# Properties

 Label 1152.3.b.f.703.3 Level $1152$ Weight $3$ Character 1152.703 Analytic conductor $31.390$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1152.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.3897264543$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 703.3 Root $$-1.22474 - 1.22474i$$ of defining polynomial Character $$\chi$$ $$=$$ 1152.703 Dual form 1152.3.b.f.703.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{5} -9.79796i q^{7} +O(q^{10})$$ $$q+2.00000i q^{5} -9.79796i q^{7} -19.5959 q^{11} +21.0000 q^{25} +50.0000i q^{29} +48.9898i q^{31} +19.5959 q^{35} -47.0000 q^{49} +94.0000i q^{53} -39.1918i q^{55} +117.576 q^{59} +50.0000 q^{73} +192.000i q^{77} -146.969i q^{79} -97.9796 q^{83} -190.000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 84q^{25} - 188q^{49} + 200q^{73} - 760q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 2.00000i 0.400000i 0.979796 + 0.200000i $$0.0640942\pi$$
−0.979796 + 0.200000i $$0.935906\pi$$
$$6$$ 0 0
$$7$$ − 9.79796i − 1.39971i −0.714286 0.699854i $$-0.753248\pi$$
0.714286 0.699854i $$-0.246752\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −19.5959 −1.78145 −0.890724 0.454545i $$-0.849802\pi$$
−0.890724 + 0.454545i $$0.849802\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 21.0000 0.840000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 50.0000i 1.72414i 0.506791 + 0.862069i $$0.330832\pi$$
−0.506791 + 0.862069i $$0.669168\pi$$
$$30$$ 0 0
$$31$$ 48.9898i 1.58032i 0.612903 + 0.790158i $$0.290002\pi$$
−0.612903 + 0.790158i $$0.709998\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 19.5959 0.559883
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −47.0000 −0.959184
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 94.0000i 1.77358i 0.462168 + 0.886792i $$0.347072\pi$$
−0.462168 + 0.886792i $$0.652928\pi$$
$$54$$ 0 0
$$55$$ − 39.1918i − 0.712579i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 117.576 1.99281 0.996403 0.0847458i $$-0.0270078\pi$$
0.996403 + 0.0847458i $$0.0270078\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 50.0000 0.684932 0.342466 0.939530i $$-0.388738\pi$$
0.342466 + 0.939530i $$0.388738\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 192.000i 2.49351i
$$78$$ 0 0
$$79$$ − 146.969i − 1.86037i −0.367089 0.930186i $$-0.619645\pi$$
0.367089 0.930186i $$-0.380355\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −97.9796 −1.18048 −0.590238 0.807229i $$-0.700966\pi$$
−0.590238 + 0.807229i $$0.700966\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −190.000 −1.95876 −0.979381 0.202020i $$-0.935249\pi$$
−0.979381 + 0.202020i $$0.935249\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 190.000i 1.88119i 0.339533 + 0.940594i $$0.389731\pi$$
−0.339533 + 0.940594i $$0.610269\pi$$
$$102$$ 0 0
$$103$$ 205.757i 1.99764i 0.0485437 + 0.998821i $$0.484542\pi$$
−0.0485437 + 0.998821i $$0.515458\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −195.959 −1.83139 −0.915697 0.401869i $$-0.868361\pi$$
−0.915697 + 0.401869i $$0.868361\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 263.000 2.17355
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 92.0000i 0.736000i
$$126$$ 0 0
$$127$$ 107.778i 0.848642i 0.905512 + 0.424321i $$0.139487\pi$$
−0.905512 + 0.424321i $$0.860513\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 78.3837 0.598349 0.299174 0.954198i $$-0.403289\pi$$
0.299174 + 0.954198i $$0.403289\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −100.000 −0.689655
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 290.000i 1.94631i 0.230153 + 0.973154i $$0.426077\pi$$
−0.230153 + 0.973154i $$0.573923\pi$$
$$150$$ 0 0
$$151$$ − 48.9898i − 0.324436i −0.986755 0.162218i $$-0.948135\pi$$
0.986755 0.162218i $$-0.0518647\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −97.9796 −0.632126
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 169.000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 46.0000i 0.265896i 0.991123 + 0.132948i $$0.0424443\pi$$
−0.991123 + 0.132948i $$0.957556\pi$$
$$174$$ 0 0
$$175$$ − 205.757i − 1.17576i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 274.343 1.53264 0.766321 0.642458i $$-0.222085\pi$$
