LMFDB - Embedded newform 1280.3.h.m.1279.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(1279,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1280.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.h (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:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.741637881856000000000000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 9x^{12} + 41x^{8} - 144x^{4} + 256 $ x^16 - 9x^12 + 41x^8 - 144*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{42} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.9
Root			$-0.493908 - 1.32516i$ of defining polynomial
Character	$\chi$	$=$	1280.1279
Dual form			1280.3.h.m.1279.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24849 q^{3} +(-4.89329 - 1.02749i) q^{5} +6.40747 q^{7} -3.94427 q^{9} +O(q^{10})$	$q+2.24849 q^{3} +(-4.89329 - 1.02749i) q^{5} +6.40747 q^{7} -3.94427 q^{9} -8.47214i q^{11} +17.4100i q^{13} +(-11.0025 - 2.31030i) q^{15} +19.0496i q^{17} -18.3607i q^{19} +14.4072 q^{21} -2.29753 q^{23} +(22.8885 + 10.0556i) q^{25} -29.1051 q^{27} +4.62059 q^{29} +43.7669i q^{31} -19.0496i q^{33} +(-31.3536 - 6.58359i) q^{35} +24.5452i q^{37} +39.1463i q^{39} -32.9443 q^{41} +35.8506 q^{43} +(19.3005 + 4.05269i) q^{45} +76.0475 q^{47} -7.94427 q^{49} +42.8328i q^{51} +64.5599i q^{53} +(-8.70500 + 41.4566i) q^{55} -41.2839i q^{57} -21.1935i q^{59} -29.3597 q^{61} -25.2728 q^{63} +(17.8885 - 85.1922i) q^{65} +85.0669 q^{67} -5.16598 q^{69} +73.6720i q^{71} +21.1727i q^{73} +(51.4648 + 22.6099i) q^{75} -54.2850i q^{77} +112.818i q^{79} -29.9443 q^{81} -40.3476 q^{83} +(19.5732 - 93.2150i) q^{85} +10.3894 q^{87} +57.5542 q^{89} +111.554i q^{91} +98.4096i q^{93} +(-18.8653 + 89.8441i) q^{95} +187.311i q^{97} +33.4164i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 80 q^{9}+O(q^{10}) $ 16 * q + 80 * q^9	$ 16 q + 80 q^{9} + 80 q^{25} - 384 q^{41} + 16 q^{49} - 336 q^{81} - 224 q^{89}+O(q^{100}) $ 16 * q + 80 * q^9 + 80 * q^25 - 384 * q^41 + 16 * q^49 - 336 * q^81 - 224 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	2.24849			0.749498			0.374749	−	0.927126i	$-0.377729\pi$
$3$	2.24849			0.749498			0.374749	+	0.927126i	$0.377729\pi$
$4$		0			0
$5$	−4.89329	−	1.02749i	−0.978658	−	0.205497i
$5$	−4.89329	−	1.02749i	−0.978658	−	0.205497i
$6$		0			0
$7$	6.40747			0.915353			0.457677	−	0.889119i	$-0.348682\pi$
$7$	6.40747			0.915353			0.457677	+	0.889119i	$0.348682\pi$
$8$		0			0
$9$	−3.94427			−0.438252
$10$		0			0
$11$		−	8.47214i		−	0.770194i	−0.922876	−	0.385097i	$-0.874168\pi$
$11$		−	8.47214i		−	0.770194i	0.922876	−	0.385097i	$-0.125832\pi$
$12$		0			0
$13$			17.4100i			1.33923i	0.742708	+	0.669616i	$0.233541\pi$
$13$			17.4100i			1.33923i	−0.742708	+	0.669616i	$0.766459\pi$
$14$		0			0
$15$	−11.0025	−	2.31030i	−0.733502	−	0.154020i
$16$		0			0
$17$			19.0496i			1.12056i	0.828303	+	0.560281i	$0.189307\pi$
$17$			19.0496i			1.12056i	−0.828303	+	0.560281i	$0.810693\pi$
$18$		0			0
$19$		−	18.3607i		−	0.966352i	−0.875523	−	0.483176i	$-0.839483\pi$
$19$		−	18.3607i		−	0.966352i	0.875523	−	0.483176i	$-0.160517\pi$
$20$		0			0
$21$	14.4072			0.686056
$22$		0			0
$23$	−2.29753			−0.0998926			−0.0499463	−	0.998752i	$-0.515905\pi$
$23$	−2.29753			−0.0998926			−0.0499463	+	0.998752i	$0.515905\pi$
$24$		0			0
$25$	22.8885	+	10.0556i	0.915542	+	0.402223i
$26$		0			0
$27$	−29.1051			−1.07797
$28$		0			0
$29$	4.62059			0.159331			0.0796654	−	0.996822i	$-0.474615\pi$
$29$	4.62059			0.159331			0.0796654	+	0.996822i	$0.474615\pi$
$30$		0			0
$31$			43.7669i			1.41184i	0.708294	+	0.705918i	$0.249465\pi$
$31$			43.7669i			1.41184i	−0.708294	+	0.705918i	$0.750535\pi$
$32$		0			0
$33$		−	19.0496i		−	0.577259i
$34$		0			0
$35$	−31.3536	−	6.58359i	−0.895818	−	0.188103i
$36$		0			0
$37$			24.5452i			0.663382i	0.943388	+	0.331691i	$0.107619\pi$
$37$			24.5452i			0.663382i	−0.943388	+	0.331691i	$0.892381\pi$
$38$		0			0
$39$			39.1463i			1.00375i
$40$		0			0
$41$	−32.9443			−0.803519			−0.401759	−	0.915745i	$-0.631601\pi$
$41$	−32.9443			−0.803519			−0.401759	+	0.915745i	$0.631601\pi$
$42$		0			0
$43$	35.8506			0.833735			0.416868	−	0.908967i	$-0.363128\pi$
$43$	35.8506			0.833735			0.416868	+	0.908967i	$0.363128\pi$
$44$		0			0
$45$	19.3005	+	4.05269i	0.428899	+	0.0900597i
$46$		0			0
$47$	76.0475			1.61803			0.809016	−	0.587787i	$-0.200001\pi$
$47$	76.0475			1.61803			0.809016	+	0.587787i	$0.200001\pi$
$48$		0			0
$49$	−7.94427			−0.162128
$50$		0			0
$51$			42.8328i			0.839859i
$52$		0			0
$53$			64.5599i			1.21811i	0.793128	+	0.609055i	$0.208451\pi$
$53$			64.5599i			1.21811i	−0.793128	+	0.609055i	$0.791549\pi$
$54$		0			0
$55$	−8.70500	+	41.4566i	−0.158273	+	0.753756i
$56$		0			0
$57$		−	41.2839i		−	0.724279i
$58$		0			0
$59$		−	21.1935i		−	0.359212i	−0.983739	−	0.179606i	$-0.942518\pi$
$59$		−	21.1935i		−	0.359212i	0.983739	−	0.179606i	$-0.0574823\pi$
$60$		0			0
$61$	−29.3597			−0.481307			−0.240654	−	0.970611i	$-0.577362\pi$
$61$	−29.3597			−0.481307			−0.240654	+	0.970611i	$0.577362\pi$
$62$		0			0
$63$	−25.2728			−0.401156
$64$		0			0
$65$	17.8885	−	85.1922i	0.275208	−	1.31065i
$66$		0			0
$67$	85.0669			1.26965			0.634827	−	0.772654i	$-0.281071\pi$
$67$	85.0669			1.26965			0.634827	+	0.772654i	$0.281071\pi$
$68$		0			0
$69$	−5.16598			−0.0748693
$70$		0			0
$71$			73.6720i			1.03763i	0.854885	+	0.518817i	$0.173627\pi$
$71$			73.6720i			1.03763i	−0.854885	+	0.518817i	$0.826373\pi$
$72$		0			0
$73$			21.1727i			0.290038i	0.989429	+	0.145019i	$0.0463243\pi$
$73$			21.1727i			0.290038i	−0.989429	+	0.145019i	$0.953676\pi$
$74$		0			0
$75$	51.4648	+	22.6099i	0.686197	+	0.301465i
$76$		0			0
$77$		−	54.2850i		−	0.705000i
