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Properties

Label 300.3.l.c.107.2
Level $300$
Weight $3$
Character 300.107
Analytic conductor $8.174$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [300,3,Mod(107,300)] 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("300.107"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,-16,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 107.2
Root \(1.87083 + 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 300.107
Dual form 300.3.l.c.143.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +(2.87083 + 0.870829i) q^{3} -4.00000 q^{4} +(1.74166 - 5.74166i) q^{6} +(-3.74166 + 3.74166i) q^{7} +8.00000i q^{8} +(7.48331 + 5.00000i) q^{9} +14.9666 q^{11} +(-11.4833 - 3.48331i) q^{12} +(14.9666 - 14.9666i) q^{13} +(7.48331 + 7.48331i) q^{14} +16.0000 q^{16} +(12.0000 - 12.0000i) q^{17} +(10.0000 - 14.9666i) q^{18} -8.00000 q^{19} +(-14.0000 + 7.48331i) q^{21} -29.9333i q^{22} +(2.00000 - 2.00000i) q^{23} +(-6.96663 + 22.9666i) q^{24} +(-29.9333 - 29.9333i) q^{26} +(17.1292 + 20.8708i) q^{27} +(14.9666 - 14.9666i) q^{28} +14.9666 q^{29} +40.0000i q^{31} -32.0000i q^{32} +(42.9666 + 13.0334i) q^{33} +(-24.0000 - 24.0000i) q^{34} +(-29.9333 - 20.0000i) q^{36} +(-14.9666 - 14.9666i) q^{37} +16.0000i q^{38} +(56.0000 - 29.9333i) q^{39} +74.8331i q^{41} +(14.9666 + 28.0000i) q^{42} +(3.74166 + 3.74166i) q^{43} -59.8665 q^{44} +(-4.00000 - 4.00000i) q^{46} +(-42.0000 - 42.0000i) q^{47} +(45.9333 + 13.9333i) q^{48} +21.0000i q^{49} +(44.8999 - 24.0000i) q^{51} +(-59.8665 + 59.8665i) q^{52} +(-32.0000 - 32.0000i) q^{53} +(41.7417 - 34.2583i) q^{54} +(-29.9333 - 29.9333i) q^{56} +(-22.9666 - 6.96663i) q^{57} -29.9333i q^{58} -74.8331i q^{59} -78.0000 q^{61} +80.0000 q^{62} +(-46.7083 + 9.29171i) q^{63} -64.0000 q^{64} +(26.0667 - 85.9333i) q^{66} +(33.6749 - 33.6749i) q^{67} +(-48.0000 + 48.0000i) q^{68} +(7.48331 - 4.00000i) q^{69} +89.7998 q^{71} +(-40.0000 + 59.8665i) q^{72} +(-59.8665 + 59.8665i) q^{73} +(-29.9333 + 29.9333i) q^{74} +32.0000 q^{76} +(-56.0000 + 56.0000i) q^{77} +(-59.8665 - 112.000i) q^{78} -88.0000 q^{79} +(31.0000 + 74.8331i) q^{81} +149.666 q^{82} +(42.0000 - 42.0000i) q^{83} +(56.0000 - 29.9333i) q^{84} +(7.48331 - 7.48331i) q^{86} +(42.9666 + 13.0334i) q^{87} +119.733i q^{88} +89.7998 q^{89} +112.000i q^{91} +(-8.00000 + 8.00000i) q^{92} +(-34.8331 + 114.833i) q^{93} +(-84.0000 + 84.0000i) q^{94} +(27.8665 - 91.8665i) q^{96} +(59.8665 + 59.8665i) q^{97} +42.0000 q^{98} +(112.000 + 74.8331i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{12} + 64 q^{16} + 48 q^{17} + 40 q^{18} - 32 q^{19} - 56 q^{21} + 8 q^{23} + 32 q^{24} + 76 q^{27} + 112 q^{33} - 96 q^{34} + 224 q^{39} - 16 q^{46} - 168 q^{47}+ \cdots + 448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 1.00000i
\(3\) 2.87083 + 0.870829i 0.956943 + 0.290276i
\(4\) −4.00000 −1.00000
\(5\) 0 0
\(6\) 1.74166 − 5.74166i 0.290276 − 0.956943i
\(7\) −3.74166 + 3.74166i −0.534522 + 0.534522i −0.921915 − 0.387392i \(-0.873376\pi\)
