# Properties

 Label 75.3.c.d.26.2 Level $75$ Weight $3$ Character 75.26 Analytic conductor $2.044$ Analytic rank $0$ Dimension $2$ CM discriminant -15 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.04360198270$$ 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: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 26.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 75.26 Dual form 75.3.c.d.26.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +3.00000i q^{3} +3.00000 q^{4} -3.00000 q^{6} +7.00000i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +3.00000i q^{3} +3.00000 q^{4} -3.00000 q^{6} +7.00000i q^{8} -9.00000 q^{9} +9.00000i q^{12} +5.00000 q^{16} -14.0000i q^{17} -9.00000i q^{18} +22.0000 q^{19} -34.0000i q^{23} -21.0000 q^{24} -27.0000i q^{27} +2.00000 q^{31} +33.0000i q^{32} +14.0000 q^{34} -27.0000 q^{36} +22.0000i q^{38} +34.0000 q^{46} -14.0000i q^{47} +15.0000i q^{48} -49.0000 q^{49} +42.0000 q^{51} +86.0000i q^{53} +27.0000 q^{54} +66.0000i q^{57} -118.000 q^{61} +2.00000i q^{62} -13.0000 q^{64} -42.0000i q^{68} +102.000 q^{69} -63.0000i q^{72} +66.0000 q^{76} -98.0000 q^{79} +81.0000 q^{81} -154.000i q^{83} -102.000i q^{92} +6.00000i q^{93} +14.0000 q^{94} -99.0000 q^{96} -49.0000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{4} - 6 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^4 - 6 * q^6 - 18 * q^9 $$2 q + 6 q^{4} - 6 q^{6} - 18 q^{9} + 10 q^{16} + 44 q^{19} - 42 q^{24} + 4 q^{31} + 28 q^{34} - 54 q^{36} + 68 q^{46} - 98 q^{49} + 84 q^{51} + 54 q^{54} - 236 q^{61} - 26 q^{64} + 204 q^{69} + 132 q^{76} - 196 q^{79} + 162 q^{81} + 28 q^{94} - 198 q^{96}+O(q^{100})$$ 2 * q + 6 * q^4 - 6 * q^6 - 18 * q^9 + 10 * q^16 + 44 * q^19 - 42 * q^24 + 4 * q^31 + 28 * q^34 - 54 * q^36 + 68 * q^46 - 98 * q^49 + 84 * q^51 + 54 * q^54 - 236 * q^61 - 26 * q^64 + 204 * q^69 + 132 * q^76 - 196 * q^79 + 162 * q^81 + 28 * q^94 - 198 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.500000i 0.968246 + 0.250000i $$0.0804306\pi$$
−0.968246 + 0.250000i $$0.919569\pi$$
$$3$$ 3.00000i 1.00000i
$$4$$ 3.00000 0.750000
$$5$$ 0 0
$$6$$ −3.00000 −0.500000
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 7.00000i 0.875000i
$$9$$ −9.00000 −1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 9.00000i 0.750000i
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 5.00000 0.312500
$$17$$ − 14.0000i − 0.823529i −0.911290 0.411765i $$-0.864913\pi$$
0.911290 0.411765i $$-0.135087\pi$$
$$18$$ − 9.00000i − 0.500000i
$$19$$ 22.0000 1.15789 0.578947 0.815365i $$-0.303464\pi$$
0.578947 + 0.815365i $$0.303464\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 34.0000i − 1.47826i −0.673562 0.739130i $$-0.735237\pi$$
0.673562 0.739130i $$-0.264763\pi$$
$$24$$ −21.0000 −0.875000
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 27.0000i − 1.00000i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.0645161 0.0322581 0.999480i $$-0.489730\pi$$
0.0322581 + 0.999480i $$0.489730\pi$$
$$32$$ 33.0000i 1.03125i
$$33$$ 0 0
$$34$$ 14.0000 0.411765
$$35$$ 0 0
$$36$$ −27.0000 −0.750000
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 22.0000i 0.578947i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 34.0000 0.739130
$$47$$ − 14.0000i − 0.297872i −0.988847 0.148936i $$-0.952415\pi$$
0.988847 0.148936i $$-0.0475849\pi$$
$$48$$ 15.0000i 0.312500i
$$49$$ −49.0000 −1.00000
$$50$$ 0 0
$$51$$ 42.0000 0.823529
$$52$$ 0 0
$$53$$ 86.0000i 1.62264i 0.584601 + 0.811321i $$0.301251\pi$$
−0.584601 + 0.811321i $$0.698749\pi$$
$$54$$ 27.0000 0.500000
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 66.0000i 1.15789i
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ −118.000 −1.93443 −0.967213 0.253966i $$-0.918265\pi$$
−0.967213 + 0.253966i $$0.918265\pi$$
$$62$$ 2.00000i 0.0322581i
$$63$$ 0 0
$$64$$ −13.0000 −0.203125
