# Properties

 Label 2025.2.b.n.649.1 Level $2025$ Weight $2$ Character 2025.649 Analytic conductor $16.170$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Learn more

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 649.1 Root $$-2.63372i$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.649 Dual form 2025.2.b.n.649.8

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.63372i q^{2} -4.93650 q^{4} +1.79743i q^{7} +7.73393i q^{8} +O(q^{10})$$ $$q-2.63372i q^{2} -4.93650 q^{4} +1.79743i q^{7} +7.73393i q^{8} +1.80812 q^{11} +1.97183i q^{13} +4.73393 q^{14} +10.4960 q^{16} -4.80812i q^{17} -2.96467 q^{19} -4.76210i q^{22} -1.73393i q^{23} +5.19325 q^{26} -8.87300i q^{28} +7.36765 q^{29} -2.62303 q^{31} -12.1758i q^{32} -12.6633 q^{34} -11.6351i q^{37} +7.80812i q^{38} -2.46648 q^{41} -7.27814i q^{43} -8.92580 q^{44} -4.56668 q^{46} -6.29208i q^{47} +3.76926 q^{49} -9.73393i q^{52} +1.72540i q^{53} -13.9012 q^{56} -19.4044i q^{58} -11.0260 q^{59} -12.6704 q^{61} +6.90833i q^{62} -11.0756 q^{64} -9.10374i q^{67} +23.7353i q^{68} +1.27460 q^{71} -3.58770i q^{73} -30.6436 q^{74} +14.6351 q^{76} +3.24997i q^{77} -2.11090 q^{79} +6.49602i q^{82} +1.09883i q^{83} -19.1686 q^{86} +13.9839i q^{88} +13.2935 q^{89} -3.54422 q^{91} +8.55953i q^{92} -16.5716 q^{94} +3.83276i q^{97} -9.92718i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.63372i − 1.86232i −0.364606 0.931162i $$-0.618796\pi$$
0.364606 0.931162i $$-0.381204\pi$$
$$3$$ 0 0
$$4$$ −4.93650 −2.46825
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.79743i 0.679364i 0.940540 + 0.339682i $$0.110319\pi$$
−0.940540 + 0.339682i $$0.889681\pi$$
$$8$$ 7.73393i 2.73436i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.80812 0.545170 0.272585 0.962132i $$-0.412121\pi$$
0.272585 + 0.962132i $$0.412121\pi$$
$$12$$ 0 0
$$13$$ 1.97183i 0.546887i 0.961888 + 0.273443i $$0.0881626\pi$$
−0.961888 + 0.273443i $$0.911837\pi$$
$$14$$ 4.73393 1.26520
$$15$$ 0 0
$$16$$ 10.4960 2.62401
$$17$$ − 4.80812i − 1.16614i −0.812421 0.583071i $$-0.801851\pi$$
0.812421 0.583071i $$-0.198149\pi$$
$$18$$ 0 0
$$19$$ −2.96467 −0.680142 −0.340071 0.940400i $$-0.610451\pi$$
−0.340071 + 0.940400i $$0.610451\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.76210i − 1.01528i
$$23$$ − 1.73393i − 0.361549i −0.983525 0.180774i $$-0.942140\pi$$
0.983525 0.180774i $$-0.0578604\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 5.19325 1.01848
$$27$$ 0 0
$$28$$ − 8.87300i − 1.67684i
$$29$$ 7.36765 1.36814 0.684069 0.729417i $$-0.260209\pi$$
0.684069 + 0.729417i $$0.260209\pi$$
$$30$$ 0 0
$$31$$ −2.62303 −0.471110 −0.235555 0.971861i $$-0.575691\pi$$
−0.235555 + 0.971861i $$0.575691\pi$$
$$32$$ − 12.1758i − 2.15239i
$$33$$ 0 0
$$34$$ −12.6633 −2.17173
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 11.6351i − 1.91280i −0.292063 0.956399i $$-0.594342\pi$$
0.292063 0.956399i $$-0.405658\pi$$
$$38$$ 7.80812i 1.26664i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.46648 −0.385199 −0.192600 0.981277i $$-0.561692\pi$$
−0.192600 + 0.981277i $$0.561692\pi$$
$$42$$ 0 0
$$43$$ − 7.27814i − 1.10991i −0.831882 0.554953i $$-0.812736\pi$$
0.831882 0.554953i $$-0.187264\pi$$
$$44$$ −8.92580 −1.34562
$$45$$ 0 0
$$46$$ −4.56668 −0.673321
$$47$$ − 6.29208i − 0.917794i −0.888489 0.458897i $$-0.848245\pi$$
0.888489 0.458897i $$-0.151755\pi$$
$$48$$ 0 0
$$49$$ 3.76926 0.538465
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 9.73393i − 1.34985i
$$53$$ 1.72540i 0.237001i 0.992954 + 0.118501i $$0.0378088\pi$$
−0.992954 + 0.118501i $$0.962191\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −13.9012 −1.85762
$$57$$ 0 0
$$58$$ − 19.4044i − 2.54792i
$$59$$ −11.0260 −1.43546 −0.717732 0.696320i $$-0.754820\pi$$
−0.717732 + 0.696320i $$0.754820\pi$$
$$60$$ 0 0
$$61$$ −12.6704 −1.62228 −0.811141 0.584851i $$-0.801153\pi$$
−0.811141 + 0.584851i $$0.801153\pi$$
$$62$$ 6.90833i 0.877358i
$$63$$ 0 0
$$64$$ −11.0756 −1.38445
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 9.10374i − 1.11220i −0.831116 0.556100i $$-0.812297\pi$$
0.831116 0.556100i $$-0.187703\pi$$
$$68$$ 23.7353i 2.87833i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.27460 0.151268 0.0756338 0.997136i $$-0.475902\pi$$
0.0756338 + 0.997136i $$0.475902\pi$$
$$72$$ 0 0
$$73$$ − 3.58770i − 0.419908i −0.977711 0.209954i $$-0.932669\pi$$
0.977711 0.209954i $$-0.0673315\pi$$
$$74$$ −30.6436 −3.56225
$$75$$ 0 0
$$76$$ 14.6351 1.67876
$$77$$ 3.24997i 0.370369i
$$78$$ 0 0
$$79$$ −2.11090 −0.237495 −0.118747 0.992924i $$-0.537888\pi$$
−0.118747 + 0.992924i $$0.537888\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 6.49602i 0.717366i
$$83$$ 1.09883i 0.120612i 0.998180 + 0.0603061i $$0.0192077\pi$$
