# Properties

 Label 2205.4.a.ba.1.2 Level $2205$ Weight $4$ Character 2205.1 Self dual yes Analytic conductor $130.099$ 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] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.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: $$130.099211563$$ 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 315) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2205.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.56155 q^{2} -1.43845 q^{4} -5.00000 q^{5} -24.1771 q^{8} +O(q^{10})$$ $$q+2.56155 q^{2} -1.43845 q^{4} -5.00000 q^{5} -24.1771 q^{8} -12.8078 q^{10} +6.24621 q^{11} +56.3542 q^{13} -50.4233 q^{16} -24.6004 q^{17} +90.7083 q^{19} +7.19224 q^{20} +16.0000 q^{22} -69.8617 q^{23} +25.0000 q^{25} +144.354 q^{26} -228.847 q^{29} +67.8920 q^{31} +64.2547 q^{32} -63.0152 q^{34} -58.8466 q^{37} +232.354 q^{38} +120.885 q^{40} -19.2007 q^{41} +365.218 q^{43} -8.98485 q^{44} -178.955 q^{46} +195.153 q^{47} +64.0388 q^{50} -81.0625 q^{52} -511.201 q^{53} -31.2311 q^{55} -586.203 q^{58} +284.000 q^{59} +123.460 q^{61} +173.909 q^{62} +567.978 q^{64} -281.771 q^{65} +144.968 q^{67} +35.3863 q^{68} -73.0284 q^{71} -638.850 q^{73} -150.739 q^{74} -130.479 q^{76} +976.189 q^{79} +252.116 q^{80} -49.1837 q^{82} +484.466 q^{83} +123.002 q^{85} +935.525 q^{86} -151.015 q^{88} -1017.30 q^{89} +100.492 q^{92} +499.896 q^{94} -453.542 q^{95} -1806.67 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 7 q^{4} - 10 q^{5} - 3 q^{8}+O(q^{10})$$ 2 * q + q^2 - 7 * q^4 - 10 * q^5 - 3 * q^8 $$2 q + q^{2} - 7 q^{4} - 10 q^{5} - 3 q^{8} - 5 q^{10} - 4 q^{11} + 22 q^{13} - 39 q^{16} + 58 q^{17} + 35 q^{20} + 32 q^{22} - 82 q^{23} + 50 q^{25} + 198 q^{26} - 334 q^{29} + 210 q^{31} - 123 q^{32} - 192 q^{34} + 6 q^{37} + 374 q^{38} + 15 q^{40} + 176 q^{41} + 46 q^{43} + 48 q^{44} - 160 q^{46} + 514 q^{47} + 25 q^{50} + 110 q^{52} - 808 q^{53} + 20 q^{55} - 422 q^{58} + 568 q^{59} + 618 q^{61} - 48 q^{62} + 769 q^{64} - 110 q^{65} + 694 q^{67} - 424 q^{68} - 814 q^{71} - 82 q^{73} - 252 q^{74} + 374 q^{76} + 600 q^{79} + 195 q^{80} - 354 q^{82} - 268 q^{83} - 290 q^{85} + 1434 q^{86} - 368 q^{88} - 72 q^{89} + 168 q^{92} + 2 q^{94} - 1626 q^{97}+O(q^{100})$$ 2 * q + q^2 - 7 * q^4 - 10 * q^5 - 3 * q^8 - 5 * q^10 - 4 * q^11 + 22 * q^13 - 39 * q^16 + 58 * q^17 + 35 * q^20 + 32 * q^22 - 82 * q^23 + 50 * q^25 + 198 * q^26 - 334 * q^29 + 210 * q^31 - 123 * q^32 - 192 * q^34 + 6 * q^37 + 374 * q^38 + 15 * q^40 + 176 * q^41 + 46 * q^43 + 48 * q^44 - 160 * q^46 + 514 * q^47 + 25 * q^50 + 110 * q^52 - 808 * q^53 + 20 * q^55 - 422 * q^58 + 568 * q^59 + 618 * q^61 - 48 * q^62 + 769 * q^64 - 110 * q^65 + 694 * q^67 - 424 * q^68 - 814 * q^71 - 82 * q^73 - 252 * q^74 + 374 * q^76 + 600 * q^79 + 195 * q^80 - 354 * q^82 - 268 * q^83 - 290 * q^85 + 1434 * q^86 - 368 * q^88 - 72 * q^89 + 168 * q^92 + 2 * q^94 - 1626 * q^97

## 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
$$7$$ 0 0
$$8$$ −24.1771 −1.06849
$$9$$ 0 0
$$10$$ −12.8078 −0.405017
$$11$$ 6.24621 0.171209 0.0856047 0.996329i $$-0.472718\pi$$
0.0856047 + 0.996329i $$0.472718\pi$$
$$12$$ 0 0
$$13$$ 56.3542 1.20229 0.601147 0.799138i $$-0.294710\pi$$
0.601147 + 0.799138i $$0.294710\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −50.4233 −0.787864
$$17$$ −24.6004 −0.350969 −0.175484 0.984482i $$-0.556149\pi$$
−0.175484 + 0.984482i $$0.556149\pi$$
$$18$$ 0 0
$$19$$ 90.7083 1.09526 0.547629 0.836721i $$-0.315530\pi$$
0.547629 + 0.836721i $$0.315530\pi$$
$$20$$ 7.19224 0.0804116
$$21$$ 0 0
$$22$$ 16.0000 0.155055
$$23$$ −69.8617 −0.633356 −0.316678 0.948533i $$-0.602567\pi$$
−0.316678 + 0.948533i $$0.602567\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 144.354 1.08885
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −228.847 −1.46537 −0.732685 0.680568i $$-0.761733\pi$$
−0.732685 + 0.680568i $$0.761733\pi$$
$$30$$ 0 0
$$31$$ 67.8920 0.393347 0.196674 0.980469i $$-0.436986\pi$$
0.196674 + 0.980469i $$0.436986\pi$$
$$32$$ 64.2547 0.354961
$$33$$ 0 0
$$34$$ −63.0152 −0.317853
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −58.8466 −0.261468 −0.130734 0.991417i $$-0.541733\pi$$
−0.130734 + 0.991417i $$0.541733\pi$$
$$38$$ 232.354 0.991916
$$39$$ 0 0
$$40$$ 120.885 0.477842
$$41$$ −19.2007 −0.0731379 −0.0365689 0.999331i $$-0.511643\pi$$
−0.0365689 + 0.999331i $$0.511643\pi$$
$$42$$ 0 0
$$43$$ 365.218 1.29524 0.647618 0.761965i $$-0.275765\pi$$
0.647618 + 0.761965i $$0.275765\pi$$
$$44$$ −8.98485 −0.0307845
$$45$$ 0 0
$$46$$ −178.955 −0.573596
$$47$$ 195.153 0.605661 0.302830 0.953044i $$-0.402068\pi$$
0.302830 + 0.953044i $$0.402068\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 64.0388 0.181129
