# Properties

 Label 450.4.c.f.199.2 Level $450$ Weight $4$ Character 450.199 Analytic conductor $26.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.5508595026$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 450.199 Dual form 450.4.c.f.199.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} -4.00000 q^{4} +14.0000i q^{7} -8.00000i q^{8} +O(q^{10})$$ $$q+2.00000i q^{2} -4.00000 q^{4} +14.0000i q^{7} -8.00000i q^{8} -6.00000 q^{11} -68.0000i q^{13} -28.0000 q^{14} +16.0000 q^{16} -78.0000i q^{17} -44.0000 q^{19} -12.0000i q^{22} +120.000i q^{23} +136.000 q^{26} -56.0000i q^{28} +126.000 q^{29} -244.000 q^{31} +32.0000i q^{32} +156.000 q^{34} -304.000i q^{37} -88.0000i q^{38} +480.000 q^{41} -104.000i q^{43} +24.0000 q^{44} -240.000 q^{46} -600.000i q^{47} +147.000 q^{49} +272.000i q^{52} -258.000i q^{53} +112.000 q^{56} +252.000i q^{58} +534.000 q^{59} +362.000 q^{61} -488.000i q^{62} -64.0000 q^{64} -268.000i q^{67} +312.000i q^{68} +972.000 q^{71} -470.000i q^{73} +608.000 q^{74} +176.000 q^{76} -84.0000i q^{77} -1244.00 q^{79} +960.000i q^{82} +396.000i q^{83} +208.000 q^{86} +48.0000i q^{88} -972.000 q^{89} +952.000 q^{91} -480.000i q^{92} +1200.00 q^{94} -46.0000i q^{97} +294.000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} - 12 q^{11} - 56 q^{14} + 32 q^{16} - 88 q^{19} + 272 q^{26} + 252 q^{29} - 488 q^{31} + 312 q^{34} + 960 q^{41} + 48 q^{44} - 480 q^{46} + 294 q^{49} + 224 q^{56} + 1068 q^{59} + 724 q^{61} - 128 q^{64} + 1944 q^{71} + 1216 q^{74} + 352 q^{76} - 2488 q^{79} + 416 q^{86} - 1944 q^{89} + 1904 q^{91} + 2400 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 - 12 * q^11 - 56 * q^14 + 32 * q^16 - 88 * q^19 + 272 * q^26 + 252 * q^29 - 488 * q^31 + 312 * q^34 + 960 * q^41 + 48 * q^44 - 480 * q^46 + 294 * q^49 + 224 * q^56 + 1068 * q^59 + 724 * q^61 - 128 * q^64 + 1944 * q^71 + 1216 * q^74 + 352 * q^76 - 2488 * q^79 + 416 * q^86 - 1944 * q^89 + 1904 * q^91 + 2400 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

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

## 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 0.707107i
$$3$$ 0 0
$$4$$ −4.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 14.0000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 8.00000i − 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −6.00000 −0.164461 −0.0822304 0.996613i $$-0.526204\pi$$
−0.0822304 + 0.996613i $$0.526204\pi$$
$$12$$ 0 0
$$13$$ − 68.0000i − 1.45075i −0.688352 0.725377i $$-0.741665\pi$$
0.688352 0.725377i $$-0.258335\pi$$
$$14$$ −28.0000 −0.534522
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ − 78.0000i − 1.11281i −0.830911 0.556405i $$-0.812180\pi$$
0.830911 0.556405i $$-0.187820\pi$$
$$18$$ 0 0
$$19$$ −44.0000 −0.531279 −0.265639 0.964072i $$-0.585583\pi$$
−0.265639 + 0.964072i $$0.585583\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 12.0000i − 0.116291i
$$23$$ 120.000i 1.08790i 0.839117 + 0.543951i $$0.183072\pi$$
−0.839117 + 0.543951i $$0.816928\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 136.000 1.02584
$$27$$ 0 0
$$28$$ − 56.0000i − 0.377964i
$$29$$ 126.000 0.806814 0.403407 0.915021i $$-0.367826\pi$$
0.403407 + 0.915021i $$0.367826\pi$$
$$30$$ 0 0
$$31$$ −244.000 −1.41367 −0.706834 0.707380i $$-0.749877\pi$$
−0.706834 + 0.707380i $$0.749877\pi$$
$$32$$ 32.0000i 0.176777i
$$33$$ 0 0
$$34$$ 156.000 0.786876
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 304.000i − 1.35074i −0.737480 0.675369i $$-0.763984\pi$$
0.737480 0.675369i $$-0.236016\pi$$
$$38$$ − 88.0000i − 0.375671i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 480.000 1.82838 0.914188 0.405291i $$-0.132830\pi$$
0.914188 + 0.405291i $$0.132830\pi$$
$$42$$ 0 0
$$43$$ − 104.000i − 0.368834i −0.982848 0.184417i $$-0.940960\pi$$
0.982848 0.184417i $$-0.0590396\pi$$
$$44$$ 24.0000 0.0822304
$$45$$ 0 0
$$46$$ −240.000 −0.769262
$$47$$ − 600.000i − 1.86211i −0.364884 0.931053i $$-0.618891\pi$$
0.364884 0.931053i $$-0.381109\pi$$
$$48$$ 0 0
$$49$$ 147.000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 272.000i 0.725377i
$$53$$ − 258.000i − 0.668661i −0.942456 0.334330i $$-0.891490\pi$$
0.942456 0.334330i $$-0.108510\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 112.000 0.267261
$$57$$ 0 0
$$58$$ 252.000i 0.570504i
$$59$$ 534.000 1.17832 0.589160 0.808016i $$-0.299459\pi$$
0.589160 + 0.808016i $$0.299459\pi$$
$$60$$ 0 0
$$61$$ 362.000 0.759825 0.379913 0.925022i $$-0.375954\pi$$
0.379913 + 0.925022i $$0.375954\pi$$
$$62$$ − 488.000i − 0.999614i
$$63$$ 0 0
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 268.000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 312.000i 0.556405i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 972.000 1.62472 0.812360 0.583156i $$-0.198182\pi$$
