# Properties

 Label 1904.2.a.q.1.2 Level $1904$ Weight $2$ Character 1904.1 Self dual yes Analytic conductor $15.204$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1904,2,Mod(1,1904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1904, 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("1904.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1904 = 2^{4} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1904.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: $$15.2035165449$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.5225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + x + 11$$ x^4 - x^3 - 8*x^2 + x + 11 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 952) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.48718$$ of defining polynomial Character $$\chi$$ $$=$$ 1904.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.48718 q^{3} +4.02435 q^{5} +1.00000 q^{7} +3.18609 q^{9} +O(q^{10})$$ $$q-2.48718 q^{3} +4.02435 q^{5} +1.00000 q^{7} +3.18609 q^{9} +1.23607 q^{11} -6.47214 q^{13} -10.0093 q^{15} +1.00000 q^{17} -6.97437 q^{19} -2.48718 q^{21} -7.07433 q^{23} +11.1954 q^{25} -0.462835 q^{27} -6.31040 q^{29} -4.88824 q^{31} -3.07433 q^{33} +4.02435 q^{35} -2.63387 q^{37} +16.0974 q^{39} -10.1954 q^{41} +2.69762 q^{43} +12.8219 q^{45} +3.67652 q^{47} +1.00000 q^{49} -2.48718 q^{51} -1.39047 q^{53} +4.97437 q^{55} +17.3465 q^{57} -5.07433 q^{59} +3.93369 q^{61} +3.18609 q^{63} -26.0461 q^{65} +5.63259 q^{67} +17.5952 q^{69} +6.84431 q^{71} +0.246650 q^{73} -27.8450 q^{75} +1.23607 q^{77} +11.6765 q^{79} -8.40711 q^{81} +6.65089 q^{83} +4.02435 q^{85} +15.6951 q^{87} +2.27872 q^{89} -6.47214 q^{91} +12.1580 q^{93} -28.0673 q^{95} +10.7069 q^{97} +3.93822 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{3} - q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + 7 * q^9 $$4 q - 3 q^{3} - q^{5} + 4 q^{7} + 7 q^{9} - 4 q^{11} - 8 q^{13} - 12 q^{15} + 4 q^{17} - 14 q^{19} - 3 q^{21} - 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} + 8 q^{33} - q^{35} - 4 q^{37} - 4 q^{39} - 7 q^{41} - 19 q^{43} - 2 q^{45} - 8 q^{47} + 4 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{55} + 44 q^{57} - 23 q^{61} + 7 q^{63} - 8 q^{65} - 15 q^{67} - 2 q^{69} - 2 q^{71} - 5 q^{73} - 10 q^{75} - 4 q^{77} + 24 q^{79} - 8 q^{81} - 10 q^{83} - q^{85} - 16 q^{87} - 16 q^{89} - 8 q^{91} - 20 q^{93} - 22 q^{95} - 15 q^{97} - 2 q^{99}+O(q^{100})$$ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + 7 * q^9 - 4 * q^11 - 8 * q^13 - 12 * q^15 + 4 * q^17 - 14 * q^19 - 3 * q^21 - 8 * q^23 + 11 * q^25 - 12 * q^27 + 4 * q^29 - 5 * q^31 + 8 * q^33 - q^35 - 4 * q^37 - 4 * q^39 - 7 * q^41 - 19 * q^43 - 2 * q^45 - 8 * q^47 + 4 * q^49 - 3 * q^51 + 5 * q^53 + 6 * q^55 + 44 * q^57 - 23 * q^61 + 7 * q^63 - 8 * q^65 - 15 * q^67 - 2 * q^69 - 2 * q^71 - 5 * q^73 - 10 * q^75 - 4 * q^77 + 24 * q^79 - 8 * q^81 - 10 * q^83 - q^85 - 16 * q^87 - 16 * q^89 - 8 * q^91 - 20 * q^93 - 22 * q^95 - 15 * q^97 - 2 * 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
$$3$$ −2.48718 −1.43598 −0.717988 0.696055i $$-0.754937\pi$$
−0.717988 + 0.696055i $$0.754937\pi$$
$$4$$ 0 0
$$5$$ 4.02435 1.79974 0.899872 0.436154i $$-0.143660\pi$$
0.899872 + 0.436154i $$0.143660\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 3.18609 1.06203
$$10$$ 0 0
$$11$$ 1.23607 0.372689 0.186344 0.982485i $$-0.440336\pi$$
0.186344 + 0.982485i $$0.440336\pi$$
$$12$$ 0 0
$$13$$ −6.47214 −1.79505 −0.897524 0.440966i $$-0.854636\pi$$
−0.897524 + 0.440966i $$0.854636\pi$$
$$14$$ 0 0
$$15$$ −10.0093 −2.58439
$$16$$ 0 0
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ −6.97437 −1.60003 −0.800015 0.599980i $$-0.795175\pi$$
−0.800015 + 0.599980i $$0.795175\pi$$
$$20$$ 0 0
$$21$$ −2.48718 −0.542748
$$22$$ 0 0
$$23$$ −7.07433 −1.47510 −0.737550 0.675293i $$-0.764017\pi$$
−0.737550 + 0.675293i $$0.764017\pi$$
$$24$$ 0 0
$$25$$ 11.1954 2.23908
$$26$$ 0 0
$$27$$ −0.462835 −0.0890727
$$28$$ 0 0
$$29$$ −6.31040 −1.17181 −0.585906 0.810379i $$-0.699261\pi$$
−0.585906 + 0.810379i $$0.699261\pi$$
$$30$$ 0 0
$$31$$ −4.88824 −0.877954 −0.438977 0.898498i $$-0.644659\pi$$
−0.438977 + 0.898498i $$0.644659\pi$$
$$32$$ 0 0
$$33$$ −3.07433 −0.535172
$$34$$ 0 0
$$35$$ 4.02435 0.680239
$$36$$ 0 0
$$37$$ −2.63387 −0.433006 −0.216503 0.976282i $$-0.569465\pi$$
