# Properties

 Label 1840.2.a.j.1.2 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ 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] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.302776 q^{3} +1.00000 q^{5} -3.30278 q^{7} -2.90833 q^{9} +O(q^{10})$$ $$q+0.302776 q^{3} +1.00000 q^{5} -3.30278 q^{7} -2.90833 q^{9} +1.69722 q^{11} +3.30278 q^{13} +0.302776 q^{15} +6.90833 q^{17} -5.90833 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.78890 q^{27} -2.60555 q^{29} +7.90833 q^{31} +0.513878 q^{33} -3.30278 q^{35} +8.00000 q^{37} +1.00000 q^{39} +0.908327 q^{41} +9.21110 q^{43} -2.90833 q^{45} +2.60555 q^{47} +3.90833 q^{49} +2.09167 q^{51} -11.2111 q^{53} +1.69722 q^{55} -1.78890 q^{57} +3.39445 q^{59} +11.5139 q^{61} +9.60555 q^{63} +3.30278 q^{65} +4.00000 q^{67} +0.302776 q^{69} +16.3028 q^{71} -5.81665 q^{73} +0.302776 q^{75} -5.60555 q^{77} +14.4222 q^{79} +8.18335 q^{81} -11.2111 q^{83} +6.90833 q^{85} -0.788897 q^{87} -10.9083 q^{91} +2.39445 q^{93} -5.90833 q^{95} +6.30278 q^{97} -4.93608 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 5 * q^9 $$2 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 5 q^{9} + 7 q^{11} + 3 q^{13} - 3 q^{15} + 3 q^{17} - q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 18 q^{27} + 2 q^{29} + 5 q^{31} - 17 q^{33} - 3 q^{35} + 16 q^{37} + 2 q^{39} - 9 q^{41} + 4 q^{43} + 5 q^{45} - 2 q^{47} - 3 q^{49} + 15 q^{51} - 8 q^{53} + 7 q^{55} - 18 q^{57} + 14 q^{59} + 5 q^{61} + 12 q^{63} + 3 q^{65} + 8 q^{67} - 3 q^{69} + 29 q^{71} + 10 q^{73} - 3 q^{75} - 4 q^{77} + 38 q^{81} - 8 q^{83} + 3 q^{85} - 16 q^{87} - 11 q^{91} + 12 q^{93} - q^{95} + 9 q^{97} + 37 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 5 * q^9 + 7 * q^11 + 3 * q^13 - 3 * q^15 + 3 * q^17 - q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 18 * q^27 + 2 * q^29 + 5 * q^31 - 17 * q^33 - 3 * q^35 + 16 * q^37 + 2 * q^39 - 9 * q^41 + 4 * q^43 + 5 * q^45 - 2 * q^47 - 3 * q^49 + 15 * q^51 - 8 * q^53 + 7 * q^55 - 18 * q^57 + 14 * q^59 + 5 * q^61 + 12 * q^63 + 3 * q^65 + 8 * q^67 - 3 * q^69 + 29 * q^71 + 10 * q^73 - 3 * q^75 - 4 * q^77 + 38 * q^81 - 8 * q^83 + 3 * q^85 - 16 * q^87 - 11 * q^91 + 12 * q^93 - q^95 + 9 * q^97 + 37 * 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
$$3$$ 0.302776 0.174808 0.0874038 0.996173i $$-0.472143\pi$$
0.0874038 + 0.996173i $$0.472143\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −3.30278 −1.24833 −0.624166 0.781292i $$-0.714561\pi$$
−0.624166 + 0.781292i $$0.714561\pi$$
$$8$$ 0 0
$$9$$ −2.90833 −0.969442
$$10$$ 0 0
$$11$$ 1.69722 0.511732 0.255866 0.966712i $$-0.417639\pi$$
0.255866 + 0.966712i $$0.417639\pi$$
$$12$$ 0 0
$$13$$ 3.30278 0.916025 0.458013 0.888946i $$-0.348561\pi$$
0.458013 + 0.888946i $$0.348561\pi$$
$$14$$ 0 0
$$15$$ 0.302776 0.0781763
$$16$$ 0 0
$$17$$ 6.90833 1.67552 0.837758 0.546042i $$-0.183866\pi$$
0.837758 + 0.546042i $$0.183866\pi$$
$$18$$ 0 0
$$19$$ −5.90833 −1.35546 −0.677732 0.735309i $$-0.737037\pi$$
−0.677732 + 0.735309i $$0.737037\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.78890 −0.344273
$$28$$ 0 0
$$29$$ −2.60555 −0.483839 −0.241919 0.970296i $$-0.577777\pi$$
−0.241919 + 0.970296i $$0.577777\pi$$
$$30$$ 0 0
$$31$$ 7.90833 1.42038 0.710189 0.704011i $$-0.248610\pi$$
0.710189 + 0.704011i $$0.248610\pi$$
$$32$$ 0 0
$$33$$ 0.513878 0.0894547
$$34$$ 0 0
$$35$$ −3.30278 −0.558271
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 0.908327 0.141857 0.0709284 0.997481i $$-0.477404\pi$$
0.0709284 + 0.997481i $$0.477404\pi$$
$$42$$ 0 0
$$43$$ 9.21110 1.40468 0.702340 0.711842i $$-0.252139\pi$$
0.702340 + 0.711842i $$0.252139\pi$$
$$44$$ 0 0
$$45$$ −2.90833 −0.433548
$$46$$ 0 0
$$47$$ 2.60555 0.380059 0.190029 0.981778i $$-0.439142\pi$$
0.190029 + 0.981778i $$0.439142\pi$$
$$48$$ 0 0
$$49$$ 3.90833 0.558332
$$50$$ 0 0
$$51$$ 2.09167 0.292893
$$52$$ 0 0
$$53$$ −11.2111 −1.53996 −0.769982 0.638066i $$-0.779735\pi$$
−0.769982 + 0.638066i $$0.779735\pi$$
$$54$$ 0 0
$$55$$ 1.69722 0.228854
$$56$$ 0 0
$$57$$ −1.78890 −0.236945
$$58$$ 0 0
$$59$$ 3.39445 0.441920 0.220960 0.975283i $$-0.429081\pi$$
0.220960 + 0.975283i $$0.429081\pi$$
