# Properties

 Label 1575.4.a.z.1.1 Level $1575$ Weight $4$ Character 1575.1 Self dual yes Analytic conductor $92.928$ Analytic rank $0$ 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] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.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: $$92.9280082590$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.58579 q^{2} -1.31371 q^{4} +7.00000 q^{7} -24.0833 q^{8} +O(q^{10})$$ $$q+2.58579 q^{2} -1.31371 q^{4} +7.00000 q^{7} -24.0833 q^{8} -38.2548 q^{11} -19.3431 q^{13} +18.1005 q^{14} -51.7645 q^{16} -87.2254 q^{17} -44.2254 q^{19} -98.9188 q^{22} +218.167 q^{23} -50.0172 q^{26} -9.19596 q^{28} +46.9411 q^{29} +194.558 q^{31} +58.8141 q^{32} -225.546 q^{34} -366.853 q^{37} -114.357 q^{38} +339.362 q^{41} +226.167 q^{43} +50.2557 q^{44} +564.132 q^{46} +11.6762 q^{47} +49.0000 q^{49} +25.4113 q^{52} -209.019 q^{53} -168.583 q^{56} +121.380 q^{58} +616.000 q^{59} +320.735 q^{61} +503.087 q^{62} +566.197 q^{64} -14.5097 q^{67} +114.589 q^{68} +952.000 q^{71} -824.489 q^{73} -948.603 q^{74} +58.0993 q^{76} -267.784 q^{77} +156.275 q^{79} +877.519 q^{82} -1036.53 q^{83} +584.818 q^{86} +921.301 q^{88} +170.225 q^{89} -135.402 q^{91} -286.607 q^{92} +30.1921 q^{94} -1059.87 q^{97} +126.704 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8}+O(q^{10})$$ 2 * q + 8 * q^2 + 20 * q^4 + 14 * q^7 + 48 * q^8 $$2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8} + 14 q^{11} - 50 q^{13} + 56 q^{14} + 168 q^{16} - 50 q^{17} + 36 q^{19} + 184 q^{22} + 244 q^{23} - 216 q^{26} + 140 q^{28} + 26 q^{29} - 120 q^{31} + 672 q^{32} - 24 q^{34} - 564 q^{37} + 320 q^{38} + 328 q^{41} + 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} + 98 q^{49} - 628 q^{52} - 56 q^{53} + 336 q^{56} + 8 q^{58} + 1232 q^{59} + 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} + 1768 q^{76} + 98 q^{77} + 1014 q^{79} + 816 q^{82} - 376 q^{83} + 768 q^{86} + 4688 q^{88} + 216 q^{89} - 350 q^{91} + 264 q^{92} - 1928 q^{94} - 2742 q^{97} + 392 q^{98}+O(q^{100})$$ 2 * q + 8 * q^2 + 20 * q^4 + 14 * q^7 + 48 * q^8 + 14 * q^11 - 50 * q^13 + 56 * q^14 + 168 * q^16 - 50 * q^17 + 36 * q^19 + 184 * q^22 + 244 * q^23 - 216 * q^26 + 140 * q^28 + 26 * q^29 - 120 * q^31 + 672 * q^32 - 24 * q^34 - 564 * q^37 + 320 * q^38 + 328 * q^41 + 260 * q^43 + 1164 * q^44 + 704 * q^46 - 350 * q^47 + 98 * q^49 - 628 * q^52 - 56 * q^53 + 336 * q^56 + 8 * q^58 + 1232 * q^59 + 336 * q^61 - 1200 * q^62 + 2128 * q^64 + 152 * q^67 + 908 * q^68 + 1904 * q^71 - 676 * q^73 - 2016 * q^74 + 1768 * q^76 + 98 * q^77 + 1014 * q^79 + 816 * q^82 - 376 * q^83 + 768 * q^86 + 4688 * q^88 + 216 * q^89 - 350 * q^91 + 264 * q^92 - 1928 * q^94 - 2742 * q^97 + 392 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.58579 0.914214 0.457107 0.889412i $$-0.348886\pi$$
0.457107 + 0.889412i $$0.348886\pi$$
$$3$$ 0 0
$$4$$ −1.31371 −0.164214
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ −24.0833 −1.06434
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −38.2548 −1.04857 −0.524285 0.851543i $$-0.675667\pi$$
−0.524285 + 0.851543i $$0.675667\pi$$
$$12$$ 0 0
$$13$$ −19.3431 −0.412679 −0.206339 0.978480i $$-0.566155\pi$$
−0.206339 + 0.978480i $$0.566155\pi$$
$$14$$ 18.1005 0.345540
$$15$$ 0 0
$$16$$ −51.7645 −0.808820
$$17$$ −87.2254 −1.24443 −0.622214 0.782847i $$-0.713767\pi$$
−0.622214 + 0.782847i $$0.713767\pi$$
$$18$$ 0 0
$$19$$ −44.2254 −0.534000 −0.267000 0.963697i $$-0.586032\pi$$
−0.267000 + 0.963697i $$0.586032\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −98.9188 −0.958617
$$23$$ 218.167 1.97786 0.988932 0.148371i $$-0.0474028\pi$$
0.988932 + 0.148371i $$0.0474028\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −50.0172 −0.377276
$$27$$ 0 0
$$28$$ −9.19596 −0.0620669
$$29$$ 46.9411 0.300578 0.150289 0.988642i $$-0.451980\pi$$
0.150289 + 0.988642i $$0.451980\pi$$
$$30$$ 0 0
$$31$$ 194.558 1.12722 0.563609 0.826042i $$-0.309413\pi$$
0.563609 + 0.826042i $$0.309413\pi$$
$$32$$ 58.8141 0.324905
$$33$$ 0 0
$$34$$ −225.546 −1.13767
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −366.853 −1.63001 −0.815003 0.579457i $$-0.803265\pi$$
−0.815003 + 0.579457i $$0.803265\pi$$
$$38$$ −114.357 −0.488190
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 339.362 1.29267 0.646336 0.763053i $$-0.276301\pi$$
0.646336 + 0.763053i $$0.276301\pi$$
$$42$$ 0 0
$$43$$ 226.167 0.802095 0.401047 0.916057i $$-0.368646\pi$$
0.401047 + 0.916057i $$0.368646\pi$$
$$44$$ 50.2557 0.172189
$$45$$ 0 0
$$46$$ 564.132 1.80819
$$47$$ 11.6762 0.0362372 0.0181186 0.999836i $$-0.494232\pi$$
0.0181186 + 0.999836i $$0.494232\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 25.4113 0.0677674
$$53$$ −209.019 −0.541717 −0.270859 0.962619i $$-0.587308\pi$$
