# Properties

 Label 400.6.c.o.49.3 Level $400$ Weight $6$ Character 400.49 Analytic conductor $64.154$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{241})$$ Defining polynomial: $$x^{4} + 121x^{2} + 3600$$ x^4 + 121*x^2 + 3600 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.3 Root $$-8.26209i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.49 Dual form 400.6.c.o.49.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+11.5242i q^{3} +27.0483i q^{7} +110.193 q^{9} +O(q^{10})$$ $$q+11.5242i q^{3} +27.0483i q^{7} +110.193 q^{9} -226.008 q^{11} +511.257i q^{13} -387.387i q^{17} -1335.93 q^{19} -311.710 q^{21} -545.369i q^{23} +4070.26i q^{27} +4637.58 q^{29} -2991.56 q^{31} -2604.55i q^{33} +1263.70i q^{37} -5891.82 q^{39} -17197.6 q^{41} +16592.0i q^{43} +13036.0i q^{47} +16075.4 q^{49} +4464.31 q^{51} -28994.7i q^{53} -15395.4i q^{57} -34429.9 q^{59} -24149.1 q^{61} +2980.55i q^{63} +29389.7i q^{67} +6284.93 q^{69} -9064.32 q^{71} -55528.7i q^{73} -6113.13i q^{77} -101587. q^{79} -20129.4 q^{81} -73240.4i q^{83} +53444.3i q^{87} +42498.5 q^{89} -13828.7 q^{91} -34475.2i q^{93} -10565.9i q^{97} -24904.6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 56 q^{9}+O(q^{10})$$ 4 * q - 56 * q^9 $$4 q - 56 q^{9} + 400 q^{11} - 1680 q^{19} - 1992 q^{21} + 9360 q^{29} + 10016 q^{31} - 54864 q^{39} - 10668 q^{41} + 63308 q^{49} + 13200 q^{51} - 163552 q^{59} - 93864 q^{61} + 110088 q^{69} - 14896 q^{71} - 216208 q^{79} - 187324 q^{81} - 141980 q^{89} - 104992 q^{91} - 167552 q^{99}+O(q^{100})$$ 4 * q - 56 * q^9 + 400 * q^11 - 1680 * q^19 - 1992 * q^21 + 9360 * q^29 + 10016 * q^31 - 54864 * q^39 - 10668 * q^41 + 63308 * q^49 + 13200 * q^51 - 163552 * q^59 - 93864 * q^61 + 110088 * q^69 - 14896 * q^71 - 216208 * q^79 - 187324 * q^81 - 141980 * q^89 - 104992 * q^91 - 167552 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 11.5242i 0.739276i 0.929176 + 0.369638i $$0.120518\pi$$
−0.929176 + 0.369638i $$0.879482\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 27.0483i 0.208639i 0.994544 + 0.104320i $$0.0332665\pi$$
−0.994544 + 0.104320i $$0.966734\pi$$
$$8$$ 0 0
$$9$$ 110.193 0.453471
$$10$$ 0 0
$$11$$ −226.008 −0.563173 −0.281586 0.959536i $$-0.590861\pi$$
−0.281586 + 0.959536i $$0.590861\pi$$
$$12$$ 0 0
$$13$$ 511.257i 0.839037i 0.907747 + 0.419518i $$0.137801\pi$$
−0.907747 + 0.419518i $$0.862199\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 387.387i − 0.325104i −0.986700 0.162552i $$-0.948027\pi$$
0.986700 0.162552i $$-0.0519725\pi$$
$$18$$ 0 0
$$19$$ −1335.93 −0.848982 −0.424491 0.905432i $$-0.639547\pi$$
−0.424491 + 0.905432i $$0.639547\pi$$
$$20$$ 0 0
$$21$$ −311.710 −0.154242
$$22$$ 0 0
$$23$$ − 545.369i − 0.214967i −0.994207 0.107483i $$-0.965721\pi$$
0.994207 0.107483i $$-0.0342792\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4070.26i 1.07452i
$$28$$ 0 0
$$29$$ 4637.58 1.02399 0.511996 0.858988i $$-0.328906\pi$$
0.511996 + 0.858988i $$0.328906\pi$$
$$30$$ 0 0
$$31$$ −2991.56 −0.559105 −0.279552 0.960130i $$-0.590186\pi$$
−0.279552 + 0.960130i $$0.590186\pi$$
$$32$$ 0 0
$$33$$ − 2604.55i − 0.416340i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1263.70i 0.151754i 0.997117 + 0.0758770i $$0.0241756\pi$$
−0.997117 + 0.0758770i $$0.975824\pi$$
$$38$$ 0 0
$$39$$ −5891.82 −0.620280
$$40$$ 0 0
$$41$$ −17197.6 −1.59775 −0.798875 0.601497i $$-0.794571\pi$$
−0.798875 + 0.601497i $$0.794571\pi$$
$$42$$ 0 0
$$43$$ 16592.0i 1.36845i 0.729272 + 0.684223i $$0.239859\pi$$
−0.729272 + 0.684223i $$0.760141\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 13036.0i 0.860795i 0.902639 + 0.430397i $$0.141627\pi$$
−0.902639 + 0.430397i $$0.858373\pi$$
$$48$$ 0 0
$$49$$ 16075.4 0.956470
$$50$$ 0 0
$$51$$ 4464.31 0.240342
$$52$$ 0 0
$$53$$ − 28994.7i − 1.41785i −0.705286 0.708923i $$-0.749181\pi$$
0.705286 0.708923i $$-0.250819\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 15395.4i − 0.627632i
$$58$$ 0 0
$$59$$ −34429.9 −1.28768 −0.643838 0.765162i $$-0.722659\pi$$
−0.643838 + 0.765162i $$0.722659\pi$$
$$60$$ 0 0
$$61$$ −24149.1 −0.830952 −0.415476 0.909604i $$-0.636385\pi$$
−0.415476 + 0.909604i $$0.636385\pi$$
$$62$$ 0 0
$$63$$ 2980.55i 0.0946117i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 29389.7i 0.799849i 0.916548 + 0.399925i $$0.130964\pi$$
