# Properties

 Label 3344.2.a.ba.1.4 Level $3344$ Weight $2$ Character 3344.1 Self dual yes Analytic conductor $26.702$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3344 = 2^{4} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3344.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$26.7019744359$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ x^7 - x^6 - 14*x^5 + 10*x^4 + 59*x^3 - 27*x^2 - 66*x + 30 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.61330$$ of defining polynomial Character $$\chi$$ $$=$$ 3344.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.19599 q^{3} +4.07680 q^{5} -3.61829 q^{7} -1.56960 q^{9} +O(q^{10})$$ $$q-1.19599 q^{3} +4.07680 q^{5} -3.61829 q^{7} -1.56960 q^{9} +1.00000 q^{11} -1.47857 q^{13} -4.87582 q^{15} -3.27003 q^{17} -1.00000 q^{19} +4.32745 q^{21} +7.45793 q^{23} +11.6203 q^{25} +5.46521 q^{27} +1.02535 q^{29} -1.64921 q^{31} -1.19599 q^{33} -14.7511 q^{35} -6.71293 q^{37} +1.76836 q^{39} -3.92451 q^{41} -5.38113 q^{43} -6.39896 q^{45} +3.71597 q^{47} +6.09205 q^{49} +3.91093 q^{51} -0.102902 q^{53} +4.07680 q^{55} +1.19599 q^{57} -13.2986 q^{59} -6.49664 q^{61} +5.67929 q^{63} -6.02783 q^{65} +3.70989 q^{67} -8.91962 q^{69} -6.32968 q^{71} -1.37759 q^{73} -13.8978 q^{75} -3.61829 q^{77} -13.6725 q^{79} -1.82753 q^{81} -5.44061 q^{83} -13.3313 q^{85} -1.22631 q^{87} +12.1357 q^{89} +5.34990 q^{91} +1.97244 q^{93} -4.07680 q^{95} -13.7910 q^{97} -1.56960 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10})$$ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 $$7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100})$$ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * 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
$$3$$ −1.19599 −0.690506 −0.345253 0.938510i $$-0.612207\pi$$
−0.345253 + 0.938510i $$0.612207\pi$$
$$4$$ 0 0
$$5$$ 4.07680 1.82320 0.911600 0.411078i $$-0.134847\pi$$
0.911600 + 0.411078i $$0.134847\pi$$
$$6$$ 0 0
$$7$$ −3.61829 −1.36759 −0.683793 0.729676i $$-0.739671\pi$$
−0.683793 + 0.729676i $$0.739671\pi$$
$$8$$ 0 0
$$9$$ −1.56960 −0.523201
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −1.47857 −0.410081 −0.205041 0.978753i $$-0.565733\pi$$
−0.205041 + 0.978753i $$0.565733\pi$$
$$14$$ 0 0
$$15$$ −4.87582 −1.25893
$$16$$ 0 0
$$17$$ −3.27003 −0.793099 −0.396549 0.918013i $$-0.629792\pi$$
−0.396549 + 0.918013i $$0.629792\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 4.32745 0.944327
$$22$$ 0 0
$$23$$ 7.45793 1.55509 0.777543 0.628830i $$-0.216466\pi$$
0.777543 + 0.628830i $$0.216466\pi$$
$$24$$ 0 0
$$25$$ 11.6203 2.32406
$$26$$ 0 0
$$27$$ 5.46521 1.05178
$$28$$ 0 0
$$29$$ 1.02535 0.190403 0.0952013 0.995458i $$-0.469651\pi$$
0.0952013 + 0.995458i $$0.469651\pi$$
$$30$$ 0 0
$$31$$ −1.64921 −0.296207 −0.148104 0.988972i $$-0.547317\pi$$
−0.148104 + 0.988972i $$0.547317\pi$$
$$32$$ 0 0
$$33$$ −1.19599 −0.208195
$$34$$ 0 0
$$35$$ −14.7511 −2.49338
$$36$$ 0 0
$$37$$ −6.71293 −1.10360 −0.551799 0.833977i $$-0.686059\pi$$
−0.551799 + 0.833977i $$0.686059\pi$$
$$38$$ 0 0
$$39$$ 1.76836 0.283164
$$40$$ 0 0
$$41$$ −3.92451 −0.612905 −0.306453 0.951886i $$-0.599142\pi$$
−0.306453 + 0.951886i $$0.599142\pi$$
$$42$$ 0 0
$$43$$ −5.38113 −0.820614 −0.410307 0.911947i $$-0.634578\pi$$
−0.410307 + 0.911947i $$0.634578\pi$$
$$44$$ 0 0
$$45$$ −6.39896 −0.953901
$$46$$ 0 0
$$47$$ 3.71597 0.542030 0.271015 0.962575i $$-0.412641\pi$$
0.271015 + 0.962575i $$0.412641\pi$$
$$48$$ 0 0
$$49$$ 6.09205 0.870292
$$50$$ 0 0
$$51$$ 3.91093 0.547640
$$52$$ 0 0
$$53$$ −0.102902 −0.0141347 −0.00706733 0.999975i $$-0.502250\pi$$
−0.00706733 + 0.999975i $$0.502250\pi$$
$$54$$ 0 0
$$55$$ 4.07680 0.549716
$$56$$ 0 0
$$57$$ 1.19599 0.158413
$$58$$ 0 0
$$59$$ −13.2986 −1.73134 −0.865668 0.500619i $$-0.833106\pi$$
−0.865668 + 0.500619i $$0.833106\pi$$
$$60$$ 0 0
$$61$$ −6.49664 −0.831809 −0.415905 0.909408i $$-0.636535\pi$$
−0.415905 + 0.909408i $$0.636535\pi$$
$$62$$ 0 0
$$63$$ 5.67929 0.715523
$$64$$ 0 0
$$65$$ −6.02783 −0.747661
$$66$$ 0 0
$$67$$ 3.70989 0.453235 0.226618 0.973984i $$-0.427233\pi$$
0.226618 + 0.973984i $$0.427233\pi$$
$$68$$ 0 0
$$69$$ −8.91962 −1.07380
$$70$$ 0 0
