LMFDB - Embedded newform 1008.3.dc.b.305.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(305,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))

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

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

chi := DirichletCharacter("1008.305");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.dc (of order $6$, degree $2$, 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:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			305.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1008.305
Dual form			1008.3.dc.b.737.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.67423 + 2.12132i) q^{5} +(3.50000 + 6.06218i) q^{7} +O(q^{10})$	$q+(-3.67423 + 2.12132i) q^{5} +(3.50000 + 6.06218i) q^{7} +(11.0227 + 6.36396i) q^{11} -1.00000 q^{13} +(-14.6969 - 8.48528i) q^{17} +(11.5000 + 19.9186i) q^{19} +(-14.6969 + 8.48528i) q^{23} +(-3.50000 + 6.06218i) q^{25} -33.9411i q^{29} +(23.5000 - 40.7032i) q^{31} +(-25.7196 - 14.8492i) q^{35} +(27.5000 + 47.6314i) q^{37} +46.6690i q^{41} -23.0000 q^{43} +(-3.67423 + 2.12132i) q^{47} +(-24.5000 + 42.4352i) q^{49} +(-44.0908 - 25.4558i) q^{53} -54.0000 q^{55} +(-73.4847 - 42.4264i) q^{59} +(-52.0000 - 90.0666i) q^{61} +(3.67423 - 2.12132i) q^{65} +(-48.5000 + 84.0045i) q^{67} +97.5807i q^{71} +(-32.5000 + 56.2917i) q^{73} +89.0955i q^{77} +(56.5000 + 97.8609i) q^{79} -29.6985i q^{83} +72.0000 q^{85} +(-117.576 + 67.8823i) q^{89} +(-3.50000 - 6.06218i) q^{91} +(-84.5074 - 48.7904i) q^{95} +104.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{7}+O(q^{10}) $ 4 * q + 14 * q^7	$ 4 q + 14 q^{7} - 4 q^{13} + 46 q^{19} - 14 q^{25} + 94 q^{31} + 110 q^{37} - 92 q^{43} - 98 q^{49} - 216 q^{55} - 208 q^{61} - 194 q^{67} - 130 q^{73} + 226 q^{79} + 288 q^{85} - 14 q^{91} + 416 q^{97}+O(q^{100}) $ 4 * q + 14 * q^7 - 4 * q^13 + 46 * q^19 - 14 * q^25 + 94 * q^31 + 110 * q^37 - 92 * q^43 - 98 * q^49 - 216 * q^55 - 208 * q^61 - 194 * q^67 - 130 * q^73 + 226 * q^79 + 288 * q^85 - 14 * q^91 + 416 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.67423	+	2.12132i	−0.734847	+	0.424264i	−0.820193	−	0.572087i	$-0.806134\pi$
$5$	−3.67423	+	2.12132i	−0.734847	+	0.424264i	0.0853458	+	0.996351i	$0.472801\pi$
$6$		0			0
$7$	3.50000	+	6.06218i	0.500000	+	0.866025i
$7$	3.50000	+	6.06218i	0.500000	+	0.866025i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	11.0227	+	6.36396i	1.00206	+	0.578542i	0.908858	−	0.417105i	$-0.136955\pi$
$11$	11.0227	+	6.36396i	1.00206	+	0.578542i	0.0932057	+	0.995647i	$0.470289\pi$
$12$		0			0
$13$	−1.00000			−0.0769231			−0.0384615	−	0.999260i	$-0.512246\pi$
$13$	−1.00000			−0.0769231			−0.0384615	+	0.999260i	$0.512246\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−14.6969	−	8.48528i	−0.864526	−	0.499134i	0.000999453	−	1.00000i	$-0.499682\pi$
$17$	−14.6969	−	8.48528i	−0.864526	−	0.499134i	−0.865525	+	0.500865i	$0.833015\pi$
$18$		0			0
$19$	11.5000	+	19.9186i	0.605263	+	1.04835i	0.992010	+	0.126161i	$0.0402654\pi$
$19$	11.5000	+	19.9186i	0.605263	+	1.04835i	−0.386747	+	0.922186i	$0.626401\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−14.6969	+	8.48528i	−0.638997	+	0.368925i	−0.784228	−	0.620473i	$-0.786941\pi$
$23$	−14.6969	+	8.48528i	−0.638997	+	0.368925i	0.145231	+	0.989398i	$0.453608\pi$
$24$		0			0
$25$	−3.50000	+	6.06218i	−0.140000	+	0.242487i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	33.9411i		−	1.17038i	−0.810895	−	0.585192i	$-0.801019\pi$
$29$		−	33.9411i		−	1.17038i	0.810895	−	0.585192i	$-0.198981\pi$
$30$		0			0
$31$	23.5000	−	40.7032i	0.758065	−	1.31301i	−0.185772	−	0.982593i	$-0.559479\pi$
$31$	23.5000	−	40.7032i	0.758065	−	1.31301i	0.943836	−	0.330413i	$-0.107188\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−25.7196	−	14.8492i	−0.734847	−	0.424264i
$36$		0			0
$37$	27.5000	+	47.6314i	0.743243	+	1.28734i	0.951011	+	0.309157i	$0.100047\pi$
$37$	27.5000	+	47.6314i	0.743243	+	1.28734i	−0.207768	+	0.978178i	$0.566620\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			46.6690i			1.13827i	0.822244	+	0.569135i	$0.192722\pi$
$41$			46.6690i			1.13827i	−0.822244	+	0.569135i	$0.807278\pi$
$42$		0			0
$43$	−23.0000			−0.534884			−0.267442	−	0.963574i	$-0.586178\pi$
$43$	−23.0000			−0.534884			−0.267442	+	0.963574i	$0.586178\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.67423	+	2.12132i	−0.0781752	+	0.0451345i	−0.538578	−	0.842576i	$-0.681038\pi$
$47$	−3.67423	+	2.12132i	−0.0781752	+	0.0451345i	0.460403	+	0.887710i	$0.347705\pi$
$48$		0			0
$49$	−24.5000	+	42.4352i	−0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−44.0908	−	25.4558i	−0.831902	−	0.480299i	0.0226013	−	0.999745i	$-0.492805\pi$
$53$	−44.0908	−	25.4558i	−0.831902	−	0.480299i	−0.854504	+	0.519446i	$0.826138\pi$
$54$		0			0
$55$	−54.0000			−0.981818
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−73.4847	−	42.4264i	−1.24550	−	0.719092i	−0.275294	−	0.961360i	$-0.588775\pi$
$59$	−73.4847	−	42.4264i	−1.24550	−	0.719092i	−0.970209	+	0.242268i	$0.922109\pi$
$60$		0			0
$61$	−52.0000	−	90.0666i	−0.852459	−	1.47650i	−0.878982	−	0.476854i	$-0.841777\pi$
$61$	−52.0000	−	90.0666i	−0.852459	−	1.47650i	0.0265234	−	0.999648i	$-0.491556\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.67423	−	2.12132i	0.0565267	−	0.0326357i
$66$		0			0
$67$	−48.5000	+	84.0045i	−0.723881	+	1.25380i	0.235553	+	0.971862i	$0.424310\pi$
$67$	−48.5000	+	84.0045i	−0.723881	+	1.25380i	−0.959433	+	0.281936i	$0.909023\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			97.5807i			1.37438i	0.726479	+	0.687188i	$0.241155\pi$
$71$			97.5807i			1.37438i	−0.726479	+	0.687188i	$0.758845\pi$
$72$		0			0
$73$	−32.5000	+	56.2917i	−0.445205	+	0.771119i	−0.998067	−	0.0621550i	$-0.980203\pi$
$73$	−32.5000	+	56.2917i	−0.445205	+	0.771119i	0.552861	+	0.833273i	$0.313536\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			89.0955i			1.15708i
