# Properties

 Label 3332.2.a.h.1.2 Level $3332$ Weight $2$ Character 3332.1 Self dual yes Analytic conductor $26.606$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$26.6061539535$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 3332.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.732051 q^{3} -3.46410 q^{5} -2.46410 q^{9} +O(q^{10})$$ $$q+0.732051 q^{3} -3.46410 q^{5} -2.46410 q^{9} -4.73205 q^{11} +1.46410 q^{13} -2.53590 q^{15} +1.00000 q^{17} -5.46410 q^{19} -4.73205 q^{23} +7.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} +6.19615 q^{31} -3.46410 q^{33} +11.4641 q^{37} +1.07180 q^{39} +6.00000 q^{41} +12.3923 q^{43} +8.53590 q^{45} -6.92820 q^{47} +0.732051 q^{51} -0.928203 q^{53} +16.3923 q^{55} -4.00000 q^{57} -9.46410 q^{59} +7.46410 q^{61} -5.07180 q^{65} +1.07180 q^{67} -3.46410 q^{69} +2.19615 q^{71} -2.00000 q^{73} +5.12436 q^{75} -1.80385 q^{79} +4.46410 q^{81} +9.46410 q^{83} -3.46410 q^{85} -2.53590 q^{87} -9.46410 q^{89} +4.53590 q^{93} +18.9282 q^{95} -8.92820 q^{97} +11.6603 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} - 6 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 4 q^{19} - 6 q^{23} + 14 q^{25} - 8 q^{27} + 2 q^{31} + 16 q^{37} + 16 q^{39} + 12 q^{41} + 4 q^{43} + 24 q^{45} - 2 q^{51} + 12 q^{53} + 12 q^{55} - 8 q^{57} - 12 q^{59} + 8 q^{61} - 24 q^{65} + 16 q^{67} - 6 q^{71} - 4 q^{73} - 14 q^{75} - 14 q^{79} + 2 q^{81} + 12 q^{83} - 12 q^{87} - 12 q^{89} + 16 q^{93} + 24 q^{95} - 4 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 4 * q^19 - 6 * q^23 + 14 * q^25 - 8 * q^27 + 2 * q^31 + 16 * q^37 + 16 * q^39 + 12 * q^41 + 4 * q^43 + 24 * q^45 - 2 * q^51 + 12 * q^53 + 12 * q^55 - 8 * q^57 - 12 * q^59 + 8 * q^61 - 24 * q^65 + 16 * q^67 - 6 * q^71 - 4 * q^73 - 14 * q^75 - 14 * q^79 + 2 * q^81 + 12 * q^83 - 12 * q^87 - 12 * q^89 + 16 * q^93 + 24 * q^95 - 4 * q^97 + 6 * q^99

## 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
$$3$$ 0.732051 0.422650 0.211325 0.977416i $$-0.432222\pi$$
0.211325 + 0.977416i $$0.432222\pi$$
$$4$$ 0 0
$$5$$ −3.46410 −1.54919 −0.774597 0.632456i $$-0.782047\pi$$
−0.774597 + 0.632456i $$0.782047\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.46410 −0.821367
$$10$$ 0 0
$$11$$ −4.73205 −1.42677 −0.713384 0.700774i $$-0.752838\pi$$
−0.713384 + 0.700774i $$0.752838\pi$$
$$12$$ 0 0
$$13$$ 1.46410 0.406069 0.203034 0.979172i $$-0.434920\pi$$
0.203034 + 0.979172i $$0.434920\pi$$
$$14$$ 0 0
$$15$$ −2.53590 −0.654766
$$16$$ 0 0
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ −5.46410 −1.25355 −0.626775 0.779200i $$-0.715626\pi$$
−0.626775 + 0.779200i $$0.715626\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.73205 −0.986701 −0.493350 0.869831i $$-0.664228\pi$$
−0.493350 + 0.869831i $$0.664228\pi$$
$$24$$ 0 0
$$25$$ 7.00000 1.40000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ −3.46410 −0.643268 −0.321634 0.946864i $$-0.604232\pi$$
−0.321634 + 0.946864i $$0.604232\pi$$
$$30$$ 0 0
$$31$$ 6.19615 1.11286 0.556431 0.830894i $$-0.312170\pi$$
0.556431 + 0.830894i $$0.312170\pi$$
$$32$$ 0 0
$$33$$ −3.46410 −0.603023
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 11.4641 1.88469 0.942343 0.334648i $$-0.108617\pi$$
0.942343 + 0.334648i $$0.108617\pi$$
$$38$$ 0 0
$$39$$ 1.07180 0.171625
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 12.3923 1.88981 0.944904 0.327346i $$-0.106154\pi$$
0.944904 + 0.327346i $$0.106154\pi$$
$$44$$ 0 0
$$45$$ 8.53590 1.27246
$$46$$ 0 0
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0.732051 0.102508
$$52$$ 0 0
$$53$$ −0.928203 −0.127499 −0.0637493 0.997966i $$-0.520306\pi$$
−0.0637493 + 0.997966i $$0.520306\pi$$
$$54$$ 0 0
$$55$$ 16.3923 2.21034
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ −9.46410 −1.23212 −0.616061 0.787699i $$-0.711272\pi$$
−0.616061 + 0.787699i $$0.711272\pi$$
$$60$$ 0 0
$$61$$ 7.46410 0.955680 0.477840 0.878447i $$-0.341420\pi$$
0.477840 + 0.878447i $$0.341420\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −5.07180 −0.629079
$$66$$ 0 0
$$67$$ 1.07180 0.130941 0.0654704 0.997855i $$-0.479145\pi$$
0.0654704 + 0.997855i $$0.479145\pi$$
$$68$$ 0 0
$$69$$ −3.46410 −0.417029
$$70$$ 0 0
