# Properties

 Label 8550.2.a.co.1.1 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $0$ 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] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.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: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 950) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.91223$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -4.22547 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -4.22547 q^{7} +1.00000 q^{8} +5.13770 q^{11} -3.16784 q^{13} -4.22547 q^{14} +1.00000 q^{16} -6.48108 q^{17} -1.00000 q^{19} +5.13770 q^{22} +7.56885 q^{23} -3.16784 q^{26} -4.22547 q^{28} -0.832162 q^{29} -4.51122 q^{31} +1.00000 q^{32} -6.48108 q^{34} -0.137699 q^{37} -1.00000 q^{38} +11.6489 q^{41} +2.51122 q^{43} +5.13770 q^{44} +7.56885 q^{46} -5.96216 q^{47} +10.8546 q^{49} -3.16784 q^{52} +0.225470 q^{53} -4.22547 q^{56} -0.832162 q^{58} -5.39331 q^{59} +14.4509 q^{61} -4.51122 q^{62} +1.00000 q^{64} +4.11021 q^{67} -6.48108 q^{68} -3.82446 q^{71} -4.70655 q^{73} -0.137699 q^{74} -1.00000 q^{76} -21.7092 q^{77} +10.6265 q^{79} +11.6489 q^{82} -12.0999 q^{83} +2.51122 q^{86} +5.13770 q^{88} +10.0000 q^{89} +13.3856 q^{91} +7.56885 q^{92} -5.96216 q^{94} -3.93972 q^{97} +10.8546 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 $$3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 2 q^{13} - 2 q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 2 q^{22} + 14 q^{23} + 2 q^{26} - 2 q^{28} - 14 q^{29} - 4 q^{31} + 3 q^{32} - 4 q^{34} + 17 q^{37} - 3 q^{38} + 8 q^{41} - 2 q^{43} - 2 q^{44} + 14 q^{46} + 13 q^{47} + 25 q^{49} + 2 q^{52} - 10 q^{53} - 2 q^{56} - 14 q^{58} + 6 q^{59} + 22 q^{61} - 4 q^{62} + 3 q^{64} - 4 q^{68} + 2 q^{71} + 12 q^{73} + 17 q^{74} - 3 q^{76} - 50 q^{77} + 24 q^{79} + 8 q^{82} + 12 q^{83} - 2 q^{86} - 2 q^{88} + 30 q^{89} - 7 q^{91} + 14 q^{92} + 13 q^{94} + 25 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 2 * q^13 - 2 * q^14 + 3 * q^16 - 4 * q^17 - 3 * q^19 - 2 * q^22 + 14 * q^23 + 2 * q^26 - 2 * q^28 - 14 * q^29 - 4 * q^31 + 3 * q^32 - 4 * q^34 + 17 * q^37 - 3 * q^38 + 8 * q^41 - 2 * q^43 - 2 * q^44 + 14 * q^46 + 13 * q^47 + 25 * q^49 + 2 * q^52 - 10 * q^53 - 2 * q^56 - 14 * q^58 + 6 * q^59 + 22 * q^61 - 4 * q^62 + 3 * q^64 - 4 * q^68 + 2 * q^71 + 12 * q^73 + 17 * q^74 - 3 * q^76 - 50 * q^77 + 24 * q^79 + 8 * q^82 + 12 * q^83 - 2 * q^86 - 2 * q^88 + 30 * q^89 - 7 * q^91 + 14 * q^92 + 13 * q^94 + 25 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.22547 −1.59708 −0.798539 0.601943i $$-0.794393\pi$$
−0.798539 + 0.601943i $$0.794393\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.13770 1.54907 0.774537 0.632528i $$-0.217983\pi$$
0.774537 + 0.632528i $$0.217983\pi$$
$$12$$ 0 0
$$13$$ −3.16784 −0.878600 −0.439300 0.898340i $$-0.644774\pi$$
−0.439300 + 0.898340i $$0.644774\pi$$
$$14$$ −4.22547 −1.12930
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.48108 −1.57189 −0.785946 0.618295i $$-0.787824\pi$$
−0.785946 + 0.618295i $$0.787824\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 5.13770 1.09536
$$23$$ 7.56885 1.57821 0.789107 0.614256i $$-0.210544\pi$$
0.789107 + 0.614256i $$0.210544\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −3.16784 −0.621264
$$27$$ 0 0
$$28$$ −4.22547 −0.798539
$$29$$ −0.832162 −0.154529 −0.0772643 0.997011i $$-0.524619\pi$$
−0.0772643 + 0.997011i $$0.524619\pi$$
$$30$$ 0 0
$$31$$ −4.51122 −0.810239 −0.405119 0.914264i $$-0.632770\pi$$
−0.405119 + 0.914264i $$0.632770\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −6.48108 −1.11150
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −0.137699 −0.0226376 −0.0113188 0.999936i $$-0.503603\pi$$
−0.0113188 + 0.999936i $$0.503603\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.6489 1.81926 0.909628 0.415425i $$-0.136367\pi$$
0.909628 + 0.415425i $$0.136367\pi$$
$$42$$ 0 0
$$43$$ 2.51122 0.382957 0.191479 0.981497i $$-0.438672\pi$$
0.191479 + 0.981497i $$0.438672\pi$$
$$44$$ 5.13770 0.774537
$$45$$ 0 0
$$46$$ 7.56885 1.11597
$$47$$ −5.96216 −0.869670 −0.434835 0.900510i $$-0.643193\pi$$
−0.434835 + 0.900510i $$0.643193\pi$$
$$48$$ 0 0
$$49$$ 10.8546 1.55066
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −3.16784 −0.439300
$$53$$ 0.225470 0.0309707 0.0154853 0.999880i $$-0.495071\pi$$
0.0154853 + 0.999880i $$0.495071\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −4.22547 −0.564652
$$57$$ 0 0
$$58$$ −0.832162 −0.109268
$$59$$ −5.39331 −0.702149 −0.351074 0.936348i $$-0.614184\pi$$
−0.351074 + 0.936348i $$0.614184\pi$$
$$60$$ 0 0
$$61$$ 14.4509 1.85025 0.925127 0.379659i $$-0.123959\pi$$