0.766321 + 0.642458i $$0.222085\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ −290.000 −1.50259 −0.751295 0.659966i $$-0.770571\pi$$
−0.751295 + 0.659966i $$0.770571\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 194.000i − 0.984772i −0.870377 0.492386i $$-0.836125\pi$$
0.870377 0.492386i $$-0.163875\pi$$
$$198$$ 0 0
$$199$$ − 342.929i − 1.72326i −0.507538 0.861630i $$-0.669444\pi$$
0.507538 0.861630i $$-0.330556\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 489.898 2.41329
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 480.000 2.21198
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 382.120i 1.71354i 0.515695 + 0.856772i $$0.327534\pi$$
−0.515695 + 0.856772i $$0.672466\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −293.939 −1.29488 −0.647442 0.762115i $$-0.724161\pi$$
−0.647442 + 0.762115i $$0.724161\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −382.000 −1.58506 −0.792531 0.609831i $$-0.791237\pi$$
−0.792531 + 0.609831i $$0.791237\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 94.0000i − 0.383673i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −176.363 −0.702642 −0.351321 0.936255i $$-0.614267\pi$$
−0.351321 + 0.936255i $$0.614267\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ −188.000 −0.709434
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 430.000i − 1.59851i −0.600990 0.799257i $$-0.705227\pi$$
0.600990 0.799257i $$-0.294773\pi$$
$$270$$ 0 0
$$271$$ 538.888i 1.98852i 0.107011 + 0.994258i $$0.465872\pi$$
−0.107011 + 0.994258i $$0.534128\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −411.514 −1.49642
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −289.000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 386.000i − 1.31741i −0.752403 0.658703i $$-0.771105\pi$$
0.752403 0.658703i $$-0.228895\pi$$
$$294$$ 0 0
$$295$$ 235.151i 0.797122i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 530.000 1.69329 0.846645 0.532158i $$-0.178619\pi$$
0.846645 + 0.532158i $$0.178619\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 334.000i − 1.05363i −0.849981 0.526814i $$-0.823386\pi$$
0.849981 0.526814i $$-0.176614\pi$$
$$318$$ 0 0
$$319$$ − 979.796i − 3.07146i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 190.000 0.563798 0.281899 0.959444i $$-0.409036\pi$$
0.281899 + 0.959444i $$0.409036\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 960.000i − 2.81525i
$$342$$ 0 0
$$343$$ − 19.5959i − 0.0571310i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −685.857 −1.97653 −0.988267 0.152738i $$-0.951191\pi$$
−0.988267 + 0.152738i $$0.951191\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ −361.000 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 100.000i 0.273973i
$$366$$ 0 0
$$367$$ − 186.161i − 0.507251i −0.967302 0.253626i $$-0.918377\pi$$
0.967302 0.253626i $$-0.0816231\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 921.008 2.48250
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ −384.000 −0.997403
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 190.000i − 0.488432i −0.969721 0.244216i $$-0.921469\pi$$
0.969721 0.244216i $$-0.0785306\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 293.939 0.744149
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 718.000 1.75550 0.877751 0.479118i $$-0.159043\pi$$
0.877751 + 0.479118i $$0.159043\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 1152.00i − 2.78935i
$$414$$ 0 0
$$415$$ − 195.959i − 0.472191i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 411.514 0.982134 0.491067 0.871122i $$-0.336607\pi$$
0.491067 + 0.871122i $$0.336607\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 670.000 1.54734 0.773672 0.633586i $$-0.218418\pi$$
0.773672 + 0.633586i $$0.218418\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 832.827i 1.89710i 0.316629 + 0.948550i $$0.397449\pi$$
−0.316629 + 0.948550i $$0.602551\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 881.816 1.99056 0.995278 0.0970655i $$-0.0309456\pi$$
0.995278 + 0.0970655i $$0.0309456\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −530.000 −1.15974 −0.579869 0.814710i $$-0.696896\pi$$