−0.998180 + 0.0603061i $$0.980792\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −19.1686 −2.06701
$$87$$ 0 0
$$88$$ 13.9839i 1.49069i
$$89$$ 13.2935 1.40910 0.704552 0.709653i $$-0.251148\pi$$
0.704552 + 0.709653i $$0.251148\pi$$
$$90$$ 0 0
$$91$$ −3.54422 −0.371535
$$92$$ 8.55953i 0.892392i
$$93$$ 0 0
$$94$$ −16.5716 −1.70923
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.83276i 0.389157i 0.980887 + 0.194579i $$0.0623340\pi$$
−0.980887 + 0.194579i $$0.937666\pi$$
$$98$$ − 9.92718i − 1.00280i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.55237 0.651985 0.325993 0.945372i $$-0.394302\pi$$
0.325993 + 0.945372i $$0.394302\pi$$
$$102$$ 0 0
$$103$$ − 8.07557i − 0.795710i −0.917448 0.397855i $$-0.869755\pi$$
0.917448 0.397855i $$-0.130245\pi$$
$$104$$ −15.2500 −1.49538
$$105$$ 0 0
$$106$$ 4.54422 0.441373
$$107$$ 8.97674i 0.867814i 0.900958 + 0.433907i $$0.142865\pi$$
−0.900958 + 0.433907i $$0.857135\pi$$
$$108$$ 0 0
$$109$$ 6.34164 0.607419 0.303710 0.952765i $$-0.401775\pi$$
0.303710 + 0.952765i $$0.401775\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 18.8658i 1.78265i
$$113$$ − 14.9025i − 1.40191i −0.713204 0.700957i $$-0.752757\pi$$
0.713204 0.700957i $$-0.247243\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −36.3704 −3.37691
$$117$$ 0 0
$$118$$ 29.0394i 2.67330i
$$119$$ 8.64225 0.792234
$$120$$ 0 0
$$121$$ −7.73069 −0.702790
$$122$$ 33.3704i 3.02121i
$$123$$ 0 0
$$124$$ 12.9486 1.16282
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 3.62303i − 0.321492i −0.986996 0.160746i $$-0.948610\pi$$
0.986996 0.160746i $$-0.0513899\pi$$
$$128$$ 4.81844i 0.425894i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 7.29345 0.637232 0.318616 0.947884i $$-0.396782\pi$$
0.318616 + 0.947884i $$0.396782\pi$$
$$132$$ 0 0
$$133$$ − 5.32878i − 0.462064i
$$134$$ −23.9767 −2.07127
$$135$$ 0 0
$$136$$ 37.1857 3.18865
$$137$$ 7.12621i 0.608833i 0.952539 + 0.304417i $$0.0984615\pi$$
−0.952539 + 0.304417i $$0.901539\pi$$
$$138$$ 0 0
$$139$$ 14.7107 1.24774 0.623871 0.781527i $$-0.285559\pi$$
0.623871 + 0.781527i $$0.285559\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 3.35695i − 0.281709i
$$143$$ 3.56531i 0.298146i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −9.44900 −0.782005
$$147$$ 0 0
$$148$$ 57.4366i 4.72126i
$$149$$ 0.565309 0.0463119 0.0231560 0.999732i $$-0.492629\pi$$
0.0231560 + 0.999732i $$0.492629\pi$$
$$150$$ 0 0
$$151$$ 0.153385 0.0124823 0.00624115 0.999981i $$-0.498013\pi$$
0.00624115 + 0.999981i $$0.498013\pi$$
$$152$$ − 22.9285i − 1.85975i
$$153$$ 0 0
$$154$$ 8.55953 0.689746
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.4607i 0.914663i 0.889296 + 0.457332i $$0.151195\pi$$
−0.889296 + 0.457332i $$0.848805\pi$$
$$158$$ 5.55953i 0.442292i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.11661 0.245623
$$162$$ 0 0
$$163$$ − 22.0595i − 1.72783i −0.503637 0.863915i $$-0.668005\pi$$
0.503637 0.863915i $$-0.331995\pi$$
$$164$$ 12.1758 0.950768
$$165$$ 0 0
$$166$$ 2.89401 0.224619
$$167$$ − 17.0684i − 1.32079i −0.750917 0.660397i $$-0.770388\pi$$
0.750917 0.660397i $$-0.229612\pi$$
$$168$$ 0 0
$$169$$ 9.11189 0.700915
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 35.9285i 2.73953i
$$173$$ 11.9447i 0.908135i 0.890967 + 0.454067i $$0.150028\pi$$
−0.890967 + 0.454067i $$0.849972\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 18.9781 1.43053
$$177$$ 0 0
$$178$$ − 35.0113i − 2.62421i
$$179$$ −8.54921 −0.638998 −0.319499 0.947587i $$-0.603515\pi$$
−0.319499 + 0.947587i $$0.603515\pi$$
$$180$$ 0 0
$$181$$ −10.5524 −0.784351 −0.392176 0.919890i $$-0.628277\pi$$
−0.392176 + 0.919890i $$0.628277\pi$$
$$182$$ 9.33449i 0.691918i
$$183$$ 0 0
$$184$$ 13.4101 0.988603
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 8.69368i − 0.635745i
$$188$$ 31.0608i 2.26534i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.3372 −1.25448 −0.627239 0.778827i $$-0.715815\pi$$
−0.627239 + 0.778827i $$0.715815\pi$$
$$192$$ 0 0
$$193$$ 1.55953i 0.112257i 0.998424 + 0.0561286i $$0.0178757\pi$$
−0.998424 + 0.0561286i $$0.982124\pi$$
$$194$$ 10.0944 0.724737
$$195$$ 0 0
$$196$$ −18.6069 −1.32907
$$197$$ − 17.9767i − 1.28079i −0.768046 0.640395i $$-0.778771\pi$$
0.768046 0.640395i $$-0.221229\pi$$
$$198$$ 0 0
$$199$$ −11.0225 −0.781362 −0.390681 0.920526i $$-0.627760\pi$$
−0.390681 + 0.920526i $$0.627760\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 17.2571i − 1.21421i
$$203$$ 13.2428i 0.929463i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −21.2688 −1.48187
$$207$$ 0 0
$$208$$ 20.6964i 1.43503i
$$209$$ −5.36049 −0.370793
$$210$$ 0 0
$$211$$ −23.9285 −1.64731 −0.823655 0.567092i $$-0.808068\pi$$
−0.823655 + 0.567092i $$0.808068\pi$$
$$212$$ − 8.51742i − 0.584979i
$$213$$ 0 0
$$214$$ 23.6423 1.61615
