LMFDB - Embedded newform 960.3.p.b.799.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(799,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("960.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.p (of order $2$, degree $1$, 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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.8
Root			$-0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	960.799
Dual form			960.3.p.b.799.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +(4.83465 - 1.27520i) q^{5} +7.58258 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +(4.83465 - 1.27520i) q^{5} +7.58258 q^{7} -3.00000 q^{9} -18.4249 q^{11} +13.1334 q^{13} +(2.20871 + 8.37386i) q^{15} +28.7477i q^{17} +20.7846 q^{19} +13.1334i q^{21} -24.0000 q^{23} +(21.7477 - 12.3303i) q^{25} -5.19615i q^{27} +18.2342i q^{29} +31.4955i q^{31} -31.9129i q^{33} +(36.6591 - 9.66930i) q^{35} +18.6156 q^{37} +22.7477i q^{39} +72.9909 q^{41} -5.48220i q^{43} +(-14.5040 + 3.82560i) q^{45} +30.9909 q^{47} +8.49545 q^{49} -49.7925 q^{51} +11.1153 q^{53} +(-89.0780 + 23.4955i) q^{55} +36.0000i q^{57} -28.2451 q^{59} -116.032i q^{61} -22.7477 q^{63} +(63.4955 - 16.7477i) q^{65} -99.5850i q^{67} -41.5692i q^{69} +54.9909i q^{71} +77.6697i q^{73} +(21.3567 + 37.6682i) q^{75} -139.708 q^{77} +150.486i q^{79} +9.00000 q^{81} +40.1232i q^{83} +(36.6591 + 138.985i) q^{85} -31.5826 q^{87} +91.9818 q^{89} +99.5850 q^{91} -54.5517 q^{93} +(100.486 - 26.5045i) q^{95} -39.8258i q^{97} +55.2747 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{7} - 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^7 - 24 * q^9	$ 8 q + 24 q^{7} - 24 q^{9} + 36 q^{15} - 192 q^{23} + 64 q^{25} + 144 q^{41} - 192 q^{47} - 152 q^{49} - 456 q^{55} - 72 q^{63} + 288 q^{65} + 72 q^{81} - 216 q^{87} - 144 q^{89} + 144 q^{95}+O(q^{100}) $ 8 * q + 24 * q^7 - 24 * q^9 + 36 * q^15 - 192 * q^23 + 64 * q^25 + 144 * q^41 - 192 * q^47 - 152 * q^49 - 456 * q^55 - 72 * q^63 + 288 * q^65 + 72 * q^81 - 216 * q^87 - 144 * q^89 + 144 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			1.73205i			0.577350i
$3$			1.73205i			0.577350i
$4$		0			0
$5$	4.83465	−	1.27520i	0.966930	−	0.255040i
$5$	4.83465	−	1.27520i	0.966930	−	0.255040i
$6$		0			0
$7$	7.58258			1.08323			0.541613	−	0.840628i	$-0.317814\pi$
$7$	7.58258			1.08323			0.541613	+	0.840628i	$0.317814\pi$
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	−18.4249			−1.67499			−0.837496	−	0.546444i	$-0.815981\pi$
$11$	−18.4249			−1.67499			−0.837496	+	0.546444i	$0.815981\pi$
$12$		0			0
$13$	13.1334			1.01026			0.505131	−	0.863043i	$-0.331444\pi$
$13$	13.1334			1.01026			0.505131	+	0.863043i	$0.331444\pi$
$14$		0			0
$15$	2.20871	+	8.37386i	0.147247	+	0.558258i
$16$		0			0
$17$			28.7477i			1.69104i	0.533942	+	0.845521i	$0.320710\pi$
$17$			28.7477i			1.69104i	−0.533942	+	0.845521i	$0.679290\pi$
$18$		0			0
$19$	20.7846			1.09393			0.546963	−	0.837157i	$-0.315784\pi$
$19$	20.7846			1.09393			0.546963	+	0.837157i	$0.315784\pi$
$20$		0			0
$21$			13.1334i			0.625400i
$22$		0			0
$23$	−24.0000			−1.04348			−0.521739	−	0.853105i	$-0.674717\pi$
$23$	−24.0000			−1.04348			−0.521739	+	0.853105i	$0.674717\pi$
$24$		0			0
$25$	21.7477	−	12.3303i	0.869909	−	0.493212i
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$			18.2342i			0.628766i	0.949296	+	0.314383i	$0.101798\pi$
$29$			18.2342i			0.628766i	−0.949296	+	0.314383i	$0.898202\pi$
$30$		0			0
$31$			31.4955i			1.01598i	0.861362	+	0.507991i	$0.169612\pi$
$31$			31.4955i			1.01598i	−0.861362	+	0.507991i	$0.830388\pi$
$32$		0			0
$33$		−	31.9129i		−	0.967057i
$34$		0			0
$35$	36.6591	−	9.66930i	1.04740	−	0.276266i
$36$		0			0
$37$	18.6156			0.503125			0.251562	−	0.967841i	$-0.419056\pi$
$37$	18.6156			0.503125			0.251562	+	0.967841i	$0.419056\pi$
$38$		0			0
$39$			22.7477i			0.583275i
$40$		0			0
$41$	72.9909			1.78027			0.890133	−	0.455701i	$-0.150611\pi$
$41$	72.9909			1.78027			0.890133	+	0.455701i	$0.150611\pi$
$42$		0			0
$43$		−	5.48220i		−	0.127493i	−0.997966	−	0.0637465i	$-0.979695\pi$
$43$		−	5.48220i		−	0.127493i	0.997966	−	0.0637465i	$-0.0203049\pi$
$44$		0			0
$45$	−14.5040	+	3.82560i	−0.322310	+	0.0850134i
$46$		0			0
$47$	30.9909			0.659381			0.329691	−	0.944089i	$-0.393056\pi$
$47$	30.9909			0.659381			0.329691	+	0.944089i	$0.393056\pi$
$48$		0			0
$49$	8.49545			0.173377
$50$		0			0
$51$	−49.7925			−0.976324
$52$		0			0
$53$	11.1153			0.209723			0.104861	−	0.994487i	$-0.466560\pi$
$53$	11.1153			0.209723			0.104861	+	0.994487i	$0.466560\pi$
$54$		0			0
$55$	−89.0780	+	23.4955i	−1.61960	+	0.427190i
$56$		0			0
$57$			36.0000i			0.631579i
$58$		0			0
$59$	−28.2451			−0.478731			−0.239365	−	0.970930i	$-0.576939\pi$
$59$	−28.2451			−0.478731			−0.239365	+	0.970930i	$0.576939\pi$
$60$		0			0
$61$		−	116.032i		−	1.90216i	−0.308947	−	0.951079i	$-0.599977\pi$
$61$		−	116.032i		−	1.90216i	0.308947	−	0.951079i	$-0.400023\pi$
$62$		0			0
$63$	−22.7477			−0.361075
$64$		0			0
$65$	63.4955	−	16.7477i	0.976853	−	0.257657i
$66$		0			0
$67$		−	99.5850i		−	1.48634i	−0.669100	−	0.743172i	$-0.733320\pi$
$67$		−	99.5850i		−	1.48634i	0.669100	−	0.743172i	$-0.266680\pi$
$68$		0			0
$69$		−	41.5692i		−	0.602452i
$70$		0			0
$71$			54.9909i			0.774520i	0.921971	+	0.387260i	$0.126578\pi$
$71$			54.9909i			0.774520i	−0.921971	+	0.387260i	$0.873422\pi$
$72$		0			0
$73$			77.6697i			1.06397i	0.846754	+	0.531984i	$0.178553\pi$
$73$			77.6697i			1.06397i	−0.846754	+	0.531984i	$0.821447\pi$