$$71$$ −6.32968 −0.751194 −0.375597 0.926783i $$-0.622562\pi$$
−0.375597 + 0.926783i $$0.622562\pi$$
$$72$$ 0 0
$$73$$ −1.37759 −0.161235 −0.0806173 0.996745i $$-0.525689\pi$$
−0.0806173 + 0.996745i $$0.525689\pi$$
$$74$$ 0 0
$$75$$ −13.8978 −1.60478
$$76$$ 0 0
$$77$$ −3.61829 −0.412343
$$78$$ 0 0
$$79$$ −13.6725 −1.53828 −0.769141 0.639079i $$-0.779316\pi$$
−0.769141 + 0.639079i $$0.779316\pi$$
$$80$$ 0 0
$$81$$ −1.82753 −0.203059
$$82$$ 0 0
$$83$$ −5.44061 −0.597184 −0.298592 0.954381i $$-0.596517\pi$$
−0.298592 + 0.954381i $$0.596517\pi$$
$$84$$ 0 0
$$85$$ −13.3313 −1.44598
$$86$$ 0 0
$$87$$ −1.22631 −0.131474
$$88$$ 0 0
$$89$$ 12.1357 1.28638 0.643191 0.765706i $$-0.277610\pi$$
0.643191 + 0.765706i $$0.277610\pi$$
$$90$$ 0 0
$$91$$ 5.34990 0.560822
$$92$$ 0 0
$$93$$ 1.97244 0.204533
$$94$$ 0 0
$$95$$ −4.07680 −0.418271
$$96$$ 0 0
$$97$$ −13.7910 −1.40026 −0.700131 0.714014i $$-0.746875\pi$$
−0.700131 + 0.714014i $$0.746875\pi$$
$$98$$ 0 0
$$99$$ −1.56960 −0.157751
$$100$$ 0 0
$$101$$ −11.0029 −1.09483 −0.547413 0.836863i $$-0.684387\pi$$
−0.547413 + 0.836863i $$0.684387\pi$$
$$102$$ 0 0
$$103$$ 4.99191 0.491867 0.245934 0.969287i $$-0.420906\pi$$
0.245934 + 0.969287i $$0.420906\pi$$
$$104$$ 0 0
$$105$$ 17.6421 1.72170
$$106$$ 0 0
$$107$$ −7.31345 −0.707018 −0.353509 0.935431i $$-0.615012\pi$$
−0.353509 + 0.935431i $$0.615012\pi$$
$$108$$ 0 0
$$109$$ −1.44482 −0.138389 −0.0691944 0.997603i $$-0.522043\pi$$
−0.0691944 + 0.997603i $$0.522043\pi$$
$$110$$ 0 0
$$111$$ 8.02861 0.762042
$$112$$ 0 0
$$113$$ −12.0369 −1.13234 −0.566169 0.824289i $$-0.691575\pi$$
−0.566169 + 0.824289i $$0.691575\pi$$
$$114$$ 0 0
$$115$$ 30.4045 2.83523
$$116$$ 0 0
$$117$$ 2.32077 0.214555
$$118$$ 0 0
$$119$$ 11.8319 1.08463
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 4.69368 0.423215
$$124$$ 0 0
$$125$$ 26.9897 2.41403
$$126$$ 0 0
$$127$$ 4.69692 0.416784 0.208392 0.978045i $$-0.433177\pi$$
0.208392 + 0.978045i $$0.433177\pi$$
$$128$$ 0 0
$$129$$ 6.43578 0.566639
$$130$$ 0 0
$$131$$ −3.74466 −0.327173 −0.163586 0.986529i $$-0.552306\pi$$
−0.163586 + 0.986529i $$0.552306\pi$$
$$132$$ 0 0
$$133$$ 3.61829 0.313746
$$134$$ 0 0
$$135$$ 22.2806 1.91761
$$136$$ 0 0
$$137$$ −15.7595 −1.34643 −0.673213 0.739449i $$-0.735086\pi$$
−0.673213 + 0.739449i $$0.735086\pi$$
$$138$$ 0 0
$$139$$ 2.52822 0.214440 0.107220 0.994235i $$-0.465805\pi$$
0.107220 + 0.994235i $$0.465805\pi$$
$$140$$ 0 0
$$141$$ −4.44427 −0.374275
$$142$$ 0 0
$$143$$ −1.47857 −0.123644
$$144$$ 0 0
$$145$$ 4.18015 0.347142
$$146$$ 0 0
$$147$$ −7.28604 −0.600942
$$148$$ 0 0
$$149$$ 1.84902 0.151477 0.0757387 0.997128i $$-0.475869\pi$$
0.0757387 + 0.997128i $$0.475869\pi$$
$$150$$ 0 0
$$151$$ 15.3184 1.24659 0.623296 0.781986i $$-0.285793\pi$$
0.623296 + 0.781986i $$0.285793\pi$$
$$152$$ 0 0
$$153$$ 5.13265 0.414950
$$154$$ 0 0
$$155$$ −6.72351 −0.540045
$$156$$ 0 0
$$157$$ 24.6631 1.96833 0.984165 0.177254i $$-0.0567215\pi$$
0.984165 + 0.177254i $$0.0567215\pi$$
$$158$$ 0 0
$$159$$ 0.123070 0.00976006
$$160$$ 0 0
$$161$$ −26.9850 −2.12671
$$162$$ 0 0
$$163$$ 3.72149 0.291490 0.145745 0.989322i $$-0.453442\pi$$
0.145745 + 0.989322i $$0.453442\pi$$
$$164$$ 0 0
$$165$$ −4.87582 −0.379582
$$166$$ 0 0
$$167$$ −2.64758 −0.204876 −0.102438 0.994739i $$-0.532664\pi$$
−0.102438 + 0.994739i $$0.532664\pi$$
$$168$$ 0 0
$$169$$ −10.8138 −0.831833
$$170$$ 0 0
$$171$$ 1.56960 0.120031
$$172$$ 0 0
$$173$$ 7.34552 0.558469 0.279235 0.960223i $$-0.409919\pi$$
0.279235 + 0.960223i $$0.409919\pi$$
$$174$$ 0 0
$$175$$ −42.0457 −3.17835
$$176$$ 0 0
$$177$$ 15.9051 1.19550
$$178$$ 0 0
$$179$$ −9.55394 −0.714095 −0.357047 0.934086i $$-0.616217\pi$$
−0.357047 + 0.934086i $$0.616217\pi$$
$$180$$ 0 0
$$181$$ 6.02638 0.447937 0.223969 0.974596i $$-0.428099\pi$$
0.223969 + 0.974596i $$0.428099\pi$$
$$182$$ 0 0
$$183$$ 7.76993 0.574369
$$184$$ 0 0
$$185$$ −27.3673 −2.01208
$$186$$ 0 0
$$187$$ −3.27003 −0.239128
$$188$$ 0 0
$$189$$ −19.7747 −1.43840
$$190$$ 0 0
$$191$$ −17.7069 −1.28123 −0.640613 0.767864i $$-0.721320\pi$$
−0.640613 + 0.767864i $$0.721320\pi$$
$$192$$ 0 0
$$193$$ 3.69348 0.265863 0.132931 0.991125i $$-0.457561\pi$$
0.132931 + 0.991125i $$0.457561\pi$$
$$194$$ 0 0