$$71$$ 2.19615 0.260635 0.130318 0.991472i $$-0.458400\pi$$
0.130318 + 0.991472i $$0.458400\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 5.12436 0.591710
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.80385 −0.202949 −0.101474 0.994838i $$-0.532356\pi$$
−0.101474 + 0.994838i $$0.532356\pi$$
$$80$$ 0 0
$$81$$ 4.46410 0.496011
$$82$$ 0 0
$$83$$ 9.46410 1.03882 0.519410 0.854525i $$-0.326152\pi$$
0.519410 + 0.854525i $$0.326152\pi$$
$$84$$ 0 0
$$85$$ −3.46410 −0.375735
$$86$$ 0 0
$$87$$ −2.53590 −0.271877
$$88$$ 0 0
$$89$$ −9.46410 −1.00319 −0.501596 0.865102i $$-0.667254\pi$$
−0.501596 + 0.865102i $$0.667254\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.53590 0.470351
$$94$$ 0 0
$$95$$ 18.9282 1.94199
$$96$$ 0 0
$$97$$ −8.92820 −0.906522 −0.453261 0.891378i $$-0.649739\pi$$
−0.453261 + 0.891378i $$0.649739\pi$$
$$98$$ 0 0
$$99$$ 11.6603 1.17190
$$100$$ 0 0
$$101$$ 9.46410 0.941713 0.470857 0.882210i $$-0.343945\pi$$
0.470857 + 0.882210i $$0.343945\pi$$
$$102$$ 0 0
$$103$$ −2.92820 −0.288524 −0.144262 0.989539i $$-0.546081\pi$$
−0.144262 + 0.989539i $$0.546081\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.26795 0.702619 0.351310 0.936259i $$-0.385736\pi$$
0.351310 + 0.936259i $$0.385736\pi$$
$$108$$ 0 0
$$109$$ 6.39230 0.612272 0.306136 0.951988i $$-0.400964\pi$$
0.306136 + 0.951988i $$0.400964\pi$$
$$110$$ 0 0
$$111$$ 8.39230 0.796562
$$112$$ 0 0
$$113$$ −7.85641 −0.739069 −0.369534 0.929217i $$-0.620483\pi$$
−0.369534 + 0.929217i $$0.620483\pi$$
$$114$$ 0 0
$$115$$ 16.3923 1.52859
$$116$$ 0 0
$$117$$ −3.60770 −0.333532
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.3923 1.03566
$$122$$ 0 0
$$123$$ 4.39230 0.396041
$$124$$ 0 0
$$125$$ −6.92820 −0.619677
$$126$$ 0 0
$$127$$ −20.3923 −1.80952 −0.904762 0.425917i $$-0.859952\pi$$
−0.904762 + 0.425917i $$0.859952\pi$$
$$128$$ 0 0
$$129$$ 9.07180 0.798727
$$130$$ 0 0
$$131$$ 11.6603 1.01876 0.509381 0.860541i $$-0.329875\pi$$
0.509381 + 0.860541i $$0.329875\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 13.8564 1.19257
$$136$$ 0 0
$$137$$ 9.46410 0.808573 0.404286 0.914632i $$-0.367520\pi$$
0.404286 + 0.914632i $$0.367520\pi$$
$$138$$ 0 0
$$139$$ 11.2679 0.955735 0.477867 0.878432i $$-0.341410\pi$$
0.477867 + 0.878432i $$0.341410\pi$$
$$140$$ 0 0
$$141$$ −5.07180 −0.427122
$$142$$ 0 0
$$143$$ −6.92820 −0.579365
$$144$$ 0 0
$$145$$ 12.0000 0.996546
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −6.53590 −0.531884 −0.265942 0.963989i $$-0.585683\pi$$
−0.265942 + 0.963989i $$0.585683\pi$$
$$152$$ 0 0
$$153$$ −2.46410 −0.199211
$$154$$ 0 0
$$155$$ −21.4641 −1.72404
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ −0.679492 −0.0538872
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.5885 −0.829352 −0.414676 0.909969i $$-0.636105\pi$$
−0.414676 + 0.909969i $$0.636105\pi$$
$$164$$ 0 0
$$165$$ 12.0000 0.934199
$$166$$ 0 0
$$167$$ 11.6603 0.902298 0.451149 0.892449i $$-0.351014\pi$$
0.451149 + 0.892449i $$0.351014\pi$$
$$168$$ 0 0
$$169$$ −10.8564 −0.835108
$$170$$ 0 0
$$171$$ 13.4641 1.02963
$$172$$ 0 0
$$173$$ 8.53590 0.648972 0.324486 0.945890i $$-0.394809\pi$$
0.324486 + 0.945890i $$0.394809\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −6.92820 −0.520756
$$178$$ 0 0
$$179$$ 21.4641 1.60430 0.802151 0.597121i $$-0.203689\pi$$
0.802151 + 0.597121i $$0.203689\pi$$
$$180$$ 0 0
$$181$$ −11.4641 −0.852120 −0.426060 0.904695i $$-0.640099\pi$$
−0.426060 + 0.904695i $$0.640099\pi$$
$$182$$ 0 0
$$183$$ 5.46410 0.403918
$$184$$ 0 0
$$185$$ −39.7128 −2.91974
$$186$$ 0 0
$$187$$ −4.73205 −0.346042
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 7.07180 0.509039 0.254520 0.967068i $$-0.418083\pi$$
0.254520 + 0.967068i $$0.418083\pi$$
$$194$$ 0 0
$$195$$ −3.71281 −0.265880
$$196$$ 0 0
$$197$$ −3.46410 −0.246807 −0.123404 0.992357i $$-0.539381\pi$$
−0.123404 + 0.992357i $$0.539381\pi$$
$$198$$ 0 0
$$199$$ 6.19615 0.439234 0.219617 0.975586i $$-0.429519\pi$$
0.219617 + 0.975586i $$0.429519\pi$$
$$200$$ 0 0
$$201$$ 0.784610 0.0553421
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −20.7846 −1.45166
$$206$$ 0 0
$$207$$ 11.6603 0.810444
$$208$$ 0 0
$$209$$ 25.8564 1.78853
$$210$$ 0 0