−0.216503 + 0.976282i $$0.569465\pi$$
$$38$$ 0 0
$$39$$ 16.0974 2.57765
$$40$$ 0 0
$$41$$ −10.1954 −1.59225 −0.796126 0.605131i $$-0.793121\pi$$
−0.796126 + 0.605131i $$0.793121\pi$$
$$42$$ 0 0
$$43$$ 2.69762 0.411384 0.205692 0.978617i $$-0.434056\pi$$
0.205692 + 0.978617i $$0.434056\pi$$
$$44$$ 0 0
$$45$$ 12.8219 1.91138
$$46$$ 0 0
$$47$$ 3.67652 0.536276 0.268138 0.963381i $$-0.413592\pi$$
0.268138 + 0.963381i $$0.413592\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.48718 −0.348276
$$52$$ 0 0
$$53$$ −1.39047 −0.190996 −0.0954982 0.995430i $$-0.530444\pi$$
−0.0954982 + 0.995430i $$0.530444\pi$$
$$54$$ 0 0
$$55$$ 4.97437 0.670744
$$56$$ 0 0
$$57$$ 17.3465 2.29761
$$58$$ 0 0
$$59$$ −5.07433 −0.660621 −0.330311 0.943872i $$-0.607154\pi$$
−0.330311 + 0.943872i $$0.607154\pi$$
$$60$$ 0 0
$$61$$ 3.93369 0.503657 0.251829 0.967772i $$-0.418968\pi$$
0.251829 + 0.967772i $$0.418968\pi$$
$$62$$ 0 0
$$63$$ 3.18609 0.401409
$$64$$ 0 0
$$65$$ −26.0461 −3.23063
$$66$$ 0 0
$$67$$ 5.63259 0.688131 0.344065 0.938946i $$-0.388196\pi$$
0.344065 + 0.938946i $$0.388196\pi$$
$$68$$ 0 0
$$69$$ 17.5952 2.11821
$$70$$ 0 0
$$71$$ 6.84431 0.812270 0.406135 0.913813i $$-0.366876\pi$$
0.406135 + 0.913813i $$0.366876\pi$$
$$72$$ 0 0
$$73$$ 0.246650 0.0288682 0.0144341 0.999896i $$-0.495405\pi$$
0.0144341 + 0.999896i $$0.495405\pi$$
$$74$$ 0 0
$$75$$ −27.8450 −3.21526
$$76$$ 0 0
$$77$$ 1.23607 0.140863
$$78$$ 0 0
$$79$$ 11.6765 1.31371 0.656856 0.754016i $$-0.271886\pi$$
0.656856 + 0.754016i $$0.271886\pi$$
$$80$$ 0 0
$$81$$ −8.40711 −0.934123
$$82$$ 0 0
$$83$$ 6.65089 0.730030 0.365015 0.931002i $$-0.381064\pi$$
0.365015 + 0.931002i $$0.381064\pi$$
$$84$$ 0 0
$$85$$ 4.02435 0.436502
$$86$$ 0 0
$$87$$ 15.6951 1.68269
$$88$$ 0 0
$$89$$ 2.27872 0.241543 0.120772 0.992680i $$-0.461463\pi$$
0.120772 + 0.992680i $$0.461463\pi$$
$$90$$ 0 0
$$91$$ −6.47214 −0.678464
$$92$$ 0 0
$$93$$ 12.1580 1.26072
$$94$$ 0 0
$$95$$ −28.0673 −2.87964
$$96$$ 0 0
$$97$$ 10.7069 1.08712 0.543562 0.839369i $$-0.317075\pi$$
0.543562 + 0.839369i $$0.317075\pi$$
$$98$$ 0 0
$$99$$ 3.93822 0.395806
$$100$$ 0 0
$$101$$ −16.7931 −1.67097 −0.835486 0.549512i $$-0.814813\pi$$
−0.835486 + 0.549512i $$0.814813\pi$$
$$102$$ 0 0
$$103$$ −9.57656 −0.943607 −0.471803 0.881704i $$-0.656397\pi$$
−0.471803 + 0.881704i $$0.656397\pi$$
$$104$$ 0 0
$$105$$ −10.0093 −0.976808
$$106$$ 0 0
$$107$$ −16.9613 −1.63971 −0.819855 0.572571i $$-0.805946\pi$$
−0.819855 + 0.572571i $$0.805946\pi$$
$$108$$ 0 0
$$109$$ −19.6082 −1.87813 −0.939065 0.343741i $$-0.888306\pi$$
−0.939065 + 0.343741i $$0.888306\pi$$
$$110$$ 0 0
$$111$$ 6.55093 0.621787
$$112$$ 0 0
$$113$$ −13.4952 −1.26952 −0.634761 0.772709i $$-0.718901\pi$$
−0.634761 + 0.772709i $$0.718901\pi$$
$$114$$ 0 0
$$115$$ −28.4696 −2.65480
$$116$$ 0 0
$$117$$ −20.6208 −1.90639
$$118$$ 0 0
$$119$$ 1.00000 0.0916698
$$120$$ 0 0
$$121$$ −9.47214 −0.861103
$$122$$ 0 0
$$123$$ 25.3578 2.28644
$$124$$ 0 0
$$125$$ 24.9324 2.23002
$$126$$ 0 0
$$127$$ −4.36484 −0.387317 −0.193659 0.981069i $$-0.562035\pi$$
−0.193659 + 0.981069i $$0.562035\pi$$
$$128$$ 0 0
$$129$$ −6.70948 −0.590737
$$130$$ 0 0
$$131$$ −10.6826 −0.933341 −0.466670 0.884431i $$-0.654547\pi$$
−0.466670 + 0.884431i $$0.654547\pi$$
$$132$$ 0 0
$$133$$ −6.97437 −0.604755
$$134$$ 0 0
$$135$$ −1.86261 −0.160308
$$136$$ 0 0
$$137$$ 18.1954 1.55454 0.777268 0.629169i $$-0.216605\pi$$
0.777268 + 0.629169i $$0.216605\pi$$
$$138$$ 0 0
$$139$$ −13.8430 −1.17415 −0.587075 0.809532i $$-0.699721\pi$$
−0.587075 + 0.809532i $$0.699721\pi$$
$$140$$ 0 0
$$141$$ −9.14419 −0.770080
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −25.3952 −2.10896
$$146$$ 0 0
$$147$$ −2.48718 −0.205140
$$148$$ 0 0
$$149$$ −3.42987 −0.280986 −0.140493 0.990082i $$-0.544869\pi$$
−0.140493 + 0.990082i $$0.544869\pi$$
$$150$$ 0 0
$$151$$ 9.16976 0.746224 0.373112 0.927786i $$-0.378291\pi$$
0.373112 + 0.927786i $$0.378291\pi$$
$$152$$ 0 0
$$153$$ 3.18609 0.257580
$$154$$ 0 0
$$155$$ −19.6720 −1.58009
$$156$$ 0 0
$$157$$ −6.70215 −0.534890 −0.267445 0.963573i $$-0.586179\pi$$
−0.267445 + 0.963573i $$0.586179\pi$$
$$158$$ 0 0
$$159$$ 3.45837 0.274266
$$160$$ 0 0
$$161$$ −7.07433 −0.557535
$$162$$ 0 0
$$163$$ 14.4104 1.12871 0.564353 0.825533i $$-0.309126\pi$$
0.564353 + 0.825533i $$0.309126\pi$$
$$164$$ 0 0
$$165$$ −12.3722 −0.963173