$$51$$ 0 0
$$52$$ −81.0625 −0.216180
$$53$$ −511.201 −1.32488 −0.662442 0.749113i $$-0.730480\pi$$
−0.662442 + 0.749113i $$0.730480\pi$$
$$54$$ 0 0
$$55$$ −31.2311 −0.0765672
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −586.203 −1.32711
$$59$$ 284.000 0.626672 0.313336 0.949642i $$-0.398553\pi$$
0.313336 + 0.949642i $$0.398553\pi$$
$$60$$ 0 0
$$61$$ 123.460 0.259139 0.129569 0.991570i $$-0.458641\pi$$
0.129569 + 0.991570i $$0.458641\pi$$
$$62$$ 173.909 0.356233
$$63$$ 0 0
$$64$$ 567.978 1.10933
$$65$$ −281.771 −0.537683
$$66$$ 0 0
$$67$$ 144.968 0.264338 0.132169 0.991227i $$-0.457806\pi$$
0.132169 + 0.991227i $$0.457806\pi$$
$$68$$ 35.3863 0.0631062
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −73.0284 −0.122069 −0.0610344 0.998136i $$-0.519440\pi$$
−0.0610344 + 0.998136i $$0.519440\pi$$
$$72$$ 0 0
$$73$$ −638.850 −1.02427 −0.512135 0.858905i $$-0.671145\pi$$
−0.512135 + 0.858905i $$0.671145\pi$$
$$74$$ −150.739 −0.236797
$$75$$ 0 0
$$76$$ −130.479 −0.196934
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 976.189 1.39025 0.695126 0.718888i $$-0.255349\pi$$
0.695126 + 0.718888i $$0.255349\pi$$
$$80$$ 252.116 0.352343
$$81$$ 0 0
$$82$$ −49.1837 −0.0662370
$$83$$ 484.466 0.640687 0.320344 0.947301i $$-0.396202\pi$$
0.320344 + 0.947301i $$0.396202\pi$$
$$84$$ 0 0
$$85$$ 123.002 0.156958
$$86$$ 935.525 1.17303
$$87$$ 0 0
$$88$$ −151.015 −0.182935
$$89$$ −1017.30 −1.21161 −0.605806 0.795612i $$-0.707149\pi$$
−0.605806 + 0.795612i $$0.707149\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 100.492 0.113881
$$93$$ 0 0
$$94$$ 499.896 0.548514
$$95$$ −453.542 −0.489815
$$96$$ 0 0
$$97$$ −1806.67 −1.89113 −0.945564 0.325437i $$-0.894489\pi$$
−0.945564 + 0.325437i $$0.894489\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −35.9612 −0.0359612
$$101$$ −483.053 −0.475897 −0.237948 0.971278i $$-0.576475\pi$$
−0.237948 + 0.971278i $$0.576475\pi$$
$$102$$ 0 0
$$103$$ 339.049 0.324345 0.162172 0.986762i $$-0.448150\pi$$
0.162172 + 0.986762i $$0.448150\pi$$
$$104$$ −1362.48 −1.28464
$$105$$ 0 0
$$106$$ −1309.47 −1.19987
$$107$$ 450.847 0.407336 0.203668 0.979040i $$-0.434714\pi$$
0.203668 + 0.979040i $$0.434714\pi$$
$$108$$ 0 0
$$109$$ −1841.70 −1.61838 −0.809189 0.587548i $$-0.800093\pi$$
−0.809189 + 0.587548i $$0.800093\pi$$
$$110$$ −80.0000 −0.0693427
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1874.72 −1.56069 −0.780347 0.625347i $$-0.784958\pi$$
−0.780347 + 0.625347i $$0.784958\pi$$
$$114$$ 0 0
$$115$$ 349.309 0.283245
$$116$$ 329.184 0.263482
$$117$$ 0 0
$$118$$ 727.481 0.567543
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1291.98 −0.970687
$$122$$ 316.250 0.234688
$$123$$ 0 0
$$124$$ −97.6591 −0.0707262
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −38.0984 −0.0266196 −0.0133098 0.999911i $$-0.504237\pi$$
−0.0133098 + 0.999911i $$0.504237\pi$$
$$128$$ 940.868 0.649702
$$129$$ 0 0
$$130$$ −721.771 −0.486950
$$131$$ −1551.51 −1.03478 −0.517389 0.855750i $$-0.673096\pi$$
−0.517389 + 0.855750i $$0.673096\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 371.343 0.239396
$$135$$ 0 0
$$136$$ 594.765 0.375005
$$137$$ 1203.24 0.750364 0.375182 0.926951i $$-0.377580\pi$$
0.375182 + 0.926951i $$0.377580\pi$$
$$138$$ 0 0
$$139$$ −1897.00 −1.15756 −0.578781 0.815483i $$-0.696471\pi$$
−0.578781 + 0.815483i $$0.696471\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −187.066 −0.110551
$$143$$ 352.000 0.205844
$$144$$ 0 0
$$145$$ 1144.23 0.655334
$$146$$ −1636.45 −0.927626
$$147$$ 0 0
$$148$$ 84.6477 0.0470135
$$149$$ −704.888 −0.387562 −0.193781 0.981045i $$-0.562075\pi$$
−0.193781 + 0.981045i $$0.562075\pi$$
$$150$$ 0 0
$$151$$ −3035.21 −1.63578 −0.817888 0.575378i $$-0.804855\pi$$
−0.817888 + 0.575378i $$0.804855\pi$$
$$152$$ −2193.06 −1.17027
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −339.460 −0.175910
$$156$$ 0 0
$$157$$ −2713.65 −1.37944 −0.689722 0.724074i $$-0.742267\pi$$
−0.689722 + 0.724074i $$0.742267\pi$$
$$158$$ 2500.56 1.25908
$$159$$ 0 0
$$160$$ −321.274 −0.158743
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −465.259 −0.223570 −0.111785 0.993732i $$-0.535657\pi$$
−0.111785 + 0.993732i $$0.535657\pi$$
$$164$$ 27.6193 0.0131506
$$165$$ 0 0
$$166$$ 1240.98 0.580236
$$167$$ −4156.06 −1.92578 −0.962891 0.269892i $$-0.913012\pi$$
−0.962891 + 0.269892i $$0.913012\pi$$
$$168$$ 0 0
$$169$$ 978.792 0.445513
$$170$$ 315.076 0.142148
$$171$$ 0 0
$$172$$ −525.346 −0.232891
$$173$$ 4241.17 1.86387 0.931936 0.362622i $$-0.118119\pi$$
0.931936 + 0.362622i $$0.118119\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −314.955 −0.134890
$$177$$ 0 0