$$211$$ 19.6603 1.35347 0.676734 0.736228i $$-0.263395\pi$$
0.676734 + 0.736228i $$0.263395\pi$$
$$212$$ 0 0
$$213$$ 1.60770 0.110157
$$214$$ 0 0
$$215$$ −42.9282 −2.92768
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1.46410 −0.0989348
$$220$$ 0 0
$$221$$ 1.46410 0.0984861
$$222$$ 0 0
$$223$$ 8.39230 0.561990 0.280995 0.959709i $$-0.409336\pi$$
0.280995 + 0.959709i $$0.409336\pi$$
$$224$$ 0 0
$$225$$ −17.2487 −1.14991
$$226$$ 0 0
$$227$$ 9.80385 0.650704 0.325352 0.945593i $$-0.394517\pi$$
0.325352 + 0.945593i $$0.394517\pi$$
$$228$$ 0 0
$$229$$ −0.392305 −0.0259242 −0.0129621 0.999916i $$-0.504126\pi$$
−0.0129621 + 0.999916i $$0.504126\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 24.0000 1.56559
$$236$$ 0 0
$$237$$ −1.32051 −0.0857762
$$238$$ 0 0
$$239$$ −20.7846 −1.34444 −0.672222 0.740349i $$-0.734660\pi$$
−0.672222 + 0.740349i $$0.734660\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 15.2679 0.979439
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ 6.92820 0.439057
$$250$$ 0 0
$$251$$ 6.92820 0.437304 0.218652 0.975803i $$-0.429834\pi$$
0.218652 + 0.975803i $$0.429834\pi$$
$$252$$ 0 0
$$253$$ 22.3923 1.40779
$$254$$ 0 0
$$255$$ −2.53590 −0.158804
$$256$$ 0 0
$$257$$ 7.60770 0.474555 0.237277 0.971442i $$-0.423745\pi$$
0.237277 + 0.971442i $$0.423745\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 8.53590 0.528359
$$262$$ 0 0
$$263$$ −23.3205 −1.43800 −0.719002 0.695008i $$-0.755401\pi$$
−0.719002 + 0.695008i $$0.755401\pi$$
$$264$$ 0 0
$$265$$ 3.21539 0.197520
$$266$$ 0 0
$$267$$ −6.92820 −0.423999
$$268$$ 0 0
$$269$$ 20.5359 1.25210 0.626048 0.779785i $$-0.284671\pi$$
0.626048 + 0.779785i $$0.284671\pi$$
$$270$$ 0 0
$$271$$ −14.9282 −0.906824 −0.453412 0.891301i $$-0.649793\pi$$
−0.453412 + 0.891301i $$0.649793\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −33.1244 −1.99747
$$276$$ 0 0
$$277$$ −14.3923 −0.864750 −0.432375 0.901694i $$-0.642324\pi$$
−0.432375 + 0.901694i $$0.642324\pi$$
$$278$$ 0 0
$$279$$ −15.2679 −0.914068
$$280$$ 0 0
$$281$$ 7.85641 0.468674 0.234337 0.972155i $$-0.424708\pi$$
0.234337 + 0.972155i $$0.424708\pi$$
$$282$$ 0 0
$$283$$ 10.5885 0.629418 0.314709 0.949188i $$-0.398093\pi$$
0.314709 + 0.949188i $$0.398093\pi$$
$$284$$ 0 0
$$285$$ 13.8564 0.820783
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −6.53590 −0.383141
$$292$$ 0 0
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ 32.7846 1.90879
$$296$$ 0 0
$$297$$ 18.9282 1.09833
$$298$$ 0 0
$$299$$ −6.92820 −0.400668
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.92820 0.398015
$$304$$ 0 0
$$305$$ −25.8564 −1.48053
$$306$$ 0 0
$$307$$ −16.7846 −0.957948 −0.478974 0.877829i $$-0.658991\pi$$
−0.478974 + 0.877829i $$0.658991\pi$$
$$308$$ 0 0
$$309$$ −2.14359 −0.121945
$$310$$ 0 0
$$311$$ −21.1244 −1.19785 −0.598926 0.800804i $$-0.704406\pi$$
−0.598926 + 0.800804i $$0.704406\pi$$
$$312$$ 0 0
$$313$$ −7.07180 −0.399722 −0.199861 0.979824i $$-0.564049\pi$$
−0.199861 + 0.979824i $$0.564049\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 15.4641 0.868550 0.434275 0.900780i $$-0.357005\pi$$
0.434275 + 0.900780i $$0.357005\pi$$
$$318$$ 0 0
$$319$$ 16.3923 0.917793
$$320$$ 0 0
$$321$$ 5.32051 0.296962
$$322$$ 0 0
$$323$$ −5.46410 −0.304031
$$324$$ 0 0
$$325$$ 10.2487 0.568496
$$326$$ 0 0
$$327$$ 4.67949 0.258776
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −15.3205 −0.842091 −0.421046 0.907039i $$-0.638337\pi$$
−0.421046 + 0.907039i $$0.638337\pi$$
$$332$$ 0 0
$$333$$ −28.2487 −1.54802
$$334$$ 0 0
$$335$$ −3.71281 −0.202853
$$336$$ 0 0
$$337$$ 34.7846 1.89484 0.947419 0.319995i $$-0.103681\pi$$
0.947419 + 0.319995i $$0.103681\pi$$
$$338$$ 0 0
$$339$$ −5.75129 −0.312367
$$340$$ 0 0
$$341$$ −29.3205 −1.58779
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ 0 0
$$347$$ 4.05256 0.217553 0.108776 0.994066i $$-0.465307\pi$$
0.108776 + 0.994066i $$0.465307\pi$$
$$348$$ 0 0
$$349$$ 30.7846 1.64786 0.823931 0.566690i $$-0.191776\pi$$
0.823931 + 0.566690i $$0.191776\pi$$
$$350$$ 0 0
$$351$$ −5.85641 −0.312592
$$352$$ 0 0
$$353$$ −7.85641 −0.418154 −0.209077 0.977899i $$-0.567046\pi$$
−0.209077 + 0.977899i $$0.567046\pi$$
$$354$$ 0 0
