LMFDB - Embedded newform 900.3.l.b.793.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(757,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 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("900.757");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.l (of order $4$, degree $2$, 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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			793.2
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	900.793
Dual form			900.3.l.b.757.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.55051 + 2.55051i) q^{7} +O(q^{10})$	$q+(-2.55051 + 2.55051i) q^{7} -8.24745 q^{11} +(12.2474 + 12.2474i) q^{13} +(-12.4495 + 12.4495i) q^{17} -34.4949i q^{19} +(-17.3485 - 17.3485i) q^{23} -9.75255i q^{29} +28.4949 q^{31} +(7.34847 - 7.34847i) q^{37} -74.4949 q^{41} +(-34.8990 - 34.8990i) q^{43} +(-22.0454 + 22.0454i) q^{47} +35.9898i q^{49} +(-64.6969 - 64.6969i) q^{53} -15.2577i q^{59} -53.5051 q^{61} +(-4.69694 + 4.69694i) q^{67} +117.980 q^{71} +(-34.1918 - 34.1918i) q^{73} +(21.0352 - 21.0352i) q^{77} -0.494897i q^{79} +(-18.3587 - 18.3587i) q^{83} -136.969i q^{89} -62.4745 q^{91} +(-94.5959 + 94.5959i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{7}+O(q^{10}) $ 4 * q - 20 * q^7	$ 4 q - 20 q^{7} + 16 q^{11} - 40 q^{17} - 40 q^{23} + 16 q^{31} - 200 q^{41} - 120 q^{43} - 200 q^{53} - 312 q^{61} + 40 q^{67} + 80 q^{71} + 20 q^{73} - 200 q^{77} - 240 q^{83} + 240 q^{91} - 300 q^{97}+O(q^{100}) $ 4 * q - 20 * q^7 + 16 * q^11 - 40 * q^17 - 40 * q^23 + 16 * q^31 - 200 * q^41 - 120 * q^43 - 200 * q^53 - 312 * q^61 + 40 * q^67 + 80 * q^71 + 20 * q^73 - 200 * q^77 - 240 * q^83 + 240 * q^91 - 300 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.55051	+	2.55051i	−0.364359	+	0.364359i	−0.865415	−	0.501056i	$-0.832945\pi$
$7$	−2.55051	+	2.55051i	−0.364359	+	0.364359i	0.501056	+	0.865415i	$0.332945\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.24745			−0.749768			−0.374884	−	0.927072i	$-0.622317\pi$
$11$	−8.24745			−0.749768			−0.374884	+	0.927072i	$0.622317\pi$
$12$		0			0
$13$	12.2474	+	12.2474i	0.942111	+	0.942111i	0.998414	−	0.0563023i	$-0.0179311\pi$
$13$	12.2474	+	12.2474i	0.942111	+	0.942111i	−0.0563023	+	0.998414i	$0.517931\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.4495	+	12.4495i	−0.732323	+	0.732323i	−0.971079	−	0.238757i	$-0.923260\pi$
$17$	−12.4495	+	12.4495i	−0.732323	+	0.732323i	0.238757	+	0.971079i	$0.423260\pi$
$18$		0			0
$19$		−	34.4949i		−	1.81552i	−0.419489	−	0.907760i	$-0.637791\pi$
$19$		−	34.4949i		−	1.81552i	0.419489	−	0.907760i	$-0.362209\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−17.3485	−	17.3485i	−0.754281	−	0.754281i	0.220994	−	0.975275i	$-0.429070\pi$
$23$	−17.3485	−	17.3485i	−0.754281	−	0.754281i	−0.975275	+	0.220994i	$0.929070\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	9.75255i		−	0.336295i	−0.985762	−	0.168147i	$-0.946222\pi$
$29$		−	9.75255i		−	0.336295i	0.985762	−	0.168147i	$-0.0537785\pi$
$30$		0			0
$31$	28.4949			0.919190			0.459595	−	0.888129i	$-0.347995\pi$
$31$	28.4949			0.919190			0.459595	+	0.888129i	$0.347995\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.34847	−	7.34847i	0.198607	−	0.198607i	−0.600795	−	0.799403i	$-0.705149\pi$
$37$	7.34847	−	7.34847i	0.198607	−	0.198607i	0.799403	+	0.600795i	$0.205149\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−74.4949			−1.81695			−0.908474	−	0.417941i	$-0.862752\pi$
$41$	−74.4949			−1.81695			−0.908474	+	0.417941i	$0.862752\pi$
$42$		0			0
$43$	−34.8990	−	34.8990i	−0.811604	−	0.811604i	0.173270	−	0.984874i	$-0.444567\pi$
$43$	−34.8990	−	34.8990i	−0.811604	−	0.811604i	−0.984874	+	0.173270i	$0.944567\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−22.0454	+	22.0454i	−0.469051	+	0.469051i	−0.901607	−	0.432556i	$-0.857612\pi$
$47$	−22.0454	+	22.0454i	−0.469051	+	0.469051i	0.432556	+	0.901607i	$0.357612\pi$
$48$		0			0
$49$			35.9898i			0.734486i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−64.6969	−	64.6969i	−1.22070	−	1.22070i	−0.967385	−	0.253312i	$-0.918480\pi$
$53$	−64.6969	−	64.6969i	−1.22070	−	1.22070i	−0.253312	−	0.967385i	$-0.581520\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	15.2577i		−	0.258604i	−0.991605	−	0.129302i	$-0.958726\pi$
$59$		−	15.2577i		−	0.258604i	0.991605	−	0.129302i	$-0.0412737\pi$
$60$		0			0
$61$	−53.5051			−0.877133			−0.438566	−	0.898699i	$-0.644514\pi$
$61$	−53.5051			−0.877133			−0.438566	+	0.898699i	$0.644514\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.69694	+	4.69694i	−0.0701036	+	0.0701036i	−0.741289	−	0.671186i	$-0.765785\pi$
$67$	−4.69694	+	4.69694i	−0.0701036	+	0.0701036i	0.671186	+	0.741289i	$0.265785\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	117.980			1.66168			0.830842	−	0.556508i	$-0.187859\pi$
$71$	117.980			1.66168			0.830842	+	0.556508i	$0.187859\pi$
$72$		0			0
$73$	−34.1918	−	34.1918i	−0.468381	−	0.468381i	0.433009	−	0.901390i	$-0.357452\pi$
$73$	−34.1918	−	34.1918i	−0.468381	−	0.468381i	−0.901390	+	0.433009i	$0.857452\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	21.0352	−	21.0352i	0.273184	−	0.273184i
$78$		0			0
$79$		−	0.494897i		−	0.00626452i	−0.999995	−	0.00313226i	$-0.999003\pi$
