# Properties

 Label 495.4.a.d.1.2 Level $495$ Weight $4$ Character 495.1 Self dual yes Analytic conductor $29.206$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 495.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: $$29.2059454528$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 495.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +6.24621 q^{7} -24.1771 q^{8} +O(q^{10})$$ $$q+2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +6.24621 q^{7} -24.1771 q^{8} +12.8078 q^{10} +11.0000 q^{11} -49.1231 q^{13} +16.0000 q^{14} -50.4233 q^{16} -82.7083 q^{17} -130.354 q^{19} -7.19224 q^{20} +28.1771 q^{22} +185.693 q^{23} +25.0000 q^{25} -125.831 q^{26} -8.98485 q^{28} +8.90720 q^{29} +5.26137 q^{31} +64.2547 q^{32} -211.862 q^{34} +31.2311 q^{35} -416.894 q^{37} -333.909 q^{38} -120.885 q^{40} +298.479 q^{41} -513.633 q^{43} -15.8229 q^{44} +475.663 q^{46} -557.295 q^{47} -303.985 q^{49} +64.0388 q^{50} +70.6610 q^{52} +168.064 q^{53} +55.0000 q^{55} -151.015 q^{56} +22.8163 q^{58} -618.773 q^{59} +786.405 q^{61} +13.4773 q^{62} +567.978 q^{64} -245.616 q^{65} -339.015 q^{67} +118.972 q^{68} +80.0000 q^{70} -1120.71 q^{71} -123.430 q^{73} -1067.90 q^{74} +187.508 q^{76} +68.7083 q^{77} -309.835 q^{79} -252.116 q^{80} +764.570 q^{82} +1021.22 q^{83} -413.542 q^{85} -1315.70 q^{86} -265.948 q^{88} +141.879 q^{89} -306.833 q^{91} -267.110 q^{92} -1427.54 q^{94} -651.771 q^{95} +798.345 q^{97} -778.673 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10})$$ 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 4 * q^7 - 3 * q^8 $$2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8} + 5 q^{10} + 22 q^{11} - 90 q^{13} + 32 q^{14} - 39 q^{16} + 16 q^{17} - 170 q^{19} - 35 q^{20} + 11 q^{22} + 124 q^{23} + 50 q^{25} - 62 q^{26} + 48 q^{28} + 158 q^{29} + 60 q^{31} - 123 q^{32} - 366 q^{34} - 20 q^{35} - 372 q^{37} - 272 q^{38} - 15 q^{40} - 38 q^{41} - 516 q^{43} - 77 q^{44} + 572 q^{46} - 224 q^{47} - 542 q^{49} + 25 q^{50} + 298 q^{52} - 472 q^{53} + 110 q^{55} - 368 q^{56} - 210 q^{58} - 248 q^{59} + 72 q^{61} - 72 q^{62} + 769 q^{64} - 450 q^{65} - 744 q^{67} - 430 q^{68} + 160 q^{70} - 2060 q^{71} - 486 q^{73} - 1138 q^{74} + 408 q^{76} - 44 q^{77} + 642 q^{79} - 195 q^{80} + 1290 q^{82} + 286 q^{83} + 80 q^{85} - 1312 q^{86} - 33 q^{88} - 244 q^{89} + 112 q^{91} + 76 q^{92} - 1948 q^{94} - 850 q^{95} - 168 q^{97} - 407 q^{98}+O(q^{100})$$ 2 * q + q^2 - 7 * q^4 + 10 * q^5 - 4 * q^7 - 3 * q^8 + 5 * q^10 + 22 * q^11 - 90 * q^13 + 32 * q^14 - 39 * q^16 + 16 * q^17 - 170 * q^19 - 35 * q^20 + 11 * q^22 + 124 * q^23 + 50 * q^25 - 62 * q^26 + 48 * q^28 + 158 * q^29 + 60 * q^31 - 123 * q^32 - 366 * q^34 - 20 * q^35 - 372 * q^37 - 272 * q^38 - 15 * q^40 - 38 * q^41 - 516 * q^43 - 77 * q^44 + 572 * q^46 - 224 * q^47 - 542 * q^49 + 25 * q^50 + 298 * q^52 - 472 * q^53 + 110 * q^55 - 368 * q^56 - 210 * q^58 - 248 * q^59 + 72 * q^61 - 72 * q^62 + 769 * q^64 - 450 * q^65 - 744 * q^67 - 430 * q^68 + 160 * q^70 - 2060 * q^71 - 486 * q^73 - 1138 * q^74 + 408 * q^76 - 44 * q^77 + 642 * q^79 - 195 * q^80 + 1290 * q^82 + 286 * q^83 + 80 * q^85 - 1312 * q^86 - 33 * q^88 - 244 * q^89 + 112 * q^91 + 76 * q^92 - 1948 * q^94 - 850 * q^95 - 168 * q^97 - 407 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.56155 0.905646 0.452823 0.891601i $$-0.350417\pi$$
0.452823 + 0.891601i $$0.350417\pi$$
$$3$$ 0 0
$$4$$ −1.43845 −0.179806
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 6.24621 0.337264 0.168632 0.985679i $$-0.446065\pi$$
0.168632 + 0.985679i $$0.446065\pi$$
$$8$$ −24.1771 −1.06849
$$9$$ 0 0
$$10$$ 12.8078 0.405017
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ −49.1231 −1.04802 −0.524011 0.851711i $$-0.675565\pi$$
−0.524011 + 0.851711i $$0.675565\pi$$
$$14$$ 16.0000 0.305441
$$15$$ 0 0
$$16$$ −50.4233 −0.787864
$$17$$ −82.7083 −1.17998 −0.589992 0.807409i $$-0.700869\pi$$
−0.589992 + 0.807409i $$0.700869\pi$$
$$18$$ 0 0
$$19$$ −130.354 −1.57396 −0.786981 0.616977i $$-0.788357\pi$$
−0.786981 + 0.616977i $$0.788357\pi$$
$$20$$ −7.19224 −0.0804116
$$21$$ 0 0
$$22$$ 28.1771 0.273062
$$23$$ 185.693 1.68347 0.841733 0.539895i $$-0.181536\pi$$
0.841733 + 0.539895i $$0.181536\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −125.831 −0.949137
$$27$$ 0 0
$$28$$ −8.98485 −0.0606420
$$29$$ 8.90720 0.0570354 0.0285177 0.999593i $$-0.490921\pi$$
0.0285177 + 0.999593i $$0.490921\pi$$
$$30$$ 0 0
$$31$$ 5.26137 0.0304829 0.0152414 0.999884i $$-0.495148\pi$$
0.0152414 + 0.999884i $$0.495148\pi$$
$$32$$ 64.2547 0.354961
$$33$$ 0 0
$$34$$ −211.862 −1.06865
$$35$$ 31.2311 0.150829
$$36$$ 0 0
$$37$$ −416.894 −1.85235 −0.926175 0.377094i $$-0.876923\pi$$