$$60$$ 0 0
$$61$$ 11.5139 1.47420 0.737101 0.675783i $$-0.236194\pi$$
0.737101 + 0.675783i $$0.236194\pi$$
$$62$$ 0 0
$$63$$ 9.60555 1.21019
$$64$$ 0 0
$$65$$ 3.30278 0.409659
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0.302776 0.0364499
$$70$$ 0 0
$$71$$ 16.3028 1.93478 0.967392 0.253285i $$-0.0815110\pi$$
0.967392 + 0.253285i $$0.0815110\pi$$
$$72$$ 0 0
$$73$$ −5.81665 −0.680788 −0.340394 0.940283i $$-0.610560\pi$$
−0.340394 + 0.940283i $$0.610560\pi$$
$$74$$ 0 0
$$75$$ 0.302776 0.0349615
$$76$$ 0 0
$$77$$ −5.60555 −0.638812
$$78$$ 0 0
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ 0 0
$$81$$ 8.18335 0.909261
$$82$$ 0 0
$$83$$ −11.2111 −1.23058 −0.615289 0.788301i $$-0.710961\pi$$
−0.615289 + 0.788301i $$0.710961\pi$$
$$84$$ 0 0
$$85$$ 6.90833 0.749313
$$86$$ 0 0
$$87$$ −0.788897 −0.0845787
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −10.9083 −1.14350
$$92$$ 0 0
$$93$$ 2.39445 0.248293
$$94$$ 0 0
$$95$$ −5.90833 −0.606182
$$96$$ 0 0
$$97$$ 6.30278 0.639950 0.319975 0.947426i $$-0.396325\pi$$
0.319975 + 0.947426i $$0.396325\pi$$
$$98$$ 0 0
$$99$$ −4.93608 −0.496095
$$100$$ 0 0
$$101$$ 2.60555 0.259262 0.129631 0.991562i $$-0.458621\pi$$
0.129631 + 0.991562i $$0.458621\pi$$
$$102$$ 0 0
$$103$$ −8.11943 −0.800031 −0.400016 0.916508i $$-0.630995\pi$$
−0.400016 + 0.916508i $$0.630995\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ 2.60555 0.251888 0.125944 0.992037i $$-0.459804\pi$$
0.125944 + 0.992037i $$0.459804\pi$$
$$108$$ 0 0
$$109$$ 1.48612 0.142345 0.0711723 0.997464i $$-0.477326\pi$$
0.0711723 + 0.997464i $$0.477326\pi$$
$$110$$ 0 0
$$111$$ 2.42221 0.229906
$$112$$ 0 0
$$113$$ −16.4222 −1.54487 −0.772436 0.635093i $$-0.780962\pi$$
−0.772436 + 0.635093i $$0.780962\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ −9.60555 −0.888034
$$118$$ 0 0
$$119$$ −22.8167 −2.09160
$$120$$ 0 0
$$121$$ −8.11943 −0.738130
$$122$$ 0 0
$$123$$ 0.275019 0.0247977
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −9.81665 −0.871087 −0.435544 0.900168i $$-0.643444\pi$$
−0.435544 + 0.900168i $$0.643444\pi$$
$$128$$ 0 0
$$129$$ 2.78890 0.245549
$$130$$ 0 0
$$131$$ 11.2111 0.979519 0.489759 0.871858i $$-0.337085\pi$$
0.489759 + 0.871858i $$0.337085\pi$$
$$132$$ 0 0
$$133$$ 19.5139 1.69207
$$134$$ 0 0
$$135$$ −1.78890 −0.153964
$$136$$ 0 0
$$137$$ 3.90833 0.333911 0.166955 0.985964i $$-0.446606\pi$$
0.166955 + 0.985964i $$0.446606\pi$$
$$138$$ 0 0
$$139$$ 12.6056 1.06919 0.534594 0.845109i $$-0.320464\pi$$
0.534594 + 0.845109i $$0.320464\pi$$
$$140$$ 0 0
$$141$$ 0.788897 0.0664372
$$142$$ 0 0
$$143$$ 5.60555 0.468760
$$144$$ 0 0
$$145$$ −2.60555 −0.216379
$$146$$ 0 0
$$147$$ 1.18335 0.0976007
$$148$$ 0 0
$$149$$ 13.3028 1.08981 0.544903 0.838499i $$-0.316567\pi$$
0.544903 + 0.838499i $$0.316567\pi$$
$$150$$ 0 0
$$151$$ −8.90833 −0.724949 −0.362475 0.931994i $$-0.618068\pi$$
−0.362475 + 0.931994i $$0.618068\pi$$
$$152$$ 0 0
$$153$$ −20.0917 −1.62432
$$154$$ 0 0
$$155$$ 7.90833 0.635212
$$156$$ 0 0
$$157$$ −18.6056 −1.48488 −0.742442 0.669910i $$-0.766333\pi$$
−0.742442 + 0.669910i $$0.766333\pi$$
$$158$$ 0 0
$$159$$ −3.39445 −0.269197
$$160$$ 0 0
$$161$$ −3.30278 −0.260295
$$162$$ 0 0
$$163$$ −9.30278 −0.728650 −0.364325 0.931272i $$-0.618700\pi$$
−0.364325 + 0.931272i $$0.618700\pi$$
$$164$$ 0 0
$$165$$ 0.513878 0.0400054
$$166$$ 0 0
$$167$$ −6.78890 −0.525341 −0.262670 0.964886i $$-0.584603\pi$$
−0.262670 + 0.964886i $$0.584603\pi$$
$$168$$ 0 0
$$169$$ −2.09167 −0.160898
$$170$$ 0 0
$$171$$ 17.1833 1.31404
$$172$$ 0 0
$$173$$ 19.6972 1.49755 0.748776 0.662823i $$-0.230642\pi$$
0.748776 + 0.662823i $$0.230642\pi$$
$$174$$ 0 0
$$175$$ −3.30278 −0.249666
$$176$$ 0 0
$$177$$ 1.02776 0.0772509
$$178$$ 0 0
$$179$$ −9.39445 −0.702174 −0.351087 0.936343i $$-0.614188\pi$$
−0.351087 + 0.936343i $$0.614188\pi$$
$$180$$ 0 0
$$181$$ 17.1194 1.27248 0.636239 0.771492i $$-0.280489\pi$$
0.636239 + 0.771492i $$0.280489\pi$$
$$182$$ 0 0
$$183$$ 3.48612 0.257702
$$184$$ 0 0
$$185$$ 8.00000 0.588172
$$186$$ 0 0
$$187$$ 11.7250 0.857416
$$188$$ 0 0
$$189$$ 5.90833 0.429768
$$190$$ 0 0