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 4.71470i − 0.320055i
$$218$$ − 16.7021i − 1.13121i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 9.48079 0.637747
$$222$$ 0 0
$$223$$ 21.7107i 1.45385i 0.686715 + 0.726927i $$0.259052\pi$$
−0.686715 + 0.726927i $$0.740948\pi$$
$$224$$ 21.8851 1.46226
$$225$$ 0 0
$$226$$ −39.2492 −2.61082
$$227$$ − 14.1002i − 0.935863i −0.883765 0.467932i $$-0.844999\pi$$
0.883765 0.467932i $$-0.155001\pi$$
$$228$$ 0 0
$$229$$ −3.67758 −0.243021 −0.121511 0.992590i $$-0.538774\pi$$
−0.121511 + 0.992590i $$0.538774\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 56.9809i 3.74098i
$$233$$ − 5.34164i − 0.349943i −0.984574 0.174971i $$-0.944017\pi$$
0.984574 0.174971i $$-0.0559833\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 54.4299 3.54308
$$237$$ 0 0
$$238$$ − 22.7613i − 1.47540i
$$239$$ 22.0335 1.42523 0.712613 0.701557i $$-0.247512\pi$$
0.712613 + 0.701557i $$0.247512\pi$$
$$240$$ 0 0
$$241$$ −18.6472 −1.20117 −0.600585 0.799561i $$-0.705066\pi$$
−0.600585 + 0.799561i $$0.705066\pi$$
$$242$$ 20.3605i 1.30882i
$$243$$ 0 0
$$244$$ 62.5475 4.00420
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 5.84582i − 0.371961i
$$248$$ − 20.2863i − 1.28818i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 14.6929 0.927407 0.463704 0.885990i $$-0.346520\pi$$
0.463704 + 0.885990i $$0.346520\pi$$
$$252$$ 0 0
$$253$$ − 3.13515i − 0.197105i
$$254$$ −9.54205 −0.598721
$$255$$ 0 0
$$256$$ −9.46070 −0.591294
$$257$$ − 22.2089i − 1.38536i −0.721247 0.692678i $$-0.756431\pi$$
0.721247 0.692678i $$-0.243569\pi$$
$$258$$ 0 0
$$259$$ 20.9132 1.29949
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 19.2089i − 1.18673i
$$263$$ 5.74001i 0.353944i 0.984216 + 0.176972i $$0.0566302\pi$$
−0.984216 + 0.176972i $$0.943370\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −14.0345 −0.860513
$$267$$ 0 0
$$268$$ 44.9406i 2.74519i
$$269$$ 15.6162 0.952139 0.476070 0.879408i $$-0.342061\pi$$
0.476070 + 0.879408i $$0.342061\pi$$
$$270$$ 0 0
$$271$$ −6.75315 −0.410225 −0.205112 0.978738i $$-0.565756\pi$$
−0.205112 + 0.978738i $$0.565756\pi$$
$$272$$ − 50.4662i − 3.05996i
$$273$$ 0 0
$$274$$ 18.7685 1.13384
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 30.2966i − 1.82034i −0.414230 0.910172i $$-0.635949\pi$$
0.414230 0.910172i $$-0.364051\pi$$
$$278$$ − 38.7438i − 2.32370i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.6351 1.11168 0.555838 0.831290i $$-0.312397\pi$$
0.555838 + 0.831290i $$0.312397\pi$$
$$282$$ 0 0
$$283$$ − 5.67366i − 0.337264i −0.985679 0.168632i $$-0.946065\pi$$
0.985679 0.168632i $$-0.0539350\pi$$
$$284$$ −6.29208 −0.373366
$$285$$ 0 0
$$286$$ 9.39004 0.555245
$$287$$ − 4.43332i − 0.261690i
$$288$$ 0 0
$$289$$ −6.11806 −0.359886
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 17.7107i 1.03644i
$$293$$ 18.2773i 1.06777i 0.845556 + 0.533887i $$0.179269\pi$$
−0.845556 + 0.533887i $$0.820731\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 89.9850 5.23027
$$297$$ 0 0
$$298$$ − 1.48887i − 0.0862478i
$$299$$ 3.41900 0.197726
$$300$$ 0 0
$$301$$ 13.0819 0.754030
$$302$$ − 0.403974i − 0.0232461i
$$303$$ 0 0
$$304$$ −31.1173 −1.78470
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 15.5050i 0.884915i 0.896789 + 0.442458i $$0.145893\pi$$
−0.896789 + 0.442458i $$0.854107\pi$$
$$308$$ − 16.0435i − 0.914162i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 30.4464 1.72646 0.863228 0.504814i $$-0.168439\pi$$
0.863228 + 0.504814i $$0.168439\pi$$
$$312$$ 0 0
$$313$$ − 6.94936i − 0.392801i −0.980524 0.196401i $$-0.937075\pi$$
0.980524 0.196401i $$-0.0629253\pi$$
$$314$$ 30.1843 1.70340
$$315$$ 0 0
$$316$$ 10.4205 0.586196
$$317$$ 16.1451i 0.906797i 0.891308 + 0.453398i $$0.149788\pi$$
−0.891308 + 0.453398i $$0.850212\pi$$
$$318$$ 0 0
$$319$$ 13.3216 0.745868
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 8.20828i − 0.457430i
$$323$$ 14.2545i 0.793142i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −58.0985 −3.21778
$$327$$ 0 0
$$328$$ − 19.0756i − 1.05327i
$$329$$ 11.3096 0.623516
$$330$$ 0 0
$$331$$ 12.6222 0.693781 0.346890 0.937906i $$-0.387238\pi$$
0.346890 + 0.937906i $$0.387238\pi$$
$$332$$ − 5.42437i − 0.297701i
$$333$$ 0 0
$$334$$ −44.9535 −2.45975
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.92040i 0.376978i 0.982075 + 0.188489i $$0.0603590\pi$$
−0.982075 + 0.188489i $$0.939641\pi$$
$$338$$ − 23.9982i − 1.30533i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −4.74276 −0.256835
$$342$$ 0 0
$$343$$ 19.3570i 1.04518i
$$344$$ 56.2886 3.03488
$$345$$ 0 0
$$346$$ 31.4589 1.69124
$$347$$ − 13.8063i − 0.741163i −0.928800 0.370581i $$-0.879158\pi$$
0.928800 0.370581i $$-0.120842\pi$$
$$348$$ 0 0
$$349$$ −6.56768 −0.351560 −0.175780 0.984429i $$-0.556245\pi$$
