# Properties

 Label 225.6.b.e.199.2 Level $225$ Weight $6$ Character 225.199 Analytic conductor $36.086$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.0863594579$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 225.199 Dual form 225.6.b.e.199.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} +28.0000 q^{4} -192.000i q^{7} +120.000i q^{8} +O(q^{10})$$ $$q+2.00000i q^{2} +28.0000 q^{4} -192.000i q^{7} +120.000i q^{8} +148.000 q^{11} +286.000i q^{13} +384.000 q^{14} +656.000 q^{16} -1678.00i q^{17} -1060.00 q^{19} +296.000i q^{22} -2976.00i q^{23} -572.000 q^{26} -5376.00i q^{28} -3410.00 q^{29} -2448.00 q^{31} +5152.00i q^{32} +3356.00 q^{34} -182.000i q^{37} -2120.00i q^{38} +9398.00 q^{41} -1244.00i q^{43} +4144.00 q^{44} +5952.00 q^{46} -12088.0i q^{47} -20057.0 q^{49} +8008.00i q^{52} -23846.0i q^{53} +23040.0 q^{56} -6820.00i q^{58} -20020.0 q^{59} +32302.0 q^{61} -4896.00i q^{62} +10688.0 q^{64} -60972.0i q^{67} -46984.0i q^{68} +32648.0 q^{71} -38774.0i q^{73} +364.000 q^{74} -29680.0 q^{76} -28416.0i q^{77} +33360.0 q^{79} +18796.0i q^{82} -16716.0i q^{83} +2488.00 q^{86} +17760.0i q^{88} +101370. q^{89} +54912.0 q^{91} -83328.0i q^{92} +24176.0 q^{94} +119038. i q^{97} -40114.0i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 56 q^{4} + O(q^{10})$$ $$2 q + 56 q^{4} + 296 q^{11} + 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1144 q^{26} - 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 18796 q^{41} + 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 46080 q^{56} - 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 65296 q^{71} + 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 4976 q^{86} + 202740 q^{89} + 109824 q^{91} + 48352 q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 0.353553i 0.984251 + 0.176777i $$0.0565670\pi$$
−0.984251 + 0.176777i $$0.943433\pi$$
$$3$$ 0 0
$$4$$ 28.0000 0.875000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 192.000i − 1.48100i −0.672054 0.740502i $$-0.734588\pi$$
0.672054 0.740502i $$-0.265412\pi$$
$$8$$ 120.000i 0.662913i
$$9$$ 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
$$13$$ 286.000i 0.469362i 0.972072 + 0.234681i $$0.0754045\pi$$
−0.972072 + 0.234681i $$0.924595\pi$$
$$14$$ 384.000 0.523614
$$15$$ 0 0
$$16$$ 656.000 0.640625
$$17$$ − 1678.00i − 1.40822i −0.710092 0.704109i $$-0.751347\pi$$
0.710092 0.704109i $$-0.248653\pi$$
$$18$$ 0 0
$$19$$ −1060.00 −0.673631 −0.336815 0.941571i $$-0.609350\pi$$
−0.336815 + 0.941571i $$0.609350\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 296.000i 0.130387i
$$23$$ − 2976.00i − 1.17304i −0.809934 0.586521i $$-0.800497\pi$$
0.809934 0.586521i $$-0.199503\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −572.000 −0.165944
$$27$$ 0 0
$$28$$ − 5376.00i − 1.29588i
$$29$$ −3410.00 −0.752938 −0.376469 0.926429i $$-0.622862\pi$$
−0.376469 + 0.926429i $$0.622862\pi$$
$$30$$ 0 0
$$31$$ −2448.00 −0.457517 −0.228758 0.973483i $$-0.573467\pi$$
−0.228758 + 0.973483i $$0.573467\pi$$
$$32$$ 5152.00i 0.889408i
$$33$$ 0 0
$$34$$ 3356.00 0.497880
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 182.000i − 0.0218558i −0.999940 0.0109279i $$-0.996521\pi$$
0.999940 0.0109279i $$-0.00347853\pi$$
$$38$$ − 2120.00i − 0.238164i
$$39$$ 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
$$43$$ − 1244.00i − 0.102600i −0.998683 0.0513002i $$-0.983663\pi$$
0.998683 0.0513002i $$-0.0163365\pi$$
$$44$$ 4144.00 0.322692
$$45$$ 0 0
$$46$$ 5952.00 0.414733
$$47$$ − 12088.0i − 0.798196i −0.916908 0.399098i $$-0.869323\pi$$
0.916908 0.399098i $$-0.130677\pi$$
$$48$$ 0 0
$$49$$ −20057.0 −1.19337
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 8008.00i 0.410691i
$$53$$ − 23846.0i − 1.16607i −0.812446 0.583037i $$-0.801864\pi$$
0.812446 0.583037i $$-0.198136\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 23040.0 0.981776
$$57$$ 0 0
$$58$$ − 6820.00i − 0.266204i
$$59$$ −20020.0 −0.748745 −0.374373 0.927278i $$-0.622142\pi$$
−0.374373 + 0.927278i $$0.622142\pi$$
$$60$$ 0 0
$$61$$ 32302.0 1.11149 0.555744 0.831353i $$-0.312433\pi$$
0.555744 + 0.831353i $$0.312433\pi$$
$$62$$ − 4896.00i − 0.161757i
$$63$$ 0 0
$$64$$ 10688.0 0.326172
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 60972.0i − 1.65937i −0.558231 0.829685i $$-0.688520\pi$$
0.558231 0.829685i $$-0.311480\pi$$
$$68$$ − 46984.0i − 1.23219i
$$69$$ 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
$$73$$ − 38774.0i − 0.851596i −0.904818 0.425798i $$-0.859993\pi$$
0.904818 0.425798i $$-0.140007\pi$$
$$74$$ 364.000 0.00772720