0.812360 + 0.583156i $$0.198182\pi$$
$$72$$ 0 0
$$73$$ − 470.000i − 0.753553i −0.926304 0.376776i $$-0.877033\pi$$
0.926304 0.376776i $$-0.122967\pi$$
$$74$$ 608.000 0.955116
$$75$$ 0 0
$$76$$ 176.000 0.265639
$$77$$ − 84.0000i − 0.124321i
$$78$$ 0 0
$$79$$ −1244.00 −1.77166 −0.885829 0.464012i $$-0.846409\pi$$
−0.885829 + 0.464012i $$0.846409\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 960.000i 1.29286i
$$83$$ 396.000i 0.523695i 0.965109 + 0.261847i $$0.0843317\pi$$
−0.965109 + 0.261847i $$0.915668\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 208.000 0.260805
$$87$$ 0 0
$$88$$ 48.0000i 0.0581456i
$$89$$ −972.000 −1.15766 −0.578830 0.815448i $$-0.696491\pi$$
−0.578830 + 0.815448i $$0.696491\pi$$
$$90$$ 0 0
$$91$$ 952.000 1.09667
$$92$$ − 480.000i − 0.543951i
$$93$$ 0 0
$$94$$ 1200.00 1.31671
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 46.0000i − 0.0481504i −0.999710 0.0240752i $$-0.992336\pi$$
0.999710 0.0240752i $$-0.00766412\pi$$
$$98$$ 294.000i 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1506.00 1.48369 0.741845 0.670572i $$-0.233951\pi$$
0.741845 + 0.670572i $$0.233951\pi$$
$$102$$ 0 0
$$103$$ 1474.00i 1.41007i 0.709171 + 0.705037i $$0.249069\pi$$
−0.709171 + 0.705037i $$0.750931\pi$$
$$104$$ −544.000 −0.512919
$$105$$ 0 0
$$106$$ 516.000 0.472815
$$107$$ − 924.000i − 0.834827i −0.908717 0.417413i $$-0.862937\pi$$
0.908717 0.417413i $$-0.137063\pi$$
$$108$$ 0 0
$$109$$ −698.000 −0.613360 −0.306680 0.951813i $$-0.599218\pi$$
−0.306680 + 0.951813i $$0.599218\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 224.000i 0.188982i
$$113$$ − 222.000i − 0.184814i −0.995721 0.0924071i $$-0.970544\pi$$
0.995721 0.0924071i $$-0.0294561\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −504.000 −0.403407
$$117$$ 0 0
$$118$$ 1068.00i 0.833198i
$$119$$ 1092.00 0.841206
$$120$$ 0 0
$$121$$ −1295.00 −0.972953
$$122$$ 724.000i 0.537278i
$$123$$ 0 0
$$124$$ 976.000 0.706834
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 1906.00i − 1.33173i −0.746071 0.665867i $$-0.768062\pi$$
0.746071 0.665867i $$-0.231938\pi$$
$$128$$ − 128.000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2874.00 −1.91681 −0.958407 0.285406i $$-0.907872\pi$$
−0.958407 + 0.285406i $$0.907872\pi$$
$$132$$ 0 0
$$133$$ − 616.000i − 0.401609i
$$134$$ 536.000 0.345547
$$135$$ 0 0
$$136$$ −624.000 −0.393438
$$137$$ 798.000i 0.497648i 0.968549 + 0.248824i $$0.0800440\pi$$
−0.968549 + 0.248824i $$0.919956\pi$$
$$138$$ 0 0
$$139$$ 700.000 0.427146 0.213573 0.976927i $$-0.431490\pi$$
0.213573 + 0.976927i $$0.431490\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1944.00i 1.14885i
$$143$$ 408.000i 0.238592i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 940.000 0.532842
$$147$$ 0 0
$$148$$ 1216.00i 0.675369i
$$149$$ 114.000 0.0626795 0.0313397 0.999509i $$-0.490023\pi$$
0.0313397 + 0.999509i $$0.490023\pi$$
$$150$$ 0 0
$$151$$ 1064.00 0.573424 0.286712 0.958017i $$-0.407438\pi$$
0.286712 + 0.958017i $$0.407438\pi$$
$$152$$ 352.000i 0.187835i
$$153$$ 0 0
$$154$$ 168.000 0.0879080
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 1948.00i − 0.990238i −0.868825 0.495119i $$-0.835125\pi$$
0.868825 0.495119i $$-0.164875\pi$$
$$158$$ − 2488.00i − 1.25275i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1680.00 −0.822376
$$162$$ 0 0
$$163$$ − 2060.00i − 0.989887i −0.868925 0.494944i $$-0.835189\pi$$
0.868925 0.494944i $$-0.164811\pi$$
$$164$$ −1920.00 −0.914188
$$165$$ 0 0
$$166$$ −792.000 −0.370308
$$167$$ 1248.00i 0.578282i 0.957286 + 0.289141i $$0.0933697\pi$$
−0.957286 + 0.289141i $$0.906630\pi$$
$$168$$ 0 0
$$169$$ −2427.00 −1.10469
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 416.000i 0.184417i
$$173$$ − 1146.00i − 0.503634i −0.967775 0.251817i $$-0.918972\pi$$
0.967775 0.251817i $$-0.0810282\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −96.0000 −0.0411152
$$177$$ 0 0
$$178$$ − 1944.00i − 0.818590i
$$179$$ −1146.00 −0.478525 −0.239263 0.970955i $$-0.576906\pi$$
−0.239263 + 0.970955i $$0.576906\pi$$
$$180$$ 0 0
$$181$$ −118.000 −0.0484579 −0.0242289 0.999706i $$-0.507713\pi$$
−0.0242289 + 0.999706i $$0.507713\pi$$
$$182$$ 1904.00i 0.775461i
$$183$$ 0 0
$$184$$ 960.000 0.384631
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 468.000i 0.183014i
$$188$$ 2400.00i 0.931053i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1692.00 −0.640989 −0.320494 0.947250i $$-0.603849\pi$$
−0.320494 + 0.947250i $$0.603849\pi$$
$$192$$ 0 0
$$193$$ − 3350.00i − 1.24942i −0.780856 0.624711i $$-0.785217\pi$$
0.780856 0.624711i $$-0.214783\pi$$
$$194$$ 92.0000 0.0340475
$$195$$ 0 0
$$196$$ −588.000 −0.214286
$$197$$ 3606.00i 1.30415i 0.758156 + 0.652073i $$0.226101\pi$$
−0.758156 + 0.652073i $$0.773899\pi$$
