# Properties

 Label 450.4.c.g.199.1 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 199.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 450.199 Dual form 450.4.c.g.199.2

## $q$-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 $$-0.473796\pi$$
0.0822304 + 0.996613i $$0.473796\pi$$
$$12$$ 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 $$0.187820\pi$$
−0.830911 + 0.556405i $$0.812180\pi$$
$$18$$ 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 $$-0.816928\pi$$
0.839117 0.543951i $$-0.183072\pi$$
$$24$$ 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 $$-0.632174\pi$$
−0.403407 + 0.915021i $$0.632174\pi$$
$$30$$ 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 $$-0.867170\pi$$
−0.914188 + 0.405291i $$0.867170\pi$$
$$42$$ 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 $$0.381109\pi$$
−0.364884 + 0.931053i $$0.618891\pi$$
$$48$$ 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 $$0.108510\pi$$
−0.942456 + 0.334330i $$0.891490\pi$$
$$54$$ 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 $$-0.700541\pi$$
−0.589160 + 0.808016i $$0.700541\pi$$
$$60$$ 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 $$-0.801818\pi$$
−0.812360 + 0.583156i $$0.801818\pi$$
$$72$$ 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 $$-0.915668\pi$$
0.965109 0.261847i $$-0.0843317\pi$$
$$84$$ 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 $$-0.303509\pi$$
0.578830 + 0.815448i $$0.303509\pi$$
$$90$$ 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 $$-0.766049\pi$$
−0.741845 + 0.670572i $$0.766049\pi$$
$$102$$ 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 $$0.137063\pi$$
−0.908717 + 0.417413i $$0.862937\pi$$
$$108$$ 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 $$0.0294561\pi$$
−0.995721 + 0.0924071i $$0.970544\pi$$
$$114$$ 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 $$-0.0921285\pi$$
0.958407 + 0.285406i $$0.0921285\pi$$
$$132$$ 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 $$-0.919956\pi$$
0.968549 0.248824i $$-0.0800440\pi$$
$$138$$ 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 $$-0.509977\pi$$
−0.0313397 + 0.999509i $$0.509977\pi$$
$$150$$ 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 $$-0.906630\pi$$
0.957286 0.289141i $$-0.0933697\pi$$
$$168$$ 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 $$0.0810282\pi$$
−0.967775 + 0.251817i $$0.918972\pi$$
$$174$$ 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 $$-0.423094\pi$$
0.239263 + 0.970955i $$0.423094\pi$$
$$180$$ 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 $$-0.396151\pi$$
0.320494 + 0.947250i $$0.396151\pi$$
$$192$$ 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 $$-0.773899\pi$$
0.758156 0.652073i $$-0.226101\pi$$
$$198$$ 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 $$-0.885887\pi$$
0.936425 0.350867i $$-0.114113\pi$$
$$228$$ 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 $$0.000805488\pi$$
−0.999997 + 0.00253051i $$0.999195\pi$$
$$234$$ 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 $$-0.207954\pi$$
0.794078 + 0.607816i $$0.207954\pi$$
$$240$$ 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 $$-0.480056\pi$$
0.0626165 + 0.998038i $$0.480056\pi$$
$$252$$ 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 $$0.0248254\pi$$
−0.996960 + 0.0779122i $$0.975175\pi$$
$$258$$ 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 $$0.383781\pi$$
−0.357055 + 0.934084i $$0.616219\pi$$
$$264$$ 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 $$-0.341354\pi$$
0.478022 + 0.878348i $$0.341354\pi$$
$$270$$ 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 $$-0.544338\pi$$
−0.138841 + 0.990315i $$0.544338\pi$$
$$282$$ 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 $$-0.825559\pi$$
0.853556 0.521000i $$-0.174441\pi$$
$$294$$ 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 $$-0.813103\pi$$
−0.832521 + 0.553993i $$0.813103\pi$$
$$312$$ 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 $$0.237948\pi$$
−0.733366 + 0.679834i $$0.762052\pi$$
$$318$$ 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 $$0.294257\pi$$
−0.602286 + 0.798280i $$0.705743\pi$$
$$348$$ 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 $$0.00791985\pi$$
−0.999690 + 0.0248784i $$0.992080\pi$$
$$354$$ 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 $$-0.280231\pi$$
0.636864 + 0.770976i $$0.280231\pi$$
$$360$$ 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 $$0.0903907\pi$$
−0.959950 + 0.280170i $$0.909609\pi$$
$$384$$ 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 $$-0.835387\pi$$
−0.869233 + 0.494402i $$0.835387\pi$$
$$390$$ 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 $$-0.546064\pi$$
−0.144209 + 0.989547i $$0.546064\pi$$
$$402$$ 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 $$-0.561735\pi$$
−0.192731 + 0.981252i $$0.561735\pi$$
$$420$$ 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 $$-0.617940\pi$$
−0.362100 + 0.932139i $$0.617940\pi$$
$$432$$ 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 $$-0.914163\pi$$
0.963861 0.266407i $$-0.0858366\pi$$
$$444$$ 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 $$-0.293259\pi$$
0.604784 + 0.796389i $$0.293259\pi$$
$$450$$ 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 $$-0.692161\pi$$
−0.567685 + 0.823246i $$0.692161\pi$$
