# Properties

 Label 405.2.a.i.1.3 Level $405$ Weight $2$ Character 405.1 Self dual yes Analytic conductor $3.234$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.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: $$3.23394128186$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-2.08613$$ of defining polynomial Character $$\chi$$ $$=$$ 405.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.08613 q^{2} +2.35194 q^{4} -1.00000 q^{5} +4.08613 q^{7} +0.734191 q^{8} +O(q^{10})$$ $$q+2.08613 q^{2} +2.35194 q^{4} -1.00000 q^{5} +4.08613 q^{7} +0.734191 q^{8} -2.08613 q^{10} +1.35194 q^{11} +0.648061 q^{13} +8.52420 q^{14} -3.17226 q^{16} +1.35194 q^{17} +0.648061 q^{19} -2.35194 q^{20} +2.82032 q^{22} -4.79001 q^{23} +1.00000 q^{25} +1.35194 q^{26} +9.61033 q^{28} -3.87614 q^{29} -7.69646 q^{31} -8.08613 q^{32} +2.82032 q^{34} -4.08613 q^{35} +7.52420 q^{37} +1.35194 q^{38} -0.734191 q^{40} +0.179679 q^{41} -0.820321 q^{43} +3.17968 q^{44} -9.99258 q^{46} -10.9065 q^{47} +9.69646 q^{49} +2.08613 q^{50} +1.52420 q^{52} -4.17226 q^{53} -1.35194 q^{55} +3.00000 q^{56} -8.08613 q^{58} -4.17226 q^{59} -3.82032 q^{61} -16.0558 q^{62} -10.5242 q^{64} -0.648061 q^{65} +8.14195 q^{67} +3.17968 q^{68} -8.52420 q^{70} +6.11644 q^{71} -12.3445 q^{73} +15.6965 q^{74} +1.52420 q^{76} +5.52420 q^{77} +10.3445 q^{79} +3.17226 q^{80} +0.374833 q^{82} +12.2584 q^{83} -1.35194 q^{85} -1.71130 q^{86} +0.992582 q^{88} +3.00000 q^{89} +2.64806 q^{91} -11.2658 q^{92} -22.7523 q^{94} -0.648061 q^{95} -13.5800 q^{97} +20.2281 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 5 * q^7 - 3 * q^8 $$3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29} + 8 q^{31} - 17 q^{32} - 4 q^{34} - 5 q^{35} + 6 q^{37} + 2 q^{38} + 3 q^{40} + 13 q^{41} + 10 q^{43} + 22 q^{44} - 3 q^{46} - 13 q^{47} - 2 q^{49} - q^{50} - 12 q^{52} + 2 q^{53} - 2 q^{55} + 9 q^{56} - 17 q^{58} + 2 q^{59} + q^{61} - 42 q^{62} - 15 q^{64} - 4 q^{65} + 11 q^{67} + 22 q^{68} - 9 q^{70} + 10 q^{71} - 8 q^{73} + 16 q^{74} - 12 q^{76} + 2 q^{79} - 5 q^{80} - 29 q^{82} + 15 q^{83} - 2 q^{85} - 28 q^{86} - 24 q^{88} + 9 q^{89} + 10 q^{91} - 39 q^{92} - 31 q^{94} - 4 q^{95} - 18 q^{97} + 40 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 5 * q^7 - 3 * q^8 + q^10 + 2 * q^11 + 4 * q^13 + 9 * q^14 + 5 * q^16 + 2 * q^17 + 4 * q^19 - 5 * q^20 - 4 * q^22 - 3 * q^23 + 3 * q^25 + 2 * q^26 + 5 * q^28 + 7 * q^29 + 8 * q^31 - 17 * q^32 - 4 * q^34 - 5 * q^35 + 6 * q^37 + 2 * q^38 + 3 * q^40 + 13 * q^41 + 10 * q^43 + 22 * q^44 - 3 * q^46 - 13 * q^47 - 2 * q^49 - q^50 - 12 * q^52 + 2 * q^53 - 2 * q^55 + 9 * q^56 - 17 * q^58 + 2 * q^59 + q^61 - 42 * q^62 - 15 * q^64 - 4 * q^65 + 11 * q^67 + 22 * q^68 - 9 * q^70 + 10 * q^71 - 8 * q^73 + 16 * q^74 - 12 * q^76 + 2 * q^79 - 5 * q^80 - 29 * q^82 + 15 * q^83 - 2 * q^85 - 28 * q^86 - 24 * q^88 + 9 * q^89 + 10 * q^91 - 39 * q^92 - 31 * q^94 - 4 * q^95 - 18 * q^97 + 40 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.08613 1.47512 0.737558 0.675283i $$-0.235979\pi$$
0.737558 + 0.675283i $$0.235979\pi$$
$$3$$ 0 0
$$4$$ 2.35194 1.17597
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.08613 1.54441 0.772206 0.635372i $$-0.219153\pi$$
0.772206 + 0.635372i $$0.219153\pi$$
$$8$$ 0.734191 0.259576
$$9$$ 0 0
$$10$$ −2.08613 −0.659692
$$11$$ 1.35194 0.407625 0.203813 0.979010i $$-0.434667\pi$$
0.203813 + 0.979010i $$0.434667\pi$$
$$12$$ 0 0
$$13$$ 0.648061 0.179740 0.0898699 0.995954i $$-0.471355\pi$$
0.0898699 + 0.995954i $$0.471355\pi$$
$$14$$ 8.52420 2.27819
$$15$$ 0 0
$$16$$ −3.17226 −0.793065
$$17$$ 1.35194 0.327893 0.163947 0.986469i $$-0.447577\pi$$
0.163947 + 0.986469i $$0.447577\pi$$
$$18$$ 0 0
$$19$$ 0.648061 0.148675 0.0743377 0.997233i $$-0.476316\pi$$
0.0743377 + 0.997233i $$0.476316\pi$$
$$20$$ −2.35194 −0.525910
$$21$$ 0 0
$$22$$ 2.82032 0.601294
$$23$$ −4.79001 −0.998786 −0.499393 0.866376i $$-0.666444\pi$$
−0.499393 + 0.866376i $$0.666444\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.35194 0.265137
$$27$$ 0 0
$$28$$ 9.61033 1.81618
$$29$$ −3.87614 −0.719781 −0.359890 0.932995i $$-0.617186\pi$$
−0.359890 + 0.932995i $$0.617186\pi$$
$$30$$ 0 0
$$31$$ −7.69646 −1.38233 −0.691163 0.722699i $$-0.742901\pi$$
−0.691163 + 0.722699i $$0.742901\pi$$
$$32$$ −8.08613 −1.42944
$$33$$ 0 0
$$34$$ 2.82032 0.483681
$$35$$ −4.08613 −0.690682
$$36$$ 0 0
$$37$$ 7.52420 1.23697 0.618485 0.785796i $$-0.287747\pi$$
0.618485 + 0.785796i $$0.287747\pi$$
$$38$$ 1.35194 0.219313
$$39$$ 0 0
$$40$$ −0.734191 −0.116086
$$41$$ 0.179679 0.0280611 0.0140306 0.999902i $$-0.495534\pi$$
0.0140306 + 0.999902i $$0.495534\pi$$
$$42$$ 0 0
$$43$$ −0.820321 −0.125098 −0.0625489 0.998042i $$-0.519923\pi$$
−0.0625489 + 0.998042i $$0.519923\pi$$
$$44$$ 3.17968 0.479355
$$45$$ 0 0
$$46$$ −9.99258 −1.47333