$$198$$ 0 0
$$199$$ −2696.00 −0.960374 −0.480187 0.877166i $$-0.659431\pi$$
−0.480187 + 0.877166i $$0.659431\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 3012.00i 1.04913i
$$203$$ 1764.00i 0.609894i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2948.00 −0.997072
$$207$$ 0 0
$$208$$ − 1088.00i − 0.362689i
$$209$$ 264.000 0.0873745
$$210$$ 0 0
$$211$$ −4.00000 −0.00130508 −0.000652539 1.00000i $$-0.500208\pi$$
−0.000652539 1.00000i $$0.500208\pi$$
$$212$$ 1032.00i 0.334330i
$$213$$ 0 0
$$214$$ 1848.00 0.590312
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 3416.00i − 1.06863i
$$218$$ − 1396.00i − 0.433711i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5304.00 −1.61441
$$222$$ 0 0
$$223$$ 1162.00i 0.348938i 0.984663 + 0.174469i $$0.0558210\pi$$
−0.984663 + 0.174469i $$0.944179\pi$$
$$224$$ −448.000 −0.133631
$$225$$ 0 0
$$226$$ 444.000 0.130683
$$227$$ 2400.00i 0.701734i 0.936425 + 0.350867i $$0.114113\pi$$
−0.936425 + 0.350867i $$0.885887\pi$$
$$228$$ 0 0
$$229$$ 2314.00 0.667744 0.333872 0.942618i $$-0.391645\pi$$
0.333872 + 0.942618i $$0.391645\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 1008.00i − 0.285252i
$$233$$ − 18.0000i − 0.00506103i −0.999997 0.00253051i $$-0.999195\pi$$
0.999997 0.00253051i $$-0.000805488\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −2136.00 −0.589160
$$237$$ 0 0
$$238$$ 2184.00i 0.594822i
$$239$$ −5868.00 −1.58816 −0.794078 0.607816i $$-0.792046\pi$$
−0.794078 + 0.607816i $$0.792046\pi$$
$$240$$ 0 0
$$241$$ −4330.00 −1.15734 −0.578672 0.815560i $$-0.696429\pi$$
−0.578672 + 0.815560i $$0.696429\pi$$
$$242$$ − 2590.00i − 0.687981i
$$243$$ 0 0
$$244$$ −1448.00 −0.379913
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2992.00i 0.770755i
$$248$$ 1952.00i 0.499807i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −498.000 −0.125233 −0.0626165 0.998038i $$-0.519944\pi$$
−0.0626165 + 0.998038i $$0.519944\pi$$
$$252$$ 0 0
$$253$$ − 720.000i − 0.178917i
$$254$$ 3812.00 0.941678
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ − 642.000i − 0.155824i −0.996960 0.0779122i $$-0.975175\pi$$
0.996960 0.0779122i $$-0.0248254\pi$$
$$258$$ 0 0
$$259$$ 4256.00 1.02106
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 5748.00i − 1.35539i
$$263$$ − 7968.00i − 1.86817i −0.357055 0.934084i $$-0.616219\pi$$
0.357055 0.934084i $$-0.383781\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 1232.00 0.283980
$$267$$ 0 0
$$268$$ 1072.00i 0.244339i
$$269$$ −4218.00 −0.956045 −0.478022 0.878348i $$-0.658646\pi$$
−0.478022 + 0.878348i $$0.658646\pi$$
$$270$$ 0 0
$$271$$ 848.000 0.190082 0.0950412 0.995473i $$-0.469702\pi$$
0.0950412 + 0.995473i $$0.469702\pi$$
$$272$$ − 1248.00i − 0.278203i
$$273$$ 0 0
$$274$$ −1596.00 −0.351890
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 1504.00i − 0.326233i −0.986607 0.163117i $$-0.947845\pi$$
0.986607 0.163117i $$-0.0521547\pi$$
$$278$$ 1400.00i 0.302037i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1308.00 0.277682 0.138841 0.990315i $$-0.455662\pi$$
0.138841 + 0.990315i $$0.455662\pi$$
$$282$$ 0 0
$$283$$ 5932.00i 1.24601i 0.782218 + 0.623005i $$0.214088\pi$$
−0.782218 + 0.623005i $$0.785912\pi$$
$$284$$ −3888.00 −0.812360
$$285$$ 0 0
$$286$$ −816.000 −0.168710
$$287$$ 6720.00i 1.38212i
$$288$$ 0 0
$$289$$ −1171.00 −0.238347
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 1880.00i 0.376776i
$$293$$ 5226.00i 1.04200i 0.853556 + 0.521000i $$0.174441\pi$$
−0.853556 + 0.521000i $$0.825559\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −2432.00 −0.477558
$$297$$ 0 0
$$298$$ 228.000i 0.0443211i
$$299$$ 8160.00 1.57828
$$300$$ 0 0
$$301$$ 1456.00 0.278812
$$302$$ 2128.00i 0.405472i
$$303$$ 0 0
$$304$$ −704.000 −0.132820
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4448.00i 0.826908i 0.910525 + 0.413454i $$0.135678\pi$$
−0.910525 + 0.413454i $$0.864322\pi$$
$$308$$ 336.000i 0.0621603i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 9132.00 1.66504 0.832521 0.553993i $$-0.186897\pi$$
0.832521 + 0.553993i $$0.186897\pi$$
$$312$$ 0 0
$$313$$ 2170.00i 0.391871i 0.980617 + 0.195936i $$0.0627743\pi$$
−0.980617 + 0.195936i $$0.937226\pi$$
$$314$$ 3896.00 0.700204
$$315$$ 0 0
$$316$$ 4976.00 0.885829
$$317$$ − 7674.00i − 1.35967i −0.733366 0.679834i $$-0.762052\pi$$
0.733366 0.679834i $$-0.237948\pi$$
$$318$$ 0 0
$$319$$ −756.000 −0.132689
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 3360.00i − 0.581508i
$$323$$ 3432.00i 0.591212i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4120.00 0.699956
$$327$$ 0 0
$$328$$ − 3840.00i − 0.646428i
$$329$$ 8400.00 1.40762
$$330$$ 0 0
$$331$$ 9596.00 1.59349 0.796743 0.604318i $$-0.206554\pi$$
0.796743 + 0.604318i $$0.206554\pi$$
$$332$$ − 1584.00i − 0.261847i