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ − 42.0000i − 0.617647i
$$69$$ 102.000 1.47826
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ − 63.0000i − 0.875000i
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 66.0000 0.868421
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −98.0000 −1.24051 −0.620253 0.784402i $$-0.712970\pi$$
−0.620253 + 0.784402i $$0.712970\pi$$
$$80$$ 0 0
$$81$$ 81.0000 1.00000
$$82$$ 0 0
$$83$$ − 154.000i − 1.85542i −0.373300 0.927711i $$-0.621774\pi$$
0.373300 0.927711i $$-0.378226\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 102.000i − 1.10870i
$$93$$ 6.00000i 0.0645161i
$$94$$ 14.0000 0.148936
$$95$$ 0 0
$$96$$ −99.0000 −1.03125
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ − 49.0000i − 0.500000i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 42.0000i 0.411765i
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −86.0000 −0.811321
$$107$$ 106.000i 0.990654i 0.868707 + 0.495327i $$0.164952\pi$$
−0.868707 + 0.495327i $$0.835048\pi$$
$$108$$ − 81.0000i − 0.750000i
$$109$$ 22.0000 0.201835 0.100917 0.994895i $$-0.467822\pi$$
0.100917 + 0.994895i $$0.467822\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 206.000i 1.82301i 0.411290 + 0.911504i $$0.365078\pi$$
−0.411290 + 0.911504i $$0.634922\pi$$
$$114$$ −66.0000 −0.578947
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 121.000 1.00000
$$122$$ − 118.000i − 0.967213i
$$123$$ 0 0
$$124$$ 6.00000 0.0483871
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 119.000i 0.929688i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 98.0000 0.720588
$$137$$ 226.000i 1.64964i 0.565399 + 0.824818i $$0.308722\pi$$
−0.565399 + 0.824818i $$0.691278\pi$$
$$138$$ 102.000i 0.739130i
$$139$$ 262.000 1.88489 0.942446 0.334358i $$-0.108520\pi$$
0.942446 + 0.334358i $$0.108520\pi$$
$$140$$ 0 0
$$141$$ 42.0000 0.297872
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −45.0000 −0.312500
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 147.000i − 1.00000i
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ −238.000 −1.57616 −0.788079 0.615574i $$-0.788924\pi$$
−0.788079 + 0.615574i $$0.788924\pi$$
$$152$$ 154.000i 1.01316i
$$153$$ 126.000i 0.823529i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ − 98.0000i − 0.620253i
$$159$$ −258.000 −1.62264
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 81.0000i 0.500000i
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 154.000 0.927711
$$167$$ − 254.000i − 1.52096i −0.649362 0.760479i $$-0.724964\pi$$
0.649362 0.760479i $$-0.275036\pi$$
$$168$$ 0 0
$$169$$ −169.000 −1.00000
$$170$$ 0 0
$$171$$ −198.000 −1.15789
$$172$$ 0 0
$$173$$ − 154.000i − 0.890173i −0.895487 0.445087i $$-0.853173\pi$$
0.895487 0.445087i $$-0.146827\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ 122.000 0.674033 0.337017 0.941499i $$-0.390582\pi$$
0.337017 + 0.941499i $$0.390582\pi$$
$$182$$ 0 0
$$183$$ − 354.000i − 1.93443i
$$184$$ 238.000 1.29348
$$185$$ 0 0
$$186$$ −6.00000 −0.0322581
$$187$$ 0 0
$$188$$ − 42.0000i − 0.223404i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ − 39.0000i − 0.203125i
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −147.000 −0.750000
$$197$$ − 374.000i − 1.89848i −0.314557 0.949239i $$-0.601856\pi$$
0.314557 0.949239i $$-0.398144\pi$$
$$198$$ 0 0
$$199$$ 142.000 0.713568 0.356784 0.934187i $$-0.383873\pi$$
0.356784 + 0.934187i $$0.383873\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 126.000 0.617647
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 306.000i 1.47826i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 362.000 1.71564 0.857820 0.513950i $$-0.171818\pi$$
0.857820 + 0.513950i $$0.171818\pi$$
$$212$$ 258.000i 1.21698i
$$213$$ 0 0
$$214$$ −106.000 −0.495327
$$215$$ 0 0
$$216$$ 189.000 0.875000
$$217$$ 0 0
