Properties

Label 7623.2.a.cz
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{8} q^{5} - q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{8} q^{5} - q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( -2 + \beta_{4} + \beta_{6} ) q^{10} + ( -2 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{13} -\beta_{1} q^{14} + ( 3 + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{10} ) q^{16} + ( -\beta_{1} - \beta_{7} - \beta_{8} ) q^{17} + ( -1 - \beta_{4} ) q^{19} + ( -2 \beta_{1} - \beta_{3} + \beta_{8} + \beta_{9} ) q^{20} + ( -\beta_{1} - \beta_{3} ) q^{23} + ( 3 - \beta_{2} - \beta_{6} - \beta_{10} ) q^{25} + ( -4 \beta_{1} - \beta_{3} - \beta_{8} - \beta_{9} ) q^{26} + ( -2 - \beta_{2} ) q^{28} + ( -\beta_{8} - \beta_{11} ) q^{29} + ( -2 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{31} + ( 3 \beta_{1} + \beta_{3} + 2 \beta_{7} - \beta_{8} + \beta_{11} ) q^{32} + ( -2 - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} - \beta_{6} - \beta_{10} ) q^{34} -\beta_{8} q^{35} + ( -\beta_{2} + 2 \beta_{4} ) q^{37} + ( -\beta_{1} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{38} + ( -5 - 3 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{40} + ( -2 \beta_{1} - \beta_{3} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{41} + ( -1 + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{43} + ( -3 - 3 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{10} ) q^{46} + ( \beta_{3} + \beta_{9} ) q^{47} + q^{49} + ( \beta_{1} - 2 \beta_{7} - 3 \beta_{8} - \beta_{11} ) q^{50} + ( -9 - 5 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{10} ) q^{52} + ( -\beta_{1} - \beta_{3} + \beta_{7} - 2 \beta_{8} ) q^{53} + ( -2 \beta_{1} - \beta_{3} ) q^{56} + ( 2 - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{10} ) q^{58} + ( -\beta_{1} + \beta_{3} + \beta_{7} + \beta_{8} ) q^{59} + ( -3 + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{7} + 2 \beta_{8} ) q^{62} + ( 7 + 4 \beta_{2} - 8 \beta_{5} - \beta_{6} + 3 \beta_{10} ) q^{64} + ( -\beta_{1} + 2 \beta_{7} - 2 \beta_{8} ) q^{65} + ( 2 \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{10} ) q^{67} + ( -4 \beta_{1} - \beta_{3} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{68} + ( 2 - \beta_{4} - \beta_{6} ) q^{70} + ( \beta_{3} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{71} + ( -5 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{73} + ( -2 \beta_{1} - \beta_{3} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{74} + ( -\beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{76} + ( 1 - \beta_{2} - 2 \beta_{5} - 3 \beta_{6} + \beta_{10} ) q^{79} + ( -7 \beta_{1} - \beta_{3} - \beta_{7} ) q^{80} + ( -7 - 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{10} ) q^{82} + ( -\beta_{1} - \beta_{3} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{83} + ( -6 + 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{10} ) q^{85} + ( -\beta_{1} + \beta_{3} - 2 \beta_{7} + 2 \beta_{9} ) q^{86} + ( 2 \beta_{1} + \beta_{8} + 2 \beta_{9} ) q^{89} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} ) q^{91} + ( -7 \beta_{1} - 2 \beta_{3} - 2 \beta_{7} + \beta_{8} - \beta_{11} ) q^{92} + ( -1 + 3 \beta_{2} + 3 \beta_{4} - 5 \beta_{5} - \beta_{6} + 2 \beta_{10} ) q^{94} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{7} - 2 \beta_{8} ) q^{95} + ( -2 + 3 \beta_{2} + \beta_{4} - 3 \beta_{5} + \beta_{10} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 20q^{4} - 12q^{7} + O(q^{10}) \) \( 12q + 20q^{4} - 12q^{7} - 20q^{10} - 20q^{13} + 28q^{16} - 12q^{19} + 32q^{25} - 20q^{28} - 16q^{31} - 24q^{34} + 4q^{37} - 48q^{40} - 16q^{43} - 24q^{46} + 12q^{49} - 96q^{52} + 20q^{58} - 44q^{61} + 76q^{64} + 20q^{70} - 52q^{73} + 8q^{79} - 68q^{82} - 72q^{85} + 20q^{91} - 20q^{94} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 22 x^{10} + 181 x^{8} - 692 x^{6} + 1240 x^{4} - 936 x^{2} + 244\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{8} - 17 \nu^{6} + 94 \nu^{4} - 186 \nu^{2} + 92 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{10} - 21 \nu^{8} + 160 \nu^{6} - 536 \nu^{4} + 748 \nu^{2} - 308 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{10} - 21 \nu^{8} + 164 \nu^{6} - 588 \nu^{4} + 924 \nu^{2} - 412 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} + 21 \nu^{9} - 160 \nu^{7} + 536 \nu^{5} - 748 \nu^{3} + 308 \nu \)\()/8\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{11} - 21 \nu^{9} + 162 \nu^{7} - 562 \nu^{5} + 840 \nu^{3} - 384 \nu \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{11} + 22 \nu^{9} - 178 \nu^{7} + 643 \nu^{5} - 980 \nu^{3} + 438 \nu \)\()/4\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{10} - 63 \nu^{8} + 484 \nu^{6} - 1652 \nu^{4} + 2356 \nu^{2} - 956 \)\()/8\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{11} - 63 \nu^{9} + 482 \nu^{7} - 1626 \nu^{5} + 2264 \nu^{3} - 880 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10} - \beta_{6} - 2 \beta_{5} + 8 \beta_{2} + 23\)
\(\nu^{5}\)\(=\)\(\beta_{11} - \beta_{8} + 2 \beta_{7} + 9 \beta_{3} + 39 \beta_{1}\)
\(\nu^{6}\)\(=\)\(13 \beta_{10} - 11 \beta_{6} - 28 \beta_{5} + 60 \beta_{2} + 149\)
\(\nu^{7}\)\(=\)\(13 \beta_{11} - 9 \beta_{8} + 30 \beta_{7} + 71 \beta_{3} + 269 \beta_{1}\)
\(\nu^{8}\)\(=\)\(127 \beta_{10} - 93 \beta_{6} - 288 \beta_{5} + 4 \beta_{4} + 454 \beta_{2} + 1023\)
\(\nu^{9}\)\(=\)\(127 \beta_{11} + 4 \beta_{9} - 55 \beta_{8} + 318 \beta_{7} + 547 \beta_{3} + 1931 \beta_{1}\)
\(\nu^{10}\)\(=\)\(1123 \beta_{10} - 729 \beta_{6} - 2632 \beta_{5} + 84 \beta_{4} + 3474 \beta_{2} + 7287\)
\(\nu^{11}\)\(=\)\(1123 \beta_{11} + 84 \beta_{9} - 251 \beta_{8} + 2942 \beta_{7} + 4203 \beta_{3} + 14235 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.79401
−2.41714
−2.09771
−1.60257
−0.870105
−0.790737
0.790737
0.870105
1.60257
2.09771
2.41714
2.79401
−2.79401 0 5.80650 1.62657 0 −1.00000 −10.6354 0 −4.54465
1.2 −2.41714 0 3.84255 4.01054 0 −1.00000 −4.45369 0 −9.69401
1.3 −2.09771 0 2.40037 −2.53213 0 −1.00000 −0.839859 0 5.31167
1.4 −1.60257 0 0.568238 −1.76461 0 −1.00000 2.29450 0 2.82792
1.5 −0.870105 0 −1.24292 0.709862 0 −1.00000 2.82168 0 −0.617654
1.6 −0.790737 0 −1.37474 4.15216 0 −1.00000 2.66853 0 −3.28327
1.7 0.790737 0 −1.37474 −4.15216 0 −1.00000 −2.66853 0 −3.28327