$78$		0			0
$79$	56.5000	+	97.8609i	0.715190	+	1.23875i	0.962886	+	0.269907i	$0.0869931\pi$
$79$	56.5000	+	97.8609i	0.715190	+	1.23875i	−0.247696	+	0.968838i	$0.579674\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	29.6985i		−	0.357813i	−0.983866	−	0.178907i	$-0.942744\pi$
$83$		−	29.6985i		−	0.357813i	0.983866	−	0.178907i	$-0.0572560\pi$
$84$		0			0
$85$	72.0000			0.847059
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−117.576	+	67.8823i	−1.32107	+	0.762722i	−0.983900	−	0.178721i	$-0.942804\pi$
$89$	−117.576	+	67.8823i	−1.32107	+	0.762722i	−0.337173	+	0.941443i	$0.609471\pi$
$90$		0			0
$91$	−3.50000	−	6.06218i	−0.0384615	−	0.0666173i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−84.5074	−	48.7904i	−0.889552	−	0.513583i
$96$		0			0
$97$	104.000			1.07216			0.536082	−	0.844166i	$-0.319904\pi$
$97$	104.000			1.07216			0.536082	+	0.844166i	$0.319904\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−128.598	−	74.2462i	−1.27325	−	0.735111i	−0.297651	−	0.954675i	$-0.596203\pi$
$101$	−128.598	−	74.2462i	−1.27325	−	0.735111i	−0.975598	+	0.219564i	$0.929537\pi$
$102$		0			0
$103$	59.5000	+	103.057i	0.577670	+	1.00055i	0.995746	+	0.0921416i	$0.0293712\pi$
$103$	59.5000	+	103.057i	0.577670	+	1.00055i	−0.418076	+	0.908412i	$0.637295\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−102.879	+	59.3970i	−0.961482	+	0.555112i	−0.896629	−	0.442783i	$-0.853991\pi$
$107$	−102.879	+	59.3970i	−0.961482	+	0.555112i	−0.0648531	+	0.997895i	$0.520658\pi$
$108$		0			0
$109$	24.5000	−	42.4352i	0.224771	−	0.389314i	−0.731480	−	0.681863i	$-0.761170\pi$
$109$	24.5000	−	42.4352i	0.224771	−	0.389314i	0.956251	+	0.292549i	$0.0945034\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			97.5807i			0.863546i	0.901982	+	0.431773i	$0.142112\pi$
$113$			97.5807i			0.863546i	−0.901982	+	0.431773i	$0.857888\pi$
$114$		0			0
$115$	36.0000	−	62.3538i	0.313043	−	0.542207i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	118.794i		−	0.998268i
$120$		0			0
$121$	20.5000	+	35.5070i	0.169421	+	0.293447i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	135.765i		−	1.08612i
$126$		0			0
$127$	−113.000			−0.889764			−0.444882	−	0.895589i	$-0.646754\pi$
$127$	−113.000			−0.889764			−0.444882	+	0.895589i	$0.646754\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	18.3712	−	10.6066i	0.140238	−	0.0809664i	−0.428239	−	0.903665i	$-0.640866\pi$
$131$	18.3712	−	10.6066i	0.140238	−	0.0809664i	0.568477	+	0.822699i	$0.307533\pi$
$132$		0			0
$133$	−80.5000	+	139.430i	−0.605263	+	1.04835i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	58.7878	+	33.9411i	0.429108	+	0.247745i	0.698966	−	0.715154i	$-0.253644\pi$
$137$	58.7878	+	33.9411i	0.429108	+	0.247745i	−0.269859	+	0.962900i	$0.586977\pi$
$138$		0			0
$139$	103.000			0.741007			0.370504	−	0.928831i	$-0.379185\pi$
$139$	103.000			0.741007			0.370504	+	0.928831i	$0.379185\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.0227	−	6.36396i	−0.0770818	−	0.0445032i
$144$		0			0
$145$	72.0000	+	124.708i	0.496552	+	0.860053i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	146.969	−	84.8528i	0.986372	−	0.569482i	0.0821839	−	0.996617i	$-0.473811\pi$
$149$	146.969	−	84.8528i	0.986372	−	0.569482i	0.904188	+	0.427135i	$0.140477\pi$
$150$		0			0
$151$	52.0000	−	90.0666i	0.344371	−	0.596468i	−0.640868	−	0.767651i	$-0.721426\pi$
$151$	52.0000	−	90.0666i	0.344371	−	0.596468i	0.985239	+	0.171183i	$0.0547589\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			199.404i			1.28648i
$156$		0			0
$157$	−76.0000	+	131.636i	−0.484076	+	0.838445i	−0.999833	−	0.0182904i	$-0.994178\pi$
$157$	−76.0000	+	131.636i	−0.484076	+	0.838445i	0.515756	+	0.856735i	$0.327511\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−102.879	−	59.3970i	−0.638997	−	0.368925i
$162$		0			0
$163$	28.0000	+	48.4974i	0.171779	+	0.297530i	0.939042	−	0.343803i	$-0.111715\pi$
$163$	28.0000	+	48.4974i	0.171779	+	0.297530i	−0.767263	+	0.641333i	$0.778382\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.24264i		−	0.0254050i	−0.999919	−	0.0127025i	$-0.995957\pi$
$167$		−	4.24264i		−	0.0254050i	0.999919	−	0.0127025i	$-0.00404345\pi$
$168$		0			0
$169$	−168.000			−0.994083
$170$		0			0
$171$		0			0
$172$		0			0
$173$	132.272	−	76.3675i	0.764581	−	0.441431i	−0.0663573	−	0.997796i	$-0.521138\pi$
$173$	132.272	−	76.3675i	0.764581	−	0.441431i	0.830938	+	0.556365i	$0.187804\pi$
$174$		0			0
$175$	−49.0000			−0.280000
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.67423	−	2.12132i	−0.0205265	−	0.0118510i	0.489702	−	0.871890i	$-0.337106\pi$
$179$	−3.67423	−	2.12132i	−0.0205265	−	0.0118510i	−0.510228	+	0.860039i	$0.670439\pi$
$180$		0			0
$181$	−55.0000			−0.303867			−0.151934	−	0.988391i	$-0.548550\pi$
$181$	−55.0000			−0.303867			−0.151934	+	0.988391i	$0.548550\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−202.083	−	116.673i	−1.09234	−	0.630663i
$186$		0			0
$187$	−108.000	−	187.061i	−0.577540	−	1.00033i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−55.1135	+	31.8198i	−0.288552	+	0.166596i	−0.637289	−	0.770625i	$-0.719944\pi$
$191$	−55.1135	+	31.8198i	−0.288552	+	0.166596i	0.348736	+	0.937221i	$0.386611\pi$
$192$		0			0
$193$	75.5000	−	130.770i	0.391192	−	0.677564i	−0.601415	−	0.798937i	$-0.705396\pi$
$193$	75.5000	−	130.770i	0.391192	−	0.677564i	0.992607	+	0.121373i	$0.0387296\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	16.9706i		−	0.0861450i	−0.999072	−	0.0430725i	$-0.986285\pi$
$197$		−	16.9706i		−	0.0861450i	0.999072	−	0.0430725i	$-0.0137146\pi$
$198$		0			0
$199$	−80.0000	+	138.564i	−0.402010	+	0.696302i	−0.993968	−	0.109667i	$-0.965022\pi$
$199$	−80.0000	+	138.564i	−0.402010	+	0.696302i	0.591958	+	0.805969i	$0.298355\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	205.757	−	118.794i	1.01358	−	0.585192i