0.925127 + 0.379659i $$0.123959\pi$$
$$62$$ −4.51122 −0.572925
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.11021 0.502142 0.251071 0.967969i $$-0.419217\pi$$
0.251071 + 0.967969i $$0.419217\pi$$
$$68$$ −6.48108 −0.785946
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.82446 −0.453880 −0.226940 0.973909i $$-0.572872\pi$$
−0.226940 + 0.973909i $$0.572872\pi$$
$$72$$ 0 0
$$73$$ −4.70655 −0.550860 −0.275430 0.961321i $$-0.588820\pi$$
−0.275430 + 0.961321i $$0.588820\pi$$
$$74$$ −0.137699 −0.0160072
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −21.7092 −2.47399
$$78$$ 0 0
$$79$$ 10.6265 1.19557 0.597786 0.801655i $$-0.296047\pi$$
0.597786 + 0.801655i $$0.296047\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 11.6489 1.28641
$$83$$ −12.0999 −1.32813 −0.664066 0.747674i $$-0.731171\pi$$
−0.664066 + 0.747674i $$0.731171\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.51122 0.270792
$$87$$ 0 0
$$88$$ 5.13770 0.547681
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 13.3856 1.40319
$$92$$ 7.56885 0.789107
$$93$$ 0 0
$$94$$ −5.96216 −0.614950
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −3.93972 −0.400018 −0.200009 0.979794i $$-0.564097\pi$$
−0.200009 + 0.979794i $$0.564097\pi$$
$$98$$ 10.8546 1.09648
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −3.19798 −0.318211 −0.159105 0.987262i $$-0.550861\pi$$
−0.159105 + 0.987262i $$0.550861\pi$$
$$102$$ 0 0
$$103$$ 10.6868 1.05300 0.526499 0.850176i $$-0.323504\pi$$
0.526499 + 0.850176i $$0.323504\pi$$
$$104$$ −3.16784 −0.310632
$$105$$ 0 0
$$106$$ 0.225470 0.0218996
$$107$$ 10.8168 1.04570 0.522848 0.852426i $$-0.324870\pi$$
0.522848 + 0.852426i $$0.324870\pi$$
$$108$$ 0 0
$$109$$ 9.24791 0.885789 0.442894 0.896574i $$-0.353952\pi$$
0.442894 + 0.896574i $$0.353952\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −4.22547 −0.399269
$$113$$ 17.6489 1.66027 0.830135 0.557562i $$-0.188263\pi$$
0.830135 + 0.557562i $$0.188263\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −0.832162 −0.0772643
$$117$$ 0 0
$$118$$ −5.39331 −0.496494
$$119$$ 27.3856 2.51043
$$120$$ 0 0
$$121$$ 15.3960 1.39963
$$122$$ 14.4509 1.30833
$$123$$ 0 0
$$124$$ −4.51122 −0.405119
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.7866 1.13463 0.567314 0.823501i $$-0.307982\pi$$
0.567314 + 0.823501i $$0.307982\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2.11526 0.184811 0.0924057 0.995721i $$-0.470544\pi$$
0.0924057 + 0.995721i $$0.470544\pi$$
$$132$$ 0 0
$$133$$ 4.22547 0.366395
$$134$$ 4.11021 0.355068
$$135$$ 0 0
$$136$$ −6.48108 −0.555748
$$137$$ −5.36317 −0.458206 −0.229103 0.973402i $$-0.573579\pi$$
−0.229103 + 0.973402i $$0.573579\pi$$
$$138$$ 0 0
$$139$$ −10.1601 −0.861771 −0.430886 0.902407i $$-0.641799\pi$$
−0.430886 + 0.902407i $$0.641799\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −3.82446 −0.320941
$$143$$ −16.2754 −1.36102
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −4.70655 −0.389517
$$147$$ 0 0
$$148$$ −0.137699 −0.0113188
$$149$$ 5.93972 0.486601 0.243301 0.969951i $$-0.421770\pi$$
0.243301 + 0.969951i $$0.421770\pi$$
$$150$$ 0 0
$$151$$ −15.2978 −1.24492 −0.622460 0.782652i $$-0.713867\pi$$
−0.622460 + 0.782652i $$0.713867\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ −21.7092 −1.74938
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 12.7866 1.02048 0.510242 0.860031i $$-0.329556\pi$$
0.510242 + 0.860031i $$0.329556\pi$$
$$158$$ 10.6265 0.845397
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −31.9819 −2.52053
$$162$$ 0 0
$$163$$ 11.4734 0.898664 0.449332 0.893365i $$-0.351662\pi$$
0.449332 + 0.893365i $$0.351662\pi$$
$$164$$ 11.6489 0.909628
$$165$$ 0 0
$$166$$ −12.0999 −0.939131
$$167$$ 19.4131 1.50223 0.751115 0.660171i $$-0.229516\pi$$
0.751115 + 0.660171i $$0.229516\pi$$
$$168$$ 0 0
$$169$$ −2.96480 −0.228062
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2.51122 0.191479
$$173$$ 9.78662 0.744063 0.372031 0.928220i $$-0.378661\pi$$
0.372031 + 0.928220i $$0.378661\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 5.13770 0.387269
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ 6.82446 0.510084 0.255042 0.966930i $$-0.417911\pi$$
0.255042 + 0.966930i $$0.417911\pi$$
$$180$$ 0 0
$$181$$ −0.137699 −0.0102351 −0.00511755 0.999987i $$-0.501629\pi$$
−0.00511755 + 0.999987i $$0.501629\pi$$
$$182$$ 13.3856 0.992207
$$183$$ 0 0
$$184$$ 7.56885 0.557983
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −33.2978 −2.43498
$$188$$ −5.96216 −0.434835
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −19.3779 −1.40214 −0.701068 0.713095i $$-0.747293\pi$$
