# Properties

 Label 475.4.b.c.324.2 Level $475$ Weight $4$ Character 475.324 Analytic conductor $28.026$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,4,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 475.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$28.0259072527$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 324.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 475.324 Dual form 475.4.b.c.324.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{2} -5.00000i q^{3} -1.00000 q^{4} +15.0000 q^{6} -11.0000i q^{7} +21.0000i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{2} -5.00000i q^{3} -1.00000 q^{4} +15.0000 q^{6} -11.0000i q^{7} +21.0000i q^{8} +2.00000 q^{9} -54.0000 q^{11} +5.00000i q^{12} +11.0000i q^{13} +33.0000 q^{14} -71.0000 q^{16} +93.0000i q^{17} +6.00000i q^{18} -19.0000 q^{19} -55.0000 q^{21} -162.000i q^{22} +183.000i q^{23} +105.000 q^{24} -33.0000 q^{26} -145.000i q^{27} +11.0000i q^{28} +249.000 q^{29} +56.0000 q^{31} -45.0000i q^{32} +270.000i q^{33} -279.000 q^{34} -2.00000 q^{36} +250.000i q^{37} -57.0000i q^{38} +55.0000 q^{39} +240.000 q^{41} -165.000i q^{42} -196.000i q^{43} +54.0000 q^{44} -549.000 q^{46} +168.000i q^{47} +355.000i q^{48} +222.000 q^{49} +465.000 q^{51} -11.0000i q^{52} +435.000i q^{53} +435.000 q^{54} +231.000 q^{56} +95.0000i q^{57} +747.000i q^{58} -195.000 q^{59} -358.000 q^{61} +168.000i q^{62} -22.0000i q^{63} -433.000 q^{64} -810.000 q^{66} +961.000i q^{67} -93.0000i q^{68} +915.000 q^{69} -246.000 q^{71} +42.0000i q^{72} +353.000i q^{73} -750.000 q^{74} +19.0000 q^{76} +594.000i q^{77} +165.000i q^{78} +34.0000 q^{79} -671.000 q^{81} +720.000i q^{82} +234.000i q^{83} +55.0000 q^{84} +588.000 q^{86} -1245.00i q^{87} -1134.00i q^{88} +168.000 q^{89} +121.000 q^{91} -183.000i q^{92} -280.000i q^{93} -504.000 q^{94} -225.000 q^{96} -758.000i q^{97} +666.000i q^{98} -108.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 30 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 30 * q^6 + 4 * q^9 $$2 q - 2 q^{4} + 30 q^{6} + 4 q^{9} - 108 q^{11} + 66 q^{14} - 142 q^{16} - 38 q^{19} - 110 q^{21} + 210 q^{24} - 66 q^{26} + 498 q^{29} + 112 q^{31} - 558 q^{34} - 4 q^{36} + 110 q^{39} + 480 q^{41} + 108 q^{44} - 1098 q^{46} + 444 q^{49} + 930 q^{51} + 870 q^{54} + 462 q^{56} - 390 q^{59} - 716 q^{61} - 866 q^{64} - 1620 q^{66} + 1830 q^{69} - 492 q^{71} - 1500 q^{74} + 38 q^{76} + 68 q^{79} - 1342 q^{81} + 110 q^{84} + 1176 q^{86} + 336 q^{89} + 242 q^{91} - 1008 q^{94} - 450 q^{96} - 216 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 30 * q^6 + 4 * q^9 - 108 * q^11 + 66 * q^14 - 142 * q^16 - 38 * q^19 - 110 * q^21 + 210 * q^24 - 66 * q^26 + 498 * q^29 + 112 * q^31 - 558 * q^34 - 4 * q^36 + 110 * q^39 + 480 * q^41 + 108 * q^44 - 1098 * q^46 + 444 * q^49 + 930 * q^51 + 870 * q^54 + 462 * q^56 - 390 * q^59 - 716 * q^61 - 866 * q^64 - 1620 * q^66 + 1830 * q^69 - 492 * q^71 - 1500 * q^74 + 38 * q^76 + 68 * q^79 - 1342 * q^81 + 110 * q^84 + 1176 * q^86 + 336 * q^89 + 242 * q^91 - 1008 * q^94 - 450 * q^96 - 216 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.00000i 1.06066i 0.847791 + 0.530330i $$0.177932\pi$$
−0.847791 + 0.530330i $$0.822068\pi$$
$$3$$ − 5.00000i − 0.962250i −0.876652 0.481125i $$-0.840228\pi$$
0.876652 0.481125i $$-0.159772\pi$$
$$4$$ −1.00000 −0.125000
$$5$$ 0 0
$$6$$ 15.0000 1.02062
$$7$$ − 11.0000i − 0.593944i −0.954886 0.296972i $$-0.904023\pi$$
0.954886 0.296972i $$-0.0959768\pi$$
$$8$$ 21.0000i 0.928078i
$$9$$ 2.00000 0.0740741
$$10$$ 0 0
$$11$$ −54.0000 −1.48015 −0.740073 0.672526i $$-0.765209\pi$$
−0.740073 + 0.672526i $$0.765209\pi$$
$$12$$ 5.00000i 0.120281i
$$13$$ 11.0000i 0.234681i 0.993092 + 0.117340i $$0.0374369\pi$$
−0.993092 + 0.117340i $$0.962563\pi$$
$$14$$ 33.0000 0.629973
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 93.0000i 1.32681i 0.748259 + 0.663406i $$0.230890\pi$$
−0.748259 + 0.663406i $$0.769110\pi$$
$$18$$ 6.00000i 0.0785674i
$$19$$ −19.0000 −0.229416
$$20$$ 0 0
$$21$$ −55.0000 −0.571523
$$22$$ − 162.000i − 1.56993i
$$23$$ 183.000i 1.65905i 0.558470 + 0.829525i $$0.311389\pi$$
−0.558470 + 0.829525i $$0.688611\pi$$
$$24$$ 105.000 0.893043
$$25$$ 0 0
$$26$$ −33.0000 −0.248917
$$27$$ − 145.000i − 1.03353i
$$28$$ 11.0000i 0.0742430i
$$29$$ 249.000 1.59442 0.797209 0.603703i $$-0.206309\pi$$
0.797209 + 0.603703i $$0.206309\pi$$
$$30$$ 0 0
$$31$$ 56.0000 0.324448 0.162224 0.986754i $$-0.448133\pi$$
0.162224 + 0.986754i $$0.448133\pi$$
$$32$$ − 45.0000i − 0.248592i
$$33$$ 270.000i 1.42427i
$$34$$ −279.000 −1.40730
$$35$$ 0 0
$$36$$ −2.00000 −0.00925926
$$37$$ 250.000i 1.11080i 0.831582 + 0.555402i $$0.187436\pi$$
−0.831582 + 0.555402i $$0.812564\pi$$
$$38$$ − 57.0000i − 0.243332i
$$39$$ 55.0000 0.225822
$$40$$ 0 0
$$41$$ 240.000 0.914188 0.457094 0.889418i $$-0.348890\pi$$
0.457094 + 0.889418i $$0.348890\pi$$
$$42$$ − 165.000i − 0.606192i
$$43$$ − 196.000i − 0.695110i −0.937660 0.347555i $$-0.887012\pi$$
0.937660 0.347555i $$-0.112988\pi$$
$$44$$ 54.0000 0.185018
$$45$$ 0 0
$$46$$ −549.000 −1.75969
