# Properties

 Label 6840.2.a.bg.1.2 Level $6840$ Weight $2$ Character 6840.1 Self dual yes Analytic conductor $54.618$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6840.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: $$54.6176749826$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.86081$$ of defining polynomial Character $$\chi$$ $$=$$ 6840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -1.39821 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} -1.39821 q^{7} +4.32340 q^{13} -0.601793 q^{17} +1.00000 q^{19} -6.04502 q^{23} +1.00000 q^{25} -4.60179 q^{29} +2.79641 q^{31} +1.39821 q^{35} +1.07480 q^{37} +5.44322 q^{41} -8.64681 q^{43} +1.85039 q^{47} -5.04502 q^{49} +3.11982 q^{53} +6.69182 q^{59} -2.64681 q^{61} -4.32340 q^{65} -14.4134 q^{67} +5.59283 q^{71} +12.6918 q^{73} -4.64681 q^{79} +1.20359 q^{83} +0.601793 q^{85} -9.44322 q^{89} -6.04502 q^{91} -1.00000 q^{95} +4.51803 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 - q^7 $$3 q - 3 q^{5} - q^{7} + 5 q^{13} - 5 q^{17} + 3 q^{19} + q^{23} + 3 q^{25} - 17 q^{29} + 2 q^{31} + q^{35} + 8 q^{37} - 6 q^{41} - 10 q^{43} - 4 q^{47} + 4 q^{49} - 5 q^{53} - 15 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} + 4 q^{71} + 3 q^{73} + 2 q^{79} + 10 q^{83} + 5 q^{85} - 6 q^{89} + q^{91} - 3 q^{95} - 4 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 - q^7 + 5 * q^13 - 5 * q^17 + 3 * q^19 + q^23 + 3 * q^25 - 17 * q^29 + 2 * q^31 + q^35 + 8 * q^37 - 6 * q^41 - 10 * q^43 - 4 * q^47 + 4 * q^49 - 5 * q^53 - 15 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 + 4 * q^71 + 3 * q^73 + 2 * q^79 + 10 * q^83 + 5 * q^85 - 6 * q^89 + q^91 - 3 * q^95 - 4 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.39821 −0.528473 −0.264236 0.964458i $$-0.585120\pi$$
−0.264236 + 0.964458i $$0.585120\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 4.32340 1.19910 0.599548 0.800339i $$-0.295347\pi$$
0.599548 + 0.800339i $$0.295347\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −0.601793 −0.145956 −0.0729781 0.997334i $$-0.523250\pi$$
−0.0729781 + 0.997334i $$0.523250\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.04502 −1.26047 −0.630236 0.776403i $$-0.717042\pi$$
−0.630236 + 0.776403i $$0.717042\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −4.60179 −0.854531 −0.427266 0.904126i $$-0.640523\pi$$
−0.427266 + 0.904126i $$0.640523\pi$$
$$30$$ 0 0
$$31$$ 2.79641 0.502251 0.251125 0.967955i $$-0.419199\pi$$
0.251125 + 0.967955i $$0.419199\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 1.39821 0.236340
$$36$$ 0 0
$$37$$ 1.07480 0.176697 0.0883483 0.996090i $$-0.471841\pi$$
0.0883483 + 0.996090i $$0.471841\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.44322 0.850089 0.425044 0.905173i $$-0.360259\pi$$
0.425044 + 0.905173i $$0.360259\pi$$
$$42$$ 0 0
$$43$$ −8.64681 −1.31863 −0.659313 0.751869i $$-0.729153\pi$$
−0.659313 + 0.751869i $$0.729153\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.85039 0.269908 0.134954 0.990852i $$-0.456911\pi$$
0.134954 + 0.990852i $$0.456911\pi$$
$$48$$ 0 0
$$49$$ −5.04502 −0.720717
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 3.11982 0.428540 0.214270 0.976774i $$-0.431263\pi$$
0.214270 + 0.976774i $$0.431263\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 6.69182 0.871201 0.435601 0.900140i $$-0.356536\pi$$
0.435601 + 0.900140i $$0.356536\pi$$
$$60$$ 0 0
$$61$$ −2.64681 −0.338889 −0.169445 0.985540i $$-0.554197\pi$$
−0.169445 + 0.985540i $$0.554197\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −4.32340 −0.536252
$$66$$ 0 0
$$67$$ −14.4134 −1.76088 −0.880441 0.474156i $$-0.842753\pi$$
−0.880441 + 0.474156i $$0.842753\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.59283 0.663747 0.331873 0.943324i $$-0.392319\pi$$
0.331873 + 0.943324i $$0.392319\pi$$
$$72$$ 0 0