−0.926175 + 0.377094i $$0.876923\pi$$
$$38$$ −333.909 −1.42545
$$39$$ 0 0
$$40$$ −120.885 −0.477842
$$41$$ 298.479 1.13694 0.568471 0.822703i $$-0.307535\pi$$
0.568471 + 0.822703i $$0.307535\pi$$
$$42$$ 0 0
$$43$$ −513.633 −1.82159 −0.910793 0.412863i $$-0.864529\pi$$
−0.910793 + 0.412863i $$0.864529\pi$$
$$44$$ −15.8229 −0.0542135
$$45$$ 0 0
$$46$$ 475.663 1.52462
$$47$$ −557.295 −1.72957 −0.864786 0.502140i $$-0.832546\pi$$
−0.864786 + 0.502140i $$0.832546\pi$$
$$48$$ 0 0
$$49$$ −303.985 −0.886253
$$50$$ 64.0388 0.181129
$$51$$ 0 0
$$52$$ 70.6610 0.188441
$$53$$ 168.064 0.435574 0.217787 0.975996i $$-0.430116\pi$$
0.217787 + 0.975996i $$0.430116\pi$$
$$54$$ 0 0
$$55$$ 55.0000 0.134840
$$56$$ −151.015 −0.360362
$$57$$ 0 0
$$58$$ 22.8163 0.0516539
$$59$$ −618.773 −1.36538 −0.682689 0.730709i $$-0.739190\pi$$
−0.682689 + 0.730709i $$0.739190\pi$$
$$60$$ 0 0
$$61$$ 786.405 1.65064 0.825319 0.564667i $$-0.190996\pi$$
0.825319 + 0.564667i $$0.190996\pi$$
$$62$$ 13.4773 0.0276067
$$63$$ 0 0
$$64$$ 567.978 1.10933
$$65$$ −245.616 −0.468690
$$66$$ 0 0
$$67$$ −339.015 −0.618169 −0.309084 0.951035i $$-0.600023\pi$$
−0.309084 + 0.951035i $$0.600023\pi$$
$$68$$ 118.972 0.212168
$$69$$ 0 0
$$70$$ 80.0000 0.136598
$$71$$ −1120.71 −1.87329 −0.936645 0.350280i $$-0.886087\pi$$
−0.936645 + 0.350280i $$0.886087\pi$$
$$72$$ 0 0
$$73$$ −123.430 −0.197896 −0.0989478 0.995093i $$-0.531548\pi$$
−0.0989478 + 0.995093i $$0.531548\pi$$
$$74$$ −1067.90 −1.67757
$$75$$ 0 0
$$76$$ 187.508 0.283008
$$77$$ 68.7083 0.101689
$$78$$ 0 0
$$79$$ −309.835 −0.441255 −0.220628 0.975358i $$-0.570811\pi$$
−0.220628 + 0.975358i $$0.570811\pi$$
$$80$$ −252.116 −0.352343
$$81$$ 0 0
$$82$$ 764.570 1.02967
$$83$$ 1021.22 1.35053 0.675263 0.737577i $$-0.264030\pi$$
0.675263 + 0.737577i $$0.264030\pi$$
$$84$$ 0 0
$$85$$ −413.542 −0.527705
$$86$$ −1315.70 −1.64971
$$87$$ 0 0
$$88$$ −265.948 −0.322161
$$89$$ 141.879 0.168979 0.0844894 0.996424i $$-0.473074\pi$$
0.0844894 + 0.996424i $$0.473074\pi$$
$$90$$ 0 0
$$91$$ −306.833 −0.353460
$$92$$ −267.110 −0.302697
$$93$$ 0 0
$$94$$ −1427.54 −1.56638
$$95$$ −651.771 −0.703898
$$96$$ 0 0
$$97$$ 798.345 0.835666 0.417833 0.908524i $$-0.362790\pi$$
0.417833 + 0.908524i $$0.362790\pi$$
$$98$$ −778.673 −0.802631
$$99$$ 0 0
$$100$$ −35.9612 −0.0359612
$$101$$ −241.400 −0.237823 −0.118912 0.992905i $$-0.537941\pi$$
−0.118912 + 0.992905i $$0.537941\pi$$
$$102$$ 0 0
$$103$$ 1168.38 1.11771 0.558853 0.829267i $$-0.311242\pi$$
0.558853 + 0.829267i $$0.311242\pi$$
$$104$$ 1187.65 1.11980
$$105$$ 0 0
$$106$$ 430.506 0.394476
$$107$$ 2106.82 1.90350 0.951748 0.306882i $$-0.0992857\pi$$
0.951748 + 0.306882i $$0.0992857\pi$$
$$108$$ 0 0
$$109$$ 493.792 0.433914 0.216957 0.976181i $$-0.430387\pi$$
0.216957 + 0.976181i $$0.430387\pi$$
$$110$$ 140.885 0.122117
$$111$$ 0 0
$$112$$ −314.955 −0.265718
$$113$$ −170.000 −0.141524 −0.0707622 0.997493i $$-0.522543\pi$$
−0.0707622 + 0.997493i $$0.522543\pi$$
$$114$$ 0 0
$$115$$ 928.466 0.752869
$$116$$ −12.8125 −0.0102553
$$117$$ 0 0
$$118$$ −1585.02 −1.23655
$$119$$ −516.614 −0.397966
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 2014.42 1.49489
$$123$$ 0 0
$$124$$ −7.56820 −0.00548100
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −948.182 −0.662500 −0.331250 0.943543i $$-0.607470\pi$$
−0.331250 + 0.943543i $$0.607470\pi$$
$$128$$ 940.868 0.649702
$$129$$ 0 0
$$130$$ −629.157 −0.424467
$$131$$ 1484.84 0.990312 0.495156 0.868804i $$-0.335111\pi$$
0.495156 + 0.868804i $$0.335111\pi$$
$$132$$ 0 0
$$133$$ −814.220 −0.530841
$$134$$ −868.405 −0.559842
$$135$$ 0 0
$$136$$ 1999.65 1.26080
$$137$$ −684.928 −0.427134 −0.213567 0.976928i $$-0.568508\pi$$
−0.213567 + 0.976928i $$0.568508\pi$$
$$138$$ 0 0
$$139$$ 830.483 0.506767 0.253384 0.967366i $$-0.418457\pi$$
0.253384 + 0.967366i $$0.418457\pi$$
$$140$$ −44.9242 −0.0271199
$$141$$ 0 0
$$142$$ −2870.75 −1.69654
$$143$$ −540.354 −0.315991
$$144$$ 0 0
$$145$$ 44.5360 0.0255070
$$146$$ −316.172 −0.179223
$$147$$ 0 0
$$148$$ 599.680 0.333063
$$149$$ 1213.64 0.667285 0.333642 0.942700i $$-0.391722\pi$$
0.333642 + 0.942700i $$0.391722\pi$$
$$150$$ 0 0
$$151$$ 30.8466 0.0166242 0.00831212 0.999965i $$-0.497354\pi$$
0.00831212 + 0.999965i $$0.497354\pi$$
$$152$$ 3151.58 1.68176
$$153$$ 0 0
$$154$$ 176.000 0.0920941
$$155$$ 26.3068 0.0136324
$$156$$ 0 0
$$157$$ 345.239 0.175497 0.0877485 0.996143i $$-0.472033\pi$$
0.0877485 + 0.996143i $$0.472033\pi$$
$$158$$ −793.659 −0.399621
$$159$$ 0 0
$$160$$ 321.274 0.158743
$$161$$ 1159.88 0.567772
$$162$$ 0 0
$$163$$ 1921.49 0.923331 0.461665 0.887054i $$-0.347252\pi$$
0.461665 + 0.887054i $$0.347252\pi$$
$$164$$ −429.346 −0.204429
$$165$$ 0 0
$$166$$ 2615.91 1.22310
