Properties

Label 17.4.d.a
Level $17$
Weight $4$
Character orbit 17.d
Analytic conductor $1.003$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00303247010\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 54x^{10} + 1085x^{8} + 9836x^{6} + 38276x^{4} + 49664x^{2} + 16384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{3}) q^{2} + \beta_{8} q^{3} + ( - \beta_{9} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{4} + ( - \beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{5} + (\beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{6}+ \cdots + ( - 4 \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + 5 \beta_{2} + \cdots - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_{3}) q^{2} + \beta_{8} q^{3} + ( - \beta_{9} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{4} + ( - \beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{5} + (\beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{6}+ \cdots + (17 \beta_{11} - 93 \beta_{10} - 536 \beta_{9} + 47 \beta_{8} + 72 \beta_{7} + \cdots + 272) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 4 q^{3} - 20 q^{5} + 20 q^{6} - 4 q^{7} + 28 q^{8} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} - 4 q^{3} - 20 q^{5} + 20 q^{6} - 4 q^{7} + 28 q^{8} - 64 q^{9} - 116 q^{10} + 40 q^{11} + 56 q^{12} - 132 q^{14} + 244 q^{15} + 184 q^{16} + 52 q^{17} - 12 q^{19} + 572 q^{20} - 620 q^{22} - 276 q^{23} - 184 q^{24} - 464 q^{25} - 708 q^{26} - 664 q^{27} + 452 q^{28} + 632 q^{29} + 188 q^{31} + 700 q^{32} + 1400 q^{33} + 764 q^{34} - 632 q^{35} + 524 q^{36} + 940 q^{37} - 1112 q^{39} - 1864 q^{40} + 176 q^{41} + 48 q^{42} - 1360 q^{43} - 1364 q^{44} - 32 q^{45} + 452 q^{46} - 540 q^{48} + 1044 q^{49} + 2856 q^{50} + 340 q^{51} + 792 q^{52} - 360 q^{53} - 244 q^{54} - 1788 q^{56} - 148 q^{57} - 360 q^{58} - 584 q^{59} - 1792 q^{60} - 1052 q^{61} - 380 q^{62} + 1752 q^{63} + 404 q^{65} + 1372 q^{66} + 1080 q^{67} + 2532 q^{68} - 344 q^{69} + 2072 q^{70} + 28 q^{71} + 824 q^{73} - 2292 q^{74} + 400 q^{75} + 1328 q^{76} - 1252 q^{77} + 1128 q^{78} - 196 q^{79} - 904 q^{80} - 1528 q^{82} - 1008 q^{83} - 4768 q^{84} - 2824 q^{85} - 1200 q^{86} - 2516 q^{87} - 56 q^{88} - 860 q^{90} + 2456 q^{91} + 396 q^{92} - 836 q^{93} + 6360 q^{94} + 2172 q^{95} + 1668 q^{96} - 904 q^{97} + 3280 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 54x^{10} + 1085x^{8} + 9836x^{6} + 38276x^{4} + 49664x^{2} + 16384 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 408 \nu^{10} + 10 \nu^{9} - 17680 \nu^{8} - 1725 \nu^{7} - 268600 \nu^{6} - 73516 \nu^{5} - 1728288 \nu^{4} - 729732 \nu^{3} - 4623456 \nu^{2} + \cdots - 2889728 ) / 1392640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 408 \nu^{10} + 10 \nu^{9} + 17680 \nu^{8} - 1725 \nu^{7} + 268600 \nu^{6} - 73516 \nu^{5} + 1728288 \nu^{4} - 729732 \nu^{3} + 4623456 \nu^{2} + \cdots + 2889728 ) / 1392640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 41\nu^{8} + 569\nu^{6} + 3051\nu^{4} + 5498\nu^{2} + 2432 ) / 544 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 241 \nu^{11} - 280 \nu^{10} + 19350 \nu^{9} + 2800 \nu^{8} + 502765 \nu^{7} + 387400 \nu^{6} + 5292396 \nu^{5} + 5701600 \nu^{4} + 20629572 \nu^{3} + \cdots + 12462080 ) / 1392640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 241 \nu^{11} - 280 \nu^{10} - 19350 \nu^{9} + 2800 \nu^{8} - 502765 \nu^{7} + 387400 \nu^{6} - 5292396 \nu^{5} + 5701600 \nu^{4} - 20629572 \nu^{3} + \cdots + 12462080 ) / 1392640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 325 \nu^{11} + 872 \nu^{10} - 18510 \nu^{9} + 34800 \nu^{8} - 386545 \nu^{7} + 459720 \nu^{6} - 3460060 \nu^{5} + 2176992 \nu^{4} - 10989460 \nu^{3} + \cdots - 5347328 ) / 1392640 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 325 \nu^{11} - 872 \nu^{10} - 18510 \nu^{9} - 34800 \nu^{8} - 386545 \nu^{7} - 459720 \nu^{6} - 3460060 \nu^{5} - 2176992 \nu^{4} - 10989460 \nu^{3} + \cdots + 5347328 ) / 1392640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\nu^{11} + 898\nu^{9} + 15367\nu^{7} + 114052\nu^{5} + 336716\nu^{3} + 239872\nu ) / 69632 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 51 \nu^{11} + 8 \nu^{10} - 2210 \nu^{9} - 80 \nu^{8} - 33575 \nu^{7} - 7960 \nu^{6} - 216036 \nu^{5} - 86432 \nu^{4} - 577932 \nu^{3} - 211424 \nu^{2} - 361216 \nu + 2048 ) / 174080 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 51 \nu^{11} - 8 \nu^{10} - 2210 \nu^{9} + 80 \nu^{8} - 33575 \nu^{7} + 7960 \nu^{6} - 216036 \nu^{5} + 86432 \nu^{4} - 577932 \nu^{3} + 211424 \nu^{2} - 361216 \nu - 2048 ) / 174080 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + 2\beta_{10} + 6\beta_{9} + \beta_{8} + \beta_{7} - \beta_{3} - \beta_{2} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -16\beta_{8} + 16\beta_{7} - \beta_{6} - \beta_{5} - 21\beta_{4} + 31\beta_{3} - 31\beta_{2} + 106 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 50 \beta_{11} - 50 \beta_{10} - 138 \beta_{9} - 15 \beta_{8} - 15 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 19 \beta_{3} + 19 \beta_{2} + 189 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 28 \beta_{11} + 28 \beta_{10} + 250 \beta_{8} - 250 \beta_{7} + 31 \beta_{6} + 31 \beta_{5} + 373 \beta_{4} - 619 \beta_{3} + 619 \beta_{2} - 1574 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 990 \beta_{11} + 990 \beta_{10} + 2602 \beta_{9} + 191 \beta_{8} + 191 \beta_{7} + 120 \beta_{6} - 120 \beta_{5} - 535 \beta_{3} - 535 \beta_{2} - 2885 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1072 \beta_{11} - 1072 \beta_{10} - 3946 \beta_{8} + 3946 \beta_{7} - 679 \beta_{6} - 679 \beta_{5} - 6349 \beta_{4} + 11187 \beta_{3} - 11187 \beta_{2} + 24438 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 18242 \beta_{11} - 18242 \beta_{10} - 46402 \beta_{9} - 2195 \beta_{8} - 2195 \beta_{7} - 2796 \beta_{6} + 2796 \beta_{5} + 13567 \beta_{3} + 13567 \beta_{2} + 45213 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 28020 \beta_{11} + 28020 \beta_{10} + 62854 \beta_{8} - 62854 \beta_{7} + 13251 \beta_{6} + 13251 \beta_{5} + 107189 \beta_{4} - 195539 \beta_{3} + 195539 \beta_{2} - 388206 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 326166 \beta_{11} + 326166 \beta_{10} + 814346 \beta_{9} + 21583 \beta_{8} + 21583 \beta_{7} + 59104 \beta_{6} - 59104 \beta_{5} - 304847 \beta_{3} - 304847 \beta_{2} - 720309 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
3.86166i
0.705468i
4.15292i
3.68604i
1.22788i
2.49971i
3.86166i
0.705468i
4.15292i
3.68604i
1.22788i
2.49971i
−3.43772 3.43772i −4.67995 + 1.93850i 15.6358i −7.10390 17.1503i 22.7523 + 9.42432i 5.36561 12.9537i 26.2496 26.2496i −0.947753 + 0.947753i −34.5367 + 83.3791i
2.2 −1.20595 1.20595i 4.10553 1.70057i 5.09138i 2.60601 + 6.29147i −7.00185 2.90026i −5.31013 + 12.8198i −15.7875 + 15.7875i −5.12843 + 5.12843i 4.44447 10.7299i
2.3 2.22945 + 2.22945i −1.83980 + 0.762069i 1.94089i −1.91633 4.62643i −5.80073 2.40274i 1.06584 2.57316i 13.5085 13.5085i −16.2878 + 16.2878i 6.04203 14.5867i
8.1 −1.89932 + 1.89932i −1.65755 + 4.00167i 0.785167i 1.92782 + 0.798529i −4.45224 10.7487i 23.0956 9.56650i −16.6858 16.6858i 5.82599 + 5.82599i −5.17821 + 2.14488i
8.2 −0.161134 + 0.161134i 3.15299 7.61199i 7.94807i 2.54200 + 1.05293i 0.718496 + 1.73460i −19.8837 + 8.23610i −2.56978 2.56978i −28.9092 28.9092i −0.579266 + 0.239940i
