# Properties

 Label 3332.2.a.h.1.1 Level $3332$ Weight $2$ Character 3332.1 Self dual yes Analytic conductor $26.606$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

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 1.1 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)

 $$f(q)$$ $$=$$ $$q-2.73205 q^{3} +3.46410 q^{5} +4.46410 q^{9} +O(q^{10})$$ $$q-2.73205 q^{3} +3.46410 q^{5} +4.46410 q^{9} -1.26795 q^{11} -5.46410 q^{13} -9.46410 q^{15} +1.00000 q^{17} +1.46410 q^{19} -1.26795 q^{23} +7.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -4.19615 q^{31} +3.46410 q^{33} +4.53590 q^{37} +14.9282 q^{39} +6.00000 q^{41} -8.39230 q^{43} +15.4641 q^{45} +6.92820 q^{47} -2.73205 q^{51} +12.9282 q^{53} -4.39230 q^{55} -4.00000 q^{57} -2.53590 q^{59} +0.535898 q^{61} -18.9282 q^{65} +14.9282 q^{67} +3.46410 q^{69} -8.19615 q^{71} -2.00000 q^{73} -19.1244 q^{75} -12.1962 q^{79} -2.46410 q^{81} +2.53590 q^{83} +3.46410 q^{85} -9.46410 q^{87} -2.53590 q^{89} +11.4641 q^{93} +5.07180 q^{95} +4.92820 q^{97} -5.66025 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 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
$$3$$ −2.73205 −1.57735 −0.788675 0.614810i $$-0.789233\pi$$
−0.788675 + 0.614810i $$0.789233\pi$$
$$4$$ 0 0
$$5$$ 3.46410 1.54919 0.774597 0.632456i $$-0.217953\pi$$
0.774597 + 0.632456i $$0.217953\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 4.46410 1.48803
$$10$$ 0 0
$$11$$ −1.26795 −0.382301 −0.191151 0.981561i $$-0.561222\pi$$
−0.191151 + 0.981561i $$0.561222\pi$$
$$12$$ 0 0
$$13$$ −5.46410 −1.51547 −0.757735 0.652563i $$-0.773694\pi$$
−0.757735 + 0.652563i $$0.773694\pi$$
$$14$$ 0 0
$$15$$ −9.46410 −2.44362
$$16$$ 0 0
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ 1.46410 0.335888 0.167944 0.985797i $$-0.446287\pi$$
0.167944 + 0.985797i $$0.446287\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.26795 −0.264386 −0.132193 0.991224i $$-0.542202\pi$$
−0.132193 + 0.991224i $$0.542202\pi$$
$$24$$ 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 $$-0.395768\pi$$
0.321634 + 0.946864i $$0.395768\pi$$
$$30$$ 0 0
$$31$$ −4.19615 −0.753651 −0.376826 0.926284i $$-0.622984\pi$$
−0.376826 + 0.926284i $$0.622984\pi$$
$$32$$ 0 0
$$33$$ 3.46410 0.603023
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.53590 0.745697 0.372849 0.927892i $$-0.378381\pi$$
0.372849 + 0.927892i $$0.378381\pi$$
$$38$$ 0 0
$$39$$ 14.9282 2.39043
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −8.39230 −1.27981 −0.639907 0.768452i $$-0.721027\pi$$
−0.639907 + 0.768452i $$0.721027\pi$$
$$44$$ 0 0
$$45$$ 15.4641 2.30525
$$46$$ 0 0
$$47$$ 6.92820 1.01058 0.505291 0.862949i $$-0.331385\pi$$
0.505291 + 0.862949i $$0.331385\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.73205 −0.382564
$$52$$ 0 0
$$53$$ 12.9282 1.77583 0.887913 0.460012i $$-0.152155\pi$$
0.887913 + 0.460012i $$0.152155\pi$$
$$54$$ 0 0
$$55$$ −4.39230 −0.592258
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ −2.53590 −0.330146 −0.165073 0.986281i $$-0.552786\pi$$
−0.165073 + 0.986281i $$0.552786\pi$$
$$60$$ 0 0
$$61$$ 0.535898 0.0686148 0.0343074 0.999411i $$-0.489077\pi$$
0.0343074 + 0.999411i $$0.489077\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −18.9282 −2.34775
$$66$$ 0 0
$$67$$ 14.9282 1.82377 0.911885 0.410445i $$-0.134627\pi$$
0.911885 + 0.410445i $$0.134627\pi$$
$$68$$ 0 0
$$69$$ 3.46410 0.417029
$$70$$ 0 0
$$71$$ −8.19615 −0.972704 −0.486352 0.873763i $$-0.661673\pi$$
−0.486352 + 0.873763i $$0.661673\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ −19.1244 −2.20829
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.1962 −1.37217 −0.686087 0.727519i $$-0.740673\pi$$
−0.686087 + 0.727519i $$0.740673\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ 0 0
$$83$$ 2.53590 0.278351 0.139176 0.990268i $$-0.455555\pi$$
0.139176 + 0.990268i $$0.455555\pi$$
$$84$$ 0 0
$$85$$ 3.46410 0.375735
$$86$$ 0 0
$$87$$ −9.46410 −1.01466
$$88$$ 0 0
$$89$$ −2.53590 −0.268805 −0.134402 0.990927i $$-0.542911\pi$$
