LMFDB - Embedded newform 756.4.k.h.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(109,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.109"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.k (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,7,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 198 x^{14} + 15603 x^{12} + 623882 x^{10} + 13296387 x^{8} + 143321550 x^{6} + 644261986 x^{4} + \cdots + 9753129 $ x^16 + 198x^14 + 15603x^12 + 623882x^10 + 13296387x^8 + 143321550x^6 + 644261986x^4 + 674526288*x^2 + 9753129
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$2.87126i$ of defining polynomial
Character	$\chi$	$=$	756.109
Dual form			756.4.k.h.541.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.20902 - 15.9505i) q^{5} +(-14.4823 + 11.5440i) q^{7} +(-13.0045 + 22.5245i) q^{11} -38.8473 q^{13} +(7.38515 - 12.7915i) q^{17} +(26.4892 + 45.8806i) q^{19} +(-56.7287 - 98.2569i) q^{23} +(-107.112 + 185.524i) q^{25} +119.708 q^{29} +(-83.5340 + 144.685i) q^{31} +(317.500 + 124.690i) q^{35} +(-39.6722 - 68.7143i) q^{37} -41.8167 q^{41} +190.219 q^{43} +(231.991 + 401.821i) q^{47} +(76.4715 - 334.367i) q^{49} +(-2.74684 + 4.75766i) q^{53} +479.036 q^{55} +(334.488 - 579.351i) q^{59} +(-469.420 - 813.059i) q^{61} +(357.745 + 619.633i) q^{65} +(-138.918 + 240.613i) q^{67} +357.478 q^{71} +(508.704 - 881.101i) q^{73} +(-71.6883 - 476.330i) q^{77} +(382.659 + 662.784i) q^{79} +1421.16 q^{83} -272.040 q^{85} +(385.995 + 668.564i) q^{89} +(562.596 - 448.453i) q^{91} +(487.879 - 845.030i) q^{95} +575.835 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 7 q^{5} - 11 q^{7} - 43 q^{11} - 52 q^{13} + 38 q^{17} + 32 q^{19} - 118 q^{23} - 143 q^{25} + 160 q^{29} + 5 q^{31} + 161 q^{35} - 136 q^{37} - 216 q^{41} + 800 q^{43} - 78 q^{47} + 253 q^{49} + 105 q^{53}+ \cdots - 838 q^{97}+O(q^{100}) $ 16 * q + 7 * q^5 - 11 * q^7 - 43 * q^11 - 52 * q^13 + 38 * q^17 + 32 * q^19 - 118 * q^23 - 143 * q^25 + 160 * q^29 + 5 * q^31 + 161 * q^35 - 136 * q^37 - 216 * q^41 + 800 * q^43 - 78 * q^47 + 253 * q^49 + 105 * q^53 - 948 * q^55 + 574 * q^59 - 424 * q^61 - 408 * q^65 - 88 * q^67 - 856 * q^71 + 644 * q^73 - 656 * q^77 + 266 * q^79 + 2054 * q^83 + 384 * q^85 + 732 * q^89 + 1022 * q^91 - 364 * q^95 - 838 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	−9.20902	−	15.9505i	−0.823680	−	1.42666i	−0.902924	−	0.429800i	$-0.858584\pi$
$5$	−9.20902	−	15.9505i	−0.823680	−	1.42666i	0.0792444	−	0.996855i	$-0.474749\pi$
$6$		0			0
$7$	−14.4823	+	11.5440i	−0.781968	+	0.623318i
$7$	−14.4823	+	11.5440i	−0.781968	+	0.623318i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.0045	+	22.5245i	−0.356456	+	0.617400i	−0.987366	−	0.158456i	$-0.949348\pi$
$11$	−13.0045	+	22.5245i	−0.356456	+	0.617400i	0.630910	+	0.775856i	$0.282682\pi$
$12$		0			0
$13$	−38.8473			−0.828792			−0.414396	−	0.910097i	$-0.636007\pi$
$13$	−38.8473			−0.828792			−0.414396	+	0.910097i	$0.636007\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.38515	−	12.7915i	0.105362	−	0.182493i	−0.808524	−	0.588464i	$-0.799733\pi$
$17$	7.38515	−	12.7915i	0.105362	−	0.182493i	0.913886	+	0.405970i	$0.133066\pi$
$18$		0			0
$19$	26.4892	+	45.8806i	0.319844	+	0.553986i	0.980455	−	0.196742i	$-0.0630362\pi$
$19$	26.4892	+	45.8806i	0.319844	+	0.553986i	−0.660611	+	0.750728i	$0.729703\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−56.7287	−	98.2569i	−0.514293	−	0.890782i	−0.999862	−	0.0165836i	$-0.994721\pi$
$23$	−56.7287	−	98.2569i	−0.514293	−	0.890782i	0.485569	−	0.874198i	$-0.338612\pi$
$24$		0			0
$25$	−107.112	+	185.524i	−0.856897	+	1.48419i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	119.708			0.766525			0.383262	−	0.923639i	$-0.374800\pi$
$29$	119.708			0.766525			0.383262	+	0.923639i	$0.374800\pi$
$30$		0			0
$31$	−83.5340	+	144.685i	−0.483973	+	0.838265i	−0.999831	−	0.0184088i	$-0.994140\pi$
$31$	−83.5340	+	144.685i	−0.483973	+	0.838265i	0.515858	+	0.856674i	$0.327473\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	317.500	+	124.690i	1.53335	+	0.602185i
$36$		0			0
$37$	−39.6722	−	68.7143i	−0.176272	−	0.305312i	0.764329	−	0.644827i	$-0.223071\pi$
$37$	−39.6722	−	68.7143i	−0.176272	−	0.305312i	−0.940601	+	0.339515i	$0.889737\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−41.8167			−0.159285			−0.0796424	−	0.996824i	$-0.525378\pi$
$41$	−41.8167			−0.159285			−0.0796424	+	0.996824i	$0.525378\pi$
$42$		0			0
$43$	190.219			0.674609			0.337304	−	0.941396i	$-0.390485\pi$
$43$	190.219			0.674609			0.337304	+	0.941396i	$0.390485\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	231.991	+	401.821i	0.719987	+	1.24705i	0.961004	+	0.276533i	$0.0891856\pi$
$47$	231.991	+	401.821i	0.719987	+	1.24705i	−0.241017	+	0.970521i	$0.577481\pi$
$48$		0			0
$49$	76.4715	−	334.367i	0.222949	−	0.974830i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.74684	+	4.75766i	−0.00711901	+	0.0123305i	−0.869563	−	0.493822i	$-0.835599\pi$
$53$	−2.74684	+	4.75766i	−0.00711901	+	0.0123305i	0.862444	+	0.506153i	$0.168933\pi$
$54$		0			0
$55$	479.036			1.17442
$56$		0			0
$57$		0			0
$58$		0			0
$59$	334.488	−	579.351i	0.738079	−	1.27839i	−0.215280	−	0.976552i	$-0.569066\pi$
$59$	334.488	−	579.351i	0.738079	−	1.27839i	0.953359	−	0.301838i	$-0.0976003\pi$
$60$		0			0
$61$	−469.420	−	813.059i	−0.985296	−	1.70658i	−0.640613	−	0.767864i	$-0.721320\pi$
$61$	−469.420	−	813.059i	−0.985296	−	1.70658i	−0.344683	−	0.938719i	$-0.612014\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	357.745	+	619.633i	0.682659	+	1.18240i
$66$		0			0
$67$	−138.918	+	240.613i	−0.253306	+	0.438740i	−0.964434	−	0.264323i	$-0.914851\pi$
$67$	−138.918	+	240.613i	−0.253306	+	0.438740i	0.711128	+	0.703063i	$0.248185\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	357.478			0.597532			0.298766	−	0.954326i	$-0.403425\pi$