$$333$$ 0 0
$$334$$ −2496.00 −0.408907
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 12158.0i 1.96525i 0.185608 + 0.982624i $$0.440574\pi$$
−0.185608 + 0.982624i $$0.559426\pi$$
$$338$$ − 4854.00i − 0.781133i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1464.00 0.232493
$$342$$ 0 0
$$343$$ 6860.00i 1.07990i
$$344$$ −832.000 −0.130402
$$345$$ 0 0
$$346$$ 2292.00 0.356123
$$347$$ − 10320.0i − 1.59656i −0.602286 0.798280i $$-0.705743\pi$$
0.602286 0.798280i $$-0.294257\pi$$
$$348$$ 0 0
$$349$$ 2158.00 0.330989 0.165494 0.986211i $$-0.447078\pi$$
0.165494 + 0.986211i $$0.447078\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 192.000i − 0.0290728i
$$353$$ − 330.000i − 0.0497567i −0.999690 0.0248784i $$-0.992080\pi$$
0.999690 0.0248784i $$-0.00791985\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 3888.00 0.578830
$$357$$ 0 0
$$358$$ − 2292.00i − 0.338369i
$$359$$ −8664.00 −1.27373 −0.636864 0.770976i $$-0.719769\pi$$
−0.636864 + 0.770976i $$0.719769\pi$$
$$360$$ 0 0
$$361$$ −4923.00 −0.717743
$$362$$ − 236.000i − 0.0342649i
$$363$$ 0 0
$$364$$ −3808.00 −0.548334
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3782.00i 0.537926i 0.963151 + 0.268963i $$0.0866809\pi$$
−0.963151 + 0.268963i $$0.913319\pi$$
$$368$$ 1920.00i 0.271975i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3612.00 0.505460
$$372$$ 0 0
$$373$$ − 11276.0i − 1.56528i −0.622475 0.782640i $$-0.713873\pi$$
0.622475 0.782640i $$-0.286127\pi$$
$$374$$ −936.000 −0.129410
$$375$$ 0 0
$$376$$ −4800.00 −0.658354
$$377$$ − 8568.00i − 1.17049i
$$378$$ 0 0
$$379$$ −980.000 −0.132821 −0.0664106 0.997792i $$-0.521155\pi$$
−0.0664106 + 0.997792i $$0.521155\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 3384.00i − 0.453247i
$$383$$ − 4200.00i − 0.560339i −0.959950 0.280170i $$-0.909609\pi$$
0.959950 0.280170i $$-0.0903907\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 6700.00 0.883474
$$387$$ 0 0
$$388$$ 184.000i 0.0240752i
$$389$$ 13338.0 1.73847 0.869233 0.494402i $$-0.164613\pi$$
0.869233 + 0.494402i $$0.164613\pi$$
$$390$$ 0 0
$$391$$ 9360.00 1.21063
$$392$$ − 1176.00i − 0.151523i
$$393$$ 0 0
$$394$$ −7212.00 −0.922171
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 7192.00i − 0.909209i −0.890693 0.454605i $$-0.849781\pi$$
0.890693 0.454605i $$-0.150219\pi$$
$$398$$ − 5392.00i − 0.679087i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2316.00 0.288418 0.144209 0.989547i $$-0.453936\pi$$
0.144209 + 0.989547i $$0.453936\pi$$
$$402$$ 0 0
$$403$$ 16592.0i 2.05088i
$$404$$ −6024.00 −0.741845
$$405$$ 0 0
$$406$$ −3528.00 −0.431260
$$407$$ 1824.00i 0.222143i
$$408$$ 0 0
$$409$$ 12358.0 1.49404 0.747022 0.664800i $$-0.231483\pi$$
0.747022 + 0.664800i $$0.231483\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 5896.00i − 0.705037i
$$413$$ 7476.00i 0.890726i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2176.00 0.256460
$$417$$ 0 0
$$418$$ 528.000i 0.0617831i
$$419$$ 3306.00 0.385462 0.192731 0.981252i $$-0.438265\pi$$
0.192731 + 0.981252i $$0.438265\pi$$
$$420$$ 0 0
$$421$$ −14506.0 −1.67929 −0.839643 0.543139i $$-0.817236\pi$$
−0.839643 + 0.543139i $$0.817236\pi$$
$$422$$ − 8.00000i 0 0.000922829i
$$423$$ 0 0
$$424$$ −2064.00 −0.236407
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 5068.00i 0.574374i
$$428$$ 3696.00i 0.417413i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6480.00 0.724201 0.362100 0.932139i $$-0.382060\pi$$
0.362100 + 0.932139i $$0.382060\pi$$
$$432$$ 0 0
$$433$$ − 11894.0i − 1.32007i −0.751236 0.660034i $$-0.770542\pi$$
0.751236 0.660034i $$-0.229458\pi$$
$$434$$ 6832.00 0.755637
$$435$$ 0 0
$$436$$ 2792.00 0.306680
$$437$$ − 5280.00i − 0.577979i
$$438$$ 0 0
$$439$$ 12688.0 1.37942 0.689710 0.724086i $$-0.257738\pi$$
0.689710 + 0.724086i $$0.257738\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 10608.0i − 1.14156i
$$443$$ 4968.00i 0.532814i 0.963861 + 0.266407i $$0.0858366\pi$$
−0.963861 + 0.266407i $$0.914163\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −2324.00 −0.246737
$$447$$ 0 0
$$448$$ − 896.000i − 0.0944911i
$$449$$ −11508.0 −1.20957 −0.604784 0.796389i $$-0.706741\pi$$
−0.604784 + 0.796389i $$0.706741\pi$$
$$450$$ 0 0
$$451$$ −2880.00 −0.300696
$$452$$ 888.000i 0.0924071i
$$453$$ 0 0
$$454$$ −4800.00 −0.496201
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1082.00i 0.110752i 0.998466 + 0.0553762i $$0.0176358\pi$$
−0.998466 + 0.0553762i $$0.982364\pi$$
$$458$$ 4628.00i 0.472166i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 11238.0 1.13537 0.567685 0.823246i $$-0.307839\pi$$
0.567685 + 0.823246i $$0.307839\pi$$
$$462$$ 0 0
$$463$$ 2302.00i 0.231065i 0.993304 + 0.115532i $$0.0368574\pi$$
−0.993304 + 0.115532i $$0.963143\pi$$
$$464$$ 2016.00 0.201704