$$178$$ −2605.87 −1.09729
$$179$$ −2940.35 −1.22778 −0.613889 0.789392i $$-0.710396\pi$$
−0.613889 + 0.789392i $$0.710396\pi$$
$$180$$ 0 0
$$181$$ 1986.35 0.815716 0.407858 0.913045i $$-0.366276\pi$$
0.407858 + 0.913045i $$0.366276\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1689.05 0.676732
$$185$$ 294.233 0.116932
$$186$$ 0 0
$$187$$ −153.659 −0.0600891
$$188$$ −280.718 −0.108901
$$189$$ 0 0
$$190$$ −1161.77 −0.443598
$$191$$ 1615.88 0.612150 0.306075 0.952007i $$-0.400984\pi$$
0.306075 + 0.952007i $$0.400984\pi$$
$$192$$ 0 0
$$193$$ −2052.44 −0.765482 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$194$$ −4627.88 −1.71269
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4468.58 −1.61611 −0.808054 0.589108i $$-0.799479\pi$$
−0.808054 + 0.589108i $$0.799479\pi$$
$$198$$ 0 0
$$199$$ −1543.07 −0.549675 −0.274838 0.961491i $$-0.588624\pi$$
−0.274838 + 0.961491i $$0.588624\pi$$
$$200$$ −604.427 −0.213697
$$201$$ 0 0
$$202$$ −1237.37 −0.430994
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 96.0037 0.0327083
$$206$$ 868.492 0.293741
$$207$$ 0 0
$$208$$ −2841.56 −0.947245
$$209$$ 566.583 0.187519
$$210$$ 0 0
$$211$$ 1284.02 0.418937 0.209469 0.977815i $$-0.432827\pi$$
0.209469 + 0.977815i $$0.432827\pi$$
$$212$$ 735.335 0.238222
$$213$$ 0 0
$$214$$ 1154.87 0.368902
$$215$$ −1826.09 −0.579248
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −4717.62 −1.46568
$$219$$ 0 0
$$220$$ 44.9242 0.0137672
$$221$$ −1386.33 −0.421968
$$222$$ 0 0
$$223$$ −3815.31 −1.14571 −0.572853 0.819658i $$-0.694163\pi$$
−0.572853 + 0.819658i $$0.694163\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −4802.18 −1.41344
$$227$$ 2271.53 0.664172 0.332086 0.943249i $$-0.392248\pi$$
0.332086 + 0.943249i $$0.392248\pi$$
$$228$$ 0 0
$$229$$ 2367.54 0.683195 0.341598 0.939846i $$-0.389032\pi$$
0.341598 + 0.939846i $$0.389032\pi$$
$$230$$ 894.773 0.256520
$$231$$ 0 0
$$232$$ 5532.84 1.56573
$$233$$ 1617.71 0.454849 0.227425 0.973796i $$-0.426970\pi$$
0.227425 + 0.973796i $$0.426970\pi$$
$$234$$ 0 0
$$235$$ −975.767 −0.270860
$$236$$ −408.519 −0.112679
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1935.29 −0.523781 −0.261891 0.965098i $$-0.584346\pi$$
−0.261891 + 0.965098i $$0.584346\pi$$
$$240$$ 0 0
$$241$$ −477.901 −0.127736 −0.0638679 0.997958i $$-0.520344\pi$$
−0.0638679 + 0.997958i $$0.520344\pi$$
$$242$$ −3309.49 −0.879099
$$243$$ 0 0
$$244$$ −177.591 −0.0465947
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5111.79 1.31682
$$248$$ −1641.43 −0.420286
$$249$$ 0 0
$$250$$ −320.194 −0.0810034
$$251$$ 4769.70 1.19944 0.599722 0.800208i $$-0.295278\pi$$
0.599722 + 0.800208i $$0.295278\pi$$
$$252$$ 0 0
$$253$$ −436.371 −0.108436
$$254$$ −97.5910 −0.0241079
$$255$$ 0 0
$$256$$ −2133.74 −0.520933
$$257$$ 682.524 0.165660 0.0828302 0.996564i $$-0.473604\pi$$
0.0828302 + 0.996564i $$0.473604\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 405.312 0.0966785
$$261$$ 0 0
$$262$$ −3974.27 −0.937142
$$263$$ −3029.11 −0.710202 −0.355101 0.934828i $$-0.615553\pi$$
−0.355101 + 0.934828i $$0.615553\pi$$
$$264$$ 0 0
$$265$$ 2556.00 0.592506
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −208.529 −0.0475295
$$269$$ 6187.33 1.40241 0.701205 0.712960i $$-0.252646\pi$$
0.701205 + 0.712960i $$0.252646\pi$$
$$270$$ 0 0
$$271$$ 7558.90 1.69436 0.847178 0.531309i $$-0.178300\pi$$
0.847178 + 0.531309i $$0.178300\pi$$
$$272$$ 1240.43 0.276516
$$273$$ 0 0
$$274$$ 3082.17 0.679564
$$275$$ 156.155 0.0342419
$$276$$ 0 0
$$277$$ 3685.36 0.799393 0.399697 0.916647i $$-0.369115\pi$$
0.399697 + 0.916647i $$0.369115\pi$$
$$278$$ −4859.26 −1.04834
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −7969.11 −1.69180 −0.845902 0.533338i $$-0.820937\pi$$
−0.845902 + 0.533338i $$0.820937\pi$$
$$282$$ 0 0
$$283$$ 2479.73 0.520864 0.260432 0.965492i $$-0.416135\pi$$
0.260432 + 0.965492i $$0.416135\pi$$
$$284$$ 105.048 0.0219487
$$285$$ 0 0
$$286$$ 901.667 0.186422
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4307.82 −0.876821
$$290$$ 2931.01 0.593500
$$291$$ 0 0
$$292$$ 918.952 0.184170
$$293$$ −5950.02 −1.18636 −0.593181 0.805069i $$-0.702128\pi$$
−0.593181 + 0.805069i $$0.702128\pi$$
$$294$$ 0 0
$$295$$ −1420.00 −0.280256
$$296$$ 1422.74 0.279375
$$297$$ 0 0
$$298$$ −1805.61 −0.350994
$$299$$ −3937.00 −0.761480
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −7774.86 −1.48143
$$303$$ 0 0
$$304$$ −4573.81 −0.862915
$$305$$ −617.301 −0.115890
$$306$$ 0 0
$$307$$ 3129.90 0.581865 0.290933 0.956744i $$-0.406034\pi$$
0.290933 + 0.956744i $$0.406034\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −869.545 −0.159312
$$311$$ 7261.25 1.32395 0.661973 0.749527i $$-0.269719\pi$$