$79$		−	0.494897i		−	0.00626452i	0.999995	−	0.00313226i	$-0.000997032\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−18.3587	−	18.3587i	−0.221189	−	0.221189i	0.587810	−	0.808999i	$-0.299990\pi$
$83$	−18.3587	−	18.3587i	−0.221189	−	0.221189i	−0.808999	+	0.587810i	$0.799990\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	136.969i		−	1.53898i	−0.638658	−	0.769491i	$-0.720510\pi$
$89$		−	136.969i		−	1.53898i	0.638658	−	0.769491i	$-0.279490\pi$
$90$		0			0
$91$	−62.4745			−0.686533
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−94.5959	+	94.5959i	−0.975216	+	0.975216i	−0.999700	−	0.0244846i	$-0.992206\pi$
$97$	−94.5959	+	94.5959i	−0.975216	+	0.975216i	0.0244846	+	0.999700i	$0.492206\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−70.2474			−0.695519			−0.347760	−	0.937584i	$-0.613057\pi$
$101$	−70.2474			−0.695519			−0.347760	+	0.937584i	$0.613057\pi$
$102$		0			0
$103$	−86.8434	−	86.8434i	−0.843139	−	0.843139i	0.146126	−	0.989266i	$-0.453319\pi$
$103$	−86.8434	−	86.8434i	−0.843139	−	0.843139i	−0.989266	+	0.146126i	$0.953319\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−96.3383	+	96.3383i	−0.900358	+	0.900358i	−0.995467	−	0.0951092i	$-0.969680\pi$
$107$	−96.3383	+	96.3383i	−0.900358	+	0.900358i	0.0951092	+	0.995467i	$0.469680\pi$
$108$		0			0
$109$		−	12.5153i		−	0.114819i	−0.998351	−	0.0574097i	$-0.981716\pi$
$109$		−	12.5153i		−	0.114819i	0.998351	−	0.0574097i	$-0.0182841\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	66.9444	+	66.9444i	0.592428	+	0.592428i	0.938287	−	0.345858i	$-0.112412\pi$
$113$	66.9444	+	66.9444i	0.592428	+	0.592428i	−0.345858	+	0.938287i	$0.612412\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	63.5051i		−	0.533656i
$120$		0			0
$121$	−52.9796			−0.437848
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.55051	+	2.55051i	−0.0200828	+	0.0200828i	−0.717077	−	0.696994i	$-0.754520\pi$
$127$	−2.55051	+	2.55051i	−0.0200828	+	0.0200828i	0.696994	+	0.717077i	$0.254520\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−44.2474			−0.337767			−0.168883	−	0.985636i	$-0.554016\pi$
$131$	−44.2474			−0.337767			−0.168883	+	0.985636i	$0.554016\pi$
$132$		0			0
$133$	87.9796	+	87.9796i	0.661501	+	0.661501i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−18.3587	+	18.3587i	−0.134005	+	0.134005i	−0.770928	−	0.636923i	$-0.780207\pi$
$137$	−18.3587	+	18.3587i	−0.134005	+	0.134005i	0.636923	+	0.770928i	$0.280207\pi$
$138$		0			0
$139$		−	219.980i		−	1.58259i	−0.611437	−	0.791293i	$-0.709408\pi$
$139$		−	219.980i		−	1.58259i	0.611437	−	0.791293i	$-0.290592\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−101.010	−	101.010i	−0.706365	−	0.706365i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	28.2679i		−	0.189717i	−0.995491	−	0.0948586i	$-0.969760\pi$
$149$		−	28.2679i		−	0.189717i	0.995491	−	0.0948586i	$-0.0302399\pi$
$150$		0			0
$151$	−61.0102			−0.404041			−0.202021	−	0.979381i	$-0.564751\pi$
$151$	−61.0102			−0.404041			−0.202021	+	0.979381i	$0.564751\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	22.6515	−	22.6515i	0.144277	−	0.144277i	−0.631279	−	0.775556i	$-0.717470\pi$
$157$	22.6515	−	22.6515i	0.144277	−	0.144277i	0.775556	+	0.631279i	$0.217470\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	88.4949			0.549658
$162$		0			0
$163$	88.9898	+	88.9898i	0.545950	+	0.545950i	0.925267	−	0.379317i	$-0.123841\pi$
$163$	88.9898	+	88.9898i	0.545950	+	0.545950i	−0.379317	+	0.925267i	$0.623841\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	171.641	−	171.641i	1.02779	−	1.02779i	0.0281898	−	0.999603i	$-0.491026\pi$
$167$	171.641	−	171.641i	1.02779	−	1.02779i	0.999603	−	0.0281898i	$-0.00897427\pi$
$168$		0			0
$169$			131.000i			0.775148i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.6311	−	10.6311i	−0.0614516	−	0.0614516i	0.675713	−	0.737165i	$-0.263836\pi$
$173$	−10.6311	−	10.6311i	−0.0614516	−	0.0614516i	−0.737165	+	0.675713i	$0.763836\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			100.247i			0.560042i	0.959994	+	0.280021i	$0.0903414\pi$
$179$			100.247i			0.560042i	−0.959994	+	0.280021i	$0.909659\pi$
$180$		0			0
$181$	259.444			1.43339			0.716696	−	0.697386i	$-0.245654\pi$
$181$	259.444			1.43339			0.716696	+	0.697386i	$0.245654\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	102.677	−	102.677i	0.549072	−	0.549072i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−77.4847			−0.405679			−0.202840	−	0.979212i	$-0.565017\pi$
$191$	−77.4847			−0.405679			−0.202840	+	0.979212i	$0.565017\pi$
$192$		0			0
$193$	−83.5857	−	83.5857i	−0.433087	−	0.433087i	0.456590	−	0.889677i	$-0.349070\pi$
$193$	−83.5857	−	83.5857i	−0.433087	−	0.433087i	−0.889677	+	0.456590i	$0.849070\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−200.252	+	200.252i	−1.01651	+	1.01651i	−0.0166464	+	0.999861i	$0.505299\pi$
$197$	−200.252	+	200.252i	−1.01651	+	1.01651i	−0.999861	+	0.0166464i	$0.994701\pi$
$198$		0			0
$199$			162.990i			0.819044i	0.912300	+	0.409522i	$0.134305\pi$
$199$			162.990i			0.819044i	−0.912300	+	0.409522i	$0.865695\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.8740	+	24.8740i	0.122532	+	0.122532i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			284.495i			1.36122i