$$355$$ −7.60770 −0.403775
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 11.3205 0.597474 0.298737 0.954336i $$-0.403435\pi$$
0.298737 + 0.954336i $$0.403435\pi$$
$$360$$ 0 0
$$361$$ 10.8564 0.571390
$$362$$ 0 0
$$363$$ 8.33975 0.437723
$$364$$ 0 0
$$365$$ 6.92820 0.362639
$$366$$ 0 0
$$367$$ −19.6603 −1.02626 −0.513128 0.858312i $$-0.671514\pi$$
−0.513128 + 0.858312i $$0.671514\pi$$
$$368$$ 0 0
$$369$$ −14.7846 −0.769656
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 12.3923 0.641649 0.320825 0.947139i $$-0.396040\pi$$
0.320825 + 0.947139i $$0.396040\pi$$
$$374$$ 0 0
$$375$$ −5.07180 −0.261906
$$376$$ 0 0
$$377$$ −5.07180 −0.261211
$$378$$ 0 0
$$379$$ −8.73205 −0.448535 −0.224268 0.974528i $$-0.571999\pi$$
−0.224268 + 0.974528i $$0.571999\pi$$
$$380$$ 0 0
$$381$$ −14.9282 −0.764795
$$382$$ 0 0
$$383$$ 9.46410 0.483593 0.241797 0.970327i $$-0.422263\pi$$
0.241797 + 0.970327i $$0.422263\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −30.5359 −1.55223
$$388$$ 0 0
$$389$$ −25.1769 −1.27652 −0.638260 0.769821i $$-0.720346\pi$$
−0.638260 + 0.769821i $$0.720346\pi$$
$$390$$ 0 0
$$391$$ −4.73205 −0.239310
$$392$$ 0 0
$$393$$ 8.53590 0.430579
$$394$$ 0 0
$$395$$ 6.24871 0.314407
$$396$$ 0 0
$$397$$ 26.3923 1.32459 0.662296 0.749242i $$-0.269582\pi$$
0.662296 + 0.749242i $$0.269582\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4.14359 0.206921 0.103461 0.994634i $$-0.467008\pi$$
0.103461 + 0.994634i $$0.467008\pi$$
$$402$$ 0 0
$$403$$ 9.07180 0.451898
$$404$$ 0 0
$$405$$ −15.4641 −0.768417
$$406$$ 0 0
$$407$$ −54.2487 −2.68901
$$408$$ 0 0
$$409$$ 37.7128 1.86478 0.932389 0.361456i $$-0.117720\pi$$
0.932389 + 0.361456i $$0.117720\pi$$
$$410$$ 0 0
$$411$$ 6.92820 0.341743
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −32.7846 −1.60933
$$416$$ 0 0
$$417$$ 8.24871 0.403941
$$418$$ 0 0
$$419$$ −6.58846 −0.321867 −0.160934 0.986965i $$-0.551450\pi$$
−0.160934 + 0.986965i $$0.551450\pi$$
$$420$$ 0 0
$$421$$ 26.2487 1.27928 0.639642 0.768673i $$-0.279083\pi$$
0.639642 + 0.768673i $$0.279083\pi$$
$$422$$ 0 0
$$423$$ 17.0718 0.830059
$$424$$ 0 0
$$425$$ 7.00000 0.339550
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −5.07180 −0.244869
$$430$$ 0 0
$$431$$ 0.339746 0.0163650 0.00818249 0.999967i $$-0.497395\pi$$
0.00818249 + 0.999967i $$0.497395\pi$$
$$432$$ 0 0
$$433$$ −24.3923 −1.17222 −0.586110 0.810232i $$-0.699341\pi$$
−0.586110 + 0.810232i $$0.699341\pi$$
$$434$$ 0 0
$$435$$ 8.78461 0.421190
$$436$$ 0 0
$$437$$ 25.8564 1.23688
$$438$$ 0 0
$$439$$ 13.8038 0.658822 0.329411 0.944187i $$-0.393150\pi$$
0.329411 + 0.944187i $$0.393150\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 20.7846 0.987507 0.493753 0.869602i $$-0.335625\pi$$
0.493753 + 0.869602i $$0.335625\pi$$
$$444$$ 0 0
$$445$$ 32.7846 1.55414
$$446$$ 0 0
$$447$$ 4.39230 0.207749
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −28.3923 −1.33694
$$452$$ 0 0
$$453$$ −4.78461 −0.224801
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0.392305 0.0183512 0.00917562 0.999958i $$-0.497079\pi$$
0.00917562 + 0.999958i $$0.497079\pi$$
$$458$$ 0 0
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ 14.7846 0.688588 0.344294 0.938862i $$-0.388118\pi$$
0.344294 + 0.938862i $$0.388118\pi$$
$$462$$ 0 0
$$463$$ −29.8564 −1.38754 −0.693772 0.720194i $$-0.744053\pi$$
−0.693772 + 0.720194i $$0.744053\pi$$
$$464$$ 0 0
$$465$$ −15.7128 −0.728664
$$466$$ 0 0
$$467$$ −21.4641 −0.993240 −0.496620 0.867968i $$-0.665426\pi$$
−0.496620 + 0.867968i $$0.665426\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −1.46410 −0.0674622
$$472$$ 0 0
$$473$$ −58.6410 −2.69632
$$474$$ 0 0
$$475$$ −38.2487 −1.75497
$$476$$ 0 0
$$477$$ 2.28719 0.104723
$$478$$ 0 0
$$479$$ 0.339746 0.0155234 0.00776169 0.999970i $$-0.497529\pi$$
0.00776169 + 0.999970i $$0.497529\pi$$
$$480$$ 0 0
$$481$$ 16.7846 0.765312
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 30.9282 1.40438
$$486$$ 0 0
$$487$$ −32.0526 −1.45244 −0.726220 0.687462i $$-0.758725\pi$$
−0.726220 + 0.687462i $$0.758725\pi$$
$$488$$ 0 0
$$489$$ −7.75129 −0.350525
$$490$$ 0 0
$$491$$ 25.1769 1.13622 0.568109 0.822953i $$-0.307675\pi$$
0.568109 + 0.822953i $$0.307675\pi$$