0.661973 + 0.749527i $$0.269719\pi$$
$$312$$ 0 0
$$313$$ 2310.83 0.417303 0.208652 0.977990i $$-0.433093\pi$$
0.208652 + 0.977990i $$0.433093\pi$$
$$314$$ −6951.16 −1.24929
$$315$$ 0 0
$$316$$ −1404.20 −0.249975
$$317$$ −4701.40 −0.832987 −0.416494 0.909139i $$-0.636741\pi$$
−0.416494 + 0.909139i $$0.636741\pi$$
$$318$$ 0 0
$$319$$ −1429.42 −0.250885
$$320$$ −2839.89 −0.496109
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2231.46 −0.384401
$$324$$ 0 0
$$325$$ 1408.85 0.240459
$$326$$ −1191.79 −0.202475
$$327$$ 0 0
$$328$$ 464.218 0.0781468
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1366.18 0.226864 0.113432 0.993546i $$-0.463816\pi$$
0.113432 + 0.993546i $$0.463816\pi$$
$$332$$ −696.879 −0.115199
$$333$$ 0 0
$$334$$ −10646.0 −1.74408
$$335$$ −724.839 −0.118215
$$336$$ 0 0
$$337$$ −740.632 −0.119718 −0.0598588 0.998207i $$-0.519065\pi$$
−0.0598588 + 0.998207i $$0.519065\pi$$
$$338$$ 2507.23 0.403477
$$339$$ 0 0
$$340$$ −176.932 −0.0282220
$$341$$ 424.068 0.0673448
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −8829.90 −1.38394
$$345$$ 0 0
$$346$$ 10864.0 1.68801
$$347$$ 2605.56 0.403094 0.201547 0.979479i $$-0.435403\pi$$
0.201547 + 0.979479i $$0.435403\pi$$
$$348$$ 0 0
$$349$$ −4665.07 −0.715517 −0.357758 0.933814i $$-0.616459\pi$$
−0.357758 + 0.933814i $$0.616459\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 401.349 0.0607726
$$353$$ 2964.75 0.447019 0.223509 0.974702i $$-0.428249\pi$$
0.223509 + 0.974702i $$0.428249\pi$$
$$354$$ 0 0
$$355$$ 365.142 0.0545908
$$356$$ 1463.33 0.217855
$$357$$ 0 0
$$358$$ −7531.87 −1.11193
$$359$$ −4267.55 −0.627389 −0.313695 0.949524i $$-0.601567\pi$$
−0.313695 + 0.949524i $$0.601567\pi$$
$$360$$ 0 0
$$361$$ 1369.00 0.199592
$$362$$ 5088.15 0.738749
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3194.25 0.458068
$$366$$ 0 0
$$367$$ 9280.33 1.31997 0.659985 0.751279i $$-0.270562\pi$$
0.659985 + 0.751279i $$0.270562\pi$$
$$368$$ 3522.66 0.498998
$$369$$ 0 0
$$370$$ 753.693 0.105899
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10781.1 1.49657 0.748286 0.663376i $$-0.230877\pi$$
0.748286 + 0.663376i $$0.230877\pi$$
$$374$$ −393.606 −0.0544195
$$375$$ 0 0
$$376$$ −4718.24 −0.647140
$$377$$ −12896.5 −1.76181
$$378$$ 0 0
$$379$$ −5914.16 −0.801557 −0.400779 0.916175i $$-0.631260\pi$$
−0.400779 + 0.916175i $$0.631260\pi$$
$$380$$ 652.396 0.0880716
$$381$$ 0 0
$$382$$ 4139.15 0.554391
$$383$$ −11513.9 −1.53612 −0.768059 0.640379i $$-0.778777\pi$$
−0.768059 + 0.640379i $$0.778777\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5257.44 −0.693256
$$387$$ 0 0
$$388$$ 2598.80 0.340036
$$389$$ −5399.73 −0.703797 −0.351898 0.936038i $$-0.614464\pi$$
−0.351898 + 0.936038i $$0.614464\pi$$
$$390$$ 0 0
$$391$$ 1718.62 0.222288
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −11446.5 −1.46362
$$395$$ −4880.95 −0.621739
$$396$$ 0 0
$$397$$ −2622.13 −0.331488 −0.165744 0.986169i $$-0.553003\pi$$
−0.165744 + 0.986169i $$0.553003\pi$$
$$398$$ −3952.66 −0.497811
$$399$$ 0 0
$$400$$ −1260.58 −0.157573
$$401$$ 11119.1 1.38469 0.692344 0.721568i $$-0.256578\pi$$
0.692344 + 0.721568i $$0.256578\pi$$
$$402$$ 0 0
$$403$$ 3826.00 0.472920
$$404$$ 694.846 0.0855690
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −367.568 −0.0447658
$$408$$ 0 0
$$409$$ 6589.18 0.796611 0.398305 0.917253i $$-0.369598\pi$$
0.398305 + 0.917253i $$0.369598\pi$$
$$410$$ 245.919 0.0296221
$$411$$ 0 0
$$412$$ −487.704 −0.0583191
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −2422.33 −0.286524
$$416$$ 3621.02 0.426767
$$417$$ 0 0
$$418$$ 1451.33 0.169825
$$419$$ −11871.6 −1.38416 −0.692081 0.721820i $$-0.743306\pi$$
−0.692081 + 0.721820i $$0.743306\pi$$
$$420$$ 0 0
$$421$$ −1731.57 −0.200455 −0.100227 0.994965i $$-0.531957\pi$$
−0.100227 + 0.994965i $$0.531957\pi$$
$$422$$ 3289.09 0.379409
$$423$$ 0 0
$$424$$ 12359.3 1.41562
$$425$$ −615.009 −0.0701937
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −648.519 −0.0732415
$$429$$ 0 0
$$430$$ −4677.62 −0.524593
$$431$$ −10653.8 −1.19066 −0.595330 0.803481i $$-0.702979\pi$$
−0.595330 + 0.803481i $$0.702979\pi$$
$$432$$ 0 0
$$433$$ −2642.01 −0.293226 −0.146613 0.989194i $$-0.546837\pi$$
−0.146613 + 0.989194i $$0.546837\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2649.19 0.290994
$$437$$ −6337.04 −0.693688
$$438$$ 0 0
$$439$$ −8858.21 −0.963051 −0.481525 0.876432i $$-0.659917\pi$$
−0.481525 + 0.876432i $$0.659917\pi$$
$$440$$ 755.076 0.0818110
$$441$$ 0 0
$$442$$ −3551.17 −0.382153
$$443$$ 6621.73 0.710176 0.355088 0.934833i $$-0.384451\pi$$
0.355088 + 0.934833i $$0.384451\pi$$
$$444$$ 0 0
$$445$$ 5086.50 0.541849
$$446$$ −9773.13 −1.03760