−0.270859 + 0.962619i $$0.587308\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −168.583 −0.402283
$$57$$ 0 0
$$58$$ 121.380 0.274792
$$59$$ 616.000 1.35926 0.679630 0.733555i $$-0.262140\pi$$
0.679630 + 0.733555i $$0.262140\pi$$
$$60$$ 0 0
$$61$$ 320.735 0.673212 0.336606 0.941646i $$-0.390721\pi$$
0.336606 + 0.941646i $$0.390721\pi$$
$$62$$ 503.087 1.03052
$$63$$ 0 0
$$64$$ 566.197 1.10585
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −14.5097 −0.0264573 −0.0132286 0.999912i $$-0.504211\pi$$
−0.0132286 + 0.999912i $$0.504211\pi$$
$$68$$ 114.589 0.204352
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 952.000 1.59129 0.795645 0.605763i $$-0.207132\pi$$
0.795645 + 0.605763i $$0.207132\pi$$
$$72$$ 0 0
$$73$$ −824.489 −1.32191 −0.660953 0.750427i $$-0.729848\pi$$
−0.660953 + 0.750427i $$0.729848\pi$$
$$74$$ −948.603 −1.49017
$$75$$ 0 0
$$76$$ 58.0993 0.0876901
$$77$$ −267.784 −0.396322
$$78$$ 0 0
$$79$$ 156.275 0.222561 0.111280 0.993789i $$-0.464505\pi$$
0.111280 + 0.993789i $$0.464505\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 877.519 1.18178
$$83$$ −1036.53 −1.37077 −0.685384 0.728182i $$-0.740366\pi$$
−0.685384 + 0.728182i $$0.740366\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 584.818 0.733286
$$87$$ 0 0
$$88$$ 921.301 1.11603
$$89$$ 170.225 0.202740 0.101370 0.994849i $$-0.467677\pi$$
0.101370 + 0.994849i $$0.467677\pi$$
$$90$$ 0 0
$$91$$ −135.402 −0.155978
$$92$$ −286.607 −0.324792
$$93$$ 0 0
$$94$$ 30.1921 0.0331285
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1059.87 −1.10942 −0.554710 0.832044i $$-0.687171\pi$$
−0.554710 + 0.832044i $$0.687171\pi$$
$$98$$ 126.704 0.130602
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 241.833 0.238251 0.119125 0.992879i $$-0.461991\pi$$
0.119125 + 0.992879i $$0.461991\pi$$
$$102$$ 0 0
$$103$$ 1679.58 1.60673 0.803367 0.595484i $$-0.203040\pi$$
0.803367 + 0.595484i $$0.203040\pi$$
$$104$$ 465.846 0.439230
$$105$$ 0 0
$$106$$ −540.479 −0.495245
$$107$$ 1506.88 1.36146 0.680728 0.732537i $$-0.261664\pi$$
0.680728 + 0.732537i $$0.261664\pi$$
$$108$$ 0 0
$$109$$ −1252.41 −1.10054 −0.550271 0.834986i $$-0.685476\pi$$
−0.550271 + 0.834986i $$0.685476\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −362.352 −0.305705
$$113$$ 1370.20 1.14069 0.570345 0.821405i $$-0.306810\pi$$
0.570345 + 0.821405i $$0.306810\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −61.6670 −0.0493589
$$117$$ 0 0
$$118$$ 1592.84 1.24265
$$119$$ −610.578 −0.470349
$$120$$ 0 0
$$121$$ 132.432 0.0994984
$$122$$ 829.352 0.615459
$$123$$ 0 0
$$124$$ −255.593 −0.185104
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1213.49 −0.847873 −0.423936 0.905692i $$-0.639352\pi$$
−0.423936 + 0.905692i $$0.639352\pi$$
$$128$$ 993.551 0.686081
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1982.42 1.32217 0.661087 0.750309i $$-0.270096\pi$$
0.661087 + 0.750309i $$0.270096\pi$$
$$132$$ 0 0
$$133$$ −309.578 −0.201833
$$134$$ −37.5189 −0.0241876
$$135$$ 0 0
$$136$$ 2100.67 1.32449
$$137$$ 2210.95 1.37879 0.689394 0.724386i $$-0.257877\pi$$
0.689394 + 0.724386i $$0.257877\pi$$
$$138$$ 0 0
$$139$$ 528.039 0.322213 0.161107 0.986937i $$-0.448494\pi$$
0.161107 + 0.986937i $$0.448494\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2461.67 1.45478
$$143$$ 739.969 0.432722
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2131.95 −1.20851
$$147$$ 0 0
$$148$$ 481.938 0.267669
$$149$$ 328.372 0.180545 0.0902727 0.995917i $$-0.471226\pi$$
0.0902727 + 0.995917i $$0.471226\pi$$
$$150$$ 0 0
$$151$$ 1029.43 0.554793 0.277396 0.960756i $$-0.410528\pi$$
0.277396 + 0.960756i $$0.410528\pi$$
$$152$$ 1065.09 0.568358
$$153$$ 0 0
$$154$$ −692.432 −0.362323
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −525.098 −0.266926 −0.133463 0.991054i $$-0.542610\pi$$
−0.133463 + 0.991054i $$0.542610\pi$$
$$158$$ 404.094 0.203468
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1527.17 0.747562
$$162$$ 0 0
$$163$$ −1002.63 −0.481790 −0.240895 0.970551i $$-0.577441\pi$$
−0.240895 + 0.970551i $$0.577441\pi$$
$$164$$ −445.823 −0.212274
$$165$$ 0 0
$$166$$ −2680.24 −1.25317
$$167$$ −359.422 −0.166544 −0.0832722 0.996527i $$-0.526537\pi$$
−0.0832722 + 0.996527i $$0.526537\pi$$
$$168$$ 0 0
$$169$$ −1822.84 −0.829696
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −297.117 −0.131715
$$173$$ 3293.65 1.44747 0.723733 0.690080i $$-0.242425\pi$$
0.723733 + 0.690080i $$0.242425\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1980.24 0.848104
$$177$$ 0 0
$$178$$ 440.167 0.185348
$$179$$ −2978.82 −1.24384 −0.621921 0.783080i $$-0.713647\pi$$
−0.621921 + 0.783080i $$0.713647\pi$$
$$180$$ 0 0
$$181$$ 1462.31 0.600514 0.300257 0.953858i $$-0.402928\pi$$
0.300257 + 0.953858i $$0.402928\pi$$
$$182$$ −350.121 −0.142597
$$183$$ 0 0