$$75$$ 0 0
$$76$$ −29680.0 −0.589427
$$77$$ − 28416.0i − 0.546180i
$$78$$ 0 0
$$79$$ 33360.0 0.601393 0.300696 0.953720i $$-0.402781\pi$$
0.300696 + 0.953720i $$0.402781\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 18796.0i 0.308696i
$$83$$ − 16716.0i − 0.266340i −0.991093 0.133170i $$-0.957484\pi$$
0.991093 0.133170i $$-0.0425157\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2488.00 0.0362747
$$87$$ 0 0
$$88$$ 17760.0i 0.244476i
$$89$$ 101370. 1.35655 0.678273 0.734810i $$-0.262729\pi$$
0.678273 + 0.734810i $$0.262729\pi$$
$$90$$ 0 0
$$91$$ 54912.0 0.695126
$$92$$ − 83328.0i − 1.02641i
$$93$$ 0 0
$$94$$ 24176.0 0.282205
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 119038.i 1.28457i 0.766468 + 0.642283i $$0.222013\pi$$
−0.766468 + 0.642283i $$0.777987\pi$$
$$98$$ − 40114.0i − 0.421921i
$$99$$ 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
$$103$$ − 19504.0i − 0.181147i −0.995890 0.0905734i $$-0.971130\pi$$
0.995890 0.0905734i $$-0.0288700\pi$$
$$104$$ −34320.0 −0.311146
$$105$$ 0 0
$$106$$ 47692.0 0.412269
$$107$$ 158292.i 1.33659i 0.743895 + 0.668297i $$0.232976\pi$$
−0.743895 + 0.668297i $$0.767024\pi$$
$$108$$ 0 0
$$109$$ −36830.0 −0.296917 −0.148459 0.988919i $$-0.547431\pi$$
−0.148459 + 0.988919i $$0.547431\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 125952.i − 0.948768i
$$113$$ − 11186.0i − 0.0824098i −0.999151 0.0412049i $$-0.986880\pi$$
0.999151 0.0412049i $$-0.0131196\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −95480.0 −0.658821
$$117$$ 0 0
$$118$$ − 40040.0i − 0.264721i
$$119$$ −322176. −2.08557
$$120$$ 0 0
$$121$$ −139147. −0.863993
$$122$$ 64604.0i 0.392970i
$$123$$ 0 0
$$124$$ −68544.0 −0.400327
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 70552.0i − 0.388150i −0.980987 0.194075i $$-0.937829\pi$$
0.980987 0.194075i $$-0.0621706\pi$$
$$128$$ 186240.i 1.00473i
$$129$$ 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.i 0.997650i
$$134$$ 121944. 0.586676
$$135$$ 0 0
$$136$$ 201360. 0.933525
$$137$$ − 144918.i − 0.659661i −0.944040 0.329831i $$-0.893008\pi$$
0.944040 0.329831i $$-0.106992\pi$$
$$138$$ 0 0
$$139$$ −112220. −0.492644 −0.246322 0.969188i $$-0.579222\pi$$
−0.246322 + 0.969188i $$0.579222\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 65296.0i 0.271748i
$$143$$ 42328.0i 0.173096i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 77548.0 0.301085
$$147$$ 0 0
$$148$$ − 5096.00i − 0.0191238i
$$149$$ 403750. 1.48986 0.744932 0.667140i $$-0.232482\pi$$
0.744932 + 0.667140i $$0.232482\pi$$
$$150$$ 0 0
$$151$$ −446648. −1.59413 −0.797064 0.603895i $$-0.793615\pi$$
−0.797064 + 0.603895i $$0.793615\pi$$
$$152$$ − 127200.i − 0.446558i
$$153$$ 0 0
$$154$$ 56832.0 0.193104
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 262258.i 0.849141i 0.905395 + 0.424570i $$0.139575\pi$$
−0.905395 + 0.424570i $$0.860425\pi$$
$$158$$ 66720.0i 0.212625i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −571392. −1.73728
$$162$$ 0 0
$$163$$ − 154564.i − 0.455658i −0.973701 0.227829i $$-0.926837\pi$$
0.973701 0.227829i $$-0.0731628\pi$$
$$164$$ 263144. 0.763983
$$165$$ 0 0
$$166$$ 33432.0 0.0941656
$$167$$ 396672.i 1.10063i 0.834958 + 0.550314i $$0.185492\pi$$
−0.834958 + 0.550314i $$0.814508\pi$$
$$168$$ 0 0
$$169$$ 289497. 0.779700
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 34832.0i − 0.0897754i
$$173$$ 573474.i 1.45680i 0.685155 + 0.728398i $$0.259735\pi$$
−0.685155 + 0.728398i $$0.740265\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 97088.0 0.236257
$$177$$ 0 0
$$178$$ 202740.i 0.479611i
$$179$$ −594460. −1.38672 −0.693362 0.720589i $$-0.743871\pi$$
−0.693362 + 0.720589i $$0.743871\pi$$
$$180$$ 0 0
$$181$$ −107098. −0.242988 −0.121494 0.992592i $$-0.538769\pi$$
−0.121494 + 0.992592i $$0.538769\pi$$
$$182$$ 109824.i 0.245764i
$$183$$ 0 0
$$184$$ 357120. 0.777624
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 248344.i − 0.519337i
$$188$$ − 338464.i − 0.698422i
$$189$$ 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
$$193$$ 52706.0i 0.101851i 0.998702 + 0.0509257i $$0.0162172\pi$$
−0.998702 + 0.0509257i $$0.983783\pi$$
$$194$$ −238076. −0.454163
$$195$$ 0 0
$$196$$ −561596. −1.04420
$$197$$ 455862.i 0.836889i 0.908242 + 0.418444i $$0.137425\pi$$
−0.908242 + 0.418444i $$0.862575\pi$$
$$198$$ 0 0
$$199$$ −865000. −1.54840 −0.774200 0.632940i $$-0.781848\pi$$
−0.774200 + 0.632940i $$0.781848\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 179796.i 0.310028i
$$203$$ 654720.i 1.11510i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 39008.0 0.0640451
