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Properties

Label 450.4.c.k.199.2
Level $450$
Weight $4$
Character 450.199
Analytic conductor $26.551$
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] = [450,4,Mod(199,450)] 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("450.199"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,0,0,0,0,0,120,0,0,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5508595026\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.4.c.k.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -4.00000 q^{4} +32.0000i q^{7} -8.00000i q^{8} +60.0000 q^{11} +34.0000i q^{13} -64.0000 q^{14} +16.0000 q^{16} -42.0000i q^{17} +76.0000 q^{19} +120.000i q^{22} -68.0000 q^{26} -128.000i q^{28} +6.00000 q^{29} -232.000 q^{31} +32.0000i q^{32} +84.0000 q^{34} +134.000i q^{37} +152.000i q^{38} -234.000 q^{41} +412.000i q^{43} -240.000 q^{44} +360.000i q^{47} -681.000 q^{49} -136.000i q^{52} +222.000i q^{53} +256.000 q^{56} +12.0000i q^{58} +660.000 q^{59} -490.000 q^{61} -464.000i q^{62} -64.0000 q^{64} +812.000i q^{67} +168.000i q^{68} -120.000 q^{71} -746.000i q^{73} -268.000 q^{74} -304.000 q^{76} +1920.00i q^{77} -152.000 q^{79} -468.000i q^{82} -804.000i q^{83} -824.000 q^{86} -480.000i q^{88} -678.000 q^{89} -1088.00 q^{91} -720.000 q^{94} +194.000i q^{97} -1362.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 120 q^{11} - 128 q^{14} + 32 q^{16} + 152 q^{19} - 136 q^{26} + 12 q^{29} - 464 q^{31} + 168 q^{34} - 468 q^{41} - 480 q^{44} - 1362 q^{49} + 512 q^{56} + 1320 q^{59} - 980 q^{61} - 128 q^{64}+ \cdots - 1440 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 32.0000i 1.72784i 0.503631 + 0.863919i \(0.331997\pi\)
−0.503631 + 0.863919i \(0.668003\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 60.0000 1.64461 0.822304 − 0.569049i \(-0.192689\pi\)
0.822304 + 0.569049i \(0.192689\pi\)
\(12\) 0 0
\(13\) 34.0000i 0.725377i 0.931910 + 0.362689i \(0.118141\pi\)
−0.931910 + 0.362689i \(0.881859\pi\)
\(14\) −64.0000 −1.22177
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 42.0000i − 0.599206i −0.954064 − 0.299603i \(-0.903146\pi\)
0.954064 − 0.299603i \(-0.0968542\pi\)
\(18\) 0 0
\(19\) 76.0000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 120.000i 1.16291i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −68.0000 −0.512919
\(27\) 0 0
\(28\) − 128.000i − 0.863919i
\(29\) 6.00000 0.0384197 0.0192099 − 0.999815i \(-0.493885\pi\)
0.0192099 + 0.999815i \(0.493885\pi\)
\(30\) 0 0
\(31\) −232.000 −1.34414 −0.672071 − 0.740486i \(-0.734595\pi\)
−0.672071 + 0.740486i \(0.734595\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 0 0
\(34\) 84.0000 0.423702
\(35\) 0 0
\(36\) 0 0
\(37\) 134.000i 0.595391i 0.954661 + 0.297695i \(0.0962180\pi\)
−0.954661 + 0.297695i \(0.903782\pi\)
\(38\) 152.000i 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) −234.000 −0.891333 −0.445667 − 0.895199i \(-0.647033\pi\)
−0.445667 + 0.895199i \(0.647033\pi\)
\(42\) 0 0
\(43\) 412.000i 1.46115i 0.682833 + 0.730575i \(0.260748\pi\)
−0.682833 + 0.730575i \(0.739252\pi\)
\(44\) −240.000 −0.822304
\(45\) 0 0
\(46\) 0 0