$$73$$ 12.6918 1.48547 0.742733 0.669588i $$-0.233529\pi$$
0.742733 + 0.669588i $$0.233529\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.64681 −0.522807 −0.261403 0.965230i $$-0.584185\pi$$
−0.261403 + 0.965230i $$0.584185\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.20359 0.132111 0.0660553 0.997816i $$-0.478959\pi$$
0.0660553 + 0.997816i $$0.478959\pi$$
$$84$$ 0 0
$$85$$ 0.601793 0.0652736
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −9.44322 −1.00098 −0.500490 0.865742i $$-0.666847\pi$$
−0.500490 + 0.865742i $$0.666847\pi$$
$$90$$ 0 0
$$91$$ −6.04502 −0.633690
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 4.51803 0.458736 0.229368 0.973340i $$-0.426334\pi$$
0.229368 + 0.973340i $$0.426334\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 5.05398 0.502890 0.251445 0.967872i $$-0.419094\pi$$
0.251445 + 0.967872i $$0.419094\pi$$
$$102$$ 0 0
$$103$$ −11.0748 −1.09123 −0.545616 0.838035i $$-0.683704\pi$$
−0.545616 + 0.838035i $$0.683704\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.17380 0.403496 0.201748 0.979437i $$-0.435338\pi$$
0.201748 + 0.979437i $$0.435338\pi$$
$$108$$ 0 0
$$109$$ 14.8414 1.42155 0.710776 0.703419i $$-0.248344\pi$$
0.710776 + 0.703419i $$0.248344\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.72161 −0.161956 −0.0809778 0.996716i $$-0.525804\pi$$
−0.0809778 + 0.996716i $$0.525804\pi$$
$$114$$ 0 0
$$115$$ 6.04502 0.563701
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0.841431 0.0771338
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −2.12878 −0.188899 −0.0944494 0.995530i $$-0.530109\pi$$
−0.0944494 + 0.995530i $$0.530109\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1.59283 −0.139166 −0.0695831 0.997576i $$-0.522167\pi$$
−0.0695831 + 0.997576i $$0.522167\pi$$
$$132$$ 0 0
$$133$$ −1.39821 −0.121240
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −9.89541 −0.845422 −0.422711 0.906265i $$-0.638921\pi$$
−0.422711 + 0.906265i $$0.638921\pi$$
$$138$$ 0 0
$$139$$ 3.70079 0.313897 0.156948 0.987607i $$-0.449834\pi$$
0.156948 + 0.987607i $$0.449834\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 4.60179 0.382158
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −18.3476 −1.50309 −0.751547 0.659680i $$-0.770692\pi$$
−0.751547 + 0.659680i $$0.770692\pi$$
$$150$$ 0 0
$$151$$ 13.9404 1.13446 0.567228 0.823561i $$-0.308016\pi$$
0.567228 + 0.823561i $$0.308016\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2.79641 −0.224613
$$156$$ 0 0
$$157$$ −0.149606 −0.0119399 −0.00596994 0.999982i $$-0.501900\pi$$
−0.00596994 + 0.999982i $$0.501900\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.45219 0.666126
$$162$$ 0 0
$$163$$ −24.6468 −1.93049 −0.965244 0.261352i $$-0.915832\pi$$
−0.965244 + 0.261352i $$0.915832\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 18.8656 1.45987 0.729933 0.683519i $$-0.239551\pi$$
0.729933 + 0.683519i $$0.239551\pi$$
$$168$$ 0 0
$$169$$ 5.69182 0.437833
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.36842 0.484182 0.242091 0.970254i $$-0.422167\pi$$
0.242091 + 0.970254i $$0.422167\pi$$
$$174$$ 0 0
$$175$$ −1.39821 −0.105695
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −11.4432 −0.855307 −0.427653 0.903943i $$-0.640660\pi$$
−0.427653 + 0.903943i $$0.640660\pi$$
$$180$$ 0 0
$$181$$ −3.29362 −0.244813 −0.122406 0.992480i $$-0.539061\pi$$
−0.122406 + 0.992480i $$0.539061\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1.07480 −0.0790211
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −20.5422 −1.48638 −0.743191 0.669079i $$-0.766689\pi$$
−0.743191 + 0.669079i $$0.766689\pi$$
$$192$$ 0 0
$$193$$ −19.6620 −1.41530 −0.707652 0.706561i $$-0.750246\pi$$
−0.707652 + 0.706561i $$0.750246\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ −9.74580 −0.690862 −0.345431 0.938444i $$-0.612267\pi$$