$$218$$ 22.0000i 0.100917i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −206.000 −0.911504
$$227$$ − 134.000i − 0.590308i −0.955450 0.295154i $$-0.904629\pi$$
0.955450 0.295154i $$-0.0953710\pi$$
$$228$$ 198.000i 0.868421i
$$229$$ −218.000 −0.951965 −0.475983 0.879455i $$-0.657907\pi$$
−0.475983 + 0.879455i $$0.657907\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 34.0000i − 0.145923i −0.997335 0.0729614i $$-0.976755\pi$$
0.997335 0.0729614i $$-0.0232450\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 294.000i − 1.24051i
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −478.000 −1.98340 −0.991701 0.128564i $$-0.958963\pi$$
−0.991701 + 0.128564i $$0.958963\pi$$
$$242$$ 121.000i 0.500000i
$$243$$ 243.000i 1.00000i
$$244$$ −354.000 −1.45082
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 14.0000i 0.0564516i
$$249$$ 462.000 1.85542
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −171.000 −0.667969
$$257$$ 466.000i 1.81323i 0.421959 + 0.906615i $$0.361343\pi$$
−0.421959 + 0.906615i $$0.638657\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 446.000i 1.69582i 0.530142 + 0.847909i $$0.322139\pi$$
−0.530142 + 0.847909i $$0.677861\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 482.000 1.77860 0.889299 0.457326i $$-0.151193\pi$$
0.889299 + 0.457326i $$0.151193\pi$$
$$272$$ − 70.0000i − 0.257353i
$$273$$ 0 0
$$274$$ −226.000 −0.824818
$$275$$ 0 0
$$276$$ 306.000 1.10870
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 262.000i 0.942446i
$$279$$ −18.0000 −0.0645161
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 42.0000i 0.148936i
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ − 297.000i − 1.03125i
$$289$$ 93.0000 0.321799
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 394.000i − 1.34471i −0.740229 0.672355i $$-0.765283\pi$$
0.740229 0.672355i $$-0.234717\pi$$
$$294$$ 147.000 0.500000
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 238.000i − 0.788079i
$$303$$ 0 0
$$304$$ 110.000 0.361842
$$305$$ 0 0
$$306$$ −126.000 −0.411765
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −294.000 −0.930380
$$317$$ − 134.000i − 0.422713i −0.977409 0.211356i $$-0.932212\pi$$
0.977409 0.211356i $$-0.0677881\pi$$
$$318$$ − 258.000i − 0.811321i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −318.000 −0.990654
$$322$$ 0 0
$$323$$ − 308.000i − 0.953560i
$$324$$ 243.000 0.750000
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 66.0000i 0.201835i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 122.000 0.368580 0.184290 0.982872i $$-0.441001\pi$$
0.184290 + 0.982872i $$0.441001\pi$$
$$332$$ − 462.000i − 1.39157i
$$333$$ 0 0
$$334$$ 254.000 0.760479
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ − 169.000i − 0.500000i
$$339$$ −618.000 −1.82301
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 198.000i − 0.578947i
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 154.000 0.445087
$$347$$ 586.000i 1.68876i 0.535744 + 0.844380i $$0.320031\pi$$
−0.535744 + 0.844380i $$0.679969\pi$$
$$348$$ 0 0
$$349$$ −458.000 −1.31232 −0.656160 0.754621i $$-0.727821\pi$$
−0.656160 + 0.754621i $$0.727821\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 274.000i − 0.776204i −0.921616 0.388102i $$-0.873131\pi$$
0.921616 0.388102i $$-0.126869\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 123.000 0.340720
$$362$$ 122.000i 0.337017i
$$363$$ 363.000i 1.00000i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 354.000 0.967213
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ − 170.000i − 0.461957i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 18.0000i 0.0483871i
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 98.0000 0.260638
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 742.000 1.95778 0.978892 0.204379i $$-0.0655175\pi$$
0.978892 + 0.204379i $$0.0655175\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 686.000i 1.79112i 0.444938 + 0.895561i $$0.353226\pi$$