−0.916548 + 0.399925i $$0.869036\pi$$
$$68$$ 0 0
$$69$$ 6284.93 0.158920
$$70$$ 0 0
$$71$$ −9064.32 −0.213397 −0.106699 0.994291i $$-0.534028\pi$$
−0.106699 + 0.994291i $$0.534028\pi$$
$$72$$ 0 0
$$73$$ − 55528.7i − 1.21958i −0.792563 0.609791i $$-0.791254\pi$$
0.792563 0.609791i $$-0.208746\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 6113.13i − 0.117500i
$$78$$ 0 0
$$79$$ −101587. −1.83135 −0.915673 0.401924i $$-0.868342\pi$$
−0.915673 + 0.401924i $$0.868342\pi$$
$$80$$ 0 0
$$81$$ −20129.4 −0.340893
$$82$$ 0 0
$$83$$ − 73240.4i − 1.16696i −0.812128 0.583479i $$-0.801691\pi$$
0.812128 0.583479i $$-0.198309\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 53444.3i 0.757012i
$$88$$ 0 0
$$89$$ 42498.5 0.568719 0.284360 0.958718i $$-0.408219\pi$$
0.284360 + 0.958718i $$0.408219\pi$$
$$90$$ 0 0
$$91$$ −13828.7 −0.175056
$$92$$ 0 0
$$93$$ − 34475.2i − 0.413333i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10565.9i − 0.114019i −0.998374 0.0570093i $$-0.981844\pi$$
0.998374 0.0570093i $$-0.0181565\pi$$
$$98$$ 0 0
$$99$$ −24904.6 −0.255382
$$100$$ 0 0
$$101$$ 46486.0 0.453439 0.226719 0.973960i $$-0.427200\pi$$
0.226719 + 0.973960i $$0.427200\pi$$
$$102$$ 0 0
$$103$$ − 119526.i − 1.11012i −0.831811 0.555059i $$-0.812696\pi$$
0.831811 0.555059i $$-0.187304\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 22922.4i − 0.193553i −0.995306 0.0967765i $$-0.969147\pi$$
0.995306 0.0967765i $$-0.0308532\pi$$
$$108$$ 0 0
$$109$$ −210121. −1.69396 −0.846980 0.531624i $$-0.821582\pi$$
−0.846980 + 0.531624i $$0.821582\pi$$
$$110$$ 0 0
$$111$$ −14563.1 −0.112188
$$112$$ 0 0
$$113$$ − 203886.i − 1.50208i −0.660258 0.751039i $$-0.729553\pi$$
0.660258 0.751039i $$-0.270447\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 56337.2i 0.380479i
$$118$$ 0 0
$$119$$ 10478.2 0.0678294
$$120$$ 0 0
$$121$$ −109972. −0.682837
$$122$$ 0 0
$$123$$ − 198188.i − 1.18118i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 175022.i − 0.962905i −0.876472 0.481453i $$-0.840109\pi$$
0.876472 0.481453i $$-0.159891\pi$$
$$128$$ 0 0
$$129$$ −191209. −1.01166
$$130$$ 0 0
$$131$$ −149218. −0.759701 −0.379850 0.925048i $$-0.624025\pi$$
−0.379850 + 0.925048i $$0.624025\pi$$
$$132$$ 0 0
$$133$$ − 36134.6i − 0.177131i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 212180.i 0.965836i 0.875666 + 0.482918i $$0.160423\pi$$
−0.875666 + 0.482918i $$0.839577\pi$$
$$138$$ 0 0
$$139$$ 77082.5 0.338391 0.169195 0.985583i $$-0.445883\pi$$
0.169195 + 0.985583i $$0.445883\pi$$
$$140$$ 0 0
$$141$$ −150229. −0.636365
$$142$$ 0 0
$$143$$ − 115548.i − 0.472522i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 185256.i 0.707095i
$$148$$ 0 0
$$149$$ −470139. −1.73485 −0.867423 0.497571i $$-0.834225\pi$$
−0.867423 + 0.497571i $$0.834225\pi$$
$$150$$ 0 0
$$151$$ −311118. −1.11041 −0.555205 0.831714i $$-0.687360\pi$$
−0.555205 + 0.831714i $$0.687360\pi$$
$$152$$ 0 0
$$153$$ − 42687.5i − 0.147425i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 543649.i 1.76023i 0.474760 + 0.880115i $$0.342535\pi$$
−0.474760 + 0.880115i $$0.657465\pi$$
$$158$$ 0 0
$$159$$ 334140. 1.04818
$$160$$ 0 0
$$161$$ 14751.3 0.0448504
$$162$$ 0 0
$$163$$ − 298673.i − 0.880495i −0.897876 0.440248i $$-0.854891\pi$$
0.897876 0.440248i $$-0.145109\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 65486.6i − 0.181703i −0.995864 0.0908513i $$-0.971041\pi$$
0.995864 0.0908513i $$-0.0289588\pi$$
$$168$$ 0 0
$$169$$ 109909. 0.296017
$$170$$ 0 0
$$171$$ −147210. −0.384989
$$172$$ 0 0
$$173$$ 355092.i 0.902040i 0.892514 + 0.451020i $$0.148940\pi$$
−0.892514 + 0.451020i $$0.851060\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 396777.i − 0.951947i
$$178$$ 0 0
$$179$$ −39265.7 −0.0915970 −0.0457985 0.998951i $$-0.514583\pi$$
−0.0457985 + 0.998951i $$0.514583\pi$$
$$180$$ 0 0
$$181$$ 99638.2 0.226063 0.113031 0.993591i $$-0.463944\pi$$
0.113031 + 0.993591i $$0.463944\pi$$
$$182$$ 0 0
$$183$$ − 278298.i − 0.614303i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 87552.4i 0.183090i
$$188$$ 0 0
$$189$$ −110094. −0.224186
$$190$$ 0 0
$$191$$ −380336. −0.754369 −0.377185 0.926138i $$-0.623108\pi$$
−0.377185 + 0.926138i $$0.623108\pi$$
$$192$$ 0 0
$$193$$ 126401.i 0.244262i 0.992514 + 0.122131i $$0.0389728\pi$$
