# Properties

 Label 225.4.a.o.1.1 Level $225$ Weight $4$ Character 225.1 Self dual yes Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 225.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.70156 q^{2} -5.10469 q^{4} -22.2094 q^{7} +22.2984 q^{8} +O(q^{10})$$ $$q-1.70156 q^{2} -5.10469 q^{4} -22.2094 q^{7} +22.2984 q^{8} +1.79063 q^{11} -58.2094 q^{13} +37.7906 q^{14} +2.89531 q^{16} +18.9844 q^{17} +104.837 q^{19} -3.04686 q^{22} -49.6125 q^{23} +99.0469 q^{26} +113.372 q^{28} +293.466 q^{29} +64.4187 q^{31} -183.314 q^{32} -32.3031 q^{34} +19.8844 q^{37} -178.388 q^{38} +165.581 q^{41} +247.350 q^{43} -9.14059 q^{44} +84.4187 q^{46} +384.544 q^{47} +150.256 q^{49} +297.141 q^{52} +463.528 q^{53} -495.234 q^{56} -499.350 q^{58} +73.7906 q^{59} -137.350 q^{61} -109.612 q^{62} +288.758 q^{64} -173.906 q^{67} -96.9093 q^{68} +594.281 q^{71} -320.231 q^{73} -33.8345 q^{74} -535.163 q^{76} -39.7687 q^{77} -770.469 q^{79} -281.747 q^{82} +173.925 q^{83} -420.881 q^{86} +39.9282 q^{88} -1019.02 q^{89} +1292.79 q^{91} +253.256 q^{92} -654.325 q^{94} -384.375 q^{97} -255.670 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 9 q^{4} - 6 q^{7} + 51 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 9 * q^4 - 6 * q^7 + 51 * q^8 $$2 q + 3 q^{2} + 9 q^{4} - 6 q^{7} + 51 q^{8} + 42 q^{11} - 78 q^{13} + 114 q^{14} + 25 q^{16} + 102 q^{17} + 56 q^{19} + 186 q^{22} - 48 q^{23} + 6 q^{26} + 342 q^{28} + 318 q^{29} + 52 q^{31} - 309 q^{32} + 358 q^{34} - 306 q^{37} - 408 q^{38} + 408 q^{41} - 120 q^{43} + 558 q^{44} + 92 q^{46} + 180 q^{47} + 70 q^{49} + 18 q^{52} + 402 q^{53} - 30 q^{56} - 384 q^{58} + 186 q^{59} + 340 q^{61} - 168 q^{62} - 479 q^{64} - 732 q^{67} + 1074 q^{68} + 36 q^{71} - 1332 q^{73} - 1566 q^{74} - 1224 q^{76} + 612 q^{77} + 380 q^{79} + 858 q^{82} - 984 q^{83} - 2148 q^{86} + 1194 q^{88} - 1116 q^{89} + 972 q^{91} + 276 q^{92} - 1616 q^{94} + 768 q^{97} - 633 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 + 9 * q^4 - 6 * q^7 + 51 * q^8 + 42 * q^11 - 78 * q^13 + 114 * q^14 + 25 * q^16 + 102 * q^17 + 56 * q^19 + 186 * q^22 - 48 * q^23 + 6 * q^26 + 342 * q^28 + 318 * q^29 + 52 * q^31 - 309 * q^32 + 358 * q^34 - 306 * q^37 - 408 * q^38 + 408 * q^41 - 120 * q^43 + 558 * q^44 + 92 * q^46 + 180 * q^47 + 70 * q^49 + 18 * q^52 + 402 * q^53 - 30 * q^56 - 384 * q^58 + 186 * q^59 + 340 * q^61 - 168 * q^62 - 479 * q^64 - 732 * q^67 + 1074 * q^68 + 36 * q^71 - 1332 * q^73 - 1566 * q^74 - 1224 * q^76 + 612 * q^77 + 380 * q^79 + 858 * q^82 - 984 * q^83 - 2148 * q^86 + 1194 * q^88 - 1116 * q^89 + 972 * q^91 + 276 * q^92 - 1616 * q^94 + 768 * q^97 - 633 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.70156 −0.601593 −0.300797 0.953688i $$-0.597253\pi$$
−0.300797 + 0.953688i $$0.597253\pi$$
$$3$$ 0 0
$$4$$ −5.10469 −0.638086
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −22.2094 −1.19919 −0.599597 0.800302i $$-0.704672\pi$$
−0.599597 + 0.800302i $$0.704672\pi$$
$$8$$ 22.2984 0.985461
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.79063 0.0490813 0.0245407 0.999699i $$-0.492188\pi$$
0.0245407 + 0.999699i $$0.492188\pi$$
$$12$$ 0 0
$$13$$ −58.2094 −1.24188 −0.620938 0.783860i $$-0.713248\pi$$
−0.620938 + 0.783860i $$0.713248\pi$$
$$14$$ 37.7906 0.721426
$$15$$ 0 0
$$16$$ 2.89531 0.0452393
$$17$$ 18.9844 0.270846 0.135423 0.990788i $$-0.456761\pi$$
0.135423 + 0.990788i $$0.456761\pi$$
$$18$$ 0 0
$$19$$ 104.837 1.26586 0.632931 0.774208i $$-0.281852\pi$$
0.632931 + 0.774208i $$0.281852\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.04686 −0.0295270
$$23$$ −49.6125 −0.449779 −0.224890 0.974384i $$-0.572202\pi$$
−0.224890 + 0.974384i $$0.572202\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 99.0469 0.747103
$$27$$ 0 0
$$28$$ 113.372 0.765188
$$29$$ 293.466 1.87914 0.939572 0.342350i $$-0.111223\pi$$
0.939572 + 0.342350i $$0.111223\pi$$
$$30$$ 0 0
$$31$$ 64.4187 0.373224 0.186612 0.982434i $$-0.440249\pi$$
0.186612 + 0.982434i $$0.440249\pi$$
$$32$$ −183.314 −1.01268
$$33$$ 0 0
$$34$$ −32.3031 −0.162939
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 19.8844 0.0883505 0.0441752 0.999024i $$-0.485934\pi$$
0.0441752 + 0.999024i $$0.485934\pi$$
$$38$$ −178.388 −0.761534
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 165.581 0.630718 0.315359 0.948972i $$-0.397875\pi$$
0.315359 + 0.948972i $$0.397875\pi$$
$$42$$ 0 0
$$43$$ 247.350 0.877221 0.438611 0.898677i $$-0.355471\pi$$
0.438611 + 0.898677i $$0.355471\pi$$
$$44$$ −9.14059 −0.0313181
$$45$$ 0 0
$$46$$ 84.4187 0.270584
$$47$$ 384.544 1.19344 0.596718 0.802451i $$-0.296471\pi$$
0.596718 + 0.802451i $$0.296471\pi$$
$$48$$ 0 0
$$49$$ 150.256 0.438065
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 297.141 0.792423
$$53$$ 463.528 1.20133 0.600665 0.799501i $$-0.294903\pi$$
