# Properties

 Label 304.5.e.b.113.2 Level $304$ Weight $5$ Character 304.113 Self dual yes Analytic conductor $31.424$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.4244687775$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 113.2 Root $$-3.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 304.113

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+18.4743 q^{5} +20.1238 q^{7} +81.0000 q^{9} +O(q^{10})$$ $$q+18.4743 q^{5} +20.1238 q^{7} +81.0000 q^{9} -59.8762 q^{11} +572.866 q^{17} -361.000 q^{19} +158.000 q^{23} -283.702 q^{25} +371.771 q^{35} +2726.10 q^{43} +1496.41 q^{45} +4284.04 q^{47} -1996.03 q^{49} -1106.17 q^{55} +4248.75 q^{61} +1630.02 q^{63} +8130.81 q^{73} -1204.94 q^{77} +6561.00 q^{81} +5678.00 q^{83} +10583.3 q^{85} -6669.21 q^{95} -4849.98 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 31q^{5} - 73q^{7} + 162q^{9} + O(q^{10})$$ $$2q - 31q^{5} - 73q^{7} + 162q^{9} - 233q^{11} + 353q^{17} - 722q^{19} + 316q^{23} + 1539q^{25} + 4979q^{35} + 3527q^{43} - 2511q^{45} + 1207q^{47} + 4275q^{49} + 7459q^{55} - 3167q^{61} - 5913q^{63} + 10033q^{73} + 14917q^{77} + 13122q^{81} + 11356q^{83} + 21461q^{85} + 11191q^{95} - 18873q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 0 0
$$5$$ 18.4743 0.738970 0.369485 0.929237i $$-0.379534\pi$$
0.369485 + 0.929237i $$0.379534\pi$$
$$6$$ 0 0
$$7$$ 20.1238 0.410689 0.205344 0.978690i $$-0.434169\pi$$
0.205344 + 0.978690i $$0.434169\pi$$
$$8$$ 0 0
$$9$$ 81.0000 1.00000
$$10$$ 0 0
$$11$$ −59.8762 −0.494845 −0.247422 0.968908i $$-0.579584\pi$$
−0.247422 + 0.968908i $$0.579584\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 572.866 1.98224 0.991118 0.132984i $$-0.0424559\pi$$
0.991118 + 0.132984i $$0.0424559\pi$$
$$18$$ 0 0
$$19$$ −361.000 −1.00000
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 158.000 0.298677 0.149338 0.988786i $$-0.452286\pi$$
0.149338 + 0.988786i $$0.452286\pi$$
$$24$$ 0 0
$$25$$ −283.702 −0.453923
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 371.771 0.303487
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 2726.10 1.47437 0.737183 0.675693i $$-0.236155\pi$$
0.737183 + 0.675693i $$0.236155\pi$$
$$44$$ 0 0
$$45$$ 1496.41 0.738970
$$46$$ 0 0
$$47$$ 4284.04 1.93936 0.969680 0.244380i $$-0.0785844\pi$$
0.969680 + 0.244380i $$0.0785844\pi$$
$$48$$ 0 0
$$49$$ −1996.03 −0.831335
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ −1106.17 −0.365676
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 4248.75 1.14183 0.570915 0.821009i $$-0.306589\pi$$
0.570915 + 0.821009i $$0.306589\pi$$
$$62$$ 0 0
$$63$$ 1630.02 0.410689
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 8130.81 1.52577 0.762883 0.646537i $$-0.223783\pi$$
0.762883 + 0.646537i $$0.223783\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1204.94 −0.203227
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 6561.00 1.00000
$$82$$ 0 0
$$83$$ 5678.00 0.824213 0.412106 0.911136i $$-0.364793\pi$$
0.412106 + 0.911136i $$0.364793\pi$$
$$84$$ 0 0
$$85$$ 10583.3 1.46481
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6669.21 −0.738970
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ −4849.98 −0.494845
$$100$$ 0 0
$$101$$ −9998.00 −0.980100 −0.490050 0.871694i $$-0.663021\pi$$
−0.490050 + 0.871694i $$0.663021\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 2918.93 0.220713
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 11528.2 0.814083
$$120$$ 0 0
$$121$$ −11055.8 −0.755128
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −16787.6 −1.07441
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 33665.5 1.96175 0.980873 0.194651i $$-0.0623574\pi$$
0.980873 + 0.194651i $$0.0623574\pi$$
$$132$$ 0 0
$$133$$ −7264.68 −0.410689
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 8396.39 0.447354 0.223677 0.974663i $$-0.428194\pi$$
0.223677 + 0.974663i $$0.428194\pi$$
$$138$$ 0 0
