# Properties

 Label 2025.2.b.n.649.7 Level $2025$ Weight $2$ Character 2025.649 Analytic conductor $16.170$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.34810603776.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9$$ x^8 + 12*x^6 + 42*x^4 + 49*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 225) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.7 Root $$1.63372i$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.649 Dual form 2025.2.b.n.649.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.63372i q^{2} -0.669052 q^{4} +0.505348i q^{7} +2.17440i q^{8} +O(q^{10})$$ $$q+1.63372i q^{2} -0.669052 q^{4} +0.505348i q^{7} +2.17440i q^{8} +3.10020 q^{11} +6.23927i q^{13} -0.825599 q^{14} -4.89047 q^{16} -6.10020i q^{17} +5.57022 q^{19} +5.06487i q^{22} +3.82560i q^{23} -10.1932 q^{26} -0.338104i q^{28} -2.45932 q^{29} +4.22858 q^{31} -3.64088i q^{32} +9.96604 q^{34} +6.72677i q^{37} +9.10020i q^{38} -5.44185 q^{41} -1.32741i q^{43} -2.07420 q^{44} -6.24997 q^{46} -3.70792i q^{47} +6.74462 q^{49} -4.17440i q^{52} -2.54205i q^{53} -1.09883 q^{56} -4.01785i q^{58} -2.88232 q^{59} -2.84345 q^{61} +6.90833i q^{62} -3.83276 q^{64} +2.40652i q^{67} +4.08135i q^{68} +5.54205 q^{71} +11.7988i q^{73} -10.9897 q^{74} -3.72677 q^{76} +1.56668i q^{77} -3.40298 q^{79} -8.89047i q^{82} +13.9012i q^{83} +2.16862 q^{86} +6.74108i q^{88} -3.38513 q^{89} -3.15301 q^{91} -2.55953i q^{92} +6.05772 q^{94} +11.0756i q^{97} +11.0188i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 8 * q - 8 * q^4 $$8 q - 8 q^{4} - 2 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} - 20 q^{26} - 2 q^{29} - 8 q^{31} - 18 q^{34} - 10 q^{41} - 44 q^{44} + 6 q^{49} - 60 q^{56} - 34 q^{59} - 26 q^{61} - 38 q^{64} - 16 q^{71} - 80 q^{74} + 22 q^{76} + 14 q^{79} - 68 q^{86} + 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100})$$ 8 * q - 8 * q^4 - 2 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 - 20 * q^26 - 2 * q^29 - 8 * q^31 - 18 * q^34 - 10 * q^41 - 44 * q^44 + 6 * q^49 - 60 * q^56 - 34 * q^59 - 26 * q^61 - 38 * q^64 - 16 * q^71 - 80 * q^74 + 22 * q^76 + 14 * q^79 - 68 * q^86 + 18 * q^89 - 34 * q^91 - 6 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.63372i 1.15522i 0.816314 + 0.577608i $$0.196014\pi$$
−0.816314 + 0.577608i $$0.803986\pi$$
$$3$$ 0 0
$$4$$ −0.669052 −0.334526
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.505348i 0.191004i 0.995429 + 0.0955019i $$0.0304456\pi$$
−0.995429 + 0.0955019i $$0.969554\pi$$
$$8$$ 2.17440i 0.768767i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.10020 0.934746 0.467373 0.884060i $$-0.345200\pi$$
0.467373 + 0.884060i $$0.345200\pi$$
$$12$$ 0 0
$$13$$ 6.23927i 1.73046i 0.501372 + 0.865232i $$0.332829\pi$$
−0.501372 + 0.865232i $$0.667171\pi$$
$$14$$ −0.825599 −0.220651
$$15$$ 0 0
$$16$$ −4.89047 −1.22262
$$17$$ − 6.10020i − 1.47952i −0.672873 0.739758i $$-0.734940\pi$$
0.672873 0.739758i $$-0.265060\pi$$
$$18$$ 0 0
$$19$$ 5.57022 1.27790 0.638948 0.769250i $$-0.279370\pi$$
0.638948 + 0.769250i $$0.279370\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 5.06487i 1.07983i
$$23$$ 3.82560i 0.797693i 0.917018 + 0.398846i $$0.130589\pi$$
−0.917018 + 0.398846i $$0.869411\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −10.1932 −1.99906
$$27$$ 0 0
$$28$$ − 0.338104i − 0.0638957i
$$29$$ −2.45932 −0.456685 −0.228342 0.973581i $$-0.573331\pi$$
−0.228342 + 0.973581i $$0.573331\pi$$
$$30$$ 0 0
$$31$$ 4.22858 0.759475 0.379738 0.925094i $$-0.376014\pi$$
0.379738 + 0.925094i $$0.376014\pi$$
$$32$$ − 3.64088i − 0.643623i
$$33$$ 0 0
$$34$$ 9.96604 1.70916
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.72677i 1.10587i 0.833223 + 0.552937i $$0.186493\pi$$
−0.833223 + 0.552937i $$0.813507\pi$$
$$38$$ 9.10020i 1.47625i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.44185 −0.849874 −0.424937 0.905223i $$-0.639704\pi$$
−0.424937 + 0.905223i $$0.639704\pi$$
$$42$$ 0 0
$$43$$ − 1.32741i − 0.202428i −0.994865 0.101214i $$-0.967727\pi$$
0.994865 0.101214i $$-0.0322726\pi$$
$$44$$ −2.07420 −0.312697
$$45$$ 0 0
$$46$$ −6.24997 −0.921508
$$47$$ − 3.70792i − 0.540856i −0.962740 0.270428i $$-0.912835\pi$$
0.962740 0.270428i $$-0.0871652\pi$$
$$48$$ 0 0
$$49$$ 6.74462 0.963518
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 4.17440i − 0.578885i
$$53$$ − 2.54205i − 0.349177i −0.984641 0.174589i $$-0.944140\pi$$
0.984641 0.174589i $$-0.0558596\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.09883 −0.146837
$$57$$ 0 0
$$58$$ − 4.01785i − 0.527570i
$$59$$ −2.88232 −0.375246 −0.187623 0.982241i $$-0.560078\pi$$
−0.187623 + 0.982241i $$0.560078\pi$$
$$60$$ 0 0
$$61$$ −2.84345 −0.364067 −0.182033 0.983292i $$-0.558268\pi$$
−0.182033 + 0.983292i $$0.558268\pi$$
$$62$$ 6.90833i 0.877358i
$$63$$ 0 0
$$64$$ −3.83276 −0.479095
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.40652i 0.294003i 0.989136 + 0.147002i $$0.0469622\pi$$