$74$		0			0
$75$	21.3567	+	37.6682i	0.284756	+	0.502242i
$76$		0			0
$77$	−139.708			−1.81439
$78$		0			0
$79$			150.486i			1.90489i	0.304710	+	0.952445i	$0.401441\pi$
$79$			150.486i			1.90489i	−0.304710	+	0.952445i	$0.598559\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			40.1232i			0.483412i	0.970349	+	0.241706i	$0.0777070\pi$
$83$			40.1232i			0.483412i	−0.970349	+	0.241706i	$0.922293\pi$
$84$		0			0
$85$	36.6591	+	138.985i	0.431284	+	1.63512i
$86$		0			0
$87$	−31.5826			−0.363018
$88$		0			0
$89$	91.9818			1.03350			0.516752	−	0.856135i	$-0.327141\pi$
$89$	91.9818			1.03350			0.516752	+	0.856135i	$0.327141\pi$
$90$		0			0
$91$	99.5850			1.09434
$92$		0			0
$93$	−54.5517			−0.586578
$94$		0			0
$95$	100.486	−	26.5045i	1.05775	−	0.278995i
$96$		0			0
$97$		−	39.8258i		−	0.410575i	−0.978702	−	0.205287i	$-0.934187\pi$
$97$		−	39.8258i		−	0.410575i	0.978702	−	0.205287i	$-0.0658129\pi$
$98$		0			0
$99$	55.2747			0.558331
$100$		0			0
$101$		−	33.1552i		−	0.328269i	−0.986438	−	0.164135i	$-0.947517\pi$
$101$		−	33.1552i		−	0.328269i	0.986438	−	0.164135i	$-0.0524832\pi$
$102$		0			0
$103$	132.730			1.28864			0.644318	−	0.764758i	$-0.277141\pi$
$103$	132.730			1.28864			0.644318	+	0.764758i	$0.277141\pi$
$104$		0			0
$105$	16.7477	+	63.4955i	0.159502	+	0.604719i
$106$		0			0
$107$			92.3550i			0.863131i	0.902082	+	0.431566i	$0.142039\pi$
$107$			92.3550i			0.863131i	−0.902082	+	0.431566i	$0.857961\pi$
$108$		0			0
$109$			120.370i			1.10431i	0.833742	+	0.552154i	$0.186194\pi$
$109$			120.370i			1.10431i	−0.833742	+	0.552154i	$0.813806\pi$
$110$		0			0
$111$			32.2432i			0.290479i
$112$		0			0
$113$			110.243i			0.975603i	0.872954	+	0.487802i	$0.162201\pi$
$113$			110.243i			0.975603i	−0.872954	+	0.487802i	$0.837799\pi$
$114$		0			0
$115$	−116.032	+	30.6048i	−1.00897	+	0.266129i
$116$		0			0
$117$	−39.4002			−0.336754
$118$		0			0
$119$			217.982i			1.83178i
$120$		0			0
$121$	218.477			1.80560
$122$		0			0
$123$			126.424i			1.02784i
$124$		0			0
$125$	89.4191	−	87.3454i	0.715353	−	0.698764i
$126$		0			0
$127$	−170.573			−1.34310			−0.671549	−	0.740960i	$-0.734371\pi$
$127$	−170.573			−1.34310			−0.671549	+	0.740960i	$0.734371\pi$
$128$		0			0
$129$	9.49545			0.0736082
$130$		0			0
$131$	−107.427			−0.820053			−0.410027	−	0.912074i	$-0.634481\pi$
$131$	−107.427			−0.820053			−0.410027	+	0.912074i	$0.634481\pi$
$132$		0			0
$133$	157.601			1.18497
$134$		0			0
$135$	−6.62614	−	25.1216i	−0.0490825	−	0.186086i
$136$		0			0
$137$		−	124.225i		−	0.906752i	−0.891319	−	0.453376i	$-0.850220\pi$
$137$		−	124.225i		−	0.906752i	0.891319	−	0.453376i	$-0.149780\pi$
$138$		0			0
$139$	−111.694			−0.803551			−0.401776	−	0.915738i	$-0.631607\pi$
$139$	−111.694			−0.803551			−0.401776	+	0.915738i	$0.631607\pi$
$140$		0			0
$141$			53.6778i			0.380694i
$142$		0			0
$143$	−241.982			−1.69218
$144$		0			0
$145$	23.2523	+	88.1561i	0.160361	+	0.607973i
$146$		0			0
$147$			14.7146i			0.100099i
$148$		0			0
$149$		−	59.8034i		−	0.401365i	−0.979656	−	0.200683i	$-0.935684\pi$
$149$		−	59.8034i		−	0.401365i	0.979656	−	0.200683i	$-0.0643160\pi$
$150$		0			0
$151$		−	64.5045i		−	0.427182i	−0.976923	−	0.213591i	$-0.931484\pi$
$151$		−	64.5045i		−	0.427182i	0.976923	−	0.213591i	$-0.0685161\pi$
$152$		0			0
$153$		−	86.2432i		−	0.563681i
$154$		0			0
$155$	40.1630	+	152.270i	0.259116	+	0.982384i
$156$		0			0
$157$	73.4376			0.467756			0.233878	−	0.972266i	$-0.424858\pi$
$157$	73.4376			0.467756			0.233878	+	0.972266i	$0.424858\pi$
$158$		0			0
$159$			19.2523i			0.121083i
$160$		0			0
$161$	−181.982			−1.13032
$162$		0			0
$163$		−	158.745i		−	0.973896i	−0.873431	−	0.486948i	$-0.838110\pi$
$163$		−	158.745i		−	0.973896i	0.873431	−	0.486948i	$-0.161890\pi$
$164$		0			0
$165$	−40.6953	−	154.288i	−0.246638	−	0.935077i
$166$		0			0
$167$	−119.477			−0.715433			−0.357716	−	0.933830i	$-0.616445\pi$
$167$	−119.477			−0.715433			−0.357716	+	0.933830i	$0.616445\pi$
$168$		0			0
$169$	3.48636			0.0206294
$170$		0			0
$171$	−62.3538			−0.364642
$172$		0			0
$173$	−46.9005			−0.271101			−0.135551	−	0.990770i	$-0.543280\pi$
$173$	−46.9005			−0.271101			−0.135551	+	0.990770i	$0.543280\pi$
$174$		0			0
$175$	164.904	−	93.4955i	0.942307	−	0.534260i
$176$		0			0
$177$		−	48.9220i		−	0.276395i
$178$		0			0
$179$	144.277			0.806015			0.403008	−	0.915197i	$-0.367965\pi$
$179$	144.277			0.806015			0.403008	+	0.915197i	$0.367965\pi$
$180$		0			0
$181$			170.615i			0.942624i	0.881967	+	0.471312i	$0.156219\pi$
$181$			170.615i			0.942624i	−0.881967	+	0.471312i	$0.843781\pi$
$182$		0			0
$183$	200.973			1.09821
$184$		0			0
$185$	90.0000	−	23.7386i	0.486486	−	0.128317i
$186$		0			0
$187$		−	529.674i		−	2.83248i
$188$		0			0
$189$		−	39.4002i		−	0.208467i
$190$		0			0
$191$			53.0091i			0.277535i	0.990325	+	0.138767i	$0.0443140\pi$
$191$			53.0091i			0.277535i	−0.990325	+	0.138767i	$0.955686\pi$
$192$		0			0
$193$		−	272.835i		−	1.41365i	−0.707387	−	0.706826i	$-0.750126\pi$
$193$		−	272.835i		−	1.41365i	0.707387	−	0.706826i	$-0.249874\pi$
$194$		0			0
$195$	29.0079	+	109.977i	0.148759	+	0.563986i
$196$		0			0
$197$	−333.547			−1.69313			−0.846566	−	0.532284i	$-0.821334\pi$
$197$	−333.547			−1.69313			−0.846566	+	0.532284i	$0.821334\pi$
$198$		0			0
$199$		−	73.4773i		−	0.369233i	−0.982811	−	0.184616i	$-0.940896\pi$
$199$		−	73.4773i		−	0.369233i	0.982811	−	0.184616i	$-0.0591042\pi$
$200$		0			0
$201$	172.486			0.858141
$202$		0			0
$203$			138.262i			0.681095i