$$492$$ 0 0
$$493$$ −3.46410 −0.156015
$$494$$ 0 0
$$495$$ −40.3923 −1.81550
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −26.9808 −1.20782 −0.603912 0.797051i $$-0.706392\pi$$
−0.603912 + 0.797051i $$0.706392\pi$$
$$500$$ 0 0
$$501$$ 8.53590 0.381356
$$502$$ 0 0
$$503$$ −2.87564 −0.128219 −0.0641093 0.997943i $$-0.520421\pi$$
−0.0641093 + 0.997943i $$0.520421\pi$$
$$504$$ 0 0
$$505$$ −32.7846 −1.45890
$$506$$ 0 0
$$507$$ −7.94744 −0.352958
$$508$$ 0 0
$$509$$ −33.7128 −1.49429 −0.747147 0.664659i $$-0.768577\pi$$
−0.747147 + 0.664659i $$0.768577\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 21.8564 0.964984
$$514$$ 0 0
$$515$$ 10.1436 0.446980
$$516$$ 0 0
$$517$$ 32.7846 1.44187
$$518$$ 0 0
$$519$$ 6.24871 0.274288
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ −28.7846 −1.25866 −0.629332 0.777137i $$-0.716671\pi$$
−0.629332 + 0.777137i $$0.716671\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.19615 0.269909
$$528$$ 0 0
$$529$$ −0.607695 −0.0264215
$$530$$ 0 0
$$531$$ 23.3205 1.01202
$$532$$ 0 0
$$533$$ 8.78461 0.380504
$$534$$ 0 0
$$535$$ −25.1769 −1.08849
$$536$$ 0 0
$$537$$ 15.7128 0.678058
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2.39230 −0.102853 −0.0514266 0.998677i $$-0.516377\pi$$
−0.0514266 + 0.998677i $$0.516377\pi$$
$$542$$ 0 0
$$543$$ −8.39230 −0.360148
$$544$$ 0 0
$$545$$ −22.1436 −0.948527
$$546$$ 0 0
$$547$$ 39.2679 1.67898 0.839488 0.543378i $$-0.182855\pi$$
0.839488 + 0.543378i $$0.182855\pi$$
$$548$$ 0 0
$$549$$ −18.3923 −0.784964
$$550$$ 0 0
$$551$$ 18.9282 0.806369
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −29.0718 −1.23403
$$556$$ 0 0
$$557$$ −33.4641 −1.41792 −0.708960 0.705249i $$-0.750835\pi$$
−0.708960 + 0.705249i $$0.750835\pi$$
$$558$$ 0 0
$$559$$ 18.1436 0.767392
$$560$$ 0 0
$$561$$ −3.46410 −0.146254
$$562$$ 0 0
$$563$$ 0.679492 0.0286372 0.0143186 0.999897i $$-0.495442\pi$$
0.0143186 + 0.999897i $$0.495442\pi$$
$$564$$ 0 0
$$565$$ 27.2154 1.14496
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 19.8564 0.832424 0.416212 0.909268i $$-0.363357\pi$$
0.416212 + 0.909268i $$0.363357\pi$$
$$570$$ 0 0
$$571$$ −13.1244 −0.549237 −0.274619 0.961553i $$-0.588552\pi$$
−0.274619 + 0.961553i $$0.588552\pi$$
$$572$$ 0 0
$$573$$ 8.78461 0.366982
$$574$$ 0 0
$$575$$ −33.1244 −1.38138
$$576$$ 0 0
$$577$$ 6.53590 0.272093 0.136047 0.990702i $$-0.456560\pi$$
0.136047 + 0.990702i $$0.456560\pi$$
$$578$$ 0 0
$$579$$ 5.17691 0.215145
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 4.39230 0.181911
$$584$$ 0 0
$$585$$ 12.4974 0.516705
$$586$$ 0 0
$$587$$ −7.60770 −0.314003 −0.157002 0.987598i $$-0.550183\pi$$
−0.157002 + 0.987598i $$0.550183\pi$$
$$588$$ 0 0
$$589$$ −33.8564 −1.39503
$$590$$ 0 0
$$591$$ −2.53590 −0.104313
$$592$$ 0 0
$$593$$ 31.8564 1.30819 0.654093 0.756414i $$-0.273051\pi$$
0.654093 + 0.756414i $$0.273051\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.53590 0.185642
$$598$$ 0 0
$$599$$ 27.7128 1.13231 0.566157 0.824297i $$-0.308429\pi$$
0.566157 + 0.824297i $$0.308429\pi$$
$$600$$ 0 0
$$601$$ 30.7846 1.25573 0.627865 0.778322i $$-0.283929\pi$$
0.627865 + 0.778322i $$0.283929\pi$$
$$602$$ 0 0
$$603$$ −2.64102 −0.107550
$$604$$ 0 0
$$605$$ −39.4641 −1.60444
$$606$$ 0 0
$$607$$ 35.2679 1.43148 0.715741 0.698366i $$-0.246089\pi$$
0.715741 + 0.698366i $$0.246089\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −10.1436 −0.410366
$$612$$ 0 0
$$613$$ −20.1436 −0.813592 −0.406796 0.913519i $$-0.633354\pi$$
−0.406796 + 0.913519i $$0.633354\pi$$
$$614$$ 0 0
$$615$$ −15.2154 −0.613544
$$616$$ 0 0
$$617$$ 47.5692 1.91506 0.957532 0.288326i $$-0.0930987\pi$$
0.957532 + 0.288326i $$0.0930987\pi$$
$$618$$ 0 0
$$619$$ −22.1962 −0.892139 −0.446069 0.894998i $$-0.647177\pi$$
−0.446069 + 0.894998i $$0.647177\pi$$
$$620$$ 0 0
$$621$$ 18.9282 0.759563
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 18.9282 0.755920
$$628$$ 0 0
$$629$$ 11.4641 0.457104
$$630$$ 0 0
$$631$$ 34.5359 1.37485 0.687426 0.726254i $$-0.258740\pi$$
0.687426 + 0.726254i $$0.258740\pi$$
$$632$$ 0 0
$$633$$ 14.3923 0.572043
$$634$$ 0 0
$$635$$ 70.6410 2.80330
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −5.41154 −0.214077