$71$	357.478			0.597532			0.298766	+	0.954326i	$0.403425\pi$
$72$		0			0
$73$	508.704	−	881.101i	0.815607	−	1.41267i	−0.0932841	−	0.995640i	$-0.529736\pi$
$73$	508.704	−	881.101i	0.815607	−	1.41267i	0.908891	−	0.417033i	$-0.136930\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−71.6883	−	476.330i	−0.106099	−	0.704972i
$78$		0			0
$79$	382.659	+	662.784i	0.544968	+	0.943912i	0.998609	+	0.0527278i	$0.0167916\pi$
$79$	382.659	+	662.784i	0.544968	+	0.943912i	−0.453641	+	0.891185i	$0.649875\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1421.16			1.87943			0.939715	−	0.341958i	$-0.111090\pi$
$83$	1421.16			1.87943			0.939715	+	0.341958i	$0.111090\pi$
$84$		0			0
$85$	−272.040			−0.347140
$86$		0			0
$87$		0			0
$88$		0			0
$89$	385.995	+	668.564i	0.459724	+	0.796265i	0.998946	−	0.0458982i	$-0.0146150\pi$
$89$	385.995	+	668.564i	0.459724	+	0.796265i	−0.539222	+	0.842164i	$0.681282\pi$
$90$		0			0
$91$	562.596	−	448.453i	0.648089	−	0.516601i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	487.879	−	845.030i	0.526898	−	0.912614i
$96$		0			0
$97$	575.835			0.602755			0.301377	−	0.953505i	$-0.402554\pi$
$97$	575.835			0.602755			0.301377	+	0.953505i	$0.402554\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−65.4556	+	113.372i	−0.0644858	+	0.111693i	−0.896466	−	0.443113i	$-0.853874\pi$
$101$	−65.4556	+	113.372i	−0.0644858	+	0.111693i	0.831980	+	0.554806i	$0.187207\pi$
$102$		0			0
$103$	−216.365	−	374.756i	−0.206982	−	0.358503i	0.743781	−	0.668424i	$-0.233031\pi$
$103$	−216.365	−	374.756i	−0.206982	−	0.358503i	−0.950762	+	0.309921i	$0.899697\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−324.252	−	561.622i	−0.292959	−	0.507421i	0.681549	−	0.731773i	$-0.261307\pi$
$107$	−324.252	−	561.622i	−0.292959	−	0.507421i	−0.974508	+	0.224352i	$0.927973\pi$
$108$		0			0
$109$	−799.218	+	1384.29i	−0.702304	+	1.21643i	0.265351	+	0.964152i	$0.414512\pi$
$109$	−799.218	+	1384.29i	−0.702304	+	1.21643i	−0.967656	+	0.252275i	$0.918821\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	578.851			0.481891			0.240946	−	0.970539i	$-0.422542\pi$
$113$	578.851			0.481891			0.240946	+	0.970539i	$0.422542\pi$
$114$		0			0
$115$	−1044.83	+	1809.70i	−0.847226	+	1.46744i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	40.7111	+	270.503i	0.0313612	+	0.208378i
$120$		0			0
$121$	327.264	+	566.838i	0.245878	+	0.425874i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1643.33			1.17587
$126$		0			0
$127$	−1975.83			−1.38052			−0.690261	−	0.723560i	$-0.742504\pi$
$127$	−1975.83			−1.38052			−0.690261	+	0.723560i	$0.742504\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	760.267	+	1316.82i	0.507060	+	0.878254i	0.999967	+	0.00817157i	$0.00260112\pi$
$131$	760.267	+	1316.82i	0.507060	+	0.878254i	−0.492907	+	0.870082i	$0.664066\pi$
$132$		0			0
$133$	−913.269	−	358.663i	−0.595417	−	0.233835i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	26.4749	−	45.8559i	0.0165102	−	0.0285966i	−0.857652	−	0.514230i	$-0.828078\pi$
$137$	26.4749	−	45.8559i	0.0165102	−	0.0285966i	0.874162	+	0.485634i	$0.161411\pi$
$138$		0			0
$139$	2874.08			1.75378			0.876892	−	0.480687i	$-0.159613\pi$
$139$	2874.08			1.75378			0.876892	+	0.480687i	$0.159613\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	505.191	−	875.016i	0.295428	−	0.511696i
$144$		0			0
$145$	−1102.39	−	1909.40i	−0.631371	−	1.09357i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1173.48	+	2032.53i	0.645203	+	1.11752i	0.984255	+	0.176757i	$0.0565607\pi$
$149$	1173.48	+	2032.53i	0.645203	+	1.11752i	−0.339051	+	0.940768i	$0.610106\pi$
$150$		0			0
$151$	−1674.37	+	2900.10i	−0.902375	+	1.56296i	−0.0779717	+	0.996956i	$0.524844\pi$
$151$	−1674.37	+	2900.10i	−0.902375	+	1.56296i	−0.824403	+	0.566003i	$0.808489\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3077.07			1.59455
$156$		0			0
$157$	−260.731	+	451.600i	−0.132539	+	0.229564i	−0.924655	−	0.380807i	$-0.875646\pi$
$157$	−260.731	+	451.600i	−0.132539	+	0.229564i	0.792116	+	0.610371i	$0.208980\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1955.84	+	768.105i	0.957401	+	0.375995i
$162$		0			0
$163$	473.335	+	819.840i	0.227451	+	0.393956i	0.957052	−	0.289917i	$-0.0936277\pi$
$163$	473.335	+	819.840i	0.227451	+	0.393956i	−0.729601	+	0.683873i	$0.760294\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2668.53			1.23651			0.618255	−	0.785978i	$-0.287840\pi$
$167$	2668.53			1.23651			0.618255	+	0.785978i	$0.287840\pi$
$168$		0			0
$169$	−687.889			−0.313104
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1488.14	−	2577.53i	−0.653994	−	1.13275i	−0.982145	−	0.188126i	$-0.939759\pi$
$173$	−1488.14	−	2577.53i	−0.653994	−	1.13275i	0.328151	−	0.944625i	$-0.393575\pi$
$174$		0			0
$175$	−590.462	−	3923.30i	−0.255056	−	1.69471i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1156.33	+	2002.82i	−0.482839	+	0.836302i	−0.999806	−	0.0197036i	$-0.993728\pi$
$179$	−1156.33	+	2002.82i	−0.482839	+	0.836302i	0.516967	+	0.856005i	$0.327061\pi$
$180$		0			0
$181$	4597.15			1.88787			0.943933	−	0.330137i	$-0.107095\pi$
$181$	4597.15			1.88787			0.943933	+	0.330137i	$0.107095\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−730.684	+	1265.58i	−0.290384	+	0.502959i
$186$		0			0
$187$	192.081	+	332.694i	0.0751141	+	0.130102i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1248.82	−	2163.02i	−0.473097	−	0.819428i	0.526429	−	0.850219i	$-0.323531\pi$
$191$	−1248.82	−	2163.02i	−0.473097	−	0.819428i	−0.999526	+	0.0307913i	$0.990197\pi$
$192$		0			0
$193$	−428.481	+	742.151i	−0.159807	+	0.276794i	−0.934799	−	0.355177i	$-0.884421\pi$
$193$	−428.481	+	742.151i	−0.159807	+	0.276794i	0.774992	+	0.631971i	$0.217754\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1870.96			0.676652			0.338326	−	0.941029i	$-0.390139\pi$