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −13081.7 −1.37497 −0.687487 0.726196i $$-0.741286\pi$$
−0.687487 + 0.726196i $$0.741286\pi$$
$$450$$ 0 0
$$451$$ −119.932 −0.0125219
$$452$$ 2696.68 0.280622
$$453$$ 0 0
$$454$$ 5818.65 0.601504
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12167.0 1.24540 0.622699 0.782461i $$-0.286036\pi$$
0.622699 + 0.782461i $$0.286036\pi$$
$$458$$ 6064.59 0.618733
$$459$$ 0 0
$$460$$ −502.462 −0.0509292
$$461$$ 11283.8 1.14000 0.570000 0.821645i $$-0.306943\pi$$
0.570000 + 0.821645i $$0.306943\pi$$
$$462$$ 0 0
$$463$$ 16542.9 1.66051 0.830254 0.557385i $$-0.188195\pi$$
0.830254 + 0.557385i $$0.188195\pi$$
$$464$$ 11539.2 1.15451
$$465$$ 0 0
$$466$$ 4143.85 0.411932
$$467$$ −10266.9 −1.01734 −0.508668 0.860963i $$-0.669862\pi$$
−0.508668 + 0.860963i $$0.669862\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −2499.48 −0.245303
$$471$$ 0 0
$$472$$ −6866.29 −0.669590
$$473$$ 2281.23 0.221757
$$474$$ 0 0
$$475$$ 2267.71 0.219052
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −4957.36 −0.474360
$$479$$ −7967.98 −0.760055 −0.380027 0.924975i $$-0.624085\pi$$
−0.380027 + 0.924975i $$0.624085\pi$$
$$480$$ 0 0
$$481$$ −3316.25 −0.314362
$$482$$ −1224.17 −0.115683
$$483$$ 0 0
$$484$$ 1858.45 0.174535
$$485$$ 9033.34 0.845738
$$486$$ 0 0
$$487$$ 9956.62 0.926443 0.463221 0.886243i $$-0.346694\pi$$
0.463221 + 0.886243i $$0.346694\pi$$
$$488$$ −2984.91 −0.276886
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 18660.8 1.71518 0.857589 0.514336i $$-0.171962\pi$$
0.857589 + 0.514336i $$0.171962\pi$$
$$492$$ 0 0
$$493$$ 5629.71 0.514299
$$494$$ 13094.1 1.19258
$$495$$ 0 0
$$496$$ −3423.34 −0.309904
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4074.21 0.365504 0.182752 0.983159i $$-0.441499\pi$$
0.182752 + 0.983159i $$0.441499\pi$$
$$500$$ 179.806 0.0160823
$$501$$ 0 0
$$502$$ 12217.8 1.08627
$$503$$ −4255.51 −0.377224 −0.188612 0.982052i $$-0.560399\pi$$
−0.188612 + 0.982052i $$0.560399\pi$$
$$504$$ 0 0
$$505$$ 2415.26 0.212827
$$506$$ −1117.79 −0.0982050
$$507$$ 0 0
$$508$$ 54.8025 0.00478636
$$509$$ 10171.8 0.885771 0.442885 0.896578i $$-0.353955\pi$$
0.442885 + 0.896578i $$0.353955\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −12992.6 −1.12148
$$513$$ 0 0
$$514$$ 1748.32 0.150030
$$515$$ −1695.25 −0.145051
$$516$$ 0 0
$$517$$ 1218.97 0.103695
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 6812.40 0.574506
$$521$$ 1680.39 0.141303 0.0706517 0.997501i $$-0.477492\pi$$
0.0706517 + 0.997501i $$0.477492\pi$$
$$522$$ 0 0
$$523$$ −13211.8 −1.10461 −0.552305 0.833642i $$-0.686251\pi$$
−0.552305 + 0.833642i $$0.686251\pi$$
$$524$$ 2231.76 0.186059
$$525$$ 0 0
$$526$$ −7759.23 −0.643191
$$527$$ −1670.17 −0.138053
$$528$$ 0 0
$$529$$ −7286.34 −0.598861
$$530$$ 6547.34 0.536600
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1082.04 −0.0879333
$$534$$ 0 0
$$535$$ −2254.23 −0.182166
$$536$$ −3504.90 −0.282441
$$537$$ 0 0
$$538$$ 15849.2 1.27009
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 9650.84 0.766954 0.383477 0.923551i $$-0.374727\pi$$
0.383477 + 0.923551i $$0.374727\pi$$
$$542$$ 19362.5 1.53449
$$543$$ 0 0
$$544$$ −1580.69 −0.124580
$$545$$ 9208.52 0.723761
$$546$$ 0 0
$$547$$ −23864.3 −1.86538 −0.932689 0.360682i $$-0.882544\pi$$
−0.932689 + 0.360682i $$0.882544\pi$$
$$548$$ −1730.80 −0.134920
$$549$$ 0 0
$$550$$ 400.000 0.0310110
$$551$$ −20758.3 −1.60496
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 9440.25 0.723967
$$555$$ 0 0
$$556$$ 2728.73 0.208136
$$557$$ 2314.22 0.176044 0.0880221 0.996119i $$-0.471945\pi$$
0.0880221 + 0.996119i $$0.471945\pi$$
$$558$$ 0 0
$$559$$ 20581.5 1.55726
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −20413.3 −1.53218
$$563$$ −7017.86 −0.525342 −0.262671 0.964885i $$-0.584603\pi$$
−0.262671 + 0.964885i $$0.584603\pi$$
$$564$$ 0 0
$$565$$ 9373.58 0.697964
$$566$$ 6351.95 0.471718
$$567$$ 0 0
$$568$$ 1765.61 0.130429
$$569$$ −5302.37 −0.390662 −0.195331 0.980737i $$-0.562578\pi$$
−0.195331 + 0.980737i $$0.562578\pi$$
$$570$$ 0 0
$$571$$ 17767.2 1.30216 0.651082 0.759008i $$-0.274315\pi$$
0.651082 + 0.759008i $$0.274315\pi$$
$$572$$ −506.333 −0.0370120
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1746.54 −0.126671
$$576$$ 0 0
$$577$$ 6089.57 0.439363 0.219681 0.975572i $$-0.429498\pi$$
0.219681 + 0.975572i $$0.429498\pi$$
$$578$$ −11034.7 −0.794089
$$579$$ 0 0
$$580$$ −1645.92 −0.117833
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −3193.07 −0.226833
$$584$$ 15445.5 1.09442
$$585$$ 0 0
$$586$$ −15241.3 −1.07442
$$587$$ −26543.9 −1.86641 −0.933205 0.359344i $$-0.883000\pi$$
−0.933205 + 0.359344i $$0.883000\pi$$
$$588$$ 0 0