$$47$$ 168.000i 0.521390i 0.965421 + 0.260695i $$0.0839517\pi$$
−0.965421 + 0.260695i $$0.916048\pi$$
$$48$$ 355.000i 1.06750i
$$49$$ 222.000 0.647230
$$50$$ 0 0
$$51$$ 465.000 1.27673
$$52$$ − 11.0000i − 0.0293351i
$$53$$ 435.000i 1.12739i 0.825982 + 0.563697i $$0.190621\pi$$
−0.825982 + 0.563697i $$0.809379\pi$$
$$54$$ 435.000 1.09622
$$55$$ 0 0
$$56$$ 231.000 0.551226
$$57$$ 95.0000i 0.220755i
$$58$$ 747.000i 1.69114i
$$59$$ −195.000 −0.430285 −0.215143 0.976583i $$-0.569022\pi$$
−0.215143 + 0.976583i $$0.569022\pi$$
$$60$$ 0 0
$$61$$ −358.000 −0.751430 −0.375715 0.926735i $$-0.622603\pi$$
−0.375715 + 0.926735i $$0.622603\pi$$
$$62$$ 168.000i 0.344129i
$$63$$ − 22.0000i − 0.0439959i
$$64$$ −433.000 −0.845703
$$65$$ 0 0
$$66$$ −810.000 −1.51067
$$67$$ 961.000i 1.75231i 0.482029 + 0.876155i $$0.339900\pi$$
−0.482029 + 0.876155i $$0.660100\pi$$
$$68$$ − 93.0000i − 0.165852i
$$69$$ 915.000 1.59642
$$70$$ 0 0
$$71$$ −246.000 −0.411195 −0.205597 0.978637i $$-0.565914\pi$$
−0.205597 + 0.978637i $$0.565914\pi$$
$$72$$ 42.0000i 0.0687465i
$$73$$ 353.000i 0.565966i 0.959125 + 0.282983i $$0.0913240\pi$$
−0.959125 + 0.282983i $$0.908676\pi$$
$$74$$ −750.000 −1.17819
$$75$$ 0 0
$$76$$ 19.0000 0.0286770
$$77$$ 594.000i 0.879124i
$$78$$ 165.000i 0.239520i
$$79$$ 34.0000 0.0484215 0.0242108 0.999707i $$-0.492293\pi$$
0.0242108 + 0.999707i $$0.492293\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 720.000i 0.969643i
$$83$$ 234.000i 0.309456i 0.987957 + 0.154728i $$0.0494501\pi$$
−0.987957 + 0.154728i $$0.950550\pi$$
$$84$$ 55.0000 0.0714404
$$85$$ 0 0
$$86$$ 588.000 0.737275
$$87$$ − 1245.00i − 1.53423i
$$88$$ − 1134.00i − 1.37369i
$$89$$ 168.000 0.200089 0.100045 0.994983i $$-0.468101\pi$$
0.100045 + 0.994983i $$0.468101\pi$$
$$90$$ 0 0
$$91$$ 121.000 0.139387
$$92$$ − 183.000i − 0.207381i
$$93$$ − 280.000i − 0.312201i
$$94$$ −504.000 −0.553017
$$95$$ 0 0
$$96$$ −225.000 −0.239208
$$97$$ − 758.000i − 0.793435i −0.917941 0.396718i $$-0.870149\pi$$
0.917941 0.396718i $$-0.129851\pi$$
$$98$$ 666.000i 0.686491i
$$99$$ −108.000 −0.109640
$$100$$ 0 0
$$101$$ −726.000 −0.715245 −0.357622 0.933866i $$-0.616412\pi$$
−0.357622 + 0.933866i $$0.616412\pi$$
$$102$$ 1395.00i 1.35417i
$$103$$ 2.00000i 0.00191326i 1.00000 0.000956630i $$0.000304505\pi$$
−1.00000 0.000956630i $$0.999695\pi$$
$$104$$ −231.000 −0.217802
$$105$$ 0 0
$$106$$ −1305.00 −1.19578
$$107$$ − 1413.00i − 1.27663i −0.769773 0.638317i $$-0.779631\pi$$
0.769773 0.638317i $$-0.220369\pi$$
$$108$$ 145.000i 0.129191i
$$109$$ −389.000 −0.341830 −0.170915 0.985286i $$-0.554672\pi$$
−0.170915 + 0.985286i $$0.554672\pi$$
$$110$$ 0 0
$$111$$ 1250.00 1.06887
$$112$$ 781.000i 0.658907i
$$113$$ 342.000i 0.284714i 0.989815 + 0.142357i $$0.0454681\pi$$
−0.989815 + 0.142357i $$0.954532\pi$$
$$114$$ −285.000 −0.234146
$$115$$ 0 0
$$116$$ −249.000 −0.199302
$$117$$ 22.0000i 0.0173838i
$$118$$ − 585.000i − 0.456387i
$$119$$ 1023.00 0.788053
$$120$$ 0 0
$$121$$ 1585.00 1.19083
$$122$$ − 1074.00i − 0.797011i
$$123$$ − 1200.00i − 0.879678i
$$124$$ −56.0000 −0.0405560
$$125$$ 0 0
$$126$$ 66.0000 0.0466647
$$127$$ 1150.00i 0.803512i 0.915747 + 0.401756i $$0.131600\pi$$
−0.915747 + 0.401756i $$0.868400\pi$$
$$128$$ − 1659.00i − 1.14560i
$$129$$ −980.000 −0.668870
$$130$$ 0 0
$$131$$ −1452.00 −0.968411 −0.484205 0.874954i $$-0.660891\pi$$
−0.484205 + 0.874954i $$0.660891\pi$$
$$132$$ − 270.000i − 0.178034i
$$133$$ 209.000i 0.136260i
$$134$$ −2883.00 −1.85861
$$135$$ 0 0
$$136$$ −1953.00 −1.23139
$$137$$ 1689.00i 1.05329i 0.850085 + 0.526646i $$0.176551\pi$$
−0.850085 + 0.526646i $$0.823449\pi$$
$$138$$ 2745.00i 1.69326i
$$139$$ −2144.00 −1.30829 −0.654143 0.756371i $$-0.726970\pi$$
−0.654143 + 0.756371i $$0.726970\pi$$
$$140$$ 0 0
$$141$$ 840.000 0.501708
$$142$$ − 738.000i − 0.436138i
$$143$$ − 594.000i − 0.347362i
$$144$$ −142.000 −0.0821759
$$145$$ 0 0
$$146$$ −1059.00 −0.600298
$$147$$ − 1110.00i − 0.622798i
$$148$$ − 250.000i − 0.138850i
$$149$$ 3000.00 1.64946 0.824730 0.565527i $$-0.191327\pi$$
0.824730 + 0.565527i $$0.191327\pi$$
$$150$$ 0 0
$$151$$ −1006.00 −0.542166 −0.271083 0.962556i $$-0.587382\pi$$
−0.271083 + 0.962556i $$0.587382\pi$$
$$152$$ − 399.000i − 0.212916i
$$153$$ 186.000i 0.0982824i
$$154$$ −1782.00 −0.932452
$$155$$ 0 0
$$156$$ −55.0000 −0.0282277
$$157$$ − 2846.00i − 1.44672i −0.690469 0.723362i $$-0.742596\pi$$
0.690469 0.723362i $$-0.257404\pi$$
$$158$$ 102.000i 0.0513588i
$$159$$ 2175.00 1.08483
$$160$$ 0 0
$$161$$ 2013.00 0.985383
$$162$$ − 2013.00i − 0.976273i
$$163$$ − 1600.00i − 0.768845i −0.923157 0.384422i $$-0.874401\pi$$
0.923157 0.384422i $$-0.125599\pi$$
$$164$$ −240.000 −0.114273
$$165$$ 0 0
$$166$$ −702.000 −0.328228
$$167$$ 2004.00i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ − 1155.00i − 0.530418i
$$169$$ 2076.00 0.944925
$$170$$ 0 0
$$171$$ −38.0000 −0.0169938
$$172$$ 196.000i 0.0868887i
$$173$$ − 462.000i − 0.203036i −0.994834 0.101518i $$-0.967630\pi$$
0.994834 0.101518i $$-0.0323699\pi$$
$$174$$ 3735.00 1.62730
$$175$$ 0 0
$$176$$ 3834.00 1.64204
$$177$$ 975.000i 0.414042i
$$178$$ 504.000i 0.212227i