0.600665 + 0.799501i $$0.294903\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −495.234 −1.18176
$$57$$ 0 0
$$58$$ −499.350 −1.13048
$$59$$ 73.7906 0.162826 0.0814129 0.996680i $$-0.474057\pi$$
0.0814129 + 0.996680i $$0.474057\pi$$
$$60$$ 0 0
$$61$$ −137.350 −0.288293 −0.144146 0.989556i $$-0.546044\pi$$
−0.144146 + 0.989556i $$0.546044\pi$$
$$62$$ −109.612 −0.224529
$$63$$ 0 0
$$64$$ 288.758 0.563980
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −173.906 −0.317105 −0.158552 0.987351i $$-0.550683\pi$$
−0.158552 + 0.987351i $$0.550683\pi$$
$$68$$ −96.9093 −0.172823
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 594.281 0.993355 0.496677 0.867935i $$-0.334553\pi$$
0.496677 + 0.867935i $$0.334553\pi$$
$$72$$ 0 0
$$73$$ −320.231 −0.513428 −0.256714 0.966487i $$-0.582640\pi$$
−0.256714 + 0.966487i $$0.582640\pi$$
$$74$$ −33.8345 −0.0531510
$$75$$ 0 0
$$76$$ −535.163 −0.807728
$$77$$ −39.7687 −0.0588580
$$78$$ 0 0
$$79$$ −770.469 −1.09727 −0.548636 0.836061i $$-0.684853\pi$$
−0.548636 + 0.836061i $$0.684853\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −281.747 −0.379436
$$83$$ 173.925 0.230009 0.115004 0.993365i $$-0.463312\pi$$
0.115004 + 0.993365i $$0.463312\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −420.881 −0.527730
$$87$$ 0 0
$$88$$ 39.9282 0.0483677
$$89$$ −1019.02 −1.21367 −0.606834 0.794829i $$-0.707561\pi$$
−0.606834 + 0.794829i $$0.707561\pi$$
$$90$$ 0 0
$$91$$ 1292.79 1.48925
$$92$$ 253.256 0.286998
$$93$$ 0 0
$$94$$ −654.325 −0.717962
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −384.375 −0.402344 −0.201172 0.979556i $$-0.564475\pi$$
−0.201172 + 0.979556i $$0.564475\pi$$
$$98$$ −255.670 −0.263537
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −34.4906 −0.0339796 −0.0169898 0.999856i $$-0.505408\pi$$
−0.0169898 + 0.999856i $$0.505408\pi$$
$$102$$ 0 0
$$103$$ 1756.30 1.68013 0.840066 0.542484i $$-0.182516\pi$$
0.840066 + 0.542484i $$0.182516\pi$$
$$104$$ −1297.98 −1.22382
$$105$$ 0 0
$$106$$ −788.722 −0.722712
$$107$$ −1361.74 −1.23032 −0.615159 0.788403i $$-0.710908\pi$$
−0.615159 + 0.788403i $$0.710908\pi$$
$$108$$ 0 0
$$109$$ 321.119 0.282180 0.141090 0.989997i $$-0.454939\pi$$
0.141090 + 0.989997i $$0.454939\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −64.3031 −0.0542506
$$113$$ −1582.25 −1.31721 −0.658607 0.752487i $$-0.728854\pi$$
−0.658607 + 0.752487i $$0.728854\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1498.05 −1.19906
$$117$$ 0 0
$$118$$ −125.559 −0.0979549
$$119$$ −421.631 −0.324797
$$120$$ 0 0
$$121$$ −1327.79 −0.997591
$$122$$ 233.709 0.173435
$$123$$ 0 0
$$124$$ −328.837 −0.238149
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1197.14 0.836449 0.418225 0.908344i $$-0.362652\pi$$
0.418225 + 0.908344i $$0.362652\pi$$
$$128$$ 975.173 0.673390
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 321.647 0.214522 0.107261 0.994231i $$-0.465792\pi$$
0.107261 + 0.994231i $$0.465792\pi$$
$$132$$ 0 0
$$133$$ −2328.37 −1.51801
$$134$$ 295.912 0.190768
$$135$$ 0 0
$$136$$ 423.322 0.266909
$$137$$ 354.291 0.220942 0.110471 0.993879i $$-0.464764\pi$$
0.110471 + 0.993879i $$0.464764\pi$$
$$138$$ 0 0
$$139$$ 77.2562 0.0471424 0.0235712 0.999722i $$-0.492496\pi$$
0.0235712 + 0.999722i $$0.492496\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1011.21 −0.597595
$$143$$ −104.231 −0.0609529
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 544.893 0.308875
$$147$$ 0 0
$$148$$ −101.503 −0.0563752
$$149$$ 1705.38 0.937651 0.468826 0.883291i $$-0.344677\pi$$
0.468826 + 0.883291i $$0.344677\pi$$
$$150$$ 0 0
$$151$$ 758.281 0.408663 0.204331 0.978902i $$-0.434498\pi$$
0.204331 + 0.978902i $$0.434498\pi$$
$$152$$ 2337.71 1.24746
$$153$$ 0 0
$$154$$ 67.6689 0.0354086
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1769.05 0.899273 0.449636 0.893212i $$-0.351554\pi$$
0.449636 + 0.893212i $$0.351554\pi$$
$$158$$ 1311.00 0.660111
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1101.86 0.539372
$$162$$ 0 0
$$163$$ 881.719 0.423690 0.211845 0.977303i $$-0.432053\pi$$
0.211845 + 0.977303i $$0.432053\pi$$
$$164$$ −845.240 −0.402452
$$165$$ 0 0
$$166$$ −295.944 −0.138372
$$167$$ −216.900 −0.100504 −0.0502522 0.998737i $$-0.516003\pi$$
−0.0502522 + 0.998737i $$0.516003\pi$$
$$168$$ 0 0
$$169$$ 1191.33 0.542254
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1262.64 −0.559742
$$173$$ 4125.91 1.81322 0.906610 0.421970i $$-0.138661\pi$$
0.906610 + 0.421970i $$0.138661\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 5.18443 0.00222040
$$177$$ 0 0
$$178$$ 1733.93 0.730134
$$179$$ 3213.14 1.34168 0.670842 0.741600i $$-0.265933\pi$$
0.670842 + 0.741600i $$0.265933\pi$$
$$180$$ 0 0
$$181$$ 3394.42 1.39395 0.696976 0.717095i $$-0.254529\pi$$
0.696976 + 0.717095i $$0.254529\pi$$
