# Properties

 Label 45.6.a.b.1.1 Level $45$ Weight $6$ Character 45.1 Self dual yes Analytic conductor $7.217$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 45.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.21727189158$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 45.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -28.0000 q^{4} -25.0000 q^{5} +192.000 q^{7} +120.000 q^{8} +O(q^{10})$$ $$q-2.00000 q^{2} -28.0000 q^{4} -25.0000 q^{5} +192.000 q^{7} +120.000 q^{8} +50.0000 q^{10} +148.000 q^{11} +286.000 q^{13} -384.000 q^{14} +656.000 q^{16} +1678.00 q^{17} +1060.00 q^{19} +700.000 q^{20} -296.000 q^{22} -2976.00 q^{23} +625.000 q^{25} -572.000 q^{26} -5376.00 q^{28} +3410.00 q^{29} -2448.00 q^{31} -5152.00 q^{32} -3356.00 q^{34} -4800.00 q^{35} +182.000 q^{37} -2120.00 q^{38} -3000.00 q^{40} +9398.00 q^{41} -1244.00 q^{43} -4144.00 q^{44} +5952.00 q^{46} +12088.0 q^{47} +20057.0 q^{49} -1250.00 q^{50} -8008.00 q^{52} -23846.0 q^{53} -3700.00 q^{55} +23040.0 q^{56} -6820.00 q^{58} +20020.0 q^{59} +32302.0 q^{61} +4896.00 q^{62} -10688.0 q^{64} -7150.00 q^{65} +60972.0 q^{67} -46984.0 q^{68} +9600.00 q^{70} +32648.0 q^{71} -38774.0 q^{73} -364.000 q^{74} -29680.0 q^{76} +28416.0 q^{77} -33360.0 q^{79} -16400.0 q^{80} -18796.0 q^{82} -16716.0 q^{83} -41950.0 q^{85} +2488.00 q^{86} +17760.0 q^{88} -101370. q^{89} +54912.0 q^{91} +83328.0 q^{92} -24176.0 q^{94} -26500.0 q^{95} -119038. q^{97} -40114.0 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.353553 −0.176777 0.984251i $$-0.556567\pi$$
−0.176777 + 0.984251i $$0.556567\pi$$
$$3$$ 0 0
$$4$$ −28.0000 −0.875000
$$5$$ −25.0000 −0.447214
$$6$$ 0 0
$$7$$ 192.000 1.48100 0.740502 0.672054i $$-0.234588\pi$$
0.740502 + 0.672054i $$0.234588\pi$$
$$8$$ 120.000 0.662913
$$9$$ 0 0
$$10$$ 50.0000 0.158114
$$11$$ 148.000 0.368791 0.184395 0.982852i $$-0.440967\pi$$
0.184395 + 0.982852i $$0.440967\pi$$
$$12$$ 0 0
$$13$$ 286.000 0.469362 0.234681 0.972072i $$-0.424595\pi$$
0.234681 + 0.972072i $$0.424595\pi$$
$$14$$ −384.000 −0.523614
$$15$$ 0 0
$$16$$ 656.000 0.640625
$$17$$ 1678.00 1.40822 0.704109 0.710092i $$-0.251347\pi$$
0.704109 + 0.710092i $$0.251347\pi$$
$$18$$ 0 0
$$19$$ 1060.00 0.673631 0.336815 0.941571i $$-0.390650\pi$$
0.336815 + 0.941571i $$0.390650\pi$$
$$20$$ 700.000 0.391312
$$21$$ 0 0
$$22$$ −296.000 −0.130387
$$23$$ −2976.00 −1.17304 −0.586521 0.809934i $$-0.699503\pi$$
−0.586521 + 0.809934i $$0.699503\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ −572.000 −0.165944
$$27$$ 0 0
$$28$$ −5376.00 −1.29588
$$29$$ 3410.00 0.752938 0.376469 0.926429i $$-0.377138\pi$$
0.376469 + 0.926429i $$0.377138\pi$$
$$30$$ 0 0
$$31$$ −2448.00 −0.457517 −0.228758 0.973483i $$-0.573467\pi$$
−0.228758 + 0.973483i $$0.573467\pi$$
$$32$$ −5152.00 −0.889408
$$33$$ 0 0
$$34$$ −3356.00 −0.497880
$$35$$ −4800.00 −0.662325
$$36$$ 0 0
$$37$$ 182.000 0.0218558 0.0109279 0.999940i $$-0.496521\pi$$
0.0109279 + 0.999940i $$0.496521\pi$$
$$38$$ −2120.00 −0.238164
$$39$$ 0 0
$$40$$ −3000.00 −0.296464
$$41$$ 9398.00 0.873124 0.436562 0.899674i $$-0.356196\pi$$
0.436562 + 0.899674i $$0.356196\pi$$
$$42$$ 0 0
$$43$$ −1244.00 −0.102600 −0.0513002 0.998683i $$-0.516337\pi$$
−0.0513002 + 0.998683i $$0.516337\pi$$
$$44$$ −4144.00 −0.322692
$$45$$ 0 0
$$46$$ 5952.00 0.414733
$$47$$ 12088.0 0.798196 0.399098 0.916908i $$-0.369323\pi$$
0.399098 + 0.916908i $$0.369323\pi$$
$$48$$ 0 0
$$49$$ 20057.0 1.19337
$$50$$ −1250.00 −0.0707107
$$51$$ 0 0
$$52$$ −8008.00 −0.410691
$$53$$ −23846.0 −1.16607 −0.583037 0.812446i $$-0.698136\pi$$
−0.583037 + 0.812446i $$0.698136\pi$$
$$54$$ 0 0
$$55$$ −3700.00 −0.164928
$$56$$ 23040.0 0.981776
$$57$$ 0 0
$$58$$ −6820.00 −0.266204
$$59$$ 20020.0 0.748745 0.374373 0.927278i $$-0.377858\pi$$
0.374373 + 0.927278i $$0.377858\pi$$
$$60$$ 0 0
$$61$$ 32302.0 1.11149 0.555744 0.831353i $$-0.312433\pi$$
0.555744 + 0.831353i $$0.312433\pi$$
$$62$$ 4896.00 0.161757
$$63$$ 0 0
$$64$$ −10688.0 −0.326172
$$65$$ −7150.00 −0.209905
$$66$$ 0 0
$$67$$ 60972.0 1.65937 0.829685 0.558231i $$-0.188520\pi$$
0.829685 + 0.558231i $$0.188520\pi$$
$$68$$ −46984.0 −1.23219
$$69$$ 0 0
$$70$$ 9600.00 0.234167
$$71$$ 32648.0 0.768618 0.384309 0.923204i $$-0.374440\pi$$
0.384309 + 0.923204i $$0.374440\pi$$
$$72$$ 0 0
$$73$$ −38774.0 −0.851596 −0.425798 0.904818i $$-0.640007\pi$$
−0.425798 + 0.904818i $$0.640007\pi$$
$$74$$ −364.000 −0.00772720
$$75$$ 0 0
$$76$$ −29680.0 −0.589427
$$77$$ 28416.0 0.546180
$$78$$ 0 0
$$79$$ −33360.0 −0.601393 −0.300696 0.953720i $$-0.597219\pi$$
−0.300696 + 0.953720i $$0.597219\pi$$
$$80$$ −16400.0 −0.286496
$$81$$ 0 0
$$82$$ −18796.0 −0.308696