$78$		0			0
$79$			112.818i			1.42808i	0.700105	+	0.714040i	$0.253137\pi$
$79$			112.818i			1.42808i	−0.700105	+	0.714040i	$0.746863\pi$
$80$		0			0
$81$	−29.9443			−0.369682
$82$		0			0
$83$	−40.3476			−0.486116			−0.243058	−	0.970012i	$-0.578150\pi$
$83$	−40.3476			−0.486116			−0.243058	+	0.970012i	$0.578150\pi$
$84$		0			0
$85$	19.5732	−	93.2150i	0.230272	−	1.09665i
$86$		0			0
$87$	10.3894			0.119418
$88$		0			0
$89$	57.5542			0.646676			0.323338	−	0.946284i	$-0.395195\pi$
$89$	57.5542			0.646676			0.323338	+	0.946284i	$0.395195\pi$
$90$		0			0
$91$			111.554i			1.22587i
$92$		0			0
$93$			98.4096i			1.05817i
$94$		0			0
$95$	−18.8653	+	89.8441i	−0.198583	+	0.945727i
$96$		0			0
$97$			187.311i			1.93104i	0.260332	+	0.965519i	$0.416168\pi$
$97$			187.311i			1.93104i	−0.260332	+	0.965519i	$0.583832\pi$
$98$		0			0
$99$			33.4164i			0.337539i
$100$		0			0
$101$	−53.0081			−0.524833			−0.262416	−	0.964955i	$-0.584519\pi$
$101$	−53.0081			−0.524833			−0.262416	+	0.964955i	$0.584519\pi$
$102$		0			0
$103$	74.5922			0.724196			0.362098	−	0.932140i	$-0.382061\pi$
$103$	74.5922			0.724196			0.362098	+	0.932140i	$0.382061\pi$
$104$		0			0
$105$	−70.4984	−	14.8032i	−0.671414	−	0.140983i
$106$		0			0
$107$	−64.9557			−0.607063			−0.303531	−	0.952821i	$-0.598166\pi$
$107$	−64.9557			−0.607063			−0.303531	+	0.952821i	$0.598166\pi$
$108$		0			0
$109$	10.8774			0.0997922			0.0498961	−	0.998754i	$-0.484111\pi$
$109$	10.8774			0.0997922			0.0498961	+	0.998754i	$0.484111\pi$
$110$		0			0
$111$			55.1896i			0.497204i
$112$		0			0
$113$		−	35.9759i		−	0.318371i	−0.987249	−	0.159185i	$-0.949113\pi$
$113$		−	35.9759i		−	0.318371i	0.987249	−	0.159185i	$-0.0508868\pi$
$114$		0			0
$115$	11.2425	+	2.36068i	0.0977606	+	0.0205277i
$116$		0			0
$117$		−	68.6698i		−	0.586921i
$118$		0			0
$119$			122.060i			1.02571i
$120$		0			0
$121$	49.2229			0.406801
$122$		0			0
$123$	−74.0750			−0.602236
$124$		0			0
$125$	−101.668	−	72.7225i	−0.813346	−	0.581780i
$126$		0			0
$127$	−99.5079			−0.783527			−0.391763	−	0.920066i	$-0.628135\pi$
$127$	−99.5079			−0.783527			−0.391763	+	0.920066i	$0.628135\pi$
$128$		0			0
$129$	80.6099			0.624883
$130$		0			0
$131$		−	6.13777i		−	0.0468532i	−0.999726	−	0.0234266i	$-0.992542\pi$
$131$		−	6.13777i		−	0.0468532i	0.999726	−	0.0234266i	$-0.00745760\pi$
$132$		0			0
$133$		−	117.646i		−	0.884553i
$134$		0			0
$135$	142.420	+	29.9051i	1.05496	+	0.221519i
$136$		0			0
$137$			146.027i			1.06589i	0.846150	+	0.532945i	$0.178915\pi$
$137$			146.027i			1.06589i	−0.846150	+	0.532945i	$0.821085\pi$
$138$		0			0
$139$		−	70.5836i		−	0.507796i	−0.967231	−	0.253898i	$-0.918287\pi$
$139$		−	70.5836i		−	0.507796i	0.967231	−	0.253898i	$-0.0817127\pi$
$140$		0			0
$141$	170.992			1.21271
$142$		0			0
$143$	147.500			1.03147
$144$		0			0
$145$	−22.6099	−	4.74760i	−0.155930	−	0.0327421i
$146$		0			0
$147$	−17.8627			−0.121515
$148$		0			0
$149$	30.4505			0.204366			0.102183	−	0.994766i	$-0.467417\pi$
$149$	30.4505			0.204366			0.102183	+	0.994766i	$0.467417\pi$
$150$		0			0
$151$			214.214i			1.41864i	0.704889	+	0.709318i	$0.250997\pi$
$151$			214.214i			1.41864i	−0.704889	+	0.709318i	$0.749003\pi$
$152$		0			0
$153$		−	75.1366i		−	0.491089i
$154$		0			0
$155$	44.9699	−	214.164i	0.290128	−	1.38170i
$156$		0			0
$157$		−	73.6355i		−	0.469016i	−0.972114	−	0.234508i	$-0.924652\pi$
$157$		−	73.6355i		−	0.469016i	0.972114	−	0.234508i	$-0.0753478\pi$
$158$		0			0
$159$			145.162i			0.912972i
$160$		0			0
$161$	−14.7214			−0.0914370
$162$		0			0
$163$	−37.9738			−0.232968			−0.116484	−	0.993193i	$-0.537162\pi$
$163$	−37.9738			−0.232968			−0.116484	+	0.993193i	$0.537162\pi$
$164$		0			0
$165$	−19.5732	+	93.2150i	−0.118625	+	0.564939i
$166$		0			0
$167$	−157.047			−0.940402			−0.470201	−	0.882559i	$-0.655819\pi$
$167$	−157.047			−0.940402			−0.470201	+	0.882559i	$0.655819\pi$
$168$		0			0
$169$	−134.108			−0.793541
$170$		0			0
$171$			72.4195i			0.423506i
$172$		0			0
$173$		−	196.705i		−	1.13702i	−0.822676	−	0.568511i	$-0.807520\pi$
$173$		−	196.705i		−	1.13702i	0.822676	−	0.568511i	$-0.192480\pi$
$174$		0			0
$175$	146.658	+	64.4308i	0.838044	+	0.368176i
$176$		0			0
$177$		−	47.6535i		−	0.269229i
$178$		0			0
$179$		−	205.193i		−	1.14633i	−0.819439	−	0.573166i	$-0.805715\pi$
$179$		−	205.193i		−	1.14633i	0.819439	−	0.573166i	$-0.194285\pi$
$180$		0			0
$181$	25.2845			0.139693			0.0698467	−	0.997558i	$-0.477749\pi$
$181$	25.2845			0.139693			0.0698467	+	0.997558i	$0.477749\pi$
$182$		0			0
$183$	−66.0152			−0.360739
$184$		0			0
$185$	25.2198	−	120.107i	0.136323	−	0.649224i
$186$		0			0
$187$	161.390			0.863050
$188$		0			0
$189$	−186.490			−0.986721
$190$		0			0
$191$			106.016i			0.555059i	0.960717	+	0.277529i	$0.0895156\pi$
$191$			106.016i			0.555059i	−0.960717	+	0.277529i	$0.910484\pi$
$192$		0			0
$193$		−	111.113i		−	0.575713i	−0.957674	−	0.287856i	$-0.907057\pi$
$193$		−	111.113i		−	0.575713i	0.957674	−	0.287856i	$-0.0929425\pi$
$194$		0			0
$195$	40.2223	−	191.554i	0.206268	−	0.982329i
$196$		0			0
$197$			17.6390i			0.0895383i	0.998997	+	0.0447692i	$0.0142552\pi$
$197$			17.6390i			0.0895383i	−0.998997	+	0.0447692i	$0.985745\pi$
$198$		0			0
$199$			55.1896i			0.277335i	0.990339	+	0.138667i	$0.0442819\pi$
$199$			55.1896i			0.277335i	−0.990339	+	0.138667i	$0.955718\pi$
$200$		0			0
$201$	191.272			0.951604
$202$		0			0
$203$	29.6063			0.145844
$204$		0			0
$205$	161.206	+	33.8498i	0.786370	+	0.165121i
$206$		0			0
$207$	9.06208			0.0437782
$208$		0			0
$209$	−155.554			−0.744278
$210$		0			0
$211$			257.416i			1.21998i	0.792408	+	0.609991i	$0.208827\pi$
$211$			257.416i			1.21998i	−0.792408	+	0.609991i	$0.791173\pi$
