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Properties

Label 200.2.k.g.107.1
Level $200$
Weight $2$
Character 200.107
Analytic conductor $1.597$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [200,2,Mod(43,200)] 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("200.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6}, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{4} + 16 \) 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 107.1
Root \(-1.40294 + 0.178197i\) of defining polynomial
Character \(\chi\) \(=\) 200.107
Dual form 200.2.k.g.43.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.40294 + 0.178197i) q^{2} +(-1.22474 + 1.22474i) q^{3} +(1.93649 - 0.500000i) q^{4} +(1.50000 - 1.93649i) q^{6} +(3.16228 - 3.16228i) q^{7} +(-2.62769 + 1.04655i) q^{8} -1.00000 q^{11} +(-1.75934 + 2.98408i) q^{12} +(3.16228 + 3.16228i) q^{13} +(-3.87298 + 5.00000i) q^{14} +(3.50000 - 1.93649i) q^{16} +(3.67423 + 3.67423i) q^{17} +3.00000i q^{19} +7.74597i q^{21} +(1.40294 - 0.178197i) q^{22} +(1.93649 - 4.50000i) q^{24} +(-5.00000 - 3.87298i) q^{26} +(-3.67423 - 3.67423i) q^{27} +(4.54259 - 7.70486i) q^{28} +7.74597 q^{29} +(-4.56522 + 3.34047i) q^{32} +(1.22474 - 1.22474i) q^{33} +(-5.80948 - 4.50000i) q^{34} +(-3.16228 + 3.16228i) q^{37} +(-0.534591 - 4.20883i) q^{38} -7.74597 q^{39} -1.00000 q^{41} +(-1.38031 - 10.8671i) q^{42} +(2.44949 - 2.44949i) q^{43} +(-1.93649 + 0.500000i) q^{44} +(3.16228 - 3.16228i) q^{47} +(-1.91490 + 6.65832i) q^{48} -13.0000i q^{49} -9.00000 q^{51} +(7.70486 + 4.54259i) q^{52} +(-6.32456 - 6.32456i) q^{53} +(5.80948 + 4.50000i) q^{54} +(-5.00000 + 11.6190i) q^{56} +(-3.67423 - 3.67423i) q^{57} +(-10.8671 + 1.38031i) q^{58} +4.00000i q^{59} -7.74597i q^{61} +(5.80948 - 5.50000i) q^{64} +(-1.50000 + 1.93649i) q^{66} +(3.67423 + 3.67423i) q^{67} +(8.95224 + 5.27801i) q^{68} -7.74597i q^{71} +(-1.22474 + 1.22474i) q^{73} +(3.87298 - 5.00000i) q^{74} +(1.50000 + 5.80948i) q^{76} +(-3.16228 + 3.16228i) q^{77} +(10.8671 - 1.38031i) q^{78} -7.74597 q^{79} +9.00000 q^{81} +(1.40294 - 0.178197i) q^{82} +(-1.22474 + 1.22474i) q^{83} +(3.87298 + 15.0000i) q^{84} +(-3.00000 + 3.87298i) q^{86} +(-9.48683 + 9.48683i) q^{87} +(2.62769 - 1.04655i) q^{88} +13.0000i q^{89} +20.0000 q^{91} +(-3.87298 + 5.00000i) q^{94} +(1.50000 - 9.68246i) q^{96} +(4.89898 + 4.89898i) q^{97} +(2.31656 + 18.2382i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{6} - 8 q^{11} + 28 q^{16} - 40 q^{26} - 8 q^{41} - 72 q^{51} - 40 q^{56} - 12 q^{66} + 12 q^{76} + 72 q^{81} - 24 q^{86} + 160 q^{91} + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40294 + 0.178197i −0.992030 + 0.126004i
\(3\) −1.22474 + 1.22474i −0.707107 + 0.707107i −0.965926 − 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(4\) 1.93649 − 0.500000i 0.968246 − 0.250000i
\(5\) 0 0