−0.579869 + 0.814710i $$0.696896\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 530.000i − 1.14967i −0.818268 0.574837i $$-0.805065\pi$$
0.818268 0.574837i $$-0.194935\pi$$
$$462$$ 0 0
$$463$$ 891.614i 1.92573i 0.269978 + 0.962866i $$0.412983\pi$$
−0.269978 + 0.962866i $$0.587017\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −685.857 −1.46864 −0.734322 0.678801i $$-0.762500\pi$$
−0.734322 + 0.678801i $$0.762500\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 380.000i − 0.783505i
$$486$$ 0 0
$$487$$ − 88.1816i − 0.181071i −0.995893 0.0905356i $$-0.971142\pi$$
0.995893 0.0905356i $$-0.0288579\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −862.220 −1.75605 −0.878025 0.478615i $$-0.841139\pi$$
−0.878025 + 0.478615i $$0.841139\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ −380.000 −0.752475
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1010.00i 1.98428i 0.125121 + 0.992141i $$0.460068\pi$$
−0.125121 + 0.992141i $$0.539932\pi$$
$$510$$ 0 0
$$511$$ − 489.898i − 0.958704i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −411.514 −0.799057
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 529.000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 391.918i − 0.732558i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 921.008 1.70873
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −1440.00 −2.60398
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 914.000i − 1.64093i −0.571694 0.820467i $$-0.693714\pi$$
0.571694 0.820467i $$-0.306286\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1077.78 −1.91434 −0.957172 0.289520i $$-0.906504\pi$$
−0.957172 + 0.289520i $$0.906504\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 290.000 0.502600 0.251300 0.967909i $$-0.419142\pi$$
0.251300 + 0.967909i $$0.419142\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 960.000i 1.65232i
$$582$$ 0 0
$$583$$ − 1842.02i − 3.15955i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −783.837 −1.33533 −0.667663 0.744463i $$-0.732705\pi$$
−0.667663 + 0.744463i $$0.732705\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 1198.00 1.99334 0.996672 0.0815138i $$-0.0259755\pi$$
0.996672 + 0.0815138i $$0.0259755\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 526.000i 0.869421i
$$606$$ 0 0
$$607$$ 969.998i 1.59802i 0.601318 + 0.799010i $$0.294643\pi$$
−0.601318 + 0.799010i $$0.705357\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 341.000 0.545600
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ − 244.949i − 0.388192i −0.980983 0.194096i $$-0.937823\pi$$
0.980983 0.194096i $$-0.0621773\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −215.555 −0.339457
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ −2304.00 −3.55008
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1006.00i 1.54058i 0.637693 + 0.770291i $$0.279889\pi$$
−0.637693 + 0.770291i $$0.720111\pi$$
$$654$$ 0 0
$$655$$ 156.767i 0.239339i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1097.37 1.66521 0.832603 0.553869i $$-0.186849\pi$$
0.832603 + 0.553869i $$0.186849\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 190.000 0.282318 0.141159 0.989987i $$-0.454917\pi$$
0.141159 + 0.989987i $$0.454917\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 1346.00i 1.98818i 0.108545 + 0.994092i $$0.465381\pi$$
−0.108545 + 0.994092i $$0.534619\pi$$
$$678$$ 0 0
$$679$$ 1861.61i 2.74170i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 293.939 0.430364 0.215182 0.976574i $$-0.430965\pi$$
0.215182 + 0.976574i $$0.430965\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 50.0000i 0.0713267i 0.999364 + 0.0356633i $$0.0113544\pi$$
−0.999364 + 0.0356633i $$0.988646\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1861.61 2.63311
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 2016.00 2.79612
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1050.00i 1.44828i
$$726$$ 0 0
$$727$$ 107.778i 0.148250i 0.997249 + 0.0741249i $$0.0236163\pi$$
−0.997249 + 0.0741249i $$0.976384\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ −580.000 −0.778523
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 1920.00i 2.56342i
$$750$$ 0 0
$$751$$ − 538.888i − 0.717560i −0.933422 0.358780i $$-0.883193\pi$$