$$47$$ −10.9065 −1.59087 −0.795435 0.606039i $$-0.792757\pi$$
−0.795435 + 0.606039i $$0.792757\pi$$
$$48$$ 0 0
$$49$$ 9.69646 1.38521
$$50$$ 2.08613 0.295023
$$51$$ 0 0
$$52$$ 1.52420 0.211368
$$53$$ −4.17226 −0.573104 −0.286552 0.958065i $$-0.592509\pi$$
−0.286552 + 0.958065i $$0.592509\pi$$
$$54$$ 0 0
$$55$$ −1.35194 −0.182295
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ −8.08613 −1.06176
$$59$$ −4.17226 −0.543182 −0.271591 0.962413i $$-0.587550\pi$$
−0.271591 + 0.962413i $$0.587550\pi$$
$$60$$ 0 0
$$61$$ −3.82032 −0.489142 −0.244571 0.969631i $$-0.578647\pi$$
−0.244571 + 0.969631i $$0.578647\pi$$
$$62$$ −16.0558 −2.03909
$$63$$ 0 0
$$64$$ −10.5242 −1.31552
$$65$$ −0.648061 −0.0803820
$$66$$ 0 0
$$67$$ 8.14195 0.994697 0.497349 0.867551i $$-0.334307\pi$$
0.497349 + 0.867551i $$0.334307\pi$$
$$68$$ 3.17968 0.385593
$$69$$ 0 0
$$70$$ −8.52420 −1.01884
$$71$$ 6.11644 0.725888 0.362944 0.931811i $$-0.381772\pi$$
0.362944 + 0.931811i $$0.381772\pi$$
$$72$$ 0 0
$$73$$ −12.3445 −1.44482 −0.722408 0.691467i $$-0.756965\pi$$
−0.722408 + 0.691467i $$0.756965\pi$$
$$74$$ 15.6965 1.82468
$$75$$ 0 0
$$76$$ 1.52420 0.174838
$$77$$ 5.52420 0.629541
$$78$$ 0 0
$$79$$ 10.3445 1.16385 0.581925 0.813243i $$-0.302300\pi$$
0.581925 + 0.813243i $$0.302300\pi$$
$$80$$ 3.17226 0.354669
$$81$$ 0 0
$$82$$ 0.374833 0.0413934
$$83$$ 12.2584 1.34553 0.672767 0.739855i $$-0.265106\pi$$
0.672767 + 0.739855i $$0.265106\pi$$
$$84$$ 0 0
$$85$$ −1.35194 −0.146638
$$86$$ −1.71130 −0.184534
$$87$$ 0 0
$$88$$ 0.992582 0.105810
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ 2.64806 0.277592
$$92$$ −11.2658 −1.17454
$$93$$ 0 0
$$94$$ −22.7523 −2.34672
$$95$$ −0.648061 −0.0664896
$$96$$ 0 0
$$97$$ −13.5800 −1.37884 −0.689421 0.724361i $$-0.742135\pi$$
−0.689421 + 0.724361i $$0.742135\pi$$
$$98$$ 20.2281 2.04334
$$99$$ 0 0
$$100$$ 2.35194 0.235194
$$101$$ 1.46838 0.146109 0.0730547 0.997328i $$-0.476725\pi$$
0.0730547 + 0.997328i $$0.476725\pi$$
$$102$$ 0 0
$$103$$ 7.52420 0.741381 0.370691 0.928756i $$-0.379121\pi$$
0.370691 + 0.928756i $$0.379121\pi$$
$$104$$ 0.475800 0.0466561
$$105$$ 0 0
$$106$$ −8.70388 −0.845395
$$107$$ −1.20999 −0.116974 −0.0584871 0.998288i $$-0.518628\pi$$
−0.0584871 + 0.998288i $$0.518628\pi$$
$$108$$ 0 0
$$109$$ 14.1042 1.35094 0.675469 0.737388i $$-0.263941\pi$$
0.675469 + 0.737388i $$0.263941\pi$$
$$110$$ −2.82032 −0.268907
$$111$$ 0 0
$$112$$ −12.9623 −1.22482
$$113$$ 11.9245 1.12177 0.560883 0.827895i $$-0.310462\pi$$
0.560883 + 0.827895i $$0.310462\pi$$
$$114$$ 0 0
$$115$$ 4.79001 0.446671
$$116$$ −9.11644 −0.846440
$$117$$ 0 0
$$118$$ −8.70388 −0.801257
$$119$$ 5.52420 0.506403
$$120$$ 0 0
$$121$$ −9.17226 −0.833842
$$122$$ −7.96969 −0.721542
$$123$$ 0 0
$$124$$ −18.1016 −1.62557
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −7.07871 −0.628134 −0.314067 0.949401i $$-0.601692\pi$$
−0.314067 + 0.949401i $$0.601692\pi$$
$$128$$ −5.78259 −0.511114
$$129$$ 0 0
$$130$$ −1.35194 −0.118573
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 2.64806 0.229616
$$134$$ 16.9852 1.46729
$$135$$ 0 0
$$136$$ 0.992582 0.0851132
$$137$$ 7.46838 0.638067 0.319033 0.947743i $$-0.396642\pi$$
0.319033 + 0.947743i $$0.396642\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ −9.61033 −0.812221
$$141$$ 0 0
$$142$$ 12.7597 1.07077
$$143$$ 0.876139 0.0732664
$$144$$ 0 0
$$145$$ 3.87614 0.321896
$$146$$ −25.7523 −2.13127
$$147$$ 0 0
$$148$$ 17.6965 1.45464
$$149$$ 10.5848 0.867143 0.433571 0.901119i $$-0.357253\pi$$
0.433571 + 0.901119i $$0.357253\pi$$
$$150$$ 0 0
$$151$$ 17.6965 1.44012 0.720059 0.693913i $$-0.244115\pi$$
0.720059 + 0.693913i $$0.244115\pi$$
$$152$$ 0.475800 0.0385925
$$153$$ 0 0
$$154$$ 11.5242 0.928646
$$155$$ 7.69646 0.618195
$$156$$ 0 0
$$157$$ −2.53162 −0.202045 −0.101023 0.994884i $$-0.532211\pi$$
−0.101023 + 0.994884i $$0.532211\pi$$
$$158$$ 21.5800 1.71681
$$159$$ 0 0
$$160$$ 8.08613 0.639265
$$161$$ −19.5726 −1.54254
$$162$$ 0 0
$$163$$ 8.47580 0.663876 0.331938 0.943301i $$-0.392298\pi$$
0.331938 + 0.943301i $$0.392298\pi$$
$$164$$ 0.422594 0.0329990
$$165$$ 0 0
$$166$$ 25.5726 1.98482
$$167$$ −12.7342 −0.985401 −0.492701 0.870199i $$-0.663990\pi$$
−0.492701 + 0.870199i $$0.663990\pi$$
$$168$$ 0 0
$$169$$ −12.5800 −0.967694
$$170$$ −2.82032 −0.216309
$$171$$ 0 0
$$172$$ −1.92935 −0.147111
$$173$$ −23.0484 −1.75234 −0.876169 0.482005i $$-0.839909\pi$$
−0.876169 + 0.482005i $$0.839909\pi$$
$$174$$ 0 0
$$175$$ 4.08613 0.308882
$$176$$ −4.28870 −0.323273
$$177$$ 0 0
$$178$$ 6.25839 0.469086
$$179$$ −2.22808 −0.166534 −0.0832672 0.996527i $$-0.526535\pi$$
−0.0832672 + 0.996527i $$0.526535\pi$$
$$180$$ 0 0
$$181$$ 0.468382 0.0348146 0.0174073 0.999848i $$-0.494459\pi$$
0.0174073 + 0.999848i $$0.494459\pi$$
$$182$$ 5.52420 0.409481
$$183$$ 0 0