8.3 2.47467 2.47467i −1.08123 + 2.61032i 4.24796i −8.05561 3.33674i 3.78400 + 9.13537i −6.33320 + 2.62330i 9.28506 + 9.28506i 13.4472 + 13.4472i −28.1923 + 11.6776i
9.1 −3.43772 + 3.43772i −4.67995 1.93850i 15.6358i −7.10390 + 17.1503i 22.7523 9.42432i 5.36561 + 12.9537i 26.2496 + 26.2496i −0.947753 0.947753i −34.5367 83.3791i
9.2 −1.20595 + 1.20595i 4.10553 + 1.70057i 5.09138i 2.60601 6.29147i −7.00185 + 2.90026i −5.31013 12.8198i −15.7875 15.7875i −5.12843 5.12843i 4.44447 + 10.7299i
9.3 2.22945 2.22945i −1.83980 0.762069i 1.94089i −1.91633 + 4.62643i −5.80073 + 2.40274i 1.06584 + 2.57316i 13.5085 + 13.5085i −16.2878 16.2878i 6.04203 + 14.5867i
15.1 −1.89932 1.89932i −1.65755 4.00167i 0.785167i 1.92782 0.798529i −4.45224 + 10.7487i 23.0956 + 9.56650i −16.6858 + 16.6858i 5.82599 5.82599i −5.17821 2.14488i
15.2 −0.161134 0.161134i 3.15299 + 7.61199i 7.94807i 2.54200 1.05293i 0.718496 1.73460i −19.8837 8.23610i −2.56978 + 2.56978i −28.9092 + 28.9092i −0.579266 0.239940i
15.3 2.47467 + 2.47467i −1.08123 2.61032i 4.24796i −8.05561 + 3.33674i 3.78400 9.13537i −6.33320 2.62330i 9.28506 9.28506i 13.4472 13.4472i −28.1923 11.6776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.4.d.a 12
3.b odd 2 1 153.4.l.a 12
17.d even 8 1 inner 17.4.d.a 12
17.e odd 16 2 289.4.a.g 12
17.e odd 16 2 289.4.b.e 12
51.g odd 8 1 153.4.l.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.d.a 12 1.a even 1 1 trivial
17.4.d.a 12 17.d even 8 1 inner
153.4.l.a 12 3.b odd 2 1
153.4.l.a 12 51.g odd 8 1
289.4.a.g 12 17.e odd 16 2
289.4.b.e 12 17.e odd 16 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + 8 T^{10} - 20 T^{9} + \cdots + 3136 \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + 40 T^{10} + \cdots + 20428832 \) Copy content Toggle raw display
$5$ \( T^{12} + 20 T^{11} + \cdots + 1004236928 \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} + \cdots + 3993906708992 \) Copy content Toggle raw display
$11$ \( T^{12} - 40 T^{11} + \cdots + 44\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{12} + 11916 T^{10} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} - 52 T^{11} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{12} + 12 T^{11} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{12} + 276 T^{11} + \cdots + 78\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{12} - 632 T^{11} + \cdots + 73\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{12} - 188 T^{11} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{12} - 940 T^{11} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{12} - 176 T^{11} + \cdots + 36\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{12} + 1360 T^{11} + \cdots + 65\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{12} + 665464 T^{10} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + 360 T^{11} + \cdots + 95\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + 584 T^{11} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{12} + 1052 T^{11} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( (T^{6} - 540 T^{5} + \cdots + 61\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 28 T^{11} + \cdots + 32\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{12} - 824 T^{11} + \cdots + 99\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{12} + 196 T^{11} + \cdots + 93\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{12} + 1008 T^{11} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + 2740280 T^{10} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{12} + 904 T^{11} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display
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