LMFDB - Embedded newform 260.2.d.a.129.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(129,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("260.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$260 = 2^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	260.d (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:	$2.07611045255$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12745506816.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9$ x^8 + 16x^6 + 79x^4 + 120*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{5}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.5
Root			$1.65070i$ of defining polynomial
Character	$\chi$	$=$	260.129
Dual form			260.2.d.a.129.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.646084i q^{3} +(-1.73205 + 1.41421i) q^{5} +0.913701 q^{7} +2.58258 q^{9} +3.94748i q^{11} +(-2.64575 + 2.44949i) q^{13} +(-0.913701 - 1.11905i) q^{15} +6.19115i q^{17} +1.11905i q^{19} +0.590327i q^{21} -5.54506i q^{23} +(1.00000 - 4.89898i) q^{25} +3.60681i q^{27} +1.58258 q^{29} -9.60433i q^{31} -2.55040 q^{33} +(-1.58258 + 1.29217i) q^{35} +7.84190 q^{37} +(-1.58258 - 1.70938i) q^{39} +5.06653i q^{41} -6.83723i q^{43} +(-4.47315 + 3.65231i) q^{45} +9.66930 q^{47} -6.16515 q^{49} -4.00000 q^{51} +1.29217i q^{53} +(-5.58258 - 6.83723i) q^{55} -0.723000 q^{57} -9.01400i q^{59} -5.58258 q^{61} +2.35970 q^{63} +(1.11847 - 7.98430i) q^{65} +7.84190 q^{67} +3.58258 q^{69} -6.77590i q^{71} +7.84190 q^{73} +(3.16515 + 0.646084i) q^{75} +3.60681i q^{77} -7.16515 q^{79} +5.41742 q^{81} -16.5975 q^{83} +(-8.75560 - 10.7234i) q^{85} +1.02248i q^{87} +11.3137i q^{89} +(-2.41742 + 2.23810i) q^{91} +6.20520 q^{93} +(-1.58258 - 1.93825i) q^{95} +1.63670 q^{97} +10.1947i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 16 q^{9} + 8 q^{25} - 24 q^{29} + 24 q^{35} + 24 q^{39} + 24 q^{49} - 32 q^{51} - 8 q^{55} - 8 q^{61} - 8 q^{69} - 48 q^{75} + 16 q^{79} + 80 q^{81} - 56 q^{91} + 24 q^{95}+O(q^{100})$ 8 * q - 16 * q^9 + 8 * q^25 - 24 * q^29 + 24 * q^35 + 24 * q^39 + 24 * q^49 - 32 * q^51 - 8 * q^55 - 8 * q^61 - 8 * q^69 - 48 * q^75 + 16 * q^79 + 80 * q^81 - 56 * q^91 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$ .

$n$	$41$	$131$	$157$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$			0.646084i			0.373017i	0.982453	+	0.186508i	$0.0597171\pi$
$3$			0.646084i			0.373017i	−0.982453	+	0.186508i	$0.940283\pi$
$4$		0			0
$5$	−1.73205	+	1.41421i	−0.774597	+	0.632456i
$5$	−1.73205	+	1.41421i	−0.774597	+	0.632456i
$6$		0			0
$7$	0.913701			0.345346			0.172673	−	0.984979i	$-0.444760\pi$
$7$	0.913701			0.345346			0.172673	+	0.984979i	$0.444760\pi$
$8$		0			0
$9$	2.58258			0.860859
$10$		0			0
$11$			3.94748i			1.19021i	0.803648	+	0.595105i	$0.202889\pi$
$11$			3.94748i			1.19021i	−0.803648	+	0.595105i	$0.797111\pi$
$12$		0			0
$13$	−2.64575	+	2.44949i	−0.733799	+	0.679366i
$13$	−2.64575	+	2.44949i	−0.733799	+	0.679366i
$14$		0			0
$15$	−0.913701	−	1.11905i	−0.235916	−	0.288937i
$16$		0			0
$17$			6.19115i			1.50157i	0.660545	+	0.750787i	$0.270325\pi$
$17$			6.19115i			1.50157i	−0.660545	+	0.750787i	$0.729675\pi$
$18$		0			0
$19$			1.11905i			0.256728i	0.991727	+	0.128364i	$0.0409725\pi$
$19$			1.11905i			0.256728i	−0.991727	+	0.128364i	$0.959027\pi$
$20$		0			0
$21$			0.590327i			0.128820i
$22$		0			0
$23$		−	5.54506i		−	1.15623i	−0.815957	−	0.578113i	$-0.803789\pi$
$23$		−	5.54506i		−	1.15623i	0.815957	−	0.578113i	$-0.196211\pi$
$24$		0			0
$25$	1.00000	−	4.89898i	0.200000	−	0.979796i
$26$		0			0
$27$			3.60681i			0.694131i
$28$		0			0
$29$	1.58258			0.293877			0.146938	−	0.989146i	$-0.453058\pi$
$29$	1.58258			0.293877			0.146938	+	0.989146i	$0.453058\pi$
$30$		0			0
$31$		−	9.60433i		−	1.72499i	−0.506067	−	0.862494i	$-0.668901\pi$
$31$		−	9.60433i		−	1.72499i	0.506067	−	0.862494i	$-0.331099\pi$
$32$		0			0
$33$	−2.55040			−0.443968
$34$		0			0
$35$	−1.58258	+	1.29217i	−0.267504	+	0.218416i
$36$		0			0
$37$	7.84190			1.28920			0.644601	−	0.764520i	$-0.277024\pi$
$37$	7.84190			1.28920			0.644601	+	0.764520i	$0.277024\pi$
$38$		0			0
$39$	−1.58258	−	1.70938i	−0.253415	−	0.273719i
$40$		0			0
$41$			5.06653i			0.791259i	0.918410	+	0.395629i	$0.129473\pi$
$41$			5.06653i			0.791259i	−0.918410	+	0.395629i	$0.870527\pi$
$42$		0			0
$43$		−	6.83723i		−	1.04267i	−0.853353	−	0.521334i	$-0.825435\pi$
$43$		−	6.83723i		−	1.04267i	0.853353	−	0.521334i	$-0.174565\pi$
$44$		0			0
$45$	−4.47315	+	3.65231i	−0.666818	+	0.544455i
$46$		0			0
$47$	9.66930			1.41041			0.705207	−	0.709002i	$-0.250854\pi$
$47$	9.66930			1.41041			0.705207	+	0.709002i	$0.250854\pi$
$48$		0			0
$49$	−6.16515			−0.880736
$50$		0			0
$51$	−4.00000			−0.560112
$52$		0			0
$53$			1.29217i			0.177493i	0.996054	+	0.0887464i	$0.0282861\pi$
$53$			1.29217i			0.177493i	−0.996054	+	0.0887464i	$0.971714\pi$
$54$		0			0
$55$	−5.58258	−	6.83723i	−0.752754	−	0.921932i
$56$		0			0
$57$	−0.723000			−0.0957637
$58$		0			0
$59$		−	9.01400i		−	1.17352i	−0.809760	−	0.586762i	$-0.800403\pi$
$59$		−	9.01400i		−	1.17352i	0.809760	−	0.586762i	$-0.199597\pi$
$60$		0			0
$61$	−5.58258			−0.714776			−0.357388	−	0.933956i	$-0.616333\pi$
$61$	−5.58258			−0.714776			−0.357388	+	0.933956i	$0.616333\pi$
$62$		0			0
$63$	2.35970			0.297294
$64$		0			0
$65$	1.11847	−	7.98430i	0.138730	−	0.990330i
$66$		0			0
$67$	7.84190			0.958041			0.479021	−	0.877804i	$-0.340992\pi$
