# Properties

 Label 225.6.a.f.1.1 Level $225$ Weight $6$ Character 225.1 Self dual yes Analytic conductor $36.086$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.0863594579$$ 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 $$\chi$$ $$=$$ 225.1

## $q$-expansion

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

## 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 $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ 0 0
$$4$$ −28.0000 −0.875000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −192.000 −1.48100 −0.740502 0.672054i $$-0.765412\pi$$
−0.740502 + 0.672054i $$0.765412\pi$$
$$8$$ −120.000 −0.662913
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 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 $$-0.575405\pi$$
−0.234681 + 0.972072i $$0.575405\pi$$
$$14$$ −384.000 −0.523614
$$15$$ 0 0
$$16$$ 656.000 0.640625
$$17$$ −1678.00 −1.40822 −0.704109 0.710092i $$-0.748653\pi$$
−0.704109 + 0.710092i $$0.748653\pi$$
$$18$$ 0 0
$$19$$ 1060.00 0.673631 0.336815 0.941571i $$-0.390650\pi$$
0.336815 + 0.941571i $$0.390650\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 296.000 0.130387
$$23$$ 2976.00 1.17304 0.586521 0.809934i $$-0.300497\pi$$
0.586521 + 0.809934i $$0.300497\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −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
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −182.000 −0.0218558 −0.0109279 0.999940i $$-0.503479\pi$$
−0.0109279 + 0.999940i $$0.503479\pi$$
$$38$$ 2120.00 0.238164
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 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 $$-0.483663\pi$$
0.0513002 + 0.998683i $$0.483663\pi$$
$$44$$ −4144.00 −0.322692
$$45$$ 0 0
$$46$$ 5952.00 0.414733
$$47$$ −12088.0 −0.798196 −0.399098 0.916908i $$-0.630677\pi$$
−0.399098 + 0.916908i $$0.630677\pi$$
$$48$$ 0 0
$$49$$ 20057.0 1.19337
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 8008.00 0.410691
$$53$$ 23846.0 1.16607 0.583037 0.812446i $$-0.301864\pi$$
0.583037 + 0.812446i $$0.301864\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 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
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −60972.0 −1.65937 −0.829685 0.558231i $$-0.811480\pi$$
−0.829685 + 0.558231i $$0.811480\pi$$
$$68$$ 46984.0 1.23219
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 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 $$-0.359993\pi$$
0.425798 + 0.904818i $$0.359993\pi$$
$$74$$ −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 $$0.597219\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 18796.0 0.308696
$$83$$ 16716.0 0.266340 0.133170 0.991093i $$-0.457484\pi$$
0.133170 + 0.991093i $$0.457484\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 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
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 119038. 1.28457 0.642283 0.766468i $$-0.277987\pi$$
0.642283 + 0.766468i $$0.277987\pi$$
$$98$$ 40114.0 0.421921
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 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 $$-0.471130\pi$$
0.0905734 + 0.995890i $$0.471130\pi$$
$$104$$ 34320.0 0.311146
$$105$$ 0 0
$$106$$ 47692.0 0.412269
$$107$$ 158292. 1.33659 0.668297 0.743895i $$-0.267024\pi$$
0.668297 + 0.743895i $$0.267024\pi$$
$$108$$ 0 0
$$109$$ 36830.0 0.296917 0.148459 0.988919i $$-0.452569\pi$$
0.148459 + 0.988919i $$0.452569\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −125952. −0.948768
$$113$$ 11186.0 0.0824098 0.0412049 0.999151i $$-0.486880\pi$$
0.0412049 + 0.999151i $$0.486880\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −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
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −70552.0 −0.388150 −0.194075 0.980987i $$-0.562171\pi$$
−0.194075 + 0.980987i $$0.562171\pi$$
$$128$$ −186240. −1.00473
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −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 $$-0.606992\pi$$
−0.329831 + 0.944040i $$0.606992\pi$$
$$138$$ 0 0
$$139$$ 112220. 0.492644 0.246322 0.969188i $$-0.420778\pi$$