$204$		0			0
$205$	352.886	−	93.0780i	1.72139	−	0.454039i
$206$		0			0
$207$	72.0000			0.347826
$208$		0			0
$209$	−382.955			−1.83232
$210$		0			0
$211$	286.647			1.35851			0.679257	−	0.733900i	$-0.262302\pi$
$211$	286.647			1.35851			0.679257	+	0.733900i	$0.262302\pi$
$212$		0			0
$213$	−95.2470			−0.447169
$214$		0			0
$215$	−6.99091	−	26.5045i	−0.0325159	−	0.123277i
$216$		0			0
$217$			238.817i			1.10054i
$218$		0			0
$219$	−134.528			−0.614282
$220$		0			0
$221$			377.556i			1.70840i
$222$		0			0
$223$	−60.7295			−0.272330			−0.136165	−	0.990686i	$-0.543478\pi$
$223$	−60.7295			−0.272330			−0.136165	+	0.990686i	$0.543478\pi$
$224$		0			0
$225$	−65.2432	+	36.9909i	−0.289970	+	0.164404i
$226$		0			0
$227$		−	328.216i		−	1.44588i	−0.690909	−	0.722942i	$-0.742789\pi$
$227$		−	328.216i		−	1.44588i	0.690909	−	0.722942i	$-0.257211\pi$
$228$		0			0
$229$		−	103.018i		−	0.449859i	−0.974375	−	0.224929i	$-0.927785\pi$
$229$		−	103.018i		−	0.449859i	0.974375	−	0.224929i	$-0.0722151\pi$
$230$		0			0
$231$		−	241.982i		−	1.04754i
$232$		0			0
$233$			249.234i			1.06967i	0.844955	+	0.534837i	$0.179627\pi$
$233$			249.234i			1.06967i	−0.844955	+	0.534837i	$0.820373\pi$
$234$		0			0
$235$	149.830	−	39.5196i	0.637576	−	0.168169i
$236$		0			0
$237$	−260.650			−1.09979
$238$		0			0
$239$		−	272.973i		−	1.14215i	−0.820900	−	0.571073i	$-0.806527\pi$
$239$		−	272.973i		−	1.14215i	0.820900	−	0.571073i	$-0.193473\pi$
$240$		0			0
$241$	−315.945			−1.31098			−0.655488	−	0.755205i	$-0.727537\pi$
$241$	−315.945			−1.31098			−0.655488	+	0.755205i	$0.727537\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$	41.0726	−	10.8334i	0.167643	−	0.0442180i
$246$		0			0
$247$	272.973			1.10515
$248$		0			0
$249$	−69.4955			−0.279098
$250$		0			0
$251$	125.781			0.501118			0.250559	−	0.968101i	$-0.419386\pi$
$251$	125.781			0.501118			0.250559	+	0.968101i	$0.419386\pi$
$252$		0			0
$253$	442.198			1.74782
$254$		0			0
$255$	−240.730	+	63.4955i	−0.944037	+	0.249002i
$256$		0			0
$257$		−	149.270i		−	0.580819i	−0.956902	−	0.290409i	$-0.906208\pi$
$257$		−	149.270i		−	0.580819i	0.956902	−	0.290409i	$-0.0937915\pi$
$258$		0			0
$259$	141.154			0.544997
$260$		0			0
$261$		−	54.7026i		−	0.209589i
$262$		0			0
$263$	96.5227			0.367007			0.183503	−	0.983019i	$-0.441256\pi$
$263$	96.5227			0.367007			0.183503	+	0.983019i	$0.441256\pi$
$264$		0			0
$265$	53.7386	−	14.1742i	0.202787	−	0.0534877i
$266$		0			0
$267$			159.317i			0.596694i
$268$		0			0
$269$		−	361.752i		−	1.34480i	−0.740186	−	0.672402i	$-0.765263\pi$
$269$		−	361.752i		−	1.34480i	0.740186	−	0.672402i	$-0.234737\pi$
$270$		0			0
$271$		−	145.477i		−	0.536816i	−0.963305	−	0.268408i	$-0.913502\pi$
$271$		−	145.477i		−	0.536816i	0.963305	−	0.268408i	$-0.0864976\pi$
$272$		0			0
$273$			172.486i			0.631818i
$274$		0			0
$275$	−400.700	+	227.185i	−1.45709	+	0.826126i
$276$		0			0
$277$	−102.659			−0.370612			−0.185306	−	0.982681i	$-0.559328\pi$
$277$	−102.659			−0.370612			−0.185306	+	0.982681i	$0.559328\pi$
$278$		0			0
$279$		−	94.4864i		−	0.338661i
$280$		0			0
$281$	−107.009			−0.380815			−0.190408	−	0.981705i	$-0.560981\pi$
$281$	−107.009			−0.380815			−0.190408	+	0.981705i	$0.560981\pi$
$282$		0			0
$283$		−	239.834i		−	0.847470i	−0.905786	−	0.423735i	$-0.860719\pi$
$283$		−	239.834i		−	0.847470i	0.905786	−	0.423735i	$-0.139281\pi$
$284$		0			0
$285$	45.9072	+	174.047i	0.161078	+	0.610693i
$286$		0			0
$287$	553.459			1.92843
$288$		0			0
$289$	−537.432			−1.85963
$290$		0			0
$291$	68.9802			0.237045
$292$		0			0
$293$	−11.6559			−0.0397813			−0.0198906	−	0.999802i	$-0.506332\pi$
$293$	−11.6559			−0.0397813			−0.0198906	+	0.999802i	$0.506332\pi$
$294$		0			0
$295$	−136.555	+	36.0182i	−0.462899	+	0.122096i
$296$		0			0
$297$			95.7386i			0.322352i
$298$		0			0
$299$	−315.202			−1.05419
$300$		0			0
$301$		−	41.5692i		−	0.138104i
$302$		0			0
$303$	57.4265			0.189526
$304$		0			0
$305$	−147.964	−	560.973i	−0.485127	−	1.83925i
$306$		0			0
$307$		−	289.840i		−	0.944105i	−0.881570	−	0.472053i	$-0.843513\pi$
$307$		−	289.840i		−	0.944105i	0.881570	−	0.472053i	$-0.156487\pi$
$308$		0			0
$309$			229.894i			0.743995i
$310$		0			0
$311$		−	37.9818i		−	0.122128i	−0.998134	−	0.0610640i	$-0.980551\pi$
$311$		−	37.9818i		−	0.122128i	0.998134	−	0.0610640i	$-0.0194494\pi$
$312$		0			0
$313$		−	326.120i		−	1.04192i	−0.853582	−	0.520958i	$-0.825575\pi$
$313$		−	326.120i		−	1.04192i	0.853582	−	0.520958i	$-0.174425\pi$
$314$		0			0
$315$	−109.977	+	29.0079i	−0.349134	+	0.0920886i
$316$		0			0
$317$	−264.869			−0.835548			−0.417774	−	0.908551i	$-0.637190\pi$
$317$	−264.869			−0.835548			−0.417774	+	0.908551i	$0.637190\pi$
$318$		0			0
$319$		−	335.964i		−	1.05318i
$320$		0			0
$321$	−159.964			−0.498329
$322$		0			0
$323$			597.510i			1.84988i
$324$		0			0
$325$	285.622	−	161.939i	0.878836	−	0.498273i
$326$		0			0
$327$	−208.486			−0.637573
$328$		0			0
$329$	234.991			0.714258
$330$		0			0
$331$	−77.8950			−0.235332			−0.117666	−	0.993053i	$-0.537541\pi$
$331$	−77.8950			−0.235332			−0.117666	+	0.993053i	$0.537541\pi$
$332$		0			0
$333$	−55.8468			−0.167708
$334$		0			0
$335$	−126.991	−	481.459i	−0.379077	−	1.43719i
$336$		0			0
$337$			77.9455i			0.231292i	0.993290	+	0.115646i	$0.0368938\pi$
$337$			77.9455i			0.231292i	−0.993290	+	0.115646i	$0.963106\pi$
$338$		0			0
$339$	−190.947			−0.563265
$340$		0			0
$341$		−	580.301i		−	1.70176i
$342$		0			0
$343$	−307.129			−0.895419
$344$		0			0
$345$	−53.0091	−	200.973i	−0.153650	−	0.582530i
