LMFDB - Embedded newform 1040.2.f.g.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(129,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1040.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.f (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:	$8.30444181021$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 $ x^10 + 11x^8 + 36x^6 + 42x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$-1.56822i$ of defining polynomial
Character	$\chi$	$=$	1040.129
Dual form			1040.2.f.g.129.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14926i q^{3} +(1.82442 - 1.29287i) q^{5} -2.12171 q^{7} -1.61930 q^{9} +O(q^{10})$	$q-2.14926i q^{3} +(1.82442 - 1.29287i) q^{5} -2.12171 q^{7} -1.61930 q^{9} +0.796315i q^{11} +(3.57359 - 0.479016i) q^{13} +(-2.77870 - 3.92114i) q^{15} -4.56010i q^{17} -0.220019i q^{19} +4.56010i q^{21} -4.73499i q^{23} +(1.65699 - 4.71745i) q^{25} -2.96747i q^{27} -8.17671 q^{29} -1.17805i q^{31} +1.71149 q^{33} +(-3.87088 + 2.74309i) q^{35} +0.121710 q^{37} +(-1.02953 - 7.68056i) q^{39} +4.87481i q^{41} -3.16559i q^{43} +(-2.95429 + 2.09354i) q^{45} +2.12171 q^{47} -2.49835 q^{49} -9.80082 q^{51} +11.0043i q^{53} +(1.02953 + 1.45281i) q^{55} -0.472877 q^{57} +3.72871i q^{59} -3.36438 q^{61} +3.43569 q^{63} +(5.90041 - 5.49410i) q^{65} -7.43569 q^{67} -10.1767 q^{69} +15.2081i q^{71} +5.43569 q^{73} +(-10.1390 - 3.56130i) q^{75} -1.68955i q^{77} +9.05575 q^{79} -11.2358 q^{81} +14.1563 q^{83} +(-5.89560 - 8.31952i) q^{85} +17.5738i q^{87} -15.6846i q^{89} +(-7.58212 + 1.01633i) q^{91} -2.53193 q^{93} +(-0.284455 - 0.401406i) q^{95} +2.09143 q^{97} -1.28948i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) $ 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9	$ 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61} - 24 q^{63} - 5 q^{65} - 16 q^{67} - 12 q^{69} - 4 q^{73} - 21 q^{75} - 28 q^{79} + 22 q^{81} + 4 q^{83} - 25 q^{85} - 2 q^{91} + 12 q^{93} + 10 q^{95} + 72 q^{97}+O(q^{100}) $ 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9 + 2 * q^13 + 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 + 3 * q^35 - 22 * q^37 + 12 * q^39 - 4 * q^45 - 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 - 12 * q^57 - 16 * q^61 - 24 * q^63 - 5 * q^65 - 16 * q^67 - 12 * q^69 - 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 + 4 * q^83 - 25 * q^85 - 2 * q^91 + 12 * q^93 + 10 * q^95 + 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$		−	2.14926i		−	1.24087i	−0.784256	−	0.620437i	$-0.786955\pi$
$3$		−	2.14926i		−	1.24087i	0.784256	−	0.620437i	$-0.213045\pi$
$4$		0			0
$5$	1.82442	−	1.29287i	0.815904	−	0.578188i
$5$	1.82442	−	1.29287i	0.815904	−	0.578188i
$6$		0			0
$7$	−2.12171			−0.801931			−0.400966	−	0.916093i	$-0.631325\pi$
$7$	−2.12171			−0.801931			−0.400966	+	0.916093i	$0.631325\pi$
$8$		0			0
$9$	−1.61930			−0.539768
$10$		0			0
$11$			0.796315i			0.240098i	0.992768	+	0.120049i	$0.0383052\pi$
$11$			0.796315i			0.240098i	−0.992768	+	0.120049i	$0.961695\pi$
$12$		0			0
$13$	3.57359	−	0.479016i	0.991135	−	0.132855i
$13$	3.57359	−	0.479016i	0.991135	−	0.132855i
$14$		0			0
$15$	−2.77870	−	3.92114i	−0.717458	−	1.01243i
$16$		0			0
$17$		−	4.56010i		−	1.10599i	−0.833186	−	0.552993i	$-0.813486\pi$
$17$		−	4.56010i		−	1.10599i	0.833186	−	0.552993i	$-0.186514\pi$
$18$		0			0
$19$		−	0.220019i		−	0.0504758i	−0.999681	−	0.0252379i	$-0.991966\pi$
$19$		−	0.220019i		−	0.0504758i	0.999681	−	0.0252379i	$-0.00803432\pi$
$20$		0			0
$21$			4.56010i			0.995095i
$22$		0			0
$23$		−	4.73499i		−	0.987314i	−0.869657	−	0.493657i	$-0.835660\pi$
$23$		−	4.73499i		−	0.987314i	0.869657	−	0.493657i	$-0.164340\pi$
$24$		0			0
$25$	1.65699	−	4.71745i	0.331398	−	0.943491i
$26$		0			0
$27$		−	2.96747i		−	0.571090i
$28$		0			0
$29$	−8.17671			−1.51838			−0.759188	−	0.650871i	$-0.774404\pi$
$29$	−8.17671			−1.51838			−0.759188	+	0.650871i	$0.774404\pi$
$30$		0			0
$31$		−	1.17805i		−	0.211584i	−0.994388	−	0.105792i	$-0.966262\pi$
$31$		−	1.17805i		−	0.211584i	0.994388	−	0.105792i	$-0.0337378\pi$
$32$		0			0
$33$	1.71149			0.297931
$34$		0			0
$35$	−3.87088	+	2.74309i	−0.654299	+	0.463667i
$36$		0			0
$37$	0.121710			0.0200090			0.0100045	−	0.999950i	$-0.496815\pi$
$37$	0.121710			0.0200090			0.0100045	+	0.999950i	$0.496815\pi$
$38$		0			0
$39$	−1.02953	−	7.68056i	−0.164857	−	1.22987i
$40$		0			0
$41$			4.87481i			0.761317i	0.924716	+	0.380659i	$0.124303\pi$
$41$			4.87481i			0.761317i	−0.924716	+	0.380659i	$0.875697\pi$
$42$		0			0
$43$		−	3.16559i		−	0.482748i	−0.970432	−	0.241374i	$-0.922402\pi$
$43$		−	3.16559i		−	0.482748i	0.970432	−	0.241374i	$-0.0775981\pi$
$44$		0			0
$45$	−2.95429	+	2.09354i	−0.440399	+	0.312087i
$46$		0			0
$47$	2.12171			0.309483			0.154742	−	0.987955i	$-0.450545\pi$
$47$	2.12171			0.309483			0.154742	+	0.987955i	$0.450545\pi$
$48$		0			0
$49$	−2.49835			−0.356907
$50$		0			0
$51$	−9.80082			−1.37239
$52$		0			0
$53$			11.0043i			1.51156i	0.654827	+	0.755779i	$0.272742\pi$
$53$			11.0043i			1.51156i	−0.654827	+	0.755779i	$0.727258\pi$
$54$		0			0
$55$	1.02953	+	1.45281i	0.138822	+	0.195897i
$56$		0			0
$57$	−0.472877			−0.0626341
$58$		0			0
$59$			3.72871i			0.485437i	0.970097	+	0.242719i	$0.0780392\pi$
$59$			3.72871i			0.485437i	−0.970097	+	0.242719i	$0.921961\pi$
$60$		0			0
$61$	−3.36438			−0.430764			−0.215382	−	0.976530i	$-0.569100\pi$
$61$	−3.36438			−0.430764			−0.215382	+	0.976530i	$0.569100\pi$
$62$		0			0
$63$	3.43569			0.432857
$64$		0			0
$65$	5.90041	−	5.49410i	0.731856	−	0.681459i
$66$		0			0
$67$	−7.43569			−0.908415			−0.454207	−	0.890896i	$-0.650077\pi$
$67$	−7.43569			−0.908415			−0.454207	+	0.890896i	$0.650077\pi$
$68$		0			0
$69$	−10.1767			−1.22513