$$166$$ 0 0
$$167$$ 3.25112 0.251579 0.125789 0.992057i $$-0.459854\pi$$
0.125789 + 0.992057i $$0.459854\pi$$
$$168$$ 0 0
$$169$$ 28.8885 2.22220
$$170$$ 0 0
$$171$$ −22.2210 −1.69928
$$172$$ 0 0
$$173$$ 4.25717 0.323666 0.161833 0.986818i $$-0.448259\pi$$
0.161833 + 0.986818i $$0.448259\pi$$
$$174$$ 0 0
$$175$$ 11.1954 0.846292
$$176$$ 0 0
$$177$$ 12.6208 0.948637
$$178$$ 0 0
$$179$$ 2.72772 0.203879 0.101940 0.994791i $$-0.467495\pi$$
0.101940 + 0.994791i $$0.467495\pi$$
$$180$$ 0 0
$$181$$ 3.15471 0.234488 0.117244 0.993103i $$-0.462594\pi$$
0.117244 + 0.993103i $$0.462594\pi$$
$$182$$ 0 0
$$183$$ −9.78381 −0.723240
$$184$$ 0 0
$$185$$ −10.5996 −0.779300
$$186$$ 0 0
$$187$$ 1.23607 0.0903902
$$188$$ 0 0
$$189$$ −0.462835 −0.0336663
$$190$$ 0 0
$$191$$ −3.53263 −0.255612 −0.127806 0.991799i $$-0.540794\pi$$
−0.127806 + 0.991799i $$0.540794\pi$$
$$192$$ 0 0
$$193$$ 16.2275 1.16808 0.584039 0.811726i $$-0.301472\pi$$
0.584039 + 0.811726i $$0.301472\pi$$
$$194$$ 0 0
$$195$$ 64.7816 4.63910
$$196$$ 0 0
$$197$$ 17.7569 1.26513 0.632563 0.774509i $$-0.282003\pi$$
0.632563 + 0.774509i $$0.282003\pi$$
$$198$$ 0 0
$$199$$ 15.1486 1.07386 0.536928 0.843628i $$-0.319585\pi$$
0.536928 + 0.843628i $$0.319585\pi$$
$$200$$ 0 0
$$201$$ −14.0093 −0.988140
$$202$$ 0 0
$$203$$ −6.31040 −0.442903
$$204$$ 0 0
$$205$$ −41.0298 −2.86565
$$206$$ 0 0
$$207$$ −22.5394 −1.56660
$$208$$ 0 0
$$209$$ −8.62079 −0.596313
$$210$$ 0 0
$$211$$ −10.5826 −0.728537 −0.364269 0.931294i $$-0.618681\pi$$
−0.364269 + 0.931294i $$0.618681\pi$$
$$212$$ 0 0
$$213$$ −17.0231 −1.16640
$$214$$ 0 0
$$215$$ 10.8562 0.740385
$$216$$ 0 0
$$217$$ −4.88824 −0.331835
$$218$$ 0 0
$$219$$ −0.613463 −0.0414540
$$220$$ 0 0
$$221$$ −6.47214 −0.435363
$$222$$ 0 0
$$223$$ 3.95130 0.264599 0.132299 0.991210i $$-0.457764\pi$$
0.132299 + 0.991210i $$0.457764\pi$$
$$224$$ 0 0
$$225$$ 35.6695 2.37797
$$226$$ 0 0
$$227$$ −14.1031 −0.936059 −0.468029 0.883713i $$-0.655036\pi$$
−0.468029 + 0.883713i $$0.655036\pi$$
$$228$$ 0 0
$$229$$ −17.5766 −1.16149 −0.580746 0.814085i $$-0.697239\pi$$
−0.580746 + 0.814085i $$0.697239\pi$$
$$230$$ 0 0
$$231$$ −3.07433 −0.202276
$$232$$ 0 0
$$233$$ 15.5766 1.02045 0.510227 0.860040i $$-0.329561\pi$$
0.510227 + 0.860040i $$0.329561\pi$$
$$234$$ 0 0
$$235$$ 14.7956 0.965159
$$236$$ 0 0
$$237$$ −29.0417 −1.88646
$$238$$ 0 0
$$239$$ 5.16976 0.334404 0.167202 0.985923i $$-0.446527\pi$$
0.167202 + 0.985923i $$0.446527\pi$$
$$240$$ 0 0
$$241$$ −2.45830 −0.158353 −0.0791766 0.996861i $$-0.525229\pi$$
−0.0791766 + 0.996861i $$0.525229\pi$$
$$242$$ 0 0
$$243$$ 22.2985 1.43045
$$244$$ 0 0
$$245$$ 4.02435 0.257106
$$246$$ 0 0
$$247$$ 45.1391 2.87213
$$248$$ 0 0
$$249$$ −16.5420 −1.04831
$$250$$ 0 0
$$251$$ 17.3863 1.09741 0.548707 0.836015i $$-0.315120\pi$$
0.548707 + 0.836015i $$0.315120\pi$$
$$252$$ 0 0
$$253$$ −8.74435 −0.549753
$$254$$ 0 0
$$255$$ −10.0093 −0.626807
$$256$$ 0 0
$$257$$ −10.2787 −0.641169 −0.320584 0.947220i $$-0.603879\pi$$
−0.320584 + 0.947220i $$0.603879\pi$$
$$258$$ 0 0
$$259$$ −2.63387 −0.163661
$$260$$ 0 0
$$261$$ −20.1055 −1.24450
$$262$$ 0 0
$$263$$ 10.1487 0.625793 0.312897 0.949787i $$-0.398701\pi$$
0.312897 + 0.949787i $$0.398701\pi$$
$$264$$ 0 0
$$265$$ −5.59576 −0.343745
$$266$$ 0 0
$$267$$ −5.66759 −0.346851
$$268$$ 0 0
$$269$$ −2.47055 −0.150632 −0.0753161 0.997160i $$-0.523997\pi$$
−0.0753161 + 0.997160i $$0.523997\pi$$
$$270$$ 0 0
$$271$$ −21.0864 −1.28091 −0.640455 0.767996i $$-0.721254\pi$$
−0.640455 + 0.767996i $$0.721254\pi$$
$$272$$ 0 0
$$273$$ 16.0974 0.974259
$$274$$ 0 0
$$275$$ 13.8383 0.834479
$$276$$ 0 0
$$277$$ 25.2335 1.51613 0.758067 0.652176i $$-0.226144\pi$$
0.758067 + 0.652176i $$0.226144\pi$$
$$278$$ 0 0
$$279$$ −15.5744 −0.932413
$$280$$ 0 0
$$281$$ 27.5391 1.64285 0.821423 0.570319i $$-0.193180\pi$$
0.821423 + 0.570319i $$0.193180\pi$$
$$282$$ 0 0
$$283$$ −26.9385 −1.60133 −0.800665 0.599113i $$-0.795520\pi$$
−0.800665 + 0.599113i $$0.795520\pi$$
$$284$$ 0 0
$$285$$ 69.8086 4.13510
$$286$$ 0 0
$$287$$ −10.1954 −0.601815
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −26.6301 −1.56108
$$292$$ 0 0
$$293$$ 9.20439 0.537726 0.268863 0.963178i $$-0.413352\pi$$
0.268863 + 0.963178i $$0.413352\pi$$
$$294$$ 0 0
$$295$$ −20.4209 −1.18895
$$296$$ 0 0
$$297$$ −0.572096 −0.0331964
$$298$$ 0 0
$$299$$ 45.7860 2.64787