−0.175780 + 0.984429i $$0.556245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 22.0153i − 1.17342i
$$353$$ − 3.52499i − 0.187616i −0.995590 0.0938082i $$-0.970096\pi$$
0.995590 0.0938082i $$-0.0299040\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −65.6231 −3.47802
$$357$$ 0 0
$$358$$ 22.5162i 1.19002i
$$359$$ −22.9285 −1.21012 −0.605061 0.796179i $$-0.706851\pi$$
−0.605061 + 0.796179i $$0.706851\pi$$
$$360$$ 0 0
$$361$$ −10.2107 −0.537407
$$362$$ 27.7920i 1.46072i
$$363$$ 0 0
$$364$$ 17.4960 0.917041
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 4.17931i − 0.218158i −0.994033 0.109079i $$-0.965210\pi$$
0.994033 0.109079i $$-0.0347902\pi$$
$$368$$ − 18.1993i − 0.948706i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3.10127 −0.161010
$$372$$ 0 0
$$373$$ 6.84091i 0.354209i 0.984192 + 0.177104i $$0.0566730\pi$$
−0.984192 + 0.177104i $$0.943327\pi$$
$$374$$ −22.8968 −1.18396
$$375$$ 0 0
$$376$$ 48.6625 2.50958
$$377$$ 14.5277i 0.748217i
$$378$$ 0 0
$$379$$ 12.7764 0.656280 0.328140 0.944629i $$-0.393578\pi$$
0.328140 + 0.944629i $$0.393578\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 45.6615i 2.33624i
$$383$$ 7.53459i 0.385000i 0.981297 + 0.192500i $$0.0616595\pi$$
−0.981297 + 0.192500i $$0.938341\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4.10736 0.209059
$$387$$ 0 0
$$388$$ − 18.9204i − 0.960538i
$$389$$ −5.45175 −0.276415 −0.138207 0.990403i $$-0.544134\pi$$
−0.138207 + 0.990403i $$0.544134\pi$$
$$390$$ 0 0
$$391$$ −8.33693 −0.421617
$$392$$ 29.1511i 1.47235i
$$393$$ 0 0
$$394$$ −47.3458 −2.38525
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5.64549i 0.283339i 0.989914 + 0.141670i $$0.0452470\pi$$
−0.989914 + 0.141670i $$0.954753\pi$$
$$398$$ 29.0301i 1.45515i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5.50418 −0.274865 −0.137433 0.990511i $$-0.543885\pi$$
−0.137433 + 0.990511i $$0.543885\pi$$
$$402$$ 0 0
$$403$$ − 5.17216i − 0.257643i
$$404$$ −32.3458 −1.60926
$$405$$ 0 0
$$406$$ 34.8779 1.73096
$$407$$ − 21.0377i − 1.04280i
$$408$$ 0 0
$$409$$ −32.8530 −1.62448 −0.812238 0.583327i $$-0.801751\pi$$
−0.812238 + 0.583327i $$0.801751\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 39.8650i 1.96401i
$$413$$ − 19.8184i − 0.975202i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 24.0085 1.17712
$$417$$ 0 0
$$418$$ 14.1181i 0.690537i
$$419$$ −22.8591 −1.11674 −0.558369 0.829593i $$-0.688573\pi$$
−0.558369 + 0.829593i $$0.688573\pi$$
$$420$$ 0 0
$$421$$ 17.9414 0.874411 0.437205 0.899362i $$-0.355968\pi$$
0.437205 + 0.899362i $$0.355968\pi$$
$$422$$ 63.0212i 3.06782i
$$423$$ 0 0
$$424$$ −13.3441 −0.648046
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 22.7742i − 1.10212i
$$428$$ − 44.3137i − 2.14198i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6.18871 −0.298100 −0.149050 0.988830i $$-0.547622\pi$$
−0.149050 + 0.988830i $$0.547622\pi$$
$$432$$ 0 0
$$433$$ − 3.11806i − 0.149844i −0.997189 0.0749221i $$-0.976129\pi$$
0.997189 0.0749221i $$-0.0238708\pi$$
$$434$$ −12.4172 −0.596045
$$435$$ 0 0
$$436$$ −31.3055 −1.49926
$$437$$ 5.14052i 0.245904i
$$438$$ 0 0
$$439$$ −13.5099 −0.644792 −0.322396 0.946605i $$-0.604488\pi$$
−0.322396 + 0.946605i $$0.604488\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 24.9698i − 1.18769i
$$443$$ − 24.2773i − 1.15345i −0.816938 0.576726i $$-0.804330\pi$$
0.816938 0.576726i $$-0.195670\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 57.1799 2.70755
$$447$$ 0 0
$$448$$ − 19.9075i − 0.940543i
$$449$$ 24.1437 1.13941 0.569705 0.821849i $$-0.307057\pi$$
0.569705 + 0.821849i $$0.307057\pi$$
$$450$$ 0 0
$$451$$ −4.45970 −0.209999
$$452$$ 73.5664i 3.46027i
$$453$$ 0 0
$$454$$ −37.1360 −1.74288
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2.82157i 0.131987i 0.997820 + 0.0659937i $$0.0210217\pi$$
−0.997820 + 0.0659937i $$0.978978\pi$$
$$458$$ 9.68573i 0.452585i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 21.4572 0.999363 0.499681 0.866209i $$-0.333450\pi$$
0.499681 + 0.866209i $$0.333450\pi$$
$$462$$ 0 0
$$463$$ − 19.8033i − 0.920339i −0.887831 0.460170i $$-0.847789\pi$$
0.887831 0.460170i $$-0.152211\pi$$
$$464$$ 77.3310 3.59000
$$465$$ 0 0
$$466$$ −14.0684 −0.651707
$$467$$ − 22.7210i − 1.05140i −0.850669 0.525701i $$-0.823803\pi$$
0.850669 0.525701i $$-0.176197\pi$$
$$468$$ 0 0
$$469$$ 16.3633 0.755588
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 85.2743i − 3.92507i
$$473$$ − 13.1598i − 0.605088i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −42.6625 −1.95543
$$477$$ 0 0
$$478$$ − 58.0300i − 2.65423i
$$479$$ −21.2880 −0.972672 −0.486336 0.873772i $$-0.661667\pi$$
−0.486336 + 0.873772i $$0.661667\pi$$
$$480$$ 0 0
$$481$$ 22.9424 1.04608
$$482$$ 49.1115i 2.23697i
$$483$$ 0 0