$$167$$ −172.297 −0.0798369 −0.0399185 0.999203i $$-0.512710\pi$$
−0.0399185 + 0.999203i $$0.512710\pi$$
$$168$$ 0 0
$$169$$ 216.080 0.0983521
$$170$$ −1059.31 −0.477913
$$171$$ 0 0
$$172$$ 738.833 0.327532
$$173$$ 1025.29 0.450587 0.225293 0.974291i $$-0.427666\pi$$
0.225293 + 0.974291i $$0.427666\pi$$
$$174$$ 0 0
$$175$$ 156.155 0.0674527
$$176$$ −554.656 −0.237550
$$177$$ 0 0
$$178$$ 363.430 0.153035
$$179$$ −1658.29 −0.692437 −0.346219 0.938154i $$-0.612534\pi$$
−0.346219 + 0.938154i $$0.612534\pi$$
$$180$$ 0 0
$$181$$ −2021.81 −0.830275 −0.415137 0.909759i $$-0.636266\pi$$
−0.415137 + 0.909759i $$0.636266\pi$$
$$182$$ −785.970 −0.320110
$$183$$ 0 0
$$184$$ −4489.52 −1.79876
$$185$$ −2084.47 −0.828396
$$186$$ 0 0
$$187$$ −909.792 −0.355778
$$188$$ 801.640 0.310987
$$189$$ 0 0
$$190$$ −1669.55 −0.637482
$$191$$ −1440.22 −0.545607 −0.272803 0.962070i $$-0.587951\pi$$
−0.272803 + 0.962070i $$0.587951\pi$$
$$192$$ 0 0
$$193$$ −2798.05 −1.04356 −0.521782 0.853079i $$-0.674733\pi$$
−0.521782 + 0.853079i $$0.674733\pi$$
$$194$$ 2045.00 0.756817
$$195$$ 0 0
$$196$$ 437.266 0.159354
$$197$$ −458.943 −0.165982 −0.0829908 0.996550i $$-0.526447\pi$$
−0.0829908 + 0.996550i $$0.526447\pi$$
$$198$$ 0 0
$$199$$ −2371.04 −0.844615 −0.422308 0.906453i $$-0.638780\pi$$
−0.422308 + 0.906453i $$0.638780\pi$$
$$200$$ −604.427 −0.213697
$$201$$ 0 0
$$202$$ −618.358 −0.215384
$$203$$ 55.6363 0.0192360
$$204$$ 0 0
$$205$$ 1492.40 0.508456
$$206$$ 2992.86 1.01225
$$207$$ 0 0
$$208$$ 2476.95 0.825699
$$209$$ −1433.90 −0.474568
$$210$$ 0 0
$$211$$ 4319.87 1.40944 0.704721 0.709484i $$-0.251072\pi$$
0.704721 + 0.709484i $$0.251072\pi$$
$$212$$ −241.752 −0.0783187
$$213$$ 0 0
$$214$$ 5396.73 1.72389
$$215$$ −2568.16 −0.814638
$$216$$ 0 0
$$217$$ 32.8636 0.0102808
$$218$$ 1264.87 0.392973
$$219$$ 0 0
$$220$$ −79.1146 −0.0242450
$$221$$ 4062.89 1.23665
$$222$$ 0 0
$$223$$ −3837.73 −1.15244 −0.576219 0.817295i $$-0.695473\pi$$
−0.576219 + 0.817295i $$0.695473\pi$$
$$224$$ 401.349 0.119715
$$225$$ 0 0
$$226$$ −435.464 −0.128171
$$227$$ 5003.71 1.46303 0.731515 0.681825i $$-0.238813\pi$$
0.731515 + 0.681825i $$0.238813\pi$$
$$228$$ 0 0
$$229$$ −277.375 −0.0800412 −0.0400206 0.999199i $$-0.512742\pi$$
−0.0400206 + 0.999199i $$0.512742\pi$$
$$230$$ 2378.31 0.681832
$$231$$ 0 0
$$232$$ −215.350 −0.0609415
$$233$$ −2269.91 −0.638225 −0.319113 0.947717i $$-0.603385\pi$$
−0.319113 + 0.947717i $$0.603385\pi$$
$$234$$ 0 0
$$235$$ −2786.48 −0.773488
$$236$$ 890.072 0.245503
$$237$$ 0 0
$$238$$ −1323.33 −0.360416
$$239$$ 1617.11 0.437665 0.218832 0.975762i $$-0.429775\pi$$
0.218832 + 0.975762i $$0.429775\pi$$
$$240$$ 0 0
$$241$$ 5646.63 1.50926 0.754629 0.656151i $$-0.227817\pi$$
0.754629 + 0.656151i $$0.227817\pi$$
$$242$$ 309.948 0.0823314
$$243$$ 0 0
$$244$$ −1131.20 −0.296794
$$245$$ −1519.92 −0.396344
$$246$$ 0 0
$$247$$ 6403.40 1.64955
$$248$$ −127.204 −0.0325705
$$249$$ 0 0
$$250$$ 320.194 0.0810034
$$251$$ 6217.61 1.56355 0.781777 0.623558i $$-0.214313\pi$$
0.781777 + 0.623558i $$0.214313\pi$$
$$252$$ 0 0
$$253$$ 2042.62 0.507584
$$254$$ −2428.82 −0.599991
$$255$$ 0 0
$$256$$ −2133.74 −0.520933
$$257$$ −7712.75 −1.87202 −0.936008 0.351980i $$-0.885509\pi$$
−0.936008 + 0.351980i $$0.885509\pi$$
$$258$$ 0 0
$$259$$ −2604.01 −0.624730
$$260$$ 353.305 0.0842732
$$261$$ 0 0
$$262$$ 3803.49 0.896871
$$263$$ −206.347 −0.0483798 −0.0241899 0.999707i $$-0.507701\pi$$
−0.0241899 + 0.999707i $$0.507701\pi$$
$$264$$ 0 0
$$265$$ 840.322 0.194795
$$266$$ −2085.67 −0.480753
$$267$$ 0 0
$$268$$ 487.655 0.111150
$$269$$ −1712.47 −0.388146 −0.194073 0.980987i $$-0.562170\pi$$
−0.194073 + 0.980987i $$0.562170\pi$$
$$270$$ 0 0
$$271$$ −477.081 −0.106940 −0.0534698 0.998569i $$-0.517028\pi$$
−0.0534698 + 0.998569i $$0.517028\pi$$
$$272$$ 4170.43 0.929666
$$273$$ 0 0
$$274$$ −1754.48 −0.386832
$$275$$ 275.000 0.0603023
$$276$$ 0 0
$$277$$ 4283.48 0.929130 0.464565 0.885539i $$-0.346211\pi$$
0.464565 + 0.885539i $$0.346211\pi$$
$$278$$ 2127.33 0.458952
$$279$$ 0 0
$$280$$ −755.076 −0.161159
$$281$$ −3477.79 −0.738319 −0.369160 0.929366i $$-0.620354\pi$$
−0.369160 + 0.929366i $$0.620354\pi$$
$$282$$ 0 0
$$283$$ −6568.27 −1.37966 −0.689829 0.723973i $$-0.742314\pi$$
−0.689829 + 0.723973i $$0.742314\pi$$
$$284$$ 1612.08 0.336829
$$285$$ 0 0
$$286$$ −1384.15 −0.286176
$$287$$ 1864.36 0.383449
$$288$$ 0 0
$$289$$ 1927.67 0.392360
$$290$$ 114.081 0.0231003
$$291$$ 0 0
$$292$$ 177.547 0.0355828
$$293$$ −8352.29 −1.66534 −0.832672 0.553766i $$-0.813190\pi$$
−0.832672 + 0.553766i $$0.813190\pi$$
$$294$$ 0 0
$$295$$ −3093.86 −0.610616
$$296$$ 10079.3 1.97921
$$297$$ 0 0
$$298$$ 3108.81 0.604324
$$299$$ −9121.83 −1.76431
$$300$$ 0 0