$$83$$ −16716.0 −0.266340 −0.133170 0.991093i $$-0.542516\pi$$
−0.133170 + 0.991093i $$0.542516\pi$$
$$84$$ 0 0
$$85$$ −41950.0 −0.629774
$$86$$ 2488.00 0.0362747
$$87$$ 0 0
$$88$$ 17760.0 0.244476
$$89$$ −101370. −1.35655 −0.678273 0.734810i $$-0.737271\pi$$
−0.678273 + 0.734810i $$0.737271\pi$$
$$90$$ 0 0
$$91$$ 54912.0 0.695126
$$92$$ 83328.0 1.02641
$$93$$ 0 0
$$94$$ −24176.0 −0.282205
$$95$$ −26500.0 −0.301257
$$96$$ 0 0
$$97$$ −119038. −1.28457 −0.642283 0.766468i $$-0.722013\pi$$
−0.642283 + 0.766468i $$0.722013\pi$$
$$98$$ −40114.0 −0.421921
$$99$$ 0 0
$$100$$ −17500.0 −0.175000
$$101$$ 89898.0 0.876893 0.438446 0.898757i $$-0.355529\pi$$
0.438446 + 0.898757i $$0.355529\pi$$
$$102$$ 0 0
$$103$$ −19504.0 −0.181147 −0.0905734 0.995890i $$-0.528870\pi$$
−0.0905734 + 0.995890i $$0.528870\pi$$
$$104$$ 34320.0 0.311146
$$105$$ 0 0
$$106$$ 47692.0 0.412269
$$107$$ −158292. −1.33659 −0.668297 0.743895i $$-0.732976\pi$$
−0.668297 + 0.743895i $$0.732976\pi$$
$$108$$ 0 0
$$109$$ 36830.0 0.296917 0.148459 0.988919i $$-0.452569\pi$$
0.148459 + 0.988919i $$0.452569\pi$$
$$110$$ 7400.00 0.0583109
$$111$$ 0 0
$$112$$ 125952. 0.948768
$$113$$ −11186.0 −0.0824098 −0.0412049 0.999151i $$-0.513120\pi$$
−0.0412049 + 0.999151i $$0.513120\pi$$
$$114$$ 0 0
$$115$$ 74400.0 0.524600
$$116$$ −95480.0 −0.658821
$$117$$ 0 0
$$118$$ −40040.0 −0.264721
$$119$$ 322176. 2.08557
$$120$$ 0 0
$$121$$ −139147. −0.863993
$$122$$ −64604.0 −0.392970
$$123$$ 0 0
$$124$$ 68544.0 0.400327
$$125$$ −15625.0 −0.0894427
$$126$$ 0 0
$$127$$ 70552.0 0.388150 0.194075 0.980987i $$-0.437829\pi$$
0.194075 + 0.980987i $$0.437829\pi$$
$$128$$ 186240. 1.00473
$$129$$ 0 0
$$130$$ 14300.0 0.0742126
$$131$$ −76452.0 −0.389234 −0.194617 0.980879i $$-0.562346\pi$$
−0.194617 + 0.980879i $$0.562346\pi$$
$$132$$ 0 0
$$133$$ 203520. 0.997650
$$134$$ −121944. −0.586676
$$135$$ 0 0
$$136$$ 201360. 0.933525
$$137$$ 144918. 0.659661 0.329831 0.944040i $$-0.393008\pi$$
0.329831 + 0.944040i $$0.393008\pi$$
$$138$$ 0 0
$$139$$ 112220. 0.492644 0.246322 0.969188i $$-0.420778\pi$$
0.246322 + 0.969188i $$0.420778\pi$$
$$140$$ 134400. 0.579534
$$141$$ 0 0
$$142$$ −65296.0 −0.271748
$$143$$ 42328.0 0.173096
$$144$$ 0 0
$$145$$ −85250.0 −0.336724
$$146$$ 77548.0 0.301085
$$147$$ 0 0
$$148$$ −5096.00 −0.0191238
$$149$$ −403750. −1.48986 −0.744932 0.667140i $$-0.767518\pi$$
−0.744932 + 0.667140i $$0.767518\pi$$
$$150$$ 0 0
$$151$$ −446648. −1.59413 −0.797064 0.603895i $$-0.793615\pi$$
−0.797064 + 0.603895i $$0.793615\pi$$
$$152$$ 127200. 0.446558
$$153$$ 0 0
$$154$$ −56832.0 −0.193104
$$155$$ 61200.0 0.204608
$$156$$ 0 0
$$157$$ −262258. −0.849141 −0.424570 0.905395i $$-0.639575\pi$$
−0.424570 + 0.905395i $$0.639575\pi$$
$$158$$ 66720.0 0.212625
$$159$$ 0 0
$$160$$ 128800. 0.397755
$$161$$ −571392. −1.73728
$$162$$ 0 0
$$163$$ −154564. −0.455658 −0.227829 0.973701i $$-0.573163\pi$$
−0.227829 + 0.973701i $$0.573163\pi$$
$$164$$ −263144. −0.763983
$$165$$ 0 0
$$166$$ 33432.0 0.0941656
$$167$$ −396672. −1.10063 −0.550314 0.834958i $$-0.685492\pi$$
−0.550314 + 0.834958i $$0.685492\pi$$
$$168$$ 0 0
$$169$$ −289497. −0.779700
$$170$$ 83900.0 0.222659
$$171$$ 0 0
$$172$$ 34832.0 0.0897754
$$173$$ 573474. 1.45680 0.728398 0.685155i $$-0.240265\pi$$
0.728398 + 0.685155i $$0.240265\pi$$
$$174$$ 0 0
$$175$$ 120000. 0.296201
$$176$$ 97088.0 0.236257
$$177$$ 0 0
$$178$$ 202740. 0.479611
$$179$$ 594460. 1.38672 0.693362 0.720589i $$-0.256129\pi$$
0.693362 + 0.720589i $$0.256129\pi$$
$$180$$ 0 0
$$181$$ −107098. −0.242988 −0.121494 0.992592i $$-0.538769\pi$$
−0.121494 + 0.992592i $$0.538769\pi$$
$$182$$ −109824. −0.245764
$$183$$ 0 0
$$184$$ −357120. −0.777624
$$185$$ −4550.00 −0.00977422
$$186$$ 0 0
$$187$$ 248344. 0.519337
$$188$$ −338464. −0.698422
$$189$$ 0 0
$$190$$ 53000.0 0.106510
$$191$$ −469552. −0.931323 −0.465661 0.884963i $$-0.654184\pi$$
−0.465661 + 0.884963i $$0.654184\pi$$
$$192$$ 0 0
$$193$$ 52706.0 0.101851 0.0509257 0.998702i $$-0.483783\pi$$
0.0509257 + 0.998702i $$0.483783\pi$$
$$194$$ 238076. 0.454163
$$195$$ 0 0
$$196$$ −561596. −1.04420
$$197$$ −455862. −0.836889 −0.418444 0.908242i $$-0.637425\pi$$
−0.418444 + 0.908242i $$0.637425\pi$$
$$198$$ 0 0
$$199$$ 865000. 1.54840 0.774200 0.632940i $$-0.218152\pi$$
0.774200 + 0.632940i $$0.218152\pi$$
$$200$$ 75000.0 0.132583
$$201$$ 0 0
$$202$$ −179796. −0.310028
$$203$$ 654720. 1.11510
$$204$$ 0 0
$$205$$ −234950. −0.390473
$$206$$ 39008.0 0.0640451
$$207$$ 0 0
$$208$$ 187616. 0.300685
$$209$$ 156880. 0.248429
$$210$$ 0 0
$$211$$ 1.10565e6 1.70967 0.854835 0.518900i $$-0.173658\pi$$
0.854835 + 0.518900i $$0.173658\pi$$
$$212$$ 667688. 1.02031