−0.989136 + 0.147002i $$0.953038\pi$$
$$68$$ 4.08135i 0.494937i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.54205 0.657720 0.328860 0.944379i $$-0.393336\pi$$
0.328860 + 0.944379i $$0.393336\pi$$
$$72$$ 0 0
$$73$$ 11.7988i 1.38095i 0.723359 + 0.690473i $$0.242597\pi$$
−0.723359 + 0.690473i $$0.757403\pi$$
$$74$$ −10.9897 −1.27752
$$75$$ 0 0
$$76$$ −3.72677 −0.427490
$$77$$ 1.56668i 0.178540i
$$78$$ 0 0
$$79$$ −3.40298 −0.382865 −0.191433 0.981506i $$-0.561313\pi$$
−0.191433 + 0.981506i $$0.561313\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 8.89047i − 0.981789i
$$83$$ 13.9012i 1.52585i 0.646486 + 0.762926i $$0.276238\pi$$
−0.646486 + 0.762926i $$0.723762\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.16862 0.233848
$$87$$ 0 0
$$88$$ 6.74108i 0.718602i
$$89$$ −3.38513 −0.358823 −0.179411 0.983774i $$-0.557419\pi$$
−0.179411 + 0.983774i $$0.557419\pi$$
$$90$$ 0 0
$$91$$ −3.15301 −0.330525
$$92$$ − 2.55953i − 0.266849i
$$93$$ 0 0
$$94$$ 6.05772 0.624806
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 11.0756i 1.12455i 0.826949 + 0.562277i $$0.190074\pi$$
−0.826949 + 0.562277i $$0.809926\pi$$
$$98$$ 11.0188i 1.11307i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −17.3690 −1.72828 −0.864141 0.503249i $$-0.832138\pi$$
−0.864141 + 0.503249i $$0.832138\pi$$
$$102$$ 0 0
$$103$$ − 0.832756i − 0.0820539i −0.999158 0.0410269i $$-0.986937\pi$$
0.999158 0.0410269i $$-0.0130629\pi$$
$$104$$ −13.5667 −1.33032
$$105$$ 0 0
$$106$$ 4.15301 0.403376
$$107$$ − 11.0684i − 1.07002i −0.844844 0.535012i $$-0.820307\pi$$
0.844844 0.535012i $$-0.179693\pi$$
$$108$$ 0 0
$$109$$ 4.65836 0.446190 0.223095 0.974797i $$-0.428384\pi$$
0.223095 + 0.974797i $$0.428384\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 2.47139i − 0.233525i
$$113$$ 11.9942i 1.12832i 0.825665 + 0.564160i $$0.190800\pi$$
−0.825665 + 0.564160i $$0.809200\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.64542 0.152773
$$117$$ 0 0
$$118$$ − 4.70892i − 0.433491i
$$119$$ 3.08273 0.282593
$$120$$ 0 0
$$121$$ −1.38874 −0.126249
$$122$$ − 4.64542i − 0.420576i
$$123$$ 0 0
$$124$$ −2.82914 −0.254064
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 3.22858i 0.286490i 0.989687 + 0.143245i $$0.0457537\pi$$
−0.989687 + 0.143245i $$0.954246\pi$$
$$128$$ − 13.5434i − 1.19708i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −9.38513 −0.819982 −0.409991 0.912090i $$-0.634468\pi$$
−0.409991 + 0.912090i $$0.634468\pi$$
$$132$$ 0 0
$$133$$ 2.81490i 0.244083i
$$134$$ −3.93159 −0.339637
$$135$$ 0 0
$$136$$ 13.2643 1.13740
$$137$$ − 2.30955i − 0.197319i −0.995121 0.0986593i $$-0.968545\pi$$
0.995121 0.0986593i $$-0.0314554\pi$$
$$138$$ 0 0
$$139$$ −10.8940 −0.924018 −0.462009 0.886875i $$-0.652871\pi$$
−0.462009 + 0.886875i $$0.652871\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 9.05418i 0.759810i
$$143$$ 19.3430i 1.61754i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −19.2760 −1.59529
$$147$$ 0 0
$$148$$ − 4.50056i − 0.369944i
$$149$$ 16.3430 1.33887 0.669436 0.742870i $$-0.266536\pi$$
0.669436 + 0.742870i $$0.266536\pi$$
$$150$$ 0 0
$$151$$ 22.7827 1.85403 0.927015 0.375025i $$-0.122366\pi$$
0.927015 + 0.375025i $$0.122366\pi$$
$$152$$ 12.1119i 0.982404i
$$153$$ 0 0
$$154$$ −2.55953 −0.206252
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 12.4607i − 0.994472i −0.867615 0.497236i $$-0.834348\pi$$
0.867615 0.497236i $$-0.165652\pi$$
$$158$$ − 5.55953i − 0.442292i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1.93326 −0.152362
$$162$$ 0 0
$$163$$ − 7.57384i − 0.593229i −0.954997 0.296614i $$-0.904142\pi$$
0.954997 0.296614i $$-0.0958576\pi$$
$$164$$ 3.64088 0.284305
$$165$$ 0 0
$$166$$ −22.7107 −1.76269
$$167$$ 2.97674i 0.230347i 0.993345 + 0.115174i $$0.0367424\pi$$
−0.993345 + 0.115174i $$0.963258\pi$$
$$168$$ 0 0
$$169$$ −25.9285 −1.99450
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0.888105i 0.0677174i
$$173$$ − 15.8530i − 1.20528i −0.798013 0.602640i $$-0.794116\pi$$
0.798013 0.602640i $$-0.205884\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −15.1615 −1.14284
$$177$$ 0 0
$$178$$ − 5.53036i − 0.414518i
$$179$$ −17.0841 −1.27693 −0.638463 0.769653i $$-0.720429\pi$$
−0.638463 + 0.769653i $$0.720429\pi$$
$$180$$ 0 0
$$181$$ 13.3690 0.993712 0.496856 0.867833i $$-0.334488\pi$$
0.496856 + 0.867833i $$0.334488\pi$$
$$182$$ − 5.15114i − 0.381828i
$$183$$ 0 0
$$184$$ −8.31839 −0.613240
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 18.9119i − 1.38297i
$$188$$ 2.48079i 0.180930i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 25.3372 1.83334 0.916669 0.399647i $$-0.130867\pi$$
0.916669 + 0.399647i $$0.130867\pi$$
$$192$$ 0 0
$$193$$ − 9.55953i − 0.688110i −0.938950 0.344055i $$-0.888199\pi$$
0.938950 0.344055i $$-0.111801\pi$$
$$194$$ −18.0944 −1.29910
$$195$$ 0 0
$$196$$ −4.51250 −0.322322