$$301$$ −3208.26 −0.614355
$$302$$ 79.0152 0.0150557
$$303$$ 0 0
$$304$$ 6572.89 1.24007
$$305$$ 3932.03 0.738187
$$306$$ 0 0
$$307$$ 5383.89 1.00090 0.500448 0.865767i $$-0.333169\pi$$
0.500448 + 0.865767i $$0.333169\pi$$
$$308$$ −98.8333 −0.0182843
$$309$$ 0 0
$$310$$ 67.3863 0.0123461
$$311$$ 1790.41 0.326447 0.163223 0.986589i $$-0.447811\pi$$
0.163223 + 0.986589i $$0.447811\pi$$
$$312$$ 0 0
$$313$$ −809.076 −0.146108 −0.0730538 0.997328i $$-0.523274\pi$$
−0.0730538 + 0.997328i $$0.523274\pi$$
$$314$$ 884.347 0.158938
$$315$$ 0 0
$$316$$ 445.682 0.0793403
$$317$$ −10744.5 −1.90370 −0.951849 0.306567i $$-0.900820\pi$$
−0.951849 + 0.306567i $$0.900820\pi$$
$$318$$ 0 0
$$319$$ 97.9792 0.0171968
$$320$$ 2839.89 0.496109
$$321$$ 0 0
$$322$$ 2971.09 0.514200
$$323$$ 10781.4 1.85725
$$324$$ 0 0
$$325$$ −1228.08 −0.209605
$$326$$ 4922.00 0.836210
$$327$$ 0 0
$$328$$ −7216.35 −1.21481
$$329$$ −3480.98 −0.583322
$$330$$ 0 0
$$331$$ 3399.12 0.564449 0.282224 0.959348i $$-0.408928\pi$$
0.282224 + 0.959348i $$0.408928\pi$$
$$332$$ −1468.97 −0.242832
$$333$$ 0 0
$$334$$ −441.349 −0.0723039
$$335$$ −1695.08 −0.276453
$$336$$ 0 0
$$337$$ −11840.0 −1.91384 −0.956919 0.290356i $$-0.906226\pi$$
−0.956919 + 0.290356i $$0.906226\pi$$
$$338$$ 553.499 0.0890721
$$339$$ 0 0
$$340$$ 594.858 0.0948844
$$341$$ 57.8750 0.00919093
$$342$$ 0 0
$$343$$ −4041.20 −0.636165
$$344$$ 12418.1 1.94634
$$345$$ 0 0
$$346$$ 2626.34 0.408072
$$347$$ 2076.67 0.321272 0.160636 0.987014i $$-0.448645\pi$$
0.160636 + 0.987014i $$0.448645\pi$$
$$348$$ 0 0
$$349$$ 5837.37 0.895322 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$350$$ 400.000 0.0610883
$$351$$ 0 0
$$352$$ 706.802 0.107025
$$353$$ 2423.64 0.365431 0.182715 0.983166i $$-0.441511\pi$$
0.182715 + 0.983166i $$0.441511\pi$$
$$354$$ 0 0
$$355$$ −5603.54 −0.837761
$$356$$ −204.085 −0.0303834
$$357$$ 0 0
$$358$$ −4247.79 −0.627103
$$359$$ 3882.22 0.570740 0.285370 0.958417i $$-0.407884\pi$$
0.285370 + 0.958417i $$0.407884\pi$$
$$360$$ 0 0
$$361$$ 10133.2 1.47736
$$362$$ −5178.96 −0.751935
$$363$$ 0 0
$$364$$ 441.363 0.0635542
$$365$$ −617.150 −0.0885016
$$366$$ 0 0
$$367$$ 5666.65 0.805986 0.402993 0.915203i $$-0.367970\pi$$
0.402993 + 0.915203i $$0.367970\pi$$
$$368$$ −9363.26 −1.32634
$$369$$ 0 0
$$370$$ −5339.48 −0.750233
$$371$$ 1049.77 0.146903
$$372$$ 0 0
$$373$$ 174.771 0.0242608 0.0121304 0.999926i $$-0.496139\pi$$
0.0121304 + 0.999926i $$0.496139\pi$$
$$374$$ −2330.48 −0.322209
$$375$$ 0 0
$$376$$ 13473.8 1.84802
$$377$$ −437.550 −0.0597744
$$378$$ 0 0
$$379$$ 252.686 0.0342470 0.0171235 0.999853i $$-0.494549\pi$$
0.0171235 + 0.999853i $$0.494549\pi$$
$$380$$ 937.538 0.126565
$$381$$ 0 0
$$382$$ −3689.21 −0.494126
$$383$$ 11014.5 1.46950 0.734748 0.678340i $$-0.237300\pi$$
0.734748 + 0.678340i $$0.237300\pi$$
$$384$$ 0 0
$$385$$ 343.542 0.0454766
$$386$$ −7167.35 −0.945099
$$387$$ 0 0
$$388$$ −1148.38 −0.150258
$$389$$ −8099.40 −1.05567 −0.527835 0.849347i $$-0.676996\pi$$
−0.527835 + 0.849347i $$0.676996\pi$$
$$390$$ 0 0
$$391$$ −15358.4 −1.98646
$$392$$ 7349.47 0.946949
$$393$$ 0 0
$$394$$ −1175.61 −0.150320
$$395$$ −1549.18 −0.197335
$$396$$ 0 0
$$397$$ −424.353 −0.0536465 −0.0268232 0.999640i $$-0.508539\pi$$
−0.0268232 + 0.999640i $$0.508539\pi$$
$$398$$ −6073.54 −0.764922
$$399$$ 0 0
$$400$$ −1260.58 −0.157573
$$401$$ 5904.18 0.735263 0.367632 0.929972i $$-0.380169\pi$$
0.367632 + 0.929972i $$0.380169\pi$$
$$402$$ 0 0
$$403$$ −258.455 −0.0319468
$$404$$ 347.241 0.0427620
$$405$$ 0 0
$$406$$ 142.515 0.0174210
$$407$$ −4585.83 −0.558504
$$408$$ 0 0
$$409$$ 1370.47 0.165686 0.0828430 0.996563i $$-0.473600\pi$$
0.0828430 + 0.996563i $$0.473600\pi$$
$$410$$ 3822.85 0.460481
$$411$$ 0 0
$$412$$ −1680.65 −0.200970
$$413$$ −3864.98 −0.460493
$$414$$ 0 0
$$415$$ 5106.11 0.603973
$$416$$ −3156.39 −0.372007
$$417$$ 0 0
$$418$$ −3673.00 −0.429790
$$419$$ −1268.73 −0.147927 −0.0739635 0.997261i $$-0.523565\pi$$
−0.0739635 + 0.997261i $$0.523565\pi$$
$$420$$ 0 0
$$421$$ −12241.9 −1.41719 −0.708594 0.705617i $$-0.750670\pi$$
−0.708594 + 0.705617i $$0.750670\pi$$
$$422$$ 11065.6 1.27646
$$423$$ 0 0
$$424$$ −4063.31 −0.465405
$$425$$ −2067.71 −0.235997
$$426$$ 0 0
$$427$$ 4912.05 0.556700
$$428$$ −3030.55 −0.342260
$$429$$ 0 0
$$430$$ −6578.48 −0.737774
$$431$$ −8050.11 −0.899675 −0.449838 0.893110i $$-0.648518\pi$$
−0.449838 + 0.893110i $$0.648518\pi$$
$$432$$ 0 0
$$433$$ −16565.7 −1.83856 −0.919282 0.393600i $$-0.871230\pi$$
−0.919282 + 0.393600i $$0.871230\pi$$
$$434$$ 84.1819 0.00931073
$$435$$ 0 0
$$436$$ −710.293 −0.0780203
$$437$$ −24205.9 −2.64971
$$438$$ 0 0
$$439$$ 4705.80 0.511607 0.255804 0.966729i $$-0.417660\pi$$
0.255804 + 0.966729i $$0.417660\pi$$