0.246322 + 0.969188i $$0.420778\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 65296.0 0.271748
$$143$$ −42328.0 −0.173096
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 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
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 262258. 0.849141 0.424570 0.905395i $$-0.360425\pi$$
0.424570 + 0.905395i $$0.360425\pi$$
$$158$$ −66720.0 −0.212625
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −571392. −1.73728
$$162$$ 0 0
$$163$$ 154564. 0.455658 0.227829 0.973701i $$-0.426837\pi$$
0.227829 + 0.973701i $$0.426837\pi$$
$$164$$ −263144. −0.763983
$$165$$ 0 0
$$166$$ 33432.0 0.0941656
$$167$$ 396672. 1.10063 0.550314 0.834958i $$-0.314508\pi$$
0.550314 + 0.834958i $$0.314508\pi$$
$$168$$ 0 0
$$169$$ −289497. −0.779700
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −34832.0 −0.0897754
$$173$$ −573474. −1.45680 −0.728398 0.685155i $$-0.759735\pi$$
−0.728398 + 0.685155i $$0.759735\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 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
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −248344. −0.519337
$$188$$ 338464. 0.698422
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −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 $$-0.516217\pi$$
−0.0509257 + 0.998702i $$0.516217\pi$$
$$194$$ 238076. 0.454163
$$195$$ 0 0
$$196$$ −561596. −1.04420
$$197$$ 455862. 0.836889 0.418444 0.908242i $$-0.362575\pi$$
0.418444 + 0.908242i $$0.362575\pi$$
$$198$$ 0 0
$$199$$ 865000. 1.54840 0.774200 0.632940i $$-0.218152\pi$$
0.774200 + 0.632940i $$0.218152\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 179796. 0.310028
$$203$$ −654720. −1.11510
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 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
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 470016. 0.677584
$$218$$ 73660.0 0.104976
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 479908. 0.660963
$$222$$ 0 0
$$223$$ −1.12158e6 −1.51031 −0.755156 0.655545i $$-0.772439\pi$$
−0.755156 + 0.655545i $$0.772439\pi$$
$$224$$ −989184. −1.31722
$$225$$ 0 0
$$226$$ 22372.0 0.0291363
$$227$$ −23348.0 −0.0300736 −0.0150368 0.999887i $$-0.504787\pi$$
−0.0150368 + 0.999887i $$0.504787\pi$$
$$228$$ 0 0
$$229$$ −596010. −0.751043 −0.375522 0.926814i $$-0.622536\pi$$
−0.375522 + 0.926814i $$0.622536\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −409200. −0.499132
$$233$$ −485334. −0.585667 −0.292834 0.956163i $$-0.594598\pi$$
−0.292834 + 0.956163i $$0.594598\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −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
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −303160. −0.316176
$$248$$ 293760. 0.303294
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 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 $$-0.288419\pi$$
0.616823 + 0.787102i $$0.288419\pi$$
$$258$$ 0 0
$$259$$ 34944.0 0.0323685
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −152904. −0.137615
$$263$$ 2.12834e6 1.89736 0.948682 0.316231i $$-0.102417\pi$$
0.948682 + 0.316231i $$0.102417\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −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
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 110298. 0.0863711 0.0431855 0.999067i $$-0.486249\pi$$
0.0431855 + 0.999067i $$0.486249\pi$$
$$278$$ 224440. 0.174176
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 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 $$-0.460695\pi$$
0.123166 + 0.992386i $$0.460695\pi$$
$$284$$ −914144. −0.672541
$$285$$ 0 0
$$286$$ −84656.0 −0.0611988
$$287$$ −1.80442e6 −1.29310
$$288$$ 0 0
$$289$$ 1.39583e6 0.983076
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −1.08567e6 −0.745146
$$293$$ 2.19481e6 1.49358 0.746788 0.665063i $$-0.231595\pi$$
0.746788 + 0.665063i $$0.231595\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 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
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.37751e6 1.43971 0.719857 0.694123i $$-0.244207\pi$$
0.719857 + 0.694123i $$0.244207\pi$$