$$484$$ 38.1625 1.73466
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 9.58690i 0.434424i 0.976124 + 0.217212i $$0.0696963\pi$$
−0.976124 + 0.217212i $$0.930304\pi$$
$$488$$ − 97.9921i − 4.43590i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −37.8443 −1.70789 −0.853945 0.520363i $$-0.825797\pi$$
−0.853945 + 0.520363i $$0.825797\pi$$
$$492$$ 0 0
$$493$$ − 35.4246i − 1.59544i
$$494$$ −15.3963 −0.692711
$$495$$ 0 0
$$496$$ −27.5314 −1.23619
$$497$$ 2.29101i 0.102766i
$$498$$ 0 0
$$499$$ −16.9253 −0.757681 −0.378840 0.925462i $$-0.623677\pi$$
−0.378840 + 0.925462i $$0.623677\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 38.6970i − 1.72713i
$$503$$ 40.4168i 1.80210i 0.433719 + 0.901048i $$0.357201\pi$$
−0.433719 + 0.901048i $$0.642799\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −8.25713 −0.367074
$$507$$ 0 0
$$508$$ 17.8851i 0.793522i
$$509$$ 41.4067 1.83532 0.917660 0.397366i $$-0.130076\pi$$
0.917660 + 0.397366i $$0.130076\pi$$
$$510$$ 0 0
$$511$$ 6.44863 0.285270
$$512$$ 34.5537i 1.52707i
$$513$$ 0 0
$$514$$ −58.4922 −2.57998
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 11.3769i − 0.500354i
$$518$$ − 55.0797i − 2.42006i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −17.0301 −0.746103 −0.373052 0.927811i $$-0.621689\pi$$
−0.373052 + 0.927811i $$0.621689\pi$$
$$522$$ 0 0
$$523$$ 9.57651i 0.418751i 0.977835 + 0.209376i $$0.0671432\pi$$
−0.977835 + 0.209376i $$0.932857\pi$$
$$524$$ −36.0041 −1.57285
$$525$$ 0 0
$$526$$ 15.1176 0.659159
$$527$$ 12.6118i 0.549380i
$$528$$ 0 0
$$529$$ 19.9935 0.869283
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 26.3055i 1.14049i
$$533$$ − 4.86347i − 0.210660i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 70.4077 3.04115
$$537$$ 0 0
$$538$$ − 41.1289i − 1.77319i
$$539$$ 6.81528 0.293555
$$540$$ 0 0
$$541$$ −0.833751 −0.0358458 −0.0179229 0.999839i $$-0.505705\pi$$
−0.0179229 + 0.999839i $$0.505705\pi$$
$$542$$ 17.7859i 0.763971i
$$543$$ 0 0
$$544$$ −58.5426 −2.50999
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.3270i 1.21117i 0.795779 + 0.605587i $$0.207062\pi$$
−0.795779 + 0.605587i $$0.792938\pi$$
$$548$$ − 35.1785i − 1.50275i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −21.8427 −0.930529
$$552$$ 0 0
$$553$$ − 3.79419i − 0.161345i
$$554$$ −79.7928 −3.39007
$$555$$ 0 0
$$556$$ −72.6192 −3.07974
$$557$$ − 11.5042i − 0.487448i −0.969845 0.243724i $$-0.921631\pi$$
0.969845 0.243724i $$-0.0783690\pi$$
$$558$$ 0 0
$$559$$ 14.3512 0.606993
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 49.0797i − 2.07030i
$$563$$ 33.0059i 1.39103i 0.718510 + 0.695517i $$0.244824\pi$$
−0.718510 + 0.695517i $$0.755176\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −14.9429 −0.628095
$$567$$ 0 0
$$568$$ 9.85769i 0.413619i
$$569$$ 27.0088 1.13227 0.566135 0.824313i $$-0.308438\pi$$
0.566135 + 0.824313i $$0.308438\pi$$
$$570$$ 0 0
$$571$$ −24.4244 −1.02213 −0.511064 0.859543i $$-0.670749\pi$$
−0.511064 + 0.859543i $$0.670749\pi$$
$$572$$ − 17.6001i − 0.735899i
$$573$$ 0 0
$$574$$ −11.6761 −0.487352
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 14.7976i − 0.616033i −0.951381 0.308017i $$-0.900335\pi$$
0.951381 0.308017i $$-0.0996653\pi$$
$$578$$ 16.1133i 0.670224i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1.97507 −0.0819396
$$582$$ 0 0
$$583$$ 3.11973i 0.129206i
$$584$$ 27.7470 1.14818
$$585$$ 0 0
$$586$$ 48.1375 1.98854
$$587$$ − 30.5780i − 1.26209i −0.775747 0.631044i $$-0.782627\pi$$
0.775747 0.631044i $$-0.217373\pi$$
$$588$$ 0 0
$$589$$ 7.77641 0.320421
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 122.122i − 5.01919i
$$593$$ − 5.09990i − 0.209428i −0.994502 0.104714i $$-0.966607\pi$$
0.994502 0.104714i $$-0.0333927\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −2.79065 −0.114309
$$597$$ 0 0
$$598$$ − 9.00471i − 0.368230i
$$599$$ 0.565309 0.0230979 0.0115490 0.999933i $$-0.496324\pi$$
0.0115490 + 0.999933i $$0.496324\pi$$
$$600$$ 0 0
$$601$$ −11.0096 −0.449091 −0.224546 0.974464i $$-0.572090\pi$$
−0.224546 + 0.974464i $$0.572090\pi$$
$$602$$ − 34.4542i − 1.40425i
$$603$$ 0 0
$$604$$ −0.757185 −0.0308094
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 19.0983i 0.775174i 0.921833 + 0.387587i $$0.126691\pi$$
−0.921833 + 0.387587i $$0.873309\pi$$
$$608$$ 36.0972i 1.46393i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.4069 0.501929
$$612$$ 0 0
$$613$$ 9.33918i 0.377206i 0.982053 + 0.188603i $$0.0603959\pi$$
−0.982053 + 0.188603i $$0.939604\pi$$
$$614$$ 40.8358 1.64800
$$615$$ 0 0
$$616$$ −25.1350 −1.01272
$$617$$ 24.4154i 0.982928i 0.870898 + 0.491464i $$0.163538\pi$$
−0.870898 + 0.491464i $$0.836462\pi$$
$$618$$ 0 0
$$619$$ −39.4863 −1.58709 −0.793544 0.608513i $$-0.791766\pi$$