$$182$$ −2199.77 −0.895921
$$183$$ 0 0
$$184$$ −1106.28 −0.443240
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 33.9939 0.0132935
$$188$$ −1962.98 −0.761514
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3467.49 1.31361 0.656804 0.754062i $$-0.271908\pi$$
0.656804 + 0.754062i $$0.271908\pi$$
$$192$$ 0 0
$$193$$ −1792.14 −0.668401 −0.334200 0.942502i $$-0.608466\pi$$
−0.334200 + 0.942502i $$0.608466\pi$$
$$194$$ 654.038 0.242047
$$195$$ 0 0
$$196$$ −767.011 −0.279523
$$197$$ 1678.19 0.606935 0.303467 0.952842i $$-0.401856\pi$$
0.303467 + 0.952842i $$0.401856\pi$$
$$198$$ 0 0
$$199$$ 3108.23 1.10722 0.553610 0.832776i $$-0.313250\pi$$
0.553610 + 0.832776i $$0.313250\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 58.6878 0.0204419
$$203$$ −6517.69 −2.25346
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2988.46 −1.01076
$$207$$ 0 0
$$208$$ −168.534 −0.0561815
$$209$$ 187.725 0.0621301
$$210$$ 0 0
$$211$$ −4473.27 −1.45949 −0.729745 0.683719i $$-0.760361\pi$$
−0.729745 + 0.683719i $$0.760361\pi$$
$$212$$ −2366.17 −0.766551
$$213$$ 0 0
$$214$$ 2317.08 0.740151
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1430.70 −0.447568
$$218$$ −546.403 −0.169757
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1105.07 −0.336357
$$222$$ 0 0
$$223$$ 1753.42 0.526535 0.263268 0.964723i $$-0.415200\pi$$
0.263268 + 0.964723i $$0.415200\pi$$
$$224$$ 4071.29 1.21440
$$225$$ 0 0
$$226$$ 2692.29 0.792427
$$227$$ −936.900 −0.273939 −0.136970 0.990575i $$-0.543736\pi$$
−0.136970 + 0.990575i $$0.543736\pi$$
$$228$$ 0 0
$$229$$ −2582.06 −0.745096 −0.372548 0.928013i $$-0.621516\pi$$
−0.372548 + 0.928013i $$0.621516\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6543.82 1.85182
$$233$$ 2295.01 0.645284 0.322642 0.946521i $$-0.395429\pi$$
0.322642 + 0.946521i $$0.395429\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −376.678 −0.103897
$$237$$ 0 0
$$238$$ 717.432 0.195396
$$239$$ 2294.01 0.620866 0.310433 0.950595i $$-0.399526\pi$$
0.310433 + 0.950595i $$0.399526\pi$$
$$240$$ 0 0
$$241$$ 382.287 0.102180 0.0510898 0.998694i $$-0.483731\pi$$
0.0510898 + 0.998694i $$0.483731\pi$$
$$242$$ 2259.32 0.600144
$$243$$ 0 0
$$244$$ 701.128 0.183956
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6102.52 −1.57204
$$248$$ 1436.44 0.367798
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2259.98 0.568322 0.284161 0.958777i $$-0.408285\pi$$
0.284161 + 0.958777i $$0.408285\pi$$
$$252$$ 0 0
$$253$$ −88.8375 −0.0220758
$$254$$ −2037.01 −0.503202
$$255$$ 0 0
$$256$$ −3969.38 −0.969087
$$257$$ −92.7843 −0.0225203 −0.0112602 0.999937i $$-0.503584\pi$$
−0.0112602 + 0.999937i $$0.503584\pi$$
$$258$$ 0 0
$$259$$ −441.619 −0.105949
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −547.302 −0.129055
$$263$$ −568.312 −0.133246 −0.0666229 0.997778i $$-0.521222\pi$$
−0.0666229 + 0.997778i $$0.521222\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 3961.87 0.913226
$$267$$ 0 0
$$268$$ 887.737 0.202340
$$269$$ −7582.41 −1.71862 −0.859309 0.511458i $$-0.829106\pi$$
−0.859309 + 0.511458i $$0.829106\pi$$
$$270$$ 0 0
$$271$$ 7943.69 1.78061 0.890304 0.455366i $$-0.150492\pi$$
0.890304 + 0.455366i $$0.150492\pi$$
$$272$$ 54.9657 0.0122529
$$273$$ 0 0
$$274$$ −602.847 −0.132917
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6823.00 −1.47998 −0.739990 0.672618i $$-0.765170\pi$$
−0.739990 + 0.672618i $$0.765170\pi$$
$$278$$ −131.456 −0.0283605
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3315.86 −0.703942 −0.351971 0.936011i $$-0.614488\pi$$
−0.351971 + 0.936011i $$0.614488\pi$$
$$282$$ 0 0
$$283$$ −6602.76 −1.38690 −0.693451 0.720504i $$-0.743910\pi$$
−0.693451 + 0.720504i $$0.743910\pi$$
$$284$$ −3033.62 −0.633846
$$285$$ 0 0
$$286$$ 177.356 0.0366688
$$287$$ −3677.46 −0.756353
$$288$$ 0 0
$$289$$ −4552.59 −0.926642
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 1634.68 0.327611
$$293$$ 5814.14 1.15927 0.579634 0.814877i $$-0.303195\pi$$
0.579634 + 0.814877i $$0.303195\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 443.390 0.0870660
$$297$$ 0 0
$$298$$ −2901.81 −0.564084
$$299$$ 2887.91 0.558570
$$300$$ 0 0
$$301$$ −5493.49 −1.05196
$$302$$ −1290.26 −0.245849
$$303$$ 0 0
$$304$$ 303.537 0.0572667
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 8124.86 1.51046 0.755229 0.655462i $$-0.227526\pi$$
0.755229 + 0.655462i $$0.227526\pi$$
$$308$$ 203.007 0.0375564
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −7336.26 −1.33762 −0.668812 0.743432i $$-0.733197\pi$$
−0.668812 + 0.743432i $$0.733197\pi$$
$$312$$ 0 0
$$313$$ 2202.66 0.397768 0.198884 0.980023i $$-0.436268\pi$$
0.198884 + 0.980023i $$0.436268\pi$$
$$314$$ −3010.15 −0.540996
$$315$$ 0 0
$$316$$ 3933.00 0.700154
$$317$$ 10008.9 1.77336 0.886679 0.462386i $$-0.153007\pi$$