$210$		0			0
$211$	207.980			0.985685			0.492843	−	0.870118i	$-0.335958\pi$
$211$	207.980			0.985685			0.492843	+	0.870118i	$0.335958\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−72.6765	+	72.6765i	−0.334915	+	0.334915i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−304.949			−1.37986
$222$		0			0
$223$	−29.7219	−	29.7219i	−0.133282	−	0.133282i	0.637318	−	0.770601i	$-0.280044\pi$
$223$	−29.7219	−	29.7219i	−0.133282	−	0.133282i	−0.770601	+	0.637318i	$0.780044\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	197.980	−	197.980i	0.872157	−	0.872157i	−0.120550	−	0.992707i	$-0.538466\pi$
$227$	197.980	−	197.980i	0.872157	−	0.872157i	0.992707	+	0.120550i	$0.0384659\pi$
$228$		0			0
$229$		−	8.96938i		−	0.0391676i	−0.999808	−	0.0195838i	$-0.993766\pi$
$229$		−	8.96938i		−	0.0391676i	0.999808	−	0.0195838i	$-0.00623412\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	28.7628	+	28.7628i	0.123445	+	0.123445i	0.766130	−	0.642685i	$-0.222180\pi$
$233$	28.7628	+	28.7628i	0.123445	+	0.123445i	−0.642685	+	0.766130i	$0.722180\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			242.969i			1.01661i	0.861178	+	0.508304i	$0.169727\pi$
$239$			242.969i			1.01661i	−0.861178	+	0.508304i	$0.830273\pi$
$240$		0			0
$241$	−32.0000			−0.132780			−0.0663900	−	0.997794i	$-0.521148\pi$
$241$	−32.0000			−0.132780			−0.0663900	+	0.997794i	$0.521148\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	422.474	−	422.474i	1.71042	−	1.71042i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	250.763			0.999055			0.499527	−	0.866298i	$-0.333507\pi$
$251$	250.763			0.999055			0.499527	+	0.866298i	$0.333507\pi$
$252$		0			0
$253$	143.081	+	143.081i	0.565536	+	0.565536i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−73.8638	+	73.8638i	−0.287408	+	0.287408i	−0.836054	−	0.548647i	$-0.815143\pi$
$257$	−73.8638	+	73.8638i	−0.287408	+	0.287408i	0.548647	+	0.836054i	$0.315143\pi$
$258$		0			0
$259$			37.4847i			0.144729i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	49.1668	+	49.1668i	0.186946	+	0.186946i	0.794374	−	0.607428i	$-0.207799\pi$
$263$	49.1668	+	49.1668i	0.186946	+	0.186946i	−0.607428	+	0.794374i	$0.707799\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			278.742i			1.03622i	0.855315	+	0.518108i	$0.173364\pi$
$269$			278.742i			1.03622i	−0.855315	+	0.518108i	$0.826636\pi$
$270$		0			0
$271$	−66.0000			−0.243542			−0.121771	−	0.992558i	$-0.538857\pi$
$271$	−66.0000			−0.243542			−0.121771	+	0.992558i	$0.538857\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−211.843	+	211.843i	−0.764777	+	0.764777i	−0.977182	−	0.212404i	$-0.931871\pi$
$277$	−211.843	+	211.843i	−0.764777	+	0.764777i	0.212404	+	0.977182i	$0.431871\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	312.434			1.11186			0.555932	−	0.831228i	$-0.312362\pi$
$281$	312.434			1.11186			0.555932	+	0.831228i	$0.312362\pi$
$282$		0			0
$283$	204.747	+	204.747i	0.723487	+	0.723487i	0.969314	−	0.245826i	$-0.0790593\pi$
$283$	204.747	+	204.747i	0.723487	+	0.723487i	−0.245826	+	0.969314i	$0.579059\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	190.000	−	190.000i	0.662021	−	0.662021i
$288$		0			0
$289$		−	20.9796i		−	0.0725937i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	325.151	+	325.151i	1.10973	+	1.10973i	0.993185	+	0.116545i	$0.0371819\pi$
$293$	325.151	+	325.151i	1.10973	+	1.10973i	0.116545	+	0.993185i	$0.462818\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	424.949i		−	1.42123i
$300$		0			0
$301$	178.020			0.591430
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−179.394	+	179.394i	−0.584345	+	0.584345i	−0.936094	−	0.351749i	$-0.885587\pi$
$307$	−179.394	+	179.394i	−0.584345	+	0.584345i	0.351749	+	0.936094i	$0.385587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	213.980			0.688037			0.344019	−	0.938963i	$-0.388212\pi$
$311$	213.980			0.688037			0.344019	+	0.938963i	$0.388212\pi$
$312$		0			0
$313$	186.414	+	186.414i	0.595573	+	0.595573i	0.939131	−	0.343558i	$-0.111632\pi$
$313$	186.414	+	186.414i	0.595573	+	0.595573i	−0.343558	+	0.939131i	$0.611632\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	57.5505	−	57.5505i	0.181547	−	0.181547i	−0.610482	−	0.792030i	$-0.709024\pi$
$317$	57.5505	−	57.5505i	0.181547	−	0.181547i	0.792030	+	0.610482i	$0.209024\pi$
$318$		0			0
$319$			80.4337i			0.252143i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	429.444	+	429.444i	1.32955	+	1.32955i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	112.454i		−	0.341806i
$330$		0			0
$331$	−214.413			−0.647774			−0.323887	−	0.946096i	$-0.604990\pi$
$331$	−214.413			−0.647774			−0.323887	+	0.946096i	$0.604990\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	94.5959	−	94.5959i	0.280700	−	0.280700i	−0.552688	−	0.833388i	$-0.686398\pi$
$337$	94.5959	−	94.5959i	0.280700	−	0.280700i	0.833388	+	0.552688i	$0.186398\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−235.010			−0.689179
$342$		0			0
$343$	−216.767	−	216.767i	−0.631975	−	0.631975i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	226.919	−	226.919i	0.653946	−	0.653946i	−0.299995	−	0.953941i	$-0.596985\pi$
