Properties

Label 12.19.c.a
Level 12
Weight 19
Character orbit 12.c
Analytic conductor 24.646
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(24.6463365252\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{19}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 3989 - \beta_{1} ) q^{3} \) \( + ( -4 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 1837394 + 110 \beta_{1} - \beta_{5} ) q^{7} \) \( + ( -5679831 - 4221 \beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 3989 - \beta_{1} ) q^{3} \) \( + ( -4 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 1837394 + 110 \beta_{1} - \beta_{5} ) q^{7} \) \( + ( -5679831 - 4221 \beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{9} \) \( + ( -23209 \beta_{1} - 302 \beta_{2} - 6 \beta_{3} - 31 \beta_{4} ) q^{11} \) \( + ( 295932266 + 36346 \beta_{1} - 18 \beta_{4} - 218 \beta_{5} ) q^{13} \) \( + ( -1642819680 - 12384 \beta_{1} + 13182 \beta_{2} + 30 \beta_{3} - 12 \beta_{4} + 639 \beta_{5} ) q^{15} \) \( + ( 239092 \beta_{1} - 49090 \beta_{2} + 210 \beta_{3} + 32 \beta_{4} ) q^{17} \) \( + ( -33672422134 + 3021823 \beta_{1} - 4869 \beta_{4} + 2938 \beta_{5} ) q^{19} \) \( + ( -33359720006 - 2136500 \beta_{1} + 294489 \beta_{2} - 378 \beta_{3} - 20844 \beta_{4} - 11988 \beta_{5} ) q^{21} \) \( + ( -59189414 \beta_{1} - 393040 \beta_{2} - 3360 \beta_{3} - 78434 \beta_{4} ) q^{23} \) \( + ( -993004183655 + 118601514 \beta_{1} - 172242 \beta_{4} - 2466 \beta_{5} ) q^{25} \) \( + ( -1361296003779 - 9324018 \beta_{1} - 1766850 \beta_{2} + 2310 \beta_{3} - 406167 \beta_{4} + 71802 \beta_{5} ) q^{27} \) \( + ( -583567108 \beta_{1} + 5452951 \beta_{2} + 32148 \beta_{3} - 732192 \beta_{4} ) q^{29} \) \( + ( -9343245323806 + 1172086452 \beta_{1} - 1680606 \beta_{4} - 159183 \beta_{5} ) q^{31} \) \( + ( -8579832069600 - 94951215 \beta_{1} - 9125040 \beta_{2} - 2625 \beta_{3} - 2813619 \beta_{4} + 32661 \beta_{5} ) q^{33} \) \( + ( -3883853798 \beta_{1} - 3950638 \beta_{2} - 200550 \beta_{3} - 5035550 \beta_{4} ) q^{35} \) \( + ( -27932684252518 + 3238663886 \beta_{1} - 4875318 \beta_{4} + 1006178 \beta_{5} ) q^{37} \) \( + ( -12281349860174 - 410436728 \beta_{1} + 64224846 \beta_{2} - 69282 \beta_{3} - 4478382 \beta_{4} - 2652750 \beta_{5} ) q^{39} \) \( + ( 3204810968 \beta_{1} - 74934746 \beta_{2} + 818832 \beta_{3} + 3676592 \beta_{4} ) q^{41} \) \( + ( 74765321129306 - 5975608869 \beta_{1} + 8777727 \beta_{4} - 497178 \beta_{5} ) q^{43} \) \( + ( -2153060291520 + 1677496230 \beta_{1} + 11298159 \beta_{2} + 600390 \beta_{3} + 27225162 \beta_{4} + 14968746 \beta_{5} ) q^{45} \) \( + ( 29661301884 \beta_{1} + 19268340 \beta_{2} - 1934940 \beta_{3} + 38787004 \beta_{4} ) q^{47} \) \( + ( 635885566591635 - 50012121370 \beta_{1} + 75894210 \beta_{4} - 19338190 \beta_{5} ) q^{49} \) \( + ( 96546394200960 + 821926188 \beta_{1} - 468596994 \beta_{2} - 2465562 \beta_{3} + 95354400 \beta_{4} - 36962964 \beta_{5} ) q^{51} \) \( + ( 91008069644 \beta_{1} + 1181305525 \beta_{2} + 623520 \beta_{3} + 124098144 \beta_{4} ) q^{53} \) \( + ( 1378505836889280 - 94263847074 \beta_{1} + 122540922 \beta_{4} + 91620306 \beta_{5} ) q^{55} \) \( + ( -1256803429634606 + 21215720305 \beta_{1} - 858109680 \beta_{2} + 4660065 \beta_{3} + 78987177 \beta_{4} + 24572241 \beta_{5} ) q^{57} \) \( + ( 35259873871 \beta_{1} - 2339109352 \beta_{2} + 12896184 \beta_{3} + 32158709 \beta_{4} ) q^{59} \) \( + ( 2225078699447114 + 196657240566 \beta_{1} - 