# Properties

 Label 2240.4.a.bn.1.1 Level $2240$ Weight $4$ Character 2240.1 Self dual yes Analytic conductor $132.164$ 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] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2240.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: $$132.164278413$$ Analytic rank: $$1$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 $$\chi$$ $$=$$ 2240.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.65685 q^{3} +5.00000 q^{5} -7.00000 q^{7} +17.3137 q^{9} +O(q^{10})$$ $$q-6.65685 q^{3} +5.00000 q^{5} -7.00000 q^{7} +17.3137 q^{9} -38.2548 q^{11} -19.3431 q^{13} -33.2843 q^{15} -87.2254 q^{17} +44.2254 q^{19} +46.5980 q^{21} +218.167 q^{23} +25.0000 q^{25} +64.4802 q^{27} +46.9411 q^{29} +194.558 q^{31} +254.657 q^{33} -35.0000 q^{35} -366.853 q^{37} +128.765 q^{39} -339.362 q^{41} +226.167 q^{43} +86.5685 q^{45} +11.6762 q^{47} +49.0000 q^{49} +580.647 q^{51} +209.019 q^{53} -191.274 q^{55} -294.402 q^{57} +616.000 q^{59} -320.735 q^{61} -121.196 q^{63} -96.7157 q^{65} -14.5097 q^{67} -1452.30 q^{69} -952.000 q^{71} +824.489 q^{73} -166.421 q^{75} +267.784 q^{77} +156.275 q^{79} -896.706 q^{81} +1036.53 q^{83} -436.127 q^{85} -312.480 q^{87} -170.225 q^{89} +135.402 q^{91} -1295.15 q^{93} +221.127 q^{95} +1059.87 q^{97} -662.333 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 10 q^{5} - 14 q^{7} + 12 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 10 * q^5 - 14 * q^7 + 12 * q^9 $$2 q - 2 q^{3} + 10 q^{5} - 14 q^{7} + 12 q^{9} + 14 q^{11} - 50 q^{13} - 10 q^{15} - 50 q^{17} - 36 q^{19} + 14 q^{21} + 244 q^{23} + 50 q^{25} - 86 q^{27} + 26 q^{29} - 120 q^{31} + 498 q^{33} - 70 q^{35} - 564 q^{37} - 14 q^{39} - 328 q^{41} + 260 q^{43} + 60 q^{45} - 350 q^{47} + 98 q^{49} + 754 q^{51} + 56 q^{53} + 70 q^{55} - 668 q^{57} + 1232 q^{59} - 336 q^{61} - 84 q^{63} - 250 q^{65} + 152 q^{67} - 1332 q^{69} - 1904 q^{71} + 676 q^{73} - 50 q^{75} - 98 q^{77} + 1014 q^{79} - 1454 q^{81} + 376 q^{83} - 250 q^{85} - 410 q^{87} - 216 q^{89} + 350 q^{91} - 2760 q^{93} - 180 q^{95} + 2742 q^{97} - 940 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 10 * q^5 - 14 * q^7 + 12 * q^9 + 14 * q^11 - 50 * q^13 - 10 * q^15 - 50 * q^17 - 36 * q^19 + 14 * q^21 + 244 * q^23 + 50 * q^25 - 86 * q^27 + 26 * q^29 - 120 * q^31 + 498 * q^33 - 70 * q^35 - 564 * q^37 - 14 * q^39 - 328 * q^41 + 260 * q^43 + 60 * q^45 - 350 * q^47 + 98 * q^49 + 754 * q^51 + 56 * q^53 + 70 * q^55 - 668 * q^57 + 1232 * q^59 - 336 * q^61 - 84 * q^63 - 250 * q^65 + 152 * q^67 - 1332 * q^69 - 1904 * q^71 + 676 * q^73 - 50 * q^75 - 98 * q^77 + 1014 * q^79 - 1454 * q^81 + 376 * q^83 - 250 * q^85 - 410 * q^87 - 216 * q^89 + 350 * q^91 - 2760 * q^93 - 180 * q^95 + 2742 * q^97 - 940 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −6.65685 −1.28111 −0.640556 0.767911i $$-0.721296\pi$$
−0.640556 + 0.767911i $$0.721296\pi$$
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ 0 0
$$9$$ 17.3137 0.641248
$$10$$ 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 $$0.566155\pi$$
$$14$$ 0 0
$$15$$ −33.2843 −0.572931
$$16$$ 0 0
$$17$$ −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 $$-0.413968\pi$$
0.267000 + 0.963697i $$0.413968\pi$$
$$20$$ 0 0
$$21$$ 46.5980 0.484215
$$22$$ 0 0
$$23$$ 218.167 1.97786 0.988932 0.148371i $$-0.0474028\pi$$
0.988932 + 0.148371i $$0.0474028\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 64.4802 0.459601
$$28$$ 0 0
$$29$$ 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 $$0.309413\pi$$
$$32$$ 0 0
$$33$$ 254.657 1.34334
$$34$$ 0 0
$$35$$ −35.0000 −0.169031
$$36$$ 0 0
$$37$$ −366.853 −1.63001 −0.815003 0.579457i $$-0.803265\pi$$
−0.815003 + 0.579457i $$0.803265\pi$$
$$38$$ 0 0
$$39$$ 128.765 0.528688
$$40$$ 0 0
$$41$$ −339.362 −1.29267 −0.646336 0.763053i $$-0.723699\pi$$
−0.646336 + 0.763053i $$0.723699\pi$$
$$42$$ 0 0
$$43$$ 226.167 0.802095 0.401047 0.916057i $$-0.368646\pi$$