−0.701068 + 0.713095i $$0.747293\pi$$
$$192$$ 0 0
$$193$$ 5.42851 0.390752 0.195376 0.980728i $$-0.437407\pi$$
0.195376 + 0.980728i $$0.437407\pi$$
$$194$$ −3.93972 −0.282856
$$195$$ 0 0
$$196$$ 10.8546 0.775328
$$197$$ −15.6489 −1.11494 −0.557470 0.830197i $$-0.688228\pi$$
−0.557470 + 0.830197i $$0.688228\pi$$
$$198$$ 0 0
$$199$$ 18.0499 1.27953 0.639763 0.768572i $$-0.279033\pi$$
0.639763 + 0.768572i $$0.279033\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −3.19798 −0.225009
$$203$$ 3.51628 0.246794
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 10.6868 0.744582
$$207$$ 0 0
$$208$$ −3.16784 −0.219650
$$209$$ −5.13770 −0.355382
$$210$$ 0 0
$$211$$ −7.50857 −0.516911 −0.258456 0.966023i $$-0.583214\pi$$
−0.258456 + 0.966023i $$0.583214\pi$$
$$212$$ 0.225470 0.0154853
$$213$$ 0 0
$$214$$ 10.8168 0.739418
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 19.0620 1.29401
$$218$$ 9.24791 0.626347
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 20.5310 1.38106
$$222$$ 0 0
$$223$$ −27.8091 −1.86223 −0.931116 0.364723i $$-0.881164\pi$$
−0.931116 + 0.364723i $$0.881164\pi$$
$$224$$ −4.22547 −0.282326
$$225$$ 0 0
$$226$$ 17.6489 1.17399
$$227$$ 8.91223 0.591525 0.295763 0.955261i $$-0.404426\pi$$
0.295763 + 0.955261i $$0.404426\pi$$
$$228$$ 0 0
$$229$$ −13.6489 −0.901946 −0.450973 0.892538i $$-0.648923\pi$$
−0.450973 + 0.892538i $$0.648923\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −0.832162 −0.0546341
$$233$$ 23.2754 1.52482 0.762411 0.647093i $$-0.224015\pi$$
0.762411 + 0.647093i $$0.224015\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −5.39331 −0.351074
$$237$$ 0 0
$$238$$ 27.3856 1.77515
$$239$$ −3.72898 −0.241208 −0.120604 0.992701i $$-0.538483\pi$$
−0.120604 + 0.992701i $$0.538483\pi$$
$$240$$ 0 0
$$241$$ −1.48878 −0.0959009 −0.0479505 0.998850i $$-0.515269\pi$$
−0.0479505 + 0.998850i $$0.515269\pi$$
$$242$$ 15.3960 0.989689
$$243$$ 0 0
$$244$$ 14.4509 0.925127
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.16784 0.201565
$$248$$ −4.51122 −0.286463
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 8.78662 0.554606 0.277303 0.960782i $$-0.410559\pi$$
0.277303 + 0.960782i $$0.410559\pi$$
$$252$$ 0 0
$$253$$ 38.8865 2.44477
$$254$$ 12.7866 0.802304
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 2.74175 0.171025 0.0855127 0.996337i $$-0.472747\pi$$
0.0855127 + 0.996337i $$0.472747\pi$$
$$258$$ 0 0
$$259$$ 0.581844 0.0361540
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2.11526 0.130681
$$263$$ 2.74704 0.169390 0.0846948 0.996407i $$-0.473008\pi$$
0.0846948 + 0.996407i $$0.473008\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.22547 0.259080
$$267$$ 0 0
$$268$$ 4.11021 0.251071
$$269$$ 9.74175 0.593965 0.296982 0.954883i $$-0.404020\pi$$
0.296982 + 0.954883i $$0.404020\pi$$
$$270$$ 0 0
$$271$$ 26.3555 1.60098 0.800490 0.599346i $$-0.204573\pi$$
0.800490 + 0.599346i $$0.204573\pi$$
$$272$$ −6.48108 −0.392973
$$273$$ 0 0
$$274$$ −5.36317 −0.324001
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0.962158 0.0578104 0.0289052 0.999582i $$-0.490798\pi$$
0.0289052 + 0.999582i $$0.490798\pi$$
$$278$$ −10.1601 −0.609364
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.6714 0.875219 0.437610 0.899165i $$-0.355825\pi$$
0.437610 + 0.899165i $$0.355825\pi$$
$$282$$ 0 0
$$283$$ 26.1601 1.55506 0.777529 0.628847i $$-0.216473\pi$$
0.777529 + 0.628847i $$0.216473\pi$$
$$284$$ −3.82446 −0.226940
$$285$$ 0 0
$$286$$ −16.2754 −0.962384
$$287$$ −49.2221 −2.90549
$$288$$ 0 0
$$289$$ 25.0044 1.47085
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4.70655 −0.275430
$$293$$ −22.4657 −1.31246 −0.656229 0.754562i $$-0.727850\pi$$
−0.656229 + 0.754562i $$0.727850\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −0.137699 −0.00800360
$$297$$ 0 0
$$298$$ 5.93972 0.344079
$$299$$ −23.9769 −1.38662
$$300$$ 0 0
$$301$$ −10.6111 −0.611612
$$302$$ −15.2978 −0.880291
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −17.2204 −0.982821 −0.491410 0.870928i $$-0.663518\pi$$
−0.491410 + 0.870928i $$0.663518\pi$$
$$308$$ −21.7092 −1.23700
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7.87439 0.446516 0.223258 0.974759i $$-0.428331\pi$$
0.223258 + 0.974759i $$0.428331\pi$$
$$312$$ 0 0
$$313$$ 25.1678 1.42257 0.711285 0.702904i $$-0.248113\pi$$
0.711285 + 0.702904i $$0.248113\pi$$
$$314$$ 12.7866 0.721590
$$315$$ 0 0
$$316$$ 10.6265 0.597786
$$317$$ −23.9045 −1.34261 −0.671306 0.741180i $$-0.734266\pi$$
−0.671306 + 0.741180i $$0.734266\pi$$
$$318$$ 0 0
$$319$$ −4.27540 −0.239376
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −31.9819 −1.78228
$$323$$ 6.48108 0.360617