−0.992514 + 0.122131i $$0.961027\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 326136.i 0.598733i 0.954138 + 0.299366i $$0.0967753\pi$$
−0.954138 + 0.299366i $$0.903225\pi$$
$$198$$ 0 0
$$199$$ −498559. −0.892451 −0.446225 0.894921i $$-0.647232\pi$$
−0.446225 + 0.894921i $$0.647232\pi$$
$$200$$ 0 0
$$201$$ −338692. −0.591309
$$202$$ 0 0
$$203$$ 125439.i 0.213645i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 60096.1i − 0.0974811i
$$208$$ 0 0
$$209$$ 301930. 0.478123
$$210$$ 0 0
$$211$$ 310763. 0.480532 0.240266 0.970707i $$-0.422765\pi$$
0.240266 + 0.970707i $$0.422765\pi$$
$$212$$ 0 0
$$213$$ − 104459.i − 0.157760i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 80916.7i − 0.116651i
$$218$$ 0 0
$$219$$ 639923. 0.901607
$$220$$ 0 0
$$221$$ 198054. 0.272774
$$222$$ 0 0
$$223$$ 627285.i 0.844700i 0.906433 + 0.422350i $$0.138795\pi$$
−0.906433 + 0.422350i $$0.861205\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 450131.i 0.579794i 0.957058 + 0.289897i $$0.0936211\pi$$
−0.957058 + 0.289897i $$0.906379\pi$$
$$228$$ 0 0
$$229$$ 461485. 0.581526 0.290763 0.956795i $$-0.406091\pi$$
0.290763 + 0.956795i $$0.406091\pi$$
$$230$$ 0 0
$$231$$ 70448.8 0.0868648
$$232$$ 0 0
$$233$$ − 675812.i − 0.815523i −0.913089 0.407761i $$-0.866310\pi$$
0.913089 0.407761i $$-0.133690\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 1.17071e6i − 1.35387i
$$238$$ 0 0
$$239$$ −1.00730e6 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$240$$ 0 0
$$241$$ −232091. −0.257404 −0.128702 0.991683i $$-0.541081\pi$$
−0.128702 + 0.991683i $$0.541081\pi$$
$$242$$ 0 0
$$243$$ 757099.i 0.822502i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 683002.i − 0.712327i
$$248$$ 0 0
$$249$$ 844035. 0.862704
$$250$$ 0 0
$$251$$ 1.77096e6 1.77429 0.887143 0.461494i $$-0.152686\pi$$
0.887143 + 0.461494i $$0.152686\pi$$
$$252$$ 0 0
$$253$$ 123258.i 0.121063i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 528458.i − 0.499088i −0.968363 0.249544i $$-0.919719\pi$$
0.968363 0.249544i $$-0.0802808\pi$$
$$258$$ 0 0
$$259$$ −34181.0 −0.0316618
$$260$$ 0 0
$$261$$ 511030. 0.464350
$$262$$ 0 0
$$263$$ 1.69907e6i 1.51469i 0.653016 + 0.757344i $$0.273503\pi$$
−0.653016 + 0.757344i $$0.726497\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 489760.i 0.420441i
$$268$$ 0 0
$$269$$ 648393. 0.546333 0.273167 0.961967i $$-0.411929\pi$$
0.273167 + 0.961967i $$0.411929\pi$$
$$270$$ 0 0
$$271$$ 1.94947e6 1.61248 0.806239 0.591590i $$-0.201500\pi$$
0.806239 + 0.591590i $$0.201500\pi$$
$$272$$ 0 0
$$273$$ − 159364.i − 0.129415i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 313195.i − 0.245254i −0.992453 0.122627i $$-0.960868\pi$$
0.992453 0.122627i $$-0.0391319\pi$$
$$278$$ 0 0
$$279$$ −329650. −0.253538
$$280$$ 0 0
$$281$$ 1.72743e6 1.30507 0.652537 0.757757i $$-0.273705\pi$$
0.652537 + 0.757757i $$0.273705\pi$$
$$282$$ 0 0
$$283$$ 205142.i 0.152261i 0.997098 + 0.0761306i $$0.0242566\pi$$
−0.997098 + 0.0761306i $$0.975743\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 465167.i − 0.333353i
$$288$$ 0 0
$$289$$ 1.26979e6 0.894307
$$290$$ 0 0
$$291$$ 121763. 0.0842912
$$292$$ 0 0
$$293$$ 390309.i 0.265607i 0.991142 + 0.132804i $$0.0423979\pi$$
−0.991142 + 0.132804i $$0.957602\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 919911.i − 0.605138i
$$298$$ 0 0
$$299$$ 278824. 0.180365
$$300$$ 0 0
$$301$$ −448787. −0.285511
$$302$$ 0 0
$$303$$ 535713.i 0.335217i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.10708e6i 0.670401i 0.942147 + 0.335201i $$0.108804\pi$$
−0.942147 + 0.335201i $$0.891196\pi$$
$$308$$ 0 0
$$309$$ 1.37744e6 0.820684
$$310$$ 0 0
$$311$$ −2.19106e6 −1.28455 −0.642277 0.766472i $$-0.722010\pi$$
−0.642277 + 0.766472i $$0.722010\pi$$
$$312$$ 0 0
$$313$$ 2.89050e6i 1.66768i 0.552006 + 0.833840i $$0.313862\pi$$
−0.552006 + 0.833840i $$0.686138\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 1.46652e6i − 0.819671i −0.912160 0.409835i $$-0.865586\pi$$
0.912160 0.409835i $$-0.134414\pi$$
$$318$$ 0 0
$$319$$ −1.04813e6 −0.576684
$$320$$ 0 0
$$321$$ 264161. 0.143089
$$322$$ 0 0
$$323$$ 517520.i 0.276008i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2.42147e6i − 1.25230i
$$328$$ 0 0
$$329$$ −352602. −0.179595
$$330$$ 0 0
