Properties

 Label 176.10.a.e Level $176$ Weight $10$ Character orbit 176.a Self dual yes Analytic conductor $90.646$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 176.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$90.6463071648$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{889})$$ Defining polynomial: $$x^{2} - x - 222$$ x^2 - x - 222 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{889})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (9 \beta + 6) q^{3} + ( - 97 \beta - 212) q^{5} + (330 \beta + 3580) q^{7} + (189 \beta - 1665) q^{9}+O(q^{10})$$ q + (9*b + 6) * q^3 + (-97*b - 212) * q^5 + (330*b + 3580) * q^7 + (189*b - 1665) * q^9 $$q + (9 \beta + 6) q^{3} + ( - 97 \beta - 212) q^{5} + (330 \beta + 3580) q^{7} + (189 \beta - 1665) q^{9} - 14641 q^{11} + ( - 490 \beta + 75402) q^{13} + ( - 3363 \beta - 195078) q^{15} + ( - 6412 \beta + 348442) q^{17} + (36812 \beta - 274012) q^{19} + (37170 \beta + 680820) q^{21} + ( - 44659 \beta - 415046) q^{23} + (50537 \beta + 180617) q^{25} + ( - 189297 \beta + 249534) q^{27} + ( - 252246 \beta - 1349406) q^{29} + (367213 \beta + 2725746) q^{31} + ( - 131769 \beta - 87846) q^{33} + ( - 449230 \beta - 7865180) q^{35} + (403577 \beta + 1127664) q^{37} + (671268 \beta - 526608) q^{39} + (228846 \beta + 6599574) q^{41} + (1121834 \beta + 8349464) q^{43} + (103104 \beta - 3716946) q^{45} + ( - 2071176 \beta - 26986464) q^{47} + (2471700 \beta - 3361407) q^{49} + (3039798 \beta - 10720524) q^{51} + (2579204 \beta + 47131774) q^{53} + (1420177 \beta + 3103892) q^{55} + ( - 1913928 \beta + 71906304) q^{57} + (8232723 \beta + 55451730) q^{59} + (11223238 \beta - 50823782) q^{61} + (189540 \beta + 7885440) q^{63} + ( - 7162584 \beta - 5433564) q^{65} + ( - 5360809 \beta + 150652850) q^{67} + ( - 4405299 \beta - 91718958) q^{69} + (393879 \beta + 161279694) q^{71} + ( - 1597174 \beta - 127189170) q^{73} + (2383608 \beta + 102056628) q^{75} + ( - 4831530 \beta - 52414780) q^{77} + ( - 19867086 \beta + 10378372) q^{79} + ( - 4313736 \beta - 343946007) q^{81} + ( - 40242910 \beta + 158970976) q^{83} + ( - 31817566 \beta + 64206304) q^{85} + ( - 15928344 \beta - 512083944) q^{87} + ( - 14637263 \beta + 689154740) q^{89} + (22966760 \beta + 234041760) q^{91} + (30039909 \beta + 750046050) q^{93} + (15204256 \beta - 734619064) q^{95} + ( - 39733637 \beta + 719083840) q^{97} + ( - 2767149 \beta + 24377265) q^{99}+O(q^{100})$$ q + (9*b + 6) * q^3 + (-97*b - 212) * q^5 + (330*b + 3580) * q^7 + (189*b - 1665) * q^9 - 14641 * q^11 + (-490*b + 75402) * q^13 + (-3363*b - 195078) * q^15 + (-6412*b + 348442) * q^17 + (36812*b - 274012) * q^19 + (37170*b + 680820) * q^21 + (-44659*b - 415046) * q^23 + (50537*b + 180617) * q^25 + (-189297*b + 249534) * q^27 + (-252246*b - 1349406) * q^29 + (367213*b + 2725746) * q^31 + (-131769*b - 87846) * q^33 + (-449230*b - 7865180) * q^35 + (403577*b + 1127664) * q^37 + (671268*b - 526608) * q^39 + (228846*b + 6599574) * q^41 + (1121834*b + 8349464) * q^43 + (103104*b - 3716946) * q^45 + (-2071176*b - 26986464) * q^47 + (2471700*b - 3361407) * q^49 + (3039798*b - 10720524) * q^51 + (2579204*b + 47131774) * q^53 + (1420177*b + 3103892) * q^55 + (-1913928*b + 71906304) * q^57 + (8232723*b + 55451730) * q^59 + (11223238*b - 50823782) * q^61 + (189540*b + 7885440) * q^63 + (-7162584*b - 5433564) * q^65 + (-5360809*b + 150652850) * q^67 + (-4405299*b - 91718958) * q^69 + (393879*b + 161279694) * q^71 + (-1597174*b - 127189170) * q^73 + (2383608*b + 102056628) * q^75 + (-4831530*b - 52414780) * q^77 + (-19867086*b + 10378372) * q^79 + (-4313736*b - 343946007) * q^81 + (-40242910*b + 158970976) * q^83 + (-31817566*b + 64206304) * q^85 + (-15928344*b - 512083944) * q^87 + (-14637263*b + 689154740) * q^89 + (22966760*b + 234041760) * q^91 + (30039909*b + 750046050) * q^93 + (15204256*b - 734619064) * q^95 + (-39733637*b + 719083840) * q^97 + (-2767149*b + 24377265) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 21 q^{3} - 521 q^{5} + 7490 q^{7} - 3141 q^{9}+O(q^{10})$$ 2 * q + 21 * q^3 - 521 * q^5 + 7490 * q^7 - 3141 * q^9 $$2 q + 21 q^{3} - 521 q^{5} + 7490 q^{7} - 3141 q^{9} - 29282 q^{11} + 150314 q^{13} - 393519 q^{15} + 690472 q^{17} - 511212 q^{19} + 1398810 q^{21} - 874751 q^{23} + 411771 q^{25} + 309771 q^{27} - 2951058 q^{29} + 5818705 q^{31} - 307461 q^{33} - 16179590 q^{35} + 2658905 q^{37} - 381948 q^{39} + 13427994 q^{41} + 17820762 q^{43} - 7330788 q^{45} - 56044104 q^{47} - 4251114 q^{49} - 18401250 q^{51} + 96842752 q^{53} + 7627961 q^{55} + 141898680 q^{57} + 119136183 q^{59} - 90424326 q^{61} + 15960420 q^{63} - 18029712 q^{65} + 295944891 q^{67} - 187843215 q^{69} + 322953267 q^{71} - 255975514 q^{73} + 206496864 q^{75} - 109661090 q^{77} + 889658 q^{79} - 692205750 q^{81} + 277699042 q^{83} + 96595042 q^{85} - 1040096232 q^{87} + 1363672217 q^{89} + 491050280 q^{91} + 1530132009 q^{93} - 1454033872 q^{95} + 1398434043 q^{97} + 45987381 q^{99}+O(q^{100})$$ 2 * q + 21 * q^3 - 521 * q^5 + 7490 * q^7 - 3141 * q^9 - 29282 * q^11 + 150314 * q^13 - 393519 * q^15 + 690472 * q^17 - 511212 * q^19 + 1398810 * q^21 - 874751 * q^23 + 411771 * q^25 + 309771 * q^27 - 2951058 * q^29 + 5818705 * q^31 - 307461 * q^33 - 16179590 * q^35 + 2658905 * q^37 - 381948 * q^39 + 13427994 * q^41 + 17820762 * q^43 - 7330788 * q^45 - 56044104 * q^47 - 4251114 * q^49 - 18401250 * q^51 + 96842752 * q^53 + 7627961 * q^55 + 141898680 * q^57 + 119136183 * q^59 - 90424326 * q^61 + 15960420 * q^63 - 18029712 * q^65 + 295944891 * q^67 - 187843215 * q^69 + 322953267 * q^71 - 255975514 * q^73 + 206496864 * q^75 - 109661090 * q^77 + 889658 * q^79 - 692205750 * q^81 + 277699042 * q^83 + 96595042 * q^85 - 1040096232 * q^87 + 1363672217 * q^89 + 491050280 * q^91 + 1530132009 * q^93 - 1454033872 * q^95 + 1398434043 * q^97 + 45987381 * q^99

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −14.4081 15.4081
0 −123.672 0 1185.58 0 −1174.66 0 −4388.12 0
1.2 0 144.672 0 −1706.58 0 8664.66 0 1247.12 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.10.a.e 2
4.b odd 2 1 22.10.a.d 2
12.b even 2 1 198.10.a.n 2
44.c even 2 1 242.10.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.a.d 2 4.b odd 2 1
176.10.a.e 2 1.a even 1 1 trivial
198.10.a.n 2 12.b even 2 1
242.10.a.e 2 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 21T_{3} - 17892$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(176))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 21T - 17892$$
$5$ $$T^{2} + 521 T - 2023290$$
$7$ $$T^{2} - 7490 T - 10178000$$
$11$ $$(T + 14641)^{2}$$
$13$ $$T^{2} - 150314 T + 5595212424$$
$17$ $$T^{2} - 690472 T + 110050366092$$
$19$ $$T^{2} + 511212 T - 235841735968$$
$23$ $$T^{2} + 874751 T - 251963912952$$
$29$ $$T^{2} + 2951058 T - 11964147063840$$
$31$ $$T^{2} - 5818705 T - 21505055373504$$
$37$ $$T^{2} - 2658905 T - 34431390323214$$
$41$ $$T^{2} - 13427994 T + 33438413932128$$
$43$ $$T^{2} + \cdots - 200309296545160$$
$47$ $$T^{2} + \cdots - 168165989315712$$
$53$ $$T^{2} + \cdots + 866157473672220$$
$59$ $$T^{2} - 119136183 T - 11\!\cdots\!48$$
$61$ $$T^{2} + 90424326 T - 25\!\cdots\!60$$
$67$ $$T^{2} - 295944891 T + 15\!\cdots\!68$$
$71$ $$T^{2} - 322953267 T + 26\!\cdots\!60$$
$73$ $$T^{2} + 255975514 T + 15\!\cdots\!08$$
$79$ $$T^{2} - 889658 T - 87\!\cdots\!20$$
$83$ $$T^{2} - 277699042 T - 34\!\cdots\!84$$
$89$ $$T^{2} - 1363672217 T + 41\!\cdots\!62$$
$97$ $$T^{2} - 1398434043 T + 13\!\cdots\!02$$