0.401047 + 0.916057i $$0.368646\pi$$
$$44$$ 0 0
$$45$$ 86.5685 0.286775
$$46$$ 0 0
$$47$$ 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
$$51$$ 580.647 1.59425
$$52$$ 0 0
$$53$$ 209.019 0.541717 0.270859 0.962619i $$-0.412692\pi$$
0.270859 + 0.962619i $$0.412692\pi$$
$$54$$ 0 0
$$55$$ −191.274 −0.468935
$$56$$ 0 0
$$57$$ −294.402 −0.684114
$$58$$ 0 0
$$59$$ 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 $$-0.609279\pi$$
−0.336606 + 0.941646i $$0.609279\pi$$
$$62$$ 0 0
$$63$$ −121.196 −0.242369
$$64$$ 0 0
$$65$$ −96.7157 −0.184556
$$66$$ 0 0
$$67$$ −14.5097 −0.0264573 −0.0132286 0.999912i $$-0.504211\pi$$
−0.0132286 + 0.999912i $$0.504211\pi$$
$$68$$ 0 0
$$69$$ −1452.30 −2.53387
$$70$$ 0 0
$$71$$ −952.000 −1.59129 −0.795645 0.605763i $$-0.792868\pi$$
−0.795645 + 0.605763i $$0.792868\pi$$
$$72$$ 0 0
$$73$$ 824.489 1.32191 0.660953 0.750427i $$-0.270152\pi$$
0.660953 + 0.750427i $$0.270152\pi$$
$$74$$ 0 0
$$75$$ −166.421 −0.256222
$$76$$ 0 0
$$77$$ 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
$$81$$ −896.706 −1.23005
$$82$$ 0 0
$$83$$ 1036.53 1.37077 0.685384 0.728182i $$-0.259634\pi$$
0.685384 + 0.728182i $$0.259634\pi$$
$$84$$ 0 0
$$85$$ −436.127 −0.556525
$$86$$ 0 0
$$87$$ −312.480 −0.385074
$$88$$ 0 0
$$89$$ −170.225 −0.202740 −0.101370 0.994849i $$-0.532323\pi$$
−0.101370 + 0.994849i $$0.532323\pi$$
$$90$$ 0 0
$$91$$ 135.402 0.155978
$$92$$ 0 0
$$93$$ −1295.15 −1.44409
$$94$$ 0 0
$$95$$ 221.127 0.238812
$$96$$ 0 0
$$97$$ 1059.87 1.10942 0.554710 0.832044i $$-0.312829\pi$$
0.554710 + 0.832044i $$0.312829\pi$$
$$98$$ 0 0
$$99$$ −662.333 −0.672394
$$100$$ 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 $$-0.796960\pi$$
−0.803367 + 0.595484i $$0.796960\pi$$
$$104$$ 0 0
$$105$$ 232.990 0.216547
$$106$$ 0 0
$$107$$ −1506.88 −1.36146 −0.680728 0.732537i $$-0.738336\pi$$
−0.680728 + 0.732537i $$0.738336\pi$$
$$108$$ 0 0
$$109$$ 1252.41 1.10054 0.550271 0.834986i $$-0.314524\pi$$
0.550271 + 0.834986i $$0.314524\pi$$
$$110$$ 0 0
$$111$$ 2442.09 2.08822
$$112$$ 0 0
$$113$$ 1370.20 1.14069 0.570345 0.821405i $$-0.306810\pi$$
0.570345 + 0.821405i $$0.306810\pi$$
$$114$$ 0 0
$$115$$ 1090.83 0.884528
$$116$$ 0 0
$$117$$ −334.902 −0.264630
$$118$$ 0 0
$$119$$ 610.578 0.470349
$$120$$ 0 0
$$121$$ 132.432 0.0994984
$$122$$ 0 0
$$123$$ 2259.09 1.65606
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1213.49 0.847873 0.423936 0.905692i $$-0.360648\pi$$
0.423936 + 0.905692i $$0.360648\pi$$
$$128$$ 0 0
$$129$$ −1505.56 −1.02757
$$130$$ 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
$$134$$ 0 0
$$135$$ 322.401 0.205540
$$136$$ 0 0
$$137$$ 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 $$-0.551506\pi$$
−0.161107 + 0.986937i $$0.551506\pi$$
$$140$$ 0 0
$$141$$ −77.7267 −0.0464239
$$142$$ 0 0
$$143$$ 739.969 0.432722
$$144$$ 0 0
$$145$$ 234.706 0.134422
$$146$$ 0 0
$$147$$ −326.186 −0.183016
$$148$$ 0 0
$$149$$ 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 $$0.410528\pi$$
$$152$$ 0 0
$$153$$ −1510.20 −0.797987
$$154$$ 0 0
$$155$$ 972.792 0.504107
$$156$$ 0 0
$$157$$ −525.098 −0.266926 −0.133463 0.991054i $$-0.542610\pi$$
−0.133463 + 0.991054i $$0.542610\pi$$
$$158$$ 0 0
$$159$$ −1391.41 −0.694001
$$160$$ 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 $$0.577441\pi$$
$$164$$ 0 0
$$165$$ 1273.28 0.600758
$$166$$ 0 0
$$167$$ −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
$$171$$ 765.706 0.342427
$$172$$ 0 0
$$173$$ −3293.65 −1.44747 −0.723733 0.690080i $$-0.757575\pi$$
−0.723733 + 0.690080i $$0.757575\pi$$
$$174$$ 0 0
$$175$$ −175.000 −0.0755929
$$176$$ 0 0
$$177$$ −4100.62 −1.74137
$$178$$ 0 0
$$179$$ −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 $$-0.597072\pi$$
−0.300257 + 0.953858i $$0.597072\pi$$
$$182$$ 0 0
$$183$$ 2135.09 0.862460
$$184$$ 0 0
$$185$$ −1834.26 −0.728961
$$186$$ 0 0
$$187$$ 3336.79 1.30487
$$188$$ 0 0
$$189$$ −451.362 −0.173713
$$190$$ 0 0