$$191$$ 8.60555 0.622676 0.311338 0.950299i $$-0.399223\pi$$
0.311338 + 0.950299i $$0.399223\pi$$
$$192$$ 0 0
$$193$$ −17.8167 −1.28247 −0.641235 0.767344i $$-0.721578\pi$$
−0.641235 + 0.767344i $$0.721578\pi$$
$$194$$ 0 0
$$195$$ 1.00000 0.0716115
$$196$$ 0 0
$$197$$ 4.30278 0.306560 0.153280 0.988183i $$-0.451016\pi$$
0.153280 + 0.988183i $$0.451016\pi$$
$$198$$ 0 0
$$199$$ 20.4222 1.44769 0.723846 0.689962i $$-0.242373\pi$$
0.723846 + 0.689962i $$0.242373\pi$$
$$200$$ 0 0
$$201$$ 1.21110 0.0854246
$$202$$ 0 0
$$203$$ 8.60555 0.603991
$$204$$ 0 0
$$205$$ 0.908327 0.0634403
$$206$$ 0 0
$$207$$ −2.90833 −0.202143
$$208$$ 0 0
$$209$$ −10.0278 −0.693634
$$210$$ 0 0
$$211$$ −7.21110 −0.496433 −0.248216 0.968705i $$-0.579844\pi$$
−0.248216 + 0.968705i $$0.579844\pi$$
$$212$$ 0 0
$$213$$ 4.93608 0.338215
$$214$$ 0 0
$$215$$ 9.21110 0.628192
$$216$$ 0 0
$$217$$ −26.1194 −1.77310
$$218$$ 0 0
$$219$$ −1.76114 −0.119007
$$220$$ 0 0
$$221$$ 22.8167 1.53481
$$222$$ 0 0
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ −2.90833 −0.193888
$$226$$ 0 0
$$227$$ 14.6056 0.969404 0.484702 0.874679i $$-0.338928\pi$$
0.484702 + 0.874679i $$0.338928\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ −1.69722 −0.111669
$$232$$ 0 0
$$233$$ 25.8167 1.69131 0.845653 0.533734i $$-0.179212\pi$$
0.845653 + 0.533734i $$0.179212\pi$$
$$234$$ 0 0
$$235$$ 2.60555 0.169967
$$236$$ 0 0
$$237$$ 4.36669 0.283647
$$238$$ 0 0
$$239$$ −5.21110 −0.337078 −0.168539 0.985695i $$-0.553905\pi$$
−0.168539 + 0.985695i $$0.553905\pi$$
$$240$$ 0 0
$$241$$ −14.4222 −0.929016 −0.464508 0.885569i $$-0.653769\pi$$
−0.464508 + 0.885569i $$0.653769\pi$$
$$242$$ 0 0
$$243$$ 7.84441 0.503219
$$244$$ 0 0
$$245$$ 3.90833 0.249694
$$246$$ 0 0
$$247$$ −19.5139 −1.24164
$$248$$ 0 0
$$249$$ −3.39445 −0.215114
$$250$$ 0 0
$$251$$ −12.5139 −0.789869 −0.394934 0.918709i $$-0.629233\pi$$
−0.394934 + 0.918709i $$0.629233\pi$$
$$252$$ 0 0
$$253$$ 1.69722 0.106704
$$254$$ 0 0
$$255$$ 2.09167 0.130986
$$256$$ 0 0
$$257$$ −1.81665 −0.113320 −0.0566599 0.998394i $$-0.518045\pi$$
−0.0566599 + 0.998394i $$0.518045\pi$$
$$258$$ 0 0
$$259$$ −26.4222 −1.64180
$$260$$ 0 0
$$261$$ 7.57779 0.469054
$$262$$ 0 0
$$263$$ −3.51388 −0.216675 −0.108338 0.994114i $$-0.534553\pi$$
−0.108338 + 0.994114i $$0.534553\pi$$
$$264$$ 0 0
$$265$$ −11.2111 −0.688693
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 4.18335 0.255063 0.127532 0.991835i $$-0.459295\pi$$
0.127532 + 0.991835i $$0.459295\pi$$
$$270$$ 0 0
$$271$$ 2.69722 0.163845 0.0819224 0.996639i $$-0.473894\pi$$
0.0819224 + 0.996639i $$0.473894\pi$$
$$272$$ 0 0
$$273$$ −3.30278 −0.199893
$$274$$ 0 0
$$275$$ 1.69722 0.102346
$$276$$ 0 0
$$277$$ −27.2111 −1.63496 −0.817478 0.575959i $$-0.804629\pi$$
−0.817478 + 0.575959i $$0.804629\pi$$
$$278$$ 0 0
$$279$$ −23.0000 −1.37697
$$280$$ 0 0
$$281$$ −26.6056 −1.58715 −0.793577 0.608470i $$-0.791784\pi$$
−0.793577 + 0.608470i $$0.791784\pi$$
$$282$$ 0 0
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ 0 0
$$285$$ −1.78890 −0.105965
$$286$$ 0 0
$$287$$ −3.00000 −0.177084
$$288$$ 0 0
$$289$$ 30.7250 1.80735
$$290$$ 0 0
$$291$$ 1.90833 0.111868
$$292$$ 0 0
$$293$$ 23.2111 1.35601 0.678004 0.735059i $$-0.262845\pi$$
0.678004 + 0.735059i $$0.262845\pi$$
$$294$$ 0 0
$$295$$ 3.39445 0.197632
$$296$$ 0 0
$$297$$ −3.03616 −0.176176
$$298$$ 0 0
$$299$$ 3.30278 0.191004
$$300$$ 0 0
$$301$$ −30.4222 −1.75351
$$302$$ 0 0
$$303$$ 0.788897 0.0453210
$$304$$ 0 0
$$305$$ 11.5139 0.659283
$$306$$ 0 0
$$307$$ 11.6972 0.667596 0.333798 0.942645i $$-0.391670\pi$$
0.333798 + 0.942645i $$0.391670\pi$$
$$308$$ 0 0
$$309$$ −2.45837 −0.139852
$$310$$ 0 0
$$311$$ −22.4222 −1.27145 −0.635723 0.771917i $$-0.719298\pi$$
−0.635723 + 0.771917i $$0.719298\pi$$
$$312$$ 0 0
$$313$$ 19.7250 1.11492 0.557461 0.830203i $$-0.311776\pi$$
0.557461 + 0.830203i $$0.311776\pi$$
$$314$$ 0 0
$$315$$ 9.60555 0.541212
$$316$$ 0 0
$$317$$ 17.7250 0.995534 0.497767 0.867311i $$-0.334153\pi$$
0.497767 + 0.867311i $$0.334153\pi$$
$$318$$ 0 0
$$319$$ −4.42221 −0.247596
$$320$$ 0 0
$$321$$ 0.788897 0.0440320
$$322$$ 0 0
$$323$$ −40.8167 −2.27110
$$324$$ 0 0
$$325$$ 3.30278 0.183205