$346$		0			0
$347$		−	342.676i		−	0.987538i	−0.869593	−	0.493769i	$-0.835619\pi$
$347$		−	342.676i		−	0.987538i	0.869593	−	0.493769i	$-0.164381\pi$
$348$		0			0
$349$			365.447i			1.04713i	0.851987	+	0.523563i	$0.175398\pi$
$349$			365.447i			1.04713i	−0.851987	+	0.523563i	$0.824602\pi$
$350$		0			0
$351$		−	68.2432i		−	0.194425i
$352$		0			0
$353$		−	183.289i		−	0.519231i	−0.965712	−	0.259616i	$-0.916404\pi$
$353$		−	183.289i		−	0.519231i	0.965712	−	0.259616i	$-0.0835959\pi$
$354$		0			0
$355$	70.1244	+	265.862i	0.197534	+	0.748907i
$356$		0			0
$357$	−377.556			−1.05758
$358$		0			0
$359$		−	509.945i		−	1.42046i	−0.703969	−	0.710230i	$-0.748591\pi$
$359$		−	509.945i		−	1.42046i	0.703969	−	0.710230i	$-0.251409\pi$
$360$		0			0
$361$	71.0000			0.196676
$362$		0			0
$363$			378.414i			1.04246i
$364$		0			0
$365$	99.0444	+	375.506i	0.271355	+	1.02878i
$366$		0			0
$367$	30.1235			0.0820803			0.0410402	−	0.999157i	$-0.486933\pi$
$367$	30.1235			0.0820803			0.0410402	+	0.999157i	$0.486933\pi$
$368$		0			0
$369$	−218.973			−0.593422
$370$		0			0
$371$	84.2826			0.227177
$372$		0			0
$373$	602.873			1.61628			0.808141	−	0.588989i	$-0.200474\pi$
$373$	602.873			1.61628			0.808141	+	0.588989i	$0.200474\pi$
$374$		0			0
$375$	151.287	+	154.878i	0.403431	+	0.413009i
$376$		0			0
$377$			239.477i			0.635218i
$378$		0			0
$379$	336.892			0.888896			0.444448	−	0.895805i	$-0.353400\pi$
$379$	336.892			0.888896			0.444448	+	0.895805i	$0.353400\pi$
$380$		0			0
$381$		−	295.442i		−	0.775438i
$382$		0			0
$383$	270.468			0.706183			0.353092	−	0.935589i	$-0.385130\pi$
$383$	270.468			0.706183			0.353092	+	0.935589i	$0.385130\pi$
$384$		0			0
$385$	−675.441	+	178.156i	−1.75439	+	0.462743i
$386$		0			0
$387$			16.4466i			0.0424977i
$388$		0			0
$389$			721.956i			1.85593i	0.372670	+	0.927964i	$0.378442\pi$
$389$			721.956i			1.85593i	−0.372670	+	0.927964i	$0.621558\pi$
$390$		0			0
$391$		−	689.945i		−	1.76457i
$392$		0			0
$393$		−	186.069i		−	0.473458i
$394$		0			0
$395$	191.900	+	727.549i	0.485824	+	1.84190i
$396$		0			0
$397$	−486.841			−1.22630			−0.613150	−	0.789966i	$-0.710098\pi$
$397$	−486.841			−1.22630			−0.613150	+	0.789966i	$0.710098\pi$
$398$		0			0
$399$			272.973i			0.684142i
$400$		0			0
$401$	−688.955			−1.71809			−0.859046	−	0.511899i	$-0.828942\pi$
$401$	−688.955			−1.71809			−0.859046	+	0.511899i	$0.828942\pi$
$402$		0			0
$403$			413.643i			1.02641i
$404$		0			0
$405$	43.5119	−	11.4768i	0.107437	−	0.0283378i
$406$		0			0
$407$	−342.991			−0.842730
$408$		0			0
$409$	412.468			1.00848			0.504240	−	0.863564i	$-0.331773\pi$
$409$	412.468			1.00848			0.504240	+	0.863564i	$0.331773\pi$
$410$		0			0
$411$	215.164			0.523513
$412$		0			0
$413$	−214.171			−0.518573
$414$		0			0
$415$	51.1652	+	193.982i	0.123290	+	0.467426i
$416$		0			0
$417$		−	193.459i		−	0.463931i
$418$		0			0
$419$	189.279			0.451739			0.225869	−	0.974158i	$-0.427478\pi$
$419$	189.279			0.451739			0.225869	+	0.974158i	$0.427478\pi$
$420$		0			0
$421$			593.172i			1.40896i	0.709724	+	0.704480i	$0.248820\pi$
$421$			593.172i			1.40896i	−0.709724	+	0.704480i	$0.751180\pi$
$422$		0			0
$423$	−92.9727			−0.219794
$424$		0			0
$425$	354.468	+	625.198i	0.834043	+	1.47105i
$426$		0			0
$427$		−	879.819i		−	2.06047i
$428$		0			0
$429$		−	419.125i		−	0.976981i
$430$		0			0
$431$			411.027i			0.953660i	0.878996	+	0.476830i	$0.158214\pi$
$431$			411.027i			0.953660i	−0.878996	+	0.476830i	$0.841786\pi$
$432$		0			0
$433$			159.303i			0.367905i	0.982935	+	0.183953i	$0.0588893\pi$
$433$			159.303i			0.367905i	−0.982935	+	0.183953i	$0.941111\pi$
$434$		0			0
$435$	−152.691	+	40.2741i	−0.351013	+	0.0925842i
$436$		0			0
$437$	−498.831			−1.14149
$438$		0			0
$439$			409.441i			0.932667i	0.884609	+	0.466334i	$0.154425\pi$
$439$			409.441i			0.932667i	−0.884609	+	0.466334i	$0.845575\pi$
$440$		0			0
$441$	−25.4864			−0.0577922
$442$		0			0
$443$			13.0140i			0.0293770i	0.999892	+	0.0146885i	$0.00467566\pi$
$443$			13.0140i			0.0293770i	−0.999892	+	0.0146885i	$0.995324\pi$
$444$		0			0
$445$	444.700	−	117.295i	0.999326	−	0.263585i
$446$		0			0
$447$	103.583			0.231728
$448$		0			0
$449$	450.000			1.00223			0.501114	−	0.865382i	$-0.332924\pi$
$449$	450.000			1.00223			0.501114	+	0.865382i	$0.332924\pi$
$450$		0			0
$451$	−1344.85			−2.98193
$452$		0			0
$453$	111.725			0.246634
$454$		0			0
$455$	481.459	−	126.991i	1.05815	−	0.279101i
$456$		0			0
$457$		−	642.744i		−	1.40644i	−0.710971	−	0.703221i	$-0.751744\pi$
$457$		−	642.744i		−	1.40644i	0.710971	−	0.703221i	$-0.248256\pi$
$458$		0			0
$459$	149.378			0.325441
$460$		0			0
$461$		−	534.894i		−	1.16029i	−0.814513	−	0.580146i	$-0.802996\pi$
$461$		−	534.894i		−	1.16029i	0.814513	−	0.580146i	$-0.197004\pi$
$462$		0			0
$463$	455.023			0.982772			0.491386	−	0.870942i	$-0.336490\pi$
$463$	455.023			0.982772			0.491386	+	0.870942i	$0.336490\pi$
$464$		0			0
$465$	−263.739	+	69.5644i	−0.567180	+	0.149601i
$466$		0			0
$467$		−	580.523i		−	1.24309i	−0.783378	−	0.621545i	$-0.786505\pi$
$467$		−	580.523i		−	1.24309i	0.783378	−	0.621545i	$-0.213495\pi$
$468$		0			0
$469$		−	755.111i		−	1.61005i
$470$		0			0
$471$			127.198i			0.270059i
$472$		0			0
$473$			101.009i			0.213550i
$474$		0			0
$475$	452.018	−	256.281i	0.951617	−	0.539538i
$476$		0			0
$477$	−33.3459			−0.0699076
$478$		0			0
$479$		−	320.036i		−	0.668134i	−0.942549	−	0.334067i	$-0.891579\pi$
$479$		−	320.036i		−	0.668134i	0.942549	−	0.334067i	$-0.108421\pi$
