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 $1$

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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: \( x^{2} - x - 222 \) Copy content Toggle raw display
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.

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

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:

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 \(S_{10}^{\mathrm{new}}(\Gamma_0(176))\). Copy content Toggle raw display

Hecke characteristic polynomials

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