$$191$$ −374.923 −0.142034 −0.0710169 0.997475i $$-0.522624\pi$$
−0.0710169 + 0.997475i $$0.522624\pi$$
$$192$$ 0 0
$$193$$ 733.028 0.273391 0.136696 0.990613i $$-0.456352\pi$$
0.136696 + 0.990613i $$0.456352\pi$$
$$194$$ 0 0
$$195$$ 643.823 0.236436
$$196$$ 0 0
$$197$$ 2093.24 0.757043 0.378521 0.925593i $$-0.376433\pi$$
0.378521 + 0.925593i $$0.376433\pi$$
$$198$$ 0 0
$$199$$ 2865.04 1.02059 0.510295 0.860000i $$-0.329536\pi$$
0.510295 + 0.860000i $$0.329536\pi$$
$$200$$ 0 0
$$201$$ 96.5887 0.0338948
$$202$$ 0 0
$$203$$ −328.588 −0.113608
$$204$$ 0 0
$$205$$ −1696.81 −0.578100
$$206$$ 0 0
$$207$$ 3777.27 1.26830
$$208$$ 0 0
$$209$$ −1691.84 −0.559936
$$210$$ 0 0
$$211$$ −5643.65 −1.84135 −0.920674 0.390331i $$-0.872360\pi$$
−0.920674 + 0.390331i $$0.872360\pi$$
$$212$$ 0 0
$$213$$ 6337.33 2.03862
$$214$$ 0 0
$$215$$ 1130.83 0.358708
$$216$$ 0 0
$$217$$ −1361.91 −0.426048
$$218$$ 0 0
$$219$$ −5488.51 −1.69351
$$220$$ 0 0
$$221$$ 1687.21 0.513549
$$222$$ 0 0
$$223$$ −6369.16 −1.91260 −0.956302 0.292381i $$-0.905552\pi$$
−0.956302 + 0.292381i $$0.905552\pi$$
$$224$$ 0 0
$$225$$ 432.843 0.128250
$$226$$ 0 0
$$227$$ 1015.67 0.296972 0.148486 0.988914i $$-0.452560\pi$$
0.148486 + 0.988914i $$0.452560\pi$$
$$228$$ 0 0
$$229$$ −4108.35 −1.18554 −0.592768 0.805373i $$-0.701965\pi$$
−0.592768 + 0.805373i $$0.701965\pi$$
$$230$$ 0 0
$$231$$ −1782.60 −0.507733
$$232$$ 0 0
$$233$$ 608.431 0.171071 0.0855357 0.996335i $$-0.472740\pi$$
0.0855357 + 0.996335i $$0.472740\pi$$
$$234$$ 0 0
$$235$$ 58.3810 0.0162058
$$236$$ 0 0
$$237$$ −1040.30 −0.285126
$$238$$ 0 0
$$239$$ −5054.44 −1.36797 −0.683985 0.729496i $$-0.739755\pi$$
−0.683985 + 0.729496i $$0.739755\pi$$
$$240$$ 0 0
$$241$$ 4.86782 0.00130109 0.000650547 1.00000i $$-0.499793\pi$$
0.000650547 1.00000i $$0.499793\pi$$
$$242$$ 0 0
$$243$$ 4228.27 1.11623
$$244$$ 0 0
$$245$$ 245.000 0.0638877
$$246$$ 0 0
$$247$$ −855.458 −0.220370
$$248$$ 0 0
$$249$$ −6900.02 −1.75611
$$250$$ 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
$$254$$ 0 0
$$255$$ 2903.23 0.712971
$$256$$ 0 0
$$257$$ −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
$$261$$ 812.725 0.192745
$$262$$ 0 0
$$263$$ −1199.09 −0.281138 −0.140569 0.990071i $$-0.544893\pi$$
−0.140569 + 0.990071i $$0.544893\pi$$
$$264$$ 0 0
$$265$$ 1045.10 0.242263
$$266$$ 0 0
$$267$$ 1133.17 0.259733
$$268$$ 0 0
$$269$$ −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 $$0.532024\pi$$
$$272$$ 0 0
$$273$$ −901.352 −0.199825
$$274$$ 0 0
$$275$$ −956.371 −0.209714
$$276$$ 0 0
$$277$$ 386.562 0.0838492 0.0419246 0.999121i $$-0.486651\pi$$
0.0419246 + 0.999121i $$0.486651\pi$$
$$278$$ 0 0
$$279$$ 3368.53 0.722826
$$280$$ 0 0
$$281$$ −3335.10 −0.708025 −0.354013 0.935241i $$-0.615183\pi$$
−0.354013 + 0.935241i $$0.615183\pi$$
$$282$$ 0 0
$$283$$ −5412.26 −1.13684 −0.568419 0.822739i $$-0.692445\pi$$
−0.568419 + 0.822739i $$0.692445\pi$$
$$284$$ 0 0
$$285$$ −1472.01 −0.305945
$$286$$ 0 0
$$287$$ 2375.54 0.488584
$$288$$ 0 0
$$289$$ 2695.27 0.548600
$$290$$ 0 0
$$291$$ −7055.42 −1.42129
$$292$$ 0 0
$$293$$ −282.211 −0.0562695 −0.0281347 0.999604i $$-0.508957\pi$$
−0.0281347 + 0.999604i $$0.508957\pi$$
$$294$$ 0 0
$$295$$ 3080.00 0.607880
$$296$$ 0 0
$$297$$ −2466.68 −0.481924
$$298$$ 0 0
$$299$$ −4220.03 −0.816222
$$300$$ 0 0
$$301$$ −1583.17 −0.303163
$$302$$ 0 0
$$303$$ −1609.85 −0.305226
$$304$$ 0 0
$$305$$ −1603.68 −0.301069
$$306$$ 0 0
$$307$$ −1919.67 −0.356878 −0.178439 0.983951i $$-0.557105\pi$$
−0.178439 + 0.983951i $$0.557105\pi$$
$$308$$ 0 0
$$309$$ 11180.7 2.05841
$$310$$ 0 0
$$311$$ 1213.31 0.221223 0.110612 0.993864i $$-0.464719\pi$$
0.110612 + 0.993864i $$0.464719\pi$$
$$312$$ 0 0
$$313$$ −1434.00 −0.258960 −0.129480 0.991582i $$-0.541331\pi$$
−0.129480 + 0.991582i $$0.541331\pi$$
$$314$$ 0 0
$$315$$ −605.980 −0.108391
$$316$$ 0 0
$$317$$ −6496.95 −1.15112 −0.575560 0.817760i $$-0.695216\pi$$
−0.575560 + 0.817760i $$0.695216\pi$$
$$318$$ 0 0