$212$		0			0
$213$			165.651i			0.777705i
$214$		0			0
$215$	−175.427	−	36.8360i	−0.815941	−	0.171330i
$216$		0			0
$217$			280.435i			1.29233i
$218$		0			0
$219$			47.6068i			0.217383i
$220$		0			0
$221$	−331.653			−1.50069
$222$		0			0
$223$	−91.2880			−0.409363			−0.204682	−	0.978829i	$-0.565616\pi$
$223$	−91.2880			−0.409363			−0.204682	+	0.978829i	$0.565616\pi$
$224$		0			0
$225$	−90.2786	−	39.6619i	−0.401238	−	0.176275i
$226$		0			0
$227$	−306.791			−1.35150			−0.675750	−	0.737130i	$-0.736180\pi$
$227$	−306.791			−1.35150			−0.675750	+	0.737130i	$0.736180\pi$
$228$		0			0
$229$	−73.6720			−0.321712			−0.160856	−	0.986978i	$-0.551425\pi$
$229$	−73.6720			−0.321712			−0.160856	+	0.986978i	$0.551425\pi$
$230$		0			0
$231$		−	122.060i		−	0.528396i
$232$		0			0
$233$		−	207.422i		−	0.890223i	−0.895475	−	0.445111i	$-0.853164\pi$
$233$		−	207.422i		−	0.890223i	0.895475	−	0.445111i	$-0.146836\pi$
$234$		0			0
$235$	−372.122	−	78.1378i	−1.58350	−	0.332501i
$236$		0			0
$237$			253.671i			1.07034i
$238$		0			0
$239$		−	246.558i		−	1.03162i	−0.856702	−	0.515812i	$-0.827490\pi$
$239$		−	246.558i		−	1.03162i	0.856702	−	0.515812i	$-0.172510\pi$
$240$		0			0
$241$	285.495			1.18463			0.592314	−	0.805707i	$-0.298215\pi$
$241$	285.495			1.18463			0.592314	+	0.805707i	$0.298215\pi$
$242$		0			0
$243$	194.617			0.800891
$244$		0			0
$245$	38.8736	+	8.16263i	0.158668	+	0.0333169i
$246$		0			0
$247$	319.660			1.29417
$248$		0			0
$249$	−90.7214			−0.364343
$250$		0			0
$251$			133.639i			0.532428i	0.963914	+	0.266214i	$0.0857727\pi$
$251$			133.639i			0.532428i	−0.963914	+	0.266214i	$0.914227\pi$
$252$		0			0
$253$			19.4650i			0.0769367i
$254$		0			0
$255$	44.0101	−	209.593i	0.172589	−	0.821935i
$256$		0			0
$257$		−	68.8262i		−	0.267806i	−0.990994	−	0.133903i	$-0.957249\pi$
$257$		−	68.8262i		−	0.267806i	0.990994	−	0.133903i	$-0.0427511\pi$
$258$		0			0
$259$			157.272i			0.607229i
$260$		0			0
$261$	−18.2249			−0.0698271
$262$		0			0
$263$	128.278			0.487747			0.243874	−	0.969807i	$-0.421582\pi$
$263$	128.278			0.487747			0.243874	+	0.969807i	$0.421582\pi$
$264$		0			0
$265$	66.3344	−	315.910i	0.250318	−	1.19211i
$266$		0			0
$267$	129.410			0.484683
$268$		0			0
$269$	516.507			1.92010			0.960050	−	0.279828i	$-0.0902773\pi$
$269$	516.507			1.92010			0.960050	+	0.279828i	$0.0902773\pi$
$270$		0			0
$271$			62.2493i			0.229702i	0.993383	+	0.114851i	$0.0366391\pi$
$271$			62.2493i			0.229702i	−0.993383	+	0.114851i	$0.963361\pi$
$272$		0			0
$273$			250.829i			0.918787i
$274$		0			0
$275$	85.1922	−	193.915i	0.309790	−	0.705145i
$276$		0			0
$277$		−	53.3148i		−	0.192472i	−0.995359	−	0.0962360i	$-0.969320\pi$
$277$		−	53.3148i		−	0.192472i	0.995359	−	0.0962360i	$-0.0306804\pi$
$278$		0			0
$279$		−	172.629i		−	0.618740i
$280$		0			0
$281$	−370.269			−1.31768			−0.658842	−	0.752281i	$-0.728954\pi$
$281$	−370.269			−1.31768			−0.658842	+	0.752281i	$0.728954\pi$
$282$		0			0
$283$	5.99366			0.0211790			0.0105895	−	0.999944i	$-0.496629\pi$
$283$	5.99366			0.0211790			0.0105895	+	0.999944i	$0.496629\pi$
$284$		0			0
$285$	−42.4186	+	202.014i	−0.148837	+	0.708821i
$286$		0			0
$287$	−211.090			−0.735504
$288$		0			0
$289$	−73.8854			−0.255659
$290$		0			0
$291$			421.167i			1.44731i
$292$		0			0
$293$		−	223.534i		−	0.762914i	−0.924386	−	0.381457i	$-0.875422\pi$
$293$		−	223.534i		−	0.762914i	0.924386	−	0.381457i	$-0.124578\pi$
$294$		0			0
$295$	−21.7760	+	103.706i	−0.0738170	+	0.351545i
$296$		0			0
$297$			246.583i			0.830244i
$298$		0			0
$299$		−	40.0000i		−	0.133779i
$300$		0			0
$301$	229.712			0.763162
$302$		0			0
$303$	−119.188			−0.393361
$304$		0			0
$305$	143.666	+	30.1667i	0.471035	+	0.0989073i
$306$		0			0
$307$	44.8446			0.146074			0.0730368	−	0.997329i	$-0.476731\pi$
$307$	44.8446			0.146074			0.0730368	+	0.997329i	$0.476731\pi$
$308$		0			0
$309$	167.720			0.542783
$310$		0			0
$311$		−	476.815i		−	1.53317i	−0.642144	−	0.766584i	$-0.721955\pi$
$311$		−	476.815i		−	1.53317i	0.642144	−	0.766584i	$-0.278045\pi$
$312$		0			0
$313$			158.766i			0.507240i	0.967304	+	0.253620i	$0.0816212\pi$
$313$			158.766i			0.507240i	−0.967304	+	0.253620i	$0.918379\pi$
$314$		0			0
$315$	123.667	+	25.9675i	0.392594	+	0.0824364i
$316$		0			0
$317$			425.319i			1.34170i	0.741593	+	0.670850i	$0.234071\pi$
$317$			425.319i			1.34170i	−0.741593	+	0.670850i	$0.765929\pi$
$318$		0			0
$319$		−	39.1463i		−	0.122716i
$320$		0			0
$321$	−146.053			−0.454993
$322$		0			0
$323$	349.763			1.08286
$324$		0			0
$325$	−175.068	+	398.490i	−0.538670	+	1.22612i
$326$		0			0
$327$	24.4577			0.0747941
$328$		0			0
$329$	487.272			1.48107
$330$		0			0
$331$			570.912i			1.72481i	0.506220	+	0.862404i	$0.331042\pi$
$331$			570.912i			1.72481i	−0.506220	+	0.862404i	$0.668958\pi$
$332$		0			0
$333$		−	96.8128i		−	0.290729i
$334$		0			0
$335$	−416.257	−	87.4051i	−1.24256	−	0.260911i
$336$		0			0
$337$			53.9639i			0.160130i	0.996790	+	0.0800651i	$0.0255128\pi$
$337$			53.9639i			0.160130i	−0.996790	+	0.0800651i	$0.974487\pi$
$338$		0			0
$339$		−	80.8916i		−	0.238618i
$340$		0			0
$341$	370.799			1.08739
$342$		0			0
$343$	−364.869			−1.06376
$344$		0			0
$345$	25.2786	+	5.30798i	0.0732714	+	0.0153854i
$346$		0			0
$347$	107.802			0.310670			0.155335	−	0.987862i	$-0.450354\pi$
$347$	107.802			0.310670			0.155335	+	0.987862i	$0.450354\pi$
$348$		0			0
$349$	−541.246			−1.55085			−0.775424	−	0.631441i	$-0.782464\pi$
$349$	−541.246			−1.55085			−0.775424	+	0.631441i	$0.782464\pi$
$350$		0			0
$351$		−	506.720i		−	1.44365i
$352$		0			0
$353$			82.5678i			0.233903i	0.993138	+	0.116952i	$0.0373122\pi$
$353$			82.5678i			0.233903i	−0.993138	+	0.116952i	$0.962688\pi$