$347$	226.919	−	226.919i	0.653946	−	0.653946i	0.953941	+	0.299995i	$0.0969849\pi$
$348$		0			0
$349$		−	182.454i		−	0.522791i	−0.965232	−	0.261396i	$-0.915817\pi$
$349$		−	182.454i		−	0.522791i	0.965232	−	0.261396i	$-0.0841827\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	263.914	+	263.914i	0.747631	+	0.747631i	0.974034	−	0.226403i	$-0.0726965\pi$
$353$	263.914	+	263.914i	0.747631	+	0.747631i	−0.226403	+	0.974034i	$0.572697\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	285.485i		−	0.795222i	−0.917554	−	0.397611i	$-0.869839\pi$
$359$		−	285.485i		−	0.795222i	0.917554	−	0.397611i	$-0.130161\pi$
$360$		0			0
$361$	−828.898			−2.29612
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−418.964	+	418.964i	−1.14159	+	1.14159i	−0.153431	+	0.988159i	$0.549032\pi$
$367$	−418.964	+	418.964i	−1.14159	+	1.14159i	−0.988159	+	0.153431i	$0.950968\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	330.020			0.889543
$372$		0			0
$373$	−283.106	−	283.106i	−0.758996	−	0.758996i	0.217143	−	0.976140i	$-0.430326\pi$
$373$	−283.106	−	283.106i	−0.758996	−	0.758996i	−0.976140	+	0.217143i	$0.930326\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	119.444	−	119.444i	0.316827	−	0.316827i
$378$		0			0
$379$			194.000i			0.511873i	0.966694	+	0.255937i	$0.0823839\pi$
$379$			194.000i			0.511873i	−0.966694	+	0.255937i	$0.917616\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−531.489	−	531.489i	−1.38770	−	1.38770i	−0.830129	−	0.557572i	$-0.811733\pi$
$383$	−531.489	−	531.489i	−1.38770	−	1.38770i	−0.557572	−	0.830129i	$-0.688267\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			479.196i			1.23187i	0.787798	+	0.615934i	$0.211221\pi$
$389$			479.196i			1.23187i	−0.787798	+	0.615934i	$0.788779\pi$
$390$		0			0
$391$	431.959			1.10475
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	389.267	−	389.267i	0.980521	−	0.980521i	−0.0192929	−	0.999814i	$-0.506142\pi$
$397$	389.267	−	389.267i	0.980521	−	0.980521i	0.999814	+	0.0192929i	$0.00614150\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	603.898			1.50598			0.752990	−	0.658032i	$-0.228611\pi$
$401$	603.898			1.50598			0.752990	+	0.658032i	$0.228611\pi$
$402$		0			0
$403$	348.990	+	348.990i	0.865980	+	0.865980i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−60.6061	+	60.6061i	−0.148909	+	0.148909i
$408$		0			0
$409$			183.959i			0.449778i	0.974384	+	0.224889i	$0.0722019\pi$
$409$			183.959i			0.449778i	−0.974384	+	0.224889i	$0.927798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	38.9148	+	38.9148i	0.0942247	+	0.0942247i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			197.814i			0.472109i	0.971740	+	0.236055i	$0.0758544\pi$
$419$			197.814i			0.472109i	−0.971740	+	0.236055i	$0.924146\pi$
$420$		0			0
$421$	−114.041			−0.270881			−0.135440	−	0.990785i	$-0.543245\pi$
$421$	−114.041			−0.270881			−0.135440	+	0.990785i	$0.543245\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	136.465	−	136.465i	0.319591	−	0.319591i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−348.454			−0.808478			−0.404239	−	0.914653i	$-0.632464\pi$
$431$	−348.454			−0.808478			−0.404239	+	0.914653i	$0.632464\pi$
$432$		0			0
$433$	−60.1010	−	60.1010i	−0.138801	−	0.138801i	0.634292	−	0.773094i	$-0.281292\pi$
$433$	−60.1010	−	60.1010i	−0.138801	−	0.138801i	−0.773094	+	0.634292i	$0.781292\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−598.434	+	598.434i	−1.36941	+	1.36941i
$438$		0			0
$439$			556.929i			1.26863i	0.773075	+	0.634315i	$0.218718\pi$
$439$			556.929i			1.26863i	−0.773075	+	0.634315i	$0.781282\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	95.5551	+	95.5551i	0.215700	+	0.215700i	0.806684	−	0.590984i	$-0.201260\pi$
$443$	95.5551	+	95.5551i	0.215700	+	0.215700i	−0.590984	+	0.806684i	$0.701260\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			213.423i			0.475331i	0.971347	+	0.237665i	$0.0763821\pi$
$449$			213.423i			0.475331i	−0.971347	+	0.237665i	$0.923618\pi$
$450$		0			0
$451$	614.393			1.36229
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	38.9388	−	38.9388i	0.0852052	−	0.0852052i	−0.663220	−	0.748425i	$-0.730810\pi$
$457$	38.9388	−	38.9388i	0.0852052	−	0.0852052i	0.748425	+	0.663220i	$0.230810\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−509.712			−1.10567			−0.552833	−	0.833292i	$-0.686453\pi$
$461$	−509.712			−1.10567			−0.552833	+	0.833292i	$0.686453\pi$
$462$		0			0
$463$	−492.955	−	492.955i	−1.06470	−	1.06470i	−0.997757	−	0.0669397i	$-0.978676\pi$
$463$	−492.955	−	492.955i	−1.06470	−	1.06470i	−0.0669397	−	0.997757i	$-0.521324\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	647.044	−	647.044i	1.38553	−	1.38553i	0.551085	−	0.834449i	$-0.314214\pi$
$467$	647.044	−	647.044i	1.38553	−	1.38553i	0.834449	−	0.551085i	$-0.185786\pi$
$468$		0			0
$469$		−	23.9592i		−	0.0510857i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	287.828	+	287.828i	0.608515	+	0.608515i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	198.969i		−	0.415385i	−0.978194	−	0.207692i	$-0.933405\pi$
$479$		−	198.969i		−	0.415385i	0.978194	−	0.207692i	$-0.0665953\pi$
$480$		0			0
$481$	180.000			0.374220
