# Properties

 Label 2352.2.b.k.1567.1 Level $2352$ Weight $2$ Character 2352.1567 Analytic conductor $18.781$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.1 Root $$-0.662827 - 0.382683i$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1567 Dual form 2352.2.b.k.1567.8

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -4.29725i q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -4.29725i q^{5} +1.00000 q^{9} +5.06262i q^{11} -3.37849i q^{13} +4.29725i q^{15} -2.76652i q^{17} -4.70319 q^{19} -4.83120i q^{23} -13.4663 q^{25} -1.00000 q^{27} +2.46054 q^{29} +5.69764 q^{31} -5.06262i q^{33} -2.33530 q^{37} +3.37849i q^{39} +5.14822i q^{41} -13.0199i q^{43} -4.29725i q^{45} -5.35449 q^{47} +2.76652i q^{51} -4.22371 q^{53} +21.7553 q^{55} +4.70319 q^{57} -9.60498 q^{59} +3.87698i q^{61} -14.5182 q^{65} -4.76342i q^{67} +4.83120i q^{69} +12.3181i q^{71} -11.5102i q^{73} +13.4663 q^{75} +13.9166i q^{79} +1.00000 q^{81} -7.32191 q^{83} -11.8884 q^{85} -2.46054 q^{87} +14.0448i q^{89} -5.69764 q^{93} +20.2108i q^{95} +14.5716i q^{97} +5.06262i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{3} + 8q^{9} + O(q^{10})$$ $$8q - 8q^{3} + 8q^{9} - 24q^{25} - 8q^{27} + 16q^{29} + 32q^{31} + 16q^{47} + 16q^{53} + 64q^{55} - 48q^{59} - 16q^{65} + 24q^{75} + 8q^{81} - 64q^{85} - 16q^{87} - 32q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ − 4.29725i − 1.92179i −0.276916 0.960894i $$-0.589312\pi$$
0.276916 0.960894i $$-0.410688\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.06262i 1.52644i 0.646141 + 0.763218i $$0.276382\pi$$
−0.646141 + 0.763218i $$0.723618\pi$$
$$12$$ 0 0
$$13$$ − 3.37849i − 0.937025i −0.883457 0.468513i $$-0.844790\pi$$
0.883457 0.468513i $$-0.155210\pi$$
$$14$$ 0 0
$$15$$ 4.29725i 1.10954i
$$16$$ 0 0
$$17$$ − 2.76652i − 0.670978i −0.942044 0.335489i $$-0.891098\pi$$
0.942044 0.335489i $$-0.108902\pi$$
$$18$$ 0 0
$$19$$ −4.70319 −1.07898 −0.539492 0.841990i $$-0.681384\pi$$
−0.539492 + 0.841990i $$0.681384\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 4.83120i − 1.00737i −0.863886 0.503687i $$-0.831976\pi$$
0.863886 0.503687i $$-0.168024\pi$$
$$24$$ 0 0
$$25$$ −13.4663 −2.69327
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 2.46054 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$30$$ 0 0
$$31$$ 5.69764 1.02333 0.511663 0.859186i $$-0.329030\pi$$
0.511663 + 0.859186i $$0.329030\pi$$
$$32$$ 0 0
$$33$$ − 5.06262i − 0.881288i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.33530 −0.383921 −0.191961 0.981403i $$-0.561485\pi$$
−0.191961 + 0.981403i $$0.561485\pi$$
$$38$$ 0 0
$$39$$ 3.37849i 0.540992i
$$40$$ 0 0
$$41$$ 5.14822i 0.804018i 0.915636 + 0.402009i $$0.131688\pi$$
−0.915636 + 0.402009i $$0.868312\pi$$
$$42$$ 0 0
$$43$$ − 13.0199i − 1.98552i −0.120117 0.992760i $$-0.538327\pi$$
0.120117 0.992760i $$-0.461673\pi$$
$$44$$ 0 0
$$45$$ − 4.29725i − 0.640596i
$$46$$ 0 0
$$47$$ −5.35449 −0.781033 −0.390517 0.920596i $$-0.627704\pi$$
−0.390517 + 0.920596i $$0.627704\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.76652i 0.387390i
$$52$$ 0 0
$$53$$ −4.22371 −0.580171 −0.290085 0.957001i $$-0.593684\pi$$
−0.290085 + 0.957001i $$0.593684\pi$$
$$54$$ 0 0
$$55$$ 21.7553 2.93349
$$56$$ 0 0
$$57$$ 4.70319 0.622952
$$58$$ 0 0
$$59$$ −9.60498 −1.25046 −0.625231 0.780440i $$-0.714995\pi$$
−0.625231 + 0.780440i $$0.714995\pi$$
$$60$$ 0 0
$$61$$ 3.87698i 0.496397i 0.968709 + 0.248198i $$0.0798385\pi$$
−0.968709 + 0.248198i $$0.920162\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −14.5182 −1.80076
$$66$$ 0 0
$$67$$ − 4.76342i − 0.581944i −0.956732 0.290972i $$-0.906021\pi$$
0.956732 0.290972i $$-0.0939787\pi$$
$$68$$ 0 0
$$69$$ 4.83120i 0.581608i
$$70$$ 0 0
$$71$$ 12.3181i 1.46189i 0.682437 + 0.730944i $$0.260920\pi$$
−0.682437 + 0.730944i $$0.739080\pi$$
$$72$$ 0 0
$$73$$ − 11.5102i − 1.34716i −0.739113 0.673581i $$-0.764755\pi$$
0.739113 0.673581i $$-0.235245\pi$$
$$74$$ 0 0
$$75$$ 13.4663 1.55496
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 13.9166i 1.56574i 0.622185 + 0.782870i $$0.286245\pi$$
−0.622185 + 0.782870i $$0.713755\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −7.32191 −0.803685 −0.401842 0.915709i $$-0.631630\pi$$