−0.134402 + 0.990927i $$0.542911\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 11.4641 1.18877
$$94$$ 0 0
$$95$$ 5.07180 0.520355
$$96$$ 0 0
$$97$$ 4.92820 0.500383 0.250192 0.968196i $$-0.419506\pi$$
0.250192 + 0.968196i $$0.419506\pi$$
$$98$$ 0 0
$$99$$ −5.66025 −0.568877
$$100$$ 0 0
$$101$$ 2.53590 0.252331 0.126166 0.992009i $$-0.459733\pi$$
0.126166 + 0.992009i $$0.459733\pi$$
$$102$$ 0 0
$$103$$ 10.9282 1.07679 0.538394 0.842693i $$-0.319031\pi$$
0.538394 + 0.842693i $$0.319031\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 10.7321 1.03751 0.518753 0.854924i $$-0.326396\pi$$
0.518753 + 0.854924i $$0.326396\pi$$
$$108$$ 0 0
$$109$$ −14.3923 −1.37853 −0.689266 0.724508i $$-0.742067\pi$$
−0.689266 + 0.724508i $$0.742067\pi$$
$$110$$ 0 0
$$111$$ −12.3923 −1.17623
$$112$$ 0 0
$$113$$ 19.8564 1.86793 0.933967 0.357360i $$-0.116323\pi$$
0.933967 + 0.357360i $$0.116323\pi$$
$$114$$ 0 0
$$115$$ −4.39230 −0.409585
$$116$$ 0 0
$$117$$ −24.3923 −2.25507
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −9.39230 −0.853846
$$122$$ 0 0
$$123$$ −16.3923 −1.47804
$$124$$ 0 0
$$125$$ 6.92820 0.619677
$$126$$ 0 0
$$127$$ 0.392305 0.0348114 0.0174057 0.999849i $$-0.494459\pi$$
0.0174057 + 0.999849i $$0.494459\pi$$
$$128$$ 0 0
$$129$$ 22.9282 2.01872
$$130$$ 0 0
$$131$$ −5.66025 −0.494539 −0.247269 0.968947i $$-0.579533\pi$$
−0.247269 + 0.968947i $$0.579533\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −13.8564 −1.19257
$$136$$ 0 0
$$137$$ 2.53590 0.216656 0.108328 0.994115i $$-0.465450\pi$$
0.108328 + 0.994115i $$0.465450\pi$$
$$138$$ 0 0
$$139$$ 14.7321 1.24956 0.624778 0.780802i $$-0.285189\pi$$
0.624778 + 0.780802i $$0.285189\pi$$
$$140$$ 0 0
$$141$$ −18.9282 −1.59404
$$142$$ 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
$$151$$ −13.4641 −1.09569 −0.547847 0.836579i $$-0.684552\pi$$
−0.547847 + 0.836579i $$0.684552\pi$$
$$152$$ 0 0
$$153$$ 4.46410 0.360901
$$154$$ 0 0
$$155$$ −14.5359 −1.16755
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ −35.3205 −2.80110
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.5885 1.61261 0.806306 0.591498i $$-0.201463\pi$$
0.806306 + 0.591498i $$0.201463\pi$$
$$164$$ 0 0
$$165$$ 12.0000 0.934199
$$166$$ 0 0
$$167$$ −5.66025 −0.438004 −0.219002 0.975724i $$-0.570280\pi$$
−0.219002 + 0.975724i $$0.570280\pi$$
$$168$$ 0 0
$$169$$ 16.8564 1.29665
$$170$$ 0 0
$$171$$ 6.53590 0.499813
$$172$$ 0 0
$$173$$ 15.4641 1.17571 0.587857 0.808965i $$-0.299972\pi$$
0.587857 + 0.808965i $$0.299972\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.92820 0.520756
$$178$$ 0 0
$$179$$ 14.5359 1.08646 0.543232 0.839583i $$-0.317200\pi$$
0.543232 + 0.839583i $$0.317200\pi$$
$$180$$ 0 0
$$181$$ −4.53590 −0.337151 −0.168575 0.985689i $$-0.553917\pi$$
−0.168575 + 0.985689i $$0.553917\pi$$
$$182$$ 0 0
$$183$$ −1.46410 −0.108230
$$184$$ 0 0
$$185$$ 15.7128 1.15523
$$186$$ 0 0
$$187$$ −1.26795 −0.0927216
$$188$$ 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
$$193$$ 20.9282 1.50645 0.753223 0.657766i $$-0.228498\pi$$
0.753223 + 0.657766i $$0.228498\pi$$
$$194$$ 0 0
$$195$$ 51.7128 3.70323
$$196$$ 0 0
$$197$$ 3.46410 0.246807 0.123404 0.992357i $$-0.460619\pi$$
0.123404 + 0.992357i $$0.460619\pi$$
$$198$$ 0 0
$$199$$ −4.19615 −0.297457 −0.148729 0.988878i $$-0.547518\pi$$
−0.148729 + 0.988878i $$0.547518\pi$$
$$200$$ 0 0
$$201$$ −40.7846 −2.87672
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 20.7846 1.45166
$$206$$ 0 0
$$207$$ −5.66025 −0.393415
$$208$$ 0 0
$$209$$ −1.85641 −0.128410
$$210$$ 0 0
$$211$$ 2.33975 0.161075 0.0805374 0.996752i $$-0.474336\pi$$
0.0805374 + 0.996752i $$0.474336\pi$$
$$212$$ 0 0
$$213$$ 22.3923 1.53430
$$214$$ 0 0
$$215$$ −29.0718 −1.98268
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 5.46410 0.369230
$$220$$ 0 0
$$221$$ −5.46410 −0.367555
$$222$$ 0 0
$$223$$ −12.3923 −0.829850 −0.414925 0.909856i $$-0.636192\pi$$
−0.414925 + 0.909856i $$0.636192\pi$$
$$224$$ 0 0
$$225$$ 31.2487 2.08325
$$226$$ 0 0
$$227$$ 20.1962 1.34047 0.670233 0.742151i $$-0.266194\pi$$
0.670233 + 0.742151i $$0.266194\pi$$