$$207$$ 0 0
$$208$$ 187616.i 0.300685i
$$209$$ −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.i − 1.02031i
$$213$$ 0 0
$$214$$ −316584. −0.472557
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 470016.i 0.677584i
$$218$$ − 73660.0i − 0.104976i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 479908. 0.660963
$$222$$ 0 0
$$223$$ 1.12158e6i 1.51031i 0.655545 + 0.755156i $$0.272439\pi$$
−0.655545 + 0.755156i $$0.727561\pi$$
$$224$$ 989184. 1.31722
$$225$$ 0 0
$$226$$ 22372.0 0.0291363
$$227$$ − 23348.0i − 0.0300736i −0.999887 0.0150368i $$-0.995213\pi$$
0.999887 0.0150368i $$-0.00478654\pi$$
$$228$$ 0 0
$$229$$ 596010. 0.751043 0.375522 0.926814i $$-0.377464\pi$$
0.375522 + 0.926814i $$0.377464\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 409200.i − 0.499132i
$$233$$ 485334.i 0.585667i 0.956163 + 0.292834i $$0.0945982\pi$$
−0.956163 + 0.292834i $$0.905402\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −560560. −0.655152
$$237$$ 0 0
$$238$$ − 644352.i − 0.737362i
$$239$$ −48880.0 −0.0553524 −0.0276762 0.999617i $$-0.508811\pi$$
−0.0276762 + 0.999617i $$0.508811\pi$$
$$240$$ 0 0
$$241$$ −110798. −0.122882 −0.0614411 0.998111i $$-0.519570\pi$$
−0.0614411 + 0.998111i $$0.519570\pi$$
$$242$$ − 278294.i − 0.305468i
$$243$$ 0 0
$$244$$ 904456. 0.972552
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 303160.i − 0.316176i
$$248$$ − 293760.i − 0.303294i
$$249$$ 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.i − 0.432607i
$$254$$ 141104. 0.137232
$$255$$ 0 0
$$256$$ −30464.0 −0.0290527
$$257$$ 1.30624e6i 1.23365i 0.787102 + 0.616823i $$0.211581\pi$$
−0.787102 + 0.616823i $$0.788419\pi$$
$$258$$ 0 0
$$259$$ −34944.0 −0.0323685
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 152904.i − 0.137615i
$$263$$ − 2.12834e6i − 1.89736i −0.316231 0.948682i $$-0.602417\pi$$
0.316231 0.948682i $$-0.397583\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −407040. −0.352722
$$267$$ 0 0
$$268$$ − 1.70722e6i − 1.45195i
$$269$$ −1.44109e6 −1.21426 −0.607128 0.794604i $$-0.707679\pi$$
−0.607128 + 0.794604i $$0.707679\pi$$
$$270$$ 0 0
$$271$$ −93248.0 −0.0771288 −0.0385644 0.999256i $$-0.512278\pi$$
−0.0385644 + 0.999256i $$0.512278\pi$$
$$272$$ − 1.10077e6i − 0.902139i
$$273$$ 0 0
$$274$$ 289836. 0.233225
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 110298.i 0.0863711i 0.999067 + 0.0431855i $$0.0137507\pi$$
−0.999067 + 0.0431855i $$0.986249\pi$$
$$278$$ − 224440.i − 0.174176i
$$279$$ 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.i − 0.246332i −0.992386 0.123166i $$-0.960695\pi$$
0.992386 0.123166i $$-0.0393047\pi$$
$$284$$ 914144. 0.672541
$$285$$ 0 0
$$286$$ −84656.0 −0.0611988
$$287$$ − 1.80442e6i − 1.29310i
$$288$$ 0 0
$$289$$ −1.39583e6 −0.983076
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 1.08567e6i − 0.745146i
$$293$$ − 2.19481e6i − 1.49358i −0.665063 0.746788i $$-0.731595\pi$$
0.665063 0.746788i $$-0.268405\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 21840.0 0.0144885
$$297$$ 0 0
$$298$$ 807500.i 0.526747i
$$299$$ 851136. 0.550581
$$300$$ 0 0
$$301$$ −238848. −0.151952
$$302$$ − 893296.i − 0.563609i
$$303$$ 0 0
$$304$$ −695360. −0.431545
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.37751e6i 1.43971i 0.694123 + 0.719857i $$0.255793\pi$$
−0.694123 + 0.719857i $$0.744207\pi$$
$$308$$ − 795648.i − 0.477908i
$$309$$ 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
$$313$$ − 1.42941e6i − 0.824702i −0.911025 0.412351i $$-0.864708\pi$$
0.911025 0.412351i $$-0.135292\pi$$
$$314$$ −524516. −0.300217
$$315$$ 0 0
$$316$$ 934080. 0.526219
$$317$$ 2.12462e6i 1.18750i 0.804650 + 0.593750i $$0.202353\pi$$
−0.804650 + 0.593750i $$0.797647\pi$$
$$318$$ 0 0
$$319$$ −504680. −0.277677
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 1.14278e6i − 0.614221i
$$323$$ 1.77868e6i 0.948618i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 309128. 0.161100
$$327$$ 0 0
$$328$$ 1.12776e6i 0.578805i
$$329$$ −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.i − 0.233048i
$$333$$ 0 0
$$334$$ −793344. −0.389131
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 2.40008e6i − 1.15120i −0.817731 0.575601i $$-0.804768\pi$$
0.817731 0.575601i $$-0.195232\pi$$
$$338$$ 578994.i 0.275665i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −362304. −0.168728
$$342$$ 0 0
$$343$$ 624000.i 0.286384i
$$344$$ 149280. 0.0680151
$$345$$ 0 0
$$346$$ −1.14695e6 −0.515055
$$347$$ 1.77741e6i 0.792436i 0.918156 + 0.396218i $$0.129678\pi$$