−0.401842 + 0.915709i $$0.631630\pi$$
$$84$$ 0 0
$$85$$ −11.8884 −1.28948
$$86$$ 0 0
$$87$$ −2.46054 −0.263798
$$88$$ 0 0
$$89$$ 14.0448i 1.48874i 0.667765 + 0.744372i $$0.267251\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −5.69764 −0.590818
$$94$$ 0 0
$$95$$ 20.2108i 2.07358i
$$96$$ 0 0
$$97$$ 14.5716i 1.47952i 0.672868 + 0.739762i $$0.265062\pi$$
−0.672868 + 0.739762i $$0.734938\pi$$
$$98$$ 0 0
$$99$$ 5.06262i 0.508812i
$$100$$ 0 0
$$101$$ − 13.1087i − 1.30436i −0.758063 0.652181i $$-0.773854\pi$$
0.758063 0.652181i $$-0.226146\pi$$
$$102$$ 0 0
$$103$$ −0.209698 −0.0206622 −0.0103311 0.999947i $$-0.503289\pi$$
−0.0103311 + 0.999947i $$0.503289\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 6.42657i 0.621280i 0.950528 + 0.310640i $$0.100543\pi$$
−0.950528 + 0.310640i $$0.899457\pi$$
$$108$$ 0 0
$$109$$ 0.940024 0.0900379 0.0450190 0.998986i $$-0.485665\pi$$
0.0450190 + 0.998986i $$0.485665\pi$$
$$110$$ 0 0
$$111$$ 2.33530 0.221657
$$112$$ 0 0
$$113$$ 1.09821 0.103311 0.0516554 0.998665i $$-0.483550\pi$$
0.0516554 + 0.998665i $$0.483550\pi$$
$$114$$ 0 0
$$115$$ −20.7609 −1.93596
$$116$$ 0 0
$$117$$ − 3.37849i − 0.312342i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −14.6301 −1.33001
$$122$$ 0 0
$$123$$ − 5.14822i − 0.464200i
$$124$$ 0 0
$$125$$ 36.3820i 3.25411i
$$126$$ 0 0
$$127$$ 5.22625i 0.463755i 0.972745 + 0.231877i $$0.0744868\pi$$
−0.972745 + 0.231877i $$0.925513\pi$$
$$128$$ 0 0
$$129$$ 13.0199i 1.14634i
$$130$$ 0 0
$$131$$ −0.875015 −0.0764504 −0.0382252 0.999269i $$-0.512170\pi$$
−0.0382252 + 0.999269i $$0.512170\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 4.29725i 0.369848i
$$136$$ 0 0
$$137$$ 0.236838 0.0202344 0.0101172 0.999949i $$-0.496780\pi$$
0.0101172 + 0.999949i $$0.496780\pi$$
$$138$$ 0 0
$$139$$ 3.00555 0.254927 0.127464 0.991843i $$-0.459316\pi$$
0.127464 + 0.991843i $$0.459316\pi$$
$$140$$ 0 0
$$141$$ 5.35449 0.450930
$$142$$ 0 0
$$143$$ 17.1040 1.43031
$$144$$ 0 0
$$145$$ − 10.5736i − 0.878087i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −14.8017 −1.21260 −0.606299 0.795237i $$-0.707347\pi$$
−0.606299 + 0.795237i $$0.707347\pi$$
$$150$$ 0 0
$$151$$ − 12.3654i − 1.00628i −0.864205 0.503140i $$-0.832178\pi$$
0.864205 0.503140i $$-0.167822\pi$$
$$152$$ 0 0
$$153$$ − 2.76652i − 0.223659i
$$154$$ 0 0
$$155$$ − 24.4842i − 1.96662i
$$156$$ 0 0
$$157$$ − 10.8906i − 0.869164i −0.900632 0.434582i $$-0.856896\pi$$
0.900632 0.434582i $$-0.143104\pi$$
$$158$$ 0 0
$$159$$ 4.22371 0.334962
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1.51024i 0.118291i 0.998249 + 0.0591455i $$0.0188376\pi$$
−0.998249 + 0.0591455i $$0.981162\pi$$
$$164$$ 0 0
$$165$$ −21.7553 −1.69365
$$166$$ 0 0
$$167$$ −10.3333 −0.799612 −0.399806 0.916600i $$-0.630922\pi$$
−0.399806 + 0.916600i $$0.630922\pi$$
$$168$$ 0 0
$$169$$ 1.58579 0.121984
$$170$$ 0 0
$$171$$ −4.70319 −0.359662
$$172$$ 0 0
$$173$$ 3.35642i 0.255184i 0.991827 + 0.127592i $$0.0407248\pi$$
−0.991827 + 0.127592i $$0.959275\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 9.60498 0.721954
$$178$$ 0 0
$$179$$ 13.2589i 0.991018i 0.868603 + 0.495509i $$0.165018\pi$$
−0.868603 + 0.495509i $$0.834982\pi$$
$$180$$ 0 0
$$181$$ − 11.8519i − 0.880946i −0.897766 0.440473i $$-0.854811\pi$$
0.897766 0.440473i $$-0.145189\pi$$
$$182$$ 0 0
$$183$$ − 3.87698i − 0.286595i
$$184$$ 0 0
$$185$$ 10.0354i 0.737816i
$$186$$ 0 0
$$187$$ 14.0058 1.02421
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 1.34660i − 0.0974368i −0.998813 0.0487184i $$-0.984486\pi$$
0.998813 0.0487184i $$-0.0155137\pi$$
$$192$$ 0 0
$$193$$ −26.1154 −1.87982 −0.939912 0.341416i $$-0.889094\pi$$
−0.939912 + 0.341416i $$0.889094\pi$$
$$194$$ 0 0
$$195$$ 14.5182 1.03967
$$196$$ 0 0
$$197$$ −7.49083 −0.533699 −0.266850 0.963738i $$-0.585983\pi$$
−0.266850 + 0.963738i $$0.585983\pi$$
$$198$$ 0 0
$$199$$ −4.49349 −0.318535 −0.159267 0.987235i $$-0.550913\pi$$
−0.159267 + 0.987235i $$0.550913\pi$$
$$200$$ 0 0
$$201$$ 4.76342i 0.335986i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 22.1232 1.54515
$$206$$ 0 0
$$207$$ − 4.83120i − 0.335792i
$$208$$ 0 0
$$209$$ − 23.8104i − 1.64700i
$$210$$ 0 0
$$211$$ 5.93122i 0.408322i 0.978937 + 0.204161i $$0.0654466\pi$$