$$465$$ 0 0
$$466$$ 36.0000 0.00357869
$$467$$ − 15876.0i − 1.57313i −0.617505 0.786567i $$-0.711856\pi$$
0.617505 0.786567i $$-0.288144\pi$$
$$468$$ 0 0
$$469$$ 3752.00 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 4272.00i − 0.416599i
$$473$$ 624.000i 0.0606587i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4368.00 −0.420603
$$477$$ 0 0
$$478$$ − 11736.0i − 1.12300i
$$479$$ 4644.00 0.442985 0.221492 0.975162i $$-0.428907\pi$$
0.221492 + 0.975162i $$0.428907\pi$$
$$480$$ 0 0
$$481$$ −20672.0 −1.95959
$$482$$ − 8660.00i − 0.818366i
$$483$$ 0 0
$$484$$ 5180.00 0.486476
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2426.00i 0.225734i 0.993610 + 0.112867i $$0.0360034\pi$$
−0.993610 + 0.112867i $$0.963997\pi$$
$$488$$ − 2896.00i − 0.268639i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −234.000 −0.0215077 −0.0107538 0.999942i $$-0.503423\pi$$
−0.0107538 + 0.999942i $$0.503423\pi$$
$$492$$ 0 0
$$493$$ − 9828.00i − 0.897831i
$$494$$ −5984.00 −0.545006
$$495$$ 0 0
$$496$$ −3904.00 −0.353417
$$497$$ 13608.0i 1.22817i
$$498$$ 0 0
$$499$$ −14204.0 −1.27427 −0.637133 0.770754i $$-0.719880\pi$$
−0.637133 + 0.770754i $$0.719880\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 996.000i − 0.0885531i
$$503$$ 4920.00i 0.436127i 0.975935 + 0.218064i $$0.0699740\pi$$
−0.975935 + 0.218064i $$0.930026\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1440.00 0.126513
$$507$$ 0 0
$$508$$ 7624.00i 0.665867i
$$509$$ −4458.00 −0.388207 −0.194104 0.980981i $$-0.562180\pi$$
−0.194104 + 0.980981i $$0.562180\pi$$
$$510$$ 0 0
$$511$$ 6580.00 0.569632
$$512$$ 512.000i 0.0441942i
$$513$$ 0 0
$$514$$ 1284.00 0.110184
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3600.00i 0.306243i
$$518$$ 8512.00i 0.722000i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4212.00 −0.354186 −0.177093 0.984194i $$-0.556669\pi$$
−0.177093 + 0.984194i $$0.556669\pi$$
$$522$$ 0 0
$$523$$ 11212.0i 0.937412i 0.883354 + 0.468706i $$0.155280\pi$$
−0.883354 + 0.468706i $$0.844720\pi$$
$$524$$ 11496.0 0.958407
$$525$$ 0 0
$$526$$ 15936.0 1.32099
$$527$$ 19032.0i 1.57314i
$$528$$ 0 0
$$529$$ −2233.00 −0.183529
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2464.00i 0.200804i
$$533$$ − 32640.0i − 2.65252i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −2144.00 −0.172774
$$537$$ 0 0
$$538$$ − 8436.00i − 0.676026i
$$539$$ −882.000 −0.0704832
$$540$$ 0 0
$$541$$ 14018.0 1.11401 0.557006 0.830508i $$-0.311950\pi$$
0.557006 + 0.830508i $$0.311950\pi$$
$$542$$ 1696.00i 0.134409i
$$543$$ 0 0
$$544$$ 2496.00 0.196719
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 18200.0i 1.42262i 0.702876 + 0.711312i $$0.251899\pi$$
−0.702876 + 0.711312i $$0.748101\pi$$
$$548$$ − 3192.00i − 0.248824i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5544.00 −0.428643
$$552$$ 0 0
$$553$$ − 17416.0i − 1.33925i
$$554$$ 3008.00 0.230682
$$555$$ 0 0
$$556$$ −2800.00 −0.213573
$$557$$ 11826.0i 0.899612i 0.893126 + 0.449806i $$0.148507\pi$$
−0.893126 + 0.449806i $$0.851493\pi$$
$$558$$ 0 0
$$559$$ −7072.00 −0.535087
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2616.00i 0.196351i
$$563$$ − 2952.00i − 0.220980i −0.993877 0.110490i $$-0.964758\pi$$
0.993877 0.110490i $$-0.0352421\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −11864.0 −0.881062
$$567$$ 0 0
$$568$$ − 7776.00i − 0.574426i
$$569$$ 3084.00 0.227220 0.113610 0.993525i $$-0.463759\pi$$
0.113610 + 0.993525i $$0.463759\pi$$
$$570$$ 0 0
$$571$$ −4756.00 −0.348568 −0.174284 0.984695i $$-0.555761\pi$$
−0.174284 + 0.984695i $$0.555761\pi$$
$$572$$ − 1632.00i − 0.119296i
$$573$$ 0 0
$$574$$ −13440.0 −0.977308
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 11014.0i − 0.794660i −0.917676 0.397330i $$-0.869937\pi$$
0.917676 0.397330i $$-0.130063\pi$$
$$578$$ − 2342.00i − 0.168537i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −5544.00 −0.395876
$$582$$ 0 0
$$583$$ 1548.00i 0.109968i
$$584$$ −3760.00 −0.266421
$$585$$ 0 0
$$586$$ −10452.0 −0.736806
$$587$$ 852.000i 0.0599077i 0.999551 + 0.0299538i $$0.00953603\pi$$
−0.999551 + 0.0299538i $$0.990464\pi$$
$$588$$ 0 0
$$589$$ 10736.0 0.751051
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 4864.00i − 0.337684i
$$593$$ 15546.0i 1.07656i 0.842767 + 0.538278i $$0.180925\pi$$
−0.842767 + 0.538278i $$0.819075\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −456.000 −0.0313397
$$597$$ 0 0
$$598$$ 16320.0i 1.11601i
$$599$$ −8616.00 −0.587713 −0.293857 0.955850i $$-0.594939\pi$$
−0.293857 + 0.955850i $$0.594939\pi$$
$$600$$ 0 0
$$601$$ 17510.0 1.18843 0.594216 0.804305i $$-0.297462\pi$$
0.594216 + 0.804305i $$0.297462\pi$$
$$602$$ 2912.00i 0.197150i
$$603$$ 0 0
$$604$$ −4256.00 −0.286712
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 13894.0i − 0.929061i −0.885557 0.464531i $$-0.846223\pi$$