$$331$$ 2.23500e6 1.12126 0.560632 0.828065i $$-0.310558\pi$$
0.560632 + 0.828065i $$0.310558\pi$$
$$332$$ 0 0
$$333$$ 139251.i 0.0688160i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.48851e6i 0.713966i 0.934111 + 0.356983i $$0.116195\pi$$
−0.934111 + 0.356983i $$0.883805\pi$$
$$338$$ 0 0
$$339$$ 2.34962e6 1.11045
$$340$$ 0 0
$$341$$ 676115. 0.314872
$$342$$ 0 0
$$343$$ 889414.i 0.408196i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.34245e6i 1.04435i 0.852837 + 0.522177i $$0.174880\pi$$
−0.852837 + 0.522177i $$0.825120\pi$$
$$348$$ 0 0
$$349$$ 119901. 0.0526940 0.0263470 0.999653i $$-0.491613\pi$$
0.0263470 + 0.999653i $$0.491613\pi$$
$$350$$ 0 0
$$351$$ −2.08095e6 −0.901559
$$352$$ 0 0
$$353$$ 2.49528e6i 1.06582i 0.846173 + 0.532909i $$0.178901\pi$$
−0.846173 + 0.532909i $$0.821099\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 120752.i 0.0501447i
$$358$$ 0 0
$$359$$ 3.43398e6 1.40625 0.703124 0.711068i $$-0.251788\pi$$
0.703124 + 0.711068i $$0.251788\pi$$
$$360$$ 0 0
$$361$$ −691400. −0.279230
$$362$$ 0 0
$$363$$ − 1.26733e6i − 0.504805i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 4.67310e6i − 1.81109i −0.424249 0.905545i $$-0.639462\pi$$
0.424249 0.905545i $$-0.360538\pi$$
$$368$$ 0 0
$$369$$ −1.89507e6 −0.724533
$$370$$ 0 0
$$371$$ 784259. 0.295818
$$372$$ 0 0
$$373$$ 2.03766e6i 0.758333i 0.925328 + 0.379167i $$0.123789\pi$$
−0.925328 + 0.379167i $$0.876211\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2.37099e6i 0.859166i
$$378$$ 0 0
$$379$$ 1.78978e6 0.640032 0.320016 0.947412i $$-0.396312\pi$$
0.320016 + 0.947412i $$0.396312\pi$$
$$380$$ 0 0
$$381$$ 2.01698e6 0.711853
$$382$$ 0 0
$$383$$ 2.04521e6i 0.712429i 0.934404 + 0.356214i $$0.115933\pi$$
−0.934404 + 0.356214i $$0.884067\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.82833e6i 0.620551i
$$388$$ 0 0
$$389$$ −234191. −0.0784687 −0.0392344 0.999230i $$-0.512492\pi$$
−0.0392344 + 0.999230i $$0.512492\pi$$
$$390$$ 0 0
$$391$$ −211269. −0.0698865
$$392$$ 0 0
$$393$$ − 1.71961e6i − 0.561629i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.81706e6i 0.897058i 0.893768 + 0.448529i $$0.148052\pi$$
−0.893768 + 0.448529i $$0.851948\pi$$
$$398$$ 0 0
$$399$$ 416421. 0.130949
$$400$$ 0 0
$$401$$ 735044. 0.228272 0.114136 0.993465i $$-0.463590\pi$$
0.114136 + 0.993465i $$0.463590\pi$$
$$402$$ 0 0
$$403$$ − 1.52946e6i − 0.469109i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 285606.i − 0.0854636i
$$408$$ 0 0
$$409$$ −6.08894e6 −1.79984 −0.899918 0.436058i $$-0.856374\pi$$
−0.899918 + 0.436058i $$0.856374\pi$$
$$410$$ 0 0
$$411$$ −2.44520e6 −0.714020
$$412$$ 0 0
$$413$$ − 931273.i − 0.268659i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 888312.i 0.250164i
$$418$$ 0 0
$$419$$ 5.65964e6 1.57490 0.787452 0.616376i $$-0.211400\pi$$
0.787452 + 0.616376i $$0.211400\pi$$
$$420$$ 0 0
$$421$$ −6.34902e6 −1.74583 −0.872914 0.487873i $$-0.837773\pi$$
−0.872914 + 0.487873i $$0.837773\pi$$
$$422$$ 0 0
$$423$$ 1.43648e6i 0.390345i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 653192.i − 0.173369i
$$428$$ 0 0
$$429$$ 1.33160e6 0.349325
$$430$$ 0 0
$$431$$ −3.03516e6 −0.787024 −0.393512 0.919320i $$-0.628740\pi$$
−0.393512 + 0.919320i $$0.628740\pi$$
$$432$$ 0 0
$$433$$ 3.50633e6i 0.898739i 0.893346 + 0.449369i $$0.148351\pi$$
−0.893346 + 0.449369i $$0.851649\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 728573.i 0.182503i
$$438$$ 0 0
$$439$$ −6.08849e6 −1.50782 −0.753908 0.656980i $$-0.771834\pi$$
−0.753908 + 0.656980i $$0.771834\pi$$
$$440$$ 0 0
$$441$$ 1.77140e6 0.433731
$$442$$ 0 0
$$443$$ − 6.03357e6i − 1.46071i −0.683066 0.730357i $$-0.739354\pi$$
0.683066 0.730357i $$-0.260646\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 5.41797e6i − 1.28253i
$$448$$ 0 0
$$449$$ −3.97943e6 −0.931548 −0.465774 0.884904i $$-0.654224\pi$$
−0.465774 + 0.884904i $$0.654224\pi$$
$$450$$ 0 0
$$451$$ 3.88680e6 0.899809
$$452$$ 0 0
$$453$$ − 3.58538e6i − 0.820899i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 430533.i − 0.0964308i −0.998837 0.0482154i $$-0.984647\pi$$
0.998837 0.0482154i $$-0.0153534\pi$$
$$458$$ 0 0
$$459$$ 1.57677e6 0.349330
$$460$$ 0 0
$$461$$ 3.80383e6 0.833622 0.416811 0.908993i $$-0.363148\pi$$
0.416811 + 0.908993i $$0.363148\pi$$