$70$		0			0
$71$			15.2081i			1.80487i	0.430823	+	0.902437i	$0.358223\pi$
$71$			15.2081i			1.80487i	−0.430823	+	0.902437i	$0.641777\pi$
$72$		0			0
$73$	5.43569			0.636200			0.318100	−	0.948057i	$-0.396955\pi$
$73$	5.43569			0.636200			0.318100	+	0.948057i	$0.396955\pi$
$74$		0			0
$75$	−10.1390	−	3.56130i	−1.17075	−	0.411224i
$76$		0			0
$77$		−	1.68955i		−	0.192542i
$78$		0			0
$79$	9.05575			1.01885			0.509426	−	0.860515i	$-0.329858\pi$
$79$	9.05575			1.01885			0.509426	+	0.860515i	$0.329858\pi$
$80$		0			0
$81$	−11.2358			−1.24842
$82$		0			0
$83$	14.1563			1.55386			0.776930	−	0.629587i	$-0.216776\pi$
$83$	14.1563			1.55386			0.776930	+	0.629587i	$0.216776\pi$
$84$		0			0
$85$	−5.89560	−	8.31952i	−0.639468	−	0.902379i
$86$		0			0
$87$			17.5738i			1.88411i
$88$		0			0
$89$		−	15.6846i		−	1.66256i	−0.555854	−	0.831280i	$-0.687609\pi$
$89$		−	15.6846i		−	1.66256i	0.555854	−	0.831280i	$-0.312391\pi$
$90$		0			0
$91$	−7.58212	+	1.01633i	−0.794822	+	0.106541i
$92$		0			0
$93$	−2.53193			−0.262549
$94$		0			0
$95$	−0.284455	−	0.401406i	−0.0291845	−	0.0411834i
$96$		0			0
$97$	2.09143			0.212352			0.106176	−	0.994347i	$-0.466139\pi$
$97$	2.09143			0.212352			0.106176	+	0.994347i	$0.466139\pi$
$98$		0			0
$99$		−	1.28948i		−	0.129597i
$100$		0			0
$101$	6.29285			0.626162			0.313081	−	0.949726i	$-0.398639\pi$
$101$	6.29285			0.626162			0.313081	+	0.949726i	$0.398639\pi$
$102$		0			0
$103$			11.9659i			1.17904i	0.807755	+	0.589518i	$0.200682\pi$
$103$			11.9659i			1.17904i	−0.807755	+	0.589518i	$0.799318\pi$
$104$		0			0
$105$	5.89560	+	8.31952i	0.575352	+	0.811902i
$106$		0			0
$107$		−	14.5166i		−	1.40337i	−0.712488	−	0.701685i	$-0.752432\pi$
$107$		−	14.5166i		−	1.40337i	0.712488	−	0.701685i	$-0.247568\pi$
$108$		0			0
$109$		−	7.92736i		−	0.759303i	−0.925130	−	0.379652i	$-0.876044\pi$
$109$		−	7.92736i		−	0.759303i	0.925130	−	0.379652i	$-0.123956\pi$
$110$		0			0
$111$		−	0.261586i		−	0.0248287i
$112$		0			0
$113$		−	14.0651i		−	1.32314i	−0.749884	−	0.661569i	$-0.769891\pi$
$113$		−	14.0651i		−	1.32314i	0.749884	−	0.661569i	$-0.230109\pi$
$114$		0			0
$115$	−6.12171	−	8.63859i	−0.570852	−	0.805553i
$116$		0			0
$117$	−5.78673	+	0.775673i	−0.534983	+	0.0717110i
$118$		0			0
$119$			9.67521i			0.886925i
$120$		0			0
$121$	10.3659			0.942353
$122$		0			0
$123$	10.4772			0.944699
$124$		0			0
$125$	−3.07600	−	10.7489i	−0.275125	−	0.961408i
$126$		0			0
$127$		−	12.0754i		−	1.07152i	−0.844371	−	0.535758i	$-0.820026\pi$
$127$		−	12.0754i		−	1.07152i	0.844371	−	0.535758i	$-0.179974\pi$
$128$		0			0
$129$	−6.80367			−0.599029
$130$		0			0
$131$	−7.22229			−0.631014			−0.315507	−	0.948923i	$-0.602175\pi$
$131$	−7.22229			−0.631014			−0.315507	+	0.948923i	$0.602175\pi$
$132$		0			0
$133$			0.466816i			0.0404781i
$134$		0			0
$135$	−3.83654	−	5.41390i	−0.330197	−	0.465954i
$136$		0			0
$137$	−14.6883			−1.25490			−0.627452	−	0.778656i	$-0.715902\pi$
$137$	−14.6883			−1.25490			−0.627452	+	0.778656i	$0.715902\pi$
$138$		0			0
$139$	17.8503			1.51404			0.757020	−	0.653392i	$-0.226655\pi$
$139$	17.8503			1.51404			0.757020	+	0.653392i	$0.226655\pi$
$140$		0			0
$141$		−	4.56010i		−	0.384030i
$142$		0			0
$143$	0.381448	+	2.84570i	0.0318983	+	0.237970i
$144$		0			0
$145$	−14.9177	+	10.5714i	−1.23885	+	0.877906i
$146$		0			0
$147$			5.36959i			0.442876i
$148$		0			0
$149$			16.6689i			1.36557i	0.730618	+	0.682786i	$0.239232\pi$
$149$			16.6689i			1.36557i	−0.730618	+	0.682786i	$0.760768\pi$
$150$		0			0
$151$		−	0.999600i		−	0.0813463i	−0.999173	−	0.0406732i	$-0.987050\pi$
$151$		−	0.999600i		−	0.0813463i	0.999173	−	0.0406732i	$-0.0129502\pi$
$152$		0			0
$153$			7.38419i			0.596976i
$154$		0			0
$155$	−1.52306	−	2.14926i	−0.122335	−	0.172632i
$156$		0			0
$157$			10.7778i			0.860162i	0.902790	+	0.430081i	$0.141515\pi$
$157$			10.7778i			0.860162i	−0.902790	+	0.430081i	$0.858485\pi$
$158$		0			0
$159$	23.6511			1.87565
$160$		0			0
$161$			10.0463i			0.791757i
$162$		0			0
$163$	−2.91622			−0.228416			−0.114208	−	0.993457i	$-0.536433\pi$
$163$	−2.91622			−0.228416			−0.114208	+	0.993457i	$0.536433\pi$
$164$		0			0
$165$	3.12246	−	2.21272i	0.243083	−	0.172260i
$166$		0			0
$167$	19.9225			1.54165			0.770826	−	0.637046i	$-0.219844\pi$
$167$	19.9225			1.54165			0.770826	+	0.637046i	$0.219844\pi$
$168$		0			0
$169$	12.5411	−	3.42362i	0.964699	−	0.263355i
$170$		0			0
$171$			0.356277i			0.0272452i
$172$		0			0
$173$		−	17.6853i		−	1.34459i	−0.740285	−	0.672293i	$-0.765309\pi$
$173$		−	17.6853i		−	1.34459i	0.740285	−	0.672293i	$-0.234691\pi$
$174$		0			0
$175$	−3.51566	+	10.0091i	−0.265759	+	0.756615i
$176$		0			0
$177$	8.01396			0.602366
$178$		0			0
$179$	−9.28135			−0.693721			−0.346860	−	0.937917i	$-0.612752\pi$
$179$	−9.28135			−0.693721			−0.346860	+	0.937917i	$0.612752\pi$
$180$		0			0
$181$	21.9022			1.62797			0.813987	−	0.580883i	$-0.197293\pi$
$181$	21.9022			1.62797			0.813987	+	0.580883i	$0.197293\pi$
$182$		0			0
$183$			7.23091i			0.534524i
$184$		0			0
$185$	0.222050	−	0.157355i	0.0163254	−	0.0115690i
$186$		0			0
$187$	3.63128			0.265545
$188$		0			0
$189$			6.29611i			0.457975i
$190$		0			0
$191$	23.0822			1.67017			0.835084	−	0.550123i	$-0.185419\pi$
$191$	23.0822			1.67017			0.835084	+	0.550123i	$0.185419\pi$
$192$		0			0
$193$	18.0186			1.29700			0.648502	−	0.761213i	$-0.275396\pi$
$193$	18.0186			1.29700			0.648502	+	0.761213i	$0.275396\pi$
$194$		0			0
$195$	−11.8082	−	12.6815i	−0.845605	−	0.908141i
$196$		0			0
$197$	−7.75883			−0.552794			−0.276397	−	0.961044i	$-0.589140\pi$
$197$	−7.75883			−0.552794			−0.276397	+	0.961044i	$0.589140\pi$
$198$		0			0
$199$	10.9641			0.777221			0.388611	−	0.921402i	$-0.372955\pi$