$$228$$ 0 0
$$229$$ 20.3923 1.34756 0.673781 0.738931i $$-0.264669\pi$$
0.673781 + 0.738931i $$0.264669\pi$$
$$230$$ 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
$$237$$ 33.3205 2.16440
$$238$$ 0 0
$$239$$ 20.7846 1.34444 0.672222 0.740349i $$-0.265340\pi$$
0.672222 + 0.740349i $$0.265340\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 18.7321 1.20166
$$244$$ 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 $$-0.570166\pi$$
−0.218652 + 0.975803i $$0.570166\pi$$
$$252$$ 0 0
$$253$$ 1.60770 0.101075
$$254$$ 0 0
$$255$$ −9.46410 −0.592665
$$256$$ 0 0
$$257$$ 28.3923 1.77106 0.885532 0.464579i $$-0.153794\pi$$
0.885532 + 0.464579i $$0.153794\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 15.4641 0.957204
$$262$$ 0 0
$$263$$ 11.3205 0.698052 0.349026 0.937113i $$-0.386512\pi$$
0.349026 + 0.937113i $$0.386512\pi$$
$$264$$ 0 0
$$265$$ 44.7846 2.75110
$$266$$ 0 0
$$267$$ 6.92820 0.423999
$$268$$ 0 0
$$269$$ 27.4641 1.67452 0.837258 0.546808i $$-0.184157\pi$$
0.837258 + 0.546808i $$0.184157\pi$$
$$270$$ 0 0
$$271$$ −1.07180 −0.0651070 −0.0325535 0.999470i $$-0.510364\pi$$
−0.0325535 + 0.999470i $$0.510364\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −8.87564 −0.535221
$$276$$ 0 0
$$277$$ 6.39230 0.384076 0.192038 0.981387i $$-0.438490\pi$$
0.192038 + 0.981387i $$0.438490\pi$$
$$278$$ 0 0
$$279$$ −18.7321 −1.12146
$$280$$ 0 0
$$281$$ −19.8564 −1.18453 −0.592267 0.805742i $$-0.701767\pi$$
−0.592267 + 0.805742i $$0.701767\pi$$
$$282$$ 0 0
$$283$$ −20.5885 −1.22386 −0.611928 0.790913i $$-0.709606\pi$$
−0.611928 + 0.790913i $$0.709606\pi$$
$$284$$ 0 0
$$285$$ −13.8564 −0.820783
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −13.4641 −0.789280
$$292$$ 0 0
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ −8.78461 −0.511460
$$296$$ 0 0
$$297$$ 5.07180 0.294295
$$298$$ 0 0
$$299$$ 6.92820 0.400668
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −6.92820 −0.398015
$$304$$ 0 0
$$305$$ 1.85641 0.106298
$$306$$ 0 0
$$307$$ 24.7846 1.41453 0.707266 0.706947i $$-0.249928\pi$$
0.707266 + 0.706947i $$0.249928\pi$$
$$308$$ 0 0
$$309$$ −29.8564 −1.69847
$$310$$ 0 0
$$311$$ 3.12436 0.177166 0.0885830 0.996069i $$-0.471766\pi$$
0.0885830 + 0.996069i $$0.471766\pi$$
$$312$$ 0 0
$$313$$ −20.9282 −1.18293 −0.591466 0.806330i $$-0.701451\pi$$
−0.591466 + 0.806330i $$0.701451\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.53590 0.479424 0.239712 0.970844i $$-0.422947\pi$$
0.239712 + 0.970844i $$0.422947\pi$$
$$318$$ 0 0
$$319$$ −4.39230 −0.245922
$$320$$ 0 0
$$321$$ −29.3205 −1.63651
$$322$$ 0 0
$$323$$ 1.46410 0.0814648
$$324$$ 0 0
$$325$$ −38.2487 −2.12166
$$326$$ 0 0
$$327$$ 39.3205 2.17443
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 19.3205 1.06195 0.530976 0.847387i $$-0.321826\pi$$
0.530976 + 0.847387i $$0.321826\pi$$
$$332$$ 0 0
$$333$$ 20.2487 1.10962
$$334$$ 0 0
$$335$$ 51.7128 2.82537
$$336$$ 0 0
$$337$$ −6.78461 −0.369581 −0.184791 0.982778i $$-0.559161\pi$$
−0.184791 + 0.982778i $$0.559161\pi$$
$$338$$ 0 0
$$339$$ −54.2487 −2.94639
$$340$$ 0 0
$$341$$ 5.32051 0.288122
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ 0 0
$$347$$ −34.0526 −1.82804 −0.914019 0.405672i $$-0.867037\pi$$
−0.914019 + 0.405672i $$0.867037\pi$$
$$348$$ 0 0
$$349$$ −10.7846 −0.577287 −0.288643 0.957437i $$-0.593204\pi$$
−0.288643 + 0.957437i $$0.593204\pi$$
$$350$$ 0 0
$$351$$ 21.8564 1.16661
$$352$$ 0 0
$$353$$ 19.8564 1.05685 0.528425 0.848980i $$-0.322783\pi$$
0.528425 + 0.848980i $$0.322783\pi$$
$$354$$ 0 0
$$355$$ −28.3923 −1.50691
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −23.3205 −1.23081 −0.615405 0.788211i $$-0.711007\pi$$
−0.615405 + 0.788211i $$0.711007\pi$$
$$360$$ 0 0
$$361$$ −16.8564 −0.887179
$$362$$ 0 0
$$363$$ 25.6603 1.34681
$$364$$ 0 0
$$365$$ −6.92820 −0.362639
$$366$$ 0 0
$$367$$ −2.33975 −0.122134 −0.0610669 0.998134i $$-0.519450\pi$$
−0.0610669 + 0.998134i $$0.519450\pi$$
$$368$$ 0 0
$$369$$ 26.7846 1.39435
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −8.39230 −0.434537 −0.217269 0.976112i $$-0.569715\pi$$
