# Properties

 Label 983.2.c.a Level 983 Weight 2 Character orbit 983.c Analytic conductor 7.849 Analytic rank 0 Dimension 39690 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ $$=$$ $$983$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 983.c (of order $$491$$, degree $$490$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.84929451869$$ Analytic rank: $$0$$ Dimension: $$39690$$ Relative dimension: $$81$$ over $$\Q(\zeta_{491})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{491}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$39690q - 489q^{2} - 487q^{3} - 565q^{4} - 485q^{5} - 477q^{6} - 489q^{7} - 485q^{8} - 558q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$39690q - 489q^{2} - 487q^{3} - 565q^{4} - 485q^{5} - 477q^{6} - 489q^{7} - 485q^{8} - 558q^{9} - 471q^{10} - 477q^{11} - 463q^{12} - 483q^{13} - 473q^{14} - 461q^{15} - 543q^{16} - 481q^{17} - 457q^{18} - 473q^{19} - 449q^{20} - 449q^{21} - 459q^{22} - 473q^{23} - 405q^{24} - 552q^{25} - 437q^{26} - 451q^{27} - 455q^{28} - 455q^{29} - 379q^{30} - 459q^{31} - 437q^{32} - 447q^{33} - 423q^{34} - 439q^{35} - 459q^{36} - 475q^{37} - 423q^{38} - 439q^{39} - 413q^{40} - 455q^{41} - 407q^{42} - 457q^{43} - 413q^{44} - 397q^{45} - 415q^{46} - 467q^{47} - 393q^{48} - 524q^{49} - 429q^{50} - 403q^{51} - 391q^{52} - 431q^{53} - 379q^{54} - 415q^{55} - 371q^{56} - 447q^{57} - 431q^{58} - 439q^{59} - 261q^{60} - 431q^{61} - 383q^{62} - 445q^{63} - 447q^{64} - 415q^{65} - 345q^{66} - 439q^{67} - 409q^{68} - 367q^{69} - 345q^{70} - 415q^{71} - 341q^{72} - 465q^{73} - 359q^{74} - 339q^{75} - 361q^{76} - 363q^{77} - 341q^{78} - 423q^{79} - 249q^{80} - 420q^{81} - 371q^{82} - 433q^{83} - 215q^{84} - 409q^{85} - 369q^{86} - 329q^{87} - 255q^{88} - 393q^{89} - 251q^{90} - 373q^{91} - 387q^{92} - 369q^{93} - 317q^{94} - 393q^{95} - 149q^{96} - 415q^{97} - 307q^{98} - 299q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −0.882111 + 2.63433i 2.23553 + 1.25595i −4.56485 3.44317i −1.70964 + 2.57870i −5.28058 + 4.78124i 0.512184 1.13021i 8.51597 5.84418i 1.85965 + 3.05328i −5.28506 6.77846i
2.2 −0.882011 + 2.63403i −2.15731 1.21201i −4.56345 3.44211i −0.627390 + 0.946311i 5.09524 4.61342i 1.88896 4.16828i 8.51098 5.84075i 1.62450 + 2.66720i −1.93925 2.48722i
2.3 −0.873868 + 2.60971i −2.35753 1.32449i −4.45024 3.35672i 0.433776 0.654278i 5.51671 4.99503i −1.45490 + 3.21047i 8.11063 5.56601i 2.24312 + 3.68287i 1.32841 + 1.70378i
2.4 −0.862858 + 2.57683i 1.07786 + 0.605560i −4.29883 3.24251i 1.73663 2.61941i −2.49047 + 2.25496i −1.58443 + 3.49630i 7.58350 5.20426i −0.765444 1.25675i 5.25132 + 6.73519i
2.5 −0.815103 + 2.43422i 1.99272 + 1.11954i −3.66431 2.76391i 0.269938 0.407156i −4.34946 + 3.93817i 0.302286 0.667041i 5.48157 3.76178i 1.15702 + 1.89966i 0.771079 + 0.988963i
2.6 −0.805572 + 2.40575i −0.153383 0.0861729i −3.54199 2.67165i −0.680394 + 1.02626i 0.330872 0.299584i 1.03475 2.28334i 5.09698 3.49786i −1.54443 2.53574i −1.92082 2.46358i