$$197$$ 2.06841i 0.147368i 0.997282 + 0.0736842i $$0.0234757\pi$$
−0.997282 + 0.0736842i $$0.976524\pi$$
$$198$$ 0 0
$$199$$ −13.0970 −0.928419 −0.464210 0.885725i $$-0.653662\pi$$
−0.464210 + 0.885725i $$0.653662\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 28.3762i − 1.99654i
$$203$$ − 1.24281i − 0.0872285i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 1.36049 0.0947900
$$207$$ 0 0
$$208$$ − 30.5130i − 2.11570i
$$209$$ 17.2688 1.19451
$$210$$ 0 0
$$211$$ 11.1119 0.764974 0.382487 0.923961i $$-0.375068\pi$$
0.382487 + 0.923961i $$0.375068\pi$$
$$212$$ 1.70076i 0.116809i
$$213$$ 0 0
$$214$$ 18.0827 1.23611
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.13690i 0.145063i
$$218$$ 7.61046i 0.515446i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 38.0608 2.56025
$$222$$ 0 0
$$223$$ − 3.89401i − 0.260762i −0.991464 0.130381i $$-0.958380\pi$$
0.991464 0.130381i $$-0.0416201\pi$$
$$224$$ 1.83991 0.122934
$$225$$ 0 0
$$226$$ −19.5952 −1.30346
$$227$$ − 12.8081i − 0.850105i −0.905169 0.425053i $$-0.860256\pi$$
0.905169 0.425053i $$-0.139744\pi$$
$$228$$ 0 0
$$229$$ −6.65295 −0.439639 −0.219820 0.975541i $$-0.570547\pi$$
−0.219820 + 0.975541i $$0.570547\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 5.34755i − 0.351084i
$$233$$ − 3.65836i − 0.239667i −0.992794 0.119833i $$-0.961764\pi$$
0.992794 0.119833i $$-0.0382360\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 1.92842 0.125530
$$237$$ 0 0
$$238$$ 5.03632i 0.326456i
$$239$$ 15.6915 1.01500 0.507500 0.861652i $$-0.330570\pi$$
0.507500 + 0.861652i $$0.330570\pi$$
$$240$$ 0 0
$$241$$ 11.2250 0.723063 0.361532 0.932360i $$-0.382254\pi$$
0.361532 + 0.932360i $$0.382254\pi$$
$$242$$ − 2.26882i − 0.145845i
$$243$$ 0 0
$$244$$ 1.90242 0.121790
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 34.7541i 2.21135i
$$248$$ 9.19462i 0.583859i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6.94042 0.438075 0.219038 0.975716i $$-0.429708\pi$$
0.219038 + 0.975716i $$0.429708\pi$$
$$252$$ 0 0
$$253$$ 11.8601i 0.745640i
$$254$$ −5.27460 −0.330958
$$255$$ 0 0
$$256$$ 14.4607 0.903794
$$257$$ − 18.3327i − 1.14356i −0.820406 0.571781i $$-0.806253\pi$$
0.820406 0.571781i $$-0.193747\pi$$
$$258$$ 0 0
$$259$$ −3.39936 −0.211226
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 15.3327i − 0.947257i
$$263$$ 16.0766i 0.991328i 0.868514 + 0.495664i $$0.165075\pi$$
−0.868514 + 0.495664i $$0.834925\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.59877 −0.281969
$$267$$ 0 0
$$268$$ − 1.61009i − 0.0983517i
$$269$$ 18.2004 1.10970 0.554849 0.831951i $$-0.312776\pi$$
0.554849 + 0.831951i $$0.312776\pi$$
$$270$$ 0 0
$$271$$ −2.48571 −0.150996 −0.0754979 0.997146i $$-0.524055\pi$$
−0.0754979 + 0.997146i $$0.524055\pi$$
$$272$$ 29.8329i 1.80888i
$$273$$ 0 0
$$274$$ 3.77317 0.227946
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 7.66726i − 0.460681i −0.973110 0.230341i $$-0.926016\pi$$
0.973110 0.230341i $$-0.0739840\pi$$
$$278$$ − 17.7978i − 1.06744i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0.273230 0.0162996 0.00814978 0.999967i $$-0.497406\pi$$
0.00814978 + 0.999967i $$0.497406\pi$$
$$282$$ 0 0
$$283$$ 3.37089i 0.200379i 0.994968 + 0.100189i $$0.0319448\pi$$
−0.994968 + 0.100189i $$0.968055\pi$$
$$284$$ −3.70792 −0.220025
$$285$$ 0 0
$$286$$ −31.6011 −1.86861
$$287$$ − 2.75003i − 0.162329i
$$288$$ 0 0
$$289$$ −20.2125 −1.18897
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 7.89401i − 0.461962i
$$293$$ − 5.64404i − 0.329728i −0.986316 0.164864i $$-0.947281\pi$$
0.986316 0.164864i $$-0.0527186\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −14.6267 −0.850159
$$297$$ 0 0
$$298$$ 26.7000i 1.54669i
$$299$$ −23.8690 −1.38038
$$300$$ 0 0
$$301$$ 0.670803 0.0386645
$$302$$ 37.2206i 2.14181i
$$303$$ 0 0
$$304$$ −27.2410 −1.56238
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 5.44105i − 0.310537i −0.987872 0.155269i $$-0.950376\pi$$
0.987872 0.155269i $$-0.0496243\pi$$
$$308$$ − 1.04819i − 0.0597263i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −19.0797 −1.08191 −0.540955 0.841052i $$-0.681937\pi$$
−0.540955 + 0.841052i $$0.681937\pi$$
$$312$$ 0 0
$$313$$ − 9.14231i − 0.516754i −0.966044 0.258377i $$-0.916812\pi$$
0.966044 0.258377i $$-0.0831877\pi$$
$$314$$ 20.3573 1.14883
$$315$$ 0 0
$$316$$ 2.27677 0.128078
$$317$$ − 14.2367i − 0.799614i −0.916599 0.399807i $$-0.869077\pi$$
0.916599 0.399807i $$-0.130923\pi$$
$$318$$ 0 0
$$319$$ −7.62440 −0.426884
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 3.15841i − 0.176011i
$$323$$ − 33.9795i − 1.89067i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12.3736 0.685308
$$327$$ 0 0
$$328$$ − 11.8328i − 0.653355i
$$329$$ 1.87379 0.103305
$$330$$ 0 0
$$331$$ −12.2000 −0.670574 −0.335287 0.942116i $$-0.608833\pi$$
−0.335287 + 0.942116i $$0.608833\pi$$
$$332$$ − 9.30061i − 0.510437i
$$333$$ 0 0
$$334$$ −4.86317 −0.266101
