# Properties

 Label 495.6.a.a.1.2 Level $495$ Weight $6$ Character 495.1 Self dual yes Analytic conductor $79.390$ Analytic rank $1$ 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] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 495.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$79.3899908074$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.307532.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 76x + 168$$ x^3 - x^2 - 76*x + 168 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.30119$$ of defining polynomial Character $$\chi$$ $$=$$ 495.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.30119 q^{2} -13.4998 q^{4} +25.0000 q^{5} -148.216 q^{7} +195.703 q^{8} +O(q^{10})$$ $$q-4.30119 q^{2} -13.4998 q^{4} +25.0000 q^{5} -148.216 q^{7} +195.703 q^{8} -107.530 q^{10} -121.000 q^{11} +234.137 q^{13} +637.504 q^{14} -409.764 q^{16} -1031.76 q^{17} +1266.29 q^{19} -337.494 q^{20} +520.444 q^{22} -384.710 q^{23} +625.000 q^{25} -1007.07 q^{26} +2000.88 q^{28} +4484.85 q^{29} -997.235 q^{31} -4500.03 q^{32} +4437.78 q^{34} -3705.39 q^{35} +5168.11 q^{37} -5446.56 q^{38} +4892.58 q^{40} -2259.17 q^{41} +15818.9 q^{43} +1633.47 q^{44} +1654.71 q^{46} -12033.2 q^{47} +5160.90 q^{49} -2688.24 q^{50} -3160.79 q^{52} +3851.67 q^{53} -3025.00 q^{55} -29006.3 q^{56} -19290.2 q^{58} +20261.0 q^{59} +2006.88 q^{61} +4289.30 q^{62} +32467.9 q^{64} +5853.41 q^{65} +40945.6 q^{67} +13928.5 q^{68} +15937.6 q^{70} -16970.8 q^{71} -56640.5 q^{73} -22229.0 q^{74} -17094.6 q^{76} +17934.1 q^{77} +58507.9 q^{79} -10244.1 q^{80} +9717.12 q^{82} +52243.8 q^{83} -25793.9 q^{85} -68040.2 q^{86} -23680.1 q^{88} -55114.4 q^{89} -34702.7 q^{91} +5193.50 q^{92} +51757.0 q^{94} +31657.3 q^{95} +99383.8 q^{97} -22198.0 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8}+O(q^{10})$$ 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8 $$3 q - 7 q^{2} + 73 q^{4} + 75 q^{5} + 92 q^{7} - 231 q^{8} - 175 q^{10} - 363 q^{11} - 90 q^{13} + 784 q^{14} - 415 q^{16} - 1934 q^{17} + 2084 q^{19} + 1825 q^{20} + 847 q^{22} - 1220 q^{23} + 1875 q^{25} - 17062 q^{26} + 11120 q^{28} - 4402 q^{29} - 10688 q^{31} - 12439 q^{32} - 4094 q^{34} + 2300 q^{35} - 8190 q^{37} - 13792 q^{38} - 5775 q^{40} - 5974 q^{41} + 18868 q^{43} - 8833 q^{44} + 46220 q^{46} - 55500 q^{47} + 1907 q^{49} - 4375 q^{50} + 27330 q^{52} - 9206 q^{53} - 9075 q^{55} - 73248 q^{56} + 15366 q^{58} + 59196 q^{59} + 79902 q^{61} - 64616 q^{62} + 2129 q^{64} - 2250 q^{65} + 4468 q^{67} + 1218 q^{68} + 19600 q^{70} + 75164 q^{71} - 61290 q^{73} + 56766 q^{74} + 37816 q^{76} - 11132 q^{77} - 83564 q^{79} - 10375 q^{80} + 147410 q^{82} - 74764 q^{83} - 48350 q^{85} + 253432 q^{86} + 27951 q^{88} - 37342 q^{89} - 126488 q^{91} - 148164 q^{92} + 59252 q^{94} + 52100 q^{95} + 33486 q^{97} + 95249 q^{98}+O(q^{100})$$ 3 * q - 7 * q^2 + 73 * q^4 + 75 * q^5 + 92 * q^7 - 231 * q^8 - 175 * q^10 - 363 * q^11 - 90 * q^13 + 784 * q^14 - 415 * q^16 - 1934 * q^17 + 2084 * q^19 + 1825 * q^20 + 847 * q^22 - 1220 * q^23 + 1875 * q^25 - 17062 * q^26 + 11120 * q^28 - 4402 * q^29 - 10688 * q^31 - 12439 * q^32 - 4094 * q^34 + 2300 * q^35 - 8190 * q^37 - 13792 * q^38 - 5775 * q^40 - 5974 * q^41 + 18868 * q^43 - 8833 * q^44 + 46220 * q^46 - 55500 * q^47 + 1907 * q^49 - 4375 * q^50 + 27330 * q^52 - 9206 * q^53 - 9075 * q^55 - 73248 * q^56 + 15366 * q^58 + 59196 * q^59 + 79902 * q^61 - 64616 * q^62 + 2129 * q^64 - 2250 * q^65 + 4468 * q^67 + 1218 * q^68 + 19600 * q^70 + 75164 * q^71 - 61290 * q^73 + 56766 * q^74 + 37816 * q^76 - 11132 * q^77 - 83564 * q^79 - 10375 * q^80 + 147410 * q^82 - 74764 * q^83 - 48350 * q^85 + 253432 * q^86 + 27951 * q^88 - 37342 * q^89 - 126488 * q^91 - 148164 * q^92 + 59252 * q^94 + 52100 * q^95 + 33486 * q^97 + 95249 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.30119 −0.760350 −0.380175 0.924915i $$-0.624136\pi$$
−0.380175 + 0.924915i $$0.624136\pi$$
$$3$$ 0 0
$$4$$ −13.4998 −0.421868
$$5$$ 25.0000 0.447214
$$6$$ 0 0
$$7$$ −148.216 −1.14327 −0.571635 0.820508i $$-0.693691\pi$$
−0.571635 + 0.820508i $$0.693691\pi$$
$$8$$ 195.703 1.08112
$$9$$ 0 0
$$10$$ −107.530 −0.340039
$$11$$ −121.000 −0.301511
$$12$$ 0 0
$$13$$ 234.137 0.384247 0.192124 0.981371i $$-0.438463\pi$$
0.192124 + 0.981371i $$0.438463\pi$$
$$14$$ 637.504 0.869286
$$15$$ 0 0
$$16$$ −409.764 −0.400160
$$17$$ −1031.76 −0.865875 −0.432937 0.901424i $$-0.642523\pi$$
−0.432937 + 0.901424i $$0.642523\pi$$
$$18$$ 0 0
$$19$$ 1266.29 0.804729 0.402365 0.915480i $$-0.368188\pi$$
0.402365 + 0.915480i $$0.368188\pi$$
$$20$$ −337.494 −0.188665
$$21$$ 0 0
$$22$$ 520.444 0.229254
$$23$$ −384.710 −0.151640 −0.0758201 0.997122i $$-0.524157\pi$$
−0.0758201 + 0.997122i $$0.524157\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ −1007.07 −0.292163
$$27$$ 0 0
$$28$$ 2000.88 0.482309