−0.217269 + 0.976112i $$0.569715\pi$$
$$374$$ 0 0
$$375$$ −18.9282 −0.977448
$$376$$ 0 0
$$377$$ −18.9282 −0.974852
$$378$$ 0 0
$$379$$ −5.26795 −0.270596 −0.135298 0.990805i $$-0.543199\pi$$
−0.135298 + 0.990805i $$0.543199\pi$$
$$380$$ 0 0
$$381$$ −1.07180 −0.0549098
$$382$$ 0 0
$$383$$ 2.53590 0.129578 0.0647892 0.997899i $$-0.479363\pi$$
0.0647892 + 0.997899i $$0.479363\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −37.4641 −1.90441
$$388$$ 0 0
$$389$$ 37.1769 1.88494 0.942472 0.334285i $$-0.108495\pi$$
0.942472 + 0.334285i $$0.108495\pi$$
$$390$$ 0 0
$$391$$ −1.26795 −0.0641229
$$392$$ 0 0
$$393$$ 15.4641 0.780061
$$394$$ 0 0
$$395$$ −42.2487 −2.12576
$$396$$ 0 0
$$397$$ 5.60770 0.281442 0.140721 0.990049i $$-0.455058\pi$$
0.140721 + 0.990049i $$0.455058\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 31.8564 1.59083 0.795417 0.606063i $$-0.207252\pi$$
0.795417 + 0.606063i $$0.207252\pi$$
$$402$$ 0 0
$$403$$ 22.9282 1.14214
$$404$$ 0 0
$$405$$ −8.53590 −0.424152
$$406$$ 0 0
$$407$$ −5.75129 −0.285081
$$408$$ 0 0
$$409$$ −17.7128 −0.875842 −0.437921 0.899013i $$-0.644285\pi$$
−0.437921 + 0.899013i $$0.644285\pi$$
$$410$$ 0 0
$$411$$ −6.92820 −0.341743
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 8.78461 0.431220
$$416$$ 0 0
$$417$$ −40.2487 −1.97099
$$418$$ 0 0
$$419$$ 24.5885 1.20122 0.600612 0.799540i $$-0.294924\pi$$
0.600612 + 0.799540i $$0.294924\pi$$
$$420$$ 0 0
$$421$$ −22.2487 −1.08434 −0.542168 0.840270i $$-0.682396\pi$$
−0.542168 + 0.840270i $$0.682396\pi$$
$$422$$ 0 0
$$423$$ 30.9282 1.50378
$$424$$ 0 0
$$425$$ 7.00000 0.339550
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −18.9282 −0.913862
$$430$$ 0 0
$$431$$ 17.6603 0.850665 0.425332 0.905037i $$-0.360157\pi$$
0.425332 + 0.905037i $$0.360157\pi$$
$$432$$ 0 0
$$433$$ −3.60770 −0.173375 −0.0866874 0.996236i $$-0.527628\pi$$
−0.0866874 + 0.996236i $$0.527628\pi$$
$$434$$ 0 0
$$435$$ −32.7846 −1.57190
$$436$$ 0 0
$$437$$ −1.85641 −0.0888040
$$438$$ 0 0
$$439$$ 24.1962 1.15482 0.577410 0.816455i $$-0.304064\pi$$
0.577410 + 0.816455i $$0.304064\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −20.7846 −0.987507 −0.493753 0.869602i $$-0.664375\pi$$
−0.493753 + 0.869602i $$0.664375\pi$$
$$444$$ 0 0
$$445$$ −8.78461 −0.416430
$$446$$ 0 0
$$447$$ −16.3923 −0.775329
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −7.60770 −0.358232
$$452$$ 0 0
$$453$$ 36.7846 1.72829
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −20.3923 −0.953912 −0.476956 0.878927i $$-0.658260\pi$$
−0.476956 + 0.878927i $$0.658260\pi$$
$$458$$ 0 0
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ −26.7846 −1.24748 −0.623742 0.781630i $$-0.714388\pi$$
−0.623742 + 0.781630i $$0.714388\pi$$
$$462$$ 0 0
$$463$$ −2.14359 −0.0996212 −0.0498106 0.998759i $$-0.515862\pi$$
−0.0498106 + 0.998759i $$0.515862\pi$$
$$464$$ 0 0
$$465$$ 39.7128 1.84164
$$466$$ 0 0
$$467$$ −14.5359 −0.672641 −0.336321 0.941748i $$-0.609183\pi$$
−0.336321 + 0.941748i $$0.609183\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 5.46410 0.251773
$$472$$ 0 0
$$473$$ 10.6410 0.489274
$$474$$ 0 0
$$475$$ 10.2487 0.470243
$$476$$ 0 0
$$477$$ 57.7128 2.64249
$$478$$ 0 0
$$479$$ 17.6603 0.806918 0.403459 0.914998i $$-0.367808\pi$$
0.403459 + 0.914998i $$0.367808\pi$$
$$480$$ 0 0
$$481$$ −24.7846 −1.13008
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 17.0718 0.775190
$$486$$ 0 0
$$487$$ 6.05256 0.274268 0.137134 0.990553i $$-0.456211\pi$$
0.137134 + 0.990553i $$0.456211\pi$$
$$488$$ 0 0
$$489$$ −56.2487 −2.54365
$$490$$ 0 0
$$491$$ −37.1769 −1.67777 −0.838885 0.544308i $$-0.816792\pi$$
−0.838885 + 0.544308i $$0.816792\pi$$
$$492$$ 0 0
$$493$$ 3.46410 0.156015
$$494$$ 0 0
$$495$$ −19.6077 −0.881300
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 24.9808 1.11829 0.559146 0.829069i $$-0.311129\pi$$
0.559146 + 0.829069i $$0.311129\pi$$
$$500$$ 0 0
$$501$$ 15.4641 0.690885
$$502$$ 0 0
$$503$$ −27.1244 −1.20942 −0.604708 0.796448i $$-0.706710\pi$$