−0.978937 + 0.204161i $$0.934553\pi$$
$$212$$ 0 0
$$213$$ − 12.3181i − 0.844022i
$$214$$ 0 0
$$215$$ −55.9498 −3.81575
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 11.5102i 0.777784i
$$220$$ 0 0
$$221$$ −9.34665 −0.628724
$$222$$ 0 0
$$223$$ −20.6163 −1.38057 −0.690286 0.723537i $$-0.742515\pi$$
−0.690286 + 0.723537i $$0.742515\pi$$
$$224$$ 0 0
$$225$$ −13.4663 −0.897756
$$226$$ 0 0
$$227$$ −20.4911 −1.36004 −0.680021 0.733193i $$-0.738029\pi$$
−0.680021 + 0.733193i $$0.738029\pi$$
$$228$$ 0 0
$$229$$ − 29.0409i − 1.91908i −0.281576 0.959539i $$-0.590857\pi$$
0.281576 0.959539i $$-0.409143\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 17.7364 1.16195 0.580975 0.813922i $$-0.302672\pi$$
0.580975 + 0.813922i $$0.302672\pi$$
$$234$$ 0 0
$$235$$ 23.0096i 1.50098i
$$236$$ 0 0
$$237$$ − 13.9166i − 0.903981i
$$238$$ 0 0
$$239$$ − 20.4248i − 1.32117i −0.750750 0.660586i $$-0.770308\pi$$
0.750750 0.660586i $$-0.229692\pi$$
$$240$$ 0 0
$$241$$ 3.25742i 0.209829i 0.994481 + 0.104915i $$0.0334569\pi$$
−0.994481 + 0.104915i $$0.966543\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 15.8897i 1.01104i
$$248$$ 0 0
$$249$$ 7.32191 0.464007
$$250$$ 0 0
$$251$$ −3.95633 −0.249721 −0.124861 0.992174i $$-0.539848\pi$$
−0.124861 + 0.992174i $$0.539848\pi$$
$$252$$ 0 0
$$253$$ 24.4585 1.53769
$$254$$ 0 0
$$255$$ 11.8884 0.744481
$$256$$ 0 0
$$257$$ − 5.14822i − 0.321137i −0.987025 0.160569i $$-0.948667\pi$$
0.987025 0.160569i $$-0.0513328\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.46054 0.152304
$$262$$ 0 0
$$263$$ 0.0785021i 0.00484065i 0.999997 + 0.00242032i $$0.000770414\pi$$
−0.999997 + 0.00242032i $$0.999230\pi$$
$$264$$ 0 0
$$265$$ 18.1503i 1.11497i
$$266$$ 0 0
$$267$$ − 14.0448i − 0.859527i
$$268$$ 0 0
$$269$$ − 0.883296i − 0.0538555i −0.999637 0.0269277i $$-0.991428\pi$$
0.999637 0.0269277i $$-0.00857240\pi$$
$$270$$ 0 0
$$271$$ 19.3084 1.17290 0.586451 0.809984i $$-0.300524\pi$$
0.586451 + 0.809984i $$0.300524\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 68.1749i − 4.11110i
$$276$$ 0 0
$$277$$ 16.9327 1.01739 0.508694 0.860948i $$-0.330129\pi$$
0.508694 + 0.860948i $$0.330129\pi$$
$$278$$ 0 0
$$279$$ 5.69764 0.341109
$$280$$ 0 0
$$281$$ 5.05417 0.301507 0.150753 0.988571i $$-0.451830\pi$$
0.150753 + 0.988571i $$0.451830\pi$$
$$282$$ 0 0
$$283$$ 19.1156 1.13631 0.568153 0.822923i $$-0.307658\pi$$
0.568153 + 0.822923i $$0.307658\pi$$
$$284$$ 0 0
$$285$$ − 20.2108i − 1.19718i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 9.34639 0.549788
$$290$$ 0 0
$$291$$ − 14.5716i − 0.854204i
$$292$$ 0 0
$$293$$ 2.37837i 0.138946i 0.997584 + 0.0694730i $$0.0221318\pi$$
−0.997584 + 0.0694730i $$0.977868\pi$$
$$294$$ 0 0
$$295$$ 41.2750i 2.40312i
$$296$$ 0 0
$$297$$ − 5.06262i − 0.293763i
$$298$$ 0 0
$$299$$ −16.3222 −0.943936
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 13.1087i 0.753073i
$$304$$ 0 0
$$305$$ 16.6604 0.953969
$$306$$ 0 0
$$307$$ −4.89599 −0.279429 −0.139714 0.990192i $$-0.544618\pi$$
−0.139714 + 0.990192i $$0.544618\pi$$
$$308$$ 0 0
$$309$$ 0.209698 0.0119293
$$310$$ 0 0
$$311$$ −10.4800 −0.594266 −0.297133 0.954836i $$-0.596030\pi$$
−0.297133 + 0.954836i $$0.596030\pi$$
$$312$$ 0 0
$$313$$ − 9.60172i − 0.542722i −0.962478 0.271361i $$-0.912526\pi$$
0.962478 0.271361i $$-0.0874737\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −16.7934 −0.943214 −0.471607 0.881809i $$-0.656326\pi$$
−0.471607 + 0.881809i $$0.656326\pi$$
$$318$$ 0 0
$$319$$ 12.4568i 0.697446i
$$320$$ 0 0
$$321$$ − 6.42657i − 0.358696i
$$322$$ 0 0
$$323$$ 13.0114i 0.723976i
$$324$$ 0 0
$$325$$ 45.4960i 2.52366i
$$326$$ 0 0
$$327$$ −0.940024 −0.0519834
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 26.8362i − 1.47505i −0.675318 0.737526i $$-0.735994\pi$$
0.675318 0.737526i $$-0.264006\pi$$
$$332$$ 0 0
$$333$$ −2.33530 −0.127974
$$334$$ 0 0
$$335$$ −20.4696 −1.11837
$$336$$ 0 0
$$337$$ 23.0827 1.25739 0.628697 0.777651i $$-0.283589\pi$$
0.628697 + 0.777651i $$0.283589\pi$$
$$338$$ 0 0
$$339$$ −1.09821 −0.0596465
$$340$$ 0 0
$$341$$ 28.8450i 1.56204i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 20.7609 1.11773
$$346$$ 0 0
$$347$$ 1.30690i 0.0701580i 0.999385 + 0.0350790i $$0.0111683\pi$$
