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Properties

Label 300.3.l.d.143.2
Level $300$
Weight $3$
Character 300.143
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 = [4,8,-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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 143.2
Root \(1.87083 - 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 300.143
Dual form 300.3.l.d.107.2

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +(0.870829 - 2.87083i) q^{3} +4.00000 q^{4} +(1.74166 - 5.74166i) q^{6} +(-3.74166 - 3.74166i) q^{7} +8.00000 q^{8} +(-7.48331 - 5.00000i) q^{9} +14.9666 q^{11} +(3.48331 - 11.4833i) q^{12} +(-14.9666 - 14.9666i) q^{13} +(-7.48331 - 7.48331i) q^{14} +16.0000 q^{16} +(12.0000 + 12.0000i) q^{17} +(-14.9666 - 10.0000i) q^{18} +8.00000 q^{19} +(-14.0000 + 7.48331i) q^{21} +29.9333 q^{22} +(-2.00000 - 2.00000i) q^{23} +(6.96663 - 22.9666i) q^{24} +(-29.9333 - 29.9333i) q^{26} +(-20.8708 + 17.1292i) q^{27} +(-14.9666 - 14.9666i) q^{28} -14.9666 q^{29} +40.0000i q^{31} +32.0000 q^{32} +(13.0334 - 42.9666i) q^{33} +(24.0000 + 24.0000i) q^{34} +(-29.9333 - 20.0000i) q^{36} +(14.9666 - 14.9666i) q^{37} +16.0000 q^{38} +(-56.0000 + 29.9333i) q^{39} +74.8331i q^{41} +(-28.0000 + 14.9666i) q^{42} +(3.74166 - 3.74166i) q^{43} +59.8665 q^{44} +(-4.00000 - 4.00000i) q^{46} +(42.0000 - 42.0000i) q^{47} +(13.9333 - 45.9333i) q^{48} -21.0000i q^{49} +(44.8999 - 24.0000i) q^{51} +(-59.8665 - 59.8665i) q^{52} +(-32.0000 + 32.0000i) q^{53} +(-41.7417 + 34.2583i) q^{54} +(-29.9333 - 29.9333i) q^{56} +(6.96663 - 22.9666i) q^{57} -29.9333 q^{58} +74.8331i q^{59} -78.0000 q^{61} +80.0000i q^{62} +(9.29171 + 46.7083i) q^{63} +64.0000 q^{64} +(26.0667 - 85.9333i) q^{66} +(33.6749 + 33.6749i) q^{67} +(48.0000 + 48.0000i) q^{68} +(-7.48331 + 4.00000i) q^{69} +89.7998 q^{71} +(-59.8665 - 40.0000i) q^{72} +(59.8665 + 59.8665i) q^{73} +(29.9333 - 29.9333i) q^{74} +32.0000 q^{76} +(-56.0000 - 56.0000i) q^{77} +(-112.000 + 59.8665i) q^{78} +88.0000 q^{79} +(31.0000 + 74.8331i) q^{81} +149.666i q^{82} +(-42.0000 - 42.0000i) q^{83} +(-56.0000 + 29.9333i) q^{84} +(7.48331 - 7.48331i) q^{86} +(-13.0334 + 42.9666i) q^{87} +119.733 q^{88} -89.7998 q^{89} +112.000i q^{91} +(-8.00000 - 8.00000i) q^{92} +(114.833 + 34.8331i) q^{93} +(84.0000 - 84.0000i) q^{94} +(27.8665 - 91.8665i) q^{96} +(-59.8665 + 59.8665i) q^{97} -42.0000i q^{98} +(-112.000 - 74.8331i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} - 4 q^{3} + 16 q^{4} - 8 q^{6} + 32 q^{8} - 16 q^{12} + 64 q^{16} + 48 q^{17} + 32 q^{19} - 56 q^{21} - 8 q^{23} - 32 q^{24} - 76 q^{27} + 128 q^{32} + 112 q^{33} + 96 q^{34} + 64 q^{38} - 224 q^{39}+ \cdots - 448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.00000