$$589$$ 6158.37 0.430817
$$590$$ −3637.40 −0.253813
$$591$$ 0 0
$$592$$ 2967.24 0.206001
$$593$$ 16365.0 1.13327 0.566635 0.823969i $$-0.308245\pi$$
0.566635 + 0.823969i $$0.308245\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1013.94 0.0696859
$$597$$ 0 0
$$598$$ −10084.8 −0.689631
$$599$$ 17516.3 1.19482 0.597411 0.801935i $$-0.296196\pi$$
0.597411 + 0.801935i $$0.296196\pi$$
$$600$$ 0 0
$$601$$ −8693.80 −0.590062 −0.295031 0.955488i $$-0.595330\pi$$
−0.295031 + 0.955488i $$0.595330\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 4365.99 0.294122
$$605$$ 6459.92 0.434105
$$606$$ 0 0
$$607$$ 18096.9 1.21010 0.605051 0.796186i $$-0.293153\pi$$
0.605051 + 0.796186i $$0.293153\pi$$
$$608$$ 5828.44 0.388774
$$609$$ 0 0
$$610$$ −1581.25 −0.104956
$$611$$ 10997.7 0.728183
$$612$$ 0 0
$$613$$ 4641.61 0.305828 0.152914 0.988239i $$-0.451134\pi$$
0.152914 + 0.988239i $$0.451134\pi$$
$$614$$ 8017.40 0.526964
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −14676.1 −0.957600 −0.478800 0.877924i $$-0.658928\pi$$
−0.478800 + 0.877924i $$0.658928\pi$$
$$618$$ 0 0
$$619$$ 19645.3 1.27563 0.637813 0.770192i $$-0.279839\pi$$
0.637813 + 0.770192i $$0.279839\pi$$
$$620$$ 488.296 0.0316297
$$621$$ 0 0
$$622$$ 18600.1 1.19903
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 5919.32 0.377929
$$627$$ 0 0
$$628$$ 3903.44 0.248032
$$629$$ 1447.65 0.0917671
$$630$$ 0 0
$$631$$ 26231.2 1.65491 0.827456 0.561531i $$-0.189788\pi$$
0.827456 + 0.561531i $$0.189788\pi$$
$$632$$ −23601.4 −1.48546
$$633$$ 0 0
$$634$$ −12042.9 −0.754391
$$635$$ 190.492 0.0119046
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −3661.55 −0.227213
$$639$$ 0 0
$$640$$ −4704.34 −0.290555
$$641$$ −30882.4 −1.90293 −0.951466 0.307754i $$-0.900423\pi$$
−0.951466 + 0.307754i $$0.900423\pi$$
$$642$$ 0 0
$$643$$ 6216.88 0.381290 0.190645 0.981659i $$-0.438942\pi$$
0.190645 + 0.981659i $$0.438942\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −5716.00 −0.348132
$$647$$ −21210.4 −1.28882 −0.644410 0.764680i $$-0.722897\pi$$
−0.644410 + 0.764680i $$0.722897\pi$$
$$648$$ 0 0
$$649$$ 1773.92 0.107292
$$650$$ 3608.85 0.217771
$$651$$ 0 0
$$652$$ 669.251 0.0401992
$$653$$ −32938.7 −1.97395 −0.986977 0.160864i $$-0.948572\pi$$
−0.986977 + 0.160864i $$0.948572\pi$$
$$654$$ 0 0
$$655$$ 7757.54 0.462767
$$656$$ 968.165 0.0576227
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −9543.51 −0.564131 −0.282066 0.959395i $$-0.591020\pi$$
−0.282066 + 0.959395i $$0.591020\pi$$
$$660$$ 0 0
$$661$$ −13274.5 −0.781116 −0.390558 0.920578i $$-0.627718\pi$$
−0.390558 + 0.920578i $$0.627718\pi$$
$$662$$ 3499.55 0.205459
$$663$$ 0 0
$$664$$ −11713.0 −0.684565
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 15987.6 0.928101
$$668$$ 5978.27 0.346267
$$669$$ 0 0
$$670$$ −1856.71 −0.107061
$$671$$ 771.159 0.0443670
$$672$$ 0 0
$$673$$ −13575.3 −0.777545 −0.388772 0.921334i $$-0.627101\pi$$
−0.388772 + 0.921334i $$0.627101\pi$$
$$674$$ −1897.17 −0.108422
$$675$$ 0 0
$$676$$ −1407.94 −0.0801058
$$677$$ 12020.7 0.682414 0.341207 0.939988i $$-0.389164\pi$$
0.341207 + 0.939988i $$0.389164\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −2973.83 −0.167707
$$681$$ 0 0
$$682$$ 1086.27 0.0609905
$$683$$ −18391.9 −1.03038 −0.515188 0.857077i $$-0.672278\pi$$
−0.515188 + 0.857077i $$0.672278\pi$$
$$684$$ 0 0
$$685$$ −6016.21 −0.335573
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −18415.5 −1.02047
$$689$$ −28808.3 −1.59290
$$690$$ 0 0
$$691$$ −15594.1 −0.858508 −0.429254 0.903184i $$-0.641223\pi$$
−0.429254 + 0.903184i $$0.641223\pi$$
$$692$$ −6100.69 −0.335135
$$693$$ 0 0
$$694$$ 6674.28 0.365061
$$695$$ 9484.98 0.517677
$$696$$ 0 0
$$697$$ 472.346 0.0256691
$$698$$ −11949.8 −0.648004
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 31093.7 1.67531 0.837656 0.546198i $$-0.183925\pi$$
0.837656 + 0.546198i $$0.183925\pi$$
$$702$$ 0 0
$$703$$ −5337.88 −0.286375
$$704$$ 3547.71 0.189928
$$705$$ 0 0
$$706$$ 7594.36 0.404841
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −2494.67 −0.132143 −0.0660714 0.997815i $$-0.521047\pi$$
−0.0660714 + 0.997815i $$0.521047\pi$$
$$710$$ 935.331 0.0494399
$$711$$ 0 0
$$712$$ 24595.3 1.29459
$$713$$ −4743.06 −0.249129
$$714$$ 0 0
$$715$$ −1760.00 −0.0920563
$$716$$ 4229.54 0.220762
$$717$$ 0 0
$$718$$ −10931.6 −0.568192
$$719$$ 34467.1 1.78777 0.893885 0.448295i $$-0.147969\pi$$
0.893885 + 0.448295i $$0.147969\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3506.77 0.180759
$$723$$ 0 0
$$724$$ −2857.27 −0.146670
$$725$$ −5721.16 −0.293074
$$726$$ 0 0
$$727$$ −9314.97 −0.475204 −0.237602 0.971363i $$-0.576361\pi$$