$$319$$ −1795.72 −0.315176
$$320$$ 0 0
$$321$$ 10031.1 1.74418
$$322$$ 0 0
$$323$$ −3857.58 −0.664524
$$324$$ 0 0
$$325$$ −483.579 −0.0825357
$$326$$ 0 0
$$327$$ −8337.11 −1.40992
$$328$$ 0 0
$$329$$ −81.7333 −0.0136964
$$330$$ 0 0
$$331$$ 9683.88 1.60808 0.804039 0.594576i $$-0.202680\pi$$
0.804039 + 0.594576i $$0.202680\pi$$
$$332$$ 0 0
$$333$$ −6351.58 −1.04524
$$334$$ 0 0
$$335$$ −72.5483 −0.0118321
$$336$$ 0 0
$$337$$ 29.1319 0.00470895 0.00235447 0.999997i $$-0.499251\pi$$
0.00235447 + 0.999997i $$0.499251\pi$$
$$338$$ 0 0
$$339$$ −9121.24 −1.46135
$$340$$ 0 0
$$341$$ −7442.80 −1.18197
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 0 0
$$345$$ −7261.51 −1.13318
$$346$$ 0 0
$$347$$ 7848.58 1.21422 0.607110 0.794618i $$-0.292329\pi$$
0.607110 + 0.794618i $$0.292329\pi$$
$$348$$ 0 0
$$349$$ 10269.6 1.57513 0.787567 0.616229i $$-0.211341\pi$$
0.787567 + 0.616229i $$0.211341\pi$$
$$350$$ 0 0
$$351$$ −1247.25 −0.189668
$$352$$ 0 0
$$353$$ 2799.93 0.422168 0.211084 0.977468i $$-0.432301\pi$$
0.211084 + 0.977468i $$0.432301\pi$$
$$354$$ 0 0
$$355$$ −4760.00 −0.711647
$$356$$ 0 0
$$357$$ −4064.53 −0.602570
$$358$$ 0 0
$$359$$ −3163.29 −0.465048 −0.232524 0.972591i $$-0.574698\pi$$
−0.232524 + 0.972591i $$0.574698\pi$$
$$360$$ 0 0
$$361$$ −4903.11 −0.714844
$$362$$ 0 0
$$363$$ −881.583 −0.127469
$$364$$ 0 0
$$365$$ 4122.45 0.591175
$$366$$ 0 0
$$367$$ 3182.85 0.452706 0.226353 0.974045i $$-0.427320\pi$$
0.226353 + 0.974045i $$0.427320\pi$$
$$368$$ 0 0
$$369$$ −5875.62 −0.828923
$$370$$ 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 $$0.441901\pi$$
$$374$$ 0 0
$$375$$ −832.107 −0.114586
$$376$$ 0 0
$$377$$ −907.989 −0.124042
$$378$$ 0 0
$$379$$ 672.434 0.0911362 0.0455681 0.998961i $$-0.485490\pi$$
0.0455681 + 0.998961i $$0.485490\pi$$
$$380$$ 0 0
$$381$$ −8078.03 −1.08622
$$382$$ 0 0
$$383$$ 1169.86 0.156075 0.0780377 0.996950i $$-0.475135\pi$$
0.0780377 + 0.996950i $$0.475135\pi$$
$$384$$ 0 0
$$385$$ 1338.92 0.177241
$$386$$ 0 0
$$387$$ 3915.78 0.514342
$$388$$ 0 0
$$389$$ 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
$$392$$ 0 0
$$393$$ −13196.7 −1.69385
$$394$$ 0 0
$$395$$ 781.375 0.0995323
$$396$$ 0 0
$$397$$ 1985.93 0.251060 0.125530 0.992090i $$-0.459937\pi$$
0.125530 + 0.992090i $$0.459937\pi$$
$$398$$ 0 0
$$399$$ 2060.81 0.258571
$$400$$ 0 0
$$401$$ −4172.38 −0.519597 −0.259799 0.965663i $$-0.583656\pi$$
−0.259799 + 0.965663i $$0.583656\pi$$
$$402$$ 0 0
$$403$$ −3763.37 −0.465178
$$404$$ 0 0
$$405$$ −4483.53 −0.550095
$$406$$ 0 0
$$407$$ 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
$$411$$ −14718.0 −1.76638
$$412$$ 0 0
$$413$$ −4312.00 −0.513752
$$414$$ 0 0
$$415$$ 5182.64 0.613026
$$416$$ 0 0
$$417$$ 3515.08 0.412791
$$418$$ 0 0
$$419$$ −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 $$-0.786963\pi$$
−0.784272 + 0.620417i $$0.786963\pi$$
$$422$$ 0 0
$$423$$ 202.158 0.0232370
$$424$$ 0 0
$$425$$ −2180.63 −0.248885
$$426$$ 0 0
$$427$$ 2245.15 0.254450
$$428$$ 0 0
$$429$$ −4925.86 −0.554366
$$430$$ 0 0
$$431$$ −6429.25 −0.718530 −0.359265 0.933236i $$-0.616973\pi$$
−0.359265 + 0.933236i $$0.616973\pi$$
$$432$$ 0 0
$$433$$ 8022.03 0.890333 0.445166 0.895448i $$-0.353145\pi$$
0.445166 + 0.895448i $$0.353145\pi$$
$$434$$ 0 0
$$435$$ −1562.40 −0.172210
$$436$$ 0 0
$$437$$ 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
$$441$$ 848.372 0.0916069
$$442$$ 0 0
$$443$$ 5486.21 0.588392 0.294196 0.955745i $$-0.404948\pi$$
0.294196 + 0.955745i $$0.404948\pi$$
$$444$$ 0 0
$$445$$ −851.127 −0.0906681
$$446$$ 0 0
$$447$$ −2185.92 −0.231299
$$448$$ 0 0
$$449$$ −7232.67 −0.760203 −0.380101 0.924945i $$-0.624111\pi$$
−0.380101 + 0.924945i $$0.624111\pi$$
$$450$$ 0 0
$$451$$ 12982.3 1.35546
$$452$$ 0 0
$$453$$ −6852.76 −0.710752
$$454$$ 0 0
$$455$$ 677.010 0.0697554
$$456$$ 0 0
$$457$$ −2900.51 −0.296893 −0.148446 0.988920i $$-0.547427\pi$$
−0.148446 + 0.988920i $$0.547427\pi$$
$$458$$ 0 0