$$184$$ −5254.16 −2.10512
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3336.79 1.30487
$$188$$ −15.3391 −0.00595064
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 374.923 0.142034 0.0710169 0.997475i $$-0.477376\pi$$
0.0710169 + 0.997475i $$0.477376\pi$$
$$192$$ 0 0
$$193$$ −733.028 −0.273391 −0.136696 0.990613i $$-0.543648\pi$$
−0.136696 + 0.990613i $$0.543648\pi$$
$$194$$ −2740.61 −1.01425
$$195$$ 0 0
$$196$$ −64.3717 −0.0234591
$$197$$ −2093.24 −0.757043 −0.378521 0.925593i $$-0.623567\pi$$
−0.378521 + 0.925593i $$0.623567\pi$$
$$198$$ 0 0
$$199$$ 2865.04 1.02059 0.510295 0.860000i $$-0.329536\pi$$
0.510295 + 0.860000i $$0.329536\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 625.330 0.217812
$$203$$ 328.588 0.113608
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 4343.03 1.46890
$$207$$ 0 0
$$208$$ 1001.29 0.333783
$$209$$ 1691.84 0.559936
$$210$$ 0 0
$$211$$ 5643.65 1.84135 0.920674 0.390331i $$-0.127640\pi$$
0.920674 + 0.390331i $$0.127640\pi$$
$$212$$ 274.590 0.0889573
$$213$$ 0 0
$$214$$ 3896.47 1.24466
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1361.91 0.426048
$$218$$ −3238.46 −1.00613
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1687.21 0.513549
$$222$$ 0 0
$$223$$ 6369.16 1.91260 0.956302 0.292381i $$-0.0944477\pi$$
0.956302 + 0.292381i $$0.0944477\pi$$
$$224$$ 411.699 0.122803
$$225$$ 0 0
$$226$$ 3543.05 1.04283
$$227$$ −1015.67 −0.296972 −0.148486 0.988914i $$-0.547440\pi$$
−0.148486 + 0.988914i $$0.547440\pi$$
$$228$$ 0 0
$$229$$ 4108.35 1.18554 0.592768 0.805373i $$-0.298035\pi$$
0.592768 + 0.805373i $$0.298035\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1130.50 −0.319917
$$233$$ 608.431 0.171071 0.0855357 0.996335i $$-0.472740\pi$$
0.0855357 + 0.996335i $$0.472740\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −809.244 −0.223209
$$237$$ 0 0
$$238$$ −1578.82 −0.430000
$$239$$ 5054.44 1.36797 0.683985 0.729496i $$-0.260245\pi$$
0.683985 + 0.729496i $$0.260245\pi$$
$$240$$ 0 0
$$241$$ 4.86782 0.00130109 0.000650547 1.00000i $$-0.499793\pi$$
0.000650547 1.00000i $$0.499793\pi$$
$$242$$ 342.442 0.0909628
$$243$$ 0 0
$$244$$ −421.352 −0.110551
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 855.458 0.220370
$$248$$ −4685.60 −1.19974
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 547.921 0.137787 0.0688934 0.997624i $$-0.478053\pi$$
0.0688934 + 0.997624i $$0.478053\pi$$
$$252$$ 0 0
$$253$$ −8345.92 −2.07393
$$254$$ −3137.83 −0.775137
$$255$$ 0 0
$$256$$ −1960.46 −0.478629
$$257$$ −1774.61 −0.430729 −0.215364 0.976534i $$-0.569094\pi$$
−0.215364 + 0.976534i $$0.569094\pi$$
$$258$$ 0 0
$$259$$ −2567.97 −0.616084
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 5126.11 1.20875
$$263$$ −1199.09 −0.281138 −0.140569 0.990071i $$-0.544893\pi$$
−0.140569 + 0.990071i $$0.544893\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −800.502 −0.184519
$$267$$ 0 0
$$268$$ 19.0615 0.00434464
$$269$$ −3250.29 −0.736706 −0.368353 0.929686i $$-0.620078\pi$$
−0.368353 + 0.929686i $$0.620078\pi$$
$$270$$ 0 0
$$271$$ −896.143 −0.200874 −0.100437 0.994943i $$-0.532024\pi$$
−0.100437 + 0.994943i $$0.532024\pi$$
$$272$$ 4515.18 1.00652
$$273$$ 0 0
$$274$$ 5717.04 1.26051
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 386.562 0.0838492 0.0419246 0.999121i $$-0.486651\pi$$
0.0419246 + 0.999121i $$0.486651\pi$$
$$278$$ 1365.40 0.294572
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 3335.10 0.708025 0.354013 0.935241i $$-0.384817\pi$$
0.354013 + 0.935241i $$0.384817\pi$$
$$282$$ 0 0
$$283$$ −5412.26 −1.13684 −0.568419 0.822739i $$-0.692445\pi$$
−0.568419 + 0.822739i $$0.692445\pi$$
$$284$$ −1250.65 −0.261311
$$285$$ 0 0
$$286$$ 1913.40 0.395601
$$287$$ 2375.54 0.488584
$$288$$ 0 0
$$289$$ 2695.27 0.548600
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 1083.14 0.217075
$$293$$ 282.211 0.0562695 0.0281347 0.999604i $$-0.491043\pi$$
0.0281347 + 0.999604i $$0.491043\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 8835.01 1.73488
$$297$$ 0 0
$$298$$ 849.099 0.165057
$$299$$ −4220.03 −0.816222
$$300$$ 0 0
$$301$$ 1583.17 0.303163
$$302$$ 2661.88 0.507199
$$303$$ 0 0
$$304$$ 2289.31 0.431910
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −1919.67 −0.356878 −0.178439 0.983951i $$-0.557105\pi$$
−0.178439 + 0.983951i $$0.557105\pi$$
$$308$$ 351.790 0.0650815
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −1213.31 −0.221223 −0.110612 0.993864i $$-0.535281\pi$$
−0.110612 + 0.993864i $$0.535281\pi$$
$$312$$ 0 0
$$313$$ 1434.00 0.258960 0.129480 0.991582i $$-0.458669\pi$$
0.129480 + 0.991582i $$0.458669\pi$$
$$314$$ −1357.79 −0.244028
$$315$$ 0 0
$$316$$ −205.300 −0.0365475
$$317$$ 6496.95 1.15112 0.575560 0.817760i $$-0.304784\pi$$
0.575560 + 0.817760i $$0.304784\pi$$