\(3\) 0.870829 − 2.87083i 0.290276 − 0.956943i
\(4\) 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 0.387392 − 0.921915i \(-0.373376\pi\)
−0.921915 + 0.387392i \(0.873376\pi\)
\(8\) 8.00000 1.00000
\(9\) −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 \(0.261849\pi\)
\(12\) 3.48331 − 11.4833i 0.290276 − 0.956943i
\(13\) −14.9666 − 14.9666i −1.15128 − 1.15128i −0.986296 − 0.164983i \(-0.947243\pi\)
−0.164983 − 0.986296i \(-0.552757\pi\)
\(14\) −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 0.965667 − 0.259784i \(-0.0836515\pi\)
−0.259784 + 0.965667i \(0.583651\pi\)
\(18\) −14.9666 − 10.0000i −0.831479 − 0.555556i
\(19\) 8.00000 0.421053 0.210526 − 0.977588i \(-0.432482\pi\)
0.210526 + 0.977588i \(0.432482\pi\)
\(20\) 0 0
\(21\) −14.0000 + 7.48331i −0.666667 + 0.356348i
\(22\) 29.9333 1.36060
\(23\) −2.00000 − 2.00000i −0.0869565 − 0.0869565i 0.662291 − 0.749247i \(-0.269584\pi\)
−0.749247 + 0.662291i \(0.769584\pi\)
\(24\) 6.96663 − 22.9666i 0.290276 − 0.956943i
\(25\) 0 0
\(26\) −29.9333 − 29.9333i −1.15128 − 1.15128i
\(27\) −20.8708 + 17.1292i −0.772994 + 0.634414i
\(28\) −14.9666 − 14.9666i −0.534522 − 0.534522i
\(29\) −14.9666 −0.516091 −0.258045 − 0.966133i \(-0.583078\pi\)
−0.258045 + 0.966133i \(0.583078\pi\)
\(30\) 0 0
\(31\) 40.0000i 1.29032i 0.764046 + 0.645161i \(0.223210\pi\)
−0.764046 + 0.645161i \(0.776790\pi\)
\(32\) 32.0000 1.00000
\(33\) 13.0334 − 42.9666i 0.394951 − 1.30202i
\(34\) 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 \(-0.657665\pi\)
0.879817 + 0.475313i \(0.157665\pi\)
\(38\) 16.0000 0.421053
\(39\) −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 \(0.634072\pi\)
\(42\) −28.0000 + 14.9666i −0.666667 + 0.356348i
\(43\) 3.74166 − 3.74166i 0.0870153 − 0.0870153i −0.662259 − 0.749275i \(-0.730402\pi\)
0.749275 + 0.662259i \(0.230402\pi\)
\(44\) 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 \(-0.532282\pi\)
0.994862 + 0.101245i \(0.0322825\pi\)
\(48\) 13.9333 − 45.9333i 0.290276 − 0.956943i
\(49\) − 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 −0.941312 − 0.337538i \(-0.890406\pi\)
0.337538 + 0.941312i \(0.390406\pi\)
\(54\) −41.7417 + 34.2583i −0.772994 + 0.634414i
\(55\) 0 0
\(56\) −29.9333 − 29.9333i −0.534522 − 0.534522i
\(57\) 6.96663 − 22.9666i 0.122222 − 0.402923i
\(58\) −29.9333 −0.516091
\(59\) 74.8331i 1.26836i 0.773186 + 0.634179i \(0.218662\pi\)
−0.773186 + 0.634179i \(0.781338\pi\)
\(60\) 0 0
\(61\) −78.0000 −1.27869 −0.639344 − 0.768921i \(-0.720794\pi\)
−0.639344 + 0.768921i \(0.720794\pi\)