$$184$$ −3.51678 −0.259261
$$185$$ −7.52420 −0.553190
$$186$$ 0 0
$$187$$ 1.82774 0.133658
$$188$$ −25.6513 −1.87081
$$189$$ 0 0
$$190$$ −1.35194 −0.0980800
$$191$$ 20.2281 1.46365 0.731826 0.681491i $$-0.238668\pi$$
0.731826 + 0.681491i $$0.238668\pi$$
$$192$$ 0 0
$$193$$ −19.9293 −1.43455 −0.717273 0.696792i $$-0.754610\pi$$
−0.717273 + 0.696792i $$0.754610\pi$$
$$194$$ −28.3297 −2.03395
$$195$$ 0 0
$$196$$ 22.8055 1.62896
$$197$$ 15.5800 1.11003 0.555015 0.831840i $$-0.312712\pi$$
0.555015 + 0.831840i $$0.312712\pi$$
$$198$$ 0 0
$$199$$ 3.58482 0.254121 0.127061 0.991895i $$-0.459446\pi$$
0.127061 + 0.991895i $$0.459446\pi$$
$$200$$ 0.734191 0.0519151
$$201$$ 0 0
$$202$$ 3.06324 0.215529
$$203$$ −15.8384 −1.11164
$$204$$ 0 0
$$205$$ −0.179679 −0.0125493
$$206$$ 15.6965 1.09362
$$207$$ 0 0
$$208$$ −2.05582 −0.142545
$$209$$ 0.876139 0.0606038
$$210$$ 0 0
$$211$$ 14.9926 1.03213 0.516066 0.856549i $$-0.327396\pi$$
0.516066 + 0.856549i $$0.327396\pi$$
$$212$$ −9.81290 −0.673953
$$213$$ 0 0
$$214$$ −2.52420 −0.172551
$$215$$ 0.820321 0.0559454
$$216$$ 0 0
$$217$$ −31.4487 −2.13488
$$218$$ 29.4232 1.99279
$$219$$ 0 0
$$220$$ −3.17968 −0.214374
$$221$$ 0.876139 0.0589355
$$222$$ 0 0
$$223$$ −26.8310 −1.79674 −0.898368 0.439244i $$-0.855246\pi$$
−0.898368 + 0.439244i $$0.855246\pi$$
$$224$$ −33.0410 −2.20764
$$225$$ 0 0
$$226$$ 24.8761 1.65474
$$227$$ −1.35194 −0.0897314 −0.0448657 0.998993i $$-0.514286\pi$$
−0.0448657 + 0.998993i $$0.514286\pi$$
$$228$$ 0 0
$$229$$ −8.23550 −0.544217 −0.272108 0.962267i $$-0.587721\pi$$
−0.272108 + 0.962267i $$0.587721\pi$$
$$230$$ 9.99258 0.658891
$$231$$ 0 0
$$232$$ −2.84583 −0.186838
$$233$$ −8.58744 −0.562582 −0.281291 0.959623i $$-0.590763\pi$$
−0.281291 + 0.959623i $$0.590763\pi$$
$$234$$ 0 0
$$235$$ 10.9065 0.711458
$$236$$ −9.81290 −0.638766
$$237$$ 0 0
$$238$$ 11.5242 0.747003
$$239$$ 23.9245 1.54755 0.773775 0.633461i $$-0.218366\pi$$
0.773775 + 0.633461i $$0.218366\pi$$
$$240$$ 0 0
$$241$$ −6.24030 −0.401973 −0.200987 0.979594i $$-0.564415\pi$$
−0.200987 + 0.979594i $$0.564415\pi$$
$$242$$ −19.1345 −1.23001
$$243$$ 0 0
$$244$$ −8.98516 −0.575216
$$245$$ −9.69646 −0.619484
$$246$$ 0 0
$$247$$ 0.419983 0.0267229
$$248$$ −5.65067 −0.358818
$$249$$ 0 0
$$250$$ −2.08613 −0.131938
$$251$$ 28.5726 1.80349 0.901743 0.432272i $$-0.142288\pi$$
0.901743 + 0.432272i $$0.142288\pi$$
$$252$$ 0 0
$$253$$ −6.47580 −0.407130
$$254$$ −14.7671 −0.926571
$$255$$ 0 0
$$256$$ 8.98516 0.561573
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 30.7449 1.91039
$$260$$ −1.52420 −0.0945268
$$261$$ 0 0
$$262$$ 12.5168 0.773289
$$263$$ −31.8687 −1.96511 −0.982555 0.185974i $$-0.940456\pi$$
−0.982555 + 0.185974i $$0.940456\pi$$
$$264$$ 0 0
$$265$$ 4.17226 0.256300
$$266$$ 5.52420 0.338710
$$267$$ 0 0
$$268$$ 19.1494 1.16973
$$269$$ 31.4971 1.92041 0.960207 0.279289i $$-0.0900987\pi$$
0.960207 + 0.279289i $$0.0900987\pi$$
$$270$$ 0 0
$$271$$ −3.24030 −0.196834 −0.0984172 0.995145i $$-0.531378\pi$$
−0.0984172 + 0.995145i $$0.531378\pi$$
$$272$$ −4.28870 −0.260041
$$273$$ 0 0
$$274$$ 15.5800 0.941223
$$275$$ 1.35194 0.0815250
$$276$$ 0 0
$$277$$ −5.58482 −0.335560 −0.167780 0.985824i $$-0.553660\pi$$
−0.167780 + 0.985824i $$0.553660\pi$$
$$278$$ 16.6890 1.00094
$$279$$ 0 0
$$280$$ −3.00000 −0.179284
$$281$$ 24.1042 1.43794 0.718969 0.695043i $$-0.244615\pi$$
0.718969 + 0.695043i $$0.244615\pi$$
$$282$$ 0 0
$$283$$ −10.5423 −0.626674 −0.313337 0.949642i $$-0.601447\pi$$
−0.313337 + 0.949642i $$0.601447\pi$$
$$284$$ 14.3855 0.853622
$$285$$ 0 0
$$286$$ 1.82774 0.108077
$$287$$ 0.734191 0.0433379
$$288$$ 0 0
$$289$$ −15.1723 −0.892486
$$290$$ 8.08613 0.474834
$$291$$ 0 0
$$292$$ −29.0336 −1.69906
$$293$$ −18.9926 −1.10956 −0.554779 0.831998i $$-0.687197\pi$$
−0.554779 + 0.831998i $$0.687197\pi$$
$$294$$ 0 0
$$295$$ 4.17226 0.242918
$$296$$ 5.52420 0.321088
$$297$$ 0 0
$$298$$ 22.0813 1.27914
$$299$$ −3.10422 −0.179521
$$300$$ 0 0
$$301$$ −3.35194 −0.193203
$$302$$ 36.9171 2.12434
$$303$$ 0 0
$$304$$ −2.05582 −0.117909
$$305$$ 3.82032 0.218751
$$306$$ 0 0
$$307$$ 29.4791 1.68246 0.841229 0.540679i $$-0.181833\pi$$
0.841229 + 0.540679i $$0.181833\pi$$
$$308$$ 12.9926 0.740321
$$309$$ 0 0
$$310$$ 16.0558 0.911909
$$311$$ 9.41256 0.533738 0.266869 0.963733i $$-0.414011\pi$$
0.266869 + 0.963733i $$0.414011\pi$$
$$312$$ 0 0
$$313$$ 11.6210 0.656858 0.328429 0.944529i $$-0.393481\pi$$
0.328429 + 0.944529i $$0.393481\pi$$
$$314$$ −5.28128 −0.298040
$$315$$ 0 0
$$316$$ 24.3297 1.36865
$$317$$ 9.17968 0.515582 0.257791 0.966201i $$-0.417005\pi$$
0.257791 + 0.966201i $$0.417005\pi$$
$$318$$ 0 0
$$319$$ −5.24030 −0.293401
$$320$$ 10.5242 0.588321
$$321$$ 0 0
$$322$$ −40.8310 −2.27542
$$323$$ 0.876139 0.0487497
$$324$$ 0 0
$$325$$ 0.648061 0.0359479
$$326$$ 17.6816 0.979295