−0.999385 + 0.0350790i $$0.988832\pi$$
$$348$$ 0 0
$$349$$ − 26.5489i − 1.42113i −0.703633 0.710564i $$-0.748440\pi$$
0.703633 0.710564i $$-0.251560\pi$$
$$350$$ 0 0
$$351$$ 3.37849i 0.180331i
$$352$$ 0 0
$$353$$ − 16.8893i − 0.898928i −0.893298 0.449464i $$-0.851615\pi$$
0.893298 0.449464i $$-0.148385\pi$$
$$354$$ 0 0
$$355$$ 52.9339 2.80944
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 33.5901i 1.77282i 0.462904 + 0.886408i $$0.346807\pi$$
−0.462904 + 0.886408i $$0.653193\pi$$
$$360$$ 0 0
$$361$$ 3.11995 0.164208
$$362$$ 0 0
$$363$$ 14.6301 0.767880
$$364$$ 0 0
$$365$$ −49.4620 −2.58896
$$366$$ 0 0
$$367$$ 7.35978 0.384178 0.192089 0.981378i $$-0.438474\pi$$
0.192089 + 0.981378i $$0.438474\pi$$
$$368$$ 0 0
$$369$$ 5.14822i 0.268006i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.63008 0.187958 0.0939791 0.995574i $$-0.470041\pi$$
0.0939791 + 0.995574i $$0.470041\pi$$
$$374$$ 0 0
$$375$$ − 36.3820i − 1.87876i
$$376$$ 0 0
$$377$$ − 8.31293i − 0.428138i
$$378$$ 0 0
$$379$$ 13.6647i 0.701908i 0.936393 + 0.350954i $$0.114143\pi$$
−0.936393 + 0.350954i $$0.885857\pi$$
$$380$$ 0 0
$$381$$ − 5.22625i − 0.267749i
$$382$$ 0 0
$$383$$ 13.6569 0.697833 0.348916 0.937154i $$-0.386550\pi$$
0.348916 + 0.937154i $$0.386550\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 13.0199i − 0.661840i
$$388$$ 0 0
$$389$$ 26.9080 1.36429 0.682144 0.731218i $$-0.261048\pi$$
0.682144 + 0.731218i $$0.261048\pi$$
$$390$$ 0 0
$$391$$ −13.3656 −0.675927
$$392$$ 0 0
$$393$$ 0.875015 0.0441387
$$394$$ 0 0
$$395$$ 59.8031 3.00902
$$396$$ 0 0
$$397$$ 36.7242i 1.84314i 0.388216 + 0.921568i $$0.373091\pi$$
−0.388216 + 0.921568i $$0.626909\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −27.6816 −1.38235 −0.691176 0.722686i $$-0.742907\pi$$
−0.691176 + 0.722686i $$0.742907\pi$$
$$402$$ 0 0
$$403$$ − 19.2494i − 0.958883i
$$404$$ 0 0
$$405$$ − 4.29725i − 0.213532i
$$406$$ 0 0
$$407$$ − 11.8227i − 0.586032i
$$408$$ 0 0
$$409$$ 8.42820i 0.416747i 0.978049 + 0.208374i $$0.0668170\pi$$
−0.978049 + 0.208374i $$0.933183\pi$$
$$410$$ 0 0
$$411$$ −0.236838 −0.0116824
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 31.4641i 1.54451i
$$416$$ 0 0
$$417$$ −3.00555 −0.147182
$$418$$ 0 0
$$419$$ 18.3029 0.894154 0.447077 0.894495i $$-0.352465\pi$$
0.447077 + 0.894495i $$0.352465\pi$$
$$420$$ 0 0
$$421$$ 22.6274 1.10279 0.551396 0.834243i $$-0.314095\pi$$
0.551396 + 0.834243i $$0.314095\pi$$
$$422$$ 0 0
$$423$$ −5.35449 −0.260344
$$424$$ 0 0
$$425$$ 37.2549i 1.80713i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −17.1040 −0.825789
$$430$$ 0 0
$$431$$ − 4.44906i − 0.214304i −0.994243 0.107152i $$-0.965827\pi$$
0.994243 0.107152i $$-0.0341731\pi$$
$$432$$ 0 0
$$433$$ 1.66205i 0.0798730i 0.999202 + 0.0399365i $$0.0127156\pi$$
−0.999202 + 0.0399365i $$0.987284\pi$$
$$434$$ 0 0
$$435$$ 10.5736i 0.506964i
$$436$$ 0 0
$$437$$ 22.7220i 1.08694i
$$438$$ 0 0
$$439$$ −18.7551 −0.895130 −0.447565 0.894251i $$-0.647709\pi$$
−0.447565 + 0.894251i $$0.647709\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 33.0314i − 1.56937i −0.619896 0.784684i $$-0.712825\pi$$
0.619896 0.784684i $$-0.287175\pi$$
$$444$$ 0 0
$$445$$ 60.3539 2.86105
$$446$$ 0 0
$$447$$ 14.8017 0.700094
$$448$$ 0 0
$$449$$ 9.29441 0.438630 0.219315 0.975654i $$-0.429618\pi$$
0.219315 + 0.975654i $$0.429618\pi$$
$$450$$ 0 0
$$451$$ −26.0635 −1.22728
$$452$$ 0 0
$$453$$ 12.3654i 0.580976i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −27.4448 −1.28381 −0.641906 0.766784i $$-0.721856\pi$$
−0.641906 + 0.766784i $$0.721856\pi$$
$$458$$ 0 0
$$459$$ 2.76652i 0.129130i
$$460$$ 0 0
$$461$$ 4.28209i 0.199437i 0.995016 + 0.0997183i $$0.0317942\pi$$
−0.995016 + 0.0997183i $$0.968206\pi$$
$$462$$ 0 0
$$463$$ − 16.1278i − 0.749521i −0.927122 0.374760i $$-0.877725\pi$$
0.927122 0.374760i $$-0.122275\pi$$
$$464$$ 0 0
$$465$$ 24.4842i 1.13543i
$$466$$ 0 0
$$467$$ −27.5403 −1.27441 −0.637207 0.770692i $$-0.719910\pi$$
−0.637207 + 0.770692i $$0.719910\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 10.8906i 0.501812i
$$472$$ 0 0
$$473$$ 65.9149 3.03077
$$474$$ 0 0
$$475$$ 63.3347 2.90600
$$476$$ 0 0
$$477$$ −4.22371 −0.193390
$$478$$ 0 0
$$479$$ 19.5509 0.893304 0.446652 0.894708i $$-0.352616\pi$$