−0.345431 + 0.938444i $$0.612267\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 6.43426 0.451597
$$204$$ 0 0
$$205$$ −5.44322 −0.380171
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −9.09899 −0.626401 −0.313200 0.949687i $$-0.601401\pi$$
−0.313200 + 0.949687i $$0.601401\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 8.64681 0.589707
$$216$$ 0 0
$$217$$ −3.90997 −0.265426
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.60179 −0.175016
$$222$$ 0 0
$$223$$ −13.8712 −0.928885 −0.464443 0.885603i $$-0.653745\pi$$
−0.464443 + 0.885603i $$0.653745\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −10.3234 −0.685188 −0.342594 0.939483i $$-0.611306\pi$$
−0.342594 + 0.939483i $$0.611306\pi$$
$$228$$ 0 0
$$229$$ −14.7368 −0.973838 −0.486919 0.873447i $$-0.661879\pi$$
−0.486919 + 0.873447i $$0.661879\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −13.1857 −0.863821 −0.431911 0.901916i $$-0.642160\pi$$
−0.431911 + 0.901916i $$0.642160\pi$$
$$234$$ 0 0
$$235$$ −1.85039 −0.120706
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0.452186 0.0292495 0.0146247 0.999893i $$-0.495345\pi$$
0.0146247 + 0.999893i $$0.495345\pi$$
$$240$$ 0 0
$$241$$ 13.5333 0.871754 0.435877 0.900006i $$-0.356438\pi$$
0.435877 + 0.900006i $$0.356438\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 5.04502 0.322314
$$246$$ 0 0
$$247$$ 4.32340 0.275092
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −21.2936 −1.34404 −0.672021 0.740532i $$-0.734573\pi$$
−0.672021 + 0.740532i $$0.734573\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 28.3989 1.77147 0.885737 0.464188i $$-0.153654\pi$$
0.885737 + 0.464188i $$0.153654\pi$$
$$258$$ 0 0
$$259$$ −1.50280 −0.0933793
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −11.1857 −0.689737 −0.344869 0.938651i $$-0.612077\pi$$
−0.344869 + 0.938651i $$0.612077\pi$$
$$264$$ 0 0
$$265$$ −3.11982 −0.191649
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −19.5928 −1.19460 −0.597298 0.802019i $$-0.703759\pi$$
−0.597298 + 0.802019i $$0.703759\pi$$
$$270$$ 0 0
$$271$$ 5.95498 0.361740 0.180870 0.983507i $$-0.442109\pi$$
0.180870 + 0.983507i $$0.442109\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −23.9404 −1.43844 −0.719220 0.694782i $$-0.755501\pi$$
−0.719220 + 0.694782i $$0.755501\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.7368 −0.640506 −0.320253 0.947332i $$-0.603768\pi$$
−0.320253 + 0.947332i $$0.603768\pi$$
$$282$$ 0 0
$$283$$ −16.0900 −0.956453 −0.478227 0.878237i $$-0.658720\pi$$
−0.478227 + 0.878237i $$0.658720\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −7.61076 −0.449249
$$288$$ 0 0
$$289$$ −16.6378 −0.978697
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −14.5630 −0.850782 −0.425391 0.905010i $$-0.639863\pi$$
−0.425391 + 0.905010i $$0.639863\pi$$
$$294$$ 0 0
$$295$$ −6.69182 −0.389613
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −26.1350 −1.51143
$$300$$ 0 0
$$301$$ 12.0900 0.696858
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 2.64681 0.151556
$$306$$ 0 0
$$307$$ 2.77559 0.158411 0.0792057 0.996858i $$-0.474762\pi$$
0.0792057 + 0.996858i $$0.474762\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7.63785 0.433102 0.216551 0.976271i $$-0.430519\pi$$
0.216551 + 0.976271i $$0.430519\pi$$
$$312$$ 0 0
$$313$$ −32.5243 −1.83838 −0.919191 0.393812i $$-0.871156\pi$$
−0.919191 + 0.393812i $$0.871156\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −10.8206 −0.607746 −0.303873 0.952713i $$-0.598280\pi$$
−0.303873 + 0.952713i $$0.598280\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −0.601793 −0.0334846
$$324$$ 0 0
$$325$$ 4.32340 0.239819
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −2.58723 −0.142639
$$330$$ 0 0
$$331$$ 14.6918 0.807536 0.403768 0.914861i $$-0.367700\pi$$