$$459$$ −5624.31 −0.571940
$$460$$ 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 $$-0.898605\pi$$
−0.949693 + 0.313183i $$0.898605\pi$$
$$464$$ 0 0
$$465$$ −6475.74 −0.645817
$$466$$ 0 0
$$467$$ 6776.71 0.671496 0.335748 0.941952i $$-0.391011\pi$$
0.335748 + 0.941952i $$0.391011\pi$$
$$468$$ 0 0
$$469$$ 101.568 0.00999991
$$470$$ 0 0
$$471$$ 3495.50 0.341962
$$472$$ 0 0
$$473$$ −8651.96 −0.841052
$$474$$ 0 0
$$475$$ 1105.63 0.106800
$$476$$ 0 0
$$477$$ 3618.90 0.347375
$$478$$ 0 0
$$479$$ 2397.32 0.228677 0.114338 0.993442i $$-0.463525\pi$$
0.114338 + 0.993442i $$0.463525\pi$$
$$480$$ 0 0
$$481$$ 7096.09 0.672669
$$482$$ 0 0
$$483$$ 10166.1 0.957711
$$484$$ 0 0
$$485$$ 5299.37 0.496148
$$486$$ 0 0
$$487$$ 5586.17 0.519781 0.259890 0.965638i $$-0.416314\pi$$
0.259890 + 0.965638i $$0.416314\pi$$
$$488$$ 0 0
$$489$$ 6674.33 0.617227
$$490$$ 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
$$494$$ 0 0
$$495$$ −3311.67 −0.300704
$$496$$ 0 0
$$497$$ 6664.00 0.601451
$$498$$ 0 0
$$499$$ −598.965 −0.0537342 −0.0268671 0.999639i $$-0.508553\pi$$
−0.0268671 + 0.999639i $$0.508553\pi$$
$$500$$ 0 0
$$501$$ 2392.62 0.213362
$$502$$ 0 0
$$503$$ 4426.76 0.392405 0.196202 0.980563i $$-0.437139\pi$$
0.196202 + 0.980563i $$0.437139\pi$$
$$504$$ 0 0
$$505$$ 1209.17 0.106549
$$506$$ 0 0
$$507$$ 12134.4 1.06293
$$508$$ 0 0
$$509$$ −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
$$512$$ 0 0
$$513$$ 2851.66 0.245427
$$514$$ 0 0
$$515$$ −8397.88 −0.718553
$$516$$ 0 0
$$517$$ −446.671 −0.0379972
$$518$$ 0 0
$$519$$ 21925.4 1.85437
$$520$$ 0 0
$$521$$ 8662.79 0.728453 0.364226 0.931310i $$-0.381333\pi$$
0.364226 + 0.931310i $$0.381333\pi$$
$$522$$ 0 0
$$523$$ 7770.40 0.649667 0.324833 0.945771i $$-0.394692\pi$$
0.324833 + 0.945771i $$0.394692\pi$$
$$524$$ 0 0
$$525$$ 1164.95 0.0968430
$$526$$ 0 0
$$527$$ −16970.4 −1.40274
$$528$$ 0 0
$$529$$ 35429.6 2.91194
$$530$$ 0 0
$$531$$ 10665.2 0.871624
$$532$$ 0 0
$$533$$ 6564.34 0.533458
$$534$$ 0 0
$$535$$ −7534.41 −0.608861
$$536$$ 0 0
$$537$$ 19829.6 1.59350
$$538$$ 0 0
$$539$$ −1874.49 −0.149796
$$540$$ 0 0
$$541$$ −21641.0 −1.71981 −0.859906 0.510453i $$-0.829478\pi$$
−0.859906 + 0.510453i $$0.829478\pi$$
$$542$$ 0 0
$$543$$ 9734.41 0.769325
$$544$$ 0 0
$$545$$ 6262.05 0.492177
$$546$$ 0 0
$$547$$ −7489.29 −0.585409 −0.292705 0.956203i $$-0.594555\pi$$
−0.292705 + 0.956203i $$0.594555\pi$$
$$548$$ 0 0
$$549$$ −5553.11 −0.431696
$$550$$ 0 0
$$551$$ 2075.99 0.160508
$$552$$ 0 0
$$553$$ −1093.93 −0.0841201
$$554$$ 0 0
$$555$$ 12210.4 0.933881
$$556$$ 0 0
$$557$$ −25297.9 −1.92443 −0.962214 0.272295i $$-0.912217\pi$$
−0.962214 + 0.272295i $$0.912217\pi$$
$$558$$ 0 0
$$559$$ −4374.77 −0.331007
$$560$$ 0 0
$$561$$ −22212.5 −1.67168
$$562$$ 0 0
$$563$$ −15661.3 −1.17237 −0.586186 0.810177i $$-0.699371\pi$$
−0.586186 + 0.810177i $$0.699371\pi$$
$$564$$ 0 0
$$565$$ 6851.02 0.510132
$$566$$ 0 0
$$567$$ 6276.94 0.464915
$$568$$ 0 0
$$569$$ −9982.75 −0.735498 −0.367749 0.929925i $$-0.619871\pi$$
−0.367749 + 0.929925i $$0.619871\pi$$
$$570$$ 0 0
$$571$$ 11583.6 0.848966 0.424483 0.905436i $$-0.360456\pi$$
0.424483 + 0.905436i $$0.360456\pi$$
$$572$$ 0 0
$$573$$ 2495.81 0.181961
$$574$$ 0 0
$$575$$ 5454.16 0.395573
$$576$$ 0 0
$$577$$ −595.378 −0.0429565 −0.0214783 0.999769i $$-0.506837\pi$$
−0.0214783 + 0.999769i $$0.506837\pi$$
$$578$$ 0 0
$$579$$ −4879.66 −0.350245
$$580$$ 0 0
$$581$$ −7255.70 −0.518102
$$582$$ 0 0
$$583$$ −7996.00 −0.568028
$$584$$ 0 0
$$585$$ −1674.51 −0.118346
$$586$$ 0 0
$$587$$ 15750.3 1.10747 0.553736 0.832693i $$-0.313202\pi$$
0.553736 + 0.832693i $$0.313202\pi$$
$$588$$ 0 0
$$589$$ 8604.42 0.601934
$$590$$ 0 0
$$591$$ −13934.4 −0.969856
$$592$$ 0 0
$$593$$ −417.878 −0.0289379 −0.0144690 0.999895i $$-0.504606\pi$$
−0.0144690 + 0.999895i $$0.504606\pi$$
$$594$$ 0 0
$$595$$ 3052.89 0.210347
$$596$$ 0 0
$$597$$ −19072.2 −1.30749
$$598$$ 0 0
$$599$$ −19997.3 −1.36406 −0.682028 0.731326i $$-0.738902\pi$$