0.933422 0.358780i $$-0.116807\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 97.9796 0.129774
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 862.000 1.12094 0.560468 0.828176i $$-0.310621\pi$$
0.560468 + 0.828176i $$0.310621\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 1154.00i − 1.49288i −0.665450 0.746442i $$-0.731760\pi$$
0.665450 0.746442i $$-0.268240\pi$$
$$774$$ 0 0
$$775$$ 1028.79i 1.32747i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1294.00i 1.62359i 0.583944 + 0.811794i $$0.301509\pi$$
−0.583944 + 0.811794i $$0.698491\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −979.796 −1.22017
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 670.000i 0.816078i 0.912965 + 0.408039i $$0.133787\pi$$
−0.912965 + 0.408039i $$0.866213\pi$$
$$822$$ 0 0
$$823$$ − 1577.47i − 1.91673i −0.285541 0.958367i $$-0.592173\pi$$
0.285541 0.958367i $$-0.407827\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −587.878 −0.710856 −0.355428 0.934704i $$-0.615665\pi$$
−0.355428 + 0.934704i $$0.615665\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ −1659.00 −1.97265
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 338.000i 0.400000i
$$846$$ 0 0
$$847$$ − 2576.86i − 3.04234i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ −92.0000 −0.106358
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 2880.00i 3.31415i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 901.412 1.03019
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 1056.00 1.18785
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 548.686i 0.613057i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −2449.49 −2.72468
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ 1920.00 2.10296
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 768.000i − 0.837514i
$$918$$ 0 0
$$919$$ 1714.64i 1.86577i 0.360174 + 0.932885i $$0.382717\pi$$
−0.360174 + 0.932885i $$0.617283\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1490.00 −1.59018 −0.795091 0.606491i $$-0.792577\pi$$
−0.795091 + 0.606491i $$0.792577\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 430.000i 0.456961i 0.973549 + 0.228480i $$0.0733757\pi$$
−0.973549 + 0.228480i $$0.926624\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1371.71 −1.44848 −0.724242 0.689546i $$-0.757810\pi$$
−0.724242 + 0.689546i $$0.757810\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −1439.00 −1.49740
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 580.000i − 0.601036i
$$966$$ 0 0
$$967$$ − 303.737i − 0.314102i −0.987590 0.157051i $$-0.949801\pi$$
0.987590 0.157051i $$-0.0501987\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 215.555 0.221993 0.110996 0.993821i $$-0.464596\pi$$
0.110996 + 0.993821i $$0.464596\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 388.000 0.393909
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 1420.70i 1.43361i 0.697275 + 0.716803i $$0.254395\pi$$
−0.697275 + 0.716803i $$0.745605\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 685.857 0.689304
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1152.3.b.f.703.3 yes 4
3.2 odd 2 inner 1152.3.b.f.703.1 4
4.3 odd 2 inner 1152.3.b.f.703.4 yes 4
8.3 odd 2 inner 1152.3.b.f.703.2 yes 4
8.5 even 2 inner 1152.3.b.f.703.1 4
12.11 even 2 inner 1152.3.b.f.703.2 yes 4
16.3 odd 4 2304.3.g.l.1279.1 2
16.5 even 4 2304.3.g.i.1279.2 2
16.11 odd 4 2304.3.g.i.1279.1 2
16.13 even 4 2304.3.g.l.1279.2 2
24.5 odd 2 CM 1152.3.b.f.703.3 yes 4
24.11 even 2 inner 1152.3.b.f.703.4 yes 4
48.5 odd 4 2304.3.g.l.1279.2 2
48.11 even 4 2304.3.g.l.1279.1 2
48.29 odd 4 2304.3.g.i.1279.2 2
48.35 even 4 2304.3.g.i.1279.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.b.f.703.1 4 3.2 odd 2 inner
1152.3.b.f.703.1 4 8.5 even 2 inner
1152.3.b.f.703.2 yes 4 8.3 odd 2 inner
1152.3.b.f.703.2 yes 4 12.11 even 2 inner
1152.3.b.f.703.3 yes 4 1.1 even 1 trivial
1152.3.b.f.703.3 yes 4 24.5 odd 2 CM
1152.3.b.f.703.4 yes 4 4.3 odd 2 inner
1152.3.b.f.703.4 yes 4 24.11 even 2 inner
2304.3.g.i.1279.1 2 16.11 odd 4
2304.3.g.i.1279.1 2 48.35 even 4
2304.3.g.i.1279.2 2 16.5 even 4
2304.3.g.i.1279.2 2 48.29 odd 4
2304.3.g.l.1279.1 2 16.3 odd 4
2304.3.g.l.1279.1 2 48.11 even 4
2304.3.g.l.1279.2 2 16.13 even 4
2304.3.g.l.1279.2 2 48.5 odd 4