$$308$$ 795648. 0.477908
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 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 $$-0.364708\pi$$
0.412351 + 0.911025i $$0.364708\pi$$
$$314$$ 524516. 0.300217
$$315$$ 0 0
$$316$$ 934080. 0.526219
$$317$$ 2.12462e6 1.18750 0.593750 0.804650i $$-0.297647\pi$$
0.593750 + 0.804650i $$0.297647\pi$$
$$318$$ 0 0
$$319$$ 504680. 0.277677
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1.14278e6 −0.614221
$$323$$ −1.77868e6 −0.948618
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 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
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −2.40008e6 −1.15120 −0.575601 0.817731i $$-0.695232\pi$$
−0.575601 + 0.817731i $$0.695232\pi$$
$$338$$ −578994. −0.275665
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −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 $$-0.370322\pi$$
0.396218 + 0.918156i $$0.370322\pi$$
$$348$$ 0 0
$$349$$ −2.14805e6 −0.944019 −0.472010 0.881593i $$-0.656471\pi$$
−0.472010 + 0.881593i $$0.656471\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 762496. 0.328005
$$353$$ −661854. −0.282700 −0.141350 0.989960i $$-0.545144\pi$$
−0.141350 + 0.989960i $$0.545144\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 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
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.49993e6 0.581307 0.290653 0.956828i $$-0.406127\pi$$
0.290653 + 0.956828i $$0.406127\pi$$
$$368$$ 1.95226e6 0.751480
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −4.57843e6 −1.72696
$$372$$ 0 0
$$373$$ 2.23807e6 0.832918 0.416459 0.909154i $$-0.363271\pi$$
0.416459 + 0.909154i $$0.363271\pi$$
$$374$$ −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 $$0.308916\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −939104. −0.329272
$$383$$ 342216. 0.119207 0.0596037 0.998222i $$-0.481016\pi$$
0.0596037 + 0.998222i $$0.481016\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −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
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5.45674e6 1.73763 0.868814 0.495138i $$-0.164883\pi$$
0.868814 + 0.495138i $$0.164883\pi$$
$$398$$ 1.73000e6 0.547442
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −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 $$0.631279\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −546112. −0.158503
$$413$$ −3.84384e6 −1.10889
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −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
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −6.20198e6 −1.64612
$$428$$ −4.43218e6 −1.16952
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 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 $$-0.880892\pi$$
−0.930804 + 0.365518i $$0.880892\pi$$
$$434$$ 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 $$0.733675\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 959816. 0.233686
$$443$$ −6.51524e6 −1.57733 −0.788663 0.614826i $$-0.789226\pi$$
−0.788663 + 0.614826i $$0.789226\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −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
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.22084e6 −0.273444 −0.136722 0.990609i $$-0.543657\pi$$
−0.136722 + 0.990609i $$0.543657\pi$$
$$458$$ −1.19202e6 −0.265534
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 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 $$-0.570371\pi$$
−0.219280 + 0.975662i $$0.570371\pi$$
$$464$$ 2.23696e6 0.482351
$$465$$ 0 0
$$466$$ −970668. −0.207065
$$467$$ 3.25097e6 0.689797 0.344898 0.938640i $$-0.387913\pi$$
0.344898 + 0.938640i $$0.387913\pi$$
$$468$$ 0 0
$$469$$ 1.17066e7 2.45753
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −2.40240e6 −0.496353
$$473$$ 184112. 0.0378381
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −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
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 8.53197e6 1.63015 0.815074 0.579357i $$-0.196696\pi$$
0.815074 + 0.579357i $$0.196696\pi$$
$$488$$ −3.87624e6 −0.736819