−0.604708 + 0.796448i $$0.706710\pi$$
$$504$$ 0 0
$$505$$ 8.78461 0.390910
$$506$$ 0 0
$$507$$ −46.0526 −2.04527
$$508$$ 0 0
$$509$$ 21.7128 0.962404 0.481202 0.876610i $$-0.340200\pi$$
0.481202 + 0.876610i $$0.340200\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −5.85641 −0.258567
$$514$$ 0 0
$$515$$ 37.8564 1.66815
$$516$$ 0 0
$$517$$ −8.78461 −0.386347
$$518$$ 0 0
$$519$$ −42.2487 −1.85451
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 12.7846 0.559032 0.279516 0.960141i $$-0.409826\pi$$
0.279516 + 0.960141i $$0.409826\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.19615 −0.182787
$$528$$ 0 0
$$529$$ −21.3923 −0.930100
$$530$$ 0 0
$$531$$ −11.3205 −0.491268
$$532$$ 0 0
$$533$$ −32.7846 −1.42006
$$534$$ 0 0
$$535$$ 37.1769 1.60730
$$536$$ 0 0
$$537$$ −39.7128 −1.71373
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 18.3923 0.790747 0.395373 0.918520i $$-0.370615\pi$$
0.395373 + 0.918520i $$0.370615\pi$$
$$542$$ 0 0
$$543$$ 12.3923 0.531805
$$544$$ 0 0
$$545$$ −49.8564 −2.13561
$$546$$ 0 0
$$547$$ 42.7321 1.82709 0.913545 0.406737i $$-0.133333\pi$$
0.913545 + 0.406737i $$0.133333\pi$$
$$548$$ 0 0
$$549$$ 2.39230 0.102101
$$550$$ 0 0
$$551$$ 5.07180 0.216066
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −42.9282 −1.82220
$$556$$ 0 0
$$557$$ −26.5359 −1.12436 −0.562181 0.827014i $$-0.690038\pi$$
−0.562181 + 0.827014i $$0.690038\pi$$
$$558$$ 0 0
$$559$$ 45.8564 1.93952
$$560$$ 0 0
$$561$$ 3.46410 0.146254
$$562$$ 0 0
$$563$$ 35.3205 1.48858 0.744291 0.667855i $$-0.232788\pi$$
0.744291 + 0.667855i $$0.232788\pi$$
$$564$$ 0 0
$$565$$ 68.7846 2.89379
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −7.85641 −0.329358 −0.164679 0.986347i $$-0.552659\pi$$
−0.164679 + 0.986347i $$0.552659\pi$$
$$570$$ 0 0
$$571$$ 11.1244 0.465540 0.232770 0.972532i $$-0.425221\pi$$
0.232770 + 0.972532i $$0.425221\pi$$
$$572$$ 0 0
$$573$$ −32.7846 −1.36960
$$574$$ 0 0
$$575$$ −8.87564 −0.370140
$$576$$ 0 0
$$577$$ 13.4641 0.560518 0.280259 0.959924i $$-0.409580\pi$$
0.280259 + 0.959924i $$0.409580\pi$$
$$578$$ 0 0
$$579$$ −57.1769 −2.37619
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −16.3923 −0.678900
$$584$$ 0 0
$$585$$ −84.4974 −3.49354
$$586$$ 0 0
$$587$$ −28.3923 −1.17188 −0.585938 0.810356i $$-0.699274\pi$$
−0.585938 + 0.810356i $$0.699274\pi$$
$$588$$ 0 0
$$589$$ −6.14359 −0.253142
$$590$$ 0 0
$$591$$ −9.46410 −0.389301
$$592$$ 0 0
$$593$$ 4.14359 0.170157 0.0850785 0.996374i $$-0.472886\pi$$
0.0850785 + 0.996374i $$0.472886\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11.4641 0.469194
$$598$$ 0 0
$$599$$ −27.7128 −1.13231 −0.566157 0.824297i $$-0.691571\pi$$
−0.566157 + 0.824297i $$0.691571\pi$$
$$600$$ 0 0
$$601$$ −10.7846 −0.439913 −0.219957 0.975510i $$-0.570592\pi$$
−0.219957 + 0.975510i $$0.570592\pi$$
$$602$$ 0 0
$$603$$ 66.6410 2.71383
$$604$$ 0 0
$$605$$ −32.5359 −1.32277
$$606$$ 0 0
$$607$$ 38.7321 1.57209 0.786043 0.618172i $$-0.212127\pi$$
0.786043 + 0.618172i $$0.212127\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −37.8564 −1.53151
$$612$$ 0 0
$$613$$ −47.8564 −1.93290 −0.966451 0.256851i $$-0.917315\pi$$
−0.966451 + 0.256851i $$0.917315\pi$$
$$614$$ 0 0
$$615$$ −56.7846 −2.28978
$$616$$ 0 0
$$617$$ −35.5692 −1.43196 −0.715981 0.698119i $$-0.754020\pi$$
−0.715981 + 0.698119i $$0.754020\pi$$
$$618$$ 0 0
$$619$$ −11.8038 −0.474437 −0.237218 0.971456i $$-0.576236\pi$$
−0.237218 + 0.971456i $$0.576236\pi$$
$$620$$ 0 0
$$621$$ 5.07180 0.203524
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 5.07180 0.202548
$$628$$ 0 0
$$629$$ 4.53590 0.180858
$$630$$ 0 0
$$631$$ 41.4641 1.65066 0.825330 0.564651i $$-0.190989\pi$$
0.825330 + 0.564651i $$0.190989\pi$$
$$632$$ 0 0
$$633$$ −6.39230 −0.254071
$$634$$ 0 0
$$635$$ 1.35898 0.0539296
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −36.5885 −1.44742
$$640$$ 0 0
$$641$$ −12.9282 −0.510633 −0.255317 0.966857i $$-0.582180\pi$$
−0.255317 + 0.966857i $$0.582180\pi$$
$$642$$ 0 0
$$643$$ −32.5885 −1.28516 −0.642582 0.766217i $$-0.722137\pi$$