$$327$$ 0 0
$$328$$ 0.131919 0.00728398
$$329$$ −44.5652 −2.45696
$$330$$ 0 0
$$331$$ −7.22066 −0.396883 −0.198442 0.980113i $$-0.563588\pi$$
−0.198442 + 0.980113i $$0.563588\pi$$
$$332$$ 28.8310 1.58231
$$333$$ 0 0
$$334$$ −26.5652 −1.45358
$$335$$ −8.14195 −0.444842
$$336$$ 0 0
$$337$$ 2.28390 0.124412 0.0622059 0.998063i $$-0.480186\pi$$
0.0622059 + 0.998063i $$0.480186\pi$$
$$338$$ −26.2436 −1.42746
$$339$$ 0 0
$$340$$ −3.17968 −0.172442
$$341$$ −10.4051 −0.563470
$$342$$ 0 0
$$343$$ 11.0181 0.594921
$$344$$ −0.602272 −0.0324724
$$345$$ 0 0
$$346$$ −48.0820 −2.58490
$$347$$ 0.708686 0.0380443 0.0190221 0.999819i $$-0.493945\pi$$
0.0190221 + 0.999819i $$0.493945\pi$$
$$348$$ 0 0
$$349$$ −21.3445 −1.14255 −0.571273 0.820760i $$-0.693550\pi$$
−0.571273 + 0.820760i $$0.693550\pi$$
$$350$$ 8.52420 0.455638
$$351$$ 0 0
$$352$$ −10.9320 −0.582675
$$353$$ 10.0968 0.537398 0.268699 0.963224i $$-0.413406\pi$$
0.268699 + 0.963224i $$0.413406\pi$$
$$354$$ 0 0
$$355$$ −6.11644 −0.324627
$$356$$ 7.05582 0.373958
$$357$$ 0 0
$$358$$ −4.64806 −0.245658
$$359$$ −30.5578 −1.61278 −0.806388 0.591386i $$-0.798581\pi$$
−0.806388 + 0.591386i $$0.798581\pi$$
$$360$$ 0 0
$$361$$ −18.5800 −0.977896
$$362$$ 0.977106 0.0513555
$$363$$ 0 0
$$364$$ 6.22808 0.326440
$$365$$ 12.3445 0.646142
$$366$$ 0 0
$$367$$ −7.17968 −0.374776 −0.187388 0.982286i $$-0.560002\pi$$
−0.187388 + 0.982286i $$0.560002\pi$$
$$368$$ 15.1952 0.792102
$$369$$ 0 0
$$370$$ −15.6965 −0.816020
$$371$$ −17.0484 −0.885109
$$372$$ 0 0
$$373$$ −21.9245 −1.13521 −0.567605 0.823301i $$-0.692130\pi$$
−0.567605 + 0.823301i $$0.692130\pi$$
$$374$$ 3.81290 0.197161
$$375$$ 0 0
$$376$$ −8.00742 −0.412951
$$377$$ −2.51197 −0.129373
$$378$$ 0 0
$$379$$ −17.3929 −0.893414 −0.446707 0.894680i $$-0.647403\pi$$
−0.446707 + 0.894680i $$0.647403\pi$$
$$380$$ −1.52420 −0.0781898
$$381$$ 0 0
$$382$$ 42.1984 2.15906
$$383$$ −0.475800 −0.0243123 −0.0121561 0.999926i $$-0.503870\pi$$
−0.0121561 + 0.999926i $$0.503870\pi$$
$$384$$ 0 0
$$385$$ −5.52420 −0.281539
$$386$$ −41.5752 −2.11612
$$387$$ 0 0
$$388$$ −31.9394 −1.62148
$$389$$ 5.58744 0.283294 0.141647 0.989917i $$-0.454760\pi$$
0.141647 + 0.989917i $$0.454760\pi$$
$$390$$ 0 0
$$391$$ −6.47580 −0.327495
$$392$$ 7.11905 0.359567
$$393$$ 0 0
$$394$$ 32.5019 1.63742
$$395$$ −10.3445 −0.520489
$$396$$ 0 0
$$397$$ 3.75228 0.188321 0.0941607 0.995557i $$-0.469983\pi$$
0.0941607 + 0.995557i $$0.469983\pi$$
$$398$$ 7.47841 0.374859
$$399$$ 0 0
$$400$$ −3.17226 −0.158613
$$401$$ −23.5652 −1.17679 −0.588394 0.808574i $$-0.700240\pi$$
−0.588394 + 0.808574i $$0.700240\pi$$
$$402$$ 0 0
$$403$$ −4.98777 −0.248459
$$404$$ 3.45355 0.171820
$$405$$ 0 0
$$406$$ −33.0410 −1.63980
$$407$$ 10.1723 0.504220
$$408$$ 0 0
$$409$$ 1.04840 0.0518400 0.0259200 0.999664i $$-0.491748\pi$$
0.0259200 + 0.999664i $$0.491748\pi$$
$$410$$ −0.374833 −0.0185117
$$411$$ 0 0
$$412$$ 17.6965 0.871842
$$413$$ −17.0484 −0.838897
$$414$$ 0 0
$$415$$ −12.2584 −0.601741
$$416$$ −5.24030 −0.256927
$$417$$ 0 0
$$418$$ 1.82774 0.0893977
$$419$$ 25.9197 1.26626 0.633131 0.774045i $$-0.281769\pi$$
0.633131 + 0.774045i $$0.281769\pi$$
$$420$$ 0 0
$$421$$ 7.64064 0.372382 0.186191 0.982514i $$-0.440386\pi$$
0.186191 + 0.982514i $$0.440386\pi$$
$$422$$ 31.2765 1.52252
$$423$$ 0 0
$$424$$ −3.06324 −0.148764
$$425$$ 1.35194 0.0655787
$$426$$ 0 0
$$427$$ −15.6103 −0.755437
$$428$$ −2.84583 −0.137558
$$429$$ 0 0
$$430$$ 1.71130 0.0825261
$$431$$ −7.98516 −0.384632 −0.192316 0.981333i $$-0.561600\pi$$
−0.192316 + 0.981333i $$0.561600\pi$$
$$432$$ 0 0
$$433$$ −12.5120 −0.601287 −0.300644 0.953737i $$-0.597201\pi$$
−0.300644 + 0.953737i $$0.597201\pi$$
$$434$$ −65.6062 −3.14920
$$435$$ 0 0
$$436$$ 33.1723 1.58866
$$437$$ −3.10422 −0.148495
$$438$$ 0 0
$$439$$ −8.76450 −0.418307 −0.209153 0.977883i $$-0.567071\pi$$
−0.209153 + 0.977883i $$0.567071\pi$$
$$440$$ −0.992582 −0.0473195
$$441$$ 0 0
$$442$$ 1.82774 0.0869367
$$443$$ −3.67095 −0.174412 −0.0872062 0.996190i $$-0.527794\pi$$
−0.0872062 + 0.996190i $$0.527794\pi$$
$$444$$ 0 0
$$445$$ −3.00000 −0.142214
$$446$$ −55.9729 −2.65040
$$447$$ 0 0
$$448$$ −43.0032 −2.03171
$$449$$ −28.1723 −1.32953 −0.664766 0.747052i $$-0.731469\pi$$
−0.664766 + 0.747052i $$0.731469\pi$$
$$450$$ 0 0
$$451$$ 0.242915 0.0114384
$$452$$ 28.0458 1.31916
$$453$$ 0 0
$$454$$ −2.82032 −0.132364
$$455$$ −2.64806 −0.124143
$$456$$ 0 0
$$457$$ 35.2616 1.64947 0.824735 0.565519i $$-0.191324\pi$$
0.824735 + 0.565519i $$0.191324\pi$$
$$458$$ −17.1803 −0.802784
$$459$$ 0 0
$$460$$ 11.2658 0.525271
$$461$$ −34.6768 −1.61506 −0.807530 0.589826i $$-0.799196\pi$$
−0.807530 + 0.589826i $$0.799196\pi$$
$$462$$ 0 0
$$463$$ 7.44874 0.346172 0.173086 0.984907i $$-0.444626\pi$$
0.173086 + 0.984907i $$0.444626\pi$$