$67$	7.84190			0.958041			0.479021	+	0.877804i	$0.340992\pi$
$68$		0			0
$69$	3.58258			0.431291
$70$		0			0
$71$		−	6.77590i		−	0.804152i	−0.915606	−	0.402076i	$-0.868289\pi$
$71$		−	6.77590i		−	0.804152i	0.915606	−	0.402076i	$-0.131711\pi$
$72$		0			0
$73$	7.84190			0.917825			0.458913	−	0.888481i	$-0.348239\pi$
$73$	7.84190			0.917825			0.458913	+	0.888481i	$0.348239\pi$
$74$		0			0
$75$	3.16515	+	0.646084i	0.365480	+	0.0746033i
$76$		0			0
$77$			3.60681i			0.411034i
$78$		0			0
$79$	−7.16515			−0.806143			−0.403071	−	0.915169i	$-0.632057\pi$
$79$	−7.16515			−0.806143			−0.403071	+	0.915169i	$0.632057\pi$
$80$		0			0
$81$	5.41742			0.601936
$82$		0			0
$83$	−16.5975			−1.82181			−0.910907	−	0.412613i	$-0.864616\pi$
$83$	−16.5975			−1.82181			−0.910907	+	0.412613i	$0.864616\pi$
$84$		0			0
$85$	−8.75560	−	10.7234i	−0.949679	−	1.16311i
$86$		0			0
$87$			1.02248i			0.109621i
$88$		0			0
$89$			11.3137i			1.19925i	0.800281	+	0.599625i	$0.204684\pi$
$89$			11.3137i			1.19925i	−0.800281	+	0.599625i	$0.795316\pi$
$90$		0			0
$91$	−2.41742	+	2.23810i	−0.253415	+	0.234617i
$92$		0			0
$93$	6.20520			0.643450
$94$		0			0
$95$	−1.58258	−	1.93825i	−0.162369	−	0.198860i
$96$		0			0
$97$	1.63670			0.166182			0.0830909	−	0.996542i	$-0.473521\pi$
$97$	1.63670			0.166182			0.0830909	+	0.996542i	$0.473521\pi$
$98$		0			0
$99$			10.1947i			1.02460i
$100$		0			0
$101$	9.16515			0.911967			0.455983	−	0.889988i	$-0.349288\pi$
$101$	9.16515			0.911967			0.455983	+	0.889988i	$0.349288\pi$
$102$		0			0
$103$			1.93825i			0.190982i	0.995430	+	0.0954908i	$0.0304420\pi$
$103$			1.93825i			0.190982i	−0.995430	+	0.0954908i	$0.969558\pi$
$104$		0			0
$105$	−0.834849	−	1.02248i	−0.0814729	−	0.0997835i
$106$		0			0
$107$			8.12940i			0.785899i	0.919560	+	0.392949i	$0.128545\pi$
$107$			8.12940i			0.785899i	−0.919560	+	0.392949i	$0.871455\pi$
$108$		0			0
$109$		−	8.48528i		−	0.812743i	−0.913708	−	0.406371i	$-0.866794\pi$
$109$		−	8.48528i		−	0.812743i	0.913708	−	0.406371i	$-0.133206\pi$
$110$		0			0
$111$			5.06653i			0.480893i
$112$		0			0
$113$		−	3.60681i		−	0.339300i	−0.985504	−	0.169650i	$-0.945736\pi$
$113$		−	3.60681i		−	0.339300i	0.985504	−	0.169650i	$-0.0542638\pi$
$114$		0			0
$115$	7.84190	+	9.60433i	0.731261	+	0.895609i
$116$		0			0
$117$	−6.83285	+	6.32599i	−0.631697	+	0.584838i
$118$		0			0
$119$			5.65685i			0.518563i
$120$		0			0
$121$	−4.58258			−0.416598
$122$		0			0
$123$	−3.27340			−0.295153
$124$		0			0
$125$	5.19615	+	9.89949i	0.464758	+	0.885438i
$126$		0			0
$127$			11.7362i			1.04142i	0.853734	+	0.520710i	$0.174333\pi$
$127$			11.7362i			1.04142i	−0.853734	+	0.520710i	$0.825667\pi$
$128$		0			0
$129$	4.41742			0.388933
$130$		0			0
$131$	15.1652			1.32499			0.662493	−	0.749068i	$-0.269499\pi$
$131$	15.1652			1.32499			0.662493	+	0.749068i	$0.269499\pi$
$132$		0			0
$133$			1.02248i			0.0886600i
$134$		0			0
$135$	−5.10080	−	6.24718i	−0.439007	−	0.537672i
$136$		0			0
$137$	−10.3923			−0.887875			−0.443937	−	0.896058i	$-0.646419\pi$
$137$	−10.3923			−0.887875			−0.443937	+	0.896058i	$0.646419\pi$
$138$		0			0
$139$	11.1652			0.947016			0.473508	−	0.880790i	$-0.342988\pi$
$139$	11.1652			0.947016			0.473508	+	0.880790i	$0.342988\pi$
$140$		0			0
$141$			6.24718i			0.526108i
$142$		0			0
$143$	−9.66930	−	10.4440i	−0.808588	−	0.873375i
$144$		0			0
$145$	−2.74110	+	2.23810i	−0.227636	+	0.185864i
$146$		0			0
$147$		−	3.98320i		−	0.328529i
$148$		0			0
$149$			2.82843i			0.231714i	0.993266	+	0.115857i	$0.0369614\pi$
$149$			2.82843i			0.231714i	−0.993266	+	0.115857i	$0.963039\pi$
$150$		0			0
$151$			3.35715i			0.273201i	0.990626	+	0.136600i	$0.0436177\pi$
$151$			3.35715i			0.273201i	−0.990626	+	0.136600i	$0.956382\pi$
$152$		0			0
$153$			15.9891i			1.29264i
$154$		0			0
$155$	13.5826	+	16.6352i	1.09098	+	1.33617i
$156$		0			0
$157$		−	22.4499i		−	1.79170i	−0.444356	−	0.895850i	$-0.646567\pi$
$157$		−	22.4499i		−	1.79170i	0.444356	−	0.895850i	$-0.353433\pi$
$158$		0			0
$159$	−0.834849			−0.0662078
$160$		0			0
$161$		−	5.06653i		−	0.399298i
$162$		0			0
$163$	−6.01450			−0.471092			−0.235546	−	0.971863i	$-0.575688\pi$
$163$	−6.01450			−0.471092			−0.235546	+	0.971863i	$0.575688\pi$
$164$		0			0
$165$	4.41742	−	3.60681i	0.343896	−	0.280790i
$166$		0			0
$167$	16.5975			1.28435			0.642177	−	0.766557i	$-0.278031\pi$
$167$	16.5975			1.28435			0.642177	+	0.766557i	$0.278031\pi$
$168$		0			0
$169$	1.00000	−	12.9615i	0.0769231	−	0.997037i
$170$		0			0
$171$			2.89003i			0.221006i
$172$		0			0
$173$			11.0901i			0.843167i	0.906790	+	0.421583i	$0.138526\pi$
$173$			11.0901i			0.843167i	−0.906790	+	0.421583i	$0.861474\pi$
$174$		0			0
$175$	0.913701	−	4.47620i	0.0690693	−	0.338369i
$176$		0			0
$177$	5.82380			0.437744
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−14.4174			−1.07164			−0.535819	−	0.844333i	$-0.679997\pi$
$181$	−14.4174			−1.07164			−0.535819	+	0.844333i	$0.679997\pi$
$182$		0			0
$183$		−	3.60681i		−	0.266623i
$184$		0			0
$185$	−13.5826	+	11.0901i	−0.998611	+	0.815362i
$186$		0			0
$187$	−24.4394			−1.78719
$188$		0			0
$189$			3.29555i			0.239716i
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$	20.9753			1.50984			0.754918	−	0.655819i	$-0.227677\pi$
$193$	20.9753			1.50984			0.754918	+	0.655819i	$0.227677\pi$
$194$		0			0
$195$	5.15853	+	0.722628i	0.369410	+	0.0517485i
$196$		0			0
$197$	−8.94630			−0.637398			−0.318699	−	0.947856i	$-0.603246\pi$