$$326$$ 0 0
$$327$$ 0.449961 0.0248829
$$328$$ 0 0
$$329$$ −8.60555 −0.474439
$$330$$ 0 0
$$331$$ −16.6056 −0.912724 −0.456362 0.889794i $$-0.650848\pi$$
−0.456362 + 0.889794i $$0.650848\pi$$
$$332$$ 0 0
$$333$$ −23.2666 −1.27500
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −22.5139 −1.22641 −0.613205 0.789924i $$-0.710120\pi$$
−0.613205 + 0.789924i $$0.710120\pi$$
$$338$$ 0 0
$$339$$ −4.97224 −0.270055
$$340$$ 0 0
$$341$$ 13.4222 0.726853
$$342$$ 0 0
$$343$$ 10.2111 0.551348
$$344$$ 0 0
$$345$$ 0.302776 0.0163009
$$346$$ 0 0
$$347$$ −28.5416 −1.53220 −0.766098 0.642724i $$-0.777804\pi$$
−0.766098 + 0.642724i $$0.777804\pi$$
$$348$$ 0 0
$$349$$ −27.2111 −1.45658 −0.728288 0.685271i $$-0.759684\pi$$
−0.728288 + 0.685271i $$0.759684\pi$$
$$350$$ 0 0
$$351$$ −5.90833 −0.315363
$$352$$ 0 0
$$353$$ −10.4222 −0.554718 −0.277359 0.960766i $$-0.589459\pi$$
−0.277359 + 0.960766i $$0.589459\pi$$
$$354$$ 0 0
$$355$$ 16.3028 0.865261
$$356$$ 0 0
$$357$$ −6.90833 −0.365627
$$358$$ 0 0
$$359$$ −11.2111 −0.591699 −0.295850 0.955235i $$-0.595603\pi$$
−0.295850 + 0.955235i $$0.595603\pi$$
$$360$$ 0 0
$$361$$ 15.9083 0.837280
$$362$$ 0 0
$$363$$ −2.45837 −0.129031
$$364$$ 0 0
$$365$$ −5.81665 −0.304458
$$366$$ 0 0
$$367$$ −14.7889 −0.771974 −0.385987 0.922504i $$-0.626139\pi$$
−0.385987 + 0.922504i $$0.626139\pi$$
$$368$$ 0 0
$$369$$ −2.64171 −0.137522
$$370$$ 0 0
$$371$$ 37.0278 1.92239
$$372$$ 0 0
$$373$$ 4.60555 0.238466 0.119233 0.992866i $$-0.461956\pi$$
0.119233 + 0.992866i $$0.461956\pi$$
$$374$$ 0 0
$$375$$ 0.302776 0.0156353
$$376$$ 0 0
$$377$$ −8.60555 −0.443208
$$378$$ 0 0
$$379$$ −14.9083 −0.765789 −0.382895 0.923792i $$-0.625073\pi$$
−0.382895 + 0.923792i $$0.625073\pi$$
$$380$$ 0 0
$$381$$ −2.97224 −0.152273
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −5.60555 −0.285685
$$386$$ 0 0
$$387$$ −26.7889 −1.36176
$$388$$ 0 0
$$389$$ −25.9361 −1.31501 −0.657506 0.753449i $$-0.728388\pi$$
−0.657506 + 0.753449i $$0.728388\pi$$
$$390$$ 0 0
$$391$$ 6.90833 0.349369
$$392$$ 0 0
$$393$$ 3.39445 0.171227
$$394$$ 0 0
$$395$$ 14.4222 0.725660
$$396$$ 0 0
$$397$$ 10.7250 0.538271 0.269136 0.963102i $$-0.413262\pi$$
0.269136 + 0.963102i $$0.413262\pi$$
$$398$$ 0 0
$$399$$ 5.90833 0.295786
$$400$$ 0 0
$$401$$ −8.60555 −0.429741 −0.214870 0.976643i $$-0.568933\pi$$
−0.214870 + 0.976643i $$0.568933\pi$$
$$402$$ 0 0
$$403$$ 26.1194 1.30110
$$404$$ 0 0
$$405$$ 8.18335 0.406634
$$406$$ 0 0
$$407$$ 13.5778 0.673026
$$408$$ 0 0
$$409$$ −25.9083 −1.28108 −0.640542 0.767923i $$-0.721290\pi$$
−0.640542 + 0.767923i $$0.721290\pi$$
$$410$$ 0 0
$$411$$ 1.18335 0.0583702
$$412$$ 0 0
$$413$$ −11.2111 −0.551662
$$414$$ 0 0
$$415$$ −11.2111 −0.550331
$$416$$ 0 0
$$417$$ 3.81665 0.186902
$$418$$ 0 0
$$419$$ −3.63331 −0.177499 −0.0887493 0.996054i $$-0.528287\pi$$
−0.0887493 + 0.996054i $$0.528287\pi$$
$$420$$ 0 0
$$421$$ 30.6972 1.49609 0.748046 0.663647i $$-0.230992\pi$$
0.748046 + 0.663647i $$0.230992\pi$$
$$422$$ 0 0
$$423$$ −7.57779 −0.368445
$$424$$ 0 0
$$425$$ 6.90833 0.335103
$$426$$ 0 0
$$427$$ −38.0278 −1.84029
$$428$$ 0 0
$$429$$ 1.69722 0.0819428
$$430$$ 0 0
$$431$$ 30.2389 1.45655 0.728277 0.685283i $$-0.240321\pi$$
0.728277 + 0.685283i $$0.240321\pi$$
$$432$$ 0 0
$$433$$ −24.0917 −1.15777 −0.578886 0.815409i $$-0.696512\pi$$
−0.578886 + 0.815409i $$0.696512\pi$$
$$434$$ 0 0
$$435$$ −0.788897 −0.0378247
$$436$$ 0 0
$$437$$ −5.90833 −0.282634
$$438$$ 0 0
$$439$$ 14.6972 0.701460 0.350730 0.936477i $$-0.385933\pi$$
0.350730 + 0.936477i $$0.385933\pi$$
$$440$$ 0 0
$$441$$ −11.3667 −0.541271
$$442$$ 0 0
$$443$$ −17.4861 −0.830791 −0.415395 0.909641i $$-0.636357\pi$$
−0.415395 + 0.909641i $$0.636357\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 4.02776 0.190506
$$448$$ 0 0
$$449$$ −2.09167 −0.0987122 −0.0493561 0.998781i $$-0.515717\pi$$
−0.0493561 + 0.998781i $$0.515717\pi$$
$$450$$ 0 0
$$451$$ 1.54163 0.0725927
$$452$$ 0 0
$$453$$ −2.69722 −0.126727
$$454$$ 0 0
$$455$$ −10.9083 −0.511390
$$456$$ 0 0
$$457$$ −32.4222 −1.51665 −0.758323 0.651879i $$-0.773981\pi$$
−0.758323 + 0.651879i $$0.773981\pi$$
$$458$$ 0 0
$$459$$ −12.3583 −0.576836
