Properties

Label 671.2.h.b
Level $671$
Weight $2$
Character orbit 671.h
Analytic conductor $5.358$
Analytic rank $0$
Dimension $236$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [671,2,Mod(70,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.70");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.h (of order \(5\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(59\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 236 q + 2 q^{2} - 2 q^{3} - 62 q^{4} - 6 q^{5} + 9 q^{6} + q^{7} + 12 q^{8} - 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 236 q + 2 q^{2} - 2 q^{3} - 62 q^{4} - 6 q^{5} + 9 q^{6} + q^{7} + 12 q^{8} - 59 q^{9} + 16 q^{10} - 15 q^{11} - 17 q^{12} - 8 q^{13} - 16 q^{14} - 20 q^{15} - 60 q^{16} - q^{17} - 12 q^{18} + 7 q^{19} + 26 q^{20} + 11 q^{22} - 10 q^{23} - 18 q^{24} - 43 q^{25} + 15 q^{26} + 10 q^{27} + 30 q^{28} - 31 q^{29} + 9 q^{30} + 28 q^{31} - 64 q^{32} + 18 q^{33} + 11 q^{34} + 27 q^{35} + 3 q^{36} + 10 q^{37} + 34 q^{38} - 20 q^{39} - 29 q^{40} - q^{41} - 20 q^{42} + 28 q^{43} + 52 q^{44} + 66 q^{45} + 28 q^{46} - 8 q^{47} + 30 q^{48} - 44 q^{49} + 18 q^{50} - 56 q^{51} - 40 q^{52} - 9 q^{53} - 56 q^{54} + 11 q^{55} - 25 q^{56} + 24 q^{57} + 77 q^{58} + 3 q^{59} + 16 q^{60} + 5 q^{61} + 54 q^{62} + 12 q^{63} - 10 q^{64} + 32 q^{65} + 80 q^{66} + 29 q^{67} + 80 q^{68} + 42 q^{69} - 75 q^{70} - 54 q^{71} - 198 q^{72} + 12 q^{73} + 18 q^{74} + 39 q^{75} + 19 q^{76} - 33 q^{77} + 15 q^{78} - 20 q^{79} + 112 q^{80} - 39 q^{81} - 64 q^{82} - 98 q^{83} - 124 q^{84} - 26 q^{85} - 142 q^{86} + 59 q^{87} - 186 q^{88} - 64 q^{89} - 179 q^{90} - 41 q^{91} + 36 q^{92} + 31 q^{93} - 4 q^{94} - 196 q^{95} - 79 q^{96} + 28 q^{97} - 7 q^{98} + 37 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1 −0.869346 2.67557i 2.15515 1.56581i −4.78489 + 3.47642i 0.984705 + 0.715430i −6.06301 4.40504i 3.79531 + 2.75745i 8.90919 + 6.47291i 1.26587 3.89595i 1.05813 3.25660i
70.2 −0.853051 2.62542i −0.947545 + 0.688432i −4.54710 + 3.30366i −3.22100 2.34019i 2.61573 + 1.90044i −2.13293 1.54966i 8.08578 + 5.87467i −0.503148 + 1.54853i −3.39631 + 10.4528i
70.3 −0.845205 2.60127i 0.160163 0.116365i −4.43422 + 3.22165i 1.41160 + 1.02559i −0.438068 0.318275i −2.29738 1.66914i 7.70268 + 5.59632i −0.914940 + 2.81589i 1.47474 4.53879i
70.4 −0.791807 2.43693i −2.09599 + 1.52283i −3.69364 + 2.68359i 3.32471 + 2.41555i 5.37064 + 3.90200i 0.868100 + 0.630711i 5.31841 + 3.86405i 1.14713 3.53049i 3.25399 10.0147i
70.5 −0.748831 2.30466i −1.16304 + 0.844998i −3.13269 + 2.27603i 0.674317 + 0.489920i 2.81836 + 2.04766i 1.46363 + 1.06339i 3.67042 + 2.66672i −0.288411 + 0.887638i 0.624151 1.92094i
70.6 −0.723495 2.22669i 0.360320 0.261787i −2.81666 + 2.04643i −1.15632 0.840113i −0.843609 0.612918i 1.47092 + 1.06869i 2.80633 + 2.03892i −0.865753 + 2.66452i −1.03408 + 3.18258i
70.7 −0.711166 2.18874i 1.70599 1.23947i −2.66681 + 1.93755i 0.461971 + 0.335641i −3.92613 2.85250i −1.71777 1.24803i 2.41364 + 1.75361i 0.447050 1.37588i 0.406095 1.24983i