−0.793544 + 0.608513i $$0.791766\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 80.1874i − 3.21522i
$$623$$ 23.8940i 0.957293i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −18.3027 −0.731523
$$627$$ 0 0
$$628$$ − 56.5757i − 2.25762i
$$629$$ −55.9430 −2.23059
$$630$$ 0 0
$$631$$ 42.1634 1.67850 0.839249 0.543747i $$-0.182995\pi$$
0.839249 + 0.543747i $$0.182995\pi$$
$$632$$ − 16.3255i − 0.649395i
$$633$$ 0 0
$$634$$ 42.5216 1.68875
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 7.43232i 0.294479i
$$638$$ − 35.0855i − 1.38905i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 35.3155 1.39488 0.697438 0.716645i $$-0.254323\pi$$
0.697438 + 0.716645i $$0.254323\pi$$
$$642$$ 0 0
$$643$$ 14.1954i 0.559813i 0.960027 + 0.279906i $$0.0903034\pi$$
−0.960027 + 0.279906i $$0.909697\pi$$
$$644$$ −15.3851 −0.606259
$$645$$ 0 0
$$646$$ 37.5424 1.47709
$$647$$ − 17.4897i − 0.687593i −0.939044 0.343796i $$-0.888287\pi$$
0.939044 0.343796i $$-0.111713\pi$$
$$648$$ 0 0
$$649$$ −19.9364 −0.782572
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 108.897i 4.26472i
$$653$$ − 10.9772i − 0.429569i −0.976661 0.214785i $$-0.931095\pi$$
0.976661 0.214785i $$-0.0689050\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −25.8882 −1.01077
$$657$$ 0 0
$$658$$ − 29.7862i − 1.16119i
$$659$$ 15.7876 0.614998 0.307499 0.951548i $$-0.400508\pi$$
0.307499 + 0.951548i $$0.400508\pi$$
$$660$$ 0 0
$$661$$ 49.8932 1.94062 0.970311 0.241862i $$-0.0777581\pi$$
0.970311 + 0.241862i $$0.0777581\pi$$
$$662$$ − 33.2435i − 1.29204i
$$663$$ 0 0
$$664$$ −8.49827 −0.329797
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 12.7750i − 0.494649i
$$668$$ 84.2582i 3.26005i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −22.9097 −0.884419
$$672$$ 0 0
$$673$$ 28.8395i 1.11168i 0.831289 + 0.555840i $$0.187603\pi$$
−0.831289 + 0.555840i $$0.812397\pi$$
$$674$$ 18.2264 0.702055
$$675$$ 0 0
$$676$$ −44.9809 −1.73003
$$677$$ − 10.4636i − 0.402150i −0.979576 0.201075i $$-0.935557\pi$$
0.979576 0.201075i $$-0.0644434\pi$$
$$678$$ 0 0
$$679$$ −6.88910 −0.264379
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 12.4911i 0.478309i
$$683$$ − 16.1875i − 0.619396i −0.950835 0.309698i $$-0.899772\pi$$
0.950835 0.309698i $$-0.100228\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 50.9809 1.94646
$$687$$ 0 0
$$688$$ − 76.3916i − 2.91240i
$$689$$ −3.40218 −0.129613
$$690$$ 0 0
$$691$$ 9.88362 0.375991 0.187995 0.982170i $$-0.439801\pi$$
0.187995 + 0.982170i $$0.439801\pi$$
$$692$$ − 58.9648i − 2.24150i
$$693$$ 0 0
$$694$$ −36.3621 −1.38029
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 11.8591i 0.449197i
$$698$$ 17.2974i 0.654718i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 43.9692 1.66069 0.830346 0.557248i $$-0.188143\pi$$
0.830346 + 0.557248i $$0.188143\pi$$
$$702$$ 0 0
$$703$$ 34.4942i 1.30097i
$$704$$ −20.0260 −0.754758
$$705$$ 0 0
$$706$$ −9.28385 −0.349402
$$707$$ 11.7774i 0.442935i
$$708$$ 0 0
$$709$$ −25.2260 −0.947384 −0.473692 0.880691i $$-0.657079\pi$$
−0.473692 + 0.880691i $$0.657079\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 102.811i 3.85299i
$$713$$ 4.54814i 0.170329i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 42.2032 1.57721
$$717$$ 0 0
$$718$$ 60.3875i 2.25364i
$$719$$ 36.8600 1.37465 0.687323 0.726352i $$-0.258786\pi$$
0.687323 + 0.726352i $$0.258786\pi$$
$$720$$ 0 0
$$721$$ 14.5153 0.540576
$$722$$ 26.8922i 1.00082i
$$723$$ 0 0
$$724$$ 52.0918 1.93598
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 38.2451i 1.41843i 0.704990 + 0.709217i $$0.250951\pi$$
−0.704990 + 0.709217i $$0.749049\pi$$
$$728$$ − 27.4107i − 1.01591i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −34.9942 −1.29431
$$732$$ 0 0
$$733$$ 39.5832i 1.46204i 0.682356 + 0.731020i $$0.260955\pi$$
−0.682356 + 0.731020i $$0.739045\pi$$
$$734$$ −11.0072 −0.406282
$$735$$ 0 0
$$736$$ −21.1119 −0.778195
$$737$$ − 16.4607i − 0.606338i
$$738$$ 0 0
$$739$$ 8.24773 0.303398 0.151699 0.988427i $$-0.451526\pi$$
0.151699 + 0.988427i $$0.451526\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 8.16790i 0.299853i
$$743$$ 1.30818i 0.0479925i 0.999712 + 0.0239963i $$0.00763898\pi$$
−0.999712 + 0.0239963i $$0.992361\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 18.0171 0.659651
$$747$$ 0 0
$$748$$ 42.9164i 1.56918i
$$749$$ −16.1350 −0.589562
$$750$$ 0 0
$$751$$ 28.4468 1.03804 0.519020 0.854762i $$-0.326297\pi$$
0.519020 + 0.854762i $$0.326297\pi$$
$$752$$ − 66.0418i − 2.40830i
$$753$$ 0 0
$$754$$ 38.2620 1.39342
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.2012i 1.38845i 0.719760 + 0.694223i $$0.244252\pi$$
−0.719760 + 0.694223i $$0.755748\pi$$
$$758$$ − 33.6495i − 1.22221i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 22.1904 0.804401 0.402200 0.915552i $$-0.368245\pi$$