$204$		0			0
$205$	−99.0000	−	171.473i	−0.482927	−	0.836454i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			292.742i			1.40068i
$210$		0			0
$211$	208.000			0.985782			0.492891	−	0.870091i	$-0.335940\pi$
$211$	208.000			0.985782			0.492891	+	0.870091i	$0.335940\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	84.5074	−	48.7904i	0.393058	−	0.226932i
$216$		0			0
$217$	329.000			1.51613
$218$		0			0
$219$		0			0
$220$		0			0
$221$	14.6969	+	8.48528i	0.0665020	+	0.0383949i
$222$		0			0
$223$	40.0000			0.179372			0.0896861	−	0.995970i	$-0.471414\pi$
$223$	40.0000			0.179372			0.0896861	+	0.995970i	$0.471414\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	194.734	+	112.430i	0.857861	+	0.495286i	0.863295	−	0.504699i	$-0.168397\pi$
$227$	194.734	+	112.430i	0.857861	+	0.495286i	−0.00543445	+	0.999985i	$0.501730\pi$
$228$		0			0
$229$	−188.500	−	326.492i	−0.823144	−	1.42573i	−0.903329	−	0.428947i	$-0.858885\pi$
$229$	−188.500	−	326.492i	−0.823144	−	1.42573i	0.0801853	−	0.996780i	$-0.474449\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−297.613	+	171.827i	−1.27731	+	0.737455i	−0.976353	−	0.216183i	$-0.930639\pi$
$233$	−297.613	+	171.827i	−1.27731	+	0.737455i	−0.300956	+	0.953638i	$0.597306\pi$
$234$		0			0
$235$	9.00000	−	15.5885i	0.0382979	−	0.0663339i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	12.7279i		−	0.0532549i	−0.999645	−	0.0266275i	$-0.991523\pi$
$239$		−	12.7279i		−	0.0532549i	0.999645	−	0.0266275i	$-0.00847678\pi$
$240$		0			0
$241$	65.0000	−	112.583i	0.269710	−	0.467151i	−0.699077	−	0.715046i	$-0.746406\pi$
$241$	65.0000	−	112.583i	0.269710	−	0.467151i	0.968787	+	0.247896i	$0.0797390\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	207.889i		−	0.848528i
$246$		0			0
$247$	−11.5000	−	19.9186i	−0.0465587	−	0.0806420i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	50.9117i		−	0.202835i	−0.994844	−	0.101418i	$-0.967662\pi$
$251$		−	50.9117i		−	0.202835i	0.994844	−	0.101418i	$-0.0323379\pi$
$252$		0			0
$253$	−216.000			−0.853755
$254$		0			0
$255$		0			0
$256$		0			0
$257$	407.840	−	235.467i	1.58693	−	0.916212i	0.593117	−	0.805117i	$-0.297897\pi$
$257$	407.840	−	235.467i	1.58693	−	0.916212i	0.993810	−	0.111096i	$-0.0354360\pi$
$258$		0			0
$259$	−192.500	+	333.420i	−0.743243	+	1.28734i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	88.1816	+	50.9117i	0.335291	+	0.193581i	0.658188	−	0.752854i	$-0.271323\pi$
$263$	88.1816	+	50.9117i	0.335291	+	0.193581i	−0.322897	+	0.946434i	$0.604657\pi$
$264$		0			0
$265$	216.000			0.815094
$266$		0			0
$267$		0			0
$268$		0			0
$269$	246.174	+	142.128i	0.915144	+	0.528359i	0.882083	−	0.471095i	$-0.156141\pi$
$269$	246.174	+	142.128i	0.915144	+	0.528359i	0.0330613	+	0.999453i	$0.489474\pi$
$270$		0			0
$271$	−260.000	−	450.333i	−0.959410	−	1.66175i	−0.723938	−	0.689865i	$-0.757670\pi$
$271$	−260.000	−	450.333i	−0.959410	−	1.66175i	−0.235471	−	0.971881i	$-0.575663\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−77.1589	+	44.5477i	−0.280578	+	0.161992i
$276$		0			0
$277$	−56.5000	+	97.8609i	−0.203971	+	0.353288i	−0.949804	−	0.312844i	$-0.898718\pi$
$277$	−56.5000	+	97.8609i	−0.203971	+	0.353288i	0.745833	+	0.666133i	$0.232052\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			458.205i			1.63062i	0.579022	+	0.815312i	$0.303434\pi$
$281$			458.205i			1.63062i	−0.579022	+	0.815312i	$0.696566\pi$
$282$		0			0
$283$	44.5000	−	77.0763i	0.157244	−	0.272354i	−0.776630	−	0.629957i	$-0.783072\pi$
$283$	44.5000	−	77.0763i	0.157244	−	0.272354i	0.933874	+	0.357603i	$0.116406\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−282.916	+	163.342i	−0.985770	+	0.569135i
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.00173010	−	0.00299663i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			67.8823i			0.231680i	0.993268	+	0.115840i	$0.0369560\pi$
$293$			67.8823i			0.231680i	−0.993268	+	0.115840i	$0.963044\pi$
$294$		0			0
$295$	360.000			1.22034
$296$		0			0
$297$		0			0
$298$		0			0
$299$	14.6969	−	8.48528i	0.0491536	−	0.0283789i
$300$		0			0
$301$	−80.5000	−	139.430i	−0.267442	−	0.463223i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	382.120	+	220.617i	1.25285	+	0.723335i
$306$		0			0
$307$	−233.000			−0.758958			−0.379479	−	0.925200i	$-0.623897\pi$
$307$	−233.000			−0.758958			−0.379479	+	0.925200i	$0.623897\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	216.780	+	125.158i	0.697041	+	0.402437i	0.806244	−	0.591582i	$-0.201497\pi$
$311$	216.780	+	125.158i	0.697041	+	0.402437i	−0.109203	+	0.994019i	$0.534830\pi$
$312$		0			0
$313$	75.5000	+	130.770i	0.241214	+	0.417795i	0.961060	−	0.276338i	$-0.0891211\pi$
$313$	75.5000	+	130.770i	0.241214	+	0.417795i	−0.719846	+	0.694133i	$0.755788\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	352.727	−	203.647i	1.11270	−	0.642419i	0.173174	−	0.984891i	$-0.444598\pi$
$317$	352.727	−	203.647i	1.11270	−	0.642419i	0.939528	+	0.342472i	$0.111264\pi$
$318$		0			0
$319$	216.000	−	374.123i	0.677116	−	1.17280i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	390.323i		−	1.20843i
$324$		0			0
$325$	3.50000	−	6.06218i	0.0107692	−	0.0186529i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−25.7196	−	14.8492i	−0.0781752	−	0.0451345i
$330$		0			0
$331$	−36.5000	−	63.2199i	−0.110272	−	0.190997i	0.805608	−	0.592449i	$-0.201839\pi$
$331$	−36.5000	−	63.2199i	−0.110272	−	0.190997i	−0.915880	+	0.401452i	$0.868505\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	411.536i		−	1.22847i
$336$		0			0
$337$	527.000			1.56380			0.781899	−	0.623405i	$-0.214251\pi$
$337$	527.000			1.56380			0.781899	+	0.623405i	$0.214251\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	518.067	−	299.106i	1.51926	−	0.877144i
$342$		0			0
$343$	−343.000			−1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	367.423	+	212.132i	1.05886	+	0.611332i	0.925116	−	0.379685i	$-0.123968\pi$