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	34.6209	−	34.6209i	0.0710902	−	0.0710902i	−0.670668	−	0.741758i	$-0.733992\pi$
$487$	34.6209	−	34.6209i	0.0710902	−	0.0710902i	0.741758	+	0.670668i	$0.233992\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	155.217			0.316124			0.158062	−	0.987429i	$-0.449475\pi$
$491$	155.217			0.316124			0.158062	+	0.987429i	$0.449475\pi$
$492$		0			0
$493$	121.414	+	121.414i	0.246276	+	0.246276i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−300.908	+	300.908i	−0.605449	+	0.605449i
$498$		0			0
$499$		−	547.444i		−	1.09708i	−0.836124	−	0.548541i	$-0.815183\pi$
$499$		−	547.444i		−	1.09708i	0.836124	−	0.548541i	$-0.184817\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−526.590	−	526.590i	−1.04690	−	1.04690i	−0.998845	−	0.0480545i	$-0.984698\pi$
$503$	−526.590	−	526.590i	−1.04690	−	1.04690i	−0.0480545	−	0.998845i	$-0.515302\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	115.773i		−	0.227452i	−0.993512	−	0.113726i	$-0.963721\pi$
$509$		−	115.773i		−	0.227452i	0.993512	−	0.113726i	$-0.0362786\pi$
$510$		0			0
$511$	174.413			0.341318
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	181.818	−	181.818i	0.351680	−	0.351680i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−791.444			−1.51909			−0.759543	−	0.650457i	$-0.774577\pi$
$521$	−791.444			−1.51909			−0.759543	+	0.650457i	$0.774577\pi$
$522$		0			0
$523$	−93.6867	−	93.6867i	−0.179133	−	0.179133i	0.611845	−	0.790978i	$-0.290428\pi$
$523$	−93.6867	−	93.6867i	−0.179133	−	0.179133i	−0.790978	+	0.611845i	$0.790428\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−354.747	+	354.747i	−0.673144	+	0.673144i
$528$		0			0
$529$			72.9388i			0.137880i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−912.372	−	912.372i	−1.71177	−	1.71177i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	296.824i		−	0.550694i
$540$		0			0
$541$	−359.526			−0.664557			−0.332279	−	0.943181i	$-0.607817\pi$
$541$	−359.526			−0.664557			−0.332279	+	0.943181i	$0.607817\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−164.647	+	164.647i	−0.301000	+	0.301000i	−0.841405	−	0.540405i	$-0.818271\pi$
$547$	−164.647	+	164.647i	−0.301000	+	0.301000i	0.540405	+	0.841405i	$0.318271\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−336.413			−0.610550
$552$		0			0
$553$	1.26224	+	1.26224i	0.00228253	+	0.00228253i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	139.040	−	139.040i	0.249623	−	0.249623i	−0.571193	−	0.820816i	$-0.693519\pi$
$557$	139.040	−	139.040i	0.249623	−	0.249623i	0.820816	+	0.571193i	$0.193519\pi$
$558$		0			0
$559$		−	854.847i		−	1.52924i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−787.423	−	787.423i	−1.39862	−	1.39862i	−0.804024	−	0.594596i	$-0.797312\pi$
$563$	−787.423	−	787.423i	−1.39862	−	1.39862i	−0.594596	−	0.804024i	$-0.702688\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			4.92856i			0.00866180i	0.999991	+	0.00433090i	$0.00137857\pi$
$569$			4.92856i			0.00866180i	−0.999991	+	0.00433090i	$0.998621\pi$
$570$		0			0
$571$	205.505			0.359904			0.179952	−	0.983675i	$-0.442406\pi$
$571$	205.505			0.359904			0.179952	+	0.983675i	$0.442406\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−689.999	+	689.999i	−1.19584	+	1.19584i	−0.220438	+	0.975401i	$0.570749\pi$
$577$	−689.999	+	689.999i	−1.19584	+	1.19584i	−0.975401	+	0.220438i	$0.929251\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	93.6480			0.161184
$582$		0			0
$583$	533.585	+	533.585i	0.915240	+	0.915240i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−672.347	+	672.347i	−1.14540	+	1.14540i	−0.157949	+	0.987447i	$0.550488\pi$
$587$	−672.347	+	672.347i	−1.14540	+	1.14540i	−0.987447	+	0.157949i	$0.949512\pi$
$588$		0			0
$589$		−	982.929i		−	1.66881i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	344.570	+	344.570i	0.581062	+	0.581062i	0.935195	−	0.354133i	$-0.115224\pi$
$593$	344.570	+	344.570i	0.581062	+	0.581062i	−0.354133	+	0.935195i	$0.615224\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	739.464i		−	1.23450i	−0.786768	−	0.617249i	$-0.788247\pi$
$599$		−	739.464i		−	1.23450i	0.786768	−	0.617249i	$-0.211753\pi$
$600$		0			0
$601$	−642.908			−1.06973			−0.534865	−	0.844937i	$-0.679638\pi$
$601$	−642.908			−1.06973			−0.534865	+	0.844937i	$0.679638\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	362.803	−	362.803i	0.597698	−	0.597698i	−0.342002	−	0.939699i	$-0.611105\pi$
$607$	362.803	−	362.803i	0.597698	−	0.597698i	0.939699	+	0.342002i	$0.111105\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−540.000			−0.883797
$612$		0			0
$613$	721.943	+	721.943i	1.17772	+	1.17772i	0.980323	+	0.197398i	$0.0632492\pi$
$613$	721.943	+	721.943i	1.17772	+	1.17772i	0.197398	+	0.980323i	$0.436751\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	72.9036	−	72.9036i	0.118158	−	0.118158i	−0.645555	−	0.763713i	$-0.723374\pi$
$617$	72.9036	−	72.9036i	0.118158	−	0.118158i	0.763713	+	0.645555i	$0.223374\pi$
$618$		0			0
$619$			228.061i			0.368435i	0.982886	+	0.184217i	$0.0589751\pi$
$619$			228.061i			0.368435i	−0.982886	+	0.184217i	$0.941025\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	349.342	+	349.342i	0.560741	+	0.560741i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			182.969i			0.290889i