−0.682028 + 0.731326i $$0.738902\pi$$
$$600$$ 0 0
$$601$$ −15992.6 −1.08545 −0.542723 0.839912i $$-0.682607\pi$$
−0.542723 + 0.839912i $$0.682607\pi$$
$$602$$ 0 0
$$603$$ −251.216 −0.0169657
$$604$$ 0 0
$$605$$ 662.162 0.0444970
$$606$$ 0 0
$$607$$ 14159.2 0.946793 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$608$$ 0 0
$$609$$ 2187.36 0.145544
$$610$$ 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 $$0.451264\pi$$
$$614$$ 0 0
$$615$$ 11295.4 0.740611
$$616$$ 0 0
$$617$$ −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 $$-0.631556\pi$$
−0.401629 + 0.915803i $$0.631556\pi$$
$$620$$ 0 0
$$621$$ 14067.4 0.909028
$$622$$ 0 0
$$623$$ 1191.58 0.0766285
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 11262.3 0.717341
$$628$$ 0 0
$$629$$ 31998.9 2.02842
$$630$$ 0 0
$$631$$ 13980.2 0.882002 0.441001 0.897507i $$-0.354623\pi$$
0.441001 + 0.897507i $$0.354623\pi$$
$$632$$ 0 0
$$633$$ 37568.9 2.35897
$$634$$ 0 0
$$635$$ 6067.45 0.379180
$$636$$ 0 0
$$637$$ −947.814 −0.0589541
$$638$$ 0 0
$$639$$ −16482.7 −1.02041
$$640$$ 0 0
$$641$$ −16060.9 −0.989655 −0.494828 0.868991i $$-0.664769\pi$$
−0.494828 + 0.868991i $$0.664769\pi$$
$$642$$ 0 0
$$643$$ 4502.17 0.276125 0.138063 0.990424i $$-0.455913\pi$$
0.138063 + 0.990424i $$0.455913\pi$$
$$644$$ 0 0
$$645$$ −7527.79 −0.459545
$$646$$ 0 0
$$647$$ 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
$$651$$ 9066.03 0.545815
$$652$$ 0 0
$$653$$ −13013.6 −0.779882 −0.389941 0.920840i $$-0.627505\pi$$
−0.389941 + 0.920840i $$0.627505\pi$$
$$654$$ 0 0
$$655$$ 9912.09 0.591294
$$656$$ 0 0
$$657$$ 14275.0 0.847671
$$658$$ 0 0
$$659$$ −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 $$-0.412106\pi$$
0.272632 + 0.962118i $$0.412106\pi$$
$$662$$ 0 0
$$663$$ −11231.5 −0.657914
$$664$$ 0 0
$$665$$ −1547.89 −0.0902625
$$666$$ 0 0
$$667$$ 10241.0 0.594501
$$668$$ 0 0
$$669$$ 42398.6 2.45026
$$670$$ 0 0
$$671$$ 12269.7 0.705909
$$672$$ 0 0
$$673$$ −25067.2 −1.43576 −0.717882 0.696164i $$-0.754888\pi$$
−0.717882 + 0.696164i $$0.754888\pi$$
$$674$$ 0 0
$$675$$ 1612.01 0.0919202
$$676$$ 0 0
$$677$$ 22409.6 1.27219 0.636093 0.771613i $$-0.280550\pi$$
0.636093 + 0.771613i $$0.280550\pi$$
$$678$$ 0 0
$$679$$ −7419.11 −0.419322
$$680$$ 0 0
$$681$$ −6761.20 −0.380455
$$682$$ 0 0
$$683$$ 8757.53 0.490626 0.245313 0.969444i $$-0.421109\pi$$
0.245313 + 0.969444i $$0.421109\pi$$
$$684$$ 0 0
$$685$$ 11054.7 0.616613
$$686$$ 0 0
$$687$$ 27348.7 1.51880
$$688$$ 0 0
$$689$$ −4043.09 −0.223555
$$690$$ 0 0
$$691$$ −8468.42 −0.466214 −0.233107 0.972451i $$-0.574889\pi$$
−0.233107 + 0.972451i $$0.574889\pi$$
$$692$$ 0 0
$$693$$ 4636.33 0.254141
$$694$$ 0 0
$$695$$ −2640.19 −0.144098
$$696$$ 0 0
$$697$$ 29601.0 1.60864
$$698$$ 0 0
$$699$$ −4050.23 −0.219162
$$700$$ 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
$$704$$ 0 0
$$705$$ −388.633 −0.0207614
$$706$$ 0 0
$$707$$ −1692.83 −0.0900503
$$708$$ 0 0
$$709$$ −19903.0 −1.05426 −0.527131 0.849784i $$-0.676732\pi$$
−0.527131 + 0.849784i $$0.676732\pi$$
$$710$$ 0 0
$$711$$ 2705.70 0.142717
$$712$$ 0 0
$$713$$ 42446.1 2.22948
$$714$$ 0 0
$$715$$ 3699.84 0.193519
$$716$$ 0 0
$$717$$ 33646.7 1.75252
$$718$$ 0 0
$$719$$ −11073.1 −0.574347 −0.287174 0.957879i $$-0.592716\pi$$
−0.287174 + 0.957879i $$0.592716\pi$$
$$720$$ 0 0
$$721$$ 11757.0 0.607288
$$722$$ 0 0
$$723$$ −32.4043 −0.00166685
$$724$$ 0 0
$$725$$ 1173.53 0.0601155
$$726$$ 0 0
$$727$$ 31652.7 1.61476 0.807382 0.590029i $$-0.200884\pi$$
0.807382 + 0.590029i $$0.200884\pi$$
$$728$$ 0 0
$$729$$ −3935.94 −0.199967
$$730$$ 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 $$0.640523\pi$$
$$734$$ 0 0
$$735$$ −1630.93 −0.0818473
$$736$$ 0 0
$$737$$ 555.065 0.0277423
$$738$$ 0 0
$$739$$ 11616.6 0.578245 0.289123 0.957292i $$-0.406636\pi$$
0.289123 + 0.957292i $$0.406636\pi$$
$$740$$ 0 0
$$741$$ 5694.66 0.282319
$$742$$ 0 0
$$743$$ 15928.0 0.786464 0.393232 0.919439i $$-0.371357\pi$$