1.8 0.870105 0 −1.24292 −0.709862 0 −1.00000 −2.82168 0 −0.617654
1.9 1.60257 0 0.568238 1.76461 0 −1.00000 −2.29450 0 2.82792
1.10 2.09771 0 2.40037 2.53213 0 −1.00000 0.839859 0 5.31167
1.11 2.41714 0 3.84255 −4.01054 0 −1.00000 4.45369 0 −9.69401
1.12 2.79401 0 5.80650 −1.62657 0 −1.00000 10.6354 0 −4.54465
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.cz 12
3.b odd 2 1 inner 7623.2.a.cz 12
11.b odd 2 1 7623.2.a.da yes 12
33.d even 2 1 7623.2.a.da yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.cz 12 1.a even 1 1 trivial
7623.2.a.cz 12 3.b odd 2 1 inner
7623.2.a.da yes 12 11.b odd 2 1
7623.2.a.da yes 12 33.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{12} - 22 T_{2}^{10} + 181 T_{2}^{8} - 692 T_{2}^{6} + 1240 T_{2}^{4} - 936 T_{2}^{2} + 244 \)
\( T_{5}^{12} - 46 T_{5}^{10} + 751 T_{5}^{8} - 5300 T_{5}^{6} + 16771 T_{5}^{4} - 21846 T_{5}^{2} + 7381 \)
\( T_{13}^{6} + 10 T_{13}^{5} + 5 T_{13}^{4} - 150 T_{13}^{3} - 174 T_{13}^{2} + 260 T_{13} - 50 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 5 T^{4} + 4 T^{6} + 8 T^{8} - 24 T^{10} - 12 T^{12} - 96 T^{14} + 128 T^{16} + 256 T^{18} + 1280 T^{20} + 2048 T^{22} + 4096 T^{24} \)
$3$ \( \)
$5$ \( 1 + 14 T^{2} + 101 T^{4} + 490 T^{6} + 2846 T^{8} + 20574 T^{10} + 123321 T^{12} + 514350 T^{14} + 1778750 T^{16} + 7656250 T^{18} + 39453125 T^{20} + 136718750 T^{22} + 244140625 T^{24} \)
$7$ \( ( 1 + T )^{12} \)
$11$ \( \)
$13$ \( ( 1 + 10 T + 83 T^{2} + 500 T^{3} + 2621 T^{4} + 11310 T^{5} + 44436 T^{6} + 147030 T^{7} + 442949 T^{8} + 1098500 T^{9} + 2370563 T^{10} + 3712930 T^{11} + 4826809 T^{12} )^{2} \)
$17$ \( 1 + 90 T^{2} + 4261 T^{4} + 146590 T^{6} + 4020830 T^{8} + 89661290 T^{10} + 1660695689 T^{12} + 25912112810 T^{14} + 335823742430 T^{16} + 3538326239710 T^{18} + 29723702456101 T^{20} + 181439451040410 T^{22} + 582622237229761 T^{24} \)
$19$ \( ( 1 + 6 T + 106 T^{2} + 530 T^{3} + 4853 T^{4} + 19396 T^{5} + 121592 T^{6} + 368524 T^{7} + 1751933 T^{8} + 3635270 T^{9} + 13814026 T^{10} + 14856594 T^{11} + 47045881 T^{12} )^{2} \)
$23$ \( 1 + 168 T^{2} + 13726 T^{4} + 723160 T^{6} + 27775475 T^{8} + 839711408 T^{10} + 21022991420 T^{12} + 444207334832 T^{14} + 7772716699475 T^{16} + 107053633489240 T^{18} + 1074896583967006 T^{20} + 6959653883893032 T^{22} + 21914624432020321 T^{24} \)
$29$ \( 1 + 158 T^{2} + 12523 T^{4} + 676610 T^{6} + 28938587 T^{8} + 1051503368 T^{10} + 32911943606 T^{12} + 884314332488 T^{14} + 20467712751947 T^{16} + 402463407221810 T^{18} + 6264585829510603 T^{20} + 66471742861431758 T^{22} + 353814783205469041 T^{24} \)
$31$ \( ( 1 + 8 T + 110 T^{2} + 688 T^{3} + 4823 T^{4} + 26760 T^{5} + 147540 T^{6} + 829560 T^{7} + 4634903 T^{8} + 20496208 T^{9} + 101587310 T^{10} + 229033208 T^{11} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 - 2 T + 91 T^{2} - 314 T^{3} + 5891 T^{4} - 17852 T^{5} + 259334 T^{6} - 660524 T^{7} + 8064779 T^{8} - 15905042 T^{9} + 170548651 T^{10} - 138687914 T^{11} + 2565726409 T^{12} )^{2} \)
$41$ \( 1 + 178 T^{2} + 13639 T^{4} + 560026 T^{6} + 12799187 T^{8} + 170859508 T^{10} + 2999880794 T^{12} + 287214832948 T^{14} + 36167443456307 T^{16} + 2660181877670266 T^{18} + 108906395199981319 T^{20} + 2389233357207127378 T^{22} + 22563490300366186081 T^{24} \)
