# Properties

 Label 83.13.b.b Level $83$ Weight $13$ Character orbit 83.b Self dual yes Analytic conductor $75.861$ Analytic rank $0$ Dimension $2$ CM discriminant -83 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$83$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 83.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.8614868339$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{249})$$ Defining polynomial: $$x^{2} - x - 62$$ x^2 - x - 62 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(-1 + 5\sqrt{249})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 29 \beta + 294) q^{3} + 4096 q^{4} + (915 \beta + 116246) q^{7} + ( - 17893 \beta + 863591) q^{9}+O(q^{10})$$ q + (-29*b + 294) * q^3 + 4096 * q^4 + (915*b + 116246) * q^7 + (-17893*b + 863591) * q^9 $$q + ( - 29 \beta + 294) q^{3} + 4096 q^{4} + (915 \beta + 116246) q^{7} + ( - 17893 \beta + 863591) q^{9} + (33459 \beta + 1616006) q^{11} + ( - 118784 \beta + 1204224) q^{12} + 16777216 q^{16} + (819195 \beta + 15722966) q^{17} + ( - 3075589 \beta - 7112136) q^{21} - 231011422 q^{23} + 244140625 q^{25} + ( - 15411789 \beta + 905055832) q^{27} + (3747840 \beta + 476143616) q^{28} + (14431227 \beta - 488546170) q^{29} + ( - 38902557 \beta + 31361078) q^{31} + ( - 36056917 \beta - 1034698152) q^{33} + ( - 73289728 \beta + 3537268736) q^{36} + ( - 24141909 \beta + 2494043174) q^{37} - 9474786718 q^{41} + (137048064 \beta + 6619160576) q^{44} + ( - 486539264 \beta + 4932501504) q^{48} + (211892955 \beta + 974567415) q^{49} + ( - 191366029 \beta - 32342803176) q^{51} + (1512658419 \beta + 25092046406) q^{59} + (1231426923 \beta - 42601298362) q^{61} + ( - 1273431818 \beta + 74914019566) q^{63} + 68719476736 q^{64} + (3355422720 \beta + 64401268736) q^{68} + (6699331238 \beta - 67917358068) q^{69} + ( - 7080078125 \beta + 71777343750) q^{75} + (5337505419 \beta + 235491150136) q^{77} + ( - 21715553162 \beta + 502580316813) q^{81} + 326940373369 q^{83} + ( - 12597612544 \beta - 29131309056) q^{84} + (18829125251 \beta - 794827261128) q^{87} - 946222784512 q^{92} + ( - 13474997173 \beta + 1764659139000) q^{93} + (578377798 \beta + 464019221374) q^{99}+O(q^{100})$$ q + (-29*b + 294) * q^3 + 4096 * q^4 + (915*b + 116246) * q^7 + (-17893*b + 863591) * q^9 + (33459*b + 1616006) * q^11 + (-118784*b + 1204224) * q^12 + 16777216 * q^16 + (819195*b + 15722966) * q^17 + (-3075589*b - 7112136) * q^21 - 231011422 * q^23 + 244140625 * q^25 + (-15411789*b + 905055832) * q^27 + (3747840*b + 476143616) * q^28 + (14431227*b - 488546170) * q^29 + (-38902557*b + 31361078) * q^31 + (-36056917*b - 1034698152) * q^33 + (-73289728*b + 3537268736) * q^36 + (-24141909*b + 2494043174) * q^37 - 9474786718 * q^41 + (137048064*b + 6619160576) * q^44 + (-486539264*b + 4932501504) * q^48 + (211892955*b + 974567415) * q^49 + (-191366029*b - 32342803176) * q^51 + (1512658419*b + 25092046406) * q^59 + (1231426923*b - 42601298362) * q^61 + (-1273431818*b + 74914019566) * q^63 + 68719476736 * q^64 + (3355422720*b + 64401268736) * q^68 + (6699331238*b - 67917358068) * q^69 + (-7080078125*b + 71777343750) * q^75 + (5337505419*b + 235491150136) * q^77 + (-21715553162*b + 502580316813) * q^81 + 326940373369 * q^83 + (-12597612544*b - 