0.393232 + 0.919439i $$0.371357\pi$$
$$744$$ 0 0
$$745$$ 1641.86 0.0807423
$$746$$ 0 0
$$747$$ 17946.1 0.879003
$$748$$ 0 0
$$749$$ 10548.2 0.514582
$$750$$ 0 0
$$751$$ 25571.9 1.24252 0.621260 0.783604i $$-0.286621\pi$$
0.621260 + 0.783604i $$0.286621\pi$$
$$752$$ 0 0
$$753$$ −3647.43 −0.176520
$$754$$ 0 0
$$755$$ 5147.14 0.248111
$$756$$ 0 0
$$757$$ −6202.41 −0.297794 −0.148897 0.988853i $$-0.547572\pi$$
−0.148897 + 0.988853i $$0.547572\pi$$
$$758$$ 0 0
$$759$$ 55557.6 2.65693
$$760$$ 0 0
$$761$$ −29199.1 −1.39089 −0.695444 0.718580i $$-0.744792\pi$$
−0.695444 + 0.718580i $$0.744792\pi$$
$$762$$ 0 0
$$763$$ −8766.87 −0.415966
$$764$$ 0 0
$$765$$ −7550.98 −0.356871
$$766$$ 0 0
$$767$$ −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
$$771$$ 11813.3 0.551812
$$772$$ 0 0
$$773$$ 25544.8 1.18859 0.594296 0.804246i $$-0.297431\pi$$
0.594296 + 0.804246i $$0.297431\pi$$
$$774$$ 0 0
$$775$$ 4863.96 0.225443
$$776$$ 0 0
$$777$$ −17094.6 −0.789273
$$778$$ 0 0
$$779$$ −15008.4 −0.690286
$$780$$ 0 0
$$781$$ 36418.6 1.66858
$$782$$ 0 0
$$783$$ 3026.77 0.138146
$$784$$ 0 0
$$785$$ −2625.49 −0.119373
$$786$$ 0 0
$$787$$ 37223.2 1.68598 0.842989 0.537931i $$-0.180794\pi$$
0.842989 + 0.537931i $$0.180794\pi$$
$$788$$ 0 0
$$789$$ 7982.19 0.360169
$$790$$ 0 0
$$791$$ −9591.42 −0.431140
$$792$$ 0 0
$$793$$ 6204.03 0.277820
$$794$$ 0 0
$$795$$ −6957.06 −0.310366
$$796$$ 0 0
$$797$$ 40384.6 1.79485 0.897425 0.441168i $$-0.145436\pi$$
0.897425 + 0.441168i $$0.145436\pi$$
$$798$$ 0 0
$$799$$ −1018.46 −0.0450945
$$800$$ 0 0
$$801$$ −2947.23 −0.130007
$$802$$ 0 0
$$803$$ −31540.7 −1.38611
$$804$$ 0 0
$$805$$ −7635.83 −0.334320
$$806$$ 0 0
$$807$$ 21636.7 0.943803
$$808$$ 0 0
$$809$$ −1955.76 −0.0849948 −0.0424974 0.999097i $$-0.513531\pi$$
−0.0424974 + 0.999097i $$0.513531\pi$$
$$810$$ 0 0
$$811$$ 34301.8 1.48520 0.742600 0.669735i $$-0.233592\pi$$
0.742600 + 0.669735i $$0.233592\pi$$
$$812$$ 0 0
$$813$$ 5965.49 0.257342
$$814$$ 0 0
$$815$$ −5013.13 −0.215463
$$816$$ 0 0
$$817$$ 10002.3 0.428319
$$818$$ 0 0
$$819$$ 2344.31 0.100021
$$820$$ 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 $$-0.650620\pi$$
−0.455724 + 0.890121i $$0.650620\pi$$
$$824$$ 0 0
$$825$$ 6366.42 0.268667
$$826$$ 0 0
$$827$$ −35220.6 −1.48094 −0.740471 0.672088i $$-0.765398\pi$$
−0.740471 + 0.672088i $$0.765398\pi$$
$$828$$ 0 0
$$829$$ −31365.5 −1.31408 −0.657039 0.753857i $$-0.728191\pi$$
−0.657039 + 0.753857i $$0.728191\pi$$
$$830$$ 0 0
$$831$$ −2573.28 −0.107420
$$832$$ 0 0
$$833$$ −4274.04 −0.177775
$$834$$ 0 0
$$835$$ −1797.11 −0.0744810
$$836$$ 0 0
$$837$$ 12545.2 0.518070
$$838$$ 0 0
$$839$$ −28287.1 −1.16398 −0.581990 0.813196i $$-0.697726\pi$$
−0.581990 + 0.813196i $$0.697726\pi$$
$$840$$ 0 0
$$841$$ −22185.5 −0.909653
$$842$$ 0 0
$$843$$ 22201.2 0.907060
$$844$$ 0 0
$$845$$ −9114.21 −0.371051
$$846$$ 0 0
$$847$$ −927.026 −0.0376068
$$848$$ 0 0
$$849$$ 36028.6 1.45642
$$850$$ 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 $$0.560449\pi$$
$$854$$ 0 0
$$855$$ 3828.53 0.153138
$$856$$ 0 0
$$857$$ −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 $$-0.539811\pi$$
−0.124743 + 0.992189i $$0.539811\pi$$
$$860$$ 0 0
$$861$$ −15813.6 −0.625931
$$862$$ 0 0
$$863$$ −4757.13 −0.187642 −0.0938208 0.995589i $$-0.529908\pi$$
−0.0938208 + 0.995589i $$0.529908\pi$$
$$864$$ 0 0
$$865$$ −16468.3 −0.647327
$$866$$ 0 0
$$867$$ −17942.0 −0.702818
$$868$$ 0 0
$$869$$ −5978.28 −0.233371
$$870$$ 0 0
$$871$$ 280.663 0.0109184
$$872$$ 0 0
$$873$$ 18350.3 0.711414
$$874$$ 0 0
$$875$$ −875.000 −0.0338062
$$876$$ 0 0
$$877$$ 30240.5 1.16437 0.582184 0.813057i $$-0.302198\pi$$
0.582184 + 0.813057i $$0.302198\pi$$
$$878$$ 0 0
$$879$$ 1878.64 0.0720875
$$880$$ 0 0
$$881$$ 44875.5 1.71611 0.858056 0.513556i $$-0.171672\pi$$
0.858056 + 0.513556i $$0.171672\pi$$
$$882$$ 0 0
$$883$$ −4892.13 −0.186448 −0.0932238 0.995645i $$-0.529717\pi$$