0.402200 + 0.915552i $$0.368245\pi$$
$$762$$ 0 0
$$763$$ 11.3986i 0.412659i
$$764$$ 85.5852 3.09637
$$765$$ 0 0
$$766$$ 19.8440 0.716994
$$767$$ − 21.7414i − 0.785036i
$$768$$ 0 0
$$769$$ 16.9130 0.609900 0.304950 0.952368i $$-0.401360\pi$$
0.304950 + 0.952368i $$0.401360\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 7.69860i − 0.277079i
$$773$$ − 38.6464i − 1.39001i −0.719003 0.695007i $$-0.755401\pi$$
0.719003 0.695007i $$-0.244599\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −29.6423 −1.06409
$$777$$ 0 0
$$778$$ 14.3584i 0.514774i
$$779$$ 7.31230 0.261990
$$780$$ 0 0
$$781$$ 2.30464 0.0824665
$$782$$ 21.9572i 0.785187i
$$783$$ 0 0
$$784$$ 39.5622 1.41294
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 10.9147i − 0.389067i −0.980896 0.194533i $$-0.937681\pi$$
0.980896 0.194533i $$-0.0623192\pi$$
$$788$$ 88.7422i 3.16131i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 26.7862 0.952409
$$792$$ 0 0
$$793$$ − 24.9839i − 0.887204i
$$794$$ 14.8687 0.527669
$$795$$ 0 0
$$796$$ 54.4124 1.92860
$$797$$ 2.92550i 0.103627i 0.998657 + 0.0518133i $$0.0165001\pi$$
−0.998657 + 0.0518133i $$0.983500\pi$$
$$798$$ 0 0
$$799$$ −30.2531 −1.07028
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 14.4965i 0.511888i
$$803$$ − 6.48700i − 0.228921i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −13.6220 −0.479816
$$807$$ 0 0
$$808$$ 50.6755i 1.78276i
$$809$$ 37.9241 1.33334 0.666671 0.745352i $$-0.267719\pi$$
0.666671 + 0.745352i $$0.267719\pi$$
$$810$$ 0 0
$$811$$ 19.5050 0.684912 0.342456 0.939534i $$-0.388741\pi$$
0.342456 + 0.939534i $$0.388741\pi$$
$$812$$ − 65.3731i − 2.29415i
$$813$$ 0 0
$$814$$ −55.4075 −1.94203
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 21.5773i 0.754894i
$$818$$ 86.5257i 3.02530i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 44.7524 1.56187 0.780934 0.624613i $$-0.214743\pi$$
0.780934 + 0.624613i $$0.214743\pi$$
$$822$$ 0 0
$$823$$ − 35.8994i − 1.25137i −0.780074 0.625687i $$-0.784819\pi$$
0.780074 0.625687i $$-0.215181\pi$$
$$824$$ 62.4559 2.17575
$$825$$ 0 0
$$826$$ −52.1963 −1.81614
$$827$$ 16.2717i 0.565822i 0.959146 + 0.282911i $$0.0913001\pi$$
−0.959146 + 0.282911i $$0.908700\pi$$
$$828$$ 0 0
$$829$$ 20.6592 0.717525 0.358762 0.933429i $$-0.383199\pi$$
0.358762 + 0.933429i $$0.383199\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 21.8391i − 0.757135i
$$833$$ − 18.1230i − 0.627926i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 26.4621 0.915210
$$837$$ 0 0
$$838$$ 60.2044i 2.07973i
$$839$$ 12.2367 0.422459 0.211229 0.977437i $$-0.432253\pi$$
0.211229 + 0.977437i $$0.432253\pi$$
$$840$$ 0 0
$$841$$ 25.2823 0.871802
$$842$$ − 47.2527i − 1.62844i
$$843$$ 0 0
$$844$$ 118.123 4.06597
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 13.8953i − 0.477450i
$$848$$ 18.1098i 0.621893i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −20.1744 −0.691570
$$852$$ 0 0
$$853$$ 16.4293i 0.562528i 0.959630 + 0.281264i $$0.0907537\pi$$
−0.959630 + 0.281264i $$0.909246\pi$$
$$854$$ −59.9809 −2.05250
$$855$$ 0 0
$$856$$ −69.4255 −2.37291
$$857$$ − 49.0897i − 1.67687i −0.545000 0.838436i $$-0.683470\pi$$
0.545000 0.838436i $$-0.316530\pi$$
$$858$$ 0 0
$$859$$ −41.0908 −1.40200 −0.700999 0.713162i $$-0.747262\pi$$
−0.700999 + 0.713162i $$0.747262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 16.2994i 0.555158i
$$863$$ 47.8366i 1.62838i 0.580601 + 0.814188i $$0.302817\pi$$
−0.580601 + 0.814188i $$0.697183\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −8.21210 −0.279058
$$867$$ 0 0
$$868$$ 23.2741i 0.789975i
$$869$$ −3.81677 −0.129475
$$870$$ 0 0
$$871$$ 17.9510 0.608247
$$872$$ 49.0458i 1.66090i
$$873$$ 0 0
$$874$$ 13.5387 0.457954
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 11.3772i 0.384180i 0.981377 + 0.192090i $$0.0615265\pi$$
−0.981377 + 0.192090i $$0.938473\pi$$
$$878$$ 35.5813i 1.20081i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −43.9924 −1.48214 −0.741071 0.671426i $$-0.765682\pi$$
−0.741071 + 0.671426i $$0.765682\pi$$
$$882$$ 0 0
$$883$$ 37.6820i 1.26810i 0.773291 + 0.634051i $$0.218609\pi$$
−0.773291 + 0.634051i $$0.781391\pi$$
$$884$$ −46.8019 −1.57412
$$885$$ 0 0
$$886$$ −63.9398 −2.14810
$$887$$ 33.2833i 1.11754i 0.829322 + 0.558771i $$0.188727\pi$$
−0.829322 + 0.558771i $$0.811273\pi$$
$$888$$ 0 0
$$889$$ 6.51213 0.218410
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 107.175i − 3.58847i
$$893$$ 18.6539i 0.624230i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −8.66080 −0.289337
$$897$$ 0 0
$$898$$ − 63.5878i − 2.12195i
$$899$$ −19.3255 −0.644543
$$900$$ 0 0
$$901$$ 8.29592 0.276377
$$902$$ 11.7456i 0.391086i
$$903$$ 0 0
$$904$$ 115.255 3.83333
