# Properties

 Label 4.11.b.a Level 4 Weight 11 Character orbit 4.b Analytic conductor 2.541 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4 = 2^{2}$$ Weight: $$k$$ = $$11$$ Character orbit: $$[\chi]$$ = 4.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.54142901069$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.26777625.2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{12}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -3 + \beta_{1} ) q^{2}$$ $$+ ( -2 \beta_{1} + \beta_{2} ) q^{3}$$ $$+ ( 4 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -390 + 64 \beta_{1} + 4 \beta_{3} ) q^{5}$$ $$+ ( 1800 + 8 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} ) q^{6}$$ $$+ ( -460 \beta_{1} - 26 \beta_{2} + 32 \beta_{3} ) q^{7}$$ $$+ ( -9072 - 16 \beta_{1} - 48 \beta_{2} - 52 \beta_{3} ) q^{8}$$ $$+ ( -7191 + 1152 \beta_{1} + 72 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -3 + \beta_{1} ) q^{2}$$ $$+ ( -2 \beta_{1} + \beta_{2} ) q^{3}$$ $$+ ( 4 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -390 + 64 \beta_{1} + 4 \beta_{3} ) q^{5}$$ $$+ ( 1800 + 8 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} ) q^{6}$$ $$+ ( -460 \beta_{1} - 26 \beta_{2} + 32 \beta_{3} ) q^{7}$$ $$+ ( -9072 - 16 \beta_{1} - 48 \beta_{2} - 52 \beta_{3} ) q^{8}$$ $$+ ( -7191 + 1152 \beta_{1} + 72 \beta_{3} ) q^{9}$$ $$+ ( 65810 - 262 \beta_{1} + 256 \beta_{2} - 64 \beta_{3} ) q^{10}$$ $$+ ( 418 \beta_{1} + 303 \beta_{2} - 64 \beta_{3} ) q^{11}$$ $$+ ( -228960 + 1120 \beta_{1} + 160 \beta_{2} + 216 \beta_{3} ) q^{12}$$ $$+ ( 53066 - 7872 \beta_{1} - 492 \beta_{3} ) q^{13}$$ $$+ ( 475440 + 1840 \beta_{1} - 1632 \beta_{2} + 824 \beta_{3} ) q^{14}$$ $$+ ( 14860 \beta_{1} - 2054 \beta_{2} - 672 \beta_{3} ) q^{15}$$ $$+ ( -903104 - 11840 \beta_{1} + 320 \beta_{2} + 688 \beta_{3} ) q^{16}$$ $$+ ( -42846 - 2432 \beta_{1} - 152 \beta_{3} ) q^{17}$$ $$+ ( 1185093 - 4887 \beta_{1} + 4608 \beta_{2} - 1152 \beta_{3} ) q^{18}$$ $$+ ( -38874 \beta_{1} + 8685 \beta_{2} + 1344 \beta_{3} ) q^{19}$$ $$+ ( -777240 + 64536 \beta_{1} - 3096 \beta_{2} - 3322 \beta_{3} ) q^{20}$$ $$+ ( -120960 + 80640 \beta_{1} + 5040 \beta_{3} ) q^{21}$$ $$+ ( -499080 - 1672 \beta_{1} - 752 \beta_{2} - 4660 \beta_{3} ) q^{22}$$ $$+ ( -53956 \beta_{1} - 21918 \beta_{2} + 6112 \beta_{3} ) q^{23}$$ $$+ ( 4475520 - 218752 \beta_{1} + 3200 \beta_{2} - 3360 \beta_{3} ) q^{24}$$ $$+ ( -1339605 - 49920 \beta_{1} - 3120 \beta_{3} ) q^{25}$$ $$+ ( -8109918 + 37322 \beta_{1} - 31488 \beta_{2} + 7872 \beta_{3} ) q^{26}$$ $$+ ( 149724 \beta_{1} + 21906 \beta_{2} - 12096 \beta_{3} ) q^{27}$$ $$+ ( 9024960 + 503360 \beta_{1} + 20416 \beta_{2} + 21008 \beta_{3} ) q^{28}$$ $$+ ( 7511658 - 311744 \beta_{1} - 19484 \beta_{3} ) q^{29}$$ $$+ ( -14664240 - 59440 \beta_{1} + 75872 \beta_{2} + 13896 \beta_{3} ) q^{30}$$ $$+ ( 388624 \beta_{1} + 49400 \beta_{2} - 30464 \beta_{3} ) q^{31}$$ $$+ ( 14514432 - 856320 \beta_{1} - 49920 \beta_{2} + 7360 \beta_{3} ) q^{32}$$ $$+ ( -16384320 + 127872 \beta_{1} + 7992 \beta_{3} ) q^{33}$$ $$+ ( -2327782 - 47710 \beta_{1} - 9728 \beta_{2} + 2432 \beta_{3} ) q^{34}$$ $$+ ( -514360 \beta_{1} - 208740 \beta_{2} + 58240 \beta_{3} ) q^{35}$$ $$+ ( -13991004 + 1162332 \beta_{1} - 56412 \beta_{2} - 59625 \beta_{3} ) q^{36}$$ $$+ ( 33602234 + 525120 \beta_{1} + 32820 \beta_{3} ) q^{37}$$ $$+ ( 37567080 + 155496 \beta_{1} - 224976 \beta_{2} - 82716 \beta_{3} ) q^{38}$$ $$+ ( -1837972 \beta_{1} + 257738 \beta_{2} + 82656 \beta_{3} ) q^{39}$$ $$+ ( -57482080 - 1019808 \beta_{1} + 282912 \beta_{2} - 21192 \beta_{3} ) q^{40}$$ $$+ ( -85045038 + 1336576 \beta_{1} + 83536 \beta_{3} ) q^{41}$$ $$+ ( 81809280 + 40320 \beta_{1} + 322560 \beta_{2} - 80640 \beta_{3} ) q^{42}$$ $$+ ( 2132210 \beta_{1} + 63367 \beta_{2} - 141184 \beta_{3} ) q^{43}$$ $$+ ( -75518880 - 725600 \beta_{1} - 672 \beta_{2} + 12200 \beta_{3} ) q^{44}$$ $$+ ( 151735050 - 909504 \beta_{1} - 56844 \beta_{3} ) q^{45}$$ $$+ ( 60295440 + 215824 \beta_{1} - 40480 \beta_{2} + 360808 \beta_{3} ) q^{46}$$ $$+ ( 1903192 \beta_{1} - 388908 \beta_{2} - 70336 \beta_{3} ) q^{47}$$ $$+ ( -61171200 + 4552192 \beta_{1} - 900608 \beta_{2} + 173952 \beta_{3} ) q^{48}$$ $$+ ( -201230351 - 8117760 \beta_{1} - 507360 \beta_{3} ) q^{49}$$ $$+ ( -46400385 - 1439445 \beta_{1} - 199680 \beta_{2} + 49920 \beta_{3} ) q^{50}$$ $$+ ( -449348 \beta_{1} + 20386 \beta_{2} + 25536 \beta_{3} ) q^{51}$$ $$+ ( 95620904 - 7958312 \beta_{1} + 401192 \beta_{2} + 403510 \beta_{3} ) q^{52}$$ $$+ ( 359392986 + 4582720 \beta_{1} + 286420 \beta_{3} ) q^{53}$$ $$+ ( -157975920 - 598896 \beta_{1} + 423648 \beta_{2} - 456408 \beta_{3} ) q^{54}$$ $$+ ( 4758580 \beta_{1} - 98074 \beta_{2} - 285152 \beta_{3} ) q^{55}$$ $$+ ( 348122880 + 9652480 \beta_{1} + 1850112 \beta_{2} - 789184 \beta_{3} ) q^{56}$$ $$+ ( -652708800 + 14649984 \beta_{1} + 915624 \beta_{3} ) q^{57}$$ $$+ ( -337396414 + 6888170 \beta_{1} - 1246976 \beta_{2} + 311744 \beta_{3} ) q^{58}$$ $$+ ( -18650662 \beta_{1} + 2601747 \beta_{2} + 840448 \beta_{3} ) q^{59}$$ $$+ ( 405771840 - 13482560 \beta_{1} - 844736 \beta_{2} - 1002768 \beta_{3} ) q^{60}$$ $$+ ( 853020842 - 10302144 \beta_{1} - 643884 \beta_{3} ) q^{61}$$ $$+ ( -408252480 - 1554496 \beta_{1} + 1159296 \beta_{2} - 1080224 \beta_{3} ) q^{62}$$ $$+ ( -9179820 \beta_{1} - 