$$29$$ 4484.85 0.990268 0.495134 0.868817i $$-0.335119\pi$$
0.495134 + 0.868817i $$0.335119\pi$$
$$30$$ 0 0
$$31$$ −997.235 −0.186377 −0.0931887 0.995648i $$-0.529706\pi$$
−0.0931887 + 0.995648i $$0.529706\pi$$
$$32$$ −4500.03 −0.776856
$$33$$ 0 0
$$34$$ 4437.78 0.658368
$$35$$ −3705.39 −0.511286
$$36$$ 0 0
$$37$$ 5168.11 0.620622 0.310311 0.950635i $$-0.399567\pi$$
0.310311 + 0.950635i $$0.399567\pi$$
$$38$$ −5446.56 −0.611876
$$39$$ 0 0
$$40$$ 4892.58 0.483490
$$41$$ −2259.17 −0.209889 −0.104944 0.994478i $$-0.533466\pi$$
−0.104944 + 0.994478i $$0.533466\pi$$
$$42$$ 0 0
$$43$$ 15818.9 1.30469 0.652343 0.757924i $$-0.273786\pi$$
0.652343 + 0.757924i $$0.273786\pi$$
$$44$$ 1633.47 0.127198
$$45$$ 0 0
$$46$$ 1654.71 0.115300
$$47$$ −12033.2 −0.794577 −0.397289 0.917694i $$-0.630049\pi$$
−0.397289 + 0.917694i $$0.630049\pi$$
$$48$$ 0 0
$$49$$ 5160.90 0.307069
$$50$$ −2688.24 −0.152070
$$51$$ 0 0
$$52$$ −3160.79 −0.162102
$$53$$ 3851.67 0.188347 0.0941737 0.995556i $$-0.469979\pi$$
0.0941737 + 0.995556i $$0.469979\pi$$
$$54$$ 0 0
$$55$$ −3025.00 −0.134840
$$56$$ −29006.3 −1.23601
$$57$$ 0 0
$$58$$ −19290.2 −0.752950
$$59$$ 20261.0 0.757760 0.378880 0.925446i $$-0.376309\pi$$
0.378880 + 0.925446i $$0.376309\pi$$
$$60$$ 0 0
$$61$$ 2006.88 0.0690553 0.0345276 0.999404i $$-0.489007\pi$$
0.0345276 + 0.999404i $$0.489007\pi$$
$$62$$ 4289.30 0.141712
$$63$$ 0 0
$$64$$ 32467.9 0.990842
$$65$$ 5853.41 0.171841
$$66$$ 0 0
$$67$$ 40945.6 1.11435 0.557173 0.830397i $$-0.311886\pi$$
0.557173 + 0.830397i $$0.311886\pi$$
$$68$$ 13928.5 0.365285
$$69$$ 0 0
$$70$$ 15937.6 0.388757
$$71$$ −16970.8 −0.399537 −0.199769 0.979843i $$-0.564019\pi$$
−0.199769 + 0.979843i $$0.564019\pi$$
$$72$$ 0 0
$$73$$ −56640.5 −1.24400 −0.621999 0.783018i $$-0.713679\pi$$
−0.621999 + 0.783018i $$0.713679\pi$$
$$74$$ −22229.0 −0.471890
$$75$$ 0 0
$$76$$ −17094.6 −0.339489
$$77$$ 17934.1 0.344709
$$78$$ 0 0
$$79$$ 58507.9 1.05474 0.527372 0.849635i $$-0.323178\pi$$
0.527372 + 0.849635i $$0.323178\pi$$
$$80$$ −10244.1 −0.178957
$$81$$ 0 0
$$82$$ 9717.12 0.159589
$$83$$ 52243.8 0.832415 0.416207 0.909270i $$-0.363359\pi$$
0.416207 + 0.909270i $$0.363359\pi$$
$$84$$ 0 0
$$85$$ −25793.9 −0.387231
$$86$$ −68040.2 −0.992018
$$87$$ 0 0
$$88$$ −23680.1 −0.325969
$$89$$ −55114.4 −0.737548 −0.368774 0.929519i $$-0.620222\pi$$
−0.368774 + 0.929519i $$0.620222\pi$$
$$90$$ 0 0
$$91$$ −34702.7 −0.439299
$$92$$ 5193.50 0.0639721
$$93$$ 0 0
$$94$$ 51757.0 0.604157
$$95$$ 31657.3 0.359886
$$96$$ 0 0
$$97$$ 99383.8 1.07247 0.536237 0.844068i $$-0.319846\pi$$
0.536237 + 0.844068i $$0.319846\pi$$
$$98$$ −22198.0 −0.233480
$$99$$ 0 0
$$100$$ −8437.35 −0.0843735
$$101$$ 25334.3 0.247118 0.123559 0.992337i $$-0.460569\pi$$
0.123559 + 0.992337i $$0.460569\pi$$
$$102$$ 0 0
$$103$$ −93216.6 −0.865766 −0.432883 0.901450i $$-0.642504\pi$$
−0.432883 + 0.901450i $$0.642504\pi$$
$$104$$ 45821.3 0.415416
$$105$$ 0 0
$$106$$ −16566.8 −0.143210
$$107$$ −205830. −1.73799 −0.868997 0.494817i $$-0.835235\pi$$
−0.868997 + 0.494817i $$0.835235\pi$$
$$108$$ 0 0
$$109$$ −155901. −1.25684 −0.628422 0.777872i $$-0.716299\pi$$
−0.628422 + 0.777872i $$0.716299\pi$$
$$110$$ 13011.1 0.102526
$$111$$ 0 0
$$112$$ 60733.4 0.457491
$$113$$ −40304.5 −0.296932 −0.148466 0.988917i $$-0.547434\pi$$
−0.148466 + 0.988917i $$0.547434\pi$$
$$114$$ 0 0
$$115$$ −9617.75 −0.0678155
$$116$$ −60544.4 −0.417762
$$117$$ 0 0
$$118$$ −87146.6 −0.576163
$$119$$ 152923. 0.989929
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −8631.97 −0.0525062
$$123$$ 0 0
$$124$$ 13462.4 0.0786266
$$125$$ 15625.0 0.0894427
$$126$$ 0 0
$$127$$ 120825. 0.664733 0.332367 0.943150i $$-0.392153\pi$$
0.332367 + 0.943150i $$0.392153\pi$$
$$128$$ 4350.25 0.0234687
$$129$$ 0 0
$$130$$ −25176.6 −0.130659
$$131$$ −266474. −1.35668 −0.678338 0.734750i $$-0.737299\pi$$
−0.678338 + 0.734750i $$0.737299\pi$$
$$132$$ 0 0
$$133$$ −187684. −0.920023
$$134$$ −176115. −0.847292
$$135$$ 0 0
$$136$$ −201918. −0.936112
$$137$$ −17176.5 −0.0781866 −0.0390933 0.999236i $$-0.512447\pi$$
−0.0390933 + 0.999236i $$0.512447\pi$$
$$138$$ 0 0
$$139$$ −5031.99 −0.0220903 −0.0110452 0.999939i $$-0.503516\pi$$
−0.0110452 + 0.999939i $$0.503516\pi$$
$$140$$ 50021.9 0.215695
$$141$$ 0 0
$$142$$ 72994.7 0.303788
$$143$$ −28330.5 −0.115855
$$144$$ 0 0
$$145$$ 112121. 0.442861
$$146$$ 243621. 0.945874
$$147$$ 0 0
$$148$$ −69768.3 −0.261820
$$149$$ 37407.9 0.138038 0.0690188 0.997615i $$-0.478013\pi$$
0.0690188 + 0.997615i $$0.478013\pi$$
$$150$$ 0 0
$$151$$ −225853. −0.806091 −0.403046 0.915180i $$-0.632048\pi$$
−0.403046 + 0.915180i $$0.632048\pi$$
$$152$$ 247817. 0.870006
$$153$$ 0 0
$$154$$ −77138.0 −0.262100
$$155$$ −24930.9 −0.0833505
$$156$$ 0 0
$$157$$ −205537. −0.665489 −0.332744 0.943017i $$-0.607975\pi$$
−0.332744 + 0.943017i $$0.607975\pi$$