$197$	−8.94630			−0.637398			−0.318699	+	0.947856i	$0.603246\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$			5.06653i			0.357365i
$202$		0			0
$203$	1.44600			0.101489
$204$		0			0
$205$	−7.16515	−	8.77548i	−0.500436	−	0.612906i
$206$		0			0
$207$		−	14.3205i		−	0.995347i
$208$		0			0
$209$	−4.41742			−0.305560
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$	4.37780			0.299962
$214$		0			0
$215$	9.66930	+	11.8424i	0.659441	+	0.807647i
$216$		0			0
$217$		−	8.77548i		−	0.595719i
$218$		0			0
$219$			5.06653i			0.342364i
$220$		0			0
$221$	−15.1652	−	16.3802i	−1.02012	−	1.10185i
$222$		0			0
$223$	7.84190			0.525133			0.262566	−	0.964914i	$-0.415431\pi$
$223$	7.84190			0.525133			0.262566	+	0.964914i	$0.415431\pi$
$224$		0			0
$225$	2.58258	−	12.6520i	0.172172	−	0.843466i
$226$		0			0
$227$	−11.1153			−0.737749			−0.368874	−	0.929479i	$-0.620257\pi$
$227$	−11.1153			−0.737749			−0.368874	+	0.929479i	$0.620257\pi$
$228$		0			0
$229$			2.23810i			0.147898i	0.997262	+	0.0739489i	$0.0235602\pi$
$229$			2.23810i			0.147898i	−0.997262	+	0.0739489i	$0.976440\pi$
$230$		0			0
$231$	−2.33030			−0.153323
$232$		0			0
$233$		−	2.58434i		−	0.169305i	−0.996411	−	0.0846527i	$-0.973022\pi$
$233$		−	2.58434i		−	0.169305i	0.996411	−	0.0846527i	$-0.0269781\pi$
$234$		0			0
$235$	−16.7477	+	13.6745i	−1.09250	+	0.892024i
$236$		0			0
$237$		−	4.62929i		−	0.300705i
$238$		0			0
$239$		−	10.7850i		−	0.697623i	−0.937193	−	0.348811i	$-0.886585\pi$
$239$		−	10.7850i		−	0.697623i	0.937193	−	0.348811i	$-0.113415\pi$
$240$		0			0
$241$		−	27.6939i		−	1.78392i	−0.452111	−	0.891962i	$-0.649329\pi$
$241$		−	27.6939i		−	1.78392i	0.452111	−	0.891962i	$-0.350671\pi$
$242$		0			0
$243$			14.3205i			0.918663i
$244$		0			0
$245$	10.6784	−	8.71884i	0.682215	−	0.557026i
$246$		0			0
$247$	−2.74110	−	2.96073i	−0.174412	−	0.188387i
$248$		0			0
$249$		−	10.7234i		−	0.679567i
$250$		0			0
$251$	3.16515			0.199783			0.0998913	−	0.994998i	$-0.468150\pi$
$251$	3.16515			0.199783			0.0998913	+	0.994998i	$0.468150\pi$
$252$		0			0
$253$	21.8890			1.37615
$254$		0			0
$255$	6.92820	−	5.65685i	0.433861	−	0.354246i
$256$		0			0
$257$		−	22.1803i		−	1.38357i	−0.722105	−	0.691783i	$-0.756825\pi$
$257$		−	22.1803i		−	1.38357i	0.722105	−	0.691783i	$-0.243175\pi$
$258$		0			0
$259$	7.16515			0.445221
$260$		0			0
$261$	4.08712			0.252986
$262$		0			0
$263$		−	20.2420i		−	1.24818i	−0.781354	−	0.624088i	$-0.785471\pi$
$263$		−	20.2420i		−	1.24818i	0.781354	−	0.624088i	$-0.214529\pi$
$264$		0			0
$265$	−1.82740	−	2.23810i	−0.112256	−	0.137485i
$266$		0			0
$267$	−7.30960			−0.447341
$268$		0			0
$269$	−9.16515			−0.558809			−0.279405	−	0.960173i	$-0.590137\pi$
$269$	−9.16515			−0.558809			−0.279405	+	0.960173i	$0.590137\pi$
$270$		0			0
$271$			15.8515i			0.962911i	0.876471	+	0.481455i	$0.159892\pi$
$271$			15.8515i			0.962911i	−0.876471	+	0.481455i	$0.840108\pi$
$272$		0			0
$273$	−1.44600	−	1.56186i	−0.0875159	−	0.0945280i
$274$		0			0
$275$	19.3386	+	3.94748i	1.16616	+	0.238042i
$276$		0			0
$277$		−	5.92146i		−	0.355786i	−0.984050	−	0.177893i	$-0.943072\pi$
$277$		−	5.92146i		−	0.355786i	0.984050	−	0.177893i	$-0.0569281\pi$
$278$		0			0
$279$		−	24.8039i		−	1.48497i
$280$		0			0
$281$			26.5133i			1.58165i	0.612042	+	0.790825i	$0.290348\pi$
$281$			26.5133i			1.58165i	−0.612042	+	0.790825i	$0.709652\pi$
$282$		0			0
$283$		−	6.83723i		−	0.406431i	−0.979134	−	0.203216i	$-0.934861\pi$
$283$		−	6.83723i		−	0.406431i	0.979134	−	0.203216i	$-0.0651392\pi$
$284$		0			0
$285$	1.25227	−	1.02248i	0.0741782	−	0.0605663i
$286$		0			0
$287$			4.62929i			0.273258i
$288$		0			0
$289$	−21.3303			−1.25472
$290$		0			0
$291$			1.05745i			0.0619886i
$292$		0			0
$293$	−9.66930			−0.564887			−0.282443	−	0.959284i	$-0.591145\pi$
$293$	−9.66930			−0.564887			−0.282443	+	0.959284i	$0.591145\pi$
$294$		0			0
$295$	12.7477	+	15.6127i	0.742201	+	0.909007i
$296$		0			0
$297$	−14.2378			−0.826161
$298$		0			0
$299$	13.5826	+	14.6709i	0.785501	+	0.848438i
$300$		0			0
$301$		−	6.24718i		−	0.360082i
$302$		0			0
$303$			5.92146i			0.340179i
$304$		0			0
$305$	9.66930	−	7.89495i	0.553663	−	0.452064i
$306$		0			0
$307$	−18.4249			−1.05157			−0.525783	−	0.850619i	$-0.676227\pi$
$307$	−18.4249			−1.05157			−0.525783	+	0.850619i	$0.676227\pi$
$308$		0			0
$309$	−1.25227			−0.0712393
$310$		0			0
$311$	−30.3303			−1.71987			−0.859937	−	0.510400i	$-0.829497\pi$
$311$	−30.3303			−1.71987			−0.859937	+	0.510400i	$0.829497\pi$
$312$		0			0
$313$			18.5734i			1.04983i	0.851154	+	0.524916i	$0.175903\pi$
$313$			18.5734i			1.04983i	−0.851154	+	0.524916i	$0.824097\pi$
$314$		0			0
$315$	−4.08712	+	3.33712i	−0.230283	+	0.188025i
$316$		0			0
$317$	4.18710			0.235171			0.117586	−	0.993063i	$-0.462485\pi$
$317$	4.18710			0.235171			0.117586	+	0.993063i	$0.462485\pi$
$318$		0			0
$319$			6.24718i			0.349775i
$320$		0			0
$321$	−5.25227			−0.293153
$322$		0			0
$323$	−6.92820			−0.385496
$324$		0			0
$325$	9.35425	+	15.4110i	0.518880	+	0.854847i
$326$		0			0
$327$	5.48220			0.303167
$328$		0			0
$329$	8.83485			0.487081
$330$		0			0
$331$			5.12813i			0.281868i	0.990019	+	0.140934i	$0.0450105\pi$
$331$			5.12813i			0.281868i	−0.990019	+	0.140934i	$0.954990\pi$
$332$		0			0
$333$	20.2523			1.10982
$334$		0			0
$335$	−13.5826	+	11.0901i	−0.742095	+	0.605918i
$336$		0			0
$337$		−	1.02248i		−	0.0556978i	−0.999612	−	0.0278489i	$-0.991134\pi$