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −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 $$0.698343\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 3.28750e6 0.582245
$$503$$ 8.61770e6 1.51870 0.759349 0.650684i $$-0.225518\pi$$
0.759349 + 0.650684i $$0.225518\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 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
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.78902e6 −0.294367
$$518$$ 69888.0 0.0114440
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −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 $$-0.314213\pi$$
0.551087 + 0.834448i $$0.314213\pi$$
$$524$$ 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
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 5.69856e6 0.872943
$$533$$ −2.68783e6 −0.409811
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 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
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.26544e7 −1.80831 −0.904157 0.427201i $$-0.859500\pi$$
−0.904157 + 0.427201i $$0.859500\pi$$
$$548$$ 4.05770e6 0.577204
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 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 $$-0.660569\pi$$
−0.483319 + 0.875444i $$0.660569\pi$$
$$558$$ 0 0
$$559$$ −355784. −0.0481567
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 384396. 0.0513379
$$563$$ 846636. 0.112571 0.0562854 0.998415i $$-0.482074\pi$$
0.0562854 + 0.998415i $$0.482074\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 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
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.86080e6 0.357724 0.178862 0.983874i $$-0.442758\pi$$
0.178862 + 0.983874i $$0.442758\pi$$
$$578$$ 2.79165e6 0.347570
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −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 $$-0.632274\pi$$
−0.403694 + 0.914894i $$0.632274\pi$$
$$588$$ 0 0
$$589$$ −2.59488e6 −0.308197
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −119392. −0.0140014
$$593$$ −1.78609e6 −0.208578 −0.104289 0.994547i $$-0.533257\pi$$
−0.104289 + 0.994547i $$0.533257\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 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
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −7.07999e6 −0.779940 −0.389970 0.920828i $$-0.627515\pi$$
−0.389970 + 0.920828i $$0.627515\pi$$
$$608$$ 5.46112e6 0.599132
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3.45717e6 0.374643
$$612$$ 0 0
$$613$$ −5.09609e6 −0.547754 −0.273877 0.961765i $$-0.588306\pi$$
−0.273877 + 0.961765i $$0.588306\pi$$
$$614$$ 4.75502e6 0.509016
$$615$$ 0 0
$$616$$ 3.40992e6 0.362070
$$617$$ −1.30003e7 −1.37480 −0.687400 0.726279i $$-0.741248\pi$$
−0.687400 + 0.726279i $$0.741248\pi$$
$$618$$ 0 0
$$619$$ 4.84406e6 0.508139 0.254070 0.967186i $$-0.418231\pi$$
0.254070 + 0.967186i $$0.418231\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 4.74610e6 0.491882
$$623$$ 1.94630e7 2.00905
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 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
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −5.73630e6 −0.560123
$$638$$ 1.00936e6 0.0981735
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −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 $$-0.187391\pi$$
0.831659 + 0.555287i $$0.187391\pi$$
$$644$$ 1.59990e7 1.52012
$$645$$ 0 0
$$646$$ −3.55736e6 −0.335387
$$647$$ −4.25469e6 −0.399583 −0.199792 0.979838i $$-0.564026\pi$$
−0.199792 + 0.979838i $$0.564026\pi$$
$$648$$ 0 0
$$649$$ 2.96296e6 0.276130
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.32779e6 −0.398701
$$653$$ 3.01085e6 0.276316 0.138158 0.990410i $$-0.455882\pi$$
0.138158 + 0.990410i $$0.455882\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 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
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.01482e7 0.883228
$$668$$ −1.11068e7 −0.963049
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.78070e6 0.409907
$$672$$ 0 0
$$673$$ −5.77063e6 −0.491117 −0.245559 0.969382i $$-0.578971\pi$$
−0.245559 + 0.969382i $$0.578971\pi$$
$$674$$ −4.80016e6 −0.407011