$$158$$ −251654. −0.801974
$$159$$ 0 0
$$160$$ −112501. −0.347420
$$161$$ 57020.1 0.173366
$$162$$ 0 0
$$163$$ −639398. −1.88496 −0.942481 0.334261i $$-0.891513\pi$$
−0.942481 + 0.334261i $$0.891513\pi$$
$$164$$ 30498.3 0.0885453
$$165$$ 0 0
$$166$$ −224711. −0.632927
$$167$$ −416990. −1.15700 −0.578502 0.815681i $$-0.696363\pi$$
−0.578502 + 0.815681i $$0.696363\pi$$
$$168$$ 0 0
$$169$$ −316473. −0.852354
$$170$$ 110945. 0.294431
$$171$$ 0 0
$$172$$ −213552. −0.550405
$$173$$ −608651. −1.54615 −0.773077 0.634312i $$-0.781284\pi$$
−0.773077 + 0.634312i $$0.781284\pi$$
$$174$$ 0 0
$$175$$ −92634.8 −0.228654
$$176$$ 49581.4 0.120653
$$177$$ 0 0
$$178$$ 237058. 0.560795
$$179$$ −511566. −1.19335 −0.596676 0.802482i $$-0.703512\pi$$
−0.596676 + 0.802482i $$0.703512\pi$$
$$180$$ 0 0
$$181$$ 56369.0 0.127892 0.0639460 0.997953i $$-0.479631\pi$$
0.0639460 + 0.997953i $$0.479631\pi$$
$$182$$ 149263. 0.334021
$$183$$ 0 0
$$184$$ −75289.0 −0.163941
$$185$$ 129203. 0.277551
$$186$$ 0 0
$$187$$ 124843. 0.261071
$$188$$ 162445. 0.335206
$$189$$ 0 0
$$190$$ −136164. −0.273639
$$191$$ −45986.9 −0.0912117 −0.0456059 0.998960i $$-0.514522\pi$$
−0.0456059 + 0.998960i $$0.514522\pi$$
$$192$$ 0 0
$$193$$ 360889. 0.697398 0.348699 0.937235i $$-0.386624\pi$$
0.348699 + 0.937235i $$0.386624\pi$$
$$194$$ −427469. −0.815455
$$195$$ 0 0
$$196$$ −69671.0 −0.129542
$$197$$ −978922. −1.79714 −0.898571 0.438828i $$-0.855394\pi$$
−0.898571 + 0.438828i $$0.855394\pi$$
$$198$$ 0 0
$$199$$ 902328. 1.61522 0.807610 0.589717i $$-0.200761\pi$$
0.807610 + 0.589717i $$0.200761\pi$$
$$200$$ 122314. 0.216223
$$201$$ 0 0
$$202$$ −108968. −0.187897
$$203$$ −664725. −1.13214
$$204$$ 0 0
$$205$$ −56479.3 −0.0938652
$$206$$ 400942. 0.658285
$$207$$ 0 0
$$208$$ −95940.7 −0.153760
$$209$$ −153221. −0.242635
$$210$$ 0 0
$$211$$ 318392. 0.492330 0.246165 0.969228i $$-0.420830\pi$$
0.246165 + 0.969228i $$0.420830\pi$$
$$212$$ −51996.7 −0.0794577
$$213$$ 0 0
$$214$$ 885312. 1.32148
$$215$$ 395473. 0.583473
$$216$$ 0 0
$$217$$ 147806. 0.213080
$$218$$ 670558. 0.955642
$$219$$ 0 0
$$220$$ 40836.8 0.0568846
$$221$$ −241572. −0.332710
$$222$$ 0 0
$$223$$ 661352. 0.890575 0.445287 0.895388i $$-0.353101\pi$$
0.445287 + 0.895388i $$0.353101\pi$$
$$224$$ 666975. 0.888157
$$225$$ 0 0
$$226$$ 173357. 0.225773
$$227$$ −606704. −0.781470 −0.390735 0.920503i $$-0.627779\pi$$
−0.390735 + 0.920503i $$0.627779\pi$$
$$228$$ 0 0
$$229$$ −352377. −0.444036 −0.222018 0.975043i $$-0.571264\pi$$
−0.222018 + 0.975043i $$0.571264\pi$$
$$230$$ 41367.8 0.0515635
$$231$$ 0 0
$$232$$ 877699. 1.07060
$$233$$ −1.49288e6 −1.80151 −0.900755 0.434328i $$-0.856986\pi$$
−0.900755 + 0.434328i $$0.856986\pi$$
$$234$$ 0 0
$$235$$ −300830. −0.355346
$$236$$ −273519. −0.319675
$$237$$ 0 0
$$238$$ −657749. −0.752693
$$239$$ −388141. −0.439536 −0.219768 0.975552i $$-0.570530\pi$$
−0.219768 + 0.975552i $$0.570530\pi$$
$$240$$ 0 0
$$241$$ −75337.2 −0.0835540 −0.0417770 0.999127i $$-0.513302\pi$$
−0.0417770 + 0.999127i $$0.513302\pi$$
$$242$$ −62973.7 −0.0691227
$$243$$ 0 0
$$244$$ −27092.4 −0.0291322
$$245$$ 129023. 0.137325
$$246$$ 0 0
$$247$$ 296485. 0.309215
$$248$$ −195162. −0.201496
$$249$$ 0 0
$$250$$ −67206.1 −0.0680078
$$251$$ 346975. 0.347627 0.173814 0.984779i $$-0.444391\pi$$
0.173814 + 0.984779i $$0.444391\pi$$
$$252$$ 0 0
$$253$$ 46549.9 0.0457212
$$254$$ −519691. −0.505430
$$255$$ 0 0
$$256$$ −1.05768e6 −1.00869
$$257$$ −1.97267e6 −1.86304 −0.931518 0.363696i $$-0.881515\pi$$
−0.931518 + 0.363696i $$0.881515\pi$$
$$258$$ 0 0
$$259$$ −765995. −0.709539
$$260$$ −79019.7 −0.0724940
$$261$$ 0 0
$$262$$ 1.14615e6 1.03155
$$263$$ 190986. 0.170260 0.0851300 0.996370i $$-0.472869\pi$$
0.0851300 + 0.996370i $$0.472869\pi$$
$$264$$ 0 0
$$265$$ 96291.8 0.0842316
$$266$$ 807266. 0.699540
$$267$$ 0 0
$$268$$ −552755. −0.470106
$$269$$ −933912. −0.786911 −0.393455 0.919344i $$-0.628720\pi$$
−0.393455 + 0.919344i $$0.628720\pi$$
$$270$$ 0 0
$$271$$ −1.17171e6 −0.969163 −0.484582 0.874746i $$-0.661028\pi$$
−0.484582 + 0.874746i $$0.661028\pi$$
$$272$$ 422776. 0.346488
$$273$$ 0 0
$$274$$ 73879.3 0.0594492
$$275$$ −75625.0 −0.0603023
$$276$$ 0 0
$$277$$ −472512. −0.370010 −0.185005 0.982738i $$-0.559230\pi$$
−0.185005 + 0.982738i $$0.559230\pi$$
$$278$$ 21643.5 0.0167964
$$279$$ 0 0
$$280$$ −725157. −0.552760
$$281$$ 2.05374e6 1.55160 0.775798 0.630981i $$-0.217348\pi$$
0.775798 + 0.630981i $$0.217348\pi$$
$$282$$ 0 0
$$283$$ 465578. 0.345562 0.172781 0.984960i $$-0.444725\pi$$
0.172781 + 0.984960i $$0.444725\pi$$
$$284$$ 229102. 0.168552
$$285$$ 0 0
$$286$$ 121855. 0.0880903
$$287$$ 334845. 0.239960
$$288$$ 0 0
$$289$$ −355335. −0.250261
$$290$$ −482255. −0.336730
$$291$$ 0 0
$$292$$ 764633. 0.524803
$$293$$ 1.19434e6 0.812751 0.406376 0.913706i $$-0.366792\pi$$
0.406376 + 0.913706i $$0.366792\pi$$