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 11.4734 0.635451
$$327$$ 0 0
$$328$$ 11.6489 0.643204
$$329$$ 25.1929 1.38893
$$330$$ 0 0
$$331$$ 30.7565 1.69053 0.845264 0.534348i $$-0.179443\pi$$
0.845264 + 0.534348i $$0.179443\pi$$
$$332$$ −12.0999 −0.664066
$$333$$ 0 0
$$334$$ 19.4131 1.06224
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −13.4734 −0.733942 −0.366971 0.930232i $$-0.619605\pi$$
−0.366971 + 0.930232i $$0.619605\pi$$
$$338$$ −2.96480 −0.161264
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −23.1773 −1.25512
$$342$$ 0 0
$$343$$ −16.2875 −0.879441
$$344$$ 2.51122 0.135396
$$345$$ 0 0
$$346$$ 9.78662 0.526132
$$347$$ 28.9468 1.55394 0.776971 0.629536i $$-0.216755\pi$$
0.776971 + 0.629536i $$0.216755\pi$$
$$348$$ 0 0
$$349$$ 27.9243 1.49475 0.747377 0.664400i $$-0.231313\pi$$
0.747377 + 0.664400i $$0.231313\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.13770 0.273840
$$353$$ 28.3099 1.50678 0.753392 0.657571i $$-0.228416\pi$$
0.753392 + 0.657571i $$0.228416\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ 6.82446 0.360684
$$359$$ −2.60163 −0.137309 −0.0686545 0.997640i $$-0.521871\pi$$
−0.0686545 + 0.997640i $$0.521871\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −0.137699 −0.00723731
$$363$$ 0 0
$$364$$ 13.3856 0.701596
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −11.6489 −0.608069 −0.304034 0.952661i $$-0.598334\pi$$
−0.304034 + 0.952661i $$0.598334\pi$$
$$368$$ 7.56885 0.394554
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −0.952717 −0.0494626
$$372$$ 0 0
$$373$$ 12.8064 0.663091 0.331545 0.943439i $$-0.392430\pi$$
0.331545 + 0.943439i $$0.392430\pi$$
$$374$$ −33.2978 −1.72179
$$375$$ 0 0
$$376$$ −5.96216 −0.307475
$$377$$ 2.63615 0.135769
$$378$$ 0 0
$$379$$ −20.9122 −1.07419 −0.537095 0.843522i $$-0.680478\pi$$
−0.537095 + 0.843522i $$0.680478\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −19.3779 −0.991460
$$383$$ 10.3511 0.528916 0.264458 0.964397i $$-0.414807\pi$$
0.264458 + 0.964397i $$0.414807\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 5.42851 0.276304
$$387$$ 0 0
$$388$$ −3.93972 −0.200009
$$389$$ −6.56620 −0.332920 −0.166460 0.986048i $$-0.553234\pi$$
−0.166460 + 0.986048i $$0.553234\pi$$
$$390$$ 0 0
$$391$$ −49.0543 −2.48078
$$392$$ 10.8546 0.548240
$$393$$ 0 0
$$394$$ −15.6489 −0.788381
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −23.5284 −1.18085 −0.590427 0.807091i $$-0.701041\pi$$
−0.590427 + 0.807091i $$0.701041\pi$$
$$398$$ 18.0499 0.904761
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.22041 −0.310633 −0.155316 0.987865i $$-0.549640\pi$$
−0.155316 + 0.987865i $$0.549640\pi$$
$$402$$ 0 0
$$403$$ 14.2908 0.711876
$$404$$ −3.19798 −0.159105
$$405$$ 0 0
$$406$$ 3.51628 0.174510
$$407$$ −0.707457 −0.0350674
$$408$$ 0 0
$$409$$ −34.4905 −1.70545 −0.852723 0.522363i $$-0.825051\pi$$
−0.852723 + 0.522363i $$0.825051\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 10.6868 0.526499
$$413$$ 22.7893 1.12139
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −3.16784 −0.155316
$$417$$ 0 0
$$418$$ −5.13770 −0.251293
$$419$$ 10.6265 0.519138 0.259569 0.965725i $$-0.416420\pi$$
0.259569 + 0.965725i $$0.416420\pi$$
$$420$$ 0 0
$$421$$ −1.98021 −0.0965095 −0.0482548 0.998835i $$-0.515366\pi$$
−0.0482548 + 0.998835i $$0.515366\pi$$
$$422$$ −7.50857 −0.365512
$$423$$ 0 0
$$424$$ 0.225470 0.0109498
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −61.0620 −2.95500
$$428$$ 10.8168 0.522848
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −15.0774 −0.726254 −0.363127 0.931740i $$-0.618291\pi$$
−0.363127 + 0.931740i $$0.618291\pi$$
$$432$$ 0 0
$$433$$ −13.5337 −0.650386 −0.325193 0.945648i $$-0.605429\pi$$
−0.325193 + 0.945648i $$0.605429\pi$$
$$434$$ 19.0620 0.915006
$$435$$ 0 0
$$436$$ 9.24791 0.442894
$$437$$ −7.56885 −0.362067
$$438$$ 0 0
$$439$$ 39.1773 1.86983 0.934915 0.354872i $$-0.115476\pi$$
0.934915 + 0.354872i $$0.115476\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 20.5310 0.976560
$$443$$ 19.3132 0.917600 0.458800 0.888540i $$-0.348279\pi$$
0.458800 + 0.888540i $$0.348279\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −27.8091 −1.31680
$$447$$ 0 0
$$448$$ −4.22547 −0.199635
$$449$$ −41.4131 −1.95440 −0.977202 0.212310i $$-0.931901\pi$$
−0.977202 + 0.212310i $$0.931901\pi$$
$$450$$ 0 0
$$451$$ 59.8486 2.81816
$$452$$ 17.6489 0.830135
$$453$$ 0 0
$$454$$ 8.91223 0.418272
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 37.8392 1.77004 0.885021 0.465551i $$-0.154144\pi$$
0.885021 + 0.465551i $$0.154144\pi$$
$$458$$ −13.6489 −0.637772