$$213$$ 0 0
$$214$$ 316584. 0.472557
$$215$$ 31100.0 0.0458843
$$216$$ 0 0
$$217$$ −470016. −0.677584
$$218$$ −73660.0 −0.104976
$$219$$ 0 0
$$220$$ 103600. 0.144312
$$221$$ 479908. 0.660963
$$222$$ 0 0
$$223$$ 1.12158e6 1.51031 0.755156 0.655545i $$-0.227561\pi$$
0.755156 + 0.655545i $$0.227561\pi$$
$$224$$ −989184. −1.31722
$$225$$ 0 0
$$226$$ 22372.0 0.0291363
$$227$$ 23348.0 0.0300736 0.0150368 0.999887i $$-0.495213\pi$$
0.0150368 + 0.999887i $$0.495213\pi$$
$$228$$ 0 0
$$229$$ −596010. −0.751043 −0.375522 0.926814i $$-0.622536\pi$$
−0.375522 + 0.926814i $$0.622536\pi$$
$$230$$ −148800. −0.185474
$$231$$ 0 0
$$232$$ 409200. 0.499132
$$233$$ 485334. 0.585667 0.292834 0.956163i $$-0.405402\pi$$
0.292834 + 0.956163i $$0.405402\pi$$
$$234$$ 0 0
$$235$$ −302200. −0.356964
$$236$$ −560560. −0.655152
$$237$$ 0 0
$$238$$ −644352. −0.737362
$$239$$ 48880.0 0.0553524 0.0276762 0.999617i $$-0.491189\pi$$
0.0276762 + 0.999617i $$0.491189\pi$$
$$240$$ 0 0
$$241$$ −110798. −0.122882 −0.0614411 0.998111i $$-0.519570\pi$$
−0.0614411 + 0.998111i $$0.519570\pi$$
$$242$$ 278294. 0.305468
$$243$$ 0 0
$$244$$ −904456. −0.972552
$$245$$ −501425. −0.533692
$$246$$ 0 0
$$247$$ 303160. 0.316176
$$248$$ −293760. −0.303294
$$249$$ 0 0
$$250$$ 31250.0 0.0316228
$$251$$ 1.64375e6 1.64684 0.823419 0.567434i $$-0.192064\pi$$
0.823419 + 0.567434i $$0.192064\pi$$
$$252$$ 0 0
$$253$$ −440448. −0.432607
$$254$$ −141104. −0.137232
$$255$$ 0 0
$$256$$ −30464.0 −0.0290527
$$257$$ −1.30624e6 −1.23365 −0.616823 0.787102i $$-0.711581\pi$$
−0.616823 + 0.787102i $$0.711581\pi$$
$$258$$ 0 0
$$259$$ 34944.0 0.0323685
$$260$$ 200200. 0.183667
$$261$$ 0 0
$$262$$ 152904. 0.137615
$$263$$ −2.12834e6 −1.89736 −0.948682 0.316231i $$-0.897583\pi$$
−0.948682 + 0.316231i $$0.897583\pi$$
$$264$$ 0 0
$$265$$ 596150. 0.521484
$$266$$ −407040. −0.352722
$$267$$ 0 0
$$268$$ −1.70722e6 −1.45195
$$269$$ 1.44109e6 1.21426 0.607128 0.794604i $$-0.292321\pi$$
0.607128 + 0.794604i $$0.292321\pi$$
$$270$$ 0 0
$$271$$ −93248.0 −0.0771288 −0.0385644 0.999256i $$-0.512278\pi$$
−0.0385644 + 0.999256i $$0.512278\pi$$
$$272$$ 1.10077e6 0.902139
$$273$$ 0 0
$$274$$ −289836. −0.233225
$$275$$ 92500.0 0.0737581
$$276$$ 0 0
$$277$$ −110298. −0.0863711 −0.0431855 0.999067i $$-0.513751\pi$$
−0.0431855 + 0.999067i $$0.513751\pi$$
$$278$$ −224440. −0.174176
$$279$$ 0 0
$$280$$ −576000. −0.439064
$$281$$ 192198. 0.145205 0.0726027 0.997361i $$-0.476869\pi$$
0.0726027 + 0.997361i $$0.476869\pi$$
$$282$$ 0 0
$$283$$ −331884. −0.246332 −0.123166 0.992386i $$-0.539305\pi$$
−0.123166 + 0.992386i $$0.539305\pi$$
$$284$$ −914144. −0.672541
$$285$$ 0 0
$$286$$ −84656.0 −0.0611988
$$287$$ 1.80442e6 1.29310
$$288$$ 0 0
$$289$$ 1.39583e6 0.983076
$$290$$ 170500. 0.119050
$$291$$ 0 0
$$292$$ 1.08567e6 0.745146
$$293$$ −2.19481e6 −1.49358 −0.746788 0.665063i $$-0.768405\pi$$
−0.746788 + 0.665063i $$0.768405\pi$$
$$294$$ 0 0
$$295$$ −500500. −0.334849
$$296$$ 21840.0 0.0144885
$$297$$ 0 0
$$298$$ 807500. 0.526747
$$299$$ −851136. −0.550581
$$300$$ 0 0
$$301$$ −238848. −0.151952
$$302$$ 893296. 0.563609
$$303$$ 0 0
$$304$$ 695360. 0.431545
$$305$$ −807550. −0.497073
$$306$$ 0 0
$$307$$ −2.37751e6 −1.43971 −0.719857 0.694123i $$-0.755793\pi$$
−0.719857 + 0.694123i $$0.755793\pi$$
$$308$$ −795648. −0.477908
$$309$$ 0 0
$$310$$ −122400. −0.0723398
$$311$$ 2.37305e6 1.39125 0.695626 0.718405i $$-0.255127\pi$$
0.695626 + 0.718405i $$0.255127\pi$$
$$312$$ 0 0
$$313$$ −1.42941e6 −0.824702 −0.412351 0.911025i $$-0.635292\pi$$
−0.412351 + 0.911025i $$0.635292\pi$$
$$314$$ 524516. 0.300217
$$315$$ 0 0
$$316$$ 934080. 0.526219
$$317$$ −2.12462e6 −1.18750 −0.593750 0.804650i $$-0.702353\pi$$
−0.593750 + 0.804650i $$0.702353\pi$$
$$318$$ 0 0
$$319$$ 504680. 0.277677
$$320$$ 267200. 0.145868
$$321$$ 0 0
$$322$$ 1.14278e6 0.614221
$$323$$ 1.77868e6 0.948618
$$324$$ 0 0
$$325$$ 178750. 0.0938723
$$326$$ 309128. 0.161100
$$327$$ 0 0
$$328$$ 1.12776e6 0.578805
$$329$$ 2.32090e6 1.18213
$$330$$ 0 0
$$331$$ 3.09985e6 1.55515 0.777573 0.628793i $$-0.216451\pi$$
0.777573 + 0.628793i $$0.216451\pi$$
$$332$$ 468048. 0.233048
$$333$$ 0 0
$$334$$ 793344. 0.389131
$$335$$ −1.52430e6 −0.742093
$$336$$ 0 0
$$337$$ 2.40008e6 1.15120 0.575601 0.817731i $$-0.304768\pi$$
0.575601 + 0.817731i $$0.304768\pi$$
$$338$$ 578994. 0.275665
$$339$$ 0 0
$$340$$ 1.17460e6 0.551052
$$341$$ −362304. −0.168728
$$342$$ 0 0
$$343$$ 624000. 0.286384
$$344$$ −149280. −0.0680151
$$345$$ 0 0
$$346$$ −1.14695e6 −0.515055
$$347$$ −1.77741e6 −0.792436 −0.396218 0.918156i $$-0.629678\pi$$
−0.396218 + 0.918156i $$0.629678\pi$$
$$348$$ 0 0