$$195$$ 7.20924 0.516264
$$196$$ 0 0
$$197$$ −25.7789 −1.83667 −0.918336 0.395802i $$-0.870467\pi$$
−0.918336 + 0.395802i $$0.870467\pi$$
$$198$$ 0 0
$$199$$ −18.8953 −1.33945 −0.669726 0.742608i $$-0.733589\pi$$
−0.669726 + 0.742608i $$0.733589\pi$$
$$200$$ 0 0
$$201$$ −4.43700 −0.312962
$$202$$ 0 0
$$203$$ −3.71002 −0.260392
$$204$$ 0 0
$$205$$ −15.9994 −1.11745
$$206$$ 0 0
$$207$$ −11.7060 −0.813623
$$208$$ 0 0
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 10.1993 0.702150 0.351075 0.936347i $$-0.385816\pi$$
0.351075 + 0.936347i $$0.385816\pi$$
$$212$$ 0 0
$$213$$ 7.57024 0.518704
$$214$$ 0 0
$$215$$ −21.9378 −1.49614
$$216$$ 0 0
$$217$$ 5.96733 0.405089
$$218$$ 0 0
$$219$$ 1.64759 0.111334
$$220$$ 0 0
$$221$$ 4.83497 0.325235
$$222$$ 0 0
$$223$$ 0.262700 0.0175917 0.00879584 0.999961i $$-0.497200\pi$$
0.00879584 + 0.999961i $$0.497200\pi$$
$$224$$ 0 0
$$225$$ −18.2393 −1.21595
$$226$$ 0 0
$$227$$ −27.3256 −1.81367 −0.906834 0.421489i $$-0.861508\pi$$
−0.906834 + 0.421489i $$0.861508\pi$$
$$228$$ 0 0
$$229$$ −5.53171 −0.365546 −0.182773 0.983155i $$-0.558507\pi$$
−0.182773 + 0.983155i $$0.558507\pi$$
$$230$$ 0 0
$$231$$ 4.32745 0.284725
$$232$$ 0 0
$$233$$ 27.4733 1.79984 0.899918 0.436059i $$-0.143626\pi$$
0.899918 + 0.436059i $$0.143626\pi$$
$$234$$ 0 0
$$235$$ 15.1493 0.988229
$$236$$ 0 0
$$237$$ 16.3523 1.06219
$$238$$ 0 0
$$239$$ −1.40339 −0.0907781 −0.0453890 0.998969i $$-0.514453\pi$$
−0.0453890 + 0.998969i $$0.514453\pi$$
$$240$$ 0 0
$$241$$ −20.7696 −1.33789 −0.668944 0.743312i $$-0.733254\pi$$
−0.668944 + 0.743312i $$0.733254\pi$$
$$242$$ 0 0
$$243$$ −14.2099 −0.911566
$$244$$ 0 0
$$245$$ 24.8361 1.58672
$$246$$ 0 0
$$247$$ 1.47857 0.0940791
$$248$$ 0 0
$$249$$ 6.50692 0.412359
$$250$$ 0 0
$$251$$ 1.29936 0.0820148 0.0410074 0.999159i $$-0.486943\pi$$
0.0410074 + 0.999159i $$0.486943\pi$$
$$252$$ 0 0
$$253$$ 7.45793 0.468876
$$254$$ 0 0
$$255$$ 15.9441 0.998457
$$256$$ 0 0
$$257$$ 3.41219 0.212847 0.106423 0.994321i $$-0.466060\pi$$
0.106423 + 0.994321i $$0.466060\pi$$
$$258$$ 0 0
$$259$$ 24.2893 1.50927
$$260$$ 0 0
$$261$$ −1.60939 −0.0996189
$$262$$ 0 0
$$263$$ −14.2418 −0.878189 −0.439094 0.898441i $$-0.644701\pi$$
−0.439094 + 0.898441i $$0.644701\pi$$
$$264$$ 0 0
$$265$$ −0.419510 −0.0257703
$$266$$ 0 0
$$267$$ −14.5142 −0.888254
$$268$$ 0 0
$$269$$ 5.39477 0.328925 0.164462 0.986383i $$-0.447411\pi$$
0.164462 + 0.986383i $$0.447411\pi$$
$$270$$ 0 0
$$271$$ 11.0624 0.671995 0.335998 0.941863i $$-0.390927\pi$$
0.335998 + 0.941863i $$0.390927\pi$$
$$272$$ 0 0
$$273$$ −6.39843 −0.387251
$$274$$ 0 0
$$275$$ 11.6203 0.700731
$$276$$ 0 0
$$277$$ 14.0808 0.846036 0.423018 0.906121i $$-0.360971\pi$$
0.423018 + 0.906121i $$0.360971\pi$$
$$278$$ 0 0
$$279$$ 2.58861 0.154976
$$280$$ 0 0
$$281$$ −10.8199 −0.645459 −0.322729 0.946491i $$-0.604600\pi$$
−0.322729 + 0.946491i $$0.604600\pi$$
$$282$$ 0 0
$$283$$ −1.90947 −0.113506 −0.0567532 0.998388i $$-0.518075\pi$$
−0.0567532 + 0.998388i $$0.518075\pi$$
$$284$$ 0 0
$$285$$ 4.87582 0.288819
$$286$$ 0 0
$$287$$ 14.2000 0.838201
$$288$$ 0 0
$$289$$ −6.30690 −0.370994
$$290$$ 0 0
$$291$$ 16.4939 0.966890
$$292$$ 0 0
$$293$$ 3.63550 0.212388 0.106194 0.994345i $$-0.466133\pi$$
0.106194 + 0.994345i $$0.466133\pi$$
$$294$$ 0 0
$$295$$ −54.2159 −3.15657
$$296$$ 0 0
$$297$$ 5.46521 0.317124
$$298$$ 0 0
$$299$$ −11.0271 −0.637712
$$300$$ 0 0
$$301$$ 19.4705 1.12226
$$302$$ 0 0
$$303$$ 13.1593 0.755984
$$304$$ 0 0
$$305$$ −26.4855 −1.51656
$$306$$ 0 0
$$307$$ 23.5329 1.34310 0.671548 0.740961i $$-0.265630\pi$$
0.671548 + 0.740961i $$0.265630\pi$$
$$308$$ 0 0
$$309$$ −5.97028 −0.339637
$$310$$ 0 0
$$311$$ 16.0026 0.907425 0.453713 0.891148i $$-0.350099\pi$$
0.453713 + 0.891148i $$0.350099\pi$$
$$312$$ 0 0
$$313$$ 16.0034 0.904566 0.452283 0.891875i $$-0.350610\pi$$
0.452283 + 0.891875i $$0.350610\pi$$
$$314$$ 0 0
$$315$$ 23.1533 1.30454
$$316$$ 0 0
$$317$$ −20.8766 −1.17255 −0.586273 0.810113i $$-0.699405\pi$$
−0.586273 + 0.810113i $$0.699405\pi$$
$$318$$ 0 0
$$319$$ 1.02535 0.0574086
$$320$$ 0 0
$$321$$ 8.74683 0.488200
$$322$$ 0 0
$$323$$ 3.27003 0.181949
$$324$$ 0 0
$$325$$ −17.1814 −0.953054
$$326$$ 0 0