$$318$$ 0 0
$$319$$ −1795.72 −0.315176
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 3948.92 0.683432
$$323$$ 3857.58 0.664524
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −2592.58 −0.440459
$$327$$ 0 0
$$328$$ −8172.96 −1.37584
$$329$$ 81.7333 0.0136964
$$330$$ 0 0
$$331$$ −9683.88 −1.60808 −0.804039 0.594576i $$-0.797320\pi$$
−0.804039 + 0.594576i $$0.797320\pi$$
$$332$$ 1361.70 0.225099
$$333$$ 0 0
$$334$$ −929.389 −0.152257
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −29.1319 −0.00470895 −0.00235447 0.999997i $$-0.500749\pi$$
−0.00235447 + 0.999997i $$0.500749\pi$$
$$338$$ −4713.48 −0.758520
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −7442.80 −1.18197
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ −5446.83 −0.853701
$$345$$ 0 0
$$346$$ 8516.68 1.32329
$$347$$ −7848.58 −1.21422 −0.607110 0.794618i $$-0.707671\pi$$
−0.607110 + 0.794618i $$0.707671\pi$$
$$348$$ 0 0
$$349$$ −10269.6 −1.57513 −0.787567 0.616229i $$-0.788659\pi$$
−0.787567 + 0.616229i $$0.788659\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2249.93 −0.340686
$$353$$ 2799.93 0.422168 0.211084 0.977468i $$-0.432301\pi$$
0.211084 + 0.977468i $$0.432301\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −223.627 −0.0332927
$$357$$ 0 0
$$358$$ −7702.60 −1.13714
$$359$$ 3163.29 0.465048 0.232524 0.972591i $$-0.425302\pi$$
0.232524 + 0.972591i $$0.425302\pi$$
$$360$$ 0 0
$$361$$ −4903.11 −0.714844
$$362$$ 3781.23 0.548998
$$363$$ 0 0
$$364$$ 177.879 0.0256137
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3182.85 −0.452706 −0.226353 0.974045i $$-0.572680\pi$$
−0.226353 + 0.974045i $$0.572680\pi$$
$$368$$ −11293.3 −1.59974
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1463.14 −0.204750
$$372$$ 0 0
$$373$$ 2615.14 0.363021 0.181510 0.983389i $$-0.441901\pi$$
0.181510 + 0.983389i $$0.441901\pi$$
$$374$$ 8628.23 1.19293
$$375$$ 0 0
$$376$$ −281.201 −0.0385687
$$377$$ −907.989 −0.124042
$$378$$ 0 0
$$379$$ −672.434 −0.0911362 −0.0455681 0.998961i $$-0.514510\pi$$
−0.0455681 + 0.998961i $$0.514510\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 969.470 0.129849
$$383$$ 1169.86 0.156075 0.0780377 0.996950i $$-0.475135\pi$$
0.0780377 + 0.996950i $$0.475135\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1895.45 −0.249938
$$387$$ 0 0
$$388$$ 1392.36 0.182182
$$389$$ 1122.22 0.146269 0.0731347 0.997322i $$-0.476700\pi$$
0.0731347 + 0.997322i $$0.476700\pi$$
$$390$$ 0 0
$$391$$ −19029.7 −2.46131
$$392$$ −1180.08 −0.152049
$$393$$ 0 0
$$394$$ −5412.68 −0.692099
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1985.93 0.251060 0.125530 0.992090i $$-0.459937\pi$$
0.125530 + 0.992090i $$0.459937\pi$$
$$398$$ 7408.38 0.933037
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4172.38 0.519597 0.259799 0.965663i $$-0.416344\pi$$
0.259799 + 0.965663i $$0.416344\pi$$
$$402$$ 0 0
$$403$$ −3763.37 −0.465178
$$404$$ −317.699 −0.0391240
$$405$$ 0 0
$$406$$ 849.658 0.103862
$$407$$ 14033.9 1.70918
$$408$$ 0 0
$$409$$ −11700.8 −1.41459 −0.707295 0.706919i $$-0.750085\pi$$
−0.707295 + 0.706919i $$0.750085\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −2206.47 −0.263848
$$413$$ 4312.00 0.513752
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1137.65 −0.134082
$$417$$ 0 0
$$418$$ 4374.72 0.511901
$$419$$ −2733.20 −0.318677 −0.159339 0.987224i $$-0.550936\pi$$
−0.159339 + 0.987224i $$0.550936\pi$$
$$420$$ 0 0
$$421$$ 13549.4 1.56854 0.784272 0.620417i $$-0.213037\pi$$
0.784272 + 0.620417i $$0.213037\pi$$
$$422$$ 14593.3 1.68339
$$423$$ 0 0
$$424$$ 5033.87 0.576571
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2245.15 0.254450
$$428$$ −1979.60 −0.223569
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6429.25 0.718530 0.359265 0.933236i $$-0.383027\pi$$
0.359265 + 0.933236i $$0.383027\pi$$
$$432$$ 0 0
$$433$$ −8022.03 −0.890333 −0.445166 0.895448i $$-0.646855\pi$$
−0.445166 + 0.895448i $$0.646855\pi$$
$$434$$ 3521.61 0.389499
$$435$$ 0 0
$$436$$ 1645.30 0.180724
$$437$$ −9648.50 −1.05618
$$438$$ 0 0
$$439$$ −5569.88 −0.605549 −0.302774 0.953062i $$-0.597913\pi$$
−0.302774 + 0.953062i $$0.597913\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 4362.77 0.469493
$$443$$ −5486.21 −0.588392 −0.294196 0.955745i $$-0.595052\pi$$
−0.294196 + 0.955745i $$0.595052\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 16469.3 1.74853
$$447$$ 0 0
$$448$$ 3963.38 0.417973
$$449$$ 7232.67 0.760203 0.380101 0.924945i $$-0.375889\pi$$
0.380101 + 0.924945i $$0.375889\pi$$
$$450$$ 0 0
$$451$$ −12982.3 −1.35546
$$452$$ −1800.05 −0.187317
$$453$$ 0 0
$$454$$ −2626.32 −0.271496
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2900.51 0.296893 0.148446 0.988920i $$-0.452573\pi$$
0.148446 + 0.988920i $$0.452573\pi$$