−0.444938 + 0.895561i $$0.646774\pi$$
$$384$$ −357.000 −0.929688
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ −476.000 −1.21739
$$392$$ − 343.000i − 0.875000i
$$393$$ 0 0
$$394$$ 374.000 0.949239
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ 142.000i 0.356784i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 294.000i 0.720588i
$$409$$ 142.000 0.347188 0.173594 0.984817i $$-0.444462\pi$$
0.173594 + 0.984817i $$0.444462\pi$$
$$410$$ 0 0
$$411$$ −678.000 −1.64964
$$412$$ 0 0
$$413$$ 0 0
$$414$$ −306.000 −0.739130
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 786.000i 1.88489i
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 602.000 1.42993 0.714964 0.699161i $$-0.246443\pi$$
0.714964 + 0.699161i $$0.246443\pi$$
$$422$$ 362.000i 0.857820i
$$423$$ 126.000i 0.297872i
$$424$$ −602.000 −1.41981
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 318.000i 0.742991i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ − 135.000i − 0.312500i
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 66.0000 0.151376
$$437$$ − 748.000i − 1.71167i
$$438$$ 0 0
$$439$$ 622.000 1.41686 0.708428 0.705783i $$-0.249405\pi$$
0.708428 + 0.705783i $$0.249405\pi$$
$$440$$ 0 0
$$441$$ 441.000 1.00000
$$442$$ 0 0
$$443$$ 566.000i 1.27765i 0.769351 + 0.638826i $$0.220580\pi$$
−0.769351 + 0.638826i $$0.779420\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 618.000i 1.36726i
$$453$$ − 714.000i − 1.57616i
$$454$$ 134.000 0.295154
$$455$$ 0 0
$$456$$ −462.000 −1.01316
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ − 218.000i − 0.475983i
$$459$$ −378.000 −0.823529
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 34.0000 0.0729614
$$467$$ 346.000i 0.740899i 0.928853 + 0.370450i $$0.120796\pi$$
−0.928853 + 0.370450i $$0.879204\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 294.000 0.620253
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 774.000i − 1.62264i
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ − 478.000i − 0.991701i
$$483$$ 0 0
$$484$$ 363.000 0.750000
$$485$$ 0 0
$$486$$ −243.000 −0.500000
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ − 826.000i − 1.69262i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 10.0000 0.0201613
$$497$$ 0 0
$$498$$ 462.000i 0.927711i
$$499$$ −938.000 −1.87976 −0.939880 0.341506i $$-0.889063\pi$$
−0.939880 + 0.341506i $$0.889063\pi$$
$$500$$ 0 0
$$501$$ 762.000 1.52096
$$502$$ 0 0
$$503$$ − 994.000i − 1.97614i −0.153995 0.988072i $$-0.549214\pi$$
0.153995 0.988072i $$-0.450786\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 507.000i − 1.00000i
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 305.000i 0.595703i
$$513$$ − 594.000i − 1.15789i
$$514$$ −466.000 −0.906615
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 462.000 0.890173
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −446.000 −0.847909
$$527$$ − 28.0000i − 0.0531309i
$$528$$ 0 0
$$529$$ −627.000 −1.18526
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1078.00 −1.99261 −0.996303 0.0859072i $$-0.972621\pi$$
−0.996303 + 0.0859072i $$0.972621\pi$$
$$542$$ 482.000i 0.889299i
$$543$$ 366.000i 0.674033i
$$544$$ 462.000 0.849265
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 678.000i 1.23723i
$$549$$ 1062.00 1.93443
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 714.000i 1.29348i
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 786.000 1.41367
$$557$$ − 614.000i − 1.10233i −0.834395 0.551167i $$-0.814183\pi$$
0.834395 0.551167i $$-0.185817\pi$$
$$558$$ − 18.0000i − 0.0322581i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 154.000i − 0.273535i −0.990603 0.136767i $$-0.956329\pi$$
0.990603 0.136767i $$-0.0436713\pi$$
$$564$$ 126.000 0.223404
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ −358.000 −0.626970 −0.313485 0.949593i $$-0.601497\pi$$