$197$	1870.96			0.676652			0.338326	+	0.941029i	$0.390139\pi$
$198$		0			0
$199$	703.294	−	1218.14i	0.250528	−	0.433928i	−0.713143	−	0.701019i	$-0.752729\pi$
$199$	703.294	−	1218.14i	0.250528	−	0.433928i	0.963671	+	0.267091i	$0.0860623\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1733.64	+	1381.91i	−0.599398	+	0.477789i
$204$		0			0
$205$	385.091	+	666.997i	0.131200	+	0.227244i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1377.92			−0.456041
$210$		0			0
$211$	1251.20			0.408229			0.204115	−	0.978947i	$-0.434568\pi$
$211$	1251.20			0.408229			0.204115	+	0.978947i	$0.434568\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1751.73	−	3034.09i	−0.555661	−	0.962434i
$216$		0			0
$217$	−460.486	−	3059.69i	−0.144055	−	0.957166i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−286.893	+	496.913i	−0.0873236	+	0.151249i
$222$		0			0
$223$	−466.516			−0.140091			−0.0700454	−	0.997544i	$-0.522314\pi$
$223$	−466.516			−0.140091			−0.0700454	+	0.997544i	$0.522314\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	857.742	−	1485.65i	0.250794	−	0.434389i	−0.712950	−	0.701215i	$-0.752642\pi$
$227$	857.742	−	1485.65i	0.250794	−	0.434389i	0.963745	+	0.266826i	$0.0859748\pi$
$228$		0			0
$229$	257.353	+	445.749i	0.0742637	+	0.128629i	0.900766	−	0.434305i	$-0.143006\pi$
$229$	257.353	+	445.749i	0.0742637	+	0.128629i	−0.826502	+	0.562934i	$0.809673\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1118.55	−	1937.38i	−0.314499	−	0.544729i	0.664832	−	0.746993i	$-0.268503\pi$
$233$	−1118.55	−	1937.38i	−0.314499	−	0.544729i	−0.979331	+	0.202264i	$0.935170\pi$
$234$		0			0
$235$	4272.82	−	7400.75i	1.18608	−	2.05435i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1874.12			−0.507226			−0.253613	−	0.967306i	$-0.581619\pi$
$239$	−1874.12			−0.507226			−0.253613	+	0.967306i	$0.581619\pi$
$240$		0			0
$241$	−1146.52	+	1985.83i	−0.306447	+	0.530782i	−0.977582	−	0.210553i	$-0.932474\pi$
$241$	−1146.52	+	1985.83i	−0.306447	+	0.530782i	0.671135	+	0.741335i	$0.265807\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6037.54	+	1859.43i	−1.57439	+	0.484876i
$246$		0			0
$247$	−1029.03	−	1782.34i	−0.265084	−	0.459139i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5461.66			−1.37345			−0.686727	−	0.726915i	$-0.740953\pi$
$251$	−5461.66			−1.37345			−0.686727	+	0.726915i	$0.740953\pi$
$252$		0			0
$253$	2950.92			0.733291
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3545.62	+	6141.19i	0.860582	+	1.49057i	0.871368	+	0.490630i	$0.163233\pi$
$257$	3545.62	+	6141.19i	0.860582	+	1.49057i	−0.0107861	+	0.999942i	$0.503433\pi$
$258$		0			0
$259$	1367.78	+	537.161i	0.328146	+	0.128871i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2664.26	−	4614.64i	0.624660	−	1.08194i	−0.363946	−	0.931420i	$-0.618571\pi$
$263$	2664.26	−	4614.64i	0.624660	−	1.08194i	0.988606	−	0.150523i	$-0.0480958\pi$
$264$		0			0
$265$	101.183			0.0234551
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2218.85	−	3843.17i	0.502922	−	0.871086i	−0.497073	−	0.867709i	$-0.665592\pi$
$269$	2218.85	−	3843.17i	0.502922	−	0.871086i	0.999994	−	0.00337684i	$-0.00107488\pi$
$270$		0			0
$271$	−2623.08	−	4543.31i	−0.587974	−	1.01840i	−0.994498	−	0.104760i	$-0.966593\pi$
$271$	−2623.08	−	4543.31i	−0.587974	−	1.01840i	0.406524	−	0.913640i	$-0.366741\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2785.89	−	4825.30i	−0.610892	−	1.05810i
$276$		0			0
$277$	−3988.95	+	6909.06i	−0.865245	+	1.49865i	0.00155937	+	0.999999i	$0.499504\pi$
$277$	−3988.95	+	6909.06i	−0.865245	+	1.49865i	−0.866804	+	0.498649i	$0.833830\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6427.68			−1.36457			−0.682283	−	0.731088i	$-0.739013\pi$
$281$	−6427.68			−1.36457			−0.682283	+	0.731088i	$0.739013\pi$
$282$		0			0
$283$	−1288.18	+	2231.19i	−0.270581	+	0.468660i	−0.969011	−	0.247019i	$-0.920549\pi$
$283$	−1288.18	+	2231.19i	−0.270581	+	0.468660i	0.698430	+	0.715679i	$0.253882\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	605.601	−	482.733i	0.124556	−	0.0992851i
$288$		0			0
$289$	2347.42	+	4065.85i	0.477798	+	0.827570i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3543.33			0.706498			0.353249	−	0.935529i	$-0.385077\pi$
$293$	3543.33			0.706498			0.353249	+	0.935529i	$0.385077\pi$
$294$		0			0
$295$	−12321.2			−2.43176
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2203.75	+	3817.01i	0.426242	+	0.738273i
$300$		0			0
$301$	−2754.80	+	2195.89i	−0.527523	+	0.420496i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8645.80	+	14975.0i	−1.62314	+	2.81136i
$306$		0			0
$307$	1346.45			0.250312			0.125156	−	0.992137i	$-0.460057\pi$
$307$	1346.45			0.250312			0.125156	+	0.992137i	$0.460057\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1992.03	+	3450.30i	−0.363208	+	0.629095i	−0.988487	−	0.151306i	$-0.951652\pi$
$311$	−1992.03	+	3450.30i	−0.363208	+	0.629095i	0.625279	+	0.780402i	$0.284985\pi$
$312$		0			0
$313$	5338.78	+	9247.04i	0.964108	+	1.66988i	0.711992	+	0.702187i	$0.247793\pi$
$313$	5338.78	+	9247.04i	0.964108	+	1.66988i	0.252116	+	0.967697i	$0.418874\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1626.62	−	2817.39i	−0.288203	−	0.499182i	0.685178	−	0.728376i	$-0.259724\pi$
$317$	−1626.62	−	2817.39i	−0.288203	−	0.499182i	−0.973381	+	0.229194i	$0.926391\pi$
$318$		0			0
$319$	−1556.75	+	2696.36i	−0.273232	+	0.473252i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	782.506			0.134798
$324$		0			0
$325$	4161.01	−	7207.09i	0.710189	−	1.23008i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7998.38	−	3141.16i	−1.34032	−	0.526376i
$330$		0			0
$331$	1834.08	+	3176.71i	0.304562	+	0.527516i	0.977164	−	0.212488i	$-0.0681567\pi$
$331$	1834.08	+	3176.71i	0.304562	+	0.527516i	−0.672602	+	0.740004i	$0.734823\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5117.19			0.834574
$336$		0			0
