# Properties

 Label 12.17.c.b Level 12 Weight 17 Character orbit 12.c Analytic conductor 19.479 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$17$$ Character orbit: $$[\chi]$$ = 12.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$19.4789452628$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{11}\cdot 5$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -3105 - \beta_{1} ) q^{3}$$ $$+ ( -8 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5}$$ $$+ ( -262570 + 462 \beta_{1} - 77 \beta_{3} ) q^{7}$$ $$+ ( -2530359 + 4428 \beta_{1} + 27 \beta_{2} - 657 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -3105 - \beta_{1} ) q^{3}$$ $$+ ( -8 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5}$$ $$+ ( -262570 + 462 \beta_{1} - 77 \beta_{3} ) q^{7}$$ $$+ ( -2530359 + 4428 \beta_{1} + 27 \beta_{2} - 657 \beta_{3} ) q^{9}$$ $$+ ( -68 \beta_{1} - 292 \beta_{2} - 103 \beta_{3} ) q^{11}$$ $$+ ( -68715790 - 65208 \beta_{1} + 10868 \beta_{3} ) q^{13}$$ $$+ ( -266318280 + 19008 \beta_{1} + 6156 \beta_{2} + 27351 \beta_{3} ) q^{15}$$ $$+ ( 475376 \beta_{1} - 13640 \beta_{2} + 35068 \beta_{3} ) q^{17}$$ $$+ ( -10337184946 + 194370 \beta_{1} - 32395 \beta_{3} ) q^{19}$$ $$+ ( -13900098366 + 1497496 \beta_{1} - 18711 \beta_{2} + 455301 \beta_{3} ) q^{21}$$ $$+ ( 15330632 \beta_{1} + 114160 \beta_{2} + 1315606 \beta_{3} ) q^{23}$$ $$+ ( -131409689135 - 28014120 \beta_{1} + 4669020 \beta_{3} ) q^{25}$$ $$+ ( -114356556945 + 12844251 \beta_{1} - 335340 \beta_{2} + 3376971 \beta_{3} ) q^{27}$$ $$+ ( 73218376 \beta_{1} + 181709 \beta_{2} + 6162101 \beta_{3} ) q^{29}$$ $$+ ( -428084121946 + 46484550 \beta_{1} - 7747425 \beta_{3} ) q^{31}$$ $$+ ( -238324680 - 1981692 \beta_{1} + 1751625 \beta_{2} + 9104049 \beta_{3} ) q^{33}$$ $$+ ( -121292248 \beta_{1} - 3966116 \beta_{2} - 11429726 \beta_{3} ) q^{35}$$ $$+ ( 1314667670930 - 166270344 \beta_{1} + 27711724 \beta_{3} ) q^{37}$$ $$+ ( 2290333053294 - 105585194 \beta_{1} + 2640924 \beta_{2} - 64262484 \beta_{3} ) q^{39}$$ $$+ ( -1024582064 \beta_{1} + 10630994 \beta_{2} - 81838174 \beta_{3} ) q^{41}$$ $$+ ( 4742542432430 + 1485494802 \beta_{1} - 247582467 \beta_{3} ) q^{43}$$ $$+ ( 7696697027760 - 318858768 \beta_{1} - 35363061 \beta_{2} - 230015781 \beta_{3} ) q^{45}$$ $$+ ( -1482840528 \beta_{1} + 1863960 \beta_{2} - 122948724 \beta_{3} ) q^{47}$$ $$+ ( -12768473357325 - 242614680 \beta_{1} + 40435780 \beta_{3} ) q^{49}$$ $$+ ( 16347134374560 - 1681842096 \beta_{1} + 72131796 \beta_{2} + 661078044 \beta_{3} ) q^{51}$$ $$+ ( 6521633368 \beta_{1} + 2390555 \beta_{2} + 544266299 \beta_{3} ) q^{53}$$ $$+ ( -82021278501360 - 7857409320 \beta_{1} + 1309568220 \beta_{3} ) q^{55}$$ $$+ ( 25905989422170 + 10856735956 \beta_{1} - 7871985 \beta_{2} + 191551635 \beta_{3} ) q^{57}$$ $$+ ( 11146798748 \beta_{1} - 268706288 \beta_{2} + 839331133 \beta_{3} ) q^{59}$$ $$+ ( -246049263264046 - 307878120 \beta_{1} + 51313020 \beta_{3} ) q^{61}$$ $$+ ( 183429393805350 + 711620910 \beta_{1} + 109105920 \beta_{2} + 659686797 \beta_{3} ) q^{63}$$ $$+ ( 17965739792 \beta_{1} + 665564614 \beta_{2} + 1718999854 \beta_{3} ) q^{65}$$ $$+ ( -199502864250370 + 4695287562 \beta_{1} - 782547927 \beta_{3} ) q^{67}$$ $$+ ( 523228498963920 - 50049979272 \beta_{1} - 994720338 \beta_{2} + 3981760038 \beta_{3} ) q^{69}$$ $$+ ( -96672468232 \beta_{1} + 67234312 \beta_{2} - 8033627582 \beta_{3} ) q^{71}$$ $$+ ( -532037079878590 + 12239273376 \beta_{1} - 2039878896 \beta_{3} ) q^{73}$$ $$+ ( 1300317931492335 + 56527946375 \beta_{1} + 1134571860 \beta_{2} - 27607915260 \beta_{3} ) q^{75}$$ $$+ ( -45094266448 \beta_{1} - 1094451050 \beta_{2} - 4122672554 \beta_{3} ) q^{77}$$ $$+ ( -487453484173018 + 169093966470 \beta_{1} - 28182327745 \beta_{3} ) q^{79}$$ $$+ ( 1435313770082721 + 11182148496 \beta_{1} + 1945822014 \beta_{2} + 12056139306 \beta_{3} ) q^{81}$$ $$+ ( -171905078796 \beta_{1} - 2341342740 \beta_{2} - 15105870813 \beta_{3} ) q^{83}$$ $$+ ( -3640234033846080 - 