−0.237602 + 0.971363i $$0.576361\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 8182.24 0.414847
$$731$$ −8984.49 −0.454588
$$732$$ 0 0
$$733$$ −16146.3 −0.813611 −0.406805 0.913515i $$-0.633357\pi$$
−0.406805 + 0.913515i $$0.633357\pi$$
$$734$$ 23772.1 1.19543
$$735$$ 0 0
$$736$$ −4488.95 −0.224816
$$737$$ 905.500 0.0452571
$$738$$ 0 0
$$739$$ −36749.1 −1.82928 −0.914640 0.404268i $$-0.867526\pi$$
−0.914640 + 0.404268i $$0.867526\pi$$
$$740$$ −423.239 −0.0210251
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 2527.09 0.124778 0.0623890 0.998052i $$-0.480128\pi$$
0.0623890 + 0.998052i $$0.480128\pi$$
$$744$$ 0 0
$$745$$ 3524.44 0.173323
$$746$$ 27616.2 1.35536
$$747$$ 0 0
$$748$$ 221.031 0.0108044
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 15828.0 0.769070 0.384535 0.923111i $$-0.374362\pi$$
0.384535 + 0.923111i $$0.374362\pi$$
$$752$$ −9840.28 −0.477178
$$753$$ 0 0
$$754$$ −33035.0 −1.59557
$$755$$ 15176.1 0.731541
$$756$$ 0 0
$$757$$ −20845.9 −1.00087 −0.500435 0.865774i $$-0.666826\pi$$
−0.500435 + 0.865774i $$0.666826\pi$$
$$758$$ −15149.4 −0.725927
$$759$$ 0 0
$$760$$ 10965.3 0.523360
$$761$$ 2420.55 0.115302 0.0576511 0.998337i $$-0.481639\pi$$
0.0576511 + 0.998337i $$0.481639\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2324.35 −0.110068
$$765$$ 0 0
$$766$$ −29493.5 −1.39118
$$767$$ 16004.6 0.753445
$$768$$ 0 0
$$769$$ −22646.3 −1.06196 −0.530980 0.847384i $$-0.678176\pi$$
−0.530980 + 0.847384i $$0.678176\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2952.33 0.137638
$$773$$ 27620.2 1.28516 0.642580 0.766219i $$-0.277864\pi$$
0.642580 + 0.766219i $$0.277864\pi$$
$$774$$ 0 0
$$775$$ 1697.30 0.0786695
$$776$$ 43680.0 2.02064
$$777$$ 0 0
$$778$$ −13831.7 −0.637391
$$779$$ −1741.67 −0.0801049
$$780$$ 0 0
$$781$$ −456.151 −0.0208993
$$782$$ 4402.35 0.201314
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 13568.2 0.616906
$$786$$ 0 0
$$787$$ −14767.1 −0.668859 −0.334429 0.942421i $$-0.608544\pi$$
−0.334429 + 0.942421i $$0.608544\pi$$
$$788$$ 6427.82 0.290586
$$789$$ 0 0
$$790$$ −12502.8 −0.563076
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 6957.50 0.311561
$$794$$ −6716.71 −0.300211
$$795$$ 0 0
$$796$$ 2219.62 0.0988348
$$797$$ −7549.97 −0.335551 −0.167775 0.985825i $$-0.553658\pi$$
−0.167775 + 0.985825i $$0.553658\pi$$
$$798$$ 0 0
$$799$$ −4800.85 −0.212568
$$800$$ 1606.37 0.0709921
$$801$$ 0 0
$$802$$ 28482.1 1.25404
$$803$$ −3990.39 −0.175365
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 9800.50 0.428298
$$807$$ 0 0
$$808$$ 11678.8 0.508489
$$809$$ −17920.0 −0.778783 −0.389391 0.921072i $$-0.627315\pi$$
−0.389391 + 0.921072i $$0.627315\pi$$
$$810$$ 0 0
$$811$$ −25536.6 −1.10569 −0.552843 0.833285i $$-0.686457\pi$$
−0.552843 + 0.833285i $$0.686457\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −941.545 −0.0405420
$$815$$ 2326.30 0.0999836
$$816$$ 0 0
$$817$$ 33128.3 1.41862
$$818$$ 16878.5 0.721447
$$819$$ 0 0
$$820$$ −138.096 −0.00588114
$$821$$ 13688.8 0.581904 0.290952 0.956738i $$-0.406028\pi$$
0.290952 + 0.956738i $$0.406028\pi$$
$$822$$ 0 0
$$823$$ −15102.9 −0.639678 −0.319839 0.947472i $$-0.603629\pi$$
−0.319839 + 0.947472i $$0.603629\pi$$
$$824$$ −8197.22 −0.346558
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36290.3 1.52592 0.762961 0.646445i $$-0.223745\pi$$
0.762961 + 0.646445i $$0.223745\pi$$
$$828$$ 0 0
$$829$$ 39405.1 1.65090 0.825449 0.564477i $$-0.190922\pi$$
0.825449 + 0.564477i $$0.190922\pi$$
$$830$$ −6204.92 −0.259489
$$831$$ 0 0
$$832$$ 32007.9 1.33374
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 20780.3 0.861236
$$836$$ −815.000 −0.0337170
$$837$$ 0 0
$$838$$ −30409.6 −1.25356
$$839$$ −33093.9 −1.36177 −0.680886 0.732389i $$-0.738405\pi$$
−0.680886 + 0.732389i $$0.738405\pi$$
$$840$$ 0 0
$$841$$ 27981.8 1.14731
$$842$$ −4435.50 −0.181541
$$843$$ 0 0
$$844$$ −1847.00 −0.0753274
$$845$$ −4893.96 −0.199239
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 25776.4 1.04383
$$849$$ 0 0
$$850$$ −1575.38 −0.0635706
$$851$$ 4111.12 0.165602
$$852$$ 0 0
$$853$$ −29441.6 −1.18178 −0.590892 0.806750i $$-0.701224\pi$$
−0.590892 + 0.806750i $$0.701224\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −10900.2 −0.435233
$$857$$ −16012.9 −0.638260 −0.319130 0.947711i $$-0.603391\pi$$
−0.319130 + 0.947711i $$0.603391\pi$$
$$858$$ 0 0
$$859$$ 13404.3 0.532421 0.266211 0.963915i $$-0.414228\pi$$
0.266211 + 0.963915i $$0.414228\pi$$
$$860$$ 2626.73 0.104152
$$861$$ 0 0
$$862$$ −27290.2 −1.07832
$$863$$ −2058.73 −0.0812050 −0.0406025 0.999175i $$-0.512928\pi$$
−0.0406025 + 0.999175i $$0.512928\pi$$
$$864$$ 0 0
$$865$$ −21205.8 −0.833549
$$866$$ −6767.66 −0.265559