\(62\) 80.0000i 1.29032i
\(63\) 9.29171 + 46.7083i 0.147488 + 0.741401i
\(64\) 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 0.912248 − 0.409638i \(-0.134345\pi\)
−0.409638 + 0.912248i \(0.634345\pi\)
\(68\) 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 \(0.282073\pi\)
\(72\) −59.8665 − 40.0000i −0.831479 − 0.555556i
\(73\) 59.8665 + 59.8665i 0.820089 + 0.820089i 0.986120 − 0.166031i \(-0.0530952\pi\)
−0.166031 + 0.986120i \(0.553095\pi\)
\(74\) 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
\(78\) −112.000 + 59.8665i −1.43590 + 0.767519i
\(79\) 88.0000 1.11392 0.556962 − 0.830538i \(-0.311967\pi\)
0.556962 + 0.830538i \(0.311967\pi\)
\(80\) 0 0
\(81\) 31.0000 + 74.8331i 0.382716 + 0.923866i
\(82\) 149.666i 1.82520i
\(83\) −42.0000 − 42.0000i −0.506024 − 0.506024i 0.407279 − 0.913304i \(-0.366477\pi\)
−0.913304 + 0.407279i \(0.866477\pi\)
\(84\) −56.0000 + 29.9333i −0.666667 + 0.356348i
\(85\) 0 0
\(86\) 7.48331 − 7.48331i 0.0870153 − 0.0870153i
\(87\) −13.0334 + 42.9666i −0.149809 + 0.493869i
\(88\) 119.733 1.36060
\(89\) −89.7998 −1.00899 −0.504493 − 0.863416i \(-0.668321\pi\)
−0.504493 + 0.863416i \(0.668321\pi\)
\(90\) 0 0
\(91\) 112.000i 1.23077i
\(92\) −8.00000 − 8.00000i −0.0869565 − 0.0869565i
\(93\) 114.833 + 34.8331i 1.23477 + 0.374550i
\(94\) 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 \(-0.893751\pi\)
0.327627 + 0.944807i \(0.393751\pi\)
\(98\) − 42.0000i − 0.428571i
\(99\) −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.d.143.2 yes 4
3.2 odd 2 300.3.l.a.143.2 yes 4
4.3 odd 2 300.3.l.c.143.1 yes 4
5.2 odd 4 300.3.l.b.107.1 yes 4
5.3 odd 4 300.3.l.c.107.2 yes 4
5.4 even 2 300.3.l.a.143.1 yes 4
12.11 even 2 300.3.l.b.143.1 yes 4
15.2 even 4 300.3.l.c.107.1 yes 4
15.8 even 4 300.3.l.b.107.2 yes 4
15.14 odd 2 inner 300.3.l.d.143.1 yes 4
20.3 even 4 inner 300.3.l.d.107.1 yes 4
20.7 even 4 300.3.l.a.107.2 yes 4
20.19 odd 2 300.3.l.b.143.2 yes 4
60.23 odd 4 300.3.l.a.107.1 ✓ 4
60.47 odd 4 inner 300.3.l.d.107.2 yes 4
60.59 even 2 300.3.l.c.143.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.l.a.107.1 ✓ 4 60.23 odd 4
300.3.l.a.107.2 yes 4 20.7 even 4
300.3.l.a.143.1 yes 4 5.4 even 2
300.3.l.a.143.2 yes 4 3.2 odd 2
300.3.l.b.107.1 yes 4 5.2 odd 4
300.3.l.b.107.2 yes 4 15.8 even 4
300.3.l.b.143.1 yes 4 12.11 even 2
300.3.l.b.143.2 yes 4 20.19 odd 2
300.3.l.c.107.1 yes 4 15.2 even 4
300.3.l.c.107.2 yes 4 5.3 odd 4
300.3.l.c.143.1 yes 4 4.3 odd 2
300.3.l.c.143.2 yes 4 60.59 even 2
300.3.l.d.107.1 yes 4 20.3 even 4 inner
300.3.l.d.107.2 yes 4 60.47 odd 4 inner
300.3.l.d.143.1 yes 4 15.14 odd 2 inner
300.3.l.d.143.2 yes 4 1.1 even 1 trivial