$$464$$ 12.2961 0.570833
$$465$$ 0 0
$$466$$ −17.9145 −0.829874
$$467$$ −29.9655 −1.38664 −0.693319 0.720630i $$-0.743852\pi$$
−0.693319 + 0.720630i $$0.743852\pi$$
$$468$$ 0 0
$$469$$ 33.2691 1.53622
$$470$$ 22.7523 1.04948
$$471$$ 0 0
$$472$$ −3.06324 −0.140997
$$473$$ −1.10902 −0.0509930
$$474$$ 0 0
$$475$$ 0.648061 0.0297351
$$476$$ 12.9926 0.595514
$$477$$ 0 0
$$478$$ 49.9097 2.28282
$$479$$ 7.98516 0.364851 0.182426 0.983220i $$-0.441605\pi$$
0.182426 + 0.983220i $$0.441605\pi$$
$$480$$ 0 0
$$481$$ 4.87614 0.222333
$$482$$ −13.0181 −0.592958
$$483$$ 0 0
$$484$$ −21.5726 −0.980573
$$485$$ 13.5800 0.616637
$$486$$ 0 0
$$487$$ −11.9442 −0.541243 −0.270621 0.962686i $$-0.587229\pi$$
−0.270621 + 0.962686i $$0.587229\pi$$
$$488$$ −2.80485 −0.126969
$$489$$ 0 0
$$490$$ −20.2281 −0.913811
$$491$$ 9.22066 0.416123 0.208061 0.978116i $$-0.433285\pi$$
0.208061 + 0.978116i $$0.433285\pi$$
$$492$$ 0 0
$$493$$ −5.24030 −0.236011
$$494$$ 0.876139 0.0394193
$$495$$ 0 0
$$496$$ 24.4152 1.09627
$$497$$ 24.9926 1.12107
$$498$$ 0 0
$$499$$ 30.1723 1.35070 0.675348 0.737499i $$-0.263993\pi$$
0.675348 + 0.737499i $$0.263993\pi$$
$$500$$ −2.35194 −0.105182
$$501$$ 0 0
$$502$$ 59.6062 2.66035
$$503$$ 10.5981 0.472546 0.236273 0.971687i $$-0.424074\pi$$
0.236273 + 0.971687i $$0.424074\pi$$
$$504$$ 0 0
$$505$$ −1.46838 −0.0653421
$$506$$ −13.5094 −0.600564
$$507$$ 0 0
$$508$$ −16.6487 −0.738667
$$509$$ −28.7523 −1.27442 −0.637211 0.770689i $$-0.719912\pi$$
−0.637211 + 0.770689i $$0.719912\pi$$
$$510$$ 0 0
$$511$$ −50.4413 −2.23139
$$512$$ 30.3094 1.33950
$$513$$ 0 0
$$514$$ 37.5503 1.65627
$$515$$ −7.52420 −0.331556
$$516$$ 0 0
$$517$$ −14.7449 −0.648478
$$518$$ 64.1378 2.81805
$$519$$ 0 0
$$520$$ −0.475800 −0.0208652
$$521$$ 36.0942 1.58132 0.790658 0.612259i $$-0.209739\pi$$
0.790658 + 0.612259i $$0.209739\pi$$
$$522$$ 0 0
$$523$$ 11.1297 0.486669 0.243334 0.969942i $$-0.421759\pi$$
0.243334 + 0.969942i $$0.421759\pi$$
$$524$$ 14.1116 0.616470
$$525$$ 0 0
$$526$$ −66.4823 −2.89877
$$527$$ −10.4051 −0.453255
$$528$$ 0 0
$$529$$ −0.0558176 −0.00242685
$$530$$ 8.70388 0.378072
$$531$$ 0 0
$$532$$ 6.22808 0.270021
$$533$$ 0.116443 0.00504370
$$534$$ 0 0
$$535$$ 1.20999 0.0523125
$$536$$ 5.97774 0.258199
$$537$$ 0 0
$$538$$ 65.7071 2.83284
$$539$$ 13.1090 0.564646
$$540$$ 0 0
$$541$$ −34.7374 −1.49348 −0.746740 0.665116i $$-0.768382\pi$$
−0.746740 + 0.665116i $$0.768382\pi$$
$$542$$ −6.75970 −0.290354
$$543$$ 0 0
$$544$$ −10.9320 −0.468704
$$545$$ −14.1042 −0.604158
$$546$$ 0 0
$$547$$ −2.71455 −0.116066 −0.0580328 0.998315i $$-0.518483\pi$$
−0.0580328 + 0.998315i $$0.518483\pi$$
$$548$$ 17.5652 0.750347
$$549$$ 0 0
$$550$$ 2.82032 0.120259
$$551$$ −2.51197 −0.107014
$$552$$ 0 0
$$553$$ 42.2691 1.79746
$$554$$ −11.6507 −0.494990
$$555$$ 0 0
$$556$$ 18.8155 0.797956
$$557$$ 8.93676 0.378663 0.189331 0.981913i $$-0.439368\pi$$
0.189331 + 0.981913i $$0.439368\pi$$
$$558$$ 0 0
$$559$$ −0.531618 −0.0224850
$$560$$ 12.9623 0.547756
$$561$$ 0 0
$$562$$ 50.2845 2.12113
$$563$$ 9.36261 0.394587 0.197293 0.980344i $$-0.436785\pi$$
0.197293 + 0.980344i $$0.436785\pi$$
$$564$$ 0 0
$$565$$ −11.9245 −0.501669
$$566$$ −21.9926 −0.924417
$$567$$ 0 0
$$568$$ 4.49064 0.188423
$$569$$ −35.8735 −1.50390 −0.751948 0.659222i $$-0.770886\pi$$
−0.751948 + 0.659222i $$0.770886\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 2.06063 0.0861591
$$573$$ 0 0
$$574$$ 1.53162 0.0639285
$$575$$ −4.79001 −0.199757
$$576$$ 0 0
$$577$$ 1.35675 0.0564821 0.0282411 0.999601i $$-0.491009\pi$$
0.0282411 + 0.999601i $$0.491009\pi$$
$$578$$ −31.6513 −1.31652
$$579$$ 0 0
$$580$$ 9.11644 0.378540
$$581$$ 50.0894 2.07806
$$582$$ 0 0
$$583$$ −5.64064 −0.233612
$$584$$ −9.06324 −0.375039
$$585$$ 0 0
$$586$$ −39.6210 −1.63673
$$587$$ −28.7900 −1.18829 −0.594145 0.804358i $$-0.702510\pi$$
−0.594145 + 0.804358i $$0.702510\pi$$
$$588$$ 0 0
$$589$$ −4.98777 −0.205518
$$590$$ 8.70388 0.358333
$$591$$ 0 0
$$592$$ −23.8687 −0.980998
$$593$$ −30.9171 −1.26961 −0.634807 0.772671i $$-0.718920\pi$$
−0.634807 + 0.772671i $$0.718920\pi$$
$$594$$ 0 0
$$595$$ −5.52420 −0.226470
$$596$$ 24.8949 1.01973
$$597$$ 0 0
$$598$$ −6.47580 −0.264815
$$599$$ −1.39292 −0.0569132 −0.0284566 0.999595i $$-0.509059\pi$$
−0.0284566 + 0.999595i $$0.509059\pi$$
$$600$$ 0 0
$$601$$ 8.82513 0.359985 0.179992 0.983668i $$-0.442393\pi$$
0.179992 + 0.983668i $$0.442393\pi$$
$$602$$ −6.99258 −0.284996
$$603$$ 0 0
$$604$$ 41.6210 1.69353
$$605$$ 9.17226 0.372905
$$606$$ 0 0
$$607$$ −2.15678 −0.0875412 −0.0437706 0.999042i $$-0.513937\pi$$
−0.0437706 + 0.999042i $$0.513937\pi$$
$$608$$ −5.24030 −0.212522
$$609$$ 0 0
$$610$$ 7.96969 0.322683
$$611$$ −7.06804 −0.285942
$$612$$ 0 0
$$613$$ −9.57521 −0.386739 −0.193370 0.981126i $$-0.561942\pi$$
−0.193370 + 0.981126i $$0.561942\pi$$