$$462$$ 0 0
$$463$$ − 7.62853e6i − 1.65382i −0.562333 0.826911i $$-0.690096\pi$$
0.562333 0.826911i $$-0.309904\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 9.05102e6i 1.92046i 0.279209 + 0.960230i $$0.409928\pi$$
−0.279209 + 0.960230i $$0.590072\pi$$
$$468$$ 0 0
$$469$$ −794943. −0.166880
$$470$$ 0 0
$$471$$ −6.26511e6 −1.30130
$$472$$ 0 0
$$473$$ − 3.74992e6i − 0.770672i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 3.19502e6i − 0.642952i
$$478$$ 0 0
$$479$$ −8.51621e6 −1.69593 −0.847964 0.530054i $$-0.822172\pi$$
−0.847964 + 0.530054i $$0.822172\pi$$
$$480$$ 0 0
$$481$$ −646076. −0.127327
$$482$$ 0 0
$$483$$ 169997.i 0.0331569i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 2.65770e6i − 0.507789i −0.967232 0.253895i $$-0.918288\pi$$
0.967232 0.253895i $$-0.0817117\pi$$
$$488$$ 0 0
$$489$$ 3.44196e6 0.650929
$$490$$ 0 0
$$491$$ 5.05116e6 0.945557 0.472779 0.881181i $$-0.343251\pi$$
0.472779 + 0.881181i $$0.343251\pi$$
$$492$$ 0 0
$$493$$ − 1.79654e6i − 0.332904i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 245175.i − 0.0445230i
$$498$$ 0 0
$$499$$ −2.15003e6 −0.386538 −0.193269 0.981146i $$-0.561909\pi$$
−0.193269 + 0.981146i $$0.561909\pi$$
$$500$$ 0 0
$$501$$ 754679. 0.134328
$$502$$ 0 0
$$503$$ 1.79475e6i 0.316289i 0.987416 + 0.158144i $$0.0505511\pi$$
−0.987416 + 0.158144i $$0.949449\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.26661e6i 0.218839i
$$508$$ 0 0
$$509$$ −697940. −0.119405 −0.0597027 0.998216i $$-0.519015\pi$$
−0.0597027 + 0.998216i $$0.519015\pi$$
$$510$$ 0 0
$$511$$ 1.50196e6 0.254452
$$512$$ 0 0
$$513$$ − 5.43757e6i − 0.912245i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 2.94624e6i − 0.484776i
$$518$$ 0 0
$$519$$ −4.09214e6 −0.666857
$$520$$ 0 0
$$521$$ −1.14928e6 −0.185495 −0.0927475 0.995690i $$-0.529565\pi$$
−0.0927475 + 0.995690i $$0.529565\pi$$
$$522$$ 0 0
$$523$$ − 7.22071e6i − 1.15432i −0.816632 0.577159i $$-0.804161\pi$$
0.816632 0.577159i $$-0.195839\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.15889e6i 0.181767i
$$528$$ 0 0
$$529$$ 6.13892e6 0.953789
$$530$$ 0 0
$$531$$ −3.79395e6 −0.583923
$$532$$ 0 0
$$533$$ − 8.79241e6i − 1.34057i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 452505.i − 0.0677155i
$$538$$ 0 0
$$539$$ −3.63316e6 −0.538657
$$540$$ 0 0
$$541$$ −1.24720e7 −1.83207 −0.916035 0.401099i $$-0.868628\pi$$
−0.916035 + 0.401099i $$0.868628\pi$$
$$542$$ 0 0
$$543$$ 1.14825e6i 0.167123i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.58966e6i 1.22746i 0.789516 + 0.613730i $$0.210332\pi$$
−0.789516 + 0.613730i $$0.789668\pi$$
$$548$$ 0 0
$$549$$ −2.66107e6 −0.376812
$$550$$ 0 0
$$551$$ −6.19546e6 −0.869350
$$552$$ 0 0
$$553$$ − 2.74776e6i − 0.382090i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3.38207e6i 0.461896i 0.972966 + 0.230948i $$0.0741827\pi$$
−0.972966 + 0.230948i $$0.925817\pi$$
$$558$$ 0 0
$$559$$ −8.48278e6 −1.14818
$$560$$ 0 0
$$561$$ −1.00897e6 −0.135354
$$562$$ 0 0
$$563$$ 1.50247e6i 0.199772i 0.994999 + 0.0998859i $$0.0318478\pi$$
−0.994999 + 0.0998859i $$0.968152\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 544468.i − 0.0711237i
$$568$$ 0 0
$$569$$ −1.87838e6 −0.243222 −0.121611 0.992578i $$-0.538806\pi$$
−0.121611 + 0.992578i $$0.538806\pi$$
$$570$$ 0 0
$$571$$ 1.49628e7 1.92053 0.960267 0.279082i $$-0.0900301\pi$$
0.960267 + 0.279082i $$0.0900301\pi$$
$$572$$ 0 0
$$573$$ − 4.38306e6i − 0.557687i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 4.03660e6i 0.504750i 0.967630 + 0.252375i $$0.0812117\pi$$
−0.967630 + 0.252375i $$0.918788\pi$$
$$578$$ 0 0
$$579$$ −1.45666e6 −0.180577
$$580$$ 0 0
$$581$$ 1.98103e6 0.243473
$$582$$ 0 0
$$583$$ 6.55303e6i 0.798492i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1.57983e7i 1.89241i 0.323574 + 0.946203i $$0.395115\pi$$
−0.323574 + 0.946203i $$0.604885\pi$$
$$588$$ 0 0
$$589$$ 3.99650e6 0.474670
$$590$$ 0 0
$$591$$ −3.75845e6 −0.442629
$$592$$ 0 0
$$593$$ − 1.37947e7i − 1.61092i −0.592647 0.805462i $$-0.701917\pi$$
0.592647 0.805462i $$-0.298083\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 5.74549e6i − 0.659768i
$$598$$ 0 0
$$599$$ −814106. −0.0927073 −0.0463537 0.998925i $$-0.514760\pi$$
−0.0463537 + 0.998925i $$0.514760\pi$$
$$600$$ 0 0
