Properties

 Label 4.7.b.a Level 4 Weight 7 Character orbit 4.b Analytic conductor 0.920 Analytic rank 0 Dimension 2 CM No Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$4 = 2^{2}$$ Weight: $$k$$ = $$7$$ Character orbit: $$[\chi]$$ = 4.b (of order $$2$$ and degree $$1$$)

Newform invariants

 Self dual: No Analytic conductor: $$0.920216334479$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-15})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 2 + \beta ) q^{2}$$ $$-4 \beta q^{3}$$ $$+ ( -56 + 4 \beta ) q^{4}$$ $$+ 10 q^{5}$$ $$+ ( 240 - 8 \beta ) q^{6}$$ $$+ 40 \beta q^{7}$$ $$+ ( -352 - 48 \beta ) q^{8}$$ $$-231 q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 2 + \beta ) q^{2}$$ $$-4 \beta q^{3}$$ $$+ ( -56 + 4 \beta ) q^{4}$$ $$+ 10 q^{5}$$ $$+ ( 240 - 8 \beta ) q^{6}$$ $$+ 40 \beta q^{7}$$ $$+ ( -352 - 48 \beta ) q^{8}$$ $$-231 q^{9}$$ $$+ ( 20 + 10 \beta ) q^{10}$$ $$-124 \beta q^{11}$$ $$+ ( 960 + 224 \beta ) q^{12}$$ $$+ 1466 q^{13}$$ $$+ ( -2400 + 80 \beta ) q^{14}$$ $$-40 \beta q^{15}$$ $$+ ( 2176 - 448 \beta ) q^{16}$$ $$-4766 q^{17}$$ $$+ ( -462 - 231 \beta ) q^{18}$$ $$+ 972 \beta q^{19}$$ $$+ ( -560 + 40 \beta ) q^{20}$$ $$+ 9600 q^{21}$$ $$+ ( 7440 - 248 \beta ) q^{22}$$ $$-1352 \beta q^{23}$$ $$+ ( -11520 + 1408 \beta ) q^{24}$$ $$-15525 q^{25}$$ $$+ ( 2932 + 1466 \beta ) q^{26}$$ $$-1992 \beta q^{27}$$ $$+ ( -9600 - 2240 \beta ) q^{28}$$ $$+ 25498 q^{29}$$ $$+ ( 2400 - 80 \beta ) q^{30}$$ $$+ 5408 \beta q^{31}$$ $$+ ( 31232 + 1280 \beta ) q^{32}$$ $$-29760 q^{33}$$ $$+ ( -9532 - 4766 \beta ) q^{34}$$ $$+ 400 \beta q^{35}$$ $$+ ( 12936 - 924 \beta ) q^{36}$$ $$+ 1994 q^{37}$$ $$+ ( -58320 + 1944 \beta ) q^{38}$$ $$-5864 \beta q^{39}$$ $$+ ( -3520 - 480 \beta ) q^{40}$$ $$+ 29362 q^{41}$$ $$+ ( 19200 + 9600 \beta ) q^{42}$$ $$-2780 \beta q^{43}$$ $$+ ( 29760 + 6944 \beta ) q^{44}$$ $$-2310 q^{45}$$ $$+ ( 81120 - 2704 \beta ) q^{46}$$ $$-976 \beta q^{47}$$ $$+ ( -107520 - 8704 \beta ) q^{48}$$ $$+ 21649 q^{49}$$ $$+ ( -31050 - 15525 \beta ) q^{50}$$ $$+ 19064 \beta q^{51}$$ $$+ ( -82096 + 5864 \beta ) q^{52}$$ $$-192854 q^{53}$$ $$+ ( 119520 - 3984 \beta ) q^{54}$$ $$-1240 \beta q^{55}$$ $$+ ( 115200 - 14080 \beta ) q^{56}$$ $$+ 233280 q^{57}$$ $$+ ( 50996 + 25498 \beta ) q^{58}$$ $$-10124 \beta q^{59}$$ $$+ ( 9600 + 2240 \beta ) q^{60}$$ $$-10918 q^{61}$$ $$+ ( -324480 + 10816 \beta ) q^{62}$$ $$-9240 \beta q^{63}$$ $$+ ( -14336 + 33792 \beta ) q^{64}$$ $$+ 14660 q^{65}$$ $$+ ( -59520 - 29760 \beta ) q^{66}$$ $$-50884 \beta q^{67}$$ $$+ ( 266896 - 19064 \beta ) q^{68}$$ $$-324480 q^{69}$$ $$+ ( -24000 + 800 \beta ) q^{70}$$ $$+ 68712 \beta q^{71}$$ $$+ ( 81312 + 11088 \beta ) q^{72}$$ $$+ 288626 q^{73}$$ $$+ ( 3988 + 1994 \beta ) q^{74}$$ $$+ 62100 \beta q^{75}$$ $$+ ( -233280 - 54432 \beta ) q^{76}$$ $$+ 297600 q^{77}$$ $$+ ( 351840 - 11728 \beta ) q^{78}$$ $$-40112 \beta q^{79}$$ $$+ ( 21760 - 4480 \beta ) q^{80}$$ $$-646479 q^{81}$$ $$+ ( 58724 + 29362 \beta ) q^{82}$$ $$-26356 \beta q^{83}$$ $$+ ( -537600 + 38400 \beta ) q^{84}$$ $$-47660 q^{85}$$ $$+ ( 166800 - 5560 \beta ) q^{86}$$ $$-101992 \beta q^{87}$$ $$+ ( -357120 + 43648 \beta ) q^{88}$$ $$+ 310738 q^{89}$$ $$+ ( -4620 - 2310 \beta ) q^{90}$$ $$+ 58640 \beta q^{91}$$ $$+ ( 324480 + 75712 \beta ) q^{92}$$ $$+ 1297920 q^{93}$$ $$+ ( 58560 - 1952 \beta ) q^{94}$$ $$+ 9720 \beta q^{95}$$ $$+ ( 307200 - 124928 \beta ) q^{96}$$ $$-1457086 q^{97}$$ $$+ ( 43298 + 21649 \beta ) q^{98}$$ $$+ 28644 \beta q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 112q^{4}$$ $$\mathstrut +\mathstrut 20q^{5}$$ $$\mathstrut +\mathstrut 480q^{6}$$ $$\mathstrut -\mathstrut 704q^{8}$$ $$\mathstrut -\mathstrut 462q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 112q^{4}$$ $$\mathstrut +\mathstrut 20q^{5}$$ $$\mathstrut +\mathstrut 480q^{6}$$ $$\mathstrut -\mathstrut 704q^{8}$$ $$\mathstrut -\mathstrut 462q^{9}$$ $$\mathstrut +\mathstrut 40q^{10}$$ $$\mathstrut +\mathstrut 1920q^{12}$$ $$\mathstrut +\mathstrut 2932q^{13}$$ $$\mathstrut -\mathstrut 4800q^{14}$$ $$\mathstrut +\mathstrut 4352q^{16}$$ $$\mathstrut -\mathstrut 9532q^{17}$$ $$\mathstrut -\mathstrut 924q^{18}$$ $$\mathstrut -\mathstrut 1120q^{20}$$ $$\mathstrut +\mathstrut 19200q^{21}$$ $$\mathstrut +\mathstrut 14880q^{22}$$ $$\mathstrut -\mathstrut 23040q^{24}$$ $$\mathstrut -\mathstrut 31050q^{25}$$ $$\mathstrut +\mathstrut 5864q^{26}$$ $$\mathstrut -\mathstrut 19200q^{28}$$ $$\mathstrut +\mathstrut 50996q^{29}$$ $$\mathstrut +\mathstrut 4800q^{30}$$ $$\mathstrut +\mathstrut 62464q^{32}$$ $$\mathstrut -\mathstrut 59520q^{33}$$ $$\mathstrut -\mathstrut 19064q^{34}$$ $$\mathstrut +\mathstrut 25872q^{36}$$ $$\mathstrut +\mathstrut 