−0.313485 + 0.949593i $$0.601497\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 117.000 0.203125
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 93.0000i 0.160900i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 394.000 0.672355
$$587$$ − 854.000i − 1.45486i −0.686184 0.727428i $$-0.740716\pi$$
0.686184 0.727428i $$-0.259284\pi$$
$$588$$ − 441.000i − 0.750000i
$$589$$ 44.0000 0.0747029
$$590$$ 0 0
$$591$$ 1122.00 1.89848
$$592$$ 0 0
$$593$$ 1166.00i 1.96627i 0.182873 + 0.983137i $$0.441460\pi$$
−0.182873 + 0.983137i $$0.558540\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 426.000i 0.713568i
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 242.000 0.402662 0.201331 0.979523i $$-0.435473\pi$$
0.201331 + 0.979523i $$0.435473\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −714.000 −1.18212
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 726.000i 1.19408i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 378.000i 0.617647i
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1186.00i 1.92220i 0.276193 + 0.961102i $$0.410927\pi$$
−0.276193 + 0.961102i $$0.589073\pi$$
$$618$$ 0 0
$$619$$ −698.000 −1.12763 −0.563813 0.825903i $$-0.690666\pi$$
−0.563813 + 0.825903i $$0.690666\pi$$
$$620$$ 0 0
$$621$$ −918.000 −1.47826
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −238.000 −0.377179 −0.188590 0.982056i $$-0.560392\pi$$
−0.188590 + 0.982056i $$0.560392\pi$$
$$632$$ − 686.000i − 1.08544i
$$633$$ 1086.00i 1.71564i
$$634$$ 134.000 0.211356
$$635$$ 0 0
$$636$$ −774.000 −1.21698
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ − 318.000i − 0.495327i
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 308.000 0.476780
$$647$$ 706.000i 1.09119i 0.838049 + 0.545595i $$0.183696\pi$$
−0.838049 + 0.545595i $$0.816304\pi$$
$$648$$ 567.000i 0.875000i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 1114.00i − 1.70597i −0.521933 0.852986i $$-0.674789\pi$$
0.521933 0.852986i $$-0.325211\pi$$
$$654$$ −66.0000 −0.100917
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ −838.000 −1.26778 −0.633888 0.773425i $$-0.718542\pi$$
−0.633888 + 0.773425i $$0.718542\pi$$
$$662$$ 122.000i 0.184290i
$$663$$ 0 0
$$664$$ 1078.00 1.62349
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ − 762.000i − 1.14072i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −507.000 −0.750000
$$677$$ − 374.000i − 0.552437i −0.961095 0.276219i $$-0.910919\pi$$
0.961095 0.276219i $$-0.0890814\pi$$
$$678$$ − 618.000i − 0.911504i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 402.000 0.590308
$$682$$ 0 0
$$683$$ 86.0000i 0.125915i 0.998016 + 0.0629575i $$0.0200533\pi$$
−0.998016 + 0.0629575i $$0.979947\pi$$
$$684$$ −594.000 −0.868421
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 654.000i − 0.951965i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1322.00 1.91317 0.956585 0.291455i $$-0.0941392\pi$$
0.956585 + 0.291455i $$0.0941392\pi$$
$$692$$ − 462.000i − 0.667630i
$$693$$ 0 0
$$694$$ −586.000 −0.844380
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ − 458.000i − 0.656160i
$$699$$ 102.000 0.145923
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 274.000 0.388102
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 742.000 1.04654 0.523272 0.852166i $$-0.324711\pi$$
0.523272 + 0.852166i $$0.324711\pi$$
$$710$$ 0 0
$$711$$ 882.000 1.24051
$$712$$ 0 0
$$713$$ − 68.0000i − 0.0953717i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 123.000i 0.170360i
$$723$$ − 1434.00i − 1.98340i
$$724$$ 366.000 0.505525
$$725$$ 0 0
$$726$$ −363.000 −0.500000
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ −729.000 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ − 1062.00i − 1.45082i
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 1122.00 1.52446