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 4.58986i − 0.250026i −0.992155 0.125013i $$-0.960103\pi$$
0.992155 0.125013i $$-0.0398972\pi$$
$$338$$ − 42.3601i − 2.30408i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 13.1094 0.709917
$$342$$ 0 0
$$343$$ 6.94582i 0.375039i
$$344$$ 2.88632 0.155620
$$345$$ 0 0
$$346$$ 25.8994 1.39236
$$347$$ − 33.4603i − 1.79624i −0.439748 0.898121i $$-0.644932\pi$$
0.439748 0.898121i $$-0.355068\pi$$
$$348$$ 0 0
$$349$$ 28.0816 1.50317 0.751586 0.659636i $$-0.229289\pi$$
0.751586 + 0.659636i $$0.229289\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 11.2875i − 0.601624i
$$353$$ − 1.84170i − 0.0980239i −0.998798 0.0490119i $$-0.984393\pi$$
0.998798 0.0490119i $$-0.0156072\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2.26483 0.120036
$$357$$ 0 0
$$358$$ − 27.9107i − 1.47513i
$$359$$ 12.1119 0.639241 0.319621 0.947546i $$-0.396445\pi$$
0.319621 + 0.947546i $$0.396445\pi$$
$$360$$ 0 0
$$361$$ 12.0274 0.633020
$$362$$ 21.8413i 1.14795i
$$363$$ 0 0
$$364$$ 2.10953 0.110569
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 14.5738i 0.760744i 0.924834 + 0.380372i $$0.124204\pi$$
−0.924834 + 0.380372i $$0.875796\pi$$
$$368$$ − 18.7090i − 0.975274i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1.28462 0.0666942
$$372$$ 0 0
$$373$$ − 9.44646i − 0.489119i −0.969634 0.244560i $$-0.921357\pi$$
0.969634 0.244560i $$-0.0786434\pi$$
$$374$$ 30.8968 1.59763
$$375$$ 0 0
$$376$$ 8.06251 0.415792
$$377$$ − 15.3444i − 0.790276i
$$378$$ 0 0
$$379$$ 28.5541 1.46673 0.733363 0.679837i $$-0.237949\pi$$
0.733363 + 0.679837i $$0.237949\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 41.3940i 2.11790i
$$383$$ 1.46541i 0.0748788i 0.999299 + 0.0374394i $$0.0119201\pi$$
−0.999299 + 0.0374394i $$0.988080\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 15.6176 0.794916
$$387$$ 0 0
$$388$$ − 7.41014i − 0.376193i
$$389$$ 12.9101 0.654569 0.327284 0.944926i $$-0.393866\pi$$
0.327284 + 0.944926i $$0.393866\pi$$
$$390$$ 0 0
$$391$$ 23.3369 1.18020
$$392$$ 14.6655i 0.740720i
$$393$$ 0 0
$$394$$ −3.37922 −0.170242
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0.868386i 0.0435831i 0.999763 + 0.0217915i $$0.00693701\pi$$
−0.999763 + 0.0217915i $$0.993063\pi$$
$$398$$ − 21.3968i − 1.07253i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 33.4125 1.66854 0.834270 0.551356i $$-0.185889\pi$$
0.834270 + 0.551356i $$0.185889\pi$$
$$402$$ 0 0
$$403$$ 26.3833i 1.31424i
$$404$$ 11.6208 0.578156
$$405$$ 0 0
$$406$$ 2.03042 0.100768
$$407$$ 20.8544i 1.03371i
$$408$$ 0 0
$$409$$ −5.05535 −0.249971 −0.124985 0.992159i $$-0.539888\pi$$
−0.124985 + 0.992159i $$0.539888\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0.557157i 0.0274492i
$$413$$ − 1.45658i − 0.0716734i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 22.7165 1.11377
$$417$$ 0 0
$$418$$ 28.2125i 1.37992i
$$419$$ −10.9576 −0.535313 −0.267657 0.963514i $$-0.586249\pi$$
−0.267657 + 0.963514i $$0.586249\pi$$
$$420$$ 0 0
$$421$$ −10.6386 −0.518495 −0.259248 0.965811i $$-0.583475\pi$$
−0.259248 + 0.965811i $$0.583475\pi$$
$$422$$ 18.1538i 0.883711i
$$423$$ 0 0
$$424$$ 5.52744 0.268436
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 1.43693i − 0.0695381i
$$428$$ 7.40535i 0.357951i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −37.3529 −1.79923 −0.899613 0.436687i $$-0.856152\pi$$
−0.899613 + 0.436687i $$0.856152\pi$$
$$432$$ 0 0
$$433$$ − 17.2125i − 0.827179i −0.910464 0.413589i $$-0.864275\pi$$
0.910464 0.413589i $$-0.135725\pi$$
$$434$$ −3.49111 −0.167579
$$435$$ 0 0
$$436$$ −3.11668 −0.149262
$$437$$ 21.3094i 1.01937i
$$438$$ 0 0
$$439$$ 31.7487 1.51528 0.757642 0.652670i $$-0.226351\pi$$
0.757642 + 0.652670i $$0.226351\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 62.1809i 2.95764i
$$443$$ − 0.355958i − 0.0169121i −0.999964 0.00845603i $$-0.997308\pi$$
0.999964 0.00845603i $$-0.00269167\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 6.36174 0.301237
$$447$$ 0 0
$$448$$ − 1.93688i − 0.0915088i
$$449$$ 7.85632 0.370762 0.185381 0.982667i $$-0.440648\pi$$
0.185381 + 0.982667i $$0.440648\pi$$
$$450$$ 0 0
$$451$$ −16.8708 −0.794416
$$452$$ − 8.02476i − 0.377453i
$$453$$ 0 0
$$454$$ 20.9249 0.982056
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 21.4910i − 1.00531i −0.864488 0.502654i $$-0.832357\pi$$
0.864488 0.502654i $$-0.167643\pi$$
$$458$$ − 10.8691i − 0.507879i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 40.9927 1.90922 0.954611 0.297856i $$-0.0962715\pi$$
0.954611 + 0.297856i $$0.0962715\pi$$
$$462$$ 0 0
$$463$$ 42.1339i 1.95813i 0.203555 + 0.979063i $$0.434750\pi$$
−0.203555 + 0.979063i $$0.565250\pi$$
$$464$$ 12.0273 0.558351
$$465$$ 0 0
$$466$$ 5.97674 0.276867
$$467$$ 22.5376i 1.04292i 0.853276 + 0.521459i $$0.174612\pi$$
−0.853276 + 0.521459i $$0.825388\pi$$
$$468$$ 0 0
$$469$$ −1.21613 −0.0561557