262553598 \beta_{4} - 148026534 \beta_{5} ) q^{61} \) \( + ( -6791496589216734 - 35012491290 \beta_{1} + 7088320280 \beta_{2} + 4763080 \beta_{3} - 430625704 \beta_{4} + 58518177 \beta_{5} ) q^{63} \) \( + ( -846749477792 \beta_{1} - 1207119802 \beta_{2} - 44507220 \beta_{3} - 1099546480 \beta_{4} ) q^{65} \) \( + ( 6260075051272874 + 714271618977 \beta_{1} - 1021896351 \beta_{4} - 111171144 \beta_{5} ) q^{67} \) \( + ( -21938830096173120 - 238331010186 \beta_{1} - 8150781732 \beta_{2} - 52710546 \beta_{3} - 1795554330 \beta_{4} + 8774118 \beta_{5} ) q^{69} \) \( + ( -710204892746 \beta_{1} - 4266141148 \beta_{2} + 49516356 \beta_{3} - 948803974 \beta_{4} ) q^{71} \) \( + ( 24903029029279058 + 1016019215808 \beta_{1} - 1572898464 \beta_{4} + 586927536 \beta_{5} ) q^{73} \) \( + ( -47998484095086835 + 520521069833 \beta_{1} + 977338710 \beta_{2} + 124632270 \beta_{3} + 576420786 \beta_{4} - 406255662 \beta_{5} ) q^{75} \) \( + ( 950024344 \beta_{1} + 44179182890 \beta_{2} + 86091600 \beta_{3} + 220833824 \beta_{4} ) q^{77} \) \( + ( 107753688936456578 - 2285100318476 \beta_{1} + 3286571778 \beta_{4} + 247504609 \beta_{5} ) q^{79} \) \( + ( -109353387722503599 + 226739154096 \beta_{1} - 42455713536 \beta_{2} - 28675278 \beta_{3} + 2600814348 \beta_{4} - 354125412 \beta_{5} ) q^{81} \) \( + ( 5574822398157 \beta_{1} - 54478176150 \beta_{2} - 345347310 \beta_{3} + 6986523427 \beta_{4} ) q^{83} \) \( + ( 236057585831527680 - 3799585636344 \beta_{1} + 5863502232 \beta_{4} - 2078549064 \beta_{5} ) q^{85} \) \( + ( -217752287202727200 - 2302878230280 \beta_{1} + 98133052470 \beta_{2} - 525743610 \beta_{3} + 11755085532 \beta_{4} + 4240022157 \beta_{5} ) q^{87} \) \( + ( 5500255664428 \beta_{1} - 9471569476 \beta_{2} + 290635842 \beta_{3} + 7052366672 \beta_{4} ) q^{89} \) \( + ( 494492834726491636 - 11108214865608 \beta_{1} + 16793805924 \beta_{4} - 3901180038 \beta_{5} ) q^{91} \) \( + ( -472451433140287094 + 4692699948490 \beta_{1} + 49327966035 \beta_{2} + 1164990600 \beta_{3} + 2807798418 \beta_{4} - 5583771126 \beta_{5} ) q^{93} \) \( + ( 11391901528966 \beta_{1} - 81928309504 \beta_{2} + 376245840 \beta_{3} + 14308471010 \beta_{4} ) q^{95} \) \( + ( 497433050730376514 + 14060510920294 \beta_{1} - 23933153502 \beta_{4} + 21650595778 \beta_{5} ) q^{97} \) \( + ( -780204924039879360 + 517016038293 \beta_{1} - 132154263468 \beta_{2} - 258163884 \beta_{3} - 440126721 \beta_{4} - 5341921866 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 23934q^{3} \) \(\mathstrut +\mathstrut 11024364q^{7} \) \(\mathstrut -\mathstrut 34078986q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 23934q^{3} \) \(\mathstrut +\mathstrut 11024364q^{7} \) \(\mathstrut -\mathstrut 34078986q^{9} \) \(\mathstrut +\mathstrut 1775593596q^{13} \) \(\mathstrut -\mathstrut 9856918080q^{15} \) \(\mathstrut -\mathstrut 202034532804q^{19} \) \(\mathstrut -\mathstrut 200158320036q^{21} \) \(\mathstrut -\mathstrut 5958025101930q^{25} \) \(\mathstrut -\mathstrut 8167776022674q^{27} \) \(\mathstrut -\mathstrut 56059471942836q^{31} \) \(\mathstrut -\mathstrut 51478992417600q^{33} \) \(\mathstrut -\mathstrut 167596105515108q^{37} \) \(\mathstrut -\mathstrut 73688099161044q^{39} \) \(\mathstrut +\mathstrut 448591926775836q^{43} \) \(\mathstrut -\mathstrut 12918361749120q^{45} \) \(\mathstrut +\mathstrut 3815313399549810q^{49} \) \(\mathstrut +\mathstrut 579278365205760q^{51} \) \(\mathstrut +\mathstrut 