$$614$$ 61.4971 2.48182
$$615$$ 0 0
$$616$$ 4.05582 0.163414
$$617$$ 37.6768 1.51681 0.758406 0.651783i $$-0.225979\pi$$
0.758406 + 0.651783i $$0.225979\pi$$
$$618$$ 0 0
$$619$$ 17.1042 0.687477 0.343738 0.939065i $$-0.388307\pi$$
0.343738 + 0.939065i $$0.388307\pi$$
$$620$$ 18.1016 0.726978
$$621$$ 0 0
$$622$$ 19.6358 0.787325
$$623$$ 12.2584 0.491122
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 24.2429 0.968942
$$627$$ 0 0
$$628$$ −5.95421 −0.237599
$$629$$ 10.1723 0.405595
$$630$$ 0 0
$$631$$ 33.1090 1.31805 0.659025 0.752121i $$-0.270969\pi$$
0.659025 + 0.752121i $$0.270969\pi$$
$$632$$ 7.59485 0.302107
$$633$$ 0 0
$$634$$ 19.1500 0.760544
$$635$$ 7.07871 0.280910
$$636$$ 0 0
$$637$$ 6.28390 0.248977
$$638$$ −10.9320 −0.432800
$$639$$ 0 0
$$640$$ 5.78259 0.228577
$$641$$ 23.1526 0.914473 0.457237 0.889345i $$-0.348839\pi$$
0.457237 + 0.889345i $$0.348839\pi$$
$$642$$ 0 0
$$643$$ 43.0639 1.69827 0.849137 0.528173i $$-0.177123\pi$$
0.849137 + 0.528173i $$0.177123\pi$$
$$644$$ −46.0336 −1.81398
$$645$$ 0 0
$$646$$ 1.82774 0.0719115
$$647$$ 20.6439 0.811595 0.405798 0.913963i $$-0.366994\pi$$
0.405798 + 0.913963i $$0.366994\pi$$
$$648$$ 0 0
$$649$$ −5.64064 −0.221415
$$650$$ 1.35194 0.0530274
$$651$$ 0 0
$$652$$ 19.9346 0.780698
$$653$$ −6.83516 −0.267480 −0.133740 0.991016i $$-0.542699\pi$$
−0.133740 + 0.991016i $$0.542699\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ −0.569988 −0.0222543
$$657$$ 0 0
$$658$$ −92.9688 −3.62430
$$659$$ 26.8613 1.04637 0.523184 0.852220i $$-0.324744\pi$$
0.523184 + 0.852220i $$0.324744\pi$$
$$660$$ 0 0
$$661$$ −2.12125 −0.0825071 −0.0412535 0.999149i $$-0.513135\pi$$
−0.0412535 + 0.999149i $$0.513135\pi$$
$$662$$ −15.0632 −0.585449
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ −2.64806 −0.102687
$$666$$ 0 0
$$667$$ 18.5667 0.718907
$$668$$ −29.9500 −1.15880
$$669$$ 0 0
$$670$$ −16.9852 −0.656194
$$671$$ −5.16484 −0.199387
$$672$$ 0 0
$$673$$ 34.8203 1.34222 0.671112 0.741356i $$-0.265817\pi$$
0.671112 + 0.741356i $$0.265817\pi$$
$$674$$ 4.76450 0.183522
$$675$$ 0 0
$$676$$ −29.5874 −1.13798
$$677$$ 24.6842 0.948692 0.474346 0.880338i $$-0.342685\pi$$
0.474346 + 0.880338i $$0.342685\pi$$
$$678$$ 0 0
$$679$$ −55.4897 −2.12950
$$680$$ −0.992582 −0.0380638
$$681$$ 0 0
$$682$$ −21.7065 −0.831184
$$683$$ 38.4610 1.47167 0.735834 0.677162i $$-0.236790\pi$$
0.735834 + 0.677162i $$0.236790\pi$$
$$684$$ 0 0
$$685$$ −7.46838 −0.285352
$$686$$ 22.9852 0.877578
$$687$$ 0 0
$$688$$ 2.60227 0.0992107
$$689$$ −2.70388 −0.103010
$$690$$ 0 0
$$691$$ 0.480608 0.0182832 0.00914159 0.999958i $$-0.497090\pi$$
0.00914159 + 0.999958i $$0.497090\pi$$
$$692$$ −54.2084 −2.06070
$$693$$ 0 0
$$694$$ 1.47841 0.0561197
$$695$$ −8.00000 −0.303457
$$696$$ 0 0
$$697$$ 0.242915 0.00920105
$$698$$ −44.5274 −1.68539
$$699$$ 0 0
$$700$$ 9.61033 0.363236
$$701$$ 18.1797 0.686637 0.343318 0.939219i $$-0.388449\pi$$
0.343318 + 0.939219i $$0.388449\pi$$
$$702$$ 0 0
$$703$$ 4.87614 0.183907
$$704$$ −14.2281 −0.536241
$$705$$ 0 0
$$706$$ 21.0632 0.792725
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ 7.18710 0.269917 0.134959 0.990851i $$-0.456910\pi$$
0.134959 + 0.990851i $$0.456910\pi$$
$$710$$ −12.7597 −0.478863
$$711$$ 0 0
$$712$$ 2.20257 0.0825449
$$713$$ 36.8661 1.38065
$$714$$ 0 0
$$715$$ −0.876139 −0.0327657
$$716$$ −5.24030 −0.195839
$$717$$ 0 0
$$718$$ −63.7475 −2.37903
$$719$$ −12.5168 −0.466797 −0.233399 0.972381i $$-0.574985\pi$$
−0.233399 + 0.972381i $$0.574985\pi$$
$$720$$ 0 0
$$721$$ 30.7449 1.14500
$$722$$ −38.7603 −1.44251
$$723$$ 0 0
$$724$$ 1.10161 0.0409409
$$725$$ −3.87614 −0.143956
$$726$$ 0 0
$$727$$ 8.42584 0.312497 0.156249 0.987718i $$-0.450060\pi$$
0.156249 + 0.987718i $$0.450060\pi$$
$$728$$ 1.94418 0.0720562
$$729$$ 0 0
$$730$$ 25.7523 0.953135
$$731$$ −1.10902 −0.0410187
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ −14.9777 −0.552839
$$735$$ 0 0
$$736$$ 38.7326 1.42770
$$737$$ 11.0074 0.405463
$$738$$ 0 0
$$739$$ −1.81290 −0.0666887 −0.0333444 0.999444i $$-0.510616\pi$$
−0.0333444 + 0.999444i $$0.510616\pi$$
$$740$$ −17.6965 −0.650535
$$741$$ 0 0
$$742$$ −35.5652 −1.30564
$$743$$ −20.1371 −0.738760 −0.369380 0.929278i $$-0.620430\pi$$
−0.369380 + 0.929278i $$0.620430\pi$$
$$744$$ 0 0
$$745$$ −10.5848 −0.387798
$$746$$ −45.7374 −1.67457
$$747$$ 0 0
$$748$$ 4.29873 0.157177
$$749$$ −4.94418 −0.180656
$$750$$ 0 0
$$751$$ 12.2132 0.445668 0.222834 0.974856i $$-0.428469\pi$$
0.222834 + 0.974856i $$0.428469\pi$$
$$752$$ 34.5981 1.26166
$$753$$ 0 0
$$754$$ −5.24030 −0.190841
$$755$$ −17.6965 −0.644040
$$756$$ 0 0
$$757$$ 52.9533 1.92462 0.962310 0.271955i $$-0.0876701\pi$$
0.962310 + 0.271955i $$0.0876701\pi$$
$$758$$ −36.2839 −1.31789
$$759$$ 0 0
$$760$$ −0.475800 −0.0172591
$$761$$ 18.4535 0.668940 0.334470 0.942406i $$-0.391443\pi$$