$$460$$ 0 0
$$461$$ 10.1833 0.474286 0.237143 0.971475i $$-0.423789\pi$$
0.237143 + 0.971475i $$0.423789\pi$$
$$462$$ 0 0
$$463$$ −17.6333 −0.819489 −0.409745 0.912200i $$-0.634382\pi$$
−0.409745 + 0.912200i $$0.634382\pi$$
$$464$$ 0 0
$$465$$ 2.39445 0.111040
$$466$$ 0 0
$$467$$ 1.81665 0.0840647 0.0420324 0.999116i $$-0.486617\pi$$
0.0420324 + 0.999116i $$0.486617\pi$$
$$468$$ 0 0
$$469$$ −13.2111 −0.610032
$$470$$ 0 0
$$471$$ −5.63331 −0.259569
$$472$$ 0 0
$$473$$ 15.6333 0.718820
$$474$$ 0 0
$$475$$ −5.90833 −0.271093
$$476$$ 0 0
$$477$$ 32.6056 1.49291
$$478$$ 0 0
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 0 0
$$481$$ 26.4222 1.20475
$$482$$ 0 0
$$483$$ −1.00000 −0.0455016
$$484$$ 0 0
$$485$$ 6.30278 0.286194
$$486$$ 0 0
$$487$$ −9.81665 −0.444835 −0.222418 0.974952i $$-0.571395\pi$$
−0.222418 + 0.974952i $$0.571395\pi$$
$$488$$ 0 0
$$489$$ −2.81665 −0.127373
$$490$$ 0 0
$$491$$ 4.18335 0.188792 0.0943959 0.995535i $$-0.469908\pi$$
0.0943959 + 0.995535i $$0.469908\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ 0 0
$$495$$ −4.93608 −0.221860
$$496$$ 0 0
$$497$$ −53.8444 −2.41525
$$498$$ 0 0
$$499$$ 31.6333 1.41610 0.708051 0.706162i $$-0.249575\pi$$
0.708051 + 0.706162i $$0.249575\pi$$
$$500$$ 0 0
$$501$$ −2.05551 −0.0918335
$$502$$ 0 0
$$503$$ −29.7250 −1.32537 −0.662686 0.748898i $$-0.730583\pi$$
−0.662686 + 0.748898i $$0.730583\pi$$
$$504$$ 0 0
$$505$$ 2.60555 0.115946
$$506$$ 0 0
$$507$$ −0.633308 −0.0281262
$$508$$ 0 0
$$509$$ −35.4500 −1.57129 −0.785646 0.618676i $$-0.787669\pi$$
−0.785646 + 0.618676i $$0.787669\pi$$
$$510$$ 0 0
$$511$$ 19.2111 0.849849
$$512$$ 0 0
$$513$$ 10.5694 0.466650
$$514$$ 0 0
$$515$$ −8.11943 −0.357785
$$516$$ 0 0
$$517$$ 4.42221 0.194488
$$518$$ 0 0
$$519$$ 5.96384 0.261784
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 20.4222 0.893001 0.446500 0.894783i $$-0.352670\pi$$
0.446500 + 0.894783i $$0.352670\pi$$
$$524$$ 0 0
$$525$$ −1.00000 −0.0436436
$$526$$ 0 0
$$527$$ 54.6333 2.37986
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −9.87217 −0.428416
$$532$$ 0 0
$$533$$ 3.00000 0.129944
$$534$$ 0 0
$$535$$ 2.60555 0.112648
$$536$$ 0 0
$$537$$ −2.84441 −0.122745
$$538$$ 0 0
$$539$$ 6.63331 0.285717
$$540$$ 0 0
$$541$$ 28.8444 1.24012 0.620059 0.784555i $$-0.287109\pi$$
0.620059 + 0.784555i $$0.287109\pi$$
$$542$$ 0 0
$$543$$ 5.18335 0.222439
$$544$$ 0 0
$$545$$ 1.48612 0.0636585
$$546$$ 0 0
$$547$$ 10.5139 0.449541 0.224770 0.974412i $$-0.427837\pi$$
0.224770 + 0.974412i $$0.427837\pi$$
$$548$$ 0 0
$$549$$ −33.4861 −1.42915
$$550$$ 0 0
$$551$$ 15.3944 0.655826
$$552$$ 0 0
$$553$$ −47.6333 −2.02557
$$554$$ 0 0
$$555$$ 2.42221 0.102817
$$556$$ 0 0
$$557$$ 22.4222 0.950059 0.475030 0.879970i $$-0.342437\pi$$
0.475030 + 0.879970i $$0.342437\pi$$
$$558$$ 0 0
$$559$$ 30.4222 1.28672
$$560$$ 0 0
$$561$$ 3.55004 0.149883
$$562$$ 0 0
$$563$$ 3.63331 0.153126 0.0765628 0.997065i $$-0.475605\pi$$
0.0765628 + 0.997065i $$0.475605\pi$$
$$564$$ 0 0
$$565$$ −16.4222 −0.690887
$$566$$ 0 0
$$567$$ −27.0278 −1.13506
$$568$$ 0 0
$$569$$ 28.4222 1.19152 0.595760 0.803162i $$-0.296851\pi$$
0.595760 + 0.803162i $$0.296851\pi$$
$$570$$ 0 0
$$571$$ 16.1194 0.674577 0.337289 0.941401i $$-0.390490\pi$$
0.337289 + 0.941401i $$0.390490\pi$$
$$572$$ 0 0
$$573$$ 2.60555 0.108848
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ −5.39445 −0.224186
$$580$$ 0 0
$$581$$ 37.0278 1.53617
$$582$$ 0 0
$$583$$ −19.0278 −0.788049
$$584$$ 0 0
$$585$$ −9.60555 −0.397141
$$586$$ 0 0
$$587$$ −16.5416 −0.682746 −0.341373 0.939928i $$-0.610892\pi$$
−0.341373 + 0.939928i $$0.610892\pi$$
$$588$$ 0 0
$$589$$ −46.7250 −1.92527
$$590$$ 0 0
$$591$$ 1.30278 0.0535890
$$592$$ 0 0
$$593$$ −1.81665 −0.0746010 −0.0373005 0.999304i $$-0.511876\pi$$
−0.0373005 + 0.999304i $$0.511876\pi$$
$$594$$ 0 0
$$595$$ −22.8167 −0.935392
$$596$$ 0 0
$$597$$ 6.18335 0.253068
$$598$$ 0 0
$$599$$ 35.3305 1.44357 0.721783 0.692119i $$-0.243323\pi$$
0.721783 + 0.692119i $$0.243323\pi$$
$$600$$ 0 0
$$601$$ 42.9361 1.75140 0.875700 0.482856i $$-0.160401\pi$$
0.875700 + 0.482856i $$0.160401\pi$$
$$602$$ 0 0
$$603$$ −11.6333 −0.473745