0.446652 + 0.894708i $$0.352616\pi$$
$$480$$ 0 0
$$481$$ 7.88980i 0.359744i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 62.6179 2.84333
$$486$$ 0 0
$$487$$ − 37.7917i − 1.71250i −0.516558 0.856252i $$-0.672787\pi$$
0.516558 0.856252i $$-0.327213\pi$$
$$488$$ 0 0
$$489$$ − 1.51024i − 0.0682954i
$$490$$ 0 0
$$491$$ − 29.2605i − 1.32051i −0.751042 0.660254i $$-0.770449\pi$$
0.751042 0.660254i $$-0.229551\pi$$
$$492$$ 0 0
$$493$$ − 6.80713i − 0.306578i
$$494$$ 0 0
$$495$$ 21.7553 0.977829
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 12.6422i − 0.565944i −0.959128 0.282972i $$-0.908680\pi$$
0.959128 0.282972i $$-0.0913203\pi$$
$$500$$ 0 0
$$501$$ 10.3333 0.461656
$$502$$ 0 0
$$503$$ −27.0714 −1.20706 −0.603528 0.797342i $$-0.706239\pi$$
−0.603528 + 0.797342i $$0.706239\pi$$
$$504$$ 0 0
$$505$$ −56.3312 −2.50671
$$506$$ 0 0
$$507$$ −1.58579 −0.0704272
$$508$$ 0 0
$$509$$ 11.6983i 0.518520i 0.965808 + 0.259260i $$0.0834786\pi$$
−0.965808 + 0.259260i $$0.916521\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 4.70319 0.207651
$$514$$ 0 0
$$515$$ 0.901124i 0.0397083i
$$516$$ 0 0
$$517$$ − 27.1077i − 1.19220i
$$518$$ 0 0
$$519$$ − 3.35642i − 0.147330i
$$520$$ 0 0
$$521$$ − 7.20643i − 0.315719i −0.987462 0.157860i $$-0.949541\pi$$
0.987462 0.157860i $$-0.0504594\pi$$
$$522$$ 0 0
$$523$$ 17.7145 0.774602 0.387301 0.921953i $$-0.373407\pi$$
0.387301 + 0.921953i $$0.373407\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 15.7626i − 0.686630i
$$528$$ 0 0
$$529$$ −0.340485 −0.0148037
$$530$$ 0 0
$$531$$ −9.60498 −0.416821
$$532$$ 0 0
$$533$$ 17.3932 0.753385
$$534$$ 0 0
$$535$$ 27.6166 1.19397
$$536$$ 0 0
$$537$$ − 13.2589i − 0.572165i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 23.1180 0.993921 0.496961 0.867773i $$-0.334449\pi$$
0.496961 + 0.867773i $$0.334449\pi$$
$$542$$ 0 0
$$543$$ 11.8519i 0.508615i
$$544$$ 0 0
$$545$$ − 4.03951i − 0.173034i
$$546$$ 0 0
$$547$$ 16.0524i 0.686351i 0.939271 + 0.343176i $$0.111503\pi$$
−0.939271 + 0.343176i $$0.888497\pi$$
$$548$$ 0 0
$$549$$ 3.87698i 0.165466i
$$550$$ 0 0
$$551$$ −11.5724 −0.493001
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ − 10.0354i − 0.425978i
$$556$$ 0 0
$$557$$ −12.5808 −0.533067 −0.266533 0.963826i $$-0.585878\pi$$
−0.266533 + 0.963826i $$0.585878\pi$$
$$558$$ 0 0
$$559$$ −43.9877 −1.86048
$$560$$ 0 0
$$561$$ −14.0058 −0.591325
$$562$$ 0 0
$$563$$ −36.5948 −1.54229 −0.771144 0.636660i $$-0.780315\pi$$
−0.771144 + 0.636660i $$0.780315\pi$$
$$564$$ 0 0
$$565$$ − 4.71927i − 0.198541i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −24.4989 −1.02705 −0.513524 0.858075i $$-0.671660\pi$$
−0.513524 + 0.858075i $$0.671660\pi$$
$$570$$ 0 0
$$571$$ − 32.7924i − 1.37232i −0.727451 0.686159i $$-0.759295\pi$$
0.727451 0.686159i $$-0.240705\pi$$
$$572$$ 0 0
$$573$$ 1.34660i 0.0562552i
$$574$$ 0 0
$$575$$ 65.0586i 2.71313i
$$576$$ 0 0
$$577$$ − 19.2140i − 0.799888i −0.916540 0.399944i $$-0.869030\pi$$
0.916540 0.399944i $$-0.130970\pi$$
$$578$$ 0 0
$$579$$ 26.1154 1.08532
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 21.3830i − 0.885594i
$$584$$ 0 0
$$585$$ −14.5182 −0.600255
$$586$$ 0 0
$$587$$ −24.1451 −0.996573 −0.498286 0.867012i $$-0.666037\pi$$
−0.498286 + 0.867012i $$0.666037\pi$$
$$588$$ 0 0
$$589$$ −26.7971 −1.10415
$$590$$ 0 0
$$591$$ 7.49083 0.308131
$$592$$ 0 0
$$593$$ 2.54582i 0.104544i 0.998633 + 0.0522722i $$0.0166463\pi$$
−0.998633 + 0.0522722i $$0.983354\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.49349 0.183906
$$598$$ 0 0
$$599$$ − 14.9899i − 0.612469i −0.951956 0.306234i $$-0.900931\pi$$
0.951956 0.306234i $$-0.0990691\pi$$
$$600$$ 0 0
$$601$$ − 24.3343i − 0.992618i −0.868146 0.496309i $$-0.834688\pi$$
0.868146 0.496309i $$-0.165312\pi$$
$$602$$ 0 0
$$603$$ − 4.76342i − 0.193981i
$$604$$ 0 0
$$605$$ 62.8691i 2.55599i
$$606$$ 0 0
$$607$$ −8.91288 −0.361763 −0.180881 0.983505i $$-0.557895\pi$$
−0.180881 + 0.983505i $$0.557895\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 18.0901i 0.731848i
$$612$$ 0 0
$$613$$ −17.1748 −0.693685 −0.346842 0.937923i $$-0.612746\pi$$
−0.346842 + 0.937923i $$0.612746\pi$$
$$614$$ 0 0
$$615$$ −22.1232 −0.892094
$$616$$ 0 0