0.387392 + 0.921915i \(0.373376\pi\)
\(8\) 8.00000i 1.00000i
\(9\) 7.48331 + 5.00000i 0.831479 + 0.555556i
\(10\) 0 0
\(11\) 14.9666 1.36060 0.680301 − 0.732933i \(-0.261849\pi\)
0.680301 + 0.732933i \(0.261849\pi\)
\(12\) −11.4833 − 3.48331i −0.956943 − 0.290276i
\(13\) 14.9666 − 14.9666i 1.15128 − 1.15128i 0.164983 − 0.986296i \(-0.447243\pi\)
0.986296 − 0.164983i \(-0.0527568\pi\)
\(14\) 7.48331 + 7.48331i 0.534522 + 0.534522i
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 12.0000 − 12.0000i 0.705882 − 0.705882i −0.259784 − 0.965667i \(-0.583651\pi\)
0.965667 + 0.259784i \(0.0836515\pi\)
\(18\) 10.0000 − 14.9666i 0.555556 − 0.831479i
\(19\) −8.00000 −0.421053 −0.210526 − 0.977588i \(-0.567518\pi\)
−0.210526 + 0.977588i \(0.567518\pi\)
\(20\) 0 0
\(21\) −14.0000 + 7.48331i −0.666667 + 0.356348i
\(22\) − 29.9333i − 1.36060i
\(23\) 2.00000 − 2.00000i 0.0869565 − 0.0869565i −0.662291 − 0.749247i \(-0.730416\pi\)
0.749247 + 0.662291i \(0.230416\pi\)
\(24\) −6.96663 + 22.9666i −0.290276 + 0.956943i
\(25\) 0 0
\(26\) −29.9333 − 29.9333i −1.15128 − 1.15128i
\(27\) 17.1292 + 20.8708i 0.634414 + 0.772994i
\(28\) 14.9666 − 14.9666i 0.534522 − 0.534522i
\(29\) 14.9666 0.516091 0.258045 − 0.966133i \(-0.416922\pi\)
0.258045 + 0.966133i \(0.416922\pi\)
\(30\) 0 0
\(31\) 40.0000i 1.29032i 0.764046 + 0.645161i \(0.223210\pi\)
−0.764046 + 0.645161i \(0.776790\pi\)
\(32\) − 32.0000i − 1.00000i
\(33\) 42.9666 + 13.0334i 1.30202 + 0.394951i
\(34\) −24.0000 − 24.0000i −0.705882 − 0.705882i
\(35\) 0 0
\(36\) −29.9333 − 20.0000i −0.831479 − 0.555556i
\(37\) −14.9666 − 14.9666i −0.404504 − 0.404504i 0.475313 − 0.879817i \(-0.342335\pi\)
−0.879817 + 0.475313i \(0.842335\pi\)
\(38\) 16.0000i 0.421053i
\(39\) 56.0000 − 29.9333i 1.43590 − 0.767519i
\(40\) 0 0
\(41\) 74.8331i 1.82520i 0.408855 + 0.912599i \(0.365928\pi\)
−0.408855 + 0.912599i \(0.634072\pi\)
\(42\) 14.9666 + 28.0000i 0.356348 + 0.666667i
\(43\) 3.74166 + 3.74166i 0.0870153 + 0.0870153i 0.749275 − 0.662259i \(-0.230402\pi\)
−0.662259 + 0.749275i \(0.730402\pi\)
\(44\) −59.8665 −1.36060
\(45\) 0 0
\(46\) −4.00000 − 4.00000i −0.0869565 − 0.0869565i
\(47\) −42.0000 − 42.0000i −0.893617 − 0.893617i 0.101245 − 0.994862i \(-0.467718\pi\)
−0.994862 + 0.101245i \(0.967718\pi\)
\(48\) 45.9333 + 13.9333i 0.956943 + 0.290276i
\(49\) 21.0000i 0.428571i
\(50\) 0 0
\(51\) 44.8999 − 24.0000i 0.880390 − 0.470588i
\(52\) −59.8665 + 59.8665i −1.15128 + 1.15128i
\(53\) −32.0000 − 32.0000i −0.603774 − 0.603774i 0.337538 − 0.941312i \(-0.390406\pi\)
−0.941312 + 0.337538i \(0.890406\pi\)
\(54\) 41.7417 − 34.2583i 0.772994 − 0.634414i
\(55\) 0 0
\(56\) −29.9333 − 29.9333i −0.534522 − 0.534522i
\(57\) −22.9666 − 6.96663i −0.402923 − 0.122222i
\(58\) − 29.9333i − 0.516091i
\(59\) − 74.8331i − 1.26836i −0.773186 − 0.634179i \(-0.781338\pi\)
0.773186 − 0.634179i \(-0.218662\pi\)
\(60\) 0 0
\(61\) −78.0000 −1.27869 −0.639344 − 0.768921i \(-0.720794\pi\)
−0.639344 + 0.768921i \(0.720794\pi\)
\(62\) 80.0000 1.29032