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 6.26732i − 0.288477i
$$473$$ − 4.11523i − 0.189219i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.06251 −0.0945348
$$477$$ 0 0
$$478$$ 25.6356i 1.17255i
$$479$$ 33.2880 1.52097 0.760483 0.649358i $$-0.224962\pi$$
0.760483 + 0.649358i $$0.224962\pi$$
$$480$$ 0 0
$$481$$ −41.9702 −1.91367
$$482$$ 18.3385i 0.835295i
$$483$$ 0 0
$$484$$ 0.929141 0.0422337
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 23.7703i − 1.07713i −0.842583 0.538566i $$-0.818966\pi$$
0.842583 0.538566i $$-0.181034\pi$$
$$488$$ − 6.18281i − 0.279882i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −4.60563 −0.207849 −0.103925 0.994585i $$-0.533140\pi$$
−0.103925 + 0.994585i $$0.533140\pi$$
$$492$$ 0 0
$$493$$ 15.0024i 0.675673i
$$494$$ −56.7787 −2.55459
$$495$$ 0 0
$$496$$ −20.6797 −0.928548
$$497$$ 2.80067i 0.125627i
$$498$$ 0 0
$$499$$ 18.8976 0.845971 0.422985 0.906137i $$-0.360982\pi$$
0.422985 + 0.906137i $$0.360982\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 11.3387i 0.506072i
$$503$$ 35.7581i 1.59438i 0.603731 + 0.797188i $$0.293680\pi$$
−0.603731 + 0.797188i $$0.706320\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −19.3762 −0.861376
$$507$$ 0 0
$$508$$ − 2.16009i − 0.0958384i
$$509$$ −24.4067 −1.08181 −0.540904 0.841084i $$-0.681918\pi$$
−0.540904 + 0.841084i $$0.681918\pi$$
$$510$$ 0 0
$$511$$ −5.96250 −0.263766
$$512$$ − 3.46207i − 0.153003i
$$513$$ 0 0
$$514$$ 29.9506 1.32106
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 11.4953i − 0.505563i
$$518$$ − 5.55362i − 0.244012i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 33.3968 1.46314 0.731571 0.681766i $$-0.238788\pi$$
0.731571 + 0.681766i $$0.238788\pi$$
$$522$$ 0 0
$$523$$ − 37.3654i − 1.63388i −0.576726 0.816938i $$-0.695670\pi$$
0.576726 0.816938i $$-0.304330\pi$$
$$524$$ 6.27914 0.274305
$$525$$ 0 0
$$526$$ −26.2648 −1.14520
$$527$$ − 25.7952i − 1.12366i
$$528$$ 0 0
$$529$$ 8.36479 0.363686
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 1.88332i − 0.0816521i
$$533$$ − 33.9532i − 1.47068i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −5.23274 −0.226020
$$537$$ 0 0
$$538$$ 29.7344i 1.28194i
$$539$$ 20.9097 0.900645
$$540$$ 0 0
$$541$$ 28.2560 1.21482 0.607409 0.794389i $$-0.292209\pi$$
0.607409 + 0.794389i $$0.292209\pi$$
$$542$$ − 4.06096i − 0.174433i
$$543$$ 0 0
$$544$$ −22.2101 −0.952250
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 38.5452i 1.64807i 0.566537 + 0.824036i $$0.308283\pi$$
−0.566537 + 0.824036i $$0.691717\pi$$
$$548$$ 1.54521i 0.0660082i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −13.6990 −0.583596
$$552$$ 0 0
$$553$$ − 1.71969i − 0.0731286i
$$554$$ 12.5262 0.532187
$$555$$ 0 0
$$556$$ 7.28866 0.309108
$$557$$ 27.4125i 1.16151i 0.814080 + 0.580753i $$0.197242\pi$$
−0.814080 + 0.580753i $$0.802758\pi$$
$$558$$ 0 0
$$559$$ 8.28206 0.350294
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0.446383i 0.0188295i
$$563$$ − 27.6392i − 1.16485i −0.812883 0.582427i $$-0.802103\pi$$
0.812883 0.582427i $$-0.197897\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −5.50710 −0.231481
$$567$$ 0 0
$$568$$ 12.0506i 0.505634i
$$569$$ 14.7161 0.616933 0.308467 0.951235i $$-0.400184\pi$$
0.308467 + 0.951235i $$0.400184\pi$$
$$570$$ 0 0
$$571$$ −28.3006 −1.18434 −0.592172 0.805812i $$-0.701729\pi$$
−0.592172 + 0.805812i $$0.701729\pi$$
$$572$$ − 12.9415i − 0.541111i
$$573$$ 0 0
$$574$$ 4.49279 0.187525
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 40.7976i 1.69843i 0.528049 + 0.849214i $$0.322924\pi$$
−0.528049 + 0.849214i $$0.677076\pi$$
$$578$$ − 33.0216i − 1.37352i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −7.02493 −0.291443
$$582$$ 0 0
$$583$$ − 7.88087i − 0.326392i
$$584$$ −25.6553 −1.06162
$$585$$ 0 0
$$586$$ 9.22080 0.380908
$$587$$ − 2.78033i − 0.114756i −0.998353 0.0573782i $$-0.981726\pi$$
0.998353 0.0573782i $$-0.0182741\pi$$
$$588$$ 0 0
$$589$$ 23.5541 0.970531
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 32.8971i − 1.35206i
$$593$$ − 14.8084i − 0.608109i −0.952655 0.304055i $$-0.901659\pi$$
0.952655 0.304055i $$-0.0983405\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.9343 −0.447888
$$597$$ 0 0
$$598$$ − 38.9953i − 1.59464i
$$599$$ 16.3430 0.667758 0.333879 0.942616i $$-0.391642\pi$$
0.333879 + 0.942616i $$0.391642\pi$$
$$600$$ 0 0
$$601$$ −6.62371 −0.270187 −0.135093 0.990833i $$-0.543133\pi$$
−0.135093 + 0.990833i $$0.543133\pi$$
$$602$$ 1.09591i 0.0446658i
$$603$$ 0 0
$$604$$ −15.2428 −0.620221
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 30.3094i − 1.23022i −0.788442 0.615110i $$-0.789112\pi$$
0.788442 0.615110i $$-0.210888\pi$$
$$608$$ − 20.2805i − 0.822483i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 23.1347 0.935931
$$612$$ 0 0
$$613$$ 14.7803i 0.596969i 0.954415 + 0.298484i $$0.0964811\pi$$
−0.954415 + 0.298484i $$0.903519\pi$$