$$300$$ 0 0
$$301$$ 2.69762 0.155488
$$302$$ 0 0
$$303$$ 41.7674 2.39948
$$304$$ 0 0
$$305$$ 15.8305 0.906454
$$306$$ 0 0
$$307$$ 5.84878 0.333807 0.166904 0.985973i $$-0.446623\pi$$
0.166904 + 0.985973i $$0.446623\pi$$
$$308$$ 0 0
$$309$$ 23.8187 1.35500
$$310$$ 0 0
$$311$$ −30.5282 −1.73109 −0.865547 0.500828i $$-0.833029\pi$$
−0.865547 + 0.500828i $$0.833029\pi$$
$$312$$ 0 0
$$313$$ −17.2880 −0.977173 −0.488586 0.872515i $$-0.662487\pi$$
−0.488586 + 0.872515i $$0.662487\pi$$
$$314$$ 0 0
$$315$$ 12.8219 0.722434
$$316$$ 0 0
$$317$$ 4.99395 0.280488 0.140244 0.990117i $$-0.455211\pi$$
0.140244 + 0.990117i $$0.455211\pi$$
$$318$$ 0 0
$$319$$ −7.80008 −0.436721
$$320$$ 0 0
$$321$$ 42.1859 2.35459
$$322$$ 0 0
$$323$$ −6.97437 −0.388064
$$324$$ 0 0
$$325$$ −72.4581 −4.01925
$$326$$ 0 0
$$327$$ 48.7693 2.69695
$$328$$ 0 0
$$329$$ 3.67652 0.202693
$$330$$ 0 0
$$331$$ −27.5211 −1.51270 −0.756349 0.654168i $$-0.773019\pi$$
−0.756349 + 0.654168i $$0.773019\pi$$
$$332$$ 0 0
$$333$$ −8.39176 −0.459865
$$334$$ 0 0
$$335$$ 22.6675 1.23846
$$336$$ 0 0
$$337$$ −19.6439 −1.07007 −0.535035 0.844830i $$-0.679701\pi$$
−0.535035 + 0.844830i $$0.679701\pi$$
$$338$$ 0 0
$$339$$ 33.5651 1.82300
$$340$$ 0 0
$$341$$ −6.04220 −0.327203
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 70.8091 3.81223
$$346$$ 0 0
$$347$$ −3.53838 −0.189950 −0.0949751 0.995480i $$-0.530277\pi$$
−0.0949751 + 0.995480i $$0.530277\pi$$
$$348$$ 0 0
$$349$$ −15.9699 −0.854849 −0.427425 0.904051i $$-0.640579\pi$$
−0.427425 + 0.904051i $$0.640579\pi$$
$$350$$ 0 0
$$351$$ 2.99553 0.159890
$$352$$ 0 0
$$353$$ −17.3440 −0.923127 −0.461564 0.887107i $$-0.652711\pi$$
−0.461564 + 0.887107i $$0.652711\pi$$
$$354$$ 0 0
$$355$$ 27.5439 1.46188
$$356$$ 0 0
$$357$$ −2.48718 −0.131636
$$358$$ 0 0
$$359$$ −15.2579 −0.805279 −0.402639 0.915359i $$-0.631907\pi$$
−0.402639 + 0.915359i $$0.631907\pi$$
$$360$$ 0 0
$$361$$ 29.6418 1.56010
$$362$$ 0 0
$$363$$ 23.5590 1.23652
$$364$$ 0 0
$$365$$ 0.992604 0.0519553
$$366$$ 0 0
$$367$$ 20.8162 1.08660 0.543298 0.839540i $$-0.317175\pi$$
0.543298 + 0.839540i $$0.317175\pi$$
$$368$$ 0 0
$$369$$ −32.4834 −1.69102
$$370$$ 0 0
$$371$$ −1.39047 −0.0721899
$$372$$ 0 0
$$373$$ −18.0929 −0.936813 −0.468407 0.883513i $$-0.655172\pi$$
−0.468407 + 0.883513i $$0.655172\pi$$
$$374$$ 0 0
$$375$$ −62.0115 −3.20226
$$376$$ 0 0
$$377$$ 40.8417 2.10346
$$378$$ 0 0
$$379$$ 32.2265 1.65536 0.827681 0.561198i $$-0.189659\pi$$
0.827681 + 0.561198i $$0.189659\pi$$
$$380$$ 0 0
$$381$$ 10.8562 0.556179
$$382$$ 0 0
$$383$$ −5.77648 −0.295164 −0.147582 0.989050i $$-0.547149\pi$$
−0.147582 + 0.989050i $$0.547149\pi$$
$$384$$ 0 0
$$385$$ 4.97437 0.253517
$$386$$ 0 0
$$387$$ 8.59486 0.436901
$$388$$ 0 0
$$389$$ −16.6070 −0.842006 −0.421003 0.907059i $$-0.638322\pi$$
−0.421003 + 0.907059i $$0.638322\pi$$
$$390$$ 0 0
$$391$$ −7.07433 −0.357764
$$392$$ 0 0
$$393$$ 26.5695 1.34026
$$394$$ 0 0
$$395$$ 46.9904 2.36434
$$396$$ 0 0
$$397$$ −9.80083 −0.491890 −0.245945 0.969284i $$-0.579098\pi$$
−0.245945 + 0.969284i $$0.579098\pi$$
$$398$$ 0 0
$$399$$ 17.3465 0.868413
$$400$$ 0 0
$$401$$ 26.4696 1.32183 0.660914 0.750462i $$-0.270169\pi$$
0.660914 + 0.750462i $$0.270169\pi$$
$$402$$ 0 0
$$403$$ 31.6374 1.57597
$$404$$ 0 0
$$405$$ −33.8331 −1.68118
$$406$$ 0 0
$$407$$ −3.25565 −0.161376
$$408$$ 0 0
$$409$$ 22.9296 1.13380 0.566898 0.823788i $$-0.308143\pi$$
0.566898 + 0.823788i $$0.308143\pi$$
$$410$$ 0 0
$$411$$ −45.2553 −2.23228
$$412$$ 0 0
$$413$$ −5.07433 −0.249691
$$414$$ 0 0
$$415$$ 26.7655 1.31387
$$416$$ 0 0
$$417$$ 34.4302 1.68605
$$418$$ 0 0
$$419$$ −5.11334 −0.249803 −0.124902 0.992169i $$-0.539862\pi$$
−0.124902 + 0.992169i $$0.539862\pi$$
$$420$$ 0 0
$$421$$ −22.2928 −1.08648 −0.543242 0.839576i $$-0.682803\pi$$
−0.543242 + 0.839576i $$0.682803\pi$$
$$422$$ 0 0
$$423$$ 11.7137 0.569541
$$424$$ 0 0
$$425$$ 11.1954 0.543056
$$426$$ 0 0
$$427$$ 3.93369 0.190365
$$428$$ 0 0
$$429$$ 19.8975 0.960659
$$430$$ 0 0
$$431$$ −16.4510 −0.792415 −0.396208 0.918161i $$-0.629674\pi$$
−0.396208 + 0.918161i $$0.629674\pi$$
$$432$$ 0 0
$$433$$ 10.1999 0.490177 0.245088 0.969501i $$-0.421183\pi$$
0.245088 + 0.969501i $$0.421183\pi$$
$$434$$ 0 0
$$435$$ 63.1627 3.02842
$$436$$ 0 0
$$437$$ 49.3390 2.36020
$$438$$ 0 0
$$439$$ −18.4074 −0.878538 −0.439269 0.898356i $$-0.644762\pi$$