$$179$$ −720.000 −0.300644 −0.150322 0.988637i $$-0.548031\pi$$
−0.150322 + 0.988637i $$0.548031\pi$$
$$180$$ 0 0
$$181$$ −2338.00 −0.960122 −0.480061 0.877235i $$-0.659386\pi$$
−0.480061 + 0.877235i $$0.659386\pi$$
$$182$$ 363.000i 0.147843i
$$183$$ 1790.00i 0.723063i
$$184$$ −3843.00 −1.53973
$$185$$ 0 0
$$186$$ 840.000 0.331139
$$187$$ − 5022.00i − 1.96388i
$$188$$ − 168.000i − 0.0651737i
$$189$$ −1595.00 −0.613858
$$190$$ 0 0
$$191$$ 2871.00 1.08763 0.543817 0.839204i $$-0.316978\pi$$
0.543817 + 0.839204i $$0.316978\pi$$
$$192$$ 2165.00i 0.813778i
$$193$$ 1658.00i 0.618370i 0.951002 + 0.309185i $$0.100056\pi$$
−0.951002 + 0.309185i $$0.899944\pi$$
$$194$$ 2274.00 0.841565
$$195$$ 0 0
$$196$$ −222.000 −0.0809038
$$197$$ 4176.00i 1.51029i 0.655556 + 0.755146i $$0.272434\pi$$
−0.655556 + 0.755146i $$0.727566\pi$$
$$198$$ − 324.000i − 0.116291i
$$199$$ 241.000 0.0858494 0.0429247 0.999078i $$-0.486332\pi$$
0.0429247 + 0.999078i $$0.486332\pi$$
$$200$$ 0 0
$$201$$ 4805.00 1.68616
$$202$$ − 2178.00i − 0.758631i
$$203$$ − 2739.00i − 0.946996i
$$204$$ −465.000 −0.159591
$$205$$ 0 0
$$206$$ −6.00000 −0.00202932
$$207$$ 366.000i 0.122893i
$$208$$ − 781.000i − 0.260349i
$$209$$ 1026.00 0.339569
$$210$$ 0 0
$$211$$ −745.000 −0.243071 −0.121535 0.992587i $$-0.538782\pi$$
−0.121535 + 0.992587i $$0.538782\pi$$
$$212$$ − 435.000i − 0.140924i
$$213$$ 1230.00i 0.395672i
$$214$$ 4239.00 1.35408
$$215$$ 0 0
$$216$$ 3045.00 0.959194
$$217$$ − 616.000i − 0.192704i
$$218$$ − 1167.00i − 0.362565i
$$219$$ 1765.00 0.544601
$$220$$ 0 0
$$221$$ −1023.00 −0.311377
$$222$$ 3750.00i 1.13371i
$$223$$ − 1978.00i − 0.593976i −0.954881 0.296988i $$-0.904018\pi$$
0.954881 0.296988i $$-0.0959822\pi$$
$$224$$ −495.000 −0.147650
$$225$$ 0 0
$$226$$ −1026.00 −0.301985
$$227$$ − 5355.00i − 1.56574i −0.622183 0.782872i $$-0.713754\pi$$
0.622183 0.782872i $$-0.286246\pi$$
$$228$$ − 95.0000i − 0.0275944i
$$229$$ 6370.00 1.83817 0.919086 0.394057i $$-0.128929\pi$$
0.919086 + 0.394057i $$0.128929\pi$$
$$230$$ 0 0
$$231$$ 2970.00 0.845938
$$232$$ 5229.00i 1.47974i
$$233$$ − 2838.00i − 0.797955i −0.916961 0.398978i $$-0.869365\pi$$
0.916961 0.398978i $$-0.130635\pi$$
$$234$$ −66.0000 −0.0184383
$$235$$ 0 0
$$236$$ 195.000 0.0537857
$$237$$ − 170.000i − 0.0465936i
$$238$$ 3069.00i 0.835856i
$$239$$ 369.000 0.0998687 0.0499344 0.998753i $$-0.484099\pi$$
0.0499344 + 0.998753i $$0.484099\pi$$
$$240$$ 0 0
$$241$$ 6608.00 1.76622 0.883109 0.469167i $$-0.155446\pi$$
0.883109 + 0.469167i $$0.155446\pi$$
$$242$$ 4755.00i 1.26307i
$$243$$ − 560.000i − 0.147835i
$$244$$ 358.000 0.0939287
$$245$$ 0 0
$$246$$ 3600.00 0.933039
$$247$$ − 209.000i − 0.0538395i
$$248$$ 1176.00i 0.301113i
$$249$$ 1170.00 0.297774
$$250$$ 0 0
$$251$$ 4674.00 1.17538 0.587690 0.809086i $$-0.300038\pi$$
0.587690 + 0.809086i $$0.300038\pi$$
$$252$$ 22.0000i 0.00549948i
$$253$$ − 9882.00i − 2.45564i
$$254$$ −3450.00 −0.852253
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ − 4512.00i − 1.09514i −0.836760 0.547570i $$-0.815553\pi$$
0.836760 0.547570i $$-0.184447\pi$$
$$258$$ − 2940.00i − 0.709443i
$$259$$ 2750.00 0.659756
$$260$$ 0 0
$$261$$ 498.000 0.118105
$$262$$ − 4356.00i − 1.02715i
$$263$$ 3768.00i 0.883440i 0.897153 + 0.441720i $$0.145632\pi$$
−0.897153 + 0.441720i $$0.854368\pi$$
$$264$$ −5670.00 −1.32183
$$265$$ 0 0
$$266$$ −627.000 −0.144526
$$267$$ − 840.000i − 0.192536i
$$268$$ − 961.000i − 0.219039i
$$269$$ −4758.00 −1.07844 −0.539220 0.842165i $$-0.681281\pi$$
−0.539220 + 0.842165i $$0.681281\pi$$
$$270$$ 0 0
$$271$$ −2041.00 −0.457498 −0.228749 0.973485i $$-0.573463\pi$$
−0.228749 + 0.973485i $$0.573463\pi$$
$$272$$ − 6603.00i − 1.47193i
$$273$$ − 605.000i − 0.134126i
$$274$$ −5067.00 −1.11718
$$275$$ 0 0
$$276$$ −915.000 −0.199553
$$277$$ − 1964.00i − 0.426012i −0.977051 0.213006i $$-0.931675\pi$$
0.977051 0.213006i $$-0.0683254\pi$$
$$278$$ − 6432.00i − 1.38765i
$$279$$ 112.000 0.0240332
$$280$$ 0 0
$$281$$ −5496.00 −1.16678 −0.583388 0.812194i $$-0.698273\pi$$
−0.583388 + 0.812194i $$0.698273\pi$$
$$282$$ 2520.00i 0.532141i
$$283$$ 3098.00i 0.650731i 0.945588 + 0.325366i $$0.105487\pi$$
−0.945588 + 0.325366i $$0.894513\pi$$
$$284$$ 246.000 0.0513993
$$285$$ 0 0
$$286$$ 1782.00 0.368433
$$287$$ − 2640.00i − 0.542977i
$$288$$ − 90.0000i − 0.0184142i
$$289$$ −3736.00 −0.760432
$$290$$ 0 0
$$291$$ −3790.00 −0.763484
$$292$$ − 353.000i − 0.0707458i
$$293$$ 117.000i 0.0233284i 0.999932 + 0.0116642i $$0.00371291\pi$$
−0.999932 + 0.0116642i $$0.996287\pi$$
$$294$$ 3330.00 0.660577
$$295$$ 0 0
$$296$$ −5250.00 −1.03091
$$297$$ 7830.00i 1.52977i
$$298$$ 9000.00i 1.74952i
$$299$$ −2013.00 −0.389347
$$300$$ 0 0
$$301$$ −2156.00 −0.412856
$$302$$ − 3018.00i − 0.575054i
$$303$$ 3630.00i 0.688244i
$$304$$ 1349.00 0.254508
$$305$$ 0 0
$$306$$ −558.000 −0.104244
$$307$$ 1420.00i 0.263986i 0.991251 + 0.131993i $$0.0421376\pi$$
−0.991251 + 0.131993i $$0.957862\pi$$
$$308$$ − 594.000i − 0.109891i
$$309$$ 10.0000 0.00184104
$$310$$ 0 0
$$311$$ −6561.00 −1.19627 −0.598135 0.801395i $$-0.704091\pi$$
−0.598135 + 0.801395i $$0.704091\pi$$