$337$	1661.63			0.268589			0.134295	−	0.990941i	$-0.457123\pi$
$337$	1661.63			0.268589			0.134295	+	0.990941i	$0.457123\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2172.64	−	3763.13i	−0.345030	−	0.597609i
$342$		0			0
$343$	2752.45	+	5725.17i	0.433290	+	0.901254i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5557.24	−	9625.41i	0.859735	−	1.48910i	−0.0124468	−	0.999923i	$-0.503962\pi$
$347$	5557.24	−	9625.41i	0.859735	−	1.48910i	0.872182	−	0.489182i	$-0.162705\pi$
$348$		0			0
$349$	−10559.9			−1.61965			−0.809824	−	0.586672i	$-0.800438\pi$
$349$	−10559.9			−1.61965			−0.809824	+	0.586672i	$0.800438\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5365.94	+	9294.09i	−0.809066	+	1.40134i	0.104445	+	0.994531i	$0.466693\pi$
$353$	−5365.94	+	9294.09i	−0.809066	+	1.40134i	−0.913511	+	0.406813i	$0.866640\pi$
$354$		0			0
$355$	−3292.02	−	5701.95i	−0.492175	−	0.852473i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4103.54	−	7107.53i	−0.603277	−	1.04491i	−0.992321	−	0.123687i	$-0.960528\pi$
$359$	−4103.54	−	7107.53i	−0.603277	−	1.04491i	0.389044	−	0.921219i	$-0.372805\pi$
$360$		0			0
$361$	2026.15	−	3509.39i	0.295400	−	0.511648i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−18738.7			−2.68720
$366$		0			0
$367$	−4358.37	+	7548.92i	−0.619905	+	1.07371i	0.369598	+	0.929192i	$0.379496\pi$
$367$	−4358.37	+	7548.92i	−0.619905	+	1.07371i	−0.989503	+	0.144515i	$0.953838\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−15.1421	−	100.611i	−0.00211897	−	0.0140794i
$372$		0			0
$373$	−323.277	−	559.933i	−0.0448758	−	0.0777272i	0.842715	−	0.538360i	$-0.180956\pi$
$373$	−323.277	−	559.933i	−0.0448758	−	0.0777272i	−0.887591	+	0.460633i	$0.847623\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4650.33			−0.635290
$378$		0			0
$379$	11061.9			1.49924			0.749620	−	0.661869i	$-0.230236\pi$
$379$	11061.9			1.49924			0.749620	+	0.661869i	$0.230236\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2004.98	−	3472.73i	−0.267493	−	0.463311i	0.700721	−	0.713436i	$-0.252862\pi$
$383$	−2004.98	−	3472.73i	−0.267493	−	0.463311i	−0.968214	+	0.250124i	$0.919529\pi$
$384$		0			0
$385$	−6937.52	+	5530.00i	−0.918361	+	0.732038i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7016.17	+	12152.4i	−0.914484	+	1.58393i	−0.106828	+	0.994277i	$0.534070\pi$
$389$	−7016.17	+	12152.4i	−0.914484	+	1.58393i	−0.807655	+	0.589655i	$0.799264\pi$
$390$		0			0
$391$	−1675.80			−0.216749
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7047.82	−	12207.2i	0.897758	−	1.55496i
$396$		0			0
$397$	1309.44	+	2268.02i	0.165539	+	0.286722i	0.936847	−	0.349741i	$-0.113730\pi$
$397$	1309.44	+	2268.02i	0.165539	+	0.286722i	−0.771308	+	0.636463i	$0.780397\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7325.29	−	12687.8i	−0.912239	−	1.58004i	−0.810895	−	0.585192i	$-0.801019\pi$
$401$	−7325.29	−	12687.8i	−0.912239	−	1.58004i	−0.101344	−	0.994851i	$-0.532314\pi$
$402$		0			0
$403$	3245.07	−	5620.63i	0.401113	−	0.694748i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2063.67			0.251333
$408$		0			0
$409$	7107.94	−	12311.3i	0.859328	−	1.48840i	−0.0132425	−	0.999912i	$-0.504215\pi$
$409$	7107.94	−	12311.3i	0.859328	−	1.48840i	0.872571	−	0.488488i	$-0.162451\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1843.89	+	12251.6i	0.219689	+	1.45972i
$414$		0			0
$415$	−13087.5	−	22668.2i	−1.54805	−	2.68130i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	338.020			0.0394114			0.0197057	−	0.999806i	$-0.493727\pi$
$419$	338.020			0.0394114			0.0197057	+	0.999806i	$0.493727\pi$
$420$		0			0
$421$	10292.1			1.19146			0.595729	−	0.803185i	$-0.296863\pi$
$421$	10292.1			1.19146			0.595729	+	0.803185i	$0.296863\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1582.08	+	2740.24i	0.180569	+	0.312756i
$426$		0			0
$427$	16184.2	+	6355.94i	1.83421	+	0.720341i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8352.98	+	14467.8i	−0.933524	+	1.61691i	−0.156279	+	0.987713i	$0.549950\pi$
$431$	−8352.98	+	14467.8i	−0.933524	+	1.61691i	−0.777245	+	0.629198i	$0.783384\pi$
$432$		0			0
$433$	−7821.01			−0.868022			−0.434011	−	0.900908i	$-0.642902\pi$
$433$	−7821.01			−0.868022			−0.434011	+	0.900908i	$0.642902\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3005.39	−	5205.49i	0.328987	−	0.569822i
$438$		0			0
$439$	3410.80	+	5907.67i	0.370817	+	0.642273i	0.989691	−	0.143216i	$-0.0457445\pi$
$439$	3410.80	+	5907.67i	0.370817	+	0.642273i	−0.618875	+	0.785490i	$0.712411\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1887.84	−	3269.84i	−0.202470	−	0.350688i	0.746854	−	0.664988i	$-0.231563\pi$
$443$	−1887.84	−	3269.84i	−0.202470	−	0.350688i	−0.949324	+	0.314300i	$0.898230\pi$
$444$		0			0
$445$	7109.28	−	12313.6i	0.757331	−	1.31174i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12178.7			1.28006			0.640029	−	0.768351i	$-0.278922\pi$
$449$	12178.7			1.28006			0.640029	+	0.768351i	$0.278922\pi$
$450$		0			0
$451$	543.807	−	941.901i	0.0567780	−	0.0983424i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12334.0	−	4843.87i	−1.27083	−	0.499086i
$456$		0			0
$457$	−2714.69	−	4701.98i	−0.277873	−	0.481289i	0.692983	−	0.720954i	$-0.256296\pi$
$457$	−2714.69	−	4701.98i	−0.277873	−	0.481289i	−0.970856	+	0.239664i	$0.922963\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2455.46			0.248074			0.124037	−	0.992278i	$-0.460416\pi$
$461$	2455.46			0.248074			0.124037	+	0.992278i	$0.460416\pi$
$462$		0			0
$463$	1868.81			0.187583			0.0937916	−	0.995592i	$-0.470101\pi$
$463$	1868.81			0.187583			0.0937916	+	0.995592i	$0.470101\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4642.12	−	8040.39i	−0.459982	−	0.796712i	0.538977	−	0.842320i	$-0.318811\pi$
$467$	−4642.12	−	8040.39i	−0.459982	−	0.796712i	−0.998959	+	0.0456078i	$0.985478\pi$
$468$		0			0