$354$		0			0
$355$	75.6970	−	360.498i	0.213231	−	1.01549i
$356$		0			0
$357$			274.450i			0.768768i
$358$		0			0
$359$		−	306.111i		−	0.852676i	−0.904564	−	0.426338i	$-0.859803\pi$
$359$		−	306.111i		−	0.852676i	0.904564	−	0.426338i	$-0.140197\pi$
$360$		0			0
$361$	23.8854			0.0661646
$362$		0			0
$363$	110.677			0.304897
$364$		0			0
$365$	21.7547	−	103.604i	0.0596019	−	0.283847i
$366$		0			0
$367$	415.543			1.13227			0.566134	−	0.824313i	$-0.308438\pi$
$367$	415.543			1.13227			0.566134	+	0.824313i	$0.308438\pi$
$368$		0			0
$369$	129.941			0.352144
$370$		0			0
$371$			413.666i			1.11500i
$372$		0			0
$373$		−	237.575i		−	0.636931i	−0.947934	−	0.318465i	$-0.896833\pi$
$373$		−	237.575i		−	0.636931i	0.947934	−	0.318465i	$-0.103167\pi$
$374$		0			0
$375$	−228.601	−	163.516i	−0.609602	−	0.436043i
$376$		0			0
$377$			80.4446i			0.213381i
$378$		0			0
$379$		−	303.135i		−	0.799828i	−0.916553	−	0.399914i	$-0.869040\pi$
$379$		−	303.135i		−	0.799828i	0.916553	−	0.399914i	$-0.130960\pi$
$380$		0			0
$381$	−223.743			−0.587252
$382$		0			0
$383$	282.084			0.736512			0.368256	−	0.929725i	$-0.379955\pi$
$383$	282.084			0.736512			0.368256	+	0.929725i	$0.379955\pi$
$384$		0			0
$385$	−55.7771	+	265.632i	−0.144876	+	0.689954i
$386$		0			0
$387$	−141.405			−0.365386
$388$		0			0
$389$	−437.124			−1.12371			−0.561856	−	0.827235i	$-0.689912\pi$
$389$	−437.124			−1.12371			−0.561856	+	0.827235i	$0.689912\pi$
$390$		0			0
$391$		−	43.7669i		−	0.111936i
$392$		0			0
$393$		−	13.8007i		−	0.0351164i
$394$		0			0
$395$	115.919	−	552.053i	0.293467	−	1.39760i
$396$		0			0
$397$		−	529.779i		−	1.33446i	−0.744854	−	0.667228i	$-0.767481\pi$
$397$		−	529.779i		−	1.33446i	0.744854	−	0.667228i	$-0.232519\pi$
$398$		0			0
$399$		−	264.525i		−	0.662971i
$400$		0			0
$401$	−280.663			−0.699907			−0.349953	−	0.936767i	$-0.613803\pi$
$401$	−280.663			−0.699907			−0.349953	+	0.936767i	$0.613803\pi$
$402$		0			0
$403$	−761.982			−1.89077
$404$		0			0
$405$	146.526	+	30.7673i	0.361792	+	0.0759687i
$406$		0			0
$407$	207.950			0.510933
$408$		0			0
$409$	−245.613			−0.600521			−0.300260	−	0.953857i	$-0.597074\pi$
$409$	−245.613			−0.600521			−0.300260	+	0.953857i	$0.597074\pi$
$410$		0			0
$411$			328.341i			0.798882i
$412$		0			0
$413$		−	135.797i		−	0.328806i
$414$		0			0
$415$	197.432	+	41.4566i	0.475741	+	0.0998954i
$416$		0			0
$417$		−	158.707i		−	0.380592i
$418$		0			0
$419$		−	528.184i		−	1.26058i	−0.776359	−	0.630291i	$-0.782935\pi$
$419$		−	528.184i		−	1.26058i	0.776359	−	0.630291i	$-0.217065\pi$
$420$		0			0
$421$	−377.313			−0.896231			−0.448116	−	0.893976i	$-0.647905\pi$
$421$	−377.313			−0.896231			−0.448116	+	0.893976i	$0.647905\pi$
$422$		0			0
$423$	−299.952			−0.709106
$424$		0			0
$425$	−191.554	+	436.017i	−0.450716	+	1.02592i
$426$		0			0
$427$	−188.122			−0.440566
$428$		0			0
$429$	331.653			0.773084
$430$		0			0
$431$		−	610.298i		−	1.41600i	−0.706211	−	0.708002i	$-0.749597\pi$
$431$		−	610.298i		−	1.41600i	0.706211	−	0.708002i	$-0.250403\pi$
$432$		0			0
$433$		−	753.548i		−	1.74030i	−0.492790	−	0.870148i	$-0.664023\pi$
$433$		−	753.548i		−	1.74030i	0.492790	−	0.870148i	$-0.335977\pi$
$434$		0			0
$435$	−50.8382	−	10.6749i	−0.116870	−	0.0245401i
$436$		0			0
$437$			42.1842i			0.0965313i
$438$		0			0
$439$		−	182.127i		−	0.414868i	−0.978249	−	0.207434i	$-0.933489\pi$
$439$		−	182.127i		−	0.414868i	0.978249	−	0.207434i	$-0.0665113\pi$
$440$		0			0
$441$	31.3344			0.0710530
$442$		0			0
$443$	−715.766			−1.61572			−0.807862	−	0.589371i	$-0.799376\pi$
$443$	−715.766			−1.61572			−0.807862	+	0.589371i	$0.799376\pi$
$444$		0			0
$445$	−281.629	−	59.1361i	−0.632875	−	0.132890i
$446$		0			0
$447$	68.4678			0.153172
$448$		0			0
$449$	580.158			1.29211			0.646056	−	0.763290i	$-0.276417\pi$
$449$	580.158			1.29211			0.646056	+	0.763290i	$0.276417\pi$
$450$		0			0
$451$			279.108i			0.618866i
$452$		0			0
$453$			481.659i			1.06326i
$454$		0			0
$455$	114.620	−	545.867i	0.251913	−	1.19971i
$456$		0			0
$457$		−	766.228i		−	1.67665i	−0.545172	−	0.838324i	$-0.683536\pi$
$457$		−	766.228i		−	1.67665i	0.545172	−	0.838324i	$-0.316464\pi$
$458$		0			0
$459$		−	554.440i		−	1.20793i
$460$		0			0
$461$	692.953			1.50315			0.751576	−	0.659646i	$-0.229294\pi$
$461$	692.953			1.50315			0.751576	+	0.659646i	$0.229294\pi$
$462$		0			0
$463$	−620.865			−1.34096			−0.670480	−	0.741927i	$-0.733912\pi$
$463$	−620.865			−1.34096			−0.670480	+	0.741927i	$0.733912\pi$
$464$		0			0
$465$	101.115	−	481.547i	0.217451	−	1.03558i
$466$		0			0
$467$	369.999			0.792289			0.396145	−	0.918188i	$-0.370348\pi$
$467$	369.999			0.792289			0.396145	+	0.918188i	$0.370348\pi$
$468$		0			0
$469$	545.064			1.16218
$470$		0			0
$471$		−	165.569i		−	0.351526i
$472$		0			0
$473$		−	303.731i		−	0.642138i
$474$		0			0
$475$	184.627	−	420.249i	0.388689	−	0.884735i
$476$		0			0
$477$		−	254.642i		−	0.533840i
$478$		0			0
$479$		−	691.029i		−	1.44265i	−0.692597	−	0.721325i	$-0.743533\pi$
$479$		−	691.029i		−	1.44265i	0.692597	−	0.721325i	$-0.256467\pi$
$480$		0			0
$481$	−427.331			−0.888423
$482$		0			0
$483$	−33.1009			−0.0685319
$484$		0			0
$485$	192.459	−	916.565i	0.396823	−	1.88983i
$486$		0			0
$487$	−850.564			−1.74654			−0.873269	−	0.487239i	$-0.838004\pi$
$487$	−850.564			−1.74654			−0.873269	+	0.487239i	$0.838004\pi$
$488$		0			0
$489$	−85.3839			−0.174609
$490$		0			0
$491$			666.020i			1.35646i	0.734851	+	0.678228i	$0.237252\pi$
$491$			666.020i			1.35646i	−0.734851	+	0.678228i	$0.762748\pi$
$492$		0			0
$493$			88.0203i			0.178540i
$494$		0			0
$495$	34.3349	−	163.516i	0.0693634	−	0.330336i
$496$		0			0
$497$			472.052i			0.949802i