$$312$$ 1155.00i 0.209580i
$$313$$ − 1483.00i − 0.267809i −0.990994 0.133904i $$-0.957249\pi$$
0.990994 0.133904i $$-0.0427515\pi$$
$$314$$ 8538.00 1.53448
$$315$$ 0 0
$$316$$ −34.0000 −0.00605269
$$317$$ 1239.00i 0.219524i 0.993958 + 0.109762i $$0.0350089\pi$$
−0.993958 + 0.109762i $$0.964991\pi$$
$$318$$ 6525.00i 1.15064i
$$319$$ −13446.0 −2.35997
$$320$$ 0 0
$$321$$ −7065.00 −1.22844
$$322$$ 6039.00i 1.04516i
$$323$$ − 1767.00i − 0.304392i
$$324$$ 671.000 0.115055
$$325$$ 0 0
$$326$$ 4800.00 0.815483
$$327$$ 1945.00i 0.328926i
$$328$$ 5040.00i 0.848437i
$$329$$ 1848.00 0.309676
$$330$$ 0 0
$$331$$ −8899.00 −1.47774 −0.738872 0.673846i $$-0.764641\pi$$
−0.738872 + 0.673846i $$0.764641\pi$$
$$332$$ − 234.000i − 0.0386820i
$$333$$ 500.000i 0.0822818i
$$334$$ −6012.00 −0.984916
$$335$$ 0 0
$$336$$ 3905.00 0.634033
$$337$$ − 5816.00i − 0.940112i −0.882637 0.470056i $$-0.844234\pi$$
0.882637 0.470056i $$-0.155766\pi$$
$$338$$ 6228.00i 1.00224i
$$339$$ 1710.00 0.273966
$$340$$ 0 0
$$341$$ −3024.00 −0.480231
$$342$$ − 114.000i − 0.0180246i
$$343$$ − 6215.00i − 0.978363i
$$344$$ 4116.00 0.645116
$$345$$ 0 0
$$346$$ 1386.00 0.215352
$$347$$ 1578.00i 0.244125i 0.992522 + 0.122063i $$0.0389509\pi$$
−0.992522 + 0.122063i $$0.961049\pi$$
$$348$$ 1245.00i 0.191779i
$$349$$ −1658.00 −0.254300 −0.127150 0.991883i $$-0.540583\pi$$
−0.127150 + 0.991883i $$0.540583\pi$$
$$350$$ 0 0
$$351$$ 1595.00 0.242549
$$352$$ 2430.00i 0.367953i
$$353$$ − 11367.0i − 1.71389i −0.515405 0.856947i $$-0.672359\pi$$
0.515405 0.856947i $$-0.327641\pi$$
$$354$$ −2925.00 −0.439158
$$355$$ 0 0
$$356$$ −168.000 −0.0250112
$$357$$ − 5115.00i − 0.758304i
$$358$$ − 2160.00i − 0.318881i
$$359$$ −2553.00 −0.375326 −0.187663 0.982233i $$-0.560091\pi$$
−0.187663 + 0.982233i $$0.560091\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ − 7014.00i − 1.01836i
$$363$$ − 7925.00i − 1.14588i
$$364$$ −121.000 −0.0174234
$$365$$ 0 0
$$366$$ −5370.00 −0.766925
$$367$$ 196.000i 0.0278777i 0.999903 + 0.0139389i $$0.00443702\pi$$
−0.999903 + 0.0139389i $$0.995563\pi$$
$$368$$ − 12993.0i − 1.84051i
$$369$$ 480.000 0.0677176
$$370$$ 0 0
$$371$$ 4785.00 0.669609
$$372$$ 280.000i 0.0390251i
$$373$$ 9353.00i 1.29834i 0.760644 + 0.649169i $$0.224883\pi$$
−0.760644 + 0.649169i $$0.775117\pi$$
$$374$$ 15066.0 2.08301
$$375$$ 0 0
$$376$$ −3528.00 −0.483890
$$377$$ 2739.00i 0.374180i
$$378$$ − 4785.00i − 0.651095i
$$379$$ −3827.00 −0.518680 −0.259340 0.965786i $$-0.583505\pi$$
−0.259340 + 0.965786i $$0.583505\pi$$
$$380$$ 0 0
$$381$$ 5750.00 0.773180
$$382$$ 8613.00i 1.15361i
$$383$$ 5694.00i 0.759660i 0.925056 + 0.379830i $$0.124018\pi$$
−0.925056 + 0.379830i $$0.875982\pi$$
$$384$$ −8295.00 −1.10235
$$385$$ 0 0
$$386$$ −4974.00 −0.655881
$$387$$ − 392.000i − 0.0514896i
$$388$$ 758.000i 0.0991794i
$$389$$ −1290.00 −0.168138 −0.0840689 0.996460i $$-0.526792\pi$$
−0.0840689 + 0.996460i $$0.526792\pi$$
$$390$$ 0 0
$$391$$ −17019.0 −2.20125
$$392$$ 4662.00i 0.600680i
$$393$$ 7260.00i 0.931854i
$$394$$ −12528.0 −1.60191
$$395$$ 0 0
$$396$$ 108.000 0.0137051
$$397$$ − 6536.00i − 0.826278i −0.910668 0.413139i $$-0.864432\pi$$
0.910668 0.413139i $$-0.135568\pi$$
$$398$$ 723.000i 0.0910571i
$$399$$ 1045.00 0.131116
$$400$$ 0 0
$$401$$ 2328.00 0.289912 0.144956 0.989438i $$-0.453696\pi$$
0.144956 + 0.989438i $$0.453696\pi$$
$$402$$ 14415.0i 1.78844i
$$403$$ 616.000i 0.0761418i
$$404$$ 726.000 0.0894056
$$405$$ 0 0
$$406$$ 8217.00 1.00444
$$407$$ − 13500.0i − 1.64415i
$$408$$ 9765.00i 1.18490i
$$409$$ 6676.00 0.807107 0.403554 0.914956i $$-0.367775\pi$$
0.403554 + 0.914956i $$0.367775\pi$$
$$410$$ 0 0
$$411$$ 8445.00 1.01353
$$412$$ − 2.00000i 0 0.000239158i
$$413$$ 2145.00i 0.255565i
$$414$$ −1098.00 −0.130347
$$415$$ 0 0
$$416$$ 495.000 0.0583398
$$417$$ 10720.0i 1.25890i
$$418$$ 3078.00i 0.360167i
$$419$$ 8136.00 0.948615 0.474307 0.880359i $$-0.342699\pi$$
0.474307 + 0.880359i $$0.342699\pi$$
$$420$$ 0 0
$$421$$ −8665.00 −1.00310 −0.501551 0.865128i $$-0.667237\pi$$
−0.501551 + 0.865128i $$0.667237\pi$$
$$422$$ − 2235.00i − 0.257815i
$$423$$ 336.000i 0.0386215i
$$424$$ −9135.00 −1.04631
$$425$$ 0 0
$$426$$ −3690.00 −0.419674
$$427$$ 3938.00i 0.446307i
$$428$$ 1413.00i 0.159579i
$$429$$ −2970.00 −0.334249
$$430$$ 0 0
$$431$$ 750.000 0.0838196 0.0419098 0.999121i $$-0.486656\pi$$
0.0419098 + 0.999121i $$0.486656\pi$$
$$432$$ 10295.0i 1.14657i
$$433$$ − 4858.00i − 0.539170i −0.962977 0.269585i $$-0.913113\pi$$
0.962977 0.269585i $$-0.0868865\pi$$
$$434$$ 1848.00 0.204394
$$435$$ 0 0
$$436$$ 389.000 0.0427287
$$437$$ − 3477.00i − 0.380612i
$$438$$ 5295.00i 0.577637i
$$439$$ −6500.00 −0.706670 −0.353335 0.935497i $$-0.614952\pi$$
−0.353335 + 0.935497i $$0.614952\pi$$
$$440$$ 0 0
$$441$$ 444.000 0.0479430
$$442$$ − 3069.00i − 0.330266i
$$443$$ 3486.00i 0.373871i 0.982372 + 0.186936i $$0.0598555\pi$$
−0.982372 + 0.186936i $$0.940144\pi$$
$$444$$ −1250.00 −0.133609
$$445$$ 0 0
$$446$$ 5934.00 0.630007
$$447$$ − 15000.0i − 1.58719i
$$448$$ 4763.00i 0.502300i
$$449$$ 15030.0 1.57975 0.789877 0.613265i $$-0.210144\pi$$