−0.0932238 + 0.995645i $$0.529717\pi$$
$$884$$ 0 0
$$885$$ −20503.1 −0.778762
$$886$$ 0 0
$$887$$ 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
$$891$$ 34303.3 1.28979
$$892$$ 0 0
$$893$$ 516.384 0.0193507
$$894$$ 0 0
$$895$$ −14894.1 −0.556263
$$896$$ 0 0
$$897$$ 28092.1 1.04567
$$898$$ 0 0
$$899$$ 9132.79 0.338816
$$900$$ 0 0
$$901$$ −18231.8 −0.674128
$$902$$ 0 0
$$903$$ 10538.9 0.388386
$$904$$ 0 0
$$905$$ −7311.57 −0.268558
$$906$$ 0 0
$$907$$ −23689.1 −0.867238 −0.433619 0.901096i $$-0.642764\pi$$
−0.433619 + 0.901096i $$0.642764\pi$$
$$908$$ 0 0
$$909$$ 4187.03 0.152778
$$910$$ 0 0
$$911$$ −13877.3 −0.504692 −0.252346 0.967637i $$-0.581202\pi$$
−0.252346 + 0.967637i $$0.581202\pi$$
$$912$$ 0 0
$$913$$ −39652.2 −1.43735
$$914$$ 0 0
$$915$$ 10675.4 0.385704
$$916$$ 0 0
$$917$$ −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
$$921$$ 12779.0 0.457201
$$922$$ 0 0
$$923$$ 18414.7 0.656692
$$924$$ 0 0
$$925$$ −9171.32 −0.326001
$$926$$ 0 0
$$927$$ −29079.7 −1.03032
$$928$$ 0 0
$$929$$ 16668.4 0.588668 0.294334 0.955703i $$-0.404902\pi$$
0.294334 + 0.955703i $$0.404902\pi$$
$$930$$ 0 0
$$931$$ 2167.04 0.0762857
$$932$$ 0 0
$$933$$ −8076.82 −0.283412
$$934$$ 0 0
$$935$$ 16684.0 0.583555
$$936$$ 0 0
$$937$$ 30384.9 1.05937 0.529685 0.848194i $$-0.322310\pi$$
0.529685 + 0.848194i $$0.322310\pi$$
$$938$$ 0 0
$$939$$ 9545.94 0.331757
$$940$$ 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
$$944$$ 0 0
$$945$$ −2256.81 −0.0776867
$$946$$ 0 0
$$947$$ −1788.41 −0.0613681 −0.0306840 0.999529i $$-0.509769\pi$$
−0.0306840 + 0.999529i $$0.509769\pi$$
$$948$$ 0 0
$$949$$ −15948.2 −0.545523
$$950$$ 0 0
$$951$$ 43249.2 1.47471
$$952$$ 0 0
$$953$$ 8578.60 0.291593 0.145796 0.989315i $$-0.453426\pi$$
0.145796 + 0.989315i $$0.453426\pi$$
$$954$$ 0 0
$$955$$ −1874.61 −0.0635194
$$956$$ 0 0
$$957$$ 11953.9 0.403776
$$958$$ 0 0
$$959$$ −15476.6 −0.521133
$$960$$ 0 0
$$961$$ 8061.99 0.270618
$$962$$ 0 0
$$963$$ −26089.7 −0.873031
$$964$$ 0 0
$$965$$ 3665.14 0.122264
$$966$$ 0 0
$$967$$ −55459.3 −1.84431 −0.922156 0.386818i $$-0.873574\pi$$
−0.922156 + 0.386818i $$0.873574\pi$$
$$968$$ 0 0
$$969$$ 25679.3 0.851330
$$970$$ 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
$$974$$ 0 0
$$975$$ 3219.11 0.105738
$$976$$ 0 0
$$977$$ 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
$$981$$ 21683.9 0.705721
$$982$$ 0 0
$$983$$ 7817.11 0.253639 0.126819 0.991926i $$-0.459523\pi$$
0.126819 + 0.991926i $$0.459523\pi$$
$$984$$ 0 0
$$985$$ 10466.2 0.338560
$$986$$ 0 0
$$987$$ 544.087 0.0175466
$$988$$ 0 0
$$989$$ 49342.0 1.58643
$$990$$ 0 0
$$991$$ 24501.6 0.785386 0.392693 0.919670i $$-0.371543\pi$$
0.392693 + 0.919670i $$0.371543\pi$$
$$992$$ 0 0
$$993$$ −64464.2 −2.06013
$$994$$ 0 0
$$995$$ 14325.2 0.456421
$$996$$ 0 0
$$997$$ 50696.0 1.61039 0.805195 0.593010i $$-0.202060\pi$$
0.805195 + 0.593010i $$0.202060\pi$$
$$998$$ 0 0
$$999$$ −23654.8 −0.749152
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 2240.4.a.bn.1.1 2
4.3 odd 2 2240.4.a.bo.1.2 2
8.3 odd 2 560.4.a.r.1.1 2
8.5 even 2 35.4.a.b.1.1 2
24.5 odd 2 315.4.a.f.1.2 2
40.13 odd 4 175.4.b.c.99.2 4
40.29 even 2 175.4.a.c.1.2 2
40.37 odd 4 175.4.b.c.99.3 4
56.5 odd 6 245.4.e.i.116.2 4
56.13 odd 2 245.4.a.k.1.1 2
56.37 even 6 245.4.e.h.116.2 4
56.45 odd 6 245.4.e.i.226.2 4
56.53 even 6 245.4.e.h.226.2 4
120.29 odd 2 1575.4.a.z.1.1 2
168.125 even 2 2205.4.a.u.1.2 2
280.69 odd 2 1225.4.a.m.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 8.5 even 2
175.4.a.c.1.2 2 40.29 even 2
175.4.b.c.99.2 4 40.13 odd 4
175.4.b.c.99.3 4 40.37 odd 4
245.4.a.k.1.1 2 56.13 odd 2
245.4.e.h.116.2 4 56.37 even 6
245.4.e.h.226.2 4 56.53 even 6
245.4.e.i.116.2 4 56.5 odd 6
245.4.e.i.226.2 4 56.45 odd 6
315.4.a.f.1.2 2 24.5 odd 2
560.4.a.r.1.1 2 8.3 odd 2
1225.4.a.m.1.2 2 280.69 odd 2
1575.4.a.z.1.1 2 120.29 odd 2
2205.4.a.u.1.2 2 168.125 even 2
2240.4.a.bn.1.1 2 1.1 even 1 trivial
2240.4.a.bo.1.2 2 4.3 odd 2