$347$	367.423	+	212.132i	1.05886	+	0.611332i	0.133741	+	0.991016i	$0.457301\pi$
$348$		0			0
$349$	−112.000			−0.320917			−0.160458	−	0.987043i	$-0.551297\pi$
$349$	−112.000			−0.320917			−0.160458	+	0.987043i	$0.551297\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−268.219	−	154.856i	−0.759828	−	0.438687i	0.0694063	−	0.997588i	$-0.477890\pi$
$353$	−268.219	−	154.856i	−0.759828	−	0.438687i	−0.829234	+	0.558902i	$0.811223\pi$
$354$		0			0
$355$	−207.000	−	358.535i	−0.583099	−	1.00996i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	440.908	−	254.558i	1.22816	−	0.709076i	0.261512	−	0.965200i	$-0.415779\pi$
$359$	440.908	−	254.558i	1.22816	−	0.709076i	0.966644	+	0.256124i	$0.0824454\pi$
$360$		0			0
$361$	−84.0000	+	145.492i	−0.232687	+	0.403026i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	275.772i		−	0.755539i
$366$		0			0
$367$	−159.500	+	276.262i	−0.434605	+	0.752758i	−0.997263	−	0.0739317i	$-0.976445\pi$
$367$	−159.500	+	276.262i	−0.434605	+	0.752758i	0.562658	+	0.826689i	$0.309779\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	356.382i		−	0.960598i
$372$		0			0
$373$	−104.500	−	180.999i	−0.280161	−	0.485253i	0.691263	−	0.722603i	$-0.257054\pi$
$373$	−104.500	−	180.999i	−0.280161	−	0.485253i	−0.971424	+	0.237350i	$0.923721\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			33.9411i			0.0900295i
$378$		0			0
$379$	433.000			1.14248			0.571240	−	0.820783i	$-0.306463\pi$
$379$	433.000			1.14248			0.571240	+	0.820783i	$0.306463\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	338.030	−	195.161i	0.882584	−	0.509560i	0.0110743	−	0.999939i	$-0.496475\pi$
$383$	338.030	−	195.161i	0.882584	−	0.509560i	0.871509	+	0.490379i	$0.163142\pi$
$384$		0			0
$385$	−189.000	−	327.358i	−0.490909	−	0.850279i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	319.658	+	184.555i	0.821744	+	0.474434i	0.851018	−	0.525137i	$-0.175986\pi$
$389$	319.658	+	184.555i	0.821744	+	0.474434i	−0.0292735	+	0.999571i	$0.509319\pi$
$390$		0			0
$391$	288.000			0.736573
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−415.189	−	239.709i	−1.05111	−	0.606859i
$396$		0			0
$397$	−56.5000	−	97.8609i	−0.142317	−	0.246501i	0.786052	−	0.618161i	$-0.212122\pi$
$397$	−56.5000	−	97.8609i	−0.142317	−	0.246501i	−0.928369	+	0.371660i	$0.878789\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−308.636	+	178.191i	−0.769665	+	0.444366i	−0.832755	−	0.553641i	$-0.813238\pi$
$401$	−308.636	+	178.191i	−0.769665	+	0.444366i	0.0630900	+	0.998008i	$0.479904\pi$
$402$		0			0
$403$	−23.5000	+	40.7032i	−0.0583127	+	0.101000i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			700.036i			1.71999i
$408$		0			0
$409$	159.500	−	276.262i	0.389976	−	0.675457i	−0.602470	−	0.798141i	$-0.705817\pi$
$409$	159.500	−	276.262i	0.389976	−	0.675457i	0.992446	+	0.122684i	$0.0391501\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	593.970i		−	1.43818i
$414$		0			0
$415$	63.0000	+	109.119i	0.151807	+	0.262938i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			767.918i			1.83274i	0.400333	+	0.916370i	$0.368895\pi$
$419$			767.918i			1.83274i	−0.400333	+	0.916370i	$0.631105\pi$
$420$		0			0
$421$	65.0000			0.154394			0.0771971	−	0.997016i	$-0.475403\pi$
$421$	65.0000			0.154394			0.0771971	+	0.997016i	$0.475403\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	102.879	−	59.3970i	0.242067	−	0.139758i
$426$		0			0
$427$	364.000	−	630.466i	0.852459	−	1.47650i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	415.189	+	239.709i	0.963314	+	0.556170i	0.897192	−	0.441642i	$-0.145604\pi$
$431$	415.189	+	239.709i	0.963314	+	0.556170i	0.0661229	+	0.997811i	$0.478937\pi$
$432$		0			0
$433$	−367.000			−0.847575			−0.423788	−	0.905762i	$-0.639300\pi$
$433$	−367.000			−0.847575			−0.423788	+	0.905762i	$0.639300\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−338.030	−	195.161i	−0.773523	−	0.446594i
$438$		0			0
$439$	268.000	+	464.190i	0.610478	+	1.05738i	0.991160	+	0.132673i	$0.0423561\pi$
$439$	268.000	+	464.190i	0.610478	+	1.05738i	−0.380681	+	0.924706i	$0.624311\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	73.4847	−	42.4264i	0.165880	−	0.0957707i	−0.414762	−	0.909930i	$-0.636135\pi$
$443$	73.4847	−	42.4264i	0.165880	−	0.0957707i	0.580642	+	0.814159i	$0.302802\pi$
$444$		0			0
$445$	288.000	−	498.831i	0.647191	−	1.12097i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			615.183i			1.37012i	0.728488	+	0.685059i	$0.240224\pi$
$449$			615.183i			1.37012i	−0.728488	+	0.685059i	$0.759776\pi$
$450$		0			0
$451$	−297.000	+	514.419i	−0.658537	+	1.14062i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	25.7196	+	14.8492i	0.0565267	+	0.0326357i
$456$		0			0
$457$	231.500	+	400.970i	0.506565	+	0.877396i	0.999971	+	0.00759675i	$0.00241814\pi$
$457$	231.500	+	400.970i	0.506565	+	0.877396i	−0.493407	+	0.869799i	$0.664249\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	271.529i		−	0.589000i	−0.955651	−	0.294500i	$-0.904847\pi$
$461$		−	271.529i		−	0.589000i	0.955651	−	0.294500i	$-0.0951531\pi$
$462$		0			0
$463$	−47.0000			−0.101512			−0.0507559	−	0.998711i	$-0.516163\pi$
$463$	−47.0000			−0.101512			−0.0507559	+	0.998711i	$0.516163\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	642.991	−	371.231i	1.37685	−	0.794927i	0.385075	−	0.922885i	$-0.374176\pi$
$467$	642.991	−	371.231i	1.37685	−	0.794927i	0.991780	+	0.127958i	$0.0408423\pi$
$468$		0			0
$469$	−679.000			−1.44776
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−253.522	−	146.371i	−0.535988	−	0.309453i
$474$		0			0
$475$	−161.000			−0.338947
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−367.423	−	212.132i	−0.767064	−	0.442864i	0.0647625	−	0.997901i	$-0.479371\pi$
$479$	−367.423	−	212.132i	−0.767064	−	0.442864i	−0.831826	+	0.555036i	$0.812704\pi$
$480$		0			0