$$294$$ 0 0
$$295$$ 506526. 0.338881
$$296$$ 1.01141e6 0.670965
$$297$$ 0 0
$$298$$ −160898. −0.104957
$$299$$ −90074.7 −0.0582673
$$300$$ 0 0
$$301$$ −2.34461e6 −1.49161
$$302$$ 971438. 0.612912
$$303$$ 0 0
$$304$$ −518880. −0.322020
$$305$$ 50172.0 0.0308825
$$306$$ 0 0
$$307$$ 1.72251e6 1.04308 0.521538 0.853228i $$-0.325359\pi$$
0.521538 + 0.853228i $$0.325359\pi$$
$$308$$ −242106. −0.145422
$$309$$ 0 0
$$310$$ 107232. 0.0633755
$$311$$ 2.60616e6 1.52792 0.763960 0.645263i $$-0.223252\pi$$
0.763960 + 0.645263i $$0.223252\pi$$
$$312$$ 0 0
$$313$$ 1.22311e6 0.705677 0.352839 0.935684i $$-0.385216\pi$$
0.352839 + 0.935684i $$0.385216\pi$$
$$314$$ 884053. 0.506005
$$315$$ 0 0
$$316$$ −789843. −0.444962
$$317$$ 700043. 0.391270 0.195635 0.980677i $$-0.437323\pi$$
0.195635 + 0.980677i $$0.437323\pi$$
$$318$$ 0 0
$$319$$ −542667. −0.298577
$$320$$ 811698. 0.443118
$$321$$ 0 0
$$322$$ −245254. −0.131819
$$323$$ −1.30650e6 −0.696794
$$324$$ 0 0
$$325$$ 146335. 0.0768495
$$326$$ 2.75017e6 1.43323
$$327$$ 0 0
$$328$$ −442127. −0.226914
$$329$$ 1.78351e6 0.908417
$$330$$ 0 0
$$331$$ 2.78738e6 1.39839 0.699193 0.714933i $$-0.253543\pi$$
0.699193 + 0.714933i $$0.253543\pi$$
$$332$$ −705279. −0.351169
$$333$$ 0 0
$$334$$ 1.79355e6 0.879728
$$335$$ 1.02364e6 0.498350
$$336$$ 0 0
$$337$$ −795333. −0.381482 −0.190741 0.981640i $$-0.561089\pi$$
−0.190741 + 0.981640i $$0.561089\pi$$
$$338$$ 1.36121e6 0.648087
$$339$$ 0 0
$$340$$ 348212. 0.163360
$$341$$ 120665. 0.0561949
$$342$$ 0 0
$$343$$ 1.72613e6 0.792208
$$344$$ 3.09581e6 1.41052
$$345$$ 0 0
$$346$$ 2.61792e6 1.17562
$$347$$ 3.80533e6 1.69656 0.848278 0.529551i $$-0.177639\pi$$
0.848278 + 0.529551i $$0.177639\pi$$
$$348$$ 0 0
$$349$$ −798008. −0.350706 −0.175353 0.984506i $$-0.556107\pi$$
−0.175353 + 0.984506i $$0.556107\pi$$
$$350$$ 398440. 0.173857
$$351$$ 0 0
$$352$$ 544503. 0.234231
$$353$$ −2.48978e6 −1.06347 −0.531733 0.846912i $$-0.678459\pi$$
−0.531733 + 0.846912i $$0.678459\pi$$
$$354$$ 0 0
$$355$$ −424271. −0.178678
$$356$$ 744032. 0.311148
$$357$$ 0 0
$$358$$ 2.20034e6 0.907366
$$359$$ 2.15148e6 0.881050 0.440525 0.897740i $$-0.354792\pi$$
0.440525 + 0.897740i $$0.354792\pi$$
$$360$$ 0 0
$$361$$ −872605. −0.352411
$$362$$ −242454. −0.0972427
$$363$$ 0 0
$$364$$ 468479. 0.185326
$$365$$ −1.41601e6 −0.556333
$$366$$ 0 0
$$367$$ 1.47515e6 0.571705 0.285853 0.958274i $$-0.407723\pi$$
0.285853 + 0.958274i $$0.407723\pi$$
$$368$$ 157640. 0.0606803
$$369$$ 0 0
$$370$$ −555725. −0.211036
$$371$$ −570879. −0.215332
$$372$$ 0 0
$$373$$ −4.71290e6 −1.75394 −0.876972 0.480541i $$-0.840440\pi$$
−0.876972 + 0.480541i $$0.840440\pi$$
$$374$$ −536972. −0.198505
$$375$$ 0 0
$$376$$ −2.35493e6 −0.859031
$$377$$ 1.05007e6 0.380508
$$378$$ 0 0
$$379$$ 4.47900e6 1.60171 0.800853 0.598861i $$-0.204380\pi$$
0.800853 + 0.598861i $$0.204380\pi$$
$$380$$ −427366. −0.151824
$$381$$ 0 0
$$382$$ 197798. 0.0693528
$$383$$ 57968.7 0.0201928 0.0100964 0.999949i $$-0.496786\pi$$
0.0100964 + 0.999949i $$0.496786\pi$$
$$384$$ 0 0
$$385$$ 448353. 0.154159
$$386$$ −1.55225e6 −0.530267
$$387$$ 0 0
$$388$$ −1.34166e6 −0.452442
$$389$$ 4.53325e6 1.51892 0.759461 0.650553i $$-0.225463\pi$$
0.759461 + 0.650553i $$0.225463\pi$$
$$390$$ 0 0
$$391$$ 396927. 0.131301
$$392$$ 1.01000e6 0.331977
$$393$$ 0 0
$$394$$ 4.21053e6 1.36646
$$395$$ 1.46270e6 0.471695
$$396$$ 0 0
$$397$$ 621573. 0.197932 0.0989660 0.995091i $$-0.468446\pi$$
0.0989660 + 0.995091i $$0.468446\pi$$
$$398$$ −3.88108e6 −1.22813
$$399$$ 0 0
$$400$$ −256102. −0.0800320
$$401$$ −954326. −0.296371 −0.148186 0.988960i $$-0.547343\pi$$
−0.148186 + 0.988960i $$0.547343\pi$$
$$402$$ 0 0
$$403$$ −233489. −0.0716150
$$404$$ −342007. −0.104251
$$405$$ 0 0
$$406$$ 2.85911e6 0.860826
$$407$$ −625341. −0.187125
$$408$$ 0 0
$$409$$ 3.19864e6 0.945491 0.472746 0.881199i $$-0.343263\pi$$
0.472746 + 0.881199i $$0.343263\pi$$
$$410$$ 242928. 0.0713704
$$411$$ 0 0
$$412$$ 1.25840e6 0.365239
$$413$$ −3.00301e6 −0.866325
$$414$$ 0 0
$$415$$ 1.30610e6 0.372267
$$416$$ −1.05362e6 −0.298505
$$417$$ 0 0
$$418$$ 659034. 0.184487
$$419$$ 6.60122e6 1.83691 0.918457 0.395520i $$-0.129436\pi$$
0.918457 + 0.395520i $$0.129436\pi$$
$$420$$ 0 0
$$421$$ −4.22510e6 −1.16180 −0.580901 0.813974i $$-0.697300\pi$$
−0.580901 + 0.813974i $$0.697300\pi$$
$$422$$ −1.36946e6 −0.374343
$$423$$ 0 0
$$424$$ 753785. 0.203626
$$425$$ −644848. −0.173175
$$426$$ 0 0
$$427$$ −297451. −0.0789489
$$428$$ 2.77865e6 0.733204
$$429$$ 0 0
$$430$$ −1.70100e6 −0.443644
$$431$$ −4.72300e6 −1.22469 −0.612344 0.790592i $$-0.709773\pi$$
−0.612344 + 0.790592i $$0.709773\pi$$
$$432$$ 0 0
$$433$$ −882129. −0.226106 −0.113053 0.993589i $$-0.536063\pi$$
−0.113053 + 0.993589i $$0.536063\pi$$
$$434$$ −635741. −0.162015
$$435$$ 0 0
$$436$$ 2.10462e6 0.530222
$$437$$ −487155. −0.122029