$$604$$ 0 0
$$605$$ −8.11943 −0.330102
$$606$$ 0 0
$$607$$ −46.0555 −1.86934 −0.934668 0.355522i $$-0.884303\pi$$
−0.934668 + 0.355522i $$0.884303\pi$$
$$608$$ 0 0
$$609$$ 2.60555 0.105582
$$610$$ 0 0
$$611$$ 8.60555 0.348143
$$612$$ 0 0
$$613$$ 3.57779 0.144506 0.0722529 0.997386i $$-0.476981\pi$$
0.0722529 + 0.997386i $$0.476981\pi$$
$$614$$ 0 0
$$615$$ 0.275019 0.0110898
$$616$$ 0 0
$$617$$ 18.9083 0.761221 0.380610 0.924736i $$-0.375714\pi$$
0.380610 + 0.924736i $$0.375714\pi$$
$$618$$ 0 0
$$619$$ 12.3305 0.495606 0.247803 0.968810i $$-0.420291\pi$$
0.247803 + 0.968810i $$0.420291\pi$$
$$620$$ 0 0
$$621$$ −1.78890 −0.0717860
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −3.03616 −0.121253
$$628$$ 0 0
$$629$$ 55.2666 2.20362
$$630$$ 0 0
$$631$$ −23.3944 −0.931318 −0.465659 0.884964i $$-0.654183\pi$$
−0.465659 + 0.884964i $$0.654183\pi$$
$$632$$ 0 0
$$633$$ −2.18335 −0.0867802
$$634$$ 0 0
$$635$$ −9.81665 −0.389562
$$636$$ 0 0
$$637$$ 12.9083 0.511447
$$638$$ 0 0
$$639$$ −47.4138 −1.87566
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ 0 0
$$643$$ 34.2389 1.35025 0.675124 0.737704i $$-0.264090\pi$$
0.675124 + 0.737704i $$0.264090\pi$$
$$644$$ 0 0
$$645$$ 2.78890 0.109813
$$646$$ 0 0
$$647$$ −26.8444 −1.05536 −0.527681 0.849442i $$-0.676938\pi$$
−0.527681 + 0.849442i $$0.676938\pi$$
$$648$$ 0 0
$$649$$ 5.76114 0.226145
$$650$$ 0 0
$$651$$ −7.90833 −0.309952
$$652$$ 0 0
$$653$$ 41.7250 1.63282 0.816412 0.577469i $$-0.195960\pi$$
0.816412 + 0.577469i $$0.195960\pi$$
$$654$$ 0 0
$$655$$ 11.2111 0.438054
$$656$$ 0 0
$$657$$ 16.9167 0.659985
$$658$$ 0 0
$$659$$ −15.6333 −0.608987 −0.304494 0.952514i $$-0.598487\pi$$
−0.304494 + 0.952514i $$0.598487\pi$$
$$660$$ 0 0
$$661$$ −34.9083 −1.35778 −0.678888 0.734242i $$-0.737538\pi$$
−0.678888 + 0.734242i $$0.737538\pi$$
$$662$$ 0 0
$$663$$ 6.90833 0.268297
$$664$$ 0 0
$$665$$ 19.5139 0.756716
$$666$$ 0 0
$$667$$ −2.60555 −0.100887
$$668$$ 0 0
$$669$$ 1.21110 0.0468239
$$670$$ 0 0
$$671$$ 19.5416 0.754396
$$672$$ 0 0
$$673$$ −37.6333 −1.45066 −0.725329 0.688403i $$-0.758312\pi$$
−0.725329 + 0.688403i $$0.758312\pi$$
$$674$$ 0 0
$$675$$ −1.78890 −0.0688547
$$676$$ 0 0
$$677$$ 16.4222 0.631157 0.315578 0.948900i $$-0.397802\pi$$
0.315578 + 0.948900i $$0.397802\pi$$
$$678$$ 0 0
$$679$$ −20.8167 −0.798870
$$680$$ 0 0
$$681$$ 4.42221 0.169459
$$682$$ 0 0
$$683$$ 0.275019 0.0105233 0.00526166 0.999986i $$-0.498325\pi$$
0.00526166 + 0.999986i $$0.498325\pi$$
$$684$$ 0 0
$$685$$ 3.90833 0.149329
$$686$$ 0 0
$$687$$ 0.605551 0.0231032
$$688$$ 0 0
$$689$$ −37.0278 −1.41065
$$690$$ 0 0
$$691$$ −51.8167 −1.97120 −0.985599 0.169098i $$-0.945914\pi$$
−0.985599 + 0.169098i $$0.945914\pi$$
$$692$$ 0 0
$$693$$ 16.3028 0.619291
$$694$$ 0 0
$$695$$ 12.6056 0.478156
$$696$$ 0 0
$$697$$ 6.27502 0.237683
$$698$$ 0 0
$$699$$ 7.81665 0.295653
$$700$$ 0 0
$$701$$ 32.0917 1.21209 0.606043 0.795432i $$-0.292756\pi$$
0.606043 + 0.795432i $$0.292756\pi$$
$$702$$ 0 0
$$703$$ −47.2666 −1.78269
$$704$$ 0 0
$$705$$ 0.788897 0.0297116
$$706$$ 0 0
$$707$$ −8.60555 −0.323645
$$708$$ 0 0
$$709$$ −15.8806 −0.596407 −0.298204 0.954502i $$-0.596387\pi$$
−0.298204 + 0.954502i $$0.596387\pi$$
$$710$$ 0 0
$$711$$ −41.9445 −1.57304
$$712$$ 0 0
$$713$$ 7.90833 0.296169
$$714$$ 0 0
$$715$$ 5.60555 0.209636
$$716$$ 0 0
$$717$$ −1.57779 −0.0589238
$$718$$ 0 0
$$719$$ −10.6972 −0.398939 −0.199470 0.979904i $$-0.563922\pi$$
−0.199470 + 0.979904i $$0.563922\pi$$
$$720$$ 0 0
$$721$$ 26.8167 0.998704
$$722$$ 0 0
$$723$$ −4.36669 −0.162399
$$724$$ 0 0
$$725$$ −2.60555 −0.0967677
$$726$$ 0 0
$$727$$ −2.90833 −0.107864 −0.0539319 0.998545i $$-0.517175\pi$$
−0.0539319 + 0.998545i $$0.517175\pi$$
$$728$$ 0 0
$$729$$ −22.1749 −0.821294
$$730$$ 0 0
$$731$$ 63.6333 2.35356
$$732$$ 0 0
$$733$$ 29.6333 1.09453 0.547266 0.836959i $$-0.315669\pi$$
0.547266 + 0.836959i $$0.315669\pi$$
$$734$$ 0 0
$$735$$ 1.18335 0.0436484
$$736$$ 0 0
$$737$$ 6.78890 0.250072
$$738$$ 0 0
$$739$$ −35.6333 −1.31079 −0.655396 0.755285i $$-0.727498\pi$$
−0.655396 + 0.755285i $$0.727498\pi$$
$$740$$ 0 0
$$741$$ −5.90833 −0.217048
$$742$$ 0 0