$630$		0			0
$631$	−86.4337			−0.136979			−0.0684894	−	0.997652i	$-0.521818\pi$
$631$	−86.4337			−0.136979			−0.0684894	+	0.997652i	$0.521818\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−440.783	+	440.783i	−0.691967	+	0.691967i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−340.061			−0.530517			−0.265258	−	0.964177i	$-0.585457\pi$
$641$	−340.061			−0.530517			−0.265258	+	0.964177i	$0.585457\pi$
$642$		0			0
$643$	−769.040	−	769.040i	−1.19602	−	1.19602i	−0.975349	−	0.220670i	$-0.929176\pi$
$643$	−769.040	−	769.040i	−1.19602	−	1.19602i	−0.220670	−	0.975349i	$-0.570824\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−602.854	+	602.854i	−0.931767	+	0.931767i	−0.997816	−	0.0660489i	$-0.978961\pi$
$647$	−602.854	+	602.854i	−0.931767	+	0.931767i	0.0660489	+	0.997816i	$0.478961\pi$
$648$		0			0
$649$			125.837i			0.193893i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−75.8842	−	75.8842i	−0.116209	−	0.116209i	0.646611	−	0.762820i	$-0.276186\pi$
$653$	−75.8842	−	75.8842i	−0.116209	−	0.116209i	−0.762820	+	0.646611i	$0.776186\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	61.1964i		−	0.0928626i	−0.998921	−	0.0464313i	$-0.985215\pi$
$659$		−	61.1964i		−	0.0928626i	0.998921	−	0.0464313i	$-0.0147849\pi$
$660$		0			0
$661$	1158.45			1.75258			0.876289	−	0.481786i	$-0.160012\pi$
$661$	1158.45			1.75258			0.876289	+	0.481786i	$0.160012\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−169.192	+	169.192i	−0.253661	+	0.253661i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	441.281			0.657646
$672$		0			0
$673$	−711.413	−	711.413i	−1.05708	−	1.05708i	−0.998269	−	0.0588084i	$-0.981270\pi$
$673$	−711.413	−	711.413i	−1.05708	−	1.05708i	−0.0588084	−	0.998269i	$-0.518730\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	351.943	−	351.943i	0.519857	−	0.519857i	−0.397671	−	0.917528i	$-0.630181\pi$
$677$	351.943	−	351.943i	0.519857	−	0.519857i	0.917528	+	0.397671i	$0.130181\pi$
$678$		0			0
$679$		−	482.536i		−	0.710656i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	277.905	+	277.905i	0.406888	+	0.406888i	0.880652	−	0.473764i	$-0.157105\pi$
$683$	277.905	+	277.905i	0.406888	+	0.406888i	−0.473764	+	0.880652i	$0.657105\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1584.74i		−	2.30007i
$690$		0			0
$691$	−1097.51			−1.58829			−0.794143	−	0.607731i	$-0.792080\pi$
$691$	−1097.51			−1.58829			−0.794143	+	0.607731i	$0.792080\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	927.423	−	927.423i	1.33059	−	1.33059i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	354.783			0.506110			0.253055	−	0.967452i	$-0.418565\pi$
$701$	354.783			0.506110			0.253055	+	0.967452i	$0.418565\pi$
$702$		0			0
$703$	−253.485	−	253.485i	−0.360576	−	0.360576i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	179.167	−	179.167i	0.253418	−	0.253418i
$708$		0			0
$709$			368.061i			0.519127i	0.965726	+	0.259564i	$0.0835787\pi$
$709$			368.061i			0.519127i	−0.965726	+	0.259564i	$0.916421\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−494.343	−	494.343i	−0.693328	−	0.693328i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			382.969i			0.532642i	0.963884	+	0.266321i	$0.0858081\pi$
$719$			382.969i			0.532642i	−0.963884	+	0.266321i	$0.914192\pi$
$720$		0			0
$721$	442.990			0.614410
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−837.297	+	837.297i	−1.15172	+	1.15172i	−0.165507	+	0.986209i	$0.552926\pi$
$727$	−837.297	+	837.297i	−1.15172	+	1.15172i	−0.986209	+	0.165507i	$0.947074\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	868.949			1.18871
$732$		0			0
$733$	29.8230	+	29.8230i	0.0406862	+	0.0406862i	0.727157	−	0.686471i	$-0.240841\pi$
$733$	29.8230	+	29.8230i	0.0406862	+	0.0406862i	−0.686471	+	0.727157i	$0.740841\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	38.7378	−	38.7378i	0.0525614	−	0.0525614i
$738$		0			0
$739$			1183.28i			1.60119i	0.599205	+	0.800596i	$0.295484\pi$
$739$			1183.28i			1.60119i	−0.599205	+	0.800596i	$0.704516\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	65.7821	+	65.7821i	0.0885358	+	0.0885358i	0.749988	−	0.661452i	$-0.230059\pi$
$743$	65.7821	+	65.7821i	0.0885358	+	0.0885358i	−0.661452	+	0.749988i	$0.730059\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	491.423i		−	0.656106i
$750$		0			0
$751$	−850.270			−1.13218			−0.566092	−	0.824342i	$-0.691545\pi$
$751$	−850.270			−1.13218			−0.566092	+	0.824342i	$0.691545\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−145.832	+	145.832i	−0.192645	+	0.192645i	−0.796838	−	0.604193i	$-0.793496\pi$
$757$	−145.832	+	145.832i	−0.192645	+	0.192645i	0.604193	+	0.796838i	$0.293496\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−157.505			−0.206971			−0.103486	−	0.994631i	$-0.533000\pi$
$761$	−157.505			−0.206971			−0.103486	+	0.994631i	$0.533000\pi$
$762$		0			0
$763$	31.9204	+	31.9204i	0.0418354	+	0.0418354i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	186.867	−	186.867i	0.243634	−	0.243634i
$768$		0			0
$769$			559.031i			0.726958i	0.931602	+	0.363479i	$0.118411\pi$
$769$			559.031i			0.726958i	−0.931602	+	0.363479i	$0.881589\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−348.965	−	348.965i	−0.451442	−	0.451442i	0.444391	−	0.895833i	$-0.353420\pi$