−0.642582 + 0.766217i $$0.722137\pi$$
$$644$$ 0 0
$$645$$ 79.4256 3.12738
$$646$$ 0 0
$$647$$ −17.0718 −0.671162 −0.335581 0.942011i $$-0.608933\pi$$
−0.335581 + 0.942011i $$0.608933\pi$$
$$648$$ 0 0
$$649$$ 3.21539 0.126215
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −10.3923 −0.406682 −0.203341 0.979108i $$-0.565180\pi$$
−0.203341 + 0.979108i $$0.565180\pi$$
$$654$$ 0 0
$$655$$ −19.6077 −0.766136
$$656$$ 0 0
$$657$$ −8.92820 −0.348322
$$658$$ 0 0
$$659$$ 42.9282 1.67225 0.836123 0.548542i $$-0.184817\pi$$
0.836123 + 0.548542i $$0.184817\pi$$
$$660$$ 0 0
$$661$$ 15.0718 0.586225 0.293112 0.956078i $$-0.405309\pi$$
0.293112 + 0.956078i $$0.405309\pi$$
$$662$$ 0 0
$$663$$ 14.9282 0.579763
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.39230 −0.170071
$$668$$ 0 0
$$669$$ 33.8564 1.30896
$$670$$ 0 0
$$671$$ −0.679492 −0.0262315
$$672$$ 0 0
$$673$$ 0.143594 0.00553512 0.00276756 0.999996i $$-0.499119\pi$$
0.00276756 + 0.999996i $$0.499119\pi$$
$$674$$ 0 0
$$675$$ −28.0000 −1.07772
$$676$$ 0 0
$$677$$ 24.2487 0.931954 0.465977 0.884797i $$-0.345703\pi$$
0.465977 + 0.884797i $$0.345703\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −55.1769 −2.11438
$$682$$ 0 0
$$683$$ −46.0526 −1.76215 −0.881076 0.472975i $$-0.843180\pi$$
−0.881076 + 0.472975i $$0.843180\pi$$
$$684$$ 0 0
$$685$$ 8.78461 0.335643
$$686$$ 0 0
$$687$$ −55.7128 −2.12558
$$688$$ 0 0
$$689$$ −70.6410 −2.69121
$$690$$ 0 0
$$691$$ 36.1962 1.37697 0.688483 0.725252i $$-0.258277\pi$$
0.688483 + 0.725252i $$0.258277\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 51.0333 1.93580
$$696$$ 0 0
$$697$$ 6.00000 0.227266
$$698$$ 0 0
$$699$$ −16.3923 −0.620014
$$700$$ 0 0
$$701$$ −16.3923 −0.619129 −0.309564 0.950878i $$-0.600183\pi$$
−0.309564 + 0.950878i $$0.600183\pi$$
$$702$$ 0 0
$$703$$ 6.64102 0.250471
$$704$$ 0 0
$$705$$ −65.5692 −2.46948
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −16.2487 −0.610233 −0.305117 0.952315i $$-0.598695\pi$$
−0.305117 + 0.952315i $$0.598695\pi$$
$$710$$ 0 0
$$711$$ −54.4449 −2.04184
$$712$$ 0 0
$$713$$ 5.32051 0.199255
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ 0 0
$$717$$ −56.7846 −2.12066
$$718$$ 0 0
$$719$$ −43.5167 −1.62290 −0.811449 0.584424i $$-0.801321\pi$$
−0.811449 + 0.584424i $$0.801321\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 5.46410 0.203212
$$724$$ 0 0
$$725$$ 24.2487 0.900575
$$726$$ 0 0
$$727$$ −4.78461 −0.177451 −0.0887257 0.996056i $$-0.528279\pi$$
−0.0887257 + 0.996056i $$0.528279\pi$$
$$728$$ 0 0
$$729$$ −43.7846 −1.62165
$$730$$ 0 0
$$731$$ −8.39230 −0.310401
$$732$$ 0 0
$$733$$ 18.7846 0.693825 0.346913 0.937897i $$-0.387230\pi$$
0.346913 + 0.937897i $$0.387230\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −18.9282 −0.697229
$$738$$ 0 0
$$739$$ 12.3923 0.455858 0.227929 0.973678i $$-0.426805\pi$$
0.227929 + 0.973678i $$0.426805\pi$$
$$740$$ 0 0
$$741$$ 21.8564 0.802915
$$742$$ 0 0
$$743$$ 23.9090 0.877135 0.438567 0.898698i $$-0.355486\pi$$
0.438567 + 0.898698i $$0.355486\pi$$
$$744$$ 0 0
$$745$$ 20.7846 0.761489
$$746$$ 0 0
$$747$$ 11.3205 0.414196
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 18.7321 0.683542 0.341771 0.939783i $$-0.388973\pi$$
0.341771 + 0.939783i $$0.388973\pi$$
$$752$$ 0 0
$$753$$ 18.9282 0.689782
$$754$$ 0 0
$$755$$ −46.6410 −1.69744
$$756$$ 0 0
$$757$$ 17.4641 0.634744 0.317372 0.948301i $$-0.397200\pi$$
0.317372 + 0.948301i $$0.397200\pi$$
$$758$$ 0 0
$$759$$ −4.39230 −0.159431
$$760$$ 0 0
$$761$$ −16.3923 −0.594221 −0.297110 0.954843i $$-0.596023\pi$$
−0.297110 + 0.954843i $$0.596023\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 15.4641 0.559106
$$766$$ 0 0
$$767$$ 13.8564 0.500326
$$768$$ 0 0
$$769$$ −0.392305 −0.0141469 −0.00707344 0.999975i $$-0.502252\pi$$
−0.00707344 + 0.999975i $$0.502252\pi$$
$$770$$ 0 0
$$771$$ −77.5692 −2.79359
$$772$$ 0 0
$$773$$ 0.679492 0.0244396 0.0122198 0.999925i $$-0.496110\pi$$
0.0122198 + 0.999925i $$0.496110\pi$$
$$774$$ 0 0
$$775$$ −29.3731 −1.05511
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 8.78461 0.314741