0.886679 + 0.462386i $$0.153007\pi$$
$$318$$ 0 0
$$319$$ 525.488 0.0922309
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1874.89 −0.324483
$$323$$ 1990.27 0.342854
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −1500.30 −0.254889
$$327$$ 0 0
$$328$$ 3692.20 0.621548
$$329$$ −8540.47 −1.43116
$$330$$ 0 0
$$331$$ −8695.94 −1.44402 −0.722012 0.691881i $$-0.756782\pi$$
−0.722012 + 0.691881i $$0.756782\pi$$
$$332$$ −887.832 −0.146765
$$333$$ 0 0
$$334$$ 369.069 0.0604627
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7400.61 1.19625 0.598126 0.801402i $$-0.295912\pi$$
0.598126 + 0.801402i $$0.295912\pi$$
$$338$$ −2027.12 −0.326216
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 115.350 0.0183183
$$342$$ 0 0
$$343$$ 4280.72 0.673869
$$344$$ 5515.52 0.864467
$$345$$ 0 0
$$346$$ −7020.49 −1.09082
$$347$$ 7841.44 1.21311 0.606557 0.795040i $$-0.292550\pi$$
0.606557 + 0.795040i $$0.292550\pi$$
$$348$$ 0 0
$$349$$ −4961.26 −0.760946 −0.380473 0.924792i $$-0.624239\pi$$
−0.380473 + 0.924792i $$0.624239\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −328.247 −0.0497035
$$353$$ −12163.0 −1.83392 −0.916959 0.398981i $$-0.869364\pi$$
−0.916959 + 0.398981i $$0.869364\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 5201.80 0.774424
$$357$$ 0 0
$$358$$ −5467.36 −0.807148
$$359$$ −5193.79 −0.763559 −0.381779 0.924253i $$-0.624689\pi$$
−0.381779 + 0.924253i $$0.624689\pi$$
$$360$$ 0 0
$$361$$ 4131.90 0.602406
$$362$$ −5775.81 −0.838591
$$363$$ 0 0
$$364$$ −6599.31 −0.950268
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 6086.09 0.865644 0.432822 0.901479i $$-0.357518\pi$$
0.432822 + 0.901479i $$0.357518\pi$$
$$368$$ −143.644 −0.0203477
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −10294.7 −1.44063
$$372$$ 0 0
$$373$$ 10581.9 1.46893 0.734466 0.678646i $$-0.237433\pi$$
0.734466 + 0.678646i $$0.237433\pi$$
$$374$$ −57.8428 −0.00799727
$$375$$ 0 0
$$376$$ 8574.72 1.17608
$$377$$ −17082.4 −2.33366
$$378$$ 0 0
$$379$$ 11655.2 1.57964 0.789822 0.613336i $$-0.210173\pi$$
0.789822 + 0.613336i $$0.210173\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −5900.16 −0.790257
$$383$$ 6364.97 0.849177 0.424588 0.905387i $$-0.360419\pi$$
0.424588 + 0.905387i $$0.360419\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3049.44 0.402105
$$387$$ 0 0
$$388$$ 1962.11 0.256730
$$389$$ −6134.33 −0.799545 −0.399773 0.916614i $$-0.630911\pi$$
−0.399773 + 0.916614i $$0.630911\pi$$
$$390$$ 0 0
$$391$$ −941.862 −0.121821
$$392$$ 3350.48 0.431696
$$393$$ 0 0
$$394$$ −2855.55 −0.365128
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −9746.46 −1.23214 −0.616072 0.787690i $$-0.711277\pi$$
−0.616072 + 0.787690i $$0.711277\pi$$
$$398$$ −5288.85 −0.666096
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1306.44 0.162695 0.0813474 0.996686i $$-0.474078\pi$$
0.0813474 + 0.996686i $$0.474078\pi$$
$$402$$ 0 0
$$403$$ −3749.77 −0.463498
$$404$$ 176.063 0.0216819
$$405$$ 0 0
$$406$$ 11090.2 1.35566
$$407$$ 35.6055 0.00433636
$$408$$ 0 0
$$409$$ −3876.93 −0.468709 −0.234354 0.972151i $$-0.575298\pi$$
−0.234354 + 0.972151i $$0.575298\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −8965.38 −1.07207
$$413$$ −1638.84 −0.195260
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10670.6 1.25762
$$417$$ 0 0
$$418$$ −319.426 −0.0373771
$$419$$ 16022.5 1.86814 0.934071 0.357088i $$-0.116230\pi$$
0.934071 + 0.357088i $$0.116230\pi$$
$$420$$ 0 0
$$421$$ −8119.73 −0.939980 −0.469990 0.882672i $$-0.655742\pi$$
−0.469990 + 0.882672i $$0.655742\pi$$
$$422$$ 7611.54 0.878019
$$423$$ 0 0
$$424$$ 10336.0 1.18386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3050.46 0.345719
$$428$$ 6951.24 0.785049
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5713.99 0.638592 0.319296 0.947655i $$-0.396554\pi$$
0.319296 + 0.947655i $$0.396554\pi$$
$$432$$ 0 0
$$433$$ −6251.34 −0.693811 −0.346906 0.937900i $$-0.612768\pi$$
−0.346906 + 0.937900i $$0.612768\pi$$
$$434$$ 2434.42 0.269254
$$435$$ 0 0
$$436$$ −1639.21 −0.180055
$$437$$ −5201.25 −0.569358
$$438$$ 0 0
$$439$$ −4230.97 −0.459984 −0.229992 0.973192i $$-0.573870\pi$$
−0.229992 + 0.973192i $$0.573870\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 1880.34 0.202350
$$443$$ 6314.29 0.677203 0.338601 0.940930i $$-0.390046\pi$$
0.338601 + 0.940930i $$0.390046\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −2983.55 −0.316760
$$447$$ 0 0
$$448$$ −6413.13 −0.676321
$$449$$ 9349.71 0.982717 0.491358 0.870957i $$-0.336501\pi$$
0.491358 + 0.870957i $$0.336501\pi$$
$$450$$ 0 0
$$451$$ 296.494 0.0309565
$$452$$ 8076.87 0.840496
$$453$$ 0 0
$$454$$ 1594.19 0.164800
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9547.46 0.977268 0.488634 0.872489i $$-0.337495\pi$$
0.488634 + 0.872489i $$0.337495\pi$$
$$458$$ 4393.53 0.448245