$$327$$ 1.72800 0.0955584
$$328$$ 0 0
$$329$$ −13.4455 −0.741273
$$330$$ 0 0
$$331$$ −15.1136 −0.830721 −0.415360 0.909657i $$-0.636345\pi$$
−0.415360 + 0.909657i $$0.636345\pi$$
$$332$$ 0 0
$$333$$ 10.5366 0.577404
$$334$$ 0 0
$$335$$ 15.1245 0.826339
$$336$$ 0 0
$$337$$ −12.2766 −0.668751 −0.334376 0.942440i $$-0.608525\pi$$
−0.334376 + 0.942440i $$0.608525\pi$$
$$338$$ 0 0
$$339$$ 14.3961 0.781886
$$340$$ 0 0
$$341$$ −1.64921 −0.0893098
$$342$$ 0 0
$$343$$ 3.28525 0.177387
$$344$$ 0 0
$$345$$ −36.3635 −1.95775
$$346$$ 0 0
$$347$$ 30.9067 1.65916 0.829580 0.558387i $$-0.188580\pi$$
0.829580 + 0.558387i $$0.188580\pi$$
$$348$$ 0 0
$$349$$ −23.9024 −1.27947 −0.639733 0.768597i $$-0.720955\pi$$
−0.639733 + 0.768597i $$0.720955\pi$$
$$350$$ 0 0
$$351$$ −8.08069 −0.431315
$$352$$ 0 0
$$353$$ 15.0158 0.799210 0.399605 0.916687i $$-0.369147\pi$$
0.399605 + 0.916687i $$0.369147\pi$$
$$354$$ 0 0
$$355$$ −25.8048 −1.36958
$$356$$ 0 0
$$357$$ −14.1509 −0.748945
$$358$$ 0 0
$$359$$ −14.0826 −0.743251 −0.371626 0.928383i $$-0.621200\pi$$
−0.371626 + 0.928383i $$0.621200\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −1.19599 −0.0627733
$$364$$ 0 0
$$365$$ −5.61616 −0.293963
$$366$$ 0 0
$$367$$ −21.3142 −1.11259 −0.556296 0.830984i $$-0.687778\pi$$
−0.556296 + 0.830984i $$0.687778\pi$$
$$368$$ 0 0
$$369$$ 6.15992 0.320673
$$370$$ 0 0
$$371$$ 0.372329 0.0193304
$$372$$ 0 0
$$373$$ −2.42088 −0.125349 −0.0626743 0.998034i $$-0.519963\pi$$
−0.0626743 + 0.998034i $$0.519963\pi$$
$$374$$ 0 0
$$375$$ −32.2794 −1.66690
$$376$$ 0 0
$$377$$ −1.51605 −0.0780806
$$378$$ 0 0
$$379$$ −8.87535 −0.455896 −0.227948 0.973673i $$-0.573202\pi$$
−0.227948 + 0.973673i $$0.573202\pi$$
$$380$$ 0 0
$$381$$ −5.61748 −0.287792
$$382$$ 0 0
$$383$$ 3.54065 0.180919 0.0904595 0.995900i $$-0.471166\pi$$
0.0904595 + 0.995900i $$0.471166\pi$$
$$384$$ 0 0
$$385$$ −14.7511 −0.751784
$$386$$ 0 0
$$387$$ 8.44624 0.429346
$$388$$ 0 0
$$389$$ −16.7041 −0.846933 −0.423466 0.905912i $$-0.639187\pi$$
−0.423466 + 0.905912i $$0.639187\pi$$
$$390$$ 0 0
$$391$$ −24.3877 −1.23334
$$392$$ 0 0
$$393$$ 4.47858 0.225915
$$394$$ 0 0
$$395$$ −55.7403 −2.80460
$$396$$ 0 0
$$397$$ 28.2073 1.41568 0.707841 0.706372i $$-0.249669\pi$$
0.707841 + 0.706372i $$0.249669\pi$$
$$398$$ 0 0
$$399$$ −4.32745 −0.216643
$$400$$ 0 0
$$401$$ −36.2078 −1.80813 −0.904065 0.427395i $$-0.859431\pi$$
−0.904065 + 0.427395i $$0.859431\pi$$
$$402$$ 0 0
$$403$$ 2.43847 0.121469
$$404$$ 0 0
$$405$$ −7.45049 −0.370218
$$406$$ 0 0
$$407$$ −6.71293 −0.332748
$$408$$ 0 0
$$409$$ 36.9236 1.82576 0.912878 0.408233i $$-0.133855\pi$$
0.912878 + 0.408233i $$0.133855\pi$$
$$410$$ 0 0
$$411$$ 18.8482 0.929715
$$412$$ 0 0
$$413$$ 48.1184 2.36775
$$414$$ 0 0
$$415$$ −22.1803 −1.08879
$$416$$ 0 0
$$417$$ −3.02373 −0.148072
$$418$$ 0 0
$$419$$ −18.0690 −0.882726 −0.441363 0.897329i $$-0.645505\pi$$
−0.441363 + 0.897329i $$0.645505\pi$$
$$420$$ 0 0
$$421$$ −8.57629 −0.417983 −0.208991 0.977918i $$-0.567018\pi$$
−0.208991 + 0.977918i $$0.567018\pi$$
$$422$$ 0 0
$$423$$ −5.83260 −0.283591
$$424$$ 0 0
$$425$$ −37.9987 −1.84321
$$426$$ 0 0
$$427$$ 23.5067 1.13757
$$428$$ 0 0
$$429$$ 1.76836 0.0853771
$$430$$ 0 0
$$431$$ −4.28147 −0.206231 −0.103116 0.994669i $$-0.532881\pi$$
−0.103116 + 0.994669i $$0.532881\pi$$
$$432$$ 0 0
$$433$$ 18.2035 0.874804 0.437402 0.899266i $$-0.355899\pi$$
0.437402 + 0.899266i $$0.355899\pi$$
$$434$$ 0 0
$$435$$ −4.99942 −0.239704
$$436$$ 0 0
$$437$$ −7.45793 −0.356761
$$438$$ 0 0
$$439$$ −29.4442 −1.40529 −0.702647 0.711538i $$-0.747999\pi$$
−0.702647 + 0.711538i $$0.747999\pi$$
$$440$$ 0 0
$$441$$ −9.56210 −0.455338
$$442$$ 0 0
$$443$$ 24.3633 1.15753 0.578767 0.815493i $$-0.303534\pi$$
0.578767 + 0.815493i $$0.303534\pi$$
$$444$$ 0 0
$$445$$ 49.4748 2.34533
$$446$$ 0 0
$$447$$ −2.21141 −0.104596
$$448$$ 0 0
$$449$$ 38.7776 1.83003 0.915015 0.403421i $$-0.132179\pi$$
0.915015 + 0.403421i $$0.132179\pi$$
$$450$$ 0 0
$$451$$ −3.92451 −0.184798
$$452$$ 0 0
$$453$$ −18.3206 −0.860779
$$454$$ 0 0
$$455$$ 21.8105 1.02249
$$456$$ 0 0
$$457$$ −7.47672 −0.349746 −0.174873 0.984591i $$-0.555952\pi$$
−0.174873 + 0.984591i $$0.555952\pi$$
$$458$$ 0 0