$469$	−765.794	−	5088.29i	−0.0753967	−	0.500971i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2473.71	+	4284.60i	−0.240468	+	0.416503i
$474$		0			0
$475$	−11349.2			−1.09629
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8485.77	−	14697.8i	0.809446	−	1.40200i	−0.103802	−	0.994598i	$-0.533101\pi$
$479$	8485.77	−	14697.8i	0.809446	−	1.40200i	0.913248	−	0.407404i	$-0.133566\pi$
$480$		0			0
$481$	1541.16	+	2669.36i	0.146093	+	0.253041i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5302.88	−	9184.86i	−0.496477	−	0.859923i
$486$		0			0
$487$	3549.67	−	6148.20i	0.330289	−	0.572077i	−0.652279	−	0.757979i	$-0.726187\pi$
$487$	3549.67	−	6148.20i	0.330289	−	0.572077i	0.982568	+	0.185901i	$0.0595205\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7030.40			0.646187			0.323093	−	0.946367i	$-0.395277\pi$
$491$	7030.40			0.646187			0.323093	+	0.946367i	$0.395277\pi$
$492$		0			0
$493$	884.062	−	1531.24i	0.0807630	−	0.139886i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5177.08	+	4126.73i	−0.467251	+	0.372453i
$498$		0			0
$499$	3274.41	+	5671.45i	0.293753	+	0.508796i	0.974694	−	0.223543i	$-0.0717622\pi$
$499$	3274.41	+	5671.45i	0.293753	+	0.508796i	−0.680941	+	0.732338i	$0.738429\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9867.01			0.874648			0.437324	−	0.899304i	$-0.355926\pi$
$503$	9867.01			0.874648			0.437324	+	0.899304i	$0.355926\pi$
$504$		0			0
$505$	2411.13			0.212463
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10051.7	+	17410.1i	0.875314	+	1.51609i	0.856428	+	0.516266i	$0.172679\pi$
$509$	10051.7	+	17410.1i	0.875314	+	1.51609i	0.0188856	+	0.999822i	$0.493988\pi$
$510$		0			0
$511$	2804.26	+	18632.8i	0.242766	+	1.61305i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3985.03	+	6902.27i	−0.340973	+	0.590583i
$516$		0			0
$517$	−12067.7			−1.02657
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7163.21	+	12407.0i	−0.602353	+	1.04331i	0.390111	+	0.920768i	$0.372437\pi$
$521$	−7163.21	+	12407.0i	−0.602353	+	1.04331i	−0.992464	+	0.122538i	$0.960897\pi$
$522$		0			0
$523$	7730.53	+	13389.7i	0.646334	+	1.11948i	0.983992	+	0.178214i	$0.0570320\pi$
$523$	7730.53	+	13389.7i	0.646334	+	1.11948i	−0.337658	+	0.941269i	$0.609635\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1233.82	+	2137.04i	0.101985	+	0.176643i
$528$		0			0
$529$	−352.780	+	611.032i	−0.0289948	+	0.0502204i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1624.47			0.132014
$534$		0			0
$535$	−5972.09	+	10344.0i	−0.482610	+	0.835904i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6536.97	+	6070.77i	0.522388	+	0.485133i
$540$		0			0
$541$	4501.18	+	7796.27i	0.357709	+	0.619571i	0.987578	−	0.157131i	$-0.0502245\pi$
$541$	4501.18	+	7796.27i	0.357709	+	0.619571i	−0.629868	+	0.776702i	$0.716891\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	29440.0			2.31390
$546$		0			0
$547$	6721.75			0.525413			0.262707	−	0.964876i	$-0.415385\pi$
$547$	6721.75			0.525413			0.262707	+	0.964876i	$0.415385\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3170.97	+	5492.27i	0.245168	+	0.424644i
$552$		0			0
$553$	−13193.0	−	5181.20i	−1.01451	−	0.398421i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1340.99	−	2322.66i	0.102010	−	0.176686i	−0.810503	−	0.585735i	$-0.800806\pi$
$557$	1340.99	−	2322.66i	0.102010	−	0.176686i	0.912513	+	0.409049i	$0.134139\pi$
$558$		0			0
$559$	−7389.50			−0.559110
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−108.719	+	188.307i	−0.00813849	+	0.0140963i	−0.870066	−	0.492935i	$-0.835924\pi$
$563$	−108.719	+	188.307i	−0.00813849	+	0.0140963i	0.861927	+	0.507032i	$0.169257\pi$
$564$		0			0
$565$	−5330.65	−	9232.96i	−0.396924	−	0.687493i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1269.67	−	2199.14i	−0.0935458	−	0.162026i	0.815455	−	0.578821i	$-0.196487\pi$
$569$	−1269.67	−	2199.14i	−0.0935458	−	0.162026i	−0.909001	+	0.416794i	$0.863153\pi$
$570$		0			0
$571$	−4859.12	+	8416.25i	−0.356126	+	0.616828i	−0.987310	−	0.158805i	$-0.949236\pi$
$571$	−4859.12	+	8416.25i	−0.356126	+	0.616828i	0.631184	+	0.775633i	$0.282569\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24305.3			1.76278
$576$		0			0
$577$	802.457	−	1389.90i	0.0578973	−	0.100281i	−0.835624	−	0.549302i	$-0.814894\pi$
$577$	802.457	−	1389.90i	0.0578973	−	0.100281i	0.893521	+	0.449021i	$0.148227\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−20581.6	+	16405.9i	−1.46966	+	1.17148i
$582$		0			0
$583$	−71.4427	−	123.742i	−0.00507522	−	0.00879054i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21537.1			1.51436			0.757182	−	0.653204i	$-0.226575\pi$
$587$	21537.1			1.51436			0.757182	+	0.653204i	$0.226575\pi$
$588$		0			0
$589$	−8850.99			−0.619183
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5306.74	−	9191.54i	−0.367490	−	0.636512i	0.621682	−	0.783269i	$-0.286449\pi$
$593$	−5306.74	−	9191.54i	−0.367490	−	0.636512i	−0.989172	+	0.146758i	$0.953116\pi$
$594$		0			0
$595$	3939.75	−	3140.43i	0.271452	−	0.216378i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11075.8	−	19183.8i	0.755500	−	1.30856i	−0.189626	−	0.981856i	$-0.560727\pi$
$599$	11075.8	−	19183.8i	0.755500	−	1.30856i	0.945125	−	0.326708i	$-0.105939\pi$
$600$		0			0
$601$	−19358.2			−1.31387			−0.656935	−	0.753947i	$-0.728147\pi$
$601$	−19358.2			−1.31387			−0.656935	+	0.753947i	$0.728147\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6027.57	−	10440.0i	0.405050	−	0.701568i
$606$		0			0
$607$	3572.92	+	6188.48i	0.238913	+	0.413810i	0.960403	−	0.278616i	$-0.0898755\pi$
$607$	3572.92	+	6188.48i	0.238913	+	0.413810i	−0.721489	+	0.692425i	$0.756542\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9012.23	−	15609.6i	−0.596720	−	1.03355i
$612$		0			0
$613$	6.03941	−	10.4606i	0.000397927	−	0.000689230i	−0.865826	−	0.500345i	$-0.833207\pi$