0.885557 0.464531i $$-0.153777\pi$$
$$608$$ − 1408.00i − 0.0939177i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −40800.0 −2.70146
$$612$$ 0 0
$$613$$ 6496.00i 0.428011i 0.976832 + 0.214006i $$0.0686511\pi$$
−0.976832 + 0.214006i $$0.931349\pi$$
$$614$$ −8896.00 −0.584712
$$615$$ 0 0
$$616$$ −672.000 −0.0439540
$$617$$ 570.000i 0.0371918i 0.999827 + 0.0185959i $$0.00591960\pi$$
−0.999827 + 0.0185959i $$0.994080\pi$$
$$618$$ 0 0
$$619$$ 2140.00 0.138956 0.0694781 0.997583i $$-0.477867\pi$$
0.0694781 + 0.997583i $$0.477867\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18264.0i 1.17736i
$$623$$ − 13608.0i − 0.875109i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −4340.00 −0.277095
$$627$$ 0 0
$$628$$ 7792.00i 0.495119i
$$629$$ −23712.0 −1.50312
$$630$$ 0 0
$$631$$ 14660.0 0.924890 0.462445 0.886648i $$-0.346972\pi$$
0.462445 + 0.886648i $$0.346972\pi$$
$$632$$ 9952.00i 0.626375i
$$633$$ 0 0
$$634$$ 15348.0 0.961431
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 9996.00i − 0.621752i
$$638$$ − 1512.00i − 0.0938255i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 456.000 0.0280982 0.0140491 0.999901i $$-0.495528\pi$$
0.0140491 + 0.999901i $$0.495528\pi$$
$$642$$ 0 0
$$643$$ 23452.0i 1.43835i 0.694831 + 0.719173i $$0.255479\pi$$
−0.694831 + 0.719173i $$0.744521\pi$$
$$644$$ 6720.00 0.411188
$$645$$ 0 0
$$646$$ −6864.00 −0.418050
$$647$$ − 7224.00i − 0.438956i −0.975617 0.219478i $$-0.929565\pi$$
0.975617 0.219478i $$-0.0704355\pi$$
$$648$$ 0 0
$$649$$ −3204.00 −0.193787
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 8240.00i 0.494944i
$$653$$ 19146.0i 1.14738i 0.819072 + 0.573691i $$0.194489\pi$$
−0.819072 + 0.573691i $$0.805511\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 7680.00 0.457094
$$657$$ 0 0
$$658$$ 16800.0i 0.995338i
$$659$$ 27810.0 1.64389 0.821945 0.569567i $$-0.192889\pi$$
0.821945 + 0.569567i $$0.192889\pi$$
$$660$$ 0 0
$$661$$ −30598.0 −1.80049 −0.900245 0.435383i $$-0.856613\pi$$
−0.900245 + 0.435383i $$0.856613\pi$$
$$662$$ 19192.0i 1.12676i
$$663$$ 0 0
$$664$$ 3168.00 0.185154
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 15120.0i 0.877734i
$$668$$ − 4992.00i − 0.289141i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2172.00 −0.124961
$$672$$ 0 0
$$673$$ 3778.00i 0.216391i 0.994130 + 0.108196i $$0.0345073\pi$$
−0.994130 + 0.108196i $$0.965493\pi$$
$$674$$ −24316.0 −1.38964
$$675$$ 0 0
$$676$$ 9708.00 0.552344
$$677$$ 27198.0i 1.54402i 0.635608 + 0.772012i $$0.280749\pi$$
−0.635608 + 0.772012i $$0.719251\pi$$
$$678$$ 0 0
$$679$$ 644.000 0.0363983
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 2928.00i 0.164397i
$$683$$ 32316.0i 1.81045i 0.424933 + 0.905225i $$0.360298\pi$$
−0.424933 + 0.905225i $$0.639702\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13720.0 −0.763604
$$687$$ 0 0
$$688$$ − 1664.00i − 0.0922084i
$$689$$ −17544.0 −0.970063
$$690$$ 0 0
$$691$$ 29324.0 1.61438 0.807191 0.590291i $$-0.200987\pi$$
0.807191 + 0.590291i $$0.200987\pi$$
$$692$$ 4584.00i 0.251817i
$$693$$ 0 0
$$694$$ 20640.0 1.12894
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 37440.0i − 2.03464i
$$698$$ 4316.00i 0.234044i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −22782.0 −1.22748 −0.613741 0.789508i $$-0.710336\pi$$
−0.613741 + 0.789508i $$0.710336\pi$$
$$702$$ 0 0
$$703$$ 13376.0i 0.717618i
$$704$$ 384.000 0.0205576
$$705$$ 0 0
$$706$$ 660.000 0.0351833
$$707$$ 21084.0i 1.12156i
$$708$$ 0 0
$$709$$ −26054.0 −1.38008 −0.690041 0.723770i $$-0.742408\pi$$
−0.690041 + 0.723770i $$0.742408\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 7776.00i 0.409295i
$$713$$ − 29280.0i − 1.53793i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4584.00 0.239263
$$717$$ 0 0
$$718$$ − 17328.0i − 0.900662i
$$719$$ −5976.00 −0.309968 −0.154984 0.987917i $$-0.549533\pi$$
−0.154984 + 0.987917i $$0.549533\pi$$
$$720$$ 0 0
$$721$$ −20636.0 −1.06592
$$722$$ − 9846.00i − 0.507521i
$$723$$ 0 0
$$724$$ 472.000 0.0242289
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 5110.00i − 0.260687i −0.991469 0.130343i $$-0.958392\pi$$
0.991469 0.130343i $$-0.0416080\pi$$
$$728$$ − 7616.00i − 0.387730i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8112.00 −0.410442
$$732$$ 0 0
$$733$$ − 17336.0i − 0.873560i −0.899568 0.436780i $$-0.856119\pi$$
0.899568 0.436780i $$-0.143881\pi$$
$$734$$ −7564.00 −0.380371
$$735$$ 0 0
$$736$$ −3840.00 −0.192316
$$737$$ 1608.00i 0.0803683i
$$738$$ 0 0
$$739$$ 13660.0 0.679961 0.339981 0.940432i $$-0.389580\pi$$
0.339981 + 0.940432i $$0.389580\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 7224.00i 0.357414i
$$743$$ 1320.00i 0.0651765i 0.999469 + 0.0325882i $$0.0103750\pi$$