$$458$$ 10623.3 1.08383
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6073.57 −0.613611 −0.306805 0.951772i $$-0.599260\pi$$
−0.306805 + 0.951772i $$0.599260\pi$$
$$462$$ 0 0
$$463$$ 18922.8 1.89939 0.949693 0.313183i $$-0.101395\pi$$
0.949693 + 0.313183i $$0.101395\pi$$
$$464$$ −2429.88 −0.243113
$$465$$ 0 0
$$466$$ 1573.27 0.156396
$$467$$ −6776.71 −0.671496 −0.335748 0.941952i $$-0.608989\pi$$
−0.335748 + 0.941952i $$0.608989\pi$$
$$468$$ 0 0
$$469$$ −101.568 −0.00999991
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −14835.3 −1.44672
$$473$$ −8651.96 −0.841052
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 802.121 0.0772377
$$477$$ 0 0
$$478$$ 13069.7 1.25062
$$479$$ −2397.32 −0.228677 −0.114338 0.993442i $$-0.536475\pi$$
−0.114338 + 0.993442i $$0.536475\pi$$
$$480$$ 0 0
$$481$$ 7096.09 0.672669
$$482$$ 12.5871 0.00118948
$$483$$ 0 0
$$484$$ −173.977 −0.0163390
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −5586.17 −0.519781 −0.259890 0.965638i $$-0.583686\pi$$
−0.259890 + 0.965638i $$0.583686\pi$$
$$488$$ −7724.35 −0.716526
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −537.392 −0.0493934 −0.0246967 0.999695i $$-0.507862\pi$$
−0.0246967 + 0.999695i $$0.507862\pi$$
$$492$$ 0 0
$$493$$ −4094.46 −0.374047
$$494$$ 2212.03 0.201466
$$495$$ 0 0
$$496$$ −10071.2 −0.911716
$$497$$ 6664.00 0.601451
$$498$$ 0 0
$$499$$ 598.965 0.0537342 0.0268671 0.999639i $$-0.491447\pi$$
0.0268671 + 0.999639i $$0.491447\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 1416.81 0.125966
$$503$$ 4426.76 0.392405 0.196202 0.980563i $$-0.437139\pi$$
0.196202 + 0.980563i $$0.437139\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −21580.8 −1.89601
$$507$$ 0 0
$$508$$ 1594.17 0.139232
$$509$$ −17727.7 −1.54374 −0.771872 0.635779i $$-0.780679\pi$$
−0.771872 + 0.635779i $$0.780679\pi$$
$$510$$ 0 0
$$511$$ −5771.43 −0.499634
$$512$$ −13017.7 −1.12365
$$513$$ 0 0
$$514$$ −4588.77 −0.393778
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −446.671 −0.0379972
$$518$$ −6640.22 −0.563233
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −8662.79 −0.728453 −0.364226 0.931310i $$-0.618667\pi$$
−0.364226 + 0.931310i $$0.618667\pi$$
$$522$$ 0 0
$$523$$ 7770.40 0.649667 0.324833 0.945771i $$-0.394692\pi$$
0.324833 + 0.945771i $$0.394692\pi$$
$$524$$ −2604.32 −0.217119
$$525$$ 0 0
$$526$$ −3100.60 −0.257020
$$527$$ −16970.4 −1.40274
$$528$$ 0 0
$$529$$ 35429.6 2.91194
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 406.695 0.0331437
$$533$$ −6564.34 −0.533458
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 349.440 0.0281595
$$537$$ 0 0
$$538$$ −8404.56 −0.673506
$$539$$ −1874.49 −0.149796
$$540$$ 0 0
$$541$$ 21641.0 1.71981 0.859906 0.510453i $$-0.170522\pi$$
0.859906 + 0.510453i $$0.170522\pi$$
$$542$$ −2317.23 −0.183642
$$543$$ 0 0
$$544$$ −5130.09 −0.404321
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −7489.29 −0.585409 −0.292705 0.956203i $$-0.594555\pi$$
−0.292705 + 0.956203i $$0.594555\pi$$
$$548$$ −2904.54 −0.226416
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2075.99 −0.160508
$$552$$ 0 0
$$553$$ 1093.93 0.0841201
$$554$$ 999.566 0.0766561
$$555$$ 0 0
$$556$$ −693.689 −0.0529118
$$557$$ 25297.9 1.92443 0.962214 0.272295i $$-0.0877826\pi$$
0.962214 + 0.272295i $$0.0877826\pi$$
$$558$$ 0 0
$$559$$ −4374.77 −0.331007
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 8623.85 0.647286
$$563$$ 15661.3 1.17237 0.586186 0.810177i $$-0.300629\pi$$
0.586186 + 0.810177i $$0.300629\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −13994.9 −1.03931
$$567$$ 0 0
$$568$$ −22927.3 −1.69367
$$569$$ 9982.75 0.735498 0.367749 0.929925i $$-0.380129\pi$$
0.367749 + 0.929925i $$0.380129\pi$$
$$570$$ 0 0
$$571$$ −11583.6 −0.848966 −0.424483 0.905436i $$-0.639544\pi$$
−0.424483 + 0.905436i $$0.639544\pi$$
$$572$$ −972.103 −0.0710589
$$573$$ 0 0
$$574$$ 6142.63 0.446670
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 595.378 0.0429565 0.0214783 0.999769i $$-0.493163\pi$$
0.0214783 + 0.999769i $$0.493163\pi$$
$$578$$ 6969.39 0.501537
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −7255.70 −0.518102
$$582$$ 0 0
$$583$$ 7996.00 0.568028
$$584$$ 19856.4 1.40696
$$585$$ 0 0
$$586$$ 729.738 0.0514423
$$587$$ −15750.3 −1.10747 −0.553736 0.832693i $$-0.686798\pi$$
−0.553736 + 0.832693i $$0.686798\pi$$
$$588$$ 0 0
$$589$$ −8604.42 −0.601934
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 18990.0 1.31838
$$593$$ −417.878 −0.0289379 −0.0144690 0.999895i $$-0.504606\pi$$
−0.0144690 + 0.999895i $$0.504606\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −431.385 −0.0296480
$$597$$ 0 0
$$598$$ −10912.1 −0.746201
$$599$$ 19997.3 1.36406 0.682028 0.731326i $$-0.261098\pi$$
0.682028 + 0.731326i $$0.261098\pi$$
$$600$$ 0 0