0.403768 + 0.914861i $$0.367700\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 14.4134 0.787490
$$336$$ 0 0
$$337$$ −1.07480 −0.0585483 −0.0292741 0.999571i $$-0.509320\pi$$
−0.0292741 + 0.999571i $$0.509320\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 16.8414 0.909352
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −32.0900 −1.72268 −0.861342 0.508026i $$-0.830375\pi$$
−0.861342 + 0.508026i $$0.830375\pi$$
$$348$$ 0 0
$$349$$ 31.4737 1.68475 0.842374 0.538894i $$-0.181158\pi$$
0.842374 + 0.538894i $$0.181158\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 11.4882 0.611457 0.305729 0.952119i $$-0.401100\pi$$
0.305729 + 0.952119i $$0.401100\pi$$
$$354$$ 0 0
$$355$$ −5.59283 −0.296837
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −26.0450 −1.37460 −0.687302 0.726372i $$-0.741205\pi$$
−0.687302 + 0.726372i $$0.741205\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −12.6918 −0.664320
$$366$$ 0 0
$$367$$ −21.5512 −1.12496 −0.562481 0.826810i $$-0.690153\pi$$
−0.562481 + 0.826810i $$0.690153\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −4.36215 −0.226472
$$372$$ 0 0
$$373$$ 24.9702 1.29291 0.646454 0.762953i $$-0.276251\pi$$
0.646454 + 0.762953i $$0.276251\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −19.8954 −1.02467
$$378$$ 0 0
$$379$$ 28.6323 1.47074 0.735370 0.677666i $$-0.237008\pi$$
0.735370 + 0.677666i $$0.237008\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −17.9612 −0.917777 −0.458889 0.888494i $$-0.651752\pi$$
−0.458889 + 0.888494i $$0.651752\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 6.09003 0.308777 0.154388 0.988010i $$-0.450659\pi$$
0.154388 + 0.988010i $$0.450659\pi$$
$$390$$ 0 0
$$391$$ 3.63785 0.183974
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 4.64681 0.233806
$$396$$ 0 0
$$397$$ 21.8325 1.09574 0.547870 0.836563i $$-0.315439\pi$$
0.547870 + 0.836563i $$0.315439\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12.7064 0.634526 0.317263 0.948338i $$-0.397236\pi$$
0.317263 + 0.948338i $$0.397236\pi$$
$$402$$ 0 0
$$403$$ 12.0900 0.602247
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 16.8864 0.834981 0.417491 0.908681i $$-0.362910\pi$$
0.417491 + 0.908681i $$0.362910\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −9.35656 −0.460406
$$414$$ 0 0
$$415$$ −1.20359 −0.0590817
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −27.7908 −1.35767 −0.678835 0.734291i $$-0.737515\pi$$
−0.678835 + 0.734291i $$0.737515\pi$$
$$420$$ 0 0
$$421$$ 15.0090 0.731492 0.365746 0.930715i $$-0.380814\pi$$
0.365746 + 0.930715i $$0.380814\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −0.601793 −0.0291912
$$426$$ 0 0
$$427$$ 3.70079 0.179094
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0.736841 0.0354924 0.0177462 0.999843i $$-0.494351\pi$$
0.0177462 + 0.999843i $$0.494351\pi$$
$$432$$ 0 0
$$433$$ 31.4945 1.51353 0.756765 0.653687i $$-0.226779\pi$$
0.756765 + 0.653687i $$0.226779\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.04502 −0.289172
$$438$$ 0 0
$$439$$ −23.2340 −1.10890 −0.554450 0.832217i $$-0.687071\pi$$
−0.554450 + 0.832217i $$0.687071\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −16.4793 −0.782954 −0.391477 0.920188i $$-0.628036\pi$$
−0.391477 + 0.920188i $$0.628036\pi$$
$$444$$ 0 0
$$445$$ 9.44322 0.447652
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2.29921 0.108507 0.0542533 0.998527i $$-0.482722\pi$$
0.0542533 + 0.998527i $$0.482722\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 6.04502 0.283395
$$456$$ 0 0
$$457$$ −20.0450 −0.937666 −0.468833 0.883287i $$-0.655325\pi$$
−0.468833 + 0.883287i $$0.655325\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −32.5872 −1.51774 −0.758869 0.651243i $$-0.774248\pi$$
−0.758869 + 0.651243i $$0.774248\pi$$
$$462$$ 0 0
$$463$$ 2.58723 0.120239 0.0601195 0.998191i $$-0.480852\pi$$