$480$		0			0
$481$	244.486			0.508288
$482$		0			0
$483$		−	315.202i		−	0.652592i
$484$		0			0
$485$	−50.7858	−	192.544i	−0.104713	−	0.396997i
$486$		0			0
$487$	147.757			0.303402			0.151701	−	0.988426i	$-0.451525\pi$
$487$	147.757			0.303402			0.151701	+	0.988426i	$0.451525\pi$
$488$		0			0
$489$	274.955			0.562279
$490$		0			0
$491$	−207.775			−0.423167			−0.211583	−	0.977360i	$-0.567862\pi$
$491$	−207.775			−0.423167			−0.211583	+	0.977360i	$0.567862\pi$
$492$		0			0
$493$	−524.192			−1.06327
$494$		0			0
$495$	267.234	−	70.4864i	0.539867	−	0.142397i
$496$		0			0
$497$			416.973i			0.838979i
$498$		0			0
$499$	77.8950			0.156102			0.0780511	−	0.996949i	$-0.475130\pi$
$499$	77.8950			0.156102			0.0780511	+	0.996949i	$0.475130\pi$
$500$		0			0
$501$		−	206.941i		−	0.413055i
$502$		0			0
$503$	−243.027			−0.483156			−0.241578	−	0.970381i	$-0.577665\pi$
$503$	−243.027			−0.483156			−0.241578	+	0.970381i	$0.577665\pi$
$504$		0			0
$505$	−42.2795	−	160.294i	−0.0837219	−	0.317414i
$506$		0			0
$507$			6.03856i			0.0119104i
$508$		0			0
$509$		−	411.331i		−	0.808116i	−0.914733	−	0.404058i	$-0.867599\pi$
$509$		−	411.331i		−	0.808116i	0.914733	−	0.404058i	$-0.132401\pi$
$510$		0			0
$511$			588.936i			1.15252i
$512$		0			0
$513$		−	108.000i		−	0.210526i
$514$		0			0
$515$	641.701	−	169.257i	1.24602	−	0.328654i
$516$		0			0
$517$	−571.005			−1.10446
$518$		0			0
$519$		−	81.2341i		−	0.156520i
$520$		0			0
$521$	−247.045			−0.474176			−0.237088	−	0.971488i	$-0.576193\pi$
$521$	−247.045			−0.474176			−0.237088	+	0.971488i	$0.576193\pi$
$522$		0			0
$523$		−	465.271i		−	0.889619i	−0.895625	−	0.444810i	$-0.853271\pi$
$523$		−	465.271i		−	0.889619i	0.895625	−	0.444810i	$-0.146729\pi$
$524$		0			0
$525$	161.939	+	285.622i	0.308455	+	0.544041i
$526$		0			0
$527$	−905.423			−1.71807
$528$		0			0
$529$	47.0000			0.0888469
$530$		0			0
$531$	84.7353			0.159577
$532$		0			0
$533$	958.619			1.79854
$534$		0			0
$535$	117.771	+	446.505i	0.220133	+	0.834588i
$536$		0			0
$537$			249.895i			0.465353i
$538$		0			0
$539$	−156.528			−0.290404
$540$		0			0
$541$			494.493i			0.914034i	0.889458	+	0.457017i	$0.151082\pi$
$541$			494.493i			0.914034i	−0.889458	+	0.457017i	$0.848918\pi$
$542$		0			0
$543$	−295.514			−0.544224
$544$		0			0
$545$	153.495	+	581.945i	0.281643	+	1.06779i
$546$		0			0
$547$			84.5215i			0.154518i	0.997011	+	0.0772591i	$0.0246169\pi$
$547$			84.5215i			0.154518i	−0.997011	+	0.0772591i	$0.975383\pi$
$548$		0			0
$549$			348.095i			0.634053i
$550$		0			0
$551$			378.991i			0.687824i
$552$		0			0
$553$			1141.07i			2.06343i
$554$		0			0
$555$	41.1165	+	155.885i	0.0740838	+	0.280873i
$556$		0			0
$557$	−293.424			−0.526793			−0.263397	−	0.964688i	$-0.584843\pi$
$557$	−293.424			−0.526793			−0.263397	+	0.964688i	$0.584843\pi$
$558$		0			0
$559$		−	72.0000i		−	0.128801i
$560$		0			0
$561$	917.423			1.63533
$562$		0			0
$563$		−	67.2324i		−	0.119418i	−0.998216	−	0.0597091i	$-0.980983\pi$
$563$		−	67.2324i		−	0.119418i	0.998216	−	0.0597091i	$-0.0190173\pi$
$564$		0			0
$565$	140.582	+	532.987i	0.248818	+	0.943341i
$566$		0			0
$567$	68.2432			0.120358
$568$		0			0
$569$	815.945			1.43400			0.717000	−	0.697074i	$-0.245515\pi$
$569$	815.945			1.43400			0.717000	+	0.697074i	$0.245515\pi$
$570$		0			0
$571$	13.9194			0.0243772			0.0121886	−	0.999926i	$-0.496120\pi$
$571$	13.9194			0.0243772			0.0121886	+	0.999926i	$0.496120\pi$
$572$		0			0
$573$	−91.8144			−0.160235
$574$		0			0
$575$	−521.945	+	295.927i	−0.907731	+	0.514656i
$576$		0			0
$577$		−	811.405i		−	1.40625i	−0.711068	−	0.703124i	$-0.751788\pi$
$577$		−	811.405i		−	1.40625i	0.711068	−	0.703124i	$-0.248212\pi$
$578$		0			0
$579$	472.564			0.816172
$580$		0			0
$581$			304.237i			0.523644i
$582$		0			0
$583$	−204.798			−0.351284
$584$		0			0
$585$	−190.486	+	50.2432i	−0.325618	+	0.0858858i
$586$		0			0
$587$		−	338.878i		−	0.577306i	−0.957434	−	0.288653i	$-0.906793\pi$
$587$		−	338.878i		−	0.577306i	0.957434	−	0.288653i	$-0.0932073\pi$
$588$		0			0
$589$			654.621i			1.11141i
$590$		0			0
$591$		−	577.720i		−	0.977530i
$592$		0			0
$593$		−	932.602i		−	1.57269i	−0.617791	−	0.786343i	$-0.711972\pi$
$593$		−	932.602i		−	1.57269i	0.617791	−	0.786343i	$-0.288028\pi$
$594$		0			0
$595$	277.971	+	1053.87i	0.467177	+	1.77120i
$596$		0			0
$597$	127.266			0.213176
$598$		0			0
$599$			513.909i			0.857945i	0.903318	+	0.428973i	$0.141124\pi$
$599$			513.909i			0.857945i	−0.903318	+	0.428973i	$0.858876\pi$
$600$		0			0
$601$	538.450			0.895923			0.447962	−	0.894053i	$-0.352150\pi$
$601$	538.450			0.895923			0.447962	+	0.894053i	$0.352150\pi$
$602$		0			0
$603$			298.755i			0.495448i
$604$		0			0
$605$	1056.26	−	278.602i	1.74589	−	0.460500i
$606$		0			0
$607$	−1011.76			−1.66682			−0.833408	−	0.552659i	$-0.813613\pi$
$607$	−1011.76			−1.66682			−0.833408	+	0.552659i	$0.813613\pi$
$608$		0			0
$609$	−239.477			−0.393230
$610$		0			0
$611$	407.016			0.666148
$612$		0			0
$613$	690.350			1.12618			0.563091	−	0.826395i	$-0.309612\pi$
$613$	690.350			1.12618			0.563091	+	0.826395i	$0.309612\pi$
$614$		0			0
$615$	161.216	+	611.216i	0.262140	+	0.993847i
$616$		0			0
$617$		−	1153.09i		−	1.86886i	−0.356143	−	0.934432i	$-0.615908\pi$
$617$		−	1153.09i		−	1.86886i	0.356143	−	0.934432i	$-0.384092\pi$
$618$		0			0
$619$	−70.1244			−0.113287			−0.0566433	−	0.998394i	$-0.518040\pi$
$619$	−70.1244			−0.113287			−0.0566433	+	0.998394i	$0.518040\pi$
$620$		0			0
$621$			124.708i			0.200817i