70.8 −0.694635 2.13787i 0.233103 0.169359i −2.46992 + 1.79450i −2.36051 1.71501i −0.523988 0.380700i 3.46385 + 2.51663i 1.91494 + 1.39129i −0.901397 + 2.77421i −2.02678 + 6.23777i
70.9 −0.667832 2.05538i −2.48560 + 1.80590i −2.16054 + 1.56972i −0.411576 0.299027i 5.37176 + 3.90281i −3.21015 2.33231i 1.17243 + 0.851823i 1.98991 6.12431i −0.339750 + 1.04564i
70.10 −0.665082 2.04691i 0.235143 0.170841i −2.12948 + 1.54716i 2.09182 + 1.51980i −0.506086 0.367693i −0.996170 0.723760i 1.10077 + 0.799754i −0.900946 + 2.77283i 1.71966 5.29257i
70.11 −0.625954 1.92649i −2.12502 + 1.54392i −1.70151 + 1.23622i −1.76588 1.28299i 4.30450 + 3.12740i 1.77767 + 1.29155i 0.169083 + 0.122846i 1.20497 3.70852i −1.36630 + 4.20504i
70.12 −0.588103 1.81000i 2.66323 1.93495i −1.31219 + 0.953359i 2.95435 + 2.14646i −5.06851 3.68249i −2.27051 1.64962i −0.582072 0.422900i 2.42172 7.45327i 2.14762 6.60970i
70.13 −0.549055 1.68982i −1.21899 + 0.885650i −0.935991 + 0.680037i 0.281486 + 0.204512i 2.16588 + 1.57360i 0.593936 + 0.431520i −1.21184 0.880454i −0.225485 + 0.693970i 0.191036 0.587949i
70.14 −0.536577 1.65141i 2.11318 1.53531i −0.821217 + 0.596649i 1.72377 + 1.25239i −3.66932 2.66592i 2.10792 + 1.53149i −1.38359 1.00524i 1.18128 3.63561i 1.14328 3.51865i
70.15 −0.536210 1.65028i −0.389829 + 0.283228i −0.817882 + 0.594226i −2.94146 2.13710i 0.676436 + 0.491460i −3.08149 2.23883i −1.38843 1.00876i −0.855302 + 2.63235i −1.94958 + 6.00018i
70.16 −0.447883 1.37844i 1.22213 0.887928i −0.0814705 + 0.0591918i −1.35535 0.984717i −1.77133 1.28695i −0.969669 0.704506i −2.22706 1.61806i −0.221870 + 0.682846i −0.750339 + 2.30931i
70.17 −0.428672 1.31932i −0.369969 + 0.268798i 0.0612004 0.0444647i 2.08561 + 1.51529i 0.513225 + 0.372880i −3.54951 2.57887i −2.32945 1.69245i −0.862426 + 2.65428i 1.10510 3.40114i
70.18 −0.427484 1.31566i 1.72167 1.25087i 0.0698112 0.0507208i −0.553752 0.402324i −2.38171 1.73041i 2.59238 + 1.88347i −2.33491 1.69641i 0.472432 1.45400i −0.292602 + 0.900537i
70.19 −0.418710 1.28866i −1.45739 + 1.05885i 0.132718 0.0964256i 2.39305 + 1.73865i 1.97472 + 1.43472i 3.32854 + 2.41832i −2.37222 1.72352i 0.0757533 0.233145i 1.23853 3.81180i
70.20 −0.417408 1.28465i 2.67326 1.94224i 0.141939 0.103125i −3.12493 2.27039i −3.61094 2.62350i 1.24958 + 0.907872i −2.37730 1.72721i 2.44699 7.53107i −1.61229 + 4.96212i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.59
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.h even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.h.b 236
11.c even 5 1 671.2.m.b yes 236
61.e even 5 1 671.2.m.b yes 236
671.h even 5 1 inner 671.2.h.b 236
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.h.b 236 1.a even 1 1 trivial
671.2.h.b 236 671.h even 5 1 inner
671.2.m.b yes 236 11.c even 5 1
671.2.m.b yes 236 61.e even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{236} - 2 T_{2}^{235} + 92 T_{2}^{234} - 188 T_{2}^{233} + 4455 T_{2}^{232} - 9182 T_{2}^{231} + 150963 T_{2}^{230} - 311394 T_{2}^{229} + 4018926 T_{2}^{228} - 8259045 T_{2}^{227} + 89420855 T_{2}^{226} + \cdots + 102191881 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display