−0.999469 + 0.0325882i $$0.989625\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 22552.0 1.10682
$$747$$ 0 0
$$748$$ − 1872.00i − 0.0915068i
$$749$$ 12936.0 0.631070
$$750$$ 0 0
$$751$$ 15860.0 0.770625 0.385313 0.922786i $$-0.374094\pi$$
0.385313 + 0.922786i $$0.374094\pi$$
$$752$$ − 9600.00i − 0.465527i
$$753$$ 0 0
$$754$$ 17136.0 0.827661
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22160.0i 1.06396i 0.846756 + 0.531981i $$0.178552\pi$$
−0.846756 + 0.531981i $$0.821448\pi$$
$$758$$ − 1960.00i − 0.0939187i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −13116.0 −0.624776 −0.312388 0.949955i $$-0.601129\pi$$
−0.312388 + 0.949955i $$0.601129\pi$$
$$762$$ 0 0
$$763$$ − 9772.00i − 0.463657i
$$764$$ 6768.00 0.320494
$$765$$ 0 0
$$766$$ 8400.00 0.396220
$$767$$ − 36312.0i − 1.70945i
$$768$$ 0 0
$$769$$ −32846.0 −1.54026 −0.770128 0.637889i $$-0.779808\pi$$
−0.770128 + 0.637889i $$0.779808\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 13400.0i 0.624711i
$$773$$ − 11982.0i − 0.557520i −0.960361 0.278760i $$-0.910077\pi$$
0.960361 0.278760i $$-0.0899233\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −368.000 −0.0170238
$$777$$ 0 0
$$778$$ 26676.0i 1.22928i
$$779$$ −21120.0 −0.971377
$$780$$ 0 0
$$781$$ −5832.00 −0.267203
$$782$$ 18720.0i 0.856043i
$$783$$ 0 0
$$784$$ 2352.00 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 21076.0i − 0.954610i −0.878738 0.477305i $$-0.841614\pi$$
0.878738 0.477305i $$-0.158386\pi$$
$$788$$ − 14424.0i − 0.652073i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3108.00 0.139706
$$792$$ 0 0
$$793$$ − 24616.0i − 1.10232i
$$794$$ 14384.0 0.642908
$$795$$ 0 0
$$796$$ 10784.0 0.480187
$$797$$ − 22086.0i − 0.981589i −0.871275 0.490794i $$-0.836707\pi$$
0.871275 0.490794i $$-0.163293\pi$$
$$798$$ 0 0
$$799$$ −46800.0 −2.07217
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 4632.00i 0.203942i
$$803$$ 2820.00i 0.123930i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −33184.0 −1.45019
$$807$$ 0 0
$$808$$ − 12048.0i − 0.524563i
$$809$$ 21384.0 0.929322 0.464661 0.885489i $$-0.346176\pi$$
0.464661 + 0.885489i $$0.346176\pi$$
$$810$$ 0 0
$$811$$ 5228.00 0.226362 0.113181 0.993574i $$-0.463896\pi$$
0.113181 + 0.993574i $$0.463896\pi$$
$$812$$ − 7056.00i − 0.304947i
$$813$$ 0 0
$$814$$ −3648.00 −0.157079
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4576.00i 0.195953i
$$818$$ 24716.0i 1.05645i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 38010.0 1.61578 0.807892 0.589331i $$-0.200609\pi$$
0.807892 + 0.589331i $$0.200609\pi$$
$$822$$ 0 0
$$823$$ − 38642.0i − 1.63667i −0.574745 0.818333i $$-0.694899\pi$$
0.574745 0.818333i $$-0.305101\pi$$
$$824$$ 11792.0 0.498536
$$825$$ 0 0
$$826$$ −14952.0 −0.629839
$$827$$ 15432.0i 0.648879i 0.945906 + 0.324440i $$0.105176\pi$$
−0.945906 + 0.324440i $$0.894824\pi$$
$$828$$ 0 0
$$829$$ 3886.00 0.162806 0.0814031 0.996681i $$-0.474060\pi$$
0.0814031 + 0.996681i $$0.474060\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 4352.00i 0.181344i
$$833$$ − 11466.0i − 0.476919i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −1056.00 −0.0436872
$$837$$ 0 0
$$838$$ 6612.00i 0.272563i
$$839$$ 27552.0 1.13373 0.566866 0.823810i $$-0.308156\pi$$
0.566866 + 0.823810i $$0.308156\pi$$
$$840$$ 0 0
$$841$$ −8513.00 −0.349051
$$842$$ − 29012.0i − 1.18743i
$$843$$ 0 0
$$844$$ 16.0000 0.000652539 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 18130.0i − 0.735483i
$$848$$ − 4128.00i − 0.167165i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 36480.0 1.46947
$$852$$ 0 0
$$853$$ − 15104.0i − 0.606273i −0.952947 0.303137i $$-0.901966\pi$$
0.952947 0.303137i $$-0.0980339\pi$$
$$854$$ −10136.0 −0.406144
$$855$$ 0 0
$$856$$ −7392.00 −0.295156
$$857$$ 12306.0i 0.490508i 0.969459 + 0.245254i $$0.0788713\pi$$
−0.969459 + 0.245254i $$0.921129\pi$$
$$858$$ 0 0
$$859$$ 47500.0 1.88670 0.943352 0.331793i $$-0.107654\pi$$
0.943352 + 0.331793i $$0.107654\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 12960.0i 0.512087i
$$863$$ 4272.00i 0.168506i 0.996444 + 0.0842529i $$0.0268504\pi$$
−0.996444 + 0.0842529i $$0.973150\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 23788.0 0.933429
$$867$$ 0 0
$$868$$ 13664.0i 0.534316i
$$869$$ 7464.00 0.291368
$$870$$ 0 0
$$871$$ −18224.0 −0.708951
$$872$$ 5584.00i 0.216856i
$$873$$ 0 0
$$874$$ 10560.0 0.408693
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 27796.0i − 1.07024i −0.844775 0.535122i $$-0.820266\pi$$
0.844775 0.535122i $$-0.179734\pi$$
$$878$$ 25376.0i 0.975397i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 39996.0 1.52951 0.764756 0.644320i $$-0.222860\pi$$
0.764756 + 0.644320i $$0.222860\pi$$
$$882$$ 0 0
$$883$$ 3772.00i 0.143758i 0.997413 + 0.0718788i $$0.0228995\pi$$