$622$		0			0
$623$	697.459			1.11952
$624$		0			0
$625$	320.927	−	536.312i	0.513484	−	0.858099i
$626$		0			0
$627$		−	663.297i		−	1.05789i
$628$		0			0
$629$			535.156i			0.850805i
$630$		0			0
$631$			445.368i			0.705813i	0.935659	+	0.352907i	$0.114807\pi$
$631$			445.368i			0.705813i	−0.935659	+	0.352907i	$0.885193\pi$
$632$		0			0
$633$			496.486i			0.784339i
$634$		0			0
$635$	−824.663	+	217.515i	−1.29868	+	0.342544i
$636$		0			0
$637$	111.574			0.175156
$638$		0			0
$639$		−	164.973i		−	0.258173i
$640$		0			0
$641$	−410.036			−0.639682			−0.319841	−	0.947471i	$-0.603630\pi$
$641$	−410.036			−0.639682			−0.319841	+	0.947471i	$0.603630\pi$
$642$		0			0
$643$			843.971i			1.31255i	0.754521	+	0.656276i	$0.227869\pi$
$643$			843.971i			1.31255i	−0.754521	+	0.656276i	$0.772131\pi$
$644$		0			0
$645$	45.9072	−	12.1086i	0.0711740	−	0.0187730i
$646$		0			0
$647$	979.927			1.51457			0.757285	−	0.653084i	$-0.226525\pi$
$647$	979.927			1.51457			0.757285	+	0.653084i	$0.226525\pi$
$648$		0			0
$649$	520.414			0.801870
$650$		0			0
$651$	−413.643			−0.635396
$652$		0			0
$653$	−350.899			−0.537365			−0.268682	−	0.963229i	$-0.586588\pi$
$653$	−350.899			−0.537365			−0.268682	+	0.963229i	$0.586588\pi$
$654$		0			0
$655$	−519.372	+	136.991i	−0.792934	+	0.209146i
$656$		0			0
$657$		−	233.009i		−	0.354656i
$658$		0			0
$659$	−659.126			−1.00019			−0.500096	−	0.865970i	$-0.666702\pi$
$659$	−659.126			−1.00019			−0.500096	+	0.865970i	$0.666702\pi$
$660$		0			0
$661$		−	988.985i		−	1.49620i	−0.663589	−	0.748098i	$-0.730967\pi$
$661$		−	988.985i		−	1.49620i	0.663589	−	0.748098i	$-0.269033\pi$
$662$		0			0
$663$	−653.945			−0.986343
$664$		0			0
$665$	761.945	−	200.973i	1.14578	−	0.302215i
$666$		0			0
$667$		−	437.621i		−	0.656103i
$668$		0			0
$669$		−	105.187i		−	0.157230i
$670$		0			0
$671$			2137.87i			3.18610i
$672$		0			0
$673$			981.633i			1.45859i	0.684198	+	0.729297i	$0.260153\pi$
$673$			981.633i			1.45859i	−0.684198	+	0.729297i	$0.739847\pi$
$674$		0			0
$675$	−64.0701	−	113.005i	−0.0949187	−	0.167414i
$676$		0			0
$677$	−204.866			−0.302609			−0.151304	−	0.988487i	$-0.548347\pi$
$677$	−204.866			−0.302609			−0.151304	+	0.988487i	$0.548347\pi$
$678$		0			0
$679$		−	301.982i		−	0.444745i
$680$		0			0
$681$	568.486			0.834782
$682$		0			0
$683$			177.480i			0.259854i	0.991524	+	0.129927i	$0.0414743\pi$
$683$			177.480i			0.259854i	−0.991524	+	0.129927i	$0.958526\pi$
$684$		0			0
$685$	−158.412	−	600.585i	−0.231258	−	0.876766i
$686$		0			0
$687$	178.432			0.259726
$688$		0			0
$689$	145.982			0.211875
$690$		0			0
$691$	1122.18			1.62399			0.811997	−	0.583662i	$-0.198380\pi$
$691$	1122.18			1.62399			0.811997	+	0.583662i	$0.198380\pi$
$692$		0			0
$693$	419.125			0.604798
$694$		0			0
$695$	−540.000	+	142.432i	−0.776978	+	0.204938i
$696$		0			0
$697$			2098.32i			3.01051i
$698$		0			0
$699$	−431.686			−0.617577
$700$		0			0
$701$			159.627i			0.227714i	0.993497	+	0.113857i	$0.0363205\pi$
$701$			159.627i			0.227714i	−0.993497	+	0.113857i	$0.963679\pi$
$702$		0			0
$703$	386.918			0.550381
$704$		0			0
$705$	68.4500	+	259.514i	0.0970922	+	0.368104i
$706$		0			0
$707$		−	251.402i		−	0.355590i
$708$		0			0
$709$		−	917.050i		−	1.29344i	−0.762727	−	0.646721i	$-0.776140\pi$
$709$		−	917.050i		−	1.29344i	0.762727	−	0.646721i	$-0.223860\pi$
$710$		0			0
$711$		−	451.459i		−	0.634964i
$712$		0			0
$713$		−	755.891i		−	1.06016i
$714$		0			0
$715$	−1169.90	+	308.575i	−1.63622	+	0.431574i
$716$		0			0
$717$	472.803			0.659418
$718$		0			0
$719$			134.918i			0.187647i	0.995589	+	0.0938235i	$0.0299089\pi$
$719$			134.918i			0.187647i	−0.995589	+	0.0938235i	$0.970091\pi$
$720$		0			0
$721$	1006.43			1.39588
$722$		0			0
$723$		−	547.234i		−	0.756893i
$724$		0			0
$725$	224.833	+	396.553i	0.310115	+	0.546969i
$726$		0			0
$727$	235.336			0.323708			0.161854	−	0.986815i	$-0.448253\pi$
$727$	235.336			0.323708			0.161854	+	0.986815i	$0.448253\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$	157.601			0.215596
$732$		0			0
$733$	−1089.83			−1.48681			−0.743406	−	0.668840i	$-0.766791\pi$
$733$	−1089.83			−1.48681			−0.743406	+	0.668840i	$0.766791\pi$
$734$		0			0
$735$	18.7640	+	71.1398i	0.0255293	+	0.0967888i
$736$		0			0
$737$			1834.85i			2.48961i
$738$		0			0
$739$	−602.754			−0.815634			−0.407817	−	0.913064i	$-0.633710\pi$
$739$	−602.754			−0.815634			−0.407817	+	0.913064i	$0.633710\pi$
$740$		0			0
$741$			472.803i			0.638060i
$742$		0			0
$743$	−93.4955			−0.125835			−0.0629175	−	0.998019i	$-0.520041\pi$
$743$	−93.4955			−0.125835			−0.0629175	+	0.998019i	$0.520041\pi$
$744$		0			0
$745$	−76.2614	−	289.129i	−0.102364	−	0.388092i
$746$		0			0
$747$		−	120.370i		−	0.161137i
$748$		0			0
$749$			700.289i			0.934965i
$750$		0			0
$751$		−	658.450i		−	0.876764i	−0.898789	−	0.438382i	$-0.855552\pi$
$751$		−	658.450i		−	0.876764i	0.898789	−	0.438382i	$-0.144448\pi$
$752$		0			0
$753$			217.858i			0.289320i
$754$		0			0
$755$	−82.2562	−	311.857i	−0.108949	−	0.413056i
$756$		0			0
$757$	−1135.98			−1.50063			−0.750317	−	0.661078i	$-0.770099\pi$
$757$	−1135.98			−1.50063			−0.750317	+	0.661078i	$0.770099\pi$
$758$		0			0
$759$			765.909i			1.00910i
$760$		0			0
$761$	−535.045			−0.703082			−0.351541	−	0.936172i	$-0.614342\pi$
$761$	−535.045			−0.703082			−0.351541	+	0.936172i	$0.614342\pi$
$762$		0			0
$763$			912.712i			1.19621i
$764$		0			0
$765$	−109.977	−	416.956i	−0.143761	−	0.545040i
$766$		0			0
$767$	−370.955			−0.483643
$768$		0			0