$$614$$ 8.88918 0.358738
$$615$$ 0 0
$$616$$ −3.40660 −0.137256
$$617$$ 33.8512i 1.36280i 0.731912 + 0.681399i $$0.238628\pi$$
−0.731912 + 0.681399i $$0.761372\pi$$
$$618$$ 0 0
$$619$$ −11.6887 −0.469807 −0.234903 0.972019i $$-0.575477\pi$$
−0.234903 + 0.972019i $$0.575477\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 31.1709i − 1.24984i
$$623$$ − 1.71067i − 0.0685364i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 14.9360 0.596963
$$627$$ 0 0
$$628$$ 8.33686i 0.332677i
$$629$$ 41.0347 1.63616
$$630$$ 0 0
$$631$$ −38.1357 −1.51816 −0.759078 0.650999i $$-0.774350\pi$$
−0.759078 + 0.650999i $$0.774350\pi$$
$$632$$ − 7.39944i − 0.294334i
$$633$$ 0 0
$$634$$ 23.2589 0.923728
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 42.0816i 1.66733i
$$638$$ − 12.4562i − 0.493144i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −34.7654 −1.37315 −0.686576 0.727058i $$-0.740887\pi$$
−0.686576 + 0.727058i $$0.740887\pi$$
$$642$$ 0 0
$$643$$ 2.68515i 0.105892i 0.998597 + 0.0529461i $$0.0168611\pi$$
−0.998597 + 0.0529461i $$0.983139\pi$$
$$644$$ 1.29345 0.0509692
$$645$$ 0 0
$$646$$ 55.5131 2.18413
$$647$$ − 40.5103i − 1.59262i −0.604887 0.796311i $$-0.706782\pi$$
0.604887 0.796311i $$-0.293218\pi$$
$$648$$ 0 0
$$649$$ −8.93578 −0.350760
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 5.06729i 0.198451i
$$653$$ 13.3354i 0.521856i 0.965358 + 0.260928i $$0.0840286\pi$$
−0.965358 + 0.260928i $$0.915971\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 26.6132 1.03907
$$657$$ 0 0
$$658$$ 3.06126i 0.119340i
$$659$$ −31.1543 −1.21360 −0.606800 0.794855i $$-0.707547\pi$$
−0.606800 + 0.794855i $$0.707547\pi$$
$$660$$ 0 0
$$661$$ 6.31788 0.245737 0.122869 0.992423i $$-0.460791\pi$$
0.122869 + 0.992423i $$0.460791\pi$$
$$662$$ − 19.9315i − 0.774659i
$$663$$ 0 0
$$664$$ −30.2267 −1.17302
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 9.40838i − 0.364294i
$$668$$ − 1.99160i − 0.0770571i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.81528 −0.340310
$$672$$ 0 0
$$673$$ − 6.59220i − 0.254110i −0.991896 0.127055i $$-0.959447\pi$$
0.991896 0.127055i $$-0.0405525\pi$$
$$674$$ 7.49857 0.288834
$$675$$ 0 0
$$676$$ 17.3476 0.667214
$$677$$ − 34.8947i − 1.34111i −0.741859 0.670556i $$-0.766056\pi$$
0.741859 0.670556i $$-0.233944\pi$$
$$678$$ 0 0
$$679$$ −5.59702 −0.214794
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 21.4172i 0.820108i
$$683$$ 26.0958i 0.998528i 0.866450 + 0.499264i $$0.166396\pi$$
−0.866450 + 0.499264i $$0.833604\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −11.3476 −0.433252
$$687$$ 0 0
$$688$$ 6.49165i 0.247492i
$$689$$ 15.8606 0.604239
$$690$$ 0 0
$$691$$ −29.3058 −1.11485 −0.557423 0.830229i $$-0.688210\pi$$
−0.557423 + 0.830229i $$0.688210\pi$$
$$692$$ 10.6065i 0.403197i
$$693$$ 0 0
$$694$$ 54.6648 2.07505
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 33.1964i 1.25740i
$$698$$ 45.8775i 1.73649i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 15.3891 0.581239 0.290620 0.956839i $$-0.406139\pi$$
0.290620 + 0.956839i $$0.406139\pi$$
$$702$$ 0 0
$$703$$ 37.4696i 1.41319i
$$704$$ −11.8823 −0.447832
$$705$$ 0 0
$$706$$ 3.00883 0.113239
$$707$$ − 8.77741i − 0.330108i
$$708$$ 0 0
$$709$$ 7.73991 0.290678 0.145339 0.989382i $$-0.453573\pi$$
0.145339 + 0.989382i $$0.453573\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 7.36062i − 0.275851i
$$713$$ 16.1768i 0.605828i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 11.4302 0.427165
$$717$$ 0 0
$$718$$ 19.7875i 0.738462i
$$719$$ 15.1316 0.564313 0.282156 0.959368i $$-0.408950\pi$$
0.282156 + 0.959368i $$0.408950\pi$$
$$720$$ 0 0
$$721$$ 0.420832 0.0156726
$$722$$ 19.6494i 0.731275i
$$723$$ 0 0
$$724$$ −8.94457 −0.332422
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 0.161876i − 0.00600366i −0.999995 0.00300183i $$-0.999044\pi$$
0.999995 0.00300183i $$-0.000955513\pi$$
$$728$$ − 6.85590i − 0.254097i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8.09746 −0.299495
$$732$$ 0 0
$$733$$ − 50.0332i − 1.84802i −0.382371 0.924009i $$-0.624893\pi$$
0.382371 0.924009i $$-0.375107\pi$$
$$734$$ −23.8095 −0.878825
$$735$$ 0 0
$$736$$ 13.9285 0.513413
$$737$$ 7.46070i 0.274818i
$$738$$ 0 0
$$739$$ −30.5505 −1.12382 −0.561909 0.827199i $$-0.689933\pi$$
−0.561909 + 0.827199i $$0.689933\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 2.09872i 0.0770463i
$$743$$ 5.96684i 0.218902i 0.993992 + 0.109451i $$0.0349093\pi$$
−0.993992 + 0.109451i $$0.965091\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 15.4329 0.565039
$$747$$ 0 0
$$748$$ 12.6530i 0.462640i
$$749$$ 5.59340 0.204379
$$750$$ 0 0
$$751$$ 34.3976 1.25519 0.627593 0.778542i $$-0.284040\pi$$
0.627593 + 0.778542i $$0.284040\pi$$
$$752$$ 18.1335i 0.661260i
$$753$$ 0 0
$$754$$ 25.0685 0.912941
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 40.6873i − 1.47881i −0.673263 0.739403i $$-0.735108\pi$$