$$349$$ −2.14805e6 −0.944019 −0.472010 0.881593i $$-0.656471\pi$$
−0.472010 + 0.881593i $$0.656471\pi$$
$$350$$ −240000. −0.104723
$$351$$ 0 0
$$352$$ −762496. −0.328005
$$353$$ 661854. 0.282700 0.141350 0.989960i $$-0.454856\pi$$
0.141350 + 0.989960i $$0.454856\pi$$
$$354$$ 0 0
$$355$$ −816200. −0.343737
$$356$$ 2.83836e6 1.18698
$$357$$ 0 0
$$358$$ −1.18892e6 −0.490281
$$359$$ 259320. 0.106194 0.0530970 0.998589i $$-0.483091\pi$$
0.0530970 + 0.998589i $$0.483091\pi$$
$$360$$ 0 0
$$361$$ −1.35250e6 −0.546222
$$362$$ 214196. 0.0859093
$$363$$ 0 0
$$364$$ −1.53754e6 −0.608236
$$365$$ 969350. 0.380845
$$366$$ 0 0
$$367$$ −1.49993e6 −0.581307 −0.290653 0.956828i $$-0.593873\pi$$
−0.290653 + 0.956828i $$0.593873\pi$$
$$368$$ −1.95226e6 −0.751480
$$369$$ 0 0
$$370$$ 9100.00 0.00345571
$$371$$ −4.57843e6 −1.72696
$$372$$ 0 0
$$373$$ −2.23807e6 −0.832918 −0.416459 0.909154i $$-0.636729\pi$$
−0.416459 + 0.909154i $$0.636729\pi$$
$$374$$ −496688. −0.183614
$$375$$ 0 0
$$376$$ 1.45056e6 0.529135
$$377$$ 975260. 0.353400
$$378$$ 0 0
$$379$$ 3.15934e6 1.12979 0.564896 0.825162i $$-0.308916\pi$$
0.564896 + 0.825162i $$0.308916\pi$$
$$380$$ 742000. 0.263600
$$381$$ 0 0
$$382$$ 939104. 0.329272
$$383$$ −342216. −0.119207 −0.0596037 0.998222i $$-0.518984\pi$$
−0.0596037 + 0.998222i $$0.518984\pi$$
$$384$$ 0 0
$$385$$ −710400. −0.244259
$$386$$ −105412. −0.0360099
$$387$$ 0 0
$$388$$ 3.33306e6 1.12399
$$389$$ −88470.0 −0.0296430 −0.0148215 0.999890i $$-0.504718\pi$$
−0.0148215 + 0.999890i $$0.504718\pi$$
$$390$$ 0 0
$$391$$ −4.99373e6 −1.65190
$$392$$ 2.40684e6 0.791101
$$393$$ 0 0
$$394$$ 911724. 0.295885
$$395$$ 834000. 0.268951
$$396$$ 0 0
$$397$$ −5.45674e6 −1.73763 −0.868814 0.495138i $$-0.835117\pi$$
−0.868814 + 0.495138i $$0.835117\pi$$
$$398$$ −1.73000e6 −0.547442
$$399$$ 0 0
$$400$$ 410000. 0.128125
$$401$$ −4.04680e6 −1.25676 −0.628378 0.777908i $$-0.716281\pi$$
−0.628378 + 0.777908i $$0.716281\pi$$
$$402$$ 0 0
$$403$$ −700128. −0.214741
$$404$$ −2.51714e6 −0.767281
$$405$$ 0 0
$$406$$ −1.30944e6 −0.394249
$$407$$ 26936.0 0.00806022
$$408$$ 0 0
$$409$$ −2.71207e6 −0.801664 −0.400832 0.916151i $$-0.631279\pi$$
−0.400832 + 0.916151i $$0.631279\pi$$
$$410$$ 469900. 0.138053
$$411$$ 0 0
$$412$$ 546112. 0.158503
$$413$$ 3.84384e6 1.10889
$$414$$ 0 0
$$415$$ 417900. 0.119111
$$416$$ −1.47347e6 −0.417454
$$417$$ 0 0
$$418$$ −313760. −0.0878328
$$419$$ −3.71746e6 −1.03445 −0.517227 0.855848i $$-0.673036\pi$$
−0.517227 + 0.855848i $$0.673036\pi$$
$$420$$ 0 0
$$421$$ 3.55250e6 0.976853 0.488426 0.872605i $$-0.337571\pi$$
0.488426 + 0.872605i $$0.337571\pi$$
$$422$$ −2.21130e6 −0.604460
$$423$$ 0 0
$$424$$ −2.86152e6 −0.773005
$$425$$ 1.04875e6 0.281643
$$426$$ 0 0
$$427$$ 6.20198e6 1.64612
$$428$$ 4.43218e6 1.16952
$$429$$ 0 0
$$430$$ −62200.0 −0.0162226
$$431$$ 4.06205e6 1.05330 0.526650 0.850082i $$-0.323448\pi$$
0.526650 + 0.850082i $$0.323448\pi$$
$$432$$ 0 0
$$433$$ 7.26287e6 1.86161 0.930804 0.365518i $$-0.119108\pi$$
0.930804 + 0.365518i $$0.119108\pi$$
$$434$$ 940032. 0.239562
$$435$$ 0 0
$$436$$ −1.03124e6 −0.259803
$$437$$ −3.15456e6 −0.790197
$$438$$ 0 0
$$439$$ −5.41028e6 −1.33986 −0.669928 0.742426i $$-0.733675\pi$$
−0.669928 + 0.742426i $$0.733675\pi$$
$$440$$ −444000. −0.109333
$$441$$ 0 0
$$442$$ −959816. −0.233686
$$443$$ 6.51524e6 1.57733 0.788663 0.614826i $$-0.210774\pi$$
0.788663 + 0.614826i $$0.210774\pi$$
$$444$$ 0 0
$$445$$ 2.53425e6 0.606666
$$446$$ −2.24315e6 −0.533976
$$447$$ 0 0
$$448$$ −2.05210e6 −0.483062
$$449$$ 509950. 0.119375 0.0596873 0.998217i $$-0.480990\pi$$
0.0596873 + 0.998217i $$0.480990\pi$$
$$450$$ 0 0
$$451$$ 1.39090e6 0.322000
$$452$$ 313208. 0.0721085
$$453$$ 0 0
$$454$$ −46696.0 −0.0106326
$$455$$ −1.37280e6 −0.310870
$$456$$ 0 0
$$457$$ 1.22084e6 0.273444 0.136722 0.990609i $$-0.456343\pi$$
0.136722 + 0.990609i $$0.456343\pi$$
$$458$$ 1.19202e6 0.265534
$$459$$ 0 0
$$460$$ −2.08320e6 −0.459025
$$461$$ 4.07210e6 0.892413 0.446207 0.894930i $$-0.352775\pi$$
0.446207 + 0.894930i $$0.352775\pi$$
$$462$$ 0 0
$$463$$ 2.02294e6 0.438561 0.219280 0.975662i $$-0.429629\pi$$
0.219280 + 0.975662i $$0.429629\pi$$
$$464$$ 2.23696e6 0.482351
$$465$$ 0 0
$$466$$ −970668. −0.207065
$$467$$ −3.25097e6 −0.689797 −0.344898 0.938640i $$-0.612087\pi$$
−0.344898 + 0.938640i $$0.612087\pi$$
$$468$$ 0 0
$$469$$ 1.17066e7 2.45753
$$470$$ 604400. 0.126206
$$471$$ 0 0
$$472$$ 2.40240e6 0.496353
$$473$$ −184112. −0.0378381
$$474$$ 0 0
$$475$$ 662500. 0.134726
$$476$$ −9.02093e6 −1.82488
$$477$$ 0 0
$$478$$ −97760.0 −0.0195700
$$479$$ 3.27936e6 0.653056 0.326528 0.945188i $$-0.394121\pi$$
0.326528 + 0.945188i $$0.394121\pi$$