0.789877 + 0.613265i $$0.210144\pi$$
$$450$$ 0 0
$$451$$ −12960.0 −1.35313
$$452$$ − 342.000i − 0.0355892i
$$453$$ 5030.00i 0.521700i
$$454$$ 16065.0 1.66072
$$455$$ 0 0
$$456$$ −1995.00 −0.204878
$$457$$ 2959.00i 0.302880i 0.988466 + 0.151440i $$0.0483910\pi$$
−0.988466 + 0.151440i $$0.951609\pi$$
$$458$$ 19110.0i 1.94968i
$$459$$ 13485.0 1.37130
$$460$$ 0 0
$$461$$ −156.000 −0.0157606 −0.00788031 0.999969i $$-0.502508\pi$$
−0.00788031 + 0.999969i $$0.502508\pi$$
$$462$$ 8910.00i 0.897253i
$$463$$ 4484.00i 0.450085i 0.974349 + 0.225042i $$0.0722520\pi$$
−0.974349 + 0.225042i $$0.927748\pi$$
$$464$$ −17679.0 −1.76881
$$465$$ 0 0
$$466$$ 8514.00 0.846359
$$467$$ − 8766.00i − 0.868613i −0.900765 0.434306i $$-0.856994\pi$$
0.900765 0.434306i $$-0.143006\pi$$
$$468$$ − 22.0000i − 0.00217297i
$$469$$ 10571.0 1.04077
$$470$$ 0 0
$$471$$ −14230.0 −1.39211
$$472$$ − 4095.00i − 0.399338i
$$473$$ 10584.0i 1.02886i
$$474$$ 510.000 0.0494200
$$475$$ 0 0
$$476$$ −1023.00 −0.0985066
$$477$$ 870.000i 0.0835106i
$$478$$ 1107.00i 0.105927i
$$479$$ 18996.0 1.81200 0.906001 0.423275i $$-0.139119\pi$$
0.906001 + 0.423275i $$0.139119\pi$$
$$480$$ 0 0
$$481$$ −2750.00 −0.260684
$$482$$ 19824.0i 1.87336i
$$483$$ − 10065.0i − 0.948185i
$$484$$ −1585.00 −0.148854
$$485$$ 0 0
$$486$$ 1680.00 0.156803
$$487$$ 7450.00i 0.693207i 0.938012 + 0.346603i $$0.112665\pi$$
−0.938012 + 0.346603i $$0.887335\pi$$
$$488$$ − 7518.00i − 0.697385i
$$489$$ −8000.00 −0.739821
$$490$$ 0 0
$$491$$ 6180.00 0.568023 0.284012 0.958821i $$-0.408335\pi$$
0.284012 + 0.958821i $$0.408335\pi$$
$$492$$ 1200.00i 0.109960i
$$493$$ 23157.0i 2.11549i
$$494$$ 627.000 0.0571054
$$495$$ 0 0
$$496$$ −3976.00 −0.359935
$$497$$ 2706.00i 0.244227i
$$498$$ 3510.00i 0.315837i
$$499$$ −2576.00 −0.231097 −0.115549 0.993302i $$-0.536863\pi$$
−0.115549 + 0.993302i $$0.536863\pi$$
$$500$$ 0 0
$$501$$ 10020.0 0.893534
$$502$$ 14022.0i 1.24668i
$$503$$ − 10545.0i − 0.934748i −0.884060 0.467374i $$-0.845200\pi$$
0.884060 0.467374i $$-0.154800\pi$$
$$504$$ 462.000 0.0408316
$$505$$ 0 0
$$506$$ 29646.0 2.60460
$$507$$ − 10380.0i − 0.909254i
$$508$$ − 1150.00i − 0.100439i
$$509$$ 14694.0 1.27957 0.639784 0.768555i $$-0.279024\pi$$
0.639784 + 0.768555i $$0.279024\pi$$
$$510$$ 0 0
$$511$$ 3883.00 0.336152
$$512$$ − 8733.00i − 0.753804i
$$513$$ 2755.00i 0.237108i
$$514$$ 13536.0 1.16157
$$515$$ 0 0
$$516$$ 980.000 0.0836087
$$517$$ − 9072.00i − 0.771733i
$$518$$ 8250.00i 0.699776i
$$519$$ −2310.00 −0.195371
$$520$$ 0 0
$$521$$ 10332.0 0.868816 0.434408 0.900716i $$-0.356958\pi$$
0.434408 + 0.900716i $$0.356958\pi$$
$$522$$ 1494.00i 0.125269i
$$523$$ 10937.0i 0.914420i 0.889359 + 0.457210i $$0.151151\pi$$
−0.889359 + 0.457210i $$0.848849\pi$$
$$524$$ 1452.00 0.121051
$$525$$ 0 0
$$526$$ −11304.0 −0.937030
$$527$$ 5208.00i 0.430482i
$$528$$ − 19170.0i − 1.58005i
$$529$$ −21322.0 −1.75245
$$530$$ 0 0
$$531$$ −390.000 −0.0318730
$$532$$ − 209.000i − 0.0170325i
$$533$$ 2640.00i 0.214542i
$$534$$ 2520.00 0.204215
$$535$$ 0 0
$$536$$ −20181.0 −1.62628
$$537$$ 3600.00i 0.289295i
$$538$$ − 14274.0i − 1.14386i
$$539$$ −11988.0 −0.957996
$$540$$ 0 0
$$541$$ 18578.0 1.47640 0.738198 0.674584i $$-0.235677\pi$$
0.738198 + 0.674584i $$0.235677\pi$$
$$542$$ − 6123.00i − 0.485250i
$$543$$ 11690.0i 0.923878i
$$544$$ 4185.00 0.329835
$$545$$ 0 0
$$546$$ 1815.00 0.142262
$$547$$ − 21404.0i − 1.67307i −0.547914 0.836535i $$-0.684578\pi$$
0.547914 0.836535i $$-0.315422\pi$$
$$548$$ − 1689.00i − 0.131662i
$$549$$ −716.000 −0.0556614
$$550$$ 0 0
$$551$$ −4731.00 −0.365785
$$552$$ 19215.0i 1.48160i
$$553$$ − 374.000i − 0.0287597i
$$554$$ 5892.00 0.451854
$$555$$ 0 0
$$556$$ 2144.00 0.163536
$$557$$ 3948.00i 0.300327i 0.988661 + 0.150163i $$0.0479800\pi$$
−0.988661 + 0.150163i $$0.952020\pi$$
$$558$$ 336.000i 0.0254911i
$$559$$ 2156.00 0.163129
$$560$$ 0 0
$$561$$ −25110.0 −1.88974
$$562$$ − 16488.0i − 1.23755i
$$563$$ 5724.00i 0.428486i 0.976780 + 0.214243i $$0.0687285\pi$$
−0.976780 + 0.214243i $$0.931271\pi$$
$$564$$ −840.000 −0.0627134
$$565$$ 0 0
$$566$$ −9294.00 −0.690205
$$567$$ 7381.00i 0.546689i
$$568$$ − 5166.00i − 0.381621i
$$569$$ 20592.0 1.51716 0.758578 0.651582i $$-0.225895\pi$$
0.758578 + 0.651582i $$0.225895\pi$$
$$570$$ 0 0
$$571$$ 20684.0 1.51593 0.757967 0.652293i $$-0.226193\pi$$
0.757967 + 0.652293i $$0.226193\pi$$
$$572$$ 594.000i 0.0434203i
$$573$$ − 14355.0i − 1.04658i
$$574$$ 7920.00 0.575914
$$575$$ 0 0
$$576$$ −866.000 −0.0626447
$$577$$ 19573.0i 1.41219i 0.708116 + 0.706096i $$0.249545\pi$$
−0.708116 + 0.706096i $$0.750455\pi$$
$$578$$ − 11208.0i − 0.806559i
$$579$$ 8290.00 0.595027
$$580$$ 0 0
$$581$$ 2574.00 0.183800
$$582$$ − 11370.0i − 0.809797i
$$583$$ − 23490.0i − 1.66871i
$$584$$ −7413.00 −0.525260
$$585$$ 0 0
$$586$$ −351.000 −0.0247435
$$587$$ − 13524.0i − 0.950929i −0.879735 0.475464i $$-0.842280\pi$$
0.879735 0.475464i $$-0.157720\pi$$
$$588$$ 1110.00i 0.0778497i
$$589$$ −1064.00 −0.0744335
$$590$$ 0 0
$$591$$ 20880.0 1.45328
$$592$$ − 17750.0i − 1.23230i
$$593$$ 8994.00i 0.622832i 0.950274 + 0.311416i $$0.100803\pi$$