\(47\) 360.000i 1.11726i 0.829416 + 0.558632i \(0.188674\pi\)
−0.829416 + 0.558632i \(0.811326\pi\)
\(48\) 0 0
\(49\) −681.000 −1.98542
\(50\) 0 0
\(51\) 0 0
\(52\) − 136.000i − 0.362689i
\(53\) 222.000i 0.575359i 0.957727 + 0.287680i \(0.0928838\pi\)
−0.957727 + 0.287680i \(0.907116\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 256.000 0.610883
\(57\) 0 0
\(58\) 12.0000i 0.0271668i
\(59\) 660.000 1.45635 0.728175 − 0.685391i \(-0.240369\pi\)
0.728175 + 0.685391i \(0.240369\pi\)
\(60\) 0 0
\(61\) −490.000 −1.02849 −0.514246 − 0.857642i \(-0.671928\pi\)
−0.514246 + 0.857642i \(0.671928\pi\)
\(62\) − 464.000i − 0.950453i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 812.000i 1.48062i 0.672265 + 0.740310i \(0.265321\pi\)
−0.672265 + 0.740310i \(0.734679\pi\)
\(68\) 168.000i 0.299603i
\(69\) 0 0
\(70\) 0 0
\(71\) −120.000 −0.200583 −0.100291 − 0.994958i \(-0.531978\pi\)
−0.100291 + 0.994958i \(0.531978\pi\)
\(72\) 0 0
\(73\) − 746.000i − 1.19606i −0.801472 − 0.598032i \(-0.795949\pi\)
0.801472 − 0.598032i \(-0.204051\pi\)
\(74\) −268.000 −0.421005
\(75\) 0 0
\(76\) −304.000 −0.458831
\(77\) 1920.00i 2.84161i
\(78\) 0 0
\(79\) −152.000 −0.216473 −0.108236 − 0.994125i \(-0.534520\pi\)
−0.108236 + 0.994125i \(0.534520\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 468.000i − 0.630268i
\(83\) − 804.000i − 1.06326i −0.846977 − 0.531629i \(-0.821580\pi\)
0.846977 − 0.531629i \(-0.178420\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −824.000 −1.03319
\(87\) 0 0
\(88\) − 480.000i − 0.581456i
\(89\) −678.000 −0.807504 −0.403752 − 0.914868i \(-0.632294\pi\)
−0.403752 + 0.914868i \(0.632294\pi\)
\(90\) 0 0
\(91\) −1088.00 −1.25333
\(92\) 0 0
\(93\) 0 0
\(94\) −720.000 −0.790025
\(95\) 0 0
\(96\) 0 0
\(97\) 194.000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 1362.00i − 1.40391i
\(99\) 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 450.4.c.k.199.2 2
3.2 odd 2 150.4.c.a.49.1 2
5.2 odd 4 450.4.a.b.1.1 1
5.3 odd 4 90.4.a.d.1.1 1
5.4 even 2 inner 450.4.c.k.199.1 2
12.11 even 2 1200.4.f.u.49.2 2
15.2 even 4 150.4.a.e.1.1 1
15.8 even 4 30.4.a.a.1.1 ✓ 1
15.14 odd 2 150.4.c.a.49.2 2
20.3 even 4 720.4.a.b.1.1 1
45.13 odd 12 810.4.e.e.541.1 2
45.23 even 12 810.4.e.m.541.1 2
45.38 even 12 810.4.e.m.271.1 2
45.43 odd 12 810.4.e.e.271.1 2
60.23 odd 4 240.4.a.c.1.1 1
60.47 odd 4 1200.4.a.bk.1.1 1
60.59 even 2 1200.4.f.u.49.1 2
105.83 odd 4 1470.4.a.a.1.1 1
120.53 even 4 960.4.a.j.1.1 1
120.83 odd 4 960.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.a.1.1 ✓ 1 15.8 even 4
90.4.a.d.1.1 1 5.3 odd 4
150.4.a.e.1.1 1 15.2 even 4
150.4.c.a.49.1 2 3.2 odd 2
150.4.c.a.49.2 2 15.14 odd 2
240.4.a.c.1.1 1 60.23 odd 4
450.4.a.b.1.1 1 5.2 odd 4
450.4.c.k.199.1 2 5.4 even 2 inner
450.4.c.k.199.2 2 1.1 even 1 trivial
720.4.a.b.1.1 1 20.3 even 4
810.4.e.e.271.1 2 45.43 odd 12
810.4.e.e.541.1 2 45.13 odd 12
810.4.e.m.271.1 2 45.38 even 12
810.4.e.m.541.1 2 45.23 even 12
960.4.a.j.1.1 1 120.53 even 4
960.4.a.s.1.1 1 120.83 odd 4
1200.4.a.bk.1.1 1 60.47 odd 4
1200.4.f.u.49.1 2 60.59 even 2
1200.4.f.u.49.2 2 12.11 even 2
1470.4.a.a.1.1 1 105.83 odd 4