3752874 \beta_{2} + 1042848 \beta_{3} ) q^{63}$$ $$+ ( -9092096 + 15424512 \beta_{1} - 3025920 \beta_{2} + 1555200 \beta_{3} ) q^{64}$$ $$+ ( -1038387900 + 6466304 \beta_{1} + 404144 \beta_{3} ) q^{65}$$ $$+ ( 178303680 - 16128576 \beta_{1} + 511488 \beta_{2} - 127872 \beta_{3} ) q^{66}$$ $$+ ( 38524318 \beta_{1} - 2655439 \beta_{2} - 2075840 \beta_{3} ) q^{67}$$ $$+ ( 29304456 - 2221704 \beta_{1} - 113016 \beta_{2} + 183902 \beta_{3} ) q^{68}$$ $$+ ( 1099797120 - 4126464 \beta_{1} - 257904 \beta_{3} ) q^{69}$$ $$+ ( 574744800 + 2057440 \beta_{1} - 387520 \beta_{2} + 3436720 \beta_{3} ) q^{70}$$ $$+ ( 2756436 \beta_{1} + 8102742 \beta_{2} - 1185120 \beta_{3} ) q^{71}$$ $$+ ( -1033126128 - 18353808 \beta_{1} + 5100624 \beta_{2} - 372564 \beta_{3} ) q^{72}$$ $$+ ( -747127534 - 55984512 \beta_{1} - 3499032 \beta_{3} ) q^{73}$$ $$+ ( 429564498 + 34652474 \beta_{1} + 2100480 \beta_{2} - 525120 \beta_{3} ) q^{74}$$ $$+ ( -8303190 \beta_{1} - 41685 \beta_{2} + 524160 \beta_{3} ) q^{75}$$ $$+ ( -1859493600 + 32091360 \beta_{1} + 2421792 \beta_{2} + 2994168 \beta_{3} ) q^{76}$$ $$+ ( 937050240 + 36476160 \beta_{1} + 2279760 \beta_{3} ) q^{77}$$ $$+ ( 1812874320 + 7351888 \beta_{1} - 9413792 \beta_{2} - 1770360 \beta_{3} ) q^{78}$$ $$+ ( 15125864 \beta_{1} - 2386100 \beta_{2} - 647104 \beta_{3} ) q^{79}$$ $$+ ( 339799680 - 55782016 \beta_{1} - 6342528 \beta_{2} - 2940960 \beta_{3} ) q^{80}$$ $$+ ( -1178945199 + 51456384 \beta_{1} + 3216024 \beta_{3} ) q^{81}$$ $$+ ( 1605076874 - 82371886 \beta_{1} + 5346304 \beta_{2} - 1336576 \beta_{3} ) q^{82}$$ $$+ ( -54031418 \beta_{1} - 17655267 \beta_{2} + 5583872 \beta_{3} ) q^{83}$$ $$+ ( -977840640 + 79833600 \beta_{1} - 2419200 \beta_{2} - 4556160 \beta_{3} ) q^{84}$$ $$+ ( -297699020 - 1793664 \beta_{1} - 112104 \beta_{3} ) q^{85}$$ $$+ ( -2190062280 - 8528840 \beta_{1} + 8021904 \beta_{2} - 3019348 \beta_{3} ) q^{86}$$ $$+ ( -83606996 \beta_{1} + 15617002 \beta_{2} + 3273312 \beta_{3} ) q^{87}$$ $$+ ( 427109760 - 74211712 \beta_{1} - 2897024 \beta_{2} + 735008 \beta_{3} ) q^{88}$$ $$+ ( 1318680498 + 32133760 \beta_{1} + 2008360 \beta_{3} ) q^{89}$$ $$+ ( -1373804190 + 149916042 \beta_{1} - 3638016 \beta_{2} + 909504 \beta_{3} ) q^{90}$$ $$+ ( 60922120 \beta_{1} + 25542524 \beta_{2} - 7000448 \beta_{3} ) q^{91}$$ $$+ ( 5605097280 + 77155520 \beta_{1} + 1187136 \beta_{2} + 350896 \beta_{3} ) q^{92}$$ $$+ ( -1517529600 - 48374784 \beta_{1} - 3023424 \beta_{3} ) q^{93}$$ $$+ ( -1847917920 - 7612768 \beta_{1} + 10724032 \beta_{2} + 3541520 \beta_{3} ) q^{94}$$ $$+ ( 123683100 \beta_{1} - 28849038 \beta_{2} - 4124064 \beta_{3} ) q^{95}$$ $$+ ( 1373644800 - 62777344 \beta_{1} + 25413632 \beta_{2} + 8056320 \beta_{3} ) q^{96}$$ $$+ ( 3585799874 - 47068032 \beta_{1} - 2941752 \beta_{3} ) q^{97}$$ $$+ ( -7595246547 - 217465871 \beta_{1} - 32471040 \beta_{2} + 8117760 \beta_{3} ) q^{98}$$ $$+ ( 85582962 \beta_{1} - 1817145 \beta_{2} - 5121792 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 12q^{2}$$ $$\mathstrut +\mathstrut 16q^{4}$$ $$\mathstrut -\mathstrut 1560q^{5}$$ $$\mathstrut +\mathstrut 7200q^{6}$$ $$\mathstrut -\mathstrut 36288q^{8}$$ $$\mathstrut -\mathstrut 28764q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 12q^{2}$$ $$\mathstrut +\mathstrut 16q^{4}$$ $$\mathstrut -\mathstrut 1560q^{5}$$ $$\mathstrut +\mathstrut 7200q^{6}$$ $$\mathstrut -\mathstrut 36288q^{8}$$ $$\mathstrut -\mathstrut 28764q^{9}$$ $$\mathstrut +\mathstrut 263240q^{10}$$ $$\mathstrut -\mathstrut 915840q^{12}$$ $$\mathstrut +\mathstrut 212264q^{13}$$ $$\mathstrut +\mathstrut 1901760q^{14}$$ $$\mathstrut -\mathstrut 3612416q^{16}$$ $$\mathstrut -\mathstrut 171384q^{17}$$ $$\mathstrut +\mathstrut 4740372q^{18}$$ $$\mathstrut -\mathstrut 3108960q^{20}$$ $$\mathstrut -\mathstrut 483840q^{21}$$ $$\mathstrut -\mathstrut 1996320q^{22}$$ $$\mathstrut +\mathstrut 17902080q^{24}$$ $$\mathstrut -\mathstrut 5358420q^{25}$$ $$\mathstrut -\mathstrut 32439672q^{26}$$ $$\mathstrut +\mathstrut 36099840q^{28}$$ $$\mathstrut +\mathstrut 30046632q^{29}$$ $$\mathstrut -\mathstrut 58656960q^{30}$$ $$\mathstrut +\mathstrut 58057728q^{32}$$ $$\mathstrut -\mathstrut 65537280q^{33}$$ $$\mathstrut -\mathstrut 9311128q^{34}$$ $$\mathstrut -\mathstrut 55964016q^{36}$$ $$\mathstrut +\mathstrut 134408936q^{37}$$ $$\mathstrut +\mathstrut 150268320q^{38}$$ $$\mathstrut -\mathstrut 229928320q^{40}$$ $$\mathstrut -\mathstrut 340180152q^{41}$$ $$\mathstrut +\mathstrut 327237120q^{42}$$ $$\mathstrut -\mathstrut 302075520q^{44}$$ $$\mathstrut +\mathstrut 606940200q^{45}$$ $$\mathstrut +\mathstrut 241181760q^{46}$$ $$\mathstrut -\mathstrut 244684800q^{48}$$ $$\mathstrut -\mathstrut 804921404q^{49}$$ $$\mathstrut -\mathstrut 185601540q^{50}$$ $$\mathstrut +\mathstrut 382483616q^{52}$$ $$\mathstrut +\mathstrut 1437571944q^{53}$$ $$\mathstrut -\mathstrut 631903680q^{54}$$ $$\mathstrut +\mathstrut 1392491520q^{56}$$ $$\mathstrut -\mathstrut 2610835200q^{57}$$ $$\mathstrut -\mathstrut 1349585656q^{58}$$ $$\mathstrut +\mathstrut 1623087360q^{60}$$ $$\mathstrut +\mathstrut 3412083368q^{61}$$ $$\mathstrut -\mathstrut 1633009920q^{62}$$ $$\mathstrut -\mathstrut 36368384q^{64}$$ $$\mathstrut -\mathstrut 4153551600q^{65}$$ $$\mathstrut +\mathstrut 713214720q^{66}$$ $$\mathstrut +\mathstrut 117217824q^{68}$$ $$\mathstrut +\mathstrut 4399188480q^{69}$$ $$\mathstrut +\mathstrut 2298979200q^{70}$$ $$\mathstrut -\mathstrut 4132504512q^{72}$$ $$\mathstrut -\mathstrut 2988510136q^{73}$$ $$\mathstrut +\mathstrut 1718257992q^{74}$$ $$\mathstrut -\mathstrut 7437974400q^{76}$$ $$\mathstrut +\mathstrut 3748200960q^{77}$$ $$\mathstrut +\mathstrut 7251497280q^{78}$$ $$\mathstrut +\mathstrut 1359198720q^{80}$$ $$\mathstrut -\mathstrut 4715780796q^{81}$$ $$\mathstrut +\mathstrut 6420307496q^{82}$$ $$\mathstrut -\mathstrut 3911362560q^{84}$$ $$\mathstrut -\mathstrut 1190796080q^{85}$$ $$\mathstrut -\mathstrut 8760249120q^{86}$$ $$\mathstrut +\mathstrut 1708439040q^{88}$$ $$\mathstrut +\mathstrut 5274721992q^{89}$$ $$\mathstrut -\mathstrut 5495216760q^{90}$$ $$\mathstrut +\mathstrut 22420389120q^{92}$$ $$\mathstrut -\mathstrut 6070118400q^{93}$$ $$\mathstrut -\mathstrut 7391671680q^{94}$$ $$\mathstrut +\mathstrut 5494579200q^{96}$$ $$\mathstrut +\mathstrut 14343199496q^{97}$$ $$\mathstrut -\mathstrut 30380986188q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut -\mathstrut$$ $$2$$ $$x^{3}\mathstrut +\mathstrut$$ $$59$$ $$x^{2}\mathstrut -\mathstrut$$ $$58$$ $$x\mathstrut +\mathstrut$$ $$336$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} + 4 \nu^{2} + 78 \nu + 161$$$$)/7$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{3} + 8 \nu^{2} + 492 \nu + 154$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$-32 \nu^{3} + 160 \nu^{2} - 1472 \nu + 3920$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$24$$$$)/48$$ $$\nu^{2}$$ $$=$$ $$($$$$3$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$44$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2736$$$$)/96$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$3$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$41$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$202$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2064$$$$)/48$$

## 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 + 7.15697i 0.5 − 7.15697i 0.5 − 2.50555i 0.5 + 2.50555i
−25.4722 19.3692i 343.535i 273.666 + 986.754i −3266.44 6654.00 8750.58i 10902.2i 12141.8 30435.5i −58967.0 83203.5 + 63268.4i
3.2 −25.4722 + 19.3692i 343.535i 273.666 986.754i −3266.44 6654.00 + 8750.58i 10902.2i 12141.8 + 30435.5i −58967.0 83203.5 63268.4i
3.3 19.4722 25.3936i 120.267i −265.666 988.937i 2486.44 −3054.00 2341.85i 29129.9i −30285.8 12510.6i 44585.0 48416.5 63139.6i
3.4 19.4722 + 25.3936i 120.267i −265.666 + 988.937i 2486.44 −3054.00 + 2341.85i 29129.9i −30285.8 + 12510.6i 44585.0 48416.5 + 63139.6i
 $$n$$: 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 $$S_{11}^{\mathrm{new}}(4, [\chi])$$.