$43$ \( ( 1 + 8 T + 152 T^{2} + 1072 T^{3} + 11384 T^{4} + 75288 T^{5} + 582042 T^{6} + 3237384 T^{7} + 21049016 T^{8} + 85231504 T^{9} + 519657752 T^{10} + 1176067544 T^{11} + 6321363049 T^{12} )^{2} \)
$47$ \( 1 + 332 T^{2} + 57032 T^{4} + 6587500 T^{6} + 563615924 T^{8} + 37403435100 T^{10} + 1967941380078 T^{12} + 82624188135900 T^{14} + 2750265915640244 T^{16} + 71008080979787500 T^{18} + 1358005300893553352 T^{20} + 17462911902295576268 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 382 T^{2} + 66487 T^{4} + 7011646 T^{6} + 511558763 T^{8} + 29260051156 T^{10} + 1541317998842 T^{12} + 82191483697204 T^{14} + 4036444699835003 T^{16} + 155408654052708334 T^{18} + 4139460036380158807 T^{20} + 66807013679625984718 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 388 T^{2} + 79864 T^{4} + 11127724 T^{6} + 1159312964 T^{8} + 94699812532 T^{10} + 6212827842398 T^{12} + 329650047423892 T^{14} + 14047813696768004 T^{16} + 469373336529763084 T^{18} + 11726466068831492344 T^{20} + \)\(19\!\cdots\!88\)\( T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( ( 1 + 22 T + 418 T^{2} + 5566 T^{3} + 62885 T^{4} + 613252 T^{5} + 5052104 T^{6} + 37408372 T^{7} + 233995085 T^{8} + 1263376246 T^{9} + 5787561538 T^{10} + 18581118622 T^{11} + 51520374361 T^{12} )^{2} \)
$67$ \( ( 1 + 196 T^{2} + 64 T^{3} + 21428 T^{4} + 1352 T^{5} + 1671734 T^{6} + 90584 T^{7} + 96190292 T^{8} + 19248832 T^{9} + 3949619716 T^{10} + 90458382169 T^{12} )^{2} \)
$71$ \( 1 + 72 T^{2} + 12670 T^{4} + 721048 T^{6} + 105137939 T^{8} + 4570036592 T^{10} + 536793730748 T^{12} + 23037554460272 T^{14} + 2671731766865459 T^{16} + 92366453520669208 T^{18} + 8181697240883791870 T^{20} + \)\(23\!\cdots\!72\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( ( 1 + 26 T + 622 T^{2} + 9206 T^{3} + 127301 T^{4} + 1316864 T^{5} + 12781136 T^{6} + 96131072 T^{7} + 678387029 T^{8} + 3581290502 T^{9} + 17663705902 T^{10} + 53899861418 T^{11} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 - 4 T + 106 T^{2} + 596 T^{3} + 1187 T^{4} + 48536 T^{5} + 293876 T^{6} + 3834344 T^{7} + 7408067 T^{8} + 293851244 T^{9} + 4128708586 T^{10} - 12308225596 T^{11} + 243087455521 T^{12} )^{2} \)
$83$ \( 1 + 260 T^{2} + 54704 T^{4} + 7982020 T^{6} + 1013286620 T^{8} + 104330092020 T^{10} + 9478839943518 T^{12} + 718730003925780 T^{14} + 48088881676965020 T^{16} + 2609644599038825380 T^{18} + \)\(12\!\cdots\!64\)\( T^{20} + \)\(40\!\cdots\!40\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 + 422 T^{2} + 99085 T^{4} + 17225714 T^{6} + 2365521806 T^{8} + 269042562326 T^{10} + 25988928108833 T^{12} + 2131086136184246 T^{14} + 148418139242807246 T^{16} + 8560857581444971154 T^{18} + \)\(39\!\cdots\!85\)\( T^{20} + \)\(13\!\cdots\!22\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( ( 1 + 16 T + 459 T^{2} + 3698 T^{3} + 62861 T^{4} + 231678 T^{5} + 5290852 T^{6} + 22472766 T^{7} + 591459149 T^{8} + 3375064754 T^{9} + 40634939979 T^{10} + 137397444112 T^{11} + 832972004929 T^{12} )^{2} \)
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