$337$		−	1.02248i		−	0.0556978i	0.999612	−	0.0278489i	$-0.00886573\pi$
$338$		0			0
$339$	2.33030			0.126565
$340$		0			0
$341$	37.9129			2.05310
$342$		0			0
$343$	−12.0290			−0.649505
$344$		0			0
$345$	−6.20520	+	5.06653i	−0.334077	+	0.272773i
$346$		0			0
$347$		−	24.1185i		−	1.29475i	−0.762172	−	0.647375i	$-0.775867\pi$
$347$		−	24.1185i		−	1.29475i	0.762172	−	0.647375i	$-0.224133\pi$
$348$		0			0
$349$		−	6.71430i		−	0.359408i	−0.983721	−	0.179704i	$-0.942486\pi$
$349$		−	6.71430i		−	0.359408i	0.983721	−	0.179704i	$-0.0575140\pi$
$350$		0			0
$351$	−8.83485	−	9.54273i	−0.471569	−	0.509353i
$352$		0			0
$353$	−9.66930			−0.514645			−0.257323	−	0.966326i	$-0.582840\pi$
$353$	−9.66930			−0.514645			−0.257323	+	0.966326i	$0.582840\pi$
$354$		0			0
$355$	9.58258	+	11.7362i	0.508590	+	0.622893i
$356$		0			0
$357$	−3.65480			−0.193433
$358$		0			0
$359$		−	2.29970i		−	0.121374i	−0.998157	−	0.0606869i	$-0.980671\pi$
$359$		−	2.29970i		−	0.121374i	0.998157	−	0.0606869i	$-0.0193291\pi$
$360$		0			0
$361$	17.7477			0.934091
$362$		0			0
$363$		−	2.96073i		−	0.155398i
$364$		0			0
$365$	−13.5826	+	11.0901i	−0.710945	+	0.580484i
$366$		0			0
$367$			1.93825i			0.101176i	0.998720	+	0.0505880i	$0.0161095\pi$
$367$			1.93825i			0.101176i	−0.998720	+	0.0505880i	$0.983890\pi$
$368$		0			0
$369$			13.0847i			0.681162i
$370$		0			0
$371$			1.18065i			0.0612965i
$372$		0			0
$373$			14.6969i			0.760979i	0.924785	+	0.380489i	$0.124244\pi$
$373$			14.6969i			0.760979i	−0.924785	+	0.380489i	$0.875756\pi$
$374$		0			0
$375$	−6.39590	+	3.35715i	−0.330283	+	0.173362i
$376$		0			0
$377$	−4.18710	+	3.87650i	−0.215647	+	0.199650i
$378$		0			0
$379$			31.0511i			1.59499i	0.603327	+	0.797494i	$0.293841\pi$
$379$			31.0511i			1.59499i	−0.603327	+	0.797494i	$0.706159\pi$
$380$		0			0
$381$	−7.58258			−0.388467
$382$		0			0
$383$	−29.0079			−1.48224			−0.741118	−	0.671375i	$-0.765704\pi$
$383$	−29.0079			−1.48224			−0.741118	+	0.671375i	$0.765704\pi$
$384$		0			0
$385$	−5.10080	−	6.24718i	−0.259961	−	0.318386i
$386$		0			0
$387$		−	17.6577i		−	0.897590i
$388$		0			0
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$	34.3303			1.73616
$392$		0			0
$393$			9.79796i			0.494242i
$394$		0			0
$395$	12.4104	−	10.1331i	0.624435	−	0.509849i
$396$		0			0
$397$	−19.8709			−0.997292			−0.498646	−	0.866806i	$-0.666169\pi$
$397$	−19.8709			−0.997292			−0.498646	+	0.866806i	$0.666169\pi$
$398$		0			0
$399$	−0.660606			−0.0330716
$400$		0			0
$401$		−	22.6274i		−	1.12996i	−0.825105	−	0.564980i	$-0.808884\pi$
$401$		−	22.6274i		−	1.12996i	0.825105	−	0.564980i	$-0.191116\pi$
$402$		0			0
$403$	23.5257	+	25.4107i	1.17190	+	1.26580i
$404$		0			0
$405$	−9.38325	+	7.66139i	−0.466258	+	0.380698i
$406$		0			0
$407$			30.9557i			1.53442i
$408$		0			0
$409$			32.1701i			1.59071i	0.606143	+	0.795356i	$0.292716\pi$
$409$			32.1701i			1.59071i	−0.606143	+	0.795356i	$0.707284\pi$
$410$		0			0
$411$		−	6.71430i		−	0.331192i
$412$		0			0
$413$		−	8.23610i		−	0.405272i
$414$		0			0
$415$	28.7477	−	23.4724i	1.41117	−	1.15222i
$416$		0			0
$417$			7.21362i			0.353253i
$418$		0			0
$419$	−21.4955			−1.05012			−0.525061	−	0.851065i	$-0.675957\pi$
$419$	−21.4955			−1.05012			−0.525061	+	0.851065i	$0.675957\pi$
$420$		0			0
$421$			34.4082i			1.67696i	0.544936	+	0.838478i	$0.316554\pi$
$421$			34.4082i			1.67696i	−0.544936	+	0.838478i	$0.683446\pi$
$422$		0			0
$423$	24.9717			1.21417
$424$		0			0
$425$	30.3303	+	6.19115i	1.47124	+	0.300315i
$426$		0			0
$427$	−5.10080			−0.246845
$428$		0			0
$429$	6.74773	−	6.24718i	0.325783	−	0.301617i
$430$		0			0
$431$		−	10.7850i		−	0.519494i	−0.965677	−	0.259747i	$-0.916361\pi$
$431$		−	10.7850i		−	0.519494i	0.965677	−	0.259747i	$-0.0836392\pi$
$432$		0			0
$433$		−	9.79796i		−	0.470860i	−0.971891	−	0.235430i	$-0.924350\pi$
$433$		−	9.79796i		−	0.470860i	0.971891	−	0.235430i	$-0.0756498\pi$
$434$		0			0
$435$	−1.44600	−	1.77098i	−0.0693304	−	0.0849121i
$436$		0			0
$437$	6.20520			0.296835
$438$		0			0
$439$	17.4955			0.835012			0.417506	−	0.908674i	$-0.362904\pi$
$439$	17.4955			0.835012			0.417506	+	0.908674i	$0.362904\pi$
$440$		0			0
$441$	−15.9220			−0.758189
$442$		0			0
$443$		−	6.56754i		−	0.312033i	−0.987754	−	0.156017i	$-0.950135\pi$
$443$		−	6.56754i		−	0.312033i	0.987754	−	0.156017i	$-0.0498653\pi$
$444$		0			0
$445$	−16.0000	−	19.5959i	−0.758473	−	0.928936i
$446$		0			0
$447$	−1.82740			−0.0864331
$448$		0			0
$449$		−	3.88587i		−	0.183386i	−0.995787	−	0.0916928i	$-0.970772\pi$
$449$		−	3.88587i		−	0.183386i	0.995787	−	0.0916928i	$-0.0292278\pi$
$450$		0			0
$451$	−20.0000			−0.941763
$452$		0			0
$453$	−2.16900			−0.101909
$454$		0			0
$455$	1.02195	−	7.29526i	0.0479098	−	0.342007i
$456$		0			0
$457$	−12.2197			−0.571614			−0.285807	−	0.958287i	$-0.592262\pi$
$457$	−12.2197			−0.571614			−0.285807	+	0.958287i	$0.592262\pi$
$458$		0			0
$459$	−22.3303			−1.04229
$460$		0			0
$461$			13.5518i			0.631171i	0.948897	+	0.315585i	$0.102201\pi$
$461$			13.5518i			0.631171i	−0.948897	+	0.315585i	$0.897799\pi$
$462$		0			0
$463$	−33.7273			−1.56744			−0.783721	−	0.621113i	$-0.786681\pi$
$463$	−33.7273			−1.56744			−0.783721	+	0.621113i	$0.786681\pi$
$464$		0			0
$465$	−10.7477	+	8.77548i	−0.498414	+	0.406953i
$466$		0			0
$467$			5.27537i			0.244115i	0.992523	+	0.122058i	$0.0389492\pi$
$467$			5.27537i			0.244115i	−0.992523	+	0.122058i	$0.961051\pi$
$468$		0			0
$469$	7.16515			0.330856
$470$		0			0