0.0601195 + 0.998191i $$0.480852\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0.299213 0.0138459 0.00692295 0.999976i $$-0.497796\pi$$
0.00692295 + 0.999976i $$0.497796\pi$$
$$468$$ 0 0
$$469$$ 20.1530 0.930578
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 35.0665 1.60223 0.801115 0.598511i $$-0.204241\pi$$
0.801115 + 0.598511i $$0.204241\pi$$
$$480$$ 0 0
$$481$$ 4.64681 0.211876
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.51803 −0.205153
$$486$$ 0 0
$$487$$ −41.7521 −1.89197 −0.945983 0.324215i $$-0.894900\pi$$
−0.945983 + 0.324215i $$0.894900\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 30.9765 1.39795 0.698974 0.715147i $$-0.253640\pi$$
0.698974 + 0.715147i $$0.253640\pi$$
$$492$$ 0 0
$$493$$ 2.76932 0.124724
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −7.81994 −0.350772
$$498$$ 0 0
$$499$$ −3.90997 −0.175034 −0.0875171 0.996163i $$-0.527893\pi$$
−0.0875171 + 0.996163i $$0.527893\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −16.4522 −0.733567 −0.366783 0.930306i $$-0.619541\pi$$
−0.366783 + 0.930306i $$0.619541\pi$$
$$504$$ 0 0
$$505$$ −5.05398 −0.224899
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 25.4432 1.12775 0.563876 0.825860i $$-0.309310\pi$$
0.563876 + 0.825860i $$0.309310\pi$$
$$510$$ 0 0
$$511$$ −17.7458 −0.785028
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 11.0748 0.488014
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 27.3836 1.19970 0.599850 0.800113i $$-0.295227\pi$$
0.599850 + 0.800113i $$0.295227\pi$$
$$522$$ 0 0
$$523$$ 41.5574 1.81718 0.908590 0.417689i $$-0.137160\pi$$
0.908590 + 0.417689i $$0.137160\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1.68286 −0.0733066
$$528$$ 0 0
$$529$$ 13.5422 0.588792
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 23.5333 1.01934
$$534$$ 0 0
$$535$$ −4.17380 −0.180449
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.6468 0.629715 0.314858 0.949139i $$-0.398043\pi$$
0.314858 + 0.949139i $$0.398043\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −14.8414 −0.635737
$$546$$ 0 0
$$547$$ −24.9252 −1.06572 −0.532862 0.846202i $$-0.678884\pi$$
−0.532862 + 0.846202i $$0.678884\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.60179 −0.196043
$$552$$ 0 0
$$553$$ 6.49720 0.276289
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 44.9348 1.90395 0.951975 0.306176i $$-0.0990496\pi$$
0.951975 + 0.306176i $$0.0990496\pi$$
$$558$$ 0 0
$$559$$ −37.3836 −1.58116
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 10.2605 0.432427 0.216213 0.976346i $$-0.430629\pi$$
0.216213 + 0.976346i $$0.430629\pi$$
$$564$$ 0 0
$$565$$ 1.72161 0.0724287
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −24.6773 −1.03452 −0.517262 0.855827i $$-0.673049\pi$$
−0.517262 + 0.855827i $$0.673049\pi$$
$$570$$ 0 0
$$571$$ −10.9765 −0.459351 −0.229676 0.973267i $$-0.573767\pi$$
−0.229676 + 0.973267i $$0.573767\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −6.04502 −0.252095
$$576$$ 0 0
$$577$$ −45.1295 −1.87876 −0.939382 0.342873i $$-0.888600\pi$$
−0.939382 + 0.342873i $$0.888600\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1.68286 −0.0698169
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −7.90997 −0.326479 −0.163240 0.986586i $$-0.552194\pi$$
−0.163240 + 0.986586i $$0.552194\pi$$
$$588$$ 0 0
$$589$$ 2.79641 0.115224
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 7.29362 0.299513 0.149756 0.988723i $$-0.452151\pi$$
0.149756 + 0.988723i $$0.452151\pi$$
$$594$$ 0 0
$$595$$ −0.841431 −0.0344953
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −20.6468 −0.843606 −0.421803 0.906688i $$-0.638603\pi$$
−0.421803 + 0.906688i $$0.638603\pi$$
$$600$$ 0 0
$$601$$ 45.1440 1.84146 0.920731 0.390197i $$-0.127593\pi$$
0.920731 + 0.390197i $$0.127593\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 11.0000 0.447214