$$601$$ −15992.6 −1.08545 −0.542723 0.839912i $$-0.682607\pi$$
−0.542723 + 0.839912i $$0.682607\pi$$
$$602$$ 4093.73 0.277156
$$603$$ 0 0
$$604$$ −1352.37 −0.0911045
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −14159.2 −0.946793 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$608$$ −2601.08 −0.173499
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −225.854 −0.0149543
$$612$$ 0 0
$$613$$ 4629.41 0.305025 0.152512 0.988302i $$-0.451264\pi$$
0.152512 + 0.988302i $$0.451264\pi$$
$$614$$ −4963.87 −0.326263
$$615$$ 0 0
$$616$$ 6449.11 0.421821
$$617$$ −23165.3 −1.51151 −0.755753 0.654857i $$-0.772729\pi$$
−0.755753 + 0.654857i $$0.772729\pi$$
$$618$$ 0 0
$$619$$ 12370.6 0.803258 0.401629 0.915803i $$-0.368444\pi$$
0.401629 + 0.915803i $$0.368444\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −3137.36 −0.202245
$$623$$ 1191.58 0.0766285
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 3708.02 0.236745
$$627$$ 0 0
$$628$$ 689.826 0.0438329
$$629$$ 31998.9 2.02842
$$630$$ 0 0
$$631$$ 13980.2 0.882002 0.441001 0.897507i $$-0.354623\pi$$
0.441001 + 0.897507i $$0.354623\pi$$
$$632$$ −3763.61 −0.236880
$$633$$ 0 0
$$634$$ 16799.7 1.05237
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −947.814 −0.0589541
$$638$$ −4643.36 −0.288139
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 16060.9 0.989655 0.494828 0.868991i $$-0.335231\pi$$
0.494828 + 0.868991i $$0.335231\pi$$
$$642$$ 0 0
$$643$$ 4502.17 0.276125 0.138063 0.990424i $$-0.455913\pi$$
0.138063 + 0.990424i $$0.455913\pi$$
$$644$$ −2006.25 −0.122760
$$645$$ 0 0
$$646$$ 9974.87 0.607517
$$647$$ 29414.8 1.78735 0.893675 0.448715i $$-0.148118\pi$$
0.893675 + 0.448715i $$0.148118\pi$$
$$648$$ 0 0
$$649$$ −23565.0 −1.42528
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 1317.16 0.0791164
$$653$$ 13013.6 0.779882 0.389941 0.920840i $$-0.372495\pi$$
0.389941 + 0.920840i $$0.372495\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −17566.9 −1.04554
$$657$$ 0 0
$$658$$ 211.345 0.0125214
$$659$$ −23474.2 −1.38759 −0.693797 0.720171i $$-0.744063\pi$$
−0.693797 + 0.720171i $$0.744063\pi$$
$$660$$ 0 0
$$661$$ −9266.36 −0.545264 −0.272632 0.962118i $$-0.587894\pi$$
−0.272632 + 0.962118i $$0.587894\pi$$
$$662$$ −25040.4 −1.47013
$$663$$ 0 0
$$664$$ 24963.0 1.45896
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10241.0 0.594501
$$668$$ 472.176 0.0273489
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −12269.7 −0.705909
$$672$$ 0 0
$$673$$ 25067.2 1.43576 0.717882 0.696164i $$-0.245112\pi$$
0.717882 + 0.696164i $$0.245112\pi$$
$$674$$ −75.3288 −0.00430498
$$675$$ 0 0
$$676$$ 2394.68 0.136247
$$677$$ −22409.6 −1.27219 −0.636093 0.771613i $$-0.719450\pi$$
−0.636093 + 0.771613i $$0.719450\pi$$
$$678$$ 0 0
$$679$$ −7419.11 −0.419322
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −19245.5 −1.08057
$$683$$ −8757.53 −0.490626 −0.245313 0.969444i $$-0.578891\pi$$
−0.245313 + 0.969444i $$0.578891\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 886.925 0.0493629
$$687$$ 0 0
$$688$$ −11707.4 −0.648750
$$689$$ 4043.09 0.223555
$$690$$ 0 0
$$691$$ 8468.42 0.466214 0.233107 0.972451i $$-0.425111\pi$$
0.233107 + 0.972451i $$0.425111\pi$$
$$692$$ −4326.90 −0.237694
$$693$$ 0 0
$$694$$ −20294.8 −1.11006
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −29601.0 −1.60864
$$698$$ −26555.1 −1.44001
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −15996.9 −0.861906 −0.430953 0.902374i $$-0.641823\pi$$
−0.430953 + 0.902374i $$0.641823\pi$$
$$702$$ 0 0
$$703$$ 16224.2 0.870423
$$704$$ −21659.8 −1.15956
$$705$$ 0 0
$$706$$ 7240.03 0.385952
$$707$$ 1692.83 0.0900503
$$708$$ 0 0
$$709$$ 19903.0 1.05426 0.527131 0.849784i $$-0.323268\pi$$
0.527131 + 0.849784i $$0.323268\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −4099.58 −0.215784
$$713$$ 42446.1 2.22948
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 3913.30 0.204256
$$717$$ 0 0
$$718$$ 8179.59 0.425153
$$719$$ 11073.1 0.574347 0.287174 0.957879i $$-0.407284\pi$$
0.287174 + 0.957879i $$0.407284\pi$$
$$720$$ 0 0
$$721$$ 11757.0 0.607288
$$722$$ −12678.4 −0.653520
$$723$$ 0 0
$$724$$ −1921.06 −0.0986125
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −31652.7 −1.61476 −0.807382 0.590029i $$-0.799116\pi$$
−0.807382 + 0.590029i $$0.799116\pi$$
$$728$$ 3260.92 0.166013
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −19727.5 −0.998149
$$732$$ 0 0
$$733$$ −16958.3 −0.854528 −0.427264 0.904127i $$-0.640523\pi$$
−0.427264 + 0.904127i $$0.640523\pi$$
$$734$$ −8230.16 −0.413870
$$735$$ 0 0
$$736$$ 12831.3 0.642618
$$737$$ 555.065 0.0277423
$$738$$ 0 0
$$739$$ −11616.6 −0.578245 −0.289123 0.957292i $$-0.593364\pi$$
−0.289123 + 0.957292i $$0.593364\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −3783.36 −0.187185