8271035021335680q^{55} \) \(\mathstrut -\mathstrut 7540820577807636q^{57} \) \(\mathstrut +\mathstrut 13350472196682684q^{61} \) \(\mathstrut -\mathstrut 40748979535300404q^{63} \) \(\mathstrut +\mathstrut 37560450307637244q^{67} \) \(\mathstrut -\mathstrut 131632980577038720q^{69} \) \(\mathstrut +\mathstrut 149418174175674348q^{73} \) \(\mathstrut -\mathstrut 287990904570521010q^{75} \) \(\mathstrut +\mathstrut 646522133618739468q^{79} \) \(\mathstrut -\mathstrut 656120326335021594q^{81} \) \(\mathstrut +\mathstrut 1416345514989166080q^{85} \) \(\mathstrut -\mathstrut 1306513723216363200q^{87} \) \(\mathstrut +\mathstrut 2966957008358949816q^{91} \) \(\mathstrut -\mathstrut 2834708598841722564q^{93} \) \(\mathstrut +\mathstrut 2984598304382259084q^{97} \) \(\mathstrut -\mathstrut 4681229544239276160q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut +\mathstrut \) \(132762\) \(x^{4}\mathstrut +\mathstrut \) \(1042140330\) \(x^{2}\mathstrut +\mathstrut \) \(1430023595000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} - 140 \nu^{4} + 152362 \nu^{3} - 21330680 \nu^{2} + 4028435530 \nu - 218699406800 \)\()/14586075\)
\(\beta_{2}\)\(=\)\((\)\( 2494 \nu^{5} - 560 \nu^{4} + 322837228 \nu^{3} - 85322720 \nu^{2} + 1562923520620 \nu - 874797627200 \)\()/14586075\)
\(\beta_{3}\)\(=\)\((\)\( 90352 \nu^{5} + 12040 \nu^{4} + 12337371424 \nu^{3} + 1834438480 \nu^{2} + 115582517582560 \nu + 18808148984800 \)\()/14586075\)
\(\beta_{4}\)\(=\)\((\)\( 229 \nu^{5} + 35980 \nu^{4} + 34890898 \nu^{3} + 5481984760 \nu^{2} + 922511736370 \nu + 56205747547600 \)\()/4862025\)
\(\beta_{5}\)\(=\)\((\)\( 22 \nu^{5} + 588112 \nu^{4} + 3351964 \nu^{3} + 75203213344 \nu^{2} + 88625581660 \nu + 281335063656640 \)\()/2917215\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(58\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(75\) \(\beta_{2}\mathstrut +\mathstrut \) \(44160\) \(\beta_{1}\)\()/7464960\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(189\) \(\beta_{5}\mathstrut +\mathstrut \) \(2737\) \(\beta_{4}\mathstrut -\mathstrut \) \(1859529\) \(\beta_{1}\mathstrut -\mathstrut \) \(41294292480\)\()/933120\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(2678018\) \(\beta_{4}\mathstrut +\mathstrut \) \(222663\) \(\beta_{3}\mathstrut -\mathstrut \) \(6519135\) \(\beta_{2}\mathstrut -\mathstrut \) \(2019526320\) \(\beta_{1}\)\()/3732480\)
\(\nu^{4}\)\(=\)\((\)\(14398209\) \(\beta_{5}\mathstrut -\mathstrut \) \(175167797\) \(\beta_{4}\mathstrut +\mathstrut \) \(118756473549\) \(\beta_{1}\mathstrut +\mathstrut \) \(2417009100871680\)\()/466560\)
\(\nu^{5}\)\(=\)\((\)\(109514633382\) \(\beta_{4}\mathstrut -\mathstrut \) \(9294242237\) \(\beta_{3}\mathstrut +\mathstrut \) \(280734038165\) \(\beta_{2}\mathstrut +\mathstrut \) \(82513541352480\) \(\beta_{1}\)\()/1244160\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
352.821i
352.821i
80.7259i
80.7259i
41.9860i
41.9860i
0 −14327.9 13495.6i 0 480487.i 0 −7.51877e6 0 2.31580e7 + 3.86728e8i 0
5.2 0 −14327.9 + 13495.6i 0 480487.i 0 −7.51877e6 0 2.31580e7 3.86728e8i 0
5.3 0 9860.33 17035.1i 0 2.47556e6i 0 6.41846e7 0 −1.92968e8 3.35943e8i 0
5.4 0 9860.33 + 17035.1i 0 2.47556e6i 0 6.41846e7 0 −1.92968e8 + 3.35943e8i 0
5.5 0 16434.6 10831.7i 0 2.83969e6i 0 −5.11537e7 0 1.52771e8 3.56028e8i 0
5.6 0 16434.6 + 10831.7i 0 2.83969e6i 0 −5.11537e7 0 1.52771e8 + 3.56028e8i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{19}^{\mathrm{new}}(12, [\chi])\).