$769$	505.441			0.657270			0.328635	−	0.944457i	$-0.393411\pi$
$769$	505.441			0.657270			0.328635	+	0.944457i	$0.393411\pi$
$770$		0			0
$771$	258.544			0.335336
$772$		0			0
$773$	280.234			0.362528			0.181264	−	0.983434i	$-0.441981\pi$
$773$	280.234			0.362528			0.181264	+	0.983434i	$0.441981\pi$
$774$		0			0
$775$	388.348	+	684.955i	0.501095	+	0.883812i
$776$		0			0
$777$			244.486i			0.314654i
$778$		0			0
$779$	1517.09			1.94748
$780$		0			0
$781$		−	1013.20i		−	1.29731i
$782$		0			0
$783$	94.7477			0.121006
$784$		0			0
$785$	355.045	−	93.6477i	0.452287	−	0.119296i
$786$		0			0
$787$			1235.92i			1.57042i	0.619227	+	0.785212i	$0.287446\pi$
$787$			1235.92i			1.57042i	−0.619227	+	0.785212i	$0.712554\pi$
$788$		0			0
$789$			167.182i			0.211891i
$790$		0			0
$791$			835.927i			1.05680i
$792$		0			0
$793$		−	1523.89i		−	1.92168i
$794$		0			0
$795$	24.5505	+	93.0780i	0.0308811	+	0.117079i
$796$		0			0
$797$	−134.918			−0.169282			−0.0846409	−	0.996412i	$-0.526974\pi$
$797$	−134.918			−0.169282			−0.0846409	+	0.996412i	$0.526974\pi$
$798$		0			0
$799$			890.918i			1.11504i
$800$		0			0
$801$	−275.945			−0.344501
$802$		0			0
$803$		−	1431.06i		−	1.78214i
$804$		0			0
$805$	−879.819	+	232.063i	−1.09294	+	0.288277i
$806$		0			0
$807$	626.573			0.776423
$808$		0			0
$809$	−402.936			−0.498067			−0.249034	−	0.968495i	$-0.580113\pi$
$809$	−402.936			−0.498067			−0.249034	+	0.968495i	$0.580113\pi$
$810$		0			0
$811$	1263.33			1.55775			0.778874	−	0.627180i	$-0.215791\pi$
$811$	1263.33			1.55775			0.778874	+	0.627180i	$0.215791\pi$
$812$		0			0
$813$	251.974			0.309931
$814$		0			0
$815$	−202.432	−	767.477i	−0.248383	−	0.941690i
$816$		0			0
$817$		−	113.945i		−	0.139468i
$818$		0			0
$819$	−298.755			−0.364780
$820$		0			0
$821$			1074.82i			1.30916i	0.755995	+	0.654578i	$0.227154\pi$
$821$			1074.82i			1.30916i	−0.755995	+	0.654578i	$0.772846\pi$
$822$		0			0
$823$	−557.078			−0.676887			−0.338444	−	0.940987i	$-0.609900\pi$
$823$	−557.078			−0.676887			−0.338444	+	0.940987i	$0.609900\pi$
$824$		0			0
$825$	−393.495	−	694.033i	−0.476964	−	0.841252i
$826$		0			0
$827$			1139.53i			1.37791i	0.724804	+	0.688955i	$0.241930\pi$
$827$			1139.53i			1.37791i	−0.724804	+	0.688955i	$0.758070\pi$
$828$		0			0
$829$			201.697i			0.243302i	0.992573	+	0.121651i	$0.0388189\pi$
$829$			201.697i			0.243302i	−0.992573	+	0.121651i	$0.961181\pi$
$830$		0			0
$831$		−	177.811i		−	0.213973i
$832$		0			0
$833$			244.225i			0.293187i
$834$		0			0
$835$	−577.631	+	152.357i	−0.691774	+	0.182464i
$836$		0			0
$837$	163.655			0.195526
$838$		0			0
$839$		−	661.873i		−	0.788883i	−0.918921	−	0.394441i	$-0.870938\pi$
$839$		−	661.873i		−	0.788883i	0.918921	−	0.394441i	$-0.129062\pi$
$840$		0			0
$841$	508.514			0.604654
$842$		0			0
$843$		−	185.345i		−	0.219864i
$844$		0			0
$845$	16.8554	−	4.44581i	0.0199472	−	0.00526132i
$846$		0			0
$847$	1656.62			1.95587
$848$		0			0
$849$	415.405			0.489287
$850$		0			0
$851$	−446.775			−0.525000
$852$		0			0
$853$	−1008.56			−1.18236			−0.591182	−	0.806538i	$-0.701338\pi$
$853$	−1008.56			−1.18236			−0.591182	+	0.806538i	$0.701338\pi$
$854$		0			0
$855$	−301.459	+	79.5136i	−0.352584	+	0.0929984i
$856$		0			0
$857$		−	197.793i		−	0.230797i	−0.993319	−	0.115399i	$-0.963185\pi$
$857$		−	197.793i		−	0.230797i	0.993319	−	0.115399i	$-0.0368145\pi$
$858$		0			0
$859$	61.4484			0.0715348			0.0357674	−	0.999360i	$-0.488612\pi$
$859$	61.4484			0.0715348			0.0357674	+	0.999360i	$0.488612\pi$
$860$		0			0
$861$			958.619i			1.11338i
$862$		0			0
$863$	−800.450			−0.927520			−0.463760	−	0.885961i	$-0.653500\pi$
$863$	−800.450			−0.927520			−0.463760	+	0.885961i	$0.653500\pi$
$864$		0			0
$865$	−226.748	+	59.8076i	−0.262136	+	0.0691417i
$866$		0			0
$867$		−	930.859i		−	1.07366i
$868$		0			0
$869$		−	2772.70i		−	3.19068i
$870$		0			0
$871$		−	1307.89i		−	1.50160i
$872$		0			0
$873$			119.477i			0.136858i
$874$		0			0
$875$	678.027	−	662.303i	0.774888	−	0.756918i
$876$		0			0
$877$	−463.341			−0.528325			−0.264162	−	0.964478i	$-0.585095\pi$
$877$	−463.341			−0.528325			−0.264162	+	0.964478i	$0.585095\pi$
$878$		0			0
$879$		−	20.1886i		−	0.0229677i
$880$		0			0
$881$	−719.009			−0.816128			−0.408064	−	0.912953i	$-0.633796\pi$
$881$	−719.009			−0.816128			−0.408064	+	0.912953i	$0.633796\pi$
$882$		0			0
$883$			1014.82i			1.14929i	0.818402	+	0.574646i	$0.194860\pi$
$883$			1014.82i			1.14929i	−0.818402	+	0.574646i	$0.805140\pi$
$884$		0			0
$885$	−62.3853	−	236.521i	−0.0704919	−	0.267255i
$886$		0			0
$887$	−284.559			−0.320811			−0.160405	−	0.987051i	$-0.551280\pi$
$887$	−284.559			−0.320811			−0.160405	+	0.987051i	$0.551280\pi$
$888$		0			0
$889$	−1293.39			−1.45488
$890$		0			0
$891$	−165.824			−0.186110
$892$		0			0
$893$	644.134			0.721315
$894$		0			0
$895$	697.528	−	183.982i	0.779361	−	0.205566i
$896$		0			0
$897$		−	545.945i		−	0.608635i
$898$		0			0
$899$	−574.295			−0.638815
$900$		0			0
$901$			319.540i			0.354650i
$902$		0			0
$903$	72.0000			0.0797342
$904$		0			0
$905$	217.568	+	824.864i	0.240407	+	0.911452i
$906$		0			0
$907$		−	93.1975i		−	0.102754i	−0.998679	−	0.0513768i	$-0.983639\pi$
$907$		−	93.1975i		−	0.102754i	0.998679	−	0.0513768i	$-0.0163609\pi$
$908$		0			0
$909$			99.4656i			0.109423i
$910$		0			0
$911$			932.036i			1.02309i	0.859256	+	0.511546i	$0.170927\pi$
$911$			932.036i			1.02309i	−0.859256	+	0.511546i	$0.829073\pi$
$912$		0			0
$913$		−	739.267i		−	0.809712i
$914$		0			0
$915$	971.633	−	256.281i	1.06189	−	0.280088i