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 1462.00 1.97835 0.989175 0.146744i $$-0.0468792\pi$$
0.989175 + 0.146744i $$0.0468792\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 514.000i − 0.691790i −0.938273 0.345895i $$-0.887575\pi$$
0.938273 0.345895i $$-0.112425\pi$$
$$744$$ −42.0000 −0.0564516
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 1386.00i 1.85542i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1438.00 −1.91478 −0.957390 0.288798i $$-0.906745\pi$$
−0.957390 + 0.288798i $$0.906745\pi$$
$$752$$ − 70.0000i − 0.0930851i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ 742.000i 0.978892i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −686.000 −0.895561
$$767$$ 0 0
$$768$$ − 513.000i − 0.667969i
$$769$$ −578.000 −0.751625 −0.375813 0.926696i $$-0.622636\pi$$
−0.375813 + 0.926696i $$0.622636\pi$$
$$770$$ 0 0
$$771$$ −1398.00 −1.81323
$$772$$ 0 0
$$773$$ 1526.00i 1.97413i 0.160330 + 0.987063i $$0.448744\pi$$
−0.160330 + 0.987063i $$0.551256\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 476.000i − 0.608696i
$$783$$ 0 0
$$784$$ −245.000 −0.312500
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ − 1122.00i − 1.42386i
$$789$$ −1338.00 −1.69582
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 426.000 0.535176
$$797$$ 826.000i 1.03639i 0.855264 + 0.518193i $$0.173395\pi$$
−0.855264 + 0.518193i $$0.826605\pi$$
$$798$$ 0 0
$$799$$ −196.000 −0.245307
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 1082.00 1.33416 0.667078 0.744988i $$-0.267545\pi$$
0.667078 + 0.744988i $$0.267545\pi$$
$$812$$ 0 0
$$813$$ 1446.00i 1.77860i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 210.000 0.257353
$$817$$ 0 0
$$818$$ 142.000i 0.173594i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ − 678.000i − 0.824818i
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 374.000i − 0.452237i −0.974100 0.226119i $$-0.927396\pi$$
0.974100 0.226119i $$-0.0726037\pi$$
$$828$$ 918.000i 1.10870i
$$829$$ 502.000 0.605549 0.302774 0.953062i $$-0.402087\pi$$
0.302774 + 0.953062i $$0.402087\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 686.000i 0.823529i
$$834$$ −786.000 −0.942446
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 54.0000i − 0.0645161i
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 841.000 1.00000
$$842$$ 602.000i 0.714964i
$$843$$ 0 0
$$844$$ 1086.00 1.28673
$$845$$ 0 0
$$846$$ −126.000 −0.148936
$$847$$ 0 0
$$848$$ 430.000i 0.507075i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −742.000 −0.866822
$$857$$ 1666.00i 1.94399i 0.235000 + 0.971995i $$0.424491\pi$$
−0.235000 + 0.971995i $$0.575509\pi$$
$$858$$ 0 0
$$859$$ −218.000 −0.253783 −0.126892 0.991917i $$-0.540500\pi$$
−0.126892 + 0.991917i $$0.540500\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 274.000i − 0.317497i −0.987319 0.158749i $$-0.949254\pi$$
0.987319 0.158749i $$-0.0507459\pi$$
$$864$$ 891.000 1.03125
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 279.000i 0.321799i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 154.000i 0.176606i
$$873$$ 0 0
$$874$$ 748.000 0.855835
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ 622.000i 0.708428i
$$879$$ 1182.00 1.34471
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 441.000i 0.500000i
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −566.000 −0.638826
$$887$$ − 1694.00i − 1.90981i −0.296914 0.954904i $$-0.595958\pi$$
0.296914 0.954904i $$-0.404042\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 308.000i − 0.344905i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 1204.00 1.33629
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −1442.00 −1.59513
$$905$$ 0 0
$$906$$ 714.000 0.788079
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ − 402.000i − 0.442731i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 330.000i 0.361842i