$773$	−348.965	−	348.965i	−0.451442	−	0.451442i	−0.895833	+	0.444391i	$0.853420\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			2569.69i			3.29871i
$780$		0			0
$781$	−973.031			−1.24588
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	821.414	−	821.414i	1.04373	−	1.04373i	0.0447293	−	0.998999i	$-0.485757\pi$
$787$	821.414	−	821.414i	1.04373	−	1.04373i	0.998999	−	0.0447293i	$-0.0142425\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−341.485			−0.431713
$792$		0			0
$793$	−655.301	−	655.301i	−0.826357	−	0.826357i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−649.065	+	649.065i	−0.814385	+	0.814385i	−0.985288	−	0.170903i	$-0.945332\pi$
$797$	−649.065	+	649.065i	−0.814385	+	0.814385i	0.170903	+	0.985288i	$0.445332\pi$
$798$		0			0
$799$		−	548.908i		−	0.686994i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	281.995	+	281.995i	0.351177	+	0.351177i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	909.918i		−	1.12474i	−0.826884	−	0.562372i	$-0.809889\pi$
$809$		−	909.918i		−	1.12474i	0.826884	−	0.562372i	$-0.190111\pi$
$810$		0			0
$811$	−652.929			−0.805091			−0.402545	−	0.915400i	$-0.631874\pi$
$811$	−652.929			−0.805091			−0.402545	+	0.915400i	$0.631874\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1203.84	+	1203.84i	−1.47348	+	1.47348i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	808.064			0.984243			0.492122	−	0.870526i	$-0.336222\pi$
$821$	808.064			0.984243			0.492122	+	0.870526i	$0.336222\pi$
$822$		0			0
$823$	−333.863	−	333.863i	−0.405666	−	0.405666i	0.474558	−	0.880224i	$-0.342608\pi$
$823$	−333.863	−	333.863i	−0.405666	−	0.405666i	−0.880224	+	0.474558i	$0.842608\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	481.162	−	481.162i	0.581817	−	0.581817i	−0.353586	−	0.935402i	$-0.615038\pi$
$827$	481.162	−	481.162i	0.581817	−	0.581817i	0.935402	+	0.353586i	$0.115038\pi$
$828$		0			0
$829$			1296.35i			1.56375i	0.623433	+	0.781877i	$0.285738\pi$
$829$			1296.35i			1.56375i	−0.623433	+	0.781877i	$0.714262\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−448.055	−	448.055i	−0.537881	−	0.537881i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	978.515i		−	1.16629i	−0.812369	−	0.583144i	$-0.801822\pi$
$839$		−	978.515i		−	1.16629i	0.812369	−	0.583144i	$-0.198178\pi$
$840$		0			0
$841$	745.888			0.886906
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	135.125	−	135.125i	0.159534	−	0.159534i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−254.969			−0.299611
$852$		0			0
$853$	−300.127	−	300.127i	−0.351849	−	0.351849i	0.508948	−	0.860797i	$-0.330034\pi$
$853$	−300.127	−	300.127i	−0.351849	−	0.351849i	−0.860797	+	0.508948i	$0.830034\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	879.823	−	879.823i	1.02663	−	1.02663i	0.0269957	−	0.999636i	$-0.491406\pi$
$857$	879.823	−	879.823i	1.02663	−	1.02663i	0.999636	−	0.0269957i	$-0.00859403\pi$
$858$		0			0
$859$			128.888i			0.150044i	0.997182	+	0.0750220i	$0.0239027\pi$
$859$			128.888i			0.150044i	−0.997182	+	0.0750220i	$0.976097\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	204.418	+	204.418i	0.236869	+	0.236869i	0.815552	−	0.578683i	$-0.196433\pi$
$863$	204.418	+	204.418i	0.236869	+	0.236869i	−0.578683	+	0.815552i	$0.696433\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			4.08164i			0.00469694i
$870$		0			0
$871$	−115.051			−0.132091
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	570.025	−	570.025i	0.649971	−	0.649971i	−0.303014	−	0.952986i	$-0.597993\pi$
$877$	570.025	−	570.025i	0.649971	−	0.649971i	0.952986	+	0.303014i	$0.0979931\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1235.86			1.40279			0.701395	−	0.712773i	$-0.252561\pi$
$881$	1235.86			1.40279			0.701395	+	0.712773i	$0.252561\pi$
$882$		0			0
$883$	713.131	+	713.131i	0.807622	+	0.807622i	0.984274	−	0.176651i	$-0.0565264\pi$
$883$	713.131	+	713.131i	0.807622	+	0.807622i	−0.176651	+	0.984274i	$0.556526\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	343.510	−	343.510i	0.387271	−	0.387271i	−0.486442	−	0.873713i	$-0.661705\pi$
$887$	343.510	−	343.510i	0.387271	−	0.387271i	0.873713	+	0.486442i	$0.161705\pi$
$888$		0			0
$889$		−	13.0102i		−	0.0146347i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	760.454	+	760.454i	0.851572	+	0.851572i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	277.898i		−	0.309119i
$900$		0			0
$901$	1610.89			1.78789
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	918.938	−	918.938i	1.01316	−	1.01316i	0.0132496	−	0.999912i	$-0.495782\pi$
$907$	918.938	−	918.938i	1.01316	−	1.01316i	0.999912	−	0.0132496i	$-0.00421759\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−788.454			−0.865482			−0.432741	−	0.901518i	$-0.642454\pi$
$911$	−788.454			−0.865482			−0.432741	+	0.901518i	$0.642454\pi$
$912$		0			0
$913$	151.412	+	151.412i	0.165840	+	0.165840i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	112.854	−	112.854i	0.123068	−	0.123068i
$918$		0			0
$919$		−	1644.33i		−	1.78926i	−0.446806	−	0.894631i	$-0.647439\pi$