0.673263 0.739403i $$-0.264892\pi$$
$$758$$ 46.6495i 1.69439i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 28.2596 1.02441 0.512204 0.858864i $$-0.328829\pi$$
0.512204 + 0.858864i $$0.328829\pi$$
$$762$$ 0 0
$$763$$ 2.35409i 0.0852239i
$$764$$ −16.9519 −0.613299
$$765$$ 0 0
$$766$$ −2.39407 −0.0865013
$$767$$ − 17.9836i − 0.649350i
$$768$$ 0 0
$$769$$ 46.9036 1.69139 0.845694 0.533668i $$-0.179187\pi$$
0.845694 + 0.533668i $$0.179187\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 6.39582i 0.230191i
$$773$$ 9.19641i 0.330772i 0.986229 + 0.165386i $$0.0528870\pi$$
−0.986229 + 0.165386i $$0.947113\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −24.0827 −0.864520
$$777$$ 0 0
$$778$$ 21.0916i 0.756169i
$$779$$ −30.3123 −1.08605
$$780$$ 0 0
$$781$$ 17.1815 0.614802
$$782$$ 38.1261i 1.36339i
$$783$$ 0 0
$$784$$ −32.9844 −1.17801
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 5.74637i − 0.204836i −0.994741 0.102418i $$-0.967342\pi$$
0.994741 0.102418i $$-0.0326579\pi$$
$$788$$ − 1.38388i − 0.0492986i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.06126 −0.215514
$$792$$ 0 0
$$793$$ − 17.7411i − 0.630004i
$$794$$ −1.41870 −0.0503479
$$795$$ 0 0
$$796$$ 8.76255 0.310580
$$797$$ 7.07450i 0.250592i 0.992119 + 0.125296i $$0.0399880\pi$$
−0.992119 + 0.125296i $$0.960012\pi$$
$$798$$ 0 0
$$799$$ −22.6191 −0.800205
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 54.5868i 1.92753i
$$803$$ 36.5787i 1.29083i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −43.1029 −1.51824
$$807$$ 0 0
$$808$$ − 37.7672i − 1.32865i
$$809$$ −38.1075 −1.33979 −0.669894 0.742457i $$-0.733660\pi$$
−0.669894 + 0.742457i $$0.733660\pi$$
$$810$$ 0 0
$$811$$ −1.44105 −0.0506022 −0.0253011 0.999680i $$-0.508054\pi$$
−0.0253011 + 0.999680i $$0.508054\pi$$
$$812$$ 0.831508i 0.0291802i
$$813$$ 0 0
$$814$$ −34.0702 −1.19416
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 7.39396i − 0.258682i
$$818$$ − 8.25904i − 0.288771i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 22.5143 0.785753 0.392876 0.919591i $$-0.371480\pi$$
0.392876 + 0.919591i $$0.371480\pi$$
$$822$$ 0 0
$$823$$ − 41.4589i − 1.44517i −0.691284 0.722583i $$-0.742955\pi$$
0.691284 0.722583i $$-0.257045\pi$$
$$824$$ 1.81075 0.0630803
$$825$$ 0 0
$$826$$ 2.37964 0.0827984
$$827$$ − 27.8133i − 0.967164i −0.875299 0.483582i $$-0.839336\pi$$
0.875299 0.483582i $$-0.160664\pi$$
$$828$$ 0 0
$$829$$ −20.7232 −0.719745 −0.359872 0.933002i $$-0.617180\pi$$
−0.359872 + 0.933002i $$0.617180\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 23.9136i − 0.829056i
$$833$$ − 41.1436i − 1.42554i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −11.5537 −0.399595
$$837$$ 0 0
$$838$$ − 17.9017i − 0.618403i
$$839$$ −18.1451 −0.626437 −0.313218 0.949681i $$-0.601407\pi$$
−0.313218 + 0.949681i $$0.601407\pi$$
$$840$$ 0 0
$$841$$ −22.9517 −0.791439
$$842$$ − 17.3806i − 0.598975i
$$843$$ 0 0
$$844$$ −7.43444 −0.255904
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 0.701798i − 0.0241141i
$$848$$ 12.4318i 0.426911i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −25.7339 −0.882148
$$852$$ 0 0
$$853$$ − 4.00708i − 0.137200i −0.997644 0.0685999i $$-0.978147\pi$$
0.997644 0.0685999i $$-0.0218532\pi$$
$$854$$ 2.34755 0.0803316
$$855$$ 0 0
$$856$$ 24.0672 0.822599
$$857$$ 9.08971i 0.310499i 0.987875 + 0.155249i $$0.0496181\pi$$
−0.987875 + 0.155249i $$0.950382\pi$$
$$858$$ 0 0
$$859$$ −16.3870 −0.559116 −0.279558 0.960129i $$-0.590188\pi$$
−0.279558 + 0.960129i $$0.590188\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 61.0243i − 2.07850i
$$863$$ 23.7967i 0.810050i 0.914306 + 0.405025i $$0.132737\pi$$
−0.914306 + 0.405025i $$0.867263\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 28.1204 0.955571
$$867$$ 0 0
$$868$$ − 1.42970i − 0.0485272i
$$869$$ −10.5499 −0.357882
$$870$$ 0 0
$$871$$ −15.0149 −0.508762
$$872$$ 10.1291i 0.343016i
$$873$$ 0 0
$$874$$ −34.8137 −1.17759
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 36.0744i − 1.21815i −0.793114 0.609073i $$-0.791542\pi$$
0.793114 0.609073i $$-0.208458\pi$$
$$878$$ 51.8687i 1.75048i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −35.4575 −1.19459 −0.597297 0.802020i $$-0.703759\pi$$
−0.597297 + 0.802020i $$0.703759\pi$$
$$882$$ 0 0
$$883$$ − 39.1320i − 1.31690i −0.752626 0.658448i $$-0.771213\pi$$
0.752626 0.658448i $$-0.228787\pi$$
$$884$$ −25.4647 −0.856470
$$885$$ 0 0
$$886$$ 0.581537 0.0195371
$$887$$ − 51.2833i − 1.72192i −0.508670 0.860962i $$-0.669863\pi$$
0.508670 0.860962i $$-0.330137\pi$$
$$888$$ 0 0
$$889$$ −1.63156 −0.0547207
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.60530i 0.0872318i
$$893$$ − 20.6539i − 0.691158i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 6.84415 0.228647
$$897$$ 0 0
$$898$$ 12.8350i 0.428311i
$$899$$ −10.3994 −0.346841
$$900$$ 0 0
$$901$$ −15.5070 −0.516614