$$743$$ 32.3305 1.18609 0.593046 0.805169i $$-0.297925\pi$$
0.593046 + 0.805169i $$0.297925\pi$$
$$744$$ 0 0
$$745$$ 13.3028 0.487376
$$746$$ 0 0
$$747$$ 32.6056 1.19297
$$748$$ 0 0
$$749$$ −8.60555 −0.314440
$$750$$ 0 0
$$751$$ −21.8167 −0.796101 −0.398051 0.917364i $$-0.630313\pi$$
−0.398051 + 0.917364i $$0.630313\pi$$
$$752$$ 0 0
$$753$$ −3.78890 −0.138075
$$754$$ 0 0
$$755$$ −8.90833 −0.324207
$$756$$ 0 0
$$757$$ 13.2111 0.480166 0.240083 0.970752i $$-0.422825\pi$$
0.240083 + 0.970752i $$0.422825\pi$$
$$758$$ 0 0
$$759$$ 0.513878 0.0186526
$$760$$ 0 0
$$761$$ −49.5416 −1.79588 −0.897941 0.440115i $$-0.854938\pi$$
−0.897941 + 0.440115i $$0.854938\pi$$
$$762$$ 0 0
$$763$$ −4.90833 −0.177693
$$764$$ 0 0
$$765$$ −20.0917 −0.726416
$$766$$ 0 0
$$767$$ 11.2111 0.404809
$$768$$ 0 0
$$769$$ 45.2666 1.63236 0.816178 0.577801i $$-0.196089\pi$$
0.816178 + 0.577801i $$0.196089\pi$$
$$770$$ 0 0
$$771$$ −0.550039 −0.0198092
$$772$$ 0 0
$$773$$ 12.0000 0.431610 0.215805 0.976436i $$-0.430762\pi$$
0.215805 + 0.976436i $$0.430762\pi$$
$$774$$ 0 0
$$775$$ 7.90833 0.284075
$$776$$ 0 0
$$777$$ −8.00000 −0.286998
$$778$$ 0 0
$$779$$ −5.36669 −0.192282
$$780$$ 0 0
$$781$$ 27.6695 0.990091
$$782$$ 0 0
$$783$$ 4.66106 0.166573
$$784$$ 0 0
$$785$$ −18.6056 −0.664061
$$786$$ 0 0
$$787$$ −37.4500 −1.33495 −0.667473 0.744634i $$-0.732624\pi$$
−0.667473 + 0.744634i $$0.732624\pi$$
$$788$$ 0 0
$$789$$ −1.06392 −0.0378764
$$790$$ 0 0
$$791$$ 54.2389 1.92851
$$792$$ 0 0
$$793$$ 38.0278 1.35041
$$794$$ 0 0
$$795$$ −3.39445 −0.120389
$$796$$ 0 0
$$797$$ 1.81665 0.0643492 0.0321746 0.999482i $$-0.489757\pi$$
0.0321746 + 0.999482i $$0.489757\pi$$
$$798$$ 0 0
$$799$$ 18.0000 0.636794
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −9.87217 −0.348381
$$804$$ 0 0
$$805$$ −3.30278 −0.116408
$$806$$ 0 0
$$807$$ 1.26662 0.0445870
$$808$$ 0 0
$$809$$ 50.7250 1.78340 0.891698 0.452631i $$-0.149515\pi$$
0.891698 + 0.452631i $$0.149515\pi$$
$$810$$ 0 0
$$811$$ 41.0278 1.44068 0.720340 0.693621i $$-0.243986\pi$$
0.720340 + 0.693621i $$0.243986\pi$$
$$812$$ 0 0
$$813$$ 0.816654 0.0286413
$$814$$ 0 0
$$815$$ −9.30278 −0.325862
$$816$$ 0 0
$$817$$ −54.4222 −1.90399
$$818$$ 0 0
$$819$$ 31.7250 1.10856
$$820$$ 0 0
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 0 0
$$823$$ 15.2111 0.530226 0.265113 0.964217i $$-0.414591\pi$$
0.265113 + 0.964217i $$0.414591\pi$$
$$824$$ 0 0
$$825$$ 0.513878 0.0178909
$$826$$ 0 0
$$827$$ 29.4500 1.02408 0.512038 0.858963i $$-0.328891\pi$$
0.512038 + 0.858963i $$0.328891\pi$$
$$828$$ 0 0
$$829$$ 31.2111 1.08401 0.542003 0.840376i $$-0.317666\pi$$
0.542003 + 0.840376i $$0.317666\pi$$
$$830$$ 0 0
$$831$$ −8.23886 −0.285803
$$832$$ 0 0
$$833$$ 27.0000 0.935495
$$834$$ 0 0
$$835$$ −6.78890 −0.234939
$$836$$ 0 0
$$837$$ −14.1472 −0.488998
$$838$$ 0 0
$$839$$ 43.8167 1.51272 0.756359 0.654156i $$-0.226976\pi$$
0.756359 + 0.654156i $$0.226976\pi$$
$$840$$ 0 0
$$841$$ −22.2111 −0.765900
$$842$$ 0 0
$$843$$ −8.05551 −0.277447
$$844$$ 0 0
$$845$$ −2.09167 −0.0719557
$$846$$ 0 0
$$847$$ 26.8167 0.921431
$$848$$ 0 0
$$849$$ −0.605551 −0.0207825
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ −21.7250 −0.743849 −0.371925 0.928263i $$-0.621302\pi$$
−0.371925 + 0.928263i $$0.621302\pi$$
$$854$$ 0 0
$$855$$ 17.1833 0.587658
$$856$$ 0 0
$$857$$ 9.63331 0.329068 0.164534 0.986371i $$-0.447388\pi$$
0.164534 + 0.986371i $$0.447388\pi$$
$$858$$ 0 0
$$859$$ 35.8167 1.22205 0.611024 0.791612i $$-0.290758\pi$$
0.611024 + 0.791612i $$0.290758\pi$$
$$860$$ 0 0
$$861$$ −0.908327 −0.0309557
$$862$$ 0 0
$$863$$ 41.4500 1.41097 0.705487 0.708723i $$-0.250729\pi$$
0.705487 + 0.708723i $$0.250729\pi$$
$$864$$ 0 0
$$865$$ 19.6972 0.669726
$$866$$ 0 0
$$867$$ 9.30278 0.315939
$$868$$ 0 0
$$869$$ 24.4777 0.830350
$$870$$ 0 0
$$871$$ 13.2111 0.447641
$$872$$ 0 0
$$873$$ −18.3305 −0.620395
$$874$$ 0 0
$$875$$ −3.30278 −0.111654
$$876$$ 0 0
$$877$$ −48.1749 −1.62675 −0.813376 0.581738i $$-0.802373\pi$$
−0.813376 + 0.581738i $$0.802373\pi$$
$$878$$ 0 0
$$879$$ 7.02776 0.237040
$$880$$ 0 0
$$881$$ 55.2666 1.86198 0.930990 0.365045i $$-0.118946\pi$$
0.930990 + 0.365045i $$0.118946\pi$$