3988q^{37}$$ $$\mathstrut -\mathstrut 116640q^{38}$$ $$\mathstrut -\mathstrut 7040q^{40}$$ $$\mathstrut +\mathstrut 58724q^{41}$$ $$\mathstrut +\mathstrut 38400q^{42}$$ $$\mathstrut +\mathstrut 59520q^{44}$$ $$\mathstrut -\mathstrut 4620q^{45}$$ $$\mathstrut +\mathstrut 162240q^{46}$$ $$\mathstrut -\mathstrut 215040q^{48}$$ $$\mathstrut +\mathstrut 43298q^{49}$$ $$\mathstrut -\mathstrut 62100q^{50}$$ $$\mathstrut -\mathstrut 164192q^{52}$$ $$\mathstrut -\mathstrut 385708q^{53}$$ $$\mathstrut +\mathstrut 239040q^{54}$$ $$\mathstrut +\mathstrut 230400q^{56}$$ $$\mathstrut +\mathstrut 466560q^{57}$$ $$\mathstrut +\mathstrut 101992q^{58}$$ $$\mathstrut +\mathstrut 19200q^{60}$$ $$\mathstrut -\mathstrut 21836q^{61}$$ $$\mathstrut -\mathstrut 648960q^{62}$$ $$\mathstrut -\mathstrut 28672q^{64}$$ $$\mathstrut +\mathstrut 29320q^{65}$$ $$\mathstrut -\mathstrut 119040q^{66}$$ $$\mathstrut +\mathstrut 533792q^{68}$$ $$\mathstrut -\mathstrut 648960q^{69}$$ $$\mathstrut -\mathstrut 48000q^{70}$$ $$\mathstrut +\mathstrut 162624q^{72}$$ $$\mathstrut +\mathstrut 577252q^{73}$$ $$\mathstrut +\mathstrut 7976q^{74}$$ $$\mathstrut -\mathstrut 466560q^{76}$$ $$\mathstrut +\mathstrut 595200q^{77}$$ $$\mathstrut +\mathstrut 703680q^{78}$$ $$\mathstrut +\mathstrut 43520q^{80}$$ $$\mathstrut -\mathstrut 1292958q^{81}$$ $$\mathstrut +\mathstrut 117448q^{82}$$ $$\mathstrut -\mathstrut 1075200q^{84}$$ $$\mathstrut -\mathstrut 95320q^{85}$$ $$\mathstrut +\mathstrut 333600q^{86}$$ $$\mathstrut -\mathstrut 714240q^{88}$$ $$\mathstrut +\mathstrut 621476q^{89}$$ $$\mathstrut -\mathstrut 9240q^{90}$$ $$\mathstrut +\mathstrut 648960q^{92}$$ $$\mathstrut +\mathstrut 2595840q^{93}$$ $$\mathstrut +\mathstrut 117120q^{94}$$ $$\mathstrut +\mathstrut 614400q^{96}$$ $$\mathstrut -\mathstrut 2914172q^{97}$$ $$\mathstrut +\mathstrut 86596q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.5 − 1.93649i 0.5 + 1.93649i
2.00000 7.74597i 30.9839i −56.0000 30.9839i 10.0000 240.000 + 61.9677i 309.839i −352.000 + 371.806i −231.000 20.0000 77.4597i
3.2 2.00000 + 7.74597i 30.9839i −56.0000 + 30.9839i 10.0000 240.000 61.9677i 309.839i −352.000 371.806i −231.000 20.0000 + 77.4597i
 $$n$$: 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
4.b Odd 1 yes

Hecke kernels

There are no other newforms in $$S_{7}^{\mathrm{new}}(4, [\chi])$$.