$$440$$ −1329.74 −0.144075
$$441$$ 0 0
$$442$$ 10407.3 1.11997
$$443$$ −15094.0 −1.61882 −0.809408 0.587246i $$-0.800212\pi$$
−0.809408 + 0.587246i $$0.800212\pi$$
$$444$$ 0 0
$$445$$ 709.394 0.0755696
$$446$$ −9830.56 −1.04370
$$447$$ 0 0
$$448$$ 3547.71 0.374138
$$449$$ −973.478 −0.102319 −0.0511595 0.998690i $$-0.516292\pi$$
−0.0511595 + 0.998690i $$0.516292\pi$$
$$450$$ 0 0
$$451$$ 3283.27 0.342801
$$452$$ 244.536 0.0254469
$$453$$ 0 0
$$454$$ 12817.3 1.32499
$$455$$ −1534.17 −0.158072
$$456$$ 0 0
$$457$$ 62.6577 0.00641358 0.00320679 0.999995i $$-0.498979\pi$$
0.00320679 + 0.999995i $$0.498979\pi$$
$$458$$ −710.510 −0.0724890
$$459$$ 0 0
$$460$$ −1335.55 −0.135370
$$461$$ 11866.2 1.19884 0.599419 0.800436i $$-0.295398\pi$$
0.599419 + 0.800436i $$0.295398\pi$$
$$462$$ 0 0
$$463$$ −13144.8 −1.31942 −0.659711 0.751519i $$-0.729321\pi$$
−0.659711 + 0.751519i $$0.729321\pi$$
$$464$$ −449.131 −0.0449361
$$465$$ 0 0
$$466$$ −5814.48 −0.578006
$$467$$ 10176.8 1.00840 0.504201 0.863586i $$-0.331787\pi$$
0.504201 + 0.863586i $$0.331787\pi$$
$$468$$ 0 0
$$469$$ −2117.56 −0.208486
$$470$$ −7137.71 −0.700506
$$471$$ 0 0
$$472$$ 14960.1 1.45889
$$473$$ −5649.96 −0.549229
$$474$$ 0 0
$$475$$ −3258.85 −0.314793
$$476$$ 743.121 0.0715565
$$477$$ 0 0
$$478$$ 4142.30 0.396369
$$479$$ −3431.25 −0.327302 −0.163651 0.986518i $$-0.552327\pi$$
−0.163651 + 0.986518i $$0.552327\pi$$
$$480$$ 0 0
$$481$$ 20479.1 1.94130
$$482$$ 14464.1 1.36685
$$483$$ 0 0
$$484$$ −174.052 −0.0163460
$$485$$ 3991.72 0.373721
$$486$$ 0 0
$$487$$ −2833.20 −0.263624 −0.131812 0.991275i $$-0.542079\pi$$
−0.131812 + 0.991275i $$0.542079\pi$$
$$488$$ −19013.0 −1.76368
$$489$$ 0 0
$$490$$ −3893.37 −0.358948
$$491$$ 2667.29 0.245159 0.122580 0.992459i $$-0.460883\pi$$
0.122580 + 0.992459i $$0.460883\pi$$
$$492$$ 0 0
$$493$$ −736.700 −0.0673008
$$494$$ 16402.7 1.49391
$$495$$ 0 0
$$496$$ −265.295 −0.0240164
$$497$$ −7000.18 −0.631793
$$498$$ 0 0
$$499$$ −11137.0 −0.999120 −0.499560 0.866279i $$-0.666505\pi$$
−0.499560 + 0.866279i $$0.666505\pi$$
$$500$$ −179.806 −0.0160823
$$501$$ 0 0
$$502$$ 15926.7 1.41603
$$503$$ −8780.30 −0.778319 −0.389159 0.921170i $$-0.627234\pi$$
−0.389159 + 0.921170i $$0.627234\pi$$
$$504$$ 0 0
$$505$$ −1207.00 −0.106358
$$506$$ 5232.29 0.459691
$$507$$ 0 0
$$508$$ 1363.91 0.119121
$$509$$ −13597.4 −1.18408 −0.592039 0.805910i $$-0.701677\pi$$
−0.592039 + 0.805910i $$0.701677\pi$$
$$510$$ 0 0
$$511$$ −770.969 −0.0667430
$$512$$ −12992.6 −1.12148
$$513$$ 0 0
$$514$$ −19756.6 −1.69538
$$515$$ 5841.89 0.499854
$$516$$ 0 0
$$517$$ −6130.25 −0.521486
$$518$$ −6670.30 −0.565784
$$519$$ 0 0
$$520$$ 5938.27 0.500789
$$521$$ −14001.3 −1.17736 −0.588682 0.808364i $$-0.700353\pi$$
−0.588682 + 0.808364i $$0.700353\pi$$
$$522$$ 0 0
$$523$$ −14749.8 −1.23320 −0.616602 0.787275i $$-0.711491\pi$$
−0.616602 + 0.787275i $$0.711491\pi$$
$$524$$ −2135.86 −0.178064
$$525$$ 0 0
$$526$$ −528.568 −0.0438149
$$527$$ −435.159 −0.0359693
$$528$$ 0 0
$$529$$ 22315.0 1.83406
$$530$$ 2152.53 0.176415
$$531$$ 0 0
$$532$$ 1171.21 0.0954483
$$533$$ −14662.2 −1.19154
$$534$$ 0 0
$$535$$ 10534.1 0.851269
$$536$$ 8196.40 0.660505
$$537$$ 0 0
$$538$$ −4386.59 −0.351523
$$539$$ −3343.83 −0.267215
$$540$$ 0 0
$$541$$ 1484.06 0.117939 0.0589694 0.998260i $$-0.481219\pi$$
0.0589694 + 0.998260i $$0.481219\pi$$
$$542$$ −1222.07 −0.0968493
$$543$$ 0 0
$$544$$ −5314.40 −0.418847
$$545$$ 2468.96 0.194052
$$546$$ 0 0
$$547$$ −16562.2 −1.29460 −0.647302 0.762234i $$-0.724103\pi$$
−0.647302 + 0.762234i $$0.724103\pi$$
$$548$$ 985.233 0.0768012
$$549$$ 0 0
$$550$$ 704.427 0.0546125
$$551$$ −1161.09 −0.0897716
$$552$$ 0 0
$$553$$ −1935.30 −0.148819
$$554$$ 10972.3 0.841463
$$555$$ 0 0
$$556$$ −1194.61 −0.0911197
$$557$$ 8821.52 0.671059 0.335529 0.942030i $$-0.391085\pi$$
0.335529 + 0.942030i $$0.391085\pi$$
$$558$$ 0 0
$$559$$ 25231.2 1.90906
$$560$$ −1574.77 −0.118833
$$561$$ 0 0
$$562$$ −8908.55 −0.668656
$$563$$ 5985.53 0.448064 0.224032 0.974582i $$-0.428078\pi$$
0.224032 + 0.974582i $$0.428078\pi$$
$$564$$ 0 0
$$565$$ −850.000 −0.0632916
$$566$$ −16825.0 −1.24948
$$567$$ 0 0
$$568$$ 27095.5 2.00158
$$569$$ 3453.08 0.254413 0.127206 0.991876i $$-0.459399\pi$$
0.127206 + 0.991876i $$0.459399\pi$$
$$570$$ 0 0
$$571$$ −21484.5 −1.57460 −0.787302 0.616568i $$-0.788523\pi$$
−0.787302 + 0.616568i $$0.788523\pi$$
$$572$$ 777.271 0.0568170
$$573$$ 0 0
$$574$$ 4775.67 0.347269
$$575$$ 4642.33 0.336693
$$576$$ 0 0
$$577$$ −13294.4 −0.959189 −0.479594 0.877490i $$-0.659216\pi$$
−0.479594 + 0.877490i $$0.659216\pi$$
$$578$$ 4937.82 0.355340
$$579$$ 0 0
$$580$$ −64.0627 −0.00458631
$$581$$ 6378.77 0.455483
$$582$$ 0 0
$$583$$ 1848.71 0.131330