$$743$$ 15928.0 0.786464 0.393232 0.919439i $$-0.371357\pi$$
0.393232 + 0.919439i $$0.371357\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 6762.19 0.331879
$$747$$ 0 0
$$748$$ −4383.57 −0.214277
$$749$$ 10548.2 0.514582
$$750$$ 0 0
$$751$$ 25571.9 1.24252 0.621260 0.783604i $$-0.286621\pi$$
0.621260 + 0.783604i $$0.286621\pi$$
$$752$$ −604.412 −0.0293094
$$753$$ 0 0
$$754$$ −2347.87 −0.113401
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6202.41 −0.297794 −0.148897 0.988853i $$-0.547572\pi$$
−0.148897 + 0.988853i $$0.547572\pi$$
$$758$$ −1738.77 −0.0833179
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 29199.1 1.39089 0.695444 0.718580i $$-0.255208\pi$$
0.695444 + 0.718580i $$0.255208\pi$$
$$762$$ 0 0
$$763$$ −8766.87 −0.415966
$$764$$ −492.539 −0.0233239
$$765$$ 0 0
$$766$$ 3025.00 0.142686
$$767$$ −11915.4 −0.560938
$$768$$ 0 0
$$769$$ 21838.2 1.02407 0.512033 0.858966i $$-0.328892\pi$$
0.512033 + 0.858966i $$0.328892\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 962.985 0.0448945
$$773$$ −25544.8 −1.18859 −0.594296 0.804246i $$-0.702569\pi$$
−0.594296 + 0.804246i $$0.702569\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 25525.2 1.18080
$$777$$ 0 0
$$778$$ 2901.82 0.133721
$$779$$ −15008.4 −0.690286
$$780$$ 0 0
$$781$$ −36418.6 −1.66858
$$782$$ −49206.6 −2.25016
$$783$$ 0 0
$$784$$ −2536.46 −0.115546
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 37223.2 1.68598 0.842989 0.537931i $$-0.180794\pi$$
0.842989 + 0.537931i $$0.180794\pi$$
$$788$$ 2749.91 0.124317
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9591.42 0.431140
$$792$$ 0 0
$$793$$ −6204.03 −0.277820
$$794$$ 5135.18 0.229522
$$795$$ 0 0
$$796$$ −3763.83 −0.167595
$$797$$ −40384.6 −1.79485 −0.897425 0.441168i $$-0.854564\pi$$
−0.897425 + 0.441168i $$0.854564\pi$$
$$798$$ 0 0
$$799$$ −1018.46 −0.0450945
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 10788.9 0.475023
$$803$$ 31540.7 1.38611
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −9731.28 −0.425272
$$807$$ 0 0
$$808$$ −5824.14 −0.253580
$$809$$ 1955.76 0.0849948 0.0424974 0.999097i $$-0.486469\pi$$
0.0424974 + 0.999097i $$0.486469\pi$$
$$810$$ 0 0
$$811$$ −34301.8 −1.48520 −0.742600 0.669735i $$-0.766408\pi$$
−0.742600 + 0.669735i $$0.766408\pi$$
$$812$$ −431.669 −0.0186559
$$813$$ 0 0
$$814$$ 36288.7 1.56255
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −10002.3 −0.428319
$$818$$ −30255.8 −1.29324
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 13665.6 0.580918 0.290459 0.956887i $$-0.406192\pi$$
0.290459 + 0.956887i $$0.406192\pi$$
$$822$$ 0 0
$$823$$ 21519.5 0.911449 0.455724 0.890121i $$-0.349380\pi$$
0.455724 + 0.890121i $$0.349380\pi$$
$$824$$ −40449.7 −1.71011
$$825$$ 0 0
$$826$$ 11149.9 0.469679
$$827$$ 35220.6 1.48094 0.740471 0.672088i $$-0.234602\pi$$
0.740471 + 0.672088i $$0.234602\pi$$
$$828$$ 0 0
$$829$$ 31365.5 1.31408 0.657039 0.753857i $$-0.271809\pi$$
0.657039 + 0.753857i $$0.271809\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −10952.0 −0.456362
$$833$$ −4274.04 −0.177775
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −2222.58 −0.0919491
$$837$$ 0 0
$$838$$ −7067.48 −0.291339
$$839$$ 28287.1 1.16398 0.581990 0.813196i $$-0.302274\pi$$
0.581990 + 0.813196i $$0.302274\pi$$
$$840$$ 0 0
$$841$$ −22185.5 −0.909653
$$842$$ 35035.8 1.43398
$$843$$ 0 0
$$844$$ −7414.11 −0.302374
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 927.026 0.0376068
$$848$$ 10819.8 0.438152
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −80035.0 −3.22393
$$852$$ 0 0
$$853$$ −9405.41 −0.377533 −0.188766 0.982022i $$-0.560449\pi$$
−0.188766 + 0.982022i $$0.560449\pi$$
$$854$$ 5805.47 0.232622
$$855$$ 0 0
$$856$$ −36290.6 −1.44905
$$857$$ −27966.9 −1.11474 −0.557369 0.830265i $$-0.688189\pi$$
−0.557369 + 0.830265i $$0.688189\pi$$
$$858$$ 0 0
$$859$$ 6281.11 0.249486 0.124743 0.992189i $$-0.460189\pi$$
0.124743 + 0.992189i $$0.460189\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 16624.7 0.656890
$$863$$ −4757.13 −0.187642 −0.0938208 0.995589i $$-0.529908\pi$$
−0.0938208 + 0.995589i $$0.529908\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −20743.3 −0.813954
$$867$$ 0 0
$$868$$ −1789.15 −0.0699629
$$869$$ −5978.28 −0.233371
$$870$$ 0 0
$$871$$ 280.663 0.0109184
$$872$$ 30162.1 1.17135
$$873$$ 0 0
$$874$$ −24949.0 −0.965574
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30240.5 1.16437 0.582184 0.813057i $$-0.302198\pi$$
0.582184 + 0.813057i $$0.302198\pi$$
$$878$$ −14402.5 −0.553601
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −44875.5 −1.71611 −0.858056 0.513556i $$-0.828328\pi$$
−0.858056 + 0.513556i $$0.828328\pi$$
$$882$$ 0 0
$$883$$ −4892.13 −0.186448 −0.0932238 0.995645i $$-0.529717\pi$$