−0.439269 + 0.898356i $$0.644762\pi$$
$$440$$ 0 0
$$441$$ 3.18609 0.151718
$$442$$ 0 0
$$443$$ −1.43047 −0.0679635 −0.0339817 0.999422i $$-0.510819\pi$$
−0.0339817 + 0.999422i $$0.510819\pi$$
$$444$$ 0 0
$$445$$ 9.17035 0.434716
$$446$$ 0 0
$$447$$ 8.53073 0.403490
$$448$$ 0 0
$$449$$ 3.72778 0.175925 0.0879625 0.996124i $$-0.471964\pi$$
0.0879625 + 0.996124i $$0.471964\pi$$
$$450$$ 0 0
$$451$$ −12.6022 −0.593414
$$452$$ 0 0
$$453$$ −22.8069 −1.07156
$$454$$ 0 0
$$455$$ −26.0461 −1.22106
$$456$$ 0 0
$$457$$ 19.7601 0.924338 0.462169 0.886792i $$-0.347071\pi$$
0.462169 + 0.886792i $$0.347071\pi$$
$$458$$ 0 0
$$459$$ −0.462835 −0.0216033
$$460$$ 0 0
$$461$$ −4.40227 −0.205034 −0.102517 0.994731i $$-0.532690\pi$$
−0.102517 + 0.994731i $$0.532690\pi$$
$$462$$ 0 0
$$463$$ −4.51887 −0.210010 −0.105005 0.994472i $$-0.533486\pi$$
−0.105005 + 0.994472i $$0.533486\pi$$
$$464$$ 0 0
$$465$$ 48.9279 2.26898
$$466$$ 0 0
$$467$$ −1.51893 −0.0702877 −0.0351438 0.999382i $$-0.511189\pi$$
−0.0351438 + 0.999382i $$0.511189\pi$$
$$468$$ 0 0
$$469$$ 5.63259 0.260089
$$470$$ 0 0
$$471$$ 16.6695 0.768090
$$472$$ 0 0
$$473$$ 3.33444 0.153318
$$474$$ 0 0
$$475$$ −78.0808 −3.58259
$$476$$ 0 0
$$477$$ −4.43018 −0.202844
$$478$$ 0 0
$$479$$ 39.8418 1.82042 0.910209 0.414148i $$-0.135920\pi$$
0.910209 + 0.414148i $$0.135920\pi$$
$$480$$ 0 0
$$481$$ 17.0468 0.777267
$$482$$ 0 0
$$483$$ 17.5952 0.800608
$$484$$ 0 0
$$485$$ 43.0884 1.95654
$$486$$ 0 0
$$487$$ −9.19532 −0.416680 −0.208340 0.978056i $$-0.566806\pi$$
−0.208340 + 0.978056i $$0.566806\pi$$
$$488$$ 0 0
$$489$$ −35.8412 −1.62080
$$490$$ 0 0
$$491$$ −26.5282 −1.19720 −0.598600 0.801048i $$-0.704276\pi$$
−0.598600 + 0.801048i $$0.704276\pi$$
$$492$$ 0 0
$$493$$ −6.31040 −0.284206
$$494$$ 0 0
$$495$$ 15.8488 0.712350
$$496$$ 0 0
$$497$$ 6.84431 0.307009
$$498$$ 0 0
$$499$$ −28.9312 −1.29514 −0.647569 0.762007i $$-0.724214\pi$$
−0.647569 + 0.762007i $$0.724214\pi$$
$$500$$ 0 0
$$501$$ −8.08613 −0.361262
$$502$$ 0 0
$$503$$ −31.1977 −1.39103 −0.695517 0.718509i $$-0.744825\pi$$
−0.695517 + 0.718509i $$0.744825\pi$$
$$504$$ 0 0
$$505$$ −67.5811 −3.00732
$$506$$ 0 0
$$507$$ −71.8511 −3.19102
$$508$$ 0 0
$$509$$ 3.87888 0.171928 0.0859641 0.996298i $$-0.472603\pi$$
0.0859641 + 0.996298i $$0.472603\pi$$
$$510$$ 0 0
$$511$$ 0.246650 0.0109111
$$512$$ 0 0
$$513$$ 3.22798 0.142519
$$514$$ 0 0
$$515$$ −38.5394 −1.69825
$$516$$ 0 0
$$517$$ 4.54443 0.199864
$$518$$ 0 0
$$519$$ −10.5884 −0.464777
$$520$$ 0 0
$$521$$ 44.1050 1.93227 0.966137 0.258030i $$-0.0830734\pi$$
0.966137 + 0.258030i $$0.0830734\pi$$
$$522$$ 0 0
$$523$$ 0.735418 0.0321576 0.0160788 0.999871i $$-0.494882\pi$$
0.0160788 + 0.999871i $$0.494882\pi$$
$$524$$ 0 0
$$525$$ −27.8450 −1.21526
$$526$$ 0 0
$$527$$ −4.88824 −0.212935
$$528$$ 0 0
$$529$$ 27.0461 1.17592
$$530$$ 0 0
$$531$$ −16.1673 −0.701599
$$532$$ 0 0
$$533$$ 65.9859 2.85817
$$534$$ 0 0
$$535$$ −68.2582 −2.95106
$$536$$ 0 0
$$537$$ −6.78434 −0.292766
$$538$$ 0 0
$$539$$ 1.23607 0.0532412
$$540$$ 0 0
$$541$$ 5.95682 0.256104 0.128052 0.991767i $$-0.459128\pi$$
0.128052 + 0.991767i $$0.459128\pi$$
$$542$$ 0 0
$$543$$ −7.84635 −0.336719
$$544$$ 0 0
$$545$$ −78.9104 −3.38015
$$546$$ 0 0
$$547$$ 39.8844 1.70533 0.852667 0.522455i $$-0.174984\pi$$
0.852667 + 0.522455i $$0.174984\pi$$
$$548$$ 0 0
$$549$$ 12.5331 0.534899
$$550$$ 0 0
$$551$$ 44.0110 1.87493
$$552$$ 0 0
$$553$$ 11.6765 0.496536
$$554$$ 0 0
$$555$$ 26.3632 1.11906
$$556$$ 0 0
$$557$$ 29.5740 1.25309 0.626545 0.779385i $$-0.284468\pi$$
0.626545 + 0.779385i $$0.284468\pi$$
$$558$$ 0 0
$$559$$ −17.4594 −0.738453
$$560$$ 0 0
$$561$$ −3.07433 −0.129798
$$562$$ 0 0
$$563$$ −43.7900 −1.84553 −0.922763 0.385367i $$-0.874075\pi$$
−0.922763 + 0.385367i $$0.874075\pi$$
$$564$$ 0 0
$$565$$ −54.3094 −2.28481
$$566$$ 0 0
$$567$$ −8.40711 −0.353065
$$568$$ 0 0
$$569$$ −44.3088 −1.85752 −0.928761 0.370678i $$-0.879125\pi$$
−0.928761 + 0.370678i $$0.879125\pi$$
$$570$$ 0 0
$$571$$ 25.9543 1.08615 0.543076 0.839684i $$-0.317260\pi$$
0.543076 + 0.839684i $$0.317260\pi$$
$$572$$ 0 0
$$573$$ 8.78631 0.367053
$$574$$ 0 0
$$575$$ −79.1999 −3.30286
$$576$$ 0 0
$$577$$ −11.1442 −0.463939 −0.231969 0.972723i $$-0.574517\pi$$
−0.231969 + 0.972723i $$0.574517\pi$$
$$578$$ 0 0
$$579$$ −40.3607 −1.67733
$$580$$ 0 0
$$581$$ 6.65089 0.275926
$$582$$ 0 0