$$459$$ −17.8714 −0.834165
$$460$$ 0 0
$$461$$ 15.9782 0.744181 0.372091 0.928196i $$-0.378641\pi$$
0.372091 + 0.928196i $$0.378641\pi$$
$$462$$ 0 0
$$463$$ 6.10221 0.283594 0.141797 0.989896i $$-0.454712\pi$$
0.141797 + 0.989896i $$0.454712\pi$$
$$464$$ 0 0
$$465$$ 8.04126 0.372904
$$466$$ 0 0
$$467$$ 26.6373 1.23263 0.616313 0.787502i $$-0.288626\pi$$
0.616313 + 0.787502i $$0.288626\pi$$
$$468$$ 0 0
$$469$$ −13.4235 −0.619838
$$470$$ 0 0
$$471$$ −29.4969 −1.35914
$$472$$ 0 0
$$473$$ −5.38113 −0.247424
$$474$$ 0 0
$$475$$ −11.6203 −0.533176
$$476$$ 0 0
$$477$$ 0.161515 0.00739527
$$478$$ 0 0
$$479$$ 16.2094 0.740626 0.370313 0.928907i $$-0.379250\pi$$
0.370313 + 0.928907i $$0.379250\pi$$
$$480$$ 0 0
$$481$$ 9.92553 0.452565
$$482$$ 0 0
$$483$$ 32.2738 1.46851
$$484$$ 0 0
$$485$$ −56.2231 −2.55296
$$486$$ 0 0
$$487$$ 4.82448 0.218618 0.109309 0.994008i $$-0.465136\pi$$
0.109309 + 0.994008i $$0.465136\pi$$
$$488$$ 0 0
$$489$$ −4.45088 −0.201276
$$490$$ 0 0
$$491$$ 8.53579 0.385215 0.192607 0.981276i $$-0.438306\pi$$
0.192607 + 0.981276i $$0.438306\pi$$
$$492$$ 0 0
$$493$$ −3.35293 −0.151008
$$494$$ 0 0
$$495$$ −6.39896 −0.287612
$$496$$ 0 0
$$497$$ 22.9026 1.02732
$$498$$ 0 0
$$499$$ 13.4325 0.601319 0.300660 0.953732i $$-0.402793\pi$$
0.300660 + 0.953732i $$0.402793\pi$$
$$500$$ 0 0
$$501$$ 3.16648 0.141468
$$502$$ 0 0
$$503$$ 1.66487 0.0742327 0.0371163 0.999311i $$-0.488183\pi$$
0.0371163 + 0.999311i $$0.488183\pi$$
$$504$$ 0 0
$$505$$ −44.8565 −1.99609
$$506$$ 0 0
$$507$$ 12.9333 0.574386
$$508$$ 0 0
$$509$$ −14.9684 −0.663463 −0.331731 0.943374i $$-0.607633\pi$$
−0.331731 + 0.943374i $$0.607633\pi$$
$$510$$ 0 0
$$511$$ 4.98452 0.220502
$$512$$ 0 0
$$513$$ −5.46521 −0.241295
$$514$$ 0 0
$$515$$ 20.3510 0.896772
$$516$$ 0 0
$$517$$ 3.71597 0.163428
$$518$$ 0 0
$$519$$ −8.78518 −0.385627
$$520$$ 0 0
$$521$$ −7.89123 −0.345721 −0.172861 0.984946i $$-0.555301\pi$$
−0.172861 + 0.984946i $$0.555301\pi$$
$$522$$ 0 0
$$523$$ 22.3062 0.975381 0.487690 0.873017i $$-0.337840\pi$$
0.487690 + 0.873017i $$0.337840\pi$$
$$524$$ 0 0
$$525$$ 50.2863 2.19467
$$526$$ 0 0
$$527$$ 5.39297 0.234922
$$528$$ 0 0
$$529$$ 32.6207 1.41829
$$530$$ 0 0
$$531$$ 20.8736 0.905837
$$532$$ 0 0
$$533$$ 5.80266 0.251341
$$534$$ 0 0
$$535$$ −29.8155 −1.28904
$$536$$ 0 0
$$537$$ 11.4264 0.493087
$$538$$ 0 0
$$539$$ 6.09205 0.262403
$$540$$ 0 0
$$541$$ 38.9694 1.67542 0.837712 0.546112i $$-0.183893\pi$$
0.837712 + 0.546112i $$0.183893\pi$$
$$542$$ 0 0
$$543$$ −7.20750 −0.309304
$$544$$ 0 0
$$545$$ −5.89025 −0.252311
$$546$$ 0 0
$$547$$ −37.7503 −1.61409 −0.807043 0.590492i $$-0.798934\pi$$
−0.807043 + 0.590492i $$0.798934\pi$$
$$548$$ 0 0
$$549$$ 10.1971 0.435204
$$550$$ 0 0
$$551$$ −1.02535 −0.0436814
$$552$$ 0 0
$$553$$ 49.4713 2.10373
$$554$$ 0 0
$$555$$ 32.7310 1.38935
$$556$$ 0 0
$$557$$ −3.17436 −0.134502 −0.0672511 0.997736i $$-0.521423\pi$$
−0.0672511 + 0.997736i $$0.521423\pi$$
$$558$$ 0 0
$$559$$ 7.95637 0.336519
$$560$$ 0 0
$$561$$ 3.91093 0.165120
$$562$$ 0 0
$$563$$ 19.9431 0.840503 0.420252 0.907408i $$-0.361942\pi$$
0.420252 + 0.907408i $$0.361942\pi$$
$$564$$ 0 0
$$565$$ −49.0721 −2.06448
$$566$$ 0 0
$$567$$ 6.61255 0.277701
$$568$$ 0 0
$$569$$ 36.6424 1.53613 0.768064 0.640374i $$-0.221220\pi$$
0.768064 + 0.640374i $$0.221220\pi$$
$$570$$ 0 0
$$571$$ −11.5300 −0.482515 −0.241258 0.970461i $$-0.577560\pi$$
−0.241258 + 0.970461i $$0.577560\pi$$
$$572$$ 0 0
$$573$$ 21.1773 0.884694
$$574$$ 0 0
$$575$$ 86.6634 3.61411
$$576$$ 0 0
$$577$$ 28.5590 1.18893 0.594463 0.804123i $$-0.297365\pi$$
0.594463 + 0.804123i $$0.297365\pi$$
$$578$$ 0 0
$$579$$ −4.41737 −0.183580
$$580$$ 0 0
$$581$$ 19.6857 0.816701
$$582$$ 0 0
$$583$$ −0.102902 −0.00426176
$$584$$ 0 0
$$585$$ 9.46131 0.391177
$$586$$ 0 0
$$587$$ −18.1461 −0.748969 −0.374484 0.927233i $$-0.622180\pi$$
−0.374484 + 0.927233i $$0.622180\pi$$
$$588$$ 0 0
$$589$$ 1.64921 0.0679546
$$590$$ 0 0
$$591$$ 30.8314 1.26823
$$592$$ 0 0
$$593$$ 15.3085 0.628644 0.314322 0.949316i $$-0.398223\pi$$
0.314322 + 0.949316i $$0.398223\pi$$
$$594$$ 0 0
$$595$$ 48.2364 1.97750
$$596$$ 0 0
$$597$$ 22.5986 0.924900
$$598$$ 0 0
$$599$$ −17.8806 −0.730580 −0.365290 0.930894i $$-0.619030\pi$$