−0.918156 + 0.396218i $$0.870322\pi$$
$$348$$ 0 0
$$349$$ 2.14805e6 0.944019 0.472010 0.881593i $$-0.343529\pi$$
0.472010 + 0.881593i $$0.343529\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 762496.i 0.328005i
$$353$$ 661854.i 0.282700i 0.989960 + 0.141350i $$0.0451443\pi$$
−0.989960 + 0.141350i $$0.954856\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2.83836e6 1.18698
$$357$$ 0 0
$$358$$ − 1.18892e6i − 0.490281i
$$359$$ −259320. −0.106194 −0.0530970 0.998589i $$-0.516909\pi$$
−0.0530970 + 0.998589i $$0.516909\pi$$
$$360$$ 0 0
$$361$$ −1.35250e6 −0.546222
$$362$$ − 214196.i − 0.0859093i
$$363$$ 0 0
$$364$$ 1.53754e6 0.608236
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.49993e6i 0.581307i 0.956828 + 0.290653i $$0.0938726\pi$$
−0.956828 + 0.290653i $$0.906127\pi$$
$$368$$ − 1.95226e6i − 0.751480i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −4.57843e6 −1.72696
$$372$$ 0 0
$$373$$ − 2.23807e6i − 0.832918i −0.909154 0.416459i $$-0.863271\pi$$
0.909154 0.416459i $$-0.136729\pi$$
$$374$$ 496688. 0.183614
$$375$$ 0 0
$$376$$ 1.45056e6 0.529135
$$377$$ − 975260.i − 0.353400i
$$378$$ 0 0
$$379$$ −3.15934e6 −1.12979 −0.564896 0.825162i $$-0.691084\pi$$
−0.564896 + 0.825162i $$0.691084\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 939104.i − 0.329272i
$$383$$ − 342216.i − 0.119207i −0.998222 0.0596037i $$-0.981016\pi$$
0.998222 0.0596037i $$-0.0189837\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −105412. −0.0360099
$$387$$ 0 0
$$388$$ 3.33306e6i 1.12399i
$$389$$ 88470.0 0.0296430 0.0148215 0.999890i $$-0.495282\pi$$
0.0148215 + 0.999890i $$0.495282\pi$$
$$390$$ 0 0
$$391$$ −4.99373e6 −1.65190
$$392$$ − 2.40684e6i − 0.791101i
$$393$$ 0 0
$$394$$ −911724. −0.295885
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5.45674e6i 1.73763i 0.495138 + 0.868814i $$0.335117\pi$$
−0.495138 + 0.868814i $$0.664883\pi$$
$$398$$ − 1.73000e6i − 0.547442i
$$399$$ 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.i − 0.214741i
$$404$$ 2.51714e6 0.767281
$$405$$ 0 0
$$406$$ −1.30944e6 −0.394249
$$407$$ − 26936.0i − 0.00806022i
$$408$$ 0 0
$$409$$ 2.71207e6 0.801664 0.400832 0.916151i $$-0.368721\pi$$
0.400832 + 0.916151i $$0.368721\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 546112.i − 0.158503i
$$413$$ 3.84384e6i 1.10889i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.47347e6 −0.417454
$$417$$ 0 0
$$418$$ − 313760.i − 0.0878328i
$$419$$ 3.71746e6 1.03445 0.517227 0.855848i $$-0.326964\pi$$
0.517227 + 0.855848i $$0.326964\pi$$
$$420$$ 0 0
$$421$$ 3.55250e6 0.976853 0.488426 0.872605i $$-0.337571\pi$$
0.488426 + 0.872605i $$0.337571\pi$$
$$422$$ 2.21130e6i 0.604460i
$$423$$ 0 0
$$424$$ 2.86152e6 0.773005
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 6.20198e6i − 1.64612i
$$428$$ 4.43218e6i 1.16952i
$$429$$ 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
$$433$$ 7.26287e6i 1.86161i 0.365518 + 0.930804i $$0.380892\pi$$
−0.365518 + 0.930804i $$0.619108\pi$$
$$434$$ −940032. −0.239562
$$435$$ 0 0
$$436$$ −1.03124e6 −0.259803
$$437$$ 3.15456e6i 0.790197i
$$438$$ 0 0
$$439$$ 5.41028e6 1.33986 0.669928 0.742426i $$-0.266325\pi$$
0.669928 + 0.742426i $$0.266325\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 959816.i 0.233686i
$$443$$ 6.51524e6i 1.57733i 0.614826 + 0.788663i $$0.289226\pi$$
−0.614826 + 0.788663i $$0.710774\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −2.24315e6 −0.533976
$$447$$ 0 0
$$448$$ − 2.05210e6i − 0.483062i
$$449$$ −509950. −0.119375 −0.0596873 0.998217i $$-0.519010\pi$$
−0.0596873 + 0.998217i $$0.519010\pi$$
$$450$$ 0 0
$$451$$ 1.39090e6 0.322000
$$452$$ − 313208.i − 0.0721085i
$$453$$ 0 0
$$454$$ 46696.0 0.0106326
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 1.22084e6i − 0.273444i −0.990609 0.136722i $$-0.956343\pi$$
0.990609 0.136722i $$-0.0436568\pi$$
$$458$$ 1.19202e6i 0.265534i
$$459$$ 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
$$463$$ 2.02294e6i 0.438561i 0.975662 + 0.219280i $$0.0703709\pi$$
−0.975662 + 0.219280i $$0.929629\pi$$
$$464$$ −2.23696e6 −0.482351
$$465$$ 0 0
$$466$$ −970668. −0.207065
$$467$$ 3.25097e6i 0.689797i 0.938640 + 0.344898i $$0.112087\pi$$
−0.938640 + 0.344898i $$0.887913\pi$$
$$468$$ 0 0
$$469$$ −1.17066e7 −2.45753
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 2.40240e6i − 0.496353i
$$473$$ − 184112.i − 0.0378381i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −9.02093e6 −1.82488
$$477$$ 0 0
$$478$$ − 97760.0i − 0.0195700i
$$479$$ −3.27936e6 −0.653056 −0.326528 0.945188i $$-0.605879\pi$$