$$584$$ 2984.18 0.211449
$$585$$ 0 0
$$586$$ −21394.8 −1.50821
$$587$$ −6695.73 −0.470805 −0.235402 0.971898i $$-0.575641\pi$$
−0.235402 + 0.971898i $$0.575641\pi$$
$$588$$ 0 0
$$589$$ −685.841 −0.0479789
$$590$$ −7925.09 −0.553002
$$591$$ 0 0
$$592$$ 21021.2 1.45940
$$593$$ 10239.6 0.709088 0.354544 0.935039i $$-0.384636\pi$$
0.354544 + 0.935039i $$0.384636\pi$$
$$594$$ 0 0
$$595$$ −2583.07 −0.177976
$$596$$ −1745.76 −0.119982
$$597$$ 0 0
$$598$$ −23366.0 −1.59784
$$599$$ 23890.8 1.62963 0.814817 0.579719i $$-0.196838\pi$$
0.814817 + 0.579719i $$0.196838\pi$$
$$600$$ 0 0
$$601$$ −11343.8 −0.769920 −0.384960 0.922933i $$-0.625785\pi$$
−0.384960 + 0.922933i $$0.625785\pi$$
$$602$$ −8218.12 −0.556388
$$603$$ 0 0
$$604$$ −44.3712 −0.00298914
$$605$$ 605.000 0.0406558
$$606$$ 0 0
$$607$$ −26032.5 −1.74074 −0.870369 0.492399i $$-0.836120\pi$$
−0.870369 + 0.492399i $$0.836120\pi$$
$$608$$ −8375.87 −0.558695
$$609$$ 0 0
$$610$$ 10072.1 0.668536
$$611$$ 27376.1 1.81263
$$612$$ 0 0
$$613$$ −4568.13 −0.300987 −0.150493 0.988611i $$-0.548086\pi$$
−0.150493 + 0.988611i $$0.548086\pi$$
$$614$$ 13791.1 0.906457
$$615$$ 0 0
$$616$$ −1661.17 −0.108653
$$617$$ 12755.9 0.832308 0.416154 0.909294i $$-0.363378\pi$$
0.416154 + 0.909294i $$0.363378\pi$$
$$618$$ 0 0
$$619$$ −1138.94 −0.0739545 −0.0369772 0.999316i $$-0.511773\pi$$
−0.0369772 + 0.999316i $$0.511773\pi$$
$$620$$ −37.8410 −0.00245118
$$621$$ 0 0
$$622$$ 4586.24 0.295645
$$623$$ 886.205 0.0569904
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −2072.49 −0.132322
$$627$$ 0 0
$$628$$ −496.607 −0.0315554
$$629$$ 34480.6 2.18574
$$630$$ 0 0
$$631$$ 7997.36 0.504548 0.252274 0.967656i $$-0.418822\pi$$
0.252274 + 0.967656i $$0.418822\pi$$
$$632$$ 7490.91 0.471475
$$633$$ 0 0
$$634$$ −27522.7 −1.72408
$$635$$ −4740.91 −0.296279
$$636$$ 0 0
$$637$$ 14932.7 0.928814
$$638$$ 250.979 0.0155742
$$639$$ 0 0
$$640$$ 4704.34 0.290555
$$641$$ 573.115 0.0353146 0.0176573 0.999844i $$-0.494379\pi$$
0.0176573 + 0.999844i $$0.494379\pi$$
$$642$$ 0 0
$$643$$ −16027.8 −0.983009 −0.491504 0.870875i $$-0.663553\pi$$
−0.491504 + 0.870875i $$0.663553\pi$$
$$644$$ −1668.42 −0.102089
$$645$$ 0 0
$$646$$ 27617.1 1.68201
$$647$$ 2622.74 0.159367 0.0796837 0.996820i $$-0.474609\pi$$
0.0796837 + 0.996820i $$0.474609\pi$$
$$648$$ 0 0
$$649$$ −6806.50 −0.411677
$$650$$ −3145.79 −0.189827
$$651$$ 0 0
$$652$$ −2763.97 −0.166020
$$653$$ −3102.00 −0.185897 −0.0929484 0.995671i $$-0.529629\pi$$
−0.0929484 + 0.995671i $$0.529629\pi$$
$$654$$ 0 0
$$655$$ 7424.19 0.442881
$$656$$ −15050.3 −0.895755
$$657$$ 0 0
$$658$$ −8916.73 −0.528283
$$659$$ 20840.2 1.23190 0.615948 0.787787i $$-0.288773\pi$$
0.615948 + 0.787787i $$0.288773\pi$$
$$660$$ 0 0
$$661$$ 18242.9 1.07348 0.536738 0.843749i $$-0.319657\pi$$
0.536738 + 0.843749i $$0.319657\pi$$
$$662$$ 8707.03 0.511191
$$663$$ 0 0
$$664$$ −24690.2 −1.44302
$$665$$ −4071.10 −0.237399
$$666$$ 0 0
$$667$$ 1654.01 0.0960171
$$668$$ 247.841 0.0143551
$$669$$ 0 0
$$670$$ −4342.03 −0.250369
$$671$$ 8650.46 0.497686
$$672$$ 0 0
$$673$$ −12746.6 −0.730084 −0.365042 0.930991i $$-0.618945\pi$$
−0.365042 + 0.930991i $$0.618945\pi$$
$$674$$ −30328.7 −1.73326
$$675$$ 0 0
$$676$$ −310.819 −0.0176843
$$677$$ 7683.11 0.436168 0.218084 0.975930i $$-0.430019\pi$$
0.218084 + 0.975930i $$0.430019\pi$$
$$678$$ 0 0
$$679$$ 4986.63 0.281840
$$680$$ 9998.23 0.563845
$$681$$ 0 0
$$682$$ 148.250 0.00832373
$$683$$ 21397.1 1.19874 0.599368 0.800473i $$-0.295418\pi$$
0.599368 + 0.800473i $$0.295418\pi$$
$$684$$ 0 0
$$685$$ −3424.64 −0.191020
$$686$$ −10351.8 −0.576140
$$687$$ 0 0
$$688$$ 25899.0 1.43516
$$689$$ −8255.84 −0.456491
$$690$$ 0 0
$$691$$ 26137.5 1.43895 0.719477 0.694516i $$-0.244381\pi$$
0.719477 + 0.694516i $$0.244381\pi$$
$$692$$ −1474.83 −0.0810181
$$693$$ 0 0
$$694$$ 5319.50 0.290959
$$695$$ 4152.41 0.226633
$$696$$ 0 0
$$697$$ −24686.7 −1.34157
$$698$$ 14952.7 0.810845
$$699$$ 0 0
$$700$$ −224.621 −0.0121284
$$701$$ −13382.4 −0.721036 −0.360518 0.932752i $$-0.617400\pi$$
−0.360518 + 0.932752i $$0.617400\pi$$
$$702$$ 0 0
$$703$$ 54343.9 2.91553
$$704$$ 6247.76 0.334476
$$705$$ 0 0
$$706$$ 6208.27 0.330951
$$707$$ −1507.83 −0.0802092
$$708$$ 0 0
$$709$$ 18164.6 0.962179 0.481090 0.876671i $$-0.340241\pi$$
0.481090 + 0.876671i $$0.340241\pi$$
$$710$$ −14353.8 −0.758715
$$711$$ 0 0
$$712$$ −3430.21 −0.180552
$$713$$ 977.000 0.0513169
$$714$$ 0 0
$$715$$ −2701.77 −0.141315
$$716$$ 2385.36 0.124504
$$717$$ 0 0
$$718$$ 9944.50 0.516888
$$719$$ −9665.62 −0.501344 −0.250672 0.968072i $$-0.580652\pi$$
−0.250672 + 0.968072i $$0.580652\pi$$
$$720$$ 0 0
$$721$$ 7297.94 0.376962
$$722$$ 25956.7 1.33796
$$723$$ 0 0
$$724$$ 2908.26 0.149288