$$617$$ 35.9980 1.44922 0.724612 0.689157i $$-0.242019\pi$$
0.724612 + 0.689157i $$0.242019\pi$$
$$618$$ 0 0
$$619$$ 4.11123 0.165244 0.0826222 0.996581i $$-0.473671\pi$$
0.0826222 + 0.996581i $$0.473671\pi$$
$$620$$ 0 0
$$621$$ 4.83120i 0.193869i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 89.0108 3.56043
$$626$$ 0 0
$$627$$ 23.8104i 0.950897i
$$628$$ 0 0
$$629$$ 6.46065i 0.257603i
$$630$$ 0 0
$$631$$ − 34.8970i − 1.38923i −0.719383 0.694613i $$-0.755575\pi$$
0.719383 0.694613i $$-0.244425\pi$$
$$632$$ 0 0
$$633$$ − 5.93122i − 0.235745i
$$634$$ 0 0
$$635$$ 22.4585 0.891239
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 12.3181i 0.487296i
$$640$$ 0 0
$$641$$ 7.32735 0.289413 0.144707 0.989475i $$-0.453776\pi$$
0.144707 + 0.989475i $$0.453776\pi$$
$$642$$ 0 0
$$643$$ −9.24275 −0.364498 −0.182249 0.983252i $$-0.558338\pi$$
−0.182249 + 0.983252i $$0.558338\pi$$
$$644$$ 0 0
$$645$$ 55.9498 2.20302
$$646$$ 0 0
$$647$$ 15.1803 0.596798 0.298399 0.954441i $$-0.403547\pi$$
0.298399 + 0.954441i $$0.403547\pi$$
$$648$$ 0 0
$$649$$ − 48.6263i − 1.90875i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −2.12849 −0.0832943 −0.0416471 0.999132i $$-0.513261\pi$$
−0.0416471 + 0.999132i $$0.513261\pi$$
$$654$$ 0 0
$$655$$ 3.76016i 0.146922i
$$656$$ 0 0
$$657$$ − 11.5102i − 0.449054i
$$658$$ 0 0
$$659$$ 13.0520i 0.508436i 0.967147 + 0.254218i $$0.0818180\pi$$
−0.967147 + 0.254218i $$0.918182\pi$$
$$660$$ 0 0
$$661$$ − 19.2786i − 0.749851i −0.927055 0.374926i $$-0.877668\pi$$
0.927055 0.374926i $$-0.122332\pi$$
$$662$$ 0 0
$$663$$ 9.34665 0.362994
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 11.8874i − 0.460281i
$$668$$ 0 0
$$669$$ 20.6163 0.797073
$$670$$ 0 0
$$671$$ −19.6277 −0.757718
$$672$$ 0 0
$$673$$ −7.08216 −0.272997 −0.136499 0.990640i $$-0.543585\pi$$
−0.136499 + 0.990640i $$0.543585\pi$$
$$674$$ 0 0
$$675$$ 13.4663 0.518320
$$676$$ 0 0
$$677$$ − 36.5042i − 1.40297i −0.712685 0.701485i $$-0.752521\pi$$
0.712685 0.701485i $$-0.247479\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 20.4911 0.785220
$$682$$ 0 0
$$683$$ 30.1206i 1.15253i 0.817261 + 0.576267i $$0.195491\pi$$
−0.817261 + 0.576267i $$0.804509\pi$$
$$684$$ 0 0
$$685$$ − 1.01775i − 0.0388863i
$$686$$ 0 0
$$687$$ 29.0409i 1.10798i
$$688$$ 0 0
$$689$$ 14.2698i 0.543635i
$$690$$ 0 0
$$691$$ −39.8113 −1.51449 −0.757247 0.653129i $$-0.773456\pi$$
−0.757247 + 0.653129i $$0.773456\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 12.9156i − 0.489916i
$$696$$ 0 0
$$697$$ 14.2426 0.539478
$$698$$ 0 0
$$699$$ −17.7364 −0.670852
$$700$$ 0 0
$$701$$ 28.0795 1.06055 0.530275 0.847826i $$-0.322089\pi$$
0.530275 + 0.847826i $$0.322089\pi$$
$$702$$ 0 0
$$703$$ 10.9834 0.414245
$$704$$ 0 0
$$705$$ − 23.0096i − 0.866591i
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −36.9475 −1.38759 −0.693797 0.720170i $$-0.744064\pi$$
−0.693797 + 0.720170i $$0.744064\pi$$
$$710$$ 0 0
$$711$$ 13.9166i 0.521913i
$$712$$ 0 0
$$713$$ − 27.5264i − 1.03087i
$$714$$ 0 0
$$715$$ − 73.5002i − 2.74875i
$$716$$ 0 0
$$717$$ 20.4248i 0.762779i
$$718$$ 0 0
$$719$$ −18.7090 −0.697728 −0.348864 0.937173i $$-0.613432\pi$$
−0.348864 + 0.937173i $$0.613432\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ − 3.25742i − 0.121145i
$$724$$ 0 0
$$725$$ −33.1345 −1.23059
$$726$$ 0 0
$$727$$ 31.2078 1.15743 0.578716 0.815529i $$-0.303554\pi$$
0.578716 + 0.815529i $$0.303554\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −36.0198 −1.33224
$$732$$ 0 0
$$733$$ 4.06897i 0.150291i 0.997173 + 0.0751455i $$0.0239421\pi$$
−0.997173 + 0.0751455i $$0.976058\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24.1154 0.888301
$$738$$ 0 0
$$739$$ − 39.9025i − 1.46784i −0.679237 0.733919i $$-0.737689\pi$$
0.679237 0.733919i $$-0.262311\pi$$
$$740$$ 0 0
$$741$$ − 15.8897i − 0.583722i
$$742$$ 0 0
$$743$$ − 41.7530i − 1.53177i −0.642979 0.765884i $$-0.722302\pi$$
0.642979 0.765884i $$-0.277698\pi$$
$$744$$ 0 0
$$745$$ 63.6064i 2.33036i
$$746$$ 0 0
$$747$$ −7.32191 −0.267895
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 9.09927i 0.332037i 0.986123 + 0.166019i $$0.0530911\pi$$
−0.986123 + 0.166019i $$0.946909\pi$$
$$752$$ 0 0
$$753$$ 3.95633 0.144177
$$754$$ 0 0
$$755$$ −53.1371 −1.93386
$$756$$ 0 0