$$640$$ 0 0
$$641$$ 0.928203 0.0366618 0.0183309 0.999832i $$-0.494165\pi$$
0.0183309 + 0.999832i $$0.494165\pi$$
$$642$$ 0 0
$$643$$ −1.41154 −0.0556658 −0.0278329 0.999613i $$-0.508861\pi$$
−0.0278329 + 0.999613i $$0.508861\pi$$
$$644$$ 0 0
$$645$$ −31.4256 −1.23738
$$646$$ 0 0
$$647$$ −30.9282 −1.21591 −0.607957 0.793970i $$-0.708011\pi$$
−0.607957 + 0.793970i $$0.708011\pi$$
$$648$$ 0 0
$$649$$ 44.7846 1.75795
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 10.3923 0.406682 0.203341 0.979108i $$-0.434820\pi$$
0.203341 + 0.979108i $$0.434820\pi$$
$$654$$ 0 0
$$655$$ −40.3923 −1.57826
$$656$$ 0 0
$$657$$ 4.92820 0.192268
$$658$$ 0 0
$$659$$ 29.0718 1.13248 0.566238 0.824242i $$-0.308398\pi$$
0.566238 + 0.824242i $$0.308398\pi$$
$$660$$ 0 0
$$661$$ 28.9282 1.12518 0.562588 0.826737i $$-0.309806\pi$$
0.562588 + 0.826737i $$0.309806\pi$$
$$662$$ 0 0
$$663$$ 1.07180 0.0416251
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.3923 0.634713
$$668$$ 0 0
$$669$$ 6.14359 0.237525
$$670$$ 0 0
$$671$$ −35.3205 −1.36353
$$672$$ 0 0
$$673$$ 27.8564 1.07379 0.536893 0.843650i $$-0.319598\pi$$
0.536893 + 0.843650i $$0.319598\pi$$
$$674$$ 0 0
$$675$$ −28.0000 −1.07772
$$676$$ 0 0
$$677$$ −24.2487 −0.931954 −0.465977 0.884797i $$-0.654297\pi$$
−0.465977 + 0.884797i $$0.654297\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 7.17691 0.275020
$$682$$ 0 0
$$683$$ −7.94744 −0.304100 −0.152050 0.988373i $$-0.548588\pi$$
−0.152050 + 0.988373i $$0.548588\pi$$
$$684$$ 0 0
$$685$$ −32.7846 −1.25264
$$686$$ 0 0
$$687$$ −0.287187 −0.0109569
$$688$$ 0 0
$$689$$ −1.35898 −0.0517732
$$690$$ 0 0
$$691$$ 25.8038 0.981625 0.490812 0.871265i $$-0.336700\pi$$
0.490812 + 0.871265i $$0.336700\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −39.0333 −1.48062
$$696$$ 0 0
$$697$$ 6.00000 0.227266
$$698$$ 0 0
$$699$$ 4.39230 0.166132
$$700$$ 0 0
$$701$$ 4.39230 0.165895 0.0829475 0.996554i $$-0.473567\pi$$
0.0829475 + 0.996554i $$0.473567\pi$$
$$702$$ 0 0
$$703$$ −62.6410 −2.36255
$$704$$ 0 0
$$705$$ 17.5692 0.661695
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 32.2487 1.21113 0.605563 0.795797i $$-0.292948\pi$$
0.605563 + 0.795797i $$0.292948\pi$$
$$710$$ 0 0
$$711$$ 4.44486 0.166695
$$712$$ 0 0
$$713$$ −29.3205 −1.09806
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ 0 0
$$717$$ −15.2154 −0.568229
$$718$$ 0 0
$$719$$ 1.51666 0.0565619 0.0282809 0.999600i $$-0.490997\pi$$
0.0282809 + 0.999600i $$0.490997\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −1.46410 −0.0544505
$$724$$ 0 0
$$725$$ −24.2487 −0.900575
$$726$$ 0 0
$$727$$ 36.7846 1.36427 0.682133 0.731228i $$-0.261053\pi$$
0.682133 + 0.731228i $$0.261053\pi$$
$$728$$ 0 0
$$729$$ −2.21539 −0.0820515
$$730$$ 0 0
$$731$$ 12.3923 0.458346
$$732$$ 0 0
$$733$$ −22.7846 −0.841569 −0.420784 0.907161i $$-0.638245\pi$$
−0.420784 + 0.907161i $$0.638245\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −5.07180 −0.186822
$$738$$ 0 0
$$739$$ −8.39230 −0.308716 −0.154358 0.988015i $$-0.549331\pi$$
−0.154358 + 0.988015i $$0.549331\pi$$
$$740$$ 0 0
$$741$$ −5.85641 −0.215140
$$742$$ 0 0
$$743$$ −41.9090 −1.53749 −0.768745 0.639555i $$-0.779119\pi$$
−0.768745 + 0.639555i $$0.779119\pi$$
$$744$$ 0 0
$$745$$ −20.7846 −0.761489
$$746$$ 0 0
$$747$$ −23.3205 −0.853253
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 15.2679 0.557135 0.278568 0.960417i $$-0.410140\pi$$
0.278568 + 0.960417i $$0.410140\pi$$
$$752$$ 0 0
$$753$$ 5.07180 0.184827
$$754$$ 0 0
$$755$$ 22.6410 0.823991
$$756$$ 0 0
$$757$$ 10.5359 0.382934 0.191467 0.981499i $$-0.438676\pi$$
0.191467 + 0.981499i $$0.438676\pi$$
$$758$$ 0 0
$$759$$ 16.3923 0.595003
$$760$$ 0 0
$$761$$ 4.39230 0.159221 0.0796105 0.996826i $$-0.474632\pi$$
0.0796105 + 0.996826i $$0.474632\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 8.53590 0.308616
$$766$$ 0 0
$$767$$ −13.8564 −0.500326
$$768$$ 0 0
$$769$$ 20.3923 0.735365 0.367683 0.929951i $$-0.380151\pi$$
0.367683 + 0.929951i $$0.380151\pi$$
$$770$$ 0 0
$$771$$ 5.56922 0.200571
$$772$$ 0 0
$$773$$ 35.3205 1.27039 0.635195 0.772352i $$-0.280920\pi$$
0.635195 + 0.772352i $$0.280920\pi$$
$$774$$ 0 0
$$775$$ 43.3731 1.55801