0.334470 + 0.942406i $$0.391443\pi$$
$$762$$ 0 0
$$763$$ 57.6317 2.08641
$$764$$ 47.5752 1.72121
$$765$$ 0 0
$$766$$ −0.992582 −0.0358634
$$767$$ −2.70388 −0.0976314
$$768$$ 0 0
$$769$$ −4.45355 −0.160599 −0.0802995 0.996771i $$-0.525588\pi$$
−0.0802995 + 0.996771i $$0.525588\pi$$
$$770$$ −11.5242 −0.415303
$$771$$ 0 0
$$772$$ −46.8726 −1.68698
$$773$$ −38.9368 −1.40046 −0.700229 0.713918i $$-0.746919\pi$$
−0.700229 + 0.713918i $$0.746919\pi$$
$$774$$ 0 0
$$775$$ −7.69646 −0.276465
$$776$$ −9.97033 −0.357914
$$777$$ 0 0
$$778$$ 11.6561 0.417892
$$779$$ 0.116443 0.00417200
$$780$$ 0 0
$$781$$ 8.26906 0.295890
$$782$$ −13.5094 −0.483094
$$783$$ 0 0
$$784$$ −30.7597 −1.09856
$$785$$ 2.53162 0.0903573
$$786$$ 0 0
$$787$$ 34.2281 1.22010 0.610050 0.792363i $$-0.291150\pi$$
0.610050 + 0.792363i $$0.291150\pi$$
$$788$$ 36.6433 1.30536
$$789$$ 0 0
$$790$$ −21.5800 −0.767783
$$791$$ 48.7252 1.73247
$$792$$ 0 0
$$793$$ −2.47580 −0.0879183
$$794$$ 7.82774 0.277796
$$795$$ 0 0
$$796$$ 8.43129 0.298839
$$797$$ −23.9655 −0.848902 −0.424451 0.905451i $$-0.639533\pi$$
−0.424451 + 0.905451i $$0.639533\pi$$
$$798$$ 0 0
$$799$$ −14.7449 −0.521636
$$800$$ −8.08613 −0.285888
$$801$$ 0 0
$$802$$ −49.1600 −1.73590
$$803$$ −16.6890 −0.588943
$$804$$ 0 0
$$805$$ 19.5726 0.689843
$$806$$ −10.4051 −0.366506
$$807$$ 0 0
$$808$$ 1.07807 0.0379265
$$809$$ 0.283896 0.00998124 0.00499062 0.999988i $$-0.498411\pi$$
0.00499062 + 0.999988i $$0.498411\pi$$
$$810$$ 0 0
$$811$$ 32.4413 1.13917 0.569584 0.821933i $$-0.307104\pi$$
0.569584 + 0.821933i $$0.307104\pi$$
$$812$$ −37.2510 −1.30725
$$813$$ 0 0
$$814$$ 21.2207 0.743784
$$815$$ −8.47580 −0.296894
$$816$$ 0 0
$$817$$ −0.531618 −0.0185990
$$818$$ 2.18710 0.0764701
$$819$$ 0 0
$$820$$ −0.422594 −0.0147576
$$821$$ −41.6694 −1.45427 −0.727136 0.686493i $$-0.759149\pi$$
−0.727136 + 0.686493i $$0.759149\pi$$
$$822$$ 0 0
$$823$$ −19.3626 −0.674938 −0.337469 0.941337i $$-0.609571\pi$$
−0.337469 + 0.941337i $$0.609571\pi$$
$$824$$ 5.52420 0.192445
$$825$$ 0 0
$$826$$ −35.5652 −1.23747
$$827$$ −18.8097 −0.654076 −0.327038 0.945011i $$-0.606050\pi$$
−0.327038 + 0.945011i $$0.606050\pi$$
$$828$$ 0 0
$$829$$ −33.1016 −1.14967 −0.574833 0.818271i $$-0.694933\pi$$
−0.574833 + 0.818271i $$0.694933\pi$$
$$830$$ −25.5726 −0.887638
$$831$$ 0 0
$$832$$ −6.82032 −0.236452
$$833$$ 13.1090 0.454201
$$834$$ 0 0
$$835$$ 12.7342 0.440685
$$836$$ 2.06063 0.0712682
$$837$$ 0 0
$$838$$ 54.0719 1.86788
$$839$$ 39.6965 1.37047 0.685237 0.728320i $$-0.259699\pi$$
0.685237 + 0.728320i $$0.259699\pi$$
$$840$$ 0 0
$$841$$ −13.9755 −0.481915
$$842$$ 15.9394 0.549307
$$843$$ 0 0
$$844$$ 35.2616 1.21376
$$845$$ 12.5800 0.432766
$$846$$ 0 0
$$847$$ −37.4791 −1.28780
$$848$$ 13.2355 0.454509
$$849$$ 0 0
$$850$$ 2.82032 0.0967362
$$851$$ −36.0410 −1.23547
$$852$$ 0 0
$$853$$ −4.11644 −0.140944 −0.0704722 0.997514i $$-0.522451\pi$$
−0.0704722 + 0.997514i $$0.522451\pi$$
$$854$$ −32.5652 −1.11436
$$855$$ 0 0
$$856$$ −0.888365 −0.0303637
$$857$$ −14.7449 −0.503675 −0.251837 0.967770i $$-0.581035\pi$$
−0.251837 + 0.967770i $$0.581035\pi$$
$$858$$ 0 0
$$859$$ −37.8539 −1.29156 −0.645779 0.763524i $$-0.723467\pi$$
−0.645779 + 0.763524i $$0.723467\pi$$
$$860$$ 1.92935 0.0657901
$$861$$ 0 0
$$862$$ −16.6581 −0.567377
$$863$$ −26.7704 −0.911274 −0.455637 0.890166i $$-0.650588\pi$$
−0.455637 + 0.890166i $$0.650588\pi$$
$$864$$ 0 0
$$865$$ 23.0484 0.783669
$$866$$ −26.1016 −0.886969
$$867$$ 0 0
$$868$$ −73.9655 −2.51055
$$869$$ 13.9852 0.474414
$$870$$ 0 0
$$871$$ 5.27648 0.178787
$$872$$ 10.3552 0.350671
$$873$$ 0 0
$$874$$ −6.47580 −0.219047
$$875$$ −4.08613 −0.138136
$$876$$ 0 0
$$877$$ −46.9681 −1.58600 −0.793001 0.609221i $$-0.791482\pi$$
−0.793001 + 0.609221i $$0.791482\pi$$
$$878$$ −18.2839 −0.617052
$$879$$ 0 0
$$880$$ 4.28870 0.144572
$$881$$ 19.8055 0.667264 0.333632 0.942703i $$-0.391726\pi$$
0.333632 + 0.942703i $$0.391726\pi$$
$$882$$ 0 0
$$883$$ −6.20257 −0.208733 −0.104367 0.994539i $$-0.533282\pi$$
−0.104367 + 0.994539i $$0.533282\pi$$
$$884$$ 2.06063 0.0693063
$$885$$ 0 0
$$886$$ −7.65809 −0.257279
$$887$$ −13.4274 −0.450848 −0.225424 0.974261i $$-0.572377\pi$$
−0.225424 + 0.974261i $$0.572377\pi$$
$$888$$ 0 0
$$889$$ −28.9245 −0.970098
$$890$$ −6.25839 −0.209782
$$891$$ 0 0
$$892$$ −63.1049 −2.11291
$$893$$ −7.06804 −0.236523
$$894$$ 0 0
$$895$$ 2.22808 0.0744764
$$896$$ −23.6284 −0.789370
$$897$$ 0 0
$$898$$ −58.7710 −1.96121
$$899$$ 29.8325 0.994971
$$900$$ 0 0
$$901$$ −5.64064 −0.187917
$$902$$ 0.506752 0.0168730
$$903$$ 0 0
$$904$$ 8.75489 0.291183
$$905$$ −0.468382 −0.0155695
$$906$$ 0 0
$$907$$ 0.673566 0.0223654 0.0111827 0.999937i $$-0.496440\pi$$
0.0111827 + 0.999937i $$0.496440\pi$$
$$908$$ −3.17968 −0.105521
$$909$$ 0 0
$$910$$ −5.52420 −0.183125