$498$		0			0
$499$		−	400.184i		−	0.801972i	−0.916084	−	0.400986i	$-0.868668\pi$
$499$		−	400.184i		−	0.801972i	0.916084	−	0.400986i	$-0.131332\pi$
$500$		0			0
$501$	−353.120			−0.704830
$502$		0			0
$503$	655.914			1.30400			0.652002	−	0.758217i	$-0.273929\pi$
$503$	655.914			1.30400			0.652002	+	0.758217i	$0.273929\pi$
$504$		0			0
$505$	259.384	+	54.4651i	0.513631	+	0.107852i
$506$		0			0
$507$	−301.542			−0.594757
$508$		0			0
$509$	−617.357			−1.21288			−0.606441	−	0.795128i	$-0.707403\pi$
$509$	−617.357			−1.21288			−0.606441	+	0.795128i	$0.707403\pi$
$510$		0			0
$511$			135.664i			0.265487i
$512$		0			0
$513$			534.390i			1.04170i
$514$		0			0
$515$	−365.001	−	76.6424i	−0.708740	−	0.148820i
$516$		0			0
$517$		−	644.285i		−	1.24620i
$518$		0			0
$519$		−	442.290i		−	0.852196i
$520$		0			0
$521$	41.7771			0.0801863			0.0400932	−	0.999196i	$-0.487235\pi$
$521$	41.7771			0.0801863			0.0400932	+	0.999196i	$0.487235\pi$
$522$		0			0
$523$	926.623			1.77175			0.885873	−	0.463927i	$-0.153560\pi$
$523$	926.623			1.77175			0.885873	+	0.463927i	$0.153560\pi$
$524$		0			0
$525$	329.759	+	144.872i	0.628113	+	0.275947i
$526$		0			0
$527$	−833.740			−1.58205
$528$		0			0
$529$	−523.721			−0.990021
$530$		0			0
$531$			83.5929i			0.157425i
$532$		0			0
$533$		−	573.560i		−	1.07610i
$534$		0			0
$535$	317.847	+	66.7411i	0.594107	+	0.124750i
$536$		0			0
$537$		−	461.376i		−	0.859174i
$538$		0			0
$539$			67.3050i			0.124870i
$540$		0			0
$541$	−638.021			−1.17934			−0.589668	−	0.807645i	$-0.700742\pi$
$541$	−638.021			−1.17934			−0.589668	+	0.807645i	$0.700742\pi$
$542$		0			0
$543$	56.8521			0.104700
$544$		0			0
$545$	−53.2260	−	11.1763i	−0.0976624	−	0.0205070i
$546$		0			0
$547$	880.150			1.60905			0.804525	−	0.593919i	$-0.202420\pi$
$547$	880.150			1.60905			0.804525	+	0.593919i	$0.202420\pi$
$548$		0			0
$549$	115.803			0.210934
$550$		0			0
$551$		−	84.8373i		−	0.153970i
$552$		0			0
$553$			722.881i			1.30720i
$554$		0			0
$555$	56.7066	−	270.059i	0.102174	−	0.486592i
$556$		0			0
$557$		−	44.6368i		−	0.0801379i	−0.999197	−	0.0400689i	$-0.987242\pi$
$557$		−	44.6368i		−	0.0801379i	0.999197	−	0.0400689i	$-0.0127578\pi$
$558$		0			0
$559$			624.159i			1.11656i
$560$		0			0
$561$	362.885			0.646855
$562$		0			0
$563$	140.402			0.249382			0.124691	−	0.992196i	$-0.460206\pi$
$563$	140.402			0.249382			0.124691	+	0.992196i	$0.460206\pi$
$564$		0			0
$565$	−36.9648	+	176.041i	−0.0654244	+	0.311576i
$566$		0			0
$567$	−191.867			−0.338390
$568$		0			0
$569$	−190.715			−0.335176			−0.167588	−	0.985857i	$-0.553598\pi$
$569$	−190.715			−0.335176			−0.167588	+	0.985857i	$0.553598\pi$
$570$		0			0
$571$			539.305i			0.944492i	0.881467	+	0.472246i	$0.156557\pi$
$571$			539.305i			0.944492i	−0.881467	+	0.472246i	$0.843443\pi$
$572$		0			0
$573$			238.377i			0.416015i
$574$		0			0
$575$	−52.5871	−	23.1030i	−0.0914558	−	0.0401791i
$576$		0			0
$577$		−	205.240i		−	0.355701i	−0.984058	−	0.177851i	$-0.943086\pi$
$577$		−	205.240i		−	0.355701i	0.984058	−	0.177851i	$-0.0569144\pi$
$578$		0			0
$579$		−	249.836i		−	0.431496i
$580$		0			0
$581$	−258.526			−0.444968
$582$		0			0
$583$	546.960			0.938182
$584$		0			0
$585$	−70.5573	+	336.021i	−0.120611	+	0.574395i
$586$		0			0
$587$	−547.756			−0.933144			−0.466572	−	0.884483i	$-0.654511\pi$
$587$	−547.756			−0.933144			−0.466572	+	0.884483i	$0.654511\pi$
$588$		0			0
$589$	803.590			1.36433
$590$		0			0
$591$			39.6613i			0.0671088i
$592$		0			0
$593$			485.793i			0.819213i	0.912262	+	0.409606i	$0.134334\pi$
$593$			485.793i			0.819213i	−0.912262	+	0.409606i	$0.865666\pi$
$594$		0			0
$595$	125.414	−	597.272i	0.210781	−	1.00382i
$596$		0			0
$597$			124.094i			0.207862i
$598$		0			0
$599$		−	151.965i		−	0.253697i	−0.991922	−	0.126849i	$-0.959514\pi$
$599$		−	151.965i		−	0.253697i	0.991922	−	0.126849i	$-0.0404862\pi$
$600$		0			0
$601$	572.158			0.952010			0.476005	−	0.879443i	$-0.342084\pi$
$601$	572.158			0.952010			0.476005	+	0.879443i	$0.342084\pi$
$602$		0			0
$603$	−335.527			−0.556429
$604$		0			0
$605$	−240.862	−	50.5759i	−0.398119	−	0.0835965i
$606$		0			0
$607$	528.708			0.871018			0.435509	−	0.900184i	$-0.356568\pi$
$607$	528.708			0.871018			0.435509	+	0.900184i	$0.356568\pi$
$608$		0			0
$609$	66.5697			0.109310
$610$		0			0
$611$			1323.99i			2.16692i
$612$		0			0
$613$			497.586i			0.811723i	0.913934	+	0.405862i	$0.133029\pi$
$613$			497.586i			0.811723i	−0.913934	+	0.405862i	$0.866971\pi$
$614$		0			0
$615$	362.470	+	76.1111i	0.589383	+	0.123758i
$616$		0			0
$617$		−	437.019i		−	0.708297i	−0.935189	−	0.354148i	$-0.884771\pi$
$617$		−	437.019i		−	0.708297i	0.935189	−	0.354148i	$-0.115229\pi$
$618$		0			0
$619$		−	145.416i		−	0.234921i	−0.993078	−	0.117461i	$-0.962525\pi$
$619$		−	145.416i		−	0.234921i	0.993078	−	0.117461i	$-0.0374754\pi$
$620$		0			0
$621$	66.8699			0.107681
$622$		0			0
$623$	368.777			0.591937
$624$		0			0
$625$	422.771	+	460.315i	0.676433	+	0.736504i
$626$		0			0
$627$	−349.763			−0.557835
$628$		0			0
$629$	−467.574			−0.743361
$630$		0			0
$631$			573.075i			0.908202i	0.890950	+	0.454101i	$0.150039\pi$
$631$			573.075i			0.908202i	−0.890950	+	0.454101i	$0.849961\pi$
$632$		0			0
$633$			578.799i			0.914375i
$634$		0			0
$635$	486.921	+	102.243i	0.766805	+	0.161013i
$636$		0			0
$637$		−	138.310i		−	0.217127i
$638$		0			0
$639$		−	290.582i		−	0.454746i
$640$		0			0
$641$	668.158			1.04237			0.521184	−	0.853444i	$-0.325491\pi$
$641$	668.158			1.04237			0.521184	+	0.853444i	$0.325491\pi$
$642$		0			0
$643$	−34.5976			−0.0538065			−0.0269032	−	0.999638i	$-0.508565\pi$
$643$	−34.5976			−0.0538065			−0.0269032	+	0.999638i	$0.508565\pi$