$471$	14.5045			0.668334
$472$		0			0
$473$	26.9898			1.24099
$474$		0			0
$475$	5.48220	+	1.11905i	0.251541	+	0.0513455i
$476$		0			0
$477$			3.33712i			0.152796i
$478$		0			0
$479$		−	6.77590i		−	0.309599i	−0.987946	−	0.154799i	$-0.950527\pi$
$479$		−	6.77590i		−	0.309599i	0.987946	−	0.154799i	$-0.0494732\pi$
$480$		0			0
$481$	−20.7477	+	19.2087i	−0.946015	+	0.875840i
$482$		0			0
$483$	3.27340			0.148945
$484$		0			0
$485$	−2.83485	+	2.31464i	−0.128724	+	0.105103i
$486$		0			0
$487$	14.7701			0.669297			0.334649	−	0.942343i	$-0.391382\pi$
$487$	14.7701			0.669297			0.334649	+	0.942343i	$0.391382\pi$
$488$		0			0
$489$		−	3.88587i		−	0.175725i
$490$		0			0
$491$	−6.33030			−0.285683			−0.142841	−	0.989746i	$-0.545624\pi$
$491$	−6.33030			−0.285683			−0.142841	+	0.989746i	$0.545624\pi$
$492$		0			0
$493$			9.79796i			0.441278i
$494$		0			0
$495$	−14.4174	−	17.6577i	−0.648015	−	0.793653i
$496$		0			0
$497$		−	6.19115i		−	0.277711i
$498$		0			0
$499$		−	20.3277i		−	0.909993i	−0.890493	−	0.454997i	$-0.849640\pi$
$499$		−	20.3277i		−	0.909993i	0.890493	−	0.454997i	$-0.150360\pi$
$500$		0			0
$501$			10.7234i			0.479085i
$502$		0			0
$503$		−	42.6919i		−	1.90354i	−0.306813	−	0.951770i	$-0.599263\pi$
$503$		−	42.6919i		−	1.90354i	0.306813	−	0.951770i	$-0.400737\pi$
$504$		0			0
$505$	−15.8745	+	12.9615i	−0.706406	+	0.576778i
$506$		0			0
$507$	8.37420	+	0.646084i	0.371911	+	0.0286936i
$508$		0			0
$509$		−	24.8655i		−	1.10214i	−0.834457	−	0.551072i	$-0.814219\pi$
$509$		−	24.8655i		−	1.10214i	0.834457	−	0.551072i	$-0.185781\pi$
$510$		0			0
$511$	7.16515			0.316968
$512$		0			0
$513$	−4.03620			−0.178203
$514$		0			0
$515$	−2.74110	−	3.35715i	−0.120787	−	0.147934i
$516$		0			0
$517$			38.1694i			1.67869i
$518$		0			0
$519$	−7.16515			−0.314515
$520$		0			0
$521$	22.4174			0.982125			0.491063	−	0.871124i	$-0.336609\pi$
$521$	22.4174			0.982125			0.491063	+	0.871124i	$0.336609\pi$
$522$		0			0
$523$			35.2086i			1.53957i	0.638306	+	0.769783i	$0.279636\pi$
$523$			35.2086i			1.53957i	−0.638306	+	0.769783i	$0.720364\pi$
$524$		0			0
$525$	2.89200	+	0.590327i	0.126217	+	0.0257640i
$526$		0			0
$527$	59.4618			2.59020
$528$		0			0
$529$	−7.74773			−0.336858
$530$		0			0
$531$		−	23.2793i		−	1.01024i
$532$		0			0
$533$	−12.4104	−	13.4048i	−0.537554	−	0.580625i
$534$		0			0
$535$	−11.4967	−	14.0805i	−0.497046	−	0.608754i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	24.3368i		−	1.04826i
$540$		0			0
$541$		−	45.1316i		−	1.94036i	−0.242385	−	0.970180i	$-0.577930\pi$
$541$		−	45.1316i		−	1.94036i	0.242385	−	0.970180i	$-0.422070\pi$
$542$		0			0
$543$		−	9.31486i		−	0.399739i
$544$		0			0
$545$	12.0000	+	14.6969i	0.514024	+	0.629548i
$546$		0			0
$547$		−	26.4331i		−	1.13020i	−0.825022	−	0.565100i	$-0.808838\pi$
$547$		−	26.4331i		−	1.13020i	0.825022	−	0.565100i	$-0.191162\pi$
$548$		0			0
$549$	−14.4174			−0.615321
$550$		0			0
$551$			1.77098i			0.0754463i
$552$		0			0
$553$	−6.54680			−0.278398
$554$		0			0
$555$	−7.16515	−	8.77548i	−0.304144	−	0.372498i
$556$		0			0
$557$	−23.5257			−0.996816			−0.498408	−	0.866942i	$-0.666082\pi$
$557$	−23.5257			−0.996816			−0.498408	+	0.866942i	$0.666082\pi$
$558$		0			0
$559$	16.7477	+	18.0896i	0.708353	+	0.765109i
$560$		0			0
$561$		−	15.7899i		−	0.666650i
$562$		0			0
$563$		−	46.5684i		−	1.96263i	−0.192418	−	0.981313i	$-0.561633\pi$
$563$		−	46.5684i		−	1.96263i	0.192418	−	0.981313i	$-0.438367\pi$
$564$		0			0
$565$	5.10080	+	6.24718i	0.214592	+	0.262821i
$566$		0			0
$567$	4.94990			0.207876
$568$		0			0
$569$	−10.4174			−0.436721			−0.218361	−	0.975868i	$-0.570071\pi$
$569$	−10.4174			−0.436721			−0.218361	+	0.975868i	$0.570071\pi$
$570$		0			0
$571$	−37.4955			−1.56914			−0.784568	−	0.620043i	$-0.787115\pi$
$571$	−37.4955			−1.56914			−0.784568	+	0.620043i	$0.787115\pi$
$572$		0			0
$573$		−	7.75301i		−	0.323886i
$574$		0			0
$575$	−27.1652	−	5.54506i	−1.13287	−	0.231245i
$576$		0			0
$577$	46.5191			1.93662			0.968308	−	0.249758i	$-0.0803512\pi$
$577$	46.5191			1.93662			0.968308	+	0.249758i	$0.0803512\pi$
$578$		0			0
$579$			13.5518i			0.563194i
$580$		0			0
$581$	−15.1652			−0.629156
$582$		0			0
$583$	−5.10080			−0.211254
$584$		0			0
$585$	2.88854	−	20.6201i	0.119427	−	0.852534i
$586$		0			0
$587$	11.1153			0.458778			0.229389	−	0.973335i	$-0.426327\pi$
$587$	11.1153			0.458778			0.229389	+	0.973335i	$0.426327\pi$
$588$		0			0
$589$	10.7477			0.442852
$590$		0			0
$591$		−	5.78006i		−	0.237760i
$592$		0			0
$593$	−15.8745			−0.651888			−0.325944	−	0.945389i	$-0.605682\pi$
$593$	−15.8745			−0.651888			−0.325944	+	0.945389i	$0.605682\pi$
$594$		0			0
$595$	−8.00000	−	9.79796i	−0.327968	−	0.401677i
$596$		0			0
$597$		−	2.58434i		−	0.105770i
$598$		0			0
$599$	−42.3303			−1.72957			−0.864785	−	0.502143i	$-0.832545\pi$
$599$	−42.3303			−1.72957			−0.864785	+	0.502143i	$0.832545\pi$
$600$		0			0
$601$	44.3303			1.80827			0.904135	−	0.427246i	$-0.140516\pi$
$601$	44.3303			1.80827			0.904135	+	0.427246i	$0.140516\pi$
$602$		0			0
$603$	20.2523			0.824738
$604$		0			0
$605$	7.93725	−	6.48074i	0.322695	−	0.263480i
$606$		0			0
$607$			34.1862i			1.38757i	0.720181	+	0.693787i	$0.244059\pi$
$607$			34.1862i			1.38757i	−0.720181	+	0.693787i	$0.755941\pi$
$608$		0			0
$609$			0.934237i			0.0378572i
$610$		0			0
$611$	−25.5826	+	23.6849i	−1.03496	+	0.958187i
$612$		0			0
$613$	−24.6301			−0.994801			−0.497400	−	0.867521i	$-0.665712\pi$