$$583$$ −1.71872 −0.0711822
$$584$$ 0 0
$$585$$ −82.9853 −3.43102
$$586$$ 0 0
$$587$$ −19.6676 −0.811768 −0.405884 0.913925i $$-0.633036\pi$$
−0.405884 + 0.913925i $$0.633036\pi$$
$$588$$ 0 0
$$589$$ 34.0924 1.40475
$$590$$ 0 0
$$591$$ −44.1647 −1.81669
$$592$$ 0 0
$$593$$ −28.1396 −1.15555 −0.577777 0.816194i $$-0.696080\pi$$
−0.577777 + 0.816194i $$0.696080\pi$$
$$594$$ 0 0
$$595$$ 4.02435 0.164982
$$596$$ 0 0
$$597$$ −37.6774 −1.54203
$$598$$ 0 0
$$599$$ −2.43434 −0.0994645 −0.0497322 0.998763i $$-0.515837\pi$$
−0.0497322 + 0.998763i $$0.515837\pi$$
$$600$$ 0 0
$$601$$ −15.8462 −0.646381 −0.323190 0.946334i $$-0.604755\pi$$
−0.323190 + 0.946334i $$0.604755\pi$$
$$602$$ 0 0
$$603$$ 17.9459 0.730815
$$604$$ 0 0
$$605$$ −38.1192 −1.54977
$$606$$ 0 0
$$607$$ −18.1413 −0.736334 −0.368167 0.929760i $$-0.620014\pi$$
−0.368167 + 0.929760i $$0.620014\pi$$
$$608$$ 0 0
$$609$$ 15.6951 0.635999
$$610$$ 0 0
$$611$$ −23.7950 −0.962641
$$612$$ 0 0
$$613$$ 14.9702 0.604641 0.302320 0.953206i $$-0.402239\pi$$
0.302320 + 0.953206i $$0.402239\pi$$
$$614$$ 0 0
$$615$$ 102.049 4.11500
$$616$$ 0 0
$$617$$ 7.43440 0.299298 0.149649 0.988739i $$-0.452186\pi$$
0.149649 + 0.988739i $$0.452186\pi$$
$$618$$ 0 0
$$619$$ 12.9548 0.520697 0.260348 0.965515i $$-0.416163\pi$$
0.260348 + 0.965515i $$0.416163\pi$$
$$620$$ 0 0
$$621$$ 3.27425 0.131391
$$622$$ 0 0
$$623$$ 2.27872 0.0912948
$$624$$ 0 0
$$625$$ 44.3598 1.77439
$$626$$ 0 0
$$627$$ 21.4415 0.856291
$$628$$ 0 0
$$629$$ −2.63387 −0.105019
$$630$$ 0 0
$$631$$ −0.899738 −0.0358180 −0.0179090 0.999840i $$-0.505701\pi$$
−0.0179090 + 0.999840i $$0.505701\pi$$
$$632$$ 0 0
$$633$$ 26.3209 1.04616
$$634$$ 0 0
$$635$$ −17.5657 −0.697072
$$636$$ 0 0
$$637$$ −6.47214 −0.256435
$$638$$ 0 0
$$639$$ 21.8066 0.862655
$$640$$ 0 0
$$641$$ −14.8322 −0.585837 −0.292919 0.956137i $$-0.594626\pi$$
−0.292919 + 0.956137i $$0.594626\pi$$
$$642$$ 0 0
$$643$$ 17.5168 0.690793 0.345397 0.938457i $$-0.387744\pi$$
0.345397 + 0.938457i $$0.387744\pi$$
$$644$$ 0 0
$$645$$ −27.0013 −1.06318
$$646$$ 0 0
$$647$$ −1.05829 −0.0416057 −0.0208028 0.999784i $$-0.506622\pi$$
−0.0208028 + 0.999784i $$0.506622\pi$$
$$648$$ 0 0
$$649$$ −6.27222 −0.246206
$$650$$ 0 0
$$651$$ 12.1580 0.476508
$$652$$ 0 0
$$653$$ 35.2848 1.38080 0.690400 0.723428i $$-0.257435\pi$$
0.690400 + 0.723428i $$0.257435\pi$$
$$654$$ 0 0
$$655$$ −42.9904 −1.67977
$$656$$ 0 0
$$657$$ 0.785848 0.0306588
$$658$$ 0 0
$$659$$ 20.1047 0.783169 0.391585 0.920142i $$-0.371927\pi$$
0.391585 + 0.920142i $$0.371927\pi$$
$$660$$ 0 0
$$661$$ 12.6541 0.492186 0.246093 0.969246i $$-0.420853\pi$$
0.246093 + 0.969246i $$0.420853\pi$$
$$662$$ 0 0
$$663$$ 16.0974 0.625171
$$664$$ 0 0
$$665$$ −28.0673 −1.08840
$$666$$ 0 0
$$667$$ 44.6418 1.72854
$$668$$ 0 0
$$669$$ −9.82762 −0.379958
$$670$$ 0 0
$$671$$ 4.86231 0.187707
$$672$$ 0 0
$$673$$ −21.6342 −0.833937 −0.416968 0.908921i $$-0.636907\pi$$
−0.416968 + 0.908921i $$0.636907\pi$$
$$674$$ 0 0
$$675$$ −5.18162 −0.199441
$$676$$ 0 0
$$677$$ 27.9965 1.07599 0.537996 0.842948i $$-0.319182\pi$$
0.537996 + 0.842948i $$0.319182\pi$$
$$678$$ 0 0
$$679$$ 10.7069 0.410894
$$680$$ 0 0
$$681$$ 35.0771 1.34416
$$682$$ 0 0
$$683$$ 40.2225 1.53907 0.769536 0.638603i $$-0.220487\pi$$
0.769536 + 0.638603i $$0.220487\pi$$
$$684$$ 0 0
$$685$$ 73.2246 2.79777
$$686$$ 0 0
$$687$$ 43.7162 1.66788
$$688$$ 0 0
$$689$$ 8.99934 0.342848
$$690$$ 0 0
$$691$$ 30.7414 1.16946 0.584729 0.811229i $$-0.301201\pi$$
0.584729 + 0.811229i $$0.301201\pi$$
$$692$$ 0 0
$$693$$ 3.93822 0.149601
$$694$$ 0 0
$$695$$ −55.7092 −2.11317
$$696$$ 0 0
$$697$$ −10.1954 −0.386178
$$698$$ 0 0
$$699$$ −38.7418 −1.46535
$$700$$ 0 0
$$701$$ 24.3696 0.920428 0.460214 0.887808i $$-0.347773\pi$$
0.460214 + 0.887808i $$0.347773\pi$$
$$702$$ 0 0
$$703$$ 18.3696 0.692823
$$704$$ 0 0
$$705$$ −36.7994 −1.38595
$$706$$ 0 0
$$707$$ −16.7931 −0.631568
$$708$$ 0 0
$$709$$ −22.2957 −0.837334 −0.418667 0.908140i $$-0.637503\pi$$
−0.418667 + 0.908140i $$0.637503\pi$$
$$710$$ 0 0
$$711$$ 37.2024 1.39520
$$712$$ 0 0
$$713$$ 34.5810 1.29507
$$714$$ 0 0
$$715$$ −32.1948 −1.20402
$$716$$ 0 0
$$717$$ −12.8581 −0.480196
$$718$$ 0 0
$$719$$ 14.2466 0.531310 0.265655 0.964068i $$-0.414412\pi$$
0.265655 + 0.964068i $$0.414412\pi$$
$$720$$ 0 0
$$721$$ −9.57656 −0.356650
$$722$$ 0 0
$$723$$ 6.11426 0.227392