\(6\) 1.50000 − 1.93649i 0.612372 − 0.790569i
\(7\) 3.16228 − 3.16228i 1.19523 − 1.19523i 0.219650 − 0.975579i \(-0.429509\pi\)
0.975579 − 0.219650i \(-0.0704915\pi\)
\(8\) −2.62769 + 1.04655i −0.929028 + 0.370011i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −1.75934 + 2.98408i −0.507877 + 0.861430i
\(13\) 3.16228 + 3.16228i 0.877058 + 0.877058i 0.993229 − 0.116171i \(-0.0370621\pi\)
−0.116171 + 0.993229i \(0.537062\pi\)
\(14\) −3.87298 + 5.00000i −1.03510 + 1.33631i
\(15\) 0 0
\(16\) 3.50000 − 1.93649i 0.875000 − 0.484123i
\(17\) 3.67423 + 3.67423i 0.891133 + 0.891133i 0.994630 − 0.103497i \(-0.0330032\pi\)
−0.103497 + 0.994630i \(0.533003\pi\)
\(18\) 0 0
\(19\) 3.00000i 0.688247i 0.938924 + 0.344124i \(0.111824\pi\)
−0.938924 + 0.344124i \(0.888176\pi\)
\(20\) 0 0
\(21\) 7.74597i 1.69031i
\(22\) 1.40294 − 0.178197i 0.299108 − 0.0379917i
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 1.93649 − 4.50000i 0.395285 − 0.918559i
\(25\) 0 0
\(26\) −5.00000 − 3.87298i −0.980581 − 0.759555i
\(27\) −3.67423 − 3.67423i −0.707107 − 0.707107i
\(28\) 4.54259 − 7.70486i 0.858468 − 1.45608i
\(29\) 7.74597 1.43839 0.719195 − 0.694808i \(-0.244511\pi\)
0.719195 + 0.694808i \(0.244511\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −4.56522 + 3.34047i −0.807024 + 0.590518i
\(33\) 1.22474 − 1.22474i 0.213201 − 0.213201i
\(34\) −5.80948 − 4.50000i −0.996317 − 0.771744i
\(35\) 0 0
\(36\) 0 0
\(37\) −3.16228 + 3.16228i −0.519875 + 0.519875i −0.917534 − 0.397658i \(-0.869823\pi\)
0.397658 + 0.917534i \(0.369823\pi\)
\(38\) −0.534591 − 4.20883i −0.0867221 − 0.682762i
\(39\) −7.74597 −1.24035
\(40\) 0 0
\(41\) −1.00000 −0.156174 −0.0780869 − 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) −1.38031 − 10.8671i −0.212986 − 1.67684i
\(43\) 2.44949 − 2.44949i 0.373544 − 0.373544i −0.495222 − 0.868766i \(-0.664913\pi\)
0.868766 + 0.495222i \(0.164913\pi\)
\(44\) −1.93649 + 0.500000i −0.291937 + 0.0753778i
\(45\) 0 0
\(46\) 0 0
\(47\) 3.16228 − 3.16228i 0.461266 − 0.461266i −0.437805 − 0.899070i \(-0.644244\pi\)
0.899070 + 0.437805i \(0.144244\pi\)
\(48\) −1.91490 + 6.65832i −0.276392 + 0.961045i
\(49\) − 13.0000i − 1.85714i
\(50\) 0 0
\(51\) −9.00000 −1.26025
\(52\) 7.70486 + 4.54259i 1.06847 + 0.629943i
\(53\) −6.32456 − 6.32456i −0.868744 − 0.868744i 0.123589 − 0.992333i \(-0.460560\pi\)
−0.992333 + 0.123589i \(0.960560\pi\)
\(54\) 5.80948 + 4.50000i 0.790569 + 0.612372i
\(55\) 0 0
\(56\) −5.00000 + 11.6190i −0.668153 + 1.55265i
\(57\) −3.67423 − 3.67423i −0.486664 − 0.486664i
\(58\) −10.8671 + 1.38031i −1.42693 + 0.181243i
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) − 7.74597i − 0.991769i −0.868388 − 0.495885i \(-0.834844\pi\)
0.868388 − 0.495885i \(-0.165156\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.80948 − 5.50000i 0.726184 − 0.687500i
\(65\) 0 0