$481$	−27.5000	−	47.6314i	−0.0571726	−	0.0990258i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−382.120	+	220.617i	−0.787877	+	0.454881i
$486$		0			0
$487$	119.500	−	206.980i	0.245380	−	0.425010i	−0.716858	−	0.697219i	$-0.754421\pi$
$487$	119.500	−	206.980i	0.245380	−	0.425010i	0.962238	+	0.272208i	$0.0877540\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			203.647i			0.414759i	0.978261	+	0.207380i	$0.0664935\pi$
$491$			203.647i			0.414759i	−0.978261	+	0.207380i	$0.933506\pi$
$492$		0			0
$493$	−288.000	+	498.831i	−0.584178	+	1.01183i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−591.552	+	341.533i	−1.19025	+	0.687188i
$498$		0			0
$499$	−24.5000	−	42.4352i	−0.0490982	−	0.0850406i	0.840432	−	0.541917i	$-0.182301\pi$
$499$	−24.5000	−	42.4352i	−0.0490982	−	0.0850406i	−0.889530	+	0.456877i	$0.848968\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			589.727i			1.17242i	0.810159	+	0.586210i	$0.199381\pi$
$503$			589.727i			1.17242i	−0.810159	+	0.586210i	$0.800619\pi$
$504$		0			0
$505$	630.000			1.24752
$506$		0			0
$507$		0			0
$508$		0			0
$509$	209.431	−	120.915i	0.411457	−	0.237555i	−0.279959	−	0.960012i	$-0.590321\pi$
$509$	209.431	−	120.915i	0.411457	−	0.237555i	0.691415	+	0.722457i	$0.256987\pi$
$510$		0			0
$511$	−455.000			−0.890411
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−437.234	−	252.437i	−0.848998	−	0.490169i
$516$		0			0
$517$	−54.0000			−0.104449
$518$		0			0
$519$		0			0
$520$		0			0
$521$	734.847	+	424.264i	1.41045	+	0.814326i	0.995431	−	0.0954841i	$-0.0304399\pi$
$521$	734.847	+	424.264i	1.41045	+	0.814326i	0.415024	+	0.909811i	$0.363773\pi$
$522$		0			0
$523$	−195.500	−	338.616i	−0.373805	−	0.647449i	0.616342	−	0.787478i	$-0.288614\pi$
$523$	−195.500	−	338.616i	−0.373805	−	0.647449i	−0.990147	+	0.140029i	$0.955280\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−690.756	+	398.808i	−1.31073	+	0.756752i
$528$		0			0
$529$	−120.500	+	208.712i	−0.227788	+	0.394541i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	46.6690i		−	0.0875592i
$534$		0			0
$535$	252.000	−	436.477i	0.471028	−	0.815844i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−540.112	+	311.834i	−1.00206	+	0.578542i
$540$		0			0
$541$	36.5000	+	63.2199i	0.0674677	+	0.116857i	0.897786	−	0.440432i	$-0.145175\pi$
$541$	36.5000	+	63.2199i	0.0674677	+	0.116857i	−0.830318	+	0.557289i	$0.811841\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			207.889i			0.381448i
$546$		0			0
$547$	−32.0000			−0.0585009			−0.0292505	−	0.999572i	$-0.509312\pi$
$547$	−32.0000			−0.0585009			−0.0292505	+	0.999572i	$0.509312\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	676.059	−	390.323i	1.22697	−	0.708390i
$552$		0			0
$553$	−395.500	+	685.026i	−0.715190	+	1.23875i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−40.4166	−	23.3345i	−0.0725612	−	0.0418932i	0.463280	−	0.886212i	$-0.346672\pi$
$557$	−40.4166	−	23.3345i	−0.0725612	−	0.0418932i	−0.535842	+	0.844319i	$0.680006\pi$
$558$		0			0
$559$	23.0000			0.0411449
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−767.915	−	443.356i	−1.36397	−	0.787488i	−0.373820	−	0.927501i	$-0.621952\pi$
$563$	−767.915	−	443.356i	−1.36397	−	0.787488i	−0.990150	+	0.140013i	$0.955286\pi$
$564$		0			0
$565$	−207.000	−	358.535i	−0.366372	−	0.634574i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	275.568	−	159.099i	0.484302	−	0.279612i	−0.237906	−	0.971288i	$-0.576461\pi$
$569$	275.568	−	159.099i	0.484302	−	0.279612i	0.722207	+	0.691677i	$0.243128\pi$
$570$		0			0
$571$	−267.500	+	463.324i	−0.468476	+	0.811425i	−0.999351	−	0.0360256i	$-0.988530\pi$
$571$	−267.500	+	463.324i	−0.468476	+	0.811425i	0.530875	+	0.847450i	$0.321864\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	118.794i		−	0.206598i
$576$		0			0
$577$	−452.500	+	783.753i	−0.784229	+	1.35832i	0.145230	+	0.989398i	$0.453608\pi$
$577$	−452.500	+	783.753i	−0.784229	+	1.35832i	−0.929459	+	0.368926i	$0.879726\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	180.037	−	103.945i	0.309875	−	0.178907i
$582$		0			0
$583$	−324.000	−	561.184i	−0.555746	−	0.962581i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			576.999i			0.982963i	0.870888	+	0.491481i	$0.163544\pi$
$587$			576.999i			0.982963i	−0.870888	+	0.491481i	$0.836456\pi$
$588$		0			0
$589$	1081.00			1.83531
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−481.325	+	277.893i	−0.811677	+	0.468622i	−0.847538	−	0.530735i	$-0.821916\pi$
$593$	−481.325	+	277.893i	−0.811677	+	0.468622i	0.0358606	+	0.999357i	$0.488583\pi$
$594$		0			0
$595$	252.000	+	436.477i	0.423529	+	0.733574i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−176.363	−	101.823i	−0.294429	−	0.169989i	0.345508	−	0.938416i	$-0.387707\pi$
$599$	−176.363	−	101.823i	−0.294429	−	0.169989i	−0.639938	+	0.768427i	$0.721040\pi$
$600$		0			0
$601$	143.000			0.237937			0.118968	−	0.992898i	$-0.462041\pi$
$601$	143.000			0.237937			0.118968	+	0.992898i	$0.462041\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−150.644	−	86.9741i	−0.248998	−	0.143759i
$606$		0			0
$607$	299.500	+	518.749i	0.493410	+	0.854612i	0.999971	−	0.00759259i	$-0.00241682\pi$
$607$	299.500	+	518.749i	0.493410	+	0.854612i	−0.506561	+	0.862204i	$0.669083\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.67423	−	2.12132i	0.00601348	−	0.00347188i
$612$		0			0
$613$	152.000	−	263.272i	0.247961	−	0.429481i	−0.714999	−	0.699125i	$-0.753573\pi$
$613$	152.000	−	263.272i	0.247961	−	0.429481i	0.962960	+	0.269645i	$0.0869062\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			292.742i			0.474461i	0.971453	+	0.237230i	$0.0762396\pi$
$617$			292.742i			0.474461i	−0.971453	+	0.237230i	$0.923760\pi$
$618$		0			0
$619$	−171.500	+	297.047i	−0.277060	+	0.479882i	−0.970653	−	0.240486i	$-0.922693\pi$
$619$	−171.500	+	297.047i	−0.277060	+	0.479882i	0.693593	+	0.720367i	$0.256027\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−823.029	−	475.176i	−1.32107	−	0.762722i