$$724$$ 0 0
$$725$$ −70.6474 −2.62378
$$726$$ 0 0
$$727$$ −26.2762 −0.974529 −0.487264 0.873255i $$-0.662005\pi$$
−0.487264 + 0.873255i $$0.662005\pi$$
$$728$$ 0 0
$$729$$ −30.2393 −1.11997
$$730$$ 0 0
$$731$$ 2.69762 0.0997752
$$732$$ 0 0
$$733$$ −29.7464 −1.09871 −0.549354 0.835590i $$-0.685126\pi$$
−0.549354 + 0.835590i $$0.685126\pi$$
$$734$$ 0 0
$$735$$ −10.0093 −0.369199
$$736$$ 0 0
$$737$$ 6.96227 0.256458
$$738$$ 0 0
$$739$$ −20.0628 −0.738021 −0.369010 0.929425i $$-0.620303\pi$$
−0.369010 + 0.929425i $$0.620303\pi$$
$$740$$ 0 0
$$741$$ −112.269 −4.12431
$$742$$ 0 0
$$743$$ 10.0948 0.370344 0.185172 0.982706i $$-0.440716\pi$$
0.185172 + 0.982706i $$0.440716\pi$$
$$744$$ 0 0
$$745$$ −13.8030 −0.505703
$$746$$ 0 0
$$747$$ 21.1903 0.775314
$$748$$ 0 0
$$749$$ −16.9613 −0.619752
$$750$$ 0 0
$$751$$ 2.58103 0.0941831 0.0470916 0.998891i $$-0.485005\pi$$
0.0470916 + 0.998891i $$0.485005\pi$$
$$752$$ 0 0
$$753$$ −43.2430 −1.57586
$$754$$ 0 0
$$755$$ 36.9023 1.34301
$$756$$ 0 0
$$757$$ 42.6072 1.54859 0.774293 0.632828i $$-0.218106\pi$$
0.774293 + 0.632828i $$0.218106\pi$$
$$758$$ 0 0
$$759$$ 21.7488 0.789432
$$760$$ 0 0
$$761$$ −6.81361 −0.246993 −0.123497 0.992345i $$-0.539411\pi$$
−0.123497 + 0.992345i $$0.539411\pi$$
$$762$$ 0 0
$$763$$ −19.6082 −0.709866
$$764$$ 0 0
$$765$$ 12.8219 0.463578
$$766$$ 0 0
$$767$$ 32.8417 1.18585
$$768$$ 0 0
$$769$$ −9.62782 −0.347188 −0.173594 0.984817i $$-0.555538\pi$$
−0.173594 + 0.984817i $$0.555538\pi$$
$$770$$ 0 0
$$771$$ 25.5651 0.920703
$$772$$ 0 0
$$773$$ −32.6842 −1.17557 −0.587784 0.809018i $$-0.699999\pi$$
−0.587784 + 0.809018i $$0.699999\pi$$
$$774$$ 0 0
$$775$$ −54.7258 −1.96581
$$776$$ 0 0
$$777$$ 6.55093 0.235013
$$778$$ 0 0
$$779$$ 71.1064 2.54765
$$780$$ 0 0
$$781$$ 8.46003 0.302724
$$782$$ 0 0
$$783$$ 2.92067 0.104376
$$784$$ 0 0
$$785$$ −26.9718 −0.962665
$$786$$ 0 0
$$787$$ 40.6653 1.44956 0.724782 0.688979i $$-0.241941\pi$$
0.724782 + 0.688979i $$0.241941\pi$$
$$788$$ 0 0
$$789$$ −25.2416 −0.898624
$$790$$ 0 0
$$791$$ −13.4952 −0.479834
$$792$$ 0 0
$$793$$ −25.4594 −0.904089
$$794$$ 0 0
$$795$$ 13.9177 0.493609
$$796$$ 0 0
$$797$$ 19.0019 0.673082 0.336541 0.941669i $$-0.390743\pi$$
0.336541 + 0.941669i $$0.390743\pi$$
$$798$$ 0 0
$$799$$ 3.67652 0.130066
$$800$$ 0 0
$$801$$ 7.26019 0.256526
$$802$$ 0 0
$$803$$ 0.304876 0.0107588
$$804$$ 0 0
$$805$$ −28.4696 −1.00342
$$806$$ 0 0
$$807$$ 6.14472 0.216304
$$808$$ 0 0
$$809$$ −7.95386 −0.279643 −0.139821 0.990177i $$-0.544653\pi$$
−0.139821 + 0.990177i $$0.544653\pi$$
$$810$$ 0 0
$$811$$ 40.6097 1.42600 0.712999 0.701165i $$-0.247336\pi$$
0.712999 + 0.701165i $$0.247336\pi$$
$$812$$ 0 0
$$813$$ 52.4458 1.83936
$$814$$ 0 0
$$815$$ 57.9923 2.03138
$$816$$ 0 0
$$817$$ −18.8142 −0.658226
$$818$$ 0 0
$$819$$ −20.6208 −0.720549
$$820$$ 0 0
$$821$$ 50.7051 1.76962 0.884810 0.465951i $$-0.154288\pi$$
0.884810 + 0.465951i $$0.154288\pi$$
$$822$$ 0 0
$$823$$ −46.3922 −1.61713 −0.808564 0.588408i $$-0.799755\pi$$
−0.808564 + 0.588408i $$0.799755\pi$$
$$824$$ 0 0
$$825$$ −34.4183 −1.19829
$$826$$ 0 0
$$827$$ 34.6166 1.20374 0.601869 0.798595i $$-0.294423\pi$$
0.601869 + 0.798595i $$0.294423\pi$$
$$828$$ 0 0
$$829$$ 2.04614 0.0710653 0.0355326 0.999369i $$-0.488687\pi$$
0.0355326 + 0.999369i $$0.488687\pi$$
$$830$$ 0 0
$$831$$ −62.7604 −2.17713
$$832$$ 0 0
$$833$$ 1.00000 0.0346479
$$834$$ 0 0
$$835$$ 13.0836 0.452778
$$836$$ 0 0
$$837$$ 2.26245 0.0782017
$$838$$ 0 0
$$839$$ −2.11462 −0.0730049 −0.0365024 0.999334i $$-0.511622\pi$$
−0.0365024 + 0.999334i $$0.511622\pi$$
$$840$$ 0 0
$$841$$ 10.8211 0.373142
$$842$$ 0 0
$$843$$ −68.4949 −2.35909
$$844$$ 0 0
$$845$$ 116.258 3.99938
$$846$$ 0 0
$$847$$ −9.47214 −0.325466
$$848$$ 0 0
$$849$$ 67.0011 2.29947
$$850$$ 0 0
$$851$$ 18.6329 0.638727
$$852$$ 0 0
$$853$$ 22.7338 0.778392 0.389196 0.921155i $$-0.372753\pi$$
0.389196 + 0.921155i $$0.372753\pi$$
$$854$$ 0 0
$$855$$ −89.4249 −3.05827
$$856$$ 0 0
$$857$$ 2.03743 0.0695973 0.0347986 0.999394i $$-0.488921\pi$$
0.0347986 + 0.999394i $$0.488921\pi$$
$$858$$ 0 0
$$859$$ 15.8648 0.541301 0.270650 0.962678i $$-0.412761\pi$$
0.270650 + 0.962678i $$0.412761\pi$$
$$860$$ 0 0
$$861$$ 25.3578 0.864192
$$862$$ 0 0
$$863$$ 14.8857 0.506714 0.253357 0.967373i $$-0.418465\pi$$
0.253357 + 0.967373i $$0.418465\pi$$
$$864$$ 0 0
$$865$$ 17.1323 0.582517