−0.326528 + 0.945188i $$0.605879\pi$$
$$480$$ 0 0
$$481$$ 52052.0 0.0102583
$$482$$ − 221596.i − 0.0434455i
$$483$$ 0 0
$$484$$ −3.89612e6 −0.755994
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 8.53197e6i 1.63015i 0.579357 + 0.815074i $$0.303304\pi$$
−0.579357 + 0.815074i $$0.696696\pi$$
$$488$$ 3.87624e6i 0.736819i
$$489$$ 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
$$493$$ 5.72198e6i 1.06030i
$$494$$ 606320. 0.111785
$$495$$ 0 0
$$496$$ −1.60589e6 −0.293097
$$497$$ − 6.26842e6i − 1.13833i
$$498$$ 0 0
$$499$$ 6.49190e6 1.16713 0.583567 0.812065i $$-0.301657\pi$$
0.583567 + 0.812065i $$0.301657\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 3.28750e6i 0.582245i
$$503$$ − 8.61770e6i − 1.51870i −0.650684 0.759349i $$-0.725518\pi$$
0.650684 0.759349i $$-0.274482\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 880896. 0.152950
$$507$$ 0 0
$$508$$ − 1.97546e6i − 0.339632i
$$509$$ 2.67323e6 0.457343 0.228671 0.973504i $$-0.426562\pi$$
0.228671 + 0.973504i $$0.426562\pi$$
$$510$$ 0 0
$$511$$ −7.44461e6 −1.26122
$$512$$ 5.89875e6i 0.994455i
$$513$$ 0 0
$$514$$ −2.61248e6 −0.436160
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 1.78902e6i − 0.294367i
$$518$$ − 69888.0i − 0.0114440i
$$519$$ 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
$$523$$ − 6.89452e6i − 1.10217i −0.834448 0.551087i $$-0.814213\pi$$
0.834448 0.551087i $$-0.185787\pi$$
$$524$$ −2.14066e6 −0.340580
$$525$$ 0 0
$$526$$ 4.25667e6 0.670820
$$527$$ 4.10774e6i 0.644283i
$$528$$ 0 0
$$529$$ −2.42023e6 −0.376026
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 5.69856e6i 0.872943i
$$533$$ 2.68783e6i 0.409811i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 7.31664e6 1.10002
$$537$$ 0 0
$$538$$ − 2.88218e6i − 0.429304i
$$539$$ −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.i − 0.0272691i
$$543$$ 0 0
$$544$$ 8.64506e6 1.25248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 1.26544e7i − 1.80831i −0.427201 0.904157i $$-0.640500\pi$$
0.427201 0.904157i $$-0.359500\pi$$
$$548$$ − 4.05770e6i − 0.577204i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.61460e6 0.507202
$$552$$ 0 0
$$553$$ − 6.40512e6i − 0.890665i
$$554$$ −220596. −0.0305368
$$555$$ 0 0
$$556$$ −3.14216e6 −0.431064
$$557$$ − 7.07786e6i − 0.966638i −0.875444 0.483319i $$-0.839431\pi$$
0.875444 0.483319i $$-0.160569\pi$$
$$558$$ 0 0
$$559$$ 355784. 0.0481567
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 384396.i 0.0513379i
$$563$$ − 846636.i − 0.112571i −0.998415 0.0562854i $$-0.982074\pi$$
0.998415 0.0562854i $$-0.0179257\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 663768. 0.0870914
$$567$$ 0 0
$$568$$ 3.91776e6i 0.509527i
$$569$$ 4.96041e6 0.642299 0.321149 0.947029i $$-0.395931\pi$$
0.321149 + 0.947029i $$0.395931\pi$$
$$570$$ 0 0
$$571$$ 8.96505e6 1.15070 0.575351 0.817907i $$-0.304866\pi$$
0.575351 + 0.817907i $$0.304866\pi$$
$$572$$ 1.18518e6i 0.151459i
$$573$$ 0 0
$$574$$ 3.60883e6 0.457180
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.86080e6i 0.357724i 0.983874 + 0.178862i $$0.0572415\pi$$
−0.983874 + 0.178862i $$0.942758\pi$$
$$578$$ − 2.79165e6i − 0.347570i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.20947e6 −0.394451
$$582$$ 0 0
$$583$$ − 3.52921e6i − 0.430037i
$$584$$ 4.65288e6 0.564534
$$585$$ 0 0
$$586$$ 4.38961e6 0.528059
$$587$$ − 6.74027e6i − 0.807387i −0.914894 0.403694i $$-0.867726\pi$$
0.914894 0.403694i $$-0.132274\pi$$
$$588$$ 0 0
$$589$$ 2.59488e6 0.308197
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 119392.i − 0.0140014i
$$593$$ 1.78609e6i 0.208578i 0.994547 + 0.104289i $$0.0332566\pi$$
−0.994547 + 0.104289i $$0.966743\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1.13050e7 1.30363
$$597$$ 0 0
$$598$$ 1.70227e6i 0.194660i
$$599$$ 4.94620e6 0.563254 0.281627 0.959524i $$-0.409126\pi$$
0.281627 + 0.959524i $$0.409126\pi$$
$$600$$ 0 0
$$601$$ −4.58100e6 −0.517337 −0.258669 0.965966i $$-0.583284\pi$$
−0.258669 + 0.965966i $$0.583284\pi$$
$$602$$ − 477696.i − 0.0537230i
$$603$$ 0 0
$$604$$ −1.25061e7 −1.39486
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 7.07999e6i − 0.779940i −0.920828 0.389970i $$-0.872485\pi$$
0.920828 0.389970i $$-0.127515\pi$$
$$608$$ − 5.46112e6i − 0.599132i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3.45717e6 0.374643
$$612$$ 0 0
$$613$$ 5.09609e6i 0.547754i 0.961765 + 0.273877i $$0.0883061\pi$$
−0.961765 + 0.273877i $$0.911694\pi$$
$$614$$ −4.75502e6 −0.509016
$$615$$ 0 0
$$616$$ 3.40992e6 0.362070
$$617$$ − 1.30003e7i − 1.37480i −0.726279 0.687400i $$-0.758752\pi$$