$624$		0			0
$625$	200.500	+	347.276i	0.320800	+	0.555642i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	933.381i		−	1.48391i
$630$		0			0
$631$	−272.000			−0.431062			−0.215531	−	0.976497i	$-0.569148\pi$
$631$	−272.000			−0.431062			−0.215531	+	0.976497i	$0.569148\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	415.189	−	239.709i	0.653840	−	0.377495i
$636$		0			0
$637$	24.5000	−	42.4352i	0.0384615	−	0.0666173i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−367.423	−	212.132i	−0.573204	−	0.330939i	0.185224	−	0.982696i	$-0.440699\pi$
$641$	−367.423	−	212.132i	−0.573204	−	0.330939i	−0.758428	+	0.651757i	$0.774032\pi$
$642$		0			0
$643$	679.000			1.05599			0.527994	−	0.849248i	$-0.322944\pi$
$643$	679.000			1.05599			0.527994	+	0.849248i	$0.322944\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−834.051	−	481.540i	−1.28911	−	0.744265i	−0.310611	−	0.950537i	$-0.600534\pi$
$647$	−834.051	−	481.540i	−1.28911	−	0.744265i	−0.978495	+	0.206272i	$0.933867\pi$
$648$		0			0
$649$	−540.000	−	935.307i	−0.832049	−	1.44115i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−775.264	+	447.599i	−1.18723	+	0.685450i	−0.957677	−	0.287846i	$-0.907061\pi$
$653$	−775.264	+	447.599i	−1.18723	+	0.685450i	−0.229557	+	0.973295i	$0.573728\pi$
$654$		0			0
$655$	−45.0000	+	77.9423i	−0.0687023	+	0.118996i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			16.9706i			0.0257520i	0.999917	+	0.0128760i	$0.00409867\pi$
$659$			16.9706i			0.0257520i	−0.999917	+	0.0128760i	$0.995901\pi$
$660$		0			0
$661$	216.500	−	374.989i	0.327534	−	0.567306i	−0.654488	−	0.756072i	$-0.727116\pi$
$661$	216.500	−	374.989i	0.327534	−	0.567306i	0.982022	+	0.188767i	$0.0604491\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	683.065i		−	1.02717i
$666$		0			0
$667$	288.000	+	498.831i	0.431784	+	0.747872i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1323.70i		−	1.97273i
$672$		0			0
$673$	737.000			1.09510			0.547548	−	0.836774i	$-0.315561\pi$
$673$	737.000			1.09510			0.547548	+	0.836774i	$0.315561\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−117.576	+	67.8823i	−0.173671	+	0.100269i	−0.584316	−	0.811526i	$-0.698637\pi$
$677$	−117.576	+	67.8823i	−0.173671	+	0.100269i	0.410644	+	0.911796i	$0.365304\pi$
$678$		0			0
$679$	364.000	+	630.466i	0.536082	+	0.928522i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1102.27	+	636.396i	1.61387	+	0.931766i	0.988463	+	0.151464i	$0.0483988\pi$
$683$	1102.27	+	636.396i	1.61387	+	0.931766i	0.625403	+	0.780302i	$0.284935\pi$
$684$		0			0
$685$	−288.000			−0.420438
$686$		0			0
$687$		0			0
$688$		0			0
$689$	44.0908	+	25.4558i	0.0639925	+	0.0369461i
$690$		0			0
$691$	−123.500	−	213.908i	−0.178726	−	0.309563i	0.762718	−	0.646731i	$-0.223864\pi$
$691$	−123.500	−	213.908i	−0.178726	−	0.309563i	−0.941445	+	0.337168i	$0.890531\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−378.446	+	218.496i	−0.544527	+	0.314383i
$696$		0			0
$697$	396.000	−	685.892i	0.568149	−	0.984063i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			322.441i			0.459972i	0.973194	+	0.229986i	$0.0738681\pi$
$701$			322.441i			0.459972i	−0.973194	+	0.229986i	$0.926132\pi$
$702$		0			0
$703$	−632.500	+	1095.52i	−0.899716	+	1.55835i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1039.45i		−	1.47022i
$708$		0			0
$709$	524.000	+	907.595i	0.739069	+	1.28011i	0.952915	+	0.303238i	$0.0980677\pi$
$709$	524.000	+	907.595i	0.739069	+	1.28011i	−0.213846	+	0.976867i	$0.568599\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			797.616i			1.11868i
$714$		0			0
$715$	54.0000			0.0755245
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−216.780	+	125.158i	−0.301502	+	0.174072i	−0.643117	−	0.765768i	$-0.722359\pi$
$719$	−216.780	+	125.158i	−0.301502	+	0.174072i	0.341616	+	0.939840i	$0.389026\pi$
$720$		0			0
$721$	−416.500	+	721.399i	−0.577670	+	1.00055i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	205.757	+	118.794i	0.283803	+	0.163854i
$726$		0			0
$727$	−641.000			−0.881706			−0.440853	−	0.897579i	$-0.645324\pi$
$727$	−641.000			−0.881706			−0.440853	+	0.897579i	$0.645324\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	338.030	+	195.161i	0.462421	+	0.266979i
$732$		0			0
$733$	12.5000	+	21.6506i	0.0170532	+	0.0295370i	0.874426	−	0.485159i	$-0.161238\pi$
$733$	12.5000	+	21.6506i	0.0170532	+	0.0295370i	−0.857373	+	0.514696i	$0.827905\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1069.20	+	617.304i	−1.45075	+	0.837591i
$738$		0			0
$739$	−267.500	+	463.324i	−0.361976	+	0.626960i	−0.988286	−	0.152614i	$-0.951231\pi$
$739$	−267.500	+	463.324i	−0.361976	+	0.626960i	0.626310	+	0.779574i	$0.284564\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			937.624i			1.26194i	0.775806	+	0.630971i	$0.217344\pi$
$743$			937.624i			1.26194i	−0.775806	+	0.630971i	$0.782656\pi$
$744$		0			0
$745$	−360.000	+	623.538i	−0.483221	+	0.836964i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−720.150	−	415.779i	−0.961482	−	0.555112i
$750$		0			0
$751$	20.5000	+	35.5070i	0.0272969	+	0.0472797i	0.879351	−	0.476174i	$-0.157977\pi$
$751$	20.5000	+	35.5070i	0.0272969	+	0.0472797i	−0.852054	+	0.523454i	$0.824643\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			441.235i			0.584417i
$756$		0			0
$757$	−1114.00			−1.47160			−0.735799	−	0.677200i	$-0.763193\pi$
$757$	−1114.00			−1.47160			−0.735799	+	0.677200i	$0.763193\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	249.848	−	144.250i	0.328315	−	0.189553i	−0.326778	−	0.945101i	$-0.605963\pi$
$761$	249.848	−	144.250i	0.328315	−	0.189553i	0.655093	+	0.755548i	$0.272629\pi$
$762$		0			0
$763$	343.000			0.449541
$764$		0			0
$765$		0			0
$766$		0			0
$767$	73.4847	+	42.4264i	0.0958079	+	0.0553147i
$768$		0			0
$769$	1127.00			1.46554			0.732770	−	0.680477i	$-0.238227\pi$