$613$	6.03941	−	10.4606i	0.000397927	−	0.000689230i	0.866224	+	0.499655i	$0.166540\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12639.1			0.824689			0.412344	−	0.911028i	$-0.364710\pi$
$617$	12639.1			0.824689			0.412344	+	0.911028i	$0.364710\pi$
$618$		0			0
$619$	−8716.62	+	15097.6i	−0.565994	+	0.980331i	0.430962	+	0.902370i	$0.358174\pi$
$619$	−8716.62	+	15097.6i	−0.565994	+	0.980331i	−0.996956	+	0.0779607i	$0.975159\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−13308.0	−	5226.38i	−0.855816	−	0.336100i
$624$		0			0
$625$	−1744.49	−	3021.54i	−0.111647	−	0.193379i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1171.94			−0.0742899
$630$		0			0
$631$	−9418.42			−0.594202			−0.297101	−	0.954846i	$-0.596020\pi$
$631$	−9418.42			−0.594202			−0.297101	+	0.954846i	$0.596020\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18195.4	+	31515.4i	1.13711	+	1.96953i
$636$		0			0
$637$	−2970.71	+	12989.2i	−0.184778	+	0.807931i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12314.8	−	21329.8i	0.758820	−	1.31431i	−0.184633	−	0.982808i	$-0.559110\pi$
$641$	12314.8	−	21329.8i	0.758820	−	1.31431i	0.943453	−	0.331507i	$-0.107557\pi$
$642$		0			0
$643$	27161.2			1.66583			0.832917	−	0.553398i	$-0.186669\pi$
$643$	27161.2			1.66583			0.832917	+	0.553398i	$0.186669\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8024.87	−	13899.5i	0.487620	−	0.844583i	−0.512278	−	0.858820i	$-0.671198\pi$
$647$	8024.87	−	13899.5i	0.487620	−	0.844583i	0.999899	+	0.0142363i	$0.00453169\pi$
$648$		0			0
$649$	8699.73	+	15068.4i	0.526185	+	0.911380i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6580.18	+	11397.2i	0.394337	+	0.683012i	0.993016	−	0.117977i	$-0.0376408\pi$
$653$	6580.18	+	11397.2i	0.394337	+	0.683012i	−0.598679	+	0.800989i	$0.704308\pi$
$654$		0			0
$655$	14002.6	−	24253.3i	0.835310	−	1.44680i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4473.74			−0.264449			−0.132225	−	0.991220i	$-0.542212\pi$
$659$	−4473.74			−0.264449			−0.132225	+	0.991220i	$0.542212\pi$
$660$		0			0
$661$	11220.0	−	19433.5i	0.660220	−	1.14353i	−0.320337	−	0.947304i	$-0.603796\pi$
$661$	11220.0	−	19433.5i	0.660220	−	1.14353i	0.980558	−	0.196231i	$-0.0628704\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2689.46	+	17870.0i	0.156831	+	1.04206i
$666$		0			0
$667$	−6790.87	−	11762.1i	−0.394218	−	0.682806i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	24418.4			1.40486
$672$		0			0
$673$	−4623.02			−0.264791			−0.132396	−	0.991197i	$-0.542267\pi$
$673$	−4623.02			−0.264791			−0.132396	+	0.991197i	$0.542267\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8282.30	+	14345.4i	0.470184	+	0.814383i	0.999419	−	0.0340926i	$-0.0108541\pi$
$677$	8282.30	+	14345.4i	0.470184	+	0.814383i	−0.529234	+	0.848476i	$0.677521\pi$
$678$		0			0
$679$	−8339.39	+	6647.45i	−0.471335	+	0.375708i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14894.4	+	25797.9i	−0.834434	+	1.44528i	0.0600558	+	0.998195i	$0.480872\pi$
$683$	−14894.4	+	25797.9i	−0.834434	+	1.44528i	−0.894490	+	0.447088i	$0.852461\pi$
$684$		0			0
$685$	−975.231			−0.0543966
$686$		0			0
$687$		0			0
$688$		0			0
$689$	106.707	−	184.822i	0.00590017	−	0.0102194i
$690$		0			0
$691$	969.425	+	1679.09i	0.0533700	+	0.0924395i	0.891476	−	0.453068i	$-0.149670\pi$
$691$	969.425	+	1679.09i	0.0533700	+	0.0924395i	−0.838106	+	0.545507i	$0.816337\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−26467.4	−	45842.9i	−1.44456	−	2.50205i
$696$		0			0
$697$	−308.823	+	534.897i	−0.0167826	+	0.0290684i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8227.44			0.443290			0.221645	−	0.975127i	$-0.428857\pi$
$701$	8227.44			0.443290			0.221645	+	0.975127i	$0.428857\pi$
$702$		0			0
$703$	2101.77	−	3640.37i	0.112759	−	0.195305i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−360.828	−	2397.51i	−0.0191942	−	0.127535i
$708$		0			0
$709$	6258.32	+	10839.7i	0.331504	+	0.574182i	0.982807	−	0.184636i	$-0.0591106\pi$
$709$	6258.32	+	10839.7i	0.331504	+	0.574182i	−0.651303	+	0.758818i	$0.725777\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18955.1			0.995615
$714$		0			0
$715$	−18609.2			−0.973351
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6764.68	−	11716.8i	−0.350876	−	0.607735i	0.635527	−	0.772079i	$-0.280783\pi$
$719$	−6764.68	−	11716.8i	−0.350876	−	0.607735i	−0.986403	+	0.164343i	$0.947449\pi$
$720$		0			0
$721$	7459.65	+	2929.59i	0.385314	+	0.151322i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12822.2	+	22208.7i	−0.656833	+	1.13767i
$726$		0			0
$727$	30114.9			1.53631			0.768157	−	0.640261i	$-0.221174\pi$
$727$	30114.9			1.53631			0.768157	+	0.640261i	$0.221174\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1404.80	−	2433.18i	0.0710784	−	0.123111i
$732$		0			0
$733$	−6608.62	−	11446.5i	−0.333008	−	0.576787i	0.650092	−	0.759856i	$-0.274730\pi$
$733$	−6608.62	−	11446.5i	−0.333008	−	0.576787i	−0.983100	+	0.183068i	$0.941397\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3613.13	−	6258.12i	−0.180585	−	0.312783i
$738$		0			0
$739$	4637.30	−	8032.05i	0.230834	−	0.399815i	−0.727220	−	0.686404i	$-0.759188\pi$
$739$	4637.30	−	8032.05i	0.230834	−	0.399815i	0.958054	+	0.286589i	$0.0925214\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8881.33			0.438526			0.219263	−	0.975666i	$-0.429635\pi$
$743$	8881.33			0.438526			0.219263	+	0.975666i	$0.429635\pi$
$744$		0			0
$745$	21613.2	−	37435.2i	1.06288	−	1.84097i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11179.3	+	4390.37i	0.545370	+	0.214180i
$750$		0			0
$751$	726.002	+	1257.47i	0.0352759	+	0.0610996i	0.883124	−	0.469139i	$-0.155436\pi$
$751$	726.002	+	1257.47i	0.0352759	+	0.0610996i	−0.847848	+	0.530239i	$0.822102\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	61677.4			2.97307
$756$		0			0
$757$	25800.7			1.23876			0.619382	−	0.785090i	$-0.287383\pi$