\(66\) −1.50000 + 1.93649i −0.184637 + 0.238366i
\(67\) 3.67423 + 3.67423i 0.448879 + 0.448879i 0.894982 − 0.446103i \(-0.147188\pi\)
−0.446103 + 0.894982i \(0.647188\pi\)
\(68\) 8.95224 + 5.27801i 1.08562 + 0.640052i
\(69\) 0 0
\(70\) 0 0
\(71\) − 7.74597i − 0.919277i −0.888106 − 0.459639i \(-0.847979\pi\)
0.888106 − 0.459639i \(-0.152021\pi\)
\(72\) 0 0
\(73\) −1.22474 + 1.22474i −0.143346 + 0.143346i −0.775138 − 0.631792i \(-0.782320\pi\)
0.631792 + 0.775138i \(0.282320\pi\)
\(74\) 3.87298 − 5.00000i 0.450225 − 0.581238i
\(75\) 0 0
\(76\) 1.50000 + 5.80948i 0.172062 + 0.666392i
\(77\) −3.16228 + 3.16228i −0.360375 + 0.360375i
\(78\) 10.8671 − 1.38031i 1.23046 − 0.156289i
\(79\) −7.74597 −0.871489 −0.435745 − 0.900070i \(-0.643515\pi\)
−0.435745 + 0.900070i \(0.643515\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 1.40294 − 0.178197i 0.154929 − 0.0196786i
\(83\) −1.22474 + 1.22474i −0.134433 + 0.134433i −0.771121 − 0.636688i \(-0.780304\pi\)
0.636688 + 0.771121i \(0.280304\pi\)
\(84\) 3.87298 + 15.0000i 0.422577 + 1.63663i
\(85\) 0 0
\(86\) −3.00000 + 3.87298i −0.323498 + 0.417635i
\(87\) −9.48683 + 9.48683i −1.01710 + 1.01710i
\(88\) 2.62769 − 1.04655i 0.280112 − 0.111562i
\(89\) 13.0000i 1.37800i 0.724763 + 0.688999i \(0.241949\pi\)
−0.724763 + 0.688999i \(0.758051\pi\)
\(90\) 0 0
\(91\) 20.0000 2.09657
\(92\) 0 0
\(93\) 0 0
\(94\) −3.87298 + 5.00000i −0.399468 + 0.515711i
\(95\) 0 0
\(96\) 1.50000 − 9.68246i 0.153093 − 0.988212i
\(97\) 4.89898 + 4.89898i 0.497416 + 0.497416i 0.910633 − 0.413217i \(-0.135595\pi\)
−0.413217 + 0.910633i \(0.635595\pi\)
\(98\) 2.31656 + 18.2382i 0.234008 + 1.84234i
\(99\) 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 200.2.k.g.107.1 yes 8
4.3 odd 2 800.2.o.h.207.3 8
5.2 odd 4 inner 200.2.k.g.43.2 yes 8
5.3 odd 4 inner 200.2.k.g.43.3 yes 8
5.4 even 2 inner 200.2.k.g.107.4 yes 8
8.3 odd 2 inner 200.2.k.g.107.3 yes 8
8.5 even 2 800.2.o.h.207.4 8
20.3 even 4 800.2.o.h.143.4 8
20.7 even 4 800.2.o.h.143.1 8
20.19 odd 2 800.2.o.h.207.2 8
40.3 even 4 inner 200.2.k.g.43.1 ✓ 8
40.13 odd 4 800.2.o.h.143.3 8
40.19 odd 2 inner 200.2.k.g.107.2 yes 8
40.27 even 4 inner 200.2.k.g.43.4 yes 8
40.29 even 2 800.2.o.h.207.1 8
40.37 odd 4 800.2.o.h.143.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.k.g.43.1 ✓ 8 40.3 even 4 inner
200.2.k.g.43.2 yes 8 5.2 odd 4 inner
200.2.k.g.43.3 yes 8 5.3 odd 4 inner
200.2.k.g.43.4 yes 8 40.27 even 4 inner
200.2.k.g.107.1 yes 8 1.1 even 1 trivial
200.2.k.g.107.2 yes 8 40.19 odd 2 inner
200.2.k.g.107.3 yes 8 8.3 odd 2 inner
200.2.k.g.107.4 yes 8 5.4 even 2 inner
800.2.o.h.143.1 8 20.7 even 4
800.2.o.h.143.2 8 40.37 odd 4
800.2.o.h.143.3 8 40.13 odd 4
800.2.o.h.143.4 8 20.3 even 4
800.2.o.h.207.1 8 40.29 even 2
800.2.o.h.207.2 8 20.19 odd 2
800.2.o.h.207.3 8 4.3 odd 2
800.2.o.h.207.4 8 8.5 even 2