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −32.7846 −1.17463
$$780$$ 0 0
$$781$$ −10.3923 −0.371866
$$782$$ 0 0
$$783$$ 13.8564 0.495188
$$784$$ 0 0
$$785$$ 6.92820 0.247278
$$786$$ 0 0
$$787$$ 1.12436 0.0400790 0.0200395 0.999799i $$-0.493621\pi$$
0.0200395 + 0.999799i $$0.493621\pi$$
$$788$$ 0 0
$$789$$ −17.0718 −0.607772
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 10.9282 0.388072
$$794$$ 0 0
$$795$$ 2.35383 0.0834817
$$796$$ 0 0
$$797$$ −38.7846 −1.37382 −0.686911 0.726742i $$-0.741034\pi$$
−0.686911 + 0.726742i $$0.741034\pi$$
$$798$$ 0 0
$$799$$ −6.92820 −0.245102
$$800$$ 0 0
$$801$$ 23.3205 0.823990
$$802$$ 0 0
$$803$$ 9.46410 0.333981
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 15.0333 0.529198
$$808$$ 0 0
$$809$$ 19.8564 0.698114 0.349057 0.937101i $$-0.386502\pi$$
0.349057 + 0.937101i $$0.386502\pi$$
$$810$$ 0 0
$$811$$ −35.3731 −1.24212 −0.621058 0.783764i $$-0.713297\pi$$
−0.621058 + 0.783764i $$0.713297\pi$$
$$812$$ 0 0
$$813$$ −10.9282 −0.383269
$$814$$ 0 0
$$815$$ 36.6795 1.28483
$$816$$ 0 0
$$817$$ −67.7128 −2.36897
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −45.0333 −1.57167 −0.785837 0.618434i $$-0.787767\pi$$
−0.785837 + 0.618434i $$0.787767\pi$$
$$822$$ 0 0
$$823$$ −15.6603 −0.545882 −0.272941 0.962031i $$-0.587996\pi$$
−0.272941 + 0.962031i $$0.587996\pi$$
$$824$$ 0 0
$$825$$ −24.2487 −0.844232
$$826$$ 0 0
$$827$$ 25.5167 0.887301 0.443651 0.896200i $$-0.353683\pi$$
0.443651 + 0.896200i $$0.353683\pi$$
$$828$$ 0 0
$$829$$ −12.1436 −0.421764 −0.210882 0.977511i $$-0.567634\pi$$
−0.210882 + 0.977511i $$0.567634\pi$$
$$830$$ 0 0
$$831$$ −10.5359 −0.365486
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −40.3923 −1.39783
$$836$$ 0 0
$$837$$ −24.7846 −0.856681
$$838$$ 0 0
$$839$$ −49.5167 −1.70950 −0.854752 0.519036i $$-0.826291\pi$$
−0.854752 + 0.519036i $$0.826291\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ 0 0
$$843$$ 5.75129 0.198085
$$844$$ 0 0
$$845$$ 37.6077 1.29374
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 7.75129 0.266024
$$850$$ 0 0
$$851$$ −54.2487 −1.85962
$$852$$ 0 0
$$853$$ 40.2487 1.37809 0.689045 0.724719i $$-0.258030\pi$$
0.689045 + 0.724719i $$0.258030\pi$$
$$854$$ 0 0
$$855$$ −46.6410 −1.59509
$$856$$ 0 0
$$857$$ 52.6410 1.79818 0.899091 0.437761i $$-0.144228\pi$$
0.899091 + 0.437761i $$0.144228\pi$$
$$858$$ 0 0
$$859$$ −7.32051 −0.249773 −0.124886 0.992171i $$-0.539857\pi$$
−0.124886 + 0.992171i $$0.539857\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 37.8564 1.28865 0.644324 0.764753i $$-0.277139\pi$$
0.644324 + 0.764753i $$0.277139\pi$$
$$864$$ 0 0
$$865$$ −29.5692 −1.00538
$$866$$ 0 0
$$867$$ 0.732051 0.0248617
$$868$$ 0 0
$$869$$ 8.53590 0.289561
$$870$$ 0 0
$$871$$ 1.56922 0.0531710
$$872$$ 0 0
$$873$$ 22.0000 0.744587
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30.3923 1.02628 0.513138 0.858306i $$-0.328483\pi$$
0.513138 + 0.858306i $$0.328483\pi$$
$$878$$ 0 0
$$879$$ −21.9615 −0.740744
$$880$$ 0 0
$$881$$ −2.78461 −0.0938159 −0.0469079 0.998899i $$-0.514937\pi$$
−0.0469079 + 0.998899i $$0.514937\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 0 0
$$885$$ 24.0000 0.806751
$$886$$ 0 0
$$887$$ −42.5885 −1.42998 −0.714990 0.699134i $$-0.753569\pi$$
−0.714990 + 0.699134i $$0.753569\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −21.1244 −0.707693
$$892$$ 0 0
$$893$$ 37.8564 1.26682
$$894$$ 0 0
$$895$$ −74.3538 −2.48537
$$896$$ 0 0
$$897$$ −5.07180 −0.169342
$$898$$ 0 0
$$899$$ −21.4641 −0.715868
$$900$$ 0 0
$$901$$ −0.928203 −0.0309229
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 39.7128 1.32010
$$906$$ 0 0
$$907$$ −24.4449 −0.811678 −0.405839 0.913945i $$-0.633021\pi$$
−0.405839 + 0.913945i $$0.633021\pi$$
$$908$$ 0 0
$$909$$ −23.3205 −0.773492
$$910$$ 0 0
$$911$$ −26.1962 −0.867917 −0.433959 0.900933i $$-0.642884\pi$$
−0.433959 + 0.900933i $$0.642884\pi$$
$$912$$ 0 0
$$913$$ −44.7846 −1.48215
$$914$$ 0 0
$$915$$ −18.9282 −0.625747
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −52.0000 −1.71532 −0.857661 0.514216i $$-0.828083\pi$$
−0.857661 + 0.514216i $$0.828083\pi$$
$$920$$ 0 0
$$921$$ −12.2872 −0.404877
$$922$$ 0 0