$$438$$ 0 0
$$439$$ −3.07514e6 −0.761560 −0.380780 0.924666i $$-0.624344\pi$$
−0.380780 + 0.924666i $$0.624344\pi$$
$$440$$ −592002. −0.145778
$$441$$ 0 0
$$442$$ 1.03905e6 0.252976
$$443$$ −3.40381e6 −0.824055 −0.412028 0.911171i $$-0.635179\pi$$
−0.412028 + 0.911171i $$0.635179\pi$$
$$444$$ 0 0
$$445$$ −1.37786e6 −0.329841
$$446$$ −2.84460e6 −0.677149
$$447$$ 0 0
$$448$$ −4.81226e6 −1.13280
$$449$$ 605757. 0.141802 0.0709010 0.997483i $$-0.477413\pi$$
0.0709010 + 0.997483i $$0.477413\pi$$
$$450$$ 0 0
$$451$$ 273360. 0.0632839
$$452$$ 544102. 0.125266
$$453$$ 0 0
$$454$$ 2.60955e6 0.594191
$$455$$ −867568. −0.196460
$$456$$ 0 0
$$457$$ −5.46329e6 −1.22367 −0.611834 0.790986i $$-0.709568\pi$$
−0.611834 + 0.790986i $$0.709568\pi$$
$$458$$ 1.51564e6 0.337623
$$459$$ 0 0
$$460$$ 129837. 0.0286092
$$461$$ 2.84223e6 0.622883 0.311441 0.950265i $$-0.399188\pi$$
0.311441 + 0.950265i $$0.399188\pi$$
$$462$$ 0 0
$$463$$ −8.51652e6 −1.84633 −0.923166 0.384401i $$-0.874408\pi$$
−0.923166 + 0.384401i $$0.874408\pi$$
$$464$$ −1.83773e6 −0.396266
$$465$$ 0 0
$$466$$ 6.42118e6 1.36978
$$467$$ −7.00140e6 −1.48557 −0.742784 0.669531i $$-0.766495\pi$$
−0.742784 + 0.669531i $$0.766495\pi$$
$$468$$ 0 0
$$469$$ −6.06877e6 −1.27400
$$470$$ 1.29393e6 0.270187
$$471$$ 0 0
$$472$$ 3.96515e6 0.819228
$$473$$ −1.91409e6 −0.393377
$$474$$ 0 0
$$475$$ 791432. 0.160946
$$476$$ −2.06442e6 −0.417619
$$477$$ 0 0
$$478$$ 1.66947e6 0.334201
$$479$$ −1.06826e6 −0.212734 −0.106367 0.994327i $$-0.533922\pi$$
−0.106367 + 0.994327i $$0.533922\pi$$
$$480$$ 0 0
$$481$$ 1.21004e6 0.238472
$$482$$ 324040. 0.0635303
$$483$$ 0 0
$$484$$ −197650. −0.0383516
$$485$$ 2.48460e6 0.479625
$$486$$ 0 0
$$487$$ 1.00308e6 0.191652 0.0958260 0.995398i $$-0.469451\pi$$
0.0958260 + 0.995398i $$0.469451\pi$$
$$488$$ 392753. 0.0746569
$$489$$ 0 0
$$490$$ −554950. −0.104415
$$491$$ −9.23719e6 −1.72916 −0.864582 0.502491i $$-0.832417\pi$$
−0.864582 + 0.502491i $$0.832417\pi$$
$$492$$ 0 0
$$493$$ −4.62727e6 −0.857448
$$494$$ −1.27524e6 −0.235112
$$495$$ 0 0
$$496$$ 408631. 0.0745807
$$497$$ 2.51534e6 0.456779
$$498$$ 0 0
$$499$$ 4.31532e6 0.775822 0.387911 0.921697i $$-0.373197\pi$$
0.387911 + 0.921697i $$0.373197\pi$$
$$500$$ −210934. −0.0377330
$$501$$ 0 0
$$502$$ −1.49240e6 −0.264318
$$503$$ −3.28052e6 −0.578127 −0.289064 0.957310i $$-0.593344\pi$$
−0.289064 + 0.957310i $$0.593344\pi$$
$$504$$ 0 0
$$505$$ 633357. 0.110515
$$506$$ −200220. −0.0347641
$$507$$ 0 0
$$508$$ −1.63111e6 −0.280430
$$509$$ 2.50882e6 0.429215 0.214608 0.976700i $$-0.431153\pi$$
0.214608 + 0.976700i $$0.431153\pi$$
$$510$$ 0 0
$$511$$ 8.39501e6 1.42223
$$512$$ 4.41009e6 0.743486
$$513$$ 0 0
$$514$$ 8.48482e6 1.41656
$$515$$ −2.33042e6 −0.387182
$$516$$ 0 0
$$517$$ 1.45602e6 0.239574
$$518$$ 3.29469e6 0.539498
$$519$$ 0 0
$$520$$ 1.14553e6 0.185780
$$521$$ 3.63347e6 0.586444 0.293222 0.956044i $$-0.405272\pi$$
0.293222 + 0.956044i $$0.405272\pi$$
$$522$$ 0 0
$$523$$ −7.95990e6 −1.27249 −0.636243 0.771488i $$-0.719513\pi$$
−0.636243 + 0.771488i $$0.719513\pi$$
$$524$$ 3.59733e6 0.572338
$$525$$ 0 0
$$526$$ −821468. −0.129457
$$527$$ 1.02890e6 0.161379
$$528$$ 0 0
$$529$$ −6.28834e6 −0.977005
$$530$$ −414170. −0.0640455
$$531$$ 0 0
$$532$$ 2.53369e6 0.388128
$$533$$ −528954. −0.0806492
$$534$$ 0 0
$$535$$ −5.14574e6 −0.777255
$$536$$ 8.01317e6 1.20474
$$537$$ 0 0
$$538$$ 4.01693e6 0.598328
$$539$$ −624469. −0.0925847
$$540$$ 0 0
$$541$$ 1.20412e6 0.176879 0.0884395 0.996082i $$-0.471812\pi$$
0.0884395 + 0.996082i $$0.471812\pi$$
$$542$$ 5.03975e6 0.736903
$$543$$ 0 0
$$544$$ 4.64293e6 0.672660
$$545$$ −3.89752e6 −0.562078
$$546$$ 0 0
$$547$$ −1.04446e7 −1.49254 −0.746268 0.665646i $$-0.768156\pi$$
−0.746268 + 0.665646i $$0.768156\pi$$
$$548$$ 231878. 0.0329844
$$549$$ 0 0
$$550$$ 325277. 0.0458508
$$551$$ 5.67912e6 0.796897
$$552$$ 0 0
$$553$$ −8.67179e6 −1.20586
$$554$$ 2.03236e6 0.281337
$$555$$ 0 0
$$556$$ 67930.6 0.00931920
$$557$$ −8.45756e6 −1.15507 −0.577534 0.816367i $$-0.695985\pi$$
−0.577534 + 0.816367i $$0.695985\pi$$
$$558$$ 0 0
$$559$$ 3.70379e6 0.501322
$$560$$ 1.51834e6 0.204596
$$561$$ 0 0
$$562$$ −8.83350e6 −1.17976
$$563$$ −9.96372e6 −1.32480 −0.662401 0.749150i $$-0.730462\pi$$
−0.662401 + 0.749150i $$0.730462\pi$$
$$564$$ 0 0
$$565$$ −1.00761e6 −0.132792
$$566$$ −2.00254e6 −0.262748
$$567$$ 0 0
$$568$$ −3.32124e6 −0.431946
$$569$$ 1.05157e7 1.36163 0.680813 0.732458i $$-0.261627\pi$$
0.680813 + 0.732458i $$0.261627\pi$$
$$570$$ 0 0
$$571$$ 9.02900e6 1.15891 0.579454 0.815005i $$-0.303266\pi$$
0.579454 + 0.815005i $$0.303266\pi$$
$$572$$ 382456. 0.0488755
$$573$$ 0 0
$$574$$ −1.44023e6 −0.182453
$$575$$ −240444. −0.0303280
$$576$$ 0 0
$$577$$ 6.65688e6 0.832398 0.416199 0.909273i $$-0.363362\pi$$
0.416199 + 0.909273i $$0.363362\pi$$
$$578$$ 1.52836e6 0.190286