$$882$$ 0 0
$$883$$ −8.27502 −0.278477 −0.139238 0.990259i $$-0.544465\pi$$
−0.139238 + 0.990259i $$0.544465\pi$$
$$884$$ 0 0
$$885$$ 1.02776 0.0345477
$$886$$ 0 0
$$887$$ −27.6333 −0.927836 −0.463918 0.885878i $$-0.653557\pi$$
−0.463918 + 0.885878i $$0.653557\pi$$
$$888$$ 0 0
$$889$$ 32.4222 1.08741
$$890$$ 0 0
$$891$$ 13.8890 0.465298
$$892$$ 0 0
$$893$$ −15.3944 −0.515156
$$894$$ 0 0
$$895$$ −9.39445 −0.314022
$$896$$ 0 0
$$897$$ 1.00000 0.0333890
$$898$$ 0 0
$$899$$ −20.6056 −0.687234
$$900$$ 0 0
$$901$$ −77.4500 −2.58023
$$902$$ 0 0
$$903$$ −9.21110 −0.306526
$$904$$ 0 0
$$905$$ 17.1194 0.569069
$$906$$ 0 0
$$907$$ −48.6611 −1.61576 −0.807882 0.589344i $$-0.799386\pi$$
−0.807882 + 0.589344i $$0.799386\pi$$
$$908$$ 0 0
$$909$$ −7.57779 −0.251340
$$910$$ 0 0
$$911$$ 4.18335 0.138600 0.0693002 0.997596i $$-0.477923\pi$$
0.0693002 + 0.997596i $$0.477923\pi$$
$$912$$ 0 0
$$913$$ −19.0278 −0.629727
$$914$$ 0 0
$$915$$ 3.48612 0.115248
$$916$$ 0 0
$$917$$ −37.0278 −1.22276
$$918$$ 0 0
$$919$$ −44.0000 −1.45143 −0.725713 0.687998i $$-0.758490\pi$$
−0.725713 + 0.687998i $$0.758490\pi$$
$$920$$ 0 0
$$921$$ 3.54163 0.116701
$$922$$ 0 0
$$923$$ 53.8444 1.77231
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 0 0
$$927$$ 23.6140 0.775584
$$928$$ 0 0
$$929$$ −14.3667 −0.471356 −0.235678 0.971831i $$-0.575731\pi$$
−0.235678 + 0.971831i $$0.575731\pi$$
$$930$$ 0 0
$$931$$ −23.0917 −0.756799
$$932$$ 0 0
$$933$$ −6.78890 −0.222259
$$934$$ 0 0
$$935$$ 11.7250 0.383448
$$936$$ 0 0
$$937$$ 37.9638 1.24022 0.620112 0.784513i $$-0.287087\pi$$
0.620112 + 0.784513i $$0.287087\pi$$
$$938$$ 0 0
$$939$$ 5.97224 0.194897
$$940$$ 0 0
$$941$$ 25.9361 0.845492 0.422746 0.906248i $$-0.361066\pi$$
0.422746 + 0.906248i $$0.361066\pi$$
$$942$$ 0 0
$$943$$ 0.908327 0.0295792
$$944$$ 0 0
$$945$$ 5.90833 0.192198
$$946$$ 0 0
$$947$$ 4.93608 0.160401 0.0802006 0.996779i $$-0.474444\pi$$
0.0802006 + 0.996779i $$0.474444\pi$$
$$948$$ 0 0
$$949$$ −19.2111 −0.623619
$$950$$ 0 0
$$951$$ 5.36669 0.174027
$$952$$ 0 0
$$953$$ −41.3305 −1.33883 −0.669414 0.742890i $$-0.733455\pi$$
−0.669414 + 0.742890i $$0.733455\pi$$
$$954$$ 0 0
$$955$$ 8.60555 0.278469
$$956$$ 0 0
$$957$$ −1.33894 −0.0432817
$$958$$ 0 0
$$959$$ −12.9083 −0.416832
$$960$$ 0 0
$$961$$ 31.5416 1.01747
$$962$$ 0 0
$$963$$ −7.57779 −0.244191
$$964$$ 0 0
$$965$$ −17.8167 −0.573538
$$966$$ 0 0
$$967$$ 12.6056 0.405367 0.202684 0.979244i $$-0.435034\pi$$
0.202684 + 0.979244i $$0.435034\pi$$
$$968$$ 0 0
$$969$$ −12.3583 −0.397005
$$970$$ 0 0
$$971$$ 17.0917 0.548498 0.274249 0.961659i $$-0.411571\pi$$
0.274249 + 0.961659i $$0.411571\pi$$
$$972$$ 0 0
$$973$$ −41.6333 −1.33470
$$974$$ 0 0
$$975$$ 1.00000 0.0320256
$$976$$ 0 0
$$977$$ 6.51388 0.208397 0.104199 0.994556i $$-0.466772\pi$$
0.104199 + 0.994556i $$0.466772\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −4.32213 −0.137995
$$982$$ 0 0
$$983$$ −34.5416 −1.10171 −0.550854 0.834602i $$-0.685698\pi$$
−0.550854 + 0.834602i $$0.685698\pi$$
$$984$$ 0 0
$$985$$ 4.30278 0.137098
$$986$$ 0 0
$$987$$ −2.60555 −0.0829356
$$988$$ 0 0
$$989$$ 9.21110 0.292896
$$990$$ 0 0
$$991$$ 15.3305 0.486990 0.243495 0.969902i $$-0.421706\pi$$
0.243495 + 0.969902i $$0.421706\pi$$
$$992$$ 0 0
$$993$$ −5.02776 −0.159551
$$994$$ 0 0
$$995$$ 20.4222 0.647427
$$996$$ 0 0
$$997$$ −16.7889 −0.531710 −0.265855 0.964013i $$-0.585654\pi$$
−0.265855 + 0.964013i $$0.585654\pi$$
$$998$$ 0 0
$$999$$ −14.3112 −0.452786
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 1840.2.a.j.1.2 2
4.3 odd 2 230.2.a.b.1.1 2
5.4 even 2 9200.2.a.ca.1.1 2
8.3 odd 2 7360.2.a.bc.1.2 2
8.5 even 2 7360.2.a.bu.1.1 2
12.11 even 2 2070.2.a.w.1.2 2
20.3 even 4 1150.2.b.f.599.3 4
20.7 even 4 1150.2.b.f.599.2 4
20.19 odd 2 1150.2.a.m.1.2 2
92.91 even 2 5290.2.a.j.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 2 4.3 odd 2
1150.2.a.m.1.2 2 20.19 odd 2
1150.2.b.f.599.2 4 20.7 even 4
1150.2.b.f.599.3 4 20.3 even 4
1840.2.a.j.1.2 2 1.1 even 1 trivial
2070.2.a.w.1.2 2 12.11 even 2
5290.2.a.j.1.1 2 92.91 even 2
7360.2.a.bc.1.2 2 8.3 odd 2
7360.2.a.bu.1.1 2 8.5 even 2
9200.2.a.ca.1.1 2 5.4 even 2