−0.365290 + 0.930894i $$0.619030\pi$$
$$600$$ 0 0
$$601$$ −32.5803 −1.32898 −0.664489 0.747298i $$-0.731351\pi$$
−0.664489 + 0.747298i $$0.731351\pi$$
$$602$$ 0 0
$$603$$ −5.82306 −0.237133
$$604$$ 0 0
$$605$$ 4.07680 0.165746
$$606$$ 0 0
$$607$$ −43.6494 −1.77167 −0.885837 0.463997i $$-0.846415\pi$$
−0.885837 + 0.463997i $$0.846415\pi$$
$$608$$ 0 0
$$609$$ 4.43715 0.179802
$$610$$ 0 0
$$611$$ −5.49432 −0.222276
$$612$$ 0 0
$$613$$ 0.843061 0.0340509 0.0170254 0.999855i $$-0.494580\pi$$
0.0170254 + 0.999855i $$0.494580\pi$$
$$614$$ 0 0
$$615$$ 19.1352 0.771606
$$616$$ 0 0
$$617$$ 4.89882 0.197219 0.0986094 0.995126i $$-0.468561\pi$$
0.0986094 + 0.995126i $$0.468561\pi$$
$$618$$ 0 0
$$619$$ 14.8704 0.597691 0.298846 0.954301i $$-0.403398\pi$$
0.298846 + 0.954301i $$0.403398\pi$$
$$620$$ 0 0
$$621$$ 40.7591 1.63561
$$622$$ 0 0
$$623$$ −43.9105 −1.75924
$$624$$ 0 0
$$625$$ 51.9299 2.07720
$$626$$ 0 0
$$627$$ 1.19599 0.0477633
$$628$$ 0 0
$$629$$ 21.9515 0.875263
$$630$$ 0 0
$$631$$ −2.83922 −0.113027 −0.0565137 0.998402i $$-0.517998\pi$$
−0.0565137 + 0.998402i $$0.517998\pi$$
$$632$$ 0 0
$$633$$ −12.1983 −0.484839
$$634$$ 0 0
$$635$$ 19.1484 0.759881
$$636$$ 0 0
$$637$$ −9.00751 −0.356891
$$638$$ 0 0
$$639$$ 9.93508 0.393026
$$640$$ 0 0
$$641$$ 20.6746 0.816599 0.408299 0.912848i $$-0.366122\pi$$
0.408299 + 0.912848i $$0.366122\pi$$
$$642$$ 0 0
$$643$$ 24.6254 0.971130 0.485565 0.874201i $$-0.338614\pi$$
0.485565 + 0.874201i $$0.338614\pi$$
$$644$$ 0 0
$$645$$ 26.2374 1.03310
$$646$$ 0 0
$$647$$ −45.3626 −1.78339 −0.891693 0.452640i $$-0.850482\pi$$
−0.891693 + 0.452640i $$0.850482\pi$$
$$648$$ 0 0
$$649$$ −13.2986 −0.522017
$$650$$ 0 0
$$651$$ −7.13688 −0.279716
$$652$$ 0 0
$$653$$ 26.1012 1.02142 0.510710 0.859753i $$-0.329383\pi$$
0.510710 + 0.859753i $$0.329383\pi$$
$$654$$ 0 0
$$655$$ −15.2662 −0.596501
$$656$$ 0 0
$$657$$ 2.16227 0.0843582
$$658$$ 0 0
$$659$$ 45.8507 1.78609 0.893045 0.449967i $$-0.148564\pi$$
0.893045 + 0.449967i $$0.148564\pi$$
$$660$$ 0 0
$$661$$ −15.4225 −0.599864 −0.299932 0.953961i $$-0.596964\pi$$
−0.299932 + 0.953961i $$0.596964\pi$$
$$662$$ 0 0
$$663$$ −5.78258 −0.224577
$$664$$ 0 0
$$665$$ 14.7511 0.572022
$$666$$ 0 0
$$667$$ 7.64698 0.296092
$$668$$ 0 0
$$669$$ −0.314187 −0.0121472
$$670$$ 0 0
$$671$$ −6.49664 −0.250800
$$672$$ 0 0
$$673$$ −37.8633 −1.45952 −0.729762 0.683702i $$-0.760369\pi$$
−0.729762 + 0.683702i $$0.760369\pi$$
$$674$$ 0 0
$$675$$ 63.5074 2.44440
$$676$$ 0 0
$$677$$ −25.3510 −0.974320 −0.487160 0.873313i $$-0.661967\pi$$
−0.487160 + 0.873313i $$0.661967\pi$$
$$678$$ 0 0
$$679$$ 49.8998 1.91498
$$680$$ 0 0
$$681$$ 32.6812 1.25235
$$682$$ 0 0
$$683$$ 21.7513 0.832290 0.416145 0.909298i $$-0.363381\pi$$
0.416145 + 0.909298i $$0.363381\pi$$
$$684$$ 0 0
$$685$$ −64.2484 −2.45480
$$686$$ 0 0
$$687$$ 6.61588 0.252412
$$688$$ 0 0
$$689$$ 0.152147 0.00579636
$$690$$ 0 0
$$691$$ −17.3051 −0.658318 −0.329159 0.944275i $$-0.606765\pi$$
−0.329159 + 0.944275i $$0.606765\pi$$
$$692$$ 0 0
$$693$$ 5.67929 0.215738
$$694$$ 0 0
$$695$$ 10.3070 0.390968
$$696$$ 0 0
$$697$$ 12.8333 0.486095
$$698$$ 0 0
$$699$$ −32.8578 −1.24280
$$700$$ 0 0
$$701$$ −29.6923 −1.12146 −0.560732 0.827997i $$-0.689480\pi$$
−0.560732 + 0.827997i $$0.689480\pi$$
$$702$$ 0 0
$$703$$ 6.71293 0.253183
$$704$$ 0 0
$$705$$ −18.1184 −0.682378
$$706$$ 0 0
$$707$$ 39.8116 1.49727
$$708$$ 0 0
$$709$$ −21.4898 −0.807067 −0.403534 0.914965i $$-0.632218\pi$$
−0.403534 + 0.914965i $$0.632218\pi$$
$$710$$ 0 0
$$711$$ 21.4605 0.804831
$$712$$ 0 0
$$713$$ −12.2997 −0.460627
$$714$$ 0 0
$$715$$ −6.02783 −0.225428
$$716$$ 0 0
$$717$$ 1.67845 0.0626828
$$718$$ 0 0
$$719$$ −9.61388 −0.358537 −0.179269 0.983800i $$-0.557373\pi$$
−0.179269 + 0.983800i $$0.557373\pi$$
$$720$$ 0 0
$$721$$ −18.0622 −0.672671
$$722$$ 0 0
$$723$$ 24.8403 0.923821
$$724$$ 0 0
$$725$$ 11.9149 0.442507
$$726$$ 0 0
$$727$$ −6.84046 −0.253699 −0.126849 0.991922i $$-0.540486\pi$$
−0.126849 + 0.991922i $$0.540486\pi$$
$$728$$ 0 0
$$729$$ 22.4775 0.832501
$$730$$ 0 0
$$731$$ 17.5965 0.650828
$$732$$ 0 0
$$733$$ 38.2277 1.41197 0.705987 0.708225i $$-0.250504\pi$$
0.705987 + 0.708225i $$0.250504\pi$$