$$911$$ 7.90970 0.262060 0.131030 0.991378i $$-0.458172\pi$$
0.131030 + 0.991378i $$0.458172\pi$$
$$912$$ 0 0
$$913$$ 16.5726 0.548473
$$914$$ 73.5604 2.43316
$$915$$ 0 0
$$916$$ −19.3694 −0.639983
$$917$$ 24.5168 0.809615
$$918$$ 0 0
$$919$$ −8.58263 −0.283115 −0.141557 0.989930i $$-0.545211\pi$$
−0.141557 + 0.989930i $$0.545211\pi$$
$$920$$ 3.51678 0.115945
$$921$$ 0 0
$$922$$ −72.3404 −2.38240
$$923$$ 3.96383 0.130471
$$924$$ 0 0
$$925$$ 7.52420 0.247394
$$926$$ 15.5390 0.510644
$$927$$ 0 0
$$928$$ 31.3430 1.02888
$$929$$ −29.6162 −0.971676 −0.485838 0.874049i $$-0.661485\pi$$
−0.485838 + 0.874049i $$0.661485\pi$$
$$930$$ 0 0
$$931$$ 6.28390 0.205946
$$932$$ −20.1971 −0.661579
$$933$$ 0 0
$$934$$ −62.5120 −2.04545
$$935$$ −1.82774 −0.0597735
$$936$$ 0 0
$$937$$ −15.2058 −0.496753 −0.248376 0.968664i $$-0.579897\pi$$
−0.248376 + 0.968664i $$0.579897\pi$$
$$938$$ 69.4036 2.26611
$$939$$ 0 0
$$940$$ 25.6513 0.836654
$$941$$ −5.65287 −0.184278 −0.0921391 0.995746i $$-0.529370\pi$$
−0.0921391 + 0.995746i $$0.529370\pi$$
$$942$$ 0 0
$$943$$ −0.860663 −0.0280270
$$944$$ 13.2355 0.430779
$$945$$ 0 0
$$946$$ −2.31357 −0.0752206
$$947$$ 40.3962 1.31270 0.656350 0.754457i $$-0.272100\pi$$
0.656350 + 0.754457i $$0.272100\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 1.35194 0.0438627
$$951$$ 0 0
$$952$$ 4.05582 0.131450
$$953$$ 22.9320 0.742839 0.371419 0.928465i $$-0.378871\pi$$
0.371419 + 0.928465i $$0.378871\pi$$
$$954$$ 0 0
$$955$$ −20.2281 −0.654565
$$956$$ 56.2691 1.81987
$$957$$ 0 0
$$958$$ 16.6581 0.538198
$$959$$ 30.5168 0.985438
$$960$$ 0 0
$$961$$ 28.2355 0.910822
$$962$$ 10.1723 0.327967
$$963$$ 0 0
$$964$$ −14.6768 −0.472708
$$965$$ 19.9293 0.641548
$$966$$ 0 0
$$967$$ 10.3700 0.333478 0.166739 0.986001i $$-0.446676\pi$$
0.166739 + 0.986001i $$0.446676\pi$$
$$968$$ −6.73419 −0.216445
$$969$$ 0 0
$$970$$ 28.3297 0.909611
$$971$$ −48.0410 −1.54171 −0.770854 0.637012i $$-0.780170\pi$$
−0.770854 + 0.637012i $$0.780170\pi$$
$$972$$ 0 0
$$973$$ 32.6890 1.04796
$$974$$ −24.9171 −0.798396
$$975$$ 0 0
$$976$$ 12.1191 0.387921
$$977$$ −27.0532 −0.865509 −0.432754 0.901512i $$-0.642458\pi$$
−0.432754 + 0.901512i $$0.642458\pi$$
$$978$$ 0 0
$$979$$ 4.05582 0.129624
$$980$$ −22.8055 −0.728494
$$981$$ 0 0
$$982$$ 19.2355 0.613829
$$983$$ 22.4817 0.717054 0.358527 0.933519i $$-0.383279\pi$$
0.358527 + 0.933519i $$0.383279\pi$$
$$984$$ 0 0
$$985$$ −15.5800 −0.496421
$$986$$ −10.9320 −0.348144
$$987$$ 0 0
$$988$$ 0.987774 0.0314253
$$989$$ 3.92935 0.124946
$$990$$ 0 0
$$991$$ −26.5316 −0.842805 −0.421402 0.906874i $$-0.638462\pi$$
−0.421402 + 0.906874i $$0.638462\pi$$
$$992$$ 62.2346 1.97595
$$993$$ 0 0
$$994$$ 52.1378 1.65371
$$995$$ −3.58482 −0.113647
$$996$$ 0 0
$$997$$ −28.3659 −0.898356 −0.449178 0.893442i $$-0.648283\pi$$
−0.449178 + 0.893442i $$0.648283\pi$$
$$998$$ 62.9433 1.99243
$$999$$ 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 405.2.a.i.1.3 3
3.2 odd 2 405.2.a.j.1.1 3
4.3 odd 2 6480.2.a.bs.1.1 3
5.2 odd 4 2025.2.b.m.649.5 6
5.3 odd 4 2025.2.b.m.649.2 6
5.4 even 2 2025.2.a.o.1.1 3
9.2 odd 6 45.2.e.b.31.3 yes 6
9.4 even 3 135.2.e.b.46.1 6
9.5 odd 6 45.2.e.b.16.3 6
9.7 even 3 135.2.e.b.91.1 6
12.11 even 2 6480.2.a.bv.1.1 3
15.2 even 4 2025.2.b.l.649.2 6
15.8 even 4 2025.2.b.l.649.5 6
15.14 odd 2 2025.2.a.n.1.3 3
36.7 odd 6 2160.2.q.k.1441.3 6
36.11 even 6 720.2.q.i.481.3 6
36.23 even 6 720.2.q.i.241.3 6
36.31 odd 6 2160.2.q.k.721.3 6
45.2 even 12 225.2.k.b.49.2 12
45.4 even 6 675.2.e.b.451.3 6
45.7 odd 12 675.2.k.b.199.5 12
45.13 odd 12 675.2.k.b.424.5 12
45.14 odd 6 225.2.e.b.151.1 6
45.22 odd 12 675.2.k.b.424.2 12
45.23 even 12 225.2.k.b.124.2 12
45.29 odd 6 225.2.e.b.76.1 6
45.32 even 12 225.2.k.b.124.5 12
45.34 even 6 675.2.e.b.226.3 6
45.38 even 12 225.2.k.b.49.5 12
45.43 odd 12 675.2.k.b.199.2 12

By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.3 6 9.5 odd 6
45.2.e.b.31.3 yes 6 9.2 odd 6
135.2.e.b.46.1 6 9.4 even 3
135.2.e.b.91.1 6 9.7 even 3
225.2.e.b.76.1 6 45.29 odd 6
225.2.e.b.151.1 6 45.14 odd 6
225.2.k.b.49.2 12 45.2 even 12
225.2.k.b.49.5 12 45.38 even 12
225.2.k.b.124.2 12 45.23 even 12
225.2.k.b.124.5 12 45.32 even 12
405.2.a.i.1.3 3 1.1 even 1 trivial
405.2.a.j.1.1 3 3.2 odd 2
675.2.e.b.226.3 6 45.34 even 6
675.2.e.b.451.3 6 45.4 even 6
675.2.k.b.199.2 12 45.43 odd 12
675.2.k.b.199.5 12 45.7 odd 12
675.2.k.b.424.2 12 45.22 odd 12
675.2.k.b.424.5 12 45.13 odd 12
720.2.q.i.241.3 6 36.23 even 6
720.2.q.i.481.3 6 36.11 even 6
2025.2.a.n.1.3 3 15.14 odd 2
2025.2.a.o.1.1 3 5.4 even 2
2025.2.b.l.649.2 6 15.2 even 4
2025.2.b.l.649.5 6 15.8 even 4
2025.2.b.m.649.2 6 5.3 odd 4
2025.2.b.m.649.5 6 5.2 odd 4
2160.2.q.k.721.3 6 36.31 odd 6
2160.2.q.k.1441.3 6 36.7 odd 6
6480.2.a.bs.1.1 3 4.3 odd 2
6480.2.a.bv.1.1 3 12.11 even 2