$$757$$ −0.902155 −0.0327894 −0.0163947 0.999866i $$-0.505219\pi$$
−0.0163947 + 0.999866i $$0.505219\pi$$
$$758$$ 0 0
$$759$$ −24.4585 −0.887787
$$760$$ 0 0
$$761$$ − 23.7781i − 0.861955i −0.902363 0.430978i $$-0.858169\pi$$
0.902363 0.430978i $$-0.141831\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −11.8884 −0.429826
$$766$$ 0 0
$$767$$ 32.4503i 1.17171i
$$768$$ 0 0
$$769$$ − 16.3028i − 0.587895i −0.955822 0.293948i $$-0.905031\pi$$
0.955822 0.293948i $$-0.0949691\pi$$
$$770$$ 0 0
$$771$$ 5.14822i 0.185409i
$$772$$ 0 0
$$773$$ − 25.0881i − 0.902358i −0.892434 0.451179i $$-0.851004\pi$$
0.892434 0.451179i $$-0.148996\pi$$
$$774$$ 0 0
$$775$$ −76.7264 −2.75609
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 24.2131i − 0.867523i
$$780$$ 0 0
$$781$$ −62.3618 −2.23148
$$782$$ 0 0
$$783$$ −2.46054 −0.0879327
$$784$$ 0 0
$$785$$ −46.7996 −1.67035
$$786$$ 0 0
$$787$$ 4.47130 0.159385 0.0796924 0.996820i $$-0.474606\pi$$
0.0796924 + 0.996820i $$0.474606\pi$$
$$788$$ 0 0
$$789$$ − 0.0785021i − 0.00279475i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 13.0984 0.465136
$$794$$ 0 0
$$795$$ − 18.1503i − 0.643725i
$$796$$ 0 0
$$797$$ − 6.62320i − 0.234606i −0.993096 0.117303i $$-0.962575\pi$$
0.993096 0.117303i $$-0.0374248\pi$$
$$798$$ 0 0
$$799$$ 14.8133i 0.524056i
$$800$$ 0 0
$$801$$ 14.0448i 0.496248i
$$802$$ 0 0
$$803$$ 58.2715 2.05636
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0.883296i 0.0310935i
$$808$$ 0 0
$$809$$ −8.37044 −0.294289 −0.147144 0.989115i $$-0.547008\pi$$
−0.147144 + 0.989115i $$0.547008\pi$$
$$810$$ 0 0
$$811$$ 1.20944 0.0424692 0.0212346 0.999775i $$-0.493240\pi$$
0.0212346 + 0.999775i $$0.493240\pi$$
$$812$$ 0 0
$$813$$ −19.3084 −0.677176
$$814$$ 0 0
$$815$$ 6.48988 0.227330
$$816$$ 0 0
$$817$$ 61.2351i 2.14235i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 51.2519 1.78871 0.894353 0.447363i $$-0.147637\pi$$
0.894353 + 0.447363i $$0.147637\pi$$
$$822$$ 0 0
$$823$$ 31.5546i 1.09992i 0.835189 + 0.549962i $$0.185358\pi$$
−0.835189 + 0.549962i $$0.814642\pi$$
$$824$$ 0 0
$$825$$ 68.1749i 2.37355i
$$826$$ 0 0
$$827$$ − 23.4800i − 0.816480i −0.912875 0.408240i $$-0.866143\pi$$
0.912875 0.408240i $$-0.133857\pi$$
$$828$$ 0 0
$$829$$ − 17.2639i − 0.599599i −0.954002 0.299800i $$-0.903080\pi$$
0.954002 0.299800i $$-0.0969199\pi$$
$$830$$ 0 0
$$831$$ −16.9327 −0.587389
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 44.4046i 1.53668i
$$836$$ 0 0
$$837$$ −5.69764 −0.196939
$$838$$ 0 0
$$839$$ 37.4781 1.29389 0.646943 0.762538i $$-0.276047\pi$$
0.646943 + 0.762538i $$0.276047\pi$$
$$840$$ 0 0
$$841$$ −22.9457 −0.791232
$$842$$ 0 0
$$843$$ −5.05417 −0.174075
$$844$$ 0 0
$$845$$ − 6.81452i − 0.234427i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −19.1156 −0.656046
$$850$$ 0 0
$$851$$ 11.2823i 0.386753i
$$852$$ 0 0
$$853$$ − 9.50115i − 0.325313i −0.986683 0.162657i $$-0.947994\pi$$
0.986683 0.162657i $$-0.0520063\pi$$
$$854$$ 0 0
$$855$$ 20.2108i 0.691193i
$$856$$ 0 0
$$857$$ − 32.7797i − 1.11973i −0.828583 0.559866i $$-0.810853\pi$$
0.828583 0.559866i $$-0.189147\pi$$
$$858$$ 0 0
$$859$$ −5.37306 −0.183327 −0.0916633 0.995790i $$-0.529218\pi$$
−0.0916633 + 0.995790i $$0.529218\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 29.6824i 1.01040i 0.863003 + 0.505200i $$0.168581\pi$$
−0.863003 + 0.505200i $$0.831419\pi$$
$$864$$ 0 0
$$865$$ 14.4234 0.490409
$$866$$ 0 0
$$867$$ −9.34639 −0.317420
$$868$$ 0 0
$$869$$ −70.4544 −2.39000
$$870$$ 0 0
$$871$$ −16.0932 −0.545296
$$872$$ 0 0
$$873$$ 14.5716i 0.493175i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −30.3242 −1.02398 −0.511988 0.858993i $$-0.671091\pi$$
−0.511988 + 0.858993i $$0.671091\pi$$
$$878$$ 0 0
$$879$$ − 2.37837i − 0.0802205i
$$880$$ 0 0
$$881$$ − 16.5012i − 0.555939i −0.960590 0.277969i $$-0.910339\pi$$
0.960590 0.277969i $$-0.0896614\pi$$
$$882$$ 0 0
$$883$$ − 22.5909i − 0.760245i −0.924936 0.380122i $$-0.875882\pi$$
0.924936 0.380122i $$-0.124118\pi$$
$$884$$ 0 0
$$885$$ − 41.2750i − 1.38744i
$$886$$ 0 0
$$887$$ 38.6036 1.29618 0.648090 0.761563i $$-0.275568\pi$$
0.648090 + 0.761563i $$0.275568\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 5.06262i 0.169604i
$$892$$ 0 0
$$893$$ 25.1832 0.842723