0.726279 0.687400i $$-0.241248\pi$$
$$618$$ 0 0
$$619$$ −4.84406e6 −0.508139 −0.254070 0.967186i $$-0.581769\pi$$
−0.254070 + 0.967186i $$0.581769\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 4.74610e6i 0.491882i
$$623$$ − 1.94630e7i − 2.00905i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 2.85883e6 0.291576
$$627$$ 0 0
$$628$$ 7.34322e6i 0.742998i
$$629$$ −305396. −0.0307777
$$630$$ 0 0
$$631$$ 6.22775e6 0.622670 0.311335 0.950300i $$-0.399224\pi$$
0.311335 + 0.950300i $$0.399224\pi$$
$$632$$ 4.00320e6i 0.398671i
$$633$$ 0 0
$$634$$ −4.24924e6 −0.419845
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 5.73630e6i − 0.560123i
$$638$$ − 1.00936e6i − 0.0981735i
$$639$$ 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
$$643$$ − 1.74382e7i − 1.66332i −0.555287 0.831659i $$-0.687391\pi$$
0.555287 0.831659i $$-0.312609\pi$$
$$644$$ −1.59990e7 −1.52012
$$645$$ 0 0
$$646$$ −3.55736e6 −0.335387
$$647$$ − 4.25469e6i − 0.399583i −0.979838 0.199792i $$-0.935974\pi$$
0.979838 0.199792i $$-0.0640265\pi$$
$$648$$ 0 0
$$649$$ −2.96296e6 −0.276130
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 4.32779e6i − 0.398701i
$$653$$ − 3.01085e6i − 0.276316i −0.990410 0.138158i $$-0.955882\pi$$
0.990410 0.138158i $$-0.0441181\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.16509e6 0.559345
$$657$$ 0 0
$$658$$ − 4.64179e6i − 0.417947i
$$659$$ −8.11462e6 −0.727871 −0.363936 0.931424i $$-0.618567\pi$$
−0.363936 + 0.931424i $$0.618567\pi$$
$$660$$ 0 0
$$661$$ 2.47370e6 0.220213 0.110107 0.993920i $$-0.464881\pi$$
0.110107 + 0.993920i $$0.464881\pi$$
$$662$$ 6.19970e6i 0.549827i
$$663$$ 0 0
$$664$$ 2.00592e6 0.176560
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.01482e7i 0.883228i
$$668$$ 1.11068e7i 0.963049i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.78070e6 0.409907
$$672$$ 0 0
$$673$$ 5.77063e6i 0.491117i 0.969382 + 0.245559i $$0.0789714\pi$$
−0.969382 + 0.245559i $$0.921029\pi$$
$$674$$ 4.80016e6 0.407011
$$675$$ 0 0
$$676$$ 8.10592e6 0.682237
$$677$$ 1.67197e7i 1.40203i 0.713147 + 0.701014i $$0.247269\pi$$
−0.713147 + 0.701014i $$0.752731\pi$$
$$678$$ 0 0
$$679$$ 2.28553e7 1.90245
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 724608.i − 0.0596544i
$$683$$ − 7.14532e6i − 0.586097i −0.956098 0.293049i $$-0.905330\pi$$
0.956098 0.293049i $$-0.0946698\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.24800e6 −0.101252
$$687$$ 0 0
$$688$$ − 816064.i − 0.0657284i
$$689$$ 6.81996e6 0.547310
$$690$$ 0 0
$$691$$ −8.78395e6 −0.699833 −0.349917 0.936781i $$-0.613790\pi$$
−0.349917 + 0.936781i $$0.613790\pi$$
$$692$$ 1.60573e7i 1.27470i
$$693$$ 0 0
$$694$$ −3.55482e6 −0.280169
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 1.57698e7i − 1.22955i
$$698$$ 4.29610e6i 0.333761i
$$699$$ 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.i 0.0147228i
$$704$$ 1.58182e6 0.120289
$$705$$ 0 0
$$706$$ −1.32371e6 −0.0999495
$$707$$ − 1.72604e7i − 1.29868i
$$708$$ 0 0
$$709$$ 1.91354e7 1.42962 0.714811 0.699318i $$-0.246513\pi$$
0.714811 + 0.699318i $$0.246513\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 1.21644e7i 0.899271i
$$713$$ 7.28525e6i 0.536686i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −1.66449e7 −1.21338
$$717$$ 0 0
$$718$$ − 518640.i − 0.0375452i
$$719$$ 1.02934e7 0.742566 0.371283 0.928520i $$-0.378918\pi$$
0.371283 + 0.928520i $$0.378918\pi$$
$$720$$ 0 0
$$721$$ −3.74477e6 −0.268279
$$722$$ − 2.70500e6i − 0.193119i
$$723$$ 0 0
$$724$$ −2.99874e6 −0.212615
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.93264e7i 1.35618i 0.734981 + 0.678088i $$0.237191\pi$$
−0.734981 + 0.678088i $$0.762809\pi$$
$$728$$ 6.58944e6i 0.460808i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.08743e6 −0.144484
$$732$$ 0 0
$$733$$ 5.26197e6i 0.361733i 0.983508 + 0.180866i $$0.0578902\pi$$
−0.983508 + 0.180866i $$0.942110\pi$$
$$734$$ −2.99986e6 −0.205523
$$735$$ 0 0
$$736$$ 1.53324e7 1.04331
$$737$$ − 9.02386e6i − 0.611961i
$$738$$ 0 0
$$739$$ −2.82944e7 −1.90585 −0.952927 0.303199i $$-0.901945\pi$$
−0.952927 + 0.303199i $$0.901945\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 9.15686e6i − 0.610572i
$$743$$ − 2.09863e7i − 1.39464i −0.716759 0.697321i $$-0.754375\pi$$
0.716759 0.697321i $$-0.245625\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.47615e6 0.294481
$$747$$ 0 0
$$748$$ − 6.95363e6i − 0.454420i
$$749$$ 3.03921e7 1.97950
$$750$$ 0 0
$$751$$ −1.89668e7 −1.22714 −0.613572 0.789639i $$-0.710268\pi$$
−0.613572 + 0.789639i $$0.710268\pi$$