298944660960 \beta_{1} + 49824110160 \beta_{3} ) q^{85}$$ $$+ ( 2501511669878760 - 241784506656 \beta_{1} - 2571835860 \beta_{2} + 30392135037 \beta_{3} ) q^{87}$$ $$+ ( 765134253824 \beta_{1} + 4970953426 \beta_{2} + 65418172294 \beta_{3} ) q^{89}$$ $$+ ( -2860638443146484 - 14625030420 \beta_{1} + 2437505070 \beta_{3} ) q^{91}$$ $$+ ( -151399958177070 + 552337324096 \beta_{1} - 1882624275 \beta_{2} + 45810524025 \beta_{3} ) q^{93}$$ $$+ ( 30784362488 \beta_{1} + 8558116336 \beta_{2} + 5418068986 \beta_{3} ) q^{95}$$ $$+ ( -7962992039410510 - 149161958328 \beta_{1} + 24860326388 \beta_{3} ) q^{97}$$ $$+ ( 2214600258602160 - 188177303628 \beta_{1} - 9755600472 \beta_{2} - 72908835723 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 12420q^{3}$$ $$\mathstrut -\mathstrut 1050280q^{7}$$ $$\mathstrut -\mathstrut 10121436q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 12420q^{3}$$ $$\mathstrut -\mathstrut 1050280q^{7}$$ $$\mathstrut -\mathstrut 10121436q^{9}$$ $$\mathstrut -\mathstrut 274863160q^{13}$$ $$\mathstrut -\mathstrut 1065273120q^{15}$$ $$\mathstrut -\mathstrut 41348739784q^{19}$$ $$\mathstrut -\mathstrut 55600393464q^{21}$$ $$\mathstrut -\mathstrut 525638756540q^{25}$$ $$\mathstrut -\mathstrut 457426227780q^{27}$$ $$\mathstrut -\mathstrut 1712336487784q^{31}$$ $$\mathstrut -\mathstrut 953298720q^{33}$$ $$\mathstrut +\mathstrut 5258670683720q^{37}$$ $$\mathstrut +\mathstrut 9161332213176q^{39}$$ $$\mathstrut +\mathstrut 18970169729720q^{43}$$ $$\mathstrut +\mathstrut 30786788111040q^{45}$$ $$\mathstrut -\mathstrut 51073893429300q^{49}$$ $$\mathstrut +\mathstrut 65388537498240q^{51}$$ $$\mathstrut -\mathstrut 328085114005440q^{55}$$ $$\mathstrut +\mathstrut 103623957688680q^{57}$$ $$\mathstrut -\mathstrut 984197053056184q^{61}$$ $$\mathstrut +\mathstrut 733717575221400q^{63}$$ $$\mathstrut -\mathstrut 798011457001480q^{67}$$ $$\mathstrut +\mathstrut 2092913995855680q^{69}$$ $$\mathstrut -\mathstrut 2128148319514360q^{73}$$ $$\mathstrut +\mathstrut 5201271725969340q^{75}$$ $$\mathstrut -\mathstrut 1949813936692072q^{79}$$ $$\mathstrut +\mathstrut 5741255080330884q^{81}$$ $$\mathstrut -\mathstrut 14560936135384320q^{85}$$ $$\mathstrut +\mathstrut 10006046679515040q^{87}$$ $$\mathstrut -\mathstrut 11442553772585936q^{91}$$ $$\mathstrut -\mathstrut 605599832708280q^{93}$$ $$\mathstrut -\mathstrut 31851968157642040q^{97}$$ $$\mathstrut +\mathstrut 8858401034408640q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$15630$$ $$x^{2}\mathstrut +\mathstrut$$ $$12922000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$54 \nu^{2} + 6210 \nu + 422010$$$$)/115$$ $$\beta_{2}$$ $$=$$ $$($$$$216 \nu^{3} + 216 \nu^{2} + 3004560 \nu + 1688040$$$$)/115$$ $$\beta_{3}$$ $$=$$ $$($$$$-648 \nu^{2} + 37260 \nu - 5064120$$$$)/115$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$12$$ $$\beta_{1}$$$$)/972$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$115$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$690$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$7596180$$$$)/972$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$27590$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1035$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$335220$$ $$\beta_{1}$$$$)/1944$$

## 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
 29.5942i − 29.5942i 121.467i − 121.467i
0 −6363.40 1598.09i 0 746888.i 0 4.25357e6 0 3.79389e7 + 2.03386e7i 0
5.2 0 −6363.40 + 1598.09i 0 746888.i 0 4.25357e6 0 3.79389e7 2.03386e7i 0
5.3 0 153.398 6559.21i 0 100768.i 0 −4.77871e6 0 −4.29997e7 2.01234e6i 0
5.4 0 153.398 + 6559.21i 0 100768.i 0 −4.77871e6 0 −4.29997e7 + 2.01234e6i 0
 $$n$$: e.g. 2-40 or 990-1000 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

This newform can be constructed as the kernel of the linear operator $$T_{5}^{4}$$ $$\mathstrut +\mathstrut 567995159520 T_{5}^{2}$$ $$\mathstrut +\mathstrut$$$$56\!\cdots\!00$$ acting on $$S_{17}^{\mathrm{new}}(12, [\chi])$$.