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6237.23 −0.630145 −0.315073 0.949068i $$-0.602029\pi$$
−0.315073 + 0.949068i $$0.602029\pi$$
$$462$$ 0 0
$$463$$ −6469.98 −0.649428 −0.324714 0.945812i $$-0.605268\pi$$
−0.324714 + 0.945812i $$0.605268\pi$$
$$464$$ 849.675 0.0850111
$$465$$ 0 0
$$466$$ −3905.10 −0.388198
$$467$$ 7206.64 0.714097 0.357049 0.934086i $$-0.383783\pi$$
0.357049 + 0.934086i $$0.383783\pi$$
$$468$$ 0 0
$$469$$ 3862.35 0.380270
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 1645.42 0.160458
$$473$$ 442.912 0.0430552
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2152.29 0.207248
$$477$$ 0 0
$$478$$ −3903.39 −0.373509
$$479$$ 10851.8 1.03514 0.517571 0.855640i $$-0.326836\pi$$
0.517571 + 0.855640i $$0.326836\pi$$
$$480$$ 0 0
$$481$$ −1157.46 −0.109720
$$482$$ −650.485 −0.0614705
$$483$$ 0 0
$$484$$ 6777.97 0.636549
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −12757.1 −1.18702 −0.593510 0.804827i $$-0.702258\pi$$
−0.593510 + 0.804827i $$0.702258\pi$$
$$488$$ −3062.69 −0.284101
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7016.52 0.644911 0.322455 0.946585i $$-0.395492\pi$$
0.322455 + 0.946585i $$0.395492\pi$$
$$492$$ 0 0
$$493$$ 5571.26 0.508960
$$494$$ 10383.8 0.945729
$$495$$ 0 0
$$496$$ 186.512 0.0168844
$$497$$ −13198.6 −1.19122
$$498$$ 0 0
$$499$$ 11372.3 1.02023 0.510113 0.860107i $$-0.329604\pi$$
0.510113 + 0.860107i $$0.329604\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −3845.50 −0.341899
$$503$$ −5587.37 −0.495285 −0.247643 0.968851i $$-0.579656\pi$$
−0.247643 + 0.968851i $$0.579656\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 151.163 0.0132806
$$507$$ 0 0
$$508$$ −6111.03 −0.533726
$$509$$ −16256.7 −1.41565 −0.707825 0.706388i $$-0.750324\pi$$
−0.707825 + 0.706388i $$0.750324\pi$$
$$510$$ 0 0
$$511$$ 7112.14 0.615699
$$512$$ −1047.24 −0.0903943
$$513$$ 0 0
$$514$$ 157.878 0.0135481
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 688.574 0.0585754
$$518$$ 751.442 0.0637384
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −19748.4 −1.66064 −0.830320 0.557286i $$-0.811843\pi$$
−0.830320 + 0.557286i $$0.811843\pi$$
$$522$$ 0 0
$$523$$ 7843.44 0.655774 0.327887 0.944717i $$-0.393663\pi$$
0.327887 + 0.944717i $$0.393663\pi$$
$$524$$ −1641.91 −0.136884
$$525$$ 0 0
$$526$$ 967.019 0.0801597
$$527$$ 1222.95 0.101086
$$528$$ 0 0
$$529$$ −9705.60 −0.797699
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 11885.6 0.968622
$$533$$ −9638.38 −0.783273
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −3877.84 −0.312495
$$537$$ 0 0
$$538$$ 12902.0 1.03391
$$539$$ 269.053 0.0215008
$$540$$ 0 0
$$541$$ 7383.29 0.586751 0.293376 0.955997i $$-0.405221\pi$$
0.293376 + 0.955997i $$0.405221\pi$$
$$542$$ −13516.7 −1.07120
$$543$$ 0 0
$$544$$ −3480.10 −0.274280
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3354.90 0.262240 0.131120 0.991367i $$-0.458143\pi$$
0.131120 + 0.991367i $$0.458143\pi$$
$$548$$ −1808.54 −0.140980
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 30766.2 2.37874
$$552$$ 0 0
$$553$$ 17111.6 1.31584
$$554$$ 11609.8 0.890346
$$555$$ 0 0
$$556$$ −394.369 −0.0300809
$$557$$ 20771.8 1.58012 0.790061 0.613028i $$-0.210049\pi$$
0.790061 + 0.613028i $$0.210049\pi$$
$$558$$ 0 0
$$559$$ −14398.1 −1.08940
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5642.15 0.423487
$$563$$ 7194.86 0.538592 0.269296 0.963057i $$-0.413209\pi$$
0.269296 + 0.963057i $$0.413209\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 11235.0 0.834350
$$567$$ 0 0
$$568$$ 13251.5 0.978913
$$569$$ −11549.5 −0.850931 −0.425466 0.904975i $$-0.639890\pi$$
−0.425466 + 0.904975i $$0.639890\pi$$
$$570$$ 0 0
$$571$$ 1482.54 0.108655 0.0543277 0.998523i $$-0.482698\pi$$
0.0543277 + 0.998523i $$0.482698\pi$$
$$572$$ 532.068 0.0388932
$$573$$ 0 0
$$574$$ 6257.42 0.455017
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 15264.0 1.10130 0.550649 0.834737i $$-0.314380\pi$$
0.550649 + 0.834737i $$0.314380\pi$$
$$578$$ 7746.52 0.557462
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3862.76 −0.275825
$$582$$ 0 0
$$583$$ 830.006 0.0589628
$$584$$ −7140.66 −0.505963
$$585$$ 0 0
$$586$$ −9893.12 −0.697408
$$587$$ 1736.89 0.122128 0.0610639 0.998134i $$-0.480551\pi$$
0.0610639 + 0.998134i $$0.480551\pi$$
$$588$$ 0 0
$$589$$ 6753.50 0.472450
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 57.5714 0.00399691
$$593$$ −11764.8 −0.814707 −0.407353 0.913271i $$-0.633548\pi$$
−0.407353 + 0.913271i $$0.633548\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −8705.42 −0.598302
$$597$$ 0 0
$$598$$ −4913.96 −0.336032
$$599$$ 9451.99 0.644737 0.322369 0.946614i $$-0.395521\pi$$
0.322369 + 0.946614i $$0.395521\pi$$
$$600$$ 0 0
$$601$$ −3131.93 −0.212569 −0.106285 0.994336i $$-0.533895\pi$$