$$752$$ − 7.92973e6i − 0.511345i
$$753$$ 0 0
$$754$$ 1.95052e6 0.124946
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.08257e7i 0.686617i 0.939223 + 0.343309i $$0.111548\pi$$
−0.939223 + 0.343309i $$0.888452\pi$$
$$758$$ − 6.31868e6i − 0.399442i
$$759$$ 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
$$763$$ 7.07136e6i 0.439736i
$$764$$ −1.31475e7 −0.814908
$$765$$ 0 0
$$766$$ 684432. 0.0421462
$$767$$ − 5.72572e6i − 0.351432i
$$768$$ 0 0
$$769$$ 1.57826e7 0.962415 0.481208 0.876607i $$-0.340198\pi$$
0.481208 + 0.876607i $$0.340198\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.47577e6i 0.0891199i
$$773$$ 2.44049e7i 1.46902i 0.678598 + 0.734510i $$0.262588\pi$$
−0.678598 + 0.734510i $$0.737412\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −1.42846e7 −0.851555
$$777$$ 0 0
$$778$$ 176940.i 0.0104804i
$$779$$ −9.96188e6 −0.588163
$$780$$ 0 0
$$781$$ 4.83190e6 0.283459
$$782$$ − 9.98746e6i − 0.584034i
$$783$$ 0 0
$$784$$ −1.31574e7 −0.764504
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 3.37607e7i − 1.94301i −0.237019 0.971505i $$-0.576170\pi$$
0.237019 0.971505i $$-0.423830\pi$$
$$788$$ 1.27641e7i 0.732278i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −2.14771e6 −0.122049
$$792$$ 0 0
$$793$$ 9.23837e6i 0.521690i
$$794$$ −1.09135e7 −0.614344
$$795$$ 0 0
$$796$$ −2.42200e7 −1.35485
$$797$$ 2.19885e7i 1.22617i 0.790019 + 0.613083i $$0.210071\pi$$
−0.790019 + 0.613083i $$0.789929\pi$$
$$798$$ 0 0
$$799$$ −2.02837e7 −1.12403
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 8.09360e6i − 0.444330i
$$803$$ − 5.73855e6i − 0.314061i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 1.40026e6 0.0759224
$$807$$ 0 0
$$808$$ 1.07878e7i 0.581303i
$$809$$ −2.93597e7 −1.57717 −0.788587 0.614923i $$-0.789187\pi$$
−0.788587 + 0.614923i $$0.789187\pi$$
$$810$$ 0 0
$$811$$ 3.17703e7 1.69617 0.848083 0.529863i $$-0.177757\pi$$
0.848083 + 0.529863i $$0.177757\pi$$
$$812$$ 1.83322e7i 0.975716i
$$813$$ 0 0
$$814$$ 53872.0 0.00284972
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.31864e6i 0.0691148i
$$818$$ 5.42414e6i 0.283431i
$$819$$ 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
$$823$$ − 1.25866e7i − 0.647753i −0.946099 0.323877i $$-0.895014\pi$$
0.946099 0.323877i $$-0.104986\pi$$
$$824$$ 2.34048e6 0.120084
$$825$$ 0 0
$$826$$ −7.68768e6 −0.392053
$$827$$ − 8.72355e6i − 0.443537i −0.975099 0.221768i $$-0.928817\pi$$
0.975099 0.221768i $$-0.0711828\pi$$
$$828$$ 0 0
$$829$$ 1.06178e7 0.536597 0.268299 0.963336i $$-0.413539\pi$$
0.268299 + 0.963336i $$0.413539\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 3.05677e6i 0.153093i
$$833$$ 3.36556e7i 1.68053i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −4.39264e6 −0.217375
$$837$$ 0 0
$$838$$ 7.43492e6i 0.365735i
$$839$$ 1.67765e7 0.822805 0.411403 0.911454i $$-0.365039\pi$$
0.411403 + 0.911454i $$0.365039\pi$$
$$840$$ 0 0
$$841$$ −8.88305e6 −0.433084
$$842$$ 7.10500e6i 0.345370i
$$843$$ 0 0
$$844$$ 3.09583e7 1.49596
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.67162e7i 1.27958i
$$848$$ − 1.56430e7i − 0.747016i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −541632. −0.0256378
$$852$$ 0 0
$$853$$ − 2.20186e7i − 1.03613i −0.855340 0.518067i $$-0.826652\pi$$
0.855340 0.518067i $$-0.173348\pi$$
$$854$$ 1.24040e7 0.581991
$$855$$ 0 0
$$856$$ −1.89950e7 −0.886045
$$857$$ 3.16676e7i 1.47287i 0.676510 + 0.736434i $$0.263492\pi$$
−0.676510 + 0.736434i $$0.736508\pi$$
$$858$$ 0 0
$$859$$ −1.58064e7 −0.730886 −0.365443 0.930834i $$-0.619082\pi$$
−0.365443 + 0.930834i $$0.619082\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8.12410e6i 0.372398i
$$863$$ 1.44287e7i 0.659476i 0.944072 + 0.329738i $$0.106960\pi$$
−0.944072 + 0.329738i $$0.893040\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −1.45257e7 −0.658178
$$867$$ 0 0
$$868$$ 1.31604e7i 0.592886i
$$869$$ 4.93728e6 0.221788
$$870$$ 0 0
$$871$$ 1.74380e7 0.778845
$$872$$ − 4.41960e6i − 0.196830i
$$873$$ 0 0
$$874$$ −6.30912e6 −0.279377
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 247902.i − 0.0108838i −0.999985 0.00544191i $$-0.998268\pi$$
0.999985 0.00544191i $$-0.00173222\pi$$
$$878$$ 1.08206e7i 0.473711i
$$879$$ 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
$$883$$ 4.18015e7i 1.80422i 0.431503 + 0.902112i $$0.357984\pi$$
−0.431503 + 0.902112i $$0.642016\pi$$
$$884$$ 1.34374e7 0.578343
$$885$$ 0 0
$$886$$ −1.30305e7 −0.557669
$$887$$ − 2.10476e7i − 0.898241i −0.893471 0.449120i $$-0.851737\pi$$
0.893471 0.449120i $$-0.148263\pi$$
$$888$$ 0 0