$$894$$ 0 0
$$895$$ 56.9769 1.90453
$$896$$ 0 0
$$897$$ 16.3222 0.544981
$$898$$ 0 0
$$899$$ 14.0193 0.467570
$$900$$ 0 0
$$901$$ 11.6849i 0.389282i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −50.9307 −1.69299
$$906$$ 0 0
$$907$$ − 36.6743i − 1.21775i −0.793266 0.608875i $$-0.791621\pi$$
0.793266 0.608875i $$-0.208379\pi$$
$$908$$ 0 0
$$909$$ − 13.1087i − 0.434787i
$$910$$ 0 0
$$911$$ 52.6835i 1.74548i 0.488185 + 0.872740i $$0.337659\pi$$
−0.488185 + 0.872740i $$0.662341\pi$$
$$912$$ 0 0
$$913$$ − 37.0680i − 1.22677i
$$914$$ 0 0
$$915$$ −16.6604 −0.550775
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ − 5.11618i − 0.168767i −0.996433 0.0843836i $$-0.973108\pi$$
0.996433 0.0843836i $$-0.0268921\pi$$
$$920$$ 0 0
$$921$$ 4.89599 0.161328
$$922$$ 0 0
$$923$$ 41.6166 1.36983
$$924$$ 0 0
$$925$$ 31.4480 1.03400
$$926$$ 0 0
$$927$$ −0.209698 −0.00688738
$$928$$ 0 0
$$929$$ − 35.3694i − 1.16043i −0.814462 0.580217i $$-0.802968\pi$$
0.814462 0.580217i $$-0.197032\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 10.4800 0.343100
$$934$$ 0 0
$$935$$ − 60.1864i − 1.96831i
$$936$$ 0 0
$$937$$ 23.6290i 0.771924i 0.922515 + 0.385962i $$0.126130\pi$$
−0.922515 + 0.385962i $$0.873870\pi$$
$$938$$ 0 0
$$939$$ 9.60172i 0.313340i
$$940$$ 0 0
$$941$$ 19.2187i 0.626510i 0.949669 + 0.313255i $$0.101419\pi$$
−0.949669 + 0.313255i $$0.898581\pi$$
$$942$$ 0 0
$$943$$ 24.8721 0.809947
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.8159i 0.903894i 0.892045 + 0.451947i $$0.149270\pi$$
−0.892045 + 0.451947i $$0.850730\pi$$
$$948$$ 0 0
$$949$$ −38.8870 −1.26232
$$950$$ 0 0
$$951$$ 16.7934 0.544565
$$952$$ 0 0
$$953$$ 56.8087 1.84021 0.920107 0.391668i $$-0.128102\pi$$
0.920107 + 0.391668i $$0.128102\pi$$
$$954$$ 0 0
$$955$$ −5.78669 −0.187253
$$956$$ 0 0
$$957$$ − 12.4568i − 0.402671i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.46310 0.0471967
$$962$$ 0 0
$$963$$ 6.42657i 0.207093i
$$964$$ 0 0
$$965$$ 112.224i 3.61262i
$$966$$ 0 0
$$967$$ − 20.3860i − 0.655570i −0.944752 0.327785i $$-0.893698\pi$$
0.944752 0.327785i $$-0.106302\pi$$
$$968$$ 0 0
$$969$$ − 13.0114i − 0.417987i
$$970$$ 0 0
$$971$$ −11.2287 −0.360347 −0.180174 0.983635i $$-0.557666\pi$$
−0.180174 + 0.983635i $$0.557666\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ − 45.4960i − 1.45704i
$$976$$ 0 0
$$977$$ −12.1833 −0.389778 −0.194889 0.980825i $$-0.562435\pi$$
−0.194889 + 0.980825i $$0.562435\pi$$
$$978$$ 0 0
$$979$$ −71.1033 −2.27247
$$980$$ 0 0
$$981$$ 0.940024 0.0300126
$$982$$ 0 0
$$983$$ 9.06374 0.289088 0.144544 0.989498i $$-0.453828\pi$$
0.144544 + 0.989498i $$0.453828\pi$$
$$984$$ 0 0
$$985$$ 32.1899i 1.02566i
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −62.9018 −2.00016
$$990$$ 0 0
$$991$$ 5.37158i 0.170634i 0.996354 + 0.0853170i $$0.0271903\pi$$
−0.996354 + 0.0853170i $$0.972810\pi$$
$$992$$ 0 0
$$993$$ 26.8362i 0.851622i
$$994$$ 0 0
$$995$$ 19.3096i 0.612157i
$$996$$ 0 0
$$997$$ 14.3440i 0.454278i 0.973862 + 0.227139i $$0.0729372\pi$$
−0.973862 + 0.227139i $$0.927063\pi$$
$$998$$ 0 0
$$999$$ 2.33530 0.0738857
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 2352.2.b.k.1567.1 8
3.2 odd 2 7056.2.b.x.1567.8 8
4.3 odd 2 2352.2.b.l.1567.1 yes 8
7.2 even 3 2352.2.bl.r.31.1 8
7.3 odd 6 2352.2.bl.q.607.1 8
7.4 even 3 2352.2.bl.t.607.4 8
7.5 odd 6 2352.2.bl.o.31.4 8
7.6 odd 2 2352.2.b.l.1567.8 yes 8
12.11 even 2 7056.2.b.w.1567.8 8
21.20 even 2 7056.2.b.w.1567.1 8
28.3 even 6 2352.2.bl.r.607.1 8
28.11 odd 6 2352.2.bl.o.607.4 8
28.19 even 6 2352.2.bl.t.31.4 8
28.23 odd 6 2352.2.bl.q.31.1 8
28.27 even 2 inner 2352.2.b.k.1567.8 yes 8
84.83 odd 2 7056.2.b.x.1567.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.1 8 1.1 even 1 trivial
2352.2.b.k.1567.8 yes 8 28.27 even 2 inner
2352.2.b.l.1567.1 yes 8 4.3 odd 2
2352.2.b.l.1567.8 yes 8 7.6 odd 2
2352.2.bl.o.31.4 8 7.5 odd 6
2352.2.bl.o.607.4 8 28.11 odd 6
2352.2.bl.q.31.1 8 28.23 odd 6
2352.2.bl.q.607.1 8 7.3 odd 6
2352.2.bl.r.31.1 8 7.2 even 3
2352.2.bl.r.607.1 8 28.3 even 6
2352.2.bl.t.31.4 8 28.19 even 6
2352.2.bl.t.607.4 8 7.4 even 3
7056.2.b.w.1567.1 8 21.20 even 2
7056.2.b.w.1567.8 8 12.11 even 2
7056.2.b.x.1567.1 8 84.83 odd 2
7056.2.b.x.1567.8 8 3.2 odd 2