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1.58864 −0.0739903 −0.0369952 0.999315i $$-0.511779\pi$$
−0.0369952 + 0.999315i $$0.511779\pi$$
$$462$$ 0 0
$$463$$ −40.3581 −1.87560 −0.937800 0.347175i $$-0.887141\pi$$
−0.937800 + 0.347175i $$0.887141\pi$$
$$464$$ −0.832162 −0.0386322
$$465$$ 0 0
$$466$$ 23.2754 1.07821
$$467$$ −39.5130 −1.82844 −0.914221 0.405217i $$-0.867196\pi$$
−0.914221 + 0.405217i $$0.867196\pi$$
$$468$$ 0 0
$$469$$ −17.3676 −0.801959
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −5.39331 −0.248247
$$473$$ 12.9019 0.593229
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 27.3856 1.25522
$$477$$ 0 0
$$478$$ −3.72898 −0.170560
$$479$$ −7.61107 −0.347759 −0.173879 0.984767i $$-0.555630\pi$$
−0.173879 + 0.984767i $$0.555630\pi$$
$$480$$ 0 0
$$481$$ 0.436209 0.0198894
$$482$$ −1.48878 −0.0678122
$$483$$ 0 0
$$484$$ 15.3960 0.699816
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 20.3907 0.923989 0.461995 0.886883i $$-0.347134\pi$$
0.461995 + 0.886883i $$0.347134\pi$$
$$488$$ 14.4509 0.654163
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −25.0224 −1.12925 −0.564623 0.825349i $$-0.690979\pi$$
−0.564623 + 0.825349i $$0.690979\pi$$
$$492$$ 0 0
$$493$$ 5.39331 0.242902
$$494$$ 3.16784 0.142528
$$495$$ 0 0
$$496$$ −4.51122 −0.202560
$$497$$ 16.1601 0.724881
$$498$$ 0 0
$$499$$ 22.6111 1.01221 0.506105 0.862472i $$-0.331085\pi$$
0.506105 + 0.862472i $$0.331085\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 8.78662 0.392166
$$503$$ 11.5035 0.512916 0.256458 0.966555i $$-0.417444\pi$$
0.256458 + 0.966555i $$0.417444\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 38.8865 1.72871
$$507$$ 0 0
$$508$$ 12.7866 0.567314
$$509$$ −9.11526 −0.404027 −0.202013 0.979383i $$-0.564748\pi$$
−0.202013 + 0.979383i $$0.564748\pi$$
$$510$$ 0 0
$$511$$ 19.8874 0.879766
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 2.74175 0.120933
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −30.6318 −1.34718
$$518$$ 0.581844 0.0255648
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −15.4888 −0.678576 −0.339288 0.940683i $$-0.610186\pi$$
−0.339288 + 0.940683i $$0.610186\pi$$
$$522$$ 0 0
$$523$$ −4.34073 −0.189807 −0.0949035 0.995486i $$-0.530254\pi$$
−0.0949035 + 0.995486i $$0.530254\pi$$
$$524$$ 2.11526 0.0924057
$$525$$ 0 0
$$526$$ 2.74704 0.119776
$$527$$ 29.2376 1.27361
$$528$$ 0 0
$$529$$ 34.2875 1.49076
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.22547 0.183197
$$533$$ −36.9019 −1.59840
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 4.11021 0.177534
$$537$$ 0 0
$$538$$ 9.74175 0.419996
$$539$$ 55.7677 2.40208
$$540$$ 0 0
$$541$$ 19.5491 0.840480 0.420240 0.907413i $$-0.361946\pi$$
0.420240 + 0.907413i $$0.361946\pi$$
$$542$$ 26.3555 1.13206
$$543$$ 0 0
$$544$$ −6.48108 −0.277874
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 41.8038 1.78740 0.893700 0.448665i $$-0.148100\pi$$
0.893700 + 0.448665i $$0.148100\pi$$
$$548$$ −5.36317 −0.229103
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.832162 0.0354513
$$552$$ 0 0
$$553$$ −44.9019 −1.90942
$$554$$ 0.962158 0.0408782
$$555$$ 0 0
$$556$$ −10.1601 −0.430886
$$557$$ −9.76418 −0.413722 −0.206861 0.978370i $$-0.566325\pi$$
−0.206861 + 0.978370i $$0.566325\pi$$
$$558$$ 0 0
$$559$$ −7.95513 −0.336466
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 14.6714 0.618874
$$563$$ 11.4509 0.482600 0.241300 0.970451i $$-0.422426\pi$$
0.241300 + 0.970451i $$0.422426\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 26.1601 1.09959
$$567$$ 0 0
$$568$$ −3.82446 −0.160471
$$569$$ 13.4338 0.563174 0.281587 0.959536i $$-0.409139\pi$$
0.281587 + 0.959536i $$0.409139\pi$$
$$570$$ 0 0
$$571$$ 37.6335 1.57491 0.787457 0.616370i $$-0.211397\pi$$
0.787457 + 0.616370i $$0.211397\pi$$
$$572$$ −16.2754 −0.680509
$$573$$ 0 0
$$574$$ −49.2221 −2.05449
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −1.56885 −0.0653121 −0.0326560 0.999467i $$-0.510397\pi$$
−0.0326560 + 0.999467i $$0.510397\pi$$
$$578$$ 25.0044 1.04005
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 51.1276 2.12113
$$582$$ 0 0
$$583$$ 1.15840 0.0479759
$$584$$ −4.70655 −0.194758
$$585$$ 0 0
$$586$$ −22.4657 −0.928048
$$587$$ 25.8693 1.06774 0.533871 0.845566i $$-0.320737\pi$$
0.533871 + 0.845566i $$0.320737\pi$$
$$588$$ 0 0
$$589$$ 4.51122 0.185881
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −0.137699 −0.00565940
$$593$$ −28.9243 −1.18778 −0.593890 0.804547i $$-0.702408\pi$$
−0.593890 + 0.804547i $$0.702408\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 5.93972 0.243301
$$597$$ 0 0
$$598$$ −23.9769 −0.980488
$$599$$ −41.3581 −1.68985 −0.844923 0.534887i $$-0.820354\pi$$