−0.950274 + 0.311416i $$0.899197\pi$$
$$594$$ −23490.0 −1.62257
$$595$$ 0 0
$$596$$ −3000.00 −0.206183
$$597$$ − 1205.00i − 0.0826087i
$$598$$ − 6039.00i − 0.412965i
$$599$$ −10128.0 −0.690850 −0.345425 0.938446i $$-0.612265\pi$$
−0.345425 + 0.938446i $$0.612265\pi$$
$$600$$ 0 0
$$601$$ −22696.0 −1.54041 −0.770207 0.637794i $$-0.779847\pi$$
−0.770207 + 0.637794i $$0.779847\pi$$
$$602$$ − 6468.00i − 0.437900i
$$603$$ 1922.00i 0.129801i
$$604$$ 1006.00 0.0677708
$$605$$ 0 0
$$606$$ −10890.0 −0.729993
$$607$$ 5182.00i 0.346509i 0.984877 + 0.173254i $$0.0554283\pi$$
−0.984877 + 0.173254i $$0.944572\pi$$
$$608$$ 855.000i 0.0570310i
$$609$$ −13695.0 −0.911247
$$610$$ 0 0
$$611$$ −1848.00 −0.122360
$$612$$ − 186.000i − 0.0122853i
$$613$$ 10082.0i 0.664287i 0.943229 + 0.332144i $$0.107772\pi$$
−0.943229 + 0.332144i $$0.892228\pi$$
$$614$$ −4260.00 −0.279999
$$615$$ 0 0
$$616$$ −12474.0 −0.815896
$$617$$ 12174.0i 0.794338i 0.917745 + 0.397169i $$0.130007\pi$$
−0.917745 + 0.397169i $$0.869993\pi$$
$$618$$ 30.0000i 0.00195271i
$$619$$ −7490.00 −0.486347 −0.243173 0.969983i $$-0.578188\pi$$
−0.243173 + 0.969983i $$0.578188\pi$$
$$620$$ 0 0
$$621$$ 26535.0 1.71467
$$622$$ − 19683.0i − 1.26884i
$$623$$ − 1848.00i − 0.118842i
$$624$$ −3905.00 −0.250521
$$625$$ 0 0
$$626$$ 4449.00 0.284054
$$627$$ − 5130.00i − 0.326750i
$$628$$ 2846.00i 0.180840i
$$629$$ −23250.0 −1.47383
$$630$$ 0 0
$$631$$ 11072.0 0.698525 0.349263 0.937025i $$-0.386432\pi$$
0.349263 + 0.937025i $$0.386432\pi$$
$$632$$ 714.000i 0.0449389i
$$633$$ 3725.00i 0.233895i
$$634$$ −3717.00 −0.232841
$$635$$ 0 0
$$636$$ −2175.00 −0.135604
$$637$$ 2442.00i 0.151893i
$$638$$ − 40338.0i − 2.50313i
$$639$$ −492.000 −0.0304589
$$640$$ 0 0
$$641$$ −18894.0 −1.16422 −0.582112 0.813108i $$-0.697774\pi$$
−0.582112 + 0.813108i $$0.697774\pi$$
$$642$$ − 21195.0i − 1.30296i
$$643$$ − 19834.0i − 1.21645i −0.793765 0.608224i $$-0.791882\pi$$
0.793765 0.608224i $$-0.208118\pi$$
$$644$$ −2013.00 −0.123173
$$645$$ 0 0
$$646$$ 5301.00 0.322856
$$647$$ − 3375.00i − 0.205077i −0.994729 0.102539i $$-0.967303\pi$$
0.994729 0.102539i $$-0.0326965\pi$$
$$648$$ − 14091.0i − 0.854239i
$$649$$ 10530.0 0.636885
$$650$$ 0 0
$$651$$ −3080.00 −0.185430
$$652$$ 1600.00i 0.0961056i
$$653$$ − 24948.0i − 1.49509i −0.664214 0.747543i $$-0.731234\pi$$
0.664214 0.747543i $$-0.268766\pi$$
$$654$$ −5835.00 −0.348879
$$655$$ 0 0
$$656$$ −17040.0 −1.01418
$$657$$ 706.000i 0.0419234i
$$658$$ 5544.00i 0.328461i
$$659$$ 9879.00 0.583962 0.291981 0.956424i $$-0.405686\pi$$
0.291981 + 0.956424i $$0.405686\pi$$
$$660$$ 0 0
$$661$$ −14155.0 −0.832928 −0.416464 0.909152i $$-0.636731\pi$$
−0.416464 + 0.909152i $$0.636731\pi$$
$$662$$ − 26697.0i − 1.56738i
$$663$$ 5115.00i 0.299623i
$$664$$ −4914.00 −0.287199
$$665$$ 0 0
$$666$$ −1500.00 −0.0872730
$$667$$ 45567.0i 2.64522i
$$668$$ − 2004.00i − 0.116073i
$$669$$ −9890.00 −0.571554
$$670$$ 0 0
$$671$$ 19332.0 1.11223
$$672$$ 2475.00i 0.142076i
$$673$$ 8948.00i 0.512511i 0.966609 + 0.256256i $$0.0824889\pi$$
−0.966609 + 0.256256i $$0.917511\pi$$
$$674$$ 17448.0 0.997139
$$675$$ 0 0
$$676$$ −2076.00 −0.118116
$$677$$ 11511.0i 0.653477i 0.945115 + 0.326738i $$0.105950\pi$$
−0.945115 + 0.326738i $$0.894050\pi$$
$$678$$ 5130.00i 0.290585i
$$679$$ −8338.00 −0.471256
$$680$$ 0 0
$$681$$ −26775.0 −1.50664
$$682$$ − 9072.00i − 0.509362i
$$683$$ − 10476.0i − 0.586900i −0.955974 0.293450i $$-0.905197\pi$$
0.955974 0.293450i $$-0.0948035\pi$$
$$684$$ 38.0000 0.00212422
$$685$$ 0 0
$$686$$ 18645.0 1.03771
$$687$$ − 31850.0i − 1.76878i
$$688$$ 13916.0i 0.771137i
$$689$$ −4785.00 −0.264578
$$690$$ 0 0
$$691$$ 30098.0 1.65699 0.828496 0.559995i $$-0.189197\pi$$
0.828496 + 0.559995i $$0.189197\pi$$
$$692$$ 462.000i 0.0253795i
$$693$$ 1188.00i 0.0651203i
$$694$$ −4734.00 −0.258934
$$695$$ 0 0
$$696$$ 26145.0 1.42388
$$697$$ 22320.0i 1.21296i
$$698$$ − 4974.00i − 0.269726i
$$699$$ −14190.0 −0.767833
$$700$$ 0 0
$$701$$ −14700.0 −0.792028 −0.396014 0.918245i $$-0.629607\pi$$
−0.396014 + 0.918245i $$0.629607\pi$$
$$702$$ 4785.00i 0.257262i
$$703$$ − 4750.00i − 0.254836i
$$704$$ 23382.0 1.25176
$$705$$ 0 0
$$706$$ 34101.0 1.81786
$$707$$ 7986.00i 0.424815i
$$708$$ − 975.000i − 0.0517553i
$$709$$ −31178.0 −1.65150 −0.825751 0.564035i $$-0.809248\pi$$
−0.825751 + 0.564035i $$0.809248\pi$$
$$710$$ 0 0
$$711$$ 68.0000 0.00358678
$$712$$ 3528.00i 0.185699i
$$713$$ 10248.0i 0.538276i
$$714$$ 15345.0 0.804303
$$715$$ 0 0
$$716$$ 720.000 0.0375805
$$717$$ − 1845.00i − 0.0960987i
$$718$$ − 7659.00i − 0.398094i
$$719$$ 33285.0 1.72645 0.863227 0.504815i $$-0.168439\pi$$
0.863227 + 0.504815i $$0.168439\pi$$
$$720$$ 0 0
$$721$$ 22.0000 0.00113637
$$722$$ 1083.00i 0.0558242i
$$723$$ − 33040.0i − 1.69954i
$$724$$ 2338.00 0.120015
$$725$$ 0 0
$$726$$ 23775.0 1.21539
$$727$$ 34729.0i 1.77170i 0.463970 + 0.885851i $$0.346425\pi$$
−0.463970 + 0.885851i $$0.653575\pi$$
$$728$$ 2541.00i 0.129362i
$$729$$ −20917.0 −1.06269
$$730$$ 0 0
$$731$$ 18228.0 0.922280
$$732$$ − 1790.00i − 0.0903829i
$$733$$ 4196.00i 0.211436i 0.994396 + 0.105718i $$0.0337141\pi$$