$$601$$ −1.55613e7 −1.75736 −0.878679 0.477413i $$-0.841574\pi$$
−0.878679 + 0.477413i $$0.841574\pi$$
$$602$$ 0 0
$$603$$ 3.23855e6i 0.362708i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 1.63191e7i − 1.79773i −0.438226 0.898865i $$-0.644393\pi$$
0.438226 0.898865i $$-0.355607\pi$$
$$608$$ 0 0
$$609$$ −1.44558e6 −0.157942
$$610$$ 0 0
$$611$$ −6.66475e6 −0.722239
$$612$$ 0 0
$$613$$ 5.38233e6i 0.578520i 0.957250 + 0.289260i $$0.0934093\pi$$
−0.957250 + 0.289260i $$0.906591\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 5.83565e6i − 0.617130i −0.951203 0.308565i $$-0.900151\pi$$
0.951203 0.308565i $$-0.0998487\pi$$
$$618$$ 0 0
$$619$$ −3.36289e6 −0.352765 −0.176383 0.984322i $$-0.556440\pi$$
−0.176383 + 0.984322i $$0.556440\pi$$
$$620$$ 0 0
$$621$$ 2.21980e6 0.230985
$$622$$ 0 0
$$623$$ 1.14951e6i 0.118657i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 3.47949e6i 0.353465i
$$628$$ 0 0
$$629$$ 489541. 0.0493358
$$630$$ 0 0
$$631$$ −8.10425e6 −0.810288 −0.405144 0.914253i $$-0.632779\pi$$
−0.405144 + 0.914253i $$0.632779\pi$$
$$632$$ 0 0
$$633$$ 3.58128e6i 0.355246i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 8.21866e6i 0.802513i
$$638$$ 0 0
$$639$$ −998828. −0.0967695
$$640$$ 0 0
$$641$$ 7.08523e6 0.681097 0.340548 0.940227i $$-0.389387\pi$$
0.340548 + 0.940227i $$0.389387\pi$$
$$642$$ 0 0
$$643$$ − 5.72946e6i − 0.546495i −0.961944 0.273247i $$-0.911902\pi$$
0.961944 0.273247i $$-0.0880978\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.17751e6i 0.204503i 0.994759 + 0.102251i $$0.0326046\pi$$
−0.994759 + 0.102251i $$0.967395\pi$$
$$648$$ 0 0
$$649$$ 7.78143e6 0.725183
$$650$$ 0 0
$$651$$ 932498. 0.0862374
$$652$$ 0 0
$$653$$ 1.42048e7i 1.30363i 0.758379 + 0.651813i $$0.225991\pi$$
−0.758379 + 0.651813i $$0.774009\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 6.11890e6i − 0.553044i
$$658$$ 0 0
$$659$$ 1.54033e7 1.38166 0.690829 0.723018i $$-0.257246\pi$$
0.690829 + 0.723018i $$0.257246\pi$$
$$660$$ 0 0
$$661$$ −544048. −0.0484322 −0.0242161 0.999707i $$-0.507709\pi$$
−0.0242161 + 0.999707i $$0.507709\pi$$
$$662$$ 0 0
$$663$$ 2.28241e6i 0.201656i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 2.52919e6i − 0.220124i
$$668$$ 0 0
$$669$$ −7.22894e6 −0.624467
$$670$$ 0 0
$$671$$ 5.45787e6 0.467969
$$672$$ 0 0
$$673$$ 1.37093e7i 1.16675i 0.812203 + 0.583374i $$0.198268\pi$$
−0.812203 + 0.583374i $$0.801732\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 1.49910e7i 1.25707i 0.777781 + 0.628536i $$0.216345\pi$$
−0.777781 + 0.628536i $$0.783655\pi$$
$$678$$ 0 0
$$679$$ 285789. 0.0237887
$$680$$ 0 0
$$681$$ −5.18738e6 −0.428628
$$682$$ 0 0
$$683$$ − 3.47996e6i − 0.285445i −0.989763 0.142723i $$-0.954414\pi$$
0.989763 0.142723i $$-0.0455857\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 5.31824e6i 0.429908i
$$688$$ 0 0
$$689$$ 1.48237e7 1.18962
$$690$$ 0 0
$$691$$ −5.09436e6 −0.405877 −0.202939 0.979191i $$-0.565049\pi$$
−0.202939 + 0.979191i $$0.565049\pi$$
$$692$$ 0 0
$$693$$ − 673627.i − 0.0532827i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6.66213e6i 0.519435i
$$698$$ 0 0
$$699$$ 7.78817e6 0.602896
$$700$$ 0 0
$$701$$ 1.75839e7 1.35151 0.675755 0.737126i $$-0.263818\pi$$
0.675755 + 0.737126i $$0.263818\pi$$
$$702$$ 0 0
$$703$$ − 1.68821e6i − 0.128836i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.25737e6i 0.0946051i
$$708$$ 0 0
$$709$$ −1.35383e7 −1.01146 −0.505730 0.862692i $$-0.668777\pi$$
−0.505730 + 0.862692i $$0.668777\pi$$
$$710$$ 0 0
$$711$$ −1.11942e7 −0.830462
$$712$$ 0 0
$$713$$ 1.63150e6i 0.120189i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 1.16083e7i − 0.843275i
$$718$$ 0 0
$$719$$ 1.27911e7 0.922756 0.461378 0.887204i $$-0.347355\pi$$
0.461378 + 0.887204i $$0.347355\pi$$
$$720$$ 0 0
$$721$$ 3.23298e6 0.231614
$$722$$ 0 0
$$723$$ − 2.67466e6i − 0.190293i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 2.11509e7i − 1.48420i −0.670289 0.742100i $$-0.733830\pi$$
0.670289 0.742100i $$-0.266170\pi$$
$$728$$ 0 0
$$729$$ −1.36164e7 −0.948949
$$730$$ 0 0
$$731$$ 6.42753e6 0.444888
$$732$$ 0 0
$$733$$ 1.23015e7i 0.845667i 0.906207 + 0.422833i $$0.138964\pi$$
−0.906207 + 0.422833i $$0.861036\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 6.64230e6i − 0.450453i
$$738$$ 0 0