−0.106285 + 0.994336i $$0.533895\pi$$
$$602$$ 9347.51 0.632851
$$603$$ 0 0
$$604$$ −3870.79 −0.260762
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22700.8 −1.51795 −0.758975 0.651120i $$-0.774300\pi$$
−0.758975 + 0.651120i $$0.774300\pi$$
$$608$$ −19218.2 −1.28191
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −22384.0 −1.48210
$$612$$ 0 0
$$613$$ −28911.6 −1.90494 −0.952471 0.304629i $$-0.901468\pi$$
−0.952471 + 0.304629i $$0.901468\pi$$
$$614$$ −13825.0 −0.908681
$$615$$ 0 0
$$616$$ −886.780 −0.0580023
$$617$$ −5566.87 −0.363231 −0.181616 0.983370i $$-0.558133\pi$$
−0.181616 + 0.983370i $$0.558133\pi$$
$$618$$ 0 0
$$619$$ 4150.32 0.269492 0.134746 0.990880i $$-0.456978\pi$$
0.134746 + 0.990880i $$0.456978\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 12483.1 0.804705
$$623$$ 22631.9 1.45542
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −3747.96 −0.239295
$$627$$ 0 0
$$628$$ −9030.46 −0.573813
$$629$$ 377.492 0.0239294
$$630$$ 0 0
$$631$$ −4090.09 −0.258041 −0.129021 0.991642i $$-0.541183\pi$$
−0.129021 + 0.991642i $$0.541183\pi$$
$$632$$ −17180.2 −1.08132
$$633$$ 0 0
$$634$$ −17030.7 −1.06684
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −8746.32 −0.544022
$$638$$ −894.150 −0.0554855
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −3909.35 −0.240890 −0.120445 0.992720i $$-0.538432\pi$$
−0.120445 + 0.992720i $$0.538432\pi$$
$$642$$ 0 0
$$643$$ −30539.5 −1.87303 −0.936516 0.350624i $$-0.885969\pi$$
−0.936516 + 0.350624i $$0.885969\pi$$
$$644$$ −5624.66 −0.344166
$$645$$ 0 0
$$646$$ −3386.58 −0.206259
$$647$$ −12707.7 −0.772167 −0.386083 0.922464i $$-0.626172\pi$$
−0.386083 + 0.922464i $$0.626172\pi$$
$$648$$ 0 0
$$649$$ 132.132 0.00799170
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4500.90 −0.270351
$$653$$ 12777.6 0.765737 0.382869 0.923803i $$-0.374936\pi$$
0.382869 + 0.923803i $$0.374936\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 479.410 0.0285332
$$657$$ 0 0
$$658$$ 14532.1 0.860976
$$659$$ −23563.5 −1.39287 −0.696435 0.717620i $$-0.745232\pi$$
−0.696435 + 0.717620i $$0.745232\pi$$
$$660$$ 0 0
$$661$$ −4361.31 −0.256634 −0.128317 0.991733i $$-0.540958\pi$$
−0.128317 + 0.991733i $$0.540958\pi$$
$$662$$ 14796.7 0.868715
$$663$$ 0 0
$$664$$ 3878.25 0.226665
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −14559.6 −0.845200
$$668$$ 1107.21 0.0641304
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −245.943 −0.0141498
$$672$$ 0 0
$$673$$ 8203.52 0.469870 0.234935 0.972011i $$-0.424512\pi$$
0.234935 + 0.972011i $$0.424512\pi$$
$$674$$ −12592.6 −0.719657
$$675$$ 0 0
$$676$$ −6081.37 −0.346004
$$677$$ 28057.1 1.59279 0.796397 0.604774i $$-0.206737\pi$$
0.796397 + 0.604774i $$0.206737\pi$$
$$678$$ 0 0
$$679$$ 8536.73 0.482488
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −196.275 −0.0110202
$$683$$ −3344.62 −0.187377 −0.0936885 0.995602i $$-0.529866\pi$$
−0.0936885 + 0.995602i $$0.529866\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −7283.91 −0.405395
$$687$$ 0 0
$$688$$ 716.156 0.0396849
$$689$$ −26981.7 −1.49190
$$690$$ 0 0
$$691$$ 12964.8 0.713757 0.356879 0.934151i $$-0.383841\pi$$
0.356879 + 0.934151i $$0.383841\pi$$
$$692$$ −21061.5 −1.15699
$$693$$ 0 0
$$694$$ −13342.7 −0.729801
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 3143.46 0.170828
$$698$$ 8441.90 0.457780
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 16162.1 0.870806 0.435403 0.900236i $$-0.356606\pi$$
0.435403 + 0.900236i $$0.356606\pi$$
$$702$$ 0 0
$$703$$ 2084.63 0.111839
$$704$$ 517.058 0.0276809
$$705$$ 0 0
$$706$$ 20696.2 1.10327
$$707$$ 766.014 0.0407481
$$708$$ 0 0
$$709$$ −14244.4 −0.754529 −0.377265 0.926105i $$-0.623135\pi$$
−0.377265 + 0.926105i $$0.623135\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −22722.7 −1.19602
$$713$$ −3195.97 −0.167868
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −16402.1 −0.856109
$$717$$ 0 0
$$718$$ 8837.55 0.459352
$$719$$ 27638.5 1.43358 0.716790 0.697289i $$-0.245611\pi$$
0.716790 + 0.697289i $$0.245611\pi$$
$$720$$ 0 0
$$721$$ −39006.4 −2.01480
$$722$$ −7030.68 −0.362403
$$723$$ 0 0
$$724$$ −17327.4 −0.889460
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2525.52 0.128840 0.0644199 0.997923i $$-0.479480\pi$$
0.0644199 + 0.997923i $$0.479480\pi$$
$$728$$ 28827.3 1.46760
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 4695.79 0.237592
$$732$$ 0 0
$$733$$ 8400.27 0.423289 0.211645 0.977347i $$-0.432118\pi$$
0.211645 + 0.977347i $$0.432118\pi$$
$$734$$ −10355.9 −0.520765
$$735$$ 0 0
$$736$$ 9094.67 0.455481
$$737$$ −311.401 −0.0155639
$$738$$ 0 0
$$739$$ −19689.1 −0.980074 −0.490037 0.871702i $$-0.663017\pi$$
−0.490037 + 0.871702i $$0.663017\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 17517.0 0.866671
$$743$$ −22526.6 −1.11227 −0.556137 0.831091i $$-0.687717\pi$$