−0.994396 + 0.105718i $$0.966286\pi$$
$$734$$ −588.000 −0.0295688
$$735$$ 0 0
$$736$$ 8235.00 0.412427
$$737$$ − 51894.0i − 2.59368i
$$738$$ 1440.00i 0.0718254i
$$739$$ 10744.0 0.534810 0.267405 0.963584i $$-0.413834\pi$$
0.267405 + 0.963584i $$0.413834\pi$$
$$740$$ 0 0
$$741$$ −1045.00 −0.0518071
$$742$$ 14355.0i 0.710227i
$$743$$ − 2208.00i − 0.109022i −0.998513 0.0545112i $$-0.982640\pi$$
0.998513 0.0545112i $$-0.0173601\pi$$
$$744$$ 5880.00 0.289746
$$745$$ 0 0
$$746$$ −28059.0 −1.37710
$$747$$ 468.000i 0.0229227i
$$748$$ 5022.00i 0.245485i
$$749$$ −15543.0 −0.758249
$$750$$ 0 0
$$751$$ 13160.0 0.639434 0.319717 0.947513i $$-0.396412\pi$$
0.319717 + 0.947513i $$0.396412\pi$$
$$752$$ − 11928.0i − 0.578417i
$$753$$ − 23370.0i − 1.13101i
$$754$$ −8217.00 −0.396877
$$755$$ 0 0
$$756$$ 1595.00 0.0767323
$$757$$ − 758.000i − 0.0363936i −0.999834 0.0181968i $$-0.994207\pi$$
0.999834 0.0181968i $$-0.00579255\pi$$
$$758$$ − 11481.0i − 0.550143i
$$759$$ −49410.0 −2.36294
$$760$$ 0 0
$$761$$ 4851.00 0.231076 0.115538 0.993303i $$-0.463141\pi$$
0.115538 + 0.993303i $$0.463141\pi$$
$$762$$ 17250.0i 0.820081i
$$763$$ 4279.00i 0.203028i
$$764$$ −2871.00 −0.135954
$$765$$ 0 0
$$766$$ −17082.0 −0.805741
$$767$$ − 2145.00i − 0.100980i
$$768$$ − 7565.00i − 0.355441i
$$769$$ 33091.0 1.55175 0.775873 0.630890i $$-0.217310\pi$$
0.775873 + 0.630890i $$0.217310\pi$$
$$770$$ 0 0
$$771$$ −22560.0 −1.05380
$$772$$ − 1658.00i − 0.0772963i
$$773$$ 42357.0i 1.97086i 0.170079 + 0.985430i $$0.445598\pi$$
−0.170079 + 0.985430i $$0.554402\pi$$
$$774$$ 1176.00 0.0546130
$$775$$ 0 0
$$776$$ 15918.0 0.736370
$$777$$ − 13750.0i − 0.634850i
$$778$$ − 3870.00i − 0.178337i
$$779$$ −4560.00 −0.209729
$$780$$ 0 0
$$781$$ 13284.0 0.608629
$$782$$ − 51057.0i − 2.33478i
$$783$$ − 36105.0i − 1.64788i
$$784$$ −15762.0 −0.718021
$$785$$ 0 0
$$786$$ −21780.0 −0.988380
$$787$$ 39877.0i 1.80618i 0.429454 + 0.903089i $$0.358706\pi$$
−0.429454 + 0.903089i $$0.641294\pi$$
$$788$$ − 4176.00i − 0.188787i
$$789$$ 18840.0 0.850091
$$790$$ 0 0
$$791$$ 3762.00 0.169104
$$792$$ − 2268.00i − 0.101755i
$$793$$ − 3938.00i − 0.176346i
$$794$$ 19608.0 0.876400
$$795$$ 0 0
$$796$$ −241.000 −0.0107312
$$797$$ 30033.0i 1.33478i 0.744706 + 0.667392i $$0.232590\pi$$
−0.744706 + 0.667392i $$0.767410\pi$$
$$798$$ 3135.00i 0.139070i
$$799$$ −15624.0 −0.691786
$$800$$ 0 0
$$801$$ 336.000 0.0148214
$$802$$ 6984.00i 0.307498i
$$803$$ − 19062.0i − 0.837713i
$$804$$ −4805.00 −0.210770
$$805$$ 0 0
$$806$$ −1848.00 −0.0807606
$$807$$ 23790.0i 1.03773i
$$808$$ − 15246.0i − 0.663802i
$$809$$ −585.000 −0.0254234 −0.0127117 0.999919i $$-0.504046\pi$$
−0.0127117 + 0.999919i $$0.504046\pi$$
$$810$$ 0 0
$$811$$ 28361.0 1.22798 0.613989 0.789315i $$-0.289564\pi$$
0.613989 + 0.789315i $$0.289564\pi$$
$$812$$ 2739.00i 0.118374i
$$813$$ 10205.0i 0.440228i
$$814$$ 40500.0 1.74389
$$815$$ 0 0
$$816$$ −33015.0 −1.41637
$$817$$ 3724.00i 0.159469i
$$818$$ 20028.0i 0.856067i
$$819$$ 242.000 0.0103250
$$820$$ 0 0
$$821$$ 25068.0 1.06563 0.532813 0.846233i $$-0.321135\pi$$
0.532813 + 0.846233i $$0.321135\pi$$
$$822$$ 25335.0i 1.07501i
$$823$$ 10901.0i 0.461707i 0.972989 + 0.230854i $$0.0741518\pi$$
−0.972989 + 0.230854i $$0.925848\pi$$
$$824$$ −42.0000 −0.00177565
$$825$$ 0 0
$$826$$ −6435.00 −0.271068
$$827$$ − 12027.0i − 0.505707i −0.967505 0.252854i $$-0.918631\pi$$
0.967505 0.252854i $$-0.0813691\pi$$
$$828$$ − 366.000i − 0.0153616i
$$829$$ 19339.0 0.810219 0.405109 0.914268i $$-0.367233\pi$$
0.405109 + 0.914268i $$0.367233\pi$$
$$830$$ 0 0
$$831$$ −9820.00 −0.409930
$$832$$ − 4763.00i − 0.198470i
$$833$$ 20646.0i 0.858753i
$$834$$ −32160.0 −1.33526
$$835$$ 0 0
$$836$$ −1026.00 −0.0424461
$$837$$ − 8120.00i − 0.335326i
$$838$$ 24408.0i 1.00616i
$$839$$ 13188.0 0.542670 0.271335 0.962485i $$-0.412535\pi$$
0.271335 + 0.962485i $$0.412535\pi$$
$$840$$ 0 0
$$841$$ 37612.0 1.54217
$$842$$ − 25995.0i − 1.06395i
$$843$$ 27480.0i 1.12273i
$$844$$ 745.000 0.0303838
$$845$$ 0 0
$$846$$ −1008.00 −0.0409642
$$847$$ − 17435.0i − 0.707289i
$$848$$ − 30885.0i − 1.25070i
$$849$$ 15490.0 0.626167
$$850$$ 0 0
$$851$$ −45750.0 −1.84288
$$852$$ − 1230.00i − 0.0494590i
$$853$$ − 4678.00i − 0.187775i −0.995583 0.0938873i $$-0.970071\pi$$
0.995583 0.0938873i $$-0.0299293\pi$$
$$854$$ −11814.0 −0.473380
$$855$$ 0 0
$$856$$ 29673.0 1.18482
$$857$$ − 15252.0i − 0.607933i −0.952683 0.303966i $$-0.901689\pi$$
0.952683 0.303966i $$-0.0983111\pi$$
$$858$$ − 8910.00i − 0.354525i
$$859$$ 610.000 0.0242293 0.0121146 0.999927i $$-0.496144\pi$$
0.0121146 + 0.999927i $$0.496144\pi$$
$$860$$ 0 0
$$861$$ −13200.0 −0.522479
$$862$$ 2250.00i 0.0889041i
$$863$$ 774.000i 0.0305299i 0.999883 + 0.0152649i $$0.00485917\pi$$
−0.999883 + 0.0152649i $$0.995141\pi$$
$$864$$ −6525.00 −0.256927
$$865$$ 0 0
$$866$$ 14574.0 0.571876
$$867$$ 18680.0i 0.731726i
$$868$$ 616.000i 0.0240880i
$$869$$ −1836.00 −0.0716709
$$870$$ 0 0
$$871$$ −10571.0 −0.411234
$$872$$ − 8169.00i − 0.317245i
$$873$$ − 1516.00i − 0.0587730i
$$874$$ 10431.0 0.403700
$$875$$ 0 0
$$876$$ −1765.00 −0.0680751