$644$		0			0
$645$	−394.448	−	82.8256i	−0.611547	−	0.128412i
$646$		0			0
$647$	−279.402			−0.431843			−0.215921	−	0.976411i	$-0.569276\pi$
$647$	−279.402			−0.431843			−0.215921	+	0.976411i	$0.569276\pi$
$648$		0			0
$649$	−179.554			−0.276663
$650$		0			0
$651$			630.557i			0.968598i
$652$		0			0
$653$			432.852i			0.662866i	0.943479	+	0.331433i	$0.107532\pi$
$653$			432.852i			0.662866i	−0.943479	+	0.331433i	$0.892468\pi$
$654$		0			0
$655$	−6.30647	+	30.0339i	−0.00962820	+	0.0458532i
$656$		0			0
$657$		−	83.5111i		−	0.127110i
$658$		0			0
$659$			840.788i			1.27585i	0.770097	+	0.637927i	$0.220208\pi$
$659$			840.788i			1.27585i	−0.770097	+	0.637927i	$0.779792\pi$
$660$		0			0
$661$	1019.96			1.54305			0.771524	−	0.636200i	$-0.219495\pi$
$661$	1019.96			1.54305			0.771524	+	0.636200i	$0.219495\pi$
$662$		0			0
$663$	−745.720			−1.12477
$664$		0			0
$665$	−120.879	+	575.674i	−0.181773	+	0.865675i
$666$		0			0
$667$	−10.6160			−0.0159160
$668$		0			0
$669$	−205.261			−0.306817
$670$		0			0
$671$			248.740i			0.370700i
$672$		0			0
$673$		−	574.671i		−	0.853895i	−0.904276	−	0.426948i	$-0.859589\pi$
$673$		−	574.671i		−	0.853895i	0.904276	−	0.426948i	$-0.140411\pi$
$674$		0			0
$675$	−666.174	−	292.669i	−0.986924	−	0.433583i
$676$		0			0
$677$			794.069i			1.17292i	0.809977	+	0.586461i	$0.199479\pi$
$677$			794.069i			1.17292i	−0.809977	+	0.586461i	$0.800521\pi$
$678$		0			0
$679$			1200.19i			1.76758i
$680$		0			0
$681$	−689.817			−1.01295
$682$		0			0
$683$	758.613			1.11071			0.555353	−	0.831614i	$-0.312583\pi$
$683$	758.613			1.11071			0.555353	+	0.831614i	$0.312583\pi$
$684$		0			0
$685$	150.041	−	714.551i	0.219037	−	1.04314i
$686$		0			0
$687$	−165.651			−0.241122
$688$		0			0
$689$	−1123.99			−1.63133
$690$		0			0
$691$			1050.96i			1.52093i	0.649377	+	0.760466i	$0.275030\pi$
$691$			1050.96i			1.52093i	−0.649377	+	0.760466i	$0.724970\pi$
$692$		0			0
$693$			214.115i			0.308968i
$694$		0			0
$695$	−72.5237	+	345.386i	−0.104351	+	0.496958i
$696$		0			0
$697$		−	627.574i		−	0.900393i
$698$		0			0
$699$		−	466.387i		−	0.667220i
$700$		0			0
$701$	−246.846			−0.352134			−0.176067	−	0.984378i	$-0.556338\pi$
$701$	−246.846			−0.352134			−0.176067	+	0.984378i	$0.556338\pi$
$702$		0			0
$703$	450.666			0.641061
$704$		0			0
$705$	−836.715	−	175.692i	−1.18683	−	0.249209i
$706$		0			0
$707$	−339.648			−0.480407
$708$		0			0
$709$	−430.352			−0.606984			−0.303492	−	0.952834i	$-0.598153\pi$
$709$	−430.352			−0.606984			−0.303492	+	0.952834i	$0.598153\pi$
$710$		0			0
$711$		−	444.986i		−	0.625860i
$712$		0			0
$713$		−	100.556i		−	0.141032i
$714$		0			0
$715$	−721.760	−	151.554i	−1.00945	−	0.211964i
$716$		0			0
$717$		−	554.385i		−	0.773200i
$718$		0			0
$719$			223.713i			0.311144i	0.987825	+	0.155572i	$0.0497221\pi$
$719$			223.713i			0.311144i	−0.987825	+	0.155572i	$0.950278\pi$
$720$		0			0
$721$	477.947			0.662895
$722$		0			0
$723$	641.935			0.887877
$724$		0			0
$725$	105.759	+	46.4627i	0.145874	+	0.0640865i
$726$		0			0
$727$	237.535			0.326733			0.163366	−	0.986565i	$-0.447765\pi$
$727$	237.535			0.326733			0.163366	+	0.986565i	$0.447765\pi$
$728$		0			0
$729$	707.093			0.969949
$730$		0			0
$731$			682.938i			0.934252i
$732$		0			0
$733$			706.446i			0.963774i	0.876233	+	0.481887i	$0.160048\pi$
$733$			706.446i			0.963774i	−0.876233	+	0.481887i	$0.839952\pi$
$734$		0			0
$735$	87.4071	+	18.3536i	0.118921	+	0.0249709i
$736$		0			0
$737$		−	720.698i		−	0.977881i
$738$		0			0
$739$		−	769.862i		−	1.04176i	−0.853629	−	0.520881i	$-0.825603\pi$
$739$		−	769.862i		−	1.04176i	0.853629	−	0.520881i	$-0.174397\pi$
$740$		0			0
$741$	718.753			0.969977
$742$		0			0
$743$	−794.965			−1.06994			−0.534970	−	0.844871i	$-0.679677\pi$
$743$	−794.965			−1.06994			−0.534970	+	0.844871i	$0.679677\pi$
$744$		0			0
$745$	−149.003	−	31.2875i	−0.200004	−	0.0419966i
$746$		0			0
$747$	159.142			0.213041
$748$		0			0
$749$	−416.202			−0.555677
$750$		0			0
$751$			1178.75i			1.56958i	0.619764	+	0.784789i	$0.287228\pi$
$751$			1178.75i			1.56958i	−0.619764	+	0.784789i	$0.712772\pi$
$752$		0			0
$753$			300.487i			0.399054i
$754$		0			0
$755$	220.102	−	1048.21i	0.291526	−	1.38836i
$756$		0			0
$757$			268.056i			0.354103i	0.984202	+	0.177052i	$0.0566560\pi$
$757$			268.056i			0.354103i	−0.984202	+	0.177052i	$0.943344\pi$
$758$		0			0
$759$			43.7669i			0.0576639i
$760$		0			0
$761$	587.758			0.772350			0.386175	−	0.922426i	$-0.373796\pi$
$761$	587.758			0.772350			0.386175	+	0.922426i	$0.373796\pi$
$762$		0			0
$763$	69.6964			0.0913452
$764$		0			0
$765$	−77.2018	+	367.665i	−0.100917	+	0.480608i
$766$		0			0
$767$	368.979			0.481068
$768$		0			0
$769$	802.210			1.04319			0.521593	−	0.853194i	$-0.325338\pi$
$769$	802.210			1.04319			0.521593	+	0.853194i	$0.325338\pi$
$770$		0			0
$771$		−	154.755i		−	0.200720i
$772$		0			0
$773$			1133.36i			1.46619i	0.680128	+	0.733093i	$0.261924\pi$
$773$			1133.36i			1.46619i	−0.680128	+	0.733093i	$0.738076\pi$
$774$		0			0
$775$	−440.101	+	1001.76i	−0.567873	+	1.29259i
$776$		0			0
$777$			353.626i			0.455117i
$778$		0			0
$779$			604.879i			0.776482i
$780$		0			0
$781$	624.159			0.799180
$782$		0			0
$783$	−134.483			−0.171754
$784$		0			0
$785$	−75.6594	+	360.320i	−0.0963814	+	0.459006i
$786$		0			0
$787$	−200.234			−0.254427			−0.127214	−	0.991875i	$-0.540603\pi$
$787$	−200.234			−0.254427			−0.127214	+	0.991875i	$0.540603\pi$
$788$		0			0
$789$	288.431			0.365566
$790$		0			0
$791$		−	230.515i		−	0.291422i
$792$		0			0
$793$		−	511.153i		−	0.644581i
$794$		0			0
$795$	149.152	−	710.322i	0.187613	−	0.893487i
$796$		0			0
$797$			841.219i			1.05548i	0.849406	+	0.527741i	$0.176961\pi$