$199$	10.9641			0.777221			0.388611	+	0.921402i	$0.372955\pi$
$200$		0			0
$201$			15.9812i			1.12723i
$202$		0			0
$203$	17.3486			1.21763
$204$		0			0
$205$	6.30248	+	8.89368i	0.440184	+	0.621162i
$206$		0			0
$207$			7.66739i			0.532920i
$208$		0			0
$209$	0.175204			0.0121191
$210$		0			0
$211$	9.28135			0.638954			0.319477	−	0.947594i	$-0.396493\pi$
$211$	9.28135			0.638954			0.319477	+	0.947594i	$0.396493\pi$
$212$		0			0
$213$	32.6862			2.23962
$214$		0			0
$215$	−4.09269	−	5.77536i	−0.279119	−	0.393876i
$216$		0			0
$217$			2.49948i			0.169676i
$218$		0			0
$219$		−	11.6827i		−	0.789444i
$220$		0			0
$221$	−2.18436	−	16.2959i	−0.146936	−	1.09618i
$222$		0			0
$223$	−8.71704			−0.583736			−0.291868	−	0.956459i	$-0.594277\pi$
$223$	−8.71704			−0.583736			−0.291868	+	0.956459i	$0.594277\pi$
$224$		0			0
$225$	−2.68317	+	7.63899i	−0.178878	+	0.509266i
$226$		0			0
$227$	−26.6758			−1.77054			−0.885268	−	0.465082i	$-0.846025\pi$
$227$	−26.6758			−1.77054			−0.885268	+	0.465082i	$0.846025\pi$
$228$		0			0
$229$			28.6735i			1.89480i	0.320058	+	0.947398i	$0.396298\pi$
$229$			28.6735i			1.89480i	−0.320058	+	0.947398i	$0.603702\pi$
$230$		0			0
$231$	−3.63128			−0.238920
$232$		0			0
$233$			1.07818i			0.0706341i	0.999376	+	0.0353171i	$0.0112441\pi$
$233$			1.07818i			0.0706341i	−0.999376	+	0.0353171i	$0.988756\pi$
$234$		0			0
$235$	3.87088	−	2.74309i	0.252509	−	0.178939i
$236$		0			0
$237$		−	19.4631i		−	1.26427i
$238$		0			0
$239$			20.8728i			1.35015i	0.737751	+	0.675073i	$0.235888\pi$
$239$			20.8728i			1.35015i	−0.737751	+	0.675073i	$0.764112\pi$
$240$		0			0
$241$			0.466816i			0.0300703i	0.999887	+	0.0150351i	$0.00478601\pi$
$241$			0.466816i			0.0300703i	−0.999887	+	0.0150351i	$0.995214\pi$
$242$		0			0
$243$			15.2461i			0.978040i
$244$		0			0
$245$	−4.55802	+	3.23003i	−0.291201	+	0.206359i
$246$		0			0
$247$	−0.105393	−	0.786257i	−0.00670597	−	0.0500283i
$248$		0			0
$249$		−	30.4256i		−	1.92814i
$250$		0			0
$251$	−2.42778			−0.153240			−0.0766201	−	0.997060i	$-0.524413\pi$
$251$	−2.42778			−0.153240			−0.0766201	+	0.997060i	$0.524413\pi$
$252$		0			0
$253$	3.77054			0.237052
$254$		0			0
$255$	−17.8808	+	12.6712i	−1.11974	+	0.793499i
$256$		0			0
$257$			17.8618i			1.11419i	0.830450	+	0.557094i	$0.188084\pi$
$257$			17.8618i			1.11419i	−0.830450	+	0.557094i	$0.811916\pi$
$258$		0			0
$259$	−0.258234			−0.0160459
$260$		0			0
$261$	13.2406			0.819571
$262$		0			0
$263$			25.7344i			1.58685i	0.608668	+	0.793425i	$0.291704\pi$
$263$			25.7344i			1.58685i	−0.608668	+	0.793425i	$0.708296\pi$
$264$		0			0
$265$	14.2271	+	20.0764i	0.873964	+	1.23329i
$266$		0			0
$267$	−33.7101			−2.06303
$268$		0			0
$269$	−6.41097			−0.390884			−0.195442	−	0.980715i	$-0.562614\pi$
$269$	−6.41097			−0.390884			−0.195442	+	0.980715i	$0.562614\pi$
$270$		0			0
$271$		−	21.3789i		−	1.29868i	−0.760500	−	0.649338i	$-0.775046\pi$
$271$		−	21.3789i		−	1.29868i	0.760500	−	0.649338i	$-0.224954\pi$
$272$		0			0
$273$	2.18436	+	16.2959i	0.132204	+	0.986274i
$274$		0			0
$275$	3.75658	+	1.31949i	0.226530	+	0.0795681i
$276$		0			0
$277$			6.71292i			0.403340i	0.979454	+	0.201670i	$0.0646369\pi$
$277$			6.71292i			0.403340i	−0.979454	+	0.201670i	$0.935363\pi$
$278$		0			0
$279$			1.90762i			0.114206i
$280$		0			0
$281$		−	24.5081i		−	1.46203i	−0.682361	−	0.731015i	$-0.739047\pi$
$281$		−	24.5081i		−	1.46203i	0.682361	−	0.731015i	$-0.260953\pi$
$282$		0			0
$283$		−	3.55044i		−	0.211052i	−0.994417	−	0.105526i	$-0.966347\pi$
$283$		−	3.55044i		−	0.211052i	0.994417	−	0.105526i	$-0.0336526\pi$
$284$		0			0
$285$	−0.862724	+	0.611367i	−0.0511034	+	0.0362142i
$286$		0			0
$287$		−	10.3429i		−	0.610524i
$288$		0			0
$289$	−3.79451			−0.223206
$290$		0			0
$291$		−	4.49502i		−	0.263503i
$292$		0			0
$293$	−1.36844			−0.0799450			−0.0399725	−	0.999201i	$-0.512727\pi$
$293$	−1.36844			−0.0799450			−0.0399725	+	0.999201i	$0.512727\pi$
$294$		0			0
$295$	4.82073	+	6.80273i	0.280674	+	0.396070i
$296$		0			0
$297$	2.36304			0.137118
$298$		0			0
$299$	−2.26814	−	16.9209i	−0.131170	−	0.978562i
$300$		0			0
$301$			6.71647i			0.387131i
$302$		0			0
$303$		−	13.5250i		−	0.776989i
$304$		0			0
$305$	−6.13803	+	4.34969i	−0.351462	+	0.249063i
$306$		0			0
$307$	8.92584			0.509425			0.254712	−	0.967017i	$-0.418019\pi$
$307$	8.92584			0.509425			0.254712	+	0.967017i	$0.418019\pi$
$308$		0			0
$309$	25.7178			1.46303
$310$		0			0
$311$	13.1052			0.743127			0.371563	−	0.928408i	$-0.378822\pi$
$311$	13.1052			0.743127			0.371563	+	0.928408i	$0.378822\pi$
$312$		0			0
$313$			12.5277i			0.708108i	0.935225	+	0.354054i	$0.115197\pi$
$313$			12.5277i			0.708108i	−0.935225	+	0.354054i	$0.884803\pi$
$314$		0			0
$315$	6.26814	−	4.44189i	0.353170	−	0.250272i
$316$		0			0
$317$	−0.455813			−0.0256010			−0.0128005	−	0.999918i	$-0.504075\pi$
$317$	−0.455813			−0.0256010			−0.0128005	+	0.999918i	$0.504075\pi$
$318$		0			0
$319$		−	6.51124i		−	0.364559i
$320$		0			0
$321$	−31.1998			−1.74140
$322$		0			0
$323$	−1.00331			−0.0558255
$324$		0			0
$325$	3.66167	−	17.6520i	0.203113	−	0.979155i
$326$		0			0
$327$	−17.0379			−0.942200
$328$		0			0
$329$	−4.50165			−0.248184
$330$		0			0
$331$		−	3.67559i		−	0.202029i	−0.994885	−	0.101014i	$-0.967791\pi$
$331$		−	3.67559i		−	0.202029i	0.994885	−	0.101014i	$-0.0322088\pi$
$332$		0			0
$333$	−0.197086			−0.0108002
$334$		0			0
$335$	−13.5658	+	9.61336i	−0.741179	+	0.525234i
$336$		0			0
$337$			16.2959i			0.887696i	0.896102	+	0.443848i	$0.146387\pi$
$337$			16.2959i			0.887696i	−0.896102	+	0.443848i	$0.853613\pi$
$338$		0			0
$339$	−30.2296			−1.64185
$340$		0			0
$341$	0.938100			0.0508010
$342$		0			0
$343$	20.1527			1.08815
$344$		0			0