2.7 −0.782295 + 2.33624i −0.0510472 0.0286790i −3.24932 2.45089i −1.37935 + 2.08052i 0.106935 0.0968230i −1.52243 + 3.35948i 4.20501 2.88573i −1.55875 2.55925i −3.78153 4.85008i
2.8 −0.752960 + 2.24863i −1.39225 0.782183i −2.89269 2.18189i 0.987618 1.48965i 2.80715 2.54170i −0.299880 + 0.661733i 3.17392 2.17814i −0.233998 0.384191i 2.60605 + 3.34244i
2.9 −0.752267 + 2.24656i −0.810068 0.455108i −2.88443 2.17566i 2.05128 3.09401i 1.63182 1.47751i 0.245279 0.541247i 3.15078 2.16226i −1.11145 1.82484i 5.40777 + 6.93585i
2.10 −0.696245 + 2.07926i −2.48076 1.39373i −2.24185 1.69098i 1.99175 3.00421i 4.62514 4.18777i 1.53695 3.39151i 1.46097 1.00260i 2.65117 + 4.35284i 4.85979 + 6.23303i
2.11 −0.695622 + 2.07740i 1.13868 + 0.639727i −2.23498 1.68580i 0.512099 0.772414i −2.12106 + 1.92049i 1.35903 2.99891i 1.44412 0.991043i −0.673191 1.10528i 1.24839 + 1.60114i
2.12 −0.691522 + 2.06516i −1.31722 0.740034i −2.18995 1.65183i −2.30250 + 3.47293i 2.43917 2.20852i 0.363807 0.802796i 1.33432 0.915691i −0.373111 0.612595i −5.57991 7.15662i
2.13 −0.682299 + 2.03761i −2.29977 1.29204i −2.08961 1.57615i −0.572361 + 0.863308i 4.20182 3.80448i −1.13382 + 2.50196i 1.09385 0.750669i 2.05904 + 3.38064i −1.36857 1.75528i
2.14 −0.668269 + 1.99571i 2.16322 + 1.21533i −1.93958 1.46298i −1.46604 + 2.21127i −3.87105 + 3.50500i −1.51734 + 3.34825i 0.745241 0.511430i 1.64195 + 2.69585i −3.43336 4.40352i
2.15 −0.649824 + 1.94063i 2.50384 + 1.40669i −1.74706 1.31777i 2.25729 3.40473i −4.35692 + 3.94491i 1.07843 2.37972i 0.317759 0.218066i 2.72988 + 4.48207i 5.14048 + 6.59303i
2.16 −0.612989 + 1.83062i 1.04506 + 0.587130i −1.37872 1.03994i 1.11482 1.68151i −1.71543 + 1.55321i −1.16764 + 2.57657i −0.434644 + 0.298279i −0.813104 1.33500i 2.39485 + 3.07156i
2.17 −0.585179 + 1.74757i −2.62769 1.47627i −1.11487 0.840921i −1.01017 + 1.52367i 4.11756 3.72820i 0.654151 1.44349i −0.917117 + 0.629381i 3.16483 + 5.19619i −2.07160 2.65697i
2.18 −0.578509 + 1.72765i 2.91461 + 1.63747i −1.05340 0.794559i −0.586439 + 0.884543i −4.51512 + 4.08815i −0.437511 + 0.965436i −1.02232 + 0.701577i 4.25313 + 6.98303i −1.18893 1.52488i
2.19 −0.555704 + 1.65955i 0.352885 + 0.198256i −0.848581 0.640067i −0.636799 + 0.960502i −0.525115 + 0.475459i −1.20682 + 2.66304i −1.35223 + 0.927979i −1.47531 2.42225i −1.24013 1.59055i
2.20 −0.529551 + 1.58145i 0.208046 + 0.116883i −0.623831 0.470542i −1.17513 + 1.77249i −0.295015 + 0.267117i 1.73067 3.81900i −1.67570 + 1.14996i −1.53091 2.51354i −2.18080 2.79703i
See next 80 embeddings (of 39690 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 978.81 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
983.c even 491 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 983.2.c.a 39690
983.c even 491 1 inner 983.2.c.a 39690

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
983.2.c.a 39690 1.a even 1 1 trivial
983.2.c.a 39690 983.c even 491 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(983, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database