$$725$$ 222.680 0.0114071
$$726$$ 0 0
$$727$$ −29779.6 −1.51921 −0.759605 0.650385i $$-0.774608\pi$$
−0.759605 + 0.650385i $$0.774608\pi$$
$$728$$ 7418.33 0.377667
$$729$$ 0 0
$$730$$ −1580.86 −0.0801511
$$731$$ 42481.7 2.14944
$$732$$ 0 0
$$733$$ 35029.5 1.76513 0.882567 0.470187i $$-0.155814\pi$$
0.882567 + 0.470187i $$0.155814\pi$$
$$734$$ 14515.4 0.729937
$$735$$ 0 0
$$736$$ 11931.7 0.597564
$$737$$ −3729.17 −0.186385
$$738$$ 0 0
$$739$$ 23297.3 1.15968 0.579842 0.814729i $$-0.303114\pi$$
0.579842 + 0.814729i $$0.303114\pi$$
$$740$$ 2998.40 0.148950
$$741$$ 0 0
$$742$$ 2689.03 0.133042
$$743$$ −21570.4 −1.06506 −0.532530 0.846411i $$-0.678759\pi$$
−0.532530 + 0.846411i $$0.678759\pi$$
$$744$$ 0 0
$$745$$ 6068.21 0.298419
$$746$$ 447.684 0.0219717
$$747$$ 0 0
$$748$$ 1308.69 0.0639710
$$749$$ 13159.6 0.641980
$$750$$ 0 0
$$751$$ 28554.8 1.38746 0.693729 0.720236i $$-0.255966\pi$$
0.693729 + 0.720236i $$0.255966\pi$$
$$752$$ 28100.7 1.36267
$$753$$ 0 0
$$754$$ −1120.81 −0.0541344
$$755$$ 154.233 0.00743458
$$756$$ 0 0
$$757$$ 7812.81 0.375114 0.187557 0.982254i $$-0.439943\pi$$
0.187557 + 0.982254i $$0.439943\pi$$
$$758$$ 647.268 0.0310156
$$759$$ 0 0
$$760$$ 15757.9 0.752105
$$761$$ −2875.13 −0.136956 −0.0684778 0.997653i $$-0.521814\pi$$
−0.0684778 + 0.997653i $$0.521814\pi$$
$$762$$ 0 0
$$763$$ 3084.33 0.146344
$$764$$ 2071.69 0.0981033
$$765$$ 0 0
$$766$$ 28214.3 1.33084
$$767$$ 30396.0 1.43095
$$768$$ 0 0
$$769$$ −27657.7 −1.29696 −0.648479 0.761233i $$-0.724594\pi$$
−0.648479 + 0.761233i $$0.724594\pi$$
$$770$$ 880.000 0.0411857
$$771$$ 0 0
$$772$$ 4024.84 0.187639
$$773$$ 3929.35 0.182832 0.0914160 0.995813i $$-0.470861\pi$$
0.0914160 + 0.995813i $$0.470861\pi$$
$$774$$ 0 0
$$775$$ 131.534 0.00609658
$$776$$ −19301.6 −0.892898
$$777$$ 0 0
$$778$$ −20747.0 −0.956064
$$779$$ −38908.0 −1.78950
$$780$$ 0 0
$$781$$ −12327.8 −0.564818
$$782$$ −39341.3 −1.79903
$$783$$ 0 0
$$784$$ 15327.9 0.698247
$$785$$ 1726.19 0.0784847
$$786$$ 0 0
$$787$$ −21125.7 −0.956860 −0.478430 0.878126i $$-0.658794\pi$$
−0.478430 + 0.878126i $$0.658794\pi$$
$$788$$ 660.166 0.0298445
$$789$$ 0 0
$$790$$ −3968.30 −0.178716
$$791$$ −1061.86 −0.0477310
$$792$$ 0 0
$$793$$ −38630.7 −1.72991
$$794$$ −1087.00 −0.0485847
$$795$$ 0 0
$$796$$ 3410.61 0.151867
$$797$$ −11696.3 −0.519828 −0.259914 0.965632i $$-0.583694\pi$$
−0.259914 + 0.965632i $$0.583694\pi$$
$$798$$ 0 0
$$799$$ 46093.0 2.04087
$$800$$ 1606.37 0.0709921
$$801$$ 0 0
$$802$$ 15123.9 0.665888
$$803$$ −1357.73 −0.0596678
$$804$$ 0 0
$$805$$ 5799.39 0.253915
$$806$$ −662.045 −0.0289324
$$807$$ 0 0
$$808$$ 5836.34 0.254111
$$809$$ −14310.2 −0.621902 −0.310951 0.950426i $$-0.600648\pi$$
−0.310951 + 0.950426i $$0.600648\pi$$
$$810$$ 0 0
$$811$$ 21697.9 0.939477 0.469739 0.882806i $$-0.344348\pi$$
0.469739 + 0.882806i $$0.344348\pi$$
$$812$$ −80.0299 −0.00345874
$$813$$ 0 0
$$814$$ −11746.9 −0.505807
$$815$$ 9607.46 0.412926
$$816$$ 0 0
$$817$$ 66954.1 2.86711
$$818$$ 3510.54 0.150053
$$819$$ 0 0
$$820$$ −2146.73 −0.0914233
$$821$$ −3613.00 −0.153587 −0.0767934 0.997047i $$-0.524468\pi$$
−0.0767934 + 0.997047i $$0.524468\pi$$
$$822$$ 0 0
$$823$$ −4763.98 −0.201776 −0.100888 0.994898i $$-0.532168\pi$$
−0.100888 + 0.994898i $$0.532168\pi$$
$$824$$ −28248.0 −1.19425
$$825$$ 0 0
$$826$$ −9900.36 −0.417043
$$827$$ 33571.7 1.41161 0.705806 0.708405i $$-0.250585\pi$$
0.705806 + 0.708405i $$0.250585\pi$$
$$828$$ 0 0
$$829$$ 17980.5 0.753303 0.376652 0.926355i $$-0.377075\pi$$
0.376652 + 0.926355i $$0.377075\pi$$
$$830$$ 13079.6 0.546986
$$831$$ 0 0
$$832$$ −27900.9 −1.16261
$$833$$ 25142.1 1.04576
$$834$$ 0 0
$$835$$ −861.486 −0.0357041
$$836$$ 2062.58 0.0853301
$$837$$ 0 0
$$838$$ −3249.91 −0.133969
$$839$$ 40139.5 1.65169 0.825847 0.563895i $$-0.190698\pi$$
0.825847 + 0.563895i $$0.190698\pi$$
$$840$$ 0 0
$$841$$ −24309.7 −0.996747
$$842$$ −31358.4 −1.28347
$$843$$ 0 0
$$844$$ −6213.91 −0.253426
$$845$$ 1080.40 0.0439844
$$846$$ 0 0
$$847$$ 755.792 0.0306603
$$848$$ −8474.36 −0.343173
$$849$$ 0 0
$$850$$ −5296.54 −0.213729
$$851$$ −77414.4 −3.11837
$$852$$ 0 0
$$853$$ −15369.2 −0.616919 −0.308459 0.951237i $$-0.599813\pi$$
−0.308459 + 0.951237i $$0.599813\pi$$
$$854$$ 12582.5 0.504173
$$855$$ 0 0
$$856$$ −50936.8 −2.03386
$$857$$ −10324.9 −0.411541 −0.205770 0.978600i $$-0.565970\pi$$
−0.205770 + 0.978600i $$0.565970\pi$$
$$858$$ 0 0
$$859$$ −27112.5 −1.07691 −0.538455 0.842655i $$-0.680992\pi$$
−0.538455 + 0.842655i $$0.680992\pi$$
$$860$$ 3694.17 0.146477
$$861$$ 0 0
$$862$$ −20620.8 −0.814787
$$863$$ −30463.6 −1.20161 −0.600807 0.799394i $$-0.705154\pi$$
−0.600807 + 0.799394i $$0.705154\pi$$
$$864$$ 0 0
$$865$$ 5126.46 0.201508
$$866$$ −42434.0 −1.66509