−0.556137 + 0.831091i $$0.687717\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −18005.8 −0.883699
$$747$$ 0 0
$$748$$ −173.528 −0.00848239
$$749$$ 30243.3 1.47539
$$750$$ 0 0
$$751$$ 34691.1 1.68562 0.842808 0.538215i $$-0.180901\pi$$
0.842808 + 0.538215i $$0.180901\pi$$
$$752$$ 1113.37 0.0539902
$$753$$ 0 0
$$754$$ 29066.8 1.40392
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6619.98 −0.317843 −0.158922 0.987291i $$-0.550802\pi$$
−0.158922 + 0.987291i $$0.550802\pi$$
$$758$$ −19832.0 −0.950303
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 29368.7 1.39897 0.699483 0.714649i $$-0.253414\pi$$
0.699483 + 0.714649i $$0.253414\pi$$
$$762$$ 0 0
$$763$$ −7131.84 −0.338388
$$764$$ −17700.5 −0.838194
$$765$$ 0 0
$$766$$ −10830.4 −0.510859
$$767$$ −4295.31 −0.202209
$$768$$ 0 0
$$769$$ 32677.4 1.53235 0.766174 0.642633i $$-0.222158\pi$$
0.766174 + 0.642633i $$0.222158\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 9148.33 0.426497
$$773$$ −28047.5 −1.30504 −0.652522 0.757770i $$-0.726289\pi$$
−0.652522 + 0.757770i $$0.726289\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −8570.96 −0.396494
$$777$$ 0 0
$$778$$ 10438.0 0.481001
$$779$$ 17359.1 0.798402
$$780$$ 0 0
$$781$$ 1064.14 0.0487552
$$782$$ 1602.64 0.0732867
$$783$$ 0 0
$$784$$ 435.039 0.0198177
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22172.1 1.00426 0.502128 0.864793i $$-0.332551\pi$$
0.502128 + 0.864793i $$0.332551\pi$$
$$788$$ −8566.64 −0.387276
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 35140.7 1.57960
$$792$$ 0 0
$$793$$ 7995.06 0.358024
$$794$$ 16584.2 0.741249
$$795$$ 0 0
$$796$$ −15866.5 −0.706501
$$797$$ −24170.3 −1.07422 −0.537112 0.843511i $$-0.680485\pi$$
−0.537112 + 0.843511i $$0.680485\pi$$
$$798$$ 0 0
$$799$$ 7300.32 0.323238
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −2222.99 −0.0978761
$$803$$ −573.415 −0.0251997
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6380.47 0.278837
$$807$$ 0 0
$$808$$ −769.085 −0.0334856
$$809$$ −15304.2 −0.665102 −0.332551 0.943085i $$-0.607909\pi$$
−0.332551 + 0.943085i $$0.607909\pi$$
$$810$$ 0 0
$$811$$ −27002.2 −1.16914 −0.584572 0.811342i $$-0.698738\pi$$
−0.584572 + 0.811342i $$0.698738\pi$$
$$812$$ 33270.7 1.43790
$$813$$ 0 0
$$814$$ −60.5849 −0.00260872
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 25931.5 1.11044
$$818$$ 6596.84 0.281972
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25061.4 −1.06535 −0.532673 0.846321i $$-0.678812\pi$$
−0.532673 + 0.846321i $$0.678812\pi$$
$$822$$ 0 0
$$823$$ 24896.4 1.05448 0.527238 0.849718i $$-0.323228\pi$$
0.527238 + 0.849718i $$0.323228\pi$$
$$824$$ 39162.8 1.65571
$$825$$ 0 0
$$826$$ 2788.59 0.117467
$$827$$ −20063.2 −0.843612 −0.421806 0.906686i $$-0.638604\pi$$
−0.421806 + 0.906686i $$0.638604\pi$$
$$828$$ 0 0
$$829$$ −13884.2 −0.581687 −0.290844 0.956771i $$-0.593936\pi$$
−0.290844 + 0.956771i $$0.593936\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −16808.4 −0.700393
$$833$$ 2852.52 0.118648
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −958.277 −0.0396444
$$837$$ 0 0
$$838$$ −27263.3 −1.12386
$$839$$ 13678.1 0.562838 0.281419 0.959585i $$-0.409195\pi$$
0.281419 + 0.959585i $$0.409195\pi$$
$$840$$ 0 0
$$841$$ 61733.1 2.53118
$$842$$ 13816.2 0.565485
$$843$$ 0 0
$$844$$ 22834.6 0.931280
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 29489.5 1.19630
$$848$$ 1342.06 0.0543473
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −986.512 −0.0397382
$$852$$ 0 0
$$853$$ −29802.9 −1.19629 −0.598143 0.801390i $$-0.704094\pi$$
−0.598143 + 0.801390i $$0.704094\pi$$
$$854$$ −5190.54 −0.207982
$$855$$ 0 0
$$856$$ −30364.6 −1.21243
$$857$$ 22045.2 0.878706 0.439353 0.898314i $$-0.355208\pi$$
0.439353 + 0.898314i $$0.355208\pi$$
$$858$$ 0 0
$$859$$ 33609.5 1.33497 0.667487 0.744622i $$-0.267370\pi$$
0.667487 + 0.744622i $$0.267370\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −9722.70 −0.384172
$$863$$ 33775.6 1.33226 0.666128 0.745838i $$-0.267951\pi$$
0.666128 + 0.745838i $$0.267951\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 10637.0 0.417392
$$867$$ 0 0
$$868$$ 7303.27 0.285587
$$869$$ −1379.62 −0.0538556
$$870$$ 0 0
$$871$$ 10123.0 0.393805
$$872$$ 7160.44 0.278077
$$873$$ 0 0
$$874$$ 8850.25 0.342522
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12637.0 −0.486570 −0.243285 0.969955i $$-0.578225\pi$$
−0.243285 + 0.969955i $$0.578225\pi$$
$$878$$ 7199.25 0.276723
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 6579.45 0.251609 0.125804 0.992055i $$-0.459849\pi$$
0.125804 + 0.992055i $$0.459849\pi$$
$$882$$ 0 0
$$883$$ 50442.1 1.92244 0.961219 0.275786i $$-0.0889382\pi$$
0.961219 + 0.275786i $$0.0889382\pi$$
$$884$$ 5641.03 0.214625
$$885$$ 0 0
$$886$$ −10744.2 −0.407401