$$739$$ 1.04425e7 0.703383 0.351691 0.936116i $$-0.385607\pi$$
0.351691 + 0.936116i $$0.385607\pi$$
$$740$$ 0 0
$$741$$ 7.87103e6 0.526606
$$742$$ 0 0
$$743$$ 3.63296e6i 0.241429i 0.992687 + 0.120714i $$0.0385185\pi$$
−0.992687 + 0.120714i $$0.961481\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 8.07060e6i − 0.529181i
$$748$$ 0 0
$$749$$ 620012. 0.0403827
$$750$$ 0 0
$$751$$ 1.77346e7 1.14741 0.573707 0.819060i $$-0.305505\pi$$
0.573707 + 0.819060i $$0.305505\pi$$
$$752$$ 0 0
$$753$$ 2.04088e7i 1.31169i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 7.89094e6i 0.500483i 0.968183 + 0.250241i $$0.0805100\pi$$
−0.968183 + 0.250241i $$0.919490\pi$$
$$758$$ 0 0
$$759$$ −1.42044e6 −0.0894992
$$760$$ 0 0
$$761$$ 1.98455e7 1.24223 0.621114 0.783721i $$-0.286681\pi$$
0.621114 + 0.783721i $$0.286681\pi$$
$$762$$ 0 0
$$763$$ − 5.68343e6i − 0.353426i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1.76026e7i − 1.08041i
$$768$$ 0 0
$$769$$ −3.89976e6 −0.237806 −0.118903 0.992906i $$-0.537938\pi$$
−0.118903 + 0.992906i $$0.537938\pi$$
$$770$$ 0 0
$$771$$ 6.09004e6 0.368964
$$772$$ 0 0
$$773$$ − 2.33736e6i − 0.140695i −0.997523 0.0703474i $$-0.977589\pi$$
0.997523 0.0703474i $$-0.0224108\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 393908.i − 0.0234068i
$$778$$ 0 0
$$779$$ 2.29748e7 1.35646
$$780$$ 0 0
$$781$$ 2.04860e6 0.120180
$$782$$ 0 0
$$783$$ 1.88762e7i 1.10030i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 1.52903e7i − 0.879993i −0.897999 0.439997i $$-0.854980\pi$$
0.897999 0.439997i $$-0.145020\pi$$
$$788$$ 0 0
$$789$$ −1.95804e7 −1.11977
$$790$$ 0 0
$$791$$ 5.51479e6 0.313392
$$792$$ 0 0
$$793$$ − 1.23464e7i − 0.697199i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.25972e7i 0.702468i 0.936288 + 0.351234i $$0.114238\pi$$
−0.936288 + 0.351234i $$0.885762\pi$$
$$798$$ 0 0
$$799$$ 5.04997e6 0.279848
$$800$$ 0 0
$$801$$ 4.68305e6 0.257898
$$802$$ 0 0
$$803$$ 1.25499e7i 0.686835i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 7.47219e6i 0.403891i
$$808$$ 0 0
$$809$$ 1.13476e7 0.609584 0.304792 0.952419i $$-0.401413\pi$$
0.304792 + 0.952419i $$0.401413\pi$$
$$810$$ 0 0
$$811$$ 1.91000e7 1.01972 0.509860 0.860257i $$-0.329697\pi$$
0.509860 + 0.860257i $$0.329697\pi$$
$$812$$ 0 0
$$813$$ 2.24661e7i 1.19207i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 2.21657e7i − 1.16179i
$$818$$ 0 0
$$819$$ −1.52383e6 −0.0793827
$$820$$ 0 0
$$821$$ 1.25668e6 0.0650678 0.0325339 0.999471i $$-0.489642\pi$$
0.0325339 + 0.999471i $$0.489642\pi$$
$$822$$ 0 0
$$823$$ 6.95590e6i 0.357976i 0.983851 + 0.178988i $$0.0572823\pi$$
−0.983851 + 0.178988i $$0.942718\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 9.04638e6i − 0.459950i −0.973197 0.229975i $$-0.926136\pi$$
0.973197 0.229975i $$-0.0738645\pi$$
$$828$$ 0 0
$$829$$ 1.54026e7 0.778407 0.389203 0.921152i $$-0.372750\pi$$
0.389203 + 0.921152i $$0.372750\pi$$
$$830$$ 0 0
$$831$$ 3.60932e6 0.181310
$$832$$ 0 0
$$833$$ − 6.22739e6i − 0.310952i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 1.21764e7i − 0.600767i
$$838$$ 0 0
$$839$$ 1.75941e7 0.862903 0.431451 0.902136i $$-0.358002\pi$$
0.431451 + 0.902136i $$0.358002\pi$$
$$840$$ 0 0
$$841$$ 995979. 0.0485580
$$842$$ 0 0
$$843$$ 1.99072e7i 0.964809i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.97455e6i − 0.142466i
$$848$$ 0 0
$$849$$ −2.36410e6 −0.112563
$$850$$ 0 0
$$851$$ 689183. 0.0326220
$$852$$ 0 0
$$853$$ 2.06755e7i 0.972934i 0.873699 + 0.486467i $$0.161715\pi$$
−0.873699 + 0.486467i $$0.838285\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 2.16205e7i 1.00557i 0.864411 + 0.502786i $$0.167692\pi$$
−0.864411 + 0.502786i $$0.832308\pi$$
$$858$$ 0 0
$$859$$ −1.70061e7 −0.786362 −0.393181 0.919461i $$-0.628625\pi$$
−0.393181 + 0.919461i $$0.628625\pi$$
$$860$$ 0 0
$$861$$ 5.36067e6 0.246440
$$862$$ 0 0
$$863$$ − 2.96043e7i − 1.35309i −0.736399 0.676547i $$-0.763475\pi$$
0.736399 0.676547i $$-0.236525\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 1.46333e7i 0.661140i
$$868$$ 0 0
$$869$$ 2.29594e7 1.03136
$$870$$ 0 0
$$871$$ −1.50257e7 −0.671103
$$872$$ 0 0
$$873$$ − 1.16429e6i − 0.0517041i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 2.77691e7i − 1.21917i −0.792722 0.609583i $$-0.791337\pi$$
0.792722 0.609583i $$-0.208663\pi$$