$613$	−24.6301			−0.994801			−0.497400	+	0.867521i	$0.665712\pi$
$614$		0			0
$615$	5.66970	−	4.62929i	0.228624	−	0.186671i
$616$		0			0
$617$	28.2849			1.13871			0.569354	−	0.822093i	$-0.307193\pi$
$617$	28.2849			1.13871			0.569354	+	0.822093i	$0.307193\pi$
$618$		0			0
$619$		−	32.8221i		−	1.31923i	−0.751603	−	0.659615i	$-0.770719\pi$
$619$		−	32.8221i		−	1.31923i	0.751603	−	0.659615i	$-0.229281\pi$
$620$		0			0
$621$	20.0000			0.802572
$622$		0			0
$623$			10.3373i			0.414157i
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$		−	2.85403i		−	0.113979i
$628$		0			0
$629$			48.5504i			1.93583i
$630$		0			0
$631$			24.3368i			0.968832i	0.874838	+	0.484416i	$0.160968\pi$
$631$			24.3368i			0.968832i	−0.874838	+	0.484416i	$0.839032\pi$
$632$		0			0
$633$			12.9217i			0.513591i
$634$		0			0
$635$	−16.5975	−	20.3277i	−0.658652	−	0.806681i
$636$		0			0
$637$	16.3115	−	15.1015i	0.646283	−	0.598342i
$638$		0			0
$639$		−	17.4993i		−	0.692261i
$640$		0			0
$641$	−24.3303			−0.960989			−0.480495	−	0.876998i	$-0.659543\pi$
$641$	−24.3303			−0.960989			−0.480495	+	0.876998i	$0.659543\pi$
$642$		0			0
$643$	9.28790			0.366279			0.183140	−	0.983087i	$-0.441374\pi$
$643$	9.28790			0.366279			0.183140	+	0.983087i	$0.441374\pi$
$644$		0			0
$645$	−7.65120	+	6.24718i	−0.301266	+	0.245983i
$646$		0			0
$647$			28.7478i			1.13019i	0.825025	+	0.565096i	$0.191161\pi$
$647$			28.7478i			1.13019i	−0.825025	+	0.565096i	$0.808839\pi$
$648$		0			0
$649$	35.5826			1.39674
$650$		0			0
$651$	5.66970			0.222213
$652$		0			0
$653$		−	15.2363i		−	0.596243i	−0.954528	−	0.298122i	$-0.903640\pi$
$653$		−	15.2363i		−	0.596243i	0.954528	−	0.298122i	$-0.0963601\pi$
$654$		0			0
$655$	−26.2668	+	21.4468i	−1.02633	+	0.837994i
$656$		0			0
$657$	20.2523			0.790118
$658$		0			0
$659$	21.4955			0.837344			0.418672	−	0.908138i	$-0.362496\pi$
$659$	21.4955			0.837344			0.418672	+	0.908138i	$0.362496\pi$
$660$		0			0
$661$			8.48528i			0.330039i	0.986290	+	0.165020i	$0.0527687\pi$
$661$			8.48528i			0.330039i	−0.986290	+	0.165020i	$0.947231\pi$
$662$		0			0
$663$	10.5830	−	9.79796i	0.411010	−	0.380521i
$664$		0			0
$665$	−1.44600	−	1.77098i	−0.0560735	−	0.0686757i
$666$		0			0
$667$		−	8.77548i		−	0.339788i
$668$		0			0
$669$			5.06653i			0.195883i
$670$		0			0
$671$		−	22.0371i		−	0.850732i
$672$		0			0
$673$		−	1.02248i		−	0.0394136i	−0.999806	−	0.0197068i	$-0.993727\pi$
$673$		−	1.02248i		−	0.0394136i	0.999806	−	0.0197068i	$-0.00627327\pi$
$674$		0			0
$675$	17.6697	+	3.60681i	0.680107	+	0.138826i
$676$		0			0
$677$			28.6411i			1.10077i	0.834912	+	0.550383i	$0.185518\pi$
$677$			28.6411i			1.10077i	−0.834912	+	0.550383i	$0.814482\pi$
$678$		0			0
$679$	1.49545			0.0573903
$680$		0			0
$681$		−	7.18142i		−	0.275193i
$682$		0			0
$683$	11.1153			0.425315			0.212658	−	0.977127i	$-0.431788\pi$
$683$	11.1153			0.425315			0.212658	+	0.977127i	$0.431788\pi$
$684$		0			0
$685$	18.0000	−	14.6969i	0.687745	−	0.561541i
$686$		0			0
$687$	−1.44600			−0.0551683
$688$		0			0
$689$	−3.16515	−	3.41875i	−0.120583	−	0.130244i
$690$		0			0
$691$			26.5749i			1.01096i	0.862839	+	0.505478i	$0.168684\pi$
$691$			26.5749i			1.01096i	−0.862839	+	0.505478i	$0.831316\pi$
$692$		0			0
$693$			9.31486i			0.353842i
$694$		0			0
$695$	−19.3386	+	15.7899i	−0.733555	+	0.598945i
$696$		0			0
$697$	−31.3676			−1.18813
$698$		0			0
$699$	1.66970			0.0631537
$700$		0			0
$701$	−33.1652			−1.25263			−0.626315	−	0.779570i	$-0.715438\pi$
$701$	−33.1652			−1.25263			−0.626315	+	0.779570i	$0.715438\pi$
$702$		0			0
$703$			8.77548i			0.330974i
$704$		0			0
$705$	−8.83485	−	10.8204i	−0.332740	−	0.407521i
$706$		0			0
$707$	8.37420			0.314944
$708$		0			0
$709$		−	23.6849i		−	0.889504i	−0.895654	−	0.444752i	$-0.853292\pi$
$709$		−	23.6849i		−	0.889504i	0.895654	−	0.444752i	$-0.146708\pi$
$710$		0			0
$711$	−18.5045			−0.693975
$712$		0			0
$713$	−53.2566			−1.99448
$714$		0			0
$715$	31.5178	+	4.41515i	1.17870	+	0.165117i
$716$		0			0
$717$	6.96800			0.260225
$718$		0			0
$719$	21.4955			0.801645			0.400823	−	0.916156i	$-0.368724\pi$
$719$	21.4955			0.801645			0.400823	+	0.916156i	$0.368724\pi$
$720$		0			0
$721$			1.77098i			0.0659548i
$722$		0			0
$723$	17.8926			0.665433
$724$		0			0
$725$	1.58258	−	7.75301i	0.0587754	−	0.287939i
$726$		0			0
$727$			26.4331i			0.980351i	0.871624	+	0.490176i	$0.163067\pi$
$727$			26.4331i			0.980351i	−0.871624	+	0.490176i	$0.836933\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$	42.3303			1.56564
$732$		0			0
$733$	22.4213			0.828150			0.414075	−	0.910243i	$-0.364105\pi$
$733$	22.4213			0.828150			0.414075	+	0.910243i	$0.364105\pi$
$734$		0			0
$735$	5.63310	+	6.89911i	0.207780	+	0.254478i
$736$		0			0
$737$			30.9557i			1.14027i
$738$		0			0
$739$		−	37.7654i		−	1.38922i	−0.719385	−	0.694611i	$-0.755576\pi$
$739$		−	37.7654i		−	1.38922i	0.719385	−	0.694611i	$-0.244424\pi$
$740$		0			0
$741$	1.91288	−	1.77098i	0.0702713	−	0.0650586i
$742$		0			0
$743$	−29.0079			−1.06420			−0.532099	−	0.846682i	$-0.678596\pi$
$743$	−29.0079			−1.06420			−0.532099	+	0.846682i	$0.678596\pi$
$744$		0			0
$745$	−4.00000	−	4.89898i	−0.146549	−	0.179485i
$746$		0			0
$747$	−42.8643			−1.56832
$748$		0			0
$749$			7.42784i			0.271407i
$750$		0			0
$751$	23.1652			0.845308			0.422654	−	0.906291i	$-0.361098\pi$
$751$	23.1652			0.845308			0.422654	+	0.906291i	$0.361098\pi$
$752$		0			0