$$606$$ 0 0
$$607$$ −20.7577 −0.842528 −0.421264 0.906938i $$-0.638413\pi$$
−0.421264 + 0.906938i $$0.638413\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 30.3476 1.22573 0.612864 0.790188i $$-0.290017\pi$$
0.612864 + 0.790188i $$0.290017\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −47.2161 −1.90085 −0.950425 0.310955i $$-0.899351\pi$$
−0.950425 + 0.310955i $$0.899351\pi$$
$$618$$ 0 0
$$619$$ −22.7064 −0.912647 −0.456323 0.889814i $$-0.650834\pi$$
−0.456323 + 0.889814i $$0.650834\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 13.2036 0.528990
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −0.646809 −0.0257899
$$630$$ 0 0
$$631$$ 32.4793 1.29298 0.646490 0.762923i $$-0.276236\pi$$
0.646490 + 0.762923i $$0.276236\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2.12878 0.0844781
$$636$$ 0 0
$$637$$ −21.8116 −0.864209
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −10.9460 −0.432342 −0.216171 0.976356i $$-0.569357\pi$$
−0.216171 + 0.976356i $$0.569357\pi$$
$$642$$ 0 0
$$643$$ 4.38924 0.173095 0.0865475 0.996248i $$-0.472417\pi$$
0.0865475 + 0.996248i $$0.472417\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −16.4522 −0.646802 −0.323401 0.946262i $$-0.604826\pi$$
−0.323401 + 0.946262i $$0.604826\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.53326 −0.0600009 −0.0300005 0.999550i $$-0.509551\pi$$
−0.0300005 + 0.999550i $$0.509551\pi$$
$$654$$ 0 0
$$655$$ 1.59283 0.0622370
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −22.1767 −0.863881 −0.431941 0.901902i $$-0.642171\pi$$
−0.431941 + 0.901902i $$0.642171\pi$$
$$660$$ 0 0
$$661$$ −26.0755 −1.01422 −0.507109 0.861882i $$-0.669286\pi$$
−0.507109 + 0.861882i $$0.669286\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1.39821 0.0542202
$$666$$ 0 0
$$667$$ 27.8179 1.07711
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −30.4168 −1.17248 −0.586241 0.810137i $$-0.699393\pi$$
−0.586241 + 0.810137i $$0.699393\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 44.2043 1.69891 0.849454 0.527663i $$-0.176932\pi$$
0.849454 + 0.527663i $$0.176932\pi$$
$$678$$ 0 0
$$679$$ −6.31714 −0.242430
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −42.3989 −1.62235 −0.811174 0.584805i $$-0.801171\pi$$
−0.811174 + 0.584805i $$0.801171\pi$$
$$684$$ 0 0
$$685$$ 9.89541 0.378084
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 13.4882 0.513861
$$690$$ 0 0
$$691$$ 7.79082 0.296377 0.148188 0.988959i $$-0.452656\pi$$
0.148188 + 0.988959i $$0.452656\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.70079 −0.140379
$$696$$ 0 0
$$697$$ −3.27569 −0.124076
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 8.05957 0.304406 0.152203 0.988349i $$-0.451363\pi$$
0.152203 + 0.988349i $$0.451363\pi$$
$$702$$ 0 0
$$703$$ 1.07480 0.0405370
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −7.06651 −0.265763
$$708$$ 0 0
$$709$$ −35.2936 −1.32548 −0.662740 0.748850i $$-0.730606\pi$$
−0.662740 + 0.748850i $$0.730606\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −16.9044 −0.633074
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 48.9315 1.82484 0.912418 0.409260i $$-0.134213\pi$$
0.912418 + 0.409260i $$0.134213\pi$$
$$720$$ 0 0
$$721$$ 15.4849 0.576687
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −4.60179 −0.170906
$$726$$ 0 0
$$727$$ −3.29025 −0.122029 −0.0610143 0.998137i $$-0.519434\pi$$
−0.0610143 + 0.998137i $$0.519434\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 5.20359 0.192462
$$732$$ 0 0
$$733$$ 22.3892 0.826966 0.413483 0.910512i $$-0.364312\pi$$
0.413483 + 0.910512i $$0.364312\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 4.90437 0.180410 0.0902051 0.995923i $$-0.471248\pi$$
0.0902051 + 0.995923i $$0.471248\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −28.1592 −1.03306 −0.516531 0.856268i $$-0.672777\pi$$