$$878$$ 0 0
$$879$$ −4.49799e6 −0.196357
$$880$$ 0 0
$$881$$ 1.16824e7 0.507099 0.253550 0.967322i $$-0.418402\pi$$
0.253550 + 0.967322i $$0.418402\pi$$
$$882$$ 0 0
$$883$$ 2.66919e7i 1.15207i 0.817426 + 0.576034i $$0.195400\pi$$
−0.817426 + 0.576034i $$0.804600\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 3.95714e7i 1.68878i 0.535733 + 0.844388i $$0.320036\pi$$
−0.535733 + 0.844388i $$0.679964\pi$$
$$888$$ 0 0
$$889$$ 4.73406e6 0.200900
$$890$$ 0 0
$$891$$ 4.54940e6 0.191982
$$892$$ 0 0
$$893$$ − 1.74151e7i − 0.730799i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 3.21321e6i 0.133339i
$$898$$ 0 0
$$899$$ −1.38736e7 −0.572518
$$900$$ 0 0
$$901$$ −1.12322e7 −0.460948
$$902$$ 0 0
$$903$$ − 5.17189e6i − 0.211072i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 2.79635e7i − 1.12869i −0.825541 0.564343i $$-0.809130\pi$$
0.825541 0.564343i $$-0.190870\pi$$
$$908$$ 0 0
$$909$$ 5.12245e6 0.205621
$$910$$ 0 0
$$911$$ 1.87296e7 0.747710 0.373855 0.927487i $$-0.378036\pi$$
0.373855 + 0.927487i $$0.378036\pi$$
$$912$$ 0 0
$$913$$ 1.65529e7i 0.657199i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 4.03610e6i − 0.158503i
$$918$$ 0 0
$$919$$ −7.77613e6 −0.303721 −0.151861 0.988402i $$-0.548526\pi$$
−0.151861 + 0.988402i $$0.548526\pi$$
$$920$$ 0 0
$$921$$ −1.27582e7 −0.495612
$$922$$ 0 0
$$923$$ − 4.63420e6i − 0.179048i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 1.31710e7i − 0.503406i
$$928$$ 0 0
$$929$$ 3.88269e7 1.47602 0.738011 0.674788i $$-0.235765\pi$$
0.738011 + 0.674788i $$0.235765\pi$$
$$930$$ 0 0
$$931$$ −2.14755e7 −0.812026
$$932$$ 0 0
$$933$$ − 2.52501e7i − 0.949640i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 1.75302e7i − 0.652288i −0.945320 0.326144i $$-0.894251\pi$$
0.945320 0.326144i $$-0.105749\pi$$
$$938$$ 0 0
$$939$$ −3.33107e7 −1.23288
$$940$$ 0 0
$$941$$ −4.29638e6 −0.158172 −0.0790859 0.996868i $$-0.525200\pi$$
−0.0790859 + 0.996868i $$0.525200\pi$$
$$942$$ 0 0
$$943$$ 9.37905e6i 0.343463i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.43637e7i 1.60751i 0.594962 + 0.803754i $$0.297167\pi$$
−0.594962 + 0.803754i $$0.702833\pi$$
$$948$$ 0 0
$$949$$ 2.83895e7 1.02327
$$950$$ 0 0
$$951$$ 1.69004e7 0.605963
$$952$$ 0 0
$$953$$ − 5.71990e6i − 0.204012i −0.994784 0.102006i $$-0.967474\pi$$
0.994784 0.102006i $$-0.0325261\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 1.20788e7i − 0.426329i
$$958$$ 0 0
$$959$$ −5.73912e6 −0.201511
$$960$$ 0 0
$$961$$ −1.96797e7 −0.687402
$$962$$ 0 0
$$963$$ − 2.52589e6i − 0.0877706i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 3.00548e7i 1.03359i 0.856110 + 0.516794i $$0.172875\pi$$
−0.856110 + 0.516794i $$0.827125\pi$$
$$968$$ 0 0
$$969$$ −5.96399e6 −0.204046
$$970$$ 0 0
$$971$$ −1.89662e7 −0.645552 −0.322776 0.946475i $$-0.604616\pi$$
−0.322776 + 0.946475i $$0.604616\pi$$
$$972$$ 0 0
$$973$$ 2.08495e6i 0.0706015i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.45541e7i 0.822975i 0.911415 + 0.411488i $$0.134991\pi$$
−0.911415 + 0.411488i $$0.865009\pi$$
$$978$$ 0 0
$$979$$ −9.60498e6 −0.320287
$$980$$ 0 0
$$981$$ −2.31540e7 −0.768162
$$982$$ 0 0
$$983$$ 3.35539e7i 1.10754i 0.832670 + 0.553769i $$0.186811\pi$$
−0.832670 + 0.553769i $$0.813189\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 4.06345e6i − 0.132771i
$$988$$ 0 0
$$989$$ 9.04877e6 0.294170
$$990$$ 0 0
$$991$$ −2.05230e7 −0.663830 −0.331915 0.943309i $$-0.607695\pi$$
−0.331915 + 0.943309i $$0.607695\pi$$
$$992$$ 0 0
$$993$$ 2.57565e7i 0.828924i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.25729e7i 0.400589i 0.979736 + 0.200294i $$0.0641898\pi$$
−0.979736 + 0.200294i $$0.935810\pi$$
$$998$$ 0 0
$$999$$ −5.14359e6 −0.163062
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 400.6.c.o.49.3 4
4.3 odd 2 200.6.c.f.49.2 4
5.2 odd 4 400.6.a.r.1.2 2
5.3 odd 4 400.6.a.u.1.1 2
5.4 even 2 inner 400.6.c.o.49.2 4
20.3 even 4 200.6.a.e.1.2 2
20.7 even 4 200.6.a.f.1.1 yes 2
20.19 odd 2 200.6.c.f.49.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
200.6.a.e.1.2 2 20.3 even 4
200.6.a.f.1.1 yes 2 20.7 even 4
200.6.c.f.49.2 4 4.3 odd 2
200.6.c.f.49.3 4 20.19 odd 2
400.6.a.r.1.2 2 5.2 odd 4
400.6.a.u.1.1 2 5.3 odd 4
400.6.c.o.49.2 4 5.4 even 2 inner
400.6.c.o.49.3 4 1.1 even 1 trivial