$345$	−18.5666	+	13.1571i	−0.999590	+	0.708356i
$346$		0			0
$347$		−	0.809921i		−	0.0434788i	−0.999764	−	0.0217394i	$-0.993080\pi$
$347$		−	0.809921i		−	0.0434788i	0.999764	−	0.0217394i	$-0.00692042\pi$
$348$		0			0
$349$			17.7333i			0.949244i	0.880190	+	0.474622i	$0.157415\pi$
$349$			17.7333i			0.949244i	−0.880190	+	0.474622i	$0.842585\pi$
$350$		0			0
$351$	−1.42147	−	10.6045i	−0.0758723	−	0.566027i
$352$		0			0
$353$	1.80442			0.0960395			0.0480198	−	0.998846i	$-0.484709\pi$
$353$	1.80442			0.0960395			0.0480198	+	0.998846i	$0.484709\pi$
$354$		0			0
$355$	19.6621	+	27.7460i	1.04356	+	1.47260i
$356$		0			0
$357$	20.7945			1.10056
$358$		0			0
$359$		−	26.4760i		−	1.39735i	−0.715440	−	0.698674i	$-0.753774\pi$
$359$		−	26.4760i		−	1.39735i	0.715440	−	0.698674i	$-0.246226\pi$
$360$		0			0
$361$	18.9516			0.997452
$362$		0			0
$363$		−	22.2789i		−	1.16934i
$364$		0			0
$365$	9.91697	−	7.02763i	0.519078	−	0.367843i
$366$		0			0
$367$			30.5490i			1.59464i	0.603555	+	0.797321i	$0.293750\pi$
$367$			30.5490i			1.59464i	−0.603555	+	0.797321i	$0.706250\pi$
$368$		0			0
$369$		−	7.89380i		−	0.410935i
$370$		0			0
$371$		−	23.3480i		−	1.21217i
$372$		0			0
$373$			1.72151i			0.0891362i	0.999006	+	0.0445681i	$0.0141912\pi$
$373$			1.72151i			0.0891362i	−0.999006	+	0.0445681i	$0.985809\pi$
$374$		0			0
$375$	−23.1021	+	6.61110i	−1.19299	+	0.341396i
$376$		0			0
$377$	−29.2202	+	3.91678i	−1.50492	+	0.201724i
$378$		0			0
$379$			32.5223i			1.67056i	0.549825	+	0.835280i	$0.314695\pi$
$379$			32.5223i			1.67056i	−0.549825	+	0.835280i	$0.685305\pi$
$380$		0			0
$381$	−25.9531			−1.32962
$382$		0			0
$383$	24.7170			1.26298			0.631491	−	0.775383i	$-0.282443\pi$
$383$	24.7170			1.26298			0.631491	+	0.775383i	$0.282443\pi$
$384$		0			0
$385$	−2.18436	−	3.08244i	−0.111325	−	0.157096i
$386$		0			0
$387$			5.12605i			0.260572i
$388$		0			0
$389$	−14.1181			−0.715817			−0.357908	−	0.933757i	$-0.616510\pi$
$389$	−14.1181			−0.715817			−0.357908	+	0.933757i	$0.616510\pi$
$390$		0			0
$391$	−21.5920			−1.09196
$392$		0			0
$393$			15.5226i			0.783009i
$394$		0			0
$395$	16.5215	−	11.7079i	0.831285	−	0.589087i
$396$		0			0
$397$	−5.15964			−0.258955			−0.129477	−	0.991582i	$-0.541330\pi$
$397$	−5.15964			−0.258955			−0.129477	+	0.991582i	$0.541330\pi$
$398$		0			0
$399$	1.00331			0.0502282
$400$		0			0
$401$		−	11.7358i		−	0.586059i	−0.956103	−	0.293030i	$-0.905337\pi$
$401$		−	11.7358i		−	0.586059i	0.956103	−	0.293030i	$-0.0946634\pi$
$402$		0			0
$403$	−0.564306	−	4.20987i	−0.0281101	−	0.209709i
$404$		0			0
$405$	−20.4987	+	14.5263i	−1.01859	+	0.721820i
$406$		0			0
$407$			0.0969196i			0.00480413i
$408$		0			0
$409$			21.4394i			1.06011i	0.847963	+	0.530056i	$0.177829\pi$
$409$			21.4394i			1.06011i	−0.847963	+	0.530056i	$0.822171\pi$
$410$		0			0
$411$			31.5689i			1.55718i
$412$		0			0
$413$		−	7.91125i		−	0.389287i
$414$		0			0
$415$	25.8270	−	18.3022i	1.26780	−	0.898422i
$416$		0			0
$417$		−	38.3648i		−	1.87873i
$418$		0			0
$419$	−27.9618			−1.36602			−0.683011	−	0.730408i	$-0.739330\pi$
$419$	−27.9618			−1.36602			−0.683011	+	0.730408i	$0.739330\pi$
$420$		0			0
$421$		−	11.1126i		−	0.541596i	−0.962636	−	0.270798i	$-0.912712\pi$
$421$		−	11.1126i		−	0.541596i	0.962636	−	0.270798i	$-0.0872875\pi$
$422$		0			0
$423$	−3.43569			−0.167049
$424$		0			0
$425$	−21.5121	−	7.55605i	−1.04349	−	0.366522i
$426$		0			0
$427$	7.13824			0.345443
$428$		0			0
$429$	6.11615	−	0.819829i	0.295290	−	0.0395817i
$430$		0			0
$431$		−	14.5679i		−	0.701712i	−0.936429	−	0.350856i	$-0.885891\pi$
$431$		−	14.5679i		−	0.701712i	0.936429	−	0.350856i	$-0.114109\pi$
$432$		0			0
$433$		−	15.7268i		−	0.755779i	−0.925851	−	0.377890i	$-0.876650\pi$
$433$		−	15.7268i		−	0.755779i	0.925851	−	0.377890i	$-0.123350\pi$
$434$		0			0
$435$	22.7206	+	32.0620i	1.08937	+	1.53726i
$436$		0			0
$437$	−1.04179			−0.0498354
$438$		0			0
$439$	−7.16424			−0.341931			−0.170965	−	0.985277i	$-0.554689\pi$
$439$	−7.16424			−0.341931			−0.170965	+	0.985277i	$0.554689\pi$
$440$		0			0
$441$	4.04558			0.192647
$442$		0			0
$443$		−	19.6963i		−	0.935801i	−0.883781	−	0.467900i	$-0.845011\pi$
$443$		−	19.6963i		−	0.935801i	0.883781	−	0.467900i	$-0.154989\pi$
$444$		0			0
$445$	−20.2780	−	28.6152i	−0.961271	−	1.35649i
$446$		0			0
$447$	35.8258			1.69450
$448$		0			0
$449$		−	6.34820i		−	0.299590i	−0.988717	−	0.149795i	$-0.952139\pi$
$449$		−	6.34820i		−	0.299590i	0.988717	−	0.149795i	$-0.0478614\pi$
$450$		0			0
$451$	−3.88188			−0.182791
$452$		0			0
$453$	−2.14840			−0.100941
$454$		0			0
$455$	−12.5190	+	11.6569i	−0.586898	+	0.546483i
$456$		0			0
$457$	14.6109			0.683469			0.341735	−	0.939796i	$-0.388986\pi$
$457$	14.6109			0.683469			0.341735	+	0.939796i	$0.388986\pi$
$458$		0			0
$459$	−13.5320			−0.631618
$460$		0			0
$461$		−	2.52938i		−	0.117805i	−0.998264	−	0.0589024i	$-0.981240\pi$
$461$		−	2.52938i		−	0.117805i	0.998264	−	0.0589024i	$-0.0187601\pi$
$462$		0			0
$463$	−11.4357			−0.531462			−0.265731	−	0.964047i	$-0.585613\pi$
$463$	−11.4357			−0.531462			−0.265731	+	0.964047i	$0.585613\pi$
$464$		0			0
$465$	−4.61930	+	3.27345i	−0.214215	+	0.151803i
$466$		0			0
$467$		−	14.8931i		−	0.689172i	−0.938755	−	0.344586i	$-0.888019\pi$
$467$		−	14.8931i		−	0.689172i	0.938755	−	0.344586i	$-0.111981\pi$
$468$		0			0
$469$	15.7764			0.728486
$470$		0			0
$471$	23.1642			1.06735
$472$		0			0
$473$	2.52081			0.115907
$474$		0			0
$475$	−1.03793	−	0.364569i	−0.0476234	−	0.0167276i
$476$		0			0
$477$		−	17.8193i		−	0.815891i
$478$		0			0
$479$			32.1619i			1.46951i	0.678330	+	0.734757i	$0.262704\pi$
$479$			32.1619i			1.46951i	−0.678330	+	0.734757i	$0.737296\pi$
$480$		0			0
$481$	0.434942	−	0.0583011i	0.0198316	−	0.00265830i