$$877$$ 31039.0i 1.19511i 0.801827 + 0.597556i $$0.203861\pi$$
−0.801827 + 0.597556i $$0.796139\pi$$
$$878$$ − 19500.0i − 0.749537i
$$879$$ 585.000 0.0224477
$$880$$ 0 0
$$881$$ 33678.0 1.28790 0.643950 0.765067i $$-0.277294\pi$$
0.643950 + 0.765067i $$0.277294\pi$$
$$882$$ 1332.00i 0.0508512i
$$883$$ − 42982.0i − 1.63812i −0.573708 0.819060i $$-0.694496\pi$$
0.573708 0.819060i $$-0.305504\pi$$
$$884$$ 1023.00 0.0389222
$$885$$ 0 0
$$886$$ −10458.0 −0.396550
$$887$$ − 4494.00i − 0.170117i −0.996376 0.0850585i $$-0.972892\pi$$
0.996376 0.0850585i $$-0.0271077\pi$$
$$888$$ 26250.0i 0.991996i
$$889$$ 12650.0 0.477241
$$890$$ 0 0
$$891$$ 36234.0 1.36238
$$892$$ 1978.00i 0.0742470i
$$893$$ − 3192.00i − 0.119615i
$$894$$ 45000.0 1.68347
$$895$$ 0 0
$$896$$ −18249.0 −0.680420
$$897$$ 10065.0i 0.374649i
$$898$$ 45090.0i 1.67558i
$$899$$ 13944.0 0.517306
$$900$$ 0 0
$$901$$ −40455.0 −1.49584
$$902$$ − 38880.0i − 1.43521i
$$903$$ 10780.0i 0.397271i
$$904$$ −7182.00 −0.264236
$$905$$ 0 0
$$906$$ −15090.0 −0.553346
$$907$$ 23839.0i 0.872724i 0.899771 + 0.436362i $$0.143733\pi$$
−0.899771 + 0.436362i $$0.856267\pi$$
$$908$$ 5355.00i 0.195718i
$$909$$ −1452.00 −0.0529811
$$910$$ 0 0
$$911$$ −10332.0 −0.375757 −0.187878 0.982192i $$-0.560161\pi$$
−0.187878 + 0.982192i $$0.560161\pi$$
$$912$$ − 6745.00i − 0.244901i
$$913$$ − 12636.0i − 0.458040i
$$914$$ −8877.00 −0.321253
$$915$$ 0 0
$$916$$ −6370.00 −0.229772
$$917$$ 15972.0i 0.575182i
$$918$$ 40455.0i 1.45448i
$$919$$ 14371.0 0.515838 0.257919 0.966166i $$-0.416963\pi$$
0.257919 + 0.966166i $$0.416963\pi$$
$$920$$ 0 0
$$921$$ 7100.00 0.254021
$$922$$ − 468.000i − 0.0167167i
$$923$$ − 2706.00i − 0.0964995i
$$924$$ −2970.00 −0.105742
$$925$$ 0 0
$$926$$ −13452.0 −0.477387
$$927$$ 4.00000i 0 0.000141723i
$$928$$ − 11205.0i − 0.396360i
$$929$$ −26889.0 −0.949623 −0.474811 0.880088i $$-0.657484\pi$$
−0.474811 + 0.880088i $$0.657484\pi$$
$$930$$ 0 0
$$931$$ −4218.00 −0.148485
$$932$$ 2838.00i 0.0997444i
$$933$$ 32805.0i 1.15111i
$$934$$ 26298.0 0.921303
$$935$$ 0 0
$$936$$ −462.000 −0.0161335
$$937$$ − 785.000i − 0.0273691i −0.999906 0.0136845i $$-0.995644\pi$$
0.999906 0.0136845i $$-0.00435606\pi$$
$$938$$ 31713.0i 1.10391i
$$939$$ −7415.00 −0.257699
$$940$$ 0 0
$$941$$ −18141.0 −0.628459 −0.314229 0.949347i $$-0.601746\pi$$
−0.314229 + 0.949347i $$0.601746\pi$$
$$942$$ − 42690.0i − 1.47656i
$$943$$ 43920.0i 1.51668i
$$944$$ 13845.0 0.477348
$$945$$ 0 0
$$946$$ −31752.0 −1.09128
$$947$$ − 23100.0i − 0.792660i −0.918108 0.396330i $$-0.870284\pi$$
0.918108 0.396330i $$-0.129716\pi$$
$$948$$ 170.000i 0.00582420i
$$949$$ −3883.00 −0.132821
$$950$$ 0 0
$$951$$ 6195.00 0.211237
$$952$$ 21483.0i 0.731374i
$$953$$ 45690.0i 1.55304i 0.630094 + 0.776519i $$0.283016\pi$$
−0.630094 + 0.776519i $$0.716984\pi$$
$$954$$ −2610.00 −0.0885764
$$955$$ 0 0
$$956$$ −369.000 −0.0124836
$$957$$ 67230.0i 2.27089i
$$958$$ 56988.0i 1.92192i
$$959$$ 18579.0 0.625597
$$960$$ 0 0
$$961$$ −26655.0 −0.894733
$$962$$ − 8250.00i − 0.276498i
$$963$$ − 2826.00i − 0.0945655i
$$964$$ −6608.00 −0.220777
$$965$$ 0 0
$$966$$ 30195.0 1.00570
$$967$$ − 21584.0i − 0.717781i −0.933380 0.358891i $$-0.883155\pi$$
0.933380 0.358891i $$-0.116845\pi$$
$$968$$ 33285.0i 1.10519i
$$969$$ −8835.00 −0.292901
$$970$$ 0 0
$$971$$ −50556.0 −1.67087 −0.835437 0.549586i $$-0.814786\pi$$
−0.835437 + 0.549586i $$0.814786\pi$$
$$972$$ 560.000i 0.0184794i
$$973$$ 23584.0i 0.777049i
$$974$$ −22350.0 −0.735257
$$975$$ 0 0
$$976$$ 25418.0 0.833617
$$977$$ − 8568.00i − 0.280568i −0.990111 0.140284i $$-0.955198\pi$$
0.990111 0.140284i $$-0.0448015\pi$$
$$978$$ − 24000.0i − 0.784699i
$$979$$ −9072.00 −0.296162
$$980$$ 0 0
$$981$$ −778.000 −0.0253207
$$982$$ 18540.0i 0.602480i
$$983$$ 29706.0i 0.963860i 0.876210 + 0.481930i $$0.160064\pi$$
−0.876210 + 0.481930i $$0.839936\pi$$
$$984$$ 25200.0 0.816409
$$985$$ 0 0
$$986$$ −69471.0 −2.24382
$$987$$ − 9240.00i − 0.297986i
$$988$$ 209.000i 0.00672993i
$$989$$ 35868.0 1.15322
$$990$$ 0 0
$$991$$ 30512.0 0.978048 0.489024 0.872270i $$-0.337353\pi$$
0.489024 + 0.872270i $$0.337353\pi$$
$$992$$ − 2520.00i − 0.0806553i
$$993$$ 44495.0i 1.42196i
$$994$$ −8118.00 −0.259042
$$995$$ 0 0
$$996$$ −1170.00 −0.0372218
$$997$$ − 47756.0i − 1.51700i −0.651674 0.758499i $$-0.725933\pi$$
0.651674 0.758499i $$-0.274067\pi$$
$$998$$ − 7728.00i − 0.245116i
$$999$$ 36250.0 1.14805
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 475.4.b.c.324.2 2
5.2 odd 4 19.4.a.a.1.1 1
5.3 odd 4 475.4.a.e.1.1 1
5.4 even 2 inner 475.4.b.c.324.1 2
15.2 even 4 171.4.a.d.1.1 1
20.7 even 4 304.4.a.b.1.1 1
35.27 even 4 931.4.a.a.1.1 1
40.27 even 4 1216.4.a.a.1.1 1
40.37 odd 4 1216.4.a.f.1.1 1
55.32 even 4 2299.4.a.b.1.1 1
95.37 even 4 361.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
19.4.a.a.1.1 1 5.2 odd 4
171.4.a.d.1.1 1 15.2 even 4
304.4.a.b.1.1 1 20.7 even 4
361.4.a.b.1.1 1 95.37 even 4
475.4.a.e.1.1 1 5.3 odd 4
475.4.b.c.324.1 2 5.4 even 2 inner
475.4.b.c.324.2 2 1.1 even 1 trivial
931.4.a.a.1.1 1 35.27 even 4
1216.4.a.a.1.1 1 40.27 even 4
1216.4.a.f.1.1 1 40.37 odd 4
2299.4.a.b.1.1 1 55.32 even 4