$$139$$ −33381.1 −1.72771 −0.863856 0.503738i $$-0.831958\pi$$
−0.863856 + 0.503738i $$0.831958\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −30212.3 −1.36085 −0.680427 0.732816i $$-0.738206\pi$$
−0.680427 + 0.732816i $$0.738206\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 46402.2 1.98224
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −49198.0 −1.99594 −0.997972 0.0636620i $$-0.979722\pi$$
−0.997972 + 0.0636620i $$0.979722\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3179.55 0.122663
$$162$$ 0 0
$$163$$ −9362.00 −0.352366 −0.176183 0.984357i $$-0.556375\pi$$
−0.176183 + 0.984357i $$0.556375\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 28561.0 1.00000
$$170$$ 0 0
$$171$$ −29241.0 −1.00000
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ −5709.15 −0.186421
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 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
$$187$$ −34301.1 −0.980900
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −61997.2 −1.69944 −0.849719 0.527235i $$-0.823229\pi$$
−0.849719 + 0.527235i $$0.823229\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −69518.0 −1.79129 −0.895643 0.444774i $$-0.853284\pi$$
−0.895643 + 0.444774i $$0.853284\pi$$
$$198$$ 0 0
$$199$$ −78104.5 −1.97229 −0.986143 0.165900i $$-0.946947\pi$$
−0.986143 + 0.165900i $$0.946947\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 12798.0 0.298677
$$208$$ 0 0
$$209$$ 21615.3 0.494845
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 50362.7 1.08951
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ −22979.9 −0.453923
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 59034.7 1.12574 0.562868 0.826547i $$-0.309698\pi$$
0.562868 + 0.826547i $$0.309698\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −66110.9 −1.21776 −0.608879 0.793263i $$-0.708380\pi$$
−0.608879 + 0.793263i $$0.708380\pi$$
$$234$$ 0 0
$$235$$ 79144.5 1.43313
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 105335. 1.84406 0.922032 0.387113i $$-0.126528\pi$$
0.922032 + 0.387113i $$0.126528\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −36875.2 −0.614331
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −74357.6 −1.18026 −0.590130 0.807308i $$-0.700924\pi$$
−0.590130 + 0.807308i $$0.700924\pi$$
$$252$$ 0 0
$$253$$ −9460.45 −0.147799
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −104877. −1.51625 −0.758124 0.652111i $$-0.773884\pi$$
−0.758124 + 0.652111i $$0.773884\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 126718. 1.72544 0.862720 0.505682i $$-0.168759\pi$$
0.862720 + 0.505682i $$0.168759\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 16987.0 0.224622
$$276$$ 0 0
$$277$$ −133030. −1.73376 −0.866880 0.498517i $$-0.833878\pi$$
−0.866880 + 0.498517i $$0.833878\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 136595. 1.70554 0.852771 0.522286i $$-0.174921\pi$$
0.852771 + 0.522286i $$0.174921\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 244655. 2.92926
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 54859.5 0.605506
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 78492.4 0.843778
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5413.87 0.0559741 0.0279871 0.999608i $$-0.491090\pi$$
0.0279871 + 0.999608i $$0.491090\pi$$
$$312$$ 0 0
$$313$$ 152162. 1.55316 0.776582 0.630016i $$-0.216952\pi$$
0.776582 + 0.630016i $$0.216952\pi$$
$$314$$ 0 0
$$315$$ 30113.5 0.303487
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −206805. −1.98224
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 86211.1 0.796473
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −88484.9 −0.752109
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 201682. 1.67497 0.837487 0.546457i $$-0.184024\pi$$
0.837487 + 0.546457i $$0.184024\pi$$
$$348$$ 0 0
$$349$$ −225949. −1.85506 −0.927532 0.373745i $$-0.878074\pi$$