−0.844923 + 0.534887i $$0.820354\pi$$
$$600$$ 0 0
$$601$$ 2.68147 0.109379 0.0546897 0.998503i $$-0.482583\pi$$
0.0546897 + 0.998503i $$0.482583\pi$$
$$602$$ −10.6111 −0.432475
$$603$$ 0 0
$$604$$ −15.2978 −0.622460
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −3.31324 −0.134480 −0.0672401 0.997737i $$-0.521419\pi$$
−0.0672401 + 0.997737i $$0.521419\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 18.8871 0.764092
$$612$$ 0 0
$$613$$ 10.0603 0.406331 0.203165 0.979144i $$-0.434877\pi$$
0.203165 + 0.979144i $$0.434877\pi$$
$$614$$ −17.2204 −0.694959
$$615$$ 0 0
$$616$$ −21.7092 −0.874688
$$617$$ −27.6265 −1.11220 −0.556100 0.831115i $$-0.687703\pi$$
−0.556100 + 0.831115i $$0.687703\pi$$
$$618$$ 0 0
$$619$$ 16.6714 0.670078 0.335039 0.942204i $$-0.391250\pi$$
0.335039 + 0.942204i $$0.391250\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 7.87439 0.315734
$$623$$ −42.2547 −1.69290
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 25.1678 1.00591
$$627$$ 0 0
$$628$$ 12.7866 0.510242
$$629$$ 0.892439 0.0355839
$$630$$ 0 0
$$631$$ −0.709194 −0.0282326 −0.0141163 0.999900i $$-0.504494\pi$$
−0.0141163 + 0.999900i $$0.504494\pi$$
$$632$$ 10.6265 0.422699
$$633$$ 0 0
$$634$$ −23.9045 −0.949370
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −34.3856 −1.36241
$$638$$ −4.27540 −0.169265
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.43553 0.0961978 0.0480989 0.998843i $$-0.484684\pi$$
0.0480989 + 0.998843i $$0.484684\pi$$
$$642$$ 0 0
$$643$$ −7.70390 −0.303812 −0.151906 0.988395i $$-0.548541\pi$$
−0.151906 + 0.988395i $$0.548541\pi$$
$$644$$ −31.9819 −1.26027
$$645$$ 0 0
$$646$$ 6.48108 0.254995
$$647$$ −10.0499 −0.395103 −0.197552 0.980292i $$-0.563299\pi$$
−0.197552 + 0.980292i $$0.563299\pi$$
$$648$$ 0 0
$$649$$ −27.7092 −1.08768
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 11.4734 0.449332
$$653$$ −7.09283 −0.277564 −0.138782 0.990323i $$-0.544319\pi$$
−0.138782 + 0.990323i $$0.544319\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 11.6489 0.454814
$$657$$ 0 0
$$658$$ 25.1929 0.982122
$$659$$ 1.16346 0.0453218 0.0226609 0.999743i $$-0.492786\pi$$
0.0226609 + 0.999743i $$0.492786\pi$$
$$660$$ 0 0
$$661$$ 1.79432 0.0697909 0.0348955 0.999391i $$-0.488890\pi$$
0.0348955 + 0.999391i $$0.488890\pi$$
$$662$$ 30.7565 1.19538
$$663$$ 0 0
$$664$$ −12.0999 −0.469566
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.29851 −0.243879
$$668$$ 19.4131 0.751115
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 74.2446 2.86618
$$672$$ 0 0
$$673$$ 25.2824 0.974566 0.487283 0.873244i $$-0.337988\pi$$
0.487283 + 0.873244i $$0.337988\pi$$
$$674$$ −13.4734 −0.518975
$$675$$ 0 0
$$676$$ −2.96480 −0.114031
$$677$$ −4.89682 −0.188200 −0.0941001 0.995563i $$-0.529997\pi$$
−0.0941001 + 0.995563i $$0.529997\pi$$
$$678$$ 0 0
$$679$$ 16.6472 0.638860
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −23.1773 −0.887504
$$683$$ 10.1980 0.390215 0.195107 0.980782i $$-0.437494\pi$$
0.195107 + 0.980782i $$0.437494\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −16.2875 −0.621859
$$687$$ 0 0
$$688$$ 2.51122 0.0957393
$$689$$ −0.714253 −0.0272109
$$690$$ 0 0
$$691$$ −39.9846 −1.52109 −0.760543 0.649288i $$-0.775067\pi$$
−0.760543 + 0.649288i $$0.775067\pi$$
$$692$$ 9.78662 0.372031
$$693$$ 0 0
$$694$$ 28.9468 1.09880
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −75.4975 −2.85967
$$698$$ 27.9243 1.05695
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −4.91729 −0.185723 −0.0928617 0.995679i $$-0.529601\pi$$
−0.0928617 + 0.995679i $$0.529601\pi$$
$$702$$ 0 0
$$703$$ 0.137699 0.00519342
$$704$$ 5.13770 0.193634
$$705$$ 0 0
$$706$$ 28.3099 1.06546
$$707$$ 13.5130 0.508207
$$708$$ 0 0
$$709$$ −36.8865 −1.38530 −0.692650 0.721274i $$-0.743557\pi$$
−0.692650 + 0.721274i $$0.743557\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ −34.1447 −1.27873
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.82446 0.255042
$$717$$ 0 0
$$718$$ −2.60163 −0.0970921
$$719$$ 23.8891 0.890914 0.445457 0.895303i $$-0.353041\pi$$
0.445457 + 0.895303i $$0.353041\pi$$
$$720$$ 0 0
$$721$$ −45.1566 −1.68172
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ −0.137699 −0.00511755
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −19.6894 −0.730240 −0.365120 0.930961i $$-0.618972\pi$$
−0.365120 + 0.930961i $$0.618972\pi$$
$$728$$ 13.3856 0.496104
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −16.2754 −0.601967
$$732$$ 0 0
$$733$$ −1.76418 −0.0651615 −0.0325808 0.999469i $$-0.510373\pi$$
−0.0325808 + 0.999469i $$0.510373\pi$$
$$734$$ −11.6489 −0.429969
$$735$$ 0 0
$$736$$ 7.56885 0.278991