$$480$$ 0 0
$$481$$ 52052.0 0.0102583
$$482$$ 221596. 0.0434455
$$483$$ 0 0
$$484$$ 3.89612e6 0.755994
$$485$$ 2.97595e6 0.574475
$$486$$ 0 0
$$487$$ −8.53197e6 −1.63015 −0.815074 0.579357i $$-0.803304\pi$$
−0.815074 + 0.579357i $$0.803304\pi$$
$$488$$ 3.87624e6 0.736819
$$489$$ 0 0
$$490$$ 1.00285e6 0.188689
$$491$$ −1.51265e6 −0.283162 −0.141581 0.989927i $$-0.545219\pi$$
−0.141581 + 0.989927i $$0.545219\pi$$
$$492$$ 0 0
$$493$$ 5.72198e6 1.06030
$$494$$ −606320. −0.111785
$$495$$ 0 0
$$496$$ −1.60589e6 −0.293097
$$497$$ 6.26842e6 1.13833
$$498$$ 0 0
$$499$$ −6.49190e6 −1.16713 −0.583567 0.812065i $$-0.698343\pi$$
−0.583567 + 0.812065i $$0.698343\pi$$
$$500$$ 437500. 0.0782624
$$501$$ 0 0
$$502$$ −3.28750e6 −0.582245
$$503$$ −8.61770e6 −1.51870 −0.759349 0.650684i $$-0.774482\pi$$
−0.759349 + 0.650684i $$0.774482\pi$$
$$504$$ 0 0
$$505$$ −2.24745e6 −0.392158
$$506$$ 880896. 0.152950
$$507$$ 0 0
$$508$$ −1.97546e6 −0.339632
$$509$$ −2.67323e6 −0.457343 −0.228671 0.973504i $$-0.573438\pi$$
−0.228671 + 0.973504i $$0.573438\pi$$
$$510$$ 0 0
$$511$$ −7.44461e6 −1.26122
$$512$$ −5.89875e6 −0.994455
$$513$$ 0 0
$$514$$ 2.61248e6 0.436160
$$515$$ 487600. 0.0810113
$$516$$ 0 0
$$517$$ 1.78902e6 0.294367
$$518$$ −69888.0 −0.0114440
$$519$$ 0 0
$$520$$ −858000. −0.139149
$$521$$ −6.18500e6 −0.998264 −0.499132 0.866526i $$-0.666348\pi$$
−0.499132 + 0.866526i $$0.666348\pi$$
$$522$$ 0 0
$$523$$ −6.89452e6 −1.10217 −0.551087 0.834448i $$-0.685787\pi$$
−0.551087 + 0.834448i $$0.685787\pi$$
$$524$$ 2.14066e6 0.340580
$$525$$ 0 0
$$526$$ 4.25667e6 0.670820
$$527$$ −4.10774e6 −0.644283
$$528$$ 0 0
$$529$$ 2.42023e6 0.376026
$$530$$ −1.19230e6 −0.184372
$$531$$ 0 0
$$532$$ −5.69856e6 −0.872943
$$533$$ 2.68783e6 0.409811
$$534$$ 0 0
$$535$$ 3.95730e6 0.597743
$$536$$ 7.31664e6 1.10002
$$537$$ 0 0
$$538$$ −2.88218e6 −0.429304
$$539$$ 2.96844e6 0.440104
$$540$$ 0 0
$$541$$ 155502. 0.0228425 0.0114212 0.999935i $$-0.496364\pi$$
0.0114212 + 0.999935i $$0.496364\pi$$
$$542$$ 186496. 0.0272691
$$543$$ 0 0
$$544$$ −8.64506e6 −1.25248
$$545$$ −920750. −0.132785
$$546$$ 0 0
$$547$$ 1.26544e7 1.80831 0.904157 0.427201i $$-0.140500\pi$$
0.904157 + 0.427201i $$0.140500\pi$$
$$548$$ −4.05770e6 −0.577204
$$549$$ 0 0
$$550$$ −185000. −0.0260774
$$551$$ 3.61460e6 0.507202
$$552$$ 0 0
$$553$$ −6.40512e6 −0.890665
$$554$$ 220596. 0.0305368
$$555$$ 0 0
$$556$$ −3.14216e6 −0.431064
$$557$$ 7.07786e6 0.966638 0.483319 0.875444i $$-0.339431\pi$$
0.483319 + 0.875444i $$0.339431\pi$$
$$558$$ 0 0
$$559$$ −355784. −0.0481567
$$560$$ −3.14880e6 −0.424302
$$561$$ 0 0
$$562$$ −384396. −0.0513379
$$563$$ −846636. −0.112571 −0.0562854 0.998415i $$-0.517926\pi$$
−0.0562854 + 0.998415i $$0.517926\pi$$
$$564$$ 0 0
$$565$$ 279650. 0.0368548
$$566$$ 663768. 0.0870914
$$567$$ 0 0
$$568$$ 3.91776e6 0.509527
$$569$$ −4.96041e6 −0.642299 −0.321149 0.947029i $$-0.604069\pi$$
−0.321149 + 0.947029i $$0.604069\pi$$
$$570$$ 0 0
$$571$$ 8.96505e6 1.15070 0.575351 0.817907i $$-0.304866\pi$$
0.575351 + 0.817907i $$0.304866\pi$$
$$572$$ −1.18518e6 −0.151459
$$573$$ 0 0
$$574$$ −3.60883e6 −0.457180
$$575$$ −1.86000e6 −0.234608
$$576$$ 0 0
$$577$$ −2.86080e6 −0.357724 −0.178862 0.983874i $$-0.557242\pi$$
−0.178862 + 0.983874i $$0.557242\pi$$
$$578$$ −2.79165e6 −0.347570
$$579$$ 0 0
$$580$$ 2.38700e6 0.294634
$$581$$ −3.20947e6 −0.394451
$$582$$ 0 0
$$583$$ −3.52921e6 −0.430037
$$584$$ −4.65288e6 −0.564534
$$585$$ 0 0
$$586$$ 4.38961e6 0.528059
$$587$$ 6.74027e6 0.807387 0.403694 0.914894i $$-0.367726\pi$$
0.403694 + 0.914894i $$0.367726\pi$$
$$588$$ 0 0
$$589$$ −2.59488e6 −0.308197
$$590$$ 1.00100e6 0.118387
$$591$$ 0 0
$$592$$ 119392. 0.0140014
$$593$$ 1.78609e6 0.208578 0.104289 0.994547i $$-0.466743\pi$$
0.104289 + 0.994547i $$0.466743\pi$$
$$594$$ 0 0
$$595$$ −8.05440e6 −0.932697
$$596$$ 1.13050e7 1.30363
$$597$$ 0 0
$$598$$ 1.70227e6 0.194660
$$599$$ −4.94620e6 −0.563254 −0.281627 0.959524i $$-0.590874\pi$$
−0.281627 + 0.959524i $$0.590874\pi$$
$$600$$ 0 0
$$601$$ −4.58100e6 −0.517337 −0.258669 0.965966i $$-0.583284\pi$$
−0.258669 + 0.965966i $$0.583284\pi$$
$$602$$ 477696. 0.0537230
$$603$$ 0 0
$$604$$ 1.25061e7 1.39486
$$605$$ 3.47868e6 0.386390
$$606$$ 0 0
$$607$$ 7.07999e6 0.779940 0.389970 0.920828i $$-0.372485\pi$$
0.389970 + 0.920828i $$0.372485\pi$$
$$608$$ −5.46112e6 −0.599132
$$609$$ 0 0
$$610$$ 1.61510e6 0.175742
$$611$$ 3.45717e6 0.374643
$$612$$ 0 0
$$613$$ 5.09609e6 0.547754 0.273877 0.961765i $$-0.411694\pi$$
0.273877 + 0.961765i $$0.411694\pi$$
$$614$$ 4.75502e6 0.509016
$$615$$ 0 0
$$616$$ 3.40992e6 0.362070
$$617$$ 1.30003e7 1.37480 0.687400 0.726279i $$-0.258752\pi$$