−0.927532 + 0.373745i $$0.878074\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 10882.0 0.0873292 0.0436646 0.999046i $$-0.486097\pi$$
0.0436646 + 0.999046i $$0.486097\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −241539. −1.87412 −0.937061 0.349165i $$-0.886465\pi$$
−0.937061 + 0.349165i $$0.886465\pi$$
$$360$$ 0 0
$$361$$ 130321. 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 150211. 1.12750
$$366$$ 0 0
$$367$$ 266878. 1.98144 0.990719 0.135923i $$-0.0434001\pi$$
0.990719 + 0.135923i $$0.0434001\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ −22260.3 −0.150179
$$386$$ 0 0
$$387$$ 220814. 1.47437
$$388$$ 0 0
$$389$$ 38543.1 0.254711 0.127355 0.991857i $$-0.459351\pi$$
0.127355 + 0.991857i $$0.459351\pi$$
$$390$$ 0 0
$$391$$ 90512.9 0.592048
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −297079. −1.88491 −0.942456 0.334329i $$-0.891490\pi$$
−0.942456 + 0.334329i $$0.891490\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 121210. 0.738970
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 104897. 0.609068
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −229522. −1.30736 −0.653682 0.756770i $$-0.726776\pi$$
−0.653682 + 0.756770i $$0.726776\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 347008. 1.93936
$$424$$ 0 0
$$425$$ −162523. −0.899783
$$426$$ 0 0
$$427$$ 85500.8 0.468937
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −57038.0 −0.298677
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ −161679. −0.831335
$$442$$ 0 0
$$443$$ −160753. −0.819126 −0.409563 0.912282i $$-0.634319\pi$$
−0.409563 + 0.912282i $$0.634319\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 374513. 1.79322 0.896612 0.442817i $$-0.146021\pi$$
0.896612 + 0.442817i $$0.146021\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −199550. −0.938967 −0.469484 0.882941i $$-0.655560\pi$$
−0.469484 + 0.882941i $$0.655560\pi$$
$$462$$ 0 0
$$463$$ −279914. −1.30576 −0.652879 0.757462i $$-0.726439\pi$$
−0.652879 + 0.757462i $$0.726439\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 51999.4 0.238432 0.119216 0.992868i $$-0.461962\pi$$
0.119216 + 0.992868i $$0.461962\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −163229. −0.729583
$$474$$ 0 0
$$475$$ 102416. 0.453923
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −428482. −1.86750 −0.933752 0.357921i $$-0.883486\pi$$
−0.933752 + 0.357921i $$0.883486\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −360562. −1.49561 −0.747803 0.663921i $$-0.768891\pi$$
−0.747803 + 0.663921i $$0.768891\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −89599.7 −0.365676
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −497221. −1.99687 −0.998433 0.0559673i $$-0.982176\pi$$
−0.998433 + 0.0559673i $$0.982176\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −358882. −1.41846 −0.709228 0.704979i $$-0.750956\pi$$
−0.709228 + 0.704979i $$0.750956\pi$$
$$504$$ 0 0
$$505$$ −184706. −0.724265
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 163622. 0.626615
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −256512. −0.959682
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −254877. −0.910792
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 119515. 0.411382
$$540$$ 0 0
$$541$$ 576412. 1.96942 0.984711 0.174196i $$-0.0557327\pi$$
0.984711 + 0.174196i $$0.0557327\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 344149. 1.14183
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −94987.4 −0.306165 −0.153083 0.988213i $$-0.548920\pi$$
−0.153083 + 0.988213i $$0.548920\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 132032. 0.410689
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 442318. 1.35663 0.678317 0.734770i $$-0.262710\pi$$
0.678317 + 0.734770i $$0.262710\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −44824.9 −0.135576
$$576$$ 0 0