$797$			841.219i			1.05548i	−0.849406	+	0.527741i	$0.823039\pi$
$798$		0			0
$799$			1448.67i			1.81310i
$800$		0			0
$801$	−227.009			−0.283407
$802$		0			0
$803$	179.378			0.223385
$804$		0			0
$805$	72.0359	+	15.1260i	0.0894855	+	0.0187901i
$806$		0			0
$807$	1161.36			1.43911
$808$		0			0
$809$	−6.22291			−0.00769210			−0.00384605	−	0.999993i	$-0.501224\pi$
$809$	−6.22291			−0.00769210			−0.00384605	+	0.999993i	$0.501224\pi$
$810$		0			0
$811$		−	711.896i		−	0.877801i	−0.898536	−	0.438900i	$-0.855368\pi$
$811$		−	711.896i		−	0.877801i	0.898536	−	0.438900i	$-0.144632\pi$
$812$		0			0
$813$			139.967i			0.172161i
$814$		0			0
$815$	185.817	+	39.0176i	0.227996	+	0.0478743i
$816$		0			0
$817$		−	658.242i		−	0.805681i
$818$		0			0
$819$		−	440.000i		−	0.537241i
$820$		0			0
$821$	−580.680			−0.707284			−0.353642	−	0.935381i	$-0.615057\pi$
$821$	−580.680			−0.707284			−0.353642	+	0.935381i	$0.615057\pi$
$822$		0			0
$823$	1333.37			1.62013			0.810066	−	0.586339i	$-0.199431\pi$
$823$	1333.37			1.62013			0.810066	+	0.586339i	$0.199431\pi$
$824$		0			0
$825$	191.554	−	436.017i	0.232187	−	0.528505i
$826$		0			0
$827$	1286.25			1.55532			0.777660	−	0.628685i	$-0.216406\pi$
$827$	1286.25			1.55532			0.777660	+	0.628685i	$0.216406\pi$
$828$		0			0
$829$	567.591			0.684670			0.342335	−	0.939578i	$-0.388782\pi$
$829$	567.591			0.684670			0.342335	+	0.939578i	$0.388782\pi$
$830$		0			0
$831$		−	119.878i		−	0.144257i
$832$		0			0
$833$		−	151.335i		−	0.181674i
$834$		0			0
$835$	768.477	+	161.364i	0.920332	+	0.193250i
$836$		0			0
$837$		−	1273.84i		−	1.52191i
$838$		0			0
$839$			200.095i			0.238492i	0.992865	+	0.119246i	$0.0380477\pi$
$839$			200.095i			0.238492i	−0.992865	+	0.119246i	$0.961952\pi$
$840$		0			0
$841$	−819.650			−0.974614
$842$		0			0
$843$	−832.549			−0.987602
$844$		0			0
$845$	656.231	+	137.794i	0.776605	+	0.163070i
$846$		0			0
$847$	315.395			0.372367
$848$		0			0
$849$	13.4767			0.0158736
$850$		0			0
$851$		−	56.3932i		−	0.0662670i
$852$		0			0
$853$		−	497.128i		−	0.582800i	−0.956601	−	0.291400i	$-0.905879\pi$
$853$		−	497.128i		−	0.582800i	0.956601	−	0.291400i	$-0.0941211\pi$
$854$		0			0
$855$	74.4101	−	354.370i	0.0870293	−	0.414467i
$856$		0			0
$857$			196.865i			0.229714i	0.993382	+	0.114857i	$0.0366410\pi$
$857$			196.865i			0.229714i	−0.993382	+	0.114857i	$0.963359\pi$
$858$		0			0
$859$		−	645.980i		−	0.752014i	−0.926617	−	0.376007i	$-0.877297\pi$
$859$		−	645.980i		−	0.752014i	0.926617	−	0.376007i	$-0.122703\pi$
$860$		0			0
$861$	−474.634			−0.551259
$862$		0			0
$863$	−908.561			−1.05279			−0.526397	−	0.850239i	$-0.676457\pi$
$863$	−908.561			−1.05279			−0.526397	+	0.850239i	$0.676457\pi$
$864$		0			0
$865$	−202.111	+	962.533i	−0.233655	+	1.11276i
$866$		0			0
$867$	−166.131			−0.191616
$868$		0			0
$869$	955.812			1.09990
$870$		0			0
$871$			1481.02i			1.70036i
$872$		0			0
$873$		−	738.804i		−	0.846282i
$874$		0			0
$875$	−651.437	−	465.967i	−0.744499	−	0.532534i
$876$		0			0
$877$			1268.13i			1.44599i	0.690853	+	0.722996i	$0.257235\pi$
$877$			1268.13i			1.44599i	−0.690853	+	0.722996i	$0.742765\pi$
$878$		0			0
$879$		−	502.615i		−	0.571803i
$880$		0			0
$881$	−580.932			−0.659401			−0.329700	−	0.944086i	$-0.606948\pi$
$881$	−580.932			−0.659401			−0.329700	+	0.944086i	$0.606948\pi$
$882$		0			0
$883$	−1216.42			−1.37760			−0.688800	−	0.724951i	$-0.741862\pi$
$883$	−1216.42			−1.37760			−0.688800	+	0.724951i	$0.741862\pi$
$884$		0			0
$885$	−48.9633	+	233.182i	−0.0553257	+	0.263483i
$886$		0			0
$887$	69.8221			0.0787172			0.0393586	−	0.999225i	$-0.487469\pi$
$887$	69.8221			0.0787172			0.0393586	+	0.999225i	$0.487469\pi$
$888$		0			0
$889$	−637.594			−0.717204
$890$		0			0
$891$			253.692i			0.284727i
$892$		0			0
$893$		−	1396.28i		−	1.56359i
$894$		0			0
$895$	−210.834	+	1004.07i	−0.235568	+	1.12187i
$896$		0			0
$897$		−	89.9398i		−	0.100267i
$898$		0			0
$899$			202.229i			0.224949i
$900$		0			0
$901$	−1229.84			−1.36497
$902$		0			0
$903$	516.506			0.571989
$904$		0			0
$905$	−123.724	−	25.9795i	−0.136712	−	0.0287066i
$906$		0			0
$907$	−962.850			−1.06158			−0.530788	−	0.847504i	$-0.678104\pi$
$907$	−962.850			−1.06158			−0.530788	+	0.847504i	$0.678104\pi$
$908$		0			0
$909$	209.078			0.230009
$910$		0			0
$911$			66.6124i			0.0731201i	0.999331	+	0.0365600i	$0.0116400\pi$
$911$			66.6124i			0.0731201i	−0.999331	+	0.0365600i	$0.988360\pi$
$912$		0			0
$913$			341.830i			0.374403i
$914$		0			0
$915$	323.031	+	67.8297i	0.353040	+	0.0741308i
$916$		0			0
$917$		−	39.3276i		−	0.0428872i
$918$		0			0
$919$			1745.54i			1.89939i	0.313175	+	0.949696i	$0.398608\pi$
$919$			1745.54i			1.89939i	−0.313175	+	0.949696i	$0.601392\pi$
$920$		0			0
$921$	100.833			0.109482
$922$		0			0
$923$	−1282.63			−1.38963
$924$		0			0
$925$	−246.816	+	561.803i	−0.266828	+	0.607354i
$926$		0			0
$927$	−294.212			−0.317381
$928$		0			0
$929$	1617.72			1.74135			0.870677	−	0.491855i	$-0.163681\pi$
$929$	1617.72			1.74135			0.870677	+	0.491855i	$0.163681\pi$
$930$		0			0
$931$			145.862i			0.156673i
$932$		0			0
$933$		−	1072.12i		−	1.14911i
$934$		0			0
$935$	−789.730	−	165.826i	−0.844631	−	0.177354i
$936$		0			0
$937$			803.266i			0.857274i	0.903477	+	0.428637i	$0.141006\pi$
$937$			803.266i			0.857274i	−0.903477	+	0.428637i	$0.858994\pi$
$938$		0			0
$939$			356.984i			0.380175i
$940$		0			0
$941$	−702.194			−0.746222			−0.373111	−	0.927787i	$-0.621709\pi$
$941$	−702.194			−0.746222			−0.373111	+	0.927787i	$0.621709\pi$
$942$		0			0
$943$	75.6904			0.0802656
$944$		0			0
$945$	912.551	+	191.616i	0.965663	+	0.202769i
$946$		0			0
$947$	67.0789			0.0708331			0.0354165	−	0.999373i	$-0.488724\pi$