$$902$$ − 27.5623i − 0.917723i
$$903$$ 0 0
$$904$$ −26.0802 −0.867416
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 47.8588i − 1.58913i −0.607181 0.794563i $$-0.707700\pi$$
0.607181 0.794563i $$-0.292300\pi$$
$$908$$ 8.56930i 0.284382i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.0431 0.597793 0.298897 0.954285i $$-0.403381\pi$$
0.298897 + 0.954285i $$0.403381\pi$$
$$912$$ 0 0
$$913$$ 43.0964i 1.42628i
$$914$$ 35.1104 1.16135
$$915$$ 0 0
$$916$$ 4.45117 0.147071
$$917$$ − 4.74276i − 0.156620i
$$918$$ 0 0
$$919$$ −10.3976 −0.342984 −0.171492 0.985185i $$-0.554859\pi$$
−0.171492 + 0.985185i $$0.554859\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 66.9708i 2.20557i
$$923$$ 34.5784i 1.13816i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −68.8351 −2.26206
$$927$$ 0 0
$$928$$ 8.95410i 0.293933i
$$929$$ 36.0216 1.18183 0.590915 0.806734i $$-0.298767\pi$$
0.590915 + 0.806734i $$0.298767\pi$$
$$930$$ 0 0
$$931$$ 37.5691 1.23128
$$932$$ 2.44763i 0.0801748i
$$933$$ 0 0
$$934$$ −36.8203 −1.20480
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 24.0326i − 0.785111i −0.919728 0.392555i $$-0.871591\pi$$
0.919728 0.392555i $$-0.128409\pi$$
$$938$$ − 1.98682i − 0.0648720i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 16.6676 0.543348 0.271674 0.962389i $$-0.412423\pi$$
0.271674 + 0.962389i $$0.412423\pi$$
$$942$$ 0 0
$$943$$ − 20.8183i − 0.677938i
$$944$$ 14.0959 0.458783
$$945$$ 0 0
$$946$$ 6.72315 0.218589
$$947$$ 27.5400i 0.894928i 0.894302 + 0.447464i $$0.147673\pi$$
−0.894302 + 0.447464i $$0.852327\pi$$
$$948$$ 0 0
$$949$$ −73.6160 −2.38968
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 6.70308i 0.217248i
$$953$$ − 18.1344i − 0.587432i −0.955893 0.293716i $$-0.905108\pi$$
0.955893 0.293716i $$-0.0948920\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −10.4984 −0.339544
$$957$$ 0 0
$$958$$ 54.3833i 1.75705i
$$959$$ 1.16713 0.0376886
$$960$$ 0 0
$$961$$ −13.1191 −0.423198
$$962$$ − 68.5676i − 2.21071i
$$963$$ 0 0
$$964$$ −7.51009 −0.241884
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 36.1875i − 1.16371i −0.813292 0.581855i $$-0.802327\pi$$
0.813292 0.581855i $$-0.197673\pi$$
$$968$$ − 3.01968i − 0.0970562i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −34.6173 −1.11092 −0.555461 0.831542i $$-0.687458\pi$$
−0.555461 + 0.831542i $$0.687458\pi$$
$$972$$ 0 0
$$973$$ − 5.50527i − 0.176491i
$$974$$ 38.8340 1.24432
$$975$$ 0 0
$$976$$ 13.9058 0.445115
$$977$$ 29.4331i 0.941650i 0.882227 + 0.470825i $$0.156044\pi$$
−0.882227 + 0.470825i $$0.843956\pi$$
$$978$$ 0 0
$$979$$ −10.4946 −0.335408
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 7.52432i − 0.240111i
$$983$$ 24.4911i 0.781145i 0.920572 + 0.390573i $$0.127723\pi$$
−0.920572 + 0.390573i $$0.872277\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −24.5097 −0.780549
$$987$$ 0 0
$$988$$ − 23.2523i − 0.739755i
$$989$$ 5.07813 0.161475
$$990$$ 0 0
$$991$$ −13.2821 −0.421919 −0.210959 0.977495i $$-0.567659\pi$$
−0.210959 + 0.977495i $$0.567659\pi$$
$$992$$ − 15.3957i − 0.488815i
$$993$$ 0 0
$$994$$ −4.57551 −0.145126
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 39.1530i − 1.23999i −0.784606 0.619994i $$-0.787135\pi$$
0.784606 0.619994i $$-0.212865\pi$$
$$998$$ 30.8734i 0.977280i
$$999$$ 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 2025.2.b.n.649.7 8
3.2 odd 2 2025.2.b.o.649.2 8
5.2 odd 4 2025.2.a.y.1.1 4
5.3 odd 4 2025.2.a.q.1.4 4
5.4 even 2 inner 2025.2.b.n.649.2 8
9.2 odd 6 675.2.k.c.199.2 16
9.4 even 3 225.2.k.c.124.2 16
9.5 odd 6 675.2.k.c.424.7 16
9.7 even 3 225.2.k.c.49.7 16
15.2 even 4 2025.2.a.p.1.4 4
15.8 even 4 2025.2.a.z.1.1 4
15.14 odd 2 2025.2.b.o.649.7 8
45.2 even 12 675.2.e.e.226.1 8
45.4 even 6 225.2.k.c.124.7 16
45.7 odd 12 225.2.e.c.76.4 8
45.13 odd 12 225.2.e.e.151.1 yes 8
45.14 odd 6 675.2.k.c.424.2 16
45.22 odd 12 225.2.e.c.151.4 yes 8
45.23 even 12 675.2.e.c.451.4 8
45.29 odd 6 675.2.k.c.199.7 16
45.32 even 12 675.2.e.e.451.1 8
45.34 even 6 225.2.k.c.49.2 16
45.38 even 12 675.2.e.c.226.4 8
45.43 odd 12 225.2.e.e.76.1 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.e.c.76.4 8 45.7 odd 12
225.2.e.c.151.4 yes 8 45.22 odd 12
225.2.e.e.76.1 yes 8 45.43 odd 12
225.2.e.e.151.1 yes 8 45.13 odd 12
225.2.k.c.49.2 16 45.34 even 6
225.2.k.c.49.7 16 9.7 even 3
225.2.k.c.124.2 16 9.4 even 3
225.2.k.c.124.7 16 45.4 even 6
675.2.e.c.226.4 8 45.38 even 12
675.2.e.c.451.4 8 45.23 even 12
675.2.e.e.226.1 8 45.2 even 12
675.2.e.e.451.1 8 45.32 even 12
675.2.k.c.199.2 16 9.2 odd 6
675.2.k.c.199.7 16 45.29 odd 6
675.2.k.c.424.2 16 45.14 odd 6
675.2.k.c.424.7 16 9.5 odd 6
2025.2.a.p.1.4 4 15.2 even 4
2025.2.a.q.1.4 4 5.3 odd 4
2025.2.a.y.1.1 4 5.2 odd 4
2025.2.a.z.1.1 4 15.8 even 4
2025.2.b.n.649.2 8 5.4 even 2 inner
2025.2.b.n.649.7 8 1.1 even 1 trivial
2025.2.b.o.649.2 8 3.2 odd 2
2025.2.b.o.649.7 8 15.14 odd 2