$$780$$ 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
$$787$$ −23.1244 −0.824294 −0.412147 0.911117i $$-0.635221\pi$$
−0.412147 + 0.911117i $$0.635221\pi$$
$$788$$ 0 0
$$789$$ −30.9282 −1.10107
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −2.92820 −0.103984
$$794$$ 0 0
$$795$$ −122.354 −4.33944
$$796$$ 0 0
$$797$$ 2.78461 0.0986359 0.0493180 0.998783i $$-0.484295\pi$$
0.0493180 + 0.998783i $$0.484295\pi$$
$$798$$ 0 0
$$799$$ 6.92820 0.245102
$$800$$ 0 0
$$801$$ −11.3205 −0.399990
$$802$$ 0 0
$$803$$ 2.53590 0.0894899
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −75.0333 −2.64130
$$808$$ 0 0
$$809$$ −7.85641 −0.276217 −0.138108 0.990417i $$-0.544102\pi$$
−0.138108 + 0.990417i $$0.544102\pi$$
$$810$$ 0 0
$$811$$ 37.3731 1.31235 0.656173 0.754611i $$-0.272174\pi$$
0.656173 + 0.754611i $$0.272174\pi$$
$$812$$ 0 0
$$813$$ 2.92820 0.102697
$$814$$ 0 0
$$815$$ 71.3205 2.49825
$$816$$ 0 0
$$817$$ −12.2872 −0.429874
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 45.0333 1.57167 0.785837 0.618434i $$-0.212233\pi$$
0.785837 + 0.618434i $$0.212233\pi$$
$$822$$ 0 0
$$823$$ 1.66025 0.0578728 0.0289364 0.999581i $$-0.490788\pi$$
0.0289364 + 0.999581i $$0.490788\pi$$
$$824$$ 0 0
$$825$$ 24.2487 0.844232
$$826$$ 0 0
$$827$$ −19.5167 −0.678661 −0.339330 0.940667i $$-0.610200\pi$$
−0.339330 + 0.940667i $$0.610200\pi$$
$$828$$ 0 0
$$829$$ −39.8564 −1.38427 −0.692135 0.721768i $$-0.743330\pi$$
−0.692135 + 0.721768i $$0.743330\pi$$
$$830$$ 0 0
$$831$$ −17.4641 −0.605823
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −19.6077 −0.678552
$$836$$ 0 0
$$837$$ 16.7846 0.580161
$$838$$ 0 0
$$839$$ −4.48334 −0.154782 −0.0773910 0.997001i $$-0.524659\pi$$
−0.0773910 + 0.997001i $$0.524659\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ 0 0
$$843$$ 54.2487 1.86842
$$844$$ 0 0
$$845$$ 58.3923 2.00876
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 56.2487 1.93045
$$850$$ 0 0
$$851$$ −5.75129 −0.197152
$$852$$ 0 0
$$853$$ −8.24871 −0.282430 −0.141215 0.989979i $$-0.545101\pi$$
−0.141215 + 0.989979i $$0.545101\pi$$
$$854$$ 0 0
$$855$$ 22.6410 0.774306
$$856$$ 0 0
$$857$$ −16.6410 −0.568446 −0.284223 0.958758i $$-0.591736\pi$$
−0.284223 + 0.958758i $$0.591736\pi$$
$$858$$ 0 0
$$859$$ 27.3205 0.932164 0.466082 0.884742i $$-0.345665\pi$$
0.466082 + 0.884742i $$0.345665\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 10.1436 0.345292 0.172646 0.984984i $$-0.444768\pi$$
0.172646 + 0.984984i $$0.444768\pi$$
$$864$$ 0 0
$$865$$ 53.5692 1.82141
$$866$$ 0 0
$$867$$ −2.73205 −0.0927853
$$868$$ 0 0
$$869$$ 15.4641 0.524584
$$870$$ 0 0
$$871$$ −81.5692 −2.76387
$$872$$ 0 0
$$873$$ 22.0000 0.744587
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 9.60770 0.324429 0.162214 0.986756i $$-0.448136\pi$$
0.162214 + 0.986756i $$0.448136\pi$$
$$878$$ 0 0
$$879$$ 81.9615 2.76449
$$880$$ 0 0
$$881$$ 38.7846 1.30669 0.653343 0.757062i $$-0.273366\pi$$
0.653343 + 0.757062i $$0.273366\pi$$
$$882$$ 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
$$887$$ −11.4115 −0.383162 −0.191581 0.981477i $$-0.561361\pi$$
−0.191581 + 0.981477i $$0.561361\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 3.12436 0.104670
$$892$$ 0 0
$$893$$ 10.1436 0.339442
$$894$$ 0 0
$$895$$ 50.3538 1.68314
$$896$$ 0 0
$$897$$ −18.9282 −0.631994
$$898$$ 0 0
$$899$$ −14.5359 −0.484799
$$900$$ 0 0
$$901$$ 12.9282 0.430701
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −15.7128 −0.522312
$$906$$ 0 0
$$907$$ 34.4449 1.14372 0.571861 0.820350i $$-0.306221\pi$$
0.571861 + 0.820350i $$0.306221\pi$$
$$908$$ 0 0
$$909$$ 11.3205 0.375478
$$910$$ 0 0
$$911$$ −15.8038 −0.523605 −0.261802 0.965121i $$-0.584317\pi$$
−0.261802 + 0.965121i $$0.584317\pi$$
$$912$$ 0 0
$$913$$ −3.21539 −0.106414
$$914$$ 0 0
$$915$$ −5.07180 −0.167668
$$916$$ 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
$$921$$ −67.7128 −2.23121
$$922$$ 0 0
$$923$$ 44.7846 1.47410
$$924$$ 0 0
$$925$$ 31.7513 1.04398
$$926$$ 0 0
$$927$$ 48.7846 1.60230
$$928$$ 0 0