−0.0932238 + 0.995645i $$0.529717\pi$$
$$884$$ −2216.51 −0.0843317
$$885$$ 0 0
$$886$$ −14186.2 −0.537916
$$887$$ 1761.40 0.0666765 0.0333382 0.999444i $$-0.489386\pi$$
0.0333382 + 0.999444i $$0.489386\pi$$
$$888$$ 0 0
$$889$$ −8494.43 −0.320466
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −8367.22 −0.314075
$$893$$ −516.384 −0.0193507
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 6954.86 0.259314
$$897$$ 0 0
$$898$$ 18702.2 0.694988
$$899$$ 9132.79 0.338816
$$900$$ 0 0
$$901$$ 18231.8 0.674128
$$902$$ −33569.3 −1.23918
$$903$$ 0 0
$$904$$ −32999.0 −1.21408
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −23689.1 −0.867238 −0.433619 0.901096i $$-0.642764\pi$$
−0.433619 + 0.901096i $$0.642764\pi$$
$$908$$ 1334.30 0.0487669
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 13877.3 0.504692 0.252346 0.967637i $$-0.418798\pi$$
0.252346 + 0.967637i $$0.418798\pi$$
$$912$$ 0 0
$$913$$ 39652.2 1.43735
$$914$$ 7500.09 0.271423
$$915$$ 0 0
$$916$$ −5397.18 −0.194681
$$917$$ 13876.9 0.499735
$$918$$ 0 0
$$919$$ 14331.6 0.514426 0.257213 0.966355i $$-0.417196\pi$$
0.257213 + 0.966355i $$0.417196\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −15705.0 −0.560971
$$923$$ −18414.7 −0.656692
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 48930.2 1.73644
$$927$$ 0 0
$$928$$ 2760.80 0.0976592
$$929$$ −16668.4 −0.588668 −0.294334 0.955703i $$-0.595098\pi$$
−0.294334 + 0.955703i $$0.595098\pi$$
$$930$$ 0 0
$$931$$ −2167.04 −0.0762857
$$932$$ −799.300 −0.0280922
$$933$$ 0 0
$$934$$ −17523.1 −0.613891
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −30384.9 −1.05937 −0.529685 0.848194i $$-0.677690\pi$$
−0.529685 + 0.848194i $$0.677690\pi$$
$$938$$ −262.632 −0.00914206
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1196.35 0.0414452 0.0207226 0.999785i $$-0.493403\pi$$
0.0207226 + 0.999785i $$0.493403\pi$$
$$942$$ 0 0
$$943$$ 74037.5 2.55673
$$944$$ −31886.9 −1.09940
$$945$$ 0 0
$$946$$ −22372.1 −0.768901
$$947$$ 1788.41 0.0613681 0.0306840 0.999529i $$-0.490231\pi$$
0.0306840 + 0.999529i $$0.490231\pi$$
$$948$$ 0 0
$$949$$ 15948.2 0.545523
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 14704.7 0.500612
$$953$$ 8578.60 0.291593 0.145796 0.989315i $$-0.453426\pi$$
0.145796 + 0.989315i $$0.453426\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −6640.06 −0.224639
$$957$$ 0 0
$$958$$ −6198.95 −0.209059
$$959$$ 15476.6 0.521133
$$960$$ 0 0
$$961$$ 8061.99 0.270618
$$962$$ 18349.0 0.614963
$$963$$ 0 0
$$964$$ −6.39489 −0.000213657 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 55459.3 1.84431 0.922156 0.386818i $$-0.126426\pi$$
0.922156 + 0.386818i $$0.126426\pi$$
$$968$$ −3189.40 −0.105900
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −22047.3 −0.728662 −0.364331 0.931270i $$-0.618702\pi$$
−0.364331 + 0.931270i $$0.618702\pi$$
$$972$$ 0 0
$$973$$ 3696.27 0.121785
$$974$$ −14444.6 −0.475191
$$975$$ 0 0
$$976$$ −16602.7 −0.544507
$$977$$ 14402.3 0.471617 0.235809 0.971800i $$-0.424226\pi$$
0.235809 + 0.971800i $$0.424226\pi$$
$$978$$ 0 0
$$979$$ −6511.94 −0.212587
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −1389.58 −0.0451561
$$983$$ 7817.11 0.253639 0.126819 0.991926i $$-0.459523\pi$$
0.126819 + 0.991926i $$0.459523\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −10587.4 −0.341959
$$987$$ 0 0
$$988$$ −1123.82 −0.0361878
$$989$$ 49342.0 1.58643
$$990$$ 0 0
$$991$$ 24501.6 0.785386 0.392693 0.919670i $$-0.371543\pi$$
0.392693 + 0.919670i $$0.371543\pi$$
$$992$$ 11442.8 0.366239
$$993$$ 0 0
$$994$$ 17231.7 0.549855
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 50696.0 1.61039 0.805195 0.593010i $$-0.202060\pi$$
0.805195 + 0.593010i $$0.202060\pi$$
$$998$$ 1548.80 0.0491245
$$999$$ 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 1575.4.a.z.1.1 2
3.2 odd 2 175.4.a.c.1.2 2
5.4 even 2 315.4.a.f.1.2 2
15.2 even 4 175.4.b.c.99.2 4
15.8 even 4 175.4.b.c.99.3 4
15.14 odd 2 35.4.a.b.1.1 2
21.20 even 2 1225.4.a.m.1.2 2
35.34 odd 2 2205.4.a.u.1.2 2
60.59 even 2 560.4.a.r.1.1 2
105.44 odd 6 245.4.e.h.116.2 4
105.59 even 6 245.4.e.i.226.2 4
105.74 odd 6 245.4.e.h.226.2 4
105.89 even 6 245.4.e.i.116.2 4
105.104 even 2 245.4.a.k.1.1 2
120.29 odd 2 2240.4.a.bn.1.1 2
120.59 even 2 2240.4.a.bo.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 15.14 odd 2
175.4.a.c.1.2 2 3.2 odd 2
175.4.b.c.99.2 4 15.2 even 4
175.4.b.c.99.3 4 15.8 even 4
245.4.a.k.1.1 2 105.104 even 2
245.4.e.h.116.2 4 105.44 odd 6
245.4.e.h.226.2 4 105.74 odd 6
245.4.e.i.116.2 4 105.89 even 6
245.4.e.i.226.2 4 105.59 even 6
315.4.a.f.1.2 2 5.4 even 2
560.4.a.r.1.1 2 60.59 even 2
1225.4.a.m.1.2 2 21.20 even 2
1575.4.a.z.1.1 2 1.1 even 1 trivial
2205.4.a.u.1.2 2 35.34 odd 2
2240.4.a.bn.1.1 2 120.29 odd 2
2240.4.a.bo.1.2 2 120.59 even 2