$$675$$ 0 0
$$676$$ 8.10592e6 0.682237
$$677$$ 1.67197e7 1.40203 0.701014 0.713147i $$-0.252731\pi$$
0.701014 + 0.713147i $$0.252731\pi$$
$$678$$ 0 0
$$679$$ −2.28553e7 −1.90245
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −724608. −0.0596544
$$683$$ 7.14532e6 0.586097 0.293049 0.956098i $$-0.405330\pi$$
0.293049 + 0.956098i $$0.405330\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −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
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −1.57698e7 −1.22955
$$698$$ −4.29610e6 −0.333761
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 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 $$0.753487\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 1.21644e7 0.899271
$$713$$ −7.28525e6 −0.536686
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −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
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.93264e7 1.35618 0.678088 0.734981i $$-0.262809\pi$$
0.678088 + 0.734981i $$0.262809\pi$$
$$728$$ −6.58944e6 −0.460808
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.08743e6 −0.144484
$$732$$ 0 0
$$733$$ −5.26197e6 −0.361733 −0.180866 0.983508i $$-0.557890\pi$$
−0.180866 + 0.983508i $$0.557890\pi$$
$$734$$ 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 $$0.0980548\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −9.15686e6 −0.610572
$$743$$ 2.09863e7 1.39464 0.697321 0.716759i $$-0.254375\pi$$
0.697321 + 0.716759i $$0.254375\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 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
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.08257e7 0.686617 0.343309 0.939223i $$-0.388452\pi$$
0.343309 + 0.939223i $$0.388452\pi$$
$$758$$ 6.31868e6 0.399442
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −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 $$0.659802\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.47577e6 0.0891199
$$773$$ −2.44049e7 −1.46902 −0.734510 0.678598i $$-0.762588\pi$$
−0.734510 + 0.678598i $$0.762588\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −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
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −3.37607e7 −1.94301 −0.971505 0.237019i $$-0.923830\pi$$
−0.971505 + 0.237019i $$0.923830\pi$$
$$788$$ −1.27641e7 −0.732278
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −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 $$-0.289929\pi$$
0.613083 + 0.790019i $$0.289929\pi$$
$$798$$ 0 0
$$799$$ 2.02837e7 1.12403
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −8.09360e6 −0.444330
$$803$$ 5.73855e6 0.314061
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 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
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.31864e6 0.0691148
$$818$$ −5.42414e6 −0.283431
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 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 $$-0.395014\pi$$
0.323877 + 0.946099i $$0.395014\pi$$
$$824$$ −2.34048e6 −0.120084
$$825$$ 0 0
$$826$$ −7.68768e6 −0.392053
$$827$$ −8.72355e6 −0.443537 −0.221768 0.975099i $$-0.571183\pi$$
−0.221768 + 0.975099i $$0.571183\pi$$
$$828$$ 0 0
$$829$$ −1.06178e7 −0.536597 −0.268299 0.963336i $$-0.586461\pi$$
−0.268299 + 0.963336i $$0.586461\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 3.05677e6 0.153093
$$833$$ −3.36556e7 −1.68053
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −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
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.67162e7 1.27958
$$848$$ 1.56430e7 0.747016
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −541632. −0.0256378
$$852$$ 0 0
$$853$$ 2.20186e7 1.03613 0.518067 0.855340i $$-0.326652\pi$$
0.518067 + 0.855340i $$0.326652\pi$$
$$854$$ −1.24040e7 −0.581991
$$855$$ 0 0
$$856$$ −1.89950e7 −0.886045
$$857$$ 3.16676e7 1.47287 0.736434 0.676510i $$-0.236508\pi$$
0.736434 + 0.676510i $$0.236508\pi$$
$$858$$ 0 0
$$859$$ 1.58064e7 0.730886 0.365443 0.930834i $$-0.380918\pi$$
0.365443 + 0.930834i $$0.380918\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8.12410e6 0.372398
