# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{19}^{\mathrm{new}}(12, [\chi])$$.