0.687400 + 0.726279i $$0.258752\pi$$
$$618$$ 0 0
$$619$$ 4.84406e6 0.508139 0.254070 0.967186i $$-0.418231\pi$$
0.254070 + 0.967186i $$0.418231\pi$$
$$620$$ −1.71360e6 −0.179032
$$621$$ 0 0
$$622$$ −4.74610e6 −0.491882
$$623$$ −1.94630e7 −2.00905
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ 2.85883e6 0.291576
$$627$$ 0 0
$$628$$ 7.34322e6 0.742998
$$629$$ 305396. 0.0307777
$$630$$ 0 0
$$631$$ 6.22775e6 0.622670 0.311335 0.950300i $$-0.399224\pi$$
0.311335 + 0.950300i $$0.399224\pi$$
$$632$$ −4.00320e6 −0.398671
$$633$$ 0 0
$$634$$ 4.24924e6 0.419845
$$635$$ −1.76380e6 −0.173586
$$636$$ 0 0
$$637$$ 5.73630e6 0.560123
$$638$$ −1.00936e6 −0.0981735
$$639$$ 0 0
$$640$$ −4.65600e6 −0.449328
$$641$$ −1.53280e6 −0.147347 −0.0736734 0.997282i $$-0.523472\pi$$
−0.0736734 + 0.997282i $$0.523472\pi$$
$$642$$ 0 0
$$643$$ −1.74382e7 −1.66332 −0.831659 0.555287i $$-0.812609\pi$$
−0.831659 + 0.555287i $$0.812609\pi$$
$$644$$ 1.59990e7 1.52012
$$645$$ 0 0
$$646$$ −3.55736e6 −0.335387
$$647$$ 4.25469e6 0.399583 0.199792 0.979838i $$-0.435974\pi$$
0.199792 + 0.979838i $$0.435974\pi$$
$$648$$ 0 0
$$649$$ 2.96296e6 0.276130
$$650$$ −357500. −0.0331889
$$651$$ 0 0
$$652$$ 4.32779e6 0.398701
$$653$$ −3.01085e6 −0.276316 −0.138158 0.990410i $$-0.544118\pi$$
−0.138158 + 0.990410i $$0.544118\pi$$
$$654$$ 0 0
$$655$$ 1.91130e6 0.174071
$$656$$ 6.16509e6 0.559345
$$657$$ 0 0
$$658$$ −4.64179e6 −0.417947
$$659$$ 8.11462e6 0.727871 0.363936 0.931424i $$-0.381433\pi$$
0.363936 + 0.931424i $$0.381433\pi$$
$$660$$ 0 0
$$661$$ 2.47370e6 0.220213 0.110107 0.993920i $$-0.464881\pi$$
0.110107 + 0.993920i $$0.464881\pi$$
$$662$$ −6.19970e6 −0.549827
$$663$$ 0 0
$$664$$ −2.00592e6 −0.176560
$$665$$ −5.08800e6 −0.446162
$$666$$ 0 0
$$667$$ −1.01482e7 −0.883228
$$668$$ 1.11068e7 0.963049
$$669$$ 0 0
$$670$$ 3.04860e6 0.262370
$$671$$ 4.78070e6 0.409907
$$672$$ 0 0
$$673$$ 5.77063e6 0.491117 0.245559 0.969382i $$-0.421029\pi$$
0.245559 + 0.969382i $$0.421029\pi$$
$$674$$ −4.80016e6 −0.407011
$$675$$ 0 0
$$676$$ 8.10592e6 0.682237
$$677$$ −1.67197e7 −1.40203 −0.701014 0.713147i $$-0.747269\pi$$
−0.701014 + 0.713147i $$0.747269\pi$$
$$678$$ 0 0
$$679$$ −2.28553e7 −1.90245
$$680$$ −5.03400e6 −0.417485
$$681$$ 0 0
$$682$$ 724608. 0.0596544
$$683$$ −7.14532e6 −0.586097 −0.293049 0.956098i $$-0.594670\pi$$
−0.293049 + 0.956098i $$0.594670\pi$$
$$684$$ 0 0
$$685$$ −3.62295e6 −0.295009
$$686$$ −1.24800e6 −0.101252
$$687$$ 0 0
$$688$$ −816064. −0.0657284
$$689$$ −6.81996e6 −0.547310
$$690$$ 0 0
$$691$$ −8.78395e6 −0.699833 −0.349917 0.936781i $$-0.613790\pi$$
−0.349917 + 0.936781i $$0.613790\pi$$
$$692$$ −1.60573e7 −1.27470
$$693$$ 0 0
$$694$$ 3.55482e6 0.280169
$$695$$ −2.80550e6 −0.220317
$$696$$ 0 0
$$697$$ 1.57698e7 1.22955
$$698$$ 4.29610e6 0.333761
$$699$$ 0 0
$$700$$ −3.36000e6 −0.259176
$$701$$ 1.60141e7 1.23086 0.615428 0.788193i $$-0.288983\pi$$
0.615428 + 0.788193i $$0.288983\pi$$
$$702$$ 0 0
$$703$$ 192920. 0.0147228
$$704$$ −1.58182e6 −0.120289
$$705$$ 0 0
$$706$$ −1.32371e6 −0.0999495
$$707$$ 1.72604e7 1.29868
$$708$$ 0 0
$$709$$ −1.91354e7 −1.42962 −0.714811 0.699318i $$-0.753487\pi$$
−0.714811 + 0.699318i $$0.753487\pi$$
$$710$$ 1.63240e6 0.121529
$$711$$ 0 0
$$712$$ −1.21644e7 −0.899271
$$713$$ 7.28525e6 0.536686
$$714$$ 0 0
$$715$$ −1.05820e6 −0.0774110
$$716$$ −1.66449e7 −1.21338
$$717$$ 0 0
$$718$$ −518640. −0.0375452
$$719$$ −1.02934e7 −0.742566 −0.371283 0.928520i $$-0.621082\pi$$
−0.371283 + 0.928520i $$0.621082\pi$$
$$720$$ 0 0
$$721$$ −3.74477e6 −0.268279
$$722$$ 2.70500e6 0.193119
$$723$$ 0 0
$$724$$ 2.99874e6 0.212615
$$725$$ 2.13125e6 0.150588
$$726$$ 0 0
$$727$$ −1.93264e7 −1.35618 −0.678088 0.734981i $$-0.737191\pi$$
−0.678088 + 0.734981i $$0.737191\pi$$
$$728$$ 6.58944e6 0.460808
$$729$$ 0 0
$$730$$ −1.93870e6 −0.134649
$$731$$ −2.08743e6 −0.144484
$$732$$ 0 0
$$733$$ 5.26197e6 0.361733 0.180866 0.983508i $$-0.442110\pi$$
0.180866 + 0.983508i $$0.442110\pi$$
$$734$$ 2.99986e6 0.205523
$$735$$ 0 0
$$736$$ 1.53324e7 1.04331
$$737$$ 9.02386e6 0.611961
$$738$$ 0 0
$$739$$ 2.82944e7 1.90585 0.952927 0.303199i $$-0.0980548\pi$$
0.952927 + 0.303199i $$0.0980548\pi$$
$$740$$ 127400. 0.00855244
$$741$$ 0 0
$$742$$ 9.15686e6 0.610572
$$743$$ −2.09863e7 −1.39464 −0.697321 0.716759i $$-0.745625\pi$$
−0.697321 + 0.716759i $$0.745625\pi$$
$$744$$ 0 0
$$745$$ 1.00938e7 0.666288
$$746$$ 4.47615e6 0.294481
$$747$$ 0 0
$$748$$ −6.95363e6 −0.454420
$$749$$ −3.03921e7 −1.97950
$$750$$ 0 0
$$751$$ −1.89668e7 −1.22714 −0.613572 0.789639i $$-0.710268\pi$$
−0.613572 + 0.789639i $$0.710268\pi$$
$$752$$ 7.92973e6 0.511345
$$753$$ 0 0
$$754$$ −1.95052e6 −0.124946