$$887$$ 984.823 0.0372797 0.0186399 0.999826i $$-0.494066\pi$$
0.0186399 + 0.999826i $$0.494066\pi$$
$$888$$ 0 0
$$889$$ −26587.7 −1.00306
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −8950.64 −0.335975
$$893$$ 40314.6 1.51072
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −21658.0 −0.807525
$$897$$ 0 0
$$898$$ −15909.1 −0.591196
$$899$$ 18904.7 0.701342
$$900$$ 0 0
$$901$$ 8799.79 0.325376
$$902$$ −504.503 −0.0186232
$$903$$ 0 0
$$904$$ −35281.6 −1.29806
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −43679.9 −1.59908 −0.799541 0.600612i $$-0.794924\pi$$
−0.799541 + 0.600612i $$0.794924\pi$$
$$908$$ 4782.58 0.174797
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 10364.3 0.376930 0.188465 0.982080i $$-0.439649\pi$$
0.188465 + 0.982080i $$0.439649\pi$$
$$912$$ 0 0
$$913$$ 311.435 0.0112891
$$914$$ −16245.6 −0.587918
$$915$$ 0 0
$$916$$ 13180.6 0.475435
$$917$$ −7143.58 −0.257254
$$918$$ 0 0
$$919$$ 11451.9 0.411059 0.205530 0.978651i $$-0.434108\pi$$
0.205530 + 0.978651i $$0.434108\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 10613.0 0.379091
$$923$$ −34592.7 −1.23362
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 11009.1 0.390692
$$927$$ 0 0
$$928$$ −53796.4 −1.90297
$$929$$ 27701.8 0.978326 0.489163 0.872192i $$-0.337302\pi$$
0.489163 + 0.872192i $$0.337302\pi$$
$$930$$ 0 0
$$931$$ 15752.5 0.554529
$$932$$ −11715.3 −0.411746
$$933$$ 0 0
$$934$$ −12262.5 −0.429596
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 5878.01 0.204937 0.102469 0.994736i $$-0.467326\pi$$
0.102469 + 0.994736i $$0.467326\pi$$
$$938$$ −6572.03 −0.228768
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 28786.0 0.997234 0.498617 0.866823i $$-0.333841\pi$$
0.498617 + 0.866823i $$0.333841\pi$$
$$942$$ 0 0
$$943$$ −8214.90 −0.283684
$$944$$ 213.647 0.00736612
$$945$$ 0 0
$$946$$ −753.642 −0.0259017
$$947$$ −1695.04 −0.0581641 −0.0290821 0.999577i $$-0.509258\pi$$
−0.0290821 + 0.999577i $$0.509258\pi$$
$$948$$ 0 0
$$949$$ 18640.5 0.637613
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −9401.72 −0.320075
$$953$$ 31929.4 1.08530 0.542651 0.839958i $$-0.317420\pi$$
0.542651 + 0.839958i $$0.317420\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −11710.2 −0.396166
$$957$$ 0 0
$$958$$ −18465.1 −0.622735
$$959$$ −7868.57 −0.264952
$$960$$ 0 0
$$961$$ −25641.2 −0.860704
$$962$$ 1969.48 0.0660069
$$963$$ 0 0
$$964$$ −1951.46 −0.0651994
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 10897.1 0.362385 0.181193 0.983448i $$-0.442004\pi$$
0.181193 + 0.983448i $$0.442004\pi$$
$$968$$ −29607.7 −0.983087
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −7041.97 −0.232737 −0.116368 0.993206i $$-0.537125\pi$$
−0.116368 + 0.993206i $$0.537125\pi$$
$$972$$ 0 0
$$973$$ −1715.81 −0.0565328
$$974$$ 21707.0 0.714103
$$975$$ 0 0
$$976$$ −397.671 −0.0130422
$$977$$ −37607.6 −1.23150 −0.615749 0.787943i $$-0.711146\pi$$
−0.615749 + 0.787943i $$0.711146\pi$$
$$978$$ 0 0
$$979$$ −1824.69 −0.0595684
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −11939.0 −0.387974
$$983$$ −25297.7 −0.820826 −0.410413 0.911900i $$-0.634615\pi$$
−0.410413 + 0.911900i $$0.634615\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −9479.85 −0.306187
$$987$$ 0 0
$$988$$ 31151.5 1.00310
$$989$$ −12271.6 −0.394556
$$990$$ 0 0
$$991$$ −41686.5 −1.33624 −0.668120 0.744053i $$-0.732901\pi$$
−0.668120 + 0.744053i $$0.732901\pi$$
$$992$$ −11808.9 −0.377955
$$993$$ 0 0
$$994$$ 22458.3 0.716633
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −25465.9 −0.808939 −0.404470 0.914551i $$-0.632544\pi$$
−0.404470 + 0.914551i $$0.632544\pi$$
$$998$$ −19350.6 −0.613761
$$999$$ 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 225.4.a.o.1.1 2
3.2 odd 2 75.4.a.c.1.2 2
5.2 odd 4 45.4.b.b.19.2 4
5.3 odd 4 45.4.b.b.19.3 4
5.4 even 2 225.4.a.i.1.2 2
12.11 even 2 1200.4.a.bt.1.2 2
15.2 even 4 15.4.b.a.4.3 yes 4
15.8 even 4 15.4.b.a.4.2 4
15.14 odd 2 75.4.a.f.1.1 2
20.3 even 4 720.4.f.j.289.3 4
20.7 even 4 720.4.f.j.289.4 4
60.23 odd 4 240.4.f.f.49.3 4
60.47 odd 4 240.4.f.f.49.1 4
60.59 even 2 1200.4.a.bn.1.1 2
120.53 even 4 960.4.f.q.769.4 4
120.77 even 4 960.4.f.q.769.2 4
120.83 odd 4 960.4.f.p.769.2 4
120.107 odd 4 960.4.f.p.769.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.b.a.4.2 4 15.8 even 4
15.4.b.a.4.3 yes 4 15.2 even 4
45.4.b.b.19.2 4 5.2 odd 4
45.4.b.b.19.3 4 5.3 odd 4
75.4.a.c.1.2 2 3.2 odd 2
75.4.a.f.1.1 2 15.14 odd 2
225.4.a.i.1.2 2 5.4 even 2
225.4.a.o.1.1 2 1.1 even 1 trivial
240.4.f.f.49.1 4 60.47 odd 4
240.4.f.f.49.3 4 60.23 odd 4
720.4.f.j.289.3 4 20.3 even 4
720.4.f.j.289.4 4 20.7 even 4
960.4.f.p.769.2 4 120.83 odd 4
960.4.f.p.769.4 4 120.107 odd 4
960.4.f.q.769.2 4 120.77 even 4
960.4.f.q.769.4 4 120.53 even 4
1200.4.a.bn.1.1 2 60.59 even 2
1200.4.a.bt.1.2 2 12.11 even 2