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 14.0784i 0.467464i 0.972301 + 0.233732i $$0.0750939\pi$$
−0.972301 + 0.233732i $$0.924906\pi$$
$$908$$ 69.6056i 2.30994i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 7.31522 0.242364 0.121182 0.992630i $$-0.461332\pi$$
0.121182 + 0.992630i $$0.461332\pi$$
$$912$$ 0 0
$$913$$ 1.98682i 0.0657542i
$$914$$ 7.43123 0.245803
$$915$$ 0 0
$$916$$ 18.1544 0.599838
$$917$$ 13.1094i 0.432912i
$$918$$ 0 0
$$919$$ −4.44684 −0.146688 −0.0733438 0.997307i $$-0.523367\pi$$
−0.0733438 + 0.997307i $$0.523367\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 56.5124i − 1.86114i
$$923$$ 2.51330i 0.0827262i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −52.1565 −1.71397
$$927$$ 0 0
$$928$$ − 89.7068i − 2.94477i
$$929$$ −13.1133 −0.430232 −0.215116 0.976588i $$-0.569013\pi$$
−0.215116 + 0.976588i $$0.569013\pi$$
$$930$$ 0 0
$$931$$ −11.1746 −0.366233
$$932$$ 26.3690i 0.863746i
$$933$$ 0 0
$$934$$ −59.8408 −1.95805
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 33.5187i 1.09501i 0.836803 + 0.547504i $$0.184422\pi$$
−0.836803 + 0.547504i $$0.815578\pi$$
$$938$$ − 43.0964i − 1.40715i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1.79076 0.0583772 0.0291886 0.999574i $$-0.490708\pi$$
0.0291886 + 0.999574i $$0.490708\pi$$
$$942$$ 0 0
$$943$$ 4.27669i 0.139268i
$$944$$ −115.729 −3.76667
$$945$$ 0 0
$$946$$ −34.6592 −1.12687
$$947$$ 52.6350i 1.71041i 0.518292 + 0.855204i $$0.326568\pi$$
−0.518292 + 0.855204i $$0.673432\pi$$
$$948$$ 0 0
$$949$$ 7.07432 0.229642
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 66.8386i 2.16625i
$$953$$ − 18.4072i − 0.596268i −0.954524 0.298134i $$-0.903636\pi$$
0.954524 0.298134i $$-0.0963642\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −108.768 −3.51781
$$957$$ 0 0
$$958$$ 56.0666i 1.81143i
$$959$$ −12.8088 −0.413619
$$960$$ 0 0
$$961$$ −24.1197 −0.778056
$$962$$ − 60.4240i − 1.94815i
$$963$$ 0 0
$$964$$ 92.0517 2.96479
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 6.09579i 0.196027i 0.995185 + 0.0980137i $$0.0312489\pi$$
−0.995185 + 0.0980137i $$0.968751\pi$$
$$968$$ − 59.7886i − 1.92168i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −34.1077 −1.09457 −0.547283 0.836947i $$-0.684338\pi$$
−0.547283 + 0.836947i $$0.684338\pi$$
$$972$$ 0 0
$$973$$ 26.4414i 0.847671i
$$974$$ 25.2493 0.809038
$$975$$ 0 0
$$976$$ −132.989 −4.25688
$$977$$ − 48.7914i − 1.56098i −0.625171 0.780488i $$-0.714971\pi$$
0.625171 0.780488i $$-0.285029\pi$$
$$978$$ 0 0
$$979$$ 24.0362 0.768201
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 99.6715i 3.18065i
$$983$$ 33.4172i 1.06584i 0.846164 + 0.532922i $$0.178906\pi$$
−0.846164 + 0.532922i $$0.821094\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −93.2985 −2.97123
$$987$$ 0 0
$$988$$ 28.8579i 0.918092i
$$989$$ −12.6198 −0.401285
$$990$$ 0 0
$$991$$ −19.3512 −0.614713 −0.307356 0.951595i $$-0.599444\pi$$
−0.307356 + 0.951595i $$0.599444\pi$$
$$992$$ 31.9374i 1.01401i
$$993$$ 0 0
$$994$$ 6.03388 0.191383
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 39.5442i − 1.25238i −0.779671 0.626189i $$-0.784614\pi$$
0.779671 0.626189i $$-0.215386\pi$$
$$998$$ 44.5766i 1.41105i
$$999$$ 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.649.1 8
3.2 odd 2 2025.2.b.o.649.8 8
5.2 odd 4 2025.2.a.y.1.4 4
5.3 odd 4 2025.2.a.q.1.1 4
5.4 even 2 inner 2025.2.b.n.649.8 8
9.2 odd 6 675.2.k.c.199.8 16
9.4 even 3 225.2.k.c.124.8 16
9.5 odd 6 675.2.k.c.424.1 16
9.7 even 3 225.2.k.c.49.1 16
15.2 even 4 2025.2.a.p.1.1 4
15.8 even 4 2025.2.a.z.1.4 4
15.14 odd 2 2025.2.b.o.649.1 8
45.2 even 12 675.2.e.e.226.4 8
45.4 even 6 225.2.k.c.124.1 16
45.7 odd 12 225.2.e.c.76.1 8
45.13 odd 12 225.2.e.e.151.4 yes 8
45.14 odd 6 675.2.k.c.424.8 16
45.22 odd 12 225.2.e.c.151.1 yes 8
45.23 even 12 675.2.e.c.451.1 8
45.29 odd 6 675.2.k.c.199.1 16
45.32 even 12 675.2.e.e.451.4 8
45.34 even 6 225.2.k.c.49.8 16
45.38 even 12 675.2.e.c.226.1 8
45.43 odd 12 225.2.e.e.76.4 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.e.c.76.1 8 45.7 odd 12
225.2.e.c.151.1 yes 8 45.22 odd 12
225.2.e.e.76.4 yes 8 45.43 odd 12
225.2.e.e.151.4 yes 8 45.13 odd 12
225.2.k.c.49.1 16 9.7 even 3
225.2.k.c.49.8 16 45.34 even 6
225.2.k.c.124.1 16 45.4 even 6
225.2.k.c.124.8 16 9.4 even 3
675.2.e.c.226.1 8 45.38 even 12
675.2.e.c.451.1 8 45.23 even 12
675.2.e.e.226.4 8 45.2 even 12
675.2.e.e.451.4 8 45.32 even 12
675.2.k.c.199.1 16 45.29 odd 6
675.2.k.c.199.8 16 9.2 odd 6
675.2.k.c.424.1 16 9.5 odd 6
675.2.k.c.424.8 16 45.14 odd 6
2025.2.a.p.1.1 4 15.2 even 4
2025.2.a.q.1.1 4 5.3 odd 4
2025.2.a.y.1.4 4 5.2 odd 4
2025.2.a.z.1.4 4 15.8 even 4
2025.2.b.n.649.1 8 1.1 even 1 trivial
2025.2.b.n.649.8 8 5.4 even 2 inner
2025.2.b.o.649.1 8 15.14 odd 2
2025.2.b.o.649.8 8 3.2 odd 2