−0.516531 + 0.856268i $$0.672777\pi$$
$$744$$ 0 0
$$745$$ 18.3476 0.672204
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −5.83584 −0.213237
$$750$$ 0 0
$$751$$ 42.9348 1.56671 0.783357 0.621572i $$-0.213506\pi$$
0.783357 + 0.621572i $$0.213506\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −13.9404 −0.507344
$$756$$ 0 0
$$757$$ −25.0665 −0.911058 −0.455529 0.890221i $$-0.650550\pi$$
−0.455529 + 0.890221i $$0.650550\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.8898 1.11975 0.559877 0.828575i $$-0.310848\pi$$
0.559877 + 0.828575i $$0.310848\pi$$
$$762$$ 0 0
$$763$$ −20.7514 −0.751251
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 28.9315 1.04465
$$768$$ 0 0
$$769$$ 7.78745 0.280823 0.140411 0.990093i $$-0.455157\pi$$
0.140411 + 0.990093i $$0.455157\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −20.7126 −0.744982 −0.372491 0.928036i $$-0.621496\pi$$
−0.372491 + 0.928036i $$0.621496\pi$$
$$774$$ 0 0
$$775$$ 2.79641 0.100450
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5.44322 0.195024
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0.149606 0.00533968
$$786$$ 0 0
$$787$$ 10.5035 0.374408 0.187204 0.982321i $$-0.440057\pi$$
0.187204 + 0.982321i $$0.440057\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 2.40717 0.0855891
$$792$$ 0 0
$$793$$ −11.4432 −0.406361
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 37.8871 1.34203 0.671015 0.741444i $$-0.265858\pi$$
0.671015 + 0.741444i $$0.265858\pi$$
$$798$$ 0 0
$$799$$ −1.11355 −0.0393947
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −8.45219 −0.297900
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 33.9438 1.19340 0.596700 0.802464i $$-0.296478\pi$$
0.596700 + 0.802464i $$0.296478\pi$$
$$810$$ 0 0
$$811$$ 4.66137 0.163683 0.0818414 0.996645i $$-0.473920\pi$$
0.0818414 + 0.996645i $$0.473920\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 24.6468 0.863340
$$816$$ 0 0
$$817$$ −8.64681 −0.302514
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 26.5693 0.927275 0.463638 0.886025i $$-0.346544\pi$$
0.463638 + 0.886025i $$0.346544\pi$$
$$822$$ 0 0
$$823$$ 24.7998 0.864466 0.432233 0.901762i $$-0.357726\pi$$
0.432233 + 0.901762i $$0.357726\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.11422 0.0735188 0.0367594 0.999324i $$-0.488296\pi$$
0.0367594 + 0.999324i $$0.488296\pi$$
$$828$$ 0 0
$$829$$ 45.7279 1.58819 0.794097 0.607790i $$-0.207944\pi$$
0.794097 + 0.607790i $$0.207944\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 3.03605 0.105193
$$834$$ 0 0
$$835$$ −18.8656 −0.652872
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −15.4432 −0.533159 −0.266580 0.963813i $$-0.585894\pi$$
−0.266580 + 0.963813i $$0.585894\pi$$
$$840$$ 0 0
$$841$$ −7.82351 −0.269776
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −5.69182 −0.195805
$$846$$ 0 0
$$847$$ 15.3803 0.528473
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −6.49720 −0.222721
$$852$$ 0 0
$$853$$ 38.1801 1.30726 0.653630 0.756814i $$-0.273245\pi$$
0.653630 + 0.756814i $$0.273245\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −52.2672 −1.78541 −0.892707 0.450638i $$-0.851196\pi$$
−0.892707 + 0.450638i $$0.851196\pi$$
$$858$$ 0 0
$$859$$ −39.8809 −1.36072 −0.680359 0.732879i $$-0.738176\pi$$
−0.680359 + 0.732879i $$0.738176\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −22.9557 −0.781420 −0.390710 0.920514i $$-0.627770\pi$$
−0.390710 + 0.920514i $$0.627770\pi$$
$$864$$ 0 0
$$865$$ −6.36842 −0.216533
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −62.3151 −2.11147
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 1.39821 0.0472680
$$876$$ 0 0
$$877$$ −2.73057 −0.0922050 −0.0461025 0.998937i $$-0.514680\pi$$
−0.0461025 + 0.998937i $$0.514680\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −9.14401 −0.308070 −0.154035 0.988065i $$-0.549227\pi$$