$$867$$ 0 0
$$868$$ −47.2726 −0.00184854
$$869$$ −3408.19 −0.133044
$$870$$ 0 0
$$871$$ 16653.5 0.647855
$$872$$ −11938.4 −0.463631
$$873$$ 0 0
$$874$$ −62004.6 −2.39970
$$875$$ 780.776 0.0301658
$$876$$ 0 0
$$877$$ −5086.12 −0.195833 −0.0979167 0.995195i $$-0.531218\pi$$
−0.0979167 + 0.995195i $$0.531218\pi$$
$$878$$ 12054.2 0.463335
$$879$$ 0 0
$$880$$ −2773.28 −0.106236
$$881$$ 10625.5 0.406338 0.203169 0.979144i $$-0.434876\pi$$
0.203169 + 0.979144i $$0.434876\pi$$
$$882$$ 0 0
$$883$$ 13112.2 0.499728 0.249864 0.968281i $$-0.419614\pi$$
0.249864 + 0.968281i $$0.419614\pi$$
$$884$$ −5844.25 −0.222357
$$885$$ 0 0
$$886$$ −38664.0 −1.46607
$$887$$ 14442.8 0.546719 0.273360 0.961912i $$-0.411865\pi$$
0.273360 + 0.961912i $$0.411865\pi$$
$$888$$ 0 0
$$889$$ −5922.54 −0.223437
$$890$$ 1817.15 0.0684393
$$891$$ 0 0
$$892$$ 5520.38 0.207215
$$893$$ 72645.8 2.72228
$$894$$ 0 0
$$895$$ −8291.44 −0.309667
$$896$$ 5876.86 0.219121
$$897$$ 0 0
$$898$$ −2493.61 −0.0926648
$$899$$ 46.8641 0.00173860
$$900$$ 0 0
$$901$$ −13900.3 −0.513970
$$902$$ 8410.27 0.310456
$$903$$ 0 0
$$904$$ 4110.10 0.151217
$$905$$ −10109.0 −0.371310
$$906$$ 0 0
$$907$$ −44981.9 −1.64675 −0.823374 0.567499i $$-0.807911\pi$$
−0.823374 + 0.567499i $$0.807911\pi$$
$$908$$ −7197.57 −0.263062
$$909$$ 0 0
$$910$$ −3929.85 −0.143157
$$911$$ −6841.96 −0.248830 −0.124415 0.992230i $$-0.539705\pi$$
−0.124415 + 0.992230i $$0.539705\pi$$
$$912$$ 0 0
$$913$$ 11233.4 0.407199
$$914$$ 160.501 0.00580843
$$915$$ 0 0
$$916$$ 398.989 0.0143919
$$917$$ 9274.61 0.333996
$$918$$ 0 0
$$919$$ −4753.54 −0.170625 −0.0853127 0.996354i $$-0.527189\pi$$
−0.0853127 + 0.996354i $$0.527189\pi$$
$$920$$ −22447.6 −0.804430
$$921$$ 0 0
$$922$$ 30395.9 1.08572
$$923$$ 55052.7 1.96325
$$924$$ 0 0
$$925$$ −10422.3 −0.370470
$$926$$ −33671.2 −1.19493
$$927$$ 0 0
$$928$$ 572.330 0.0202453
$$929$$ 7507.93 0.265153 0.132576 0.991173i $$-0.457675\pi$$
0.132576 + 0.991173i $$0.457675\pi$$
$$930$$ 0 0
$$931$$ 39625.7 1.39493
$$932$$ 3265.14 0.114757
$$933$$ 0 0
$$934$$ 26068.3 0.913256
$$935$$ −4548.96 −0.159109
$$936$$ 0 0
$$937$$ 8540.47 0.297764 0.148882 0.988855i $$-0.452433\pi$$
0.148882 + 0.988855i $$0.452433\pi$$
$$938$$ −5424.24 −0.188814
$$939$$ 0 0
$$940$$ 4008.20 0.139078
$$941$$ −9101.13 −0.315290 −0.157645 0.987496i $$-0.550390\pi$$
−0.157645 + 0.987496i $$0.550390\pi$$
$$942$$ 0 0
$$943$$ 55425.5 1.91400
$$944$$ 31200.6 1.07573
$$945$$ 0 0
$$946$$ −14472.7 −0.497407
$$947$$ −47540.0 −1.63130 −0.815650 0.578546i $$-0.803620\pi$$
−0.815650 + 0.578546i $$0.803620\pi$$
$$948$$ 0 0
$$949$$ 6063.26 0.207399
$$950$$ −8347.73 −0.285091
$$951$$ 0 0
$$952$$ 12490.2 0.425221
$$953$$ −47370.7 −1.61016 −0.805082 0.593164i $$-0.797879\pi$$
−0.805082 + 0.593164i $$0.797879\pi$$
$$954$$ 0 0
$$955$$ −7201.12 −0.244003
$$956$$ −2326.12 −0.0786947
$$957$$ 0 0
$$958$$ −8789.33 −0.296420
$$959$$ −4278.20 −0.144057
$$960$$ 0 0
$$961$$ −29763.3 −0.999071
$$962$$ 52458.4 1.75813
$$963$$ 0 0
$$964$$ −8122.38 −0.271374
$$965$$ −13990.2 −0.466696
$$966$$ 0 0
$$967$$ −36171.6 −1.20290 −0.601448 0.798912i $$-0.705409\pi$$
−0.601448 + 0.798912i $$0.705409\pi$$
$$968$$ −2925.43 −0.0971351
$$969$$ 0 0
$$970$$ 10225.0 0.338459
$$971$$ 31713.2 1.04812 0.524060 0.851681i $$-0.324417\pi$$
0.524060 + 0.851681i $$0.324417\pi$$
$$972$$ 0 0
$$973$$ 5187.37 0.170914
$$974$$ −7257.40 −0.238750
$$975$$ 0 0
$$976$$ −39653.1 −1.30048
$$977$$ −22800.5 −0.746626 −0.373313 0.927706i $$-0.621778\pi$$
−0.373313 + 0.927706i $$0.621778\pi$$
$$978$$ 0 0
$$979$$ 1560.67 0.0509490
$$980$$ 2186.33 0.0712651
$$981$$ 0 0
$$982$$ 6832.41 0.222027
$$983$$ 44597.4 1.44704 0.723518 0.690305i $$-0.242524\pi$$
0.723518 + 0.690305i $$0.242524\pi$$
$$984$$ 0 0
$$985$$ −2294.72 −0.0742292
$$986$$ −1887.10 −0.0609507
$$987$$ 0 0
$$988$$ −9210.95 −0.296599
$$989$$ −95378.1 −3.06658
$$990$$ 0 0
$$991$$ −34788.1 −1.11512 −0.557558 0.830138i $$-0.688261\pi$$
−0.557558 + 0.830138i $$0.688261\pi$$
$$992$$ 338.068 0.0108202
$$993$$ 0 0
$$994$$ −17931.3 −0.572180
$$995$$ −11855.2 −0.377723
$$996$$ 0 0
$$997$$ −20360.5 −0.646765 −0.323383 0.946268i $$-0.604820\pi$$
−0.323383 + 0.946268i $$0.604820\pi$$
$$998$$ −28528.0 −0.904849
$$999$$ 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 495.4.a.d.1.2 2
3.2 odd 2 165.4.a.c.1.1 2
5.4 even 2 2475.4.a.n.1.1 2
15.2 even 4 825.4.c.j.199.1 4
15.8 even 4 825.4.c.j.199.4 4
15.14 odd 2 825.4.a.m.1.2 2
33.32 even 2 1815.4.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.1 2 3.2 odd 2
495.4.a.d.1.2 2 1.1 even 1 trivial
825.4.a.m.1.2 2 15.14 odd 2
825.4.c.j.199.1 4 15.2 even 4
825.4.c.j.199.4 4 15.8 even 4
1815.4.a.n.1.2 2 33.32 even 2
2475.4.a.n.1.1 2 5.4 even 2