$$737$$ 21.1170 0.777855
$$738$$ 0 0
$$739$$ 28.1755 1.03645 0.518227 0.855243i $$-0.326592\pi$$
0.518227 + 0.855243i $$0.326592\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −0.952717 −0.0349753
$$743$$ −5.42851 −0.199153 −0.0995763 0.995030i $$-0.531749\pi$$
−0.0995763 + 0.995030i $$0.531749\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 12.8064 0.468876
$$747$$ 0 0
$$748$$ −33.2978 −1.21749
$$749$$ −45.7059 −1.67006
$$750$$ 0 0
$$751$$ 25.8640 0.943792 0.471896 0.881654i $$-0.343570\pi$$
0.471896 + 0.881654i $$0.343570\pi$$
$$752$$ −5.96216 −0.217418
$$753$$ 0 0
$$754$$ 2.63615 0.0960031
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3.76947 0.137004 0.0685019 0.997651i $$-0.478178\pi$$
0.0685019 + 0.997651i $$0.478178\pi$$
$$758$$ −20.9122 −0.759566
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.3605 −0.665568 −0.332784 0.943003i $$-0.607988\pi$$
−0.332784 + 0.943003i $$0.607988\pi$$
$$762$$ 0 0
$$763$$ −39.0767 −1.41467
$$764$$ −19.3779 −0.701068
$$765$$ 0 0
$$766$$ 10.3511 0.374000
$$767$$ 17.0851 0.616908
$$768$$ 0 0
$$769$$ −38.1300 −1.37500 −0.687501 0.726183i $$-0.741292\pi$$
−0.687501 + 0.726183i $$0.741292\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 5.42851 0.195376
$$773$$ −29.1575 −1.04872 −0.524361 0.851496i $$-0.675696\pi$$
−0.524361 + 0.851496i $$0.675696\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −3.93972 −0.141428
$$777$$ 0 0
$$778$$ −6.56620 −0.235410
$$779$$ −11.6489 −0.417366
$$780$$ 0 0
$$781$$ −19.6489 −0.703094
$$782$$ −49.0543 −1.75418
$$783$$ 0 0
$$784$$ 10.8546 0.387664
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 47.0165 1.67596 0.837978 0.545704i $$-0.183738\pi$$
0.837978 + 0.545704i $$0.183738\pi$$
$$788$$ −15.6489 −0.557470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −74.5750 −2.65158
$$792$$ 0 0
$$793$$ −45.7782 −1.62563
$$794$$ −23.5284 −0.834990
$$795$$ 0 0
$$796$$ 18.0499 0.639763
$$797$$ −42.6359 −1.51024 −0.755121 0.655586i $$-0.772422\pi$$
−0.755121 + 0.655586i $$0.772422\pi$$
$$798$$ 0 0
$$799$$ 38.6412 1.36703
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −6.22041 −0.219650
$$803$$ −24.1808 −0.853323
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 14.2908 0.503372
$$807$$ 0 0
$$808$$ −3.19798 −0.112504
$$809$$ −49.4630 −1.73903 −0.869514 0.493909i $$-0.835568\pi$$
−0.869514 + 0.493909i $$0.835568\pi$$
$$810$$ 0 0
$$811$$ −16.7816 −0.589280 −0.294640 0.955608i $$-0.595200\pi$$
−0.294640 + 0.955608i $$0.595200\pi$$
$$812$$ 3.51628 0.123397
$$813$$ 0 0
$$814$$ −0.707457 −0.0247964
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −2.51122 −0.0878564
$$818$$ −34.4905 −1.20593
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −11.5337 −0.402527 −0.201264 0.979537i $$-0.564505\pi$$
−0.201264 + 0.979537i $$0.564505\pi$$
$$822$$ 0 0
$$823$$ −32.6309 −1.13744 −0.568720 0.822531i $$-0.692561\pi$$
−0.568720 + 0.822531i $$0.692561\pi$$
$$824$$ 10.6868 0.372291
$$825$$ 0 0
$$826$$ 22.7893 0.792940
$$827$$ 6.64650 0.231122 0.115561 0.993300i $$-0.463133\pi$$
0.115561 + 0.993300i $$0.463133\pi$$
$$828$$ 0 0
$$829$$ −30.9217 −1.07395 −0.536977 0.843597i $$-0.680434\pi$$
−0.536977 + 0.843597i $$0.680434\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −3.16784 −0.109825
$$833$$ −70.3495 −2.43747
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −5.13770 −0.177691
$$837$$ 0 0
$$838$$ 10.6265 0.367086
$$839$$ 48.7109 1.68169 0.840844 0.541277i $$-0.182059\pi$$
0.840844 + 0.541277i $$0.182059\pi$$
$$840$$ 0 0
$$841$$ −28.3075 −0.976121
$$842$$ −1.98021 −0.0682426
$$843$$ 0 0
$$844$$ −7.50857 −0.258456
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −65.0551 −2.23532
$$848$$ 0.225470 0.00774267
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.04222 −0.0357270
$$852$$ 0 0
$$853$$ −46.1447 −1.57997 −0.789983 0.613129i $$-0.789911\pi$$
−0.789983 + 0.613129i $$0.789911\pi$$
$$854$$ −61.0620 −2.08950
$$855$$ 0 0
$$856$$ 10.8168 0.369709
$$857$$ −8.40607 −0.287146 −0.143573 0.989640i $$-0.545859\pi$$
−0.143573 + 0.989640i $$0.545859\pi$$
$$858$$ 0 0
$$859$$ −8.49581 −0.289873 −0.144937 0.989441i $$-0.546298\pi$$
−0.144937 + 0.989441i $$0.546298\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −15.0774 −0.513539
$$863$$ 3.37352 0.114836 0.0574179 0.998350i $$-0.481713\pi$$
0.0574179 + 0.998350i $$0.481713\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −13.5337 −0.459892
$$867$$ 0 0
$$868$$ 19.0620 0.647007
$$869$$ 54.5957 1.85203
$$870$$ 0 0
$$871$$ −13.0205 −0.441182
$$872$$ 9.24791 0.313174
$$873$$ 0 0
$$874$$ −7.56885 −0.256020
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 13.2780 0.448368 0.224184 0.974547i $$-0.428028\pi$$