$$734$$ 0 0
$$735$$ −29.7037 −1.09564
$$736$$ 0 0
$$737$$ 3.70989 0.136656
$$738$$ 0 0
$$739$$ 17.7157 0.651682 0.325841 0.945425i $$-0.394353\pi$$
0.325841 + 0.945425i $$0.394353\pi$$
$$740$$ 0 0
$$741$$ −1.76836 −0.0649622
$$742$$ 0 0
$$743$$ −21.3028 −0.781525 −0.390763 0.920491i $$-0.627789\pi$$
−0.390763 + 0.920491i $$0.627789\pi$$
$$744$$ 0 0
$$745$$ 7.53808 0.276174
$$746$$ 0 0
$$747$$ 8.53960 0.312447
$$748$$ 0 0
$$749$$ 26.4622 0.966908
$$750$$ 0 0
$$751$$ −25.1552 −0.917926 −0.458963 0.888455i $$-0.651779\pi$$
−0.458963 + 0.888455i $$0.651779\pi$$
$$752$$ 0 0
$$753$$ −1.55402 −0.0566317
$$754$$ 0 0
$$755$$ 62.4499 2.27279
$$756$$ 0 0
$$757$$ 15.5116 0.563780 0.281890 0.959447i $$-0.409039\pi$$
0.281890 + 0.959447i $$0.409039\pi$$
$$758$$ 0 0
$$759$$ −8.91962 −0.323762
$$760$$ 0 0
$$761$$ 35.2085 1.27631 0.638154 0.769908i $$-0.279698\pi$$
0.638154 + 0.769908i $$0.279698\pi$$
$$762$$ 0 0
$$763$$ 5.22779 0.189259
$$764$$ 0 0
$$765$$ 20.9248 0.756538
$$766$$ 0 0
$$767$$ 19.6630 0.709988
$$768$$ 0 0
$$769$$ −5.57667 −0.201100 −0.100550 0.994932i $$-0.532060\pi$$
−0.100550 + 0.994932i $$0.532060\pi$$
$$770$$ 0 0
$$771$$ −4.08095 −0.146972
$$772$$ 0 0
$$773$$ 35.9175 1.29186 0.645931 0.763396i $$-0.276469\pi$$
0.645931 + 0.763396i $$0.276469\pi$$
$$774$$ 0 0
$$775$$ −19.1643 −0.688403
$$776$$ 0 0
$$777$$ −29.0499 −1.04216
$$778$$ 0 0
$$779$$ 3.92451 0.140610
$$780$$ 0 0
$$781$$ −6.32968 −0.226494
$$782$$ 0 0
$$783$$ 5.60375 0.200262
$$784$$ 0 0
$$785$$ 100.547 3.58866
$$786$$ 0 0
$$787$$ −7.53242 −0.268502 −0.134251 0.990947i $$-0.542863\pi$$
−0.134251 + 0.990947i $$0.542863\pi$$
$$788$$ 0 0
$$789$$ 17.0331 0.606395
$$790$$ 0 0
$$791$$ 43.5531 1.54857
$$792$$ 0 0
$$793$$ 9.60573 0.341110
$$794$$ 0 0
$$795$$ 0.501731 0.0177946
$$796$$ 0 0
$$797$$ −49.3837 −1.74926 −0.874629 0.484792i $$-0.838895\pi$$
−0.874629 + 0.484792i $$0.838895\pi$$
$$798$$ 0 0
$$799$$ −12.1513 −0.429883
$$800$$ 0 0
$$801$$ −19.0482 −0.673036
$$802$$ 0 0
$$803$$ −1.37759 −0.0486141
$$804$$ 0 0
$$805$$ −110.012 −3.87743
$$806$$ 0 0
$$807$$ −6.45210 −0.227125
$$808$$ 0 0
$$809$$ 34.4637 1.21168 0.605840 0.795587i $$-0.292837\pi$$
0.605840 + 0.795587i $$0.292837\pi$$
$$810$$ 0 0
$$811$$ −20.0278 −0.703272 −0.351636 0.936137i $$-0.614375\pi$$
−0.351636 + 0.936137i $$0.614375\pi$$
$$812$$ 0 0
$$813$$ −13.2306 −0.464017
$$814$$ 0 0
$$815$$ 15.1718 0.531444
$$816$$ 0 0
$$817$$ 5.38113 0.188262
$$818$$ 0 0
$$819$$ −8.39722 −0.293423
$$820$$ 0 0
$$821$$ 6.45245 0.225192 0.112596 0.993641i $$-0.464083\pi$$
0.112596 + 0.993641i $$0.464083\pi$$
$$822$$ 0 0
$$823$$ 43.0190 1.49955 0.749774 0.661694i $$-0.230162\pi$$
0.749774 + 0.661694i $$0.230162\pi$$
$$824$$ 0 0
$$825$$ −13.8978 −0.483859
$$826$$ 0 0
$$827$$ 43.1557 1.50067 0.750335 0.661058i $$-0.229892\pi$$
0.750335 + 0.661058i $$0.229892\pi$$
$$828$$ 0 0
$$829$$ 13.4937 0.468656 0.234328 0.972158i $$-0.424711\pi$$
0.234328 + 0.972158i $$0.424711\pi$$
$$830$$ 0 0
$$831$$ −16.8406 −0.584193
$$832$$ 0 0
$$833$$ −19.9212 −0.690228
$$834$$ 0 0
$$835$$ −10.7937 −0.373530
$$836$$ 0 0
$$837$$ −9.01329 −0.311545
$$838$$ 0 0
$$839$$ 14.6851 0.506988 0.253494 0.967337i $$-0.418420\pi$$
0.253494 + 0.967337i $$0.418420\pi$$
$$840$$ 0 0
$$841$$ −27.9487 −0.963747
$$842$$ 0 0
$$843$$ 12.9405 0.445693
$$844$$ 0 0
$$845$$ −44.0858 −1.51660
$$846$$ 0 0
$$847$$ −3.61829 −0.124326
$$848$$ 0 0
$$849$$ 2.28372 0.0783769
$$850$$ 0 0
$$851$$ −50.0645 −1.71619
$$852$$ 0 0
$$853$$ −51.5775 −1.76598 −0.882990 0.469392i $$-0.844473\pi$$
−0.882990 + 0.469392i $$0.844473\pi$$
$$854$$ 0 0
$$855$$ 6.39896 0.218840
$$856$$ 0 0
$$857$$ −46.6355 −1.59304 −0.796520 0.604612i $$-0.793328\pi$$
−0.796520 + 0.604612i $$0.793328\pi$$
$$858$$ 0 0
$$859$$ 18.6711 0.637048 0.318524 0.947915i $$-0.396813\pi$$
0.318524 + 0.947915i $$0.396813\pi$$
$$860$$ 0 0
$$861$$ −16.9831 −0.578783
$$862$$ 0 0
$$863$$ −43.0160 −1.46428 −0.732141 0.681153i $$-0.761479\pi$$
−0.732141 + 0.681153i $$0.761479\pi$$
$$864$$ 0 0
$$865$$ 29.9462 1.01820
$$866$$ 0 0
$$867$$ 7.54300 0.256174
$$868$$ 0 0
$$869$$ −13.6725 −0.463809
$$870$$ 0 0
$$871$$ −5.48533 −0.185863
$$872$$ 0 0
$$873$$ 21.6464 0.732619
$$874$$ 0 0