$$579$$ 0 0
$$580$$ −1.51361e6 −0.186829
$$581$$ −7.74336e6 −0.951676
$$582$$ 0 0
$$583$$ −466053. −0.0567889
$$584$$ −1.10847e7 −1.34491
$$585$$ 0 0
$$586$$ −5.13707e6 −0.617975
$$587$$ −2.20946e6 −0.264661 −0.132331 0.991206i $$-0.542246\pi$$
−0.132331 + 0.991206i $$0.542246\pi$$
$$588$$ 0 0
$$589$$ −1.26279e6 −0.149983
$$590$$ −2.17866e6 −0.257668
$$591$$ 0 0
$$592$$ −2.11770e6 −0.248348
$$593$$ 261524. 0.0305404 0.0152702 0.999883i $$-0.495139\pi$$
0.0152702 + 0.999883i $$0.495139\pi$$
$$594$$ 0 0
$$595$$ 3.82306e6 0.442710
$$596$$ −504998. −0.0582336
$$597$$ 0 0
$$598$$ 387429. 0.0443036
$$599$$ −7.09631e6 −0.808101 −0.404050 0.914737i $$-0.632398\pi$$
−0.404050 + 0.914737i $$0.632398\pi$$
$$600$$ 0 0
$$601$$ 6.04793e6 0.683000 0.341500 0.939882i $$-0.389065\pi$$
0.341500 + 0.939882i $$0.389065\pi$$
$$602$$ 1.00846e7 1.13414
$$603$$ 0 0
$$604$$ 3.04897e6 0.340064
$$605$$ 366025. 0.0406558
$$606$$ 0 0
$$607$$ −1.05203e7 −1.15892 −0.579461 0.815000i $$-0.696737\pi$$
−0.579461 + 0.815000i $$0.696737\pi$$
$$608$$ −5.69835e6 −0.625158
$$609$$ 0 0
$$610$$ −215799. −0.0234815
$$611$$ −2.81741e6 −0.305314
$$612$$ 0 0
$$613$$ −8.55961e6 −0.920032 −0.460016 0.887911i $$-0.652156\pi$$
−0.460016 + 0.887911i $$0.652156\pi$$
$$614$$ −7.40884e6 −0.793103
$$615$$ 0 0
$$616$$ 3.50976e6 0.372671
$$617$$ 6.18055e6 0.653604 0.326802 0.945093i $$-0.394029\pi$$
0.326802 + 0.945093i $$0.394029\pi$$
$$618$$ 0 0
$$619$$ −1.12540e7 −1.18053 −0.590267 0.807208i $$-0.700978\pi$$
−0.590267 + 0.807208i $$0.700978\pi$$
$$620$$ 336561. 0.0351629
$$621$$ 0 0
$$622$$ −1.12096e7 −1.16175
$$623$$ 8.16882e6 0.843217
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ −5.26085e6 −0.536562
$$627$$ 0 0
$$628$$ 2.77470e6 0.280748
$$629$$ −5.33223e6 −0.537381
$$630$$ 0 0
$$631$$ 1.17369e7 1.17349 0.586746 0.809771i $$-0.300409\pi$$
0.586746 + 0.809771i $$0.300409\pi$$
$$632$$ 1.14502e7 1.14030
$$633$$ 0 0
$$634$$ −3.01102e6 −0.297502
$$635$$ 3.02062e6 0.297278
$$636$$ 0 0
$$637$$ 1.20836e6 0.117990
$$638$$ 2.33411e6 0.227023
$$639$$ 0 0
$$640$$ 108756. 0.0104955
$$641$$ −6.80878e6 −0.654522 −0.327261 0.944934i $$-0.606126\pi$$
−0.327261 + 0.944934i $$0.606126\pi$$
$$642$$ 0 0
$$643$$ −1.16456e7 −1.11080 −0.555400 0.831583i $$-0.687435\pi$$
−0.555400 + 0.831583i $$0.687435\pi$$
$$644$$ −769758. −0.0731374
$$645$$ 0 0
$$646$$ 5.61952e6 0.529808
$$647$$ −2.61990e6 −0.246050 −0.123025 0.992404i $$-0.539260\pi$$
−0.123025 + 0.992404i $$0.539260\pi$$
$$648$$ 0 0
$$649$$ −2.45159e6 −0.228473
$$650$$ −629416. −0.0584325
$$651$$ 0 0
$$652$$ 8.63173e6 0.795204
$$653$$ −1.48492e7 −1.36276 −0.681379 0.731931i $$-0.738619\pi$$
−0.681379 + 0.731931i $$0.738619\pi$$
$$654$$ 0 0
$$655$$ −6.66184e6 −0.606724
$$656$$ 925726. 0.0839891
$$657$$ 0 0
$$658$$ −7.67120e6 −0.690715
$$659$$ 2.47470e6 0.221977 0.110989 0.993822i $$-0.464598\pi$$
0.110989 + 0.993822i $$0.464598\pi$$
$$660$$ 0 0
$$661$$ 7.61212e6 0.677645 0.338822 0.940850i $$-0.389971\pi$$
0.338822 + 0.940850i $$0.389971\pi$$
$$662$$ −1.19891e7 −1.06326
$$663$$ 0 0
$$664$$ 1.02243e7 0.899938
$$665$$ −4.69211e6 −0.411447
$$666$$ 0 0
$$667$$ −1.72537e6 −0.150164
$$668$$ 5.62927e6 0.488103
$$669$$ 0 0
$$670$$ −4.40286e6 −0.378921
$$671$$ −242833. −0.0208210
$$672$$ 0 0
$$673$$ 6.62472e6 0.563807 0.281903 0.959443i $$-0.409034\pi$$
0.281903 + 0.959443i $$0.409034\pi$$
$$674$$ 3.42088e6 0.290060
$$675$$ 0 0
$$676$$ 4.27231e6 0.359581
$$677$$ −228067. −0.0191245 −0.00956227 0.999954i $$-0.503044\pi$$
−0.00956227 + 0.999954i $$0.503044\pi$$
$$678$$ 0 0
$$679$$ −1.47302e7 −1.22613
$$680$$ −5.04795e6 −0.418642
$$681$$ 0 0
$$682$$ −519005. −0.0427278
$$683$$ 1.58370e6 0.129903 0.0649517 0.997888i $$-0.479311\pi$$
0.0649517 + 0.997888i $$0.479311\pi$$
$$684$$ 0 0
$$685$$ −429412. −0.0349661
$$686$$ −7.42443e6 −0.602356
$$687$$ 0 0
$$688$$ −6.48202e6 −0.522083
$$689$$ 901818. 0.0723720
$$690$$ 0 0
$$691$$ 2.07411e7 1.65248 0.826241 0.563317i $$-0.190475\pi$$
0.826241 + 0.563317i $$0.190475\pi$$
$$692$$ 8.21665e6 0.652273
$$693$$ 0 0
$$694$$ −1.63674e7 −1.28998
$$695$$ −125800. −0.00987910
$$696$$ 0 0
$$697$$ 2.33091e6 0.181737
$$698$$ 3.43238e6 0.266660
$$699$$ 0 0
$$700$$ 1.25055e6 0.0964618
$$701$$ 1.54594e7 1.18822 0.594109 0.804384i $$-0.297505\pi$$
0.594109 + 0.804384i $$0.297505\pi$$
$$702$$ 0 0
$$703$$ 6.54433e6 0.499433
$$704$$ −3.92862e6 −0.298750
$$705$$ 0 0
$$706$$ 1.07090e7 0.808606
$$707$$ −3.75494e6 −0.282523
$$708$$ 0 0
$$709$$ −4.71237e6 −0.352066 −0.176033 0.984384i $$-0.556327\pi$$
−0.176033 + 0.984384i $$0.556327\pi$$
$$710$$ 1.82487e6 0.135858
$$711$$ 0 0
$$712$$ −1.07861e7 −0.797376
$$713$$ 383646. 0.0282623
$$714$$ 0 0
$$715$$ −708263. −0.0518119
$$716$$ 6.90602e6 0.503437
$$717$$ 0 0
$$718$$ −9.25391e6 −0.669906
$$719$$ −2.29858e6 −0.165820 −0.0829099 0.996557i $$-0.526421\pi$$