$482$		0			0
$483$	21.5920			0.982471
$484$		0			0
$485$	3.81564	−	2.70394i	0.173259	−	0.122780i
$486$		0			0
$487$	15.7484			0.713626			0.356813	−	0.934176i	$-0.383863\pi$
$487$	15.7484			0.713626			0.356813	+	0.934176i	$0.383863\pi$
$488$		0			0
$489$			6.26770i			0.283435i
$490$		0			0
$491$	1.19587			0.0539688			0.0269844	−	0.999636i	$-0.491410\pi$
$491$	1.19587			0.0539688			0.0269844	+	0.999636i	$0.491410\pi$
$492$		0			0
$493$			37.2866i			1.67930i
$494$		0			0
$495$	−1.66712	−	2.35254i	−0.0749315	−	0.105739i
$496$		0			0
$497$		−	32.2672i		−	1.44738i
$498$		0			0
$499$			41.0855i			1.83924i	0.392812	+	0.919619i	$0.371502\pi$
$499$			41.0855i			1.83924i	−0.392812	+	0.919619i	$0.628498\pi$
$500$		0			0
$501$		−	42.8186i		−	1.91300i
$502$		0			0
$503$		−	34.9278i		−	1.55736i	−0.627424	−	0.778678i	$-0.715891\pi$
$503$		−	34.9278i		−	1.55736i	0.627424	−	0.778678i	$-0.284109\pi$
$504$		0			0
$505$	11.4808	−	8.13582i	0.510888	−	0.362039i
$506$		0			0
$507$	−7.35823	−	26.9540i	−0.326790	−	1.19707i
$508$		0			0
$509$		−	6.97956i		−	0.309364i	−0.987964	−	0.154682i	$-0.950565\pi$
$509$		−	6.97956i		−	0.309364i	0.987964	−	0.154682i	$-0.0494352\pi$
$510$		0			0
$511$	−11.5330			−0.510188
$512$		0			0
$513$	−0.652899			−0.0288262
$514$		0			0
$515$	15.4703	+	21.8308i	0.681704	+	0.961980i
$516$		0			0
$517$			1.68955i			0.0743063i
$518$		0			0
$519$	−38.0102			−1.66846
$520$		0			0
$521$	−7.80036			−0.341740			−0.170870	−	0.985294i	$-0.554658\pi$
$521$	−7.80036			−0.341740			−0.170870	+	0.985294i	$0.554658\pi$
$522$		0			0
$523$			40.2344i			1.75933i	0.475596	+	0.879664i	$0.342232\pi$
$523$			40.2344i			1.75933i	−0.475596	+	0.879664i	$0.657768\pi$
$524$		0			0
$525$	21.5121	+	7.55605i	0.938863	+	0.329773i
$526$		0			0
$527$	−5.37203			−0.234009
$528$		0			0
$529$	0.579871			0.0252118
$530$		0			0
$531$		−	6.03792i		−	0.262024i
$532$		0			0
$533$	2.33511	+	17.4206i	0.101145	+	0.754569i
$534$		0			0
$535$	−18.7680	−	26.4843i	−0.811411	−	1.14501i
$536$		0			0
$537$			19.9480i			0.860820i
$538$		0			0
$539$		−	1.98947i		−	0.0856926i
$540$		0			0
$541$			1.93217i			0.0830706i	0.999137	+	0.0415353i	$0.0132249\pi$
$541$			1.93217i			0.0830706i	−0.999137	+	0.0415353i	$0.986775\pi$
$542$		0			0
$543$		−	47.0734i		−	2.02011i
$544$		0			0
$545$	−10.2490	−	14.4628i	−0.439020	−	0.619519i
$546$		0			0
$547$		−	0.299309i		−	0.0127975i	−0.999980	−	0.00639876i	$-0.997963\pi$
$547$		−	0.299309i		−	0.0127975i	0.999980	−	0.00639876i	$-0.00203680\pi$
$548$		0			0
$549$	5.44795			0.232513
$550$		0			0
$551$			1.79903i			0.0766412i
$552$		0			0
$553$	−19.2137			−0.817049
$554$		0			0
$555$	−0.338196	−	0.477242i	−0.0143556	−	0.0202578i
$556$		0			0
$557$	−37.6211			−1.59406			−0.797028	−	0.603943i	$-0.793596\pi$
$557$	−37.6211			−1.59406			−0.797028	+	0.603943i	$0.793596\pi$
$558$		0			0
$559$	−1.51637	−	11.3125i	−0.0641356	−	0.478469i
$560$		0			0
$561$		−	7.80454i		−	0.329508i
$562$		0			0
$563$		−	7.42476i		−	0.312917i	−0.987685	−	0.156458i	$-0.949992\pi$
$563$		−	7.42476i		−	0.312917i	0.987685	−	0.156458i	$-0.0500077\pi$
$564$		0			0
$565$	−18.1844	−	25.6607i	−0.765022	−	1.07955i
$566$		0			0
$567$	23.8390			1.00115
$568$		0			0
$569$	42.0875			1.76440			0.882201	−	0.470874i	$-0.156061\pi$
$569$	42.0875			1.76440			0.882201	+	0.470874i	$0.156061\pi$
$570$		0			0
$571$	20.2286			0.846541			0.423270	−	0.906003i	$-0.360882\pi$
$571$	20.2286			0.846541			0.423270	+	0.906003i	$0.360882\pi$
$572$		0			0
$573$		−	49.6095i		−	2.07247i
$574$		0			0
$575$	−22.3371	−	7.84584i	−0.931521	−	0.327194i
$576$		0			0
$577$	−18.1986			−0.757617			−0.378809	−	0.925475i	$-0.623666\pi$
$577$	−18.1986			−0.757617			−0.378809	+	0.925475i	$0.623666\pi$
$578$		0			0
$579$		−	38.7265i		−	1.60942i
$580$		0			0
$581$	−30.0356			−1.24609
$582$		0			0
$583$	−8.76290			−0.362922
$584$		0			0
$585$	−9.55456	+	8.89662i	−0.395033	+	0.367830i
$586$		0			0
$587$	−30.4324			−1.25608			−0.628040	−	0.778181i	$-0.716142\pi$
$587$	−30.4324			−1.25608			−0.628040	+	0.778181i	$0.716142\pi$
$588$		0			0
$589$	−0.259193			−0.0106799
$590$		0			0
$591$			16.6757i			0.685947i
$592$		0			0
$593$	18.3114			0.751960			0.375980	−	0.926628i	$-0.377306\pi$
$593$	18.3114			0.751960			0.375980	+	0.926628i	$0.377306\pi$
$594$		0			0
$595$	12.5088	+	17.6516i	0.512809	+	0.723646i
$596$		0			0
$597$		−	23.5646i		−	0.964434i
$598$		0			0
$599$	−32.0692			−1.31031			−0.655157	−	0.755493i	$-0.727397\pi$
$599$	−32.0692			−1.31031			−0.655157	+	0.755493i	$0.727397\pi$
$600$		0			0
$601$	21.4255			0.873964			0.436982	−	0.899470i	$-0.356047\pi$
$601$	21.4255			0.873964			0.436982	+	0.899470i	$0.356047\pi$
$602$		0			0
$603$	12.0407			0.490333
$604$		0			0
$605$	18.9117	−	13.4017i	0.768869	−	0.544857i
$606$		0			0
$607$		−	25.7076i		−	1.04344i	−0.853117	−	0.521720i	$-0.825291\pi$
$607$		−	25.7076i		−	1.04344i	0.853117	−	0.521720i	$-0.174709\pi$
$608$		0			0
$609$		−	37.2866i		−	1.51093i
$610$		0			0
$611$	7.58212	−	1.01633i	0.306740	−	0.0411165i
$612$		0			0
$613$	−47.7114			−1.92704			−0.963522	−	0.267628i	$-0.913760\pi$
$613$	−47.7114			−1.92704			−0.963522	+	0.267628i	$0.913760\pi$
$614$		0			0
$615$	19.1148	−	13.5456i	0.770783	−	0.546213i
$616$		0			0
$617$	43.5087			1.75159			0.875797	−	0.482679i	$-0.160336\pi$
$617$	43.5087			1.75159			0.875797	+	0.482679i	$0.160336\pi$
$618$		0			0
$619$		−	27.8570i		−	1.11967i	−0.828605	−	0.559834i	$-0.810865\pi$
$619$		−	27.8570i		−	1.11967i	0.828605	−	0.559834i	$-0.189135\pi$
$620$		0			0
$621$	−14.0509			−0.563845
$622$		0			0
$623$			33.2781i			1.33326i
$624$		0			0
$625$	−19.5088	−	15.6336i	−0.780350	−	0.625343i
$626$		0			0
$627$		−	0.376559i		−	0.0150383i
$628$		0			0