$919$		−	1644.33i		−	1.78926i	0.446806	−	0.894631i	$-0.352561\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1444.95	+	1444.95i	1.56549	+	1.56549i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1021.51i			1.09957i	0.835305	+	0.549787i	$0.185291\pi$
$929$			1021.51i			1.09957i	−0.835305	+	0.549787i	$0.814709\pi$
$930$		0			0
$931$	1241.46			1.33347
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	370.555	−	370.555i	0.395470	−	0.395470i	−0.481162	−	0.876632i	$-0.659785\pi$
$937$	370.555	−	370.555i	0.395470	−	0.395470i	0.876632	+	0.481162i	$0.159785\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1524.06			−1.61962			−0.809811	−	0.586691i	$-0.800430\pi$
$941$	−1524.06			−1.61962			−0.809811	+	0.586691i	$0.800430\pi$
$942$		0			0
$943$	1292.37	+	1292.37i	1.37049	+	1.37049i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−644.545	+	644.545i	−0.680618	+	0.680618i	−0.960139	−	0.279522i	$-0.909824\pi$
$947$	−644.545	+	644.545i	−0.680618	+	0.680618i	0.279522	+	0.960139i	$0.409824\pi$
$948$		0			0
$949$		−	837.526i		−	0.882535i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−461.641	−	461.641i	−0.484409	−	0.484409i	0.422128	−	0.906536i	$-0.361283\pi$
$953$	−461.641	−	461.641i	−0.484409	−	0.484409i	−0.906536	+	0.422128i	$0.861283\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	93.6480i		−	0.0976517i
$960$		0			0
$961$	−149.041			−0.155089
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−462.196	+	462.196i	−0.477969	+	0.477969i	−0.904482	−	0.426512i	$-0.859742\pi$
$967$	−462.196	+	462.196i	−0.477969	+	0.477969i	0.426512	+	0.904482i	$0.359742\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	198.390			0.204315			0.102158	−	0.994768i	$-0.467425\pi$
$971$	198.390			0.204315			0.102158	+	0.994768i	$0.467425\pi$
$972$		0			0
$973$	561.060	+	561.060i	0.576629	+	0.576629i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	543.206	−	543.206i	0.555993	−	0.555993i	−0.372171	−	0.928164i	$-0.621386\pi$
$977$	543.206	−	543.206i	0.555993	−	0.555993i	0.928164	+	0.372171i	$0.121386\pi$
$978$		0			0
$979$			1129.65i			1.15388i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−321.387	−	321.387i	−0.326945	−	0.326945i	0.524478	−	0.851424i	$-0.324260\pi$
$983$	−321.387	−	321.387i	−0.326945	−	0.326945i	−0.851424	+	0.524478i	$0.824260\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1210.89i			1.22436i
$990$		0			0
$991$	−1543.82			−1.55784			−0.778918	−	0.627125i	$-0.784231\pi$
$991$	−1543.82			−1.55784			−0.778918	+	0.627125i	$0.784231\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−800.429	+	800.429i	−0.802838	+	0.802838i	−0.983538	−	0.180701i	$-0.942164\pi$
$997$	−800.429	+	800.429i	−0.802838	+	0.802838i	0.180701	+	0.983538i	$0.442164\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.l.b.793.2		4
3.2	odd	2		300.3.k.a.193.1		4
5.2	odd	4	inner	900.3.l.b.757.2		4
5.3	odd	4		180.3.l.b.37.2		4
5.4	even	2		180.3.l.b.73.2		4
12.11	even	2		1200.3.bg.o.193.2		4
15.2	even	4		300.3.k.a.157.1		4
15.8	even	4		60.3.k.a.37.2	yes	4
15.14	odd	2		60.3.k.a.13.2	✓	4
20.3	even	4		720.3.bh.f.577.2		4
20.19	odd	2		720.3.bh.f.433.2		4
60.23	odd	4		240.3.bg.d.97.1		4
60.47	odd	4		1200.3.bg.o.1057.2		4
60.59	even	2		240.3.bg.d.193.1		4
120.29	odd	2		960.3.bg.b.193.1		4
120.53	even	4		960.3.bg.b.577.1		4
120.59	even	2		960.3.bg.a.193.2		4
120.83	odd	4		960.3.bg.a.577.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.k.a.13.2	✓	4	15.14	odd	2
60.3.k.a.37.2	yes	4	15.8	even	4
180.3.l.b.37.2		4	5.3	odd	4
180.3.l.b.73.2		4	5.4	even	2
240.3.bg.d.97.1		4	60.23	odd	4
240.3.bg.d.193.1		4	60.59	even	2
300.3.k.a.157.1		4	15.2	even	4
300.3.k.a.193.1		4	3.2	odd	2
720.3.bh.f.433.2		4	20.19	odd	2
720.3.bh.f.577.2		4	20.3	even	4
900.3.l.b.757.2		4	5.2	odd	4	inner
900.3.l.b.793.2		4	1.1	even	1	trivial
960.3.bg.a.193.2		4	120.59	even	2
960.3.bg.a.577.2		4	120.83	odd	4
960.3.bg.b.193.1		4	120.29	odd	2
960.3.bg.b.577.1		4	120.53	even	4
1200.3.bg.o.193.2		4	12.11	even	2
1200.3.bg.o.1057.2		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.55051 + 2.55051i) q^{7} +O(q^{10})\)	\(q+(-2.55051 + 2.55051i) q^{7} -8.24745 q^{11} +(12.2474 + 12.2474i) q^{13} +(-12.4495 + 12.4495i) q^{17} -34.4949i q^{19} +(-17.3485 - 17.3485i) q^{23} -9.75255i q^{29} +28.4949 q^{31} +(7.34847 - 7.34847i) q^{37} -74.4949 q^{41} +(-34.8990 - 34.8990i) q^{43} +(-22.0454 + 22.0454i) q^{47} +35.9898i q^{49} +(-64.6969 - 64.6969i) q^{53} -15.2577i q^{59} -53.5051 q^{61} +(-4.69694 + 4.69694i) q^{67} +117.980 q^{71} +(-34.1918 - 34.1918i) q^{73} +(21.0352 - 21.0352i) q^{77} -0.494897i q^{79} +(-18.3587 - 18.3587i) q^{83} -136.969i q^{89} -62.4745 q^{91} +(-94.5959 + 94.5959i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{7}+O(q^{10}) \) 4 * q - 20 * q^7	\( 4 q - 20 q^{7} + 16 q^{11} - 40 q^{17} - 40 q^{23} + 16 q^{31} - 200 q^{41} - 120 q^{43} - 200 q^{53} - 312 q^{61} + 40 q^{67} + 80 q^{71} + 20 q^{73} - 200 q^{77} - 240 q^{83} + 240 q^{91} - 300 q^{97}+O(q^{100}) \) 4 * q - 20 * q^7 + 16 * q^11 - 40 * q^17 - 40 * q^23 + 16 * q^31 - 200 * q^41 - 120 * q^43 - 200 * q^53 - 312 * q^61 + 40 * q^67 + 80 * q^71 + 20 * q^73 - 200 * q^77 - 240 * q^83 + 240 * q^91 - 300 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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