$916$		0			0
$917$	−814.573			−0.888302
$918$		0			0
$919$			1249.44i			1.35957i	0.733413	+	0.679783i	$0.237926\pi$
$919$			1249.44i			1.35957i	−0.733413	+	0.679783i	$0.762074\pi$
$920$		0			0
$921$	502.018			0.545079
$922$		0			0
$923$			722.218i			0.782468i
$924$		0			0
$925$	404.847	−	229.536i	0.437673	−	0.248147i
$926$		0			0
$927$	−398.189			−0.429545
$928$		0			0
$929$	1488.88			1.60267			0.801336	−	0.598215i	$-0.204123\pi$
$929$	1488.88			1.60267			0.801336	+	0.598215i	$0.204123\pi$
$930$		0			0
$931$	176.575			0.189661
$932$		0			0
$933$	65.7864			0.0705106
$934$		0			0
$935$	−675.441	−	2560.79i	−0.722397	−	2.73881i
$936$		0			0
$937$		−	689.670i		−	0.736040i	−0.929818	−	0.368020i	$-0.880036\pi$
$937$		−	689.670i		−	0.736040i	0.929818	−	0.368020i	$-0.119964\pi$
$938$		0			0
$939$	564.856			0.601550
$940$		0			0
$941$			884.134i			0.939568i	0.882781	+	0.469784i	$0.155668\pi$
$941$			884.134i			0.939568i	−0.882781	+	0.469784i	$0.844332\pi$
$942$		0			0
$943$	−1751.78			−1.85767
$944$		0			0
$945$	−50.2432	−	190.486i	−0.0531674	−	0.201573i
$946$		0			0
$947$		−	1716.80i		−	1.81288i	−0.422334	−	0.906440i	$-0.638789\pi$
$947$		−	1716.80i		−	1.81288i	0.422334	−	0.906440i	$-0.361211\pi$
$948$		0			0
$949$			1020.07i			1.07489i
$950$		0			0
$951$		−	458.766i		−	0.482404i
$952$		0			0
$953$			980.080i			1.02842i	0.857666	+	0.514208i	$0.171914\pi$
$953$			980.080i			1.02842i	−0.857666	+	0.514208i	$0.828086\pi$
$954$		0			0
$955$	67.5972	+	256.281i	0.0707824	+	0.268357i
$956$		0			0
$957$	581.906			0.608052
$958$		0			0
$959$		−	941.945i		−	0.982216i
$960$		0			0
$961$	−30.9636			−0.0322202
$962$		0			0
$963$		−	277.065i		−	0.287710i
$964$		0			0
$965$	−347.919	−	1319.06i	−0.360538	−	1.36690i
$966$		0			0
$967$	−1573.83			−1.62754			−0.813771	−	0.581186i	$-0.802589\pi$
$967$	−1573.83			−1.62754			−0.813771	+	0.581186i	$0.802589\pi$
$968$		0			0
$969$	−1034.92			−1.06803
$970$		0			0
$971$	−1139.13			−1.17315			−0.586573	−	0.809896i	$-0.699523\pi$
$971$	−1139.13			−1.17315			−0.586573	+	0.809896i	$0.699523\pi$
$972$		0			0
$973$	−846.926			−0.870427
$974$		0			0
$975$	280.486	+	494.711i	0.287678	+	0.507396i
$976$		0			0
$977$		−	474.207i		−	0.485370i	−0.970105	−	0.242685i	$-0.921972\pi$
$977$		−	474.207i		−	0.485370i	0.970105	−	0.242685i	$-0.0780282\pi$
$978$		0			0
$979$	−1694.76			−1.73111
$980$		0			0
$981$		−	361.109i		−	0.368103i
$982$		0			0
$983$	1625.31			1.65342			0.826711	−	0.562627i	$-0.190209\pi$
$983$	1625.31			1.65342			0.826711	+	0.562627i	$0.190209\pi$
$984$		0			0
$985$	−1612.58	+	425.339i	−1.63714	+	0.431817i
$986$		0			0
$987$			407.016i			0.412377i
$988$		0			0
$989$			131.573i			0.133036i
$990$		0			0
$991$		−	1321.44i		−	1.33344i	−0.745307	−	0.666721i	$-0.767697\pi$
$991$		−	1321.44i		−	1.33344i	0.745307	−	0.666721i	$-0.232303\pi$
$992$		0			0
$993$		−	134.918i		−	0.135869i
$994$		0			0
$995$	−93.6983	−	355.237i	−0.0941691	−	0.357022i
$996$		0			0
$997$	−198.145			−0.198742			−0.0993708	−	0.995050i	$-0.531683\pi$
$997$	−198.145			−0.198742			−0.0993708	+	0.995050i	$0.531683\pi$
$998$		0			0
$999$		−	96.7295i		−	0.0968264i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.p.b.799.8	yes	8
4.3	odd	2		960.3.p.a.799.4	yes	8
5.4	even	2		960.3.p.a.799.1	✓	8
8.3	odd	2		960.3.p.a.799.5	yes	8
8.5	even	2	inner	960.3.p.b.799.1	yes	8
20.19	odd	2	inner	960.3.p.b.799.5	yes	8
40.19	odd	2	inner	960.3.p.b.799.4	yes	8
40.29	even	2		960.3.p.a.799.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.3.p.a.799.1	✓	8	5.4	even	2
960.3.p.a.799.4	yes	8	4.3	odd	2
960.3.p.a.799.5	yes	8	8.3	odd	2
960.3.p.a.799.8	yes	8	40.29	even	2
960.3.p.b.799.1	yes	8	8.5	even	2	inner
960.3.p.b.799.4	yes	8	40.19	odd	2	inner
960.3.p.b.799.5	yes	8	20.19	odd	2	inner
960.3.p.b.799.8	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +(4.83465 - 1.27520i) q^{5} +7.58258 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +(4.83465 - 1.27520i) q^{5} +7.58258 q^{7} -3.00000 q^{9} -18.4249 q^{11} +13.1334 q^{13} +(2.20871 + 8.37386i) q^{15} +28.7477i q^{17} +20.7846 q^{19} +13.1334i q^{21} -24.0000 q^{23} +(21.7477 - 12.3303i) q^{25} -5.19615i q^{27} +18.2342i q^{29} +31.4955i q^{31} -31.9129i q^{33} +(36.6591 - 9.66930i) q^{35} +18.6156 q^{37} +22.7477i q^{39} +72.9909 q^{41} -5.48220i q^{43} +(-14.5040 + 3.82560i) q^{45} +30.9909 q^{47} +8.49545 q^{49} -49.7925 q^{51} +11.1153 q^{53} +(-89.0780 + 23.4955i) q^{55} +36.0000i q^{57} -28.2451 q^{59} -116.032i q^{61} -22.7477 q^{63} +(63.4955 - 16.7477i) q^{65} -99.5850i q^{67} -41.5692i q^{69} +54.9909i q^{71} +77.6697i q^{73} +(21.3567 + 37.6682i) q^{75} -139.708 q^{77} +150.486i q^{79} +9.00000 q^{81} +40.1232i q^{83} +(36.6591 + 138.985i) q^{85} -31.5826 q^{87} +91.9818 q^{89} +99.5850 q^{91} -54.5517 q^{93} +(100.486 - 26.5045i) q^{95} -39.8258i q^{97} +55.2747 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{7} - 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^7 - 24 * q^9	\( 8 q + 24 q^{7} - 24 q^{9} + 36 q^{15} - 192 q^{23} + 64 q^{25} + 144 q^{41} - 192 q^{47} - 152 q^{49} - 456 q^{55} - 72 q^{63} + 288 q^{65} + 72 q^{81} - 216 q^{87} - 144 q^{89} + 144 q^{95}+O(q^{100}) \) 8 * q + 24 * q^7 - 24 * q^9 + 36 * q^15 - 192 * q^23 + 64 * q^25 + 144 * q^41 - 192 * q^47 - 152 * q^49 - 456 * q^55 - 72 * q^63 + 288 * q^65 + 72 * q^81 - 216 * q^87 - 144 * q^89 + 144 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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