$$875$$ −97.6565 −3.30139
$$876$$ 0 0
$$877$$ 56.0429 1.89243 0.946217 0.323534i $$-0.104871\pi$$
0.946217 + 0.323534i $$0.104871\pi$$
$$878$$ 0 0
$$879$$ −4.34803 −0.146655
$$880$$ 0 0
$$881$$ 17.9947 0.606257 0.303128 0.952950i $$-0.401969\pi$$
0.303128 + 0.952950i $$0.401969\pi$$
$$882$$ 0 0
$$883$$ −1.99550 −0.0671540 −0.0335770 0.999436i $$-0.510690\pi$$
−0.0335770 + 0.999436i $$0.510690\pi$$
$$884$$ 0 0
$$885$$ 64.8418 2.17963
$$886$$ 0 0
$$887$$ −7.20927 −0.242063 −0.121032 0.992649i $$-0.538620\pi$$
−0.121032 + 0.992649i $$0.538620\pi$$
$$888$$ 0 0
$$889$$ −16.9948 −0.569988
$$890$$ 0 0
$$891$$ −1.82753 −0.0612246
$$892$$ 0 0
$$893$$ −3.71597 −0.124350
$$894$$ 0 0
$$895$$ −38.9495 −1.30194
$$896$$ 0 0
$$897$$ 13.1883 0.440344
$$898$$ 0 0
$$899$$ −1.69102 −0.0563986
$$900$$ 0 0
$$901$$ 0.336492 0.0112102
$$902$$ 0 0
$$903$$ −23.2866 −0.774928
$$904$$ 0 0
$$905$$ 24.5684 0.816680
$$906$$ 0 0
$$907$$ 25.7832 0.856116 0.428058 0.903751i $$-0.359198\pi$$
0.428058 + 0.903751i $$0.359198\pi$$
$$908$$ 0 0
$$909$$ 17.2701 0.572814
$$910$$ 0 0
$$911$$ −31.1334 −1.03149 −0.515747 0.856741i $$-0.672486\pi$$
−0.515747 + 0.856741i $$0.672486\pi$$
$$912$$ 0 0
$$913$$ −5.44061 −0.180058
$$914$$ 0 0
$$915$$ 31.6764 1.04719
$$916$$ 0 0
$$917$$ 13.5493 0.447437
$$918$$ 0 0
$$919$$ −5.42725 −0.179029 −0.0895143 0.995986i $$-0.528531\pi$$
−0.0895143 + 0.995986i $$0.528531\pi$$
$$920$$ 0 0
$$921$$ −28.1452 −0.927416
$$922$$ 0 0
$$923$$ 9.35887 0.308051
$$924$$ 0 0
$$925$$ −78.0063 −2.56483
$$926$$ 0 0
$$927$$ −7.83531 −0.257345
$$928$$ 0 0
$$929$$ −21.6025 −0.708756 −0.354378 0.935102i $$-0.615307\pi$$
−0.354378 + 0.935102i $$0.615307\pi$$
$$930$$ 0 0
$$931$$ −6.09205 −0.199659
$$932$$ 0 0
$$933$$ −19.1390 −0.626583
$$934$$ 0 0
$$935$$ −13.3313 −0.435979
$$936$$ 0 0
$$937$$ 31.6840 1.03507 0.517536 0.855661i $$-0.326849\pi$$
0.517536 + 0.855661i $$0.326849\pi$$
$$938$$ 0 0
$$939$$ −19.1399 −0.624608
$$940$$ 0 0
$$941$$ −5.63987 −0.183854 −0.0919272 0.995766i $$-0.529303\pi$$
−0.0919272 + 0.995766i $$0.529303\pi$$
$$942$$ 0 0
$$943$$ −29.2687 −0.953120
$$944$$ 0 0
$$945$$ −80.6176 −2.62249
$$946$$ 0 0
$$947$$ 51.7369 1.68122 0.840611 0.541639i $$-0.182196\pi$$
0.840611 + 0.541639i $$0.182196\pi$$
$$948$$ 0 0
$$949$$ 2.03686 0.0661194
$$950$$ 0 0
$$951$$ 24.9682 0.809650
$$952$$ 0 0
$$953$$ −16.2553 −0.526561 −0.263281 0.964719i $$-0.584804\pi$$
−0.263281 + 0.964719i $$0.584804\pi$$
$$954$$ 0 0
$$955$$ −72.1874 −2.33593
$$956$$ 0 0
$$957$$ −1.22631 −0.0396410
$$958$$ 0 0
$$959$$ 57.0225 1.84135
$$960$$ 0 0
$$961$$ −28.2801 −0.912261
$$962$$ 0 0
$$963$$ 11.4792 0.369913
$$964$$ 0 0
$$965$$ 15.0576 0.484721
$$966$$ 0 0
$$967$$ 54.5004 1.75261 0.876307 0.481754i $$-0.160000\pi$$
0.876307 + 0.481754i $$0.160000\pi$$
$$968$$ 0 0
$$969$$ −3.91093 −0.125637
$$970$$ 0 0
$$971$$ 34.2679 1.09971 0.549855 0.835260i $$-0.314683\pi$$
0.549855 + 0.835260i $$0.314683\pi$$
$$972$$ 0 0
$$973$$ −9.14783 −0.293266
$$974$$ 0 0
$$975$$ 20.5488 0.658090
$$976$$ 0 0
$$977$$ 55.5644 1.77766 0.888831 0.458234i $$-0.151518\pi$$
0.888831 + 0.458234i $$0.151518\pi$$
$$978$$ 0 0
$$979$$ 12.1357 0.387858
$$980$$ 0 0
$$981$$ 2.26780 0.0724052
$$982$$ 0 0
$$983$$ 41.3971 1.32036 0.660181 0.751106i $$-0.270479\pi$$
0.660181 + 0.751106i $$0.270479\pi$$
$$984$$ 0 0
$$985$$ −105.095 −3.34862
$$986$$ 0 0
$$987$$ 16.0807 0.511853
$$988$$ 0 0
$$989$$ −40.1321 −1.27613
$$990$$ 0 0
$$991$$ −60.8219 −1.93207 −0.966036 0.258406i $$-0.916803\pi$$
−0.966036 + 0.258406i $$0.916803\pi$$
$$992$$ 0 0
$$993$$ 18.0758 0.573618
$$994$$ 0 0
$$995$$ −77.0324 −2.44209
$$996$$ 0 0
$$997$$ 26.0627 0.825414 0.412707 0.910864i $$-0.364583\pi$$
0.412707 + 0.910864i $$0.364583\pi$$
$$998$$ 0 0
$$999$$ −36.6876 −1.16074
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 3344.2.a.ba.1.4 7
4.3 odd 2 209.2.a.d.1.2 7
12.11 even 2 1881.2.a.p.1.6 7
20.19 odd 2 5225.2.a.n.1.6 7
44.43 even 2 2299.2.a.q.1.6 7
76.75 even 2 3971.2.a.i.1.6 7

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.2 7 4.3 odd 2
1881.2.a.p.1.6 7 12.11 even 2
2299.2.a.q.1.6 7 44.43 even 2
3344.2.a.ba.1.4 7 1.1 even 1 trivial
3971.2.a.i.1.6 7 76.75 even 2
5225.2.a.n.1.6 7 20.19 odd 2