$753$			2.04495i			0.0745222i
$754$		0			0
$755$	−4.74773	−	5.81475i	−0.172787	−	0.211621i
$756$		0			0
$757$			41.0234i			1.49102i	0.666494	+	0.745510i	$0.267794\pi$
$757$			41.0234i			1.49102i	−0.666494	+	0.745510i	$0.732206\pi$
$758$		0			0
$759$			14.1421i			0.513327i
$760$		0			0
$761$			11.3137i			0.410122i	0.978749	+	0.205061i	$0.0657392\pi$
$761$			11.3137i			0.410122i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$		−	7.75301i		−	0.280678i
$764$		0			0
$765$	−22.6120	−	27.6939i	−0.817539	−	1.00128i
$766$		0			0
$767$	22.0797	+	23.8488i	0.797252	+	0.861131i
$768$		0			0
$769$			8.95240i			0.322832i	0.986886	+	0.161416i	$0.0516061\pi$
$769$			8.95240i			0.322832i	−0.986886	+	0.161416i	$0.948394\pi$
$770$		0			0
$771$	14.3303			0.516093
$772$		0			0
$773$	−9.66930			−0.347781			−0.173890	−	0.984765i	$-0.555634\pi$
$773$	−9.66930			−0.347781			−0.173890	+	0.984765i	$0.555634\pi$
$774$		0			0
$775$	−47.0514	−	9.60433i	−1.69014	−	0.344998i
$776$		0			0
$777$			4.62929i			0.166075i
$778$		0			0
$779$	−5.66970			−0.203138
$780$		0			0
$781$	26.7477			0.957109
$782$		0			0
$783$			5.70805i			0.203989i
$784$		0			0
$785$	31.7490	+	38.8844i	1.13317	+	1.38785i
$786$		0			0
$787$	−18.4249			−0.656777			−0.328389	−	0.944543i	$-0.606506\pi$
$787$	−18.4249			−0.656777			−0.328389	+	0.944543i	$0.606506\pi$
$788$		0			0
$789$	13.0780			0.465590
$790$		0			0
$791$		−	3.29555i		−	0.117176i
$792$		0			0
$793$	14.7701	−	13.6745i	0.524502	−	0.485594i
$794$		0			0
$795$	1.44600	−	1.18065i	0.0512843	−	0.0418735i
$796$		0			0
$797$		−	25.0343i		−	0.886760i	−0.896334	−	0.443380i	$-0.853779\pi$
$797$		−	25.0343i		−	0.886760i	0.896334	−	0.443380i	$-0.146221\pi$
$798$		0			0
$799$			59.8641i			2.11784i
$800$		0			0
$801$			29.2185i			1.03239i
$802$		0			0
$803$			30.9557i			1.09240i
$804$		0			0
$805$	7.16515	+	8.77548i	0.252538	+	0.309295i
$806$		0			0
$807$		−	5.92146i		−	0.208445i
$808$		0			0
$809$	4.74773			0.166921			0.0834606	−	0.996511i	$-0.473403\pi$
$809$	4.74773			0.166921			0.0834606	+	0.996511i	$0.473403\pi$
$810$		0			0
$811$		−	33.2892i		−	1.16894i	−0.811415	−	0.584471i	$-0.801302\pi$
$811$		−	33.2892i		−	1.16894i	0.811415	−	0.584471i	$-0.198698\pi$
$812$		0			0
$813$	−10.2414			−0.359182
$814$		0			0
$815$	10.4174	−	8.50579i	0.364906	−	0.297945i
$816$		0			0
$817$	7.65120			0.267682
$818$		0			0
$819$	−6.24318	+	5.78006i	−0.218154	+	0.201972i
$820$		0			0
$821$			19.7990i			0.690990i	0.938421	+	0.345495i	$0.112289\pi$
$821$			19.7990i			0.690990i	−0.938421	+	0.345495i	$0.887711\pi$
$822$		0			0
$823$			50.9280i			1.77524i	0.460576	+	0.887620i	$0.347643\pi$
$823$			50.9280i			1.77524i	−0.460576	+	0.887620i	$0.652357\pi$
$824$		0			0
$825$	−2.55040	+	12.4944i	−0.0887936	+	0.434998i
$826$		0			0
$827$	15.1515			0.526870			0.263435	−	0.964677i	$-0.415145\pi$
$827$	15.1515			0.526870			0.263435	+	0.964677i	$0.415145\pi$
$828$		0			0
$829$	−42.2432			−1.46717			−0.733583	−	0.679600i	$-0.762154\pi$
$829$	−42.2432			−1.46717			−0.733583	+	0.679600i	$0.762154\pi$
$830$		0			0
$831$	3.82576			0.132714
$832$		0			0
$833$		−	38.1694i		−	1.32249i
$834$		0			0
$835$	−28.7477	+	23.4724i	−0.994856	+	0.812297i
$836$		0			0
$837$	34.6410			1.19737
$838$		0			0
$839$			44.6029i			1.53986i	0.638126	+	0.769932i	$0.279710\pi$
$839$			44.6029i			1.53986i	−0.638126	+	0.769932i	$0.720290\pi$
$840$		0			0
$841$	−26.4955			−0.913636
$842$		0			0
$843$	−17.1298			−0.589982
$844$		0			0
$845$	16.5983	+	23.8642i	0.570997	+	0.820952i
$846$		0			0
$847$	−4.18710			−0.143871
$848$		0			0
$849$	4.41742			0.151606
$850$		0			0
$851$		−	43.4839i		−	1.49061i
$852$		0			0
$853$	−33.7273			−1.15480			−0.577401	−	0.816461i	$-0.695933\pi$
$853$	−33.7273			−1.15480			−0.577401	+	0.816461i	$0.695933\pi$
$854$		0			0
$855$	−4.08712	−	5.00568i	−0.139777	−	0.171191i
$856$		0			0
$857$		−	14.4272i		−	0.492825i	−0.969165	−	0.246413i	$-0.920748\pi$
$857$		−	14.4272i		−	0.492825i	0.969165	−	0.246413i	$-0.0792519\pi$
$858$		0			0
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$	−2.99091			−0.101930
$862$		0			0
$863$	−9.66930			−0.329147			−0.164573	−	0.986365i	$-0.552625\pi$
$863$	−9.66930			−0.329147			−0.164573	+	0.986365i	$0.552625\pi$
$864$		0			0
$865$	−15.6838	−	19.2087i	−0.533265	−	0.653114i
$866$		0			0
$867$		−	13.7812i		−	0.468033i
$868$		0			0
$869$		−	28.2843i		−	0.959478i
$870$		0			0
$871$	−20.7477	+	19.2087i	−0.703010	+	0.650861i
$872$		0			0
$873$	4.22690			0.143059
$874$		0			0
$875$	4.74773	+	9.04517i	0.160502	+	0.305783i
$876$		0			0
$877$	41.7599			1.41013			0.705066	−	0.709142i	$-0.250917\pi$
$877$	41.7599			1.41013			0.705066	+	0.709142i	$0.250917\pi$
$878$		0			0
$879$		−	6.24718i		−	0.210712i
$880$		0			0
$881$	13.5826			0.457609			0.228804	−	0.973472i	$-0.426518\pi$
$881$	13.5826			0.457609			0.228804	+	0.973472i	$0.426518\pi$
$882$		0			0
$883$		−	24.3882i		−	0.820728i	−0.911922	−	0.410364i	$-0.865402\pi$
$883$		−	24.3882i		−	0.820728i	0.911922	−	0.410364i	$-0.134598\pi$
$884$		0			0
$885$	−10.0871	+	8.23610i	−0.339075	+	0.276853i
$886$		0			0
$887$		−	8.39909i		−	0.282014i	−0.990009	−	0.141007i	$-0.954966\pi$
$887$		−	8.39909i		−	0.282014i	0.990009	−	0.141007i	$-0.0450340\pi$
$888$		0			0
$889$			10.7234i			0.359651i
$890$		0			0
$891$			21.3852i			0.716430i
$892$		0			0
$893$			10.8204i			0.362092i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−9.47860	+	8.77548i	−0.316481	+	0.293005i