29131309056) * q^84 + (18829125251*b - 794827261128) * q^87 - 946222784512 * q^92 + (-13474997173*b + 1764659139000) * q^93 + (578377798*b + 464019221374) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 617 q^{3} + 8192 q^{4} + 231577 q^{7} + 1745075 q^{9}+O(q^{10})$$ 2 * q + 617 * q^3 + 8192 * q^4 + 231577 * q^7 + 1745075 * q^9 $$2 q + 617 q^{3} + 8192 q^{4} + 231577 q^{7} + 1745075 q^{9} + 3198553 q^{11} + 2527232 q^{12} + 33554432 q^{16} + 30626737 q^{17} - 11148683 q^{21} - 462022844 q^{23} + 488281250 q^{25} + 1825523453 q^{27} + 948539392 q^{28} - 991523567 q^{29} + 101624713 q^{31} - 2033339387 q^{33} + 7147827200 q^{36} + 5012228257 q^{37} - 18949573436 q^{41} + 13101273088 q^{44} + 10351542272 q^{48} + 1737241875 q^{49} - 64494240323 q^{51} + 48671434393 q^{59} - 86434023647 q^{61} + 151101470950 q^{63} + 137438953472 q^{64} + 125447114752 q^{68} - 142534047374 q^{69} + 150634765625 q^{75} + 465644794853 q^{77} + 1026876186788 q^{81} + 653880746738 q^{83} - 45665005568 q^{84} - 1608483647507 q^{87} - 1892445569024 q^{92} + 3542793275173 q^{93} + 927460064950 q^{99}+O(q^{100})$$ 2 * q + 617 * q^3 + 8192 * q^4 + 231577 * q^7 + 1745075 * q^9 + 3198553 * q^11 + 2527232 * q^12 + 33554432 * q^16 + 30626737 * q^17 - 11148683 * q^21 - 462022844 * q^23 + 488281250 * q^25 + 1825523453 * q^27 + 948539392 * q^28 - 991523567 * q^29 + 101624713 * q^31 - 2033339387 * q^33 + 7147827200 * q^36 + 5012228257 * q^37 - 18949573436 * q^41 + 13101273088 * q^44 + 10351542272 * q^48 + 1737241875 * q^49 - 64494240323 * q^51 + 48671434393 * q^59 - 86434023647 * q^61 + 151101470950 * q^63 + 137438953472 * q^64 + 125447114752 * q^68 - 142534047374 * q^69 + 150634765625 * q^75 + 465644794853 * q^77 + 1026876186788 * q^81 + 653880746738 * q^83 - 45665005568 * q^84 - 1608483647507 * q^87 - 1892445569024 * q^92 + 3542793275173 * q^93 + 927460064950 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/83\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 8.38987 −7.38987
0 −835.531 4096.00 0 0 151885. 0 166671. 0
82.2 0 1452.53 4096.00 0 0 79692.4 0 1.57840e6 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
83.b odd 2 1 CM by $$\Q(\sqrt{-83})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 83.13.b.b 2
83.b odd 2 1 CM 83.13.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.13.b.b 2 1.a even 1 1 trivial
83.13.b.b 2 83.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{13}^{\mathrm{new}}(83, [\chi])$$:

 $$T_{2}$$ T2 $$T_{3}^{2} - 617T_{3} - 1213634$$ T3^2 - 617*T3 - 1213634

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 617 T - 1213634$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 231577 T + 12104045326$$
$11$ $$T^{2} - 3198553 T + 815456163646$$
$13$ $$T^{2}$$
$17$ $$T^{2} + \cdots - 809869692422114$$
$19$ $$T^{2}$$
$23$ $$(T + 231011422)^{2}$$
$29$ $$T^{2} + 991523567 T - 78\!\cdots\!34$$
$31$ $$T^{2} - 101624713 T - 23\!\cdots\!14$$
$37$ $$T^{2} - 5012228257 T + 53\!\cdots\!06$$
$41$ $$(T + 9474786718)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 48671434393 T - 29\!\cdots\!94$$
$61$ $$T^{2} + 86434023647 T - 49\!\cdots\!54$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$(T - 326940373369)^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$