$629$		−	0.555010i		−	0.0221297i
$630$		0			0
$631$			26.4973i			1.05484i	0.849605	+	0.527420i	$0.176840\pi$
$631$			26.4973i			1.05484i	−0.849605	+	0.527420i	$0.823160\pi$
$632$		0			0
$633$		−	19.9480i		−	0.792862i
$634$		0			0
$635$	−15.6119	−	22.0305i	−0.619538	−	0.874255i
$636$		0			0
$637$	−8.92806	+	1.19675i	−0.353743	+	0.0474169i
$638$		0			0
$639$		−	24.6266i		−	0.974213i
$640$		0			0
$641$	−8.96406			−0.354059			−0.177029	−	0.984206i	$-0.556649\pi$
$641$	−8.96406			−0.354059			−0.177029	+	0.984206i	$0.556649\pi$
$642$		0			0
$643$	4.95848			0.195543			0.0977716	−	0.995209i	$-0.468829\pi$
$643$	4.95848			0.195543			0.0977716	+	0.995209i	$0.468829\pi$
$644$		0			0
$645$	−12.4127	+	8.79623i	−0.488750	+	0.346351i
$646$		0			0
$647$			16.3038i			0.640967i	0.947254	+	0.320483i	$0.103845\pi$
$647$			16.3038i			0.640967i	−0.947254	+	0.320483i	$0.896155\pi$
$648$		0			0
$649$	−2.96923			−0.116553
$650$		0			0
$651$	5.37203			0.210547
$652$		0			0
$653$			16.8293i			0.658583i	0.944228	+	0.329291i	$0.106810\pi$
$653$			16.8293i			0.658583i	−0.944228	+	0.329291i	$0.893190\pi$
$654$		0			0
$655$	−13.1765	+	9.33746i	−0.514847	+	0.364845i
$656$		0			0
$657$	−8.80204			−0.343400
$658$		0			0
$659$	33.0231			1.28640			0.643199	−	0.765699i	$-0.277607\pi$
$659$	33.0231			1.28640			0.643199	+	0.765699i	$0.277607\pi$
$660$		0			0
$661$			39.0519i			1.51894i	0.650541	+	0.759472i	$0.274542\pi$
$661$			39.0519i			1.51894i	−0.650541	+	0.759472i	$0.725458\pi$
$662$		0			0
$663$	−35.0241	+	4.69475i	−1.36022	+	0.182329i
$664$		0			0
$665$	0.603531	+	0.851667i	0.0234039	+	0.0330262i
$666$		0			0
$667$			38.7166i			1.49911i
$668$		0			0
$669$			18.7352i			0.724343i
$670$		0			0
$671$		−	2.67910i		−	0.103426i
$672$		0			0
$673$		−	41.8467i		−	1.61307i	−0.591185	−	0.806536i	$-0.701340\pi$
$673$		−	41.8467i		−	1.61307i	0.591185	−	0.806536i	$-0.298660\pi$
$674$		0			0
$675$	−13.9989	−	4.91707i	−0.538818	−	0.189258i
$676$		0			0
$677$			42.6066i			1.63751i	0.574146	+	0.818753i	$0.305334\pi$
$677$			42.6066i			1.63751i	−0.574146	+	0.818753i	$0.694666\pi$
$678$		0			0
$679$	−4.43741			−0.170292
$680$		0			0
$681$			57.3332i			2.19701i
$682$		0			0
$683$	−15.5212			−0.593901			−0.296951	−	0.954893i	$-0.595970\pi$
$683$	−15.5212			−0.593901			−0.296951	+	0.954893i	$0.595970\pi$
$684$		0			0
$685$	−26.7975	+	18.9900i	−1.02388	+	0.725569i
$686$		0			0
$687$	61.6266			2.35120
$688$		0			0
$689$	5.27124	+	39.3249i	0.200818	+	1.49816i
$690$		0			0
$691$		−	38.8594i		−	1.47828i	−0.673551	−	0.739140i	$-0.735232\pi$
$691$		−	38.8594i		−	1.47828i	0.673551	−	0.739140i	$-0.264768\pi$
$692$		0			0
$693$			2.73589i			0.103928i
$694$		0			0
$695$	32.5663	−	23.0780i	1.23531	−	0.875399i
$696$		0			0
$697$	22.2296			0.842007
$698$		0			0
$699$	2.31729			0.0876480
$700$		0			0
$701$	−30.3813			−1.14749			−0.573744	−	0.819035i	$-0.694509\pi$
$701$	−30.3813			−1.14749			−0.573744	+	0.819035i	$0.694509\pi$
$702$		0			0
$703$		−	0.0267785i		−	0.00100997i
$704$		0			0
$705$	−5.89560	−	8.31952i	−0.222041	−	0.313331i
$706$		0			0
$707$	−13.3516			−0.502139
$708$		0			0
$709$		−	30.3070i		−	1.13820i	−0.822267	−	0.569102i	$-0.807291\pi$
$709$		−	30.3070i		−	1.13820i	0.822267	−	0.569102i	$-0.192709\pi$
$710$		0			0
$711$	−14.6640			−0.549944
$712$		0			0
$713$	−5.57806			−0.208900
$714$		0			0
$715$	4.37503	+	4.69859i	0.163617	+	0.175717i
$716$		0			0
$717$	44.8609			1.67536
$718$		0			0
$719$	−42.1802			−1.57306			−0.786528	−	0.617555i	$-0.788123\pi$
$719$	−42.1802			−1.57306			−0.786528	+	0.617555i	$0.788123\pi$
$720$		0			0
$721$		−	25.3882i		−	0.945505i
$722$		0			0
$723$	1.00331			0.0373134
$724$		0			0
$725$	−13.5487	+	38.5733i	−0.503188	+	1.43257i
$726$		0			0
$727$		−	33.6752i		−	1.24894i	−0.781047	−	0.624472i	$-0.785314\pi$
$727$		−	33.6752i		−	1.24894i	0.781047	−	0.624472i	$-0.214686\pi$
$728$		0			0
$729$	−0.939438			−0.0347940
$730$		0			0
$731$	−14.4354			−0.533913
$732$		0			0
$733$	32.8827			1.21455			0.607276	−	0.794491i	$-0.292262\pi$
$733$	32.8827			1.21455			0.607276	+	0.794491i	$0.292262\pi$
$734$		0			0
$735$	6.94216	+	9.79636i	0.256065	+	0.361344i
$736$		0			0
$737$		−	5.92115i		−	0.218109i
$738$		0			0
$739$			14.8051i			0.544615i	0.962210	+	0.272307i	$0.0877867\pi$
$739$			14.8051i			0.544615i	−0.962210	+	0.272307i	$0.912213\pi$
$740$		0			0
$741$	−1.68987	+	0.226516i	−0.0620789	+	0.00832126i
$742$		0			0
$743$	−23.5884			−0.865376			−0.432688	−	0.901544i	$-0.642435\pi$
$743$	−23.5884			−0.865376			−0.432688	+	0.901544i	$0.642435\pi$
$744$		0			0
$745$	21.5507	+	30.4111i	0.789557	+	1.11418i
$746$		0			0
$747$	−22.9234			−0.838724
$748$		0			0
$749$			30.7999i			1.12541i
$750$		0			0
$751$	−40.5393			−1.47930			−0.739650	−	0.672992i	$-0.765009\pi$
$751$	−40.5393			−1.47930			−0.739650	+	0.672992i	$0.765009\pi$
$752$		0			0
$753$			5.21793i			0.190152i
$754$		0			0
$755$	−1.29235	−	1.82369i	−0.0470334	−	0.0663708i
$756$		0			0
$757$			43.8231i			1.59278i	0.604784	+	0.796389i	$0.293259\pi$
$757$			43.8231i			1.59278i	−0.604784	+	0.796389i	$0.706741\pi$
$758$		0			0
$759$		−	8.10387i		−	0.294152i
$760$		0			0
$761$			20.5134i			0.743609i	0.928311	+	0.371804i	$0.121261\pi$
$761$			20.5134i			0.743609i	−0.928311	+	0.371804i	$0.878739\pi$
$762$		0			0
$763$			16.8196i			0.608909i
$764$		0			0
$765$	9.54677	+	13.4718i	0.345164	+	0.487075i
$766$		0			0
$767$	1.78611	+	13.3249i	0.0644929	+	0.481134i
$768$		0			0
$769$		−	38.6434i		−	1.39352i	−0.717306	−	0.696758i	$-0.754625\pi$
$769$		−	38.6434i		−	1.39352i	0.717306	−	0.696758i	$-0.245375\pi$
$770$		0			0
$771$	38.3895			1.38257
$772$		0			0
$773$	37.2827			1.34097			0.670484	−	0.741924i	$-0.266087\pi$