$$923$$ 3.21539 0.105836
$$924$$ 0 0
$$925$$ 80.2487 2.63856
$$926$$ 0 0
$$927$$ 7.21539 0.236985
$$928$$ 0 0
$$929$$ −45.7128 −1.49979 −0.749894 0.661558i $$-0.769896\pi$$
−0.749894 + 0.661558i $$0.769896\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −15.4641 −0.506272
$$934$$ 0 0
$$935$$ 16.3923 0.536086
$$936$$ 0 0
$$937$$ −27.8564 −0.910029 −0.455015 0.890484i $$-0.650366\pi$$
−0.455015 + 0.890484i $$0.650366\pi$$
$$938$$ 0 0
$$939$$ −5.17691 −0.168942
$$940$$ 0 0
$$941$$ −6.67949 −0.217745 −0.108873 0.994056i $$-0.534724\pi$$
−0.108873 + 0.994056i $$0.534724\pi$$
$$942$$ 0 0
$$943$$ −28.3923 −0.924581
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 52.7321 1.71356 0.856781 0.515681i $$-0.172461\pi$$
0.856781 + 0.515681i $$0.172461\pi$$
$$948$$ 0 0
$$949$$ −2.92820 −0.0950535
$$950$$ 0 0
$$951$$ 11.3205 0.367093
$$952$$ 0 0
$$953$$ −42.2487 −1.36857 −0.684285 0.729215i $$-0.739886\pi$$
−0.684285 + 0.729215i $$0.739886\pi$$
$$954$$ 0 0
$$955$$ −41.5692 −1.34515
$$956$$ 0 0
$$957$$ 12.0000 0.387905
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 7.39230 0.238461
$$962$$ 0 0
$$963$$ −17.9090 −0.577108
$$964$$ 0 0
$$965$$ −24.4974 −0.788600
$$966$$ 0 0
$$967$$ 48.3923 1.55619 0.778096 0.628146i $$-0.216186\pi$$
0.778096 + 0.628146i $$0.216186\pi$$
$$968$$ 0 0
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ 56.1051 1.80050 0.900249 0.435374i $$-0.143384\pi$$
0.900249 + 0.435374i $$0.143384\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 7.50258 0.240275
$$976$$ 0 0
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ 44.7846 1.43132
$$980$$ 0 0
$$981$$ −15.7513 −0.502900
$$982$$ 0 0
$$983$$ −29.9090 −0.953948 −0.476974 0.878917i $$-0.658266\pi$$
−0.476974 + 0.878917i $$0.658266\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −58.6410 −1.86468
$$990$$ 0 0
$$991$$ 10.1962 0.323891 0.161946 0.986800i $$-0.448223\pi$$
0.161946 + 0.986800i $$0.448223\pi$$
$$992$$ 0 0
$$993$$ −11.2154 −0.355910
$$994$$ 0 0
$$995$$ −21.4641 −0.680458
$$996$$ 0 0
$$997$$ −42.3923 −1.34258 −0.671289 0.741196i $$-0.734259\pi$$
−0.671289 + 0.741196i $$0.734259\pi$$
$$998$$ 0 0
$$999$$ −45.8564 −1.45083
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 3332.2.a.h.1.2 2
7.6 odd 2 68.2.a.a.1.1 2
21.20 even 2 612.2.a.e.1.1 2
28.27 even 2 272.2.a.e.1.2 2
35.13 even 4 1700.2.e.c.749.2 4
35.27 even 4 1700.2.e.c.749.3 4
35.34 odd 2 1700.2.a.d.1.2 2
56.13 odd 2 1088.2.a.p.1.2 2
56.27 even 2 1088.2.a.t.1.1 2
77.76 even 2 8228.2.a.k.1.1 2
84.83 odd 2 2448.2.a.y.1.1 2
119.6 even 16 1156.2.h.f.733.2 16
119.13 odd 4 1156.2.b.c.577.2 4
119.20 even 16 1156.2.h.f.757.2 16
119.27 even 16 1156.2.h.f.1001.2 16
119.41 even 16 1156.2.h.f.1001.3 16
119.48 even 16 1156.2.h.f.757.3 16
119.55 odd 4 1156.2.b.c.577.3 4
119.62 even 16 1156.2.h.f.733.3 16
119.76 odd 8 1156.2.e.d.829.3 8
119.83 odd 8 1156.2.e.d.905.3 8
119.90 even 16 1156.2.h.f.977.3 16
119.97 even 16 1156.2.h.f.977.2 16
119.104 odd 8 1156.2.e.d.905.2 8
119.111 odd 8 1156.2.e.d.829.2 8
119.118 odd 2 1156.2.a.a.1.2 2
140.139 even 2 6800.2.a.bh.1.1 2
168.83 odd 2 9792.2.a.cs.1.2 2
168.125 even 2 9792.2.a.cr.1.2 2
476.475 even 2 4624.2.a.x.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
68.2.a.a.1.1 2 7.6 odd 2
272.2.a.e.1.2 2 28.27 even 2
612.2.a.e.1.1 2 21.20 even 2
1088.2.a.p.1.2 2 56.13 odd 2
1088.2.a.t.1.1 2 56.27 even 2
1156.2.a.a.1.2 2 119.118 odd 2
1156.2.b.c.577.2 4 119.13 odd 4
1156.2.b.c.577.3 4 119.55 odd 4
1156.2.e.d.829.2 8 119.111 odd 8
1156.2.e.d.829.3 8 119.76 odd 8
1156.2.e.d.905.2 8 119.104 odd 8
1156.2.e.d.905.3 8 119.83 odd 8
1156.2.h.f.733.2 16 119.6 even 16
1156.2.h.f.733.3 16 119.62 even 16
1156.2.h.f.757.2 16 119.20 even 16
1156.2.h.f.757.3 16 119.48 even 16
1156.2.h.f.977.2 16 119.97 even 16
1156.2.h.f.977.3 16 119.90 even 16
1156.2.h.f.1001.2 16 119.27 even 16
1156.2.h.f.1001.3 16 119.41 even 16
1700.2.a.d.1.2 2 35.34 odd 2
1700.2.e.c.749.2 4 35.13 even 4
1700.2.e.c.749.3 4 35.27 even 4
2448.2.a.y.1.1 2 84.83 odd 2
3332.2.a.h.1.2 2 1.1 even 1 trivial
4624.2.a.x.1.1 2 476.475 even 2
6800.2.a.bh.1.1 2 140.139 even 2
8228.2.a.k.1.1 2 77.76 even 2
9792.2.a.cr.1.2 2 168.125 even 2
9792.2.a.cs.1.2 2 168.83 odd 2