$898$		0			0
$899$		−	15.1996i		−	0.506934i
$900$		0			0
$901$	−8.00000			−0.266519
$902$		0			0
$903$	4.03620			0.134316
$904$		0			0
$905$	24.9717	−	20.3893i	0.830088	−	0.677764i
$906$		0			0
$907$		−	50.9280i		−	1.69104i	−0.533945	−	0.845519i	$-0.679291\pi$
$907$		−	50.9280i		−	1.69104i	0.533945	−	0.845519i	$-0.320709\pi$
$908$		0			0
$909$	23.6697			0.785074
$910$		0			0
$911$	54.3303			1.80004			0.900022	−	0.435845i	$-0.143551\pi$
$911$	54.3303			1.80004			0.900022	+	0.435845i	$0.143551\pi$
$912$		0			0
$913$		−	65.5183i		−	2.16834i
$914$		0			0
$915$	5.10080	+	6.24718i	0.168627	+	0.206525i
$916$		0			0
$917$	13.8564			0.457579
$918$		0			0
$919$	50.3303			1.66024			0.830122	−	0.557582i	$-0.188271\pi$
$919$	50.3303			1.66024			0.830122	+	0.557582i	$0.188271\pi$
$920$		0			0
$921$		−	11.9040i		−	0.392251i
$922$		0			0
$923$	16.5975	+	17.9274i	0.546314	+	0.590086i
$924$		0			0
$925$	7.84190	−	38.4173i	0.257840	−	1.26315i
$926$		0			0
$927$			5.00568i			0.164408i
$928$		0			0
$929$		−	46.7794i		−	1.53478i	−0.641179	−	0.767391i	$-0.721555\pi$
$929$		−	46.7794i		−	1.53478i	0.641179	−	0.767391i	$-0.278445\pi$
$930$		0			0
$931$		−	6.89911i		−	0.226109i
$932$		0			0
$933$		−	19.5959i		−	0.641542i
$934$		0			0
$935$	42.3303	−	34.5625i	1.38435	−	1.13032i
$936$		0			0
$937$			37.1469i			1.21354i	0.794879	+	0.606768i	$0.207534\pi$
$937$			37.1469i			1.21354i	−0.794879	+	0.606768i	$0.792466\pi$
$938$		0			0
$939$	−12.0000			−0.391605
$940$		0			0
$941$			18.0280i			0.587696i	0.955852	+	0.293848i	$0.0949360\pi$
$941$			18.0280i			0.587696i	−0.955852	+	0.293848i	$0.905064\pi$
$942$		0			0
$943$	28.0942			0.914873
$944$		0			0
$945$	−4.66061	−	5.70805i	−0.151610	−	0.185683i
$946$		0			0
$947$	−51.2385			−1.66503			−0.832514	−	0.554004i	$-0.813099\pi$
$947$	−51.2385			−1.66503			−0.832514	+	0.554004i	$0.813099\pi$
$948$		0			0
$949$	−20.7477	+	19.2087i	−0.673500	+	0.623540i
$950$		0			0
$951$			2.70522i			0.0877227i
$952$		0			0
$953$			22.7196i			0.735961i	0.929833	+	0.367981i	$0.119951\pi$
$953$			22.7196i			0.735961i	−0.929833	+	0.367981i	$0.880049\pi$
$954$		0			0
$955$	20.7846	−	16.9706i	0.672574	−	0.549155i
$956$		0			0
$957$	−4.03620			−0.130472
$958$		0			0
$959$	−9.49545			−0.306624
$960$		0			0
$961$	−61.2432			−1.97559
$962$		0			0
$963$			20.9948i			0.676548i
$964$		0			0
$965$	−36.3303	+	29.6636i	−1.16951	+	0.954904i
$966$		0			0
$967$	−7.46050			−0.239914			−0.119957	−	0.992779i	$-0.538276\pi$
$967$	−7.46050			−0.239914			−0.119957	+	0.992779i	$0.538276\pi$
$968$		0			0
$969$		−	4.47620i		−	0.143796i
$970$		0			0
$971$	−24.6606			−0.791396			−0.395698	−	0.918381i	$-0.629497\pi$
$971$	−24.6606			−0.791396			−0.395698	+	0.918381i	$0.629497\pi$
$972$		0			0
$973$	10.2016			0.327048
$974$		0			0
$975$	−9.95678	+	6.04363i	−0.318872	+	0.193551i
$976$		0			0
$977$	−20.6337			−0.660131			−0.330065	−	0.943958i	$-0.607071\pi$
$977$	−20.6337			−0.660131			−0.330065	+	0.943958i	$0.607071\pi$
$978$		0			0
$979$	−44.6606			−1.42736
$980$		0			0
$981$		−	21.9139i		−	0.699656i
$982$		0			0
$983$	−20.6337			−0.658113			−0.329057	−	0.944310i	$-0.606731\pi$
$983$	−20.6337			−0.658113			−0.329057	+	0.944310i	$0.606731\pi$
$984$		0			0
$985$	15.4955	−	12.6520i	0.493726	−	0.403126i
$986$		0			0
$987$			5.70805i			0.181689i
$988$		0			0
$989$	−37.9129			−1.20556
$990$		0			0
$991$	29.4955			0.936954			0.468477	−	0.883476i	$-0.344803\pi$
$991$	29.4955			0.936954			0.468477	+	0.883476i	$0.344803\pi$
$992$		0			0
$993$	−3.31320			−0.105141
$994$		0			0
$995$	6.92820	−	5.65685i	0.219639	−	0.179334i
$996$		0			0
$997$		−	52.8663i		−	1.67429i	−0.546979	−	0.837146i	$-0.684222\pi$
$997$		−	52.8663i		−	1.67429i	0.546979	−	0.837146i	$-0.315778\pi$
$998$		0			0
$999$			28.2843i			0.894875i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	260.2.d.a.129.5	yes	8
3.2	odd	2		2340.2.j.d.649.6		8
4.3	odd	2		1040.2.f.e.129.3		8
5.2	odd	4		1300.2.f.f.701.6		8
5.3	odd	4		1300.2.f.f.701.3		8
5.4	even	2	inner	260.2.d.a.129.4	yes	8
13.5	odd	4		3380.2.c.d.2029.6		8
13.8	odd	4		3380.2.c.d.2029.5		8
13.12	even	2	inner	260.2.d.a.129.6	yes	8
15.14	odd	2		2340.2.j.d.649.1		8
20.19	odd	2		1040.2.f.e.129.6		8
39.38	odd	2		2340.2.j.d.649.3		8
52.51	odd	2		1040.2.f.e.129.4		8
65.12	odd	4		1300.2.f.f.701.5		8
65.34	odd	4		3380.2.c.d.2029.3		8
65.38	odd	4		1300.2.f.f.701.4		8
65.44	odd	4		3380.2.c.d.2029.4		8
65.64	even	2	inner	260.2.d.a.129.3	✓	8
195.194	odd	2		2340.2.j.d.649.8		8
260.259	odd	2		1040.2.f.e.129.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.d.a.129.3	✓	8	65.64	even	2	inner
260.2.d.a.129.4	yes	8	5.4	even	2	inner
260.2.d.a.129.5	yes	8	1.1	even	1	trivial
260.2.d.a.129.6	yes	8	13.12	even	2	inner
1040.2.f.e.129.3		8	4.3	odd	2
1040.2.f.e.129.4		8	52.51	odd	2
1040.2.f.e.129.5		8	260.259	odd	2
1040.2.f.e.129.6		8	20.19	odd	2
1300.2.f.f.701.3		8	5.3	odd	4
1300.2.f.f.701.4		8	65.38	odd	4
1300.2.f.f.701.5		8	65.12	odd	4
1300.2.f.f.701.6		8	5.2	odd	4
2340.2.j.d.649.1		8	15.14	odd	2
2340.2.j.d.649.3		8	39.38	odd	2
2340.2.j.d.649.6		8	3.2	odd	2
2340.2.j.d.649.8		8	195.194	odd	2
3380.2.c.d.2029.3		8	65.34	odd	4
3380.2.c.d.2029.4		8	65.44	odd	4
3380.2.c.d.2029.5		8	13.8	odd	4
3380.2.c.d.2029.6		8	13.5	odd	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	260.2.d.a.129.5

Level	$260$
Weight	$2$
Character	260.129
Analytic conductor	$2.076$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$