$$889$$ −1.35460e7 −0.574852
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 3.14041e7i 1.32152i
$$893$$ 1.28133e7i 0.537690i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 3.57581e7 1.48800
$$897$$ 0 0
$$898$$ − 1.01990e6i − 0.0422053i
$$899$$ 8.34768e6 0.344482
$$900$$ 0 0
$$901$$ −4.00136e7 −1.64208
$$902$$ 2.78181e6i 0.113844i
$$903$$ 0 0
$$904$$ 1.34232e6 0.0546305
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 7.48309e6i − 0.302039i −0.988531 0.151019i $$-0.951744\pi$$
0.988531 0.151019i $$-0.0482556\pi$$
$$908$$ − 653744.i − 0.0263144i
$$909$$ 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
$$913$$ − 2.47397e6i − 0.0982239i
$$914$$ 2.44168e6 0.0966772
$$915$$ 0 0
$$916$$ 1.66883e7 0.657163
$$917$$ 1.46788e7i 0.576457i
$$918$$ 0 0
$$919$$ 1.68976e7 0.659990 0.329995 0.943983i $$-0.392953\pi$$
0.329995 + 0.943983i $$0.392953\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 8.14420e6i 0.315516i
$$923$$ 9.33733e6i 0.360760i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.04587e6 −0.155055
$$927$$ 0 0
$$928$$ − 1.75683e7i − 0.669669i
$$929$$ −1.28653e7 −0.489081 −0.244541 0.969639i $$-0.578637\pi$$
−0.244541 + 0.969639i $$0.578637\pi$$
$$930$$ 0 0
$$931$$ 2.12604e7 0.803892
$$932$$ 1.35894e7i 0.512459i
$$933$$ 0 0
$$934$$ −6.50194e6 −0.243880
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 1.06887e7i − 0.397718i −0.980028 0.198859i $$-0.936276\pi$$
0.980028 0.198859i $$-0.0637236\pi$$
$$938$$ − 2.34132e7i − 0.868870i
$$939$$ 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
$$943$$ − 2.79684e7i − 1.02421i
$$944$$ −1.31331e7 −0.479665
$$945$$ 0 0
$$946$$ 368224. 0.0133778
$$947$$ − 1.70892e7i − 0.619222i −0.950863 0.309611i $$-0.899801\pi$$
0.950863 0.309611i $$-0.100199\pi$$
$$948$$ 0 0
$$949$$ 1.10894e7 0.399706
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 3.86611e7i − 1.38255i
$$953$$ − 2.22259e7i − 0.792735i −0.918092 0.396367i $$-0.870271\pi$$
0.918092 0.396367i $$-0.129729\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1.36864e6 −0.0484333
$$957$$ 0 0
$$958$$ − 6.55872e6i − 0.230890i
$$959$$ −2.78243e7 −0.976961
$$960$$ 0 0
$$961$$ −2.26364e7 −0.790678
$$962$$ 104104.i 0.00362685i
$$963$$ 0 0
$$964$$ −3.10234e6 −0.107522
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 2.41551e7i − 0.830696i −0.909663 0.415348i $$-0.863660\pi$$
0.909663 0.415348i $$-0.136340\pi$$
$$968$$ − 1.66976e7i − 0.572752i
$$969$$ 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
$$973$$ 2.15462e7i 0.729608i
$$974$$ −1.70639e7 −0.576344
$$975$$ 0 0
$$976$$ 2.11901e7 0.712047
$$977$$ − 1.56612e7i − 0.524915i −0.964944 0.262457i $$-0.915467\pi$$
0.964944 0.262457i $$-0.0845329\pi$$
$$978$$ 0 0
$$979$$ 1.50028e7 0.500281
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 3.02530e6i − 0.100113i
$$983$$ 1.63420e7i 0.539412i 0.962943 + 0.269706i $$0.0869266\pi$$
−0.962943 + 0.269706i $$0.913073\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −1.14440e7 −0.374873
$$987$$ 0 0
$$988$$ − 8.48848e6i − 0.276654i
$$989$$ −3.70214e6 −0.120355
$$990$$ 0 0
$$991$$ 1.37576e7 0.444997 0.222498 0.974933i $$-0.428579\pi$$
0.222498 + 0.974933i $$0.428579\pi$$
$$992$$ − 1.26121e7i − 0.406919i
$$993$$ 0 0
$$994$$ 1.25368e7 0.402459
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.29097e7i 0.411320i 0.978624 + 0.205660i $$0.0659341\pi$$
−0.978624 + 0.205660i $$0.934066\pi$$
$$998$$ 1.29838e7i 0.412644i
$$999$$ 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.b.e.199.2 2
3.2 odd 2 25.6.b.a.24.1 2
5.2 odd 4 45.6.a.b.1.1 1
5.3 odd 4 225.6.a.f.1.1 1
5.4 even 2 inner 225.6.b.e.199.1 2
12.11 even 2 400.6.c.j.49.2 2
15.2 even 4 5.6.a.a.1.1 1
15.8 even 4 25.6.a.a.1.1 1
15.14 odd 2 25.6.b.a.24.2 2
20.7 even 4 720.6.a.a.1.1 1
60.23 odd 4 400.6.a.g.1.1 1
60.47 odd 4 80.6.a.e.1.1 1
60.59 even 2 400.6.c.j.49.1 2
105.62 odd 4 245.6.a.b.1.1 1
120.77 even 4 320.6.a.j.1.1 1
120.107 odd 4 320.6.a.g.1.1 1
165.32 odd 4 605.6.a.a.1.1 1
195.77 even 4 845.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 15.2 even 4
25.6.a.a.1.1 1 15.8 even 4
25.6.b.a.24.1 2 3.2 odd 2
25.6.b.a.24.2 2 15.14 odd 2
45.6.a.b.1.1 1 5.2 odd 4
80.6.a.e.1.1 1 60.47 odd 4
225.6.a.f.1.1 1 5.3 odd 4
225.6.b.e.199.1 2 5.4 even 2 inner
225.6.b.e.199.2 2 1.1 even 1 trivial
245.6.a.b.1.1 1 105.62 odd 4
320.6.a.g.1.1 1 120.107 odd 4
320.6.a.j.1.1 1 120.77 even 4
400.6.a.g.1.1 1 60.23 odd 4
400.6.c.j.49.1 2 60.59 even 2
400.6.c.j.49.2 2 12.11 even 2
605.6.a.a.1.1 1 165.32 odd 4
720.6.a.a.1.1 1 20.7 even 4
845.6.a.b.1.1 1 195.77 even 4