−0.0829099 + 0.996557i $$0.526421\pi$$
$$720$$ 0 0
$$721$$ 1.38162e7 0.989805
$$722$$ 3.75324e6 0.267956
$$723$$ 0 0
$$724$$ −760968. −0.0539535
$$725$$ 2.80303e6 0.198054
$$726$$ 0 0
$$727$$ 2.81705e7 1.97678 0.988391 0.151933i $$-0.0485496\pi$$
0.988391 + 0.151933i $$0.0485496\pi$$
$$728$$ −6.79143e6 −0.474934
$$729$$ 0 0
$$730$$ 6.09054e6 0.423008
$$731$$ −1.63213e7 −1.12969
$$732$$ 0 0
$$733$$ −1.32094e7 −0.908078 −0.454039 0.890982i $$-0.650017\pi$$
−0.454039 + 0.890982i $$0.650017\pi$$
$$734$$ −6.34492e6 −0.434696
$$735$$ 0 0
$$736$$ 1.73121e6 0.117802
$$737$$ −4.95441e6 −0.335988
$$738$$ 0 0
$$739$$ −6.63649e6 −0.447021 −0.223510 0.974702i $$-0.571752\pi$$
−0.223510 + 0.974702i $$0.571752\pi$$
$$740$$ −1.74421e6 −0.117090
$$741$$ 0 0
$$742$$ 2.45546e6 0.163728
$$743$$ 7.93013e6 0.526997 0.263499 0.964660i $$-0.415124\pi$$
0.263499 + 0.964660i $$0.415124\pi$$
$$744$$ 0 0
$$745$$ 935197. 0.0617323
$$746$$ 2.02711e7 1.33361
$$747$$ 0 0
$$748$$ −1.68535e6 −0.110137
$$749$$ 3.05072e7 1.98700
$$750$$ 0 0
$$751$$ −1.43446e7 −0.928086 −0.464043 0.885813i $$-0.653602\pi$$
−0.464043 + 0.885813i $$0.653602\pi$$
$$752$$ 4.93076e6 0.317958
$$753$$ 0 0
$$754$$ −4.51654e6 −0.289319
$$755$$ −5.64633e6 −0.360495
$$756$$ 0 0
$$757$$ 8.38496e6 0.531816 0.265908 0.963998i $$-0.414328\pi$$
0.265908 + 0.963998i $$0.414328\pi$$
$$758$$ −1.92650e7 −1.21786
$$759$$ 0 0
$$760$$ 6.19543e6 0.389079
$$761$$ −2.68159e7 −1.67853 −0.839267 0.543719i $$-0.817016\pi$$
−0.839267 + 0.543719i $$0.817016\pi$$
$$762$$ 0 0
$$763$$ 2.31069e7 1.43691
$$764$$ 620812. 0.0384793
$$765$$ 0 0
$$766$$ −249334. −0.0153536
$$767$$ 4.74385e6 0.291167
$$768$$ 0 0
$$769$$ −2.82141e7 −1.72049 −0.860243 0.509884i $$-0.829688\pi$$
−0.860243 + 0.509884i $$0.829688\pi$$
$$770$$ −1.92845e6 −0.117215
$$771$$ 0 0
$$772$$ −4.87192e6 −0.294210
$$773$$ −2.41553e7 −1.45400 −0.727000 0.686637i $$-0.759086\pi$$
−0.727000 + 0.686637i $$0.759086\pi$$
$$774$$ 0 0
$$775$$ −623272. −0.0372755
$$776$$ 1.94497e7 1.15947
$$777$$ 0 0
$$778$$ −1.94983e7 −1.15491
$$779$$ −2.86077e6 −0.168904
$$780$$ 0 0
$$781$$ 2.05347e6 0.120465
$$782$$ −1.70726e6 −0.0998350
$$783$$ 0 0
$$784$$ −2.11475e6 −0.122877
$$785$$ −5.13842e6 −0.297616
$$786$$ 0 0
$$787$$ 1.96829e7 1.13280 0.566400 0.824131i $$-0.308336\pi$$
0.566400 + 0.824131i $$0.308336\pi$$
$$788$$ 1.32152e7 0.758156
$$789$$ 0 0
$$790$$ −6.29134e6 −0.358654
$$791$$ 5.97376e6 0.339474
$$792$$ 0 0
$$793$$ 469884. 0.0265343
$$794$$ −2.67351e6 −0.150498
$$795$$ 0 0
$$796$$ −1.21812e7 −0.681409
$$797$$ −2.41989e6 −0.134943 −0.0674714 0.997721i $$-0.521493\pi$$
−0.0674714 + 0.997721i $$0.521493\pi$$
$$798$$ 0 0
$$799$$ 1.24153e7 0.688004
$$800$$ −2.81252e6 −0.155371
$$801$$ 0 0
$$802$$ 4.10474e6 0.225346
$$803$$ 6.85350e6 0.375080
$$804$$ 0 0
$$805$$ 1.42550e6 0.0775315
$$806$$ 1.00428e6 0.0544525
$$807$$ 0 0
$$808$$ 4.95800e6 0.267164
$$809$$ −2.06594e7 −1.10981 −0.554903 0.831915i $$-0.687245\pi$$
−0.554903 + 0.831915i $$0.687245\pi$$
$$810$$ 0 0
$$811$$ 1.38880e7 0.741461 0.370730 0.928741i $$-0.379107\pi$$
0.370730 + 0.928741i $$0.379107\pi$$
$$812$$ 8.97363e6 0.477615
$$813$$ 0 0
$$814$$ 2.68971e6 0.142280
$$815$$ −1.59850e7 −0.842980
$$816$$ 0 0
$$817$$ 2.00314e7 1.04992
$$818$$ −1.37580e7 −0.718904
$$819$$ 0 0
$$820$$ 762457. 0.0395987
$$821$$ −5.26859e6 −0.272795 −0.136398 0.990654i $$-0.543552\pi$$
−0.136398 + 0.990654i $$0.543552\pi$$
$$822$$ 0 0
$$823$$ 3.61766e7 1.86178 0.930890 0.365299i $$-0.119033\pi$$
0.930890 + 0.365299i $$0.119033\pi$$
$$824$$ −1.82428e7 −0.935994
$$825$$ 0 0
$$826$$ 1.29165e7 0.658710
$$827$$ −3.01917e7 −1.53505 −0.767527 0.641017i $$-0.778513\pi$$
−0.767527 + 0.641017i $$0.778513\pi$$
$$828$$ 0 0
$$829$$ 2.93884e7 1.48522 0.742608 0.669727i $$-0.233589\pi$$
0.742608 + 0.669727i $$0.233589\pi$$
$$830$$ −5.61777e6 −0.283053
$$831$$ 0 0
$$832$$ 7.60193e6 0.380728
$$833$$ −5.32479e6 −0.265883
$$834$$ 0 0
$$835$$ −1.04248e7 −0.517428
$$836$$ 2.06845e6 0.102360
$$837$$ 0 0
$$838$$ −2.83931e7 −1.39670
$$839$$ −1.57845e7 −0.774150 −0.387075 0.922048i $$-0.626515\pi$$
−0.387075 + 0.922048i $$0.626515\pi$$
$$840$$ 0 0
$$841$$ −397286. −0.0193692
$$842$$ 1.81730e7 0.883376
$$843$$ 0 0
$$844$$ −4.29822e6 −0.207698
$$845$$ −7.91183e6 −0.381184
$$846$$ 0 0
$$847$$ −2.17003e6 −0.103934
$$848$$ −1.57828e6 −0.0753691
$$849$$ 0 0
$$850$$ 2.77361e6 0.131674
$$851$$ −1.98822e6 −0.0941112
$$852$$ 0 0
$$853$$ 2.68137e7 1.26178 0.630891 0.775871i $$-0.282689\pi$$
0.630891 + 0.775871i $$0.282689\pi$$
$$854$$ 1.27939e6 0.0600288
$$855$$ 0 0
$$856$$ −4.02815e7 −1.87898
$$857$$ 2.59115e7 1.20515 0.602574 0.798063i $$-0.294142\pi$$
0.602574 + 0.798063i $$0.294142\pi$$
$$858$$ 0 0
$$859$$ 1.20430e7 0.556870 0.278435 0.960455i $$-0.410184\pi$$
0.278435 + 0.960455i $$0.410184\pi$$
$$860$$ −5.33879e6 −0.246148