$$755$$ 1.11662e7 0.712915
$$756$$ 0 0
$$757$$ −1.08257e7 −0.686617 −0.343309 0.939223i $$-0.611548\pi$$
−0.343309 + 0.939223i $$0.611548\pi$$
$$758$$ −6.31868e6 −0.399442
$$759$$ 0 0
$$760$$ −3.18000e6 −0.199707
$$761$$ −1.90534e7 −1.19264 −0.596322 0.802745i $$-0.703372\pi$$
−0.596322 + 0.802745i $$0.703372\pi$$
$$762$$ 0 0
$$763$$ 7.07136e6 0.439736
$$764$$ 1.31475e7 0.814908
$$765$$ 0 0
$$766$$ 684432. 0.0421462
$$767$$ 5.72572e6 0.351432
$$768$$ 0 0
$$769$$ −1.57826e7 −0.962415 −0.481208 0.876607i $$-0.659802\pi$$
−0.481208 + 0.876607i $$0.659802\pi$$
$$770$$ 1.42080e6 0.0863587
$$771$$ 0 0
$$772$$ −1.47577e6 −0.0891199
$$773$$ 2.44049e7 1.46902 0.734510 0.678598i $$-0.237412\pi$$
0.734510 + 0.678598i $$0.237412\pi$$
$$774$$ 0 0
$$775$$ −1.53000e6 −0.0915034
$$776$$ −1.42846e7 −0.851555
$$777$$ 0 0
$$778$$ 176940. 0.0104804
$$779$$ 9.96188e6 0.588163
$$780$$ 0 0
$$781$$ 4.83190e6 0.283459
$$782$$ 9.98746e6 0.584034
$$783$$ 0 0
$$784$$ 1.31574e7 0.764504
$$785$$ 6.55645e6 0.379747
$$786$$ 0 0
$$787$$ 3.37607e7 1.94301 0.971505 0.237019i $$-0.0761704\pi$$
0.971505 + 0.237019i $$0.0761704\pi$$
$$788$$ 1.27641e7 0.732278
$$789$$ 0 0
$$790$$ −1.66800e6 −0.0950886
$$791$$ −2.14771e6 −0.122049
$$792$$ 0 0
$$793$$ 9.23837e6 0.521690
$$794$$ 1.09135e7 0.614344
$$795$$ 0 0
$$796$$ −2.42200e7 −1.35485
$$797$$ −2.19885e7 −1.22617 −0.613083 0.790019i $$-0.710071\pi$$
−0.613083 + 0.790019i $$0.710071\pi$$
$$798$$ 0 0
$$799$$ 2.02837e7 1.12403
$$800$$ −3.22000e6 −0.177882
$$801$$ 0 0
$$802$$ 8.09360e6 0.444330
$$803$$ −5.73855e6 −0.314061
$$804$$ 0 0
$$805$$ 1.42848e7 0.776935
$$806$$ 1.40026e6 0.0759224
$$807$$ 0 0
$$808$$ 1.07878e7 0.581303
$$809$$ 2.93597e7 1.57717 0.788587 0.614923i $$-0.210813\pi$$
0.788587 + 0.614923i $$0.210813\pi$$
$$810$$ 0 0
$$811$$ 3.17703e7 1.69617 0.848083 0.529863i $$-0.177757\pi$$
0.848083 + 0.529863i $$0.177757\pi$$
$$812$$ −1.83322e7 −0.975716
$$813$$ 0 0
$$814$$ −53872.0 −0.00284972
$$815$$ 3.86410e6 0.203777
$$816$$ 0 0
$$817$$ −1.31864e6 −0.0691148
$$818$$ 5.42414e6 0.283431
$$819$$ 0 0
$$820$$ 6.57860e6 0.341664
$$821$$ 2.71430e6 0.140540 0.0702699 0.997528i $$-0.477614\pi$$
0.0702699 + 0.997528i $$0.477614\pi$$
$$822$$ 0 0
$$823$$ −1.25866e7 −0.647753 −0.323877 0.946099i $$-0.604986\pi$$
−0.323877 + 0.946099i $$0.604986\pi$$
$$824$$ −2.34048e6 −0.120084
$$825$$ 0 0
$$826$$ −7.68768e6 −0.392053
$$827$$ 8.72355e6 0.443537 0.221768 0.975099i $$-0.428817\pi$$
0.221768 + 0.975099i $$0.428817\pi$$
$$828$$ 0 0
$$829$$ −1.06178e7 −0.536597 −0.268299 0.963336i $$-0.586461\pi$$
−0.268299 + 0.963336i $$0.586461\pi$$
$$830$$ −835800. −0.0421121
$$831$$ 0 0
$$832$$ −3.05677e6 −0.153093
$$833$$ 3.36556e7 1.68053
$$834$$ 0 0
$$835$$ 9.91680e6 0.492216
$$836$$ −4.39264e6 −0.217375
$$837$$ 0 0
$$838$$ 7.43492e6 0.365735
$$839$$ −1.67765e7 −0.822805 −0.411403 0.911454i $$-0.634961\pi$$
−0.411403 + 0.911454i $$0.634961\pi$$
$$840$$ 0 0
$$841$$ −8.88305e6 −0.433084
$$842$$ −7.10500e6 −0.345370
$$843$$ 0 0
$$844$$ −3.09583e7 −1.49596
$$845$$ 7.23742e6 0.348692
$$846$$ 0 0
$$847$$ −2.67162e7 −1.27958
$$848$$ −1.56430e7 −0.747016
$$849$$ 0 0
$$850$$ −2.09750e6 −0.0995760
$$851$$ −541632. −0.0256378
$$852$$ 0 0
$$853$$ −2.20186e7 −1.03613 −0.518067 0.855340i $$-0.673348\pi$$
−0.518067 + 0.855340i $$0.673348\pi$$
$$854$$ −1.24040e7 −0.581991
$$855$$ 0 0
$$856$$ −1.89950e7 −0.886045
$$857$$ −3.16676e7 −1.47287 −0.736434 0.676510i $$-0.763492\pi$$
−0.736434 + 0.676510i $$0.763492\pi$$
$$858$$ 0 0
$$859$$ 1.58064e7 0.730886 0.365443 0.930834i $$-0.380918\pi$$
0.365443 + 0.930834i $$0.380918\pi$$
$$860$$ −870800. −0.0401488
$$861$$ 0 0
$$862$$ −8.12410e6 −0.372398
$$863$$ 1.44287e7 0.659476 0.329738 0.944072i $$-0.393040\pi$$
0.329738 + 0.944072i $$0.393040\pi$$
$$864$$ 0 0
$$865$$ −1.43368e7 −0.651499
$$866$$ −1.45257e7 −0.658178
$$867$$ 0 0
$$868$$ 1.31604e7 0.592886
$$869$$ −4.93728e6 −0.221788
$$870$$ 0 0
$$871$$ 1.74380e7 0.778845
$$872$$ 4.41960e6 0.196830
$$873$$ 0 0
$$874$$ 6.30912e6 0.279377
$$875$$ −3.00000e6 −0.132465
$$876$$ 0 0
$$877$$ 247902. 0.0108838 0.00544191 0.999985i $$-0.498268\pi$$
0.00544191 + 0.999985i $$0.498268\pi$$
$$878$$ 1.08206e7 0.473711
$$879$$ 0 0
$$880$$ −2.42720e6 −0.105657
$$881$$ −4.10268e7 −1.78085 −0.890426 0.455128i $$-0.849594\pi$$
−0.890426 + 0.455128i $$0.849594\pi$$
$$882$$ 0 0
$$883$$ 4.18015e7 1.80422 0.902112 0.431503i $$-0.142016\pi$$
0.902112 + 0.431503i $$0.142016\pi$$
$$884$$ −1.34374e7 −0.578343
$$885$$ 0 0
$$886$$ −1.30305e7 −0.557669
$$887$$ 2.10476e7 0.898241 0.449120 0.893471i $$-0.351737\pi$$
0.449120 + 0.893471i $$0.351737\pi$$
$$888$$ 0 0
$$889$$ 1.35460e7 0.574852