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −654.000 −0.713974
$$917$$ 0 0
$$918$$ − 378.000i − 0.411765i
$$919$$ −1298.00 −1.41240 −0.706202 0.708010i $$-0.749593\pi$$
−0.706202 + 0.708010i $$0.749593\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ −1078.00 −1.15789
$$932$$ − 102.000i − 0.109442i
$$933$$ 0 0
$$934$$ −346.000 −0.370450
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 1574.00i − 1.66209i −0.556205 0.831045i $$-0.687743\pi$$
0.556205 0.831045i $$-0.312257\pi$$
$$948$$ − 882.000i − 0.930380i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 402.000 0.422713
$$952$$ 0 0
$$953$$ − 1474.00i − 1.54669i −0.633983 0.773347i $$-0.718581\pi$$
0.633983 0.773347i $$-0.281419\pi$$
$$954$$ 774.000 0.811321
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −957.000 −0.995838
$$962$$ 0 0
$$963$$ − 954.000i − 0.990654i
$$964$$ −1434.00 −1.48755
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 847.000i 0.875000i
$$969$$ 924.000 0.953560
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 729.000i 0.750000i
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −590.000 −0.604508
$$977$$ − 1934.00i − 1.97953i −0.142710 0.989765i $$-0.545582\pi$$
0.142710 0.989765i $$-0.454418\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −198.000 −0.201835
$$982$$ 0 0
$$983$$ − 1954.00i − 1.98779i −0.110319 0.993896i $$-0.535187\pi$$
0.110319 0.993896i $$-0.464813\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −958.000 −0.966700 −0.483350 0.875427i $$-0.660580\pi$$
−0.483350 + 0.875427i $$0.660580\pi$$
$$992$$ 66.0000i 0.0665323i
$$993$$ 366.000i 0.368580i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 1386.00 1.39157
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ − 938.000i − 0.939880i
$$999$$ 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 75.3.c.d.26.2 2
3.2 odd 2 inner 75.3.c.d.26.1 2
4.3 odd 2 1200.3.l.l.401.1 2
5.2 odd 4 15.3.d.a.14.1 1
5.3 odd 4 15.3.d.b.14.1 yes 1
5.4 even 2 inner 75.3.c.d.26.1 2
12.11 even 2 1200.3.l.l.401.2 2
15.2 even 4 15.3.d.b.14.1 yes 1
15.8 even 4 15.3.d.a.14.1 1
15.14 odd 2 CM 75.3.c.d.26.2 2
20.3 even 4 240.3.c.b.209.1 1
20.7 even 4 240.3.c.a.209.1 1
20.19 odd 2 1200.3.l.l.401.2 2
40.3 even 4 960.3.c.a.449.1 1
40.13 odd 4 960.3.c.c.449.1 1
40.27 even 4 960.3.c.d.449.1 1
40.37 odd 4 960.3.c.b.449.1 1
45.2 even 12 405.3.h.a.134.1 2
45.7 odd 12 405.3.h.b.134.1 2
45.13 odd 12 405.3.h.a.269.1 2
45.22 odd 12 405.3.h.b.269.1 2
45.23 even 12 405.3.h.b.269.1 2
45.32 even 12 405.3.h.a.269.1 2
45.38 even 12 405.3.h.b.134.1 2
45.43 odd 12 405.3.h.a.134.1 2
60.23 odd 4 240.3.c.a.209.1 1
60.47 odd 4 240.3.c.b.209.1 1
60.59 even 2 1200.3.l.l.401.1 2
120.53 even 4 960.3.c.b.449.1 1
120.77 even 4 960.3.c.c.449.1 1
120.83 odd 4 960.3.c.d.449.1 1
120.107 odd 4 960.3.c.a.449.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.3.d.a.14.1 1 5.2 odd 4
15.3.d.a.14.1 1 15.8 even 4
15.3.d.b.14.1 yes 1 5.3 odd 4
15.3.d.b.14.1 yes 1 15.2 even 4
75.3.c.d.26.1 2 3.2 odd 2 inner
75.3.c.d.26.1 2 5.4 even 2 inner
75.3.c.d.26.2 2 1.1 even 1 trivial
75.3.c.d.26.2 2 15.14 odd 2 CM
240.3.c.a.209.1 1 20.7 even 4
240.3.c.a.209.1 1 60.23 odd 4
240.3.c.b.209.1 1 20.3 even 4
240.3.c.b.209.1 1 60.47 odd 4
405.3.h.a.134.1 2 45.2 even 12
405.3.h.a.134.1 2 45.43 odd 12
405.3.h.a.269.1 2 45.13 odd 12
405.3.h.a.269.1 2 45.32 even 12
405.3.h.b.134.1 2 45.7 odd 12
405.3.h.b.134.1 2 45.38 even 12
405.3.h.b.269.1 2 45.22 odd 12
405.3.h.b.269.1 2 45.23 even 12
960.3.c.a.449.1 1 40.3 even 4
960.3.c.a.449.1 1 120.107 odd 4
960.3.c.b.449.1 1 40.37 odd 4
960.3.c.b.449.1 1 120.53 even 4
960.3.c.c.449.1 1 40.13 odd 4
960.3.c.c.449.1 1 120.77 even 4
960.3.c.d.449.1 1 40.27 even 4
960.3.c.d.449.1 1 120.83 odd 4
1200.3.l.l.401.1 2 4.3 odd 2
1200.3.l.l.401.1 2 60.59 even 2
1200.3.l.l.401.2 2 12.11 even 2
1200.3.l.l.401.2 2 20.19 odd 2