$$866$$ 0 0
$$867$$ −2.48718 −0.0844692
$$868$$ 0 0
$$869$$ 14.4330 0.489605
$$870$$ 0 0
$$871$$ −36.4549 −1.23523
$$872$$ 0 0
$$873$$ 34.1132 1.15456
$$874$$ 0 0
$$875$$ 24.9324 0.842869
$$876$$ 0 0
$$877$$ −3.66844 −0.123874 −0.0619372 0.998080i $$-0.519728\pi$$
−0.0619372 + 0.998080i $$0.519728\pi$$
$$878$$ 0 0
$$879$$ −22.8930 −0.772162
$$880$$ 0 0
$$881$$ −39.9695 −1.34661 −0.673304 0.739366i $$-0.735126\pi$$
−0.673304 + 0.739366i $$0.735126\pi$$
$$882$$ 0 0
$$883$$ 17.1028 0.575556 0.287778 0.957697i $$-0.407083\pi$$
0.287778 + 0.957697i $$0.407083\pi$$
$$884$$ 0 0
$$885$$ 50.7905 1.70730
$$886$$ 0 0
$$887$$ 26.5622 0.891871 0.445936 0.895065i $$-0.352871\pi$$
0.445936 + 0.895065i $$0.352871\pi$$
$$888$$ 0 0
$$889$$ −4.36484 −0.146392
$$890$$ 0 0
$$891$$ −10.3918 −0.348137
$$892$$ 0 0
$$893$$ −25.6414 −0.858058
$$894$$ 0 0
$$895$$ 10.9773 0.366931
$$896$$ 0 0
$$897$$ −113.878 −3.80229
$$898$$ 0 0
$$899$$ 30.8467 1.02880
$$900$$ 0 0
$$901$$ −1.39047 −0.0463234
$$902$$ 0 0
$$903$$ −6.70948 −0.223278
$$904$$ 0 0
$$905$$ 12.6957 0.422018
$$906$$ 0 0
$$907$$ −42.8523 −1.42289 −0.711443 0.702744i $$-0.751958\pi$$
−0.711443 + 0.702744i $$0.751958\pi$$
$$908$$ 0 0
$$909$$ −53.5041 −1.77462
$$910$$ 0 0
$$911$$ 37.1141 1.22964 0.614822 0.788666i $$-0.289228\pi$$
0.614822 + 0.788666i $$0.289228\pi$$
$$912$$ 0 0
$$913$$ 8.22096 0.272074
$$914$$ 0 0
$$915$$ −39.3735 −1.30165
$$916$$ 0 0
$$917$$ −10.6826 −0.352770
$$918$$ 0 0
$$919$$ −23.7655 −0.783950 −0.391975 0.919976i $$-0.628208\pi$$
−0.391975 + 0.919976i $$0.628208\pi$$
$$920$$ 0 0
$$921$$ −14.5470 −0.479340
$$922$$ 0 0
$$923$$ −44.2973 −1.45806
$$924$$ 0 0
$$925$$ −29.4872 −0.969535
$$926$$ 0 0
$$927$$ −30.5118 −1.00214
$$928$$ 0 0
$$929$$ 24.7467 0.811912 0.405956 0.913893i $$-0.366939\pi$$
0.405956 + 0.913893i $$0.366939\pi$$
$$930$$ 0 0
$$931$$ −6.97437 −0.228576
$$932$$ 0 0
$$933$$ 75.9292 2.48581
$$934$$ 0 0
$$935$$ 4.97437 0.162679
$$936$$ 0 0
$$937$$ 26.9577 0.880671 0.440336 0.897833i $$-0.354859\pi$$
0.440336 + 0.897833i $$0.354859\pi$$
$$938$$ 0 0
$$939$$ 42.9983 1.40320
$$940$$ 0 0
$$941$$ −30.6992 −1.00077 −0.500383 0.865804i $$-0.666807\pi$$
−0.500383 + 0.865804i $$0.666807\pi$$
$$942$$ 0 0
$$943$$ 72.1255 2.34873
$$944$$ 0 0
$$945$$ −1.86261 −0.0605907
$$946$$ 0 0
$$947$$ −28.3892 −0.922525 −0.461262 0.887264i $$-0.652603\pi$$
−0.461262 + 0.887264i $$0.652603\pi$$
$$948$$ 0 0
$$949$$ −1.59635 −0.0518197
$$950$$ 0 0
$$951$$ −12.4209 −0.402774
$$952$$ 0 0
$$953$$ 39.1278 1.26747 0.633737 0.773549i $$-0.281520\pi$$
0.633737 + 0.773549i $$0.281520\pi$$
$$954$$ 0 0
$$955$$ −14.2166 −0.460037
$$956$$ 0 0
$$957$$ 19.4002 0.627121
$$958$$ 0 0
$$959$$ 18.1954 0.587560
$$960$$ 0 0
$$961$$ −7.10510 −0.229197
$$962$$ 0 0
$$963$$ −54.0402 −1.74142
$$964$$ 0 0
$$965$$ 65.3050 2.10224
$$966$$ 0 0
$$967$$ −39.8719 −1.28219 −0.641097 0.767460i $$-0.721520\pi$$
−0.641097 + 0.767460i $$0.721520\pi$$
$$968$$ 0 0
$$969$$ 17.3465 0.557251
$$970$$ 0 0
$$971$$ −16.8891 −0.541996 −0.270998 0.962580i $$-0.587354\pi$$
−0.270998 + 0.962580i $$0.587354\pi$$
$$972$$ 0 0
$$973$$ −13.8430 −0.443787
$$974$$ 0 0
$$975$$ 180.217 5.77155
$$976$$ 0 0
$$977$$ 21.4487 0.686205 0.343103 0.939298i $$-0.388522\pi$$
0.343103 + 0.939298i $$0.388522\pi$$
$$978$$ 0 0
$$979$$ 2.81665 0.0900205
$$980$$ 0 0
$$981$$ −62.4736 −1.99463
$$982$$ 0 0
$$983$$ 48.8004 1.55649 0.778245 0.627960i $$-0.216110\pi$$
0.778245 + 0.627960i $$0.216110\pi$$
$$984$$ 0 0
$$985$$ 71.4600 2.27690
$$986$$ 0 0
$$987$$ −9.14419 −0.291063
$$988$$ 0 0
$$989$$ −19.0839 −0.606832
$$990$$ 0 0
$$991$$ −5.00203 −0.158895 −0.0794474 0.996839i $$-0.525316\pi$$
−0.0794474 + 0.996839i $$0.525316\pi$$
$$992$$ 0 0
$$993$$ 68.4502 2.17220
$$994$$ 0 0
$$995$$ 60.9632 1.93266
$$996$$ 0 0
$$997$$ −54.7966 −1.73543 −0.867713 0.497066i $$-0.834411\pi$$
−0.867713 + 0.497066i $$0.834411\pi$$
$$998$$ 0 0
$$999$$ 1.21905 0.0385690
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 1904.2.a.q.1.2 4
4.3 odd 2 952.2.a.g.1.3 4
8.3 odd 2 7616.2.a.bj.1.2 4
8.5 even 2 7616.2.a.bp.1.3 4
12.11 even 2 8568.2.a.bj.1.1 4
28.27 even 2 6664.2.a.o.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.3 4 4.3 odd 2
1904.2.a.q.1.2 4 1.1 even 1 trivial
6664.2.a.o.1.2 4 28.27 even 2
7616.2.a.bj.1.2 4 8.3 odd 2
7616.2.a.bp.1.3 4 8.5 even 2
8568.2.a.bj.1.1 4 12.11 even 2