$$577$$ −179948. −0.540500 −0.270250 0.962790i $$-0.587106\pi$$
−0.270250 + 0.962790i $$0.587106\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 114263. 0.338495
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −38758.7 −0.112485 −0.0562423 0.998417i $$-0.517912\pi$$
−0.0562423 + 0.998417i $$0.517912\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −702398. −1.99744 −0.998720 0.0505740i $$-0.983895\pi$$
−0.998720 + 0.0505740i $$0.983895\pi$$
$$594$$ 0 0
$$595$$ 212975. 0.601583
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −204248. −0.558017
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 28243.5 0.0751620 0.0375810 0.999294i $$-0.488035\pi$$
0.0375810 + 0.999294i $$0.488035\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −761325. −1.99986 −0.999930 0.0118526i $$-0.996227\pi$$
−0.999930 + 0.0118526i $$0.996227\pi$$
$$618$$ 0 0
$$619$$ 328078. 0.856241 0.428120 0.903722i $$-0.359176\pi$$
0.428120 + 0.903722i $$0.359176\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −132825. −0.340031
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −506384. −1.27181 −0.635903 0.771769i $$-0.719372\pi$$
−0.635903 + 0.771769i $$0.719372\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ −410564. −0.993023 −0.496511 0.868030i $$-0.665386\pi$$
−0.496511 + 0.868030i $$0.665386\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 795843. 1.90116 0.950580 0.310480i $$-0.100490\pi$$
0.950580 + 0.310480i $$0.100490\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 762372. 1.78789 0.893945 0.448178i $$-0.147927\pi$$
0.893945 + 0.448178i $$0.147927\pi$$
$$654$$ 0 0
$$655$$ 621945. 1.44967
$$656$$ 0 0
$$657$$ 658595. 1.52577
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −134209. −0.303487
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −254399. −0.565029
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 155117. 0.330581
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −278468. −0.583202 −0.291601 0.956540i $$-0.594188\pi$$
−0.291601 + 0.956540i $$0.594188\pi$$
$$692$$ 0 0
$$693$$ −97599.7 −0.203227
$$694$$ 0 0
$$695$$ −616692. −1.27673
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 222802. 0.453402 0.226701 0.973964i $$-0.427206\pi$$
0.226701 + 0.973964i $$0.427206\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −201197. −0.402516
$$708$$ 0 0
$$709$$ 731762. 1.45572 0.727859 0.685727i $$-0.240515\pi$$
0.727859 + 0.685727i $$0.240515\pi$$
$$710$$ 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
$$719$$ 837359. 1.61977 0.809886 0.586588i $$-0.199529\pi$$
0.809886 + 0.586588i $$0.199529\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −631766. −1.19533 −0.597664 0.801747i $$-0.703904\pi$$
−0.597664 + 0.801747i $$0.703904\pi$$
$$728$$ 0 0
$$729$$ 531441. 1.00000
$$730$$ 0 0
$$731$$ 1.56169e6 2.92254
$$732$$ 0 0
$$733$$ 538322. 1.00192 0.500961 0.865470i $$-0.332980\pi$$
0.500961 + 0.865470i $$0.332980\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −1.04695e6 −1.91707 −0.958536 0.284970i $$-0.908016\pi$$
−0.958536 + 0.284970i $$0.908016\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ −558150. −1.00563
$$746$$ 0 0
$$747$$ 459918. 0.824213
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.14504e6 1.99815 0.999076 0.0429669i $$-0.0136810\pi$$
0.999076 + 0.0429669i $$0.0136810\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −238330. −0.411537 −0.205769 0.978601i $$-0.565969\pi$$
−0.205769 + 0.978601i $$0.565969\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 857246. 1.46481
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 1.04962e6 1.77493 0.887465 0.460875i $$-0.152464\pi$$
0.887465 + 0.460875i $$0.152464\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 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
$$785$$ −908896. −1.47494
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 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