$757$	25800.7			1.23876			0.619382	+	0.785090i	$0.287383\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	638.768	+	1106.38i	0.0304275	+	0.0527019i	0.880838	−	0.473418i	$-0.156980\pi$
$761$	638.768	+	1106.38i	0.0304275	+	0.0527019i	−0.850411	+	0.526119i	$0.823646\pi$
$762$		0			0
$763$	−4405.74	−	29273.8i	−0.209041	−	1.38897i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12994.0	+	22506.2i	−0.611714	+	1.05952i
$768$		0			0
$769$	14389.0			0.674746			0.337373	−	0.941371i	$-0.390462\pi$
$769$	14389.0			0.674746			0.337373	+	0.941371i	$0.390462\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16465.4	−	28518.8i	0.766129	−	1.32698i	−0.173518	−	0.984831i	$-0.555513\pi$
$773$	16465.4	−	28518.8i	0.766129	−	1.32698i	0.939647	−	0.342144i	$-0.111153\pi$
$774$		0			0
$775$	−17895.0	−	30995.1i	−0.829429	−	1.43661i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1107.69	−	1918.58i	−0.0509462	−	0.0882415i
$780$		0			0
$781$	−4648.83	+	8052.01i	−0.212994	+	0.368916i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9604.32			0.436679
$786$		0			0
$787$	−14446.7	+	25022.5i	−0.654346	+	1.13336i	0.327711	+	0.944778i	$0.393723\pi$
$787$	−14446.7	+	25022.5i	−0.654346	+	1.13336i	−0.982057	+	0.188583i	$0.939611\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8383.07	+	6682.26i	−0.376824	+	0.300372i
$792$		0			0
$793$	18235.7	+	31585.1i	0.816606	+	1.41440i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7033.32			0.312588			0.156294	−	0.987711i	$-0.450045\pi$
$797$	7033.32			0.312588			0.156294	+	0.987711i	$0.450045\pi$
$798$		0			0
$799$	6853.16			0.303438
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13230.9	+	22916.6i	0.581456	+	1.00711i
$804$		0			0
$805$	−5759.69	−	38270.1i	−0.252177	−	1.67558i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16579.7	+	28716.8i	−0.720531	+	1.24800i	0.240256	+	0.970710i	$0.422769\pi$
$809$	−16579.7	+	28716.8i	−0.720531	+	1.24800i	−0.960787	+	0.277287i	$0.910565\pi$
$810$		0			0
$811$	3646.25			0.157876			0.0789379	−	0.996880i	$-0.474847\pi$
$811$	3646.25			0.157876			0.0789379	+	0.996880i	$0.474847\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8717.90	−	15099.8i	0.374693	−	0.648987i
$816$		0			0
$817$	5038.75	+	8727.37i	0.215769	+	0.373723i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9933.28	+	17204.9i	0.422258	+	0.731372i	0.996160	−	0.0875518i	$-0.0279043\pi$
$821$	9933.28	+	17204.9i	0.422258	+	0.731372i	−0.573902	+	0.818924i	$0.694571\pi$
$822$		0			0
$823$	−12888.2	+	22323.0i	−0.545873	+	0.945480i	0.452678	+	0.891674i	$0.350469\pi$
$823$	−12888.2	+	22323.0i	−0.545873	+	0.945480i	−0.998551	+	0.0538060i	$0.982865\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33812.2			1.42172			0.710862	−	0.703331i	$-0.248305\pi$
$827$	33812.2			1.42172			0.710862	+	0.703331i	$0.248305\pi$
$828$		0			0
$829$	−21072.4	+	36498.5i	−0.882841	+	1.52913i	−0.0346729	+	0.999399i	$0.511039\pi$
$829$	−21072.4	+	36498.5i	−0.882841	+	1.52913i	−0.848168	+	0.529727i	$0.822294\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3712.28	−	3447.53i	−0.154409	−	0.143397i
$834$		0			0
$835$	−24574.6	−	42564.4i	−1.01849	−	1.76407i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3779.26			−0.155512			−0.0777559	−	0.996972i	$-0.524775\pi$
$839$	−3779.26			−0.155512			−0.0777559	+	0.996972i	$0.524775\pi$
$840$		0			0
$841$	−10059.0			−0.412440
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6334.78	+	10972.2i	0.257897	+	0.446691i
$846$		0			0
$847$	−11283.1	−	4431.15i	−0.457724	−	0.179759i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4501.10	+	7796.14i	−0.181311	+	0.314040i
$852$		0			0
$853$	8513.08			0.341714			0.170857	−	0.985296i	$-0.445346\pi$
$853$	8513.08			0.341714			0.170857	+	0.985296i	$0.445346\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15026.2	−	26026.2i	0.598933	−	1.03738i	−0.394046	−	0.919091i	$-0.628925\pi$
$857$	15026.2	−	26026.2i	0.598933	−	1.03738i	0.992979	−	0.118291i	$-0.0377417\pi$
$858$		0			0
$859$	−9093.32	−	15750.1i	−0.361188	−	0.625595i	0.626969	−	0.779044i	$-0.284295\pi$
$859$	−9093.32	−	15750.1i	−0.361188	−	0.625595i	−0.988157	+	0.153449i	$0.950962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7276.88	−	12603.9i	−0.287031	−	0.497152i	0.686069	−	0.727537i	$-0.259335\pi$
$863$	−7276.88	−	12603.9i	−0.287031	−	0.497152i	−0.973100	+	0.230384i	$0.926002\pi$
$864$		0			0
$865$	−27408.6	+	47473.0i	−1.07736	+	1.86605i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−19905.2			−0.777028
$870$		0			0
$871$	5396.59	−	9347.16i	0.209938	−	0.363624i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−23799.2	+	18970.7i	−0.919496	+	0.732944i
$876$		0			0
$877$	23862.6	+	41331.3i	0.918795	+	1.59140i	0.801247	+	0.598333i	$0.204170\pi$
$877$	23862.6	+	41331.3i	0.918795	+	1.59140i	0.117548	+	0.993067i	$0.462497\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27523.8			−1.05255			−0.526277	−	0.850313i	$-0.676413\pi$
$881$	−27523.8			−1.05255			−0.526277	+	0.850313i	$0.676413\pi$
$882$		0			0
$883$	−47523.2			−1.81119			−0.905596	−	0.424141i	$-0.860576\pi$
$883$	−47523.2			−1.81119			−0.905596	+	0.424141i	$0.860576\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14080.6	−	24388.3i	−0.533010	−	0.923201i	−0.999257	−	0.0385459i	$-0.987727\pi$
$887$	−14080.6	−	24388.3i	−0.533010	−	0.923201i	0.466247	−	0.884655i	$-0.345606\pi$
$888$		0			0
$889$	28614.4	−	22809.0i	1.07952	−	0.860504i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12290.5	+	21287.8i	−0.460567	+	0.797725i
$894$		0			0
$895$	42594.7			1.59082
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9999.69	+	17320.0i	−0.370977	+	0.642551i
$900$		0			0
$901$	40.5716	+	70.2721i	0.00150015	+	0.00259834i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−42335.3	−	73326.9i	−1.55500	−	2.69333i