$$929$$ 9.71281 0.318667 0.159334 0.987225i $$-0.449065\pi$$
0.159334 + 0.987225i $$0.449065\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −8.53590 −0.279453
$$934$$ 0 0
$$935$$ −4.39230 −0.143644
$$936$$ 0 0
$$937$$ −0.143594 −0.00469100 −0.00234550 0.999997i $$-0.500747\pi$$
−0.00234550 + 0.999997i $$0.500747\pi$$
$$938$$ 0 0
$$939$$ 57.1769 1.86590
$$940$$ 0 0
$$941$$ −41.3205 −1.34701 −0.673505 0.739183i $$-0.735212\pi$$
−0.673505 + 0.739183i $$0.735212\pi$$
$$942$$ 0 0
$$943$$ −7.60770 −0.247741
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 49.2679 1.60099 0.800497 0.599337i $$-0.204569\pi$$
0.800497 + 0.599337i $$0.204569\pi$$
$$948$$ 0 0
$$949$$ 10.9282 0.354744
$$950$$ 0 0
$$951$$ −23.3205 −0.756219
$$952$$ 0 0
$$953$$ 6.24871 0.202416 0.101208 0.994865i $$-0.467729\pi$$
0.101208 + 0.994865i $$0.467729\pi$$
$$954$$ 0 0
$$955$$ 41.5692 1.34515
$$956$$ 0 0
$$957$$ 12.0000 0.387905
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −13.3923 −0.432010
$$962$$ 0 0
$$963$$ 47.9090 1.54384
$$964$$ 0 0
$$965$$ 72.4974 2.33377
$$966$$ 0 0
$$967$$ 27.6077 0.887804 0.443902 0.896075i $$-0.353594\pi$$
0.443902 + 0.896075i $$0.353594\pi$$
$$968$$ 0 0
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ −20.1051 −0.645204 −0.322602 0.946535i $$-0.604558\pi$$
−0.322602 + 0.946535i $$0.604558\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 104.497 3.34660
$$976$$ 0 0
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ 3.21539 0.102764
$$980$$ 0 0
$$981$$ −64.2487 −2.05130
$$982$$ 0 0
$$983$$ 35.9090 1.14532 0.572659 0.819794i $$-0.305912\pi$$
0.572659 + 0.819794i $$0.305912\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 10.6410 0.338365
$$990$$ 0 0
$$991$$ −0.196152 −0.00623099 −0.00311549 0.999995i $$-0.500992\pi$$
−0.00311549 + 0.999995i $$0.500992\pi$$
$$992$$ 0 0
$$993$$ −52.7846 −1.67507
$$994$$ 0 0
$$995$$ −14.5359 −0.460819
$$996$$ 0 0
$$997$$ −21.6077 −0.684323 −0.342161 0.939641i $$-0.611159\pi$$
−0.342161 + 0.939641i $$0.611159\pi$$
$$998$$ 0 0
$$999$$ −18.1436 −0.574038
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.1.1 2
7.6 odd 2 68.2.a.a.1.2 2
21.20 even 2 612.2.a.e.1.2 2
28.27 even 2 272.2.a.e.1.1 2
35.13 even 4 1700.2.e.c.749.4 4
35.27 even 4 1700.2.e.c.749.1 4
35.34 odd 2 1700.2.a.d.1.1 2
56.13 odd 2 1088.2.a.p.1.1 2
56.27 even 2 1088.2.a.t.1.2 2
77.76 even 2 8228.2.a.k.1.2 2
84.83 odd 2 2448.2.a.y.1.2 2
119.6 even 16 1156.2.h.f.733.4 16
119.13 odd 4 1156.2.b.c.577.4 4
119.20 even 16 1156.2.h.f.757.4 16
119.27 even 16 1156.2.h.f.1001.4 16
119.41 even 16 1156.2.h.f.1001.1 16
119.48 even 16 1156.2.h.f.757.1 16
119.55 odd 4 1156.2.b.c.577.1 4
119.62 even 16 1156.2.h.f.733.1 16
119.76 odd 8 1156.2.e.d.829.1 8
119.83 odd 8 1156.2.e.d.905.1 8
119.90 even 16 1156.2.h.f.977.1 16
119.97 even 16 1156.2.h.f.977.4 16
119.104 odd 8 1156.2.e.d.905.4 8
119.111 odd 8 1156.2.e.d.829.4 8
119.118 odd 2 1156.2.a.a.1.1 2
140.139 even 2 6800.2.a.bh.1.2 2
168.83 odd 2 9792.2.a.cs.1.1 2
168.125 even 2 9792.2.a.cr.1.1 2
476.475 even 2 4624.2.a.x.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
68.2.a.a.1.2 2 7.6 odd 2
272.2.a.e.1.1 2 28.27 even 2
612.2.a.e.1.2 2 21.20 even 2
1088.2.a.p.1.1 2 56.13 odd 2
1088.2.a.t.1.2 2 56.27 even 2
1156.2.a.a.1.1 2 119.118 odd 2
1156.2.b.c.577.1 4 119.55 odd 4
1156.2.b.c.577.4 4 119.13 odd 4
1156.2.e.d.829.1 8 119.76 odd 8
1156.2.e.d.829.4 8 119.111 odd 8
1156.2.e.d.905.1 8 119.83 odd 8
1156.2.e.d.905.4 8 119.104 odd 8
1156.2.h.f.733.1 16 119.62 even 16
1156.2.h.f.733.4 16 119.6 even 16
1156.2.h.f.757.1 16 119.48 even 16
1156.2.h.f.757.4 16 119.20 even 16
1156.2.h.f.977.1 16 119.90 even 16
1156.2.h.f.977.4 16 119.97 even 16
1156.2.h.f.1001.1 16 119.41 even 16
1156.2.h.f.1001.4 16 119.27 even 16
1700.2.a.d.1.1 2 35.34 odd 2
1700.2.e.c.749.1 4 35.27 even 4
1700.2.e.c.749.4 4 35.13 even 4
2448.2.a.y.1.2 2 84.83 odd 2
3332.2.a.h.1.1 2 1.1 even 1 trivial
4624.2.a.x.1.2 2 476.475 even 2
6800.2.a.bh.1.2 2 140.139 even 2
8228.2.a.k.1.2 2 77.76 even 2
9792.2.a.cr.1.1 2 168.125 even 2
9792.2.a.cs.1.1 2 168.83 odd 2