$769$	1127.00			1.46554			0.732770	+	0.680477i	$0.238227\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−187.386	−	108.187i	−0.242414	−	0.139958i	0.373872	−	0.927480i	$-0.378030\pi$
$773$	−187.386	−	108.187i	−0.242414	−	0.139958i	−0.616286	+	0.787523i	$0.711363\pi$
$774$		0			0
$775$	164.500	+	284.922i	0.212258	+	0.367642i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−929.581	+	536.694i	−1.19330	+	0.688953i
$780$		0			0
$781$	−621.000	+	1075.60i	−0.795134	+	1.37721i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	644.881i		−	0.821505i
$786$		0			0
$787$	85.0000	−	147.224i	0.108005	−	0.187070i	−0.806957	−	0.590610i	$-0.798887\pi$
$787$	85.0000	−	147.224i	0.108005	−	0.187070i	0.914962	+	0.403540i	$0.132220\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−591.552	+	341.533i	−0.747853	+	0.431773i
$792$		0			0
$793$	52.0000	+	90.0666i	0.0655738	+	0.113577i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1170.97i			1.46922i	0.678489	+	0.734610i	$0.262635\pi$
$797$			1170.97i			1.46922i	−0.678489	+	0.734610i	$0.737365\pi$
$798$		0			0
$799$	72.0000			0.0901126
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−716.476	+	413.657i	−0.892249	+	0.515140i
$804$		0			0
$805$	504.000			0.626087
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−290.265	−	167.584i	−0.358794	−	0.207150i	0.309757	−	0.950816i	$-0.399752\pi$
$809$	−290.265	−	167.584i	−0.358794	−	0.207150i	−0.668552	+	0.743666i	$0.733085\pi$
$810$		0			0
$811$	1114.00			1.37361			0.686806	−	0.726840i	$-0.259012\pi$
$811$	1114.00			1.37361			0.686806	+	0.726840i	$0.259012\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−205.757	−	118.794i	−0.252463	−	0.145759i
$816$		0			0
$817$	−264.500	−	458.127i	−0.323745	−	0.560743i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	84.5074	−	48.7904i	0.102932	−	0.0594280i	−0.447650	−	0.894209i	$-0.647739\pi$
$821$	84.5074	−	48.7904i	0.102932	−	0.0594280i	0.550582	+	0.834781i	$0.314406\pi$
$822$		0			0
$823$	796.000	−	1378.71i	0.967193	−	1.67523i	0.263590	−	0.964635i	$-0.415093\pi$
$823$	796.000	−	1378.71i	0.967193	−	1.67523i	0.703603	−	0.710593i	$-0.251573\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1022.48i			1.23637i	0.786033	+	0.618184i	$0.212131\pi$
$827$			1022.48i			1.23637i	−0.786033	+	0.618184i	$0.787869\pi$
$828$		0			0
$829$	−191.500	+	331.688i	−0.231001	+	0.400106i	−0.958103	−	0.286424i	$-0.907533\pi$
$829$	−191.500	+	331.688i	−0.231001	+	0.400106i	0.727102	+	0.686530i	$0.240867\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	720.150	−	415.779i	0.864526	−	0.499134i
$834$		0			0
$835$	9.00000	+	15.5885i	0.0107784	+	0.0186688i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			576.999i			0.687722i	0.939020	+	0.343861i	$0.111735\pi$
$839$			576.999i			0.687722i	−0.939020	+	0.343861i	$0.888265\pi$
$840$		0			0
$841$	−311.000			−0.369798
$842$		0			0
$843$		0			0
$844$		0			0
$845$	617.271	−	356.382i	0.730499	−	0.421754i
$846$		0			0
$847$	−143.500	+	248.549i	−0.169421	+	0.293447i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−808.332	−	466.690i	−0.949861	−	0.548402i
$852$		0			0
$853$	527.000			0.617819			0.308910	−	0.951091i	$-0.400036\pi$
$853$	527.000			0.617819			0.308910	+	0.951091i	$0.400036\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1146.36	−	661.852i	−1.33764	−	0.772289i	−0.351187	−	0.936305i	$-0.614222\pi$
$857$	−1146.36	−	661.852i	−1.33764	−	0.772289i	−0.986458	+	0.164016i	$0.947555\pi$
$858$		0			0
$859$	−707.000	−	1224.56i	−0.823050	−	1.42556i	−0.903400	−	0.428798i	$-0.858937\pi$
$859$	−707.000	−	1224.56i	−0.823050	−	1.42556i	0.0803504	−	0.996767i	$-0.474396\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1186.78	−	685.186i	1.37518	−	0.793959i	0.383603	−	0.923498i	$-0.374683\pi$
$863$	1186.78	−	685.186i	1.37518	−	0.793959i	0.991574	+	0.129539i	$0.0413499\pi$
$864$		0			0
$865$	−324.000	+	561.184i	−0.374566	+	0.648768i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1438.26i			1.65507i
$870$		0			0
$871$	48.5000	−	84.0045i	0.0556831	−	0.0964460i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	823.029	−	475.176i	0.940604	−	0.543058i
$876$		0			0
$877$	−112.000	−	193.990i	−0.127708	−	0.221197i	0.795080	−	0.606504i	$-0.207429\pi$
$877$	−112.000	−	193.990i	−0.127708	−	0.221197i	−0.922788	+	0.385307i	$0.874095\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1612.20i		−	1.82997i	−0.403488	−	0.914985i	$-0.632202\pi$
$881$		−	1612.20i		−	1.82997i	0.403488	−	0.914985i	$-0.367798\pi$
$882$		0			0
$883$	−329.000			−0.372593			−0.186297	−	0.982494i	$-0.559649\pi$
$883$	−329.000			−0.372593			−0.186297	+	0.982494i	$0.559649\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	47.7650	−	27.5772i	0.0538501	−	0.0310904i	−0.472833	−	0.881152i	$-0.656769\pi$
$887$	47.7650	−	27.5772i	0.0538501	−	0.0310904i	0.526683	+	0.850062i	$0.323435\pi$
$888$		0			0
$889$	−395.500	−	685.026i	−0.444882	−	0.770558i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−84.5074	−	48.7904i	−0.0946331	−	0.0546365i
$894$		0			0
$895$	18.0000			0.0201117
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1381.51	−	797.616i	−1.53672	−	0.887226i
$900$		0			0
$901$	432.000	+	748.246i	0.479467	+	0.830462i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	202.083	−	116.673i	0.223296	−	0.128920i
$906$		0			0
$907$	155.500	−	269.334i	0.171444	−	0.296950i	−0.767481	−	0.641072i	$-0.778490\pi$
$907$	155.500	−	269.334i	0.171444	−	0.296950i	0.938925	+	0.344122i	$0.111823\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			356.382i			0.391198i	0.980684	+	0.195599i	$0.0626652\pi$
$911$			356.382i			0.391198i	−0.980684	+	0.195599i	$0.937335\pi$
$912$		0			0
$913$	189.000	−	327.358i	0.207010	−	0.358552i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	128.598	+	74.2462i	0.140238	+	0.0809664i
$918$		0			0