$$462$$ 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 $$0.288144\pi$$
−0.617505 + 0.786567i $$0.711856\pi$$
$$468$$ 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 $$-0.571093\pi$$
−0.221492 + 0.975162i $$0.571093\pi$$
$$480$$ 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 $$-0.496577\pi$$
0.0107538 + 0.999942i $$0.496577\pi$$
$$492$$ 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 $$-0.930026\pi$$
0.975935 0.218064i $$-0.0699740\pi$$
$$504$$ 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 $$-0.437820\pi$$
0.194104 + 0.980981i $$0.437820\pi$$
$$510$$ 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 $$-0.443331\pi$$
0.177093 + 0.984194i $$0.443331\pi$$
$$522$$ 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 $$-0.851493\pi$$
0.893126 0.449806i $$-0.148507\pi$$
$$558$$ 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 $$0.0352421\pi$$
−0.993877 + 0.110490i $$0.964758\pi$$
$$564$$ 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 $$-0.536241\pi$$
−0.113610 + 0.993525i $$0.536241\pi$$
$$570$$ 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 $$-0.990464\pi$$
0.999551 0.0299538i $$-0.00953603\pi$$
$$588$$ 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 $$-0.819075\pi$$
0.842767 0.538278i $$-0.180925\pi$$
$$594$$ 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 $$-0.405061\pi$$
0.293857 + 0.955850i $$0.405061\pi$$
$$600$$ 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 $$-0.994080\pi$$
0.999827 0.0185959i $$-0.00591960\pi$$
$$618$$ 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 $$-0.504472\pi$$
−0.0140491 + 0.999901i $$0.504472\pi$$
$$642$$ 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 $$0.0704355\pi$$
−0.975617 + 0.219478i $$0.929565\pi$$
$$648$$ 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 $$-0.805511\pi$$
0.819072 0.573691i $$-0.194489\pi$$
$$654$$ 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 $$-0.807111\pi$$
−0.821945 + 0.569567i $$0.807111\pi$$
$$660$$ 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 $$-0.719251\pi$$
0.635608 0.772012i $$-0.280749\pi$$
$$678$$ 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 $$-0.639702\pi$$
0.424933 0.905225i $$-0.360298\pi$$
$$684$$ 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 $$-0.289664\pi$$
0.613741 + 0.789508i $$0.289664\pi$$
$$702$$ 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 $$-0.450467\pi$$
0.154984 + 0.987917i $$0.450467\pi$$
$$720$$ 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 $$-0.989625\pi$$
0.999469 0.0325882i $$-0.0103750\pi$$
$$744$$ 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 $$-0.398871\pi$$
0.312388 + 0.949955i $$0.398871\pi$$
$$762$$ 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 $$0.0899233\pi$$
−0.960361 + 0.278760i $$0.910077\pi$$
$$774$$ 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 $$0.163293\pi$$
−0.871275 + 0.490794i $$0.836707\pi$$
$$798$$ 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 $$-0.653824\pi$$
−0.464661 + 0.885489i $$0.653824\pi$$
$$810$$ 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 $$-0.799391\pi$$
−0.807892 + 0.589331i $$0.799391\pi$$
$$822$$ 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 $$-0.894824\pi$$
0.945906 0.324440i $$-0.105176\pi$$
$$828$$ 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 $$-0.691844\pi$$
−0.566866 + 0.823810i $$0.691844\pi$$
$$840$$ 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 $$-0.921129\pi$$
0.969459 0.245254i $$-0.0788713\pi$$
$$858$$ 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 $$-0.973150\pi$$
0.996444 0.0842529i $$-0.0268504\pi$$
$$864$$ 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 $$-0.777140\pi$$
−0.764756 + 0.644320i $$0.777140\pi$$
$$882$$ 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 $$-0.965083\pi$$
0.993990 0.109474i $$-0.0349168\pi$$
$$888$$ 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 $$-0.697118\pi$$
−0.580437 + 0.814305i $$0.697118\pi$$
$$912$$ 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 $$-0.507892\pi$$
−0.0247921 + 0.999693i $$0.507892\pi$$
$$930$$ 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 $$-0.437303\pi$$
0.195698 + 0.980664i $$0.437303\pi$$
$$942$$ 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 $$0.165568\pi$$
−0.867746 + 0.497008i $$0.834432\pi$$
$$948$$ 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 $$-0.710726\pi$$
0.614707 0.788755i $$-0.289274\pi$$
$$954$$ 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 $$-0.597492\pi$$
−0.301515 + 0.953461i $$0.597492\pi$$
$$972$$ 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 $$0.139960\pi$$
−0.904881 + 0.425665i $$0.860040\pi$$
$$978$$ 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 $$-0.923797\pi$$
0.971480 0.237120i $$-0.0762035\pi$$
$$984$$ 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.g.199.1 2
3.2 odd 2 450.4.c.f.199.2 2
5.2 odd 4 450.4.a.m.1.1 1
5.3 odd 4 90.4.a.b.1.1 1
5.4 even 2 inner 450.4.c.g.199.2 2
15.2 even 4 450.4.a.c.1.1 1
15.8 even 4 90.4.a.e.1.1 yes 1
15.14 odd 2 450.4.c.f.199.1 2
20.3 even 4 720.4.a.e.1.1 1
45.13 odd 12 810.4.e.u.541.1 2
45.23 even 12 810.4.e.a.541.1 2
45.38 even 12 810.4.e.a.271.1 2
45.43 odd 12 810.4.e.u.271.1 2
60.23 odd 4 720.4.a.t.1.1 1

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