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 6097.48 0.238024
$$870$$ 0 0
$$871$$ 8169.54 0.317812
$$872$$ 44527.0 1.72922
$$873$$ 0 0
$$874$$ −16232.7 −0.628236
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −16477.0 −0.634422 −0.317211 0.948355i $$-0.602746\pi$$
−0.317211 + 0.948355i $$0.602746\pi$$
$$878$$ −22690.8 −0.872183
$$879$$ 0 0
$$880$$ 1574.77 0.0603245
$$881$$ 43307.7 1.65616 0.828079 0.560612i $$-0.189434\pi$$
0.828079 + 0.560612i $$0.189434\pi$$
$$882$$ 0 0
$$883$$ 15197.9 0.579217 0.289608 0.957145i $$-0.406475\pi$$
0.289608 + 0.957145i $$0.406475\pi$$
$$884$$ 1994.17 0.0758723
$$885$$ 0 0
$$886$$ 16961.9 0.643168
$$887$$ 44953.6 1.70168 0.850842 0.525422i $$-0.176093\pi$$
0.850842 + 0.525422i $$0.176093\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 13029.3 0.490724
$$891$$ 0 0
$$892$$ 5488.13 0.206005
$$893$$ 17702.0 0.663355
$$894$$ 0 0
$$895$$ 14701.8 0.549079
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −33509.5 −1.24524
$$899$$ −15536.9 −0.576400
$$900$$ 0 0
$$901$$ 12575.7 0.464993
$$902$$ −307.212 −0.0113404
$$903$$ 0 0
$$904$$ 45325.2 1.66758
$$905$$ −9931.77 −0.364799
$$906$$ 0 0
$$907$$ −38388.7 −1.40537 −0.702687 0.711499i $$-0.748017\pi$$
−0.702687 + 0.711499i $$0.748017\pi$$
$$908$$ −3267.48 −0.119422
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 46222.1 1.68102 0.840508 0.541799i $$-0.182257\pi$$
0.840508 + 0.541799i $$0.182257\pi$$
$$912$$ 0 0
$$913$$ 3026.08 0.109692
$$914$$ 31166.3 1.12789
$$915$$ 0 0
$$916$$ −3405.59 −0.122842
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −30946.4 −1.11080 −0.555402 0.831582i $$-0.687436\pi$$
−0.555402 + 0.831582i $$0.687436\pi$$
$$920$$ −8445.26 −0.302644
$$921$$ 0 0
$$922$$ 28904.1 1.03244
$$923$$ −4115.46 −0.146763
$$924$$ 0 0
$$925$$ −1471.16 −0.0522936
$$926$$ 42375.6 1.50383
$$927$$ 0 0
$$928$$ −14704.5 −0.520149
$$929$$ −1907.48 −0.0673652 −0.0336826 0.999433i $$-0.510724\pi$$
−0.0336826 + 0.999433i $$0.510724\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −2326.99 −0.0817845
$$933$$ 0 0
$$934$$ −26299.2 −0.921347
$$935$$ 768.296 0.0268727
$$936$$ 0 0
$$937$$ 3334.99 0.116275 0.0581374 0.998309i $$-0.481484\pi$$
0.0581374 + 0.998309i $$0.481484\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 1403.59 0.0487022
$$941$$ 9632.00 0.333681 0.166841 0.985984i $$-0.446643\pi$$
0.166841 + 0.985984i $$0.446643\pi$$
$$942$$ 0 0
$$943$$ 1341.40 0.0463223
$$944$$ −14320.2 −0.493732
$$945$$ 0 0
$$946$$ 5843.48 0.200833
$$947$$ −53606.0 −1.83945 −0.919727 0.392559i $$-0.871590\pi$$
−0.919727 + 0.392559i $$0.871590\pi$$
$$948$$ 0 0
$$949$$ −36001.9 −1.23148
$$950$$ 5808.85 0.198383
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 28433.6 0.966481 0.483240 0.875488i $$-0.339460\pi$$
0.483240 + 0.875488i $$0.339460\pi$$
$$954$$ 0 0
$$955$$ −8079.38 −0.273762
$$956$$ 2783.82 0.0941790
$$957$$ 0 0
$$958$$ −20410.4 −0.688341
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −25181.7 −0.845278
$$962$$ −8494.75 −0.284700
$$963$$ 0 0
$$964$$ 687.436 0.0229676
$$965$$ 10262.2 0.342334
$$966$$ 0 0
$$967$$ 33877.6 1.12661 0.563304 0.826250i $$-0.309530\pi$$
0.563304 + 0.826250i $$0.309530\pi$$
$$968$$ 31236.4 1.03717
$$969$$ 0 0
$$970$$ 23139.4 0.765939
$$971$$ 10208.6 0.337394 0.168697 0.985668i $$-0.446044\pi$$
0.168697 + 0.985668i $$0.446044\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 25504.4 0.839029
$$975$$ 0 0
$$976$$ −6225.27 −0.204166
$$977$$ −35478.1 −1.16177 −0.580883 0.813987i $$-0.697293\pi$$
−0.580883 + 0.813987i $$0.697293\pi$$
$$978$$ 0 0
$$979$$ −6354.27 −0.207439
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 47800.7 1.55334
$$983$$ −54435.8 −1.76626 −0.883130 0.469128i $$-0.844568\pi$$
−0.883130 + 0.469128i $$0.844568\pi$$
$$984$$ 0 0
$$985$$ 22342.9 0.722746
$$986$$ 14420.8 0.465773
$$987$$ 0 0
$$988$$ −7353.04 −0.236773
$$989$$ −25514.7 −0.820346
$$990$$ 0 0
$$991$$ −47319.6 −1.51681 −0.758404 0.651784i $$-0.774021\pi$$
−0.758404 + 0.651784i $$0.774021\pi$$
$$992$$ 4362.38 0.139623
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 7715.35 0.245822
$$996$$ 0 0
$$997$$ −26273.1 −0.834580 −0.417290 0.908773i $$-0.637020\pi$$
−0.417290 + 0.908773i $$0.637020\pi$$
$$998$$ 10436.3 0.331017
$$999$$ 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 2205.4.a.ba.1.2 2
3.2 odd 2 2205.4.a.y.1.1 2
7.6 odd 2 315.4.a.j.1.2 yes 2
21.20 even 2 315.4.a.h.1.1 2
35.34 odd 2 1575.4.a.r.1.1 2
105.104 even 2 1575.4.a.u.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
315.4.a.h.1.1 2 21.20 even 2
315.4.a.j.1.2 yes 2 7.6 odd 2
1575.4.a.r.1.1 2 35.34 odd 2
1575.4.a.u.1.2 2 105.104 even 2
2205.4.a.y.1.1 2 3.2 odd 2
2205.4.a.ba.1.2 2 1.1 even 1 trivial