$$890$$ −5.06850e6 −0.214489
$$891$$ 0 0
$$892$$ −3.14041e7 −1.32152
$$893$$ 1.28133e7 0.537690
$$894$$ 0 0
$$895$$ −1.48615e7 −0.620162
$$896$$ 3.57581e7 1.48800
$$897$$ 0 0
$$898$$ −1.01990e6 −0.0422053
$$899$$ −8.34768e6 −0.344482
$$900$$ 0 0
$$901$$ −4.00136e7 −1.64208
$$902$$ −2.78181e6 −0.113844
$$903$$ 0 0
$$904$$ −1.34232e6 −0.0546305
$$905$$ 2.67745e6 0.108668
$$906$$ 0 0
$$907$$ 7.48309e6 0.302039 0.151019 0.988531i $$-0.451744\pi$$
0.151019 + 0.988531i $$0.451744\pi$$
$$908$$ −653744. −0.0263144
$$909$$ 0 0
$$910$$ 2.74560e6 0.109909
$$911$$ 6.63165e6 0.264744 0.132372 0.991200i $$-0.457741\pi$$
0.132372 + 0.991200i $$0.457741\pi$$
$$912$$ 0 0
$$913$$ −2.47397e6 −0.0982239
$$914$$ −2.44168e6 −0.0966772
$$915$$ 0 0
$$916$$ 1.66883e7 0.657163
$$917$$ −1.46788e7 −0.576457
$$918$$ 0 0
$$919$$ −1.68976e7 −0.659990 −0.329995 0.943983i $$-0.607047\pi$$
−0.329995 + 0.943983i $$0.607047\pi$$
$$920$$ 8.92800e6 0.347764
$$921$$ 0 0
$$922$$ −8.14420e6 −0.315516
$$923$$ 9.33733e6 0.360760
$$924$$ 0 0
$$925$$ 113750. 0.00437116
$$926$$ −4.04587e6 −0.155055
$$927$$ 0 0
$$928$$ −1.75683e7 −0.669669
$$929$$ 1.28653e7 0.489081 0.244541 0.969639i $$-0.421363\pi$$
0.244541 + 0.969639i $$0.421363\pi$$
$$930$$ 0 0
$$931$$ 2.12604e7 0.803892
$$932$$ −1.35894e7 −0.512459
$$933$$ 0 0
$$934$$ 6.50194e6 0.243880
$$935$$ −6.20860e6 −0.232255
$$936$$ 0 0
$$937$$ 1.06887e7 0.397718 0.198859 0.980028i $$-0.436276\pi$$
0.198859 + 0.980028i $$0.436276\pi$$
$$938$$ −2.34132e7 −0.868870
$$939$$ 0 0
$$940$$ 8.46160e6 0.312344
$$941$$ −2.82455e7 −1.03986 −0.519930 0.854209i $$-0.674042\pi$$
−0.519930 + 0.854209i $$0.674042\pi$$
$$942$$ 0 0
$$943$$ −2.79684e7 −1.02421
$$944$$ 1.31331e7 0.479665
$$945$$ 0 0
$$946$$ 368224. 0.0133778
$$947$$ 1.70892e7 0.619222 0.309611 0.950863i $$-0.399801\pi$$
0.309611 + 0.950863i $$0.399801\pi$$
$$948$$ 0 0
$$949$$ −1.10894e7 −0.399706
$$950$$ −1.32500e6 −0.0476329
$$951$$ 0 0
$$952$$ 3.86611e7 1.38255
$$953$$ −2.22259e7 −0.792735 −0.396367 0.918092i $$-0.629729\pi$$
−0.396367 + 0.918092i $$0.629729\pi$$
$$954$$ 0 0
$$955$$ 1.17388e7 0.416500
$$956$$ −1.36864e6 −0.0484333
$$957$$ 0 0
$$958$$ −6.55872e6 −0.230890
$$959$$ 2.78243e7 0.976961
$$960$$ 0 0
$$961$$ −2.26364e7 −0.790678
$$962$$ −104104. −0.00362685
$$963$$ 0 0
$$964$$ 3.10234e6 0.107522
$$965$$ −1.31765e6 −0.0455493
$$966$$ 0 0
$$967$$ 2.41551e7 0.830696 0.415348 0.909663i $$-0.363660\pi$$
0.415348 + 0.909663i $$0.363660\pi$$
$$968$$ −1.66976e7 −0.572752
$$969$$ 0 0
$$970$$ −5.95190e6 −0.203108
$$971$$ 5.48313e7 1.86630 0.933149 0.359491i $$-0.117050\pi$$
0.933149 + 0.359491i $$0.117050\pi$$
$$972$$ 0 0
$$973$$ 2.15462e7 0.729608
$$974$$ 1.70639e7 0.576344
$$975$$ 0 0
$$976$$ 2.11901e7 0.712047
$$977$$ 1.56612e7 0.524915 0.262457 0.964944i $$-0.415467\pi$$
0.262457 + 0.964944i $$0.415467\pi$$
$$978$$ 0 0
$$979$$ −1.50028e7 −0.500281
$$980$$ 1.40399e7 0.466981
$$981$$ 0 0
$$982$$ 3.02530e6 0.100113
$$983$$ 1.63420e7 0.539412 0.269706 0.962943i $$-0.413073\pi$$
0.269706 + 0.962943i $$0.413073\pi$$
$$984$$ 0 0
$$985$$ 1.13966e7 0.374268
$$986$$ −1.14440e7 −0.374873
$$987$$ 0 0
$$988$$ −8.48848e6 −0.276654
$$989$$ 3.70214e6 0.120355
$$990$$ 0 0
$$991$$ 1.37576e7 0.444997 0.222498 0.974933i $$-0.428579\pi$$
0.222498 + 0.974933i $$0.428579\pi$$
$$992$$ 1.26121e7 0.406919
$$993$$ 0 0
$$994$$ −1.25368e7 −0.402459
$$995$$ −2.16250e7 −0.692466
$$996$$ 0 0
$$997$$ −1.29097e7 −0.411320 −0.205660 0.978624i $$-0.565934\pi$$
−0.205660 + 0.978624i $$0.565934\pi$$
$$998$$ 1.29838e7 0.412644
$$999$$ 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 45.6.a.b.1.1 1
3.2 odd 2 5.6.a.a.1.1 1
4.3 odd 2 720.6.a.a.1.1 1
5.2 odd 4 225.6.b.e.199.1 2
5.3 odd 4 225.6.b.e.199.2 2
5.4 even 2 225.6.a.f.1.1 1
12.11 even 2 80.6.a.e.1.1 1
15.2 even 4 25.6.b.a.24.2 2
15.8 even 4 25.6.b.a.24.1 2
15.14 odd 2 25.6.a.a.1.1 1
21.20 even 2 245.6.a.b.1.1 1
24.5 odd 2 320.6.a.j.1.1 1
24.11 even 2 320.6.a.g.1.1 1
33.32 even 2 605.6.a.a.1.1 1
39.38 odd 2 845.6.a.b.1.1 1
60.23 odd 4 400.6.c.j.49.2 2
60.47 odd 4 400.6.c.j.49.1 2
60.59 even 2 400.6.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 3.2 odd 2
25.6.a.a.1.1 1 15.14 odd 2
25.6.b.a.24.1 2 15.8 even 4
25.6.b.a.24.2 2 15.2 even 4
45.6.a.b.1.1 1 1.1 even 1 trivial
80.6.a.e.1.1 1 12.11 even 2
225.6.a.f.1.1 1 5.4 even 2
225.6.b.e.199.1 2 5.2 odd 4
225.6.b.e.199.2 2 5.3 odd 4
245.6.a.b.1.1 1 21.20 even 2
320.6.a.g.1.1 1 24.11 even 2
320.6.a.j.1.1 1 24.5 odd 2
400.6.a.g.1.1 1 60.59 even 2
400.6.c.j.49.1 2 60.47 odd 4
400.6.c.j.49.2 2 60.23 odd 4
605.6.a.a.1.1 1 33.32 even 2
720.6.a.a.1.1 1 4.3 odd 2
845.6.a.b.1.1 1 39.38 odd 2