$947$	67.0789			0.0708331			0.0354165	+	0.999373i	$0.488724\pi$
$948$		0			0
$949$	−368.618			−0.388427
$950$		0			0
$951$			956.327i			1.00560i
$952$		0			0
$953$			97.4301i			0.102235i	0.998693	+	0.0511176i	$0.0162783\pi$
$953$			97.4301i			0.102235i	−0.998693	+	0.0511176i	$0.983722\pi$
$954$		0			0
$955$	108.930	−	518.768i	0.114063	−	0.543212i
$956$		0			0
$957$		−	88.0203i		−	0.0919752i
$958$		0			0
$959$			935.663i			0.975666i
$960$		0			0
$961$	−954.542			−0.993280
$962$		0			0
$963$	256.203			0.266047
$964$		0			0
$965$	−114.167	+	543.706i	−0.118307	+	0.563426i
$966$		0			0
$967$	1727.52			1.78647			0.893236	−	0.449588i	$-0.148429\pi$
$967$	1727.52			1.78647			0.893236	+	0.449588i	$0.148429\pi$
$968$		0			0
$969$	786.440			0.811599
$970$		0			0
$971$			379.593i			0.390930i	0.980711	+	0.195465i	$0.0626216\pi$
$971$			379.593i			0.390930i	−0.980711	+	0.195465i	$0.937378\pi$
$972$		0			0
$973$		−	452.263i		−	0.464812i
$974$		0			0
$975$	−393.639	+	896.002i	−0.403732	+	0.918976i
$976$		0			0
$977$		−	1676.42i		−	1.71589i	−0.513746	−	0.857943i	$-0.671742\pi$
$977$		−	1676.42i		−	1.71589i	0.513746	−	0.857943i	$-0.328258\pi$
$978$		0			0
$979$		−	487.607i		−	0.498066i
$980$		0			0
$981$	−42.9032			−0.0437342
$982$		0			0
$983$	−1108.29			−1.12746			−0.563729	−	0.825960i	$-0.690634\pi$
$983$	−1108.29			−1.12746			−0.563729	+	0.825960i	$0.690634\pi$
$984$		0			0
$985$	18.1239	−	86.3129i	0.0183999	−	0.0876274i
$986$		0			0
$987$	1095.63			1.11006
$988$		0			0
$989$	−82.3678			−0.0832839
$990$		0			0
$991$			1382.06i			1.39461i	0.716775	+	0.697305i	$0.245618\pi$
$991$			1382.06i			1.39461i	−0.716775	+	0.697305i	$0.754382\pi$
$992$		0			0
$993$			1283.69i			1.29274i
$994$		0			0
$995$	56.7066	−	270.059i	0.0569916	−	0.271416i
$996$		0			0
$997$			391.981i			0.393161i	0.980488	+	0.196580i	$0.0629837\pi$
$997$			391.981i			0.393161i	−0.980488	+	0.196580i	$0.937016\pi$
$998$		0			0
$999$		−	714.390i		−	0.715105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.h.m.1279.9		16
4.3	odd	2	inner	1280.3.h.m.1279.5		16
5.4	even	2	inner	1280.3.h.m.1279.6		16
8.3	odd	2	inner	1280.3.h.m.1279.12		16
8.5	even	2	inner	1280.3.h.m.1279.8		16
16.3	odd	4		40.3.e.c.19.5	yes	8
16.5	even	4		40.3.e.c.19.3	✓	8
16.11	odd	4		160.3.e.c.79.6		8
16.13	even	4		160.3.e.c.79.5		8
20.19	odd	2	inner	1280.3.h.m.1279.10		16
40.19	odd	2	inner	1280.3.h.m.1279.7		16
40.29	even	2	inner	1280.3.h.m.1279.11		16
48.5	odd	4		360.3.p.g.19.6		8
48.11	even	4		1440.3.p.g.559.4		8
48.29	odd	4		1440.3.p.g.559.5		8
48.35	even	4		360.3.p.g.19.4		8
80.3	even	4		200.3.g.h.51.1		8
80.13	odd	4		800.3.g.h.751.4		8
80.19	odd	4		40.3.e.c.19.4	yes	8
80.27	even	4		800.3.g.h.751.6		8
80.29	even	4		160.3.e.c.79.4		8
80.37	odd	4		200.3.g.h.51.7		8
80.43	even	4		800.3.g.h.751.3		8
80.53	odd	4		200.3.g.h.51.2		8
80.59	odd	4		160.3.e.c.79.3		8
80.67	even	4		200.3.g.h.51.8		8
80.69	even	4		40.3.e.c.19.6	yes	8
80.77	odd	4		800.3.g.h.751.5		8
240.29	odd	4		1440.3.p.g.559.3		8
240.59	even	4		1440.3.p.g.559.6		8
240.149	odd	4		360.3.p.g.19.3		8
240.179	even	4		360.3.p.g.19.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.e.c.19.3	✓	8	16.5	even	4
40.3.e.c.19.4	yes	8	80.19	odd	4
40.3.e.c.19.5	yes	8	16.3	odd	4
40.3.e.c.19.6	yes	8	80.69	even	4
160.3.e.c.79.3		8	80.59	odd	4
160.3.e.c.79.4		8	80.29	even	4
160.3.e.c.79.5		8	16.13	even	4
160.3.e.c.79.6		8	16.11	odd	4
200.3.g.h.51.1		8	80.3	even	4
200.3.g.h.51.2		8	80.53	odd	4
200.3.g.h.51.7		8	80.37	odd	4
200.3.g.h.51.8		8	80.67	even	4
360.3.p.g.19.3		8	240.149	odd	4
360.3.p.g.19.4		8	48.35	even	4
360.3.p.g.19.5		8	240.179	even	4
360.3.p.g.19.6		8	48.5	odd	4
800.3.g.h.751.3		8	80.43	even	4
800.3.g.h.751.4		8	80.13	odd	4
800.3.g.h.751.5		8	80.77	odd	4
800.3.g.h.751.6		8	80.27	even	4
1280.3.h.m.1279.5		16	4.3	odd	2	inner
1280.3.h.m.1279.6		16	5.4	even	2	inner
1280.3.h.m.1279.7		16	40.19	odd	2	inner
1280.3.h.m.1279.8		16	8.5	even	2	inner
1280.3.h.m.1279.9		16	1.1	even	1	trivial
1280.3.h.m.1279.10		16	20.19	odd	2	inner
1280.3.h.m.1279.11		16	40.29	even	2	inner
1280.3.h.m.1279.12		16	8.3	odd	2	inner
1440.3.p.g.559.3		8	240.29	odd	4
1440.3.p.g.559.4		8	48.11	even	4
1440.3.p.g.559.5		8	48.29	odd	4
1440.3.p.g.559.6		8	240.59	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.24849 q^{3} +(-4.89329 - 1.02749i) q^{5} +6.40747 q^{7} -3.94427 q^{9} +O(q^{10})\)	\(q+2.24849 q^{3} +(-4.89329 - 1.02749i) q^{5} +6.40747 q^{7} -3.94427 q^{9} -8.47214i q^{11} +17.4100i q^{13} +(-11.0025 - 2.31030i) q^{15} +19.0496i q^{17} -18.3607i q^{19} +14.4072 q^{21} -2.29753 q^{23} +(22.8885 + 10.0556i) q^{25} -29.1051 q^{27} +4.62059 q^{29} +43.7669i q^{31} -19.0496i q^{33} +(-31.3536 - 6.58359i) q^{35} +24.5452i q^{37} +39.1463i q^{39} -32.9443 q^{41} +35.8506 q^{43} +(19.3005 + 4.05269i) q^{45} +76.0475 q^{47} -7.94427 q^{49} +42.8328i q^{51} +64.5599i q^{53} +(-8.70500 + 41.4566i) q^{55} -41.2839i q^{57} -21.1935i q^{59} -29.3597 q^{61} -25.2728 q^{63} +(17.8885 - 85.1922i) q^{65} +85.0669 q^{67} -5.16598 q^{69} +73.6720i q^{71} +21.1727i q^{73} +(51.4648 + 22.6099i) q^{75} -54.2850i q^{77} +112.818i q^{79} -29.9443 q^{81} -40.3476 q^{83} +(19.5732 - 93.2150i) q^{85} +10.3894 q^{87} +57.5542 q^{89} +111.554i q^{91} +98.4096i q^{93} +(-18.8653 + 89.8441i) q^{95} +187.311i q^{97} +33.4164i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 80 q^{9}+O(q^{10}) \) 16 * q + 80 * q^9	\( 16 q + 80 q^{9} + 80 q^{25} - 384 q^{41} + 16 q^{49} - 336 q^{81} - 224 q^{89}+O(q^{100}) \) 16 * q + 80 * q^9 + 80 * q^25 - 384 * q^41 + 16 * q^49 - 336 * q^81 - 224 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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