$773$	37.2827			1.34097			0.670484	+	0.741924i	$0.266087\pi$
$774$		0			0
$775$	−5.55740	−	1.95202i	−0.199628	−	0.0701187i
$776$		0			0
$777$			0.555010i			0.0199109i
$778$		0			0
$779$	1.07255			0.0384281
$780$		0			0
$781$	−12.1105			−0.433346
$782$		0			0
$783$			24.2641i			0.867129i
$784$		0			0
$785$	13.9342	+	19.6632i	0.497335	+	0.701809i
$786$		0			0
$787$	−48.7190			−1.73665			−0.868323	−	0.495999i	$-0.834802\pi$
$787$	−48.7190			−1.73665			−0.868323	+	0.495999i	$0.834802\pi$
$788$		0			0
$789$	55.3098			1.96908
$790$		0			0
$791$			29.8422i			1.06107i
$792$		0			0
$793$	−12.0229	+	1.61159i	−0.426946	+	0.0572293i
$794$		0			0
$795$	43.1494	−	30.5777i	1.53035	−	1.08448i
$796$		0			0
$797$		−	0.328613i		−	0.0116401i	−0.999983	−	0.00582003i	$-0.998147\pi$
$797$		−	0.328613i		−	0.0116401i	0.999983	−	0.00582003i	$-0.00185258\pi$
$798$		0			0
$799$		−	9.67521i		−	0.342284i
$800$		0			0
$801$			25.3981i			0.897397i
$802$		0			0
$803$			4.32852i			0.152750i
$804$		0			0
$805$	12.9885	+	18.3286i	0.457784	+	0.645998i
$806$		0			0
$807$			13.7788i			0.485037i
$808$		0			0
$809$	24.2689			0.853249			0.426624	−	0.904429i	$-0.359703\pi$
$809$	24.2689			0.853249			0.426624	+	0.904429i	$0.359703\pi$
$810$		0			0
$811$			42.7416i			1.50086i	0.660950	+	0.750430i	$0.270153\pi$
$811$			42.7416i			1.50086i	−0.660950	+	0.750430i	$0.729847\pi$
$812$		0			0
$813$	−45.9488			−1.61149
$814$		0			0
$815$	−5.32040	+	3.77028i	−0.186365	+	0.132067i
$816$		0			0
$817$	−0.696489			−0.0243671
$818$		0			0
$819$	12.2778	−	1.64575i	0.429020	−	0.0575073i
$820$		0			0
$821$			3.18294i			0.111085i	0.998456	+	0.0555426i	$0.0176889\pi$
$821$			3.18294i			0.111085i	−0.998456	+	0.0555426i	$0.982311\pi$
$822$		0			0
$823$		−	13.8891i		−	0.484143i	−0.970258	−	0.242072i	$-0.922173\pi$
$823$		−	13.8891i		−	0.484143i	0.970258	−	0.242072i	$-0.0778269\pi$
$824$		0			0
$825$	2.83592	−	8.07385i	0.0987340	−	0.281096i
$826$		0			0
$827$	−2.31427			−0.0804751			−0.0402375	−	0.999190i	$-0.512811\pi$
$827$	−2.31427			−0.0804751			−0.0402375	+	0.999190i	$0.512811\pi$
$828$		0			0
$829$	−36.5581			−1.26971			−0.634857	−	0.772630i	$-0.718941\pi$
$829$	−36.5581			−1.26971			−0.634857	+	0.772630i	$0.718941\pi$
$830$		0			0
$831$	14.4278			0.500494
$832$		0			0
$833$			11.3927i			0.394734i
$834$		0			0
$835$	36.3470	−	25.7572i	1.25784	−	0.891364i
$836$		0			0
$837$	−3.49583			−0.120834
$838$		0			0
$839$			11.0809i			0.382557i	0.981536	+	0.191278i	$0.0612633\pi$
$839$			11.0809i			0.382557i	−0.981536	+	0.191278i	$0.938737\pi$
$840$		0			0
$841$	37.8586			1.30547
$842$		0			0
$843$	−52.6742			−1.81420
$844$		0			0
$845$	18.4539	−	22.4601i	0.634833	−	0.772649i
$846$		0			0
$847$	−21.9934			−0.755702
$848$		0			0
$849$	−7.63081			−0.261889
$850$		0			0
$851$		−	0.576296i		−	0.0197552i
$852$		0			0
$853$	9.12502			0.312435			0.156217	−	0.987723i	$-0.450070\pi$
$853$	9.12502			0.312435			0.156217	+	0.987723i	$0.450070\pi$
$854$		0			0
$855$	0.460619	+	0.649998i	0.0157528	+	0.0222295i
$856$		0			0
$857$		−	4.81456i		−	0.164462i	−0.996613	−	0.0822312i	$-0.973795\pi$
$857$		−	4.81456i		−	0.164462i	0.996613	−	0.0822312i	$-0.0262046\pi$
$858$		0			0
$859$	42.1481			1.43807			0.719037	−	0.694971i	$-0.244583\pi$
$859$	42.1481			1.43807			0.719037	+	0.694971i	$0.244583\pi$
$860$		0			0
$861$	−22.2296			−0.757583
$862$		0			0
$863$	−49.8252			−1.69607			−0.848035	−	0.529940i	$-0.822215\pi$
$863$	−49.8252			−1.69607			−0.848035	+	0.529940i	$0.822215\pi$
$864$		0			0
$865$	−22.8647	−	32.2653i	−0.777423	−	1.09705i
$866$		0			0
$867$			8.15537i			0.276971i
$868$		0			0
$869$			7.21123i			0.244624i
$870$		0			0
$871$	−26.5721	+	3.56182i	−0.900362	+	0.120688i
$872$		0			0
$873$	−3.38666			−0.114621
$874$		0			0
$875$	6.52637	+	22.8060i	0.220632	+	0.770983i
$876$		0			0
$877$	43.2831			1.46157			0.730784	−	0.682609i	$-0.239155\pi$
$877$	43.2831			1.46157			0.730784	+	0.682609i	$0.239155\pi$
$878$		0			0
$879$			2.94112i			0.0992017i
$880$		0			0
$881$	−20.2113			−0.680937			−0.340468	−	0.940256i	$-0.610586\pi$
$881$	−20.2113			−0.680937			−0.340468	+	0.940256i	$0.610586\pi$
$882$		0			0
$883$			20.3203i			0.683833i	0.939730	+	0.341916i	$0.111076\pi$
$883$			20.3203i			0.683833i	−0.939730	+	0.341916i	$0.888924\pi$
$884$		0			0
$885$	14.6208	−	10.3610i	0.491473	−	0.348281i
$886$		0			0
$887$		−	37.6498i		−	1.26416i	−0.774904	−	0.632079i	$-0.782202\pi$
$887$		−	37.6498i		−	1.26416i	0.774904	−	0.632079i	$-0.217798\pi$
$888$		0			0
$889$			25.6205i			0.859283i
$890$		0			0
$891$		−	8.94721i		−	0.299743i
$892$		0			0
$893$		−	0.466816i		−	0.0156214i
$894$		0			0
$895$	−16.9330	+	11.9995i	−0.566009	+	0.401101i
$896$		0			0
$897$	−36.3674	+	4.87481i	−1.21427	+	0.162765i
$898$		0			0
$899$			9.63258i			0.321265i
$900$		0			0
$901$	50.1807			1.67176
$902$		0			0
$903$	14.4354			0.480380
$904$		0			0
$905$	39.9587	−	28.3166i	1.32827	−	0.941275i
$906$		0			0
$907$			20.9901i			0.696965i	0.937315	+	0.348482i	$0.113303\pi$
$907$			20.9901i			0.696965i	−0.937315	+	0.348482i	$0.886697\pi$
$908$		0			0
$909$	−10.1900			−0.337982
$910$		0			0
$911$	20.5887			0.682135			0.341067	−	0.940039i	$-0.389212\pi$
$911$	20.5887			0.682135			0.341067	+	0.940039i	$0.389212\pi$
$912$		0			0
$913$			11.2729i			0.373078i
$914$		0			0
$915$	9.34861	+	13.1922i	0.309055	+	0.436121i
$916$		0			0
$917$	15.3236			0.506030
$918$		0			0
$919$	−3.06293			−0.101037			−0.0505183	−	0.998723i	$-0.516087\pi$
$919$	−3.06293			−0.101037			−0.0505183	+	0.998723i	$0.516087\pi$
$920$		0			0
$921$		−	19.1839i		−	0.632132i
$922$		0			0
$923$	7.28494	+	54.3476i	0.239787	+	1.78887i
$924$		0			0
$925$	0.201673	−	0.574162i	0.00663096	−	0.0188783i
$926$		0			0
$927$		−	19.3764i		−	0.636406i