$919$	−72.5000	−	125.574i	−0.0788901	−	0.136642i	0.823881	−	0.566762i	$-0.191804\pi$
$919$	−72.5000	−	125.574i	−0.0788901	−	0.136642i	−0.902771	+	0.430121i	$0.858471\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	97.5807i		−	0.105721i
$924$		0			0
$925$	−385.000			−0.416216
$926$		0			0
$927$		0			0
$928$		0			0
$929$	268.219	−	154.856i	0.288718	−	0.166691i	−0.348645	−	0.937255i	$-0.613358\pi$
$929$	268.219	−	154.856i	0.288718	−	0.166691i	0.637364	+	0.770563i	$0.280025\pi$
$930$		0			0
$931$	−1127.00			−1.21053
$932$		0			0
$933$		0			0
$934$		0			0
$935$	793.635	+	458.205i	0.848807	+	0.490059i
$936$		0			0
$937$	−487.000			−0.519744			−0.259872	−	0.965643i	$-0.583680\pi$
$937$	−487.000			−0.519744			−0.259872	+	0.965643i	$0.583680\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1190.45	−	687.308i	−1.26509	−	0.730401i	−0.291037	−	0.956712i	$-0.594000\pi$
$941$	−1190.45	−	687.308i	−1.26509	−	0.730401i	−0.974055	+	0.226310i	$0.927334\pi$
$942$		0			0
$943$	−396.000	−	685.892i	−0.419936	−	0.727351i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−165.341	+	95.4594i	−0.174594	+	0.100802i	−0.584750	−	0.811213i	$-0.698808\pi$
$947$	−165.341	+	95.4594i	−0.174594	+	0.100802i	0.410156	+	0.912015i	$0.365474\pi$
$948$		0			0
$949$	32.5000	−	56.2917i	0.0342466	−	0.0593168i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1035.20i			1.08626i	0.839649	+	0.543129i	$0.182761\pi$
$953$			1035.20i			1.08626i	−0.839649	+	0.543129i	$0.817239\pi$
$954$		0			0
$955$	135.000	−	233.827i	0.141361	−	0.244845i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			475.176i			0.495491i
$960$		0			0
$961$	−624.000	−	1080.80i	−0.649324	−	1.12466i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			640.639i			0.663874i
$966$		0			0
$967$	1681.00			1.73837			0.869183	−	0.494490i	$-0.164645\pi$
$967$	1681.00			1.73837			0.869183	+	0.494490i	$0.164645\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1043.48	−	602.455i	1.07465	−	0.620448i	0.145200	−	0.989402i	$-0.453617\pi$
$971$	1043.48	−	602.455i	1.07465	−	0.620448i	0.929448	+	0.368954i	$0.120284\pi$
$972$		0			0
$973$	360.500	+	624.404i	0.370504	+	0.641731i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1017.76	+	587.606i	1.04172	+	0.601439i	0.920321	−	0.391165i	$-0.127928\pi$
$977$	1017.76	+	587.606i	1.04172	+	0.601439i	0.121402	+	0.992603i	$0.461261\pi$
$978$		0			0
$979$	−1728.00			−1.76507
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1660.75	+	958.837i	1.68948	+	0.975419i	0.954917	+	0.296873i	$0.0959439\pi$
$983$	1660.75	+	958.837i	1.68948	+	0.975419i	0.734558	+	0.678546i	$0.237389\pi$
$984$		0			0
$985$	36.0000	+	62.3538i	0.0365482	+	0.0633034i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	338.030	−	195.161i	0.341789	−	0.197332i
$990$		0			0
$991$	−312.500	+	541.266i	−0.315338	+	0.546182i	−0.979509	−	0.201399i	$-0.935451\pi$
$991$	−312.500	+	541.266i	−0.315338	+	0.546182i	0.664171	+	0.747580i	$0.268785\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	678.823i		−	0.682234i
$996$		0			0
$997$	471.500	−	816.662i	0.472919	−	0.819119i	−0.526601	−	0.850113i	$-0.676534\pi$
$997$	471.500	−	816.662i	0.472919	−	0.819119i	0.999520	+	0.0309934i	$0.00986707\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.dc.b.305.1		4
3.2	odd	2	inner	1008.3.dc.b.305.2		4
4.3	odd	2		126.3.s.a.53.1	✓	4
7.2	even	3	inner	1008.3.dc.b.737.2		4
12.11	even	2		126.3.s.a.53.2	yes	4
21.2	odd	6	inner	1008.3.dc.b.737.1		4
28.3	even	6		882.3.b.e.197.1		2
28.11	odd	6		882.3.b.b.197.1		2
28.19	even	6		882.3.s.a.863.2		4
28.23	odd	6		126.3.s.a.107.2	yes	4
28.27	even	2		882.3.s.a.557.1		4
84.11	even	6		882.3.b.b.197.2		2
84.23	even	6		126.3.s.a.107.1	yes	4
84.47	odd	6		882.3.s.a.863.1		4
84.59	odd	6		882.3.b.e.197.2		2
84.83	odd	2		882.3.s.a.557.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.s.a.53.1	✓	4	4.3	odd	2
126.3.s.a.53.2	yes	4	12.11	even	2
126.3.s.a.107.1	yes	4	84.23	even	6
126.3.s.a.107.2	yes	4	28.23	odd	6
882.3.b.b.197.1		2	28.11	odd	6
882.3.b.b.197.2		2	84.11	even	6
882.3.b.e.197.1		2	28.3	even	6
882.3.b.e.197.2		2	84.59	odd	6
882.3.s.a.557.1		4	28.27	even	2
882.3.s.a.557.2		4	84.83	odd	2
882.3.s.a.863.1		4	84.47	odd	6
882.3.s.a.863.2		4	28.19	even	6
1008.3.dc.b.305.1		4	1.1	even	1	trivial
1008.3.dc.b.305.2		4	3.2	odd	2	inner
1008.3.dc.b.737.1		4	21.2	odd	6	inner
1008.3.dc.b.737.2		4	7.2	even	3	inner

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.67423 + 2.12132i) q^{5} +(3.50000 + 6.06218i) q^{7} +O(q^{10})\)	\(q+(-3.67423 + 2.12132i) q^{5} +(3.50000 + 6.06218i) q^{7} +(11.0227 + 6.36396i) q^{11} -1.00000 q^{13} +(-14.6969 - 8.48528i) q^{17} +(11.5000 + 19.9186i) q^{19} +(-14.6969 + 8.48528i) q^{23} +(-3.50000 + 6.06218i) q^{25} -33.9411i q^{29} +(23.5000 - 40.7032i) q^{31} +(-25.7196 - 14.8492i) q^{35} +(27.5000 + 47.6314i) q^{37} +46.6690i q^{41} -23.0000 q^{43} +(-3.67423 + 2.12132i) q^{47} +(-24.5000 + 42.4352i) q^{49} +(-44.0908 - 25.4558i) q^{53} -54.0000 q^{55} +(-73.4847 - 42.4264i) q^{59} +(-52.0000 - 90.0666i) q^{61} +(3.67423 - 2.12132i) q^{65} +(-48.5000 + 84.0045i) q^{67} +97.5807i q^{71} +(-32.5000 + 56.2917i) q^{73} +89.0955i q^{77} +(56.5000 + 97.8609i) q^{79} -29.6985i q^{83} +72.0000 q^{85} +(-117.576 + 67.8823i) q^{89} +(-3.50000 - 6.06218i) q^{91} +(-84.5074 - 48.7904i) q^{95} +104.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{7}+O(q^{10}) \) 4 * q + 14 * q^7	\( 4 q + 14 q^{7} - 4 q^{13} + 46 q^{19} - 14 q^{25} + 94 q^{31} + 110 q^{37} - 92 q^{43} - 98 q^{49} - 216 q^{55} - 208 q^{61} - 194 q^{67} - 130 q^{73} + 226 q^{79} + 288 q^{85} - 14 q^{91} + 416 q^{97}+O(q^{100}) \) 4 * q + 14 * q^7 - 4 * q^13 + 46 * q^19 - 14 * q^25 + 94 * q^31 + 110 * q^37 - 92 * q^43 - 98 * q^49 - 216 * q^55 - 208 * q^61 - 194 * q^67 - 130 * q^73 + 226 * q^79 + 288 * q^85 - 14 * q^91 + 416 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more