−0.997413 + 0.0718788i $$0.977101\pi$$
$$884$$ 21216.0 0.807207
$$885$$ 0 0
$$886$$ −9936.00 −0.376757
$$887$$ 5784.00i 0.218949i 0.993990 + 0.109474i $$0.0349168\pi$$
−0.993990 + 0.109474i $$0.965083\pi$$
$$888$$ 0 0
$$889$$ 26684.0 1.00670
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 4648.00i − 0.174469i
$$893$$ 26400.0i 0.989297i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1792.00 0.0668153
$$897$$ 0 0
$$898$$ − 23016.0i − 0.855294i
$$899$$ −30744.0 −1.14057
$$900$$ 0 0
$$901$$ −20124.0 −0.744093
$$902$$ − 5760.00i − 0.212624i
$$903$$ 0 0
$$904$$ −1776.00 −0.0653417
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 8440.00i − 0.308981i −0.987994 0.154490i $$-0.950626\pi$$
0.987994 0.154490i $$-0.0493736\pi$$
$$908$$ − 9600.00i − 0.350867i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 31920.0 1.16087 0.580437 0.814305i $$-0.302882\pi$$
0.580437 + 0.814305i $$0.302882\pi$$
$$912$$ 0 0
$$913$$ − 2376.00i − 0.0861272i
$$914$$ −2164.00 −0.0783137
$$915$$ 0 0
$$916$$ −9256.00 −0.333872
$$917$$ − 40236.0i − 1.44897i
$$918$$ 0 0
$$919$$ −34652.0 −1.24381 −0.621906 0.783092i $$-0.713642\pi$$
−0.621906 + 0.783092i $$0.713642\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 22476.0i 0.802828i
$$923$$ − 66096.0i − 2.35707i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4604.00 −0.163388
$$927$$ 0 0
$$928$$ 4032.00i 0.142626i
$$929$$ 1404.00 0.0495842 0.0247921 0.999693i $$-0.492108\pi$$
0.0247921 + 0.999693i $$0.492108\pi$$
$$930$$ 0 0
$$931$$ −6468.00 −0.227691
$$932$$ 72.0000i 0.00253051i
$$933$$ 0 0
$$934$$ 31752.0 1.11237
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 7654.00i − 0.266857i −0.991058 0.133429i $$-0.957401\pi$$
0.991058 0.133429i $$-0.0425987\pi$$
$$938$$ 7504.00i 0.261209i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −11298.0 −0.391397 −0.195698 0.980664i $$-0.562697\pi$$
−0.195698 + 0.980664i $$0.562697\pi$$
$$942$$ 0 0
$$943$$ 57600.0i 1.98909i
$$944$$ 8544.00 0.294580
$$945$$ 0 0
$$946$$ −1248.00 −0.0428922
$$947$$ − 28968.0i − 0.994016i −0.867746 0.497008i $$-0.834432\pi$$
0.867746 0.497008i $$-0.165568\pi$$
$$948$$ 0 0
$$949$$ −31960.0 −1.09322
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 8736.00i − 0.297411i
$$953$$ 46410.0i 1.57751i 0.614707 + 0.788755i $$0.289274\pi$$
−0.614707 + 0.788755i $$0.710726\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 23472.0 0.794078
$$957$$ 0 0
$$958$$ 9288.00i 0.313238i
$$959$$ −11172.0 −0.376186
$$960$$ 0 0
$$961$$ 29745.0 0.998456
$$962$$ − 41344.0i − 1.38564i
$$963$$ 0 0
$$964$$ 17320.0 0.578672
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 41506.0i − 1.38029i −0.723670 0.690146i $$-0.757546\pi$$
0.723670 0.690146i $$-0.242454\pi$$
$$968$$ 10360.0i 0.343991i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 18246.0 0.603030 0.301515 0.953461i $$-0.402508\pi$$
0.301515 + 0.953461i $$0.402508\pi$$
$$972$$ 0 0
$$973$$ 9800.00i 0.322892i
$$974$$ −4852.00 −0.159618
$$975$$ 0 0
$$976$$ 5792.00 0.189956
$$977$$ − 25998.0i − 0.851330i −0.904881 0.425665i $$-0.860040\pi$$
0.904881 0.425665i $$-0.139960\pi$$
$$978$$ 0 0
$$979$$ 5832.00 0.190390
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 468.000i − 0.0152082i
$$983$$ 14616.0i 0.474240i 0.971480 + 0.237120i $$0.0762035\pi$$
−0.971480 + 0.237120i $$0.923797\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 19656.0 0.634863
$$987$$ 0 0
$$988$$ − 11968.0i − 0.385377i
$$989$$ 12480.0 0.401255
$$990$$ 0 0
$$991$$ −2968.00 −0.0951379 −0.0475689 0.998868i $$-0.515147\pi$$
−0.0475689 + 0.998868i $$0.515147\pi$$
$$992$$ − 7808.00i − 0.249903i
$$993$$ 0 0
$$994$$ −27216.0 −0.868450
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 9052.00i − 0.287542i −0.989611 0.143771i $$-0.954077\pi$$
0.989611 0.143771i $$-0.0459229\pi$$
$$998$$ − 28408.0i − 0.901042i
$$999$$ 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 450.4.c.f.199.2 2
3.2 odd 2 450.4.c.g.199.1 2
5.2 odd 4 450.4.a.c.1.1 1
5.3 odd 4 90.4.a.e.1.1 yes 1
5.4 even 2 inner 450.4.c.f.199.1 2
15.2 even 4 450.4.a.m.1.1 1
15.8 even 4 90.4.a.b.1.1 1
15.14 odd 2 450.4.c.g.199.2 2
20.3 even 4 720.4.a.t.1.1 1
45.13 odd 12 810.4.e.a.541.1 2
45.23 even 12 810.4.e.u.541.1 2
45.38 even 12 810.4.e.u.271.1 2
45.43 odd 12 810.4.e.a.271.1 2
60.23 odd 4 720.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
90.4.a.b.1.1 1 15.8 even 4
90.4.a.e.1.1 yes 1 5.3 odd 4
450.4.a.c.1.1 1 5.2 odd 4
450.4.a.m.1.1 1 15.2 even 4
450.4.c.f.199.1 2 5.4 even 2 inner
450.4.c.f.199.2 2 1.1 even 1 trivial
450.4.c.g.199.1 2 3.2 odd 2
450.4.c.g.199.2 2 15.14 odd 2
720.4.a.e.1.1 1 60.23 odd 4
720.4.a.t.1.1 1 20.3 even 4
810.4.e.a.271.1 2 45.43 odd 12
810.4.e.a.541.1 2 45.13 odd 12
810.4.e.u.271.1 2 45.38 even 12
810.4.e.u.541.1 2 45.23 even 12