\(63\) −46.7083 + 9.29171i −0.741401 + 0.147488i
\(64\) −64.0000 −1.00000
\(65\) 0 0
\(66\) 26.0667 − 85.9333i 0.394951 − 1.30202i
\(67\) 33.6749 − 33.6749i 0.502611 − 0.502611i −0.409638 − 0.912248i \(-0.634345\pi\)
0.912248 + 0.409638i \(0.134345\pi\)
\(68\) −48.0000 + 48.0000i −0.705882 + 0.705882i
\(69\) 7.48331 − 4.00000i 0.108454 − 0.0579710i
\(70\) 0 0
\(71\) 89.7998 1.26479 0.632393 − 0.774648i \(-0.282073\pi\)
0.632393 + 0.774648i \(0.282073\pi\)
\(72\) −40.0000 + 59.8665i −0.555556 + 0.831479i
\(73\) −59.8665 + 59.8665i −0.820089 + 0.820089i −0.986120 − 0.166031i \(-0.946905\pi\)
0.166031 + 0.986120i \(0.446905\pi\)
\(74\) −29.9333 + 29.9333i −0.404504 + 0.404504i
\(75\) 0 0
\(76\) 32.0000 0.421053
\(77\) −56.0000 + 56.0000i −0.727273 + 0.727273i
\(78\) −59.8665 − 112.000i −0.767519 − 1.43590i
\(79\) −88.0000 −1.11392 −0.556962 − 0.830538i \(-0.688033\pi\)
−0.556962 + 0.830538i \(0.688033\pi\)
\(80\) 0 0
\(81\) 31.0000 + 74.8331i 0.382716 + 0.923866i
\(82\) 149.666 1.82520
\(83\) 42.0000 − 42.0000i 0.506024 − 0.506024i −0.407279 − 0.913304i \(-0.633523\pi\)
0.913304 + 0.407279i \(0.133523\pi\)
\(84\) 56.0000 − 29.9333i 0.666667 − 0.356348i
\(85\) 0 0
\(86\) 7.48331 − 7.48331i 0.0870153 − 0.0870153i
\(87\) 42.9666 + 13.0334i 0.493869 + 0.149809i
\(88\) 119.733i 1.36060i
\(89\) 89.7998 1.00899 0.504493 − 0.863416i \(-0.331679\pi\)
0.504493 + 0.863416i \(0.331679\pi\)
\(90\) 0 0
\(91\) 112.000i 1.23077i
\(92\) −8.00000 + 8.00000i −0.0869565 + 0.0869565i
\(93\) −34.8331 + 114.833i −0.374550 + 1.23477i
\(94\) −84.0000 + 84.0000i −0.893617 + 0.893617i
\(95\) 0 0
\(96\) 27.8665 − 91.8665i 0.290276 − 0.956943i
\(97\) 59.8665 + 59.8665i 0.617181 + 0.617181i 0.944807 − 0.327627i \(-0.106249\pi\)
−0.327627 + 0.944807i \(0.606249\pi\)
\(98\) 42.0000 0.428571
\(99\) 112.000 + 74.8331i 1.13131 + 0.755890i
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 300.3.l.c.107.2 yes 4
3.2 odd 2 300.3.l.b.107.2 yes 4
4.3 odd 2 300.3.l.d.107.1 yes 4
5.2 odd 4 300.3.l.d.143.2 yes 4
5.3 odd 4 300.3.l.a.143.1 yes 4
5.4 even 2 300.3.l.b.107.1 yes 4
12.11 even 2 300.3.l.a.107.1 ✓ 4
15.2 even 4 300.3.l.a.143.2 yes 4
15.8 even 4 300.3.l.d.143.1 yes 4
15.14 odd 2 inner 300.3.l.c.107.1 yes 4
20.3 even 4 300.3.l.b.143.2 yes 4
20.7 even 4 inner 300.3.l.c.143.1 yes 4
20.19 odd 2 300.3.l.a.107.2 yes 4
60.23 odd 4 inner 300.3.l.c.143.2 yes 4
60.47 odd 4 300.3.l.b.143.1 yes 4
60.59 even 2 300.3.l.d.107.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.l.a.107.1 ✓ 4 12.11 even 2
300.3.l.a.107.2 yes 4 20.19 odd 2
300.3.l.a.143.1 yes 4 5.3 odd 4
300.3.l.a.143.2 yes 4 15.2 even 4
300.3.l.b.107.1 yes 4 5.4 even 2
300.3.l.b.107.2 yes 4 3.2 odd 2
300.3.l.b.143.1 yes 4 60.47 odd 4
300.3.l.b.143.2 yes 4 20.3 even 4
300.3.l.c.107.1 yes 4 15.14 odd 2 inner
300.3.l.c.107.2 yes 4 1.1 even 1 trivial
300.3.l.c.143.1 yes 4 20.7 even 4 inner
300.3.l.c.143.2 yes 4 60.23 odd 4 inner
300.3.l.d.107.1 yes 4 4.3 odd 2
300.3.l.d.107.2 yes 4 60.59 even 2
300.3.l.d.143.1 yes 4 15.8 even 4
300.3.l.d.143.2 yes 4 5.2 odd 4