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 2.45418e6 3.84427
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −486842. −0.755018
$$804$$ 0 0
$$805$$ 58739.9 0.0906445
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −223470. −0.341446 −0.170723 0.985319i $$-0.554610\pi$$
−0.170723 + 0.985319i $$0.554610\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −172956. −0.260388
$$816$$ 0 0
$$817$$ −984124. −1.47437
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.17432e6 −1.74220 −0.871101 0.491104i $$-0.836594\pi$$
−0.871101 + 0.491104i $$0.836594\pi$$
$$822$$ 0 0
$$823$$ −143923. −0.212486 −0.106243 0.994340i $$-0.533882\pi$$
−0.106243 + 0.994340i $$0.533882\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −1.14346e6 −1.64790
$$834$$ 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
$$841$$ 707281. 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 527643. 0.738970
$$846$$ 0 0
$$847$$ −222485. −0.310123
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −394318. −0.541937 −0.270968 0.962588i $$-0.587344\pi$$
−0.270968 + 0.962588i $$0.587344\pi$$
$$854$$ 0 0
$$855$$ −540206. −0.738970
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 1.47565e6 1.99985 0.999927 0.0120966i $$-0.00385055\pi$$
0.999927 + 0.0120966i $$0.00385055\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −337829. −0.441247
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1.41242e6 1.81975 0.909877 0.414879i $$-0.136176\pi$$
0.909877 + 0.414879i $$0.136176\pi$$
$$882$$ 0 0
$$883$$ 694207. 0.890365 0.445182 0.895440i $$-0.353139\pi$$
0.445182 + 0.895440i $$0.353139\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −392848. −0.494845
$$892$$ 0 0
$$893$$ −1.54654e6 −1.93936
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ −809838. −0.980100
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ −339977. −0.407857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 677477. 0.805667
$$918$$ 0 0
$$919$$ −1.41552e6 −1.67604 −0.838022 0.545636i $$-0.816288\pi$$
−0.838022 + 0.545636i $$0.816288\pi$$
$$920$$ 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
$$929$$ −1.31392e6 −1.52243 −0.761214 0.648501i $$-0.775396\pi$$
−0.761214 + 0.648501i $$0.775396\pi$$
$$930$$ 0 0
$$931$$ 720568. 0.831335
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −633687. −0.724856
$$936$$ 0 0
$$937$$ 1.35678e6 1.54536 0.772682 0.634794i $$-0.218915\pi$$
0.772682 + 0.634794i $$0.218915\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1.55528e6 −1.73424 −0.867120 0.498099i $$-0.834031\pi$$
−0.867120 + 0.498099i $$0.834031\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ −1.14535e6 −1.25583
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 168967. 0.183723
$$960$$ 0 0
$$961$$ 923521. 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1.33392e6 −1.42652 −0.713259 0.700900i $$-0.752782\pi$$
−0.713259 + 0.700900i $$0.752782\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 0 0
$$973$$ −671754. −0.709553
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 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
$$985$$ −1.28429e6 −1.32371
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 430724. 0.440359
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −1.44292e6 −1.45746
$$996$$ 0 0
$$997$$ −486609. −0.489542 −0.244771 0.969581i $$-0.578713\pi$$
−0.244771 + 0.969581i $$0.578713\pi$$
$$998$$ 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 304.5.e.b.113.2 2
4.3 odd 2 76.5.c.a.37.2 2
12.11 even 2 684.5.h.b.37.1 2
19.18 odd 2 CM 304.5.e.b.113.2 2
76.75 even 2 76.5.c.a.37.2 2
228.227 odd 2 684.5.h.b.37.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.c.a.37.2 2 4.3 odd 2
76.5.c.a.37.2 2 76.75 even 2
304.5.e.b.113.2 2 1.1 even 1 trivial
304.5.e.b.113.2 2 19.18 odd 2 CM
684.5.h.b.37.1 2 12.11 even 2
684.5.h.b.37.1 2 228.227 odd 2