0.224184 + 0.974547i $$0.428028\pi$$
$$878$$ 39.1773 1.32217
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 26.6489 0.897825 0.448912 0.893576i $$-0.351812\pi$$
0.448912 + 0.893576i $$0.351812\pi$$
$$882$$ 0 0
$$883$$ 47.7884 1.60821 0.804103 0.594490i $$-0.202646\pi$$
0.804103 + 0.594490i $$0.202646\pi$$
$$884$$ 20.5310 0.690532
$$885$$ 0 0
$$886$$ 19.3132 0.648841
$$887$$ −36.3304 −1.21985 −0.609927 0.792457i $$-0.708801\pi$$
−0.609927 + 0.792457i $$0.708801\pi$$
$$888$$ 0 0
$$889$$ −54.0295 −1.81209
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −27.8091 −0.931116
$$893$$ 5.96216 0.199516
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −4.22547 −0.141163
$$897$$ 0 0
$$898$$ −41.4131 −1.38197
$$899$$ 3.75406 0.125205
$$900$$ 0 0
$$901$$ −1.46129 −0.0486826
$$902$$ 59.8486 1.99274
$$903$$ 0 0
$$904$$ 17.6489 0.586994
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 39.3873 1.30784 0.653918 0.756566i $$-0.273124\pi$$
0.653918 + 0.756566i $$0.273124\pi$$
$$908$$ 8.91223 0.295763
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 20.5561 0.681054 0.340527 0.940235i $$-0.389395\pi$$
0.340527 + 0.940235i $$0.389395\pi$$
$$912$$ 0 0
$$913$$ −62.1654 −2.05738
$$914$$ 37.8392 1.25161
$$915$$ 0 0
$$916$$ −13.6489 −0.450973
$$917$$ −8.93799 −0.295158
$$918$$ 0 0
$$919$$ 12.4054 0.409216 0.204608 0.978844i $$-0.434408\pi$$
0.204608 + 0.978844i $$0.434408\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −1.58864 −0.0523190
$$923$$ 12.1153 0.398779
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −40.3581 −1.32625
$$927$$ 0 0
$$928$$ −0.832162 −0.0273171
$$929$$ 31.3330 1.02800 0.514002 0.857789i $$-0.328162\pi$$
0.514002 + 0.857789i $$0.328162\pi$$
$$930$$ 0 0
$$931$$ −10.8546 −0.355745
$$932$$ 23.2754 0.762411
$$933$$ 0 0
$$934$$ −39.5130 −1.29290
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −7.27299 −0.237598 −0.118799 0.992918i $$-0.537904\pi$$
−0.118799 + 0.992918i $$0.537904\pi$$
$$938$$ −17.3676 −0.567071
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −6.72128 −0.219107 −0.109554 0.993981i $$-0.534942\pi$$
−0.109554 + 0.993981i $$0.534942\pi$$
$$942$$ 0 0
$$943$$ 88.1689 2.87117
$$944$$ −5.39331 −0.175537
$$945$$ 0 0
$$946$$ 12.9019 0.419476
$$947$$ 10.0757 0.327416 0.163708 0.986509i $$-0.447655\pi$$
0.163708 + 0.986509i $$0.447655\pi$$
$$948$$ 0 0
$$949$$ 14.9096 0.483986
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 27.3856 0.887573
$$953$$ −8.14473 −0.263834 −0.131917 0.991261i $$-0.542113\pi$$
−0.131917 + 0.991261i $$0.542113\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −3.72898 −0.120604
$$957$$ 0 0
$$958$$ −7.61107 −0.245903
$$959$$ 22.6619 0.731791
$$960$$ 0 0
$$961$$ −10.6489 −0.343513
$$962$$ 0.436209 0.0140639
$$963$$ 0 0
$$964$$ −1.48878 −0.0479505
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −47.1773 −1.51712 −0.758560 0.651604i $$-0.774097\pi$$
−0.758560 + 0.651604i $$0.774097\pi$$
$$968$$ 15.3960 0.494845
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −0.230528 −0.00739801 −0.00369901 0.999993i $$-0.501177\pi$$
−0.00369901 + 0.999993i $$0.501177\pi$$
$$972$$ 0 0
$$973$$ 42.9313 1.37632
$$974$$ 20.3907 0.653359
$$975$$ 0 0
$$976$$ 14.4509 0.462563
$$977$$ −5.80905 −0.185848 −0.0929240 0.995673i $$-0.529621\pi$$
−0.0929240 + 0.995673i $$0.529621\pi$$
$$978$$ 0 0
$$979$$ 51.3770 1.64202
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −25.0224 −0.798498
$$983$$ −22.3511 −0.712889 −0.356444 0.934317i $$-0.616011\pi$$
−0.356444 + 0.934317i $$0.616011\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 5.39331 0.171758
$$987$$ 0 0
$$988$$ 3.16784 0.100782
$$989$$ 19.0070 0.604388
$$990$$ 0 0
$$991$$ −34.9415 −1.10995 −0.554976 0.831866i $$-0.687273\pi$$
−0.554976 + 0.831866i $$0.687273\pi$$
$$992$$ −4.51122 −0.143231
$$993$$ 0 0
$$994$$ 16.1601 0.512568
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 9.35282 0.296207 0.148103 0.988972i $$-0.452683\pi$$
0.148103 + 0.988972i $$0.452683\pi$$
$$998$$ 22.6111 0.715741
$$999$$ 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 8550.2.a.co.1.1 3
3.2 odd 2 950.2.a.k.1.2 3
5.4 even 2 8550.2.a.cj.1.3 3
12.11 even 2 7600.2.a.cb.1.2 3
15.2 even 4 950.2.b.g.799.2 6
15.8 even 4 950.2.b.g.799.5 6
15.14 odd 2 950.2.a.m.1.2 yes 3
60.59 even 2 7600.2.a.bm.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.2 3 3.2 odd 2
950.2.a.m.1.2 yes 3 15.14 odd 2
950.2.b.g.799.2 6 15.2 even 4
950.2.b.g.799.5 6 15.8 even 4
7600.2.a.bm.1.2 3 60.59 even 2
7600.2.a.cb.1.2 3 12.11 even 2
8550.2.a.cj.1.3 3 5.4 even 2
8550.2.a.co.1.1 3 1.1 even 1 trivial