0643188i \(0.0204874\pi\)
−0.997929 + 0.0643188i \(0.979513\pi\)
\(48\) − 132.342i − 0.397958i
\(49\) −179.674 −0.523831
\(50\) 0 0
\(51\) −611.042 −1.67771
\(52\) 16.8910i 0.0450455i
\(53\) 428.612i 1.11084i 0.831571 + 0.555418i \(0.187442\pi\)
−0.831571 + 0.555418i \(0.812558\pi\)
\(54\) 238.483 0.600988
\(55\) 0 0
\(56\) 182.897 0.436439
\(57\) 588.581i 1.36771i
\(58\) 55.7969i 0.126319i
\(59\) 164.858 0.363774 0.181887 − 0.983319i \(-0.441779\pi\)
0.181887 + 0.983319i \(0.441779\pi\)
\(60\) 0 0
\(61\) 188.248 0.395125 0.197563 − 0.980290i \(-0.436697\pi\)
0.197563 + 0.980290i \(0.436697\pi\)
\(62\) 273.722i 0.560690i
\(63\) 946.857i 1.89354i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 16.3442 0.0304823
\(67\) − 932.167i − 1.69974i −0.526995 − 0.849868i \(-0.676682\pi\)
0.526995 − 0.849868i \(-0.323318\pi\)
\(68\) 295.496i 0.526973i
\(69\) 190.242 0.331920
\(70\) 0 0
\(71\) −263.336 −0.440173 −0.220086 − 0.975480i \(-0.570634\pi\)
−0.220086 + 0.975480i \(0.570634\pi\)
\(72\) − 331.329i − 0.542326i
\(73\) 900.401i 1.44362i 0.692094 + 0.721808i \(0.256689\pi\)
−0.692094 + 0.721808i \(0.743311\pi\)
\(74\) 403.437 0.633764
\(75\) 0 0
\(76\) 284.634 0.429602
\(77\) 22.5876i 0.0334299i
\(78\) 69.8563i 0.101406i
\(79\) 956.182 1.36176 0.680879 − 0.732396i \(-0.261598\pi\)
0.680879 + 0.732396i \(0.261598\pi\)
\(80\) 0 0
\(81\) −131.942 −0.180990
\(82\) 408.279i 0.549841i
\(83\) 194.877i 0.257717i 0.991663 + 0.128858i \(0.0411313\pi\)
−0.991663 + 0.128858i \(0.958869\pi\)
\(84\) 756.405 0.982507
\(85\) 0 0
\(86\) −108.205 −0.135675
\(87\) 230.759i 0.284368i
\(88\) − 7.90397i − 0.00957461i
\(89\) 213.751 0.254580 0.127290 − 0.991866i \(-0.459372\pi\)
0.127290 + 0.991866i \(0.459372\pi\)
\(90\) 0 0
\(91\) −96.5410 −0.111211
\(92\) − 92.0000i − 0.104257i
\(93\) 1132.03i 1.26222i
\(94\) 82.8981 0.0909605
\(95\) 0 0
\(96\) −264.685 −0.281399
\(97\) − 628.762i − 0.658156i −0.944303 − 0.329078i \(-0.893262\pi\)
0.944303 − 0.329078i \(-0.106738\pi\)
\(98\) 359.348i 0.370404i
\(99\) 40.9189 0.0415405
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 1150.4.b.r.599.1 10
5.2 odd 4 1150.4.a.v.1.1 yes 5
5.3 odd 4 1150.4.a.q.1.5 ✓ 5
5.4 even 2 inner 1150.4.b.r.599.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.q.1.5 ✓ 5 5.3 odd 4
1150.4.a.v.1.1 yes 5 5.2 odd 4
1150.4.b.r.599.1 10 1.1 even 1 trivial
1150.4.b.r.599.10 10 5.4 even 2 inner