$$863$$ −1.44287e7 −0.659476 −0.329738 0.944072i $$-0.606960\pi$$
−0.329738 + 0.944072i $$0.606960\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −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
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −247902. −0.0108838 −0.00544191 0.999985i $$-0.501732\pi$$
−0.00544191 + 0.999985i $$0.501732\pi$$
$$878$$ −1.08206e7 −0.473711
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −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 $$-0.857984\pi$$
−0.902112 + 0.431503i $$0.857984\pi$$
$$884$$ −1.34374e7 −0.578343
$$885$$ 0 0
$$886$$ −1.30305e7 −0.557669
$$887$$ −2.10476e7 −0.898241 −0.449120 0.893471i $$-0.648263\pi$$
−0.449120 + 0.893471i $$0.648263\pi$$
$$888$$ 0 0
$$889$$ 1.35460e7 0.574852
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 3.14041e7 1.32152
$$893$$ −1.28133e7 −0.537690
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 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
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −7.48309e6 −0.302039 −0.151019 0.988531i $$-0.548256\pi$$
−0.151019 + 0.988531i $$0.548256\pi$$
$$908$$ 653744. 0.0263144
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 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 $$0.607047\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 8.14420e6 0.315516
$$923$$ −9.33733e6 −0.360760
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −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
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1.06887e7 −0.397718 −0.198859 0.980028i $$-0.563724\pi$$
−0.198859 + 0.980028i $$0.563724\pi$$
$$938$$ 2.34132e7 0.868870
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −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 $$-0.600199\pi$$
−0.309611 + 0.950863i $$0.600199\pi$$
$$948$$ 0 0
$$949$$ −1.10894e7 −0.399706
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −3.86611e7 −1.38255
$$953$$ 2.22259e7 0.792735 0.396367 0.918092i $$-0.370271\pi$$
0.396367 + 0.918092i $$0.370271\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −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
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −2.41551e7 −0.830696 −0.415348 0.909663i $$-0.636340\pi$$
−0.415348 + 0.909663i $$0.636340\pi$$
$$968$$ 1.66976e7 0.572752
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 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 $$-0.584533\pi$$
−0.262457 + 0.964944i $$0.584533\pi$$
$$978$$ 0 0
$$979$$ −1.50028e7 −0.500281
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −3.02530e6 −0.100113
$$983$$ −1.63420e7 −0.539412 −0.269706 0.962943i $$-0.586927\pi$$
−0.269706 + 0.962943i $$0.586927\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −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
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.29097e7 0.411320 0.205660 0.978624i $$-0.434066\pi$$
0.205660 + 0.978624i $$0.434066\pi$$
$$998$$ −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 225.6.a.f.1.1 1
3.2 odd 2 25.6.a.a.1.1 1
5.2 odd 4 225.6.b.e.199.2 2
5.3 odd 4 225.6.b.e.199.1 2
5.4 even 2 45.6.a.b.1.1 1
12.11 even 2 400.6.a.g.1.1 1
15.2 even 4 25.6.b.a.24.1 2
15.8 even 4 25.6.b.a.24.2 2
15.14 odd 2 5.6.a.a.1.1 1
20.19 odd 2 720.6.a.a.1.1 1
60.23 odd 4 400.6.c.j.49.1 2
60.47 odd 4 400.6.c.j.49.2 2
60.59 even 2 80.6.a.e.1.1 1
105.104 even 2 245.6.a.b.1.1 1
120.29 odd 2 320.6.a.j.1.1 1
120.59 even 2 320.6.a.g.1.1 1
165.164 even 2 605.6.a.a.1.1 1
195.194 odd 2 845.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 15.14 odd 2
25.6.a.a.1.1 1 3.2 odd 2
25.6.b.a.24.1 2 15.2 even 4
25.6.b.a.24.2 2 15.8 even 4
45.6.a.b.1.1 1 5.4 even 2
80.6.a.e.1.1 1 60.59 even 2
225.6.a.f.1.1 1 1.1 even 1 trivial
225.6.b.e.199.1 2 5.3 odd 4
225.6.b.e.199.2 2 5.2 odd 4
245.6.a.b.1.1 1 105.104 even 2
320.6.a.g.1.1 1 120.59 even 2
320.6.a.j.1.1 1 120.29 odd 2
400.6.a.g.1.1 1 12.11 even 2
400.6.c.j.49.1 2 60.23 odd 4
400.6.c.j.49.2 2 60.47 odd 4
605.6.a.a.1.1 1 165.164 even 2
720.6.a.a.1.1 1 20.19 odd 2
845.6.a.b.1.1 1 195.194 odd 2