$928$		0			0
$929$			12.2657i			0.402424i	0.979548	+	0.201212i	$0.0644880\pi$
$929$			12.2657i			0.402424i	−0.979548	+	0.201212i	$0.935512\pi$
$930$		0			0
$931$			0.549683i			0.0180151i
$932$		0			0
$933$		−	28.1664i		−	0.922127i
$934$		0			0
$935$	6.62496	−	4.69475i	0.216659	−	0.153535i
$936$		0			0
$937$			4.98472i			0.162844i	0.996680	+	0.0814219i	$0.0259461\pi$
$937$			4.98472i			0.162844i	−0.996680	+	0.0814219i	$0.974054\pi$
$938$		0			0
$939$	26.9253			0.878673
$940$		0			0
$941$			7.53068i			0.245493i	0.992438	+	0.122747i	$0.0391703\pi$
$941$			7.53068i			0.245493i	−0.992438	+	0.122747i	$0.960830\pi$
$942$		0			0
$943$	23.0822			0.751659
$944$		0			0
$945$	8.14003	+	11.4867i	0.264795	+	0.373663i
$946$		0			0
$947$	−33.7019			−1.09517			−0.547583	−	0.836752i	$-0.684452\pi$
$947$	−33.7019			−1.09517			−0.547583	+	0.836752i	$0.684452\pi$
$948$		0			0
$949$	19.4249	−	2.60379i	0.630560	−	0.0845225i
$950$		0			0
$951$			0.979660i			0.0317677i
$952$		0			0
$953$		−	31.0905i		−	1.00712i	−0.863960	−	0.503560i	$-0.832023\pi$
$953$		−	31.0905i		−	1.00712i	0.863960	−	0.503560i	$-0.167977\pi$
$954$		0			0
$955$	42.1115	−	29.8422i	1.36270	−	0.965670i
$956$		0			0
$957$	−13.9943			−0.452372
$958$		0			0
$959$	31.1642			1.00635
$960$		0			0
$961$	29.6122			0.955232
$962$		0			0
$963$			23.5067i			0.757494i
$964$		0			0
$965$	32.8734	−	23.2956i	1.05823	−	0.749912i
$966$		0			0
$967$	52.8189			1.69854			0.849271	−	0.527957i	$-0.177042\pi$
$967$	52.8189			1.69854			0.849271	+	0.527957i	$0.177042\pi$
$968$		0			0
$969$			2.15637i			0.0692724i
$970$		0			0
$971$	−22.8041			−0.731819			−0.365910	−	0.930650i	$-0.619242\pi$
$971$	−22.8041			−0.731819			−0.365910	+	0.930650i	$0.619242\pi$
$972$		0			0
$973$	−37.8731			−1.21416
$974$		0			0
$975$	−37.9386	−	7.86987i	−1.21501	−	0.252038i
$976$		0			0
$977$	−0.156331			−0.00500148			−0.00250074	−	0.999997i	$-0.500796\pi$
$977$	−0.156331			−0.00500148			−0.00250074	+	0.999997i	$0.500796\pi$
$978$		0			0
$979$	12.4898			0.399177
$980$		0			0
$981$			12.8368i			0.409848i
$982$		0			0
$983$	−23.5630			−0.751542			−0.375771	−	0.926712i	$-0.622622\pi$
$983$	−23.5630			−0.751542			−0.375771	+	0.926712i	$0.622622\pi$
$984$		0			0
$985$	−14.1553	+	10.0311i	−0.451027	+	0.319618i
$986$		0			0
$987$			9.67521i			0.307965i
$988$		0			0
$989$	−14.9890			−0.476624
$990$		0			0
$991$	−14.0102			−0.445048			−0.222524	−	0.974927i	$-0.571430\pi$
$991$	−14.0102			−0.445048			−0.222524	+	0.974927i	$0.571430\pi$
$992$		0			0
$993$	−7.89979			−0.250692
$994$		0			0
$995$	20.0030	−	14.1751i	0.634138	−	0.449380i
$996$		0			0
$997$			0.625270i			0.0198025i	0.999951	+	0.00990124i	$0.00315172\pi$
$997$			0.625270i			0.0198025i	−0.999951	+	0.00990124i	$0.996848\pi$
$998$		0			0
$999$		−	0.361171i		−	0.0114269i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.f.g.129.2		10
4.3	odd	2		520.2.f.b.129.9	yes	10
5.4	even	2		1040.2.f.f.129.9		10
13.12	even	2		1040.2.f.f.129.2		10
20.3	even	4		2600.2.k.f.2001.3		20
20.7	even	4		2600.2.k.f.2001.18		20
20.19	odd	2		520.2.f.a.129.2	✓	10
52.51	odd	2		520.2.f.a.129.9	yes	10
65.64	even	2	inner	1040.2.f.g.129.9		10
260.103	even	4		2600.2.k.f.2001.4		20
260.207	even	4		2600.2.k.f.2001.17		20
260.259	odd	2		520.2.f.b.129.2	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.f.a.129.2	✓	10	20.19	odd	2
520.2.f.a.129.9	yes	10	52.51	odd	2
520.2.f.b.129.2	yes	10	260.259	odd	2
520.2.f.b.129.9	yes	10	4.3	odd	2
1040.2.f.f.129.2		10	13.12	even	2
1040.2.f.f.129.9		10	5.4	even	2
1040.2.f.g.129.2		10	1.1	even	1	trivial
1040.2.f.g.129.9		10	65.64	even	2	inner
2600.2.k.f.2001.3		20	20.3	even	4
2600.2.k.f.2001.4		20	260.103	even	4
2600.2.k.f.2001.17		20	260.207	even	4
2600.2.k.f.2001.18		20	20.7	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.14926i q^{3} +(1.82442 - 1.29287i) q^{5} -2.12171 q^{7} -1.61930 q^{9} +O(q^{10})\)	\(q-2.14926i q^{3} +(1.82442 - 1.29287i) q^{5} -2.12171 q^{7} -1.61930 q^{9} +0.796315i q^{11} +(3.57359 - 0.479016i) q^{13} +(-2.77870 - 3.92114i) q^{15} -4.56010i q^{17} -0.220019i q^{19} +4.56010i q^{21} -4.73499i q^{23} +(1.65699 - 4.71745i) q^{25} -2.96747i q^{27} -8.17671 q^{29} -1.17805i q^{31} +1.71149 q^{33} +(-3.87088 + 2.74309i) q^{35} +0.121710 q^{37} +(-1.02953 - 7.68056i) q^{39} +4.87481i q^{41} -3.16559i q^{43} +(-2.95429 + 2.09354i) q^{45} +2.12171 q^{47} -2.49835 q^{49} -9.80082 q^{51} +11.0043i q^{53} +(1.02953 + 1.45281i) q^{55} -0.472877 q^{57} +3.72871i q^{59} -3.36438 q^{61} +3.43569 q^{63} +(5.90041 - 5.49410i) q^{65} -7.43569 q^{67} -10.1767 q^{69} +15.2081i q^{71} +5.43569 q^{73} +(-10.1390 - 3.56130i) q^{75} -1.68955i q^{77} +9.05575 q^{79} -11.2358 q^{81} +14.1563 q^{83} +(-5.89560 - 8.31952i) q^{85} +17.5738i q^{87} -15.6846i q^{89} +(-7.58212 + 1.01633i) q^{91} -2.53193 q^{93} +(-0.284455 - 0.401406i) q^{95} +2.09143 q^{97} -1.28948i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) \) 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9	\( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61} - 24 q^{63} - 5 q^{65} - 16 q^{67} - 12 q^{69} - 4 q^{73} - 21 q^{75} - 28 q^{79} + 22 q^{81} + 4 q^{83} - 25 q^{85} - 2 q^{91} + 12 q^{93} + 10 q^{95} + 72 q^{97}+O(q^{100}) \) 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9 + 2 * q^13 + 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 + 3 * q^35 - 22 * q^37 + 12 * q^39 - 4 * q^45 - 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 - 12 * q^57 - 16 * q^61 - 24 * q^63 - 5 * q^65 - 16 * q^67 - 12 * q^69 - 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 + 4 * q^83 - 25 * q^85 - 2 * q^91 + 12 * q^93 + 10 * q^95 + 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more