−0.154035 + 0.988065i $$0.549227\pi$$
$$882$$ 0 0
$$883$$ −11.0540 −0.371996 −0.185998 0.982550i $$-0.559552\pi$$
−0.185998 + 0.982550i $$0.559552\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −37.1469 −1.24727 −0.623636 0.781715i $$-0.714345\pi$$
−0.623636 + 0.781715i $$0.714345\pi$$
$$888$$ 0 0
$$889$$ 2.97648 0.0998279
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 1.85039 0.0619211
$$894$$ 0 0
$$895$$ 11.4432 0.382505
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −12.8685 −0.429189
$$900$$ 0 0
$$901$$ −1.87748 −0.0625481
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 3.29362 0.109484
$$906$$ 0 0
$$907$$ 4.00627 0.133026 0.0665129 0.997786i $$-0.478813\pi$$
0.0665129 + 0.997786i $$0.478813\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −21.2161 −0.702921 −0.351461 0.936203i $$-0.614315\pi$$
−0.351461 + 0.936203i $$0.614315\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 2.22711 0.0735455
$$918$$ 0 0
$$919$$ −23.5478 −0.776771 −0.388385 0.921497i $$-0.626967\pi$$
−0.388385 + 0.921497i $$0.626967\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 24.1801 0.795896
$$924$$ 0 0
$$925$$ 1.07480 0.0353393
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 42.7223 1.40167 0.700836 0.713322i $$-0.252810\pi$$
0.700836 + 0.713322i $$0.252810\pi$$
$$930$$ 0 0
$$931$$ −5.04502 −0.165344
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −31.1890 −1.01890 −0.509451 0.860500i $$-0.670151\pi$$
−0.509451 + 0.860500i $$0.670151\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −20.2251 −0.659319 −0.329659 0.944100i $$-0.606934\pi$$
−0.329659 + 0.944100i $$0.606934\pi$$
$$942$$ 0 0
$$943$$ −32.9044 −1.07151
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 20.1801 0.655764 0.327882 0.944719i $$-0.393665\pi$$
0.327882 + 0.944719i $$0.393665\pi$$
$$948$$ 0 0
$$949$$ 54.8719 1.78122
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −20.6856 −0.670071 −0.335035 0.942206i $$-0.608748\pi$$
−0.335035 + 0.942206i $$0.608748\pi$$
$$954$$ 0 0
$$955$$ 20.5422 0.664731
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 13.8358 0.446782
$$960$$ 0 0
$$961$$ −23.1801 −0.747744
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 19.6620 0.632943
$$966$$ 0 0
$$967$$ −53.9530 −1.73501 −0.867505 0.497428i $$-0.834278\pi$$
−0.867505 + 0.497428i $$0.834278\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 9.41277 0.302070 0.151035 0.988528i $$-0.451739\pi$$
0.151035 + 0.988528i $$0.451739\pi$$
$$972$$ 0 0
$$973$$ −5.17447 −0.165886
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 4.21881 0.134972 0.0674859 0.997720i $$-0.478502\pi$$
0.0674859 + 0.997720i $$0.478502\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 10.3088 0.328801 0.164401 0.986394i $$-0.447431\pi$$
0.164401 + 0.986394i $$0.447431\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 52.2701 1.66209
$$990$$ 0 0
$$991$$ 54.6356 1.73556 0.867779 0.496951i $$-0.165547\pi$$
0.867779 + 0.496951i $$0.165547\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 9.74580 0.308963
$$996$$ 0 0
$$997$$ 22.9169 0.725786 0.362893 0.931831i $$-0.381789\pi$$
0.362893 + 0.931831i $$0.381789\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6840.2.a.bg.1.2 3
3.2 odd 2 760.2.a.j.1.3 3
12.11 even 2 1520.2.a.s.1.1 3
15.2 even 4 3800.2.d.l.3649.2 6
15.8 even 4 3800.2.d.l.3649.5 6
15.14 odd 2 3800.2.a.x.1.1 3
24.5 odd 2 6080.2.a.bv.1.1 3
24.11 even 2 6080.2.a.bq.1.3 3
60.59 even 2 7600.2.a.bq.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.3 3 3.2 odd 2
1520.2.a.s.1.1 3 12.11 even 2
3800.2.a.x.1.1 3 15.14 odd 2
3800.2.d.l.3649.2 6 15.2 even 4
3800.2.d.l.3649.5 6 15.8 even 4
6080.2.a.bq.1.3 3 24.11 even 2
6080.2.a.bv.1.1 3 24.5 odd 2
6840.2.a.bg.1.2 3 1.1 even 1 trivial
7600.2.a.bq.1.3 3 60.59 even 2