$$861$$ 0 0
$$862$$ 2.03145e7 0.931191
$$863$$ −2.28033e7 −1.04225 −0.521125 0.853481i $$-0.674487\pi$$
−0.521125 + 0.853481i $$0.674487\pi$$
$$864$$ 0 0
$$865$$ −1.52163e7 −0.691461
$$866$$ 3.79420e6 0.171920
$$867$$ 0 0
$$868$$ −1.99534e6 −0.0898915
$$869$$ −7.07945e6 −0.318017
$$870$$ 0 0
$$871$$ 9.58685e6 0.428184
$$872$$ −3.05102e7 −1.35880
$$873$$ 0 0
$$874$$ 2.09535e6 0.0927849
$$875$$ −2.31587e6 −0.102257
$$876$$ 0 0
$$877$$ −3.10668e7 −1.36395 −0.681974 0.731376i $$-0.738878\pi$$
−0.681974 + 0.731376i $$0.738878\pi$$
$$878$$ 1.32268e7 0.579052
$$879$$ 0 0
$$880$$ 1.23954e6 0.0539575
$$881$$ 3.45208e7 1.49844 0.749222 0.662319i $$-0.230427\pi$$
0.749222 + 0.662319i $$0.230427\pi$$
$$882$$ 0 0
$$883$$ −3.29428e7 −1.42186 −0.710932 0.703260i $$-0.751727\pi$$
−0.710932 + 0.703260i $$0.751727\pi$$
$$884$$ 3.26117e6 0.140360
$$885$$ 0 0
$$886$$ 1.46404e7 0.626571
$$887$$ 2.11009e7 0.900519 0.450260 0.892898i $$-0.351331\pi$$
0.450260 + 0.892898i $$0.351331\pi$$
$$888$$ 0 0
$$889$$ −1.79082e7 −0.759970
$$890$$ 5.92644e6 0.250795
$$891$$ 0 0
$$892$$ −8.92810e6 −0.375705
$$893$$ −1.52375e7 −0.639419
$$894$$ 0 0
$$895$$ −1.27891e7 −0.533684
$$896$$ −644775. −0.0268311
$$897$$ 0 0
$$898$$ −2.60547e6 −0.107819
$$899$$ −4.47245e6 −0.184564
$$900$$ 0 0
$$901$$ −3.97399e6 −0.163085
$$902$$ −1.17577e6 −0.0481179
$$903$$ 0 0
$$904$$ −7.88772e6 −0.321019
$$905$$ 1.40922e6 0.0571951
$$906$$ 0 0
$$907$$ 1.30426e7 0.526438 0.263219 0.964736i $$-0.415216\pi$$
0.263219 + 0.964736i $$0.415216\pi$$
$$908$$ 8.19036e6 0.329677
$$909$$ 0 0
$$910$$ 3.73157e6 0.149379
$$911$$ 4.09652e7 1.63538 0.817690 0.575658i $$-0.195254\pi$$
0.817690 + 0.575658i $$0.195254\pi$$
$$912$$ 0 0
$$913$$ −6.32150e6 −0.250982
$$914$$ 2.34987e7 0.930417
$$915$$ 0 0
$$916$$ 4.75700e6 0.187325
$$917$$ 3.94956e7 1.55105
$$918$$ 0 0
$$919$$ 5.59367e6 0.218478 0.109239 0.994016i $$-0.465159\pi$$
0.109239 + 0.994016i $$0.465159\pi$$
$$920$$ −1.88222e6 −0.0733165
$$921$$ 0 0
$$922$$ −1.22250e7 −0.473609
$$923$$ −3.97349e6 −0.153521
$$924$$ 0 0
$$925$$ 3.23007e6 0.124124
$$926$$ 3.66312e7 1.40386
$$927$$ 0 0
$$928$$ −2.01819e7 −0.769295
$$929$$ 4.15645e7 1.58010 0.790048 0.613045i $$-0.210056\pi$$
0.790048 + 0.613045i $$0.210056\pi$$
$$930$$ 0 0
$$931$$ 6.53520e6 0.247107
$$932$$ 2.01536e7 0.759999
$$933$$ 0 0
$$934$$ 3.01144e7 1.12955
$$935$$ 3.12106e6 0.116755
$$936$$ 0 0
$$937$$ −1.62578e7 −0.604941 −0.302470 0.953159i $$-0.597811\pi$$
−0.302470 + 0.953159i $$0.597811\pi$$
$$938$$ 2.61030e7 0.968685
$$939$$ 0 0
$$940$$ 4.06113e6 0.149909
$$941$$ −1.33538e7 −0.491622 −0.245811 0.969318i $$-0.579054\pi$$
−0.245811 + 0.969318i $$0.579054\pi$$
$$942$$ 0 0
$$943$$ 869126. 0.0318276
$$944$$ −8.30224e6 −0.303225
$$945$$ 0 0
$$946$$ 8.23286e6 0.299105
$$947$$ −1.36642e7 −0.495117 −0.247559 0.968873i $$-0.579628\pi$$
−0.247559 + 0.968873i $$0.579628\pi$$
$$948$$ 0 0
$$949$$ −1.32616e7 −0.478003
$$950$$ −3.40410e6 −0.122375
$$951$$ 0 0
$$952$$ 2.99274e7 1.07023
$$953$$ 2.92858e7 1.04454 0.522270 0.852780i $$-0.325085\pi$$
0.522270 + 0.852780i $$0.325085\pi$$
$$954$$ 0 0
$$955$$ −1.14967e6 −0.0407911
$$956$$ 5.23981e6 0.185426
$$957$$ 0 0
$$958$$ 4.59478e6 0.161752
$$959$$ 2.54582e6 0.0893885
$$960$$ 0 0
$$961$$ −2.76347e7 −0.965263
$$962$$ −5.20462e6 −0.181323
$$963$$ 0 0
$$964$$ 1.01704e6 0.0352487
$$965$$ 9.02223e6 0.311886
$$966$$ 0 0
$$967$$ −4.41255e7 −1.51748 −0.758741 0.651393i $$-0.774185\pi$$
−0.758741 + 0.651393i $$0.774185\pi$$
$$968$$ 2.86529e6 0.0982834
$$969$$ 0 0
$$970$$ −1.06867e7 −0.364683
$$971$$ −2.28334e7 −0.777183 −0.388592 0.921410i $$-0.627038\pi$$
−0.388592 + 0.921410i $$0.627038\pi$$
$$972$$ 0 0
$$973$$ 745819. 0.0252552
$$974$$ −4.31444e6 −0.145723
$$975$$ 0 0
$$976$$ −822347. −0.0276332
$$977$$ 5.35594e7 1.79515 0.897573 0.440866i $$-0.145329\pi$$
0.897573 + 0.440866i $$0.145329\pi$$
$$978$$ 0 0
$$979$$ 6.66884e6 0.222379
$$980$$ −1.74177e6 −0.0579331
$$981$$ 0 0
$$982$$ 3.97309e7 1.31477
$$983$$ 2.25516e7 0.744378 0.372189 0.928157i $$-0.378607\pi$$
0.372189 + 0.928157i $$0.378607\pi$$
$$984$$ 0 0
$$985$$ −2.44730e7 −0.803706
$$986$$ 1.99028e7 0.651961
$$987$$ 0 0
$$988$$ −4.00248e6 −0.130448
$$989$$ −6.08570e6 −0.197843
$$990$$ 0 0
$$991$$ 7.10889e6 0.229942 0.114971 0.993369i $$-0.463323\pi$$
0.114971 + 0.993369i $$0.463323\pi$$
$$992$$ 4.48758e6 0.144788
$$993$$ 0 0
$$994$$ −1.08190e7 −0.347312
$$995$$ 2.25582e7 0.722348
$$996$$ 0 0
$$997$$ 3.59868e7 1.14658 0.573291 0.819352i $$-0.305666\pi$$
0.573291 + 0.819352i $$0.305666\pi$$
$$998$$ −1.85610e7 −0.589896
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 495.6.a.a.1.2 3
3.2 odd 2 165.6.a.e.1.2 3
15.14 odd 2 825.6.a.f.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.e.1.2 3 3.2 odd 2
495.6.a.a.1.2 3 1.1 even 1 trivial
825.6.a.f.1.2 3 15.14 odd 2