$906$		0			0
$907$	−1652.12	+	2861.56i	−0.0604827	+	0.104759i	−0.894681	−	0.446705i	$-0.852597\pi$
$907$	−1652.12	+	2861.56i	−0.0604827	+	0.104759i	0.834199	+	0.551464i	$0.185931\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3368.71			0.122514			0.0612570	−	0.998122i	$-0.480489\pi$
$911$	3368.71			0.122514			0.0612570	+	0.998122i	$0.480489\pi$
$912$		0			0
$913$	−18481.5	+	32011.0i	−0.669934	+	1.16036i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−26211.8	−	10294.0i	−0.943936	−	0.370707i
$918$		0			0
$919$	3823.38	+	6622.29i	0.137238	+	0.237703i	0.926450	−	0.376417i	$-0.122844\pi$
$919$	3823.38	+	6622.29i	0.137238	+	0.237703i	−0.789212	+	0.614121i	$0.789511\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−13887.0			−0.495230
$924$		0			0
$925$	16997.5			0.604188
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4476.85	+	7754.13i	0.158106	+	0.273848i	0.934186	−	0.356787i	$-0.116128\pi$
$929$	4476.85	+	7754.13i	0.158106	+	0.273848i	−0.776080	+	0.630635i	$0.782794\pi$
$930$		0			0
$931$	17366.6	−	5348.54i	0.611351	−	0.188283i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3537.75	−	6127.57i	0.123740	−	0.214324i
$936$		0			0
$937$	−15549.6			−0.542138			−0.271069	−	0.962560i	$-0.587377\pi$
$937$	−15549.6			−0.542138			−0.271069	+	0.962560i	$0.587377\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3109.91	−	5386.51i	0.107736	−	0.186605i	−0.807116	−	0.590392i	$-0.798973\pi$
$941$	3109.91	−	5386.51i	0.107736	−	0.186605i	0.914853	+	0.403787i	$0.132306\pi$
$942$		0			0
$943$	2372.21	+	4108.78i	0.0819191	+	0.141888i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13131.1	−	22743.8i	−0.450586	−	0.780438i	0.547837	−	0.836585i	$-0.315452\pi$
$947$	−13131.1	−	22743.8i	−0.450586	−	0.780438i	−0.998422	+	0.0561478i	$0.982118\pi$
$948$		0			0
$949$	−19761.8	+	34228.4i	−0.675969	+	1.17081i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−55927.4			−1.90101			−0.950507	−	0.310703i	$-0.899436\pi$
$953$	−55927.4			−1.90101			−0.950507	+	0.310703i	$0.899436\pi$
$954$		0			0
$955$	−23000.8	+	39838.6i	−0.779361	+	1.34989i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	145.944	+	969.723i	0.00491428	+	0.0326527i
$960$		0			0
$961$	939.631	+	1627.49i	0.0315408	+	0.0546302i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15783.6			0.526519
$966$		0			0
$967$	10414.7			0.346345			0.173172	−	0.984892i	$-0.444598\pi$
$967$	10414.7			0.346345			0.173172	+	0.984892i	$0.444598\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21062.0	+	36480.4i	0.696098	+	1.20568i	0.969809	+	0.243865i	$0.0784152\pi$
$971$	21062.0	+	36480.4i	0.696098	+	1.20568i	−0.273712	+	0.961812i	$0.588251\pi$
$972$		0			0
$973$	−41623.1	+	33178.4i	−1.37140	+	1.09317i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3692.63	−	6395.82i	0.120919	−	0.209437i	−0.799211	−	0.601050i	$-0.794749\pi$
$977$	3692.63	−	6395.82i	0.120919	−	0.209437i	0.920130	+	0.391613i	$0.128083\pi$
$978$		0			0
$979$	−20078.8			−0.655485
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−8882.20	+	15384.4i	−0.288197	+	0.499173i	−0.973380	−	0.229199i	$-0.926389\pi$
$983$	−8882.20	+	15384.4i	−0.288197	+	0.499173i	0.685182	+	0.728372i	$0.259723\pi$
$984$		0			0
$985$	−17229.7	−	29842.7i	−0.557344	−	0.965349i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−10790.9	−	18690.4i	−0.346947	−	0.600929i
$990$		0			0
$991$	−12966.0	+	22457.8i	−0.415620	+	0.719874i	−0.995493	−	0.0948321i	$-0.969769\pi$
$991$	−12966.0	+	22457.8i	−0.415620	+	0.719874i	0.579874	+	0.814706i	$0.303102\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−25906.6			−0.825421
$996$		0			0
$997$	14169.0	−	24541.4i	0.450086	−	0.779572i	−0.548305	−	0.836279i	$-0.684727\pi$
$997$	14169.0	−	24541.4i	0.450086	−	0.779572i	0.998391	+	0.0567067i	$0.0180600\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.k.h.109.1	yes	16
3.2	odd	2		756.4.k.f.109.8	✓	16
7.2	even	3	inner	756.4.k.h.541.1	yes	16
21.2	odd	6		756.4.k.f.541.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.k.f.109.8	✓	16	3.2	odd	2
756.4.k.f.541.8	yes	16	21.2	odd	6
756.4.k.h.109.1	yes	16	1.1	even	1	trivial
756.4.k.h.541.1	yes	16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-9.20902 - 15.9505i) q^{5} +(-14.4823 + 11.5440i) q^{7} +(-13.0045 + 22.5245i) q^{11} -38.8473 q^{13} +(7.38515 - 12.7915i) q^{17} +(26.4892 + 45.8806i) q^{19} +(-56.7287 - 98.2569i) q^{23} +(-107.112 + 185.524i) q^{25} +119.708 q^{29} +(-83.5340 + 144.685i) q^{31} +(317.500 + 124.690i) q^{35} +(-39.6722 - 68.7143i) q^{37} -41.8167 q^{41} +190.219 q^{43} +(231.991 + 401.821i) q^{47} +(76.4715 - 334.367i) q^{49} +(-2.74684 + 4.75766i) q^{53} +479.036 q^{55} +(334.488 - 579.351i) q^{59} +(-469.420 - 813.059i) q^{61} +(357.745 + 619.633i) q^{65} +(-138.918 + 240.613i) q^{67} +357.478 q^{71} +(508.704 - 881.101i) q^{73} +(-71.6883 - 476.330i) q^{77} +(382.659 + 662.784i) q^{79} +1421.16 q^{83} -272.040 q^{85} +(385.995 + 668.564i) q^{89} +(562.596 - 448.453i) q^{91} +(487.879 - 845.030i) q^{95} +575.835 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 7 q^{5} - 11 q^{7} - 43 q^{11} - 52 q^{13} + 38 q^{17} + 32 q^{19} - 118 q^{23} - 143 q^{25} + 160 q^{29} + 5 q^{31} + 161 q^{35} - 136 q^{37} - 216 q^{41} + 800 q^{43} - 78 q^{47} + 253 q^{49} + 105 q^{53}+ \cdots - 838 q^{97}+O(q^{100}) \) 16 * q + 7 * q^5 - 11 * q^7 - 43 * q^11 - 52 * q^13 + 38 * q^17 + 32 * q^19 - 118 * q^23 - 143 * q^25 + 160 * q^29 + 5 * q^31 + 161 * q^35 - 136 * q^37 - 216 * q^41 + 800 * q^43 - 78 * q^47 + 253 * q^49 + 105 * q^53 - 948 * q^55 + 574 * q^59 - 424 * q^61 - 408 * q^65 - 88 * q^67 - 856 * q^71 + 644 * q^73 - 656 * q^77 + 266 * q^79 + 2054 * q^83 + 384 * q^85 + 732 * q^89 + 1022 * q^91 - 364 * q^95 - 838 * q^97

Introduction

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